PSI - Issue 78

Andrea Nettis et al. / Procedia Structural Integrity 78 (2026) 1412–1419

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4. Discussion of results The EAL calculated by using the CSM and NLTHA are compared in Fig. 2 in the case of a cloud analysis involving the power-law functional form. The average value of EAL computed by using the two approaches is also indicated to determine global comparisons for all the bridge realisations. Generally, the EAL is lower for the longitudinal direction with respect to the transverse one. This is a useful remark to consider when developing a simplified loss-based design framework, which should be more driven by the response of the bridge in the most critical direction, as proposed by Gentile and Calvi for buildings (Gentile and Calvi, 2023). Fig. 2 also shows that, as expected, short piers introduce a higher bridge vulnerability with respect to taller piers. In the longitudinal direction, the CSM provides satisfying accuracy with respect to NLTHA. Particularly, in this case, it overestimates the NLTHA-based EAL with a relative error on the average EAL estimation equal to 12% within the case-study dataset. The highest discrepancies are associated with bridge realisations whose DS is governed by tall piers (i.e. B3, B333, B33333). However, such case studies are associated with negligible EAL values (i.e., lower than 10 −5 ) and, therefore, such relative errors are not deemed unsatisfactory, since they correspond to particularly low absolute-term errors, unlikely to affect the effectiveness of future loss-based design procedures. In the transverse direction, the average overestimation of the CSM is equal to 75%. This unsatisfactory result is likely caused by the relevant overestimation of the fragility dispersion by the CSM inducing overestimations in the probability of damage for low IM values which are characterised by a relevant annual probability of exceedance. In both directions, the CSM produces EAL estimations which are on the safe side compared to NLTHA. This is a desirable feature of an approximate loss-based design procedure.

Fig. 2. Expected Annual Losses computed by CSM and NLTHA considering a power-law functional form within the cloud analysis.

Fig. 3 illustrates the EAL calculated using CSM and NLTHA within a multi-stripe analysis. Also, the EAL computed via cloud-based NLTHAs are included to compare the approximation related to the simplified seismic response analysis strategy and the assumptions of the fragility analysis approach. Considering these analysis strategies, Fig. 4 reports the probabilistic seismic demand models, fragility curves for DS1 and DS4 and the vulnerability curves for the case B222. In the longitudinal direction, the multi-stripe CSM exhibits an overestimation with respect to the average NLTHA-based EAL equal to 40%. Although this error is relevant, it is lower than the discrepancies observed when comparing CSM and NLTHA-based estimates adopting the cloud analysis to estimate demand. Indeed, the use of cloud analysis induces an overestimation on the average EAL equal to 60%. This finding proves that, for case studies consistent with this parametric analysis, the approximation induced by the choice of the demand and fragility analysis approach (i.e., cloud vs multi-stripe) are more influential than the one introduced by using a simplified seismic response analysis strategy (i.e., CSM vs NLTHA). Fig. 3b illustrates that the errors related to use of the CSM increase substantially in the transverse direction reaching overestimations higher than 100%. Such inaccuracies are lower than the ones derived by the cloud-based NLTHA with respect to the multi-stripe analysis, proving that in this direction, the approximations related to the simplified analysis approach are more severe than the ones associated with the fragility analysis assumptions. Indeed, as illustrated in Fig. 4d, for the analysis in the transverse direction, a severe