Transactions on Modelling and Simulation vol 10, 1995 WIT Press, ISSN X

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1 Numerical modeling of unreinforced masonry building C. Gavarini," F. Mollaioli," G. Valente* " Dipartimento di Ingegneria Strutturale e Geotecnica, Universitd degli Studi di Roma "La Sapienza", Roma, Italy * Dipartimento di Ingegneria delle Strutture, delle Acque e del Terreno, Universitd de L'Aquila, Monteluco di Roio, L'Aquila, Italy Abstract Research on the dynamic characteristics of masonry structural components has significantly lagged behind that of other construction materials in spite of masonry is the oldest and most traditional of those currently in use. For this reason the main purpose of this research is to implement numerical methods for existing masonry structures by the comparison with experimental tests. 1 Introduction Finite element modelling of masonry buildings is often motivated by the scarcity of data, which is simultaneously the main source of uncertainty on the quality of numerical results. In fact a very meaningful application of computational models is represented by the interpretation of structural behaviour that is achieved by calibration of some known parameter of the model until some significant feature is correctly reproduced in the numerical results. For this reason extensive numerical computations must be carried out in combination with the experimental investigation of structural behaviour. This paper focuses on the calibration of structural models for the evaluation of resistance of masonry walls, as a function of material properties and load combinations. The data arising from this research will be a support to the development of a wide numerical analysis for the determination of the seismic vulnerability of existing masonry buildings. In this paper we are referred to experimental test of brick masonry prototype building held at the University of Pavia, [3]. The two principal walls

2 596 Computational Methods and Experimental Measurements (a window wall and a door wall, called respectively, B and D) of the two floor building, shown in Fig.l, have been analyzed in plane stress condition. Firstly a monotonical analysis has been performed utilizing for the comparison the simulation of simple stress conditions, the onset of cracking and the results at the ultimate load condition; then a cyclic analysis with a planned displacement history, including the number and magnitude of displacement cycles and the force distribution that were maintened in the walls, has been executed. 2 Code employed and monotonic loading The employed code is similar to ADINA and has been implemented taking nonlocal strains and rough crack model into account. The maximum aggregate size is assumed equal to the thickness of brick (da 55 mm ). For the masonry walls, low tensile strength and fracture energy have been considered. Cracking occurs when one of the tensile principal stresses tends to overcome cr*. A crack plane develops at a right angle to the previous principal direction and it conserves its orientation for the whole loading process. Subsequent crack planes could be only orthogonal to thefirst.after cracking the normal coefficient in local stiffness matrix is abolished and shear coefficient is multiplied by a costant retention factor less than unity. The nonlinear uniaxial compressive behaviour is defined by initial Young's modulus EQ 7 crushing point (7(6c, o"c), ultimate point U(ec,o'c). In multiaxial compressive state, the crushing and ultimate points of uniaxial test may be enhanced relating to the projection on the biaxial failure envelope in <Ji<J2 plane. Unloading from a compressive state is parallel to the initial Young's modulus. Isoparametric plane elements with a maximum of eight nodes and four integration points in each direction are used. The tangent stiffness matrix is referred to: 1. principal stress direction before tensile failure; 2. cracks coordinate system after tensile failure, where it is imposed The assumed mechanical features are: v 0.17 Poisson's coefficient; E = 1700 N/mnf Young's modulus; 7 = 1700 N/mnf weight density; o~c 6.2 N/mm^ uniaxial compressive streght; p 20 N/mm^floorvertical load; at = N/mm^ uniaxial tensile strenght. Further conditions are: equal horizontal loads at the floors; uniform horizontal displacements for each floor. The ultimate cracking patterns of the monotonical analysis is represented in Fig. 2 for walls B and D. For wall B a cooperation of the orthogonal walls for a depth 1250 mm has been supposed and the principal results are Pcr=4.01 kn, Pu= kn, 61 = 4.37 mm, 82 = 7.02 mm (where

