Estimation of relations among hysteretic response measures and design parameters for RC rectangular shear walls

Abstract

Seismic design of RC structures requires estimation of structural member behavioral measures as functions of design parameters. In this study, the relations among cyclic behavioral measures and design parameters have been investigated for rectangular RC shear walls using numerical simulations calibrated based on the published laboratory tests. The OpenSEES numerical simulations modeling of plastic hinge hysteretic behavior of RC shear walls and estimation of empirical relations among wall hysteretic indices and design parameters are presented. The principal design parameters considered were wall dimensions, axial force, reinforcement ratios, and end-element design parameters. The estimated hysteretic response measures are wall effective stiffness, yield and ultimate curvatures, plastic moment capacity, yield and ultimate displacements, flexural shear capacity, and dissipated energy. Using results of numerous analyses, the empirical relations among wall cyclic behavioral measures and design parameters are developed and their accuracy is investigated.

Keywords

RC wall hysteretic measures / RC wall design parameters / empirical relations / numerical simulations / RC rectangular wall plastic hinge

Introduction

Recent major earthquakes have represented that the seismic risk of buildings in urban areas is on rise. More reliable seismic specifications for seismic evaluation, design, and retrofit of structures can effectively reduce this trend. In the past two decades, the philosophy of seismic design codes has shifted from strength-based design to ductility-based design. Empirical force reduction factors or R-factors have been utilized to compute the lateral strength of the structure while limiting the displacement ductility demands. The R-based approach is suited for the buildings with implicit requirements of ductility and regularity. However, such requirements have not been well-defined. Recently, developments in performance-based design have led to the belief that each building is an individual entity. A building can be designed to follow a desired performance during a seismic event, as long as the building characteristics, serviceability, reparability, damage intensity, and damage distribution after an earthquake could be evaluated. Although this achievement is no longer farfetched, a side effect of such diversified researches is the addition of numerous aspects to the design process [¹] which have been already hidden in the current design process. These aspects are regarding to determination of seismic hazard, structural performance, damage spread, safety hazards, and also financial and life losses.

Nonlinear-time-history analysis is the most appropriate type of structural analysis. But it is time consuming, expensive and also in need of advanced structural dynamics and seismology knowledge. Therefore, nonlinear static analysis has been developed to decrease such design process difficulties. On the other hand, performing a linear dynamic analysis needs the effective stiffness of each structural member to be known. For evaluation of a designed structure or performance of an existing building is required, one needs to know the local and global demand to capacity ratio, such as ductility demand ratio, which provides information about the performance level of the structure. Dominant modes of deterioration, possible collapse mechanisms, their occurrence possibility assessment, and behavior of a structural component or system under seismic loading requires estimation of behavioral measures. For example, seismic collapse resistance of a structural system depends on the strength, stiffness, and deformation capacity of individual structural components and the overall seismic resistance system. The mentioned behavioral measures and the design parameters are mutually related. So, the designer had better have a prior knowledge about these relations. However, in well-known design codes, such as ACI [²], very simple relations are adopted. One example is the ratio of effective stiffness to gross stiffness of 0.35 and 0.70 recommended for RC beams and columns, respectively. Depending on the state of stress in the wall, this ratio for RC walls is set to be between 0.35 and 0.70. These recommended ratios are not derived after comprehensive researches and may not lead to the best seismic designs. FEMA P695 [³] has attempted to use part of PEER tests to derive an empirical relation for estimating cyclic flexural behavior of RC columns as a function of column axial load ratio, dimensions, and percentage of longitudinal reinforcement. The estimations based on the FEMA P695 formula don't compare well with the PEER experimental data. In 2012, Li [⁴] proposed a model based on wall sectional dimensions, axial load ratio, and aspect ratio for determination of initial stiffness of columns and walls for linear analysis. Based on 17,000 numerical simulations, Sakhayi [⁵] developed empirical relations for hysteretic responses of circular RC columns with FRP jackets. The literature review shows that a definite process for cyclic behavioral measures of RC walls doesn’t exist.

