57 Borzi, Calvi, Faccioli, Bommer

Inelastic spectra for displacement-based seismic design

B. Borzia,1, G.M. Calvib, A.S. Elnashaia,*, E. Facciolic, J.J. Bommera

aDepartment of Civil and Environmental Engineering, Imperial College, London SW7 2BU, UKbDipartimento di Meccanica Strutturale, UniversitaÂ degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy

cDipartimento di Ingegneria Strutturale, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milan, Italy

Accepted 6 September 2000

Abstract

In recognition of the emergence of displacement-based seismic design as a potentially more rational approach than force-based techniques,

this paper addresses derivation of inelastic displacement spectra and associated topics. A well-constrained earthquake strong-motion dataset

is used to derive inelastic displacement spectra, displacement reduction factors and ductility±damping relationships. These are in a format

amenable for use in design and assessment of structures with a wide range of response characteristics. q 2001 Elsevier Science Ltd. All

rights reserved.

Keywords: Attenuation relationship; Displacement spectrum; Displacement-based approach; Ductility; Damping; Displacement modi®cation factor

1. Introduction

Force-based seismic design remains, in spite of its short-

comings, the method widely used in codes. However, defor-

mation-based (Calvi and Kingsley [1], Kowalsky et al. [2],

Priestley et al. [3]) or deformation-controlled (Panagiotakos

and Fardis [4]) procedures have recently been proposed. The

new approach utilises displacement spectra expressed as

spectral ordinates versus effective period (period at maxi-

mum displacement) to quantify the demand imposed on

structural systems.

The origins of displacement-based design may be traced

to work published as early as the 1960s, where comments on

the displacements of inelastic systems and their relationship

to their elastic counterparts were made (e.g. Muto et al.,

1960, as reported by Moehle [5]). However, it was the

work of Sozen and his associates (Gulkan and Sozen [6],

Shibata and Sozen [7]) that developed the concept of a

substitute structure. The substitute structure is a single

degree of freedom elastic system, the characteristics of

which represent the inelastic system. This concept enables

the use of an elastic displacement spectrum in design, while

using the displacement capacity of an inelastic system.

Various contributions were made towards the develop-

ment of displacement-based seismic design since the early

work mentioned above. However, it was in the 1990s that

formal proposals were made to implement the emerging

ideas into a design procedure. The earliest is that by Moehle

and his co-workers (Moehle [5], Qi and Moehle [8]). There-

after, a complete and workable procedure for seismic design

of structures that sets aside forces and relies on displace-

ment as a primary design quantity was proposed by

Kowalsky et al. [2] for single degree of freedom systems

(such as bridge piers). A parallel paper on multi-degree of

freedom systems is due to Calvi and Kingsley [1]. The steps

comprising the design process for SDOF systems are given

below for simplicity:

a. A target displacement for the structure is selected, based

on the type of structure and the governing limit state.

b. Knowing the yield displacement, and the material and

structural system, a value of equivalent damping is deter-

mined.

c. Displacement spectra representative of the seismo-

tectonic environment are used. The inputs are the target

displacement and the equivalent damping. The output is

an effective period of vibration.

d. The structure is dimensioned to give an effective period,

taking into account reduced stiffness consistent with the

level of deformation, equal to that obtained from the displa-

cement spectra.

e. If the effective period is not suf®ciently close to the

required period, go to step b. above and repeat until

convergence.

Soil Dynamics and Earthquake Engineering 21 (2001) 47±61

0267-7261/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.

PII: S0267-7261(00)00075-0

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* Corresponding author. Tel.: 144-20-7594-6058; fax: 144-20-7225-

2716.

E-mail address: [email protected] (A.S. Elnashai).1 Present address: EQE International Ltd, Warrington WA3 6WJ, UK.

It is clear from the above that whereas displacement-

based design is the logical framework for seismic

design, since the primary source of seismic energy

dissipation is inelastic deformation, it imposes new

requirements for veri®able design. Primarily, accurate,

representative and parametrically described displace-

ment spectra are essential ingredients that have only

very recently become available (Bommer et al. [9],

Bommer and Elnashai [10], Tolis and Faccioli [11]).

In the study by Bommer et al. [9], it was shown that

a well-balanced catalogue for acceleration studies may

be unsuitable for displacement purposes. The ordinates

of acceleration response spectra are insensitive to the

processing applied to the accelerograms, whereas the

in¯uence of the ®lter parameters on the displacement

spectra is pronounced.

In this paper attenuation coef®cients for inelastic displa-

cement constant ductility spectra are calculated. Attenuation

coef®cients for residual displacements were also taken into

account. This is of great importance in the assessment of

displacement-based design situations, hence the spectra are

presented as ratio of maximum response displacement. By

means of a comparison between elastic and inelastic displa-

cement spectral ordinates a reduction coef®cient taking into

account the effects of real inelastic behaviour in a simpli®ed

method of design based on displacement is calculated. For

this coef®cient, herein denoted h -factor, regression coef®-

cients were evaluated (Borzi et al. [12], Borzi [13]) for all

magnitudes, distances and site conditions. Further, average

values obtained constitute a valid reference for new code

implementations for simpli®ed methods based on displace-

ments.

2. Input motion

2.1. Strong-motion dataset selection and processing

The dataset employed for the de®nition of inelastic

constant ductility spectra was presented by Bommer et al.

[9] and used for the derivation of frequency-dependent

attenuation equations for ordinates of displacement

response spectra. All records were ®ltered individually at

Imperial College (London, UK) and used to derive displa-

cement spectra for different levels of damping, from 5 to

30%. Each record was individually processed using an ellip-

tical ®lter and a long-period cut-off that resulted in a physi-

cally reasonable displacement time-history (Bommer and

Elnashai [10]). This ®ltering process removes permanent

displacements, caused by ground deformation or fault slip,

in order not to mix these phenomena with transitory ground

vibrations. The dataset has been adapted from that

employed by Ambraseys et al. [14,15] to derive attenuation

relationships for ordinates of acceleration response spectra.

