A comparative analysis of energy dissipation and equivalent viscous damping of RC columns subjected to uniaxial and biaxial loading

A comparative analysis of energy dissipation and equivalent viscous damping of RC columns subjected to uniaxial and biaxial loading Hugo Rodrigues a, ⇑ , Humberto Varum a , António Arêde b , Aníbal Costa a a Departamento de Engenharia Civil, Universidade de Aveiro, Portugal b Departamento de Engenharia Civil, Faculdade de Engenharia, Universidade do Porto, Portugal a r t i c l e i n f o Article history: Received 27 June 2011 Revised 6 October 2011 Accepted 2 November 2011 Keywords: RC columns Hysteretic behaviour Biaxial testing Energy dissipation Viscous damping a b s t r a c t The hysteretic behaviour of RC columns has been object of many experimental studies over the past years. However, the majority of these studies are focused on unidirectional loading. An experimental program was carried out where 24 columns were tested for different loading histories, under uniaxial and biaxial conditions. The experimental results are presented in this paper and are discussed in terms of global column behaviour, and particularly with regards to energy dissipation and damping capacity. The energy dissipation capacity of the columns was evaluated in terms of cumulative dissipated energy, comparing uniaxial and biaxial test results, and individual cycle dissipated energy. Ultimately, an equation relating the normalised dissipated energy with the displacement ductility is proposed. The equivalent viscous damping was analysed by comparing the uniaxial with biaxial test results, demonstrating the high influence of the load path in the biaxial response of RC columns. Proposals for estimating the equivalent viscous damping given by other authors are compared with the experimental results. Finally, simplified expressions are proposed to estimate equivalent viscous damping in RC columns under biaxial loading. Ó2011 Elsevier Ltd. All rights reserved. 1. Introduction The behaviour of reinforced concrete (RC) elements subjected to axial loading in conjunction with cyclic biaxial bending is recognised as a very important research topic for building structures in earthquake prone regions. Previous experimental work agrees that biaxial horizontal cyclic loading can increase the strength and stiffness degradation, when compared to uniaxial loading. In addition, the failure mechanism of RC columns is found to be highly dependent on the load path and history and strongly affects both the ductility and energy dissipation capacity of the columns [1,2]. Energy dissipation is a fundamental structural property of RC elements when subjected to seismic demands. For RC structures designed to accommodate damage without collapse due to a seismic event, the input energy can be dissipated through RC element’s hysteretic response, without a significant reduction in strength [3]. Nonlinear static methods, for assessment or design, use energy dissipation capacity related parameters to evaluate the inelastic earthquake response of structures and to describe the strength and stiffness degradation of RC elements subjected to cyclic loading [4]. Viscous damping is used to characterise the energy dissipation capacity of RC elements and is one of the key parameters for the application of displacement based design (DBD) methods [5]. DBD methods can be based on the substitute-structure concept, developed by Shibata and Sozen [6], which represents the structure intended for design or assessment purposes by the secant stiffness to maximum displacement response and equivalent viscous damping representing the combined effects of elastic and hysteretic damping [7]. Energy dissipation and the equivalent viscous damping have been correlated with displacement ductility for uniaxial stress. The current work intends to compare energy dissipation and equivalent viscous damping on RC columns subjected to uniaxial and biaxial loads. Finally, consideration is given to whether the available formulas relating viscous damping with the displacement ductility that have been proposed for uniaxial demands are applicable to biaxial loading. 