PSI - Issue 70

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

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correlation, the corresponding DS values for µ ϕ -r are computed for each damage state. These derived values are presented in Table 1, enabling a comprehensive fragility assessment based on both µ ϕ -m and µ ϕ -r demands.

Fig. 3. Relationship between µ ϕ -m and µ ϕ -r .

5. Fragility Curve Development Fragility curves are probabilistic tools that quantify the likelihood of a structural component or system reaching or exceeding predefined DS under various levels of seismic intensity. These curves are constructed using PSDMs, which establish statistical relationships between selected EDPs and seismic Intensity Measures (IMs) (Cornell et al., 2002; Ramanathan, 2012; Dukes, 2013; Mangalathu, 2017; Soleimani, 2017; Dong and Frangopol, 2015). In the present study, the average first-mode pseudo-spectral acceleration ( AvgSa ), expressed in units of g , is employed as the IM. This choice is informed by prior research highlighting AvgSa(g) ’s robustness in capturing structural response characteristics across a range of vibration periods. Further, AvgSa(g) is defined as the geometric mean of the pseudo spectral accelerations over a specified range of periods, typically from 0.5 T₁ to 1.5 T₁ , where T₁ denotes the fundamental period of the structure. The selected range is discretized at appropriate time intervals ( n ) to ensure accurate representation of spectral behaviour. The mathematical expression for AvgSa(g) is given as follows (Bianchini et al. , 2010; Kohrangi et al. , 2016; Anand et al. , 2024; Aditya et al. , 2025). ( ) = [∏ − =1 ] ( 1⁄ ) (2) Here, S a-Ti denotes the pseudo-spectral acceleration values corresponding to 10 periods of interest, ranging from 0.5 T 1 (0.37 s) to 1.5 T 1 (1.10 s). Further, to derive the fragility curves, both the demand ( D ) or EDP and capacity ( C ) or S c are assumed to follow a lognormal distribution (Srivastava et al ., 2022; Srivastava et al ., 2024; Nielson and DesRoches, 2007). Subsequently, the fragility curves can be formulated using the following expression. [ > | ]= [ ( / ) √ 2 | + 2 ] (3) Here, Φ [•] represents the standard normal cumulative distribution function. S D and S C are the median estimates of demand and capacity, respectively. β D|IM denotes the logarithmic standard deviation of the demand for a given IM, while β C represents the logarithmic standard deviation of the capacity with a value of 0.35 across all considered cases (Padgett, 2007; Ramanathan, 2012; Dukes, 2013; Mangalathu, 2017; Soleimani, 2017). The left-hand side of the equation expresses the conditional probability that the demand exceeds the capacity for a given IM level. NLTHA results provided paired values of IM and corresponding demand for each ground motion record is used to construct PSDMs, for developing the demand side of the fragility curves (Baker, 2015; Stefanidou and Kappos, 2016; Soleimani et al ., 2017; Pandikkadavath et al ., 2022; Jithiya et al ., 2021; Jithiya et al . 2022; Pang et al ., 2023). These models establish a linear relationship between the S D and IM in logarithmic transformed space as shown below. ( ) = ( ) + ( ) (4) Where, ln(a) represents the intercept on the vertical axis, while b is the slope obtained through linear regression analysis. The corresponding β D|IM , required for the fragility function, can be calculated using the following expression. | = √∑ [ ( ) − ( )] 2 = 1 −2 (5)