Experimental study and numerical model calibration of full-scale superimposed reinforced concrete walls with I-shaped cross sections

Abstract

The superimposed reinforced concrete wall in which both the walls and slabs are semi-precast superimposed reinforced concrete components has been widely used to construct high-rise residential buildings in some seismic regions of China. This article aims to investigate the seismic performance and reveal the inherent damage mechanism of this wall. Quasi-static tests of two full-scale superimposed reinforced concrete walls with I-shaped cross sections, consisting of the walls in orthogonal directions and two T-shaped cast-in-place boundary elements, were conducted. Through the test, the behavior of the horizontal joints between the wall panels and the foundation; the behavior of the vertical connections between the wall panels of orthogonal direction; the reliability of the connections between precast and cast-in-place concrete; and the lateral load, deformation, and energy dissipation capacities of the specimens are evaluated. In addition, a refined numerical model based on the multi-spring model was adopted to assess the seismic performance of the superimposed reinforced concrete walls with I-shaped cross sections. The reliability of this model was validated through comparison with the experimental data. This study offers valuable experimental data and numerical model references for future seismic performance assessments of superimposed reinforced concrete wall structures.

Keywords

I-shaped cross section numerical analysis quasi-static test semi-precast superimposed reinforced concrete wall

Introduction

In recent years, the total area of the completed urban residential buildings in China is growing by up to 0.6 billion square meters each year (Tian et al., 2015). Most of these residential buildings are high-rise reinforced concrete (RC) shear wall structures and are constructed using a conventional cast-in-place manner. However, the labor cost has recently increased significantly in China, and energy conservation as well as environmental protection has attracted increasing attention. As a result, research on the precast concrete structures has become increasingly popular (Jiang et al., 2011; Lu et al., 2008; Qian et al., 2011; Sun et al., 2015; Xiao et al., 2015; Xue et al., 2011).

Among these structures, the superimposed RC wall structure in which both the walls and slabs are semi-precast superimposed RC components has been widely used to construct high-rise residential buildings in some seismic regions of China. A sketch and a construction process of this type of structure are presented in Figures 1 and 2, respectively. The superimposed RC wall is composed of two precast side panels and cast-in-place concrete in the cavity, while the superimposed RC slab is composed of one precast bottom slab and the upper cast-in-place surface. Truss bars comprising one upper chord bar, two lower chord bars, and continuous inclined web bars are prefabricated within the precast panels and slabs to connect the precast and cast-in-place concrete together and enhance the shear resistance at the connection interface. In this structure, the superimposed components are connected by reinforcing bars passing through the joints, such as joints between walls or walls and foundations, and those between walls and slabs (Figure 1(b)). At the edge and the intersection part of walls in orthogonal directions, the boundary elements are typically constructed using cast-in-place concrete to increase the integrity of the structure and ensure that the seismic behavior of the structure is similar to that of the cast-in-place walls (Figure 2(b)).

Figure 1.

Sketch of the superimposed RC wall structures: (a) three-dimensional sketch and (b) two-dimensional sketch.

Figure 2.

Construction of the superimposed RC wall: (a) erection of the wall panel and (b) cast-in-place concrete boundary elements.

This structure is usually considered as a “monolithic” precast concrete structure, which means that the corresponding seismic performance and seismic design method are considered to be similar to their cast-in-place RC counterpart. However, further detailed research is required to verify whether the superimposed RC wall structure could achieve the same seismic performance as cast-in-place RC wall structure. A literature review indicates that considerable research on “monolithic” precast RC walls has been conducted for the conventional fully precast walls (Clough et al., 1989; Holden et al., 2003; Kang et al., 2013; Smith et al., 2013; Soudki et al., 1995). In contrast, research on the seismic performance of a superimposed RC wall structure is rarely reported, because this structure was mostly adopted to construct low-rise and mid-rise buildings in the non-seismic regions of some European countries (such as Germany and the United Kingdom). After this type of structure was introduced to the seismic regions of China, the authors of this article and a number of researchers have conducted some seismic experimental studies (Chong et al., 2010; Wang et al., 2012; Xiao et al., 2015); however, most of these were limited to single wall panels. Unlike single wall panels, in real RC wall structures, walls usually have transverse components attached as the flanges, forming T-shaped or I-shaped cross-section shear walls. Research on this type of component, which is particularly valuable for further understanding of the seismic performance of superimposed RC wall high-rise structures, is currently unavailable.

