DSpace at VNU: Ultimate Displacement of Reinforced Concrete Columns with Light Transverse Reinforcement

This article was downloaded by: ["University at Buffalo Libraries"] On: 20 January 2015, At: 05:43 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Earthquake Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ueqe20 Ultimate Displacement of Reinforced Concrete Columns with Light Transverse Reinforcement a Cao Thanh Ngoc Tran & Bing Li b a Department of Civil Engineering at International University , Vietnam National University – Ho Chi Minh City , Vietnam b School of Civil and Environmental Engineering at Nanyang Technological University , Singapore Accepted author version posted online: 25 Sep 2012.Published online: 08 Jan 2013 To cite this article: Cao Thanh Ngoc Tran & Bing Li (2013) Ultimate Displacement of Reinforced Concrete Columns with Light Transverse Reinforcement, Journal of Earthquake Engineering, 17:2, 282-300, DOI: 10.1080/13632469.2012.730117 To link to this article: http://dx.doi.org/10.1080/13632469.2012.730117 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content This article may be used for research, teaching, and private study purposes Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/termsand-conditions Journal of Earthquake Engineering, 17:282–300, 2013 Copyright © A S Elnashai & N N Ambraseys ISSN: 1363-2469 print / 1559-808X online DOI: 10.1080/13632469.2012.730117 Ultimate Displacement of Reinforced Concrete Columns with Light Transverse Reinforcement CAO THANH NGOC TRAN1 and BING LI2 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Department of Civil Engineering at International University, Vietnam National University – Ho Chi Minh City, Vietnam School of Civil and Environmental Engineering at Nanyang Technological University, Singapore This article presents the analytical and experimental investigations carried out on reinforced concrete (RC) columns with light transverse reinforcement A semi-empirical model is developed to estimate the ultimate displacement (displacement at axial failure) of RC columns with light transverse reinforcement subjected to simulated seismic loading The developed model is calibrated using the collected data of RC columns tested to the point of axial failure A series of experiments is conducted on five RC columns with light transverse reinforcement to validate the applicability and accuracy of the developed model It is concluded from the study that the mean ratios of the experimental to predicted ultimate displacement and its coefficient of variation are 1.077 and 0.194, respectively, showing a good correlation between the developed model and the experimental data Keywords Reinforced Concrete; Seismic Loading; Ultimate Displacement; Axial Failure; Column Introduction Structures consisting of reinforced concrete (RC) columns with light transverse reinforcement are very common in regions of low to moderate seismicity, and are the predominant structural system in Singapore Recent post-earthquake investigations [EERI, 1999a, 1999b, 1999c; Moehle, 1991] have indicated that extensive damage occurred in such RC columns as a result of excessive shear deformation, leading to shear failure, axial failure, and full collapse of structures The literature reviews conducted have shown that extensive research studies have been carried out in various countries on ductile columns throughout past decades However, there are limited research studies related to RC columns with light transverse reinforcement Only a few researchers, namely Yoshimura and Yamanaka [2000], Lynn [2001], Sezen [2002], Nakamura and Yoshimura [2002], Yoshimura et al [2003], and Ousalem [2003] studied the seismic behavior of RC columns with light transverse reinforcement up till its axial failure point (the point at which the column is unable to sustain its applied axial load) This has resulted in a limited understanding of the collapse mechanisms of RC columns with light transverse reinforcement The Nanyang Technological University (NTU) in Singapore conducted a research project with an aim to attain a better understanding of the collapse mechanisms of RC columns with light transverse reinforcement A simple model is developed and presented Received April 2012; accepted September 2012 Address correspondence to Cao Thanh Ngoc Tran, Department of Civil Engineering, International University, Vietnam National University – Ho Chi Minh City, Vietnam E-mail: tctngoc@hcmiu.edu.vn 282 Ultimate