INTRODUCTION
As the need for the seismic design of civil structures increases, many experimental and analytical studies predicting the nonlinear response of structures according to the load increase and computing the ultimate resistance under extreme load conditions have been performed (ASCE, 1982). Due to the advantage of its rigidity, shear walls which offer great resistance for lateral loads have been widely adopted in building structures. Nevertheless, the design method mentioned in design codes does not always give a reasonable calculation since the shear wall capacity, as predicted on the basis of the truss analogy concept, often overestimates the structural behavior established by experiment. Therefore, nonlinear finite element analysis is definitely required to more exactly evaluate ultimate resisting capacity and loaddeformation behavior.
In designing reinforced concrete framewall buildings, designers may choose nonplanar wall section, such as L, T, C etc., as opposed to planar shapes, such as rectangular or barbell. Responses of nonplanar walls with at least one crosssectional principal axis that is not a symmetry axis are typically governed by unsymmetrical bending and would be influenced by inelastic biaxial interaction more significantly than those of planar walls. A related problem arises in evaluating the seismic vulnerability of framewall construction possessing significant plan irregularity (Palermo and Vecchio, 2004).
It is necessary for an enhanced evaluation of structural behavior to model each constituent material and interaction between reinforcing steel and concrete appropriately. For the reinforced concrete finite element analysis of plane stress dominated structures such as shear walls, these phenomena should be considered in the analytical model: the strength criterion of concrete subjected to various combinations of biaxial stresses, variation in material properties before and after cracking, concrete cracking and crack propagation and the tensionstiffening behavior of reinforced concrete composite material.
Many mathematical models for the mechanical behavior of concrete are in use in the numerical analysis of Reinforced Concrete (RC) structures (ASCE, 1982). Among concrete constitutive relations, the orthotropic model that was developed based on the concept of equivalent uniaxial strain strikes a balance between accuracy and economy. Hence, it is being widely used for numerical analysis of reinforced concrete structures by many researchers (ASCE, 1982; Chen, 1982; Zienkiewicz et al., 2005).
Research significance: In this study, the first objective is to describe an analysis tool which is based on layered nonlinear finite element method (NONLACS2) and investigate nonlinear behavior of flanged shear walls. The second objective is presentation of analytical model with compressive strength degradation developed with Kwak RC panels (Kwak and Kim, 2006) and Vecchio`s experimental study (Vecchio, 1999) in addition to the use of a model and a description of the strain softening region of concrete and tensionstiffening effect. The developed finiteelement model is validated by comparison with test results from several idealized orthogonally reinforced concrete shear walls tested by Vecchio and Palermo (2002). In addition, to assess the applicability of the material model under different stress conditions, loaddisplacement relations and crack patterns are compared with flanged shear wall test results.
Nonlinear finite element program: A nonlinear finite element analysis program, NONLACS2 (NONLinear Analysis of Concrete and Steel Structures), developed by Kheyroddin^{ }(1996), is used to analyze the selected R.C. shear walls. The program employs a layered finite element approach and can be used to predict the nonlinear behavior of any plain, reinforced or prestressed concrete, steel, or composite concretesteel structure that is composed of thin plate members with plane stress conditions. This includes beams, slabs (plates), shells, folded plates, box girders, shear walls, or any combination of these structural elements. Timedependent effects such as creep and shrinkage can also be considered.
Concrete properties: The concrete behaves differently under different
types and combinations of stress conditions due to the progressive microcracking
at the interface between the mortar and the aggregates (transition zone). The
propagation of these cracks under the applied loads contributes to the nonlinear
behavior of the concrete. As shown in Fig. 1a, the uniaxial
stressstrain curve of concrete adopted in this study, is made of two parts.
The ascending branch up to the peak compressive strength is represented by the
equation proposed by Ashour and Morley (1993):
Where:
E_{0} 
= 
Initial modulus of elasticity of the concrete 
E_{sc} 
= 
Secant modulus of the concrete at the peak stress 
σ 
= 
Stress 
ε 
= 
Strain 
ε_{max} 
= 
Strain at peak stress 
The descending or the strainsoftening branch is idealized by the Bazant model
(Bazant et al., 1986):
Where, σ_{c} is compressive strength of the concrete. For uniaxially
loaded concrete, σ_{c} is equal to f´_{c}.
For analysis of most plane stress problems, concrete is assumed to behave as
a stressinduced orthotropic material. In this study the orthotropic constitutive
relationship developed by Darwin and Pecknold (1977) is used for modeling the
concrete using the smeared cracking idealization. The constitutive matrix, D,
is given by:
in which, E_{1} and E_{2} are the tangent moduli in the directions
of the material orthotropy and v is the Poisson`s ratio. The orthotropic material
directions coincide with the principal stress directions for the uncracked concrete
and these directions are parallel and normal to the cracks for the cracked concrete.
The concept of the "equivalent uniaxial strain" developed by Darwin and Pecknold
(1977) is utilized to relate the increments of stress and strain in the principal
directions. Therefore, stressstrain curves similar to the uniaxial stressstrain
curves can be used to formulate the required stressstrain curves in each principal
direction.
The strength of concrete, σ_{c} and the values of E_{1},
E_{2} and v are functions of the level of stress and the stress combinations.
The concrete strength when subjected to biaxial stresses is determined using
the failure envelope developed by Kupfer et al. (1969). The values of
E_{1} and E_{2} for a given stress ratio (α = σ_{1}/σ_{2})
are found as the slopes of the σ_{1}ε_{1} and σ_{2}ε_{2}
curves, respectively. For the descending branches of both compression and tension
stressstrain curves, E_{i} is set equal to a very small number, 0.0001,
to avoid computational problems associated with a negative and zero values for
E_{i}. The concrete is considered to be crushed, when the equivalent
compressive strain in the principal directions exceeds the ultimate compressive
strain of the concrete, ε_{cu}. For determination of the concrete
ultimate compressive strain, ε_{cu}, two models for unconfined
high and normalstrength concrete (Pastor, 1986) and confined concrete (Chung
et al., 2002) are implemented into the program.

