# Nonlinear dynamic analysis of shear walls

### APPROPRIATE MODELING TECHNIQUE FOR NONLINEAR DYNAMIC ANALYSIS OF SHEAR WALLS

### CHAPTER 1

### INTRODUCTION

### 1.1 Background and problem statement

Reinforced concrete (RC) shear walls are widely used in medium to high rise buildings as these walls are very effective in providing resistance and stiffness against the lateral loads imposed by wind and/or earthquake. These walls also provide sufficient ductility and very good lateral drift control which prevent from the undesirable brittle failure against the strong lateral loads, especially during an earthquake. Their high plane stiffness and strength make them well suited for bracing buildings of up to 35 stories, while simultaneously carrying gravity loading. The term shear wall is perhaps a misnomer, since wall panel is primarily influenced by the tension on one side and the compression on the other side, also creating a moment. Hence, the term shear wall is really a short and stubby flexural wall (Bryan Stafford Smith and Alex Coull 1991).

In recent years, seismic design methodologies have put the limitation in the maximum drift of the structure during strong earthquake. So it is essential to protect the other non-seismic structural components in the building that resist gravity forces only. From the point of view of economy and drift control, RC walls become imperative for tall buildings (Paulay and Priestley, 1992).

RC walls in multistory buildings are slender, and behave essentially as vertical cantilever walls. These RC walls are expected to deform well into inelastic range and dissipate the energy input by the base motion through stable hysteretic behaviour of structural components. The inelastic behavior of the entire wall is dependent on the plastic hinge zone at the wall base, where large rotations and yielding of reinforcement takes place. As a result, the entire structural systems stiffness, strength, ductility, and means of dissipating energy are wholly contingent on the response of this region.

The proper modeling of shear walls and other forms of structural walls has been a matter of concern for the structural engineers from a long time. There are several types of analysis methods available for analyzing shear walls structure. The analysis can be made in linear elastic and nonlinear condition. The elastic method of analysis consists of Continuous Connection Method (CCM), Transfer Matrix Method, Wide Column Analogy (WCA) or frame analysis, Finite Element Method and Discrete Force Method. But, these elastic models cannot predict the real hysteretic behaviour of the structure if it is subjected to intense lateral loadings, like earthquakes. So, it necessitates the proper inelastic model that can closely predict the real hysteretic behaviour of the structure. Extensive research, both analytical and experimental, has been conducted to simulate the hysteretic behaviour in flexure and shear of RC walls. From the research, different hysteretic models are developed that are capable of predicting the inelastic behaviour of RC walls. Prediction of the inelastic wall response requires accurate, effective, and strong analytical models that incorporate important material characteristics and behavioral response features. These characteristics include stiffness degradation, strength softening and deterioration, and pinching behaviour. This model should be capable of taking into account the effects of both the axial force and the bending moment on the shear behaviour with sufficient accuracy.

The principal effects of stiffness degradation are:

- an increase in the flexibility and period of vibration of the undamaged structure during large deformation reversals,
- a decrease in energy dissipation capacity,
- a significant redistribution of internal forces which could lead to excessive deformations in some regions.

Many analytical models have been proposed for predicting the nonlinear response of RC structural walls. They can be classified into two broad groups:

- Detailed models derived using mechanics of solids, which are based on a detailed interpretation of local behaviour (microscopic approach) and
- Models overall behaviour with reasonable accuracy (macroscopic approach).

There are several macroscopic models available for the analysis of RC walls. Equivalent beam model, equivalent truss model, three-vertical-line-element model (TVLEM) (kabeyasawa et al., 1983), axial-element-in-series model (AESM) (Vulcano and Bertero, 1986), modified axial-element-in-series model (Vulcano and Bertero, 1988), multiple-vertical-line-element model (MVLEM) (Vulcano and et al., 1988) etc. are some of the well known macroscopic models. These models represent the wall as a set of nonlinear translation and/or rotational springs connected by rigid beams. These macroscopic models are practical and efficient, although their application is restricted based on simplifying assumptions upon which the model is based.

On the other hand, Microscopic finite element models offer a powerful analytical tool for simulating the nonlinear behaviour of RC structures. But their efficiency, reliability and practicality are questionable due to complexities involved in developing the model and interpreting the results. Also the computation is generally very time-consuming and requires a large storage: thus, in practice the use of microscopic finite element models is restricted to the analysis of isolated or coupled walls.

