Es2017fiberbeam-columnmodelfordiagonallyreinforcedconcretecouplingbeamsincorporatingshearandreinforcementslipeffects.pdf

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Fiber beam-column model for diagonally reinforced concrete coupling beams incorporating shear and reinforcement slip effects Article  in  Engineering Structures · December 2017 DOI: 10.1016/j.engstruct.2017.10.035

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This manuscript was published at: Ding R, Tao M X, Nie X*, Mo Y L. Fiber beam-column model for diagonally reinforced concrete coupling beams incorporating shear and reinforcement slip effects. Engineering Structures, 2017, 153: 191-204. The final publication is available at the journal website. The researchers can also privately get the final publication version via sending Email or ResearchGate message to Prof. Mu-Xuan Tao ([email protected]).

Fiber Beam-Column Model for Diagonally Reinforced Concrete Coupling Beams Incorporating Shear and Reinforcement Slip Effects Ran Ding a, Mu-Xuan Tao b, Xin Nie c* and Y.L. Mo d Abstract: Due to the improved energy-dissipation and deformation capacity compared to the conventionally reinforced concrete (RC) coupling beams, diagonally RC coupling beams are recommended by the ACI 318 code especially for a span-to-depth ratio of less than two and thus acquire more and more applications in coupled wall and core tube systems for tall buildings. This paper proposes a sufficiently accurate and efficient displacement-based fiber beam-column model for the nonlinear seismic analysis of diagonally RC coupling beams with span-to-depth ratios ranging between one and five. The model is developed on the platform of a general FEA package MSC.Marc. First, the conventional fiber beam-column element is modified to consider the flexural contribution of diagonal bars. Then the new section shear force-shear distortion and slip deformation rules are proposed and incorporated into the modified fiber element, respectively, since both the shear and reinforcement slip are critical mechanisms influencing the seismic performance of the beam. The equations for critical model parameters including the cracked shear stiffness and chord rotation limit are developed and verified based on the results of sixteen test specimens collected from previous research. The model is utilized to simulate the collected specimens together

a. Postdoctoral researcher, Key Lab. of Civil Engineering Safety and Durability of China Education Ministry, Dept. of Civil Engineering, Tsinghua University, Beijing, China 100084. b. Associate Professor, Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Beijing 100084, China. c. Postdoctoral researcher, Key Lab. of Civil Engineering Safety and Durability of China Education Ministry, Dept. of Civil Engineering, Tsinghua University, Beijing, China 100084. Email: [email protected] d. Professor, Dept. of Civil and Environmental Engineering, University of Houston, Houston, USA 77204. 1

with a coupled wall system and proves to be a powerful tool for the nonlinear seismic analysis of diagonally RC coupling beams and coupled walls with satisfied accuracy, efficiency and modeling convenience. Author Keywords: fiber beam-column model; diagonally reinforced concrete coupling beams; seismic analysis; hysteretic behavior; nonlinear shear behavior; reinforcement slip

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Nomenclature A Asd Asv b d

area of the coupling beam section sectional area of diagonal reinforcements sectional area of section confining transverse reinforcement width of the coupling beam effective depth of the coupling beam

Dd

diameter of diagonal bars

Vre Vsd,i Vssd,i Vun

Dl E f c'

diameter of top or bottom longitudinal bars concrete elastic modulus cylinder concrete compressive strength

α β δext

fsd fyd fyl G h

current stress of the diagonal bars yield strength of the diagonal bars yield strength of top or bottom longitudinal bars concrete shear modulus = E/2(1+v) depth of the coupling beam

δimit δm δre δt δun

i

δun,m

kre kt l l/h le

counter tracing successive cycle numbers in one direction reaching current maximum strain shape control parameters for monotonic tensile stress-strain curves of rebar cracked shear stiffness initial shear stiffness of the shear force-shear strain skeleton curve reloading reference slope slope before the reloading turning point length or span of the coupling beam span-to-depth ratio of the coupling beam length of elastic region for the bar anchorage

lpy

length of inelastic region for the bar anchorage

γun,m

nd nl

number of diagonal bars in each bundle number of top or bottom horizontal bars

θext ρsv

k1 k2 and k3

kcr kini

nsv

s Vcr Vm

εs εy γcr γlimit γm γre γt γun

ρsd

number of legs of sectional transverse rebars

v

3

spacing of section confining transverse rebars shear force of the shear cracking point shear force of unloading starting point from the skeleton curve shear force of reloading beginning point shear force of strength deterioration point shear force of unloading starting point from the other curves inclination angle of diagonal bars the shear stiffness reduction factor slippage of the bar caused by strain accumulation along the development length slip corresponding to γimit slip deformation of strength deterioration point slip of reloading beginning point slip of reloading turning point slip deformation of unloading starting point from other curves slip deformation of unloading starting point from the skeleton curve current strain of the diagonal bars yield strain of the diagonal bars shear strain of the cracking point at the shear skeleton curve drift ratio at strength degradation shear strain of the strength deterioration point shear strain of reloading beginning point shear strain of reloading turning point shear strain of unloading starting point from other curves shear strain of unloading starting point from the skeleton curve angle of the slip-induced crack at the beam ends ratio of section confining transverse reinforcement, ρsv = nsvAsv/bs ratio of diagonal reinforcements, ρsd = ndAsd/bh concrete Poisson’s ratio = 0.17

1. Introduction Nowadays reinforced concrete coupled walls and core tubes have become the most popular structural system for tall buildings due to their efficient resistance to seismic actions and compatibility with architectural requirements. In a coupled wall structural system the coupling beams are key components, which contribute significantly to structural stiffness under frequent earthquakes and energy dissipation during severe earthquakes. Numerous efforts have been devoted to investigating the seismic behavior and improving the seismic performance of RC coupling beams [1-14]. Diagonal confinement

α

Section stirups

Diagonal bar

Longitudinal bar

(a) Conventional layout

Longitudinal bar

(b) Traditional diagonal layout 1200 800

Force (kN)

Sectional confineme nt

α

Diagonal bar

Longitudinal bar

400 0 -400

-1200

-12

-8

-4

0 4 Drift ratio (%)

8

12

(d) Typical hysteretic curves

(c) Improved diagonal layout

Figure 1.

