Pushover Analysis for Structure Containing RC Walls

The 2nd International Conference on Urban Disaster Reduction, Taipei, Taiwan November, 27-29, 2007

Pushover Analysis for Structure Containing RC Walls

Chung- Yue Wang1 and Shaing-Yung Ho2

Abstract In this paper, a method for the determination of the parameters of plastic hinge properties (PHP) for structure containing RC wall in the pushover analysis is proposed. Nonlinear relationship between the lateral shear force and lateral deformation of RC wall is calculated first by the Response-2000 and Membrane-2000 code. The PHP (plastic hinge properties) value of each parameter for the pushover analysis function of SAP2000 or ETABS is defined as the product of two parameters α andβ . Values ofα at states of cracking, ultimate strength and failure of the concrete wall under shear loading can be determined respectively from the calculations by Response-2000. While the corresponding β value of each PHP parameter is obtained from the regression equations calibrated from the experimental results of pushover tests of RC frame-wall specimens. The accuracy of this newly proposed method is verified by other experimental results. It shows that the presented method can effectively assist engineers to conduct the performance design of structure containing RC shear wall using the SAP2000 or ETABS codes. Keywords: pushover analysis, RC wall, plastic hinge properties Corresponding Author Name: Chung-Yue Wang Address: Dept. of Civil Engineering, National Central University, Chungli, Taiwan 32054, ROC E-mail: [email protected] Tel: 886-3-4227151 ext. 34127 Fax: 886-3-4252960

. 1. INTRODUCTION

1.1 Pushover Analysis Pushover analysis is a static, nonlinear procedure in which the magnitude of the

structural loading is incrementally increased in accordance with a certain predefined pattern. Static pushover analysis is an attempt by the structural engineering profession to evaluate the real strength of the structure and it promises to be a useful and effective tool for performance based design.

The ATC-40 (1996) documents have developed modeling procedures, acceptance criteria and analysis procedures for pushover analysis. These documents define

1 Professor, Department of Civil Engineering, National Central University, Chungli, Taiwan 32054, ROC 2 Associate Professor, Department of Construction Engineering, Vanung University, Chungli, Taiwan 32061, ROC


force-deformation criteria for hinges used in pushover analysis. As shown in Figure 1, five points labeled A, B, C, D, and E are used to define the force-displacement behavior of the hinge and three points labeled IO, LS, and CP are used to define the acceptance criteria for the hinge. The IO, the LS and the CP stand for Immediate Occupancy, Life safety and Collapse Prevention, respectively. These are informational measures that are reported in the analysis results and used for performance-based design.

Figure 2 illustrates a typical representation of capacity curve of RC structure containing shear wall. It is clear to be observed that the pattern of the experimental cure is continuous and is different from the general Skelton model shown in Fig. 1. As shown in that figure, the response is linear to an effective crack point, B, followed by cracking (with concrete cracking) to yield point C, followed by yielding (possibly with strain hardening) to ultimate point D, followed by final collapse and loss of gravity load capacity at point E.

Figure 1: Force-displacement curve defined for the plastic hinge in the pushover analysis.

1.2 SAP2000 SAP2000 is a well known and widely accepted, general-purpose, three-dimensional

structural analysis program. The pushover analysis module has been installed into the SAP2000. In the procedure of the pushover analysis, the assignment of the values of plastic hinge properties (PHP) strongly affects the prediction of the capacity curve of RC structure.

SAP2000 program includes several built-in default hinge properties that are based on average values from ATC-40 for concrete members. These built-in properties can be useful for preliminary analyses, but user-defined properties are recommended for final analyses (Habibullah and Pyle, 1998). Yielding and post-yielding behavior can be modeled using discrete user-defined hinges. Currently SAP2000 allows hinges can only be introduced into frame elements; the PHP properties can be assigned to a frame element at any location along it.

The authors have been developed a dual parameters method to define the PHP properties of RC frame structure for the pushover analysis (Ho and Wang, 2006). The purpose of this paper is to extend the application of this method to the RC structures containing RC shear wall. In order to use the functions provided by the SAP2000 code, the RC shear wall is treated as a wide, flat column. Modeling a RC wall as a wide and flat column (frame elements) not only can consider the steel reinforcements in RC elements exactly, but also can assign the PHP of RC walls according to its plastic behavior. In SAP2000, the default properties are available for hinges in the following degrees of freedom:

1. Axial (P) 2. Major shear (V2)

ED

C

B

A

FOR

CE

DISPLACEMENT

IO LS

CP


3. Major moment (M3) 4. Coupled P-M2-M3 (PMM)

Since the wall or wall segment behavior is governed by shear, it is more appropriate to use shear drift ratio as the deformation measure. Shear drift ratio capacities are defined in ATC-40 Table 9-11. In the present method, the major shear (V2) is adopted as the PHP type of RC walls.