Fig. 1: 
Uniaxial stressstrain curves 
For elimination of the numerical difficulties after crushing (ε>ε_{cu})
and cracking of the concrete (ε>ε_{tu}), a small amount
of compressive and tensile stress as a fraction of concrete strength, γ_{c}f`_{c}
and γ_{t}f`_{t}, is assigned (optional) at a high level
of stress (Fig. 1a), where parameters γ_{c} and
γ_{t} define the remaining compressive and tensile strength factors,
respectively.
Crack modeling techniques: Cracking of the concrete is one of the important
aspects of material nonlinear behavior of the concrete. Besides reducing the
stiffness of the structure, cracks have resulted in redistribution of stresses
to the reinforcing steel as well as increasing the bond stress at the steelconcrete
interface (Sundara et al., 2002). Cracking of the concrete is idealized
using the fixed smeared cracking model and is assumed to occur when the principal
tensile stress at a point (usually a Gauss integration point) exceeds the tensile
strength of the concrete. After cracking, the axes of orthotropy are aligned
parallel and orthogonal to the crack. The elastic modulus perpendicular to the
crack direction is reduced to a very small value close to zero and the Poisson
effect is ignored. The effect of the crack is smeared within the element by
modifying the [D] matrix. If σ_{1} exceeds the tensile strength
of concrete, f^{`}_{t}, the material stiffness matrix is defined
as (one crack is opened):
Where, 0<β<=1.0
Once one crack is formed, the principal directions are not allowed to rotate
and a second crack can form only when σ_{2}>f^{`}_{t},
in a direction perpendicular to the first crack. Then,
Where, 0<β<=1.0
The shear retention factor, β, with a value of less than unity, serves
to eliminate the numerical difficulties that arise if the shear modulus is reduced
to zero and more importantly, it accounts for the fact that cracked concrete
can still transfer shear forces through aggregate interlock and dowel action.
Due to the bond between the concrete and the steel reinforcement, a redistribution
of the tensile stress from the concrete to the reinforcement will occur (MartínPérez
and Pantazopoulou, 2001). In fact, the concrete is able to resist tension between
the cracks in the direction normal to the crack; this phenomenon is termed tensionstiffening.
The tensionstiffening effect is idealized by assuming the ascending and the
descending branches of the tensile stressstrain curve that is described in
the next section. For evaluation of an appropriate value of the ultimate tensile
strain of the concrete, ε_{tu} and elimination of mesh size dependency
phenomenon, Kheyroddin et al. (1997) proposed the following simple formula:
Where:
h 
= 
Width of the element in mm 
ε_{cr} 
= 
Concrete cracking strain 
For elimination of the element size effect phenomenon, both the new proposed
model and the crack band model, based on fracture mechanics, proposed by Bazant
and Oh (1983) have been implemented into the NONLACS2 program.
Reinforcing bar properties: The reinforcing bars are modeled as an elastic
strainhardening material as shown in Fig. 1b. The reinforcing
bars can be modeled either as smeared layers or as individual bars. In both
cases, perfect bond is assumed between the steel and the concrete.
Finite element formulation: The element library includes plane membrane,
plate bending, one dimensional bar, shear connecter, spring boundary elements
as well as a facet shell element, which is a combination of the plane membrane
and the plate bending elements. Figure 2 shows some of these
elements and the associated degrees of freedom. The two nodes, three degrees
of freedom per node one dimensional bar element is used to model uniaxial truss
members, unbonded prestressed tendons and shear connectors. The program employs
a layered finite element approach. The structure is idealized as an assemblage
of thin constant thickness plate elements with each element subdivided into
a number of imaginary layers as shown in Fig. 2c. A layer
can be either of concrete, smeared reinforcing steel or a continuous steel plate.
The number of layers depends on the behavior of the structure being analyzed.
Each layer is assumed to be in a state of plane stress and can assume any stateuncracked,
partially cracked, fully cracked, nonyielded, yielded and crusheddepending
on the stress or strain conditions.
Nonlinear analysis method: Nonlinear analysis is performed using an
incrementaliterative tangent stiffness approach and the element stiffness is
obtained by adding the stiffness contributions of all layers at each Gauss quadrature
point. The change in the material stiffness matrix during loading necessitates
an incremental solution procedure with a tangent stiffness scheme that using
piecewise linearization has been adopted in the NONLACS2 program.
Tensionstiffening model: Cracking in concrete will develop and propagate in the direction normal to that of the major principal strain starting from the section where a crack first originates. Even after cracking, however, concrete is still partially capable of resisting tensile forces due to the bond between concrete and reinforcement. This phenomenon which results from crack formation and the bond between steel and its surrounding concrete is defined as the tension stiffening effect. This effect can be adequately accounted for by increasing the average stiffness of the element which has relatively large dimensions when compared with the size of the cracked section. An increase in the tensile stiffness of concrete can be accomplished by using a stressstrain relation which includes a descending branch in the tension region.
Many experimental studies to predict the postcracking behavior of reinforced concrete structures have been conducted and several analytical models using the fracture energy concept or bond mechanisms have been developed (Gupta and Maestrini, 1989; Hegemier et al., 1985). In this study, based on the force equilibriums, compatibility conditions and the bondstressslip relationship between reinforcement and the surrounding concrete in the principal tensile direction, a descending branch to define the postcracking stressstrain relation of concrete is proposed. Differently from most of the analytical models which consider the tension stiffening effect in a uniaxial stress state along the reinforcing steel (Gupta and Maestrini, 1990; Choi and Cheung, 1996), the effect according to the reinforcing steel in both directions are directly included in the softening branch of concrete and the cracking which is not normal to the reinforcing steel can be simulated effectively.
Force equilibriums: A cracked reinforced concrete element subjected
to membrane stresses is shown in Fig. 3a and a part of the
element along the inclined crack faces with crack spacing of 2a can be taken
as the free body diagram Fig. 3b. If the z direction is normal
to the crack surface, it makes an angle θ with the x direction and coincides
with the principal tensile strain axis.