Diagonal layout (CB20-1) Conventional layout (CB20-2)

-800

Comparison of RC coupling beams with different reinforcement layouts

Among all kinds of RC coupling beams, in practice the beams with conventional and diagonal reinforcement layouts are the most widely used. Compared with the coupling beams constructed with reinforcement scheme similar to frame beams(Figure 1[a]), it is confirmed by many test results that the diagonal reinforcement layout (Figure 1[b]) that was first introduced by Paulay and Binney [3] benefits the coupling beam with significantly a higher deformation and 4

energy-dissipation capacity [3-11], as illustrated by Figure 1(d). In addition, ACI 318-14 [15] introduced an alternative detailing option, as shown in Figure 1(c), where transverse reinforcement is placed to confine the entire beam cross section instead of the diagonal bar bundles. Consequently the construction of a diagonally RC coupling beam is greatly simplified. Recently, the nonlinear dynamic analysis has gained increasing attention and application in the performance-based seismic design of tall buildings, which requires nonlinear models for all structural members with adequate accuracy, efficiency and numerical stability. Displacement-based fiber beam-column elements have been successfully used to model the frame beams and columns due to their simplicity and accuracy [16,17]. However, the diagonally RC coupling beams featured with complex hysteretic behavior cannot be reasonably simulated by the conventional fiber model. It is thus necessary to develop an enhanced fiber model for the seismic structural analysis of coupling beams in tall buildings that is capable of considering the nonlinear shear and reinforcement slip effects and still keeps the simplicity and convenience of the original fiber model. To date, several researchers have reported macro models for diagonally RC coupling beams based on springs, hinges and line elements [18-21]. The prior research has greatly advanced the modeling of diagonally reinforced coupling beams. However, there remains a significant gap in the literature. Previous models of coupling beams have not fully considered the three important deformation modes, which are the shear, flexure and slip effects, with adequately verified model parameters by a large number of tests reported in literature. Therefore the applicability of the models remains doubtful. In addition, the models cannot be directly applied into the framework of fiber models. We have previously proposed a fiber beam-column element for conventionally RC

5

coupling beams [22]. However, the model can consider neither the contribution of diagonal bars nor the reinforcement slip mechanism and the higher deformation and energy-dissipation capacity of diagonally RC coupling beams, which are the objects of this study. Therefore, this study aims to develop a fiber beam-column model for diagonally RC coupling beams based on a comprehensive representation of the complex deformation mechanism, which can be directly applied to the seismic analysis of high-rise structures by the general FEA software. It is worth noting that a total of sixteen test specimens collected from seven different research groups are summarized to validate adequately the proposed model and calibrate critical model parameters so that the model can accommodate beams with a broad range of span-to-depth ratios from one to five. First, the fiber model is modified to consider the flexural contribution of diagonal bars. Then the modified fiber element is combined with the proposed section shear force versus shear strain and slip deformation laws in the general FEA software MSC.Marc [23]. The methods for critical parameters including shear stiffness after cracking and chord rotation limit are also developed and verified by the collected test results. Finally, the model is used to simulate the seismic performance of all test specimens and a RC coupled wall specimen, and the predicted hysteretic behavior is compared to the measured results. 2. Review of diagonally reinforced coupling beam tests A number of tests have been conducted to assess the seismic behavior of diagonally RC coupling beams [4-11]. In this study, sixteen test specimens are reviewed and the key parameters are summarized in Table 1. As shown in Figure 1(b) and (c), two typical reinforcement layouts exist in the literature. The first is proposed by Paulay and Binney [3] originally and recommended by

6

ACI 318-05 [24], where the stirrups are placed to confine both the diagonal bars and the horizontal bars. The second is an improved scheme and specified in ACI 318-14 [15], where stirrups only surround the horizontal bars, giving rise to a much more convenient construction process. Comparative tests of the two layouts have been conducted by Naish et al. [4], and it has been proven that the seismic performance of beams detailed based on ACI 318-14 [15] is equivalent or slightly better than beams designed as per ACI 318-05 [24]. Table 1. Database of diagonally RC coupling beam specimens Reference specimen

b

h

l

(mm) (mm) (mm)

l/h

α

Dd

(°) (mm)

nd

fyd

Dl

(MPa) (mm)

nl

fyl

fc’

(MPa) (MPa)

Vtest

Vmodel

(kN)

(kN)

Diagonal

Section

stirrup (mm) stirrup(mm)

CB-24F 304.8 381 914.4 2.40 15.7 22.2

6

483 9.525 3

483

47.3

769.3

780.1

-1

D9.5@76

CB-24D 304.8 381 914.4 2.40 15.7 22.2

6

483 9.525 3

483

47.3

708.4

780.1

D9.5@63

D6.4@63

CB-33F 304.8 457.2 1524 3.33 12.3 22.2

6

483 9.525 3

483

47.3

556.2

591.8

-

D9.5@76

CB-33D 304.8 457.2 1524 3.33 12.3 22.2

6

483 9.525 3

483

47.3

536.7

591.8

D9.5@63

D6.4@63

Naish [4]

CB10

250 500 500 1.00 26.0

25

4

486

10

3

468

34.5

1449.9

1401.3

-

D13@100

CB20

300 500 1000 2.00 16.0

29

4

466

13

3

502

52.1

1068.6

1062.9

-

D13@100

CB30DA 300 500 1500 3.00 8.8

32

4

465

13

2

441

39.7

701.3

680.9

D10@150

D13@200

CB30DB 300 500 1500 3.00 8.8

32

4

465

10

3

475

38.4

732.9

701.5

-

D10@100

CCB11 120 600 600 1.00 31.0

8

6

517

8

2

517

37.8

350.5

363.7

D6@60

D8@140

254 356 914 2.57 13.0 25.4

4

431

19

2

418

37.6

615.4

620.9

D9.5@76

D6.4@76

Lim [5,6]

Kwan and Zhao [7] Fortney [8]

DCB1

Galano and

P07

150 400 600 1.50 22.1

10

4

567

6

2

567

54.0

238.6

251.6

-

D6@140

Vignoli [9]

P12

150 400 600 1.50 19.7

10

4

567

6

2

567

41.6

242.2

245.9

D6@90

D6@180

CB2A

130 500 500 1.00 26.0

10

4

504

6

3

281

28.5

287.5

300.3

D6@50

D6@120

CB2B

130 300 500 1.67 18.0

10

4

504

6

3

281

26.3

172.0

162.0

D6@50

D6@120

9.5

2

487

53.3

46.0

C6

101.6 169.4 423.4 2.50 18.2

5

2

492

18.1

62.73

-

D4.9@34

57.3

37.1

34.9

5

2

490

23.9

-

D4.9@34

27.9

24.9

Tassios [10]

12.72

1

408

9.5

2

569

Shiu [11] C8

101.6 169.4 845.8 5.00 9.6 12.7

1

433

Notes: 1. For specimens following the recommended transverse reinforcement layout of ACI 318-14, diagonal stirrups are removed; 2. The diagonal bars are unsymmetrical for Specimens C6 and C8; 3. The measured maximum shear forces are different in two directions due to the unsymmetrical layout of diagonal bars.

It is concluded from the test results that the failure of the coupling beam is mainly attributed to the crushing of concrete at the ends, and the buckling of diagonal bars under compression as well as

7

fracture under tension. The deformation consists of flexure, shear and reinforcement slip mechanisms, which are closely related to the span-to-depth ratio l/h. For l/h <2.0, shear deformation will dominate the overall behavior [5], whereas if l/h >2.0, beam-wall interface rotation caused by reinforcement slip constitutes most of the total deformation [4,6]. It is worth noting that the shear-sliding deformation mechanism and shear-tension failure mode both typical in conventionally RC coupling beams [1,2] are avoided in diagonally RC coupling beams, due to the contribution of diagonal bars. In addition, typical measured load-displacement curves exhibit much better fully rounded hysteretic loops without pinching effects and much larger ultimate deformation (as much as 5%-8% beam chord rotation) than conventional coupling beams [4,5]. The effective stiffness is reported to be approximately 15%-20% of the initial bending stiffness EcIg generally and can be even lower than 10% of EcIg [4,25]. Based on the above summary and analysis, the proposed model should incorporate the three distinct deformation mechanisms and give a precise prediction to the beam strength, effective stiffness, deformation and energy dissipation capacity.

