Evaluation of load–deformation behavior of reinforced concrete shear walls with continuous or lap-spliced bars in plastic hinge zone

Abstract

This article presents an analysis procedure for evaluation of load–deformation behavior of reinforced concrete shear walls with continuous or lap-spliced bar connections in plastic hinge zones under horizontal loads. For the shear walls with continuous bars, the lateral deformations caused by flexure, shear, and reinforcement slip are evaluated by considering their interaction. The flexural deformation is calculated by conventional fiber model. The shear mechanism is based on modified compression field theory with a softened smeared cracked reinforced concrete membrane element. Both the flexural and shear deformations are estimated separately in the plastic hinge and non-plastic hinge regions. In addition, an approach is proposed for analysis of plastic hinge length based on fracture energies of materials. For the shear walls with lap-spliced bars, due to its complicated behavior and mechanism, a simple way to deal with the lap splice is proposed. The equations regarding bond-slip of the lap splice with minimum spliced length are established and the stress and strain states of lap splices with different spliced lengths are analyzed on the basis of equilibrium of forces with a mean bond stress model. Finally, the validity of the proposed analysis procedure is confirmed by comparing the analytical results with previous experimental data.

Keywords

continuous bar flexure lap splice load–deformation response plastic hinge shear shear wall

Introduction

Due to the high lateral stiffness of shear walls to resist wind and earthquake action, it is effective to use shear walls for buildings as the first lateral load resisting system. There are many experimental studies performed to investigate the seismic performance and deformational characteristics of reinforced concrete (RC) shear walls (Almeida et al., 2016; Dazio et al., 2009; Kang et al., 2013; Syed, 2013; Zhang et al., 2011). The shear walls usually provide a good response under seismic action due to their dimensions and in-plane position arrangement. The axial, flexural, shear, strain penetration of the longitudinal reinforcing bars into the anchorage and sliding shear mechanisms are often used to explain the deformation behavior of shear walls with different detailing and aspect ratios under axial and horizontal loads, especially for the combined flexural and shear behavior. Shear walls with different sections, reinforcement layouts, and aspect ratios lead to different failure modes, namely, bending failure and shear failure. To analyze a flexure-dominant shear wall (slender member), many fiber beam-column elements have been developed in the past several decades. The fiber method is mainly based on the traditional Euler–Bernoulli beam theory that plane sections remain plane. However, it may not be appropriate when axial, bending, and shear actions are coupled in RC members with low or middle aspect ratios.

Many studies have attempted to cope with the axial-bending-shear coupling by adopting the Timoshenko beam theory or a series of uniaxial concrete elements in a so-called multiple vertical line element model (MVLEM; Orakcal et al., 2004). Other models have been used to determine the shear behavior of concrete members by developing rotating (or fixed) angle softened truss model (RA-STM or FA-STM; Hsu, 1988; Hsu and Zhang, 1996), distributed stress field model (Vecchio, 2000), and axial-shear-flexure interaction (ASFI) method (Mostafaei and Vecchio; Mostafaei and Kabeyasawa, 2007). The developed ASFI method was derived through both equilibrium and compatibility of flexure and shear mechanisms that employed the plane section assumption and the modified compression field theory (MCFT; Vecchio and Collins, 1986). However, this method was mainly employed in RC columns with an inflection point, and the plastic hinge length $L_{p}$ was simply taken as the height of cross section. The plastic hinge length $L_{p}$ that formed at the ends of the shear wall (also including other concrete members, such as beams and columns) significantly influences the flexural deformation capacity of a shear wall that can be evaluated by double integrating curvature distribution over the length of the shear wall. Recently, according to the experimental results of shear walls conducted by Dazio et al. (2009) and Beyer et al. (2011), it has been found that shear strains are concentrated in the plastic hinge zone. So the magnitude of the shear deformations not only depends on the horizontal load but also depends on the plastic hinge length. Therefore, it may be reasonable to estimate the shear deformation by perceiving the shear deformations within and outside the plastic hinge regions, respectively. Various equations have been proposed to calculate the plastic hinge length. For instance, the plastic hinge length can be simply assumed as half the member’s length to obtain a safe estimate of displacement capacity from curvature capacity (Bohl and Adebar, 2011)

L_{p} = L_{w} / 2

(1)

where $L_{w}$ is the wall length. Another general form of $L_{p}$ was expressed as (Panagiotakos and Fardis, 2011; Paulay and Uzumeri, 1975)

L_{p} = α L_{w} + β d_{e} + ξ f_{s y} d_{b}

(2)

where $α$ , $β$ , and $ξ$ are alternative constants; $d_{e}$ is the depth of the cross section; and $f_{sy}$ and $d_{b}$ are the yield strength and diameter of reinforcement, respectively. However, this type of expression requires extensive experimental verification which may not be available completely.

For fixing the reinforcement conveniently, a lap-spliced joint can be used in the concrete members. Biskinis and Fardis (2010a; 2010b) developed ultimate and yield deformation models of flexure-controlled members with continuous bars and lap-spliced bars, respectively, in the plastic hinge region. The models were mainly derived with empirical expressions according to a large databank of cyclic and monotonic tests. Tastani et al. (2015) established a series of field equations governing lap-spliced behavior. The presented model is mainly used when the bars are elastic (namely, $ε_{s} \leq ε_{sy}$ , where $ε_{sy}$ is the yield strain of steel bar). The object of this article is to propose a procedure for evaluation of the load–deformation behavior of RC shear walls with continuous or lap-spliced bars in the plastic hinge zone with focus on flexure and shear interaction, plastic hinge development for the member response, and deformation characteristic (bond-slip) of lap splice. The analytical results of the proposed procedure will be compared with previously obtained results.

