Vol.1 No.1 EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION June 2002

Article ID: 1671-3664 2002 01-0094-06

Static inelastic analysis of RC shear walls

Chen Qin ( ) and Qian Jiaru ( )

Department of Civil Engineering, Tsinghua University, Beijing 100084,China

Abstract:A macro-model of a reinforced concrete (RC) shear wall is developed for static inelastic analysis. The model is composed of RC column elements and RC membrane elements. The column elements are used to model the boundary zone and the membrane elements are used to model the wall panel. Various types of constitutive relationships of concrete could be adopted for the two kinds of elements. To perform analysis, the wall is divided into layers along its height. Two adjacent layers are connected with a rigid beam. There are only three unknown displacement components for each layer. A method called single degree of freedom compensation is adopted to solve the peak value of the capacity curve. The post-peak stage analysis is performed using a forced iteration approach. The macro-model developed in the study and the complete process analysis methodology are verified by the experimental and static inelastic analytical results of four RC shear wall specimens.

Keywords: RC shear wall; macro-model; static inelastic analysis; experiment

1 Introduction Reinforced concrete (RC) shear walls are the main

vertical structural members in high-rise buildings to resist lateral loads caused by wind and earthquake effects. In order to perform static inelastic analysis of building structures, a simple and rational analytical model of a shear wall should be developed. At the present time, various types of finite element models (Lefas and Kotsovos, 1990; Kotsovos et al., 1992; Vecchio, 1992) and macro-models (Park and Reinhorm, 1987; Linde and Bachmann, 1994; Koumousis and Peppas, 1992) have been proposed. The finite element model has the advantage of accuracy, but among the existing finite element models of a shear wall, the effect of concrete confinement of boundary zone on the wall deformation capacity is seldom considered. On the other hand, since the degrees of freedom of finite element models are large, it is not practical to adopt finite element procedures for analysis of building structures. Though the degrees of freedom of a macro-model of a shear wall are smaller and it can be applied to the analysis of building structures, most existing macro-model analysis procedures have not been extended to the post-peak stage, so the true post-yield deformation capacity of the wall cannot be predicted.

In this study, a new macro-model of a shear wall for static inelastic analysis is developed. The proposed Correspondence to: Chen Qin, Department of Civil

Engineering, Tsinghua University, Beijing 100084, China. Tel. (010)62788620, E-mail:chenqin99@mails.tsinghua.edu.cn

Ph D candidate Professor Supported by: National Natural Science Foundation of China,

Grant number 59895410

model has the advantages of a finite element model but with fewer degrees of freedom. The constitutive models of cracked concrete and confined concrete are adopted for the wall panel and the boundary zone, respectively. A method named single degree of freedom compensation is adopted to perform the approaching-peak analysis and the post-peak analysis is performed using the forced iteration approach. In this way, the inelastic deformation capacity and the ductility of shear walls can be studied.

The accuracy and applicability of the newly developed macro-model are verified by the experimental and analytical results of four RC shear wall specimens

2 Analytical model 2.1 Basic elements of the model

An isolated RC shear wall is composed of two parts: wall panel and a boundary zone. The wall panel is reinforced with vertical and horizontal distributed reinforcements and it is in plane stress state. The boundary zone, which could be a rectangular section, flanged section or barbell section, is mainly subjected to tension or compression. It is concentrically reinforced with deformed reinforcements and its concrete could be confined with stirrups or unconfined. Correspondingly, a static inelastic analysis model of RC shear wall developed in this study is also composed of two kinds of basic element: RC membrane element and RC column element. The former is used to model the wall panel and the later is adopted to model the boundary zone. 2.2 Establishment of analytical model

Two basic assumptions are adopted for static

No.1 Chen Qin et al. : Static inelastic analysis of RC shear walls 95

inelastic analysis of shear walls with the proposedmodel. First, the cross section remains plane afterdeformation, and second, the stress in the horizontaldirection is equal to 0, i.e., 0x .

