PSI - Issue 44

702 Fabrizio Paolacci et al. / Procedia Structural Integrity 44 (2023) 697–704 Fabrizio Paolacci et al./ Structural Integrity Procedia 00 (2022) 000–000 where Φ ( ∙ ) is the standard normal cumulative distribution function, LS m is the median estimate of the structural limit state, D m is the median estimate of the demand, β d|IM is the dispersion of the demand conditioned on the IM, and β LS is the dispersion of the structural limit state. The proposed methodology aims to study the seismic structural behaviour of the bridges taking in account the ductility fail mechanisms. Along this vein, the ultimate displacement (D u ) of the structure was considered as structural limit state (LS m ), while the maximum displacement (D max ) expected in the site where the bridge is located was taken in account as demand (D m ). Parameters D u , used to build the fragility curve, is herein evaluated by means of a pushover analysis, where D u corresponds to the ultimate displacement of the capacity curve of the structure, as shown in Fig. 4a. Along this vein, the bridge is modelled with a simplified Finite Element Model where the piers are modelled as beam elements and desk modelled as concentrated mass. The pushover capacity curve obtained can be approximated to a bilinear capacity curve as described in Günay 2008 and shown in Fig. 4a. Once the linear pushover capacity curve of the structure has been carried out, D u is quickly evaluated. The demand parameter D max could be evaluated from the spectra relative to the PGA considered and the pushover capacity curve. Along this way, the maximum elastic displacement of the structure D e,max is carried out graphing in the same plane the acceleration-displacement seismic response spectra (ADSR), relative to the specific limit states considered with a fixed PGA, and the bilinear pushover curve. The intersection point between the extension of the slope line of the capacity curve and the spectra represent the elastic maximum displacement of the structure, as shown in Fig. 4b. The plastic displacement D max , relative to the PGA considered, can be evaluated from D e,max , depending if the structure is in the equal displacement or equal energy field, following the well-known theory of the method N2 described in the codes. Nonetheless, a simplified procedure to evaluate the D max relative to the PGA of interest is herein considered. Following the suggestion of Pinto et al. 1996, the value of the demand D max , can be quickly evaluated with Eq. (4): (T) = ( ) 2 ∙ (4) Where R(T) is the value of spectral acceleration relative to the first mode of the bridge normalized with respect to the PGA of the limit state considered, = 2 is the pulsation of the structure relative to the first period and PGA is the specific value of acceleration that we want to know the relative displacement. Finally, the values of the dispersion β d|IM and β LS are both assumed 0.2 following the suggestions described in Pinto et al. 1996. The last step is to evaluate the linear equation of the function that better approximate the hazard curve in the PGA range of interest, Fig. 3a, in order to evaluate the parameter for the λ ( LS ) evaluation. SDOF system curve 6

(a)

(b) Fig. 4 a) Evaluation of D max b) Bilinear pushover capacity curve

4.2. Application to the case study The proposed methodology has been applied to a significant case study. The selected case study is a 3-spans bridge with a total length of 27 m and a maximum height of 2.95m with frame piers. A simplified FEM model of the pier has been developed in Midas Gen environment in order to carry out a pushover analysis. Beams elements has been used