The above mentioned macroscopic elements were developed mainly based on the experiments conducted in the squat RC walls and some slender walls. Because of the limitations on the laboratory testing of the high rise buildings RC walls, precise nonlinear modeling for predicting the nonlinear behaviour of these RC walls still has not been fully solved.

### 1.2 Objectives

The main objectives of this study are mentioned below:

- To identify the set of modeling parameters that lead to the reliable simulation of nonlinear behaviour of RC slender wall for performing the nonlinear static and nonlinear direct integration time history analysis.
- To present the various nonlinear modeling techniques that is easily adaptable to wall configurations.
- To compare the effectiveness of each model in terms of efficiency, practicality and reliability.

### 1.3 Scope of the study and limitations

The study mainly includes the modeling of various nonlinear features that are typically found in RC walls. Different macroscopic and microscopic finite element models are used to model the nonlinear behaviour of RC wall. The purpose of this nonlinear modeling is to capture or to estimate the capacity of the structure to deform and the corresponding force demand. The nonlinear modeling requires following:

- The materials properties as used are modeled correctly throughout their entire loading history, including the effects if confinement etc.
- The cross-section dimensions and reinforcement distribution is correctly represented while determining the action-deformation relationships.
- The effects of hysteresis and degradation are included either explicitly or at least indirectly.
- The assumptions about hinge locations and yield zones or hinge lengths are realistic.

Modeling and analysis of RC wall are conducted in commonly available platform SAP 2000 (CSI 2009 Version 14.1.0) to determine the nonlinear response of RC wall. This study is limited to the modeling of the planar and box shear walls and also the interaction between flexure and shear is not taken into account. The other limitations include itself with the software as the software has limited hysteretic model only. P-? effect is also ignored in the analysis.

### CHAPTER 2

### LITERATURE REVIEW

### 2.1 General

In the last two decades, much effort has been carried out to simulate the inelastic response of RC structural walls subjected to large cyclic deformation reversals. Numerous analytical models incorporating information from experimental investigations and on-field observations of the hysteretic behavior of RC structural walls have been proposed. These range from the simple two-component model with bilinear hysteretic law to refined fiber or layer models based on sophisticated descriptions of the cyclic stress-strain behavior of concrete and reinforcing steel. These analytical models could be broadly classified into two groups as:

- Macroscopic models
- Microscopic finite element models

### 2.2 Review of previous studies on macroscopic wall model

Macroscopic models refer to those models that are derived based on a simplified idealization, which are capable of predicting a specific overall behaviour with reasonable accuracy. Some of the macroscopic models are explained below

### 2.2.1 Equivalent truss model

An equivalent truss system is used to represent the wall member. Hiraishi H. (1983) introduced a non-prismatic truss member based on the experimental results. The stress along the height of the boundary column in tension determines the cross sectional area of this truss member. Under the cyclic load, it is difficult to define the structural topology and geometry of the truss members, so this technique is limited to a monotonic loading.

### 2.2.2 Equivalent beam model

It is one of the most common and primitive model for predicting the wall hysteretic behaviour. In this approach, the entire wall is replaced by an equivalent line element placed at the centroidal axis and is connected by rigid links on beam girder. A beam-column model is common to replace the two dimensional wall. In this model, a nonlinear rotational spring at each end of the elastic flexural element is provided in order to account the inelastic behaviour at critical regions (figure 2.1). Furthermore; to account for the fixed-end rotation at any connection interface, an extra nonlinear rotational spring can be provided. To simulate the more realistic behaviour of wall, certain improvements have been introduced in this single beam-column element. The improvements include the descritization of wall member into multiple springs (Takayanagi and Schnobrich, 1976), varying inelastic regions (Keshavarzian and Schnobrich, 1984) and specific inelastic shear behaviour (Aristizabal, 1983). This model has still one big limitation associated with the shifting of neutral axis which is due to the assumption that the rotations occur around points of the centroidal axis of the wall. Thus, the important features of the experimentally observed behaviour (figure 2.2), like neutral axis fluctuation of the wall cross section, wall rocking, outriggering interaction with the frame surrounding the wall in a frame wall-structure (Kabeyasawa et al., 1983) etc. are disregarded in this single beam-column model.

### 2.2.3 Multi-Component-in-Parallel Model (MCPM)

In order to obtain a more refined description of the flexural wall behaviour, Vulcano, Bertero, and Colotti (1988) proposed the multi-component-in-parallel model (MCPM, also referred to as multiple-vertical-line element model MVLEM).