Fig.1 illustrates how to use two kinds of basicelement to model the shear wall. The wall is dividedinto five layers along its height. For one layer, the boundary zone at each end is modeled by two RCcolumn elements, and the wall panel is modeled withfour RC membrane elements. In total, the wall shownin Fig.1 has 20 RC column elements and 20 RCmembrane elements.

Fig. 1 Modeling of RC shear wall

By the plane section assumption, it is consideredthat there is a rigid beam between two adjacent layers.With each rigid beam, there are three degrees offreedom at its center point: {ug g w g }, i.e.,horizontal displacement, vertical displacement androtation, respectively.

According to the second assumption, thehorizontal strain at any point of a membrane elementis the same: x . If there are membraneelements in one layer, the number of unknownhorizontal strain for one layer is . Then the totaldegrees of freedom of one layer of the model are

.

k

k

k33 Stiffness matrix of layer 3.1 Stiffness matrix of membrane element

The stiffness matrix K e of the membrane elementcan be derived from the virtual work theory:

eyxt ddTe DBBK (1)

in which, B is the shape function matrix, D is thematerial stress-strain relation matrix and t is thethickness of the wall panel.

Fig. 2 shows the local coordinate system of the RC membrane element. The displacement patterns of any point of the membrane element are as follows:

432

210 aaaaau 3 (2a)

3210 bbbbv (2b)

in which, u is horizontal displacement, v isvertical displacement, and are horizontal andvertical local coordinate values, respectively, and ,0

Fig.2 Coordinate system of RC membrane element

1a , , , ,b , ,b , are the parameters tobe determined.

2a 3a 4a 0 1b 2 3b

By taking the derivative of the displacement andtransformation of the coordinate, the followingrelations are obtained:

BU (3)

in which is the strain vector, ( x , y and xy are normal strain in the x direction,normal strain in direction and shear strain in yxy plane, respectively), U is the displacementvector, ,and is the shape functionmatrix obtained from Eq.3.

T}v{uU B

For the membrane element, it is considered that thevertical and horizontal distributed reinforcements are smeared in concrete. The material stress-strain relation matrix of the membrane element is

nsc DDD (4)

in which, is the stress-strain relation matrix of

concrete, is the stress-strain relation matrix of

reinforcement in one direction, and is the sum

of the stress-strain relation matrices of reinforcements

embedded in concrete in n directions. In this

study, n equals 2.

cD

sD

nsD

The displacement increment of the layer of themembrane element can be transformed to that of the membrane element in the same layer by their geometrical relations (Fig.3)

gg dd uBu (5)

in which, is the displacement increment at the center point of the top and bottom edges of themembrane element, ={d

; is the displacementincrement at the center point of the layer, =

…

ud

2 dg1 d

uddu

g2 d

1ug

g2w

2ud

j

1vd 2vdgu

1wdg1{d u

wd

d v

T}g1w d

du d T...} ; and

is the transfer matrix between the twodisplacement increment vectors. The relationshipsbetween the force increment and the displacementincrement of the layer of the membrane element in theglobal coordinate system are

gB

a

EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.196

ggeg ddT

uBKBXk

g (6)

in which, d = 1 1 2

2 2

gXg

g{d X g1dY gd M gd X

gdY d M 1d X 2Xd … Td }kX is the forceincrement of the layer, jXd is the horizontal force of the membrane element corresponding to x . By the second assumption, jXd =0.

