Report - PEER - University of California, Berkeley
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Report - PEER - University of California, Berkeley

Report - PEER - University of California, Berkeley Report - PEER - University of California, Berkeley

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2.2 Modal ScalingThe principal aim of IRSA is to estimate the above-defined modal displacementincrements and accordingly the other modal response quantities of interest during anincremental application of a piecewise linear RSA. Hence a reasonable estimation ofrelative values of modal response increments, which may be called modal scaling,constitutes the most critical part of the development of IRSA.An appropriate modal scaling procedure is proposed for IRSA in its inceptionstage (Aydinoglu 2003) where inelastic spectral displacements associated with theinstantaneous configuration of the structure are used to scale the modal displacementincrements. Interestingly, such a scaling procedure paves the way for adopting theequal displacement rule in practical applications where seismic input is defined viasmoothed elastic response spectrum. According to this simple and well-known rule,spectral displacement of an inelastic SDOF system and that of the correspondingelastic system are assumed practically equal to each other provided that the effectiveinitial period is longer than the characteristic period of the elastic response spectrum.The characteristic period is approximately defined as the transition period fromthe constant acceleration segment to the constant velocity segment of the spectrum.For periods shorter than the characteristic period, elastic spectral displacement isamplified using a displacement modification factor, i.e., C 1 coefficient given inFEMA 356 (ASCE 2000). However such a situation is seldom encountered in mid- tohigh-rise buildings and long bridges with tall piers involving multi-mode response. Insuch structures, effective initial periods of the first few modes are likely to be longerthan the characteristic period and therefore those modes automatically qualify for theequal displacement rule. On the other hand, effective post-yield slopes of the modalcapacity diagrams get steeper and steeper in higher modes with gradually diminishinginelastic behavior (Fig. 1). Thus it can be comfortably assumed that inelastic modaldisplacement response in higher modes would not be different from the correspondingmodal elastic response. Hence, smoothed elastic response spectrum may be used in itsentirety for scaling modal displacements without any modification. As a reasonablefurther simplification for practice, elastic periods calculated in the first pushover stepmay be considered in lieu of the initial periods, the latter of which are estimatedapproximately from the bi-linearization of the modal capacity diagrams (Fig. 1b).(a)(b)Figure 1. (a) Modal capacity diagrams, (b) scaling with equal displacement rule.349

When equal displacement rule is employed, the scaling procedure applicable tomodal displacement increments is simply expressed as (Aydinoglu 2003)(i) (i) (1)∆ dn= ∆F% Sden(6)(i)where ∆F % is an incremental scale factor, which is applicable to all modes at the(1)(i)’th pushover step. Sdenrepresents the initial elastic spectral displacement definedat the first step (Fig. 1b), which is taken equal to the inelastic spectral displacementassociated with the instantaneous configuration of the structure at any pushover step.Modal displacement at the end of the same pushover step can then be written as(i) (i) (1)dn= F% Sden(7)(i)in which F % represents the cumulative scale factor with a maximum value of unity:% (i) (i 1) (i)= % −+ ∆ % ≤1(8)F F FNote that Eqs.6,7 actually represent a monotonic scaling of the elastic responsespectrum progressively at each pushover step, which may be regarded analogous tothe scaling of an individual earthquake record as applied in the Incremental DynamicAnalysis (IDA) procedure (Vamvatsikos and Cornell 2002). The spectrum scalingcorresponding to the first yield and an intermediate step are indicated in Fig. 1b.It is worth warning that equal displacement rule may not be valid at near-faultsituations with forward directivity effect. On the other hand, legitimacy of the rulewith P-delta effects is another important issue addressed elsewhere (Aydinoglu 2004).It needs to be stressed that IRSA is a displacement-controlled procedure andtherefore the above-mentioned monotonic spectrum scaling applies to spectraldisplacements only, not to the spectral pseudo-accelerations. If required however, a(i)compatible modal pseudo-acceleration increment, ∆ a n, corresponding to theincrement of scaled modal displacement can be defined from Eqs.4,6 as(ω )∆ a = ∆ F% S ; S = S(9)(ω )(i) 2(i) (i) (i) (i) n (1)n ain ain (1) 2 aenn(i)where Sainrepresents compatible inelastic spectral pseudo-acceleration and S(1)aenrefers to initial elastic spectral pseudo-acceleration corresponding to the elastic(1)spectral displacement, Sden, defined at the first pushover step.At this point, it may be worthwhile to point out the main difference of IRSA froman essentially similar incremental response spectrum analysis procedure developed byGupta and Kunnath (2000). Note that the latter is a load-controlled procedure wheremodal pseudo-acceleration increments have been scaled at each step to define staticequivalentseismic load vector increments using instantaneous elastic spectral(i)pseudo-accelerations, Saen. The key point is the incompatibility of distribution of theso-defined instantaneous static-equivalent seismic loads with the resulting nonlinearinstantaneous displacement response (see Eq.11 below for compatible seismic loads).350

2.2 Modal ScalingThe principal aim <strong>of</strong> IRSA is to estimate the above-defined modal displacementincrements and accordingly the other modal response quantities <strong>of</strong> interest during anincremental application <strong>of</strong> a piecewise linear RSA. Hence a reasonable estimation <strong>of</strong>relative values <strong>of</strong> modal response increments, which may be called modal scaling,constitutes the most critical part <strong>of</strong> the development <strong>of</strong> IRSA.An appropriate modal scaling procedure is proposed for IRSA in its inceptionstage (Aydinoglu 2003) where inelastic spectral displacements associated with theinstantaneous configuration <strong>of</strong> the structure are used to scale the modal displacementincrements. Interestingly, such a scaling procedure paves the way for adopting theequal displacement rule in practical applications where seismic input is defined viasmoothed elastic response spectrum. According to this simple and well-known rule,spectral displacement <strong>of</strong> an inelastic SDOF system and that <strong>of</strong> the correspondingelastic system are assumed practically equal to each other provided that the effectiveinitial period is longer than the characteristic period <strong>of</strong> the elastic response spectrum.The characteristic period is approximately defined as the transition period fromthe constant acceleration segment to the constant velocity segment <strong>of</strong> the spectrum.For periods shorter than the characteristic period, elastic spectral displacement isamplified using a displacement modification factor, i.e., C 1 coefficient given inFEMA 356 (ASCE 2000). However such a situation is seldom encountered in mid- tohigh-rise buildings and long bridges with tall piers involving multi-mode response. Insuch structures, effective initial periods <strong>of</strong> the first few modes are likely to be longerthan the characteristic period and therefore those modes automatically qualify for theequal displacement rule. On the other hand, effective post-yield slopes <strong>of</strong> the modalcapacity diagrams get steeper and steeper in higher modes with gradually diminishinginelastic behavior (Fig. 1). Thus it can be comfortably assumed that inelastic modaldisplacement response in higher modes would not be different from the correspondingmodal elastic response. Hence, smoothed elastic response spectrum may be used in itsentirety for scaling modal displacements without any modification. As a reasonablefurther simplification for practice, elastic periods calculated in the first pushover stepmay be considered in lieu <strong>of</strong> the initial periods, the latter <strong>of</strong> which are estimatedapproximately from the bi-linearization <strong>of</strong> the modal capacity diagrams (Fig. 1b).(a)(b)Figure 1. (a) Modal capacity diagrams, (b) scaling with equal displacement rule.349

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