Analytical model for seismic simulation of reinforced concrete coupled shear walls

Engineering Structures 168 (2018) 819–837

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Analytical model for seismic simulation of reinforced concrete coupled shear walls

T

⁎

Ran Dinga, Mu-Xuan Taob, Xin Niea, , Y.L. Moc a

Key Lab. of Civil Engineering Safety and Durability of China Education Ministry, Dept. of Civil Engineering, Tsinghua University, Beijing 100084, China Beijing Engineering Research Center of Steel and Concrete Composite Structures, Tsinghua University, Beijing 100084, China c Dept. of Civil and Environmental Engineering, University of Houston, Houston 77204, USA b

A R T I C LE I N FO

A B S T R A C T

Keywords: Coupled wall Conventionally and diagonally reinforced coupling beams Seismic analysis Analytical models Degree of coupling Parametric analyses

Reinforced concrete coupled walls are widely used as the main seismic resistant structural system in high-rise buildings. This paper proposes a new mixed beam-shell model for the seismic analysis of reinforced concrete coupled walls with sufficient efficiency and accuracy on the platform of general finite element software MSC.Marc. Boundary elements at the ends of wall piers are simulated by conventional fiber beam-column elements, while the web of the wall pier is modeled by the layered shell element. Coupling beams are simulated by non-conventional fiber beam-column elements, which can not only take into account the shear and shear-sliding deformation together with various failure modes of conventionally reinforced beams, but also the shear and rebar slip deformation of diagonally reinforced beams. RBE2 link elements are utilized to connect the coupling beams to the wall piers. Eight test specimens reported in the literature are used to validate the proposed model. The mechanism of the coupled wall is thoroughly investigated in terms of the beam deformation, base shear and moment distribution as well as axial force of the wall piers. Furthermore, parametric analyses on specimens with different degrees of coupling and types of reinforcement layouts of coupling beams are conducted. Based on the analyses, the influences of the complex behavior and various modeling parameters of coupling beams on the behavior of coupled wall are revealed quantitatively. As a conclusion of the parametric analyses results, it is recommended that the complicated behavior of coupling beams be accurately considered for most cases in the seismic analysis of coupled wall systems.

1. Introduction To meet the architectural or other practical requirements, openings are commonly placed in the reinforced concrete (RC) structural walls, thus forming the coupled wall system, which has been widely used in modern high-rise buildings as the main lateral force resisting system. Due to the coupling effects of the beams between the adjacent wall piers, the structure can bear a much larger base moment and shear force with an obviously increased structural stiffness compared to the individual wall piers. In addition, during major earthquakes, the coupling beams can dissipate a large portion of energy and protect the wall piers from severe damage. Some experimental research has been conducted to investigate the seismic behavior of the coupled wall system [1–5]. However, compared to numerous tests on individual shear walls [6–10] and coupling beams [11–18], tests on RC coupled walls are quite limited. Previous tests have recognized the influences of different types of coupling beams and degrees of coupling and have revealed the redistribution of base shear

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and moment among wall piers. Among these experimental programs, Lehman et al. [1] completed a test program on one piece of planar coupled wall specimen featured with high-resolution response and damage data, which provides valuable information for the analytical modeling research. Nowadays, structural analytical models with sufficient accuracy and efficiency are widely and urgently needed to conduct large numbers of elasto-plastic dynamic time-history analyses of different structural systems, so as to support and promote the research on the structural seismic performance and performance-based design method. To date, for the analytical modeling of the important RC coupled wall system, there remains a significant gap in the past research mainly due to the complicated seismic behavior of coupling beams [4,19–21]. Two modeling approaches can be summarized from previous research as shown in Fig. 1: (1) to get higher calculation efficiency, the wall piers and coupling beams are both simulated by beam elements, as shown in Fig. 1(a); (2) to obtain better accuracy, the wall piers and coupling beams are modeled by complex a 2D membrane element or a 3D shell

Corresponding author. E-mail address: [email protected] (X. Nie).

https://doi.org/10.1016/j.engstruct.2018.05.003 Received 14 October 2017; Received in revised form 26 February 2018; Accepted 1 May 2018 0141-0296/ © 2018 Published by Elsevier Ltd.

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Beam-column element with shear and slip springs Flexural spread plastic element elastic plastic elastic

Traditional fiber beam-column element with modified effective stiffness

Slip spring

Shear spring

Distributed or lumped fiber beamcolumn element for wall piers

Axial spring ...

Shell or membrane element for wall piers

...

Shear spring Multiple-vertical-line element for wall piers

Rigid offset

Shell or membrane element for coupling beams

(a) beam-column element

(b) shell or membrane element

Fig. 1. Two modeling approaches for RC coupled wall structures. y Coupling beam

Tied node

Coupling beam

dx,i=0 dy,i=0 x Retained node

Coupling beam

Shear element Slip or sliding element Coupling beam

Web of shear wall

Web of shear wall layered shell element

Web of shear wall Coupling beam

Beam‐wall connection

RBE2 link

Coupling beam

Node sharing between beam-column and layered shell element

Coupling beam traditional fiber beamcolumn element with offset hc/2

hc Confined boundary element

Web of shear wall layered shell element

Confined boundary element

Confined boundary element

(a) schem ati c dia gram of t ypica l coupl ed wall

Confined boundary element

(b) mode ling stra tegy of c oupled wal l

Fig. 2. Proposed modeling strategy for coupled wall structures.

element, as shown in Fig. 1(b). The first approach cannot capture the complex shear-sliding-slip behavior of coupling beams. On the other hand, despite the fact that several improved beam elements are proposed to simulate the complicated compression-bending-shear or tension-bending-shear behavior of wall piers such as the multiple-verticalline element [22,23], it is quite difficult to determine the parameters of the model. In addition, the damage state of the wall piers cannot be visually presented to help the designers and researchers easily understand the seismic performance of structures with shear walls or core tubes. The second approach requires too much calculation time due to the complex 2D concrete constitutive relationships; thus it is not

suitable for large-scale structural time-history dynamic analysis. What is more, the complex shear, sliding and slip behavior of coupling beams may still not be reasonably reflected by the membrane or shell models. Therefore, this study aims to propose a mixed beam-shell model for the seismic analysis of the RC coupled wall system with both satisfactory accuracy and efficiency as well as to reveal the influence of the complex coupling beam behavior on the seismic behavior of coupled walls. The authors once proposed two non-conventional fiber beamcolumn elements for conventionally and diagonally RC coupling beams [24,25], respectively. The elements can accurately consider the complex deformation mechanism and failure modes of RC coupling beams 820

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element. It should be noted that two other methods may also be used to model the boundary confined elements: (i) changing the thickness of horizontal and vertical reinforcement layers in the layered shell element to consider the increased longitudinal bars and stirrups, and adopting the constitutive relationship for confined concrete; and (ii) the truss element simulating the additional longitudinal rebar in the column and inserted into the layered shell element. However, compared to the above two methods, the proposed method in this study can greatly enhance the computational efficiency by reducing the number of shell elements, which is quite attractive in the large-scale structural analysis of high-rise building systems. It is worth noting that sometimes confined concrete might not always be only in the boundary. The majority of the tall buildings that utilize RC coupled walls have the entire sections confined, whereas in others boundary region goes quite deep. In such cases, the entire wall will be simulated by shell elements with constitutive models for confined concrete. Only in cases where confined section is limited in the range of the ends of the wall, the beam-column elements are adopted to simulate the confined boundary element. The confined sections in test specimens simulated in Section 3 are all quite limited and thus modeled by the traditional fiber beam-column element. On the other hand, the confined boundary element is considered as an axial-flexure member while the shear-compression interaction is ignored, because the shear contribution of concrete boundary is not as much as the web of the wall due to the limited dimension. Moreover, the height-to-width ratio for individual wall pier is usually very large in actual structures, thus the effects of shear force may not be as significant as those of the bending moment and axial force. In recent years, the authors’ research team has been devoted to the development of a conventional displacement-based fiber beam-column element suitable for various RC, steel and steel-concrete composite structural members [27,28] in MSC.Marc based on the two-node beam element No. 98 with one integration point in the middle. The uniaxial constitutive laws of rebar and concrete are shown in Fig. 3 and more detailed information can be found in [27,28]. The model for confined concrete follows that proposed by Mander et al. [33] and is applied to the confined part in the boundary element while the unconfined concrete model is applied to the cover concrete. The hysteretic model can reasonably capture the strength and stiffness degradation phenomena under arbitrary possible complex cyclic loading paths of unconfined and confined concrete materials. On the other hand, the rebar model can reflect the curved Bauschinger effect of steel.

