Numerical modeling of flanged RC shear walls considering shear deformation and coupled flexural-shear effects

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Department of Civil Engineering, Kunming University of Science and Technology, Kunming 650500, P. R. China)

Abstract: A new flexibility-based nonlinear finite element model that considers shear deformation and coupled flexural-shear effects is proposed in this article. Cyclic loading tests of T-shaped and L-shaped shear wall specimens were conducted to verify the validity of the proposed model. All specimens exhibited a flexural failure mode characterized by the crushing of the concrete and the buckling of the reinforcement at the free web boundary. Closer stirrups and longer confined boundary elements should be used in the free web end to prevent premature failure when compressed. The seismic design of the boundary element at the web-flange junction could be relaxed, as no concrete spalling was observed at the web-flange junction. The ductility decreased as the shear span ratio decreased. The specimens exhibited higher strength and stiffness but lower ductility when the flange was in tension. Numerical simulation of the cyclic loading test of the RC flanged shear wall was conducted based on the proposed model, demonstrating that the model efficiently simulates the nonlinear response of RC flanged shear walls, as proved by satisfactory agreement between the analytical results and the test results.

Keywords:shear wall; quasi-static test; fiber model; shear deformation；coupled flexural-shear effects

1 Introduction

Reinforced concrete walls are effective in resisting lateral loads placed on high-rise buildings[1]. They provide enough strength and deformation capacity to meet the demands of strong earthquake ground motions[2-4]A great deal of research has been performed to investigate the seismic behavior of RC walls[5-8]. Previous tests have been conducted on either rectangular shear walls or flanged shear walls with ordinary strength reinforcement. However, this paper focuses on the experimental study of the seismic behavior of flange RC shear walls with high strength stirrups. Six flanged RC shear wall specimens with high strength stirrups were tested to failure under cyclic loading. The effects of the axial load ratio, aspect ratio, and confinement on the failure mode, ductility capacity, hysteretic behavior, and energy dissipating capacity were investigated.

Simulation of the nonlinear response of RC shear walls requires the use of a reliable computational model. Two-dimensional continuum plane stress or shell elements are usually used for this purpose, but such elements are computationally expensive. Previous studies have shown that beam-column elements can be used to simulate the nonlinear behavior of RC shear walls[9-10]In regards to this, much research has emerged in recent decades in the field of stiffness-based fiber beam-column elements[11-12]. So far, most stiffness-based element models become less accurate in highly nonlinear situations, since the displacement field is approximated through the assignment of the displacement interpolation function. In most cases, cubic Hermitian polynomials are used for the displacement interpolation functions. Simple numerical examples presented indicate that the conventional displacement formulation is unable to establish solutions associated with softening behavior, when used at the section or member level. This problem can be solved with the use of more elements per member; however, this leads to much less efficiency. Most efforts in the nonlinear analysis of RC structures have demonstrated that flexibility-based element methods offer greater accuracy and computational efficiency[13], since the element formulations satisfy the equilibrium of the bending moment, the axial force and the shear force along the element[14-15].

Shear deformation accounts for a large portion of the total deformation of flanged RC shear walls under lateral loading. Flexibility-based finite element methods ignore the effect of shear deformation, which is suitable for slender RC members but less applicable to shear walls with small slenderness ratios. A flexibility-based nonlinear finite element model that considers shear deformation and coupled flexural-shear effects, which offers greater accuracy and computational efficiency, since fewer elements are needed along the length of the member, is proposed in this study. Furthermore, cyclic loading tests of flanged shear wall specimens are carried out to validate the analytical results. Lastly, seismic behaviors of flanged RC walls are summarized, and key issues related to seismic design are discussed.

2 Fiber element model

The fiber element model is shown in Fig.1. The RC shear wall is divided into many cross sections along the vertical direction, and every cross section is divided into many reinforcement and concrete fibers along the horizontal direction. Each fiber is characterized by its cross-section area and its basic uniaxial nonlinear stress-strain relationship for concrete and reinforcement. Element degrees of freedom (DOFs) are shown in Fig.2.

