seismic assessment of reinforced concrete frame structures
TRANSCRIPT
Seismic Assessment of Reinforced Concrete Frame Structures with a New Flexibility Based Element
António José Coelho Dias Arêde
Thesis submitted to the Faculdade de Engenharia da Universidade do Porto in candidature for the degree of Doutor in
Civil Engineering
FACULDADE DE ENGENHARIA
UNIVERSIDADE DO PORTO
October 1997
Publication financially supported by the Junta Nacional de Investigação Científica e Tecnológica
Il faut avoir le courage de dire des
choses imparfaites, de renoncer au
mérite d’avoir fait tout ce qu’on pouvait
faire, d’avoir dit tout ce qu’on pouvait
dire, enfin de sacrifier son amour-propre
au désir d’être utile et d’améliorer la
marche du progrès.
Lavoisier
ACKNOWLEDGEMENTS
I wish to express my sincere gratitude to the following persons and institutions, which have
contributed to the accomplishment of this milestone of my academic education and profes-
sional career:
• To Professor Raimundo Delgado, my supervisor, whose collaboration and friendship went
far beyond the academic and technical field. His comments and suggestions have always
played a very important role in the course of this study.
• To Artur Pinto, for the opportunity of staying at the ELSA laboratory of the Joint Research
Centre at Ispra, for his extremely valuable support and patient guidance, and, very specially,
for the friendship and confidence. Last but not the least, I thank him and his family very
much for the solidarity in many circumstances, which has definitely contributed for my
social and human integration in Italy.
• To João Guedes, my housemate, for the friendship, understanding and tolerance shown dur-
ing three and half years of social and professional life. Also, to Alfredo Campos Costa, for
the precious help, for the fruitful discussions and the encouraging words in some difficult
situations.
• To Pierre Pegon, for the excellent suggestions and advice in the model development and
implementation.
• To Professor Couto Marques for the precision and remarkable patience in revising the text.
• To all ELSA staff for the friendly and welcoming environment, with particular reference to
Professor Jean Donea for the confidence and the establishment of further cooperation links.
• To my friends and colleagues Rui Faria and Nelson Vila Pouca, for having so promptly
released me from my teaching activity, which was determinant for my intense dedication to
thesis writing.
• To my parents, brothers and nephews for all the care, patience and understanding during all
this long period of absence. To a large extent, I owe them the happy end of this and other
steps of my life.
• To my very good friends Tó Viana and Luísa who have always known how to be supportive
and encouraging in all situations. They are the living proof that the Friendship is not a void
word.
• To all my Ispra friends with whom I have shared excellent moments and who have helped
me feel at home in a foreign country.
Part of the research reported in this thesis was financially supported by the Human Capital and
Mobility programme of the European Commission, under the PREC8 (“Prenormative
Research in Support of Eurocode 8”) project. A one-year grant and the publication of this the-
sis were supported by the Portuguese Board for Scientific and Technological Research (JNICT
- “Junta Nacional de Investigação Científica e Tecnológica”). Both financial supports are
gratefully acknowledged.
Abstract
The present thesis focuses on the development of a global element model for the non-linear analysis ofreinforced concrete (RC) frame structures when subjected to monotonic or cyclic loads. The model isvalidated with the results of a broad experimental campaign on a full scale structure and is intensivelyapplied to the seismic behaviour assessment of structures designed according to Eurocode 8.
An innovative flexibility-based member model is presented, where the flexibility formulation isadopted to avoid the difficulties in accounting for the modifications in the kinematic shape functions ofthe classic stiffness formulation due to progressive variation of the element stiffness during the loadinghistory. The flexibility formulation makes use of force shape functions which are strictly derived fromequilibrium conditions and, thus, remain exact regardless of the element state.
The non-linear behaviour is controlled by a section moment-curvature model of Takeda type with trilin-ear skeleton curves. Besides the member-end sections and one mid-span fixed section, a number ofmoving control sections are monitored in order to constantly define and update the uncracked, crackedand yielded zones inside the element. For a given moment distribution (corresponding to imposed end-section rotations) the location of the moving control sections is first updated. The flexibility and thecurvature distributions along the element are then defined according to the referred section models,leading, by integration, to the element flexibility matrix and the end-section rotations, respectively. Aninternal iterative scheme is required to ensure that the curvature distribution in the element leads to end-section rotations compatible with the imposed ones and, at convergence, both the plastic zone lengthsand the progressive softening due to cracking become automatically defined.
Such a modelling strategy allows the element state along its full length to be updated at each load stepand, consequently, provides an adequate simulation of both the global structural stiffness and the evolu-tion of dynamic characteristics during the seismic response. The model is implemented in a generalpurpose computer code for finite element static and dynamic structural analysis, together with an auxil-iary procedure for the trilinear skeleton curve definition based on an efficient algorithm specificallydesigned to avoid the usual fibre discretization of the section.
The model is used to simulate the seismic response of a four-storey full scale RC frame structurepseudo-dynamically tested for two different earthquake levels and quasi-statically tested with increas-ing intensity cyclic load up to a near-failure stage. The tests, carried out at the ELSA laboratory of theJoint Research Centre at Ispra (Italy), are fully described and the test results are compared against thenumerical simulations. This provides an excellent means of checking whether the model is able todescribe the quasi-static or dynamic structural behaviour throughout distinct stages, while keeping agood compromise between computational efficiency and result quality.
The non-linear seismic analysis of a set of RC frame structures designed according to Eurocode 8 iscarried out in the framework of a European-wide prenormative research programme in support of thatdesign code (PREC8). Structures consist of two basic configurations (one regular and another irregu-lar), designed for several combinations of ductility class and design acceleration. Numerical modellingis done by means of the proposed flexibility element model and the seismic analysis is performed con-sidering several accelerograms fitting the EC8 spectrum and scaled by increasing intensities. The struc-tural responses are analysed by relative comparison between trial cases, focusing on issues such as:overstrength, cracking and yielding patterns, local ductility and damage distribution, drift and damage.An exercise of system reliability analysis is also presented in order to estimate bounds of failure proba-bility of the various structures, which are then compared between trial cases in order to find out theinfluence of the design parameters on the structural safety.
Resumo
Na presente tese procede-se ao desenvolvimento de um modelo de elemento global destinado à análisenão-linear de estruturas em pórtico de betão armado, quando sujeitas a acções monotónicas ou alterna-das. O modelo é validado por meio da simulação de uma vasta campanha de ensaios experimentaissobre uma estrutura testada à escala real, e é aplicado na verificação do comportamento sísmico deestruturas projectadas de acordo com o Eurocódigo 8 (EC8).
Apresenta-se um modelo inovador, desenvolvido com base na formulação de flexibilidade a fim decontornar as dificuldades associadas às modificações das funções de forma cinemáticas da formulaçãoclássica de rigidez, que são originadas pela alteração progressiva da rigidez do elemento durante ahistória de carga. A formulação de flexibilidade baseia-se em funções de forma de força que são obtidasexclusivamente por condições de equilíbrio, pelo que se mantêm exactas independentemente do estadodo elemento.
O comportamento não-linear é controlado por um modelo de secção do tipo Takeda, com curvas basetrilineares em termos de momento-curvatura. Para além das secções de extremidade e de uma secçãocentral fixa, são ainda controladas certas secções móveis que permitem definir e actualizar constante-mente as zonas plastificadas, fendilhadas e não-fendilhadas dentro do elemento. Para uma dada dis-tribuição de momentos (correspondente a rotações impostas nas extremidades), a posição das secçõesmóveis é devidamente actualizada. Subsequentemente, as distribuições de flexibilidade e de curvaturaao longo do elemento são definidas de acordo com os modelos de secção e, por integração, obtêm-se amatriz de flexibilidade e as rotações de extremidade, respectivamente. Um processo iterativo internoencarrega-se de garantir que uma dada distribuição de curvaturas origina rotações de extremidade com-patíveis com as rotações impostas e, atingida a convergência, tanto os comprimentos das zonas plásti-cas como a progressiva perda de rigidez devida à fendilhação vêm automaticamente definidos.
Esta técnica de modelação permite a actualização do estado do elemento em todo o seu comprimento acada passo de carga, conduzindo assim a uma simulação adequada da rigidez global da estrutura e daevolução das suas características dinâmicas durante a resposta sísmica. O modelo é incorporado numprograma geral de análise estrutural estática e dinâmica por elementos finitos, conjuntamente com umprocedimento numérico auxiliar para a definição da curva base trilinear baseado num algoritmo eficazespecificamente desenvolvido para evitar a discretização da secção em fibras.
O modelo de elemento é usado para reproduzir numericamente a resposta dum modelo físico à escalareal duma estrutura em pórtico de betão armado com quatro pisos, testado com recurso ao métodopseudo-dinâmico para duas intensidades sísmicas diferentes, e também sob acções cíclicas de intensid-ades gradualmente crescentes até um estado próximo da ruína. Os testes, realizados no laboratórioELSA do centro Comum de Investigação em Ispra (Itália), são detalhadamente descritos e os resultadossão comparados com os das simulações numéricas. Tal comparação fornece num excelente meio de val-idação que permite verificar se o modelo numérico é adequado para descrever o comportamento estru-tural quase-estático ou dinâmico ao longo de diversas fases, garantindo um bom compromisso entreeficiência computacional e qualidade de resultados.
Apresenta-se a análise sísmica não-linear de algumas estruturas em pórtico de betão armado projecta-das segundo o EC8, no quadro de um programa europeu de investigação de validação deste eurocódigo(PREC8). As estruturas consistem em duas configurações básicas, uma regular e outra irregular, pro-jectadas para várias classes de ductilidade e acelerações de projecto. A modelação numérica é feitacom o elemento global proposto e a análise sísmica é realizada com vários acelerogramas, gerados porforma a ajustarem-se ao espectro de projecto do EC8, e escalados com intensidades crescentes. As res-postas estruturais são analisadas por comparação relativa entre os diversos casos, concentrando especi-ficamente a atenção em aspectos como: a sobre-resistência, distribuições de fendilhação, plastificação,exigência de ductilidade e dano local, "drift" e dano global. É incluído também um exercício de cálculode limites das probabilidades de ruína, que são comparados entre as várias estruturas a fim de averiguara influência de alguns parâmetros de projecto na segurança estrutural.
Resumé
Cette thèse se consacre au développement d’un modèle d’élément global pour l’analyse non-linéairedes structures formées de portiques en béton armé sous chargements monotones ou cycliques. Lemodèle est d’abord validé à l’aide des résultats d’un large ensemble d’essais expérimentaux effectuéessur une structure à l’échelle réelle, puis appliqué massivement à la vérification du comportement sis-mique des structures dimensionnées avec l’Eurocode 8 (EC8).
Un nouveau modèle d’élément est présenté, dont la formulation de flexibilité a été choisie pour éviterles difficultés associées aux modifications des fonctions de forme cinématiques dans la formulationclassique de rigidité, modifications qui résultent de la variation progressive de la raideur de l’élémentpendant la réponse. La formulation de flexibilité utilise des fonctions de forme des forces, strictementobtenues à partir des conditions d’équilibre, et donc exactes indépendamment de l’état de l’élément.
Le comportement non-linéaire est controlé par des lois moment-courbure du type Takeda, basées surdes courbes enveloppes trilinéaires. Outre les deux sections d’extrémité et une section centrale fixe,quelques sections mobiles sont aussi contrôlées pour définir et actualiser constamment les zones plasti-fiées, fissurées et non-fissurées dans l’élément. Pour une certaine distribution de moments fléchissants(correspondant à des rotations imposées dans les noeuds), la position des sections mobiles est d’abordactualisée. Puis, les distributions de flexibilité et courbure dans l’élément sont définies à l’aide desmodèles de section. Finalement, la matrice de flexibilité et les rotations aux extrémités peuvent êtredéduites par intégration. Des itérations internes sont nécessaires pour assurer que la distribution decourbure soit compatible avec les rotations imposées aux extrémités; lors que la convergence estatteinte, les zones plastiques et l’adoucissement progressif dû à la fissuration sont automatiquementdéfinis.
Ce type de modélisation permet l’actualisation de l’état de l’élément dans toute sa longueur à chaquepas de charge, ce qui fournit une simulation convenable de la raideur globale de la structure ansi que del’évolution de ses caractéristiques dynamiques pendant la réponse sismique. Le modèle a été introduitdans un code de calcul général d’éléments finis pour l’analyse non-linéaire, statique ou dynamique, destructures, et une procédure auxiliaire a été écrite pour la définition des courbes enveloppes trilinéairesbasée sur un algoritme spécifique afin d’éviter des discrétisations à fibres.
Le modèle d’élément est utilisé pour la simulation numérique de la réponse sismique d’une structure àquatre étages formée de portiques en béton armé, testée avec la méthode pseudo-dynamique sousl’action de séismes de deux intensités différentes, ainsi que sous chargement cyclique quasi-statiquejusqu’à un état proche de la ruine. Les tests, qui ont été faits au laboratoire ELSA du Centre Communde Recherche à Ispra (Italie), sont complètement décrits et les résultats sont comparés avec ceux dessimulations numériques. Cela constitue un excellent moyen de vérifier si le modèle peut reproduire laréponse quasi-statique ou dynamique dans diverses phases de comportement, en conservant un bon rap-port efficace computationelle versus qualité des résultats.
L’analyse sismique non-linéaire de quelques structures en béton armé, conçues et dimensionées selonl’EC8, est faite dans le cadre d’un programme européen de recherche prénormative (PREC8). Deuxconfigurations structurelles sont considérées (une regulière et une autre irregulière), dimensionnéespour différentes combinaisons de classe de ductilité et d’accélération de projet. La modélisationnumérique est faite avec l’élément développé et quelques accélérogrammes compatibles avec le spectrede l’EC8 sont utilisés (aux intensités croissantes) pour l’analyse sismique. Les réponses structurellessont comparées entre les différents cas, en ce qui concerne la réserve de résistance, les distributions defissuration, plastification, ductilité et endommagement local, ainsi que les "drifts" et l’endommagementglobal. Un exercice de calcul des limites de la probabilité de ruine est aussi fait, pour vérifier l’influ-ence de quelques paramètres de projet sur la sécurité structurelle.
TABLE OF CONTENTS
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxix
Chapter 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 Chapter organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Hysteretic behaviour of reinforced concrete members . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Numerical modelling strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3.2 Member type models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Distributed inelasticity member models . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 General flexibility formulation for beam-column elements . . . . . . . . . . . . . . . . . . . . . . 332.4.1 Conventions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4.2 Relations between spaces of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.4.3 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4.3.1 Stiffness method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.4.3.2 Flexibility method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.4.4 The element state determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.4.1 General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4.4.2 Nodal forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.4.4.3 Element loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.4.4.4 Remarks on the global non-linear algorithm . . . . . . . . . . . . . . . . . . . . 592.4.4.5 Control sections and numerical integration . . . . . . . . . . . . . . . . . . . . 61
2.5 Concluding summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 3
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL. . . . . . 63
3.1 General comments and innovative features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.2 Basic assumptions and remarks on convention and notation . . . . . . . . . . . . . . . . . . . . . 66
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3.3 Trilinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Control sections and element zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.2 Cracking sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.4.3 Yielding sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.4.4 Null moment sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 Behaviour of the control sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.5.1 Modified trilinear model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.5.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.5.1.2 Motivation for model modification . . . . . . . . . . . . . . . . . . . . . . . . . 863.5.1.3 Proposed model modifications . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.5.2 Transition from uncracked to cracked section behaviour . . . . . . . . . . . . . . . . . . . 933.5.3 State evolution of control sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
3.5.3.1 Internal moving sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.5.3.2 Fixed sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.6 Element state determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6.2 Flexibility distribution within the element . . . . . . . . . . . . . . . . . . . . . . . . . 1013.6.3 Element flexibility matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1033.6.4 Displacement residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.6.5 Element applied loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1113.6.6 Behaviour of plastic end zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.6.6.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.6.6.2 Plastic zone splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1153.6.6.3 Event-to-event scheme in the element iterative process . . . . . . . . . . . . . 1183.6.6.4 Evolution of curvatures in fixed plastic zones . . . . . . . . . . . . . . . . . . 120
3.6.7 Integration of deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.6.8 Convergence criteria for the element iterative process . . . . . . . . . . . . . . . . . . . 1243.6.9 Convergence problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.6.9.1 Difficult or no- convergence situations . . . . . . . . . . . . . . . . . . . . . . 1253.6.9.2 Line search scheme for element iterations . . . . . . . . . . . . . . . . . . . . 128
3.7 Summary of the non-linear algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.7.1 General structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.7.2 Element state determination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
3.8 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Chapter 4
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION . . . . . . . . . . . 141
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1414.2 Implementation in the computer code CASTEM2000 . . . . . . . . . . . . . . . . . . . . . . . . 142
4.2.1 Basics of CASTEM2000 and main implementation needs . . . . . . . . . . . . . . . . . 1424.2.2 Flexibility based element implementations . . . . . . . . . . . . . . . . . . . . . . . . . 1494.2.3 Definition of skeleton curves for RC global section modelling . . . . . . . . . . . . . . 153
4.2.3.1 Type of sections, notations and conventions . . . . . . . . . . . . . . . . . . . 1534.2.3.2 Material models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1554.2.3.3 Linear behaviour: the cracking point . . . . . . . . . . . . . . . . . . . . . . . 158
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4.2.3.4 Turning points for non-linear behaviour . . . . . . . . . . . . . . . . . . . . . 1604.2.3.5 Remarks on implementation and validation . . . . . . . . . . . . . . . . . . . 175
4.3 Flexibility-based element validation at the single member level . . . . . . . . . . . . . . . . . . 1764.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1764.3.2 Specimen characteristics and test description . . . . . . . . . . . . . . . . . . . . . . . 1764.3.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.3.4 Remarks on model validation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA . . . . . . . . . . . . . . . . . . . . 191
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1915.2 The Pseudo-Dynamic test method. An overview . . . . . . . . . . . . . . . . . . . . . . . . . . 192
5.2.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1925.2.2 Time integration techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
5.2.2.1 Newmark explicit algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 1955.2.2.2 α-implicit algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1965.2.2.3 Mixed explicit-implicit algorithms . . . . . . . . . . . . . . . . . . . . . . . . 199
5.2.3 Substructuring in the PSD method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2005.2.4 Applications of the PSD method at the ELSA laboratory . . . . . . . . . . . . . . . . . 202
5.3 Structure design and layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2035.4 Material properties and reduced scale member tests . . . . . . . . . . . . . . . . . . . . . . . . . 2055.5 Full-scale tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
5.5.1 PSD test setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2085.5.2 The input accelerogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2115.5.3 Preliminary tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2135.5.4 Seismic tests on the bare frame structure . . . . . . . . . . . . . . . . . . . . . . . . . 214
5.5.4.1 Low level test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2145.5.4.2 High level test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
5.5.5 Infilled frame seismic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2215.5.5.1 Uniformly infilled configuration . . . . . . . . . . . . . . . . . . . . . . . . . 2225.5.5.2 Soft-storey configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
5.5.6 Final cyclic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2265.6 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
Chapter 6
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING. . . . . . . . . . . . . . . . . . . . . . . 243
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2436.2 Modelling assumptions and data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.2.1 Structure modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2446.2.1.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2446.2.1.2 Collaborating slab width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2456.2.1.3 Mass and vertical static loads . . . . . . . . . . . . . . . . . . . . . . . . . . 248
6.2.2 Cross-section characteristics and material properties . . . . . . . . . . . . . . . . . . . 249
xviii TABLE OF CONTENTS
6.2.2.1 Cross-sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2496.2.2.2 Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2526.2.2.3 Steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
6.2.3 Skeleton curves for the section model . . . . . . . . . . . . . . . . . . . . . . . . . . . 2536.2.4 Hysteretic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
6.2.4.1 Unloading stiffness degradation . . . . . . . . . . . . . . . . . . . . . . . . . 2556.2.4.2 Pinching effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2566.2.4.3 Strength degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
6.2.5 Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2576.2.6 Modelling of infills . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
6.2.6.1 Setup of numerical tools for infill modelling . . . . . . . . . . . . . . . . . . . 2596.2.6.2 Application to a single frame . . . . . . . . . . . . . . . . . . . . . . . . . . . 2616.2.6.3 Application to the full-scale structure . . . . . . . . . . . . . . . . . . . . . . 2626.2.6.4 Infill panels in the present study . . . . . . . . . . . . . . . . . . . . . . . . . 263
6.3 Damage quantification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2646.3.1 General overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2646.3.2 The Park and Ang damage index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
6.3.2.1 The damage parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2696.3.2.2 Yielding rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2716.3.2.3 Ultimate rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2736.3.2.4 Hysteretic dissipated energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.4 Analysis of results from numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.4.2 Procedure for static analytical simulation of the tests . . . . . . . . . . . . . . . . . . . 2806.4.3 Static analysis by flexibility modelling versus experimental tests . . . . . . . . . . . . . 284
6.4.3.1 Pushover analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2846.4.3.2 Bare frame seismic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2866.4.3.3 Infilled frame tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2976.4.3.4 Final cyclic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3016.4.3.5 Summary of static analysis results . . . . . . . . . . . . . . . . . . . . . . . . 308
6.4.4 Dynamic analysis by flexibility modelling versus experimental tests . . . . . . . . . . . 3096.4.4.1 Comparison of structural frequencies . . . . . . . . . . . . . . . . . . . . . . . 3106.4.4.2 Low-level test on the bare structure . . . . . . . . . . . . . . . . . . . . . . . 3116.4.4.3 High level test on the bare structure . . . . . . . . . . . . . . . . . . . . . . . 3166.4.4.4 Remarks on energy comparison . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.4.4.5 Summary of dynamic analysis results . . . . . . . . . . . . . . . . . . . . . . 325
6.4.5 Flexibility element versus fixed length plastic hinge (F.H.) modelling . . . . . . . . . . 3266.4.5.1 Assumptions for F.H. modelling . . . . . . . . . . . . . . . . . . . . . . . . . 3266.4.5.2 Discussion on F.H. modelling and comparison with flexibility analysis results . 3286.4.5.3 Summary of F.H. and flexibility modelling comparison . . . . . . . . . . . . . 335
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Chapter 7
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 . . . 341
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3417.2 The PREC8 project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
TABLE OF CONTENTS xix
7.2.1 Basics of EC8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3427.2.2 The RC frame structure topic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
7.3 The building configurations 2 and 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.3.1 General comments and structure layout . . . . . . . . . . . . . . . . . . . . . . . . . . 3467.3.2 Vertical static loads and seismic action . . . . . . . . . . . . . . . . . . . . . . . . . . 3497.3.3 Structure modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
7.3.3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3517.3.3.2 Mass, damping and natural frequencies . . . . . . . . . . . . . . . . . . . . . 3527.3.3.3 Moment-curvature constitutive relations for global section behaviour . . . . . 354
7.4 Non-linear seismic analysis of building configurations 2 and 6 . . . . . . . . . . . . . . . . . . . 3597.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3597.4.2 Structural strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3597.4.3 Cracking, yielding and damage patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 3647.4.4 Ductility demand and damage distribution in elevation . . . . . . . . . . . . . . . . . . 3707.4.5 Overall analysis of response parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 3767.4.6 Safety assessment by probabilities of failure. An exercise . . . . . . . . . . . . . . . . 383
7.4.6.1 Methodology and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 3847.4.6.2 Comparative analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
7.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
Chapter 8
FINAL REMARKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397
8.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3978.2 Future developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
Appendix A
Linear Elastic Timoshenko Beam Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
A.1 Section formulation and constitutive relationship . . . . . . . . . . . . . . . . . . . . . . . . . . 421A.2 Element flexibility matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422A.3 Element displacements as integrated deformations . . . . . . . . . . . . . . . . . . . . . . . . . 426
Appendix B
Trilinear Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Appendix C
Internal Force Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
C.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437C.2 Element applied forces in three directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437C.3 Element applied forces only in the non-linear bending plane . . . . . . . . . . . . . . . . . . . . 439C.4 Moving section abscissas and respective derivatives . . . . . . . . . . . . . . . . . . . . . . . . 441
xx TABLE OF CONTENTS
Appendix D
The Event-to-Event Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Appendix E
Non-Linear Dynamic and Static Analysis Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
LIST OF FIGURES
Chapter 2
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART . . 13
Figure 2.1 Typical global response diagrams of beams and columns for monotonic loading . . . . . . . . 16
Figure 2.2 Cyclic global response examples (Carvalho (1993)) . . . . . . . . . . . . . . . . . . . . . . . 17
Figure 2.3 Member modelling: a) two component model (Clough et al. (1965)) and b) one component model (Giberson (1967)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2.4 Spaces of variables and axis systems: from the global to the local element level. . . . . . . . . 34
Figure 2.5 Space of variables at the local section level.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Figure 2.6 Element applied loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
Figure 2.7 Main tasks of the classical stiffness based state determination process . . . . . . . . . . . . . . 42
Figure 2.8 State determination of flexibility based elements: from the global to the local level . . . . . . . 46
Figure 2.9 Flowchart for the element state determination of flexibility based elements . . . . . . . . . . . 47
Figure 2.10 Details of element and section state determination for flexibility based elements . . . . . . . . 49
Figure 2.11 Details of element and section state determination for flexibility based elements with the application of element loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Figure 2.12 Sequence for the application of incremental element and nodal loads . . . . . . . . . . . . . . 54
Figure 2.13 Details of the element state determination for first internal iteration of the first N-R iteration, in the presence of element loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Figure 2.14 Details of the element state determination for the (j>1) internal iterations of the first N-R iteration, in the presence of element loads. . . . . . . . . . . . . . . . . . . . . . . . . . 56
Figure 2.15 State determination of flexibility based elements for displacements corrections relative to the step beginning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 3
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL. . . . . . 63
Figure 3.1 Adjustment of local section axis system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Figure 3.2 Primary or skeleton trilinear curve for the global section model. . . . . . . . . . . . . . . . . . 71
Figure 3.3 Element control sections: fixed sections (E1, E2 and H) and moving section (M) . . . . . . . . 73
Figure 3.4 Distinction and evolution between cracking and cracked sections . . . . . . . . . . . . . . . . 75
xxii LIST OF FIGURES
Figure 3.5 General layout of assumed cracking sections and local abscissas for no element loads or concentrated force applied in H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Figure 3.6 Cracking sections and local abscissas for parabolic moment distribution.. . . . . . . . . . . . 77
Figure 3.7 Examples of restricted cracking sections in the presence of distributed force . . . . . . . . . . 79
Figure 3.8 Definition of cracking moment directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Figure 3.9 General layout of assumed yielding and cracking sections . . . . . . . . . . . . . . . . . . . 83
Figure 3.10 Locations of null moment sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Figure 3.11 Evolution of cracking and yielding section points in the model diagram . . . . . . . . . . . . 85
Figure 3.12 Comparison of fibre and trilinear section modelling formulations. Section, member and model data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Figure 3.13 Comparison of fibre and trilinear section modelling formulations. Local and global response for two axial load levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Figure 3.14 Effects of tension-softening in the fibre formulation. . . . . . . . . . . . . . . . . . . . . . . 90
Figure 3.15 Hysteretic rules for the modified trilinear model. . . . . . . . . . . . . . . . . . . . . . . . . 92
Figure 3.16 Cracking transition in the case of non-uniform moment distribution . . . . . . . . . . . . . . 94
Figure 3.17 Cracking transition in the case of uniform moment distribution . . . . . . . . . . . . . . . . . 95
Figure 3.18 Rules for progressive transition of the cracking plateau transition . . . . . . . . . . . . . . . 96
Figure 3.19 Examples of flexibility distributions for loading and unloading cases. . . . . . . . . . . . . . 102
Figure 3.20 Derivation of additional flexibility terms due to moving sections . . . . . . . . . . . . . . . . 106
Figure 3.21 Monotonic development of plastic zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 3.22 Plastic zone splitting in fixed and variable length parts . . . . . . . . . . . . . . . . . . . . . 115
Figure 3.23 Flexibility distributions in plastic zones with no further yielding development . . . . . . . . . 116
Figure 3.24 Flexibility distributions in plastic zones with further yielding development . . . . . . . . . . 117
Figure 3.25 Application of the event-to-event scheme to the element state determination . . . . . . . . . . 119
Figure 3.26 Total curvature evolution for non-monotonic loading . . . . . . . . . . . . . . . . . . . . . . 121
Figure 3.27 Cracking transition and the role of additional flexibility terms . . . . . . . . . . . . . . . . . 126
Figure 3.28 Typical cases generating convergence problems. . . . . . . . . . . . . . . . . . . . . . . . . 128
Figure 3.29 Interpolation for line search scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Figure 3.30 Flow chart for structure state determination . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
LIST OF FIGURES xxiii
Chapter 4
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION . . . . . . . . . . 141
Figure 4.1 Illustrative example for GIBIANE input . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Figure 4.2 Typical moment-curvature diagram and trilinear approximation . . . . . . . . . . . . . . . . 153
Figure 4.3 Types of sections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
Figure 4.4 Reference axis system convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Figure 4.5 Steel stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
Figure 4.6 Concrete stress-strain relationship . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Figure 4.7 Imminent cracking condition. Section, applied forces and internal strains . . . . . . . . . . . 159
Figure 4.8 Generic situation for the unified procedure. Forces and internal strains . . . . . . . . . . . . 161
Figure 4.9 Rectangular section split into unconfined and confined zones . . . . . . . . . . . . . . . . . 163
Figure 4.10 T-shape section split into unconfined and confined zones . . . . . . . . . . . . . . . . . . . 163
Figure 4.11 Influence of d0 in the development of stress zones . . . . . . . . . . . . . . . . . . . . . . . 164
Figure 4.12 Influence of d0 in the development of uniform geometry zones . . . . . . . . . . . . . . . . 164
Figure 4.13 Development of integration zones and adjustable widths bi and bi . . . . . . . . . . . . . . . 165
Figure 4.14 Yielding criteria at the tensioned steel or at compressed confined concrete . . . . . . . . . . 172
Figure 4.15 Ultimate criteria at the tensioned steel or at compressed confined concrete . . . . . . . . . . 173
Figure 4.16 Supplementary point. Criterion related to the most tensioned steel . . . . . . . . . . . . . . . 174
Figure 4.17 Section layout S2 and schematic representation of tested cantilever beams . . . . . . . . . . 177
Figure 4.18 Cyclic sequences (V5 and V6) of imposed displacements for LNEC and KT beams. . . . . . 179
Figure 4.19 Comparison of numerical and experimental monotonic M-ϕ curves for LNEC and KT beam sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Figure 4.20 LNEC and KT beams S2: monotonic tests V1 and V2 . . . . . . . . . . . . . . . . . . . . . 183
Figure 4.21 LNEC beam S2: tests V5 and V6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Figure 4.22 KT beam S2: tests V5 and V6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA . . . . . . . . . . . . . . . . . . . . 191
Figure 5.1 Implementation of the explicit PSD method. . . . . . . . . . . . . . . . . . . . . . . . . . . 196
Figure 5.2 Implementation of the implicit PSD method. . . . . . . . . . . . . . . . . . . . . . . . . . . 198
xxiv LIST OF FIGURES
Figure 5.3 General layout of the 4-storey RC building tested at ELSA (dimensions in metres) . . . . . . 204
Figure 5.4 Input accelerogram (Friuli-like) and elastic response spectra . . . . . . . . . . . . . . . . . . 212
Figure 5.5 Time histories of storey displacements, relative inter-storey drift, total storey-shear and respective peak value profiles for Low and High level tests . . . . . . . . . . . . . . . . 215
Figure 5.6 Shear-drift diagrams at each storey, for Low and High level tests. . . . . . . . . . . . . . . . 216
Figure 5.7 Time histories of dissipated energy and base shear - top displacement diagrams for Low and High level tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
Figure 5.8 Detail of the 2nd storey experimental shear-drift diagram for the high level test . . . . . . . . 220
Figure 5.9 Spatial distribution of rotations in the critical zones, for the high level test. . . . . . . . . . . 221
Figure 5.10 Time histories of storey displacements, relative inter-storey drift, total storey-shear and respective peak value profiles for both configurations of infilled frame tests. . . . . . . . 223
Figure 5.11 Shear-drift diagrams at each storey for both configurations of infilled frame tests . . . . . . . 224
Figure 5.12 Time history of the imposed top displacement for the final cyclic tests . . . . . . . . . . . . . 227
Figure 5.13 Base shear - top displacement diagrams and curves of total deformation energy for the final cyclic tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Figure 5.14 Storey profiles of peak values of displacement, inter-storey drift and inter-storey shear for the final cyclic tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
Chapter 6
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING . . . . . . . . . . . . . . . . . . . . . . . 243
Figure 6.1 Mesh for the structural analysis using flexibility global elements (dimensions in m) . . . . . . 245
Figure 6.2 Typical beam and column cross-sections for both external and internal frames. . . . . . . . . 249
Figure 6.3 Schematic reinforcement layout for the beams. . . . . . . . . . . . . . . . . . . . . . . . . . 250
Figure 6.4 Schematic reinforcement layout for the columns . . . . . . . . . . . . . . . . . . . . . . . . 251
Figure 6.5 Moment-curvature diagram for a column section. Comparison of trilinear curve and fibre analysis results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
Figure 6.6 Diagonal strut model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Figure 6.7 Equivalence of element in anti-symmetric bending with cantilever elements . . . . . . . . . . 270
Figure 6.8 Yielding chord rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
Figure 6.9 Estimation of ultimate chord rotation by curvature integration . . . . . . . . . . . . . . . . . 275
Figure 6.10 Example of inconsistency in energy splitting between element end sections: a) Beam and deformed shape, b) bending moments and c) curvature diagrams. . . . . . . . . 278
Figure 6.11 Storey displacement prescription and unloading to zero actuator forces . . . . . . . . . . . . 282
LIST OF FIGURES xxv
Figure 6.12 Introduction of the infill panel diagonal struts. . . . . . . . . . . . . . . . . . . . . . . . . . 283
Figure 6.13 Unloading of infilled frame configuration: a) removal of actuators and b), removal of infill panels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Figure 6.14 Pushover analysis: inverted triangular force distribution. . . . . . . . . . . . . . . . . . . . . 285
Figure 6.15 Pushover analysis: base shear-top displacement and storey shear-drift diagrams. . . . . . . . 285
Figure 6.16 0.4S7 test. Static analysis versus experimental storey results . . . . . . . . . . . . . . . . . 288
Figure 6.17 0.4S7 test. Static analysis versus experimental results . . . . . . . . . . . . . . . . . . . . . 289
Figure 6.18 1.5S7 test. Static analysis versus experimental storey results . . . . . . . . . . . . . . . . . 292
Figure 6.19 1.5S7 test. Static analysis versus experimental results . . . . . . . . . . . . . . . . . . . . . 293
Figure 6.20 1.5S7 test - Static analysis. Spatial distributions of peak values . . . . . . . . . . . . . . . . 294
Figure 6.21 Uniformly infilled test. Static analysis versus experimental results . . . . . . . . . . . . . . . 298
Figure 6.22 Soft-storey test. Static analysis versus experimental results. . . . . . . . . . . . . . . . . . . 299
Figure 6.23 Shear-drift diagrams for the Ductility 3 phase of final test. Effects of considering or neglecting infilled frame tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
Figure 6.24 Inter-storey drift profiles for 1.5S7, soft-storey and final Duct. 3 tests . . . . . . . . . . . . . 303
Figure 6.25 Final cyclic test, Ductilities 5 and 8: first and second storey shear-drift diagrams . . . . . . . 304
Figure 6.26 Final cyclic test: total energy diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
Figure 6.27 Final cyclic test Duct. 8: results for modified unloading stiffness degradation . . . . . . . . . 306
Figure 6.28 Final cyclic test: profiles of peak values of storey shear . . . . . . . . . . . . . . . . . . . . 307
Figure 6.29 Influence of assumed displacements different from the applied ones . . . . . . . . . . . . . . 308
Figure 6.30 0.4S7 test. Dynamic analysis with 1.8% viscous damping versus experimental storey results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Figure 6.31 0.4S7 test. Dynamic analysis with 1.8% viscous damping vs. experimental results . . . . . . 314
Figure 6.32 0.4S7 test. Dynamic analysis with no viscous damping versus experimental results . . . . . . 315
Figure 6.33 1.5S7 test. Dynamic analysis with 1.8% viscous damping vs. experimental results . . . . . . 317
Figure 6.34 1.5S7 test. Dynamic analysis with 1.8% viscous damping and modified pinching versus experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Figure 6.35 1.5S7 test. Dynamic analysis with zero viscous damping versus experimental storey results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
Figure 6.36 1.5S7 test. Dynamic analysis with no viscous damping versus experimental results . . . . . . 321
Figure 6.37 Total input energy for experimental and numerical analysis . . . . . . . . . . . . . . . . . . 323
Figure 6.38 Relative absorbed energy for experimental and numerical analysis. . . . . . . . . . . . . . . 324
xxvi LIST OF FIGURES
Figure 6.39 Member discretization for fixed length plastic hinge (F.H.) analysis . . . . . . . . . . . . . . 326
Figure 6.40 0.4S7 test. Static analysis with F.H. modelling . . . . . . . . . . . . . . . . . . . . . . . . . 329
Figure 6.41 1.5S7 test. Static analysis with F.H. modelling . . . . . . . . . . . . . . . . . . . . . . . . . 330
Figure 6.42 0.4S7 test. Dynamic analysis with F.H. modelling and 1.8% viscous damping . . . . . . . . . 332
Figure 6.43 0.4S7 test. Dynamic analysis with F.H. modelling and zero viscous damping . . . . . . . . . 333
Figure 6.44 1.5S7 test. Dynamic analysis with F.H. modelling and zero viscous damping . . . . . . . . . 334
Chapter 7
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 . . . 341
Figure 7.1 Basic configurations of the eight storey trial cases (PREC8) . . . . . . . . . . . . . . . . . . 347
Figure 7.2 Artificial accelerograms (S1...S4) and response spectra (5% damping) fitting the EC8 response spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
Figure 7.3 Structural systems of planar frame associations . . . . . . . . . . . . . . . . . . . . . . . . . 352
Figure 7.4 Approximation of the post-yielding branch of the (M-ϕ) diagrams in columns . . . . . . . . . 355
Figure 7.5 Beam section moment-curvature diagrams of all cases with configuration 6 in direction YY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
Figure 7.6 Column section moment-curvature diagrams of all cases with configuration 6 in direction YY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
Figure 7.7 Global overstrength. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
Figure 7.8 Base shear - top displacement curve for the C2_15L case, direction XX. Definition of global yielding force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
Figure 7.9 Global hardening factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
Figure 7.10 Cracking pattern: Configuration 6, Direction X under earthquake S1 for intensity 1.0 and 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
Figure 7.11 Positive rotation ductility pattern: Configuration 6, Direction X under earthquake S1 for intensity 1.0 and 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Figure 7.12 Damage pattern: Configuration 6, Direction X under earthquake S1 for intensity 1.0 and 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
Figure 7.13 Damage pattern: Configuration 2, Direction X under earthquake S1 for intensity 1.0 and 2.0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369
Figure 7.14 Column rotation ductility profiles (maxima) . . . . . . . . . . . . . . . . . . . . . . . . . . . 372
Figure 7.15 Beam rotation ductility profiles (maxima) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
Figure 7.16 Column damage profiles (maxima) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
Figure 7.17 Beam damage profiles (maxima). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
LIST OF FIGURES xxvii
Figure 7.18 Total drift (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
Figure 7.19 Maximum inter-storey drift (%) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
Figure 7.20 Sensitivity coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
Figure 7.21 Maximum damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
Figure 7.22 Global (average) damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382
Figure 7.23 Local probability of failure at the hinge level . . . . . . . . . . . . . . . . . . . . . . . . . . 384
Figure 7.24 High and medium seismicity hazard curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
Figure 7.25 Vulnerability curve fitting to non-linear analysis results . . . . . . . . . . . . . . . . . . . . 388
Figure 7.26 Bounds of annual probability of failure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
Appendix A
Linear Elastic Timoshenko Beam Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421
Figure A.1 Distribution of flexibility properties along the element . . . . . . . . . . . . . . . . . . . . . 423
Appendix B
Trilinear Model Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431
Figure B.1 Hysteretic rules of the trilinear model. General loading path . . . . . . . . . . . . . . . . . . 431
Figure B.2 Trilinear model. The pinching effect and interior cycles . . . . . . . . . . . . . . . . . . . . 433
Figure B.3 Trilinear model. Strength deterioration rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 434
Appendix C
Internal Force Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437
Figure C.1 Element applied loads in the non-linear bending plan; simplified notation . . . . . . . . . . . 440
Appendix D
The Event-to-Event Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
Figure D.1 The event-to-event scheme for stiffness based problems . . . . . . . . . . . . . . . . . . . . 445
LIST OF TABLES
Chapter 3FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL. . . . . . 63
Table 3.1 Definition of cracking section abscissas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Chapter 4NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION . . . . . . . . . . 141
Table 4.1 Mechanical properties of steel and concrete of LNEC and KT beams . . . . . . . . . . . . . 178
Chapter 5THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA . . . . . . . . . . . . . . . . . . . . 191
Table 5.1 Mean cube (fcm,cub) and cylinder (fcm) compressive and tensile (fctm) strengths of concrete . . 205
Table 5.2 Mean tensile properties of steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Table 5.3 Frequencies (Hz) for all testing cases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
Chapter 6ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING. . . . . . . . . . . . . . . . . . . . . . . 243
Table 6.1 Floor mass values and vertical loads on beams . . . . . . . . . . . . . . . . . . . . . . . . . 248
Table 6.2 Mean tensile properties of steel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
Table 6.3 Numerical simulations performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
Table 6.4 Structural frequencies. Comparison of measured values with those calculated by flexibility discretization310
Chapter 7SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8. . . 341
Table 7.1 Trial cases, design behaviour factors and earthquake intensities . . . . . . . . . . . . . . . . 348
Table 7.2 Member cross-sectional dimensions (m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Table 7.3 Floor masses. Adopted values and design values (in brackets) . . . . . . . . . . . . . . . . . 353
Table 7.4 Frequencies (Hz) for all cases (design values in brackets) . . . . . . . . . . . . . . . . . . . 353
Table 7.5 Design base shear force ratio to structure weight (seismic coefficient) . . . . . . . . . . . . . 359
Table 7.6 Overstrength factors at yielding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362
Chapter 1
INTRODUCTION
1.1 General
Earthquake Engineering is an ever challenging research field, due to the wide variety of issues
involved. These can be as different as the seismic event generation and the characterization of
seismic motion at a given site, or the dynamic behaviour of structures and their non-linear
response, or the definition of measures to assess the structural reliability under seismic events.
In this context, a given structure can be regarded as a sort of “operator” mapping the seismic
input action into effects that are compared against some performance limits which are estab-
lished according to socially and economically acceptable costs.
It follows that, regardless of the seismic input and the response reliability analysis, an adequate
knowledge of the structural dynamic behaviour appears of major importance for the seismic
assessment of structures. The peculiar features of the reinforced concrete (RC) behaviour, par-
ticularly under cyclic loading, introduce increased levels of complexity in the non-linear struc-
tural response, for which significant experimental testing would be desirable, with particular
interest on large or full scale specimens.
Because of the extremely high cost of large scale tests, either dynamic or pseudo-dynamic,
they are often restricted to a few cases that may be considered representative of certain types of
structures. Thus, recourse has to be made to numerical modelling of the structural hysteretic
response by means of non-linear behaviour models, which are calibrated on the basis of widely
available results from experiments on single reinforced concrete members or sub-assemblages;
such calibration can be complemented, or even checked, by comparing the numerical results of
the structural response with available experimental evidence from large or full scale tests.
2 Chapter 1
The importance of modelling the non-linear hysteretic behaviour of RC structures arises from
the present seismic design philosophy, according to which an adequate seismic behaviour of a
given structure should fulfil the following requirements:
• under the action of low intensity earthquakes (A), that may occur several times during the
structure lifetime, no structural or non-structural damage should be detected;
• the response to moderate intensity seismic events (B), typically those occurring once in the
structure lifetime, may lead to slight (but insignificant) structural damage associated with
visible non-structural damage;
• for major earthquakes (C), thus with very low probability of occurrence during the structure
lifetime, significant but repairable structural damage can be expected as long as no partial
nor total collapse of the structure occurs.
From the above, it follows that the structural response will be distinct for each level of seismic
intensity. Typically, for the intensity level A an essentially linear elastic behaviour shall be
expected, while for intensity B the structure will exhibit apparent cracking on structural mem-
bers and, possibly, some incipient and localized yielding. On the other hand, for earthquakes of
intensity C, significant inelastic behaviour is likely to develop in a generalized fashion
throughout the critical zones of the structure where hysteretic dissipation capability must be
provided to cope with the seismic induced energy.
According to that seismic design philosophy, the intensity levels A and B may be assigned lin-
ear elastic behaviour, preferably associated with dynamic analysis for the seismic response
prediction. However, recent studies point out the need to account for the non-linear effects aris-
ing from cracking of reinforced concrete members, even for such levels of seismic action, in
order to obtain adequate estimates of the structural response (Fardis and Panagiotakos (1997),
Calvi and Pinto (1996)); indeed, this corresponds to the usual problem of deciding which stiff-
ness shall be used for linear elastic analysis (uncracked, cracked or some intermediate state).
The analysis of seismic effects arising from earthquakes of intensity level C should be ideally
performed by means of non-linear behaviour models, able to account for the exploitation of
high ductility (generically defined as the inelastic deformation capacity without significant loss
of strength) and for the capacity of energy dissipation through stable hysteretic mechanisms.
However, such type of analyses (particularly in the dynamic context) are not yet possible in
INTRODUCTION 3
current seismic design practice because of their complexity. Thus, seismic design codes allow
linear elastic analysis to be performed for a seismic action q times lower than the design one,
where q is the so-called behaviour factor, through which the non-linear effects are approxi-
mately taken into account. For RC structures the q-factor typically varies between 2 and 5, and
basically it means that the structure must provide the strength for forces q times lower than
those resulting from the linear elastic analysis with the design earthquake and, simultaneously,
have the capacity of accommodating inelastic deformations of the order of those obtained in
that elastic analysis. Such inelastic deformation capacity shall be guaranteed in the critical
zones bound to behave non-linearly, which have to be provided with a certain amount of avail-
able ductility; the larger is the q-factor the more available ductility is required in those critical
zones.
Nevertheless, such simplified analysis require some issues to be investigated, viz the calibra-
tion of q factors and their relation with ductility required in dissipative zones of the structure,
as well as the influence of the structure geometry and the distributions of mass and stiffness on
the response to high intensity earthquakes. As suggested by Fardis (1991), q-factor calibration
can be made through parametric studies aiming at checking whether or not a structure will be
able to accommodate permanent, yet repairable, damage under a seismic input q times stronger
than the one used to obtain its design load-effects by linear elastic analysis.
Parametric studies in the seismic analysis context typically require large computational effort
as a result of time domain analyses, generally for several earthquakes characterizing the seis-
mic input and, quite often, for several intensities of the seismic action. Thus, adequate analysis
models have to be carefully chosen, through which the various non-linear behaviour mecha-
nisms can be simulated with acceptable realism, at a manageable computational cost.
For RC frame structures, constituting the structural type the present work refers to, member
models are deemed particularly suitable to carry out seismic non-linear analysis, since they
provide the best compromise of response detail, efficiency and simplicity. Each member
(beam, column) is typically discretized by a single element in which the behaviour is most
often controlled by phenomenological models at certain critical sections; a description of dam-
age and non-linear effects along the member can be obtained, whose detail depends on the for-
mulation underlying the element model. Therefore, besides the global structure behaviour,
member models provide some insight into the member response and major goals of parametric
4 Chapter 1
studies can be achieved, such as the local ductility or damage assessment and their spatial dis-
tribution throughout the structure.
Many member models have been proposed to date, some of them assuming the inelastic behav-
iour lumped in “point hinges” at member ends, while others consider the inelasticity spreading
along the member in order to better describe the actual behaviour. Most member models were
cast in the framework of the classical stiffness method using cubic hermitian polynomials to
approximate the element displacement field as for linear elastic elements. However, for distrib-
uted inelasticity elements this approach deviates from the actual behaviour because the cubic
representation of displacements becomes less adequate as the member stiffness distribution
changes due to inelasticity development. Upon recognition of this problem, some attempts
were proposed for improved representation of internal deformations along the element but the
most consistent approaches were developed in the flexibility formulation context (Ciampi and
Carlesimo (1986), Taucer et al. (1991), Spacone et al. (1992)).
In flexibility based elements use is made of force interpolation functions strictly derived from
equilibrium conditions. This constitutes a major advantage over stiffness based elements
because such functions are exact regardless of the damaged state of the member and, addition-
ally, they can be straightforwardly derived for frame elements. However, there is a price to
pay: since no displacement interpolation functions are available, the element state determina-
tion cannot be directly performed. An internal iterative scheme is thus required to compute the
element resisting forces associated with the imposed displacements at element nodes; during
that iterative scheme the state determination is performed at a few control sections and both the
flexibility matrix and the displacements of the element are obtained from integration of the
section flexibility and deformation distributions, respectively. The scheme is driven by gradual
elimination of residual displacements while strictly preserving equilibrium along the element.
The formulation is general in the sense that, provided the force interpolation functions are
defined, it can be used with any type of section modelling technique, viz the fibre model and
global section models of phenomenological or differential type; in addition, it is very suitable
to accommodate associations in series of several elements behaving non-linearly, each
accounting for a specific source of non-linearity.
In a few words, not only the formulation of flexibility based elements appears a rather promis-
INTRODUCTION 5
ing and elegant technique for frame analysis, as it also brought back to light the somewhat
“forgotten” role of the force method in structural analysis.
1.2 Objectives
With the advent of flexibility based elements a few elements were proposed with considerable
level of sophistication and quite good adequacy for describing the non-linear hysteretic behav-
iour of frame members. Particular reference is made to the fibre beam-column element (Taucer
et al. (1991)), the first consistently developed and implemented in the flexibility formulation
framework; however, despite the model ability to describe complex hysteretic behaviour, its
use may easily become prohibitive in seismic analysis due to the extremely high computational
and data management cost.
Other flexibility based models were proposed by Spacone et al. (1992) and Filippou et al.
(1992), in which global section constitutive laws are adopted to control the RC non-linear
behaviour. In the former, a differential constitutive relationship is used to monitor several pre-
defined control sections; in the latter, a phenomenological behaviour law is adopted and the
spread of post-yielding inelasticity is considered through progressive development of plastic
zones. However, neither of them accounts for the cracking spread along the member because
the non-linear response is assumed to start from the cracked stage.
In the present work, special attention was devoted to the development and implementation of a
new flexibility based element model, which is robust, efficient and economic, both in discreti-
zation and computation time requirements, while being able to adequately describe the cyclic
response of RC members for different behaviour stages - uncracked, cracked and yielded.
The proposed element is particularly suitable for the analysis of members under cyclic defor-
mation reversals in bending, possibly combined with low levels of axial load. Bi-axial bending
is considered, though assuming non-linear behaviour only for one bending plane; the other is
assumed to behave linearly. Moreover, the actual behaviour is approximated by uncoupling the
effects of axial force from the bending behaviour, as a result of the assumption of low axial
force values.
In line with this new element proposal, the following objectives were envisaged:
• To develop the element model in the framework of the flexibility formulation, approxi-
6 Chapter 1
mately following the basic steps for the element state determination as proposed in previous
pioneering works (Taucer et al. (1991), Spacone et al. (1992)). The element model should
fulfil a basic discretization feature of member models, i.e. one structural member is to be
modelled by only one element, thus requiring due consideration of element applied loads
(distributed or concentrated). The non-linear behaviour should be controlled at the global
section level with a multi-linear phenomenological model and the number of control sec-
tions should be the minimum possible at each load step.
• To implement the element model in a general purpose computer code for the non-linear
static and dynamic analysis of structures.
• To develop and to implement in the same code, a new algorithm for the definition of the
basic curves of the section model for the most common cases of rectangular and T-shape RC
sections.
• To apply the proposed element to the numerical analysis of a four-storey full-scale RC
building pseudo-dynamically tested under two different earthquake loading levels and
quasi-statically loaded up to failure by means of cyclic tests of increasing intensity. This
should serve to check the model ability to simulate the global structure behaviour through-
out distinct stages, both in quasi-static or dynamic conditions. On the other hand, an addi-
tional insight on the structure response could be obtained, in particular concerning the
distribution and quantification of damage in the various members.
Another major aim of the present work is to provide some contribution to the seismic behav-
iour assessment of reinforced concrete frame structures. In view of the forthcoming approval
of Eurocode 8 (EC8) as a european standard, particular concern is devoted to assess the seis-
mic performance of structures designed according to that code. Therefore, as part of a euro-
pean-wide project of “Prenormative Research in Support of Eurocode 8” (PREC8) set up by
the European Commission and National Authorities, the numerical seismic analysis of some
EC8 designed building structures (9 out of a total of 26) was sought in order to assess “the
interrelation between a number of design parameters used in EC8, which, in a combined form,
influence the non-linear behaviour of structures subjected to earthquake motion” (Carvalho et
al. (1996), Pinto and Calvi (1996)).
1.3 Chapter organization
INTRODUCTION 7
The present Thesis is organized in eight Chapters and five Appendices, covering the descrip-
tion of the new flexibility based element development and implementation (Chapters 2 to 4)
and its application to the seismic analysis of RC frame structures (Chapters 5 to 7). The
Appendices include further details on specific topics which are deemed unnecessary for an
adequate reading of the global development and implementation schemes.
Following the introduction, Chapter 2 focuses on the modelling techniques of RC frames
under earthquake horizontal actions. The typical features of structural member response to
monotonic and cyclic loading are recalled in order to highlight the basics underlying the deri-
vation of phenomenological models. Numerical tools for non-linear modelling of reinforced
concrete buildings are briefly reviewed and special attention is given to member models; the
evolution of distributed inelasticity member models and the most relevant phenomenological
hysteretic models developed to date are addressed. The formulation of the most recent flexibil-
ity based elements is recognized as rather promising and suitable for frame member modelling,
for which it has been chosen as the framework for the element model development to be pur-
sued herein. Therefore, the general flexibility formulation for frame elements is addressed in
contrast with the classical stiffness method and the element state determination is identified as
the most critical task because displacement shape functions are not available. The iterative
scheme required for the element state determination is then presented as proposed in previous
pioneering works; no specific requirement is considered for the section model and due account
is taken of distributed or concentrated loads that may be applied to the element.
Chapter 3 is devoted to the full description of the new flexibility element development. A glo-
bal section constitutive law is adopted, consisting of a multi-linear step wise model based on
trilinear envelope curves and a number of hysteretic rules; it is basically a Takeda-type model
controlling the moment-curvature relationship, which accounts for the cracking and yielding
stages and for other typical phenomena of reinforced concrete, viz the stiffness and strength
degradation as well as “pinching”.
Two types of control sections are considered: the fixed control sections, consisting of the two
element end and one mid-span sections, and the moving sections, accounting for the so-called
yielding, cracking and null-moment sections. For a given load step, yielding sections define
the transition between sections having already yielded and those still in the pre-yielding range;
cracking sections establish the transition between cracked and uncracked sections, and null-
8 Chapter 1
moment sections are considered to allow for possible different behaviour in positive and nega-
tive bending directions. The set of fixed and moving sections defines distinct zones (yielded,
cracked and uncracked) which change during the loading history and allow to perform the inte-
grations inherent in the element state determination. Although an existing Takeda-type model
is used, a modification has to be considered consisting in a special transition from uncracked to
full-cracked behaviour in order to provide an approximate control of cracked zones.
The inclusion of uniformly distributed or concentrated element applied forces is discussed and
the development is cast for that purpose. However, the consideration of distributed forces was
found to require a rather cumbersome implementation scheme for the present stage of the algo-
rithm development; thus, it was decided to approximately account for the distributed forces by
means of an equivalent concentrated force applied in the mid-span section.
The element state determination is carried out by means of an internal element iterative
scheme, during which the flexibility distributions are defined in the distinct zones in order to
obtain the element flexibility matrix; the moving nature of cracking sections, where a curva-
ture discontinuity is considered, has been found to introduce additional flexibility contribu-
tions which are duly taken into account in the element matrix. The restricted number of fixed
control sections and the use of moving ones, requires particular attention to the evaluation of
displacement residuals which are computed through a different scheme than the one used in the
general flexibility formulation. Furthermore, an event-to-event scheme inside the element iter-
ative process has been found adequate to carefully control the non-linear behaviour of the
yielded end-zones. In addition, some convergence problems related with the cracking transi-
tion required the inclusion of a line-search scheme during internal iterations.
The basic steps of the general non-linear algorithm for static and dynamic analysis are summa-
rized, outlining the full sequence of steps of the element state determination. At the structure
level, the Newton-Raphson method is used to solve the non-linear equation system and the
classic Newmark scheme is adopted for the time integration of the dynamic equilibrium equa-
tions.
Chapter 4 describes the main interventions for the implementation of the new element model
in the general purpose computer code CASTEM2000. Because this object-oriented code
exhibits particular features, very different from traditional specific purpose programs, a brief
INTRODUCTION 9
review of CASTEM2000 basics is given, illustrated by a rather simple structural analysis prob-
lem; the major concern is to introduce some key concepts such as objects, commands, opera-
tors and procedures that integrate the code environment.
As an auxiliary tool to perform a pre-processing task, a new algorithm was developed and
implemented to define the trilinear envelope curves for rectangular and T-shape sections.
Instead of making a fibre-type analysis, which can be regarded as a general technique for bend-
ing analysis of RC sections, an algorithm was specifically designed for rectangular and T-
shape sections based on a systematization of the internal equilibrium conditions for such type
of sections. Realistic material models are used, viz a bilinear one for steel (with strain harden-
ing) and a parabolic/linear-softening/residual-plateau model for concrete; confined and uncon-
fined zones are duly distinguished in the section. The cracking point in the moment curvature
envelope is defined by classic closed form expressions, while the remaining turning points
(yielding and ultimate) are obtained by a unified process, though with different criteria for the
point definition.
A few validation tests at the single element level are also reported in Chapter 4, by comparison
with experimental results from RC members tested under monotonic and cyclic loading; the
main features of the model response can be identified and possible limitations can be detected
which should be taken into account when analysing complete structures.
Chapter 5 presents the experimental tests in the four-storey full-scale RC building carried out
in the European Laboratory for Structural Assessment (ELSA) of the Joint Research Centre
(JRC), Ispra, Italy, where the author has developed the present work. Included in a wider
research programme involving several european partners, this testing campaign aimed at a
comparison between the actual and the expected design behaviour of a high ductility structure
designed according to Eurocodes 2 and 8. The experimental activity consisted of preliminary
tests (free vibration and stiffness) for dynamic characterization of the structure, followed by
unidirectional seismic tests pseudo-dynamically performed in the bare structure for two inten-
sity levels (0.4 and 1.5) of the reference earthquake, artificially generated to fit the EC8 elastic
spectrum used in the design. Further seismic tests at 1.5 intensity were carried-out in the struc-
ture with unreinforced masonry infills in the external frames, both in a totally infilled configu-
ration and in a partially infilled one where the first storey was kept bare in order to simulate
soft-storey conditions. Finally, a quasi-static cyclic test was performed in the bare structure up
10 Chapter 1
to a near-failure stage.
A brief description of the pseudo-dynamic testing technique is included and a few comments
are made on the structure design and layout, the characterization tests of material properties
and the reduced scale member tests prior to the full-scale ones. The results of tests on the com-
plete structure are thoroughly described in order to provide a global view of the structural per-
formance throughout different behaviour stages; this was deemed important to support the
discussion of numerical simulations in the subsequent chapter.
Chapter 6 presents and discusses the application of the proposed element to the numerical sim-
ulation of the full-scale tests performed in the four-storey structure at the ELSA laboratory.
Because these tests covered different behaviour stages (pre-yielding and post-yielding with
increasing ductility levels up to failure) and due to their essentially static nature (from a strict
experimental standpoint), they provide an excellent means of calibration and assessment of
numerical models, through either static simulations with the experimentally imposed displace-
ment or dynamic calculations with the accelerogram used in the experiment. All the performed
tests (except the preliminary ones) are simulated, both in the bare and the infilled structure
configurations. Therefore, the presentation and discussion of modelling assumptions includes
also references to the infill panel modelling which is accomplished by pairs of diagonal struts
ruled by uniaxial force-deformation models whose parameters are identified by means of bi-
dimensional refined analysis.
For its importance in the seismic assessment of structures, the damage quantification is briefly
addressed, referring and discussing some available proposals of damage indices. The widely
used Park and Ang index is adopted to calculate damage via the chord rotation at each element
end section. Particular attention is devoted to the quantification of parameters required in that
index, viz the ultimate rotation and the dissipated energy by hysteresis.
The numerical simulations are divided into static and dynamic ones. The former are performed
by applying the displacements actually imposed in the experiment and provide structural
strength responses to be compared with experimental results without involving additional
dynamic effects. Static simulations are carried out for all tests (the seismic ones in both bare
and infilled structures, and the final cyclic test) and particular care is taken to follow the actual
test sequence as close as possible; the real conditions of load application, including the unload-
INTRODUCTION 11
ing phases between tests, are simulated by adequate adaptation of boundary conditions, for
which the modularity and object-oriented features of CASTEM2000 proved to be rather suita-
ble. Static pushover analyses are performed under inverted triangular distribution of monoton-
ically increasing forces in order to provide estimates for the maximum base-shear and the
global yielding displacement. The quality of static simulation results is discussed through com-
parison with the experimental ones, mainly in terms of storey shear forces, shear-drift dia-
grams and dissipated energy, as well as rotations in beams. Additional analytical response
variables, such as rotation ductility and damage, are also presented even though no experimen-
tal counterpart is available. Results are sequentially analysed, first for the bare structure seis-
mic tests, then for the infilled structure and finally for the quasi-static cyclic test. Dynamic
simulations are restricted to the bare structure seismic tests and preceded by the comparison of
calculated and experimentally measured structural frequencies for several testing stages.
Results of dynamic calculations are compared with experimental ones, mainly in terms of sto-
rey displacements, shear forces and shear-drift diagrams; evolutions of the total dissipated
energy are also compared. The viscous damping characterization is discussed at length for both
seismic tests (0.4 and 1.5) in view of the dissipation capacity of the model for the different
behaviour stages and some comments are made on the validity of comparing analytical and
experimental dissipated energy in the dynamic context.
Finally the proposed flexibility element modelling strategy is compared with a more classical
one, in which each member is discretized by one linear elastic element and two non-linear ele-
ments to simulate the plastic hinge zones at the member ends. Besides the presentation of the
assumptions for this modelling strategy, results are compared for the seismic tests on the bare
structures, through both static and dynamic simulations; mainly the quality of results is com-
pared (by means of the same response variables as before), while no specific comparison is
made concerning model efficiency due to the different discretization needs in the two model-
ling options.
Chapter 7 is devoted to the presentation and discussion of the numerical seismic assessment of
some RC frame structures included in the PREC8 programme, resulting from our activities in
the JRC. Because the major scope is concerned with the implications of EC8 provisions on the
seismic behaviour of building structures, a brief review of EC8 basics is included mainly to
recall the principles of design philosophy that must be kept in mind for result analysis. The
analysed structures have two basic configurations, one regular and another irregular, both with
12 Chapter 1
eight storeys and a rectangular plan of 15x20 m2. Each configuration was designed for two
design accelerations (0.15g and 0.30g) and different ductility classes; in addition, the irregular
configuration was also designed using the simplified static analysis method allowed in EC8.
Nine trial case structures are modelled with the proposed flexibility element, following model-
ling assumptions similar to those of the building analysed in Chapter 6.
Non-linear dynamic planar analyses are performed for each trial case in both horizontal direc-
tions, under the action of four accelerograms (artificially generated to fit the EC8 elastic spec-
trum) with a number of increasing intensities. Additionally, pushover static analyses are
carried out to allow the quantification of structural global overstrength in terms of the base-
shear force, whose results are extensively discussed and compared between the trial cases. The
spread of seismic effects on the structures is analysed by means of cracking patterns and spatial
distributions of ductility and damage. The overall structural response is described by global
parameters such as the total drift, inter-storey drift and damage (maximum and global aver-
age). Where pertinent, comparisons are made with EC8 limits (namely for the serviceability
limit state), though the most systematic comparative analysis is made between trial cases
according to the different design accelerations and ductility classes.
It is also included an attempt of system reliability analysis of the various structures, aiming at
estimates of probability of failure, mainly for comparative purposes between the different trial
cases rather than an absolute evaluation of the structural safety. The quantification of probabil-
ities of failure is based on the damage values in the critical zones (plastic hinges); the adopted
methodology and assumptions for this complex topic are described on the basis of previous
studies addressing the computation of local hinge probabilities of failure, the probabilistic
quantification of the seismic intensity, the damage capacity characterization and the computa-
tion of estimates for the system probability of failure. Upper and lower bounds of the latter are
calculated and compared between the analysed structures aiming at an assessment of design
parameter (e.g. ductility class, design acceleration) influence on the structural safety.
Finally, the most relevant results and conclusions of this work are summarized in Chapter 8,
where suggestions for future research are also pointed out in line with the developments made
in this thesis.
Chapter 2
MODELLING OF REINFORCED CONCRETE FRAME
STRUCTURES - STATE OF THE ART
2.1 Introduction
It is currently well established the need of adequate non-linear models to account for the hys-
teretic behaviour of RC frame structures. The major purposes are related with the calibration of
some design parameters, such as q-factors and the inherent ductility requirements in structural
dissipative zones, as well as possible irregularities related with both the structure geometry and
the distributions of mass and stiffness. These calibration studies basically aim at giving support
to simplified procedures (e.g. linear elastic analysis using q-factors, rules for irregularity clas-
sification, detailing provisions to assure a certain ductile capacity), constituting sets of practi-
cal tools for current seismic design.
Under high intensity earthquakes, the structural behaviour of buildings is usually controlled by
its resistance to horizontal actions, from which the inelastic behaviour concentrates in the end
zones of structural members. However, besides other reasons related with design options and
structural configurations, the spatial distribution of such zones is also affected by the possible
preponderance of gravity loads, which, therefore, shall be taken into account.
Moreover, recent studies which point out the damaging potential of vertical ground motion
(Elnashai and Papazoglou (1995), Papazoglou and Elnashai (1996)), have shown in some cases
the lack of conservatism of vertical earthquake forces as estimated by current code spectra. In
addition, proposals for the inclusion of vertical motion effects in seismic design can be found
in the literature (e.g. Elnashai and Papazoglou (1997)). Despite this recent evidence and the
problem relevance for the seismic response of structures, particularly in near-source regions,
14 Chapter 2
the present work will focus only on the effects of the horizontal seismic components.
The scope of the present chapter is to provide a brief, yet comprehensive, description of mod-
elling techniques for the non-linear hysteretic behaviour of reinforced concrete frame struc-
tures, cast in the form of a historical review, from which the present state-of-the-art can be
addressed. Thus, the main features of structural member behaviour under monotonic and cyclic
loading are first recalled in 2.2, as they constitute the basic evidence for derivation of most
phenomenological behaviour models.
Numerical modelling strategies for reinforced concrete buildings are discussed in 2.3, with
particular emphasis on member models, which can provide sufficient detail of structural
response while keeping with manageable algorithms of analysis. Some of the most relevant
phenomenological hysteretic models are referred and the evolution of distributed inelasticity
member models is described, since, by contrast with point-hinge models, they allow to closely
follow the actual force distributions and the stiffness modifications along the member.
Among the most recently developed member models, the flexibility based ones appear as a
very elegant and suitable option for frame member modelling. Indeed, the flexibility formula-
tion allows to account for inelasticity spread in a quite natural way, while keeping complete
freedom for the choice of section behaviour model. In view of the member model developed in
the present work, which belongs to the flexibility based model family, the general flexibility
formulation for beam-column elements is described with some detail in 2.4, yet mainly based
on previous works available in the literature.
2.2 Hysteretic behaviour of reinforced concrete members
The global behaviour of a given structure reflects the behaviour of its members and corre-
sponding interconnections. Hence, experimental evidence from tests on single reinforced con-
crete members is of major importance for understanding the complete structure behaviour,
under either monotonic or cyclic loading conditions. Even if experimental tests do not exactly
match the actual conditions of members in a real structure, the source of relevant non-linear
phenomena remains essentially the same and, therefore, experimental results provide
extremely valuable information on non-linear response mechanisms.
In this context, the present section deals with the presentation of the main features of rein-
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 15
forced concrete member behaviour, as they are commonly found in experimental tests of single
elements (e.g. cantilever or simply supported beams or columns) or beam-column sub-assem-
blages. Although widely known within the seismic (or generally cyclic) structural behaviour
context, such features are briefly recalled herein for completeness purposes in view of the sub-
sequent sections. Excessive detail is avoided, in particular that related with the specific behav-
iour of constitutive materials (steel and concrete) and their interaction, since they can be found
in standard literature (Park and Paulay (1975), Paulay and Priestley (1992)) or in available
broad reviews (Coelho (1992), Costa (1989)). Furthermore, only uniaxial bending response is
considered as it exhibits the most relevant behaviour features.
For adequate understanding of the hysteretic behaviour, the member response under monotonic
loading shall be addressed first. Two typical behaviour types are usually found: that of beam
members, hence with null or rather low level of axial force, and that of columns, normally with
non-negligible axial force level.
Figure 2.1 shows illustrative global response diagrams (e.g. tip force-displacement of a canti-
lever specimen) for both behaviour types, where, after an initial quasi-linear branch up to the
cracking force , another branch follows in correspondence with the cracked behaviour of
a certain zone along the member. In the vicinity of the highly non-linear response range (i.e.,
nearby the yielding threshold ), the behaviour is very different in beams and columns: the
former case is mainly controlled by the steel behaviour (for which a schematic steel stress-
strain diagram is included), while the latter reflects the typical compressive behaviour of con-
crete due to the predominance of section compression forces required for axial force equilib-
rium. By increasing the axial force, the ultimate displacement reduces due to the decreasing of
the section ductile capacity; in turn, the yielding threshold may increase or reduce, depending
on whether the axial force is less or greater than the so-called balanced value which defines the
transition from yielding controlled by the tensile steel towards yielding dominated by the com-
pressed concrete.
Monotonic response diagrams are also found to constitute envelopes of cyclic response, which
makes them rather useful for the definition of analysis models. Examples of cyclic behaviour
are included in Figure 2.2, referring to quasi-static cyclic tests (Carvalho (1993)) on a uniform
section cantilever beam of 1.5m span (as illustrated) under applied tip-displacement sequences
and with null axial force.
Fc( )
Fy( )
16 Chapter 2
Figure 2.1 Typical global response diagrams of beams and columns for monotonic loading
These diagrams exhibit the most typical features of the hysteretic response of reinforced con-
crete members; particularly referring to Figure 2.2-a), the following is highlighted:
• The load path 0-1 follows the monotonic response, clearly showing stiffness reductions due
to concrete cracking at C- and to reinforcement yielding at Y-.
• Unloading along 1-2 exhibits stiffness close to the initial cracked one (i.e., the slope of a
line connecting the origin with the yielding point Y-), indeed a behaviour similar to that of
the steel.
• The first reloading (inversion of force sign) follows the path 2-3, initially with a high stiff-
ness branch, after which the previously compressed but still uncracked section zones,
become progressively cracked, thus leading to stiffness reduction at point C+; this diagram
part develops as if the positive envelope were shifted backwards to the residual displace-
ment at point 2, therefore reaching yielding of bottom reinforcement still for negative dis-
placement at point Y+.
• Instead, for the subsequent reloading phases (6-7 or 10-11 in the positive direction and 4-5 or
8-9 in the negative one) apparent stiffness drops occur, which are generated by two distinct
(but mixed) phenomena:
- full-depth open cracks due to plastic elongation of steel, cause the section resisting
moments to be provided by a steel couple without any concrete contribution, which,
combined with bond deterioration between concrete and steel in the vicinity of crack
F
d
Fy
Fc fs
εs
Steel
F
d
Fy
Fc fc
εc Concrete
a) Beam (null or low axial force) b) Column
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 17
lips, induce increased deformations to reach the same force of previous cycles;
- the typical Bauschinger effect exhibited by steel under cyclic loading, due to which, and
after post-yielding load reversal, a much pronounced non-linear response is obtained for
steel stress levels significantly lower than the monotonic yielding stress.
Figure 2.2 Cyclic global response examples (Carvalho (1993))
-50
-40
-30
-20
-10
0
10
20
30
40
50
-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140
Displ (mm)
Forc
e (k
N)
b) Test S1-V5
-50
-40
-30
-20
-10
0
10
20
30
40
50
-140 -120 -100 -80 -60 -40 -20 0 20 40 60 80 100 120 140
Displ (mm)
Forc
e (k
N)
0 42
3
1
7
11
59
610
8
12
13
15
C-
C+
a) Test S1-V4
Y+
Y-
F+
0.2
4φ123φ12
0.3
1.5
δ+
Concrete: C25/30Steel: B500s
(Tempcore)
18 Chapter 2
• Reloading in the negative direction shows the so-called pinching effect: open cracks may
eventually close in the first reloading phase, hence activating again compressive stresses in
the concrete and increasing the stiffness as shown in the second reloading phase; crack clo-
sure is actually achieved in the negative direction because the larger top steel area can gen-
erate sufficient tensile forces to balance forces in the steel yielding in compression (which,
in turn, are necessary to compensate the previous plastic elongation). However, the same
does not occur for positive reloading and, consequently, cracks may never close again for
that direction. Furthermore, it is worth mentioning that, in symmetrically reinforced sec-
tions, due to the plastic elongation of steel, the load is likely to be carried very largely just
by the couple of steel forces; therefore, should significant axial compressive force be
present, higher compressive stresses are demanded and cracks tend to close earlier, which
means that the pinching effect may also become apparent in columns, particularly if large
excursions in the post-yielding range are likely to occur.
• For the same level of peak displacements, the resisting force is progressively decreasing as
a result of buckling of reinforcing bars, which started developing in cycle peaks 5 and 9 and
finally led to failure. This phenomenon of strength decay may also be triggered-off, or
aggravated, by less effective force transfer from steel to concrete due to bond deterioration
or by degradation of the concrete compressive strength, both causes arising from the cyclic
effect of crack opening and closing and of bars pulling in and out from concrete.
Another noteworthy feature is related with the unloading stiffness degradation, quite apparent
in Figure 2.2-b). The auxiliary dashed lines connecting the starting and end points of unloading
branches show decreasing slope for increasing inelastic excursions, in agreement with the pro-
gressive reduction of average stiffness of cycles for larger amplitude; actually, Figure 2.2-b)
shows reloading branches typically pointing at the maximum deformation reached in the previ-
ous cycle, which lead to that average stiffness reduction.
Finally, it must be pointed out that the above referred pinching effect may be also generated by
strong influence of shear forces or by reinforcement slippage.
For members having low shear span ratio ( , where l is the distance between maximum and
null moment sections and d is the effective depth of the section), say below 2.0-2.5, high shear
stresses have to be transferred across plastic zones and the member stiffness may become
reduced due to shear deformations; these can assume particular importance when cracks are
l d⁄
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 19
open over the entire depth of the section, causing shear forces to be transferred by aggregate
interlock (a rather brittle mechanism and prone to sliding deformations) and by dowel action
(which may severely affect the steel-concrete bond in the vicinity of cracks).
Reinforcement slippage inside the concrete core, due to bond deterioration, is also responsible
for pinched diagrams, particularly for beam-column joints. Slippage of tensioned steel bars
induce higher deformations taking place without (or with low) increase of resistance due to
lack of bond mechanism, i.e. at very low (or null) stiffness; therefore, only after the slippage
deformations of the previous cycle have been sufficiently recovered to produce crack closure,
the stiffness may increase again due to re-activation of the compressive stresses on concrete.
The described features are the most relevant ones as shown by experimental evidence and have
constituted the basis for development of the so-called phenomenological analysis models for
numerical simulation of member behaviour. In these kind of models the observed response fea-
tures, viz:
• the unloading stiffness degradation,
• the reloading stiffness deterioration, due to both the Bauschinger effect and the open cracks,
• the pinching effect, resulting from crack closure and related with reinforcement asymmetry,
high shear forces or reinforcement pull-out and slippage and
• the strength deterioration caused by concrete degradation in compression or buckling of
reinforcement,
are approximately simulated by means of a set of rules relying upon a skeleton, or basic, curve
(typically the monotonic envelope or even others based on that one). These rules are usually
set up in an empirical basis and calibrated with experimental results.
2.3 Numerical modelling strategies
2.3.1 General
Numerical analysis of reinforced concrete building structures under seismic loading can be
performed, as any other type of structure, by means of the Finite Element Method. An ade-
quate response assessment requires realistic non-linear behaviour modelling of the constituent
material, which, for reinforced concrete, is better accomplished by adopting separate models
for concrete, steel and, eventually, their interaction through bond.
20 Chapter 2
In the most general three-dimensional analysis, the behaviour of structural members is
described by integration of the stress-strain relations of the constituent materials at the control
point level (typically the Gauss points). However, despite the computer power available nowa-
days and the large progress made to date in the field of constitutive modelling of materials, this
refined analysis may easily become prohibitive for multi-storey structures, particularly if time
step dynamic calculations have to be performed for seismic assessment purposes. Indeed, time
domain seismic analysis often requires several calculations for different input motions in order
to draw reliable conclusions about the seismic behaviour, which means that computational
demands may increase exponentially. Therefore, this type of detailed modelling has been
restricted to the analysis of individual members or sub-assemblages, mainly to have a better
understanding of some local phenomena rather than the global response.
On the other hand, the specific behaviour of building structures under horizontal actions (those
of major concern in the present work), for which the horizontal degrees-of-freedom play an
important role, permits the idealization of simplified structural models for the assessment of
global response parameters.
Typically, relatively simple models consisting of few degrees-of-freedom are used to simulate
the behaviour of groups of structural members, or even the entire structure. That is the case of
the so-called storey-models, as the shear-beam model, consisting of one horizontal displace-
ment per storey and having the behaviour governed by relations between the storey shear force
and the corresponding inter-storey drift (which can also be used for spatial analysis by adopt-
ing two horizontal translations and one rotation). Such type of models can be used when the
inelastic behaviour source is mainly located in columns, typically in the so-called weak-col-
umn strong-beam frame systems. The inter-storey behaviour is simulated by non-linear phe-
nomenological models where cracking and yielding are included along with several hysteretic
rules taking into account the main features referred in 2.2. In the limit, an even simpler model,
involving only a single degree-of-freedom in each horizontal direction and ruled by base shear
- top displacement relationships, can be adopted for structures fairly controlled by its first
mode of vibration.
However, the use of such simplified models is restricted to regular structures, both in stiffness
and mass distribution terms, and only global response features can be obtained, viz time histo-
ries and peak values of horizontal displacements or base shear. Further detail concerning ine-
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 21
lastic effects and damage in individual members, obviously cannot be predicted.
In between these two extreme types of modelling strategies, the so-called member-type models
appear as the best available compromise of structural response detail and efficiency/simplicity.
A single element in the model is used for each member in the structure (a beam, a column, the
inter-storey portion of a shear-wall, etc.), and the behaviour is most often described by phe-
nomenological models, although stress-strain constitutive relationships at the material level
can be also used along with some member type modelling.
Although not accounting for minor details of geometry and exact reinforcement arrangement,
member models provide a description of the seismic damage and non-linear effects along the
member length, the accuracy and detail of which being dependent of the powerfulness of the
underlying formulation.
Member type models are worldwide used for non-linear dynamic analysis of reinforced con-
crete building structures, because computational requirements are quite manageable (even for
three-dimensional analysis) in view of the outcome they can provide. For this reason, and since
the model developed in this work fits in this modelling strategy, member models are further
discussed next in order to establish the framework of the present developments.
2.3.2 Member type models
The simplest and earliest member models have been first developed for uniaxial bending dom-
inated elements, where shear forces are not of major importance. This appears quite reasonable
because, on one hand the flexural behaviour has been always better understood than the shear
one, and, on the other hand, the earthquake-resistant design trends emerged in the past 25
years, being based on the weak-beam strong-column philosophy, lead to relatively slender
beams mainly controlled by uniaxial bending rather than shear.
However, other structural members, viz the first floor columns where inelastic behaviour is
likely to occur (even for the weak-beam strong-column type of structures), involve interaction
of axial load and bending (possibly biaxial) which may be important to incorporate in the
model. Soon this aspect has attracted the researchers attention and some models were proposed
over the past two decades to take it into account, particularly if significant fluctuation of axial
load is likely to occur during the seismic response. Similarly, the bending-shear interaction
22 Chapter 2
problem has been investigated, although fewer research work can be found due to the problem
complexity in the seismic analysis context.
In the present work, emphasis is put in uniaxial bending behaviour where the presence of static
axial force (due to gravity loads) is taken into account, but not its variation during the
response; the complete interaction problem is thoroughly addressed and the most significant
models are reviewed in Fardis (1991). Furthermore, any inelastic shear behaviour interaction is
not considered herein.
Member modelling comprises two main issues:
• the element model, where assumptions are made concerning the stiffness distribution and
the force or displacement fields along the member, which allow more or less simplified for-
mulations;
• a hysteretic behaviour model, defining the generalized force-displacement relationships to
be adopted in the inelastic zones of the member.
In the context of building structure response to horizontal actions, particularly seismic ones,
the inelastic behaviour is typically located at and near the member ends, where bending
moments are maximum, and the inelastic zone development is directly related to the moment
distribution along the member.
Thus, two basic models proposed in the 1960s have set up the framework for member model-
ling, both assuming the inelastic behaviour lumped in “point hinges” at member ends:
• The so-called two-component model proposed by Clough et al. (1965) consists of two beam
elements associated in parallel as shown in Figure 2.3-a). One is an elastic-perfectly-plastic
beam and accounts for the elastic behaviour: prior to yielding at one of the member ends, it
behaves elastically, the stiffness matrix being that of the classical elastic beam; after the
onset of yielding at one end, the tangent stiffness matrix contribution is that of a beam with
a hinge at that end and, when yielding occurs also at the other end, the stiffness contribution
becomes that of a beam with hinges at both ends. The other element behaves continuously
elastic, having the stiffness equal to that assumed in the post-yielding range ( , where k is
the elastic stiffness and p is the fraction attributed to hardening), which means that the first
component has to account for the remaining elastic stiffness prior to yielding.
The superposition of diagrams yields an elastoplastic behaviour and, because the first com-
pk
1 p–( )k
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 23
ponent unloads/reloads parallel to the elastic branch and the second component remains
elastic, the resulting hysteretic behaviour is bilinear. Due to the parallel association of com-
ponents, this model is a stiffness based one and stiffness matrix contributions can be
directly obtained and added together.
Figure 2.3 Member modelling: a) two component model (Clough et al. (1965)) and b) one com-
ponent model (Giberson (1967))
• The one-component model, as introduced by Giberson (1967) and shown in Figure 2.3-b),
is an association in series of one elastic element with two non-linear rotation springs at each
end where the inelasticity is lumped; the former accounts for elastic deformations and the
latter are governed by a non-linear behaviour model to account for inelastic rotation compo-
nents. The fact that point hinges (zero length) are considered leads to inelastic rotation at a
given hinge uniquely determined as a function of the moment at that hinge (no coupling of
deformations) and the corresponding flexibility matrix becomes diagonal. By assuming the
inflection point at mid-span, the equivalence between each half member and a cantilever
beam allows the inelastic rotation to be estimated by curvature integration and lumped into
each hinge; hence, the moment-rotation relationship can be derived and used for the hinge
state determination. According to the association in series, the inelastic flexibility matrix
associated with the non-linear springs is then added to the flexibility matrix of the linear
elastic element.
MA MB
θA θB1 p–( )k
pk
Elastic sections turning to plastic for θ θy>
θA
θB
θAe
θBe
M
θ
M
θ
1 p–( )k
pk
1
M
θk
1
+ =
θAp
θBp
θA θBIF
Elasticnon-linearrotational springs
a) Two-component model
b) One-component model
24 Chapter 2
The two component model, although conceptually very simple, has the major drawback of not
being able to simulate stiffness degradation; instead, in the one-component model this problem
is overcome by the use of a hysteretic behaviour model at each hinge, incorporating rules for
that and other features described in 2.2. However, the one-component model is not problem-
free: the lumped inelasticity is just an approximation of the real spreading of the inelastic zone
as the moment distribution changes, and the moment-rotation relationships are derived under
the assumption of inflection point at mid-span which is not satisfied in many cases (e.g. col-
umns, beams with gravity loads or members with different contents of top and bottom rein-
forcement).
Despite these limitations, the uncoupling between inelastic rotations of member-ends and the
possibility of using any hysteretic model have rendered the one-component model rather popu-
lar and widely used. Additionally, its simplicity, low computational cost and numerical stabil-
ity and robustness have motivated the development of several hysteretic models used for both
local (moment-curvature) and global (force-displacement or moment-rotation) behaviour anal-
ysis.
Some of the most representative phenomenological models developed to date are briefly
referred next, following the chronological order:
• The Clough and Johnston model (Clough and Johnston (1966)) is based on a bilinear basic
curve with unloading parallel to initial stiffness, but including degradation of reloading
stiffness by pointing to the maximum deformation previously reached.
• Anagnostopoulos (1972) has complemented the previous model with the ability of degrad-
ing unloading stiffness according to the maximum reached deformation.
• The Takeda model (Takeda et al. (1970)) is based on a trilinear basic envelope, accounting
for elastic, cracked and hardening stages, which may be different in the two loading direc-
tions. It includes unloading stiffness degradation after the onset of cracking (thus account-
ing for dissipation already at the cracked stage) according to the maximum reached
deformation, as well as the same rule for reloading stiffness deterioration as for the previous
models. Small amplitude cycles are taken into account but neither pinching nor strength
deterioration are considered. Although requiring a wide set of hysteretic rules (16), this
model has become a reference milestone of phenomenological models.
• Otani (1974) and Litton (1975) have proposed simplifications of the Takeda model, by
using bilinear basic curves and a smaller set of slightly modified hysteretic rules.
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 25
• The Banon model (Banon (1980)) is also based on a bilinear curve, with hysteretic rules
similar to those of the Takeda model but already including the modelling of the pinching
effect.
• Roufaiel and Meyer have developed another model (Meyer et al. (1983), Roufaiel and
Meyer (1987)) which includes the features of the previous one and also takes into account
the strength deterioration due to cyclic effect beyond a certain critical deformation (related
with a critical compressive strain in concrete).
• Costa and Campos Costa have proposed a model (Costa and Campos Costa (1987), Costa
(1989)) which considers a trilinear basic curve (different in positive and negative direc-
tions), along with hysteresis rules similar to the Takeda model (slightly modified), including
the pinching effect and the strength degradation. Before yielding the model behaves biline-
arly without stiffness and strength reduction and small cycles are taken into account by
monitoring relative maximum deformations.
• The Park et al. (1987b) model basically includes the feature of the previous one, although
with slightly different rules: unloading stiffness degradation is achieved by using a common
target point for unloading branches and is activated after cracking occurrence; strength deg-
radation is taken into account with the concept of damage contribution from dissipated
energy as introduced by Park et al. (1984).
This type of models arise from the knowledge of experimentally observed phenomena and typ-
ically reproduce the stiffness variations during the loading process by means of piecewise lin-
ear type curves. Hysteretic rules are usually based on the history of some response parameters,
allowing the state determination to be performed without additional effort.
Other type of models provide continuous hysteretic relations, of which the Ramberg-Osgood
model is an example; it has been derived from the steel stress-strain constitutive law intro-
duced by Ramberg and Osgood (1943) and adapted by Jennings (1963) for the hysteretic
behaviour of reinforced concrete members.
Additionally, it is worth referring the so-called rate-type or differential constitutive models,
derived from the endochronic theory formulation in Ozdemir (1981) and Brancaleoni et al.
(1983), which is formally identical to viscoelasticity but where time is replaced by a deforma-
tion measure called “intrinsic time”. Examples are the Bouc-Wen model (Wen (1980)), widely
used in non-linear stochastic analysis, and the Spacone model (Spacone et al. (1992)), where
26 Chapter 2
the tangent stiffness is approximated by a continuous function of the current state and of model
parameters; this means that a non-linear incremental constitutive law has to be previously
obtained from integration of the differential relation (expressing the tangent stiffness) within
each load step. In the context of usual stiffness based finite element programs, this constitutes
a major drawback, but, as outlined by Spacone et al. (1992), it fits quite naturally in the flexi-
bility based algorithm described later in this chapter; nevertheless, the large set of required
model parameters to be selected, still represents a major problem in the adoption of such type
of models.
Most of the above referred hysteretic models have been largely used along with the one-com-
ponent model due to its versatility and simplicity. However, the already mentioned limitations
inherent in such member model, particularly those related with the difficulty of defining
moment-rotation laws and the inconsistency arising from the assumption of a fixed inflection
point at mid-span, led to the development of distributed inelasticity member models in order to
more closely follow the member behaviour coherently with its actual characteristics and load-
ing conditions.
2.3.3 Distributed inelasticity member models
The generalization of the one-component model to account for inelasticity spread along the
member was first introduced by Otani (1974). In his proposal, the non-linear rotational springs
for member inelastic behaviour are replaced by two inelastic finite length elements. However,
two other non-linear rotational springs are additionally considered (with zero length) to repre-
sent the fixed-end rotations at the beam-column interface due to reinforcement slippage inside
the concrete core of the joint. Upon recognition of the inconsistency of the one-component
model resulting from considering a fixed inflection point at mid-span when applied to generic
linear moment distributions, Otani has derived the inelastic flexibility matrix associated with
the two non-linear elements as a function of the current location of the inflection point. Each
element part at each side of the inflection point is identified with an equivalent cantilever,
where the free-end displacement and the rotation - fixed-end moment relations are prescribed.
However, by adopting some simplifying approximations, the approach results in a non-sym-
metric flexibility matrix, unless the further assumption of lumping inelasticity at member-ends
is introduced. Therefore, the proposal ends up in the lumped inelasticity group, although the
inelastic stiffness takes into account the variation of location of the inflection point. Despite
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 27
the non-significant final outcome of the model concerning spread of inelasticity, it was the first
to recognize the importance of fixed-end rotations in predicting the seismic response of rein-
forced-concrete structures.
Soleimani et al. have proposed the first model actually taking into account the gradual spread
of inelasticity along the member length as a function of the loading history (Soleimani et al.
(1979)). Inelastic zones gradually increase from the beam-column interface, while the rest of
the member remains elastic. The behaviour of inelastic zones is controlled by a moment-curva-
ture relationship of Clough type (Clough and Johnston (1966)) prescribed at the member end
sections, where point hinges are also considered to account for fixed-end rotations at the beam-
column interface. These are related to curvature at the corresponding end section by means of
an “effective length” factor assumed constant during the response.
Several other authors have proposed slightly different models along the same trend-line
(Arzoumanidis and Meyer (1981), Roufaiel and Meyer (1983), Roufaiel and Meyer (1987)),
basically to introduce refinements in the hysteretic model, viz the pinching effect accounting
for shear interaction, the strength degradation and the (constant) axial force interaction with
the basic curve of the model.
The Soleimani proposal was further elaborated by Filippou and Issa (1988), and cast in a more
complete context. The member is subdivided into different sub-elements, each accounting
exclusively for a single effect, namely: the elastic behaviour, assumed cracked before yielding
of reinforcement, the inelastic behaviour due to bending (either concentrated in point hinges or
spreading along the member) and the fixed-end rotations at the beam-column interface. Sub-
elements are associated in series, for which the relevant contributions to the flexibility matrix
are summed-up; the same applies for contributions of member-end rotations arising from each
sub-element.
The point hinge idealizations used in this model (for concentrated inelastic bending or fixed-
end rotations) are based on a bilinear moment-rotation envelope with constant post-yielding
stiffness. For inelastic bending, that stiffness is an approximation of the non-linear post-yield-
ing relation, calculated for a pre-defined ultimate moment capacity, and the hysteretic behav-
iour follows a Clough type model. For the fixed-end rotation, the moment-rotation relation is
derived using the detailed model proposed by Filippou et al. (1983), according to which a
28 Chapter 2
beam-column joint model, representing a particular connection under analysis, is subjected to
monotonic increasing beam end moments to give rise to the concentrated rotations due to bar
pull-out at the beam-column interface.
The spread of inelastic bending is considered along member end zones, with non-decreasing
lengths, where the yielding moment is exceeded. An average stiffness is assumed in these
zones entirely determined by the corresponding end section behaviour, which is governed by a
modified Clough type model, and a special algorithm is used to control the advancement of
inelastic zones.
This model was further improved in a subsequent study by Filippou et al. (1992) to include
another sub-element accounting for shear distortions in inelastic zones and shear sliding at the
beam-column interface. Additionally, constant axial force - bending moment interaction was
also included in the basic curve of the model.
Takayanagi and Schnobrich (1976,1979) have proposed another type of member model, by
dividing the element into several short sub-elements (finite length springs) along the member
axis, each of them governed by a non-linear moment-rotation (or curvature) model. Properties
(namely stiffness) are assumed constant along each sub-element length which is controlled by
the moment at its mid-point. The axial force - bending moment interaction is taken into
account by a limit surface for each spring and static condensation is used to lump the element
behaviour into the element end springs; this model fits in the so-called multi-slice model type.
However, it may suffer from the problem of unbalanced forces developing in internal sub-ele-
ments whose degrees-of-freedom are not explicitly controlled in the global non-linear algo-
rithm scheme; indeed, since the elimination of residual forces does not take them into account,
the internal resisting moments calculated via deformations at each sub-element do not neces-
sarily match the applied moments obtained via equilibrium conditions; consequently, equilib-
rium is locally violated, which often introduces numerical instability.
This model, as well as others with distributed inelasticity (Hellesland and Scordelis (1981) ,
Mari and Scordelis (1984)), are typically based on the classical stiffness method using the well
known cubic Hermitian polynomials to approximate displacements along the element. Both
the element stiffness matrix and the nodal equivalent forces are obtained by integration of the
section stiffness and force distributions, respectively, duly weighted by deformation shape
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 29
functions derived from the Hermitian polynomials, as for the linear elastic elements.
However, cubic polynomials give the exact solution for prismatic members having uniform
linear elastic properties and without loads applied in the span (thus, only linearly distributed
moments originating linear distribution of curvatures). Once the non-linear behaviour becomes
significant, the curvature distribution may fairly deviate from the linear one, which means that
the cubic representation of displacements is no longer adequate and numerical problems are
very likely to appear. In the general context of the Finite Element Method, the problem is over-
come by mesh refinement in the potential inelastic regions, but this does not fit with the scope
of member modelling.
Some proposals have been made with improved representation of internal deformations.
Menegotto and Pinto (1977) proposed to achieve this improvement by combined approxima-
tion of both the section deformations and flexibilities, the latter being assumed linearly distrib-
uted between a few controlled sections (thus, equivalent to hyperbolic stiffness distribution).
The recognition of the non-adequacy of cubic polynomials to approximate inelastic member
deformations led to the improvement of displacement interpolation by introducing the so-
called variable interpolation functions. These were proposed first by Mahasuverachai (1982)
for piping and tubular structures, and adapted to reinforced concrete members by Kaba and
Mahin (1984) along with section layer discretization. Typically, these functions are derived
from force interpolation polynomials (obtained from equilibrium conditions, independently of
the element state) corrected by the current flexibility distribution and the flexibility matrix of
the member. A mixed approach is used, where both deformation and force interpolation func-
tions are adopted. However, the model is found to contain inconsistencies (Taucer et al.
(1991)) that give rise to numerical problems; particularly, the state determination is such that
equilibrium between section resisting forces (along the member) and section applied forces is
not satisfied.