while retain the high efficiency of the beam element, thus they are adopted in the new model to simulate the coupling beams. The layered shell element has been widely used to model the shear walls with enough accuracy and visualized damage state [21,26]; therefore it is assembled in the proposed model to simulate the behavior of wall piers. To further improve the efficiency of the model, the boundary elements of wall piers are modeled with traditional fiber beam-column elements [27,28]. The proposed model is verified by eight test specimens reported in the literature [1–4]. Based on the new model, the structural mechanism such as the beam chord rotation, base shear and moment distribution, and additional axial forces in the wall piers, is thoroughly investigated. In addition, a detailed parametric analysis is conducted to evaluate how the complex factors in the coupling beam affect the structural seismic behavior of the coupled wall system. Based on the analysis results, the significance of the proposed model is further demonstrated. 2. Proposed mixed beam-shell model for RC coupled walls As shown in Fig. 2, a RC coupled wall system is composed of two individual wall piers linked by several coupling beams. The wall piers can be further divided into two parts: the webs and the confined boundary elements. Different components are modeled by different finite elements developed on the platform of general software MSC.Marc [29], which will be illustrated in the following sections together with the assembling technique. It should be noted that the model proposed and analyses conducted in this study are two-dimensional, potential three-dimensional effects such as the out-of-plane deformation are not considered. 2.1. Layered shell element for shear wall webs The webs of individual walls are modeled with the layered shell element provided by the MSC.Marc [29], which can capture the compression-bending-shear or tension-bending-shear behavior of wall piers reasonably. Because the shear walls are reinforced with horizontal and vertical distributed rebar, the layered material model is adopted to simulate the constitutive relationship of RC shear walls [21,26,30]. The shell is divided into several layers according to the actual arrangement of concrete and rebar. The thickness of the rebar layer is equal to the total thickness of the wall multiplied by the corresponding rebar ratio (the total area of a layer of horizontal or vertical rebar divided by the sectional area of a piece of wall). Horizontal and vertical rebar can be simplified to be an orthotropic material with mutually perpendicular principal axes while the concrete can be modeled with an isotropic material [21,26,30]. The material model for rebar is an elasto-plastic kinematic hardening model with a von Mises yield surface and an associated plastic flow rule. The modulus of elasticity (Es) and Poisson’s ratio (νs), are 206 GPa and 0.3, respectively. The equivalent stress – plastic strain relation is derived from a trilinear uniaxial stress-strain curve, where the strain corresponding to the beginning of hardening is 0.025 and the hardening modulus is 0.005Es. The constitutive relationship of concrete uses an elasto-plastic isotropic hardening model for compression. The Rüsch curve is adopted as the uniaxial compressive stress-strain relationship [31]. The Poisson’ ratio vc, the peak and extreme compressive strain are 0.17, 0.002 and 0.0035, respectively. The smeared crack model and crack band model proposed by Bazant and Oh [32] are used to describe the cracking behavior of concrete. The detailed parameters for the above material models have been suggested by Nie et al. [30].

2.3. Fiber beam-column model for RC coupling beams As shown in Fig. 4(a), two typical types of RC coupling beams are considered in this study, i.e. the conventional and diagonal reinforcement layouts. The former one is modeled with the shear element and shear sliding element proposed by Ding et al. [24], while the latter one is simulated by the shear element and reinforcement slip element proposed by Ding et al. [25], as shown in Fig. 4(b). The model consists of two kinds of elements assembled in a series to model the nonlinear shear and shear sliding behavior for conventionally RC coupling beams and the nonlinear shear and reinforcement slip behavior for diagonally RC coupling beams. The shear element is developed by combining new section models with the traditional fiber beam-column element to consider the nonlinear shear force – shear distortion relationships, as shown in Figs. 5(a) and 6(a) [24]. The shear sliding/rebar slip element at the beam ends is developed by combining another new section model into the conventional fiber model to consider the nonlinear shear force – shear sliding/rebar slip relationships, as shown in Figs. 5(b) and 6(b) [25]. The four different failure modes for conventionally RC coupling beams including the shear tension (ST), shear compression (SC), shear sliding (SS) and flexure (F) modes can be reflected by the model. As shown in Fig. 7, the shear tension failure is the opening up of main

2.2. Traditional fiber beam-column model for confined boundary element The confined boundary elements at the two ends of the wall piers act as rectangular RC columns with longitudinal and transverse rebar. Therefore, they are simulated with the traditional fiber beam-column 821

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ft

cu

( t0

unconfined

(

un, un)

(

Hardening modulus Eh t t b1 , b1 )

fy

tu

c0, fc0)

confined ( cc,fcc)

-fy Hardening modulus Eh

(b) rebar

(a) concrete

Fig. 3. Constitutive models for concrete (unconfined and confined) and rebar.

2.4. Assembly of coupled wall model

diagonal cracks caused by insufficient shear capacity; the shear compression failure is featured with the severe damage and spalling of concrete in the plastic hinge region with significant shear effects; the shear sliding failure refers to obvious sliding movement along fulldepth, through cracks near the beam-wall interface; and the flexure failure is the excessive local rotations of the plastic hinges. It is demonstrated that the ST mode is caused by the fact that the shear tension capacity [Vus in Fig. 5(a)] is lower than the flexural capacity. On the other hand, for SC, SS and F modes, the shear tension capacity is larger than the flexural capacity. Therefore the load capacity is equal to the flexural capacity, but the ultimate chord rotations [δlimit in Fig. 5(b)] where the shear force begins to degrade are different. Corresponding equations are proposed for the δlimit in SS and SC modes which can be found in [24], while no ultimate chord rotation is applied to the flexure mode, which is automatically considered by the flexural response with the one-dimensional stress-strain constitutive relationships for rebar and concrete. The contribution of diagonal reinforcement in the diagonally RC coupling beam is also successfully incorporated in the model. In addition, the bearing capacity is equal to the flexural capacity since the shear tension capacity is always larger than the flexural capacity due to the existence of diagonal bars. The two coupling beam models have both been calibrated and verified by numerous tests with sufficient precision [24,25].

Stirups

The above three types of elements are then assembled to model a coupled wall system as illustrated in Fig. 2. The fiber beam-column elements modeling the confined boundary elements are linked to the layered shell elements modeling the wall webs through the share-node approach to make sure they can work together. To make sure the confined boundary elements are in their exact position, the beamcolumn element is offset by a distance equal to half of the height of the column section, as shown in Fig. 2. To transfer the internal forces in the coupling beam reliably and rationally to the wall, constraint equations of degrees-of-freedoms (DOFs) are defined between the end nodes of the coupling beams and the wall nodes in the range of the beam height, as illustrated in Fig. 2. The RBE2 link element provided by MSC.Marc is used to impose the constraints. 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 the end nodes. The yaxis is kept perpendicular to the x-axis. The corresponding wall nodes are defined as the tied nodes and the relative displacements in the x and y directions dx,i and dy,i are kept as zero so that the axial force, bending moment and shear force can be transferred between the coupling beams and the wall piers.

Longitudinal bar

Conventional layout

Internal fiber element considering nonlinear shear effects

Diagonal bars

Spread plasticity fiber element for flexure

Horizontal bars

node

Shear sliding/ reinforcement slip element at the end

node

nonlinear shear hinge incorporated in the fiber model

confinement hoops and ties

Diagonal layout (a) Typical RC coupling be ams

(b) Coupling beam models

Fig. 4. Modeling strategy for RC coupling beams proposed by the authors [24,25]. 822

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V

V Vus Vm Vsd,1

G

J1

un,m

O I1

F1 C1

E1

I

T W

un,m

B1

Vun Vsd,1 Vm

M

re

A1

un,m

Z un,m

un,m

O

pW

Vcr N

Z

R

W L

D J

B C

B1 pP

I

K

unloading point

S

T

U C1 V

pB1 un,m

Vcr un,m

unloading point

Y

strength deterioration point

Descending branch

H

D1

Vp

un,m res

Q

F

P

E

Vs,1

L1

0.8

A

Vs,2 Y

p un,m res re

J Vcr

A1

X

V

C

D1 G1

B

p

U

G

Backbone curve

res

S

K1 D

p

N

H1

H

Vp

un

re un,m

R

P

E

0.8

F

A

Vcr

un,m

Q

limit

Vs,4 Vs,3

strength deterioration point

reloading starting point

reloading starting point

Vssd Vun,m

reloading pinching point

Vus

X M

(a) Shear force-shear deformation relations

reloading pinching point

(b) Shear force- shear sliding displacement relations

Fig. 5. Proposed section shear rules for conventionally RC coupling beams [24].

