A restoring force model for steel fiber reinforced concrete shear walls

Engineering Structures 75 (2014) 469–476

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A restoring force model for steel fiber reinforced concrete shear walls Jun Zhao ⇑, Huahua Dun Zhengzhou University, School of Civil Engineering, 100 Kexue Avenue, Zhengzhou, Henan, PR China

a r t i c l e

i n f o

Article history: Received 29 November 2013 Revised 5 June 2014 Accepted 10 June 2014 Available online 4 July 2014 Keywords: Steel fiber reinforced concrete Shear wall Restoring force model Skeleton curve Hysteretic loop

a b s t r a c t A cyclic restoring force model is a useful tool for seismic analysis of reinforced concrete shear walls in high-rise building structures. In this paper, five steel fiber reinforced concrete shear walls were tested under horizontal reverse cyclic load with constant axial load. Crack load, ultimate load, lateral displacement, steel strain and concrete strain were measured and the skeleton curves and hysteretic loops were plotted. By analyzing the shape of the typical skeleton curve and hysteretic loop, the feature points which could represent the restoring force models of the curves were defined. After the determination of the feature points and the stiffness, the calculated skeleton curves and the hysteretic loops were obtained and found to agree closely with those of the experimental specimens. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Because of the frequent occurrence of earthquakes and the extensive application of high-rise buildings, we have been giving greater and greater attention to seismic performance, especially on the restoring force capacity of shear wall structures [1,2]. Due to the high brittleness and low tensile strength of concretes, normal reinforced concretes have low crack resistance, low ultimate strength, and weak deformability and energy dissipative capacities, which indicate weak seismic capacity. Because of this, reinforced concrete shear walls are liable to crack and fail under reversed cyclic load, and need to be repaired or strengthened to improve the crack resistant and seismic performance [3–5]. A variety of methods have been used to resolve these defects such as slit concrete shear wall [6,7], shear wall with brace [8–10], steel and concrete composite shear wall [11–17], and shear wall with damper [18,19]. The purposes of the above technical measures are to improve the seismic performance of reinforced concrete shear walls. But among the above methods, some may reduce the crack resistances, the ultimate strengths or the initial stiffness of shear walls, others may result in complicated and high cost construction. One effective approach to increase the ductility and tensile property of concrete is to add steel fibers into the concrete to obtain steel fiber reinforced concrete. Much research work has been done in recent years on the performance of steel fiber reinforced concrete and its structures. Short steel fibers dispersed ⇑ Corresponding author. Tel.: +86 013838097292. E-mail address: [email protected] (J. Zhao). http://dx.doi.org/10.1016/j.engstruct.2014.06.013 0141-0296/Ó 2014 Elsevier Ltd. All rights reserved.

uniformly in concrete can substantially improve the tensile strength, ductility, impact resistance and fatigue resistance of concrete. Therefore, flexural strength, shear strength, crack resistance, load bearing capacity after cracking and toughness of concrete structures can be improved [20–22]. Thus the closely spaced steel fibers throughout shear walls may help to control cracks and dissipate energies, which are expected to enhance the seismic performance of the members. For the past few years, the authors have performed experimental research on shear walls with steel fibers, which showed that steel fibers can simultaneously improve the crack resistance, ultimate capacity, ductility and energy dissipation capacity of reinforced concrete shear walls [23,24]. In this paper, five steel fiber reinforced concrete shear walls are made to experimentally and theoretically research the restoring force performances. A simple restoring force model will be obtained based on the experimental skeleton curves and hysteretic loops of steel fiber reinforced concrete shear walls [25–32]. 2. General test situation Five steel fiber reinforced concrete shear walls with identification numbers BSWA-10-50, BSWA-15-50, BSWA-20-50, BSWA10-30, and BSWA-10-70 were tested. The meanings of the identification numbers are as follows. The first number represents the volume fraction of steel fibers and the second number represents the concrete’s strength grade. For example, BSWA-10-50 indicates a specimen with a volume fraction of 1.0% and a strength grade of C50.

