PSI - Issue 84

Enrique García-Macías et al. / Procedia Structural Integrity 84 (2026) 837–844

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which are considered in the subsequent analyses. With regard to the loading conditions, the dynamic response of the bridge has been studied under the passage of the A1 train of the HSML-A model of Eurocode.

Fig. 3. Continuous three-span bridge proposed in Eurocode 1 (CEN (2003)).

Fig. 4. Displacement (a,c) and acceleration (b,c) series at mid-span of a simply supported beam traversed by the A1 train travelling at 160 km/h (a,b) and 277 km/h (c,d) (semi-analytic: ∆ = 1 ms; HT: 10 Hz cut-off frequency, (19 )/ reduced Hilbert integral, ∆ = 2 ms ). Figures 4(a,b) and (c,d) show the displacement and acceleration time histories at mid-span of the central span for train speeds of 160 km/h and 277 km/h, respectively, together with the sampling points employed in the maximum identification procedure. The solid blue curves correspond to the semi-analytic solution, while the light-green curves represent the instantaneous envelopes obtained via the HT. The green markers denote the initial coarse sampling of the response, the orange points indicate the function evaluations performed by Brent’s method during the interval reduction stage, and the red markers correspond to the locally refined evaluations in the vicinity of the identified extrema. As observed in the displacement responses (Figs. 4(a,c)), the optimisation algorithm concentrates function evaluations around the regions of interest, progressively refining the estimate of the peak response. A similar behaviour is observed for the acceleration responses (Figs. 4(b,d)), despite their highly oscillatory nature. In this case, the dense clustering of evaluation points around the extrema highlights the effectiveness of the Brent-based strategy in handling strongly non-convex objective functions.