PSI - Issue 84

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

844

single post-processing point, which combines an HT-based envelope with a reduced sampling interval of 2 ms. As observed in Fig. 6(a), the displacement response exhibits a forced-vibration phase followed by free-vibration decay. The instantaneous envelope closely bounds the oscillatory response and enables accurate localisation of the peak displacement, yielding a maximum value of 2.053 mm, which is virtually identical to the reference semi-analytic result. A similar behaviour is observed for the acceleration response in Fig. 6(b), which is strongly oscillatory during the train passage and rapidly decays afterwards. Despite the complexity of the signal, the proposed approach accurately identifies the region of maximum response, with a peak acceleration of 0.499 m/s², identical to the semi-analytic solution. From a computational standpoint, the proposed framework significantly improves efficiency. While the reference semi-analytic evaluation required approximately 7.86 min, the maximum identification procedure was completed in 15 s, corresponding to an approximately 31-fold reduction in computational time, while maintaining reliable peak response estimates for this large-scale structure. It should be noted that the computational cost of the proposed approach scales with the number of post-processing points. Consequently, its efficiency may decrease for a large number of response locations, although further investigation is required to assess this aspect. 4. Conclusions This work proposes an optimisation-based strategy for the rapid identification of maximum dynamic responses in high-speed railway bridges, exploiting the closed-form time-domain expression of the semi-analytic solution. By formulating the maximum response estimation as a scalar optimisation problem in time, the method avoids uniformly fine time discretisation and concentrates response evaluations in regions of interest. The use of Brent’s derivative-free algorithm enables robust maximum identification, significantly reducing the number of function evaluations. Numerical results demonstrate that the proposed approach achieves accurate peak estimates with a substantial reduction in computational cost compared to conventional time-sampling strategies, while maintaining excellent agreement with reference solutions. While the efficiency depends on the number of post-processing points, the results indicate strong potential for practical integration in high-speed railway bridge design. Acknowledgements The authors gratefully acknowledge Dr. Pedro Museros (Universitat Politècnica de València) for his contributions to the original development of the semi-analytical solution and Mr. Alejandro Castillo-Linares (ACL-Estructuras) for providing real data from the Rodenillo viaduct. References European Commission. Sustainable and Smart Mobility Strategy – Putting European Transport on Track for the Future. COM(2020) 789 final, Brussels, 2020. Gu, Gunmo. "Resonance in long-span railway bridges carrying TGV trains." Computers & Structures 152 (2015): 185-199. European Committee for Standardization (CEN), Eurocode 1: Actions on structures - part 2: Traffic loads on bridges, EN 1991-2 Eurocode 1 (2003). Ouyang, Huajiang. "Moving-load dynamic problems: A tutorial (with a brief overview)." Mechanical Systems and Signal Processing 25.6 (2011): 2039-2060. Frýba, Ladislav. Vibration of solids and structures under moving loads. Vol. 1. Springer science & business media, 2013. König, Paul, et al. "Dynamic analysis of railway bridges exposed to high-speed trains considering the vehicle–track–bridge–soil interaction." Acta Mechanica 232.11 (2021): 4583-4608. Martínez-Castro, A. E., P. Museros, and A. Castillo-Linares. "Semi-analytic solution in the time domain for non-uniform multi-span Bernoulli– Euler beams traversed by moving loads." Journal of Sound and Vibration 294.1-2 (2006): 278-297. Huang, Norden E., and Nii O. Attoh-Okine. The Hilbert-Huang transform in engineering. CrC Press, 2005. Schreier, Peter J., and Louis L. Scharf. Statistical signal processing of complex-valued data: the theory of improper and noncircular signals . Cambridge university press, 2010. Feldman, Michael. "Non-linear system vibration analysis using Hilbert transform--I. Free vibration analysis method'Freevib'." Mechanical systems and signal processing 8.2 (1994): 119-127. Martínez-Castro, Alejandro E., and García-Macías, Enrique. "Train-speed sensitivity approach for maximum response envelopes in dynamics of railway bridges." Journal of Sound and Vibration 452 (2019): 13-33. García-Macías, Enrique, and Martínez-Castro, A. E. "Hilbert transform-based semi-analytic meta-model for maximum response envelopes in dynamics of railway bridges." Journal of Sound and Vibration 487 (2020): 115618.