钢筋混凝土柱初始刚度

1. INTRODUCTION In recent years, earthquake design philosophy has shifted from a traditional force-based approach toward a displacement-based ideology. The assumed initial stiffness of reinforced concrete (RC) columns could affect the estimation of the displacement and displacement ductility, which are crucial in displacement-based design. In addition, the assumed initial stiffness properties of columns also affect the estimation of the fundamental period and distribution of internal forces of structures. Therefore, an accurate evaluation of the initial stiffness of columns becomes an inevitable requirement. Literature reviews show that there is a considerable amount of uncertainty regarding the estimation of the initial stiffness of columns when subjected to seismic loads. Current design codes often employ a stiffness reduction factor to deal with this uncertainty. In an attempt to address these uncertainties, the study presented within this paper is devoted to developing a rational method to determine the initial stiffness of RC columns when subjected to seismic loads. A comprehensive parametric study based on the proposed Advances in Structural Engineering Vol. 15 No. 2 2012 265 Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios Cao Thanh Ngoc Tran1 and Bing Li2,* 1Department of Civil Engineering, International University, Vietnam National University, Ho Chi Minh City, Vietnam 2School of Civil and Environment Engineering, Nanyang Technological University, Singapore 639798 (Received: 1 December 2010; Received revised form: 17 May 2011; Accepted: 4 June 2011) Abstract: The estimation of the initial stiffness of columns subjected to seismic loadings has long been a matter of considerable uncertainty. This paper reports a study that is devoted to addressing this uncertainty by developing a rational method to determine the initial stiffness of RC columns when subjected to seismic loads. A comprehensive parametric study based on a proposed method is initially carried out to investigate the influences of several critical parameters. A simple equation is then proposed to estimate the initial stiffness of RC columns. The applicability and accuracy of the proposed method and equation are then verified with the experimental data obtained from literature studies. Key words: reinforced concrete, column initial stiffness, stiffness ratio. method was carried out to investigate the influences of several critical parameters. A simple equation to estimate the initial stiffness of RC columns is also proposed within this paper. The applicability and accuracy of the proposed method and equation are then verified with the experimental data obtained from the literature. 2. DEFINING INITIAL STIFFNESS OF RC COLUMNS There are two methods as illustrated in Figure 1(a) that are commonly utilized to determine the initial stiffness of RC columns (Ki). In the first method, the initial stiffness of RC columns are estimated by using the secant of the shear force versus lateral displacement relationship passing through the point at which the applied force reaches 75% of the flexural strength (0.75 Vu). In the second method, the column is loaded until either the first yield occurs in the longitudinal reinforcement or the maximum compressive strain of concrete reaches 0.002 at a critical section of the column. This corresponds to point A in Figure 1(a). Generally, the two approaches give similar values. In this study, the later approach was adopted. *Corresponding author. Email address: cbli@ntu.edu.sg; Tel: +65-6790-5292. Associate Editor: J.G. Dai. However, the above mentioned definition cannot be used for columns whose shear strengths do not substantially exceed its theoretical yield force. For these columns, defined as those whose maximum measured shear force was less than 107% of the theoretical yield force, the effective stiffness was defined based on a point on the measured force-displacement envelope with a shear force equal to 0.8 Vmax as illustrated in Figure 1(b) (Elwood et al. 2009). Assuming the column is fixed against rotation at both ends and has a linear variation in curvature over the height of the column, the measured effective moment of inertia can be determined as: (1)I L K Ee i c = 3 12 The stiffness ratio (κ) is defined as follows: (2) where Ig is the moment of inertia of the gross section; Ki is the initial stiffness of columns and L is the height of columns and Ec is the elastic modulus of concrete. 3. REVIEW OF EXISTING INITIAL STIFFNESS MODELS 3.1. ACI 318-08 (2008) ACI 318-08 (2008) recommends the following options for estimating member stiffness for the determination of lateral deflection of building systems subjected to factored lateral loads: (a) 0.35 EIg for members with an axial load ratio of less than 0.10 and 0.70 EIg for members with an axial load ratio of more than or equal to 0.10; or (b) 0.50 EIg for all members. 