Displacement of RC Columns 283 within this article to estimate the lateral displacement at axial failure (or ultimate lateral displacement) of RC columns with light transverse reinforcement under seismic loading The developed model is calibrated using the collected data of RC columns tested to the point of axial failure The applicability and accuracy of the proposed model are then verified by using the experimental results obtained from the tests conducted on five RC columns detailed with light transverse reinforcement at NTU Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Proposed Model Available literatures on post-earthquake investigations [EERI, 1999a, 1999b, 1999c; Moehle, 1991] highlight various types of failures of RC columns with light transverse reinforcement One of the most critical modes of failure in these types of columns is often caused by the formation of a steep shear crack, as illustrated in Figs and The sliding between the surfaces of this shear crack results in an excessive shear deformation of the columns This leads to a sudden loss of axial capacity Any axial load supported by such damaged columns must be transferred through the obvious shear failure plane, as shown in Fig FIGURE Damaged columns during 1995 Kobe Earthquake Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 284 C T Ngoc Tran and B Li FIGURE Damaged column during 2004 Niigata Earthquake (color figure available online) 2.1 Basic Assumptions The following summarizes the basic assumptions employed in the proposed model ● ● ● ● The applied axial load at the point of axial failure is transferred through the shear failure plane, as illustrated in Fig The angle of the shear failure plane of 60◦ as defined by Priestley et al [1994] is adopted The shear demand in columns is considered to be negligible and therefore ignored at the point of axial failure [Lynn, 2001] When the shear strength degrades corresponding to a displacement ductility of and for bidirectional and unidirectional lateral loading, respectively [Priestley, 1994], the additional deformation of the columns is assumed to be only contributed from the sliding between cracking surfaces as shown in Fig 2.2 Derivation of the Proposed Model At the point of axial failure as shown in Fig 3, the external and internal works developed by the column are given as: Wext = P × ∗ av (1) Ultimate Displacement of RC Columns P 285 * a * av fsl Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Vc Ld fyt fsl FIGURE Assumed failure plane at the point of axial failure Wint = Wc + Wsv + Wsl , (2) where Wext and W int are the external and internal work done, respectively; P is the applied column axial load; Wc , Wsv , and Wsl are the internal works done by the concrete, transverse reinforcement, and longitudinal reinforcement, respectively; and ∗av are the vertical displacement due to the sliding between cracking surfaces at the point of axial failure The internal work done by the longitudinal reinforcement, transverse reinforcement, and concrete are calculated as: Wsl = (ρl bhfsl ) × Wsv = dc tan θ Ast fyt × s ∗ a = Wc = Vc × ∗ av (3) dc tan θ Ast fyt × s ∗ a = Vc × ∗ av ∗ av cot θ , cot θ = dc fyt Ast s ∗ av (4) (5) where ρ l is the longitudinal reinforcement ratio; b and h are the width and depth of columns, respectively; fsl is the axial strength of the longitudinal reinforcement at axial failure; fyt is the yield strength of transverse reinforcement; dc is the depth of core (centerline to centerline of ties); s is the spacing of transverse reinforcement; Ast is the total transverse reinforcement area within spacing s; θ is the angle of shear crack; and Vc is the shear force carried by concrete ∗a is the horizontal displacement due to the sliding between cracking surfaces at the point of axial failure 286 C T Ngoc Tran and B Li By substituting Eqs (3), (4), and (5) into (2) and equating Eqs (1) and (2), it gives: P ∗ av ∗ av = ρl bhfsl + dc fyt Ast s ∗ av + Vc ∗ av cot θ (6a) or P = ρl bhfsl + dc fyt Ast + Vc cot θ s (6b) Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Equation (6) can be rewritten as: P = Psl + Pst + Pc (7) Psl = ρl bhfsl (8) dc fyt Ast s (9) Pc = Vc cot θ , (10) in which Pst = where Psl , Pst , and Pc are the axial strength contributed by longitudinal reinforcement, transverse reinforcement, and concrete at the point of axial failure respectively The axial strength of longitudinal reinforcing bars at axial failure can be calculated as follows: fsl = (P − Pst − Pc ) ρl bh (11) Hence, the ratio of the axial strength of longitudinal reinforcing bars at axial failure to the yield strength of longitudinal reinforcement (ηsl ) is given by: ηsl = (P − Pst − Pc ) ρl bhfyl (12) Based on the model [12] of Priestley et al [1994], the shear force carried by concrete is calculated as follows: Vc = k fc 0.8Ag (13) in which Ag is the cross-sectional area and the parameter k depends on the displacement ductility demand, as defined in Fig The horizontal displacement at the top of columns at the point of axial failure is calculated as: a = ∗ a +2 y for Unidirectional Lateral Loading (14) Hence, the horizontal displacement due to the sliding between cracking surfaces at the point of axial failure is given as: ∗ a = a −2 y for Unidirectional Lateral Loading, (15) where y is the yield displacement defined as follows: a secant was drawn to intersect the shear force-lateral displacement relation at the yield force This line was extended to the Ultimate Displacement of RC Columns k(MPa) k(psi) 0.29 3.5 Bidirectional Lateral Loading Unidirectional Lateral Loading 0.1 1.2 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 287 Member Displacement Ductility FIGURE Definition of parameter k intersection with a horizontal line corresponding to the flexural strength, and then projected onto the horizontal axis to obtain the yield displacement [Priestley et al., 2007] The damaged length (Ld ), as shown in Fig 3, is given by: Ld = h tan θ (16) The ratio of the horizontal displacement due to the sliding between cracking surfaces at axial failure to the damaged length (δa∗ ) is given as follows: δa∗ = a−2 h tan θ y × 100% (17) 2.3 Calibration of the Proposed Model A database consisting of 47 RC columns tested to the point of axial failure was been constructed Details of these RC columns are shown in Table These columns encompass a wide range of cross sectional sizes, material properties, and column axial loads These columns were subjected to a combination of a axial load and unidirectional cyclic loadings to simulate earthquake actions Table shows the experimental ratios of (ηsl )exp and δa∗ exp for each of the test column in the database, which are calculated based on Eqs (12) and (17), respectively Figure plots (ηsl )exp vs δa∗ exp for all test columns in the database Based on the database, an empirical equation was then developed to relate the ratio of the axial strength of longitudinal reinforcing bars to the yield strength of the longitudinal reinforcing bars (ηsl ) to the ratio of the horizontal displacement due to the sliding between cracking surfaces to the damaged length (δa∗ ) as follows: ηsl = 0.2046 × δa∗ + (18) As shown in Fig 5, there are three specimens with large amplitude of abscissa away from the proposed curve It is to be noted that these specimens have a moderate axial load ratio in a range of 0.18–0.23 For such columns, the proposed curve predicts lower ultimate displacements as comparing with the experimental results As comparing with the specimens of the same authors, these specimens have the highest transverse reinforcement ratio This may have attributed to the uncertainties of the proposed curve 288 457 457 Sezen [2002] 2CLD12 457 2CHD12 457 392 2946 392 2946 503 503 503 503 1512 1512 1512 1512 457 457 457 457 457 457 305 305 632 75 632 75 632 75 803 180 803 180 803 180 21.1 667 305 21.1 2669 305 25.6 33.1 25.6 33.1 25.7 27.6 27.6 25.6 2946 2946 2946 2946 2946 2946 2946 2946 457 457 457 457 457 457 457 457 Lynn [2001] 3CLH18 457 2CLH18 457 3SLH18 457 2SLH18 457 2CMH18 457 3CMH18 457 3CMD12 457 3SMD12 457 393 397 393 397 397 393 393 393 27.0 27.0 27.0 27.0 27.0 27.0 Yoshimura and Yamanaka [2000] FS0 300 300 255 600 FS0 300 300 255 600 FS0 300 300 255 600 S1 400 400 350 600 S2A 400 400 350 600 S2A 400 400 350 600 245 245 142 142 142 142 142 142 245 245 157 157 157 157 157 157 0.18 0.18 0.07 0.07 0.07 0.07 0.07 0.07 0.18 0.18 0.70 0.70 0.70 0.22 0.22 0.22 Longitudinal Reinforcement 469 469 400 400 400 400 400 400 400 400 355 355 355 355 355 355 8 8 8 8 8 12 12 12 16 16 16 28.7 28.7 32.3 25.4 32.3 25.4 25.4 32.3 32.3 32.3 19.0 19.0 19.0 22.0 22.0 22.0 469 469 335 335 335 335 335 335 335 335 387 387 387 547 547 547 2.83 7.17 6.66 9.11 0.00 2.72 2.72 2.72 8.99 8.18 5.36 5.96 6.88 5.06 exp 0.167 0.293 0.185 0.293 0.961 0.623 0.587 0.589 0.32 0.32 0.32 0.197 0.197 0.197 0.633 0.405 0.423 0.349 1.000 0.643 0.643 0.643 0.352 0.374 0.477 0.45 0.415 0.491 0.264 0.723 0.437 0.84 0.961 