Shear walls with continuous bars at the connection

For estimating the response of a typical RC shear wall, the total displacement $Δ_{t}$ at the top of a cantilever wall is mainly composed of four components that are caused by flexure $(Δ_{f})$ , bond-slip at the end of wall ( $Δ_{θ}$ , due to fixed end rotation), shear $(Δ_{s})$ , and sliding $(Δ_{sl})$ , respectively. The sliding displacement is usually relatively small; hence, it is not considered, unless it occurs in the shear wall with low aspect ratio

Δ_{t} = Δ_{f} + Δ_{θ} + Δ_{s}

(3)

The components of the total response are represented by several springs and are illustrated in Figure 1. At the pre-yield stage, three springs in series are used to model the three components of lateral response. At the post-yield stage, the shear wall is divided into two sections that are plastic hinge region and elastic region (non-plastic hinge region). And five springs in series located at characteristic positions along the two sections are used to model the three components of lateral response. The shear and flexural deformations are calculated by the MCFT and fiber models. Their combined interaction in the analytical procedure is based on the concept of the ASFI approach (Mostafaei and Kabeyasawa, 2007). For the flexural deformation analysis, the inclined cracked concrete in the web of shear wall will change the concrete strength which is considered by employing compression softening factor based on MCFT for an element within or outside the plastic hinge region. For the shear deformation analysis, both equilibriums of axial stresses caused by flexure, shear and applied axial load, and compatibility of strains will be taken into account. Then, the flexural and shear behavior analyses can be performed separately in a simple way.

Figure 1.

Components of the total deformation response represented by springs: (a) a test specimen in the previous experiment, (b) a flexure-type deformation of shear wall, (c) components of the total response at pre-yield stage with three springs, and (d) components of the total response at the post-yield stage with five springs.

Flexural deformation

In the fiber model analysis, flexural deformations are evaluated through integrating the curvature along the height of cantilever shear wall. Generally, the shear wall curvature distribution is idealized and illustrated in Figure 2. Before yielding of reinforcement, the curvature shows a linear and elastic curvature distribution. After yielding, the curvature shows a nonlinear curvature distribution, which assumes that the regions of inelastic curvature are concentrated at the end of shear wall with a fixed length. The yield curvature can be calculated as follows

ϕ_{y} = ε_{s y} / (L_{w} - y_{s c} - c_{y})

(4)

where $ε_{sy}$ is the yield strain of reinforcement, $L_{w}$ is the cross section height (shear wall length), $y_{sc}$ is the centroidal point of the tensile reinforcement in the boundary element, and $c_{y}$ is the length of compression region when the reinforcement yields. Then, the lateral yield load $V_{y}$ and flexural bending moment $M_{y}$ can be calculated according to section analysis.

Figure 2.

Section stress and strain distributions of concrete and reinforcement and curvature distribution assumed along the shear wall axis.

After yielding, the lateral load is evaluated as

V = (\sum_{i = 1}^{m} f_{s i} A_{s i} y_{s i} + \sum_{j = 1}^{n_{c o r e}} f_{c c o r e j} A_{c o r e j} y_{c o r e j} + \sum_{k = 1}^{n} f_{c k} A_{c k} y_{c k}) / H

(5)

where $f_{si}$ , $A_{si}$ , and $y_{si}$ are the reinforcement stress, area, and distance from the center of the cross section of steel bar fiber i, respectively; m is the number of steel bar fibers, including tensile and compressive steel bars; $f_{ccorej}$ , $A_{corej}$ , and $y_{corej}$ are the confined concrete stress, area and distance from the center of the cross section, in the boundary elements, of fiber j, respectively; $n_{core}$ is the number of confined concrete fibers, $f_{ck}$ , $A_{ck}$ , and $y_{ck}$ are the concrete stress, area, and distance of other concrete fiber k, respectively; and n is the number of the unconfined concrete fibers. All the stresses $f_{si}$ , $f_{ccorej}$ , and $f_{ck}$ can be evaluated according to section analysis with a given depth of the compressing zone, c, and curvature, $ϕ$ , and corresponding constitutive laws with several iteration steps. The constitutive laws of the reinforcement and the concrete are shown in Figure 3. For the reinforcement, a bilinear stress–strain relationship is adopted and expressed as follows

\begin{array}{l} f_{s} = ε_{s} E_{s}, f_{s} \leq f_{s y} \\ f_{s} = f_{s y} + E_{s h} (ε_{s} - ε_{s y}), f_{s} > f_{s y} \end{array}

(6)

where $f_{s}$ is the reinforcement stress, $ε_{s}$ is the reinforcement strain, $E_{s}$ is the elastic modulus, $E_{sh}$ is the plastic modulus and can be estimated by $(f_{su} - f_{sy}) / (ε_{su} - ε_{sy})$ , where $ε_{su}$ is the ultimate reinforcement strain, and $f_{su}$ is the ultimate reinforcement stress. For the concrete considering the confined effect by the stirrups, the stress–strain relationship can be determined based on available confinement models such as the modified Kent and Park (Park et al., 1982) model, illustrated as follows.

Figure 3.

Constitutive laws of the materials: (a) confined and unconfined concrete including tensile and compressive constitutive laws and (b) reinforcement with a bilinear stress–strain relationship.

The stress–strain relationship of compressed concrete according to the pre-peak branch of a Hognestad (1951) parabola

\begin{array}{l} f_{c} = f_{p} [2 ε_{c} / ε_{p} - {(ε_{c} / ε_{p})}^{2}], - ε_{c} \leq - ε_{p} \\ f_{c} = f_{p} [1 - Z_{m} (- ε_{c} - (- ε_{p}))], - ε_{c} > - ε_{p} \end{array}

(7)

where

f_{p} = K β f_{c}', ε_{p} = K ε_{c}'

K = 1 + ρ_{wv} f_{yy} / f_{c}', Z_{m} = \frac{0.5}{\frac{3 + 0.29 f_{c}'}{145 f_{c}' - 1000} + 0.75 ρ_{wv} \sqrt{\frac{h ″}{s_{h}} - K ε_{c}'}}

