Nonlinear finite element dynamic analysis of squat shear wall with openings

Abstract

Shear wall has been considered as a major lateral load-resisting element inmultistoried building located in wind- or earthquake-prone zone. The behavior ofshear wall under various loading conditions has been the subject of intenseresearch for the last few decades. The behavior of shear walls without openingsis completely well understood and well documented in literature. The use ofsquat shear wall has been found in many low-rise buildings. On the other hand,squat shear walls may also be provided with openings due to the functionalrequirement such as placement of doors/windows in the building. The size andlocation of the opening play a significant role in the response of the shearwall. Even though it is intuitively known that the size of opening hassignificant effects on the behavior of a shear wall, it is desirable to know thelimiting size of opening in the shear wall, beyond which the shear walls mayfail or become unserviceable, especially when subjected to severe earthquakeground motions. In this study, the materially nonlinear dynamic response of theshear wall, with and without openings for different damping ratios, subjected toEL Centro earthquake has been captured. For dynamic analysis, constantacceleration Newmark β method of direct time integration has been used.From the study, it was observed that the presence of opening results in severedisplacements and stresses on the shear wall and also results in stressconcentration near the opening tip. Hence, the presence of damping has beenconsidered to be vital for large opening under severe dynamic loadingconditions.

Introduction

The reinforced concrete tall buildings are subjected to lateral loads such as windand earthquake. In order to resist the lateral load, shear walls are provided in theframed structures as a lateral load resisting element (Rahimian 2011 Kim and Lee 2008, Kuang and Ho 2008). The importance of shear wall in mitigating the damage toreinforced concrete structures is well documented in the literature (Kuang and Ho2008). The reinforced concrete and masonry shear wallshave been in use for many decades as a major structural configuration to resist windand earthquake forces. In general, masonry shear walls are modeled using no-tensionassumption given the fact that masonry hardly resists tensile forces (Corbi 2013; Baratta and Corbi 2010). Theassumption of no-tension results in the tremendous reduction in the computationaleffort in masonry structures (Baratta et al. 2008, Baratta andCorbi 2010; Baratta and Corbi 2012). Theshear wall should possess sufficient strength and stiffness under any loadingconditions. The shear walls are generally classified on the basis of aspect ratio(height/width ratio). The shear walls with aspect ratio between 1 and 3 aregenerally considered to be of squat type and shear walls with aspect ratio greaterthan 3 are considered to be of slender type. In general, the structural response ofshear wall depends strongly on the type of loading, aspect ratio of shear wall,size, and location of the openings in the shear wall. The squat shear wallsgenerally fail in racking/shear mode whereas the slender shear walls fail in aflexural mode. In the case of low-rise shear walls, the racking deformation, a kindof shear deformation, becomes predominant and substantially contributes to theoverall deformation. The squat shear walls are frequently used in low-risemultistoried buildings as well as in high-rise buildings where the shear walls areextended up to few stories only. The importance of overturning of rigid blocks andracking deformation was described by Baratta et al. (2012,2013), as squat shear walls are more vulnerable tooverturning. Various experimental and analytical studies have been performed toinvestigate the response of solid shear wall under monotonic loading conditions(Lefas et al. 1990; Derecho et al. 1979; Mullapudi et al. 2009; Paknahad et al.2007). Sometimes openings are provided in the shear walldue to functional requirements. The structural behavior of the shear wall withopening becomes complex due to the stress concentration near the openings(Neuenhofer 2006). Many experimental investigations have beenperformed on reinforced concrete shear walls with and without openings subjected tosevere dynamic earthquake loading conditions (Mo 1988; Ricciet al. 2012; Gasparani et al. 2013).The aspect ratio of shear wall plays a significant role on the structural responseof shear wall with openings. The elastic analysis of shear wall can be performedusing (a) continuous connection method (CCM), (b) transfer matrix method, (c) widecolumn analogy (WCA) or frame analysis, (d) finite element method, and (e) discreteforce method. Neuenhofer (2006) has observed that for thesame opening area, the reduction in stiffness for squat and slender shear walls are50% and 20%, respectively. Thus, the aspect ratio becomes critical for squat shearwalls. Few analytical studies have been made on the response of shear wall withopenings (Neuenhofer 2006; MacLeod 1970; Rosman 1964; Schwaighofer 1967; Taylor et al. 1988). Rosman (1964) developed an approximate linear elastic approach usinglaminar analysis, based on different assumptions to analyze the shear wall with onerow and two rows of openings. Schwaighofer (1967) used thisapproach to analyze the shear wall with three rows of openings and observed that theRosman's theory predicts the behavior of shear wall with three rows of openings alsowith sufficient accuracy. Conventional methods such as CCM, transfer matrix method,and WCA cannot be used to analyze the complex structure such as shear walls withopenings. Moreover, it was found that the conventional methods result in remarkablypoor results for the squat shear walls where the mode of failure is predominantlyshear. It was also shown in literature that the hand calculation eitherunderestimates/overestimates the response in predicting the response of shear wallwith openings (Neuenhofer 2006). In order to make realisticpredictions of strength, stiffness, and seismic energy dissipation capacity, it isessential to use proper numerical technique. The finite element analysis has beenthe most versatile and successfully employed method of analysis in the past toaccurately predict the structural behavior of reinforced concrete shear wall inlinear as well as nonlinear range under any severe loading conditions. With theadvent in computing facilities, finite element method has gained an enormouspopularity among the structural engineering community, especially in the nonlineardynamic analysis. The nonlinearity of the structure may be due to the geometry ormaterial. Since shear wall is a huge structure, the deformation of the shear wallwas assumed to be in control and hence, the geometric nonlinearity has not beenconsidered. In the present study, the two-dimensional analysis of the shear wallusing finite element methods has been conducted in order to accurately predict themodes of failure. In this study, nonlinear dynamic finite element two-dimensionalanalysis of reinforced concrete shear wall with and without openings has beencarried out using nine-node degenerated shell element with five degrees of freedomsat each node.

In order to investigate the influence of opening sizes on the elastic response ofsquat shear wall, a shear wall with dimensions of 3.6 m high, 3.6 m wideand 0.2-m-thick shear wall, subjected to EL Centro earthquake loading acting overthe period of 31.18 s, has been considered. The total earthquake loadingspreading over 31.18 s has been discretized into an interval of 0.02 s,thus resulting in 1,558 data points. The concentrated mass of 1,000 kN is located atthe top of the shear wall. Nevertheless, in order to investigate the post-earthquakeeffect on shear wall, the displacement response of shear wall has been captureduntil 40 s. Since every structure possesses some inherent damping, a minimumdamping ratio of 2.5% has been considered. The Rayleigh damping has been employedwith stiffness proportionality only to a circular frequency of 10 rad/second. Inorder to investigate the size of opening on structural response of shear wall, threedifferent sizes of openings are considered, namely, (a) small opening(1.2 m × 1.2 m), (b) medium opening(1.2 m × 2.4 m), and (c) large opening(2.4 m × 2.4 m). Newmark β method of direct timeintegration with constant acceleration scheme, consistent mass matrix, and Rayleighdamping, have been adopted to calculate the dynamic response at discrete timeintervals. The analysis has also been carried out for different ratios 5%, 7.5%, and10%. The results of the shear wall with openings are compared with the solid shearwall.

