Material Modelling In The Seismic Response Analysis For The Design Of Rc Framed Structures Two similar continuum plasticity material models are used to examine the influence of material modelling on the seismic response of reinforced concrete frame structures. In the first model reinforced concrete is modelled as a homogenised material using an isotropic Drucker–Prager yield criterion. In the second model, also based on the Drucker–Prager criterion, concrete and reinforcement are included separately. While the latter considers strain softening in tension the former does not. The seismic input is provided using the Eurocode 8 elastic spectrum and five compatible acceleration histories. The results show that the design response from response history analyses (RHAs) is significantly different for the two models. The influence of compression hardening and strength enhancement with strain rate is also examined for the two models. It is found that the effect of these parameters is relatively small. In recent years there has been considerable research in nonlinear static analysis (NSA) or pushover procedures for seismic design. The NSA response is frequently compared with that obtained using RHA, which also uses the same material models, to verify the accuracy of the static procedure. A number of features exhibited by reinforced concrete during dynamic or cyclic loading cannot be easily included in a static procedure. The design NSA and RHA responses for the two material models are compared. The NSA procedures considered are the Displacement Coefficient Method and the Capacity Spectrum Method. A comparison of RHA and NSA procedures shows that there can be a significant difference in local design response even though the target deformation values at the control node are close. Moreover, the difference between the mean peak RHA response and the pushover response is not independent of the material model.
Engineering Structures 27 (2005) 1014–1023 www.elsevier.com/locate/engstruct Material modelling in the seismic response analysis for the design of RC framed structures Pankaj Pankaj∗, Ermiao Lin School of Engineering and Electronics, The University of Edinburgh, Edinburgh, UK Received 14 June 2004; received in revised form February 2005; accepted February 2005 Available online March 2005 Abstract Two similar continuum plasticity material models are used to examine the influence of material modelling on the seismic response of reinforced concrete frame structures In the first model reinforced concrete is modelled as a homogenised material using an isotropic Drucker–Prager yield criterion In the second model, also based on the Drucker–Prager criterion, concrete and reinforcement are included separately While the latter considers strain softening in tension the former does not The seismic input is provided using the Eurocode elastic spectrum and five compatible acceleration histories The results show that the design response from response history analyses (RHAs) is significantly different for the two models The influence of compression hardening and strength enhancement with strain rate is also examined for the two models It is found that the effect of these parameters is relatively small In recent years there has been considerable research in nonlinear static analysis (NSA) or pushover procedures for seismic design The NSA response is frequently compared with that obtained using RHA, which also uses the same material models, to verify the accuracy of the static procedure A number of features exhibited by reinforced concrete during dynamic or cyclic loading cannot be easily included in a static procedure The design NSA and RHA responses for the two material models are compared The NSA procedures considered are the Displacement Coefficient Method and the Capacity Spectrum Method A comparison of RHA and NSA procedures shows that there can be a significant difference in local design response even though the target deformation values at the control node are close Moreover, the difference between the mean peak RHA response and the pushover response is not independent of the material model © 2005 Elsevier Ltd All rights reserved Keywords: Seismic design; Continuum plasticity; Response history analysis; Pushover methods Introduction Economic considerations and the seismic design philosophy dictate that building structures be able to resist major earthquakes without collapse but with some structural damage Therefore it is imperative that seismic design is based on nonlinear analysis of structures For the nonlinear analysis of reinforced concrete structures a variety of models have been considered [1,2] These include: linear elasticfracture models; hypoelastic models; continuum plasticity models; hysteretic plastic and degrading stiffness models; ∗ Corresponding address: School of Engineering and Electronics, The University of Edinburgh, Alexander Graham Bell Building, Edinburgh EH9 3JL, UK Tel.: +44 131 6505800; fax: +44 131 6506781 E-mail address: Pankaj@ed.ac.uk (P Pankaj) 0141-0296/$ - see front matter © 2005 Elsevier Ltd All rights reserved doi:10.1016/j.engstruct.2005.02.003 and continuum damage models The most commonly used models for RC frame structures are hysteretic plastic and degrading