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2.1. Specimen details The specimens’ height-to-width ratio was 2 and the sectional dimensions were H  L  B = 1800 mm  900 mm  200 mm. In addition, the shear walls had a top beam and a base girder. The top beam was used to simulate the restricting effects from the floor and served as a load point. The base girder was used to simulate a rigid foundation and fix the shear wall on the floor of the laboratory. The detailed specimen dimensions and the layout of steel bars were displayed in Fig. 1 and Table 1. Shear-cut type steel fibers with length, diameter and aspect ratio of 32.4 mm, 0.56 mm and 57.4, respectively, were used. 2.2. Test scheme The horizontal load was applied cyclically by the hydraulic servo load system on the longitudinal axis of the top beam as shown in Fig. 2. The vertical load applied by lifting jack reached design load and was kept constant. The ratio of axial compressive force to axial compressive ultimate capacity of section was 0.1. Rolling shafts between lifting jack and reaction beam assured smooth horizontal slipping of specimens Load control mode was adopted first. After the longitudinal steel bars yielded, the displacement control mode was used [33]. 2.3. Test results The compressive and tensile strengths of steel fiber reinforced concrete are listed in Table 1. The mechanical indexes of the steel bars are listed in Table 2. Through the above experiment, the crack load, ultimate load and lateral displacement were measured and the hysteretic loops and skeleton curves were plotted. 3. Feature points of skeleton curve The typical skeleton curve of steel fiber reinforced concrete shear walls is shown in Fig. 3, which shows that the curve has a slight decline after the ultimate load. In this paper, a four linear

skeleton curve is used to relatively coincide with the experimental skeleton curve. Thus, using the feature points, namely the crack point, yield point, ultimate load point and ultimate displacement point, we can transform the curvilinear skeleton curve of continuously changing stiffness into the multi-linear skeleton curve of constant slop in each line segment. To obtain skeleton curves that take the effects of steel fibers into account, the four feature points were determined based on the experimental data. 3.1. Crack point The calculated models of steel fiber reinforced concrete shear walls were similar to that of normal concrete shear walls. For simplicity, the crack moments of steel fiber reinforced concrete shear walls under the combination of M and N can be calculated by Eqs. (1) and (2) based on the calculation formula of normal reinforced concrete shear walls in consideration of the effect of steel fibers. Comparisons of test and calculated crack moment values are listed in Table 3.

 Mfcr ¼

cft W o þ

 N W o bfcr ¼ F fcr H AWO

bfcr ¼ 1  0:58kf þ

0:717k2f

ð1Þ ð2Þ

where kf is the steel fiber content characteristic parameters, kf ¼ lf =df , lf the length of steel fiber, df is the equivalent diameter of steel fiber; Ffcr the horizontal cracking load on top of steel fiber reinforced concrete shear walls, namely the value of the applied load in the test; H the height of steel fiber reinforced concrete shear walls; c the sectional plastic impact factor, according to the code for design of concrete structures in China (GB50010-2010), the value is c ¼ ð0:7 þ 120=hÞcm , cm is 1.55 for the rectangular cross-section, here c = 1.29; ft the design values of tensile strength of ordinary concrete corresponding to the strength of steel fiber reinforced concrete; Wo the elastic resistance moment of the transformed section; N the applied axial force, with pressure taken as positive; AWO the transformed sectional area of the shear walls; bfcr is the comprehensive influence coefficient of steel fibers on the cross-sectional

Fig. 1. Detailed specimen dimensions and layout of steel bars.

