Seismic Analysis of Masonry Buildings

A Three-Dimensional Cyclic Meso-Scale Numerical Procedure for Simulation of Unreinforced Masonry Structures

Three-dimensional (3D) cyclic analysis and constitutive material model are needed to better understand the behavior of unreinforced masonry (URM) buildings under earthquake excitations. So far, most of the existing constitutive material models applied to the field of masonry structures have focused on two-dimensional modeling and monotonic loading. In addition, most of the studies have used implicit dynamic procedures in the time domain. Based on the inherent features of implicit formulations for nonlinear problems, a number of iterations are required at each time step to achieve convergence, which leads to intensive computations and lack of convergence in some cases such as cyclic loadings.

In this paper, a 3D cyclic constitutive material model implemented within an explicit analysis procedure is proposed, which can be used to model large deformation behavior of masonry walls. A rigorous constitutive material model is proposed and validated with available experimental data from previous researches, and for the attributes for which experimental data is not readily available; a number of new experimental tests has been conducted by the authors. The material model is implemented in a user-defined subroutine and compiled with ABAQUS (VUMAT). The subroutine is then tested by several numerical examples on a single element under cyclic normal and transverse deformations to examine the behavior of the material model. Moreover, several analyses are conducted and the numerical results are compared with experimental data to assess the robustness and predictive capabilities of the proposed material model and the numerical solution algorithms.

Bidirectional Behavior of Unreinforced Masonry Walls

Most of the studies related to the modeling of masonry structures have by far investigated either the in-plane or the out-of-plane behavior. However, seismic loads mostly impose simultaneous in-plane and out-of-plane demands on load-bearing masonry walls. Thus, there is a need to reconsider design equations of unreinforced masonry walls by taking into account bidirectional effects. The intent of this study is to investigate the bidirectional behavior of an unreinforced masonry wall with a typical aspect ratio under different displacement-controlled loading directions making use of finite element analysis. For this purpose, the response of the wall is evaluated with the increase of the top displacement having different directions. A set of 19 monotonic and three cyclic loading analyses are performed and the results are discussed in terms of the variation of failure modes and load-displacement diagrams. Moreover, the results of walls capacity in each loading condition are compared with the corresponding load in the previously developed analytical interaction curves by the authors and those of the FEMA356 formulations. The results indicate that the direction of the resultant force of the wall is initially proportional to the ratio of stiffness in the in-plane and the out-of-plane directions. However, with the increase of damage, the resultant force direction inclines towards the wall’s longitudinal direction regardless of the direction of the imposed displacement. Finally, recommendations are made for applicability of FEMA356 formulations under different bidirectional loading conditions.   

 Simplified Interaction Curves for In-Plane and Out-of-Plane Behaviors of Unreinforced Masonry Walls

Different types of macro-elements have been proposed over the past decades to simulate the behavior of unreinforced masonry (URM) structures under seismic loads. In many of these macro-elements URM walls are replaced with beam elements with different hysteretic behaviors. The effect of out-of-plane load or change of gravity load due to the over turning moment is usually not considered in the behavior of proposed macro-elements. To investigate the importance of these factors, this paper presents interaction curves for bidirectional loadings imparted on unreinforced masonry walls. Two parameters are systematically changed to derive the interaction curves for a wall with particular dimensions, including compressive traction atop the wall to represent gravity loading, and loading angle that represents combination of in-plane and out-of-plane earthquake loading. Interaction curves are developed considering various possible failure modes for bricks and mortar, including tension, crushing and a combination of shear and compression/tension failures. The derived interaction curves show the initiation of failure of URM walls as a function of compressive traction and loading angle. Several examples are presented for URM walls with different aspect ratios to aid in understanding the effects of various parameters on the derived interaction curves. Finally for a specific case, the derived interaction curve is compared with nonlinear finite element results and FEMA356. The results show that, as a simplified method, the derived interaction curves can be used in the preliminary evaluation of URM walls under bidirectional loadings.

Out-of-Plane Strength Reduction of Unreinforced Masonry Walls Due to In-plane Damage

There are numerous studies on the behavior of Unreinforced Masonry (URM) walls in both in-plane (IP) and out-of-plane (OP) directions; however, few aimed at understanding the simultaneous contribution of these intrinsic responses during earthquakes. Undoubtedly, even a strong URM wall shows weakened capacity in the OP direction due to minor cracks and other damages in the IP direction and this capacity reduction has not yet been accounted for in seismic codes. In this study, performance of three URM walls is evaluated by several numerical analyses in terms of the OP capacity reduction due to IP displacements and failure modes. Several parameters influencing the OP capacity have been studied including aspect ratio, roof boundary condition, IP displacement and IP loading patterns. The results indicate that reduction in the OP capacity of URM walls varies from negligible in flexible roofs to very high in rigid diaphragms especially under high IP displacements. Moreover, IP loading pattern is more important in walls with higher aspect ratios due to their IP failure modes.

On the Bidirectional Loading of Unreinforced Masonry Walls: Experimental and Numerical Investigation

This paper describes the results of an experimental study that focused on bidirectional behavior of unreinforced masonry walls and on verification of the related proposed design equations. The tests were conducted in both the in-plane direction and bidirectional considering monotonic and cyclic quasi-static loading protocols. Various boundary conditions, representing possible wall-roof connections, were also considered for different walls to investigate the influence of rotation of the top plane of the wall on the failure modes. The results of the tests were recorded with a host of high precision data acquisition systems, showing three dimensional displacements of a grid on the surface of the wall. A companion paper presents the finite element modeling of the walls and compares the obtained simulation results with those of the experimental data. The experimental and numerical results are finally used to investigate the adequacy of ASCE41 empirical equations and some recommendations are made.