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    PUBLICATION NUMBER P148

    Modelling o fSteel Structures

    fo rComputer Analysis

    Published b y :The Steel Construction Institute

    Silwood ParkAscot

    Berkshire SL5 7QNTelephone: 01344 23345Fax: 01344 22944

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    0 1995 The Steel Construction InstituteApart from any fair dealing for the purposes of research or private study or criticism or review, as permittedunder heCopyrightDesignsandPatentsAct, 1988, thispublicationmaynotbe eproduced,stored, ortransmitted, in any form or by any means, without the prior permission in writing of the publishers, or in thecase of reprographic reproduction only n accordance with the terms of the licences issued by the K CopyrightLicensing Agency, or in accordance with the terms of licences issuedy the appropriate Reproduction RightsOrganisation outside the UK.Enquiries concerning reproduction outside the terms stated here should be sent to The Steel ConstructionInstitute, at the address given on the title page.Although care has been taken to ensure, to the best of our knowledge, that all data and information containedherein are accurate to the extent that they relate to either matters of fact or accepted practice or matters ofopinion at the time of publication, the authors and any other participant of the Eureka 130 CIMsteel project,assume no responsibility for any errors in or misinterpretations of such data and/or information or any loss ordamage arising from or related to their use.

    PublicationNumber: P148ISBN 1 85942 025 7British Library Cataloguing-in-Publication Data.A catalogue record for this book is available from the British Library.

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    FOREWORDTh is guide has been produced as part of the Eureka 130 CIM steel project, and s he result ofcollaboration betw een me mb ers of the following organisations:Ove Arup and PartnersCSC (UK) LimitedQSEThe Steel Construction Institute.Principal authors of the document are:Dr David Brohn (QSE)David B rown (SCI)Dr Richard Henderson (Ove Arup andPartners)Alan R athbone (CSC (UK) Limited)Valuable com ment w as received from:Dr K F Chung (SCI)David Cunliffe (J N Rowen Limited)Charles King (SCI)Dr Mark Lawson (SCI)Professor D avid Nethercot (University of Nottingham)Alan Pottage (Ward Structures Limited)Peter Purvey (Taywood Engineering Limited)

    The guide is primarily aime d at structural designers using readily available analysis software. It isrecognised that the use of a powerful program can be counter-productive unless approached w ith anunderstanding of real structural behavio ur and practical construction. This guide, th eref ore , offersadvice on the modelling of common steel structures for analysis by computer.References to BS 5950: Part I and DD E W 1993-1-1 I 99 2 (Eurocode 3) have been made withpermission of BSI . Complete copies of the Standards can be obta ined from BSI Customer Services,389 Chiswick High Road, London, W 4 4AL.

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    FOREWORDSUMMARY123

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    INTRODUCTIONSCOPEA QUALITATIVE UNDERSTANDING OF STRUCTURAL BEHAVIOUR3.1nderstandingtructural behaviour3.2Thenfluenceof tructuralormonabrication3.3 RecommendationsFRAME ANALYSIS4.1ypesfnalysis4.2 Elasticnalysis4.3 Plasticnalysis4.4 Elastic-plasticnalysis4.5ummaryf analysis types4.7nalysisroblems4.6nclusion fecond-order ffectsn analysis4.8 RecommendationsMODELLING OF FRAMED STRUCTURES5.1ntroduction5.2wo dimensionalodelling5.3 Threeimensionalodelling5.4Decompositionowo dimensionalrames5.5ecommendationsMODELLING OF BEAM AND COLUMN FRAMES6.1 Basic frameeometry6.2 Compositerames6.3 Shear deflectionTRUSSES AND LATTICE GIRDERS7.1nalysisodels7.2ointccentricity7.3 Practicaletailing7.4russnalysis andesignrocedurePORTAL FRAMES8.1igid-plasticndlastic-plasticnalysis8.2 Rigorous implementation fhe lastic-plasticmethod8.3 Haunches8.4 Portalases8.5 Valley supports

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    MEMBERS9.1urved members9.2 Tapered members9.3tiffnessftublements9.4 Castellatednd cellular membersMODELLING OF CONNECTIONS10.1onnectionehaviour10.2 Rigorousapproach to he modelling of connections10.3tiffnessimits10.4Assessmentofactual onnection tiffness10.5Modelling f semi-rigid connection behaviour10.6 Recommendations - connection behaviour10.7Modelling of connections10.8 Bracingonnections10.9 Recommendations - modelling f onnectionsMODELLING OF SUPPORTS1.1otationalixity11.2Horizontaland erticalixity1.3 RecommendationsMODELLING OF LOADS12.1 General12.2Modellingfoads12.3 Load types12.4 Load combinations12.5 RecommendationsINPUT AND OUTPUT CHECKSMEMBER DESIGN14.114.214.314.414.514.614.7

    APPENDIX A:APPENDIX B:APPENDIX C:c.1c.2REFERENCESBIBLIOGRAPHY

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    IntroductionRestraint to bucklingServiceability checksEffective lengthsDestabilising oadsMinimum weightDesign optionsSelf assessment exerciseP-delta effectsNotional horizontal forcesAmplified sway methodAmplified sway method - worked example

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    This docum ent gives guidance on the creation of computer models for steel structures with orthodo xdetails and conn ections in order to produce sa fe, cost effe ctive, real structure s. It is prim arily aimedat stru ctu ral eng inee rs using readily available analysis softw are. It highlights the importance of aqualitative understanding of structural response both during the creation of the analysis model andwhilst appraising the analysis output.After a gene ral ntrodu ction to the elastic, plastic and elastic-plastic analysis of two and threedimensional frames, separate chap ters address the modelling of:a simpleeam and columnramesa trusses and lattice girdersa portalramesa curved,aperedndonomogeneousmembersconnectionsa supportsa loadsIt also provides guidance on simple checks to ensure the analysis is correct and an overview ofmem ber d esign for the less experienced designer.This gu ide is limited to the m odelling of general building and plant structures of normal proportionsun der static loading. Offshore structures, masts, bridges, shells and plates are not covered, nor isgrillage analysis. T he guide concentrates on first order analysis programs. Second order analysis isdiscussed, but both the analysis and the type of stru cture requ iring second ord er analysis ar e outsidethe scope of this document.

    Modelisation des st ructures en acier pour une analyse par ordinateur

    Ce document apporte une guidance pour la rkalistionde moddes nformatiques de structures enacier,utilisant des dktails d assemblages classiques, a$n d obtenir undimensionnement skcuritaire,kconomique et rkaliste. I1 est principalement destini a m ingknieurs de structures utilisant deslogicielsde structures existants. I1 met en vidence l importance dune bonnecomprkhensionqualitative d u comportement de la structure, tant lors de Iklaboration du mod2le informatique quelors de la vtrification des rksultats de lanalyse.Apr2s une introduction gknkrale consacrte a l analyse tlastique, plastique et tlasto-plastiquedossatures planes ou a trois dimensions, diffkrents chapitres sont consacrts ci la modklisation:a despoutres simples et des p o te a u des cadresdes treillis et des poutres en treillisa desortiquesa des ltments courbes, a hauteur variable ou mixtesa des assemblagesa des appuisa des charges

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    Ce guide es t limitk a la modklisation de structures classiques de dtiments de d imensions habituelles,soumis a un chargement statique. Les structures minces, les pylbnes, les ponts, les coques et plaquesainsi que les grillages de poutres ne sontpas pris en considkration. Le guide est surtout consacrk auxprogrammes d 'analyse du premier ordre. Les analyses du second ordre sont bridvement discutkesma is, tant cette analyse que l'ktude des structures qui nkcessitent une analyse du second ordredkpassent le cadre de ce guide.

    Modellieren von stahlbauten fur die computer-berechnungZusammenfassungDiese Publikation gibt Anleitungen zumEntwurf von Computer-Modellen fu r Stahlbauten mitgewohnlichen Details und Verb indungen, um sichere, wirtschaflliche, reale Tragwerke herzustellen .Sie ist vorwiegend an Tragwerksplaner gerichtet, die entsprechende Software einsetzen. Sieunterstreicht die Bedeutung, die Antwort des Tragwerks qualitativzu verstehen, sowohl beim Entwurfdes Berechnungsmodells als auch bei der Beurteilung der Ergebnisse.Nach einer allgemeinen Einjkhrung in die elastische, plastische und elastisch-plastischeBerechnungvon zwei- und drei-dimesionalen Tragwerken, befassen sich separate Kapitel mit der Modellierungvon:m einfachen Tragernndtiitzenm Raumtragwerkenndachwerktragernm Rahmenm gekrummten, gevouteten undnhomogenen Bauteilenm Verbindungenm Aupagernm BelastungenS e gibt Anleitungen fu r einfache Kontrollen der Berechnungund einen Uberblick iiber dieBauteilbemessung fur den weniger geiibten ngenieur.Dieser Leitfaden beschrankt sich auf die Modellierung gewohnlicher Tragwerke mit normalenAbmessungen unter statischen Lasten. Tragwerke der Bereiche Ogshore, Maste, Brucken, Schalen,Platten und Tragerroste werden nicht behandelt. Der Leiqaden konzentnert sich auf die Berechnungnach Theorie Erster Ordnung. Die Theorie Zweiter Ordnung wird angesprochen , aber sowohl dieBerechnung als auch die Tragwerksarten, die eine Berechnungnach Theorie Zweiter Ordnungerforderlich machen, sind in d iesem Dokument nicht erJaJst.