β_{c} = \frac{1}{0.8 - 0.34 ε_{1} / ε_{c}'} \leq 1.0

The stress–strain relationship of concrete in tension is given by

\begin{matrix} f_{c} = E_{c} ε_{1} ε_{c} \leq ε_{t}' \\ f_{c} = f_{t}' / (1 + \sqrt{200 ε_{1}}) ε_{c} > ε_{t}' \end{matrix}

(8)

where $f_{c}$ is the concrete stress, $ε_{c}$ is the concrete strain, $f_{p}$ is the compression strength considering the confined effect by the stirrups and concrete stress–strain softening due to the inclined cracking in the web element, $ε_{p}$ is the concrete strain corresponding to $f_{p}$ , K and $Z_{m}$ are the confinement parameters, $f_{c}'$ and $ε_{c}'$ are the maximum concrete cylinder strength and strain, respectively, $ε_{c}'$ is taken as 0.002, $ρ_{wv}$ is the volume ratio of stirrups to confined concrete core measured to the outside of the peripheral stirrups, $h ″$ is the confined concrete core width measured to the outside of stirrups, $s_{h}$ is the spacing of stirrups measured from center to center, $β_{c}$ is the compression softening factor which varies with shear strains and will be calculated in the following section, $ε_{1}$ is the principle tensile strain, and $f_{t}'$ and $ε_{t}'$ are the maximum concrete tensile stress and corresponding strain.

After yielding, the flexural deformation can be calculated as follows

\begin{array}{l} Δ_{f} = \int_{0}^{H} x ϕ (x) d x = Δ_{f y} + Δ_{f p} = 1 / 3 ϕ_{y} H^{2} \\ + (ϕ_{p} - ϕ_{y}) L_{p} (H - \frac{L_{p}}{2}) \end{array}

(9)

where H is the height of shear wall, $ϕ (x)$ is the section curvature at the height of x along the axis of the shear wall, $ϕ_{y}$ is the yield curvature at the base of wall, $ϕ_{p}$ is the curvature at the plastic hinge region after yielding of reinforcement. $Δ_{fy}$ is the yield flexural deformation, and $Δ_{fp}$ is the flexural deformation caused by the plastic hinge region of shear wall. Then, the chord flexural drift ratio (i.e. flexural rotation $γ_{f}$ ) can be evaluated as $γ_{f} = Δ_{f} / H$ .

The plastic hinge length is considered to be constant after yielding of reinforcement. In fact, this is an approximate method, because the plastic hinge length usually increases with an increase of applied load at early stage, and then tends to constant when the reinforcement strains are large and are developed in a certain concentrated length (Dazio et al, 2009). There are several reasons to adopt the constant plastic length for this study. First, constant length plastic hinge models are currently available in commercial and research software and, thus, can be used immediately by practicing and research engineers (Berry et al., 2008). Second, according to the vertical strains along the height of the shear walls in relevant literatures (Dazio et al., 2009; Tran and Wallace, 2015), the major deformations were often lumped in a certain region at the plastic stage. Although the plastic region will spread along the height of the specimens, the value of the increasing plastic strain at the top of plastic hinge region was not large. Third, the subsequent section of this study will show that the fixed length plastic hinge model can be used to predict shear wall deformation behavior accurately for a range of demand levels, which is the primary objective. Various well-known empirical models for $L_{p}$ have been proposed in the past decades (Hines et al., 2004; Mun and Yang, 2015; Sasani and Der Kiureghian, 2001). Many of the empirical models were based on regression analysis according to the tests, and which need cost and time involved in large tests due to the high nonlinearity of materials and the interaction effect between constituent materials in the plastic hinge region (Zhao et al., 2012). Therefore, a new method will be provided to determine the plastic hinge length.

In this study, it is assumed that the energy of the applied lateral load in the member, which is produced by flexural behavior when the member is in the post-yield stage, is equal to the sum of fracture energies of materials including reinforcement and concrete in the plastic hinge region. In other words, the energy from the external applied load producing flexural deformation is dissipated by the materials of plastic hinge region. This assumption is reasonable because most of the external energy is dissipated by flexural behavior of member (Eom and Park, 2013), and the flexural deformations are mainly attributed to the plastic hinge region at the post-yield stage

E_{no} = E_{nm} = E_{nc} + E_{ns}

(10)

where $E_{no}$ is the energy from applied lateral load after yielding of reinforcement, $E_{nm}$ is the energy dissipation of the member by materials in the plastic hinge region, and $E_{nc}$ and $E_{ns}$ are the fracture energies of concrete and steel bar, respectively. $E_{ns}$ corresponds to the total longitudinal reinforcement (including the tensile and compressive reinforcement) after yielding. The bond-slip in the footing also dissipates external energy which equals the total strain energy stored in the steel and concrete (Eleftheriou et al., 2017). However, the bond-slip energy in the footing is not considered for calculating the total external energy, because the hinge length $L_{p}$ was mainly used for calculating the flexural and shear deformations. It is the fact that, in most cases, $E_{ns}$ is obviously larger than $E_{nc}$ . However, ignoring $E_{nc}$ will overestimate plastic hinge length $L_{p}$ , but the fracture energy of tensile concrete is not taken into account in this study due to its small contribution to the fracture energy, because the tensile strength and strain of the concrete are very small compared with that of concrete in compression. The fracture energy of reinforcement per unit area, $G_{fs}$ , along $L_{p}$ (see Figure 4) is defined as follows

G_{fs} = \sum \int_{ε_{y}}^{ε_{s} (x)} L_{p} f_{s} d ε_{s}, ε_{s} (x) \geq ε_{sy}

(11)

Figure 4.

Energy dissipation in the plastic hinge region of a member: (a) a plastic hinge energy dissipation mechanism, (b) fracture energy of reinforcement, and (c) fracture energy of concrete in compression.

