PSI - Issue 78

Joud Habib et al. / Procedia Structural Integrity 78 (2026) 799–806

804

4. Derivation of fragility curves 4.1. Conditional probit regression (CP)

This model is a modified version of the standard probit approach proposed by David et al. (2000), designed to avoid overlapping fragility curves for increasing damage states within a relevant IM range, specifically from 0.1g to 8g. Instead of directly estimating the traditional regression coefficients — the slope , and intercept , — the model first estimates the coefficients ( , , , ) which correspond to the values of the linear predictor , at ln(0.1g) and ln(8g), respectively. The predictor , represents inverse of the probability of reaching or exceeding conditional on initial damage as defined in Equation (3). Here represents the initial damage state ; represents the damage state where the probability is evaluated and starts with +1 and ends at 4. These coefficients are used to calculate , and , as explained in equation (4), Fig. 4. (a) shows these coefficients for different linear predictors. −1 [ ( ≥ | , )] = , = , + , ( ) (3) , = , − , (8)− (0.1) , , = , − , × (0.1 ) (4) To ensure the logical order and avoid intersections of the fragility curves, the model imposes the following constraints on the coefficients: , ≥ , +1 , , ≥ , +1 and , ≤ +1, , , ≤ +1, . Before fitting this model, damage is converted into a binary variable, 1 if EDP equals or exceeds the threshold of where the probability is been evaluated and 0 otherwise . The probability of reaching or exceeding damage state , given that the initial damage state is and the intensity measure is , is defined as the standard normal cumulative distribution function evaluated at the corresponding linear predictor. Since the regression is applied to each damage state separately, it must be performed sequentially, starting from the initial damage state 0 to determine the first coefficients 0,1 and 0,1 , and so on.

Fig. 4. Linear predictors for; (a) CP; (b) OPCDS.

4.2. Ordinal probit regression on combined datasets with different slopes (OPCDS) This model is a modified version of the ordinal regression; it is applied on a combined dataset containing all the datasets with different initial damage states and it employs four distinct slopes for the predictors instead of a single average slope. The linear predictor η i,j underlying the ordinal regression is defined in equation (5), where represents the initial damage state starts with 0 and ends at 3, represents the damage state where the probability of being at or below is evaluated and starts with and ends at 3, 0, represents the intercept component associated with the initial