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SEISMIC BEHAVIOR OF REINFORCED CONCRETE WALLS FOR A PERFORMANCE BASED DESIGN Conference Paper · August 2014

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SEISMIC BEHAVIOR OF REINFORCED CONCRETE WALLS FOR A PERFORMANCE BASED DESIGN Susumu KONO1, Msanori TANI2, Tomohisa MUKAI2, Hiroshi FUKUYAMA2, Rafik TALEB3 and Masanobu SAKASHITA4 ABSTRACT A nationwide research project has been carried out since 2008 to revise the Japanese Building Standard. Until 2014 more than fifty topics were studied covering various research areas such as clarification of seismic behaviors of structural components, definition of design load from tsunami, development of materials to enhance fire protection and energy saving, and etc. Among these topics, ten topics covered structural behaviors of reinforced concrete members. Behavior of structural walls under seismic loading is one of the main issues investigated in the field of reinforced concrete structures. The authors carried out a series of quasi-static cyclic tests on reinforced concrete walls in 2011 and 2012 in order to study the ultimate drift capacity and progress of damage. Test variables involved wall section geometry, reinforcement details of section end, and axial force level. Based on the experiments and accompanying numerical studies, following conclusions were found. The ultimate drift capacity generally increased as axial load level decreased, the size of confined end region increased, the amount of confining reinforcement increased, and shear span ratio increased. The ultimate drift capacities were simulated by combining a flexural fiber model and a tri-linear shear model. When the plastic hinge length was properly optimized so that the simulated ultimate drift capacity was sufficiently accurate for design, the determined plastic hinge length was larger than the values computed by Priestley et al.’s formula but smaller than half the wall length reported by Wallace and Moehle.

INTRODUCTION The ministry of land, infrastructure, transport and tourism (MLIT) in Japan is equivalent to the ministry of construction in other countries. It has been promoting multiple nationwide research projects to improve the Japanese building standard [1] since 2008 including several projects on reinforced concrete structural walls. The authors carried out a two-year project for developing displacement based design criteria of reinforced concrete structural walls with a collaboration with other universities. Some of the topics include: 1. Evaluation of the ultimate displacement capacity so that the ductility factor of the building can be propertly determined. The effects of section configuration (such as T-shape or Lshape), boundary columns on the ultimate drift capacity are clarified. 2. Evaluation of damage at the serviceability and repairability limit states by quantifiying cracking and crushing of concrete, and yielding of reinforcement. 1

Professor, Tokyo Institute of Technology, Yokohama, Japan. [email protected] Building Research Institute, Tsukuba, Japan. [email protected], t_mukai@ , fukuyama@ 3 Ph.D. candidate, Tokyo Institute of Technology, Yokohama, Japan. [email protected] 4 Assistant Professor, Kyoto University, Kyoto, Japan. [email protected] 2

1

3.

4.

Evaluation of effects of number of cycles on the damage and hysteresis of loaddisplacement relations so that the numerical model is developed for long-period earthquakes. Evaluation of the effect of out-of-plane displacement on the in-plane performance.

Among these topics, this paper discusses the first topic on the ultimate deformation. The Architectural Institute of Japan (AIJ) published the revised version of reinforced concrete building standard (AIJ RC Standard) [4] in 2010. The 2010 AIJ RC standard relaxed requirements on boundary columns which had been mandatory in the 1999 AIJ RC standard [5]. The 2010 standard allows the design of rectangular section walls with good confinement at section end regions. The end region, which is supposed to act equivalently as a boundary column, should be well confined so that structural walls with a rectangular section have similar seismic performance to those with boundary columns. In the two-year project, nine structural wall specimens were tested by varying geometry of boundary columns, reinforcement details of end regions, and axial force level. The paper describes the experimental setups and results, and accompanying numerical simulations. Then the characteristic points of backbone curves were simulated using existing procedures by paying special attention to the ultimate displacement capacity.