Methods

Geometric modeling

The displacement-based finite element method has been considered to be the mostpopular choice because of its simplicity and ease with which the computationscan be performed. The use of shell element to model moderately thick structureslike shear wall is well documented in the literature (Liu and Teng 2008). Nevertheless, the general shell theory based on theclassical approach has been found to be complex in the finite elementformulation. On the other hand, the degenerated shell element (Ahmad et al.1970, Kant et al. 1994) derivedfrom the three-dimensional element has been quite successful in modelingmoderately thick structures because of their simplicity and has circumvented theuse of classical shell theory. The degenerated shell element (Figure 1) is based on the assumption that the normal to the middlesurface remains straight but not necessarily normal after deformation. Also, thestresses normal to the middle surface are considered to be negligible. However,when the thickness of element reduces, the degenerated shell element hassuffered from shear locking and membrane locking when subjected to fullnumerical integration. The shear locking and membrane locking are the parasiticshear stresses and membrane stresses present in the finite element solution. Inorder to alleviate locking problems, the reduced integration technique has beensuggested and adopted by many authors (Zienkiewicz et al. 1971; Paswey and Clough 1971). However, theuse of reduced integration resulted in spurious mechanisms or zero energy modesin some cases. The reduced integration ignores the high-ranked terms ininterpolated shear strain by numerical integration, thus introducing the chanceof development of spurious or zero energy modes in the element. The selectiveintegration, wherein different integration orders are used to integrate thebending, shear, and membrane terms of stiffness matrix, avoids the locking inmost of the cases. The assumed strain approach has been successfully adopted bymany researchers (Huang 1989; Bathe 2006) as an alternative to avoid locking. In the assumedstrain-based degenerated shell elements, the transverse shear strain andmembrane strains are interpolated from the assumed sampling points(Figure 2) obtained from the compatibilityrequirement between flexural and shear strain fields respectively.

Figure 1

Geometry of nine-node degenerated shell element.

Figure 2

Sampling point locations for assumed shear/membrane strains.

Thus, the assumed strain approach allows the use of full integration, avoidingthe risk of zero energy modes. In this element, five degrees of freedom areconsidered at each node, comprising three translations and two rotations of thenormal. The formulation of degenerated isoparametric shell element is completelydescribed by Huang (1989). The geometry of thedegenerated shell element can be conveniently represented by the coordinates andnormal vectors of the middle surface. In general, the geometry and kinematics ofdeformations are described by using different coordinate systems. The Cartesiancoordinate system is used to define the geometry of the structure, nodalcoordinates and displacements, global stiffness matrix, and applied load vector.The coordinates of a point within an element are obtained by interpolating thenodal coordinates through the element shape functions:

$$\left\{\begin{array}{l}x\\ y\\ z\end{array}\right\}={\displaystyle \sum _{k=1}^9{N}_k\left(\xi ,\eta \right){\left\{\begin{array}{l}{x}_k\\ y_k\\ z_k\end{array}\right\}}_{\mathrm{mid}}+}{\displaystyle \sum _{k=1}^9{N}_k}\left(\xi ,\eta \right)\frac{\zeta h_k}{2}\left\{\begin{array}{l}{V}_{3k}^x\\ V_{3k}^y\\ V_{3k}^z\end{array}\right\}.$$

(1)

The displacements at any point inside the finite element can also be expressedby

$$\left\{\begin{array}{c}\hfill u\hfill \\ \hfill v\hfill \\ \hfill w\hfill \end{array}\right\}={\displaystyle \sum _{k=1}^n{N}_k{\left\{\begin{array}{c}\hfill u_k\hfill \\ \hfill v_k\hfill \\ \hfill w_k\hfill \end{array}\right\}}_{\mathrm{mid}}+{\displaystyle \sum _{k=1}^n{N}_k\zeta \frac{h_k}{2}\left[\begin{array}{cc}\hfill v_{1k}^x\hfill & \hfill -v_{2k}^x\hfill \\ \hfill v_{1k}^y\hfill & \hfill -v_{2k}^y\hfill \\ \hfill v_{1k}^z\hfill & \hfill -v_{2k}^z\hfill \end{array}\right]\left[\begin{array}{c}\hfill {\beta }_{1k}\hfill \\ \hfill {\beta }_{2k}\hfill \end{array}\right]}}$$

(2)

In the above expression, N _k (ξ, η)  are the element shapefunctions and h _k is the shell thickness at node k. V _ik are the natural coordinates at any node k under consideration. Anodal coordinate system is defined at each nodal point with origin situated atthe reference mid-surface. The vector V_3k is constructed from the nodal coordinates at top and bottom surfaces andis expressed as

$${\mathbf{v}}_{\mathbf{3}k}={\left\{\begin{array}{l}{x}_k\\ y_k\\ z_k\end{array}\right\}}_{\mathit{top}}-{\left\{\begin{array}{l}{x}_k\\ y_k\\ z_k\end{array}\right\}}_{\mathit{bottom}}.$$

(3)

v_3k defines the direction of the normal at any node ‘k’,which is not necessarily perpendicular to the mid-surface. The major advantageof the definition of v_3k with normal but not necessarily perpendicular to mid-surface is thatthere are no gaps or overlaps along element boundaries. The element shapefunctions are calculated in the natural coordinate system as

$$\left.\begin{array}{l}{N}_1=\frac{1}{4}\xi \left(1+\xi \right)\eta \left(1+\eta \right)\\ N_2=\frac{1}{2}\left(1+\xi \right)\left(1-\xi \right)\eta \left(1+\eta \right)\\ N_3=-\frac{1}{4}\left(1-\xi \right)\eta \left(1+\eta \right)\\ N_4=-\frac{1}{2}\xi \left(1-\xi \right)\left(1+\eta \right)\left(1-\eta \right)\\ N_5=\frac{1}{4}\left(1-\xi \right)\eta \left(1-\eta \right)\\ N_6=-\frac{1}{2}\left(1+\xi \right)\left(1-\xi \right)\eta \left(1-\eta \right)\\ N_7=-\frac{1}{4}\xi \left(1+\xi \right)\eta \left(1-\eta \right)\\ N_8=\frac{1}{2}\xi \left(1+\xi \right)\left(1+\eta \right)\left(1-\eta \right)\\ N_9=\left(1+\xi \right)\left(1-\xi \right)\left(1+\eta \right)\left(1-\eta \right)\end{array}\right\}.$$

(4)

Once the displacements are determined, the strains and stresses are calculatedusing strain–displacement matrix and material constitutive matrix,respectively. Since the resultant stresses in the z-direction(out-of-plane direction) are considered to be 0, there are only five independentstrains. The strain condition at a point is defined by the vector

$$ϵ=\left[{ϵ}_x{ϵ}_y{\gamma }_{\mathit{xy}}{\gamma }_{\mathit{xz}}{\gamma }_{\mathit{yz}}\right].$$

(5)