According to Orakcal, K., Massone, L. M. and Wallace, J. W., 2006, the flexural response of a wall member was simulated by introducing the number of multi-uniaxial-element-in-parallel model connected by horizontal rigid beams at the top and bottom floor levels which is shown in figure 2.3. Here the axial stiffnesses K1 and K2 of the boundary columns are represented by the two external elements but the axial and flexural stiffness of the central panel is represented by two or more interior elements, with axial stiffnesses K3,. , Kn. The stiffnesses of all the vertical elements are computed from the material constitutive law and the assigned tributary area which is shown in figure 2.4. The nonlinear shear response of the wall member is represented by horizontal spring, with stiffness KH computed from the material constitutive law and the effective shear area. The value of parameter c varied between 0 and 1 depending on the distribution of curvature along the inter story height h.

Even though the multiple-vertical-line-element model (MVLEM) is very effective in predicting the flexural response of the wall, it is difficult in assigning the refined constitutive laws of the wall materials.

### 2.2.4 Panel-Wall Macro Element (PWME)

The prediction of the overall (shear and flexural) behavior of RC walls for both monotonic and reversed cyclic loading is improved by introducing the 2D panel-wall in between the boundary elements proposed by Kabeyasawa (1997). A two dimensional nonlinear panel-wall (figure 2.5) is introduced to replace the horizontal, vertical and rotational springs at the wall centre line. This model is found to be unstable if the wall is subjected to high axial load and significant cyclic nonlinear shear deformations.

### 2.3 Review of previous studies on microscopic wall models

These are more detailed models derived from solid mechanics which are based on a detailed interpretation of local behaviour. Since the constitutive model of the concrete has great influence in the nonlinear behaviour of RC walls, so the macroscopic models that have been discussed earlier cannot give satisfying behaviour under complicated stressed condition of concrete.

Bazant et al. proposed the micro-plane model which considered the microstructure of the material. Microplanes are the set of planes at any orientation in the material microstructure and the constitutive law was formulated to relate the stress and strain components on these microplanes. The stress and strain components on a particular microplane are called the microscopic stress and strain components. The overall macroscopic behaviour is obtained by superimposing the effects of all these microplanes. The micro-plane model provides an efficient and powerful numerical tool for the development implementation of constitutive law of any kind of material.

Z. W. Miao, X. Z. Lu1, J. J. Jiang and L. P. Ye (China, 2006) proposed a multi-layer shell element model based on the composite material mechanics principles for simulating the coupled in-plane/out-plane bending or the coupled in-plane bending-shear nonlinear behaviors of RC shear wall. The multi-layer shell element consists of many layers with different material properties and different thickness which is shown in figure 2.6.The longitudinal rebars are smeared one or more layer which is shown in figure 2.7. By using finite element analysis, the axial stress strain and curvature of one element at the middle layer can be determined. The strains and curvatures of other layers are computed by assuming the plane section remains plane. Thus the overall behaviour of RC wall is simulated using this material constitutive law.

### 2.4 Nonlinear flexural stiffness model for RC shear walls

A high-rise RC shear wall deforms primarily due to flexural behaviour under the lateral movement caused by earthquake. The nonlinear behaviour of RC walls can be established by determining the bending moment-curvature of the RC section under a certain axial loading (0.10fc Ag). Various section analysis tools are available to compute the bending moment-curvature of the RC section. Among them, Response-2000 (Bentz, 2000) is one of the popular tools to build the moment-curvature response of the RC section.

The nonlinear flexural stiffness model for RC shear walls can be simplified by using a tri-linear bending-moment-curvature as proposed by Ahmed M. M. Ibrahim (The University of British Columbia, 2000). It is proposed that the bending moment-curvature response of a wall section including the effect of axial compression acting on the wall, the amount of vertical reinforcement and the state of cracking (i.e., tensioning effect) can be represented by a simple trilinear relationship. Two important points could be notified in this trilinear curve, one is cracking of concrete and another is yielding of the vertical reinforcement which are associated with the change of slopes (figure 2.8).

### 2.5 Nonlinear shear response in RC shear walls:

Gerin and Adebar (2004) proposed the trilinear shear response model for the RC section based on the underlying principle of the general model which is shown in figure 2.9. The simplified model, developed from the general model, provides an estimate of the envelope of the shear response including the cracked-section shear stiffness and the maximum shear strain. The model can treat fully-cracked or initially uncracked sections Here, the backbone of the shear response is assumed to be elastic-plastic.