0.969 0.913 0.916 0.909 0.856 0.671 0.438 0.475 0.401 (ηsl ) (ηsl )exp (ηsl )pro (η )exp sl pro 146.0 13.7 0.196 0.263 0.745 55.0 0.86 0.991 0.851 1.165 61.0 91.3 91.3 106.6 30.3 61.0 61.0 61.0 54.6 50.4 31.8 51.6 52.8 40.2 fyt fyl ( a )exp db (MPa) nbars (mm) (MPa) (mm) δa∗ Transverse Reinforcement b h d L fc P s Ast ρ st Specimen (mm) (mm) (mm) (mm) (MPa) (kN) (mm) (mm2 ) (%) Column Section TABLE Database of RC columns tested to the point of axial failure Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 289 255 255 255 255 255 Yoshimura et al [2003] No.1 300 300 No.2 300 300 No.3 300 300 No.4 300 300 No.5 300 300 1200 1200 1200 1200 1200 30.7 30.7 30.7 30.7 30.7 25.2 25.2 25.2 25.2 25.2 25.2 600 600 600 600 600 600 255 255 255 255 255 255 552 552 552 828 967 430 430 657 657 430 430 429 429 876 876 100 150 200 100 100 100 100 100 100 100 100 100 100 100 100 20.9 1491 305 21.1 667 305 Yoshimura et al [2003] 2M 300 300 2C 300 300 3M 300 300 3C 300 300 2M13 300 300 2C13 300 300 392 2946 392 2946 26.5 26.5 26.5 26.5 457 457 Nakumura and Yoshimura [2002] N18M 300 300 255 900 N18C 300 300 255 900 N27M 300 300 255 900 N27C 300 300 255 900 2CVD12 457 2CLD12M 457 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 245 245 0.19 0.13 0.09 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.19 0.18 0.18 392 392 392 392 392 392 392 392 392 392 392 375 375 375 375 469 469 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 8 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 13.0 13.0 16.0 16.0 16.0 16.0 28.7 28.7 402 402 402 402 402 396 396 396 396 350 350 380 380 380 380 469 469 160.8 28.0 64.8 9.87 24.0 1.81 24.0 1.87 24.0 1.79 67.2 12.0 46.8 8.05 33.6 5.37 31.8 5.02 24.6 3.93 18.0 2.66 92.7 16.3 185.4 34.2 42.3 6.42 27.0 3.48 0.488 0.507 0.501 0.758 0.898 0.37 0.37 0.607 0.607 0.634 0.634 0.386 0.386 0.874 0.874 3.297 1.532 0.686 1.048 1.227 1.276 0.979 1.273 1.231 1.142 0.978 1.678 3.088 2.023 1.497 (Continued) 0.148 0.331 0.73 0.723 0.732 0.29 0.378 0.477 0.493 0.555 0.648 0.23 0.125 0.432 0.584 86.0 5.61 0.535 0.465 1.151 161.0 15.6 0.196 0.239 0.82 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 290 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 300 Ousalem [2003] C1 300 C4 300 C8 300 C12 300 D1 300 D16 300 D11 300 D12 300 D13 300 D14 300 D15 300 D5 300 No.6 No.7 260 260 260 260 260 260 260 260 260 260 260 260 900 900 900 900 600 600 900 900 900 900 900 750 255 1200 255 1200 13.0 13.0 20.0 20.0 27.7 26.1 28.2 28.2 26.1 26.1 26.1 28.5 30.7 30.7 364 160 364 75 486 75 324 75 540 50 540 50 540 150 540 150 540 50 540 50 540 50 540 50 552 100 552 150 39.3 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 56.5 113 56.5 56.5 56.5 0.08 0.25 0.25 0.25 0.38 0.38 0.13 0.13 0.38 0.38 0.75 0.38 0.19 0.13 Longitudinal Reinforcement 587 384 384 384 398 398 398 398 398 398 398 398 392 392 12 12 12 12 12 12 16 16 16 16 16 12 12 12 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 13.0 16.0 13.0 13.0 63.6 24.0 (ηsl ) (ηsl )exp (ηsl )pro (η )exp sl pro 0.917 0.462 0.78 0.276 0.572 0.576 0.758 0.746 0.53 0.238 0.143 0.396 0.59 1.097 0.928 1.54 0.984 0.979 0.661 0.676 0.798 1.777 2.098 0.972 1.131 0.609 9.77 0.726 0.333 2.18 2.37 0.755 0.674 1.12 exp 9.1 0.44 0.541 36.2 5.68 0.507 13.6 1.38 0.724 72.6 12.8 0.425 24.3 3.66 0.563 24.0 3.60 0.564 16.9 1.56 0.501 17.5 1.66 0.504 31.5 4.33 0.423 90.5 15.7 0.423 161.0 29.2 0.3 46.1 7.46 0.385 Mean Coefficient of Variation 340 340 340 340 447 447 447 447 447 447 447 431 409 409 fyt fyl ( a )exp db (MPa) nbars (mm) (MPa) (mm) δa∗ Transverse Reinforcement b h d L fc P s Ast ρ st Specimen (mm) (mm) (mm) (mm) (MPa) (kN) (mm) (mm2 ) (%) Column Section TABLE (Continued) Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Ultimate Displacement of RC Columns 291 0.9 0.8 0.7 ηsl 0.6 0.5 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 0.4 0.3 Experimental 0.2 Proposed 0.1 0 10 20 30 * δa(%) 40 50 60 FIGURE Relationship between ηsl and δa∗ Experimental Investigation An experimental program carried out on RC columns with light transverse reinforcement subjected to cyclic loading was conducted to verify the developed semi-empirical model Five 1/2-scale RC columns with light transverse reinforcement were tested up to the point of axial failure to investigate the collapse mechanism of such columns when subjected to seismic loading 3.1 Specimens and Test Procedure Figure 6, along with Table 2, provides the dimensions and the details of reinforcements within the test specimens