Since geometric nonlinearity is not considered, the five strain components arerelated to displacements of only first order:

$$\mathbf{ϵ}=\left[\begin{array}{c}\hfill {ϵ}_{x^{\prime }}\hfill \\ \hfill {ϵ}_{y^{\prime }}\hfill \\ \hfill {\gamma }_{x^{\prime }{y}^{\prime }}\hfill \\ \hfill {\gamma }_{x^{\prime }{z}^{\prime }}\hfill \\ \hfill {\gamma }_{y^{\prime }{z}^{\prime }}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill \frac{\partial u^{\prime }}{\partial x^{\prime }}\hfill \\ \hfill \frac{\partial v^{\prime }}{\partial y^{\prime }}\hfill \\ \hfill \frac{\partial u^{\prime }}{\partial y^{\prime }}+\frac{\partial v^{\prime }}{\partial x^{\prime }}\hfill \\ \hfill \frac{\partial u^{\prime }}{\partial z^{\prime }}+\frac{\partial w^{\prime }}{\partial x^{\prime }}\hfill \\ \hfill \frac{\partial v^{\prime }}{\partial z^{\prime }}+\frac{\partial w^{\prime }}{\partial y^{\prime }}\hfill \end{array}\right].$$

(6)

The transformation matrix has been used to convert the local coordinate systeminto global coordinate system.

$$\left[\begin{array}{ccc}\hfill \frac{\partial u^{\prime }}{\partial x^{\prime }}\hfill & \hfill \frac{\partial v^{\prime }}{\partial x^{\prime }}\hfill & \hfill \frac{\partial w^{\prime }}{\partial x^{\prime }}\hfill \\ \hfill \frac{\partial u^{\prime }}{\partial y^{\prime }}\hfill & \hfill \frac{\partial v^{\prime }}{\partial y^{\prime }}\hfill & \hfill \frac{\partial w^{\prime }}{\partial y^{\prime }}\hfill \\ \hfill \frac{\partial u^{\prime }}{\partial z^{\prime }}\hfill & \hfill \frac{\partial v^{\prime }}{\partial z^{\prime }}\hfill & \hfill \frac{\partial w^{\prime }}{\partial z^{\prime }}\hfill \end{array}\right]={\left[\mathbf{T}\right]}^T\left[\begin{array}{ccc}\hfill \frac{\partial u}{\partial x}\hfill & \hfill \frac{\partial v}{\partial x}\hfill & \hfill \frac{\partial w}{\partial x}\hfill \\ \hfill \frac{\partial u}{\partial y}\hfill & \hfill \frac{\partial v}{\partial y}\hfill & \hfill \frac{\partial w}{\partial y}\hfill \\ \hfill \frac{\partial u}{\partial z}\hfill & \hfill \frac{\partial v}{\partial z}\hfill & \hfill \frac{\partial w}{\partial z}\hfill \end{array}\right]\left[\mathbf{T}\right]$$

(7)

In order to transform the Cartesian coordinate system into natural coordinatesystem, the Jacobian matrix has been employed:

$$\left[\begin{array}{ccc}\hfill \frac{\partial u}{\partial x}\hfill & \hfill \frac{\partial v}{\partial x}\hfill & \hfill \frac{\partial w}{\partial x}\hfill \\ \hfill \frac{\partial u}{\partial y}\hfill & \hfill \frac{\partial v}{\partial y}\hfill & \hfill \frac{\partial w}{\partial y}\hfill \\ \hfill \frac{\partial u}{\partial z}\hfill & \hfill \frac{\partial v}{\partial z}\hfill & \hfill \frac{\partial w}{\partial z}\hfill \end{array}\right]=J^{-1}\left[\begin{array}{ccc}\hfill \frac{\partial u}{\partial \xi }\hfill & \hfill \frac{\partial v}{\partial \xi }\hfill & \hfill \frac{\partial w}{\partial \xi }\hfill \\ \hfill \frac{\partial u}{\partial \eta }\hfill & \hfill \frac{\partial v}{\partial \eta }\hfill & \hfill \frac{\partial w}{\partial \eta }\hfill \\ \hfill \frac{\partial u}{\partial \zeta }\hfill & \hfill \frac{\partial v}{\partial \zeta }\hfill & \hfill \frac{\partial w}{\partial \zeta }\hfill \end{array}\right],$$

(8)

where

$$J=\left[\begin{array}{ccc}\hfill \frac{\partial x}{\partial \xi }\hfill & \hfill \frac{\partial y}{\partial \xi }\hfill & \hfill \frac{\partial z}{\partial \xi }\hfill \\ \hfill \frac{\partial x}{\partial \eta }\hfill & \hfill \frac{\partial y}{\partial \eta }\hfill & \hfill \frac{\partial z}{\partial \eta }\hfill \\ \hfill \frac{\partial x}{\partial \varsigma }\hfill & \hfill \frac{\partial y}{\partial \varsigma }\hfill & \hfill \frac{\partial z}{\partial \varsigma }\hfill \end{array}\right].$$

(9)

The strain–displacement matrix [B] relates the strain and displacementcomponents as

$$ϵ=\mathbf{B}\delta$$

(10)

$$\left\{\delta \right\}={\left[uvw\alpha \beta \right]}^T.$$

(11)

The numerical integration has to be resorted to in order to evaluate the elementstiffness matrix for an isoparametric degenerated shell element

$${\displaystyle \iint \mathit{dx}\mathit{dy}={\displaystyle \underset{-1}{\overset{+1}{\int }}{\displaystyle \underset{-1}{\overset{+1}{\int }}det\left|J\right|\mathit{d\xi }\mathit{d\eta }}.}}$$

(12)

The layered element formulation (Teng et al. 2005) allowsthe integration through the element thickness, which is divided into severalconcrete and steel layers. Each layer is assumed to have one integration pointat its mid-surface. The steel layers are used to model the in-planereinforcement only. The assumed transverse shear strain fields, interpolated atthe six appropriately located sampling points, as shown in Figure 2, are

$$\begin{array}{c}\hfill {\overline{\gamma }}_{\mathit{\xi \zeta }}={\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^2{P}_i\left(\eta \right)Q_j\left(\xi \right){\gamma }_{\mathit{\xi \zeta }}^{\mathit{ij}}}}\hfill \\ \hfill {\overline{\gamma }}_{\mathit{\eta \zeta }}={\displaystyle \sum _{i=1}^3{\displaystyle \sum _{j=1}^2{P}_i\left(\xi \right)Q_j\left(\eta \right){\gamma }_{\mathit{\eta \varsigma }}^{\mathit{ij}}}}\hfill \end{array}.$$

(13)

In the above equation, ${\gamma }_{\mathit{\xi \varsigma }}^{\mathit{ij}}$ and ${\gamma }_{\mathit{\eta \zeta }}^{\mathit{ij}}$ are the shear strains obtained from Lagrangian shapefunctions.