As described above, this research conducts quasi-static tests of two full-scale superimposed RC walls with I-shaped cross sections to investigate the corresponding seismic performance, including (1) the behavior of the horizontal joints between the wall panels and the foundation; (2) the behavior of the vertical connections between wall panels of orthogonal directions; and (3) the reliability of the connections between precast and cast-in-place concrete in the superimposed components. Note that superimposed slabs are fabricated on the top of the walls to better replicate the real engineering practice; in addition, the behavior of the connections between walls and slabs is also evaluated. The failure mode, lateral load bearing capacity, deformation capacity, and energy dissipation capacity of superimposed shear walls with I-shaped cross sections are obtained and compared to those of the cast-in-place shear walls. In addition, numerical simulation of these specimens is also performed using the multi-spring (MS) model in CANNY; good agreement is achieved between the experimental data and the simulation results, and the determination of the associated plastic hinge length is calibrated and recommended.

Quasi-static experiments

Design of full-scale specimens

To accurately understand the seismic behavior of superimposed wall in detail, full-scale test specimens were adopted here. The tested superimposed RC walls with I-shaped cross sections are composed of one web wall and two transverse flange walls. The dimensions of two full-scale specimens, named as SW1 and SW2, are identical, as shown in Figure 3. Reinforcement details of the specimens are also presented in Figure 3. The white blank regions represent the cast-in-place parts, while the grey filled regions represent the precast parts. Note that the area of protruding reinforcement bars is almost equal to that of the longitudinal reinforcement bars in the precast panels, which is consistent with engineering practice.

Figure 3.

Dimensions and reinforcement details of the specimens (mm): (a) elevation of the specimen, (b) details of the foundation, and (c) details of the 1-1 section.

The lengths of web wall and width of flange wall are 2000 and 1800 mm, respectively. The aspect ratio of the specimens is A_r = H_w/l_w = 1.97, where H_w = 3940 mm is calculated from the base of the wall panel to the loading point of the lateral force, and l_w = 2000 mm is the length of the web wall. The thicknesses of the web and flange wall are both 200 mm, including a 100-mm cast-in-place concrete layer and two side precast layers with a thickness of 50 mm.

To improve the deformation and energy dissipation capacities, the boundary elements should be constructed at both ends of the shear walls according to the Chinese Code for Seismic Design of Buildings (GB50011-2010, 2010). Regarding the walls with attached transverse walls, the boundary elements are usually T-shaped columns, as shown in Figure 3(c). Specifically, the boundary elements are classified into two types in the code, namely, the confining and ordinary boundary elements, which are applicable to the walls with higher and lower design axial loads, respectively. In this study, the T-shaped columns, constructed using cast-in-place concrete as previously mentioned, are designed as the confining boundary elements to achieve a better seismic performance. It is well acknowledged that the ductility of shear walls can be substantially improved if the boundary element is well confined by stirrups; this improvement is attributed to the fact that the confinement effect provided by the stirrups can efficiently improve the ductility of concrete. In this study, different stirrup ratios (i.e. 1.57% for SW1 and 2.27% for SW2) in the boundary columns are adopted to investigate the confinement effect.

Note that the precast wall and slab panels are both constructed in Germany, because the product line of precast panels in China was not available during the study. The concrete strength class is C35/45 according to Deutsche Norm of Plain, Reinforced and Prestressed Concrete Structures (DIN 1045-1, 2008). The concrete with a compressive strength class of C30 according to Chinese Code for Design of Concrete Structures (GB50010-2010, 2010) is adopted for all the cast-in-place parts. The compressive strength of each type of concrete was measured through the cube tests of three 150 mm × 150 mm × 150 mm specimens. The strength value of every specimen (f_{cu, 1}, f_{cu, 2}, and f_{cu, 3}), the mean values (f_{cu, m}), and corresponding standard deviations (σ) are listed in Table 1. Three specimens for each type of reinforcing steel bar were adopted for the tensile tests, and the characteristic properties are summarized in Table 2.