The integral result of equation (11) is the area of post-yield stress–strain curve multiplied by $L_{p}$ and can be expressed as

G_{fs} = \sum \frac{1}{2} [(ε_{s} (x) - ε_{sy}) (f_{s} (x) + f_{sy}) L_{p}], ε_{s} (x) \geq ε_{y}

(12)

where $ε_{s} (x)$ and $f_{s} (x)$ are the reinforcement strain and stress of each steel bar fiber at the distance x along the section height from the center line of section, respectively. The fracture energy of compressive concrete per unit area $G_{fci}$ of each compressive concrete fiber is defined as follows

G_{fci} = \int_{ε_{c 1}}^{ε_{c} (x)} L_{d} f_{c} d ε_{c}, ε_{c} (x) \geq ε_{c}'

(13)

The result of equation (13) is the area of the post-peak concrete stress–strain curve multiplied by $L_{d}$ and can be expressed as

\begin{array}{l} G_{f c i} = 1.25 t [f_{c^{'}} (ε_{c^{'}} - ε_{c 1}) + (f_{c} (x) + f_{c^{'}}) (ε_{c} (x) - ε_{c^{'}}) \\ - (f_{c} (x) (ε_{c} (x) - ε_{c 2})], ε_{c} (x) \geq ε_{c^{'}} \end{array}

(14)

where $ε_{c} (x)$ and $f_{c} (x)$ are the concrete strain and stress of each compressive concrete fiber at the distance x along the section height from the center line of the section, respectively; $L_{d}$ is the damage length of compressive concrete which is not necessarily equal to $L_{p}$ ; and $ε_{c 1}$ and $ε_{c 2}$ are the residual strains when the concrete is unloaded at maximum stress, $f_{c}'$ , and at stress $f_{c} (x)$ , respectively. For the confined concrete, equation (14) can be revised by substituting $f_{c}'$ and $ε_{c}'$ for $f_{p}$ and $ε_{p}$ , respectively. According to Markeset and Hillerborg’s (Hsu and Zhang, 1996) uniaxial compression tests on plain concrete with various dimensions and concrete strength grade, the failure length of concrete is usually 2.5 times the shorter side of a compression section length (Markeset and Hillerborg, 1995). Therefore, in this study, the damage length $L_{d}$ is taken as $2.5 t$ , where t is the thickness of shear wall. Then, $L_{p}$ can be calculated as

L_{p} = \frac{(E_{no} - \sum G_{fci} A_{ci})}{\sum \frac{1}{2} [(ε_{s} (x) - ε_{sy}) (f_{s} (x) + f_{sy})] A_{sj}}

(15)

where $A_{ci}$ is the compression area of fiber i, and $A_{sj}$ is the reinforcement area of fiber j. The calculated $L_{p}$ is an approximate value because the energies are assumed not to be dissipated by other elastic parts of concrete and reinforcement (the other elastic parts are regarded as a rigid body, see Figure 4). To avoid deviation of the calculated results from the design practice, $L_{p}$ should be restrained to a limited range. In accordance with the plastic hinge length equation of beams, Paulay and Uzumeri (1975) proposed a modified equation for $L_{p}$ in shear walls

L_{p} = α 0.8 L_{w} + β H

(16)

Then, an upper-bound evaluation (α = 0.5 and β = 0.1) resulted from Joint ACI-ASCE Committee 428 (1968) recommendations and a lower-bound evaluation (α = 0.25 and β = 0.044) resulted from Paulay and Priestley’s recommendations (Paulay and Priestley, 1993) are adopted in this study

0.2 L_{w} + 0.044 H \leq L_{p} \leq 0.4 L_{w} + 0.1 H

(17)

Shear deformation

After selection and calculation of the constitutive models and flexural deformations, respectively, the next steps are estimation of shear deformation and implementation of a combined flexure-shear model into the proposed procedure for evaluation of the load–deformation behavior.

The shear mechanism is based on the MCFT model that utilizes softened smeared cracked RC membrane element. The shear deformation mechanism is shown in Figure 5. The shear deformations of the shear wall are concerned with average strains (axial strain $ε_{x}$ , transverse strain $ε_{y}$ , and shear strain $γ_{s}$ ). After yielding of the reinforcement, the shear strain distribution along the shear wall axis is no longer linear. The strains are larger and concentrated in the plastic zone at the end of shear wall and are small above the plastic zone. So the shear deformation is divided into two parts. Assume that the axial, transverse, and shear strains are identical in the plastic hinge zone; at the top part of shear wall, the axial strain distribution is linear. Due to a nonuniform distribution of axial strain (linear distribution according to plane section assumption) along the wall length, another assumption is that the average axial strain is taken as the centroidal strain of the cross section. The centroidal strain $ε_{0}$ can be expressed as

ε_{0} = (\frac{L_{w}}{2} - c) ϕ

(18)

where $ϕ$ is curvature of cross section and c is the depth of compressing zone. Based on the equilibrium and compatibility conditions of the MCFT (Vecchio, 2000; Vecchio and Collins, 1986), the stresses and strains of a cracked element need to meet the following required relationships

\tan^{2} θ = \frac{f_{c 1} - f_{cx}}{f_{c 1} - f_{cy}} = \frac{ε_{y} - ε_{2}}{ε_{x} - ε_{2}}

(19)

ε_{1} = ε_{x} + ε_{y} - ε_{2}

(20)

where $θ$ is the rotated crack angle that is between the x axis and $1$ axis; $f_{c 1}$ is the principle tensile stress, $f_{cx}$ and $f_{cy}$ are the stresses in the x and y directions, respectively; $ε_{x}$ and $ε_{y}$ are the axial (x direction) and transverse strains (y direction), respectively; and $ε_{1}$ and $ε_{2}$ are the principle tensile and compressive strains, respectively. The principle tensile strain $ε_{1}$ is used to determine the compression softening factor $β_{c}$ and concrete tensile strength. $ε_{y}$ in equation (20) can be determined as (Mostafaei and Vecchio, 2008)