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J. Zhao, H. Dun / Engineering Structures 75 (2014) 469–476 Table 1 Main specimen parameters. Identification number (mm)

BSWA-10-50 BSWA-15-50 BSWA-20-50 BSWA-10-30 BSWA-10-70

Dimensions

Reinforcements

Steel fiber reinforced concrete strength (MPa)

hw

bw

h

t

r

s

u

ffcu

fft

900 900 900 900 900

200 200 200 200 200

1800 1800 1800 1800 1800

4£14 4£14 4£14 4£14 4£14

6£12 6£12 6£12 6£12 6£12

£14@150 £14@150 £14@150 £14@150 £14@150

£8@150 £8@150 £8@150 £8@150 £8@150

53.8 57.0 53.7 33.8 75.5

5.20 6.31 6.54 4.40 6.61

hw, bw—height and thickness of cross section; h—height of specimens; ffcu—compressive strength of steel fiber reinforced concrete; fft—tensile strength of steel fiber reinforced concrete.

Fig. 2. Specimen load setup.

Table 2 Mechanical index of steel bars. Type

Diameter, d (mm)

Yield strength, fy (N/mm2)

Tensile strength, fu (N/mm2)

Elastic modulus, Es (GPa)

Elongation (%)

HRB335 HRB335 HPB235

14 12 8

373.5 404.5 340.0

555.5 570.5 428.0

200 200 210

31.4 31.5 30.0

K¼

1 H3 =3EI þ lH=GA

ð3Þ

where EI is the bending stiffness of shear wall; G the shear modulus of the member, G ¼ 0:4E; A the cross-sectional area of the member; and l is the unevenly distributed coefficient of shear stress, l = 1.2. Taking into account the effects of steel fibers on the stiffness of shear walls and the plasticity of concrete, the initial stiffness of steel fiber reinforced concrete shear walls before cracking can be revised by Eq. (3) and expressed as follows.

K¼

1

cf ðH3 =3EI þ lH=GAÞ

ð4Þ

where cf = 3.1. Fig. 3. The skeleton curve of BSWA-10-30.

3.2. Yield point performance of the reinforced concrete shear wall, which can be calculated by Eq. (2) obtained from the numerical regression of test data. Through known material and structural mechanics, we can obtain the elastic stiffness of cantilever shear walls, shown as the following formula:

The yield load and yield stiffness of steel fiber reinforced concrete shear walls can be expressed as the ratio of ultimate load and the ratio of elastic stiffness, respectively. Then the yield point of shear wall is determined as follows.

F y ¼ a1 F u ; K y ¼ b1 K

ð5Þ

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Table 3 Test and calculated crack and ultimate values (kN m). Moment

BSWA-10-50

BSWA-15-50

BSWA-20-50

BSWA-10-30

BSWA-10-70

Test crack moment Calculated crack moment Test ultimate moment Calculated ultimate moment

270 244.9 516.7 530.8

288 305.4 558.6 593.3

360 353.75 554 618.6

180 194.47 426.7 429.9

324 317.24 627.7 636.1

Fig. 4. Calculated model of ultimate moment.

Fig. 5. The test and calculated skeleton curves of steel fiber reinforced concrete shear walls.

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where Fy is the yield load; Fu the ultimate load; Ky is the yield stiffness. Carrying on the data fitting to the experimental results, we can obtain b1 ¼ 0:29 and a1 ¼ 0:82.

3.3. Ultimate load point The ultimate load and stiffness should be determined to locate the ultimate point. The ultimate moments of steel fiber reinforced concrete shear walls can be calculated by Eqs. (6) and (7). According to the failure mode and the measured strains of steel bars, the stress distribution of a cross section of shear wall at ultimate state is shown as Fig. 4, which can be used to deduce the calculated formulas for load-carrying capacity of shear wall. From Fig. 4, the equations of equilibrium can be obtained as follows.

N ¼ a1 fc bw x 

Asw fyw ðhw  hb  1:5xÞ  fftb bw xt hw0

Fig. 7. Typical hysteresis loop and feature points.