Table 1.

Compressive strength of the concrete (MPa).

Concrete type	f _{cu, 1}	f _{cu, 2}	f _{cu, 3}	f _{cu, m}	σ
Cast-in-place concrete, C30	26.0	22.8	29.2	26.0	4.5
Precast concrete, C35/45	47.5	48.0	47.5	47.7	0.4

Table 2.

Mechanical properties of the reinforcing steels.

Diameter d (mm)	Yield strength f_y (MPa)	Ultimate strength f_u (MPa)	Elongation ratio A (%)
8	398.1	495.6	34.5
10	407.6	499.4	27.0

Test setup, loading procedures, and instrumentation

The test setup of the superimposed walls is illustrated in Figure 4. Because full scale is adopted for both specimens, the design axial loads at the top of the specimens would be rather large and beyond the capacity of the loading actuator. The axial load was therefore neglected during the test. It should be noted that neglecting the axial load is on the safe side as far as the shear and flexural bearing capability of the specimens being concerned. In fact, shear slip along the horizontal joints between the wall panels and the foundations were observed during the test, and the lack of axial force is considered to be one of the main reasons. This will be discussed in detail in section “Shear slip.” The cyclic lateral load was applied by a horizontal actuator as shown in Figure 4. Before the specimens yielded, the lateral load was force controlled and one cycle was performed at each force level. Five levels were considered in this phase: 20%, 40%, 60%, 80%, and 100% times the predicted yield load V_y of the specimen. After yielding of the specimens, the loading process was controlled by displacement and three cycles were repeated at each displacement level. The displacement loading amplitudes had an increment of Δ_y, which is the predicted yield displacement of the specimen. Take SW1 for an example, the loading protocols are shown in Figure 5. Both specimens were designed to be continuously loaded until the lateral load bearing capacity decreases to 85% of the peak load, according to the Chinese Specification for Seismic Testing Methods of Building (JGJ 101-96, 1996). Besides, the actuator’s pushing and pulling directions are defined as the positive and negative directions, respectively, in the article.

Figure 4.

Test setup of the superimposed walls: (a) diagram of the experimental device and (b) photograph of the experimental device.

Figure 5.

Loading protocols of SW1.

The lateral load was monitored through the load cell equipped in the actuator, and the horizontal displacement at the top of the specimens was measured using a linear variable differential transformer (LVDT). Several vertical LVDTs were installed at the horizontal joints to measure the opening width of the joint, and two lateral LVDTs were also adopted to measure the shear slip between the wall and foundation. Electrical resistance strain gauges were placed on the longitudinal reinforcements at the horizontal joints between walls and foundations.

Test observations and analysis of the experimental data

Failure mode

Because only the stirrup ratios in the boundary element are different for SW1 and SW2, a minor difference was observed between the response of SW1 and SW2. Therefore, only the experimental phenomena of SW1 are presented here in detail.

When the lateral load reached 150 kN (i.e. approximately 25% of the maximum load), the first horizontal crack appeared at the wall–foundation joint, where the precast and cast-in-place concrete connected with a relatively lower tension strength (Figure 6). Next, the crack rapidly extended from the outermost fiber in tension toward the neutral axis of the wall. At the following cycles, the cracks gradually propagated beyond the center of the wall panel and intersected with those developed in previous cycles. As the test continued, because the horizontal joint between the panel and the foundation had less reinforcements compared to the adjacent sections and therefore acted as the weakest part of the specimen, the wall exhibited rocking characteristic, that is, the wall behaved essentially as a rigid body dominated by gap opening at the horizontal joint. At further cycles, although new flexural cracks formed gradually within a range from the wall bottom to the height of nearly 2100 mm, and several cracks developed into inclined flexural-shear cracks (Figure 7), it is noticeable that the width of all the new cracks was quite small and the gap formed at the wall–foundation joint was the only major crack during the test, which is quite different from the cast-in-place RC walls. The outermost longitudinal rebars in the boundary elements first yielded in tension at 300 kN, which was approximately 51% of the maximum load (according drift ratio equaled to 0.12%). When the drift ratio of the specimen reached 0.75%, spalling of the concrete cover first occurred at the edge of the T-shaped boundary element at the wall bottom, as shown in Figure 8. When the drift ratio reached 1.25%, the outermost longitudinal bars near the wall–foundation joint, which had buckled in the previous cycles, began to fracture as shown in Figure 9. As the lateral displacement increased, concrete crushing proceeded to the entire cast-in-place concrete boundary element (Figure 10). Meanwhile, two precast side panels basically remained undamaged due to the higher concrete strength, as mentioned above. When the lateral load decreased to approximately 85% of the maximum load, the correspondingly drift ratio in the negative and positive directions reached 1.52% and 1.67%, respectively. At this point, the test was terminated. Note that the connection of the superimposed walls and slabs performed well, and negligible damage was observed.