Fig. 3: 
Descriptions for (a) cracked RC planar element and (b) force
equilibrium in the arbitrary section 
Since the applied principal tensile force of RC membrane element, T, is carried
partly by the concrete matrix (F_{c}) and partly by the reinforcing
steel (F_{s}), the following force equilibrium equation is obtained:
Using the equivalent steel modulus of elasticity, E_{s,eq} calculated
from the force equilibrium with respect to the principal tensile direction and
expressed in terms of the reinforcement ratio and elastic modulus of steel in
each direction, the force component carried by steel can be rearranged as:
Where:
ε_{s1} 
= 
Steel strain about the z direction 
A_{c1} 
= 
Crosssection area of concrete perpendicular to the z direction and as
before, θ is the angle between the direction normal to the crack and
the global x direction. 
Also, the force component carried by concrete can be expressed by:
Where, ε_{1u} means the equivalent uniaxial strain along the z
direction. E_{2}, representing the elastic modulus of concrete in the
orthogonal direction to the z direction, varies according to the loading history,
while E_{1} in the principal tension direction (z direction) is the
same as the initial elastic modulus of concrete E_{c} and does not change
with increasing principal tensile strain because the concrete between cracks
still remains as an uncracked elastic material.
Reinforcing bars transfer tensile stresses to the concrete through the bond
stresses along the surface between reinforcements and surrounding concrete,
so that an infinitesimal element of length dz is taken out from the intact concrete
between cracks to obtain the equilibrium equations for the concrete and steel.
Figure 4 represents the free body diagrams at the steel and
concrete interfaces.
Based on the force equilibrium at the steel interface, the steel force variation
can be derived in terms of the bond stresses:
in which p is the perimeter of a reinforcing bar, n is the number of bars placed
within an infinitesimal length dz, f_{b} is the bond stress at the steelconcrete
interface and the subscripts x and y denote the x and y directions, respectively.