The proposal is further developed by Zeris (1986) and Zeris and Mahin (1988), namely by
extending it to biaxial bending through fibre modelling and by improving the element state
determination process. The model succeeded to overcome the numerical problems found with
the Kaba and Mahin proposal and, particularly, equilibrium is guaranteed by a specific iterative
procedure. The response of elements with softening behaviour that could not be analysed with
30 Chapter 2
previous models, was finally simulated successfully. The proposed model has shown satisfac-
tory performance (Taucer et al. (1991)) but the element state determination was found not very
clear, surely due to its derivation from ad-hoc corrections of the Kaba and Mahin model. Nev-
ertheless, these two last proposals, by involving the use of force interpolation functions, have
pointed-out the promising framework of the flexibility approach.
The formulation of non-linear flexibility based frame elements has been cast in the form of a
unified and general theory (Taucer et al. (1991), Spacone et al. (1992) and Spacone (1994))
derived from the framework of mixed finite element methods (Zienkiewicz and Taylor (1989)).
Use is made of the two field mixed method, in order to address the element state determination
(indeed the most critical task in the flexibility based formulation) and to clarify its strong con-
nection with numerical implementation. Thus, a consistent and elegant element state determi-
nation scheme is proposed, whose insertion in classical stiffness based finite element programs
appears rather straightforward. Moreover, the formulation is suitable to accommodate any kind
of section constitutive relationship, either a global one (e.g. moment-curvature at the section
level) of phenomenological or differential type, or an implicit one arising from integration of
the local stress-strain behaviour at the fibre level.
The general formulation for flexibility based elements typically requires a few control sections
distributed along the element whose behaviour is monitored. Only force interpolation func-
tions are used, because they have the major advantage of being exact regardless of the dam-
aged state of the member. The element flexibility matrix is obtained from integration of the
flexibility distribution (known at the control sections) and an internal iterative scheme is used
to obtain the element resisting forces for imposed displacements in the element nodes; such
scheme is driven by residual displacements to be gradually eliminated while strictly preserving
equilibrium along the element.
The formulation has been associated with fibre discretization of the section by Taucer et al.
(1991) to develop a general three-dimensional beam-column element where biaxial bending
with axial force interaction is automatically taken into account. In a parallel work by Spacone
et al. (1992), a differential constitutive relation is adopted for uniaxial bending, taking into
account strength and stiffness degradation and pinching through a measure of accumulated
damage; since the interaction of varying axial force is not considered, the model is mainly
appropriate for beams or columns with low axial load level. However, that includes also the
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 31
fixed-end rotations modelling by means of rotational point hinges at the member ends. The
most remarkable aspect is the natural and elegant way how the proposed element state determi-
nation scheme can accommodate the association in series of two or more elements behaving
non-linearly; this problem had been already addressed in a previous work by Filippou and Issa
(1988), but the solution proposed there was more cumbersome and tricky. Therefore, in our
opinion, Spacone proposal appears more clear in dealing with the association in series of any
number of non-linear elements and rather suitable to incorporate the wide set of sub-elements
proposed in Filippou et al. (1992) for separate simulation of each non-linear behaviour source.
For its importance in the present work context, the general formulation of flexibility based ele-
ments is further discussed in 2.4, following the basic steps of previous works (Taucer et al.
(1991), Spacone et al. (1992)).
Finally, although not strictly fitting in the member model family, it is worth referring another
recent modelling strategy found in Izzuddin (1991), Izzuddin et al. (1994) and Karayannis et
al. (1994). It follows a displacement based formulation, and classifies the non-linearity sources
in two basic types: one for the effects of concrete cracking and the non-linear compressive
response of concrete, and another for the effects of open cracks, concrete crushing (and con-
finement) and the post-yielding behaviour of steel. The first source, although non-linear, is
assigned an elastic behaviour, but a higher order element is proposed whose formulation is
based on a quartic shape function for transverse displacements. The second source is modelled
by a classical cubic Hermitian element, for which the non-linear behaviour is controlled at two
Gauss points by means of a layer approach, separately accounting for the behaviour of steel
and of unconfined and confined concrete. The analysis is performed by an adaptive procedure:
all elements start behaving elastically and modelled by a single elastic quartic element; as ine-
lasticity takes place in critical regions, inelastic cubic layered elements are gradually inserted
as needed. Automatic mesh refinement techniques are required, with particular care on setting
up initial conditions for new elements when they are inserted; however, the model appears to
be quite efficient and accurate when compared to results from discretizations with inelastic
layered elements inserted at the beginning of the analysis.
The above review is meant to cover, in a historical perspective, the currently available range of
member models for uniaxial bending conditions. Several others can be found in the literature
to account for varying axial load interaction with bending (eventually biaxial), which were not
32 Chapter 2
referred herein since that is not relevant for the developments envisaged in the present work.
This problem was investigated by Coelho (1992) through the analysis of some frame structures
where the column axial load variation was approximately taken into account. Despite some-
what significant fluctuations of axial loads in external columns (due to overturning moments),
the results in terms of maximum storey displacements and ductility demands were just slightly
affected, showing some trend for lower values comparatively to the assumption of constant
axial loads. Thus, the overall conclusion pointed to a not very significant influence of that
problem on the global response. For this reason, and because the consideration of that issue in
the model developed herein would lead to extremely complicated numerical schemes, it has
been decided to restrict the present stage of development to the interaction of static axial force
- bending moment, i.e. just on the skeleton curve of the adopted phenomenological model.
The review has culminated with the formulation of flexibility based elements, which, in our
opinion, appears as the most promising and elegant formulation developed to date. To a certain
extent, this formulation brings back to evidence the important role the force method plays in
structural analysis (particularly of frame structures), that was somewhat “forgotten” due to the
advent and massive application of the displacement based Finite Element Methods. Both tech-
niques are clearly shown to be compatible (Spacone et al. (1992)), thus opening the doors to
further developments of complementary tools for structural modelling.
Indeed, it appears quite convenient that, for an integrated discretization of a complete struc-
ture, several types of elements can be mixed-up in order to better simulate the behaviour of
each constituent. As a typical example, for the complete numerical model of a building struc-
ture it is desirable to consider adequate plate or shell elements to simulate slabs, plane ele-
ments to model walls, member models for beams and columns, joint elements for beam-
column joints (of which a very interesting proposal is reported in Monti et al. (1993)), founda-
tion elements to include soil-structure interaction and, finally, to make them all compatible in a
unique analysis model. Fortunately, that is the trend of general purpose computer codes emerg-
ing over the last decade, an example of which is CASTEM2000 (CEA (1990)) where the ele-
ment model developed in the present work has been implemented (further references to the
code are included in Chapter 4).
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 33
2.4 General flexibility formulation for beam-column elements
For a comprehensive perspective of the new approach developed in the present work, the gen-
eral flexibility formulation for beam-column elements is reviewed in the following sections.
The description is mainly based on previous work (Taucer et al. (1991), Spacone et al. (1992),
Spacone (1994)) where further details can be readily found. Slightly different conventions and
notation are adopted as introduced in 2.4.1 and rigid zones are considered at the element ends,
called rigid lengths for briefness. Furthermore, an attempt is made in 2.4.2 to clearly distin-
guish all the involved variable spaces, ranging from the global structure to the local section
levels.
The most relevant topics of the theoretical background for the flexibility formulation are
recalled in 2.4.3 in contrast with the classical stiffness method, highlighting the difficulties
associated with the element state determination in the flexibility context arising from the non-
availability of displacement shape functions.
Special attention is given to the element state determination in 2.4.4, as the most delicate task
in the flexibility formulation, and, after the presentation of the algorithm to handle only nodal
loads, the process to include also element applied loads is described as an extension and adap-
tation of the previous one. However, the presentation is made without restricting references,
neither to the type of section model (fibre or global) nor to the type of element applied loads, in
order to keep its generality.
Finally, it is worth recalling that this formulation can be also included in the context of classi-
cal stiffness based finite element computer codes, despite its somewhat higher implementation
cost. Indeed, given an input of nodal displacement increment for each element, the algorithm
provides the usual output of the state determination of any classical finite element, i.e., the cor-
responding increment of element restoring forces and the updated stiffness matrix, ready for
assembly in the global structure restoring force vector and stiffness matrix, respectively.
2.4.1 Conventions and notation
Figures 2.4 and 2.5 show the three different reference frames used in the present study: the glo-
bal coordinate system (X,Y,Z), the element local system (x,y,z) and the section local reference
system (xs ,ys ,zs).
34 Chapter 2
Associated with these axis systems, the force, displacement and generalized stress and strain
vectors are written according to the following general rule: boldface upper case letters denote
force and generalized stress vectors (for which Q and S are the reserved letters), while bold-
face lower case letters are used for displacement and generalized strain vectors (denoted by u
and e).
Figure 2.4 Spaces of variables and axis systems: from the global to the local element level
Figure 2.4 shows all the variable spaces used in the analysis at global and element level,
whereas Figure 2.5 refers to the variables at section level. Translation and rotation displace-
ment components are denoted by uξ and θξ, respectively, where ξ stands for the associated axis;
similar notation, Fξ and Μξ, is used for the corresponding force and moment components. The
Z
Y
X
zy
x
1
2
1
2
x
E1
E2x
E1
E2
Ll1 l2
FZG
MZG
FYG
MYG
FXG
MXG
QG uG( , )QE uE( , )
a) Global Reference System andSpace of Global Nodal Variables
b) Element Global Space of Variables
Q1E
Q2E
Qe ue( , )
c) Element Local Variables
Q1e
Q2e
zy
Q1f
Q2f
Qf uf( , )
d) Flexible Element Local Variables(with rigid body modes)
Q u( , )
e) Flexible Element Local Variables(without rigid body modes)
zy
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 35
positive direction of each component is the same of the corresponding axis. Superscripts are
used to identify the variable space and subscripts refer to a specific element node (for the total
element, i.e. with rigid lengths included) or an end section (for the flexible element part).
Figure 2.5 Space of variables at the local section level.
According to Figure 2.4-a), the force and displacement vectors, at the global level are written
as follows
(2.1)
while at the element level with rigid body modes, the following notation is adopted
(2.2)
The superscript η in Eq. (2.2) holds for E, e and f, identifying the complete variable spaces of
the element, viz that with axes parallel to the global system (Figure 2.4-b)), the one referring to
the local system (Figure 2.4-c)) and the one of the flexible element part (Figure 2.4-d)).
The basic system without rigid body modes is characterized by the degrees of freedom (d.o.f.)
shown in Figure 2.4-e), and the corresponding force and displacement vectors are defined by
(2.3)
At the section level, the generalized stress and strain and the displacement vectors are written
in the local axis system (Figure 2.5) and given by
zy
xE1 xs
zs ysMz
My
Mx
Vz Vy
Nxns
QGFX
G FYG FZ
G MXG MY
G MZG
T= and uG
uXG uY
G uZG θX
G θYG θZ
GT
=
Qη Q1η
Q2η⎩ ⎭
⎨ ⎬⎧ ⎫
= and uη u1η
u2η⎩ ⎭
⎨ ⎬⎧ ⎫
= with
Qiη
FXi
η FYi
η FZi
η MXi
η MYi
η MZi
ηT
= uiη
uXi
η uYi
η uZi
η θXi
η θYi
η θZi
η
i 1 2,=( )
T=
Q Fx1Mx1
My1Mz1
My2Mz2
T= and u ux1
θx1θy1
θz1θy2
θz2
T=
36 Chapter 2
(2.4)
The local system (xs ,ys ,zs) is considered with the xs co-linear with the element x axis, but the
remaining local axes may be oriented in any other direction; if this is the case, then a local
orthogonal transformation must be performed over the relevant vectors in order to conform
with the element (x,y,z) reference system. The positive unit normal ns defines the local system
positive directions and, consequently, the positive directions for stresses, strains and displace-
ments. However, in this section and for the sake of simplicity, these two reference systems are
considered parallel and the subscript s is suppressed.
The usual meaning of the generalized stresses and strains is adopted, namely: axial force (N)
and strain (ε), transverse forces (V) and distortion rotations (β), twisting and bending moments
(M) and associated curvatures (ϕ). Shear distortions (β) are included although they depend on
the adopted beam formulation (e.g., in the Bernoulli case those distortions are neglected).
The externally applied actions can be the following:
• nodal actions, included in the and vectors, and, therefore, consisting of imposed
forces or displacements;
• element loads, usually forces and moments imposed along the element, either distributed or
concentrated, as schematically shown in Figure 2.6; in the present study, element loads are
considered only in the flexible part, since this corresponds to the most common situation.
Figure 2.6 Element applied loads
S S x( ) NxsVys
VzsMxs
MysMzs
T= =
e e x( ) εxsβys
βzsϕxs
ϕysϕzs
T= =
a a x( ) uxsuys
uzsθxs
θysθzs
T= =
QG uG
zy
xE1 E2
Lx
Pz
µz
Py
µy
Px µxpz
py
px
mz
my
mx
ns
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 37
The vectors of distributed and concentrated element loads are given, respectively, by
(2.5)
and, in the most general case, the load distribution can be of any type and applied wherever
desired.
However, moments applied in the element are rarely considered for practical situations; hence,
only the following reduced vectors are used in this study
(2.6)
For equilibrium purposes (discussed in the following sections), force and moment resultants of
the element loads have to be defined. Between the flexible end section E1 and a generic section
with abscissa x, the vector of resultants (denoted by for force components and by for
moment ones) includes the contribution of all the loads applied in that zone and is written as
(2.7)
where moments are taken relative to the generic section. In particular, for the whole flexible
element, the total resultant vector is defined as
(2.8)
2.4.2 Relations between spaces of variables
The above mentioned spaces of variables are inter-related through topologic, geometric, equi-
librium and compatibility type conditions.
The topologic conditions relate the global nodal space with the element global space
and are stated by means of well known procedures of matrix structural analysis.
p p x( ) px py pz mx my mz
T= =
P P x( ) Px Py Pz µx µy µz
T= =
p p x( ) px py pz
T= =
P P x( ) Px Py Pz
T= =
R R x( ) x y z x y z
T= =
RL R L( )xL
yL
zL
xL
yL
zL
T= =
QG uG( , )
QE uE( , )
38 Chapter 2
The geometric conditions consist on axis system rotations and define the relation between the
global and the local spaces of variables at the element level; as already men-
tioned, this type of relation may have to be also applied at the section level if non-coincident
reference systems are used.
The transformation between the element local space and the flexible element space
is derived from equilibrium conditions of the rigid lengths and can be written as
(2.9)
with the transformation matrix given by
(2.10)
where, according to Figure 2.4-b), l1 and l2 stand for the element rigid lengths and I is the
(3x3) identity matrix.
Element rigid body modes are fixed by imposing the appropriate boundary conditions shown
in Figure 2.4-e) and the flexible element behaviour can be described by the reduced space of
variables . The equilibrium conditions, involving the fixing support reactions, the
forces and the total resultants of element loads , allow the following relation to be obtained
(2.11)
where the transformation matrix and the vector of element loads contribution , are
given, respectively, by
QE uE( , ) Qe ue( , )
Qe ue( , )
Qf uf( , )
Qe Tr Qf⋅= and uf TrT
ue⋅=
Tr
Tr
I[ ] 0[ ] 0[ ] 0[ ]
T1r[ ] I[ ] 0[ ] 0[ ]
0[ ] 0[ ] I[ ] 0[ ]
0[ ] 0[ ] T2r[ ] I[ ]
= with
T1r
0 0 00 0 l1–
0 l1 0
=
T2r
0 0 00 0 l2
0 l– 2 0
=
Q u( , ) Q
RL
Qf Tb Q⋅ Qpf+=
Tb Qpf
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 39
(2.12)
Assuming small deflections, the relation between displacement vectors with and without rigid
body modes is given by
(2.13)
Following a very similar procedure, the relation between the element forces Q and the generic
section stress can be obtained. Actually, writing the equilibrium equations for the portion
of element between the end section E1 and the generic section at abscissa x, including the
resultants of element loads applied up to x, leads to the following relation
(2.14)
where the matrix and the vector (with the contribution of the element applied
loads) are given by
(2.15)
In turn, the compatibility relation between displacements u and the section deformations
is given by the following integral form
Tb
1 0 0 0 0 00 0 0 1 L⁄ 0 1 L⁄0 0 1 L⁄– 0 1 L⁄– 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 01– 0 0 0 0 0
0 0 0 1 L⁄– 0 1 L⁄–0 0 1 L⁄ 0 1 L⁄ 00 1– 0 0 0 00 0 0 0 1 00 0 0 0 0 1
= and Qpf
0
zL L⁄
yL L⁄–
000
xL–
yL– z
L L⁄–
zL– y
L L⁄+
xL–
00⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
u TbT
uf⋅=
S x( )
S x( ) b x( ) Q⋅ Sp x( )+=
b x( ) Sp x( )
b x( )
1– 0 0 0 0 00 0 0 1 L⁄– 0 1 L⁄–0 0 1 L⁄ 0 1 L⁄ 00 1– 0 0 0 00 0 x L⁄ 1–( ) 0 x L⁄ 00 0 0 x L⁄ 1–( ) 0 x L⁄
= and Sp x( )
x–
zL L⁄– y–
yL L⁄ z–
x–
x yL L⁄( ) y–
x zL L⁄( ) z–⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
e x( )
40 Chapter 2
(2.16)
which can be derived from the virtual force principle.
2.4.3 Theoretical background
This section presents the theoretical bases for the general flexibility formulation of beam-col-
umn elements, written in the incremental form of non-linear analysis. The formulation
accounts for the presence of nodal loads only, since the case of element loads can be dealt with
equivalent nodal forces, conveniently taken into account as explained later.
At the section level, the force-deformation relation can be stated as desired: either by means of
a fibre discretization (Taucer et al. (1991)), each fibre being controlled by a stress-strain con-
stitutive material law, or by a global section constitutive relation in terms of generalized forces
and displacements (Spacone et al. (1992)).
The flexibility method is presented after a preliminary review of the classical stiffness method
in order to recall their main differences in the analysis procedure, particularly for non-linear
behaviour, and to identify the phases where adaptations are needed for integration of the flexi-
bility method in classical stiffness based finite element codes.
2.4.3.1 Stiffness method
The formulation based on the stiffness method requires the adoption of displacement shape
functions to approximate the kinematics inside the element in terms of member end displace-
ments with rigid body modes. Denoting by ∆ the increments of the relevant quantities, the
approximation of the displacement and deformation fields is done by
(2.17)
where is a differential operator, depending on the section formulation, is the matrix of
the displacement shape functions and contains the deformation shape functions. Eqs.
(2.17) represent the compatibility conditions between section deformations (and displace-
u bT x( ) e x( )⋅ xd0
L
∫=
∆a x( ) N x( ) ∆uf⋅=∆e x( ) ∂ ∆a x( )⋅= ⎭
⎬⎫
∆e x( ) B x( ) ∆uf⋅=⇒ with B x( ) ∂ N x( )⋅=
∂ N x( )
B x( )
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 41
ments) and member end displacements .
Increments of section forces and deformations can be related by the local stiffness matrix
according to the linearized form
(2.18)
and the equilibrium between increments of element forces and section forces can
be stated with the principle of virtual displacements by
(2.19)
written upon an equilibrated distribution of forces. Substitution of Eq. (2.18) in Eq. (2.19),
which must hold for any choice of the virtual displacement vector , leads to the classical
stiffness relationship at the element level , where the stiffness matrix is
readily computed by
(2.20)
Eq. (2.19) can be also written to yield the element resisting forces
(2.21)
in equilibrium with a given distribution of section resisting forces .
The set of analytical expressions is now complete for the classical stiffness method. The non-
linear behaviour requires an iterative procedure at the global structural degrees of freedom, for
which the Newton-Rapshon scheme is most often adopted. The typical sequence of tasks is
schematically described in the flowchart of Figure 2.7 and develops as follows:
• for a given force increment at the structure level , or an iteration corrective incre-
ment, an estimate of displacement increment is obtained using a global stiffness
matrix (initial, tangent or other, depending on the selected variant of the Newton-Raphson
method);
∆uf
k x( )
∆S x( ) k x( ) ∆e x( )⋅=
∆Qf ∆S x( )
δufT
∆Qf⋅ δeT x( ) ∆S x( )⋅ xd0
L
∫=
δuf
∆Qf Kf ∆uf⋅=( )
Kf BT x( ) k x( ) B x( )⋅ ⋅ xd0
L
∫=
Qrf( )
Qrf BT x( ) Sr x( )⋅ xd
0
L
∫=
Sr x( )( )
∆QG( )
∆uG( )
42 Chapter 2
• increments of element displacements are obtained from the structural ones ,
by means of successive transformations according to 2.4.2, and the element state determina-
tion starts (as highlighted in Figure 2.7);
• section deformation increments are then computed by Eq. (2.17) and allow the
deformations to be updated;
• constitutive laws of sections are usually written in terms of , thus permitting to directly
obtain the local stiffness matrix and the section resisting forces , which are then
passed to the element level according to Eqs. (2.20) and (2.21); the element state determina-
tion is thus complete;
• the element resisting force vectors and the stiffness matrices are transferred to
the structure level (again according to relations given in 2.4.2), in order to check equilib-
rium against external loads and, if required, proceed with further corrections.
Figure 2.7 Main tasks of the classical stiffness based state determination process
These steps constitute the well known procedure at the core of the widely used stiffness based
methods of finite element analysis, for which further details are deemed unnecessary. How-
ever, as noted before, the deformation shape functions are generally derived for linear
elastic behaviour of prismatic members and, therefore, remain valid only for such conditions.
Thus, unless a more refined discretization is adopted for the member, the approximation
obtained with just one element per member in the non-linear range may become rather crude.
Typically, for monotonic analysis, this approximation leads to a solution stiffer than the “cor-
rect” one because the functions are not able to describe the more complex deformed
∆uf( ) ∆uG( )
∆e x( )( )
e x( )
e x( )
k x( ) Sr x( )
Qrf( ) Kf( )
(Increment, or iteration correction)Unbalanced forces (structure level) Increment of structure
displacements
Increment of elementdisplacements
Increment of sectiondeformations
Section restoringforces
Element restoringforces
Structure restoringforces
Deformation shapefunctions
Section model
Integration using thedeformation shape functions
Elem
ent S
tate
Det
erm
inat
ion
1
2
3
4
B x( )
B x( )
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 43
shapes resulting from the stiffness decrease in the element non-linear zones. Note that, “cor-
rect” solution shall be understood within the assumptions of the section formulation and the
behaviour law.
2.4.3.2 Flexibility method
In the flexibility method, no displacement shape functions are assumed and the formulation is
cast in the element space without rigid body modes. As stated before, the element equilibrium
conditions (for the case of no element applied loads) lead to the total and incremental force
fields expressed by
(2.22)
where contains the force interpolation functions given by Eqs. (2.15).
The linearized section constitutive law can be also written in the inverse form of Eq. (2.18) as
(2.23)
Note that, as well as the operator, also the local flexibility and stiffness matrices
depend only on the section formulation. Since the Timoshenko beam formulation has been
adopted for the element developed in the present work, the corresponding matrix is given
in Appendix A, for the particular case of linear elastic behaviour. The elastic section flexibility
matrix consists only of diagonal constant terms; if non-linearity is to be included, then non-
constant terms do appear corresponding to the non-linear deformation components.
The compatibility between end displacement increments and the local section deforma-
tions is conveyed by application of the virtual force principle
(2.24)
upon an initial state of compatible deformations. Substituting Eqs. (2.22) and (2.23) in Eq.
(2.24), which must hold for an arbitrary choice of the virtual force vector , the flexibility
relationship at the element level is derived , where the flexibility matrix is
given by
S x( ) b x( ) Q⋅= and ∆S x( ) b x( ) ∆Q⋅=
b x( )
∆e x( ) f x( ) ∆S x( )⋅=
∂ f x( ) k x( )
f x( )
∆u
∆e x( )
δQT ∆u⋅ δST x( ) ∆e x( )⋅ xd0
L
∫=
δQ
∆u F ∆Q⋅=( )
44 Chapter 2
(2.25)
Note that Eq. (2.24) can be readily modified to obtain the element displacements in terms of
section deformations as given by Eq. (2.16).
The stiffness matrix is obtained by inversion and is then transformed to the flexible
element with rigid body modes by
(2.26)
In turn, the stiffness matrix in the element local space is then obtained by
(2.27)
upon which another reference system transformation must be performed to obtain the matrix
ready to be assembled in the global structural stiffness matrix .
This method elicits the following comments:
• the equilibrium at the element and section levels is strictly ensured by Eqs. (2.22), when no
element loads are included, because these equations remain unchanged regardless of the lin-
ear or non-linear behaviour; this means that the force interpolation functions are
always exact, contrarily to the deformation shape functions , which are only strictly
valid for linear behaviour of prismatic members;
• however, the above referred advantage of the flexibility method is counteracted by signifi-
cant difficulties on the numerical implementation, which have surely contributed to the
strong preference by displacement based methods in numerical modelling so far; actually,
finite element analysis programs are typically designed for imposing displacements (rather
than forces) at the member ends and for evaluating the corresponding resisting forces
(instead of displacements), for which deformation shape functions are usually required.
This final comment highlights the fact that the flexibility element state determination cannot be
directly included in the sequence shown in Figure 2.7. Actually, it requires more steps than the
stiffness case and an iterative scheme has to be enforced at the element level, as described in
the next section.
F bT x( ) f x( ) b x( )⋅ ⋅ xd0
L
∫=
u
e x( )
K F 1–=( )
Kf Tb K TbT
⋅ ⋅=
Ke
Ke Tr Kf TrT
⋅ ⋅=
KE KG
b x( )
B x( )
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 45
2.4.4 The element state determination
2.4.4.1 General comments
The flexibility element state determination scheme is extensively described by Spacone et al.
(1992) and Taucer et al. (1991), in the flexibility method context and also in the framework of
the two field mixed method (Zienkiewicz and Taylor (1989)) as a more general formulation.
Both methods lead to the same result concerning the element force-displacement relation,
although, according to the referred authors, the latter method leads to a more consistent imple-
mentation of the most difficult task, i.e., the element state determination.
In the present work the intrinsic formalism of the mixed method was deemed unnecessary,
because, in our opinion, a good understanding of that formulation requires a previous knowl-
edge of the state determination sequence. After all, that is for sure the reason why the presenta-
tion of the mixed method approach in Spacone et al. (1992) is preceded by the description of
the state determination.
The element state determination in classical stiffness based computer programs consists in the
evaluation of the element stiffness matrix and restoring force vector . In the flexibil-
ity method the stiffness matrix can be readily obtained from the flexibility one as described
above, but the force vector , corresponding to given element displacements , cannot
be directly obtained due to the lack of deformation shape functions; therefore, a special proce-
dure has to be adopted, which is better understood in the context of the non-linear analysis
scheme.
The non-linear analysis algorithm at the structure level often consists of the classical incremen-
tal-iterative process schematically shown in Figure 2.8-a). The outer process refers to the
application of external loads, performed in a sequence of increments, and is denoted by super-
script k. The inner process, identified by superscript n, consists on the iterative process (of
Newton-Raphson type) necessary for completion of each load increment k.
The specific features of the present formulation, for which no explicit displacement shape
functions are available, require another iterative process at the element level for the state deter-
mination within each Newton-Raphson (N-R) iteration n. This innermost process, labelled
with superscript j and also included in Figure 2.8, yields the element restoring forces corre-
Kf( ) Qrf( )
Qrf( ) uf( )
Qn
46 Chapter 2
sponding to the element displacements . Figure 2.8-b) shows the internal iteration evolution
in terms of element forces and displacements for each N-R iteration n, while Figure 2.8-c)
schematically illustrates the corresponding development in terms of section forces and defor-
mations. In the present work, only this innermost iterative process (internal iteration scheme)
will be described with some detail in terms of the reduced variable space , which can be
related with the complete space ones by Eqs. (2.11) and (2.13).
Figure 2.8 State determination of flexibility based elements: from the global to the local level
un
Q u( , )
Qf uf( , )
Q
A
B
D
EFC
A
B
DC
A
B
D
EFC
S
Element loop ( j )for n-th N-Riteration
Internal loops ( j ) for each N-R iteration
ELEMENT
SECTION
un-1 uun un+1 un+2 uk
en-1 en en+1 e
Qn
Sn
A
B
D
E
F(load increment k)
A, B, D constitute the Newton-Raphson iteration n
STRUCTURE
uG( )n
uG( )n 1+
QG( )k
QG
uG( )n 1–
a)
c)
b)
uG
QG( )n
QG( )n 1–
QG( )k 1–
uG( )k
ej
Sj
Sn-1
Qn-1
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 47
This special procedure in the flexibility formulation aims at satisfying both force equilibrium
and deformation compatibility with the imposed displacements at the element level. Therefore,
the procedure has to account for the possible existence of element applied loads, since it modi-
fies both the equilibrium conditions and the distribution of deformations along the element.
Thus, for the sake of simplicity, two situations are distinguished concerning applied loads: the
case of nodal forces only, which is introduced first in 2.4.4.2, and the case with element loads
also included, which is explained in 2.4.4.3 as an extension of the previous one.
2.4.4.2 Nodal forces
The special procedure for element state determination is better understood by means of a pre-
liminary comparison with the usual process of classical stiffness based state determination.
Referring to Figure 2.7, the non-existence of explicitly defined displacement shape functions
for flexibility based elements does not allow to perform tasks 2 and 4. Instead, the transition to
section level must be done in terms of forces, following the sequence described in Figure 2.9,
where the first and the last tasks are labelled with 1 and 4, respectively, in order to highlight
the correspondence with the scheme of Figure 2.7.
Figure 2.9 Flowchart for the element state determination of flexibility based elements
The increment of end section forces (in brief element forces) is first computed from the incre-
ment of element displacements using the last stiffness matrix; then, with the force shape func-
tions, the increment of section forces can be obtained. At this point, section deformations
should be calculated, but difficulties arise because non-linear section force-deformation rela-
Incr. of element forces
Increment or residuals of element displacements
Incr. of section deformations
Section restoring forces,flexibility and residual forces
Section residual deformations
Force shapefunctions
Section model
Integration weighted
Incr. of section forcesCurrent section flexibility
Updated section flexibility
by force shape functions
Current element stiffness
Element restoring forces4
1
48 Chapter 2
tionships are usually expressed in terms of deformations. Additionally, since the section
applied forces modify the section stiffness, the associated element stiffness matrix is no more
the one used to obtain these forces. The problem is solved by recourse to a non-linear iterative
procedure at the element level, which is stopped when element residual displacements vanish,
as required by nodal kinematic compatibility.
The process evolution, at both element and section levels, is illustrated in Figure 2.10 for the
generic n-th Newton-Raphson iteration where the element resisting forces , in correspond-
ence with the element displacements , are to be obtained. Figure 2.10 also
includes the initial conditions (j=1) and the main expressions involved; for section variables
and expressions, the reference to the generic abscissa (x) was suppressed for simplicity.
The process starts at point A with the application of the first element force increment
, obtained from the given element displacement increments and using the
initial tangent flexibility matrix in the inverse form; the element state point moves to position
B. The section force increments are computed using the force shape functions
and the prediction of section deformation increments is given by ,
using the local flexibility matrix compatible with the previously used element flexibility
matrix. Both force and deformation vectors can be updated to and , respectively,
and the section state is transferred to point B.
The section model allows for the restoring forces and the new tangent flexibility
to be obtained. However, force residuals are likely to appear until the state point B does not
reach the model curve. Since the equilibrium has to be satisfied, these force residuals
cannot be allowed and so the section has to deform more; using the section
force-deformation relation linearized about the updated state, the residual section deformations
can be given by .
If these residual deformations were allowed to take place, then, also at the element level, resid-
ual displacements would appear and be computed by , according to
Eq. (2.16). In such case, at both the section and the element levels, the point B (corresponding
to the final state for iteration j=1) would move to position B’; however, the kinematic compat-
ibility at the element level prevents displacements to go beyond .
Qn
un un-1 ∆un+=
∆Q1 F0[ ]1–
∆u1⋅=
∆S1 x( ) b x( )
∆e1 x( ) f0 x( ) ∆S1 x( )⋅=
S1 x( ) e1 x( )
Sr1 x( ) f1 x( )
S1 x( ) Sr1 x( )–( )
∆er1 x( ) f1 x( ) S1 x( ) Sr
1 x( )–( )⋅=
∆ur1 bT x( ) ∆er
1 x( )⋅ xd0
L∫=
un
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 49
Figure 2.10 Details of element and section state determination for flexibility based elements
This means that corrective element forces must be applied in order to restore compatibility by
pushing the point B’ back to the allowed displacements. Based on the updated element tangent
flexibility matrix (see Eq. (2.25)), corrective forces can be obtained by ,
constituting the increment of element forces for the second iteration (j=2); thereby, the
state point moves now to position C. Again, through the force shape functions, the section
force increments are obtained and applied starting from the previous
force level ; the corresponding deformation increments are superimposed
Q
u
A
B
D
C
A
B
DC
e
S
ELEMENT
SECTION ...(x)
B’
C’
B’C’
Sr1 Sn
Sn-1
∆S1
∆S2
∆S3
en-1 en∆e1
f0
f1
f2
un∆un
QnQn-1
∆Q1
∆Q2
∆Q3
F0F1
F2 Initial conditions ( j=1 ): F0 Fn-1= ∆u1 ∆un=
Generic iteration:
∆Qj Fj-1[ ]1–
∆uj ∆urj-1–( )⋅=
Qn Qn-1 ∆Qj
j 1=
converg.
∑+=
Initial conditions ( j=1 ): f0 fn-1= ∆er
0 0=
Generic iteration:∆Sj b ∆Qj⋅=
∆ej fj-1 ∆Sj⋅ ∆erj-1+=
Sn Sn-1 ∆Sj
j 1=
converg.
∑+=
en en-1 ∆ej
j 1=
converg.
∑+=
S1
S2
S3
e1 ...∆er
j fj Sj Srj–( )⋅=
∆urj-1 bT x( ) ∆er
j-1⋅ xd0
L
∫=
∆uj 1> 0=
un-1
∆er2
∆ur1
∆ur2
∆er1
F1 F1[ ]1–
∆ur1–( )⋅
∆Q2
∆S2 x( ) b x( ) ∆Q2⋅=
S1 x( ) f1 x( ) ∆S2 x( )⋅
50 Chapter 2
to the previous residual deformations , thus moving the section state point to position
C. At this stage the new restoring forces and tangent flexibility matrices are com-
puted and, the process is repeated as for the first iteration: residual deformations are calculated
and then integrated to yield new residual displacements, which in turn generate new element
corrective forces to start the third iteration and so on. The process stops when an adopted
measure of residuals (in terms of section forces or deformations, or element displacements for
example) is less than a pre-defined tolerance.
In Figure 2.10, convergence is supposed to be achieved at point D, which gives the element
restoring forces for the given displacements ; from then on, the Newton-Raphson proc-
ess, at the structure level, can proceed to the next iteration n+1.
Two aspects of this process should be underlined:
• at the element level, after the given initial displacement increments have been applied (j=1),
the displacements remain fixed while forces are repeatedly corrected; for this reason the
points B, C and D, representing the element state during iteration loops j>1, lie in the same
vertical line as shown at the top of Figure 2.10;
• at the section level, both deformations and forces are continuously updated and so, the state
points B, C and D cannot lie in the same vertical line; this complies with the force equilib-
rium that must be strictly satisfied in all sections, through the exact force shape functions,
and with the kinematic compatibility conditions to be satisfied in integral form as expressed
by Eq. (2.16).
2.4.4.3 Element loads
Element loads (concentrated or distributed along the span) cannot be included in the same way
as in the usual stiffness based methods, since the displacement shape functions (from which
the equivalent nodal forces are usually obtained) are not explicitly known and do not remain
constant throughout the non-linear process.
Therefore, for each load increment a set of equivalent nodal forces due to element loads is
implicitly computed and included at the element level by an iterative process (similar to the
one used when only nodal loads are present), which ensures that both the equilibrium and the
compatibility conditions are satisfied.
∆er1 x( )
Sr2 x( ) f2 x( )
Qn un
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 51
As in Figure 2.10, the process sequence at both the element and the section levels is shown in
Figure 2.11, for the generic n-th N-R iteration of the k-th load increment, and follows almost
the same steps as proposed by Taucer et al. (1991). It is assumed that a prescribed increment
of element loads is included simultaneously with other nodal loads and remains constant
during the N-R scheme.
Figure 2.11 Details of element and section state determination for flexibility based elements
with the application of element loads.
When each step starts (n=1), the whole structure, considered as a discrete system in the global
nodal variable space, does not “see” the element loads to be applied in that step. Thus, the
process begins with the application of the nodal load increments only, generating increments of
nodal displacements which are then transferred to the element level as described before.
∆p
Q
u
A
B
D
C
A
B
DC
e
S
B’’C’
B’C’
Sr1 Sn
Sn-1
∆SQ1
∆S2
∆S3
en-1 en
∆e1
∆er2
f0 f1
f2
un∆unun-1
QnQn-1
∆Q1
∆ur1
∆ur2
∆Q2
∆Q3
F0
F1F2
S1
S2
S3
e1 ...
B’’∆Sp
∆S1
B’
∆ur1( )Q ∆ur( )p
∆er1( )Q
∆er1
∆er( )p
ELEMENT
SECTION
52 Chapter 2
Exactly as in Figure 2.10, the element and section state points start from position A up to posi-
tion B (see Figure 2.11), where the section state determination takes place and the local equi-
librium has to be controlled. Here, the main difference appears with respect to the process
described before: according to Eq. (2.14), part of the section forces are due to the element
applied loads (denoted by in Figure 2.11) which have to be superimposed to those aris-
ing from nodal forces (referred to as , where the superscript stands for the internal iter-
ation j=1 and the subscript identifies these internal forces source). Hence, with the total
increment of internal force for the first iteration , the section
force is updated to the level , not coincident with that of point B.
When equilibrium is controlled at each section, it is apparent that unbalanced forces, given by
, include now an explicit term due the element loads. For the equilib-
rium to be satisfied, residual deformations are considered as before
, using the updated section flexibility ; residual deformations
consist of a term due to nodal loads and another due to element loads , as
shown in Figure 2.11. These residuals enforce the section point B to move to the new position
B” at a different level from the point B’ associated only with the nodal load contribution.
At the element level, displacement residuals can be obtained in the same way as in 2.4.4.2,
which allows for the computation of corrective forces, using the updated flexibility matrix ,
in order to start the second iteration. The application of these element loads moves the
state point B” to the position C, restoring displacement compatibility. Again, at the section
level, the corrective increment of forces is applied from the previous
force level, which, in this case, corresponds to point B”. From this stage on, the process fol-
lows exactly the same steps as for the case with only nodal loads, since the element loads are
already applied and only corrective nodal forces remain to be imposed until the displacement
compatibility is reached.
At the convergence stage (at points D in Figure 2.11) the section forces, deformations and flex-
ibility are obtained, taking into account the whole increment of element loads . Also at the
element level, the restoring forces are such that the imposed displacements are satisfied while
accounting for the influence of . This means that for the next Newton-Raphson iterations
(n>1) the increment of element loads must not be included again, since the element state deter-
mination starts from a stage in which all the increment is already considered.
∆Sp x( )
∆SQ1 x( )
∆S1 x( ) ∆SQ1 x( ) ∆Sp x( )+=( )
S1 x( )
∆Su1 x( ) S1 x( ) Sr
1 x( )–=
∆er1 x( ) f1 x( ) ∆Su
1 x( )⋅=( ) f1 x( )
∆er1 x( )Q( ) ∆er x( )p( )
F1
∆Q2
∆S2 x( ) b x( ) ∆Q2⋅=
∆p
∆p
∆p
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 53
It should be emphasized that, the influence of element applied loads in the residual displace-
ments is taken into account only after the state determination is performed. Actually, one could
think of performing the state determination for a predictor deformation increment
including the contribution of ; obviously, both the resisting force and the flexi-
bility would be different, as well as the section residuals . However, due to the
influence of , the integration of deformation residuals would not lead to the adequate
displacement residuals, since they refer to a deformation state which is no longer coherent with
the target displacements . Therefore, if such a strategy were adopted, a further correction
would be needed at the residual displacement level, in order to refer them to the right target
displacements. In our opinion, this leads to a cumbersome process for residuals evaluation, and
thus, the option of including only after the state determination appears to be prefera-
ble.
The scheme for this sequence of incremental element and nodal load application is illustrated
in Figure 2.12 for the common case of planar frame structures where only vertical uniformly
distributed loads are considered together with other nodal forces acting in the frame plane.
The load increment k consists of global nodal forces or displacements and of ele-
ment loads as indicated. For simplicity the initial state of the increment is associated with
the superscript 0, although it actually corresponds to the final state of the previous step k-1. At
the element level, the reduced variable space includes only end section bending moments
and rotations .
For the first N-R iteration (n=1) the whole increment is applied simultaneously with the
rotation increments (obtained directly from ), starting from a stage with
element loads and end section moments and rotations. It is assumed that does not
vary with displacement corrections.
During the first internal iteration (j=1), the increments are applied, through
the corresponding first approximation of moments , and the increment is included
in the residual corrective phase, as explained above. For the next internal iterations (j>1) the
level of element loads is already updated to and only corrective end section
moments are due. The same happens for the following N-R iterations (n>1).
∆e1 x( )
∆Sp x( ) Sr1 x( )
f1 x( ) ∆er1 x( )
∆Sp x( )
un
∆Sp x( )
∆QG ∆uG( , )
∆p
M
θ
∆p
∆θn=1 ∆uG( )n=1 p0
M θ( , )0 ∆p
∆θj=1 ∆θn=1=
∆Mj=1 ∆p
p p0 ∆p+=
∆Mj
54 Chapter 2
Figure 2.12 Sequence for the application of incremental element and nodal loads
Given the fact that substantial differences appear in the process only for the first N-R and inter-
nal iterations (n=1 and j=1), the detailed evolution for that stage is shown in Figure 2.13 for
arbitrary distributions of moment, flexibility and curvature.
∆θn
1 2
p0
QG uG( , )0
1 2
∆p
∆QG ∆uG( , )
1 2
p
QG uG( , )
+ =
Load increment k ;
NEWTON-RAPHSON ITERATIONS
p0
∆p
+
+
M θ( , )0
∆θn=1
p0
∆p
+
+
M θ( , )0
∆Mj=1
p p0 ∆p+=
+M θ( , )n-1
INTERNAL ITERATIONS
+M θ( , )j-1
∆Mj
p p0 ∆p+=
+M θ( , )j-1
∆Mj
p p0 ∆p+=
j 1≥j 1>j 1=
n 1>n 1=
Initial state 0 k 1–( )≡
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 55
The distinct tasks are identified, such as the predicting and updating phase, the section state
determination, the equilibrium checking and the local residuals computation. Only flexural
non-linearity is considered and, for simplicity, starting only from the left end section. A clear
distinction is made between internal force distributions due to element loads and nodal loads,
and it is evidenced that effects of are included only when section equilibrium is checked.
Figure 2.13 Details of the element state determination for first internal iteration of the first N-R
iteration, in the presence of element loads
∆p
+
+ =
ELEMENT LEVEL
+
+ =
+ =
=
-MQ
0 x( )
M0 x( )
Mp0 x( )
f0 x( )
∆MQ1 x( )
∆ϕQ1 x( )
MQ1 x( )
ϕ1 x( )
MQ1 x( )
∆Mp x( )
ϕ0 x( )
M1 x( )
Mr1 x( )
f1 x( )
∆Mu1 x( )
∆ϕr1 x( ) f1 x( )∆Mu
1 x( )=
∆θr1 x( )
Sect
ion
Stat
eDe
term
inat
ion
Section Equilibrium
j 1=
n 1=N-R iteration
Internal iteration
∆ϕQ1 x( ) f0 x( )∆MQ
1 x( )=
F1
Section Residuals
Predicting and updating phase
Load increment k ; Initial state 0 k 1–( )≡
⇒
⇒
56 Chapter 2
Figure 2.14 shows the same type of evolution, but for the subsequent internal iterations (j>1),
in order to give a comprehensive view of how the process develops. The same scheme is also
valid for all the successive internal iterations of the (n>1) N-R iterations; it should be noted
that no more effects of are considered at this stage (n>1) and that, during the predicting
phase, the previous residual deformations are included.
Figure 2.14 Details of the element state determination for the (j>1) internal iterations of the first
N-R iteration, in the presence of element loads
∆p
ELEMENT LEVEL
+ =
+ =
=
+
M1 x( )
f1 x( )
∆MQ2 x( )
∆ϕQ2 x( )
M2 x( )
ϕ2 x( )ϕ1 x( )
M2 x( ) Mr2 x( )
f2 x( )
∆Mu2 x( )
∆θr2 x( )
Sect
ion
Stat
eDe
term
inat
ion
Section Equilibrium
∆ϕr1 x( )
N-R iteration
Internal iteration j 1>
∆ϕQ2 x( ) f1 x( )∆MQ
2 x( )=
∆ϕr2 x( ) f1 x( )∆Mu
1 x( )=
F2
Section Residuals
Predicting and updating phase
n 1=
Load increment k ; Initial state 0 k 1–( )≡
⇒
⇒
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 57
At each N-R iteration, after the element restoring forces are obtained, they have to be
transformed into the flexible element local space of variables. Eq. (2.11) expresses the required
transformation, where the element loads have now a relevant contribution to the force
components not belonging to the element reduced space. Additionally, these element loads
shall be in accordance with those used in the state determination for the N-R iteration. There-
fore, if total restoring forces are to be transferred, then also the total load shall be used;
on the contrary, in the case of restoring force increments , then also the increment
must be considered in the transformation.
The restoring force vector is transferred to the element global space through the rela-
tions given in 2.4.2, and assembled in the global nodal restoring force vector. Finally global
unbalanced forces can be obtained and, then, the next iteration starts.
It is worth mentioning that vectors contain the nodal forces equivalent to the applied ele-
ment loads for the present state of the element. They have been implicitly obtained during the
internal iterative process and, obviously, they change when the flexibility distribution changes.
The special case, in which element loads are applied only in the first increment without any
other global nodal forces, helps to understand how the process develops. In fact, these condi-
tions mean that, for the first N-R iteration (n=1), the element state determination is performed
aiming at null element displacements in order to satisfy nodal kinematic compatibility. How-
ever, nodal element restoring forces are not null and are likely to generate non-zero global
nodal forces. Since these must be zero, unbalanced forces do appear and generate displacement
corrections for the next N-R iterations (n>1). When convergence is reached, the final state of
the structure is characterized by zero nodal loads associated with non-zero nodal displacements
due to the element loads.
A slightly different version for the application of element loads was proposed by Spacone
(Spacone (1994) and Spacone et al. (1996)) and is briefly presented next because of its clarity.
For a given N-R iteration n, should the displacement increments be applied simultane-
ously with an element load increment , the element force increments for the first internal
iteration (j=1) must be given by
(2.28)
Qn
Qpf( )
Qn p
∆Qn ∆p
Qf( )n
Qf( )n
∆un
∆p
∆Qj=1 Kj=0 ∆un⋅ ∆Qp+=
58 Chapter 2
where and is the vector of element forces necessary to ensure the
desired displacement increment . Actually, represents the element fixed end forces
due to , conveniently updated for the current state of the element.
According to Eq. (2.14), the section force and deformation increments are given by
(2.29)
(2.30)
The application of the virtual force principle, leads to the following expression
(2.31)
whose validity requires that
(2.32)
Therefore, can be computed before the element iterations start and be included in the sec-
tion incremental forces at the first iteration (j=1) according to
(2.33)
The section state determination is carried out for the predictor deformations
(2.34)
whose compatibility with the target displacements is ensured by the way was
obtained. For the subsequent iterations (j>1) the algorithm proceeds as before, i.e., only cor-
rections due to nodal forces have to be made until displacement compatibility is reached.
This procedure appears to treat more clearly the element load application, because it highlights
more rationally the fixed-end forces and avoids the somewhat tricky process of considering
those load effects only after the section state determination.
Kj=0 Fj=0[ ]1–
= ∆Qp
∆un ∆Qp
∆p
∆S x( )j=1 b x( ) ∆Qj=1⋅ ∆Sp x( )+=
∆e x( )j=1 f x( )j=0 ∆S x( )j=1⋅=
∆un bT x( ) f x( )j=0 b x( ) Kj=0 ∆un⋅ ∆Qp+[ ]⋅ ∆Sp x( )+{ }⋅ ⋅ xd0
L
∫=
∆Qp K– j=0 bT x( ) f x( )j=0 ∆Sp x( )⋅ ⋅ xd0
L
∫⋅=
∆Qp
∆S x( )j=1 b x( ) Kj=0 ∆un⋅ ∆Qp+[ ]⋅ ∆Sp x( )+=
e x( )j=1 e x( )j=0 f x( )j=0 ∆S x( )j=1⋅+=
un ∆Qp
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 59
2.4.4.4 Remarks on the global non-linear algorithm
The element state determination described so far has been cast within a global N-R algorithm
whereby the element incremental forces (as well as the incremental section forces and
deformations) are calculated from iterative displacements rather than incremental ones. In
other words, iteration n is performed by considering the iterative correction starting from
displacements and forces at the end of the previous iteration n-1, for which equilibrium has not
been reached yet.
For materials having path-dependent properties, such as crack opening and closing and crush-
ing in concrete, it is emphasized (Argyris et al. (1978)) that, along with the adoption of small
increments, the evaluation of incremental restoring forces (or stresses) be done from incremen-
tal deformations (i.e., those referred to the step beginning as an equilibrated state). This
requires another scheme as shown in Figure 2.15 (slightly different from that of Figure 2.8)
where the state determination is performed for a total displacement increment referring to
the displacement at the step beginning, which is transformed into element forces
with the initial stiffness matrix of that step. Section forces and deformations are then succes-
sively obtained and corrected following the same procedure as before.
The whole process for the element state determination remains almost unchanged, provided
the updating of force and deformation vectors (at both element and section levels) is performed
only when convergence is reached also at the structure level.
However, a specific remark is due concerning the application of element loads. Actually, for a
given step where an increment is to be applied, the fact that every N-R iteration n is
referred to the step beginning, requires the contribution of to be included in the first inter-
nal iteration j for all N-R iterations. This arises from the fact that this algorithm does not
“memorize” the loading history from preceding iterations, but only from the previous step
increments. Therefore, in the element load application sequence schematically illustrated in
Figure 2.12, the leftmost part referring to the first N-R iteration becomes valid for any
nth N-R iteration, whereas the rightmost part is not applicable. Accordingly, the illustrative
details of the element state determination included in Figures 2.13 and 2.14 are also valid for
any nth N-R iteration rather than only for the first one.
∆Qn
∆un
∆un
∆u0n
u0 uk-1=( )
∆p
∆p
n 1=( )
60 Chapter 2
Figure 2.15 State determination of flexibility based elements for displacements corrections rel-
ative to the step beginning
Q
A
B
DC
S
Element loop ( j )for n-th N-Riteration
Internal loops ( j ) for each N-R iteration
ELEMENT
SECTION
u0 uu1 un uk
e0 en e
Qk
Sn
uG( )2… uG( )
n…
QG( )k
QG
uG( )1
c)
b)
uG
QG( )n
QG( )1
QG( )k 1–
uG( )k
e1
S1
S0 Sk-1=
QG( )2
uG( )k 1–
uG( )0
=
(load increment k)
STRUCTUREa)
Q1
Qn
Q0 Qk-1=
Bn
B1Cn
C1 Dn
D1
Dn
D1
D2
B
A
A
Bn
B1Cn
C1DnD1
A
MODELLING OF REINFORCED CONCRETE FRAME STRUCTURES - STATE OF THE ART 61
2.4.4.5 Control sections and numerical integration
All the element integrations, for calculation of both the flexibility matrices and the residual
displacement vectors, are numerically performed. A set of control sections is pre-defined to be
monitored during the analysis, whose location is chosen according to the expected distribution
of non-linearity along the element. Since the inelastic behaviour often concentrates at member
ends, control sections shall also concentrate in these zones.
Additionally, the end sections have also to be monitored and, therefore, the Gauss-Lobatto
scheme appears more adequate (Spacone et al. (1992)) than the Gauss-Legendre method. The
number of control sections to be considered is chosen, as usual, according to the degree of pol-
ynomials to be integrated; it is recalled that, the Gauss-Lobato scheme with m integration
points leads to exact integration of polynomials of degree up to (2m-3).
2.5 Concluding summary
The present chapter has described the framework for the hysteretic behaviour modelling of
reinforced concrete frame structures, starting with a preliminary identification of the main fea-
tures of member behaviour under monotonic and cyclic conditions. Particularly, cyclic load
effects such as the stiffness degradation (for unloading and reloading stages), the pinching
effect and the strength deterioration have been presented in order to highlight the most relevant
aspects based on which the phenomenological models are derived.
A good compromise between details of structural response and model manageability can be
obtained with member models, whose evolution over the last three decades has been presented
in a historical perspective. The major requirements for member modelling have been identi-
fied, viz the hysteretic model and the element model.
The first developments of member models in uniaxial bending conditions have been recalled,
namely with reference to the two-component and the one-component (or point hinge) models,
and some of the most relevant phenomenological hysteretic models have been briefly referred.
Member models with distributed inelasticity gained increasing acceptance for non-linear frame
analysis upon recognition of the point-hinge model limitations to adapt to actual force and
stiffness distributions along the member. Thus, the evolution has been described, from the ear-
62 Chapter 2
liest attempt by Otani (1974) to the recent proposal in the flexibility context by Taucer et al.
(1991) and Spacone et al. (1992), reflecting the continuous effort of researchers to develop
member models that closely follow the spread of inelastic effects along the member while
accounting for the various sources of non-linear phenomena.
The advent of flexibility based formulations has been highlighted and emphasis has been put
on its adequacy for frame member modelling, as a rather suitable formulation to account for
inelasticity spread, to accommodate different types of section model and to easily incorporate
the association in series of non-linear sub-elements.
Since the member model developed in the present work belongs to the family of flexibility
based models, attention has been focused on the general flexibility formulation for beam-col-
umn elements as described in previous work. The main issues of the theoretical bases have
been addressed and the steps for the element state determination in flexibility models (thus in
the absence of deformation shape functions) have been presented, first for the case of nodal
loads only and then for the element applied loads also; additionally, its insertion in classical
stiffness based finite element algorithms has been discussed.
Chapter 3
FLEXIBILITY BASED ELEMENT WITH
MULTI-LINEAR GLOBAL SECTION MODEL
3.1 General comments and innovative features
The importance of adequately modelling non-linear behaviour has been discussed in Chapter
2, where member models have been recognized particularly suitable for seismic non-linear
analysis of reinforced concrete frame structures.
The state-of-the-art review presented in the previous chapter illustrated the great amount of
work developed in frame modelling, from which very powerful models became available for
non-linear behaviour simulation. The identification of the involved phenomena and the succes-
sive proposals to take them into account, led to models with remarkable level of sophistication.
Despite their unquestionable value, highly sophisticated models may easily become incompat-
ible with the needs of massive calculations often encountered in seismic analysis.
In this context, a major concern of the present work has been the research and development of
a modelling tool with a moderate level of sophistication, particularly in what concerns non-lin-
ear behaviour. Thus, while retaining the advanced features of recent sophisticated member
models (Taucer et al. (1991), Spacone et al. (1992), Spacone (1994)), a new element model has
been sought, which provides robustness, efficiency/economy (in terms of discretization and
computation time) and the ability to trace the non-linear behaviour right from the early stages
(such as the post-cracking/pre-yielding phase, typically corresponding to serviceability limit
states) up to advanced inelasticity development due to large ductility demands. Particularly, the
non-linear behaviour in the pre-yielding phase (very often neglected due to its reduced influ-
ence in extensive yielding stages) has been recently reported as important for an adequate
64 Chapter 3
assessment of the structural dynamic characteristics (Fardis and Panagiotakos (1997), Calvi
and Pinto (1996)) and of the seismic response for serviceability states (as required in modern
seismic design codes, e.g. EC8).
Therefore, in order to contribute for member model development or improvement, the follow-
ing basic issues were deemed necessary:
• The ability to adequately describe the member stiffness, in particular its evolution through-
out the various behaviour stages during the response; this is indeed an important issue in the
seismic analysis context, where a good assessment of structural frequencies of vibration has
a direct influence on the dynamic response.
• The non-linear behaviour modelling should not go beyond the global section level, so as to
ensure a reasonably low cost of seismic analysis computations.
• In order to achieve discretization and computation economy and ease of usage (preferably
comparable to that of linear analysis with classical beam elements), each structural member
(beam, column or inter-storey piece of wall) should be modelled, as much as possible, by
only one element.
• The number of control sections in the element should be reduced to the minimum required,
yet ensuring an adequate description of properties along the member and a sufficiently
accurate simulation of the response.
• The possibility of incorporating new developments, allowing for a better adjustment to the
actual member behaviour, namely by including other non-linearity sources not yet consid-
ered (e.g. effects of bar slippage, inelastic shear).
• The ease of incorporation, without major pre-requirements, in classical non-linear analysis
algorithms typically based on standard or modified Newton-Raphson methods.
The general flexibility element introduced in 2.4, is an excellent means for closely tracing out
the structural non-linear response and, with the adoption of the fibre model (Taucer et al.
(1991)) for reproducing the local section behaviour, a very powerful modelling tool is
obtained. However, such a methodology can hardly fulfil the requirement of efficiency and
economy for the analysis of building structures with large number of elements, specially when
only the global element behaviour is of interest.
The use of global models to describe section behaviour appears as a logic and natural option in
order to improve computation time performance of the flexibility element formulation, and, as
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 65
mentioned above, it has been adopted in a previous work (Spacone et al. (1992)) where a dif-
ferential section constitutive law is considered. However, in that proposal, the behaviour law
still had to be controlled at several control sections defined a priori and the effects of cracking
development were not properly taken into account because sections were assumed cracked
from the beginning.
To some extent, the need of several sections, chosen according to an expected element behav-
iour and distribution of non-linearity, is similar to the usual mesh refinement in classical stiff-
ness based finite elements. Therefore, in terms of time economy, the advantage of flexibility
elements over the stiffness ones is not evident, since control sections may be specified in zones
where non-linearity is not likely to appear.
In the present study, a global section constitutive law is also adopted; however, an effort is
made to restrict the number of pre-defined control sections, yet adequately tracing out the dam-
aging process along the element. For this purpose a multi-linear step wise model is adopted,
based on a trilinear envelope curve (for each direction of deformation) along with additional
hysteretic rules. The cracking and yielding section stages are taken into account and the effects
of other RC typical phenomena are also included, such as stiffness and strength degradation
and pinching (due to crack closing, bond slip or non-linear shear deformations). It will be
shown that such a behaviour modelling option, together with some simplifying assumptions,
allows for the definition of special control sections which move during the loading process and
reduce the need for pre-defined locations.
After the statement of some basic assumptions and the adjustment of convention and notation
in 3.2, the adopted global section model is presented in 3.3 referring to a moment-curvature
Takeda-type model based on a trilinear skeleton curve. Control sections and distinct zones con-
sidered along the element are introduced in 3.4; these are assumed of two types, viz the fixed
and the moving ones, the former consisting of the element end sections and a mid-span one,
while the latter stand for the yielding, cracking and null-moment sections. In view of the mov-
ing section development, the way of considering element applied loads is also discussed in 3.4,
where a simplifying assumption is introduced. The behaviour of control sections is addressed
in 3.5, where a modification is considered in the trilinear model, in order to make possible the
approximate control of cracked zones by means of a special transition from uncracked to fully-
cracked behaviour.
66 Chapter 3
The element state determination is thoroughly detailed in 3.6. The flexibility distribution along
the element is presented and due account is taken of the influence of moving cracking sections
on the element flexibility matrix. Although based on the flexibility formulation, the specific
features of this element, arising from the combination of fixed and moving control sections,
require particular care on the evaluation of displacement residuals. The evolution of non-linear
behaviour in the element end zones is carefully addressed and an event-to-event scheme is
adopted for adequate control of residuals. Convergence criteria are stated and convergence
problems related with the cracking transition led to the adoption of a line-search scheme per-
formed inside the element iterative process.
The non-linear algorithm is summarized in 3.7, where the main steps of the general non-linear
scheme for static and dynamic analysis are first recalled to highlight the element state determi-
nation stage; then, the sequence of steps for that element stage is described according to the
preceding sections. The main features of the developed member model are recalled in 3.8.
3.2 Basic assumptions and remarks on convention and notation
The basic notation and conventions to be used next are essentially the same as adopted in 2.4
for the general flexibility formulation. However, some adjustments are introduced below, aim-
ing at notation simplification and better adaptation to the specific features of the element.
Referring to Figure 2.5, non-linear behaviour is only assumed in one bending plane (xz or xy),
while all the remaining deformation components are considered linear. This assumption is
acceptable if the effects of interaction between internal forces are negligible as in the case of
orthogonal beams in space frame structures. However, it can become a major drawback of the
formulation if, for example, columns are to be analysed in bi-axial bending with axial force
interaction. Note, however, that such limitation is not exclusive of the present modelling strat-
egy, since it mainly arises from the option for a global section model.
The previous assumption allows to omit the subscripts referring to directions of element and
section forces, since the relevant aspects for non-linear behaviour only refer to one bending
direction (by default assumed to be the y direction). Therefore, from now on, bending
moments, curvatures and rotations will be simply denoted, respectively, by M, ϕ, and θ; for the
related shear force and distortion the notation simplifies to V and β.
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 67
The moving control sections depend on the moment distribution along the member and, there-
fore, element applied loads have to be pre-defined. In the present work these loads consist of a
uniformly distributed force vector (denoted by p) and a concentrated force vector (P) at the
span section H defined in 3.4. However, for the next developments, the relevant loads reduce
to those acting in the plane where non-linear behaviour is considered (thus, by default in direc-
tion z) and are simply denoted by p and P, respectively, for the distributed and concentrated
forces. If additional concentrated forces or different intensities of p need to be considered
along the member, a more refined discretization must be adopted.
By contrast with the general methodology introduced in 2.4.4.3, element load application will
be considered only in the first loading step and without any other load type. Actually, this cor-
responds to the common loading situation, for which the vertical static forces, due to self-
weight and live loads, are installed prior to other lateral actions (e.g. wind or earthquake
induced loading). Significant simplification of the numerical implementation can be achieved
with this option, but it is worth mentioning that, from the formal and conceptual standpoints,
the inclusion of element loads after the first step is straightforward.
The behaviour control is to be performed at the section level, in terms of generalized stress and
strain; thus, the internal force convention must be clearly defined and the section axis system
must be adjusted accordingly. For non-linear bending in plane xz (usually vertical) as assumed
by default, the section bending moments are considered positive when producing tensile strain
at the bottom fibres; the moment-curvature relationships are defined accordingly.
For the local section reference system assumed in 2.4, the above requirement is not satisfied,
since positive moments produce tensile strain at the top fibres. Thus, a rotation of the axis sys-
tem is introduced as shown in Figure 3.1, inducing a change of sign on the y and z components.
Figure 3.1 Adjustment of local section axis system
E1
zy
Mzs
ns ns2
xs xs2ns1
xs1ys
zs
Mys
zs2
ys2
ys1
zs1 E2
x
68 Chapter 3
The matrix and the vector of Eq. (2.14) are redefined accordingly, leading to
(3.1)
which hold for the internal force expressions in the rotated axis system, given by
(3.2)
and associated with the conjugate deformations denoted by .
Figure 3.1 also includes the end section axis systems associated with the respective positive
normal unit vectors ( or ), as commonly used in bending analysis. This allows to define
end section internal forces satisfying the above convention and to transform the forces and dis-
placements of the element reduced space into the local end section axis system.
Denoting by and the transformed forces and displacements, where the subscript es
stands for end sections, the following relations apply
(3.3)
as well as their inverse forms, since the matrix is invertible due to the orthogonal character
of the transformation. refers only to the reduced space degrees of freedom and is obtained
from the total matrix of the axis system transformation, thus having the expression
(3.4)
such that .
b x( ) Sp x( )
bs x( )
1– 0 0 0 0 00 0 0 1 L⁄ 0 1 L⁄0 0 1– L⁄ 0 1– L⁄ 00 1– 0 0 0 00 0 1 x L⁄–( ) 0 x– L⁄ 00 0 0 1 x L⁄–( ) 0 x L⁄–
= and Spsx( )
x–
zL L⁄ y+
– yL L⁄ z+
x–
x– yL L⁄( ) y+
x– zL L⁄( ) z+⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Ss x( )
Ss x( ) bs x( ) Q⋅ Spsx( )+=
es x( )
ns1ns2
Q u( , )
Qes ues
Q Tes Qes⋅= ues TesT u⋅=
Tes
Tes
Tes
1– 0 0 0 0 00 1– 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1– 00 0 0 0 0 1–
=
TesT Tes
-1 Tes= =
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 69
Introducing Eqs. (3.3) in Eq. (3.2), the internal forces can be expressed in terms of
by
(3.5)
where . Therefore, the shape function matrix , relating the generic
section internal forces with those of the end sections in the rotated reference frames shown in
Figure 3.1, is given by
(3.6)
The expression equivalent to Eq. (2.16), in the present axis systems, is written
(3.7)
The advantage of working with these rotated reference frames is related to the fact that all
forces and deformations become directly compliant with the global section behaviour law.
However, transformations to the element axis system (x,y,z) have to be performed, since all the
remaining relations have been defined referring to it. Force and deformation transformations
are carried out using Eqs. (3.3), whereas the flexibility matrix is transformed by
(3.8)
leading to the same matrix as given by Eq. (2.25).
From now on, if no other reference is made, internal forces and deformations are assumed rel-
ative to the local axis systems above introduced.
For convenience purposes in later developments, the following decomposition of and is
introduced
Ss x( ) Qes
Ss x( ) bes x( ) Qes⋅ Spsx( )+=
bes x( ) bs x( ) Tes⋅= bes x( )
bes x( )
1 0 0 0 0 00 0 0 1 L⁄ 0 1– L⁄0 0 1– L⁄ 0 1 L⁄ 00 1 0 0 0 00 0 1 x L⁄–( ) 0 x L⁄ 00 0 0 1 x L⁄–( ) 0 x L⁄
=
ues besT x( ) es x( )⋅ xd
0
L
∫=
F TesT Fes Tes⋅ ⋅= with Fes bes
T x( ) f x( ) bes x( )⋅ ⋅ xd0
L
∫=
u ues
70 Chapter 3
(3.9)
where each partial contribution has non-zero value only for one type of displacement compo-
nent; for these contributions are given as follows
(3.10)
Similar expressions apply to and an identical decomposition can be applied to and .
3.3 Trilinear model
In the present study a global section model has been adopted to reproduce the non-linear
moment-curvature constitutive relation. It is mainly based on an existing trilinear-type model
(Kunnath et al. (1992)), consisting on a primary or skeleton curve, shown in Figure 3.2, com-
bined with three control parameters for the hysteretic rules in each bending direction. It
appears to be a quite versatile Takeda-type model, since a wide variety of hysteretic features
can be reproduced with an adequate choice of the trilinear curve and the control parameters.
The skeleton curve is defined by typical turning points associated with characteristic stages of
the RC section behaviour, namely the cracking (C) and yielding (Y) points, and a post-yielding
branch. The post-yielding stiffness can be indirectly obtained by establishing a ultimate state
point (U) associated to a pre-defined section limit state. All these entities can be different for
positive and negative bending directions indicated by the corresponding superscripts.
The turning points, as well as the ultimate state point, can be identified by means of well
known cracking, yielding and ultimate state criteria along with a fibre-type section discretiza-
tion, thus accounting for the behaviour of steel and concrete through adequate material models.
In this method the section is divided into fibres or layers, to which the specific material models
are assigned, so as to reproduce their uniaxial behaviour. Such formulation is implemented in
u uux u
θx uθy u
θz+ + +=
ues uesux ues
θx uesθy ues
θz+ + +=
u
uux
ux1
00000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= uθx
0θx1
0000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= uθy
00θy1
0θy2
0⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= uθz
000θz1
0θz2⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
ues Q Qes
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 71
the general purpose computer code CASTEM2000 (CEA (1990)), and details can be found in
Guedes et al. (1994).
A general fibre-based procedure can be established for all kind of sections, but, since rectangu-
lar and T-shaped sections are of particular interest, a specific algorithm has been written, based
on analytical expressions. Therefore, a very efficient procedure has been developed (Arêde and
Pinto (1996)) and implemented in CASTEM2000, allowing to account for the confinement of
the concrete core and the strain hardening of steel. Since this corresponds to a pre-processing
task, at the level of input data preparation, the description of that procedure is presented in the
context of implementation notes in 4.2.
Figure 3.2 Primary or skeleton trilinear curve for the global section model.
The hysteretic behaviour is controlled by a set of rules governing the loading, unloading and
reloading phases. Loading is considered when the state point moves along the primary curve,
while unloading occurs for load intensity decreasing without sign inversion. Reloading is asso-
ciated with increasing load intensity, generally after sign inversion and before reaching again
the primary curve. Hysteretic rules can account for the following phenomena:
• unloading stiffness degradation for increasing inelastic deformations;
• the pinching effect, related with the reduction of the hysteresis loops area, due to pro-
nounced decrease of reloading stiffness followed by some stiffness recovering beyond a
certain threshold of plastic deformation;
My+
Mc-
My-
Mc+
ϕ
M
ϕc+
ϕc-ϕy
-k0
1kc
+
kp-
kc-
kp+
ϕy+
C+
Y+
C-
Y-
O
72 Chapter 3
• strength degradation due to deterioration of concrete resisting capacity, as a consequence of
cyclic loading along with increasing plastic deformation.
An exhaustive model description is deemed unnecessary in this work, since the model is in
existance and has been implemented and tested (Kunnath et al. (1990), Kunnath et al. (1992));
however, some details are included in Appendix B, in order to clarify its most relevant fea-
tures, capabilities and limitations.
This model has been chosen because it is quite versatile in reproducing several familiar phe-
nomena of RC section behaviour; nevertheless, other models could be adopted with different
rules for stiffness and strength degradation or the pinching effect, since they do not signifi-
cantly affect the development strategy underlying the global element model.
3.4 Control sections and element zones
3.4.1 Definitions
The control sections, involved in the flexibility element state determination, are introduced in
this chapter. Between control sections, distinct element zones are defined in order to perform
integration for element flexibility and displacement residual computation.
The element is first assumed divided into two parts where the section properties are considered
uniform and equal to those of the corresponding end section; this assumption requires a span
control section where the transition of properties is considered. It is also assumed that yielding
can occur only in zones adjacent to the end sections, whereas cracking is admitted at any span
section. Therefore, in cases where plastic sections may be expected along the span, a subdivi-
sion into more elements should be adopted.
Two groups of control sections are considered:
• Fixed sections: this group includes the element end sections E1 and E2 and a span section H,
dividing the element into the two parts mentioned above and shown in Figure 3.3. End sec-
tions are fully controlled by the adopted global section model; therefore, the corresponding
cracking and yielding stages, as well as pinching and degradation of stiffness or strength
can be taken into account. The span section H is controlled by a simplified model (see 3.5)
allowing for the cracking stage to be considered; since it is a transition section, a double
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 73
control is performed because distinct properties may be assigned to its left and right sides.
This ensures that sections belonging to one specific part of the element are governed
according to the adequate properties and model.
Figure 3.3 Element control sections: fixed sections (E1, E2 and H) and moving section (M)
• Moving sections: this section type is associated to special points in the moment-curvature
relation (from now on referred to as M-ϕ curve), which correspond to the cracking ,
yielding and null moment (O) points, as shown in Figure 3.2. Since they have fixed
moment values, the corresponding section position along the element has to change when-
ever the moment distribution is modified. Actually, the movement of these sections is only
possible because their points in the M-ϕ curve are known and univocally defined.
The treatment of fixed sections is quite normal, requiring only the respective model to be
defined. By contrast, the positions of moving sections need to be found, prior to their state
determination, which can be readily achieved once the moment distribution is known.
Denoting by the moment for a specific moving section, then its position is associated with
the abscissa (see Figure 3.3, where the usual representation of positive moments down-
wards is adopted) which can be obtained from
(3.11)
where the left-hand term is the moment distribution, written in terms of end section moments
and , relative to the respective axis systems as defined in Figure 3.1, and of the ele-
ment applied loads p generically referring to distributed or concentrated loads.
E1
M*
E2Hx*
+- M x( )
xM
sH1sH2
s1 s2
C+ C-( , )
Y+ Y-( , )
M*
x*
M x* p ME1ME2,, ,( ) M*=
ME1ME2
74 Chapter 3
For convenience, local abscissas ( and ) are also introduced (see Figure 3.3) in association
with each end section or element part ( for the left part and for the right
one), as this helps to clarify the scheme for determination of internal moving sections.
Since these sections will be always labelled relating to the index i of the element part they
belong to, the subscript i in is suppressed whenever that abscissa refers to a specific internal
moving section; similarly, for the span section H, refers to its local abscissa in the element
part . According to this convention, Eq. (3.11) can be also written in terms of local abscis-
sas
(3.12)
and, depending on several factors, it may have none, one or more than one solution, as dis-
cussed later on. In any case, it represents the general procedure to obtain moving section posi-
tions, whichever they are, and its solutions are included in Appendix C, for the cases of linear
and parabolic moment distributions, when a given moment is imposed. Due to the impor-
tance of moving sections, they are further detailed in the following sections, separately accord-
ing to their nature.
3.4.2 Cracking sections
Cracking sections are associated to each end section and can be obtained through Eq. (3.12)
by imposing , where stands for the positive or negative cracking moment at
the end section . The following assumptions are considered and commented later in 3.5.3.1:
• once a section has reached cracking, it will not revert back the uncracked state; therefore,
these sections can move along the element in a unique direction;
• if cracking occurs for a certain direction of bending, then the section is also assumed in
cracking conditions for the opposite direction.
Along with these assumptions, the concepts of cracking and cracked sections must be distin-
guished: in a cracking section, the cracking moment has been reached but never exceeded,
whereas in a cracked section that threshold has been previously overcome.
Therefore, depending on the evolution of the bending moment diagram from one step to
another, a cracking section can either remain as a cracking one, or evolve to a cracked section.
s1 s2
Ei EiH i 1= i 2=
si
sHi
EiH
M si* p ME1
ME2,, ,( ) M*=
M*
Ei
M* Mci
+/-= Mci
+/-
Ei
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 75
Figure 3.4 illustrates these possible situations referring to the evolution of a cracking section
, generated by the moment diagram at the k-th load step.
Figure 3.4 Distinction and evolution between cracking and cracked sections
If this diagram evolves to one of those indicated by for the step k+1, the cracking
section turns to a cracked one, and the new cracking section shifts to the right side (in cor-
respondence with the small circles). On the contrary, if is the new diagram,
remains as the cracking section.
The number of cracking sections likely to develop depends on the element applied loads and
must be discussed accordingly.
For the case of no element loads (linear bending moment diagram in the whole element), or for
a concentrated force applied on section H (linear distribution of moments in each element
part), at most two cracking sections can develop along each element part, as shown in Figures
3.5-a) and b).
These sections are considered of two types:
• outermost or extreme cracking sections ( or ), which monotonically move apart from
the corresponding end section ( or ) and are associated with bending moment diagram
decreasing (in absolute value) from the end section towards H or the null-moment section;
• innermost or span cracking sections ( or ), which, by contrast, move towards the
respective end sections and correspond to a moment diagram increasing (in absolute value)
from the null-moment section towards H.
It must be noticed that, in case null-moment sections do not exist inside the element, the above
definitions are still valid if reference to null-moment sections is replaced by reference to end
sections.
Ck Mk x( )
Mc1
-
+
-
Mk x( )
Mc1
+ Ck
Mk+1 x( )
Mk+1 x( )Mk+1 x( )′
Ck evolves to cracked section
Ck remains cracking section
Mk 1+ x( )
Ck
Mk 1+ x( )′ Ck
C1 C2
E1 E2
C3 C4
76 Chapter 3
Figure 3.5 General layout of assumed cracking sections and local abscissas for no element
loads or concentrated force applied in H
For each element part , the extreme and span cracking sections are simply denoted by
and and the corresponding local abscissas are and .
In the case of uniformly distributed force, either with or without a concentrated force applied
on section H simultaneously, the bending moment distribution is parabolic and a maximum of
four cracking sections can be defined in each element part as shown in Figure 3.6.
Actually, the location where cracking starts in this case cannot be easily anticipated due to the
superposition of moments generated by end section loads and by the distributed force. Depend-
E1
Mc1
-
E2H
sc1
+- M x( )
x
Mc1
+ Mc2
+
Mc2
-
C1 C2C3 C4
sc2
Uncracked
Crackeds1 s2
sH1sH2
E1
Mc1
-
E2H
sc1
+- M x( )
x
Mc1
+ Mc2
+
Mc2
-
C1 C2C3 C4
sc3sc2
sc4
Uncracked
Cracked
a) No element loads
b) Concentrated force applied in H
EiH Ci
Ci+2 sCisCi 2+
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 77
ing on the relative importance of these two loading types, cracking does not necessarily start at
the end sections Ei or the span section H; instead it may develop first at any span section where
the maximum moment is found. Therefore, two other cracking sections have to be introduced
( and ) in order to adequately bound the uncracked and cracked zones likely to
develop.
Figure 3.6 Cracking sections and local abscissas for parabolic moment distribution.
However, the total of eight cracking sections in the whole element can hardly develop simulta-
neously. Since cracking moments are usually almost uniform along the element, the full devel-
opment of the four sections in one element part (as assumed in Figure 3.6) would induce full
cracking of the other part, for which only the respective end section and the section H should
be monitored. Furthermore, such a moment diagram is very unlikely in the context of structural
response under cyclic lateral loads.
Sections and are activated only when the maximum moment section does not occur
in the fixed control sections (i.e. when vanishes in one of the element parts), which
is more expectable for low intensity of lateral loads (thus, less relevant end section moments),
precisely when cracking often initiates and is of more importance. On the other hand, these
sections are mainly relevant to define the cracked zones in the internal part of the element,
which are less important for the structural response to lateral loads; additionally, in the context
of cyclic loading and due to the unique moving direction rule of cracking sections, and
can be overridden by the development of and in subsequent steps of loading.
Ci+4 Ci+6
sHi
Ei
Mci
-
H
sci
M si( )Mci
+
Ci Ci+6Ci+2 Ci+4
sci 2+
Uncracked
+
-
Crackedsci 4+
sci 6+
MEi
MH
si
Ci+4 Ci+6
d M dsi⁄
Ci+4
Ci+6 Ci Ci+2
78 Chapter 3
If lateral load effects are predominant over those generated by the distributed force p during
the cracking initiation phase, the sign of tends to be uniform in each element part
(thus for maximum moment occurring in the fixed control sections) and cracked zones can be
adequately bounded by sections and , only.
For example, in the case of a cantilever beam subjected to a distributed load p and a tip force F,
schematically illustrated in Figures 3.7-a), sections and cannot develop along the
element; only cracking sections and are allowed to appear, and the latter only for the
step k´. Also, in the case of a simply supported beam shown in Figures 3.7-b), under the action
of alternating end section applied moments and a constant distributed force p adequately
selected to prevent the development of the maximum moment in the span, sections and
are sufficient to control the cracked zones likely to form. It is worth mentioning that, the
algorithm presented herein has been applied in these two cases, for a distributed force p, and it
has performed quite well under the restrictive assumption of sections and only.
Obviously, the relative predominance of lateral loads or distributed force cannot be easily
anticipated and the development of cracking sections and should be taken into
account. However, from the implementation standpoint, the control of such sections was found
quite cumbersome for the present stage of algorithm development and likely to lead to a very
heavy numerical scheme. In fact, for this yet unexplored modelling strategy, a very accurate
description of cracking zones in the central part of the element was deemed of lower priority,
because:
• It stands only for a temporary stage of the loading history and occurs in element locations
less relevant for the response to lateral cyclic loads.
• It is also affected by several other assumptions such as: the fact that cracking in one bending
direction induces automatically cracked behaviour in the opposite direction, which for the
central zone may not be true due to the local predominance of element applied loads; the
consideration of cracking moments and post-cracking stiffness involving only the section
behaviour, without taking into account effects of tension stiffening; the uncertainty related
with the collaborating slab width to be considered in the section modelling; the approxima-
tion of distributed force by a uniform diagram, which for very common cases is quite debat-
able (as for example when bi-directionally reinforced concrete slabs lead to force
distributions in the beams better described by a bi-triangular diagram, rather than a uniform
one).
d M dsi⁄
Ci Ci+2
Ci+4 Ci+6
C1 C4
Ci
Ci+2
Ci Ci+2
Ci+4 Ci+6
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 79
Figure 3.7 Examples of restricted cracking sections in the presence of distributed force
Despite a first implementation phase in which the uniformly distributed force p was effectively
considered along with the control of sections and only, in order to assess the algorithm
ability to treat simple and adequately chosen non-critical cases (as those referred above and
shown in Figures 3.7), the aforementioned reasons justify the adoption of a simplified strategy
for considering the effects of p without developing sections and .
The approximate procedure consists in replacing the uniformly distributed force p by a concen-
trated and equivalent one Peq applied in span section H, usually located at or near the half-
span. Several equivalence criteria can be considered, but the most logical appears to be the one
based on equality between elastic fixed-end moments due to p and those due to Peq. Note that
p is usually the first load to be applied (normally arising from dead and live loads), when the
structure behaviour is elastic or quasi-elastic. With this approximation, the control of cracking
sections reduces to that already described for the concentrated force case shown in Figure 3.5-
b).
Mc1
C1
a) Cantilever beam
E1 E2H
M x( )
ME1
b) Simply supported beam
p
F
ME1ME2
Mc1
C1
E1 E2H
M x( )'
ME2'
Mc2
C4
Step k
Step k´>k
p
Mc1
- Mc2
-
Mc2
+Mc1
+
C1
E1 E2H
M x( )Step k
C2
C4
Mc1
- Mc2
-
Mc2
+Mc1
+
C1
E1 E2H
M x( )
Step k´>k
C3 Full cracked
Ci Ci+2
Ci+4 Ci+6
80 Chapter 3
It is important to note that, from the implementation standpoint, this simplification is quite sig-
nificant due to the particular features of the model adaptations to be referred later (see 3.5). In
fact, such features (particularly the cracking curvature discontinuity), along with the moving
character of cracking sections and the possibility of full cracking in one or both element parts,
require special care for the control of cracking sections and their contributions to the element
flexibility (see 3.6.2); therefore, the more cracking sections are involved, the more elaborated
is the procedure to control them all.
However, in a future phase of this element model development, the consideration of parabolic
moment diagrams and the inherent control of sections and , shall be included in
order to render the algorithm more general. For this reason, the formulation described in this
study already explicitly includes the force p, both in the derived expressions and in some fig-
ures, although the implemented algorithm corresponds to the simplified version above men-
tioned. Therefore, the expressions effectively used for the cases where distributed load exists
in the element, correspond to and .
It is noteworthy that, in each element part, cracking section locations are bounded by the corre-
sponding end section and the span section H; therefore, in case they are found out of the
respective element part, they are assumed coincident with the closest bound and removed from
the control process.
In order to comply with the definition of cracking sections as stated above, an adequate search-
ing scheme is enforced independently for each element part , using the respective local
abscissa (thus, expressing the moment distribution as ). Such scheme consists of the
following steps:
• First, the adequate cracking moment directions are set according to Figure 3.6 and to the
following definitions of and as described in Figure 3.8:
is the sign of cracking moment allowing to obtain (and possibly )
is the sign of allowing to obtain (and possibly ).
• The local abscissas of cracking sections are then obtained from the defining equations and
according to some restrictive conditions summarized in Table 3.1, where the superscripts
and in take the signs as defined above.
The solutions of the defining equations are detailed in Appendix C and the restrictive condi-
Ci+4 Ci+6
p 0= P Peq=
Ei
EiH
si M si( )
b b
b MciCi Ci 6+
b b–= MciCi 2+ Ci 4+
b
b Mci
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 81
tions are introduced to establish whether or not a given cracking section can be defined for the
current moment distribution (e.g., the abscissa of section is provided by the solution(s) of
but it can only be considered as a valid solution for if is
positive and if holds in case the section has been previously activated).
Figure 3.8 Definition of cracking moment directions
Table 3.1 Definition of cracking section abscissas
Section Defining Equation Restrictive Conditions
Ci 2+
M si( ) Mci
b= sCi 2+d M dsi⁄( )
Ci 2+
sCi+2sCi+6
≤ Ci 6+
b sign MEi( )= if d M
dsi-----------
Ei
0<
b sign– MEi( )= if d M
dsi-----------
Ei
0>
Mci
- M si( )
+
-
MEi
EiMci
+ +
-
MEi
Ei
Mci
+
-
MEi
Ei
Mci
-
+MEi
Ei
M si( )
M si( )
M si( )
si si
Ci M si( ) Mci
b= d Mdsi
-----------Ci
0<
Ci+6 M si( ) Mci
b= d Mdsi
-----------Ci 6+
0>
Ci+2 M si( ) Mci
b= d Mdsi
-----------Ci 2+
0> sCi+2sCi+6
≤
Ci+4 M si( ) Mci
b= d Mdsi
-----------Ci 4+
0< sCi+4sCi
≥
82 Chapter 3
The consideration of cracking sections, leads to a set of cracked and uncracked zones with a
general configuration as the one shown in Figure 3.6. Thus, for a complete development of
these zones, the following classification holds in each element part:
Obviously, when sections and are not considered, the third and fourth zones do not
exist and is replaced by in the fifth zone.
3.4.3 Yielding sections
Yielding sections are obtained in the same way as the cracking sections, but the process
becomes simpler since yielding is not admitted at span sections. Thus, yielding sections,
denoted by appear associated to the end sections , when is imposed in Eqs.
(3.11) or (3.12); as for the cracking sections, refers to the positive or negative yielding
moment at the end section .
The general layout of the assumed yielding and cracking sections is, as shown in Figure 3.9,
almost analogous to that of Figure 3.5-b). As for cracking sections, the following assumptions
are considered and discussed later in 3.5.3.1:
• the yielded section behaviour is irreversible, i.e. the section stiffness prior to yielding can-
not be recovered, thus enforcing also yielding sections to move in a unique direction along
the element;
• if yielding takes place for a certain bending direction, then a yielded behaviour is also
enforced for loading in the opposite direction.
Again, a clear distinction between yielding and yielded sections is due as for the cracking sec-
tions: in the former case, the yielding threshold has never been exceeded, while for the latter
the yielding conditions have been previously reached. Positions of yielding sections are found
by means of the same procedure as adopted for the extreme cracking sections, replacing the
cracking moments by the yielding ones; therefore, no further explanation should be required.
Since cracking moments are always lower than yielding ones (at least for normally designed
sections), yielding sections will be always behind cracking ones (referring to local abscissas).
cracked uncracked cracked uncracked cracked
Ei Ci– Ci Ci+2– Ci+2 Ci+4– Ci+4 Ci+6– Ci+6 H–
Ci+4 Ci+6
Ci+6 Ci+2
Yi Ei M* Myi
+/-=
Myi
+/-
Ei
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 83
Figure 3.9 General layout of assumed yielding and cracking sections
Therefore, for complete development of cracking and yielding sections under the assumption
of concentrated load applied at H, at most a new zone has to be considered per element part,
leading to an updated set as follows:
3.4.4 Null moment sections
So far, most of the schematic examples illustrated in previous figures have shown null moment
sections lying in uncracked zones. Since these zones behave linear elastically, the control of
such null moment sections is easy from the computational standpoint.
However, the same does not hold true if these moving sections lie on cracked or yielded zones,
since, for positive and negative bending directions, distinct behaviours are likely to appear in
post-cracking sections. Due to the irreversibility of cracking and yielding section progression,
at a certain load step, the null moment sections may fall into a previously cracked or yielded
zone. That is the case illustrated in Figure 3.10, where both bending moment diagrams and the
corresponding cracking sections are included for two consecutive steps of loading (k-1 and k).
For simplicity, neither the yielded zones nor the span cracking section for the right element
part (coincident with H), are included.
yielded cracked uncracked cracked
E1
Mc1
-
E2
H
+
-
M x( )
x
Mc1
+Mc2
+
Mc2
-
C1 C2C3 C4sy2
Uncracked Cracked
My1
-
My2
-
Y2Y1
Yieldedsy1
Ei Yi– Yi Ci– Ci Ci+2– Ci+2 H–
C4
84 Chapter 3
Figure 3.10 Locations of null moment sections
It can be seen that, while the null moment section for the step k-1 is lying in an uncracked zone
, for the step k the null moment section falls in the cracked zone .
Since in this zone, sections with positive moments may exhibit different behaviour from those
in negative bending, the transition section must be controlled in order to correctly account
for stiffness variations appearing there.
As it will be seen in the following sections, the simplifications introduced at the section model-
ling level (see 3.5) suggest that, null moment section control is simple only if it lies in a
cracked zone; by contrast, the control is much more demanding in the yielded zone. Therefore,
in the present work, null moment sections are activated and controlled only when they exist in
cracked zones.
Null moment section locations are obtained by a similar but simpler procedure to that cracking
and yielding sections. Imposing in Eqs. (3.11) or (3.12), solutions can be obtained
and checked if they are located inside element parts. However, this must be carried out only
after cracking and yielding sections have been found, in order to check if null moment sections
have to be controlled or not.
E1
Mc1
-
E2
H
+
-
Mk-1 x( )
x
Mc1
+Mc2
+
Mc2
-
Uncracked
Cracked
C1k-1
C2k
C2k-1
Mk x( )
C3k
C3k-1
C1kOk
Ok-1
Cracked at step k Cracked at step k-1
C1k-1 C– 3
k-1[ ] Ok E1 C– 1k[ ]
Ok
M* 0=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 85
3.5 Behaviour of the control sections
3.5.1 Modified trilinear model
3.5.1.1 General
As aforementioned, the control of internal moving sections is only practicable if their repre-
sentative points in the M-ϕ curve are known without the need for tracing out all the behaviour
history. This fact becomes clearer by considering Figure 3.11, which illustrates examples of
possible evolutions of cracking and yielding section points along the trilinear model curve. For
both sections, it is assumed that at the load step k their points in the curve are and , coin-
ciding with the curve turning points. For subsequent steps k+1, k+2,..., these section points
occupy the positions and (j=1,2,...), corresponding to a sequence of loading, unloading
and reloading.
Figure 3.11 Evolution of cracking and yielding section points in the model diagram
It is apparent that passing from to presents no problem for the section state determina-
tion, since all the necessary data are available in the primary curve (previous point and stiff-
ness, as well as new stiffness). However, once an unloading step occurs ( to ), the new
stiffness becomes dependent on the inversion point, which, furthermore, becomes necessary
also for subsequent reloading steps. This point, although lying on the basic curve, must be
stored for each step and for all cracking sections continuously appearing. The same reasoning
holds for yielding sections.
Obviously, such requirements are practically impossible to be achieved and thus some simpli-
fying assumptions must be included, in order to perform the state determination of internal sec-
tions using only the primary curve data and the section moment (or curvature) for the step
under analysis.
C0 Y0
Cj Yj
ϕ
M
C0
Y0
C1
C2
Y1
Y2
C0 C1
C1 C2
86 Chapter 3
3.5.1.2 Motivation for model modification
The simplifications adopted in this work arise from the recognition of the detailed shape of M-
ϕ curves for RC sections. Actually, the trilinear curve is just a possible approach to the real
one, and its validity depends on several factors related to the characteristics of material model,
the section geometry, the steel contents and, last but not least, the acting axial force.
The real diagram M-ϕ of a given section is very difficult, if not impossible, to obtain by exper-
imental means, since the measurement of curvatures is always performed over a finite length.
Any M-ϕ curve experimentally obtained is an average diagram over a certain zone, rather than
the effective local measure corresponding to a well defined section. Thus, in this context, the
concept of real M-ϕ curve is not strictly meaningful.
Nevertheless, a good approximation to the actual section moment-curvature behaviour can be
achieved using the above referred fibre model (Vaz (1992), Guedes et al. (1994)) for the dis-
cretization and analysis of sections. In the fibre model formulation as implemented in the com-
puter code CASTEM2000, the main features of the adopted concrete and steel models (for
monotonic loading) are very similar to those assumed in the procedure for the M-ϕ primary
curve definition as used in the present study (see 4.2). However, two differences are found that
should be referred:
• the post-yielding behaviour of steel is more refined in the fibre formulation context, since a
yielding “plateau” is considered and a fourth degree polynomial is used for the hardenning
range; however, with slight modifications in model parameters, it can be forced to fit the
bilinear approach used in Arêde and Pinto (1996);
• the concrete tensile behaviour in the fibre formulation includes a linear softening branch
after the tensile strength is reached, whereas in the present study the strength drops immedi-
ately to zero once cracking occurs; again, the two assumptions can be made to fit by ade-
quate choice of model parameters, but the inclusion of concrete tension softening may be
appropriate as discussed later.
The aforementioned fibre formulation allows to obtain the section M-ϕ curves and, in associa-
tion with a linear Timoshenko finite element discretization (Guedes et al. (1994)), the non-lin-
ear behaviour of beam/column structural components can be traced out. On that basis, a
sensitivity study of section M-ϕ curves can be found in Arêde and Pinto (1996), aiming at the
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 87
validation of the proposed procedure to obtain the trilinear primary curve. Several RC sections
have been analysed by the fibre model and compared with the trilinear M-ϕ curve, for varying
conditions related to the total amount of longitudinal reinforcing steel, the ratio of compression
to tension steel contents, the slab width participating in beam behaviour and the axial force.
A detailed description of that study is out of the scope of the present work, but, since the axial
force has been recognized as the most important factor affecting the typical shape of M-ϕ
curves, an example of a symmetrically reinforced concrete section with two levels of axial load
is analysed below. The section geometry, the reinforcement and the basic parameters for the
material models are illustrated in Figure 3.12; more details about the concrete model can be
found in 4.2. The axial load level is defined by the normalized axial force ν, expressed in the
usual way as , and has been adopted with the values 0 and 0.1. Figure 3.12
includes also the scheme of a cantilever beam with uniform section, discretized with constant
curvature Timoshenko elements, in order to obtain the force-displacement diagrams for both
section modelling strategies: the fibre formulation and the trilinear curve.
Figure 3.12 Comparison of fibre and trilinear section modelling formulations. Section, member
and model data
ν N bdfc0( )⁄=
b=0.45m
h=0.
45m
d=0.
41m
4 φ 20
4 φ 20
φ 10 @.075
fc
εcεc0
fc0
fctEc
Confined
Unconfinedfcc
0.2*fcc
fc0 = 44.8 MPa
fct = 4.48 MPa
εc0= 2 (*10-3)
Ec = 33.7 GPa
fs
fy
εsεy εu
Esh
Concrete Steelfy= 570 MPaεy= 2.85 (*10-3)
Esh= 0.87 GPa
εu= 100 (*10-3)
L=3.5m
1 5 8 3
N
No. and length of Timoshenko elements:
Cantilever beam
Section
.5 2.0m
a) Section detailing
c) Concrete and steel model data
F
u
.05
b) Member layout and discretization
88 Chapter 3
The M-ϕ curves obtained from the application of both section formulations, as well as the cor-
responding cantilever tip force-displacement diagrams F-u, are compared in Figure 3.13 for
the two levels of axial force.
Figure 3.13 Comparison of fibre and trilinear section modelling formulations. Local and global
response for two axial load levels
The following aspects can be observed at the section level:
• For zero axial force (the usual case of beams) the post-cracking behaviour exhibits a tempo-
rary decrease of resisting moment due to a progressive transfer of stress from cracked con-
ν = 0.0 ν = 0.1
Displ.(m).
Force (kN)
Displ.(m)
Force (kN)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 X1.E-2
0.
20.
40.
60.
80.
100.