Figure 6.

First flexural crack occurred at the horizontal joint between wall panel and foundation.

Figure 7.

Crack pattern of SW1 after test (mm).

Figure 8.

Spalling of the concrete cover at the edge of the boundary element.

Figure 9.

Gap opening and fracture of the protruding reinforcement bar.

Figure 10.

Concrete crushing of the entire boundary element.

Figure 7 presents the crack distribution and damage modes of SW1 at the end of the test. The dashed lines represent the boundary lines between the precast and the cast-in-place concrete. The flexural-shear cracks extended smoothly from the cast-in-place boundary elements to the precast panels, and neither obvious cracks nor shear slip was observed at the interface between precast and cast-in-place concrete, indicating that these two parts can work together as a single entity. Note that inelastic deformation concentrated near the gap between the wall and foundation, leading to a relatively small concrete crushing region of approximately 200–300 mm height from the bottom of the walls.

Deformation mode and opening width of the gap

As shown in Figure 11, three components contribute to the lateral deformation of the specimens: opening of the gap at wall–foundation joint, shear slip at the wall–foundation joint, and deformation of the wall itself. The contribution of these components varied at different stages of the test.

Figure 11.

Deformation mode of the specimens.

The relationship curves of the maximum opening width of the gap and the top lateral displacement at each cycle are presented in Figure 12. It can be found that the curves exhibit an obvious bilinear characteristic and therefore can be divided into two stages. At the first stage (i.e. the lateral displacement was less than approximately 20 mm), the maximum opening width was relatively small. After the protruding reinforcement steels yielded, the curves approached to the second stage. The opening of the gap gradually dominated the lateral deformation of the specimens, and the maximum width of the gap increased much faster than during the first stage.

Figure 12.

Relationship curves of the maximum opening width of the gap w_max and lateral top displacement Δ at each cycle.

Shear slip

At the wall–foundation joint, a certain extent of shear slip was observed, as shown in Figure 13. The relationship curve of the maximum shear slip and top lateral displacement at each cycle of SW1, which also exhibits a bilinear characteristic, is presented in Figure 14. The top lateral force and shear slip hysteresis curve of SW1 are shown in Figure 15. The shear capacity and lateral stiffness of the wall were relatively high in the elastic stage because the shear force applied on the wall–foundation joint was efficiently resisted by the friction force on the compressive region of the contact surface. After yielding of the protruding reinforcement, the opening width of the joint increased gradually. When reloading started, the gap could not be closed by the self-weight, and the lateral shear force was mainly resisted by the reinforcing bars, inducing a low resistance and reduced stiffness. Furthermore, an increase in both stiffness and strength was observed until the closing of the gap, when the friction mechanism of shear transfer was enabled.

Figure 13.

Lateral shear slip of the flange wall.

Figure 14.

Shear slip Δ_s–top displacement Δ relation of SW1.

Figure 15.

Lateral load F–shear slip Δ_s hysteresis loop of SW1.

In addition to the lack of axial load, the shape of wall cross section also has a certain impact on the shear slip. Figure 16 shows the effective friction area in the compressive region of a single wall panel and an I-shaped wall. c and A_f represent the effective height of the compressive region and the effective friction area, respectively; b_w is the thickness of the wall or the web wall; and b is the thickness of the flange wall. Under identical bending moment and axial force, the I-shaped wall will have a much less effective height of compressive region than the single wall panel, especially when the width of flange is quite large. Accordingly, the I-shaped wall will have a much smaller effective friction area and lower friction force to resist the shear slip.