The interpolating functions P _i (z) and Q _j (z) are

$$\begin{array}{l}{P}_1\left(z\right)=\frac{z}{2}.\left(z+1\right),P_2\left(z\right)=1-z^2,P_3\left(z\right)=\frac{z}{2}.\left(z-1\right)\\ Q_1\left(z\right)=\frac{1}{2}\left(1+\sqrt{3}.z\right),Q_2\left(z\right)=\frac{1}{2}\left(1-\sqrt{3}.z\right).\end{array}$$

(14)

Hence, it can be observed that ${\overline{\gamma }}_{\mathit{\xi \zeta }}$ is linear in ξ direction and quadratic inη direction, while ${\overline{\gamma }}_{\mathit{\eta \zeta }}$ is linear in η direction and quadratic inξ direction. The polynomial terms for curvature of nine-nodeLagrangian element, κ _ξ and κ _η , are the same as the assumed shear strain as given by

$$\begin{array}{l}{\kappa }_{\xi }=\frac{\partial {\theta }_{\xi }\left(1,\xi ,\eta ,\mathit{\xi \eta },{\xi }^2,{\xi }^2\eta ,{\eta }^2,\xi {\eta }^2,{\xi }^2{\eta }^2\right)}{\partial \xi }\\ {\kappa }_{\xi }={\kappa }_{\xi }\left(1,\xi ,\eta ,\mathit{\xi \eta },{\eta }^2,\xi {\eta }^2\right)\end{array}$$

(15)

$$\begin{array}{l}{\kappa }_{\eta }=\frac{\partial {\theta }_{\eta }\left(1,\xi ,\eta ,\mathit{\xi \eta },{\xi }^2,{\xi }^2\eta ,{\eta }^2,\xi {\eta }^2,{\xi }^2{\eta }^2\right)}{\partial \eta }\\ {\kappa }_{\eta }=\left(1,\xi ,\eta ,\mathit{\xi \eta },{\xi }^2,{\xi }^2{\eta }^2\right)\end{array}$$

(16)

$$\begin{array}{l}{\overline{\gamma }}_{\mathit{\xi \zeta }}={\overline{\gamma }}_{\mathit{\xi \zeta }}\left(1,\xi ,\eta ,\mathit{\xi \eta },{\eta }^2,\xi {\eta }^2\right)\\ {\overline{\gamma }}_{\mathit{\eta \zeta }}={\overline{\gamma }}_{\mathit{\eta \zeta }}\left(1,\xi ,\eta ,\mathit{\xi \eta },{\eta }^2,\xi {\eta }^2\right).\end{array}$$

(17)

The original shear strain obtained from the Lagrange shape functionsγ _ξζ and γ _ηζ is

$$\begin{array}{l}{\gamma }_{\mathit{\xi \zeta }}={\theta }_{\xi }+\frac{\partial w}{\partial \xi }={\gamma }_{\mathit{\xi \zeta }}\left(1,\xi ,\eta ,\mathit{\xi \eta },{\xi }^2,{\xi }^2\eta ,{\eta }^2,\xi {\eta }^2,{\xi }^2{\eta }^2\right)\\ {\gamma }_{\mathit{\eta \zeta }}={\theta }_{\eta }+\frac{\partial w}{\partial \eta }={\gamma }_{\mathit{\eta \zeta }}\left(1,\xi ,\eta ,\mathit{\xi \eta },{\xi }^2,{\xi }^2\eta ,{\eta }^2,\xi {\eta }^2,{\xi }^2{\eta }^2\right).\end{array}$$

(18)

In the above equation, θ _ξ and θ _η are the rotating normal and w is the transverse displacement ofthe element. It can be clearly seen that the original shear strain and assumedshear strain are not compatible and hence the shear locking exists for very thinshell cases. The appropriately chosen polynomial terms and sampling pointsensure the elimination of risk of spurious zero energy modes. The assumed straincan be considered as a special case of integration scheme wherein for function ${\overline{\gamma }}_{\mathit{\xi \zeta }}$, full integration is employed in η direction andreduced integration is employed in ξ direction. On the other hand,for function ${\overline{\gamma }}_{\mathit{\eta \zeta }}$, reduced integration is employed in η directionand full integration is employed in ξ direction. The membrane andshear strains are interpolated from identical sampling points, even though themembrane strains are expressed in orthogonal curvilinear coordinate system andtransverse shear strains are expressed in natural coordinate system. The nextsection briefly deals with the material modeling of concrete and reinforcingsteel.

Material modeling

The modeling of material play may a crucial role in achieving the correctresponse. The presence of nonlinearity may add another dimension of complexityto it. Many material models have been developed in the past over the years suchas compression field theory (CFT) proposed by Collins and Mitchell (1980) and modified compression field theory (MCFT) proposedby Vecchio and Collins (1986) to model the crackedresponse of concrete in compression. The nonlinearities in the structure mayaccurately be estimated and incorporated in the solution algorithm. The accuracyof the solution algorithm depends strongly on the prediction of second-ordereffects that cause nonlinearities, such as tension stiffening, compressionsoftening, and stress transfer nonlinearities around cracks. Thesenonlinearities are usually incorporated in the constitutive modeling of thereinforced concrete. In order to incorporate geometric nonlinearity, thesecond-order terms of strains are included. In this study, only materialnonlinearity has been considered. The subsequent sections describe the modelingof concrete in compression and tension, and modeling of steel briefly.

Concrete modeling in tension

The presence of crack in concrete has much influence on the response ofnonlinear behavior of reinforced concrete structures. The crack in theconcrete is assumed to occur when the tensile stress exceeds the tensilestrength. The cracking of concrete results in the loss of continuity in theload transfer and, hence, the stresses in both concrete as well as steelreinforcement differ significantly. Therefore, the analysis of concretefracture has been very important in order to predict the response ofstructure precisely. The numerical simulation of concrete fracture can berepresented either by discrete crack proposed by Ngo and Scordelis (1967) or by smeared crack proposed by Rashid (1968). The objective of discrete crack is to simulatethe initiation and propagation of dominant cracks present in the structure.In the case of discrete crack approach, nodes are disassociated due to thepresence of cracks and therefore, the structure requires frequentrenumbering of nodes, which may render huge computational cost.Nevertheless, when the structure's behavior has been dominated by only fewdominant cracks, the discrete modeling of cracking seems the only choice. Onthe other hand, the smeared crack approach smears out the cracks over thecontinuum, captures the deterioration process through the constitutiverelationship, and reduces the computational cost and time drastically.