This is a high-quality dataset in terms of both accelerograms

that have been individually corrected, and information

regarding the recording stations and earthquake character-

istics. The original dataset of Ambraseys et al. [14,15] has

been modi®ed in the following ways:

1. records for small magnitude earthquakes have been

removed;

2. some site classi®cations have been corrected in the light

of new information;

3. two new strong-motion records have been added;

4. all of the original records have been ®ltered with indivi-

dually selected cut-offs.

The necessity of eliminating all the records generated by

earthquakes of magnitude less than Ms 5.5 is due to the

particular interest in long-period radiation. In Fig. 1 the

distribution of records comprising the dataset with regard

to magnitude, distance and site classi®cation are

B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±6148

Fig. 1. Distribution of records for: (a) rock; (b) stiff; and (c) soft soil.

demonstrated to emphasise credentials of this set of natural

records.

While the source distance and the surface-wave magni-

tude are available for all the accelerograms, for three

records the local site geology is unknown. For the remaining

180, the percentages of distribution in the three site group-

ings of rock, stiff and soft soil are 25.0, 51.1 and 23.9%,

respectively. These percentages are close to the distribution

of the original dataset, being 25.5, 54.3 and 20.2%, respec-

tively. For two records only one component of the motion is

known. Therefore a total of 364 accelerograms were

processed.

It is acknowledged that site intensity could be used for the

selection of strong-motion datasets. However, this has its

own problems, and is not considered to be superior to the

selection based on magnitude, distance and site condition.

2.2. Attenuation model and regression analysis

Studies concerning the evaluation of seismic hazard

related to earthquakes utilise predictive models commonly

referred to as attenuation relationships. These models gener-

ally express values of strong-motion parameters as a func-

tion of source characteristics, propagation path and local site

geology.

In order to de®ne elastic spectra, a common approach to

perform a hazard analysis is to de®ne the hazard in terms of

the peak ground acceleration (PGA), which anchors the

zero-period ordinate for a standard spectral shape. A criti-

cism of this approach is that the resulting spectra do not

correspond to the same seismic hazard for all periods (i.e.

non-uniform hazard spectra). To obtain spectra charac-

terised according to the same seismic hazard for every spec-

tral ordinate (uniform hazard spectra) the de®nition of

period-dependent attenuation relationships are proposed.

The attenuation model used in this work is that of Ambra-

seys et al. [14,15] employed to de®ne elastic acceleration

spectra. The formulation of the attenuation relationship is:

log�y� � C1 1 C2Ms 1 C4 log�r�1 CaSa 1 CsSs 1 sP �1�where y is the strong-motion parameter to be predicted, Ms is

the surface wave magnitude and:

r ��d 2 1 h2

0

q�2�

in which d is the shortest distance from the station to the

surface projection of the fault rupture in km and h0 is a

regression constant. The h0 coef®cient takes into account

that the fault projection is not necessarily the source of

the peak motion, but it does not represent explicitly the

effect of the focal depth on the motion. The coef®cients

C1, C2, C4, Ca, Cs and h0 are determined by regression. In

this attenuation model three soil conditions are distin-

guished by the average shear wave velocity. When the

shear wave velocity is higher than 750 m/s the soil is clas-

si®ed as rock. Soft soil has a shear wave velocity is less than

360 m/s, whilst stiff soil is assumed in the intermediate

range of shear wave velocity. In the regression model Sa

takes the value of 1 for soft soil conditions and 0 otherwise,

while Ss takes the value of 1 for stiff soil conditions and 0

otherwise. Finally, P is a parameter that is multiplied by the

standard deviation s and takes the value of 0 when the mean

value of log(y) is calculated and 1 for the 84-percentile

value of log(y).

3. Structural models

3.1. Elastic perfectly-plastic model

In order to determine the in¯uence of magnitude, distance

and soil condition on inelastic response spectra, attenuation

relationships have been de®ned using an elastic perfectly-

plastic (EPP) response model. The EPP model was

employed since it is the simplest form of inelastic force-

resistance as well as being the basis for early relationships

between seismic motion and response modi®cation factors.

Moreover, by virtue of its two parameters de®nition: level of

force-resistance and stiffness, few structural characteristics

are included, hence the in¯uence of strong-motion records

may be better visualised. The stiffness corresponds to the

period of vibration for which the spectral ordinate has to be

calculated and the resistance is derived iteratively. In this

work inelastic constant ductility spectra were obtained.

Therefore the resistance of the system corresponds to the

resistance for which the system has a required ductility

equal to the target ductility. The ensuing inelastic spectra

would re¯ect solely the characteristics of the input motion.

3.2. Hysteretic hardening±softening model

In order to investigate the in¯uence of the response char-

acteristics of structures on inelastic displacement spectra, a

hysteretic hardening-softening model (HHS) was used

(Ozcebe and Saatcioglu [16]). The structural model is char-

acterised by the de®nition of a primary curve, unloading and

reloading rules. The primary curve for a hysteretic force±

displacement relationship is de®ned as the envelope curve

under cyclic load reversals. For non-degrading models the

primary curve is taken as the response curve under mono-

tonic loading. In this model the primary curve is used to

de®ne the limits for member strength. Two points on the

primary curve have to be de®ned. It is essential to de®ne

cracking and yield loads (Vcr and Vy) and the corresponding

displacements (D cr and D y), as shown in Fig. 2. If, for exam-

ple, this model was used to describe the hysteretic behaviour

of reinforced concrete members, the cracking load would

correspond to the spreading of cracks in the concrete and the

yielding load would be related with the load at which the

strain in bars is equal to the yield strain of steel or some

other criterion can be selected by the user. Unloading and

reloading branches of the HHS model have been established

through a statistical analysis of experimental data

B. Borzi et al. / Soil Dynamics and Earthquake Engineering 21 (2001) 47±61 49

(Saatcioglu and co-workers [17,18]). The load reversal rules

are brie¯y described below.