120.
140.
Curv.(m-1)
Mom.(kN.m)
.0 .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 x1.E-2
0.
.5 1.0 1.5 2.0
2.5 3.0 3.5 4.0 4.5
5.0 x1.E2
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 X1.E-2
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
Fibre Trilin Tril_mod
a) Moment- curvature diagrams
b) Tip force-displacement curves
ν = 0.0 ν = 0.1
Fibre Trilin
Curv.(m-1)
Mom.(kN.m)
.0 .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 x1.E-2
.0 .5 1.0 1.5
2.0 2.5 3.0 3.5 4.0
4.5 5.0 x1.E2
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 89
crete fibres to uncracked ones and to the steel; after a certain point, the influence of
tensioned steel becomes dominant and the resistance increases again with a stiffness that
tends gradually to the fully-cracked section stiffness (i.e., when all the concrete in the ten-
sioned area is considered inactive). In such case it can be seen that a trilinear curve does not
fit well in the post-cracking zone, where the fully-cracked stiffness (pointing from the ori-
gin to the yielding point) would be more appropriate.
• For the case of , the transition phase due to cracking is not so pronounced because
the acting axial force implies higher compressive forces in the section; therefore, the ten-
sioned concrete area is less important to the section equilibrium and the stress transfer
becomes less visible during the transition phase. The trilinear curve appears now to be a
good approximation.
The force-displacement results agree with the above comments. Actually, for the null axial
force case, the “lack” of curvature shown by the trilinear curve between the cracking and yield-
ing points, leads to significant lower displacements although with tangent stiffness not very
different from that of the fibre formulation.
By contrast, the good fitting of M-ϕ curves when the axial load is present, leads also to an
excellent agreement of force-displacement curves.
It is important to note that, for , the immediate post-cracking strength drop in the M-ϕ
curve is much more pronounced than in the F-u response of the element. In the case of the
results shown, this is just due to the fact that cracking is not occurring simultaneously in all
sections. However, two other phenomena, not considered in the above results, can also contrib-
ute to that fact, namely:
• The inclusion of concrete tension-softening with finite slope, instead of a sudden elimina-
tion of tensile strength whenever the peak tensile strain is exceeded (i.e., an infinite tension-
softening); this aspect is strictly related to the local section behaviour modelling and tends
to increase the cracking moment and to smooth the strength drop (already at the M-ϕ level).
This is shown in Figure 3.14, where the section fibre modelling includes now a linear sof-
tening branch with finite slope for the tension behaviour model, also illustrated in the M-ϕ
diagram. The dashed vertical line in the tension model corresponds to the assumption made
in the case of Figure 3.13. However, the cracked stiffness still fits quite well the M-ϕ curve
in the post-cracking phase.
ν 0.1=
ν 0=
90 Chapter 3
Figure 3.14 Effects of tension-softening in the fibre formulation
• The fact that, in a given element zone, bending moments have exceeded the cracking
moment, does not imply all the sections to be effectively cracked; due to bond between rein-
forcement and the surrounding concrete, gradual redistribution of internal forces take place
from concrete to steel leading to a crack pattern that tends to stabilize at a certain finite
spacing (Feenstra (1993)). Therefore, a cracked zone has a higher stiffness than if consid-
ered with a uniform cracked stiffness as taken from the M-ϕ curve. This phenomenon,
known as tension-stiffening, is related with the behaviour of the global element (or part of
it) rather than with the local section behaviour; thus, its effect appears only in the global
response and tends to smooth even more the post-cracking transition phase.
Feenstra (1993) discusses the superposition of these two effects in RC members, though in the
stress-strain context, which can be also considered in the M-ϕ space for a clear understanding
of the cracking transition phase. However, the tension-stiffening effect is often indirectly mod-
elled by means of tension softening in the concrete model (Figueiras (1983), Póvoas (1991)).
3.5.1.3 Proposed model modifications
In order to partially account for the above described behaviour, a modified M-ϕ curve would
be desirable. On the one hand, the post-cracking strength drop has very localized effect which,
furthermore, is negligible when global element behaviour is sought; on the other hand, it is par-
tially counterbalanced by finite tension-softening of concrete and by the tension-stiffening
effect along the member. It is clear, however, that a transition from the uncracked to the fully-
cracked stiffness has to be included (at least for cases with zero or low axial force level) if the
ν = 0.0
Displ.(m).
Force (kN)
Curv.(m-1)
Mom.(kN.m)
.0 .3 .6 .9 1.2 1.5 1.8 2.1 2.4 2.7 3.0 x1.E-2
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 x1.E2
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 x1.E-2
0.
10.
20.
30.
40.
50.
60.
70.
80.
90.
fct
3εctεct εc
TensionModel
Fibre Trilin Tril_mod
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 91
post-cracking behaviour is to be adequately traced out. With this in mind, a modified M-ϕ pri-
mary curve is proposed as follows:
• sections behave linearly with elastic stiffness up to the cracking moment;
• at imminent cracking, the transition is enforced from uncracked to fully-cracked stiffness, at
constant moment (the cracking one); this means that discontinuities arise in both stiffness
(or flexibility) and curvature distributions along the member;
• for incipient yielding, sections are likely to change to the post-yielding branch as in the
usual trilinear case, thus inducing discontinuity only in the stiffness distribution.
The corresponding primary curve is also included in the M-ϕ diagrams of Figures 3.13 and
3.14 (under the reference of Tril_mod) and, as already pointed out, an excellent fit with the
fibre formulation curve is obtained when axial force is negligible. If the axial force becomes
significant, the two curves do not agree so well. A possible solution to approximately over-
come this drawback could be the adoption of an intermediate post-cracking stiffness (between
the uncracked and the fully-cracked one), estimated in such a way to compensate the excess of
member displacements caused by the use of the fully-cracked branch. Indeed, this was tried for
some column cases for which the force-displacement response obtained by the proposed ele-
ment model, with an intermediate stiffness, tended to the fibre modelling response. However,
any estimate of such a stiffness depends on the expected level of end section deformation and
the distribution of curvatures along the element. Since this could not provide a general criteria,
it was decided to keep the fully-cracked stiffness for post-cracking behaviour in the present
stage of model development, although bearing in mind that column deformations might
become somewhat overestimated when the behaviour is mostly in the post-cracking range.
The hysteretic rules associated to the modified trilinear M-ϕ curve are shown in Figure 3.15
and can be stated as follows:
• loading, unloading and reloading of uncracked sections take place along the linear elastic
branch (1) with flexibility ;
• for cracked sections, i.e. those for which (or ), where
is the maximum moment (positive or negative) experienced up to the present load
stage, loading occurs along branches 2 (or 3) up to yielding, while unloading and reloading
are done pointing to and from the origin along branches 6 (or 7) with flexibility (or );
this means that sections between the cracking and yielding ones are not allowed to have
residual deformations at zero moment;
f0
Mc+ Mmax My
+≤< My- Mmax Mc
-<≤
Mmax
fy+ fy
--
92 Chapter 3
• the case of yielded sections, i.e. for which (or ), is treated with
almost the same rules of the trilinear model described in Appendix B, namely concerning
unloading and reloading with possible pinching or strength degradation; for subsequent pur-
poses, the following generic notation is adopted for the flexibility of yielded sections:
for loading, for unloading and for reloading.
Two relevant differences, relative to the rules stated in Appendix B, shall be referred:
• for unloading in yielded zones, the common point is now on the lines (or ), in
order to cope with the fact that a given yielding section unloads pointing to the origin; there-
fore, continuity of stiffness distribution exists at the yielding section for unloading condi-
tions, but the same parameter, as defined in Appendix B, leads now to a greater amount of
stiffness degradation;
• after a section has yielded for a given bending direction (e.g. ), but not actually for
the opposite direction (e.g. ), then, for reloading in this opposite direction, the
maximum point is taken at the yielding point ( in Figure B.2) unless the pinching
effect has to be also considered; in fact, this procedure is similar to those adopted in several
other models based on a bilinear primary curve (e.g. Filippou et al. (1992), Coelho (1992)).
Figure 3.15 Hysteretic rules for the modified trilinear model
The modified trilinear model features allow an immediate knowledge of each internal section
Mmax My+> Mmax My
-<
fp+/-
fu+/- fr
+/-
OY+ OY-
α
M My+>
Mmax My->
E- Y-≡
f0
fy+
fp+
Mc-
Mc+
ϕ
M
C+
My+
Mc-
My-
Mc+
ϕ
M
ϕy- ϕy
+
C+
Y+
C-
Y-
Cc+
Cc-
C-
1
1
3
7
2
6
My+
My-
ϕ
M
ϕy- ϕy
+
Y+
Y-
7
6
5
4
fy-
fu+
fp-
fu-
a) Uncracked sections b) Cracked sections c) Yielded sections
o o o
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 93
state point in the M-ϕ diagram, without the need for all the previous loading history. Actually,
with the exception of yielded sections (i.e., between end and yielding sections), the state deter-
mination for any internal section is straightforward and can be done in terms of either input
moments or input curvatures. This is obviously a consequence of the one-to-one nature of the
assumed M-ϕ curve and is further detailed in the next section.
Finally it must be underlined that, despite the fact that model modifications have been intro-
duced reporting to internal moving section behaviour, the modified model is adopted also for
the fixed sections, namely the end sections and the span (H) one, because they are likely to
pass by internal state phases similar to those of moving sections. The difference lies only on
their fixed character, which allows for their loading history to be monitored.
3.5.2 Transition from uncracked to cracked section behaviour
The transition from uncracked to fully-cracked behaviour is performed along a constant
moment plateau, herein designated by cracking plateau, which, according to 3.4, is assumed
uniform in each element part ( and ).
If one looks at the behaviour of an isolated section following such a model, this plateau
assumption appears to be a major drawback since it introduces an indetermination when the
cracking moment is reached. However, if the element moment distribution is taken into
account, it becomes apparent that such indetermination can hardly occur in most practical situ-
ations. Referring to a given element part the two following cases can take place:
• Non-uniform bending moment distribution
The cracking moment is found at a finite number of sections where a “jump” in the curva-
ture distribution is assumed, as shown in Figure 3.16 for linear moment distribution.
It can be seen that no state points exist between and in the M-ϕ diagram, regardless of
the way of crossing the plateau; for this reason the plateau is traced as a dashed line.
The left and right sides ( and ) of the cracking section have their state points in the M-
ϕ diagram coincident with and , respectively. The curvature diagram exhibits discon-
tinuity between these two points, which introduces a special contribution in the element
flexibility matrix due to the section that is changing with the applied moments. This
aspect is crucial to successfully pass the cracking plateau and is detailed in 3.6.2 for the cor-
rect flexibility derivation.
E1H E2H
EiH
C Cc
CL CR
Cc C
C
94 Chapter 3
Therefore, if the influence of the cracking plateau is adequately taken into account, it can be
used for the cracking transition with no risk of indetermination because, in this loading
case, no section state point really exists in the plateau.
Figure 3.16 Cracking transition in the case of non-uniform moment distribution
• Uniform bending moment distribution
This loading case, although not so common in practical situations, is more troublesome
when the cracking plateau has to be overtaken. The case of may arise either
due to combinations of end section applied moments and applied force at H exactly match-
ing the moment in that element part, or due to rotations imposed at the end sections
such that the required uniform curvature lies between and , as shown in Figure
3.17. The former case leads to an indeterminate solution, while the solution for the latter is
readily known ( ), but can it be numerically obtained only if a special iterative procedure
is adopted to bypass the zero stiffness problem. It is clear, though, that all sections have
their state points in the cracking plateau; curvature discontinuities do not appear and the
whole element part is controlled by the end section.
In order to handle both situations of moment distribution, a specific algorithm can be adopted
for the cracking plateau transition based on the following:
• Instead of assuming a constant moment plateau, a very small non-zero stiffness is assigned
to the cracking transition; this is meant to avoid the solution indetermination when, for
example, a constant moment is applied.
CCL CR
Left Right
ϕc ϕ si( )
Mc
ϕ
M
C
L
oϕcL
ϕcR
Cc
R
Mc
M si( )
C
∆ϕcc
∆ϕcc
Ei H
si
HEi
M si( ) Mc=
Mc
ϕa( ) ϕc ϕcc
Mc
Mc
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 95
• Any section whose state point lies on the cracking plateau is governed by the fully-cracked
flexibility , for loading (i.e, above the plateau) and a secant flexibility or for unload-
ing or reloading below the plateau.
Figure 3.17 Cracking transition in the case of uniform moment distribution
These last procedures are schematically illustrated in Figure 3.18, where, for simplicity, the
assumed non-zero stiffness of the plateau is not shown. The iterative process for searching the
solution across the plateau is identical to the section state determination procedure shown in
Figure 2.10, but the updated flexibility is always . Two possible situations are considered:
the case where convergence is reached for curvature beyond the plateau (Figure 3.18-a)) and
that for convergence occurring inside the plateau (Figure 3.18-b)).
Unloading from the plateau occurs pointing to the origin (see Figure 3.18-c)) and eventual
reloading on the opposite direction is done aiming at a plateau point with a fraction of curva-
ture “jump” equal to the maximum one previously attained . While
this maximum curvature “jump” is not exceeded, the unloading and reloading flexibilities
remain unchanged; further increase of that “jump” implies the flexibility to be re-activated .
Attention must be drawn to the fact that the scheme of Figure 3.18-a) could be present in all
loading cases (uniform ones included), whereas those of Figures 3.18-b) and 3.18-c) exclu-
sively refer to the uniform loading case reaching the cracking moment or unloading from it.
This procedure has been actually implemented in the present work context, but very poor con-
vergence performance was obtained for current loading cases, i.e., typically consisting of non-
uniform bending moment diagrams. Therefore, it was found preferable to distinguish the two
cases of moment distribution and to treat them accordingly:
fy fu fr
ϕcϕ si( ) ϕa=
Mc
ϕ
MC
oϕcc
ϕcc
CcMc
Ei
ϕcc
ϕa
M si( ) Mc=
Ei
Hsi
H
fy
∆ϕ* ∆ϕCc
-⁄( ) ∆ϕE+ ∆ϕCc
+⁄( )
fy
96 Chapter 3
• If a uniform moment distribution is found for a given element part, the scheme shown in
Figure 3.18 is adopted for the cracking plateau transition; it is noteworthy that, in such a
case, no internal moving sections need to be activated because the behaviour of the whole
element part is controlled by the respective end section.
• Conversely, for the most usual case of non-uniform moment distribution, the direct algo-
rithm is used, where the total curvature “jump” is considered in all the cracked zone sec-
tions for the cracking plateau transition (see Figure 3.16).
Figure 3.18 Rules for progressive transition of the cracking plateau transition
This strategy avoids penalizing convergence performance in the large majority of cases, but
requires extra implementation effort since the following issues have to be taken into account:
• Cases of bending moment distribution close to uniform (low rate of variation along the ele-
ment), may lead to sudden cracking along rather large zones, thus developing significant
displacement residuals to be eliminated; as this may cause non-convergence problems (typ-
ically characterized by iterative solutions “jumping” below and above the correct solution,
but never being able to reach it), numerical solution guiding schemes have to be adopted,
such as the line search technique (Simons and Powell (1982), Criesfield (1982), Marques
(1986)) in order to successfully achieve convergence in the element iterative scheme.
• Both procedures to overcome the cracking plateau (Figures 3.16 and 3.18) have to be made
compatible in the same algorithm.
Mc
ϕ
M
C
oϕcc
ϕc
Cc
fy
f0
fy
Mc+
ϕ
M
C
oϕcc
+ϕc+
Cc
fy+
f0
Mc
ϕ
M
C
oϕcc
ϕc
Cc
fy
f0
fy
b) Convergence INSIDE the plateau
fy-
fu+
fr-
∆ϕE+
E
∆ϕ*
c) Unloading from or
fr- fu
+ f0–
fy+ f0–
---------------⎝ ⎠⎜ ⎟⎛ ⎞
fy- f0–( )=fu
+ ϕE+
Mc+
--------= =>
a) Convergence BEYOND the plateau
reloading to the plateau
∆ϕCc
+
∆ϕCc
-
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 97
3.5.3 State evolution of control sections
3.5.3.1 Internal moving sections
The fact that internal moving sections (cracking, yielding and null-moment sections) are asso-
ciated to turning points in the model curve, where flexibility and/or curvature discontinuities
may occur, implies the left and right sides of each section to be monitored independently. For
this purpose, an adequate convention must be adopted for the section side identification; since
internal sections are associated to end-ones, it is assumed that the left side of an internal section
is the side existing in the interval between that section and the corresponding end-one, while
the right side is the one existing outside that interval.
With this convention and recalling the concepts of cracking and cracked section, as well as of
yielding and yielded sections, it follows that:
• a cracking section has one uncracked side and another side behaving as cracked;
• a yielding section presents cracked behaviour in one side and yielded in the other.
This distinction is important because each section side follows different paths for subsequent
loading evolutions. Consider the step for the k-th external load increment; at the beginning of
that step, cracking and yielding sections refer to the previous step k-1 and are denoted by
and , respectively, while at the end, the corresponding updated sections are and .
Depending on the local effect of the load increment (loading, unloading or reloading), the fol-
lowing state evolution cases are likely to appear (referring to Figure 3.15). For cracking sec-
tions, and, for simplicity, assuming the load increment small enough to avoid “jumping”
directly from uncracked to yielded behaviour, the following situations hold:
• unloading or reloading without generation of new (=> ):
the uncracked side remains elastic (branch 1) and
the cracked side follows the lines (branches 6 or 7);
• loading or reloading with generation of new (=> ):
the uncracked side proceeds from elastic (branch 1) to cracked (branches 2 or 3) and
the cracked side remains along the lines (branches 2 or 3);
Ck-1
Yk-1 Ck Yk
Ck-1 Ck Ck Ck-1≡
OY+/-
Ck-1 Ck Ck Ck-1≠
OY+/-
98 Chapter 3
• (just generated):
the uncracked side remains elastic (branch 1) and matches point , whereas
the cracked side moves from uncracked behaviour to the lines (branches 2 or 3),
matching point ;
In turn, for yielding sections the following cases are considered:
• unloading or reloading without generation of new (=> ):
both the cracked and the yielded side follow the lines (branches 6 or 7);
• loading or reloading with generation of new (=> ):
the cracked side proceeds from lines (branches 2 or 3) to yielded behaviour
(branches 4 or 5) and
the yielded side remains along the post-yielding lines (branches 4 or 5);
• (just generated):
both the cracked and the yielded sides match point , being the cracked side on lines
(branches 2 or 3) and the yielded one on the post-yielding lines (branches 4 or 5).
It will be shown later that only the current step (k) configuration of internal moving sections is
of interest for the state determination and for residual computation. Therefore, the behaviour of
the step k-1 moving sections is not needed to be controlled, which renders the numerical imple-
mentation easier and clearer.
For the load step under analysis, all the cracked sections exist between the cracking and the
yielding or the span (H) sections (see Figure 3.9). In any case, their state points lie on the
lines, likely to have different slopes in the positive and the negative direction. Since
these lines point to the origin, the element stiffness distribution may change exactly at the null-
moment section (O), if it happens to fall in a cracked zone. As already mentioned in 3.4.4, if
the section O lies on a uncracked zone its control is not required because the elastic stiffness
does not change with the bending direction. Conversely, if it shows up in a yielded zone, it can-
not be easily controlled since it is a moving section without monitored loading path. Therefore,
the null-moment section is controlled as a cracked section, following the model labelled
cracked in Figure 3.15-b), and is only activated if falling in a cracked zone.
Ck
C+/-
OY+/-
Cc+/-
Yk-1 Yk Yk Yk-1≡
OY+/-
Yk-1 Yk Yk Yk-1≠
OY+/-
Yk
Y+/-
OY+/-
OY+/-
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 99
The yielded sections, i.e. those between the yielding and the end sections, appear to be the
most problematic to control, since no simplified and clear model is assigned to them and their
loading history is impracticable to be known. Therefore, as specifically discussed later, yielded
sections will be approximately controlled according to the behaviour of the associated end and
yielding sections simultaneously.
At this point some comments are pertinent about the assumptions concerning the cracking and
yielding development as stated in 3.4.2 and 3.4.3, respectively.
The irreversibility of cracking is a very acceptable assumption in view of the impossibility of
recovering the loss of concrete strength and the inherent stiffness decrease and permanent
deformations. However, for sections in the pre-yielding range and having cracked only for one
bending direction, some temporary stiffness recovery can be observed until cracking occurs for
the opposite bending direction (this is apparent in the results shown in Figure 2.2); after that,
the stiffness progressively decreases towards the fully-cracked stage. This effect, as well as the
permanent deformations actually developing in the pre-yielding range, cannot be taken into
account by the adopted model due to both the irreversibility assumption and the origin-oriented
stiffness considered after cracking.
The temporary recovery of post-cracking stiffness is not of major importance in the cyclic
loading context, because cracking load reversals are very likely to occur in the zones where
stiffness variations are more relevant for the global element behaviour; this sustains both the
irreversibility issue and the assumption that a given cracking section for one bending direction,
is also a cracking one for the opposite direction. In turn, the non consideration of permanent
(residual) deformations leads to the underestimation of energy dissipation in the pre-yielding
range. Despite these shortcomings, the referred assumptions were still considered in order to
assure one-to-one model diagrams that make feasible the control of cracking sections; how-
ever, it is recognized that future improvements should be made on the energy dissipation issue.
Concerning the yielding irreversibility, it is also apparent that, upon load reversals, the pre-
yielding (fully-cracked) stiffness cannot be recovered (again as evidenced in Figure 2.2).
Moreover, if yielding sections were allowed to move back, it would require the control of a
range of sections in the post-yielding branch, which, as referred above, is practically not
achievable. A similar reasoning sustains the need to assume that a yielding section for one
100 Chapter 3
bending direction behaves also as a yielding one for the opposite direction.
3.5.3.2 Fixed sections
The one-side end sections ( and ) are controlled, as explained in 3.5.1, by means of the
curves shown in Figure 3.15, complemented with the cracking plateau transition shown in Fig-
ure 3.18 for the uniform bending moment distributions. After yielding, the hysteretic rules
described in Appendix B apply with two slight modifications referred to in 3.5.1.
The span section H is controlled as the end sections are, both in its left and right sides with the
appropriate model and properties, but the behaviour is limited to uncracked and cracked stages.
However, for uniform bending moment distributions, the explicit control of section H is not
performed because it is taken into account by the behaviour of end section(s).
3.6 Element state determination
3.6.1 General
The general procedure for the element state determination has been outlined and illustrated in
2.4.4, for fixed sections fully controlled by a general model, the basic goal being the calcula-
tion of the total restoring forces (or just the increment ) and the updated stiffness
matrix for a given increment of displacements in the Newton-Raphson iteration n.
According to the notation used in 2.4.4 and the expressions included in Figure 2.10, the whole
process is essentially based on the condensed form expression
(3.13)
where the applied displacements and the residual ones are given by
(3.14)
and the flexibility matrix is obtained according to
E1 E2
Qn ∆Qn
Kn ∆un
∆Qj Fj-1[ ]1–
∆uj ∆urj-1–( )⋅=
∆u…( ) ∆ur…( )
∆uj=1 ∆un= ; ∆ur0 0=
∆uj>1 0 = ; ∆urj-1 bT x( ) ∆er
j-1 x( )⋅ xd0
L
∫=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 101
(3.15)
The final outcome is thus expressed as
(3.16)
where the increment of restoring forces is given by
(3.17)
The major tasks of the element state determination in each iteration j consist on updating the
flexibility distribution and the respective matrix , and on computing residual displace-
ments , in order to check convergence and to set up the next iteration. Both issues are
addressed in the following sections, accounting for the specific features of the element.
3.6.2 Flexibility distribution within the element
The definition of the section flexibility distribution along the element is a key issue of the flex-
ibility formulation, namely aiming at:
• obtaining the element tangent flexibility matrix for the current load step,
• the computation of deformation residuals in the current step and
• the section state determination in the subsequent load step.
If the non-linear behaviour refers only to some of the internal force and deformation compo-
nents, the flexibility distribution needs to be established only for such components since the
other contributions remain constant for the whole loading process. That is exactly the present
case, in which only one bending direction (see 3.1) contributes non-linearly for the flexibility
matrix; the remaining constant contributions are included in Appendix A and just need to be
added to the total flexibility matrix of the element.
The state determination is assumed to be already performed for all control sections, which
means that each moving section location is known, all the uncracked, cracked and yielded
Fj-1 bT x( ) fj-1 x( ) b x( )⋅⋅ xd0
L
∫=
Qn Qn-1 ∆Qn+= ; Kn Fj=converg[ ]1–
=
∆Qn ∆Qj
j 1=
converg
∑=
fj x( ) Fj
∆urj
102 Chapter 3
zones are defined by the adequate internal sections, and each control section flexibility is
updated according to the respective models.
The flexibility distribution is then defined in each zone by linear functions between the bound-
ing section sides lying inside the zone, as shown in Figure 3.19 where two examples of flexi-
bility distributions are included. Figure 3.19-a) refers to the development of yielded, cracked
and uncracked zones associated with and resulting from increasing nodal and element
applied loads. In turn, Figure 3.19-b) shows a typical unloading situation where the three types
of zones are also present, although developed by a previous linear moment distribution ;
for the current distribution none of the moving sections develops further and it is appar-
ent that all the control sections are actually unloading.
Figure 3.19 Examples of flexibility distributions for loading and unloading cases
According to the evolution of the control section state described in 3.5, the possible flexibility
diagrams along the element consist of:
• uniform distributions in uncracked and cracked zones, as a result of the model assumptions;
• uniform distributions in yielded zones, when end sections are loading and yielded zones are
developing (case of Figure 3.19-a));
• linear approximations to the yielded zone flexibility distributions, when end sections are
unloading or reloading and yielding sections are not moving (case of Figure 3.19-b)).
M x( )
M0 x( )
M x( )
Mc1
- Mc2
-
Mc2
+Mc1
+ HC1
C4C3
a) Loading bilinear moment diagram b) Unloading linear moment diagram
C2
E1 E2Y1 Y2
My2
-
My1
-
fy1
- fy2
-fy1
+fy2
+
f01 f02
fp2
-fp1
-
M x( )
f x( )
HC1
C2E1 E2
Y1
Y2
M0 x( )
M x( )
fy1
- fy2
+
f01 f02fu2
+fu1
-
f x( )fp1
-
fp2
+
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 103
The flexibility distributions illustrated in Figure 3.19 cover the possible cases for uncracked
and cracked zones. By contrast, yielded zones may present specific problems demanding spe-
cial care, as is the case of Figure 3.19-b) where, for the sharp transition from a uniform to a lin-
ear (non-uniform) distribution when unloading occurs in a yielded zone (see the dashed and
solid line diagrams of ), recourse is made to an event-to-event scheme along with addi-
tional procedures detailed later in 3.6.6.
3.6.3 Element flexibility matrix
Once the flexibility distribution is defined, Eq. (3.8) could be used to obtain the element flexi-
bility matrix . Since only the y bending component is of interest, attention will just focus in
the corresponding non-zero matrix terms given by
(3.18)
is the bending flexibility distribution (as shown in Figure 3.19) and are the force
shape functions contributing to that bending component. These functions are labelled with sub-
scripts l and m corresponding to the end sections and can be extracted from Eq. (3.6) to be sim-
ply written as
(3.19)
The relation between end section indices (l and m) and the corresponding degrees of freedom
(l´ and m´) in the matrix is also included in Eq. (3.18), where the superscript M is labelling
the flexibility terms to highlight that only the bending deformation contribution is considered.
Actually, also the shear distortion contributes to those terms but, due to its linear behaviour, it
has not to be considered explicitly here.
Since is step-wise linearly distributed, the integrals of Eq. (3.18) are divided by zones
where that function is uniquely defined, and that expression is written as
f x( )
Fes
Fl'm'M Flm
M φl x( )fM x( )φm x( ) xd0
L
∫= = for l' 3 5,=( ) m' 3 5,=( )∧
where
l m, 1…2=( ) andl' 1 2l+=
m' 1 2m+=⎩⎨⎧
fM x( ) φl x( )
φ1 1 x L⁄–= and φ2 x L⁄=
Fes
fM x( )
104 Chapter 3
(3.20)
where the lower and upper integration limits are defined according to the control
section positions for the current load step.
However, in the present formulation, some of the integration limits may vary with the applied
moments. Recalling that element flexibility terms are defined as displacement derivatives with
respect to the moments, and that displacements result from the integration of deformations
between those varying limits, it becomes clear that contributions to the flexibility matrix may
be expected due to the internal moving sections.
In order to find out these additional contributions, one must look back at the definition of flex-
ibility terms in the present formulation. This will be restricted to the tangent flexibility
terms associated with an increment of end section moments after a previously equili-
brated moment distribution . For simplicity, no element applied loads are considered, as
their inclusion is straightforward.
The following derivation is based on an example where both yielding and cracking sections are
developed due to the applied increment of moments as shown in Figure 3.20-a). Since the
incremental form of the curvature distribution is essential for this derivation, both sets of yield-
ing and cracking sections, associated to the previous and the updated moment distributions, are
considered, i.e. and ; the corresponding flexibility distribu-
tions are included in Figure 3.20-b).
Curvature distributions shown in Figure 3.20-c) refer to the total ones, i.e. those necessary to
be provided by each section with the current flexibility in order to satisfy the equilibrium. For
clarity sake, the distributions of are also depicted, although their evaluation is not
actually needed along the whole element.
By definition, the tangent flexibility terms are given by
(3.21)
FlmM φl x( )fζ
M x( )φm x( ) xda1ζ
a2ζ
∫ζ 1=
Nzone
∑=
a1ζ a2ζ,( )
FlmM
∆MEm=1,2
M0 x( )
Y10 C1
0 C20 Y2
0, , ,( ) Y1 C1 C2 Y2, , ,( )
∆ϕζ x( )
FlmM
FlmM
∆MEm∂
∂∆θEl
M
=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 105
where the end section rotations , associated with the applied moments , result from
incremental curvature integration along the element as follows
(3.22)
Denoting , substituting Eq. (3.22) in Eq. (3.21) and recalling that
may vary with the applied moments, one obtains
(3.23)
The first term of this expression can be easily shown to be equivalent to Eq. (3.20) and the
zone division suggested by the distribution in Figure 3.20-b) would be enough for its
evaluation; instead, the second term accounts for the moving section contributions to the ele-
ment flexibility and is further detailed next.
According to Figure 3.20-a), ten integration zones are activated, the limits of which are
grouped as follows
It is apparent that only the abscissas related with may vary due to the current
increment of applied moments; therefore, denoting by the second term of Eq. (3.23), it
reduces to
(3.24)
Zone : 1 2 3 4 5 6 7 8 9 10
∆θEl
M ∆MEm
∆θEl
M φl x( )∆ϕ x( ) xd0
L
∫ φl x( )∆ϕζ x( ) xda1ζ
a2ζ
∫ζ 1=
Nzone
∑= =
glζ x( ) φl x( )∆ϕζ x( )=
a1ζ a2ζ,( )
FlmM
∆MEm∂
∂ glζ x( ) xda1ζ
a2ζ
∫ζ 1=
Nzone
∑ glζ a2ζ( )∆MEm
∂∂a2ζ glζ a1ζ( )
∆MEm∂∂a1ζ–
ζ 1=
Nzone
∑+=
fM x( )
ζ
a1ζ 0 xY1
0 xY1x
C10 xC1
xH xC2x
C20 xY2
xY2
0
a2ζ xY1
0 xY1x
C10 xC1
xH xC2x
C20 xY2
xY2
0 L
Y1 C1 C2 Y2, , ,( )
FlmM
FlmM φl xY1
( ) ∆ϕ2 xY1( ) ∆ϕ3 xY1
( )–[ ]∆MEm
∂
∂xY1 φl xC1( ) ∆ϕ4 xC1
( ) ∆ϕ5 xC1( )–[ ]
∆MEm∂
∂xC1+=
φl xC2( ) ∆ϕ6 xC2
( ) ∆ϕ7 xC2( )–[ ]
∆MEm∂∂xC2 φl xY2
( ) ∆ϕ8 xY2( ) ∆ϕ9 xY2
( )–[ ]∆MEm
∂∂xY2+ +
106 Chapter 3
Note that the terms involving curvature increments are evaluated at both the left and
right sides of the same section; therefore, these increments are assigned the subscript corre-
sponding to the zone containing each side of the section.
Figure 3.20 Derivation of additional flexibility terms due to moving sections
Eq. (3.24) shows that, if no curvature discontinuities exist, as is the case for yielding sections,
the corresponding terms involving terms vanish and it simplifies to
∆ϕζ …( )
ζ
fM x( )
HE1 E2
M0 x( )
M x( )
fy1
- fy2
+
f01 f02
fp2
-fp1
-
C1C10
C20
Y10
Y20C2
Y1
Y2
C10 C2
0Y10 Y2
0
fy1
- fy2
+
f01 f02
fp2
-fp1
-
C1 C2Y1 Y2
fM0
x( )
∆ME1
∆ME2
C1
C2
Y1
Y2
C10
C20
Y10
Y20
ϕ0 x( )
ϕ x( )
ϕy1
-
ϕy2
+
∆ϕCc1
-
∆ϕCc2
+
∆ϕ x( )
xY1
0 xY1x
C10 xC1
0
L
xY2
0xY2x
C20xC2
1 2
7 8 9
3 45 6
10
xH
x
∆ϕζ x( )
ζ =
∆ϕζ x( )
1 2
7 8 9
3 4 5
6 10ζ =
a) Moment
b) Flexibility
c) Curvature
diagrams
diagrams
diagrams
∆ϕζ …( )
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 107
(3.25)
in which the superscripts L and R have been introduced to identify the cracking section side
where the curvature increment is evaluated (following the convention stated in 3.5.3 for the left
and right sides).
From Eq. (3.25) it can be concluded that the additional terms arise due to the development
of cracking sections during the load step under analysis and to the “jump” in curvatures exist-
ing in the cracking section model. The first reason is related to the abscissa derivatives and
means that, if a cracking section is not developing, then no contribution is included in the flex-
ibility matrix. In turn, the second reason is associated with the difference of curvature incre-
ments , for , which reduces to the curvature “jump” referred in 3.5.2.
A similar reasoning can be applied to the case of full development of cracking sections and
as shown in Figure 3.19-a); the corresponding expression for the additional flexibility
terms becomes
(3.26)
where i refers to each element part and the unit factor takes the positive or negative sign for
and , respectively. Actually, Eq. (3.26) is still a particular case of the general
expression for the most complete case when , , and fully develop for para-
bolic moment distributions, which can be written in the following condensed form
(3.27)
where the counter has been introduced in accordance with Figure 3.6.
This allows to write the complete expression for the consistent tangent flexibility terms as
FlmM φl xC1
( ) ∆ϕC1
L ∆ϕC1
R–[ ]∆MEm
∂
∂xC1 φl xC2( ) ∆ϕC2
L ∆ϕC2
R–[ ]∆MEm
∂
∂xC2–=
FlmM
∆ϕCi
L ∆ϕCi
R–[ ] i 1 2,=( )
Ci
Ci+2
FlmM ξi φl xCi
( ) ∆ϕCi
L ∆ϕCi
R–[ ]∆MEm
∂
∂xCi φl xCi 2+( ) ∆ϕCi 2+
L ∆ϕCi 2+
R–[ ]∆MEm
∂
∂xCi 2++⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
2
∑=
ξi
i 1= i 2=
Ci Ci+2 Ci+4 Ci+6
FlmM ξi φl xCi κ+
( ) ∆ϕCi κ+
L ∆ϕCi κ+
R–[ ]∆MEm
∂
∂xCi κ+
κ 0 2 4 6, , ,=∑
⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
2
∑=
κ
FlmM
108 Chapter 3
(3.28)
Closed form expressions can be obtained for the analytical calculation of each zone integrals
and the included derivatives are calculated upon definition of cracking section abscissas as
stated in 3.4.2 and in Appendix C. It must be noted that the additional flexibility terms have to
be obtained according to the applied loads (both end section and element loads).
According to Eq. (3.18), the terms are identified with , leading to the (6x6) matrix
, where the superscript M has been kept to remind the non-linear bending contribution to
the total matrix . Then, the transformation given by Eq. (3.8) is applied to and the
result is accumulated with the remaining contributions included in Appendix A, referring
already to the element local axis system.
3.6.4 Displacement residuals
The procedure for displacement residual computation as described in the general flexibility
formulation (see 2.4.4), typically requires the evaluation of section residual forces in order to
define residual deformations, whose integration along the element leads to the desired dis-
placement residuals.
Such procedure could also be used in the present formulation, but the moving character and the
simplified behaviour of internal control sections suggest an alternative methodology based on
the integration of total deformations rather than residual ones.
Consider again Figure 2.10 for the n-th Newton-Raphson iteration, aiming at target displace-
ments . For any internal iteration j, total deformations are given by
(3.29)
where for (j=1) and for (j>1). Total displacements are
then obtained as
FlmM φl x( )fζ
M x( )φm x( ) xda1ζ
a2ζ
∫ζ 1=
Nzone
∑ +=
ξi φl xCi κ+( ) ∆ϕCi κ+
L ∆ϕCi κ+
R–[ ]∆MEm
∂∂xCi κ+
κ 0 2 4 6, , ,=∑
⎩ ⎭⎨ ⎬⎧ ⎫
i 1=
2
∑
FlmM Fl'm'
M
FesM
Fes FesM
un un-1 ∆un+=
etj x( ) ej-1 x( ) ∆ej x( ) ∆er
j x( )+ +=
ej-1 x( ) en-1 x( )= ej-1 x( ) etj-1 x( )=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 109
(3.30)
and, it is intuitive, that residual displacements can simply be obtained by the difference of
to the target ones . Actually, the following alternative expression
(3.31)
can be proved to be equivalent to
(3.32)
therefore, leading to another way of computing displacement residuals, provided the total
deformations at the section level are known.
Should the global N-R scheme be based on incremental iterative corrections of displacements
relative to the step beginning as suggested in 2.4.4.4, the same procedure is still valid if refer-
ences to and to are replaced by and , respectively. Accordingly, total defor-
mations are also computed by accumulation over (for j=1),
i.e., over a deformed shape corresponding to a duly converged equilibrium configuration.
Since only one bending direction is assumed to behave non-linearly and element applied loads
are only considered acting on the corresponding plane, displacement residuals are likely to
develop only for components affecting that direction. According to the notation of 3.2, the lin-
early independent vectors , and of Eq. (3.9) refer to components behaving linear
elastically and the corresponding residual displacements are null; therefore, the result of defor-
mation integration leads to the input displacements, which means that such integration is use-
less for those terms. The remaining vector includes the non-linear displacement
components, so it has to be obtained by integration of the relevant deformations (see 3.6.7),
namely the shear distortion and the bending curvature in the xz plane. The former contribution
is straightforward due to its linear behaviour, whereas the latter is more demanding because the
curvature distribution has to be adequately set up according to the non-linear behaviour.
utj bT x( ) et
j x( )⋅ xd0
L
∫=
∆urj
utj un
∆urj ut
j un– bT x( ) etj x( )⋅ xd
0
L
∫ un-1 ∆un+( )–= =
∆urj bT x( ) ∆er
j x( )⋅ xd0
L
∫=
un-1 ∆un u0 ∆u0n
etj x( ) ej-1 x( ) e0 x( ) ek-1 x( )= =
uux u
θx uθz
uθy
110 Chapter 3
This alternative methodology for residual computation has been adopted herein for its ade-
quacy in the present formulation context. The following reasons sustain this option:
• The model characteristics associated to internal moving sections are such that total curva-
tures can be readily known for given applied moments; moreover, this is valid for all
uncracked and cracked zones, which usually form a major part of the element.
• If residual curvature distribution were to be obtained, then yielding and cracking
sections of both the current and the previous iterations would have to be controlled, as
shown in Figure 3.20; note that in this figure reference is actually made to the total curva-
ture increment , which includes the predictor and the residual/corrector incre-
ments, and the whole set of zones would be required to perform integration.
• By contrast, if only total curvature distribution is of interest, there is no need for controlling
the previous iteration moving sections, due to the uniform, and readily known, flexibility
distribution along uncracked and cracked zones. This leads to an important economy from
the computation time standpoint, since significantly less internal moving sections are actu-
ally controlled.
Therefore, it is clear that for most of the element zones, residual computation via total curva-
tures is readily achieved. However, to cope with this strategy, special care has to be taken with
the yielded zones, since the total curvature distribution definition is not straightforward when
unloading or reloading situations occur in such zones. This aspect is specifically detailed in
3.6.6 where a special procedure is proposed to improve consistency between distributions of
applied moments and total curvatures, for the assumed flexibility distribution.
Additionally, the global N-R scheme adopted in this work is actually based on incremental iter-
ative corrections relative to the step beginning, which is compatible with the fact that the
cracking and yielding section development is irreversible between equilibrated steps, though
not necessarily between iterations. In other words, the section movement is irreversible with
respect to their position at the step beginning, but, between consecutive internal iterations j-1
and j of a given external N-R iteration n, their positions can change regardless of keeping or
reversing the direction of movement. Thus, once cracking and yielding sections are found for a
given internal iteration j, their irreversible movement conditions must be checked with respect
to the step beginning and, according to Figures 3.6 and 3.9, they can be expressed in terms of
local abscissas by the following relations:
∆ϕr x( )( )
∆ϕ x( )( )
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 111
where the superscript 0 stands for the step beginning and the cases of sections and
have been included for completeness, although they are not considered in this study.
3.6.5 Element applied loads
In the present work, element applied loads have been considered according to the Spacone pro-
posal (Spacone (1994)) as described by Eqs. (2.28) to (2.34) in 2.4.4.3.
Since these loads are assumed fully applied at the first load step, thus for elastic behaviour, that
proposal becomes quite suitable and efficient from the computational standpoint. In fact, the
element fixed-end forces due to element applied loads, either distributed p or concentrated
P at section H, can be readily obtained by Eq. (2.32) with the elastic flexibility distribution
, as defined in Appendix A by Eq. (A.4), and with the corresponding stiffness matrix .
Particularly, the integral of Eq. (2.32) gives the elastic displacements due to total element
applied loads in the reduced space, which can be calculated by the closed form expressions
included in Appendix A (see A.3), accounting for the type of load and the distribution of sec-
tion properties along the element.
Thereby, Eqs. (2.32) and (2.33), for the total element loads (p instead of ), can be re-written
respectively as
(3.33)
and
(3.34)
which hold only for the first internal iteration (j=1) of every N-R iteration n, according to the
adopted global N-R scheme referring to the step beginning.
For subsequent internal iterations (j>1) the contribution of is removed from Eq. (3.34), but
the distribution is kept explicitly included in order to correctly evaluate forces in inter-
sYi
j sYi
0≥ sCi
j sCi
0≥ sCi 2+
j sCi 2+
0≤ sCi 4+
j sCi 4+
0≥ sCi 6+
j sCi 6+
0≤
Ci 4+ Ci 6+
Qp
f0 x( ) K0
up
∆p
Qp K0– up⋅=
∆S x( )j=1 b x( ) K0 ∆un up–( )⋅[ ]⋅ Sp x( )+=
up
Sp x( )
112 Chapter 3
nal sections. Expressions of (and of in the local axis system) are included in
Appendix C for both the distributed load p and the concentrated force P at the span section H.
It is reminded that, once the restoring element forces are obtained in the reduced space, the
transformation to the element space with rigid body modes is performed by Eq. (2.11), where
the contribution of element applied loads must be included in every N-R iteration n of the
first step; again, expressions of are given in Appendix C for both types of element loads.
3.6.6 Behaviour of plastic end zones
3.6.6.1 General
The behaviour of yielded zones (herein also designated as plastic zones) needs to be carefully
analysed, since it is effectively controlled only at their boundary sections, i.e., end and yielding
sections, and nothing is known concerning the intermediate sections for most loading stages.
For the assessment of curvature distribution in plastic zones, the following loading case dis-
tinction is important:
i) post-yielding loading responsible for monotonic development of the plastic zone
ii) unloading, reloading or post-yielding loading for which the plastic zone has not been
continuously developing.
The loading case i) is easily handled since all the intermediate sections have their state points
lying on the model post-yielding branch. This is illustrated in Figure 3.21 for the simple case
of a cantilever with a uniformly distributed load and a vertical force applied at the tip, in which
the yielding moment has been exceeded. Both moment, flexibility and total curvature distribu-
tions are included, as well as the basic model curve (assumed uniform along the element).
For a linear moment diagram, the total curvature distribution in the plastic zone (labelled by 1
in Figure 3.21) is readily known once the state determination is performed for its boundary
sections ( and ). In turn, for the case of the parabolic moment diagram (see Figure 3.21),
the coefficient of the 2nd order term of the curvature distribution is easily shown to
depend only on the distributed load and on the post-yielding stiffness . This coefficient
remains constant as long as the distributed load does not change, while the other coefficients
( and ) can be obtained through the total curvature values at the end and yielding sec-
Sp x( ) Spsx( )
Qn
Qpf
Qpf
E1 Y1
aϕ1( )
p fp
bϕ1cϕ1
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 113
tions (respectively, and ) in order to completely define the distribution .
Figure 3.21 includes also the curvature distributions and for the remaining zones
(cracked and uncracked ones, respectively) as well as their second order polynomial coeffi-
cients; once these coefficients are known, along with curvature values at each zone boundary
sections, the curvature distribution can be completely defined.
Figure 3.21 Monotonic development of plastic zone
It is noticed that, even for subsequent loading stages as those of ii), plastic zones have to pass
through the above shown curvature diagram because yielding development is only assumed to
start from the end sections. Therefore, for any loading process involving distributed load at the
beginning, the very first coefficient is always as indicated in Figure 3.21.
The loading cases of ii) are such that total curvature distribution in plastic zones cannot be
ϕE ϕy ϕ1 x( )
ϕ2 x( ) ϕ3 x( )
Mc
C1
E1 E2Y1
My
fy f0
fp
M x( ) aMx2 bMx cM+ +=
f x( )
MEM x( )
ϕ
M
oϕcc
ϕc
fpE
ϕy
ϕccϕc
ϕy
ϕ1 x( )
ϕ1 x( ) ϕy fp M x( ) My–[ ]+=
C1Y1
ϕ2 x( ) fyM x( )=
ϕ3 x( ) f0M x( )=
ϕ1 x( )
ϕE
aϕ1fpaM=
ϕζ x( ) aϕζx2 bϕζ
x cϕζ+ +=
aϕ2fyaM=
aϕ3f0aM=
p
F
ζ=1 2 3
ζ 1 2 3, ,=( )
a) Moment
b) Flexibility
d) Curvature
diagramc) Moment-Curvature
aMp2---=
aϕ1
114 Chapter 3
explicitly derived from the model. The following general expression may be considered
(3.35)
which can be obtained from the expressions included in Figure 2.10, after particularizing for
curvatures. The predictor term, based on the previous flexibility and the increment of
applied moments , is added with the corrective term due to unbalanced moments
along with the updated flexibility . As previously stated, flexibility is assumed
linearly distributed, but information is still lacking concerning , since the state points
in the model (i.e., resisting moments ) for intermediate sections are unknown and, gen-
erally, cannot be inferred from those of end and yielding sections.
Actually, for certain simple loading cases (as the first unloading from post-yielding branch, for
instance), a relation could be derived between the assumed and the consistent ,
within the model rules; however, once loading becomes more complex, such relation turns out
to be impossible to obtain. Another possibility could be to assume a distribution law for
, which, however, should be consistent with the adopted for all loading stages.
Since this is practically impossible to be achieved, such option could not ensure solution objec-
tivity and algorithm convergence.
In the present study this problem is overcome by the use of an event-to-event type procedure
inside the iterative process for the element state determination. The event-to-event technique
(Simons and Powell (1982), Porter and Powell (1971)) is suitable for non-linear problems
behaving linearly after and before specific events. By event it is meant a stiffness variation
which can be predicted according to the model rules and the current state. The basic idea is to
closely follow the model path at all loading steps, by adapting them to the event sequence and
updating the stiffness and state accordingly, in order to eliminate unbalanced forces. This tech-
nique is further detailed in Appendix D with the help of a very simple illustrative example; its
application to the present element state determination is explained later in this section.
In the present context, the event-to-event scheme is used only for the control of sections gov-
erning the plastic zone behaviour. The main purpose is to assure that, a sub-increment being
applied at each iteration, the flexibility distribution inside plastic zones either remains
unchanged or modifies to but with negligible unbalanced moments ; in both
ϕj x( ) ϕj-1 x( ) fj-1 x( )∆Mj x( ) fj x( )∆Muj x( )+ +=
fj-1 x( )
∆Mj x( )
∆Muj x( ) fj x( )
∆Muj x( )
Mrj x( )
fj x( ) Mrj x( )
Mrj x( ) fj x( )
fj-1 x( )
fj x( ) ∆Muj x( )
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 115
cases, the following approximation holds
(3.36)
In fact, behind this approach there lies the assumption that, if flexibility remains unchanged at
the plastic zone boundary sections, the same happens for all the yielded sections and, therefore,
residuals become null. Obviously, this is enforced by the adopted linear flexibility approach
and, for complex loading situations, the realism of such option may be questionable from a
local section standpoint. Nevertheless, it should be recalled that, rather than a local section
detailed behaviour, a global element response is to be assessed; in such context, this approach
is believed to be acceptable, although it is recognized that improvements may be needed in
future developments.
3.6.6.2 Plastic zone splitting
It is important to realize that, for the flexibility distribution to remain unchanged in the
plastic zones, it is necessary that neither the boundary section flexibilities nor the zone length
change during a given sub-increment. The first requirement is accounted for with the event-to-
event scheme, whereas the second one is achieved by dividing each plastic zone into two parts
as shown in Figure 3.22. One fixed part between the end section and the yielding one
of the previous step (duly equilibrated and converged) and a variable length part,
between and the yielding section of the present iteration.
Figure 3.22 Plastic zone splitting in fixed and variable length parts
With this plastic zone division it follows that:
• variable length parts are straightforwardly handled as explained above for the monotonic
development of plastic zones (see Figure 3.21);
• the event-to-event procedure is applied to control the fixed parts, specifically at the end sec-
ϕj x( ) ϕj-1 x( ) fj-1 x( )∆Mj x( )+≈
f x( )
Ei=1,2( )
Yi=1,20( )
Yi=1,20 Yi=1,2( )
HE1 E2
M0 x( )
M x( )
Y10
Y20
Y1
Y2
∆ME1
∆ME2
Fixed Variable Length
116 Chapter 3
tions and the left sides of .
Depending on the state of boundary sections, flexibility distributions in fixed plastic zones can
develop in several possible ways as shown in Figures 3.23 and 3.24, for a given load step.
Dashed line diagrams represent the flexibility corresponding to the previous equilibrated
step, whereas solid line diagrams refer to the updated flexibility ; for simplicity it is
assumed that at most one event can occur at those boundary sections during the load step. Pos-
sible positions of those section state points in the moment-curvature diagram are also included
and small arrows indicate the post-event evolution.
Figure 3.23 Flexibility distributions in plastic zones with no further yielding development
Figure 3.23 shows some cases with no yielding development, which are briefly described next:
Yi=1,20
f0 x( )
f x( )
Y0 Y≡E
M
o
E
fy+
fp+
fu+
ϕ
YL0
Events in: E & Y0
Y0 Y≡E
o E
fy+
fr1
-
fu+
Y0
Events in: E & Y0
fy-
fr1
-fy-
Y0 Y≡E
E
fr1
-
fp-
Y0Event in: E
fy-
fp-
fy-
Y0 Y≡E
E
fr1
-
Y0
Event in: E
fy-
fy-
fr2
-
fr2
-
Y0 Y≡E
E
fy+
fp+
YL0
Event in: Y0
Y0 Y≡E
E
fy+
fp+
YL0
Events in: E & Y0
fu/r+
fu/r+
fy+ fu
+
fp+
f x( )f0 x( )
b)a)
d)c)
f)e)
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 117
• Figure 3.23-a) refers to unloading of both E and sections from the post-yielding branch,
which means events taking place at both sections. Therefore, a very small (“infinitesimal”)
sub-increment is first applied with the flexibility, just to enforce the changing to ;
the remaining displacement sub-increment is then applied with the already updated flexibil-
ity distribution. No residuals are expected as long as no more events take place, but if this
should happen, as a result of flexibility modifications, a new subdivision would be
enforced.
• Figure 3.23-b) shows the case of events taking place at both boundary sections due to
changing of moment sign. The first sub-increment is applied with until zero moments
are reached, after which flexibility can be switched to , in order to apply the remaining
sub-increment.
• Figures 3.23-c) and 3.23-d) represent reloading cases where the end section is likely to
reach again the post-yielding branch or to change stiffness due to pinching effects, whereas
remains in the same branch.
• Figures 3.23-e) and 3.23-f) refer to situations of opposite loading direction in boundary sec-
tions; while unloads, the end section either remains on the post-yielding branch (case e))
or reaches it from a reloading/unloading situation (case f)).
Figure 3.24 Flexibility distributions in plastic zones with further yielding development
Y0
f0 x( ) f x( )
f0 x( )
f x( )
Y0
Y0
Y0E
fy+
fp+
YY0E
M
o
E
fy+
fp+
ϕ
YL0
No Events
EY0Event in: Y0
EYL0
Event in: E
Y0E
E
fy+
fp+
YL0
Event in: E
fu/r+
fu/r+
Y
fu+
fu+
Y0E
fy+
fp+
Y
fu/r+
Y
fp+
fy+
b)a)
d)c)
118 Chapter 3
Some yielding development situations are illustrated in Figure 3.24, namely:
• The case a), which is the same of Figure 3.21, but it is irrelevant for events detection.
• Unloading at the end section from the post-yielding branch, while still loads along it, as
in the case b); therefore, an event at section E enforces the sub-incrementation process with
a procedure similar to the case of Figure 3.23-a).
• The situation c), where the end section remains in a reloading/unloading phase, while
changes to the post-yielding behaviour; until this event occurs in , a first sub-increment is
applied, after which the flexibility is updated.
• The case d), in which an end section event occurs when it reaches again the post-yielding
branch after a reloading/unloading situation; the left side of keeps on loading along the
same path.
3.6.6.3 Event-to-event scheme in the element iterative process
As previously mentioned, the event-to-event scheme is used in the present work, to control the
boundary sections of fixed plastic zones ( and ), because the assumed step-wise lin-
ear behaviour is suitable to be controlled by that technique. A similar scheme has been used by
Filippou and Issa (1988) but with the purpose of controlling the plastic zone development.
It is worth stressing the contrast between the event-to-event scheme used at the global structure
level (as described in Appendix D) and its application at the element level. In the first case, any
event enforces a subdivision of the global load increment and the whole incremental/iterative
process may easily become inefficient when strongly non-linear behaviour is encountered. In
turn, the element event-to-event scheme includes a very limited number of sections generating
events (four, at most) and it is activated only inside the internal element iterative process, thus
without directly affecting the global structure N-R process.
In other words, the advantage of using the event-to-event scheme herein is twofold: a) it pro-
vides a suitable means of achieving the state determination of the fixed plastic zones (within
the assumptions stated in 3.6.6.1) and b), being used at the element level, it is not likely to
affect significantly the global N-R scheme.
The sequence for the event-to-event application is schematically shown in Figure 3.25. This
scheme becomes more efficient if there is at most one event per section; therefore, prior to the
sub-incrementation for reaching an event, the auxiliary increment is made equal to the
Y0
Y0
Y0
Y0
Ei=1,2 Yi=1,20
∆ua( )
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 119
total increment and subdivided into two parts ( and ) such that no more than one
event is found at each section upon application of (see step 1.1/ of Figure 3.25).
For the application of , events are searched among the boundary sections of fixed plastic
zones. Following a procedure similar to that explained in Appendix D, increment reduction
factors are set up for those sections, and the event (V), if existing, is associated to the minimum
factor. Thus, in case of event, is first considered as the working increment and is
split into two parts:
Figure 3.25 Application of the event-to-event scheme to the element state determination
• the first sub-increment is applied with the previous flexibility, up to the turning
point, but slightly exceeding it; this enforces the flexibility variation with numerically neg-
ligible unbalanced moments at the boundary sections of plastic zones, but it does not ensure
∆u( ) ∆ub ∆ur
∆ub
∆ub
∆ub ∆uw( )
V
VTarget
Preliminary Subdivision
Events Detection in all sections
Sub-incrementation
State determination for
Set up:
Set:
Residuals can appear
∆u Total Displacement Increment=
∆ub∆ur
u0 ufub
u0 ∆u1
∆u2δu1∆u1
δ– u1
∆uw ∆u2 δu1–=
δu1( )
∆uw ∆ub=
Check Convergence for
(Max. 1 event per section)
∆ua ∆u=
∆ua
∆uw
∆u2
1/
1.1/
2/
2.1/
2.3/
2.2/
V = event closest to the beginning
∆u1
Update flexibility distribution
∆uw ∆u2 δu1–=
∆uw - NON Converged => Restart 2.1/
- Converged -> ∆ur 0=( ) STOP⇒
∆ur 0≠( ) ∆ua ∆ur=( )⇒Restart 1.1/
If
If
u1
∆u1( )
120 Chapter 3
the increment of deformations along the element to be compatible with ; indeed, modi-
fications on the stiffness and positions of the remaining internal sections (yielding and
cracking ones) may lead to element displacements different from , the difference being
designated by residual displacements ;
• the second sub-increment is applied with the updated flexibility, but corrected with
residuals resulting from the first sub-increment.
The application of is followed by the setting up of moving sections and the state determi-
nation at every section; the flexibility distribution is updated and displacement residuals
are computed. In order to respect the target displacements (see steps 2.1/ and 2.2/ of Figure
3.25), the residual correction is then applied together with , if still existing.
Thus, a new working increment is given by , where only residuals may per-
sist in case the sub-incrementation has not been done. This second and the following phases for
application of must be performed again from steps 2.1/ in order to enforce sub-incremen-
tation for further events likely to develop; the process stops when convergence is reached as
stated in 3.6.8.
According to 3.6.4, displacement residuals are calculated via integration of total curvatures;
thereby, after application of , total displacements are obtained and
the new working increment can be readily calculated from the alternative expression
, where actually stands for the updated displacements at the end of
each iteration.
After completion of , both element and section states are duly updated and, if there is still
any remaining increment part to be applied, the process restarts from step 1.1/.
It is worth mentioning that, increment subdivisions are performed directly on the applied dis-
placements, instead of the corresponding moments because these are dependent on the flexibil-
ity matrix used in the state determination.
3.6.6.4 Evolution of curvatures in fixed plastic zones
Application of Eq. (3.36), for total curvature diagram definition, is actually performed only in
the fixed plastic zones. It is apparent that accumulation of deformations is taking place from
∆u1
∆u1
δu1
∆u2( )
δu1( )
∆u1
δu1( )
δ– u1( ) ∆u2
∆uw ∆u2 δu1–=
∆uw
∆u1 u1 u0 ∆u1 δu1+ +=( )
∆uw ∆ub u1 u0–( )–= u1
∆ub
∆ur
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 121
one iteration to another, which must be reflected in the coefficients of the curvature equation.
If element applied loads are assumed to take place only at the first load step, as referred to in
3.1, then further step loading or iterative corrections consist only of end section applied loads,
leading to linear functions for . With the linear flexibility distribution approach, Eq.
(3.36) implies curvature increments to have (at most) a 2nd order term, whose coefficient is
readily obtained from the expressions of and . This is shown in Figure 3.26 for the
same example of Figure 3.21, where unloading is now taking place due to a tip force applied in
the reverse direction.
Figure 3.26 Total curvature evolution for non-monotonic loading
According to Figure 3.26, the 2nd order coefficient of the parabolic increment of curvatures
is added to the value corresponding to the previous iteration (or step), in order to
obtain the coefficient for the updated total curvature expression; the remaining coefficients
∆M x( )
f x( ) ∆M x( )
E1 E2
fy f0
∆M x( ) b∆x c∆+=
f x( )
ME ME
ϕ
M
oϕcc
ϕc
fp
E
ϕy
ϕccϕc
ϕE
ϕE ϕE0 fu ME ME
0–( )+=
ϕ2 x( ) fyM x( )=
ϕ3 x( ) f0M x( )=
ϕ1 x( ) ϕ10 x( ) f1 x( )∆M x( )+=
ϕE
p
aϕ1aϕ1
0 bfb∆+=
ϕζ x( ) aϕζx2 bϕζ
x cϕζ+ +=
aϕ2fyaM=
aϕ3f0aM=
F
M0 x( )
f1 x( ) bfx cf+=
M x( )
ϕy
Y10 C1
0
fu
Y10 C1
0
ME0
ϕE0
fuME
0
ϕE0
ζ=1 2 3
ζ 1 2 3, ,=( )
a) Moment
b) Flexibility
d) Curvature
diagramc) Moment-Curvature
aMp2---=
bfb∆( ) aϕ1
0
aϕ1
122 Chapter 3
can be derived from the curvature values at sections and .
Although the case of element applied loads in other steps (after the first one) is not considered
in the present study, it is clear from Figure 3.26 that it would lead to a 3rd order polynomial for
the curvature diagram. In such a case, the 3rd and 2nd order coefficients for the curvature incre-
ment due to an element load increment , would be computed only in terms of coeffi-
cients and of and then accumulated with those from previous steps. For the remaining
coefficients, the procedure referred to in the previous paragraph applies.
It is apparent that for no inclusion of distributed load p, the second order coefficients are
restricted to the plastic zone, having only the contribution of generated once unloading
has taken place.
3.6.7 Integration of deformations
Once the curvature distribution is adequately updated, the integration is performed over the
distributions of deformations relevant for the non-linear displacements included in . How-
ever, since both moment and curvature distributions have been referred to the local section axis
system, the vector is actually obtained first.
Looking back at Eqs. (3.9) and (3.10) and following a procedure similar to that of 3.6.3, the
two non-zero components of (i.e., the 3rd and the 5th ones ) can be given by
(3.37)
where the superscripts V and M stand for the shear and the bending contributions, respectively.
According to Eq. (3.22), the terms are given by
(3.38)
for which, the integrals of each zone can be analytically evaluated by closed form expressions.
Similarly, the shear related terms are defined by
E1 Y10
∆p( ) f x( )
∆p
aϕζ
bfb∆
uθy
uesθy
uesθy
θEl'θEl
θEl
V θEl
M+= = l 1 2,= l' 1 2l+=→( )
θEl
M
θEl
M φl x( )ϕ x( ) xd0
L
∫ φl x( )ϕζ x( ) xda1ζ
a2ζ
∫ζ 1=
Nzone
∑= =
θEl
V
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 123
(3.39)
where is the shear distortion distribution and are the shear force shape functions,
which can be drawn from Eq. (3.6) and are given by
(3.40)
Since the shear behaviour is linear, holds, where remains constant
during the loading process and is assumed as given by Eqs. (A.5) and (A.6) in Appendix A.
The distribution may include element applied loads and is expressed by Eqs. (C.7) and
(C.8) in Appendix C. Therefore, Eq. (3.39) becomes
(3.41)
where the (-) sign holds for and the (+) for , and .
According to the notation of Appendix A, the above integrals yield
(3.42)
which, for the case of uniform properties along the element, simplifies to .
After computation of as above explained, Eq. (3.3) is applied to transform it to ,
referred to the element axis system x,y,z. Then, the result is superimposed to the remaining
components of Eqs. (3.9) and (3.10) to obtain the complete displacement vector . However,
note that such operation is worthless from the algorithm standpoint, since residuals and, conse-
quently, convergence have to be checked only for the non-linear components, i.e., directly for
. On the other hand, for the element output forces, the superposition of the non-linear com-
ponents with the linear ones is relevant and has to be performed.
θEl
V φlV x( )β x( ) xd
0
L
∫=
β x( ) φlV x( )
φ1V 1 L⁄–= and φ2
V 1 L⁄=
β x( ) fV x( )V x( )= fV x( )
V x( )
θEl
V 1L---± 1
GAz( )1----------------- VE p L
2--- x–⎝ ⎠⎛ ⎞– P 1 h–( )– xd
0
hL∫
1GAz( )2
----------------- VE p L2--- x–⎝ ⎠⎛ ⎞– Ph– xd
hL
hL
∫
+⎩
⎭
⎨
⎬
⎧
⎫
=
l 1= l 2= VE ME2ME1
–( ) L⁄=
θEl
V hGAz( )1
----------------- 1 h–( )GAz( )2
-----------------+ VE1
GAz( )1----------------- 1
GAz( )2-----------------– h2 h–( ) pL
2------ P+⎝ ⎠⎛ ⎞+
⎩ ⎭⎨ ⎬⎧ ⎫
±=
θEl
V 1GAz----------VE±=
uesθy u
θy
u
uθy
124 Chapter 3
3.6.8 Convergence criteria for the element iterative process
The internal element iterative scheme aims at calculating restoring forces for a given increment
of applied displacements, by means of progressive elimination of displacement residuals. Con-
vergence is reached when these residuals are smaller than a pre-defined measure of numeri-
cally negligible displacements. Convergence is checked only in terms of rotations in the non-
linear bending plane; the corresponding tolerance for rotation residuals is denoted here by
which is controlled by a user-supplied relative tolerance (e.g. 10-4, 10-6, ...).
Therefore, convergence is considered to be reached when the following condition is satisfied
(3.43)
where stand for residuals of element end section rotations.
The value of has to be chosen according to several distinct situations likely to develop in
the element. Thereby, in the present work it is given by the following condition
(3.44)
where:
• is the maximum absolute value of rotation increments applied to the element in the
current step;
• is the maximum elastic rotation (absolute value) corresponding to the element end sec-
tion moments at the step beginning;
• is the maximum elastic rotation (absolute value) associated with anti-symmetric bending
produced by end section moments equal to the average of absolute values of cracking
moments in the element and
• is a value close to (but slightly higher than) the numerical precision of the computer.
Comparison of against is the criterion often used; however, for very small (or even
null increments) this condition is not applicable, for which the displacement measure
related with the installed stress state may be more appropriate. In the case of initial step, where
only element loads p (or P at section H) are considered, the structure is unstressed (thus,
) and no displacement increments are considered (i.e., ); therefore, the con-
θtoler ∅
θrmax max ∆θr1
∆θr2( , ) θtoler<=
∆θri=1,2
θtoler
θtoler max ∅∆θa( ) ∅θe0( ) ∅θc( ) Precis, ,{ , }=
∆θa
θe0
θc
Mc
Mc Mc1
+ Mc1
- Mc2
+ Mc2
-+ + +( ) 4⁄=( )
Precis
θrmax ∅∆θa
θe0
θe0 0= ∆θa 0=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 125
vergence check against and fails and another criterion related to the internal
resistance of the element is adopted. The rotation has been chosen as a kinematic measure
of the cracking limit and the verification is done for . Finally, none of the previous rota-
tion tolerances can be lower than the computer precision and the is imposed as the
lower bound of tolerance.
3.6.9 Convergence problems
3.6.9.1 Difficult or no- convergence situations
The particular features of the model to overcome the cracking transition may lead to difficult
convergence situations that are likely to develop depending on the moment distribution along
the element.
Simple cases, as for instance a cantilever beam loaded by a tip force or a beam under anti-sym-
metric bending, do not present problems related with cracking transition, provided the addi-
tional flexibility terms are duly incorporated in the flexibility matrix as detailed in 3.6.3.
Figure 3.27 shows such the simple case of a cantilever beam subjected to linear bending
moment distribution , behaving as uncracked for the step 0. By imposing a displacement
increment with the previous stiffness (the elastic one), the moment distribution for the first
internal iteration (j=1) becomes (see Figure 3.27-a)), inducing cracking initiation.
The corresponding flexibility and total curvature distributions (see Figures 3.27-c) and 3.27-
d)) are dictated by the section model (shown in Figure 3.27-b)) and are denoted, respectively,
by and . Total displacements can be obtained upon integration of
and the corresponding displacement residuals can be evaluated. The updated flexibility
matrix arises from integration of , corrected with the additional flexibility terms
due to cracking section movement.
The second internal iteration (j=2) is then performed by imposing the correction of moments
leading to and obtained from and the previous stiffness matrix , which
accounts for the section stiffness variation ( to ) and for the curvature “jump” in the
cracked zone developed. The presence of the additional flexibility terms ensures that the ele-
ment stiffness drops sufficiently to keep the end section state point above the crack-
ing plateau (see Figure 3.27-b)); thereby, cracking is still detected in the element and the
∅∆θa ∅θe0
θc
∅θc
Precis
M0 x( )
∆u
Mj=1 x( )
fj=1 x( ) ϕj=1 x( ) uj=1 ϕj=1 x( )
∆urj=1
Fj=1 fj=1 x( )
FlmM
Mj=2 x( ) ∆urj=1– Kj=1
f0 fy
Kj=1 Ej=2
126 Chapter 3
successive iterations tend to adjust the cracked zone until the total displacements obtained
from integration match the desired ones (i.e., ) within a pre-defined toler-
ance.
Figure 3.27 Cracking transition and the role of additional flexibility terms
However, if these additional terms were not included, the flexibility matrix would contain only
the influence of section flexibility variation from to , which could also occur if the M-ϕ
diagram followed the dashed line (parallel to the cracked branch) after point C. In these condi-
tions, the element flexibility matrix would not “recognize” the curvature “jump” along the
cracked zone, which means an inconsistency between that matrix and the developed curva-
tures. Consequently, a higher element stiffness would arise and, depending of the magnitude of
the curvature “jump” , the enforced correction to could lead to a state point
below the cracking plateau (see Figure 3.27-b)). The elastic behaviour would be
reached again, the element stiffness would turn back to the initial state and the next iteration
(j=3) would develop as the first one. A closed loop would take place, switching between two
opposite states (below and above the cracking plateau) without the possibility of reaching con-
vergence.
uconverg u0 ∆u+≅
Mc
C1
E1 E2
fy f0
Mj=1 x( )
fj=1 x( )
ME
ϕ
M
oϕcc
ϕc
C
ϕccϕc
ϕE
C1
ϕj=1 x( )
∆urj=1 ∆uj=1 ∆u– uj=1 u0 ∆u+( )–= =
a) Moment
c) Flexibility
d) Curvature
b) Moment-Curvature diagram
M0 x( )
Mj=2 x( )Ej=1
Ej=2
Ej=2( )'
fy
f0
∆MEj=2 ∆ME
j=2( )'
C1
ϕ0 x( )
∆uj=1
∆u ∆MEj=1⇒
f0 fy
∆ϕccMj=2 x( )
Ej=2( )'
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 127
Therefore, the inclusion of additional flexibility terms ensures the consistency between the ele-
ment flexibility and curvatures and avoids the above referred closed loop problem.
On the other hand, it must be realized that such terms become less relevant for the element
flexibility as the cracking sections move farther away from the end ones. Taking the simple
example under study, and considering the stage where only the section is developing, the
additional term expression (see Eq. (3.25)) reduces to
(3.45)
where the term coincides with the cracking curvature “jump” .
Particularly for the terms , which are the most affected by the cracking development next
to the end section , Eq. (3.45) can be written as
(3.46)
showing that increasing values of ( moving apart from ) tend to reduce . In turn,
the lower or higher variation of with the applied moments is expressed by the derivatives
and depends mainly of the moment diagram slope in that zone.
This means that, for an “almost flat” moment distribution in element zones crossing the crack-
ing plateau, sudden and significant development of a cracked zone may occur, whereby the
additional terms can be of less relative weight in the element flexibility matrix. Therefore,
a situation of closed loop as above referred can be triggered off by lack of flexibility for the
applied moment increment (or iterative correction).
Two typical situations of this type are shown in Figure 3.28. In the first one (see Figure 3.28-
a)) an initially uncracked element is forced to develop a large cracked zone due to small end
section increments leading to a very flat diagram; such situation requires a pronounced stiff-
ness decrease in order to compensate for the large residuals developed in the cracked zone.
Another case is shown in Figure 3.28-b) where an existing slightly cracked element part may
C1
FlmM φl xC1
( ) ∆ϕC1
L ∆ϕC1
R–[ ]∆MEm
∂∂xC1=
∆ϕC1
L ∆ϕC1
R–[ ] ∆ϕcc
F1mM
E1
F1mM 1
xC1
L-------–⎝ ⎠
⎛ ⎞∆MEm
∂
∂xC1 ∆ϕcc=
xC1C1 E1 F1m
M
xC1
∂xC1∂∆MEm⁄
F1mM
128 Chapter 3
become fully-cracked due to small variations of end section moments generating an almost flat
diagram in that element part ( and are supposed of having been generated by previous
moment distributions).
Note that such problems are not caused by inconsistencies in the formulation and a very strin-
gent subdivision of the applied increment would lead to a converged solution, but at a quite
high computational cost.
Similar convergence difficulties are reported in the literature (Criesfield (1982)) often related
with reinforced concrete cracking problems, which are solved by adopting line search tech-
niques. This has been the strategy adopted also in the present work to overcome the cracking
plateau transition. As the event-to-event scheme referred in 3.6.6, a line search procedure can
be very efficient in the present context because it is restricted to the element level rather than
involving the state determination for the entire structure.
Figure 3.28 Typical cases generating convergence problems
3.6.9.2 Line search scheme for element iterations
The main and basic features of line search schemes are briefly recalled in the next paragraphs,
devoting particular attention to the adaptation for the present element iterative algorithm and to
the criteria for the selection of the increment scaling factor.
Line search schemes are based on the recognition that the increment, or iterative correction, of
C1 C3
Mc
C1
E1 E2H
M x( )
a)
C1
M0 x( )
H
H
E1 E2
Mc
E1 E2H
M x( )
b)
M0 x( )
H
H
E1 E2C1 C3
for M0 x( )
for M x( )
for M0 x( )
for M x( )
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 129
the driving variable vector (displacements or forces), obtained by solving the equilibrium
equations, does not necessarily lead to the best configuration of equilibrium, i.e., the one with
the least residual output vector (forces or displacements). Instead, a multiple of the
driving increment may give a better estimate of the equilibrium state, where the scaling factor,
or step length parameter is chosen so as to minimize some measure of residuals. Typically,
is optimum (Criesfield (1982)) when it makes the component of in the direction of
be zero.
In displacement based non-linear algorithms, stands for displacements , successively
corrected to satisfy equilibrium with the imposed force increment , and the residuals are
the out-of-balance forces . On the contrary, for force based schemes, such as the internal
element iterative algorithm considered in this study, consists of the forces , iteratively
adjusted in order to ensure compatibility with the desired displacements ; in this case
refer to the residual displacements violating kinematic compatibility.
Particularizing for the present force based scheme, for a given element internal iteration j, the
following expression holds
(3.47)
where the scalar is estimated by imposing
(3.48)
in which represents the work done by the increment of forces in the residual displace-
ments. Eq. (3.48) can also be interpreted as imposing the stationarity of the total potential
energy in the direction of .
Since and are known for iteration j, the scalar Eq. (3.48) requires only a one-dimen-
sional search of for which a trial-and-error procedure is often adopted. Each trial value of
(say ) involves the corresponding residual calculation , often called
“extra residual calculation” or simply “line search”. Different proposals of line search schemes
are available depending on more or less sophisticated algorithms for estimation of (Cries-
field (1982), Matheis and Strang (1979), Marques (1986)).
∆a
∆r β∆a
β( )
β ∆r ∆a
∆a ∆u
∆Q
∆Qr
∆a ∆Q
∆u ∆r
∆ur
Qj Qj-1 β∆Qj+=
β
ψ ∆Qj( )T
∆ur Qj-1 β∆Qj+( )⋅ 0= =
ψ ∆Qj
∆Qj
Qj-1 ∆Qj
β β
βh ∆urh Qj-1 βh∆Qj+( )
β
130 Chapter 3
In the present study the line search scheme adopted is proposed by Criesfield (1982), accord-
ing to which upper and lower values of are first sought, bounding the root of Eq. (3.48). The
solution is then estimated by successive linear interpolations between the lower and upper
bounds.
For , residuals are readily known as they correspond to the previ-
ous iteration (having given rise to , for ); additionally, the residual evaluation for
is always done as if no line search were used. Therefore, the two work residuals
(3.49)
are set first and the corresponding sign is checked. Taking as the basic value, the ratios
are calculated and plotted against , as shown in Figure 3.29.
Figure 3.29 Interpolation for line search scheme
If is negative, the lower and upper bounds are readily known as and
, respectively, and the new estimate is interpolated between them for ;
the residuals for are evaluated and a new value of (say ) is obtained. If is negative,
the upper bound is adjusted to and a new interpolation is performed (see Figure 3.29-
a)). Instead, if is positive, the lower bound becomes while the upper one is kept as
β
β 0= ∆ur Qj-1( ) ∆urj-1=
∆Qj j 2≥
β 1=
ψ0 ∆Qj( )T
∆ur Qj-1( )⋅ ∆Qj( )T
∆urj-1⋅= = for β 0=
ψ1 ∆Qj( )T
∆ur Qj-1 ∆Qj+( )⋅ ∆Qj( )T
∆urj⋅= = for β 1=
ψ0
ρ ψ ψ0⁄= β
a) Lower bound (βL ) unchanged
β1 1=β2
ρ1
ρ2
β3
ρ3 3
21
StopLine Search
Tolsr
-Tolsr
ρ0 1=
ρ ψψ0------=
1
b) Lower bound (βL ) shifted
β1 1=
β2ρ1
ρ2
β3
ρ33
2
1
StopLine Search
Tolsr
-Tolsr
ρ0 1=
ρ ψψ0------=
1
βL βLβU βUβU βL
ρ1 βL 0=
βU β1 1= = β2 ρ 0=
β2 ρ ρ2 ρ2
βU β2=
ρ2 βL β2=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 131
(see Figure 3.29-b)), allowing a new trial to be done by interpolation using the fol-
lowing expression
(3.50)
The process stops when is lower than a pre-defined tolerance TOLSR less than unity. This
tolerance depends on the characteristics and the powerfulness of the basic iterative procedure,
and states how far the line search is allowed to go. Usually some slackness is considered ade-
quate for line searches in conjunction with N-R or modified N-R algorithms (Criesfield
(1982)), which leads to TOLSR values set within 0.3-0.5 (Marques (1986)).
If is positive, an extrapolation process has to be set first to find a pair of and bound-
ing the zero of . In order to avoid dangerous extrapolations (which could be directly done
using Eq. (3.50)), the step is shifted forward, i.e., for and , residuals are
evaluated (only for is needed) and the ratio is checked for negative value. The process is
repeated until lower and upper bounds ( and ) are found (thus, for opposite signs of
and ), such that interpolation can be performed as stated before.
Once an acceptable value of (say ) is found, the corresponding residuals are taken as the
displacement residuals for iteration j, i.e., , and the state of all sec-
tions and the element stiffness matrix are updated accordingly. The next iteration j+1 is then
set up for application of corrective forces and new line searches may
eventually take place.
3.7 Summary of the non-linear algorithm
3.7.1 General structure
The element state determination as described in the previous section is nested within two outer
processes, namely one for the incremental sequence of load application and another for the
Newton-Raphson iterative scheme within each load step.
The incremental scheme for load application corresponds to the usually assumed loading his-
tory division into load increments or time steps, arising from either static or dynamic calcula-
βU β1=
β βLβU βL–( )
1 ρU ρL⁄–-------------------------+=
ρ
ρ1 βL βU
ρ
β1 1= β2 β1 1+=
β2 ρ2
βL βU ρL
ρU
β βa
∆urj ∆ur Qj-1 βa∆Qj+( )=
∆Qj+1 Kj-1 ∆urj–( )=
132 Chapter 3
tion. It follows the well known general non-linear scheme recalled in Appendix E, where the
Newton-Raphson (or external) iterative scheme refers to each step beginning as discussed in
2.4.4.4; thus, the corresponding expressions to be referred next are included in that Appendix.
Both the incremental sequence and the N-R scheme are included in the flow chart of Figure
3.30, the former corresponding to the outer loop over the load step k, and the latter being asso-
ciated with the loop for iteration n.
A unique scheme is adopted for both static and dynamic cases, but minor adaptations are high-
lighted where appropriate. It starts by computing the initial structure tangent stiffness ,
followed by the loop for step k, where the corresponding external load vector is first set
up and, if required, corrected for dynamic behaviour in task 2. This correction corresponds to
Eq. (E.12) in Appendix E, where (standing for ) is affected with contributions of iner-
tia and damping forces from the previous step (i.e., , according to
the Appendix E notation); additionally, the dynamic contribution to the effective stiffness
matrix is set up according to Eq. (E.18) as .
The structure stiffness matrix may be updated to the tangent one at each step beginning (see
task 3), depending on the adopted type of N-R scheme (standard or modified).
The initialization for the external iterative scheme is performed in task 4, where the increment
of nodal forces is set up according to Eq. (E.21) as the difference between the exter-
nal load vector and the nodal resultant force vector equivalent to the internal
stress state at the end of the previous step. In the same task, the stiffness matrix may be
selected as the tangent one, depending on the desired N-R scheme.
Task 5 accounts for the contributions of eventual dynamic behaviour to the force increment
and to the effective stiffness matrix; these two corrections, indeed correspond to the last term
of Eq. (E.19) and to the first of Eq. (E.18), respectively.
For a given iteration n, the displacement field increment is first updated (task 6), cor-
responding to Eqs. (E.14) and (E.20), and then transferred into an operator which performs the
state determination of all elements in the structure. The concept of operator is related with the
organization of CASTEM2000 (CEA (1990)) that will be addressed in 4.2; however, the fol-
lowing flowchart steps are described by reflecting already this basic philosophy of the code.
KG( )0
Qextk( )
qk Qextk
Qdynk M ak 1–⋅ C vk 1–⋅+=
KD 1 β∆t2( )⁄( )M γ β∆t( )⁄( )C+=
∆QG( )n=1
Qextk QG( )
k-1
K*G
∆uG( )n
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 133
Figure 3.30 Flow chart for structure state determination
State determination of all elements
ielem = 1, number of elements
Set input: ∆uE( )n
∆Qf( )0
V( )0
State determination of element ielem
Output: Qf( )n
Vn
Set k 1= and assemble initial structure tangent stiffness matrix: KG( )0
Define k-th load vector: Qextk
If required: Update new structure tangent stiffness matrix: KG( )k
Set n 1=
initialize: ∆uG( )0
0= ∆QG( )n=1
Qextk QG( )
k-1–= K*
G KG( )k or KG( )
0=
Update: ∆uG( )n
∆uG( )n-1
K*G[ ]
1–∆QG( )
n⋅+=
Assemble resisting forces: QE( )n
Qf( )n
QG( )n
Element level Global assemblage
Compute structure unbalanced forces: ∆QuG( )
nQext
k QG( )n
–=
If required: Update structure tangent stiffness matrix
K*G KG( )
n=KE( )
nVn KG( )
n
Element level Global assemblageKn
sufficiently small ?∆QuG( )
n No∆QG( )
n+1∆Qu
G( )n
=
Next n
Yes
Next k
1
2
Task:
5
3
4
8
7
9
10
11
For dynamic behaviour, correct Qextk Qext
k Qdynk-1+= and define matrix KD
and
For dynamic behaviour correct ∆QG( )n
∆QG( )n
KD ∆uG( )n-1
⋅–=
K*G K*
G KD+=
;
6
134 Chapter 3
Besides the displacement field, the referred operator for task 7 requires also, as main input, the
field of element internal forces (assumed here as the forces at the flexible element end sec-
tions) and the field of element internal variables describing the element state, both fields
corresponding to the step beginning. The operator output consists of the updated fields of
forces and internal variables at the end of iteration n.
Since the element state determination contains the most innovative features in this work, this
summary will focus mainly on that, for a single N-R iteration n. However, for clarity sake, the
rest of the outer processes is still described before.
After completion of the element state determination, task 8 is accomplished by another opera-
tor which transforms the element internal forces into their equivalents in the total
element (with rigid lengths) reference system parallel to the global one, by means of transfor-
mations described in 2.4.2; these forces are then assembled to give the structure resisting force
vector .
Unbalanced forces are computed as described in task 9 of Figure 3.30 and, if required,
the tangent stiffness matrix may be updated in task 10 for the present iteration by means of
another operator. Based on the field of element internal variables , this operator performs
the calculation of each element stiffness matrix in its reduced space (without rigid body
modes) which then leads to in the total element global space by successive transforma-
tions. The global structure tangent stiffness matrix is then obtained by assembling the
element matrices . The adoption of for performing the next N-R iteration, again
depends on the choice made for the N-R scheme (standard or modified).
Convergence is checked in task 11 for sufficiently small forces and, in case it is not
reached, a new iteration n+1 is enforced for . When convergence is
reached, the next step k+1 is set up and a new increment application starts.
3.7.2 Element state determination
As shown in Figure 3.30 the input for each element state determination, for the n-th N-R itera-
tion, consists of the vectors , and referring, respectively, to the current itera-
tive increment of element displacements in its global reference system, to the flexible element
end section forces and to the element internal variables at the step beginning. Then, for each
Qf
V( )
Qf( )n
Vn
Qf( )n
QE( )n
QG( )n
∆QuG( )
n
Vn
Kn
KE( )n
KG( )n
KE( )n
KG( )n
∆QuG( )
n
∆QG( )n 1+
∆QuG( )
n=
∆uE( )n
Qf( )0
V0
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 135
element, the n-th iteration develops as follows:
Step (1) Transform displacements due to reference system rotation .
Transform to the flexible element space (see Eq. (2.9)).
Extract rigid body modes (see Eq. (2.13)).
Split into linear and non-linear components , where
stands for a (6x1) dimension vector whose unique non-zero components are the
end section rotations in the non-linear bending plane.
Step (2) Extract reduced space initial forces , which can be deducted from
Eqs. (2.11) and (2.12) showing that components in the reduced space are not
affected by the transformation to the complete space.
Split into linear and non-linear force components , where
stands also for a (6x1) dimension vector containing only as non-zero components
the end section moments in the non-linear bending plane.
Initialize the increment force vector , such that at the
end of iteration n.
Step (3) Initialize the element reduced space flexibility and stiffness matrices, respec-
tively, and , based on the internal variables .
Split into linear and non-linear component contributions .
Step (4) If element loads p exist and the load increment is the first, compute the elastic
displacements due to p and split into contributions from linear and non-linear
components .
If loads p do not exist or the increment is not the first, set and .
Step (5) Start the element state determination for and initialize counter
for splitting according to events in the plastic zones.
Step (6) Initialize element internal state (for the step beginning):
- set up moving sections ( , , and );
- set up flexibility and curvature distributions associated with the
moment one .
Step (7) Set up element integration zones and compute by integration of , duly
corrected with the contribution of elastic shear distortions .
Step (8) If set .
∆uE( )n
∆ue( )n
→
∆ue( )n
∆uf( )n
→
∆uf( )n
∆un→
∆un ∆ulinn ∆θn+= ∆θn
Qf( )0
Q0→
Q0 Qlin0 M0+= M0
∆Qn 0= Qn Q0 ∆Qn+=
F0 K0 F0[ ]1–
= V0
F0 F0 Flin Fθ0+=
up
up uplinθp+=
up 0= θp 0=
∆θ ∆θn=
ib 1= ∆θ
Yi0 Ci
0 Ci+20 Oi
0
f0 x( ) ϕ0 x( )
M0 x( )
θ0 ϕ0 x( )
β0 x( )
ib 1> θp 0=
136 Chapter 3
Search for events in plastic zone boundary sections and associated with
the application of end section moments and the distribu-
tion of element loads. Set .
If required, split according to the maximum number of events found in any
of those sections, such that at most one event will be found per section; thus,
Step (9) Initialize internal iteration counter and increment .
Step (10) Initialize Line Search counter and save .
Step (11) If set .
Compute iterative increment of end section moments with the previous stiffness
matrix .
Step (12) If , initialize or update variables and residuals of the Line Search process.
Step (13) Search for events in and associated with the application of and sub-
divide according to the event which occurs first, i.e. for the least reduction
factor found among sections having events . Thus,
Step (14) Update end section moments and the distribution along the element,
where the contribution of the element applied loads p (in case it exists) is always
included by means of the adequate expressions of (or ) as included
in Appendix C:
Step (15) Search for new moving sections , , and for the present diagram
, such that
Ei Yi0
∆M K0 ∆θ θp–( )⋅=
ib ib 1+=
∆θ
∆θ∆θb (basic)
∆θr ∆θ ∆θb–( ) (remaining) =⎝⎜⎛
→
j 1= ∆θj=1 ∆θb=
iLS 1= ∆θLS ∆θj=
j 1> θp 0=
∆Mj Kj-1 ∆θj θp–( )⋅=
j 1>
Ei Yi0 ∆Mj
∆θj
r( ) 0 r 1≤<( )
∆θj r∆θj= set ∆θj ∆θj=→
∆Mj Kj-1 r∆θj θp–( )⋅= set ∆Mj ∆Mj=→
Mj Mj x( )
Sp x( ) Spsx( )
Mj Mj-1 ∆Mj+=
Mj x( ) Mj-1 x( ) ∆Mj x( )+=
Yij Ci
j Ci+2j Oi
j
Mj x( )
sYi
j sYi
0≥ sCi
j sCi
0≥ and sCi+2
j sCi+2
0≤
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 137
Define the corresponding derivatives for the additional flexibility terms, depend-
ing upon eventual further progression of cracking sections and taking into
account possible full-cracking along one (both) element part(s).
Step (16) Perform the state determination of fixed sections ( and H) and of . Due to
the increment reduction imposed by the event-to-event procedure and to the
multi-linear character of the M-ϕ curve, the total curvature in sections
and is given by .
In the section (or sections) having generated the conditioning event, the flexibil-
ity is updated to after the event, whereas for the remaining sections the
flexibility is kept .
For the section H, the M-ϕ diagram is one-to-one and the total curvature and
the flexibility are obtained directly from .
Step (17) Perform the state determination of the moving sections , , and , for
which and are directly obtained from the with their one-to-one M-ϕ
diagrams.
Generate the additional flexibility terms due to eventual progression of and/or
for the current iteration.
Step (18) Update integration zones, based on the activated moving sections.
Step (19) Integrate the flexibility diagram to obtain the flexibility terms of associ-
ated with the non-linear component of deformation, and include the additional
contributions due to moving sections.
Accumulate with linear contributions and invert to obtain the stiffness matrix
Step (20) Integrate the total curvature diagram to obtain total rotations and
include the contributions due to elastic shear distortion.
Compute displacement residuals (necessary to assure equilibrium) given by:
Step (21) Check displacement convergence:
If is verified according to 3.6.8, proceed to Step (23).
Ei Yi0
ϕ…j Ei
Yi0 ϕ…
j ϕ…j-1 f…
j-1∆M…j+=
f…j-1 f…
j
f…j f…
j-1=
ϕHj
fHj MH
j
Yij Ci
j Ci+2j Oi
j
ϕ…j f…
j M…j
Cij
Ci+2j
fj x( ) Fθj
Fj Flin Fθj+= Kj Fj[ ]
1–=
ϕj x( ) θj
δθj
δθj θj θ0 ∆θb––=
max δθ1 δθ2( , ) θtoler<
138 Chapter 3
In case convergence is not reached:
- if , set and restart in Step (10) for next iteration ;
- if proceed to Step (22).
Step (22) Control line search residuals:
If line search tolerances are verified according to 3.6.9, accept residuals and
proceed for the next iteration restarting in Step (10) with .
If tolerances are not satisfied, search for new scaling factor , set new iteration
scaled increment and restart new line search trial in
Step (11).
Step (23) Convergence is reached for a state of internal equilibrium in the element, corre-
sponding to the sub-increment (possibly the total increment). Update ele-
ment state for the end of :
Step (24) Check if there is still a remaining sub-increment to be applied:
If , set and restart in Step (8).
If , no remaining sub-increment exists and proceed in Step (25).
Step (25) Complete with the contribution of elastic forces, which can be simply cal-
culated by due to the uncoupling of force (or displacement) compo-
nents contributing to different deformation directions. Therefore:
Step (26) Transform force increment to the complete space of the flexible element, eventu-
ally including contribution of element forces p: (see Eq. (2.11));
Compute the final forces in the flexible element reference system for the n-th
Newton-Raphson iteration, given by , and the final out-
j 1=( ) ∆θj+1 δθj–= j 1+( )
j 1>( )
δθj
j 1+( ) ∆θj+1 δθj–=
β
∆θj β∆θLS= iLS iLS 1+=
∆θb
∆θb
θnew0 θ0 ∆θb+= → set θ0 θnew
0=
∆Qnewn ∆Qn Mj=converg M0–( )+= → set
M0 Mj=converg=
∆Qn ∆Qnewn=⎝
⎜⎛
setV0 Vj=converg=
K0 Kj=converg=⎝⎜⎛
∆θr
∆θr 0≠ ∆θ ∆θr=
∆θr 0=
∆Qn
K0 ∆ulinn⋅( )
∆Qtotn ∆Qn K0 ∆ulin
n⋅( )+= → set∆Qn ∆Qtot
n=
Vn Vj=converg=⎝⎜⎛
∆Qn ∆Qf( )n
→
Qf( )n
Qf( )0
∆Qf( )n
+=
FLEXIBILITY BASED ELEMENT WITH MULTI-LINEAR GLOBAL SECTION MODEL 139
put of the element state determination is available ( and ) corresponding
to the element displacements .
The above described sequence of steps has been included in the CASTEM2000 operator which
performs the state determination of all structure elements, and, indeed, constitutes the most
significant contribution to that object-oriented and general purpose computer code. However,
several other adaptations had to be done in other operators (e.g., the one for tangent matrix cal-
culation) in order to cope with the present element specific features. Comments on such adap-
tations and further details about the code structure are included in Chapter 4.
3.8 Concluding remarks
In the present chapter the general flexibility formulation has been particularized for the case
where the global section behaviour is described by a multi-linear model based on a trilinear
skeleton curve.
Use is made of the stiffness variations at cracking and yielding points of the trilinear curve, in
order to define the so-called cracking and yielding sections. These sections move along the ele-
ment and allow its division into several zones with distinct behaviour. Thus, control sections
are considered of fixed type (the member-end and the mid-span ones) and of moving type (the
cracking, yielding and null-moment sections), which permit a continuous updating of the ele-
ment flexibility distribution during the response. Basically, the element becomes subdivided
into a number of zones, viz the plastic (adjacent to the yielded end sections), the cracked and
the uncracked zones, the two latter developing in between the plastic ones according to the
bending moment distribution.
Modifications have been introduced in an existing behaviour law, mainly in the post-cracking
range in order to make possible the control of cracked zones by means of an efficient, yet
approximate way. Specifically, a transition from uncracked to cracked behaviour is considered
localized in the section where cracking is incipient, in such a way that sections in the cracked
zones keep a linear elastic behaviour, though with reduced stiffness. The major advantage is
that the fully-cracked stiffness is progressively introduced in the element according to the
actual loading; however, at the present development stage, it does not allow energy dissipation
by hysteresis in cracked zones.
Qf( )n
Vn
uE( )n
uE( )0
∆uE( )n
+=
140 Chapter 3
The element state determination is performed by means of an internal iterative scheme where
corrections to element end-forces are successively made by elimination of element displace-
ment residuals and the element tangent flexibility (and stiffness) matrix is continuously
updated; the calculation of this matrix and those residuals constitute the major and most inno-
vative tasks in the developed scheme.
Due to the specific character of moving sections, their state determination is readily performed
once their nature and location is defined for a given distribution of moments.
The element flexibility is determined according to the instantaneous location of control sec-
tions where the flexibility is known from the corresponding model; between control sections
the flexibility is assumed linearly distributed, which is coherent with the uncracked and
cracked section models and, under certain circumstances, also with the model of yielded sec-
tions. The moving character of cracking sections, where a curvature discontinuity is assumed
to account for the uncracked/cracked behaviour transition, has been found to introduce special
contributions to the element flexibility matrix which have to be duly included in order to suc-
cessfully account for that transition.