demands from the upper seven stories were obtained based on a finite element analysis on the 10-story building subjected to a predefined lateral loading mode and then applied to the top of each wall pier using a specially designed loading apparatus developed by the NEES program. The responses of wall piers and coupling beams were monitored very carefully by extensive instrumentations for the first time ever. Therefore this specimen is particularly focused to verify adequately the proposed model in terms of not only the overall structural response but also the mechanism with respect to the interaction between the coupling beams and wall piers. As shown in Fig. 8a, layered shell element, conventional and non-conventional fiber beam-column element are adopted to model the web of shear wall, the confined boundary elements and the coupling beams, as illustrated in Section 2. The base of the specimen is fixed in the model. To facilitate the application of loading, the original 10-story shear wall structure is simulated and subjected to the predefined lateral loading mode. The typical comparisons between the measured and predicted results are plotted in Fig. 9. It can be found from Fig. 9(a) that the overall base shear – top displacement relations correlate very well. The small

3. Model application to collected test specimens Coupled wall test specimens are collected from the literature [1–4] to verify the proposed model adequately and to investigate thoroughly the structural mechanism such as the beam deformation, base shear and moment distribution, and additional axial forces in the wall piers. Finally eight planar coupled walls with various degrees of coupling (DOCs), reinforcement layouts and number of stories are summarized and simulated in the following sections, including six walls with conventionally RC coupling beams and two walls with diagonally RC coupling beams.

3.1. Lehman et al. 2013 Lehman et al. [1] reported an experiment on the seismic behavior of a coupled wall with diagonally RC coupling beams. The geometry, reinforcement details and material properties of the specimen are shown in Fig. 8. The test specimen represented the lower three stories of a typical 10-story shear wall structure. The axial, shear and moment

V Vs,3 Vs,2 V

A kcr

A

u n,m

A1 A2

cr ,Vcr )

F1 F E1

m

,Vre)

re

kt

C1

J

C2

re,0)

K

C

Vcr

t,Vt)

Q

Vs,1

Descending branch

X

m

P

,0)

pQ

re

Vs,1

J W

strength deterioration point

N

L

H S E

D pE O B C

pY

unloading point

D2 D1 D

Y

B B1B2

E

A1

un,Vun)

kini kt

F

limit

Backbone curve

G

,Vm)

G R

C1

B1

pU I

T

U

,Vre)

re

unloading point strength deterioration point

reloading starting point reloading turning point

reloading starting point

K

reloading turning point

H

V -

limit

M

(b) Shear force- slip displacement relations

(a) Shear force-shear deformation relations

Fig. 6. Proposed section shear rules for diagonally RC coupling beams [25]. 823

Vs,1

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(b) Flexure failure (F)

(a) Shear tension failure (ST)

(c) Shear sliding failure (SS)

(d) Shear compression failure (SC)

Fig. 7. Failure modes of conventionally RC coupling beams.

152

4#4 Diagonal rebars A 3#2 Horizontal reb ars 19°

1220

A

4#4 #2

#2@51mm confinement hoops and ties

A-A

12#2

Coupling beam

16#4 Longitudinal rebars

horizontal web reinforcement #2@76mm 1st floor #2@152mm 2nd&3rd floor

356

825

Layered shell element

Fixed base

762

1220

610 4574

1220

(a) Numerical model

762

#2@152mm vertical web reinforcement #2@51mm confinement ties and hoops

Wall pier

914

1220

152

1220

3965

Traditional fiber beamcolumn element

305 762

305

M2 V2 N2 1676

305

3657

610

5641

Predefined lateral load ASCE 7-05

M1 V1 N1 1676

Coupling beam element

(unit:mm)

#2 rebar fy: 522MPa; fu: 583MPa #4 rebar fy: 454MPa; fu: 721MPa fc': 39.2MPa

(b) test specimen Fig. 8. Details of the coupled wall specimen by Lehman et al. [1]. 824

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6

1000 Chord rotation of coupling beam (%)

800

Base shear (kN)

600 400 200 0 -200

-400 -600

test results

-800

model results

-1000

-3

-2

-1

0 Drift ratio (%)

1

2

3

Measured result-1st floor Measured result-2nd floor Measured result-3rd floor Predicted result-1st floor Predicted result-2nd floor Predicted result-3rd floor

5 4 3 2 1 0

(a) Base shear versus top drift ratio

0.0

0.5

1.0 1.5 2.0 Drift ratio of coupled wall (%) (b) coupling beam rotation

2.5

3000 Measured axial force-1st floor Measured axial force-2nd floor

Axial force of wall pier (kN)

2000

Measured axial force-3rd floor

1000 0

-1000

Predicted axial force-1st floor Predicted axial force-2nd floor

-2000 -3000

Predicted axial force-3rd floor

-3

-2

-1

0 1 2 Drift ratio (%) (c) Axial force of wall piers at each floor

3

Fig. 9. Comparison of measured and predicted results.

Therefore all the coupling beams keep their shear capacity until the failure of the coupled wall, which is consistent with the reported damage phenomenon. As described by Lehman et al. [1], at the final state of a −2.27% drift ratio, most of the damage was localized to the beam ends with some cover spalling while the number and width of diagonal cracks in the middle were relatively small, indicating that the coupling beams yielded but not failed. One of the most important characteristics of the coupled wall system is the additional tensile and compressive axial forces imposed on the wall piers by coupling beams. Lehman et al. [1] and Turgeon [34] estimated the shear force of coupling beams by using the strain data of the diagonal rebar. They computed the steel stresses from the steel strain gauges using the Hoehler-Stanton model [34] and obtained the tension forces in the diagonal reinforcement. Then the shear force in coupling beams can be estimated as twice the vertical component of the total tensile forces. Fig. 9(c) shows the comparison between the analytical and test results. It can be found that the proposed model gives a precise prediction for the shear force of the coupling beams and axial force of the wall pier. Fig. 10 further illustrates the predicted base moment and shear force distribution between the two wall piers which can hardly be obtained by test measurement. As explained in Fig. 10(a), the base moment consists of three components, which are the moments in the two wall piers as well as the coupling moment contributed by the additional axial forces caused by the shear forces in coupling beams. The coupling moment is calculated by multiplying the total shear forces in all the coupling beams with the centroid distance of the two wall piers. The shear force of coupling beam can be directly obtained from the results.

differences of the initial and unloading stiffness are mainly attributed to the base rotation displacement and horizontal sliding displacement between the wall base and the foundation block, which account for approximately 20% of the total displacement as reported [1]. The failure of the coupled wall is very abrupt after a 2% drift ratio due to the sudden crushing of core concrete and buckling of longitudinal rebar in the confined boundary element. This is very difficult to capture since the sudden crushing of concrete and buckling of rebar is not considered in the present constitutive models for concrete and reinforcement. However, the numerical results show that the concrete in the confined boundary element has entered the descending stage indicating large compressive strain. In addition, the rebar in the confined boundary element has yielded under compression. These predicted strains indicate the flexural failure of the wall pier. The responses of the coupling beams are then evaluated, as shown in Fig. 9(b). It is demonstrated that the predicted chord rotations of coupling beams generally match the test results. The deformation of the coupling beams at the second and third floors is obviously larger than that of the first floor. The largest chord rotation is no more than 5% while the predicted ultimate rotation is 7.791%, which is calculated by the following formula [25]:

γlimit = 0.0239 + 4.3209ρsv l/ h (ρsv l/ h ⩽ 0.0125) γlimit = 0.07791

(ρsv l/ h > 0.0125)

(1)

where ρsv is the stirrup ratio; ρsv = nAsv/bs; Asv, n and s are the area, number of legs and space of the transverse rebar, respectively and l, b and h are the span, width and height of the beam, respectively. 825

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100

80

Base shear distribution (%)

Predicted coupling moment Measured coupling moment Design coupling moment

60 40

Wall pier in compression-predicted Wall pier in tension-predicted

20

0

0.0

0.5

1.0 1.5 2.0 Drift ratio of coupled wall (%)

80

Wall pier in tension Wall pier in compression

Base moment distribution (%)

100

60 40 20 0

2.5

0.0

0.5

(a) Base moment distribution versus top drift ratio

1.0 1.5 2.0 Drift ratio of coupled wall (%)

2.5

(b) Base shear distribution versus top drift ratio

Fig. 10. Base shear and moment distribution analysis.