    Modelado de estructuras de acero para analisis numeric0 por ordenadorResumenEste documento suministrayuda para la creacion de modelos or oredenador de estructuras decerocon detalles y conexiones estandar con el objetivo de producir estructuras seguras, baratas ypracticas. Esta dirigido, en primera instancia, ingenieros estructurales que esten usando r o g r a mde analisis disponibles en la actualidad. Destaca la importancia de la comprensibn cualitativa de larespuesta estructural tanto durante la creacion del modelo como durante el analisis de Los resultadosdel calculo.

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    Despuks de una introduccibngeneral a1 andlisis eldstico, plas tic0 y elasto-plastic0 depbrticos de dosy tres dimensiones, subsiguientes capitulos tratan sobre el modelado de:m Vigasy columnas simplesm Celosiasm PorticosElementos cuwos, cuiias y elementos no homogkneosm Apoyosm ConexionesCargasTambitn sedan ejemplos de comprobaciones simples para asegurarse que el analisis es correct0 yunpanorama sobre el diseiio de elementos para 10s diseiiadores con menos experiencia.Esta guia se limita a1 modelado de estructuras generales de proporciones normales y bajo cargaestatica. La estructuras offshore, mastiles, puentes, tanques y chapas no son tratadas. Estedocumento se concentra en 10s pr og ra m s de andisis de primer orden. El calculo de segundo ordenes tra tad o, pero tanto el a nd isis como la distincion de las estructuras que requieren este ordensuperior se encuentran @era del alcance de este documento.

    Modellazione di st rut ture in acciaio per Ianalisi mediante elaboratoreSommarioQuesta pubblicazione fornisce ndicazioni pe r la modellazione a1 calcolatore di strutture in acciaiodi tip0 tradizionale con la jinalita di portare a1 dimensionamento di sistemi convenienti sia dal untodi vista della sicureua sia sotto il proj ilo dei costi. L attenzionee stata principalmente rivolta agliingegneri strutturisti che usano programmi di ca lcolo pe r lanalisi strutturale. Viene sottolineatal importanza della corretta interpretazione, almeno a livello qualitativo, della risposta strutturaleagend o sia a livello d i creazione del modello di analisi sia nella fa se di esame dei risultati ina li.Dopo un introduzionegenera le ai diversi tipi di analisi (elastica,plastica e elasto-plastica) pe r sistemiintelaiati bidimensionali e tridimensionali, in alcuni capitoli viene e trattata la modellazione di:

    travi in semplice appoggio e colonne di telaiom elementi biella e travi reticolarim portalielementi cuwi, rastremati e non omogeneim collegamentim vincoli d i appoggiocarichiVengono anche orn ite, pe r progettisti meno espeni, indicazioni di massima pe r semplici controlli attia verificare la corretteu a dei risultati dellanalisi e delle dimensioni progettate dell elementostrutturale.Questa guida e limitata alla modellazione di edijici comuni e si occupa di strutture con proporzioniin pianta regolari e soggette a carichi statici. Strutture marine, antenne, ponti, cupole e lastre nonSono trattate, come pure lanalisi a1fuoco. Inparticolare si concentra 1attenzionesui programmi dianalisi del pr im0 ordine . Viene introdotta anche lanalisi del secondo ordine anche se quest0 tip0 dianalisi com e pure il tip0 di struttura che la richiede esula dallo scopo del presente lavoro.

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    Modeller ing av stAlkonstruktioner for dataanalysSammanfattningDenna publikation ger vagledning om datamodellering av stdlkonstruktioner med konventinellaforbandch detaljer med avsikt att producera sakrakonomiska konstruktionslosningar.Publikationen ar forsta hand avsedd or konstruktorer som anvander fardigskriven program vara.I publikationen betonas vikten av en grundlaggande forstdelse fo r hur konstruktionen ar tank t attfungera bdde under skapandet av analysmodellen och vid granskandet av analysresultatet,E f t r en allmiin introduktionav begreppen elastisk,plastisk och e lastisk-plastisk analys v tvd och tredimensionella ramar folje r kapitel sorn behandlar modellering av:

    ramar med icke momenttagande orbandfackverkch ramarkrokta, avsmalnande och ickehomogenalementforbandUPP@

    hallramar

    lasterPublikationen ger ocksh tips om enkla satt att kontrolleratt analysen ar korrekt och en overblick avkonstruktion av element (balkar, pelare etc.) o r den mindre e@arne konstruktoren.Denna pub likation ar begransad till modellering av normula byggnader med rimliga proporh'onerunder statisk last. Offshorekonstruktioner, master, broar, skal och plattor ar inte inkluderade, inteheller rustbadds anulys. Publikationen ar koncentrerad pd programvara som behandlar forstaordingens effekter. Andra ordningens analys diskuterm, men bdde andra ordningens analys liksomde byggnader som kan tankas behova andra ordningens analys ar utanforad som behandlasi dennapublikation.

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    1Th e power o f comp uters and structural analysis programs has increased dramatically over the pasttwenty years. Now ev en the smallest design office has access to a personal computer with at least atw o or three-dimensional elasticanalysis prog ram . Although echnologyhas adva nced , there isincreasin g eviden ce that analysis prog ram s are being used without an understand ing of the actualbehaviour of real structures, and with an unrealistic confidence in the analysis results. The fabricationindus try repor ts increasing incidences of designs that are overly comp lex, resulting in expensivefabrica tion details and a loss of cost effectiveness. In some cases designs have been presented w hichd o not represent reality, for exam ple with support conditions that will not be achieved in practice,which, in the extreme cases, could invalidate the design.The Standing Com mittee on Structural Safety* recognised the possibility that the use o f com puters orstructura l engineering calculations m ay ead ounsafe structures and, in their Tenth Rep ort('),identified the following opportunities for unsafe design:

    Persons without adequate structural engineering knowledge or training may carry out the structuralanalysis.Ther e may be communication gaps between the design initiator and the comp uter prog ram writerand user.A program may be used out of context.Th e checking pro cess may not be sufficiently fundam ental.Th e limitations of the prog ram may not be sufficiently apparent to the user.For unusual structures, even experienced engineers may not intuitively appreciate w eaknesses inprograms for analysis or detailing.

    It is also increasingly common for analysis (and then design) to be carried out with rigid co nnec tionsthrou gho ut the stru cture. This invariably esults ncomplex connection design , extensive localstiffening in connection zones and highly expensive fabrication. Cost comparisons between simple(pinn ed) and rigid con struction were studied in the earlier phase of the CIM steel project, and thereader is referred to the publication Design fo r Manufacture G uidelines(2).This document gives guidance on the creation of computer models for steel structures with orthodoxdetails and connections, in order to produce safe, cost effective real structure s. It highlights heimp ortanc e of a qualitative understanding of structural response, both during the creation of heanalysis model and whilst appraising the analysis output.A com puter analysis prog ram should be treated as an extremely useful servant, and not allowed tobecome a bad master. An expen sive, extensively stiffened structure that results from a poor designis generally the responsibility of the structural designer, not the program!

    * The Standing Committee on Structural Safety is an independent body established by the Institutions of Civiland Structural Engineers to maintain a continuing review of building and civil engineering matters affectingsafety of structures.1

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    2 SCOPEThis guide is limited to the modelling of general building and plant structures of normal proportionsund er static loading. Offshore structures, masts, bridges, shells and plates are not covered , nor isgrillage analysis. The guide concentrates on first-order analysis program s. Second-ord er analysisis discussed , but both the analys is and the type of structure requiring second o rder analysis are outsidethe scope of this document.M any analysis programs are linked to complementary design program s, making mem ber designpossible with just a few key strokes. This publication includes a section devoted to memb er designin an attempt to alert the unwary to ome of the possible pitfalls.W ithin this guide the following distinctions should be noted:

    Analysis: Th e determination of forces andbendingmoments.Design: Th e selection andcheckingof mem ber izes.Elemen t: The link between nodes in the analysis mod el.M em ber: A component of the real structure comprising of one, or mo re elem ents.

    In this document, references are made to BS 5950: Part l(3). All references to Clause numbers referto the Clau ses of that Stand ard, unless otherwise noted.

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    3 A QUALITATIVE UNDERSTANDING OFSTRUCTURAL BEHAVIOUR

    Th e use of com puters in the analysis of tructures enables the rapid calculationof forces and momentswithin a com plex frame, by the rigorous application of proven theory and mathem atics. Analysis bycomputer offers advantages to the structural designer in speed and in the accuracy of the arithmetic.There is, however, a growing concern that reliance on computer analysis can seriously reduce thestructural designers ability to understand intuitively the real behav iour of a structure. This con cernis not n ew . The following is taken from a 1956 publication by Nervi(4):The pre-em inence iven to mathematics in our schools of engineering, a purelyanalytica l basis ofthe theory of elasticity, and its intrinsic dlfi cu lties, persuade the young student that there is limitlesspotency in theoretical calculations and give him blind fait h in their results. Under these conditionsneither students nor teachers tryo understand and ee l intuitively the physica l eality of a structure,how it moves under a load,and how the various elements of a statically indeterminate system reactamong the mse lves . Today everything is done by theoretical calculation . That student is rated bestwho best knowshow to set up and solve mathematical equations.

