Experimental Studies on the Backbone Curves of Reinforced Concrete Columns with Light Transverse Reinforcement Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved Cao Thanh Ngoc Tran, Ph.D 1; and Bing Li, Ph.D Abstract: This paper presents the investigations carried out on the backbone curves of reinforced concrete (RC) columns with light transverse reinforcement An experimental program consisting of five half-scale RC columns with light transverse reinforcement was carried out The specimens are tested to the point of axial failure to obtain the backbone curves of such columns Quasi-static cyclic loading simulating earthquake actions is applied The backbone curves obtained from the tested specimens are then compared with the existing seismic assessment guidelines The test results indicate that the initial column stiffness and ultimate displacements (displacements at axial failure) are overestimated and underestimated by some guidelines and provisions The existing initial stiffness and ultimate displacement models are briefly reviewed and compared with the experimental results The results show that the existing initial stiffness and ultimate displacement models produced better results than the existing seismic assessment guidelines DOI: 10.1061/(ASCE)CF.1943-5509.0000626 © 2014 American Society of Civil Engineers Author keywords: Reinforced concrete columns; Seismic loading; Backbone; Light transverse reinforcement Introduction A large number of existing reinforced concrete (RC) columns has not been designed following the requirements of modern seismic design codes Vital deficiencies in such columns include typical reinforcement details such as (1) lightly, widely spaced, and poorly anchored transverse reinforcement and (2) lap-splice details These are generally termed as nonseismically detailed RC columns Recent postearthquake investigations (ERRI 1999a, b, c) indicated that extensive damage occurred as a result of excessive shear deformation in nonseismically detailed RC columns, thus leading to shear failure, axial failure, and full collapse of structures Therefore, a thorough evaluation of nonseismically detailed RC columns is needed to understand the seismic behavior of these structures Extensive experimental research studies carried out on ductile columns in different countries throughout the last decades have given a better understanding on the seismic behavior of ductile columns However, there is relatively limited literature available for nonseismically detailed columns with respect to ductile detailed columns In addition, most tests of RC columns subjected to seismic loading have been terminated shortly after loss of lateral load resistance Few tests on RC columns have been carried out to the point of axial failure (Yoshimura and Yamanaka 2000; Lynn 2001; Sezen 2002; Nakamura and Yoshimura 2002; Yoshimura and Nakamura 2003;Yoshimura et al 2003; Ousalem 2006; Henkhaus et al 2009; Tran 2010; Wibowo et al 2014) This has resulted in the Lecturer, Dept of Civil Engineering at International Univ., Vietnam National Univ., Ho Chi Minh 70800, Vietnam (corresponding author) E-mail: tctngoc@hcmiu.edu.vn Associate Professor, School of Civil and Environmental Engineering, Nanyang Technological Univ., Singapore 639798 Note This manuscript was submitted on October 4, 2013; approved on May 6, 2014; published online on September 8, 2014 Discussion period open until February 8, 2015; separate discussions must be submitted for individual papers This paper is part of the Journal of Performance of Constructed Facilities, © ASCE, ISSN 0887-3828/04014126(11)/$25.00 © ASCE limited understanding of failure and collapse mechanisms of nonseismically detailed structures Therefore, further analytical and experimental studies are needed to understand the seismic behavior of nonseismically detailed columns better The main focus of this research is on the backbone curves of the RC columns with light transverse reinforcement This paper comprises of two parts The first part presents the test results obtained from an experimental program consisting of five half-scale RC columns with light transverse reinforcement The backbone curves obtained from these tests are the compared with the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Further comparisons with the existing initial stiffness (Elwood and Eberhard 2009; Tran and Li 2012) and ultimate displacement models (Elwood and Moehle 2005; Tran and Li 2013; Wibowo et al 2014) are presented in the second part of the paper The recommended backbone curves are also presented in the second part of the paper Previous Seismic Assessment Models In this section, the backbone curves based on FEMA 356 (FEMA 2000) and the provisions of ASCE 41 (Elwood et al 2007) are reviewed briefly According to FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007), the force-displacement relationship follows the general trend as shown in Fig Flexural and shear rigidity are considered in the calculation of the initial stiffness of columns in both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) Shear rigidity for rectangular