2. Test program In the experimental campaign were tested 24 rectangular RC columns with different types of geometric characteristic and reinforcement detailing and were cyclically tested for different loading histories with a constant axial force and under displacement controlled conditions. The column specimens are 1.70 m high and 0141-0296/$ - see front matter Ó2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2011.11.014 ⇑Corresponding author. Tel.: +351 234370049; fax: +351 234370094. E-mail address: [email protected] (H. Rodrigues). Engineering Structures 35 (2012) 149–164 Contents lists available at SciVerse ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct are cast in strong square concrete foundation blocks. The crosssection dimensions and the reinforcement detailing are presented in Fig. 1. The materials considered at the specimen design phase were a regular concrete class C35/45 for columns N01-N04 and C30/35 for columns N05-N24, with reinforcement steel grade of A400NR-SD, the average concrete strength obtained in tests on samples are summarised in Table 1. Fig. 2 shows the setup adopted for the experimental testing including the two independent horizontal actuators to apply the lateral loads on the column specimen, one with a capacity of 500 kN and ±150 mm stroke and the other with a capacity of 200 kN and ±100 mm stroke. A vertical 700 kN capacity actuator was used to apply the axial load. Two steel reaction frames and a concrete reaction wall form the reaction system for the three actuators. The column specimens and the reaction frames were fixed to the strong floor of the laboratory with prestressed steel bars to avoid sliding or overturning of the specimen during testing, or sliding of the reaction frame. Since the axial load actuator remains in the same position during the test while the column specimen laterally deflects, a sliding device is used (placed between the top-column and the actuator), which was built to minimise spurious friction effects. As stated previously, for all the specimens tested, a constant axial force was imposed, the values of which are included in Table 1, for both absolute and normalised axial force. In order to characterise the response of the column specimens, cyclic lateral displacements were imposed at the top of the column with steadily increasing demand levels. Three cycles were repeated for each lateral deformation demand level. This procedure allows for the understanding of the column’s behaviour, a comparison between different tests and provides information for the development and calibration of numerical models, the following nominal peak displacement levels (in mm) were considered: 3, 5, 10, 4, 12, 15, 7, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80. 3. Global columns response From the analysis of the measured displacement and shear force paths (along the Xand Ydirections) are analysed. Due to the large number of tests, only a few examples of the results are presented in Figs. 3–5, but all the discussion refers to the results of the complete testing program, detailed information about the forcedisplacement results can be found in [2]. From the experimental campaign the main finding were: From the observation of the shear-drift curves, four main branches can be identified in their envelopes, corresponding to: (i) pre-cracking response; (ii) post-cracking until the reinforcement steel yields; (iii) a plateau or post-yield hardening zone; and (iv) a softening phase. These four stages are clear in both the uniaxial and biaxial tests. However, in the biaxial tests the plateau tends to be shorter and the softening is more pronounced, i.e. a more abrupt decay of the column strength is observed with increasing lateral deformation demands. The initial column stiffness in both directions it is not significantly affected by the biaxial load path. Fig. 1. RC column specimen dimensions and reinforcement detailing: cross-sections details and specimen dimensions and general scheme of the reinforcement layout. 150 H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 As expected, when comparing the maximum strength in one specific direction of the columns for each biaxial test against the corresponding uniaxial test, lower values were obtained for all biaxial tests than uniaxial ones. The biaxial loading induces a 20–30% reduction of the maximum strength of the columns in their weak direction, Y, while reductions from 8 to 15% for the stronger direction, X. The ultimate ductility is significantly reduced in columns subjected to biaxial load paths. The strength degradation is practically zero, in the first loading cycles, increasing after displacement ductility demands of about 3. From the strength degradation analysis, more pronounced strength degradation was observed for biaxial tests when compared with corresponding uniaxial tests. 