stiffness models [e.g [3,4]] Numerical simulation of the behaviour of plain and reinforced concrete using continuum plasticity models has been a subject of intense research and the past two decades have seen the development of a plethora of diverse mathematical models for use with finite element analyses [5–9] Most of these models have been validated and used for static (or slow cyclic) analyses and there is little evidence of continuum plasticity models finding a place in the seismic analysis of framed structures This paper examines the influence of two similar continuum plasticity models, the Drucker–Prager (DP) model and the Concrete Damaged Plasticity (CDP) model, on the analytical seismic response of a framed structure While both these models are P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 1015 Fig The four-storey frame used: (a) dimension; (b) beam cross-section; (c) column cross-section essentially based on the Drucker–Prager yield criterion [10], the latter is capable of incorporating complex features such as strain softening in tension, hardening in compression and stiffness degradation The influence of material modelling on seismic response was considered earlier briefly by the authors [11] and in this paper this influence is examined in detail for a simple reinforced concrete plane frame Nonlinear response history analysis for several possible ground motions, as prescribed by a number of codes, makes seismic design of structures very complicated As a result, there has been considerable research to develop displacement based nonlinear static analysis (NSA) or pushover procedures that can provide seismic design values NSA response is frequently compared with that obtained using response history analysis (RHA), which also uses the same material models, to verify the accuracy of the static procedure A number of features exhibited by reinforced concrete during dynamic or cyclic loading (e.g progressive degradation with each cycle of loading, influence of strain rate) cannot be easily included in a static procedure Therefore it is important to examine whether the difference between the design RHA and NSA response is influenced by the choice of material models In other words, the hypothesis that the comparison between a given NSA and RHA procedure will show similar trends for different material models needs to be tested Displacement-based NSA procedures exist in several codes and guidelines in one form or the other [12–15] The existing nonlinear static techniques can be broadly divided into two categories: Displacement Coefficient Method (DCM) [13,14,16,17] and Capacity Spectrum Method (CSM) [15,18–20] The common feature of these techniques is that appropriately distributed lateral forces are applied along the height of the building, and then monotonically increased with a displacement control until a certain deformation is reached The key difference between the CSM and DCM procedures is that the former usually requires formulation in an acceleration–displacement format Theoretically, for a general nonlinear multiple degrees of freedom system, the peak seismic response (required for design) can only be approximated by a static procedure There has been considerable research directed towards improving pushover procedures so they can reflect various aspects of a nonlinear dynamic analysis For example, Chopra and Goel [16,17] proposed a modal pushover procedure to include contribution of higher modes Chopra and Goel [18] provided a method to determine a capacitydemand diagram, in which the displacement demand was determined by analysing inelastic systems in place of equivalent linear systems The suggested method used the constant-ductility design spectra and was shown to be an improvement over the ATC-40 [15] procedures Farfaj and co-workers [19,20] extended the CSM procedure to include cumulative damage and called the method N2 The method has been shown to be a significant improvement over CSM and in many studies N2 is referred as a method distinct from CSM This paper examines this difference between the design RHA and NSA response for both DCM and CSM procedures for a simple frame The test structure and material modelling The test structure used to evaluate the influence of material modelling was a single-bay, four-storey frame The reinforced concrete members were modelled using Drucker–Prager plasticity and concrete damaged plasticity In each case a number of variations were considered 2.1 The test structure The test structure is shown in Fig The total mass including live load for the frame is 97 000 kg The columns were assumed fixed at the base A damping ratio of 5% was assumed The finite element model used two-node cubic beam elements The finite element mesh comprised of four elements (for two columns) in each storey and four elements representing beams at each floor level 1016 P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 2.2 The Drucker–Prager (DP) model The Drucker–Prager criterion [10] is an approximation of the Mohr–Coulomb criterion In the principal stress space the Mohr–Coulomb criterion is an irregular hexagonal pyramid [21] Points of singularity at the intersections between the surfaces of the pyramid can cause computational difficulties, although