Crack modeling has gone through several stages due to the advancement intechnology and computing facilities. Earlier research work indicates thatthe formation of crack results in the complete reduction in stresses inperpendicular direction thus neglecting the phenomenon called tensionstiffening. With the rapid increase in extensive experimental investigationsas well as computing facilities, many finite element codes have beendeveloped for the nonlinear finite element analysis, which incorporate thetension stiffening effect. The first tension stiffening model using degradedconcrete modulus was proposed by Scanlon and Murray and subsequently, manyanalytical models have been developed such as Lin and Scordelis model, Veboand Ghali model, and Gilbert and Warner model (Nayal and Rasheed 2006). The cracks are always assumed to be formed inthe direction which is perpendicular to the direction of the maximumprincipal stress. These directions may not necessarily remain the samethroughout the analysis and loading; hence, the modeling of orientation ofcrack plays a significant role on the response of the structure. Still, dueto simplicity, many investigations have been performed using fixed crackapproach, wherein the direction of principal strain axes may remain fixedthroughout the analysis. In this study also, the direction of crack has beenconsidered to be fixed throughout the duration of the analysis. However, themodeling of aggregate interlock has not been taken very seriously. Theconstant shear retention factor or the simple function has been employed tomodel the shear transfer across the cracks. Apart from the initiation ofcrack, the propagation of crack also plays a crucial role in the response ofstructure. The prediction of crack propagation is a very difficultphenomenon due to scarcity and conflict of test results. Nevertheless, thepropagation of cracks plays a crucial role on the response of nonlinearanalysis of reinforced concrete structures. The plain concrete exhibitssoftening behavior and reinforced concrete exhibits stiffening behavior dueto the presence of active reinforcing steel. A gradual release of theconcrete stress (Figure 3) is adopted in thispresent study (Owen and Hinton 1980).

Figure 3

Tension stiffening of concrete.

The reduction in the stress is given by the following expression:

$$E_i=\alpha {f^{\prime }}_t\left(1-\frac{{ϵ}_i}{{ϵ}_m}\right)\frac{1}{{ϵ}_i};{ϵ}_t\le {ϵ}_i\le {ϵ}_m.$$

(19)

In the above equation, α and ϵ _m are the tension stiffening parameters. ϵ _m is the maximum value reached by the tensile strain at the pointconsidered; ϵ _i is the current tensile strain in material direction 1. Thecoefficient α depends on the percentage of steel in thesection. In the present study, the values of α andϵ _m are taken as 0.5 and 0.0020, respectively. It has also been reportedthat the influence of the tension stiffening constants on the response ofthe structures is generally small and hence the constant value is justifiedin the analysis (Owen and Hinton 1980). Generally,the cracked concrete can transfer shear forces through dowel action andaggregate interlock. The magnitude of shear moduli has been considerablyaffected because of extensive cracking in different directions(Table 1).

Table 1 Crack formulation

Thus, the reduced shear moduli can be put to incorporate the aggregateinterlock and dowel action. In the plain concrete, aggregate interlock isthe major shear transfer mechanism, and for reinforced concrete, dowelaction is the major shear transfer mechanism, with reinforcement ratio beingthe critical variable. In order to incorporate the aggregate interlock anddowel action, the appropriate value of cracked shear modulus (Cedolin et al.1977) has been considered in the material modelingof concrete.

Concrete modeling in compression

The theory of plasticity has been used in the compression modeling of theconcrete. The failure surface or bounding surface has been defined todemarcate plastic behavior from the elastic behavior. Failure surface is theimportant component in the concrete plasticity. Sometimes the failuresurface can be referred to as yield surface or loading surface. The materialbehaves in an elastic fashion as long as the stress lies below the failuresurface. Several failure models have been developed and reported in theliterature (Chen 1982). Nevertheless, thefive-parameter failure model proposed by Willam and Warnke (1975) seems to possess all inherent properties of the failuresurface. The failure surface (Figure 4) isconstructed using two meridians namely, compression meridian and tensionmeridian. The two meridians are pictorially depicted in a meridian plane andcross section of the failure surface is represented in the deviatoricplane.

Figure 4

Failure model for concrete.

The variations of the average shear stresses τ_mt and τ_mc along tensile (θ = 0°) andcompressive (θ = 60°) meridians areapproximated by second-order parabolic expressions in terms of the averagenormal stresses σ _m as follows:

$$\begin{array}{cc}\hfill \frac{{\tau }_{\mathrm{mt}}}{{f^{\prime }}_c}=\frac{{\rho }_t}{\sqrt{5}{f^{\prime }}_c}=a_0+a_1\left(\frac{{\sigma }_m}{{f^{\prime }}_c}\right)+a_2{\left(\frac{{\sigma }_m}{{f^{\prime }}_c}\right)}^2\hfill & \hfill \theta =0^0\hfill \\ \hfill \frac{{\tau }_{\mathrm{mc}}}{{f^{\prime }}_c}=\frac{{\rho }_c}{\sqrt{5}{f^{\prime }}_c}=b_0+b_1\left(\frac{{\sigma }_m}{{f^{\prime }}_c}\right)+b_2{\left(\frac{{\sigma }_m}{{f^{\prime }}_c}\right)}^2\hfill & \hfill \theta ={60}^0\hfill \end{array}.$$

(20)

These two meridians must intersect the hydrostatic axis at the same point ${\sigma }_m/{f^{\prime }}_c={\overline{\xi }}_0$ (corresponding to hydrostatic tension); the number ofparameters need to be determined is reduced to five. The five parameters(a₀ or b₀, a₁, a₂, b₁, b₂) are to be determined from a set of experimental data withwhich the failure surface can be constructed using second-order parabolicexpressions.

The failure surface is expressed as

$$f\left({\sigma }_m,{\tau }_m,\theta \right)=\sqrt{5}\frac{{\tau }_m}{\rho \left({\sigma }_m,\theta \right)}-1=0$$

(21)

$$\begin{array}{c}\rho \left(\theta \right)=2{\rho }_c\left({\rho }_c^2-{\rho }_t^2\right)cos\theta +{\rho }_c\left(2{\rho }_t-{\rho }_c\right)\\ \frac{{\left[4\left({\rho }_c^2-{\rho }_t^2\right){cos}^2\theta +5{\rho }_t^2-4{\rho }_t{\rho }_c\right]}^{1/2}}{4\left({\rho }_c^2-{\rho }_t^2\right){cos}^2\theta +{\left({\rho }_c-2{\rho }_t\right)}^2}.\end{array}$$

(22)

The formulation of Willam-Warnke five-parameter material model is describedin Chen (1982). Once the yield surface is reached, anyfurther increase in the loading results in plastic flow. The magnitude anddirection of the plastic strain increment are defined using the flow rulewhich is described in the next section.

Flow rule

In this method, associated flow rule is employed because of the lack ofexperimental evidence in non-associated flow rule. The plastic strainincrement expressed in terms of current stress increment is given as

$$d{ϵ}_{\mathit{ij}}^p=\mathit{d\lambda }\frac{\partial f\left(\sigma \right)}{\partial {\sigma }_{\mathit{ij}}}.$$

(23)

dλ determines the magnitude of the plastic strain increment.The gradient∂ f(σ)/∂ σ _ij defines the direction of plastic strain increment to be perpendicularto the yield surface; f(σ) is the loading conditionor the loading surfaces.

Hardening rule

The relationship between loading surfaces (or effective stress) and theplastic work (accumulated plastic strain) is represented by a hardening rule(Figure 5). The ‘Madrid parabola’is used to define the hardening rule

$$\sigma =E_oϵ-\frac{1}{2}\frac{E_0}{{ϵ}_0}{ϵ}^2.$$

(24)

Figure 5

Isotropic hardening and the corresponding uniaxialstress–strain curve.

