PSI - Issue 78

Joud Habib et al. / Procedia Structural Integrity 78 (2026) 799–806 damage state , 1, represents the intercept component related to the evaluated damage state and is the slope for damage state . and 1, are computed using the coefficients ( , ) (Fig. 4. (b)) in a similar way as in equation (3). Before fitting this model, damage is converted into a categorical integer Y= [0,1,2,3,4], representing the corresponding damage state based on EDP and thresholds. The maximum likelihood estimation (MLE) of the fitted data is used to estimate the coefficients is in equation (6), where , =1 if ∈ , and , =0 otherwise, , =1 if = , and , =0 otherwise. To preserve the logical ordering of the fragility curves and avoid overlaps, the following constraints are imposed on the intercepts 0,0 ≥ 0,1 ≥ 0,2 ≥ 0,3 and 1,0 ≤ 1,1 ≤ 1,2 ≤ 1,3 . Φ −1 ( ( ≤ | , )) = η i,j = 0, + 1, + ln( ) ; = 0,1,2,3 ; = ,…,3 (5) = ∏ ∏ ∏ ( ( = | , )) , × , 4 = 3 =0 (6) For this model, the probability of reaching or exceeding damage state , given that the initial damage state is and the intensity measure is is calculated using equation (7), where as defined before while represents the damage state where the probability of reaching or exceeding is evaluated and starts with +1 and ends at 4. ( ≥ | , ) = 1 − Φ[ 0, + 1, −1 + −1 ln( )] ; = 0,..,3 ; = + 1,…,4 (7) 5. Results For all eighteen typologies, regression coefficients for both proposed models were computed, and the corresponding fragility curves — both independent and state-dependent — were derived. To highlight the advantages of the proposed approaches, fragility curves were also developed using the standard probit regression, where overlapping among curves may occur. Fig. 5. presents an illustrative example of extensive damage fragility curves for CDL-H3-5, conditioned on different initial damage states. As shown in Fig. 5. (a) standard probit regression leads to overlapping curves, undermining the logical progression of damage states. In contrast, Fig. 5. (b) and (c) demonstrate that both proposed models produce consistently ordered fragility curves without overlap. Beyond the issue of overlap, a comparative analysis with the standard probit (SP) model — an unconstrained regression approach — shows that both proposed models successfully maintain the correct logical order of the fragility curves. While SP may result in overlapping curves due to the lack of ordering constraints, CP and OPCDS address this issue effectively. 805

Fig. 5. Extensive damage fragility curves for CDL-H3-5: (a) using SP; (b) using CP; (c) using OPCDS

In comparing the two proposed models, their relative performance varies depending on the initial damage state. For fragility curves conditioned on no or slight initial damage, the conditional probit model provides results that are closer to those of the unconstrained SP model, while still preserving the logical order. Conversely, for higher initial damage states, the ordinal probit model with distinct slopes performs better in capturing the expected progression of vulnerability. This highlights the complementary strengths of the two approaches across different damage-state scenarios. These results also underscore that prior damage significantly increases structural vulnerability during