Structural members exhibit stiffness degradation under

cyclic loading. When the number of cycles or the amplitude

of inelastic deformation increases, the system becomes

softer. Furthermore, the hysteretic behaviour is affected by

pinching. The axial load is an important parameter in

predicting pinching effects (due to the onset of crack

closure). The slope of reloading branches increases beyond

the crack load.

The slopes of the lines connecting the origin to the crack-

ing point (K1 in Fig. 2) and the yield point to the cracking

point in the opposite quadrant (K2 in Fig. 2) are used to

de®ne the unloading branches under cyclic loads. The latter

slope depends on deformation and force levels attained at

the beginning of unloading. Experimental results indicate

that if unloading starts between the cracking and the yield

load, and the yield load has not been exceeded in the rele-

vant quadrant, then unloading stiffness is bounded by K1 and

K2. In this model a linear variation between these limits was

proposed as a function of displacement ductility. If the

unloading load exceeds the yield load, the unloading

curve changes the slope to a value close to the cracking load.

4. Procedural considerations

In order to de®ne inelastic displacement spectra and

displacement modi®cation (h?, displacement ductilities of

2, 3, 4 and 6 are considered. Ductility levels higher than 6

are not included because they constitute global displace-

ment ductility and structures very rarely have a local ducti-

lity supply commensurate with global ductility above 6. The

inelastic spectra have been de®ned between 0.05 and 3 s.

For the HHS model the initial elastic period can be consid-

ered either as the stiffness before Vcr or the secant stiffness.

In the current work period corresponding to the secant stiff-

ness was considered. This is because the stiffness before Vcr

is not representative of the structural behaviour. A low

damping value of 1% was included. This viscous damping

is representative of the non-hysteretic dissipation, since

hysteretic damping is already included. Sensitivity analyses

were undertaken and have indicated that further precision of

this assumption is unwarranted.

The input parameters for the HHS model described above

are the monotonic curve and the relationship between axial

compressive force and nominal concentric axial capacity. In

order to de®ne the inelastic constant ductility spectra the

magnitude of the monotonic curve is not an input parameter.

It is de®ned in an iterative way forcing the relationship

between maximum and yield displacements to satisfy the

target ductility. To obtain the inelastic spectra and displace-

ment modi®cation factor (h ) an approximation of the

primary curve with three linear branches has been assumed

(Fig. 3). Consequently, the input parameters de®ning the

shape of the primary curve are:

1. the relationship between the cracking and the yielding

load (Vcr/Vy);

2. the relationship between the stiffness before the cracking

load and the secant stiffness (Kcr/Ky);

3. the slope of the post yield branch.


Fig. 3. Shape of primary curve used in this work.

Fig. 2. HHS model for structural members.

To select the values of parameters to be employed, exten-

sive analysis of the in¯uence of each parameter on the

inelastic spectra was undertaken. The results of a parametric

investigation indicate that the parameter with the strongest

in¯uence on inelastic spectra is the slope of the post yield

branch. Hence ®xed ratios between Vcr and Vy and between

Kcr and Ky were considered. Although the aforementioned

ratios have a large variability, constant values are assumed,

since the results of preliminary parametric investigations

using a ®ne mesh of variation show that they do not have

a signi®cant in¯uence in terms of the inelastic displacement

spectra. From the experimental results of Paulay and Priest-

ley [19], Priestley et al. [20], Calvi and Pinto [21] and Pinto

[22], it is reasonable to consider the secant stiffness at the

yield point equal to 50% of the stiffness before Vcr; The

latter is taken equal to 30% of Vy. The ratio between the

cracking and yield loads in¯uences the pinching behaviour

that does not occur often for structures with loads higher

than approximately 30% of the yielding load Vy. The consid-

ered representative slopes of the structural behaviour are:

² K3 � 0 : (elastic perfect plastic behaviour)

² K3 � 10%Ky : (hardening behaviour)

² K3 � 220%Ky : (softening behaviour)

² K3 � 230%Ky : (softening behaviour)

An axial load equal to 10% of the nominal axial load is

assumed, since the model does not account for second-order

effects. The above characteristics have been veri®ed to

cover both new structures with seismic detailing, and exist-

ing poorly detailed structures (Borzi et al. [12], Borzi [13]).

An iterative procedure was utilised for the de®nition of

spectral ordinates corresponding to target ductility. In rare

cases it has not been possible to obtain a convergent solution

with the HHS model. The percentage of spectral ordinates

that have not been considered in the regression analyses are

reported in Table 1.

It is observed that the number of spectral ordinates to be

excluded from parametric analysis for the slope of the third

branch equal to 230%Ky and ductility equal to 4, is very

high. The attenuation relationship for this combination of

parameters was therefore not considered. The above obser-

vations (non-convergence) are fully justi®ed by noting that

highly degrading systems are inherently of low ductility.

This was further investigated by comparison with a different

program for inelastic spectra available to the authors

(program inspect). The lack of convergence was observed

for the same range of degrading stiffness and high ductility,

thus con®rming that these situations correspond to structural

collapse.

