Material Modelling In The Seismic Response Analysis For The Design Of Rc Framed Structures

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.

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 3 February 2005; accepted 3 February 2005

Available online 8 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 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

© 2005 Elsevier Ltd All rights reserved

Keywords: Seismic design; Continuum plasticity; Response history analysis; Pushover methods

1 Introduction

Economic considerations and the seismic design

philos-ophy dictate that building structures be able to resist major

earthquakes without collapse but with some structural

dam-age Therefore it is imperative that seismic design is based

on nonlinear analysis of structures For the nonlinear

anal-ysis of reinforced concrete structures a variety of models

have been considered [1,2] These include: linear

elastic-fracture 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

Fig 1 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 capacity-demand diagram, in which the displacement capacity-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

2 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 1 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

of the Mohr–Coulomb criterion In the principal stress space

the Mohr–Coulomb criterion is an irregular hexagonal

pyra-mid [21] Points of singularity at the intersections between

the surfaces of the pyramid can cause computational

difficul-ties, although algorithms exist to overcome these [22] The

Drucker–Prager criterion, on the other hand, is a smooth

cir-cular cone in principal stress space In the DP model

con-sidered 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

cohe-sion and friction angle as parameters to define yield

Cohe-sion 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

proper-ties used with the DP model are given inTable 1 The

fric-tion angleβ is based on the study by Lowes [7]

Table 1

Material properties used with the DP model

Young’s modulus of reinforced concrete, E 28.6 × 109 N/m2

Poisson’s ratio of reinforced concrete,ν 0.15

Compressive yield strength, f c 20.86 × 106 N/m2

The model was used with both perfect plasticity and

hardening plasticity For hardening plasticity the hardening

modulus H c = 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 inFig 2 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 2 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 micro-cracks 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 inTable 2

In compression either perfect plasticity or hardening plas-ticity was assumed For hardening plasplas-ticity the hardening

Table 2

Material properties used with the CDP model

Young’s modulus of concrete, E c 28.6 × 109 N/m2

Young’s modulus of reinforcement, E s 20 × 10 10 N/m2

Poisson’s ratio of concrete,ν 0.15

Ratio of initial equibiaxial compressive yield stress

to initial uniaxial compressive yield stress,σb0 /σc0

1.16 Ratio of the second stress invariant on the tensile

meridian to that on the compressive meridian, K c

2/3

Compressive yield strength, f c 20.86 × 106 N/m2

Initial tensile crack stress,σt 1 1.78 × 106 N/m2

Yield stress for reinforcement, f y 460 × 10 6 N/m2

modulus H c = 0.05Ec (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

ele-ment) size dependence [e.g [25]]; for simplicity, a constant

strain softening modulus in tension of H T = −0.122Ecwas

assumed for all CDP analyses The influence of strain rate

was also considered and included as discussed for the DP

model

3 Earthquake loading

In this study the seismic excitation is prescribed using the

elastic design spectrum of Eurocode 8 [12] corresponding to

Soil Subclass B (limits of the constant spectral acceleration

branch T B = 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 8 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 8 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 3 (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 8

4 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

Fig 4 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]

5 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 4 and 5 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 5 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 3 The peak top deformation (m) in the DP structure Model Strain rate

included

Earthquake history Mean

DP-HP No 0.20 0.15 0.26 0.18 0.21 0.20

Yes 0.17 0.15 0.24 0.17 0.16 0.18 DP-PP No 0.18 0.14 0.27 0.18 0.21 0.20

Yes 0.16 0.14 0.24 0.15 0.17 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

Table 4

The peak base shear (kN) in the DP structure

Model Strain rate

included

Earthquake history Mean

Fig 6 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 inFigs 6

and 7 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 5 lists the peak top

deformations From Table 5 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

Fig 7 Base shear history in the CDP structure subjected to ground motion V.

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 5 The peak top deformation (m) in the structure modelled using CDP Model Strain rate

included

Earthquake history Mean

CDP-HP No 0.18 0.17 0.27 0.18 0.25 0.21

Yes 0.18 0.17 0.26 0.17 0.24 0.21 CDP-PP No 0.18 0.17 0.27 0.17 0.25 0.21

Yes 0.18 0.17 0.26 0.16 0.25 0.21

Table 6 The peak base shear (kN) in the structure modelled using CDP Model Strain rate

included

Earthquake history Mean

The response of a local parameter, namely the peak mo-ment at a base node (not shown), indicated a slightly higher variation due to the strain rate effect (maximum ∼9%),

Fig 8 Top deformation 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 9(base shear) andFig 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 3and5) are not significantly different;

on the other hand, the base shear values (Tables 4and 6)

Fig 9 Base shear history for different material models for excitation V.

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 inFigs 8and9 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

6 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 do so for another In this section the

Table 7

Peak responses from RHA, DCM and CSM for CDP and DP structures

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 7 (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 7 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 inFigs 11

and12 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

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

examiningFig 12it 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 11even though the target displacements are close

7 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

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

8 Notation

The following abbreviations and symbols have been used

in this paper:

CDP Concrete damaged plasticity (model) CSM Capacity spectrum method

DCM Displacement coefficient method

DP Drucker–Prager (model)

HP Hardening plasticity

PP Perfect plasticity

E Young’s modulus of reinforced concrete (for

homogenised DP model)

E c Young’s modulus of concrete (for CDP model)

E s Young’s modulus of reinforcement (for CDP

model)

f c Compressive yield strength of concrete

f y Yield stress for reinforcement (for CDP model)

H C Hardening modulus

H T Softening modulus for concrete in tension (for CDP

model)

K c Ratio of the second stress invariant on the tensile

meridian to that on the compressive meridian for concrete (for CDP model)

NSA Nonlinear static analysis RHA Response history analysis

β Friction angle

ν Poisson’s ratio of concrete

ψ Dilation angle

σ b0 /σ c0 Ratio of initial equibiaxial compressive yield stress

to initial uniaxial compressive yield stress (for CDP model)

σ t 1 Initial tensile crack stress (for CDP model)

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