This study describes a slender reinforced concrete shear wall experimental test under inplane cyclic lateral load, and the development of an analytical model which uses the fiber method approach to consider hysteretic nonlinear constitutive material models behavior. The shear wall tested had bending behavior, since the amount of longitudinal reinforcing bars produced weak bending capacity compared to the shear strength. The analytical model tries to represent global and local behavior of the wall, and its calibration is based on reaching experimental parameters like area enclosed and secant stiffness on every loop. After the analytical model was calibrated, the relation between some performance points and damage states observed during the test is studied.
Many authors have studied experimental concrete shear wall models tested in a laboratory under cyclic inplane lateral load and have developed analytical models which represent with good accuracy the global and local hysteretic experimental behavior. Park et al. (1987) contributed to the state of the art in modeling of reinforced concrete behavior and structural analysis, for developing the analytical tool IDARC (Inelastic Damage Analysis of Reinforced Concrete FrameShearWall Structures) [
Wallace and Moehle (1992) described an analytical procedure to determine the need for concrete confinement at the boundaries of structural walls in building subjected to earthquakes and concluded that confinement is not required for symmetrically reinforced rectangular wall cross sections but could be necessary at the extremities of walls having T, L, or other similarly shaped cross sections [
Ile and Reynouard (2000) proposed a constitutive model for predicting the cyclic response of reinforced concrete structures; the model adopted the concept of a smeared crack approach with orthogonal fixed cracks and assumed a plane stress condition; this model was compared with experimental results of a shear wall tested at NUPEC’s Tadotsu Engineering Laboratory and finally after several models and comparisons the numerical results showed good correlation between the predicted and the actual response [
Hidalgo et al. (2002) studied the behavior of reinforced concrete walls that exhibit the shear mode of failure; this study tested 26 fullscale specimens subjected to cyclic horizontal displacements of increasing amplitude; test parameters were the aspect ratio of the walls, the amount of vertical and horizontal distributed reinforcement, and the compressive strength of concrete. The investigation gave conclusions about the dissipation characteristics and the strength deterioration after maximum strength shown by the walls and the influence of vertical distributed reinforcement on the seismic behavior of walls [
Thomsen IV and Wallace (2004) have taken the results of experimental studies of moderatescale, slender wall specimens with rectangularshaped and Tshaped cross sections and verified the results predicted using displacementbased design and concluded that displacementbased design is both a powerful and flexible tool for assessing wall detailing requirements [
Su and Wong (2007) tested walls in the form of a slender vertical cantilever with an aspect ratio of 4, fabricated with high strength concrete and high longitudinal ratio, and identified that axial load ratio is an indispensable parameter for consideration in seismic performance assessment of reinforced concrete shear walls [
Kuang and Ho (2008) investigated the seismic behavior and displacement ductility of the shear walls; in this study largescale nonseismically detailed, squat reinforced concrete shear walls with aspect ratios of 1.0 and 1.5 are tested, as practiced in low to moderate probability of seismic occurrence regions, under reversed cyclic loading, and it is shown that an ordinary squat shear wall with nonseismic design and detailing may not have sufficient ductility to respond adequately to an unexpected moderate earthquake [
After reviewing several investigations, and with the aim of contributing with the state of the art of reinforced concrete shear walls, this study proposes a slender reinforced concrete shear wall with relation between height of the applied load and length
The proposed analytical model considers the shear wall as an inelastic frame forcebased element, with nonlinear behavior of its materials, using the following constitutive models: bilinear steel model (stl_bl); MenegottoPinto steel model (stl_mp) [
The shear wall tested was 4.40 m high and 1.5 m long. Their boundary elements have a section of 0.3 m × 0.3 m, and its web thickness was 0.15 m. In Figure
Test implementation.
According to ACI 31814 [
Reinforcement details.
The shear wall was designed using ACI 31814 [
The shear wall was instrumented with a load cell and two linear variable differential transformers (LVDT), and strain gauges were attached to the rebar and transverse reinforcement. Figure
Nomenclature.
Code  Description 

LSG#  Left strain gauge on rebar 
RSG#  Right strain gauge on rebar 
SG#  Strain gauge on transverse reinforcement 
Instrumentation of shear wall.
Figure
Loading protocol.
Hysteresis cycles.
In Figure
Test observations.
Damage state  Description 


Insignificant or minor damage. It does not need repair. Less than 30% of its height with cracks, and they are less than or equal to 1 mm thickness. 