The element residual displacements are calculated by an alternative scheme rather than that of
the general flexibility formulation; it is based on integration of total deformations instead of
residual ones, since it has been found more compatible with the control of moving sections.
The control of plastic end zones has been carefully detailed. Each of those zones is subdivided
into one fixed-length part and another with variable length (during the application of a given
load increment), in order to minimize possible inaccuracies resulting from the fact that the
plastic zone behaviour is just based on the end section and yielding section behaviour. In the
same line, an event-to-event scheme is used inside the element iterative process.
Convergence difficulties related with the uncracked/cracked behaviour transition required the
adoption of a line-search scheme, which, being internal to the element iterations, has proved to
be quite efficient.
Finally, the steps of the element iterative process are summarized for a given iteration of the
global non-linear algorithm of the structure, and the interface between both schemes is high-
lighted in a flowchart illustrating the main tasks of the structure state determination process.
Chapter 4
NUMERICAL IMPLEMENTATION,
AUXILIARY TOOLS AND VALIDATION
4.1 Introduction.
The flexibility formulation as described in the previous chapter has been implemented in the
general purpose computer code CASTEM2000 (CEA (1990)), and the main implementation
related topics are addressed in 4.2.
In view of the particular features of the code, indeed rather different from traditional specific
purpose computer codes, a brief description of CASTEM2000 basics is first provided in 4.2.1
and complemented with a rather simple example of structural analysis, in order to present the
user interface environment. The basic code structure and the intervention levels for new code
improvements are addressed. Subsequently, the most significant implementation needs are
identified within the framework of previously introduced concepts such as objects, commands,
operators and procedures. In particular, the major implementation topics related with the flexi-
bility element formulation are referred in 4.2.2 and the operators requiring the most relevant
modifications are briefly presented in order to point-out the performed interventions.
A new algorithm is presented in 4.2.3 for the definition of trilinear approximations of moment-
curvature relationships for rectangular and T-shape reinforced concrete sections. This pre-
processing task is described with some detail and particular emphasis is put on a unified proc-
ess for definition of turning points in the non-linear behaviour range; the implementation of
this auxiliary tool in CASTEM2000 has been cast in the form of a new operator.
Some validation tests at the single element level are included in 4.3 for experimentally tested
142 Chapter 4
members under monotonic and cyclic loading conditions. Rather than aiming at an exhaustive
process of model validation against experimental results, the tests presented allow to find-out
the expectable quality of numerical results and to identify model limitations that must be kept
in mind when analysing complete structures.
4.2 Implementation in the computer code CASTEM2000
4.2.1 Basics of CASTEM2000 and main implementation needs
CASTEM2000 is a multi-purpose finite element based computer code for structural analysis,
initially developed by the “Commissariat à l´Energie Atomique” (CEA) in the framework of
structural mechanics research (CEA (1990)). The need of treating several types of problems
based on different formulations (solid and fluid mechanics as well as thermal processes),
required the development of a high level tool of analysis based on a unified and powerful tech-
nique such as the finite element method.
Aiming at a unified way of handling different problems, the code has been structured follow-
ing the, nowadays increasingly adopted, object-oriented technique of programming. It is based
on a specifically developed high level language GIBIANE (or simply GIBI) (CEA (1990))
consisting on a wide set of commands and operators used to control and define the program
flow by object manipulation in a specific environment or shell.
The object-oriented features of CASTEM2000 lead to a high level of versatility and flexibility
in the sense that it can be adjusted to the particular problem to be solved. In contrast with clas-
sical codes designed for the analysis of certain well-defined type of problems, to which spe-
cific cases have to be adjusted, CASTEM2000 allows the user to build-up the program flow by
himself, to follow the analysis task-by-task, to modify the task sequence, to re-define tasks and
to check their outputs, in a word, to adapt it to his own needs.
The macrolanguage GIBI permits to define the usual operations characteristic of finite element
analysis, by means of simple instructions involving commands or operators acting on input
objects and, possibly, generating new output objects. A typical case of such operations is the
stiffness matrix assemblage: previously defined element mesh and fields of material properties
and element characteristics are provided as input objects for the operator which performs ele-
ment stiffness computation and assemblage, giving the global stiffness matrix as a new object.
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 143
Objects are defined as pieces of information grouped according to specific and well defined
rules characterizing the object type. Some of the main object types are briefly referred next;
inside brackets it is included the original french name used in the code for the object type:
• geometric types, specifically node (POINT) and mesh (MAILLAGE);
• scalar or vector field types, defined on the nodes or the elements, namely the nodal field
type (CHPOINT) and the element field type (MCHAML), respectively; the former stands
for example for displacements, forces or temperatures, whereas the latter may refer to mate-
rial properties, geometric characteristics or internal stress/strain components, defined on all
the element integration points;
• model type (MODELE), which includes references to the formulation associated with the
finite elements in a given mesh and to the constitutive behaviour model;
• stiffness type (RIGIDITE), consisting of the material stiffness or mass matrices associated
with a given mesh, or the stiffness corresponding to imposed boundary conditions;
• loading type (CHARGEMENT), containing information about a given force or displace-
ment field representing the load, and, possibly, the time description of the load process;
• integer (ENTIER), real (FLOTTANT) and string (MOT) types, standing for single numeri-
cal/string constants or variables;
• table type (TABLE), containing a set of objects of any type, identified by numerical indices
or strings.
Commands (DIRECTIVES) and operators (OPERATEURS) are used to perform operations on
input objects allowing to manipulate them (by modifying them or not) in the first case, and to
generate new objects in the second case. The available commands and operators in the GIBI
language can cover a wide range of purposes as different as:
• general operations, namely direct generation of objects (e.g. copying), object management
(e.g. listing, deleting), mathematical operations (e.g. arithmetical, logical, trigonometrical),
modification and extraction of data from objects (e.g. maximum value of a scalar field) and
flow control operations (e.g. loops, “if-then-else” type blocks, etc.);
• preparation of the analysis model, viz, geometric mesh generation, model definition (formu-
lation, type of element, type of behaviour model), material properties, element characteris-
tics, boundary conditions (fixing supports, imposed displacements, relations between
degrees of freedom) and loading;
• solution of a discretized problem, namely definition and assemblage of stiffness and mass
144 Chapter 4
matrices, solution of linear equation systems and of eigensystems;
• analysis of results, consisting of post-processing computations (stress and strain fields,
reactions, etc) and of graphical visualization.
Commands and operators can be organized following a user-defined sequence of tasks in order
to perform the desired analysis. Such sequence constitutes the so-called GIBI input for
CASTEM2000 running sessions, either in interactive or batch mode.
An illustrative example of a GIBI input is given below for the very simply structure shown in
Figure 4.1, where uniform elastic material properties and cross-section characteristics are
assumed as indicated.
Figure 4.1 Illustrative example for GIBIANE input
In the following input list, commands and operators are highlighted with boldface italic letters,
upper case words stand for the names of expected arguments and lower case, normal typeface
letters refer to “values” of such arguments.
1/ opti DIME 3 ELEM seg2;
2/ p0 = 0.0 0.0 0.0;p1 = 0.0 0.0 4.0;p2 = 3.0 0.0 4.0;
3/ c1 = droi 1 p0 p1;b1 = droi 1 p1 p2;
4/ ll_1 = c1 et b1;
5/ mo_1 = modl ll_1 mecanique elastique poutre;
6/ ma_1 = matr mo_1 YOUN 20.e6 NU 0.3;
7/ ca_1 = carb mo_1 SECT 0.36 INRY 0.0108 INRZ 0.0108 TORS 0.0216;
8/ mc_1 = ma_1 et ca_1;
9/ bll1 = bloq rota depl p0;
10/ bll2 = (bloq UY ll_1) et (bloq RX ll_1) et (bloq RZ ll_1);
11/ ri_ll = rigi mo_1 mc_1;
4.0
m
F1 = 100 kN
3.0 m
y
x
z
c1
b1
E = 20x106 kPaMaterial properties
Section Characteristics
ν = 0.3
A = 0.36 m2
Iy = Iz = 0.0108 m4
Ix = 0.0216 m4p0
p1 p2
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 145
12/ ritot = ri_ll et bll1 et bll2;
13/ f1 = forc FZ -100.0 p2;
14/ d1 = reso ri_ll f1;
15/ sd1 = defo ll_1 d1;
16/ pv = 0.0 -1.e8 0.0;
17/ trac pv sd1;
18/ s1 = sigm mc_1 d1;
19/ list s1;
The input line 1/ stands for the command opti where the use of the tridimensional space is
declared by the argument DIME with the value 3; the type of geometric supports for the finite
elements is set by the argument ELEM for the element named seg2 consisting of a two-node lin-
ear segment. The coordinates of nodes p0, p1 and p2 are set in lines 2/ and the geometric ele-
ments c1 and b1 (for the column and the beam, respectively) are defined in lines 3/ by the
operator droi stating that 1 single element is considered between the extreme nodes. The total
mesh ll_1 is then obtained in line 4/ by the operator et which performs the concatenation of the
partial meshes referring to the column c1 and the beam b1.
The model type object mo_1 is obtained in line 5/ from the operator modl which associates the
mesh ll_1 with the mechanic formulation (mecanique), the elastic behaviour (elastique) and
beam finite elements named poutre, consisting in the classical Bernoulli type element. Material
properties (assumed uniform) are stored in the element field ma_1 generated by the operator
matr where the Young modulus YOUN and the Poisson ratio NU are declared in line 6/. Addi-
tionally, the operator carb allows to define in line 7/ the element characteristics (not obtainable
from the mesh), namely the cross-section area and moments of inertia, stored in the element
field ca_1 as components labelled by their argument names (SECT, INRY, INRZ, and TORS). The
total set of material properties and characteristics is grouped in the field mc_1 by the operator
et in line 8/.
Boundary conditions associated with the fixed support are set in line 9/ by the operator bloq,
which, under the keywords rota and depl completely blocks rotations and displacements of the
node p0; the result consists of a stiffness type object bll1. Similarly, the total mesh ll_1 is pre-
vented to have displacements UY (in the yy direction) and rotations RX and RZ, as enforced by
the operator bloq in line 10/, where three boundary conditions are imposed and then concate-
nated in only one stiffness type object bll2.
146 Chapter 4
The stiffness type object ri_ll due to the elastic material behaviour is constructed in line 11/ by
the operator rigi whose arguments are the model mo_1 and the element field mc_1 of properties
and characteristics. The total structure stiffness ritot is then obtained by concatenation of the
relevant contributions in line 12/, namely the material stiffness ri_ll and the boundary condition
ones bll1 and bll2.
The applied force field f1 is defined in line 13/ by the operator forc imposing the value -100.0
for the force component FZ in the point p2.
The linear system with stiffness ritot and subjected to the force field f1 is solved by the operator
reso (a linear equation system solver) in line 14/, leading to the displacement field d1.
The phase of result analysis starts in line 15/, with the generation of the deformed shape sd1 (a
mesh type object) by the operator defo upon the original mesh ll_1 and the displacement field
d1. This deformed shape sd1 can be visualized in line 17/ by means of the command trac
requiring a viewpoint pv (previously defined in line 16/).
The operator sigm is used in line 18/ to generate the element field s1 containing the internal
stresses (in this case the end section forces), from the properties and characteristics mc_1 and
the displacement field d1; the contents of s1 is then listed in line 19/ as the result of the com-
mand list.
The above given example aims just at a general but illustrative perspective of how
CASTEM2000 works and, for complex problems, it may appear somewhat cumbersome. In
such cases the use of procedures becomes extremely advantageous; procedures are sequences
of operators cast in independent GIBI segments and acting as higher level operators to accom-
plish well defined purposes. Actually, procedures play the same role as subroutines do in com-
mon programming languages, allowing to perform rather standard and repetitive sets of tasks.
Typical examples of procedures are the implementation of the Newmark method for the inte-
gration of dynamic equilibrium equations or the non-linear incremental analysis using the
Newton-Raphson algorithm.
CASTEM2000 provides a set of built-in procedures to accomplish some usual tasks in struc-
tural analysis which cannot be handled by a single operator, but other procedures can be easily
designed and implemented by the user.
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 147
Thus, concerning tool implementation or improvement, the code offers two possible ways, viz:
• Development of procedures, written in GIBI (thus, strictly relying upon existing operators)
when the envisaged tasks do not involve new elements, models or formulations; in these
conditions, this is a low cost option from the implementation standpoint since it allows a
very fast development and on-line testing, without the need of modifications at the basic
CASTEM2000 software level.
• Development of new operators, based on existing and new subroutines constituting the code
source software; new operators are required when not yet available elements, models or for-
mulations are to be incorporated, and may be appropriate for efficiency purposes when cer-
tain algorithms, despite also implementable at GIBI level, would lead to cumbersome and
computationally heavy procedures; the implementation cost of operators is obviously
higher than that of procedures, since it requires a more in-depth knowledge of the code data
structure in order to provide the adequate operator interface.
In the present work, developments have been performed at both procedure and operator levels,
although the main contribution consisted of new improvements on existing operators in order
to incorporate the proposed flexibility based element; the most significant interventions
required for these developments refer essentially to:
• the inclusion of the new flexibility based element, for which all the affected existing opera-
tors needed to be adapted;
• the development of a new operator for the definition of skeleton curves for RC global sec-
tion modelling, indeed a quite cumbersome but crucial pre-processing task;
• slight adaptations of the general procedure for the non-linear static and dynamic analysis,
which essentially follows the scheme described in Appendix E and the flowchart included
in Figure 3.30;
• several post-processing procedures for result analysis and visualization of frame structures.
The two first interventions, being related with operator development, require a few comments
about some key issues on CASTEM2000 organization concerning finite element formulations
and models, as well as on the most relevant features of internal programming.
As far as formulations and models are concerned, the following aspects are highlighted:
• Finite elements are treated by distinguishing the geometric support and the underlying for-
mulation; therefore,
148 Chapter 4
- various types of geometric supports can be adopted simultaneously in the same structure;
- each geometric support can be assigned more than one finite element formulation,
depending on the problem type (mechanic, thermic, ...) and on specific characteristics
related with the internal distribution of the unknown variables; a typical example is the
two-node linear segment used as geometric support of both the truss element and the
Bernoulli or the Timoshenko beam elements.
• Each finite element can be used with different behaviour models, provided the necessary
consistency is assured between the element formulation and the model; an example of that
is the two-node Timoshenko finite element, which can be used either with global section
behaviour laws (for both moment-curvature and shear-distortion components) or with a
fibre discretization approach where the behaviour is controlled at each fibre level (Guedes
et al. (1994)).
• In a given structure, several zones can be defined where different combinations of one sin-
gle finite element type with one element model are considered; thus, there can be as many
zones as the number of different combinations of elements and models.
It is apparent that such features provide a high degree of versatility to handle very different
types of problems, which is further assisted by a powerful set of tools for mesh generation.
Concerning the programming features, it is worth mentioning that the source code is written in
an extended FORTRAN77 language, the so called ESOPE language, which includes a few
additional instructions for management of data structures. Basically, arrays of data are grouped
into larger data segments which are initialized, activated, de-activated or suppressed according
to the code flow needs. The required data for subroutines to perform their tasks, is made avail-
able by some of those extra instructions; once the data is no more needed, other instructions are
used to make it unavailable again.
Each operator is supported by a driver, i.e., a subroutine (written in ESOPE) where the input
and output objects (fields, models, tables, etc) are decoded into segment-based data structures
managed by ESOPE instructions. The data is then transferred to lower level subroutines where
the structural calculations are performed; typically, the lowest level subroutines just handle
data in the traditional way of FORTRAN, which renders more transparent and easy the core of
implementations where the basic structure-related operations are performed.
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 149
This means that, when new operators do not need to be developed, the implementation is quite
straightforwardly accomplished by normal FORTRAN subroutines, requiring only a careful
identification of the data to be transferred from/to the upper level CASTEM2000 subroutines
more directly related with the operator driver.
In order to cope with the above referred structure subdivision into zones, subroutines of opera-
tors dealing with objects extended over the whole structure (or part of it), typically perform the
three following nested loops:
• the outer loop over the total number of zones in the structure,
• another loop extended over all the elements in each zone and
• finally, the inner loop over all integration (Gauss) points in each element.
In some operators and for certain type of elements, eventually the inner loop may be skipped,
as in the case of the elastic stiffness evaluation of beam or truss elements where the result can
be directly derived without considering the integration points.
4.2.2 Flexibility based element implementations
The element developed in the present work required the intervention in several operators,
where a new finite element formulation and a modified (new) model had to be included.
Among the already available elements in CASTEM2000, the Bernoulli beam element is the
closest to the flexibility based one. Therefore, despite their different characteristics, the imple-
mentation scheme for the flexibility element followed that of Bernoulli element as close as
possible. With this in mind, the main similarities and differences, with respect to the Bernoulli
element implementation, are highlighted in the following comments:
• the same geometric support has been used, viz the two-node segment named seg2;
• the same internal generalized forces and deformations, as well as nodal forces and displace-
ments were considered (as defined in 2.4.1);
• a new formulation has been defined and included because
- there are no deformation shape functions and the formulation must be cast consistently;
- an internal span section H is considered, bounding the two element parts likely to have
different properties, although uniform in each of them;
- rigid lengths are admitted in the element end zones;
150 Chapter 4
- the sections where generalized internal forces and deformations are controlled (thus,
playing the role of integration Gauss points in the classical finite elements) are located at
the end points of the flexible element, not necessarily coincident with the element nodes;
• a different non-linear model is considered, although based on the existing Takeda type
model for use with the Bernoulli element;
• no inner loop is performed over the integration points because the state determination is car-
ried out for each element as a unique entity following the steps described in 3.6 and 3.7.
Consequently, a new finite element named FLD1 and a new model designated by
TAKEMF_MOMY were added to the lists of available finite elements and models, respectively.
Additionally, a new type of formulation named FLEXIBIL was included in the formulation list.
Contrarily to the available beam elements, different material properties and element character-
istics can be assigned to its left and right parts. Thus, the element fields containing these data
may have different values in the two integration sections, which for the flexibility element are
the end sections of the flexible part but do not assume the usual role of integration section.
In terms of element characteristics, the same list as for the Bernoulli element was kept (namely,
the cross-sectional areas and the moments of inertia defined in Appendix A and a vector defin-
ing the orientation of the section axis system), to which the rigid lengths in each end zone and
the relative abscissas of the span section H have been added under the keynames LORG and
XH_L.
Concerning internal variables, while the Bernoulli element requires only variables for the con-
trol of sections fully governed by the behaviour model, in the flexibility element more infor-
mation has to be stored related with the yielding and cracking development in each element
part. Therefore, three new variables were added and given the keynames XYLD, XCRK and
XCRI, respectively standing for the yielding and cracking section abscissas , and ,
as defined in 3.4 and associated with each end section. Additionally, three other variables
named FLE1, FLE2 and COF2 were also included, related with the evolution of the element flex-
ibility and the curvature distributions. FLE1 and FLE2 stand, respectively, for the diagonal and
off-diagonal non-linear terms of the flexibility matrix as defined in 3.6.3 by Eq. (3.18)
(namely, FLE1 for and , and FLE2 for and ); COF2 is the quadratic term of the
curvature distribution in each plastic end zone as defined in 3.6.6.
sYisCi
sCi 2+
F33M F55
M F35M F53
M
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 151
Operators where the most significant modifications had to be made are referred next, along
with a brief description of their syntax and purpose; the notation used for operators and objects
is the same of the example given in the previous section.
1/ Operator: rigi
Syntax: zrige = rigi zmodl zmatr
Purpose: calculation of the structure elastic stiffness with the model zmodl and the field of material
properties and element characteristics zmatr; the output is the stiffness type object zrige;
2/ Operator: sigm
Syntax: zsige = sigm zmodl zdepl zmatr
Purpose: calculation of the elastic element stresses (or internal generalized forces) associated with a
given displacement field zdepl; the output is the element field type object zsige;
3/ Operator: bsig
syntax: zforc = bsig zmodl zsigm zmatr
Purpose: to obtain and assemble the element nodal forces equivalent to a given internal stress distribu-
tion (zsigm); the output is the nodal field type object zforc;
4/ Operator: epsi
syntax: zepsi = epsi zmodl zdepl zmatr
Purpose: calculation of the element internal strains (or generalized displacements) for a given dis-
placement field zdepl; the output is the element field type object zepsi;
5/ Operator: ecou
syntax: zsig1 zvar1 zdpl1= ecou zmodl zsig0 zvar0 zdeps zddep zmatr ztabl
Purpose: to perform the state determination on non-linear systems (updating stresses and internal var-
iables) for a given increment of strains zdeps or displacements zddep imposed over an
equilibrium state characterized by stresses zsig0 and internal variables zvar0; further input
is required in the table type object ztabl containing additional information for the control of
the non-linear process; the output consists of the element field type objects zsig1, zvar1 and
zdpl1, respectively, the updated stresses, internal variables and plastic deformations (not rel-
evant for the present implementation);
6/ Operator: ktan
syntax: zrig1 = ktan zmodl zsig1 zvar1 zmatr
Purpose: calculation of the structure tangent stiffness for a state characterized by element stresses
zsig1 and internal variables zvar1; the output is the stiffness type object zrig1.
In the above referred operators, new subroutines had to be included inside the loop over the
elements in order to accomplish each operator task for the flexibility based element. Thus,
beyond the preparation of data to be transferred into/from the new subroutines, the following
152 Chapter 4
adaptations had to be made:
rigi a master subroutine (fldsti) was included (calling slave routines) to obtain the ele-
ment elastic stiffness matrix via the flexibility one according to Appendix
A, followed by inversion and by the transformations given by Eqs. (2.26) and (2.27),
and by appropriate reference system rotations;
sigi inclusion of a master subroutine (fldefe), and slave routines, where the elastic forces
are computed for given displacements , with the matrix obtained
as in the rigi operator; the transformation to forces is also performed according
to Eq. (2.11);
bsig inclusion of subroutine (fldbsg) to perform the transformation of forces to in
the element nodes (as expressed by Eq. (2.9)), followed by reference system rotation
to obtain in the global reference system;
epsi inclusion of subroutine (fldeps) to transform the element displacements , first by
reference system rotation to the element local axes, giving , and then to as
expressed by Eq. (2.9);
ecou a master subroutine (fldesd) was included to perform the element state determina-
tion for non-linear behaviour according to Chapter 3 and following the steps
described in 3.7; indeed it is the major intervention within this implementation,
where several slave subroutines are controlled, of which the main ones are:
fldisd, for preparing the input at the reduced space level (see steps 1 to 3 in the ele-
ment state determination sequence in 3.7) and
fldipe, for performing the iterative process in the element (corresponding to steps 4
to 26 of the same sequence);
ktan a master subroutine (fldstt) was included (calling slave routines) to obtain the ele-
ment tangent stiffness matrix via the flexibility one , where the non-linear
contributions are extracted from the internal variables; the linear contributions and
appropriate matrix transformations are handled as in the rigi operator.
The aforementioned interventions just provide an overview of the basic needs for the flexibil-
ity based element implementations. Naturally, the whole set of modifications and new code
segments actually developed went far beyond the simplified description made herein. How-
ever, further details are deemed unnecessary in the present work, since they can be found else-
where (Arêde and Pinto (1997)).
K( ) F( )
Q K u⋅=( ) u K
Qf
Qf Qe
QE
uE
ue uf
K( ) F( )
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 153
4.2.3 Definition of skeleton curves for RC global section modelling
The present section deals with the definition of the so-called trilinear approximation of the
moment-curvature relationship for reinforced concrete sections under uniaxial bending condi-
tions. Figure 4.2 illustrates the typical shape of this relationship (dashed line curve) and the
corresponding trilinear approximation (solid line curve) defined by the usual turning points C
(cracking), Y (yielding) and U (ultimate), as referred in 3.3.
Figure 4.2 Typical moment-curvature diagram and trilinear approximation
It has already been mentioned that the M-ϕ curve, or just the turning points for a trilinear
approximation, can be obtained by a fibre-type analysis, since this constitutes a general tech-
nique for bending analysis of reinforced concrete sections. However, the fibre or layer discreti-
zation required at the section level may become computationaly rather heavy when large
number of sections have to be analysed.
Taking into account that most of the sections in frame structures reduce to rectangular and T-
shape ones, and making a systematization of the internal equilibrium analysis in such type of
sections, it is possible to build-up an algorithm through which the section state is assessed for
each turning point without the need of section discretization. Although based on an iterative
scheme (due to the piecewise non-linear behaviour models), the algorithm was found quite
efficient and very easy to use and, furthermore, it can be seen as an alternative way to trace-out
the M-ϕ curve in the post-yielding range.
4.2.3.1 Type of sections, notations and conventions
The following study focus on the most common cross-sections of structural elements in rein-
forced concrete buildings, viz rectangular for columns and T-sections for beams; the assumed
MMu
My
Mc
ϕϕuϕyϕc
C
YU
154 Chapter 4
general layout is shown in Figure 4.3 and the meaning of section characteristics is as follows:
• stand for the width and height of the section web;
• refer to the width and height of the section slab contribution;
• ; stand for the areas and depths of the bottom and top steel layers (main
reinforcement);
• ; ;... refer to the areas and depths of the interior steel layers (secondary
reinforcement);
• is the slab reinforcement, assumed included in ;
• is the concrete cover, measured from the concrete surface to the stirrups and assumed uni-
form along the stirrup perimeter.
Figure 4.3 Types of sections
The T-shape section can be assumed either in the position shown (flange upwards) or in the
reverse (flange downwards); in any case the meaning of characteristics is always as described
above.
Concerning the forces acting in the section and the internal generalized stresses and strains, the
following convention is adopted, along with the reference axis systems shown in Figure 4.4:
• acting forces ( and ) are applied at the geometric centre of the rectangular concrete
gross section with positive directions according to the reference axis system
passing through that point.
• internal generalized strains ( and ) are referred to the central principal axis system
, containing the section centroid for linear elastic behaviour or the neutral axis
for non-linear behaviour.
b h,
bs hs,
As1d1,( ) As2
d2,( )
As1
I d1I,( ) As2
I d2I,( ) nI
AsLAs2
c
ASLAS2
AS1
hs
bs
d1 h
d2
d1 h
b
d2I
d1I
AS2
I
AS1
I
a) Rectangular section b) T-shape section
c
a2
a1
AS1
d2
AS2
b
Nx My
Cg( )
xg yg zg, ,( )
εx ϕy
x y z, ,( ) G
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 155
Figure 4.4 Reference axis system convention
Since only the bending component is of interest, in the following the subscripts refer-
ring to axes are omitted and, thus, , , and are simply denoted by N, M, and .
4.2.3.2 Material models
Steel
The steel behaviour is described by a bilinear model with strain hardening as shown in Figure
4.5, for which the control parameters consist of the yielding strain and stress , and the
hardening or plastic modulus . Identical characteristics are assumed for both tension and
compression.
The ruling expressions both in the elastic and the hardening range are also included in Figure
4.5. However, they can be combined in a unique one, very convenient for computational pur-
poses, which can be written as follows
(4.1)
where for the elastic behaviour and for the hardening range.
Figure 4.5 Steel stress-strain relationship
zg
MyNx
xg
z
x
εx
ϕy
Acting Forces Internal Strains
yg
zg z=
y G
Cg
My ϕy,( )
Nx My εx ϕy ε ϕ
εsy fsy
Esh
fs pεs 1+/- 1 p–( )εsy+[ ]Es=
p 1= p Esh Es⁄=
fs
fsy
fsy–
εsy
εsy–
εs
EshEs
εs εsy– εsy[ , ]∈ fs⇒ Esεs=
εs εsy– εsy[ , ]∉ fs⇒ 1+/-fsy Esh εs 1+/-εsy–( )+=
Elastic range
Plastic range
1+/- sign εs( )=
εsm
fsm
156 Chapter 4
The plastic tensile branch is valid up to maximum stress and strain ( and ) after which
failure is assumed to occur. Usually, is taken as the uniform strain at maximum force in
tensile tests and is calculated according to the standard ENV10080 (ENV10080 (1991)).
Concrete
The model adopted to describe the concrete behaviour is schematically illustrated in Figure 4.6
and consists of the following:
• for compression, a parabolic diagram up to the peak stress, followed by a linear descending
branch until a residual stress plateau is reached;
• for tension, a linear elastic branch, after which the stress drops to zero.
Figure 4.6 Concrete stress-strain relationship
The corresponding expressions for each branch are detailed next in terms of the stress and
strain of each turning point. For the elastic tensile branch the stress is simply given by
, where is the Young modulus of concrete (initial tangent modulus or other).
For the compressive behaviour, expressions for the three distinct branches are given by
(4.2)
where refers to the slope of the linear softening branch. The defining parameters ,
and depend on the concrete strength and the peak strain obtained from compression tests
and on the degree of concrete confinement by transversal reinforcement.
fsm εsm
εsm
fc
εc
εcm
εct
εcr
fct
fcr
fcm
fc3εc( )fc1
εc( )fc2
εc( )Zm
1
fc Ecεc= Ec
fc1εc( ) fcm 2
εc
εcm-------⎝ ⎠
⎛ ⎞ εc
εcm-------⎝ ⎠
⎛ ⎞2
–=
fc2εc( ) fcm 1 Zm εc εcm–( )–[ ]=
fc3εc( ) fcr=
Zm fcm εcm
Zm
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 157
According to Park et al. (1990), the peak stress and strain can be given by
(4.3)
in terms of the peak compressive stress and strain , resulting from cylindrical com-
pression tests. The k factor refers to the confinement degree and is expressed by
(4.4)
where stands for the volumetric confinement ratio, defined as the volume of stirrups per
unit length divided by the volume of concrete core effectively confined, and refers to the
yielding stress of transversal steel.
Following the same proposal (Park et al. (1990)), the slope can be estimated by
(4.5)
where is expressed in MPa, is the width of the confined concrete core and s is the stirrup
spacing.
For the complete definition of the softening branch, the residual stress must be also known
and is often taken as 20% of the peak stress for confined concrete.
The above introduced expressions also allow to define the diagram characteristics for uncon-
fined concrete, simply by making in Eqs. (4.4) and (4.5), along with the assumption of
a zero residual stress. Therefore, the case of unconfined concrete reduces to
(4.6)
Finally, for the tensile behaviour, the maximum tensile stress can be approximately taken
as 10% of , or according to design code prescriptions.
fcm kfc0= ; εcm kεc0
=
fc0( ) εc0
( )
k 1fsyt
fc0
------ρw+=
ρw
fsyt
Zm
Zm 0.53 0.29fc0
+145fc0
1000–-------------------------------- 3
4---ρw
b's---- kεc0
–+⎝ ⎠⎜ ⎟⎛ ⎞
⁄=
fc0b'
fcr
fcm
ρw 0=
fcm fc0= ; εcm εc0
= ; fcr 0=
Zm 0.53 0.29fc0
+145fc0
1000–-------------------------------- εc0
–⎝ ⎠⎜ ⎟⎛ ⎞
⁄=
fct( )
fc0
158 Chapter 4
4.2.3.3 Linear behaviour: the cracking point
For the cracking point (C) definition, it is usually considered that a RC section is at “immi-
nent” cracking when its extreme tension fibre reaches the cracking stress . Additionally, the
following behaviour assumptions are also considered: initially plane sections remain plane
after deformation (Navier-Bernoulli hypothesis); both concrete and steel behave linear elasti-
cally and all section fibres are homogenized in concrete, with the homogenizing coefficient
given by .
The study refers to the general T-section shown in Figure 4.7, of which rectangular sections are
a particular case. According to that layout, some auxiliary parameters are introduced next in
order to help computations:
• for the concrete section, and parameters are defined as and ,
respectively, such that for rectangular sections and ; the rectangular con-
crete section area is denoted by ;
• for the section steel, the following geometric moments are considered
- = total homogenized steel area;
- = total homogenized steel static moment relative to the top fibre axis ;
- = total homogenized steel moment of inertia relative to the top fibre axis .
Geometrically, the uncracked homogenized section is characterized by its total area , cen-
troid depth and total moment of inertia , given by
(4.7)
where the coefficients , and are defined as follows
(4.8)
and become equal to unity for the case of rectangular sections.
The total section axial stiffness is given by , whereas the axial stiffness for the section
without the flange is defined by .
fct
m Es Ec⁄=
Sb Sh Sb bs b⁄= Sh hs h⁄=
Sb 1= Sh 0=
bh( ) Ac
AsT
SsT
T yT( )
IsT
T yT( )
AT
dG IT
AT AcγA AsT+= ; dG
h2--- Ac
AT------γz
SsT
T
AT------+=
ITh2
3-----AcγI IsT
T ATdG2( )–+=
γA γz γI
γA 1 Sb 1–( )Sh+= γz 1 Sb 1–( )Sh2+= γI 1 Sb 1–( )Sh
3+=
EcAT( )
EcA( ) Ec Ac AsT+( )=
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 159
Figure 4.7 Imminent cracking condition. Section, applied forces and internal strains
The moment and curvature values defining the point (Figure 4.2) are simply obtained by
applying the imminent cracking condition along with elementary static principles, provided the
external axial force is known. For that purpose, consider the section shown in Figure 4.7
with the strain diagrams corresponding to the axial force and the bending moment . The
total strain profile is given by
(4.9)
where subscripts “ ” and “ ” refer to the “axial” and “flexural” strain contributions, respec-
tively. The strain profile arises from any eccentric action of , whereas is exclu-
sively due to the action of .
In the elastic range, the concrete stress profile can be directly obtained from Eq. (4.9) and, by
imposing the tensile strength at the most tensioned fibre, the moment for the imminent
cracking condition can be given by
(4.10)
where the coordinate and the eccentricity , as defined in Figure 4.7, are given by
and , respectively; the tensile stress limit , “corrected” for the
axial force action, is defined as
(4.11)
In these conditions, the neutral axis coordinate can be also obtained from Eq. (4.9) and the
AS2
AS1
d2
d1h y
z
Cg
GdGdg
zt
eN
zg
xg
NM
zzz
xN
MNMze
de
εfNεf
M εaN
neutral axis
Partial strain diagramsSection Applied forces
hs
bs
b
ASj
I
djI
yT
C
N
N M
ε z( ) εaN εf
N z( ) εfM z( )+ +=
a f
εfN z( ) N εf
M z( )
M
fct Mc
Mc fct'IT
zt---- NeN–=
zt eN
zt h dG–= eN dg dG–= fct'
fct' fctNAT------–=
ze
160 Chapter 4
corresponding depth (referring to the most compressed fibre) becomes
(4.12)
The cracking curvature can be obtained by
(4.13)
and the uncracked flexural stiffness becomes defined as
(4.14)
which represents the slope of the M-ϕ curve branch up to the cracking point C.
From Eq. (4.14) it is apparent that the axial force may affect the flexural stiffness due to
any eccentricity; therefore, unless the section is symmetric or the axial force is null, different
flexural stiffness is obtained for positive and negative bending directions.
The expressions obtained for , and refer to the positive direction of bending. For the
negative one, the same expressions are still valid if is replaced by , but still refers
to the top fibre which is now the most tensioned one.
4.2.3.4 Turning points for non-linear behaviour
Basics for a unified procedure
By contrast with the closed form expressions obtained for the linear behaviour, the determina-
tion of any point for the post-cracking curve part requires a more complex and elaborate proce-
dure. However, for any point to be found (yielding, ultimate or others), a unified procedure can
be adopted based on the following:
• A pre-defined criterion, strictly associated to each turning point, states a given strain for
a specified fibre located at depth (measured from the most compressed fibre) as shown
in Figure 4.8; this provides a fixed point around which the strain linear profile can twist,
thus only another point remaining to be determined, which usually corresponds to the neu-
tral axis fibre at depth .
de
de dGN
fct'AT------------zt⎝ ⎠
⎛ ⎞–=
ϕc( )
ϕcεct
h de–--------------=
EcIT( )∗Mc
ϕc------- EcIT
1 NeN
Mc---------+⎝ ⎠
⎛ ⎞-------------------------= =
EcIT
Mc ϕc de
zt zt h– de
εa
da
d0
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 161
• The equilibrium equations (for axial force and bending moment) are written for the previ-
ous paragraph conditions, but, since the material (steel and concrete) stresses depend piece-
wise non-linearly on the strain profile, one trial value of must be used to define the
adequate stress profile for each distinct fibre zones.
Actually, this means that a set of assumptions has to be made, concerning the behaviour of
characteristic zones of the section, which still must be verified for the solution of the equi-
librium equations.
Therefore, with the initial assumptions set, the equilibrium equations can be written and
solved for a new value of ; this, in turn, leads to a new set of section behaviour conditions
which must match the assumptions made a priori; iterations are performed until this coinci-
dence occurs.
• Once the value for equilibrium is found, the strain profile is defined and both curvature
and bending moment can be computed.
Using such a scheme, the only aspect that modifies from one point (e.g yielding) to another
(e.g. ultimate) is the criterion stating the specific conditions for each of them.
For the following developments, consider again the generic T-shape section shown in Figure
4.8, where a given strain is imposed at depth . For non-linear behaviour, the section is
considered fully-cracked, thus with the concrete below the neutral axis assumed inactive.
Both the applied forces ( and ) and internal ones developed in each steel layer ( ,
and ) and in the concrete active (shaded) area ( ) are indicated in Figure 4.8.
Figure 4.8 Generic situation for the unified procedure. Forces and internal strains
For these conditions, the curvature, the strain at any fibre (of generic coordinate ) and the
d0
d0
d0
εa da
N M Fs1Fs2
Fsj
I Fc
AS1
d2
d1hy
z
Cg
O
d0dg
zg
xg
NM
zz
x
da
ε z( )
εa
neutral axis
Strain diagramSection Applied forces
AS2
FS2
FS1
Fc
ASj
I
djI
FSj
I
Internal forces
dc
z
162 Chapter 4
equilibrium equations can be written in terms of the neutral axis depth as the main variable,
and are given by
(4.15)
(4.16)
(4.17)
(4.18)
where stands for the moment of each force, taken relative to the neutral axis, and is
the number of the secondary steel reinforcement layers. Each contribution from concrete and
steel for Eqs. (4.17) and (4.18) is detailed in the next sections.
Concrete contribution
Unconfined and confined zones
The concrete active area usually consists of unconfined and confined zones which have to be
considered separately due to their different behaviour models.
For both the T-shape and the rectangular sections, the geometry of that active area, either the
unconfined or the confined zone, can be always reduced to a T-shape form. This means that
contributions from both zones can be handled in the same manner, if adjustable dimensions are
adopted (flange width , web width , top fibre depth and flange thickness ) which are
assigned the adequate values to match the desired configuration.
Figures 4.9 and 4.10 show the concrete section division into unconfined and confined zones,
for T-shape and rectangular sections. Their reduction to convenient T-shape sections and the
corresponding geometric characteristics ( , , and ) are also included, with the super-
scripts “u” and “c” standing for the unconfined and confined parts, respectively.
It should be noted that in the T-shape section type (see Figure 4.10) the two lateral dashed areas
d0
ϕ ϕ d0( )εa
da d0–----------------= =
ε z( )εa
da d0–----------------z=
N Fc F+ s1Fs2
Fsj
I
j 1 nI,=∑+ +=
N dg d0–( ) M+ M Fc( ) M Fs1( ) M Fs2
( ) M Fsj
I( )j 1 nI,=∑+ + +=
M …( ) nI
bs bw dt hs
bs bw dt hs
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 163
are considered in the confined part, even if they are not enclosed by stirrups, since the presence
of the slab provides a certain confinement to the lateral zones.
Figure 4.9 Rectangular section split into unconfined and confined zones
Figure 4.10 T-shape section split into unconfined and confined zones
Formulation for a T-shape section contribution
The definition of a general formulation for the concrete contribution to the equilibrium equa-
tions is not an easy task due to the non-linear piecewise stress distribution and to the specific
features of the section geometry.
However, for a certain type of sections, it may be possible the statement of a unique formula-
tion able to handle all the distinct situations likely to appear. In order to render easier the iden-
c2
h
d0
Section
c1
b
cl cl 2cl
b b 2cl–
Unconfined Confined
dtu
dtc Unconfined
Confined
bsc b 2cl–= bw
c bsc=
hsc 0= dt
c c2=
bsu b= bw
u 2cl=
hsu c2= dt
u 0=n.a.
c2
h
d0
Section
c1
b
cl cl 2cl
bs
b 2cl–
Unconfined Confined
dtu
dtc
bsc bs 2cl–= bw
c b 2cl–=
hsc 0= dt
c c2=
bsu bs= bw
u 2cl=
hsu c2= dt
u 0=
n.a.
bs bs 2cl–
164 Chapter 4
tification of these situations, the influence of stress non-linearity and of geometry peculiarities
must be separated as much as possible; this can be accomplished by the following procedure:
• The stress distribution over the active height is divided in zones where the stress function is
uniquely defined (herein termed by “stress zones”); the development of this distribution
and, consequently, the number of “stress zones” depend on the maximum compressive
strain which, in turn, depends on the trial value as shown in Figure 4.11. Actually, the
depths and , where the deformations and can be found, are readily obtained
through Eq. (4.16) and given by
(4.19)
Figure 4.11 Influence of d0 in the development of stress zones
• The active area is also divided into zones where the geometry is uniquely defined (called
“geometric zones”); the number of these zones also depend on the value, since it influ-
ences the active area as shown in Figure 4.12 for the general T-shape section.
Figure 4.12 Influence of d0 in the development of uniform geometry zones
d0
dm dr εm εr
dm d0 zm+= with zmεm
εa----- da d0–( )=
dr d0 zr+= with zrεr
εa---- da d0–( )=
d0
dt
n.a.
d0
d0
n.a.
n.a.
zm
zt
zr fcm
fcm
fcr
z fc
fc
fczm
z z
13 2
2 1
1
dr
dm
a) 3 stress zones b) 2 stress zones c) 1 stress zone
d0
dt
d0
d0
n.a.
n.a.ds
d0 ds> 2 zones⇒ d0 ds≤ 1 zone⇒
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 165
• The intersection of the “stress zone” and “geometric zone” sets, leads to a final set of zones
where the force and moment resultants can be obtained by analytical integration (the “inte-
gration zones”). For the rectangular section, due to the geometric uniformity all over the
section, the “integration zones” coincide with the “stress zones”. In the case of the T-shape
section there is only one change in the geometric uniformity due to the flange-web transi-
tion at depth (see Figure 4.12). Therefore, there is at most one more “integration zone”
than “stress zones”, whose position depends on the trial value of as shown in Figure
4.13.
Figure 4.13 Development of integration zones and adjustable widths bi and bi
• The fact that only one change in geometry occurs for T-shape sections greatly simplifies the
integration procedure. For each “stress zone” two width values are set up, namely for the
less compressed (inferior) and for the more compressed (superior) fibres, respectively
denoted by and , where the subscript i stands for the zone number. If these widths
ds
d0
dt
ds
d0
n.a.zm
zt
zr
z
fc
2
drdm
d0
n.a.zm
zt
zr
z
fc1
3
drdm
2
dt
ds d0
n.a.zm
ztzr
z
fc
drdm d0n.a.
zm
ztzr
z
fc
1
drdm
b
bs
Section
b3 bs=
b2 b=
b1 b=
b3 bs b–( )–=
b2 0=
b1 0=
b3 bs=
b2 bs=
b1 b=
b3 0=
b2 bs b–( )–=
b1 0=
b3 bs=
b2 bs=
b1 bs=
b3 0=
b2 0=
b1 bs b–( )–=
b3 bs=
b2 bs=
b1 bs=
b3 0=
b2 0=
b1 0=
3
1
3
1
2
2
12
33
00
0
0
zs zs
zs
biinf bi
sup
166 Chapter 4
are equal, the stress integration is straightforward because no geometry change occurs in
that zone. By contrast, if these widths are different, it means that the geometry transition
fibre exists in that zone; therefore, a first integration is performed over the whole zone
height for a constant width , which is then corrected with the result from a fur-
ther integration performed between the inferior and the transition fibres, for a constant
width . Figure 4.13 includes different situations and the corresponding
widths and for each integration zone, in order to clarify this procedure.
The numerical implementation of this algorithm requires that, for each zone, some integrals be
previously obtained analytically, viz:
• the total force and moment per unit width, resulting from integration between the
inferior and the superior fibres and given by
(4.20)
• the correcting force and moment per unit width, obtained by integration between
the inferior fibre and the transition one (that may exist in the zone), and written as
(4.21)
Eq. (4.16) is used to replace in the integrals of Eqs. (4.20) and (4.21), which allows to
express them in terms of their limits and the main variable . The analytical integration is
trivial since only polynomials are involved, but the results are not included here because cum-
bersome expressions are obtained.
Therefore, once a trial value is assumed, the corresponding and depths can be calcu-
lated by Eq. (4.19) and the integration zones can be defined accordingly. It is checked which of
bi bisup=[ ]
bi bisup bi
inf–( )–=[ ]
bi bi
fi( ) mi( )
f1 f1 d0( ) fc1εc( ) zd
zm
0
∫= = ; m1 m1 d0( ) fc1εc( )z zd
zm
0
∫= =
f2 f2 d0( ) fc2εc( ) zd
zr
zm
∫= = ; m2 m2 d0( ) fc2εc( )z zd
zr
zm
∫= =
f3 f3 d0( ) fc3εc( ) zd
zt
zr
∫= = ; m3 m3 d0( ) fc3εc( )z zd
zt
zr
∫= =
fi( ) mi( )
f1 f1 d0( ) fc1εc( ) zd
zs
0
∫= = ; m1 m1 d0( ) fc1εc( )z zd
zs
0
∫= =
f2 f2 d0( ) fc2εc( ) zd
zs
zm
∫= = ; m2 m2 d0( ) fc2εc( )z zd
zs
zm
∫= =
f3 f3 d0( ) fc3εc( ) zd
zs
zr
∫= = ; m3 m3 d0( ) fc3εc( )z zd
zs
zr
∫= =
εc
d0
d0 dm dr
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 167
the four situations of Figure 4.13 is matched for the trial case values, by searching the stress
zone where the transition fibre is lying; the result of this search defines the assumption related
with the concrete behaviour, as already referred.
Note that, in case one or both depths and happen to lie above the top fibre depth ,
the corresponding zones (2 or 3) fall outside the section geometry. Therefore, their contribu-
tions must not be considered for equilibrium, which is achieved by limiting such depths to the
minimum value ; in these conditions, the result of the involved stress integrals vanishes
automatically. However, from the computational standpoint, it is more convenient that Eqs.
(4.20) and (4.21) yield expressions always in terms of the same variables and, consequently,
the following alternative expressions for the integration limits are more appropriate:
(4.22)
(4.23)
where, according to Eq. (4.19), and are still given by
(4.24)
Replacing the integration limits and by and , respectively, it is ensured that inte-
grals are expressed only in terms of (as main variable) and the parameters and .
For each zone, the total and correcting widths, respectively, and , are computed in order
to affect the corresponding contributions of force and moment integrals (per unit width) to the
concrete resultants and (see Eqs. (4.17) and (4.18)). Finally, these resultants can be
generally expressed, as functions of , by
(4.25)
(4.26)
where, only the integrals associated with non-zero width need actually to be evaluated.
dm dr dt( )
dt
zm* ξmdm 1 ξm–( )dt d0–+= with
ξm 1= if dm dt>
ξm 0= if dm dt≤⎩⎨⎧
zr* ξrdr 1 ξr–( )dt d0–+= with
ξr 1= if dr dt>
ξr 0= if dr dt≤⎩⎨⎧
dm dr
dm d0εm
εa----- da d0–( )+= and dr d0
εr
εa---- da d0–( )+=
zm zr zm* zr
*
d0 ξm ξr
bi bi
Fc M Fc( )
d0
Fc Fc d0( ) bifi bifi+( )i=1,3∑= =
M Fc( ) Mc d0( ) bimi bimi+( )i=1,3∑= =
168 Chapter 4
Unconfined and confined concrete contributions
Using the formulation and the division of zones as described above, the contribution of both
unconfined and confined parts is straightforwardly defined, since they have been reduced to
general T-shape configurations.
The geometric and material model parameters of each part just have to be introduced in the
expressions presented above and the respective contributions to and become readily
defined. Some specific conditions may have to be considered for the unconfined or the con-
fined parts, depending on the criterion assumptions and requirements for each point definition.
The global contribution of concrete to the equilibrium Eqs. (4.17) and (4.18) is finally obtained
by the superposition of the unconfined and confined concrete contributions, and given by
(4.27)
(4.28)
which are rational polynomials of . Again, the superscripts “u” and “c” refer to the uncon-
fined and confined parts, respectively.
Steel contribution
Several steel layers are considered: the principal ones (top and bottom reinforcement, to which
some point definition criteria usually refer) and the interior layers that may exist, as shown in
Figure 4.3.
The contribution of each generic layer (principal and secondary ones included) is defined
according to the steel stress-strain diagram shown in Figure 4.5 and traduced by Eq. (4.1). The
behaviour can be either in the linear elastic or in the positive or negative plastic ranges, which
is checked-out according to the trial value of .
From Eq. (4.16) the strain , at each layer at depth , is given by
(4.29)
Fc M Fc( )
Fc Fc d0( ) Fcu Fc
c+ biufi
u biufi
u+( ) bicfi
c bicfi
c+( )+[ ]i=1,3∑= = =
Mc d0( ) M Fcu( ) M Fc
c( )+ biumi
u biumi
u+( ) bicmi
c bicmi
c+( )+[ ]i=1,3∑= =
d0
d0
εsii di
εsi
εa
da d0–---------------- di d0–( )=
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 169
and the decision conditions are the following
(4.30)
The assumption concerning the steel behaviour becomes defined by the values of and .
Subsequently the stress in the generic steel layer can be expressed in terms of , and,
finally, the force and moment contributions to the equilibrium equations are given by
(4.31)
(4.32)
where is the corresponding steel area.
Solution of the equilibrium equation and corresponding section state
Using the contributions of concrete and steel as above described, Eq. (4.17) can be expressed
in terms of . A detailed analysis of each contribution shows that it consists of a rational pol-
ynomial of , which can be reduced to a polynomial form of 3rd order if all the equation is
multiplied by a factor .
Thus, the steel and concrete (unconfined and confined) contributions for the global equation
are factored by this term and the corresponding polynomial partial coefficients are obtained
accordingly. The same applies for the axial force term which leads to a 2nd order polynomial.
The final 3rd order equation is solved by the Cardan algorithm (Spiegel (1970)), thus leading to
1, 2 or 3 distinct real roots, depending on the equation coefficients.
However, the obtained roots are not necessarily solutions of the equilibrium problem. Actually,
a calculated value of will be the desired solution if it fulfils the following requirements:
• to be an admissible solution, meaning that it has to lie within a range of values whose
upper and lower bounds are pre-defined according to the criterion for the
point definition;
• to be an equilibrium solution, for which the equilibrium equation assumptions based on the
εsiε– sy εsy[ , ]∈ => pi 1= (Elastic)
εsiε– sy εsy[ , ]∉ =>
pi p=
1i+/- sign εsi
( )=⎩⎨⎧
(Plastic)
pi 1i+/-
fsid0
FsiFsi
d0( ) pi εadi d0–da d0–----------------⎝ ⎠
⎛ ⎞ 1i+/- 1 pi–( )εsy+ EsAsi
= =
M Fsi( ) Msi
d0( ) Fsidi d0–( )= =
Asi
d0
d0
da d0–( )2
d0
d0
d0max( ) d0
min( )
170 Chapter 4
trial value, still must be verified for the obtained admissible solution.
The solution searching technique can be either a “sequential” one where the whole admissible
solution range is checked up step-by-step until the equilibrium solution is found,
or an “oriented” one where new trial values are based on the equilibrium equation residuals of
the previous obtained roots. In the present work the solution is first searched by the “oriented”
technique and, only in case that it is not found after a given maximum number of iterations, the
“sequential” technique is activated; details of both techniques can be found in Arêde and Pinto
(1996).
Once the equilibrium solution is found for the neutral axis depth (say ), the section state can
be readily known. The installed curvature is obtained by Eq. (4.15) for ; the corre-
sponding moment is obtained from Eq. (4.18), where all the concrete and steel contribu-
tions are given by Eqs. (4.28) and (4.32), respectively, after being particularized for .
The deformation of any concrete or steel layer is given by Eq. (4.16) (for ) and the corre-
sponding stress is obtained according to the respective model. In particular, for the steel layers,
the force resultant can be readily computed using Eq. (4.31).
Note that all the development has been made referring to the positive bending direction. For
the negative one, the same procedure can be applied, having in mind that the negative bending
in a given section is equivalent to the positive one in the inverted section. Therefore, the fol-
lowing adjustments are due before the negative bending calculations start:
• for T-shape sections only:
is replaced by and vice-versa; is replaced by ;
• for all types of sections:
are replaced by ; are replaced by ;
are replaced by .
In such conditions, the obtained moment and curvature are positive quantities and, therefore,
the sign must be changed “a posteriori”. However, it is reminded that the value (positive)
still refers to the most compressed fibre (in this case, the bottom one) and, thus, any computed
layer deformations and forces will come already with the correct sign (positive for tension and
negative for compression).
d0
d0min d0
max,[ ]
d0*
ϕ* d0 d0*=
M*
d0*
d0*
b bs hs h hs–
As1d1,( ) As2
h d– 2,( ) As2d2,( ) As1
h d– 1,( )
djI h dj
I–
d0*
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 171
Criteria for definition of turning points
The scope of the following paragraphs is the definition of the physical conditions associated
with the characteristic points of the diagram to be determined. These points are usu-
ally the yielding and the ultimate ones, and are related to the attainment of yielding or ultimate
conditions in one of the constitutive materials. In the present study, these conditions are con-
sidered as in Pipa (1993) and are detailed next. Besides these main points, supplementary ones
can be also defined aiming at a better knowledge of the post-yielding behaviour.
For every point, a given strain is prescribed for a specific layer and a range of possible neutral
axis positions is set up to restrict solutions to the physically acceptable ones. It is worth men-
tioning that, for the present stage of algorithm development, only common situations are con-
sidered, in the sense that neither totally tensioned nor completely compressed sections are
admitted. This restriction is quite acceptable because trilinear envelopes are usually obtained
for the structure with vertical static loads, where the predominance of tensile or compressive
axial force at yielding or ultimate stages is undesirable from the seismic design point of view;
indeed, under axial tensile forces the flexural strength can be drastically reduced, while under
dominant compressive force both the strength and the ductility may be seriously affected.
Yielding point
The yielding point is defined according to one of the two following criteria:
• the most tensioned steel layer reaches the yielding strain
yielding of steel;
• the most compressed layer of confined concrete reaches the peak strain
yielding of concrete, where the superscript “c” stands for the confined model.
The first criterion generally holds for beam sections, where the axial force is null, or very low,
leading to a low level of the maximum compressive strain in the concrete. By contrast, the sec-
ond criterion applies for column sections, where the axial force may induce considerably high
levels of compressive strain; in such cases, when this criterion condition is matched, the tensile
strain in the steel is usually quite far from the yielding strain.
Figure 4.14 illustrates generic strain profiles associated to each yielding criterion; the fixed
point, in turn of which the strain diagram can twist, is indicated, as well as the boundary
M ϕ–( )
εsy
→
εcmc
→
172 Chapter 4
dashed lines for the possible strain diagrams which provide the neutral axis upper and lower
bounds ( and ).
For the first criterion, the strain is imposed at the layer depth . The maxi-
mum value of is reached when the strain at the most compressed confined concrete layer is
equal to and the corresponding strain diagram defines the transition between the two crite-
ria. Therefore, can be obtained through Eq. (4.16) and is given by
(4.33)
The minimum depth is set up at the top fibre , since it corresponds to a limit for the
acceptable solutions. For positions of neutral axis above it, all the fibres are tensioned which
corresponds to an undesirable situation of dominant tensile axial force as pointed out above.
Figure 4.14 Yielding criteria at the tensioned steel or at compressed confined concrete
Instead, for the second criterion the strain is imposed at the layer depth .
The value of corresponds now to the of the previous criterion (given by Eq. (4.33)),
while the maximum depth is now set up at the bottom fibre . Beyond this limit, all
the fibres are compressed at yielding condition, which means predominance of compressive
axial force, i.e. another undesirable solution already referred.
Both criteria are analysed using the unified procedure explained before, and the yielding point
is associated with the criterion leading to the least curvature.
d0max d0
min
εa εsy= da d1=
d0
εcmc
d0max
d0max d1εcm
c dtcεsy–
εcmc εsy–
-------------------------------=
d0min 0=( )
c2
h
d0
Section
c1
dtu
dtc
n.a.
z
da
ε z( )
εa εsy=
d0min
d0max
εcmc
d1
d0
n.a.
z
daε z( )
εsy
d0min
d0max
εa εcmc=
b) Yielding at concretea) Yielding at steel
Twisting point
Confinedconcrete
εa εcmc= da dt
c=
d0min d0
max
d0max h=
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 173
Ultimate point
The ultimate point criteria are defined in a similar way as for the yielding point, and can be
stated as follows:
• the most tensioned steel layer reaches a pre-defined maximum strain
ultimate at steel;
• the most compressed confined concrete layer reaches the residual strain
ultimate at concrete.
The maximum steel strain is defined according to 4.2.3.2, whereas is the strain when
the residual stress is first reached in the confined concrete model (see Figure 4.6) and can be
obtained imposing in Eq. (4.2).
Again, the first criterion generally holds for beam sections, while the second applies for col-
umn sections where the axial force can be important; both are shown in Figure 4.15.
Figure 4.15 Ultimate criteria at the tensioned steel or at compressed confined concrete
For the first criterion, the strain is imposed at the layer depth , and the upper
bound is still given by Eq. (4.33), but with and replaced by and , respec-
tively; the lower bound, is still set up at the top section fibre as for the yielding point.
For the second criterion the strain is imposed at the layer depth ; again the
value of corresponds to the of the previous criterion and the maximum depth is still
set up at the bottom fibre , due to the same reasons explained for the yielding point.
However, special care is taken with the unconfined part in order to eliminate the corresponding
εsm
→
εcrc
→
εsm εcrc
fc2
c εcrc( ) fcr
c=
c2
h
d0
Section
c1
dtu
dtc
n.a.
z
da
ε z( )
εa εsm=
d0min
d0max
εcrc
d1
d0
n.a.
z
daε z( )
εsm
d0min
d0max
εa εcrc=
b) Ultimate at concretea) Ultimate at steel
Twisting point
Confinedconcrete
εa εsm= da d1=
d0max εsy εcm
c εsm εcrc
d0min
εa εcrc= da dt
c=
d0min d0
max
d0max h=
174 Chapter 4
flange, which, at this deformation stage, is already spalled off.
As for the yielding point, both criteria are checked and the ultimate point is associated with the
least curvature.
Supplementary point
Although for some sections (e.g., beam ones) the post-yielding monotonic behaviour is quite
well represented by the straight line between the yielding (Y) and ultimate (U) points, for oth-
ers a more detailed knowledge is required in the post-yielding zone. This can be accomplished
by applying the unified procedure to define supplementary points between Y and U, provided
the physical conditions for them are appropriately defined.
In the following, an example of a supplementary point is included, whose definition criterion
states that the strain at the most tensioned steel layer is the average of strains corresponding to
the Y and U points. Denoting these strains by and , the imposed strain at is
given by and defines the twisting point of the strain diagram, as shown in
Figure 4.16.
Figure 4.16 Supplementary point. Criterion related to the most tensioned steel
The upper and lower bounds for are set up imposing that the strain at the most compressed
layer of confined concrete must be within the values obtained for the yielding and ultimate
conditions (say and , respectively); therefore, according to the dashed line strain dia-
grams of Figure 4.16, and are now given by
εsY εs
U da d1=
εa εsY εs
U+( ) 2⁄=
c2
h
d0
Section
c1
dtu
dtc
n.a.
z
da
ε z( )
εa εsY εs
U+( ) 2⁄=
d0min
d0max
εcU
d1
εcY
Twisting point εsY
εsUConfined
concrete
d0
εcY εc
U
d0min d0
max
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 175
(4.34)
Note that the same reasoning can be used for any point between Y and U, and, in the limit, the
post-yielding zone can be completely traced out by defining a set of supplementary points
whose imposed strain at the tension steel layer is , where ranges
between 0 and 1.
The criterion can be also related to an imposed strain at the most compressed concrete layer,
following an entirely similar strategy; in such case, and are associated with the ten-
sion steel strains for the Y and U points.
4.2.3.5 Remarks on implementation and validation
The above described algorithm has been implemented in CASTEM2000 by means of a new
operator named TRIL, supported by a specific new driver transforming input/output data struc-
tures and controlling a master subroutine TRILIN where calculations are performed.
Basically, the input consists of the material model data, the geometric characteristics, the lon-
gitudinal and transversal steel data and the axial force. The output provides the initial elastic
characteristics (axial and flexural stiffness), the M-ϕ curve defined by its turning points and
further extra results related with the internal section state for each turning point. Details on
implementation can be found in Arêde and Pinto (1996).
The algorithm validation has been carried out by analysing several examples of reinforced con-
crete sections, for which the trilinear approximations are compared against the results of the
section monotonic analysis using fibre model discretization with the same material model data.
Several section characteristics, related with reinforcement contents, slab participation and axial
force, are varied in order to assess the trilinear approximation throughout some typical cases of
sections. Results of such validation are reported in Arêde and Pinto (1996), showing that the so
obtained trilinear curves compare quite well with the M-ϕ curves obtained by fibre analysis.
d0min dt
cεa daεcY–
εa εcY–
--------------------------= and d0max dt
cεa daεcU–
εa εcU–
---------------------------=
εa λεsY 1 λ–( )εs
U+= λ
d0min d0
max
176 Chapter 4
4.3 Flexibility-based element validation at the single member level
4.3.1 General
The exhaustive validation of a global element model such as the flexibility based model devel-
oped herein, is a rather difficult and cumbersome task to perform, particularly if comparison
against experimental evidence is sought. The wide range of situations that can be considered in
one element integrated in a complete structure can hardly be reproduced and found in experi-
ments on single members or sub-assemblages where the interpretation of phenomena is often
easier.
Nevertheless, the numerical simulation of single member experiments by the flexibility ele-
ment model is quite useful because the main features of the model response can be assessed by
comparison against experimental results and any model limitations can be detected.
The present section deals with the numerical simulation of several reinforced concrete mem-
bers experimentally tested under monotonic and cyclic quasi-static loading conditions,
included in the preliminary experimental campaign of small scale tests for supporting the full-
scale tests of the four storey building performed at the ELSA laboratory (Negro et al. (1994)).
This member testing aimed at representing the beam behaviour in the full-scale structure, and
particularly, the assessment of slab participation was sought. Thus, several cantilever beams
having identical section reinforcement contents were tested, both with and without a slab
flange contributing to the beam behaviour. Tests for T-shape sections were carried-out by Pipa
and Carvalho (Pipa (1993), Carvalho (1993)) at the “Laboratório Nacional de Engenharia
Civil” (LNEC), Lisbon, while those for rectangular sections were performed by König and
Heunish (Carvalho (1993)); a full description of specimen layouts, material characteristics,
testing setup and results can be found elsewhere (Carvalho (1993)).
4.3.2 Specimen characteristics and test description
Two basic section configurations were considered, depending on the reinforcement layout: the
section type S1, assumed representative of external frame beams, and the section type S2 for
the internal frame beams, both at the first storey of the referred building where beams in the
test direction were more heavily reinforced.
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 177
In the present work, only the section type S2 members are analysed, with both the rectangular
and the T-shape configurations. The respective section layout and the schematic structural rep-
resentation of members are shown in Figure 4.17; T-shape section beams are referenced as
LNEC-beams and rectangular ones are designated KH-beams.
Figure 4.17 Section layout S2 and schematic representation of tested cantilever beams
Specimens were designed and built-up at a 2/3 reduced scale and, for both beam configura-
tions, material characteristics were taken similar to those specified for the full-scale structure.
Reinforcing bars were B500S Tempcore steel, both in the longitudinal and transversal and slab
reinforcement, for which standard tension tests were made in order to evaluate the mechanical
characteristics. Concrete was taken of the class C25/30, using the same mix in LNEC and KH
beams for conformity of mechanical characteristics, which were evaluated by standard com-
pressive strength tests on 150 mm cubes for each cast operation.
From the available data (Pipa (1993) and Carvalho (1993)), average characteristics were
extracted and adopted in the numerical simulations as summarized in Table 4.1 (the same nota-
tion is used as in 4.2.3.2).
Concerning the steel, and stand for the tensile strength and to the strain at maximum
force, respectively; identical characteristics were used for longitudinal and transversal steel
due to their similarity.
0.10
1.00
a) LNEC beam section S2 b) KH beam section S2
0.02
0.03
0.20
F+
c) Cantilever
4φ12
2φ12
4φ6 4φ6
0.30y
z
(x)M+
0.02
0.03
0.20
4φ12
2φ12
0.30y
z
(x)M+
yz
x
1.50
0.15 0.15
M+ϕ+
lp1lp2
δ+
φ6 // 0.07
φ6 // 0.07
beam
fsm εsm
178 Chapter 4
For the concrete, both the Young modulus and the tensile strength were estimated from
the characteristic strength , by means of code expressions as used in 6.2.2 for the full scale
structure. Additionally, for KH beams, the values indicated with (*) were taken equal to those
for LNEC beams due to lack of test information. It is noteworthy that, with exception of , all
listed data refer to mean values obtained from the various material samples.
Tests have consisted of monotonic ones (in both loading directions) and of quasi-static cyclic
sequences of imposed displacements as follows:
• Monotonic tests, referenced as V1 and V2, respectively for the positive and negative direc-
tions (thus, inducing tensile and compressive strains in the bottom reinforcement, respec-
tively), were performed aiming at the assessment of ultimate displacements ( and ).
• Four cyclic tests, viz V3, V4, V5 and V6, were done by applying tip displacement
sequences whose peak values were normalized by the above referred ultimate displace-
ments. These loading sequences were defined for the purpose of evaluating the influence of
the imposed displacement level and the cyclic repetition effect on the specimen behaviour.
In the present work, both monotonic tests were numerically simulated, while only the V5 and
V6 cyclic ones were analysed because they were considered representative for the purpose of
cyclic tests. According to Carvalho (1993), slightly different loading sequences were used for
the two types of beams, which can be confirmed by inspection of Figure 4.18 where the cyclic
histories of imposed displacements are depicted in terms of the monotonic ultimate displace-
ments and (also included in the figure).
The numerical simulations were not performed over each complete sequence. Indeed, for very
large imposed displacements, buckling of reinforcement bars was triggered-off, which cannot
be simulated by the numerical model; therefore, from that stage on, the test development is
meaningless from the numerical simulation standpoint and it has not been included here (the
respective breakpoint is also indicated in Figure 4.18).
Table 4.1 Mechanical properties of steel and concrete of LNEC and KH beams
Steel Concrete
(MPa) (%) (MPa) (%) (%) (MPa) (%) (MPa) (MPa) (GPa)LNEC 550 0.27 615 9.8 0.34 35.0 0.2 27.8 31.3 2.8
KH 538 0.27 632 10.7 0.45 35.8 (*)0.2 (*)27.8 (*)31.3 (*)2.8
Ec fct
fck
fck
fsy εsy fsm εsm Esh Es⁄ fc0εc0
fck Ec fct
δu+ δu
-
δu+ δu
-
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 179
Note that, since for each test a different specimen was used, slight variations of material prop-
erties are obviously expectable; therefore, this has to be kept in mind when comparing experi-
mental results with numerical simulations, actually performed with identical average
characteristics for all tests.
Figure 4.18 Cyclic sequences (V5 and V6) of imposed displacements for LNEC and KH beams
Tests have provided results for the applied force F (measured by means of force transducers)
and for the imposed displacement , as well as for the upper and lower beam face displace-
ments in the potential plastic zone lengths, indicated in Figure 4.17-c) by and , which
were obtained by displacement transducers. The plastic zone lengths refer to , where is
the beam depth, in correspondence with the often accepted plastic zone spreading between
and .
From these measurements, tip force-displacement curves and moment - average curvature dia-
grams in the plastic zones were obtained and are used herein for comparison with numerical
results.
a) LNEC beams
b) KH beamsδ δu+⁄
δ δu-⁄
δ δu-⁄
δ δu+⁄
Cycle Cycle
CycleCycle
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
δu- 154mm–=
δu+ 107mm=
δu- 132mm–=
δu+ 109mm=
V5
V5
V6
V6
Not simulated Not simulated
Not simulated Not simulated
δ
lp1lp2
h 2⁄ h
h 2⁄ h
180 Chapter 4
4.3.3 Numerical simulations
For the numerical analyses subsequently presented, the material characteristics listed in Table
4.1 were adopted for use with the steel and concrete models described in 4.2.3.2. Additionally,
the following assumptions were made:
• uniform concrete cover of 2 cm and volumetric confinement ratio of 0.8%, obtained accord-
ing to the stirrup contents shown in Figures 4.17-a) and b);
• the slab reinforcement of LNEC beams is included in the top steel layer area, having the
same properties of the main reinforcement;
• centroids of both the top and the bottom reinforcement layers are located at 3 cm from the
nearest section face.
The moment-curvature skeleton curve for each section was obtained by the TRILIN operator
presented in to 4.2.3 and, for numerical simulation using the flexibility element model, a single
element was considered in correspondence with the schematic representation shown in Figure
4.17-c) and with uniform section characteristics.
The availability of experimental measurements of average curvature in the plastic zones has
suggested a preliminary comparison of the experimental moment-curvature relationships with
the numerical trilinear approximation ones as obtained by the TRILIN operator. Additionally,
this comparison was complemented with fibre analysis of the section, performed by means of
the same numerical tool as used for the section analysis included in 3.5.1.
The corresponding monotonic moment-curvature diagrams (for both loading directions) are
depicted in Figure 4.19 and compared against experimental curves obtained from V1 and V2
tests, for the plastic zone referenced by in Figure 4.17-c). The fibre (indeed, layer) discreti-
zations of sections are also included in Figure 4.19.
From these result comparisons, the following aspects are highlighted:
• There is good agreement between the trilinear approximation and the fibre analysis curves,
for the stages where such agreement is sought (viz for the turning points); naturally, outside
such stages, deviations do occur as is apparent in the immediate post-yielding range where
the hardening effect cannot be followed by that step-wise linear curve, unless a supplemen-
tary branch is considered.
lp1
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 181
Figure 4.19 Comparison of numerical and experimental monotonic M-ϕ curves for LNEC and
KH beam sections
• The approximation for yielding moments is quite acceptable; indeed, although it may
appear somewhat crude for the negative bending of the LNEC beam, it actually fits within
the scatter of the steel yielding stress obtained from material tests (a maximum deviation of
8% from the average value of can be found in the tested sample set (Carvalho (1993))).
• Instead, yielding curvatures are significantly underestimated by the numerical analyses,
which is due to the pull out effect caused by reinforcement slippage inside the anchorage
zone and included in the deformations measured in the first plastic zone . The effect is
extensively reported in Pipa (1993) as responsible for yielding displacements about 50%
.0 .10 .20 .30 .40 .0
10.0
20.0
30.0
40.0
50.0
.0 .10 .20 .30 .40 .0
10.0
20.0
30.0
40.0
50.0
.0 -.10 -.20 -.30 -.40
-100.0
-80.0
-60.0
-40.0
-20.0
.0 .0 -.10 -.20 -.30 -.40
-100.0
-80.0
-60.0
-40.0
-20.0
.0
EXPERIMENTAL FIBRE TRILINEAR
Curv.(m-1)
Mom.(kN.m) Mom.(kN.m)
Mom.(kN.m) Mom.(kN.m)
Curv.(m-1)
Curv.(m-1) Curv.(m-1)
a) LNEC beam section S2 b) KH beam section S2
M+
M –
M –
.0 .05 0
50
.025
Zoom
fsy
lp1( )
182 Chapter 4
higher than those expectable for no pull out contribution. For both beam types, the numeri-
cal result deviations from experimental values of are more apparent for the negative
direction than for the positive one (for in the LNEC beam case, a zoom view is pro-
vided up to the near-yielding zone, in order to confirm that deviation), which is related to
the different contents of tensioned steel bars inside the same width (25cm): for only
are engaged, while are resisting the applied moment , thus leading to
higher tensile force to be transferred to the concrete in poorer bond conditions (due to the
increased density of reinforcement) and, consequently, inducing larger slippage.
• The LNEC beam sections exhibit rather different post-yielding behaviour in the two bend-
ing directions due to the slab effect. For the positive direction, compressive stresses can
spread along the slab flange within a reduced thickness, meaning that the internal force
lever arm tends to increase for curvature values above . This effect, combined with the
strain hardening of steel, leads to the significant increase of resisting moment until crushing
and spalling of the unconfined concrete cover occurs; from then on, the internal resistance
starts to decrease and tends to the ultimate moment. By contrast, for , the increase of
steel force due to hardening is counterbalanced by the reduction of internal lever arm
caused by the greater neutral axis depth required to generate the compressive force neces-
sary to satisfy equilibrium; therefore, the hardening effect in almost vanishes and the
drop of resistance due to spalling becomes much less apparent, yet still perceptible. The
fibre analysis can approximately trace out these results although anticipating the effect of
concrete cover spalling, because, rather than assuming the typical post-yielding plateau, the
steel hardening is considered starting immediately after . Thus, in the fibre modelling,
forces developed in the yielded steel bars are higher than in the experiment, requiring larger
compressive forces in the concrete and leading to increased moments; this means that
crushing and spalling of concrete cover is triggered off (in the fibre analysis) for curvature
values lower than in the experiment, quite apparent for and also detectable for .
• On the contrary, the behaviour of KH beams is quite similar for both loading directions and,
besides the already referred deviations of yielding (and post-yielding) curvatures due to pull
out effects, the numerical analyses provide good approximations well within the scatter of
material properties. In comparison to LNEC beams, it is worth noting the significant differ-
ence of (because in KH beams there is no slab reinforcement contribution) and the
lower hardening effect for in KH beams (explained by reasons similar to those for the
LNEC beam behaviour in the negative direction).
ϕy
M+
M+
2φ12 4φ12 M -
ϕy+
M -
M -
εsy
M+ M -
M -
M+
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 183
The above comments allow to accept that, within the limitations inherent in the trilinear curve,
it actually provides a good approximation of the experimental moment-curvature relationship;
however, it is apparent that, for a better fitting of the post-yielding/pre-spalling behaviour, at
least one further branch would be suitable to enhance the adequacy of a multi-linear step wise
approximation, particularly if the slab contribution to the strength is to be duly accounted for.
Results of numerical simulations using the flexibility element for monotonic tests V1 and V2
are shown in Figure 4.20 together with the experimental ones.
Figure 4.20 LNEC and KH beams S2: monotonic tests V1 and V2
EXPERIMENTAL TRILINEAR
Force (kN)
Displ.(m)
a) LNEC beam S2 b) KH beam S2
.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
.0 .02 .04 .06 .08 .10 .12
.0
-10.0
-20.0
-30.0
-40.0
-50.0
-60.0
-70.0
.0 -.02 -.04 -.06 -.08 -.10 -.12 .0
-10.0
-20.0
-30.0
-40.0
-50.0
-60.0
-70.0
.0 -.02 -.04 -.06 -.08 -.10 -.12
.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
.0 .02 .04 .06 .08 .10 .12
Force (kN)
Displ.(m)
Force (kN)
Displ.(m)
Force (kN)
Displ.(m)
V1
V2
V1
V2 .0
-70.0
.0 -.01 -.02
Zoom
184 Chapter 4
From the comparison of numerical and experimental force-displacement curves for both
LNEC and KH beams (Figure 4.20), the following can be observed:
• Results agree reasonably well, with exception of the test V1 on the LNEC beam due to the
slab effect already explained.
• Yielding forces are well estimated, thus fully agreeing with what has been said for the M-ϕ
curves.
• The underestimation of yielding displacements, caused by the missing contribution of the
pull out effect, is quite apparent. To some extent, this fact does not allow an adequate com-
parison of the numerical and experimental post-cracking branches, which would be desira-
ble in order to find out the effect of the cracking plateau and the cracking development in
the global response. Nevertheless, a zoom view of the post-cracking stage is included for
the test V2 of the LNEC beam, from which it can be observed that, despite a somewhat
underestimated yielding force, the post-cracking branch would tend to the experimental one
if the pull out contribution were included.
The results of cyclic test simulations are shown in Figures 4.21 and 4.22, respectively for
LNEC and KH beams. Comparison with experimental diagrams is included, not only in terms
of force-displacements, but also for moment - average curvature in the first plastic zone, which
required the average curvature from numerical analysis to be obtained by integration in the
plastic length in order to have consistent results. Due to problems in the instrumentation
setup, part of the experimental results for the LNEC beam - test V6 were not available; the cor-
responding part is indicated in Figures 4.21-b) by a dashed line.
Results suggest the following comments:
• The lack of pull out contribution is still apparent at the yielding stage; however, for further
cycling, the less good simulation of this effect becomes less relevant in presence of other
phenomena such as the pinching due to reinforcement asymmetry and the unloading/reload-
ing stiffness deterioration.
• The discrepancy of positive post-yielding strength for the LNEC beam, so apparent in the
monotonic test V1, actually vanishes for cyclic loading. This is related with the loading and
yielding occurrence for , prior to yielding for , which has induced significant crack-
ing in the slab and yielding in the top reinforcement. In the subsequent cycles for reverse
load direction , the slab cracks could not close again (as observed in the experiment
(Pipa (1993))), which, combined with the Baushinger effect in the top steel, led to signifi-
lp1
M - M +
M +( )
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 185
cant curvature development upon yielding for and, consequently, the hardening effect
in the positive direction became less pronounced than for monotonic loading conditions.
Therefore, it can be concluded that, for cyclic conditions, even if the apparent hardening
(with significant contribution from the compressed slab participation) happens to develop
for the very first cycle with the slab under compression, for subsequent cycles, it vanishes
as a result of the described effects.
Figure 4.21 LNEC beam S2: tests V5 and V6
M +
a) Test V5: Force-displacement and Moment-Curvature diagrams
Force (kN)
Displ.(m)
Mom. (kN.m)
Curvat.(m-1) -.08 -.06 -.04 -.02 .0 .02 .04 .06
-70.0
-60.0
-50.0
-40.0
-30.0
-20.0
-10.0
.0
10.0
20.0
30.0
-.40 -.30 -.20 -.10 .00 .10 .20 -100.0
-80.0
-60.0
-40.0
-20.0
.0
20.0
40.0
b) Test V6: Force-displacement and Moment-Curvature diagrams
Instrumentation fault Instrumentation fault
Force (kN)
Displ.(m) -.08 -.06 -.04 -.02 .0 .02 .04 .06
-70.0
-60.0
-50.0
-40.0
-30.0
-20.0
-10.0
.0
10.0
20.0
30.0 Mom. (kN.m)
Curvat.(m-1) -.40 -.30 -.20 -.10 .00 .10 .20
-100.0
-80.0
-60.0
-40.0
-20.0
.0
20.0
40.0
V5
V6
EXPERIMENTAL TRILINEAR
186 Chapter 4
Figure 4.22 KH beam S2: tests V5 and V6
• As for the monotonic tests, the yielding strength is well estimated.
• For the level of involved deformations, the rule for unloading stiffness degradation seems to
be adequate.
• Reloading towards slab under compression is done with excessively reduced stiffness in the
numerical simulation (see Figures 4.21-a)). Indeed, the rule of aiming at the previous maxi-
mum point leads to strong deterioration of this reloading stiffness, at least for the first cycles
EXPERIMENTAL TRILINEAR
a) Test V5: Force-displacement and Moment-Curvature diagrams
b) Test V6: Force-displacement and Moment-Curvature diagrams
-50.0
-40.0
-30.0
-20.0
-10.0
.0
10.0
20.0
30.0
-80.0
-60.0
-40.0
-20.0
.0
20.0
40.0
-50.0
-40.0
-30.0
-20.0
-10.0
.0
10.0
20.0
30.0
-80.0
-60.0
-40.0
-20.0
.0
20.0
40.0
Force (kN)
Displ.(m)
Mom. (kN.m)
Curvat.(m-1) -.08 -.06 -.04 -.02 .0 .02 .04 .06 -.40 -.30 -.20 -.10 .00 .10 .20
Force (kN)
Displ.(m)
Mom. (kN.m)
Curvat.(m-1) -.08 -.06 -.04 -.02 .0 .02 .04 .06 -.40 -.30 -.20 -.10 .00 .10 .20
V5
V6
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 187
at a given deformation level; for subsequent cycles with the same displacement, the stiff-
ness tends to approach the experimental one as exhibited in Figures 4.21-b).
• The stiffness for reloading with pinching effect is also overly reduced. Actually, the pinch-
ing moment (i.e., that for the crack closure point) is well approximated but the correspond-
ing deformation appears over-estimated (see, for instance, Figures 4.21-a)), thus
excessively reducing the first reloading branch stiffness. Additionally, for larger displace-
ments (say 3% to 4% of the member length), the pinching effect can be just roughly cap-
tured by two straight lines, and, rather than closely following the force-deformation curve,
the approximation should preferably aim at compensation of loop areas (i.e., dissipated
energy).
• Strength degradation in the negative direction is strongly influenced by the pinching effect.
In fact, due to the excessively low reloading stiffness induced by pinching, the obtained
resistance for imposed deformation becomes lower than the experimental one (see Figures
4.21-b) and 4.22-b), thus, somehow overlapping the appearance of strength degradation.
Instead, for the positive direction, low degradation occurs and, where visible (ex. Figures
4.21-b)), the numerical results lead to less degradation than the experiment; this may be
related with, either the lower dissipated energy or a low value of the strength degradation
parameter, or even with both aspects.
Attention is drawn to the similar shape of force-displacement and moment-curvature diagrams;
indeed, such similarity was expected since the non-linear response is mainly controlled by that
plastic zone behaviour, but it means also that corrections on the reloading stiffness and pinch-
ing modelling at the section level will have straight correspondence at the global level. In other
words, should improvements be undertaken to have results closer to the experimental ones, it
appears to be sufficient that they are done in the section model.
4.3.4 Remarks on model validation results
The above presented results show that, overall, quite reasonable numerical simulations can be
obtained with the flexibility element formulation under monotonic and cyclic conditions.
However, deviations do occur, particularly concerning the cyclic loading cases, which are
related to limitations or less adequate rules of hysteretic behaviour at the section level, with
particular emphasis for the modelling of reloading stiffness, either with or without pinching
effect included.
188 Chapter 4
As already mentioned, the flexibility element formulation was developed herein having the
main requirement that the model be based on a trilinear skeleton curve and on multi-linear step
wise features for hysteretic behaviour simulation. However, no specific restrictions were put
for hysteretic rules, which means that improvements in the section model are perfectly compat-
ible with the flexibility formulation and may be sufficient to achieve results closer to the exper-
imental ones. Furthermore, it is reminded that the major concern of the present work is more
the global element formulation than the local section model refinement, yet keeping the formu-
lation open to new improvements at the section level. For these reasons and, despite the above
highlighted limitations, the presently adopted Takeda-type model was still kept for further cal-
culations. In this context, the present validation examples serve mainly for global assessment
of the model and for identifying the major limitations to bear in mind when analysing other
cases of complete structures in the following chapters.
Finally, it is noteworthy that the validation of a global element formulation is much harder,
cumbersome and even less complete than a section model validation. Actually, it is very diffi-
cult to predict and to generate the whole range of situations likely to develop in a given ele-
ment, because it is strongly dependent on several factors, viz the moment distribution, the
cracking and yielding development, the state of end sections which directly affect the corre-
sponding plastic zone state and rarely behave independently. Even more difficult is the availa-
bility of single elements or sub-assemblages experimentally tested that had gone through
several possible cases of the element state; typically, cantilever or simply supported members
are tested under simple loading conditions not likely to generate complex internal force distri-
butions and load reversals as often developed in members integrated in complete frame struc-
tures.
Therefore, part of the validation process had to be done by artificially generating various load-
ing conditions in single elements and by checking the output of numerical analyses for reason-
able results (examples: bilinear moment distributions, possibly reaching full-cracking of one or
both element parts; loading, unloading and reloading sequences affecting the plastic zone
development and aiming at generating cases as those included in Figures 3.23 and 3.24).
However, none of such tedious local validation analyses is included herein; instead, attention is
drawn to the validation at the global structure level as included in Chapter 5, which is deemed
more meaningful in the context of global element modelling.
NUMERICAL IMPLEMENTATION, AUXILIARY TOOLS AND VALIDATION 189
4.4 Conclusions
In this chapter the most significant tasks for implementation of the flexibility element were
addressed and a new algorithm was presented for the definition of trilinear approximations of
moment-curvature relationships. Additionally, several isolated members were analysed with
the developed formulation and implemented tools, as part of the validation process at the sin-
gle element level, and results were discussed in comparison with experimental ones.
Implementations were carried out in the computer code CASTEM2000, for which a prelimi-
nary review of the basic features was included and complemented with a simple but illustrative
example of structural analysis.
CASTEM2000 operators affected by the implementation of the flexibility formulation were
presented and briefly described; the major adaptations to accommodate the new element for-
mulation were generally reported.
For the definition of moment-curvature trilinear approximations, an auxiliary tool has been
implemented as a new CASTEM2000 operator, and based on an efficient algorithm for the
analysis of rectangular and T-shape reinforced concrete sections. It has been designed to over-
come the need of fibre discretization, yet accounting for realistic material behaviour models:
steel is assumed behaving bilinearly (with strain hardening) and concrete follows a parabolic/
linear-softening/residual-plateau model type, either in confined or unconfined conditions.
While cracking points are defined by closed-form expressions, the non-linear range turning
points (viz yielding and ultimate) are treated by a common process and the distinct features
reduce to the point definition criterion; such a process offers the additional possibility of con-
sidering supplementary points by simply providing adequate definition criteria.
In a restricted context of the validation of the flexibility element model using experimentally
tested specimens, rectangular and T-shape section cantilever beams were analysed by one flex-
ibility element, both in monotonic and in cyclic loading conditions. Results were compared
with experimental ones, showing that good numerical simulations can be obtained with the
present formulation. However, modelling limitations were found, which are related to the hys-
teretic rules at the section level, namely the reloading stiffness modelling, with or without the
pinching effect. It has been recognized that corrections for such limitations are essentially
localized in the section model and do not interfere with the element formulation. However,
190 Chapter 4
such corrections were not performed herein and, therefore, the above referred limitations have
to be taken into account in the numerical analyses of the next chapters.
Chapter 5
THE 4-STOREY FULL-SCALE BUILDING
TESTED AT ELSA
5.1 Introduction
The present chapter deals with the experimental research related to the four-storey full-scale
reinforced concrete building tested in the European Laboratory for Structural Assessment
(ELSA) of the Joint Research Centre (JRC) at Ispra, Italy.
The initiative was included in the framework of a Cooperative Research Programme on the
Seismic Response of Reinforced Concrete Structures (Carvalho (1991,1992,1993)) as part of
the activities of the European Association of Structural Mechanics Laboratories (EASML).
The work carried out by the so-called “Reinforced Concrete Working Group” of the EASML
started with the first phase (Carvalho (1991)) in which: a) similarities and differences were
identified between seismic design and analysis methods in different european countries and, b)
numerical studies were performed concerning the design of some buildings using the current
drafts of Eurocode 2 (1991) (EC2) and Eurocode 8 (1994) (EC8) and concerning also non-lin-
ear analysis with artificially generated accelerograms consistent with the EC8 elastic spectrum.
The objective of the second phase (Carvalho (1992,1993)) was further assessment of the EC8
and the definition of damage indicators and failure criteria for plastic zones of structural mem-
bers. Simultaneously, more emphasis was put on testing activity, both at the member and the
complete structure levels. In this line, a full-scale high-ductility structure was designed accord-
ing to EC2 and EC8, to be pseudo-dynamically tested in the ELSA laboratory in order to allow
a detailed comparison between the actual and the intended design behaviour.
192 Chapter 5
The specific features of the Pseudo-Dynamic (PSD) method, as a hybrid numerical-experimen-
tal testing procedure, are briefly described in 5.2, including references to time integration tech-
niques (a key issue in the PSD method) and to substructuring procedures in PSD tests.
Examples of PSD testing activity in the ELSA laboratory are also provided.
The main topics concerning the structure design and layout are given in 5.3, while the charac-
terization tests of concrete and steel properties are referred in 5.4, along with the reduced scale
member tests performed in complement of the full-scale ones.
The full-scale testing activity in the four-storey building is described in 5.5. After brief refer-
ences to the specific PSD test setup, the used input accelerogram and the preliminary tests for
dynamic characterization of the structure, the main emphasis is put on the results of the seismic
tests and the final cyclic tests. Seismic tests were performed for two levels of earthquake inten-
sity on the bare structure and additional tests were carried out for two mansonry infilled frame
configurations. Final cyclic tests were performed on the bare frame structure aiming at a near
failure stage.
Finally, the main conclusions of the whole experimental activity related to the four-storey
building are summarized in 5.6.
5.2 The Pseudo-Dynamic test method. An overview
5.2.1 General
The Pseudo-Dynamic method is a hybrid testing method which combines classical experimen-
tal techniques with on-line computer simulation of structural dynamic behaviour (Donea et al.
(1990)). Displacements are quasi-statically imposed to the structure, satisfying the discrete
equations of dynamic equilibrium at each time step, and the structure restoring forces are
experimentally measured.
Since the inertia forces are analytically modelled, in a PSD experiment the expected real veloc-
ities do not need to be reproduced as it is required in shaking table tests. Therefore, the demand
for hydraulic flow power is much lower, which allows for testing structures at a larger scale
than in shaking tables. This is quite adequate for the analysis of structures made out of materi-
als, such as reinforced concrete, whose behaviour is influenced by aggregate size, micro-crack-
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 193
ing, bond between concrete and steel, etc., which are phenomena difficult to simulate in small
scale models. Moreover, the time scale being substantially amplified (about two to three orders
of magnitude), it allows for a more detailed inspection of damage evolution during the test.
Also, the ability to use substructuring techniques in PSD testing (Buchet and Pegon (1994))
makes it very attractive for structures in which, some parts can be easily modelled numerically,
while other parts are simulated by a physical specimen, only where strictly necessary. This is
the case of bridges, where the non-linear structural response to horizontal seismic forces is
mainly controlled by the piers, while the deck (difficult to fit inside the laboratory) can be ade-
quately modelled by numerical tools. Thus, the piers are physically tested, the deck is numeri-
cally simulated and adequate interface conditions are taken into account.
On the other hand, the reduced velocity of load application may have some drawbacks. If the
structure constitutive materials are strain-rate sensitive, the restoring forces for a given dis-
placement level may actually differ from those in a real time scale experiment, such as shaking
table tests. Fortunately, this seems not to be the case of reinforced concrete for which the rate
sensitivity appears of reduced importance as reported by Gutierrez et al. (1993).
Additionally, the PSD testing of structures with distributed mass is not a straightforward task,
since the inertia forces also develop in a distributed manner and would demand a similar distri-
bution of actuators. Such a requirement cannot be practically met and, therefore, more elabo-
rated PSD testing techniques and further assumptions are needed for distributed mass systems;
by contrast, shaking table testing does not present this limitation. However, for building frame
structures this problem is meaningless because the mass is typically concentrated on the floors.
It turns out that these two testing techniques appear as complementary and the adequacy of
each of them to simulate the dynamic structural response has to be judged in accordance with
the specific type of structure and material under analysis.
5.2.2 Time integration techniques
In the PSD method the non-linear behaviour modelling is pursued by means of the measure-
ment of restoring forces due to imposed displacements. The external loading, usually ground
accelerations, is incrementally applied and the dynamic equilibrium equations are solved by an
adequate integration algorithm, the inertia and damping forces being analytically modelled.
194 Chapter 5
The use of step-by-step integration methods is recalled in Appendix E within the context of
numerical non-linear analysis for the integration of
(5.1)
which is achievable by considering additional expressions specific to the adopted method (see
Eqs. (E.5) and (E.6)). In the PSD algorithm, Eq. (5.1) is used together with such expressions
(to be discussed below), in order to perform the following tasks:
• computation of displacements according to the integration method expressions;
• application of to the structure and measurement of restoring forces in the actuators, lead-
ing to ;
• calculation of and using Eq. (5.1), along with other expressions specific to the inte-
gration method, and restart for the next step.
Time integration methods proposed for PSD testing consist of explicit, implicit and hybrid
algorithms. As referred in Appendix E, for the first method, any displacement prediction for a
given step is based only on the previous step response, while for the implicit algorithms it also
depends on the response of the step under analysis.
The explicit methods, although leading to good results, have the major drawback of condi-
tional stability. This means that the adopted time step must be smaller than a threshold related
with the highest frequency of vibration, in order to achieve stability. For certain structures this
implies a greater number of steps to adequately lower the time interval, which may increase the
errors due to testing control.
By contrast, implicit methods are unconditionally stable, which makes them rather attractive
for testing very stiff structures, or with many degrees of freedom (d.o.f.). However, for struc-
tures with non-linear behaviour, either the tangent stiffness matrix must be obtained, or an iter-
ative procedure for displacement correction has to be adopted. For this reason, implicit
algorithms have been considered not usable in the PSD context, since, on the one hand the
“tangent stiffness matrix” is very difficult to assess experimentally, and, on the other hand, due
to the dependence of structural deformations on the loading history, any corrective iterative
procedure would introduce spurious hysteretic cycles.
However, during the last years, procedures have been proposed to adopt implicit algorithms in
M ak⋅ C vk⋅ rk+ + qk=
dk
dk
rk
ak vk
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 195
PSD testing, mainly based, either on slight modifications at the experimental level, or in the
application of different numerical expressions for the displacements.
In the following sections, details are given on the use and implementation of classic explicit
and implicit algorithms, and references are made to other recent proposals consisting on hybrid
explicit-implicit techniques.
5.2.2.1 Newmark explicit algorithm
The Newmark family of time integration methods is described in Appendix E, according to
which the integration of Eq. (5.1) is based on Eqs. (E.5) and (E.6), re-written here for conven-
ience
(5.2)
(5.3)
The central difference method is obtained by choosing and ; in this case the
stability condition for such explicit algorithm is which must be satisfied for the
higher natural frequency associated with the adopted structural discretization.
The displacements are therefore given by
(5.4)
and the implementation of the central difference algorithm in the PSD method involves meas-
uring the restoring forces , followed by the computation of the acceleration and velocity
vectors by Eqs. (5.2) and (5.1), leading to
(5.5)
(5.6)
At this stage the response for is known and the algorithm proceeds to the next step.
vk vk 1– 1 γ–( )ak 1– γak+[ ]∆t+=
dk dk 1– vk 1– ∆t 12--- β–⎝ ⎠⎛ ⎞ ak 1– βak+ ∆t2+ +=
β 0= γ 1 2⁄=
0 ωn∆t 2≤ ≤
ωn
dk dk 1– ∆tvk 1–∆t2
2-------ak 1–+ +=
rk ak
vk
M ∆t2-----C+⎝ ⎠
⎛ ⎞ ak qk rk– C vk 1–∆t2-----ak 1–+⎝ ⎠
⎛ ⎞–=
vk vk 1–∆t2----- ak 1– ak+[ ]+=
tk
196 Chapter 5
In Figure 5.1 a schematic illustration helps to clarify the procedure for practical implementa-
tion of the explicit PSD method.
Figure 5.1 Implementation of the explicit PSD method.
Displacements are continuously applied following a pre-defined ramp from until .
The effectively applied displacements are obtained as a feed-back signal from the displace-
ment transducers and, if they do not match the desired displacements , the correction
is enforced as a new input signal for the actuators. When is satisfied, the
step is complete and information is sent to the main computer in order to process the next
step.
5.2.2.2 α-implicit algorithm
The impossibility of simultaneously achieving second order accuracy and numerical dissipa-
tion capabilities with the unconditionally stable implicit Newmark method is pointed out in
Appendix E.