algorithms exist to overcome these [22] The Drucker–Prager criterion, on the other hand, is a smooth circular cone in principal stress space In the DP model considered in this study, reinforced concrete was treated as a homogenized continuum The criterion is pressure sensitive, which is an important feature of materials like reinforced concrete that have varying yield strengths in tension and compression The Drucker–Prager criterion uses the cohesion and friction angle as parameters to define yield Cohesion can be determined from compressive, tensile or shear tests The advantage of using a simple two-parameter model is that it provides computational transparency The properties used with the DP model are given in Table The friction angle β is based on the study by Lowes [7] Table Material properties used with the DP model Young’s modulus of reinforced concrete, E Poisson’s ratio of reinforced concrete, ν Friction angle, β Compressive yield strength, f c 28.6 × 109 N/m2 0.15 15◦ 20.86 × 106 N/m2 The model was used with both perfect plasticity and hardening plasticity For hardening plasticity the hardening modulus Hc = 0.05E, which is similar to some other studies [e.g [23]], was assumed In this model for perfect plasticity (PP) the yield surface remains unchanged with increasing plastic strain For hardening plasticity the yield surface expands isotropically No strain softening is assumed for this model To examine the influence of strain rate on dynamic response the strength amplification results of Bischoff and Perry [24] were used The authors compiled a range of tests conducted by different investigators and plotted the ratio of dynamic compressive strength to static strength against logarithm of the strain rate They found that there was no clear increase in strength up to a strain rate of about 5×10−5 At higher strain rates the strength increases linearly on the above-mentioned log-linear graph In this study the variation of strain rate was taken as shown in Fig This is similar to the upper limit suggested by Bischoff and Perry [24] 2.3 The concrete damaged plasticity model The Concrete Damaged Plasticity (CDP) model used is due to Hibbitt, Karlsson and Sorensen [8] In this study the concrete damaged plasticity was used to model concrete and the reinforcement was modelled separately Fig Assumed dynamic strength amplification using rebar elements that employed metal plasticity The CDP model is applicable for monotonic, cyclic and dynamic loading The yield criterion is based on the work by Lee and Fenves [5] and Lubliner et al [6] In biaxial compression, the criterion reduces to the Drucker–Prager criterion The material model uses two concepts, isotropic damaged elasticity in association with isotropic tensile and compressive plasticity, to represent the inelastic behaviour of concrete Both tensile cracking and compressive crushing are included in this model This means the evolution of the yield surface is controlled by both compression and tension yield parameters In the elastic regime, the response is linear Beyond the failure stress in tension, the formation of microcracks is represented macroscopically with a softening stress–strain response, which induces strain localisation The post-failure behaviour for direct straining is modelled using tension stiffening, which also allows for the effects of the reinforcement interaction with concrete In compression the model permits strain hardening prior to strain softening Thus, this material model reflects the key characteristics of concrete well The interaction of the rebar and concrete, such as bond slip, is modelled through concrete’s tension stiffening, which can simulate the load transferred across cracks through the rebar The rebar within the concrete element is defined by the fractional distances along the axes in the cross section of the element In this study, only longitudinal reinforcement was included Bars were assumed to be elastic-perfectly plastic To avoid excessive dissimilarity from the DP model discussed, strain softening in compression and stiffness degradation were not included The material properties that remain unchanged in this model are given in Table In compression either perfect plasticity or hardening plasticity was assumed For hardening plasticity the hardening P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 1017 Table Material properties used with the CDP model Young’s modulus of concrete, E c Young’s modulus of reinforcement, E s Poisson’s ratio of concrete, ν Dilation angle, ψ Ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress, σb0 /σc0 Ratio of the second stress invariant on the tensile meridian to that on the compressive meridian, K c Compressive yield strength, f c Initial tensile crack stress, σt1 Yield stress for reinforcement, f y 28.6 × 109 N/m2 20 × 1010 N/m2 0.15 15◦ 1.16 2/3 20.86 × 106 N/m2 1.78 × 106 N/m2 460 × 106 N/m2 modulus Hc = 0.05E c (for concrete) was assumed No strain softening is assumed in compression Although it is now well recognised that strain softening is not a material property and the strain softening modulus has mesh (or element) size dependence [e.g [25]]; for simplicity, a constant strain softening modulus in tension of HT = −0.122E c was assumed for all CDP analyses