Deformation Capacity and Performance-Based Seismic Design for Reinforced Concrete Shear Walls

Transcription

1 Deformation Capacity and Performance-Based Seismic Design for Reinforced Concrete Shear Walls Ying Zhou* 1, Dan Zhang 2, Zhihua Huang 3 and Dan Li 4 1 Professor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, P.R. China 2 Doctoral Candidate, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, P.R. China 3 Doctor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, P.R. China 4 Master Student, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, P.R. China Abstract Deformation capacity of reinforced concrete (RC) shear walls is mainly influenced by concrete confinement at the boundaries of shear walls, axial force ratio and wall aspect ratio. In this paper, the relationship among the wall boundary transverse reinforcement characteristic value λ vw, the axial force ratio n, the wall aspect ratio r and the ultimate displacement uw is established first. Then, the relationship between λ vw n r uw is verified against the results of 71 RC shear wall experiments conducted by eight different research institutions. Based on the established relationship, the performance-based seismic design (PBSD) method for RC shear walls is proposed. According to the method presented in the paper, the amount of transverse reinforcement at wall boundaries could be determined, if the inter-storey drift demand θ and the damage index D w are predetermined. The use of the proposed PBSD method, which may guide future RC shear wall design, is illustrated in detail by an example. Keywords: RC shear wall; deformation capacity; performance-based seismic design 1. Introduction Reinforced concrete shear walls are one of the most critical structural elements of mid-rise and high-rise structures, as they typically serve to resist the majority of lateral seismic loads. The deformation capacity of RC shear walls, measured by the ultimate deformation Δ uw, significantly influences the seismic behavior of the entire structure. Therefore, investigation of the relationship between deformation capacity and other related parameters of RC shear walls is of major importance to performance-based seismic design for RC shear walls. There are various factors that influence the deformation capacity of RC shear walls including boundary confinements, axial force ratio, wall aspect ratio (sometimes represented as shear span ratio), etc. Boundary confinement is one of the dominating parameters controlling the damage mode and deformation capacity of RC shear walls. Experiments *Contact Author: Ying Zhou, Professor, State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, 1239 Siping Road, Shanghai , P.R. China Tel: Fax: yingzhou@tongji.edu.cn ( Received April 10, 2013 ; accepted November 11, 2013 ) indicate that the strain capacity of boundary concrete could be increased by enhancing the constraint level of the boundary element (usually quantified as the wallboundary transverse reinforcement characteristic value λ vw ), which as a result improves the wall deformation ductility. The axial force ratio n is another fundamental parameter that influences the failure mode and deformation capacity of RC shear walls as it directly changes the relative compression depth of the cross section of the wall. The wall aspect ratio r is a crucial parameter among those that influence the failure mode of the structure. It is verified experimentally that flexural failure usually occurs when wall aspect ratio r is greater than 2; flexural-shear failure happens when r is between 1 and 2; and shear failure predominates when r is less than 1. The deformation capacity design of RC shear walls seeks to establish a relationship between deformation and deformation-affecting factors. In this paper, the λ vw n r uw relationship is set up so that the PBSD of RC shear walls can be realized by calculating the transverse reinforcement at the boundaries in light of the anticipated deformation demands. 2. λ vw n r uw Relationship for RC Shear Walls It was proposed by Mander et al. (1988) and summarized by Paulay and Priestly (1992) that the Journal of Asian Architecture and Building Engineering/January 2014/