In the above equation, E₀ is the initial elasticity modulus, ϵ is thetotal strain, and ϵ₀ is the total strain at peak stress f′ _c . The total strain can be divided into elastic and plastic componentsas

$$ϵ={ϵ}_e+{ϵ}_p$$

(25)

$$\left\{\dot{\sigma }\right\}=\left[D^e\right]\left(\left\{\dot{ϵ}\right\}-\left\{{\dot{ϵ}}^p\right\}\right)$$

(26)

$$\left\{{\dot{ϵ}}^p\right\}=\dot{\lambda }\left\{a\right\};\left\{a\right\}=\left\{\frac{\partial F}{\partial \sigma }\right\}.$$

(27)

In the above equation, F is the yield function and $\dot{\lambda }$ is the consistency parameter which defines the magnitudeof the plastic flow. The loading and unloading conditions (Kuhn-Tuckerconditions) can be stated as

$$\dot{\lambda }\ge 0.;F\le 0;\dot{\lambda }F=0.$$

(28)

The first of these Khun-Tucker conditions indicates that the consistencyparameter is non-negative; the second condition implies that the stressstates must lie on or within the yield surface. The third condition ensuresthat the stresses lie on the yield surface during the plastic loading. Thefollowing are the conditions:

$$\begin{array}{l}{\left\{\mathrm{a}\right\}}^{\mathrm{T}}\left\{\dot{\mathrm{\sigma }}\right\}=0\\ {\left\{\mathrm{a}\right\}}^{\mathrm{T}}\left[{\mathrm{D}}^{\mathrm{e}}\right]\left(\left\{\dot{\mathrm{ϵ}}\right\}-\left\{{\dot{\mathrm{ϵ}}}_{\mathrm{p}}\right\}\right)=0\\ {\left\{\mathrm{a}\right\}}^{\mathrm{T}}\left[{\mathrm{D}}^{\mathrm{e}}\right]\left(\left\{\dot{\mathrm{ϵ}}\right\}-\dot{\mathrm{\lambda }}\left\{\mathrm{a}\right\}\right)=0\\ \dot{\mathrm{\lambda }}=\frac{{\left\{\mathrm{a}\right\}}^{\mathrm{T}}\left[{\mathrm{D}}^{\mathrm{e}}\right]\left\{\dot{\mathrm{ϵ}}\right\}}{{\left\{\mathrm{a}\right\}}^{\mathrm{T}}\left[{\mathrm{D}}^{\mathrm{e}}\right]\left\{\mathrm{a}\right\}}.\end{array}$$

(29)

The elastoplastic constitutive matrix is given by the followingexpression:

$$\left[D_{\mathrm{ep}}\right]=\left[D\right]-\frac{\left[D\right]\left\{a\right\}{\left\{a\right\}}^T\left[D\right]}{H+{\left\{a\right\}}^T\left[D\right]\left\{a\right\}}.$$

(30)

In the above equation, a = flow vector, defined by thestress gradient of the yield function; D = constitutivematrix in elastic range. The second term in the above equation representsthe effect of degradation of the material during the plastic loading.

Modeling of reinforcement in tension and compression

In order to incorporate the effect of steel reinforcement, the layered approachis adopted in this study. The steel is modeled as a smeared layer of equivalentthickness in the natural coordinate system. The properties of the material areassumed to be constant in that layer. The bilinear stress–strain curvewith linear elastic and strain hardening region is adopted in this study forboth compression and tension. The vertical and horizontal reinforcement in theshear wall is taken as 1%; however, no ductile detailing is considered near theopenings.

Evaluation of stiffness matrix

The stiffness matrix can be calculated using the following expression

$$\left[K_m\right]={\displaystyle \underset{0}{\overset{L}{\int }}{\left[B\right]}^T\left[D\right]\left[B\right]\mathit{dV}}.$$

(31)

Dynamic analysis

The dynamic analysis of structure can be performed by three ways, namely, (a)equivalent lateral force method, (b) response spectrum method, and (c)time-history method. The equivalent lateral force method determines theequivalent dynamic effect in a static manner. The response spectrum method aimsin determining the maximum response quantity of the structure. For tall andirregular buildings, dynamic analysis by response spectrum method seems to bethe popular choice among designers. The time-history analysis of the structurehas been successfully used to analyze the structure especially of hugeimportance. Even though time-history analysis consumes time, it is the onlymethod capable of giving results closer to the actual one especially in thenonlinear regime. In the dynamic analysis, the loads are applied over a periodof time and the response is obtained at different time intervals. The equationof dynamic equilibrium at any time ‘t’ is given by Equation(1)

$$\left[M\right]\left[{\ddot{U}}^t\right]+\left[C\right]\left[{\dot{U}}^t\right]+\left[K\right]\left[U^t\right]=\left[R^t\right].$$

(32)

In the above equation, M, C and K are the mass, damping and stiffness matricesrespectively. The mass matrix can be formulated either by using consistent massapproach or by using lumped mass approach. Since damping cannot be preciselydetermined analytically, the damping can be considered proportional to mass orstiffness or both depending on the type of the problem. The direct timeintegration of the equation of motion can be performed using explicit (centraldifference scheme) and implicit (Houbolt method, Newmark β method andWilson Theta method, etc.) time integration. In the explicit time integration,the formation of complete stiffness matrix of the structure is not required andhence saves a lot of computer time and money in storing and saving those data.Moreover, in the case of all explicit time integration schemes, the iterationsare not required as the equilibrium at timet + ∆t depends on the equilibrium attime t. Nevertheless, the major drawback of explicit time integrationis that the time step (∆t) used for calculation of response hasto be smaller than the critical time step (∆t_cr) to ensure the stable solution:

$$\mathit{\Delta t}\le \Delta t_{\mathrm{cr}}=\frac{T_n}{\pi }=\frac{2}{\omega }.$$

(33)

On the other hand, implicit time integration requires the iterations to becarried out within the time step, as the solution at timet + ∆t involves the equilibrium equationat t + ∆t. The Newmark β methodconverges to various implicit and explicit schemes for different values of beta,which is called the stability parameter. In this study, forβ = 0.25, the Newmark β method converges to the constantacceleration implicit method, known as trapezoidal rule. The trapezoidal rule isunconditionally stable and hence allows larger time step to be used in thecalculation of response. Nevertheless, the time step can be made smaller fromthe accuracy point of view. The formulation of implicit Newmark β method(trapezoidal rule) is mentioned in Table 2 (Bathe2006).

