PSI - Issue 78

Yang Liu et al. / Procedia Structural Integrity 78 (2026) 2030–2037

2035

IM from small to large, Shahi, et al. (2011), Fig. 5 shows the proportion of FF, NFNP and NFPL source with respect to the total source contributions. The hazard curves of the site, for a reference life of 50 years, have been provided by the United States Geological Survey (2017). The probability of occurrence of seismic events with a given intensity IM, PH(IM=i), are determined and used into Equation (5) along with the value of s(ST=k | IM=i) shown in Fig.5. Finally, the probability of occurrence of NFPL, NFNP and FF earthquakes is computed. For the fragility analysis of the bridge, a suite of ground motions representative of NF and FF ground motions have been selected by using the PEER strong ground motion database of NGA-West2, Ancheta, et al. (2013). More details can be found in Liu et al 2024 4.3 Seismic fragility analysis The fragility analysis of the bridge has been conducted by assuming a log normal distribution of the seismic demand (D) and capacity (C), and deriving the probability of exceeding the capacity C as follows: where,  D|IM and  D|IM are respectively median and dispersion of the demand, obtained numerically through time history analysis;  C and  C are the median and the dispersion of the capacity obtainable through numerical analysis (i.e. push over analysis) or experimental investigations;  (•) is the standard normal cumulative distribution function. In this study, the record-to-record variability associated with the seismic action are considered in the probabilistic seismic demand analysis (PSDA). Moreover, the uncertainties associated with the structural response and the capacity is considered in the probabilistic seismic capacity analysis (PSCA). Accordingly, the PSDA of the bridge is performed through Multiple Stripe Analysis (MSA), Tothong, et al. (2008). A total of 144 nonlinear time-history analysis have been performed. The pier drift ratio (ratio of the maximum displacement of pier top and the total height of pier, DR) and the pier displacement ductility (ratio of the maximum displacement and the yielding displacement of the pier,  p ) have been used as Engineering Demand Parameters (EDP), Paolacci and Corritore (2023). By using the limit stated provided in Liu et al 2024 the fragility curves representing the probability of exceedance at each level of IM for NFPL, NFNP, FF condition and for the entire set of ground motions, are shown in Fig. 6. These results represent the components fragility curves and that for a complete representation of the seismic vulnerability of the bridge their combination should be used. Based on the conservative assumption of element is series, the system fragility can be expressed as:   component system 1 =1 1 n c c Fragility Fragility = − −  (7) in which, the n component is the number of components considered in the analysis, c is the specific component. By In this section, the parameters of the functionality recovery functions for the four damage states are estimated by the method presented in section 2. Empirical values for the three recovery parameters of the residual functionality, idle time and recovery time can be determined based on the evaluation of data from the real functionality recovery of bridges in California as listed in Table 4, Shinozuka, et al. (2005), HAZUS (2009). The values of the shape parameters (defined as A and s) recommended by Dong, et al. (2013) for the recovery functions, which are based on engineering experience, used. The parameter A governs the amplitude of the sinusoidal function, and the parameter s defines the flex position of the sinusoid, as defined in Bocchini et al. (2012). In addition, the target functionality is assumed as the constant and equal, for each damage state, to 100%. Fig. 7 shows the recovery empirical functions for the four damage states according to the values of Table 4. The curves of Fig. 7 have been obtained by applying Equation (7). Consequently, the expected functionality of the bridge in 50 years can be calculated applying Equation (8), which includes the effects of NF earthquakes, indicated in the following as RM. For comparison, the functionality without distinguishing NF from FF conditions is also estimated, condition indicated in the following as SM. applying Equation (7), the system fragility is obtained. 3.5. Functionality recovery model and resilience analysis 2   + 2 D IM ) ln( ) ln( − ( P D C IM  ) C D IM C m m         = (6)