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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2005; 34:145165

Published online 25 October 2004 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.414

Scaling of spectral displacement ordinates with damping ratios

Julian J. Bommer; and Rishmila Mendis

Department of Civil and Environmental Engineering; Imperial College London; London SW7 2AZ; U.K.

SUMMARY

The next generation of seismic design codes, especially those adopting the framework of performance-based design, will include the option of design based on displacements rather than forces. For direct

displacement-based design using the substitute structure approach, the spectral ordinates of displacementneed to be specied for a wide range of response periods and for several levels of damping. The codedisplacement spectra for damping values higher than the nominal value of 5% of critical will generally be obtained, as is the case in Eurocode 8 and other design codes, by applying scaling factors to the5% damped ordinates. These scaling factors are dened as functions of the damping ratio and, in somecases, the response period, but are independent of the nature of the expected ground shaking. Using both predictive equations for spectral ordinates at several damping levels and stochastic simulations,it is shown that the scaling factors for dierent damping levels vary with magnitude and distance,reecting a dependence of the scaling on the duration of shaking that increases with the damping ratio.The options for incorporating the inuence of this factor into design code specications of displacementresponse spectra are discussed. Copyright ? 2004 John Wiley & Sons, Ltd.

KEY WORDS: displacement response spectra; displacement-based design; damping ratios; seismic designcodes

1. INTRODUCTION

Recognition of the poor correlation between transient inertial forces induced by earthquakeshaking and damage to structures has led to the development of displacement-based approachesfor seismic design and assessment. Amongst the dierent methods that have been proposedfor estimating inelastic displacements in structures, many are based on the response of anequivalent elastic system (see review by Miranda and Ruiz-Garca [1]). These methods are

based on equivalent linearization, using the substitute structure concept [2, 3] in which theinelastic deformation is modelled by a reduced stiness and the hysteretic dissipation ofenergy is modelled by an increased level of viscous damping.

The adoption of direct displacement-based design (DBD) approaches places the onus on en-gineering seismologists to provide suitable input in terms of long-period spectral displacements

Correspondence to: Julian J. Bommer, Department of Civil and Environmental Engineering, Imperial CollegeLondon, London SW7 2AZ, U.K.

E-mail: [email protected]

Received 6 July 2004

Revised 20 July 2004Copyright ? 2004 John Wiley & Sons, Ltd. Accepted 21 July 2004


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146 J. J. BOMMER AND R. MENDIS

Figure 1. Displacement spectra dened in Eurocode 8.

for a range of damping ratios. Transformation of the acceleration spectra in current seismiccodes to displacement spectra will generally not produce reliable displacement ordinates at thelonger periods that become relevant to DBD [4, 5]. In order for the acceleration spectrum notto result in spectral displacements that increase monotonically with increasing response period,there must be a portion over which the acceleration decays in inverse proportion to the squareof the period. The 1990 French seismic code [6] was the rst to introduce such a decaying

branch in the acceleration spectrum for periods beyond 3:0 seconds. Eurocode 8 [7] includesspecications for displacement spectra in an informative annex, amongst the rst codes toexplicitly make provisions for displacement-based design of buildings. The EC8 displacementspectrum (Figure 1) is designed to be compatible with the acceleration spectrum, following

the proposal of Bommer et al. [8], with the control periods TD, TE and TF based on the workof Tolis and Faccioli [9].Although the original motivation for this work is related to dening input for direct DBD,

the issue of over-damped spectral displacements at intermediate and long response periods isalso of clear relevance to the design of buildings and bridges incorporating base isolation orsupplementary damping devices.

There is currently a signicant level of disagreement regarding appropriate values for thecontrol periods of the displacement spectrum. In EC8, the Type 1 spectrum (applicable inhigh seismicity areas) control period TD has a value of only 2 seconds; TE is set at 6 secondsand the spectral ordinates are expected to converge to the peak ground displacement (PGD) ata period (TF) of 10seconds. Guan et al. [10] propose displacement spectra based on Japaneseand Californian accelerograms recorded between 1989 and 2001; the control periods in their

proposal are very similar to those for the EC8 Type 1 spectrum. Other proposals assign muchlarger values to the control periods: as Faccioli et al. [5] report, the procedure of Newmark andHall [11] yields values of 10 and 30 seconds for TE and TF respectively. The 2003 proposedrevision of the NEHRP guidelines [12] also includes a constant displacement branch in thedenition of the response spectrum; maps are provided that show the variation of the periodTL, equivalent to the period TD in the EC8 formulation (Figure 1). Values of this control

period specied for the contiguous United States vary between 4 and 16 seconds, increasing