To develop a robust and sound performance-based seismic design procedure for buildings through a practical deterministic design approach, firstly, the main framework should be defined. Banan in 2010 based on procedure reported in AASHTO [⁶] define the performance-based seismic design procedure for buildings as follow:

1) Hazard study in order to specify code-based or site-specific ground motions or earthquake spectra;

2) Identification of the structure’s earthquake resisting system consisting of ductile members with plastic hinges and essentially-elastic members;

3) Performing linear dynamic analysis to estimate demand deformations;

4) Performing equivalent nonlinear static analyses to determine expected deformation capacities including P-D effect and over-strength design forces;

5) Performance-based design check of strains, curvatures, and displacements;

6) Strength-based reinforced concrete design essentially for elastic members and also shear design of ductile members;

7) Detailing of all members and components for possible location of plastic hinges and eliminating brittle failures.

Based on the proposed design framework for seismic design of RC buildings, a broad objective-oriented research program has set off in department of Civil and Environmental Engineering at Shiraz University Including some empirical relations between cyclic behavioral measures and design parameters, based on numerous numerical simulations for FRP-jacketed reinforced circular columns [⁵], RC rectangular columns [⁷], and concrete incased columns [⁸].

In this study based on numerous numerical simulations, relationship between behavioral measures and design parameters of rectangular RC shear walls will be derived to assist designers for seismic design of a building.

In the following sections, a summary of: (i) the numerical simulations for proposing a reasonably accurate model for the hysteretic behavior measures of plastic hinges of RC shear wall members, and, (ii) estimation of empirical relations among shear walls hysteretic response measures and design parameters, are presented.

Numerical simulations

Numerical modelling of RC structural walls needs selecting an accurate modelling scheme which not only gives accurate results but also can model the wall as simple as possible but not simpler. As in the members with aspect ratio greater than three it is reasonable to ignore shear deformation, the fiber element modelling is accurate modelling. Fiber elements represent the RC cross section as a combination of concrete and rebar fibers while the effect of confinement is implemented within the uniaxial material models for concrete and rebars. The material model expresses the relationships between axial stress and strain for confined and unconfined concrete and rebar fibers. For cyclic loadings, these relationships should embody the fiber behaviors under loading, unloading, and reloading.

A displacement beam element which is derived based on the displacement formulation and considers the spread of plasticity along the element [⁹,¹⁰] is used for modeling RC structural walls.

In this study, the Mander-Priestley-Park constitutive model [¹¹] is used for uniaxial unconfined and confined concrete stress–strain relationship. For internal hysteresis loops of unloading and reloading, the Karsan-Jirsa constitutive model is adopted [¹²]. Furthermore, for complete cyclic stress–strain relationship including buckling effects for steel rebars, the modified Giuffre-Menegotto-Pinto constitutive model is used [¹⁴]. For more information on the constitutive models, refer to Mander et al. (1988), Karsan and Jirsa (1969), and Dhakal and Maekawa (2002) [¹¹,¹²,¹³].

The effects of bond-slip should be included in the material models to realistically model the reinforced concrete behavior. Bond-slip significantly affects the loading and unloading behavior of an RC section within a plastic hinge [¹⁴]. The increased flexibility due to the bond-slip influences the effective stiffness, deformation, and yield displacement of an RC member. However, a fiber element's behavior is based on the linear strain distribution over the cross section. Such assumption cannot include the effects of bond-slip for a moment-curvature analysis. Therefore, in order to overcome this problem, the efficient macro solution proposed by Sharifi et al. [¹⁴] for modifying the elastic modulus of steel rebars is employed.

Verification of OpenSEES model

In order to verify the accuracy of the developed OpenSEES models, experimental test results reported for cyclic behavior of RC structural walls by Oesterele et al. [¹⁵], Pilakoutas and Elnashai [¹⁶], Thomsen and Wallace [¹⁷], and Riva et al. [¹⁸] have been used.

For a specific cantilever rectangular wall with the reported characteristics, the member properties, such as the number of elements, the number of integration points, and density of sectional mesh were calculated by minimizing an objective error function. The average error, as the objective function, defines the discrepancy between the selected response measures estimated by fiber element modeling and their corresponding experimental measures. The considered measures of the hysteretic response of the cantilever rectangular wall have been the effective stiffness of backbone curve, maximum lateral capacity, and the hysteretic energy dissipated till reaching the ultimate demand displacement, which are defined in Fig. 1.

The errors corresponding to the hysteretic response measures were summed up with equal weights to determine the total error. The total error value was minimized by a simple searching procedure to calculate member properties. To illustrate the procedure, one of the four tests by Pilakoutas and Elnashai in 1995 [¹⁶]and one of the specimens tested by Oesterle et al. [¹⁵] are discussed in detail as follows.