Fig.1 Fiber element model

Fig.2 Element degree of freedom without rigid-body modes

The element force vector and displacement vector are denoted by the first two equations

(1)

(2)

The section force and the corresponding section deformation are respectively denoted as

{F(x)}s=My(x)Mz(x)Fy(x)Fz(x)Nx(x)]T

(3)

{d(x)}s=[φy(x)φz(x)γy(x)γz(x)εx(x)]T

(4)

whereMy(x) andMz(x) are the bending moment about they-axis and thez-axis, respectively;Fy(x) andFz(x) are the shear force in theyandzdirection, respectively;Nx(x) is the sectional axial force;φy(x) andφz(x) are the curvature about they-axis andz-axis, respectively;γy(x)andγz(x) are the shear deformation in theyandzdirection, respectively;εx(x) is the axial deformation.

(5)

WhereNf(x) is the force interpolation matrix given by

(6)

The section constitutive relation can be denoted as

d{d(x)}s=[f(x)]sd{F(x)}s

(7)

where {f(x)}sis the section tangent flexibility matrix.

Using the principle of virtual force, Eq.(5) and Eq.(7) give

[f]ed{F}e=dj5i0abt0be

(8)

where [f]eis the element flexibility matrix.

(9)

Element stiffness matrix [k]ecan be obtained by the inversion of the element flexibility matrix

[k]e=[[f]e]-1

(10)

3 Constitutive relation of fiber material

3.1 Axial constitutive relation of fiber

Nonlinear characteristics of the fiber element model are mainly governed by the constitutive relation of the fiber material. The uniaxial constitutive relation of materials is used for the axial constitutive relation of the fiber. In this paper, the reinforcement constitutive model proposed by Hoehler and Stanton[16], is selected, as shown in Fig.3. The concrete constitutive model proposed by Hoshikuma[17]is used. The concrete hysteretic rule proposed by Mander[18]is selected, as shown in Fig.4.

Fig.3 Reinforcement constitutive

Fig.4 Concrete hysteretic rule

3.2 Shear stiffness of fiber

It is assumed that the reinforcement is dispersed into the concrete, and the two materials act together to resist the shear force. Thus, only the shear stiffness of the concrete fiber is considered and the shear stiffness of the reinforcement fiber is not considered. For each concrete fiber, the shear stiffness is determined by its different axial stress state. The detailed calculation methods are listed below:

1) It is assumed that the elastic shear stiffness of the fiber isGe,when the fiber is compressed. The shear stiffness of the fiber isr1Ge, when the compressive strain of the fiber reaches the yield strain of the reinforcement. The shear stiffness of the fiber isr2Ge, when the compressive strain of the fiber reaches or exceeds the peak strain of the concrete.

2) The shear stiffness of the fiber isr3Ge, when the tensile strain of the fiber reaches the cracking strain of the concrete. The shear stiffness isr4Ge, when the tensile strain of the fiber reaches the yield strain of the reinforcement. The shear stiffness isr5Ge, when the tensile strain of the fiber reaches or exceeds the ultimate tensile strain of the reinforcement.

The shear stiffness of the fiber between the two states can be obtained by interpolation. According to the latest research,r1=0.5,r2=0.02,r3=0.5,r4=0.15 andr5=0.02[19].

4 Experiment details

4.1 Specimen design and loading program

In order to verify the fiber element model of flanged RC shear walls proposed in this study, six RC flanged shear wall specimens, named L500, L650, L800, T500, T650 and T800 respectively, were tested under cyclic loading. The vertical height of the walls was 1 400 mm, and their cross-section thickness was 100 mm. The design strength grade of the concrete was C40 (nominal cubic compressive strengthfcu,k=40 MPa). The test average cubic compressive strengthfcuof the concrete measured on cubes of 150 mm size was 47.2 MPa. The mechanical properties of the reinforcement are shown in Table 1. Specimen details and properties are summarized in Table 2, where the steel ratioρsof the longitudinal reinforcement is defined as the ratio of the cross sectional area of the longitudinal reinforcement to the total wall cross sectional area, and the volumetric steel ratioρvof the boundary element at the non-flange end is defined as the ratio of the volume of the stirrups to that of the wall (Chinese GB 50010-2010 code)[20]. Dimensions and reinforcement details of the specimens are shown in Fig.5.

