Structural Analysis of Historical Constructions - Modena, Lourenço & Roca (eds) © 2005 Taylor & Francis Group, London, ISBN 04 1536 379 9

Cracking simulation in a plain structure using the finite element method

L. Pani, B. De Nicolo & Z. Odoni Department of Structural Engineering, University of Cagliari, !taly

ABSTRACT: This paper outlines the procedure developed to simulate the cracking mechanism in the stone masonry façade of Santa Chiara Church (Cagliari). The static damage growth is demonstrated, from the first crack to the fully developed crack partem. The procedure is implemented in specifically developed program, based on an iterative process and storage of intermediate results, utilising FEM. The program was first tested on simple structural models whose solution is known from the theory of Structural Mechanics, and then on a real structure. The results show a very good reliability and the manageability ofthe programo Furthermore, the program easily allows lhe designer to adjusl specific variables in order to achieve the expected results.

INTRODUCTION

Throughout the world, many designers, engineers and researches are studying in order to exploit historie masonry structures to their potential. This interest derives from the consciousness that historie masonry structures are part of the cultural heritage and as such are to be preserved and reused. For many centuries, stone structures have been built in accordance with empirical design criteria, which guaranteed high safety factors. For the careful restoration of this cultural pat­rimony there is a need for computational modelling, which can show the real mechanical behaviour of stone masonry (Buli 2001).

In order to model the structural behaviour of stone masonry, severa I hypothesis are necessary on the material constitutive law (elastic, elastic-plastic model, heterogeneous or homogeneous material , mortar/stone interaction, etc.) due to the uniqueness of each building.

Today 's commercial FEM programs do not aIways allow following, step by step, the dynamic process of crack formation. In fact it is necessary to have easy means which can perform damage simuIation of both materiaIs and structures, and allow introducing probabIe modifications of the governing parameters. In the present study, a FEM-based program has been developed, which allows investigating lhe behaviour of stone masonry subject to static damage and/or mate­rial degradation. The reliability of the program was tested with a number of applications on simple beams, in order to compare the results with those obtained according to the theory of Structural Mechanics.

2 CASE STUDY: THE FAÇADE OF SANTA CHIARA CHURCH

The Santa Chiara Church, situated in the historical quarter of Stampace in Cagliari , ltaly, was built in the second half of 13th century.

The building is the remaining part of a monastery, neglected to the ravages of time and ultimately des­troyed by the bombing during the Second World War. The church has a rectangular layout, with a barrei vault ceiling spanning 7.50 m (Fig. I). Several restoration

Afterwards, the program has been applied to a real o.ill ... ""'Iõ,-..... ,~'!iiiiõõlr~, plane structure whose geometry and cracking state is known. Figure \. Section of the church.

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Figure 2. A detail of façade at the comice.

works had been carried out both on the load-bearing walls and on the vault (Ingegno 1993).

The façade, whose average thickness is 1,35 m, is independent from the remaining building and is exclusively an vertical wall (Fig. 2).

The façade has structural problems, which led to extensive cracking and is in need for restoration. The cracks start from the architrave of the main door and from the windows, and then almost vertically towards the stone cornice. The cracks are evident both on the inner and outer side. Hence, it was necessary to determine the reason for this cracking, before a new restoration work.

2.1 The stone masonry

The façade was built with limestone blocks, with vari­able size and shape and little worked. The stones were laid with lime and soil mortar, and the interstices among the stones were filled with stone chips.

Limestone is typical of Cagliari and its hinterland. Its chemical and mechanical properties are quite vari­able. Table 1 shows the average values ofthe material properties, obtained from an extensive in-situ exper­imentai investigation of a variety ca1careous stones from Cagliari (Barroccu et a!. 1981, Concu et a!. 2003).

The architrave of lhe main door is ma de of juniper wood, and appears rather damaged.

Table I. Chemical, physical and mechanical properties of masonry.