5. Inelastic displacement spectra

The cut-off period of every accelerogram is associated

with an initial elastic period TI, which corresponds to the

secant stiffness at yield. This is less than the cut-off periods

employed in order to ®lter the records. For inelastic systems

there is doubt as to whether spectral ordinates should corre-

spond to the equivalent elastic period TE rather than the

initial elastic period TI, both of which are less than the

cut-off period. However, it was observed that even if

equivalent elastic periods are somewhat higher than the

cut-off period of 3 s, the inelastic spectral ordinates are

still valid. This was con®rmed by comparison to selected

digital records Borzi [13].

In order to study the in¯uence of input motion para-

meters on inelastic displacement spectra, attenuation coef-

®cients were calculated for the EPP model. Fig. 4a±c

show the in¯uence of magnitude, distance and soil condi-

tions on inelastic displacement spectra. They con®rm the

strong in¯uence of input motion parameters on inelastic

displacement spectra, as already demonstrated in previous

studies for elastic displacement spectra by Bommer et al.

[9] (Fig. 5).

To compare the elastic and inelastic spectra, attenuation

coef®cients were de®ned for elastic spectra. Fig. 6 shows the

in¯uence of ductility on inelastic displacement spectra. In

this representation the initial elastic period of vibration

corresponding to the secant stiffness at the yield point is

assumed. Thus the elastic and inelastic systems with differ-

ent ductility requirements are characterised by the same

initial stiffness. The results con®rm the established observa-

tion that elastic and inelastic systems with the same initial

stiffness reach similar maximum displacements. However,

the difference between elastic and inelastic spectral ordi-

nates calculated in this work is larger than in previous

ones. This is due to the fact that the damping value used

herein is small (1% of critical). Therefore, the elastic spec-

tral ordinates for periods corresponding to the soil frequency

tend to be higher than the inelastic ones. As an example, in

the work of Miranda [23] a damping of 5% of critical was

employed. In comparing elastic and inelastic displacements,

small damping values should be employed, because the

damping must represent only the dissipation of energy not

related with inelastic behaviour. However the ratio between

inelastic and elastic displacements tend to be equal to 1 only

in the long period range, as already observed in previous

studies (Miranda [23], Gupta and Sashi [24], Whittaker et al.

[25], Rahnama and Krawinkler [26]). In the short period

range the inelastic demand exceeds the elastic one as

shown in Fig. 6b. In this work the ratio between inelastic

and elastic displacement demand of a system characterised


Table 1

Percentage of ordinates excluded from regression analysis

K3 m � 2 (%) m � 3 (%) m � 4 (%) m � 6 (%)

0 0.16 0.58 1.23 3.16

10%Ky 0.03 0.13 0.27 0.80

220%Ky 1.67 5.12 12.00 ±

230%Ky 2.45 8.16 30.98 ±

by the same initial elastic period of vibration TI is not further

investigated, because in the direct displacement-based

approach an equivalent elastic period of vibration TE is

assumed. The substitute elastic systems used in previous

work and in the current one are shown in Fig. 7a and b,

respectively. Attention is drawn to the comparison of displa-

cement reached by the system shown in Fig. 7b, as discussed

hereafter.


Fig. 5. In¯uence of: (a) magnitude; and (b) distance on elastic displacement spectra obtained by Bommer et al. (1996).

Fig. 4. In¯uence of: (a) magnitude; (b) distance; and (c) site condition on inelastic displacement spectra evaluated for the EPP model.

Inelastic displacement spectra were also obtained for the

HHS model. In the case of softening behaviour high levels

of ductility were not considered. This must be done because

from a practical point of view high ductilities are not signif-

icant for softening systems. In the case of post elastic stiff-

ness equal to 220 and 230% the secant stiffness for the

yield point ductility up to 4 and 3, respectively, were

assumed.

Fig. 8 shows the in¯uence of ductility and hysteretic

behaviour on inelastic displacement spectra. For these


Fig. 7. Elastic and inelastic systems compared in: (a) previous work; and in (b) current work.

Fig. 6. In¯uence ductility on: (a) displacement spectra; and (b) ratio between inelastic and elastic displacement spectra (EPP model, Ms � 4; d � 10 km; rock

site).

Fig. 8. In¯uence of: (a) ductility and; (b) hysteretic behaviour on inelastic displacement spectra �Ms � 6; d � 10 km; soft soil).

graphs an equivalent elastic period of vibration was

assumed. This representation renders it possible to observe

that there is an appreciable difference between elastic and

inelastic spectral ordinates, but both the ductility and the

hysteretic characteristics have negligible in¯uence. Conse-

quently, a displacement spectra derived for a certain level of

ductility and hysteretic model characterisation will be an

acceptable approximation for different values of the latter

parameters.

The standard deviations s (Eq. (1)) of the logarithm of

the calculated elastic and inelastic displacement spectra

are reported in Fig. 9. From the aforementioned ®gure it

is observed that the standard deviations are almost

constant in the whole period range. Furthermore, the

uncertainties in terms of elastic and inelastic

displacement spectral ordinates are similar. As a conse-

quence, the calculated ratios between elastic and inelastic

spectral ordinates are characterised by a lower level of

uncertainty.

Veri®cation of the derived spectra is non-trivial, due to

the dearth of information in the literature on inelastic displa-

cement spectra. However, the models and methods were

veri®ed, alongside the dataset, by comparing a sub-set of

results of this study with previously published work for

force spectra and response modi®cation factors (R or q).

Details are give elsewhere [27], where comparisons have

con®rmed the validity of the results obtained for inelastic

acceleration spectra by comparison with published ones

(Vidic et al. [28], Miranda and Bertero [29], Krawinkler

and Nassar [30], Newmark and Hall [31]).


Fig. 9. Standard deviation of the calculated displacement spectra.