Tolerable damage. It needs some repair. Less than 60% of its height with cracks, and they are less than or equal to 2 mm thickness. 



Significant, substantial, or major damage. It needs to be demolished. More than 60% of its height with cracks, and many of them are more than 2 mm in thickness. Unconfined concrete has substantial spalling. 
Crack patterns.
At the end of the test, it is possible to observe a crack at the base of the structure, and it tends to be horizontal. This huge crack demonstrates a flexural failure. Table
In order to compare test observations, the FEMA 306 component damage classification is presented in Table
Component damaged classification [
Severity of damage  Description 

Insignificant  Damage does not significantly affect structural properties in spite of a minor loss of stiffness. Restoration measures are cosmetic unless the performance objective requires strict limits on nonstructural component damage in future events. 


Slight  Damage has a small effect on structural properties. Relatively minor structural restoration measures are required for restoration for most components and behavior modes. 


Moderate  Damage has an intermediate effect on structural properties. The scope of restoration measures depends on the component type and behavior mode. Measures may be relatively major in some cases. 


Heavy  Damage has a major effect on structural properties. The scope of restoration measures is generally extensive. Replacement or enhancement of some components may be required. 


Extreme  Damage has reduced structural performance to unreliable levels. The scope of restoration measures generally requires replacement or enhancement of components. 
In fact, it is possible to compare both tables and make an equivalence. At the end of the test, the shear wall has an extreme damage according to FEMA 306 [
As described above, the tested structure is a shear wall with a height of 4.20 m on the applied load and a length of 1.5 m parallel to the direction of the applied load; hence a length to height ratio of 0.36 was obtained. Shear walls up to a ratio of 0.2 can be analyzed acceptably with bending theory [
One important reason to use SeismoStruct was because it allows studying the historical performance of every section and component selected, and thus the comparison between the real and the mathematical model made can be more illustrative.
The fiber model made is an inelastic forcebased frame element joined with its base by a linear shear spring, with a static timehistory load equal to the real one; the frame element was divided into five sections to study the particular behavior of them.
This model is based on flexibility fiber elements; they are divided into longitudinal fibers with particular areas and geometric characteristics along them as shown in Figure
Element fiber model [
The fibers follow a uniaxial stressstrain linear and nonlinear relation of each particular material, and although there is not an explicit constitutive relation between sections they stem from integration of the outcomes and it is a method which can provide good results [
The integration section method used in this case was the GaussLobatto one, because it was preferable to work obtaining a flexibility matrix of all the elements, since if they are analyzed based on a force field the model can be accurate enough with few integration sections [
Also, inasmuch as the convergence being something difficult to achieve, some values such as the number of fibers and the integration section had to vary until the goal was reached. At the end the model was made with five longitudinal sections and five hundred fibers.
A model of a shear spring which takes into account all the shear deformation was included to get good accuracy between experimental and analytical results. The spring that was chosen considers the shear strength degradation [
This study uses the computational software for nonlinear analysis SeismoStruct mentioned above in which there are many constitutive concrete materials, but the shear wall model studied here considers only one concrete material named as con_ma in the SeismoStruct library. The material con_ma has the proposal of Mander et al. [
Wall concrete sections.
Wall sections
Confined concrete
Unconfined concrete
Mander et al. [
Unconfined concrete properties.
Parameter  Value  Units 