EXPERIMENTAL PROGRAM Specimens and loading setups

The experiment investigated the behavior of symmetric structural walls. Nine specimens were prepared as shown in Table 1. Figure 1 and Figure 2 show dimensions and reinforcement arrangement. “12D10”, for example, stands for twelve deformed bars with 10 mm diameter. The first seven specimens had shear strength to flexural strength ratio more than 1.25 so that they reach the flexural yielding point first and then fail in shear or flexure. The last two specimens (BC and NC) had similar capacities with respect to shear and flexure but was set to have lower sliding shear capacity by Refs. [2] and [3]. Four specimens in 2011 (BC40, BC80, NC40 and NC80) had two test variables; section configuration (barbell shape or rectagular shape) and amount of confining reinforcement at end regions. BC40 and BC80 had a barbell shape section with boundary columns. NC40 and NC80 had a rectangular section with no boundary columns although end regions had confining reinforcement. Four specimens had identical wall length, total section area, and confined area with similar flexural strengths. Based on the ultimate limit compressive strain defined by Mander et al. [6], the predicted ultimate drift angle for BC40 and NC40 exceeded 1.5% and that for BC80 and NC80 was smaller than 1.5%. Three specimens in 2012 (MC, SC and HN) were made to study buckling of vertical reinforcement at the ultimate conditions. They had different amount of confined section area and shear reinforcement and axial force levels without boundary columns. MC was set as a standard specimen among three specimens. SC had the same amount of confining shear reinforcement volume ratio, ρwv, but confined section area was halved. HN had two times larger confined section area with similar predicted ultimate drift ratio compared to MC. The axial load for HN was 1200kN and was two times larger than that for MC and SC. The last two specimens in 2012 (BC and NC) were prepared to study the sliding shear failure at the wall base. Hence, the axial load ratio (axial load divided by the product of the wall section area and concrete strength) was set as low as 0.033. Loading setup is shown in Figure 3(a). Damage process was recorded in the experiment. Since cracking is one of the main types of damage which requires large amount of repair cost, the crack width for peak load and unloaded conditions were systematically recorded. Data on cracks included the location, width, type (shear, flexure, bond) and length at the peak and unloaded stages. Many data are in an analog format and it takes time to transform them to meaningful data. This damage data will be discussed elsewhere.

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S. Kono, M. Tani and others

98

98

98

25

78 D6@200

12D10

25

D6@200

98

25

12D10

98

25

25

D4@80(Shear)

10D10

120 118

28x50 1750

D4@40(Shear) 10D10

D4@50

120

HN

D4@50

16x50 1750

150

BC

50 100 100 100 100

25

50 50 50 25

25

50 50 50 50 50 25

24x50 1750

25 100 100 250

D4@40(Shear)

50

SC

25

D4@50

D6@40(Shear)

1250 1750

250

120

150

98

100

25 70

1250 1750

50

MC

25 100 100 250

25 D6@200

25 70

250

98

250

250

D6@80(Shear)

D6@200

BC4012D10

100 100 25

25 100

D6@200

25

25 80

BC80 12D10

98

1750

NC

Figure 1. Plan view of specimens (Unit: mm).

B

1750

275 800

400

1700

400

275

275

(a) NC40(left) and NC80(right)

B

1750

800

B- B

275

(b) MC

(c) NC

Figure 2. Elevation of selected specimens (Unit: mm).

Out-of-plane movement was restrained with this device.a

Out-of-plane movement was restrained by two hydraulic jacks behind the specimen.

Displacement gages

Displacement gages

(a) Test in 2011

100 100 25

1750

25

98

12D10

60

25

98

D6@200 100

25

78

128

D6@200

D6@40(Shear) 12D10

128

12D10

60

NC40

25

D6@200 100

25

D6@80(Shear)

NC80 12D10

3

(b) Test in 2012

Figure 3. Loading setups (Unit: mm).

Table 1. Major properties and variables of wall specimens. Year Specimen

Elevation D×h （Shear span） (mm)

Confined end region Dimension (mm) b×D

Long. rebar (pgc)

216×216

8-D10 (0.91%)

2011

BC40 BC80 NC40

1750 ×2800 （3000)

94×506 NC80 MC

2012

SC

84×214 1750 ×1700 （2400)

HN BC NC

84×114 84×414

1500 x1700 (2400)

200x200 120x200

Wall panel Shear rebar (pwc)

12-D10 (1.29%) 10-D10 (2.56%) 6-D10 (2.70%) 10-D10 (1.38%) 8-D10 (1.43%) 8-D10 (2.38%)

Thickness (mm)

Vertical and Horizontal rebars (pw)

80

D6@100 Staggered (0.40%)

3-D6@40 (0.95%) 2-D6@80 (0.32%) 4-D6@40 (2.47%) 4-D6@80 (1.24%) 4-D4@80 (1.36%) 2-D4@40 (1.46%) 4-D4@40 (2.61%) D4@40 (0.53%) D4@40 (0.59%)

128

D6@100 Staggered (0.25%)

120

2-D4@50 (0.47%)

2-D4@80 (0.39%) 2-D4@80 (0.29%)

91 120

Axial force （Axial force level）

1500kN (0.11)

600kN (0.10) 1200kN (0.20) 300kN (0.030) 300kN (0.033)

1) Axial force level was defined as the ratio of axial load to the nominal axial force capacity. The nominal axial force capacity was computed using the total section area of columns and a wall panel. 2) Shear span was the distance from the contraflexure point to the top of the lower stub. 3) D4 and D10 stand for deformed bars with diameter of 4 mm and 10 mm, respectively.