    It is highly regrettable that some of the highest qualities of the human mind, such as intuition anddirect apprehension,.. have been overwhelmed by abstract and impersonal mathematicalormulas .The ability of g raduates to understand structural behaviour intuitively was researched in 1977(5) andconclusions reached that the general understanding was poor@ ). Since that study, the use of com puteranalysis has be com e the no rm, and there is no reason to believe that todays undergradu ates have anybetter intuitive understanding of structural behaviour. It is, how ever, recognised thatwith thepre-eminence of computer analysis, an intuitive understanding becomes increasingly important, bothin the creation of analysis mo dels and critically, in the appraisal of the analysis results.This intuitive understanding is termeda qualitative approach, and emb odies understanding,appreciation and intuition. A qualitative approac h involves a non -num erical consideration of thestructure and its behaviour.

    3.1 Unders tand ing s t ruc tu ral behav iourNu m erica l analysis of structures is built on an understanding of algebra, geom etry and calculus.A qualitative approach uses broa der, mo re intuitive and dynam ic reasoning skills to evaluate thebeh avio ur of any particular structure. Th e key principles involved n developing a qualitativeunderstanding of structural behaviour are:

    To consider the deformed shape of a structure.To red uce comp lex structures into statically determinate, simple systems from w hich the truestructure ma y be rebuilt.

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    Figure 1 Typicalrameor assessment

    Figure 1 demo nstrates the application of a qualitative approach. When asked to draw the approximatebending moment diagra m, m any recently qualified graduates conclude that the solution is that shownin Figure 2. This should immediately be recognised as incorr ect, as the sagging mom ent under theload can only be sustained if there is a vertical reaction at D.

    Figure 2 Commonincorrect) olutiono Figure l

    T he co rr ec t solution is easily obtained by a qualitative application of he Flexibility Me thod ofanalysis. If the obvious release is chosen as the horizontal reaction D, the resulting bending mom entdiagram f or the released structure is shown in Figure 3. The deflected shape, (Figure 4), hows thatthe reaction at D must be to the righ t, and the bending mom ent diagram is show n in Figure 5 . Thefinal solution is the combination of both bending moment diagram s, as show n in Figure 6.The basic rules to a qualitative understanding may be taught relatively quick ly, but their applicationrequires practice and improves with experience. A number of frames are included in Appendix A fo rthose who wish to develop ( or measure) their understanding by attempting to draw the bendingmoment diagrams.

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    Figure 3 Bending moment diagram forreleased structure

    Figure4 Deflected shape of releasedS ruc ture

    Figure 5 Bending moment diagram for the reaction

    \

    +Figure 6 Construction of the final bending moment diagram

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    3.2 T h e i n f l u en c eof s t ruc tu ral fo r m on abr ica tionIn addition toan understanding of structural response, the structural designer musthave anappre ciation of practical fabrication and erection of steel structu res. The work of a fabricator isprimarilyconc erned with making connections between me mb ers, and activitiesassociatedwithconnections can account for up to 60% of the value added to the plain material. Simple connectionsare therefore o ne key to economic fabrication.The process of structural design involves three stages:1) Synthesis (conceptualisation)2) Analysis3) Design (the selection and checking of member sizes)At the synthesis stage, the structural designer decides on the geom etry of the struc ture, and how it isto resist the loads w hich are applied to it. Many consequences flow from the decisions about how thestructure is to resist the applied loads:

    what analysis is necessarywhat the form of the joints in the structu re will behow easy it will be o mak ehow easy it will be to erect.

    In the pas t, analysis of structures was a laborious manual ask and decisions were made by hestructural designer to reduce the effort required in analysis. These decisions generally resulted in astructu re which was easy to cons truct, as well as easy to analyse and design. For exam ple:ma king all beam s simply supp orted avoids the need to distribute moments to other elementsextensive rationalisation reduces the design effort necessary

    W ith the advent of quick and easy analysis by comp uter, the structural designer does not need to m akesim plification s to reduce analysis effo rt, and his can lead o more complicated, expensive, realstructures.D urin g the synthesis stage of design , due regard should be given to he conseque nces of nitialdecisio ns on the fabrication and erection stages of he project. As an exam ple, the econom ies infabrication and erection of a pin-jointed braced frame, compared w ith a moment resisting frame, maymean that the simpler structure is chosen at the synthesis stage.Guidelines on fabrication processes and relative costs are given in the CIMsteel publication Designfo r Manufacture G uidelined2).The benefits of an understanding of fabrication techniques is illustratedby he truss show n inFigure 7. W hen modelling the truss for analysis, the internal mem bers may be rigidly connected atthe nodes (the usual default condition within most analysis programs) or released and modelled aspinned members.

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    12 p o i n t l o a d s of 13 .5k N

    A l l j o i n t si g i d

    12 p o i n to a d s o f 13 .5k N

    m B o t hh o r d s 1 5 0 x 1 5 0 ~ 8S H S M i d s p a ne f l e c t i o nI \VA l ln t e r n a l s 1 2 0 x 1 2 0 ~ 8SHS 16.495mmA l lo i n t s i n n e d

    Figure 7 Deflection of rigidandpinnedmodels of thesame russ

    Despite the fact that the structure is triangulated and the point loading is applied to the nodes, in therigid frame there will be bending m oments developed in the members because of the continuity at thenodes.This sim ple exam ple illustrates the following issues:

    To what e xtent should the analysis model be modified in ord er to simplify the fabrication?How does the structural designer recognise the implication of that change on the analysis model?How does the structural designer know what changes are appropriate?

    In this case the issue is whether pinning the ends of the diagonals will significantly affect thebehaviour of the truss. On e way to make such a judgem ent is to com pare the displacemen ts. In theexample analysed the central deflection differs only in the third significant figure between the pinnedand rigid models; this is because the strain energy in the bending moments in the internal membersis sm all. H ow ever, there is a significant difference in fabricating a moment connection at each node.As the pinned end model is a sim ple and cost-effective solution for fabrication, it should be the modeladopted in the analysis and design.

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    3.3 RecommendationsManyexam ples could be quoted o llustrate where a mo re appropriate analysis mod el, o r anappreciation of practical fabrication, would improve the cost effectiveness of a proposed design, ifthis were considered at he analysis stage.The structural designer mustassess if an overallimprovem ent in cost effectiveness could be achieved with a simpler m odel, or rationalisation ofsection s izes . The influence of connection details on the whole cost should never be forgotten. Aleast w eight solution is rarely the cheapes t overall, as illustrated by the form of the relationshipbetween cost and weight in Figure 8 .

    cc Min imum w e i g h t s o l u t i o n0

    1 ; ,W e i g h t

    Figure 8 Relationship between cost and weight of steel n ypical frames

    Th e main objective is for the structural designer to improve buildability. Most experiencedstructural designers do n ot have any difficulty in reaching a solution which achieves this g oal. Thoseless experienced tructural esigners, working with reduced design time (due toast trackconstruction) and fee competition, and the facility to solve by comp uter analysis the most comp lexanalys is prob lem, have little opportunity to develop a qualitative approach or to understand theimplications of design o n fabrication an d erection.A qualitative approach can be developed by training and practice. The reader wishing to practicesuch skills is referred to Appendix A.Advice on fabrication costs andcost effective details can befound n Design fur ManufactureGuidelined2).Modelling issues and best practice advice are found in the later Sections of this publication.

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    4 FRAME ANALYSIS4.1 Types o f analys isTh ere are three types of structural analysis programs in common use. These are:

    ElasticPlasticElastic-plastic

    Analysis program s d o not include checks on the adequacy of members to sustain theorces generated.This is a design activity and may be arried out b y some software packages s a further process afterthe analysis.Analysis may b e first-order o r second -order , as explained in the following sections.4.1.1 First-ordernalysisIn first-orde r analysis the stiffness of he structure is assumed o be constant and unaffected bychanges in the geom etry of the structure when it is loaded. This is the standard assumption oflinear-elastic analysis.The principle of superposition applies to this approach. W here the analysis model rem ains the sam e,the results from analyses of different sets of applied loads can be added together and the results ofindividual load cases can be scaled . The analysis results are proportional to the applied loads.

    4.1.2 Second -order nalysi sIn second -orde r problem s, the effective stiffness of the structure is changed by the action of the loadsupon it. Exam ples of this are cable structures, where a cable becomes stiffer as it straightens out, anda str ut subject to axial load as well as lateral load. In the latter examp le the deflection under thelateral load is modified by the action of the axial load. The two load-effects interact and the principleof superposition does no t apply . The flexural stiffness of an axially-loaded strut which is on the pointof buckling is effectively zero.M em ber s actin g in catenary as well as bending are similar to the cable examp le. In none of theexam ples given can the structural behaviour b e modelled using a linear-elastic analysis.Second-order effects are commonly illustrated by considering the additional displacements, forces andmoments which arise from the action of applied loads on a deflecting stru cture. Thes e are known asP-delta effects, and are demonstrated by example in Appendix B.In some circumstances a first-order analysis may be u sed to approximate the results f a second-orderanalysis, by techniques such as the Amplified Sway Metho d, o r by Extend ed Simple Design, w hichare both described in Clause 5.6.3. The Clause prescribes that in elastic design of multi-storey rigidjointed frames, either one of these approaches must be used when the frame is a sway frame, sincethe secon d ord er effects ar e significant. The Am plified Sway Method is suitable for elastic frameanalysis by computer, and a brief worked example of this approach is included in Appendix C.