Copyright ? 2004 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2005; 34:145165


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SPECTRAL DISPLACEMENT ORDINATES 147

exponentially with earthquake magnitude; values of 12 seconds correspond to regions aected by earthquakes in the New Madrid zone and on the major Californian faults, and values ashigh as 16 seconds for areas aected by earthquakes on the Cascadia subduction zone. Thereis clearly much work yet to be done to obtain reliable estimates of the long-period spectral

displacements.In seismic design codes such as EC8, the displacement spectra for damping ratios other

than 5% are obtained by applying scaling factors to the ordinates of the 5% damped spectrum.The scaling factors are dened as a function of the response period and the damping ratioonly. Several dierent proposals for these scaling factors, reviewed briey in the next sectionof this paper, have been put forward, prompting discussion as to which factors are the mostappropriate for use in code applications. The approach adopted in this paper is to explore thevariation amongst the proposed scaling factors by investigating if there are any systematicvariations in the scaling factors that would suggest that their denition in terms of response

period and damping ratio alone is insucient.

2. SPECTRAL REDUCTION FACTORS

Figure 2 compares the spectral ratios, in this case normalized to the 10% damped spectralordinates, from the studies of Newmark and Hall [11] and Wu and Hanson [13], which forthe 20% damped spectral ordinates are in good agreement. The ratios from both studies tendtowards unity at very short and very long periods, as would be expected. Both studies were

based on pseudo-velocity spectra and hence are applicable to displacement spectra. Lin andChang [14] found that whilst the ratios obtained from spectra of relative displacement showrelatively little variation with response period (and hence the same is true for pseudo-velocityand pseudo-acceleration spectra), the ratios obtained from spectra of relative velocity andabsolute acceleration display a much more pronounced variation with period. Lin and Chang

[14] also point out that the spectral ratios for dierent damping values have generally beendetermined from studies of displacement spectra and then applied to acceleration spectra forthe calculation of seismic design forces. This criticism is not relevant to the present study

Figure 2. Spectral ratios, with respect to the 10% damped ordinates, from two studies; the ratios forthe Newmark and Hall [11] study are obtained using median values.



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since we are expressly concerned with the scaling of intermediate- and long-period ordinatesof spectral displacement, SD.

In EC8, the 5% damped spectral ordinates can be scaled for higher damping values, (%),using a simple expression derived by Bommer et al. [8]:

SD()

SD(5%)=

10

5 + (1)

The scaling factor applies to the ordinates at all response periods between TB (the startof the acceleration plateau) and TE (Figure 1); at shorter and longer periods the factorincreases linearly to reach a value of 1.0 at T=0 and T=TF. Figure 3(a) compares theresulting spectral ratios, normalized to the 5% damped spectral displacements, from Equation(1) with values obtained by Lin and Chang [14] from their study of the median spectra frommore than 1000 strong-motion accelerograms. Although there is considerable divergence be-tween the two sets of curves at long periods, this reects as much as anything the uncertaintyassociated with obtaining long-period spectral displacements and the sensitivity of the long-

period spectral shape to record processing. Certainly over the period range that correspondsto the constant displacement plateau in the EC8 spectrum between 2 and 6 seconds, there isreasonable agreement with the results of Lin and Chang [14].

Equation (1) replaced a similar expression included in the original draft of EC8 [15], theorigin and derivation of which have not been documented:

SD()

SD(5%)=

7

2 + (2)

Figure 3(b) compares the ratios from Equation (2) with the values obtained by Lin andChang [14] using the 84-percentile spectral ordinates. For periods up to 6 seconds, the agree-ment is excellent, which may be fortuitous but might also give a clue as to the derivation ofEquation (2). We believe that the scaling factors adopted in seismic codes for adjusting the5% damped spectrum to other damping ratios should be based on median estimates, since the

Figure 3. Spectral ratios, with respect to the 5% damped ordinates: (a) Equation (1) compared with thevalues of Lin and Chang [14] based on median ordinates; and (b) Equation (2) compared with the valuesof Lin and Chang [14] based on 84-percentile ordinates. In both plots, the ratios for the EC8 spectra

have been obtained using the control periods for the Type 1 spectrum and site class C.