ε_{y} = \sqrt{b^{2} + c} - b

(21)

where

\begin{array}{l} b = \frac{f_{c 1}}{2 ρ_{s y} E_{s}} - \frac{ε_{2}}{2}, c = \frac{(ε_{x} - ε_{2}) (f_{c 1} - f_{c x}) + f_{c 1} ε_{2}}{ρ_{s y} E_{s}}, \\ f_{c x} = f_{x} - ρ_{s x} f_{s x} \end{array}

where $ρ_{sx}$ and $ρ_{sy}$ are the longitudinal and transverse reinforcement ratios, respectively; $f_{x}$ is the applied axial load; and $f_{sx}$ is the reinforcement stress in the x direction. The principle compressive strain $ε_{2}$ and average axial strain $ε_{x}$ with the element at the top section of shear wall can be determined as

ε_{2} = 0.5 (ε_{c} + ε_{xa}), ε_{x} = 0.5 (ε_{0} + ε_{xa})

(22)

where $ε_{c}$ is the average uniaxial concrete compressive strains corresponding to the resultant forces of the concrete stress blocks and $ε_{xa}$ is the axial strain at the top of the shear wall where the moment is zero. When the calculated elements are in the plastic hinge region, $ε_{2} = ε_{c}$ and $ε_{x} = ε_{0}$ . Then, the compression softening factor $β_{c}$ can be determined according to equation (7). At the ultimate stage, the principle tensile strains $ε_{1}$ in middle- and high-rise shear walls are usually significantly large, resulting in a fairly small magnitude of $β_{c}$ . According to the experiments on shear walls, the inclined crack mainly appeared in the web of shear wall. Therefore, the calculated $β_{c}$ will be used to compute the concrete stresses in the web rather than that in the boundary elements for the evaluation of the flexural deformations.

Figure 5.

Shear deformation mechanism: (a) shear deformations in the plastic hinge zone and the top section of the shear wall, (b) average stresses and strains in cracked element, and (c) Mohr’s circle for average strains.

Based on the compatibility of the membrane element, the shear deformation Δ_s can be determined as follows

Δ_{s} = Δ_{sp} + Δ_{sy} = γ_{sp} L_{p} + γ_{sy} (H - L_{p})

(23)

where $γ_{sp}$ and $γ_{sy}$ are the shear strains of the plastic hinge section and top section of shear wall, respectively. The shear strain $γ_{s}$ (including $γ_{sp}$ and $γ_{sy}$ ) can be expressed as

γ_{s} = 2 (ε_{x} - ε_{2}) \tan θ

(24a)

γ_{s} = ε_{x} \tan θ + \frac{ε_{y}}{\tan θ} - 2 ε_{2} / \sin 2 θ

(24b)

where $θ$ can be calculated by equation (19) and expressed as

θ = \tan^{- 1} \sqrt{\frac{ε_{y} - ε_{2}}{ε_{x} - ε_{2}}}

(25)

Deformation due to bond-slip of the reinforcement at the fixed end

When a shear wall is subjected to lateral load, the tensile reinforcement will be pulled out from the footing due to the bond-slip of steel bar so that it will cause a fixed end rotation at the wall-footing interface associated with strain penetration (Tastani and Pantazopoulou, 2013, 2015). The fixed rotation leads to additional lateral deformation $Δ_{θ}$ . The bond-slip of the steel bar is illustrated in Figure 6. Although the ratio of lateral displacement caused by the reinforcement pullout to the total lateral displacement in shear walls may not be larger than that in columns due to the higher shear wall length and plastic hinge length, it still cannot be ignored. Thus, the fixed end rotation deformation must be evaluated.

Figure 6.

Bond slip at the end of the shear wall: (a) strains and stresses along the bond anchorage length, (b) average bond strength, and (c) rotation caused by reinforcement pullout ( $θ_{pul}$ in the figure is the rotation of the cross section, $θ_{pul} = γ_{pul}$ ).

The fixed end rotation deformation caused by bond-slip is related to the bond stress along the steel bar, bond anchorage length in the footing, and the reinforcement properties. In this study, the bond stresses $τ_{b}$ at the pre-yield stage and post-yield stage are taken as $0.6 f_{c}'^{2 / 3}$ (MPa) and $0.3 f_{c}'^{2 / 3}$ (MPa), which resulted from Sigrist’s suggestion (Marti et al., 1998; Sigrist, 1995). The relationship of reinforcement bond-slip at the wall-footing interface can be estimated as follows

\begin{matrix} s_{0} = \frac{l_{sy} ε_{s}}{2} = \frac{f_{s} d_{b} ε_{s}}{8 τ_{be}}, ε_{s} \leq ε_{sy} \\ s_{0} = \frac{l_{sy} ε_{sy}}{2} + \frac{l_{sp} (ε_{s} + ε_{sy})}{2} = \frac{f_{sy} d_{b} ε_{sy}}{8 τ_{be}} \\ + \frac{(f_{s} - f_{y}) d_{b} (ε_{y} + ε_{sy})}{8 τ_{bp}}, ε_{s} > ε_{sy} \end{matrix}

(26)

where $s_{0}$ is the reinforcement slip at the starting point of anchorage; $l_{sy}$ and $l_{sp}$ are the elastic and inelastic lengths of reinforcement, respectively; $d_{b}$ is the diameter of the anchor steel bar; and $τ_{be}$ and $τ_{bp}$ are the average bond stresses at the pre-yield and post-yield stages, respectively. The lateral deformation due to the bond-slip, $Δ_{θ}$ , can be calculated by

Δ_{θ} = γ_{pul} H = \frac{s_{0} H}{L_{w} - y_{sc} - c}

(27)

where $γ_{pul}$ is the rotation caused by reinforcement pullout.

Analysis procedure

The load–deformation relationship of a shear wall with continuous bars can be evaluated by the following several steps. Considering the interaction between flexure and shear behavior in a shear wall, the procedure for calculation requires iteration. The analysis procedure is based on a displacement controlled approach. In addition, the procedure will terminate when the concrete or reinforcement reaches the limit state, whichever occurs first. Thus, the major calculated steps are introduced as follows:

Define and input structural geometries, reinforcement layout, and all material properties of concrete and steel bars. Calculate the stirrup ratio per unit volume and obtain K and $Z_{m}$ of confined and unconfined concrete.