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    4.2 Elast ic analysisElastic analysis programs are the most widely used for structural analysis and are based o n theassumption that the material wh ich is being modelled is linear-elastic. The appro priate value of theelastic modulus has to be provided in the analysis. The analysis does not take into account or chec kwhether the elem ents can actually sustain the predicted load effects or whether they fail by yielding,buckling etc.Th e majority of all software packages now allow the analysis of three dimens ional structures althoughsom e may only allow two dimensional structures to be modelled.

    4.3 Plast ic analysisPlastic analysis methods, more correc tly called rigid-plastic, were commonly used in the analysis ofplane-frame structures such as portal fram es. The stress-strain cur ve fo r the material implicit in theanalysis involves zero displacem ent up to the plastic resistance of the member, followed by con tinuousdeformation a t no increase in load (i.e . plastic collapse). Analysis programs based on this method areused to d eter mi ne th e se t of plastic hinges which form a collapse mechanism for a given system ofapplied oads, and the load factor which corresponds to the mechanism. The mechanismwhichcorres pon ds to the lowest load factor is used to design the frame.These methods do not include the calculation of displacements and were usually applied only to twodimensional tructures.For simple structur es for example single-bay portal fram es), bendingmom ents in parts of the fram e other than those in which the plastic hinges have formed can beestimatedfrom the plastic moments n the mechanism. Program s which carry out this form ofanalysis have now largely been superseded by those which perform elastic-plastic analysis.

    4.4 Elast ic-plast ic analysisFirst-order elastic-plastic analysis programs are often used for the analysis of portal fram e structure s.The se prog ram s assume that the elem ents behave elastically up to the form ation of a plastic hinge anddeform without sustaining further momen t. The software is used to predict collapse mechanisms asin the rigid-plastic method of analysis but the analysis method allows the calculation of d eflections andthe value of bending moments in all the elemen ts of the analysis mo del. This subject is dealt with indepth in Section 8 .Bo th portal fram es and multi-storey rigid fram es can be analysed using first-order elastic-plasticanalysis methods to take into account the onset and formation of plastic hinges progressively aroundthe frame, but second order effects (see Appendix B) must be properly considered. Second -ordereffects can be ignored if the fram e is made sufficiently stiff such that the effects are neg ligible.Alternatively, second-order effects may be allowed for by using the Merchant-Rankine form ula, orits modification by W ood. The latter is incorporated into Clause 5.7.3.3, and p lastically designedmulti-storey frames must comply with the provisions of this Clause.Sufficient stiffness of plastically designed portal frames is ensured by the deflection and snap-throughchecks of Clause 5S . 3 . Second-order effects in portal frames complying with the provisions of thisClause are small enough to be ignored.A second-order elastic-plastic analysis would be required to take into account frame instability effectswithout re cours e to the M erchan t-Ran kine m ethod . Few analysis prog rams a re capab le of this.

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    4.5 S u m m a r y o f analysis typesThe ypes of analysisdiscussed in the preceding Sections are llustrated in Figure 9. The linerepresenting linear ela stic-plastic b ehav iour follows the linear elastic line until the form ation of thefirst plastic hinge. This occurs at point A , where the bending moment at some point in the frameequals the plastic moment capacityof the section.

    L o a df a c t o r P l a s t i c o l l a p s eo ad

    --.L in e a re las t ic -p las t ic

    B Se c o n dr d e rlas t ic -p las t ic

    De f l e c t i o n

    Figure 9 Illustration o f analysis types

    Line ar elastic analysis may be continued past this level of loa d, althou gh the resulting stresses in themembers will exceed yield.The straigh t sections on the linear elastic-plastic curve beyond point A represent the deflection of thestructure between the formationof successive plastic hinges.The secon d-ord er elastic-plastic analysis similarly follows the line of the econd-order elastic analy sisuntil the form ation of the first hinge (P oint B in Figure 9). The curve of the second-order analysisis explained by an examination of Figure 10.

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    I p A + l k

    e + A

    Figure 10 Basic frame Figure 11 Deflectedorm

    Point oad P produ ces a bendingmoment P e . According to a inear elastic analysis , P can beincreased u ntil (for plastic and compact sections) a plastic hinge is formed , such that:Pe = MPTh e defle cted shape of the structure is shown in Figure 11. Th e lever arm of the point load isincreased, and hence for any given load, he bending moment predicted by a second -order analysisis greater than a first-order (linear) nalysis.

    Both these relationships are shown in Figure 12. This illustrates that in a second-order analysis, ahinge is predicted to form at a lower load level than in a linear analysis, which is also concluded froman inspection of Figure 9.

    L o a d

    Figure 12 Comparison of first and second-order analysis or he rame in Figure l 0

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    4.6 Inc lus ion of second-order e f fects n analys isSecond-ord er effects are particularly significant in certain s tructures, and must be included in theiranalysis.M ulti-storey frame s in con tinuous construction may need to have second-ord er effects included, if thefram e is a sway fram e, as described in Clause 5. 1. 3. For elastically designed multi-storey rigidframes which have been classified as a sway frame, a convenient method to allow for second-ordereffects is the Am plified Sw ay m ethod, described in Appendix C . For plastically designed multi-storeyframes classified as a sway frame, a simple approach is described in Clause 5.7 .3. 3.Multi-storey frames in simple construction need not be checked for classification as sway or non-sway, according to BS 5950: Part 1 . However, the Foreword to BS 5950: Part 1 notes that therecommendations apply to the majority of structures, and assumes that bracing in simple cons tructionwill be well proportioned and robu st. If the structure is of slend er prop ortions , with an unorthodoxbracing arra ngem ent, second -order effects may be significant. Second-order effects will be significantif the frame is a sway fram e as described in Clause 5.1 .3. A tall, slender building with unorthodoxbracing (e .g. particularly steep bracing, o r K bracing) is an example of a braced structure whichshould be checked for classification as a sw ay or non-sw ay fram e. If the fram e is classified as asway frame , the Amplified Sway method illustrated n Appendix C can be used to amp lify the lateralloading . In a braced frame , this will result in increased forces in the bracing members and thecolumns forming part of the bracing system.The plastic analys is of portal frames in accordance with Clause 5.5 need not include second-ordereffects. Fram es which satisfy this Clause are deemed to have geome try, stiffness and loading suchthat second-order effects are small enough to be igno red.

    4.7 Analys is p rob lemsWhen elements of widely different orders of stiffness are joined , the stiffness matrix is said to be ill-conditioned, leading to a loss of accuracy, despite the computational capability of the analysissoftw are. If a mechanism is mod elled, the stiffness matrix is singular, and usually cann ot be solved .An alysis prob lems giving unexpected results are generally linked to one of these prob lems . It simpossib le to list and desc ribe all the problems here, but by fa r the most com mon is a lack of restra intin one or m ore directions or rotations, allowing the structure to form a mechanism or to spin. Inplane- fram e analysis, this fault is uncom mon , since most programs will detect the error , and moststructu ral design ers w ill prov ide restraints in this simple case. In three-dimensiona l mo delling , thesituation is altogether more comp lex, withwhat can appe ar a bewildering array of releases ofmembers and supports.Finally, reasonable resu lts from a linear elastic analysis will only be obtained if reaso nable stiffnessesare u sed in the analysis. Frequently, users of analysis program s tend to forget that the ratio ofstiffnesses is critical. In the exam ple of Figure 13 , if the stiffness of the beam is many times highertha n that of the colum ns, excessive deflections will result and the forces will not be correc t. Anexercise in The Structural En gineed 7) has shown that it is possible to model a multi-storey structurein w hich some parts of the fram e deflect in the op posite direction to a horizontal load applied to onefloor. This illustrates the care required when defining element prope rties in the analysis mo del. Asa general guide, the ratio of stiffness of elements meeting at a node should not exceed lo5.

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    - 100,000 I,

    Figure 13 Poorly onditioned tructure

    4.8 RecommendationsFirst-or der analysis is suitable for most orthodox steelwork structu res, and is recommended forgene ral use. Seco nd-o rder analysis is generally not necessary for the analysis of straightforward,orthodox steel structur es. Me thods ar e available by which a first orde r analysis can be used to takeinto accoun t secon d-ord er effects, where this is necessary. One such m ethod, applicable to elasticdesign, is described in Appendix C.Provision is made in BS 5950: Part 1 to ensure that structures are sufficiently stiff such that secondaryeffects a re small enough to be igno red, o r to ensure that the effects are taken into account. Theseprovisions ar e described in the preceding Section.Elastic analysis is widely and successfully used for most steel structur es. Elastic-plastic analysis ofportal fram es is frequently undertaken by analysis software specifically w ritten for this ype ofstructure, and is used with equal success in pra ctice. Th e modelling and analysis of portal frames isdiscussed in Section 8 .