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inherent variability in the ground-motion predictions is already accounted for in the proba-bilistic derivation of the 5% damped spectrum. As can be appreciated from Figure 3, using theratios based on 84-percentile values will produce lower spectral ordinates for higher dampingvalues and it is not clear to us how such a choice could be rationalized for safe design.

A third variation of the scaling factor was proposed by Tolis and Faccioli [9], whose studywas based primarily on the recordings of the 1995 Hyogo-ken Nanbu (Kobe) earthquake:

SD()

SD(5%)=

15

10 + (3)

In a few other seismic design codes, alternative scaling factors are encountered, althoughit would appear that most code drafting committees have not envisaged the use of dampingratios other than the nominal value of 5% assumed for reinforced concrete. The 1990 Frenchcode [6] and the 1994 Spanish code [16] both included the following scaling factor for spectralaccelerations, SA:

SA()SA(5%)

=

5

0:4

(4)

The 1983 Portuguese seismic code [6] and the 1984 Indian seismic code [6] both includegraphical representations of acceleration spectra at more than one damping level, but do not

provide scaling factors. Figure 4 compares the ratios obtained from these graphs, averagedover the period range covered (in both cases the variation with period is small), and the ratiosobtained from Equations (1)(4).

The 2001 Caltrans Seismic Design Criteria [17] allow damping to be increased from5% to 10% if the bridge is heavily inuenced by energy dissipation at the abutments and isexpected to respond like a single-degree-of-freedom system. The spectral displacements canthen be scaled using the following equation, originally derived by Kawashima and Aizawa

[18] for spectral ordinates of absolute acceleration:

SD()

SD(5%)=

1:5

0:4 + 1

+ 0:5 (5)

Figure 4. Spectral reduction factors for dierent damping values from a number of seismic design codesand from the study of Tolis and Faccioli [9].



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The 1997 edition of the Uniform Building Code (UBC) [19] provides scaling factors ata number of discrete damping values for the design of seismically isolated structures. Thesefactors, and those from Equation (5), are also shown in Figure 4.

The curves and individual values shown in Figure 4 reect a signicant disagreement

amongst models for the scaling of 5% damped spectral displacements for higher dampingvalues. Since the studies are presumably all based on the analysis of elastic single-degree-of-freedom (SDOF) oscillators, one may assume that the dierences have not arisen fromvariations in the structural models. The logical way to explore the possible cause of thedivergence is to investigate the extent to which the scaling factors are dependent on thecharacteristics of the seismic demand.

3. FACTORS INFLUENCING SPECTRAL SCALING

In order to explore systematic inuences of characteristics of the earthquake ground motion

on the spectral scaling factors for dierent damping levels, there are two options: the rstbeing to use real accelerograms; the second being to use synthetic accelerograms generatedfrom seismological models.

To identify consistent patterns within the appreciable variability that is always displayedby strong-motion data, very large numbers of records would need to be employed since oneis unlikely to nd records that are essentially similar in all but one characteristic. For thisreason, we employ ground-motion prediction (attenuation) equations for spectral ordinates atmultiple levels of damping, derived from regression on empirical strong-motion data. Theanalyses are based on the median values from these equations, for the reasons explained

previously, which eectively provides a direct way of identifying the average behaviour oflarge numbers of strong-motion accelerograms. Four sets of predictive equations are adopted,whose characteristics are summarized in Table I. The rst two equations are derived from

western North American data, the last two from European data, although Berge-Thierry et al.[20] supplemented their data with Californian records to improve the near-source coverage oflarge magnitude earthquakes.

As stated above, the second option is to use seismological models to generate syntheticground motions to explore the variation of the spectral scaling factors with dierent seismo-logical parameters. We make use herein of the stochastic method of Boore [25] to validatethe results obtained from the predictive equations for the inuence of earthquake magnitude.

In the following three sub-sections, the inuence of magnitude, distance and site classica-tion on the spectral scaling factors inferred from these equations are explored. In Section 3.4,we briey address the special case of records aected by near-source rupture directivity.

3.1. Magnitude

Using the four predictive equations listed in Table I, median response spectra for dierentdamping levels were constructed for a rock site at 10 km from earthquakes of dierent mag-nitude. For each magnitude, the spectral ordinates were then divided by the ordinates of the5% damped spectrum and the ratios plotted (Figure 5). A consistent pattern was observed inall cases: for periods greater than about 0:3 secondswhich are the periods of relevance todirect DBD (based on equivalent linearization)the reduction of the spectral displacements



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Table I. Characteristics of ground-motion prediction equations used in this study.