cross sections is defined as 0.4Ec Ag in both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) According to FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007), flexural rigidity is related to applied column axial loads as tabulated in Table The deformation indexes (a, b) as illustrated in Fig are defined as flexural plastic hinge ratios which depend on column axial load, nominal shear stress, and details of columns The index c as 04014126-1 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Shear Force Specimen Details and Test Procedure b a Vu Vy C B Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved D E cVu A Displacement Fig Generalized force-displacement relationship in FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) defined in FEMA 356 (FEMA 2000) is 0.2, whereas per ASCE 41 (Elwood et al 2007), this index ranges from to 0.2 depending on the column axial load, nominal shear stress, and detailing of the columns According to both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines, the maximum shear force of the column is limited by its shear strength The shear strength is defined in both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) as pffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Av f yt d 0.5 f c0 P V n ẳ k1 ỵ k2 @ þ pffiffiffiffiffi0 A0.8Ag s as =d 0.5 f c Ag ð1Þ where k1 is equal to for transverse steel spacing less than or equal to d=2, k1 is equal to 0.5 for spacing exceeding d=2 but not more that d, k1 is equal to otherwise; k2 is taken as for displacement ductility less than 2, as 0.7 for displacement ductility more than 4, and varies linearly for intermediate displacement ductility; as =d shall not be taken greater than or less than 2; and λ is equal to for normal-weight concrete Experimental Studies An experimental program on RC columns with light transverse reinforcement subjected to seismic loading was conducted to study the backbone curves of such columns Five half-scale RC columns with light transverse reinforcement were tested up to the point of axial failure Table Flexural Rigidity in FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) Column axial load P ≥ 0.5f c0 Ag P ≤ 0.3f c0 Ag P ≤ 0.1f c0 Ag FEMA 356 (FEMA 2000) ASCE 41 (Elwood et al 2007) 0.7Ec I g 0.5Ec I g — 0.7Ec I g — 0.3Ec I g Note: Linear interpolation between values listed in the table shall be permitted © ASCE Fig and Table illustrate the schematic dimensions and detailing of test specimens The variables in the test specimens included column axial loads, aspect ratio, and cross-sectional shapes Longitudinal reinforcement which consisted of 8-T20 deformed bars were characterized by a yield strength f y of 408 MPa (59.2 ksi) This resulted in a ratio of longitudinal reinforcement area to the gross area of column of 2.05% The transverse reinforcement of all test specimens comprised of R6 mild steel bars with 135° bent spaced at 125 mm (4.92 in.) were characterized by a yield strength f y of 393MPa (57.0 ksi) The theoretical flexural strength V u and yield force V y of the test specimens were estimated using the material properties obtained through tests and in accordance with the recommendations provided by FEMA 356 guidelines (FEMA 2000) The nominal shear strengths V n of the test specimens were calculated based on the suggestion of FEMA 356 (FEMA 2000) The values of V u , V y , and V n of the test specimens are tabulated in Table A schematic of the loading apparatus is shown in Fig A reversible horizontal load was applied to the top of the column using a 1,000-kN (224.8-kip.) capacity actuator which was mounted onto a reaction wall The actuator was pinned at both ends to allow rotation during the test The base of the column was fixed to a strong floor with four posttensioned bolts The axial load was applied to the column using two 1,000-kN (224.8-kip.) capacity actuators through a transfer beam The column axial load was increased slowly until the designated level was achieved During each test, the column axial load was maintained by manually adjusting the vertical actuators after each load step The lateral load was applied cyclically through the horizontal actuator in a quasi-static fashion as shown in Fig The loading protocol consisting of displacement-controlled steps is illustrated in Fig The test specimens had been extensively instrumented both internally and externally Among those measurements were lateral load and displacement imposed at the top of the column, shear, and flexure deformations at the critical regions of the specimen and also the strains in the steel reinforcing bars as shown in Fig Experimental Results Cracking Patterns The crack patterns of the test specimens at shear failure (a loss of more than 20% of obtained maximum shear force) are shown in Fig Important features in crack development of the specimens are highlighted All of test specimens developed fine flexural cracks that were concentrated at both ends of the columns when loaded up to a drift ratio of 0.40% The lower the applied axial load, the more flexural cracks were observed in the columns Slight inclination was also observed