1.3 Response-2000

To reasonably model the interactions among steel reinforcements and concrete material of a member under shear, a mechanics based reinforced concrete analysis method should be selected. Response 2000 uses the Modified Compression Field Theory (MCFT) firstly proposed by Collins (Mitchell and Collins, 1974, Collins, 1978). A definitive description of the MCFT can be found in the paper published by the American Concrete Institute. (Vecchio and Collins, 1986).

Response 2000 will calculate strengths and deformations for beams and columns subjected to axial load, moment and shear based on the familiar assumption “plane sections remain plane” of engineering beam theory. Therefore, to have a reasonable analysis result, a limitation on the depth/span ratio of the frame component should exist.

Deep beams (short and flat columns) can be considered as special beams where shear force is much more predominant than bending moment. Due to the parabolic distribution of the shear stress from top to bottom of a beam section, the cross sections of a deep beam which were originally plane surfaces become warped (Timoshenko and Gere, 1972), Furthermore, the interaction between wall and columns make the behavior of RC frame containing RC wall more complicated. This is the reason of using dual parameters to characterize the PHP properties. As shown in Fig. 2, parameters

Vβ and ∆β .are used to modified the capacity curve [( IV

I αα ,∆ ), I = B, C, D, E] predicted by the Response 2000 based on the plane section remaining plane assumption. The physical meaning of the β parameter is close to the shear factor in the Timoshenko beam theory.

Figure 2 Construction of the capacity curve of RC structure containing shear walls in the

pushover analysis with SAP2000 by dual parameters.

),( DVV

DD αβαβ ∆∆

),( DV

D αα∆

),( CV

C αα∆ (2) Modification by Vβ , ∆β

(3) Experimental curve

)95.0,6.2( DVV

DE αβαβ ∆∆

),( CVV

CC αβαβ ∆∆

),( BVV

BB αβαβ ∆∆

)95.0,6.2( DV

EV

DE αααα == ∆∆

(1) Calculate α from Response2000


2. DETERMINATION OF PHP AND REGRESSIVE FORMULA OF β

The core idea of the determination of the PHP of RC wall by dual parameters is illustrated by Fig. 2. The Response is used first to obtain a reference skeleton curve. The ultimate values D

∆α and DVα are

modified by parameters ∆β and Vβ , respectively. Regression formulae for parameters ∆β and Vβ are obtained by comparing the ultimate values of predicted and experimental data. These modification coefficients are also applied to points B and C and E on the skeleton curve. Detail procedure is explained in the following sections. 2.1 Construction of the Skeleton Curve α

Experimental results of 11 RC frames containing RC walls under pushover tests were adopted to construct the regression formulae of the modification parameters ( Vβ , ∆β ). The material properties and size of these specimens were given in Table 1.

The capacity curve of RC frame containing RC wall is calculated first by the code Response 2000. As shown in Fig. 2, three characteristic points denoted as crack, yield and ultimate states are selected to form a skeleton curve for the determination of the values of PHP. However, crack or yielding point can not be decided easily from the relational curve. But, one can estimate the ultimate point according to maximum strength of the capacity curve predicted by response 2000. Therefore, the present method selected the ultimate point as the reference to scale down all other characteristic points by its modification coefficients ( Vβ , ∆β ).

One thing has to be mentioned is the definition of failure point in the pushover analysis. Experimental data show that the failure point can not be identified exactly due to a rather precarious condition at final stage of the experiment. Therefore, present method adopts the average value of experimental results at final stage to define the failure point and use the factor 0.95 and 2.6 to time the ultimate shear force and ultimate lateral displacement respectively to define the failure point E, as shown in Figs. 2 and 3.

Figure 3 Selection of the characteristic points from the response curve predicted by Response 2000. 2.2 Regression Formulae for the Parameters Vβ and ∆β

To construct the regression formula of the parameter Vβ which is used to modify the ultimate strength, another parameter Vβ is defined as the ratio of the ultimate strength of experiment and the ultimate strength predicted by Response 2000 from the previous step.