No. of Magnitude Distance Response Damping

Equations records scale=range metric=range1 parameter2 values3

Trifunac and Lee [21] 438 (N.A) R4hyp PSV 5, 10, 20%(N.A.)

Boore et al. [22] 271 Mw RJB PSV 5, 10, 20%(5.3 7.7) (2 100 km)

Bommer et al. [23] 183 Ms RJB SD 5, 10, 15,(5.5 7.9) (1 260 km) 20, 25, 30%

Berge-Thierry et al. [20] 483 Ms Rhyp PSA 5, 7, 10,(4.5 7.3) (7100 km) 20%

Notes: 1. Distance measures as dened by Abrahamson and Shedlock [24]; 2. PSV = pseudo-velocity responsespectrum, PSA = pseudo-acceleration response spectrum, SD = relative displacement response spectrum; 3. As

with some of the design codes reviewed previously, values of damping lower than 5% are not consideredin this study since they are unlikely to be relevant to displacement-based design; 4. Modied hypocentraldistance that accounts for source dimensions.

Figure 5. Variation of the ratios of 20% damped spectral ordinates to the 5% dampedordinates with earthquake magnitude from the equations of (upper) Trifunac and Lee

[21] and (lower) Bommer et al. [23].



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Figure 6. Variation of the spectral ratios (relative to the 5% damped ordinates) atT= 2:0 s with earthquake magnitude inferred from median values obtained from the

four predictive equations in Table I.

for higher damping ratios increases with the earthquake magnitude. Figure 6 shows the vari-ation of the scaling factors for a response period of 2:0 sthe longest period covered by theequations of Boore et al. [22]with magnitude for dierent damping ratios from all fourequations.

Stochastic simulations were also generated for the same scenario of a rock site at 10 kmfrom earthquakes of dierent magnitude, using the program SMSIM [26]. The same trendof decreasing spectral ratios with increasing magnitude was observed, as shown in Figure 7.

The default source, path and site parameters for coastal California were employed for thesimulations.

The variation of the spectral ratios with magnitude shown in Figure 6 clearly varies fromone equation to another, and may in some cases appear to be not very pronounced; thecurvature in the plots corresponding to Boore et al. [22] is the result of the quadratic term inmagnitude in their equations. However, the ratio of 20% to 5% spectral ordinates are similarfor all four equations and it is reasonable to assume that the pronounced gradient of the curve



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Figure 7. Variation of the spectral ratios (relative to the 5% damped ordinates) at T= 2:0 s withearthquake magnitude inferred from stochastic simulations.

for 30% obtained from the equations of Bommer et al. [23] would also be observed for theseother equations had they provided coecients for this damping level.

In plotting the ratios for magnitudes up to 8 the equations are extended slightly beyondtheir range of applicability. For the 30% damped spectrum, the scaling factors inferred fromthe equations of Bommer et al. [23] decrease by 15% as the magnitude is increased from 6to 7.5.

3.2. Source-to-site distance

The equations of Trifunac and Lee [21] include exactly the same attenuation terms for spectralordinates at all damping levels hence the variation of the spectral ratios with distance cannot

be inferred. The other three equations exhibit the same tendency at intermediate responseperiods of the scaling factors decreasing as the distance from the earthquake source increases.This is illustrated for two of the equations in Figure 8.

The spectral ratios appear to be most sensitive to changes of distance close to the earthquake

source and for the results obtained from both Boore et al. [22] and Bommer et al. [23], thesignicant variation of the ratios with distance is limited to the rst 1015km from the source.Since the strong-motion datasets are relatively sparse in terms of near-source recordings, theinferred variation of the ratios at these distances should be interpreted with some caution.

The combined eect of varying magnitude and distance on the spectral scaling factor isappreciable: using the equations of Bommer et al. [23] and considering the ratio of the 30%spectral ordinate to the 5% damped ordinate at T= 2:0 s, the ratio decreases from 0.68 for amagnitude Ms = 5:5 earthquake at 5km to 0.51 for an event ofMs = 7:5 at 50km: a reductionof 25%.

3.3. Site classication

The four predictive equations include site classication as an explanatory variable, whichallows the inuence of this parameter on the spectral ratios to be investigated, as shownin Figure 9.