3 Computational Methods and Experimental Measurements 597 Per is the load at onset of cracking, Pu is the ultimate load and #1, 62 are the corresponding displacements). Some results regarding the comparison between different numerical codes, [2] is presented in Table 1 for the door wall D (the actual code is the n.4). Code Per (kn) Table 1 Pu (kn) i (mm) (mm) Pcr=load at onset of cracking; Pu=ultimate load. 3 Cyclic loading The masonry building prototype was subjected to a sequence of test runs in which the maximum top displacement in each wall was gradually increased until the maximum desired drift for the run was achieved. Each test consisted of a standard pattern of displacement cycles as represented in Fig.3, where is also listed the sequence of target top floor drift and corresponding displacement. Before cracking occurs, the sequence consisted of a single preliminary loading cycle, two cycles at the desired maximum displacement and one degradation cycle. Following cracking, the number of cycles in each run were increased to two preliminary loading cycles, three maximum displacement cycles and two degradation cycles. The preliminary and degradation cycles had displacements equal to #A = 0.33A, or one third of the maximum run displacement, A. When tests runs were not executed consecutively, the structure was left in a zero force condition before commencing the following test. The main controlling parameter for each wall was the drift, i.e. the top floor displacement divided by the height to the top actuator (5.77 m). Drift of the two walls were equal for all tests. The displacement at the first floor level of each wall was controlled such that the applied force at the first floor were equal to the applied force at the top floor level. The fields of maximum stresses and displacements are shown in Fig. 4, while the maximum crack fields are represented in Fig.5, comparing the results of the numerical analysis with the experimental test. The maximum base shears are kn and kn for the window wall B and the door wall D respectively. The differences with the test are of 2-3 % approximately for both walls [2]. 4 Conclusion Analysis of existing masonry building needs the choice of adequate numeri-

4 598 Computational Methods and Experimental Measurements cal models; several codes for the structural analysis have been implemented in this study in order to best represent the behaviour of masonry structures. As a consequence of all the elements above analyzed is possible to carry out the following considerations: - the numerous coefficients of the Code, calibrated with the numerical and experimental comparison on simple masonry elements and prototype, show a good agreement with the experimental data in stress, strain, crack and displacement fields and in ultimate loads; - the numerical model is an effective tool for both static ( monotonic and cyclic) and dynamic parametric investigation. Seismic analysis with the accelerograms utilized for the prototype tested on shaking table will be performed and reinforced masonry will also be analyzed. References [1] Gavarini, C. Mollaioli, F. Valente, G. "Dynamic Analysis of Church Masonry", IABSE Symposium on Structural Preservation of the Architectural Heritage, Rome, Italy, 1993, vol.70. [2] Gavarini, C. Andreaus, U. Carriero, A. D'Asdia, P. D'Ayala, D. Ippoliti, L. Mollaioli, F. Valente, G. Viskovic, A. "Numerical Modeling of Unreinforced Masonry Building", Italian Seminar on Numerical Modeling of URM Building, Dept. of Structural Mechanics, Pavia, October [3] CNR-GNDT, Documento 1.1: Progetto delle Prove Sperimentali, Unita Operativa del Dipartimento di Meccanica Strutturale, Universita degli Studi di Pavia, Italy, Febbraio [4] Di Tommaso, A. Valente, G. "Pull Out Test with Fracture and Aggregate Interlock", RILEM TC-90 FMA, Round Robin Analysis of Anchor Bolts, by Elfgren, L. Delft, May [5] Gavarini, C. Mollaioli, F. Valente, G. "Prevenzione sismica per una chiesa in muratura", 6 Convegno Nazionale ANIDIS, Perugia, Ottobre [6] Valente, G. "Modelling of crack shear in concrete", EURO-C, Innsbruck, March 1994.

5 Computational Methods and Experimental Measurements 599 b) c) Figure 1: Prototype: a) Window wall B: b) Plan: c) Door wall D.

6 600 Computational Methods and Experimental Measurements Figure 2: Damaging at ultimate load for wall B and D. Preliminary Max Displacement Degrading -oa Run number 1 Drift (%) Displacement [urn] Pattern Figure 3: Standardised pattern of displacements cycles; a) Before cracking; b) After cracking; c) Sequence of target top floor drifts and corresponding displacements; d) Cyclic action 8 6(t).

7 Computational Methods and Experimental Measurements 601 Jv^::!::;: ;;;;;;:::: s!?%;;;;:;;:::::: s>^«:ti: : >: b) Figure 4: a) Maximum stress field in run n.4 before the last, amin = 1-98 N/rnm^ ; b) Maximum displacementsfieldin run n. 4 before the last.

8 602 Computational Methods and Experimental Measurements H*VKVWlit/till ^A\%VV IH/'piS' \\\\\ */fj/>it%>» 'SsfB*. \\ \ b) Figure 5: Maximum crack field. Comparison with experimental test. a) Wall B; b) Wall D.