Fig. 4: 
Free body diagrams for (a) the RC element and (b) steel and
concrete interfaces 
Assuming that the bond stresses are identical in the x and y directions, the
following equilibrium equations for the steel and concrete can be obtained:
Where:
pnf_{b}=p_{x}n_{x}+p_{y}n_{y})f_{b}

Bond slip behavior: Since the bondslip (Δ) at the steelconcrete
interface is defined by the relative displacement between the reinforcing steel
and concrete (Δ = u_{s}u_{c}), substitution of (8)
and (9) into the second order differential equation of bondslip
leads to:
Where:
After substituting Eq. 11 into 12 and then
if the linear bond stressslip relationship of f_{b} = E_{b}Δ
is assumed, the governing differential equation is obtained:
Where:
k^{2} 
= 
(pnE_{b}/A_{c1}) (1/E_{s,eq }+ 1v^{2}/E_{c}) 
E_{b} 
= 
Slip modulus 
The general solution to Eq. 13 is given by Δ = C_{1}sinhkz+C_{2}coshkz,
in which C_{1} and C_{2} are constants that have to be determined
from the boundary conditions. Because of symmetry, the slip should be zero at
the center (z = 0) of the segment and Δ(^{–}z) = ^{–}Δ(z)
must be satisfied Fig. 3b. Hence, C_{2}=0 is determined.
Integration of Eq. 11 after substituting the obtained general
solution leads to the following expression for the steel force F_{s}:
Where, the constant of integration C_{3} = T(pnE_{b}C_{1}/k)xcosh(ka)
is obtained from the boundary condition at the crack surface (F_{s}
= T at z = a). Moreover, the displacement of reinforcement is obtained from
Eq. 8.
Through the same procedure, the resisting force and displacement y the concrete
matrix are also determined as:
in which C_{1}=(T/E_{s,eq}A_{c1})(1/kcoshka) is determined
from (15) and (17).
Tensionstiffening model: ased on the obtained equations, a descending
branch in the tension region of the concrete stressstrain relation can be determined.
First, the equilibrium Eq. 7 is:
Where, the term σ_{c} s the effective tensile stress in concrete
and ε_{s1} is the average strain in the reinforcing bar. These
values can be obtained from (15) and (18).
The maximum tensile force in concrete occurs at x = 0. Hence, from Eq.
16, this maximum force can be obtained as:
The corresponding concrete stress is σ_{c,max} = F_{c,max}/A_{c1},
that is, the maximum tensile stress in concrete is directly proportional to
the applied principal tensile force T. Accordingly, σ_{c,max} converges
to the tensile strength of concrete, f`_{t} as the applied tensile force
T increases. At that point, a new crack will be formed at z = 0. After eliminating
T from (19) and (20), those are rewritten
in a nondimensional form:
Where, ε_{crack} = f_{t}`/E_{c}, n`_{x}
and n`_{y} are the modular ratios in the x and y directions, respectively.
In Eq. 22, σ_{c}/f`_{t} converges to
the value of 2/3 as the parameter ka related to crack spacing approaches zero.
However, the actual crack spacing is not narrowed any more but remains constant
after reaching a certain value. The experimental study by Campione and Mendola
(2004) indicates that the number of cracks is stabilized when the average strain
is about 0.001. Therefore, with the assumption that the linear bond stressslip
relation remains, (22) and (23) can be available
up to ε = 0.001.
Further deformation leads to the yielding of the reinforcing steel, followed
by the increase of the slip while maintaining a plateau f_{b} = τ_{b}.
For more increase of the slip, the bond stress decreases linearly to the value
of the ultimate frictional bond resistance (ASCE, 1982). In the case of constant
bond stress (f_{b} = τ_{b}) and the yielding of the reinforcing
steel, the principal tensile force carried by each material can be calculated
from Eq. 11 with appropriate boundary conditions Fig. 3b.
From Eq. 8, the displacement of steel can be expressed by
Moreover, the average strain in the reinforcing steel, ε_{s1},
can be obtained by differentiating u_{s} with respect to z and substituting
a for z, that is, ε_{s1}=(1/E_{s,eq}A_{c1})(T^{–}pnτ_{b}a/2).
Accordingly, the effective tensile stress of concrete can be calculated from