The influence of strain rate was also considered and included as discussed for the DP model Earthquake loading In this study the seismic excitation is prescribed using the elastic design spectrum of Eurocode [12] corresponding to Soil Subclass B (limits of the constant spectral acceleration branch TB = 0.15 s and TC = 0.60 s respectively) were taken with 5% critical damping and amplification factor of 2.5 The peak ground acceleration used was 0.3g The pushover analysis procedures adopted use this spectrum directly For response history analyses, to avoid the peculiarity of a particular time history, five compatible time histories are used as suggested by Eurocode For the generation of time histories, the program developed by Basu et al [26] was used The algorithm uses a target spectrum or design spectrum that is defined using straight lines on a tripartite plot The algorithm makes use of modulated filtered stationary white noise to produce an artificial accelerogram It begins with a random number generator and the amplitudes are continuously modified in the iterative process The artificially generated accelerograms have a clear rise phase, a strong motion phase and a decay phase Five acceleration time histories (called V, W, X, Y and Z) were generated A typical simulated earthquake ground acceleration history is shown in Fig 3(a) The response spectrum of this generated acceleration history is compared with the design spectrum of Eurocode in Fig 3(b) For convenience, the elastic design spectrum is normalised with respect to the peak ground acceleration The computation of the response spectrum from acceleration histories was conducted at 159 periods At each period the ratio of the computed pseudo-acceleration (spectral acceleration value Fig (a) A typical generated acceleration history and (b) its compatibility to the design spectrum from the response spectrum of the acceleration history) and the target value (spectral acceleration corresponding to the elastic design spectrum) was obtained The statistics of these spectral ratios shows that the response spectra of simulated histories match the target spectrum well All generated histories were also checked to ensure that they satisfy the requirements of Eurocode Analytical methods The RHAs were conducted using an implicit integration approach [8] The acceleration time history was generated at 0.01 s intervals, but the integration scheme provides an automatic time step adjustment based on a half step residual concept [27] A single parameter operator [28] with controllable numerical damping is used to remove high frequency noise, due to time step change [29], through the introduction of numerical damping As discussed, two pushover analysis techniques are used The DCM approach was based on FEMA 273 [13] FEMA 273 recommends that two different loading patterns be 1018 P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 Fig Top displacement history in the DP structure subjected to ground motion V considered However, in this study the loading is applied according to the first mode pattern only FEMA 273 does not provide a clear methodology for the determination of yield displacement and strength from the pushover curve The bilinear curve determined from the pushover curve is often sensitive to the target displacement This has been recognised in FEMA 274 [14] In this study an iterative process was used to evaluate the yield values Since the process is load controlled, it is often necessary to use the Riks procedure [8] to avoid problems with convergence The CSM procedure adopted is numerical (rather than graphical) based on the studies of Fajfar [20], Chopra and Goel [18] and Vidic et al [30] Influence of material modelling on dynamic response 5.1 DP material model Typical responses of the frame for excitation history V are shown in Figs and In these figures HP denotes hardening plasticity and PP denotes perfect plasticity The value ‘0’ indicates that strain rate effects are not included, while ‘001’ indicates that they are The figures show that inclusion or exclusion of strain rate or hardening makes little difference to the overall frequency content of the response However, for this model the amplitude quantities for different cases appear to suffer an influence, albeit this is not significant The values of typical peak responses were examined for all time histories The peak top deformation (Table 3) shows, as one would expect intuitively, the inclusion of strain rate effect on strength reduces the peak deformation Further, the peak value is influenced more significantly for some time histories than for others The response to excitation Z shows a 23% difference due to strain rate On the other hand the difference is only about 2% for excitation W This Fig Base shear history in the DP structure subjected to ground motion V indicates that the response induced by the peculiar nature of a time history can sometimes cause a strain rate that is sufficiently significant to affect peak response The influence of the hardening parameter also varies significantly from one excitation to another The maximum variation due to the hardening parameter is for excitation V It is interesting to note that in the dynamic environment hardening can cause either