6. h -factor determination

6.1. De®nition of h -factor

The displacement-based design approach herein

considered is based on an idealisation of an equivalent elas-

tic SDOF system representing the structure. Two basic char-

acteristics of structure play an important role in determining

the response to strong-ground motion: change in period of

vibration and energy dissipation capacity. Therefore the

maximum inelastic response can be interpreted in terms of

linear elastic analysis by means of a hypothetical elastic

structure (substitute structure). The substitute structure has

to be equivalent to the original system in terms of period of

vibration and amount of dissipated energy. This is achieved

by de®ning an equivalent or effective period TE and an

equivalent damping value jE. The latter may be related to

a reduction coef®cient (h) of the elastic spectral ordinates. It

was observed that the most representative period of vibra-

tion for the global response is the period corresponding to

the secant stiffness at maximum displacement (Gulkan and

Sozen [6]). As a consequence, the relationship between the

initial elastic period TI and the equivalent elastic period TE

is:

TE � TI

��m

1 1 am 2 a

r�3�

as shown in Fig. 10.

A modi®cation of the spectral ordinates is necessary to

include a measure of the energy dissipation capacity of the

structure. The equivalent elastic period TE is used to deter-

mine spectra in terms of periods of the substitute structure.

The relationship between the elastic and inelastic spectral

ordinates is shown in Fig. 11. In order to de®ne a reduction

coef®cient to transform the elastic displacement spectral

ordinates the following relationship is utilised:

h � SDEL�TE�SDIN�TE�

�4�

where SDEL and SDIN are, respectively, elastic and inelastic

spectral displacement ordinates corresponding to the

equivalent elastic period.

In the context of displacement-based design, h is the

coef®cient equivalent to the behaviour factor, q, in force-

based design. Deformation beyond the elastic range princi-

pally has the effect of increasing both the vibration period

and the energy dissipation. While both the stiffness degra-

dation and the dissipation of energy are considered in the

behaviour factor, the factor h is only a function of dissipa-

tion of energy. This is due to the fact that the increase of

vibration period is already taken into account in the de®ni-

tion of an equivalent elastic period of vibration TE. The

coef®cient h may be expressed as a function of damping

using for example the relationship given in Eurocode 8:

h ��2 1 j

7

s�5�

where h is equal to 1 when j is equal to 5%. In this work the

elastic displacement spectra are for j equal to 1%. Thus the

Eq. (5) above must be modi®ed as:

h ��2 1 j

7

s ��7

2 1 1

r�

��2 1 j

3

s�6�

The damping coef®cient may now be expressed as:

j � 3h 2 2 2 �7�In this study, coef®cients for the de®nition of h as a

function of magnitude, distance, soil condition and period

are obtained. Mean values for all ductility levels and both

response models (EPP and HHS) are also calculated in order

to obtain damping values pertaining to displacement-based

design of inelastically responding structures.

6.2. Attenuation relationships

Attenuation coef®cients for the displacement modi®ca-

tion factor (h) for EPP and HHS models were calculated.

Fig. 12 demonstrates the in¯uence of magnitude and

distance on h . The in¯uence of the above-listed parameters


Fig. 10. Relation between TE and TI.

Fig. 11. Relation between SDEL(TE) and SDIN(TE).

on h can be neglected. This is a consequence of having the

same dependence from input motion parameters that was

observed for elastic and inelastic spectra. Displacement

modi®cation factor is obtained as the ratio between elastic

and inelastic displacement spectra, the in¯uence of input

motion parameters on h -factor is therefore eliminated.

6.3. Average values

The mean values of h -factor for different periods, magni-

tudes, distance and soil conditions were evaluated. The

input motion parameters employed in this analysis are:

² magnitude in the range [5.5,7] with magnitude step of

0.5;

² distance less than 150 km and distance step 2 km;

² soil characteristics considered in the attenuation model.

Not all the combinations of the above parameters have

been used, because for earthquakes of low magnitude there

is no need to consider long distances. In order to select

magnitude and distance pairs of engineering signi®cance,

the following limits have been set:

² for magnitude less than 5.5 distances over 50 km are

excluded;

² for magnitude less than 6 distances over 75 km are

excluded;

² for magnitude less than 6.5 distances over 100 km are

excluded.

The mean values and the standard deviation of h obtained

above are calculated considering the de®nition of the reduc-

tion coef®cient given in Eq. (4). The standard deviations

calculated herein represent the dispersion of the average

values of h on the period, when magnitude, distance and

soil condition have changed. These values are presented in

Table 2.

Using Eq. (7) damping coef®cients jE corresponding to

the same parameters were evaluated. In Fig. 13 the inelastic

spectral ordinates obtained for the HHS model in the case of

elastic-perfectly plastic behaviour and the elastic spectral

ordinates for damping equal to jE are compared. In this

representation the inelastic spectral ordinates are those at

equivalent elastic periods TE. When elastic and inelastic

response displacements correspond to the same equivalent

elastic vibration period TE, different ductility requirements

correspond to different maximum displacements. Thus,

comparing the inelastic displacements mentioned above

with the elastic ones, different equivalent damping values

are obtained for various levels of ductility. The equivalent

damping value, which accounts for energy dissipation, is

obtained for post-yield cycles of a given amplitude through

the equation:

jE � EH

4pEEL

�8�

where EH is the energy dissipated in a full cycle of load

reversals and EEL the elastic strain energy. Eq. (8) can be

written for the EPP model as:

jE � a 1 21

m

� ��9�

where a is 0.64 for the EPP hysteretic model when all the

cycles of load reversals have the same amplitude up to the

target ductility. Using the results obtained from the above

procedure, values of a can be re-evaluated for a more realis-

tic de®nition of jE. Eq. (9) which relates the equivalent


Fig. 12. In¯uence of: (a) magnitude; and (b) distance on h-factor evaluated for the EPP model.