in the flexural cracks of the test specimens at this stage In loading to a drift ratio of 1.00%, whereas the specimens with a lower axial load developed severe shear cracking at both ends of the columns, the specimens with a higher axial load only showed a slight inclination in the flexural cracks In the subsequent loading cycles, the occurrence of a steep diagonal crack and the opening of the existing diagonal cracks resulted in a reduction in the shearresisting capacity of the test specimens There are two axial failure modes observed in the test specimens as illustrated in Fig In the first mode of axial failure, the steep diagonal crack developed in the column during the previous stages became wider This led to sliding between the crack surfaces as well 04014126-2 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil 350 350 600 T20 1200 T20 1700 Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved R6 @ 125 T20 1700 R6 @ 125 R6 @ 125 600 350 350 900 900 135 degree hook 25 900 135 degree hook 135 degree hook 25 25 350 R6 8-T20 R6 8-T20 350 490 350 350 RC-1.7-0.20 RC-1.7-0.35 SC-1.7-0.20 SC-1.7-0.35 R6 8-T20 SC-2.4-0.20 Fig Reinforcement details of test specimens (in mm) Table Summary of Test Specimens Specimen SC-1.7-0.20 SC-1.7-0.35 RC-1.7-0.20 RC-1.7-0.35 SC-2.4-0.20 f c0 (MPa) b×h (mm × mm) L (mm) P=f c0 Ag 27.5 25.5 24.5 27.1 22.6 350 × 350 1,200 250 × 490 1,700 0.20 0.35 0.20 0.35 0.20 350 × 350 Aspect ratio Vy (kN) Vu (kN) Vn (kN) 1.7 330.5 323.3 290.0 316.4 217.3 353.3 374.0 313.8 353.5 231.2 281.0 318.3 297.8 354.5 199.1 2.4 Note: mm ¼ 0.04 in:; kN ¼ 0.225 kip L-shaped Steel Frame Strong Wall Actuator Specimen Actuator Actuator Strong Floor Fig Experimental setup © ASCE 04014126-3 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil 2.5 DR=1/50 DR=1/65 1.5 DR=1/80 DR=1/125 Drift ratio (%) 0.5 DR=1/200 DR=1/1000 DR=1/500 DR=1/300 -0.5 DR=1/700 DR=1/400 10 12 DR=1/250 -1 14 16 18 20 24 26 28 30 32 34 DR=1/100 -1.5 Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved 22 DR=1/150 DR=1/70 DR=1/55 -2 -2.5 Cycle number Fig Loading protocol as the buckling of longitudinal reinforcing bars and fracturing of transverse reinforcing bars along this diagonal crack In the second mode of axial failure, crushing of concrete as well as buckling of longitudinal reinforcing bars and fracturing of transverse developed across a damaged zone This type of axial failure was only observed in Specimen RC-1.7-0.20, whereas the rest of the test specimens exhibited the first mode of axial failure Hysteretic Responses Fig shows the hysteretic responses of the test specimens The backbone curves based on FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines are also shown in Fig Typical brittle-failure hysteretic responses were observed in all test specimens The hysteretic loops of the specimens show the degradation of stiffness and load-carrying capacity during repeated cycles due to the cracking of the concrete and yielding of the reinforcing bars The pinching effect was observed in the hysteretic loops of all the 50 490 365 412 365 250 (a) 70 70 (b) Fig Typical strain gauge and linear variable displacement transducer (LVDT) locations (in mm) © ASCE Shear Strengths Table summarizes the shear strengths of the test specimens The shear strength of SC-1.7 Series specimens was enhanced by around 14%, as the column axial load was increased from 0.20 to 0.35f c0 Ag An analogous trend was observed in the specimens of RC-1.7 Series, whose shear strengths experienced an enhancement of around 13% as the applied axial load was increased from 0.20 to 0.35fc0 Ag The previous discussion clearly indicates that the column axial load was beneficial to the shear strength of the test specimens whose applied axial load was in the range of 0.20 to 0.35fc0 Ag The shear strength of Specimens SC-2.4-0.20 and SC-1.7-0.20 obtained from the tests were 218.9 kN (49.2 kip.) and 294.2 kN (66.1 kip.), respectively The increase in shear strength between Specimens SC-2.4-0.20 and SC-1.7-0.20 was 34% Thus, it can be concluded that the shear strength of the specimens in the current experimental program increased with a decrease in aspect ratio Initial Stiffness 50 250 26 test specimens The shear failure in most test specimens occurred at a drift ratio of less than 2.0% as shown in Table The initial stiffness was calculated based on a point obtained from the measured force-displacement envelope with a shear force that is equal to the theoretical yield force This theoretical yield force is defined as either the first yield that occurs within the longitudinal reinforcement or when the maximum compressive strain of the concrete attains a value of 0.002 at any critical section of the column This definition would not apply for columns whose shear strength does not substantially exceed its theoretical yield force For such columns, defined as those whose maximum measured shear force was less than 1.07 of the theoretical yield force, the initial stiffness was defined based on a point on its measured force-displacement envelope with a shear force that equates to 0.80 of the obtained