TECHNICAL NOTE

An Anomaly in the EquivalentLinearization Approach for the Estimationof Inelastic Response

Vijay Namdev Khosea) E.Aff.M.EERI, andYogendra Singhb)

E.Aff.M.EERI

The estimation of inelastic displacement is an essential step in displacement-based design. Estimation of inelastic response using response spectra is a simplerand attractive alternative to nonlinear dynamic analysis. A number of methodsbased on different approaches are now available to estimate nonlinear res-ponse from design spectra. Equivalent linearization is one of the widely usedapproaches for this purpose. The approach is known to yield reasonably accurateresults. However, an anomaly in the approach is observed, which results in sig-nificantly non-conservative estimates of inelastic displacement, particularly indisplacement-controlled spectral range. This paper illustrates the anomalyusing displacement modification factors and numerical examples of oscillatorsof different periods and ductility. The results from the equivalent linearizationapproach have been compared with the results of nonlinear dynamic analysisand yield spectrum approach. [DOI: 10.1193/031112EQS088T]

INTRODUCTION

Displacement-based design (DBD) methods for seismic actions are slowly gaining popu-larity over the conventional force-based design. The estimation of the peak displacement ofan equivalent inelastic single-degree-of-freedom (SDOF) system is an integral part of DBD.Although nonlinear dynamic (time history) analysis is the most accurate method of estimat-ing inelastic displacement, it is time-consuming and complex. Further, it requires the selec-tion of appropriate earthquake records, whereas most of the national seismic design codesspecify only design response spectra without adequate guidelines for the selection of designground motions. Under such conditions, the estimation of inelastic response (peak responseof the inelastic system) using elastic response spectrum is a simpler and attractive alternative.

Different approaches are now available to estimate peak inelastic displacement using theresponse spectrum. The equivalent linearization (EL) approach (Gülkan and Sozen 1974,Iwan 1980, Kowalsky 1994, Kwan and Billington 2003, Grant et al. 2005, Dwairi et al.2007, Priestley et al. 2007, Pennucci et al. 2011) is one of the popular approaches for esti-mating inelastic displacement. In this approach, the inelastic response is obtained from theresponse of an equivalent elastic SDOF system having an appropriate period and damping.The other approaches to estimate inelastic response include the yield spectrum (YS), or

Earthquake Spectra, Volume 30, No. 2, pages 965–972, May 2014; © 2014, Earthquake Engineering Research Institute

a) Senior Engineer, Thornton Tomasetti, Inc., Mumbai-400013, Indiab) Professor, Department of Earthquake Engineering, IIT Roorkee, Roorkee-247667, India

965

constant ductility, approach (Newmark and Hall 1982, Riddell et al. 1989, Krawinkler andNassar 1992, Miranda 1993, Vidic et al. 1994, Ordaz and Pérez-Rocha 1998, Miranda 2000,Riddell et al. 2002, Cuesta et al. 2003, Chopra and Chintanapakdee 2004, Ruiz-García andMiranda 2004) and the displacement modification approach (ASCE 41-06 2007). It can beshown that with the recently developed models, the inelastic displacements obtained usingdifferent approaches tend to converge, except an anomaly in the EL approach in long periods(displacement-controlled) spectral region. This paper illustrates the anomaly in the ELapproach by comparing displacement modifications factors (ratios of peak inelastic topeak elastic displacement) with those obtained from the YS approach and an analyticalstudy using nonlinear time history analysis. Numerical examples are also presented todemonstrate the effect of this anomaly on estimated peak inelastic displacement.

EQUIVALENT LINEARIZATION APPROACH

In the EL approach, the behavior of the inelastic system is represented by an equivalentlinear system having equivalent damping, ξeq, and an effective or equivalent period, Teq.A number of EL models are available, which provide expressions for equivalent dampingand effective periods, which either corresponds to secant stiffness at peak displacement, or anoptimal stiffness between initial and secant stiffness. Once the equivalent damping is esti-mated, damping reduction factor, Rξ, can be obtained using available relationships (Newmarkand Hall 1982, CEN 1998, CEN 2004). Recently, Pennucci et al. (2011) have observed thatthe damping reduction factor has substantial random variability and that it is more appropriateto use displacement reduction factors (DRF) obtained directly from the ductility and hyster-esis type, without estimation of equivalent damping as an intermediate step. They have alsonoted that the slope of the elastic demand spectrum between initial and effective periods has asignificant effect on the values of DRF in long-period ranges. However, the model proposedby them does not incorporate this effect.