Compressive strength, 
21,75  MPa 
Tensile strength, 
2,175  MPa 
Modulus of elasticity, 
21919,59  MPa 
Strain at peak stress, 
0,0025  — 
Specific weight, 
24  KN/m3 
Concrete confinement factors.
Sections  Factors 

Unconfined  1 
Web confinement/core  1 
Edge confinement  1,33 
Concrete constitutive model.
The fiber method approach permits recording the time series of the constitutive behavior in any transverse section fiber, so in Figure
Unconfined and confined concrete, cyclic stressstrain constitutive model.
The steel material was used as a setting parameter to calibrate the analytical model, thus there was used three steel constitutive materials presented in the SeismoStruct library. These material models have different levels of complexity, for the number and kind of data required to describe their behavior. The material models used are bilinear, DoddRestrepo, and MenegottoPinto. Each model is briefly described below.
This material is called stl_bl in SeismoStruct library, and it is the simplest and one of the most efficient models of reinforcing steel. The stl_bl permits having good accuracy; however there are other models which can represent better the uniaxial steel behavior. This material can be defined by using two points as it is shown in Figure
Bilinear steel model parameters.
Parameter  Value  Units 

Modulus of elasticity, 
228180  MPa 
Postyield modulus, 
1551,33  MPa 
Yield strength, 
418,547  MPa 
Strain hardening parameter, 
0,00679  — 
Fracture/buckling strain  0,1  — 
Specific weight  78  KN/m3 
Analytical bilinear steel model.
This model is based on macroscopic observations, and the values of stress and strain are obtained considering the instantaneous geometric properties of steel. The model focuses only on the behavior of the material, so it does not consider the buckling effect of the rebar. It also proposes a way of representing the Bauschinger effect [
DoddRestrepo steel model parameters.
Parameter  Value  Units 

Modulus of elasticity, 
228180  MPa 
Yield strength, 
418,547  MPa 
Stress at peak load, 
578,592  MPa 
Strain at initiation of strain hardening curve, 
0,01605  — 
Strain at peak load, 
0,105  — 
Strain at the intermediate point on of strain hardening curve, 
0,06052  — 
Stress at the intermediate point on of strain hardening curve, 
525,042  MPa 
Specific weight  78  KN/m3 
DoddRestrepo steel model.
This material was proposed by Menegotto and Pinto [
MenegottoPinto steel model parameters.
Parameter  Value  Units 

Modulus of elasticity, 
228180  MPa 
Yield strength, 
418,547  MPa 
Strain hardening parameter, 
0,00679  — 
Transition curve initial shape parameter  19,7  — 
Transition curve shape calibrating coeff.  18,5  — 
Transition curve shape calibrating coeff.  0,15  — 
Isotropic hardening calibrating coeff.  0  — 
Isotropic hardening calibrating coeff.  1  — 
Fracture/buckling strain  0,1  — 
Specific weight  78  KN/m3 
MenegottoPinto steel model.
Stl_mp model does not consider a yield plate in its stressstrain curve or have a strain hardening section. To include the effect of strain hardening stl_mp model uses the simplification proposed in stl_bl model where after yielding there is a line that joins the yielding point (
Both models stl_dr and stl_mp consider Bauschinger effect; however this effect can be modified in stl_mp by using some parameters listed in Table
In SeismoStruct the option to display the stressstrain behavior for any fiber is available, so in Figure
Rebar constitutive model comparison in the most strained steel fiber.
Area enclosed error.
Cycle  Model with stl_blrebar [error %]  Model with stl_drrebar [error %]  Model with stl_mprebar [error %]  Model with minor area error 

1  —  —  —  — 
2  —  —  —  — 
3  —  —  —  — 
4  —  —  —  — 
5  28,22  99,97  28,22  stl_mp 
6  68,67  31,64  68,45  stl_dr 
7  93,03  97,95  91,43  stl_mp 
8  10,39  27,92  7,77  stl_mp 
9  0,51  23,76  8,31  stl_bl 
10  20,14  16,61  9,55  stl_mp 
11  11,18  18,03  12,30  stl_mp 
Secant stiffness error.
Cycle  Model with stl_blrebar [error %]  Model with stl_drrebar [error %]  Model with stl_mprebar [error %]  Model with minor secant stiffness error 