maximum shear force The initial stiffness of all the test specimens is tabulated in Table The initial stiffness of SC-1.7 Series specimens was enhanced by around 7% as the column axial load was increased from 0.20 to 0.35f c0 Ag A similar trend was observed in the specimens of RC-1.7 Series As compared with Specimen RC-1.7-0.20, Specimen RC-1.7-0.35 experienced a 23% increase in its initial stiffness This clearly indicates that the column axial load was beneficial to the initial stiffness of the test specimens It should be noticed that 04014126-4 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved Fig Observed cracks at shear failure of test specimens: (a) SC-1.7-0.20; (b) SC-1.7-0.35; (c) RC-1.7-0.20; (d) RC-1.7-0.35; (e) SC-2.4-0.20 the reinforcement details and cross sections of the specimens in RC-1.7 Series are different from the ones in SC-1.7 Series This could attribute to the differences in the increase in the initial stiffness of the specimens in these series when the column axial load is increased The initial stiffness of Specimens SC-2.4-0.20 and SC-1.7-0.20 obtained from the tests were 12.9 kN=mm (73.7 kip:=in:) and 26.9 kN=mm (153.6 kip:=in:), respectively There was an increase in the initial stiffness of Specimens SC-1.7-0.20 and SC-2.4-0.20 of approximately 108.5% The only difference in the details of these specimens is the height of the specimens Specimen SC-2.4-0.20 is higher than Specimen SC-1.7-0.20; therefore, it is susceptible to deformation than Specimen SC-1.7-0.20 Drift Ratios at Axial Failure Fig Typical modes of axial failure in test specimens: (a) Mode 1; (b) Mode © ASCE The drift ratios at axial failure of the test specimens are tabulated in Table An increase in the column axial load ratio reduced the drift ratio at axial failure As observed from Table 3, the drift ratio at axial failure in SC-1.7 and RC-1.7 Series specimens reduced by around 14 and 30%, respectively, as the column axial load ratio was increased from 0.20 to 0.35 The effects of aspect ratio on the drift ratio at axial failure can be noticed by comparing Specimens SC-2.4-0.20 and SC-1.7-0.20 At a column axial load ratio of 0.20, the drift ratio at axial failure reduced from 2.82 to 1.82% with a decrease in the aspect ratio from 2.4 to 1.7 04014126-5 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Drift Ratio (%) -0.5 Drift Ratio (%) 0.5 -2 -1.5 -1 -0.5 0.5 1.5 89.6 300 67.2 300 67.2 200 44.8 200 44.8 100 22.4 100 22.4 0 -100 -22.4 SC-1.7-0.20 FEMA 356 -400 -24 (a) -4 -44.8 ASCE 41 Proposed Model -18 -3 -12 -6 12 Lateral Displacement (mm) -2 Drift Ratio (%) -1 18 0 -100 -22.4 SC-1.7-0.35 -200 -300 -89.6 -400 -24 -67.2 Proposed Model (b) -44.8 FEMA 356 ASCE 41 -67.2 24 Shear Force (kN) 400 Shear Force (kip) 89.6 -300 -89.6 -18 -3 -12 -6 12 Lateral Displacement (mm) -2 -1 Drift Ratio (%) 18 24 400 89.6 400 89.6 300 67.2 300 67.2 200 44.8 200 44.8 100 22.4 100 22.4 -100 -22.4 RC-1.7-0.20 FEMA 356 -200 ASCE 41 -300 -400 -68 (c) Proposed Model -34 -17 17 34 Lateral Displacement (mm) Shear Force (kN) -3 51 -2 -22.4 -100 -44.8 -200 -67.2 -300 RC-1.7-0.35 -44.8 FEMA 356 ASCE 41 -67.2 Proposed Model 68 -89.6 -400 -51 -89.6 -51 0 -34 -17 17 Lateral Displacement (mm) (d) -1 Drift Ratio (%) 51 300 67.2 200 44.8 100 22.4 0 -100 -22.4 SC-2.4-0.20 FEMA 356 -200 34 Shear Force (kip) Shear Force (kN) Shear Force (kN) 1.5 400 -200 Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved Shear Force (kip) -1 Shear Force (kip) -1.5 Shear Force (kip) Shear Force (kN) -2 -44.8 ASCE 41 Proposed Model -300 -51 (e) -67.2 -34 -17 17 Lateral Displacement (mm) 34 51 Fig Hysteretic responses of test specimens Strains in Reinforcing Bars Strain profiles from Specimen RC-1.7-0.35 were selected to illustrate the distribution of strain in both transverse and longitudinal reinforcing bars as it is not possible to present the results of all the specimens in this paper Detailed strain profiles of all test specimens have been reported elsewhere (Tran 2010) © ASCE The measured strains along the longitudinal reinforcing bars of Specimen RC-1.7-0.35 are shown in Fig The general strain profiles of Specimen RC-1.7-0.35 have a good agreement with the bending moment pattern The largest recorded tensile strain of 0.0027 was observed at Location L6 In loading to a drift ratio of 1.44%, the compressive strain at Location L6 exceeded the compressive yield strain of −0.0025 During the test, both compressive 04014126-6 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Table Summary of Test Results Specimen Shear strength (kN) Initial stiffness (kN=mm) Drift ratio at shear failure (%) Drift ratio at axial failure (%) Maximum cumulative energy dissipation (kN · m) 294.2 335.5 305.5 345.7 218.9 26.9 28.8 15.4 18.9 12.9 1.43 1.43 2.23 1.65 1.82 1.82 1.56 2.87 2.02 2.82 13.5 9.1 44.3 26.5 34.9 and tensile yielding were observed in the longitudinal reinforcing bars shear cracks along the specimen Yielding of the transverse steel bars was also observed at this stage Strains in Transverse Reinforcing Bars Displacement Decompositions The measured strains in the transverse reinforcing bars of