ð6Þ

4. Calculation expressions of skeleton curve

M fu ¼

Asw fyw ðhw  hb  1:5xÞðhw  hb þ 0:5xÞ þ fy As ðhw  hb Þ 2hwo  1 xt x  þ Nðhw0  hb Þ þ fftb bw xt hw   2 2 2

ð7Þ

where Asw, fyw is the area and yield strength of distributed longitudinal bars; As, fy the area and yield strength of longitudinal bars in embedded column; fftb the bending tensile strength of steel fiber reinforced concrete, fftb = cfft, fft the tensile strength of steel fiber reinforced concrete; x the depth of compressive zone; xt the depth of tensile zone, xt = hw - 1.25x; hw, bw, hb, hwo the depth, width, depth of embedding column and effective depth; and c is the influence coefficient of steel fiber on the tensile property, c ¼ 0:409. The comparison of test and calculated values of ultimate moments are listed in Table 3. The ultimate stiffness K u can be expressed as the ratio of initial stiffness.

K u ¼ b2 K

After determining the above feature points on the skeleton curves, the calculation expressions of skeleton curves can be obtained and the theoretical skeleton curves can then be plotted using the calculated results. Because the feature points divide the skeleton curves into four linear segments for both positive and negative load directions, the expressions of skeleton curves must be expressed respectively as follows.

F¼

D 3:1ðH3 =EI þ lH=GAÞ

F ¼ F fcr þ 0:09

F ¼ F y þ 0:03 F ¼ Fu

D ðD  Dfcr Þ ðF fcr  F  F y Þ H =EI þ lH=GA 3

D ðD  Dy Þ ðF y  F  F u ; D 6 Du Þ H =EI þ lH=GA 3

ðD P Du Þ

F ¼ F fcr þ 0:09

F ¼ F y þ 0:03

3.4. Ultimate displacement point The ultimate displacements of the five steel fiber reinforced concrete shear walls in the test are around 40 mm. To facilitate the analysis, we take the ultimate displacements as Dmax = 40 mm.

ð9Þ

ð10Þ

ð11Þ ð12Þ

ð8Þ

On the basis of test data, we can obtain b2 = 0.09.

ðF fcr  F  F fcr Þ

D H3 =EI þ lH=GA

D H3 =EI þ lH=GA

ðD þ Dfcr Þ ðF fcr  F  F y Þ

ð13Þ

ðD þ Dy Þ ðF y P F P F u ; D P Du Þ ð14Þ

F ¼ F u

ðD 6 Du Þ

Fig. 6. Hysteresis curve example of steel fiber reinforced concrete shear walls.

ð15Þ

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Table 4 The calculation of feature points. Feature points

Cracking to yield

Yield to load limit

Load limit to displacement limit

Load

Displacement

Load

Displacement

Load

Displacement

Fixed point 1 Stiffness mutation 2

0.6 Fcr Fm1

0.6Dcr Dm1

0.6Fcr Fm1

0.6Dcr Dm1

0.6Fcr Fy

0.6Dcr

Peak point 3 Pinched point 4

Fm Fcr

Dm cr Dm  F m F K

Fm 0

Dm Dm  FKmy

Fm 0

F y 0:6F cr Fm

Dm þ 0:6Dcr Dm Dm  FKmy

Note: Fm and Dm is the maximum load and displacement of number m loop.

Table 5 The calculated stiffness of hysteretic loop.

5. The characteristics of hysteretic curves

Stiffness of each segment

Phase of cracking to yield

Phase of yielding to limit

Horizontal section

12

1 k12 ¼ DF 22 F D1

1 k12 ¼ DF 22 F D1

1 k12 ¼ DF 22 F D1

23

k23 = Ky

k23 = Ky

2 k23 ¼ DF 33 F D2

34 45

k34 = K

k34 = Ky

k34 = Ky

cr k45 ¼ DF 55F D4

k45 ¼ D5F5D4

k45 ¼ D5F5D4

where Dfcr is the crack displacement and Dy is the yield displacement. According to the above formula 9–15, the feature points were calculated and the calculated skeleton curves were plotted in Fig. 5. As seen in Fig. 5, the calculated skeleton curves and the experimental skeleton curves can coincide closely with each other, which illustrates that the above feature points and suggested formula can be used to describe the skeleton curves of steel fiber reinforced concrete shear walls.