Figure 1 shows inelastic displacement response spectra (IDRS) obtained from a fewselected EL models for ductility values of 2%, 4%, 6%, and 8%, and 5% elastic damping.ASCE 7-10 (2010) design spectrum for a PGA value of 0.4 g on site class B has been con-sidered as the basis elastic spectrum, assuming the long-period transition period (also knownas the corner period between velocity-controlled and displacement-controlled spectralranges), TL, as 6 sec. In the case of all the models, elasto-plastic hysteresis type hasbeen considered, except in the case of the Pennucci et al. (2011) model, where theTakeda-Thin hysteresis type has been considered, as the model for elasto-plastic hysteresistype is not available. The spectra are normalized by maximum elastic spectral displacementand plotted against the effective period. It is observed that the Priestley et al. (2007) andPennucci et al. (2011) models estimate quite close IDRS for all ductility values. Further,the inelastic displacement estimated by Pennucci et al. model is generally conservative com-pared to other models, except for a ductility value of 2, wherein the Grant et al. (2005) modelpredicts slightly higher inelastic displacement.

As noted earlier, different EL models consider different definitions of effective periods,and hence a more rational comparison can be made by transforming the IDRS to the initialperiod using the relationships between initial and effective periods. Figure 2 shows the trans-formed displacement spectra against the initial period for ductility values of 2, 4, 6, and 8.

966 V. N. KHOSE AND Y. SINGH

An interesting observation about the transformed spectra is the dependence of corner periodson ductility and the definition of effective period. As the effective period in the EL models isa function of ductility, it increases (shifts) with ductility, but the corner period in the elasticspectra remains constant for all the values of damping. This results in the observed shift in thecorner period when the spectra are transformed to the initial period. Further, the magnitude ofshift depends on the definition of the effective period and two different classes of EL modelsbased on the two definitions of the effective period are distinctly visible in Figure 2. This shiftin the corner period leads to the anomaly in long-period spectral ranges, resulting in muchlower estimates of inelastic displacement.

THE ANOMALY

As observed earlier, the effective period of the inelastic systems shifts with ductility, butthere is no shift in the corner period in the elastic response spectra for different damping. Thisresults in an anomaly, causing the reduction of inelastic displacement in the long-period(displacement-controlled) range. This anomaly can be better illustrated by plotting displace-ment modification factors (DMFs) obtained from the EL approach, YS approach, and ana-lytical study, plotted against the initial period. As noted earlier, the Pennucci et al. (2011) andPriestley et al. (2007) models yield generally conservative results compared to other EL

Figure 1. Normalized inelastic displacement spectra obtained from various equivalent lineariza-tion models, for 0.4 g PGA, 5% elastic damping and varying ductility, μ. The spectra are normal-ized by the peak elastic spectral displacement.

ANOMALY IN THE EL APPROACH FOR THE ESTIMATIONOF INELASTIC RESPONSE 967

models, and thus these models have been chosen for comparison. Similarly, in the YSapproach, the Chopra and Chintanapakdee (2004) and Miranda (2000) models are the latestmodels and yield almost identical results, and these have been considered as representativeYS models for the comparative study.

The DMF, which is also known as inelastic displacement ratio (Miranda 2000, Chopraand Chintanapakdee 2004), is defined as the ratio of the inelastic spectral displacement toelastic spectral displacement and can serve as the common basis for the comparison of dif-ferent approaches. In the case of analytical study and YS approach, the DMF is obtaineddirectly from the estimated elastic and inelastic displacements, whereas in the case of theEL approach, the dependence of the effective period on ductility is to be considered, andthe DMF for a given ductility is expressed as:

EQ-TARGET;temp:intralink-;sec3;41;162DMFμ ¼ðΔinelasticÞTe

ðΔelasticÞT(1)

where ðΔelasticÞT is the elastic displacement at initial period, T ; and ðΔinelasticÞTeis the inelastic

displacement at corresponding effective period, Te.

Figure 2. Normalized inelastic displacement spectra transformed to initial period, obtainedfrom various equivalent linearization models, for 0.4 g PGA, 5% elastic damping, and varyingductility, μ. The spectra are normalized by the peak elastic spectral displacement.

968 V. N. KHOSE AND Y. SINGH

The analytical study has been conducted using an ensemble of five recorded groundmotions made compatible with ASCE 7-10 (2010) spectrum (Figure 3) for PGA value of0.4 g on site class B and the long-period transition period, TL, equal to 6 sec. The time his-tories have been selected from the PEER Ground Motion Database (PGMD 2011) to bestmatch with the target spectrum. In the selection of ground motions, the magnitude is chosenin the range 6.5–7, consistent with long-period transition period, TL, equal to 6 sec (FEMA2003); and shear wave velocity, VS30 is chosen as 760–1;500m∕sec (corresponding to siteclass B). The selected time histories have been further modified using a wavelet-based tooldeveloped by Mukherjee and Gupta (2002) to make them compatible with the target spectra.The nonlinear response analysis software BISPEC (2010) has been used to develop inelasticspectra for elasto-plastic hysteretic model. Average inelastic displacement spectra obtainedfrom the five spectrum-compatible ground motions have been used for obtaining the DMF.

