Report - PEER - University of California, Berkeley

2.5 Displacement Capacity as a Function of PeriodInspection of the displacement capacity equations given above renders clear that, inorder for the capacity curves to be graphically superimposed onto a period-dependentdemand curve (response spectra), as suggested in Figure 1, it is necessary to replaceall H t terms present in these capacity equations with period-dependent functions. Suchstep has been carried out by Glaister and Pinho (2003), leading to a set of capacityequations that are conceptually identical to Equations (1) to (7), but convenientlydefined in terms of effective period, rather than height. For the sake of succinctness,however, such formulae are not reproduced here, being nonetheless found in Crowleyet al. (2004).2.6 Displacement DemandDisplacement response spectra are used in this method to represent the input from theearthquake to the building class under consideration. The relationship betweenequivalent viscous damping (ξ) and ductility (µ), used to account for the energydissipated through hysteretic action at a given level of ductility, is presented in thefollowing equation:−b( i )ξ = a1− µ + ξ(10)Ewhere a and b are calibrating parameters which vary according to the characteristicsof the energy dissipation mechanisms, whilst ξ E represents the equivalent viscousdamping when the structure is within the elastic, or pre-yield, response range. Valuesof a=25, b=0.5 and ξ E =5%, suggested by Calvi (1999), are currently adopted.The equivalent viscous damping values obtained through Equation (10), fordifferent ductility levels, can then be combined with Equation (11), proposed byBommer et al. (2000) and currently implemented in EC8 (CEN, 2003), to compute areduction factor η to be applied to the 5% damped spectra at periods from thebeginning of the acceleration plateau to the end of the displacement plateau;( ζ ) ⎤1/2η = ⎡⎣10 5 + ⎦ (11)3. PROBABILISTIC FRAMEWORKThe first-order reliability method (FORM) can be used to calculate the approximatecumulative distribution function of a non-linear function of correlated randomvariables, such as the limit state displacement capacity function and limit state periodfunction. Once the cumulative distribution functions of the demand and the capacityhave been found, the calculation of the probability of exceedance of a specified limitstate can be obtained using the standard time-invariant reliability formulation (e.g.,179