Create a fiber model and differentiate the regions of confined and unconfined concrete; apply axial load and calculate the axial strain $ε_{xa}$ .

Assume ultimate lateral deformation (or drift ratio) used to calculate $L_{p}$ , it can be computed with given ultimate concrete and steel bar strains.

Assume plastic hinge length $L_{p}$ , which can be initially taken as $L_{w} / 2$ , that is, the shear wall is divided into two parts.

Consider c, $ϕ$ , the principle tensile stress $f_{c 1}$ (an initial value of $f_{c 1} = 0.145 \sqrt{f_{c}'}$ can be used) and compression softening factor $β_{c}$ . Then obtain $ε_{s}$ , $ε_{c}$ and corresponding $f_{s}$ , $f_{c}$ of each fiber on the two cross sections (yield section 2 and plastic section 1, see Figure 1) using equations (5) to (7); calculate $ε_{0}$ using equation (18).

Check the force balances of the two cross sections. If the force balances are not satisfied, then go to step 5. Compute the shear forces $V_{y}$ and V using equation (5).

Calculate $ε_{y}$ using equation (21), and $ε_{2}$ , $ε_{x}$ using equation (22), respectively, then obtain $ε_{1}$ using equation (20).

Check the convergence of $β_{c}$ and $f_{c 1}$ , if not converged go to step 5.

Calculate $θ$ using equation (25), obtain $γ_{sp}$ and $γ_{sy}$ of the two parts using equation (24), and then determine the shear deformations $Δ_{s}$ . Determine the flexural deformation $Δ_{f}$ and reinforcement pullout deformation $Δ_{θ}$ to obtain the total deformation.

Compute the fracture energies of concrete and reinforcement ( $G_{fs}$ , $G_{fc}$ ) using equations (12) and (14). Calculate $L_{p}$ and check its convergence and range, if not converged go to step 4.

Impose an increment of lateral deformation (or drift ratio) which is required to do the calculations, repeat steps 5 to 9 with calculated $L_{p}$ until convergence of the deformations.

Shear walls with lap-spliced bars at the plastic hinge zone

The behavior and deformational mechanisms of lap splice are complicated in the plastic hinge zone and are different from that of continuous bar. In some cases, it is not allowed to adopt a lap splice in the plastic hinge zone of concrete members due to the seismic defects of a lap splice with poor detailing (Kim et al., 2006). However, lap-spliced joints are still used in RC and precast concrete (PC) shear walls owing to its good economy and convenient construction (Zhi et al., 2017). Consequently, a simple way to deal with a lap splice in the plastic hinge region is proposed in this study.

A common way to cope with a lap splice is to consider the joint as a double anchorage (see Figure 7). So the mechanism of the lap splice is affected by the bond state along the spliced length. The development capacities, including bond strength and deformation capacities, of a lap splice are related to many factors such as spliced length, transverse reinforcement, cover thickness, and concrete strength. Many test studies on the bond behavior of lap splices concerned four-point beam testing where the spliced bars were placed in the constant moment zone (Chun, 2015). Therefore, the two spliced bars are subjected to equal tensile force. Other test studies investigated the mechanics of the lap splice with short length (Cho and Pincheira, 2006; Kim et al., 2006). The full tensile strength of the spliced bars cannot be developed in a lap splice with short length and it does not meet the modern design specifications. Therefore, this study mainly discusses the lap splice with minimum spliced length that can transfer the full strength of the bars at the end of the spliced length. In other words, members will not be failing due to bond failure (such as cover splitting) of the splice in the plastic hinge zone.

Figure 7.

Double anchorage and force equilibrium within the spliced length.

The deformation mechanisms of shear walls with continuous and lap-spliced bars are not the same though their lateral load capacities are identical according to limit state analysis of flexural sections and even their load–deformation behavior is somewhat similar. Considering a lap splice in the plastic hinge region of a shear wall, its bond states are illustrated in Figure 7. The initial two flexural cracks are usually formed at the opposite ends of the lap splice with increasing lateral loads due to abrupt changes of reinforcement stiffness. According to the spliced length, the stress and strain states of two spliced bars are classified into three types that are stress and strain distributions with $l_{\min} < l_{lap} < 2 l_{\min}$ , $l_{lap} = l_{\min}$ , and $l_{lap} \geq 2 l_{\min}$ (where $l_{lap}$ is the lap-spliced length). The third spliced length seldom appears in practice. The force equilibrium over a distance x is shown in Figure 7 and can be expressed as follows

f_{s 10} A_{s 1} = f_{s 1} (x) A_{s 1} + f_{s 2} (l_{lap} - x) A_{s 2} + f_{c} (x) A_{c}

(28)

where $f_{s 10}$ is the stress of spliced bar 1 ( $f_{s 20}$ is used for bar 2) at the beginning of a lap splice, respectively; $f_{s 1} (x)$ and $f_{s 2} (l_{lap} - x)$ are the stresses of two spliced bars at distance x from the starting point of lap splice (for bar 2, the distance is $l_{lap} - x$ , they are at the same horizontal location); $f_{c} (x)$ is the concrete tensile stress; $A_{s 1}$ , $A_{s 2}$ , and $A_{c}$ are the areas of bar 1, bar 2, and concrete, respectively (usually $A_{s 1} = A_{s 2}$ ). The bond stresses $τ_{b}$ are still taken as $0.6 f_{c}'^{2 / 3}$ (MPa) at the pre-yield stage and $0.3 f_{c}'^{2 / 3}$ (MPa) at post-yield stage, respectively. To simplify the calculation, the concrete tensile stress is neglected in this study, due to its small contribution to the deformation.