### 2.6 Past experiments carried out on a slender concrete wall

One of the large scale tests on a slender concrete wall was conducted in University of British Columbia (Adebar et al., 2002). The prototype wall was 73.2 m high with height to length ratio of nearly 10. The prototype was scaled down to 1/4 and was 12.8 m high which is shown in figure 2.12 and 2.13

The test wall was subjected with a uniform constant axial compression of 1500 KN (corresponds to gravity load in order of 0.10fc Ag). Four full cycles lateral loading were applied at the height of 12.2 m from the base of wall using displacement controlled hydraulic actuator which is shown in figure 2.11.

Different analytical models were proposed for simulating the nonlinear response of the above tested slender concrete wall. Some of them are briefly explained below in sections 1) and 2):

1) Lepage, S. L. Neuman and J. J. Dragovich proposed a simplified analytical model (shown in figure 2.14) consisting of linear shell elements coupled with uniaxial linear and nonlinear line elements. The proposed model was good enough to have satisfactory agreement between the analytical and experimental result which is shown in figure 2.15.

2) Perry Adebar and Ahmed M. M. Ibrahim, 2002 proposed a trilinear bending moment-curvature for capturing the nonlinear behaviour of RC wall which is shown in figure 2.16. To account for the influence of cyclic loading on tension stiffening of cracked concrete, the concept of upper-bound response for a previously uncracked wall and lower-bound response for a severely cracked wall was introduced.

### CHAPTER 3

### METHODOLOGY

### 3.1 General

There are various methods available for the analysis of structures, both elastic and inelastic. The elastic analysis procedure cannot predict the real behaviour of structure under strong seismic loading as this analysis cannot account the redistribution of forces during progressive yielding. So in order to predict the real behaviour of structure, inelastic or nonlinear analysis is required.

In the early days of computer aided analysis, the nonlinearity is considered in beams and columns of the building and the shear wall is modeled as elastic element. From the last two decades, various analytical models have been proposed based on the experimental results for predicting the nonlinear behaviour of wall which are described in sections 2.2 and 2.3. These models range from very simplified model to very refined and complex models. The refined and complex model can be used to predict the behaviour of small structure but can cause difficulty or mislead during the analysis of practical complex structure like tall buildings. So it is important to select the appropriate simplified model that can closely predict the real behaviour of even complex structures.

This study is carried out to model the shear walls of the high rise buildings with different available nonlinear modeling techniques in order to predict the nonlinear response of shear walls. The detailed methodologies of selection of shear walls, material model, structural loadings, analysis methods, hysteretic behaviour, modeling techniques of shear walls etc are described in the following sections.

### 3.2 Software platform

The nonlinear behaviour of the shear wall is performed in the currently available versatile computer software package SAP 2000 (CSI 2009 Version 14.1.0). Because of the wide variety of analysis options including nonlinear static pushover and nonlinear time history analysis and of the advanced graphics option, this software is more productive and practical in the market today. SAP 2000 provides the facility to assign the nonlinear material properties by the use of different hinges and link elements.

Out of the several frame hinges available in the SAP 2000 software, interacting P-M-M and fiber P-M-M hinges are used to simulate the nonlinearity of the shear wall. The software facilitates the two options for defining the nonlinear properties of these hinges; one is by program calculated based on the element material and sectional properties according to FEMA-356 or by user defined. Load-deformation (moment- rotation) curve of the assigned hinges follows the pattern shown in figure 3.1.

Another option to model the nonlinearities associated with the shear wall is by using the nonlinear link elements Multi linear Plastic available in Sap 2000. Each link element is composed of six degree of freedom (i.e. six separate springs) that can represent the axial, shear, torsion and pure bending. The following figure 3.2 shows the 2-D link composed of three internal springs; axial, shear and plane bending and dj2 is the location of shear spring from the joint j.

### 3.3 Features of the shear wall used for the analyses

Cantilever walls of 10, 20 and 30 stories planar and box shaped shear walls are chosen for the analyses. The structural parameters used for modeling are taken according to the realistic examples of shear walls built in high-rise buildings. The cross sections of the shear walls are shown in figure 3.3 with story height of 3.2 m which is uniform over the total height. The compressive strength of concrete, fc? is taken as 4000 Psi (27.6 N/mm2) and the yield strength of all the reinforcement is taken as 415 N/mm2.

(a) Cross section of planar shear wall

(b) Cross section of box shear wall

### 3.4 Structural loadings

### 3.4.1 Gravity load

In order to predict the nonlinear response of the shear wall against the lateral load (seismic load), it is necessary to include the effect of gravity load present at the time of lateral loading. Gravity load includes self weight of shear wall, super imposed dead load and live load. Since the shear wall is modeled separately in this study, so the super imposed dead load and live load includes the tributary area of slab that the shear wall will support.

