Prediction of inelastic structural response using an average of spectral accelerations

In probabilistic engineering assessment, the ground motion intensity measure (IM), which quantifies the severity of a seismic event, is used both as a scale factor for recorded ground motions in incremental dynamic analysis and as that parameter which defines the seismic hazard at a specified site. In this paper, the geometric mean of pseudo-spectral acceleration ordinates over a certain range of periods, Sa,avg(T1,.,Tn), or briefly Sa,avg, is used as an optimal scalar IM to predict inelastic structural response of buildings subjected to recorded ground motions. This average of spectral values is a better predictor than both the elastic pseudo-spectral acceleration at fundamental period of structure and the peak ground acceleration, especially for inelastic structural systems. Furthermore, the seismic hazard at the site in terms of Sa,avg as IM is simpler than the one performed for vector-values and inelastic IMs. Especially for multi-degree-of-freedom systems with long periods, Sa,avg is very sensitive to higher-mode effects and it is able to account degrading behaviour. Sa,avg is studied as a statistical predictor of structural response through its desirable properties and results are compared with traditional elastic-based scalar IMs. Sa has been used as a predictor and IM in seismic performance assessment, although significant variability in the structural response level has been observed for tall and long period buildings. This problem, which is named as inefficiency of Sa, is in part due to the fact that it does not reflect important higher mode spectral accelerations and any spectral shape (Abrahamson & Silva 1997). This question has been addressed by pairing it with a measure of spectral shape, ε, in a vector-valued IM (Baker & Cornell 2005a), <Sa,ε>. Nevertheless, in the light of PSDA approach, to a vector-valued IM must be associated a vector-valued PSHA to obtain the joint hazard curve, which has not yet been commonly applied, or using the conventional seismic disaggregation analysis (Bazzurro & Cornell 1999). It has been shown that the effectiveness of ε as a criterion to select ground motion records and to predict inelastic response of multi-degree-of-freedom systems is considerably greater than that of Sa alone for ordinary strong ground motions. In order to provide a good predictor of inelastic structural response for those buildings located both far from and near to earthquake source, a new scalar intensity measure based on the inelastic spectral displacement, Sdi, has been recently developed. It has been demonstrated that, for structural systems dominated by the first mode of vibration, the Sdi predictor is more convenient than <Sa,ε>, whereas, for those structures affected by higher-mode periods, it needs to be combined with a higher frequency elastic spectral displacement (Tothong & Luco 2007). Especially for practical applications, the difficulties in working with a vector-valued or inelastic IM can be a barrier which is hard to overcome. It is well-known that structural response of multi-degreeof-freedom systems or inelastic systems is sensitive to multiple periods Ti (Baker & Cornell 2005b), so an intensity measure which averages elastic spectral acceleration values over a certain range of periods might be a useful and convenient predictor of structural response of inelastic systems. The concept of averaging spectral acceleration values over a certain period range was already anticipated in federal provisions, although it is more of a rough guide based on design spectrum to choose recorded ground motion rather than to define one predictor. In fact, both for twoand three-dimensional response history analysis procedures, many codes states that response spectra for the suite of motions is not less than the design response spectrum for the site, for periods ranging from 0.2·T to 1.5·T (SEI 2005). Nevertheless, this choice of period range has never been formally evaluated. By comparing different response predictors, it was shown that correlation between damage, higher modes influence and elastic spectral acceleration were improved by averaging the spectral ordinates over an interval with bounds related to T. Depending on the multiplicity of degrees of freedom of structures, the higher vibration mode influence and the rate of degradation of the hysteresis nonlinear model, an average of spectral acceleration ordinates was studied as a efficient statistical predictor of inelastic response (Bianchini et al., in prep). The present work aims to show the desirable properties of Sa,avg(T1,.,Tn), or briefly Sa,avg, as a ground motion IM to be used in PSDA. Sa,avg was defined as the geometric mean of the spectral acceleration ordinates on a set of n periods, i.e., the n-th root of the product of n elastic spectral values, and it is applied here to demand assessment of inelastic multi-degree-of-freedom systems. It is found that Sa,avg can be used as a useful and practical predictor of structural response, compared with the most conventional elastic-based scalar IMs. 2 RESPONSE PREDICTION IN PBEE FRAMEWORK 2.1 General considerations The aim of PSDA is to assess the structural performance of a given building by probabilistic assessment of the response under ground motion records. The latter is subsequently combined with information about seismic hazard at the site, calculated using PSHA, which provides the MAF of exceeding a specified level of IM value, λIM(im) = P[IM > im]. In general, structural response can be simulated through EDPs using the chosen IMs and corresponding earthquake records. One method of calculating EDPs is through nonlinear incremental dynamic analysis (IDA), which predicts structural response under ground motion records incrementally scaled to different IM levels (Vamvatsikos & Cornell 2002). A collection of IDA curves parameterized on the same IMs and EDP and generated for the same structure under different recording defines the socalled IDA curve set. Then, median, 16%-ile and 84%-ile IDA are defined to summarize an IDA curve set. This is consistent also with the assumption to consider, for non-collapse data, the conditional response EDP (given an IM level) lognormal distributed around the mean and the standard deviation of their natural logarithms, respectively, called median and dispersion. By combination of the sitespecific ground motion hazard curve with the structural response information, the MAF of exceeding a specified level of EDP value, λEDP(edp) = P[EDP > edp], is obtained by the total probability theorem (Benjamin & Cornell 1970) as follow ( ) ( ) ( ) EDP IM EDP IM , d im edp G edp im im λ λ = ∫ (1) where GEDP|IM(edp,im) = P[EDP > edp | IM = im] means the probability of exceeding a specified EDP level, edp, given a level of IM, im, or the complementary distribution function of EDP given IM. Equation (1) represents the classical form of PSDA proposed in PBEE framework. However, in order to compare the desirable properties of different IMs (i.e., their influence on λEDP(edp) assessment), it is not necessary to assess the whole integral, but only the statistics on GEDP|IM. Figure 1. Distribution functions for (a) IM-based rule and (b) EDP-based rule of IDA curve set constituted of a suite of 40 recorded ground motions for a nonlinear single-degree-offreedom systems. The area subtended by the bold red line represents (a) GEDP|IM(edp,im) and (b) FIMcap|EDP(im’,edp). The first integrand in Equation (1) assumes the so-called IM-based rule, which provides the distribution of demand that a given level of intensity can generate in the structure. In an IDA curve set it is hard to define a value that signals collapse for all IDA curves; in other words, prescribing a single point on the IDA curves that clearly divides them in two regions (non-collapse and collapse) does not result always feasible. This difficulty can be overcome by the EDP-based rule, which provides the distribution of intensities IM that are required to produce a level of damage, edp, or a given ductility level, μ. By using an EDP-based rule instead of the IMone, the structural response hazard can be computed as