Table 2

Average (kxl) and standard deviation (s) values of h-factor

m � 2 m � 3 m � 4 m � 6

kxl s kxl s kxl s kxl s

EPP 1.83 0.07 2.23 0.13 2.45 0.18 2.65 0.23

K3 � 0 2.07 0.13 2.39 0.19 2.55 0.23 2.70 0.28

K3 � 10%Ky 1.97 0.11 2.25 0.17 2.39 0.21 2.51 0.26

K3 � 220%Ky 2.27 0.18 2.73 0.31 3.15 0.44 ± ±

K3 � 230%Ky 2.41 0.21 3.12 0.42 ± ± ± ±

damping value and the ductility factor is assumed for the

HHS model too. Therefore, differences in hysteretic beha-

viour are represented by variations in a . The equivalent

damping values jE and the corresponding a coef®cients

are reported in Table 3.

Assessment of the mean values of h lead to the following

observations:

1. Lower h is obtained with the EPP model than with the

HHS when applied to a perfectly plastic case. This is due

to the higher initial stiffness of the HHS model that uses a

secant stiffness of 50% of the initial stiffness. Therefore,

for the same maximum displacement the ductility

demand for the EPP case is twice that of HHS case,

provided the same initial stiffness is used.

2. In the hardening case lower h was obtained than in the

elasto-plastic case, because the equivalent system corre-

sponding to hardening is stiffer than that corresponding

to the elasto-plastic case.

3. In the softening case the average h value is higher than in

both the EPP and the hardening cases. This is due to the

high values of the displacement modi®cation factor for

stiff systems with softening behaviour. In terms of dissi-

pation of energy, high h values correspond to high

energy dissipation. Stiff systems tend to have a large

number of load reversals. Therefore, for softening

systems a large number of cycles of loading and reload-

ing reach the limit curve, leading to a greater dissipation

of energy.

For long period systems the mean values of h-factor

obtained above are unconservative, since they are in¯u-

enced by the high values characteristic of the response

modi®cation factor of stiff systems. In order to improve

the estimation of h -factor for long period systems, mean

values were evaluated comparing the energy of inelastic

and elastic displacement spectra, both considered for an

equivalent period TE. The displacement modi®cation factor

is thus given by the following expression:

h �

ZT MaxE

0SDEL

ZTMaxE

0SDIN

�10�

Long period ordinates become of greater importance than

short period ordinates, since the area under both the elastic

and inelastic spectra in the short period range is only a small

percentage of the total area. The results of this investigation

for the range of ductilities taken into account are given in

Tables 4 and 5.

The mean values of h found from Eq. (10) are lower than

those reported in Table 2. Therefore, the latter values are


Fig. 13. (a) Inelastic displacement spectra for different ductilities; and (b) elastic displacement spectra for equivalent viscous dampings (HHS model for K3 �0; Ms � 6; d � 10 km; soft soil).

Table 3

Equivalent damping jE and a values

m � 2 m � 3 m � 4 m � 6

jE (%) a jE (%) a jE (%) a jE (%) a

EPP 8 0.16 13 0.19 16 0.21 19 0.23

K3 � 0 11 0.22 15 0.23 18 0.23 20 0.24

K3 � 10%Ky 10 0.19 13 0.20 15 0.20 17 0.20

K3 � 220%Ky 13 0.27 20 0.31 28 0.37 ± ±

K3 � 230%Ky 15 0.31 27 0.41 ± ± ± ±

conservative as they are not in¯uenced by the behaviour of

systems in the short period range.

6.4. Period-dependent response modi®cation functions

Period-dependent h functions were evaluated. The aver-

age and the standard deviation values of h -factor obtained

changing input motion parameters, were calculated for all

periods and ductilities considered in this study. The results

of these analyses are represented in Fig. 14. The hysteretic

behaviour and the ductility have a strong in¯uence on the

value of h -factor, but this is evident only in the short period

range. For these periods h is higher for softening systems

than for perfectly elastic and hardening ones. This is due to

the fact that short period systems tend to have an increased

number of load reversals. Therefore, for softening system

hysteresis cycles reaching the post-elastic branch of the

force-displacement primary curve occur more after, thus

increasing the energy dissipation. The displacements of

both elastic and inelastic systems tend to the ground displa-

cement in the long period range, independently of the

hysteretic behaviour. In order to obtain usable values of hin the whole period range, they are expressed as a function

of period as follows:

1. a horizontal portion in the period range limited by the

equivalent elastic period in which the inelastic displace-

ment spectra reach the peak value;

2. a decaying branch in the long period range. This function

must tend to 1 when the period increases. This is reason-

able because both the elastic and inelastic displacements

converge to the peak ground displacement if the SDOF

system is in®nitely ¯exible.

The formulation of displacement modi®cation factor

proposed is:

h � �h if TE # �T

h � 1 1 � �h 2 1�� T =T� if TE . �T

(�11�

where �T is the equivalent elastic period at the border of the

increasing branch of the inelastic displacement spectra and

�h is a constant value which de®nes the displacement modi-

®cation factor up to �T : Considering the inelastic displace-

ment spectra calculated in this work, an adequate value of �T ;

for all the levels of ductilities and hysteretic behaviours

used, seems to be 2 s. Therefore the constant branch of

the functions de®ning h is operative for most practical

applications. The assumed values of �h are the mean values

calculated via Eq. (10) and reported in Table 4. As a conse-

quence of this choice the functions are conservative in the

short period range. However, this is reasonable because in

the short period range the displacements are small and there-

fore the accuracy of the estimates is not crucial. Neverthe-

less, h must be more conservative in the short period range

because the displacement spectra present an increasing

branch. Erroneous evaluation of the period of vibration

can lead to an under-estimation of the required displacement

capacity. This cannot occur in the long period range that is

characterised by practically horizontal line of the displace-

ment spectrum. Fig. 15 shows the approximate curves repre-

senting Eq. (11). It is evident that the approximate curves do

not vary signi®cantly as a function of ductility and hystere-

tic behaviour.