The four equations do not use the same site classication scheme. Trifunac and Lee [21]divide their data into basement rock, alluvium and intermediate sites, and spectral ratiosdisplay very small decreases, at intermediate periods, as one passes from stier to softer sitegeologies. Boore et al. [22] and Bommer et al. [23] use the same site classication scheme



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Figure 8. Variation of the ratios of 20% damped spectral ordinates to the 5% damped ordinates withsource-to-site distance (measured from the surface projection of the fault rupture) from the equations

of Boore et al. [22] and Bommer et al. [23] for an earthquake of magnitude 7.

based on 30m shear wave velocities, with 360m=s and 760m=s marking the boundaries betweenclasses. The results from the equations of Bommer et al. [23] do not show a consistent pattern

but it is known that the classication of a large proportion of strong-motion stations in Europeand the Middle East is highly uncertain. The same qualifying remark applies to the equationsof Berge-Thierry et al. [20]; their results indicate lower scaling factors at rock sites than atalluvium sites, but for response periods beyond 5 seconds (not shown in the plots in orderfor them to be visually comparable) the curves are inverted. Amongst the four studies, theonly one for which most recording station site classications may be considered reliable isBoore et al. [22] and hence greatest weight should perhaps be given to the pattern indicatedfrom their results: appreciably lower spectral scaling factors at soil sites than at rock sites.The number of records from rock sites in their data set, however, was limited, so even here

some caution may be in order in drawing conclusions from the results.There are some cases where site eects on strong motion have been very pronounced,

foremost amongst these being the recordings from Mexico City of the 1985 Michoacan earth-quake. Figure 10 shows one of these records, its displacement response spectra for variousdamping levels and the variation of the ratios of 30% to 5% spectral displacements withresponse period. At the dominant period of this narrow-band signal, the reduction factors arevery small, whereas at other periods the scaling factors are exceptionally high.



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Figure 9. Variation of the spectral ratios (20% to 5% damped ordinates) with site classication inferredfrom median values obtained from the four predictive equations in Table I.

3.4. Near-source directivity eects

At sites close to the fault rupture, forward rupture directivity can create large-amplitudevelocity pulses that increase the response spectral ordinates. Near-source ground motions af-fected by forward rupture directivity have been observed to be particularly damaging and theircharacterization has become a major concern for earthquake engineering. Somervilleet al. [27]derived a model for the impact of forward directivity on the acceleration spectral ordinates,leading to monotonically increasing ordinates for response periods beyond 0:6 seconds forsites aected by forward directivity during earthquakes ofMw = 6:5 or larger. Abrahamson[28] modied the model to include the inuence of distance and the decay of the amplifyingeect with separation from the fault rupture. The model has more recently been modied,using recordings from larger earthquakes that have occurred in the last few years, to repre-sent the forward-directivity eect as a narrow-band pulse whose central period increases with

earthquake magnitude [29]. Faccioli et al. [5] have incorporated the narrow-band pulse modelinto their formulation for long-period spectral displacements.Priestley [30] has proposed the following modication of the EC8 spectral scaling factor

in Equation (1) for near-source recordings exhibiting the forward directivity velocity pulse:

SD()

SD(5%)=

10

5 +

0:25(6)



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Figure 10. Top: N90 component of the recording from the soft soil site at SCTof the 19 September 1985 Michoacan, Mexico, earthquake. Middle: Displacementresponse spectra for various damping levels. Bottom: Ratio of 30% to 5% damped

ordinates compared with those obtained from Equations (1) and (2).

The proposal for using the square root of the factors dened in Equation (1) for the specialcase of near-source earthquake ground motions was based on the idea that in the near-eldregion, the velocity pulses may reduce the eectiveness of damping [30]. Figure 11 showsa classic near-source accelerogram aected by forward rupture directivity: the Lucerne recordof the 1992 Landers (California) earthquake. The gure shows the time-histories and the



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Figure 11. Fault normal component of the Lucerne accelerogram from the 1992 Landersearthquake: 5% damped pseudo-velocity response spectrum and time-histories of accelera-tion, velocity and displacement.

pseudo-velocity response spectrum of the resolved fault normal component, as analysed bySomerville et al. [27]. The velocity pulse can be seen to have a period of close to 4 seconds,and this clearly manifests on the response spectrum.



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Figure 12. Displacement spectra of the fault normal component of the Lucerne accelerogram from the1992 Landers earthquake (upper) and spectral ratios, normalized to the 5% ordinates, compared withthe ratios given by Equations (1) and (6).