Table 2. Mechanical properties of concrete and reinforcement. Specimen

BC80, BC40 NC80, NC40 MC SC HN BC NC

Comp. strength (MPa) 59.5 52.5 29.6 27.5 27.8 35.2 35.5

Concrete Young’s modulus (GPa) 30.9 30.1 33.0 29.5 31.0 28.2 28.7

Type of Reinforcing bars D6 D10

Splitting strength (MPa) 5.10 3.66 2.60 2.24 2.52 2.68 2.80

Steel Young’s Yield modulus strength (GPa) (MPa) 387 189 377 194

Tensile strength (MPa) 496 533

D4 D10

395 385

190 196

524 524

D4 D10

384 348

180 182

523 467

Test results

Damage conditions near the peak load are shown in Figure 4. Until the peak load, flexure and flexureshear type cracks spread over the wall panel. Spalling of concrete at the compressive fiber was observed at drift ratio (R) in the range of 0.5% to 1.0%. Buckling of vertical bars was seen at the end of the loading test and considered to have happened just before or at the peak load from strain gage readings. However, the impact of buckling on the degradation of load carrying capacity did not seem profound. Global wall buckling was seen at the very end of large drift ratio. In general, crushing of concrete at the compression zone mainly caused degradation of lateral load carrying capacity resulting in flexure compression failure for all specimens. Specimens with rectangular sections showed quick degradation of lateral load carrying capacity compared to those with barbell shape sections as can be seen in Figure 5. Drift at the loading point was decomposed into flexural and shear components using the displacement gages as shown in Fig. 3. The flexural component ranged from 70% to 90% although figures are not shown in this paper. The drift due to pullout of longitudinal bars from the lower stub was also measured by using vertical displacement gages set at wall base with measurement length of 50mm. Contribution of pullout to the drift ranged from 5% to 10%. The drift component from pullout was included in the flexural contribution in this paper.

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S. Kono, M. Tani and others

(a) BC40 (R=2%) (b) BC80 (R=2%)

(f) SC (R=1.5%)

(c) NC40 (R=2%) (d) NC80 (R=1.5%) (e) MC (R=0.75%)

(g) HN (R=0.75%) (h) BC (R=2%) Figure 4. Damage states at the end of loading test. 600

400

400

Lateral load (kN)

800

600

Lateral load (kN)

800

200

0

-200

-400

-400

B… BC80 B… BC40

-600

NC80 NC80 NC40 NC40

-600

-800

-800 -5

-4

-3

-2

-1

0

1

2

3

Drift ratio(%)

4

5

-5

-4

-2

-1

0

1

600

400

400

400

-200 -400

Lateral load (kN)

800

600

0

200 0 -200 -400

-600

MC

M…

-800 0

1

-600

2

SC -1

0

200 0 -200

-600

HN H…

S…

1

2

-2

-1

0

1

Drift ratio (%)

Drift ratio (%)

Drift ratio (%)

(c) MC

(d) SC

(e) HN

200

200

Shear force (kN)

300

100 0 -100 -200 BC

-300 -3

5

-800 -2

300

-4

4

-400

-800 -1

3

(b) NC40 and NC80

600

200

2

Drift ratio (%)

800

Lateral load (kN)

Lateral load (kN)

(a) BC40 and BC80

-3

800

-2

(i) NC (R=2%)

200

0

-200

Shear force (kN)

5

-2

-1 0 1 R (x10-2rad)

2

3

4

2

100 0 -100 -200 NC

-300 -4

-3

-2

(f) BC

-1 0 1 R (x10-2rad)

(g) NC Figure 5. Load-drift ratio relations.

