Three Dimensional Modeling of Masonry Structures and Interaction of In- Plane and Out-of-Plane Deformation of Masonry Walls

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1 Three Dimensional Modeling of Masonry Structures and Interaction of In- Plane and Out-of-Plane Deformation of Masonry Walls Kiarash M. Dolatshahi a and Amjad J. Aref b a Ph.D Candidate, Department of Civil, Structural and Environmental Engineering, University at Buffalo The State University of New York, NY, USA (km256@buffalo.edu) b Professor, Department of Civil, Structural and Environmental Engineering, University at Buffalo The State University of New York, NY, USA (aaref@buffalo.edu) Abstract A review of existing material models and computational procedures for modeling masonry walls reveals that such models were developed either for in-plane or out of plane deformation of walls. Most of these models are also using implicit procedures for solving the equilibrium equations in the temporal domain. In implicit analysis for modeling masonry structures, several researchers reported convergence issues, and therefore, the analysis cannot track the masonry walls up to failure. Therefore, due to the convergence issues, the obtained resultant deformation is usually limited. This paper presents the development of a comprehensive material model with explicit computational procedures, which can capture bidirectional cyclic deformation of masonry walls. Using ABAQUS, the material model is implemented in a user-defined subroutine. The material model is validated by comparing the numerical results with several well-documented experimental studies, and the results evidently show the capabilities and robustness of the proposed numerical procedure. Finally, using the developed model, a masonry wall is subjected to various loadings in different directions. Keywords: URM walls, Bidirectional loadings, Interaction Curves, Combined in-plane and out-of-plane Introduction In the field of earthquake engineering, masonry structures are some of the most complex and it is difficult to accurately predict their behavior when subjected to earthquakes. In masonry structures, unlike steel or concrete structures with the same plan configuration and number of stories, a large number of elements are engaged in the response of the structure to the earthquake, and their seismic behavior is mostly governed by nonlinear behavior. Accordingly, detailed modeling of a masonry structure using finite element methods or any other high-end computational method leads to extensive computational analyses. A cursory literature review of existing approaches to modeling masonry structures reveals that few detailed material models and analysis strategies exist, and those that do primarily focus on investigating the two-dimensional (2D) cyclic behavior of masonry components and structures (Calderini and Lagomarsino 2008; Casolo and Pena 2007; Gambarotta and Lagomarsino 1997a; b; Oliveira and Lourenco 2004). Most of the notable research that addresses the cyclic behavior of masonry structures uses implicit finite element solutions. Casting a masonry structural model using dynamic implicit methods offers some advantages as well as many technical challenges. Based on the most fundamental characteristics of implicit procedures, at each time step, iterations are required to solve a system of equations. As is well known, the behavior of masonry structures is inherently nonlinear, so by using implicit procedures, it is difficult to achieve the convergence. Most of the finite element codes used to analyze masonry structures are based on implicit formulations; thus, it is difficult and time consuming to proceed to a large displacement domain. The convergence of the system deteriorates at the point of load reversal. Moreover, most of the available models for masonry structures provide analyses in either the in-plane or out-of-plane direction. However, in reality, most loading occurs as a combination of in-plane and out-of-plane motion. Therefore, a comprehensive modeling strategy is needed to simulate the behavior of masonry structures for three-dimensional cyclic loading. Most of the research pertaining to the modeling of masonry structures refers to simulation of the effects of monotonic loading. Using meso-scale analysis, Page (1978) applied nonlinear interface elements to model the 1

2 behavior of joints in masonry walls. In this model, the behavior of the interface element consisted of tensile and shear domain. Later, Lourenco (1996) added a cap part to the model proposed by Page (1978). By adding the cap part, Lourenco modeled the nonlinear behavior of the bricks in the joints. This model was efficient in terms of computational time, and was able to capture the monotonic response of masonry walls subjected to in-plane loading. However, the model could not be used in the large displacements domain. Oliveira ( 2004) extended the strategy of Lourenco (1996) to include cyclic loading. He used line interface elements to model the joints along with linear plane elements to model bricks. This model was partially successful in modeling in-plane behavior, but it could not capture the hysteresis loops for the failure mode with diagonal cracks. In addition, using an implicit procedure made it difficult to achieve convergence in the load reversals. Many numerical and experimental investigates have been focused on the in-plane or out-of-plane behavior of URM walls; however, very few studies have investigated the bidirectional behavior of URM walls. Using a simple material model, Hashemi and mosalam (2007) and Kadysiewski and mosalam (2009) investigated the interaction of an infill URM wall. In these models they considered two degrees of freedom for an infill wall, One degree of freedom for the in-plane displacement of top face of the wall and one out-of-plane displacement for the middle of the wall. In this paper, the development of a user-define subroutine in ABAQUS (VUMAT) (2005) is presented for a new material model that provides robust predictive capabilities to assess the seismic response of masonry structures. The material models together with the three-dimensional (3D) FE models of masonry walls are presented. Afterward the presented model is used to model a masonry wall under bidirectional loadings. Model description Bricks and mortar are distinctly defined in the proposed model of masonry walls. For the bricks, solid elements (C3D8R) in ABAQUS (2005) together with plane interface elements (COH3D8) for mortar are used (Figure 1). Top face Thickness direction Cohesive element node Midsurface Bottom face Figure 1. Eight node plane interface element (ABAQUS) Each element initially behaves elastically, and upon a further increase in load and based on their related yield surface, their behavior become nonlinear. According to Figure 2, in order to create the model, bricks are expanded by half the mortar dimension and interface elements are located between the solid elements which represent the mortar. 2

