PSI - Issue 70

Edavalath Nadeem et al. / Procedia Structural Integrity 70 (2025) 19–26

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In this equation, d i denotes the peak demand value corresponding to the i th ground motion record, and N represents the total number of ground motion records considered. As mentioned in the previous section, this study considers µ ϕ m as the primary EDP, the corresponding S C values are tabulated in Table 1. However, for µ ϕ -r , the S C values corresponding to associated DSs are not promptly available. Consequently, they are determined from the DSs values of µ ϕ -m by adopting the Equation 1 derived in the previous section and the obtained values are listed in Table 1. Subsequently, the component fragility curves for µ ϕ -m and µ ϕ -r can be developed separately. Recognizing that seismic fragility curves considering both maximum and residual EDPs provide a more comprehensive assessment of a structure's vulnerability than those based on a single EDP (Wei et al ., 2023), combined fragility curves incorporating both µ ϕ -m and µ ϕ -r can be developed from the individual fragility curves corresponding to each EDP. This can be estimated using the joint probabilistic seismic demand model (JPSDM) proposed by Nielson and DesRoches (2007). However, developing a JPSDM is time-consuming and computationally demanding. Hence, the first order reliability theory can be utilized to determine the upper and lower limits of the combined fragility. The lower bound corresponds to the maximum of the fragility curves considering single EDPs, and the upper bound reflects the combination of fragilities considering both maximum and residual EDPs. These bounds can be formulated as follows (Zhang and Huo, 2009; Zheng et al ., 2018; Chen, 2020). = 1 [ ( )] ≤ ( ) ≤ 1 − ∏[1 − ( )] =1 (6) Here, m represents the number of EDPs of the vulnerable component, P(Fₖ) is the failure probability of the component and P(F com ) denotes the combined failure probability of the structure. Note that Equation 6 is valid only for a serial system where the overall system fails when any one of the components fails. The same concept can be applied here, as the structure fails if either the maximum or residual column curvature ductility demand value reaches As explained in the previous section, PSDM regression fits are developed between EDPs ( µ ϕ -m and µ ϕ -r ) and IMs ( AvgSa(g) ) in the logarithmic space. The vertical axis intercept, ln a value for µ ϕ -m and µ ϕ -r cases are obtained as +1.56 and -1.13 respectively. Similarly, the slope coefficients, for µ ϕ -m and µ ϕ -r cases are obtained as +0.76 and +1.03 respectively and the corresponding β D|IM values are noted as 0.48 and 0.92 respectively for µ ϕ -m and µ ϕ -r cases. Based on the PSDM regression fits derived, seismic fragility curves are developed for µ ϕ -m , µ ϕ -r and the combined cases. Figure 4 shows the fragility curves developed for the aforementioned cases. The exceedance probabilities of DS1, DS2, DS3 and DS4 in the case of µ ϕ -m are 0.99, 0.61, 0.19 and 0.06 at AvgSa = 1.0 g respectively. The exceedance probabilities of DS1, DS2, DS3 and DS4 in the case of µ ϕ -r are found to be 0.99, 0.43, 0.06 and 0.01 at the same IM respectively. Similarly, for the combined fragility ( µ ϕ -mr ) case, the exceedance probabilities of DS1, DS2, DS3 and DS4 are 1.00, 0.78, 0.24 and 0.07 at AvgSa(g) = 1.0 respectively. The median values of fragility ( AvgSa(g) at 0.5 exceedance probability) for all the considered cases are summarized in Table 2 and the same are shown graphically in Figure 5. The relative vulnerabilities of the structural system for a particular damage state can be assessed by comparing these median values. For a less vulnerable structure, the median value will be higher, and vice-versa. the capacity threshold value. 6. Fragility Analyses Results

(a)

(b)

(c)

Fig. 4 (a) Component fragility curves for µ ϕ -m (b) for µ ϕ -r and (c) combined fragility curves.