PSI - Issue 84

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ScienceDirect

Procedia Structural Integrity 84 (2026) 837–844

© 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Conference Abstract Extracting maximum response envelopes is a critical step to ensure vibration serviceability of high-speed railway bridges. Traditionally, this requires analysing the structural response for multiple design trains crossing the structure at different speeds. For each train-speed combination, the moving load problem must be solved to identify the maximum structural response in terms of displacements, velocities, and accelerations. These analyses are computationally demanding, which hampers the early design stages and the comparison of alternatives. To overcome these difficulties, this work presents a novel semi-analytic solution for the fast extraction of maximum response envelopes. The approach relies on the semi-analytic solution of the moving load problem, which leverages modal decomposition and the exact integration of the modal equations in the time domain over a finite element mesh. As a result, unlike traditional step-by-step integration methods, no integration time step is required, and the structural response can be sampled at any desired instant. On this basis, the identification of maxima can thus be formulated as an optimization problem. However, since the time histories of the structural response contain highly oscillating terms, this turns the identification of maxima as a non-convex optimization problem. To alleviate this issue, we adopt the closed-form expression of the Hilbert transform of the semi-analytic solution previously developed by the authors. This transform provides the instantaneous envelope of the structural response, a considerably smoother curve that renders the identification of maxima a non-convex optimization problem. The potential of the proposed approach is illustrated through a set of benchmark numerical case studies, demonstrating significant gains in computational efficiency and providing a faster pathway to extract maximum response envelopes and thereby expedite the design of high-speed railway bridges. III Fabre Conference: Existing Bridges, Viaducts, and Tunnels: Research, Innovation, and Applications Semi-Analytic Optimization of Maximum Response Envelopes in High-Speed Railway Bridges via the Hilbert Transform Enrique García-Macías a, * and Alejandro E. Martínez-Castro a a Department of Structural Mechanics and Hydraulic Engineering, University of Granada, Av. Fuentenueva sn, 18002 Granada, Spain

* Corresponding author. Tel.: +34 958241000 ext. 20668. E-mail address: enriquegm@ugr.es

2452-3216 © 2026 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the scientific committee of the Conference 10.1016/j.prostr.2026.06.107