In 1995, Pilakoutas and Elnashai [16] carried out an experimental study on four RC walls. The test specimen and the setup are shown in Fig. 2.

A comparison between the hysteresis response obtained from the experimental tests and numerical simulations is shown in Fig. 3. The estimated errors computed using these two responses are summarized in Table 1.

In 1976, Oesterle et al. [¹⁵] conducted a research on cantilever walls with rectangular, dumble, and flanged cross sections. As an example, in this study, a barbell cross sectional shaped wall of Oesterle’s study, specimen B5, is discussed in details. The B5 wall specimen cross sectional dimensions and reinforcement details are illustrated in Fig. 4. The responses derived from the numerical simulation of specimen B5, as well as the experimental data, are illustrated in Fig. 5. Three different error criteria were computed to compare the numerical simulation results with the experimental measurements (Table 2). The magnitude of these errors is less than 5% indicating the numerical simulation model compares well with the experimental data.

Some other experimental tests used for verification of the numerical simulation model are listed in Table 3. As can be seen, a wide range of longitudinal, transverse, and web reinforcement were used for calibrating the model. The provided data also shows that the estimated errors are less than 10 percent, which are reasonable to prove the numerical model calibrated.

Parametric study

Using OpenSEES [¹⁹], various types of walls were analyzed under cyclic loading with uniform and triangular loading patterns. The main design parameters varied through the parametric study are defined in Fig. 6 and their variation ranges are presented in Table 4.

The yield strength of wall transverse reinforcements was set to 412 MPa. The concrete cover of transverse rebars was 38 mm. The layout of longitudinal and transverse reinforcement satisfied the detailing requirements of ACI 318-11 provisions [²].

A wall will reach its ultimate performance if the cyclic analysis satisfies the termination criteria defining wall section failure. The wall failure criteria were defined as when: (i) a main rebar reached the state of rupture, or in other words, reached its ultimate strain, {\epsilon }_{su}, or (ii) wall concrete core was crushed and reached its ultimate compression strain, {\epsilon }_{cu}, or (iii) wall lateral capacity was reduced by 15% from its peak value. The rebar ultimate strain, {\epsilon }_{su}, was set to 0.09 to be smaller than common value for ultimate elongation of current rebars under cyclic tests. The concrete core ultimate strain, {\epsilon }_{cu}, was determined using the constitutive model proposed by Mander et al. [¹¹] based on wall end element confinement. The results of a typical cyclic analysis are shown in Fig. 7.

For a cantilever wall plastic hinge, hysteretic concrete core and rebar stress–strain relations, hysteretic moment–curvature plot, and moment–curvature elastic–plastic idealization were determined. Furthermore, wall hysteretic lateral force–displacement curve was derived. The elastic–plastic idealization of the moment–curvature curve was developed as the backbone curve of the hysteretic moment–curvature plot. The behavioral measures for both loading patterns were computed. The maximum or the minimum values corresponding to each loading pattern are reported and according to the nature of the behavior measure one has been selected as the controlling response. In this way, the proposed formula will be valid for both design and retrofit strategies.

Influence of design parameters on behavioral measures

As mentioned earlier, design parameters, such as wall dimensions, longitudinal, and transverse reinforcement percentage, axial load ratio, end element dimensions, and loading pattern were considered in this study. The design parameter will affect behavioral measures in different ways. For example, axial load ratio and end element characteristics affect the yield curvature and effective stiffness of the wall. In order to investigate the design parameters’ effects on yield curvature, a wall with constant design parameters, mentioned in Table 5, was modeled. Since higher axial load ratio will produce pre-compression stress in rebars and affect the yield curvature. The section should experience larger curvature so that the rebars yield, but the end element thickness did not have significant effect on yield curvature, as can be seen in Fig. 8. Similarly, for studying the effects of stiffness on the wall, a wall with constant design parameters mentioned in Table 6 is considered. As can be seen in Fig. 9, the increase in longitudinal end element rebar ratio and axial load ratio will directly affect the effective stiffness of the wall, which was previously mentioned by Priestley et al. [²⁰].

Empirical modeling

Calibrated model data, obtained from 5602 wall behaviors simulations, were used to develop empirical equations for estimating model parameters based on the actual wall design parameters. The functional used in the regression analysis was determined based on the variation trends in the computed data and the effects of each individual variable outcome previous researches existing equations and engineering judgment based on mechanics and expected behavior. The regression analysis was performed using the natural logarithm of the model parameters, and the logarithmic standard deviation determined the uncertainty level. Regression analysis and power method were used to eliminate the insignificant parameters from the proposed relations. Using the proposed relations, the designer can judge which parameters have significant effect on the behavioral measures.