Table 1 Yield and ultimate stresses of the reinforcement of the shear wall

Table 2 Properties of specimens

Fig.5 Dimensions and reinforcement details of RC shear

The axial load was first applied to the center of the shear wall by the vertical jack, and kept constant during the test. Fig.6 shows the test setup including loading devices. The scheme of the loading program is shown in Fig.7. The loading history was started by applying two identical displacement cycles with increments of ±2 mm up to 10 mm, followed by increments of ±5 mm up to failure. Each test continued until the specimens dropped to 85% of the peak lateral load.

4.2 Damage development and failure mode

Fig.6 Test setup

Fig.7 Loading history

In this experiment, all specimens exhibited a flexural model characterized by the crushing of the concrete and the buckling of the reinforcement at the free web boundary, as shown in Fig.8. During the test,exural cracks were first observed at the bottom of the web when the specimens were loaded to approximately half the peak lateral load. Shear cracks began to form at a drift level of 0.4%. As the loading displacement increased to a drift level of 1.3%, vertical splitting of the free web was quite extensive, and severe crushing and spalling of the concrete cover were observed. As the loading displacement increased to a drift level of 3.0%, major crushing of the concrete occurred, and the outermost longitudinal reinforcing bars started to buckle. The lateral force dropped to 85% of the peak lateral load. The failure mechanism suggested that closely spaced stirrups and longer confined boundary elements should be used in the free web end, preventing premature failure under compression. No obvious concrete spalling was observed at the web-flange junction, suggesting that the seismic design of the boundary element at the web-flange junction should be relaxed.

Fig.8 Crack patterns and failure of specimens

4.3 Loading capacity and ductility

The loading capacity and displacement ductility of the shear wall specimens are shown in Table 3. As shown, the specimens exhibited bigger loading capacity but lower ductility capacity when the flange was in tension. With the decrease of the shear span ratio, the displacement ductility decreased. The smaller the shear span ratio, the closer the stirrups and the longer the confined boundary elements should be used in the free web end to achieve the goal of the displacement ductility coefficient of 3.0. For example, specimens T800, L800. The ductility in the negative position of T800, L800 were, 2.2, 1.9, respectively, meaning that the volumetric steel ratio of the boundary element (0.60%) was not enough for specimens T800 and L800 with the aspect ratio of 1.75. The use of high strength stirrups could restrain the compressed concrete and postpone the buckling of the longitudinal rebars in the free web boundary, preventing premature failure when the web was compressed. The ultimate drift ratio of all specimens greatly exceeded the allowable inter-story drift ratio value (1/120) of the RC shear wall according to the design provisions of the Chinese GB 50011-2010 code[21].

Table 3 Loading capacity and ductility

Note:Δcis the lateral displacement when the web concrete first cracked;Fcis the lateral force when the web concrete first cracked;Fyis the yield lateral force;Δyis the yield lateral displacement;Fmis the maximum lateral force;Δmis the lateral displacement at the maximum lateral force;Fuis the ultimate lateral force;Δuis the ultimate top displacement;μis the displacement ductility coefficient;θuis the ultimate drift ratio.

4.4 Strain analysis

Fig.9 shows the arrangements of the strain gauges of the reinforcement of the shear wall specimens. Fig.10 and Fig.11 show the strain evolution process of the longitudinal reinforcements and the stirrups of the shear wall specimens, respectively. As we can see from Fig.10, the strain distribution of the longitudinal reinforcement at the peak point remained approximately linear, so the plane cross-section assumption can be used for the design of the flanged RC shear wall. The compression strains of the longitudinal reinforcement at the intersection of the web and flange were extremely small, compared with the tensile strains at the free web boundary, indicating that it is not necessary to add special confinement reinforcement to the web-flange intersection. As shown in Fig.11, the stirrups that yielded were mainly concentrated in the middle and lower parts of the web, meaning that the shear damage was mainly concentrated in the middle and lower parts of the web. All of the high-strength stirrups at the free web boundary yielded, showing that the use of high strength stirrups at the free web boundary could confine the transverse deformation of the specimens, thus preventing premature failure when the web was compressed.