180

Then DOC is the ratio between the coupling moment and the total base moment. It can be seen that the coupling moment takes up approximately 60% of the total base moment, indicating a strong coupling effect in the system. The design and estimated coupling moment are also plotted in the figure and are very close to the simulation results. The wall pier in tension carries less than 10% of the total moment while the wall pier in compression undertakes more than 30%. The shear force distribution as shown in Fig. 10(b) demonstrates that the 70–80% of the total base shear force is undertaken by the wall pier in compression. Therefore it can be concluded that the wall pier in compression is subjected to far more moment and shear force than that in tension.

150

Load (kN)

120 90 60 CS 30 0

3.2. Shiu et al. [2] Shiu et al. [2] conducted cyclic loading tests on two coupled walls with conventionally RC coupling beams. The dimension, reinforcement layouts and material properties of the coupled wall test specimens CWCS and CW-RCS are illustrated in Fig. 11. The shear walls of the two specimens are almost identical. The investigated parameters are the dimension and reinforcement ratio of the coupling beams. The beam RCS in the specimen CW-RCS is strengthened with larger section and more reinforcement when compared to that of the specimen CW-CS.

1900

CW-RCS

5490

A-A

unit: mm

D3@34 mm hoops

6@102 mm vertical web reinforcement

12#4 Longitud in al rebars

248 702 B-B 6@102 mm fc ' : 26.5 MPa horizontal web reinforcement #3 rebar fy : 483MPa ; fu : 745 MPa #4 rebar fy : 434MPa ; fu : 696 MPa D3 rebar fy : 510MPa ; fu : 586 MPa 6 rebar fy : 531MPa ; fu : 710 MPa

Fig. 11. Details of the coupled wall specimen CW-CS and CW-RCS. 826

203

170

915

1900

10

6#3 Horizontal rebars 254

D3@34 mm hoops CW-CS

B

423

7.5

D3@21 mm hoops

915

745 170

B

5.0 Drift ratio (%)

2 6 Horizontal rebars 102

915

915

915

A

2.5

The two specimens are compared in order to investigate the influence of the DOC, which is a critical index for assessing the mechanism of the coupled wall system. The coupling beams in the two specimens are first simulated and the

P A

0

Fig. 12. Predicted behavior of the coupling beams CS and RCS.

915

P

RCS

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1200

1200

800

800 Base shear (kN)

Base shear (kN)

R. Ding et al.

400 0

-400

test results model results

400 0

-400

-800

test results model results

-1200 -2.4

-1.2

-800

without coupling beam

0.0 Drift ratio (%) (a) CW-CS

1.2

without coupling beam -1200 -2.4

2.4

-1.2

0.0 Drift ratio (%) (b) CW-RCS

1.2

2.4

Fig. 13. Comparison of the predicted and measured base shear-drift ratio curves of the coupled wall specimens CW-CS and CW-RCS.

correlate well with the measured results. When the shear wall yields, most of the coupling beams have already failed and entered the descending stage, which also agrees with the observed results. Fig. 15(a) presents the value and orientation of concrete principal cracking strain of the wall piers at the peak load. The observed cracking pattern is also presented for comparison. The predicted cracking distribution correlates well with the test results. Almost the same crack pattern can be observed for the two wall piers, indicating that they behave almost independently as cantilevers due to the weak coupling effects. Fig. 14(b) shows the relationship between the shear force in the coupling beam and the top drift ratio of the wall for CW-RCS. The yielding and failure sequence of the coupling beams are similar to that of CW-CS. Shiu et al. [2] reported that the top four coupling beams yielded at around a 0.5% drift except that the sixth floor beam yielded at a drift of 0.68% while the second and first floor beams yielded at nearly 0.9% and 1.05% drifts, which generally matches the simulated results. Fig. 15(b) presents the value and orientation of concrete principal cracking strain of the wall piers at the peak load. The observed cracking pattern is also presented for comparison. The predicted cracking distribution correlates well with the test results. Obvious differences are found between the two wall piers. Horizontal cracks initiate in the wall in tension and propagate into the wall in compression. Diagonal cracks are found in the wall subjected to shear and compression. The crack distribution indicates that the two wall piers are strongly connected by the coupling beams; in other words, they work together as a single vertical cantilever beam.

skeleton curves between the shear force and chord rotation are shown in Fig. 12. It can be seen that the coupling beam in the specimen CWRCS has a much larger shear capacity than the specimen CW-CS due to increased section dimension and rebar. Thus the additional axial loads on the two individual walls are much larger in specimen CW-RCS than CW-CS. Then the predicted and measured lateral load-top drift ratio curves of the two coupled walls are presented in Fig. 13, which indicates that the analytical results match well with the test results. The simple superposition of the load-top drift ratio curves of two isolated wall piers is also plotted in the figure. It is found that the maximum load capacity of CW-CS is approximately the same as the uncoupled result due to the relatively weak coupling beams. On the other hand, the load bearing capacity of CW-RCS is about 50% higher than the uncoupled result, which indicates that the stronger coupling beam can improve the load capacity of coupled walls effectively. At the end of the loading process, the damage in the wall piers of CW-CS is still minor, while the compressive strains of vertical rebar and concrete as well as the tensile strains of horizontal rebar in the first story of compression wall pier of CW-RCS are much larger, indicating shear-compression failure of the wall pier. Fig. 14(a) illustrates the development and distribution of shear force in the coupling beams by plotting the relationship between the shear force of beam and top drift ratio of the wall for CW-CS. The top four beams were reported to yield at a 0.1% drift while the second and first floor beams were reported to yield at 0.18% and 0.34% drifts, respectively [2,34]. Thus it can be seen that the simulated results generally

200 Shear force in coupling beams (kN)

Shear force in coupling beams (kN)

30 0.1%

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10 0.18%

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1st floor 2nd floor 3rd floor 0.5

4th floor 5th floor 6th floor

1.0 Top drift ratio (%)

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50

0

2.0

(a) CW-CS

1.05%

0.5%

0

0.5

1.0 1.5 Top drift ratio (%)

(b) CW-RCS

Fig. 14. Predicted development of shear force in coupling beams of specimens CW-CS and CW-RCS. 827

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Orientation of maximum principal cracking strain

Value of maximum principal cracking strain

Observed cracking pattern

(a) Specimen CW-CS (Drift ratio=0.85%)

Orientation of maximum principal cracking strain

Value of maximum principal cracking strain

Observed cracking pattern

(b) Specimen CW-RCS (Drift ratio=0.85%) Fig. 15. Cracking pattern of the coupled wall specimens CW-CS and CW-RCS at peak load.

3.3. Santhakumar et al. [3]

Figs. 16 and 17 show the comparison of predicted and measured strain curves for vertical reinforcing bars at different locations of the wall for specimens CW-CS and CW-RCS. As can be seen from the figures, the model can reasonably predict the whole-process strain development of critical rebar in the wall. Fig. 18 shows the comparison of predicted and measured wall rotation at the first story for specimen CW-CS, and good correlation can be found from the figure.

Rebar A

Rebar C

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Rebar strain

(a) Rebar A

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Numerical results Test results 0.000

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Rebar B

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Base shear (kN)

Base shear (kN)

800

Santhakumar [3] once performed tests on two specimens with conventionally (Wall-A) and diagonally (Wall-B) RC coupling beams respectively and focused on the influence of rebar layouts in coupling beams on the seismic behavior of coupled walls. The dimension, reinforcement layouts and material properties of the specimens are illustrated in Fig. 19. The wall piers of the two specimens are identical. The

Numerical results Test results

400

0

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0.010

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0.000

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Rebar strain

0.010

-800 -0.005

(b) Re bar B

0.000

0.005

Rebar strain

0.010

-800 -0.005

(c) Re bar C

Fig. 16. Strain curves of vertical reinforcing bars for specimen CW-CS (900 mm above the base). 828

0.000

0.005

Rebar strain

(d) Re bar D

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Rebar D

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Base shear (kN)

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Numerical results 0.000

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Numerical results Tes t results

-1200 0.010 -0.010

(c) Rebar C

-0.005

0.000

Rebar strain

0.005

0.010

(d) Rebar D

Fig. 17. Strain curves of vertical reinforcing bars for specimen CW-RCS (900 mm above the base).