In order to overcome this limitation, Hilber et al. (1977) proposed another implicit method by
introducing a new parameter in Eq. (5.1) which allows for numerical dissipation. Eqs. (5.2)
and (5.3) remain unchanged, whereas Eq. (5.1) becomes
(5.7)
which reduces to Eq. (5.1) when . According to the authors, if , and are taken as
Controller
dk
dk 1– tdk
Ramp Generator
ed
+ -x
dk
rk
Main Computer
Calculate:dk
ak
Control
Algorithm
Servo-Valve
Actuator
Structure
+
Compute:
Set:dk 1– dk=vk 1– vk=ak 1– ak=
vk
d dk 1– dk
x
d
e d x–= x dk=
tk
α
Mak 1 α+( )Cvk αCvk– 1 α+( )rk αrk 1––+ + 1 α+( )qk αqk 1––=
α 0= α β γ
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 197
(5.8)
an implicit, unconditionally stable and second order accurate algorithm with numerical dissi-
pation can be obtained.
In the PSD method context, the use of this integration algorithm has the major problem of dis-
placement dependency on the unknown accelerations . However, Thewalt and Mahin
(1991) proposed a solution for this problem by means of a slight modification on the testing
sequence, such that information already existing in the analogue data flow is used to apply the
implicit displacements .
In order to explain this technique, let Eq. (5.2) be substituted in Eq. (5.7) so that accelerations
can be written as follows
(5.9)
where
(5.10)
consist of terms independent of the response for . The velocity vector can be given by
(5.11)
and replacing Eq. (5.9) in Eq. (5.3) leads to
(5.12)
where
(5.13)
13---– α 0<≤ γ 1 2α–( )
2---------------------= β 1 α–( )2
4-------------------=
dk ak
dk
ak M11– qk α+ αrk 1– Cvk– 1 α+( )rk–+[ ]=
qk α+ 1 α+( )qk αqk 1––=
vk vk 1– 1 γ–( ) 1 α+( )∆tak 1–+=
M1 M γ 1 α+( )∆tC+=
tk
vk vk γ∆tak 1–+=
dk dk Brk–=
dk dk 1– ∆tvk 1–∆t2
2------- 1 2β–( )ak 1– ∆t2βM1
1– qk α+ αrk 1– Cvk–+( )+ + +=
B ∆t2βM11– 1 α+( )=
198 Chapter 5
Note that Eq. (5.12) is implicit on displacements since the restoring forces appearing on
the second term depend on . In a numerical analysis procedure, can be expressed in terms
of by means of the tangent stiffness matrix. In the PSD test context, Thewalt and Mahin
proposed the following procedure: the explicit portion of displacements is digitally com-
puted and the corresponding signal is sent to the controllers to be applied by actuators; the
remainder is calculated in analogue form using the feedback voltages from specimen
restoring forces which are continuously measured in the load cells.
The practical implementation of this technique is illustrated in Figure 5.2 and can be described
as follows: the target displacements are defined as and computed by the first expression of
Eqs. (5.13). While in the conventional explicit method is progressively applied in ramp
form until the actually achieved displacements reach , in this implicit method the driving
signal for the controller is now modified at the analogue level to be ,
where is the restoring force which is continuously measured and is constant as given by
the second expression of Eqs. (5.13). At the end of the step, i.e., when the corrective signal
vanishes, the measured displacements match corresponding to the implicit
ones.
Figure 5.2 Implementation of the implicit PSD method.
Attention must be drawn to the fact that displacements achieved at step completion depend
on the measured restoring forces and are unknown to the main computer beforehand. During
displacement application the information updating is kept at the analogue level and is not seen
dk rk
dk rk
dk
dk
Brk–( )
dk
dk
x dk
d x– e d x Br+( )–=
r B
e
x dk Brk–= dk
Controller
dk
dk 1– tdk
Ramp Generator
ed+
- x
dk
rk
Control
Algorithm
Structure
d
Br
+-
Servo-Valve
Actuator+
Main Computer
Calculate:dk
akCompute:
Set:dk 1– dk=vk 1– vk=ak 1– ak=
vk
dk
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 199
by the computer which makes error checking more difficult than in the explicit PSD method.
It is noted that this has been the first implicit technique successfully applied in the PSD testing
of structures with several d.o.f.. However it has been achieved through modifications at the
system hardware level, which tends to limit its use in laboratories traditionally only equipped
for the explicit PSD method.
The error checking and implementation difficulties of this implicit PSD method has steered the
research during the last decade towards mixed explicit-implicit algorithms. In the next para-
graphs, the main topics and advantages of these algorithms are highlighted.
5.2.2.3 Mixed explicit-implicit algorithms
The basic idea of mixed methods is that the simplicity of the explicit PSD version is retained
and combined with the unconditional stability of an implicit method.
A typical example of these methods is the Operator-Splitting (OS) method originally proposed
by Nakashima et al. (1990) and based in an algorithm developed by Hughes et al. (1979), in
which the restoring force vector is approximated by the sum of one elastic term with a nonlin-
ear one. For convenience Eq. (5.1) is re-written in the form
(5.14)
where is the elastic stiffness matrix, is the updated stiffness matrix (tangent or secant,
depending on whether Eq. (5.14) refers to incremental or to total quantities) and is a dis-
placement predictor based only on the previous step response. Denoting by the
restoring forces due only to the predictor , and comparing Eqs. (5.14) and (5.1), it can be
concluded that are estimated by and corrected by the difference between target and pre-
dicted displacements through the elastic stiffness.
Eq. (5.14) can be also re-arranged in the form
(5.15)
which suggests its application to the PSD method as follows:
• the elastic matrix is first obtained, either numerically or experimentally;
Mak Cvk Kdk Ke dk dk–( )+ + + qk=
Ke K
dk
rk Kdk=
dk
rk rk
Mak Cvk Kedk rk Kedk–( )+ + + qk=
200 Chapter 5
• the displacement predictor is applied to the structure and the corresponding restoring
forces are measured;
• the term inside brackets can be evaluated and rearranged in the second member, leading to
an implicit equation that can be solved numerically (because is known and constant) to
obtain the corrected displacements .
This method has been presented in the context of the Newmark family integration algorithms
and, in particular, for and the classical Newmark method (with trapezoi-
dal rule for acceleration) is recovered. According to the authors, the method is unconditionally
stable, as long as the structural non-linearity is of the softening type, and with an accuracy
equivalent to that of the classical Newmark method at lower response modes.
If numerical dissipation is to be included, this method can be also used together with the above
referred -implicit algorithm as proposed by Combescure and Pegon (1994) leading to the so-
called -OS.
Following the option of using the PSD only in the explicit form, Shing et al. (1990) proposed
another method in which a numerical iterative scheme is used to apply the implicit part of the
displacements, instead of the analogue correction due to the restoring forces as proposed by
Thewalt and Mahin. Such iterative procedure is based on the initial stiffness matrix and, in
order to prevent spurious hysteretic cycles, a reduction factor is used for displacements which
reduces the risk of “overshooting”, i.e., applying displacements larger than the desired ones.
Further details about this method are out of the scope of this work, but can be extensively
found in Shing et al. (1990), where the method accuracy is proved with an error-propagation
analysis and numerical dissipation is shown to be activated due to an error-correction method
provided in the process.
5.2.3 Substructuring in the PSD method
Due to the mixed numerical-experimental character of the PSD method it has been possible to
implement substructuring techniques, through which part of the structure is analytically mod-
elled, whereas the other part is physically simulated and tested.
Such improvement has considerably opened the field of the PSD method applications, namely
to the analysis of very large structures (for instance, bridges (Pinto et al. (1996))), to the possi-
dk
rk
Ke
dk
β 1 4⁄= γ 1 2⁄=
α
α
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 201
ble simulation of asynchronous motions along structure supports (Pegon (1996)) as well as
soil-structure interaction effects, etc. Although the substructuring technique is beyond the pur-
pose of this work, a brief reference is made to the main features involved.
The use of substructuring in PSD testing requires an adequate behaviour model for the numer-
ically simulated part and an efficient time integration technique. It is noted that the integration
method assumes an increased importance in the PSD testing with substructuring, since this
usually corresponds to large numbers of tested and modelled d.o.f.. To this aim the -OS
method mentioned in 5.2.2 combines the essential requirements in this type of PSD testing,
namely: i) the unconditional stability, ii) a selective numerical damping only for the highest
frequency modes and iii) the non-iterative feature for which the implementation simplicity of
the explicit central difference method is still valid.
During a PSD test with substructuring two processes are running in parallel, corresponding to
the numerical and to the tested substructures, between which a data flow is established at pre-
cise moments of each load step. For clarity sake the whole structure d.o.f. are divided into three
types: those of the numerical substructure (SS), those of the tested substructure (TT) and those
of the interface (ST) between both substructures.
The whole set of d.o.f. is statically condensed to ST+TT and the PSD test is run only over these
d.o.f. following the usual algorithm, provided that the influence of the above condensation is
adequately included in the matrices of the process (see Buchet and Pegon (1994)). In the pre-
dictor phase, the displacement vector is split into sub-vectors corresponding to the SS and to
the ST+TT d.o.f. and sent, respectively, to the numerical substructure and to the tested one. In
each of them the associated restoring forces are evaluated and sent back, together, to provide
information for the solution of the condensed equilibrium equation in the ST+TT d.o.f.. Once
this task is concluded the whole response can be updated for the step under analysis, both in
the numerical and the tested substructures, and the process is ready to proceed to the next step.
It is noted that, with such an implementation scheme, only slight modifications are necessary
in the explicit PSD method, which are mainly related with information interchange between
both substructures. This contributes to the implementation simplicity in laboratories tradition-
ally equipped for the PSD test without substructuring.
α
202 Chapter 5
5.2.4 Applications of the PSD method at the ELSA laboratory
Since 1992 the ELSA laboratory is performing PSD tests for the seismic assessment of struc-
tures, already covering a wide range of different structural systems.
The testing activity started with a full scale moment resisting steel frame designed with EC8
(Kakaliagos (1994)) which helped to fix the test setup and proved the laboratory capabilities
for PSD testing pointed out by Donea et al. (1996). Besides the global performance of the
structure a particular aspect under investigation was the behaviour of semi-rigid beam-column
joints; details about this tests can be found in Kakaliagos (1994).
The subsequent full-scale tests were performed in the four-storey reinforced concrete structure
the present chapter refers to. These tests are extensively described and discussed in the next
sections, for which no further reference is made herein.
A broad PSD-testing campaign was undertaken in large scale RC bridge specimens, in the
framework of an integrated European programme of pre-normative research in support of EC8,
aiming at providing background and improvement for analysis and design methods. The
bridges under study had three piers with height ranging from 7 to 21 m and identical rectangu-
lar hollow cross-sections and a continuous deck 200 m long. The specimens at 1:2.5 scale were
tested with the substructuring technique, using physical models for the piers and numerical
modelling for the deck.
Two main configurations of regular and irregular bridges were considered and tested initially
for fixed support conditions and synchronous motion, with an artificial accelerogram and aim-
ing at the assessment of the influence of irregularity on the seismic ductile behaviour of bridge
piers. Later, other tests were carried out on specimens with isolation/dissipation devices in
order to check the possibility of obtaining a more homogeneous ductility demand for the piers
and in the last testing phase an asynchronous input motion was simulated at the base of the
piers; detailed information about these tests can be found in Pinto et al. (1996).
The seismic behaviour of monumental structures is increasingly drawing the attention of
researchers due to the need of preserving the historical heritage. Along this trend-line, PSD
testing of representative models of monumental structures has already started at ELSA, of
which the first specimen was a large scale (1:3) model of the facade of the Palazzo Geracci in
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 203
Sicily, reproduced in the laboratory with masonry stone blocks similar to those of the original
structure. The tests were quite successful and allowed to check the ability of the PSD setup to
test structures with high initial stiffness; reports on this tests can be found in Anthoine (1997).
Another monumental stone masonry structure to be tested is a full-scale model of the São Vice-
nte de Fora Monastery in Lisbon, of which the cloisters were considered a representative part.
A set of four arches was built in the laboratory, also with masonry adequately chosen with sim-
ilar characteristics to that of the monastery, and special devices were required for the loading
system. A preliminary study involving numerical analysis and definition of the experimental
model is detailed in Pegon and Pinto (1996).
Other examples of specimens to be pseudo-dynamically tested are a four-storey composite
steel-concrete structure designed according to Eurocode 3 (1992) (EC3), Eurocode 4 (1992)
(EC4) and EC8, as well as a two-storey RC building to be infilled with panels of hollow brick
masonry in several configurations.
Besides the specific scope associated with each test, obvious additional advantages are
obtained concerning validation of numerical models. Numerical simulations are usually per-
formed to anticipate the structural behaviour of the specimens and quite often either new mod-
els are developed or existing ones are improved. Thus the output of these large-scale tests
provides an excellent means of calibration and assessment of numerical models.
5.3 Structure design and layout
The specimen under analysis consists of a full-scale RC frame structure, four storey high and
designed in accordance with EC2 and EC8. It is a high-ductility structure (according to EC8
classification) whose general layout is shown in Figure 5.3.
The structure is symmetric in the testing direction with two identical lateral frames and a
stronger central one. In the orthogonal direction it is asymmetric due to different span lengths,
leading to a slight irregularity, which was introduced to have a more realistic building and for
possible tests in this direction (Carvalho (1993)). External columns have 40cm x 40cm cross-
section, while the central one has 45cm x 45cm. All beams are 30cm wide and 45cm high and
the solid slab thickness is 15cm.
204 Chapter 5
Figure 5.3 General layout of the 4-storey RC building tested at ELSA (dimensions in metres)
The structure was cast in place with normal-weight concrete C25/30 as specified in EC2 and
B500S Tempcore steel rebars and welded meshes. In spite of violating the requirements of
EC8 for ductility class “High” structures, concerning the strain at failure and the ratio of tensile
failure strength to yield strength, this kind of steel was used since it is gaining market in
Europe (Carvalho (1993)). Strong arguments sustained its adoption in order to profit the oppor-
tunity of a large scale test to assess the adequacy of this steel for seismic resistant structures.
However, to some extent, such option may have prevented the main scope of the test to be
achieved, i.e., an experimental contribution for the assessment of EC8 design rules. Actually, it
seems more convenient to first judge about this material adequacy by means of sub-assem-
0.80
3.00
3.50
13.3
0
3.00
3.00
0.45
0.40 x 0.40
0.45 x 0.45
10.0
0
5.00
5.00
6.00 4.0010.00
0.15
0.40 x 0.40
0.45 x 0.45
0.40 x 0.40
Rea
ctio
n W
all
Testing Floor
TestingDirection
d(+)
Z
XY
Y
X Z
a) Elevation
b) Plan view
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 205
blages tests, since the majority of such tests which are available do not include this steel and
little experimental evidence exists about its behaviour (specially concerning eventual rein-
forcement slippage in the concrete core).
Design was performed for typical loads (additional dead load of 2.0 kN/m2 for finishing and
partitions and live load of 2.0 kN/m2) and for high seismicity, assuming a peak ground acceler-
ation of 0.30g, a soil type B and an importance factor of 1 (importance category III). Since the
regularity requirements are fulfilled, both in plan and elevation, the design was performed for
high regularity and no reduction in the behaviour factor is due. For the ductility class High this
yields a behaviour factor of 5.
Two independent planar models were used for the design analysis and the simplified method
prescribed in EC8 was used to account for torsion effects. More details about design, as well as
reinforcement layout drawings can be found in Negro et al. (1994).
5.4 Material properties and reduced scale member tests
The characterization of concrete has been done by means of compressive tests on 150 mm side
cubes (Negro et al. (1994)), leading to the mean values listed in Table 5.1 for each casting
operation. Estimates of the mean cylinder compressive and tensile strengths are also included,
having been obtained from the cube strength as mentioned later in section 6.2.2. Note that the
values are clearly higher than 33 MPa, representative of a C25/30 concrete mean compressive
strength, and show a significant scatter between casting phases.
Table 5.1 Mean cube (fcm,cub) and cylinder (fcm) compressive and tensile (fctm) strengths of concrete
Casting operation of: fcm,cub (MPa) fcm (MPa) fctm (MPa)1st storey Columns 49.8 44.8 3.67
1st storey Beams 56.4 51.4 4.09
2nd storey Columns 47.6 42.6 3.50
2nd storey Beams 53.2 48.2 3.87
3rd storey Columns 32.0 27.0 2.37
3rd storey Beams 47.2 42.2 3.48
4th storey Columns 46.3 41.3 3.43
4th storey Beams 42.1 37.1 3.14
206 Chapter 5
For the steel, tensile strength tests were carried out for several diameters, representative of
those used in the specimen construction, the available results being given in Table 5.2. In par-
ticular, concerning strains, the unique available information refers to the ultimate strain meas-
ured in a reference length of (where stands for the bar diameter).
In order to support the preparation of the full scale tests, a set of reduced scale (1:2/3) monot-
onic and cyclic tests was performed on 24 RC cantilever beams of 1.50m span. These tests
were performed at LNEC (Pipa (1993)) and KH (Carvalho (1993)) and have been already
referred and used in 4.3. The specimen cross-sections were taken representative of typical situ-
ations in the whole four-storey structure, namely one type of sections for the external frame
beams and another type for the internal frame (Carvalho (1993)).
Two main issues were envisaged:
• Evaluation of the slab influence in the response of beams, for which the whole specimen set
was divided into one part with T-shape beams (tested at LNEC) in order to include a contri-
bution from the slab, and another part having identical cross-sections but with no flange
included (tested at KH).
• Investigation of the B500S Tempcore steel adequacy in the cyclic behaviour of reinforced
concrete elements, in view of its lower ductile and hardening capacities comparatively to
traditional hot-rolled steel (reductions of about 20% and 10% on ductility and hardening,
respectively, are reported by Pipa (1993)).
According to Carvalho (1993) and Pipa (1993) the main results of these tests can be summa-
rized as follows:
Table 5.2 Mean tensile properties of steel
Diameter(mm)
Area(mm2)
Yielding Stress(MPa)
Ultimate Stress(MPa)
Ultimate Strain(%)
6 29.2 566.0 633.5 23.5
8 51.4 572.5 636.1 22.3
10 80.3 545.5 618.8 27.5
12 113.1 589.7 689.4 23.0
14 153.3 583.2 667.4 22.7
16 199.2 595.7 681.0 20.6
20 310.0 553.5 660.0 23.1
26 517.2 555.6 657.3 21.6
5φ φ
5φ[ ]
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 207
• Besides the expected increase in strength at the cracking stage, the influence of the slab
width in the T-shape beams was also found at failure due to the increased section asymme-
try, which was quite apparent in the more “pinched” force-displacement diagrams of LNEC
test results comparatively to those from KH. Actually, failure was induced by the buckling
of bottom reinforcement bars after having been subjected to large strain variations, to which
the slab contribution was twofold: i) an increase of tensile strains at the bottom bars, due to
a larger but less deep area of compressed concrete and, ii) a higher strength when the top
bars are tensioned, due to the presence of slab reinforcement, which also increases the com-
pressive forces in the bottom bars and, thus, their tendency to buckle.
• The lower ductile and hardening capacities of the B500S Tempcore steel did not appear to
affect the available ductility of the specimens. Indeed, on the one hand the failure was trig-
gered mostly by buckling phenomena rather than the full exploitation of the steel ductile
capacity, and, on the other hand, the obtained plastic hinge length (mainly controlled by the
steel hardening) seemed to compare well with results of other experiments made with steel
having higher hardening capacity. However, the higher strength of the used steel indirectly
affects the greater tendency of bars to buckle, since, for the same tensile force, bars with
smaller diameter can be used.
• Quite evident pull-out effects were found due to bar slippage from the footing, contributing
to an increase of about 40% to 50% for displacements measured at yielding, relatively to
those expected if no pull-out had occurred. It must be pointed out that, the stronger the steel,
the greater the tendency of pull-out phenomena to occur, due to a lower lateral surface area
of the bars for the same tensile force to be transmitted to the concrete. Therefore, RC mem-
bers made with B500S Tempcore steel are, expectably, more prone to pull-out than those
made with lower yielding strength steel.
The overall conclusion from these tests highlighted the importance of taking the slab contribu-
tion into account in the beam response and sustained the compatibility of B500S Tempcore
steel with the EC8 requirements for ductile design of earthquake resistant structures; however,
the lack of more experimental evidence on the latter topic still recommends further research on
issues such as the anchorage conditions and the buckling phenomena.
5.5 Full-scale tests
The experimental programme related with the four-storey RC structure consisted on the fol-
208 Chapter 5
lowing phases (Negro et al. (1994), Negro et al. (1995)):
• Preliminary tests aiming at the dynamic characterization of the structure and the calibration
of the pseudo-dynamic test setup before the main non-linear tests.
• Seismic tests in the bare structure performed first for a low intensity level and then for a
high level of intensity (respectively, for 0.4 and 1.5 times the reference earthquake).
• Seismic tests on the structure with the external frames infilled with unreinforced masonry,
first in the totally infilled configuration and then without any infills on the first storey in
order to simulate a soft storey condition; both tests were performed for the high level inten-
sity (1.5).
• Final cyclic test which was carried out quasi-statically, again on the bare structure, for three
levels of increasing top-displacement, in order to assess the structure behaviour at a near-
failure stage.
All these tests were performed in the direction indicated in Figure 5.3, where the positive
direction convention for displacements is also indicated. Between each test the structure was
unloaded to zero force, since the actuators had to be removed before a new testing phase.
Details are given in the next paragraphs for each testing phase. The main global results are pre-
sented and commented, after brief references to the test setup and the input accelerogram. The
following abbreviations are adopted to identify the tests: 0.4S7 (or low level) and 1.5S7 (or
high level), standing for the low and the high level seismic tests on the bare frame, uniform
and soft-storey for the two configurations of masonry infilled frame tests and Duct.3, Duct.5
and Duct.8 for the three phases of the final cyclic test (as detailed in 5.5.6).
5.5.1 PSD test setup
The kinematic degrees of freedom considered for the PSD tests consisted of four slab horizon-
tal displacements in the testing direction, which are sufficient to adequately describe the struc-
ture behaviour for lateral actions.
No rotational d.o.f. in the floor plan, nor translational ones in the transverse direction were
considered in the equations of motion. Therefore, these d.o.f. had to be restrained in the exper-
imental setup, which has been achieved by the following:
• the storey forces necessary to drive the structure were provided by means of a pair of digit-
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 209
ally servo-controlled hydraulic actuators per floor (attached to the external frames) which
imposed the same displacement and thus restrained the floor rotation;
• in order to prevent translation in the transversal direction, another actuator was horizontally
placed in the third floor, parallel to the reaction wall, to which a zero displacement condi-
tion was imposed during the test.
Picture 5.1 shows a lateral view of the structure including the main actuator setup and the steel
reference frames for displacement measurements.
Picture 5.1 Lateral view of the structure with actuator and reference frame setup
Forces were measured by load cells placed at the end of each actuator piston rod. Piston con-
trol was done by an optical digital encoding device, through which storey-slab displacements
were measured relative to the steel reference frames mounted on the testing platform.
In correspondence with the four d.o.f., a lumped mass matrix was considered. In order to
account for the additional dead loads and factorized live loads, supposed to act simultaneously
with seismic forces, additional masses were attached to each floor (namely 24.3 ton for the
first three storeys and 26.1 ton for the top storey) and included in the mass matrix.
210 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 211
Since only four d.o.f. are present, not very stringent constraints are imposed to the integration
time step. Therefore the central difference method was adopted for PSD testing with a time
interval of 4 ms and 2 ms, respectively, for the bare frame and for the infilled frame tests.
The viscous damping factor was set to zero, since the energy in earthquake-like vibration is
essentially dissipated by hysteresis which comes automatically included in the response of the
physically tested specimen.
Details on the whole instrumentation setup can be found in Negro et al. (1994), but the main
types of measurements are briefly referred herein for completeness:
• Displacements and restoring forces of the four storeys.
• Total rotations within the potential plastic hinge zones (critical zones), at the ends of all the
beams and at the base level of columns; in beams the measurement base length was 450 mm
(the full depth of beams) and in columns two base lengths were adopted, namely 225 mm
and 450 mm.
• Distribution of rotations inside the critical zones of the second storey beams, by means of
measurements on three base lengths: 60 mm, 225 mm and 450 mm (measured from the side
of columns).
• Deformations of the top face of the slab in the testing direction, along the transversal beams
of the second storey, in order to estimate the slab contribution to the stiffness and strength of
the frames.
5.5.2 The input accelerogram
A set of ten artificial accelerograms were generated by Pinto and Pegon (1991), fitting the EC8
response spectrum for soil type B and a damping factor of 5%, for which a peak ground accel-
eration of 0.30g and 10s of duration was prescribed. Out of these artificial signals, modulated
by waveforms identified from the 1976 Friuli earthquake, two were selected for preliminary
non-linear analyses carried out by several members of the EASML (Carvalho (1993), Negro et
al. (1994)). From such analyses, one signal was indicated by all members as the most demand-
ing and, therefore, it was chosen as the reference signal for the experimental seismic tests. Des-
ignated by S7, the adopted accelerogram is shown in Figure 5.4 together with its elastic
response spectrum, which is compared with the reference response spectrum of EC8 (soil type
B and 5% damping).
∆t
212 Chapter 5
Figure 5.4 Input accelerogram (Friuli-like) and elastic response spectra
For both experimental and numerical analyses described in the present and the following chap-
ters, two levels of seismic action intensity were considered:
• the low level, corresponding to 0.4 times the design acceleration (thus, 0.12g), was chosen
to approximately represent the seismic event within the serviceability state as established in
paragraph 4.3.2 of Part 1-2 of EC8;
• the high level, associated with a peak ground acceleration of 1.5 times the design one (thus,
0.45g), was intended to induce net inelastic effects on the structure.
It is worth stressing that the intensity factor was set up to scale the design peak ground acceler-
ation (0.30g); however, because the peak acceleration of the reference signal is actually 0.38g
(thus, about 27% higher than the design acceleration), the resulting peak accelerations for the
low and high level tests reach 0.15g and 0.57g, respectively. Therefore, if test results are to be
compared with design values (e.g. for base-shear), one has to bear in mind such discrepancy
between the design assumption and the actually adopted peak ground acceleration.
Time (s)
Ground acceleration[g]
Period (s)
Spectral acceleration [g]
0.0 0.4 0.8 1.2 1.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
EC8 Spect.
2.0
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -.40 -.32 -.24 -.16 -.08 .00 .08 .16 .24 .32 .40
a) Artificially generated accelerogram S7
b) Elastic response and EC8 spectra
Max.: 0.38
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 213
5.5.3 Preliminary tests
The following different types of preliminary tests were performed: dynamic snap-back tests,
their PSD simulation and direct stiffness tests.
From the dynamic snap-back tests the main initial frequencies, mode shapes and damping fac-
tor were envisaged. They were performed by pulling the structure towards the reaction wall by
means of a steel bar designed to break at a pre-defined load which would not introduced signif-
icant cracking in the structure. Two of these tests were carried out: one by pulling the top sto-
rey and the other by applying the force on the third storey, so that the four main vibration
modes could be captured. The free vibration time histories of storey displacements allowed to
obtain frequencies from the corresponding power spectra and the average damping ratio was
estimated as 1.8%.
The dynamic snap-back tests were simulated by means of PSD testing; the main objective was
to assess the PSD implementation algorithm. A null damping factor was considered for the vis-
cous damping forces in these PSD simulations. The results compared reasonably well (Negro
et al. (1994)) and even helped to detect a fault in the acquisition system during the dynamic
testing. The time histories of storey displacements from the PSD simulation led to an average
damping ratio of 2.2%, slightly higher than the previous one.
The direct stiffness tests were performed by displacing each floor by a prescribed value (small
enough in order to avoid significant cracking) while the others remained fixed. The restoring
force measurements led to the estimation of the stiffness matrix condensed to the four degrees
of freedom (one per floor). Little asymmetry on the stiffness matrix was found, which is una-
voidable since some stiffness degradation always occurs, even if negligible, between each test-
ing step. Using an estimate of the lumped mass matrix and the experimentally obtained
stiffness matrix, the vibration frequencies were calculated (Negro et al. (1994)) and are sum-
marized in Table 5.3 along with those obtained from the snap-back tests.
Frequency values for the various tests compare well and the fact that values are decreasing
from the dynamic test to the stiffness one, can be related with the cracking progression and
possibly with the adopted mass estimate, which is used to obtain the stiffness test frequencies
but not in the dynamic test. The same reason might explain the difference of damping ratios
given above.
214 Chapter 5
It is noteworthy that the PSD simulation without viscous damping forces in the dynamic equi-
librium equations, led to energy dissipation similar to that of the dynamic test. It follows that
dissipation had to be due to the hysteretic restoring forces measured during the PSD simula-
tion, because viscous damping forces were not included. This comparison of dynamic with
PSD simulation of snap-back tests confirms the hysteretic nature of structural damping.
5.5.4 Seismic tests on the bare frame structure
After the preliminary tests the structure was subjected to seismic tests of two intensity levels
whose main results are presented and commented in this section. Results of both tests are
shown together for easier comparison between them (see Figures 5.5, 5.6 and 5.7); generally,
the left-most side stands for the low level test and the right-most side to the high level test.
Figure 5.5 includes the time histories of storey displacements, of relative inter-storey drifts
(difference of successive storey displacements divided by the inter-storey height) and of total
inter-storey shear (the total horizontal force immediately below each floor level). The storey
profiles of peak values of those time histories are also included in the same figure. The storey
shear-drift diagrams, relating the inter-storey shear with the relative inter-storey drift, are
shown in Figure 5.6. The curves of dissipated energy at each storey, which are equal to the area
enclosed by the shear-drift diagrams multiplied by the storey height, are depicted in Figure 5.7,
also including the base shear - top displacement diagrams.
5.5.4.1 Low level test
For the low level test, intended to correspond to a serviceability limit state, neither significant
damage was observed nor apparent yielding seems to have occurred. In spite of the non-availa-
bility of internal force-deformation relationships to check the local section behaviour, it can be
roughly assessed by the storey shear-drift diagrams shown in Figure 5.6-a).
Table 5.3 Frequencies (Hz) for all testing cases.
Mode DynamicSnap-Back
PSDSnap-Back
StiffnessTest
1 1.90 1.85 1.782 5.95 5.54 5.123 10.40 9.94 8.654 16.30 13.5 12.00
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 215
Figu
re 5
.5Ti
me
hist
orie
s of
sto
rey
disp
lace
men
ts, r
elat
ive
inte
r-st
orey
drif
t, to
tal s
tore
y-sh
ear a
nd re
spec
tive
peak
val
ue p
rofil
es fo
r
Low
and
Hi g
h le
vel t
estsTi
me
(s)
DIS
PL.(m
)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-.1
0
-.0
8
-.0
6
-.0
4
-.0
2
.0
.02
.04
.06
.08
.10
Tim
e (s
)
DIS
PL.(m
)
-.2
5
-.2
0
-.1
5
-.1
0
-.0
5
.0
.05
.10
.15
.20
.25
DIS
PL. (
m)
.0
.05
.10
.15
.20
.25
1
2
3
4
Low
Hig
h
STO
REY
21
34
Stor
ey:
DRI
FT (%
)
-1.0
-
.8
-.6
-
.4
-.2
.0
.2
.4
.6
.8
1.0
D
RIFT
(%)
-2.5
-2
.0
-1.5
-1
.0
-.5
.0
.5
1
.0
1.5
2
.0
2.5
.0
.50
1
.0
1.5
0
2.0
2
.50
1
2
3
4
Test:
21
34
Stor
ey:
STO
REY
Tim
e (s
)
SHEA
R (k
N)
-6.0
-4
.8
-3.6
-2
.4
-1.2
.0
1.2
2
.4
3.6
4
.8
6.0
x1
.E2
Tim
e (s
)
SHEA
R (k
N)
-1.5
-1
.2
-.9
-
.6
-.3
.0
.3
.6
.9
1
.2
1.5
x1
.E3
SHEA
R (x
1.E3
kN
)
.0
.3
0
.6
0
.9
0
1.2
0
1.5
0
1
2
3
4 ST
ORE
Y
Tim
e (s
)
Tim
e (s
)
DRI
FT (%
)
.0
.8
1.6
2.
4 3
.2
4.0
4.8
5.
6 6
.4
7.2
8.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
a) T
ime
hist
orie
s - L
ow le
vel
c) T
ime
hist
orie
s - H
igh
leve
l b
) Pea
k va
lues
216 Chapter 5
Figure 5.6 Shear-drift diagrams at each storey, for Low and High level tests
DRIFT (%)
SHEAR (kN)
-1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 -6.0 -4.8 -3.6 -2.4 -1.2 .0 1.2 2.4 3.6 4.8 6.0 x1.E2
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5 -1.2 -.9 -.6 -.3 .0 .3 .6 .9 1.2 1.5
DRIFT (%)
SHEAR (kN) x1.E3
DRIFT (%)
SHEAR (kN)
-1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 -6.0 -4.8 -3.6 -2.4 -1.2 .0 1.2 2.4 3.6 4.8 6.0 x1.E2
DRIFT (%)
SHEAR (kN)
-1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 -6.0 -4.8 -3.6 -2.4 -1.2 .0 1.2 2.4 3.6 4.8 6.0 x1.E2
DRIFT (%)
SHEAR (kN)
-1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 -6.0 -4.8 -3.6 -2.4 -1.2 .0 1.2 2.4 3.6 4.8 6.0 x1.E2
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5 -1.2 -.9 -.6 -.3 .0 .3 .6 .9 1.2 1.5
DRIFT (%)
SHEAR (kN) x1.E3
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5 -1.2 -.9 -.6 -.3 .0 .3 .6 .9 1.2 1.5
DRIFT (%)
SHEAR (kN) x1.E3
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5 -1.2 -.9 -.6 -.3 .0 .3 .6 .9 1.2 1.5
DRIFT (%)
SHEAR (kN) x1.E3
a) Low level b) High level
Storey4
Storey3
Storey1
Storey2
Low levelenvelope
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 217
Figu
re 5
.7Ti
me
hist
orie
s of
dis
sipa
ted
ener
gy a
nd b
ase
shea
r - to
p di
spla
cem
ent d
iagr
ams
for L
ow a
nd H
igh
leve
l tes
ts
Tim
e (s
)
ENER
GY
(kJ)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
3.5
7.0
10.
5
14.
0
17.
5
21.
0
24.
5
28.
0
31.
5
35.
0
Tim
e (s
)
ENER
GY
(kJ)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.3
5
.7
0
1.0
5
1.4
0
1.7
5
2.1
0
2.4
5
2.8
0
3.1
5
3.5
0
x1.E
2
STO
REY-
1
STO
REY-
2
STO
REY-
3
STO
REY-
4
TOTA
L
Low
leve
lH
igh
leve
l
TOP
DIS
PL.(m
)
BASE
SH
EAR
(kN
)
-.1
0 -
.08
-.0
6 -
.04
-.0
2
.0
.02
.0
4
.06
.0
8
.10
-6
.0
-4.8
-3
.6
-2.4
-1
.2
.0
1
.2
2.4
3
.6
4.8
6
.0
x1.E
2
TOP
DIS
PL.(m
)
BASE
SH
EAR
(kN)
-.2
5 -
.20
-.1
5 -
.10
-.0
5
.0
.05
.1
0
.15
.2
0
.25
-1
.5
-1.2
-
.9
-.6
-
.3
.0
.3
.6
.9
1.2
1
.5
x1.E
3
a) E
nerg
y di
agra
ms
b) B
ase
shea
r -
diag
ram
s
top
disp
lace
men
t
Low
leve
len
velo
pe
218 Chapter 5
For the low level test it can be seen that a major stiffness variation occurs in those diagrams,
particularly for the storeys 1, 2 and 3, corresponding to the extensive cracking produced by the
first significant displacement peak occurred around 2.2 seconds. A clear stiffness drop is
induced towards that peak and temporary large residual displacements took place after unload-
ing, which were due to member sections only partially cracked for one bending direction. After
a significant load reversal, these sections also cracked in the other bending direction and the
global section stiffness was reduced to the fully-cracked one; further unloading followed this
cracked stiffness and led to low residual deformations.
The early stage of damage corresponding to the low level test can be confirmed by comparing
the shear-drift diagrams between both level tests. For this purpose the envelopes of the low
level test diagrams are overprinted with thicker line in the corresponding high level test dia-
grams, in order to highlight the different global stiffness between the two tests. Additionally,
the amount of dissipated energy, also related to the damage state of the structure, is much lower
in the low level test, as can be qualitatively seen by the different size of shear-drift loops and
quantitatively checked in Figure 5.7-a). Note that the total final dissipated energy in the low
level test is at least one order of magnitude lower than the high level one.
Although not experimentally measured, the first mode frequency after the low level test can be
estimated from the last displacement cycles as 1.27 Hz. Figure 5.5-b) shows that the top dis-
placement reached the peak value of 3.7 cm (i.e. a total drift of 0.30%), the maximum inter-
story drift was 0.36% in the second storey and the peak base-shear was 583 kN.
Since these results arise from a seismic input approximately corresponding to the serviceability
state, it is apparent that the structure verifies the inter-storey drift requirement associated with
the EC8 serviceability limit state (paragraph 4.3.2 of Part 1-2) which is 0.4% for structures
“having non-structural elements of brittle materials attached to the structure” and 0.6% in case
such elements do not interfere with structural deformations.
Note that the maximum inter-storey drift occurs at the second storey due to the frame type
deformed shape. The first storey columns are restrained at the bottom by the foundation, while
the displacements of second storey columns mainly depend on the stiffness of the joints and
the framing beams. Thus, the more flexible boundary conditions at the second storey columns
lead to higher inter-storey drift, even for a shear force lower than for the first storey.
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 219
From the overall inspection of the structure after the test, no significant permanent cracks were
visible apart from those detected before the test and possibly due to shrinkage effects; this
aspect sustains the small residual deformations as stated above.
5.5.4.2 High level test
The reference accelerogram was scaled by a 1.5 factor for the high level test, which was
thought to apparently damage the structure. Instead, this was not the case, since not very exten-
sive and permanent damage could be found from the post-test inspection.
However, during the test, clearly visible cracks opened for the maxima deformed shapes of the
structure, in the critical regions (member end zones) of the three first storey beams and of sev-
eral columns (mostly of the first storey), and in the two first storey beam-column joints.
The cracking pattern of beams consisted of one major crack at the beam-column interface,
which remained permanently open after the test, and several other cracks, less open and with
spacing increasing towards the mid-span. These cracks were inclined, due to internal shear
forces, with the inclination angle reducing for increasing distance from the end sections. In the
columns, cracks appeared more visible at the base level, with a pattern similar to that of beams.
In the beam-column joints of the first and second storey diagonal cracking patterns clearly
developed, showing that a diagonal strut mechanism was activated to transfer forces across the
joint.
Note that a very low dissipation capacity of the structure can be detected in the shear-drift dia-
grams shown in Figure 5.6-b). A detailed analysis of such diagrams showed that some dissipa-
tion effectively occurred until the maximum drift in each direction was reached (approximately
between 3 s and 4 s), after which the cycle diagrams became very pinched for both loading
directions. This aspect is illustrated in Figure 5.8 where, for the second storey, the shear-drift
diagram was split into two parts: one until 4 s of the test duration and another from that instant
till the end. Note that this drop in dissipation capacity is also apparent from the dissipated
energy curves shown in Figure 5.7-a), where the energy increase rate clearly reduces after 4 s.
Development of permanent major cracks at beam-column interfaces suggests that yielding of
rebars took place locally and the steel-concrete bond inside the joint core might have been seri-
ously damaged, leading to significant bar-slippage. This contributes to a stiffness reduction
220 Chapter 5
while cracks are totally open across the whole section depth, during the unloading-reloading
process; after crack closure, internal forces can be transferred by a main diagonal strut, for
which bond behaviour is not relevant, and stiffness increases again. As a result, force-displace-
ment diagrams show the pinched shape apparent in Figure 5.8-b).
Figure 5.8 Detail of the 2nd storey experimental shear-drift diagram for the high level test
Besides the described cracking pattern and the local yielding of bars at the beam-column inter-
faces, no other apparent signs of serious damage could be seen in the structure, namely spalling
of concrete cover or buckling of reinforcement.
By means of a stiffness test performed after the high level test, new vibration modes were com-
puted leading to the following frequencies (Hz): 0.82, 2.79, 5.19 and 7.34. The mode shapes
did not differ too much from the virgin structure ones, but the drop of frequency values is quite
significant. Note that the fundamental frequency became less than 50% of the initial value,
which means a reduction of the corresponding modal stiffness to about 20% of the value before
the low level test. This fact sustains the occurrence of non-linear phenomena clearly beyond
the cracking phase.
The peak value of top displacement was 21.2 cm (thus, a total drift of 1.70%), while the inter-
storey drift and base shear were 2.41% and 1435 kN, respectively. The maximum inter-storey
drift still occurred at the second level and, although not meaningful as a limit state verification,
it is worth noting that it is far beyond the EC8 thresholds for the serviceability limit state.
From the rotation measurements in the critical zones, the spatial distribution of the correspond-
ing peak values was plotted as shown in Figure 5.9, where the maximum rotation obtained is
14.9 mRad, while for the low level test it was 2.51 mRad.
-8.0 0.0 8.0 -1.5
-.9
.0
.9
1.5 x1.E3
DRIFT (cm)
-8.0 0.0 8.0 -1.5
-.9
.0
.9
1.5 x1.E3SHEAR (kN) SHEAR (kN)
DRIFT (cm)
a) From 0 to 4 s b) After 4 s
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 221
Figure 5.9 Spatial distribution of rotations in the critical zones, for the high level test.
A good and almost uniform distribution of rotations was obtained among the beams of the
three first storeys, whilst the top storey beams show little contribution to the global deforma-
tion as can be confirmed in the dissipated energy curves of Figure 5.7. Note that, according to
5.5.1, for the critical regions of the second storey, rotations are split into two parts correspond-
ing to the two half lengths of each region.
Although useful for assessing the overall distribution, the results of Figure 5.9 do not help to
identify the mechanism of deformation developed, due to the lack of rotation measurements
for the columns. Since these measurements were made only at the expected plastic hinge loca-
tions, namely beam and base column ends, it is difficult to check, only on the basis of these
results, the effective formation of a weak beam - strong column mechanism.
5.5.5 Infilled frame seismic tests
Aiming at the study of the influence of infill panels on the behaviour of frame structures, two
pseudo-dynamic tests were performed with different infill lay-out. After the testing phase on
the bare structure, the external frames were infilled with light unreinforced masonry, first in a
regular configuration (uniformly infilled) and then in an irregular one, in which only the three
upper floors where infilled, simulating a structure with a “soft-storey”.
No repair was carried out before these tests because the damage induced by the previous ones
appeared to be relatively low. On the other hand, only high level intensity tests were per-
formed, because the fragile behaviour of mortar was not compatible with a low level test; the
same input signal was used as for the 1.5S7 test.
Experimental Rotations (mRad) - Max. = 14.9
Internal Frame External Frame
222 Chapter 5
The infill panels were made with typical materials, namely hollow ceramic bricks having the
dimensions of 245 x 112 x 190 (h) mm with vertical holes and mortar with average compres-
sive strength of 5 MPa. Compressive tests were performed on the blocks, parallel and orthogo-
nal to the holes, and small panel specimens were also tested to obtain the mechanical
properties of the masonry in three directions: the strongest one (parallel to the holes), the
weakest one (orthogonal to the first) and the diagonal one. From these tests, the compressive
strengths and the Young modulus in the perpendicular directions, as well as the shear modulus,
can be obtained. Details about these values can be found in Negro et al. (1995) and Combes-
cure (1996).
The infilled frame test results are briefly presented and commented in the following paragraphs
and, as before, results for the regular and the irregular configuration are displayed side by side,
the leftmost corresponding to the uniform configuration and the rightmost to the soft-storey
one. Time histories of storey displacements, relative inter-storey drifts and total inter-storey
shear are shown in Figure 5.10, which also includes the corresponding storey profiles of peak
values. The storey shear-drift diagrams are depicted in Figure 5.11.
5.5.5.1 Uniformly infilled configuration
By means of a new stiffness measurement before the PSD test, the vibration frequencies were
updated for this infilled configuration. The first mode frequency increased from 0.82 Hz, after
the bare frame tests, to 3.34 Hz as a consequence of the stiffening effect of the infill panels.
From Figure 5.10 the maximum values of top displacement, inter-storey drift and inter-storey
shear can be read as 8.0 cm, 1.12% and 2083 kN, respectively. The comparison with the corre-
sponding values for the 1.5S7 test highlights the increased stiffness, since the peak value of top
displacement reduced to less than 40% and the drift reduced to about 45%. Note that the inter-
storey drift peak value occurred in the first floor, whereas for the bare frame tests it was found
in the second floor.
The maximum base shear, as an indicator of the structure global strength, increases about 45%,
which, in spite of the important strength degradation, still can be considered as a sign of wor-
thy strength enhancement. Actually, from the comparison of shear-drift diagrams, the same
amount of drift leads to shear values for the infilled structure significantly higher than for the
bare structure.
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 223
Figu
re 5
.10
Tim
e hi
stor
ies
of s
tore
y di
spla
cem
ents
, rel
ativ
e in
ter-
stor
ey d
rift,
tota
l sto
rey-
shea
r and
resp
ectiv
e pe
ak v
alue
pro
files
for
both
con
figur
atio
ns o
f inf
illed
fram
e te
sts
Tim
e (s
)
DIS
PL.(m
)
Tim
e (s
)
DIS
PL.(m
)
DIS
PL. (
m)
1
2
3
4
Uni
form
Soft-
stor
ey
STO
REY
21
34
Stor
ey:
DRI
FT (%
)D
RIFT
(%)
1
2
3
4 Te
st:2
13
4St
orey
:
STO
REY
Tim
e (s
)
SHEA
R (k
N)
x1.E
3
Tim
e (s
)
SHEA
R (k
N)
x1.E
3
SHEA
R (x
1.E3
kN
)
1
2
3
4 ST
ORE
YTi
me
(s)
Ti
me
(s)
D
RIFT
(%)
-.2
0
-.1
6
-.1
2
-.0
8
-.0
4
.0
.04
.08
.12
.16
.20
.0
.0
2 .0
4 .0
6 .0
8 .1
0 .1
2 .1
4 .1
6 .1
8 .2
0
-4.0
-3
.2
-2.4
-1.6
-
.8
.0
.8
1.6
2
.4
3.2
4
.0
.0
.4
.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
.0
.5
1.0
1
.5
2.0
2
.50
-2
.5
-2.0
-1
.5
-1.0
-
.5
.0
.5
1.0
1
.5
2.0
2
.5
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-.2
0
-.1
6
-.1
2
-.0
8
-.0
4
.0
.04
.08
.12
.16
.20
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-4.0
-3
.2
-2.4
-1.6
-
.8
.0
.8
1.6
2
.4
3.2
4
.0
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-2.5
-2
.0
-1.5
-1
.0
-.5
.0
.5
1
.0
1.5
2
.0
2.5
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
a) T
ime
hist
orie
s - U
nifo
rm c
onfig
urat
ion
c) T
ime
hist
orie
s - S
oft-s
tore
y co
nfig
urat
ion
b) P
eak
valu
es
224 Chapter 5
Figure 5.11 Shear-drift diagrams at each storey for both configurations of infilled frame tests
DRIFT (%)
SHEAR (kN) x1.E3 a) Uniform b) Soft-storey
Storey4
Storey3
Storey1
Storey2
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
DRIFT (%)
SHEAR (kN) x1.E3
-4.0 -3.2 -2.4 -1.6 -.8 .0 .8 1.6 2.4 3.2 4.0 -2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 225
For this test no rotation measurements were taken for beams and columns, but the panel distor-
tions were measured by means of a specific instrumentation scheme detailed in Negro et al.
(1995).
Damage was found mainly in the first and second floor panels, which is also reflected in the
shear-drift diagrams of Figure 5.11. Highly pinched diagrams were obtained due to the pro-
gressive deterioration of infills. However, the pinching effect was found from a very early
deformation stage, and also in the third and fourth storeys, due to the sliding in the masonry-
concrete interface, where cracks first appeared. Crushing of the masonry at the panel corners of
the first two levels was quite extensive and even led to the complete failure of one first level
panel where a horizontal slice at mid-height fell-off. At the third and fourth storeys no crushing
or significant cracking was observed.
5.5.5.2 Soft-storey configuration
For this configuration the panels were replaced with new ones but only in the three upper sto-
reys. This led to a “soft-storey” at the first level, for which no design provisions were taken as
would be required by EC8, namely a local increase of design forces and the detailing of the
entire column length as a critical region.
The new fundamental frequency turned out to be 1.67 Hz, therefore about half of the obtained
for the regular configuration. The peak values of top displacement, inter-storey drift and shear
were, respectively, 17.6 cm, 3.55% and 1690 kN.
As expected, the deformation was mainly concentrated at the first level, with an inter-storey
drift about three times higher than the second storey one. As a result, the part of the structure
above the first floor moved almost as a rigid body on the first level columns, which can be
understood from the results and was clearly seen during the test. This is confirmed by the low
damage in three upper floors, since only in the second floor some masonry crushing appeared.
Accordingly, the energy dissipation took place essentially at the first level, due to the deforma-
tion of the bare frame during the cycle with a peak drift not experienced before. After the peaks
in both loading directions, the shear-drift diagram (see Figure 5.11-b)) became very pinched,
due to anchorage pull-out at both ends of columns and beams, and significant strength reduc-
tion occurred due to concrete spalling clearly observed in the columns.
226 Chapter 5
It is worth noting the increase of maximum base-shear (over 15%) when compared with the
bare frame test results (both the seismic and the final cyclic ones). In the bare frame tests, as
well as in the soft-storey one, the base-shear is bounded by the first storey strength, which
depends on the developed deformation mechanism. For the bare frame tests, the inflection
points in the first storey columns are closer to the upper end sections and their resistance is not
fully engaged. On the contrary, for the soft-storey test such columns tend to have a more shear-
type deflected shape, the inflection points become closer to the mid-span and higher resisting
moments develop in the upper end sections; consequently an increased shear force is generated
in that storey, but for a much higher drift likely to correspond to excessive ductility demands.
5.5.6 Final cyclic tests
The near-failure behaviour of the bare frame structure was simulated by means of the final
cyclic tests, whose main scope was the assessment of ultimate displacement and energy dissi-
pation capacities and of the damage sensitivity to cyclic loading.
Due to the significant amount of damage suffered during the irregular infilled frame test,
mainly in the two first floors, the structure had to be repaired. The repair was made in the criti-
cal regions where concrete cover spalling had occurred and where permanently open cracks
were visible. This intervention consisted on removing the damaged cover and replacing it by a
new one made out of no shrinkage mortar reinforced with stainless steel fibres.
The final tests consisted on three sets of cycles of increasing top displacement amplitude as
shown in Figure 5.12, each set with three cycles at the same peak displacement as follows:
• The first set reached the maximum displacement of 21.0 cm, in both loading directions,
equal to the one of the high level test, and aimed, on one hand at approximately re-establish-
ing the conditions existing at the end of the 1.5S7 test and, on the other hand, at checking
the structural hysteretic behaviour for that level of displacement already experienced.
• The second set drove the structure to a top displacement of 35.0 cm, also in both loading
directions.
• In the last set, a maximum displacement of about 60.0 cm was imposed to the structure in
the negative direction (pushing the structure away from the reaction-wall), whilst in the
other direction only 35.0 cm were reached due non-symmetric actuator stroke.
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 227
Figure 5.12 Time history of the imposed top displacement for the final cyclic tests
Each of these sets corresponded to a certain level of global ductility. Actually, from the base
shear - top displacement diagrams of the 0.4S7 and 1.5S7 tests, the yielding value of top dis-
placement was roughly estimated as 7 cm; thus, the first cycle set corresponds to a ductility
level of 3, the second one to a ductility of 5 and the last one attained the ductility of 8 in one
testing direction. Therefore, from now on these tests are identified by the corresponding ductil-
ity level, respectively Duct.3, Duct.5 and Duct.8, and, although they actually consisted in a
unique test, the results are split according to each ductility level; the corresponding total drifts
were 1.68%, 2.8% and 4.8%, respectively.
The driving signal was set as the top displacement according to Figure 5.12, but an inverted tri-
angular distribution of forces was actually prescribed to the actuators. The cyclic response is
illustrated in Figure 5.13 by means of the base shear - top displacement diagrams and the total
deformation energy curves for each ductility level testing phase. The base shear - top displace-
ment diagrams were preferred here because they are essentially similar to the storey shear-drift
curves. The peak value profiles of storey displacements, inter-storey drift and shear are
depicted in Figure 5.14.
For ductility level 3, a pinched force-displacement diagram was obtained as it had already
occurred for the post-peak response of the 1.5S7 and the soft-storey tests. Since displacements
did not exceed the peak values previously experienced, the deformation and strength mecha-
Step
DISPL. (m)
0 2700 5400 -.60
-.48
-.36
-.24
-.12
.0
.12
.24
.36
.48
.60
Duct. 3 Duct. 5 Duct. 8
228 Chapter 5
nisms remain essentially the same and are strongly influenced by the joint behaviour as already
explained. Therefore, the hysteresis loops still exhibit very low energy dissipation capacity,
with little strength deterioration; nevertheless, quite stable diagrams are obtained.
The maximum inter-storey drift was 2.24%, still occurring at the second storey, whilst the
maximum base shear was 1222 kN. The drift value agrees well with that of the 1.5S7 test, but
the base shear decreased, which, in spite of the repair, reflects the stiffness deterioration
induced by the soft-storey test.
The visual inspection of the specimen at the last peak for ductility level 3, showed little
spalling of concrete cover at the base section of first storey columns and diagonal cracking in
the central beam-column joints of the first and second storeys.
The progression towards ductility level 5, clearly introduced significant damage in the struc-
ture. The maximum inter-storey drift reached 3.9%, still at the second storey although very
close to that of the first storey (see Figure 5.14-b)), and the maximum base shear turned out to
be 1444 kN, therefore a similar value to that of the 1.5S7 test. The base shear - top displace-
ment diagram became much more dissipative, but quite apparent strength degradation occurred
for loops after the first. The observed behaviour for this testing phase was as follows:
• At the first positive displacement peak cracks of about 3-4 mm opened at the column base
sections and a global deformed shape of soft-storey type could be seen, showing the defor-
mation mechanism and the damage localization in the first storey due to the soft-storey test.
• The subsequent negative peak increased the crack opening at the column base sections to 5-
7 mm and showed wide cracks in beams and joints (about 3-5 mm); moreover, crushing of
concrete cover at the column bases started.
• In the following displacement peaks, some joints were fairly cracked (diagonal pattern as
shown in Picture 5.2), full depth cracks could be seen at beam-column interface sections of
the first storey and apparent spalling of concrete cover occurred, both in columns and in
some beam end zones which had not been repaired.
• In the first storey exterior joints adjacent to the shorter bay, the concrete cover spalled in the
external face of the column, i.e. in the transversal frame plane, as a result of the stress trans-
fer between the reinforcement and the concrete within the bent part of the anchorage length
(Paulay and Priestley (1992)); this is shown in Picture 5.3, actually referring to the structure
final stage but included here because this phenomenon started at ductility level 5.
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 229
Figure 5.13 Base shear - top displacement diagrams and curves of total deformation energy for
the final cyclic tests
Top Disp [m]
Base Shear (kN)
-.60 -.48 -.36 -.24 -.12 .0 .12 .24 .36 .48 .60 -1.50
-1.20
-.90
-.60
-.30
.0
.30
.60
.90
1.20
1.50 x1.E3
b) Energy
Step
ENERGY (kJ) x1.E3
ENERGY (kJ) x1.E3
ENERGY (kJ)
0 900 1800 .0
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50 x1.E3
0 900 1800 .0
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50
0 900 1800 .0
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50
Step
Step
Storey 1
Storey 2
Storey 3
Storey 4
TOTAL
Duct. 3
Duct. 5
Duct. 8
Top Disp [m]
Base Shear (kN)
-.60 -.48 -.36 -.24 -.12 .0 .12 .24 .36 .48 .60 -1.50
-1.20
-.90
-.60
-.30
.0
.30
.60
.90
1.20
1.50 x1.E3
Top Disp [m]
Base Shear (kN)
-.60 -.48 -.36 -.24 -.12 .0 .12 .24 .36 .48 .60 -1.50
-1.20
-.90
-.60
-.30
.0
.30
.60
.90
1.20
1.50 x1.E3
a) Base shear - Top displacement
230 Chapter 5
Figure 5.14 Storey profiles of peak values of displacement, inter-storey drift and inter-storey
shear for the final cyclic tests
DISPL. (m)
STOREY
.0 .06 .12 .18 .24 .30 .36 .42 .48 .54 .60
1
2
3
4
Duct. 3
Duct. 5
Duct. 8
DRIFT (%) .0 .75 1.50 2.25 3.00 3.75 4.50 5.25 6.00 6.75 7.50
SHEAR (x1.E3 kN) .0 .15 .30 .45 .60 .75 .90 1.05 1.20 1.35 1.50
Duct. 3
Duct. 5
Duct. 8
STOREY
1
2
3
4
STOREY
1
2
3
4
Duct. 3
Duct. 5
Duct. 8
a) Maximum StoreyDISPLACEMENTS
b) Maximum Inter-StoreyDRIFT
c) Maximum StoreySHEAR
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 231
Picture 5.2 Diagonal cracking pattern in the beam-column joint (final test - Ductility 5)
Picture 5.3 Cracking pattern in a 1st storey joint in the external face of the column
232 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 233
The testing phase for ductility level 8, proceeded with heavy damage in the structure. In several
critical regions of the first and second storey, concrete crushing and subsequent cover spalling
was apparent, followed by stirrup failure and buckling of rebars, some of which ruptured in the
subsequent cycles. This is evidenced in Picture 5.4-a) showing reinforcement instability at one
first storey beam, of which at least one bar was visibly ruptured at the end of the test (see Pic-
ture 5.4-b)). Similarly, Picture 5.4-c) shows the rupture of a 20 mm bar of the central column at
the base section.
The extent of slab participation for such a high deformation level is also apparent in Picture
5.4-a) showing slab cracks developed parallel to the transverse beams along their total length.
The overall cracking pattern at an exterior first storey beam is illustrated in Picture 5.5-a).
Higher cracking density and more inclined cracks are found in the shorter span (on the left
side) than in the larger one, confirming the predominancy of lateral load over the vertical load
effects in the shorter span. A detailed view of cracking near the leftmost column is shown in
Picture 5.5-b), where a permanent full depth crack can be seen at the beam-column interface.
The base shear - top displacement diagram was still found very dissipative although with even
more strength degradation, which is confirmed by the maximum base-shear of 1425 kN,
slightly lower than that for the Duct.5 and reached for a higher displacement.
The maximum inter-storey drift increased up to 7.2%, approximately uniform in the first and
second storeys, corresponding to the structure deformed shape shown in Picture 5.6 taken at
the last top displacement peak of 60 cm. From the observed damage and the strength deteriora-
tion evidenced in the force-displacement diagrams, it can be concluded that a near-failure stage
was actually reached.
During the evolution from the Duct. 3 to the Duct. 8 phases, the first storey drift increased and
the difference of drift values between the first and the second storeys was progressively
reduced, so that, at the end, an almost uniform drift was obtained in these two storeys. This
was due, not only to the damaged beam-column joint at the first storey, but also to the progres-
sive deterioration of the critical region at the column-foundation interface where plastic hinges
actually developed.
234 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 235
Pict
ure
5.4
B
uckl
ing
and
rupt
ure
of b
eam
and
col
umn
rein
forc
emen
t bar
s (fi
nal t
est -
Duc
t. 8
)
a)
b)
c)
236 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 237
Picture 5.5 Final overall cracking pattern of a 1st storey exterior beam
a) Final cracking pattern of beam
b) Detail of crack in the beam-column interface
238 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 239
Picture 5.6 General view of deformed shape at the last peak of 60 cm top displacement for the final test - Duct.8
240 Chapter 5
THE 4-STOREY FULL-SCALE BUILDING TESTED AT ELSA 241
Note that force-displacement diagrams for ductility levels of 5 and 8 are considerably more
dissipative than those of the previous tests, not only because larger drifts are reached inducing
significant residual deformations, but also due to the lower pinching exhibited by the diagrams.
This fact is thought to be related with a lower influence of the joint behaviour and rebar slip-
page on the global deformation, and can be explained as follows:
• If a joint is severely damaged, the restraining effect of the framing beams on the columns is
drastically reduced, which means that columns behave more independently of beams; there-
fore, higher drifts are obtained, inducing column rotations larger than those on the beams
due to the non-effectiveness of force transfer across the joint.
• The pinching effect is less apparent in the columns than in the beams, on one hand due to
section symmetry and, on the other hand, because they are less prone to anchorage slippage
due to the beneficial effect of the axial force; consequently, if the behaviour becomes more
controlled by the columns, at least in the two first storeys, it follows that a less marked
pinching effect appears in the shear-drift diagrams.
5.6 Concluding remarks
In the preceding sections the results of the experimental testing of the four-storey reinforced
concrete building have been presented.
Reduced scale tests on cantilever specimens seemed to sustain the use of B500S Tempcore
steel within the EC8 requirements for ductile design of earthquake resistant structures,
although problems related with anchorage of rebars need to be further investigated.
Full-scale testing has started with snap-back (free-vibration) and stiffness tests to obtain initial
vibration frequencies (1.8 Hz for the fundamental one) and viscous damping factor (1.8%) of
the structure.
Pseudo-dynamic tests were performed in the bare frame structure, first for low level intensity
and then for high intensity earthquake. The former just led to cracking throughout the three
first storeys, responsible for a clear stiffness drop, but no evidence of yielding could be found.
In turn, for the high level intensity, further cracking developed, mainly along diagonals of
beam-column joints and locally at the beam-column interfaces where yielding of reinforce-
ment has occurred; besides a significant frequency (and stiffness) drop, a clear pinching effect
242 Chapter 5
was observed in force-deformation response diagrams, related to bar-slippage inside the joints
and responsible for rather low dissipation after the response peaks. However, an almost uni-
form distribution of rotations was obtained throughout the critical zones of beams (although
with little contribution of the top storey) and no serious permanent damage was found at the
end of these tests.
The structure with external infilled frames was tested for two configurations, viz a uniformly
infilled and a partially infilled, the latter simulating a “soft-storey” at the first level. Infills con-
sisted of typical unreinforced mansonry made out of hollow ceramic bricks and have contrib-
uted for a clear increase of the initial frequency and for some enhancement of the global
strength of the structure. Serious damage could be found in the masonry panels of the two first
storeys of the uniformly infilled configuration, reflected in a very clear pinching effect in the
response diagrams. The test on the “soft-storey” configuration led to visible damage in the first
storey, above which the upper floors moved almost as a rigid body, thus leading to energy dis-
sipation mainly concentrated in that storey. This test was found critical for the structure, since
no specific design provisions were included as required by EC8 to account for the presence of
infills and, particularly, of a “soft-storey”. Consequently, significant strength reduction
occurred due to concrete spalling observed mainly in the first storey columns.
After repairing of the damaged zones, a final cyclic test was quasi-statically performed again
in the bare frame structure for three sets of cycles at increasing top displacement amplitude,
whose final deformation stage reached a global ductility factor around 8.
The final test first produced some damage signs already obtained in previous tests and progres-
sively introduced heavier damage consisting of concrete crushing and cover spalling, which
subsequently led to stirrup failure and buckling of rebars. The clear strength degradation and
the high inter-storey drifts (around 7%) in the two first storeys, exhibited for the stage at global
ductility 8 corresponded to a near-failure state of the structure at the end of the final test.
Chapter 6
ANALYSIS OF THE 4-STOREY FULL-
SCALE BUILDING
6.1 Introduction
The experimental campaign on the four-storey reinforced concrete building described in the
previous chapter, provides an excellent means of calibration and assessment of numerical mod-
els for global seismic behaviour simulation.
Indeed, the availability of different tests for distinct stages of the structure behaviour (pre-
yielding and post-yielding for increasing ductility levels up to failure) and the use of the PSD
method (which is an essentially static test from the strictly experimental point of view) have
rendered the outcome of these tests rather suitable for comparison with numerical simulations
throughout various behaviour stages, both in quasi-static or dynamic conditions.
Particularly, if only the global behaviour is sought (say at the storey, or even at the element
level) the experimental tests have provided the fundamental output to be compared against the
response obtained by global element models as the one developed in the present study. How-
ever, notwithstanding the valuable information obtained, some additional local measures
would have been desirable to assess the section behaviour in critical zones (beam and base-col-
umn plastic regions, beam-column joints, etc.) from which a better insight of the global ele-
ment behaviour could be have been obtained. Unfortunately this was not possible and the
experimental-analytical comparisons are mostly performed at the storey level, reflecting the
behaviour of the relevant elements.
In this context, the main scope of this chapter is the assessment of the flexibility element model
244 Chapter 6
(as presented in Chapter 3), along with the auxiliary procedure to characterize the local section
behaviour (described in Chapter 4), for the simulation of the global behaviour of members
integrated in a complete frame structure, by comparing the numerical response against the glo-
bal structural results from the full-scale tests.
The whole set of data and modelling assumptions for numerical simulations are described in
6.2, covering relevant aspects for the frame structure modelling and discretization, the section
behaviour, the structural damping and the mansonry infill modelling.
A key issue in the seismic assessment of structures is the quantification of damage induced by
earthquake action, in order to determine the closeness of the structure state in relation to pre-
defined limit states. To this end, a brief review of available proposals of damage indices is
included in 6.3 and the damage index adopted in the present work context is further detailed,
particularly in what concerns the involved parameter quantification.
The major concern of this chapter, i.e., the numerical simulation of the experimental campaign,
is extensively described in 6.4. A broad set of numerical analyses is included, covering static
pushover analyses (to roughly check the monotonic structural behaviour), the static and
dynamic simulations of seismic tests (on bare and infilled structure configurations) and the
analyses for the quasi-static final cyclic test. Additionally, some static and dynamic calcula-
tions were performed also with a traditional model of fixed length plastic hinges, in order to
compare with the flexibility simulations. Finally, the main conclusions of this chapter are sum-
marized in 6.5, with particular emphasis on the numerical simulations of the four-storey build-
ing response.
6.2 Modelling assumptions and data
6.2.1 Structure modelling
6.2.1.1 Discretization
For the numerical analyses presented in this work, the structure was modelled in the testing
direction by means of a planar frame association. Due to symmetry conditions, only two
frames were considered: one to simulate the internal frame and another, with double stiffness,
vertical load and mass properties, to simulate the two external frames.
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 245
Each beam and column was discretized by one flexibility global element as described in Chap-
ter 3, resulting in a very simple mesh of 40 elements and 30 nodes illustrated in Figure 6.1.
Rigid lengths were considered in the beams, measured between the column axes and the beam-
column interfaces. According to the capabilities of the flexibility element, distinct material and
geometrical properties were assigned where required to the left and right element nodes and
assumed uniform between each end section and the mid-span section.
Figure 6.1 Mesh for the structural analysis using flexibility global elements (dimensions in m)
The rigid floor diaphragm assumption was accomplished by imposing the same horizontal dis-
placement to all nodes at the same floor.
6.2.1.2 Collaborating slab width
The participation of slabs in the strength and stiffness of beams is a rather difficult issue to
quantify due to the wide range of involved factors. Usually it is measured by an equivalent
width , assumed fully collaborating with the beam deformation, which is taken uniform
along the full length of the beam or portions of it. In the following, refers only to the slab
contribution, meaning that the beam web width is not included, and the effective slab
width is given by .
A major difficulty in estimating relies on the concept itself: the equivalent slab width for
quantifying the structural stiffness is different from the one involved in beam strength calcula-
tions (Tjebbes (1994)). In the stiffness case, the slab participation along the full length of the
beam is relevant, and it is meaningful, for instance, when a good estimation of structural fre-
quencies is envisaged. For strength computation, the slab contribution must be evaluated at the
Internal Frame External Frame
X
Z
3.00
3.50
3.00
3.00
6.004.006.004.00Storey
4
3
2
1
Level
beq( )
beq
bw( )
beff bw beq+=
beq
246 Chapter 6
locations where maximum bending moments are expected and, as explained below, it is an
important issue for an adequate assessment of the structural mechanisms of failure. Therefore,
the modelling of slab participation in a given structure by means of equivalent slab width, aim-
ing at a good representation of both stiffness and strength, appears as a quite hard task to be
accomplished.
In particular, the slab contribution for the strength case is twofold:
• When the beam is bent with the top reinforcement bars in tension, the slab rebars are also
tensioned and act as an extra flexural reinforcement for the beam, up to a transverse dis-
tance from the beam that depends on the loading intensity. The so-obtained extra flexural
strength must be taken into account in order to correctly anticipate the failure mechanism
and to be coherent with the capacity design philosophy subjacent to modern design codes
(Eurocode 8 (1994)). If a strong column - weak beam mechanism is expected, the non-con-
sideration of such extra strength can indeed prevent that mechanism to develop and may
lead to a very different, and not desired, behaviour for the structural numerical model
(Paulay and Priestley (1992)).
• For compressive strain developing on the top face of the beam, the corresponding stresses
can spread along the slab and decrease the neutral axis depth. As explained for the reduced
scale tests in 5.4, this can lead to higher tensile strains in bottom bars and increase the ten-
dency to buckle in subsequent cycles. The inherent strength deterioration in cyclic behav-
iour can be analytically captured only if the collaborating slab width is included and,
obviously, if the adopted model is able to treat such a phenomenon.
From these two points, the first appears to be the most important, but the related uncertainties
are substantial. Above all, the fact that the loading intensity induces an increase of ,
because larger plastic hinge rotations near the column faces will engage more slab steel placed
farther away from the column, renders more difficult the structural modelling for analysis of
increasing earthquake intensity levels. Moreover, the effectiveness of slab steel also depends
on the existence of transversal beams; generally, the presence of such beams, implies a larger
slab participation.
Also, the zone where the slab contribution is to be assessed has influence on for strength
purposes. For plastic hinges next to external columns, the slab width is lower than for internal
plastic hinges, due to different mechanisms of membrane force transfer between the slab and
beq
beq
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 247
the beams. Additionally, the value is also affected depending on whether the slab is com-
pressed or tensioned, which complicates even more the modelling for cyclic loading.
Due to all the uncertainties regarding the adequate assessment of the equivalent slab width and,
since the whole experimental testing programme on the structure under analysis involved sev-
eral loading intensities and distinct test conditions (for which the slab participation was cer-
tainly variable), it was decided to adopt a uniform estimate for over the whole structure.
In the preliminary numerical analyses reported in Carvalho (1993) and performed by the JRC
team prior to the tests, the values of were adopted according to an approximation proposed
in Park and Paulay (1975). The slab width was taken up to a distance of measured from
the beam face to each side, where is the slab thickness. Thus, for the present case the fol-
lowing effective slab width values were obtained:
• for the external frames:
• for the internal frames:
After the bare frame tests, an in depth analysis of the measurements for the assessment of the
collaborating slab width was carried out and reported in Tjebbes (1994). As already mentioned
in 5.5.1, these measurements were made in the second storey and time histories of deforma-
tions on the top face of the slab in the testing direction were obtained. The transversal profile
of strains on the slab could be traced along the internal and one external transversal beams,
from which the portion of slab participating in the beam deformation was estimated.
The obtained values of , for the displacements peaks of both the 0.4S7 and the 1.5S7 tests,
confirmed the distinct extent of slab collaboration for the low and the high level tests, as well
as the larger slab widths next to the transverse internal frame when compared with those adja-
cent to the external transverse beams. Moreover, the values of slab width more adequate for
stiffness estimation purposes, were, as expected, lower than those for strength analysis.
However, taking an average uniform value for in both bending directions it was found
(from experimental measurements) that the above adopted widths are about 20% underesti-
mated for the strength case, whilst for the stiffness case those values are overestimated by
around 10% along the external longitudinal frames and underestimated by about 20% along the
central frame. This shows that the adopted values provide a reasonable approximation of the
average slab width for the whole structure.
beq
beq
beq
4hs
hs
beff bw 4hs+ 0.90 m= =
beff bw 2 4hs( )+ 1.50 m= =
beq
beq
248 Chapter 6
In our opinion, it would be pointless to consider a more rigorous fitting of with the experi-
mentally obtained results, namely by considering different values according to the slab loca-
tion, the loading direction or intensity, since other relevant phenomena could not be taken into
account. Among these, the following are highlighted:
• The collaborating slab width depends on the deformation level and, particularly, on the
inter-storey drift; since it varies along the height, it follows that values for the top sto-
rey, for example, are not the same as those obtained for the second floor.
• Measurements were made by displacement transducers located only on the top face of the
slab, and the resulting deformations were considered, in the calculation of , as the aver-
age membrane strains in the slab; actually, the membrane strains are lower than the top face
ones, but there was no measurement to account for that difference, which is thought to be
significant since the slab thickness is 1/3 of the total beam depth.
From the explained reasons, it was decided to keep the estimates of equivalent slab width as
used in the preliminary analyses, i.e., with the values given above.
6.2.1.3 Mass and vertical static loads
The structural mass was quantified on the basis of an average value of the concrete unit weight
of 25 kN/m3. For the dead loads, the contributions of the 0.15 m thick reinforced concrete slab
and of the beam and column gross sections were taken into account. The additional loads (fin-
ishings and factorized live loads), as used in the test and already mentioned in 5.5.1, were
included in the total mass of each floor. Table 6.1 summarizes the mass contributions from
dead and additional loads, and the total mass considered per floor.
For the mass matrix computation, each floor mass was distributed over all the nodes belonging
to the floor, leading to a concentrated mass system with no rotational inertia.
Table 6.1 Floor mass values and vertical loads on beams
Mass (103 kg) Vertical loadsFloor Dead loads Additional Total (kN/m)
4 58.5 26.1 84.6 35.0
3 64.2 24.3 88.5 34.0
2 64.2 24.3 88.5 34.0
1 65.1 24.3 89.4 34.0
beq
beq
beq
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 249
The vertical loads acting simultaneously with the seismic loads are due to the slab dead loads
and the beam self-weights, plus the factorized live loads. Due to symmetry conditions, the
same uniformly distributed vertical loads per unit length have been considered in the external
and internal frame beams, as also listed in Table 6.1. It is recognized however, that a more
refined quantification of the vertical load effects would require triangular distributions of load
per unit length.
6.2.2 Cross-section characteristics and material properties
6.2.2.1 Cross-sections
The reinforcement and cross-section details are extensively reported in Negro et al. (1994).
However, for the sake of completeness they are summarized in Figures 6.2, 6.3 and 6.4.
Figure 6.2 Typical beam and column cross-sections for both external and internal frames
For the beam cross-sections the top slab reinforcement mesh inside the slab width was consid-
ered part of the beam top reinforcement, whilst the mesh of the slab bottom face was neglected.
a) Internal frame
At
Ab
b) External frame
At
Ab
Asw
0.30 m
0.90 m1.50 m
0.30 m
0.45
m
0.15
m
# Q188 (DIN) = φ6//0.15
0.40 m
0.40
m
0.40 m
0.45
m
0.45 m0.40 m
a2
a1
a1
Beams
Columns
Lateral CentralLateral Central
250 Chapter 6
Figu
re 6
.3Sc
hem
atic
rein
forc
emen
t lay
out f
or th
e be
ams
a) In
tern
al fr
ame
3.5
2φ14
1φ12
3.5
2φ14
1φ12
5.2
2φ14
3φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
3φ12
5.2
5.2
3φ14
3φ12
5.2
3φ14
3φ12
5.2
3φ14
3φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
3φ14
1φ12
3.5
5.2
5φ14
2φ12
5.2
5φ14
2φ12
5.2
5φ14
2φ12
3.5
5φ14
3.5
5φ14
3.5
5φ14
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
3.5
5φ14
2φ12
3.5
5φ14
2φ12
3.5
5φ14
2φ12
3.5
5φ14
1φ12
3.5
5φ14
1φ12
5.2
5φ14
2φ12
5φ14
5.2
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
a 2 =
... (c
m)
a 1 =
... (c
m)
a 2 =
... (c
m)
a 1 =
... (c
m)
a 2 =
... (c
m)
a 1 =
... (c
m)
a 2 =
... (c
m)
a 1 =
... (c
m)
φ6//8
φ6//8
φ6//8
φ6//8
φ8//2
0φ8
//20
110
112.
511
2.5
110
Stirr
ups
φ(m
m)//
(cm
)
Dis
tanc
e fr
om
Col
umn
Cen
treLi
ne (c
m)
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
2φ14
1φ12
3.5
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3.5
4φ14
3φ14
1φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
3.5
3φ14
1φ12
3.5
5.2
4φ14
2φ12
5.2
5φ14
5.2
5φ14
5.2
5φ14
5.2
4φ14
5.2
4φ14
2φ12
5φ14
5.2
4φ14
5.2
4φ14
5.2
4φ14
5.2
3φ14
5.2
3φ14
3.5
5.2
6φ14
5.2
6φ14
5.2
6φ14
5.2
6φ14
5.2
6φ14
5.2
6φ14
4φ14
2φ12
5.2
3φ14
2φ12
3.5
3φ14
2φ12
3.5
3φ14
2φ12
3.5
3φ14
2φ12
3.5
3φ14
2φ12
3.5
φ6//8
φ6//8
φ6//8
φ6//8
φ8//2
0φ8
//20
110
112.
511
2.5
110
b) E
xter
nal f
ram
e
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 251
Figu
re 6
.4Sc
hem
atic
rein
forc
emen
t lay
out f
or th
e co
lum
ns
8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ20
4.0
8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ20
4.0
4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ2
04.
0
φ10/
/7.5
φ8//2
0
φ10/
/7.5
φ10/
/7.5
φ8//2
0
φ10/
/7.5
φ10/
/7.5
φ8//2
0
φ10/
/7.5
φ10/
/7.5
φ8//2
0
φ10/
/7.5
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
Stirr
ups
φ(m
m)//
(cm
)St
eel B
ars
a 1 =
a2 (
cm)
a) In
tern
al fr
ame
b) E
xter
nal f
ram
e
8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 4φ20
+ 4
φ16
4.0 4φ20
+ 4
φ16
4.0 4φ20
+ 4
φ16
4.0 4φ25
+ 4
φ16
4.25
8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 8φ16
3.8 4φ20
+ 4
φ16
4.0 4φ20
+ 4
φ16
4.0 4φ20
+ 4
φ16
4.0 4φ25
+ 4
φ16
4.25
4φ16
+ 8
φ14
3.8 4φ16
+ 8
φ14
3.8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ1
63.
8 12φ2
04.
0
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//2
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
φ10/
/10
φ8//1
0
φ10/
/10
Stirr
ups
φ(m
m)//
(cm
)St
eel B
ars
a 1 =
a2 (
cm)
252 Chapter 6
The layout of reinforcement distribution for the beams is shown in Figure 6.3, where the
amount of top and bottom steel are indicated along rows, respectively, above and below each
beam. The adopted depths for the gravity centres of the top and bottom layers (a2 and a1) are
also provided. The amount of slab reinforcement is not included in that figure as it can be
obtained from Figure 6.2. The transversal reinforcement is the same for all storeys and is indi-
cated below the frames in Figure 6.3.
Figure 6.4 includes the reinforcement for columns, where the total longitudinal steel is indi-
cated on the right side of each column and the stirrup distribution is included in the left side.
For each beam and column the steel at the end sections and at the mid-span section are given.
However, for modelling purposes, only the end section data was used since, in the flexibility
element, uniform characteristics are assumed along each left and right portion.
6.2.2.2 Concrete
The concrete characteristics have been considered in accordance with the compressive
strengths listed in Table 5.1. The values of mean cylinder compressive strength there
included were obtained by , where the conversion factor was estimated by inter-
polation between the values for the concrete classes closer to the experimental strength
. For a given class was taken as the ratio of the nominal cylinder to cube strengths
(e.g., for the class C25/30, the conversion factor was ).
The tensile strength , also listed in Table 5.1, was obtained by the EC2 expression
, where the characteristic strength (in MPa) is estimated by .
The so obtained value refers to the axial tensile strength, which was then converted to the
flexural tensile strength by the approximate factor , where h stands
for the cross-section height in the bending direction (REBAP (1984)). For the structure under
analysis the factor takes the value of 1.1.
In the absence of further testing characterization, the concrete elasticity modulus was also esti-
mated by the EC2 expression , for in MPa and in GPa. Such
value corresponds to the secant modulus at 40% of the peak compressive stress.
For the stress-strain behaviour of concrete, the diagrams shown in 4.2.3.2 were adopted.
fcm( )
fcm ξfcm cub,=
ξ
fcm cub, ξ
ξ 25 30⁄ 0.83= =
fctm( )
fctm 0.3fck2 3⁄= fck fcm 8–=
fctm
α 0.6 0.4 h4( )⁄+ 1≥=
Ecm 9.5 fck 8+( )1 3⁄= fck Ecm
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 253
According to the notation of those diagrams, the peak compressive stress for unconfined
concrete was taken with the values listed in Table 5.1 and for the corresponding strain, ,
a uniform value of 0.002 was adopted. The values were adopted for the limit of tensile
stress in the linear stress-strain diagram, together with the elastic modulus . For the
confined concrete, the confinement factor and the slope of the descending branch were
obtained from the cross-section and transverse reinforcement data. Residual stresses were
taken zero and 20% of the peak stress, respectively, for unconfined and confined concrete.
6.2.2.3 Steel
The steel data for definition of the stress-strain diagrams included in 4.2.3.2, is based on the
characteristics listed in Table 5.2. The values for yielding and ultimate stresses were assumed
constant for all the beams and equal to the average values corresponding to the bar diameters
actually used. Identical procedure was adopted for the columns and the resulting values are
listed in Table 6.2.
Since no further data was available, the elasticity modulus was assumed as 200 GPa and the
uniform strain at maximum tensile force was taken as 10%, a typical value for the
B500S Tempcore steel (Pipa (1993)), leading to the strain hardening also given in Table
6.2.
6.2.3 Skeleton curves for the section model
The trilinear skeleton curves required for the global section modelling were obtained by the
procedure detailed in 4.2.3, using the section and material data described in the previous sec-
tions.
Table 6.2 Mean tensile properties of steel
Longitudinal Transversal
(mm) (MPa) (GPa) (MPa) (%) (GPa) (mm) (MPa)
Beams12
586 200 678 10 0.956
56914 8
Columns14
577 200 669 10 0.95 810 55916
20
fc0
fcm εc0
fctm
fct( ) Ecm
k Zm
φ fsy Es fsm εsm Esh φ fsyt
εsm( )
Esh( )
M ϕ,( )
254 Chapter 6
The axial forces in columns were estimated from the static vertical loads assumed to act simul-
taneously with the seismic forces, whilst in the beams no axial forces were considered. There-
fore, a preliminary elastic analysis was performed, considering only these vertical loads, and
the resulting axial forces were assumed constant throughout all loading stages.
For beams, the trilinear curves fit quite well the results of the monotonic analysis using a fibre
discretization of the section, as shown in the examples included in 4.3 and in Arêde and Pinto
(1996).
For columns, such fitting depends on the axial force level and on the presence of internal layers
of steel between the main reinforcement layers. Specifically, the last can lead to an underesti-
mation of the yielding capacity, which can unduly advance the formation of plastic hinges in
the columns. This can be seen in Figure 6.5, where, for a lateral column section of the first sto-
rey in the external frame, the trilinear moment-curvature diagram is shown and compared with
the one obtained from fibre analysis.
Figure 6.5 Moment-curvature diagram for a column section. Comparison of trilinear curve and
fibre analysis results.
In order to have a better definition in the yielding zone, all column sections were modelled by
fibre analysis and the corresponding moment-curvature diagrams were obtained and plotted
together with the trilinear one. By visual inspection, better estimates of the yielding moments
CURVATURE (m-1) .00 .04 .08 .12 .16 .20 .24 .28 .32 .36 .40
.00
.40
.80
1.20
1.60
2.00
2.40
2.80
3.20
3.60
4.00x1.E2
Trilinear
MOMENT (kN.m)
Fibre
Original
TrilinearAdjusted
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 255
were obtained from the fibre analysis diagram, and corrected trilinear curves were adopted
with new yielding and ultimate points, but preserving the original cracked and post-yielding
stiffnesses; an example of the corrected curve is also depicted in Figure 6.5. The overall
increase of yielding moments due to this correction was about 12% of the original values.
The average ratio of post-yielding stiffness to the cracked stiffness is 0.65% for the beams and
0.7% for the columns.
6.2.4 Hysteretic behaviour
For the hysteretic behaviour modelling, the parameters controlling the unloading stiffness deg-
radation , the pinching effect and the strength degradation , had to be defined. The
following paragraphs explain the related options and assumptions.
6.2.4.1 Unloading stiffness degradation
By means of the shear-drift diagrams, the experimental results show that unloading stiffness
degradation occurs and, therefore it shall be taken into account. However, the adoption of
values is quite difficult since there are no available expressions, relating with the section and
the member characteristics. The authors of the model (Kunnath et al. (1990)) provided only a
range of values, between 2 and 4, adequate for well detailed sections, i.e. in cases where no
high degradation is expected. In other circumstances, it is suggested that the identification of
be based on experimental results.
In the present case, such direct identification is not possible, since no internal forces were
measured and, therefore, no local force-deformation diagrams are available at the element
level. Furthermore, the parameter in the context of the model adaptation as referred in 3.5, is
different from the original one, the difference being dependent on several factors such as the
cracked and the post-yielding stiffnesses, the maximum deformation reached and the amount
of degradation, i.e., the itself. Typically, the common point (as shown in Appendix B) has to
be placed farther away from the origin in the modified model (see 3.5.1) in order to achieve the
same unloading stiffness as in the original model; this means that higher values shall be
adopted.
Although the shear-drift diagrams reflect other phenomena than the stiffness degradation, they
α( ) γ( ) β( )
α
α
α
α
α
α
256 Chapter 6
still provide a means of approximately checking to which extent this issue is affecting the
response. Therefore, upon several trials with different estimates of , while keeping the other
hysteretic parameters fixed, the best agreement with experimental shear-drift and deformation
energy diagrams was obtained by adopting in the first floor beams and columns and
in the remaining members. The low value was found adequate to indirectly
account for the curved part of unloading branches near the zero force level, which reflects the
low stiffness when cracks are open. In all the subsequent calculations, namely for the infilled
frame tests and for the final cyclic tests, the same values of were adopted.
6.2.4.2 Pinching effect
The pronounced pinching exhibited in the shear-drift diagrams is thought to be the result of the
reinforcement asymmetry existing in the beam sections, which became overlapped by rebar
slippage inside the joint concrete core after the peak drifts.
The modelling of pinching due to reinforcement asymmetry was done by adopting the values
as proposed in Appendix B, i.e., given by the ratio of yielding moments of the two bending
directions, which led to values around 0.55.
By contrast, the adequate inclusion of rebar slippage contribution to the pinching effect is a
much more difficult task. It requires, either a specific modelling of the joint behaviour, such as
the one proposed by Monti et al. (1993) for example, or a point hinge modelling at each ele-
ment end section with a moment-rotation law strictly associated with the increment of rotation
due to pull-out. None of these techniques were adopted in the present study, but they are recog-
nized as important to adequately capture the observed pinching effect.
6.2.4.3 Strength degradation
Strength degradation was taken into account by means of the parameter given by the empir-
ical expression included in Appendix B. The confinement and the tension reinforcement ratios
were directly obtained from the cross-section characteristics, whilst the normalized axial force
was estimated from the static vertical loads referred to in 6.2.3. The average values for this
parameter were 2.5% for the columns and 3% and 2%, respectively for positive and negative
bending direction of beams.
α
α 1=
α 4= α 1=
α
γ
β
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 257
6.2.5 Damping
In the present work, where damping forces were to be included in the dynamic equilibrium
equation Eq. (5.1), the viscous damping matrix was considered given by the well known
Rayleigh expression (Clough and Penzien (1975)) as follows
(6.1)
where and are constants of proportionality to the mass and stiffness matrices.
This damping matrix is orthogonal to the vibration modes because the mass and stiffness
matrices also verify the orthogonality condition. The parameters and can be obtained so
as to satisfy prescribed damping values for two distinct modes, usually the first and the second.
Let these modes be characterized by the frequencies and , to which the damping factors
and are imposed. The parameters and are given by the solution of
(6.2)
and, therefore, the corresponding matrix satisfies the desired damping in the modes with
frequencies and . For modes with frequencies different from and , the damping factor
cannot be user-controlled since it becomes fixed when the parameters and are set.
The adequacy of using a viscous type damping is as a controversial issue, because if the behav-
iour models were able to “exactly” describe the hysteretic dissipation of energy, there would be
no need for considering other sources of dissipation. Thus, the use of viscous damping forces
to account for energy dissipation appears as an approximation introduced to compensate for
lack of model accuracy.
For the behaviour model used in the present work, two distinct stages can be considered as far
as dissipation is concerned, namely before and after the yielding of critical sections. Before
yielding, the model is typically low-dissipative because it is origin-oriented for unloading and
reloading phases (thus with no dissipation at all) and the energy dissipated due to the cracking
transition is rather low. It follows that viscous damping forces might be needed to simulate the
energy dissipation in dynamic calculations, which was confirmed by the comparison with
C
C amM akK+=
am ak
am ak
f1 f2
ξ1 ξ2 am ak
4πf1( ) 1– πf1
4πf2( ) 1– πf2
am
ak⎩ ⎭⎨ ⎬⎧ ⎫ ξ1
ξ2⎩ ⎭⎨ ⎬⎧ ⎫
=
C
f1 f2 f1 f2
am ak
258 Chapter 6
experimental results included in 6.4.4. Therefore, for the low level test, the numerical analysis
was performed using the experimentally obtained damping factor of 1.8% for both the first and
the second modes.
For loading levels after yielding, the model is dissipative, but several questions can be asked:
• How accurate is the dissipation capacity of the model and, if insufficient, how much of the
energy should be dissipated by viscous forces?
• If viscous damping is to be used, which factor(s) should be adopted, taking into account the
progressive damage and the modifications of the dissipation capacity of the structure? Actu-
ally, it does not seem rational to adopt the same damping factor as before yielding, because
it was referred to an almost undamaged structure and proved to be suitable for the low-dissi-
pative model stage.
These aspects are further discussed in 6.4.4, but it can be anticipated here that good results
were obtained by considering no viscous damping forces for the numerical simulations of the
high intensity tests.
The stiffness matrix and the corresponding frequencies were adequately updated for the initial
conditions of each test in order to have a damping matrix more “coherent” with the actual state
of the structure, i.e. externally unloaded but with the residual deformations and the damage
induced by the previous testing phase.
6.2.6 Modelling of infills
Infill panels were modelled based on a detailed work by Combescure (1996), where multi-
level analyses are used for the simulation of the non-linear response of infilled frames. Basi-
cally this work consisted on the following steps:
• Development and/or improvement of a set of numerical tools for the non-linear analysis of
masonry infill panels, both at local (fine) and at global levels.
• Application of these tools to the numerical analysis of a single infilled frame experimentally
tested at LNEC (Pires F. (1990)), aiming at: a) a better understanding of the behaviour and
strength mechanisms, as well as the influence of modelling strategies and related parame-
ters, and b) checking the adequacy of global modelling and the procedure for parameter
identification based on results of the local modelling.
• Application to the full-scale infilled frames under study in the present work, in order to: a)
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 259
assess the influence of boundary conditions on the infill behaviour by means of local mod-
elling of panels both in an isolated frame and in the complete structure, and b) compare the
results of the global modelling against the experimental ones.
6.2.6.1 Setup of numerical tools for infill modelling
For the local (fine) analysis each constitutive material was modelled by a specific behaviour
law, some of them already available in CASTEM 2000, and briefly described in the following
paragraphs. Further details can be found in Combescure (1996).
The model proposed by Menegotto and Pinto (1973) was used for simulating the behaviour of
steel reinforcement, geometrically supported by uniaxial bar elements.
The concrete of the frame was simulated in plane-stress conditions; a non-linear cracking
model (the so-called Ottosen model, Dahlblom and Ottosen (1991)) was considered for the
tensile behaviour, while a linear elastic model was adopted for compression, because the main
source of concrete non-linearity is considered to arise from the cracking phenomena. The
cracking model is of “fixed-crack” type with the maximum principal stress criterion for first
crack development; it accounts for the tensile softening of concrete and the anisotropy induced
by crack formation; additionally, it incorporates a regularization method (Hillerborg et al.
(1976)) to reduce the dependency of global results on the mesh-size.
The interface between the concrete frame and the infills is of great importance for the tested
structure behaviour because significant contribution for non-linearity arises from that zone.
Two types of modelling strategies were available and considered for this purpose, namely the
unilateral contact problem solution and joint elements.
• The unilateral contact between two solids is modelled in CASTEM 2000 by means of kine-
matic conditions, handled by the Lagrange multiplier method, and leads to neither tensile
nor shear force transfer if contact is deactivated.
• The available joint elements (Pegon and Pinto (1996)) can be used in 2D analysis, where the
generalized stresses are the normal and shear forces across the joint plane and the general-
ized deformations are the corresponding relative displacements. The behaviour can be sim-
ulated either by an elastic-perfectly-plastic dilatant model (both for normal and shear
forces) or by a non-dilatant elasto-plastic model with softening for tensile and shear forces.
The latter allows to consider tensile strength progressively decreasing to zero as the joint is
260 Chapter 6
opening, and thus avoids an overestimation of the panel resistance if the tensile strength is
assumed constant.
Masonry infill panels under cyclic loading, are subjected to fast and strong degradation of their
mechanical characteristics, namely the compressive strength. Obviously, also the tensile
strength degrades rapidly; however, since cracking mainly occurs at the frame-panel interfaces
where the non-linear behaviour is modelled by the joint elements, the panel behaviour can be
assumed mainly controlled by compressive stresses. Therefore, a 2D plasticity-based model
was developed (Combescure (1996)) allowing cyclic loading and strength degradation to be
taken into account in the principal stress space; two plasticity surfaces are considered, defined
according to the lowest or the highest principal stress. An internal surface accounts for the
elastic domain (with kinematic hardening), while an external one defines the maximum
strength of the material (with isotropic negative hardening); the latter is affected by the plastic
energy dissipated due to the internal surface, which simulates the effect of cyclic loading in the
strength degradation.
The global analysis consisted of the two following approaches:
• Each infill panel was modelled by a pair of diagonal struts ruled by a uniaxial force-defor-
mation behaviour law, while each member of the reinforced concrete frame was modelled
by one internal linear elastic element and two finite length plastic hinge elements with a
moment-curvature relationship as given in Appendix B.
• Each storey (frame plus infill panels) was modelled by a single shear-beam element ruled
by a shear-drift law based on a primary trilinear curve (with softening branch) and a set of
hysteretic rules.
In the present work context only the infill modelling by diagonal struts is relevant, in order to
be used together with the flexibility elements for the frame members. No tensile resistance is
assumed for diagonal struts, while for compressive behaviour the strut model is based on an
axial force-deformation multi-linear law schematically shown in Figure 6.6 where both the
model basic curve and an arbitrary cyclic path (labelled with numbers next to the arrows) are
included; the model main features are as follows:
• For monotonic loading, the initial elastic branch (OC) is followed by the cracking phase
(CP), during which cracks in the frame-panel interface take place. Then, a plastic zone (PS)
is adopted to simulate the masonry crushing at the compressed corners; it holds until a cer-
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 261
tain maximum deformation is reached and then a softening branch (SM) is enforced.