Figure 16.

Effective friction area in the compressive region of the wall.

According to ACI ITG-5.2-09, horizontal shear strength V_n of the wall–foundation interface could be calculated as µC, where C is the compressive force acting on the concrete at the interface, µ is the coefficient of friction, and shall be taken as 0.5 (ACI ITG-5.2-09, 2009). It is obvious that increasing the axial load will improve the compressive force on the concrete at the interface C, and thus lead to a higher horizontal shear strength. Besides, the effect of the cross-section shape of the wall on the horizontal shear strength will be analyzed as below.

Based on the equilibrium of the wall–foundation interface section without axial load

M = V_{n} H_{w} = C \cdot γ l_{w}

(1)

where M is the bending moment of the wall–foundation interface section; H_w and l_w are the height and length of the wall, respectively; and γ is the ratio of the lever arm length to the section length.

Consequently, for the single wall panels with rectangle cross sections, shear capacity of the wall–foundation interface could be checked through the following equation

V_{n} = \frac{γ l_{w}}{H_{w}} C \leq μ C

(2)

Take a single wall panel with the same geometric parameters as the specimens in the article for example, l_w and H_w are 2000 and 3940 mm, respectively, γ can be taken as 0.85 approximately, and µ is 0.5. It is obvious that requirement of equation (2) could be fulfilled, indicating that for the single wall panels with a relative large aspect ratio, the shear strength requirement of the wall–foundation interface could be satisfied even without axial loads, but the situation is different for the I-shaped walls. According to Figure 16, the effective friction area and the compressive force C are much smaller for the I-shaped walls; therefore, shear strength of the specimens in this study could not fulfill the requirement of equation (2), which can explain the occurrence of an obvious shear slip during the test.

Based on the above discussion, it can be concluded that the small effective friction area of wall with I-shaped cross section and lack of axial load led to the shear slip of specimens. However, it should be emphasized that although shear slip between the wall and foundation represents a source of energy dissipation and force isolation, it should not be considered as a reliable resistance mechanism, because the accumulated unrestrained slip could result in enough eccentricity to threaten the stability and integrity of a building. Further research (e.g. research on reliable design or necessary construction method) is required to address and avoid the shear slip.

Lateral load–displacement response

The lateral load versus displacement relationships of the specimens are presented in Figure 17. A certain extent of pinching phenomenon is observed after the yielding of protruding reinforcement, and this phenomenon becomes more significant as the peak displacement increases. This is attributed to the shear slip at the horizontal joints, the bond-slip of the longitudinal reinforcing bars, and the shear deformation of the wall panels. Comparison of the load–displacement skeleton curves of two specimens is presented in Figure 18. The curves of these two specimens are nearly identical, except that SW2 reaches a slightly higher ultimate displacement than SW1. This is because SW2 is designed with a higher stirrup ratio in the boundary element.

Figure 17.

Lateral load F–displacement Δ hysteresis curves: (a) SW1 and (b) SW2.

Figure 18.

Lateral load F–displacement Δ skeleton curves.

Strength and displacement ductility

To investigate whether the superimposed RC walls have as reliable bearing and deformation capacities as the cast-in-place ones, the corresponding response characteristic parameters of two specimens are calculated and listed in Table 3. F_y and Δ_y, which are calculated using the energy equivalence method (Mahin and Bertero, 1976), are the yield load and associated displacement, respectively. F_max is the peak lateral load, F_u is defined as 85% of F_max, and Δ_u is the displacement corresponding to F_u.

Table 3.

Test results.

Specimens	Directions	F_y (kN)	F_max (kN)	F_u (kN)	F_pre (kN)	F_max/F_pre	Δ_y (mm)	Δ_u(mm)	µ = Δ_u/Δ_y	θ_u = Δ_u/H_w
SW1	Positive	510.0	607.6	516.5	547.5	1.11	19.2	60.7	3.16	1/65
SW1	Negative	480.0	583.5	496.0		1.07	17.3	60.5	3.48	1/65
SW2	Positive	480.0	582.7	495.3		1.06	15.4	60.7	3.94	1/65
SW2	Negative	480.0	586.3	498.4		1.07	17.1	65.4	3.82	1/60

To compare the flexural strength of cast-in-place and superimposed RC walls, the predicted peak lateral load F_pre is calculated according to the method for the cast-in-place walls and then compared with F_max in Table 3. Note that the tested strength of materials was utilized during the calculation and the confinement effect provided by the stirrups was not considered. The F_max/F_pre ratios for the specimens are all greater than 1.0, which indicates that the specimens have reliable flexural bearing capacity, and the calculation method for the flexural strength of cast-in-place RC walls is also applicable for the superimposed RC walls.

