Keywords: Unreinforced masonry walls, macro modeling, micro modeling, in-plane loading, out of plane loading, failure mechanism, opening, aspect ratio

The aim of this thesis is to develop numerical models for evaluating the vulnerability of unreinforced masonry construction under different types of loading. Therefore, the behavior of unreinforced masonry panels under monotonic loading in both macro-and micro-scales is studied. Simulating the nonlinear behavior of the masonry wall in pre and post-peak regions and capturing its failure mechanism is the main focus of this study. First, the masonry wall in the panel is substituted by two simple bars using the so-called macro-element strategy and a tri-linear behavior is proposed to assess the ultimate strength of the wall as well as its response before and after peak. The lack of information about the failure mechanism of the masonry wall and relation between the failure mechanism and mechanical properties of the bar elements in this type of modeling lead to another description of this structure namely micro-modeling strategy. In this strategy, units and mortars are modeled separately and all inelastic behavior of the masonry wall is supposed to happen in mortars. Hence, special attention is paid to development of a reliable description of material properties for these elements using an accurate constitutive law.

Three dimensional representation of a masonry wall in this work enhances the capability of existing methods to predict the masonry behavior under both in-plane and out-of-plane loadings. Firstly, failure envelopes including tension cut-off and the Mohr-Coulomb yield surface are assigned to interface elements in GEFDyn finite element software. Then, the elstoplastic constitutive law is improved by adding a compression cap to the yield surfaces in order to include compressive failure of masonry in the interface element. In the new model, softening behavior for tensile and compressive strength as well as cohesion of mortar is considered. The ability of both models to reproduce the pre-and post-peak behavior of the masonry wall is verified by comparing the numerical results with experimental data. The importance of defining the compression failure of masonry by limiting the shear strength of the wall with its compressive strength is shown by comparing the obtained results with those of a real test. The results showed that the second model is capable to simulate the behavior of masonry wall with a good accuracy. Then, the effect of initial stresses and geometrical properties of the wall such as opening and aspect ratio and material properties of the mortar like its cohesion, tensile strength and compressive strength, on lateral strength and failure mechanism of the masonry walls are demonstrated. Moreover, in order to comprehend failure characteristics damage indexes based on the total length of cracks in different rows and columns of mortars are introduced and compared for different configurations.

The lengths of sliding in horizontal mortars and opening in vertical ones are the most important i parameters that control the behavior of the wall. Finally, the relation between different cracking profiles and contributing material properties are summarized into a table. The applicability of the 3D model is also demonstrated in simulating the behavior of masonry walls under out-ofplane loads. The crack patterns and load-displacement paths of a wall under this type of loading, using different boundary conditions, initial loading and mortar's cohesion are studied and compared together. Obtained results indicate high dependency of the results to these parameters. Finally, the effects of the surrounding walls on performance of each individual wall and that of the set are demonstrated through the analysis of the behavior of two perpendicular walls under different loading configurations.

Contents

Abstract i 4. [START_REF]FEMA, Evaluation of earthquake damaged concrete and masonrywall buildings: Basic procedures manual[END_REF] Comparison between load-displacement curves from cyclic to monotonic loading .

4.34 Shear behavior of masonry, experimental results by Van der Pluijm (1993) [START_REF] Vd | Shear behavior of bed joints[END_REF] for different confinement stresses, with C 0 = 0.

Introduction

Masonry construction was one of the first types of structures erected by ancient civilizations which is still being used in different parts of the world. Simplicity of construction procedure, availability of the constituent materials, aesthetic, durability, easy maintenance, fire protection, and insulation properties of masonry structures have made it popular throughout history. Masonry walls are used as load-bearing and infill panels in low and medium/high rise buildings respectively. High structural strength under gravity loads and significant stiffness of masonry under lateral loads have drawn attention of structural engineers to examine their structural behavior. However, the research development and practice in the industry on masonry structures have not been as advanced as concrete and steel. The reason lies behind complex composition and behavior of masonry structures. Masonry is composed of units (i.e. brick and concrete) and mortar, which have distinct material properties. Unit and mortar constitute a composite structure, exhibiting a complex behavior which is not easy to predict. Conducted research on masonry structures has been very limited, compared to concrete and steel. Therefore, there is a great need for further investigation and development of computational tools and methods to assess the behavior of such structures at different scales.

Masonry structures are home to a large population of the world, especially in the developing countries. Failure of masonry structures under earthquake and wind loads, result in fatal catastrophes frequently. The heavy masonry walls do not sway with the earthquake motion and exhibit brittle behavior, resulting in crack propagation and collapse under cyclic loading. Moreover, victims of natural disasters that live in masonry structures, get buried under heavy weight of masonry wall, which may cause immediate death or hinder search and rescue operations. Hence, evaluating the vulnerability of masonry is a key issue in risk analysis. Vulnerability can be expressed as a degree of loss of an element at risk or set of elements at risk, due to the occurrence of a natural or technological phenomenon. Two key factors can be identified as essential basis of all methods of calculating vulnerability: a robust assessment of aggression hazard (earthquake, wind, flood, drought ) and an appropriate model for calculating structural damage.

Various methods are currently used to determine the vulnerability of infrastructure, individual structures and economic or social factors related to risk posed by a hazard. These methods aim to find the level of damage induced in each element at risk for a given hazard level. The choice of the appropriate method to use depends on the size of the project, the quantity and quality of data and resources. If post-event observations and experts' opinion constitute the methods used for groups of buildings, analytical simplified models and detailed analysis of structures are convenient for individual buildings.

In earthquake engineering, a more sophisticated concept, called "Performance Based Design" has been developed in recent years. This approach reveals the need to define the status of a structure in its response to seismic excitation by a level of performance which is close to the meaning of ultimate limit states and service widely used in France and abroad. The effective use of a method based on the performance of the structure requires the definition of progressive degrees of damage related to the function of a particular system in the diagnostic approach. The objective of a damage computation is therefore to evaluate the loss of stiffness of the structure based on a sufficiently detailed analysis and to estimate the characteristics of vulnerability for a given level of earthquake or other type of aggression.

Moreover, numerous important historic sites and structures are exposed to the same aggressions while being deteriorated and often poorly maintained. In addition to the evaluation of their performance, it is important to analyze rehabilitation solutions and remedial actions for existing buildings. 1

.2 Objectives of study

This study focuses on nonlinear analysis of unreinforced masonry construction using multiscale modeling. The behavior of masonry structure under monotonic loading is investigated using finite element tools in both macro-and micro-levels. First, the overall behavior of masonry infills and their response is evaluated using macro modeling strategy in which the infill panel is substituted by two struts. As this approach necessitates experimental data from full scale tests which are not always available due to their high cost and important number of parameters to explore, a numerical approach is used to develop a virtual panel -testing simulator. This sim-ulator is based on a Finite Elements 3D micro-modeling strategy in which bricks and mortars are modeled individually. Not only the failure mechanisms of masonry walls with different geometries and material properties can be assessed by this approach but also both in-plane and out-of-plane loadings can be taken into account. The main objectives of this study are:

to develop a simple and suitable constitutive law for macro models, capable to represent the behavior of masonry infill walls in post-peak regime, to verify and develop an accurate micro-model in 3D which is able to predict the behavior of masonry wall before and after damage and give a deep insight of failure mechanisms of unreinforced masonry walls under in-plane loading. The developed constitutive law should be able to predict different failure modes and ultimate sustainable loads of the masonry with a reasonable agreement with experimental data, to investigate the relation between failure mechanism of the wall and geometrical and material properties of masonry walls, to introduce damage indexes which illustrate failure mechanisms of masonry structures under monotonic loads, to find a relation between material properties and cracking profile in the wall, to predict the behavior of masonry walls under out of plane loads, considering the effect of boundary condition and geometrical configurations on response of masonry structures.

In Chapter 2, a general overview of different strategies which have been implemented for numerical modeling of masonry construction are reviewed. First, the research development on modeling masonry infill panels using simple macro elements and the associated constitutive laws are summarized. Then, characterization of micro-modeling of masonry walls discussed. In this method, units and mortars are modeled separately using solid and interface elements. In this strategy, bricks are supposed to remain unchanged and all non linearity is assumed to happen in the mortars. Hence, defining an appropriate constitutive law which is able to reproduce different types of failure in this composite material is the main issue of this strategy. Therefore, different failure envelopes which have been implemented by researchers in addition to the advantages and disadvantages of each model are discussed. Then, the other types of models which have been used to predict the behavior of masonry wall under cyclic loading are presented.

Chapter 3 deals with the numerical modeling of masonry infill panels using macro elements. In this chapter, different failure mechanisms of an infill masonry panel are studied. Then an experimental test which has been carried out by CSTB (Centre Scientifique et Technique du Batiment) [START_REF] Brgm | Dynamiques concept, CSTB, Une approche mecanique d'evaluation de la vulnerabilite sismique des maconneries EVSIM[END_REF] is modeled using four different techniques from the literature and the deficiency of each model is discussed. Afterward, a tri-linear constitutive model to represent masonry infill wall under monotonic loading is proposed and validated by reproducing the results of two different experimental tests. Since this type of modeling can just reproduce overall behavior of the wall and is not able to predict the failure mechanisms of walls with different geometrical and material properties, a more accurate model is suggested to be developed for capturing failure characteristics of unreinforced masonry walls.

Chapter 4 introduces a 3D model of unreinforced masonry wall in which units and mortars are represented separately. In this chapter, the performance of masonry walls under in-plane loads is studied using the numerical tool. The nonlinear behavior of a masonry wall is concentrated in the mortars which are modeled using interface elements. First, a simple model which includes tension and shear failure of the joints is used. The deficiencies of the model is verified by comparing the numerical results with experimental data. Then, a more realistic model which comprehends tension, shear and compression failure of masonry is developed. In this model, shear strength of the wall is limited by compression strength of masonry through a straight line. Hence, multisurface yield function comprises tension cut-off, a Mohr-coulomb friction law and a linear compression cap. The softening behavior for tensile strength, cohesion, and compressive strength of masonry is considered. Validation of the model is proved by comparing the obtained numerical results with experimental evidence available in the literature. The importance of considering compression cap in the yield surface is demonstrated by comparing the obtained results of the first and second models. The effect of opening, aspect ratio and material properties on formed failure mechanism of the wall are studied here. Moreover, the length of cracks in different rows and columns of mortars is measured to introduce a damage index associated with different types of failure mechanisms.

Chapter 5 is allocated to study of the behavior of masonry walls under out-of-plane loading. In this chapter first, two masonry walls with different boundary conditions are subjected to outof-plane loading. The failure mechanism, crack's profile and ultimate strength of the walls are compared and the behavior of masonry wall under this kind of loading is analyzed. A parametric study has been carried out to find the relation between different material properties and damage indexes. Then, two perpendicular walls are modeled and subjected to uniform displacements in x and y directions and rotation. Analysis of the behavior of a set of walls gives a more accurate insight about the difference between in-plane and out-of-plane strength of the walls and assesses the influence of surrounding walls on total performance of a structure. Finally the influence of the size of constituent walls on general behavior of the system is pursued. The manuscript ends with a general conclusion and perspective giving some proposition for future work.

Chapter 2

Literature review 2.1 Introduction

Masonry is the art of building construction with units which are bounded together by the mortar. The simulation of such constructions is difficult because they are composed of two different materials with distinct directional properties such as units (clay bricks, concretes or stone) and mortar. Anisotropic behavior of bricks, dimensions of unit and mortar, aspect ratio, arrangement of bed and head joints, properties of unit mortar interface bond and quality of working, among others are some factors that make masonry modeling difficult. The reliability of numerical modeling highly depends on extracting correct mechanical properties of masonry components from experimental data as well as an accurate definition of boundary conditions and geometry of structure.

Masonry walls are implemented in structural systems in two different circumstances: as infill in steel or reinforced frame structures or as main load bearing element in masonry constructions. In the first system, masonry wall is surrounded by beam and columns and whenever the system is subjected to different loads, interaction between infill and frame would be important.

Experimental results show that masonry existence has a significant influence on the behavior of structural system especially under seismic loading. Therefore, masonry stiffness, strength, seismic behavior as well as its interaction with surrounding frame should be considered to have an accurate estimation of these systems. But in the second system, the behavior of masonry components namely units and mortars and their interactions are important. For assessing the behavior of masonry wall in each of these two systems, depending on the desired simplicity, level of accuracy and application field [START_REF] Schueremans1 | Continuous Assessment of Historic Structures A State of the Art of applied Research and Practice in Belgium[END_REF], two finite element modeling strategies are used to investigate the behavior of masonry constructions which are namely macro modeling and micro modeling approaches (Figure 2.1 and Figure 2.2).

Macro modeling approach used to simulate the behavior of infill masonry frames is based on representation of an infill by either a single or multiple compressive equivalent diagonal strut, see Figure 2.1(b) or an equivalent orthotropic continuous model. The main purpose of such simplified modeling is that it takes into account global behavior of a structural system without modeling all components and all possible local failure modes. This strategy is based on physical understanding of infilled frame's behavior so a few elements are chosen to represent the overall behavior of the system. Using an equivalent diagonal bar was the first time proposed by Poliakov [1957] [START_REF] Polyakov | Masonry in Framed Buildings. An Investigation into the Strength and Stiffness of Masonry Infilling[END_REF].

He suggested that a panel can be replaced by a diagonal strut in compression. [START_REF] Holmes | Steel frames with brickwork and concrete infilling[END_REF] [48] adopted his suggestion and replaced the infill with an equivalent pin-jointed diagonal strut with the same material and the same thickness as the infill. Later, many researchers improved the model. The behavior of infill masonry frames have been studied by changing diagonal strut model's variation such as using single or multiple diagonal struts, [START_REF] Stafford-Smith | Behavior of Square Infilled Frames[END_REF], [START_REF] Bertoldi | Analytical models in infilled frames[END_REF], [START_REF] Smyrou | Implementation and verification of Masonry panel model for nonlinear dynamic analysis of Infilled RC frames[END_REF], material properties namely the elastic behavior of diagonal bar [START_REF] Mainstone | On the stiffnesses and strengths of infilled frames[END_REF], [START_REF] Liauw | Nonlinear behavior of non-integral infilled frames[END_REF] and inelastic behavior of equivalent strut [START_REF] Reinhorn | Modeling of masonry infill panels for structural analysis[END_REF], [START_REF] Rodrigues | Simplified macro-model for infill masonry panels[END_REF].

When masonry wall is simulated by macro modeling strategy, the properties of units, mortar and interface between mortar and bricks are smeared out in one continuum isotropic or an-isotropic material with a relation between average masonry stress and strain. A complete macro model can reproduce an orthotropic material with different tensile and compressive strengths along the material axes as well as different inelastic behavior for each material axis [START_REF] Lourenco | Continuum model for masonry: Parameter estimation and validation[END_REF]. A great number of studies have used homogenization technique to identify an appropriate constitutive law including main information related to behavior of composite material [START_REF] Pande | Equivalent elastic moduli for brick masonry[END_REF], [START_REF] Zucchini | A micro-mechanical model for the homogenisation of masonry[END_REF] and [START_REF] Zucchini | micro-mechanical homogenisation model for masonry: Application to shear walls[END_REF]. In this method behavior of composite in elastic and inelastic regions is obtained from mechanical characteristics of its components namely blocks and mortar joints and the geometry of the assemblage [START_REF] Calderini | A micromechanical inelastic model for historical masonry[END_REF]. This modeling strategy is applicable when the dimension of a structure is large enough so that the relationship between average stress and average strain is acceptable [START_REF] Chaimoon | Modelling of unreinforced masonry walls under shear and compression[END_REF].

On the other hand, in micro-model strategy, Figure 2.1(a), for simulating infill frames' behavior, the structure is divided into different types of elements, including brick, mortar, interface between brick and mortar, interface between masonry panel and frame and the frame elements. Hence, in a micro model representing masonry, surrounding frame and infill wall components are established using a numerical method such as finite element method (FEM) or discrete element method (DEM). Then, the behavior of structural system is assessed in detail and all the possible failure mechanisms can be reproduced [START_REF] Stavridis | Finite-Element Modeling of Nonlinear Behavior of Masonry-Infilled RC Frames[END_REF], [START_REF] Mehrabi | Finite element modeling of masonry-infilled RC frames[END_REF]. Intensive computational effort is the limitation of this strategy that makes the implementation of this strategy for analysis of large structures, difficult.

The first finite element approach to model infill frames was suggested by Mallick et al [1967],

[1971] [START_REF] Mallick | The behaviour of infilled frames under static loading[END_REF], [START_REF] Mallick | Effect of openings on the lateral stiffness of infilled frames[END_REF], defining an appropriate interface element to take into account the slip between frame and infill. Different works have been done using different techniques to evaluate behavior of infill in terms of micro-modeling [START_REF] Liauw | New development in research of inlled frames[END_REF], [4], [START_REF] Dayala | Realistic shear capacity assessment of inll frames: comparison of two numerical procedures[END_REF]. Generally, infill walls were represented by 4 or 8-node rectangular elements and frames by beam elements and in some of the models, interaction between frame and models was simulated by interface elements [START_REF] Koutromanos | Numerical modeling of masonryinfilled RC frames subjected to seismic loads[END_REF]. In spite of its accuracy, this method is not useful for modeling of practical cases such as multi-bay or multistory structures. This type of modeling is very useful for calibrating simplified modeling parameters.

Micro modeling is probably the best way to understand the failure mechanism of masonry constructions. In this method bricks and mortars are modeled separately. Depending on the accuracy of the model, two different types of this approach are implemented: detailed-micro modeling and simplified micro-modeling. In detailed approach units and mortars are represented by solid continuum elements while interface between brick and mortar are represented by discontinuous interface elements which are planes of failure and slipping. In this type of modeling, for taking into account brittle behavior of blocks and non-linearity of joints, a finite element model with very fine mesh is required, Ignatakis et al. [1989] [49]. The accuracy of modeling is very high but the calculation time is the main issue with this method [START_REF] Fouchal | Contribution to the modelling of interfaces in masonry construction[END_REF]. In the second approach, bricks are modeled by solid elements whereas the properties of mortar and interface between mortar and units are lumped to the discontinuous interface elements. This model is normally used to capture basic failure modes of masonry during loading and local behavior of each component. Because of complexity of modeling, this approach is efficient for small size models and not useful for modeling of large structures because of large number of elements and cost of calculation.

Macro modeling of infill masonry frames

Considerable amount of research work has been done on implementation of macro-modeling strategy considering plasticity theory to assess the behavior of masonry wall subjected to monotonic and cyclic loads. This approach is applied for numerical modeling of large scale structures in which global behavior of the structure is important. Disadvantages of this method includes the inability to produce different failure modes and need for comprehensive experimental data to determine the properties of equivalent material.

Macro-modeling strategy for infill masonry frame implements a single global structural member, composed most often of equivalent diagonal struts instead of masonry panel. Infill masonry at high lateral load deforms in shear mode while surrounding frame represents deformation in flexural mode. So, to model the composite action between infill and frame, researchers have suggested that a diagonal strut with appropriate geometry and material characteristic is substituted with infill masonry. This method called equivalent diagonal strut approach: El-Dakhakhni et al. [2003] [30] also modeled infill by three struts. They proposed a simple method for estimating the lateral stiffness and lateral load capacity of infilled masonry steel frames which crushes in the corners as well as the internal force in steel frame member. Nonlinear behavior of both masonry infill (crushing) and steel frame (formation of plastic hings) were accounted by this method. In their method three struts one diagonal and two off-diagonal are substituted masonry infill whose force-displacement diagram was based on orthotropic behavior of masonry infill. A simplified trilinear stress-strain relation was accounted for masonry infill, For modeling the multi-story frame with infilled masonry, Decanini et al. [2004] [27] replaced each panel by two struts which were inactive in tension and just worked in compression. However combination of both struts provides a resistance against lateral loads. Hysteresis model which was adopted for masonry panel is shown in Figure 2.5. The influence of mechanical characteristic of infill was investigated by using different types of masonry with different strengths. Based on the results of studies, they showed that the global nonlinear seismic behavior of masonry infill frames can be obtained by using simple model which combines a shear type model with equivalent strut elements. Crisafulli et al. [2007] [24] developed a macro model based on mutli-strut formulation. A 4-node panel element which connected to the frame at beam-column joints was implemented to take into account compressive and shear behavior of masonry separately. Two parallel struts and a shear spring in each direction were used for taking into consideration the infill stiffness and masonry's strength. The configuration of model considers the lateral stiffness of the panel and the strength of masonry during shear failure or tension diagonal failure. As the contact length between panel and the frame decreased by load increment and cracks formed in the masonry infill, the area of equivalent strut supposed to decrease by increment of lateral displacement on the system, Figure 2.6(b). The model was able to reproduce different shear failures which were observed for masonry infills. The model was also suitable to use in large structural frames. The main limitation of this model is that as panel element was connected to frame in beam-column joints, the bending moment and shear forces of the frame could not be obtained. The previous models had 2 dimension configurations, so they were not capable to take into account the out-of-plane behavior of masonry. [START_REF] Hashemi | Seismic Evaluation of Reinforced Concrete Buildings Including Effects of Masonry Infill Walls[END_REF] [46] developed a 3D model in which interaction between frame and panel as well as in-plane and out of plane behavior of infill were taken into account. A three-dimensional strut and tie model that takes into account both in-plane and out of plane strength of infill under bi-directional loading was used. The model was composed of eight compression-only struts which were connected together with a tensiononly link element at the center of the infill panel. Results have shown that existence of in-plane forces cause decrease in out-of-plane capacity of infill whereas out-of plane forces also decrease in-plane capacity of infilled frame. [START_REF] Puglisi | Modeling of masonry of infilled frames Part I: The plastic concentrator[END_REF] showed that the conventional equivalent strut does not correspond to the real behavior of wall. The real infill is a unique element but the strut model strategy replaces masonry by two independent bars. So, a simple modification in diagonal strut system was proposed so that to consider coupling between two bars. A plastic concentrator was included at intersection of the bars to capture inelastic behavior of the wall. It also links the two bars to account for the transfer of forces between them. Presence of plastic concentrator gives a more

Puglisi et al. [2009]

Micro modeling of masonry walls

A lot of research has been done by using simplified micro modeling to investigate behavior of masonry wall under different load condition. Different researchers have developed different models to elaborate performance of masonry wall under monotonic and seismic loading. Most of research has been done to investigate the in-plane nonlinear behavior of masonry and is studied in 2 dimensions. Therefore, these models are unable to reproduce the out of plane behavior of the wall. As one of the most critical issues about the performance of the unreinforced masonry construction against seismic loading is its out-of plane behavior and great damage which may happen because of its reaction, a real and accurate model must be able to model both in plane and out of plane behavior of the wall simultaneously under monotonic and cyclic loading.

Page [1978] [82], was the first one who used micro-modeling strategy to investigate in-plane behavior of clay masonry. Brick element supposed to be elastic linear whereas nonlinear behavior of masonry occurred in mortars which were modeled by linkage elements. Low tensile strength, shear strength depending upon degree of present compression stress, nonlinear deformation for tension and shear and high compression strength were the characteristics of linkage elements. The failure criterion in tension was a linear failure envelope while a bi-linear failure criterion was used for shear, see Figure 2.10(a). If tensile or shear failure criterion was violated, failure occurs in the joint. His model was capable to reproduce nonlinear behavior of masonry and local joint failure. Softening behavior for compression was taken into account by changing shear stiffness depending on present compressive stress. Test on deep masonry beam under vertical load was used to validate numerical analysis. Stress distribution was captured but the failure load could not be evaluated because no failure was considered for brick.

Another simple micro-model was proposed by Chiostrini et al. [1989] [19], in which bricks were assumed to be elastic isotrop and gap-elements were used to model mortars. These gap elements were opened progressively due to tension failure of masonry. Tangent stiffness matrix also was changed because of geometrical non linearity. Gap condition means when gap elements open or close or when material behaves nonlinear. In their study analytical response highly depended on the position of gap elements and there were some free parameters which were chosen to reproduce experimental results. Sliding of bed joint could be seen and the bricks were not involved in their model. A complex procedure was used to simulate tensile failure in tension. Interface elements were removed when tension cracks appeared in the model and shear stiffness was assigned to zero in case of shear sliding. Lourenco et al. [1993] [59] considered gradual softening behavior after crack or slip in the model. They proposed a model which includes continuum elements as brick either elastic linear or with Von-Misses plasticity to capture compression failure in bricks and line interface elements as a plane of weakness where shear and tension damage were assumed to take place in. Softening behavior was considered for tension and shear in interface elements. Failure surface for interface consisted of two parts: parabolic tension cut-off for tension part and Coulomb failure envelope for shear with softening behavior for cohesion. A first model with elastic behavior for bricks was considered. The model was capable to show crack pattern and qualitative behavior of the experiment but as brick crushing was not taken into account after diagonal cracking load continued to increase with displacement increment. Then Von-Misses criterion for brick crushing was added into the model and analysis showed compression in bricks was limited but calculated failure load was still too high. They concluded that there is no relevant difference between the parabolic criterion and tension cut-off and dilatancy angle has a great influence on behavior of wall.

Masonry units were modeled using smeared crack elements and mortars were represented by interface element in [START_REF] Lotfi | Interface model applied to fracture of masonry structures[END_REF] [57] finite element model. The dilatancy which was observed in experiments was modeled using a dilatant interface element for mortars. In this model, the dilatancy decreased by normal stress increment and the rate of dilatancy reduced by increasing the cumulative tangential displacements. Tension and compression failures of units were obtained by using Von-Misses plasticity model in combination of Rankine tension cut-off failure surface.

A three-parameter hyperbolic yield function was considered for interface element to reproduce initiation and propagation of fracture in mortar under combination of normal and shear stresses, Figure 2.11(b). This yield function was used to provide smooth transaction between Mohr-Coulomb criterion considered for shear failure and tension cut-off assigned for tension failure. Softening behavior for shear-tension as well as in tension-compression was considered for mortars and units respectively. The comparison between experimental and analytical model showed that failure mode, ductility and crack pattern provided well, the load carrying capacity of masonry evaluated well but under higher normal loads, there was some discrepancy at the final stage of loading because of deficiency of smeared crack model and uncertainty in material parameters for mortar joints.

A sophisticated micro-model was developed by Lourenco et al. [1997] [61] which considered compression failure of masonry. In their model, bricks were assumed to remain elastic whereas a multisurface yield surface including all failure mechanisms in masonry was considered for mortars via zero thickness interface elements. Multisurface yield failure consisted of tension cutoff envelope for tension, Mohr-Coulomb failure criterion for shear and a cap for compression. Moreover an interface element was suggested in the middle of bricks as potential crack plane to capture tension failure of bricks. The main characteristic of the model was cap envelope which limited combination of shear and compression stress in the element. By adding this limitation, unit diagonal tension as well as masonry crushing was included in the model. Softening behavior was also considered for tension, shear and compression. There was a good agreement between experimental data and computational analysis in term of capturing load-displacement diagram in both pre-peak and post-peak regions.