Specimen RC-1.7-0.35 are illustrated in Fig 10 It was observed that the measured strains varied considerably as drift ratios increased The largest strain was recorded at Location T4 The strains in the transverse reinforcing bars have not reached the yield strain of 0.002 up to a drift ratio of 1.44% The largest recorded strain up to this stage was only 0.0013 In loading to a drift ratio of 1.58%, the strains in the transverse reinforcing bars increased drastically because of the growth and opening of diagonal The contribution of deformation components expressed as percentages of the total lateral displacements at the peak displacements during each displacement cycle of Specimen RC-1.7-0.35 is shown in Fig 11 Detailed displacement decompositions of all test specimens had been reported elsewhere (Tran 2010) Approximately 51.4–58.4% of the total lateral displacement was contributed by the flexural deformation component, whereas only up to 45% was accounted for the shear deformation component The shear deformation component initially grew gradually to approximately 16.3% of the total lateral displacement up to a drift ratio of 1.44% As the drift ratio was increased up to 1.73%, the corresponding shear deformation component drastically grew to approximately 45% of the total displacement L1 L2 L3 L4 L5 L6 4000 εy 3000 L6 250 L5 2000 -6 Strain (×10 ) 250 Cumulative Energy Dissipation L4 1000 -1000 L3 250 -2000 L2 εy -3000 -4000 -2.5 -2 -1.5 -1 -0.5 0.5 1.5 250 L1 2.5 Drift Ratio (%) Fig Local strains in longitudinal reinforcing bar of specimen RC-1.7-0.35 T1 4000 T2 T3 T4 T5 Table shows the comparison between the maximum cumulative energy dissipation obtained from the test specimens There was a decrease in the maximum cumulative energy dissipation recorded from both SC-1.7 and RC-1.7 Series specimens as the column axial load was increased The maximum cumulative energy dissipations obtained from SC-1.7 Series specimens was reduced by around 33%, as the column axial load was increased from 0.20 to 0.35f c0 Ag An analogous trend was observed in the specimens of RC-1.7 Series, whose maximum cumulative energy dissipations experienced a drop of around 40% as the applied axial load was increased from 0.20 to 0.35fc0 Ag It can therefore be concluded based on the test results that column axial load decreases the T6 100 Displacement Decomposition (%) T6 T5 T4 3000 εy -6 Strain (×10 ) Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved SC-1.7-0.20 SC-1.7-0.35 RC-1.7-0.20 RC-1.7-0.35 SC-2.4-0.20 2000 1000 125 0 0.5 1.5 Drift Ratio (%) T3 T2 T1 38 2.5 80 Unaccounted 60 Flexure 40 20 0 Fig 10 Local strains in transverse reinforcing bars of specimen RC-1.7-0.35 © ASCE Shear 0.5 Drift Ratio (%) 1.5 Fig 11 Displacement decompositions of Specimen RC-1.7-0.35 04014126-7 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Table Modeling Parameters FEMA 356 (FEMA 2000) Specimen SC-1.7-0.20 SC-1.7-0.35 RC-1.7-0.20 RC-1.7-0.35 SC-2.4-0.20 ASCE 41 (Elwood et al 2007) a b c Condition a 0.0040 0.0025 0.0040 0.0025 0.0043 0.0107 0.0087 0.0107 0.0087 0.0114 0.2 0.2 0.2 0.2 0.2 iii iii iii ii iii 0 0.0062 b c 0.0113 0.0075 0.0154 0.0104 0.0470 0.0113 Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved Note: Condition iii = shear failure; Condition ii = shear-flexure failure maximum cumulative energy dissipation of the test specimens The reinforcement details and cross sections of the specimens in RC-1.7 Series are different from the ones in SC-1.7 Series This could attribute to the differences in the decrease in the maximum cumulative energy dissipations of the specimens in these series when the column axial load is increased Backbone Curves of flexure, bar slip, and shear The flexural deformations are calculated by using moment-curvature analysis The bar slips are accounted for in the model using the simplified concept of the effective length of the member by Priestley et al (1996) The shear deformation of RC columns at yield force is derived by applying a method that is similar to the analogous truss model of Park and Paulay (1975) Details of the derivation of the model of Tran and Li (2012) have been reported elsewhere (Tran and Li 2012) Based on the model, Tran and Li (2012) performed a parametric study to investigate the effects of various parameters on the initial stiffness of RC columns The parameters investigated in the study of Tran and Li (2012) include transverse reinforcement ratios (ρst ), longitudinal reinforcement ratios (ρl ), yield strength of longitudinal reinforcing bars (fyl ), concrete compressive strength (fc0 ), aspect ratio (as =d), and axial load ratio (P=fc0 Ag ) Tran and Li (2012) concluded that the stiffness ratio apparently increases with an increase in aspect ratios (Ra ) and axial load ratio (Rn ) The transverse and longitudinal reinforcement ratios, yield strength of longitudinal bars, and concrete compressive strength insignificantly influenced the stiffness ratio of RC columns Based on the parametric study, Tran and Li (2012) has developed an equation to estimate the stiffness ratio of RC columns as follows: ẳ 2.043R2n ỵ 2.961Rn