The displacement ductility ratio µ_Δ = Δ_u/Δ_y and ultimate drift ratio θ_u = Δ_u/H_w of the specimens are also presented in Table 3. The ductility ratios of the two specimens vary from 3.16 to 3.94. SW2 exhibits a higher ductility than SW1 due to a higher stirrup reinforcement ratio in the boundary element. The rocking behavior of the superimposed walls leads to a relatively small inelastic region, and thus decreases the deformation capability (i.e. ultimate drift ratio). Despite this fact, θ_u of these specimens are both higher than 1/100, which is the limitation of maximum drift ratio under rare earthquake (i.e. 2%–3% probabilities of exceedance in 50 years) in Chinese Code for Seismic Design of Buildings (GB50011-2010, 2010).

Based on the above discussion, it can be concluded that the lateral load bearing capacity of superimposed RC walls is nearly identical with that of cast-in-place ones, and the corresponding deformation capacity can meet the requirements of Chinese Code, although it is relatively smaller than that of cast-in-place ones.

Numerical analysis of the quasi-static tests

The damage modes of superimposed RC walls observed in the tests exhibit a significant difference compared to those of cast-in-place RC walls. In brief, the rocking characteristics of the superimposed RC wall led to a smaller plastic region, and most damage was observed near the horizontal joints between the walls and the foundations. As a result, to simulate such a failure mode of superimposed RC walls, an appropriate numerical model with acceptable accuracy and computational efficiency is required. Currently, most researchers adopt five types of numerical models to simulate RC shear walls: (1) the concentrated plasticity model (Kabeyasawa et al., 1983); (2) the distributed plasticity fiber beam model (Martinelli and Filippou, 2009); (3) the multi-layered shell model (Lu et al., 2013a, 2013b, 2015); (4) the multi-vertical-line element model (Orakcal et al., 2004; Vulcano et al., 1988; Wallace, 2007); and (5) the three-dimensional (3D) solid model (CEB-FIP, 2008). Considerable efforts have focused on validating and calibrating the fiber beam model and the multi-layered shell model for cast-in-place RC walls; however, the rocking characteristic cannot be conveniently considered using these two models because the plane assumption is adopted. Regarding the 3D solid model, it is capable of capturing the complicated mechanical behavior occurring at the horizontal joints through the application of a contact element (Chong et al., 2014). Nevertheless, this type of model usually presents relatively low computational efficiency and stability, which are critical issues for investigating the seismic performance of superimposed RC wall structures. As an alternative, concentrated plasticity models with plastic hinges are simple and can provide reasonably good estimates of global and average local responses. This type of model may have the potential to be more beneficial for this research.

Numerical model

Among the concentrated plasticity models, the multi-axial spring model in CANNY (Li and Otani, 1993) is proven to be capable of simulating the flexural interaction among bi-directional bending moments and an axial load (Ye et al., 2002). The MS model comprises a number of uniaxial nonlinear springs representing the rebars and concrete to account for the interaction of axial-flexural responses, as shown in Figure 19. Instead of using one MS element with I-shaped cross section, which will obviously overestimate the stiffness and strength of the specimens, the web wall and two flange walls are simulated with three single walls. Two end nodes of the web wall on the top side (nodes 3 and 4 in Figure 20) are defined as “mid-nodes” of the flange walls, and the program automatically treats the displacement components of wall mid-nodes in the wall plane depending on the deformation of wall side to conform the assumption of plane-section deformation. The area contributing to corresponding displacement of the I-shaped cross-section wall is shown in Figure 21. The in-plane bending moment of the wall is composed of the moment resisted by the web wall and that conformed by axial tension and compression force of the two flange walls. The in-plane shear force is mainly resisted by the web wall, and the out-of-plane bending moment and shear force are resisted by the two flange walls.

