Nonlinear Static Pushover Analysis for Shear Wall Structures in

SAP2000 Program

Zhibin Su1, Tao Han1, Shengnan Sun1,a 1 School of Architecture & Civil Engineering, Liaocheng University, Liaocheng, Shandong, 252059,

China

a sunshengnan1982@163.com

Keywords: static pushover analysis; SAP2000 program; reinforced concrete; seismic response; seismic resistance

Abstract. To study the nonlinear mechanical characteristics of reinforced concrete shear wall

structures under rare earthquakes, a single reinforced concrete shear wall model is established in

SAP2000 program, which is simulated by nonlinear multi-layer shell element. Nonlinear static

pushover analysis of the model is presented by uniform acceleration lateral load pattern and inverted

triangle lateral load pattern. The relationship curve between base shear and top displacement of shear

wall, and the stress distribution diagrams of the concrete layer and rebar layer are obtained. It may be

concluded that, the yielding of rebar layer and the cracking of the concrete layer may be observed by

stress distribution diagrams. SAP2000 program is feasible to nonlinear simulation of shear wall

structures.

Introduction

With the development of performance-based design theory in structural seismic design, nonlinear

static pushover analysis [1-3] has gained more and more attention worldwide in earthquake

engineering [4-8]. This method is to assign a series of lateral forces along the structure height,

gradually increasing this lateral force, until the structure control point achieving the preset

displacement or the structure being pushed over. Compared with the traditional linear static analysis

method, nonlinear static pushover analysis may take the structure nonlinear mechanical

characteristics under rare earthquakes into account, get the weak links of the structural design, and

observe the ductility properties of the entire structure or members. In this way, we can obtain a perfect

design.

Reinforced concrete(RC) shear walls are common lateral force-resisting members in high-rise

buildings. Since structures will enter a nonlinear state under rare earthquakes, simulating nonlinear

mechanical characteristics of concrete reinforced shear walls accurately are an important issue for

seismic resistance. Reinforced concrete shear walls are two-dimensional bearing members. Thus the

mechanical characteristics are more complex than beams and columns that may be simplified as truss

components. How to accurately simulate the nonlinear characteristics of reinforced concrete shear

walls has been extensively studied. Common methods used to simulate RC shear walls are the

equivalent shear beam model, the three vertical truss element model, more vertical truss element

model, equivalent truss model and space thin-walled model [9]. These methods can not take the

in-plane bending–in-plane shear–out-of-plane bending coupling of RC shear wall components into

account. Therefore, for non-linear analysis of large shear wall structures, they may lead to errors.

Multi-layer shell elements which are established based on the composite material mechanical

principle are finite elements used to description nonlinear mechanical characteristics the RC shear

walls. A shell is divided into many layers by shell elements. Each layer is set to be a material layer

with thickness according to requirement. For RC shear walls, materials usually include steel and

concrete. When finite element analysis of a multi-layer shell RC shear wall is presented, the strain and

curvature of middle layer is obtained first. Then according to the thickness of each layer and its

distance to the middle layer, the strain of each layer is obtained based on plane section assumption.

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Subsequently, the corresponding stress of each layer is obtained by material constitutive equations.

Stress of the entire shell element may be obtained by integration through each layer. Multi-Layer shell

elements may take the in-plane bending–in-plane shear–out-of-plane bending coupling of RC shear

wall components into account. Thus, it may reflect the space mechanical characteristics of RC shear

walls fully.

Definition of Multi-layer Shell Element

In previous versions of SAP2000 V14, only elastic analysis of shell elements may be presented. If

nonlinear analysis is involved, nonlinearity of RC shear walls has to be simulated indirectly by frames

with equivalent stiffness or adding connection elements for engineers. It is very inconvenient. Thus, a

new element named nonlinear multi-layer shell element is added in SAP2000 V14. This element is

more realistic, reasonable and easier to simulate the nonlinear mechanical characteristics of RC shear

walls, without equaled by other components.

Multi-layer shell elements simulate the nonlinear behavior of RC shear walls by material

constitutive models of concrete and rebar directly. Thus, the material constitutive models of concrete

and reinforced is critical. There are two material constitutive models of concrete in SAP2000

program: Simple and Mander. Simple model doesn’t take the effect of stirrups on the concrete

constitutive relationship into account. Mander model may amend the constitutive model according to

stirrups. SAP2000 provides three hysteretic models: Elastic, Kinematic and Takeda. Users may

choose one according to requirement. In addition, stress-strain curves of steel or concrete may be

transformed into user-defined forms. which is facilitate for engineers to modify the material

constitutive model according to their requirement.

Modeling

In this paper, the RC shear wall model in SAP2000 program is established according to RC Shear wall

shown in Figure 2. According to the reinforcement ratio and reinforcement method of shear wall, the

concealed part and the middle part of RC shear wall are established respectively. In this model,

measured yield strength value of the φ8 rebar is 400.9 Mpa. Ultimate strength value is 519.5 Mpa.

Measured yield strength value of the Φ16 rebar is 396.8 Mpa. Ultimate strength value is 608.8 Mpa.

The cylinder compressive strength of concrete is 40.9 Mpa. Vertical load on this model is 950 kN.