Figure 12 shows the displacement response spectra for various damping levels obtainedfrom the accelerogram in Figure 11, and compares the spectral ratios with those obtainedfrom Equations (1) and (6). One should, of course, be cautious about drawing conclusionsfrom a single record, although the time history in question clearly represents a case of almostall the energy of the ground shaking being concentrated in a single velocity pulse. Theaverage ratios obtained from the accelerogram (the short-period variations are the result ofhigh-frequency excitation in the record due to a thin soil layer at the site) at most periods

agree well with those predicted by the equation of Priestley [30]. Only in the region of the period of the velocity pulse itself (4 seconds), are the ratios much closer to those predicted by the existing equation in EC8.

From Figure 12 it is clear that the velocity pulse does manifest on the 5% damped spectrumas a distinct peak in the displacements, but the eect is smoothed out by higher dampingratios. This is an area that requires further investigation, which should be undertaken in tandemwith rening models for the displacement spectra due to near-source ground motions. The



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results shown here do suggest that Equation (6) proposed by Priestley [30] for reduced spectralscaling factors for motions aected by forward directivity is an appropriate modication, butit is also necessary to ensure that the forward directivity pulse is captured in the denitionof the 5% damped spectrum.

4. INTERPRETATION AND IMPLEMENTATION OF RESULTS

The results of the previous section suggest that the divergence amongst proposed scaling fac-tors for adjusting the 5% damped spectral ordinates for higher levels of damping displayedin Figure 4, may in large part be due to dierences in the characteristics of the groundmotions employed in their derivation. Figure 3 also indicates that another cause of the diver-gence may be the choice of using median values or some other percentile. Although alreadystated previously, we again make our case against the use of values other than the median fordetermining these ratios, the fundamental point being that since they are to be applied to a

5% damped spectrum derived from probabilistic hazard analysis, the use of values other thanthe median is to double count the inuence of the scatter in the ground-motion predictionequations. Furthermore, it is in contradiction with the probabilistic approach to make anyarbitrary selection of percentiles on the basis of perceived conservatism. This is even morethe case here where the conservative choice of the ratio of 84-percentile ordinates, as pro-

posed by Lin and Chang [14], actually results in lowerand hence not conservative at allspectral displacements for higher damping values.

Instead of making subjective decisions regarding appropriate levels of aleatory variabil-ity, our approach has been to identify causes for the variation in the median values and totransform random variability into epistemic uncertainty by identifying additional explanatoryvariables to be incorporated into the prediction of the scaling factors.

4.1. Inuence of duration

From simple structural dynamics considerations, the reduction of the spectral ratios is expectedto increase with increasing duration of motion. Consider an undamped oscillator subjected toa harmonic excitation of its base: with each cycle of motion, it will accumulate more energyand hence vibrate more energetically, the maximum displacement response thus increasingmonotonically with the number of cycles. For a viscously-damped oscillator with a low levelof damping (say 5%), after the rst cycle of motion has passed, only some of the energyin the oscillator will have been dissipated by the damping and some will remain, hence withthe next cycle of motion the newly input energy will be added and the vibrations will thus

become stronger with an increasing number of cycles. However, after a certain number of

cycles, a steady-state response is reached whereby the input of energy through excitationof the base is exactly matched by the dissipation of energy through damping; the spectralresponse then remains constant as the number of cycles of motion grows. The higher thelevel of damping, the more rapidly the oscillator will reach the steady-state response and thesmaller the dierence between this level and the response to the rst cycle. Once the steadystate is reached, the ratio between the spectral displacements is equal to the reciprocal of theratio between the damping values (Figure 13).



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Figure 13. Maximum spectral displacements of damped oscillators (middle) of period T and dierentdamping levels subjected to harmonic excitation with the same period of vibration (top) and variation

of spectral ratio with number of cycles of motion (bottom).

The patterns identied in Section 3 show that the spectral ratios are strongly dependent onearthquake magnitude and source-to-site distance, and weakly dependent on the site classi-cation. All of these patterns are entirely consistent with the ratios being dependent on thestrong-motion duration. In order to be meaningful, any discussion of duration must specify

which of the 40 or so denitions encountered in the literature is being employed [31].Bommer and Martinez-Pereira [32] grouped denitions of strong-motion duration into threegeneric categories of bracketed, uniform and signicant, but also identied that the most im-

portant distinction amongst denitions is whether the time interval is determined from absolutelevels of motion or from relative proportions of the maximum amplitude or Arias intensity.Since in the present context the discussion is focused on the ratios of spectral displacementsfor oscillators with dierent levels of equivalent viscous damping, the appropriate denitions



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Figure 14. Predicted median values of signicant duration from the equations of Abrahamson and Silva[34] for the signicant duration based on 575% of the total Arias intensity.

are those based on relative measures, such as the signicant duration. This is dened as theinterval between the times at which two specied levels, such as 5% and 95% or 5% and 75%,of the total Arias intensity are reached [33]. Predictive equations for signicant duration showthat the duration of the shaking grows strongly with magnitude (an observation that holdsfor all denitions of duration) and also increases with increasing distance from the seismicsource; the increase of duration at soil sites compared to rock sites is small (Figure 14).