4. Dissipated energy 4.1. Cumulative dissipated energy Bousias et al. [8] stated that the strong coupling between the two transverse directions of columns with biaxial loading produces an apparent reduction of strength and stiffness in each of the two transverse directions when considered separately, but also an increase in the hysteretic energy dissipation. This increase is due to the larger width of the hysteresis loops in the transverse direction in the presence of a non-zero force or deflection in the orthogonal direction. Qiu et al. [9] claim that the accumulative hysteresis dissipation energy of a specimen under biaxial loading is apparently larger than that under unidirectional loading and is closely related to the loading position and path length. The cumulative hysteretic dissipation energy was evaluated for all the tests, considering the area of each loading cycle in the Xand Ydirection and then the total energy was calculated as the sum of these two parts, according to Eqs. (1)–(3). Ed X ¼ZF X d X ð1Þ Ed Y ¼ZF Y d Y ð2Þ Ed tot ¼ZF X d X þZF Y d Y ð3Þ The results in terms of evolution of cumulative dissipated energy are presented in Fig. 6 for the uniaxial and biaxial tests. In Fig. 6, for each displacement amplitude level, the plotted value of dissipated energy corresponds to the end of the third cycle. For the quadrangular load path, the maximum displacement for each cycle occurs in the path corner. It is also presented in the plots, along with an additional series (dashed line) representing the sum of the dissipated energy in the uniaxial tests of the corresponding column cross-section. From the analysis of the results it can be concluded: Comparing the two uniaxial test results, as expected a lower energy dissipation was observed for the columns tested in its weakest direction, associated with the inferior column strength in this direction (15–20% lower in the column of 30 40 cm 2 section and 60–80% in the column of 30 50 cm 2 section). The biaxial load paths induce larger amounts of dissipated energy than the correspondently uniaxial paths. However, the sum of the dissipated energy in the two unidirectional tests, the Xand Ydirections, leads to a dissipation energy evolution very close to that derived from the tests with rhombus and circular load paths. The results in terms of dissipated energy evolution for the cruciform, rhombus and circular load paths, is similar for all the columns tested. The circular load path induces an energy Table 1 Specimen specifications and loading characteristics. Series Column Geometry (cm cm) f cm (MPa) N (kN) m N/(A c f cm ) Displacement path type 1 PB01-N01 20 40 48.35 170 0.04 Uniaxial strong PB02-N02 Uniaxial weak PB12-N03 Cruciform PB12-N04 Rhombus 2 PB01-N05 30 40 21.40 300 0.12 Uniaxial strong PB02-N06 Uniaxial weak PB12-N07 Rhombus PB12-N08 Quadrangular PB12-N17 36.30 510 Circular 3 PB01-N09 30 50 24.39 300 0.08 Uniaxial strong PB02-N10 Uniaxial weak PB12-N11 Rhombus PB12-N12 Quadrangular PB12-N18 36.30 440 Circular 4 PB01-N13 30 30 21.57 210 0.1 Uniaxial strong PB12-N14 Rhombus PB12-N15 Quadrangular PB12-N16 Circular 5 PB12-N19 30 50 43.14 300 0.045 Rhombus PB12-N20 600 0.09 Rhombus 6 PB12-N21 30 40 43.14 620 0.12 Rhombus PB12-N22 Quadrangular 7 PB12-N23 30 30 36.30 650 0.2 Rhombus PB12-N24 Quadrangular f cm – mean concrete compressive strength. N– axial load. m =N/(A c f cm ) – axial load ratio. A c – area of the column cross section. H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 151 dissipation approximately 20% higher than the rhombus load path. Rhombus load paths dissipate 30% more energy than cruciform load paths. The quadrangular load path dissipates less energy than the other biaxial load paths. It should be recalled that the maximum drift demands on the quadrangular load path is reached in the path corner, corresponding to times the maximum drift reached along the Xand Yaxes. In accordance with this, the quadrangular load path dissipates 30–45% less energy when compared to the rhombus load path. However, the quadrangular load path would dissipate 40–60% more energy than the rhombus load path. Fig. 2. Testing setup: (a) General view, (b) schematic layout (plan view). 