PSI - Issue 78

Danilo D’Angela et al. / Procedia Structural Integrity 78 (2026) 1617–1624

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curves, expressed as functions of PGA* and PGV*, were defined for multiple damage states (DSs). The attainment of each DS was determined by the exceedance of the dimensionless rotational angle limit associated with related DS ( |θ max /α| DS,lim ) by the dimensionless rotational limit ( |θ max /α| ). A total of 12 DSs were introduced to comprehensively cover the full range of rocking amplitudes, from the onset of rocking to (static) overturning, including intermediate levels. The selected |θ max /α| lim values ranged from 0.01 to 1.0, through small increments. The fragility parameters, namely median (x M ) and logarithmic standard deviation ( σ ), were estimated according to Equations (6) and (7) (Porter et al., 2007), for each combination of key features, i.e., RB, DS, EDP, and loading history set. Finally, after the estimation of the fragility parameters, fitting surfaces were computed in terms of x M as a function of block parameters (R or p) and DS (expressed in terms of |θ max /α| DS,lim ), for investigated EDPs and loading history sets. x M = exp (M1 ∑ log (r i ) M i=1 ) (6) σ=√M1−1∑ [log(r i x m )] 2 M i=1 (7) 3.2. Results Capacity surfaces were constructed by fitting the fragility results in three dimensions, namely PGA* or PGV*, |θ max /α| lim , and R or p. A second-degree polynomial regression was adopted for all cases under investigation, yielding consistently high coefficients of determination (COD), typically exceeding 0.9. Fig. 3, Fig. 4, and Fig. 5 show capacity surfaces, corresponding to fragility medians (x M ) expressed in terms of dimensionless peak ground acceleration (PGA*) as a function of limit dimensionless rocking amplitude angle (|θ max /α| lim ) and semi-diagonal block dimension (R), considering strong ground motion (SGM), all floor motions (FM), and strong floor motions (SFM), respectively. The form of the fitting curves is reported in Equation (8), where Z stand for PGA* or PGV*, X to |θ max /α| lim , and Z is associated with R or p. Fitting coefficients and related COD values can be found in (D’Angela and Magliulo, 2025). In particular, the specification of the coefficient depends on the considered intensity measure (PGA* or PGV*), block dimension (R or p), and input set (SGM, FM, and SFM). | = 00 + 10 + 01 + 20 2 + 11 + 02 2 (8) The plotted surfaces serve as “universal” capacity estimation tools (Dimitrakopoulos and Paraskeva, 2015), meaning that the same functional form can represent various IMs and input sets, depending on the block geometric and dynamic parameters (R and p).