2 ultimate compressive strain ε cu of confined concrete could be obtained by the equation below: (1) In this equation, ρ v is the ratio of the volume of transverse confining steel to the volume of confined concrete core; f yh is the yield stress of transverse stirrups; and ε cm is the ultimate tension strain of transverse stirrups. The confined concrete strength is f cc = kf c, where k is a strength enhancement coefficient that depends on the transverse reinforcement characteristic value λ vw. This parameter is defined as follows: (2) According to Lu et al. (2005), when the reinforcement characteristic value is 0.023~0.523 (which is representative of most of the common design practices), parameter k varies between 1.12~1.62. Solving for ρ v in Eq. 1 and substituting it into Eq. 2 gives: whereas for the concrete strength grade C80, it is set at Linear interpolation is used to obtain a 1 for concrete with a strength between that of C50 and C80. Generally, RC shear walls are symmetrically reinforced at both ends, therefore A s = A s '. Assuming a 1 = 1.0 and l w l w0, Eq. 6 could be simplified to: Axial force ratio is defined as n = N/(f c t w l w ), therefore: Letting ρ w f yw / f c = k f, we get: Substituting Eq. 9 into Eq. 5 and rearranging gives (7) (8) (9) (10) (3) The ultimate compressive strain can be expressed as: (4) In Eq. 4, f uw is the ultimate curvature of the wall cross section; x n is the practical compression depth of cross section; x n = ξ n l w0, ξ n is the actual compression depth coefficient, l w0 is the effective height of wall cross section, and d' is the distance between the inner edge of transverse stirrups and the edge of the concrete compressive region. Substituting Eq. 4 into Eq. 3 gives the following expression: (5) In the computing model of rectangular cross section shear walls, as shown in Fig.1., the force equilibrium equation of the ultimate state is: (6) In the above equation, A s and A s ' are the tensile and compressive reinforcement area at wall boundaries; f y and f y ' are the tensile and compressive reinforcement yield stress at wall boundaries; ρ w is the distributed steel ratio; t w is the thickness of the shear wall; l w is the length of the wall; f yw is the yield stress of the compressive strength; a 1 is the equivalent stress diagram coefficient. When the concrete strength grade is less than or equivalent to C50, a 1 is taken as 1.0 Fig.1. Equilibrium Requirements for Rectangular Wall Cross Section (Wallace, 1992) When the longitudinal steel in the shear wall such as the one shown in Fig.2.(a) has just yielded, the cross section curvature can be assumed to approximately follow a linear distribution along the height of the wall, as illustrated in Fig.2.(b). When the bottom cross section reaches its ultimate state, the curvature distribution follows that illustrated in Fig.2.(c). In order to simplify the calculation of wall displacement, Park and Paulay (1975) proposed the equivalent plastic hinge length concept, which assumes that there is a section of length l pw within which the curvature is identical with the maximum plastic curvature of the 210 JAABE vol.13 no.1 January 2014 Ying Zhou

3 bottom cross section. Supposing structural components rotate with the centre point of the equivalent plastic hinge region as the plastic rotation centre before they reach the ultimate state, the ultimate deformation uw could be equated as: Fig.2. Curvature of Shear Walls (11) Kowalsky et al. (2001) put forward a theory that the plastic hinge length of shear walls could be regarded as half of the cross section height when processing the performance-based seismic design of shear walls, namely: (12) Priestley and Kowalsky (1998) recommended the shear wall yield curvature empirical formula as: (13) Thus the yield curvature of shear walls could be approximated as: (14) Substituting Eq. 12 and Eq. 14 into Eq. 11 and rearranging, gives: (15) If the reinforcement yield strain is assumed to be ε y = , Eq. 15 can be simplified to: (16) Substituting Eq. 16 into Eq. 10, gives, (17) So far the λ vw n r uw relationship has been established, and will next be verified by experiments in the section to follow. 