To study the effect of dilatancy of material and roughness of contact, Giambanco et al. [40] implemented a micro-model for masonry with fully elastic bricks and inelastic interface to predict the behavior of mortar joints in masonry structures. A material dilatancy parameter and the geometrical dilatancy related to roughness of contact were introduced into the model to study effect of roughness at structural level. Moreover a bilinear yield function comprising Mohr-Coulomb criterion for shear and tension cut-off was adopted to evaluate the shape of contact during sliding and cohesion loss process. The compression failure of material did not take into account in their model. Softening behavior was assumed for cohesion and tensile strength. The roughness of surface was also taken into account to describe the change of surface shape during sliding and loss of cohesion. So, the angle α (Figure 2.13(b)) was added to friction angle in shear yield function formulation. This angle depends on the configuration of contact surface and cohesion loss during the loading. It was shown that the roughness of contact surfaces and its evolution have a great influence on failure mode, crack pattern, ductility and post peak branch of load-displacement diagram. In case with no roughness the cracks appeared just at top and bottom of the wall and there was a clear difference between load displacement curves in postpeak region while a good agreement was obtained between results of experiments and analytical model with asperity in term of crack pattern. They concluded that introduction of brick crushing and/or joint compaction will establish the limit load. Lower bond limit analysis is an alternative to conventional finite element method for elaborating collapse load of masonry structures. Sutcliffe et al. [2001] [101] developed a new technique for calculating the lower bond limit load to investigate the behavior of masonry wall under in-plane loading. Two assumptions were made to use this method: first, the materials were assumed to be perfect-plastic, flow rule was associated with no strain hardening or softening and second, all changes in geometry of structure and deformations were negligible. In order to prevent nonlinear restrictions, linear yield surfaces were used for brick and mortars. Two separate yield function were defined for bricks and mortars. Bricks were supposed to be isotropic and homogeneous with Mohr-Coulomb yield failure curve applicable to considering tensile failure of units. A tri-linear yield surface composed of tension cut-off for tensile failure, Mohr-Coulomb for shear failure and a linear approximation of cap model of Lourenco et al. [1997] [61] for compression failure also was assumed for mortars, Figure 2.14. In theirs study softening behavior was not taken into account. Two examples were used to investigate the effectiveness of results. They showed that the model can capture collapse load with good agreement to experimental data, but the failure mode and crack patterns could not be achieved by their model.

Modeling of masonry under cyclic loads

All the researches explained above, study the in-plane behavior of unreinforced masonry shear wall under monotonic loading using micro modeling strategy. Due to the complexity of be- havior of masonry under cyclic loading few studies on the behavior of these constructions are available. Nonlinear behavior of masonry under monotonic loads consists of cracking, crushing, sliding, tension and compression and shear softening behavior but there are some other complex behaviors which should be predicted for masonry model under cyclic loading. Reloading and unloading rules, stiffness degradation in tensile and compressive regimes, energy dissipation at each cycle and between cycles, crack closure in compression and plastic strain at zero stress level are the other parameters which should be considered in masonry constitutive laws in case of cyclic loading so the extension of monotonic constitutive model to cyclic one is very intricate. Different strategies and models were used to evaluate the behavior of such construction under cyclic loading. Marco modeling, micro modeling and equivalent frame systems are the strategies which have been proposed by different researchers to assess these phenomena.

Using equivalent frame system to investigate the seismic behavior of masonry has been explored long time ago. Different researchers have used this method to carry pushover analysis and assess the behavior of such construction. In this system, masonry wall is discretized by piers and spandrels which are modeled using 2 node macro element, considering nonlinear behavior of material by defining an appropriate constitutive law. The shear and compression failure modes are considered in the model. Researchers have shown that this model can obtain overall behavior of masonry building in a reasonable fashion [START_REF] Karantoni | Effectiveness of seismic strengthening techniques for masonry buildings[END_REF], [START_REF] Magenes | Simplified non-linear seismic analysis of masonry buildings[END_REF], [START_REF] Belmouden | An equivalent frame model for seismic analysis of masonry and reinforced concrete buildings[END_REF] and [START_REF] Roca | Strength capacity of masonry wall structures by the equivalent frame method[END_REF]. Different strategies have provided to overcome the limitation of this method such as defining the reduced stiffness of the masonry, the compatibility between the walls and connecting transverse members, and then it was extended to analysis of 3D systems [START_REF] Magenes | A method for pushover analysis in seismic assessment of masonry buildings[END_REF] too.

Gambarotta et al.

[1997] [START_REF] Gambarotta | Damage models for the seismic response of brick masonry shear walls. Part I: The mortar joint model and its applications[END_REF] used a composite model to reproduce the behavior of masonry shear walls under in-plane cyclic loads. They applied a constitutive model which takes into account the damage of mortars together with brick and mortar interface decohesion due to opening and frictional sliding. Bricks were modeled by isoparametric elements which were connected by interface elements. Friction of mortars was limited by definition of a friction limit and their damage by introducing the damage conditions as shown in Figure 2.18(a). The brittle behavior in tension, energy dissipation due to friction, and stiffness degradation under compression were the characteristics of mortar joint model. Softening behavior was assumed for tensile and shear in mortar joint. Constitutive law for bricks was composed of two parts: Elastic part and post-elastic region. In post-elastic region the tensile fracture of bricks and compressive failure of masonry were reproduced. Tension failure of brick were captured by putting the interface elements in the middle of bricks while an elastic limit condition based on Von-Misses criterion assumed for brick to predict compression failure of masonry as a consequence of different transverse deformation in brick and mortar joints. Figure 2.17 shows the model response to tensile and shear strains as well as cyclic shearing strains. The model was validated by simulating two experimental tests. It was shown that the proposed model was capable to reproduce the stiffness of walls and capture the maximum lateral load as well as the hysteresis behavior of wall during cyclic loading. Different failure mechanisms also were distinguished separately. By the way the proposed model was complicated for modeling the large masonry walls with opening. Gambarotta et al. [1997] [38] then extended their model [START_REF] Gambarotta | Damage models for the seismic response of brick masonry shear walls. Part I: The mortar joint model and its applications[END_REF] to evaluate cyclic behavior of masonry by using a continuum model in which masonry wall was made of two layers, bed joint and brick and head joint. A constitutive formulation based on homogenization process was applied to the bricks. Decohesion and slipping was assumed for bed mortar interface while damage and failure of brick were considered for bricks and head joints. Model was improved by considering compressive strength of masonry and shear strength of brick see Specific separate constitutive law involving hysteresis behavior under axial and shear deformations was assigned to elements so that to reproduce post-elastic behavior such as mechanical deterioration and hysteresis energy dissipation. Coulomb-like law also was adopted to relate strength of shear spring to vertical axial loads.

This approach is effective for time history analysis of large masonry structures and reduces computational effort significantly. Constitutive law for individual springs was provided to satisfy following assumption for masonry: low strength, brittle response and rapid mechanical degradation in tension, exhibition of maximum bearing capacity with limited ductility, progressive mechanical degradation and energy dissipation after peak in compression and dependence to normal stresses and significant energy dissipation for repeated cycles in shear. They showed adoption of a discrete model consisted of rigid body and spring to model behavior of masonry under cyclic loading has some advantage in term of simplicity over a continuum model especially for defining the degradation of material under cyclic loads. The proposed model deals with uniaxial stress state at once so less computational effort needs to do for dynamical analysis of masonry. Karapitta et al.

[2011] [START_REF] Karapitta | Explicit finite-element analysis for the in-plane cyclic behavior of unreinforced masonry structures[END_REF] developed a smeared crack constitutive model to investigate the behavior of unreinforced masonry under monotonic and cyclic in-plane loading. Macro-modeling approach was considered in their study since masonry was supposed to be homogeneous with different strengths along and normal to bed joints and cracks are smeared over the areas of finite elements. Three constitutive laws for describing tension, shear and compression damage based on uni-axial stress and equivalent total strain were implemented in the model. The stiffness degradation depending on direction of loading, unloading and reloading was evaluated in the analysis. Tensile and shear stress strain curves were composed of linear elastic and softening regions, whereas linear elastic, strain hardening and strain softening were characterized for compression in masonry.

The proposed unloading-reloading regimes considered for tension, compression and shear were shown in Figure 2. [START_REF] Chrysostomou | A Six-Strut Model for Nonlinear Dynamic Analysis of Steel Infilled Frames[END_REF]. Results showed that model is capable to predict the most important charac- teristics of masonry subjected to cyclic loading such as hysteresis behavior, stiffness degradation and energy dissipation. Some improvements such as dependence of shear to compression, considering effect of Poisson ratio and coupling of tension and compression damage were proposed by authors.

In this study the behavior of masonry as infill and as shear wall will be investigated in scale of macro and micro, respectively. At first, the behavior of infill masonry panels is studied by macro-modeling strategy. In macro element modeling, two compression-only struts elements are representing masonry. The results of monotonic loading experiments on masonry infill panels which have been done by CSTB (Centre Scientifique et Technique du Btiment) are used to calibrate the diagonal strut's characteristics. In this approach, having knowledge about the behavior and failure modes of the wall is very necessary for calibrating model parameters.

As laboratory tests are costly and their results highly depend on condition of tests, micromodeling strategy is used to evaluate behavior of masonry wall under different types of loading, wall's dimensions and mechanical properties. In second part, mechanical properties of bricks and joints are considered separately. Masonry wall is modeled in 3D using solid elements as bricks whereas mortars are modeled using zero-thickness interface elements. A constitutive law is defined for material so that to simulate the behavior of masonry shear wall. Numerical results are validated by making a comparison with experimental data carried out by Raijmakers and Vermeltfoort [1992] [85], Vermeltfoort Raijmakers [1993] [START_REF] Vd | Shear behavior of bed joints[END_REF] and CUR [1994] [START_REF]Structural masonry: a experimental numerical basis for practical designrules[END_REF]. In third part, the behavior of masonry wall under out-of-plane loads is investigated. A masonry wall and two perpendicular walls are subjected to out-of-plane loads and their failure mechanisms are studied. Finally, the micro-model is improved to reproduce the response of masonry walls under cyclic loading.

Conclusion

The development of numerical modeling of masonry structures using macro and micro model strategies has been reviewed in this chapter. The accuracy of models to assess the nonlinear behavior of masonry infills using the macro elements with different geometries and mechanical properties has been evaluated. Moreover, different constitutive laws proposed by several researchers for interface element representing nonlinear behavior of masonry wall have been appraised. Studying the development of different strategies for modeling masonry construction and considering their advantages and disadvantages in this chapter, gave a good insight about the more important parameters which should be considered in the modeling and enable the author to propose the more convenient models for representing masonry in both macro and micro levels.

Chapter 3

Macro modeling 3.1 Introduction

Steel or reinforced concrete frames with masonry panels are one of the most popular types of constructions in seismic regions all over the world especially in residential and commercial buildings. Masonry panels typically consist of brick, clay tile or concrete blocks and mortars and are constructed between beam and columns of a frame. These panels are used for interior partitioning as well as exterior enclosure. They are not considered as structural elements in design process, so their stiffness and strength are ignored whereas global response of infill frames can significantly be affected by presence of infill.

Masonry wall increases the stiffness and strength of infilled frame so its performance is not the same as bare frames. Increasing the rigidity of a system will change the natural frequency of the structure therefore the applied load on structure may increase or decrease dramatically depending on the seismic spectrum values at the closeness of the bare structure natural period. Therefore masonry panel increases stiffness and strength of structural system and introduces more modes of failure mechanism associated with wall failure and wall-frame interaction. Existence of masonry infill also can change redistribution of stresses that can make unpredictable damage along frames.

A lot of research activities including experimental or analytical works have been done to investigate the behavior of infill masonry panel on frames and its influence on performance of structural system. The simplest and more practical way to simulate behavior of infilled frames has been done by macro modeling. In this strategy, masonry infill is replaced by a single or multiple compression-only diagonal struts with appropriate geometrical and mechanical characteristics.

In this section, first, different failure mechanisms of infilled frames are discussed briefly. Then, the results of CSTB tests [START_REF] Brgm | Dynamiques concept, CSTB, Une approche mecanique d'evaluation de la vulnerabilite sismique des maconneries EVSIM[END_REF] are presented and the equivalent strut model is identified to model masonry infill. Three different methods which are used in literature are selected to calibrate the characteristic of equivalent diagonal strut. Then a trilinear model is proposed as constitutive law to simulate masonry behavior. The methods are:

• FEMA 356 model [START_REF]Pre-standard and Commentary for the Seismic Rehabilitation of Buildings[END_REF] • Calibrated model proposed by Hashemi and Mosalam [2007] [46]

• Mostafaei and Kabeyasawa model [2004] [76]

• Tri-linear proposed model

The details of each model will be discussed in the following sections.

Behavior of infill frames

Modeling of the behavior of infill wall is a very complex issue due to important role of interaction between infill and masonry on performance of this construction. In most cases, when in-plane lateral loads are applied at one top of an infill frame, a diagonal truss-formed strut is formed from top loaded corner to the opposite bottom corner to resist it. In moderate loads, the infill separates from surrounding frame at non-integrated parts and in the opposite direction infill acts as diagonal, Figure 3.1. As load increases, depending on mechanical properties of infill and frame and their interaction, failure occurs finally either in infill or frame. Tension failure in windward column or shear failure in columns or beams are the most common types of fracture in the frame. If frame has sufficient strength to prevent these failures, load increment causes failure in infill masonry. For infill frames made of concrete, failure is initiated by cracking along diagonal compression and ends to collapse by crushing or crushing near loaded corners. The same phenomenon appears in masonry made of brick in addition to an alternative possibility of shear failure along mortar joints. Different failure modes for infill masonry frames can be classified into five distinct modes [5]:

• Corner crushing: This type of failure is associated with the crushing of infill at least at one loaded corner. This failure happens when masonry infill has low compressive strength. Corner crushing failure is frequently observed.

• Diagonal compression: This mode of failure is represented by crushing the infill within its central region. It can be seen in infill with high slenderness ratio that experiences out-of-plane buckling instability under in-plane loading. This type of failure occurs rarely in practical panels because in these cases, infill thickness is designed to satisfy acoustic isolation and fire protection requirement. Inertial forces in out of plane direction of wall also can cause this type of failure. The combined effect of in-plane and out-of-plane loading decrease strength of the wall in both in-plane and out-of-plane directions which increases the probability of wall failure in both directions [START_REF] Mosalam | Chapter 23: Seismic Analysis and Design of Masonry-Infilled Frames[END_REF].

• Sliding shear: This failure happens in panels with weak mortar joints in comparison to the masonry units as well as in panels with low to medium aspect ratio whose shear stress dominates its normal stress. Horizontal sliding along bed joints is the characteristic of this type of failure. This type of failure is widely seen. • Diagonal cracking: Is in the form of a diagonal crack which connects the top loaded corner to bottom opposite corner. This type of failure is experienced in frames with weak joints and infill with high compressive strength or in weak frames. Some researches do not consider this phenomenon as failure mode because the infill carries additional load after diagonal cracks. • Frame failure: When infill panel has high compressive strength, applied load is carried out by infill and transferred to surrounding frame and causes failure in columns. If in design process the effect of strong infill on surrounding frame is not considered, this type of failure is inevitable. This failure mode may also associate with frame with weak joints. It should be noted that the first two modes of failure namely corner crushing and shear sliding are important in practical point of view.

Macro element of code TREMURI

Lagomarsino et al. [2004]

[54] developed a model based on a macro mechanical element representative of the in-plane behavior masonry panels to simulate the cyclic and dynamic responses of masonry wall as well as entire buildings. They performed several types of analysis such as pushing progressive, incremental non-linear, incremental dynamic and modal, using a finite element code developed at DICAT (Dipartimento di Ingegneria delle Costruzioni dell'Ambiente e del Territorio, University of Genoa). The results of these analyzes were compared with those obtained by quasi-static tests on a prototype of a masonry building on a real scale (Magenes and Calvi [1997] [68]). In addition, [START_REF] Calderini | A micromechanical inelastic model for historical masonry[END_REF] [?] developed a damage model for numerical analysis of historical masonry buildings. Masonry is considered as a composite material manufactured by assembling bricks periodic orthogonal joints connected by mortar.

Masonry is considered as a composite material constructed by regular assembling of bricks which are connected by mortars.

Masonry elements subjected to dynamic actions have two distinct behavior in the plane of the wall: the slipping due to shear, cracking due to shear and bending. These two main failure modes have been confirmed by actual observations after earthquakes and experimental tests performed on masonry walls. Most frequently failure mode for these structures is formed by the diagonal cracks due to shear forces. This failure mechanism is developed as a combination of vertical and horizontal loads when the major principal stresses in the wall become greater than the tensile strength of the masonry. The flexural strength of a panel in masonry is conditioned by the crushing of the compressed portion.

The macro-element proposed by [START_REF] Gambarotta | On dynamic response of masonry panels[END_REF] [39] allows, using a reduced number of degrees of freedom to represent the two main failure modes of masonry structures: shear (taking into account the friction) and flexion. This approach is the main feature of macro element used in the calculation software TREMURI. Using internal variables, this model takes into account the development of damage to the interior of the element, by the deterioration of rigidity and decrease in resistance. The main simplifying assumptions of the method are the following: the structural elements are the walls and floors (or vaults). The floors are considered as rigid elements plans. The behavior of floors in bending and out of plane behavior of walls are not taken into account (out of plane behavior of the walls is considered negligible compared to the in plane behavior). TREMURI model is used as the base model to perform preliminary analysis and to verify the developed macro-element in this work. A post-processor damage has also been developed in this context.

Experiments conducted at CSTB

At first step, the results of CSTB tests are used to define and calibrate the numerical model. Figure 3.7 shows the test set-up which was used to apply lateral in-plane loads on masonry panel. Two types of loading were available: monotonic and cyclic. Force was applied to the corner of the panel and transferred through its length by a hamlet whose connection with the rest of the wall is badly identified. In addition, confinement was provided by tendons which were located on both sides of the wall. Tension of tendons was not measured. These tests were conducted using 37 walls, corresponding to 11 different categories of construction. Before each test, dimensional control and simple compression test were performed on blocks. The results of CSTB tests for wall number 3 areused as input data. The properties of

Numerical modeling of masonry

As shown in Figure 3.11, the masonry wall is replaced by two diagonal struts. So the section area obtained in each method is divided by 2 for each strut. A 2D model of the frame is developed whose beam and columns have concrete rectangular sections. Structural elements are beam and columns with nonlinear hysteresis bending moment-end rotation characteristics. Diagonal struts are represented by bar elements with non-linear behavior.

The geometrical and mechanical properties of each bar are determined using the methodology described for each model. The modulus of elasticity of masonry, the compressive strength of the blocks and the shear strength of mortar joint are the main parameters. The monotonic displacement is imposed to the top corner of the frame. The same displacements are imposed to the two top nodes in the model because of integrity action of the wall. The symmetric stress-strain curve with same resistance in tension and compression is proposed to model as masonry material property.

As mentioned before, the geometrical and mechanical characteristics of struts are obtained by 4 different methods and the results of numerical analysis are compared to experimental data to gain an insight how much the accuracy of each model is. First, strut characteristics will be obtained by FEMA 356 [START_REF]Pre-standard and Commentary for the Seismic Rehabilitation of Buildings[END_REF] formulation as described in the following section.

FEMA 356 proposed strut model

According to FEMA 356 [START_REF]Pre-standard and Commentary for the Seismic Rehabilitation of Buildings[END_REF] provision, an infill can be replaced by a single diagonal strut whose thickness and modulus of elasticity are the same as the infill ones. In the proposed model, the section of strut is equal to the thickness of strut times the strut width a which is calculated by Equation A.

θ = arctan(h inf /l inf ) (3.3)
In above formulations, h col is the column height between centerlines of the beams, r inf is the diagonal length of infill panel, E me is the expected modulus of elasticity of infill wall material, t inf is the thickness of infill panel and equivalent strut, θ is the angle whose tangent is the infill height-to-length aspect ratio which is calculated by Equation 3.4, E f e is the expected modulus of elasticity of frame material, I col is the moment of inertia of columns around the axis perpendicular to the loading direction and h inf is the height of infill wall. As the modulus of elasticity of infill material was not measured, the proposed equation by UBC code (Section 2106.2.12.1 1997 UBC) is used:

E m = 750.f ′ m < 3000ksi (3.4)
Hence, the compression strength of the masonry prism is needed for calculating the modulus of the modulus of elasticity of masonry infill. f ′ m is determined by the equation recommended by Paulay and Priestly [1992] [84]:

f ′ m = f ′ cb (f ′ tb + α. f ′ j ) U u (f ′ tb + α. f ′ cb ) (3.5)
Where: α = j 4.1×h b and U u is stress non uniformity coefficient and is equal to 1.5. f ′ cb is compression strength of brick and is equal to 9.3M P a, f ′ tb is tension strength of brick and may be determined for solid brick [START_REF] Paulay | Seismic Design of Reinforced Concrete and Masonry Buildings[END_REF] as 0.1 f ′ cb = 93M P a, f ′ j is mortar compression strength and is equal to 17.9M P a, j is mortar joint thickness which is assumed 15mm and h b is the height of masonry unit and is equal to 20cm. Parameters are calculated as: α = 0.0183, f ′ m = 7.09M P a, E m = 5325M P a, λ = 2.881 and a = 0.344m. The calculations are represented in detail in Appendix A. The expected in-plane shear strength of the unreinforced masonry infill wall is given by Equation A.6:

V ine = A n .f vie (3.6)
A n = t inf .l inf (3.7)

f vie = ν me = 0.75 [ν ts + P CE A n ] (3.8)
Where f vie is expected shear strength of masonry infill, ν ts is the average bed-joint shear strength, P CE is the expected gravity compressive force applied on the URM infill wall and A n is the area of net mortared section of the masonry infill wall and is equal to A n = 0.743m 2 . As there is no information about shear strength of the joints and applied normal stress on it, a value is supposed for shear strength of masonry infill. ν ts + P CE An is assumed to be 620kP a, ν me = 496kP a,V ine = 368.03kN . FEMA 356 provisions, provide a maximum nonlinear drift ratio d which is correspond to the sudden loss of the lateral strength of the unreinforced masonry walls in nonlinear analysis. The value of drift is is determined depending on the height of the infill, aspect ratio and frame to infill shear strength ratio. To define the value of ultimate strain for masonry material ǫ mu shown in Figure 3.12, the value of maximum nonlinear drift ratio is used. Based on FEMA Table 7-9 for the wall aspect ratio equal to 1.43 and the frame to infill strength ratio less than 0.7, the maximum nonlinear drift ratio is obtained: d = 0.35% β < 0.7 ⇒ l inf /h inf = 3.71/2.6 = 1.43 ⇒ d = 0.35% Using above calculated parameters, the material and geometric properties of the proposed diagonal strut model, namely the area of the diagonal struts A strut , yield strength of masonry f my , its corresponding strain ǫ my and ultimate strain of masonry can be calculated as:

A strut = t inf .a
(3.9)

f my = V ine A strut . r inf l inf (3.10) ǫ my = f my E me (3.11)
ǫ mu = d. h inf .l inf r 2 inf (3.12)
By substituting the value of each parameter in above formulation, the proposed stress-strain curve for material and the strut's area are obtained as: A strut = 0.0688m 2 , f my = 6.51M P a, ǫ my = 0.0012, ǫ mu = 0.001645.

Here, 2 diagonal struts are used for simulating masonry infill. The area proposed by FEMA, is divided by two and its value is allocated to each diagonal bar. Displacements are imposed to the top corners of the frame. The value of displacements increases monotonically from zero to 2cm in both two top nodes of frame and here the total displacement is 4cm. Relation between top displacements versus base shear of diagonal struts is acquired. Figure 3.13(a) shows the strain-stress curve which is defined as material properties of struts. The Figure 3.13(b) demonstrates pushover curve of the wall. As shown in this figure, black dashed line is the response of the frame, the blue dashed one is the response of the masonry wall (diagonal struts) and the red one is the total response of the system which is obtained from summation of infill and frame responses. The solid blue line is the response of CSTB test number 3. As can be seen in Figure 3.13(b) the FEMA 356 model overestimates the shear strength of the system and could not be able to predict the residual strength of the system. It should be noted that the values of normal stress and bed joint shear strength were not available and the assumed values may have led to overestimating the ultimate strength. Initial stiffness of masonry infill which is calculated by FEMA formulations is very lower than that of the experiment. This is maybe due to underestimating some parameters like modulus of elasticity of masonry. This parameter was calculated based on UBC code. According to FEMA 356 formulations, nonlinear behavior of the wall is perfect plastic and after a certain displacement, the infill masonry panel exhibits no strength in shear and collapse. Figure 3. 13(b) shows that this assumption is highly conservative and wall carries out considerable amount of shear stress in plastic region.

Calibrated model proposed by Hashemi [46]

As the results calibrated by FEMA 356 were not satisfactory, another methodology which proposed by Hashemi et al. [2007] [46] is used. The calibrated model is based on compression material properties namely maximum compressive strength of masonry f ′ m which is obtained from uniaxial compression test on masonry wall and a parabolic stress-strain curve with linear softening. In this modeling, the parameters of strut model are calibrated to be in accordance with the observed load-displacement curve of experimental results. Masonry infill strength V ine , its corresponding lateral drift ∆ y , the residual strength V res and its corresponding lateral displacement ∆ res are obtained from the base shear-displacement curve of wall 3, see Figure 3

A strut = V inf f ′ m (3.13) ǫ mo = l inf r 2 inf .∆ y (3.14) f mu = V res A strut (3.15) ǫ mu = l inf r 2 inf .∆ res (3.16)
By substituting the value of each parameters the area of strut model as well as material model are calculated: A strut = 0.0508m 2 , ǫ mo = 6.25 × 10 -4 , f mu = 2.87M P a, ǫ mu = 0.0014 and Push-over curve obtained from calibrated model formulation has a more realistic shape in plastic region respecting the FEMA 356 one. But difference between calculated maximum load and that obtained from the test is still high. The maximum normal stress in the strut assumed to be equal to the compressive strength of masonry. The area of struts is computed based on the ratio of shear strength of masonry infill in yield point and compressive strength of masonry and thus the modulus of elasticity of masonry in this case is approximately 2 times greater than those of FEMA 356. This method can predict the total schema of masonry behavior well, but underestimates the strength of the structure in both elastic and plastic regions. Hence, these equations underestimate the response of the masonry wall. By the way, the formulation is based on the pushover curve results which can be obtained from experiment data. Hence to calculate the model's parameters, we need to run the test but the aim of numerical modeling is to calculate the response of structure without performing the experiment.

f ′ m = f ′ mo = 7

Mostafaei and Kabeyasawa model [76]

In this section constitutive law for material is based on the formulation of [START_REF] Mostafaei | Effect of Infill Masonry Walls on the Seismic Response of Reinforced Concrete Buildings Subjected to the 2003 Bam Earthquake Strong Motion : A Case Study of Bam Telephone Center[END_REF]. Figure 3.16 shows the force-displacement curve which is applied to the masonry infill panel. In this study like others masonry is replaced by two struts and their combination in two opposite directions produces the lateral stiffness of the wall. The section of bar elements is determined based on FEMA 356 formulations. Shear strength at assumed yielding point V y , shear strength at maximum point V m as well as post-peak residual shear strength V p and their corresponding displacements, U y , U m and U p are the main parameters of this trilinear envelope. α in the curve represents the ratio of stiffness after yielding to the initial stiffness. The difference of this method with those of FEMA 356 and calibrated is that in this case, maximum strength of masonry is determined based on the minimum of compression failure and sliding shear failure values of masonry wall. Then, yield strength of the wall and its corresponding displacement are determined based on the geometry of Figure 3.16. The plastic strength of the masonry wall and its corresponding displacement are calculated based on some assumptions. Finally, forces and displacements are converted to stress and strain and applied to the bar elements.

V c t.l m = a.t.f ′ m . cosθ t.l m (3.17)
Thus: Vc t.lm = 0.44M P a. Maximum shear strength of infill in shear sliding failure mechanism is calculated as:

V f t.l m = τ 0 (1 -µ.tanθ) (3.18)
Where τ 0 is cohesive capacity of the mortar beds and it ranges typically between 0.098 and 1.4M P a and may be assumed typically as τ 0 = 0.04.f ′ m [START_REF] Paulay | Seismic Design of Reinforced Concrete and Masonry Buildings[END_REF] and µ is sliding friction coefficient along the bed joints and is determined based on Chen et al. [2003] [18] results as:

µ = 0.654 + 0.000515f ′ j (3.19)
Where f ′ j is the mortar block compression strength. Based on these formulations, the sliding shear failure mechanism can be obtained as:

V f
t.lm = 0.5306M P a. Therefore, the corresponding shear strength is calculated as the minimum of sliding and compression diagonal strength: V m = 0.44 × 0.2 × 3.71 = 0.326M N . The maximum displacement corresponding to the maximum lateral force is estimated by Equation 3.20, Mandan et al. [1992] [START_REF] Madan | Modeling of Masonry Infill Panels for Structural Analysis[END_REF]:

U m = ǫ ′ m .d m cosθ (3.20)
In this equation ǫ m is the masonry compression strain corresponding the maximum compression stress: ǫ m = 1.33 × 10 -3 and d m is the diagonal strut length. So the value of maximum lateral displacement is calculated as: U m = 0.0074m. The initial stiffness K 0 can be obtained by [START_REF] Madan | Modeling of Masonry Infill Panels for Structural Analysis[END_REF] [65] formulation:

K 0 = 2.( V m U m ) (3.21)
The lateral yielding force V y and its correspond displacement U y may be calculated from the geometry as shown in Figure 3.16, α supposed to be 0.2.