3.2. FEMA 356 (2000) FEMA 356 (2000) suggests the variation of effective stiffness values with the applied axial load ratio. The effective stiffness is taken as 0.50 EIg for members with an axial load ratio of less than 0.30, while a value of 0.7 EIg is adopted for members with an axial load ratio of more than 0.50. This value varies linearly for intermediate axial load ratios as illustrated in Figure 2. 3.3. ASCE 41 (2007) As shown in Figure 2, ASCE 41 (2007) recommends that the effective stiffness is taken as 0.30 EIg for members κ = ×I I e g 100% 266 Advances in Structural Engineering Vol. 15 No. 2 2012 Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios (a) A' A Initial stiffness Sh ea r f or ce 0.75 Vu Vu Vy Lateral displacement (b) (Elwood et al. 2009) Initial stiffness A Vu Sh ea r f or ce 0.80 Vmax Vy Lateral displacement Figure 1. Methods to determine initial stiffness 0 − 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 Axial load ratio f 'c Ag St iff ne ss ra tio k (% ) ACI 318-0.8 (a) ACI 318-0.8 (b) FEMA 356 ASCE 41 PP92 EE09 Figure 2. Relationships between stiffness ratio and axial load ratio of existing models with an axial load ratio of less than 0.10, as 0.7 EIg for members with an axial load ratio of more than 0.50 and varies linearly for intermediate axial load ratios. 3.4. Paulay and Priestley (1992) According to Paulay and Priestley’s recommendation (1992), the effective stiffness is taken as 0.40 EIg for members with an axial load ratio of less than −0.05, as 0.8 EIg for members with an axial load ratio of more than 0.50 and varies linearly for intermediate axial load ratios as illustrated in Figure 2. 3.5. Elwood and Eberhard (2009) Elwood and Eberhard (2009) recommend the following equation for estimating the initial stiffness of reinforced concrete columns subjected to seismic loading: (3) where db is the diameter of longitudinal reinforcing bars; a is the shear span and h is the column depth; Ag is the gross sectional area of columns and f′c is the compressive strength of concrete. Figure 2 illustrates the variation of stiffness ratio based on Elwood and Eberhard’s model (2009) versus k P A f d h h a g c b = + ′ +        ≤ ≥ 0 45 2 5 1 110 1 0 . . / .and 22 the axial load ratio for specimens with db and a equal to 25 mm and 850 mm respectively. 4. EXPERIMENTAL INVESTIGATION ON INITIAL STIFFNESS OF RC COLUMNS In this section, the experimental results obtained from testing of six RC columns conducted by Tran et al. (2009) are briefly discussed with respect to the initial stiffness of the test specimens. Four column axial loads of 0.05, 0.20, 0.35, 0.50 f′c Ag and two aspect ratios of 1.71 and 2.43 were investigated in this experimental program. Table 1 summarizes all the details of the test specimens. It is to be noted that only a brief summary of important test features that are relevant to this study are presented within this paper. Detailed information has been documented in another publication (Tran et al. 2009). The relationships between initial stiffness and the column axial load ratio obtained from all the test specimens are tabulated in Table 2. The initial stiffness of SC-1.7 Series specimens enhanced by around 9.8%, 17.6%, and 40.4% as the column axial load was increased from 0.05 to 0.20, 0.35, and 0.50 f′cAg, respectively. An analogous trend was observed in the specimens of RC-1.7 Series, whose initial stiffness experienced an enhancement of around 33.9%, 64.3% and 86.1% with an increase in the column axial load from 0.05 to 0.20, 0.35 and 0.50 f′cAg, respectively. As compared to Specimen SC-2.4-0.20, Specimen SC-2.4-0.50 experienced an Advances in Structural Engineering Vol. 15 No. 2 2012 267 Cao Thanh Ngoc Tran and Bing Li P f Ac g' Table 1. Summary of test specimens (Tran et al. 2009) Longitudinal Transverse L Specimen reinforcement reinforcement (MPa) (mm × mm) (mm) SC-2.4-0.20 0.20 SC-2.4-0.50 1700 0.50 SC-1.7-0.05 8-T20 2-R6 @ 125 0.05 SC-1.7-0.20 ρl = 2.05% ρv = 0.13% 25.0 350 × 350 0.20 SC-1.7-0.35 1200 0.35 SC-1.7-0.50 0.50 ′fc b h×× Table 2. Experimental verification of the proposed method Specimen (kN/mm) SC-2.4-0.20 12.9 0.782 0.254 0.355 0.355 0.444 0.305 0.793 SC-2.4-0.50 15.5 0.572 0.301 0.421 0.301 0.301 0.263 0.525 SC-1.7-0.05 24.5 0.918 0.319 0.223 0.223 0.372 0.236 0.560 SC-1.7-0.20 26.9 0.865 0.169 0.236 0.236 0.295 0.203 0.590 SC-1.7-0.35 28.8 0.653 0.188 0.263 0.239 0.239 0.190 0.553 SC-1.7-0.50 34.4 0.620 0.220 0.308 0.220 0.220 0.193 0.507 Mean 0.735 0.242 0.301 0.262 0.312 0.232 0.588 Coefficient of Variation 0.141 0.060 0.076 0.054 0.084 0.046 0.104 K K i i EE −− −− expK K i i PP −− −− expK K i i ASCE −− −− expK K i i FEMA −−exp − K K i i ACI b −− −− exp ( ) K K i i ACI a −− −− exp ( ) K K i p −− −− exp iKi−−exp increase in the initial stiffness of 20.2%. The aforementioned discussion clearly indicated that column axial load