The longitudinal reinforcement consisted of T20 and T25 deformed bars that were characterized with yield strengths fy of 408 MPa (59.2 ksi) and 409 MPa (59.3 ksi), respectively The transverse reinforcement of all test specimens comprised of R6 mild steel bars that were characterized with a yield strength fy of 393MPa (57.0 ksi) The theoretical flexural strengths Vu of the test specimens were estimated using the material properties obtained through tests and in accordance with the recommendations provided by ACI-ASCE Committee 352 [ACI-ASCE, 1991] The nominal shear strengths Vn of the test specimens (the shear strength of the columns at zero displacement ductility) were calculated based on Sezen’s suggestion [Sezen, 2002] The values of Vu and Vn 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 through an actuator with a 1000 kN (224.8 kip) capacity which was mounted onto a reaction wall The actuator was pinned at both ends to allow it to rotate during the test The base of the column was fixed to a strong floor with four post-tensioned bolts The column axial load was slowly applied to the specimens using actuators, each with a capacity of 1,000 kN (224.8 kip) through a transfer beam until the 292 C T Ngoc Tran and B Li 135 degree hook 350 350 25 600 R6 8-T20 135 degree hook 350 SC-2.4-0.20 SC-2.4-0.50 135 degree hook 350 T20 1700 25 350 R6 @ 125 25 R6 @ 125 350 SC-1.7-0.20 SC-1.7-0.50 R6 8-T25 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 350 S-2.4-0.30 T20 1200 R6 8-T20 600 350 900 900 FIGURE Reinforcement details of test specimens TABLE Summary of test specimens Specimen S-2.4-0.30 SC-2.4-0.20 SC-2.4-0.50 SC-1.7-0.20 SC-1.7-0.50 Longitudinal Transverse fc b×h L Reinforcement Reinforcement (MPa) (mm×mm) (mm) 8-T25 8-T20 2-R6 @ 125 49.3 22.6 24.2 27.5 26.4 350 × 350 1700 1200 P fc Ag Vu (kN) Vn (kN) 0.30 0.20 0.50 0.20 0.50 434.6 231.2 259.1 353.3 375.2 340.4 199.1 258.5 287.8 374.1 R6 = Plain round bar of mm diameter T20 = Deformed high strength bar of 20 mm diameter T25 = Deformed high strength bar of 25 mm diameter L-shaped Steel Frame Strong Wall Actuator Specimen Actuator Actuator Strong Floor FIGURE Typical experimental setup designated level was achieved During each test, the column axial load was maintained by manually adjusting the vertical actuators after each load step The test was terminated at a point when the column axial force from the vertical actuators can not be increased to the Ultimate Displacement of RC Columns 293 2.5 DR = 1/50 DR = 1/65 1.5 DR = 1/80 Drift ratio (%) DR = 1/200 0.5 DR = 1/1000 DR = 1/500 DR = 1/300 –0.5 –1 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 DR = 1/125 DR = 1/700 DR = 1/400 10 12 14 16 18 20 22 26 28 30 32 34 DR = 1/150 DR = 1/100 –1.5 DR = 1/70 DR = 1/55 –2 –2.5 24 DR = 1/250 Cycle number FIGURE Loading procedure designated level This point was defined as the axial failure of the columns The typical loading procedure is illustrated in Fig The test specimens had been extensively installed with measuring devices both internally and externally Amongst the measurements recorded were lateral loads and displacements imposed at the top of the column, shear, and flexure deformations at the critical regions of the specimens and also the strains in both transverse and longitudinal reinforcing bars 3.2 Experimental Results and Discussions 3.2.1 Cracking Patterns The final crack patterns of the test specimens are shown in Fig As it is not possible to describe the crack development for all the specimens in detail within this article, only 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%, while 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 shear inclination in the flexural cracks In the subsequent loading cycles, the occurrence of a steep shear crack and the opening of the existing shear cracks resulted in a reduction in the shear-resisting capacity of the test specimens At the axial failure, the steep shear crack developed on the column from the previous stages became wider This led to sliding between the crack surfaces as well as the buckling of longitudinal reinforcing bars and fracturing of transverse reinforcing bars along this shear crack 3.2.2 Hysteretic Responses Figure 10 shows the hysteretic responses of the test specimens The theoretical flexural strength and nominal shear strength of the specimens are also shown in Fig 10 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 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 