7. Residual inelastic displacements

Perfectly feasible designs may undergo large irreversible

displacements leading to a permanent off-set. This would

impair the function of the structure. It is therefore of prac-

tical signi®cance to derive relationships between various

inputs and response parameters and the permanent irrecov-

erable displacement of structures. This has been undertaken

and the results are herein discussed. The response charac-

teristics chosen are for softening behaviour with stiffness K3

of 220 and 230%. This is because in hardening and

perfectly plastic cases the residual displacements are less

signi®cant than in the case of softening behaviour (Kawa-

shima et al. [32], MacRae and Kawashima [33]). The


Table 4

Average (kxl) and standard deviation (s) values of h-factor calculated with

Eq. (10)

m � 2 m � 3 m � 4 m � 6

kxl s kxl s kxl s kxl s

EPP 1.53 0.11 1.66 0.15 1.72 0.17 1.74 0.18

K3 � 0 1.81 0.19 2.04 0.29 2.12 0.33 2.21 0.39

K3 � 10%Ky 1.78 0.18 2.05 0.28 2.17 0.33 2.30 0.40

K3 � 220%Ky 1.84 0.21 1.98 0.27 1.98 0.29 ± ±

K3 � 230%Ky 1.85 0.21 1.93 0.27 ± ± ± ±

Table 5

Equivalent damping jE and a values corresponding to h -factor reported in Table 4

m � 2 m � 3 m � 4 m � 6

jE (%) a jE (%) a jE (%) a jE (%) a

EPP 5 0.10 6 0.09 7 0.09 7 0.08

K3 � 0 8 0.16 10 0.16 11 0.15 13 0.15

K3 � 10%Ky 8 0.15 11 0.16 12 0.16 14 0.17

K3 � 220%Ky 8 0.16 10 0.15 10 0.13 ± ±

K3 � 230%Ky 8 0.17 9% 0.14 ± ± ± ±

regression coef®cients for the de®nition of residual displa-

cement spectra were calculated. Fig. 16 shows the ratio

between maximum inelastic displacement and residual

displacement. The residual displacements are a high percen-

tage of the maximum displacements reached by the systems.

In the aforementioned ®gure they are strongly dependent on

the ductility requirements and the post-elastic stiffness, as

observed in previous work (Kawashima et al. [32], MacRae

and Kawashima [33]). If the designer considers that such

permanent displacements are unacceptable, another itera-

tion in the displacement design cycle, described in

Section 1 above, is necessary.


Fig. 14. (a) In¯uence of ductility; and (b) hysteretic behaviour on h-factor.

Fig. 15. (a) In¯uence of ductility; and (b) hysteretic behaviour on period-dependent h -factor functions.

Fig. 16. Ratio between inelastic and residual displacement spectra for softening behaviour �Ms � 6; d � 10 km; soft site).

8. Conclusions

Seismic assessment and design based on displacements

provides a conceptually appealing alternative to conven-

tional force-based design. Since one of the commonly

used deformational quantities employed to assess the satis-

faction of limit states are displacements, the only question

that arises from considering displacement-based design is

why was it not adopted earlier. Seismic design displacement

spectra are three-parameter demand representations, the

three parameters being effective period, relative response

displacement and equivalent damping ratio. In three

recently published works (Bommer and co-workers [9,10],

Tolis and Faccioli [11]) displacement spectra (as a function

of magnitude, distance and site condition) and modi®cations

to Eurocode 8 spectrum were proposed. However, the

derived spectra would not be used unless robust and veri®-

able relationships between global ductility factor and corre-

sponding equivalent damping are derived. Also, in the latter

papers, elastic highly damped spectra were derived to repre-

sent inelastic response.

The paper introduces a new form of inelastic displace-

ment spectra and displacement reduction factor, as well as

residual displacement spectra. The results of non-linear

analyses undertaken have been employed for the de®nition

of an equivalent elastic system (substitute structure). The

equivalence between elastic and inelastic systems is inter-

preted in terms of period of vibration and dissipation capa-

city. There is a simple relation between the initial period and

equivalent elastic period of vibration, which is a function of

required ductility and primary curve shape, while for the

energy dissipation a reduction factor h has been proposed.

This may be related to equivalent elastic damping, which

can conveniently describe the energy dissipation. The

displacement modi®cation factor h for displacement-

based approaches is the reduction coef®cient corresponding

to the behaviour factor (R or q) in force-based methods. The

advantage of h over the behaviour factor is that for the

de®nition of the former a correct identi®cation of ductility

capacity and hysteretic behaviour of the structure is not

important, because their in¯uence is already taken into

account via an equivalent elastic vibration period. On the

other hand, for the behaviour factor that takes into account

the effects of plasticity for a force-based design or assess-

ment method, a strong in¯uence from the ductility has been

demonstrated. This is because the behaviour factor consid-

ers both the increase in vibration periods and the dissipation

of energy due to inelastic behaviour. The evaluation of h is

therefore an indispensable ingredient in the displacement-

based design approach that was hitherto unavailable.

Acknowledgements

The writers would like to express their gratitude to Mr

G.O. Chlimintzas for his help with the strong-motion

dataset. The regression program was kindly provided by

Dr S.K. Sarma, whilst support was given by Mr D. Lee

during the implementation of the hysteretic model. All the

above are from Imperial College. Funding for the stay of the

primary author at Imperial College was provided by the EU

network programs ICONS and NODISASTR.