After putting aside the insignificant parameters from the database, the optimized relations were derived as follows (Behavioral Measures Definition is shown in Table 7):

(1){\varphi }_{y\_\min }{L}_w=\mathrm{0.01}{t}_w^{\mathrm{0.12}}{L}_w^{-\mathrm{0.19}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.14}}{E}_t^{-\mathrm{0.15}},

(2){\varphi }_{y\max }{L}_w=\mathrm{0.004}{t}_w^{\mathrm{0.11}}{L}_w^{-\mathrm{0.05}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.12}}{E}_t^{-\mathrm{0.14}},

(3){\varphi }_{u\min }{L}_w=\mathrm{0.21}{t}_w^{-\mathrm{0.27}}{L}_w^{\mathrm{0.30}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.12}}{E}_t^{\mathrm{0.34}}{({\rho }_{\min })}^{\mathrm{0.23}},

(4){\varphi }_{u\max }{L}_w=\mathrm{0.11}{t}_w^{-\mathrm{0.30}}{L}_w^{\mathrm{0.38}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.13}}{E}_t^{\mathrm{0.36}}{({\rho }_{\min })}^{\mathrm{0.16}},

(5)\left({\displaystyle \frac{E{I}_{\min }}{E{I}_g}}\right)=\mathrm{1.46}{t}_w^{-\mathrm{0.38}}{L}_w^{\mathrm{0.56}}{(A.R)}^{\mathrm{0.02}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.23}}{E}_L^{\mathrm{0.19}}{\rho }_{\mathrm{1}}^{\mathrm{0.23}},

(6)\left({\displaystyle \frac{E{I}_{\max }}{E{I}_g}}\right)=\mathrm{0.78}{t}_w^{-\mathrm{0.41}}{L}_w^{\mathrm{0.66}}{(A.R)}^{\mathrm{0.05}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.23}}{E}_L^{\mathrm{0.17}}{\rho }_{\mathrm{1}}^{\mathrm{0.20}},

(7)\left({\displaystyle \frac{K_{\min }}{K_{Elastic}}}\right)=\mathrm{2.88}E-\mathrm{07}{t}_w^{-\mathrm{0.43}}{L}_w^{\mathrm{2.48}}{(A.R)}^{\mathrm{1.93}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.24}}{E}_L^{\mathrm{0.16}}{E}_t^{\mathrm{0.57}}{\rho }_{\mathrm{1}}^{\mathrm{0.19}},

(8)\left({\displaystyle \frac{K_{\max }}{K_{Elastic}}}\right)=\mathrm{3.075}E-\mathrm{11}{t}_w^{-\mathrm{0.40}}{L}_w^{\mathrm{3.60}}{(A.R)}^{\mathrm{3.05}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.24}}{E}_L^{\mathrm{0.18}}{E}_t^{\mathrm{0.53}}{\rho }_{\mathrm{1}}^{\mathrm{0.21}},

(9)\left({\displaystyle \frac{M_{p_{\min }}}{K_e}}\right)=\mathrm{34.57}{t}_w^{\mathrm{0.59}}{L}_w^{-\mathrm{0.6}}{(A.R)}^{\mathrm{0.002}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.28}}{E}_L^{-\mathrm{0.70}}{E}_t^{-\mathrm{0.75}}{\rho }_{\mathrm{1}}^{-\mathrm{0.77}},

(10)\left({\displaystyle \frac{M_{p_{\max }}}{K_e}}\right)=\mathrm{36.41}{t}_w^{\mathrm{0.5}}{L}_w^{-\mathrm{0.56}}{(A.R)}^{\mathrm{0.03}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.24}}{E}_L^{-\mathrm{0.64}}{E}_t^{-\mathrm{0.73}}{\rho }_{\mathrm{1}}^{-\mathrm{0.74}},

(11){\displaystyle \frac{{\mathrm{\Delta }}_{y\min }}{h}}=\mathrm{10280.1}{t}_w^{\mathrm{0.14}}{L}_w^{-\mathrm{2.07}}{(A.R)}^{-\mathrm{0.99}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.13}}{E}_t^{-\mathrm{0.14}},