28

00

50

0

beam for loading

250

250

250

280

050

0

400

160

(a) (b)

1 1

Foundation beam

2400

8@100

8@100

2

3

550 1300 550

4 16

8@1004

1

4 16 8@100 8@100

1300

160

8@100

180 940 180

1 2 34

6

(c)

Fig. 2 (a) front elevation (b) lateral elevation (c)Reinforcement cross section

The lateral load pattern in pushover analysis influences the calculation results greatly. How to

choose the lateral load pattern is not explained clearly in “Code for seismic design of buildings

(GB50011-2001)”. Japan's new “Building Standard Law (BSL2000)” emphasizes: the lateral load

pattern should be able to represent the inertia force distribution pattern of the structure under rare

earthquakes. U.S. Federal Emergency Management Agency's FEMA273 suggests that at least two

1008 Mechanical Engineering, Materials Science and Civil Engineering II

patterns should be used in a model [1]. One is inverted triangle lateral load pattern, the other is

uniform acceleration lateral load pattern. Inverted triangle lateral load pattern represents the inertia

force distribution of elastic seismic response of structure. Uniform acceleration lateral load pattern

represents the inertia force distribution after structure weak layer yielding. Some scholars believe that

uniform acceleration lateral load pattern and inverted triangle lateral load pattern represent the upper

and lower limits of structure seismic response respectively. In this paper, the two lateral load patterns

are adopted.

Inverted triangle lateral load pattern is used in norms worldwide. It is expressed as:

1

/n

i i m m b

m

F w w h V=

=

∑ (1)

In uniform acceleration lateral load pattern, the force acted by earthquake on each floor is

proportional to the gravity representative value of the floor. It is expressed as:

1

/n

i i m b

m

F w w V=

=

∑ . (2)

Where, mh is the height of the mth

floor to the ground floor; n is the total number of floors; mw and

iw are gravity representative values of the mth

floor and the ith

floor respectively; bV is the total base

shear of the structure.

Displacement control is adopted in the nonlinear static pushover analysis. The check point is the

top point of the structure. Gravity and vertical loads are maintained in the pushover analysis.

Analysis of results

Relationship between base shear force and top displacement under inverted triangle lateral load

pattern is shown in Figure 3(a). The top displacement and base shear is linear relationship in initial

phase. As the top displacement increases, the reinforcement yields and the concrete cracks. The shear

wall is pushed over finally. When the base shear force reaches maximum, maximum stress of concrete

layer and rebar layer are shown in Figure 3(b) and Figure 3(c) respectively. The concrete layer cracks

due to tension at the right of bottom and the rebar layer yields due to tension.

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0080

50

100

150

200

250

300

350

Bas

e S

hea

r F

orc

e(k

N)

Top Displacement(m) (a) (b) (c)

Fig. 3 The reaction of the shear wall under inverted triangle lateral load pattern

Relationship between base shear force and top displacement under uniform acceleration lateral

load pattern is shown in Figure 4(a). It may be seen that the rule of curve is similar to that of inverted

triangle lateral load pattern. However, the base shear force when the displacement reaches its

maximum is larger than that of inverted triangle lateral load pattern. When the base shear force

reaches maximum, maximum stress of concrete layer and rebar layer are shown in Figure 4(b) and

Figure 4(c) respectively. As we can see, the positions where the concrete cracks and the rebar yields

are same as those of inverted triangle lateral load pattern.

Applied Mechanics and Materials Vol. 470 1009

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.0080

100

200

300

400

500

600

700B

ase

Shea

r F

orc

e(kN

)

Top Displacement (a) (b) (c)

Fig. 4 The reaction of the shear wall under uniform acceleration lateral load pattern

Conclusions

Conclusions of this paper may be drawn as follows.

(1) The yielding of rebar layer and the cracking of the concrete layer may be observed by stress

distribution diagrams.

(2) SAP2000 program is feasible to nonlinear simulation of shear wall structures under rare

earthquakes.

Acknowledgements

This work was financially supported by A Project of Shandong Province Higher Educational Science

and Technology Program(Grant No.J12LG10).

References

[1] Federal Emergency Management Agency, “Guidelines and Commentary for the Seismic

Rehabilitation of Buildings,” FEMA 274, 1998.

[2] Federal Emergency Management Agency, “Prestandard and Commentary for the Seismic

Rehabilitation of Buildings,” FEMA 356, 2000.

[3] ATC, “Seismic Evaluation and Retrofit of Concrete Buildings,” Report No.ATC-40, Applied

Technology Council, Redwood City, California, 1996.

[4] Fenwick R., D. Bull, “What is the stiffness of reinforced concrete walls?” SESOC Journal, vol.

13, pp. 23-32, Feberary, 2000.

[5] Gup ta B, Kunnath S K. “Adaptive spectra-based pushover procedure for seismic evaluation of

structures,” Earthquake Spectra, vol. 16, pp. 367-391, May, 2000.

[6] KALKAN E,SASHI S K. “Adaptive modal combination procedure for nonlinear static analysis

of buildingstructures,” Journal of Structural Engineering, vol. 132, 1721-1731, November 2006.

[7] Hou Shuang, Ou Jinping. “A study of load pattern selection of pushover analysis and influence of

higher modes,” Earthquake Engineering and Engineering Vibration, vol.24, pp. 89-97, June,

2004. (In Chinese ).

[8] Adebar, P., A. M. M. Ibrahim, “Simple Nonlinear Flexural Stiffness Model for Concrete

Structural Walls,” Earthquake Spectra, vol. 18, pp. 407-426, March, 2002.

[9] Jiang Jianjing, Lu Xinzheng, Ye Lieping, “Finite Element Analysis of Concrete Structures,”

BeiJing tsinghua university press, BeiJing, 2005 (In Chinese).

1010 Mechanical Engineering, Materials Science and Civil Engineering II

Mechanical Engineering, Materials Science and Civil Engineering II 10.4028/www.scientific.net/AMM.470 Nonlinear Static Pushover Analysis for Shear Wall Structures in SAP2000 Program 10.4028/www.scientific.net/AMM.470.1007