was beneficial to the initial stiffness of test specimens. The initial stiffness of Specimens SC-2.4-0.20, SC-1.7- 0.20, SC-2.4-0.50 and SC-1.7-0.50 obtained from the tests were 12.9 kN/mm, 26.9 kN/mm, 15.5 kN/mm and 34.4 kN/mm respectively. The increase in the initial stiffness when comparing between Specimens SC-1.7-0.20 and SC-2.4-0.20 was 108.5%. Similarly, an enhancement in the initial stiffenss of 121.9% was observed in Specimen SC-1.7-0.50 as compared to Specimen SC-2.4-0.50. The initial stiffness of test columns calculated based on ACI 318-2008 (2008), FEMA 356 (2000), ASCE 41 (2007), Paulay and Priestley (1992), and Elwood and Eberhard (2009) are also all tabulated in Table 2. All these models tend to overestimate the initial stiffness of the test columns. Amongst all of these existing models, Elwood and Eberhard (2009) provides the best mean ratio of the experimental to predicted initial stiffness. However none of these models are accurate. 5. PROPOSED METHOD 5.1. Yield Force (Vy) The initial stiffness of columns is determined by applying the second method as described in the previous section. The yield force (Vy) corresponding to point A in Figure 1(a) is obtained from the yield moment (My) when the reinforcing bar closest to the tension edge of columns has reached its yield strain. Moment-curvature analysis is adopted to determine this moment. 5.2. Displacement at Yield Force (∆′y) The displacement of a column at yield force (Vy) can be considered as the sum of the displacement due to flexure, bar slip and shear. (4) where ∆′y is the displacement of a column at yield force; ∆′flex is the displacement due to flexure and bar slip at yield force; and ∆′shear is the displacement due to shear at yield force 5.2.1. Flexure deformations (∆′flex) In this proposed method, the simplified concept of an effective length of the member suggested by Priestley et al. (1996) was used to account for the displacement due to bar slip in flexure deformations. Assuming a linear variation in curvature over the height of the column, the contribution of flexural deformations and bar slips to the displacement at the yield force for RC columns with a fixed condition at both ends can be estimated as follows: ′ = ′ + ′∆ ∆ ∆y flex shear (5) where φ′y is the curvature at the yield force determined by using moment-curvature analysis and L is the clear height of columns. The strain penetration length (Lsp) is given by: (6) where fyl is the yield strength of longitudinal reinforcing bars; and db is the diameter of longitudinal reinforcing bars. 5.2.2. Shear deformations (∆′shear) The idea of utilizing the truss analogy to model cracked RC elements has been around for many years. The truss analogy is a discrete modeling of actual stress fields within RC members. The complex stress fields within structural components resulting from applied external forces are simplified into discrete compressive and tensile load paths. The analogy utilizes the general idea of concrete in compression and steel reinforcement in tension. The longitudinal reinforcement in a beam or column represents the tensile chord of a truss while the concrete in the flexural compression zone is considered as part of the longitudinal compressive chord. The transverse reinforcement serves as ties holding the longitudinal chords together. The diagonal concrete compression struts, which discretely simulate the concrete compressive stress field, are connected to the ties and longitudinal chords at rigid nodes to attain static equilibrium within the truss. The truss analogy is a very promising way to treat shear because it provides a visible representation of how forces are transferred in a RC members under an applied shear force. Park and Paulay (1975) derived a method to determine the shear stiffness by applying the truss analogy for short or deep rectangular beams of unit length. The shear stiffness is the magnitude of the shear force, when applied to a beam of unit length that will cause unit shear displacement at one end of the beam relative to the other. This model is reliable in estimating shear deformations of short or deep beams in which the influences of flexure are negligible. The behaviors of RC columns under seismic loading are much more complex because of the interaction between shear and flexure. The influences of axial strain due to flexure in estimating shear deformations of RC columns should be considered to accurately predict the initial stiffness of RC columns. By applying a method that is similar to Park and Paulay’s analogous truss model (1975), the L f dsp yl b= 0 022. ′ = ′ +( ) ∆ flex y spL Lφ 2 6 2 268 Advances in Structural Engineering Vol. 15 No. 2 2012 Initial Stiffness of Reinforced Concrete Columns with Moderate Aspect Ratios shear stiffness of RC columns is derived in this part of the paper. The effects of flexure in shear deformations are