The stiffness matrix of the layer of the membraneelement is the sum of the stiffness matrix of allmembrane elements in the layer:

lK

kl

gegT

BKBK (7)

Eq. (6) can be rewritten as the block matrix

dd

0d

2221

1211 uKKKKX

(8)

Resolving the equations, the strain increment d ofthis layer can be obtained

uKK dd 211

22 (9)

From Eq. (8) and Eq. (9)

uKKKKX d)(d 211

221211 (10)

Eq. (10) can be marked as

For the RC membrane element, the modifiedcompression field theory (MCFT) (Vecchio and Collins, 1986; Vecchio, 1989; Vecchio and Collins,1993) is adopted to determine the constitutive relationof cracked concrete. The theory treated crackedconcrete in terms of average stress and average strain with the directions of principal stress and principalstrain, and reflected the compression softening effect and tension hardening effect. A perfect elasto-plasticrelationship is adopted for the reinforcement.

uKX dd m (11)

in which Tg

2g

2g2

g1

g1

g1 ddddddd MYXMYXX

(12a)Tg

2g2

g2

g1

g1

g1 ddddddd wvuwvuu

(12b)Until now, the stiffness matrix of the layer of the

membrane element has been developed. Thenumber of the degrees of freedom has been condensedfrom

mK

k3 to 3.

gdu1

gdv1 gdw1

gdu2

gdv2 gdw2

1du

2du

1dv

2dv

1dw

2dw

d

cdv1

cdv2

3.2 Stiffness matrix of column element The RC column element of the model is a tensile

or a compressive bar. There is one unknown verticaldisplacement at each end of the RC column element.By the plane section assumption, it can be determinedwith the displacements of rigid beams (Fig.3)

uBu dd cc (13)

in which, is the shape function matrix of columnelement, is the vertical displacement incrementvector of the column element, ,and is the same as Eq. (12). The stiffness matrixof the column element in a global coordinate system is

cBcdu

Tc2

c1c }dd{d vvu

ud

Fig.3 Displacement sketch of a layer

cTcc BBK

ssss

(14)

in which, is the tensile or compressive stiffnessparameter that can be determined by the constitutiverelations of concrete and reinforcement.

s

3.3 Stiffness matrix of layer

The stiffness matrix of the layer of the model is thesum of the stiffness matrices of membrane elementsand column elements:

2

1cmlayer

iKKK (15)

in which, two means there are two column elements in one layer.

3.4 Constitutive relations of materials in this study

For the RC column element, the completestress-strain curve equations for confined concrete andunconfined concrete suggested by Guo Zhenhai (1999)are adopted for concrete, and the hardeningelasto-plastic relationship is adopted for reinforcement.

4 Static inelastic analysis process

The equilibrium equations of static inelastic analysis of a structure are

No.1 Chen Qin et al. : Static inelastic analysis of RC shear walls 97

KP g (16)

in which, is the global stiffness matrix,gK P andis the load and displacement vector, respectively. It is well known that equilibrium equations are easy to solve when the stiffness matrix is positive. When theP curve approaches its peak (Fig.4), the stiffnessmatrix is ill-conditioned. To solve the ill-conditionedlinear equations, a method called single degree of freedom compensation (SDFC) proposed by ZouJiling (2001) is used in this study. The globalstiffness matrix is decomposed into stylebefore the equilibrium equations are resolved. To keepstability of the resolution, a compensation value isintroduce which makes every diagonal element of matrix is large enough. Then matrix ismodified to C is a diagonal matrix whoseelements are d , or 0 , and the modifiedequilibrium equations can be accurately solved. The true solution can be obtainedafter iterative calculation of the modified equilibriumequations. The forced iteration approach is adopted toextend analysis to the post-peak stage. The completeprocess of static inelastic analysis of the RC shear wallis carried out in this manner.

gK

C0

T

K

)

LDL

0d

g

d,D

K g

0P

dgK( C

P

Fig.4 Capacity curve

5 Experimental verification

5.1 Specimens

Four isolated RC shear wall specimens (SW-1SW-4) were tested and analyzed to verify the shear wall model newly developed in this study. SW-1 and SW-2 were rectangular sections, SW-3 was an I section and SW-4 was a T section. The details of thespecimens are shown in Fig.5.