Eq. 18 with ε_{s1}(σ_{c} = 1/A_{c1}>=pnτ_{b}a/2).
The nondimensional parametric equations for σ_{c}/f`_{t}
and ε/ε_{crack}, as was done in the case of the linear bond
stressslip relationship, are determined as follows:
Where, f_{t}` = pnτ_{b}a/A_{c1} from Eq.
24. The principal tensile force T is given by T = f_{yx}ρ_{x}A_{x}
cos θ+f_{yy}ρ_{y}A_{y} sin θ in which
f_{y} is the yield strength of steel, ρ is the reinforcement ratio,
A is the crosssection area normal to the crack surface of concrete and the
subscripts x and y mean the x and y directions, respectively. A similar result
of σ_{c}/f_{t}` = 1/2 was introduced by Gupta and Maestrini
(1990). Substituting the relation of T into Eq. 27 yields
the following equation represented in terms of material parameters only:
in which p/s = (p_{x}/s_{x})+(p_{y}/s_{y})(l_{y}/l_{x}),
t is the thickness of the RC element, s_{x} and s_{y} are the
spacings of reinforcements and l_{x} and l_{y} are the length
of sides normal to the x and y directions, respectively.
Finally, the boundary value of the strain corresponding to σ_{c}
= 0 is calculated from the relation of T = F_{s}+F_{c} with
F_{c} = 0 through the same procedure.
τ_{b} is considered a material property and its magnitude depends
on many factors. When τ_{b} is known from measurements, the descending
branch of the tension region can be defined exactly. For computational convenience,
τ_{b} = 5 MPa generally assumed in many numerical analyses was
adopted in this study. Also, half an average crack spacing, a(S_{ave}
= 2a), is determined by using S_{max} = (cosθ/S_{x,max}+sinθ/S_{y,max})^{1}
and S_{ave} = 0.75 S_{max} introduced by Campione and Mendola
(2004).
Applications: The capability of NONLACS2 program to reliably simulate the fundamental behavior arising from elastic and inelastic flexure interaction was verified by correlating analytically simulated and measured response of three works. In each case, different forms of output (including mode of failure) were extracted and compared with experimental results to verify key aspects of the numerical model. Although some discrepancies were observed, the overall match between the analytical models and experimental tests was good.
Kwak and Kim RC shear panels: The experimental results from reinforced
concrete shear panels tested by Kwak and Kim (2006) are widely used to validate
the analytical models for reinforced concrete membrane element. These panels
were orthogonally reinforced and had identical dimensions of 890x890x70 mm.
Lateral load was imposed on the topright joint and the axial load was spread
along the top rows of joint. Figure 5 shows the configuration
of the test specimen and the finite element grid used. The finite element used
in this study is an isoparametric four node element with 2x2 Gauss integration
because all the stresses at every Gauss point are the same values and the four
node element gives more stable stress results through the loading history. The
assumed material properties were as follows: Poisson`s ratio v = 0.2, the tensile
strength of concrete, f_{c}` = 0.33√f_{c}` MPa, the elastic
modulus of steel E_{s1 } = 200,000 MPa, E_{s2 }= 0.01 E_{s1}.
A twodimensional static monotonic analysis was performed by Kwak and Kim (2006).
For modeling of shear wall in NONLACS2 program, have been used fournode shell
elements, QLC3 type as plane stress and bending. Results analyses are plotted
in Fig. 6 along with the envelope response of the twodimensional
pushover analysis and the experimental response of the specimen. In spite of
the exact predictions in the failure modes, the analysis slightly overestimates
the shear strength of the panels. In addition, the numerical results still give
more exact predictions for the shear strength of a shear dominant structure
because the prediction by code guidelines represents the large overestimate.
Totally, the result indicated that NONLACS2 programs provide reasonable results
and can be used to approximate ultimate load. An ultimate load of 1390 KN was
reported for shear panel.