ỵ 1.739ị3.023Ra þ 2.573Þ Comparison with FEMA 356 and ASCE 41 Table summarizes all the indexes (a, b, c) for all the test specimens calculated based on FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) Fig compares the backbone curves of the test specimens with analytical results obtained from FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) models The test results showed that both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines provide a good prediction of the shear strength of the test specimens However, the column initial stiffness and ultimate displacements (displacements at axial failure) were overestimated and underestimated by both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines, respectively Therefore, further work on the backbone curves is needed to accurately capture the behavior of RC columns tested to the point of axial failure Research on the initial stiffness and the ultimate displacements of RC columns had been done by Elwood and Moehle (2005), Elwood and Eberhard (2009), Tran and Li (2012, 2013), and Wibowo et al (2014) These models will be reviewed in the following part of the paper Existing Initial Stiffness Models Model of Elwood and Eberhard Elwood and Eberhard (2009) recommend the following equation for estimating the initial stiffness of reinforced concrete columns subjected to seismic loading kẳ 0.45 ỵ 2.5P=Ag fc0 ỵ 110dhb ịahs ị and 0.2 ð2Þ where db is the diameter of longitudinal reinforcing bars; as is the shear span; and h is the column depth Model of Tran and Li Tran and Li (2012) developed an analytical model for the initial stiffness of RC columns In the model of Tran and Li (2012), the yield displacement is the sum of the displacement because © ASCE ð3Þ where Ra and Rn are the aspect ratio (as =d) and axial load ratio (P=f c0 Ag ) The stiffness ratio (κ) is defined as follows: κ¼ Ie × 100% Ig ð4Þ where the measured effective moment of inertia can be determined as Ie ¼ L3 K i 12Ec ð5Þ where K i is the initial stiffness of columns; I g is the moment of inertia of the gross section; L is the height of columns; and Ec is the elastic modulus of concrete Existing Ultimate Displacement Model Model of Elwood and Moehle The model of Elwood and Moehle (2005) proposed the following equation for the drift ratio at axial failure based on the shear friction model:   ỵ tan ị2 6ị ẳ L a 100 tan ỵ PAst fyt ds c tan θÞ where dc is the depth of the core (centerline to centerline of ties); Ast is the total transverse reinforcement area within spacing s; θ is the angle of diagonal crack; and fyt is the yield strength of transverse reinforcement Model of Tran and Li Tran and Li (2013) developed an analytical model for the ultimate displacement of RC columns with light transverse reinforcement based on the energy analogy and the experimental data of 47 RC columns tested to the point of axial failure Details of the derivation of the model of Tran and Li (2013) had been reported elsewhere (Tran and Li 2013) In this model, the ultimate 04014126-8 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil a ẳ 2y ỵ a h tan 7ị  fyl df yt Ast P ẳ l bh ỵ 0.2874 ì a ỵ sin s p ỵ k f c0 0.8Ag ị cot Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved  ð8Þ where ρl is the longitudinal reinforcement ratio; fyl is the yield strength of longitudinal reinforcement, respectively; k is a parameter that depends on the displacement ductility demand; Δy is the yield displacement of columns; and Ag is the cross-sectional area There are two variables, namely δ Ãa and Δa , and two independent equations [Eqs (7) and (8)] By solving these two independent equations, the ultimate displacement, Δa , can be determined Model of Wibowo et al Wibowo et al (2014) used the curve fitting method to derive the drift ratio at axial failure as follows: δ a ¼ 51 ỵ l ị1ị ỵ 7st ỵ 5n 9ị where ρst is transverse reinforcement area ratio (Ast =bs); β is calculated as n=nb ; n is the axial load ratio; and nb is the axial load ratio at the balance point of the interaction diagram Comparison with the Existing Initial Stiffness and Ultimate Displacement Models As shown in Table 5, it was found that the initial stiffness models developed by Elwood and Eberhard (2009) and Tran and Li (2012) produced better results than the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Comparing the models between Elwood and Eberhard (2009) and Tran and Li (2012), the model of Tran and Li (2012) produced a better mean K i- exp (kN=mm) K i−Elwood (kN=mm) K i-Tran (kN=mm) K i- exp = K i-Elwood K i- exp = K i-Tran 45.6 52.1 34.8 48.3 16.3 31.1 44.1 18.2 28.6 16.5 0.590 0.553 0.442 0.391 0.793 0.554 0.156 0.865 0.653 0.846 0.661 0.782 0.761 0.100 SC-1.7-0.20 26.9 SC-1.7-0.35 28.8 RC-1.7-0.20 15.4 RC-1.7-0.35 18.9 SC-2.4-0.20 12.9 Mean Coefficient of variation Incorporating Existing Initial Stiffness and Ultimate Displacement Models to FEMA 356 Guidelines As discussed in the previous part, the existing initial stiffness (Elwood and Eberhard 2009; Tran and Li 2012) and ultimate displacement models (Elwood and Moehle 2005; Tran and Li 2013; Wibowo et al 2014) produced better