152 H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 Fig. 3. Global results of rectangular column PB12-N07 for rhombus load path. Fig. 4. Global results of rectangular column PB12-N12 for rectangular load path. H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 153 Comparing the dissipated energy of the biaxial load paths with the sum of the dissipated energy in the two unidirectional tests, the rhombus load path tends to dissipate more than 10–20% and the circular load path dissipates more than 20–40%. The lower bound of these differences is found for the column with the square cross-section. This allows the conclusion that in the assessment and design of RC structures, not considering the bending interaction between each direction in the numerical models can introduce an error about of 10–40% in terms of energy dissipation. 4.2. Individual cycle energy The energy dissipated for each individual loading cycle and the accumulated energy dissipation along each tested column was calculated. For purposes of correlation, the cycles for which relevant damage states occurred during the tests were also identified, namely the reinforcement bar buckling, conventional column collapse and bar failure. In Figs. 7 and 8 examples of the graphics obtained are presented. From the analysis of the results obtained for the 24 tested columns, the following can be concluded: In the first cycle of each peak displacement, higher energy dissipation is observed relative to the subsequent cycle with the same peak displacement. In the uniaxial tests, the reduction of the dissipated energy in the 2nd and 3rd cycles is about 10% of the energy dissipated in the 1st cycle. This reduction is more pronounced for the biaxial load path, reaching 25%. The damage induced during the first cycle reduces the stiffness and strength, reducing the energy dissipation capacity of the column in the second and third cycles (see examples in Figs. 7 and 8). A significant drop in the energy dissipation is observed after reaching the conventional rupture of the column. This effect is associated with the longitudinal bars buckling, which induces a high level of column strength degradation. 4.3. Total dissipated energy until conventional collapse According to Ohno and Nishioka [10] the total dissipated energy of a RC column is independent of the loading path. This finding is in agreement with that of Tsuno and Park [11]. However, in both studies the columns tested were all square columns (40 40 cm 2 and 55 55 cm 2 , respectively) and with axial load stresses between 0.98 and 1.96 MPa. Figs. 9 and 10 compare the total dissipated energy obtained from the test results. This total dissipated energy corresponds to the energy dissipated from the start of the test until conventional rupture is reached, referring a strength decay of 20% relative to the maximum strength [12]. From the analysis of the results, the following observations can be drawn: For square columns (N13–N16) tested with a axial load stress of 2.33 MPa, the results obtained are in agreement with those reported by Ohno and Nishioka [10] and by Tsuno and Park [11], i.e. the dissipated energy up until conventional rupture is approximately the same (see Fig. 9) with differences lower than 10%. However, for square columns (N23 and N24) with a higher level of axial load stress of 7.33 MPa this conclusion is not valid (see Fig. 9). The increase in axial load stress influences the total energy dissipated. For rectangular columns, the finding of Ohno and Nishioka [10] is not valid (see Fig. 9), the differences in strength and stiffness of the two orthogonal directions induce differences that cannot be dissociated from the biaxial coupling effect. Fig. 5. Global results of rectangular column PB12-N12 for circular load path. 154 H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 Uniaxial tests in rectangular columns tested, for the same loading history (comparing N05 with N06 and N09 with N10), show that the dissipated energy up until conventional rupture is dependent of the loading direction. The total dissipated energy in the weaker direction of the column at the point of rupture is 90% (N06) and 20% (N10) higher than the corresponding results for the tests in the strong direction (N05 and N09). The axial load ratio shows to directly influence the total energy dissipation while the level of axial load force does not. In fact the response of similar rectangular columns of different concrete class tested for similar axial load ratios [N11 ( m = 0.08 and N= 300 kN) and N19 ( m = 0.09 and N= 600 kN)] result in dissipation that is approximately the same as the total energy. On the other hand, similar rectangular columns with different concrete classes tested for equal axial loading [N11 ( m = 0.08) and N20 ( m = 0.045) both with N= 300 kN] demonstrated that the total energy dissipated is higher for the column with the lower axial load ratio (see Fig. 9). For square columns tested with different axial load ratios and different axial loads [N14 and N15 ( m = 0.1) with N= 210 kN and N23 and N24 ( m = 0.2) with N= 650 kN] similar findings were established, i.e. higher levels of axial load ratio present lower values of total dissipated energy (see Fig. 10). The non-repetition of the cycles, for the same load path considered in columns N21 and N22 when compared with columns N7 and N8 (see Fig. 11), increase the total dissipated energy (10– 20%) until reaching conventional rupture. 