Compressive force by concrete strut

Tensile force T=fsdndAsd (fsd>fyd ) α

Compressive force C=fydndAsd

Figure 2.

Basic mechanism of diagonally reinforced coupling beams

3. Modified fiber model incorporating diagonal reinforcement In recent years, the authors’ research group [16,17] has developed a conventional displacement-based fiber beam-column element for RC, steel and steel-concrete composite structural members in MSC.Marc [23] for the two-node beam element No. 98 with one integration 8

point in the middle. The uniaxial constitutive laws of steel, rebar and concrete have been discussed in detail [16,17], which will not be repeated in this paper. Based on the traditional fiber element, this research first extends the fiber model to simulate the flexural contribution of diagonal bars. l

b

A

3Al

h

ltanα

α

2Am 4Ad

Diagonal bar Longitudinal bar

A

A-A (a) Actual reinforcement layout

l A

ltanα

3Al 2Am 2Adcos(α)

Longitudinal bar A Equivalent diagonal bar

A-A (b) Proposed fiber model

Figure 3. Modified fiber model incorporating diagonal bars (nd=4)

As pointed out by Hwang et al. [5,6] and explained in Figure 2, diagonal bars not only increase the shear capacity with their vertical components but also improve the bending capacity with their horizontal components. In addition, it is repeatedly verified by many test results that the flexural strength of the diagonally RC coupling beams perfectly represents the bearing capacity of the beam, since the shear strength is sufficient to prevent the shear failure featured with unfavorable wide diagonal cracks. According to this mechanism, the horizontal component of the diagonal bars is incorporated into the modified fiber model, as described in Figure 3. Each diagonal bundle is divided into two equivalent bars, the area of which is 0.5ndAsdcosα. The equivalent bars are located at the left and right side of the section with the y-coordinate equal to the centroid of the bundle. Using this modified model, the maximum shear force sustained by the beam and the bending behavior can be reasonably predicted. Table 1 list the comparison between the simulated (Vmodel) 9

and measured maximum shear force (Vtest) of the sixteen beams. The error ranges between -13.7% and 10.3% and the average error is only 0.7% which clearly demonstrate that the modified fiber model is able to predict the beam capacity accurately. 4. Section shear force-shear strain relationship Shear deformation is a remarkable deformation component for RC coupling beams, especially for beams with small span-to-depth ratios. In a recent study for conventionally RC coupling beams [22], the authors have proposed to consider the flexure-shear deformation by incorporating the nonlinear section shear force (V)-shear strain (γ) relationship into the traditional fiber model. As illustrated in Figure 4, for the compression-flexure or tension-flexure behavior, the conventional fiber model calculates the section tangent stiffness matrix as n n  n  − E A E A y Etk Ak xk  ∑ ∑ ∑ tk k tk k k  =  k 1 =k 1 =k 1  n n  n  2 k 1  ∑ Etk Ak yk = Etk Ak yk −∑ Etk Ak xk yk  ∑ =  k 1 =k 1 =k 1  n n  n  2 Etk Ak xk  ∑  −∑ Etk Ak xk −∑ Etk Ak xk yk =  k 1 =k 1 =k 1 

(1)

where Etk is the fiber tangent stiffness of the kth fiber; Ak is the area of the kth fiber and xk and yk are the x and y coordinates of the center of the kth fiber in the section local coordination system. node

My

Y

Z

Vx Integration point

φy node

N

γy Integration point

[N, Mx, My] =k1[ε, φx, φy] [Vx, Vy, T ]T=k2[γx, γy, ω]T T

Figure 4.

Mx

Vy

X

T

T

φx

γx ε

ω

Generalized sectional stress, strain and stiffness

10

For the shear-torsion behavior, the section tangent stiffness matrix is dVx / dγ x  k2 =  0  0 

0 dVy / dγ y 0

0   0  GI p 

(2)

The section force vector D is n n  n  D  ∑ σ k Ak , Vx ( γ x ) , Vy ( γ y ) , − ∑ σ k Ak yk , ∑ σ k Ak xk , GI pω  = = = k 1= k 1 k 1 

T

(3)

By defining the nonlinear section shear force (V)-shear strain (γ) relationship which is discussed in the following two sections, the tangent shear stiffness in k2 can be calculated in each step; thus the nonlinear shear behavior is simulated. This study follows the same principle as that proposed for conventionally RC coupling beams [22], but modifies the backbone curves and hysteretic rules so that the shear behavior of diagonally RC coupling beams can be reasonably simulated. Actually, the shear stiffness after diagonal cracking, the unloading and reloading stiffness together with the strength deterioration point are all re-investigated and new equations are proposed. 4.1 Backbone curves According to the observed damage mode of diagonally reinforced coupling beams, a bilinear model is proposed as the backbone curve, which considers the significantly reduced shear stiffness after shear cracking, while neglects the load degradation because no shear failure has ever been observed and the flexural strength controls the beam capacity. As shown in Figure 5, the key parameters in the two branches are defined as:

Vcr = (0.158 f c′ + 34.4 ρsv d / l )bd ≤ 0.29 f c′bd

(4)

kini = GA / 1.2

(5)

11

kcr = β kini

(6)

= β 1.2178 ρ sd (l / h)0.5 + 0.0139

(7)

It should be noted that Equation (4) is from ACI 426 [26] and holds for SI units. The calibration of Equation (7) will be discussed in Section 7.1. V G A (γu n,m,Vm) A1(γu n,Vu n) A2

kcr

Vsd,1 Vsd,2

(γcr ,Vcr )

kini

γm

kt

F1 F B B1B2

E1

(γre,Vre)

(γre,0)

E C1

J

C2

γm

γ

backbon e u nloading point

D2 D1 D K

Figure 5.

C

Vcr

strength deterioration point reloading s tartin g poin t reloading turning point

H

Shear force-shear strain skeleton curves and hysteretic rules

4.2 Hysteretic rules The detailed description of the unloading and reloading rules can be found in [22], this section only emphasizes the difference between the conventionally and diagonally RC coupling beams. Unloading rule As shown in Figure 5(a), if Vm>Vcr(AB), or Vun>Vcr (A1B1), the unloading slope kun is calculated as: kun = k1 −

β ( k1 − k2 ) 3β kini ; k2 = 0.2 β kini (γ un,m − γ cr ) ; k1 = 120γ cr

(8)

Reloading rule For reloading in the reverse direction, reloading starting from point (γre,0) follows a bilinear curve before going back to the backbone curves. The curve passes through the reloading turning 12

point (γt, Vt) (B1C1, B2C2, EF and E1F1) and then aims at the strength deterioration reference point (γm, Vsd) (C1D1, C2D2, F1A2 and FA1). The coordinates of the two reference points are calculated as = γt

(1.15-0.15l /h ) γ re

= = k t 2.5 kre ; kre

(1 ≤ l /h ≤ 5)

Vsd,i

γ m − γ re

γ m = γ un,m = Vsd,i Vm (0.95 −

(9) (10)

(11)

γ un,m i ) 1000γ cr

(12)

Upon the first unloading from the current maximum shear strain, i is set to be 1. The value of i is counted every time the load reverses with the maximum strain ranging from 0.8γun,m to 1.2γun,m[22]. 5 Section shear force-slip deformation relationship 5.1 Slip-induced beam deformation analysis For beams with l/h larger than 2.0, it is found that the slip of diagonal reinforcement at the beam-wall interface contributes significantly to the beam deformations. As shown in Figure 6(a). the diagonal bars under tension may extend, or slip relative to the wall, due to the accumulated tensile strain over the embedment length and the slip will cause rigid-body rotation of the beam. Alsiwat and Saatcioglu [27] proposed an approach to predict reasonably the reinforcement anchorage slip and Naish et al. [18] utilized the method with a few modifications to calculate the slip-related rotations corresponding to the yield of diagonal bars.