Figure 19.

Multi-axial spring element analysis model.

Figure 20.

Cross section and 3D frame of the analysis model.

Figure 21.

Area contributed to corresponding displacement: (a) in-plane bending deformation, (b) in-plane shear deformation, and (c) out-of-plane bending deformation.

Within one MS wall element, each reinforcing bar is represented by one steel spring, and the concrete is represented by 200 concrete springs. The skeleton curve of the abovementioned uniaxial nonlinear springs for concrete and rebars is calculated using the following equations

K_{0}^{i} = \frac{E_{i} A_{i}}{p_{z}}

(3)

F_{c} = σ_{c} A_{i}, D_{c} = ε_{c} p_{z}

(4)

F_{s} = σ_{s} A_{i}, D_{s} = ε_{s} p_{z}

(5)

where $K_{0}^{i}$ and A_i are the initial axial stiffness and the area of the ith spring, respectively. E_i represents the elastic modulus of the uniaxial material defined ith spring. The axial force–displacement curve of uniaxial concrete springs (i.e. F_c–D_c curve) and rebar springs (i.e. F_s–D_s curve) are determined from the corresponding material constitutive models (i.e. σ_c–ε_c curve for concrete and σ_s–ε_s curve for steel rebar). The steel springs and concrete springs are simulated with steel material model SR3 and concrete model CS3 in CANNY, respectively, and the constitutive relationships are shown in Figure 22(a) and (b). Because the horizontal joint between foundation and the superimposed shear walls provides negligible tension resistance, the concrete behavior under tension is herein neglected. The key parameters of the spring constitutive relationship are listed in Tables 4 and 5. For the reinforcing steel, f_sy and ε_sy are the yielding stress and strain, respectively; E_s is the elastic modulus; γ and µ are parameters to define the skeleton curve before yielding; and β is the ratio of the post-yielding modulus to E_s (Figure 22(a)). For the concrete, f_c and ε_c are the axial compressive strength and the corresponding strain, respectively; E_c is the elastic modulus; ν is the parameter to define the ascending stage of the skeleton; and λ and µ are parameters to determine the ultimate point (Figure 22(b)). The confined model proposed by Mander et al. (1988) is adopted here to calculate the characteristic parameters of the confined concrete in the boundary element.

Figure 22.

Constitutive relationships adopted in the analysis: (a) reinforcing steel spring, (b) concrete spring, and (c) shear spring.

Table 4.

Key parameters of the reinforcing steels constitutive relationship.

Reinforcing steel type	E_s (MPa)	f_sy (f′_sy) (MPa)	d_sy (d′_sy) (µε)	µ (µ′)	γ (γ′)	β (β′)
D8	2.0e5	398.1	1990	1.0	1.0	0.03
D10	2.0e5	407.6	2038	1.0	1.0	0.03

Table 5.

Key parameters of the concrete constitutive relationship.

Concrete type	E_c (MPa)	f_c (MPa)	ε_c (µε)	ν	λ	µ
C30, unconfined	3.0e4	19.8	1470	0.3	0.2	4.8
C30, confined	3.0e4	20.8	2500	0.3	0.5	8.0
C35/45, unconfined	3.33e4	36.3	1730	0.3	0.15	3.5

In equations (3) to (5), p_z represents the length of the plastic region. Note that the simulation results are highly affected by the length of plastic region adopted in the MS model. The experimental results indicate that the deformation mode of the superimposed RC wall exhibited significant characteristics of in-plane rocking, leading to a relatively small length of plastic hinge, which is quite similar to the unbonded post-tensioned precast shear walls. As a result, 6% of the specimen height H_w recommended by ATC ITG-5.2-09 for the unbonded post-tensioned precast shear walls (ACI ITG-5.2-09, 2009) is also adopted for the superimposed RC wall.