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    5 MODELLING OF FRAMED STRUCTURES5.1 I n t roduc t i onIt is generally con venient to consider first the fo rm of the building frame in both sectional directions,and to identify:

    The primary structural elements which form the main frames and transfer both horizontal andvertical load to the foundations.The secondary structural elements, such as secondary beams or purlins, which transfer the loadsto the primary structural elements.The other elements, such as cladding or partitions, which only transfer loads to the primary orsecondary structural elements.

    At the sam e tim e, an y constraints on the form of the building must be identified as these m ay w elldictate ho w the structure is modelled, and in particular, which (if any) frames may be braced , andwhich must be mod elled as rigid.Th e ob jectiv e for the designer is (within the constraints of the specification and any architecturalrequirements) to provide a safe, econo mica l structure. The definition of an econ om ical structure isnot straightforward, and it may be necessary to investigate several forms of framing beforeund ertaking the detailed analysis and design. Ho we ver, it is possible to provide general guidancebased mainly on the understanding that mom ent resisting connections re significantly more expensivethan nominal pinned connections. Thus in order of preference, the designer should consider:

    Simple construction - i.e. braced frames with pinned connectionsRigid fram es in one direction, with ties and bracing in the otherRigid fram es in two directions.

    It must be em phas ised that in most cases, there is mo re than one option for the fo rm of the buildingfram e. Further advice on structural form can be found in Steel Designers Manual(*),and generalguidance on economicdetails in Design fo r Manufacture G uidelines(2).

    5.2 Two d im ens iona l mod e ll ingA num ber of advantages of two-dime nsional mod els can be dentified:

    simplicity of the analysis model;simplicity of the analysis output;a gre ater deg ree of rationalisation in mem ber design and connection design.

    The se advantages have further practical benefit during fabrication, w here econ om ic benefit is gainedwith rationalisation, repetition and less com plex connection details. Fo r these reasons, a twodimensional frame with simple ties in the other plane is recomm ended.Taking as an exam ple the two storey office shown in Figure 14, it is clear that vertical bracing wo uldnot be allowed in ev ery transverse fra me. O ne solution is to model the transverse fram es as rigid,but it ma y be possible to provide bracing on the longitudinal elevations. This would permit thestru ctur e to be modelled as a rigid frame in the transverse directio n, with ties and bracing in thelongitudinal elevation. This is a comm on form of such structure s, and is of course similar to thenormal f orm of po rtal fram e buildings.

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    Figure 14 Typical tw o storeyoffice frame

    If, for some reason, bracing cannot be accomm odated in he ongitudinalelevations due to, forexample, the presence of doors or glazing in each bay, discrete rigid frames are commonly providedin one or m or e bays to resist the longitudinal forces . The structure would still be modelled as rigidonly in the transverse direction, with a separate model for the rigid frames on the elevations.

    5.3 Thr ee d imens ional m ode l lingThree-dimensional modelling is undoubtedly a useful tool, particularly in complex structures whichcannot easily be resolved into simple two dimensional frames. With a three dimens ional mode l, it isalso possible to ransfe r the complete design model into, forexam ple, estimating software anddetailing pack ages.Three-dimensional modelling does, however, bring additional problems, with the potential to makemistakes within a large model, and with the complex analysis output which can frequently confuse,rather than elucidate. For orthodox structures, comprising tw o series of frames at approximately rightangles , two dim ensional m odelling is usually satisfactory and is recommended.Th e discussion in Section 3.2 on the influence of structural form on fabrication and erection is alsorelevant whe n considering the choice between three and two dimensional models. It is easy to defineconnections between beams and columns in orthogonal directions as rigid for analysis purposes, butthe cost of the local stiffening required to achieve such connections in reality mus t be cons idered . Ingeneral, the cost of moment connections, particularly those between beams and the webs ofcolumns,will outweigh avingsmade elsewhere in he structure.Thre e dimensional structures maybemodelled with moment resisting connections in one orthogonal direction, and pinned connections inthe other direction, if it is essential to model the entire structure.

    5.4 Dec o m p o s i t i o n t o t w o d i m en s io n al f ra m esDe com pos ition is the term used to describe the transformation of a real structure into the planefram e mod els used in analysis. The structural designer will generally produce both the structural formand the analysis model, based on intuition and experience, although general principles can beidentified.Th e firs t step is to identify the primary, secondary and other elements within the s tructure, asdescribed in Section 5.1.

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    Having identified the primary, secondary and other structural elements, the second step is to eliminatethe planar elements from the structure, such as cladding , facades and floor slabs. Th e next step is toeliminate the secondary elements, such as rails, purlins or secondary floor beam s. At eac h stage , theapplied loads are converted to equivalent loads acting on the reduced structure. Th us a roof load ona portal building will convert to a uniformly distributed load on the purlins as the roof cladding iseliminated from the mod el. As the purlins are eliminated, point loads will be applied to the rafter atpurlin positions. In this exam ple, following the recommendations of Section 12.2, the multiple pointloads representing the purlins would be mod elled as a uniformly distributed load on the rafter. Theprocess of 'decom position' is illustrated in Figure 15 .Certain o bvious bracing has been omitted from Figure 15 for clarity.

    1 Comple te bu i ld ing

    2 Pr imary and s e c o n d a r ym e m b e r s

    3 Pr im a r ym e m b e r s( F r a m e s for a n a l y s i s )

    Ve r t i c a lo ad r o mm e z z a n in e beam

    Figure 15 'Decomposition' to main frames oranalysis

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    A careful consideration of the load paths is essential, to ensure loads from out-of-plane members areincluded in the analysis of the plane fram e. Tw o comm on situations are shown in Figures 16and 17.In the frame shown in Figure16 the corner column carries axial load from the bracing system, an din Figure 17, the end reactions from the edge beams on the longitudinal elevations, and from themezzanine floor, must be included.

    Figure 16 Typicalmulti-storeyrame

    Figure 17 Industrialuilding with internal first floor

    Most frames, being modelledas an intermediate frame within the length of a structure (Figure 18),will be subjected to load effects only in the plane of the fra me . Note that the end fram es (typically),although subjected to smaller in-plane loads, may also have out of plane loading to be taken intoacco unt. Ou t of plane effects (if any) can be included during the manual design of the mem bers.

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    L e l e v a t i o nII n t e r n a l r a m eIE n d r a m e

    Figure 18 Load effects onendand ntermediate rames

    Although the secondary elements have been eliminated to produce a plane frame for analysis, thebenefit of any restraint offered by the second ary elements may be utilised during mem ber design.In two dimensional modelling, it is generally assumed that out of plane members do not producebi-axial bending or torsional effects in the primary frame. Thus the actual connections should benominally pinned to minimise the out of plane effects which will, in reality, be present in all frame s.In som e circum stances, out of plane members produ ce bending effects in the primary fram e whichmu st either be included in the analysis, or included manually at the design stage. As an examp le,brickwork supports are usually subjected to torsion from the ecce ntric load, and the connections w illbe d esigned to transfer this moment into the primary structure. To model this, a bending moment(ap plie d at a point) may be introduced in the frame loading. Alternatively, the additional mom entmay be included manually at the design stage.5.5 Recommendat ionsMostcommonsteelworkstructure s may be satisfactorily analysed as twodimensionalmodels.Exce ptions include som e plant structures and structures designed to span in two directions such asspace trusses.Two-dimensional modelling is gene rally recom me nded , as:

    it is simple r than three-dimensiona l modelling.model fram es are generally duplicated in reality, giving economy in analysis and design effort, andrationalisation and repetition in fabrication.connections to out-of-plane members are nominal pins w herever possible, avoiding com plex andstiffened conn ections.most standard profiles are intended primarily for bending about one axis.

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    6 MODELLING OF B EAMA NDCOLUMNFRAMES6.1 Bas ic f ram egeomet ryIt is normally sufficient to represent the frame by elements along the centrelines of the me mb ers, b utquestions can arise when d iscontinuities are present in reality. Two examples are shown in Figure 19.In the case of the column connection, a short, stiff horizontal member is introduced, which may berigidly co nnec ted, pinned at either end , or centrally hinged. The choice of pin position affects thelocation and size of the resulting bending m om ent, and must be reflected in the connection designbetween the two sections. In many cases, option (i) will be appr opriate, design ing the lower columnfor an ad ditional bending mom ent, and a nominally pinned connection from the upper colum n.

    Real i ty Real i ty

    Analysismodel Analysismodel

    Figure 19 Eccentriconnections

    In the second exam ple, it is not necessary to model the offset if the smaller beam remains within thedepth of the l arge r, provided that any axial loads in the beams are sm all.6.2 Co m p o s i t eramesThe readers attention is drawn to the SCI publication Commentary on BS 5950: Part 3: Section 3.1CompositeBeams (9), where comprehensive advice may be found, and from which the ollowing briefguidance has been extracted.Cu rren tly, com posite fram e des ign involves a degree of manual redistribution of bending momentsand manua l design, comp ared to bare frame analysis and design which can be (and frequently is)carriedou t totally by softw are. Composite fram e analysis is therefo re usually carried out byconsidering a suitable sub-frame, such as shown in Figure 20, rather than a complete structure.

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    Figure 20 Sub-frame for omposite frame analysis

    Composite section propertiesmust be 'transfo rme d' into an equivalent steel section by dividin g thecross-sectional area of concrete by the appropriate modular ratio, a,. The relationship betweencomposite section properties nd steel section weight is shown in Figure 21, from which propertiesfor initial analysis may be conveniently extracted.