Fig. 6: 
Comparison of experimental and analytical results of Kwak
and Kim RC shear panels 

Fig. 7: 
Measured and computed tension flange strain profile for Specimen
TW2 
Wallace and Thomsen flanged walls (1995): Wallace and Thomsen (1995)
conducted tests of Tshaped RC walls subjected to axial compression and cyclic
lateral loading. Using measured material properties and dimensions, the monotonic
and cyclic responses of Wall TW2 are calculated and compared with the test results.
The measured and calculated loaddeflection curves match well. As shown in Fig.
7, the measured strain distribution in the flange under tension also compares
well to the computed strains for various drift levels. Because the model cannot
capture local bar buckling, it was unable to predict the initiation of these
modes of failure as observed in the tests. Nevertheless, the analytical results
correlate well to the test data prior to the occurrence of these local modes
of failure.
Vecchio and Palermo flanged walls (2002): The proposed material and
tensionstiffening models were also applied to two largescale flanged shear
walls tested under static cyclic displacement by Vecchio and Palermo (2002).
The specimens were constructed with stiff top and bottom slabs. The top slabs
(2600x1440x150 mm) served to distribute the horizontal and axial load to the
walls of the structure. The bottom slab (2600x1440x300 mm), clamped to the laboratory
strong floor, simulated a rigid foundation. The slabs were reinforced with No.
30 (29.9 mm) deformed reinforcing bars at a spacing of 350 mm in each direction,
with a top and bottom layer. Two types of walls, that is, Type I, 1000 mm wide
x1000 mm high x 70 mm thick (h/l = 1) and Type II, 750 mm wide x 1500 mm high
x 65 mm thick (h/l = 2), were tested.

Fig. 8: 
Test specimen details: (a) end view and (b) side view 
The web wall was reinforced with D6 reinforcing bars, the bars were spaced
140 mm horizontally and 130 mm vertically in two parallel layers. The two flange
walls were approximately 2000 mm long, 1000 and 1500 mm high and 95 mm thick.
The flanges were also reinforced with D6 reinforcing bars, spaced 140 mm horizontally
and 130 mm vertically near the web wall and 255 mm near the tips of the flanges.
The concrete clear covers in the walls and slabs were 15 and 50 mm, respectively.
Dimensional details of the walls are shown in Fig. 8 and the
reinforcement layout for the web and flange walls are given in Fig.
9 and 10.

Fig. 9: 
Top view of wall reinforcement 
The finite element mesh, shown in Fig. 11, consisted of 540
constant strain rectangular elements. The mesh was divided into four zones:
the web wall, flange walls, top slabs and bottom slabs. For modeling of Ishaped
shear wall in NONLACS2 program, have been used fournode shell elements, QLC3
type as plane stress and bending.
The shear wall specimens were subjected to the combined action of the uniformly
distributed axial load and the horizontal load which was monotonically increased
to failure. Type I shear walls SW13 and SW16 and Type II SW21, SW22, SW24 and
SW25 were selected for the correlation study. Table 1 includes
the material properties of concrete and the loading conditions. The material
properties not mentioned in Table 1 are as follows: the uniaxial
compressive strength of concrete, f_{c}` = 0.85 f_{cu}, where
f_{cu} is the cube strength. The tensile strength of concrete, f_{t}`
= 0.33(f_{c}`)^{1/2} and the yield strength of vertical and
horizontal reinforcements were 470 MPa and 520 MPa, respectively. All these
values except f`_{t} are from the experimental data by Vecchio and Palermo
(2002).
In order to investigate the contribution of the tensionstiffening effect to
the structural response, the analytical results with and without tensionstiffening
effect are compared with the measured loaddisplacement relations in Fig.
12. It is clear from the comparison of these results with the experimental
data that the consideration of tensionstiffening effect yields a very satisfactory
agreement for the structural stiffness and ultimate capacity as in the previous
examples.