V y = V m -α.K 0 .U m 1 -α (3.22) U y = V y K 0 (3.23)
The parameters are calculated as: K 0 = 89.19M N/m, V y = 0.2475M N , U y = 0.2475 89.19 = 0.0028m. The U p and V p are calculated so that the line connecting the peak of the envelope and the point (U p ,V p ) passes through the 80% post-peak point [START_REF] Mostafaei | Effect of Infill Masonry Walls on the Seismic Response of Reinforced Concrete Buildings Subjected to the 2003 Bam Earthquake Strong Motion : A Case Study of Bam Telephone Center[END_REF]. Therefore:

V p = 0.3V m (3.24) U p = 3.5(0.015h m -U m ) + U m (3.25)
V p = 0.099M N , U p = 0.118m. In this method the area of the strut is determined according to FEMA 356 formulation so: A strut = 0.069m 2 . The proposed stress-strain envelope is obtained by:

σ m = V m A strut .cos(α) (3.26) σ y = V y A strut .cos(α) (3.27) σ p = V p A strut .cos(α) (3.28) α = tan -1 (2.7/4.21) = 0.5703 rad , l d = √ 2.7 2 + 4.21 2 = 5.256m. ǫ m = U m .cos(α) l d (3.29) ǫ y = U y .cos(α) l d (3.30) ǫ p = U p .cos(α) l d (3.31)
So, σ m = 5.68M P a, σ y = 4.26M P a, σ p = 1.7M P a, ǫ m = 0.0012, ǫ y = 0.000484 ǫ p = 0.0018. Material model is shown in Figure 3.17(a). Pushover curved obtained by this method is also shown in Figure 3.17 The performance curve shows that this model is able to predict the value of maximum shear resistance of system better than previous methods but it overestimates the correspond displacement of this point. The accuracy of this model is more than the two described before. The stiffness of this model like that of FEMA 356 and calibrated model is underestimated. In this method, the stiffness of the wall is calculated based on shear strength of the masonry and its corresponding displacement which is determined from its compressive strength. Since the value of compressive strength was not available and obtained through EquationA.5, the stiffness of wall may have been underestimated. The plastic response of the wall is overestimated using this method. That's because of the plastic region formulations.

Tri-linear proposed model

In this section, a tri-linear model for describing the material properties of replaced struts is proposed. The model is validated by simulating two experimental tests on masonry walls which have been done by CSTB [START_REF] Brgm | Dynamiques concept, CSTB, Une approche mecanique d'evaluation de la vulnerabilite sismique des maconneries EVSIM[END_REF] and Vermeltfoort [1992] [START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF]. In first part, the formulation is implemented to model CSTB test and the results of four cases are compared. Then, the proposed force-displacement curve is applied to model another experiment which has been done by Vermeltfoort [1992] [START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF].

In this method, the area of strut element is determined by FEMA 356 formulations. The maximum shear strength of masonry is distinguished as the shear sliding strength of the masonry wall. V y and its corresponding displacement are determined based on proposed material envelope as shown in Figure 3.18. In this method, V p is assumed to be 50% of maximum shear strength and U p is determined to be in accordance with the slope of the third line of proposed curve as shown in Figure 3.18. The same slope in second and third line of shear force-displacement curve is assumed.

Validation of proposed model, CSTB test

In this section, the proposed model is used to simulate the experimental test of CSTB. The proposed stress-strain curve and the numerical results are described in Figure 3.19(a) and Figure 3.19(b) respectively. The compression strength of masonry is calculated using Equa-tion A.5 like previous cases. The calculations are described in detail in Appendix A. The parameters are computed: E m = 6.36 × 10 9 P a, λ = 3.226, a = 0.3287m,

A strut = 0.0657m 2 , V m = V ini = 0.33M N , U m = 0.0047m, K 0 = 143.6M N/m, V y = 0.244M N , U y = 0.0018m, V p = 0.
165M N , U p = 0.0106m, σ m = 6.65M P a, σ y = 4.99M P a, σ p = 3.33M P a, ǫ m = 0.00075, ǫ y = 0.000288 and ǫ p = 0.0017.

Comparing the obtained results using different models

In this section, the properties of each model are summarized in Table 3.1 to compare better the results. The computed load-displacement curves in addition to tested one are drawn in Figure 3.20. Comparing the properties of different models shows that both area of strut and modulus of elasticity are important. For example the modulus of elasticity in calibrated model is two times greater than those of Mostafaei but since the section of bars is different for these two models, the same initial stiffness was obtained for both models. Determining post-peak region and plastic strength of the wall helps to have more reliable information about the performance of the wall under loads. In this study, the V p = 0.5V m was proposed and a good agreement between computational model and experimental data was found. In the macro-modeling strategy, the wall is replaced by two diagonal struts. The section of struts is calculated based on FEMA formulations. A steel frame with rectangular section equal to 5 × 10cm 2 , is supposed to surround the infill masonry. The modulus of elasticity of beams and columns are supposed to be the same as that of CSTB. Hence, the area of struts is calculated as: λ = 9.5 =⇒ a = 0.097m =⇒ A strut = a.t = 0.1×0.097 ∼ = 0.01m 2 . The cohesion of mortars, initial normal stress on the wall, compression failure of masonry together by friction angel are known and are equal to C = 3.5e5P a, σ n0 = 3e5P a, f ′ m = 10.5e6P a and φ = 36.9 o . The maximum shear sliding force of the wall can be computed by Equation 3.33:

V m = V ini = C + σ n0 .tan(φ).t inf .l inf (3.33)
So the parameters are calculated as:

V m = 5.4e4P a, E m = 900.f ′ m = 9.45GP a, ǫ ′ m = f ′ m Em = 0.001, θ = atan(1/0.99) = 45.3 o , U m = ǫ ′ m .dm cos(θ) = 0.002m, K 0 = 2 × ( 5.4e4 0.002 = 54M P a, V y = 5.4e4-0.2×54×10 6 ×0.002 1-0.2 = 40500P a, U y = 40500 54×10 6 = 0.0007m, V p = 0.5 × 5.4 × 10 4 = 27000P a, U p = 27000
0.2×54×10 6 + 0.002 = 0.0045m. The yield, maximum and plastic stresses in addition to their corresponding displacements are: σ y = 5.7M P a, σ m = 7.6M P a, σ p = 3.9M P a, ǫ y = 0.00034, ǫ m = 0.001, ǫ p = 0.0023.

The stress-strain curve of struts is shown in Using macro-model strategy, the masonry wall is replaced by two strut elements. The struts have the same properties in compression and tension. The properties of proposed model are calculated based on Figure 3.18 and corresponding equations. The ultimate shear strength of the wall is determined by this method but there is no other information about the formed cracks in the wall and the failure mechanism of the wall which are very important to know.

Conclusion

In this section, the masonry wall was modeled using macro-modeling strategy. In this method, the infill masonry is replaced with two diagonal struts. First, the masonry wall which was tested by CSTB [START_REF] Brgm | Dynamiques concept, CSTB, Une approche mecanique d'evaluation de la vulnerabilite sismique des maconneries EVSIM[END_REF] was modeled using two bar elements. The material and geometrical properties of the bar elements were calculated using three different methods from the literature: FEMA356 [START_REF]Pre-standard and Commentary for the Seismic Rehabilitation of Buildings[END_REF], calibrated model proposed by Hashemi et al. [2007] [46] and Mostafaei et al. [2007] [76]). Then a tri-linear model was suggested and successfully reproduced the experimental results.

Then the proposed model was used to simulate the response of a shear wall which was subjected to monotonic loading by Vermeltfoort [1992] [85] Vermeltfoort and Raijmakers [1993] [START_REF] Vd | Shear behavior of bed joints[END_REF] and CUR [1994] [25]). The computational results show that the trilinear model is able to properly assess the response of the masonry walls. However, data relating to the compressive strength of the masonry, the shear strength of the bed joints, the vertical load acting on the wall are necessary.

The trilinear proposed model, with its basic assumptions, is more suitable to simulate the behavior of infill panels. Rotation is not considered in this macro model so the flexural behavior of masonry panel cannot be simulated by this model. On the other hand, knowing the mechanism of failure is the main issue of researches on masonry structures and macro-modeling strategy as presented here is not capable to give any information about that. Therefore, in next sections, the behavior of masonry wall in detail using micro-models will be studied. Failure mechanism of the masonry wall will be assessed using different models considering different geometry and material characteristics.

Chapter 4

Micro modeling : in plane loading

Introduction

Unreinforced masonry wall (URM) term is devoted to masonry wall which contains no reinforcement. These types of walls are used in exterior walls to bear the loads or in interior wall to separate the volumes. Traditionally, URM using clay brick has been implemented to resist against different loads in low rise construction almost all over the world. Even today, buildings using unreinforced masonry wall are constructed in many areas because of its durability, resistance against fire, simplicity to construct and low construction cost but they have poor performance under seismic loads. Therefore, special attention has been given to evaluate their performance under different conditions by many researchers either by experimental tests or numerical models to allow more reliable use of masonry.

As masonry walls are composed of two different materials with their own special characteristics, the prediction of their performance under different load conditions, different geometries and different material characteristics are very difficult. To get a comprehensive insight into the behavior of these walls, a large number of laboratory tests involving various conditions are required. As, obtaining accurate responses from laboratory tests is costly and time consuming and needs precise testing instruments, sophisticated numerical tools have been developed to investigate the behavior of these complex constructions. As described before, one of the main approaches used to simulate the behavior of masonry walls is micro modeling strategy. Micro models try to describe the behavior of masonry in detail and capture local failure of each component and catch final failure mechanism of the wall. Thus, mortars and bricks and their interfaces are modeled separately using appropriate constitutive laws.

In this section attention will be given to micro modeling of the unreinforced masonry wall in 3D configuration. The model is composed of unit elements (bricks), which are considered as solid and rigid elements, and interface elements (mortars) as potential crack planes. Mortars are considered as weakness planes of wall so that interface elements control nonlinear behavior of masonry wall whereas bricks remain elastic. First, possible failure modes of masonry are studied. Then, a simple constitutive law is implemented for the interface to simulate reaction of wall under monotonic loading. Afterward, the computational model is improved to take into account all failure mechanisms of the wall. The effect of opening, aspect ratio and properties of mortars on behavior of masonry walls are also investigated in this chapter.

Masonry failure mechanisms

The basic types of failure mechanisms in masonry wall are characterized as unit failure mechanism, joint failure mechanism and combined mechanisms involving joints and units. Five different failure mechanisms are allocated to masonry as: tension failure of bricks, tension failure of joints, shear failure of joints diagonal tensile failure of unit and crushing failure of masonry. Figure 4.1 shows these phenomena: cracking of the units in direct tension is dealt with unit's failure, cracking of the joints and sliding of the bed or head joints (at low value of normal stress) are considered as joint's failure and diagonal tensile cracking which is happened under sufficient normal stress and masonry crushing are identified as combined failure mechanisms [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]. • Diagonal tensile failure: When the wall is subjected to high normal compressive loads and significant horizontal force is applied to the wall. This type of failure is the most common mode of failure in shear masonry walls. It usually happens for the wall with aspect ratio of 1, but it can also occur in a panel with higher aspect ratio (which is defined as the ratio of height and length of the wall, H/L) when high vertical loads also are applied to the wall. Diagonal crack pattern happens when the principal tensile stress in the mortar exceeds the tensile strength of joints.

• Sliding shear failure: This type of failure is predominant for the wall with high lateral loads and low normal stresses. When there is no sufficient compression to produce higher shear strength and the cohesion of mortar is not enough to resist lateral loads, bed joints slide and failure of wall happens. This type of failure generally happens in walls with low aspect ratio, for example H/L = 1/1.5 and often in walls with H/L = 1/1 depending on the characteristics of their components.

• Crushing failure: Crushing of masonry at the toes of wall occurs usually due to rocking deformation whenever masonry has high shear strength and is subjected to low lateral displacement. This mode of failure is most common for the walls with aspect ratio higher than one, for example H/L = 1.5/1. In the cases with low vertical loads, the wall can rock like a rigid body and finally experience crushing failure. As aspect ratio of the wall can cause a specific mode of failure, its influence on the behavior of masonry wall is discussed in this chapter using micro model of masonry.

The aim of this study is to capture all failure mechanisms of masonry walls with different geometrical and material properties under in-plane loading and to find the most important parameters which influence the behavior of masonry wall. Pushover curve and crack's profile of each analysis are drawn to compare results better.

Numerical modeling of masonry wall

GEFDyn Software [START_REF] Chouvet | GEFDyn: Logiciel d'Analyse de Comportement Mecanique des Sols par Elements Finis avec Prise en Compte du Couplage Sol-Eau-Air, Manuel scientique[END_REF], is used for modeling masonry walls and for pre and post processing of results, SDTools (Structural dynamic toolbox) software is implemented [94]. GEFDyn can be used for 3D or 2D analysis of structures and geo-structures under static or dynamic loading in linear and non-linear domain taking into account coupling of mechanical, hydraulic and thermal phenomena. It is developed for geotechnical problems with numerous non-linear rheological constitutive laws dedicated to geo-materials and specially the soil and its interaction with structures [START_REF] Chouvet | GEFDyn: Logiciel d'Analyse de Comportement Mecanique des Sols par Elements Finis avec Prise en Compte du Couplage Sol-Eau-Air, Manuel scientique[END_REF]. SDTools specializes in providing open and extensible MATLAB based solutions in experimental modal analysis and finite element modeling for vibration problems [94].In this study, the masonry wall is simulated using micro-model strategy which considers bricks and interface between mortars and bricks separately. The results of numerical modeling are validated comparing the obtained results with available experimental data.

For modeling of masonry wall, two types of elements are used. 3D solid elements for bricks and interface elements for mortars which are shown in The behavior of mortars and tension failure of bricks are represented by interface elements. This type of element permits discontinuity in displacement field. The behavior of this element is written in terms of stress and relative displacement across interface element. 2D eight nodes interface element used in this modeling is shown in Figure 4.4. Interface element formulations are described in the following section.

Interface element formulations

In finite element modeling, the interaction between two bodies can be described by [START_REF] Giambanco | Numerical analysis of masonry structures via interface models[END_REF]:

• Link element: connects two opposite nodes of element in contact [START_REF] Herrmann | Finite element analysis of contact problems[END_REF] • Thin element: continuum finite element with very fine thickness [START_REF] Griffiths | Numerical modeling of interfaces using conventional finite elements[END_REF] • Zero thickness interface element: allows displacement discontinuity between contacted nodes [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF] Interface element was first used by Goodman et al. [1968] [41] to describe discontinuity in rock masses and Page [1978] [82] was the first one who implemented interface element in modeling masonry walls. In this study, properties of mortar and interaction between units and mortars are lumped into interface element so they are source of nonlinearity in the model. An artificial thickness is considered for interface in this study to define stress state of element. The behavior of interface element is described by a relation between stresses and relative displacements between two opposite points of an element.

Interface field:

In interface element, n is normal vector from surface 1 to surface 2.

Displacement jump:

Displacement which is read in interface element is the difference between displacements of surface 2 and surface 1.

[u] = [u 2 ] -[u 1 ] (4.1)
Normal and tangential components of displacement are defined as:

[u n ] = [u].n (4.2) [u T ] = [u] -[u n ].n (4.3) |u T | = [u s ] 2 + [u t ] 2 (4.4)
Where u s and u t are the sliding components in the plane of interface.

Stress components:

Normal and tangential components of stress vector read as below:

σ n = n.σ.n (4.5) σ T = σ.n -σ n .n (4.6) σ T = (σ 2 t + σ s ) 1/2 (4.7)
Stress vector always remains continue between interface element and the two solids even when there is opening under principle of action and reaction. Therefore, there are only discontinuities in displacement field.

Simple yield surface for interface elements (Model I)

In this section the constitutive law which is implemented for the interface elements is described.

In Model I, all tension and shear failures has been assumed to take place in interface elements. Hence, the yield function is composed of a opening criterion and the plasticity criterion for describing the tension and shear behavior of masonry joints. The Mohr-Coulomb failure criterion is written in the context of non-associated plasticity. The yield function read:

f 1 = σ 2 t + σ 2 s + σ n .tanφ -C (4.8)
Where C and φ are cohesion and friction angel of joint respectively. The dilatancy angle defines as the ratio of plastic opening of interface over sliding of interface. The dilatancy angle is a function of relative plastic shear displacement and normal confining stress. By increasing the values of these two quantities, the dilatancy angle tends to be zero. This phenomenon is confirmed by experiments. The potential function which specifies the direction of plastic flow g 1 is represented in Equation 4.9:

g 1 = σ 2 t + σ 2 s + σ n .tanψ (4.9)
In tension region, whenever displacement in the element overpasses a certain value, the interface element opens and tension, shear and compression stresses in element become zero. This certain value called artificial interface thickness, e which actually is yield displacement. There is no limitation for compression stress in the interface element and it is calculated by multiplying relative displacement times the modulus of elasticity of interface element. The formulations related to the behavior of interface element in tension are represented below:

• No interpenetration between two solids: [u n ] ≥ -e ⇒ σ.n = E.∆u n
• Detachment and free surface:

[u n ] > e ⇒ σ.n = 0 • Contact and compression: [u n ] < e ⇒ σ.n = E.∆u n
Where e is thickness of interface element and F t is tensile strength of mortars:

e = F t E (4.10)
Hence:

f 2 = 0 if σ n > 0&[u n ] > e (4.11)
In elasto-plastic approach total displacement's increment in interface element can be decomposed into an elastic part and a plastic part:

[∂ t u t ] = [∂ t u e t ] + [∂ t u p t ] (4.12) [∂ t u s ] = [∂ t u e s ] + [∂ t u p s ] (4.13) [∂ t u n ] = [∂ t u e n ] + [∂ t u p n ] (4.14)
Increment of stress vector depends on relative elastic displacement as shown in Equation 4.15, Equation 4.16 and Equation 4.17 where G and E are shear and normal elastic rigidities of the interface.

[∂ t σ t ] = G t [∂ t u e t ] = G t ([∂ t u t ] -[∂ t u p t ]) (4.15) [∂ t σ s ] = G s [∂ t u e s ] = G s ([∂ t u s ] -[∂ t u p s ]) (4.16) [∂ t σ n ] = E[∂ t u e n ] = E([∂ t u n ] -[∂ t u p n ]) (4.17)
We have associated flow rules, therefore for each failure mechanism, the plastic displacement in the element obtained as:

Mechanism 1 : shear [∂ t u p t ] = λ 1 . ∂g 1 ∂σ t = λ 1 . σ t |σ T | (4.18) [∂ t u p s ] = λ 1 . ∂g 1 ∂σ s = λ 1 . σ s |σ T | (4.19) [∂ t u p n ] = λ 1 . ∂f 1 ∂σ n = λ 1 .tanψ (4.20)
In above formulation, ψ is supposed to be zero in interface elements in this study. λ is plastic multiplier and calculated by Equation 4.21.

Mechanism 2 : tension or compression

f 2 = 0 if σ n > 0
No limitation and yield function is considered for compression σ n < 0 Shear mechanism is active when:

f 1 = 0, ḟ1 = 0: ḟ1 = 0 ⇒ G. σs σ T [∂ t u s -λ 1 σs σ T ] + σt σ T [∂ t u t -λ 1 σt σ T ] + E.tanφ[∂ t u n -λ 1 tanψ] = 0 λ 1 = G (σ T .[∂ t u T ]/|σ T | + E [∂ t u n ] tanφ/(G + E tanφ tanψ) = 0 (4.21)
Given λ 1 , increment of stresses can be calculated in interface elements from the following equations.

∂ t σ t = G t [∂ t u t -λ 1 σ t /|σ T |] (4.22) ∂ t σ s = G s [∂ t u s -λ 1 σ s /|σ T |] (4.23) ∂ t σ n = E [∂ t u n -λ 1 tanψ] (4.24)

Numerical Validation

In micro modeling strategy mechanical properties of different constituents involved and the behavior, geometry and material property of each component define the global behavior of structure while in the other models such as continuous approach or macro-element strategy the identification of parameters needs a good understanding of the overall behavior of infill or wall that is very difficult to achieve. Therefore, the approach of using numerical simulations instead of laboratory tests can be interesting.

The micro-modeling strategy for masonry wall is validated by a comparison between GEFDyn results and experimental results available in the literature. The results are compared to shear wall tests carried out in the Netherlands at 1992 [START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF], [START_REF] Vd | Shear behavior of bed joints[END_REF] and [START_REF]Structural masonry: a experimental numerical basis for practical designrules[END_REF]. This experimental data is used because of the availability of most of the parameters, which are necessary to characterize in micro model.

It should be noted that multiple experiments for masonry walls with the same settings may lead to different results [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]. This is due to variabilities and uncertainties which may occur both in the material properties or test condition. The main objective of this study is to demonstrate the ability of the numerical model to reproduce the main features of the wall behavior that were observed in the tests and not just an exact simulation of experimental results. A large number of parameters for material properties, boundary conditions and geometrical properties are necessary to develop an accurate model that fits the experimental results.

Model geometry and loading

Two types of walls were considered in the masonry shear wall tests that were carried out by CUR

Boundary conditions

The steel beams are not modeled here but the top nodes of the wall are fixed in z direction and the movement of nodes at the base is prevented in all directions.

u x = d, u y = 0, u z = 0 f orz = z max u x = 0, u y = 0, u z = 0 f orz = 0

Material properties

As mentioned before, interface element allows discontinuities in displacement and its behavior is defined by an equation between stress vector and relative displacement. Equation 4.25 demonstrates the relation between displacement and stress in interface element:

σ = D.ǫ (4.25)
Where D = {E interf ace , G sinterf ace , G tinterf ace }. As mortars are assumed to be isotropic, G sinterf ace = G tinterf ace . The elastic stiffness matrix D can be calculated from the properties of units, mortars and the thickness of mortars. In this analysis, the bricks are supposed to remain elastic and their properties do not change. So, according to uniform stress distribution and serial chain connection between components, the elastic stiffness matrix D defines as [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]:

E interf ace = E u .E m h m (E u -E m ) (4.26)
G interf ace = G u .G m h m (G u -G m ) (4.27)
Where h m is the actual thickness of mortar in tested wall, E u and E m are Young's moduli and G u and G m are shear moduli of unit and mortar respectively. Multi-surface yield surface which composed of Mohr-Coulomb shear failure and tension cut-off failure, Figure 4.6 is considered as interface element failure criterion.

The properties of bricks and mortars are given in Table 4.1. These properties are obtained from the results of tension, compression and shear experimental samples. As the modulus of elasticity of the brick is much higher than those of mortar, the bricks remain elastic and mortars act as the plane of weakness. Therefore, the nonlinear characteristic of the wall is concentrated in interface elements.

Nonlinear properties of interface elements are given in Table 4.2. The friction angle, measured by tanφ, is assumed constant, i.e. tanφ = 0.75; φ = 36.9 o . This is the (residual) value obtained by Vander Pluijm (1993) [START_REF] Vd | Shear behavior of bed joints[END_REF]. The dilatancy angle that is measured by tanψ is assumed to be equal to zero. Results from Van der Pluijm(1993) studies indicate that a normal confining pressure According to Figure 4.7, when the displacement exceeds element's thickness, the interface element will be opened. The opening in the interface elements is because of tension failure, so the thickness of interface elements is calculated as the ratio between tension strength and modulus of elasticity of interface as in Equation 4.10 and reads:

e = ft E = 2.5×10 5 8.2e9 = 3e -6
An elastic dummy stiffness is allocated to interface element in the middle of bricks and its artificial thickness is calculated based on bricks properties. f t and E here are the tension strength of brick and dummy modulus of elasticity of the interface element located in the middle of bricks.

e = ft E = 2×10 6 1e15 = 2e -9
E Brick-middle G Brick-middle f t 1.0e6 1.0e6 2.0 [M P a] GP a/m M P a
The cracking pattern obtained for an initial vertical pressure of 300 kPa and 2 mm horizontal displacement at the top of the panel is given in Figure 4.13. Note that horizontal displacement is applied to all top nodes. The whole process can be understood from these figures. As shown in In the same way, if minor principle stress in a point exceeds compression strength of brick, crushing failure would be imminent. Then, by increasing lateral displacement, shear and compression stresses in elements augment and the orientation of major principal stress changes and becomes reversed. As shown in the figure, the major principle stresses are along a diagonal, so that in these locations, tension failure happened. Thus, interface elements in correspondence to the direction of major principal stresses fail in tension and make such crack pattern in the wall.

In Figure 4.17, different failure mechanisms as well as their corresponding cracks in horizontal interface, vertical interface and brick middle interface are presented. Shear failure in interface elements is demonstrated by sliding in element whereas opening of interface element represents tension failure in that element. Moreover, opening of brick-middle interface illustrates tension failure in the bricks.

By paying attention to Figure 4.17(b) it reveals that there is sliding failure at top left and bottom right of the wall, but the diagonal pattern cracks are greater so they are predominant. .17(a) also illustrates the opening in horizontal interfaces which are located at two corners plus diagonal direction. The total crack pattern also is shown in Figure 4.17 (e).

Wall with opening

In this section the wall with central opening which was subjected to a monotonic loading by Raijmakers and Vermenltfoort [1993] [START_REF] Vd | Shear behavior of bed joints[END_REF] is modeled and validated. Force displacement diagram of the model is drawn in Figure 4.21 and compared with experimental results. It should be first noted that difference between load displacement of walls J2G and J3G is originated from different crack propagations. As shown in Figure 4.19 diagonal cracks are not experienced in the wall J3G as in the wall J2G, so wall J3G fails sooner and its collapse load is 20% lower than that of wall J2G. The response of walls in terms of load-displacement and total crack pattern are different for the same walls. This difference comes from the scatter properties of material and small number of bricks constituted the wall. Failure mechanism in wall J2G is bending failure and the wall withstand higher lateral strength while in right wall (wall J3G) an early sliding of the mortars at top of the opening is the reason of failure and lower loaddisplacement curve is obtained for this wall. The absence of limitation for compressive strength of masonry and no definition for crushing failure of masonry cause the great difference between pushover curves of experiment and the model. Hence, in order to have an accurate model a limit for compression strength should be defined. This limit should also controls the shear strength in the masonry wall. Tension failure in the bricks as well as sliding and opening of the mortars are shown in detail in

Influence of aspect ratio

A parametric study by varying the height H and width L of the wall shows different patterns of cracking. The ratio H/L made an important change in the push-over curve, the fracture mechanisms and subsequently crack patterns of the wall. In this section, two series of wall with aspect ratio less than 1 and more than 1 are studied. Therefore, the length of the wall is multiplied by 0.5, 1.5 and 2. The same values are used to multiply by the height of wall. The following sections demonstrate the results of numerical modeling for aspect ratio less than 1 and greater than 1.

(a) Aspect ratio less than 1

Pushover curve and total crack pattern of the walls with aspect ratio less than 1, are summarized in Figure 4.23. Please note that the experimental curve is that of the wall J4D and is only put in the figure to give a reference for comparison. For the case H/L equal to one or less than one, the diagonal crack pattern can be found but the start and final points of the cracks depends on the length of the wall. By paying attention to the push-over curves it can be understood that the stiffness of the wall is augmented by increasing the length of elements. The jump observed in push over curves is due to tension failure initiation in the joints and loss of continuity between elements. As can be seen in push over curve, the shear strength is increased infinitely, because no limitation is assumed for compression strength in the mortar elements. In some cases, the failure of the wall is because of crushing failure of the corners, but in this section the model is not capable to capture the compressive failure neither in bricks and nor in mortars. The push over curves and normalized push over of these cases are shown in Figure 4.24 to have an accurate comparison. As can be seen in the Figure 4.24, total tendency of the push over curve is the same. For the wall with the same aspect ratio, case z = 0.5z 0 and x = 2x 0 , the longer wall bears much greater loads while both curves have the same stiffness. In normalized push over curve as shown, for the same aspect ratio, the wall with lower length has a little more strength. As a conclusion, by increasing the aspect ratio, the stiffness and maximum shear strength increase. The cracking configuration is highly dependent on this ratio. For the case H/L equal to unity or less, the crack is diagonally in a way, but the starting point and end point of the crack depend on the length of the wall.