In order to research the theoretic models of steel fiber reinforced concrete shear walls, the shape and feature of the experimental hysteretic loops were analyzed to find suitable feature points. The typical hysteretic curves of steel fiber reinforced concrete shear walls are shown in Fig. 6. It can be seen that the hysteretic curves are composed of hysteretic loops which can be divided generally into two kinds of shapes, namely fusiform shape and reversed S-shaped loops, as shown in Fig. 6. At the early stage of the load, the hysteretic loops are fusiform shape as shown in Fig. 6(a). At the latter stage of the load, the hysteretic loops are reversed S-shape as shown in Fig. 6(b). After analyzing the typical hysteretic curve, we can obtain feature points which can determine the shape of the hysteretic loop. (1) Fixed point The ascent portions of the hysteretic loops of five steel fiber reinforced concrete shear walls under both the positive and

Fig. 8. The test and calculated hysteresis curves of steel fibers reinforced concrete shear walls.

J. Zhao, H. Dun / Engineering Structures 75 (2014) 469–476

negative loading paths pass the same point, which can be defined as one feature point named the fixed point. (2) Stiffness mutation point For both fusiform and reverse S-shaped hysteretic loops, there are special points after which the slopes of segments change noticeably. The special points indicate that the stiffness of the shear walls change suddenly. We define these special points as the stiffness mutation points. (3) Peak point After the stiffness mutation points, the load increases gradually with the increment of deformation until the maximum load of the loop is reached. The point corresponding to the maximum load is defined as the peak point. (4) Pinched point After the peak point, the load begins to decline with the reduction of deformation. In the process of unloading, there exists pinched phenomenon that indicates the low residual stiffness. The point at the beginning of the pinched phenomenon is named the pinch point. 6. Determination of hysteretic curves Based on the analysis of the above feature points, the simplified typical shape of the hysteretic loop can be determined as seen in Fig. 7. When we carry on the model process of the restoring force of the steel fiber reinforced concrete shear walls, the curves with continuous changes in stiffness are simplified to segments with constant slopes for each segment. Thus, feature points and stiffness must be determined at each stage to obtain the complete hysteretic curves. In conjunction with the test data, the feature points and the stiffness for all segments can be divided into three stages: the stage from cracking to yield, the stage from yield to load limit and the stage from load limit to displacement limit. The calculated results of the feature points and stiffness are shown in Tables 4 and 5. The detailed calculation processes are in reference [27]. According to the above discussion, four typical hysteretic loops, namely almost cracking point hysteretic loop, almost yield point hysteretic loop, almost load limit point hysteretic loop, and almost displacement limit point hysteretic loop, are plotted for each specimen. As seen in Fig. 8, the calculated hysteretic loops coincide well with the experimental hysteretic loops. 7. Conclusions Through the analysis of experimental data, the restoring force models steel fiber reinforced concrete shear walls are discussed and the conclusions can be made as follows. (1) Based on the experimental skeleton curves of steel fiber reinforced concrete shear walls, the feature points, which included the crack point, the yield point, the ultimate load point and the ultimate displacement point, were analyzed. Using the regression method, the formulas of the above four points and corresponding stiffness were calculated to get the expression of the skeleton curves. (2) The four-line type curve can be used to demonstrate the skeleton curve of steel fiber reinforced concrete shear wall. The calculated models of cracking load and ultimate load

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in this paper are suitable to determine the cracking and ultimate load points in which the effects of steel fibers are taken into account. (3) According to the characteristics of the experimental hysteretic loop, the whole hysteretic loop was divided into four feature points and eight segments. The four feature points were the fixed point, stiffness mutation point, peak point and pinched point. The values of the four feature points and the stiffness of the eight segments can be calculated by analyzing the key points of the test hysteretic loops. (4) The calculated models of skeleton curves and hysteretic loops in this paper can be used to determine the restoring force models of steel fiber reinforced concrete shear wall.

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