4.4. Normalised dissipated energy vs. displacement ductility As stated by Elmenshawi and Brown [3], the relation between a RC element’s displacement ductility and dissipated energy is complex due to the sensitivity of both factors to the element variables. For each column tested, the calculated energy dissipation evolution was normalised with the total dissipated energy until the first yield point (E y ) until the column conventional failure, i.e. for a strength decay of 20% relative to the maximum strength. In Fig. 12, the evolution of the normalised dissipated energy as a function of the corresponding displacement ductility for the tested columns is represented. Results from the uniaxial and biaxial tests are represented with different mark filling. The best-fit power correlation curve for all tests results (uniaxial and biaxial) is shown in Fig. 12 and is given by expression 4. E cum E y ¼0:64 l 2:1 ð4Þ This expression is very similar to that obtained by best-fit correlation to the test results from the uniaxial and biaxial loading separately. 01234 0 25 50 75 100 125 X Y Accumulative hysteresis dissipated energy (kN.m) Maximum drift (%) PB01-N01 PB02-N02 PB12-N03 PB12-N04 0 1 2 3 4 5 0 50 100 150 200 250 300 X Y Accumulative hysteresis dissipated energy (kN.m) Maximum drift (%) PB01-N05 PB02-N06 PB12-N07 PB12-N08 PB12-N17 PB01-N05+PB02-N06 0 1 2 3 4 5 0 50 100 150 200 250 300 X Y Accumulative hysteresis dissipated energy (kN.m) Maximum drift (%) PB01-N09 PB02-N10 PB12-N11 PB12-N12 PB12-N18 PB01-N09+PB02-N10 0 1 2 3 4 5 0 50 100 150 200 X Y Accumulative hysteresis dissipated energy (kN.m) Maximum drift (%) PB01-N13 PB02-N14 PB12-N15 PB12-N16 PB01-N13x2 Fig. 6. Comparison of cumulative dissipated energy for columns with different load paths (uniaxial and biaxial loads). H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 155 As given by the proposed equation, for a displacement ductility of 4 (corresponding to the minimum required ductility to withstand a severe earthquake), the corresponding normalised dissipated energy estimated is 12. Elmenshawi and Brown [3] and Nmai and Darwin [13] have investigated this relationship for beams (with zero axial force), proposing similar equations. From this expression a value of normalised dissipated energy for the same displacement ductility is 3 time higher, around 35. This difference can be associated with the axial loading levels. As stated by Darwin and Nmai [14], the proposed equations need to be verified with other experimental results. A validated expression can be very useful to estimate the dissipated energy in the seismic design of RC elements in accordance with international codes such as ACI 318-08 [15]. 5. Equivalent viscous damping ratio 5.1. Evaluation of equivalent damping from experimental results The equivalent damping depends on the structural displacement ductility demand and the location of the plastic hinges in the elements [16]. It may be interpreted as the superposition of the elastic and hysteretic damping as shown in Expression 5, Fig. 7. Individual cycle energy and cumulative dissipated energy for a rectangular columns (N09 to N12 and N18). 156 H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 wherein the symbol ‘‘+’’ stands for superposition rather than a trivial summation. n eq ¼n 00 el þ 00 n hyst ð5Þ It is widely accepted that, for typical RC structures, the elastic damping ratio (n el ) is generally taken as 5% of the critical damping [17] and computed proportionally either to the initial (or tangent) stiffness, or to the mass, or to both stiffness and mass. By contrast, the hysteretic damping (n hyst ) depends essentially on the postyielding characteristics of the element and it is normally taken defined proportionally to the secant stiffness [18], which is directly related to the hysteretic rules generally calibrated to represent the structural response in the inelastic phase [19]. Therefore, both