294 C T Ngoc Tran and B Li (a) S-2.4-0.30 (d) SC-1.7-0.20 (b) SC-2.4-0.20 (c) SC-2.4-0.50 (e) SC-1.7-0.50 FIGURE Observed cracks at axial failure of test specimens (color figure available online) yielding of the reinforcing bars The pinching effect was observed in the hysteretic loops of all the test specimens Except for Specimen SC-1.7-0.50, none of the specimens reached their theoretical flexural strengths up until the end of the tests The shear failure in all test specimens occurred at a drift ratio of less than 2.0% 3.2.3 Shear Strengths Table summarizes the shear strengths of all test specimens The shear strength of SC-1.7 Series specimens were enhanced by around 27.7%, as the column Ultimate Displacement of RC Columns –2 –1.5 Drift Ratio (%) –0.5 0.5 –1 1.5 500 112 Vu 44.8 100 22.4 0 –100 –22.4 –200 –44.8 Vu –500 –34 –25.5 –17 –8.5 8.5 17 Lateral Displacement (mm) –89.6 –112 25.5 34 (a) 300 Vu Vn Shear Force (kN) 200 100 67.2 300 44.8 200 22.4 0 –100 –22.4 –200 SC-2.4-0.20 Vn Vu –300 –51 Drift Ratio (%) –2 Shear Force (kN) –2 Shear Force (kip) Drift Ratio (%) –1 –17 17 34 Lateral Displacement (mm) –1 –2 –1.5 1.5 0.5 1.5 67.2 44.8 100 22.4 0 –100 –22.4 SC-2.4-0.50 –44.8 Vn Vu –300 –34 –25.5 –17 –8.5 8.5 17 Lateral Displacement (mm) 51 –2 –1.5 –1 (c) Drift Ratio (%) –0.5 0.5 Vu Vn –67.2 25.5 1.5 34 89.6 67.2 300 200 44.8 200 44.8 100 22.4 100 22.4 300 0 –100 –22.4 –200 –44.8 –300 SC-1.7-0.20 Vn Vu –400 –24 –18 –12 –6 12 Lateral Displacement (mm) –67.2 –89.6 18 24 Shear Force (kip) 400 Vu Vn Shear Force (kN) 89.6 400 Shear Force (kN) –1 Vu Vn (b) Drift Ratio (%) –0.5 0.5 –0.5 –200 –44.8 –67.2 –34 –1.5 67.2 0 –22.4 –100 –44.8 –200 SC-1.7-0.50 –300 V Vun –400 –24 –18 –12 –6 12 Lateral Displacement (mm) (d) Shear Force (kip) –400 –67.2 S-2.4-0.30 Vn Shear Force (kip) Shear Force (kN) 67.2 200 –300 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 89.6 Vn 300 Shear Force (kip) 400 –3 295 –67.2 –89.6 18 24 (e) FIGURE 10 Hysteretic responses of test specimens TABLE Summary of test results Specimen S-2.4-0.30 SC-2.4-0.20 SC-2.4-0.50 SC-1.7-0.20 SC-1.7-0.50 Shear Strength (kN) Drift Ratio at Axial Failure (%) Maximum Cumulative Energy Dissipation (kNm) 357.1 218.9 237.6 294.2 375.6 1.76 2.82 1.68 1.82 1.42 29.8 34.9 26.3 13.5 4.2 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 296 C T Ngoc Tran and B Li axial load was increased from 0.20 to 0.50 fc Ag An analogous trend was observed in the specimens of SC-2.4 Series, whose shear strengths experienced an enhancement of around 8.5% as the applied axial load was increased from 0.20 to 0.50 fc Ag The above discussion clearly indicates that the column axial load was beneficial to the shear strength of the test specimens The shear strength of Specimens SC-2.4-0.20, SC-1.7-0.20, SC-2.4-0.50, and SC-1.70.50 obtained from the tests were 218.9 kN (49.2 kip), 294.2 kN (66.1 kip), 237.6 kN (53.4 kip), and 375.6 kN (84.4 kip), respectively The increase in shear strength between Specimens SC-2.4-0.20 and SC-1.7-0.20 was 34.4% Similarly, an enhancement of 58.1% was observed in the shear strength of Specimen SC-1.7-0.50 as compared to that of Specimen SC-2.4-0.50 Thus, it can be concluded that the shear strength of the test specimens increased with a decrease of its aspect ratio 3.2.4 Drift Ratios at Axial Failure The drift ratios at axial failure of the test specimens are tabulated in Table Generally, 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 SC2.4 and SC-1.7 Series specimens reduced by around 40.4% and 26.9%, respectively, as the column axial load ratio was increased from 0.20 to 0.50 The effects of aspect ratio on the drift ratio at axial failure can be noticed by comparing the SC-2.4 and SC-1.7 Series specimens 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 At a column axial load ratio of 0.50, a decrease of 20.8% was observed between the specimen of SC-2.4 and SC-1.7 Series It was concluded, based on the test results obtained from Specimens SC-2.4-0.20, SC-2.4-0.50, SC-1.7-0.20, and SC-1.7-0.50 that a decrease in their aspect ratio led to a reduction in their drift ratio at axial failure 3.2.5 Cumulative Energy Dissipation 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-2.4 and SC1.7 Series specimens as the column axial load was increased from 0.20 to 0.50 fc Ag It can therefore be concluded, based on the test results, that column axial load has detrimental effects on the maximum cumulative energy dissipation of the test specimens The effects of aspect ratio on the maximum cumulative energy dissipation can be noticed by comparing between the SC-2.4 and SC-1.7 Series specimens As shown in Table 3, at a column axial load ratio of 0.20, the maximum cumulative energy dissipation was reduced from 34.9 kN×mm (308.9 psi.