• For hysteretic behaviour, unloading or reloading is origin-oriented in the cracking phase
(thus the non-linearity is only due to stiffness decrease); once in the plastic or softening
branches, unloading is done at constant stiffness (the full-cracked one, OP) until null resist-
ance is achieved; this is kept during the sliding phase until reloading starts, whose rules can
account for the typical pinched shape of infill diagrams due to a pronounced delay of crack
closure. The strength degradation due to cyclic loading is also reproduced by affecting the
maximum restoring force previously reached by a reduction factor which depends on the
accumulated plastic deformation.
Figure 6.6 Diagonal strut model
Both the primary curve and the model parameters for hysteretic behaviour are obtained from
results of refined analyses with the local models, which provide information about the horizon-
tal shear force in the panel (T), the corresponding distortion (drift) and average vertical
panel deformation . By considering the distortion and vertical extension as the most
significant panel deformation modes, the diagonal axial force (N) can be related with T and the
corresponding axial deformation can be written in terms of and , by means of
geometrical conditions. This compatibilization of generalized forces and deformations
between the two modelling approaches allowed to define a procedure (Combescure (1996)) for
the identification of global modelling parameters based on refined modelling technique.
6.2.6.2 Application to a single frame
Experimental tests carried out at LNEC on single infilled frames were simulated by Combes-
cure (1996), using the refined modelling technique above referred, and the following conclu-
sions have been extracted:
• From all the available interface modelling tools, the joint model with softening leads to bet-
εaxial
N
C
P
M
S
816
7
3
1110
12 14
5
1
2
13
4
15
6
17 18
9
19
Sliding phase
StrengthDegradation
O
γ( )
εvertical( )
εaxial( ) γ εvertical
262 Chapter 6
ter estimates of initial stiffness and allows to simulate the stiffness decrease due to cracking
at the interfaces prior to masonry crushing in the panel corners;
• Compression diagonal struts clearly form between opposite panel corners and the maximum
strength is reached when masonry crushes in the diagonal extremities.
• In the near-failure phase, the diagonal tends to move and secondary diagonals are engaged,
due to stress transfer from the crushed corners to the less damaged neighbourhood; thus,
one has to bear in mind the fixed-diagonal model limitations for simulating highly damaged
panels.
The use of diagonal modelling, with the primary curve fitted to the results from refined analy-
sis under monotonic loading, showed shear-drift curves of cyclic tests with the same aspect of
those from the refined analysis; additionally, if strength degradation is accounted for, reasona-
ble agreement with experiments can be obtained. The procedure for obtaining the primary
curve could be tested but the identification of hysteretic parameters still lacks further experi-
mental support.
6.2.6.3 Application to the full-scale structure
The four-storey full scale structure tested at ELSA was modelled using the refined analysis
technique (Combescure (1996)) subjected to an inverted triangular force distribution on the
uniformly infilled configuration.
Additionally, several sub-structures were also modelled and subjected to horizontal displace-
ment-controlled loading; these sub-structures consisted of one or two panels with the surround-
ing frame members, respectively, for the isolated panel configuration and for the total storey
configuration, and for both the first and the second storeys.
The main aim was to compare the behaviour of a given panel in an isolated configuration with
that in the whole structure. Deformed shapes, plastic deformation distributions and shear-drift
diagrams of infill panels were compared in both situations, allowing to draw the following
main conclusions (Combescure (1996)):
• Panels considered in the structure are subjected to vertical tensile or compressive stresses,
induced by the over-turning moment, that cannot be captured by an isolated panel analysis;
hence, the compression diagonal assumption is not strictly valid. However, shear-drift dia-
grams are not very different from those in the isolated or storey configurations and they still
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 263
can be used to define the overall panel behaviour.
• Beams on the panel lower boundary play an important role on the panel behaviour and,
therefore, must be taken into account for a better assessment of the panel characteristics.
The seismic behaviour of the whole structure was then simulated by means of global model-
ling. Reinforced concrete members were modelled, as in previous studies (Pinto et al. (1994)),
by the association of two plastic hinge elements and a linear elastic one. The infills were mod-
elled by diagonal struts, the primary curve having been obtained from the local refined analysis
of the first and second storey panel sub-structures. The remaining upper storeys were consid-
ered with the same characteristics of the second one. Hysteretic behaviour and strength degra-
dation parameters were taken with the values which gave the best simulations for the LNEC
specimens. Note that diagonal characteristics were obtained by means of numerical modelling
rather than experimental results, which means that possible defects inherent to the refined
modelling technique will obviously affect the global modelling.
Results have shown a good description of the experimental base shear but with a clear under-
estimation of the top displacement. This appears to be due to stiffness over-evaluation in the
less damaged storeys (the third and fourth ones) where the analytical shear-drift diagrams are
fairly poor approximations to the experimental response.
6.2.6.4 Infill panels in the present study
The infill modelling is not an item of major concern in the present work context and is
included here mainly for the sake of completeness. Thus, the same global modelling of infills
by diagonal struts was adopted here, exactly as used in Combescure (1996) where details can
be found. However, from the results there included, it is recognized that further effort should
be put in the definition of panel characteristics, particularly those related to the initial stiffness
and interface modelling in the upper storeys. It is worth noting that the adopted diagonal char-
acteristics were based on the local (refined) analysis and, according to Combescure (1996),
they led to less good results than a cruder modelling of global storeys by shear-beams but with
parameters directly extracted from the experimental results.
A more accurate definition of diagonal characteristics is beyond the scope of this work, but this
modelling deficiency has to be kept in mind when analysing the comparison of numerical sim-
ulations with experimental results included in 6.4.
264 Chapter 6
6.3 Damage quantification
6.3.1 General overview
The structural damage induced by earthquake action must be quantitatively defined, in order to
assess how close is the state of structure in relation to a pre-defined set of limit states.
The structural damage is usually seen as the ratio of a demand quantity (obtained from the
structural response to seismic events) to the ultimate capacity (Park et al. (1984)), leading to a
damage index. The demand is expressed in terms of one or more response variables, the so
called damage parameters (e.g. displacement, curvature, energy, etc.).
It is noteworthy that damage parameters in this context differ from the concept of damage var-
iables used in Continuum Damage Mechanics, in which nonlinear behaviour is explicitly
dependent on the damage state of the structure and vice-versa. Typically, in such formulation,
the damage variable works as an “internal variable” explicitly taken into account in the consti-
tutive model, whilst in the present context, the damage parameters for damage index definition
refer to “output variables”.
Damage indices can be defined at several levels, namely at the structure, the element, the sec-
tion or even the fibre level. The level for damage definition cannot be finer than the discretiza-
tion level, in the sense that, for example, if the non-linear behaviour of a building structure is
associated to one DOF per floor, the damage index cannot be computed in terms of member
deformations or plastic hinge rotations.
The damaging process in reinforced concrete elements under earthquake loading is usually a
combination of large strain excursions with several repetitions of load reversals. Damage
parameters related to the first phenomenon are often the peak values of response variables; typ-
ical examples are the storey displacements or the plastic hinge rotations. On the other hand, the
strength and stiffness degradation induced by load reversals, particularly those of high ampli-
tude, cause failure for deformation demands lower than those under monotonic loading. There-
fore, adequate damage indices for cyclic conditions shall be based in damage parameters able
to “memorize” the history of deformation, which can be achieved by means of cumulative
measures of the response variables. The total dissipated energy or the cumulative inelastic
deformation (or ductility) are examples of such damage parameters.
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 265
Thus, damage indices can be defined in terms of peak values, or cumulative measures or even
combination of both. Extensive overviews of damage indices published in the literature are
presented by Coelho (1992) and particularly by Fardis et al. (1993) who discuss several pro-
posals for damage indices which are categorized, analysed and applied to a large set of monot-
onic and cyclic tests, under uniaxial or biaxial loading, most of them up to failure. The
comparison of test results against damage predictions using those indices allowed their assess-
ment and even led to a new proposal of an energy-based damage index. Such a detailed analy-
sis of the available indices is deemed unnecessary in the present work; nevertheless, brief
references are included next, essentially based in Fardis et al. (1993), concerning the most
widely used indices.
The typical form of damage indices based on response peak values (usually of displacements
or deformations) is the following
(6.3)
where is the damage parameter, is the available capacity and is the threshold of ,
above which the damage is supposed to start. The exponent accounts for the rate of increase
of damage index with the damage parameter and the threshold is often considered as the
yielding displacement. Obviously, if the peak value for is entered in Eq. (6.3) the maxi-
mum damage index value is obtained.
For monotonic loading such index seems adequate since Eq. (6.3) yields for
and when reaches the available capacity. By contrast, no measure of cyclic effects is
included, which renders it inadequate for assessing the damage state when cyclic deterioration
is present.
Another way of expressing a peak value based damage index is by means of the ratio of secant
flexibilities as proposed by Roufaiel and Meyer (1987), where , and
are, respectively, the secant flexibilities at the current deformation, at yielding and at ulti-
mate capacity in monotonic loading.
Attempts to define a damage index in terms of cumulative inelastic deformation have been first
based on the low and high cycle fatigue of materials and are given by
Dδ δth–δu δth–-----------------⎝ ⎠
⎛ ⎞m
=
δ δu δth δ
m
δth
δM δ
D 0= δ δth=
D 1= δ
D f fy–( ) fu fy–( )⁄= f fy
fu
266 Chapter 6
(6.4)
where is the amplitude obtained in the ith half-cycle of the response (supposing the
response previously divided into equivalent half-cycles) and b is a constant. However,
experimental evidence supports the use of Eq. (6.4) for steel but not for reinforced concrete.
Damage indices based on cumulative energy parameters can also be used and are often defined
using the dissipated energy normalized by the potential energy up to failure in monotonic load-
ing . An example of such proposal was introduced by Meyer et al. (1988), in which two
indices are calculated separately for each loading direction and then combined into a unique
one; this proposal has the particular feature of also including the energy dissipated in interior
cycles as a normalizing quantity (i.e. summed to ) and thus reducing the influence on the
damage index of the occurrence of a great number of interior cycles, as demonstrated in Coe-
lho (1992).
Experimental evidence suggests that a damage index should combine contributions from both
peak and cumulative responses. The first proposal of such a damage index was introduced by
Banon (1980) and refined by Banon and Veneziano (1982), through a non-linear combination
of , the total dissipated energy normalized by the deformation energy up to yielding, with
the so-called Flexural Damage Ratio (FDR), given by where refers to the
secant flexural flexibility at peak deformation. The study was supported by a small set of
cyclic tests, to which the proposed index was applied leading to normally distributed results
with mean 1.0 and coefficient of variation 27.5%.
Along the same trend line, Park et al. (1984) presented a mixed damage index, nowadays
widely used and known as the Park and Ang index, given by
(6.5)
or by
(6.6)
Dδi
δu-----⎝ ⎠
⎛ ⎞b
i 1=
N
∑=
δi
N
Eu
Eu
En
FDR fm fy⁄= fm
D δM
δu------ β
Qyδu----------- Ed∫+=
D δM
δu------ β δ
δu-----⎝ ⎠
⎛ ⎞ α EdEc δ( )-------------∫+=
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 267
where and have the meaning above introduced, is the yield strength, is the
incremental hysteretic energy, is the hysteretic energy per cycle at deformation , and
and are non-negative parameters. All the parameters in Eqs. (6.5) and (6.6), namely , ,
, and were empirically obtained by statistical regression of a large set of experimen-
tal tests (142 monotonic and 261 cyclic) performed up to failure. Eqs. (6.5) and (6.6), along
with the empirical expressions for the parameters, were applied to the performed test set and
the authors found log-normal distributions of with identical characteristics in both cases,
i.e., mean value 1.0 and coefficient of variation of 54% for the first and 50% for the second.
Therefore, due to its simplicity, Eq. (6.5) appears preferable for the seismic damage assess-
ment; details about the determination of the corresponding parameters, as proposed in (Park et
al. (1984,1987a), Park et al. (1987b)), are presented later in this work.
Fardis et al. (1993) proposed a new energy-based damage index, with a similar structure to that
of Park and Ang index, and given by
(6.7)
where stands for the maximum deformation energy over the response to earthquake
loading, is the total deformation energy up to failure under monotonic loading and the
remaining terms have the same meaning as for the Park and Ang index. Note the closeness of
these two indices: the term is a measure of the maximum deformation , since this
deformation is involved in the computation of that energy; similarly is essentially
dependent on the deformation at monotonic failure.
Damage assessment by Eq. (6.7) has the advantage of handling multiaxial deformations, since
energy is a scalar which can be computed in terms of force and displacement or deformation
vectors. This is of particular interest for the earthquake response in three dimensions, particu-
larly in columns due to biaxial bending plus axial force interaction. However, the so-obtained
damage index refers to the whole element and not to individual plastic hinges at element end
sections, which may become a serious drawback if mechanism-based reliability analysis is
sought (where damage must be evaluated at the plastic hinge level). To overcome this short-
coming, the authors suggest energy splitting between both element end sections, which still
involves some problems as pointed out in Fardis et al. (1993) and discussed later in 6.3.2.4.
δM δu Qy dE
Ec δ( ) δ α
β δu Qy
Ec δ( ) α β
D
DmaxEd β Ed∫+
Ed u,------------------------------------=
maxEd
Ed u,
maxEd δM
Ed u,
268 Chapter 6
Concerning the parameters involved in Eq. (6.7), namely the factor and the failure deforma-
tion under monotonic loading for computation of , the expressions for the Park and
Ang index parameters were used and their adequacy was discussed and assessed by the appli-
cation of Eq. (6.7) to the large experimental data set above mentioned (which included some
biaxial bending tests). Results of Eq. (6.7) showed quite good agreement with experimental
ones, mostly when a constant value of 0.03 was adopted instead of the expressions proposed
by Park et al. The best choice of parameter fitting led to a mean damage value of 1.05 with a
coefficient of variation of 55%, which is still a very high result scatter, demanding further
improvements on parameter assessment.
The damage index proposal given by Eq. (6.7) seems to be a very promising and consistent
option for damage quantification, the major problem being the link with subsequent reliability
studies. The output provides an overall picture of the damage distribution in the structure, and
even a global average damage value, but the lack of information about the damage state of the
critical zones (plastic hinges) appears as a weak point of the proposal which needs to be
improved.
From an assessment of damage indices performed by Fardis et al. (1993), it became clear that
the best available are the Park and Ang proposal and their own suggestion of energy-based
index described in the same study. In the present work, the Park and Ang damage index given
by Eq. (6.5) was preferred, for the following reasons:
• In spite of the promising results of the proposal by Fardis et al. (1993) and of its agreement
with experimental results (as good as the Park and Ang expression agreement), Eq. (6.5) is
still the most established and widely accepted at present in damage assessment.
• Since only planar analyses are performed, with no bending and axial force interaction, the
damage source is restricted to uniaxial bending and, therefore, the potential advantage of
using Eq. (6.7) is diminished.
• It is of low computational cost in the context of the developed computer code and, more
importantly, the damage values are directly associated with each plastic hinge as they come
from end section deformations;
In the following paragraphs details are given of the expressions adopted for the chosen peak
value measure and the specifically related parameters.
β
δu Ed u,
β
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 269
6.3.2 The Park and Ang damage index
6.3.2.1 The damage parameter
In the present study, Eq. (6.5) was used to quantify damage in terms of the chord rotation at
each element end section. Being defined as the rotation between the tangent to the element axis
at one end section and the chord connecting both end sections, it follows that chord rotations
coincide with the rotations and in the element reduced space (or basic system) as shown
in Figure 2.4-e). Particularly, since only planar analyses are performed, the rotations of interest
are , i.e. those producing bending in the frame plane; in the following, they are simply noted
by . Therefore, the damage index as used in the present work is expressed by
(6.8)
where, the yielding moment is known from the model skeleton curve and the parameter ,
the same as used in the hysteretic model for controlling the strength degradation, is given in
Appendix B by Eq. (B.4). The peak value is directly obtained from the maximum response,
for each element end section, whilst the remaining terms, namely the ultimate rotation and
the dissipated hysteretic energy , are computed as explained in 6.3.2.3 and 6.3.2.4.
Damage values are calculated for both positive and negative bending directions, thus requiring
, , and to be evaluated separately. The hysteretic energy contributes equally to the
positive and negative bending damage indices, for consistency with the definition of parameter
as stated in Appendix B. Indeed, according to Figure B.3, the incremental hysteretic energy
refers to a complete cycle (possibly with interior small ones) instead of any energy splitting
due to positive and negative deformations.
Rotations were preferred to curvatures because they can give an integrated measure of damage
associated with each end section and, most importantly, they are less sensitive to the plastic
hinge development length and to the distribution of curvatures there existing. If, by virtue of
the assumed flexibility distributions, the curvature presents a very sharp distribution in the end
section neighbourhood, then very high curvature values may be found and the damage
becomes overestimated. Moreover, in reinforced concrete elements and due to the crack devel-
opment, the curvature concept has to be understood in an “average” manner, i.e., as the rotation
θy θz
θy
θ
D θM
θu------ β
Myθu------------ Ed∫+=
My β
θM
θu
Ed∫
θM θu My β
β
270 Chapter 6
divided by the finite distance between two sections where deformations are obtained. There-
fore, the rotation damage appears closer to the physical phenomenon than curvature damage
and can be better compared with available experimental results, most often expressed in terms
of rotations or displacements.
It is worth recalling that the option for chord rotation as damage parameter is inspired on the
equivalence of an element in anti-symmetric bending with the two cantilevers having the same
chord rotations of each end section, as shown in Figure 6.7.
Figure 6.7 Equivalence of element in anti-symmetric bending with cantilever elements
Each cantilever length is given by the so-called shear-span , where stands for
the end section shear force. The tip displacement equals the displacement of the point of
inflection I (for ) relative to the tangent to the element axis at the end section and the
chord rotation becomes the cantilever drift ; however, one must bear in mind that
such equivalence is strictly valid only for anti-symmetric bending, because in the other cases
the inflection point is not necessarily lying on the undeformed axis.
Note that the chord rotations are directly known in the flexibility element context; in addition,
although the above stated analogy is not general, it is helpful to overcome the difficulty of
defining yielding and ultimate chord rotations, indeed the major drawback of adopting chord
rotation as damage parameter.
1 2
M1
M2+
-
+M2
-M1
I
Ls1
Ls2
L
Ls1
Ls2θ1
θ2
θ2
θ1δ1
δ2
δ1
δ2
1 2
LsiMi Vi⁄= Vi
δi
M 0=
θi δi Lsi⁄
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 271
6.3.2.2 Yielding rotation
The need for defining the yielding rotation is mainly due to ductility assessment purposes.
Beside the curvature ductility factor given by , where is the yielding curvature,
the chord rotation ductility factor (or simply rotation ductility) can be defined by .
Similarly to , also the yielding rotation can be associated with the limit state of tensile
reinforcement yielding at the most stressed section, in general the end sections. However,
while the attainment of that limit state is sufficient for the yielding curvature definition, the
quantification of the yielding rotation requires further knowledge about the deformation along
a certain length, which, by the analogy shown in Figure 6.7, can be assumed as the shear-span.
At this stage, two important aspects must be discussed: which shear-span shall be considered
and in what consists the deformation along that span?
The shear-span is obviously dependent on the moments installed at both end sections when
yielding is attained at the end section of interest. From the point of view of the stand-alone ele-
ment, this is very unpredictable, since it depends on factors that are unknown prior to the struc-
ture analysis, such as the initial distribution of internal forces and subsequent redistributions
due to plastic hinging, the configuration of applied loads, etc. Moreover, the presence of trans-
verse loads applied along the element complicates even more the problem, because the validity
of “splitting” the member behaviour into the two cantilevers becomes questionable. Taking
into account all the involved uncertainties causing fluctuation of the shear-span value, and
bearing in mind the essentially conventional character of the shear-span, the half member
length is considered for the shear-span of all element end sections, regardless of the bending
direction and of the existence, or not, of transversal loads.
While recognizing that such option can be questionable, as it may not be fully consistent with
the element state when the maximum value of chord rotation is reached during the response, it
still provides a means of comparing, for all the structure elements, how far the chord rotations
go with respect to a conventional measure of yielding.
Regarding the deformations along the shear-span, they include several contributions, namely
the flexural and shear, elastic and inelastic, deformations, the effects of cracks, inclined or not,
the bond-slippage effects, the tension-stiffening effect between cracks, etc. In the context of
the Park and Ang damage model (Park et al. (1984)), the authors considered the yielding rota-
µϕ ϕ ϕy⁄= ϕy
µθ θ θy⁄=
ϕy θy
272 Chapter 6
tion given by
(6.9)
where the subscripts “flex”, “slip” and “shear” indicate the source of each contribution, i.e.,
flexural, slippage and shear, respectively, and the superscript “PA” stands for Park and Ang.
The flexural contribution results from the assumption of a linear variation of curvature along
the shear-span (from at the end section to zero at the inflection point) and also includes the
elastic contribution of shear.
The term is estimated as a concentrated rotation due to reinforcement slip at the face of
the crack often occurring at the end section. A bond-slip model is adopted, similar to the one
proposed in the CEB-FIP Model Code (1990), to obtain the slip when the tension reinforce-
ment yields; the corresponding rotation is calculated by , where is an estimate
of the internal lever arm.
The contribution of is roughly estimated by an idealized shear cracking model, where
45o inclined cracks are assumed to develop, uniformly spaced by , along a certain length
from the end section to the inflection point. At each crack a constant concentrated rotation
is assumed to occur, leading to a transverse displacement of the inflection point, equal to the
rotation times the mean distance to the crack. An approximate expression is proposed for the
length of the “no shear crack zone” and thus, by summing up the contributions due to all the
cracks, the transverse displacement of the inflection point can be expressed in terms of ,
as well as the chord rotation .
Empirical expressions were obtained for by means of the large set of experimental results
used for the damage index validation (Park et al. (1984)), specifically the tests where the yield-
ing point could be identified and the experimental chord rotation could be measured.
After substitution of in the left hand side of Eq. (6.9) and upon calculation of the terms
and , the shear contribution was obtained and, subsequently, was esti-
mated allowing regression analysis.
The full set of expressions for the application of Eq. (6.9) is extensively described in Park et al.
(1984) and was later detailed and translated to more usual European notation and units by
θyPA θy flex, θy slip, θy shear,+ +=
ϕy
θy slip,
sy
θy slip, sy z⁄= z
θy shear,
z
θs
δs θs
θy shear,
θs
θyexp
θyexp
θy flex, θy slip, θy shear, θs
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 273
Fardis et al. (1993). Note that the peak response displacement as used by Park et al. (1984)
was calculated by means of force-displacement relationships tuned by the experimental results,
namely the trilinear envelopes and the hysteretic energy absorption per cycle. This led to struc-
tural responses in which all the above mentioned non-linear phenomena are included since the
model was fitted for that purpose. Therefore, both the displacement (or rotation) demand and
their yielding and ultimate values are consistently obtained.
In the present work the adopted model does not take into account the slippage and inelastic
shear contributions for the member deformations. Thus, it appears logical that for the evalua-
tion of the chord rotation ductility the yielding rotation shall include only the flexural (elastic
and inelastic) and the elastic shear contributions as for the calculation of the response values of
. According to the trilinear model used, with the adjustments for the cracking plateau as
referred in 3.5, and for the assumption of inflection point at mid-span as stated above, the
yielding chord rotation is obtained by integration of the shaded diagram of curvatures shown in
Figure 6.8 over an element with half of the total length; the steps referred in 3.6.7 are applied
for due consideration of elastic shear deformation.
Figure 6.8 Yielding chord rotation
6.3.2.3 Ultimate rotation
The ultimate chord rotation necessary for the damage index evaluation by Eq. (6.5) was
first proposed by the authors (Park et al. (1984)), as given by
(6.10)
where the ultimate rotation ductility was given by empirical expressions obtained from
regression analysis of experimental results, and calculated using Eq. (6.9). Another empir-
ical expression was proposed later (Park et al. (1987a)) for a one step calculation of ,
δM
θ
L/2 L/2
Moments Curvatures
Myϕy
ϕcc
ϕc
CY CY
Mcθy
Yielding
θu
θuPA µθ u,
PA θyPA
=
µθ u,PA
θyPA
θuPA
274 Chapter 6
directly from the member characteristics and is given by
(6.11)
where is the volumetric confinement ratio, is the normalized axial force (positive for
compressive forces), is the mechanical ratio of tension reinforcement and is the cylindri-
cal compressive strength of concrete. and refer to the shear-span and to the effective sec-
tion depth, respectively.
From the study performed by Fardis et al. (1993), it was concluded that the best agreement of
the Park and Ang index with the experimental results was achieved by using Eq. (6.11) for
definition.
However, the direct use of Eq. (6.11) for calculation of in this work does not seem very log-
ical for the same reasons pointed out before. The response value of provided by the model
does not include the whole set of non-linear phenomena implicitly taken into account in the
experimentally based expression of and, therefore, underestimated damage values are
likely to be obtained if such expression is directly used.
Actually, an alternative procedure for consistency between the analysis and the damage model
assumptions could be based on Fardis et al. (1993), in which the skeleton curve of the point
hinge model, directly expressed in terms of chord rotations, was adjusted to match and
, in the yielding and the ultimate points, respectively. However, such fitting of the skeleton
curve is not possible with the model used herein, because it is based in curvatures and no pre-
defined plastic hinge length is considered to provide “equivalence” between curvature and
rotation.
Another solution to estimate could be the integration of the curvature diagram as proposed
for the yielding rotation. The ultimate limit state, characterized by the ultimate point in the tri-
linear curve, is assumed to be attained at the end section and a linear moment diagram is
adopted between that section and the inflection point as shown in Figure 6.9; the correspond-
ing curvature diagram can be integrated to obtain the chord rotation at that end section.
However, such chord rotation is a sort of lower bound of the ultimate rotation for similar rea-
θuPA 0.0634
Ls
d-----⎝ ⎠
⎛ ⎞0.93 max ρw 0.004( , )
max ν 0.05( , )-----------------------------------
0.48ωt
-0.27fc-0.15=
ρw ν
ωt fc
Ls d
θu
θu
θ
θuPA
θyPA
θuPA
θu
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 275
sons as for the yielding case; indeed, the non-linear effects of reinforcement bond-slippage and
pull-out and diagonal cracking caused by inelastic shear are not included and they are even
more pronounced near failure. Moreover, the ultimate rotation in the context of seismic resist-
ant structures often corresponds to a certain residual strength still retained after the peak
strength deformation is exceeded. Such is exactly the assumption underlying the proposals of
Park et al. (1984) for quantification of , in which failure is conventionally defined at
0.80Mmax on the descending branch of the monotonic moment-rotation curve.
Figure 6.9 Estimation of ultimate chord rotation by curvature integration
For the structure under analysis, the ultimate rotations were first calculated by integration of
curvatures, assuming the inflection point at the mid-span section, and the obtained damage
indices were excessively high, having reached maximum values about 0.75 for the simulation
of the 1.5S7 test. According to the damage index authors (Park et al. (1987a)) such value cor-
responds to a severe degree of damage, beyond repair, with extensive spalling and crushing of
concrete and buckling of rebars. The physical appearance of the structure after that test did not
reveal such extent of damage; indeed it appeared in a fairly good shape.
For these reasons such estimate of ultimate rotations could not be accepted and, considering
that direct application of Eq. (6.11) is not suitable as above explained, an intermediate
approach was adopted as follows.
Both the ultimate and yielding rotations were first calculated, according to Park et al., by Eqs.
(6.9) and (6.11) allowing a ultimate ductility factor to be obtained as . This
factor was preferred, instead of the one included in Eq. (6.10) for which direct expressions
were also proposed in Park et al. (1984), for the above explained reasons of better adequacy of
Eq. (6.11). Then, the ultimate rotation is obtained by
θu
Ultimate
Moments Curvatures
ϕy
ϕu
θu
My
Mc
Mu
CYU CYU
L/2 L/2
µθ u, θuPA θy
PA⁄=
276 Chapter 6
(6.12)
meaning that the same ultimate ductility is kept from the empirical expressions, but applied to
the yielding rotation obtained by the same procedure as used for the response evaluation with
the adopted model. Regarding the shear-span in Eq. (6.11), the same assumption was adopted
as for the calculation, i.e., half of the total element length for each end section.
While recognizing that such damage quantification procedure is debatable due to its somewhat
heuristic nature, it must be recognized that this is still an open issue whose solution is far from
straightforward. Indeed, a more coherent approach would require the explicit quantification of
bar-slippage and inelastic shear effects, but that is beyond the present work scope. Therefore,
rather than an absolute and accurate evaluation of damage at a given critical zone, the presently
adopted damage quantification should be mostly regarded as a comparative measure between
damaged zones (or between different structures); nevertheless, the results of its application to
the four storey structure under the 1.5S7 test (as presented later in 6.4) led to much more rea-
sonable damage values (maximum values around 0.4) and, thus, have encouraged its adoption
for damage quantification purposes in the present work.
6.3.2.4 Hysteretic dissipated energy
For the computation of the energy dissipated by hysteresis, the procedure adopted by Fardis et
al. (1993) was used in this study. The section internal forces and deformations relevant for hys-
teresis dissipation consist on moments and curvatures , whose deformation energy,
upon integration along the element length , leads to the total deformation energy of the ele-
ment.
However, considering the internal moments at each element node (say and ), in equilib-
rium with the internal section moments , the following expression applies
(6.13)
The splitting of element energy into the two end sections is convenient for damage computa-
tion purposes, but it is not a straightforward task. Indeed, it is a problem of similar nature to
that of defining the adequate shear-span for computing yielding and ultimate chord rotations,
θu µθ u, θy flex, θuPA θy flex, θy
PA⁄( )= =
θy
M x( ) ϕ x( )
L
M1 M2
M x( )
Ed∫ M x( ) ϕ x( )d xd0
L
∫∫ M1 θ1d M2 θ2d+( )∫= =
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 277
since that splitting could be done by dividing the integration length into two sub-zones
between each end section and the inflection point, the result being assigned to each element
end zone. However, despite the fact that such integration is a minor problem in the present
flexibility formulation context, this way of splitting fails whenever the inflection point falls
outside the element, or when there is more than one such point (as is the case when transversal
loads are considered). Specific criteria are required to split the energy integrated between the
two end sections for the first case, or the amount of energy in between inflection points for the
second case.
Alternatively, the energy splitting can be done as given by the right-hand side integral of Eq.
(6.13), by assigning each term inside brackets to the corresponding end section, thus
avoiding the computation of the integrals along the member length. This procedure is inde-
pendent of whether the inflection point exists or not and, in general, the results agree with the
distribution of deformations. However, this is not problem-free as explained next:
• It can be shown that the result of is not coincident with the energy integration
between the end section i and the inflection point (if existing), except for perfectly anti-
symmetrical situations; nevertheless, this means that, as long as the behaviour of the two
end sections is not very different, the deformation energy can be split by this simpler proce-
dure.
• In some particular situations, unreasonable results can be obtained by , as is the case
of Figure 6.10, where the illustrated beam is bent by the action of moment , the moment
diagram being shown in Figure 6.10-b) and corresponding to the incremental curvature dia-
gram depicted in Figure 6.10-c); by integration of along the ele-
ment, one would assign a non-zero energy value to the right node given by
, whilst a null value is obtained for and, consequently, an overesti-
mation of energy for the left node given by . Note that a similar inconsistency would
occur for a flexible support at node 2, allowing positive rotation but still for negative
moment ; a negative energy would be obtained by against the evidence of posi-
tive energy as the result of integration.
In spite of these shortcomings, this way of splitting energy was preferred to the integration pro-
cedure, for the following reasons:
• The main aim is to estimate the dissipated energy by hysteresis, whose contribution to the
damage value is weighted by the factor , having low values, typically less than 0.1; there-
Mi θid( )
Mi θid
Mi θid
M1
dE x( ) M x( ) ϕ x( )d xd=
dE2 dE x( )l1
L∫= M2 θ2d
M1 θ1d
θ2d
M2 M2 θ2d
β
278 Chapter 6
fore, any errors in estimating the energy contribution do not significantly affect the result;
note that the same reason would not apply if the fully energy-based index given by Eq. (6.7)
would had been adopted.
• In view of the previous point, the non-dependency on the existence of inflection point
makes this procedure advantageous, specially if one bears in mind that the energy is well
computed; indeed, if errors exist, they affect only the splitting between the end sections, but
the total energy is assigned to the element, meaning that any underestimation of damage at
one end section is balanced by overestimation at the other end.
Figure 6.10 Example of inconsistency in energy splitting between element end sections:
a) Beam and deformed shape, b) bending moments and c) curvature diagrams.
In a complex structure, where non-linear effects may lead to redistribution of internal forces
during earthquake loading, the type of particular situations as shown in Figure 6.10 is very
unpredictable. Therefore, since it is not acceptable that a given plastic hinge ends-up with a
negative value for dissipated energy, if such is found at the end of the analysis, then the corre-
sponding energy is set to zero.
M1
dϕ x( )L
I
1
a)
2
l1
M x( )
M2
dθ1dθ2 0=
x
I
dϕ1
dϕ2
M1
M2
b)
c)
+
+
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 279
6.4 Analysis of results from numerical simulations
6.4.1 General
The results of numerical simulations with the flexibility global element model are analysed in
the following sections through systematic comparison with experimental results. The main
objective is the assessment of the numerical model ability for reproducing the behaviour of
complete structures, taking advantage of pseudo-dynamic full-scale tests as an excellent means
of analysis of the mechanical response.
Additionally, the flexibility modelling performance is also compared to other analytical model-
ling tools. Among these, the analysis by means of point hinge or fixed length plastic hinge
models is, for sure, the most widely used at present. The latter modelling strategy is more ade-
quate in view of inelasticity spreading and, therefore, it was adopted in this comparative study;
for simplicity it is briefly denoted by F.H., standing for Fixed Hinge length.
Due to the nature of the PSD testing technique, all the experimental phases (except the free-
vibration tests) consisted of quasi-statically imposing displacements at each storey of the struc-
ture, which means that, from the strict experimental point of view, the tests are essentially tra-
ditional quasi-static ones since no actual inertia or damping effects developed in the structure.
Thus, for a given testing phase, numerical static analyses can be performed for an external
loading consisting of the displacements really imposed to structure, and the obtained response
can be compared to the experimental one without involving additional dynamic effects.
For adequate analytical simulations of the dynamic behaviour, three major issues have to be
properly addressed, viz the hysteretic model, the vibration frequencies and the damping. These
issues are fairly inter-related, since a good assessment of frequencies depends on the adopted
model, and the need for explicit inclusion of viscous damping depends on the model ability to
simulate the dissipation characteristics.
Naturally, dynamic modelling is more demanding and difficult than static analysis, because,
even if a suitable model of the hysteretic behaviour is available to provide good estimates of
peak response values, a wrong assessment of frequencies or an inadequate characterization of
damping (if not included in the behaviour model) may lead to poor simulations of the structural
dynamic response.
280 Chapter 6
In view of the above issues, several analyses were performed depending on their nature (static
or dynamic), on the type of test and on the modelling option. These analyses are summarized
in Table 6.3 where the actually performed numerical simulations are indicated with a cross.
A major concern of the numerical simulations was to follow the real test sequence as closely as
possible. Therefore, the actual conditions of load application and, particularly, the unloading
phases between tests had to be adequately traced out as described in 6.4.2.
Results obtained with flexibility modelling are first discussed for static analysis in 6.4.3, in
order to address the model performance for simulating the global hysteretic behaviour under
quasi-static loading. Then the corresponding dynamic analysis results are presented in 6.4.4,
restricted to those indicated in Table 6.3.
The comparison of results from the flexibility and the F.H. modelling is presented and dis-
cussed in 6.4.5, for both static and dynamic conditions.
6.4.2 Procedure for static analytical simulation of the tests
For the static analyses, the boundary conditions had to be adapted for imposing displacements
at each storey level. Therefore, horizontal supports were considered at each floor level, as
shown in Figure 6.11-a), and time histories of the experimentally applied displacements are
prescribed at these supports. The corresponding reactions, recorded at each step, provided the
storey restoring forces from which the inter-storey shears could be obtained and compared to
the experimental ones.
At the end of the test, actuators were driven to zero force, possibly with non-zero residual dis-
placements. In order to analytically simulate this force controlled unloading phase, the hori-
Table 6.3 Numerical simulations performed
Structure Status Test
Flexibility Modelling F.H. Modelling
Static Dynamic Static Dynamic
Bare0.4*S7 x x x x
1.5*S7 x x x x
InfilledUniform x - - -
Soft-Storey x - - -
Bare Final Cyclic x - - -
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 281
zontal supports had to be removed from the analytical model, while keeping the structural
stiffness unchanged. Such task is not easily handled in most of the structural analysis computer
codes, in which boundary conditions prescribed at the beginning of the analysis cannot be
modified because the stiffness matrix becomes definitely affected by the stiffness of supports.
However, due to the modularity and object-oriented features of CASTEM2000 (CEA (1990)),
various sources of stiffness can be considered, namely the material stiffness (strictly associated
with the constitutive material) and the rigidity due to local boundary conditions (only those of
fixing supports) and to non-local ones (i.e., those consisting of relations between several
degrees of freedom). The constraints associated with the boundary conditions are handled by
the Lagrange multiplier method (Pegon and Anthoine (1994)), leading to an expanded set of
degrees of freedom consisting on the original kinematic unknowns plus the Lagrange multipli-
ers. The process works independently of the structural material stiffness in the sense that, if
any changes are introduced in the boundary conditions only the coefficients of the Lagrange
multipliers are modified and the material stiffness matrix is not affected; both stiffness contri-
butions are put together in the global matrix only when the system of equations is to be solved.
It is worth mentioning that the possibility of prescribing non-local boundary conditions was
used to simulate the assumption of rigid floor diaphragm, by setting an equality condition for
horizontal displacements (in the loading direction) of all the nodes existing on the same floor,
thereby avoiding the use of artificial rigid links often adopted to connect frames in this kind of
planar analysis. Therefore, the internal and external frames are always shown disconnected, as
their connection is intrinsically taken into account in the model.
Taking advantage of the above referred features, the unloading phase is simply performed as
schematically illustrated in Figure 6.11-b), and consists in:
• reading the last reactions at the floor horizontal supports;
• removal of these floor supports and application of the previous horizontal reactions at each
floor level, progressively decreasing to zero force at completion of the unloading phase.
Before a new testing phase starts, the horizontal supports have to be set up again in order to
apply the new horizontal displacement time histories. Note that the material stiffness is the one
corresponding to the externally unloaded structure, although with a possible internal stress
state due to plastic deformations. Thus, the “actual” state of the structure is kept between con-
282 Chapter 6
secutive testing phases, so that the whole experimental campaign can be simulated as closely
as possible.
Figure 6.11 Storey displacement prescription and unloading to zero actuator forces
The infilled frame tests required special procedures for the simulation. After the unloading
phase of the previous test (the 1.5S7 one), the diagonal struts simulating the infill panels were
added to the unloaded structure with the actual stress, strain and damage states. Again, this
operation was possible thanks to the object-oriented features of CASTEM2000.
From the code point of view, the reinforced concrete frames are represented by one (or more)
model-type object, linked to a geometrical support (the mesh-type object) and a field of mate-
rial properties and characteristics (an object of type field-by-element) in which all the model
internal variables are included. These objects are the strictly necessary to define and construct,
at any stage, the stiffness-type object consisting on the element and the structural stiffness
matrices, either the initial elastic or the tangent ones.
Therefore, other objects of the same type (model, mesh and characteristics) can be defined for
the new sub-structure consisting of the diagonal struts and the corresponding stiffness-type
object can be obtained. These objects are then “assembled” to the corresponding ones of the
bare structure and the analysis is finally performed. Note that the mesh for the infill diagonal
struts is, obviously, built-up with the same nodes of the frames. The procedure is schematically
shown in Figure 6.12 for the uniformly infilled test.
Before the analysis starts, the horizontal supports are again set up exactly as before. Since, in
d1
d2
d3
d4
1
2
3
4
R1
R2
R3
R4
di
(Experimental)Input Output
Ri
t tf t
1
2
3
4
Input Output
R4f
R3f
R2f
R1f
d4r
d3r
d2r
d1r
tu t
Rif
- residual displacements
- tangent material stiffness
and internal forces
a) Imposing storey displacements b) unloading to zero actuator forces
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 283
the code context, the boundary conditions lead also to stiffness-type objects, the inclusion of
such supports is simply done by “adding” objects of the same type. That is why, from the user
point of view, the manipulation of boundary conditions turns out to be such an easy task. Once
the adequate stiffness-type object is defined, it can be simply included in or removed from the
total stiffness-type object.
Figure 6.12 Introduction of the infill panel diagonal struts
After the uniformly infilled test was statically simulated by imposing the experimental floor
displacements, again the horizontal supports were removed and the analytical model was
driven to zero external load. However, due to internal stress state, the infill diagonal struts still
interact with the reinforced concrete frames, which means that a straight removal of the infill
stiffness would lead to an out-of-equilibrium state.
Thus, another special procedure is required for the infill removal, as shown in Figure 6.13 and
it is performed as follows:
• the internal forces at the diagonal struts are evaluated from their internal stress state;
• upon removal of the infill contribution for the global stiffness-type object, the external bare
frame is loaded at the nodes where connection with infills did exist before, by forces ;
• the equilibrium being in this way assured, the bare structure is subjected to , progres-
sively decreasing to zero.
Internal Frame External Frame
Internal Frame External Frame
Internal Frame External Frame
Object Type:
Model
Mesh
Field ofCharacteristics
mb
Mb
Cb
Stiffness Kb
Model
Mesh
Field ofCharacteristics
mi
Mi
Ci
Stiffness Ki
Object Type:
Model
Mesh
Field ofCharacteristics
mT = mb+ mi
MT = Mb+ Mi
CT = Cb+ Ci
Stiffness KT = Kb+ Ki
Bare Structure
Infill Struts
Infilled structure
Fi
Fi–
Fi–
284 Chapter 6
Figure 6.13 Unloading of infilled frame configuration: a) removal of actuators and b), removal
of infill panels
At this stage, the configuration of the non-uniformly infilled frame could be set up exactly by
the same procedure as for the uniformly infilled case: first the model, mesh, characteristic and
model-type objects were defined for the diagonal struts (now, only at the three upper floors)
and superimposed with the bare frame ones; then the floor horizontal supports were added
again and the static analysis proceeded with the experimentally imposed displacements. At the
end, the same unloading process was applied (as for the uniformly infilled case) before starting
the final cyclic test simulations.
6.4.3 Static analysis by flexibility modelling versus experimental tests
The static analyses included in this section are strictly concerned with the assessment of the
flexibility modelling ability to simulate the hysteretic behaviour of the structure. However,
besides the simulations of experimental tests, a monotonic pushover analysis was numerically
performed in order to have an estimate of both the maximum base shear and the global yielding
displacement.
Since all the following analysis are statically performed, it is worth mentioning that when the
term “time” is used, it actually refers to the step of the analysis.
6.4.3.1 Pushover analysis
The pushover analysis was carried out by applying progressively increasing static forces at
each floor, with an inverted triangular distribution as indicated in Figure 6.14, up to a maxi-
mum top displacement similar to the maximum reached during the tests (about 0.60 m).
External FrameInternal Frame External Frame
Infill StrutsInfilled structure
After testing and unloadingto zero actuator forces
Internal Frame External Frame
Bare structure
Strut internal forces Fi Unloading of
tu t
- Fiinteraction forces
a) b)
0
0
0
0
d4
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 285
Figure 6.14 Pushover analysis: inverted triangular force distribution.
The obtained diagrams of base shear-top displacement and of storey shear-drift are plotted in
Figure 6.15, clearly showing a global behaviour of trilinear type. Thick and thin solid lines cor-
respond to pushover for positive and negative displacements, respectively. Due to frame and
beam section asymmetry and to the presence of vertical loads, positive and negative pushover
analysis do not lead to coincident results; however, differences are not significant.
Figure 6.15 Pushover analysis: base shear-top displacement and storey shear-drift diagrams.
Visual estimates of the yielding point can be extracted from Figure 6.15-a) as indicated by
small circles. Depending on the definition of global yielding, different values can be obtained:
• should global yielding be considered as the first attainment of yielding at the section level,
an estimate of 8 cm seems to be adequate;
• if the yielding point is defined where the force-displacement curve clearly changes slope,
Internal Frame External Frame
F1
F2
F3
F4d4
TOP DISPL.(m)
BASE SHEAR (kN)
.0 .15 .30 .45 .60 .75 .0
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50 x1.E3
DRIFT (%)
STOREY SHEAR (kN)
.0 1.5 3.0 4.5 6.0 7.5 .0
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50 x1.E3
Storey 1
Storey 2
Storey 3
Storey 4
0.14 m
0.11 m
0.08 m
a) Base Shear - Top Displacement b) Storey shear - drift
NegativePositive
NegativePositive
286 Chapter 6
then the yielding point approaches 14 cm for the top displacement;
• finally for the intersection of the straight lines fitting the post-cracking and the post-yielding
branches, the yielding displacement becomes an intermediate value around 11 cm.
The comparison of these values with the experimental yielding estimate is difficult to perform,
because no experimental pushover test was performed. As already referred, such an estimate (7
cm) was obtained from the base shear-top displacement diagrams of the 0.4S7 and 1.5S7 tests,
therefore with the inherent difficulties of establishing an adequate envelope of the force-dis-
placement diagram. However, in spite of the unavoidable crudeness of the experimental esti-
mate, the obtained analytical values show a trend for yielding displacements higher than the
experimental ones.
By contrast, the maximum base shear appears underestimated compared to experimental
results: according to 5.5.6, the maximum base shear reached for the final cyclic tests is higher
than 1400 kN, whilst in the pushover analysis (in both directions) it is around 1300 kN. This
aspect might be partially due to the low values of post-yielding stiffness adopted for the global
section model (see 6.2.3) and also due to strength mechanisms that may be differently acti-
vated for pushover analysis and for the tests.
Attention is drawn to the fact that, as for the experimental results, also the pushover analysis
led to maximum drift at the second storey as evidenced in Figure 6.15-b) by the dashed line
connecting the storey yielding points. This fact corroborates the reasons already given in 5.5.4
for the occurrence of the maximum drift at the second storey.
6.4.3.2 Bare frame seismic tests
The simulations of the bare frame seismic tests (see 5.5.4) are described in the next paragraphs
and compared to the experimental results. When time-histories or diagrams are compared in
the same drawing, dashed lines always refer to the experimental results.
Low level test (0.4S7)
For the 0.4S7 test, time histories of the first and fourth storey shear, the corresponding shear-
drift diagrams and evolutions of absorbed energy are plotted in Figure 6.16. The base shear-top
displacement diagram and the peak value profile of storey shear are shown in Figure 6.17,
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 287
including also a plot of the cracked zones of each structural member and the spatial distribution
of the maximum positive and negative chord rotations.
From the time histories of storey shear a very good agreement is found between analytical and
experimental results. There is a less good agreement during the initial phase before the first
peaks occur, which is related with the difficulty of adequately characterizing the initial stiff-
ness of the members. The presence of an initial state of micro-cracking, related with shrinkage
effects or caused by the specimen transportation, and variations on the slab participation may
justify the higher initial stiffness obtained in the analytical simulation and evidenced in Figure
6.16-b).
However, once larger incursions in the cracked phase have occurred, member stiffness became
well estimated and the analytical response matched the experimental one, particularly in the
first storey. In the top storey, even after the peaks, the agreement was not so good, though fairly
satisfactory, which may be related with the lower drift values obtained there (less than 50% of
the lower storeys) and a more reduced extent of cracking, thus keeping the uncertainty about
the initial stiffness.
Shear-drift diagrams show that unloading from the first peak (around 2.3s) is not followed by
the analytical response, since this corresponds to member sections only cracked for one bend-
ing direction, thus having a higher stiffness than the fully cracked one as assumed in the ana-
lytical model.
Nevertheless, after large peaks in both directions, the assumption of origin-oriented unloading
and reloading stiffness seems to be quite reasonable, as no major residual drifts persist after
significant cracking is stabilized. On the other hand, this is not sustained by the absorbed
energy diagrams where significant discrepancy can be found between the analytical and the
experimental results; yet, notice must be taken that the involved energy level for this test is less
than 10% of the energy absorbed in the 1.5S7 test and, therefore, this error in energy evaluation
is expected to significantly reduce in presence of the dissipated energy at the end of a post-
yielding analysis.
288 Chapter 6
Figu
re 6
.16
0.4
S7 te
st. S
tatic
ana
lysi
s ve
rsus
exp
erim
enta
l sto
rey
resu
lts
Tim
e (s
)
SHEA
R (k
N)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-7.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5
x1
.E2
DRI
FT (x
1.E-
3)
SHEA
R (k
N)
-5.0
-4
.0
-3.0
-2
.0
-1.0
.
0
1.0
2
.0
3.0
4
.0
5.0
-7
.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5
x1
.E2
Tim
e (s
)
ENER
GY
(kJ)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.
0
Tim
e (s
)
ENER
GY
(kJ)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.
0
Stor
ey 4
Stor
ey 1
Tim
e (s
)
SHEA
R (k
N)
DRI
FT (x
1.E-
3)
SHEA
R (k
N)
.0
.
8 1
.6
2.4
3.2
4.
0 4
.8
5.6
6.4
7.
2 8
.0
-3.0
-2.4
-1.8
-1.2
-.6
.0
.6
1.2
1.8
2.4
3.0
x1
.E2
-2.0
-1
.6
-1.2
-.
8 -
.4
.0
.4
.8
1
.2
1.6
2
.0
-3.0
-2.4
-1.8
-1.2
-.6
.0
.6
1.2
1.8
2.4
3.0
x1
.E2
b) S
tore
y sh
ear-
drift
dia
gram
c) S
tore
y en
ergy
a) S
tore
y sh
ear
Anal
ytic
Expe
rim
enta
l
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 289
Figu
re 6
.17
0.4S
7 te
st. S
tatic
ana
lysi
s ve
rsus
exp
erim
enta
l res
ults
TOP
DIS
PL (x
1.E-
2 m
).
BASE
SH
EAR
(kN)
-5.0
-4.
0 -3
.0 -
2.0
-1.0
.0
1
.0
2.0
3.0
4.
0 5
.0
-7.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5
x1
.E2
SHEA
R (x
1.E2
kN
)
STO
REY
.0
1
.5
3.0
4
.5
6.0
7
.5
1
2
3
4
Neg
ativ
e =
-3.2
4Po
sitiv
e =
4.0
6
a) B
ase
shea
r - to
p di
spla
cem
ent d
iagr
amb)
Pea
k sh
ear s
tore
y pr
ofile
c) A
naly
tical
cra
ckin
g pa
ttern
d) C
hord
Rot
atio
ns. M
ax.(m
Rad
):
Anal
ytic
Expe
rim
enta
l
290 Chapter 6
It is worth analysing the sources of the discrepancy of dissipated energy (given by the inferior
peaks of diagrams): the energy “jump”, between 2s and 3s time, is associated with the non-
fully cracked stiffness of sections where the cracking moments were exceeded only for one
bending direction, and cannot be followed by the numerical model because if cracking occurs
it is assumed in both directions; after that, the dissipated energy discrepancy increases progres-
sively (although at a decreasing rate) due to residual deformations, that, indeed do exist, and
cannot be considered in the model.
Globally, a quite good agreement is achieved between the base shear - top displacement dia-
grams resulting from numerical and experimental analysis (see Figure 6.17-a)), as well as for
the peak values of storey shear shown in Figure 6.17-b), where differences between calculated
and measured values are less than 5%.
From the final cracking pattern (see Figure 6.17-c)) the following is highlighted:
• The behaviour of short-span beams is mainly controlled by the lateral action, since no cen-
tral cracking develops and the extent of cracking decreases from the lower to the upper
beams, thus following the trend of the storey shear profile.
• In the long-span beams, the influence of the vertical load is quite apparent due to the
obtained central cracking, which is less in the external frames due to their stiffness being
more than twice that of the internal one (whilst the vertical load is exactly the double);
moreover, the cracking in these beams is more uniform in height than for the short-span
beams, thus confirming the greater influence of vertical loads for the longer span beams.
• Cracking is more relevant in columns adjacent to the long span beams, due to the vertical
load effect, but is well distributed along the height.
Peak-values of chord rotations (see Figure 6.17-d) are essentially decreasing along the height,
although for the long span beams and the adjacent columns this trend is not strictly followed,
due to the vertical load effect. Note that, both maximum positive and negative peak-values are
higher than experimental ones (max. 2.51 mRad, according to Negro et al. (1994)), but they are
not directly comparable because the latter refer only to the plastic hinge zone.
However, integration of numerically obtained curvatures inside the length of the plastic zone
considered in the test, leads to maximum values of 2.1 and -3.0 mRad, therefore comparing
much better with the experimental ones. Such analytical values actually refer only to the 0.4S7
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 291
test, because the contribution of initial rotations previously installed due to vertical loads was
removed to make results comparable with the experimental ones.
High level test (1.5S7)
The analytical and experimental results for the 1.5S7 test are included in Figures 6.18 to 6.20.
For the first storey, it is apparent the quite good agreement between analytical and experimen-
tal shear responses shown in Figure 6.18-a): the highest experimental shear peak is only 10%
underestimated by the analysis. This can be related with the collaborating slab width, for which
the experimental estimates were somewhat higher than those adopted in the calculations
(therefore developing higher strength), and also with the low post-yielding stiffness in the
moment-curvature relationships. By contrast, during reloading phases, the analytical shear
overestimates the response, due to the pronounced pinched shape of the experimental shear-
drift diagram shown in Figure 6.18-b). The analytical loops become exterior to the experimen-
tal ones, thus leading to a larger amount of dissipated energy as evident from Figure 6.18-c).
Actually, the comparison of shear-drift diagrams suggests that it might be difficult to ade-
quately simulate such a pronounced pinching effect by two straight lines as adopted in the
present behaviour model; nevertheless, energy dissipation agrees reasonably well.
In the top storey, deviations between analytical and experimental shear are more evident (peak
values about 16% underestimated) and the absorbed energy obtained from the calculations is
significantly lower than the measured one; however, the effect in the overall dissipation is
minor due to the reduced contribution of that storey. A lower stiffness is exhibited by the corre-
sponding shear-drift diagram, suggesting that excessive cracking might have been obtained in
the analytical model; several factors may have contributed, namely the slab width and the way
how vertical loads were considered (note that such loads are relatively more important in the
top storey than in the others).
From the above explained, the analytic response is characterized by a slight underestimation of
storey peak shear but, overall, a reasonably good agreement of shear-drift diagrams is found,
although with some difficulties in simulating the pinching effect (see also Figure 6.19-a)) lead-
ing to slight overestimates of the global dissipated energy. However, this fact was somehow
expected, since no rebar slippage and pull-out effects are modelled, and they seem to be
responsible for the significantly pinched shape of the diagrams.
292 Chapter 6
Figu
re 6
.18
1.5
S7 te
st. S
tatic
ana
lysi
s ve
rsus
exp
erim
enta
l sto
rey
resu
lts
.0
.
8
1.6
2.4
3.
2 4
.0
4.8
5.6
6.
4 7
.2
8.0
-1
.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
x1
.E3
-2.5
-2
.0
-1.5
-1
.0
-.5
.
0
.5
1
.0
1.5
2
.0
2.5
-1
.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
x1
.E3
.0
.
8
1.6
2.4
3.
2 4
.0
4.8
5.6
6.
4 7
.2
8.0
.0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
x1
.E2
.0
.
8
1.6
2.4
3.
2 4
.0
4.8
5.6
6.
4 7
.2
8.0
.0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
x1
.E2
Tim
e (s
)
SHEA
R (k
N)
DRI
FT (%
)
SHEA
R (k
N)
Tim
e (s
)
ENER
GY
(kJ)
Stor
ey 1
Tim
e (s
)
SHEA
R (k
N)
DRI
FT (%
)
SHEA
R (k
N)
Tim
e (s
)
ENER
GY
(kJ)
Stor
ey 4
.0
.
8
1.6
2.4
3.
2 4
.0
4.8
5.6
6.
4 7
.2
8.0
-9.0
-7.2
-5.4
-3.6
-1.8
.0
1.8
3.6
5.4
7.2
9.0
x1
.E2
-1.5
-1
.2
-.9
-.
6
-.3
.
0
.3
.6
.9
1
.2
1.5
-9
.0
-7.2
-5.4
-3.6
-1.8
.0
1.8
3.6
5.4
7.2
9.0
x1
.E2
b) S
tore
y sh
ear-
drift
dia
gram
c) S
tore
y en
ergy
a) S
tore
y sh
ear
Anal
ytic
Expe
rim
enta
l
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 293
Figu
re 6
.19
1.5S
7 te
st. S
tatic
ana
lysi
s ve
rsus
exp
erim
enta
l res
ults
TOP
DIS
PL (m
)SH
EAR
(x1.
E3 k
N)
STO
REY
1
2
3
4
a) B
ase
shea
r - to
p di
spla
cem
ent d
iagr
amb)
Pea
k sh
ear s
tore
y pr
ofile
c) A
naly
tical
cra
ckin
g pa
ttern
d) A
naly
tical
yie
ldin
g pa
ttern
BASE
SH
EAR
(kN)
-.2
5 -
.2
-.15
-.1
-.
05
.0
.05
.1
.
15
.2
.25
-1
.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
x1
.E3
.0
.3
.6
.9
1.2
1
.5
Max
. Yie
ldin
g Le
ngth
: 0.2
9 m
Anal
ytic
Expe
rim
enta
l
294 Chapter 6
Figure 6.20 1.5S7 test - Static analysis. Spatial distributions of peak values
a) Chord Rotations. Max.(mRad): Negative = -21.3Positive = 24.0
b) Chord Rotation Ductility. Max.: Negative = 4.0Positive = 4.8
c) Damage. Max.: Negative = 0.36Positive = 0.41
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 295
The analytical cracking pattern included in Figure 6.19-c) shows extensive cracking progres-
sion both in beams and columns, having even reached the full length in some beams and indi-
cating the fact that this test went far beyond the cracking phase.
The yielding pattern reveals that plastic hinge formation took place mainly at the beams, where
the maximum yielding progression reached about 30 cm, and also, but less evidently at some
column end sections. Note that yielding lengths are measured in the flexible portion of each
member; for the beams this means yielding development starting at the beam-column face,
whilst for the columns the plotted yielding zones start at the beam-axis. Since the beams are 45
cm deep and the maximum plastic length in columns is visibly less than half the maximum
yielding length, it follows that, all the column yielding zones next to joints, actually do not
develop in the member. Therefore, plastic hinges in vertical elements reduce only to the bottom
sections of base-columns as expected.
The maximum plastic hinge length is 0.67h (where h is the beam depth), thus inside the range
of values often adopted (0.5h to 1.5h), and occurs in the first storey long-span beam due to the
combination of vertical load effects with the high shear at that storey. Following the decreasing
trend of shear in height, the yielding lengths reduce towards the upper storeys.
The spatial distributions of maximum chord rotation, the corresponding ductility and damage
are illustrated in Figure 6.20. Concentric lozenges are plotted for each member end section,
with diagonals proportional to the maximum positive and negative values. In order to identify
where member yielding occurs, only ductility values greater than unity are plotted.
Chord rotations are significantly higher (about 60%) than experimental rotations (see Figure
5.9), as already found for the low level test. Again, the integration of numerically obtained cur-
vatures inside the “experimental” plastic zones (45 cm both in beams and columns) leads to
20.7 mRad and -19.4 mRad, for plastic rotations, referring exclusively to the contribution of
the 1.5S7 test. These results are closer to the experimental ones, but still significantly higher,
which agrees with the overestimation of the yielding displacement given by the pushover anal-
ysis; the low values of post-yielding stiffness used in the section modelling may be responsible
for this rotation discrepancy.
However, the overall distribution of chord rotations agrees with the experimentally measured
rotations, showing decreasing values on beams along the height and higher rotations at the
296 Chapter 6
base-columns in accordance with the yielding pattern. The concrete weakness in the third floor
columns (see concrete properties in Table 5.1) is reflected in locally higher rotations, when
compared to those of the second storey columns where the maximum drift occurs.
Ductility distribution exhibits the same pattern of chord rotations and reaches the maximum
value of 4.8. This indicator of how far the structure is from yielding, serves mainly for compar-
ison between the various loading stages. Its consistency with other measures of ductility, as for
example the top displacement ductility, cannot be easily checked since no direct relation can be
derived for complex structures.
Actually, for the simplest case of a cantilever beam loaded by a vertical force at the tip, the dis-
placement ductility factor is coincident with the chord rotation ductility factor and can
be easily related with curvature ductility , either by the model adopted herein or by other
proposals such as in Paulay and Priestley (1992). However, for complex structures, the relation
between a global displacement and any member deflection or rotation is dependent on several
factors which include: the type of deformations involved (i.e., the significance of shear, joint
and any foundation deformations), the strength proportions between members, the loading pat-
tern, etc. Furthermore, the definition of yielding displacement requires the assumption of the
mechanism likely to develop, which, for subsequent loading stages can change due to succes-
sive formation of plastic hinges. It follows that any explicit relation between global and local
ductility factors appears impossible to be defined, unless parametric studies are performed.
In spite of the absence of any helpful relation between structure and element ductilities, it is
worth recalling that this 1.5S7 test roughly led to the top displacement ductility of 3. The value
obtained for the rotation ductility (4.8) confirms the expected trend in the ductile behaviour of
structures, i.e. the ductility demand of any member of a ductile chain is greater than the global
ductility demand, but no other conclusions can be easily drawn.
The damage pattern appears consistent with the rotation and ductility distributions, illustrating
the dissipative mechanism of the structure, mainly based on the three first storey beams and on
the base columns. Globally the damage is well distributed, although with higher values for the
short-span beams, and the dissipation occurs as expected from the capacity design require-
ments. According to the approximate scale of damage proposed by Park et al. (1984), relating
the damage index values of several earthquake damaged structures with the visible damage
µ∆ µθ
µϕ
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 297
observed, the maximum value obtained for this test (around 0.4) seems reasonable, although
somewhat high, when compared with the damage state registered during and after the test:
major cracks permanently open at beam-column interfaces, cracking along members and in
joints for peak deflections and no concrete crushing or reinforcement buckling.
6.4.3.3 Infilled frame tests
Results of infilled frame numerical analyses and their comparisons against experimental
results are shown in Figures 6.21 and 6.22, respectively for the uniformly infilled and the soft-
storey cases. A general overview reveals that, by contrast with the previous results for bare
frame tests, a fairly less good agreement with experimental results is obtained for the infilled
configurations.
For the uniformly infilled case, the first storey shear evolution (see Figure 6.21-a)) matches
reasonably well the experimental response and the peak value is properly estimated. In spite of
a higher stiffness for the negative loading direction, the analytical shear-drift diagram (see Fig-
ure 6.21-b)) gives a good prediction of the experimental behaviour in that storey. After the
large drift peaks have occurred in both directions, i.e. after 4.8s, the numerical and experimen-
tal responses compare very well, which is related to the fact that the panel failed (as actually
observed during the test) and the response became controlled by the frame at that storey.
In the top storey, the analytic results significantly deviate from the experimental ones. Clearly
the analytic initial stiffness is overevaluated and, since the response for this storey takes place
in the low drift range (post-elastic but pre-crushing), the shear force becomes fairly larger than
the experimental one. This is thought to be related with the adopted panel characteristics for
the upper storeys: according to Combescure (1996) they have been taken equal to those of the
second storey, under the assumption that it could be representative of the three upper storeys;
however, this is not confirmed by the results obtained here. In addition, the response in the
cracking range is mainly controlled by the masonry-frame interface behaviour, whose initial
state can hardly be assessed with reasonable accuracy. Indeed, the joint characteristics for
interface modelling were adopted with average values (Combescure (1996)), some of them
estimated from numerical simulations of other experiments independent of the present one.
Thus, it is recognized that further effort should be invested in parameter characterization,
mainly directly from the experimental results rather than from numerical refined simulations.
298 Chapter 6
Figure 6.21 Uniformly infilled test. Static analysis versus experimental results
-5.0 -4.0 -3.0 -2.0 -1.0 .0 1.0 2.0 3.0 4.0 5.0 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
Time (s)
SHEAR (kN)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 -2.5
-2.0
-1.5
-1.0
-.5
.0
.5
1.0
1.5
2.0
2.5 x1.E3 SHEAR (kN)
SHEAR (x1.E3 kN)
STOREY
.0 .5 1.0 1.5 2.0 2.5
1
2
3
4 ENERGY (kJ)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0 x1.E2
Storey 4
Storey 1
DRIFT (%)
Time (s)
-1.5 -1.2 -.9 -.6 -.3 .0 .3 .6 .9 1.2 1.5 -2.5
-2.0
-1.5
-1.0
-.5
.0
.5
1.0
1.5
2.0
2.5 x1.E3
Time (s)
SHEAR (kN)
DRIFT (x1.E-3)
SHEAR (kN)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
a) Storey shear b) Storey shear-drift diagrams
c) Peak shear storey profile d) Energy diagrams
Analytic
Experimental
Analytic (RC frame only)
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 299
Figure 6.22 Soft-storey test. Static analysis versus experimental results
SHEAR (kN)
Time (s)
SHEAR (kN)
DRIFT (x1.E-3)
SHEAR (kN)
STOREY
.0 .5 1.0 1.5 2.0 2.5
1
2
3
4
DRIFT (%)
SHEAR (x1.E3 kN)
a) Storey shear b) Storey shear-drift diagrams
c) Peak shear storey profile
Storey 4
Storey 1
d) Energy diagrams
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
-5.0 -4.0 -3.0 -2.0 -1.0 .0 1.0 2.0 3.0 4.0 5.0 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
Time (s)
SHEAR (kN)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 -2.5
-2.0
-1.5
-1.0
-.5
.0
.5
1.0
1.5
2.0
2.5 x1.E3
-5.0 -4.0 -3.0 -2.0 -1.0 .0 1.0 2.0 3.0 4.0 5.0 -2.5
-2.0
-1.5
-1.0
-.5
.0
.5
1.0
1.5
2.0
2.5 x1.E3
Time (s)
ENERGY (kJ)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0 x1.E2
Analytic
Experimental
Analytic (RC frame only)
300 Chapter 6
The storey profile of peak shear values confirms how crude is the analytical prediction even if
a good estimate of the base shear was achieved (see Figure 6.21-c)). Note the almost uniform
peak shear obtained in the three upper storeys which reflects the similar characteristics adopted
for the corresponding diagonals.
Figure 6.21-d) shows the experimental and the analytical evolutions of deformation energy; for
the analytical results the contribution of the reinforced concrete alone is also included, from
which can be seen the predominance of the infills in the overall response. The low rate of
energy increase (namely after the larger peaks) is a clear indication of the low dissipation
capacity typical of masonry panels under cyclic loading.
For the soft-storey configuration a reasonably good agreement is found between analytical and
experimental results in the first storey (see Figure 6.22-a)), where the resistance to lateral loads
is mainly controlled by the reinforced concrete frame. For the first incursions into larger drift
ranges, the strength is underestimated, as observed already for the 1.5S7 test, and this trend
remains for the post-peak response in the positive direction, apparently due to excessive degra-
dation of strength. Attention is drawn to the fact that the unloading stiffness degrades more in
the analytical prediction than in the experiment, suggesting that the controlling parameter,
which seemed adequate for the previous bare frame tests, may not be appropriate for such
higher drift range; actually, the same fact is confirmed in the final test results.
The behaviour in the top storey exhibits the same problems as for the uniformly infilled case
(see Figure 6.22-a)): the excessive initial stiffness leads to overestimation of storey shear,
which helped to distort the storey profile of peak shear values. Actually, Figure 6.22-c) shows
a strange decrease in shear at the second storey due to force transfer to the upper storeys, which
remained stiffer due to their lower damage.
The deformation energy diagrams illustrated in Figure 6.22-d) almost duplicated the values of
the uniformly infilled case. This is due to the larger engagement of the reinforced concrete at
the first storey, which gives now the main contribution as clearly shown by the analytical
energy diagrams.
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 301
6.4.3.4 Final cyclic test
Due to the extent of damage induced by the soft-storey test, some of the first storey member
end zones had to be repaired as referred in 5.5.6. This intervention cannot be easily simulated
in the analysis, thus introducing an increased degree of uncertainty and difficulty on reproduc-
ing the experimental behaviour for the final test. Additionally, at the end of the soft-storey test
the structure position had to be adjusted in order to reduce the visible residual drifts. Difficul-
ties in monitoring this operation and the resetting of displacement transducers to zero at the
beginning of the final cyclic test, prevented the initial position to be adequately assessed for
the numerical analysis.
The above referred reasons raised the question of whether the final test simulation should be
done after the infilled test analyses, or alternatively, by ignoring them and proceeding directly
from the 1.5S7 test. In order to clarify this topic, the analytical results of the Duct. 3 phase of
the final test were compared with the experiment in two distinct situations, viz by considering
or by neglecting the infilled tests in analytical simulations. The results of such a comparison in
terms of shear-drift diagrams, are shown in Figure 6.23 for both the first and the second storey.
The most relevant aspect is the better agreement with the experimental results when tests with
infills are not considered. By neglecting these tests (see Figure 6.23-a)), for both the first and
the second storeys, the response is quite well captured in the negative side where the maximum
drift was previously exceeded.
However, in the positive side, the first storey diagram exhibits further plastification because
the corresponding peak drift for Duct. 3 has not yet been reached; additionally, the peak shear
force becomes higher than the experimental one because the analytical simulation does “see”
the strength and stiffness degradation occurred during the infilled frame tests.
On the contrary, the effect of considering the tests with infills is a drastic reduction of the first
storey stiffness due to the higher drift values previously reached during the soft-storey test.
Note the comparison of peak values of inter-storey drifts shown in Figure 6.24 for the 1.5S7,
the soft-storey and the final Duct. 3 tests, from which it is apparent that, concerning the rein-
forced concrete structure, only the first storey was seriously affected by the infilled frame tests.
302 Chapter 6
Figure 6.23 Shear-drift diagrams for the Ductility 3 phase of final test. Effects of considering or
neglecting infilled frame tests.
After specimen repairing the experimental stiffness was partially recovered, but this cannot be
reproduced by the analytical simulation; consequently, results deviate quite a lot as evidenced
in Figure 6.23-b) for the first storey. Moreover, such deviation is also responsible for the
results of the second storey: both the initial stiffness and the maximum shear force are overes-
timated, which can be due to force transfer from the first to the second (and less damaged) sto-
rey.
b) Considering infillsa) Direct (neglecting infills)
Storey 1
DRIFT (%)
SHEAR (kN)
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
Storey 1
Storey 2 Storey 2
Analytic
Experimental
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 303
Figure 6.24 Inter-storey drift profiles for 1.5S7, soft-storey and final Duct. 3 tests
The aforementioned reasons indicate that analytical simulations of the final cyclic tests are bet-
ter achieved if the infilled frame tests are ignored. Even so, the uncertainty about the initial
position of the structure still persists and this has to be kept in mind when comparing analytical
and experimental results in the next paragraphs.
The first and second storey shear-drift diagrams for the Duct. 5 and 8 stages are shown in Fig-
ures 6.25-a) and 6.25-b), respectively. Together with Figure 6.23-a), the comparison of analyt-
ical and experimental shear-drift diagrams shows that the numerical modelling provides less
good predictions than for the previous tests of the bare structure.
From the above mentioned comparisons, the following aspects are highlighted:
• For Duct. 3, the strength degradation appears to be lower than the experimental one, whilst
the unloading stiffness deterioration seems reasonably modelled (note that it was tuned for
the 1.5S7 test, thus for a similar ductility level as for the present one); by contrast, the
pinching effect is not well modelled as already noted for the previous tests.
• For Duct. 5, in the first storey, further plastification occurs in both directions, but the devel-
oped force is lower than the experimental one; the unloading stiffness degrades more than in
the experimental diagram and the transition between the unloading and the reloading phases
tends to shift away from the null force zone. As for the previous ductility level, the strength
degradation is quite low when compared with the experimental results.
• Still for Duct. 5, but for the second storey, further plastification develops as for the first sto-
1.5S7
DRIFT
STOREY
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 (%)
1
2
3
4
Soft-StoreyDuct. 3
304 Chapter 6
rey, but the most relevant aspect is the strength drop occurred in the positive direction for
the second cycle. Actually, it is much larger than the degradation in the following cycles,
which may suggest that it can be related with a mistaken deformed shape of the structure at
the test beginning; the “fatter” loops appear in correspondence with the narrower ones of
the first storey, somehow showing force transfer between the two lower storeys.
Figure 6.25 Final cyclic test, Ductilities 5 and 8: first and second storey shear-drift diagrams
• The Duct. 8 level shows, for the first storey, an increase of the positive shear force without
further plastification: such paradoxical result seems to arise again from force transfer from
b) Ductility 8a) Ductility 5
Storey 1
DRIFT (%)
SHEAR (kN)
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-7.5 -6.0 -4.5 -3.0 -1.5 .0 1.5 3.0 4.5 6.0 7.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
DRIFT (%)
SHEAR (kN)
-7.5 -6.0 -4.5 -3.0 -1.5 .0 1.5 3.0 4.5 6.0 7.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
Storey 1
Storey 2 Storey 2
-4.5 -3.6 -2.7 -1.8 -.9 .0 .9 1.8 2.7 3.6 4.5
-4.5 -3.6 -2.7 -1.8 -.9 .0 .9 1.8 2.7 3.6 4.5
Analytic
Experimental
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 305
the second storey, where the peak shear force stays well below the experimental one (a sim-
ilar reason as for the Duct. 5 may justify this discrepancy). The unloading stiffness degrades
significantly more than in the experiment and the resulting diagrams become unreasonably
narrow; additionally, the unloading-reloading transition tends to shift away even more from
the null force zone, but, nevertheless, the peak shear force is well estimated.
The increasing deviation of analytical and experimental unloading stiffness for increasing duc-
tility progressively induces less energy dissipation in the numerical analysis, as shown in the
energy diagrams included in Figure 6.26. Indeed, for Duct. 3, the energy is overestimated
because the model cannot simulate the pull-out phenomena; on the other hand, a less adequate
modelling of the unloading stiffness deterioration is responsible for significant underestima-
tion of the dissipated energy for the Duct. 8 level, in which the pull-out effects become less
important because drifts have larger participation of column deformation (less prone to bar-
slippage than beams).
Figure 6.26 Final cyclic test: total energy diagrams
In order to find out the influence of the latter aspect, other calculations were performed with
increased values of the unloading stiffness degradation parameter only in the first storey
columns (i.e., instead of 1 as referred in 6.2.4, a uniform value of 4 was used for all those col-
umn sections). Figure 6.27 shows the first storey shear-drift and the total energy diagrams for
the Duct. 8 test phase, clearly evidencing that the unloading behaviour is much better simu-
lated by the numerical model (see Figure 6.27-a)); consequently, the energy evolution becomes
ENERGY
0 900 1800 .00
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50 x1.E3
Ductility 8
Ductility 5
Ductility 3
StepAnalytic
Experimental
α( )
306 Chapter 6
closer to the experimental one (see Figure 6.27-b)). Note, however, that the unloading-reload-
ing transition still occurs out of the zero force zone, thus showing no significant influence of
the parameter on that phenomenon.
Figure 6.27 Final cyclic test Duct. 8: results for modified unloading stiffness degradation
In spite of the deviations of analytical and experimental shear-drift diagrams, the peak shear
response is reasonably well estimated. The corresponding storey profiles are included in Fig-
ure 6.28, showing a fairly good agreement of results for Duct. 3; deviations do appear for Duct.
5, which are partially recovered in storeys 1 and 2 for the Duct. 8 level. Note the remarkable
agreement of base shear for all three ductility stages, significantly improved comparatively to
the 1.5S7 test (see Figure 6.19), which means less than 10% of underestimation as provided by
the analysis.
The strange shape of the Duct. 8 shear-drift diagrams suggests some comments regarding
either a possible mistaken deformed shape at the beginning of the test simulation, or assumed
displacements different from those actually applied. In fact, the first “steps” of diagrams in
Figure 6.25-b) show a sudden drop in the second storey, simultaneously with a “high stiffness”
initial loading branch in the first storey. Such effect may be due to an internal strain distribu-
tion not compatible with externally applied displacements, demanding force adjustments in
order to restore compatibility.
α
b) Total energy diagrama) 1st storey shear-drift diagram
DRIFT (%)
SHEAR (kN)
-7.5 -6.0 -4.5 -3.0 -1.5 .0 1.5 3.0 4.5 6.0 7.5 -1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 x1.E3
Storey 1
ENERGY
0 900 1800 .00
.15
.30
.45
.60
.75
.90
1.05
1.20
1.35
1.50 x1.E3
Step
Analytic
Experimental
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 307
Figure 6.28 Final cyclic test: profiles of peak values of storey shear
In other words, if by starting from an initial inter-storey deformed shape (as schematically
shown in Figure 6.29), displacements and are assumed to be applied, the corresponding
drift is as shown in Figure 6.29-b), with the same sign of the initial drift; therefore, a load-
ing or reloading process is induced. On the contrary, if the actually applied displacements are
as shown in Figure 6.29-c), the drift takes the opposite direction to the initial one and unload-
ing occurs; thus, the shear force is likely to reduce, as is apparent from Figure 6.23-b) in the
storey 2.
This fact has been specifically pointed out because the final cyclic test was interrupted at the
end of the Duct. 5 phase, in order to allow for several inspections. The interruption was intro-
duced at zero value of the top displacement, followed by structure unloading to zero forces in
the actuators. The restarting process for the Duct. 8 phase was carried out, first by restoring the
displacements to values prescribed at the interruption step and then by proceeding with the
foreseen top displacement history. During that restoring process, the effectively applied dis-
placements might not have been correctly stored and the displacement series may not exactly
match the real ones.
In fact, a detailed inspection of such series in the stop-restart phase (actually not shown in Fig-
ure 5.12) revealed some inconsistency between the first and the second storey displacements
c) Ductility 8a) Ductility 3 b) Ductility 5
SHEAR (x1.E3 kN)
STOREY
1
2
3
4
.0 .3 .6 .9 1.2 1.5 SHEAR (x1.E3 kN)
STOREY
1
2
3
4
.0 .3 .6 .9 1.2 1.5 SHEAR (x1.E3 kN)
STOREY
1
2
3
4
.0 .3 .6 .9 1.2 1.5
Analytic
Experimental
u1a u2
a
∆ua
308 Chapter 6
recorded in the restarting process, which may be the cause of the unexpected features of the
corresponding analytical shear-drift diagrams. If that had not occurred, both first and second
storey analytical diagrams of Figure 6.25-b) would be expected to shift, downwards and
upwards, respectively, in the positive force direction.
Figure 6.29 Influence of assumed displacements different from the applied ones
Finally, it must be referred that at the end of Duct. 8 stage simulation, the maximum yielding
length reached 1.33h in the first storey beams (for h the beam depth), that is twice the value for
the 1.5S7 test. In turn, the peak values of chord rotation ductility and damage reached 18.6 and
1.57, respectively, both for positive bending and occurring in the first storey short-span beams.
Such a high damage value clearly agrees with the near failure stage and the large inter-storey
drifts reached (over 7%), almost uniformly, in the two first storeys.
6.4.3.5 Summary of static analysis results
The most relevant aspects on the static simulation of the tests are briefly summarized in the
following paragraphs.
The 0.4S7 test analysis has given quite good results in terms of storey shear and shear-drift dia-
grams, mainly for stabilized cracked behaviour. However, due to the origin-oriented features of
the global section model, the dissipated energy is not so well simulated.
Identically, for the 1.5S7 bare frame test, good agreement of results was achieved for the shear
forces (maximum deviation around 10% to 15% from the experiment) and for the dissipated
energy, mainly in the storeys with higher deformations. Difficulties in modelling the pinching
due to anchorage slippage led to a slight overestimation of energy. Concerning more localized
deformations, the plastic zone rotations are somewhat overestimated, thus agreeing with the
u1a u1
r
u2a u2
r∆ua ∆ur∆u0
c) Actually applieda) Initial inter-storey b) Assumed deformed shape displacements displacements
Storey 1
Storey 2
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 309
higher yielding top displacement obtained by the analysis; such effects may be due to the low
post-yielding stiffness used for the local section modelling, which may also explain the lower
storey shear values. However, reasonable values are obtained for the damage index, in view of
the observed state of the structure, and the respective distribution corresponds to the expected
dissipation mechanism (strong column - weak beam).
For the infilled frame tests, results diverge from the experimental ones, where and while infill
contribution is important, due to overestimation of the initial stiffness of panels. The uniformly
infilled configuration, after significant cracking of the first storey panel, behaved essentially as
a soft-storey structure and the results for that storey became closer to the experimental ones,
actually as exhibited for the irregular (soft-storey) configuration.
In spite of difficulties in the final test simulation due to the repairing intervention and the resid-
ual deformations after the infilled frame tests, the results in terms of shear are well captured,
although slightly (10%) underestimated (still, the low post-yielding stiffness may be the possi-
ble cause). However, the unloading stiffness degradation was found difficult to simulate with
the present model rule; for a wide range of global ductility, meaning an even wider range of
local (curvature) ductility of which the upper bound refers to a near failure stage, the adopted
rule using a common point leads to excessive degradation of the unloading stiffness.
A general overview of the performed simulations reveals good adequacy of the model to fol-
low the evolution of the global stiffness. With the exception of the infilled frame tests, the glo-
bal average stiffness in experimental shear-drift diagrams is reasonably estimated throughout
the several ductility stages.
6.4.4 Dynamic analysis by flexibility modelling versus experimental tests
The adequacy of the presently adopted model to simulate the global hysteretic behaviour was
already assessed by means of the static analysis, and good performances were obtained, mainly
for the bare frame tests. The model adequacy to trace out the dynamic response is now
addressed in this section.
Before the presentation of the dynamic analysis results, calculated values of frequencies are
compared with those measured for several testing stages, as this is relevant for the subsequent
analysis of results.
310 Chapter 6
The characterization of damping as adopted herein is discussed simultaneously with the pres-
entation of results, restricted to the bare frame tests (the 0.4S7 and 1.5S7 ones) due to the
higher level of uncertainty of infill modelling.
6.4.4.1 Comparison of structural frequencies
The frequencies were experimentally obtained at the following testing phases: before and after
the bare frame tests (respectively, pre-0.4S7 and post-1.5S7), before the uniformly infilled test
(pre-Uniform Infill) and before the soft-storey test (pre-Soft Storey). For comparison purposes
only the fundamental frequencies are presented in Table 6.4, along with the analytical values
obtained by means of the tangent stiffness matrix at the unloaded stage.
The initial frequency (pre-0.4S7) is overestimated by the analysis, which reflects the difficul-
ties in characterizing the initial stiffness as already pointed out for the static analysis of the
0.4S7 test. However, after this test, the experimental frequency is quite well captured by the
model, as confirmed by the pair of values identified by pre-1.5S7. The analytical one corre-
sponds to the initial tangent matrix before the 1.5S7 test, whilst the experimental one was esti-
mated by the last top displacement peaks of the 0.4S7 response, because no measurements
were made between the two tests. Such a good agreement of frequencies is in direct corre-
spondence with the previously obtained agreement of static analysis results after having
exceeded the first peaks (see Figure 6.16-a). The same reasoning applies to the frequencies for
the post-1.5S7 case, which indeed do compare very well.
Table 6.4 Structural frequencies. Comparison of measured values with those calculated by flexibility discretization
Structure Status Case
1st Mode Frequency (Hz)
Measured Calculated
Bare
Pre-0.4*S7 1.78 2.23
Pre-1.5*S7 1.27* 1.29
Post-1.5*S7 0.82 0.84
Infilled
Pre-Uniform Infill 3.34 6.67
Post-Uniform Infill ** 1.00
Pre-Soft Storey 1.67 1.66
Post-Soft Storey ** 0.76
Bare Post-Final Cyclic ** 0.58
(* obtained from displacement history; ** not measured)
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 311
For the uniformly infilled configuration a very poor frequency prediction is obtained by the
analysis (twice the experimental value), which is clearly related with the overestimation of the
initial stiffness of infills already detected in the static analysis. At the end of that test no fre-
quency measurements were made, but the analytical value is included to provide an additional
indication of the stiffness decrease along the testing sequence. Note that the obtained value
(1.0 Hz) is higher than that for the post-1.5S7 case, which is due to the presence of infills only
slightly damaged in the two upper storeys although strongly damaged in the two first ones.
On the contrary, the experimentally obtained frequency for the soft-storey case is very well
predicted by the numerical model; this may be related with the first mode deflected shape
which mainly involved the first storey deformation, the remaining ones having moved as an
“almost rigid box” supported by the ground floor columns whose stiffness was well captured.
This is corroborated by the frequency agreement at the post-1.5S7 stage and is due to the fact
that the stiffness of the first storey columns has not been significantly affected by the uni-
formly infilled test in which much lower inter-storey drift was reached.
At the end of the soft-storey test, the analytical frequency (0.76 Hz) dropped below the post-
1.5S7 case, mainly as a result of damage in the first storey columns, and after the final cyclic
tests it became less than one third of the initial value.
From the above paragraphs, it can be concluded that, in spite of the difficulties in estimating
the initial frequency, the model is able to adequately follow the frequency evolution of the bare
frame during the loading process.
6.4.4.2 Low-level test on the bare structure
The dynamic analysis for the 0.4S7 tests were performed under the assumptions stated in 6.2.5,
thus for a viscous damping factor of 1.8%, and results are shown and compared with the exper-
imental ones in Figures 6.30 to 6.32.
Time histories of storey displacements are depicted in Figure 6.30-a), for both the first and the
top displacement. Identically, storey shear responses and the corresponding shear-drift dia-
grams are also included in Figures 6.30-b) and 6.30-c), respectively.
Two distinct phases can be observed in these time histories: the first, approximately until 4s, in
312 Chapter 6
which the analytical response was dominated by the overestimation of frequency and could not
reach the first peaks, both in displacement and in force; the second phase, which, after signifi-
cant cracking induced by the negative peak at 4.5s, was characterized by a stabilized frequency
allowing to closely follow the experimental response, although with slight underestimation of
peaks. This is apparent in both the first and the top storeys, and can be also observed in the
shear-drift diagrams; note that experimental drifts could not be reached in the analysis, which
can be due to the higher initial stiffness.
Note, however, that such underestimation of the experimental response appears to be quite
acceptable, since only minor deviations of relevant results are found. Storey profiles of peak
displacements, drifts and shear forces are plotted in Figures 6.31-a), b) and c), respectively,
allowing to conclude that the maximum top displacement is 15% lower than the experimental
one, whilst the peak drift (second storey) and the base shear are underestimated by 5% and
12%, respectively.
By contrast with static calculations, the energy dissipation in dynamic analysis is due to both
viscous damping and hysteretic restoring forces. Moreover, in the static analysis, the external
loading consisted of imposed storey displacements and the deformation energy could be
directly evaluated by the work done by storey reaction forces on the corresponding displace-
ments, since this total external work equals the internal deformation energy.
However, in the dynamic analysis no explicit external forces are considered at storey level (the
external loading is a base accelerogram) and, therefore, the hysteretic deformation energy has
to be evaluated either by the sum of each element energy (directly available from the flexibility
element model), or by the work done by all components of nodal hysteretic forces on the
respective displacements. The energy contribution of viscous damping has to be computed by
the latter process, since it is not defined at the element level; nodal viscous damping forces are
obtained from velocities and the corresponding work done on nodal displacements yields the
contribution to energy dissipation.
Figure 6.31-d) shows the energy responses from experimental and numerical analysis. The
numerical one is plotted both with and without the viscous damping contribution, in order to
assess its weight in the dissipated energy.
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 313
Figu
re 6
.30
0.4
S7 te
st. D
ynam
ic a
naly
sis
with
1.8
% v
isco
us d
ampi
ng v
ersu
s ex
perim
enta
l sto
rey
resu
lts
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
DIS
PL. (
m)
-5.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0
x1
.E-2
SHEA
R (k
N)
SHEA
R (k
N)
Stor
ey 4
Stor
ey 1
SHEA
R (k
N)
-5.0
-4
.0
-3.0
-2
.0
-1.0
.
0
1.0
2
.0
3.0
4
.0
5.0
-7
.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5
x1
.E2
SHEA
R (k
N)
-7.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5
x1
.E2 .0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
DIS
PL. (
m)
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
-5
.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0
x1
.E-2
Tim
e (s
)
Tim
e (s
)
Tim
e (s
)
Tim
e (s
)
DRI
FT (x
1.E-
3)
DRI
FT (x
1.E-
3)
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
-3
.0
-2.4
-1.8
-1.2
-.6
.0
.6
1.2
1.8
2.4
3.0
x1
.E2
-2.0
-1
.6
-1.2
-.
8
-.4
.
0
.4
.
8
1.2
1
.6
2.0
-3
.0
-2.4
-1.8
-1.2
-.6
.0
.6
1.2
1.8
2.4
3.0
x1
.E2
b) S
tore
y sh
ear
c) S
tore
y sh
ear-
drift
dia
gram
a)
Sto
rey
disp
lace
men
ts
Anal
ytic
Expe
rim
enta
l
314 Chapter 6
Figure 6.31 0.4S7 test. Dynamic analysis with 1.8% viscous damping vs. experimental results
It is apparent that hysteretic dissipation almost stopped after cracking has stabilized, indeed as
already found in the static analysis. On the contrary, by including the viscous damping contri-
bution, the analytical energy dissipation follows the increase rate of the experimental one. This
means that, except for the 2 to 3s time interval, during which an energy jump occurred due to
large cracking progression, the viscous damping factor of 1.8% seems adequate to compensate
the lack of hysteretic damping inherent in the model for stabilized cracking conditions.
In order to find out the influence of viscous forces in the dynamic response, an additional anal-
ysis was performed for zero viscous damping and the corresponding top displacement and
energy time histories are illustrated in Figures 6.32-a) and 6.32-b), respectively.
STOREY
.0 1.0 2.0 3.0 4.0 5.0
1
2
3
4STOREY
.0 1.0 2.0 3.0 4.0 5.0
1
2
3
4
STOREY
.0 1.5 3.0 4.5 6.0 7.5
1
2
3
4
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
ENERGY (kJ)
.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
DISPL. (x1.E-2 m)
SHEAR (x1.E2 kN)
DRIFT (x1.E-3)
Time (s)
b) Peak-drift storey profilea) Peak-displacement storey profile
d) Evolutions of total energyc) Peak-shear storey profile
Analytic
Experimental
Analytic (hysteretic only)
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 315
Figure 6.32 0.4S7 test. Dynamic analysis with no viscous damping versus experimental results
It can be observed that, up to 2s time, the analytical response remains essentially as before,
confirming the influence of a poor estimate of the initial frequency. However, the first peaks
between 2s and 3s are better captured and, therefore, larger cracking occurs earlier than for the
previous calculations. Thus, frequency becomes closer to the experimental value and the ana-
lytical response is improved in the interval 3s to 4s, after which the lack of damping leads to an
amplified response, although the peak displacement is not very much overestimated. The hys-
teretic energy evolution is identical to that of the analysis with 1.8% damping, although reach-
ing slightly higher values due to the larger displacements involved.
Comparison of Figures 6.30-a) and 6.32-a) allows to conclude that a globally better response is
obtained if 1.8% of viscous damping is included, bearing in mind that the slight underestima-
tion of peaks is related with the initial stiffness. This better agreement seems quite acceptable
and can be explained as follows:
• The experimentally measured viscous damping factor (1.8%) refers to a structural behav-
iour mainly controlled by micro-cracking, which, after accumulation of micro-cracks leads
to the formation of localized (and eventually visible) cracks typical of the post-cracking
stage.
• Therefore, the 1.8% factor can be regarded as representing the viscous equivalent to the hys-
teretic damping inherent in the cracking process, during which the steel (behaving linearly)
b) Evolutions of total energya) Top-displacement
Time (s)
DISPL. (m) .
-5.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0 x1.E-2
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Time (s)
ENERGY (kJ)
.0
2.5
5.0
7.5
10.0
12.5
15.0
17.5
20.0
22.5
25.0
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0
Analytic
Experimental
316 Chapter 6
does not contribute to energy dissipation.
• Consequently, better results can be expected if the stiffness drop, due to cracking through-
out the structure, is progressively taken into account (although without hysteretic dissipa-
tion), but in association with viscous damping equivalent to the hysteretic one actually
existing in the real structure.
Other results of dynamic analysis concerning cracking pattern and distributions of chord rota-
tion peak values are not presented since they do not significantly differ from those of static
analysis. However, it is worth mentioning that maximum positive and negative chord rotations
(3.56 mRad and -2.72 mRad, respectively) are somewhat lower than the static analysis ones, in
agreement with the lower drifts and displacements obtained from the dynamic analysis.
6.4.4.3 High level test on the bare structure
The 1.5S7 test dynamic analysis was first performed for the same conditions of the 0.4S7 test,
namely with a viscous damping factor of 1.8%; results are included in Figure 6.33.
Good agreement is found with experimental results concerning the first higher displacement
peaks (up to 3.5s) and the phase of the response (see Figure 6.33-a)). However, after these
peaks the response becomes progressively underestimated by the analysis, for which two pos-
sible reasons can be pointed out:
• Pinching is more relevant after the attainment of large peaks but the analytical model is not
able to follow that effect because it is mainly related with reinforcement slippage inside
beam-column joints. Therefore, the unloading and reloading stiffnesses are higher than the
experimental ones (see Figure 6.33-c)), loops become “fatter” than in the experiment and
the expected displacements become progressively more difficult to reach; note, however,
that the fundamental frequency from the analysis does not seem very different from the
experimental one (as apparent in Figure 6.33-a)) because the global average secant stiff-
nesses are not very different either.
• The viscous damping contribution to energy dissipation might be excessive, as suggested by
the energy curves shown in Figure 6.33-b); actually, the viscous dissipation added to the
hysteretic dissipation (inherent in the model at this post-yielding stage) led to dissipated
energy higher than the experimental one, and the analytical response became overdamped.
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 317
Figure 6.33 1.5S7 test. Dynamic analysis with 1.8% viscous damping vs. experimental results
In order to assess the sensitivity of results to the pinching effect and the amount of viscous
damping, additional analyses were performed. Specifically concerning the first issue, the
pinched shape of force-displacement diagrams was artificially made more pronounced by uni-
formly reducing the pinching moments in the local section behaviour laws to 50% of their orig-
inal value, whilst the viscous damping factor was kept the same; results of such analysis are
shown in Figure 6.34.
The top displacement time history (Figure 6.34-a)) indicates that peak values are better esti-
Time (s)
DISPL. (m) .
Time (s)
ENERGY (x1.E2 kJ)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -.25
-.20
-.15
-.10
-.05
.0
.05
.10
.15
.20
.25
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 .0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Storey 4
b) Evolutions of total energya) Top displacement
Analytic
Experimental
c) Shear-drift diagram of 1st storey
Analytic (hysteretic only)
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
x1.E3 SHEAR (kN)
DRIFT (%)
Storey 1
318 Chapter 6
mated, though slightly in excess, but the second half part of the response becomes more out-of-
phase. The modification of pinching effect can be detected in the shear-drift diagram, with
cycles narrower than in the previous analysis, and is reflected in the slight decrease of energy
dissipated by hysteresis (Figure 6.34-b).
Figure 6.34 1.5S7 test. Dynamic analysis with 1.8% viscous damping and modified pinching
versus experimental results
Note that this simple way of modifying the pinching effect intended to show the trend of
results, but cannot be seen as a meaningful result itself. Indeed, such modification could be
hardly justified, i.e., there was no specific reason for that 50% reduction and the occurrence of
Storey 1
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 -.25
-.20
-.15
-.10
-.05
.0
.05
.10
.15
.20
.25
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 .0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
Time (s)
DISPL. (m) .