an increase or decrease in the peak deformation response Comparing the peak responses from different excitation histories with the mean values shows the largest difference for the case DP-PP with strain rate effect included for the excitation history X In general, the peak values vary far more significantly when different spectrum compatible time histories are used than due to inclusion of hardening or strain rate Table The peak top deformation (m) in the DP structure Model Strain rate included Earthquake history Mean V W X Y Z DP-HP No Yes 0.20 0.17 0.15 0.15 0.26 0.24 0.18 0.17 0.21 0.16 0.20 0.18 DP-PP No Yes 0.18 0.16 0.14 0.14 0.27 0.24 0.18 0.15 0.21 0.17 0.20 0.17 The peak base shear variations were also examined (Table 4) and show that the variation of base shears for different histories is not as significant as top deformation The inclusion of hardening generally tends to increase the base shear, as does the inclusion of strain rate effect Examining the local parameter — moment at a base node again showed a significant influence of strain rate and hardening parameter for some excitation histories P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 1019 Table The peak base shear (kN) in the DP structure Model Strain rate included Earthquake history Mean V W X Y Z DP-HP No Yes 236 262 219 237 239 267 230 205 254 271 236 249 DP-PP No Yes 225 226 226 206 204 238 215 195 239 259 222 225 Fig Base shear history in the CDP structure subjected to ground motion V Fig Top deformation history in the CDP structure subjected to ground motion V 5.2 CDP material model Some typical responses of the structure subjected to excitation V and modelled using CDP are shown in Figs and The nomenclature used in these figures is similar to that used earlier, i.e HP and PP stand for hardening and perfect plasticity respectively; ‘0’ and ‘001’ indicate exclusion and inclusion of strain rate effects respectively For this model the response histories show that there is negligible influence of hardening parameter or strain rate on the design parameters Once again the peak values of various response quantities were examined For example Table lists the peak top deformations From Table it can be seen that there is little influence of strain rate for any of the five earthquakes Comparing the response between the hardening and perfect plasticity, it can be seen that the differences are again small with maximum for earthquake Y (∼5%) The major difference in the peak response is again due to different excitation histories For example the top deformation of earthquake history X is around 28% higher than the mean peak value The analysis showed that the peak strain rate during seismic excitation was around 0.004 per second However, this did not appear to influence the peak response significantly Similarly it can be seen that the influence of strain rate on base shear (Table 6) is small for different earthquake histories with the maximum of around 4% The influence of hardening parameter is even smaller Interestingly, the base shear values did not vary significantly for different earthquake histories The maximum variation was found to be around 8% from the mean This indicates that earthquake excitation histories have larger influence on top deformation than on base shear This is clearly due to the generally flat load–displacement response in the post-elastic range Table The peak top deformation (m) in the structure modelled using CDP Model Strain rate included Earthquake history Mean V W X Y Z CDP-HP No Yes 0.18 0.18 0.17 0.17 0.27 0.26 0.18 0.17 0.25 0.24 0.21 0.21 CDP-PP No Yes 0.18 0.18 0.17 0.17 0.27 0.26 0.17 0.16 0.25 0.25 0.21 0.21 Table The peak base shear (kN) in the structure modelled using CDP Model Strain rate included Earthquake history Mean V W X Y Z CDP-HP No Yes 122 125 128 129 124 125 119 123 136 138 126 128 CDP-PP No Yes 121 125 125 127 122 125 118 122 134 137 124 127 The response of a local parameter, namely the peak moment at a base node (not shown), indicated a slightly higher variation due to the strain rate effect (maximum ∼9%), 1020 P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 Fig Top deformation history for different material models for excitation V Fig Base shear history for different material models for excitation V but the influence of the hardening parameter was still found to be small (maximum ∼4%) The above results show that the hardening parameter and strain rate effects as used in this study have little influence on the peak response for the CDP model 5.3 Comparison of response for CDP and DP material models In this section, the response of the frame structure when modelled using CDP and DP is compared It should be noted that both the models are based on the Drucker–Prager criterion Although CDP and DP models come into play only in the post-elastic domain, it is important to realise that the two models are slightly different even in the elastic domain — the CDP model includes reinforcement bars separately whilst the DP model does not As a result the CDP model has slightly higher natural frequencies Figs 8–10 show the variation of typical responses for the two material models For ease in comparison, strain rate effects have not been included These figures show that the