Ys KN/m3 dry specific weight 18 Ysw KN/m3 saturated specific weight 24 n % porosity 5.6 a c MPa dry compressive strength 24.5 a cw MPa sat. compressive strength 9.8 a t MPa dry tensile strength 0.95 a tw MPa saturated tensile strength 0.32 E MPa modulus of elasticity 2070 CaC03 % content 85

The following parameters have been assumed for modelling the damaged juniper wood: Ej ll = 2000 MPa, O"t ll = 0,9 MPa, whereas the values Ejll = li 000 MPa, O"t ll = 30 MPa have been used for the undamaged condition.

3 THE BASIC FORMULATION

3.1 Modelfor masonry

In order to schematize the material behaviour, a linear elastic model is assumed. The tensile and compressive strength are assumed at the saturated state, considering the great hygroscopicity of limestone. The E-modulus of masonry is assumed about 25% lower than that of limestone (E = 1500 MPa), in order to account for the presence of mortar joints. Masonry is assumed to be an homogeneous and isotropic material.

However, to introduce some non-homogeneity, the E-modulus is assumed to randomly vary of ± 10% around the chosen value. In fact, the program allows assigning different properties to every single element.

3.2 Procedure

The procedure can be schematized as follows:

1. First step preliminary computation of stress and strain at the structure nodes, search for nodes at which the strain exceeds the cracking limit, mesh refinement in the surrounding nodes area.

2. Iteration computation of stress in the refined structure, search for more strained nodes where cracking is possible, if the last requirement is positive, triangle separa­tion at that node.

The loading condition was the self weight (dead load). The tensile strain limit of cracking for masonry is defined based on the De S. Venant-Grashof criterion.

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Furthermore, since in the hypothesis of homoge­nous and isotropic masonry a crack occurs in a plane orthogonal to the direction of the maximum tensile stress, to determine these criticai points reference was made to the principal stresses based on Mohr theory (Mastrodicasa 1993).

4 THE PROGRAM

The various components described above have been implemented on PC, in a finite element program, writ­ten ad hoc in TURBO PASCAL, using pointer-type variables.

When the data amount exceeds the RAM of the computer, data are stored in externai files , and in mixed storage (RAM and files) when data amount is relatively moderate.

The program code is modular and is divided into logical subsets, each composed of one or more proce­dures. The main subsets are explained below.

The logical procedure can also be applied to 3D structures.

The program can generate readable files for com­mercial graphical software packages: Autocad-LT reads the input data file and print the structural model, Surfer 5.0 reads the result files and plots the colour contour of stress and displacement components.

4.1 Choice of the element

The faça de is represented in the model by meshes of six-nodes triangular plane stress elements (Krishnamoorty 1994, Zienkiewicz & Taylor 2000).

This element is chosen for its geometric versatil­ity and excellent performance in stress analysis of structures with high stress concentrations. The trian­gular shape allows ali side orientation compatible with geometric shape of cracks.

Furthermore, the mid-side nodes allow separating the triangles also at the intermediate points, once the limit of crack condition has been obtained.

4.2 lnput data

The diagram in Figure 3 shows the procedure of data input.

In "Geometric data of Structure", nodes and ele­ments are automatically generated, while their orien­tation for each zone is fixed by the user, in order to detail the cracking as much as possible.

The physical and mechanical properties for each element are stored in "Material characteristic".

In order to simulate the non-homogeneity of mate­rial, it is possible to replace the values ofmaterial prop­erties with others, obtained from percentage variation with random law around fixed value.

Geometric data of Structure

Material characteristics

Boundary conditions

Loads

Figure 3. Data input procedure.

SOL VING PROCEDURE

Analysis of structure

Mesh refinement

Iterative process

Figure 4. Solving procedure.

Separated nodes

New analysis of structure

The procedure can review the material characteris­tics for each element. The constraints are specified in "Boundary conditions", while "Loads" describe how the structure is loaded.

4.3 Solving procedure

The calculation consists ofthree part (Fig. 4). In "Analysis of structure", the element stiffness

matrices and the nodal loads are calculated and

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Figure 5. Refinement ofthe mesh.