13

Stepped bond stress ub

A s fs reba r

Figure 6.

uf 0.5 fc'

ue

Diagonal reinforcement slip model in the proposed fiber model

This study aims to simulate the whole-process hysteretic response of slip deformations on the basis of the method proposed by Naish et al. [18]. If adequate embedment of the diagonal reinforcement is provided as is the case for all test specimens, the slip at the unloaded end can be neglected. As shown in Figure 6(b), the reinforcement slip is calculated using a stepped-bond stress distribution where an elastic uniform bond stress ue is assumed for elastic steel stresses and a frictional uniform bond stress uf is assumed for stresses exceeding the yield strength. The elastic bond stress ue can be calculated according to ACI Committee 408 [28] and equals to 0.87 f c′ MPa for most cases. The inelastic bond stress uf is calculated as 0.5 f c′ MPa for simplicity as suggested by Sezen et al. [29] and Pan et al. [30]. Thus the reinforcement slip can be calculated by integrating the strains over the development length that equals the area below the strain diagram in Figure 6(b).

l f D δ ext= 1.25 × ε s × e ; le= sd d ; ue= 0.87 f c′ (ε s ≤ ε y ) 2 4ue 14

(13)

l +l f D l l 2 1 δ ext= 1.25 × ε y × e + ∫ ε ( x)dx= 1.25 × ε y × e + ε y yd d (k3 − 1)( k1 + k2 ); 2 l 2 4uf 3 3 ( fsd − f yd ) Dd ; uf 0.5 f c′ (ε s > ε y ) lpy = = 4uf e

py

e

θ ext =

δ ext d−x

(14)

(15)

where δext is the bar slippage caused by strain accumulation along the development length; le and lpy represent the elastic and inelastic region lengths; θext represents angle of the slip-induced crack at the beam ends; fsd, εs and εy are the current stress, strain and yield strain of the diagonal bars. The factor 1.25 in Equation (13) is suggested by Naish et al. [18] to account for approximately the severe beam end damage due to the shear effects in coupling beams with low span-to-depth ratio and hysteretic loading influences. k1, k2 and k3 are the shape control parameters for the monotonic tensile stress-strain curves of reinforcement [16,17], as shown in Figure 6(c); x is the neutral axis depth and d is the effective depth with respect to the outmost diagonal bar. lslip Slip element at the end Spread pl asticity fi ber element for flexure

node

node

nonlinear shear hinge incorporated in the fiber model

Internal fiber element considering nonlinear shear effects

Figure 7.

Proposed coupling beam fiber model considering shear and slip deformation

Based on the above method, the slip-related rotation at the beam end can be calculated and then the corresponding additional displacement of the beam is obtained as lθext. In the proposed coupling beam model, as shown in Figure 7, the shear and flexural deformation are simulated by the internal shear element which has been developed in the above section. On the other hand, the slip 15

deformation is simulated by two additional slip elements at the ends which are similar to the internal shear element but modify the shear element by setting the shear stiffness according to the shear force (V) versus slip deformation (δext) relationship. The slip deformation, regarded as the shear strain of the slip element, is lθext/(2lslip), where lslip is the length of the slip element at the beam ends. It is worth noting that despite the fact that the slip-related additional deformation is actually the rigid-body rotation of the beam, the model assumes an equivalent deformation that is the local shear deformation at beam ends so that the shear element can be conveniently converted to the slip element in order to simulate the slip deformation with only a few modifications. Therefore, the proposed coupling beam fiber model with shear elements in the middle and slip elements at the ends in the series can reasonably reflect the deformation modes of diagonally RC coupling beams. The V - δext relationship is illustrated in detail in Sections 5.2 and 5.3. 5.2 Backbone curves

Shear force V

(δs,3,Vs,3) (δs,2,Vs,2)

(δs,1,Vs,1)

Calculated curve Simplified trilinear curve

Slip deformation δ

Figure 8.

Proposed simplified trilinear shear force-slip deformation model

The flexural analysis with the modified traditional fiber model considering diagonal reinforcement is conducted first to obtain the monotonic relationship between the shear force and the tensile stress of the diagonal bar. Then the slip deformation (δext) at each step can be calculated 16

by Equations (13)-(15), and the V - δext monotonic curve is thus obtained. However, the curve cannot be explicitly expressed, which brings great difficulty to the development and numerical stability of the element. To overcome this barrier, as shown in Figure 8, a simplified trilinear curve is suggested to represent the calculated V - δext curve with Equations (13)-(15). Therefore the V - δext backbone curves can be defined before analysis using an explicit expression, as shown in Figure 9. V Vs,3 Vs,2 Vssd,1 Vssd,2 Backbone curve

kt δm

Y (δt,Vt) Q

δpY X

Vs,1 W

δlimit F

(δu n,Vu n)

A

Z A1

V

Descending branch

Vs,2

H S Vs,1

E

C1

D δpE δpU I O B T C U J

P δpQ

(δre,0)

L(δre,Vre)

N

G R

B1

k3

Vs,1

δm

δ

k2 Vs,1 k 1

k4

Vs,1

backbon e u nloading point

δ

strength deterioration point reloading s tartin g poin t reloading turning point

K V -δlimit

Vssd,1

(a) Definition of hysteretic rules

(b) Definition of critical stiffnesses

M (δu n,m,Vm)

Figure 9.

Proposed hysteretic shear force-slip deformation model

5.3 Hysteretic rules The detailed unloading and reloading rules are similar to those of the shear force-shear sliding deformation in conventionally RC coupling beams, which can be found in [22], thus this section only emphasizes the difference between the two beams with different reinforcement layouts. Unloading rule 1. If Vm < Vs,1, and δun,m < δs,2, or Vun < Vs,1 and δun < δs,2 (CD), the unloading slope equals to k1. 2. If Vm>Vs,1 and δun,m<δs,2, or Vun>Vs,1 and δun<δs,2, unloading will follow a linear curve. The 17

stiffness is given by Equation (16) (AB and KL). Vs,2 + Vs,1 k −k kun = k1 − 1 2 (δ un,m − δ s,1 ) ; k1 = ks,1 ; k2 = δ s,2 − δ s,1 δ s,2 + δ s,1

(16)

where k1 and k2 are defined in Figure 9(b). 3. If δun,m>δs,2 and Vm>Vs,1, or Vun>Vs,1 and δun>δs,2, unloading will follow a bilinear curve. The stiffness when the shear force is larger than Vs,1 is given by Equation (17) (GH, RS, MN, VW and A1B1) and less than Vs,1 is given by Equation (18) (HI, ST, NP, WX and B1C1).