(12){\displaystyle \frac{{\mathrm{\Delta }}_{y\max }}{h}}=\mathrm{1.13}{t}_w^{\mathrm{0.14}}{L}_w^{-\mathrm{0.98}}{(A.R)}^{\mathrm{0.1}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.13}}{E}_t^{-\mathrm{0.15}},

(13){\displaystyle \frac{M_{y\min }}{M_e}}=\mathrm{27.45}{t}_w^{\mathrm{0.57}}{L}_w^{-\mathrm{0.54}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.36}}{E}_L^{-\mathrm{0.75}}{E}_t^{-\mathrm{0.75}}{\rho }_{\mathrm{1}}^{-\mathrm{0.79}},

(14){\displaystyle \frac{M_{y\max }}{M_e}}=\mathrm{19.5}{t}_w^{\mathrm{0.59}}{L}_w^{-\mathrm{0.51}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.35}}{E}_L^{-\mathrm{0.74}}{E}_t^{-\mathrm{0.77}}{\rho }_{\mathrm{1}}^{-\mathrm{0.77}},

(15){\displaystyle \frac{V_{u\min }h}{Me}}=\mathrm{0.01}{t}_w^{\mathrm{0.59}}{L}_w^{\mathrm{0.53}}{(A.R)}^{\mathrm{1.05}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.18}}{E}_L^{-\mathrm{0.46}}{E}_t^{-\mathrm{0.732}}{\rho }_{\mathrm{1}}^{-\mathrm{0.69}},

(16){\displaystyle \frac{V_{u\max }h}{Me}}=\mathrm{0.02}{t}_w^{\mathrm{0.58}}{L}_w^{\mathrm{0.49}}{(A.R)}^{\mathrm{1.02}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{\mathrm{0.15}}{E}_L^{-\mathrm{0.46}}{E}_t^{-\mathrm{0.72}}{\rho }_{\mathrm{1}}^{-\mathrm{0.69}},

(17){\displaystyle \frac{{\mathrm{\Delta }}_{u\min }}{h}}=\mathrm{9.81}e\mathrm{03}{t}_w^{-\mathrm{0.25}}{L}_w^{-\mathrm{1.67}}{(A.R)}^{-\mathrm{0.99}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.24}}{E}_t^{\mathrm{0.35}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.58}},

(18){\displaystyle \frac{{\mathrm{\Delta }}_{u\max }}{h}}=\mathrm{81.76}{t}_w^{-\mathrm{0.21}}{L}_w^{-\mathrm{0.54}}{(A.R)}^{-\mathrm{0.13}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.19}}{E}_t^{\mathrm{0.29}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.62}},

(19){\displaystyle \frac{\mathrm{BD}{\mathrm{.Energy}}_{\min }}{Elastic\hspace{0.17em}Energy}}=\mathrm{1.31}e\mathrm{06}{t}_w^{\mathrm{1.16}}{L}_w^{-\mathrm{1.34}}{(A.R)}^{\mathrm{0.12}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.14}}{E}_L^{-\mathrm{1.40}}{E}_t^{-\mathrm{1.58}}{\rho }_{\mathrm{1}}^{-\mathrm{1.66}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.81}},

(20){\displaystyle \frac{\mathrm{BD}{\mathrm{.Energy}}_{\max }}{Elastic\hspace{0.17em}Energy}}=\mathrm{3.27}\mathrm{e}+\mathrm{5}{t}_w^{\mathrm{1.00}}{L}_w^{-\mathrm{1.16}}{(A.R)}^{\mathrm{0.22}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.23}}{E}_L^{-\mathrm{1.30}}{E}_t^{-\mathrm{1.43}}{\rho }_{\mathrm{1}}^{-\mathrm{1.69}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.53}},

(21){\displaystyle \frac{D{\mathrm{.Energy}}_{\min }}{Elastic\hspace{0.17em}Energy}}=\mathrm{33.64}{t}_w^{\mathrm{1.19}}{L}_w^{-\mathrm{2.28}}{(A.R)}^{-\mathrm{0.93}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.26}}{E}_L^{-\mathrm{1.32}}{E}_t^{-\mathrm{1.59}}{\rho }_{\mathrm{1}}^{-\mathrm{1.45}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.89}},

(22){\displaystyle \frac{D{\mathrm{.Energy}}_{\max }}{Elastic\hspace{0.17em}Energy}}=\mathrm{1.07}\mathrm{e}\mathrm{6}{t}_w^{\mathrm{1.25}}{L}_w^{-\mathrm{1.12}}{(A.R)}^{\mathrm{0.17}}{\left({\displaystyle \frac{P}{P_{\mathrm{0}}}}\right)}^{-\mathrm{0.21}}{E}_L^{-\mathrm{1.34}}{E}_t^{-\mathrm{1.62}}{\rho }_{\mathrm{1}}^{-\mathrm{1.46}}{({\rho }_{\mathrm{h}})}^{\mathrm{0.95}},

In this study, design parameters are classified into two main categories based on whether they significantly affect the value of the behavioral measures or not so significantly. These two categories are presented in Table 8.