×in.) to 13.5 kN×mm (119.5 psi.×in.) as its aspect ratio decreases from 2.4 to 1.7 At a column axial load ratio of 0.50, a decrease of 84.0% was observed between the specimen of SC-2.4 and SC-1.7 Series It can be concluded based on the test results that a decrease in the aspect ratio led to a drop in the maximum cumulative energy dissipation of the test specimens 3.2.6 Displacement Decompositions The contribution of deformation components expressed as percentages of the total lateral displacements at the peak displacements during each displacement cycle of Specimen SC-2.4-0.50 is shown in Fig 11 Detailed displacement decompositions of all test specimens had been reported elsewhere [Tran, 2010] Approximately 60–72% of the total lateral displacement was contributed by the flexural deformation component, whereas only up to 41% was accounted for the shear deformation component The shear deformation component initially grew gradually to approximately 16% of the total lateral displacement up to a drift ratio of 1.57% As the Ultimate Displacement of RC Columns 297 100 80 Unaccounted 60 40 Flexure 20 0 0.5 Drift Ratio (%) 1.5 FIGURE 11 Displacement decompositions of specimen SC-2.4-0.50 drift ratio was increased up to 1.68%, the corresponding shear deformation component drastically grew to about 41% of the total displacement 3.3 Verification of the Proposed Model The displacements at axial failure obtained from the test results of five RC columns in the current experimental investigation are used to validate the developed semi-empirical model The analytical displacements at axial failure of test specimens are calculated based on Eqs (9), (10), (12), (13), (17), and (18) It was found that the average ratio of the experimental to predicted displacement at axial failure predicted by the proposed model was 1.050, as shown in Fig 12 and Table 4, showing a good correlation between the proposed equation and experimental data (Δa/L)analytical Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 Displacement Decomposition (%) Shear Proposed Equation Elwood et al.'s Equation 0 (Δa/L)experimental FIGURE 12 Comparisons between experimental and analytical ultimate displacements of various equations 298 C T Ngoc Tran and B Li TABLE Experimental verification of the proposed model Specimen ( a /L)exp S-2.4-0.30 1.76 SC-2.4-0.20 2.82 SC-2.4-0.50 1.68 SC-1.7-0.20 1.82 SC-1.7-0.50 1.42 Mean Coefficient of Variation Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 a /L)pro ( ( 1.38 3.15 1.53 2.4 1.17 ( ( a /L)Elwood 1.14 3.01 1.35 2.63 1.23 a /L)exp a /L)pro 1.278 0.897 1.100 0.757 1.214 1.050 0.218 ( ( a /L)exp a /L)Elwood 1.541 0.939 1.243 0.692 1.149 1.113 0.320 P M V dc Vd Ast fyt Ps s N Vsf Ast fyt Vd Ps FIGURE 13 Free body diagram of column after shear failure The drift ratios at axial failure obtained from Elwood and Moehle’s model [2005] are also shown in Table for comparison with the proposed method Elwood and Moehle’s model [2005] proposed the following equation for the drift ratio at axial failure based on the shear friction model with the free body diagram, as shown in Fig.13: L = a + (tan θ )2 s 100 tan θ + P Ast fyt dc tan θ where dc is the depth of core (centerline to centerline of ties) , (19) Ultimate Displacement of RC Columns 299 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 The mean ratios of the experimental to the predicted displacement at axial failure and its coefficient of variation are 1.050 and 0.218 for the proposed model, and 1.113 and 0.320 for Elwood and Moehle’s model [2005], respectively Comparing the two models with experimental data indicates that the proposed model produced better mean ratio and its coefficient of variation of the experimental to the predicted displacement at axial failure than Elwood and Moehle’s equation [2005] It is to be noted that Elwood and Moehle’s equation [2005] was developed based on the test data of columns experiencing flexureshear failures whereas all the test columns in the current experimental program experienced pure shear failures This difference in the failure mode indicates the inaccuracy of Elwood and Moehle’s equation [2005] when applied to the test columns in the current experimental