2δ k kun = s,2 2 ≥ k3

(V > V )

(17)

4δ k kun =s,2 2 ≥ k4 5δ un,m

(V ≤ V )

(18)

δ un,m

s,1

s,1

where k3 and k4 are defined in Figure 9(b). 4. If Vmδs,2 (UV), or Vunδs,2, the unloading slope can be calculated with Equation (18). Reloading rule For reloading in the reverse direction, reloading starting from point (δre, 0) follows a bilinear curve before reaching the backbone curve. The curve passes through the reloading turning point (δt, Vt) (DE, PQ, TU and XY) and then aims at the strength deterioration reference point (δm, Vssd) (EF, QR, UV and YZ). The coordinates of the two reference points are calculated as: = δt

(1.15-0.15l /h ) δ re

= k t 2.5 = kre ; kre

(1 ≤ l /h ≤ 5)

(19)

Vssd,i

(20)

δ m − δ re

δ m = δ un,m

18

(21)

 δ δ = Vssd,i Vm exp  −0.1 un,m × i − 0.015 i  un,m δ limit  δ limit 

   

(22)

It may be noted that although Equation (22) seems a little complex, it has already been programed and can be calculated automatically. Actually, the basic form of Equation (22) was first proposed by Ozcebe and Saatcioglu [31] to calculate the strength deterioration reference point in hysteretic shear models for columns, and also adopted by some other researchers such as Xu and Zhang [32] and Ding et al. [22]. In this study, the equation is found to be able to simulate reasonably the strength deterioration in the diagonally RC coupling beams after setting the two coefficients to be 0.1 and 0.015. 6 Fiber beam-column element considering shear and slip deformation As shown in Figure 7, the shear element considering nonlinear shear effect and the slip element considering the nonlinear slip mechanism are developed by incorporating the traditional fiber element with section shear force-shear strain and slip deformation relations, respectively. When the two elements are combined together to model the coupling beam, the shear force in all the elements are always kept the same. The failure of the coupling beam is mainly caused by the crushing of concrete at the end, the buckling of diagonal bars under compression and fracture under tension, which are all closely related to the large lateral displacement of the beam. Based on test data analysis, the following formula is proposed to calculate the drift ratio at strength degradation γlimit: 0.0239 + 4.3209 ρsv l / h ( ρsv l / h ≤ 0.0125) γ limit = = γ limit 0.07791

ρsv =

Asv bs

( ρsv l / h > 0.0125)

(without diagonal stirrups)

19

(23)

ρsv =

1.2Asv bs

(with diagonal stirrups)

where ρsv is the transverse reinforcement ratio and the factor 1.2 is an approximation to consider the contribution of stirrups surrounding diagonal bars. When the beam drift ratio reaches the drift limit, the load-deformation curve will go into the descending branch. As shown in Figure 9, this phenomenon is realized by the end slip elements which enter the descending stage at δlimit corresponding to the beam drift limit γlimit, while the internal shear element will unload. The slope of the descending branch of the shear force-drift ratio curve is set to be 0.1 times the initial stiffness, according to the test data analysis. Equation (23) will be discussed in detail in Section 7.2. 7 Calibration of critical model parameters 7.1 Shear stiffness after diagonal cracking In the proposed model, the nonlinear flexural deformation has been properly simulated by the modified fiber model incorporating diagonal bars in Section 3, and the reinforcement slip deformation can be reasonably considered based on the method proposed in Section 5.1. Therefore, the remaining problem is the calculation of nonlinear shear deformation, which mainly depends on the shear stiffness after diagonal cracking. By assuming that the flexure and slip deformation are properly calculated, the actual cracked shear stiffness ratio β can be acquired by matching the proposed model with measured shear force-total deformation curves. According to the test observations, cracked shear stiffness is closely related to the diagonal reinforcement ratio ρsd and span-to-depth ratio l/h. To further reveal their relationship, a key parameter ρsd(l/h)0.5 reflecting the integrated influences of ρsd and l/h is proposed, and the correlation between ρsd(l/h)0.5 and α is

20

plotted in Figure 10. A satisfying positive correlation can be found and the regression formula Equation (7) is thus proposed. 10

Chord rotation at strength degradation γlimit (%)

Shear stiffness reduction factor β=kcr/kini

0.08

0.06

0.04 y=1.2178x+0.0139 R2=0.7917 0.02

0.00 0.00

0.01

0.02

0.03

0.04

6 y=4.3209x+0.0239 R2=0.6632

4 2 0

0.05

γlimit =7.791%

8

ρsv(l/h)0.5=1.25% 0.000

Key parameter ρsd(l/h)0.5

Figure 10. Regression analysis of cracked shear stiffness ratio

Figure 11.

0.005

0.010 0.015 Key parameter ρsv(l/h)0.5

0.020

0.025

Regression analysis of drift ratio limit γlimit

7.2 Drift ratio at strength degradation Based on the large number of test results, it can be found that the drift ratio where load begins to drop (γlimit) is significantly affected by the span-to-depth ratio and transverse reinforcement ratio including the stirrups surrounding the diagonal bars and horizontal bars. Figure 11 plots the relationship between the measured value of γlimit and the unified parameter ρsv(l/h)0.5. A clear positive correlation can be found before 1.25%; thus Equation (23) is proposed by regression of data points. 8 Model application in diagonally RC coupling beam and coupled wall analysis The above proposed and calibrated model is now applied to all the test specimens listed in Table 1, to further verify the accuracy of the model. In addition, a RC coupled wall specimen is simulated to show the feasibility of the proposed model in the seismic analysis of a coupled wall structural system. 8.1 Beam tests by Lim et al. [5,6] 21

Lim et al. [5,6] reported the test results of four diagonally reinforced beams with different span-to-depth ratios--CB10, CB20, CB30DA and CB30DB--and the detailed information can be found in Table 1. Specimens CB30DA and CB30DB have different transverse reinforcement schemes, which follows the ACI318-05[23] and ACI318-14[15] specifications respectively. It is concluded that Specimen CB10 with l/h=1.0 shows significant shear behavior and fails in the shear-flexure mode. The other three specimens show obvious concrete crushing and rotation at the beam ends and all fail in the flexure mode. The behavior of Specimens CB30DA and CB30DB are basically the same before the 7.7% rotation, while after that Specimen CB30DA failed and Specimen CB30DB continues to sustain considerable shear forces until a drift ratio of 9.9%. This might be attributable to the different transverse reinforcement detailing. 1500

1500 CB10

1000

Lateral load (kN)

Lateral load (kN)

1000 500 0 -500

-1000 -1500 -12 800

8

12

600

400 200 0 -200 -400

-800 -12

Measured results Predicted res ults -8

-500

-4 0 4 Beam chord rotation (%) (c) CB30DA

Measured results Predicted res ults -8

800

CB30DA

-600

0

-1000 -12

Lateral load (kN)