Fig. 10: 
Geometries and reinforcement details of Vecchio and Palermo
flanged shear walls 

Fig. 11: 
Finite element mesh configuration 

Fig. 12: 
Influence of tensionstiffening effect in shear walls SW21
and SW22. Solution (a) with tensionstiffening and solution (b) without
tensionstiffening 

Fig. 13: 
Horizontal load versus top horizontal displacement for shear
walls 
Table 1: 
Loading conditions and material properties of shear walls 


Fig. 14: 
Cracking and crushing patterns of shear walls at ultimate
load level 
As shown in Fig. 13, the lateral stiffness of RC structures
is significantly affected by the level of axial force. This is basically caused
by the fact that the applied compressive stress reduces the tensile stress as
in the prestressed concrete structure, the confinement effect. Figure
14 shows the tensile cracking and compressive crushing pattern of specimens
SW13, SW16, SW21 and SW22 at ultimate load. This figure verifies the propagated
tensile cracks and compressive crushing failure reported by Vecchio and Palermo
(2002). Similar tensile cracking and crushing patterns have been observed by
Vecchio (1999) in his analytical runs for the same specimens.
The enhanced prediction can be obtained by considering the material nonlinearity and interaction parameters such as the fixed smeared cracking model, tension stiffening effect and bondslip characteristics in the analysis of RC shear walls and panels (Kwak and Kim, 2004; van der Put, 2007).
CONCLUSIONS
An analytical model is proposed for the nonlinear finite element analysis of RC structures. Based on the concept of equivalent uniaxial strain, a concrete material model is presented in the axes of orthotropy which coincides with the principal axes of total strain and rotation during the load history. The applicability of the proposed orthotropic constitutive model in finite element analysis is verified by comparison with the reliable experimental results of the concrete stressstrain relations and loaddisplacement relations. Moreover, the tensionstiffening model introduced in this study may give the theoretical background for Vecchio`s experimentally developed model which has been broadly adopted in many numerical analyses of RC structures.
The correlation studies between analytical results and test values and the parametric studies associated with them lead to the following conclusions: (1) the inclusion of tensionstiffening is important even in the structure dominantly affected by shear, (2) in the shear wall which is anisotropically reinforced and whose postcracking behavior is dominated by the yielding of steel, the crack angle changes according to the loading history. Hence, fixed smeared cracking model must be adopted to exactly predict the cracked structural behavior, (3) for heavily reinforced shear walls, the global response of the panels is not dominantly affected by the tensionstiffening parameter but governed by concrete crushing. Therefore, compression softening due to tensile cracking should be considered and (4) the analyses of shear walls under a horizontal load verify strength enhancement as the uniformly distributed axial load increases.
ACKNOWLEDGMENTS
The research reported in this study was made possible by the financial supports from the Semnan University by the Ministry of Science and Technology of Islamic Republic of Iran. The authors would like to express their gratitude to organization for their support.
NOTATIONS
[D] 
= 
Constitutive matrix 
E_{1},E2 
= 
The tangent moduli in the directions of the material orthotropy 
E_{i } 
= 
Tangent moduli in principal directions i; i = 1, 2 
E_{0} 
= 
The initial modulus of elasticity of the concrete 
E_{sc} 
= 
The secant modulus of the concrete at the peak stress 
E_{s} 
= 
Modulus of elasticity of reinforcement 
f`_{c} 
= 
Compressive strength of concrete 
f`_{t} 
= 
Direct tensile strength of concrete 
f_{y, }f_{sy } 
= 
Yield strength of reinforcement bar 
f_{su} 
= 
Ultimate stress of reinforcement 
h 
= 
Width of the element 
h_{w} 
= 
Wall height 
α 
= 
Loading type factor 
β 
= 
Shear retention factor 
γ_{c} 
= 
The remaining compressive strength factor 
γ_{t} 
= 
The remaining tensile strength factor 
ε 
= 
Strain 
ε_{c} 
= 
The strain of concrete 
ε_{max}, ε_{0} 
= 
The maximum compressive strain of the concrete 
ε_{cu} 
= 
The ultimate compressive strain of the concrete 
ε_{cr} 
= 
Cracking strain of concrete 
ε_{tu} 
= 
Ultimate tensile strain of concrete 
ε_{sy} 
= 
Yield strain of reinforcement 
ε_{su} 
= 
Ultimate strain of reinforcement 
v 
= 
Poisson`s ratio 
σ 
= 
Stress 
σ_{c} 
= 
Compressive strength of the concrete 
ρ_{min} 
= 
The minimum ratio of wall area to floorplan area 