PSI - Issue 44

Carlo Del Gaudio et al. / Procedia Structural Integrity 44 (2023) 259–266

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Carlo Del Gaudio et al. / Structural Integrity Procedia 00 (2022) 000–000

Resulting fragility functions pointed out the general trend of seismic vulnerability to increase with the building height and its reduction with design level and construction age. Some anomalies were however detected in the empirical seismic vulnerability of some building typologies, mainly due to data paucity affecting and limiting the reliability of the obtained results. Based on these considerations, two empirically based procedures, exploiting different data processing techniques and fitting strategies, were developed, allowing for a robust seismic fragility model overcoming issues related with limited amount of data. 3.1. Empirical-Constrained (EC) procedure The Empirical-Constrained (EC) procedure was conceived to automatically adjust any possible “anomalies” to the solution of the non-linear optimization problem, namely in the unknown parameters’ estimation of lognormal fragility curves. These anomalies could derive for example from a heterogeneous assortment of data, sometimes constituted by a potential limited number of buildings, thus resulting not adequately populated. This correction is automatically applied by imposing a set of inequality between the unknown parameters, such they can follow the observed hierarchies with the main vulnerability factors, i.e. with number of storeys, NS, construction age, CE, design type, DT, as well as with damage state, DS. A unique non-linear optimization process, searching for the minimum of the multivariate (multinomial) likelihood function is fulfilled, which solution returns the unknown parameters for the twenty-four building typologies defined in Table 1, constituted by 120 logarithmic mean, !" , characteristic of the k th typology (k = 1,…,24) and the i th DS (j = 1,…,5), and by a unique logarithmic standard deviation value, , common to all classes: ( $ , ) = argmax ())) - # ! !# ! 2 = ! | # : " $ !" ; " # ! < (3) 2 = ! | # : " = ⎩⎪ ⎨ ⎪ ⎧ 1 − Φ C log( / , ) D = 1 Φ C log( / , ) D − Φ C log( / +1, ) D ∈ [2; 4] Φ C log( / , ) D = 5 (4) where all terms are previously defined in eq 1-2. • Moreover, the solution of this non-linear optimization process must guarantee the respect of 306 linear constraints, representing the compliance with the general hierarchies observed as a function of the main vulnerability factors, i.e. with number of storeys, NS, construction age, CE, design type, DT, as well as with damage state, DS: • 96 constraints enforce greater fragility increasing the severity of DS ( !|&',)',*+ − !,-|&',)',*+ ≤ 0 ); • 90 constraints enforce greater fragility increasing the number of storeys ( &',-|!,).,*+ − &'|!,).,*+ ≤ 0 ); • 80 constraints enforce lower fragility increasing the construction age ( ).|!,&',*+ − ).,-|!,&',*+ ≤ 0 ); • 40 constraints enforce lower fragility going from gravity loads- to seismic loads designed buildings ( *+|!,&',). − *+,-|!,&',). ≤ 0 ). These constraints are effectively introduced through the following linear inequality: ∙ 2 R , : ≤ (5) where 2 R , : is the vector of the unknown parameters and A is the coefficient matrix of dimensions M (equal to the number of constraints = 306) x N (equal to the 121 unknown parameters).