2

3

4

Defining the ultimate condition as the twenty percent loss of load carrying capacity, all specimens failed in a flexural compression mode and buckling did not seem profound. When drift was increased beyond the ultimate point, the bucking of vertical reinforcement and/or global buckling of wall panel were observed. In other words, numerical modeling can be made without modeling the buckling in nine specimens. Based on this observation, the numerical model was constructed as described in the next section.

NUMERICAL SIMULATIONS OF TEST RESULTS Modeling of backbone curves

The backbone curve was simulated up to the ultimate drift by adding flexural drift from a section analysis and shear drift of a tri-linear backbone model. A flexure model in a section analysis consists of multiple fibers which represent plain concrete, confined concrete, and longitudinal reinforcement. The stress-strain relation of concrete is shown in Figure 6(a). Plain concrete of a wall panel and cover concrete was modeled with Popovics model [7] (ascending branch) and a straight line (falling branch). Confined concrete was modeled by combining Sun-Sakino model [8] and Popovics model [7]. The ultimate strain of confined concrete was computed with Mander’s model [1] as marked by diamond shape in Figure 6(a). As shown in Figure 6(c), the distribution of curvature was divided into elastic and plastic parts assuming the equivalent plastic hinge length was lp. Flexural component of drift, Rf, was obtained by integrating curvature twice. Shear component of drift, Rs, was computed based on the AIJ design guidelines [9] as stated in Appendix. Then, Rs was added to Rf to obtain the total drift, R, as Eq. (1). Interaction between flexure and shear was not considered to compute Rs and Rf. (1) Numerical results and discussions

Figure 7 compares the computed and experimental backbone curves for two representative cases. The results were obtained by optimizing an effecitve plastic hinge length, lp, so that the simulated ultimate , was computed based on Fig. drift did not exceed the test results. The ultimate flexural drift angle, 6(b) using Eq. (2). ∆

∆

∆ (2)

∆

3 1 ∆

∆

0.5

where ∆ is the ultimate drift, is the shear span, ∆ is the yield displacement, ∆ is the plastic is the yield curvature, and are the yield and ultimate moment capacities, is displacement, is the plastic curvature. ∆ was computed assuming that the effective plastic hinge length and equals to in this paper. The effective plastic hinge length, lp, was 350mm (0.2 times wall length, 4.4 times wall thickness for BC40&BC80, 2.7 times for NC40&NC80) and 600mm (0.34 times wall length, 5.0 times wall thickness). Priestley’s et al. [10] reported that the effective plastic hinge length can be approximately estimated as Eq. (3) based on experimental results on columns and beams. 0.08

0.022

(MPa)

(3)

where is the shear span, and are the diameter and yield strength of longitudinal reinforcement, respectively. The values for based on Eq. (3) are 319 mm (BC40, BC80, NC40 and NC80), 273 mm 6

S. Kono, M. Tani and others

7

(MC, SC and HN) and 265 mm (BC and NC). If these values were used, the ultimate drift capacity would have been underestimated. The optimized values are smaller than half the wall length, which Wallace and Moehle [11] reported conservative as a plastinc hinge length. Computed results for seven specimens are compared with test results in Table 3. Index “” stands for the ratio of experiment to analytical results. Index “” was computed on load for cracking, yielding and peak conditions, and was computed on drift for the ultimate condition. The simulated load values were conservative for cracking but had good accuracy for yielding and peak load. The simulated ultimate flexural drift, Ruf , agreed well since was optimized (uf ranged from 0.90 to 1.42.). The ultimate drift underestimated the test results (u ranged from 1.38 to1.67.). A note on instability

Based on the damage at the 2010 Off Maule Earthquake or the 2011 Christchurch Earthquake [12], there is still a concern that the instability may trigger a load drop. Under the loading conditions for nine specimen, instability did not have to be considered to simulate the behavior up to the ultimate condition. However, wall panels may buckle under even larger axial load level, or the vertical reinforcement may buckle when it experiences large tensile strain history before compression force is applied. Since nine specimens do not cover a spectra of various seismic loading conditions, it is still necessary to study more cases for expected seismic loadings and boundary conditions in real buildings. It is also necessary to study conditions which cause the instability in global and local sense. The mechanism of reinforcement buckling after experiencing large tensile strain is studied by Dodd and Restrepo [13] but no quantifying formula has been established to simulate the premature failure. ∆ ∆  80 Qu  BC40 BC80 NC40 Plain

NC80

Stress (MPa)

60

H 40

20

0 0

0.005

0.01 Strain (m/m)

0.015

0.02

(a) Stress-strain relations for concrete

(b) Inelastic deformation (c) Curvature distribution at the ultimate condition at the ultimate condition

Figure 6. Assumptions for a section analysis. 800 expRu expRy calRmax

600 Q(kN)

Q(kN)

600

800

expRmax calRy calRu

400 200

400

expRu expRmax expRy calRy calRmax calRu

200

0

0 0

0.5

1

R(%)

1.5

2

0

0.5

1

R(%)

1.5

(a) MC (b) HN Figure 7. Simulated backbone curves and test results for two representative specimens.