800

The span-to-depth ratios of coupling beams for specimens CW1, CW2 and CW3 are 2.0, 1.6 and 1.33, respectively. CW1 and CW3 are subjected to two lateral loading forces at the third (P/2) and roof floors (P), while CW2 is subjected to one lateral loading force at the top of the wall. The loading is first controlled by forces before yielding and then controlled by top displacement until failure. However, it can be seen in Fig. 23 that only test data in the force-controlled stage is provided for specimens CW1 and CW3 while the data in the displacement-controlled stage is missing. This can be attributed to the fact that the ratio between the loads applied at the third and roof floor was not able to be kept constant due to some problems in the displacement-controlled stage, as reported by Chen et al. [4]. In addition, since the load pattern for CW2 is different from those for CW1 and CW3, the results cannot be directly compared to evaluate the influence of the beam span-to-height ratio. Nevertheless, satisfactory agreement of the predicted and measured curves between the lateral load applied at the top and the wall drift ratio is clearly shown in Fig. 23

Base shear (kN)

400

0

-400

-800 -0.010

-0.005 0.000 0.005 Rotation of the right wall pier at first story

0.010

Fig. 18. Rotation of wall pier at first story for specimen CW-CS (900 mm above the base).

4. Parametric analysis for effects of complex coupling beam behaviors

specimens were subjected to constant gravity loading and reversed cyclic lateral loading at the third, fifth and seventh floors. The hysteretic behaviors of the two types of coupling beams used in this test are first analyzed by the proposed fiber model in Section 2.3, as illustrated in Fig. 20. It is demonstrated that the drift and energy dissipation capacity of the diagonally RC coupling beams are significantly stronger than those of the conventionally RC coupling beams with a shear sliding failure mode. Fig. 21 presents the comparison of model and test results for the two wall specimens, and satisfactory correlation is found. For specimen Wall-A in Fig. 21(a), the load capacity drops obviously at approximately a 1.5% drift ratio, when most of the coupling beams suffer shear sliding failure and the shear forces in these beams decrease obviously. Thus the coupling moment decreases and the moments carried by the two wall piers increase, which cause excessive compressive strain of the concrete and rebar in the wall piers and degradation in the strength of the coupled wall system. For specimen Wall-B in Fig. 21(b), the coupled wall and diagonally reinforced coupling beams keep their bearing capacity without deterioration until they reach a 3.5% drift ratio. In addition, remarkable pinching effects are observed for Wall-A, while the hysteretic loops are quite plump for Wall-B. Good agreement between the predicted and observed results can be found with respect to the above phenomena.

It is well known that the coupling beam is the key component in the coupled wall system, which significantly affects the structural behavior. However, due to the great difficulty in accurately capturing their complex seismic behaviors, it remains to be thoroughly investigated and clearly revealed how the complicated mechanisms and key parameters of coupling beams affect the overall responses of the coupled wall systems. Therefore, this section attempts to reveal and evaluate quantitatively these influences based on the proposed and adequately verified model so as to give useful recommendations on the analytical modeling of coupled walls. Parametric analysis for coupled walls with different DOCs and types of rebar layouts are conducted to identify the effects of different coupling beam behaviors and parameters including the four distinct deformation mechanisms (flexural, shear, shear sliding and reinforcement slip deformation), four failure modes (shear tension, shear compression, shear sliding and flexure for conventionally RC coupling beams), shear capacity, ultimate chord rotation limit and degradation slope. Four groups of models are established, distinguished by different DOCs and reinforcement layouts as listed in Table 1. The detailed information and results for each group are illustrated in the following sections. It is worth noting that the sensitivity study is mainly conducted using monotonic analysis and provides insight into the capacity and stiffness. More results on the actual seismic behavior of the structure need to investigated based on dynamic and hysteretic analyses.

3.4. Chen et al. [4] Chen et al. [4] conducted reversed cyclic tests on three 5-story coupled wall specimens and the investigated parameter was the beam span-to-depth ratio. Fig. 22 shows the dimensions, rebar layouts and material properties of the specimens. The rebar details are identical for the three specimens and the only difference is the height of the beam.

4.1. Coupled walls with conventionally RC coupling beams The 6-story specimens CW-CS and CW-RCS tested by Shiu et al. [2] are selected as the basic models to conduct parametric analysis on coupled walls with conventionally RC coupling beams with weak 829

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111.2kN

610

111.2kN

(unit:mm) 1600 380 610

E

2#3+2#2 Diagonal rebars F

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2#3 Horizontal reb ars

P1

76

D

762

E

C

762

B

B

F

Coupling beam A

F-F

F4.75@114mm stirrups

Coupling beam B

#2@76mm horizontal web reinforcement 178 2#5 Longitudinal rebars 4#2

#2@102mm horizontal web reinforcement

140

5486

127

10#5 Longitudinal rebars

127

102

102

C

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P2

E-E

#2@51mm stirrups

305

762

D

305

2#3+2#2 32°

F2.67@19mm spiral wire

127

A-A

B-B #2@152mm horizontal web reinforcement

102

762

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#2@102mm horizontal web reinforcement

A

A

127

914

127

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C-C

(vertical) #2 rebar (horizontal) #2 rebar #5 rebar (beam) #2 rebar P1=P2=P3 #3 rebar

fy: 343MPa; fy: 352MPa; fy: 305MPa; fy: 346MPa; fy: 315MPa;

fu: 487MPa fu: 498MPa fu: 472MPa fu: 487MPa fu: 431MPa

70

102

762

P3

8#5 Longitudinal rebars

127

127

127

4#5 Longitudinal rebars

D-D

fc': 31.6MPa (Shear wall A) fc': 30.0MPa (Shear wall B)

1905 Fig. 19. Details of the coupled wall specimens Wall-A and Wall-B.

150

lower than the flexural capacity. Therefore, as illustrated in Table 2, the analysis is further divided into two groups, i.e. shear tension (models CCW-W1 – CCW-W4 and CCW-S1 – CCW-S5) and the other three modes (models CCW-W6 – CCW-W11 and CCW-S6 – CCW-S12). The transverse rebar of beams in the former group is reduced to achieve the shear tension failure mode. As can be seen in Table 2, CCW-W1/S1 represent the basic complete model considering the shear tension failure mode and three deformation components. CCW-W2/S2 exclude the shear sliding deformation from CCW-W1/S1, while CCW-W3/S3 exclude both the shear and shear sliding deformation from CCW-W1/S1. In CCW-W4/S4 and CCW-S5, the degradation slopes after the occurrence of shear tension failure are doubled and tripled, respectively. On the other hand, models CCW-W6/ S6 represent the basic complete model with flexure failure mode and three deformation components. CCW-W7/S7 exclude the shear sliding deformation from CCW-W6/S6, while CCW-W8/S8 exclude both the shear and shear sliding deformation from CCW-W6/S6, which is just the traditional fiber model. CCW-W9/S9 and CCW-W10/S10 consider the shear compression and shear sliding failure modes. The main differences among the CCW-W6/S6, CCW-W9/S9 and CCW-W10/S10 for the three different failure modes are the drift ratio where load degradation begins. In CCW-W11/S11 and CCW-S12, the degradation slopes after shear compression failure are doubled and tripled based on CCW-W9/ S9, respectively. The degradation slope is changed so as to investigate the sensitivity of seismic behavior of the coupled wall system to it, because it is usually very difficult to determine this parameter due to the significant uncertainty in the failure process. For clarity and convenience, three indexes are chosen to compare the stiffness, capacity and descending stage of the beam and wall behavior, i.e. maximum bearing capacity Pu, drift ratio DRy corresponding

conventionally RC beam (CB-A)

Shear force (kN)

100

diagonally RC beam (CB-B)

50 0

-50

-100 -150 -0.12

-0.08

-0.04 0.00 0.04 Chord rotation ratio (%)

0.08

0.12

Fig. 20. Comparison of the coupling beams in Wall-A and Wall-B.

(CCW-W) and strong (CCW-S) coupling effects (DOC = 15% and 45%), respectively. The wall dimensions and reinforcement are kept the same among models in these two series to separately investigate the effects of coupling beams. The numbers for each model are used to represent both the coupled walls and their coupling beams for brevity. For example, the CCW-W series denotes both the coupled walls incorporating conventionally RC coupling beams with weak coupling effects and the corresponding coupling beams. Among the four failure modes in conventionally RC coupling beams [24], the shear tension mode is distinguished from the other modes because the transverse rebar is insufficient and the bearing capacity is 830

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test results model results -4

-2

0 Drift ratio (%)

2

0

4

-400

test results model results -4

-2

0 Drift ratio (%)

(a) Wall-A

2

4

(b) Wall-B

Fig. 21. Comparison of the predicted and measured base shear-drift ratio curves for Wall-A and Wall-B.