    - mm fE

    IA s s u m e d e c t i o n

    N o r m a lw e i g h t o n c r e t e a, =l O )

    L i g h t w e i g h t

    20 4 0 60 80 100 12040 160W e ig h t o f s e c t io n k g l m )

    Figure 21 Ratioof second mom entof area of compositesection to hatof steelsection

    An alternative approach, suitable for use only in computer programs, is to introduce for the initialanalysis the com posite section properties in the central 70%of the beam span. Bare steel properties(i.e. he crackedsection)are used to theends of eachspan, in the hogging mom entregions.Following the determination of the bending moment distribution, the extent of the property typesshould be mod ified, and the frame re-analysed. This basic approach may be mo dified as ap prop riatewhen the bending moment distributions asymm etric, for example in the case of sway frames.In both cases, the mom ents m ay be redistributed depending on the m ethod of analysis and the sec tionclassification at the su pport.

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    6.3 Shear deflectionShe ar deflection o ccu rs as a result of the shear strain in an element. It is an effect which is d istinctfrom , and additional to, the bending deflection. The Engineers theory of bending comm only usedto calculate deflections by h and ignores shear deflection. W here non-com posite beams are of theprop ortions norm ally used in the construction of conventional floors, neglecting the effect of shearin calculating beam deflection is acceptable because it is a small proportion of the bending deflection.Where beams are deepn proportion to their length, shear deflection becomes mo re significant. Oth ersituations wh ere shear deflection may be a significant proportion of the total arise in com positeconstruction where the bending stiffness of the beam may be very large because of the action of thecon crete . Th e shear stiffness of the web of the beam is unaffected, an d the defection due to shearther efo re becomes a larg er proportion of the overall deflection. The deflection of castellated andcellular beam s also includes a relatively large com pon ent d ue to the action of sh ear.If truss or V ierendeel frames are m odelled by beam elements to simplify an analysis (or in preparingschem e calculations), shear de flection must be included in deflection calculations because in this caseit is a sign ifica nt proportion of the total. This is becau se the area of the me mb ers resisting theshearing force in a truss (the internal members) is a small proportion of the total area of the truss,wh ereas for a flanged beam the area of the web is significant when comp ared to the total area.Shear deflection will be taken into account in an analysis if the beam elem ent used in modelling thestructure is formulated to do so. If an element is a shear beam, its shear area will be required asan input to the pro gra m. This is beca use cross-sectional shapes vary significantly in their shearstiffness. Sh ear area is often input as a factor which is to be applied to the cross -sectional area of theelement. The shear modulus, G, will also be required. For steel,

    EG = 2 (1 +v)

    where: E = Elastic mod ulusv = Poissons ratio (Taken as 0.30 in Clause 3.1.2.)

    A requirement to provide this information indicates that shear deflection will be taken into accountin the analysis.

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    6.3.1 Examples of shear deflect ionAs stated in Section 6 . 3 , the effect of shear deflection increases as beams become stocky, i.e. theyhave a small span/depth ratio. In Figure 22 the additional deflection due to shear is show n as apercentage of the deflection due to bending, for simply supported and cantilever I section members.M inor ap proximations have been made to arrive at the figures shown in the tables.

    U.D.L U.D.L

    < L > bSimplyuppor ted beam wit h U.D.L. Canti lever b eam wi th U.D.L.

    Span/depth Shear deflection as % of Span/depth Shear deflection as % ofratio bending deflection ratio bending deflection20 110 1 4 10 65 5 4 5 23

    Figure 22 The influence o f span/depth atioon shear deflection

    Clearly the contribution of shear deflection to the total deflectionbecomes important only in the caseof sto cky mem bers. Note that plate girders have typical span/depth ratios of between 8 and 15.Span/depth ratios for universal sections in orthodox building construction are around 20.In the examples illustrated, the contribution of sh ear deflection to the overall deflection is inverselyproportional to the square of the span/depth ratio. At small span/depth ratios, she ar deflection willexceed deflection due to bending, but both are likely to be small.

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    7 TRUSSES A N D L A TTICE GIRDERS7.1 Analys ismode lsTh ere are a variety of m odels w hich ma y be used for the analysis of a truss. These include:

    pin jointedframes.continuous cho rds and pin jointed internal (i.e. web) memb ers.rigid frames.

    The first two options are preferre d since in most situations there will be no bending mom ents to beincluded in the joint capacity ch ecks and connection design. In reality, secondary mom ents will bepresent, due to the change in geometry as the truss deflects, the actual rigidity at the connection andthe stiffness of the me mb ers. Des ign standards generally define when these seconda ry stresses maybe ignored. Clause 4.10 states that se cond ary effects may be assu me d to be insignificant if theslenderness ( h ) f the chord members in the plane of the truss is greater than 50, and that of mostof the w eb mem bers is greater than 100.An initial pin jointed analysis and preliminary mem ber sizing is therefore required, to ascertain if theconstraints on slenderness can be achieved. If, despite judicious choice of mem bers, the secondarystresses cannot be ignored, a rigid fram e analysis will be required to d etermine the bendin g mo men tsin the mem bers and at the connections.

    7.2 Jo in t eccen t r i c it yIn most analysis and design situations, it is both convenient and reasonable that the connection esign,being subsequent to both analysis and member design, is carried out in a manner consistent with theassum ptions previously mad e. T hus in most situations, the type of connection (nominal pin, momentresisting etc.) follows the assumptions of the analysis.In truss an d lattice construction how ever, the design of the joints between chord and internal membersfreque ntly dom inates the member design. Gap or overlap joints a re often used to increase jointcapacity, o r to imp rove the fabrication detail. This introdu ces eccentricity into the setting out of theelem ents, and it will gene rally be necessary to include the effect of this eccentricity in the calculationof mem ber forces and mo me nts. As the eccentricities are not known p rior to the choice of mem ber,this is usu ally do ne manua lly, following an initial analysis with no des at centreline intersections. Ifthe eccentricities are known, they may be modelled as illustrated in Figure 23.Certain types of connection have been well researched (notably those between hollow sections) andguidance exists that defines when the moment due to eccentricity at nodes may be ignored forconnectiondesignand design of some truss mem bers.The guidance given byCIDEC T(") sreproduced in Table 1. Note that the design of the comp ressio n chord must alw ays include themo men ts ue to joint eccentricity. Further advice on joint capacities isoeoundnEC 3 Annex K(") and also published by British Steel Tub es and Pipes('*).

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    G a p j o in t O v e r l a p j o in t

    T y p i c a lo d e l f o r g a p j o i n ty p i c a lo d e lo rv e r l a po i n t

    Figure 23 Truss connections

    Table 1 Guidance on when moments need to be considered for RHS truss design~~ ~~

    T y p e of M o m e n tr i ma r y P r i mar yM o m e n t su eo :o d a lc c en t r i c i t yr ans v e r s ee m b e rec onda r yf f ec t s

    load ingCompress ionho r des

    des ignY e s N o

    Des i gn o f o the rm e m b e r s

    N oes

    Des i gn of No, p r ov i ded Yes ( ' o )o , p rov ideda l i d i t yc onnec t i onsc c en t r i c i t yi m i t s n o tx c e e d e d " "r e

    7.3 Pract ica l e ta i l ingThe m odelling of truss and lattice structures is so dominated by member and connection design, thata few points of general advice are ap propriate. In truss and lattice fabrication, connection details havea very significant influence on overall cost. Details w hich involve complex cutting, or extensivestiffening to im prov e joint capacities are to be avoided. The tempting idea of using an H sectionbottom chord (web horizontal) which will better resist out of plane buckling in a reversal load case,should only be adop ted if the internal mem bers c an be satisfactorily connecte d. Th e difficult acces sfor welding, size of gusset plate and load transfer through the web should be considered.Sim ilarly, changes in section size along the length of a chord are best avoided unless a c ost effec tive,simple connection can be provided. If a complex, stiffened connec tion is necessary to transfe r axialforce and pro vide continuity of stiffness about both axes, it may be cheaper to maintain the samesection throughout.

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    In hollow section truss construction, joint capacities are improved with chords that are smaller in sizebut with thicker walls and internal members which are relatively wide.If the chords are I sections, with connections to the flanges, joint capacity is improved with thickerflanges, and internal members which are relatively narrow, compared with the width of the flange.

    7.4 Truss analys is and d es ign procedureIn summary, the analysis and design of a truss should be approached in the following sequence toobtain an efficient and economical structure.i)

    ii)iii)

    iv)

    v)

    vi)

    Determine the truss layout, span, depth, panel lengths, lateral bracing by the usual methods,but keep the number of connections to a minimum, and maintain a minimum angle of 30"between chords and internal members.De term ine loads; simplify these to equivalent loads at the nodes.De termin e axial force s in all memb ers by assuming that the joints a re pinned and that allmember centrelines intersect at nodes.Dete rmin e preliminary m ember sizes and check if secondary stresses can be ignored. Ifsecondary stresses cannot be ignored, re-configure the truss or re-analyse the truss with rigidconnections.Ch eck the joint geom etry and joint capacities. Modify the joint geometry, with particularattention to the eccentricity limits. Consider the fabrication procedu re when deciding on ajoint layout.Ch eck the effect of primary mom ents on the design of the chord m embers, using either theactual load positions or notional moments specified in the design standa rd. Add the effect ofjoint eccentricity where required, by manual methods, or by creating a new analysis modelrepresenting the actual setting out.