In addition, uniaxial nonlinear springs, which are uncoupled with the abovementioned uniaxial nonlinear concrete and steel springs, are adopted to represent the in-plane shear behavior of the three wall panels. An original-oriented constitutive model with tri-linear skeleton curve was adopted for the shear springs, as shown in Figure 22(c).The values of the constitutive model parameters, as indicated in Table 6, are all calculated by the program CANNY automatically based on the empirical equations in ACI318-2005 (2005). In Table 6, K is the initial shear stiffness, F_c is the crack strength, F_y is the yielding strength, α and β are the ratio of the post-cracking stiffness and post-yielding stiffness to K, respectively.

Table 6.

Key parameters of the shear spring constitutive relationship.

K (kN)	F_c (kN)	F_y (kN)	α	β
4.12e6	357.5	1122.6	0.16	0.001

Model calibration and numerical results

Based on the abovementioned numerical model and associated parameters determining method, numerical simulation of two superimposed RC walls is conducted. Because the improvement of concrete performance from confinement exhibits negligible differences between these specimens, the simulation results of SW1 and SW2 are almost identical, in agreement with the test results. The comparison between the experimental data and simulation result of SW1 is then presented in Figure 23, including the lateral load–displacement hysteretic curve and the skeleton curve. To facilitate the comparison of characteristic results, the experimental top displacement is recalculated through deducting the relative shear slip at the horizontal joint. Good agreement between the experimental data and the simulation is achieved for SW1, in particular, for the hysteresis loop shape and the residual displacement, thus validating the feasibility and reliability of the MS model in simulating the I-shaped superimposed RC wall with obvious in-plane rocking characteristics. Besides, the analysis results also indicated that the shear springs cracked but did not reach the capacity; therefore, shear deformation accounted for minor part of the overall response, which was consistent with the observation during the test.

Figure 23.

Comparison of the analysis and test results of SW1: (a) lateral load–displacement skeleton curve and (b) lateral load–displacement hysteresis curves.

To investigate the influence of the plastic region length on the calculation results, a plastic hinge length calculated using equation (4) proposed by Paulay and Priestley (1992) for cast-in-place RC members is also adopted in the MS model

p_{z} = 0.08 L + 0.022 d_{b} f_{y}

(6)

where L is the shear span of the shear wall, and d_b and f_y are the diameter and yielding strength of the longitudinal reinforcement steels, respectively.

A comparison among the simulation results using different plastic hinge lengths is presented in Figure 24. It can be found that the plastic hinge length of the MS model has an obvious effect on the displacement capacity and ductility of the specimen, and 0.06 H_w is more suitable for the superimposed RC wall.

Figure 24.

Comparison of the simulation results of different plastic hinge lengths.

Conclusion

To investigate the seismic performance of superimposed RC walls with I-shaped cross sections, quasi-static tests of two full-scale specimens were conducted. The failure mode, lateral load bearing capacity, deformation capacity, and energy dissipation capacity of this specimen were obtained and compared to those of cast-in-place RC walls. Specifically, (1) the failure mode of the superimposed RC wall exhibits a significant in-plane rocking characteristic. Inelastic deformation is primarily concentrated near the wall–foundation joint, while damage on the wall panel is relatively slight, leading to a small plastic region compared to cast-in-place walls. (2) The lateral load bearing capacity is nearly identical with that of cast-in-place walls; the calculation method for the flexural strength of cast-in-place RC walls is also applicable for the superimposed RC walls. (3) The deformation capacity can meet the requirements of the Chinese Code, although it is relatively smaller than that of cast-in-place RC walls. (4) A notable behavior is that shear slip appears at wall–foundation joints due to the small effective friction area and lack of axial force. This shear slip should be well addressed and avoided through reliable design or necessary configuration method. (5) Neither obvious crack nor shear slip was observed at the interface between precast and cast-in-place concrete, indicating that these two parts can work together as a single entity. (6) Stable energy dissipation capacity is observed.

In addition, a refined numerical model based on the MS model is recommended and calibrated to assess the seismic performance of the superimposed RC walls of I-shaped cross sections. Good agreement is achieved between the experimental data and the simulation results. The MS model associated with a 6% specimen height plastic hinge is recommended for this type of shear wall. The research outcome of this study will provide valuable experimental data and a reference numerical model for future research on the seismic performance of superimposed RC wall structures.

Footnotes

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: The authors are grateful for the financial support received from the National Natural Science Foundation of China (Grant Nos 51278519 and 50908071).

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