Fig.9 Arrangement of steel strain

Fig.10 Strain evolution of the longitudinal reinforcement in the web of the shear wall

Fig.11 Strain evolution of the stirrups in the web of the

4.5 Hysteretic behavior

As shown in Fig.12, the hysteresis loops of specimens in the positive direction exhibited a much wider and thicker shape, showing that higher energy dissipation capacity could be achieved compared with in the negative direction. A pinching effect could be found from the hysteresis loops of specimens in the negative direction for the effect of shear deformation. The hysteresis curves of specimens with a cross-section height-to-width ratio of 6.5 were very plump, and their energy dissipation was very good. The bearing capacity of the T-shaped specimens was greater than that of the L-shaped specimens, because the effective flange width of the T-shaped specimens was bigger than that of L-shaped specimens. Considering the uncertainty of the seismic direction, the seismic performance of the L-shaped shear wall should be far inferior to that of the T-shaped shear wall.

Fig.12 Force versus displacement hysteresis curves of shear wall

5 Model validation

Shear wall sections are composed of three parts: Cover concrete, core concrete and rebar. Discretization of section fiber is shown in Fig.13. Section discretization schemes are shown in Table 4. Using a finite analysis program based on the flexibility-based element model that considers shear deformation and coupled flexural-shear effects this paper proposed, numerical analysis of shear wall specimens was carried out to obtain the force-displacement envelopes.

Satisfactory agreement between the analytical results and the experimental results can be seen in Fig.14. It should be noted that in calculating the ultimate load, the proposed method is approximately 95% of the test results, because the boundary condition on the top of the specimens is fully free in the simulative analysis, but in the test, the vertical jack exerts friction on the top of the specimens. Furthermore, as seen in Fig.14, the section discretization of various numbers of fibers has little effect on the computational efficiency, since the element formulations strictly enforce force equilibrium.

Fig.13 Discretization of section

Table 4 Section discretization of different numbers of rebar fibers and concrete fibers

Fig.14 Force-displacement curves of specimens with different discretizations of section

6 Conclusions

In this paper,a new flexibility-based nonlinear finite element model which considers shear deformation and coupled flexural-shear effects is proposed, and cyclic loading tests of T-shaped and L-shaped shear wall specimens are carried out to verify the proposed model. The following conclusions can be drawn:

1) All shear wall specimens exhibited a flexural failure model characterized by the crushing of concrete and the buckling of steel at the free web boundary.

2) The use of high strength stirrups could effectively restrain the compressed concrete and postpone the buckling of the longitudinal rebars in the free web boundary, preventing premature failure when the web was compressed. The ultimate drift ratio of all specimens greatly exceeded the allowable inter-story drift ratio value (1/120) of RC shear walls according to the design provisions of the Chinese GB 50011-2010 code.

3) No obvious concrete spalling was observed at the web-flange junction, and the compression strains of the longitudinal reinforcement at the intersection of the web and flange were extremely small, compared with the tensile strains at the free web boundary, indicating that it is not necessary to add special confinement reinforcement to the web-flange intersection.

4) The specimens exhibited higher strength and stiffness but lower ductility capacity when the flange was in tension. The displacement ductility decreased when the shear span ratio decreased. The volumetric steel ratio of the boundary element (0.60%) was not enough for specimens with an aspect ratio under 2.0 to achieve the goal of the displacement ductility coefficient of 3.0.

5) The strain distribution of the longitudinal reinforcement at the peak point remained approximately linear, so the plane cross-section assumption can be used for the design of flanged RC shear walls.

6) A new flexibility-based fiber element model that considered shear deformation and coupled flexural-shear effects was proposed for simulating the nonlinear response of special shaped shear walls, dominated by flexural failure with a small slenderness ratio. There was a good agreement between analytical and experimental results, demonstrating that the model offers excellent accuracy and requires few elements per member, offering a more efficient alternative to the traditional flexibility-based fiber element model in the nonlinear analysis of special-shaped RC shear walls.

Acknowledgements

The authors would like to acknowledge the financial support from the National Natural Science Foundation of China (Grant No: 51568028).