The developed empirical formula have been investigated to represent the well-known trends with respect to design parameters for some of the behavioral measures. For example, Li [⁴] reported wall dimensions, axial load, and aspect ratio as important parameters for determining effective stiffness. Such observation has been obtained in this study as mentioned in Table 8 and the earliest related works on empirical formula Priestley et al. [²⁰] by and Paulay and Priestley [²¹]. Furthermore, the work by Tjhin et al. [²²] has shown that the wall yield curvature is influenced by the axial load for some range of its value. Such a trend has been captured by the yield curvature formula.

To illustrate the accuracy of the proposed formulas, a sample plot of the computed results from OpenSEES versus the results from empirical formulations, is provided in Fig. 10 and The results of correlation analysis are summarized in Table 9 which can be seen that all correlations are near unity but two formulations for yield curvature that is still report fine correlations.

Application of developed empirical formulations

Empirical formulations have a wide range of application in both fields of design and retrofit of structures. In this section, it is represented how this formulation can help designer to predict performance of a wall, and how to decide which design parameter effectively improves its performance.

The proposed formulations were applied to the shear wall under transverse cyclic loading studied by Aaleti [²³]. The wall has a rectangular cross section with end elements, and a reverse cyclic load is applied at its top (Fig. 11). The design parameters of the wall are reported in Table 10. The wall responses derived based on experimental tests and OpenSEES numerical simulations are shown in Fig. 12.

Displacement demand at the top of the wall at 2% drift was 4.9 in. The maximum ductility damage index was calculated using Eq. (23) {\mathrm{\Delta }}_y and {\mathrm{\Delta }}_u in this equation, were respectively derived from Eqs. (12) and (17).

(23)D{I}_{\mu }={\displaystyle \frac{{\mathrm{\Delta }}_d-{\mathrm{\Delta }}_y}{{\mathrm{\Delta }}_u-{\mathrm{\Delta }}_y}}

where {\mathrm{\Delta }}_d is demand displacement.

Using the proposed formula, the yield displacement, the ultimate displacement, and the ductility damage index were computed 29.01, 242.61, and 0.45, respectively. The ductility damage index definition (Eq. (23)) reveals that increasing the ultimate displacement improves the wall performance. Eq. (17) shows that by increasing the end elements' width, L_e, wall aspect ratio, {\displaystyle \frac{h}{D}}, and transverse reinforcement ratio, {\rho }_h, the ultimate displacement, {\mathrm{\Delta }}_u, will be increased. Since the length and the height of the shear wall are usually fixed, designer can change only end elements thickness and longitudinal reinforcement ratio. The effect of the end elements’ width on the ductility damage index is presented in Fig. 11.

Another parameter that improves the ultimate displacement of a wall is the transverse reinforcement ratio of the end elements which could be modified for a retrofit strategy by using steel jackets or using FRP [⁵]. Figure 13 shows the effect of the end element transverse reinforcement ratio on increasing ultimate displacement. It decreases the ductility damage index, which in turn helps the designer to select an appropriate amount of transverse reinforcement that keeps the shear wall performance in a desirable level.

This research shows that the longitudinal reinforcement ratio, minimum reinforcement ratio, and the end elements’ depth have insignificant effects on the yield and ultimate displacements of the wall. Figures 14–16 represent the variation of ductility damage index versus the reinforcement ratio and depth of the end elements.

Conclusions

A summary of numerical simulations for modeling the hysteretic behavior of plastic hinges of RC shear walls is presented. Empirical relations among wall hysteretic indices and design parameters were derived. The developed numerical models were calibrated using several published shear wall cyclic tests. A database of cyclic behavioral measures and design parameters was developed through a thorough parametric study of more than 5602 shear walls. The considered main wall design parameters were wall dimensions, axial force, reinforcement ratios and end element design parameters. The considered hysteretic response measures were effective stiffness, yield and ultimate curvatures, plastic moment capacity of a wall plastic hinge, yield and ultimate displacements, effective stiffness, flexural shear capacity, and dissipated energy of shear wall member. The developed empirical formulas with their computed accuracy and correlation are presented. Also, the use of the developed formulas for designing shear walls is illustrated.