damping types should not be directly summed up. For a perfectly symmetric hysteretic response and corresponding closed loop (as in the case of pure harmonic loading), the hysteretic equivalent damping coefficient (n hyst ) can be given accurately given by the well known Expression 6, where E D stands for the dissipated energy within a given cycle, A loop is the area of the corresponding closed loop in the total restoring force– displacement diagram and E S0 is the ‘‘elastic’’ strain energy associated with the maximum force (F max ) and displacement (D max ) reached in the loop. n hyst ¼E D 4 p E S0 ¼A loop 2 p F max D max ð6Þ However, in the case of seismic loads or even for tests performed under displacement controlled conditions, some asymmetries can be observed and the loops may not be closed, which means that the direct use of Expression 6 is less appropriate. Therefore, based on the work of Jacobsen [20] and according to the procedure proposed by Varum [21], the equivalent hysteretic damping can be evaluated for each half-cycle of the force– displacement curves as shown in Fig. 13 and described next: First, each half-cycle is identified, delimited by a pair of zero-force points. For each force–displacement half-cycle, the maximum generalised force (F max ) and the maximum generalised displacement (D max ) are evaluated, which allows calculating the ‘‘elastic’’ strain energy (E S0 ). For each half-cycle, the dissipated energy (E D ) is computed by performing the integral of the force–displacement curve leading to the A half-loop value. Finally, the equivalent damping ratio (n eq ) is computed with the Eq. (7), for each half-cycle. n hyst ¼1 p A halfÿloop F max D max ð7Þ This evaluation may be used as a first approach for estimating the hysteretic damping, and for comparing the tested columns with different cross sections and for different load paths. However, as pointed out by Dwairi et al. [22], it should be noted that an overestimation of the equivalent damping may be obtained, when it is computed proportionally to the dissipated energy and the ductility level. For each column tested, with a uniaxial or biaxial load path, the equivalent damping was calculated, according to the methodology presented, for each independent direction (Xand Y) from the Fig. 8. Individual cycle energy and cumulative dissipated energy for square columns (N13–N16). H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164 157 Different proposals, already available in the literature, for the prediction of equivalent damping of RC columns under uniaxial loading were compared with the experimental results obtained from the uniaxial tests, showing that some of these expressions do not adequately represent the results obtained. The equivalent biaxial damping was computed with the results of each biaxial test, presenting a huge dispersion. The equivalent biaxial damping is highly dependent on the load path. Two simplified expressions were proposed, based on the experimental results, allowing a rough estimation of the equivalent biaxial damping in RC columns subjected to biaxial loading. Even recognizing a possible overestimation of the equivalent damping with the adopted calculation method, these equations represent a first attempt for estimating the equivalent damping of columns under biaxial loading. However, these expressions should be corrected based on results from non-linear time history analysis and/or dynamic shaking table tests. A large number of questions are still open concerning the biaxial behaviour of RC columns, especially regarding equivalent viscous damping associated with loading path. In the present work, the expressions proposed relating the normalised dissipated energy and equivalent biaxial damping with displacement ductility constitutes a preliminary step towards this goal. However these expressions need to be checked against additional experimental results. Even so, the research work reported is expected to contribute towards a better understanding of the biaxial response of RC columns and for the calibration of suitable numerical models for the representation of the biaxial lateral response of reinforced concrete columns under cyclic loading reversals. Acknowledgments The authors acknowledge to the LESE laboratory staff, particularly Valdemar Luís, André Martins and Eng o Luís Noites, for all the support in the preparation and implementation of the testing set-up. This paper reports research developed under financial support provided by ‘‘FCT – Fundação