Figs. 27–29 plot the load-displacement relationships of beams and walls in models CCW-W6 – CCW-W11 and CCW-S6 – CCW-S12 with SS, SC and F modes. Some remarkable results are summarized as follows which can also be found in bold in Table 4: (1) for walls with low DOC, neglecting the sliding deformation (CCW-W7) leads to 12.5% less DRy and 19% more energy dissipated; neglecting both the shear and sliding deformation (CCW-W8) leads to 27.5% less DRy and 38% more energy dissipated, (2) for walls with high DOC, neglecting both the shear and sliding deformation (CCW-S8) leads to 32.8% less DRy; doubling and tripling the degradation slope of beams (CCW-S11/12) result in 8.9% and 9.1% smaller Pu together with 12.6% and 17.2% less Pd; neglecting the sliding deformation (CCW-S7) as well as neglecting both the shear and sliding deformation (CCW-W8) lead to 40% and 77% more energy dissipated.

to Pu/2 before yielding, and the force Pd corresponding to a 6.67% drift ratio (beams CCW-S6-S12) or a 5% drift ratio (the other beams) or a 1.25% drift ratio (all the walls). The calculated indexes for the behaviors of beams and coupled wall systems are summarized in Table 4. As is seen from the table, it is divided into six zones by horizontal separation lines and the first model in each zone is set as the benchmark to be compared with the other models in the zone. It should be pointed out that when calculating DRy, Pu is adopted as the maximum bearing capacity of the benchmark model. Figs. 24–26 show the shear force-drift ratio comparisons of beams and walls in models CCW-W1 - CCW-W4 and CCW-S1 - CCW-S5 with the shear tension mode. In addition, results of models CCW-W8/S8 as the traditional fiber models are added into the figures for comparison. Some noteworthy results are highlighted here which can also be found in bold in Table 4: (1) for weak beams, the traditional fiber model shows 42.3% higher Pu, 133.9% larger Pd and 85.7% less DRy (CCWW8). Similar results are also found for strong beams, (2) for walls with low DOC, the traditional fiber model leads to 10.4% higher Pu and 19.4% less DRy (CCW-W8), (3) for walls with high DOC, the traditional fiber model causes 29.3% higher Pu and 28.0% larger Pd (CCW-S8); neglecting the shear and sliding deformation results in 39.4% less DRy (CCW-S3), (4) the energy dissipation capacities for wall systems in CCW-W8/S8 are 41% and 61% larger, respectively.

70

unit:mm

700

2 8

700 700 3700

1800

A

600

2 4 @50

CW1

4 8 2 8

70

CW2 A-A 6@90

2 8

CW3 6@80

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4 8

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B 100

400 100 B-B 4 rebar fy: 797MPa 6 rebar fy: 311MPa 8 rebar fy: 278MPa fc': 41.2MPa

700

500 200

400 300

B

450 250

700

P/2

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4 8

200

200

A

300

100kN

P

1800

The geometry, together with the wall reinforcement details of the coupled wall models with diagonally RC coupling beams, resemble those in the test specimen by Lehman et al. [1]. The number of stories of the model is twelve. The area of diagonal bars in the coupling beams is adjusted to achieve weak and strong coupling effects (DOC = 25% and 50%), which are represented by the DCW-W (weak beam) and DCW-S (strong beam) series, respectively.

250

100kN

4.2. Coupled wall with diagonally RC coupling beams

400

600

Fig. 22. Details of the coupled wall specimens CW1/CW2/CW3. 831

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120

60

120

60

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Lateral load applied at the top (kN)

180 Lateral load applied at the top (kN)

Lateral load applied at the top (kN)

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(a) CW1

1.0

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test results model results

-120

-180 -1.5

-1.0

-0.5 0.0 0.5 Drift ratio (%)

(b) CW2

1.0

1.5

(c) CW3

Fig. 23. Comparison of the predicted and measured overall load-displacement curves for CW1/2/3.

Figs. 30–32 plot the load-displacement relationships of all the models for diagonally RC coupling beams and walls. Some prominent results are presented here which can also be found in bold in Table 3: (1) for walls with low DOC, the traditional fiber model (DCW-W3) leads to 19.6% less DRy; ACI 318-14 equation (DCW-W6) causes 103.8% larger DRy; neglecting the slip deformation (DCW-W2) as well as neglecting both the shear and slip deformation (DCW-W3) lead to 22% and 33% more energy dissipated, (2) for walls with high DOC, the traditional fiber model (DCW-S3) generates 36.2% less DRy together with 7.6% larger Pu; ACI 318-14 equation (DCW-S6) causes 15.7% smaller Pu and 15.5% larger DRy; neglecting the slip deformation (DCW-S2) as well as neglecting both the shear and slip deformation (DCW-S3) lead to 37% and 54% more energy dissipated.

Table 1 Definition of different model groups.

Conventional reinforcement layout Diagonal reinforcement layout

Coupling ratio ≤ 25%

Coupling ratio ≥ 45%

CCW-W

CCW-S

DCW-W

DCW-S

As listed in Table 3, Model DCW-W1/S1 represents the basic proposed model with three deformation components. DCW-W2/S2 excludes the reinforcement slip deformation from DCW-W1/S1, while DCW-W3/S3 excludes both the shear and reinforcement slip deformations from CCW-W1/S1, which are just the traditional fiber model. In DCW-W4/S4, the ultimate chord rotation ratio is reduced by half on the basis of DCW-W1/S1 and in DCW-W5/S5, the degradation slope after the ultimate chord rotation is doubled based on DCW-W4/S4. Since it is demonstrated by many researchers that the equation below for the shear strength of diagonally reinforced coupling beams in ACI 318-14 [35] obviously underestimates the actual capacity, DCW-W6/S6 employs the result of Eq. (2) to investigate the influence from this equation on the behavior of beams and coupled walls.

VnACI = 2Avd fy sinα

4.3. Summary of parametric analyses On the basis of the above parametric study, some typical and important findings are summarized here: (1) The shear sliding and reinforcement slip deformation of coupling beams are found to slightly influence the base shear-top displacement skeleton curves of the coupled walls. However, the effects on the energy dissipation capacity of walls with conventionally RC coupling beams without shear tension failure and diagonally RC coupling beams cannot be neglected. With the increase of the coupling ratio, the energy consumed can be overestimated by 20–40%; (2) With a traditional fiber model, the bearing capacity of walls incorporating conventionally RC coupling beams with shear tension failure can be obviously overestimated by 29.3%. In addition, the stiffness and consumed energy of the coupled walls are significantly increased by the traditional fiber model. In the analytical examples

(2)

where fy and Avd are the yield strength and total area of one group of diagonal bars; α is defined as the inclination angle of the diagonal bars with respect to the longitudinal axis of the beam. Similar to the analysis in Section 4.1, three indexes are chosen to compare the stiffness, capacity and descending branch, i.e. the maximum bearing capacity Pu, drift ratio corresponding to Pu/2 (DRy), and the force (Pd) corresponding to the 8% drift ratio (beams) or the 2.0% drift ratio (wall systems). Table 2 Detailed information of model series CCW. Beams in coupled walls

Failure mode ST

CCW-W1/S1 CCW-W2/S2 CCW-W3/S3 CCW-W4/S4 CCW-S5 CCW-W6/S6 CCW-W7/S7 CCW-W8/S8 CCW-W9/S9 CCW-W10/S10 CCW-W11/S11 CCW-S12

SC

Deformation components SS

F

√ √ √ √ √ √ √ √ √ √ √ √

Degradation slope

Flexure

Shear

Shear sliding

Proposed

√ √ √ √ √ √ √ √ √ √ √ √

√ √ × √ √ √ √ × √ √ √ √

√ × × √ √ √ × × √ √ √ √

√ √ √

832

Double

Triple

√ √ √ √ √ √ √ √ √

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Table 3 Definition of model series DCW. Deformation components

DCW-W1/S1 DCW-W2/S2 DCW-W3/S3 DCW-W4/S4 DCW-W5/S5 DCW-W6/S6

Degradation slope

Flexure

Shear

Rebar slip

Proposed

√ √ √ √ √ √

√ √ × √ √ √

√ × × √ √ √

√ √ √ √

Double

Ultimate chord rotation

Bearing capacity

Proposed

Half

Proposed

√ √

√ √ √ √ √

√ √ √ √

√

ACI318-14

√

√

Table 4 Summary of indexes for the behaviors of beam and wall system in all models. Model