Estimated from Response 2000

The approximation curve

Characteristic point 1 (initial cracking)

Characteristic point 2 (yielding)

Ultimate point

Failure point


Table 1 Material properties and dimensions of RC frames containing shear wall

L: Length (cm), W: width (cm), D: depth (cm), H: height (cm), fy: yielding stress (Mpa)

Year [ref.] 2004[Xiao] 2004[Xiao] 2004[Xiao] 2004[Xiao] 2004[Xiao] 2004[Xiao]

No. of specimen HWFL1 HWFL2 HWFH1 HWFH2 MWF1 MWF2

frame (W , H) 200 x340 200 x340 200 x340 200 x340 250 x240 250 x240

L x W x D 150x25x40 150x25x40 150x25x40 150x25x40 200x30x40 200x30x40

Main steel 4-#5 4-#5 4-#5 4-#5 4-#5 4-#5

fy (Mpa) 428.13 428.13 428.13 428.13 390 390

Stirrup #3@15cm #3@15cm #3@15cm #3@15cm #3@10cm #3@10cm Beam

sect

ion

fy (Mpa) 458.72 458.72 458.72 458.72 458.72 458.72


Main steel 4-#5 4-#5 4-#5 4-#5 4-#5 4-#5

fy (Mpa) 428.13 428.13 428.13 428.13 390 390

Stirrup #3@10cm #3@10cm #3@10cm #3@10cm #3@10cm #3@10cm

Col.

sect

ion

fy (Mpa) 458.72 458.72 458.72 458.72 458.72 458.72


Longitudinal #3@17cm #3@23cm #3@17cm #3@23cm #3@17cm #3@23cm

Transverse #3@23cm #3@23cm #3@23cm #3@23cm #3@23cm #3@23cm

Wal

l sec

tion

fy (Mpa) 458.72 458.72 458.72 458.72 458.72 458.72

Concrete fc’ (Mpa) 23.04 22.43 28.03 27.73 23.45 23.45

Year [ref.] 2004 [Xiao] 2004 [Xiao] 2001 [Ye] 2002 [ Qio] 2002 [ Qio]

No. of specimen LWF1 LWF2 WF15Y WF12C WF15C

frame (W , H) 320 x240 320 x240 350 x205 350 x205 350 x205

L x W x D 270x30x40 270x30x40 250x30x50 250x30x50 250x30x50

Main steel 4-#5 4-#5 8-#6 8-#6 8-#6

fy (Mpa) 390 390 571 571 571

Stirrup #3@10cm #3@10cm #3@20cm #3@20cm #3@20cm Beam

sect

ion

fy (Mpa) 458.72 458.72 446 446 446


Main steel 4-#5 4-#5 10-#6 10-#6 10-#6

fy (Mpa) 390 390 571 571 571

Stirrup #3@10cm #3@10cm #3@30cm #3@30cm #3@30cm

Col.

sect

ion

fy (Mpa) 458.72 458.72 446 446 446


Longitudinal #3@17cm #3@23cm #3@20cm #3@30cm #3@20cm

Transverse #3@23cm #3@23cm #3@20cm #3@30cm #3@20cm

Wal

l sec

tion

fy (Mpa) 458.72 458.72 446(double) 446 446( double )

Concrete fc’ (Mpa) 23.75 23.75 21.6 21 22.6


Following parameters are selected to construct the regression formula: 11. ratioK )( : the ratio of rigidity between two columns connected to wall and wall at ultimate stage. 12. wallusumu PP )/()( : ultimate strength of columns connected to wall divided by ultimate strength of

wall. 13. wH / : the height of wall divided by the width of wall. Hence, the regressive formula is in the following form. )()/()()( wwallusumuratioV HdPPcKba ×+×+×+=β (1) 1tt

Using part of the data and test results of the specimens shown in Table 1 and the values of Vβ , those coefficients of the regression equation (1) can be determined as a=0.84652, b=－2.39019, c=0.09413, d=0.07647.

The lateral displacement of the shear wall is controlled by the shearing resistance and the bending rigidity inherited in the wall system. Hence, the lateral displacement, u∆ at ultimate stage is proposed by the following equation.

cg

crww

crw

u EIHVHb

GAHVHa

322 )()( +=∆ (2)

Where crV is the lateral force at cracking stage(KN), wA is the total cross sectional area of RC wall and columns section (mm2), G is the shear modulus of concrete (Mpa), gI is the total mass moment of inertia of RC wall and side columns (mm4), cE is the elastic modulus of concrete (Mpa). Using part of the data and test results of the specimens shown in Table 1, those coefficients of the regression equation (2) can be determined as a = 0.0018214，b =－0.00001519.