Lateral load (kN)

600

-4 0 4 Beam chord rotation (%) (a) CB10

500

-750

Measured results Predicted res ults -8

CB20

8

8

12

CB30DB

400 200 0 -200 -400 Measured results Predicted res ults

-600 12

-4 0 4 Beam chord rotation (%) (b) CB20

-800 -12

-8

-4 0 4 Beam chord rotation (%) (d) CB30DB

8

12

Figure 12. Comparison of predicted and measured lateral load versus beam chord rotation curves

As shown in Figure 12, the predicted shear force versus chord rotation hysteretic relations correlate well with the experimental results. The maximum load, peak rotation, loading and unloading stiffness and the overall shape of loops are all accurately simulated. The drop of load 22

caused by the crushing of concrete and buckling of bars is generally so sudden that it is quite difficult to capture this behavior accurately. 1500

CB30DB

CB20

1000

1000

500 0 -500 Slip response Shear res ponse Total response

-750 -1000 -12

-8

0 4 -4 Beam chord rotation (%) (a) CB10 (l/h=1.0)

8

Lateral load (kN)

1000

Lateral load (kN)

Lateral load (kN)

1500

1500 CB10

500 0 -500 Slip response Shear res ponse Total response

-750 12

-1000 -12

-8

-4 0 4 Beam chord rotation (%) (b) CB20 (l/h=2.0)

8

500 0 -500 Slip response Shear res ponse Total response

-750

12

-1000 -12

-8

-4 0 4 Beam chord rotation (%) (c) CB30DB (l/h=3.0)

8

12

Figure 13. Analysis of deformation components for specimens with different span-to-depth ratios

The comparison of different response components are then analyzed based on the proposed model, as shown in Figure 13. Generally, the shear and slip deformation take up most of the total responses; thus the flexure deformation is not plotted for clarity. It is indicated that for specimens with small l/h such as CB10, the shear deformation contributes a large proportion to the total response; for specimens with moderate l/h such as CB20, the shear and slip deformation are almost the same; and for specimens with relatively large l/h such as CB30DB, slip rotation greatly overshadows the shear deformation and contributes most of the deformation. Figure 14 further summarizes the ratio of slip rotation to total deformation at flexural strength. It can be seen that the slip deformation ratio tends to increase with the span-to-depth ratio and can reach as large as 50%, which is also emphasized by Naish et al. [4,18]. The cumulative energy consumed by the specimens before the sudden failure are also calculated with the predicted hysteretic curves and compared to the measured results, as illustrated in Figure 15. It can be demonstrated that the proposed model is able to give a satisfactory prediction of the cumulative energy of diagonally reinforced coupling beams, which is of concern

23

60

1500

50

1200

Numerical results

Cumulative energy (kNm)

Slip deformation/Total deformation(%)

for structural seismic analysis.

40 30 20

Test results

900 600 300

10 0

0

0

1

2 Span-to-depth ratio l/h

3

CB10

4

Figure 14. Trends of slip rotation to total deformation ratio

Figure 15.

CB20 CB30DA Speicemen

CB30DB

Comparison of cumulative energy

8.2 Beam tests by other researchers [4,7-11] All the other twelve specimens in the database in Table 1, tested by Naish et al. [4], Kwan and Zhao [7], Fortney et al. [8], Galano and Vignoli [9], Tassios et al. [10] and Shiu et al. [11], are also simulated by the proposed model. Figure 16 plots both the numerical and experimental results, which extensively demonstrate the accuracy of the proposed model. It is worth noting that Naish [33] attempted to apply his model to coupling beams with a small span-to-depth ratio, as shown in Figure 16(g). The specimen CCB11 was tested by Kwan and Zhao [7] and its span-to-depth ratio was only 1.17. The predicted shear force-beam chord rotation curve is found to overestimate the stiffness significantly and underestimate the shear capacity, thus demonstrating that the model is not able to predict the behavior of coupling beams with small span-to-depth ratios. However, the proposed model in this paper successfully predicts the hysteretic behavior of this specimen due to the reasonable consideration of shear effects and adequately calibrated parameters.

24

1000

500

Lateral load (kN)

250 0 -250 -500 Measured results Predicted res ults

-750 -1000 -12

-8

0 4 -4 Beam chord rotation (%) (a) CB24F

8

12

0 -250 -500 Measured results Predicted res ults -8

-4 0 4 Beam chord rotation (%) (b) CB24D

8

300

12

250 0 -250 -500 Measured results Predicted res ults

-750 -1000 -12

-8

-4 0 4 Beam chord rotation (%) (d) CB33D

8

100 0 -100 -200 Measured results Predicted res ults -8

-4 0 4 Beam chord rotation (%) (e) CB2A

8

600 Lateral load (kN)

200 100 0 -100 -200

Naish et al. 2010 Measured results Proposed model

-300 -400 -12

-8

0 4 -4 Beam chord rotation (%) (g) CCB11

8

300

200 0 -200 -400 Measured results Predicted res ults

60 Lateral load (kN)

150 75 0 -75 -150 Measured results Predicted res ults

-225 -300 -12

-8

-4 0 4 Beam chord rotation (%) (j) P12

8

-8

-4 0 4 Beam chord rotation (%) (h) DCB1

8

-100 Measured results Predicted res ults -8

-4 0 4 Beam chord rotation (%) (f) CB2B

8

12

P07

150 75 0 -75 -150 Measured results Predicted res ults -8

-4 0 4 Beam chord rotation (%) (i) P07

8

12

40

20 0 -20 -40 Measured results Predicted res ults -8

C8

30

40

-80 -12

0 -50

-300 -12

12

C6

-60 12

12

CB2B

-225

80 P12

225

8

50

225

400

-800 -12

-4 0 4 Beam chord rotation (%) (b) CB33F

300 DCB1

-600 12

-8

100

-200 -12

12

Lateral load (kN)

CCB11

300

Measured results Predicted res ults

-150

800

400

-500

150

200

-400 -12

0 -250

200 CB2A

-300 12

250

-1000 -12

Lateral load (kN)

500

CB33F

500

-750

400

Lateral load (kN)

Lateral load (kN)

250

-1000 -12

CB33D

750

Lateral load (kN)

750

500

-750

1000

Lateral load (kN)

1000 CB24D

Lateral load (kN)

Lateral load (kN)

750

J

CB24F

750

Lateral load (kN)

1000

-4 0 4 Beam chord rotation (%) (k) C6

8

20 10 0 -10 -20 Measured results Predicted res ults

-30 12

-40

-12

-8

-4 4 0 Beam chord rotation (%) (l) C8

8

12

Figure 16. Comparison of predicted and measured lateral load versus beam chord rotation curves

8.3 Coupled wall tests by Cheng et al. [34] Cheng et al. [34] conducted quasi-static tests on coupled walls with diagonal coupling beams. The specimen CW-RC, as shown in Figure 17, is chosen to be simulated by the proposed coupling beam model. The traditional fiber model without shear and slip effects is also applied to the system to further demonstrate the accuracy of the proposed coupling beam model. The finite element