Indexes for beam behavior

CCW-W1 CCW-W2 CCW-W3 CCW-W4 CCW-W8 CCW-W6 CCW-W7 CCW-W8 CCW-W9 CCW-W10 CCW-W11 CCW-S1 CCW-S2 CCW-S3 CCW-S4 CCW-S5 CCW-S8 CCW-S6 CCW-S7 CCW-S8 CCW-S9 CCW-S10 CCW-S11 CCW-S12 DCW-W1 DCW-W2 DCW-W3 DCW-W4 DCW-W5 DCW-W6 DCW-S1 DCW-S2 DCW-S3 DCW-S4 DCW-S5 DCW-S6

Indexes for wall system behavior

Pu (kN)

Error (%)

DRy (‰)

Error (%)

Pd (kN)

Error (%)

Pu (kN)

Error (%)

DRy (‰)

Erro r (%)

Pd (kN)

Error (%)

18.9 18.9 18.9 18.9 26.9 25.7 25.7 26.9 24.7 25.6 24.7 122.2 122.2 122.9 122.2 122.2 188.2 171.8 176.2 188.2 158.8 162.9 158.6 158.5 49.7 54.2 54.4 46.5 46.5 22.5 119.5 144.2 145.0 114.5 114.5 78.8

– 0.0 0.0 0.0 42.3 – 0.0 4.7 −3.9 −0.4 −3.9 – 0.0 0.6 0.0 0.0 54.0 – 2.6 9.5 −7.6 −5.2 −7.7 −7.7 – 9.1 9.5 −6.4 −6.4 −54.7 – 20.7 21.3 −4.2 −4.2 −34.1

1.32 1.32 0.19 1.32 0.19 1.02 1.02 0.47 1.02 1.02 1.02 4.14 3.55 0.41 4.14 4.14 0.41 7.57 5.91 1.06 7.57 7.57 7.57 7.57 2.50 2.05 0.95 2.50 2.50 – 2.52 2.10 0.86 2.52 2.52 2.91

– 0.0 −85.7 0.0 −85.7 – 0.0 −53.5 0.0 0.0 0.0 – −14.3 −90.0 0.0 0.0 −90.0 – −21.9 −86.0 0.0 0.0 0.0 0.0 – −18.0 −62.2 0.0 0.0 – – −17.0 −66.0 0.0 0.0 15.2

11.5 11.2 10.5 3.7 26.9 25.6 26.0 26.9 20.4 25.6 15.5 100.4 100.4 92.5 81.9 63.1 187.6 170.0 174.5 187.6 124.9 137.4 87.4 67.2 46.4 53.1 53.2 32.3 16.5 18.6 107.4 141.3 142.7 76.9 46.6 68.7

– −2.6 −8.7 −67.8 133.9 – 1.6 5.1 −20.3 0.0 −39.5 – 0.0 −7.9 −18.4 −37.2 86.9 – 2.6 10.4 −26.5 −19.2 −48.6 −60.5 – 14.4 14.7 −30.4 −64.4 −59.9 – 31.6 32.9 −28.4 −56.6 −36.0

601 600 603 575 664 660 701 664 619 620 596 821 821 815 785 760 1062 994 1060 1062 916 940 905 903 392 409 411 368 346 336 541 583 583 517 515 457

– −0.2 0.3 −4.3 10.4 – 6.2 0.5 −6.2 −6.2 −9.8 – 0.0 −0.7 −4.4 −7.4 29.3 – 6.7 6.9 −7.8 −5.4 −8.9 −9.1 – 4.5 4.8 −6.2 −11.7 −14.2 – 7.6 7.6 −4.5 −4.9 −15.7

1.97 1.95 1.99 1.93 1.59 2.19 1.91 1.59 2.19 2.19 2.19 1.71 1.64 1.04 1.71 1.71 1.04 2.44 2.33 1.64 2.44 2.44 2.44 2.44 2.54 2.40 2.04 2.54 2.54 5.18 2.34 2.21 1.50 2.34 2.34 2.70

– −0.9 0.9 −1.8 −19.4 – −12.5 −27.5 0.0 0.0 0.0 – −4.3 −39.4 0.0 0.0 −39.4 – −4.5 −32.8 0.0 0.0 0.0 0.0 – −5.4 −19.6 0.0 0.0 103.8 – −5.8 −36.2 0.0 0.0 15.5

601 600 603 574 625 647 695 625 619 620 596 821 821 815 767 705 1051 971 1050 1051 911 935 848 804 392 404 405 367 338 335 536 577 577 503 473 457

– −0.2 0.3 −4.5 4.0 – 7.5 −3.4 −4.3 −4.2 −7.9 – 0.0 −0.7 −6.6 −14.1 28.0 – 8.2 8.3 −6.2 −3.7 −12.6 −17.2 – 3.0 3.3 −6.4 −13.7 −14.5 – 7.6 7.6 −6.2 −11.8 −14.8

40

200

Strong beam 150

20

10

0

200

CCW-W8 CCW-W4 Shear force (kN)

Shear force (kN)

30

CCW-W1 CCW-W2 CCW-W3

150

100 CCW-S1 CCW-S2 CCW-S3 CCW-S8

50

0

1

2 3 Drift ratio (%) (a) weak beam

4

5

Shear force (kN)

Weak beam

0

0

2.5

100

50

Strong beam

5.0 7.5 Drift ratio (%) (b) strong b eams CCW-S1/2/3/8

10

0

0

2.5

5.0 7.5 Drift ratio (%) (b) strong b eams CCW-S1/4/5

Fig. 24. Comparison of beam behaviors in models with conventionally RC beams and shear tension mode.

833

CCW-S1 CCW-S4 CCW-S5

10

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R. Ding et al.

800

1200

800

Wall system (15% DOC)

Wall system (15% DOC) Base shear (kN)

400

400

CCW-W1 CCW-W2 CCW-W3

200

0

Wall system (15% DOC)

600 Base shear (kN)

Base shear (kN)

600

800

0

0.5

1.0 Drift ratio (%) (a) CCW-W1/2/3

1.5

CCW-W1 CCW-W8 CCW-W4

200

0

2.0

0

0.5

1.0 Drift ratio (%) (b) CCW-W1/4/8

400 0 -400 -800

1.5

2.0

CCW-W1 CCW-W8

-1200 -2.4

-1.2

0.0 Drift ratio (%)

1.2

2.4

(c) hy steretic loo ps of C CW-W1/8

Fig. 25. Comparison of coupled wall behavior with weak coupling effects and shear tension mode. 1200

Wall system (45% DOC)

800

Wall system (45% DOC)

CCW-S1 CCW-S2 CCW-S3 CCW-S8

300

0.5

1.0 Drift ratio (%)

1.5

600

2.0

0

0

0.5

1.0 Drift ratio (%)

400 0

-400

CCW-S1 CCW-S4 CCW-S5

300

(a) CCW-S1/2/3/8

-800

1.5

2.0

CCW-S1 CCW-S8

-1200 -2.4

(b) CCW-S1/4/5

-1.2

0.0 Drift ratio (%)

30

Beam shear (kN)

30

20

10

CCW-W6 CCW-W7 CCW-W8 CCW-W9

Weak beam 0

0

200

2.5

5.0 Drift ratio (%)

7.5

20

10

CCW-W6 CCW-W9 CCW-W10 CCW-W11

Weak beam 0

10

(a) weak b eam CCW-W6/7/8/9

0

2.5

5.0 Drift ratio (%)

7.5

10

(b) weak b eam CCW-W6/9/10/11

200

Strong beam Beam shear (kN)

150

100

50 Strong beam 0

0

2.5

CCW-S6 CCW-S7 CCW-S8 CCW-S9

5.0 7.5 Drift ratio (%) (c) srong b eam CCW-S6/7/8/9

10

150

100 CCW-S6 CCW-S9 CCW-S10 CCW-S11 CCW-S12

50

0

0

2.5

5.0 7.5 Drift ratio (%) (d) strong b eam CCW-W6/9/10/11/12

Fig. 27. Comparison of different beam models for conventionally RC beams with other failure modes.

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1.2

(c) hy steretic loo ps of C CW-S1/8

Fig. 26. Comparison of coupled wall behavior with strong coupling effects and shear tension mode.

Beam shear (kN)

0

Base shear (kN)

600

Base shear (kN)

900

Beam shear (kN)

Base shear (kN)

900

0

1200

1200

Wall system (45% DOC)

10

2.4

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800

800

400 CCW-W6 CCW-W7 CCW-W8

200 0

0

0.5

1.0 1.5 Drift ratio (%) (a) CCW-W6/7/8

2.0

600

1200 Wall system (15% DOC)

Base shear (kN)

600

Base shear (kN)

Base shear (kN)

Wall system (15% DOC)

400

CCW-W6 CCW-W9 CCW-W10 CCW-W11

200 00

0.5

1.0 1.5 Drift ratio (%) (b) CCW-W6/9/10/11

800

Wall system (15% DOC)

400

0 -400 CCW-W7 CCW-W8 CCW-W9

-800

2.0 -1200 -2.4

-1.2

0.0 1.2 Drift ratio (%) (c) hy steretic loo ps fo r CCW-W7/8/9

2.4

Fig. 28. Comparison of coupled wall behavior with weak coupling effect and other failure modes.