For the time being, the lap splice is viewed as a double anchorage, then the stress states of the two spliced bars after yielding are illustrated in Figure 7 which is similar to the stress distribution of anchor bar in footing in Figure 6. The distributions of stresses along each spliced bar are also indicated by gray (bar 2) and magenta (bar 1) regions in Figure 8, respectively, when the reinforcement has entered the yield phase. Because of the moment gradient along the wall height, the stresses of the two spliced bars are usually not the same. However, for simplification of analysis, still assume $f_{s 10} = f_{s 20}$ due to the small spliced length compared with the actual wall height, and due to the stress concentration at the end of the lap splice. In this case, the elastic and plastic lengths of the two spliced bars are still denoted as $l_{sy}$ and $l_{sp}$ , respectively. It can be found that there is a blank area between the two regions. This means that only taking into account the anchorage bond stresses is not enough, because it does not satisfy forces equilibrium of equation (28). The residual stress $Δ f_{s}$ will be accommodated by the two spliced bars together. Assume that the stresses supported by bar 1 and bar 2 are the same, that is, $Δ f_{s} / 2$ . Therefore, the stress and strain distributions of spliced bars are divided into three parts with lengths of $l'_{sp}$ , $l_{sy 1}$ , and $l_{sy 2}$ , respectively, as illustrated in Figure 8. $l'_{sp}$ is an inelastic region which will be slightly larger than $l_{sp}$ , $l_{sy 1}$ and $l_{sy 2}$ are elastic regions. $l'_{sp}$ , $l_{sy 1}$ and $l_{sy 2}$ can be derived by geometrical relationship

\begin{matrix} l'_{sp} = \frac{l_{lap} - l_{sy}}{2} + \frac{(f_{s 1} (x) - f_{sy}) d_{b}}{8 τ_{be}} \\ l_{sy 1} = l_{lap} - l'_{sp} - l_{sy 2} \\ l_{sy 2} = l'_{sp} \end{matrix}

(29)

Figure 8.

Stress and strain states of two spliced bars: (a) normal strain and stress distributions $(l_{\min} < l_{lap} < 2 l_{\min})$ and force equilibrium at distance $x$ from the starting point of the splice, (b) critical states of strain and stress distributions $(l_{lap} = l_{\min})$ , and (c) $l_{lap} \geq 2 l_{\min}$ .

It can also be found that $l'_{sp}$ will slightly increase as the lap-spliced length increases till $l_{lap} = 2 l_{\min}$ . And $l'_{sp} = l_{sp}$ when $l_{lap} = l_{\min}$ . Then, the total bond-slip of the lap splice of each bar can be derived as follows

\begin{array}{l} s_{10} = \frac{1}{E_{s}} [(f_{s 1} (x) - f_{s y}) l_{s y 2} + (2 f_{s y} - f_{s 1} (x)) l_{s y 1}] \\ + \frac{1}{E_{s h}} [(f_{s 2} (x) - f_{s y}) {l^{'}}_{s p}] \end{array}

(30)

where $s_{10}$ is the bond-slip of bar 1. The bond-slip of bar 2, $s_{20}$ , can be calculated using the equation above by substituting $f_{s 1} (x)$ and $f_{s 2} (x)$ for $f_{s 2} (x)$ and $f_{s 1} (x)$ , respectively. In many cases $f_{s 1} (x)$ is approximately equal to $f_{s 2} (x)$ resulting in $s_{10} = s_{20}$ . Thus, the lateral deformation of a shear wall with lap-spliced bars, $Δ_{tlap}$ , can be calculated as follows

\begin{array}{l} Δ_{t l a p} = Δ_{f l a p} + Δ_{θ l a p} + Δ_{s l a p} = Δ_{f l a p} \\ + \frac{(s_{0} + s_{10}) H + s_{20} H^{'}}{L_{w} - y_{s c} - c} + Δ_{s l a p} \end{array}

(31)

where $Δ_{flap}$ , $Δ_{θ lap}$ , and $Δ_{slap}$ are the lateral deformations of the lap-spliced shear wall due to flexure, reinforcement slip, and shear, respectively, and $H' = H - l_{lap}$ . $Δ_{flap}$ can be calculated as

Δ_{flap} = 1 / 3 ϕ_{y} H'^{2} + (ϕ_{p} - ϕ_{y}) (L_{p} - l_{lap}) (H' - \frac{L_{p} - l_{lap}}{2})

(32)

Due to the complicated stress situation in the plastic lap-spliced region, $Δ_{s}$ of a shear wall with continuous bars will still be used for $Δ_{slap}$ , namely $Δ_{slap} = Δ_{s}$ . In fact, lap splices are usually adopted in the flexural member, so, the $Δ_{s} / Δ_{f}$ ratios are small for high-rise members. However, for low-rise members, $Δ_{slap}$ should be recalculated.

Verification of the proposed analytical procedure

The validity of the proposed analytical procedure for the evaluation of the deformational behavior of shear walls with continuous and lap-spliced bars is investigated in the present section. The test data are obtained from a series of experiments conducted in a previous study (Zhi et al., 2017) on cast-in-place (CIP) and precast (PC) concrete specimens with different dimensions, stirrups and reinforcement ratios, and steel bar connection ways. The information on the tested specimens is listed in Table 1. The geometry and reinforcement layout at the joint of several selected test units are shown in Figures 1 and 9. The test setup is drawn in Figure 10. The lateral loads were applied by a 1000 kN hydraulic servo control system that was mounted to a reaction wall.

Table 1.

List of the tested specimens.

Specimen	Dimensions(mm)	Length of boundarymember, l_b (mm)	Reinforcementratio of boundary member, ρ_b (%)	Reinforcementratio of totalcross section, ρ_t (%)	Length of lap splice
SW1-1	200 × 1600 × 3450	385	1.20	0.86	(continuous bars)
SW1-2 SW1-3	200 × 1600 × 3450	407	1.13	0.86	600 mm
SW1-4	200 × 1600 × 3450	407	1.13	0.86	600 mm
SW1-5 SW1-6	200 × 1600 × 3450	407	1.13	0.86	600 mm
SW2-1	200 × 1700 × 3650	415	1.94	1.15	(continuous bars)
SW2-2 SW2-3	200 × 1700 × 3650	415	1.94	1.15	600 mm

Figure 9.