3. Verification of λ vw n r uw Relationship As RC shear walls can effectively resist lateral force, they are increasingly used in high-rise buildings. Substantial experimental research on regular rectangular cross section RC shear walls has been carried out by institutions at home and abroad. In this paper, experimental data from 71 tested shear wall specimens are collected from eight institutions which are Tongji University (Zhang, 2007; Zhang and Lu, 2007; Jiang and Lu, 1997; Zhou, 2004; Zhang and Zhou, 2004), Dalian University of Technology (Li and Li, 2004), Tsinghua University (Li et al., 2002; Chen et al., 2004; Zhang, 1996), Beijing University Of Technology (Cao et al., 2008), Shenyang Architectural University (Cui et al., 2004), Hanyang University, Korea (Oh et al., 2002), Imperial College of Science and Technology, London (Lefas et al., 1990) and Clarkson University, America (Thomsen and Wallace, 2004). All the data from these experiments are tabulated in Table 1. In Fig.3., the key parameters of these 71 RC shear walls are summarized graphically. The wall aspect ratio r ranges from 0.5 to 3.0; the axial force n from 0 to 0.857; the ultimate displacement ratio θ uw ranges from 0.48% to 2.86%; the ultimate curvature φ uw ranges from 0.90 to mm -1 ; and the transverse reinforcement characteristic value λ vw,e ranges from to These data typically cover regular conditions in current construction. Following the process described in the previous section, the wall boundary transverse reinforcement characteristic value could be calculated as theoretic value λ vw,c. It is demonstrated in Fig.4. that when taking the theoretic value λ vw,c and experimental value λ vw,e of the transverse reinforcement characteristic value as x-axis and y-axis separately, the dots are uniformly and bilaterally distributed along the y=x line. The ratio of λ vw,c / λ vw,e ranges from 0.11 to 3.93, averagely as 0.95 and the variation coefficient of the ratio is The main reason for the discreteness lies in the experimental data from different institutions. According to the strength enhancement coefficient k and the data range of the ultimate tensile strain ε sm of confined transverse stirrups, the value of k/(1.4ε sm ) usually varies from 6 to 12. For simplification and a design safety margin, k/(1.4ε sm ) in Eq. 10 is taken as 20. JAABE vol.13 no.1 January 2014 Ying Zhou 211

4 Table 1. Experimental Data of Shear Walls Specimen r k f n uw (mm) θ uw (%) φ uw (10-5 mm -1 ) λ vw,e 1 SW ) SW ) SW ) SW-1 2) SW-2 2) SW1 3) SW2 3) SW3 3) SW4 3) SW5 3) SW6 3) SW7 3) SW8 3) SW9 3) SW10 3) SSW-2 4) SSW-3 4) SSW-T 4) SW11 6) SW12 6) SW13 6) SW14 6) SW15 6) SW16 6) SW17 6) SW21 6) SW22 6) SW23 6) SW24 6) SW25 6) SW26 6) SJ-1 7) SJ-2 7) SJ-3 7) SJ-4 7) SJ-5 7) SJ-6 7) SJ-7 7) SJ-8 7) SJ-9 7) CW-1 8) CW-2 8) CW-3 8) WR-20 11) WR-10 11) RW1 15) RW2 15) SW1-1 17) SW1-2 17) SW1-3 17) SW1-4 17) SW2-1 17) SW2-2 17) SW4-1 17) SW4-2 17) SW4-3 17) SW4-4 17) SW5-1 17) SW5-3 17) SW6-1 17) SW6-3 17) SW-2 18) SW-3 18) SW1-1 19) SW2-1 19) JAABE vol.13 no.1 January 2014 Ying Zhou

5 Specimen r k f n uw (mm) θ uw (%) φ uw (10-5 mm -1 ) λ vw,e 66 SW3-1 19) SW-7 20) SW-8 20) SW-9 20) SW-1 21) SW-2 21) Fig.3. Experimental Data Distribution of Shear Walls So Eq. 17 could be rewritten as: (19) Suppose that the shear wall inter-storey drift angle demand is: (20) Substituting Eq. 21 into Eq. 20 and rearranging gives Fig.4. Comparison between the Calculation Results λ vw,c and the Experimental Data λ vw,e 4. Performance-based Seismic Design of RC Shear Walls Suppose the shear wall deformation demand is w, and define the damage index of shear walls D w as below: (18) (21) Now the performance-based seismic design of shear walls is applicable using Eq. 21. It is also clear that the option of the damage index of shear walls is involved. In this paper, the damage index based on four performance levels suggested by Zhang and Lu (2007) is employed, as illustrated in Table 2. The performance-based seismic design of RC shear walls procedure is illustrated in Fig.5. and the design example is presented next. JAABE vol.13 no.1 January 2014 Ying Zhou 213