1200

1200 Wall system (45% DOC)

900

Base shear (kN)

Base shear (kN)

Wall system (45% DOC)

600 CCW-S6 CCW-S7 CCW-S8 CCW-S9

300

0

0

0.5

1200

1.0 1.5 Drift ratio (%) (a) CCW-S6/7/8/9

0

2.0

Base shear (kN)

Base shear (kN)

600

0

CCW-S6 CCW-S9 CCW-S11 CCW-S12 0.5

1.0 1.5 Drift ratio (%) (c) CCW-S6/9/11/12

CCW-S6 CCW-S9 CCW-S10

0

0.5

1200

900

0

600

300

Wall system (45% DOC)

300

900

800

2.0

Wall system (45% DOC)

400 0

-400 CCW-S7 CCW-S8 CCW-S9

-800 2.0

1.0 1.5 Drift ratio (%) (b) CCW-S6/9/10

-1200 -2.4

-1.2

0.0 1.2 Drift ratio (%) (d) hy steretic loo ps fo r CCW-S7/8/9

2.4

Fig. 29. Comparison of coupled wall behavior with strong coupling effect and other failure modes.

Based on the model, the mechanism with respect to the interaction between the coupling beams and wall piers is investigated, and parametric analyses for specimens with different DOCs and types of reinforcement layouts are conducted to evaluate the influences of various coupling beam behavior and parameters. The following conclusions can be drawn within the scope of this research:

of this study, the maximum errors of stiffness and dissipated energy can be as much as 64.9% and 77%, respectively. (3) The degradation ratio and ultimate chord rotation of coupling beams mainly affect the curves when the wall drift ratio exceeds 1.0%. In addition, the effects on the wall system are usually limited to approximately 10%, which are much less than the effects on the beam. Therefore, although these two parameters are usually very difficult to determine because of significant uncertainty, it seems that the seismic behavior of the coupled wall system is not sensitive to them. (4) The formula proposed by ACI 318-14 for shear capacity of the diagonally RC coupling beams can lead to 15% lower lateral capacity and 51% lower stiffness of the coupled wall system, which deserves special attention.

1. The proposed wall model consisting of the layered shell element, conventional fiber beam-column element, coupling beam fiber element and RBE2 link element is proven adequately to be able to capture the overall behavior of coupled wall systems by eight test specimens. 2. The proposed model is used to investigate the mechanism of the coupled wall including the cracking of wall pier, coupling beam and wall deformation, base shear force and moment distribution, wall pier axial force and strain development of rebar. It is verified by the valuable test data from Lehman et al. [1] and Shiu et al. [2] with high precision. It is demonstrated that the wall pier in compression is subjected to far more base moment and shear than that in tension.

5. Conclusions This paper proposes an accurate, efficient and practical model for RC coupled walls on the platform of general FEA software MSC.Marc. 835

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60

60 Weak beam

40

30 20 10 0

DCW-W1 DCW-W2 0

Weak beam

50

Shear force (kN)

Shear force (kN)

50

40

30 20

DCW-W1 DCW-W4 DCW-W5

10

DCW-W3 DCW-W6

4

8 Drift ratio (%) (a) weak beams DCW-W1/2/3/6

0

12

200

0

12

Strong beam

150

Shear force (kN)

Shear force (kN)

8 Drift ratio (%) (b) weak beams DCW-W1/4/5

200

Strong beam

100 50 DCW-S1 DCW-S2

0

4

0

150

100 50

DCW-S1 DCW-S4 DCW-S5

DCW-S3 DCW-S6

4

8

12

0

0

4 8 Drift ratio (%) (d) strong beams DCW-S1/4/5

Drift ratio (%) (c) strong beams DCW-S1/2/3/6

12

Fig. 30. Comparison of different beam models for diagonally RC beams. 600

600

200

DCW-W1 DCW-W2 0.0

0.5

400

400

200 DCW-W1 DCW-W4 DCW-W5

DCW-W3 DCW-W6

1.0 1.5 Drift ratio (%) (a) DCW-W1/2/3/6

2.0

2.5

0 0.0

0.5

1.0

1.5

2.0

Base shear (kN)

400

0

600 Wall systems (25% DOC)

Base shear (kN)

Base shear (kN)

Wall systems (25% DOC)

Wall systems (25% DOC)

200 0 -200 DCW-W1 DCW-W2 DCW-W3

-400 2.5 -600-2.50

-1.25

Drift ratio (%) (b) DCW-W1/4/5

0.00 1.25 Drift ratio (%) (c) DCW-W1/2/3

2.50

Fig. 31. Comparison of coupled wall behavior with weak coupling effects. 800

800

400

200

DCW-S1 DCW-S2 0.5

1.0 1.5 Drift ratio (%)

(a) DCW-S1/2/3/6

600

600

Base shear (kN)

600

0 0.0

800 Wall systems (50% DOC)

Base shear (kN)

Base shear (kN)

Wall systems (50% DOC)

400

200

2.0

2.5

0 0.0

0.5

1.0 1.5 Drift ratio (%)

2.0

400 200 0

-200 -400

DCW-S1 DCW-S4 DCW-S5

DCW-S3 DCW-S6

Wall systems (50% DOC)

2.5

-800 -2.50

(b) DCW-S1/4/5

Fig. 32. Comparison of coupled wall behavior with strong coupling effects.

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DCW-S1 DCW-S2 DCW-S3

-600 -1.25

0.00 Drift ratio (%) (c) DCW-S1/2/3

1.25

2.50

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3. Based on the parametric study conducted in this paper, it is found that neglecting the shear sliding or reinforcement slip deformation of coupling beams can moderately overestimate the dissipated energy, and the traditional fiber model can significantly overestimate the stiffness and dissipated energy of the coupled wall. In addition, the formula specified in ACI 318-14 for shear capacity of diagonally RC coupling beams can cause remarkable error to lateral capacity and stiffness of coupled walls in the studied examples. 4. According to the parametric study, the behavior of the coupled wall system is found to be not sensitive to the degradation ratio and ultimate chord rotation of coupling beams. 5. Based on the parametric analyses results, in most cases, it is recommended that the complex behavior of coupling beams be considered accurately using the proposed model in the seismic analysis of coupled wall systems.

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It should be noted that the parametric study in this paper is mainly based on monotonic analysis. More realistic dynamic and hysteretic analysis need to be conducted in the future, to further investigate the actual seismic behavior of the structure. Acknowledgments The writers gratefully acknowledge the financial supports provided by the National Key Research Program of China (2016YFC0701404) and the National Science Fund of China (Grant No. 51708328). Tsinghua University of China providing financial supports to the first and third authors to study at the University of Houston is gratefully acknowledged. References [1] Lehman DE, Turgeon JA, Birely AC, et al. Seismic behavior of a modern concrete coupled wall. J Struct Eng 2013;139(8):1371–81. [2] Shiu KN, Takayanagi TS, Corley WG. Seismic behavior of coupled wall systems. J Struct Eng 1984;110(5):1051–66. [3] Santhakumar AR. The ductility of coupled shear walls. Ph.D. dissertation, Univ. of Canterbury, Christchurch, New Zealand; 1974. [4] Lu X, Chen Y. Modeling of coupled shear walls and its experimental verification. J Struct Eng 2005;131(1):75–84. [5] Cheng MY, Fikri R, Chen CC. Experimental study of reinforced concrete and hybrid coupled shear wall systems. Eng Struct 2015;82:214–25. [6] Lowes LN, Lehman DE, Birely AC, et al. Earthquake response of slender planar concrete walls with modern detailing. Eng Struct 2012;43(10):31–47. [7] Tasnimi AA. Strength and deformation of mid-rise shear walls under load reversal. Eng Struct 2000;22(4):311–22. [8] Oh JH, Han SW, Lee LH. Effect of boundary element details on the seismic deformation capacity of structural walls. Earthq Eng Struct D 2002;31:1583–602. [9] Riva P, Meda A, Giuriani E. Cyclic behavior of a full scale RC structural wall. Eng Struct 2003;25:835–45. [10] Dazio A, Beyer K, Bachmann H. Quasi-static cyclic tests and plastic hinge analysis of

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