References

[1] Calvi GM, Kingsley GR. Displacement-based seismic design of

multi-degree-of-freedom bridge structures. Earthquake Engng Struct

Dyn 1995;24:1247±66.

[2] Kowalsky MJ, Priestley MJN, MacRae GA. Displacement-based

design of RC bridge columns in seismic regions. Earthquake Engng

Struct Dyn 1995;24:1623±43.

[3] Priestley MJN, Seible F, Calvi GM. Seismic design and retro®t of

bridges. New York: Wiley, 1996.

[4] Panagiotakos TB, Fardis MN. Deformation-controlled earthquake-

resistant design of RC buildings. Journal of Earthquake Engineering

1999;3(4):495±518.

[5] Moehle JP. Displacement-based design of RC structures subjected to

earthquakes. Earthquake Spectra 1992;8(3):403±28.

[6] Gulkan P, Sozen M. Inelastic response of reinforced concrete struc-

tures to earthquake motions. ACI J 1974;71:604±10.

[7] Shibata A, Sozen M. Substitute-structure method for seismic design in

R/C. J Struct Div, ASCE 1976;102:1±18.

[8] Qi X, Moehle JP. Displacement design approach for reinforced

concrete structures subjected to earthquakes. Report No. UCB/

EERC-91/02, 1991, p. 186.

[9] Bommer JJ, Elnashai AS, Chlimintzas GO, Lee D. Review and devel-

opment of response spectra for displacement-based seismic design.

ESEE Research Report No. 98-3, ICONS, Imperial College, London,

1998.

[10] Bommer JJ, Elnashai AS. Displacement spectra for seismic design. J

Earthquake Engng 1999:1±32.

[11] Tolis SV, Faccioli E. Displacement design spectra. J Earthquake

Engng 1999:107±25.

[12] Borzi B, Elnashai AS, Faccioli E, Calvi GM, Bommer JJ. Inelastic

spectra and ductility±damping relationships for displacement-based

seismic design. ESEE Research Report No. 98-4, ICONS, Imperial

College, London, 1998.

[13] Borzi B. Design spectra based on inelastic response. PhD thesis,

Politecnico di Milano, Italy, 1999.

[14] Ambraseys NN, Simpson KA. Prediction of vertical response spectra

in Europe. Earthquake Engng Struct Dyn 1996;25:401±12.

[15] Ambraseys NN, Simpson KA, Bommer JJ. Prediction of horizontal

response spectra in Europe. Earthquake Engng Struct Dyn

1996;25:371±400.

[16] Ozcebe G, Saatcioglu M. Hysteretic shear model for reinforced

concrete members. J Struct Engng, ASCE 1989;115:133±48.

[17] Saatcioglu M, Ozcebe G, Lee BCK. Tests of reinforced concrete

columns under uniaxial and beaxial load reversals. Department of

Civil Engineering, University of Toronto, Toronto, Canada, 1988.

[18] Saatcioglu M, Ozcebe G. Response of reinforced concrete columns to

simulated seismic loading. ACI Struct J 1989:3±12.

[19] Paulay T, Priestley MJN. Seismic design of reinforced concrete and

masonry buildings. New York: Wiley, 1992.

[20] Priestley MJN, Seible F, Calvi GM. Seismic design and retro®t of

bridges. New York: Wiley, 1996.

[21] Calvi GM, Pinto PE. Experimental and numerical investigations on

the seismic response of bridges and recommendations for code provi-

sions. ECOEST/PREC8 Report No. 4, Laboratorio Nacionale de

Engenharia Civil, Portugal, 1996, p. 137.


[22] Pinto AV. Pseudodynamic and shaking table tests on R.C. Bridges.

ECOEST & PREC8, 1996, p. 5.

[23] Miranda E. Evaluation of site dependent inelastic seismic design

spectra. J Struct Engng, ASCE 1993;119:1319±38.

[24] Gupta B, Sashi KK. Effect of hysteretic model parameters on inelastic

seismic demand. In: Proceedings of the Sixth US National Conference

on Earthquake Engineering, Oakland, California, 1998.

[25] Whittaker A, Constantinou M, Tsopelas P. Displacement estimates

for performance-based seismic design. J Struct Engng, ASCE

1998;124:905±12.

[26] Rahnama M, Krawinkler H. Effects of soft soil and hysteretic model

on seismic demands. Report No. 108, The John A. Blume Earthquake

Engineering Center, Stafford University, Stanford, California, 1993.

[27] Borzi B, Elnashai AS. Re®ned force reduction factor for seismic

design. J Engng Struct 2000 (in press).

[28] Vidic T, Fajfar P, Fischinger M. Consistent inelastic design spectra:

strength and displacement. Earthquake Engng Struct Dyn

1994;23:507±21.

[29] Miranda E, Bertero VV. Evaluation of strength reduction factor for

earthquake-resistance design. Earthquake Spectra 1994;10:357±79.

[30] Krawinkler H, Nassar AA. Seismic design based on ductility and

cumulative damage demand and capacities. In: Fajfar P, Krawinkler

H, editors. Nonlinear seismic analysis and design of reinforced

concrete buildings. New York: Elsevier, 1992.

[31] Newmark NM, Hall WJ. Earthquake spectra and design. EERI mono-

graph series, EERI, Oakland, 1982.

[32] Kawashima K, MacRae G, Hoshikuma J, Nagaya K. Residual displa-

cement response spectra. J Struct Engng, ASCE 1998;124:523±50.

[33] MacRae G, Kawashima K. Post-earthquake residual displacement of

bilinear oscillators. Earthquake Engng Struct Dyn 1997;26:701±16.


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