(b) Aspect ratio greater than 1

In this part the results of modeling for three walls with aspect ratios greater than one are presented. The first case belongs to the wall whose length is divided by 2, so it has aspect ratio equal to 2. To make a good comparison of results different cases are gathered inf Figure 4.25.

For a ratio greater than 1, the failure mechanism is because of brick crushing. The crack on the edges of the panel is dominant, but the diagonal is not clear. As there is no limit for compression

92

for mortars in this study, the push over curves increase with load increment and collapse load can not be distinguished here. Figure 4.26 shows push over curves and normalized push over curves for aspect ratio greater than 1. In normalized curve it can be found that walls with the same aspect ratio have the same pushover curve and sustain the same lateral load. Wall's stiffness has inverse relation with wall's aspect ratio. As can be seen in Figure 4.26(a) wall with aspect ratio equal to 1.5 is stiffer than walls with aspect ratio equal to 2. Overturning moment has greater value for higher wall and that causes earlier compression failure in the wall.

The results show that the aspect ratio (H/L) has a great influence on its resistance against different loads and thus on the failure mode of the wall. For the ratio greater than 1, the mechanism of grinding of bricks would be predominant, so compression failure of the toes is the predominant failure mode of the masonry. For the case with the ratio below 1, depending on normal stress and loading, horizontal sliding of the mortar would be the main mechanism of failure, but for the case with a rate approximately equal to 1, the diagonal tension is predominant.

Cyclic behavior of masonry walls

In this section, the masonry wall without opening which was tested by Raijmakers and Vermeltfoort (1992) and Vermeltfoort and Raijmakers (1993) is subjected to two types of cyclic loading in order to study the behavior of masonry subjected to earthquakes. Two types of cyclic loading are applied to the masonry wall in order to find the capability of model to capture different phenomena and failure mechanisms. The maximum amplitude of applied displacements is 2 mm. In a first test just one cycle is applied to the wall and maximum displacement is reached in primary loading and then the cycle performs whereas in the second test, the maximum displacement is reached after five cycles during which the displacement amplitude augments gradually. The characteristics of loading and the wall's response will be discussed in the following. The cracks due to sliding and opening of horizontal elements, the cracks due to tension failure of bricks and total crack pattern after loading are shown in Figure 4.28. As cyclic displacements are applied to the wall, two series of stepped cracks in wall's diagonals are formed. The length of cracks at the center of the wall is more than the other parts. As can be seen in the Figure 4.31, the crack pattern is different from the first one. In the second case, the failure is important in the middle of the wall and a rupture can be distinguished over there. This failure is because of opening of vertical mortars. The obtained load-displacement curve shows that as no degradation was assumed for the model, stiffness of the wall is not changed during loading and as the compression failure is not captured in the wall, base shear increases with displacement increment. From result depicted above it can be concluded that the number of cycles has an important influence in the failure mode of the masonry shear wall. The length of cracks and their location on the wall for different interface elements after loading are represented in Figure 4.32. In the absence of the degradation phenomenon in the modeling of material behavior, results show that even if the envelope curves of load-displacement are similar, failure mechanisms and cracking pattern are different because redistribution of loads in both cases are not the same. It is found that the loading history plays an essential role on the map of cracks. The model still needs improvement in terms of reduction in the cohesion and rigidity to the rate of degradation during cyclic loading in order to take into account the material damage. Moreover, in addition to refining compression cap and softening behavior for material's strength, the reduction of modulus of elasticity with initiating plastic displacement will be added to the model in order to take into account the stiffness degradation of the wall during cyclic loading.

Conclusion and perspective

The results show that the 3D model of this study successfully simulated the behavior of masonry wall. Numerical model was validated with comparing its response to experimental data. In GEFDyn masonry model, bricks are modeled using solid elements and mortars are represented with interface elements. A Mohr-coulomb yield function with tension cut-off is considered for interface element as material property and bricks are supposed to remain elastic. A potential crack plane is considered in the middle of bricks using interface element to simulate tensile failure of bricks. Crack pattern is captured very well and push over curve shows that the model is capable of producing initial stiffness of the wall and its behavior to first steps of non-linearity. As no limitation was supposed for compression failure of the joints and bricks and no softening for shear and tension behavior of the mortars, the load increased with displacement increment and shear stress was raised by compression augmentation. These lacks in the model cause a great difference between load-displacement diagrams of the wall with opening. Thus, the model will be developed in the next section to take into account softening behavior and compression failure in the masonry.

Some simulations were made to capture the effect of cyclic loading on the masonry wall and the number of cycles. It was found that the loading history plays a role in the map of cracks. The model still requires improvement in terms of reduction in the cohesion and rigidity based on the rate of degradation during cyclic loading to take account of material damage.

Multi-surface yield function for interface elements (Model II)

In previous section, the masonry wall was simulated using micro-modeling strategy in which bricks and mortars are modeled separately. Bricks were modeled by solid elements while mortars were represented by interface element. A simple yield surface considering tension failure and shear sliding failure of the mortars was assumed for vertical and horizontal interface elements.

The micro model was validated by comparing its response with the experimental data available in the literature [START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF], [START_REF] Vd | Shear behavior of bed joints[END_REF]. The obtained results of numerical modeling show that the crack pattern was captured well but compression failure was not taken into account. As there is no limitation considered for compression stress in the mortars, shear stress increased by compressive stress increment according to Mohr-Coulomb criterion so that in the load displacement curve, a considerable difference between the ultimate load of numerical model and experimental evidence was observed. Thus the previous model could not take into account the ultimate strength of masonry.

In this section, the micro model is improved. The softening behavior is considered in shear and tension stress-strain curve. Moreover, a linear compression cap is added to the previous yield surface to capture compression failure of the wall. Therefore, in this section a multi-surface constitutive law composed of tension cut-off, Mohr-Coulomb and linear cap representing tension, shear and compression failure of the mortars is used. The idea of compression cap was the first time introduced by Drucker-Prager [1957] [START_REF] Drucker | Soil mechanics and work hardening theories of plasticity, Transaction[END_REF] in soil mechanics. First time, [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF] [60] added a composite compression cap to the yield surface of interfaces in modeling masonry walls using micro-modeling strategy. In this section, a simplified compression cap is introduced and used to capture compression failure of the wall. Then parametric study will be done and the effect of each model parameter on the behavior of wall will be investigated. Then the failure mechanism of walls with different aspect ratios is investigated. Moreover, the behavior of wall with central opening is also simulated.

Properties of composite yield surface

Three types of failure are considered for mortars i.e. interface elements: tension failure, shear failure and compression failure. Associated flow rule is considered for tension whereas non associated flow rule is used for shear and compression. Different types of failure and their yield function formulation are described and discussed in the following sections:

Mohr-Coulmob shear criterion

Mohr-Coulomb criterion is used for defining the shear yield threshold. In this section, softening behavior is assumed for cohesion. The yield function according to Equation 4.8 reads:

f 1 = (σ s ) 2 + (σ t ) 2 + σ n .tanφ -C (4.28)
Where:

C = C 0 . exp(C 0 .|γ p |/GII) (4.29) γ p = [u p s ] 2 + [u p t ] 2 (4.30)
In the above equations, C 0 is the initial cohesion of interface, φ is friction angle, [u p s ] and [u p t ] are the shear plastic displacements in s and t directions respectively. GII is fracture energy of mode II. A non-associated flow rule is considered for shear in mortar interface. In Equation 4.31, ψ is dilatancy angle. The potential function for shear is unchanged and defined as:

g 1 = (σ s ) 2 + (σ t ) 2 + σ n .tanψ (4.31)
A lateral displacement is applied to two bricks with horizontal mortar to investigate the accuracy of formulations. There is a good accuracy between numerical modeling and actual behavior of specimen and softening behavior of mortar is reproduced well. The results are shown in

F t = F t0 . exp(-F t0 /GI. < [u p n ] + >) (4.32)
Where:

< x > + = x if x ≥ 0 < x > + = 0 if x < 0
Yield function for tension is defined as:

f 2 = σ n -F t ( 4

Compression failure criterion

Compression failure is added to the interface element formulation. Compressive strength of mortar is assumed to vary with normal plastic displacement which is expressed by an exponential function.

F c = F c0 . exp(F c0 /GIII. < [u p n ] ->) (4.34)
Where:

< x > -= 0 if x > 0 < x > -= x if x ≤ 0
In Equation 4.34, F c0 is the initial compressive strength of masonry, [u p n ] is the plastic normal relative displacement and GIII is fracture energy of mode III failure. In this study, to reproduce crushing of masonry and diagonal shear failure, the combination of compression and shear stresses is limited. As mentioned in literature review, this strategy has been used in 2D by some of researchers [START_REF] Sutcliffe | Lower bound limit analysis of unreinforced masonry shear walls[END_REF], [START_REF] Chaimoon | Modelling of unreinforced masonry walls under shear and compression[END_REF]. So the new yield function is introduced to limit shear by compression increment in interface element. Yield function is introduced:

f 3 = (σ s ) 2 + (σ t ) 2 -σ n .tanθ -F c .tanθ (4.35)
In Equation 4.35, F c is given in Equation 4.34 and θ is the new parameter which represents the slope of compression cap, shown in Figure 4.38. It has worth to note again that u s , u t and u n are the jump of displacements in different directions.

Composite yield function corners

The coupling between shear and tension as well as shear and compression are considered in the model. The coupling between them happens in the intersection of Mohr-Coulomb shear criterion and tension cut-off as well as Mohr-Coulomb and compression cap.

(a) Shear and tension are active:

Tension and shear mechanisms are active when : f 2 = 0, ḟ2 = 0 and f 1 = 0, ḟ1 = 0 so:

f 1 = σ 2 s + σ 2 t + σ n .tan(φ) -C 0 . exp(-C 0 /GII.|γ p |) = 0 ḟ1 = ∂f 1 /∂σ s . σs + ∂f 1 /∂σ t . σt + ∂f 1 /∂σ n . σn + ∂f 1 /∂u p s . up s + ∂f 1 /∂u p t . up t = 0 f 2 = σ n -F t0 . exp(-F t0 /GI < [u p n ] + >) ḟ2 = ∂f 2 /∂σ n . σn + ∂f 2 /∂[u p n ].[ up n ] [ un ] p = k [ un ] p ; [ us ] p = k [ us ] p ; [ ut ] p = k [ ut ] p
Where k denotes the number of active mechanism.

|τ | = (σ s ) 2 + (σ t ) 2 [ un ] p = λ 1 .∂g 1 /∂σ n + λ 2 .∂f 2 /∂σ n = λ 1 .tan(ψ) + λ 2 [ us ] p = λ 1 .∂g 1 /∂σ s + λ 2 .∂f 2 /∂σ s = λ 1 . σs |τ | [ ut ] p = λ 1 .∂g 1 /∂σ t + λ 2 .∂f 2 /∂σ t = λ 1 . σt |τ |
The consistency condition of active mechanisms can be written as:

A : ḟ1 = 0 ⇒ σs |τ | .G.[ us -λ 1 . σs |τ | ]+ σt |τ | .G.[ ut -λ 1 . σt |τ | ]+E.tan(φ).[ un -λ 1 .tan(ψ)-λ 2 ]+C 2 0 /GII. exp(-C 0 /GII.|γ u p t |γ p | .λ 1 . σt |τ | ] = 0 G[ σs |τ | . us + σt |τ | . ut ]+E.tan(φ). un = λ 1 .[G σ 2 s |τ | 2 +G σ 2 t |τ | 2 +Etan(φ)tan(ψ)-C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ]] + λ 2 [E.tan(φ)] G[ σs |τ | . us + σt |τ | . ut ] + Etan(φ) un = λ 1 .h 11 + λ 2 .

h 12

With: Compression and shear mechanisms are active when : f 3 = 0, ḟ3 = 0 and f 1 = 0, ḟ1 = 0. Hence:

h 11 = G + Etan(φ)tan(ψ) -C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ] h 12 = Etan(φ) B : ḟ2 = 0 ⇒ E[ un -λ 1 tan(ψ) -λ 2 ] + F 2 t0 /GI. exp(-F t0 /GI.u p n )[λ 1 tan(ψ) + λ 2 ] = 0 E un = λ 1 [Etan(ψ) -F 2 t0 /GI. exp(-F t0 /GI.u p n ).tan(ψ)] + λ 2 [E -F 2 t0 /GI. exp(-F t0 /GI.u p n )] E un = λ 1 .h 21 + λ 2 .h 22 h 21 = Etan(ψ) -F 2 t0 /GI. exp(-F t0 /GI.u p n ).tan(ψ) h 22 = E -F 2 t0 /GI.
f 1 = σ 2 s + σ 2 t + σ n .tan(φ) -C 0 . exp(-C 0 /GII.|γ p |) = 0 ḟ1 = ∂f 1 /∂σ s . σs + ∂f 1 /∂σ t . σt + ∂f 1 /∂σ n . σn + ∂f 1 /∂u p s . up s + ∂f 1 /∂u p t . up t = 0 f 3 = |τ | -σ n .tan(θ) -F c0 . exp(F c0 /GIII. < [u p n ] ->).tan(θ) ḟ3 = ∂f 3 /∂σ s . σs + ∂f 3 /∂σ t . σt + ∂f 3 /∂σ n . σn + ∂f 3 /∂u p n . up n = 0 [ un ] p = λ 1 .∂g 1 /∂σ n + λ 3 .∂g 3 /∂σ n = λ 1 .tan(ψ) -λ 3 .tan(β) [ us ] p = λ 1 .∂g 1 /∂σ s + λ 3 .∂g 3 /∂σ s = λ 1 . σs |τ | + λ 3 . σs |τ | [ ut ] p = λ 1 .∂g 1 /∂σ t + λ 3 .∂g 3 /∂σ t = λ 1 . σt |τ | + λ 3 . σt |τ | A : ḟ1 = 0 ⇒ σs |τ | .G.[ us -λ 1 . σs |τ | -λ 3 . σs |τ | ] + σt |τ | .G.[ ut -λ 1 . σt |τ | -λ 3 . σt |τ | ] + E.tan(φ).[ un -λ 1 .tan(ψ) + λ 3 .tan(β)]+C 2 0 /GII. exp(-C 0 /GII.|γ p |). u p s |γ p | .[λ 1 . σs |τ | +λ 3 . σs |τ | ]+C 2 0 /GII. exp(-C 0 /GII.|γ p |). u p t |γ p | .[λ 1 . σt |τ | + λ 3 . σt |τ | ] = 0 G[ σs |τ | . us + σt |τ | . ut ]+E.tan(φ). un = λ 1 .[G σ 2 s |τ | 2 +G σ 2 t |τ | 2 +Etan(φ)tan(ψ)-C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ]] + λ 3 [G σ 2 s |τ | 2 + G σ 2 t |τ | 2 + Etan(φ)tan(β) -C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ] = 0 G[ σs |τ | . us + σt |τ | . ut ] + Etan(φ) un = λ 1 .h 11 + λ 3 .h 13 h 11 = G + Etan(φ)tan(ψ) -C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ] h 13 = G -Etan(φ)tan(β) -C 2 0 /GII. exp(-C 0 /GII.|γ p |).[ u p s |γ p | σs |τ | + u p t |γ p | . σt |τ | ] B : ḟ3 = 0 ⇒ σs |τ | .G.[ us -λ 1 . σs |τ | -λ 3 . σs |τ | ] + σt |τ | .G.[ ut -λ 1 . σt |τ | -λ 3 . σt |τ | ] -E.tan(θ).[ un -λ 1 .tan(ψ) + λ 3 .tan(β)] -F c0 2 /GIII. exp(F c0 /GIII.u p n ).tan(θ)[λ 1 tan(ψ) -λ 3 tan(β) = 0 G[ σs |τ | . us + σt |τ | . ut ]-E.tan(θ). un = λ 1 .[G σ 2 s |τ | 2 +G σ 2 t |τ | 2 -Etan(θ)tan(ψ)+F 2 c0 /GIII. exp(F c0 /GIII.u p n ).tan(θ)tan( λ 3 [G σ 2 s |τ | 2 + G σ 2 t |τ | 2 + Etan(θ)tan(ψ) -F 2 c0 /GIII. exp(F c0 /GIII.u p n ).tan(θ)tan(β)] G[ σs |τ | . us + σt |τ | . ut ] -Etan(θ) un = λ 1 .h 31 + λ 3 .h 33 h 31 = G -Etan(θ)tan(ψ) + F 2 c0 /GIII. exp(F c0 /GIII.u p n ).tan(θ)tan(ψ) h 33 = G + Etan(θ)tan(β) -F 2 c0 /GIII. exp(F c0 /GIII.u p n ).tan(θ)tan(β)
By calculating λ 1 and λ

Numerical validation for Model II

In this section, the numerical analysis is repeated while new yield functions are used for interface elements. The difference between this model and the previous one is that the compression failure is taken into account in this model by limiting the shear stress with compressive strength increment. In order to impose this limitation some new parameters are added into the model such as: f t0 , GI, f c0 , GIII, GII , θ and β. Table 4.4 and Table 4. 5 show the values which are used for horizontal and vertical interface elements and brick-middle interfaces respectively. Modulus of elasticity and shear modulus of brick interface element and its thickness is the same as the one in the model I. In Table 4.4, C 0 represents initial cohesion of mortars. For first try, θ is given: θ = 45-φ/3 then the effect of this parameter will be studied. The horizontal displacement is applied to the wall similar to the model I. Initial normal stress on the wall is σ n = 0.3M P a. Elastic properties of mortars and bricks are given in Table 4.1.

Here, non associated flow rule is considered for compression so β = 45 o . The effect of β on There is a good agreement between the results of computational modeling and experimental data using this formulation. θ = 90 is associated with the composite yield surface with compression cut-off.

In this case shear stress do not decrease with the increment of compression stress in the failure envelope and at σ n = F c, shear stress in the interface element becomes zero. The results show although that the crack profiles are similar but pushover curves considerably diverge. Hence, the formulation used in this study with θ = 45 -φ/3 is an appropriate equation which reproduces results similar to experimental evidence.

Behavior of masonry wall with opening

In this section, the behavior of wall with opening is investigated using new model. The behavior of wall with opening was studied using model I before. As shown before, the numerical model was not capable to reproduce the maximum load which is withstood by the wall. In this section, As can be seen in

Aspect ratio effect

In this section, the behavior of masonry shear walls with different aspect ratios namely lower than 1 and greater than 1, will be analyzed. The combined yield function consisting in compression cap is used for modeling of these walls. For each case, the total crack pattern, pushover curve and deformed shape are illustrated. Then pushover curves resulted from model I and model II will be compared. It should note that in all cases the initial compressive stresses equal to p = 0.3M P a imposed to the wall before applying horizontal displacements.

(a) Aspect ratio less than 1

This category contains three different walls. In the first case, the walls height is divided by 2 whereas for second and third cases the length of wall is multiplied by 1.5 and 2. Therefore, two different aspect ratios for masonry wall namely 0. Deformed shape shows that all horizontal mortars on the first row slide. In this case as softening behavior is considered for cohesion of mortars, the shear strength of the mortars decreases by raising displacements and as a result horizontal elements experience shear sliding failure very soon. Thus, wall loses its strength under lateral displacements and in each step loses its cohesion and therefore more plastic shear displacement is undergone by the wall. As the height of the wall is divided by 2 the weight of the wall reduces and thus normal stress on the wall decreases. Therefore, the level of confinement is not sufficient to prohibit the shear sliding failure of the wall.

As shown in pushover curve Figure 4.56(b), the wall with demi-height, AS (aspect ratio)=0.5 is stiffer than tested wall with AS=1 but at d = 0.3mm the wall with AS=0.5 slips on its first row and its lateral strength decreases significantly. Comparing pushover curve of two yield functions demonstrates that since model I cannot capture the reduction of mortar's cohesion, its lateral strength increases under incremental displacements. Since the results of model II are validated by comparing experimental evidence, the obtained pushover curve using this model is reliable.

(b) H/L = H 0 /1.5L 0 , AS = 2/3
The aspect ratio of this wall is less than 1 and is equal to 2/3. This wall also is stiffer than the wall with aspect ratio equal to 1. Moreover, maximum shear load carried out by this wall is much greater than the other one. The wall slides and loses its strength in the lowest row of wall as shown in The softening behavior of mortar's cohesion and consequently decreasing shear strength of the wall cause sliding shear failure before formation of diagonal cracks in the wall. The previous yield function was not capable to reproduce this phenomenon so its results were not trustworthy. Pushover curves using different methodologies are compared in Figure 4.59. The local jump in pushover curve using model I corresponds to formation of diagonal zigzag cracks in the wall. The wall's length is multiplied by 2 in this case. Hence, the wall has aspect ratio equal to 0.5. Similar to previous cases, sliding of the wall in the lowest rows of mortars is the reason of the wall's failure and softening behavior of cohesion causes such failure. The normal stress on the wall also is insufficient to prevent sliding shear failure of mortars. Lateral load-displacement curve is shown in Figure 4.61(b). According to this figure, the initial stiffness and the maximum shear strength of the wall are about two times greater than the wall with aspect ratio 1. Sliding shear failure happens at displacement equal to d = 0.64mm and reduces lateral strength of the wall significantly.

The load displacement diagrams using model I and model II are shown in Figure 4.61 to emphasis the importance of considering accurate yield function on predicting strength of the wall. As illustrated in this figure wall slides before formation of observed diagonal cracks in the results of model I (local drop in pushover curve). Sufficient normal stress on the wall may prevent sliding shear failure of the wall and causes the wall to carry out greater lateral loads.

Obtained pushover curves for different cases are drawn in the same figure to have a better comparison. The obtained force-displacement curve for the walls with aspect ratio less than 1 are shown in Figure 4.62(a). Same aspect ratio can be obtained for two walls with different dimensions. For example AS=0.5 may correspond to the wall whose height is divided by 2 or the wall whose length is multiplied by 2. As illustrated in than the wall with aspect ratio equal to 1. Moreover, walls with lower aspect ratio (AS=0.5) has the same initial stiffness and are stiffer than another wall with aspect ratio equal to 0.67 (AS=2/3). As shown in this figure, for same aspect ratio, wall with greater dimension carries out greater lateral loads. By the way, all three cases exhibit brittle behavior and their lateral strength decreases considerably due to sliding shear failure of horizontal mortars in the first row. To make a better comparison, pushover curves are normalized. Hence, the obtained horizontal force is divided by its length and applied displacement to the wall is divided to the wall's length. The results are shown in Figure 4.62(b). As shown in this figure, wall with aspect ratio less than 1 are stiffer than wall with AS=1. Walls with lower aspect ratio withstand greater lateral strength. Moreover, for the wall with the same aspect ratio, wall with larger dimensions is stronger than another one. These walls have the brittle failure and at larger displacements exhibit much lower lateral strength than the wall with aspect ratio equal to 1.

By comparing the obtained results using model II and model I it is found that neglecting the softening behavior of cohesion in mortars may lead to the wrong results and obtaining wrong failure mechanism for the walls with aspect ratios less than 1. If sliding of the wall at the highest and lowest rows of mortars is prevented by using mortars with higher cohesion or applying sufficient vertical load to prevent such failure, diagonal cracks may appear in the wall and wall carries out much greater loads according to the response of model I. Another important parameter is the maximum length of cracks in the elements. The maximum length which is formed in the wall is 1.88 mm for first item with AS=0.5 (H = 0.5H 0 ) while its value is 1.67 mm and 1.7 mm for the wall with AS=2/3 (L = 1.5L0) and AS=0.5 (L = 2L0) respectively. Thus, for the wall with lower dimension, the length of cracks is higher than other walls with aspect ratio less than 1.

(b) Aspect ratio greater than 1

To create the wall with aspect ratio more than one, the wall's length is divided by 2 for a case and for the others the height of wall is multiplied by 1.5 and 2. Deformed shape, pushover curve and total crack pattern are shown for each case. The same cracking pattern is obtained for the wall using models I and II. The reproduced initial stiffness for both yield functions is the same and is lower than that of wall with aspect ratio equal to 1. Comparison between pushover curves in Figure 4.64 shows that the same load-displacement curve is assessed by using models I and II until d = 2mm. No crushing failure takes place in wall corners up to this displacement. The initial stiffness of this wall is changed at d = 0.17mm due to separation of the wall's corners from the wall due to opening of the horizontal joints. At this point wall loses it integrity so its stiffness decreases.

The horizontal mortars in the top left and bottom corners of the wall fail in tension and shear and no diagonal cracks appear in the wall with such properties as shown in The curves are the same using either of yield functions up to 2mm. Since compression failure happens at d = 2.67mm in some of horizontal mortars and as maximum displacement which was applied to the wall in previous study was d = 2mm, it can be concluded that the results before compression failure are the same but after that, previous model cannot reproduce the actual behavior of the wall.

(c) H/L = 2H 0 /L 0 , AS = 2
In this case, the wall's height is multiplied by 2 and d = 2mm displacement is gradually applied to the top nodes of the wall. Deformed shape, pushover curve and crack pattern are shown in Figure 4.67. Similar to previous cases, the mortars at top left and bottom right of the wall fail in shear and tension and thus slide and open. Diagonal cracks were not observed in the wall. According to pushover curve, no crushing happens in this wall up to d = 2mm. In this case, stiffness of the wall changes at d = 0.3mm because of tension failure of horizontal mortars at the corners. The stiffness of the wall in this case also is much lower that the stiffness of wall with aspect ratio equal to 1 and that is because of decreasing the shear section of the wall, see The pushover curves for different cases with aspec ratio greater than 1 are drawn in a same figure to be compared together and with that of the wall with aspect ratio equal to 1, Figure 4.69(a). As explained before, whenever plastic displacements initiate in the interface element, the slope of pushover curve changes and for all of these cases, the opening and sliding of horizontal mortars in the corners cause change of stiffness at earlier steps. According to results of Figure 4.69(a), walls with aspect ratio greater than 1, have lower stiffness than walls with aspect ratio equal to 1 and the stiffness decreases with aspect ratio increment. For example wall with aspect ratio =1.5 have larger stiffness and sustain larger lateral loads than wall with aspect ratio = 2. Walls with the same aspect ratio have the same load-displacement curve but horizontal mortars in wall with lower dimension open and slide sooner. It's worth to know that the wall with aspect ratio equal to 1.5 carries out larger lateral load than wall with aspect ratio equal to 1 at larger displacements (after d = 1.7mm to d = 4mm) although its stiffness is much lower than that of with AS¿1 because the failure mechanisms are different in these case. It behaves in flexion while in the other case shear mechanism prevails. For the wall with AS=1 the main reason of wall's damage is formation of diagonal cracks in the wall which is together with crushing of some horizontal elements at the corner while for wall with AS¿1, no zigzag cracks appear in the wall and wall fails due to crushing of horizontal elements at the corners. It should be noted that the walls with AS=2 are loaded up to d = 2mm and the crushing failure does not happen in the wall up to this displacement. It is shown that the model can capture crushing failure of horizontal mortars for AS=1.5 which were loaded up to d = 4mm while previous model was not able to reproduced that. As shown in labels, the horizontal force is divided by wall's length whereas displacement applied to the wall is divided by the height of the wall to find the strength of the wall regardless its dimensions. As shown in this figure, the initial stiffness and lateral strength of the walls decrease by aspect ratio increment. For the walls with the same aspect ratio, wall with greater dimensions carries out lower normalized lateral strength. The maximum length of crack in the wall with AS=2 and L = 0.5L 0 is 0.26 mm at d = 2mm, for the wall with AS=1.5 this parameter is 0.22 mm and for the wall with AS=2 and H = 2H 0 , it is 0.16 mm. Thus, by increasing the dimension of wall the length of opening and sliding of horizontal mortars decreases. For the wall with the same aspect ratio, the normal stress on the lower mortars is greater for the one with larger dimension because of its weight, so those elements open and slide less than another one.