3 Mortar (E m ) h b +h m Brick (E b ) Brick Solid elements Solid elements Interface elements Figure 2. Detailed model of brick and mortar In addition, in order to capture the exact behavior of the wall, bricks are divided into two parts and a potential crack is placed in the middle of the bricks. Figure 3 shows the generated FE mesh for the wall and a general view of the wall considered in subsequent examples presented in this section. A finer mesh with elements has also been used for some of analyses to check the mesh sensitivity. Checking mesh sensitivity of the results, it has been concluded that the results for fine and course meshes are in a good agreement to each other. Since the time of analysis for the fine mesh was much higher than the coarse mesh, for the rest of the analysis the coarse mesh has been used. Elastic behavior of joints (a) (b) Figure 3. (a) General mesh of the wall (b) General view of the wall The material model for joints, used in this paper, was originally developed and validated in (Dolatshahi 2011). In the recent paper, a brief description is presented. The behavior of the interface elements is initially linear, i.e.,, where t is the traction vector and k is the elastic stiffness of joints and is calculated from the properties of both the brick and mortar. In order to determine the normal and shear stiffness, brick and mortar are assumed to be two elastic springs in series. The displacement for the combination of the brick and mortar system must be equal to the displacement of the solid element-plane interface system under the same compressive and shear loads (Figure 2). Therefore, the stiffness of the system is calculated by using equations 1 and 2. 3

4 (1) (2) Where the parameters that are used in equations 1 and 2 are shown in Figure 2 Plastic behavior of joints Three distinct modes are considered for the interface elements namely, tension, shear, and tension-shear intersection (Figure 4). Based on the plasticity theory and also explicit formulations, after calculating the trial stresses, the position of these trial stresses must be checked with their respective yield surface. If the value of the trial stress is less than the yield surface, the trial stress are correct and can be considered to be in the elastic region. Otherwise, the stresses must be modified based on the related plastic domain. { ( ) τ x τ y c f t σ Figure 4. Yield surface for the joints Numerical results In this section deformed configurations and load-displacement curves for two different loading protocols are presented. In the first model the load was applied in in-plane direction and in the second one 63 degree from the inplane direction. In Fig. 5 point A shows the maximum elastic load (yielding point) and point B represents the ultimate load which the wall can carry. 4

5 Figure 5. (a) deformed shape at 6 mm *5 (b) Load-displacement curve Fig. 5 shows the deformed shaped of the wall after 6 mm in-plane deformation. As it is shown in this figure, diagonal crack is the relative failure mode for this wall. Figure 6. (a) deformed shape at 7 mm *7 (b) Load-displacement curve In the second model (Fig. 6), in addition to diagonal crack, gradually top and bottom of the wall start to separate from the support. By increasing the participation of the out-of-plane displacement, the dominated failure mode in the second model will be the separation of bottom and top of the wall (Fig. 6). Since the in-plane behavior of the wall is much more brittle and stronger than the out-of-plane behavior, even a low participation of in-plane displacement is generating a large force in the second model. 5

6 Conclusion In this paper, two walls were subjected to different monotonic loading protocols. The main intend of this study is to investigate different failure modes for various loading directions. It is shown that, for a specific unreinforced masonry wall and for a specific compressive load (120 kn), as the loading direction changes, the respective failure mode changes as well. For in-plane loading the diagonal crack will be the failure mode and gradually it changes to rocking mode for out-of-plane loading. Anywhere between in-plane and out-of-plane directions, the failure mode will be something between diagonal crack and rocking mode. References Abaqus. (2005). "Theory manual." Hibbit and Karlson and Sorensen Inc. Calderini, C., and Lagomarsino, S. (2008). "Continuum model for in-plane anisotropic inelastic behavior of masonry." Journal of Structural Engineering, 134(Compendex), Casolo, S., and Pena, F. (2007). "Rigid element model for in-plane dynamics of masonry walls considering hysteretic behaviour and damage." Earthquake Engineering and Structural Dynamics, 36(Compendex), Dolatshahi, K. M. (2011). "Numerical and experimental modeling of masonry structures," PhD, University at Buffalo, Buffalo. Gambarotta, L., and Lagomarsino, S. (1997a). "Damage models for the seismic response of brick masonry shear walls. Part i: The mortar joint model and its applications." Earthquake Engineering and Structural Dynamics, 26(Compendex), Gambarotta, L., and Lagomarsino, S. (1997b). "Damage models for the seismic response of brick masonry shear walls. Part ii: The continuum model and its applications." Earthquake Engineering and Structural Dynamics, 26(Compendex), Hashemi, A., and Mosalam, K. M. (2007). "Seismic evaluation of reinforced concrete buildings including effects of masonry infill walls." University of California, Berkeley, Berkeley. Kadysiewski, S., and Mosalam, K. M. (2009). "Modeling of unreinforced masonry infill walls considering in-plane and out-of-plane interaction." University of California, Berkeley. Lourenco, P. (1996). "Computational strategies for masonry structures." Delft Universty Press, Netherlands. Oliveira, D. V., and Lourenco, P. B. (Year). "Implementation and validation of a constitutive model for the cyclic behaviour of interface elements." Computational Mechanics in Portugal, April 17, April 19, 2003, Elsevier Ltd, Portugal, Portugal, Page, A. W. (1978). "Finite element model for masonry." 104(Compendex),