2

Table 3. Comparison of computed and experimental test results for selected specimens. Specimens

BC40 BC80 NC40 NC80 MC SC HN

expQc

(kN) (1) 443 418 328 334 284 126 407

Cracking Yielding of reinforcement c y expQy expRy calQy calQc (%) (kN) (kN) (%) (kN) ― ― (2) (3) (1)/(3) (4) (5) (6) (4)/(6) +0.12 288 1.54 562 +0.29 479 1.17 +0.08 290 1.44 487 +0.26 482 1.01 +0.07 200 1.64 478 +0.19 431 1.11 +0.09 201 1.66 467 +0.30 434 1.08 +0.11 148 1.91 444 +0.30 397 1.12 +0.05 151 0.84 420 +0.46 348 1.21 +0.10 222 1.84 586 +0.36 593 0.99 expRc

expQmax

(kN) (7) 634 633 606 598 490 461 611

Peak load Ultimate drift max expRu calRu u uf expRuf calRuf calQmax (%) (kN) (%) (%) (%) (%) ― ― ― (8) (9) (7)/(9) (10) (11) (10)/(11) (12) (13) (12)/(13) +1.41 568 1.12 +4.00 2.40 1.67 +2.99 3.21 0.93 +1.17 563 1.12 +2.00 1.29 1.55 +1.43 1.59 0.90 +1.91 567 1.07 +2.38 1.72 1.38 +1.75 1.53 1.14 +1.16 558 1.07 +1.50 1.00 1.50 +1.21 1.06 1.14 +0.73 438 1.12 +1.53 1.33 1.15 +1.25 1.26 0.99 +1.40 393 1.17 +1.81 1.15 1.57 +1.58 1.14 1.39 +0.69 646 0.95 +1.50 1.23 1.22 +1.29 0.94 1.37

expRmax

1. Values in the positive direction are shown. 2. Values for BC and NC are not shown in this table. 3. The ultimate drift ratio of experiment, capRu, was taken when the shear capacity dropped by 20% from the peak. If the loading was terminated before 20% drop occurred, the maximum recorded drift ratio was written.

CONCLUSIONS The experiment was conducted on nine reinforced concrete structural walls to study the effect of influencing factors on the ultimate drift capacity and other characteristic points of backbone curves. Considered influencing factors were reinforcement details, configuration of end regions, and axial force level. All specimens failed by crushing of concrete at the foot of walls after yielding of tensile longitudinal reinforcement. 1) The ultimate drift capacity increased when a boundary column was present, the axial force level was low, the confined region size and amount of confinement increased, and shear span ratio increased. 2) The ultimate drift was simulated conservatively by adding flexural component based on a section analysis and shear component based on the 1999 AIJ guidelines. In this simulation, the plastic hinge length was assumed to be a function of wall thickness or length. Using Priestley et al.’s formula for the effective plastic hinge length, the ultimate drift ratio was underestimated. With half the wall length as the plastic hinge length reported by Wallace and Moehle, the ultimate drift ratio was overestimated. Further study is needed to determine the plastic hinge length of walls. 3) It is necessary to study the premature failure of walls with a rectangular section under flexure.

ACKNOWLEDGMENTS The research was funded by the promotion of building standard provisions by the Ministry of Land, Infrastructure, Transportation and Tourism in 2011 and 2012. Graduate students (Mr. Kouhei Sakamoto and Mr. Kodai Toya) who contributed this project are highly appreciated for their efforts.

REFERENCES 1. Ministry of Land, Infrastructure, Transportation and Tourism. The Building Standard Law of Japan, 1950. 2. 3.

European Committee for Standardization (2004), Eurocode 8 Design of structures for earthquake resistance - Part 1: General rules, seismic actions and rules for buildings. Paulay, T., Priestley, M. J. N., and Synge, A. J. (1982), “Ductility in Earthquake Resisting Squat Shearwalls,” ACI Journal, Vol. 79, No. 4, 257-269.