Parametric study

In this section, effect of different parameters on the response of the model will be studied in order to have a better idea about the influence of each parameter on behavior of the wall and to find the most critical parameters, which change greatly the wall's performance. The aim of this study is to reproduce different failure mechanism of masonry wall and to find the reason of such failure. Hence, some parameters in horizontal and vertical interfaces are changed in order to investigate the variation in the wall's behavior. These parameters are: the cohesion C 0 , tensile strength f t0 , compression strength f c0 , normal stress on the wall σ n0 and slope of shear-compression part of yield function θ. Moreover, influence of fracture energy of modes I, II and III namely GI, GII and GIII are studied in Appendix B. Finally the influence of normal stress on the walls will be studied. In this section, four values are chosen for each parameter and results are compared to the original one.

Cohesion of mortars

Four different values are chosen for C to investigate the wall's behavior under monotonic loading: C 0 = 9.5e4P a, C 0 = 1.5e5P a, C 0 = 5.5e5P a and C 0 = 7.5e5P a whereas the tested wall has C 0 = 3.5e5P a. Therefore, the performance of these five cases can be compared here in terms of deformed shape, load-displacement curve and profile of cracks. In order to have a better comparison the same scale is used for the length of cracks. Deformed shape and crack pattern related to each case are illustrated in Figure 4.70.

As can be seen in this figure, for walls with lower values of cohesion for mortar, sliding shear failure happens at two other rows in addition to top and bottom horizontal mortars. Diagonal cracks are also formed in the wall in these cases. In the cases with C 0 = 9.5E04P a and C 0 = 1.5E05P a, first mortars at the corners of the wall open and slide. Then, diagonal cracks appear in the wall and after that zigzag cracks cause sliding failure at the horizontal mortars next to them as shown in Figure 4.70. The difference between walls with C 0 = 9.5E04P a and C 0 = 1.5E05P a is that for the first one as the cohesion of mortars is lower than the second one, the top row of mortars slides more and the length of cracks in two additional rows is lower than another one. By increasing the cohesion of mortar for the wall with C 0 = 3.5E05P a the sliding failure of middle rows disappears and diagonal crack patterns become predominant. Moreover in this case, horizontal mortars at the top and bottom of the wall slide and open. As shown in Figure 4.70(h) and (j) for the wall with high cohesion, no diagonal cracks are formed in the wall and walls fail due to shear sliding failure of top horizontal mortars. The length of cracks and their distribution in the wall decrease for the wall with C 0 = 7.5E05P a rather than the wall with C 0 = 5.5E05P a. Horizontal mortars at the lower row in the wall slide more than the other ones.

By comparing the load-displacement curves and paying attention to failure of the walls, it can be assessed that although increasing the cohesion of mortars enables the wall to undergo higher shear stresses but for the walls with high cohesion brittleness of the wall also increases. The significant drift in pushover curves of walls with C 0 = 5.5E05P a and C 0 = 7.5E05P a corresponds to the sliding shear failure of top horizontal mortars. The first reduction in stiffness of the walls with C 0 = 9.5E04P a and C 0 = 1.5E05P a corresponds to formation of diagonal cracks in the wall while the sliding shear failure of mortars adjacent to diagonal cracks causes the reduction of shear strength of the wall after d = 1.6mm and d = 1.8mm respectively. Hence, it should be noted that although the increment of cohesion increases the lateral strength of the wall but it may increase the brittleness of masonry wall under specific condition and the type of failure cannot be guessed easily. In the walls with higher tensile strength with f t0 = 4.5E05P a and f t0 = 5.5E05P a, high tensile strength of mortars prevents tension failure of mortars and their opening. Since failure happens at weakest planes, top row and bottom row of mortars undergo sliding shear failure in these two cases respectively. It should be noted that just increasing a parameter does not always lead to a better response and increasing tensile strength of mortars highly may cause the brittle failure in the wall.

Compression strength of masonry

The low value for f c0 causes the corner joints to fail in compression sooner and consequently wall loses its shear strength and undergoes sliding shear failure at its top rows. In cases 1 and 2 with lower compression strength, horizontal mortars open at the corners first. Then, diagonal cracks appear in the walls whereas after a while sliding shear failure of top row becomes predominant and walls lose their lateral strength a lot. As there is a relation between lateral strength of the wall and compression strength of masonry, increasing compressive strength of masonry causes the wall to undergo more lateral loads. Masonry wall with high compressive strength carries out higher lateral strength and crushes due to higher deformations. The crack profile is diagonal and wall experiences the compression failure. As shown in Figure 4.77(h) and (j) the diagonal cracks are formed in the wall. Moreover, the compression toes of the wall experience crushing. The length of cracks in different walls with different compressive strength shows that the sliding of horizontal elements decreases by compression strength increment in upper row of mortar.

(a) f c0 = 8.05E06P a (b) f c0 = 8.05E6P a (c) f c0 = 9.05E6P a (d) f c0 = 9.05E6P a (e) f c0 = 1.05E7P a (f) f c0 = 1.05E7P a (g) f c0 = 2.05E7P a (h) f c0 = 2.05E7P a (i) f c0 = 3.05E7P a (j) f c0 = 3.05E7P a
Opening of vertical elements increases also by using mortar elements with high compressive strength. Moreover, vertical elements also slide and open more in these cases. The length of cracks in the walls with higher compressive strength is approximately uniformly distributed. The walls with higher compressive strength have better performance because the difference between total length of opening in vertical elements and sliding in horizontal interfaces is less than other cases. Therefore, better performance belongs to a wall which uses its maximum capacity and the total length of cracks in horizontal and vertical elements are closer.

Initial normal load

In this section, effect of initial normal load on behavior of masonry wall will be investigated. Hence, different vertical stresses such as σ n0 = 0.05M P a, σ n0 = 0.2M P a, σ n0 = 0.3M P a, σ n0 = 1.21M P a and σ n0 = 2.12M P a are applied to the same wall. The pushover curves for σ n0 = 0.3M P a, σ n0 = 1.21M P a and σ n0 = 2.12M P a are compared with the experimental data in the literature [START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]. The results of first case with σ n = 0.05M P a show that when there is no sufficient normal load on the wall, diagonal cracks are not formed and the wall sustains sliding shear failure at top horizontal mortars and exhibits it in a brittle manner. Softening behavior of mortars in cohesion also causes the wall to loose more shear strength under displacements and Comparing the numerical results and experimental evidences in case 3 with σ n0 = 1.21M P a reveals that the model can reproduce the load-displacement curve and the pre-and post-peak behavior of the wall well. The crack pattern in the model is shown in Figure 4.80(h) but no horizontal cracks at the top and bottom mortars are observed in test's results whereas in numerical model top and bottom mortars slide and horizontal cracks appear at these locations. This discrepancy may happen due to difference between applying load in real case and numerical one.

In the last wall with σ n0 = 2.12M P a there is a good agreement between results and experimental data as shown in Figure 4.80(j). As shown in this figure, diagonal cracks are reproduced well and are predominant. In pushover curve the initial stiffness of the wall was not obtained well. That is because of this fact that after the crushing of element due to high compression it still carries out the loads and stresses in the element do not set to zero. The softening behavior assessed well in the model but maximum load is underestimated 8% in numerical model. It should note that in this case although the wall undergoes larger lateral loads but behavior of wall is brittle after formation of diagonal cracks in the wall and it looses more lateral strength than previous cases. As represented in Figure 4.81, by increasing the normal pressure on the wall, the stiffness of wall increases (except the last case) and wall carries out greater lateral loads. Diagonal cracks appear in the wall with higher normal stress and sliding of horizontal mortars decreases. On the other side, walls with low level of confinement for example the wall with σ n = 0.05M P a fail due to sliding shear failure of top horizontal mortars and they exhibit brittle behavior. Thus as shown in Figure 4.80, the wall with low level of confinement, the sliding failure is predominant while in the wall with higher applied normal stress the sliding of top and bottom rows are almost vanished and diagonal crack pattern becomes predominant. Increasing the normal stress prevents opening of horizontal elements and increases the shear strength of the wall. Hence, this parameter has a key role in performance of masonry wall under in-plane loading. The length of cracks in different cases are shown in Figure 4.82. Sliding of horizontal cracks decreases by increasing normal stress. Moreover according to Figure 4.82(b) opening of horizontal elements also decreases by normal stress increment and even for the wall with σ n0 = 2.12M P a negative displacements create in the elements. Sliding of vertical elements is decreased by initial load increment in the walls related with diagonal crack pattern. Figure 4.82(e) shows that the bricks in the walls with lower initial normal stresses do not experience tension failure. As explained before, the length of opening in vertical elements in addition to sliding of horizontal elements can demonstrate the type of failure and maximum lateral strength of the wall. Best performance belongs to the wall with σ n0 = 2.12M P a in which the total length of cracks in horizontal elements (sliding) and vertical elements (opening) are close but in the first case the difference between these values are high and wall exhibits brittle performance.

Captured crack patterns

In this section different failure mechanisms which were captured by numerical modeling and their characteristics are demonstrated in the following. These results are valid for the mentioned boundary condition. A collection of properties needs to enhance the behavior of wall. For example as shown before using mortars with very high cohesion value may cause the brittle behavior.

As explained in this case, the displacements applied at the top row of bricks forces the top row into nonlinear region sooner and consequently softening behavior in shear strength which is accompanied by opening of vertical elements causes sliding shear failure at top row. As a results wall exhibits brittle behavior at the top row and endures sliding failure as opposed to distributed cracks along the diameter of the wall. Opening of brick-middle interface

Conclusion

A micro-modeling strategy is presented to model the mechanical behavior of masonry walls in this chapter. In this modeling, bricks are discretized with continuum elements which are connected by interface elements representing mortars. The mortars are supposed to be the plane of weakness and the nonlinearity takes place in these elements. A bilinear yield function considering shear and tension failure of mortars was implemented. Then, the model was enhanced and a compression cap was added to the model in order to take into account compression failure of mortars. Softening was integrated in the three tensile, compression and shear mechanisms using the failure energy as the controlling parameter for each failure. The model was introduced in software GEFDyn and a large number of computations were done. The numerical results were validated by comparison with experimental evidence. The behavior of wall at different levels of loading was studied accurately. The model was able to reproduce the behavior of masonry before and after peak very well. The crucial role of compression cap in capturing the correct failure mechanism and load-displacement curve was demonstrated by comparing the computational results of wall with opening and walls with different aspect ratios using yield function with and without compression cap.

A parametric study has been done with respect to aspect ratio of the wall to investigate its influence on failure mechanism of the wall. Studies show that the aspect ratio has a great influence on failure mechanism and ultimate lateral strength of the wall. Results illustrate that walls with lower aspect ratio have greater stiffness and withstand larger lateral strength than walls with aspect ratio greater or equal to 1 but they fail sooner because of sliding of their top or bottom row. In the analyzed walls with aspect ratio less than 1, normal stress on the walls was not sufficient to prevent sliding shear failure of the wall. On the other side, the cohesion of mortars decreases by increasing displacements and for the walls with the same aspect ratio, the one with larger dimensions withstands larger stresses and its maximum strength is highly greater. Moreover, walls with larger aspect ratio than 1 have lower stiffness than walls with aspect ratio equal to 1 and the stiffness is decreased when aspect ratio increases. The failure mechanism for the wall with aspect ratio greater than 1 is crushing failure at the compressed corners which was captured for wall with AS=1.5.

Finally a parametric study has been done in order to find the importance of each parameter on wall's behavior. Cracking profiles, pushover curves and the summation of cracks' lengths in different types of elements located at different rows and columns are compared in order to find the role of each parameter in the performance of the wall and to find the most suitable condition.

Studies illustrate that combination of factors contribute and shape the structural behavior of a wall including cohesion, tension strength, compression strength of mortars and initial vertical loads which are applied to the wall. Each of the parameters should stay in a certain range in order for the wall to exhibit predictable behavior under specific loading condition. Between these parameters increasing the initial normal load will certainly improve the behavior of wall because the normal stress increases both shear strength of the wall and postpones the opening of horizontal mortars due to tension. However, in practice, this parameter depends on the location of the wall in the structure and is almost an input of design process. It is also shown that the larger total length of opening in vertical elements and lower total sliding in horizontal elements and formation of diagonal cracks in the wall cause the wall to undergo greater loads and to exhibit more ductile behavior. Hence, the removal of mortar between vertical interfaces along the diagonal of the wall may force a desirable failure mechanism with stepped diagonal cracks.

Results also show that tension failure in the bricks does not affect the results considerably. Thus, its modeling using interface element in the middle of bricks can be neglected for the simplicity.

As noted before, the boundary conditions play an important role on the behavior of the wall.

The results presented in this chapter were limited to a unique type of boundary conditions. Therefore, in the next chapter this effect is studied, analyzing the behavior of connected walls.

Introduction

As mentioned in the literature review, failure of masonry wall under out-of-plane loads is one of the most critical issues in masonry construction. Other type of loadings such as explosion, flood, avalanche or soil horizontal movement are also possible configurations. The masonry walls perpendicular to a seismic motion are subjected to out-of-plane loads and fail in out of plane direction. Out-of-plane failure of masonry constructions is the main cause of injury and loss of life when they are subjected to earthquake loads. A diagonal crack is initiated from top right to the middle of the wall which is accompanied by a horizontal sliding crack from middle to the edge of the wall. In Figure 5.1(b) damage of two perpendicular masonry walls under earthquake movement is shown. The walls are separated from their intersection and damage is mostly concentrated at the top part of the walls joint. Moreover, a diagonal cracks is also formed in the lower part of the wall I. Some horizontal crack in connection of wall I and roof can be seen whereas wall II also moved and some cracks in its common corner with wall I are appeared. Figure 5.2(a) shows the shear sliding failure of a masonry wall in a rural building in Turkey [START_REF] Celepa | Failures of masonry and concrete buildings during the March 8, 2010 Kovanclar and Palu (Elaz) Earthquakes in Turkey[END_REF].

A continuous horizontal crack is obvious in the wall as its length is much greater than its height. Earthquake movement in Turkey also caused a slight damage in a one-story masonry building in Palu, as shown in Figure 5.2(b) [START_REF] Celepa | Failures of masonry and concrete buildings during the March 8, 2010 Kovanclar and Palu (Elaz) Earthquakes in Turkey[END_REF]. Walls experienced horizontal sliding failure at their top.

In addition, a diagonal crack which is initiated from the corner of opening to the bottom of the wall is also obvious. The aim of this study is to reproduce such crack patterns in a wall due to out of plane loads and in a set of two perpendicularly jointed walls due to roof's movement. The performance of masonry wall under out-of-plane loads has not been studied considerably. In most of experimental studies, masonry wall was subjected to the uniform pressure perpendicular to its surface in order to simulate the out-of-plane load condition [1], [START_REF] Drysdale | Out-of-plane bending of concrete block walls[END_REF], [START_REF] Griffith | Out-of-plane flexural strength of unreinforced clay brick masonry walls[END_REF],[103] and the effect of different parameters such as axial compression load, mortar type, unit type, aspect ratio, boundary condition etc. on the behavior of masonry wall under different conditions were studied. But in this study the effect of out-of-plane loads on the behavior of unreinforced masonry wall will be studied. Therefore displacements are applied to the panel in different locations and effect of gradual monotonic out-of-plane loads on performance of unreinfoced masonry wall will be investigated.

In this chapter, behavior of one wall and two perpendicular walls under out-of-plane loading will be studied. In the first section behavior of a wall under different boundary conditions and different out-of-plane displacements is studied and the crack pattern analyzed. A parametric study will be done in order to assess the effect of different parameters on the behavior of a masonry wall under out-of-plane displacements. Then, two walls with same lengths will be attached and subjected to loads in different directions. A parametric study on vertical stress applied on the wall and cohesion of mortar will be done to assess the effect of each parameter on the performance of the set.

Afterward, a rigid roof is supposed to connect the walls and the influence of its movement on the behavior of each wall will be studied. For simulating the movement of the roof, all nodes at the top of walls are subjected to the same displacement in different directions. Roof rotation will be modeled by applying different displacements in x and y directions to the top nodes proportional to their distance to the roof's center of gravity. Finally, the length of the wall in y direction is multiplied by 2 to obtain the influence of wall's geometry on ultimate strength carried out by the walls, cracks profile and deformed shape.

Behavior of a wall under out-of-plane displacements

In this section, in order to study the out-of-plane behavior of unreinforced masonry walls, two simple walls are modeled and subjected to out-of plane displacements. These walls and the applied displacements on them, are schematically shown in Figure 5.3. The dimension of walls is the same as those used in chapter 3 namely: 990× 1000 [mm 2 ] and it consists of 18 courses of bricks with dimension: 210 × 52 × 100 [mm 3 ]. As shown in this figure, in Case A, the wall's right edge is subjected to out of plane displacement while the movement in x, y and z directions of its base and left edge is precluded. A vertical pre-compression equal to σ n0 = 0.3M P a is applied to the wall and the vertical degree of freedom of the top nodes is fixed. The displacement applied to the wall's top corner is twice the displacement applied to the middle of its height. In order to study another type of out of plane loading, the Case B is modeled. In this case, the horizontal and vertical movements at top and bottom nodes are precluded and equal displacements are applied to the wall's middle height at right and left edges. Similar to Case A, the wall is subjected to a vertical pre-compression equal to σ n0 = 0.3M P a. The modeling strategy is the same as that of previous chapter so that the bricks are divided to two parts, each part is modeled using 3D solid elements and at their connection, horizontal and vertical mortars are modeled by interface elements. The material properties which are the same as those of the previous chapter are shown in Table 5.1.

After modeling and capturing the results, a parametric study by changing the mortar's cohesion C 0 , the compression strength f c0 and the initial compression σ n0 will be conducted. The numerical results for each wall are presented separately.

M P a [N m/m 2 ] M P a [N m/m 2 ] MPa [N m/m 2 ] [m]

Case A

The initial modeling involves applying 2mm and 1mm displacements to the top and middle heights of the wall as demonstrated in

Parametric study on Case A

In this section, the results of parametric study will be presented in order to find the influence of some material properties on the behavior of the wall and to distinguish which parameter has more influence on the performance of the wall against out-of-plane loading. These parameters are initial cohesion of mortars, compression failure of mortars and initial normal compression applied to the wall. Two different values, one less and another greater than actual value are chosen to investigate the wall's behavior. These values are C 0 = 0.035M P a and C 0 = 3.5M P a for mortar's cohesion, σ n0 = 0.05M P a and σ n0 = 2.12M P a for initial vertical stress and F c0 = 10.5e5P a F c0 = 10.5e7P a for compressive strength of masonry. The results of main case with C 0 = 0.35M P a, σ n0 = 0.3M P a, F c0 = 10.5e6P a is also compared with other cases. For having a better comparison, maximum length of cracks fits to 0.1mm in total crack pattern's figures in all cases. As shown in Figure 5.9, for the wall with C = 3.5e4P a, the stepped cracks are more distributed in the wall and mortars in top left corners experience greater opening and sliding. Maximum lateral load supported by this wall is 2.8kN and in comparison with previous case with C = 3.5e5P a, the lateral maximum strength is decreased. It should be noted that the lateral strength of the wall under out-of plane displacements is very low in this case. Maximum crack length which is formed in the wall is 0.142mm which is greater than previous case C = 3.5e5P a with 0.138mm. For the wall with C = 3.5e6P a the mortars' cracks decreased in term of size and spreading through the wall and the maximum length of the crack was 0.125mm. Hence, by increasing the cohesion lateral strength of the wall under out-of plane displacement increases whereas the number and length of sliding elements decrease a lot. As shown in Figure 4.72 in previous chapter, r1 is the bottom row and r17 is the top row of horizontal mortars. In addiction, c1 and c8 correspond to the left and right columns in the wall.

In stresses and strains in the wall showed that formation of throughout stepped cracks from top left to bottom right of the wall at his displacement, was its main reason. These cracks were formed in the wall mainly due to opening of vertical elements and as after opening of these elements, the cohesion of mortars sets to zero, some elements loses their lateral strength and their summation remains unchanged. The cracking profiles in the wall with C = 3.5e4P a before and after peak is shown in Figure 5.11. The length of sliding in in-plane and out-of-plane directions as well as opening of horizontal interfaces for wall with different C in different rows are drawn in Figure 5.12. Length of cracks in different rows shows how they are distributed in the wall. According to Figure 5.12(a) and (b) the sliding of mortars in both directions (parallel and perpendicular to the loading) increases considerably by decrement of cohesion of mortars and the critical locations are the top and bottom rows of the wall. Figure 5.12(c) shows that opening of horizontal mortars is approximately the same for walls with different cohesions and greatest opening cracks belong to first row of mortars.

Sliding of vertical elements in both directions also is higher for walls with lower cohesion, see Figure 5.12(d) and Figure 5.12(e). As shown in Figure 5.12(f), vertical mortars at the left part of the wall open more and the length of opening decreases by cohesion increment. Tension failure of bricks also is illustrated in Figure 5.12(g). According to this figure, more bricks fail in tension in walls with higher cohesion. These bricks are located at left corner of the wall because as illustrated before, the movement of wall at its left edge is prevented. The summation of cracks' length (sliding and opening) for all types of interfaces were shown in Figure 5.12(h) in order to find which parameter causes failure in the wall due to out-of-plane displacements. As shown in this figure, as the length of opening in horizontal mortars is the same for all cases and the length of sliding of horizontal elements in out-of-plane direction and opening of vertical elements is higher for the wall with lowest cohesion, it can be concluded that these parameters play a key role to performance of the wall. The high value for sliding of mortars in out-of-plane direction and opening of vertical elements cause brittle failure in the wall. These parameters change a lot by increasing the mortar's cohesion. In this part the vertical compression which is initially applied to the wall is changed to investigate its influence on the overall behavior of the wall under out-of plane displacements. Profile of cracks for σ n0 = 0.05M P a, σ n0 = 0.3M P a and σ n0 = 2.12M P a at d top = 2mm are illustrated in Figure 5.13. As shown in Figure 5.13, the initial normal confinement has a considerable role to the behavior of wall under out-of-plane displacement. By increasing normal stress on wall, the size of cracks in the wall decreases significantly. For the wall with σ n = 0.05M P a, maximum crack's length is 0.162mm which is the greatest value among the walls with different cohesion and initial normal stresses. Studying the behavior of wall with σ n = 0.05M P a shows that at d = 0.3mm more than half of horizontal mortars located at right part of lower row open and simultaneously the vertical mortars at the top left of the wall begins to open. This situation causes the wall to undergo shear stresses in x direction and as can be seen in the following figures, the sliding of horizontal elements in x direction in this case is very larger than the wall with σ n = 0.3M P a. As a result a drop in pushover curve appears. The wall loses its lateral strength more than the other cases because the number of horizontal interface which are opened is greater than other ones due to lack of enough vertical compression to prevent their opening. Afterward, cracks form and spread through the top left corner and bottom row of horizontal mortars. Then, at d = 1.4mm stepped diagonal cracks which connect the top left cracks to the bottom row of horizontal mortars form in the wall and causes another draft in the pushover curve. The larger cracks form in the lower row of horizontal mortars due to tension failure of those elements. Maximum lateral load which is carried out by the wall is 2.52kN which is much lower than the other cases. Comparing to the case with σ n = 0.3M P a, the sliding of mortars increases and more mortars slide in both directions. Moreover, the length of mortar's opening increases by reducing the initial normal stress on the wall. In this case, just one brick fails in tension. Maximum principle stress also concentrates at the top left corner of the wall similar to the other cases.

For wall with σ n = 2.12M P a, horizontal mortars except the one at the top right corner do not slide because of high level of normal compression. At d = 0.37mm some of horizontal mortars located at the right part of the wall are compressed under out-of-plane displacements and on the other side some of vertical elements at top left corner of the wall begin to open. The combined effect of overturning moments and opening of vertical mortars causes the wall to carry out more lateral strength at this step. In this case, the sliding of horizontal elements in x direction is 0 in all rows except the top one. Maximum lateral load which is carried out by this wall is 8kN which is much more greater than the other cases. The number of horizontal elements which experience tension failure and open is much less than the other cases. Vertical elements slide and open mainly at top left corner of wall toward its middle. On the other hand by increasing the strength of horizontal and vertical mortars, more bricks fail in tension. The maximum major principle stress takes place in bottom right corner of the wall. It should be noted that in this case, no stepped cracks appear in the wall.

As shown in pushover curve, the lateral shear strength which is carried out by the wall in- creases by increment of initial vertical load applied to the wall. On the other hand, the wall with lower initial normal stress yields sooner and withstands lower lateral strength than other cases. It should be noted that the variation of stiffness in the walls with σ n = 0.05M P a and σ n = 2.12M P a depends on the movement of horizontal elements in x direction. Horizontal mortars in the wall with σ n = 0.05M P a slide considerably in x direction whereas no sliding in x direction was observed for the wall with σ n = 2.12M P a (except the movement of top left horizontal mortar in the location of applied displacement). Figure 5.15 displays the length of sliding and opening cracks in different types of elements. As shown in Figure 5.15(a) and Figure 5.15(b), the length of sliding cracks in horizontal elements decreases by increasing the initial normal stresses. The opening of horizontal mortars also decreases considerably in the upper rows by increasing the normal compression on the wall. As shown in Figure 5.15(c) for the wall with σ n0 = 2.12M P a in some rows, horizontal interfaces close. The opening of horizontal mortars at first and top rows are higher than the other rows.

Sliding of vertical interface parallel to applied displacement also considerably decreases by normal compression increment. Vertical elements at the middle of the wall slid more than ones located at the wall's edges. Sliding of vertical interfaces at z direction for the wall subjected to σ n0 = 0.05M P a is also noticeable so under low level of normal stresses, vertical mortars slide more in z direction. Opening of vertical elements in left side of the wall is greater for lower values of σ n0 whereas for wall with σ n0 = 2.12M P a opening of vertical elements in different columns does not vary a lot. Increasing the normal stress, decrease the length of opening in vertical elements too. Brick fail in tension mostly at the left part of the wall and the length of opening increase by increment of initial normal stress on the wall as shown in Figure 5.15(g). The summation of crack's length for different categories of cracks are shown in Figure 5.15(h). Opening of horizontal interfaces, opening of vertical interfaces and sliding of horizontal interface parallel to the thickness of the wall are the parameters which have greater values for the wall with lower lateral strength. As shown in Figure 5.15(a) by increasing initial normal stress, total length of opening and sliding cracks in mortars decrease considerably. It should be noted that the displacement in x direction also may change the stiffness of the wall as described before.

(c) Compressive strength of masonry

In this section, the pushover curves and total crack pattern for wall with different masonry compressive strength namely F c0 = 10.5e5P a, F c0 = 10.5e6P a and F c0 = 10.5e7P a are compared in order to assess the effect of this parameter on behavior of wall under out of plane loading. In the wall with F c0 = 10.5e5P a, the top and bottom left horizontal mortars slide a lot as shown in Figure 5. 16(a). By increasing the displacement in this case, shear strength of the wall decreases by normal stress increment and mortars slide more. As a result, the size of cracks will increase by reducing of masonry compressive strength and thus the wall carries out lower lateral strength than the other cases. The stiffness of wall in this case changes firstly at d = 0.4mm due to opening of horizontal interfaces at the first row. Then, at d = 0.8mm stepped cracks appear on the wall and wall loses its stiffness again. On the other hand, there is no difference between the cracking profiles and pushover curves in the walls with F c0 = 10.5e6P a and F c0 = 10.5e7P a for displacement up to 2mm because the displacement is not sufficient to cause the crushing in their mortars or decrease their shear strength.