6 Table 2. Performance Levels and Damage Indices of Shear Walls Function Intact Slightly Faulted Moderately Faulted Not yet Collapsed Condition Description A handful of slight cracks emerge on the surface of shear walls; the crack width is less than 1mm and the length less than 10mm. A handful of cracks emerge on the surface of shear walls; the crack width is less than 1mm and the length less than 10mm. Some major cracks emerge; the crack width is less than 2mm and the length less than 100mm. Corner concrete begins to fall off. Penetrating cracks wider than 2mm emerge on the surface; relative rotation occurs in the wall. Surface concrete falls off dramatically; longitudinal reinforcement exposed but not yet buckled. D w 0~ ~ ~ ~0.9 Fig.6. Reinforcement of a Shear Wall (mm) = Determine the wall-boundary transverse reinforcement ratio ρ v = 1.58%, and consequently determine the transverse stirrup Φ14@100 mm. Similarly, for the same RC shear wall from the previous example when the damage index is selected to be D w = 0.8 and the inter-storey drift demand calculated as θ = 1/100, engineers could determine the transverse reinforcement characteristic value λ vw = This is equivalent to wall boundary transverse reinforcement ratio ρ v = 2.75%, and therefore engineers would choose Φ18@100 mm as transverse reinforcement. Fig.5. Flow Chart of the Performance-Based Seismic Design of RC Shear Walls Given the grade C40-concrete wall with a height, width and thickness of 4000mm, 3200mm and 300mm, respectively; which has 450mm-long boundary confinement on both sides, and the reinforcement determined by the first stage of the design as illustrated in Fig.6., performance-based seismic design proceeds as below: 1. Suppose the damage index of the shear wall D w =0.4; 2. Calculate the inter-story drift demand θ of the storey in the shear wall (refer to Zhou et al. 2012), which in this example is determined to be 1/300; 3. Calculate n=0.5 and wall aspect ratio r=h w /l w =4000/3200=1.25; 4. Determine the wall-boundary transverse reinforcement characteristic value λ vw using Eq. 21: 5. Conclusions Properly designed RC shear walls largely influence the safety of buildings under earthquakes and it is important to use performance-based seismic design in RC shear wall design. The deformation capacity design of structural members is required in performancebased design, in order to build a relationship between member deformations and the deformationrelated parameters. The deformation capacity of RC shear walls is dominated by boundary confinement conditions, axial force ratio, and wall aspect ratio. In this paper, the relationship among the wall-boundary transverse reinforcement characteristic value λ vw, axial force ratio n, wall aspect ratio r and shear wall ultimate displacement uw is established, namely the λ vw n r uw equation. The theoretical values of λ vw computed with this equation are compared with 71 shear wall experiments carried out by eight research institutes to verify the proposed λ vw n r uw equation. On the basis of the equation, a performance-based seismic design method for RC shear walls is presented. Using 214 JAABE vol.13 no.1 January 2014 Ying Zhou