Table 2 Step by step Newmark β method of time integration

Formulation of mass matrix

In any dynamic analysis, the formulation of mass matrix is very important incapturing the correct response of a structure. The masses can be assumed tobe distributed over the entire finite element mesh or can be assumed to belumped at nodes. The former is known as consistent mass matrix and thelatter is known as lumped mass matrix. The mass matrix is said to beconsistent if the formulation involves the same shape functions (N _i ) as used for the determination of stiffness matrix. The consistentmass matrix contains off-diagonal terms. Nevertheless, the consistent massmatrix is computationally expensive. The consistent element mass matrix isgiven by

$$\left[M_e\right]={\displaystyle \underset{V}{\int }\rho \left[N_i\right]{\left[N_i\right]}^T\mathit{dV}}.$$

(34)

For linear or translational motion, resistance of an object to a change ofstate in motion is measured in terms of mass and is given by Newton's law asfollows:

$$F=ma.$$

(35)

On the other hand, when a rigid body is rotated, the resistance of object toa change of state in a rotating motion given by rotational inertia measuredin terms of moment of inertia. The influence of rotary inertia in the caseof thick plates is established in the literature (Huang 1989) as

$$\begin{array}{l}T={\displaystyle \sum _{i=1}^N\frac{1}{2}}{m}_i{v}_i^2={\displaystyle \sum _{i=1}^N\frac{1}{2}}{m}_i{\left(\omega r_i\right)}^2\\ =\frac{1}{2}{\omega }^2\left[{\displaystyle \sum m_i{r}_i^2}\right];I={\displaystyle \sum _{i=1}^N{m}_i{r}_i^2};T=\frac{1}{2}I{\omega }^2,\end{array}$$

(36)

where in the above expression of kinetic energy, I is the moment ofinertia and ω is the angular velocity. The above equation canbe comparable in linear terms by replacing the moment of inertia by mass andangular velocity by linear velocity. The Newton's law of rotational motionis given by

$$\tau =\mathit{I\alpha },$$

(37)

where α is the angular acceleration. Thelumped mass matrix is purely diagonal and hence computationally cheaper thanthe consistent mass matrix. Nevertheless, the diagonalization of the massmatrix from the full mass matrix results in the loss of information andaccuracy (Huang 1987). There are many ways to lumpthe mass matrices as described in the literature. Nodal quadrature, row sum,and special lumping are the three lumping procedures available to generatethe lumped mass matrices. It has found that all three methods of lumpinglead to the same mass matrix for nine-node rectangular elements. One of themost efficient means of lumping is to distribute the element mass inproportion to the diagonal terms of consistent mass matrix (Archer andWhalen 2006) and also discarding the off-diagonalelements. This way of lumping has been successfully used in many finiteelement codes in practice. The advantage of above special lumping scheme isthat the assurance of positive definiteness of mass matrix. Lumped massmatrix is preferred, if not mandatory, over consistent mass matrix in thecase of explicit time integration. The use of lumped mass matrix is mostlyemployed in lower-order elements. For higher-order elements, the use oflumped mass matrix may not be an appropriate option. Hence, in this study, aconsistent mass matrix is employed to formulate the mass matrix.

The linear inertia or translational inertia is given by the followingexpression:

$$m_{\mathit{\text{ii}}}=w_i{\displaystyle \int \mathit{\text{ρdV}}}.$$

(38)

The rotational inertia is given by the following expression:

$$\begin{array}{l}{I}_{\mathit{ii}}=w_i{\displaystyle \underset{V_e}{\int }\rho {z^{\prime }}^2\mathit{dV}}=W_i{I}^{\prime }\\ I^{\prime }={\displaystyle \underset{V_e}{\int }\rho z^{\text{'}2}\mathit{dV}},\end{array}$$

(39)

where W _i is the multiplier and ρ is the mass density andz′ is the position of the layer with respect to the axisof rotation. The multiplier is represented by the following expression:

$$W_i=\frac{{\displaystyle \underset{V_e}{\int }\rho \left[N_i\right]{\left[N_i\right]}^T\mathit{dV}}}{{\displaystyle \sum _{K=1}^n{\displaystyle \underset{V_e}{\int }\rho \left[N_k\right]{\left[N_k\right]}^T\mathit{dV}}}}.$$

(40)

Formulation of damping matrix

Mass and stiffness matrices can be represented systematically by overallgeometry and material characteristics. However, damping can only berepresented in a phenomenological manner and thus making the dynamicanalysis of structures in a state of uncertainty. Nevertheless, severalinvestigations have been done in making the representation of damping in asimplistic yet logical manner (Zareian and Medina 2010). There is no single universally accepted methodology forrepresenting damping because of the nature of the state variables whichcontrol damping. Rayleigh dissipation function assumes that the dissipationof energy takes place and can be idealized as the function of velocity.Damping matrix can be formulated analogous to mass and stiffness matrices(Duggal 2007). When Rayleigh damping is used, theresultant damping matrix is of the same size as stiffness matrix. It is alsoimportant to note that the damping matrix should be formulated from dampingratio and not from the member sizes. Rayleigh damping is being usedconveniently because of its versatility in segregating each modesindependently. The amount of damping can be set appropriately by setting thevalues of alpha and beta relevantly depending on the requirement ofincluding higher modes.

Sometimes, the additional concentrated damping is also incorporated atselected degrees of freedom in addition to Rayleigh damping. Since dampingis a function of velocity, if there is no motion, there will be no damping.It has been mentioned that visco-elastic dampers are being employed instructures to mitigate dynamic effects. Usually, viscous damping is assumedbecause closed-form solution can be easily available. Very littleinformation is available about variation of damping in linear and nonlinearsystems. However, the effect of damping is generally less than the inertialand stiffness effects in most of the practical situations. Therefore, it isreasonable to account for damping by a simplified approximation. Therepresentation of damping through viscous damping coefficient has been inuse due to simplicity and accuracy. The damping force is assumed to beproportional to velocity and the constant of proportionality being theviscous damping coefficient. Whenever the system vibrates in a fluid,viscous damping is said to occur. The damping forces are proportional tovelocity of the medium and are represented as

$$F=c\dot{u}$$

(41)

$$\left[C\right]=\left[M\right]{\displaystyle \sum _{k=0}^{k=p-1}{\left(a_k{\left[M\right]}^{-1}\left[K\right]\right)}^k}.$$

(42)

In the above equation, k = 2 yields

$$\left[C\right]=\alpha \left[M\right]+\beta \left[K\right]$$

(43)

$${\zeta }_i=\frac{\alpha }{2{\omega }_i}+\frac{\beta {\omega }_i}{2}.$$

(44)

From the above equation, it is essential to note that if β parameter is0, the higher modes of the structure will be assigned with very littledamping. When the parameter α is 0, the higher modes will beheavily damped as the damping ratio is directly proportional to thefrequency (Clough and Penzein 1993). Thus, the choiceof damping is problem-dependent. Hence, it is inevitable to perform modalanalysis to determine the different frequencies for different modes. It isalso to be noted that the damping is controlled by only two parameters.Thus, in the Cauchy series, one may include as many terms as possibledepending upon the computational efficiency. The use of proportional dampingis implemented in most of the finite element codes. The reason for the useof proportional damping is justified by the following explanation. In theequation of motion, the coupling of terms usually occurs which is reflectedin the mass and stiffness matrices. Inertia coupling is present when themass matrix is non-diagonal, and static coupling is present when thestiffness matrix is non-diagonal. The coupling of the modes usually can beavoided easily in the case of undamped free vibration, but the same is nottrue for damped vibration. Hence, in order to represent the equation ofmotion in uncoupled form, it is suggested to have a damping matrixproportional to the uncoupled mass and stiffness matrices. Thus, Rayleigh'sproportional damping has the specific advantage in the sense that theequation of motion can be uncoupled when it is proportional to mass andstiffness matrices. Thus, it is proposed to use Rayleigh damping in thisstudy.