In order to calculate the super imposed dead load, the density of concrete is taken as 2400 kg/m3. The design live load is taken as 3.5 KN/m2. For the dynamic analysis against earthquake loading, most likely live load is taken as 25% of the design live load.

### 3.4.2 Lateral load

An inverted triangular load pattern is taken as load pattern for the nonlinear static analysis of the shear wall. In order to perform the nonlinear time history analysis, a well known and mostly used for the research purpose El Centro 1940 Imperial Valley Earthquake (North-South Component) is used. The time history plot of the El Centro is shown in figure 3.4 in which the PGA of 3.42 m/s2 and duration of 31.18 seconds was recorded.

### 3.5 Analysis methods

### 3.5.1 Nonlinear Static Analysis

In order to perform the nonlinear static analysis also called pushover analysis, SAP 2000 (CSI 2009) is used. The applied lateral load pattern for the nonlinear static analysis is one of the important parameter that can affect the distribution of nonlinear behaviour of structures. There are several load patterns available for the pushover analysis like modal load pattern, inverted load pattern. In this study, inverted triangular load pattern is used for the analysis as the structure is symmetrical. The analysis is continued until the structure becomes unstable or until the predefined target displacement is reached whichever is earlier.

Since the pushover analysis cannot account accurately for the changes in dynamic response as the structure degrades in stiffness or cannot account if the higher mode effects are predominant as in case of high rise buildings, then there is need of nonlinear time history analysis which is discussed below.

### 3.5.2 Nonlinear Time History Analysis

Unlike pushover analysis, the nonlinear time history is performed using ground motion time history as each and every time step. SAP 2000 (CSI 2009) allows performing various types of time history analysis. In this study, nonlinear direct-integration time history analysis is used with Hyber-Hughes-Taylor alpha (HHT). The program allows controlling only one parameter called alpha which can vary from 0 to -1/3. If the value of alpha = 0 then the method is equivalent to Newmark method (1959) with the values of gamma = 0.5 and beta = 0.25, which is also same as the average acceleration method. More accuracy can be maintained by setting the value of alpha = 0 but it may permit excessive vibrations in the higher mode frequencies. In case of negative values of alpha, the higher mode frequencies are more severely damped. In order to ensure the solution not to be dependent upon these parameters, different values of time-step size and alpha are examined. Since the relative convergence tolerance affects the accuracy of output result, it is necessary to set the value of this parameter small enough. 5% critical damping ration is used for all the modes.

### 3.6 Hysteretic behaviour

In order to simulate the hysteretic response of the material, hinges and link element in the SAP 2000 (CSI-2009), the software facilitates different hysteretic models. The hysteretic type could be Kinematic, Takeda or Pivot. No additional parameters are required to define in case of Kinematic and Takeda hysteretic models whereas certain parameters like a1, a2, 1, 2 and ? are required to be assigned to control the degradation of the hysteretic loop.

### 3.6.1 Kinematic hysteretic behaviour

The kinematic hysteretic behaviour is developed based upon the kinematic hardening behaviour of the material. In SAP 2000 (CSI 2009), no additional parameters are required to define this hysteretic behaviour. The kinematic hysteretic model is shown in figure 3.5. In the figure 3.5, the behaviour is elastic from point 0 to point 1. When the load reverses within this path, it moves along the point 0 to point -1 without any plastic deformation. The plastic deformation starts from point 1 to point 2 due to continuous increase loading. When the load reverses within this path (point 1 to point 2), then it will unload through the shifted elastic line. During the process, as the point 2 is pushed in the next cycle, the point -2 will be pulled out by an identical amount. When the load reverses again, point 1 is pushed toward point 2 and then together pushed to point 3 pulling points -1 and -2 with them. This process is continued throughout the analysis by maintaining the slope between points 3 and -3.

### 3.6.2 Takeda hysteretic behaviour

This model is very identical with the Kinematic hysteretic model except the hysteretic loop is different. Here in this model, the degradation in the hysteretic loop is considered as described in Takeda, Sozen and Neilson (1970). As in case of Kinematic model, it is also not required to assign the additional parameters for this Takeda model in SAP 2000 (CSI 2009). The hysteretic behaviour of this model is shown in figure 3.6.