program Conclusions An analytical model is developed in this article to estimate the ultimate displacement of RC columns with light transverse reinforcement The following provides specific findings of the article ● ● A semi-empirical equation is developed to estimate the ultimate displacement of reinforced concrete columns with light transverse reinforcement by using the collected data of RC columns tested to the point of axial failure from existing research studies An experimental program consisting of five RC columns with light transverse reinforcement was carried out to validate the developed model It was found that the average ratio of the experimental to predicted displacement at axial failure predicted by the proposed method was 1.050, showing a good correlation between the proposed equation and experimental data The proposed equation may be suitable as an assessment tool to calculate the displacement at axial failure of RC columns with light transverse reinforcement References ACI-ASCE Committee 352 [1991] “Recommendations for design of beam-column joints in monolithic reinforced concrete structures,”ACI Manual of Concrete Practice, ACI 352R-91, Farmington Hills, Michigan Ngoc Tran, C T [2010] “Experimental and analytical studies on the seismic behavior of RC columns with light transverse reinforcement,” Ph.D thesis, Nanyang Technological University, Singapore EERI [1999a] “The Tehuacan, Mexico Earthquake of June 15, 1999,” EERI Special Earthquake Report Oakland, CA EERI [1999b] “The Athens, Greece Earthquake of September 7, 1999,” EERI Special Earthquake Report Oakland, CA EERI [1999c] “The Chi-Chi, Taiwan Earthquake of September 21, 1999,” EERI Special Earthquake Report Oakland, CA Elwood, K J and Moehle, J P [2005] “Axial capacity model for shear-damaged columns,” ACI Structural Journal 102(4), 578–587 Lynn, A C [2001] “Seismic evaluation of existing reinforced concrete building columns,” Ph.D thesis, Department of Civil and Environmental Engineering, University of California, Berkeley, California Moehle, J P and Mahin, S A [1991] “Observations on the behavior of reinforced concrete buildings during earthquakes,” in ACI SP-127, Earthquake-Resistant Concrete Structures Inelastic Response and Design, ed S K Ghosh (American Concrete Institute, Detroit), pp 67–89 Downloaded by ["University at Buffalo Libraries"] at 05:43 20 January 2015 300 C T Ngoc Tran and B Li Nakamura, T and Yoshimura, M [2002] “Gravity load collapse of reinforced concrete columns with brittle failure modes,” Journal of Asian Architecture and Building Engineering 1(1), 21–27 Ousalem, H [2003] “Experimental and analytical study on axial load collapse assessment and retrofit of reinforced concrete columns,” Ph.D thesis, Department of Architecture Engineering, University of Tokyo, Tokyo Priestley, M J N., Verma, R., and Xiao, Y [1994] “Seismic shear strength of reinforced concrete columns,” Journal of Structural Engineering 120(8), 2310–2329 Priestley, M J N., Calvi, G M., and Kowalsky, M J [2007] Displacement-Based Seismic Design of Structures, IUSS Press, Pavia, Italy Sezen, H [2002] “Seismic response and modeling of reinforced concrete building columns,” Ph.D thesis, Department of Civil and Environmental Engineering, University of California, Berkeley, California Yoshimura, M and Yamanaka, N [2000] “Ultimate limit state of RC columns,” The 2nd US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, Sapporo, Japan, pp 313–326 Yoshimura, M and Nakamura, T [2003] “Axial collapse of reinforced concrete short columns,” The 4th US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, Toba, Japan, pp 187–198 Yoshimura, M., Takaine, Y., and Nakamura, T [2003] “Collapse drift of reinforced concrete columns,” The 5th US-Japan Workshop on Performance-Based Earthquake Engineering Methodology for Reinforced Concrete Building Structures, Hakone, Japan, pp 239–253  ... estimate the lateral displacement at axial failure (or ultimate lateral displacement) of RC columns with light transverse reinforcement under seismic loading The developed model is calibrated... developed to estimate the ultimate displacement of reinforced concrete columns with light transverse reinforcement by using the collected data of RC columns tested to the point of axial failure from... investigations carried out on reinforced concrete (RC) columns with light transverse reinforcement A semi-empirical model is developed to estimate the ultimate displacement (displacement at axial