References

[1] Banan M R. Beyond R-factor: design theory for damage-based seismic design of RC buildings. In: Proceedings of a Turkey–Iran–US seismic workshop, 2010

[2] ACI Committee 318. Building code requirements for structural concrete (ACI 318-11) and commentary. Farmington Hills, MI, USA: American Concrete Institute, 2011

[3] FEMA-695. Quantification of building seismic performance factors. FEMA, 2009

[4] Li B. Initial stiffness of reinforced concrete columns and walls. In: World Conference on Earthquake Engineering, Lisborn, 2012

[5] Sakhayi S. Estimation of Relation Among Cyclic Behavioral Measures and Design Parameters for FRP-Jacketed Reinforced Concrete Columns. Dissertation for M.Sc degree. Civil and Environmental Engineering Department, Shiraz University, 2013

[6] AASHTO. AASHTO guide specification for LRFD bridge design. Associated of State Highway and transportation officials, Washington D. C., 2010

[7] Abazarnejad M. Determination of Data-Based Formula Relating Damage Index and Permanent Set to RC Column Design Parameters Using Cyclic Analysis. Dissertation for M.Sc degree. Civil and Environmental Engineering Department, Shiraz University, 2013

[8] Khalifeh Mehrjardi H. Estimation of Relation between Damage Index and Column Design Parameters for Concrete Encased Column. Dissertation for M.Sc degree. Civil and Environmental Engineering Department, Shiraz University, 2011

[9] Neuenhofer A, Filippou F C. Geometrically nonlinear flexibility-based frame finite element. Journal of Structural Engineering, 1998, 124(6): 704–711

[10] Neuenhofer A, Filippou F C. Evaluation of nonlinear frame finite-element models. Journal of Structural Engineering, 1997, 123(7): 958–966

[11] Mander J B, Priestley M J N, Park R. Theoretical stress-strain model for confined concrete. Journal of Structural Engineering, 1988, 114(8): 1804–1826

[12] Karsan I D, Jirsa J O. Behavior of concrete under compressive loading. Journal of Structural Division, 1969, 95: 2543–2564

[13] Sharifi A, Banan M R, Banan M R. A strain-consistent approach for determination of bounds of ductility damage index for different performance levels for seismic design of RC frame members. Engineering Structures, 2012, 37: 143–151

[14] Dhakal R P, Maekawa K. Path-dependent cyclic stress–strain relationship of reinforcing bar including buckling. Engineering Structures, 2002, 14(11): 1383–1396

[15] Oesterle R G, Fiorato A E, Johal L S, Carpenter J E, Russell H G, Coreley W G. Earthquake resistant structural walls-test of isolated walls. Natinal Science Foundation, 1976

[16] Pilakoutas K, Elnashai A. Cyclic behaviour of reinforced concrete cantilever walls, Part I: experimental results. ACI Structural Journal, 1995, 92(3): 271–281

[17] Thomsen J H, Wallace J W. Displacement-based design of reinforced concrete structural walls: an experimental investigation of walls with rectangular and T-shaped cross-section. National Science Foundation, 1995

[18] Riva P, Meda A, Giuriani E. Cyclic behavior of a full scale RC structural wall. Engineering Structures, 2003, 25(6): 835–845

[19] Mckenna F, Fenves G L. The open system for earthquake engineering simulation. University of California at berkeley, 2013

[20] Priestley M J N, Seible F, Calvi G M. Seismic Design and Retrofit of Bridges. John Wiley, 1996

[21] Paulay T, Priestley N. Seismic Design of Reinforced Concrete and Masonry Buildings. John Wiley, 1992

[22] Tjhin T N, Aschheim M A, Wallace J W. Yield displacement estimates for displacement-based seismic design of ductile reinforced concrete structural wall buildings. In: Proceedings of the 13th World Conference on Earthquake Engineering, Vancouver, B.C., Canada, 2004

[23] Aaleti S R. Behaviour of Rectangular Concrete Walls Subjected to Simulated Seismic Loading. Dissertation for PhD degree. Iowa State University, 2009

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