### 3.6.3 Pivot hysteretic behaviour

This model is identical to the Takeda model but there are additional parameters like a1, a2, 1, 2 and ? need to be assigned for controlling the degradation of the hysteretic loop which is shown in figure 3.7. This model is suitable for reinforced concrete elements and is based on the observation that unloading and reverse loading tend to be directed towards the specific controlled points, called pivot points in the force-deformation (or moment-rotation) plane. This model is fully described in Dowell, Seible and Wilson (1998).

### 3.7 Modeling techniques

### 3.7.1 Single Column Model

The simplest model of the shear wall is of course, representing it by an equivalent column at the center line of the wall section. This model is used as a reference model for the comparison of various wall models when analysis is carried out independently of the rest of the structure. This model is especially suitable for the walls with hw / lw ratio greater than 5 in which hw is the total height of wall and lw is the width or length of wall. The shear wall is represented by a column section ( lw x t) as shown in figure 3.8 where t is the thickness of shear wall. The nonlinearity is assumed to be concentrated at the base of the wall which can be modeled by using different hinges available in SAP 2000 (CSI 2009). In this study, interacting P-M2-M3 hinges and fiber P-M2-M3 hinges are used.

### 3.7.1.1 Interacting P- M2-M3 hinge

The force-deformation or moment-rotation properties of interacting P-M2-M3 hinge can be determined from the section analysis or FEMA 356 code which is shown in figure 3.9 and the values of a, b and c are tabulated in table 3.1.

### 3.7.1.2 Fiber P-M2-M3 hinge

The fiber P-M2-M3 hinge discretizes the whole cross section of the shear wall into no. of individual concrete and reinforcement fibers. The force displacement or stress-strain relationships of these fibers are taken directly from the assigned material properties of concrete and reinforcement. In SAP 2000 (CSI 2009), there are some models for defining the stress-strain relationships of concrete and reinforcement. In this study, the following models shown in figure 3.10 are used for concrete and reinforcement respectively. However, there is option in the software to manually assign these stress-strain properties also.

### 3.7.2 Fiber or frame model

For the shear walls in tall building, primarily the flexural deformation mode governs, except may be for the lower floors near the base. An equivalent frame discretized model as shown in figure 3.12 can be used to determine the non-linear response. This approach is based on the fiber modeling of the wall sections which is described in section 2.2.3. In this approach, closely spaced discretized columns representing the axial portions of the wall in the section can be used, connected at the floor level by rigid element to enforce deformation compatibility, without causing local bending. The nonlinear axial hinges or nonlinear link elements available in SAP 2000 (CSI 2009) can be added in each discretized columns at arbitrary level, representing the axial response of the corresponding portion of the wall. The properties of these axial hinges or link elements can be computed from the amount of concrete and steel present in the corresponding discretized columns and the constitutive law of the material. The shear response of the wall can also be modeled by using nonlinear link element at the center of the wall and the property of this link element is calculated as described in section 2.5. In order to simulate the hysteretic response in the hinges or link elements, appropriate hysteretic models described in section 3.6 can be used. This model, when properly defined can help to identify the extent of yielding in the wall and eliminates the need to predefine the hinge length, needed in single element models, or to determine this length through analysis.

### 3.7.3 Nonlinear layered shell model

SAP 2000 (CSI 2009) software has capability to analyze the nonlinear response of shell element, especially when subjected to in-plane stresses and deformation. The complex stress state and presence of reinforcement in possible arbitrary direction with respect to the shell elements natural coordinates possess significant difficulties in generating nonlinear force-deformation relationship. This problem can be handled easily by using layered shell element which is available in SAP 2000 (CSI 2009). Here, the entire cross section of shear wall is discretized into individual layers, these layers could be for concrete, horizontal reinforcement, vertical reinforcements etc. These layers are located by a specific distance from the reference surface and with the specified thickness as shown in figure 3.13. The material properties of each layer could be specified by the properties of concrete and steel which is shown in above figure 3.10. Each layer could be assigned as shell, membrane or plate element depending up on the requirement. The hysteretic response of the wall section can be simulated by assigning the hysteretic behaviour in the property of concrete and steel materials explicitly.

### CHAPTER 4

### EXPECTED RESULTS

The expected results in this study are as follows:

- Use of different modeling techniques of the reinforced concrete shear walls will give a more clear understanding of the nonlinear response under the seismic loading.
- It is expected to get the better result from fiber or frame model and the nonlinear layered shell model than the single column model. Single column model is just used as a reference model to compare the results.
- Results obtained from all the modeling techniques will be compared in terms of accuracy and efficiency and finally the appropriate model will be recommended that can be used efficiently and conveniently for the practical purpose.

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