Fig.5 RC shear wall speciments

The concrete cubic strength of specimens cu was25.2MPa, 22.9MPa, 34.1MPa and 34.4MPa,respectively. The yield strength y of reinforcementwith diameters of 4mm, 6mm and 10mm was631.7MPa, 451.7MPa and 395MPa, respectively.

f

f

The boundary zone of the specimens was confinedwith stirrups and the wall panel was reinforced withdistributed reinforcements. The stirrup characteristic

value v is used as a confinement parameter:

cyvv ff (17)

in which, v is the volumetric ratio of stirrup, isthe concrete compressive strength, =0.67 in thisstudy. For SW-1~SW-3,

cfcf cuf

v was 0.26, 0.28 and 0.25,respectively. For SW-4, it was 0.19 and 0.25 for the

EARTHQUAKE ENGINEERING AND ENGINEERING VIBRATION Vol.198

rectangular boundary zone and flanged boundary zone,respectively.

Strains of vertical reinforcements in boundary zones were measured with strain gauges. Lateral displacement along the wall height was measured withdisplacement transducers. Test data were recorded with the computer-based data acquisition system.

The specimens were subjected to constant vertical load and cyclic lateral load at the top. The axial force ratio of specimens was 0.44, 0.21, 0.31 and 0.31,respectively. All specimens were designed with strongshear and weak bending and failed in flexure.

5.2 Static inelastic analysis of specimens

Each specimen was divided into 11 layers alongthe height. There are eight RC membrane elementseach layer. At the rectangular end, each columnelement models half of the boundary zone. At theflanged end, one column element models the flange

and one column element models the web of boundaryzone. The complete process static inelastic analysisprogram of RC shear wall developed by this study was used to perform the analysis. 5.3 Comparison of analytical and experimental

resultsLateral force-top displacement curves, i.e.,

capacity curves, of specimens obtained by experimentand analysis are given in Fig.6. The experimentalresults are skeleton curves of hysteresis loops of specimens.

Table 1 lists the analytical and experimentalmaximum horizontal loads, i.e., the ultimate strengthof the specimens. The difference between the two results is less then 10%. Table 1 also shows the displacement ductility ratios and the ultimate rotationsof specimens. The displacement ductility ratio, ,and the ultimate rotation, , are defined as

(a)SW-1 (b)SW-2

(c)SW-3 (d)SW-4

Fig.6 Analytical and experimental lateral force-top displacement curves of specimens

No.1 Chen Qin et al. : Static inelastic analysis of RC shear walls 99

yu

(18b)

(18a)

Hu

the yield and the ultimatein which, and are

spla nt

y u

lateral di ceme at the top of specimen,respectively, and H is the displacement measureheight.

From the comparison, it is found that the staticine

Table 1 Comparison of experimental and analytical results Ultim

lastic analysis provides a good estimate of the capacity curves of the specimens.

ate lateral force /kN

xperimental Analytical

3.21 1/118

N

215.8

232.1 3.68 1/119

8.6 1/49S

N

185.6

192.9 5.58 1/72

S 328.2 4.23

N 326.1 4.35

1/101

1/99

S

N

282.0

286.4 297.9

3.48

2.92 3.43

1/120

1/166 1/110

SpecimensE Experimental Analytical Experimental Analytical

SW-1S

237.2 4.13 1/95

SW-2 201.8 6.45 1/56

SW-3 321.4 4.48 1/87

SW-4293.2 3.96 1/94

Note: S means loading from south and N means loading from north.

Conclusions

The macro-model of the RC shear wall proposedin t

References

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ffness

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6

his study simulates the wall panel and the boundaryzone with the RC membrane element and RC columnelement, respectively. It has the advantages of a finiteelement model but with fewer unknown numbers ofdisplacement. The approaching-peak analysis and thepost-peak analysis is carried out by using the singledegree of freedom compensation method and theforced iteration method, respectively. The accuracyand applicability of the proposed macro-model and the complete process static inelastic analysis of RC shearwalls are verified by comparing experimental andanalytical results.

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o