results than the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Therefore, in this part of the paper, the backbone curve of FEMA 356 (FEMA 2000) is modified based on the existing models (Elwood and Eberhard 2009; Tran and Li 2012; Elwood and Moehle 2005; Tran and Li 2013; Wibowo et al 2014) The modified backbone curve of FEMA 356 is shown in Fig 12 In this paper, the models proposed by Tran and Li (2012, 2013) are incorporated in the modified backbone curve of FEMA 356 (FEMA 2000) as a sample evaluation of the proposed modified FEMA 356 backbone curve Other models (Elwood and Eberhard 2009; Elwood and Moehle 2005; Wibowo et al 2014) could be a Vm Vp Table Comparison with the Existing Initial Models Specimen ratio of the experimental to predicted initial stiffness and its coefficient of variation than the one of Elwood and Eberhard (2009) As shown in Table 6, the mean ratios of the experimental to the predicted displacement at axial failure and its coefficient of variation are 0.959 and 0.200 for the model of Elwood and Moehle (2005), 1.085 and 0.283 for the model of Tran and Li (2012), and 0.952 and 0.161 for the model of Wibowo et al (2014), respectively Comparing the existing models with the experimental data indicates that the existing models produced better results than the existing seismic assessment guidelines [FEMA 356 (FEMA 2000); Elwood et al 2007] Among the existing models, the model of Elwood and Moehle (2005) produced a better mean ratio of the experimental to predicted initial stiffness than other models (Tran and Li 2013; Wibowo et al 2014) Shear Force displacement of RC columns Δa can be found by solving the following equations: C B Ki E A cVu Displacement a Fig 12 Proposed backbone curves Table Comparison with the Existing Ultimate Displacement Models Specimen ðΔa =LÞexp SC-1.7-0.20 1.82 SC-1.7-0.35 1.56 RC-1.7-0.20 2.87 RC-1.7-0.35 2.02 SC-2.4-0.20 2.82 Mean Coefficient of variation © ASCE ðΔa =LÞElwood ðΔa =LÞTran ðΔa =LÞWibowo 2.63 1.79 2.66 1.66 3.01 2.40 1.20 2.79 1.40 3.14 2.69 1.60 2.93 1.86 2.69 ðΔa =LÞexp = ðΔa =LÞElwood ðΔa =LÞexp = ðΔa =LÞTran ðΔa =LÞexp = ðΔa =LÞWibowo 0.692 0.87 1.078 1.215 0.939 0.959 0.200 0.757 1.299 1.027 1.445 0.897 1.085 0.283 0.677 0.973 0.979 1.083 1.049 0.952 0.161 04014126-9 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved used instead of the models of Tran and Li (2012, 2013) to verify the accuracy of the proposed modified backbone curve of FEMA 356 The Point B in Fig 12 is defined based on the values of K i and V p The initial stiffness K i is calculated based on Eq (3); V p is equal to the minimum value of the theoretical yield force V y and the nominal shear strength based on FEMA 356 model V n [FEMA 356 (FEMA 2000)] The Point C in Fig 12 is defined based on the values of a and V m ; a is defined similarly to that in the model of FEMA 356 (FEMA 2000); V m is equal to the minimum value of the theoretical flexural strength V u and the nominal shear strength based on FEMA 356 model V n [FEMA 356 (FEMA 2000)] The Point E in Fig 12 is defined based on the values of c and Δa ; a is defined similarly to the model of FEMA 356 (FEMA 2000); the ultimate displacement Δa is calculated based on Eqs (7) and (8) Comparison of available models with the test results obtained from the current experimental investigation as illustrated in Fig indicated that the modified FEMA 356 provided a better prediction of the behavior of the test specimens than FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) models The initial stiffness and the ultimate displacement were captured well by the modified model The modified method may be suitable as an assessment tool to model the backbone curves of RC columns with light transverse reinforcement Notation The following symbols are used in this paper: Ag = cross-sectional area; Ast = total transverse reinforcement area within spacing s; as =d = aspect ratio; b = width of columns; d = distance from the extreme compression fiber to centroid of tension reinforcement; f c0 = compressive strength of concrete; f yl = yield strength of longitudinal reinforcement; f yt = yield strength of transverse reinforcement; h = depth of columns; k2 = parameter depends on the displacement ductility demand; P = applied axial load; s = spacing of transverse reinforcement; V c = shear force carried by concrete; V n = nominal shear strength of columns; V u = theoretical flexural strength of columns; V y = theoretical yield force of columns; θ = angle of shear crack; and Δa = horizontal displacement of columns at the point of axial failure References Conclusions The backbone curves of the reinforced concrete columns with light transverse reinforcement were investigated using the experimental and analytical studies The conclusions drawn from the experimental and analytical investigations of the five reinforced concrete columns with light transverse reinforcement are as follows: The column axial load was found having a detrimental effect on the drift ratio at axial failure and maximum energy dissipation capacity of test specimens However, the shear strength and initial