Nonlinear solution

The numerical procedure for nonlinear analysis employs the iterative procedure tosatisfy the equilibrium at the end of the load step. Once the convergence of thesolution is achieved, the algorithm proceeds to the next step. It is alwaysdesirable to keep the load step very small especially after the onset ofnonlinear behavior. The stiffness matrix is updated at the beginning of eachload step. The convergence is said to be achieved if the out-of-balance forces,calculated as under, is less than the specified tolerance:

$${\psi }_i^n=f^n-p_i^n=f^n-{\displaystyle \underset{V}{\int }{B}^T{\sigma }_i^n\mathit{dV}<\mathrm{tolerance}\left(0.0025\right)}.$$

(45)

Results and discussion

A reinforced concrete squat square shear wall,3.6 m × 3.6 m with 0.2-m thick, subjected to EL Centroearthquake loading over the period of 31.18 s with a peak ground accelerationof +0.29 g, as shown in Figure 6, has beenconsidered. For the finite element dynamic analysis, the entire shear wall isdiscretized using the nine-node degenerated shell elements of size(0.6 m × 1.2 m). The discretizations of the shear wallfor solid and opening cases are shown in Figure 7. Thedisplacement response has been calculated for a period of 40 s for differentdamping ratios 2.5%, 5%, 7.5%, and 10%, using Newmark β method with constantacceleration scheme. In order to investigate the size of opening on structuralbehavior of shear wall, the results of shear wall with no opening are compared withshear wall with three different sizes of openings, namely, (a) small opening(1.2 m × 1.2 m), (b) medium opening(1.2 m × 2.4 m), and (c) large opening(2.4 m × 2.4 m).

Figure 6

Acceleration time history.

Figure 7

Undeformed and Deformed shapes of shear wall. Undeformed and Deformedshapes for various opening cases (on the positive side) and Undeformed andDeformed shapes for various opening cases (on the negative side).

For the finite element analysis, the Young's modulus of elasticity of the concrete istaken as 2.98 × 10¹⁰ N/m² and Poisson'sratio is taken as 0.17. The undeformed and deformed shapes of the shear walls withand without openings are shown in Figure 7. Thedisplacement and stresses (bending and shear) are grouped into positive (if thevalue is above the horizontal axis of the time history) and negative (if the valueis below the horizontal axis of the time history), and, accordingly, the graphs andtables are also presented in this study. The deformed shapes (positive and negativedeformations) of solid shear wall and shear wall with small, medium, and largeopenings are qualitatively plotted in Figure 7 in orderto highlight the mode in which the shear wall deforms. The mode of deformation ofsolid shear wall and shear wall with small opening is almost similar. On the otherhand, shear wall with medium and large openings characterizes the shear mode ofdeformation. Hence, it is concluded that the racking deformation becomes importantin the presence of medium- as well as large-sized openings. The displacement,bending stress, and shear stress time-history responses for various opening sizes,viz., no opening, small opening, medium opening, and large opening withdamping ratios ζ = 2.5%, 5%, 7.5%, and 10% are plotted inFigures 8,9,10, respectively.

Figure 8

Top displacement time history for different damping ratios. (a) Shearwall with no opening. (b) Shear wall with small opening. (c)Shear wall with medium opening. (d) Shear wall with largeopening.

Figure 9

Top bending stress time history for different damping ratios. (a)Shear wall with no opening. (b) Shear wall with small opening.(c) Shear wall with medium opening. (d) Shear wall withlarge opening.

Figure 10

Top shear stress time history for different damping ratios. (a) Shearwall with no opening. (b) Shear wall with small opening. (c)Shear wall with medium opening. (d) Shear wall with largeopening.

It is obvious from Figure 8 that the increase in theopening size of shear wall has the strong influence on the displacement response,which is measured at the top of the shear wall, as evident from various timehistories as well from Tables 3 and 4. From Figure 8, it was observed that shearwall with large opening resulted in huge displacement and the influence of dampinghas not been found to be substantially benefitting in controlling the response.Moreover, as can be seen in Figure 8, displacements werepredominantly on the negative side (referred in this study as negative displacement)and thus indicate that the structure is behaving in the one-sided cyclic fashioncharacterizing the instability of the structure. The negative displacement values(as observed from Table 4) also validate the same. Hence,it is advisable to avoid large openings in the shear wall for better structuralresponse (Figures 11 and 12). Onthe other hand, increase in damping has a significant effect on the structuralresponse.

Figure 11

Positive displacement response. (a) Influence of opening. (b)Influence of damping.

Table 3 Positive structural responses of shear wall for different openingcases

Table 4 Negative structural responses of shear wall for different openingcases

Figure 12

Negative displacement response (a) Influence of opening. (b)Influence of damping.

The bending stress responses measured at the bottom of the shear wall for variousopening cases are plotted in Figure 9. It was observedthat the bending stresses of shear wall with no openings, small opening, and mediumopening almost follow the same trend with not much reversal of stresses which are onthe negative side. However, in the case of large opening, the huge reversal ofstresses taking over the entire time history indicates that the structure is heavilystressed and such openings are to be avoided in practice. As expected, damping hasnot resulted in substantial mitigation of the structural damage for shear wall withlarge openings (Figures 13 and 14).

Figure 13

Positive bending stress response. (a) Influence of opening.(b) Influence of damping.

Figure 14

Negative bending stress response. (a) Influence of opening.(b) Influence of damping.

Figure 15

Positive shear stress response. (a) Influence of opening. (b)Influence of damping.

Figure 16

Negative shear stress response. (a) Influence of opening. (b)Influence of damping.

The shear stress responses of shear wall measured at the bottom of the shear wall forvarious opening cases are plotted in Figure 10. Theresponse history of shear stress for shear wall with various opening cases indicatesthat for large openings, the shear stresses are found to be affected by dampingratios to the great extent. Shear stresses were found to be minimal for shear wallwith no opening, and damping has a pronounced effect on the shear wall with noopening and with small opening cases (Figures 15 and 16). Since only proportional stiffness damping was usedthroughout the study, higher modes were considerably damped and hence, the responses(displacement and stresses) decayed toward the end of the time step to a greaterextent.

Conclusions

In this paper, the influence of opening sizes on the structural response of squatsquare shear wall, 3.6 m × 3.6 m and 0.2 m thicksubjected to EL Centro earthquake loading and applied over the period of31.18 s, has been investigated for different damping ratios (2.5%, 5%, 7.5%,and 10%) using nonlinear finite element analysis. In addition, the followingconclusions have been drawn:

• 1. Racking deformation becomes paramount for shear walls with medium and large openings.

• 2. The shear wall with large openings is to be avoided as it results in instability with one-sided cyclic behavior.

• 3. Higher modes have been considerably damped due to the presence of stiffness proportional damping.

Authors’ information

• G. Muthukumar is a Lecturer in the Department of Civil Engineering, BITS Pilani. Prof. Manoj Kumar is an Associate Professor & Head in the Department of Civil Engineering, BITS Pilani.