Figure 4 shows the comparison of the DMFs obtained from the EL and YS approachesand the analytical study. It can be observed that in short- and intermediate-period ranges(T < TL∕

ffiffiffi

μp

), the DMFs obtained from the three approaches are close, and the EL approachyields mostly conservative estimates for ductility ratio 4 and higher. Further, the conserva-tism in the EL approach increases with the increase in ductility. However, for longer periods(T > TL∕

ffiffiffi

μp

), the DMFs from the EL approach start decreasing and become constant beyondTL. In this range of periods, the EL approach underestimates the DMFs compared to the YSapproach and the analytical results.

NUMERICAL EXAMPLES

The effect of the anomaly in the EL approach on the estimated peak inelastic displace-ments has been illustrated using examples of oscillators having different initial periods (3, 4,5, and 6 sec) and ductility (2, 4, 6, and 8). The same EL and YS Models considered in pre-vious section are also considered for the numerical study. The analytical inelastic displace-ment is obtained from the average response to the five spectrum-compatible ground motions,as discussed earlier.

Table 1 shows the peak inelastic displacements of the oscillators, obtained using differentapproaches. The analytically obtained displacement is shown in mm, whereas the displace-ment obtained using other approaches is shown as percentage of that obtained analytically.It can be observed that for ductility values of 2–6, the EL approach estimates inelastic

Figure 3. Target elastic spectrum (5% elastic damping) considered for study.

ANOMALY IN THE EL APPROACH FOR THE ESTIMATIONOF INELASTIC RESPONSE 969

Table 1. Comparison of inelastic displacement estimated by different approaches. (Theanalytically obtained displacement is shown in mm, whereas the displacement obtained usingother approaches is shown as percentage of the analytically obtained displacement.)

Ti

(sec)

Inelastic displacement

μ ¼ 2 μ ¼ 4 μ ¼ 6 μ ¼ 8

AS*

(mm)

EL

YS§

(%)AS*

(mm)

EL

YS§

(%)AS*

(mm)

EL

YS§

(%)AS*

(mm)

EL

YS§

(%)PCK†

(%)PSC‡

(%)PCK†

(%)PSC‡

(%)PCK†

(%)PSC‡

(%)PCK†

(%)PSC‡

(%)

3 290 92 97 103 297 111 118 101 261 121 130 115 255 122 131 1184 409 87 91 97 274 120 128 146 276 115 123 145 272 115 122 1475 390 96 102 128 325 101 108 153 374 85 91 133 362 86 92 1386 567 66 70 105 440 75 80 136 423 75 80 141 443 70 75 135

*Analytical study.†Priestley et al. (2007).‡Pennucci et al. (2011).§Chopra and Chintanapakdee (2004).

Figure 4. Comparison of displacement modification factors obtained by the EL approach, the YSapproach, and analytical study for 0.4 g PGA, 5% elastic damping, and varying ductility, μ.

970 V. N. KHOSE AND Y. SINGH

displacement either close to or more conservative than the analytical results (the Pennucciet al. model yielding slightly higher estimates compared to the Priestley et al. model) forperiods up to 5 sec, whereas for periods equal to 6 sec, the inelastic displacements estimatedusing the EL approach are significantly underestimated. In the case of ductility equal to 8, theinelastic displacement from the EL approach is underestimated even for a 5-sec period. In thecase of the YS approach, the estimated inelastic displacements are either conservative orclose to the analytical results for the whole range of period and ductility, considered inthe study. It can be concluded that the EL approach underestimates inelastic displacementin the whole of the displacement-controlled and in some part of the velocity-controlled spec-tral ranges. Further, the period beyond which the EL approach underestimates inelastic dis-placement reduces with an increase in ductility.

CONCLUSION

Equivalent linearization is one of the popular approaches to estimate peak inelasticdisplacements using response spectrum. An anomaly in the EL approach has been illustrated,which results in underestimated inelastic displacement in the whole of the displacement-controlled and in some portion of the velocity-controlled spectral ranges. The anomalyhas been demonstrated by plotting the displacement modification factors and estimates ofinelastic displacement for oscillators of varying period and ductility using the EL andYS approaches and an analytical study. Further, it has been noted that the period beyondwhich the EL approach underestimates inelastic displacement decreases with an increasein ductility.

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(Received 11 March 2012; accepted 4 December 2012)

972 V. N. KHOSE AND Y. SINGH