response histories can be significantly different when two different material models are used It is also interesting to see that the peaks and troughs for the two models are similarly located It can be seen that the direction of the peak response can be different for the two models For example, the maximum top deformation in the DP model is positive whilst the same quantity for the CDP model is negative (Fig 8) The peak values also occur at different times Time history of the internal force responses shown in Fig (base shear) and Fig 10 (moment at a base node) are consistently smaller for the CDP model This is apparently because of strain softening included in the CDP model Comparing the mean peak values from the five earthquakes for the two material models, it can be seen that the mean top deformations (Tables and 5) are not significantly different; on the other hand, the base shear values (Tables and 6) Fig 10 History of moment at a base node for different material models for excitation V are almost half for the CDP models when compared to the DP models Thus the mean reflects what is observed for the excitation V in Figs and Even more dramatic variation is seen for the mean value of the moment at the base node The peak moment response from the Drucker–Prager model is about two and half times the value from the CDP model The low internal force peak responses from the CDP model are clearly due to strain softening in tension Performance of pushover procedures for different material models The performance of pushover analysis procedures is generally evaluated against response history analysis Clearly for both analysis procedures the same material model is used Thus the inherent assumption made is that if the two procedures compare well for a given material model they would so for another In this section the P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 1021 Table Peak responses from RHA, DCM and CSM for CDP and DP structures Response quantity CDP-HP RHA DCM CSM DP-HP RHA DCM CSM Deformation (m) floor Deformation (m) floor Deformation (m) floor Deformation (m) floor Base shear (kN) Moment base node (kN m) 0.209 0.181 0.130 0.061 126 132 0.206 0.173 0.121 0.055 98 69 0.201 0.168 0.116 0.051 97 70 0.199 0.145 0.086 0.032 236 358 0.182 0.134 0.079 0.028 166 329 0.179 0.131 0.077 0.027 166 328 pushover analysis procedures are evaluated with respect to response history analysis for different material models The motivation is to examine how these nonlinear static procedures perform without the inclusion of cyclic loading presented in a real seismic situation for different material models Both CDP and DP material models are considered For both models only the hardening plasticity cases are included Once again the four-storey single-bay frame discussed earlier is used Using pushover procedures the target displacement was obtained for both DCM and CSM procedures These are given in Table (deformation floor 4) along with the peak deformation obtained from RHA The RHA values are the mean of the peak deformation values from the five earthquake motions It can be seen that the target displacement from pushover procedures match the RHA values very well, more so for the CDP model than for the DP model In general the pushover values are slightly lower than the RHA values For pushover procedures the monotonically increasing lateral forces were applied based on the fundamental mode In Table typical responses for the pushover procedures are compared with the mean peak RHA values for some typical response quantities It can be seen that while the top deformation values from DCM and CSM match the RHA values closely, the error increases for deformation in lower floors for both CDP and DP structures The base shear values are underestimated by the pushover procedures by around 22% for the CDP structure and by about 30% for the DP structure The moment for a node at the base of the frame is underestimated by about 47% and 8% respectively The variation of inter-storey drifts is shown in Figs 11 and 12 It may be noted that for RHA, the drifts are not evaluated from the peak deformations, but from the peak of the time-wise variation of drifts It can be seen that the pushover procedures underestimate the drift of the lowest storey and overestimate the drifts of other storeys for the CDP model However, for the DP model the drifts are underestimated for all storeys by the pushover procedures Thus the difference in results between RHA and pushover response is not similar for the two material models These comparisons between the design response obtained using RHA and pushover analysis procedures show two important features Firstly, they show that for a given Fig 11 Heightwise variation of storey drifts for CDP-HP structure Fig 12 Heightwise variation of storey drifts for DP-HP structure material model the two design responses can be significantly different Improvement of pushover procedures so that they can accurately calculate the design response for a dynamic problem has been a subject of active research in the past decade The fact that some of the design quantities differ significantly from the RHA responses even when the evaluation of the top displacement response is relatively accurate can be partly