Figure 6. Refinement schemes.

assembled to obtain the finite element model of the structure, and the boundary condition are imposed. The procedure solves simultaneous algebraic equa­tions in order to evaluate nodal displacements with the iterative method of gradient conjugate.

Element strains are calculated from the nodal dis­placements and the element displacement fie ld inter­polation, and fina lly stresses are calculated from strains. The nodal slresses are evaluated as average ofthe stresses between adjacent elements.

The "Mesh refinement structure" allows refining the mesh around the probable cracking points obtained by the preliminary solution.

At first , the nodes exceeding the tensi le strain limit, in accordance with the De S. Venant-Grashof criterion, are identified. The zone ofthe potential cracking, con­sisting of some elements around the criticai nodes, is bounded. The sum of potential cracking zones deter­mines an area where the triangles are meshed in four triangles as showed in Figure 5.

At last, a list ofboundary element is formed to allow for graduated refinement of the mesh according to scheme "A" . This scheme provides a simpler logical

and an easier programming method, compared with scheme "B", which for triangle ratio could results be better (Fig. 6).

The values of E-modulus, sei f weight, loads and boundary conditions of each new triangle are auto­matically generated in the "Input data" procedures.

The structure is schematised by new meshes and is again calculated in "lterative process". The nodes exceeding the tensile strain limit, established with De S. Venant-Grashof criterion, are separated.

The new geometric configuration of the structure is calculated again. The iteration of each logical sub­set continues until either stabilisation of cracking or fai lure of the structure is achieved.

For comparing results given by the Structural Mechanics, the program was first tested on a simply supported beam, with a point load at midspan.

5 THE SANTA CHIARA CHURCH FAÇADE

The façade has been schematised by a mesh. The nodes at the base of the wall were assumed to be fixed . In a first application the structure has been loaded by its self-weight and by a vertical concentrated force directed downwards, applied at lhe top of cornice along the vertical axis of symmetry of the façade .

The excellent results obtained, in terms of stresses and strains, has encouraged us to continue in the search for the cause of cracking.

Different hypotheses were made to simulate the cause of cracking. For the identification of possible mechanisms we also took advantage of the experi­ence acquired in a previous work carried out on the same façade, in which a commercial programme was used instead (Pusceddu & Monni 1997). The following hypotheses have been considered:

1. settlement of the arch above the main door, 2. settlement ofthe main door juniper wood architrave, 3. vibrations caused by an organ placed in the choir

at the back the façade , whose juniper wood beams were loaded on the façade,

4. bombing during the second world war.

The hypothesis of arc settlement was neglected in the present study, since no damage is observed in the façade at that location.

The vibrations produced by the organ have been estimated taking into account the dynamic response of the compressor ofthe organ, in terms ofangle of exci­tation, damping coefficient, period and values of the spectrum. The direction of the vibrations has been assumed perpendicular to the façade. For sake of simplicity, only the first three natural frequency of vibration have been considered. The effect of bomb­ing, regarded as an impulsive force, has been analyzed using analogous parameters.

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Table 2. Limits Standards SN 640 312, Structural Cate­gory IV, Building that are particularly vulnerable or worth preserving.

Cantinuaus or steady state vibration Frequency (Hz) Max velocity (inls) 10-30 0,12 30- 60 0,12-0,2 Transient ar impact vibration source sources Frequency (Hz) Max velocity (inls) 10-60 0,3 60- 90 0,3- 0,5

0,27

.0 ,24

0,21

0,18

0,15

'0,12

0,09

0,06

0,03

. 0 .00

Figure 7. Main tensile stress (MPa) with architrave undamaged.

The reference Iimits have been assumed in accor­dance with Standard OTN 4150-3 and Standards SN 640312 (Table 2). The values ofthe natural frequencies ofthe façade are very different than the Iimit proposed for both hypotheses. However the vibration produced by organ and bomb may have also contributed to establish the actual state of cracking.

The performed analysis showed that the hypothe­sis of the "settlement of the main door juniper wood architrave" provides a cracking map equal to the actual one. This hypothesis is confirmed by Figure 7, which shows the principal tensile stresses in the façade , cal­culated assuming the juniper wood architrave to be undamaged.