para a Ciência e Tecnologia’’, Portugal, namely through the Ph.D. grants of the first author with reference SFRH/BD/63032/2009, and through the research project PTDC/ECM/102221/2008. References [1] CEB RC frames under earthquake loading, Lausanne Bulletin 220, 1996. [2] Rodrigues, H, Arêde A, Varum H, Costa AG. Experimental study on the biaxial bending cyclic behaviour of RC columns. In: Presented at the 14th European conference on earthquake engineering. Ohrid, Republic of Macedonia, 2010. [3] Elmenshawi A, Brown T. Hysteretic energy and damping capacity of flexural elements constructed with different concrete strengths. Eng Struct 2010;32:297–305. [4] Eom T-S, Park H-G. Evaluation of energy dissipation of slender reinforced concrete members and its applications. Eng Struct 2010;32:2884–93. [5] Lu B, Silva PF. Estimating equivalent viscous damping ratio for RC members under seismic and blast loadings. Mech Res Commun 2006;33:787–95. [6] Shibata A, Sozen MA. Substitute-structure method for seismic design in R/C. J Struct Div, vol. ASCE, 1976. [7] Blandon C. Equivalent viscous damping equations for direct displacement based design. master. Rose School, Pavia, 2004. [8] Bousias SN, Verzeletti G, Fardis MN, Gutierrez E. Load-path effects in column biaxial bending with axial force. J Eng Mech 1995:596–605. [9] Qiu F, Li W, Pan P, Qian J. Experimental tests on RC columns under biaxial quasi-static loading. Engrg Struct 2002;24:419–28. [10] Ohno T, Nishioka T. An experimental study on energy absorption capacity of columns in reinforced concrete structures. Proc JSCE Struct Eng/Earthquake Eng 1984;1(2):137–47. [11] Tsuno K, Park R. Experimental study of reinforced concrete bridge piers subjected to bi-directional quasi-static loading. Struct Engrg Struct JSCE 2004;21(1):11s–26s. [12] Park YJ, Ang HS. Seismic damage model for reinforced concrete. ASCE – J Struct Eng 1985;111(4):722–39. [13] Nmai CK, Darwin D. 1984. Cyclic behavior of lightly reinforced concrete beams. University of Kansas Centre for Research, SM report no. 12. [14] Darwin D, Nmai CK. Energy dissipation in RC beams under cyclic load. J Struct Eng 1986;112:1829–46. [15] ACI318-08 – Building code requirements for structural concrete (ACI 318-08) and commentary (ACI 318R-08), ed. USA: American Concrete Institute; 2008. [16] Priestley MJN. Displacement-based seismic assessment of reinforced concrete buildings. J Earthquake Eng Imperial College Lond 1997;1:157–92. [17] Priestley MJN, Calvi GM, Kowalsky MJ. Direct displacement-based seismic design of structures. Pavia: IUSS Press; 2007. [18] Otani S. Hysteresis models of reinforced concrete for earthquake response analysis. J Faculty Eng 1981;36(2):407–41. [19] Priestley MJN, Grant DN, Blandon CA. Direct displacement-based seismic design. In: Presented at the proceedings of the New Zealand society for earthquake engineering conference. Wairakei, New Zealand, 2005. [20] Jacobsen LS. ASME Trans 1930;52:169–81 [Steady forced vibrations as influenced by damping]. [21] Varum H. Seismic assessment, strengthening and repair of existing buildings, Ph.D. thesis. Aveiro: Civil Engineering Department, University of Aveiro; 2003. [22] Dwairi HM, Kowalsky MJ, Nau JM. Equivalent damping in support of direct displacement-based design. J Earthquake Eng 2007;11:512–30. 2007/07/30. [23] Rosenblueth E, Herrera I. On a kind of hysteretic damping. J Eng Mech Division ASCE 1964;90:37–48. [24] Gulkan P, Sozen MA. Inelastic responses of reinforced concrete structures to earthquakes motions. In: Proceedings of the ACI, vol. 71(12), 1974. [25] Kowalsky MJ. Displacement based design – a methodology for seismic design applied to RC bridge columns, Master’s thesis. San Diego, CA: University of California; 1994. [26] Stojadinovic B, Thewalt CR. Energy balanced hysteresis models. College of Engineering, University of California at Berkeley, 1996. [27] Lu Y, Hao H, Carydis PG, Mouzakis H. Seismic performance of RC framesdesigned for three different ductility levels. Eng Struct 2001;25:537–47. [28] Kowalsky MJ, Dwairi H. Investigation of Jacobsen’s equivalent viscous damping approach as applied to displacement-based seismic design. In: Presented at the proceedings of the 13th world conference on earthquake engineering. Vancouver, Canada, 2004. [29] Fardis M, Panagiotakos T. Hysteretic damping of reinforced concrete elements. In: Presented at the 11th world conference on earthquake engineering, Acapulco, Mexico, 1996. 164 H. Rodrigues et al. / Engineering Structures 35 (2012) 149–164