7 this method, designers could calculate the transverse reinforcement at shear wall boundaries in light of the anticipated deformation requirement, provided that the inter-storey drift demand θ and damage index D w are given. Acknowledgements This work was financially supported by the National Natural Science Foundation of China ( ), National Basic Research Program of China (2014CB049100), and Shanghai Rising-Star Program (13QA ). References 1) Cao, W. L., Sun, T. B., Yang, X. M. et al. (2008) Experimental study on seismic performance of high-rise shear wall with bidirectional single row of steel bars. World Earthquake Engineering, 24 (3), pp (In Chinese) 2) Chen, Q., Qian, J. R. and Li, G. Q. (2004) Static elastic-plastic analysis of shear walls with macro-model. China Civil Engineering Journal, 37 (3), pp (In Chinese) 3) Cui, X. G., Liu, Y. J., Guo, F. et al. (2004) Experiment analysis of reinforced concrete shear wall with the cold-rolled ribbed reinforcement. Journal of Shenyang Architectural and Civil Engineering Institute (Natural Science Edition), 20 (2), pp (In Chinese) 4) Jiang, H. J. and Lu, X. L. (1997) Experimental study on low cyclic reversed loading of vertical energy dissipation shear walls. Engineering Mechanics, A02, pp (In Chinese) 5) Kowalsky, M. J. et al. (2001) RC structural walls designed according to UBC and displacement-based methods. Journal of Structural Engineering, 127 (5), pp ) Lefas, I. D., Kotsovos, M. D. and Ambraseys, N. N. (1990) Behavior of reinforced concrete structural walls: strength, deformation characteristics and failure mechanism. ACI Structural Journal, 87 (1), pp ) Li, H. N. and Li, B. (2004) Experimental study on seismic restoring performance of reinforced concrete shear walls. Journal of Building Structures, 25 (5), pp (In Chinese) 8) Li, G. Q., Qian, J. R. and Gu, W. L. (2002) Study on seismic behavior of shear walls with cold-rolled ribbed welded steel fabric. Building Structures, 32 (10), pp (In Chinese) 9) Lu, X. L., Zhou, D. S. and Jiang, H. J. (2005) Deformation capacity and performance-based seismic design method for RC frame columns. Earthquake Engineering and Engineering Vibration, 25 (6), pp (In Chinese) 10) Mander, J. B, Priestley, M. J. N. and Park, R. (1988) Theoretical stress-strain model for confined concrete. Journal of Structural Engineering, 114 (8), pp ) Oh, Y. H., Han, S. W. and Lee, L. H. (2002) Effect of boundary element details on the seismic deformation capacity of structural walls. Earthquake Engineering and Structural Dynamics, 31 (8), pp ) Park, R. and Paulay, T. (1975) Reinforced Concrete Structures. New York: John Wiley & Sons. 13) Paulay, T. and Priestly, M. J. N. (1992) Seismic design of reinforced concrete and masonry buildings. New York: John Wiley & Sons. 14) Priestley, M. J. N and Kowalsky, M. J. (1998) Aspects of drift and ductility capacity of rectangular cantilever structural walls. Bulletin of the New Zealand National Society for Earthquake Engineering, 31 (2), pp ) Thomsen IV, J. H. and Wallace, J. W. (2004) Displacementbased design of slender reinforced concrete structural walls experimental verification. Journal of Structural Engineering, 130 (4), pp ) Wallace, J. W. (1992) New methodology for seismic design of RC shear walls. Journal of Structural Engineering, 120 (3), pp ) Zhang, H. M. (2007) Research and development of performancebased seismic design theory and method. Ph.D. dissertation of Tongji University. (In Chinese) 18) Zhang, H. M. and Lu, X. L. (2007) Experiment study on seismic behavior of reinforced concrete walls strengthened by bonded steel. Structural Engineers, 23 (1), pp (In Chinese) 19) Zhang, Z. and Zhou, K. R. (2004) Experimental study on seismic behavior of high-performance concrete shear walls with various aspect ratios. Structural Engineers, 21 (2), pp (In Chinese) 20) Zhang, Y. F. (1996) Experimental study on reinforced concrete shear walls seismic performance under high axial force ratio. Master Degree Thesis of Tsinghua University. (In Chinese) 21) Zhou, G. Q. (2004) Research on the stress-strain hysteresis relation of high-rise buildings and its property. Master Degree Thesis of Tongji University. (In Chinese) 22) Zhou, Y., Lu, X. L. and Huang, Z. H. (2012) Study on R-μ-T models in predicting the displacement demand of a hybrid structure. Journal of Asian Architecture and Building Engineering, 11 (1), pp JAABE vol.13 no.1 January 2014 Ying Zhou 215