Time (s)
ENERGY (x1.E2 kJ)
Storey 4
b) Evolutions of total energya) Top displacement
c) Shear-drift diagram of 1st storey
Analytic
Experimental
Analytic (hysteretic only)
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5 SHEAR (kN)
DRIFT (%)
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
x1.E3
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 319
pinching is not uniform throughout the structure because it is dependent on the deformation
level and is fairly related with the reinforcement slippage inside the beam-column joints; fur-
thermore, the latter aspect requires more refined tools (Monti et al. (1993)) for an adequate
modelling.
The influence of viscous damping was assessed by performing other analyses with zero damp-
ing factor, actually as adopted in the experiment; results are illustrated in Figures 6.35 to 6.36.
The top storey displacement time-history, included in Figures 6.35-a), shows a better agree-
ment with the experimental one, than in the previous analysis with viscous damping included
(compare with Figure 6.33-a)). The experimental displacements are not exceeded by the analy-
sis and the response phase deviation becomes less pronounced than that of Figure 6.34-a).
Among the analyses performed, the last two cycles of the present analytical response exhibit
the best approximation to the experimental one, although still far below.
Beside the difficulty of reproducing the pinching behaviour, higher vibration modes seem to
affect the analytical response as evidenced in time histories of the first and top storey shear
depicted in Figures 6.35-b). Mostly during the second half of the response, higher frequency
components are present that counteract the main one; these components could not be damped
out because no viscous damping is prescribed and the model does not seem to account for them
properly. Note that this effect is clearly more pronounced in the top storey shear response,
mainly ranging in the post-cracking stage where the model hysteretic features are less accurate.
Figures 6.36-a), b) and c) show, respectively, the storey profiles for displacement, drift and
shear peak values. Quite good predictions are provided by the analytical model, which just
slightly underestimate the experimental results: the maximum error is about 7% for the peak
base shear, whilst for the peak top displacement it is less than 4%. The evolution of total
energy is included in Figure 6.36-d) where a very good agreement is also found between anal-
ysis and experiment. This result also supports the thought that no viscous damping needs to be
included if the model adequately accounts for the hysteretic behaviour and the intrinsic dissi-
pation of energy.
Finally, as for the 0.4S7 test, the cracking and yielding patterns, the distributions of peak val-
ues of chord rotation, ductility and damage do not substantially differ from those of the static
analysis and, therefore, they are not presented here.
320 Chapter 6
Figu
re 6
.35
1.5
S7 te
st. D
ynam
ic a
naly
sis
with
zer
o vi
scou
s da
mpi
ng v
ersu
s ex
perim
enta
l sto
rey
resu
lts
b) S
tore
y sh
ear
c) S
tore
y sh
ear-
drift
dia
gram
a)
Sto
rey
disp
lace
men
ts
DIS
PL. (
m)
Stor
ey 4
Stor
ey 1
SHEA
R (k
N)
SHEA
R (k
N)
DIS
PL. (
m)
Tim
e (s
) Ti
me
(s)
Tim
e (s
)
DRI
FT (%
)
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
-
.25
-.2
0
-.1
5
-.1
0
-.0
5
.0
.0
5
.1
0
.1
5
.2
0
.2
5 .0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
-
.25
-.2
0
-.1
5
-.1
0
-.0
5
.0
.0
5
.1
0
.1
5
.2
0
.2
5
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
x1.E
3
-2.5
-2
.0
-1.5
-1
.0
-.5
.
0
.5
1
.0
1.5
2
.0
2.5
x1.E
3
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
SHEA
R (k
N)
SHEA
R (k
N)
Tim
e (s
)
D
RIFT
(%)
-1.5
-1
.2
-.9
-.
6
-.3
.
0
.3
.
6
.9
1
.2
1.5
-9
.0
-7.2
-5.4
-3.6
-1.8
.0
1.8
3.6
5.4
7.2
9.0
x1
.E2
.0
1
.0
2.0
3
.0
4.0
5
.0
6.0
7
.0
8.0
-9
.0
-7.2
-5.4
-3.6
-1.8
.0
1.8
3.6
5.4
7.2
9.0
x1
.E2
Anal
ytic
Expe
rim
enta
l
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 321
Figure 6.36 1.5S7 test. Dynamic analysis with no viscous damping versus experimental results
6.4.4.4 Remarks on energy comparison
So far, energy comparisons in the dynamic analysis context have been presented without any
reference made to the energy input. However, different responses obtained from the analysis
and the experiment, mean that energy input to the system is not the same in both cases. Conse-
quently, energy diagrams presented in the previous paragraphs correspond to the energy
absorbed from sources which are not necessarily coincident in the numerical analysis and in
the experiment; this aspect is further addressed in the following paragraphs.
Consider Eqs. (6.14) of the dynamic equilibrium of both the pseudo-dynamic test and the
numerical analysis
.0 .5 1.0 1.5 2.0 2.5
1
2
3
4 STOREY STOREY
STOREY ENERGY (x1.E2 kJ)
DISPL. (m)
SHEAR (x1.E3 kN)
DRIFT (%)
Time (s)
.0 .05 .10 .15 .20 .25
1
2
3
4
.0 .3 .6 .9 1.2 1.5
1
2
3
4
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 .0 .5
1.0 1.5 2.0 2.5 3.0
3.5 4.0 4.5 5.0
b) Peak drift storey profilea) Peak displacement storey profile
d) Evolution of total energyc) Peak shear storey profile
Analytic
Experimental
322 Chapter 6
(6.14)
For simplicity no viscous damping forces are included and the mass matrix is assumed the
same in the experiment and in the analysis. Both displacements , accelerations and restor-
ing forces have the subscript e or a, referring to the experiment and the analysis, respec-
tively.
The corresponding energy balance equations are obtained by integrating Eqs. (6.14) over the
displacement history up to a given instant t, and are given by
(6.15)
The first terms in Eqs. (6.15) can be simplified to and , referring to
the kinetic energy of the system at instant t, in the following denoted by and , respec-
tively. On the other hand, the second terms of the first members refer to the absorbed energy
(including the elastic and the hysteretic components) and are denoted by and . Thus,
Eqs. (6.15) can be re-written as
(6.16)
where and are the input energies, respectively the experimental and analytical ones.
In the energy diagrams shown so far, the experimental curves actually refer to the term and
the analytical ones to . Since input energy terms and are not necessarily identical
because experimental and analytical displacements are not the same, the question can be asked
whether or not the absorbed energy terms and are strictly comparable.
However, if the main aim is the assessment of energy absorption capacity inherent in the
numerical modelling (more specifically the energy dissipation), the influence of the input
energy can be “removed” by dividing Eqs. (6.16) by their second members. The following
Mu··e re+ Mu··g–=
Mu··a ra+ Mu··g–=
u u··
r
Mu··e( )T ued0
t
∫ reT ued
0
t
∫+ Mu··g–( )T ued0
t
∫=
Mu··a( )T uad0
t
∫ raT uad
0
t
∫+ Mu··g–( )T uad0
t
∫=
u· eTMu· e( ) 2⁄ u· a
TMu· a( ) 2⁄
Eek Ea
k
Eea Ea
a
Eek Ee
a+ Eei=
Eak Ea
a+ Eai=
Eei Ea
i
Eea
Eaa Ee
i Eai
Eea Ea
a
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 323
equations are obtained
(6.17)
where and refer to the relative absorbed energy and, similarly,
and are the relative kinetic energies. Since these last terms vanish when velocities approach
zero, it follows that terms tend to 1; moreover, and contain the elastic (recoverable)
and the dissipated (irrecoverable) contributions, for which the curves range between 1 and
the minimum envelope representing the relative dissipated energy.
The input energy diagrams for both for the 0.4S7 and the 1.5S7 tests were obtained and are
included in Figure 6.37.
Figure 6.37 Total input energy for experimental and numerical analysis
For the latter test, experimental and analytical diagrams do compare quite well as a result of
the good agreement of storey displacements, particularly during the time interval with larger
ground accelerations. Deviations of displacements after 5.0s time (see Figure 6.35-a)) did not
significantly affect the input energy comparison due to the lower ground accelerations.
On the contrary, for the 0.4S7 test, rather different input energy diagrams are obtained, as is
apparent in Figure 6.37-a). Note, however, that after 4.0s time, the experimental and analytical
curves are similar, but shifted by an approximately constant amount of energy. Such energy
εek εe
a+ 1=
εak εa
a+ 1=
εea Ee
a Eei⁄= εa
a Eaa Ea
i⁄= εek
εak
εa εea εa
a
εa
b) 1.5S7 testa) 0.4S7 testTime (s)
.0 .5
1.0 1.5 2.0 2.5 3.0
3.5 4.0 4.5 5.0
ENERGY (kJ)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .0 2.5
5.0 7.5
10.0 12.5 15.0
17.5 20.0
22.5 25.0
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Time (s)
ENERGY (x1.E2 kJ)
Analytic
Experimental
324 Chapter 6
deficit in the numerical analysis, arises from the underestimation of displacements (caused by
overevaluation of stiffness) during the time interval with larger ground accelerations (note the
displacement deviations in Figure 6.30-a) and the significant accelerations in Figure 5.4-a),
between 2.0s and 4.0s time).
The minimum envelopes of relative absorbed energy diagrams are included in Figure 6.38, for
the tests under analysis. As stated before, such curves represent the relative dissipated energy,
which in the 0.4S7 test includes both hysteretic and viscous components. In particular, for this
test the curve of energy dissipated only by hysteresis is also included (see Figure 6.38-a)).
Figure 6.38 Relative absorbed energy for experimental and numerical analysis
Besides the need of the viscous damping to match the energy response of the 0.4S7 test (as
already concluded before), it can be seen that analytical relative dissipated energy approxi-
mately follows the experimental one, once the major deviations of displacements are over-
come. In fact, during the initial time interval, results do not agree, but, by the end of the
analysis, the fraction of dissipated energy is almost the same in both cases, in spite of the sig-
nificantly different input energy.
For the 1.5S7 test the fraction of energy dissipated by hysteresis in the analysis globally fol-
lows quite well the experimental one and, in the end, again very similar values are obtained in
both cases. This fact was expectable in view of the good agreement between total absorbed
energy diagrams (see Figure 6.36-d)) and between the input energy ones (see Figure 6.37-b)).
This allows to conclude that, in spite of displacement responses somewhat different from the
Time (s)
Relative ENERGY
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .00 .11
.22 .33
.44 .55 .66
.77 .88
.99 1.10
Analytic
Experimental
Analytic (hysteretic only)
Time (s)
Relative ENERGY
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .00 .11
.22 .33
.44 .55 .66
.77 .88
.99 1.10
b) 1.5S7 testa) 0.4S7 test
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 325
experimental ones, the numerical modelling gives good predictions for the dissipated energy in
the dynamic analysis of the bare frame tests.
6.4.4.5 Summary of dynamic analysis results
The most relevant aspects about the dynamic simulation of the tests are briefly summarized in
the following paragraphs.
Structural frequencies obtained by the analysis of bare frame seismic tests do compare well
with the experimental ones, in spite of the initial frequency deviations due to the uncertainty on
modelling the actual initial stiffness.
The simulation of the 0.4S7 test has shown good adequacy of the model to follow the dynamic
response in the post-cracking range. Besides the initial deviations related with the initial stiff-
ness estimate, the response is very well simulated (both in frequency and in amplitude) by
including viscous damping forces characterized by the experimentally measured factor (1.8%),
equivalent to the hysteretic dissipation due to cracking.
In turn, for the 1.5S7 test, the response is very reasonably reproduced by the model, particu-
larly if no viscous damping is included (actually, as adopted in the experiment). Indeed, for the
involved deformation level (in the post-yielding range), the behaviour is mainly controlled by
the steel, whose dissipation characteristics in the non-linear range, are not likely to be simu-
lated by the viscous damping obtained for the initial undamaged state of the structure. Addi-
tionally, the hysteretic model rules are set up accounting for the steel behaviour, particularly its
dissipation properties. In this context it can be accepted that, more than useless, the inclusion
of viscous damping for post-yielding range simulations may lead to poor approximations of the
actual behaviour, if hysteretic dissipation is inherent in the model.
Finally, note that, where viscous damping was included in the foregoing analyses, its contribu-
tion was quite significant, in spite of the low factor of 1.8%. In the pre-yielding range, due to
the low hysteretic dissipation of the model, that contribution reached over 60% of the total dis-
sipated energy, while for post-yielding behaviour it was about 30%. Therefore, should viscous
damping be necessary to account for energy dissipation, its participation can be expected to
reach high levels, even for low viscous damping factors. Indeed, similar conclusions had been
drawn in previous works by Uang and Bertero (1988).
326 Chapter 6
6.4.5 Flexibility element versus fixed length plastic hinge (F.H.) modelling
The assessment of the flexibility model performance, when compared to other modelling
options, should focus on quality of results, computation efficiency and modelling simplicity
from the user point-of-view. The first issue is duly addressed in this section, but no reliable and
general conclusion can be drawn for the remaining ones, because the features of the flexibility
and the F.H. models are not strictly comparable as far as discretization is concerned (this topic
is further discussed below).
Some global results (storey displacements, inter-storey drifts and shear forces, energy dia-
grams, etc.) obtained by the F.H. model are presented along with the experimental ones, in
order to compare both flexibility and F.H. model ability to trace out the experimental response.
For this purpose, only the 0.4S7 and the 1.5S7 tests were simulated, through both static and
dynamic analysis.
6.4.5.1 Assumptions for F.H. modelling
Structural discretization for F.H. modelling was done by using beam-column elements already
available in CASTEM2000, namely:
• A two-node element with one Gauss point, thus assuming uniform bending moment and
curvature, including shear deformation according to Timoshenko formulation and support-
ing a non-linear moment-curvature behaviour law as described in Appendix B; this element
will be simply referred as the TIMO element.
• The classical Bernoulli two-node beam-element with linear elastic behaviour.
Since no global element was available in CASTEM2000 including both plastic hinge and rigid
lengths at member end zones, modelling of each beam and column had to be done by means of
the association of several elements as shown in Figure 6.39.
Figure 6.39 Member discretization for fixed length plastic hinge (F.H.) analysis
Two rigid elements were considered to account for the joint region and modelled with “infi-
1 2
Linear elastic element
lp1 lp2
Non-Linear element
lr1 lr2
Rigid element
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 327
nite” stiffness Bernoulli elements; according to the flexibility discretization, only for the
beams were considered these rigid elements.
Lengths of plastic hinge zones, each one modelled by one single TIMO element, were taken
and , respectively for beams and columns, where h stands for
the full-depth of the cross-section; smaller plastic hinge lengths were adopted for the columns
because less plastic development is expected there. Note that such option is precisely one diffi-
cult issue of the F.H. modelling, because hinge lengths depend on the expected deformation
level and their distribution must be known or assumed a priori to provide non-linear elements
in the adequate locations.
The elastic characteristics and model parameters for the non-linear elements were taken almost
the same as those adopted in the flexibility formulation for the corresponding member end sec-
tions. The only exception refers to the strength degradation parameter , adopted with signifi-
cantly lower values than those referred in 6.2.4, due to the location of the common point which
differs from that considered in the flexibility formulation model.
Actually, for the original trilinear model as described in Appendix B, the common points for
unloading branches exist in the elastic stiffness line 1 (see Figure 3.15), whereas for the modi-
fied model they lie in the fully-cracked stiffness lines 2 or 3; this means that, if the same value
of is used for the original model as for the modified one, less stiffness degradation and more
energy dissipation are obtained (due to “fatter” loops). Therefore, from the observation of the
shear-drift diagrams the storey initial stiffness were estimated using the 0.4S7 test results and,
both yielding thresholds and hypothetical common points for storey unloading stiffness were
roughly obtained from the 1.5S7 test diagrams. The order of magnitude of was estimated
and, upon several trials, a uniform value of 0.2 led to reasonable agreement of the dissipated
energy diagrams between analysis and experiment. However, it is recognized that such an
energy related criterion is not robust enough for the estimation of because other effects are
involved, such as the pinched shape of diagrams and the global stiffness.
For the internal linear element, the elastic and geometric section characteristics were taken as
the average of each end section.
Vertical loads, uniformly distributed along the flexible length, were also included by means of
their nodal equivalents.
lp1lp2
h= = lp1lp2
h 2⁄= =
α
α
α
α
328 Chapter 6
The adopted F.H. discretization, consisting of 5 and 3 elements per beam and column, respec-
tively, actually prevents any comparison of model efficiency because the inherent increase of
total number of elements and nodes requires much more computation time. In fact this was
confirmed by the CPU time for the F.H. calculation much higher than for flexibility computa-
tion; for the static simulations of 0.4S7 and 1.5S7 tests, 18 min and 20 min, respectively, were
required with the flexibility formulation, while with F.H. modelling 60 min and 89 min were
necessary to perform 400 steps of analysis.
Finally, concerning modelling simplicity, it is obvious that the present flexibility formulation
renders the discretization and data preparation tasks much easier than the F.H. model subdivi-
sion of each member in several elements. However, it is recognized that such increased sim-
plicity arises from features, such as the rigid and the plastic hinge lengths being automatically
included in only one element, which are not the main added value of the present flexibility for-
mulation; indeed they could be easily included in a F.H. formulation by means of condensation
of d.o.f. as adopted by Coelho (1992).
6.4.5.2 Discussion on F.H. modelling and comparison with flexibility analysis results
Static analysis
Results from the static analysis with the F.H. formulation are shown in Figures 6.40 and 6.41,
respectively for the 0.4S7 and 1.5S7 tests.
Concerning the 0.4S7 test, the most relevant aspect is the overestimation of storey shear, as
apparent from Figure 6.40-a) referring to the first storey. Until 2.0s, the response is similar to
that of the flexibility formulation (see Figure 6.16-a) for comparison), but then shear peaks
become visibly overestimated due to the higher stiffness involved. Clearly, the linear behav-
iour of the internal elements, with constant uncracked stiffness, prevents the F.H. analytical
response to approach the experimental one (see the shear-drift diagram in Figure 6.40-b)), con-
trarily to the results of flexibility formulation. This effect occurs in the whole structure and can
be confirmed by the generalized overestimation of storey peak shear shown in Figure 6.40-d).
However, the lack of energy dissipation inherent in the flexibility analysis in the pre-yielding
range, is partially overcome by the F.H. modelling, because the model allows for residual
deformation immediately after cracking (see Figure 6.40-b)). Consequently, the dissipated
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 329
energy becomes better simulated, as is clearly apparent from Figure 6.40-c) when compared
with Figure 6.16-c), in spite of the overestimated stiffness.
Figure 6.40 0.4S7 test. Static analysis with F.H. modelling
Results from the 1.5S7 test show a response quite similar to the flexibility modelling one,
which can be confirmed by comparing Figure 6.41-a) with Figure 6.18-a). Again, up to 2.0s
time, the first storey shear is overestimated due to the high stiffness involved (see the shear-
drift diagram in Figure 6.41-b)), after which results do agree very well with experimental ones,
as obtained previously for the flexibility analysis. Note that the main difference between the
Time (s)
Time (s)
DRIFT (x1.E-3)
Storey 1
b) 1st storey shear-drift diagram a) 1st storey shear
c) Evolution of 1st storey energy
SHEAR (kN)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 -7.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5 x1.E2
-5.0 -4.0 -3.0 -2.0 -1.0 .0 1.0 2.0 3.0 4.0 5.0 -7.5
-6.0
-4.5
-3.0
-1.5
.0
1.5
3.0
4.5
6.0
7.5 SHEAR (kN)x1.E2
ENERGY (kJ)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 .0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
8.0
9.0
10.0 STOREY
.0 1.5 3.0 4.5 6.0 7.5
1
2
3
4
SHEAR (x1.E2 kN)
d) Profile of storey peak shear
Analytic
Experimental
330 Chapter 6
two models is related to the pre-yielding behaviour; once the response enters in the post-yield-
ing range, both models have almost identical features and, therefore, the corresponding
responses are not likely to deviate. In the unloading phases, slight differences may be found
due to the adaptations of the degradation rule for unloading stiffness.
Figure 6.41 1.5S7 test. Static analysis with F.H. modelling
The initial “fatter” loops induce higher energy dissipation, but after 2.0s time the energy dia-
gram develops quite similarly to that of the flexibility analysis shown in Figure 6.18-c); the
final energy value is approximately the same, revealing that the initial energy “jump” is com-
pensated by slightly less dissipation during unloading phases.
Time (s)
Time (s)
DRIFT (%)
Storey 1
b) 1st storey shear-drift diagram a) 1st storey shear
c) Evolution of 1st storey energy
SHEAR (kN)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
x1.E3 SHEAR (kN)x1.E3
ENERGY (*1.E2 kJ)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
STOREY
1
2
3
4
SHEAR (x1.E3 kN)
d) Profile of storey peak shear
Analytic
Experimental
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
-2.5 -2.0 -1.5 -1.0 -.5 .0 .5 1.0 1.5 2.0 2.5
.0
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
2.0
.0 .3 .6 .9 1.2 1.5
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 331
Peak values of storey shear are larger than those obtained from flexibility analysis, particularly
in the three upper storeys, which may be related with the higher stiffness in the present model-
ling. However, in absolute terms, peak shear deviation from the experimental values do not
significantly differ from those of the flexibility analysis.
The overall comparison of results, obtained from static analysis with both the flexibility and
the F.H. modelling of the 0.4S7 and 1.5S7 tests, allows to conclude that the experimental
response is better estimated by the flexibility formulation, particularly in the pre-yielding
range where the progressive adaptation of member stiffness is quite important. Better approxi-
mation of storey shear is achieved, although for energy assessment purposes the F.H. formula-
tion reveals itself more adequate before yielding. However, after yielding development,
structural responses provided by both modelling strategies do not differ significantly.
Dynamic analysis
For dynamic analysis purposes, model parameters for the F.H. formulation were kept as for the
static analyses. Concerning viscous damping, the option made was to select the parameters
leading to the best results with the flexibility approach, namely 1.8% and 0%, respectively for
the 0.4S7 and 1.5S7 tests. However, as before, this aspect turned out to be critical for the good
assessment of the dynamic response.
The results of dynamic analysis for 0.4S7 are shown in Figure 6.42: displacement time histo-
ries of the first and fourth stories (Figures 6.42-a) and b)) reveal poorer approximation to the
experimental ones, when compared against the flexibility results in Figures 6.30-a). In spite of
a reasonable estimate of the top displacement, the response clearly deviates from the experi-
mental time history: right after the positive peak near 2.2s time, when significant cracking
occurred, the response by F.H. modelling stays behind the flexibility analysis due to the higher
stiffness of the linear internal elements; in the subsequent peaks the response becomes progres-
sively out-of-phase and over-damped after the highest peak.
Such over-damping effect suggests that the adopted viscous damping might not be adequate
simultaneously with the hysteretic dissipation provided by this model, mostly if one bears in
mind the quite good results obtained with the same viscous damping and no hysteretic dissipa-
tion in the cracking phase, as adopted in the flexibility formulation.
332 Chapter 6
Figure 6.42 0.4S7 test. Dynamic analysis with F.H. modelling and 1.8% viscous damping
Concerning the inter-storey drift (see Figure 6.42-c)), peak values are clearly worse estimated
in the two upper storeys and better evaluated in the first one, comparatively to the flexibility
modelling results (Figure 6.31-b)); however, the maximum peak drift (in the second storey) is
almost the same in both analyses. In turn, the peak storey shear shows that, where the drift
develops more (i.e., in the three lower storeys), the force response is quite overestimated due to
the excessive stiffness involved.
In order to check the influence of viscous damping, a similar calculation was performed but
Storey 1
d) Profile of storey peak shearb) 1st storey displacement
a) 4th storey displacement
STOREY
.0 1.0 2.0 3.0 4.0 5.0
1
2
3
4
DRIFT (x1.E-3)
c) Profile of storey peak-drift
STOREY
1
2
3
4
SHEAR (x1.E2 kN)
Analytic
Experimental
Time (s)
DISPL. (m)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
x1.E-2 Storey 4
-5.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0
Time (s)
DISPL. (m)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
x1.E-2
-5.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 333
with no viscous damping. A clear amplification of the response is obtained, as apparent from
the top displacement time history and the inter-storey drift peak profile shown in Figure 6.43.
Figure 6.43 0.4S7 test. Dynamic analysis with F.H. modelling and zero viscous damping
In spite of the hysteretic dissipation inherent in the F.H. modelling, the response amplification
due to viscous damping “removal” is much more evident than it was for the flexibility model-
ling (for comparison check Figure 6.30-a) against Figure 6.32-a) for the flexibility analyses,
and Figure 6.42-a) against Figure 6.43-a) for the F.H. calculations). This fact suggests that,
beyond the problem of adopting an adequate viscous damping factor, the length of zones where
hysteretic dissipation is taken into account may be responsible for the cruder agreement
between F.H. modelling and the experimental results. Indeed, this modelling allows dissipation
only in the plastic hinges, while the cracked zones develop further outside those lengths (as
shown in the cracking pattern included in Figure 6.17-c)), which means that dissipation may
not be sufficiently accounted for in the assumed non-linear zones.
Figure 6.44 shows the 1.5S7 test results, as previously referred, for no viscous damping. Com-
paring these results with those obtained by the flexibility formulation (see Figures 6.35-a) and
c)), the peak displacements and drifts become fairly amplified in the F.H. modelling.
Observing the top displacement response and the first storey shear-drift diagram, the first two
cycles are characterized by displacements lower than the experimental ones, due to the higher
Storey 4
b) Profile of storey peak drifta) 4th storey displacement
STOREY
1
2
3
4
DRIFT (x1.E-3)
Analytic
Experimental
Time (s)
DISPL. (m)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
x1.E-2
-5.0
-4.0
-3.0
-2.0
-1.0
.0
1.0
2.0
3.0
4.0
5.0
.0 1.0 2.0 3.0 4.0 5.0
334 Chapter 6
stiffness of the structure. Consequently, the dissipated energy in those cycles is very low and,
as a determinant factor for the subsequent cycles, this contributes to push the following peak
deformations to higher levels in order to compensate for the low energy dissipation.
Figure 6.44 1.5S7 test. Dynamic analysis with F.H. modelling and zero viscous damping
In terms of peak storey shear, results do not significantly differ from the flexibility analysis;
the slight increase, which can be observed upon comparison of Figure 6.44-d) and Figure 6.36-
c), arises from the larger development of deformations in the post-yielding range taking place
in the F.H. modelling.
Time (s) DRIFT (%)
Storey 4
b) 1st storey shear-drift diagram a) 4th storey displacement
c) Profile of storey peak drift
DISPL. (m)
.0 .8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0
SHEAR (kN)x1.E3
SHEAR (x1.E3 kN)
d) Profile of storey peak shear
Analytic
Experimental
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
.0 .3 .6 .9 1.2 1.5
-.30
-.24
-.18
-.12
-.06
.00
.06
.12
.18
.24
.30
STOREY
.0 0.7 1.4 2.1 2.8 3.5
1
2
3
4
DRIFT (%)
STOREY
1
2
3
4
Storey 1
-3.5 -2.8 -2.1 -1.4 -.7 .0 .7 1.4 2.1 2.8 3.5
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 335
6.4.5.3 Summary of F.H. and flexibility modelling comparison
In the previous section it has been found that F.H. modelling gives less good results than the
flexibility element model in the pre-yielding range of static analysis, while in the post-yielding
analysis both modelling strategies led to similar responses; the adaptive stiffness of the mem-
ber has been considered to play the most important role.
In dynamic analysis, the viscous damping selection appears a more difficult problem because
the F.H. model itself accounts for dissipation in the pre-yielding range and, therefore, both
damping sources contribute to dissipate energy. In turn, the consideration of zero viscous
damping in F.H. modelling led to overestimates of the response, because the dissipation on
actually cracked zones is not duly taken into account; additionally, in the post-yielding range
the simulation is less good than the flexibility one due to the high stiffness of the internal linear
elastic elements.
Note that better results could be expected if the stiffness of the internal elements was reduced,
but that is for sure one of the most difficult options of the F.H. modelling. First of all, one has
to have an a priori estimate of the expected deformation in order to define where and how
much the stiffness has to be reduced, which is neither easy nor straightforward. Then, if a
given analysis is likely to go through a wide range of deformation levels, a certain stiffness
reduction may be adequate up to a certain limit, but not valid from then on.
The above topic resumes the main advantage of the flexibility formulation over the traditional
fixed length plastic hinge modelling: the progressive adaptation of the member stiffness,
throughout the deformation range, allows to closely follow the overall structure stiffness and,
therefore, the vibration frequencies as determinant factors for adequate assessment of the seis-
mic response.
Finally, concerning the dissipative characteristics, it is worth mentioning that, despite the low
energy dissipation obtained with the flexibility model in the pre-yielding range, results indicate
good predictions of the experimental behaviour if the fully-cracked stiffness is assumed pro-
gressively developing along structural members in association with viscous damping forces (of
Rayleigh type) characterized by a factor estimated for the initial state of the structure. After
yielding has started, the viscous damping is deemed unnecessary as long as the structural stiff-
ness modifications are adequately traced. Note that modifying viscous damping characteristics
336 Chapter 6
depending on the deformation range, is still an inconvenient procedure for compensating a
model drawback, but it is surely less troublesome than any attempt to modify the structural
stiffness of a F.H. based model during the analysis.
6.5 Conclusions
The present chapter has focused on the numerical analyses of the four-storey full-scale build-
ing by means of the flexibility element model. In this context, all the modelling assumptions
were first described and the necessary data was defined based on the actual structure layout
and the average properties available from constituent material tests; then, both the skeleton
curves and the hysteretic behaviour parameters for global section modelling were defined.
Where damping forces needed to be explicitly included, they have been considered of viscous
type and characterized by the classical Rayleigh damping matrix proportional to the mass and
stiffness matrices. The adequacy of including explicit damping forces has been discussed in
view of the model features concerning energy dissipation by hysteresis.
Infill panels were modelled by diagonal struts according to a previous work by Combescure
(1996) on the analysis of the infilled configurations of this structure. The basic steps of such
work have been briefly recalled to address the background studies (refined non-linear analysis
and application to experimentally tested specimens) on which the diagonal strut model was
based, particularly its validation and derivation of model parameters. The difficulties of defin-
ing panel characteristics, viz those related to the initial stiffness and the panel-frame interface,
have been pointed out as they clearly affect the numerical simulations of tests.
Damage indices for the quantification of structural damage have been presented, based on the
most relevant proposals available in the literature. The need to incorporate the combined
effects of large strain excursions and several repetitions of load reversal in a unique damage
index has been highlighted. Among the proposals fulfilling this requirement, the damage index
adopted herein was chosen based on an assessment study of damage indices by Fardis et al.
(1993), according to which the well known Park and Ang index (Park et al. (1987a)) provides
one of the best agreements with experimental results. This index was considered here in terms
of chord rotations at each element end section, coincident with the element rotations in the
space without rigid body modes. The quantification of element related parameters for damage
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 337
evaluation (viz, the ultimate rotation , the dissipated hysteretic energy and the respec-
tive factor) was carefully discussed in view of the available proposals and the specific fea-
tures of the element.
A wide set of numerical analyses was performed in order to closely trace out the structure
behaviour throughout the several stages and to systematically compare it with the experimental
results.
Although without experimental counterpart, a static pushover analysis was first performed by
applying an inverted triangular distribution of storey forces to the structure, from which the
global yielding displacement and the maximum base shear were estimated. The former
appeared somewhat overevaluated while the base shear became slightly underestimated in
relation to the approximate experimental values based on seismic tests; this might be due to the
low post-yielding stiffness of the global section model and, eventually, due to strength mecha-
nisms differently activated in the pushover analysis and in the tests. However, the roughness of
the experimental estimates may also justify the discrepancy.
Static simulations of the performed tests were carried out by applying the experimentally
imposed displacements, in order to assess the modelling results without involving dynamic
effects.
The simulations of the bare frame seismic tests has shown quite good results for stabilized
cracked behaviour, although with some problems to account for energy dissipation due to the
origin-oriented features of the section model in the cracked range. Once yielding sets in (as for
the high level test), good agreement is still found between numerical and experimental results.
Energy dissipation becomes better captured, although slightly overestimated due to difficulties
in simulating the pinching effect arising from anchorage slippage. The overall distribution of
the damage index confirms the expected dissipation mechanism (strong column - weak beam)
and the maximum values (0.4) conform with the observed state of the structure.
Numerical static analyses for the infilled frame seismic tests provided poor simulations of the
experimental behaviour due to the estimates of model parameters for infill panels (particularly
the initial stiffness). However, simulations became better after and where significant cracking
occurred in infills, because the response became controlled by the reinforced concrete frame.
θu Ed∫β
338 Chapter 6
Results from the final cyclic test simulations were difficult to compare with experimental ones
due to the repair of damaged zones; nevertheless, the global strength was reasonably captured
(though slightly underestimated). However, the most relevant aspect is the difficulty in simu-
lating the unloading stiffness degradation for large ductility levels, which, for the present
model rule leads to over-degradation and, consequently, to lower energy dissipation.
In the context of dynamic analysis, the assessment of structural frequencies has shown good
agreement of numerical and experimental results for the bare frame structure.
Only the bare structure low and high level seismic tests were numerically simulated, having
shown good adequacy of the model to trace out the dynamic response in the post-cracking
range. However, viscous damping forces had to be included to compensate for the lack of hys-
teretic dissipation in the model, and the initially measured viscous damping factor (1.8%)
proved to yield very good results while the response is dominated by cracking. In turn, for the
high level test, when the response became mainly controlled by the steel behaviour, the viscous
damping became useless (even inadequate) since the dissipation features are taken into account
in the section model; therefore, the simulation of this test with zero viscous damping led to
very reasonable estimates of experimental results, concerning both displacement and energy
dissipation.
Whether or not the viscous damping should be included, its contribution to energy dissipation
has been found to reach significant levels, even for low factors. This means that viscous damp-
ing inclusion should be carefully judged upon the specific dissipative features of the model
before and after yielding.
Some numerical simulations with the flexibility element model were compared with analyses
by traditional fixed length plastic hinge modelling. The comparison has shown that flexibility
modelling leads to dynamic analysis results closer to the experimental ones, which arises from
its main advantage over traditional fixed hinge modelling, i.e., the progressive adaptation of
the member stiffness, throughout the deformation range. Since the overall structural stiffness
can be closely followed, so are the vibration frequencies and a better simulation of the dynamic
behaviour can be obtained.
Additionally, this comparison has emphasized that, as long as structural modifications are con-
veniently traced out (as with the flexibility modelling), good predictions of the experimental
ANALYSIS OF THE 4-STOREY FULL-SCALE BUILDING 339
behaviour can be obtained by adopting fully cracked stiffness along with viscous damping
forces in the pre-yielding range and by removing viscous dissipation in the post-yielding
behaviour.
Overall, the flexibility element modelling has shown good adequacy to simulate the seismic
behaviour of the structure under analysis. However, the following issues are recognized to
require further development: the element dissipation features in the pre-yielding range, the
unloading stiffness degradation, the pinching effect and the rebar slippage simulations. The
first is directly related with the flexibility element formulation proposed herein, while the sec-
ond and the third depend only of the global section model; the last issue is better accounted for
by specific elements as proposed by Filippou et al. (1992) or Monti et al. (1993).
340 Chapter 6
Chapter 7
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES
DESIGNED ACCORDING TO EC8
7.1 Introduction
The Eurocode 8 is included in the set of nine Eurocodes presently being discussed and pub-
lished as prestandards defining the common rules for structural and geotechnical design. The
EC8 is concerned with the seismic design and, due to its innovative features relatively to exist-
ing national codes, it is considered an advanced code and even “the latest word in codified
earthquake resistant design” (Carvalho et al. (1996)).
Aiming at the safe, yet economic, design of earthquake resistant structures, the innovative fea-
tures of EC8 (e.g. the establishment of a serviceability limit state for damage limitation under
seismic action, or the adoption of capacity design procedures for reinforced concrete building
structures) have been subjected to a european-wide discussion and testing process over the last
few years, in order to gather relevant information for code improvement or revision to become
a definitive european standard.
Aiming at EC8 testing and validation, a unified effort has been carried out within the frame-
work of the pioneering research project entitled “Prenormative Research in support of Euroc-
ode 8” (PREC8), co-funded by the European Commission and National Authorities.
The present chapter includes the presentation and discussion of results from the numerical seis-
mic analyses of RC frame structures performed under the PREC8 programme, as a result of
our activity at the Joint Research Centre (JRC). The major concern is in line with the project
entitled “Reinforced concrete frames and walls” which included the design of trial cases (con-
342 Chapter 7
stituting a set of typical RC buildings) and the evaluation of their non-linear seismic response,
in order to find out the implications of EC8 provisions on the seismic behaviour of building
structures. Some trial cases, viz those having the so called basic configurations 2 and 6, were
analysed by the author within the JRC team activities and the results were extensively reported
in Arêde et al. (1996). Some modifications and improvements were subsequently performed
which are already included in the results recalled in this chapter.
For completeness, a general overview of the PREC8 project is provided in 7.2, specifically
focusing on reinforced concrete frame and wall structures. Then, the building configurations 2
and 6 are presented in 7.3 where details are included concerning structural layout, trial cases,
loads, modelling assumptions and response variables to express the analysis results. The most
important results obtained from the non-linear analyses are described and discussed in 7.4,
bearing in mind the expected seismic performance of structures designed according to the
basic philosophy underlying EC8. Finally, the main conclusions of the present chapter are
summarized in 7.5, particularly focusing on the assessment of the structures analysed herein.
7.2 The PREC8 project
7.2.1 Basics of EC8
According to EC8, the basic principles of safe seismic design require that in case of seismic
events: a) human life protection is ensured, b) damage is limited and c) important structures for
civil protection (lifelines, hospitals, etc.) are kept operational. This is assumed to be accom-
plished if the structural design ensures the two following basic requirements to be fulfilled:
• no collapse of the structure under the design earthquake (having a very small probability of
occurrence) which stands for the performance assessment referring to the Ultimate Limit
State (ULS)
• the damage is kept within repairable limits when the structure is subjected to a seismic
action with a larger probability of occurrence, thus assuring the structural performance with
reference to the Serviceability Limit State (SLS)
The seismic action is represented in EC8 by the elastic response spectrum, based on the design
ground acceleration (indeed, the effective peak ground acceleration in rock or firm soil) and on
other parameters conveying the influence of local ground conditions (sub-soil classes). The
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 343
design ground acceleration is assumed, for EC8 purposes, to approximately represent the seis-
mic hazard variable in a given zone. The seismic zonation is to be defined by National Author-
ities and the hazard is assumed to be constant within each seismic zone, such that the design
ground acceleration typically corresponds to a reference return period of 475 years (equivalent
to 10% exceedance probability in 50 years).
The ULS verifications rely on the exploitation of the structural ductile capacity, which means
that a non-linear ductile response is generally assumed to take place in order to dissipate the
seismic input energy; however, the limiting case of non-dissipative structures is also consid-
ered in EC8. For design purposes the non-linear ductile response is approximately taken into
account by means of the concept of a global behaviour factor, through which a design spectrum
suitable for linear analysis methods is derived from the elastic response spectrum; the design
spectrum is thus used to characterize the design seismic action for ULS verifications.
Instead, the SLS verifications are carried out for a seismic action derived from the design one
by a reduction factor taking into account the lower return period of earthquakes associated with
that limit state.
The structural regularity issue is treated separately in terms of regularity in plan (mainly affect-
ing the requirements on structural analysis models) and of regularity in elevation (which also
influences the behaviour factor values, more reduced for irregular structures). For the more
general cases, EC8 requirements may lead to significant computational demands, such as those
related to spatial dynamic analysis where accidental eccentricities and simultaneous action of
earthquake components are to be included.
The energy dissipation issue for RC structures is particularly developed in EC8 by the adoption
of different Ductility Classes (DC), reflecting a certain trade-off between structural strength
and ductility, indeed, the two basic and simultaneous requirements in earthquake resistant
structures. According to EC8, three ductility classes can be considered: a) Ductility Class L
(DCL) corresponding to structures designed and detailed according to EC2, although supple-
mented with additional detailing rules to enhance the available ductility; b) Ductility Class M
(DCM) to which correspond structures designed, dimensioned and detailed enabling the struc-
tural behaviour to enter well within the inelastic range without brittle failures and c) Ductility
Class H (DCH) corresponding to structures whose design, dimensioning and detailing provi-
344 Chapter 7
sions ensure the development of chosen stable mechanisms allowing large hysteretic energy
dissipation.
In operational terms, ductility classes are distinguished by different behaviour factors (q),
reducing as the ductility class decreases, according to the proportionality rule of 1.00, 0.75 and
0.50, respectively for DCH, DCM and DCL. Moreover, for the DCM and DCH, Capacity
Design procedures (Paulay and Priestley (1992), Eurocode 8 (1994)) are adopted in order to
provide the structure with a more suitable mechanism for energy dissipation. Typically, for
reinforced concrete frame structures, Capacity Design aims at allowing inelastic excursions
only at beam end regions and at the base of ground floor columns, such that, despite the dam-
age occurred in those regions, the overall stability of the structure is still assured under gravity
load actions. To this end, a certain sequence of individual member design has to be followed in
order to account for the actual strength of adjacent members.
Although the practical application of EC8 as a provisional norm has been already undertaken,
some priority needs of research support were identified and a broad scientific network was set
up to accomplish a Pre-normative Research project in support of EC8 (PREC8) (Pinto and
Calvi (1996)). In view of different types of structures with specific behaviour features, distinct
topics were covered, namely: a) RC frames and walls; b) Infilled frames; c) Bridges and d)
Foundations and retaining walls.
The major concern in the present chapter focuses on the first topic, whose main scope was to
investigate “the interrelation between a number of design parameters used in EC8, which in a
combined form, influence the non-linear behaviour of structures subjected to earthquake
motions” (Carvalho et al. (1996), Pinto and Calvi (1996)). These parameters are the regularity
classification, the analysis methods, the behaviour factor values and the effects of capacity
design procedures. An additional issue which has drawn the attention of researchers aimed at
assessing how safe and economic are the structures designed according to EC8 provisions and
whether equivalent results are obtained by designing for different Ductility Classes. Further
details on this topic of the PREC8 project are included in the next section.
7.2.2 The RC frame structure topic
A large parametric study of RC frame and wall structures was performed in the PREC8 con-
text, particularly aiming at the assessment of ductility classes and respective behaviour factors.
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 345
A set of buildings was designed according to Eurocodes 2 and 8, for which other parameters
than the ductility class were also varied, in order to constitute a number of different situations.
Their seismic response was analytically evaluated for earthquakes of increasing intensity, aim-
ing at checking whether the EC8 provisions lead to a satisfactory structural response through-
out the different trial cases.
The set of buildings consisted of 26 distinct RC structures, distributed as follows according to
a specific envisaged purpose (Carvalho et al. (1996)):
• buildings for three types of uses, in order to highlight to which extent the relative impor-
tance of vertical versus horizontal loads may affect the result of capacity design provisions;
• buildings with four different heights (3, 4, 8 and 12 storeys), in order to check the influence
of the natural period of the structure and the effect of minimum design provisions likely to
prevail at the upper storeys of taller buildings;
• buildings located in two distinct seismic zones, characterized by design accelerations 0.15g
and 0.30g (respectively, standing for low/medium and medium/high seismicity), where the
adequacy of code rules should be checked and compared;
• buildings with framed structures and a central core, in order to introduce the effect of cou-
pled frame-wall behaviour and assess the outcome of code provisions for walls.
Buildings for numerical analysis were grouped in six basic configurations (1 to 6) and, with the
exception of configuration 4 (an industrial building) which had three storeys and large spans,
all structures were based on a similar rectangular plan layout with 15 m (3 bays of 5 m) by 20
m (5 bays of 4 m or 3 bays of 8/4/8 m). Storeys were typically 3 m high, except for the indus-
trial building and for configuration 6 where four columns were suppressed at the ground level
while the remaining ones are 4.5 m high (an irregular situation was sought, related to a softer
first storey and to some columns resting on first floor beams). Generally, dynamic analysis
methods were used for the design seismic analysis, but in two specific configurations addi-
tional analysis by static methods was carried out (as allowed by EC8) in order to check their
influence on the seismic performance of structures.
The whole process of building design was carried out at University of Patras (Fardis (1995)) in
a computerized way, from which it became clear that EC8 application requires computer anal-
ysis using appropriate software, particularly due to seismic action combinations and to capac-
ity design procedures, the latter introducing links between several phases of the design.
346 Chapter 7
Additionally, interesting information was gathered concerning the characteristics of EC8
designed structures and, particularly, regarding the required quantities of steel and concrete
depending on the ductility class. It was found that the total quantities of both materials is
approximately the same for the three ductility classes (Fardis (1995), Carvalho et al. (1996)),
although differently distributed: a) the ratio of column-to-beam total steel shifts from about
55%-45% for DCL, to 60%-40% for DCM and to 65%-35% for DCH, and b) the ratio of longi-
tudinal-to-transverse steel varies from about 80%-20% for DCL, to 75%-25% for DCM and to
60%-40% for DCH. These trends on the steel distributions arise from the capacity design pro-
visions of EC8, not foreseen for DCL but gaining increasing importance as the ductility class
increases.
Once the design results were available, the non-linear analysis of the 26 buildings was carried
out by six partners of the PREC8 project. Aiming at a certain control of results, each trial case
was analysed by two different teams and some basic assumptions were commonly adopted by
all partners (e.g. mean values of resistances, input accelerograms and damping quantification).
The analyses were performed separately for each horizontal direction with four distinct artifi-
cial accelerograms generated to fit the EC8 spectrum for soil type B and for peak ground accel-
erations of 0.15g or 0.30g, whichever matches the design acceleration. Each trial case was
analysed for the nominal intensity (1.0 times the peak acceleration) and also for two other
increased intensities (with scaling factors of 1.5 and 2.0) aiming at roughly tracing the vulner-
ability functions of the structures.
The JRC team was commissioned to perform the non-linear analyses of 9 trial cases with the
basic configurations 2 and 6, consisting of 8 storey frame buildings, regular in the cases of con-
figuration 2 and irregular in the other cases, which are further described in the next section.
7.3 The building configurations 2 and 6
7.3.1 General comments and structure layout
The general layout of the two basic configurations 2 and 6 is illustrated in Figure 7.1, both in
plan and in elevation, where the global coordinate system is also included.
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 347
3.00
24.00
4.00
3.00 3.00 3.00 3.00 3.00 3.00 3.00
4.00
4.00
4.00
4.00
Y
X
4.00
4.00
4.00
4.00
4.00
Col
umns
rem
oved
in
the
grou
nd fl
oor
5.005.005.00
Y
X
3.00
25.50
3.00 3.00 3.00 3.00 3.00 3.00 4.50
5.005.005.00
Figu
re 7
.1B
asic
con
figur
atio
ns o
f the
eig
ht s
tore
y tr
ial c
ases
(PR
EC8)
a) C
onfig
urat
ion
2b)
Con
figur
atio
n 6
348 Chapter 7
The structures are symmetric in both horizontal directions (XX and YY) and, while configura-
tion 2 is regular in plan and in elevation, configuration 6 exhibits two sources of irregularity in
elevation: a) the first storey appears to be softer than the remaining ones, due to its greater
height and to the absence of some columns cut off at the first storey; b) the existence of these
cut-off columns, supported by medium-long span beams, may generate itself additional
demands which can spread all over the structure.
Each configuration was designed according to EC2 and EC8, for different ductility classes (L,
M and H) and for two design accelerations (0.15g and 0.3g). Furthermore, the case of configu-
ration 6 for ductility class M and design acceleration 0.3g was also designed using the simpli-
fied static analysis of paragraph 3.3.2 of EC8, Part 1.2, herein labelled as “Mst”. The
combination of these design assumptions (ductility class, design acceleration and analysis
method) leads to the nine distinct trial cases listed in Table 7.1, which also includes the design
behaviour factors and the reference names (labels) identifying each trial case in the following
sections.
All the relevant data obtained from the design process is extensively described in Fardis
(1994), namely concerning section design forces, cross-section dimensions, reinforcement
details (longitudinal and transversal) and adopted slab widths contributing for beam strength
and stiffness; however, for completeness purposes, the cross-section dimensions of trial case
members are summarized in Table 7.2. Columns have uniform cross-sections in elevation and,
for configuration 6, the columns removed in the ground floor are referred to as “Cut-off” col-
umns. Except for the first storey beams of configuration 6 (which are more robust to support
the cut-off columns) the beam cross-sections are also uniform in elevation and almost uniform
in plan.
Table 7.1 Trial cases, design behaviour factors and earthquake intensities
Config. Design Accel.(a)
Ductility Class
Reference Name
Behaviour Factor (q)
Earthquake Intensities (*a)
20.15g
L 2_15L 2.5 - 1.0 1.5 2.0M 2_15M 3.75 - 1.0 1.5 2.0
0.30g M 2_30M 3.75 - 1.0 1.5 2.0H 2_30H 5.0 0.5 1.0 1.5 2.0
6
0.15g L 6_15L 2.0 - 1.0 1.5 2.0M 6_15M 3.0 - 1.0 1.5 2.0
0.30gM 6_30M 3.0 - 1.0 1.5 2.0
Mst 6_30Mst 3.0 - 1.0 1.5 2.0H 6_30H 4.0 0.5 1.0 1.5 2.0
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 349
7.3.2 Vertical static loads and seismic action
According to the data used in the design, the following vertical static loads, per unit area, have
been considered:
• slab self-weight: ws = 3.5 kN/m2
• finishing: wf = 2.0 kN/m2
• live load: ql = 2.0 kN/m2
Using the appropriate combination coefficients for variable actions as prescribed in EC8, the
static vertical loads to be adopted simultaneously with the seismic action have been computed
by the following rules:
• top floor: ws + wf + 0.30*ql
• other floors: ws + wf + 0.15*ql
Vertical loads uniformly distributed per unit length in beams were obtained from the previous
values with the adequate influence areas and the beam self-weight was also included. These
loads were applied prior to any seismic input in order to start the seismic analysis with the
effects of dead and live loads already taken into account (namely, in what concerns stiffness).
The seismic action was simulated by four artificial accelerograms, shown in Figure 7.2, that
were provided to all the participant teams in the PREC8 project.
Table 7.2 Member cross-sectional dimensions (m)
CaseColumns Beams (b/h)
Internal External Corner Dir. X Dir. X(long span)
Dir. Y
2_15L .60x.60 .55x.55 .50x.50 .30x.60 .30x.60 .30x.602_15M .60x.60 .55x.55 .50x.50 .25x.50 .30x.60 25x.502_30M .70x.70 .60x.60 .60x.60 .30x.60 .30x.60 .30x.602_30H .70x.70 .60x.60 .60x.60 .30x.60 .30x.60 .30x.60
Internal External Cut-off Dir. X(1st floor)
Dir. X(2-8th floors)
Dir. Y
6_15L .70x.70 .60x.60 .50x.30 .30x.80 .30x.60 .30x.606_15M .70x.70 .60x.60 .50x.30 .30x.80 .30x.60 .30x.606_30M .80x.80 .70x.70 .50x.30 .30x.80 .30x.60 .30x.606_30Mst .80x.80 .70x.70 .50x.30 .30x.80 .30x.60 .30x.606_30H .80x.80 .70x.70 .50x.30 .30x.80 .30x.60 .30x.60
350 Chapter 7
Figure 7.2 Artificial accelerograms (S1...S4) and response spectra (5% damping) fitting the
EC8 response spectrum
Time [s]
Acceleration (m/s2)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 Max.: 0.96
Time [s]
Acceleration (m/s2)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 Max.: 1.0
Time [s]
Acceleration (m/s2)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 Max.: 1.06
Time [s]
Acceleration (m/s2)
.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 -1.0 -.8 -.6 -.4 -.2 .0 .2 .4 .6 .8 1.0 Max. : 0.84
Period [s]
(m/s2)
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 .0
.5
1.0
1.5
2.0
2.5
3.0
Period [s]
(m/s2)
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 .0
.5
1.0
1.5
2.0
2.5
3.0
Period [s]
(m/s2)
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 .0
.5
1.0
1.5
2.0
2.5
3.0
Period [s]
(m/s2)
.0 .5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 .0
.5
1.0
1.5
2.0
2.5
3.0
EC8 - Soil B
S1
S2
S3
S4
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 351
The accelerograms of 10 s duration were generated to fit the EC8 response spectrum for soil
type B and 5% damping and normalized to a unitary base acceleration. The set of accelero-
grams was scaled for the design acceleration corresponding to each of the nine trial cases listed
in Table 7.1 and then factored by the intensities included in the same table. Thus, considering
all the trial cases, with all the earthquake intensities, for all accelerograms, in both directions
XX and YY independently, a total of 232 non-linear dynamic analyses was performed.
7.3.3 Structure modelling
7.3.3.1 Discretization
The frame structures shown in Figure 7.1 were discretized for independent planar analyses in
the XX and YY directions. Taking profit of the symmetry, only the association of two distinct
frames (one internal and other external) is considered in each direction of analysis with double
values of stiffness, vertical static load and mass. The corresponding structural systems are
illustrated in Figure 7.3 for both configurations 2 and 6 in the two directions of analysis.
Equal horizontal displacements were imposed to all the nodes at the same floor in order to
accomplish the assumption of rigid floor diaphragm. Each structural member was discretized
by only one flexibility global element as adopted for the analysis of the four storey building
tested at ELSA (see Chapter 6). Despite the problems found in simulating the experimental
response of the ELSA building for high ductility levels (mainly due to the modelling of
unloading stiffness degradation) the model was still adopted for the present chapter analyses
because the foreseen intensities are not expected to develop such high ductility levels as those
attained for the final cyclic tests. Thus, even with the referred modelling limitation, the model
is likely to reproduce reasonably well the structural behaviour for the expected ductility range.
Vertical static loads were considered by the approximation inherent in the flexibility element
model, i.e., the distributed load was lumped into equivalent concentrated forces at end and
mid-span sections.
Different characteristics were assigned to the left and right parts of each element, according to
the actual data of the corresponding member end section. However, these characteristics are
assumed uniform along each element part, meaning that details of span reinforcement or varia-
tion of slab width are not taken into account.
352 Chapter 7
Rigid lengths were assumed only for the beam ends (with half width of the nearest column)
and the effect of pull-out of reinforcement bars from the joint was not considered. Shear
behaviour was considered linear elastic for both beams and columns, using shear reduced areas
equal to the cross-sectional area divided by 1.2 (rectangular cross-section).
Figure 7.3 Structural systems of planar frame associations
7.3.3.2 Mass, damping and natural frequencies
The structural mass has been computed for each floor according to the vertical static loads
referred to in 7.3.2, and equally distributed by the nodes belonging to that floor. Table 7.3 sum-
marizes the total mass values for each floor of all the trial cases. The total mass of the building
is included for comparison with the value used in the design (also listed in the same table),
which shows small deviations of no more than 5%. Note that the contribution of half inter-sto-
rey height of columns is also included in the floor mass, which explains the different values for
the top floor and also for the first floor of configuration 6.
Internal Frame External Frame
X
Z
Storey level
8 7
2 1
Direction XX Direction YY
Internal Frame External Frame
a) Configuration 2
Internal Frame External Frame
Direction XX Direction YY
Internal Frame External Frame
b) Configuration 6
6 5 4 3
Storey level
8 7
2 1
6 5 4 3
X
Z
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 353
The first and second natural frequencies, in both directions of analysis, are given in Table 7.4
for all cases assumed with uncracked behaviour, including also the frequency values obtained
in the design process (Fardis (1994)). The values shown here differ slightly from the design
ones, with a general overestimation trend that agrees with the lower mass values and that may
be also related with little differences of slab width. The general agreement of frequencies is
quite acceptable and can be regarded as an “extra” global check of the structure input data.
The global inspection of the first mode frequency values (as obtained here) shows that, with
the exception of the 2_15M, 6_15L and 6_15M structures, for most cases in the XX direction,
the fundamental vibration period “falls” into the “constant” acceleration branch of the spec-
trum (check Figure 7.2) although very close to the transition point for the descending branch
(at 0.6s period or 1.67 Hz frequency). Instead, for the YY direction, the structures are more
flexible and, most of them, fall into the descending branch.
Table 7.3 Floor masses. Adopted values and design values (in brackets)
CaseMass per Floor (ton)
Total (ton)Floor 1 Floors 2 - 7 Floor 8
2_15L 270 270 255 2145 (2236)2_15M 260 260 244 2064 (2080)2_30M 284 284 262 2250 (2333)2_30H 284 284 262 2250 (2333)6_15L 296 272 256 2184 (2299)6_15M 296 272 256 2184 (2299)6_30M 316 288 263 2307 (2328)6_30Mst 316 288 263 2307 (2328)6_30H 316 288 263 2307 (2328)
Table 7.4 Frequencies (Hz) for all cases (design values in brackets)
Case1st Mode Frequency 2nd Mode Frequency
XX YY XX YY2_15L 1.71 (1.64) 1.49 (1.45) 5.20 (5.00) 4.53 (4.45)2_15M 1.52 (1.52) 1.23 (1.23) 4.67 (4.69) 3.79 (3.86)2_30M 1.86 (1.78) 1.63 (1.58) 5.69 (5.48) 4.98 (4.91)2_30H 1.85 (1.78) 1.61 (1.58) 5.66 (5.48) 4.94 (4.91)6_15L 1.63 (1.52) 1.50 (1.40) 4.91 (4.61) 4.58 (4.33)6_15M 1.61 (1.52) 1.47 (1.40) 4.86 (4.61) 4.53 (4.33)6_30M 1.80 (1.71) 1.68 (1.59) 5.45 (5.26) 5.14 (4.97)6_30Mst 1.83 (1.71) 1.72 (1.59) 5.52 (5.26) 5.21 (4.97)6_30H 1.77 (1.71) 1.63 (1.59) 5.88 (5.26) 5.05 (4.97)
354 Chapter 7
This fact is quite relevant concerning the structural seismic response because it is responsible
for part of the structural overstrength. Indeed, for structures having vibration periods near or in
the descending branch of the spectrum, the structure softening due to concrete cracking and
reinforcement yielding, increases the vibration period, which leads to seismic forces lower
than the design ones and, consequently, to lower demands in the structure comparatively to
those foreseen in the design. This constitutes the so called “demand-side” overstrength (Fardis
and Panagiotakos (1997)), which is different, in nature, from the “supply-side” overstrength
(typically related with the difference between the design values of material properties and the
mean values as used in non-linear analyses) and leads to a reduction of ductility demands.
Viscous damping was included, proportional to the mass and stiffness (Rayleigh type), with a
damping factor of 2% for both the first and the second modes, as adopted for some analyses of
the four-storey ELSA building. Despite the conclusions drawn in 6.4.4.5, pointing out that vis-
cous damping might not be adequate for hysteretic response simulation in the post-yielding
range, the factor of 2% was still kept for the present analyses because it corresponds to a fixed
data value imposed to all partners participating in the PREC8 project.
7.3.3.3 Moment-curvature constitutive relations for global section behaviour
The trilinear moment-curvature primary curves for global section modelling were obtained by
the process described in 4.2.3, for which the material properties were adopted according to typ-
ical mean values of the concrete class (C25/30) and the steel grade (B500 Tempcore) as used in
the design. Thus, according to the notation of stress-strain diagrams shown in 4.2.3.2, the fol-
lowing values were considered:
• Steel: fsy = 585 MPa; Es = 200 GPa; Esh = 1.7 GPa; εsm = 0.090;
• Concrete: fc0 = 33 MPa; εc0 = 0.0022; fct = 2.6 MPa; Ec = 30.5 GPa;
For the concrete model, the final comments of 6.2.2.2 apply: the confinement parameters were
obtained from the cross-section and transverse reinforcement data and residual stresses were
taken as 0% and 20% of the peak stress, respectively, for unconfined and confined concrete.
For M-ϕ curve evaluation, the effective slab width participating in beam strength and stiffness
was considered using values available in the design information (Fardis (1994)). Concerning
the axial force, null values were taken for the beams, while constant values due the vertical
static loads were considered for columns.
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 355
By contrast with the ELSA building, no fibre refinement was done for more accurate definition
of yielding and of post-yielding in columns (as described in 6.2.3). However, special care was
taken with the ultimate point definition for column sections where the presence of axial load
can lead to a rather curved shape of the post-yielding branch of the M-ϕ diagram. Actually, in
some cases, the ultimate point can lie on the softening part of the curve, as schematically
shown in Figure 7.4, in such a low level that the approximated post-yielding line YU has a neg-
ative slope. In order to avoid this situation an extra (intermediate) point I is determined and the
line YI is adopted for the post-yielding branch. Additionally, a modified ultimate point U’ is
considered such that the strain energy between Y and U’ is the same as that between Y and U.
The intermediate point I was taken here as the supplementary point referred to in 4.2.3.4, cor-
responding to an average strain in the most compressed concrete fibre or in the most tensioned
reinforcement between strain values at yielding and at ultimate conditions (see Figure 4.16).
Figure 7.4 Approximation of the post-yielding branch of the (M-ϕ) diagrams in columns
The set of M-ϕ curves for all the different cross-sections of configuration 6, in the direction
YY, is included in Figures 7.5 and 7.6 both for beams and columns, respectively. Although no
detailed information can be extracted from those figures for a specific cross section, they are
useful to provide an idea of the order of magnitude of the resistances involved and also of the
available curvature ductility (at least in terms of a rough estimate). The following aspects can
be pointed out from those figures:
• Ultimate curvatures for beams in positive bending (ϕ+u) are much more uniform than those
in negative bending (ϕ-u), which is due to the fact that for ϕ+
u the bottom steel layer (having
always a smaller area than the top one) is tensioned and the compression zone can spread
along the effective slab width. Thus, no concrete crushing is likely to occur for this bending
ϕ
My
Mc
M
ϕc ϕy
C
Y
Mu
ϕu
U′
U
I
356 Chapter 7
direction (positive), while for negative bending large compressions develop in the web bot-
tom fibres due to the high top steel content. It is apparent that this effect is much more criti-
cal for the lower ductility classes (i.e., DCL for 0.15g and DCM for 0.30g) because in such
cases higher design seismic forces are involved.
• Yielding moments (My) of beams decrease when the ductility class level is increased; more-
over, as expected between two ductility classes (e.g. M and L), the ratio of corresponding
yielding moments is not far from the inverse ratio of behaviour factors
. The relation between these two ratios depends on practical design reasons
(detailing) and on the relative magnitude of vertical static loads compared to the seismic
forces, but still the value of is helpful to check the trend of changes in My between duc-
tility classes. Similar reasons explain the fact that between the 0.30g and the 0.15g design
options the corresponding ratio of My values is not 2.0 as could be expected. Furthermore, it
is worth mentioning that, for the static design option 30Mst of configuration 6, the values of
My are higher than for the case 30M which is related to the simplified static procedure used
for the seismic analysis that usually overestimates the action effects.
• For columns, the ultimate curvature values and their regularity, as well as the corresponding
ductility factor, are higher for increasing ductility class levels and the same happens for the
uniformity of the post-yielding stiffness.
• The trend of My variations in columns when ductility class is varied is not as clear as it is
for beams, because, apart from the behaviour factors ratio, the capacity design magnifica-
tion factors play an important role in column design. Actually, these factors are different
from one ductility class to another and, due to its dependency on the design and the resistant
moments distribution, their effect can go either in the same or in the opposite direction of
the ductility class variation. For this reason some cases were found where the increase of
ductility class seems not to modify the column My values (configuration 2, not shown
herein) while in other cases a clear modification is noticeable (configuration 6).
Finally, concerning the hysteretic behaviour parameters, similar options were adopted as for
the four-storey ELSA building (see 6.2.4). For the unloading stiffness degradation the parame-
ter was considered for both positive and negative directions, the pinching effect was
taken into account only due to top and bottom reinforcement asymmetry ( factors were
obtained by Eq. (B.3) in Appendix B) and the strength degradation factors were calculated
using Eq. (B.4).
rm MyM My
L⁄=( )
rq qM qL⁄( )1–
=
rq
α 4.0=
γ
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 357
6_30
M_Y
6_15
M_Y
-.2
5 -.
20 -
.15
-.10
-.0
5 .
00
.05
.10
.1
5 .
20
.25
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
(x1.
e3)
6_15
L_Y
6_30
H_Y
6_30
Mst
_Y
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
-.2
5 -.
20 -
.15
-.10
-.0
5 .
00
.05
.10
.1
5 .
20
.25
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
-.2
5 -.
20 -
.15
-.10
-.0
5 .
00
.05
.10
.1
5 .
20
.25
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
-.2
5 -.
20 -
.15
-.10
-.0
5 .
00
.05
.10
.1
5 .
20
.25
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
-.2
5 -.
20 -
.15
-.10
-.0
5 .
00
.05
.10
.1
5 .
20
.25
-1.5
-1.2
-.9
-.6
-.3
.0
.3
.6
.9
1.2
1.5
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
Figu
re 7
.5B
eam
sec
tion
mom
ent-c
urva
ture
dia
gram
s of
all
case
s w
ith c
onfig
urat
ion
6 in
dire
ctio
n YY
M
358 Chapter 7
.0
0 .
02
.04
.06
.0
8 .
10
.12
.14
.1
6 .
18
.20
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
.0
0 .
02
.04
.06
.0
8 .
10
.12
.14
.1
6 .
18
.20
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
.0
0 .
02
.04
.06
.0
8 .
10
.12
.14
.1
6 .
18
.20
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
.00
.02
.0
4 .
06
.08
.10
.1
2 .
14
.16
.18
.2
0
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
.0
0 .
02
.04
.06
.0
8 .
10
.12
.14
.1
6 .
18
.20
.0
.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
6_30
M_Y
6_15
M_Y
(x
1.e3
)6_
15L
_Y
6_30
H_Y
6_30
Mst
_Y
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
(x1.
e3)
Mom
ent (
kN.m
)
Cur
vatu
re (m
-1)
Figu
re 7
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omen
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re d
iagr
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YY
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 359
7.4 Non-linear seismic analysis of building configurations 2 and 6
7.4.1 General
The results from the non-linear analysis of the structures are presented and discussed in the
present section. They are expressed by common response variables such as total base shear, top
displacement (or total drift), inter-storey drift (relative to the storey height), member displace-
ments, ductility factors and damage indices.
Both ductility and damage refer to member end chord rotations and are defined as for the 4-sto-
rey ELSA building, following the procedures and expressions described in 6.3.2.
Generally, results refer to average values calculated from the response to the four accelero-
grams considered; typically they are categorized according to the basic design assumptions
(design acceleration and ductility class) and to the action intensity.
7.4.2 Structural strength
The structural strength engaged during seismic response is an important issue to be addressed
since it can provide an insight into the global behaviour. The total peak base shear provides a
measure of the global strength and can be compared to the design forces based on which the
member dimensioning and detailing were carried out. Such forces (available from the design
information in Fardis (1994)) are given in Table 7.5 as a fraction of the total weight, showing
that, except for the configuration 2 designed for 0.30g, the design seismic forces almost coin-
cide in both directions XX and YY.
The ratio of peak base shear from the non-linear seismic analysis to the design seismic
forces is plotted in Figure 7.7 for all trial cases and intensity levels. This ratio, henceforth
Table 7.5 Design base shear force ratio to structure weight (seismic coefficient)
Case Configuration 2 Configuration 6XX YY XX YY
15L 0.138 0.134 0.169 0.16315M 0.084 0.086 0.113 0.10930M 0.190 0.205 0.233 0.23230H 0.145 0.162 0.175 0.17630Mst -- -- 0.249 0.249
Rseismmax( )
Rd( )
360 Chapter 7
denoted by , is a measure of the structure overstrength for loading conditions exceeding the
design seismic force and it can be seen that maximum values of about 2.0 can be found for
twice the design intensity.
Figure 7.7 Global overstrength
The major sources contributing for that overstrength can be categorized as follows:
• the difference between mean and design values of constitutive material strengths; mean val-
ues as used in non-linear analysis are approximately 1.3 and 2.0 times the design values,
respectively for steel and concrete;
• minimum reinforcement requirements and the unavoidable bar rounding up;
• capacity design requirements and gravity load rather than earthquake dominated design of
some sections;
• the post-yielding hardening at the section level;
• the fact that different strength mechanisms may be activated in the design and in the analy-
sis, such that, for a seismic action with the design intensity, only part of the critical zones
have yielded while the remaining ones keep behaving in the pre-yielding range (non-simul-
taneous yielding of all the critical zones that are assumed in the design).
ψm
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Global OverStrength Global OverStrength
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
ψm Rseismmax Rd⁄=( )
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 361
The first three sources typically affect the level of the global yielding threshold, while the two
last ones mostly contribute for the smoothness of the yielding transition and for the hardening
in the post-yielding range. Although all these overstrength sources are mixed up in the global
response, their influence on the yielding level can be approximately assessed by means of a
push-over analysis consisting of an inverted triangular distribution of lateral forces monotoni-
cally applied to the structures (as also performed for the 4-storey ELSA building). Thus, base
shear - top displacement diagrams were obtained for all trial cases in both directions of analy-
sis, and, despite the difficulty of defining what the global yielding point is, approximate esti-
mates were determined for the yielding base shear force according to the criterion described
next; the base shear - top displacement curve for one trial case is shown in Figure 7.8.
Figure 7.8 Base shear - top displacement curve for the C2_15L case, direction XX. Definition
of global yielding force
Push-over analysis curves were considered up to a top displacement approximately corre-
sponding to a total drift of 2%, which is an upper bound well above the maximum drift found
in the analysis (about 1.5% for twice the design intensity); such a displacement level is deemed
to correspond to a stabilized hardening range of the global response. A fictitious yielding point
Y* is defined by the intersection of straight lines approximating the pre- and post-yielding
curve zones and the global yielding point Y is taken lying on the curve and having the same
displacement as Y*; thus, the yielding base shear Ry is read at the level of point Y.
The ratio of Ry to Rd (design force), standing for the overstrength at yielding and denoted by
TOP DISPL.(m)
BASE SHEAR (kN)
.00 .05 .10 .15 .20 .25 .30 .35 .40 .45 .50 .00
.60
1.20
1.80
2.40
3.00
3.60
4.20
4.80
5.40
6.00 (x1.E3)
Y*
YRy
362 Chapter 7
, is given in Table 7.6 and shows that a significant reserve of strength is found for most
cases; values range from about 1.1 to 1.5, but with a clear trend to exceed 1.3. Average values
of are also listed, showing that the ductility class does not appear to influence the over-
strength at yielding, while for the lower design acceleration slightly larger values of are
obtained. This is consistent with the higher relative weight of gravity loads in the 0.15g cases,
whose design is more likely to be controlled by other load combinations than the seismic ones
and, therefore, to have enhanced strength to lateral loads. Finally, it is emphasized that the glo-
bal average factor is about 1.36, thus agreeing with the above referred overstrength factor
(1.3) for steel that mostly controls the yielding threshold.
The comparison of the force with Ry allows to check, in a global sense, whether the
post-yielding behaviour has been actually entered; the ratio of to Ry, herein denoted by
and designated by global hardening factor, is plotted in Figure 7.9 for all trial cases.
The factor is actually a measure of the structural hardening at the global level for the cases
where , but it also means that for cases having clearly below 1.0, no yielding has
taken place. Indeed, that is the case of the DCL structures for the design intensity, due to their
important reserve of strength and to the fact that just a low portion of seismic action is to be
absorbed by ductile behaviour (q-factors of 2.5 and 2.0); this justifies also the fact that for
twice the design intensity, the peak base shear stays below 1.2 times the yielding one, while for
the other trial cases, the hardening factor reaches values as high as 1.5.
The analysis of is also valuable to understand the global overstrength factor for the
design intensity (see Figure 7.7), whose average value is about 1.0 for DCL structures. In these
cases, an elastic response to the design intensity earthquakes would generate average base-
shear forces about q (2.5 and 2.0) times the design ones; since no yielding was found, it fol-
Table 7.6 Overstrength factors at yielding
CaseConfiguration 2 Configuration 6XX YY XX YY Average
15L 1.50 1.29 1.41 1.35 1.3915M 1.50 1.15 1.49 1.36 1.3830M 1.38 1.16 1.36 1.31 1.3030H 1.38 1.09 1.45 1.29 1.3030Mst -- -- 1.52 1.49 1.50Average 1.44 1.17 1.45 1.36 1.36
ψo
ψo
ψo
ψo
ψo Ry Rd⁄=( )
Rseismmax
Rseismmax
ψh
ψh
ψh 1> ψh
ψh ψm( )
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 363
lows that the obtained elastic force reduction to the design force level must arise from crack-
ing. Actually, the corresponding frequency drop induces a reduction of the spectral ordinate in
the hyperbolic descending branch of the elastic spectrum (see Figure 7.2), which for the DCL
structures appear to be about 40% to 50% of the original values (based on the uncracked
behaviour as assumed in the design); indeed, similar results have been reported by Fardis and
Panagiotakos (1997) for other structures within the PREC8 framework, reflecting the impor-
tant reduction of seismic demands due to structural cracked behaviour.
Figure 7.9 Global hardening factor
For structures of higher ductility classes (M and H), the higher q factors (3 to 5) lead to force
reductions (compared to the elastic ones) larger than those induced by the frequency drop due
to cracking, which means that after the cracking effect, there is still a portion of seismic
demand to be absorbed by post-yielding behaviour. This is confirmed in Figure 7.9 for almost
all DCM and DCH cases, whose factors for the design intensity exceed 1.0, except for the
6_15M structures.
From the above presented results, the most relevant topics can be summarized as follows:
• the important strength reserve, particularly at the yielding level, induces structures designed
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1 x Design
1.5 x Design
2 x Design.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1 x Design
1.5 x Design
2 x Design.
Hardenning Factor Hardenning Factor
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1 x Design
1.5 x Design
2 x Design.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1 x Design
1.5 x Design
2 x Design.
ψh Rseismmax Ry⁄=( )
ψh
364 Chapter 7
for medium seismicity (0.15g) to behave in the pre-yielding range or just at imminent yield-
ing for the design intensity;
• for cases exceeding the global yielding, the global overstrength factor ranges between 1.3
and 1.5 for the design intensity, reaching values of about 2.1 for twice the design intensity;
• part of such overstrength is due to global hardening (factors up to 1.5 were found for the
highest intensity) which shows a slight trend to increase with the ductility class level.
7.4.3 Cracking, yielding and damage patterns
In the previous section the importance of structural cracking for the seismic response was high-
lighted. The extension of cracking in structural members is illustrated in Figure 7.10 for the
response of structures with configuration 6, in the direction XX, under the action of the earth-
quake S1 with intensities 1.0 and 2.0.
It is apparent that extensive cracking is found for the design intensity, mostly in beams and
also, to some extent, in the internal columns of the irregular frame. For twice the design inten-
sity, cracking develops further (particularly in the columns) but it is clear that the most signifi-
cant stiffness drop due to cracking takes place for intensity 1.0.
Cracking patterns do not significantly differ for the analysed cases; yet a trend is found for less
cracking development in higher ductility classes (lower seismic forces arising from larger q-
factors) and for larger cracking when the design acceleration is increased from 0.15g to 0.30g.
The rotation ductility patterns are shown in Figure 7.11, while the corresponding damage pat-
terns are included in Figure 7.12 for the same structures. Only positive rotation ductility is
included since the most relevant findings are common to negative rotation ductility patterns;
additionally, only ductility values above 1.0 are plotted as this can provide an indirect and gen-
eral view of the yielding pattern. For comparison between the regular and the irregular config-
urations, Figure 7.13 includes also the damage pattern for configuration 2 in the direction XX.
For the 0.15g designed structures very low ductilities are found for the design intensity as a
consequence of the important strength reserve. However, since these values exceed 1.0, it
appears that local yielding at the element level has developed, while at the global level a pre-
yielding behaviour was found (check values for 6_15L and 6_15M cases in Figure 7.9).
These apparently contradictory findings can be explained by two different reasons:
ψh
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 365
• On the one hand, rotation ductility values very close to 1.0 (as in the 6_15L case) may cor-
respond to situations where the plastic hinge has not actually developed. Indeed, at the ele-
ment level, and particularly in beams, the moment distribution along the members may be
more influenced by gravity loads comparatively to the 0.15g earthquake action effects
(which, furthermore, become strongly reduced by cracking development). Consequently,
the inflection point location may significantly deviate from the mid-span section and situa-
tions may occur where the chord rotation exceeds the yielding rotation (for which the
inflection point is assumed at mid-span) without actual development of plastic hinge. Note,
however, that such drawback, inherent in the adopted rotation ductility definition, tends to
vanish for higher intensity of lateral loads.
• On the other hand, the global yielding is not defined at the first onset of yielding in a given
member, which means that, before global yielding, some local hinging may have already
developed. Naturally, as the global response becomes closer to the yielding threshold, this
fact becomes more evident as in the 6_15M case. Indeed, Figure 7.9 shows that this struc-
ture in direction XX reached a maximum base shear closer to the yielding one than the
6_15L case and, accordingly, the corresponding rotation ductility becomes clearly above
1.0.
For twice the design intensity (still for 0.15g designed structures), plastic hinge formation tends
to spread all over the structure, though less evidently in the larger span beams due to their
higher flexibility compared to that of shorter span beams. Except for the cut-off columns (more
slender than the remaining ones), plastic hinging is found only in beams and at the end-zones
of ground floor columns, thus agreeing with the dissipation mechanism foreseen in the design.
Structures designed for 0.30g exhibit larger ductility demands in accordance with the clear
onset of yielding for the design intensity, although somewhat low in view of the adopted
behaviour factor. Note that, for q-factors of 3.0 and 4.0, respectively for 6_30M and 6_30H
structures, ductility demands at the member level could be expected to be at least similar to or
greater than the behaviour factor; however, since a significant overstrength factor is engaged
for the design intensity, the local ductility demand becomes reduced.
Comparing to the 0.15g structures, the 6_30M and 6_30H cases show a more uniform spread
of plastic hinging, which, for intensity 2.0, engages almost all the critical zones of the beam
sidesway mechanism underlying the design philosophy.
366 Chapter 7
Figure 7.10 Cracking pattern: Configuration 6, Direction X under earthquake S1 for intensity 1.0
and 2.0
Intensity: 2.0Intensity: 1.0
6_15L
6_15M
6_30M
6_30H
Case:Internal External Internal External
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 367
Figure 7.11 Positive rotation ductility pattern: Configuration 6, Direction X under earthquake S1
for intensity 1.0 and 2.0
Max.= 2.09 Max.= 4.25
Intensity: 2.0Intensity: 1.0
6_15L
6_15M
6_30M
6_30H
Case:
Max.= 1.09 Max.= 2.55
Max.= 1.44 Max.= 2.82
Max.= 2.25 Max.= 4.62
Internal External Internal External
368 Chapter 7
Figure 7.12 Damage pattern: Configuration 6, Direction X under earthquake S1 for intensity 1.0
and 2.0
Intensity: 2.0Intensity: 1.0
6_15L
6_15M
6_30M
6_30H
Case:
Max.= 0.23 Max.= 0.49
Max.= 0.17 Max.= 0.42
Max.= 0.18 Max.= 0.33
Max.= 0.29 Max.= 0.63
Internal External Internal External
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 369
Figure 7.13 Damage pattern: Configuration 2, Direction X under earthquake S1 for intensity 1.0
and 2.0
Intensity: 2.0Intensity: 1.0
2_15L
2_15M
2_30M
2_30H
Case:
Max.= 0.30 Max.= 0.59
Max.= 0.23 Max.= 0.40
Max.= 0.15 Max.= 0.37
Max.= 0.27 Max.= 0.65
Internal External Internal External
370 Chapter 7
According to Figures 7.12 and 7.13, damage is found in almost all critical zones, even if no
local yielding has occurred there; this arises from the way how the damage index is calculated,
according to which the deformations before yielding are also taken into account. To some
extent, this can be regarded as a measure of the cracking contribution to the damage in struc-
tural members.
The ratio between maximum damage values for intensities 1.0 and 2.0 approximately follows
the corresponding ratio between rotation ductilities, actually as expected since the damage
index is mostly influenced by the peak rotation values.
The increase of design acceleration leads to larger damage values; on the other hand, for higher
ductility classes (with the same design seismic input) lower damage is obtained as a result of
better design detailing, particularly concerning transversal reinforcement, which enhances the
section (and member) ultimate ductile capacity.
Despite some significant values in the cut-off columns, the damage in configuration 6 (Figure
7.12) is better distributed in the external frame because beam spans are uniform. By contrast,
the large difference of span lengths in the internal frame cause the damage to concentrate in the
shorter central span; a similar result is obtained in the internal frame of configuration 2 (Figure
7.13). Thus, from the obtained results, the damage distribution appears more affected by the
non-uniformity of beam spans rather than the presence of cut-off columns.
Finally, it is noteworthy that higher damage is obtained in the ground floor columns of config-
uration 2 than in configuration 6, for which two reasons can be pointed out. On the one hand,
the column cross-sections of configuration 2 have smaller dimensions than those of configura-
tion 6 and, particularly for the internal frame, for approximately the same axial force; this
results in lower ultimate ductile capacity and, therefore, in higher damage (indeed, the internal
frame shows the most significant increase of base column damage, when passing from irregu-
lar to the regular configuration). On the other hand, configuration 2 has a lower first storey
height, which means that the corresponding drift (and column rotations as well) becomes
increased, thus leading to higher damage than in configuration 6.
7.4.4 Ductility demand and damage distribution in elevation
Besides the global overview of spatial distribution of response variables as given in the previ-
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 371
ous section, it is interesting to analyse the peak value distribution of meaningful control varia-
bles along the structure height. For this purpose Figures 7.14 and 7.15 show the elevation
profiles of the maximum column and beam rotation ductilities in each storey, where storey col-
umns are considered those below a given storey level. Similarly, Figures 7.16 and 7.17 include
the corresponding column and beam maximum damage profiles.
From the column ductility profiles (see Figures 7.14) it is apparent that, for the design inten-
sity, ductility values above 1.0 are found only for the 0.30g designed structures, mostly at the
base end-zones of ground floor columns and also in the cut-off columns at their base and top
storey end-zones; however, such ductilities do not exceed 1.5, showing that just incipient
yielding is enforced. Column ductility values are approximately uniform above the 2nd storey
for each ductility class and design intensity case, except for the irregular structure due to the
cut-off columns where the highest demands are concentrated.
Generally, where significant column ductilities above 1.0 are found, the higher ductility class
structures tend to develop larger ductility demands, though this effect is not strictly systematic.
For twice the design intensity, the maximum column ductility for configuration 2 is about 3.0,
occurring in the ground floor columns, whereas for configuration 6 it is about 3.7 in the cut-off
columns.
The beam ductility profiles (see Figures 7.15) confirm that just incipient yielding is obtained
for the design intensity in 0.15g structures; such incipiency is more apparent for DCL struc-
tures where the ductility just slightly exceeds 1.0. For twice the design intensity, the maximum
ductility demand in 0.15g structures is about 3.0, occurring for DCM.
As for the columns, and although not systematic, increased beam ductility demands are
obtained for higher ductility classes, particularly where more significant values are found;
additionally, the maximum demand tends to shift from the upper to the lower floors as the duc-
tility class increases.
In the 0.30g structures, maximum beam ductilities for the design intensity vary between about
2.0 to 3.0, thus confirming the clear onset of yielding; instead, for intensity 2.0, the maximum
demands reach values about 6.2 for the regular structure, whereas in the irregular one this
value slightly exceeds 7.0 in the 2nd storey beams next to the cut-off columns.
372 Chapter 7
Figure 7.14 Column rotation ductility profiles (maxima)
Design Accel.: 0.30gDesign Accel.: 0.15g
Design Accel.: 0.30gDesign Accel.: 0.15g
Direction XX
Direction YY
Direction XX
Direction YY
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um Colum nDuctility
Maxim um Colum nDuctility
Maxim um Colum nDuctility
Maxim um Colum nDuctility
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5St
orey
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um Colum nDuctility
Maxim um Colum nDuctility
Maxim um Colum nDuctility
Maxim um Colum nDuctility
a) Configuration 2
b) Configuration 6
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 373
Figure 7.15 Beam rotation ductility profiles (maxima)
Design Accel.: 0.30gDesign Accel.: 0.15g
Design Accel.: 0.30gDesign Accel.: 0.15g
Direction XX
Direction YY
Direction XX
Direction YY
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um BeamDuctility
Maxim um BeamDuctility
Maxim um BeamDuctility
Maxim um BeamDuctility
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um BeamDuctility
Maxim um BeamDuctility
Maxim um BeamDuctility
Maxim um BeamDuctility
a) Configuration 2
b) Configuration 6
374 Chapter 7
Figure 7.16 Column damage profiles (maxima)
Design Accel.: 0.30gDesign Accel.: 0.15g
Design Accel.: 0.30gDesign Accel.: 0.15g
Direction XX
Direction YY
Direction XX
Direction YY
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1St
orey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um Colum nDam age
Maxim um Colum nDam age
Maxim um Colum nDam age
Maxim um Colum nDam age
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um Colum nDam age
Maxim um Colum nDam age
Maxim um Colum nDam age
Maxim um Colum nDam age
a) Configuration 2
b) Configuration 6
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 375
Figure 7.17 Beam damage profiles (maxima)
a) Configuration 2
b) Configuration 6
Design Accel.: 0.30gDesign Accel.: 0.15g
Design Accel.: 0.30gDesign Accel.: 0.15g
Direction XX
Direction YY
Direction XX
Direction YY
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um BeamDam age
Maxim um BeamDam age
Maxim um BeamDam age
Maxim um BeamDam age
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1
2
3
4
5
6
7
8
0 0.2 0.4 0.6 0.8 1
Stor
ey
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.LowDuct.Medium
1.0 x Design Intens .1.5 x Design Intens .2.0 x Design Intens .Duct.MediumDuct.High
1.0 x Des ign Intens.1.5 x Des ign Intens.2.0 x Des ign Intens.Duct.LowDuct.Medium
1.0 x Design Intens.1.5 x Design Intens.2.0 x Design Intens.Duct.MediumDuct.High
Maxim um BeamDam age
Maxim um BeamDam age
Maxim um BeamDam age
Maxim um BeamDam age
376 Chapter 7
Column damage distribution in elevation (see Figures 7.16) essentially follows the trend of the
corresponding ductility demands and it is quite apparent that, for higher ductility classes, less
damage is observed. As already pointed out, larger damage values are found for the base col-
umns of configuration 2 compared to those of configuration 6.
For configuration 2, at the design intensity, the maximum column damage is about 0.25 for the
0.15g structures, whereas for 0.30g cases it just slightly exceeds 0.30; for twice the design
intensity these damage values approximately duplicate. In turn, for configuration 6, the maxi-
mum damage appears to concentrate in the cut-off columns (particularly in the cases designed
for higher seismicity) and for the design intensity, it is about 0.16 and 0.20, respectively for
0.15g and 0.30g cases; again these values approximately duplicate for twice the design inten-
sity.
Beam damage profiles (see Figures 7.17) generally show that higher ductility classes lead to
less damage, although some exceptions are observed for the irregular structure. Still the shift of
peak damage from the upper to the lower storeys is found when the ductility class is increased.
For 0.15g structures the maximum damage for the design intensity is always below 0.2, while
for twice that intensity it is about 0.4. For the 0.30g structures these values appear duplicated,
but the most relevant aspect is the concentration of maximum damage in a couple of storeys,
comparatively to 0.15g structures where the damage is more uniform in elevation (particularly
for configuration 2).
Damage values are typically lower in columns than in beams, following an average reference
ratio of 0.5, which is just violated at base column critical zones where significant damage can
be up to 1.5-2.0 times higher than in the first storey beams. Such high ratio is due to the fact
that maximum damage in beams rarely occurs at the first storey, but still some cases of config-
uration 2 exhibit absolute maximum column damage higher than the beam ones.
7.4.5 Overall analysis of response parameters
Rather than observing the distribution of response variables throughout each structure, the fol-
lowing paragraphs focus on the analysis and comparison of (global) response parameters.
Average estimates of peak values are obtained from the response to the four accelerograms and
the maxima over the entire structure are retained for comparison between trial cases.
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 377
The maximum total drift (ratio of top displacement to the structure height) is plotted in Figure
7.18, from which no significant and systematic differences appear between distinct ductility
classes for the same design acceleration; nevertheless, a certain trend can be observed in some
cases for larger drifts when the ductility class is increased, more visible for the 0.15g designed
cases.
In average, configuration 2 leads to drifts higher than configuration 6, as a consequence of the
slender columns of the former; however, even for twice the design intensity, low drift values
are obtained (below 1.6%) when compared to values at the near-failure stage. Indeed, as a ref-
erence, one can look back at the tests on the four storey ELSA building, that reached a total
drift of 4.8% for the final stage (see 5.5.6), to which the failure was considered imminent.
Additionally, it is worth recalling that 1.7% total drift was observed for the high level seismic
test, which, despite a nominal peak ground acceleration of 1.5 times the design one (0.30g),
actually reached almost twice the design acceleration (due to the characteristics of the artifi-
cially generated accelerogram); thus, the drift herein obtained agrees quite well with the exper-
imental evidence on a similar structure type (same design acceleration and ductility class).
Figure 7.18 Total drift (%)
The ratio of total drift at the design intensity to the corresponding design values has been found
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Total Drift (%) Total Drift (%)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
378 Chapter 7
(in average) about 1.9, resulting from the structure softening mainly caused by cracking (and
also, to less extent, due to yielding) which, on one hand, tends to amplify displacements and,
on the other hand, reduces the seismic action effects as a consequence of the spectral ordinate
reduction caused by the frequency drop.
Identical comments can be made regarding the maximum inter-storey drift over all storeys as
illustrated in the charts of Figure 7.19. The trend for higher drifts as the ductility class is raised
up becomes more apparent in these charts, but still a wide margin is available with respect to
the inter-storey drifts at failure (note that for the ELSA building values about 7% were
reached). As for the total drift, the regular structure (configuration 2), designed for 0.30g and
DCH as the ELSA building, exhibits about 2.2% inter-storey drift for twice the design inten-
sity, thus approaching the experimentally tested building result of 2.4% drift obtained for the
high level seismic test.
Figure 7.19 Maximum inter-storey drift (%)
Moreover, Eurocode 8 prescribes a serviceability limit state verification (damage control) spe-
cifically concerning the inter-storey drift. According to paragraph 4.3.2 of part 1-2 of EC8, two
limits ((a) and (b)) shall be observed for the inter-storey drift depending on whether or not
“non-structural elements of brittle materials are attached to the structure” that may interfere
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Inte r-storey Drift(%) Inter-storey Drift(%)
EC8 (a)EC8 (b)
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.5
1.0
1.5
2.0
2.5
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
EC8 (a)EC8 (b)
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 379
with structural deformations. The limits (a) and (b) are, respectively, 0.4*ν % and 0.6*ν %,
where ν stands for the reduction factor of the drift demand to account for the lower seismic
intensity associated with the serviceability limit state. For the present structure category, ν
takes the value 2.0 and the inter-storey drifts for the seismic design intensity are limited by
0.8% and 1.2%, respectively, for conditions (a) and (b). These limits are also indicated in Fig-
ure 7.19 by dashed lines (and denoted by EC8(a) and EC8(b)), showing that, for the design
intensity no case exceeds the limit EC8(b), while most of the 0.30g designed structures do not
fulfil the EC8(a) requirement.
The results of the linear elastic (uncracked) analysis (Fardis (1994)) show that, under design
conditions (say, loading and modelling assumptions) all structures are well within the EC8(a)
limit, which means that the obtained inter-storey drifts shall be compared against 0.8%. It is
apparent that such limit is verified for all the 0.15g designed structures, notwithstanding the
large cracking extent, but for the 0.30g cases it is somewhat exceeded, particularly for configu-
ration 2. This is due to the stiffness drop caused by the generalized spreading of cracking over
the whole structure and confirms the expectable non-conservatism of displacement estimates
based on uncracked behaviour as pointed out in the paragraph 3.1 of part 1-2 of EC8.
Despite this excess of inter-storey drift, the maximum sensitivity coefficient to the second
order effects (as defined in the paragraph 4.2.2 of part 1-2 of EC8) never exceeds the limit 0.1
foreseen in the code, above which the so-called P-∆ effects must be duly accounted for. Indeed,
this is confirmed in Figure 7.20 where the ratio of to the code limit (0.1) is plotted for all
trial cases; although this ratio is, in average, higher for structures of configuration 2 (due to
their relatively larger “slenderness” when compared to those of configuration 6), its maximum
value indicates that the second order overturning moments at a given storey are adequately
below the prescribed limit (i.e., 10% of the storey overturning moments caused by the seismic
inter-storey shear force).
Aiming at the analysis of the seismic action effects at the critical zone level, the maximum
damage over the entire structure is plotted in Figure 7.21 for each trial case and all seismic
intensities.
These results are not easily comparable between different cases because the maximum damage
may occur in distinct locations; note that, as apparent in Figures 7.16 and 7.17, the maximum
θ( )
θ
380 Chapter 7
base column damage is almost as high as the beam damage for some cases. However, still a
quite visible trend is found in Figure 7.21 for lower damage when the ductility class is
increased, indeed as already pointed out for the damage profiles. Additionally, for most cases,
the maximum damage obtained for the irregular structures (configuration 6) is lower than for
structures of configuration 2; this aspect essentially follows the trend already found for the
inter-storey drift (see Figure 7.19) and is mainly related with the relative magnitude of
demands which are higher in configuration 2 due to its less robust members (particularly col-
umns).
Figure 7.20 Sensitivity coefficient
For the design intensity, the maximum damage does not exceed 0.25 for 0.15g designed struc-
tures, while for the 0.30g cases about 0.35 maximum damage occurs in DCM structures that
appear to be the most critical. Particularly for the 0.30g cases, the damage extent is significant,
though within an acceptable level, having in mind the structure performance requirements
under the design intensity earthquakes.
For comparison purposes, it is recalled the damage found for the four storey ELSA building
under a nominal ground acceleration of 1.5 times the design one (0.30g). Among the present
trial cases, the 0.30g_H structure with configuration 2 in direction YY is the one resembling
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Sens itivity Coeff. / Design Value Sens itivity Coeff. / Design Value
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 381
the most with the ELSA structure, i.e. having similar beam spans and storey heights and iden-
tical design acceleration, ductility class and behaviour factor; the corresponding maximum
damage (between 0.36 and 0.53, respectively for intensities 1.5 and 2.0) does not significantly
deviate from the referred ELSA building results, for which a maximum value of 0.41 was
obtained.
Instead, for twice the design intensity, quite important damage develops: almost 0.75 maxi-
mum damage is found for the 0.30g_M structure with configuration 2 in the XX direction. It is
noteworthy that, despite the fact that somewhat larger inter-storey drifts are obtained for most
0.30g_H cases, comparatively to DCM ones, the higher ductility structures tend to develop less
damage by virtue of their enhanced ultimate ductile capacity.
Figure 7.21 Maximum damage
As stated before, some of the above findings do not appear systematic because some maximum
values under comparison do not refer to the same critical zone. Thus, in order to “remove” this
location dependency, the peak damage can be compared in terms of average values over the
entire structure. Such values are often taken as energy weighted averages (Coelho (1992),
Fardis et al. (1993)) and the same criterion was considered herein, adopting the relative energy
contribution of each critical zone to the total energy dissipated in the structure as the weighting
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Maxim um Dam age Maxim um Dam age
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
382 Chapter 7
factor. The average damage results are shown in Figure 7.22, where the trend for lower damage
associated with higher ductility class becomes more apparent and systematic.
Figure 7.22 Global (average) damage
Typically, the 0.15g structures exhibit low average damage, even for twice the design intensity,
which is mainly a consequence of their significant overstrength and demand reduction due to
stiffness and frequency decrease. However, among these structures, the DCL ones tend to yield
higher values than the DCM cases for intensity 2.0, because no stringent provisions for ductil-
ity enhancement are required for DCL as they are for DCM.