For any specimen a modification parameter ∆β can be calculated by the following equation.

responseuu )/(∆∆=∆β (3)

Where u∆ is the regressive value which was estimated from the regression equation (2), while the responseu )(∆ is the ultimate displacement calculated from Response 2000. This ∆β is used to adjust the

displacement coordinate of each characteristic point in Fig, 3.

(a) Specimen SW7 (b) Specimen HM4-3


(c) Specimen SW16 (d) Specimen PWL

Figure 4 Comparison of experimental results and predictions by the dual parameters method for the determination of PHP values of RC structures containing shear walls.

Figure 4 shows the predictions of the capacity curves of some specimens that were not used in

the regression analysis by the method and associated regression formulae proposed in this paper. Comparing with the experimental results, it shows that the proposed dual parameters method for the determination of PHP can provide quite good predictions of the capacity of the structures containing shear walls.

3. EXAMPLES

In order to evaluate the capability and accuracy of proposed method on the analysis of real larger structure, 2 bays 2 stories RC structures with and without shear walls, as shown in Fig. 5, were adopted to do the study. Figure 6 illustrates the agreement between the prediction and experimental result of each case. The location and number of shear walls strongly affect the load capacity of the structure.

Figure 5 Geometrical data and reinforcement details of a 2 bays- 2 stories frame specimen.


Figure 6 Comparisons of experimental curves and predicted curves of 2 bays-2 stories RC frames with and without RC walls.

4. Conclusions

A dual parameters method is introduced to define the plastic hinge properties (PHP) of RC wall in the pushover analysis of RC structure. The effectiveness of this simple method is verified by the agreement of the prediction curves with some additional test data. This newly proposed method is quite simple and is easy for engineers to link with commercial structural analysis code to conduct the performance design of structure under seismic loading. References: Applied Technology Council (1996)﹐“Seismic Evaluation and Retrofit of Concrete Buildings,” ATC-40﹐Redwood City,

California, pp. 8.1-8.3. Collins, M. P., (1978), “Towards a Rational Theory for RC Members in shear,” Journal of the Structure Division, ASCE,

Vol. 104, pp. 649-666. Chang, Kuo-Chun and Sen-Nann Shain, " Seismic Upgrade of RC Frames with Infilled Brick Walls," National

Center for Research on Earthquake Engineering , NCREE – 03 - 039, Taiwan (2003). Habibullah, Ashraf. and Stephen, Pyle (1998)., “Practical Three-Dimensional Nonlinear Static Pushover Analysis,”

Structure Magazine, U.S.A. , pp.1-2. Ho, S.Y. and Wang, C. Y. (2006), “Determination of the Plastic Hinge Properties of RC structure containing Brick Walls,”

Structural Engineering of Taiwan, (in press) (in Chinese). Mitchell, D. and Collins, M. P., (1974), “Diagonal Compression Field Theory – A Rational Model for Structural Concrete in

pure Torsion,” Journal of American Concrete Institute, 71, 396-408. Qio, C. Z.. (2002), A Study of Seismic Performance of Reinforced Concrete Walls which Retrofit with FRP, Master Thesis,

Dep. of Construction Engineering, National Taiwan University of Science and Technology, Taipai., Taiwan. Timoshenko S. P. and Gere J. M (1972)., Mechanics of Materials, Litton Educational Publishing Inc., New York, U. S. A., p.

126. Vecchio, F. J. and Collins, M. P. “Predicting the Response of Reinforced Concrete Beams Subjected to Shear Using

Modified Compression Field Theory.” ACI Structural Journal, Vol. 85, No. 3, 1988, pp. 258-268. Xiao, F. P. (2004), Experimental and Numerical Studies on Reinforced Concrete Framed Shear Walls, Ph. D. Dissertation,

Dep. of Civil Engineering, Nation Cheng Gong University, Tainan, Taiwan. Ye, Y. X..(2001), A Study of Seismic Performance of Reinforced Concrete Walls which Retrofit with FRP, Master Thesis,

Dep. of Construction Engineering, National Taiwan University of Science and Technology, Taipai., Taiwan.

Pushover Analysis for Structure Containing RC Walls

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