The sliding of horizontal elements in both perpendicular and parallel to the loading at first and last rows for the wall with lower compressive strength is much greater than other walls as shown in Figure 5.18. In the wall with low compressive strength, the shear strength of the wall decreases sooner with normal stress increment. Hence, elements in right top and bottom rows lose their shear strength and slide. Generally by increasing compressive strength of masonry, the sliding of horizontal cracks decreases a lot especially at top and bottom rows whereas the length of the other types of cracks increases a little and wall carries out more lateral strength.

The summation of length of cracks in different cases are displayed in Figure 5.18. Maximum length of cracks belongs to opening of vertical interface. The total length of opening of horizontal interface is also considerable. By comparing the length of cracks in different cases, it can be found that the high level of sliding cracks of horizontal mortars in x direction also can represent In this part, a masonry wall with specific boundary condition (Case A) was loaded under outof-plane displacement at two points in its right edge. The behavior of wall was studied and parametric study has been done in order to find the most important parameters in the behavior of masonry wall under out of plane loading. The results proves that the shear strength of the wall under out of plane displacements is very lower than that of in-plane and its value greatly depends on the boundary condition and position of applied loads. The cohesion of mortars and compressive strength of masonry were divided and multiplied by 10. Moreover, the wall was analyzed for 3 different values of initial normal stress. Results show that the initial normal compression has the greatest influence on behavior of the wall and after that cohesion of mortars also is an important parameter. Opening of vertical cracks is the main reason of failure in the wall. Moreover, opening and sliding of horizontal mortars in x direction (perpendicular to the loading's direction) also have considerable influence on failure of the wall. The cracking profile for most vulnerable cases is drawn in below Table 5.2. the wall and generally better performance of these walls. The length of sliding cracks in both y and z direction decreases by cohesion increment. Moreover, the distribution of cracks in different columns reduces and lower number of vertical elements slide and in vertical element Figure 5.24(d) and Figure 5.24(e). Opening of two vertical elements at the middle edges of the wall also decreases greatly by cohesion increment. There is a smooth distribution of opening cracks in different columns for the wall with C 0 = 3.5e5P a and C 0 = 3.5e6P a. Opening of brickmiddle interface elements also increases by cohesion's increment. According to Figure 5.24(h), the sliding of horizontal mortars at x and y direction and opening of vertical mortars for the wall with lowest lateral strength C 0 = 3.5e4P a is greater than the other walls. Hence, it can be concluded that these parameter are very important by increasing the cohesion of mortars their values decrease and wall reacts better under loading.

(b) Initial normal compression

In Figure 5.25, the response of wall for σ n0 = 0.05M P a, σ n0 = 0.3M P a and σ n0 = 2.12M P a are illustrated. The horizontal mortar in the end left middle of wall with σ n0 = 0.05M P a slides in x direction at earlier steps of loading and causes stiffness reduction in the wall. In this case, the opening of mortars at the top, bottom and middle rows of horizontal mortars is significant and the cracks due to sliding and opening of vertical mortar are distributed in rhomboid shape. At d = 1.27mm bricks in the middle of wall fail in tension and the stiffness of wall decreases again. As shown in Figure 5.25 the length of cracks decreases from σ n0 = 0.05M P a to σ n0 = 0.3M P a but their distribution changes for σ n0 = 2.12M P a. In this case high opening of vertical mortars in the middle of edge columns in x direction and sliding of horizontal mortars in the middle edges cause damage in the wall.

By increasing the normal stress to σ n0 = 2.12M P a the opening of horizontal mortars is decreased a lot in all rows. In this case the loaded bricks in corners rotate and separate from the wall and the rest of horizontal mortars do not displace considerable. The amount of this movement can be represented by the length of opening and sliding cracks in adjacent horizontal and vertical mortars. According to pushover curve, the stiffness of wall changes at d = 0.22mm due to sliding of the end right middle brick in x direction. At d = 0.34mm, separation of end middle bricks causes reduction of compressive forces in the end columns of bricks. It should be noted that after d = 1mm, the lateral strength of wall decreases because of large plastic deformation of some horizontal elements located in the middle of the wall. Thus, as shown in Figures 5.25 The length of cracks in different elements as well as their variation by normal stress are shown in Figure 5.27. According to these figures, opening of horizontal elements decreases by normal stress increment whereas the length of opening cracks in vertical and brick-middle interfaces decrease. It is important to know that boundary conditions and the way that displacements are applied to the wall are very important factors in the response of the wall. For example for the wall with σ n0 = 2.12M P a high level of normal loads can not prevent the movement of loaded bricks and large plastic deformations in the wall causes reduction of lateral strength in the loaded elements. Figure 5.27(h) the summation of cracks' length at different rows and columns for horizontal and vertical mortars shows that the length of sliding in horizontal mortars and opening in vertical mortars increase by initial normal stress increment because of slipping of two middle corner bricks. High normal stress on the wall can not prevent the opening of beside vertical mortars because of punctual nature of applied displacements. The tension failure of horizontal interfaces has not great influence on behavior of the wall because although its value is very high for σ n0 = 0.05M P a, its maximum lateral strength at d = 1.5mm is approximately the same with other cases. The length of sliding cracks in horizontal and vertical cracks as well as opening cracks in vertical elements and brick-middle interfaces decrease by compression strength increment as shown in Figure 5.30. In the wall with low compressive strength two horizontal interfaces at the left and right middle edges of the wall begin to crush and their lateral strength decreases and slide more. As shown in Figure 5.30(h) for the most vulnerable case, the length of sliding cracks in horizontal and opening cracks in vertical elements are more than the other cases.

The results of this section shows that the cohesion of mortars has the most important role on behavior of masonry wall under such boundary conditions and loading. The opening cracks in vertical interfaces and sliding cracks in two directions in horizontal interfaces are the greatest among all studied cases in this section. The corresponding crack pattern is shown in Table 5.3. Cracking profile Specifications Observed in:

• The cracks concentrated in the middle of the wall

• The loaded bricks rotate a lot

• Masonry wall with low cohesion

In this section, two walls with different boundary conditions and loading patterns are subjected to out of plane displacements. A parametric study for each case also has been done to investigate the influence of mortar's cohesion, initial normal stress on the wall and compressive strength of masonry on the wall's response. Results show that increasing all of these parameters enhances the performance of the wall. The level of pre-compression is the most important parameter is most important parameter for Case A because high value of normal stress prevents sliding of horizontal mortars and opening of vertical mortars whereas mortar's cohesion plays the most important role to the behavior of Case B. It should be noted that applying too high initial normal stress on the wall may cause the concentration of stress on the loaded elements and imposing considerable shear plastic deformations and the lateral strength reduction of the wall. Results show that compressive strength of masonry has the lowest influence on the behavior of wall under out-of-plane displacements for both cases. For having a masonry wall with a good performance under out-of-plane displacements, the sliding cracks of horizontal mortars in parallel and perpendicular directions to the loading and opening of vertical elements should be decreased.

Behavior of two walls under out-of-plane Loads

In this section the behavior of two perpendicular walls is investigated. In this configuration, applying in-plane loading to a wall imposes out-of-plane loading to the other wall. The two walls has the same geometry 1×1×0.1 [m 3 ] as shown in Figure 5.31. The properties of mortars at the walls' connection are the same as those of horizontal and vertical. The properties of mortars and blocks are the same as the validated wall in chapter 3. All top nodes are precluded from vertical movement and σ n0 = 0.3M P a is applied to the set. The behavior of the set is analyzed first and then the obtained results of a parametric study by changing mortar's cohesion and initial normal stress will be studied. In this study, first the top nodes of Wall 1 are subjected to the incremental in-plane displacements (x direction) up to 2mm. This loading imposes out-of-plane loads to the Wall 2. Then, the effect of cohesion and initial normal loads on the behavior of the series are investigated. Afterward, a supposed roof is considered for two walls and its movement namely the movement of all top nodes (Wall 1 + Wall 2) in x and y directions as well as its rotation will be studied. In the later one, all top nodes rotates 0.05 degree to simulate roof's movement under earthquake. Next, As shown in Figure 5.33, applied in-plane displacements to the Wall 1 cause diagonal stepped crack in the wall as predicted. This type of loading forces out of plane movement in Wall 2, see Figure 5.33(c). Due to this loading, the vertical mortars of walls' intersection open and slide as shown in Figure 5.33 and some of the bricks at the connection of walls (rows 2, 16 and 17) also move forward. Out-of-plane displacements on Wall 2 cause tension failure at its lower row of horizontal mortars and opening cracks appear on that. Total crack pattern is illustrated in Figure 5.37(f). As shown in this figure, the length of diagonal crack pattern in Wall 1 is greater than other cracks. Moreover, the loading creates cracks in the joint mortars and the length of cracks in the bottom part of the connection is greater. Base shear versus applied displacement for Wall 1, Wall 2 and the set in x direction, parallel to applied displacements, are illustrated in Figure 5.34. As shown in this figure, Wall 2 has a little strength under out of plane displacements but its existence makes the set stronger against lateral loads. The main resistance of the set comes from Wall 1 to which in-plane displacements are applied. In this part a parametric study by variation of mortar's cohesion and initial normal loads on the wall has been done. It should be noted that displacements are just applied to Wall 1 in x direction. The cohesion of all mortars are changed to C 0 = 1e5P a and C 0 = 7e5P a respectively while the effect of pre-compression on behavior of complex is investigated by setting normal stress to σ n = 0.05M P a and σ n = 2.12M P a. The pushover curves, total crack pattern and final deformed shape are represented here for different cases are represented in the following. The maximum length of cracks is set to a fixed number for all cases to have a easier comparison.

For the wall with C 0 = 1e5P a, the length of diagonal cracks increases and the mortars in intersection of walls slip and open considerably. Thus, cohesion increment causes reduction in the length of cracks in the walls as well as in the vertical mortars at the walls' connection. Pushover curves, for walls with different cohesion values are drawn in Figure 5.42. The series with C 0 = 7e5P a undergoes greater lateral strength but slipping of Wall 1 in the last case causes a brittle sliding shear failure in the system. The relative displacement between two walls also is reduced by cohesion increase. In order to have a good understanding about the pushover curve of the set with C 0 = 1e5P a, the shear forces of each node at the base of wall 1 are drawn in Figure 5.39. The summation of shear forces of these nodes makes the base shear of Wall 1. As The effect of normal stress on the behavior of complex is studied in this part. In the set with σ n0 = 0.05M P a after applying the displacements, the top and bottom rows of Wall 1 open and slide and by increasing the displacements at earlier steps. In the obtained pushover curve Figure 5.44 the first reduction in the stiffness of the set is due to these opening and sliding cracks. Thereupon, at d = 0.6mm, the Wall 1 slips on its first row of horizontal mortars and the lateral strength of the wall decreases considerably. As can be seen in pushover curve, the set exhibit brittle behavior under low normal stresses. Consequently, two walls separate and their intersection slides in the bottom part. The vertical mortars at the intersection of two walls slide in both directions at its bottom part and open at its upper part. In this case, bricks do not fail in tension. exhibits ductile performance. As shown in the figures, walls with lower value of initial normal stresses undergoes considerably lower shear strength than the wall with higher initial normal stresses and initiation of cracks in the mortars happens sooner than that of the set with higher initial normal stress. Hence, the normal stress has a great influence on the behavior of masonry walls both under in-plane and out-of-plane loading.

In this section the effect of mortar's cohesion and initial normal loads on the behavior of two perpendicular walls under specific loading condition was investigated. Obtained results showed that both cohesion of mortars and initial normal stress have very important role to the behavior of walls under in-plane and out-of-plane loadings. These parameters enhance the performance of wall under both types of loading. It was shown that the influence of normal stress on behavior of two perpendicular walls is greater than wall's cohesion. The studies show that the properties of mortars in walls' joint may have a significant influence on the performance of the system. By using the mortars with higher cohesion in the intersection of the walls the integrity of the system will be preserved under larger loads and thus the system carries out larger displacements.

Series 1: roof 's movement simulation

Presence of a rigid roof with appropriate connections to the walls in masonry constructions causes the surrounding walls to be subjected to the same displacement as that of the roof in different directions. Hence, for simulating the behavior of a system with a rigid roof, all top nodes are submitted to the same displacements as the roof's center of gravity. This section is composed of three different types of movement. First, all top nodes belong to Wall 1 and Wall 2 are subjected to the same displacements in x direction. The displacements (in-plane for Wall 1 and out-ofplane for Wall 2) are gradually applied to the set from 0 to 2mm. As the length of the series 1 in x and y direction is not the same, to study the difference arises from this variation, the same displacements are imposed to the top nodes in y direction. Finally the rotation of roof is simulated by applying different displacements in x and y directions to the top nodes. The roof's center of gravity supposed to rotate 0.05 degree anticlockwise. Maximum displacement in x and y directions are 4.8mm and 4.38mm respectively which are applied to different nodes according to their position with respect to roof's center of gravity. The results are represented in following sections. It should be noted that for all cases, the initial normal stress equal to σ n = 0.3M P a are applied to the walls.

Series 1: displacements are applied to all top nodes in x direction

The results of numerical study on the behavior of the set when all top nodes move d = 2mm in x direction are represented here. The deformed shape in different planes as well as total crack pattern is shown in The deformed shape of this set is compared to that of in previous section in which just top nodes of Wall 1 were subjected to displacement in x direction, see Figure 5.46. According to this figure, in the later one, diagonal cracks are not formed in the Wall 1, the behavior of the set is brittle and the failure mechanism is sliding shear failure while in the former one, diagonal cracks appear in the wall and the failure mechanism is tension-shear and the system undergoes greater lateral strength. walls are subjected to out-of-plane displacements in x direction (in-plane). This figure shows that the wall with higher length has lower strength against in-plane loads because of sliding shear failure of the top row of horizontal mortars. In this case, vertical mortars at the connection of walls also move significantly greater than the other case. In Figure 5.50(a) the greater length of the set carries out lower lateral loads. Hence, it can be concluded that the geometry of the walls in a set has an important influence on its performance.

Series 2: roof movement simulation

In this part, the effect of length of walls on behavior of the set is studied. Hence, as shown in the Figure 5.32, the length of Wall 2 is multiplied by 1.5. The dimension of set is: 1.6m length in y direction, 1m length in x direction and 0.1m thickness. In this section similar to the Series 1, displacements from 0 to 2mm are gradually applied to the top nodes in x direction and y directions. Afterward, supposed roof rotates 0.05 degree anticlockwise. The results of each section will be discussed and compared to the previous case. It should be noted that the properties of the different series is the same. The lateral strength of the set under loading in y direction for each individual walls and the set are shown in Figure 5.59(a). Wall 1 is loaded under out-of-plane displacements so it undergoes lower lateral resistance whereas Wall 2 is subjected to in-plane displacements and it carries out much bigger lateral displacements. The significant drop in pushover curves is due to slipping the top row of horizontal mortars of Wall 2. Figure 5.59(b) indicates that the combination of walls with larger dimensions is much more stiffer and its ultimate lateral strength is noticeably higher than that of with lower length but it exhibits the brittle behavior and the drop in the pushover curve for the walls with larger length is much greater than the one with smaller length.

Series 2: supposed roof 's center of gravity of walls rotates 0.05 degree

In this section, the supposed roof at the top of the walls rotates 0.05 degree. As the dimension of Wall 2 in Series 2 is 1.5 greater than Series 1, the total movement of walls in x and y direction is much more greater than Series 1. For example, maximum displacement in x and y directions are 7mm and 4.39mm. By applying this rotation to the supposed roof, some stepped cracks are formed in Wall 1 (as shown in Comparing the results of different series with different dimensions under the same rotation reveals that the structure with larger dimension is subjected to larger displacements with the same rotation. The larger wall carries out larger displacements while the smaller one undergoes the lower loads.

Conclusion

Studying the behavior of masonry walls under out-of-plane loading was the main purpose of this chapter. In the first part, two simple walls with different boundary conditions were subjected to out-of-plane displacement at various locations. The effects of cohesion of mortars, normal stress and compression strength of masonry on out-of-plane behavior of wall were investigated. The results can be concluded as:

• Resistance of a masonry wall under out-of-plane displacements is much lower than that of under in-plane ones. The wall's resistance depends on boundary condition and the way out-of-plane loads are applied to the wall.

• Cohesion of mortars has a significant effect on behavior of masonry wall under out-ofplane loads. By raise of cohesion, the resistance of wall under shear loads, consequently under out-of-plane displacement increases and masonry wall carries out larger out-of-plane displacements.

• Initial normal stress has a great influence on masonry wall performance under out-ofplane displacements. High level of initial stress on the wall causes horizontal mortars open later under out-of-plane displacements and on the other side, according to Mohr-Coulmob criterion, the shear strength of masonry wall directly depends on normal stress and enhances by normal stress increase.

• Compression strength of masonry affects the resistance of masonry under out-of-plane loads.

In some cases, multiplying the compression strength of masonry by 10 did not change the lateral resistance of the wall because the loads were not sufficient to cause crushing failure in the mortars.

• Opening of vertical and horizontal interfaces are the main reason of failure of masonry walls under out-of-plane loads. Thus, the tensile strength of the mortars has an important role on its behavior under these loads.

Second part of this section was attributed to the behavior of a set composed of two walls which were connected together by vertical mortars. The size of walls was the same. First, one of the walls was subjected to in-plane displacements which were acted as out-of-plane for the other wall. The properties of the walls were changed to investigate the effect of materials strength on performance of the set. The following results were obtained:

• Masonry wall shows higher structural strength when perpendicularly connected to a similar wall compared with a single wall standing by itself.

• Property of mortars in the intersection of two perpendicular walls has a great role on performance of the set. A strong intersection can transfer the loads well and decrease the damage of the set.

• Cohesion of mortars affects the behavior of walls both under in-plane and out-of-plane loads. As cohesion increases, the performance of the set and each individual wall improves and their strength under in-plane and out-of-plane displacements increases.

• Initial normal pressure imposed on the wall has a considerable effect on the behavior of masonry walls. When higher normal stress is imposed initially on the wall, its performance under in-plane and out-of-plane loads increases. Larger lateral loads are carried out by the wall depending on the geometry and boundary condition of walls. On the other side, in the wall with higher level of pre-compression, the shear failure of bottom horizontal rows can be prevented.

Afterward, a rigid roof was supposed to connect both walls and the roof's movement was simulated by moving all top nodes simultaneously. So, first, all top nodes move in x direction 2mm, then move in y direction 2mm (the length of the set in x direction was 1m whereas it is 1.1m in y direction), thereafter the same displacements at x and y direction were imposed to the top nodes and finally walls rotate 0.05 degree around the center of gravity of supposed roof. The numerical modeling shows that:

• Aspect ratio of wall subjected to the displacement has an important role on the behavior of the wall. The ultimate strength of a masonry wall under out-of-plane displacements is higher for the wall with lower aspect ratio. The same results are also obtained for the strength of the walls under in-plane displacements where a set composed of two perpendicular walls is subjected to displacements.

• Response of the wall highly depends on relative displacement applied on it. The stiffness as well as ultimate lateral strength of the set is influenced by such relative displacements.

• Roof's rotation causes diagonal cracks in the wall with lower length, sliding shear failure in the wall with larger length and separation of walls from their intersection. The results show that properties of mortars in the connection of two walls have an important role on the performance of the assembly.

Finally, the length of the perpendicular wall was changed in order to see how the geometry of different walls can influence the behavior of the set. Results show the behavior of the set as well as each individual wall, improve by using a larger wall in the system. The lateral strength of the wall in the direction parallel to larger one is greater. It was shown that the main issue in these cases is separation of walls due to sliding and opening of vertical mortars in the intersection of walls.

Summary and conclusion

Masonry construction involves intricate structures with complex behavior under different loads. These structures are wide-spread around the world constituting a great number of existing facilities as well as historical landmarks in different countries. Therefore, design, maintenance, and reinforcement of such structures against different load conditions are at the center of attention of many researchers. Safety assessment of such complex structures necessitates development of robust modeling tools and techniques that can analyze the behavior of such constructions in order to enhance their structural performance. The comprehensive literature review shows variation of experimental test results due to the effect of different parameters that contribute to the behavior of such structures. High cost of conducting experiments and dependence of the obtained data on the experimental conditions make numerical methods competitive. Macro and micro modeling, two well established numerical methods, are comprehensively investigated. A robust computational tool, developed to conduct this research, is used to analyze the masonry structure in a three dimensional environment. Unlike the past studies, the 3D analysis of masonry wall allowed studying its out-of-plane behavior as well. The performance of unreinforced masonry wall under both in-plane and out-of-plane loading were studied in detail in chapters 4 and 5.

Macro modeling: This strategy is selected in order to assess the ultimate strength of infill masonry panels. The author tested the numerical models developed in past research works to reproduce the experimental data. However, some drawbacks of those methods were revealed and the new tri-linear stress-strain curve is proposed to address those limitations. Determination of modulus of elasticity value plays a great role in obtaining meaningful results in this approach. The proposed model is justified again by comparing the obtained numerical results with experimental data available in the literature. This model is mainly used to model infill masonry panels. In the proposed macro model, flexural behavior of masonry wall can not be captured because the connection of macro-element and frame are supposed to be rigid and the relative rotation between wall and frame can not be simulated. Moreover, the failure mechanism of masonry wall and the local damages which are very important factors for the rehabilitation of masonry walls, can not be predicted by this model either. Therefore, the study on behavior of masonry walls has been further developed using micro-modeling.

Micro modeling strategy: in-plane loading A masonry wall has been modeled in three dimension configuration using micro-modeling strategy where bricks are modeled with 3D solid elements whereas interface elements are used for representing mortars. The bricks are supposed to remain unchanged whereas the nonlinear behavior of this complex composite is supposed to happen in weakness planes namely interface elements. Two different multisurface models, model I and model II were developed in the program, capable to predict tension, shear failures and tension, shear and compression failures as well as softening behavior for strength of the wall respectively. The elastic properties and crack-206 ing profile of masonry shear wall under in-plane loads were reproduced well with model I but the model was not capable to assess the maximum lateral strength of the wall. In addition, a considerable difference between the obtained ultimate lateral strengths from numerical and experimental data was found for the wall with an opening. Studies showed that considering compression failure of the mortar and limitation of shear stress in the interface element with its compressive strength has an essential role to predict the real ultimate strength of masonry wall under in-plane loads. The validity of the later model was verified by comparing its response to experimental data for both the wall without and with opening under in-plane loads. The effect of aspect ratio (ratio between wall's height and length) on performance and damage profile of the masonry wall was investigated for two different categories: AS > 1 and AS < 1. Results demonstrate that the wall with lower aspect ratio is stiffer and undergoes greater lateral strength but it exhibits brittle behavior. Failure mode in this wall was sliding shear failure of a row of horizontal mortars. Moreover, analyzing the behavior of the walls with the same aspect ratio and different dimensions shows that the lower wall carries out lower lateral load. On the other side, the failure mechanism of walls with the aspect ratio greater than 1 was compression failure of corner mortars. The pushover curves of different walls showed that the walls with greater aspect ratio have lower strength under lateral loads.

Different properties of mortars such as: its tensile strength, cohesion, compressive strength, the slope of compression cap and initial normal stress on the wall were changed and the lateral behavior of the wall and profile of cracks were drawn for each case. Then, the length of the cracks in different types of elements were measured and used as a damage index. It was shown that sliding of horizontal mortars and opening of vertical mortars are the most important issues in failure of masonry walls. Increasing the initial normal stress on the wall decreases considerably both of these parameters and improves the performance of the wall. However, the cracks being distributed smoothly in different elements, the wall exhibits more stable behavior. The tension failure of bricks and horizontal mortars do not influence considerably the response of masonry walls. Finally, the obtained cracking profiles were associated to the walls with different material characteristics.

Micro modeling strategy: out-of-plane loading The validated 3D micro-modeling is implemented for analysis of unreinforced masonry walls' behavior under out-of-plane movement. First, a masonry wall with two different boundary conditions was subjected to out-of-plane displacements. Results show that initial normal stress on the wall and cohesion of mortars have positive influence on the behavior of masonry wall under this type of loading while its compressive strength is not very important. Opening of vertical elements and sliding of horizontal ones in parallel and perpendicular directions to that of loading, are the most important damage indexes. According to the numerical modeling, the strength of masonry walls under out-of-plane actions is much lower than that of under in-plane loads. It should be noted that sometimes the concentration of loads causes the failure in the wall. The effect of surrounding walls on behavior of the masonry wall under different types of loading was investigated using two sets of two perpendicular walls with different geometries. The sets were

Effect of mode I fracture energy GI

In this section, the mode I fracture energy is changed and its behavior will be studied.

• As shown in figure, different fracture energy for mode III results approximately the same crack patterns and the difference between load displacement curves are not great. Diagonal crack pattern, sliding at top and bottom horizontal joints as well as brick tension failure can be seen in these walls. Hence, the value of GIII has not great influence on the behavior of shear wall.