1  —  —  —  — 
2  —  —  —  — 
3  —  —  —  — 
4  —  —  —  — 
5  52,38  311,44  52,38  stl_bl y stl_mp 
6  24,54  25,04  24,43  stl_mp 
7  57,44  60,19  56,22  stl_mp 
8  17,23  34,55  24,66  stl_bl 
9  12,02  2,16  0,31  stl_mp 
10  4,66  2,08  1,88  stl_mp 
11  14,24  2,54  2,32  stl_mp 
Base sheardisplacement, experimental and analytical data comparison.
With stl_bl in rebar
With stl_dr in rebar
With stl_mp in rebar
Area enclosed in every loop.
Secant stiffness in every loop.
Base shearlateral displacement and performance points.
In Table
Performance points and states of damage.
Order  Performance point  Displacement [mm]  Drift  Damage 

1st  North, 
8,10  0,0019  Slight 
2nd  South, 
7,99  0,0019  Slight 
3rd  North, 0,8 
22,81  0,0054  Moderate 
4th  South, 0,8 
16,80  0,0040  Moderate 
5th  North, 
54,81  0,0131  Extensive 
6th  South, 
109,40  0,0260  Extensive 
In Figure
Secant stiffness reduction.
Cycle 

Damage  Observations 

1  0  —  — 
2  0  —  — 
3  0  —  — 
4  0  —  First crack happened (experimental test) 
5  0,63  Slight  — 
6  0,78  Slight  — 
7  0,79  Slight  First 
8  0,90  Moderate  First 0,8 
9  0,95  Moderate  First 
10  0,97  Extensive  — 
11  0,98  Extensive  — 
Secant stiffness variation.
From 63% to 79% secant stiffness reduction there is slight damage, and from 80% to 95% secant stiffness reduction there is moderate damage. Finally from 95% secant stiffness reduction onwards there is extensive damage.
Linear and nonlinear flexure contribution are produced by the wall body modeled with the fiber method approach while the linear shear contribution is considered by a linear shear spring. In Figure
Bending and shear contribution.
First cracks appear between the lateral applied load of –four tons and + six tons; they are caused by the bending moment near the wall base. Cracks tend to be horizontal, and they are along its boundary elements and the web of the shear wall. At the end of the test, a horizontal crack developed at the bottom of the shear wall confirms a flexural failure. Furthermore, it shows cracks over 60% of its height, and cracks at the bottom have more than two millimeters of thickness.
This slender shear wall has high bending contribution on linear and nonlinear lateral displacements, being more than 88% of the total displacement on each cycle. Most of the response was produced by bending in both experimental and analytical results; however to obtain more accurate results it is necessary to include a linear shear deformation link in the analytical model.
Hysteretic nonlinear behavior on the structure tested was the result of nonlinear effects considered in concrete and rebar constitutive materials because the shear spring used in the analytical model had linear behavior in each cycle.
The yield of rebar occurs just before observing moderate damage. The reason must be because the rebar loses stiffness then forces are conveyed to concrete, and the loss of stiffness in the steel allows greater deformations in the structure.
Although the hysteretic capacity curve changes its shape with different models of steel, the secant stiffness has no significant variation with the steel models used in this study.
The steel model is one important factor for calibrating the hysteretic energy dissipation in the slender shear wall studied in this document. Steel models do not improve the calibration of static structural properties like stiffness, strength, and displacement, but they can get a better approximation of dynamic properties like hysteretic energy dissipation.
This study uses only one concrete material model because [
Performance points obtained by the analysis of uniaxial nonlinear behavior of the materials used in this document can be considered as reference points to identify the propagation of damage in slender shear walls.
The authors declare that there are no conflicts of interest regarding the publication of this article.