25

modeling concept for the coupled wall system is briefly introduced first. As shown in Figure 18(a), a coupled wall system consists of two individual wall piers, boundary confined columns and coupling beams. The individual walls can be modeled with a thick shell element provided by the MSC.MARC [23]. Because the shear walls are reinforced with horizontal and vertical distributed rebars, the layered material model can be applied as the material constitutive relationship [35,36]. The detailed parameters for the material models have been suggested by Nie et al. [35]. The confined boundary column consists of longitudinal and transverse reinforcement, which is mainly subjected to axial force. Therefore, it can be simulated with the traditional fiber beam-column element. The coupling beam is modeled with the coupling beam element proposed in this paper. 300

122.5kN 122.5kN 122.5kN 122.5kN 180

4#5 Diagonal rebars A 3#3 Horizontal rebars

4#5

19°

1500

A

300

1500

300

P

#3

#3@75mm transverse rebars

2#3 A-A

Coupling beam

P/2

260

200

1500

8#7 Longitudinal reinforcement #3@150mm web reinforcement both directions

#3@60mm transverse reinforcement

1500 825 1080

1300

450

1300

1080

confined boundary element

Wall pier

#3 rebar fy: 454MPa; fu: 684MPa #5 rebar fy: 475MPa; fu: 691MPa #7 rebar fy: 455MPa; fu: 656MPa (unit:mm)

Figure 17. Details of the coupled wall specimen

26

fc': 30MPa (base block) 35MPa (lower 2 floors) 37MPa (upper 2 floors)

y

Boundary confined column Coupling beam

Tied node

Local coordinate system x Retained node dx,i=0 dy,i=0

Coupling beam element

Shear element Slip element Shear wall （Multilayer shell element）

Fiber beamcolumn element

Offset of fiber beamcolumn element

Node sharing between beam-column and multilayer shell element

(a) scheme of coupled wall system

(b) modeling concept of coupled wall system

Figure 18. Scheme and modeling concept of coupled wall system

The above three types of elements are then combined to model a coupled wall system as illustrated in Figure 18(b). The fiber beam-column elements modeling the confined boundary column are linked to the multilayer shell elements modeling the shear wall through the share-node approach to make sure they can work together. An offset equal to half of the column width is given to the beam-column element to reflect its exact position. To assure the internal forces in the coupling beam can be reliably transferred to the wall, constraint equations of degrees-of-freedoms need to be defined between the end nodes of coupling beams and the wall nodes in the range of beam height, as illustrated in Figure 18(b). A local coordinate system is first attached to the end node of the coupling beam element, which is defined as the retained node. The local x-axis is fixed in the longitudinal direction of the coupling beam and co-rotates according to the rotation of end nodes. The y-axis is kept perpendicular to the x-axis. The constraint conditions as Equations (24) and (25) are applied to the corresponding wall nodes which are defined as the tied nodes here. Equation (24) can be used to define the deformation 27

compatibility in the local x-direction for transferring the axial force and bending moment and Equation (25) can realize the deformation compatibility in the local y-direction for transferring the shear force. d x,i = 0

(24)

d y,i = 0

(25)

where dx,i and dy,i are the displacements of the ith tied node in the x-direction and y-direction of the local coordinate system, respectively. Based on the proposed coupled wall model, the seismic behavior of Specimen CW-RC is simulated with different coupling beam models and the predicted base-moment versus roof drift ratio relationships are compared to the experimental results. The specimen is loaded horizontally with two hydraulic actuators. The force applied by the roof-floor actuator is kept twice that of the third-floor actuator. In addition, the 245kN additional vertical load is applied to the top of each wall pier using four hydraulic jacks before applying lateral displacement. Traditional fiber model Proposed model

600

Base moment (kN·m)

Lateral load (kN)

400 200 0 -200 -400

10000

10000

8000

8000

6000

6000

Base moment (kN·m)

800

4000 2000 0 -2000 -4000 -6000

-600 -800 -10

-7.5

-5

-2.5 0 2.5 5 Beam chord rotation (%)

7.5

10

-10000 -0.04

-0.02

0.00 Roof drift ratio

(a) Coupling beam

0.02

4000 2000 0 -2000 -4000 -6000

Tes t results Proposed model

-8000

Traditional fiber model Proposed model

-8000 0.04

-10000 -0.04

-0.02

0.00 Roof drift ratio

0.02

0.04

(b) Coupled wall system

Figure 19. Comparison of predicted and measured results for beams and coupled walls of specimen CW-RC

It can be seen from Figure 19(b) that the predicted results with the proposed coupling beam model are well correlated with the test results in terms of the initial stiffness, shear capacity and the overall unloading and reloading loops. However, compared to the proposed coupling beam model,

28

the predicted results with the traditional fiber model yield obviously larger initial unloading and reloading stiffness of the coupled wall system, which is attributed to the different predicted behavior of coupling beams as plotted in Figure 19(a). It can be found that the traditional fiber model greatly overestimates the initial unloading and reloading stiffness of the coupling beam because the significant shear and slip effects are neglected. 9 Conclusions Based on the summary and analysis of the experimental results, this paper proposes an accurate, efficient and practical fiber model for diagonally RC coupling beams on the platform of general FEA software MSC.Marc. It can be concluded from the research that : 1. The proposed model can accommodate diagonally RC coupling beams with a wide range of span-to-depth ratios from one to five since all the deformation components including flexure, shear and slip are considered and the critical parameters are calibrated by a total of sixteen test specimens collected from seven different research groups. 2. The modified fiber model incorporating the horizontal component of the diagonal bars is able to consider the flexural contribution of diagonal bars reasonably and give precise prediction of beam strength. 3. The proposed section shear force-shear strain and slip displacement model can reasonably represent the complex hysteretic behavior including unloading and reloading stiffness, strength and stiffness deterioration. 4. The formulas for cracked shear stiffness and chord rotation at strength degradation are proposed and verified.

29

5. The hysteretic behavior of sixteen test specimens are simulated by the proposed model and good correlation between numerical and experiment results is demonstrated in terms of the overall hysteretic loops and energy consumed. The different deformation components are analyzed and the slip rotation ratio is found to increase with the span-to-depth ratio and can be as large as 50%, whereas the shear deformation is dominant when the beam span-to-depth ratio is less than 2.0. 6. The accuracy, efficiency and applicability of the proposed model is further shown by means of the simulation of a RC coupled shear wall specimen. Acknowledgments The writers gratefully acknowledge the financial support provided by the National Science Fund of China (Grant No. 51708328 and No. 51722808) and the China Postdoctoral Science Foundation (Grant No. 2016M601039, 2017T100083). References [1] Paulay T. Coupling beams of reinforced concrete shear walls. ASCE J Struct Div 1971; 97(3): 843-862. [2] Breña SF, Ihtiyar O. Performance of conventionally reinforced coupling beams subjected to cyclic loading. J Struct Eng 2011; 137(6): 665-676. [3] Paulay T, Binney JR. Diagonally Reinforced Coupling Beams of Shear Walls. ACI Spec. Publ. 1974; 42: 579-598. [4] Naish D, Fry A, Klemencic R, Wallace J. Reinforced concrete coupling beams-Part I: testing. ACI Struct J 2013; 110(6): 1057-1066 [5] Lim E, Hwang SJ, Wang TW, Chang YH. An investigation on the seismic behavior of deep reinforced concrete coupling beams. ACI Struct J 2016; 113(2): 217-226. [6] Lim E, Hwang SJ, Cheng CH, Lin PY. Cyclic tests of reinforced concrete coupling beam with intermediate span-depth ratio. ACI Struct J 2016; 113(3): 515-524. [7] Kwan AKH, Zhao ZZ. Cyclic behaviour of deep reinforced concrete coupling beams. Proc. ICE, Struct & Build, 2002; 152(3): 283-293. 30