  87[N/mm 2 ]; tan(φ 0 ) = 1.01;tan(φ 0 ) = 0.73; GII = 0.0580.13 * σ[N mm/mm 2 ] [86] . . . . . . . . . . . . . . . . . . . . . 4.35 (a)Deformed shape (b) stress-strain curve of masonry in shear . . . . . . . . . . . 4.36 Tensile behavior, experimental results from Van der Pluijm (1992)[85], with F t 0 = 0.30[N/mm 2 ] and GI = 0.012[N mm/mm 2 ] . . . . . . . . . . . . . . . . . . . . . . 4.37 (a)Deformed shape (b) stress-strain curve of masonry in direct tension . . . . . . . 4.38 Composite yield function for 3D masonry wall configuration used in this study . . 4.39 (a) Variation of compression cap with respect to F c (b) Comparison between compression cap used in this study and that of given by Lourenco in 2D [60] . . . . . 4.40 Wall deformation under monotonic loading for model II in Plane XZ . . . . . . . . 4.41 Load displacement diagram for model II . . . . . . . . . . . . . . . . . . . . . . . 4.42 Profile of cracks at (a) local peak d = 1.1mm (b) local drop d = 1.3mm . . . . . . 4.43 Horizontal forces which are carried out by each brick, composite yield surface . . . 4.44 Comparison between numerical modeling results for yield function with and without compression cap for model II and model I, respectively . . . . . . . . . . . . . 4.45 Profile of cracking for interface elements for model II at d = 4mm . . . . . . . . . 4.46 Stress distribution at (a) d = 1mm (b) d = 2mm (c) d = 4mm . . . . . . . . . . . 4.47 Profile of cracking in the wall with different θ . . . . . . . . . . . . . . . . . . . . 4.48 Pushover curve for the wall with different θ . . . . . . . . . . . . . . . . . . . . . . 4.49 Results of analysis at displacement of 1mm . . . . . . . . . . . . . . . . . . . . . . 4.50 Results of analysis at displacement of 2mm . . . . . . . . . . . . . . . . . . . . . . 4.51 Results of analysis at displacement of 20mm . . . . . . . . . . . . . . . . . . . . . 4.52 Load displacement curve for (a) model II (b) comparison between model I and model II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.53 Results of analysis at displacement of 3mm . . . . . . . . . . . . . . . . . . . . . . 4.54 Total deformed shape (a) model II (b) model I . . . . . . . . . . . . . . . . . . . . 4.55 Profile of cracking for wall with opening at d = 20mm . . . . . . . . . . . . . . . . 4.56 Results for wall with AS = 0.5 (H = 0.5H 0 ) model II . . . . . . . . . . . . . . . . 4.57 Load-displacement curves for wall using models I and II, AS = 0.5 (H = 0.5H 0 ) . 4.58 Results for wall with AS = 2/3 (L = 1.5L 0 ) model II . . . . . . . . . . . . . . . . 4.59 Load-displacement curves for wall using models I and II, AS = 2/3 (L = 1.5L 0 ) . 4.60 Results for wall with AS = 0.5 (L = 2L 0 ) model II . . . . . . . . . . . . . . . . . . 4.61 Load-displacement curves for wall using models I and II, AS = 0.5 (L = 2L 0 ) . . . 4.62 Load-displacement diagram for different aspect ratios H/L < 1 . . . . . . . . . . 4.63 Results for wall with AS = 2 (L = 0.5L 0 ) model II . . . . . . . . . . . . . . . . . . 4.64 Load-displacement curves for wall using models I and II, AS = 2 (L = 0.5L 0 ) . . . 4.65 Results for wall with AS = 1.5 (H = 1.5H 0 ) model II . . . . . . . . . . . . . . . . 4.66 Load-displacement curves for wall using models I and II, AS = 1.5 (H = 1.5H 0 ) . 4.67 Results for wall with AS = 2 (H = 2H 0 ) model II . . . . . . . . . . . . . . . . . . 4.68 Load-displacement curves for wall using models I and II, AS = 2 (H = 2H 0 ) . . . 4.69 load-displacement diagram for different aspect ratios H/L > 1 . . . . . . . . . . . 4.70 Deformed shapes and crack patterns in the walls with different mortars' cohesion at d = 4mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.71 Pushover curves for the walls with different cohesion . . . . . . . . . . . . . . . . 4.72 Numerating rows and columns in the wall . . . . . . . . . . . . . . . . . . . . . . 4.73 Length of cracks in different types of elements for walls with different cohesion at d = 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.74 Deformed shapes and crack patterns in the walls with different tensile strengths at d = 4mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.75 Pushover curves for the walls with different tensile strengths . . . . . . . . . . . . 4.76 Length of cracks in different types of elements for walls with different tensile strengths at d = 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.77 Deformed shapes and crack patterns in the walls with different compressive strengths at d = 4mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.78 Pushover curves for the walls with different compressive strengths . . . . . . . . . 4.79 Length of cracks in different types of elements for walls with different compressive strengths at d = 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.80 Deformed shapes and crack patterns in the walls with different initial normal stresses at d = 4mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.81 Pushover curve for the wall with different σ n . . . . . . . . . . . . . . . . . . . . . 4.82 Length of cracks in different types of elements for walls with different initial vertical stresses at d = 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 (a) Failure of masonry wall, Haiti earthquake (b) Out-of-plane collapse of masonry wall, Champerico, Guatemala . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 (a) Damage of a rural masonry building at Yukar Demirci Village, Turkey [16](b) Failure of a rural masonry building in Palu, Turkey [16] . . . . . . . . . . . . . . . 5.3 Studied walls subjected to out of plane displacement . . . . . . . . . . . . . . . . 5.4 (a), (b) Deformed shape (c) total crack pattern in the wall, Case A, at d top = 2mm 5.5 Lateral displacement vs. horizontal loads in y direction, Case A . . . . . . . . . . 5.6 Cracks on the wall (a) at d = 0.91mm (b) at d = 1.03mm, Case A . . . . . . . . 5.7 Stress distribution in bricks at d = 2mm for Case A . . . . . . . . . . . . . . . . . 5.8 Crack pattern in the masonry wall d = 2mm, Case A . . . . . . . . . . . . . . . . 5.9 Cracking profile for the walls with different mortar's cohesion . . . . . . . . . . . . 5.10 Effect of mortar's cohesion on pushover curve . . . . . . . . . . . . . . . . . . . . 5.11 Crack pattern (a) before and (b) after stepped cracks for wall with C = 3.5e4P a 5.12 Length of different types of cracks in different interface elements at rows and columns of walls with different C, Case A . . . . . . . . . . . . . . . . . . . . . . 5.13 Total crack pattern for walls with different σ n0 . . . . . . . . . . . . . . . . . . . . 5.14 Influence of initial normal stress on the behavior of the wall . . . . . . . . . . . . 5.15 Length of different types of cracks in different interface elements at rows and columns of walls with different σ n0 , Case A . . . . . . . . . . . . . . . . . . . . . . 5.16 Profile of cracking for walls with different compressive strength for masonry . . . . 5.17 Influence of masonry compressive strength on the behavior of wall . . . . . . . . . 5.18 Length of different types of cracks in different interface elements at rows and columns of walls with different F c0 , Case A . . . . . . . . . . . . . . . . . . . . . . 5.19 (a), (b) Deformation of wall (c) cracking profile of the wall, Case B . . . . . . . . 5.43 Deformation and crack pattern for Series 1 with different σ n0 , displacements are applied to Wall 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 5.44 Pushover curve for different σ n0 , Series 1, Wall 1 loaded in x direction . . . . . . 189 5.45 Deformation in different planes, Series 1, d = 2mm, all top nodes move in x direction190 5.46 Crack's pattern for Series 1 (a) Wall 1 subjected to u x (b) Walls 1 and 2 subjected to u x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.47 Load displacement curves for Series 1 (a) all top nodes are loaded in x direction (b) comparison between two cases: Series 1, all nodes loaded and Wall 1 top nodes are loaded in x direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 5.48 Deformation in different planes, Series 1, d = 2mm, all top nodes move in y direction192 5.49 Load displacement curve for Series 1 (a) all top nodes are loaded in y direction (b) comparison between two cases: Series 1, all nodes move in x direction and all nodes move in y direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 5.50 Load displacement curve comparison: (a) walls subjected to in-plane displacements (b) walls are subjected to out-of-plane displacements . . . . . . . . . . . . . 193 5.51 Deformation and total crack pattern for walls subjected to 0.05 degree rotation, Series 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 5.52 Lateral load-displacement curves of walls in (a) x direction (b) y direction, Series 1, 0.05 degree rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 5.53 Crack pattern in the masonry walls ,Series 1, roof rotates 0.05 degree . . . . . . . 196 5.54 Deformation in different planes, Series 2, all top nodes move d = 2mm in x direction197 5.55 Total cracks pattern, all top nodes move 2mm in x direction (a) Series 2 (b) Series 1198 5.56 Load displacement curve, all top nodes move 2mm in x direction (a) Series 2 (b) comparison between Series 2 and Series 1 . . . . . . . . . . . . . . . . . . . . . . . 198 5.57 Deformation in different planes, Series 2, all top nodes move d = 2mm in y direction199 5.58 Total cracks pattern, all top nodes move 2mm in y direction (a) Series 2 (b) Series 1200 5.59 Load displacement curve, all top nodes move 2mm in y direction (a) Series 2 (b) comparison between Series 2 and Series 1 . . . . . . . . . . . . . . . . . . . . . . . 200 5.60 Deformation and total crack pattern for walls whose supposed roof is subjected to 0.05 degree of rotation, Series 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 5.61 Lateral load-displacement curves of walls in (a) x direction (b) y direction, Series 2, 0.05 degree rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Collapse in unreinforced masonry building (a) Westmorland California [102] (b) Port-au-Prince, Haiti [45]

Figure 2 . 1 :

 21 Figure 2.1: (a) Micro modeling approach to model infilled frame (b) macro modeling strategy for representing infilled frame [88], [52]

Figure 2 . 2 :

 22 Figure 2.2: Different Modeling strategies for masonry structures (a) masonry sample (b) detailed micro-modeling (c) simplified micro-modeling (d) macro-modeling

Figure 2 . 3 :

 23 Figure 2.3: (a) Single strut model suggested by Stafford-Smith [1966] [99] (b) six-strut model proposed by Chrysostom [1991] [21]

Figure 2 .Figure 2 . 4 :

 224 Figure 2.4: (a) Strut model by [30] (b) trilinear relations: stress-strain relation and typical force-deformation relation for struts proposed by El-Dakhakhni et al. [2003] [30]

Figure 2 . 5 :

 25 Figure 2.5: Force-displacement envelope curve of the equivalent strut, Decanini et al.[2004] [27]

Figure 2 . 6 :

 26 Figure 2.6: (a) Multi-strut model (b) variation of the strut area proposed by Crisafulli et al. [2007][START_REF] Crisafulli | Proposed macro-model for the analysis of infilled frame structures[END_REF]

Figure 2 . 7 :Figure 2 . 8 :Figure 2 . 9 :

 272829 Figure 2.7: Three-dimensional SAT model of URM infill wall proposed by Hashemi et al.[2007][46]

( a )Figure 2 . 10 :

 a210 Figure 2.10: Failure surface considered in Page [1987] [82] and Riddington et al. [1990] [91] studies, respectively

Figure 2 .Figure 2 . 12 :

 2212 Figure 2.11: (a) Yield surface used in Lourenco et al. [1983] [59] (b) hyperbolic yield criterion proposed by Lotfi et al.[1994] modelings[START_REF] Lotfi | Interface model applied to fracture of masonry structures[END_REF]

Figure 2 .

 2 Figure 2.13: (a) Yield function for interface elements (b) hyperbolic asperity model by Giambanco et al. [2001] [40]

Figure 2 . 14 :

 214 Figure 2.14: Three linear yield surface of interface element by Sutcliffe et al. [2001] [101]

Figure 2 .

 2 Figure 2.15: (a) Single masonry unit (b) piecewise linear yield surface used by Chaimoon et al. [2007] [17]

Figure 2 . 16 :

 216 Figure 2.16: Schematic representation of masonry wall modeled by equivalent frame model (a) Belmouden et al. [2009] [9] (b) Bayer et al. [2012] [11]

Figure 2 .Figure 2 . 17 :Figure 2 . 18 :

 2217218 Gambarotta et al. [1997] [START_REF] Gambarotta | Damage models for the seismic response of brick masonry shear walls. Part II: the continuum model and its applications[END_REF] then extended their model[START_REF] Gambarotta | Damage models for the seismic response of brick masonry shear walls. Part I: The mortar joint model and its applications[END_REF] to evaluate cyclic behavior of masonry by using a continuum model in which masonry wall was made of two layers, bed joint and brick and head joint. A constitutive formulation based on homogenization process was applied to the bricks. Decohesion and slipping was assumed for bed mortar interface while damage and failure of brick were considered for bricks and head joints. Model was improved by considering compressive strength of masonry and shear strength of brick see Figure2.18(b). The model was successfully implemented to analyse the behavior of large masonry structures with opening. Numerical calculations showed that model can reproduce damage distribution and cyclic behavior of structures well.

Figure 2 . 19 :

 219 Figure 2.19: Motion of auxiliary yield function proposed by[START_REF] Oliveira | Implementation and Validation of a Constitutive Model for the Cyclic Behaviour of Interface Elements[END_REF] in tension and compression[START_REF] Oliveira | Implementation and Validation of a Constitutive Model for the Cyclic Behaviour of Interface Elements[END_REF]

Figure 2 . 20 :

 220 Figure2.20: Scheme of an irregular masonry defined by four rigid elements[START_REF] Casolo | Rigid element model for in-plane dynamics of masonry walls considering hysteretic behaviour and damage[END_REF]

Figure 2 . 21 :

 221 Figure 2.21: Hysteretic behavior of the axial (left) and shear (right) connecting springs used in Casolo et al. [2007] model [14]

Figure 2 . 22 :

 222 Figure 2.22: Cyclic stress-strain relationship for unloading-reloading stages, initial loading in (a) tension (b) compression and (c) shear proposed by Karapitta et al. [2011] [51]

Figure 3 . 1 :

 31 Figure 3.1: Formation of diagonal strut[START_REF] Crisafulli | Analysis of infilled frame structures, seminar on masonry and earthen structures[END_REF]

Figure 3 . 2 :Figure 3 . 3 :

 3233 Figure 3.2: Corner Crushing failure (CC)

Figure 3 . 4 :

 34 Figure 3.4: Sliding Shear failure (SS)

Figure 3 . 5 :

 35 Figure 3.5: Diagonal Cracking failure (DCK)

Figure 3 . 6 :

 36 Figure 3.6: Frame Failure (FF)

Figure 3 . 7 :

 37 Figure 3.7: CSTB test set-up[START_REF] Brgm | Dynamiques concept, CSTB, Une approche mecanique d'evaluation de la vulnerabilite sismique des maconneries EVSIM[END_REF]

Figure 3 . 8 :

 38 Figure 3.8: Blocks and joints properties

Figure 3 . 9 :

 39 Figure 3.9: Masonry wall properties and results of experiments

Figure 3 . 10 :Figure 3 . 11 :

 310311 Figure 3.10: Crack propagation, load displacement diagram, block prototype and utilization percentage

Figure 3 . 12 :

 312 Figure 3.12: Masonry strut material model recommended by FEMA 356

Figure 3 .

 3 Figure 3.13: (a) Material stress-strain curve (b) force-displacement diagram using FEMA 356 strut model

  .

8 .

 8 Material model for calibrated model is shown in Figure 3.14. As shown in this figure, the residual resistance of masonry is taken into account.

Figure 3 . 14 :

 314 Figure 3.14: Masonry strut material model proposed by Hashemi et al. [2007][46]

  .09M P a. The material property and obtained pushover curve of structure are shown in Figure 3.15(a) and Figure 3.15(b). The detailed computations are represented in Appendix A.

Figure 3 .

 3 Figure 3.15: (a) Material stress-strain curve (b) force-displacement diagram using calibrated strut model

Figure 3 . 16 :

 316 Figure 3.16: Force-displacement envelope for conventional masonry infill walls proposed by Mostafaei et al. [2004] [76]

  (a).

Figure 3 .

 3 Figure 3.17: (a) Material stress-strain curve (b) force-displacement diagram using Mostafaei and Kabasaki [76] strut model

Figure 3 . 18 :

 318 Figure 3.18: Shear force-displacement curve proposed for diagonal struts

Figure 3 .Figure 3 .

 33 Figure 3.19: (a) Material stress-strain curve used in model (b) load-displacement curve of CSTB test using trilinear proposed model

Figure 3 . 20 :Figure 3 .

 3203 Figure 3.20: Numerical simulation of shear base force and top displacement curve in an infilled frame under monotonic loading using different methods

Figure 3 .

 3 22(a) whereasFigure 3.22(b) represents the comparison between experimental data and computational results.

Figure 3 .

 3 22(b) shows the proposed macro model can reproduce the load-displacement curve of tested wall very well. There is a very good agreement between initial stiffness, ultimate shear strength and post-peak response of the wall obtained from computational model and experimental evidence.

Figure 3 .

 3 Figure 3.22: (a) Stress-strain curve of material (b) load-displacement curve obtained for trilinearproposed model used for Vermeltfoort test[START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF]

Figure 4 . 1 :

 41 Figure 4.1: Masonry failure mechanisms: (a) joint tensile cracking (b) joint slipping (c) unit direct tensile cracking (d) unit diagonal tensile cracking (e) masonry crushing

Figure 4 . 2 :

 42 Figure 4.2: Failure modes for masonry walls subjected to in-plane loads (a) diagonal tensile failure (b) sliding shear failure (c) bending failure (crushing of masonry) [96]

Figure 4 . 3 and

 43 Figure 4.4, respectively.Interface elements also are implemented in the middle of brick to capture tensile failure of bricks. Isotropic Hexagonal iso-parametric continuum element (3D Solid volume) with Mohr-coulomb failure criterion is used to represent bricks. The material characteristics of solid elements are chosen somehow to restrict brick's behavior in elastic domain. Each brick is modeled using two solid elements and an interface element between the solids, see Figure4.5. So, the tension failure of the brick is reproduces by an interface element.

Figure 4 . 3 :

 43 Figure 4.3: GEFdyn 3D solid element with eight integration points

Figure 4 . 4 :

 44 Figure 4.4: GEFdyn zero thickness interface element [79]

Figure 4 . 5 :

 45 Figure 4.5: Potential crack planes in the middle of units are modeled by zero-thickness interface element

Figure 4 .

 4 6 represents the composite yield surface which is used in model I. As shown in this figure, the yield function is given for a 3D configuration. This yield function consists of two parts: A straight tension cut-off condition for simulating opening of cracks as well as tension failure of the mortars and the Mohr-Coulomb failure criterion for capturing the sliding of the joints and shear failure of mortars under compressive normal stresses. The behavior of interface element under normal stress and shear stress is shown in this Figure 4.7. As shown in figure, no softening behavior is considered for masonry in this model in tension and shear. No degradation is allowed in cohesion and friction angel of interface material. Whenever shear stress reaches yield value, it behaves nonlinear and shear-displacement relation obeys perfect plastic theory.

Figure 4 . 6 :Figure 4 . 7 :

 4647 Figure 4.6: Simple yield surface for interface elements

Figure 4 . 8 :

 48 Figure 4.8: Coulomb friction law, definition of dilatancy angle[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

  [1994][START_REF]Structural masonry: a experimental numerical basis for practical designrules[END_REF], Raijmakers and Vermeltfoort[1992][START_REF] Vd | Material properties of masonry and its components under tension and shear[END_REF] and Vermeltfoort and Raijmakers[1993][START_REF] Vd | Shear behavior of bed joints[END_REF]. One type is a continuous masonry wall and the other one is the wall with central opening. As described in Figure4.9, first, the vertical precompression was applied to the wall. Then, a horizontal monotonically increased displacement was imposed at the top of the wall whereas the top and bottom boundaries were kept horizontal and vertical movement at the top of the wall was precluded. The shear walls have a width of 990[mm] and height of 1000 [mm] built up with 18 horizontal rows and 16 horizontal active courses and 2 courses that are clamped in steel beams.

Figure 4 . 9 :

 49 Figure 4.9: Experimental test (a) first: Vertical loading (b) second: Horizontal loading [60]

Figure 4 . 10 :

 410 Figure 4.10: Experimental crack pattern after loading[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

Figure 4 .

 4 Figure 4.12 also represents the arrangement of units, horizontal and vertical interfaces as well as interfaces in the middle of bricks.

Figure 4 . 11 :

 411 Figure 4.11: GEFDyn model geometry and elements

Figure 4 . 12 :

 412 Figure 4.12: Different categories of interface elements used for modeling masonry wall

Figure 4 . 13 :

 413 Figure 4.13: Deformed shape of masonry wall at d = 2 mm model I

Figure 4 .

 4 [START_REF] Casolo | A simplified homogenization-discrete element model for the non-linear static analysis of masonry walls out-of-plane loaded[END_REF] due to difference between stiffness of mortars and bricks a diagonal strut parallel to the diagonal of the wall

Figure 4 .

 4 Figure 4.14: Load-Displacement diagram for model I

Figure 4 .

 4

17

 17

  (c) and Figure4.

17

 17

  Figure 4.17: Crack pattern in the masonry wall at d = 2mm

Figure 4 .

 4 [START_REF] Chen | Seismic Evaluation of RC Buildings Infilled with Brick Walls[END_REF] shows the experiment's wall which was tested and subjected to vertical pre-compression equal to 0.3 [N/mm 2 ]. The material properties and dimension of the wall is the same as the one without opening. The central opening creates two weak piers and forces the struts to be developed around the opening.

Figure 4 . 18 :

 418 Figure 4.18: Wall with central opening tested by Raijmakers and Vermenltfoort [1993] [86]

Figure 4 .

 4 Figure 4.19 shows the crack pattern of the two walls with the same properties. Diagonal stepped cracks are developed at two sides of the central opening. Tensile cracks also can be found at top and bottom of left and right small piers respectively. Finally top left and bottom right toes fail because of crushing.

Figure 4 . 19 :Figure 4 . 20 :

 419420 Figure 4.19: Experimental crack patterns for different tests[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

Figure 4

 4 d [m] Horizontal force [kN] Numerical Experimental: Wall 2J Experimental: Wall 3J

Figure 4 . 21 :

 421 Figure 4.21: Load displacement diagram of experiment and numerical model for model I

  Figure 4.22: Crack pattern in the masonry wall with opening d = 2mm for model I

Figure 4 . 23 :

 423 Figure 4.23: Deformed shape, pushover curve and crack pattern in the masonry wall for aspect ratios less than 1 at d = 2mm for model I

Normalized

  d [m] Horizontal force [kN] Experimental Numerical z=0.5z0 x=x0 model I Numerical z=z0 x=1.5x0 model I Numerical with z=z0 x=2x0 model I (a) Push-over curve Horizontal displacement d [m/m] (d/H) Normalized Horizontal force [kN/m] (F/L) Normalized Push-over curve for H/L < 1 Experimental Numerical z=0.5z0 x=x0 model I Numerical z=z0 x=1.5x0 model I Numerical z=z0 x=2x0 model I (b) Normalized push-over curve

Figure 4 .

 4 Figure 4.24: load-displacement diagram for different aspect ratios H/L < 1 for model I

  Figure 4.25: Deformed shape, pushover curve and crack pattern in the masonry wall for aspect ratios greater than 1 at d = 2mm for model I

Normalized

  Horizontal displacement d [m/m] (d/H) Normalized Horizontal force [kN/m] (F/L) Normalized Push-over curve for H/L > 1 Experimental Numerical z=z0 x=0.5x0 model I Numerical z=1.5z0 x=x0 model I Numerical z=2z0 x=x0 model I (b) Normalized push-over curve

Figure 4 .

 4 Figure 4.26: load-displacement diagram for different aspect ratios H/L > 1 for model I

Figure 4 . 27 :

 427 Figure 4.27: One cycle displacement

Figure 4 .

 4 Figure 4.28: (a) Deformed shape (b) load-displacement curve in the masonry wall for one cycle loading

  (a) Sliding of Horizontal Mortars (b) Opening of Horizontal Mortars (c) Sliding of Vertical Mortars (d) Opening of Vertical Mortars (e) Opening of Bricks (f) Total crack pattern

Figure 4 . 29 : 95 (

 42995 Figure 4.29: Crack pattern in different elements of the wall for one cycle loading at final step d = 0mm

Figure 4 . 30 :

 430 Figure 4.30: Multiple cycles displacement

Figure 4 . 32 :

 432 Figure 4.32: Crack pattern in different elements of the wall for multiple cycles loading at final step d = 0mm

Figure 4 .

 4 Figure 4.33 shows different load displacement curves obtained from numerical modeling for one cycle and multiple cycles loading and experimental data from monotonic loading. As shown in this figure, model I is not able to predict the cyclic response of the wall. The reduction of wall's stiffness because of loading-unloading process and cyclic loading as well as maximum lateral strength of the wall could not be captured in this model.

Figure 4 . 33 :

 433 Figure 4.33: Comparison between load-displacement curves from cyclic to monotonic loading

Figure 4 .

 4 34, represents the results of direct shear tests which have been done by Pluijm et al. [1993][START_REF] Vd | Shear behavior of bed joints[END_REF]. The solid curve which is defined by exponential function can match with experimental data well. Thus, in this formulation, the softening behavior of masonry is controlled by shear plastic displacement.

Figure 4 . 34 :

 434 Figure 4.34: Shear behavior of masonry, experimental results by Van der Pluijm (1993)[86] for different confinement stresses, with C 0 = 0.87[N/mm 2 ]; tan(φ 0 ) = 1.01;tan(φ 0 ) = 0.73; GII = 0.0580.13 * σ[N mm/mm 2 ] [86]

Figure 4 Figure 4 .

 44 Figure 4.35.

Figure 4 . 36 :

 436 Figure 4.36: Tensile behavior, experimental results from Van der Pluijm (1992)[85], with F t 0 = 0.30[N/mm 2 ] and GI = 0.012[N mm/mm 2 ]

. 33 )Figure 4 .

 334 Figure 4.37: (a)Deformed shape (b) stress-strain curve of masonry in direct tension

Figure 4 . 38 :Figure 4 .

 4384 Figure 4.38: Composite yield function for 3D masonry wall configuration used in this study

  (a) Wall deformation at d=2mm (b) Wall deformation at d=4mm (c) Total crack pattern at d=2mm (d) Total crack pattern at d=2mm

Figure 4 . 40 :Figure 4 . 41 :

 440441 Figure 4.40: Wall deformation under monotonic loading for model II in Plane XZ

Figure 4 . 42 :

 442 Figure 4.42: Profile of cracks at (a) local peak d = 1.1mm (b) local drop d = 1.3mm

Figure 4 .

 4 Figure 4.43 shows the summation of horizontal force at the base of each brick. The horizontal force of the wall in y label of pushover curve is the summation of all horizontal forces at the base of the wall. This figure shows the contribution of each brick in horizontal strength of the wall. The color of bricks and the corresponding pushover curve is the same. As shown in this figure,

Figure 4 . 43 :Figure 4 . 44 :

 443444 Figure 4.43: Horizontal forces which are carried out by each brick, composite yield surface

Figure 4 .Figure 4 . 46 :

 4446 Figure 4.45: Profile of cracking for interface elements for model II at d = 4mm

  (a) θ = 25 o (b) θ = 32.7 o (c) θ = 90 o

Figure 4 . 47 :

 447 Figure 4.47: Profile of cracking in the wall with different θ

Figure 4 . 48 :

 448 Figure 4.48: Pushover curve for the wall with different θ

Figure 4 . 49 :

 449 Figure 4.49: Results of analysis at displacement of 1mm

Figure 4 .

 4 49, diagonal stepped cracks initiate from the top right and bottom right corners of the opening. At this point some of horizontal elements on bottom right and top left of the wall and small piers open slightly. By increasing the applied displacements, previous cracks open more and their length increases. Thus at d = 2mm as initial diagonal pattern cannot develop to the compressed toes, two other diagonal cracks parallel to previous ones are developed to the wall's corners. As shown in Figure 4.50, the horizontal mortars located at the top of left and the bottom right of the small piers fail in tension and open and force the adjacent horizontal elements to close and sustain compression loads.

Figure 4 . 50 :

 450 Figure 4.50: Results of analysis at displacement of 2mm

Figure 4 . 51 :

 451 Figure 4.51: Results of analysis at displacement of 20mm

Figure 4 . 52 :

 452 Figure 4.52: Load displacement curve for (a) model II (b) comparison between model I and model II

Figure 4 . 53 :Figure 4 . 54 :

 453454 Figure 4.53: Results of analysis at displacement of 3mm

Figure 4 . 55 :

 455 Figure 4.55: Profile of cracking for wall with opening at d = 20mm

5 (Figure 4 . 56 :

 5456 Figure 4.56: Results for wall with AS = 0.5 (H = 0.5H 0 ) model II

Figure 4 . 57 :

 457 Figure 4.57: Load-displacement curves for wall using models I and II, AS = 0.5 (H = 0.5H 0 )

Figure 4 .

 4 58. First, the corners of wall fail in tension and open. Then at d = 0.69mm sliding shear failure is observed at all horizontal mortars in the first row and after that point the lateral strength which is carried out by the wall decreases considerably. Similar to previous case, reducing cohesion of mortar exponentially by plastic relative shear displacement and insufficient normal stress cause sliding shear failure in the wall. As shown in Figure 4.58(b), wall with L = 1.5L 0 sustains less strength than wall with L = L 0 after d = 0.93mm because of its failure.

Figure 4 . 58 :

 458 Figure 4.58: Results for wall with AS = 2/3 (L = 1.5L 0 ) model II

  .5x0 model I Numerical z=z0 x=1.5x0 model II

Figure 4 .

 4 Figure 4.59: Load-displacement curves for wall using models I and II, AS = 2/3 (L = 1.5L 0 )

Figure 4 .

 4 62(a), all three walls are stiffer 120 model II (b) Pushover curve (c) Total crack pattern

Figure 4 . 60 :

 460 Figure 4.60: Results for wall with AS = 0.5 (L = 2L 0 ) model II

Figure 4 . 61 :

 461 Figure 4.61: Load-displacement curves for wall using models I and II, AS = 0.5 (L = 2L 0 )

Normalized

  Numerical z=0.5z0 x=x0 model II Numerical z=z0 x=1.5x0 model II Numerical z=z0 x=2x0 model II (a) Push-over curve Horizontal displacement d [m/m] (d/H) Normalized Horizontal force [kN/m] (F/L) Experimental z=z0, x=x0 Numerical z=z0 x=1.5x0 model II Numerical z=0.5z0 x=x0 model II Numerical z=z0 x=2x0 model II (b) Normalized Push-over curve

Figure 4 . 62 :

 462 Figure 4.62: Load-displacement diagram for different aspect ratios H/L < 1

2 (

 2 (a) H/L = H 0 /0.5L 0 , AS =

Figure 4 . 63 :

 463 Figure 4.63: Results for wall with AS = 2 (L = 0.5L 0 ) model II

Figure 4 .Figure 4 . 64 :

 4464 Figure 4.64: Load-displacement curves for wall using models I and II, AS = 2 (L = 0.5L 0 )

  Figure 4.65(b) shows that the initial stiffness of the wall with aspect ratio greater than 1 is lower than that of the wall with aspect ratio equal to 1.