Strong-motion duration is generally not found to be strongly dependent on site classicationwhen the latter is classied only by the nature of the uppermost layers at the site [32]. Thisis not to discount the fact that duration of shaking can be appreciably prolonged at certainsoil sites, but this is interpreted to be due to two- and three-dimensional eects of basins

and the eect of trapped energy [35]. The lack of clear trends in most of the graphs inFigure 9 is therefore still consistent with duration of shaking being the underlying cause ofthe variation of the spectral ratios. Of the four equations for predicting spectral ordinatesused in this study, only Boore et al. [22] can be considered to be based on consistent andreliable site classications. The curves in Figure 9 derived from these equations do show aclear separation of the behaviour at rock and soil sites, from which it might be concludedthat since the sti and soft soil sites will include some that are in basins, the lower ratios

predicted for these sites are due to the prolongation of shaking at such locations.The eect of near-source rupture directivity, and the corresponding reduction of the spectral

scaling factors proposed by Priestley [30], is also consistent with the controlling parameter being duration. The model of Somerville et al. [27] predicts a reduction of the signicantduration (575% of Arias intensity) by 60% due to forward rupture directivity experienced

by the Lucerne accelerogram (Figure 11).Returning to Figure 4, however, there is a case that would appear to contradict this in-

terpretation: the ratios predicted by Tolis and Faccioli [9], based on data from the Mw = 6:9Kobe earthquake, are higher than those obtained by Bommer et al. [8], based on a Eu-ropean data set concentrated in the magnitude range from 5.5 to 7.0. However, there isonce again an explanation that is consistent with the interpretation of duration as being thekey parameter: the Kobe earthquake was an almost pure bi-lateral fault rupture, hence the



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duration of shaking at near-source sites was about one half of the values that would beexpected from a unilateral rupture [32]. The curves in Figure 14 would suggest that theimpact of bi-lateral rupture on the duration of shaking would have been equivalent to re-ducing the magnitude by one unit to about Mw = 6. Bi-lateral fault ruptures in major earth-

quakes are rare [36] hence such adjustments for the duration do not often need to be made.However, the 1989 Loma Prieta (California) earthquake (Mw = 6:9) was another case ofalmost pure bi-lateral rupture; recordings from this event constitute 44% of the dataset ofBoore et al. [22] hence the magnitude-dependence inferred from their equations may, in ef-fect, be even higher.

The ratios in the lower graph on Figure 13 are much lower than any of those depicted inFigure 4, conrming that although general trends may be inferred from consideration of har-monic signals, they are not comparable with non-stationary earthquake signals. Nonetheless,Figure 13 does serve to support the interpretation of duration as the controlling parame-ter. Denitions for counting the eective number of cycles of motion in accelerograms varyalmost as much as those for measuring the duration [37], but use is made here of the equationof Liu et al. [38] for the equivalent number of cycles of motion at 65% of the maximumamplitude. For a site at 10 km from the source, the equation predicts that the number of fullcycles will be 3 for Mw = 5:5, 5 for Mw = 6:5 and 10 for Mw = 7:5. According to the graph atthe bottom of Figure 13, increasing the number of cycles from 3 to 10 results in a reductionof the ratio of the 30% to the 5% damped spectral displacement of about 40%, which isabout twice the reduction inferred from the equations of Bommer et al. [23] for the sameincrease of magnitude (Figure 6). This dierence emphasizes the fact that transient earthquakeground motions are not harmonic signals and hence the spectral ratios from accelerograms willgenerally be much higher than those shown in Figure 13. However, the accelerogram fromMexico City shown in Figure 10 came very close to reaching the steady state as a result ofthe response of the lacustrine deposits underlying the city; it can be seen that at the dominant

period of about 2seconds, the spectral ratio for 30% to 5% damped ordinates was about 0.25,

tending towards the value of 0.167 for steady-state response to a harmonic excitation.

