PSI - Issue 84

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

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1. Introduction High-Speed Lines (HSLs) are central to sustainable mobility due to their lower carbon footprint and higher energy efficiency compared to road and air transport (European Commission (2020)). From an engineering standpoint, the design of high-speed rail infrastructure is particularly demanding due to the high dynamic loads induced by passing trains, especially in bridges, where dynamic amplification effects complicate compliance with ultimate and serviceability limit states (Gu (2015)). Current design codes impose limits on maximum vertical deck accelerations due to their impact on track alignment, ballast stability, and passenger comfort. To this end, multiple approaches have been proposed over the last decades to address the moving-load-induced vibration problem (see Ouyang (2011) for an extensive state-of-the-art review), ranging from analytical solutions (Frýba (2013)) to more sophisticated methods incorporating vehicle–and soil–bridge interaction effects (König (2021)). Notwithstanding the wide range of available simulation techniques, the extraction of maximum response envelopes remains an intricate task. In practice, design envelopes are derived by computing the maximum bridge response as a function of train speed. Their numerical evaluation is time-consuming due to several factors: (i) number of degrees of freedom (DOFs), (ii) number of considered vibration modes, (iii) time-step size, (iv) train speeds, (v) number of train configurations, and (vi) number of post-processing locations. It is thus apparent that the development of computationally efficient dynamic analysis techniques is of pivotal importance. Step-by-step numerical integration approaches, such as the widely used Newmark–beta method, may become computationally prohibitive in this context. As an alternative, Martínez-Castro et al. (2006) proposed a semi-analytical solution for the dynamic analysis of railway bridges subjected to massless moving loads. The key advantage of this formulation lies in the fact that, while the spatial domain is discretised using the finite element (FE) method and modal superposition, the solution remains analytical in the time domain. Exploiting this closed-form representation in time, the authors subsequently proposed two meta-models for the rapid assessment of design envelopes. The first approach is based on the sensitivity of dynamic response envelope curves with respect to train speed (Martínez-Castro and García-Macías (2019)), while the second relies on the Hilbert Transform (HT) of the semi-analytical solution (García-Macías and Martínez-Castro (2020)). These approaches achieved substantial computational reductions by subsampling the speed and the time domain, respectively. Building upon the HT-based approach, we introduce an enhanced meta-model for rapid assessment of maximum response envelopes. The method reformulates peak identification as an optimisation problem, using the instantaneous amplitude of the HT as the cost function, thereby avoiding fine time discretisation. Two case studies illustrate the approach and demonstrate substantial reductions in computation time, highlighting its potential for early-stage structural performance assessment. 2. Theoretical Background 2.1. The Semi-Analytic solution Figure 1(a) illustrates the basic configuration of the 3D FE mesh of a railway bridge. A local Cartesian coordinate system ≡ { ; , , } is defined with the origin located at the initial point of the load lane. A single moving point load travels along the bridge at a constant speed . The resulting time-dependent load can be expressed as ( , ) = ( − ) , where denotes the Dirac delta function and is time. Consider the element ≡ : ∈ [ , ] lying along the load lane, where and denote the global -coordinates of its initial and final points, respectively. Let be the local abscissa measured from the origin of the element, defined as = − . Assuming linear structural behaviour, the response to a train of loads can be obtained by superposition. Therefore, the formulation is presented for a single moving load and, for clarity, for a unit load =1 . The response corresponding to any load magnitude is then obtained by simple scaling by . Under these assumptions, the vertical displacement ( , , , ) of an arbitrary (1) where ( ) is the time-dependent modal amplitude of the -th mode, and ( , , ) the corresponding mode shape. point on element can be expressed as: ( , , , ) = ∑ ( ) ( , , ), =1