For the 0.30g structures the average damage slightly exceeds 0.45 in the DCM case of config-
uration 2 (direction XX), which is confirmed to be the most critical. Such average damage is
quite acceptable, particularly because it refers to twice the design intensity. Thus, despite some
locally higher damage, this result highlights the significant reserve of structural capacity to
withstand earthquake action beyond the design one while keeping “its structural integrity and a
residual load bearing capacity after the seismic event” (Eurocode 8 (1994)).
It is worth mentioning that, for structures showing the most critical damage (i.e. the 0.30g
structures), the local maximum values deviate from the average ones by a factor of about 1.5,
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
Global Dam age Global Dam age
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_MSt 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.15g_L 0.15g_M 0.30g_M 0.30g_H
Intensity:
1.0 x Design
1.5 x Design
2.0 x Design
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 383
which keeps approximately uniform regardless of the ductility class and of the intensity factor.
Such deviation does not appear exaggerated, particularly if one bears in mind the high degree
of redundancy of these statically indeterminate structures. Since the damage tends to locate
according to a beam sidesway mechanism, which, furthermore, does not develop at once by
virtue of the global structure post-yielding behaviour, a significant redistribution capacity can
be expected and, thus, a locally higher damage (e.g. 0.75, as herein obtained) is not likely to
strongly affect the structure safety.
The fact that structures of configuration 6, i.e., the irregular ones, typically show a trend for
similar (or even better) performance when compared to the regular ones (check for instance the
inter-storey drift and the average damage charts), suggests that the behaviour factor reduction
(0.8 as prescribed in EC8) reveals itself adequate to take into account the irregularity effects.
Moreover, although that reduction intended to account for the irregularity due to cut-off col-
umns, it ended up controlling also another irregular feature that is common to both configura-
tions 2 and 6. Actually, it was verified that the maximum damage mainly occurred in the
shorter span beams (as well as in the adjacent ground floor columns) of internal frames, contra-
rily to a more uniform distribution over the equal-span beams of the external frames. Further-
more, the damage patterns shown in Figure 7.12, suggest that the “irregularity” originated by
so different spans may be, at least, as important as the cut-off column irregularity.
Finally, a comment is due to the trial case 0.30g_Mst, for which the simplified static analysis
procedure was used in the design. This case is included mainly for comparison with the
0.30g_M structure of configuration 6, and, from the overall inspection of Figures 7.18 to 7.22,
a similar (or even better) performance is obtained for the 0.30g_Mst case. Similar (and some-
times lower) values of total drift, inter-storey drift and damage are obtained comparatively to
the 0.30g_M case, along with slightly higher global overstrength (see Figure 7.7), mostly aris-
ing from a significant overstrength at yielding (Table 7.6); these facts sustain the reliable fea-
tures of that simplified procedure.
7.4.6 Safety assessment by probabilities of failure. An exercise
In the preceding sections, the seismic performance of the structures studied herein was ana-
lysed in comparative terms by means of typical response variables. However, it is widely
accepted that, for a quantitative assessment of the structural safety recourse should be made to
384 Chapter 7
the probability of failure or, in more general terms, the probability of attaining a specific limit
state. Therefore, a first attempt was made to apply system reliability analysis to the structures
under study, in order to estimate probabilities of failure.
The complexity of the problems involved in system reliability analysis and of the correspond-
ing solution techniques would require a description and discussion at length, far beyond the
present work scope. No deep insight is envisaged into such issues, extensively addressed and
discussed by Pinto (1997), but the basic topics and assumptions are briefly presented and the
inherent difficulties are highlighted next. In this context, the basic aim is to have a measure of
safety against failure under seismic loading, mostly for comparative analysis between the dif-
ferent structures, rather than an absolute evaluation of the structural safety.
7.4.6.1 Methodology and assumptions
Local probability of failure
The computation of the global probability of failure for a given structure is commonly based
on the knowledge of local probabilities of failure in the structure critical zones where maxi-
mum damage is likely to develop (plastic hinges).
The calculation of hinge probability of failure requires a number of steps and assumptions
described next according to Figure 7.23. A main issue illustrated in that figure is the so-called
vulnerability function (Costa (1989), Duarte et al. (1990)) relating the seismic
action intensity (I) with the action effects (D), the damage index in the present case.
Figure 7.23 Local probability of failure at the hinge level
D ℑ I( )=
ℑ
pa I( )
I
I∗
D∗
pr D( )
ps D( )
D
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 385
The peak ground acceleration is selected as the action intensity, whose probabilistic quantifica-
tion is assumed described by the probability density function consistent with the site
seismicity.
The vulnerability function allows to map the probabilistic description of the action into the
space of action effects, viz the damage axis. Thus, the probability density function of damage
can be obtained, from which the probability of exceeding a given damage limit state
(say ) can be directly calculated as
(7.1)
where is the probability distribution of D.
Assuming the damage capacity described by the probability density function , the hinge
probability of failure is given by the following convolution integral (Campos Costa
(1993))
(7.2)
that can be seen as the result of the following reasoning:
• the damage capacity is less than or equal to D, the probability of which is given by
(7.3)
• the damage demand (action effect) falls in the interval , the corresponding
probability being
(7.4)
• the elementary probability of failure results from the intersection of the two above events
(7.5)
which, upon integration, leads to .
pa I( )
ps D( )
D∗
P D D∗>( ) 1 Fs D∗( )–=
Fs
pr D( )
Pf( )
Pf pr r( ) rd0
D
∫ ps D( ) Dd0
∞
∫=
P r D≤( ) Fr D( ) pr r( ) rd0
D
∫= =
D D dD+[ , ]
P D s D dD+<≤( ) ps D( ) Dd=
dPf P r D≤( )P D s D dD+<≤( ) pr r( ) rd0
D
∫ ps D( ) Dd= =
Pf
386 Chapter 7
Note that Eq. (7.2) can be seen as the result of an integration in the action effect space (or
rather, axis) because the function is used in the outermost integral; however, the same
result could be obtained in the damage capacity space by the following alternative reasoning:
• the damage capacity falls in the interval , the probability of which is given by
(7.6)
• the damage demand (action effect) is greater than D and the corresponding probability is
(7.7)
• as before, the elementary probability of failure can be obtained and, consequently, the
total hinge probability of failure is given by
(7.8)
This alternative way of calculating is more convenient because it avoids the transformation
of into . Actually, for a given intensity such that , the following
equality holds
(7.9)
and can be transformed into
(7.10)
which means that, once defined the probability distributions of both the action intensity and of
the damage capacity, the value of becomes directly given by
(7.11)
Probabilistic quantification of the seismic intensity
For the seismic intensity quantification recourse was made to broad studies by Campos Costa
ps D( )
D D dD+[ , ]
P D r D dD+<≤( ) pr D( ) Dd=
P s D>( ) 1 Fs D( )– 1 ps s( ) sd0
D
∫–= =
dPf
Pf 1 Fs D( )–[ ]pr D( ) Dd0
∞
∫=
Pf
pa I( ) ps D( ) I∗ I∗ ℑ 1– D∗( )=
P D D∗≤( ) P I I∗≤( )=
1 Fs D∗( )– 1 Fa ℑ 1– D∗( )( )–=
Pf
Pf 1 Fa ℑ 1– D( )( )–[ ]pr D( ) Dd0
∞
∫=
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 387
and Pinto (1997) and Pinto (1997) which attempt to characterize the European seismic hazard
scenarios, based on a large seismic database catalogue. The seismic hazard is categorized into
five different classes of increasing severity ranging from Very Low to High; for each seismic-
ity class, hazard estimates were obtained relating a series of the return periods (T) with the
expected peak ground accelerations and Weibull distributions were adopted to represent the
hazard curves.
For each class, the annual exceeding probability of peak ground acceleration (I) is given by
(7.12)
where the distribution parameters and are selected by imposing the hazard estimates for
return periods of 475 and 7000 years.
Two hazard curves were adopted herein, such that peak ground accelerations of 0.30g and
0.15g are obtained for the return period of 475 years (the reference one in the EC8). This corre-
sponded to the direct adoption of the High seismicity class (for approximately 0.30g) and of
scaled hazard from the Moderate High class to match the 0.15g acceleration at T=475 years.
The obtained curves are depicted in Figure 7.24 and the corresponding parameters are (αH =
0.376, βH = 0.00246) and (αM = 0.356, βM = 0.00093), respectively, for the high (0.30g) and
medium (0.15g) seismicities as considered herein.
Figure 7.24 High and medium seismicity hazard curves
P a I≥( ) 1T--- I
β---⎝ ⎠⎛ ⎞
α–exp= =
α β
10 100 1 103 1 1040
0.2
0.4
0.6
0.8
1
PGA (g)
T (years)
High Seismicity
Medium Seismicity
388 Chapter 7
Vulnerability functions and damage capacity
The vulnerability functions were fitted to the results of the non-linear analyses for the three
earthquake intensities, equal and above the design one. As schematically shown in Figure 7.25,
for a given plastic hinge, at each intensity four damage values were obtained (from the
response to the four earthquakes considered) and a curve of type
(7.13)
was fitted, where the parameters and were obtained by regression analysis; the corre-
sponding inverse becomes , with and , ready for
direct use in Eq. (7.11).
Figure 7.25 Vulnerability curve fitting to non-linear analysis results
Following an identical strategy as in previous reliability studies (e.g. Campos Costa (1993)),
for the damage capacity quantification a lognormal distribution of probability was adopted
(7.14)
where is the probability density of non-occurrence of failure for a damage value D. The
distribution parameters and can be related with the mean value and the coefficient of
variation cov by
(7.15)
ℑ
D ℑ I( ) c1Ic2= =
c1 c2
I ℑ 1– D( ) b1Db2= = b1 1 c1⁄= b2 1 c2⁄=
I(g)
xxxx
xxxx
xxxx
D ℑ
I3I2I1
pr D( ) 12πδD
------------------
Dτ----⎝ ⎠⎛ ⎞
2ln
2δ2-----------------–exp=
pr D( )
τ δ D
δ 1 cov2+( )ln= τ D δ2
2-----⎝ ⎠⎛ ⎞exp⎝ ⎠
⎛ ⎞⁄=
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 389
which, from the statistical analysis carried-out by Park et al. (1984) on a large set of experi-
mental results on reinforced concrete elements tested up to failure, can be taken approximately
as and .
System probability of failure
The system reliability, aiming at an estimate of the global probability of failure by combination
of local probabilities of failure, is one of the most complex and problematic issues in the struc-
tural safety evaluation. Notwithstanding the uncertainties related with both the seismic input
and the damage capacity characterization, the difficulty in establishing the combination of
local failure modes is a major obstacle due to its dependence on several aspects, viz the type of
loading (directly influencing the mechanisms of failure) and the correlation between action
effects and between damage capacities. Particularly, the latter aspects transform the evaluation
of the global probability of failure into a multi-variate problem formulated in a n-dimensional
space of random variables (where both action effects and resistances are included), whose ana-
lytical solution can hardly be obtained. Consequently, numerical approximations have to be
used, as for example the so called first and second order reliability bounds which consist of
estimates of lower and upper limits for the system probability of failure, rather than its “exact”
value.
In this line, structures are usually classified into series systems, or parallel systems or combina-
tions of both. The first type stands for systems where failure occurs when at least one of its
components fails, also designated by weakest-link systems (as suggested by the chain anal-
ogy). Instead, for parallel systems, failure is assumed to occur when all the components fail.
For series systems, the global probability of failure is limited by the Cornell bounds (Cor-
nell (1967), Madsen et al. (1986)):
(7.16)
where stands for the probability of failure of the ith component and n is the number of set
components. The lower bound states that the system fails when the most loaded component
fails, for full dependence between local failure events, while the upper bound, assuming com-
D 1= cov 0.5=
PF
max Pif( ) PF 1 1 Pi
f–( )i 1=
n
∏–≤ ≤
Pif
390 Chapter 7
plete independence of events, is obtained as the complementary of the system survival proba-
bility which equals the probability that all components survive
. Closer bounds can be defined for (Madsen et al. (1986), Pinto (1997)),
constituting the second order bounds, already including joint probability of events.
First order bounds for parallel systems can be written as , where the upper
bound identifies global failure with the failure of the less loaded component.
For practical purposes, statically determinate structures can be simulated as series systems, but
for the cases of major concern, i.e. statically indeterminate structures generally with high
degree of redundancy, the direct analogy with series or parallel systems is not apparent and,
most likely, not possible.
Indeed, for redundant structures, once the most loaded component fails, the redistribution
capacity allows the structure to “survive” further on, which means that, in an attempt for clas-
sifying it as a series system, the lower bound becomes overestimated: the structure
is “allowed” to have a lower value. Looking also at the upper bound, and bearing in mind
that it derives from the complementary probability of the intersection of survival events
assumed uncorrelated, the actual dependence between hinge events tends to reduce the value
of such bound. Thus, from this qualitative reasoning, it can be concluded that bounds of failure
probability for a series system constitute overestimates of bounds for a redundant structure.
On the other hand, the consideration of a redundant structure as a parallel system leads to
underestimated bounds of . Actually, to associate the system failure with the less loaded
component failure does not seem safe because of the different loading conditions that may trig-
ger the failure of distinct hinges (e.g., when the plastic hinges of structure upper storeys are
loaded to failure, typically the lower storey hinges may have already failed and endangered the
overall stability).
For these reasons, more sophisticated approaches can be adopted by means of parallel and
series systems combinations as proposed by Pinto (1997). Several mechanisms of failure are
previously identified under distinct load patterns assumed “independent” by virtue of imposed
vector orthogonality conditions. Such mechanisms are considered as parallel systems whose
failure probability is obtained from the local probabilities of failure of the relevant hinges (i.e.
only those activated for a specific mechanism). The global probability of failure is the outcome
PS 1 PF–=( )
PS 1 Pif–( )∏=( ) PF
0 PF min Pif( )≤ ≤
max Pif( )( )
PF
PF
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 391
of a series association of those mechanisms, for which second order bounds are computed tak-
ing into account the correlation between mode failure events.
In the present work context, such an elaborated procedure is considered too much demanding
(in computational terms) for the envisaged purposes. Thus, a simple methodology was adopted
in which the design assumed mechanism is considered to control the failure mode (a beam
sidesway mechanism) and, despite the above mentioned overestimation trend, the Cornell
bounds were considered for the event set consisting of plastic hinge failure in all beams and in
the base end-zones of ground floor columns. Therefore, once the local probabilities of failure
are obtained by Eq. (7.11) for all the hinges considered, the bounds of global probability of
failure can be computed according to Eq. (7.16).
7.4.6.2 Comparative analysis
The above described methodology was applied for all trial cases under analysis and the
obtained upper and lower bounds of global annual probability of failure are plotted in the loga-
rithmic scale charts of Figure 7.26.
Figure 7.26 Bounds of annual probability of failure
PF
a) Configuration 2 b) Configuration 6
Direction YY
Direction XX
Design Case
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000.15g_L 0.15g_M 0.3g_M 0.3g_H
Limits:
Upper
Low er
Design Case
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000.15g_L 0.15g_M 0.3g_M 0.3g_Mst 0.3g_H
Limits:
Upper
Low er
Design Case
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000.15g_L 0.15g_M 0.3g_M 0.3g_H
Limits:
Upper
Low er
Design Case
1.00E-06
1.00E-05
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+000.15g_L 0.15g_M 0.3g_M 0.3g_Mst 0.3g_H
Limits:
Upper
Low er
392 Chapter 7
As expected, high probabilities of failure are obtained, particularly in what concerns the upper
bound, with significant differences between the upper and lower bounds (in general over one
order of magnitude). For comparison purposes, it should be mentioned that values of annual
probability of failure around 2x10-4 are referred by Paulay and Priestley (1992) as appropriate
for the survival limit state of office buildings.
The structures designed for 0.15g exhibit some trend for lower probabilities of failure (10-4 to
10-3) than the 0.30g cases (10-3 to 10-2), particularly in the direction XX, which is coherent
with the less damage found in 0.15g structures (see Figures 7.21 and 7.22); additionally, it is
worth highlighting the reasonable uniformity of for these medium seismicity structures in
both directions XX and YY.
For 0.30g structures in the direction XX (where irregularities do exist due to both the cut-off
columns and the large span beams), the probability of failure tends to be higher than in the
direction YY where lateral loads are resisted by regular frames, which means that irregularities
contribute, as expected, for less safe solutions. Additionally, it is interesting to note that config-
uration 2, considered regular despite having adjacent beams with so different spans, shows
probabilities of failure higher than the assumed irregular structure, which sustains the ade-
quacy of the 80% reduction of the q-factor to account for irregularity in configuration 6. This
reduction, besides taking into account the cut-off column irregularity, also contributes to soften
the negative effects of the significant contrast of beam spans whose influence is quite apparent
in configuration 2.
The 0.30g_Mst and 0.30g_M cases (configuration 6) show quite similar probabilities of fail-
ure, which confirms the adequacy of the simplified static analysis procedure allowed in EC8.
Last but not the least, an interesting and important result is that, for a given design accelera-
tion, the ductility class does not seem to affect the structure reliability, which is fundamental
from the design code standpoint. Indeed, the EC8 key issue of allowing the designer to chose
between different ductility classes has to ensure that identically safe solutions are obtained,
and the present results seem to confirm the fulfilment of such requirement.
PF
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 393
7.5 Concluding remarks
This section has dealt with the presentation and discussion of a numerical seismic assessment
study of some reinforced concrete frame structures designed according to EC8. The study,
included in the framework of the PREC8 activities, has focused on the non-linear seismic anal-
ysis of two building configurations (2 and 6) designed for several combinations of ductility
class and design acceleration (0.15g and 0.30g).
Besides an overview of the PREC8 programme in order to establish the present study context,
some insight was given into particular details of the structural configurations analysed herein:
the configuration 2, considered the regular one, and the configuration 6, where irregularities
were introduced by columns removed in the ground floor and by increased first storey height.
Structures were modelled with the flexibility element developed in the present work and non-
linear dynamic analyses were performed for each trial case under the action of four accelero-
grams fitting the EC8 spectrum with some increasing intensities.
Overall, the structures have shown an important reserve of strength which has been confirmed
by the seismic analysis results and by additional pushover static analyses. The available over-
strength at yielding (about 1.36) induces the 0.15g designed structures to behave in the pre-
yielding range, or just at incipient yielding, when loaded for the design seismic intensity. Glo-
bal overstrength factors as high as 2.1 can be found for earthquake action at twice the design
intensity, resulting from material overstrength and from global hardening.
Extensive cracking develops along structural members, leading the most significant stiffness
drop to take place at the design intensity. On the other hand, for the cases where significant ine-
lastic excursions occur (0.30g designed structures for ductility classes M and H), the local duc-
tility demands at the design intensity are lower than what could be expected, which results
from the available overstrength. Plastic hinging is reasonably spread all over the structure and,
for twice the design intensity, the beam sidesway mechanism shows up quite clearly. However,
the large contrast of span lengths of internal frames in one direction leads to some damage con-
centration in the shorter span central beams, which has been found as a source of irregularity
(actually not specifically foreseen and not taken into account in the assumed regular structure).
The total drift does not significantly differ between distinct ductility classes for the same
394 Chapter 7
design acceleration, although a slight trend can be observed for larger drifts with increasing
ductility class. Yet, low drifts are obtained (1.6%) for twice the design intensity, showing a
large margin to failure. The same applies for the interstorey drift, although the EC8 limit
observed in the design is somewhat exceeded by the obtained results (for the design intensity)
due to the extent of cracking development. However, no apparent second order effects are
likely to occur since the prescribed EC8 limit for the sensitivity coefficient is never exceeded.
In general, lower average damage is obtained when the ductility class is increased for the
same design acceleration, as a clear effect of the more stringent design provisions for ductility
enhancement. Typically, the 0.15g structures lead to low damage (about 0.1 for the design
intensity and less than 0.3 for twice that intensity), while in 0.30g cases the average damage
slightly exceeds 0.45 for twice the design intensity. These values are considered quite accepta-
ble, since they are close to the damage values obtained from similar numerical simulations of
the ELSA building structure for approximately identical seismic action levels and for which no
strong damage was detected in the experiment.
From a comparative study on probabilities of failure, structures designed for a given peak
ground acceleration have been found identically safe regardless of the ductility class they
belong to. This is an important issue since it appears to confirm the uniform risk assumption
underlying the designer “freedom” to trade between strength and ductility, as allowed in EC8
through the availability of three Ductility Classes. From this point on, the advantage of one
ductility class over the others has to be judged upon economical criteria, which constitutes a
major concern of a recent study by Pinto (1997).
Based on previous items (drift, damage and probabilities of failure) the following conclusions
are further highlighted:
• The behaviour factor reduction (80%) has shown quite good adequacy to account for the
irregularity source specifically considered in the design, since both the structural perform-
ance and the safety level of the irregular structures appeared as good as those of the regular
ones. In addition, it proved to be very important in reducing the negative effects of the non-
uniformity of beam spans which was present in both the regular and the irregular configura-
tions; possibly, this irregularity source should be considered for future inclusion in the irreg-
ularity criteria of EC8.
• The simplified static analysis method allowed in EC8 led to good results and to solutions as
SEISMIC ASSESSMENT OF RC FRAME STRUCTURES DESIGNED ACCORDING TO EC8 395
safe as those obtained by the reference multi-modal analysis method.
• Structures designed for high seismicity (0.30g), not only have shown higher demands (dam-
age included), but also led to larger probabilities of failure when compared with the medium
seismicity (0.15g) structures; this is related with the less explored non-linear behaviour in
the latter structures, for which the load combinations dominated by gravity loads are more
likely to control the design.
Overall, the analysed structures have shown quite good performance, well within the basic
requirements of EC8, and the main goals of the present study were reasonably achieved,
despite the improvement needs on specific topics (e.g. damage characterization and probability
of failure quantification).
396 Chapter 7
Chapter 8
FINAL REMARKS
8.1 Summary and conclusions
The present work was devoted to i) the development of a global element model, computation-
ally efficient for the analysis of RC structures under monotonic or cyclic loads inducing non-
linear behaviour throughout various response stages, ii) the model validation against results of
experimental tests on a full-scale structure and iii) its intensive application to the seismic
assessment of structures, particular those designed according to Eurocode 8 (EC8).
The innovative model
The model was required to describe the progressive modifications of stiffness due to non-line-
arity spread inside the member, such that one structural member could be modelled by only
one element. To this end, the flexibility formulation was chosen for the present development in
view of its adequacy to overcome the non-availability of kinematic shape functions to approx-
imate the element deformation field for different non-linear stages. Despite the use of this less
common formulation, the element model is easy to incorporate in typical finite element non-
linear analysis algorithms; thus, emphasis was mainly put on the element model development
and implementation, rather than well established non-linear schemes at the global structure
level.
Since the frame structural response to lateral actions is mainly of interest, the inelastic behav-
iour was essentially assumed at the element end zones, although cracking effects due to verti-
cal static loads were also taken into account in the span. The non-linear behaviour was
controlled in terms of moment-curvature by a modified Takeda-type model prescribed at the
element end sections; the non-linearity was assumed to spread along the element by recourse to
398 Chapter 8
cracking and yielding sections which are characterized by the cracking and yielding points,
respectively, in the trilinear skeleton curve of the model. Thus, at each load step, the element
was divided into distinct behaviour zones delimited by two types of sections, viz the fixed ones
(at member ends and at mid-span) and the moving sections (the cracking, yielding and null-
moment sections). Zones can be yielded (or plastic), cracked or uncracked; while the yielded
zones are allowed to develop only near the end sections, the cracked zones are considered
wherever the bending moment has exceeded the cracking threshold(s).
For a feasible and efficient (though approximate) control of cracked section zones, which are
likely to develop along considerable portions of the element, some modifications were intro-
duced in the adopted Takeda-type model such that a sudden transition from the uncracked to
the fully-cracked secant stiffness occurs once the cracking moment is exceeded. In doing so,
the element stiffness is progressively affected by the fully-cracked behaviour of cracked zones,
though no residual (permanent) deformations due to cracking are considered there for the
present stage of model development; this means that the hysteretic energy dissipation is not
properly considered in cracked zones.
In compliance with the general flexibility formulation, an internal iterative scheme was
required to accomplish the element state determination, driven by progressive elimination of
element displacement residuals and giving rise to the element end forces and tangent flexibility
matrix. However, by contrast with the general formulation, the displacement residuals are eval-
uated by integrating total deformations rather than residual ones since this was found more
suitable for the moving nature of most control sections. The control of plastic end zones
required particular care in order to minimize possible inaccuracies arising from the fact that
their behaviour is based only on the end and the yielding section behaviours; in addition, the
cracking development with the adopted uncracked/cracked transition was found to require par-
ticular detail for a successful advancement of the incremental-iterative scheme.
Implementation and auxiliary tools
The proposed element model was implemented in the general purpose computer code
CASTEM2000, an object-oriented program where both the well known Newton-Raphson
algorithm for non-linear system solution and the classical Newmark integration scheme were
available. For the particular nature of CASTEM2000, the basic code features were briefly
FINAL REMARKS 399
reviewed and the major interventions for the new element model implementation were pointed
out. The quite cumbersome, but unavoidable task of defining the moment-curvature trilinear
basic curve was pursued by means of a specifically developed algorithm for the most common
RC sections, viz the rectangular and T-shape ones. Designed to overcome the need of fibre dis-
cretization of the section, this algorithm was also implemented in CASTEM2000 as an auxil-
iary tool, where realistic behaviour models are considered for both steel and concrete, the latter
in confined or unconfined conditions; the section equilibrium is analysed for the curve turning
points, viz cracking, yielding and ultimate (or possibly other supplementary ones) under spe-
cific and pre-established point definition criteria.
The model validation and performance
Several experimentally tested cantilever beams having rectangular and T-shape cross-sections
were modelled by one flexibility element, for its validation at the single element level. Results
showed good numerical simulations of the experimental behaviour but some modelling limita-
tions were detected in relation to the hysteretic rules of the adopted section model, which, how-
ever, can be modified without interfering with the element formulation.
The element model validation at the global structure level was carried out through the simula-
tion of experimental tests on a four-storey reinforced concrete frame structure. The full-scale
specimen was designed according to Eurocode 8 as a high ductility structure and subjected to a
series of unidirectional seismic tests (pseudo-dynamically performed) and quasi-static cyclic
ones, thoroughly described in the present work.
Under the low level seismic test (corresponding to serviceability conditions in the pre-yielding
range), the bare structure performed quite well, although generalized cracking was found and
confirmed by the significant reduction of the fundamental frequency (from 1.78 Hz to 1.27
Hz). However, no yielding evidence was observed and the maximum inter-storey drift approx-
imately agreed with the EC8 limit. Proceeding to the high level seismic test (intending to
induce net inelastic excursions), cracking developed further, particularly in the beam-column
joint panels (with a bi-diagonal pattern) and in the beam-column interface section where major
cracks formed and rebar yielding took place; low energy dissipation was exhibited after the
response peaks, as apparent from the very “pinched” force-deformation diagrams, which was
related to bar-slippage inside the joints. However, rotations in the beam end zones were reason-
400 Chapter 8
ably well distributed, except for the top-storey where rather low rotation values were found.
Despite the referred crack formation, no significant permanent damage (e.g. concrete crushing
and spalling) was visible.
The seismic tests on the structure infilled with masonry panels showed a net increase of the ini-
tial frequency and some global strength enhancement resulting from the presence of infills.
Due to the high level test, the uniformly infilled structure suffered serious damage in the two
first storey masonry panels, responsible for the very “pinched” force-deformation diagrams
and the inherent low energy dissipation. The soft-storey infilled configuration (only partially
infilled in the storeys above the first) exhibited damage concentration in the first storey where
energy dissipation mainly took place; the large ductility demands in that storey induced signif-
icant strength degradation resulting from the observed concrete cover spalling in the first sto-
rey columns, which required the repair of damaged zones before proceeding with further
testing.
The quasi-static final cyclic tests (up to a top displacement ductility factor around 8) induced
progressively heavier damage, particularly in the two lower storeys where concrete cover
crushing and spalling was observed and followed by stirrup failure and buckling of rebars. The
highly damaged state of the structure, responsible for net degradation of global strength and for
large inter-storey drifts (around 7%), showed that a near-failure stage was actually reached at
the end of the test.
Several numerical analyses of the above referred structure were performed by recourse to the
proposed flexibility element model in order to reproduce the experimental behaviour through-
out the different testing stages. For a detailed insight of the critical zones behaviour, the chord
rotation at each element end section was selected as the key parameter to compute local ductil-
ity demands and to estimate damage; the latter was calculated according to the widely used
Park and Ang damage index, for which the quantification of the relevant parameters, viz the
ultimate chord rotation, the dissipated hysteretic energy and the corresponding weighting fac-
tor, was addressed in detail.
Since all the experimental tests consisted in imposing storey displacements quasi-statically and
in measuring the restoring forces, static simulations were possible by prescribing the same dis-
placements in the numerical model, the results being compared with the experimental ones.
FINAL REMARKS 401
For the bare frame seismic tests and, particularly, once a stabilized cracked behaviour was
reached, quite good simulations were obtained despite some lack of energy dissipation result-
ing from the model specific features in the cracking range. However, for net yielding develop-
ment during the high level test, numerical results agreed very well with experimental ones,
showing good simulation of dissipated energy, yet slightly over-evaluated possibly due to the
non-consideration of the pinching effect caused by anchorage slippage phenomena. Still for the
high level test, the ductility and damage distributions highlighted the expected strong column -
weak beam dissipation mechanisms; the maximum ductility demand (4.8) appeared reasonably
low in view of the behaviour factor (5) and the intensity factor (1.5), and the maximum damage
(0.4) complied with the observed state of the structure when compared with an approximate
damage scale proposed in the literature.
For the test simulations of the infilled structure configurations, the infill panels were modelled
by diagonal struts following a previous work, in which extensive background studies were per-
formed in order to develop a diagonal strut model, to make its validation and to derive the cor-
responding parameters. The numerical static analysis made herein did not provide good
simulations of the experimental behaviour, possibly as a result of the difficulties inherent in the
estimates of infill model parameters; indeed, better simulations were obtained after and where
extensive cracking affected the infill behaviour, because the response became mostly control-
led by the reinforced concrete frames.
Despite the difficulties in simulating the final cyclic test arising from the repair of damaged
zones, the global strength was reasonably described; however, these simulations mainly served
to highlight that the present hysteretic model rule for the evolution of unloading stiffness
induced excessive degradation when large ductilities are attained, thus leading to lower energy
dissipation than the experiment.
Dynamic analysis simulations of the bare structure seismic tests were also performed, for
which the structural frequencies showed good agreement with experimental measurements.
The dynamic response in the post-cracking range was very reasonably described by the model
upon inclusion of viscous damping forces (of Rayleigh type) to compensate for the lack of
model hysteretic dissipation in that behaviour range; the viscous damping factor (1.8%), exper-
imentally measured before the low level test, proved to be adequate while the response was
dominated by cracking. When the steel behaviour mostly controlled the response, as in the
402 Chapter 8
high level test, the viscous damping became useless because the model was able to incorporate
the hysteretic dissipation; the best simulations of the experimental results were obtained with-
out viscous damping. However, it was found that, should viscous damping be included, its con-
tribution to the energy dissipation can be very significant, even for low damping factors.
The proposed element model performance was compared with that from analyses by tradi-
tional fixed length plastic hinge modelling. The comparison was restricted to the bare frame
seismic tests, having shown that, for static simulations, the traditional way of modelling leads
to results less good than the proposed element in the pre-yielding range, while both modelling
strategies give similar results for post-yielding analyses. In the dynamic context, the flexibility
element model led to better approximation of the experimental response as a result of the pro-
gressive adaptation of member stiffness, indeed the main advantage over the traditional fixed
hinge modelling. The dynamic behaviour was better simulated because the structural stiffness,
and consequently the vibration characteristics, were closely described and updated. It should
be emphasized that good simulations were obtained in the pre-yielding range by adopting the
fully-cracked section stiffness progressively introduced in the element along with viscous
damping forces, while in the post-yielding range the behaviour was better described by remov-
ing any viscous dissipation. Globally, the seismic behaviour was adequately simulated by the
proposed flexibility element modelling, though some issues are recognized to require further
development.
Analyses for Eurocode 8 validation
Aiming at the numerical seismic assessment of reinforced concrete frame structures designed
according to EC8, a number of non-linear seismic analyses of two building configurations
were carried out as part of a european-wide pre-normative research programme. Building
structures were designed for several combinations of ductility class and design accelerations
(0.15g and 0.30g, corresponding to medium and high seismicity zones, respectively); one con-
figuration was considered regular, while the other included irregularities caused by columns
removed in the ground floor and by increased first storey height. Each trial case was modelled
by recourse to the proposed flexibility element model and the non-linear analyses were carried
out under the action of four accelerograms fitting the EC8 spectrum and scaled by three
increasing intensities, at and above the design one (additional analyses at half the design inten-
sity were also performed for the high ductility class structures).
FINAL REMARKS 403
Significant overstrength was found for all structures resulting from the strength reserve of con-
stituent materials (where the partial safety factors for material properties are included), some
design requirements (minimum reinforcement, capacity design and detailing), the gravity load
influence and the post-yielding hardening at the material and the global structure levels. Over-
strength factors of about 1.36 were obtained at yielding, such that the 0.15g designed trial
cases exhibited pre-yielding behaviour, or just imminent yielding, under the design seismic
intensity; however, overstrength increases further with the seismic intensity, mainly resulting
from global hardening, and overstrength factors above 2.0 could be found for twice the design
intensity.
At the design intensity, extensive cracking was found in structural members leading to very
significant stiffness reductions, and, due to overstrength, the local ductility demands were
lower than expected in the cases where the most significant inelastic excursions occurred (i.e.
in the 0.30g designed structures). Despite the fact that a reasonable spread of plastic hinging
could be found throughout all the structures, some damage concentration appeared in the
shorter span beams of frames with contrasting span lengths; this was found to be a source of
irregularity not duly foreseen and not considered in the design of the assumed regular struc-
ture.
Both the total and the inter-storey drifts were not strongly affected by changing the ductility
class for the same design acceleration, though a slight trend was found showing larger drifts
when the ductility class is increased. However, even for twice the design intensity, low drifts
were obtained (maximum values of 1.6% and 2.3%, respectively for the total and the inter-sto-
rey drifts), at least three times below the values reached at failure in the experimentally tested
structure above referred; this fact indicated that a large margin to failure was available.
By contrast, a clear trend showed that lower average damage was obtained when the ductility
class was increased for a given design acceleration, as a consequence of the more stringent
design provisions to enhance ductility. However, the ductility class was found not to affect the
safety level of structures designed for the same peak ground acceleration, since not very differ-
ent failure probability bounds were obtained for distinct ductility classes. From the EC8 stand-
point this is a very important finding, because it contributes to ensure that, by trading between
strength and ductility, the designer obtains identically safe solutions.
404 Chapter 8
Globally, the analysed structures showed quite good performance within the EC8 require-
ments. The behaviour factor reduction to account for irregularities proved to be adequate and,
additionally, very important for reducing the negative effects induced by non-uniformity of
beam-spans. Good results and identically safe solutions were provided by the simplified static
analysis method allowed in EC8 when compared to those obtained by the reference multi-
modal analysis method. The less explored non-linear behaviour of medium seismicity (0.15g)
structures, for which the design is more likely to be controlled by load combinations domi-
nated by gravity forces, led to lower demands (including damage) and lower probabilities of
failure than structures designed for high seismicity (0.30g).
8.2 Future developments
In line with the work developed in the present thesis, a number of topics are deemed important
for further research regarding the improvement of modelling strategies and the refinement of
seismic assessment studies. Concerning the modelling issues, the following are highlighted:
• A softening branch in the post-yielding range should be included in the moment-curvature
section model, in order to allow a more realistic description of the near-failure stage; this is
of particular interest in columns due to the axial force effects. To this end, the proposed
algorithm to obtain the trilinear envelope curve could be extended for a better definition of
the post-yielding branch.
• The possibility of using an intermediate stiffness between the uncracked and the fully-
cracked ones in the moment-curvature relationship should be investigated in order to
account for the tension stiffening effect in the element cracked zones. Particularly, in the
presence of axial force, deviations between the actual post-cracking and the fully-cracked
stiffness may become significant, which should be further addressed and, possibly, compen-
sated by means of an adequate intermediate stiffness estimated on the basis of the actual
characteristics of the whole element.
• It has been clearly pointed out that both the unloading stiffness degradation and the pinch-
ing rules should be improved. A broad and systematic study of the numerical simulation of
the experimental response of single reinforced concrete members or sub-assemblages would
be extremely useful. Ideally, a reduced number of the most recent phenomenological mod-
els should be selected to provide a variety of hysteretic rules in order to use them in the
numerical simulations. On the other hand, a considerable amount of results of reinforced
FINAL REMARKS 405
concrete experimental activity is nowadays worldwide available which should be grouped
according to some basic variables such as, the section type, the steel and concrete grades,
the test type, the maximum deformation (or ductility) reached, the shear-span ratio, the
proneness to pull-out effects, etc. Such classification would contribute to clarify and to
render more consistent the fitting of hysteretic rules to the experimental evidence and the
definition of empirical relations between hysteretic parameters and the member characteris-
tics.
• In view of the difficulties in correctly modelling the hysteretic dissipation in cracked zones
continuously developing, and bearing in mind that viscous damping proved to be adequate
in the pre-yielding range when considered simultaneously with the progressive inclusion of
the fully-cracked stiffness in the member, a different and perhaps more consistent or general
way of including the viscous effect should be investigated. A possible solution could rely on
the adoption of a global damping matrix resulting from the assemblage of element damping
matrices, which in turn could be proportional to the element stiffness matrices where the
effects of cracked zone development would come directly included.
• A more tight control of yielded zones should be considered in order to closely follow the
actual flexibility and curvature distributions in such zones. For this purpose, recourse could
be made to the progressive activation of some fixed sections in the yielded zones to be fully
controlled by the same section model of the corresponding end section.
• The influence of inclined shear cracking in the yielded zones should be investigated, partic-
ularly concerning possible increase of inelasticity spread as suggested by the usual strut-
and-tie modelling for shear analysis of reinforced concrete members. The consequences of
further extending the yielded zones to account for shear and how much this extension
should be, deserve particular attention.
• The suitability of the flexibility formulation to handle associations in series of several ele-
ments should be explored to complement the proposed element with others devoted to the
simulation of rebar slippage and non-linear shear effects; the former could be taken into
account by fixed point hinges at member ends, while the latter could be described by an ele-
ment with two variable length extreme zones (where the non-linear shear effects develop)
connected by a rigid element.
Concerning the seismic assessment of reinforced concrete structures, the following topics are
emphasized:
406 Chapter 8
• Studies for a better characterization of damage should be undertaken; new proposals for
damage index computation at the critical zone level are needed and, despite the energy-
based proposals appear adequate, research is still lacking on how to split an element damage
index into its critical zones.
• The safety assessment by failure probability should be improved, particularly in what con-
cerns the probabilistic characterization of the damage capacity and the extension of vulner-
ability functions to higher levels of seismic intensity. The system reliability also requires a
deeper insight in order to obtain closer and more reliable bounds of the failure probability.
In addition, the safety assessment study should be extended to all structures considered in
the PREC8 programme, in order to check whether the present work conclusions also apply
to the other trial cases.
• Further research activity in line with the PREC8 programme should be undertaken to find
out the effects of irregularities in a systematic and objective way. To this end, not only the
safety assessment would be required, but also the quantification of structural overstrength
and effective behaviour factors would be helpful in assessing the EC8 rules for ductile seis-
mic design.
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APPENDICES
Appendix A
Linear Elastic Timoshenko Beam Formulation
A.1 Section formulation and constitutive relationship
The main expressions of the Timoshenko beam theory are briefly recalled in this appendix.
Detailing of this well known formulation is deemed unnecessary in the present work, since it
can be easily found in the literature.
Notation and conventions are the same as introduced in 2.4 and the bases for theory derivation
and application are adopted as used in Pegon (1993). Hence, referring to Figure 2.5, it is
assumed that:
• the element reference system (x,y,z) consists of the principal inertia axes and the same
applies to the local section reference frame;
• translation components of the displacement vector refer to the section centroid and
rotation components refer to the section plane.
Thus, according to the Timoshenko formulation accounting for shear deformations, the com-
ponents of the generalized strain vector are given by
(A.1)
which, for the elastic case, are related with the generalized stress vector components by
(A.2)
a x( )
e x( )
εx xddux=
ϕx xddθx=
βy xdduy θz–=
ϕy xddθy=
βz xdduz θy+=
ϕz xddθz=
S x( )
εx Nx EA( )⁄=
ϕx Mx GJx( )⁄=
βy Vy GAy( )⁄=
ϕy My EIy( )⁄=
βz Vz GAz( )⁄=
ϕz Mz EIz( )⁄=
422 Appendix A
where:
• A, and stand, respectively, for the cross-sectional area and the shear reduced areas in
both y and z directions;
• , and refer to the modified moment of polar inertia and to the moments of flexural
inertia in the y and z directions, respectively;
• E and G are the Young and distortion modulus.
All these geometric and mechanical characteristics, may vary along the element but, as for sec-
tion deformations and forces, the reference to abscissa x was suppressed for simplicity.
, and are introduced in Eq. (A.2) to account for possible cross-sectional warping, and
are usually given by
(A.3)
where and are parameters less then or equal to 1, depending on the Poisson ratio and on
the cross-section shape and dimensions (Dias da Silva (1995)).
Therefore, the section flexibility matrix can be written in the diagonal form
(A.4)
whose terms can be uniform or vary along the element according to an assumed distribution.
A.2 Element flexibility matrix
In the present work, the element is considered divided into two parts, along which the cross-
sectional properties are assumed uniform, as shown in Figure A.1; the subscript or
is assigned to properties in correspondence with the respective element part. Thus, the
flexibility distribution is defined by
(A.5)
where h is the relative abscissa of the span section H and both and are diagonal
Ay Az
Jx Iy Iz
Ay Az Jx
Ay αyA= Az αzA= Jx αyIy αzIz+=
αy αz
f x( ) diag EA( )-1 GAy( )-1 GAz( )-1 GJx( )-1 EIy( )-1 EIz( )-1, , , , ,[ ]=
i 1=
i 2=
f x( )f1 x( )
f2 x( )⎩⎨⎧
=forfor
0 x hL≤ ≤hL x L≤ ≤
f1 x( ) f2 x( )
Linear Elastic Timoshenko Beam Formulation 423
matrices of constant terms given by
(A.6)
Figure A.1 Distribution of flexibility properties along the element
The element flexibility matrix is obtained by integration of according to Eq. (2.25);
however, a slight modification is introduced in the notation of the matrix, in order to
highlight partial contributions to .
Without considering element applied loads (irrelevant for the present purposes), Eq. (2.14) can
be re-written as
(A.7)
where the row-matrices stand for each internal force component (indicated as super-
script) and are given by
(A.8)
Due to the diagonal nature of , the integral of Eq. (2.25) to obtain , leads to the follow-
ing matrix summation
fi x( ) diag EA( )i-1 GAy( )i
-1 GAz( )i-1 GJx( )i
-1 EIy( )i-1 EIz( )i
-1, , , , ,[ ]=
E1 E2HxH hL=
x
f2 x( )f1 x( )
F f x( )
b x( )
F
Nx
Vy
Vz
Mx
My
Mz⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
x( )
bNx x( )
bVy x( )
bVz x( )
bMx x( )
bMy x( )
bMz x( )
Fx1
Mx1
My1
Mz1
My2
Mz2⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
⋅=
b… x( )
bNx x( ) 1– 0 0 0 0 0=
bVy x( ) 0 0 0 1 L⁄– 0 1 L⁄–=
bVz x( ) 0 0 1 L⁄ 0 1 L⁄ 0=
bMx x( ) 0 1– 0 0 0 0=
bMy x( ) 0 0 x L⁄ 1–( ) 0 x L⁄ 0=
bMz x( ) 0 0 0 x L⁄ 1–( ) 0 x L⁄=
f x( ) F
424 Appendix A
(A.9)
each one accounting for the contribution of the internal force/deformation component indi-
cated by the superscript. The respective matrix expressions are given by
(A.10)
where the same superscript has also been included in the section flexibility terms.
Using the flexibility distribution given by Eq. (A.5) and the expressions (A.8), the six flexibil-
ity matrix contributions are written as:
(A.11)
(A.12)
(A.13)
(A.14)
F FNx F
Vy FVz F
Mx FMy F
Mz+ + + + +=
FNx b
Nx x( )[ ]T
fNx x( ) b
Nx x( )⋅ ⋅ xd0
L
∫=
FVy b
Vy x( )[ ]T
fVy x( ) b
Vy x( )⋅ ⋅ xd0
L
∫=
FVz b
Vz x( )[ ]T
fVz x( ) b
Vz x( )⋅ ⋅ xd0
L
∫=
FMx b
Mx x( )[ ]T
fMx x( ) b
Mx x( )⋅ ⋅ xd0
L
∫=
FMy b
My x( )[ ]T
fMy x( ) b
My x( )⋅ ⋅ xd0
L
∫=
FMz b
Mz x( )[ ]T
fMz x( ) b
Mz x( )⋅ ⋅ xd0
L
∫=
FNx
F11Nx L
EA( )1---------------h L
EA( )2--------------- 1 h–( )+=
FlmNx 0= for l 1≠( ) or m 1≠( )⎩
⎪⎨⎪⎧
= FMx
F22Mx L
GJx( )1-----------------h L
GJx( )2----------------- 1 h–( )+=
FlmMx 0= for l 2≠( ) or m 2≠( )⎩
⎪⎨⎪⎧
=
FVy
FlmVy L 1–
GAy( )1------------------h L 1–
GAy( )2------------------ 1 h–( )+= for l 4 6,=( ) and m 4 6,=( )
FlmVy 0= for l 4 6,≠( ) or m 4 6,≠( )⎩
⎪⎨⎪⎧
=
FVz
FlmVz L 1–
GAz( )1-----------------h L 1–
GAz( )2----------------- 1 h–( )+= for l 3 5,=( ) and m 3 5,=( )
FlmVz 0= for l 3 5,≠( ) or m 3 5,≠( )⎩
⎪⎨⎪⎧
=
FMy
F33My L
EIy( )1---------------ξa1
LEIy( )2
---------------ξa2+=
F35My F53
My LEIy( )1
---------------ξb1
LEIy( )2
---------------ξb2+= =
F55My L
EIy( )1---------------ξc1
LEIy( )2
---------------ξc2+=
FlmMy 0= for l 3 5,≠( ) or m 3 5,≠( )⎩
⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧
=
Linear Elastic Timoshenko Beam Formulation 425
(A.15)
where the coefficients , , and (for ), depend only on the element subdivision
factor h and are given by
(A.16)
For the particular case of uniform properties along the whole element ( )
any value of h is suitable because only the sum of each pair of coefficients affecting section
flexibility terms is relevant and it remains constant regardless of h.
Therefore, after summation of all contributions, the total flexibility matrix becomes
(A.17)
highlighting that force/displacement component interaction exists only between those contrib-
uting for the same direction of deformation (namely and , or and ).
FMz
F44Mz L
EIz( )1---------------ξa1
LEIz( )2
---------------ξa2+=
F46Mz F64
Mz LEIz( )1
---------------ξb1
LEIz( )2
---------------ξb2+= =
F66Mz L
EIz( )1---------------ξc1
LEIz( )2
---------------ξc2+=
FlmMz 0= for l 4 6,≠( ) or m 4 6,≠( )⎩
⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎧
=
ξaiξbi
ξcii 1 2,=
ξa1
h3
3----- h2 h–( )–=
ξb1
h3
3----- h2
2-----–=
ξc1
h3
3-----=
ξa2
1 h3–3
-------------- h2 h–( )+ 13--- ξa1
–= =
ξb2
1 h3–3
-------------- 1 h2–2
--------------– 16---– ξb1
–= =
ξc2
1 h3–3
-------------- 13--- ξc1
–= =
f x( ) f1 x( ) f2 x( )= =
F
F
LEA------- 0 0 0 0 0
0 LGJx--------- 0 0 0 0
0 0 L 1–
GAz---------- L
3EIy-----------+⎝ ⎠
⎛ ⎞ 0 L 1–
GAz---------- L
6EIy-----------–⎝ ⎠
⎛ ⎞ 0
0 0 0 L 1–
GAy---------- L
3EIz-----------+⎝ ⎠
⎛ ⎞ 0 L 1–
GAy---------- L
6EIz-----------–⎝ ⎠
⎛ ⎞
0 0 L 1–
GAz---------- L
6EIy-----------–⎝ ⎠
⎛ ⎞ 0 L 1–
GAz---------- L
3EIy-----------+⎝ ⎠
⎛ ⎞ 0
0 0 0 L 1–
GAy---------- L
6EIz-----------–⎝ ⎠
⎛ ⎞ 0 L 1–
GAy---------- L
3EIz-----------+⎝ ⎠
⎛ ⎞
=
My1My2
Mz1Mz2
426 Appendix A
A.3 Element displacements as integrated deformations
The element displacement vector , given by Eq. (2.16) as an integral of deformations, is
detailed next, for the case of element applied forces in three directions. Expressions are
obtained for the element axis system (x,y,z); therefore, the vector given by Eq. (C.3) in
Appendix C must be adopted in the following.
According to Eq. (2.14), and making use of , the elastic section deformations are given by
(A.18)
which, after substitution in Eq. (2.16), lead to , where stands for displace-
ments due to element applied loads, expressed by
(A.19)
Actually, only needs to be detailed, since the flexibility matrix is already known. Fol-
lowing a similar procedure to that used in A.2, can also be decomposed into the contribu-
tions arising from the distinct force/deformation components.
Thus, using the row-matrices, expressed by Eqs. (A.8), and substituting
by its components, Eq. (A.19) yields
(A.20)
where, again, the superscript identifies the contributing components. These are given by
(A.21)
u
Sp x( )
f x( )
e x( ) f x( ) b x( ) Q⋅ ⋅ f x( ) Sp x( )⋅+=
u F Q⋅ up+= up
up bT x( ) f x( ) Sp x( )⋅ ⋅ xd0
L
∫=
up F
up
b… x( )
ep x( ) f x( ) Sp x( )⋅=
up upNx up
Vy upVz up
Mx upMy up
Mz+ + + + +=
upNx
εx( )p– xd0
L
∫00000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= upMx
0
ϕx( )p– xd0
L
∫0000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Linear Elastic Timoshenko Beam Formulation 427
(A.22)
(A.23)
whose deformation components inside the integrals are related to internal forces by Eq. (A.2).
For the flexibility distribution given by Eq. (A.5) and the vector components (see Eq.
(C.3)), the previous integrals yield the following expressions
(A.24)
(A.25)
upVz
00βz( )pL
------------ xd0
L
∫0βz( )pL
------------ xd0
L
∫0⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= upMy
00
xL--- 1–⎝ ⎠⎛ ⎞ ϕy( )p xd
0
L
∫0
xL---⎝ ⎠⎛ ⎞ ϕy( )p xd
0
L
∫0⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
upVy
000βy( )p–L
---------------- xd0
L
∫0βy( )p–L
---------------- xd0
L
∫⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= upMz
000
xL--- 1–⎝ ⎠⎛ ⎞ ϕz( )p xd
0
L
∫0
xL---⎝ ⎠⎛ ⎞ ϕz( )p xd
0
L
∫⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Sp x( )
εx( )p– xd0
L
∫pxL2
2----------- h2
EA( )1--------------- 1 h2–( )
EA( )2-------------------+ PxL 1 h–
EA( )2---------------+=
ϕx( )p– xd0
L
∫ 0=
βy( )p–L
---------------- xd0
L
∫p– yL2
------------ηs1
GAy( )1------------------
ηs2
GAy( )2------------------+ Py
ηs1
GAy( )1------------------
ηs2
GAy( )2------------------+–=
βz( )pL
------------ xd0
L
∫pzL2
--------ηs1
GAz( )1-----------------
ηs2
GAz( )2-----------------+ Pz
ηs1
GAz( )1-----------------
ηs2
GAz( )2-----------------++=
xL--- 1–⎝ ⎠⎛ ⎞ ϕy( )p xd
0
L
∫ p– zL3 ηa1
EIy( )1---------------
ηa2
EIy( )2---------------+ PzL
2 ηc1
EIy( )1---------------
ηc2
EIy( )2---------------+–=
xL---⎝ ⎠⎛ ⎞ ϕy( )p xd
0
L
∫ p– zL3 ηb1
EIy( )1---------------
ηb2
EIy( )2---------------+ PzL2 ηd1
EIy( )1---------------
ηd2
EIy( )2---------------+–=
xL--- 1–⎝ ⎠⎛ ⎞ ϕz( )p xd
0
L
∫ pyL3 ηa1
EIz( )1---------------
ηa2
EIz( )2---------------+ PyL2 ηc1
EIz( )1---------------
ηc2
EIz( )2---------------++=
xL---⎝ ⎠⎛ ⎞ ϕz( )p xd
0
L
∫ pyL3 ηb1
EIz( )1---------------
ηb2
EIz( )2---------------+ PyL2 ηd1
EIz( )1---------------
ηd2
EIz( )2---------------++=
428 Appendix A
where the coefficients , , , and for are given by
(A.26)
with , and as expressed by Eq. (A.16). For the case of element uniform properties
with a possible concentrated force at H (thus, ), these coefficients reduce to
Adopting the following weighted average flexibilities
(A.27)
where the subscripts p (lower case) and P (upper case) stand, respectively, for the distributed
and the concentrated force contributions, the vector becomes
i
1 7/24 -1/12 1/24 1/4 11/384 -5/384 1/24 -1/48
2 1/24 -1/12 7/24 -1/4 5/384 -11/384 1/48 -1/24
1/3 -1/6 1/3 0 1/24 -1/24 1/16 -1/16
ηsiηai
ηbiηci
ηdii 1 2,=
ηs1h h2–=
ηa1h4 8⁄ h3 3⁄ h2 4⁄+–=
ηb1h4 8⁄ h3 6⁄–=
ηc11 h–( ) ξb1
–( )=
ηd11 h–( ) ξc1
–( )=
ηs2ηs1
–=
ηa21 24⁄ ηa1
–=
ηb21 24⁄– ηb1
–=
ηc2h 1 3⁄ ξa1
–( ) hξa2= =
ηd2h 1 6⁄– ξb1
–( ) hξb2= =
ξaiξbi
ξci
h 1 2⁄=
ξaiξbi
ξciηsi
ηaiηbi
ηciηdi
Σi
1EA( )p
--------------- h2
EA( )1--------------- 1 h2–( )
EA( )2-------------------+=
1GAy
----------ηs1
GAy( )1------------------
ηs2
GAy( )2------------------+=
1EIy( )p1
-----------------ηa1
EIy( )1---------------
ηa2
EIy( )2---------------+=
1EIy( )p2
-----------------ηb1
EIy( )1---------------
ηb2
EIy( )2---------------+=
1EIz( )p1
-----------------ηa1
EIz( )1---------------
ηa2
EIz( )2---------------+=
1EIz( )p2
-----------------ηb1
EIz( )1---------------
ηb2
EIz( )2---------------+=
1EA( )P
--------------- 1 h–EA( )2
---------------=
1GAz
----------ηs1
GAz( )1-----------------
ηs2
GAz( )2-----------------+=
1EIy( )P1
------------------ηc1
EIy( )1---------------
ηc2
EIy( )2---------------+=
1EIy( )P2
------------------ηd1
EIy( )1---------------
ηd2
EIy( )2---------------+=
1EIz( )P1
-----------------ηc1
EIz( )1---------------
ηc2
EIz( )2---------------+=
1EIz( )P2
-----------------ηd1
EIz( )1---------------
ηd2
EIz( )2---------------+=
up
Linear Elastic Timoshenko Beam Formulation 429
(A.28)
For the particular case of uniform element properties and this simplifies to
(A.29)
showing that the shear contributions to rotations vanish due to anti-symmetry of shear distor-
tion diagrams along the element.
Additionally, this particular case helps to estimate the concentrated load at H equivalent to the
distributed one as referred in 3.4. If the equivalence criterion consists in the equality of elastic
fixed end section moments, it is identically enforced if the equality of elastic end section dis-
placements is imposed. In this context, if a “fictitious” concentrated load vector Peq is sought,
equivalent to the “real” distributed one p, the components of Peq must be such that the right-
most terms of components in Eq. (A.29) be equal to the leftmost ones. Therefore, the fol-
lowing equivalence conditions are obtained
up
ux1( )pθx1( )pθy1( )pθz1( )pθy2( )pθz2( )p⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
pxL2
2 EA( )p------------------ Px
LEA( )P
---------------+
0
pzL
2GAz
------------- L3
EIy( )p1
-----------------– Pz1
GAz
---------- L2
EIy( )P1
------------------–+
p– yL
2GAy
-------------- L3
EIz( )p1
-----------------– Py1
GAy
---------- L2
EIz( )P1
-----------------––
pzL
2GAz
------------- L3
EIy( )p2
-----------------– Pz1
GAz
---------- L2
EIy( )P2
------------------–+
p– yL
2GAy
-------------- L3
EIz( )p2
-----------------– Py1
GAy
---------- L2
EIz( )P2
-----------------––⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= =
h 1 2⁄=
up
pxL2
2EA-----------⎝ ⎠⎛ ⎞ Px
L2EA-----------⎝ ⎠⎛ ⎞+
0
pzL– 3
24EIy--------------⎝ ⎠⎜ ⎟⎛ ⎞
PzL– 2
16EIy--------------⎝ ⎠⎜ ⎟⎛ ⎞
+
pyL3
24EIz--------------⎝ ⎠⎜ ⎟⎛ ⎞
PyL2
16EIz--------------⎝ ⎠⎜ ⎟⎛ ⎞
+
pzL3
24EIy--------------⎝ ⎠⎜ ⎟⎛ ⎞
PzL2
16EIy--------------⎝ ⎠⎜ ⎟⎛ ⎞
+
pyL– 3
24EIz--------------⎝ ⎠⎜ ⎟⎛ ⎞
PyL– 2
16EIz--------------⎝ ⎠⎜ ⎟⎛ ⎞
+⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
up
430 Appendix A
(A.30)
which have been adopted in the present study for the approximate consideration of distributed
forces.
Px( )eq pxL=
Py( )eq23---p
yL=
Pz( )eq23---p
zL=
Appendix B
Trilinear Model Details
In this appendix some details are included about of the adopted trilinear model. The primary
curve (see Figure 3.2) and some basic definitions were already introduced in 3.3; the hysteretic
behaviour rules are described in the next paragraphs.
A general loading path is illustrated in Figure B.1 to exemplify typical features of the hystere-
tic process; neither pinching nor strength degradation effects are included in this case.
Figure B.1 Hysteretic rules of the trilinear model. General loading path
5
1
3
4
2
8
6
6
9
77
8
a
e , k
d
c
b , g
f
l
j
i , oh
n
m
+
+
+
+
+
+
+
+
++
++
My+
Mc-
My-
Mc+
α-My-
–
α+My+
–
ϕy-
ϕy+
ϕ
M
A+
A-
9
E+ ME+ ϕE
+( , )
E- ME- ϕE
-( , )
Loading path:
a-b-c-d-e-f-g-h-i-j-k-l-m-n-o
k1+
k1-
k2+
k2-
1st 2nd 3rd Cycle:
432 Appendix B
Branches 1 to 5 refer to loading situations; the unloading lines are labelled with 6 and 7, while
the reloading lines are identified by 8 and 9.
Unloading lines are obtained so as to induce stiffness degradation for increasing inelastic
deformation; for this purpose, a common target point is considered ( or ) in the extrapo-
lated elastic line at a moment level or , where and control the degree of
unloading stiffness degradation. Unloading lines aim at this target point and keep this stiffness
until they reach the axis. It is apparent that low values of increase the stiffness degrada-
tion and, consequently, reduce the area enclosed by the hysteresis loops and the corresponding
dissipated energy.
Reloading branches 8 and 9, included in Figure B.1, aim at target points and , respec-
tively, where the maximum absolute value of deformation has been found in previous cycles
(for the same sign of moments).
Although these are the general rules for unloading and reloading, some special situations are
considered:
• if the section has cracked for one direction of bending, but still remains uncracked for the
other direction, then, when reloading, the target is the cracking point on that side, as is the
case of path a-b-c-d; then, loading follows along the cracked branch (2 or 3);
• in the case of unloading or reloading for interior cycles, i.e. enclosed between the outermost
lines as is the case in Figure B.2, the stiffness remains “frozen” at that of the corresponding
outermost lines (for example, the unloading branches 12 and 14 are parallel to line 6, and
the same happens between lines 13, 15 and 7); although not labelled, the positive and nega-
tive interior reloading branches are parallel to the lines 9 and 8, respectively.
The pinching effect is given by a reduction of the hysteresis loop area caused by a low reload-
ing stiffness that may increase after a certain amount of plastic deformation is recovered. The
effect is mainly due to development of cracks that remain open during almost the entire cycle;
in such case, some phases of the loading sequence are very likely to exhibit a rather low stiff-
ness, since the applied moment is only resisted by tensile and compressive forces in the steel.
Stiffness recovering may occur if further deformations go far enough to close the opened
cracks and to engage again the stiffness of the compressed concrete area.
A+ A-
α+My+
– α-My-
– α+ α-
ϕ α
E- E+
Trilinear Model Details 433
Figure B.2 Trilinear model. The pinching effect and interior cycles
This phenomenon is commonly associated with pronounced asymmetry of top and bottom
reinforcement, but other aspects may contribute to the increase of the pinching effect, such as:
• loss or accentuated deterioration of bond, inducing significant slip between concrete and
steel;
• shear deformations, with a high relative weight when compared to the flexural deforma-
tions, causing important shear stress to be transferred by aggregate interlock and dowel
action (thus, with a very low stiffness) while cracks remain open.
Modelling of pinching, when it is to be included, is performed by lowering the target point of
reloading branches immediately after crossing the axis. Instead of the point E (see Figure
B.2), lines 8 (or 9) aim at point B existing on the previous unloading line at a level (or
), until reaching the so called crack-closing deformation (or ) assumed correspond-
ing to null moment in the previous unloading phase. After this point (or ) new reloading
lines 10 or 11 aim again at the previous maximum point (or ) and the process goes on as
for the case of no pinching.
Strength degradation is likely to develop in RC sections subjected to cyclic loading, mainly
due to increasing deterioration of the concrete resisting capacity. The closing process of open
Ms- γ-My
-=
ϕ
M
ϕs-
1k1
+
ϕs+
C+
Y+
C-
Y-
B+
E+
E-
k3+
k2+
k2-
k1-
k3-
B-
9
8
6
713
11
10
12
14
15
ϕx-
ϕx+
X+
X-
Ms+ γ+My
+=
ϕ
γ+My+
γ-My- ϕx
+ ϕx-
X+ X-
E+ E-
434 Appendix B
cracks becomes less effective during cyclic loading, since the roughness of the contact surface
progressively increases. Moreover, steel-concrete bond also deteriorates along reinforcing
bars, at increasing distances from the crack lips, thus reducing the effectiveness of stress trans-
fer between steel and the surrounding concrete due to lack of local anchorage. These two
aspects lead to both stiffness and strength reduction for increasing deformations and number of
cycles, and is often referred to as softening effect.
In the present model this effect is taken into account by means of the dissipated hysteretic
energy contribution to the increase of section deformations (Kunnath et al. (1990)); the sche-
matic process is shown in Figure B.3.
Figure B.3 Trilinear model. Strength deterioration rule
The parameter is defined as the ratio of incremental damage caused by the increase of max-
imum deformation to the normalized increment of hysteretic energy, , responsible for
such deformation increase. It is given, in terms of moment and curvatures, by
(B.1)
where is the ultimate curvature under monotonic loading. The energy increment corre-
sponds to the hysteretic dissipation occurred in the cycle starting at the previous maximum
ϕ
M
C+
Y+
C-
Y-
E+
E-
∆ϕ+ β+ ∆E+
My+
----------⎝ ⎠⎜ ⎟⎛ ⎞
=
∆E+ (Dissipated hysteretic energy
T+ MT+ ϕT
+( , )
in the cycle starting at E+)
Prediction
Predictionpoint
reloading line
β
dϕm dE
βdϕm
ϕu----------⎝ ⎠
⎛ ⎞ dEϕuMy-------------⎝ ⎠
⎛ ⎞⁄dϕmMy
dE-----------------= =
ϕu dE
Trilinear Model Details 435
deformation , which is likely to include smaller internal loops also contributing to . In
fact this parameter appears in the well known Park and Ang damage index definition (Park
et al. (1984)), weighting the dissipated energy contribution to the total damage.
Using Eq. (B.1) the deformation increment can be obtained by
(B.2)
and, obviously, it is to be computed for both directions of loading.
If no pinching effect is considered, the reloading line at the prediction point aims at the previ-
ous maximum point ; this allows for an estimate of the hysteretic dissipated energy in
that cycle (including internal ones). Thereby, the increment of curvature can be computed
by Eq. (B.2) and a modified target point is defined on the primary curve at curvature
. The actual reloading branch stiffness is then obtained aiming at this new target
point. If pinching is also to be included, the procedure is affected in the same way, but only for
the second reloading branch, i.e. for the stiffness determination after the point (see Fig-
ure B.2).
It must be emphasized that this strength degradation procedure does not exactly correspond to
the softening effect, since it does not ensure an effective reduction of strength for increasing
deformation. In fact, with this methodology a strength drop is achieved between successive
cycles at the same maximum deformation, but for increasing deformations the strength is not
prevented to increase because there is no explicit softening branch in the primary curve. There-
fore, if the modelling of softening is thought to be important, such as in the case of sections
highly controlled by the compressed concrete behaviour (e.g. heavily loaded columns), another
descending branch should be included in the primary curve after the peak moment is reached.
Concerning the values of hysteretic model parameters, the following suggestions can be found
in Kunnath et al. (1990):
• For the unloading stiffness degradation, values of between 2 and 4 appear to give good
results for well detailed RC sections and the peak response is not very sensitive to values.
Note that with an origin oriented model is obtained, while setting a non-
degrading Clough-type model is achieved.
ϕm dE
β
dϕm βdEMy-------=
E+ ∆E+
∆ϕ
T+
ϕE+ ∆ϕ++
k3+ X+
α
α
α 0= α ∞=
436 Appendix B
• The pinching behaviour due to top and bottom reinforcement asymmetry (such as the beam
cases) is well simulated when the pinching moment in the stronger side has approxi-
mately the same magnitude of the weaker side yielding moment. This means that, when no
other causes for pinching are present, values can be given by
(B.3)
which implicitly assumes that pinching only occurs below the yielding moment. If other
pinching sources are to be included (such as bond-slip or high shear deformations) then
other values have to be selected depending upon the degree of pinching expected. The
authors refer that low values in the range of 0.01 to 0.1 are not unusual for cases with high
contribution of shear deformation.
• For the strength deterioration parameter an empirical expression, based on a great
amount of experimental results, is commonly used (Kunnath et al. (1990)) and given (with
slight algebraic adjustments) by
(B.4)
where ρw is the volumetric confinement ratio (i.e. the volume of closed stirrups divided by
the volume of confined concrete core), ν is the normalized axial force (taken positive if
compressive) and ωt is the mechanical ratio of tension reinforcement; typical values of
are not far from 0.05.
This model description did not intend to be exhaustive. However it is believed that the above
paragraphs and figures help to understand the main features, capabilities and limitations of the
model, which is important for the interpretation of the numerical results presented in this work.
Ms
γ
γ+ My-
My+
------- 1≤= γ- My+
My-
------- 1≤=
γ
β( )
β 0.9100ρw 0.37max ν 0.05( , ) 0.5 ωt 0.17–( )2+( )=
β
Appendix C
Internal Force Distributions
C.1 General
In this appendix, several expressions are included for internal force distribution due to both
uniformly distributed forces applied along the element and concentrated ones applied in the
span section H; only forces are considered, i.e. applied moments are not included.
The general case of non-zero force components in three directions is first introduced in C.2 and
the particular case where element loads exist only in the non-linear bending plane is further
detailed in C.3. The corresponding expressions for abscissas of internal moving sections and
for their derivatives with respect to applied moments are included in C.4.
C.2 Element applied forces in three directions
Consider again Figure 2.6, now with uniformly distributed forces p in the whole flexible ele-
ment and concentrated forces P acting in section H. With the notation adopted there, and
denoting by the relative abscissa of H, the vector of resultants expressed by
Eq. (2.7) becomes
(C.1)
h xH L⁄= R x( )
R x hL<( )
x x( )
y x( )
z x( )
x x( )
y x( )
z x( )⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫ pxx
pyx
pzx0
pzx2
2-----
pyx2
2-----–
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= = ; R x hL≥( )
pxx Px+
pyx Py+
pzx Pz+
0
pzx2
2----- Pz x hL–( )+
pyx2
2-----– Py x hL–( )–
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
438 Appendix C
while the vector defined by Eq. (2.8) is obtained by substituting L for x in Eq. (C.1).
Thereby, the vector given by Eq. (2.12), referring to the element load contribution to the
forces, can be expressed by
(C.2)
and the contributions to section internal forces, expressed both in the element axes (x,y,z)
(given by , Eq. (2.15)) and in the local axis system (given by , Eq. (3.1)), are now
written, respectively, as
(C.3)
and
(C.4)
Therefore, the expression (3.5) of the generic section internal forces , in terms of the end
section ones , can be detailed as follows
RL
Qpf
Qf
Qpf
Qp1
f
Qp2
f⎩ ⎭⎪ ⎪⎨ ⎬⎪ ⎪⎧ ⎫
= with Qp1
f
0
p– yL2--- Py 1 h–( )–
p– zL2--- Pz 1 h–( )–
000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= and Qp2
f
p– xL Px–
p– yL2--- Pyh–
p– zL2--- Pzh–
000⎩ ⎭
⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Sp x( ) Spsx( )
Sp x hL<( )
p– xx
pyL2--- x–⎝ ⎠⎛ ⎞ Py 1 h–( )+
pzL2--- x–⎝ ⎠⎛ ⎞ Pz 1 h–( )+
0
pz L x–( )x2--- Pz 1 h–( )x+
p– y L x–( )x2--- Py 1 h–( )x–
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= ; Sp x hL≥( )
p– xx Px–
pyL2--- x–⎝ ⎠⎛ ⎞ Pyh–
pzL2--- x–⎝ ⎠⎛ ⎞ Pzh–
0
pz L x–( )x2--- Pz L x–( )h+
p– y L x–( )x2--- Py L x–( )h–
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Spsx hL<( )
p– xx
p– yL2--- x–⎝ ⎠⎛ ⎞ Py 1 h–( )–
p– zL2--- x–⎝ ⎠⎛ ⎞ Pz 1 h–( )–
0
p– z L x–( )x2--- Pz 1 h–( )x–
py L x–( )x2--- Py 1 h–( )x+
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
= ; Spsx hL≥( )
p– xx Px–
p– yL2--- x–⎝ ⎠⎛ ⎞ Pyh+
p– zL2--- x–⎝ ⎠⎛ ⎞ Pzh+
0
p– z L x–( )x2--- Pz L x–( )h–
py L x–( )x2--- Py L x–( )h+
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
=
Ss x( )
Qes
Internal Force Distributions 439
(C.5)
and
(C.6)
where the superscripts and have been included in the components, in order to
remind that they consist of internal forces associated with the end section local axis systems;
Figure 3.1 helps to clarify the meaning of Eqs. (C.5) and (C.6).
C.3 Element applied forces only in the non-linear bending plane
For the particular case of element forces applied only in the non-linear bending plane, by
default xz, internal forces are readily obtained from Eqs. (C.5) and (C.6). The influence of ele-
ment loads appears only in the components and , which are likely to affect deforma-
tions in the xz plane. Due to their importance in the present work context, these components are
further detailed next, with a simplified notation: subscripts and are suppressed, the end
section moments of interest ( and ) are simply denoted by and , and the dis-
tributed and concentrated forces ( and ) are referred to as p and P, respectively.
NxsNx
E1 pxx–=
Vys
MzE1
L---------
MzE2
L---------– py
L2--- x–⎝ ⎠⎛ ⎞– Py 1 h–( )–=
Vzs
MyE1
L---------–
MyE2
L--------- pz
L2--- x–⎝ ⎠⎛ ⎞– Pz 1 h–( )–+=
MxsMx
E1=
Mys1 x
L---–⎝ ⎠
⎛ ⎞MyE1 x
L---⎝ ⎠⎛ ⎞My
E2 pz L x–( )x2---– Pz 1 h–( )x–+=
Mzs1 x
L---–⎝ ⎠
⎛ ⎞MzE1 x
L---⎝ ⎠⎛ ⎞Mz
E2 py L x–( )x2--- Py 1 h–( )x+ + +=⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛
for x hL<( )
NxsNx
E1 pxx– Px–=
Vys
MzE1
L---------
MzE2
L---------– py
L2--- x–⎝ ⎠⎛ ⎞– Pyh+=
Vzs
MyE1
L---------–
MyE2
L--------- pz
L2--- x–⎝ ⎠⎛ ⎞– Pzh+ +=
MxsMx
E1=
Mys1 x
L---–⎝ ⎠
⎛ ⎞MyE1 x
L---⎝ ⎠⎛ ⎞My
E2 pz L x–( )x2---– Pz L x–( )h–+=
Mzs1 x
L---–⎝ ⎠
⎛ ⎞MzE1 x
L---⎝ ⎠⎛ ⎞Mz
E2 py L x–( )x2--- Py L x–( )h+ + +=⎝
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎛
for x hL≥( )
E1 E2 Qes
VzsMys
ys zs
MyE1 My
E2 ME1ME2
pz Pz
440 Appendix C
Therefore, according to Figure C.1, only the following expressions are considered:
(C.7)
and
(C.8)
Figure C.1 Element applied loads in the non-linear bending plan; simplified notation
Considering the local abscissas and , the relative local abscissas of section
H are given, respectively, by
(C.9)
Introducing the transverse force , the distributions of V and M can be
also written as
(C.10)
and
V x( )ME1
L---------–
ME2
L--------- p L
2--- x–⎝ ⎠⎛ ⎞– P 1 h–( )–+=
M x( ) 1 xL---–⎝ ⎠
⎛ ⎞ME1
xL---⎝ ⎠⎛ ⎞ME2
p L x–( )x2---– P 1 h–( )x–+=
for x hL<( )
V x( )ME1
L---------–
ME2
L--------- p L
2--- x–⎝ ⎠⎛ ⎞– Ph+ +=
M x( ) 1 xL---–⎝ ⎠
⎛ ⎞ME1
xL---⎝ ⎠⎛ ⎞ME2
p L x–( )x2---– P L x–( )h–+=
for x hL≥( )
p
ME1 ME2
MV
xs1 s2
L
H
sH1sH2
P
s1 x= s2 L x–=
h1sH1
L------- h= = and h2
sH2
L------- 1 h1–= =
VE ME2ME1
–( ) L⁄=
V s1( ) VE p L2--- s1–⎝ ⎠⎛ ⎞ P 1 h1–( )+–=
M s1( ) ME1VEs1 p L s1–( )
s12---- P 1 h1–( )s1+–+=
Internal Force Distributions 441
(C.11)
for the element parts and , respectively. Eqs. (C.10) and (C.11) are very convenient
for locating internal moving sections and can be condensed in only one, as follows
(C.12)
where the sign coefficient is defined by
(C.13)
C.4 Moving section abscissas and respective derivatives
Abscissas
The location of sections associated to a specific moment are obtained according to Eq.
(3.11), in terms of the element abscissa x, or according to Eq. (3.12) referring to local abscissas
. Since is generally associated to each end section, the use of Eq. (C.12), as a particular
case of Eq. (3.12), is more adequate and, is therefore adopted herein. The moment distribution
expressed by Eq. (C.12) is slightly modified to
(C.14)
where , and the local abscissa of the section having is given by the
solution of
(C.15)
which has to be obtained separately according to the degree of the involved polynomials.
V s2( ) VE p L2--- s2–⎝ ⎠⎛ ⎞ P 1 h2–( )++=
M s2( ) ME2VEs2– p L s2–( )
s22---- P 1 h2–( )s2+–=
E1H E2H
V si( ) VE ξi p L2--- si–⎝ ⎠⎛ ⎞ P 1 hi–( )+–=
M si( ) MEiξiVEsi p L si–( )
si2--- P 1 hi–( )si+–+=
i 1 2,=( )
ξi
ξi11–⎩
⎨⎧
=forfor
i 1=i 2=
or ξi sid xd⁄=
M*
si M*
M si( ) MEiξiVEi
′ si p L si–( )si2---–+=
VEi
′ VE ξiP 1 hi–( )–= si* M*
M* M si*( ) MEi
ξiVEi
′ si* p L si
*–( )si
*
2----–+= =
442 Appendix C
Thus, for
(C.16)
which holds for non-uniform moments along , i.e. for .
For , the existence of solutions is controlled by the discriminant of the 2nd order Eq.
(C.15), which can be expressed by
(C.17)
If no real solutions exist, whereas if one or two solutions exist, given by
(C.18)
Once is obtained, it must be checked for its existance in the relevant interval ( ) in order
to be accepted as a possible solution and associated with the specific moving section under
study (cracking, yielding or null moment one). It may be transformed to the element abscissa,
according to or , respectively, for or . The subscript i has
been included inside brackets in order to identify the element part to which the moving
section is related. Actually, stands for the specific notation of each moving section, i.e. for
instance or as defined in 3.4.
Finally, it is worth mentioning that the case of uniform moments along one or both element
part(s) is not adressed because the concept of moving sections becomes meaningless once the
behaviour is fully controlled by end sections under the criteria and assumptions stated in 3.5.
Derivatives
The derivatives of moving section abscissas with respect to increments of end section
moments are detailed next, using the local abscissas as defined by Eqs. (C.16) or (C.18).
Since , it follows that , which renders
easier the task of obtaining derivatives because abscissas are written in terms of .
p 0=
si* M* MEi
–
ξiVEi
′----------------------=
EiH VEi
′ 0≠
p 0≠ ∆i( )
∆iL2--- ξi
VEi
′
p-------–
⎝ ⎠⎜ ⎟⎛ ⎞
2
2M* MEi
–( )
p---------------------------+=
∆i 0< ∆i 0≥
si* L
2--- ξi
VEi
′
p-------–
⎝ ⎠⎜ ⎟⎛ ⎞
∆i±=
si* EiH
x i( )* s1
*= x i( )* L s2
*–= i 1= i 2=
EiH
i( )
Ci κ+ Yi
x i( )*
si*
MEjMEj
0 ∆MEj+= ∂…( ) ∂∆MEj
( )⁄ ∂…( ) ∂MEj( )⁄=
MEj
Internal Force Distributions 443
Considering the relation between and , as well as the above introduced definitions of
and , and using the Krönecker symbol the following relations hold
(C.19)
(C.20)
As for the abscissas, situations of and are treated separately. For
(C.21)
(C.22)
which hold only if .
For , and in case of two existing real solutions , the general expressions for
derivatives become
(C.23)
(C.24)
whereas for the case of one double solution , Eq. (C.18) reduces to and
the respective derivatives are
(C.25)
x i( )* si
* VE
VEi
′ δij
x i( ) j,*
MEj∂∂x i( )
* MEj∂∂si
*
⎝ ⎠⎜ ⎟⎛ ⎞
si j,*= for i 1=
MEj∂∂si
*
⎝ ⎠⎜ ⎟⎛ ⎞
– si j,*–= for i 2=
⎩ ⎭⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎧ ⎫
ξisi j,*= = =
MEj∂
∂VEi
′ξjL----⎝ ⎠⎛ ⎞–= and
MEj∂
∂MEi δij=
p 0= p 0≠ p 0=
si j,* ξj M* MEi
–( ) δijVEi
′ L–
ξiVEi
′ 2L
--------------------------------------------------------=
x i( ) j,* ξisi j,
* ξj M* MEi–( ) δijVEi
′ L–
VEi
′ 2L
--------------------------------------------------------= =
VEi
′ 0≠
p 0≠ ∆i 0>( )
si j,* ξiξj
pL--------- 1
p--- 1
2--- ξi
VEi
′
pL-------–
⎝ ⎠⎜ ⎟⎛ ⎞
ξiξj δij– ∆i-1/2±=
x i( ) j,* ξisi j,
* ξjpL------ 1
p--- 1
2--- ξi
VEi
′
pL-------–
⎝ ⎠⎜ ⎟⎛ ⎞
ξj ξiδij– ∆i-1/2±= =
∆i 0=( ) si* L
2--- ξi
VEi
′
p-------–
⎝ ⎠⎜ ⎟⎛ ⎞
=
si j,* ξiξj
pL---------= and x i( ) j,
* ξisi j,* ξj
pL------= =
444 Appendix C
Appendix D
The Event-to-Event Technique
In the present appendix, the event-to-event technique (Simons and Powell (1982), Porter and
Powell (1971)) is briefly introduced, following the steps for its application in the context of
classical stiffness based structural analysis.
A very simple example is used to illustrate the process. It consists of the single frame shown in
Figure D.1-a) where the non-linear behaviour is assumed concentrated in the six indicated
plastic hinges. Each of these hinges is ruled by a piecewise non-linear moment-rotation dia-
gram, as exemplified in Figure D.1-b) for the generic hinge i.
Figure D.1 The event-to-event scheme for stiffness based problems
At the local hinge level, the initial state point is assumed as ; upon application of the rota-
tion increment , the state point moves to . In between, events (stiffness changes) may
occur (for the illustrated example two events are considered, associated with moments and
), but only the first one is of interest. Thus, the bending moment and flexibility ,
associated with the first event at each hinge, are registered and the corresponding load incre-
ment reduction factor is set up as follows:
2
3
1
4
5 6 ∆F
∆Mi
∆θi
M1V
Mi0
M2V
θi0
V1
V2
θ
M
Hi0
Hi
Events: j=1,2
Mi* first Mj
V( )[ ]hinge i=j =1,2
fi*= flexib. after Mi
*
f1
f2
a) Structure b) Hinge i
Hi0
∆θi Hi
M1V
M2V Mi
*( ) fi*( )
446 Appendix D
(D.1)
The global reduction factor (r) is searched among all the hinges and given by
(D.2)
and the hinge number (possibly more than one) where the factor r occurs is denoted by .
If the load increment is reduced to and applied to the structure, this ensures
that non-linearity is about to occur only at the hinge(s) , whereas the remaining ones behave
with the current stiffness without reaching any event. The internal force and deformation state
of each section is updated accordingly, the flexibility of hinge(s) being modified to .
The new stiffness matrix can also be calculated and the remaining load increment, given by
(D.3)
is applied, following the same process as for the original increment .
Attention is drawn to the fact that no residual forces are likely to appear in this process,
because the subdivision is enforced exactly to avoid them, i.e. to closely follow the equilibrium
path.
However, it is clear that, such apparent “advantage” may easily become a major drawback
when a significant number of plastic hinges has entered non-linear behaviour, because they all
contribute directly to the global equilibrium equations. Different constitutive relationships and
non-concomitant events among all sections obviously induce a heavy subdivision process
which is more pronounced as the number of non-linear sections increases. Mostly at critical
loading stages (like reversals or zero crossings), any small out-of-phase loading between dif-
ferent sections may enforce many small steps that, in a usual N-R process, would not be
needed. Therefore, the use of the event-to-event scheme, in the context of non-linear global
structure analysis, has to be judged carefully in view of the local constitutive relationship and,
mostly, of the number of sections involved.
riMi
* Mi0–
∆Mi--------------------=
r min ri{ } i=1,...,Nhinge( )=
ir
F∆ F∆ r r F∆=
ir
ir fir*( )
F∆ c F∆ F∆ r–=
F∆
Appendix E
Non-Linear Dynamic and Static Analysis Scheme
The analysis of both static and dynamic non-linear structural problems can be performed by
means of a common scheme, since the dynamic one is transformable into a pseudo-static prob-
lem (in each load step) if the dynamic equilibrium equations are integrated step-by-step using
well established algorithms. Hence, in this appendix, the non-linear dynamic scheme is
recalled, in the context of the Newmark method family, currently one of the most widely
accepted integration techniques, and then particularized to both the static case and the seismic
one, the latter being a specific dynamic loading type.
For notation clarity in the following, the global structure displacement vector will be sim-
ply denoted by and the general (static or dynamic) applied nodal forces by . Thus, in gen-
eral and , and the semi-discrete dynamic equilibrium equation system is
given by
(E.1)
where and stand for the global structure mass and damping matrices, respectively, and
is the restoring force vector associated with the displacement configuration ; for the
elastic case is simply given by , where refers here to the global elastic
stiffness matrix.
Let the time instants and correspond to steps and , respectively, i.e.
and , where is the time step interval, such that a common notation
holds valid for both dynamic and static analysis. The discretization of the continuous time var-
iable into finite steps allows to write Eq. (E.1) in the discrete form
uG
d q
d d t( )= q q t( )=
M d··⋅ C d·⋅ r d( )+ + q=
M C
r d( ) d
r d( ) r K0 d⋅= K0
t ∆t– t k 1– k
t ∆t– tk 1–= t tk= ∆t
t
448 Appendix E
(E.2)
where the following approximations were introduced for step
(E.3)
Step-by-step methods for the integration of Eq. (E.2) typically aim at the solution for step ,
which can be obtained either exclusively in terms of the response for the previous step ,
as in the case of explicit methods, or also in terms of the response for step , leading to
implicit algorithms.
Explicit methods are computationally more attractive but their application is often limited by
response stability conditions, which may require rather small particularly for structures
with significant contributions from high frequency vibration modes. By contrast, implicit
methods have the major advantage of unconditional stability, for which the time step is exclu-
sively chosen for accuracy purposes, but more elaborate algorithms are required.
The most common implicit methods are essentially based on the classical Newmark method
(Delgado (1984)), where an average constant acceleration is assumed in the interval
and is defined by
(E.4)
which, upon integration leads to
(E.5)
(E.6)
Parameters and are introduced to control the accuracy and stability of the method.
M ak⋅ C vk⋅ rk+ + qk=
k
dk d tk( )≈ d t( )=
vk d· tk( )≈ d· t( )=
ak d·· tk( )≈ d·· t( )=
rk r d tk( )( )≈ r d t( )( )=
qk q tk( ) q t( )= =
k
k 1–( )
k
∆t
tk 1– tk,[ ]
ak 1 γ–( )ak 1– γak+=
vk vk 1– 1 γ–( )ak 1– γak+[ ]∆t+=
dk dk 1– vk 1– ∆t 12--- β–⎝ ⎠⎛ ⎞ ak 1– βak+ ∆t2+ +=
β γ
Non-Linear Dynamic and Static Analysis Scheme 449
The particular case of and leads to the well known central difference method,
an explicit and conditionally stable one.
For the method is unconditionally stable, although the desirable second order
accuracy (i.e. second order derivatives accurately reproduced) is only achieved for .
On the other hand, numerical damping of contributions due to high frequency spurious modes
can be obtained if , but only first order accuracy can be obtained. Therefore, numerical
dissipation and second order accuracy cannot be achieved simultaneously, which is a drawback
that has motivated improvements in the method over the last two decades (Hilber et al. (1977))
in order to introduce numerical dissipation for higher spurious modes, while keeping good
response performance at the lower frequency ones. Further references to such improvements
are also included in 5.2 in the context of the experimental pseudo-dynamic method, but in the
following paragraphs only the classical version of the Newmark method is considered aiming
at a simpler presentation of the non-linear dynamic analysis scheme.
Nevertheless, Hughes (1987) has shown that the best efficiency in numerical dissipation is
obtained by adopting , for a given , which, for the case where sec-
ond order accuracy is to be ensured, gives and , indeed the most common
values in the Newmark method parameters.
Eqs. (E.5) and (E.6) can be rearranged in terms of the displacement increment
and the kinematic entities of the previous step as follows
(E.7)
(E.8)
where and depend exclusively on the response for the step and are given,
respectively, by
(E.9)
β 0= γ 1 2⁄=
2β γ 1 2⁄≥ ≥
γ 1 2⁄=
γ 1 2⁄>
β γ 1 2⁄+( )2 4⁄= γ 1 2⁄≥
γ 1 2⁄= β 1 4⁄=
∆dk dk dk 1––=
ak1
β∆t2-----------∆dk ak 1––=
vkγ
β∆t---------∆dk vk 1––=
ak 1– vk 1– k 1–( )
ak 1–1β∆t---------vk 1–
12β------ 1–⎝ ⎠⎛ ⎞ ak 1–+=
450 Appendix E
(E.10)
Introducing Eqs. (E.9) and (E.10) in Eq. (E.2), the following equilibrium equation is obtained
(E.11)
with
(E.12)
Since the restoring force vector depends non-linearly of , Eq. (E.11) must be
solved by means of an iterative process such that, when convergence is reached, the following
residual force vector vanishes to within a pre-defined tolerance
(E.13)
Thus, let be the approximation of for the iteration , for which the corre-
sponding force residuals are . Developing in Taylor series for
and denoting by the displacement variation such that
(E.14)
the following first order approximation can be obtained for the residuals corresponding to iter-
ation
(E.15)
Hence, for the linearization expressed by Eq. (E.15), the force residuals vanish if the displace-
ment variation vector is such that
(E.16)
vk 1–γβ--- 1–⎝ ⎠⎛ ⎞ vk 1–
γ2β------ 1–⎝ ⎠⎛ ⎞ ∆tak 1–+=
1β∆t2-----------M γ
β∆t---------C+⎝ ⎠
⎛ ⎞ ∆dk⋅ r dk 1– ∆dk+( )+ qk=
qk qk M ak 1–⋅ C vk 1–⋅+ +=
r d t( )( ) d t( )
Ψ ∆dk( )
Ψ Ψ ∆dk( ) qk r dk 1– ∆dk+( ) 1β∆t2-----------M γ
β∆t---------C+⎝ ⎠
⎛ ⎞ ∆dk⋅––= =
∆dkn 1– ∆dk n 1–
Ψkn 1– Ψ ∆dk
n 1–( )= Ψ ∆dkn 1–
δkn
∆dkn ∆dk
n 1– δkn+=
n
Ψ ∆dkn( ) Ψ ∆dk
n 1–( ) ∂Ψ∂ ∆dk( )-----------------
n 1–δk
n⋅+=
δkn
∂Ψ∂ ∆dk( )-----------------
n 1–– δk
n⋅ Ψ ∆dkn 1–( )=
Non-Linear Dynamic and Static Analysis Scheme 451
Taking into account that , the above expressed derivatives can
be obtained from Eq. (E.13) and, therefore, Eq. (E.16) transforms into
(E.17)
where the so-called effective stiffness matrix is given by
(E.18)
in which is the tangent stiffness matrix for the iteration of step .
Thus, according to Eq. (E.13) the residual vector is evaluated by
(E.19)
where and the displacement correction becomes
(E.20)
which can be replaced in Eq. (E.14) to provide an estimate for iteration . The process contin-
ues until the satisfaction of a given convergence criterion, which can be defined in terms of a
norm or the maximum of residual forces, or in terms of a residual displacement norm or even
in terms of residual energy (Owen and Hinton (1980), Faria (1994)).
The above stated sequence of steps constitutes the Newton-Raphson method, widely used to
solve non-linear equation systems, for which an initial estimate must be provided. As
referred in Faria (1994) this estimate is somewhat arbitrary since the subsequent corrections
(given by Eq. (E.20)) will tend to approach the real solution. Obviously, the more reasonable
the estimate, the less iterations will be needed to reach convergence; in this context, the option
for either or appear to be adequate, and the first one is in fact adopted
in CASTEM2000. Thus, replacing by in Eq. (E.19), the first estimate of residu-
als is given by
∂ …( ) ∂ ∆dk( )⁄ ∂ …( ) ∂d⁄[ ]k=
Kkn 1– δk
n⋅ Ψkn 1–=
Kkn 1–
Kkn 1– 1
β∆t2-----------M γ
β∆t---------C d∂
∂r⎝ ⎠⎛ ⎞
k
n 1–
+ +=
∂r ∂d⁄( )kn 1– n 1– k
Ψkn 1–
Ψkn 1– qk rk
n 1– 1β∆t2-----------M γ
β∆t---------C+⎝ ⎠
⎛ ⎞ ∆dkn 1–⋅––=
rkn 1– r dk 1– ∆dk
n 1–+( )=
δkn Kk
n 1–[ ]1–Ψ⋅ k
n 1–=
n
∆dk0
∆dk0 0= ∆dk ∆dk 1–=
∆dk ∆dk0 0=
452 Appendix E
(E.21)
which then provides the first displacement correction using Eq. (E.20).
The tangent stiffness matrix included in is often replaced by a stiffness matrix which is
kept constant during several steps or iterations in order to reduce the computational effort
inherent in the stiffness matrix factorization; this leads to the so-called modified Newton-
Raphson schemes which can also be easily used in CASTEM2000.
The analysis for static loading is, indeed, a particular case of the dynamic problem: since accel-
eration and velocity vectors vanish, the residual vector as given by Eq. (E.19) for the generic
iteration becomes and the matrix coincides with the tangent
stiffness matrix , while displacement corrections are still obtained by Eq. (E.20).
Additionally, for the first iteration with the force residuals match the increment of
external applied forces, i.e., , because the convergence
in the previous step imposed the satisfaction of to within a pre-defined toler-
ance (which justifies the use of in the previous expression of ).
The seismic input is a particular case of dynamic loading in which ground motions are imposed
to the structure base supports (or foundation). It can be derived directly from the D´Alembert
principle (Clough and Penzien (1975)) to yield the most common way of considering the seis-
mic input. Thus, denoting by the vector of the base translational displacement components,
the total displacement , velocity and acceleration vectors of the structure
become
(E.22)
where is the so-called pseudo-static matrix reflecting the rigid body modes due to ground
motion.
Re-writing Eq. (E.1) in terms of total kinematic entities and assuming that no other forces
are applied, it yields
Ψk0 qk r dk 1–( )–=
Kkn 1–
n 1– Ψkn 1– qk rk
n 1––= Kkn 1–( )
∂r ∂d⁄( )kn 1–
∆dk0 0=
Ψk0 qk r dk 1–( )– qk qk 1––≈ ∆qk= =
r dk 1–( ) qk 1–=
≈ Ψk0
dg
dT( ) d· T( ) d··T( )
dT d 1˜
dg⋅+=
d· T d· 1˜
d· g⋅+=
d··T d·· 1˜
d··g⋅+=
1˜
q
Non-Linear Dynamic and Static Analysis Scheme 453
(E.23)
where the contribution of restoring forces vanishes because is strictly associ-
ated with rigid body motions of the structure.
Additionally, for most cases of building structure analysis, the damping matrix is defined in
terms of the stiffness one and the contribution of also vanishes. Thus, Eq. (E.23) is
currently used in the form
(E.24)
which is equivalent to Eq. (E.1) with the dynamic force vector particularized for
. The pseudo-static matrix is the well known matrix consisting of unit val-
ues for the displacement components parallel to each of the translational components of
and zero in the remaining positions.
In the present work, this form of seismic action prescription has been adopted, even in cases
when is also defined in terms of . Indeed, although not strictly valid in such circum-
stances, this option has nevertheless been kept based on the assumption that damping forces
are essentially caused by relative motions and on the belief that, for non-linear behaviour of
reinforced concrete structures, the damping source arises mainly from hysteresis (supposed
already included in ) thus allowing to eliminate contributions involving the viscous
damping matrix .
M d··⋅ C d·⋅ r d( )+ + M– 1˜
d··g⋅ ⋅ C 1˜
d· g⋅ ⋅– r 1˜
dg⋅( )–=
r 1˜
dg⋅( ) 1˜
dg⋅
C
C 1˜
d· g⋅ ⋅
M d··⋅ C d·⋅ r d( )+ + M– 1˜
d··g⋅ ⋅=
q M– 1˜
d··g⋅ ⋅= 1˜
d··g
C M
r d( )
C
454 Appendix E