Geometry and reinforcement layout at the joint of selected test units (all dimensions in mm).

Figure 10.

Test setup of the tested specimens.

All the specimens were tested under low reversed cyclic loading and the proposed analysis approach is based on a monotonic loading approach. The effects of the hysteretic behavior are neglected for the analysis of the specimens. And the effects of the unavoidable natural construction joints between the precast elements are also neglected. The dimensions were 3450 mm tall, 1600 mm wide, and 200 mm thick for specimens SW1-1–6 (Group 1) and 3650 mm tall, 1700 mm wide, and 200 mm thick for SW2-1–3 (Group 2). The axial loads for specimens in Group 1 were 600 kN and for specimens in Group 2 were 750 kN, respectively. It was kept constant till specimens were failure. PC specimens SW2-2 and SW2-3 had higher stirrup ratios than CIP specimens SW2-1. Inversely, CIP specimens SW1-1 had a higher stirrup ratio than PC specimens SW1-2 to SW1-6. The main failure mechanism for these specimens was flexural failure with reinforcement fracture and concrete crushing. The proposed analysis process that has been based on displacement control is implemented for these specimens. The estimated load–deformation (load-drift ratio) relationships compared with the tested results are illustrated in Figure 11, resulting in consistent correlations. Consequently, it might be concluded that the proposed analysis process can be used for the estimation of load–deformation behavior of RC shear walls with continuous or lap-spliced bar in plastic hinge zones based on a displacement-controlled calculation method.

Figure 11.

Experimental and analytical results of tested members SW1-1 to SW1-6 and SW2-1 to SW 2-3: (a) SW1-1, (b) SW1-2, (c) SW1-3, (d) SW1-4, (e) SW1-5, (f) SW1-6, (g) SW2-1, (h) SW2-2, (i)SW2-3.

For the analysis of specimens SW2-1–3, their footings were fixed to the strong ground with high strength fining twisted steel bars hoping to prevent the specimens from sliding horizontally. Due to no prestress applied to the footing by four high strength fining twisted steel bars with diameter of 32 mm, the footing were rotated slightly with the lateral loading like the upper specimen. Therefore, to consider the actual situation of experiment and predict the tested results accurately, the lateral deformation resulted from footing rotation and horizontal slip of SW2-1 were also considered in this analysis, which can be calculated by strain compatibility of the flexural foundation beam according to measured data of footing and section analysis. The compression region of the foundation beam can be approximately taken as a quarter of the beam length.

In the analysis process, the ultimate strains of confined concrete and reinforcement were taken as 0.008 and 0.03, respectively. The compression softening factor $β_{c}$ was assumed as constant that is 1.0 before yielding of the steel bars because in this case $β_{c}$ may be larger than 1.0. The calculated plastic hinge lengths $L_{p}$ of the continuous-bar specimens SW1-1 and SW2-1 were 806 and 867 mm, respectively, which are approximately half of the wall lengths. The principle tensile stress $f_{c 1}$ , the depth of compressing zone c, the rotated crack angle $θ$ , and the compression softening factor $β_{c}$ all decreased with an increment as increase in lateral deformation. The ratios of shear deformation $β_{s}$ to general flexural deformation $(Δ_{θ} + Δ_{f})$ were approximately 0.25 in CIP specimens with continuous bars and approximately 0.3 in PC specimens with lap-spliced bars, respectively, at the ultimate stage. This means that the flexural deformation of a shear wall with lap-spliced bars was reduced. This is because the lap splice in the plastic hinge region of the specimen decreased the plastic length resulting from the pair of spliced bars, although there were plastic regions in the vicinity of the ends of the lapped splice. This phenomenon will be more pronounced when the spliced length is large. The shear wall dissipating the energy in case of an earthquake depends on the development of material plasticity. Hence, the lap-spliced connection is not suggested to be used in the plastic hinge region. Finally, it should be noted that, although the calculated ratios of shear deformation to general flexural deformation were in a reasonable range in the cases of these shear walls, the components (flexure $Δ_{f}$ , bond-slip $Δ_{θ}$ , and shear $Δ_{s}$ ) of lateral displacement and the magnitude of the plastic hinge length still need to be verified with more detailed tested results in the future.

Conclusion

In this study, a procedure is proposed for evaluation of the load–deformation behavior of RC shear walls with continuous or lap-spliced bars in the plastic hinge zone under horizontal loading. Determinations of three components of the total lateral deformation that are deformations due to flexure, shear, and reinforcement slip (fixed end rotation) were implemented individually by considering their interaction. The flexural behavior was analyzed by conventional fiber section analysis. The shear response was obtained from MCFT. The influence of shear on flexural behavior was achieved by incorporating a compression softening factor based on the concrete tensile strength into the flexural analysis. The shear model included the impact of the flexural cross section strains and the plastic hinge length on the shear behavior.

In addition, an approach was proposed for analysis of the plastic hinge length based on fracture energies of the materials. The energy of the lateral load applied in the member, when the member is in the post-yield stage, is assumed to be equal to the sum of the fracture energies of materials, including reinforcement and concrete in the plastic hinge region. Thus, the plastic hinge length is derived through the energy balance between the energies caused by the external lateral load and dissipated as inside fracture energies of the materials.

For the shear walls with lap-spliced bars in the plastic hinge region, a simple way to deal with the lap splice was proposed in this study due to its complicated behavior and mechanism. The stress and strain states of a lap splice with different spliced length were analyzed by force equilibrium with an average bond stress model. The established equations regarding bond-slip of the lap splice with minimum spliced length were used to calculate the total response of a lap-spliced shear wall. The validity of proposed analysis procedure was confirmed by comparing the analytical results with previously conducted experiments.

Footnotes

Acknowledgements

The authors would like to thank the anonymous reviews for their constructive comments which help the authors in depth to improve the quality of the article.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key Research and Development Program of China (2016YFC 0701703) and National Natural Science Foundation of China (51678136).

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