attributed to the choice of the loading 1022 P Pankaj, E Lin / Engineering Structures 27 (2005) 1014–1023 pattern that assumes that the response is controlled by the fundamental mode even in the post-elastic regime This is consistent with previous findings [16] The second and perhaps a more interesting feature demonstrated by the results is that the difference between pushover and RHA response is not independent of the material model In other words this means that even if pushover and RHA responses closely match for a particular material model they may be different for another The cause of these relative differences can be understood by examining the two material models used in this study The DP model essentially behaves like a bilinear force–deformation model of the kind used in previous pushover studies [16,17] During monotonically increasing lateral loading of a pushover analysis both branches of hysteretic force–deformation relationship are utilized in a manner not too different from a cyclic loading situation Thus a simple model of this kind is more likely to provide a better match between the pushover and RHA response as the key attributes of the model are captured by the pushover procedure Indeed examining Fig 12 it can be seen that the drift trend for RHA and pushover procedures are similar along the height In fact the difference is largely due to the target deformation that is underestimated by the pushover procedures (Table 7) On the other hand the CDP model presents attributes that cannot be captured by the pushover procedures used In this model, while the reinforcement behaves in a bilinear manner concrete does not During a loading cycle elements undergo compression hardening on one face and tensile strength degradation on the other, followed by tensile degradation and hardening on respective faces These complex attributes of the model are only available in a cyclic loading regime and not in a monotonically increasing lateral load procedure As a result the trend for storey drift for RHA and pushover procedures can be seen to be different along the height in Fig 11 even though the target displacements are close Conclusions This simple study shows that the influence of strain rate on the seismic analysis of reinforced concrete structures is small The inclusion of a small value of hardening parameter has negligible influence on the RHA response for the CDP model and a small influence for the DP model For a given material model the peak RHA response from different excitation histories causes significantly larger variation than does inclusion or exclusion of compression hardening and strain rate parameters However, when the RHA response of the two material models is compared a significant difference is observed In the CDP model reinforcement is included separately and it also includes strain softening in tension, while the DP model treats reinforced concrete as a homogenized continuum It is found that although the peak deformation response (represented by the mean peak RHA values) is fairly close, the internal force peak response from CDP is significantly lower than that obtained from DP A comparison of RHA response with that obtained using DCM and CSM procedures shows that there can be a significant difference in the internal force response between dynamic and static procedures even though the target deformation values at the control node match Moreover, the difference between the mean peak RHA response and the pushover response is not independent of the material model, i.e the static and dynamic procedures can yield similar values for one material model and fairly dissimilar values for another Notation The following abbreviations and symbols have been used in this paper: CDP CSM DCM DP HP PP E Ec Es fc fy HC HT Kc NSA RHA β ν ψ σb0 /σc0 σt Concrete damaged plasticity (model) Capacity spectrum method Displacement coefficient method Drucker–Prager (model) Hardening plasticity Perfect plasticity Young’s modulus of reinforced concrete (for homogenised DP model) Young’s modulus of concrete (for CDP model) Young’s modulus of reinforcement (for CDP model) Compressive yield strength of concrete Yield stress for reinforcement (for CDP model) Hardening modulus Softening modulus for concrete in tension (for CDP model) Ratio of the second stress invariant on the tensile meridian to that on the compressive meridian for concrete (for CDP model) Nonlinear static analysis Response history analysis Friction angle Poisson’s ratio of concrete Dilation angle Ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress (for CDP model) Initial tensile crack stress (for CDP model) References [1] CEB Behaviour and analysis of reinforced concrete structures under alternate actions inducing inelastic response – vol 1, General models Bull d’ Inf CEB, 210, Lausanne, 1991 [2] Penelis GG, Kappos AJ Earthquake-resistant concrete structures E&FN Spon; 1997 [3] Takeda T, Sozen MA, Nielsen NN Reinforced concrete response to simulated earthquakes J Struct Eng Div, ASCE 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[10], the latter is capable of incorporating complex features such as strain softening in tension, hardening in compression and stiffness degradation The influence of material modelling on seismic