stiffness increased with an increase in column axial load Both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines provided a good prediction of the shear strength of the test specimens However, the column initial stiffness and ultimate displacements were overestimated and underestimated by both FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) guidelines, respectively An analytical method is developed in this paper to model the backbone curves of test specimens Comparison of available models with the test results obtained from the current experimental investigation indicated that the proposed method provided a better prediction of the behavior of the test specimens than FEMA 356 (FEMA 2000) and ASCE 41 (Elwood et al 2007) models The initial stiffness and the ultimate displacement were captured well by the proposed method The proposed method may be suitable as an assessment tool to model the backbone curves of RC columns with light transverse reinforcement Acknowledgments This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.01-2013.12 © ASCE Earthquake Engineering Research Institute (EERI) (1999a) “The Tehuacan, Mexico earthquake of June 15, 1999.” EERI Special Earthquake Rep, Oakland, CA Earthquake Engineering Research Institute (EERI) (1999b) “The Athens, Greece earthquake of September 7, 1999.” EERI Special Earthquake Rep, Oakland, CA Earthquake Engineering Research Institute (EERI) (1999c) “The Chi-Chi, Taiwan earthquake of September 21, 1999.” EERI Special Earthquake Rep, Oakland, CA Elwood, K., et al (2007) “Update to ASCE/SE 41 concrete provisions.” Earthquake Spectra, 23(3), 493–523 Elwood, K., and Eberhard, M O (2009) “Effective stiffness of reinforced concrete columns.” ACI Struct J., 106(4), 476–484 Elwood, K., and Moehle, J (2005) “Axial capacity model for sheardamaged columns.” ACI Struct J., 102(4), 578–587 FEMA (2000) “Prestandard and commentary for the seismic rehabilitation of buildings.” FEMA 356, Washington, DC Henkhaus, K W., Ramerez, J A., and Pujol, S (2009) “Simultaneous shear and axial failures of reinforced concrete columns.” Improving the Seismic Performance of Existing Buildings and Other Structures, ASCE, Reston, VA, 536–546 Lynn, A C (2001) “Seismic evaluation of existing reinforced concrete building columns.” Ph.D thesis, Univ of California, Berkeley Nakamura, T., and Yoshimura, M (2002) “Gravity load collapse of reinforced concrete columns with brittle failure modes.” J Asian Archit Build Eng., 1(1), 21–27 Ousalem, H (2006) “Experimental and analytical study on axial load collapse assessment and retrofit of reinforced concrete columns.” Ph.D thesis, Univ of Tokyo, Tokyo Park, R., and Paulay, T (1975) Reinforced concrete structures, Wiley, New York Priestley, M J N., Seible, F., and Calvi, G M (1996) Seismic design and retrofit of bridge structures, Wiley, New York Sezen, H (2002) “Seismic response and modeling of reinforced concrete building columns.” Ph.D thesis, Univ of California, Berkeley Tran, C T N (2010) “Experimental and analytical studies on the seismic behavior of RC columns with light transverse reinforcement.” Ph.D thesis, Nanyang Technological Univ., Singapore, 185 Tran, C T N., and Li, B (2012) “Initial stiffness of reinforced concrete columns with moderate aspect ratio.” Adv Struct Eng., 15(2), 265–276 04014126-10 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil Yoshimura, M., Takaine, Y., and Nakamura, T (2003) “Collapse drift of reinforced concrete columns.” 5th US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, PEER, Berkekey, CA, 239–253 Yoshimura, M., and Yamanaka, N (2000) “Ultimate limit state of RC columns.” 2nd US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, PEER, Berkekey, CA, 313–326 Downloaded from ascelibrary.org by Queens University Libraries on 02/20/16 Copyright ASCE For personal use only; all rights reserved Tran, C T N., and Li, B (2013) “Ultimate displacement of reinforced concrete columns with light transverse reinforcement.” J Earthquake Eng., 17(2), 282–300 Wibowo, A., Wilson, J L., Lam, N T K., and Gad, E F (2014) “Drift performance of lightly reinforced concrete columns.” Eng Struct., 59, 522–535 Yoshimura, M., and Nakamura, T (2003) “Axial collapse of reinforced concrete short columns.” 4th US-Japan Workshop on PerformanceBased Earthquake Engineering Methodology for Reinforced Concrete Building Structures, PEER, Berkekey, CA, 187–198 © ASCE 04014126-11 J Perform Constr Facil., 2015, 29(5): 04014126 J Perform Constr Facil  ... of shear crack; and Δa = horizontal displacement of columns at the point of axial failure References Conclusions The backbone curves of the reinforced concrete columns with light transverse reinforcement. .. investigated using the experimental and analytical studies The conclusions drawn from the experimental and analytical investigations of the five reinforced concrete columns with light transverse reinforcement. .. on the backbone curves is needed to accurately capture the behavior of RC columns tested to the point of axial failure Research on the initial stiffness and the ultimate displacements of RC columns