[8] Fortney PJ, Rassati GA, Shahrooz BM. Investigation on Effect of Transverse Reinforcement on Performance of Diagonally Reinforced Coupling Beams. ACI Struct J 2008; 105(6): 781-788. [9] Galano L, Vignoli A. Seismic behavior of short coupling beams with different reinforcement layouts. ACI Struct J 2000; 97(6): 876-885. [10] Tassios TP, Moretti M, Bezas A. On the behavior and ductility of reinforced concrete coupling beams of shear walls. ACI Struct J 1996; 93(6): 711-720. [11] Shiu KN, Barney GB, Fiorato AE, Corley WG. Reversing load tests of reinforced concrete coupling beams. Proc. Central American Conf. on Earthquake Engineering, Central America, 1978. [12] Park WD, Yun HD. Seismic behaviour of coupling beams in a hybrid coupled shear walls. J Constr Steel Res 2005; 61(11): 1492-1524. [13] Gong BN, Shahrooz BM. Concrete-steel composite coupling beams. I: Component testing. J Struct Eng 2001; 127(6): 625-631. [14] Nie JG, Hu HS, Eatherton MR. Concrete filled steel plate composite coupling beams: Experimental study. J Constr Steel Res 2014; 94(1): 49-63. [15] ACI 318-14. Building code requirements for structural concrete (ACI 318-14) and commentary/reported by ACI Committee 318. Farmington Hills, Mich.: American Concrete Institute; 2014. [16] Tao MX, Nie JG. Fiber beam-column model considering slab spatial composite effect for nonlinear analysis of composite frame systems. J Struct Eng. 2014; 140(1): 04013039. [17] Tao MX, Nie JG. Element mesh, section discretization and material hysteretic laws for fiber beam-column elements of composite structural members. Mater and Struct 2015; 48(8): 2521-2544. [18] Naish D, Fry A, Klemencic R,Wallace J. Reinforced concrete coupling beams-part II: modeling. ACI Struct J 2013; 110(6):1067–1076 [19] Lu X, Chen Y. Modeling of coupled shear walls and its experimental verification. J Struct Eng 2005; 131(1):75–84 [20] Barbachyn S, Kurama Y, Novak LC. Analytical evaluation of diagonally reinforced concrete coupling beams under lateral loads. ACI Struct J 2012; 109(4):497–508. [21] Toprak AE, Bal IE, Gülay FG. Review on the macro-modeling alternatives and a proposal for modeling coupling beams in tall buildings. Bull Earthquake Eng 2015; 13: 2309-2326 [22] Ding R, Tao MX, Nie JG, Mo YL. Shear deformation and sliding-based fiber beam-column model for seismic analysis of reinforced concrete coupling beams. J Struct Eng, 2016; 142(7): 04016032.

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[23] MSC. Marc Version 2007r1 [Computer software]. MSC. Software Corp., Santa Ana, CA. [24] ACI 318-05. Building code requirements for structural concrete (ACI 318-05) and commentary/reported by ACI Committee 318. Farmington Hills, Mich.: American Concrete Institute; 2005. [25] Vu NS, Li B, Beyer K. Effective stiffness of reinforced concrete coupling beams. Eng Struct 2014; 76: 371-382. [26] ASCE-ACI Joint Task Committee 426. Shear strength of reinforced concrete members. ASCE J Struct Div 1973; 99(6): 1091-1187． [27] Alsiwat JM, Saatcioglu M. Reinforcement anchorage slip under monotonic loading. J Struct Eng 1992; 118(9): 2421-2438. [28] ACI Committee 408. Suggested development, splice, and standard hook provisions for deformed bars in tension (ACI 408.1R-79). American Concrete Institute, Farmington Hills, MI, 1979. [29] Sezen H, Setzler EJ. Reinforcement Slip in Reinforced Concrete Columns. ACI Struct J 2008; 105(3):280-289. [30] Pan WH, Tao MX, Nie JG. Fiber beam-column element model considering reinforcement anchorage slip in the footing. Bull Earthquake Eng 2017; 15(3): 991-1018. [31] Ozcebe G, Saatcioglu M. Hysteretic Shear Model for Reinforced Concrete Members. J Struct Eng 1989; 115(1): 132-148. [32] Xu SY, Zhang J. Hysteretic shear-flexure interaction model of reinforced concrete columns for seismic response assessment of bridges. Earthq Eng Struct D, 2011; 40(3):315-337. [33] Naish DAB. Testing and modeling of reinforced concrete coupling beams. Ph.D. Ann Arbor. University of California, Los Angeles; 2010. [34] Cheng MY, Fikri R, Chen CC. Experimental study of reinforced concrete and hybrid coupled shear wall systems. Eng Struct 2015; 82: 214-225. [35] Nie JG, Tao MX, Cai CS, Chen G. Modeling and investigation of elasto-plastic behavior of steel-concrete composite frame systems. J Constr Steel Res 2011; 67(12): 1973-1984. [36] Ding R, Tao MX, Zhou M, Nie JG. Seismic behavior of RC structures with absence of floor slab constraints and large mass turbine as a non-conventional TMD: a case study. Bull Earthquake Eng 2015; 13(11): 3401-3422.

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LIST OF FIGURES Fig. 1: Comparison of RC coupling beams with different reinforcement layouts Fig. 2: Basic mechanism of diagonally reinforced coupling beams Fig. 3: Modified fiber model incorporating diagonal bars (nd=4) Fig. 4: Generalized sectional stress, strain and stiffness Fig. 5: Shear force-shear strain skeleton curves and hysteretic rules Fig. 6: Diagonal reinforcement slip model in the proposed fiber model Fig. 7: Proposed coupling beam fiber model considering shear and slip deformation Fig. 8: Proposed simplified trilinear shear force-slip deformation model Fig. 9: Proposed hysteretic shear force-slip deformation model Fig. 10: Regression analysis of cracked shear stiffness ratio Fig. 11: Regression analysis of drift ratio limit γlimit Fig. 12: Comparison of predicted and measured lateral load versus beam chord rotation curves Fig. 13: Analysis of deformation components for specimens with different span-to-depth ratios Fig. 14: Trends of slip rotation to total deformation ratio Fig. 15: Comparison of cumulative energy Fig. 16: Comparison of predicted and measured lateral load versus beam chord rotation curves Fig. 17: Details of the coupled wall specimen Fig. 18: Scheme and modeling concept of coupled wall system Fig. 19: Comparison of predicted and measured results for beams and coupled walls of specimen CW-RC

33

LIST OF TABLES Table 1: Database of diagonally RC coupling beam specimens

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