4.2. Implementation in seismic design codes

A simple and direct way of capturing the magnitude- and distance-dependence of the spec-tral displacement ordinates at dierent damping levels would be to dene, through separatezonation maps, a series of design spectra with dierent damping values for DBD in futureseismic design codes. However, this would be cumbersome and would often require interpo-lation for intermediate values of damping, so it is preferable to retain the practice of deningthe 5% damped spectrum and scaling the ordinates for higher values of equivalent damping.The question is then: how to incorporate the dependence of the scaling factor on magnitudeand distance? In EC8, where the uniform hazard spectrum is approximated through the use

of two spectral shapes (Type 1 for regions of high seismicity and Type 2 for regions whereearthquakes of magnitude > 5:5 are not expected), the simple solution is to apply dierentscaling factors to the two spectra: factors even higher than those from Equation (1) could beapplied to the Type 2 spectrum, whereas for the Type 1 spectrum the scaling factors could

be appreciably lower.For code formats other than the two spectral shapes of EC8, however, there needs to be

another approach. One option would be to provide zonation maps in terms of duration but



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this is not recommended because disaggregation of the hazard would often show that theduration at a given location is dominated by a dierent earthquake scenario than that drivingthe spectral displacement at the period of interest. This would then lead to the 5% dampedspectral ordinates and the scaling factors for higher damping levels being incompatible. Since

in current code formats the magnitude and distance of the controlling events are generally notvisible to the user, use can be made of the parameters used to dene the spectral ordinates.The denition of long-period spectral displacements will require at least three parameters to

be mapped, the rst two being spectral ordinates at short and intermediate response periods,the third being either a long-period ordinate or, as in the NEHRP guidelines [12] discussedin the Introduction, the control period for the constant displacement plateau. Bommer et al.[8] proposed the alternative approach of mapping PGA, PGD and the peak ground velocity(PGV), and then dening the corner periods of the spectrum from their ratios. Since, likethe NEHRP period TL, the ratio between PGD and PGV at rock sites is a function primarilyof the earthquake magnitudeand only weakly of distance [39, 40]the adjustment to thespectral scaling factor could be dened from these values.

5. CONCLUSION

Direct displacement-based seismic design and assessment require input in the form of displace-ment response spectra over long period ranges (up to the product of the yield period and thesquare root of the ductility demand factor) and for a number of damping levels (up to about30% of critical). Spectral displacements for long periods and high damping levels are alsodirectly relevant to the design of bridges and buildings with base isolation and supplementarydamping devices. In seismic design codes, the spectra for damping levels higher than 5%are obtained by applying scaling factors to the ordinates of the 5% damped spectrum. Thesefactors have been shown to be weakly dependent on response period other than in those re-

gions where the spectral displacements converge to zero or to PGD. At intermediate response periods, the spectral scaling factors are currently dened only in terms of the damping ratioand there is signicant disagreement amongst the proposed factors.

This study has shown that the spectral scaling factors vary with seismological features: thefactors decrease with increasing magnitude, decrease with increasing distance, and to a lesserextent, increase for softer site conditions. These variations all reect a consistent trend ofthe scaling factors decreasing with increasing duration of the ground motion. We have not

presented duration-dependent scaling factors because these should be obtained concurrentlywith the derivation of displacement spectra for wide period ranges, including the eect ofnear-source rupture directivity. The purpose of this paper is to demonstrate the inuencethat duration has on the spectral scaling factors and to explore the extent and nature ofthis inuence. These ndings can be incorporated into ongoing work to dene displacement

response spectra for design, thus producing more reliable models and improved estimates ofthe design motions.

ACKNOWLEDGEMENTS

The authors are indebted to Professors Nigel Priestley and Jose Ignacio Restrepo, amongst others, fordiscussions that contributed to the motivation to undertake this work. Special thanks are also due to



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Dr Robert W. Graves of URS Corporation for providing the rotated Lucerne accelerogram. Thanks arealso due to Dr David M. Boore for his SMSIM program and to Fleur Strasser for guidance on runningsimulations. We are also grateful to Jonathan Hancock and Dr Sarada K. Sarma for useful discussionsrelated to this topic.

A rst draft of this paper was reviewed by the following people: Sinan Akkar, Juliet Bird, DavidBoore, Michele Calvi, Helen Crowley, Damian Grant, Jonathan Hancock, Mervyn Kowalsky, EduardoMiranda, Rui Pinho and Nigel Priestley. We are indebted to all of them for their insightful commentsand helpful suggestions, all of which have contributed to signicant improvement of the manuscript.

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