incorporated in the proposed model through the axial strains at the center of columns (εy,CL). Assuming that transverse reinforcing bars start resisting the applied shear force when the shear cracking starts occurring, the stress in transverse reinforcing bars at the yield force is calculated as: (7) where d is the distance from the extreme compression fiber to centroid of tension reinforcement; s is the spacing of transverse reinforcement; Ast is the total transverse steel area within spacing s; and θ is the angle of diagonal compression strut. Hence the strain in transverse reinforcing bars is: (8) where εyt is the yield strain of transverse reinforcing bars; Es is the elastic modulus of steel. Similar to Park and Paulay’s model (1975), the concrete compression stress at the yield force is given as: (9) where b is the width of columns; Lcs = d sinθ is the effective depth of the diagonal strut as shown in Figure 3. Hence the strain in the concrete compression strut is given as: f V bL y cs 2 = cosθ ε εx sy s yt f E = ≤ f V V s A dsy y cr st = −( ) tanθ (10) where Ec is the elastic modulus of concrete given as: (11) Based on Vecchio and Collins’s model (1986), the effective compressive strength of concrete is calculated as follows: (12) By applying Mohr’s circle transformation for the mean strains at the center of Section C-C as shown in Figure 4, it gives: (13) (14) (15) For the axial mean strains, compatibility requires that the plain sections remain plane. Hence the mean strain at the center of section C-C is given as: (16) where εy, top, εy, bot are the axial strains at the extreme tension and compression fibers, respectively as shown in Figure 4(b). There are six variables, namely εx , εy,CL, γxy, ε1, ε2 and θ; and six independent Eqns 8, 10, 13, 14, 15 and 16. By solving these six independent equations, the shear strain (γxy) at the center of section C-C could be determined. The column is divided into several segments along its height of the column to determine the total shear deformation at the top of the column. The mean axial strain at the center of the section is determined based on ε ε ε y CL y top y bot , , ,= + 2 tan , 2θ γ ε ε = − xy x y CL ε ε ε ε ε γ 2 2 2 2 2 2 = + − −    +     x y CL x y CL xy, , ε ε ε ε ε γ 1 2 2 2 2 2 = + + −    +     x y CL x y CL xy, , f f fce c c= + ≤ ' ' .0 8 170 1ε E fc c= 5000 ε2 2= f Ec Advances in Structural Engineering Vol. 15 No. 2 2012 269 Cao Thanh Ngoc Tran and Bing Li Diagonal strut LCS d θθ Figure 3. Diagonal strut of RC columns (Park and Paulay1975) the moment-curvature analysis. The shear strains at the lower and upper section of the segment are calculated using the above equations. Hence, the total shear displacement caused by the yield force can be calculated as follows: (17) where γ ixy and γ xyi+1 are the shear strains at the lower and upper section of the segment i; hi is the height of segment i and n is the number of segments. 5.3. Initial Stiffness Once the flexural and shear deformations at the top of columns under yield force are obtained, the initial stiffness of columns can be determined as: (18) 6. VALIDATION OF THE PROPOSED METHOD The proposed method is validated by comparing its results to the initial stiffness of six columns obtained from the experimental study previously conducted by Tran et al. (2009). It was found that the average ratio of experimental to predicted initial stiffness by the proposed method was 0.735 as tabulated in Table 2. It shows a relatively good correlation between the analytical and K V i y flex shear = ′ + ′∆ ∆ ′ = +      + = ∑∆shear xy i xy i i i n h γ γ 1 1 2 experimental results. The initial stiffness of the tested columns calculated based on ACI 318-2008 (2008), FEMA 356 (2000), ASCE 41 (2007), Paulay and Priestley (1992), and Elwood and Eberhard (2009) are also tabulated in Table 2. The mean ratio of the experimental to predicted initial stiffness and its coefficient of variation were 0.242 and 0.060, 0.301 and 0.076, 0.262 and 0.054, 0.312 and 0.084, 0.232 and 0.046, and 0.588 and 0.104 for ACI 318-2008 (2008a), ACI 318-2008 (2008b), FEMA 356 (2000), ASCE 41 (2007), Paulay and Priestley (1992), and Elwood and Eberhard (2009) respectively. Comparison of available models with experimental data indicated that the proposed method produced a better mean ratio of the experimental to predicted initial stiffness than other models. The proposed method may be suitable as an assessment tool to calculate the initial stiffness of RC columns. 7. PARAMETRIC STUDIES A parametric study conducted to improve the understanding of the effects of various parameters on the initial stiffness of RC columns is presented within this section. The parameters investigated are transverse reinforcement ratios (ρv), longitudinal reinforcement ratios (ρl), yield strength of longitudinal reinforcing bars (fyl), concrete compressive strength (f′c), aspect r