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    8 PORTALRAMESThis Sectio n aims to define those areas of analysis modelling which are m ost often used n thecomputer aided analysis and design of portal frames.Proprietary software dedicated to the analysis of portal frames generally involves an elastic analysisto chec k fram e deflection at serviceability limit state, and an elastic-plastic analysis to determine theforces an d mo me nts in the frame at ultimate limit state. These method s have largely overtaken therigid-plastic m ethod .

    8.1 Rigid-plast ic and elast ic-plast ic analysis8.1.1 The rigi d-plastic methodTh e rigid plastic method is a simplified appr oach suitable for hand calculation and graphical methods,although it is also incorporated in some softw are. . In this m ethod the fram e is assum ed not to defo rmund er load (no linear elastic component) until all hinges equired for a given mechanism have formed .Th e fram e then collapses. The design process involves compa ring a number of predeterminedme chan isms to evaluate which o ne has the lowest load factor and hence represents the maximu m loadwhich could be carried by the frame prior to collapse. In each case, the bending mom ent diagramalong the m em bers is constructed to check that the plastic mom ent is nowh ere exceeded.For simple structures such as single span frames this process is a relatively simple matter since thereare a very limited number of possible failure mechanism s.How ever for more complex frames, e .g. multi-span, steps in eaves height, sprung supports or valleybases, the number of potential failure m echanisms, particularly u nder com plex loading conditions,is vas t. Alternative appro ache s are therefore usually incorporated to quickly establish a closeapproximation to the critical mechanism without the need to try all possibilities.8.1.2 The elastic-plastic metho dThe elastic-plastic method, in addition to finding the collapse oad, determines the order in which thehinges form, the load factor associated with each hinge formation, and how the bending momentsaroun d the frame vary betw een each hinge formation. The frame is assumed to behave linearlybetween each hinge formation.The incremental approach of the method means that it can determine w hether hinges form and laterun-form i.e. hinges cease to rotate and begin unloading as a result of the necessary redistributionof moment around the frame . This phenom enon and the incremental approach is best illustrated byan example. Consider the frame in Figure 24. The elastic-plastic analysis indicates that in thisparticular exam ple, the first hinge w ould form at the sharp end of the haunch , B, at a load factor of0.88. Th is can be confirmed by a linear elastic analysis since the frame remains elastic until theform ation of the first hinge. T he corresponding mom ent at the top of the stanchion, A , is less than4 .

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    CB

    Moment = 0.54 MpMoment = Mp

    Moment = 0.94 Mp

    A ALoad actor = 0.88

    Figure 24 Incremental approach - first step

    As mo re load is introduced, the next hinge to form is at the top of the stanchion at a load factor of0.99 (Figure 25). Thus hinges now exist at positions A and B although as applied load is increasedstill further, the moment at hinge B would begin to reduce because of the continued redistribution ofmom ent around the frame. This is known as hinge reversal.

    C

    A d ' oment = 0.78 MpMoment = Mp

    A ALoad actor = 0.99

    Figure 25 Incremental approach - secondstep

    Finally the last hinge to form would be in the rafter close to the apex, C, at a collapse load factor of1.05 (Figure 26). It may be noted that at Ultimate Lim it State (load facto r = l . O ) , the moment at Bwill be very close toMP nd imp ortantly will have undergone some rotation.

    A

    C

    4 Moment = MpMoment = 0.98 Mp' oment = Mp

    ALoad actor = 1.05

    Figure 26 Increment al approach - final bendingmoments

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    Elastic-plastic analysis programs have largely replaced rigid-plastic ones for the following reasons:T h e state of the fram e can be established at any load factor rather than only at collapse. Thisallows an accurate determination of the bending moment diagram at a load factor of 1 O, i.e. atultimate limit state.Determination of the critical mechanism for more complex frames using the rigid-plastic methodis not a simple matter and may lead to slight approximations. The elastic-plastic method willalways find the critical collapse mechanism.T h e elastic-plastic m ethod has a com plete hinge form ation history, whereas the igid-plasticme thod takes account of only those hinges which exist at collapse. Ther efore any hinge whichforms, rotates, ceases to rotate and then unloads is not identified by the rigid-plastic method.

    8.2 Rigorous mplementa t ion o f the e last ic -p last ic methodIt is perfectly possible to use a straightforward elastic analysis program in a step-wise manner toproduce a pseudo elastic-plastic analysis. This is relatively easy in conceptual terms but can be verytediou s fo r anything but the simplest of fram es. The process is an aid to understanding the wayelastic-plastic analysis operates.Th e fir st ste p is to ca rry out an elastic analysis at the full design loading. It is then necessary toinvestigate the bending moment diagram around the frame and determine the point or node at whichthe ratio of the applied moment to the plastic moment of resistance of the section is the reatest. Thisis the position of the first hinge formation. A new m odel is then created w ith a pin at that point, anda pair of equal and opposite moments equal to MP of the section applied at the pin. This new modelis then reanalysed to determ ine the position of the next hinge formation. A further pin and pair ofmoments are inserted at that position, the model reconstituted and the process continued.This was the basic approach of early software for elastic-plastic analysis, although the re-creation ofthe model was incorporated internally within the software by reconstituting the stiffness matrix at eachhinge formation. Comp utationally this was found to be inefficient and, as with the hand method , didnot cope easily with complex features such as hinge reversal.

    8.3 HaunchesHaun ches ar e frequently provided at the eaves and apex connections of a portal frame. These shouldbe modelled as tapered m embers, as recommended in Section 9 . 2 .

    - i - -~ -- These e lements ha ve iden t ica l, l s e c t i o np r o p e r t i e s ,b a s e d o nthe ra f t e r haunch e lement0 ana lys is mode l node

    Ac tu a l haunch Pr ismatic equ iva len t haunch

    Figure 27 Modelling of eaves haunch

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    Eav es haun ches of normal proportions may satisfactorily be modelled with two rafter elements andone column element, evaluated at the cross-sections shown in Figure 2 7. The haunched rafter ismodelled with average section properties for lengths corresponding to V 3 and 343 of the haunch assho w n. T he top of the colum n may be modelled using the section properties of the deeper haunchsection . The assum ption that the neutral axis remains at the centre line of the rafter and does notdesc end tow ards the haunch is safe, since it tends to overestimate both the com pressio n in the bo ttomflange, and the shea r.Increas ed refinement is not justified by improved accuracy for most normally proportioned eaveshaunches . Th e equivalent elements should be connected rigidly at their intersections.Plastic hinges m ust not be allowed to form in the haunched region d uring the elastic-plastic analysis.Hence, when defining the properties for the haunch elements, the mom ent capacity should be set toa large value. (De pend ing on the software, this may be by direct input of a high mom ent capacity,or, for example, by input of a high section modulus.)Apex h aunc hes of normal proportions have no significant influence on the frame analysis, and d o notneed to be included in the analysis mo del. So-called apex haunches in propped portals or monopitchportals make a significant contribution to the frame, and should be modelled in the analysis.

    8.4 Portal basesTh e m ode lling of bases is cove red in detail in Section 11. The following points are particularlyrelevant to elastic-plastic portal frame analysis, where horizontal, vertical and rotational springstiffnesses can be combined with a mom ent capacity.

    If porta l bases are m odelled with a high mom ent capacity and relatively low rotational springstiffness, significant rotation would be necessary in order for a plastic hinge to form at the base.I f po rtal bases a re mo delled with a low mom ent capacity and relatively high rotational springstiffness, then it is likely that a hinge forming part of the final collapse mechanism will occur ata base position. Furth ermo re, it is likely that the moment at the base from the elastic analysis atServiceability Limit State will be greater than the moment capacity of the base.

    In reality, typical bases can on ly sustain an angle of rotation of less than lo , and so it is importantto chooseam om ent capacity and rotational spring stiffness which are reasonably balanced. Ifunb alanced cases occu r, then the analysis results should be checked by hand or by the progra m toensure that the base has not rotated (either elastically or plastically) by an unacceptable amount.Analysis program s ma y present a warning if a pre-set rotation limit is excee ded , or may indicate thecalculated rotation at nodes for the user to check. It is particularly im portant in the case of lowmoment capacity and relatively high rotational spring stiffness to judge whether the plastic rotationwhich is inferred can be accom mo dated by the base detail, i.e . is hebase sufficiently ductile?Em bedd ed holding do wn bolts and the welds should not be relied upo n to provide ductility.Note that a moment capacity for a base has no relevance in an elastic analysis.

    8.5 Valley supportsHit and miss valley frames a re comm on in portal frame buildings - this is where one or moreinternal stanchions in a multi-span frame are om itted in every second frame, the miss frame. Valleybeam s running longitudinally support the rafters at the m iss positions and react back nto the columnsof the adjacent frames, the hit frames.In a typical two dim ensional elastic-plastic analysis, valley supports are usually m odelled by the30

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    inclusion of vertical, horizontal and rotational spring stiffnesses at those positions. Specifying asup port mom ent capacity at a valley beam would imply plasticorsional behaviour of the valley be am,and this op tion is generally not available in proprietary p ortal frame analysis software.The behaviour of the hit and miss frames are influenced by each other an d so an iterative approachto the analysis and desig n of both is required;

    the reaction from the valley beam has to be included in the loading on the hit frame but will beunkno wn until the miss frame has been analysed and designed,the sprin g stiffness of the valley beam in the miss fram e will be unkn ow n u ntil the beam has beenthe horizontal deflections of both frames need to be similar, since in reality thesheeting, whichdesigned or a section size estimated,is very stiff, constrains the two fram es to move together.