Figure 4 . 65 :

 465 Figure 4.65: Results for wall with AS = 1.5 (H = 1.5H 0 ) model II

Figure 4 . 66 :

 466 Figure 4.66: Load-displacement curves for wall using models I and II, AS = 1.5 (H = 1.5H 0 )

Figure 4 . 4

 44 67(b). The pushover curves for model I and model II are the same as explained before, see Figure

Figure 4 . 67 :

 467 Figure 4.67: Results for wall with AS = 2 (H = 2H 0 ) model II

Figure 4 . 68 :

 468 Figure 4.68: Load-displacement curves for wall using models I and II, AS = 2 (H = 2H 0 )

Figure 4 .

 4 Figure 4.69(b) shows normalized pushover curve for different cases. As shown in labels, the horizontal force is divided by wall's length whereas displacement applied to the wall is divided by the height of the wall to find the strength of the wall regardless its dimensions. As shown in this figure, the initial stiffness and lateral strength of the walls decrease by aspect ratio increment. For the walls with the same aspect ratio, wall with greater dimensions carries out lower normalized lateral strength. The maximum length of crack in the wall with AS=2 and L = 0.5L 0 is 0.26 mm at d = 2mm, for the wall with AS=1.5 this parameter is 0.22 mm and for the wall with AS=2 and H = 2H 0 , it is 0.16 mm. Thus, by increasing the dimension of wall the length of opening and sliding of horizontal mortars decreases. For the wall with the same aspect ratio, the normal stress on the lower mortars is greater for the one with larger dimension because of its weight, so those elements open and slide less than another one.

Normalized

  Horizontal displacement d [m/m] (d/H) Normalized Horizontal force [kN/m] (F/L) Experimental z=z0, x=x0 Numerical z=z0 x=0.5x0 model II Numerical z=1.5z0 x=x0 model II Numerical z=2z0 x=x0 model II (b) Normalized Push-over curve

Figure 4 .

 4 Figure 4.69: load-displacement diagram for different aspect ratios H/L > 1

  (a) C 0 = 9.5E04P a (b) C 0 = 9.5E04P a (c) C 0 = 1.5E05P a (d) C 0 = 1.5E05P a (e) C 0 = 3.5E05P a (f) C 0 = 3.5E05P a (g) C 0 = 5.5E05P a (h) C 0 = 5.5E05P a (i) C 0 = 7.5E05P a (j) C 0 = 7.5E05P a

Figure 4 . 70 :

 470 Figure 4.70: Deformed shapes and crack patterns in the walls with different mortars' cohesion at d = 4mm

Figure 4 . 71 :

 471 Figure 4.71: Pushover curves for the walls with different cohesion

Figure 4 .

 4 Figure 4.73 shows the length of cracks in different types of elements at d = 2mm. According to Figure 4.73(a) the length of sliding cracks at the top rows are greater than the other ones in all cases.In the middle rows, sliding of cracks decreases with cohesion increment. This figure shows that for two walls with lower cohesion, the length of cracks in different rows are more similar and the maximums belong to rows 1, 5, 13 and 17. According to Figure4.73(b), the horizontal mortars near to the edges of the wall open more than the others and that opening is greater at the bottom rows because of overturning moment. The opening of horizontal interfaces first increases by cohesion increment but for two great ones its value decreases.

Figure 4 . 72 :

 472 Figure 4.72: Numerating rows and columns in the wall

Figure 4 .

 4 73(d) represents the length of opening in vertical mortars. As explained before, as the distribution of opening cracks in three first cases is approximately normal, diagonal cracks have formed in the wall. The length of opening decreases with cohesion increment. The opening of vertical elements in C 0 = 1.5e05P a is greater than for C 0 = 9.5e04P a because as explained before, the top horizontal mortars slide less and instead vertical interfaces open more. The bricks at the edges fail more in tension in cases with lower cohesion. Moreover, the length of opening reduces with cohesion increment and for the walls with C 0 = 5.5e05P a and C 0 = 7.5e05P a no bricks in the wall fail in tension. Sliding of horizontal elements together with opening of vertical elements cause failure in the walls with lower cohesion whereas high total length of sliding cracks in horizontal elements in walls with greater cohesion represents brittle failure in the wall.

Figure 4 . 73 :

 473 Figure 4.73: Length of cracks in different types of elements for walls with different cohesion at d = 2mm

  (c) and(d) show that sliding of the horizontal joints as well as opening of vertical joints at the bottom left of the wall cause failure in the wall. In this case vertical mortars open sooner and their length of opening is greater and their opening causes sliding shear failure at the adjacent horizontal mortars. The first drop in pushover curve at d = 0.9mm corresponds to the formation of diagonal cracks in the wall whereas at d = 2.88mm wall loses its lateral strength significantly due to sliding shear failure of horizontal mortars at the bottom part of the wall.

Figure 4 .

 4 [START_REF] Milani | 3D upper bond limit analysis of multi-leaf masonry walls[END_REF] represents the load-displacement curves for wall with different tensile strengths for the mortars. By increasing the tensile strength for first three cases the lateral strength of the wall increases whereas for the other cases sliding shear failure of a row of mortar causes brittle behavior of masonry wall.As shown in Figure4.76, except walls with f t0 = 8.5e4P a and f t0 = 2.5e5P a the other walls undergo sliding shear failure at the top or bottom rows of mortars. Opening of horizontal mortars at the wall's edges are greater than the middle of wall. Horizontal mortars in walls with sliding shear failure open less than the other walls. The normal distribution of opening and sliding of vertical mortars in walls with f t0 = 8.5e4P a and f t0 = 2.5e5P a represent formation of diagonal cracks in the walls. Bricks also do not fail in tension in walls with sliding shear failure as shown in Figure4.76. The tension failure of bricks decreases by increasing tensile strength of the mortars. Finally, Figure4.76(f) shows that walls with higher opening length in vertical mortars and sliding of horizontal mortars react better under monotonic loads. Cracks are distributed more smoothly in walls with higher lateral strength while the cracks are concentrated in specific elements in walls with brittle behavior.

Figure 4 . 74 :

 474 Figure 4.74: Deformed shapes and crack patterns in the walls with different tensile strengths at d = 4mm

Figure 4 . 75 :

 475 Figure 4.75: Pushover curves for the walls with different tensile strengths

Figure 4 . 76 :

 476 Figure 4.76: Length of cracks in different types of elements for walls with different tensile strengths at d = 2mm

Figure 4 . 77 :

 477 Figure 4.77: Deformed shapes and crack patterns in the walls with different compressive strengths at d = 4mm

Figure 4 . 78 :

 478 Figure 4.78: Pushover curves for the walls with different compressive strengths

Figure 4 .Figure 4 . 80 :

 4480 Figure 4.79: Length of cracks in different types of elements for walls with different compressive strengths at d = 2mm

Figure 4 . 81 :

 481 Figure 4.81: Pushover curve for the wall with different σ n

Figure 4 . 82 :•

 482 Figure 4.82: Length of cracks in different types of elements for walls with different initial vertical stresses at d = 2mm

  Figures 5.1 and Figures 5.1 show different types of cracks in perpendicular masonry walls under different earthquakes. In-plane loading in a wall resulting out of plane failure to the other one.

Figure 5 . 1 :

 51 Figure 5.1: (a) Failure of masonry wall, Haiti earthquake (b) Out-of-plane collapse of masonry wall, Champerico, Guatemala

Figure 5 .

 5 Figure 5.1(a) shows the failure of a masonry wall in Haiti. A diagonal crack is initiated from top right to the middle of the wall which is accompanied by a horizontal sliding crack from middle to the edge of the wall. In Figure5.1(b) damage of two perpendicular masonry walls under earthquake movement is shown. The walls are separated from their intersection and damage is mostly concentrated at the top part of the walls joint. Moreover, a diagonal cracks is also formed in the lower part of the wall I. Some horizontal crack in connection of wall I and roof can be seen whereas wall II also moved and some cracks in its common corner with wall I are appeared. Figure5.2(a) shows the shear sliding failure of a masonry wall in a rural building in Turkey[START_REF] Celepa | Failures of masonry and concrete buildings during the March 8, 2010 Kovanclar and Palu (Elaz) Earthquakes in Turkey[END_REF]. A continuous horizontal crack is obvious in the wall as its length is much greater than its height. Earthquake movement in Turkey also caused a slight damage in a one-story masonry building in Palu, as shown in Figure5.2(b)[START_REF] Celepa | Failures of masonry and concrete buildings during the March 8, 2010 Kovanclar and Palu (Elaz) Earthquakes in Turkey[END_REF]. Walls experienced horizontal sliding failure at their top. In addition, a diagonal crack which is initiated from the corner of opening to the bottom of the wall is also obvious. The aim of this study is to reproduce such crack patterns in a wall due to

Figure 5 . 2 :

 52 Figure 5.2: (a) Damage of a rural masonry building at Yukar Demirci Village, Turkey [16](b) Failureof a rural masonry building in Palu, Turkey[START_REF] Celepa | Failures of masonry and concrete buildings during the March 8, 2010 Kovanclar and Palu (Elaz) Earthquakes in Turkey[END_REF]

Figure 5 . 3 :

 53 Figure 5.3: Studied walls subjected to out of plane displacement

Figure 5 .

 5 3(a). The deformed shape in different views are shown in Figure 5.4(a) and (b). The stepped cracks at the lower part of the wall as well as horizontal sliding of the lower row of mortar are obvious in these figures. As shown in Figure 5.4(c), the applied out of plane displacements cause stepping cracks at lower part of the wall. In earlier steps of loading, the out of plane displacement causes some cracks due to opening of lowest row of the horizontal mortars. Moreover, the horizontal and vertical mortars at left and right top corners of the wall slid and open because of the concentration of loads. Afterward, cracks are forming a distributed pattern from left top corner to the middle of the wall. Suddenly the stepped cracks are formed in displacement equal to 1.5mm in the bottom part of the wall.

Figure 5 . 4 :Figure 5 . 5 :Figure 5 . 6 :

 545556 Figure 5.4: (a), (b) Deformed shape (c) total crack pattern in the wall, Case A, at d top = 2mm

Figure 5 . 7 :

 57 Figure 5.7: Stress distribution in bricks at d = 2mm for Case A

Figure 5 . 8 :

 58 Figure 5.8: Crack pattern in the masonry wall d = 2mm, Case A

Figure 5 . 9 :

 59 Figure 5.9: Cracking profile for the walls with different mortar's cohesion

Figure 5 .Figure 5 . 10 :

 5510 Figure 5.10: Effect of mortar's cohesion on pushover curve

Figure 5 . 11 :

 511 Figure 5.11: Crack pattern (a) before and (b) after stepped cracks for wall with C = 3.5e4P a

Figure 5 . 12 :

 512 Figure 5.12: Length of different types of cracks in different interface elements at rows and columns of walls with different C, Case A

Figure 5 . 13 :

 513 Figure 5.13: Total crack pattern for walls with different σ n0

Figure 5 . 14 :

 514 Figure 5.14: Influence of initial normal stress on the behavior of the wall

Figure 5 . 15 :Figure 5 . 16 :Figure 5 . 17 :Figure 5 . 18 :

 515516517518 Figure 5.15: Length of different types of cracks in different interface elements at rows and columns of walls with different σ n0 , Case A

•Figure 5 .

 5 Figure 5.19: (a), (b) Deformation of wall (c) cracking profile of the wall, Case B

Figure 5 .Figure 5 . 21 :Figure 5 .

 55215 Figure 5.21 shows that the horizontal mortars just right to the applied loads, slide in y and x directions. The horizontal mortars fail in tension and open at the top, middle and bottom rows. The sliding of vertical elements is not considerable compared to the horizontal mortars. The shape of opening of vertical mortars is like a rhomboid with highest value at the middle corners. As shown in the Figure 5.21(g), almost all the bricks in the middle of the wall are failed in tension.

Figure 5 .Figure 5 . 22 :Figure 5 . 23 :Figure 5 . 24 :

 5522523524 Figure 5.22: Deformation and cracking profile for different C 0 , Case B

Figure 5 . 25 :Figure 5 . 26 :

 525526 Figure 5.25: Deformation and cracks profile in Case B with different σ n0

Figure 5 . 27 :

 527 Figure 5.27: Length of different types of cracks in different interface elements at rows and columns of walls with different σ n0 , Case B

Figure 5 . 28 :Figure 5 .Figure 5 . 29 :

 5285529 Figure 5.28: Deformation and crack pattern for wall with different F c0

Figure 5 . 30 :

 530 Figure 5.30: Length of different types of cracks in different interface elements at rows and columns of walls with different F c0 , Case B

Figure 5 . 31 :

 531 Figure 5.31: Series 1: two perpendicular walls (1×1×0.1) subjected to out-of-plane loads

Figure 5 . 33 :

 533 Figure 5.33: Deformation of the set, top nodes of Wall 1 move in x direction, Series 1

Figure 5 . 34 :

 534 Figure 5.34: Load-displacement curve for the Series 1, Wall 1 moves in x direction

Figure 5 .

 5 Figure 5.37 represents the sliding and opening cracks in different elements.Figure 5.37(h) and (i) also show total cracking pattern in different views. The sliding of horizontal elements mainly

Figure 5 .Figure 5 . 35 :Figure 5 . 36 :

 5535536 Figure 5.37 represents the sliding and opening cracks in different elements.Figure 5.37(h) and (i) also show total cracking pattern in different views. The sliding of horizontal elements mainly

  (d), (e) and (f).

Figure 5 .

 5 37(g) shows that just the bricks in diagonal of Wall 1 fail in tension.

Figure 5 . 37 :

 537 Figure 5.37: Crack pattern in the masonry walls d = 2mm, Series 1

( a )

 a Cohesion of MortarsDeformed shape and total crack pattern for the Series 1 with different cohesion values are displayed in Figure5.38. The first case corresponds to walls with C 0 = 1e5P a. In this case, the diagonal cracks appear in the wall at d = 0.42mm whereas they formed at d = 1.13mm in the wall with C 0 = 3.5e5P a. Hence, formation of diagonal cracks decreases the lateral strength of the Wall 1. Thus, decrease of cohesion causes diagonal cracks to happen sooner. By increment of displacements, the size of diagonal cracks as well as cracks in the intersection of walls increases. In this case maximum cracks correspond to the sliding of top vertical mortar of the walls' joint whereas the maximum value of cracks for walls with C 0 = 3.5e5P a, happens in diagonal region. In pushover curve, y label represents the summation of horizontal force of all nodes at the base of the set in x direction. Walls slide from the top vertical mortar in intersection at d = 0.22mm and the stiffness of the set decreased. Moreover, at this point walls begin to slid at their intersection and the lateral strength of Wall 2 also reduces.

Figure 5 . 38 :Figure 5 . 39 :Figure 5 . 40 :

 538539540 Figure 5.38: Deformation and crack pattern for Series 1 with different C 0 , displacements are applied to Wall 1

Figure 5 . 41 :Figure 5 . 42 :

 541542 Figure 5.41: Minor principal stress (a) d = 1.54 (b) d = 1.72

Figure 5 . 43 :Figure 5 . 44 :

 543544 Figure 5.43: Deformation and crack pattern for Series 1 with different σ n0 , displacements are applied to Wall 1

Figure 5 .

 5 45. As illustrated in this figure, Wall 1 slips on its first row of horizontal mortars. As can be seen in the figure, walls separate from the bottom part of intersection and some stepped cracks with small length also are observed at top left corner of Wall 1. Some cracks at bottom and top rows of Wall 2 also appears but their value is negligible compared to the sliding of Wall 1. By applying the loads, the cracks are created firstly in the bottom right of Wall 1 and the top common corner of walls. Cracks grow as applied displacements increase. Then at 0.5mm stepped cracks at the Wall 1 top left corner initiate and develop to the top horizontal mortars of Wall 1. Afterward, lower joints mortars slide and grow upward. Finally Wall 1 slips completely on its lower horizontal mortars at d = 1.12mm and as a result the lateral load carried out by the system decreases as shown in load-displacement curve Fig 5.47.

Figure 5 .

 5 Figure 5.45: Deformation in different planes, Series 1, d = 2mm, all top nodes move in x direction

Figure 5 . 46 :Figure 5 . 47 :

 546547 Figure 5.46: Crack's pattern for Series 1 (a) Wall 1 subjected to u x (b) Walls 1 and 2 subjected to u x

Figure 5 . 48 :

 548 Figure 5.48: Deformation in different planes, Series 1, d = 2mm, all top nodes move in y direction

Figure 5 .Figure 5 . 49 :Figure 5 . 50 :

 5549550 Figure 5.49(b) represents a comparison between load-displacement curves of Series 1 under displacement in y and x directions. As illustrated in this figure, the set has greater lateral strength when subjected to displacements in x direction. Results show that all with higher aspect ratio subjected to in-plane displacements sliding shear failure of the loaded wall causes its brittle behavior. The strength of walls when subjected to out-of-plane loads and in-plane loads, are compared in Figure5.50.

Series 1 :

 1 roof 's center of gravity rotates 0.05 degree In this part, the center of gravity of supposed roof rotates 0.05 degree. The movement of top nodes in y and x directions is determined according to the distance of each node to the center of gravity. The final deformed shape in x, y, z and two other planes are represented in Figure5.51. According to this figure, top horizontal mortar row of Wall 2 slide, the walls separate and different series of stepped cracks also propagate in the Wall 1.

Figure 5 . 51 :Figure 5 . 52 :

 551552 Figure 5.51: Deformation and total crack pattern for walls subjected to 0.05 degree rotation, Series 1

Figure 5 . 53 :

 553 Figure 5.53: Crack pattern in the masonry walls ,Series 1, roof rotates 0.05 degree

Series 2 :

 2 displacement are applied to all top nodes in x direction In this case, all top nodes are subjected to the displacements in x direction. The obtained deformed shape and total crack pattern are shown in Figure5.54. In this case, diagonal cracks are formed in the Wall 1 due to applied in-plane displacements on it. As shown in the figures, the walls just slide in shear directions but do not separate. Therefore, the dimension of perpendicular walls has an important role to the behavior of the set. By increasing the length of perpendicular walls, the wall subjected to in-plane displacements undergoes more lateral strength and behave very better. In addition, in this case the walls do no disconnect. Cracks first are formed in Wall 1 bottom right and top corners. By increasing the loads, cracks propagate at top and greatly in bottom horizontal mortars of Wall 1 at d = 0.6mm that reduces the stiffness of Wall 1 under lateral loads, Figure5.56(a). Then stepped cracks at top left corner of Wall 1. Simultaneously, top and bottom rows of Wall 2 also crack and the vertical mortars in the walls' intersection slid on each other at d = 0.8mm, see stiffness reduction of Wall 2 at this point in Figure5.56(a). Then, diagonal cracks propagate in the Wall 1 at d = 1mm. The drop in pushover curve of the set corresponds to formation of these cracks, see Figure5.56. By increasing the displacements, the length of diagonal cracks, sliding cracks in the bottom rows of horizontal mortars in Wall 1 and sliding of vertical mortars at walls' intersection increase.

Figure 5 . 54 :

 554 Figure 5.54: Deformation in different planes, Series 2, all top nodes move d = 2mm in x direction

Figure 5 .

 5 56(b) represents Series 1 and Series 2 total pushover curves. As illustrated in the figure, the lateral strength of Series 2 is considerably higher than the Series 1.

Figure 5 . 55 :Figure 5 . 56 : 1 Series 2 :

 55555612 Figure 5.55: Total cracks pattern, all top nodes move 2mm in x direction (a) Series 2 (b) Series 1

Figure 5 . 57 :

 557 Figure 5.57: Deformation in different planes, Series 2, all top nodes move d = 2mm in y direction

Figure 5 .Figure 5 . 58 :Figure 5 . 59 : 1 (Figure 5 . 60 :

 55585591560 Figure 5.58: Total cracks pattern, all top nodes move 2mm in y direction (a) Series 2 (b) Series 1

Figure 5 . 61 :

 561 Figure 5.61: Lateral load-displacement curves of walls in (a) x direction (b) y direction, Series 2, 0.05 degree rotation

Figure 5 . 62 :

 562 Figure 5.62: Crack pattern in the masonry walls, Series 2, 0.05 rotation

Figure B. 3 :Figure B. 4 :Figure B. 6 :Figure B. 8 :

 3468 Figure B.3: Shear wall results for β = 32.80 o

GI = 8N/m 2 •Figure B. 10 : 2 0 2 Figure B. 21 :

 2102221 Figure B.10: Pushover curve for the wall with different β

Figure B. 22 : 2 • 2 • GIII = 7 .5E3N/m 2 •

 222272 Figure B.22: Profile of cracking in the wall with different GII

Figure B. 25 :

 25 Figure B.25: Shear wall results for GIII = 7.5E3N/m 2

Figure B. 27 : 2 Figure B. 28 :

 27228 Figure B.27: Pushover curve for the wall with different GIII

Table 3 .

 3 1: Properties of equivalent struts models tested FEMA 356 model Calibrated model Mostafaei model Proposed model A strut [m 2 ]

		0.0345	0.0254	0.0345	0.0323
	E m [M P a] 5.3 × 10 3 f ′ m [M P a] 7.09	11.3 × 10 3 7.09	5.3 × 10 3 7.09	6.36 × 10 3 7.09
	σ y [M P a]	6.51	-	4.26	4.99
	ǫ y [M P a]	0.0012	-	0.00048	0.00029
	σ m [M P a]	6.51	7.09	5.68	6.65
	ǫ m [M P a]	0.00165	0.000625	0.0012	0.00075
	σ p [M P a]	0	2.87	1.7	3.33
	ǫ p [M P a]	0.00165	0.0014	0.0018	0.0017

Table 4 .

 4 1: Elastic properties of bricks and mortar used in the model[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

		Brick	Mortar
	E	ν	E interf ace	G interf ace
	16700	0.15	82	36
	M P a		GP a/m	GP a/m
	between 1.0 to 2.0 [N/mm2] is enough to yield dilatancy angle to zero [60].

Table 4 .

 4 2: Inelastic properties of joints[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

	Tension		Shear		Thickness
	f t	C	tanφ	tanψ	e
	0.25	0.35	36.9 o	0	3e-6
	M P a	M P a			[m]

Table 4 . 3

 43

: Properties of the potential brick cracks

[START_REF] Lourenco | Computational strategies for Masonry structures[END_REF]

  exp(-F t0 /GI.u p n ) Therefore, there are two equations with two unknowns, namely λ 1 and λ 2 . The unknowns are calculated based on these two equations. By having the unknowns, [ un ] p , [ us ] p and [ ut ] p , the stresses can be calculated based on the following equations :

	p ) p ) σn = E( un -un σs = G( us -us σt = G( ut -ut p )	(4.37) (4.38) (4.39)
	(b) Shear and compression are active:

Table 4 .

 4 4: Inelastic properties for the horizontal and vertical interface elements for model II

		Tension			Shear			Compression		Thickness
	f t0	GI	C 0	φ	ψ	GII	f c0	GIII	θ	beta	e
	0.25 18	0.35	36.9 o 0 o	125	10.5 5e3	32.7 o 45 o	3e-6
	M P a P a.m	M P a			P a.m	MPa P a.m			[m]

Table 4 .

 4 5: Brick-middle interface element properties for model II

		Tension			Shear			Compression		Thickness
	f t0	GI	C 0	φ	ψ	GII	f c0	GIII	θ	beta	e
	2	80	2.8	45 o	45 o 50	1e15 1e15	45 o	0	2e-9
	M P a P a.m	M P a			P a.m	MPa P a.m			[m]

Table 5 .

 5 1: Inelastic properties for the horizontal and vertical interface elements

		Tension		Shear			Compression		Thickness
	f t0	GI	C 0	tanφ tanψ GII	f c0	GIII	θ	beta	e
	0.25 18	0.35	36.9 o 0	125	10.5 5e3	32.8 o 45 o	3e-6

Table 5 . 2 :

 52 The crack profile for the damaged walls, Case A

	Cracking profile	Specifications	Observed in:

Table 5 . 3 :

 53 The crack profile for the damaged walls, Case B

(a) Wall deformation d=4mm (b) Pushover curve

Acknowledgements

It is with immense gratitude that I acknowledge the great support, help, and patience of my

Acknowledgements

iii

List of Tables

Chapter 5

Micro modeling: out of plane loading effect of walls' length on performance of the set under roof's movement will be studied. Hence, the length of Wall 2 is multiplied times 1.5, see Figure 5.32.

Series 1: Top nodes of Wall 1 move in x direction

In this section, top nodes of Wall 1 are subjected to in-plane displacement gradually up to 2mm, so Wall 2 is subjected to out-of-plane displacement equal to 2mm at its connection with Wall 1.

In order to have a good understanding of what happens in the different walls, those deformations and final crack pattern are shown in Figure 5.33 in different views .

subjected to uniform loading in x and y directions and also a slight rotation. The presence of surrounding wall increases the lateral strength of the wall in in-plane direction and whatever the length of surrounding wall is greater, the system undergoes greater loads. The mortars in the joint have an essential role on the behavior of two attached walls. Increasing initial confinement enhances the performance of the set considerably. It should be noted that the same applied rotation to the supposed walls' roof causes a greater damage in the set with greater geometries.

Perspective and future work

In term of macro models, only the behavior of the walls under monotonic loads was studied, so development of constitutive law for simulating the behavior of infill under cyclic loads is necessary specially for the analysis of seismic behavior.

A comprehensive probabilistic study of different variables that affect the strength and failure mechanism of masonry walls, including but not limited to aspect ratio, opening location and size, and material properties is recommended for the future work. Investigating the effect of variation of different parameters on the behavior of such structures will lead to highly useful information that is instrumental in identifying and setting rehabilitation priorities for vulnerable structures that are often located in highly populated areas of the world.

The micro modeling strategy is very time consuming and costly and it is not applicable to model all of masonry walls in a building. Hence, predicting the failure mechanism of the wall to assign the appropriate macro-elements can be a very useful tool for analyzing masonry buildings under different loading.

Chapter 7

Appendixes

Appendix A

Macro models formulations

FEMA 356 model

Where :

h col : Column height between centerlines of the beams, r inf : Diagonal length of infill panel, E me : Expected modulus of elasticity of infill wall material, t inf : Thickness of infill panel and equivalent strut, θ: Angle whose tangent is the infill height-to-length aspect ratio, E f e : Expected modulus of elasticity of frame material, I col : Moment of inertia of columns about the axis perpendicular to the loading direction, h inf : Height of infill wall.

For the infilled RC frame of the CSTB test structure, the equation's parameters are: h col = 2.8m, r inf = 4.53m, t inf = 20cm, assumed: E f e = 24.82 × 10 9 P a, h inf = 2.6m and:

A n = t inf .l inf (A.7)

A n = 0.2 × 3.71 = 0.743m 2 ν ts + P CE An assumed : 620kPa ⇒ ν me = 0.75 × 620 = 496kP a V ine = 0.743 × 0.496 = 368.03kN A strut = 0.2 × 0.344 = 0.0688m 2 f my = 0.368 0.688 . 4.53 3.71 = 6.51M P a ε my = 6.51×10 6 5.32×10 9 = 0.0012 ε mu = 0.35% × 2.6×3. 71 4.53 2 = 0.001645

Calibrated model

By substituting the value of each parameter, the area of strut model as well as material model are calculated: Proposed model

0.2×143.6 + 0.0047 = 0.0106m σ m = 0.33×10 6 0.0657×0.8417 = 6.65M P a σ y = 0.276×10 6 0.0657×0.8417 = 4.99M P a σ p = 0.184×10 6 0.0657×0.8417 = 3.33M P a ε m = 0.0047×0.8417 The effect of mode II of fracture energy on behavior of the shear wall is studied here. Variation of mode II fracture energy affects the behavior of shear wall. As in yield surface formulation cohesion depends on mode II fracture energy, so its variation affect the wall's behavior. By increasing the GII, the mortars lose lower cohesion and consequently wall undergoes more shear strength. on the other side, reducing the fracture energy of mode II causes the mortars to loose their cohesion sooner and wall's failure happens sooner.