Figure 8 shows the result of a new calculation with the hypothesis of damaged juniper wood architrave. Stress concentrations are observed in the areas around the openings and near the arch above the main door.

0,90

0,81

0,73

0,63

0,54

0,45

0,36

0,27

0,18

0,09

0,00

Figure 8. Main tensile stress (MPa) with architrave damaged.

0,64

0,56

0,48

0,40

0,32

0,24

0,16

0,08

0,00

Figure 9. Main tensile stress (MPa) and cracks in their final state.

The principal tensile stresses were obtained afier the iterations of the previously described procedure until the stabilisation of cracking, in accordance with De S. Venant-Grashof criterion.

The principal tensile stresses are represented in Figure 9, and the cracks in their final state are pointed out.

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Figure 10. Photograph of the façade before the rehabil itation.

To show the reliability ofthe model and the utilised procedure, a photograph of the façade before the rehabi litation is reproduced in Figure 10.

6 CONCLUSION

The application to a real case has demonstrated good reliability of a masonry model and a calculation proce­dure to simulate the cracking state in a plane structure. The procedure has been implemented in a specifically developed FEM-based program o

For those who are experienced in numerical mod­elling of masonry, the adopted model may appear too simplistic. In this case we had over-concentration of stresses only in restricted areas of the structure, therefore it a relatively simple model was chosen, but detailed hypotheses (i.e. non-homogeneity) were assumed to describe lhe material behaviour (Augarde 2001).

In particular it was shown that the FEM method associated to the iterative process allows for a simu­lation of the actual degradation of both material and structure. Furthermore, the choice ofthe triangular ele­men! allows fo llowing the natural progress of cracking. The analysis of input data and results through a graph­ical display (Autocad-LT and Surfer 5.0) extends the handiness of the program, allowing a real-time inter­action ofthe designer in order to achieve the expected results.

REFERENCES

Augarde, C. 200 I . Settlement Induced Damage to masonry Building, in Buli , J.w. Computational Modelling 0/ Masorny, Srickwork and Slockwork Structures. Glasgow: Bell & Bain.

Barroccu, G. & Crespellani, T. & Loi, A. 198 1. Caratteris­tiche geologico-tecniche deI sottosuolo dell'area urbana di Cagliari. Rivista italiana di geotecnica (XV, 2).

Buli, J.w. 2001. Preface. In Computational Modelling 0/ Masorny, Srickwork and Slockwork Structures. Glasgow: Bell & Bain.

Concu, G. & De Nicolo, B. & Mistretta, F & Valdés, M. 2003. NDT ultrasonic method for ancient stone masonry diag­nosis in Cagliari (Italy); in Structural Faults and Repair; Proc. Intern. Symp., London July 1- 3, 2003. Common­wealth Institute, Kensington.

Concu, G. 2003. Controlli non distruttivi sulle murature in pietra: validazione di una metodologia sonica; in Con­ferenza Nazionale sulle Prove non Distruttive Monitorag­gio Diagnostica, 10° Congresso Nazionale dell ' AIPnD, Ravenna, 2-4 Aprile, 2003.

Kri shnamoorty, C.S. 1994. Finite element analisys. Theory and programming. N. Delhy: MacGraw.

Ingegno, A. et a!. 1993 . Santa Chiara, restauri e scoperte. Cagliari: Pisano.

Mastrodicasa, S. 1993. Dissesti statici delle strlltture edilizie. Diagnosi, Consolidamento. Istrllzioni teoriche. Milano: Hoep li .

Pusceddu, A. & Monni, M. 1997. Degrado stati co della chiesa di Santa Chiara, Ana lisi statica e dinamica con gli elementi f in iti , Degree Thesis, Cagl iari: University of study.

Standard DIN 4150. 1999. Vibration in building Part 3 Effect on structures.

Standard SN 640 312. 1983. Association of Swiss Highway Engineers .

Zienkiewicz, O.L. & Taylor, R.C. 2000. The flnite element method, Oxford: Butterworth-Heinemann.

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