4. Architectural Inst. of Japan (2010). AIJ Standards for Structural Calculation of RC Structures, 2010. 5. Architectural Inst. of Japan (1999). AIJ Standards for Structural Calculation of RC Structures, 1999． 6. Mander, J. B., Priestley, M. J. N., and Park, R. Observed Stress-Strain Behavior of Confined

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Concrete, Journal of Structural Engineering, ASCE, Vol.114, No.8, August 1988, pp. 1827-1849. 7. Popovics, S.: A numerical approach to the complete stress-strain curve of concrete, Cement Concrete Research, No. 3, pp. 583-599, 1973. 8. Sakino, K. and Sun, Y., (1994). Stress-strain curve of concrete confined by rectilinear hoop, J. Structural and Construction Engineering, Architectural Institute of Japan, No. 461, pp. 95-104. (In Japanese) 9. Architectural Institute of Japan (1999), Guidelines for Earthquake Resistant Reinforced Concrete Buildings Based on Inelastic Deformation , Sec. 3.2.8 Hysteresis characteristics for structural walls, AIJ, pp. 326-327. (In Japanese) 10. Priestley, M. J. N., Seible, F., Calvi, G. M., Seismic Design and Retrofit of Bridges, WileyInterscience, 1996. 11. John W. Wallace and Jack P. Moehle, ”Ductility and Detailing Requirements of Bearing Wall Buildings.” J. Struct. Eng., 118(6), pp. 1625–1644, 1992. 12. Architectural Institute of Japan, Reconnaissance Report on The 2010 Chile Off Maule Earthquake and Reconnaissance Report on The 2011 New Zealand Christchurch Earthquake, p. 313, 2012.9. 13. Dodd, L. L. and Restrepo, J. I., Model for Predicting Cyclic Behavior of Reinforcing Steel, Journal of Structural Engineering, ASCE, 121(3), 433-445, 1995.

APPENDIX The relation between shear force, Q , and shear strain,  , employed the backbone curve listed in AIJ design guidelines9. It is expressed as a tri-linear curve in Figure 8. Two characteristic points correspond to shear cracking and shear yielding. Shear force

Q

 3GAW

W Qy

 2 GAW

W QSC

GAW

C

 y GAW

Shear strain



y

Figure 8. Tri-linear backbone curve model for shear force – shear strain relation for walls9 Shear cracking point W

QSC is the shear cracking force and expressed as

W

QSC  

 SC  tl kW

.

 SC is the shear cracking strength of concrete and expressed as  SC 

C

 t C  t   0  .

 t is the tensile strength of concrete and expressed as C  t  0.369 B . ( C  t and  B are in kgf/cm2)  0 is the compressive stress due to axial force. (kgf/cm2) t is the wall thickness, l is the center-to-center distance of boundary columns. 31  u 1  u 2 1  v  kW is the shape factor and is expressed as kW  for rectangle section. 41  u 3 1  v  u is the ratio of internal length to the total length of shear wall and expressed as l ' DC l'  u . l ' is the internal length of wall width.  DC : Sum of depths of l '  DC l ' DC 0.73

C

two boundary columns. v 

t bC

 is a reduction factor to compute the reliable shear crack strength. It ranges from 0.77 to 1.0 and 0.77 was used in this analysis.

 C is the shear cracking strain and expressed as  C 

QSC . GAW

W

G is the elastic shear modulus and AW is the section area of the wall. Shear yielding point W

Q y is shear force for the shear cracking and is expressed as

 0.053 pte 0.23 Fc  180    be j . ( Fc is in kgf/cm2)     Q 2 . 7 p  0 . 1  W y Wh Wh 0  M Qd   0.12    2 Fc is the compressive strength of concrete (kgf/cm ). be is the width of the equivalent rectangular section which has a same length of the wall with a barbell shape section. j is the external length of the wall. pte is the steel ratio of equivalent tensile vertical reinforcement in a percent and expressed as pte  100 t . d is the effective width of the wall. pWh is the steel be d ratio of horizontal wall panel shear reinforcement assuming be is the effective width of the wall. a It is expressed as pWh  Wh . ae s aWh is the area of horizontal wall panel shear reinforcement. a e is the effective width of the wall. s is the spacing for shear reinforcement. M Q is the shear span.

 y is the shear yield strain and assumed as 0.004. Property of the third branch

The slope of the third branch is assumed as  3 GAW .  3  0.001 .

10

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