    Th e vertical spring stiffness is relatively easy to calculate, knowing (or e stimating) the section sizeof the valley beam. The horizontal deflection of the beam due to a unit point load can be calculatedusing Engineers bending theo ry, and defined by the ratio of [deflection]/[force]. The spring stiffnessis the inverse of this, in appro priate units. The horizo ntal d eflection of the valley beam will dependon the de gree of fixity assu med at the supp orts o n the hit frames.A horizontal spring tiffness can be calculated in a sim ilar m annerusing the weak ax is properties ofthe valley beam . Th e horizontal supports to the valley beam (the hit frames) are however not rigid,and the horizontal deflection calculationmust include the support deflection before determining heequivalent spring stiffness. A horizontal spring stiffness will produce a horizontal load on the valleybeam, requiring the valley beam to be designed for biaxial bending. Alternatively, bracing in theplane of the roof may be provided to the valley beam. In both cases the horizon tal r eaction s must beincluded in the d esign of the hit frame.One convenient approach when using bracing to the valley beam, is to apply an assumed horizontalsupp ort force at he valley of the miss fram e, which should be released horizontally. An equ al andopposite force is ap plied at the valley of the hit fram e, and the horizontal deflections comp ared.This approa ch is then repeated, until the two h orizontal deflections are approxima tely equal. Theanalysis of each frame including the calculated horizontal force will generate the co rrect forces,moments and deflections.

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    9 MEMBERSNorm al frame mem bers a re generally m odelled as one (or more) straight elements, with associatedsection properties. Universal beam s, universal colum ns, tees, angles, channels and hollow sectionsare m odelled on this basis. The following sections give guidance on how to model non-standardsections.

    9.1 Curvedm e m b e r sCu rved mem bers are mo delled as a series of short, straight elements. M odelling by using more,shortereleme nts, mp roves the accuracy of he results. As a general guide, a length of arccorresponding to 15 produces reasonable results.

    9.2 Taperedm e m b e r sTap ered mem bers can be simply modelled as a series of short elements, each with an inertiacorresponding to the depth of the mem ber at that position. Tw o to five such sections g ive reasonab leaccuracy.Figure 28 show s a simple cantilever, with an inertia 1at the tip, and an inertia of 4.61at the support.If the cantilever is mo delled with three , five , and ten sections, the following results are typical.Tak ing the deflection of a model with five sections as the standard, using three sections modified thetip deflection by 2 % , and ten sections by 1%, as shown in Figure 2 8 . Note hat modelling hecantilever as asingle mem ber with an average inertia of 2.81gives a tip deflection some15% differentfrom the stepped mem ber. Generally, three sections are satisfactory when modelling taperedmembers.

    N u m b e r o f e l e m e n t se f l e c t i o n at t i p (% )1 (average inertia)

    3510

    115102l0099

    Figure 28 Deflectionof a cantilever with different element inertiasSom e prog ram s nclude facilities for modelling tapered me mb ers, but they usually follow thepproachdescribed above.

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    9.3 St i f fness o f s tube lementsThe stiffness of stub elements introduced in bracing systems, frames or trusses must be carefullychosen o r the analysis may yield inaccurate results.As a general guide the prob lem can be avoided if the stiffness (UL)of elements meeting at a node donot differ by more than a factor of lo5. This will be satisfied in most cases if properties of rolledsections ar e used as stubs in preference to creating an elem ent with a massiv e inertia.So me pro gra ms hav e the facility to provide a rigid link between mem bers meeting with a sm allecc en tric ity , and it will not be necessary to determine a suitable inertia if this facility is available.Sim ilarly, mem bers mee ting in this fashion may be 'coupled', which allows the release optionsdescrib ed in Section 6.1.

    9.4 Caste l la tedand ce llu lar m emb ersM any steelwork analysis programs provide libraries of standard section properties, and may alsoinclude the section properties for castellated beams. This will allow the structural designer to includecastellated me mb ers in a frame model in the sam eway as standard sections. Whilst the fram e bendingmom ents prod uced by this a ppro ach will generally be satisfactory, the structural designer should notethat the deflection of a castellated or cellular beam will be more than that predicted by Engineer'sben ding theory . This is due to the Vierendeel effect and to sh ear deflection (see Section 6.3) .As a rule of thumb, the deflection of a cellular or castellated beam may be taken as 25% greater thanthe equivalent depth beam without openings. Additional deflection du e to the Vierendeel effectbeco mes mo re significant with multiple, long openings. As a rule of thum b, the deflection of a beamwith m ultiple, long openings m ay be taken as 35 % greater than that of the equivalent depth beamwithout openings.In some circumstances, the structural designer may conclude that the additional deflection may beign ore d, or is not critical. Alternatively an allowance may be made for the additional deflectionduring des ign and checking of the memb ers.Cas tellated and cellular beam s may themselves be modelled as a fram e, and detailed guidance willbe found in EC3 Annex N(").

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    IO MODELLING OF CONNECTIONS10.1 Connection behaviourW ithin a fram e, conn ection behaviour affects the distribution of internal forces and mo me nts and theoveralldefor matio n of the structure. In many cases, how ever, the effect of modelling a stiffconnection as fully rigid, o r a simple connection as perfectly pinned, co mp ared to mo delling the realbehaviour, is sufficiently small to be neglected. Elastic analysis prog ram s consider o nly the stiffnessof the connection and it is convenient to define three connection types as follows:RigidA conn ection wh ich is stiff enough for the effect of its flexibility on the frame bending mom entdiagra m to be neg lected. In practice, the flexibility (rotational stiffness) of a connection is not usuallydetermined. Con nections designed on a strength basis alone are generally considered to be rigid.Semi-RigidA connection wh ich is too flexible to qualify as rigid, but is not a pin. The beh aviou r of this type ofjoin t mu st b e taken into account in the frame analysis.PinnedA con nec tion w hich is sufficiently flexible to be regarded as a pin for analysis purposes.Connections with a capacity of less than 25% of the beam moment capacity, together with someductility or freedo m to rotate, are commonly regarded as pinned. In practice, certain details, typifiedby thosefound in Joints in Simple Construction, Volume are considered to be pinned, andnocalculation of the connection stiffness is attempted.Typical examples of rigid and pinned connections are shown in Figure 29, and moment-rotationcurves are illustrated in Figure 30.

    3- D r ig id Pinnedonnection

    Figure 29 Typical connections (UK practice)

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    10.2 Rigorous approach to the mo de ll ing o f connec t ionsW hilst the comm on technique of modelling connections in analysis as absolutely rigidr totally pinnedhas been successfully applied for many years, a rigorous approach would acknow ledge that all rigidconnections exhibit a degree of flexibility, and all pinned connections possess some stiffness. Toadopt a rigorous approach to joint m odelling, two questions need to be resolved by the structuraldesigner:1 . W hat ar e the limits which define a rigid, pinned o r semi-rigid connection?2. How stiff is the particular connection?These tw o issues are considered in the following two sections.10.3 St i ff ness l im i t sFigure 30 shows a number of moment-rotation curves, representing connections of varying stiffness,and shows the dividing lines between rigid, semi-rigid and pinned connections.

    (E, I, L an d M,,relate to heconnectingbeam)25 ElI T Unbracedrames)

    M(Braced rames)

    --

    0.5 ElL

    RotationEC 3

    RotationUK

    0.25 M

    Figure 30 Stiffness limits

    Unfo rtunately, there is no comm on agreement on the slope of these dividing lines. Within the UK ,the figure of 2EZ/L has been suggested as the division between rigid and semi-rigid. How ever EC3Annex J() offers 2 alternatives:8EZ/L for braced frames, and25EZ/L for unbraced frames.

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    In Annex J , the slope of the line between pinned and semi-rigid connections for both braced andunbraced fram es, is given as OSEZ/L . In the UK, however, pinned connections are usually d efinedby their mom ent capacity, no t stiffness. Connections which have a maximum capacity of less than25%MP re generally regarded as p inned, provided they have some ductility or freedom to rotate.

    10.4 As sessm ent o f ac tua l connec t ion s t i f fnessTh e only accu rate way at the present time to determine the mom ent-rotation charac teristics of aconnection isy testing. Methods f calculating connection stiffness do exist, notably inEC 3 Annex J("). Many structural designers have little confidence in the predictions made in thecurrent (1995) version ofAnnex J , when com pared to test results. Assessments of connectionstiffness are therefore usually subjective.

    10.5 Mod el l ing o f semi -r ig id connec t ion behav iourDue to the uncertainties described in Sections 1 0.3 and 10.4, it is relatively uncommon to determineconn ection stiffness prior o,orduring analysis. Inparticular, rame analysis with springsreprese nting conn ection stiffness is uncom mon , although some specialist programs can include astiffness function which varies with the applied moment. If the connection char