PSI - Issue 84

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

841

integration schemes constrained by stability-driven time steps, the semi-analytic solution only samples the response to accurately capture its maximum. Leveraging its closed-form time-domain nature, this task can be recast as an optimisation problem in which the bridge response is the objective function, substantially reducing computational cost. However, the highly oscillatory nature of the response leads to a strongly non-convex objective function (Fig. 2). By contrast, the instantaneous envelope obtained via the HT is much smoother and provides an upper bound of the maximum response, thereby improving convexity (see Fig. 2). In this framework, the objective is to determine the time instant that maximises the instantaneous envelope, ( ) = − ( ). The resulting optimisation problem is solved using Brent’s derivative-free method over a bounded interval. The initial bracketing interval is defined as: [ , ] = [ ( 0 − , ), ( 0 + , )]. (15) where 0 is an initial estimate of the maximiser, defines the half-width of the search interval, and min , max denote the admissible bounds of the time domain. Brent’s method iteratively refines the interval [ , ] by combining inverse quadratic interpolation, secant interpolation, and bisection steps, ensuring that the minimum of ( ) remains bracketed at every iteration. Interpolation-based steps are preferred whenever admissible, while bisection is used as a fallback to guarantee convergence. The iterative process is terminated when the interval length or the change in ( ) falls below a prescribed tolerance. When applied to the maximum response envelope estimation at a particular post-processing point, the following algorithmic procedure has been developed to achieve maximum computational efficiency: A. Identification of candidate extrema : A preliminary time-domain response ( ) (displacement or acceleration) is computed at discrete time instants { } between the initial min and the last time instant max . Potential local extrema are identified by detecting peaks and valleys in the sampled response signal. To limit computational effort, only the most significant extrema, ranked by amplitude, are retained. B. Selection of initial guesses: The time instants associated with the selected extrema define a set of candidate initial guesses 0 ( ) for the optimisation process. For each candidate 0 ( ) , the objective function is defined as ( ) = −| ( ) + ℋ ( )| . C. Bracketing of the optimisation interval: A local search interval is constructed around each initial guess, ∈ [max ( 0 ( ) − , min ), min ( 0 ( ) + , max )], where is chosen based on the average time discretisation of the coarse response. D. Local optimisation via Brent’s method: Brent’s method is applied independently within each bracketing interval to minimise ( ) , which can be performed in parallel for multiple initial guesses. E. Global optimum: Among all converged local solutions, the one yielding the lowest objective function value is selected, providing the time instant ˉ and the corresponding maximum response envelope value. Finally, a local search in the vicinity of the maxima is conducted to further refine the solution.

Fig. 2. Concept of the proposed optimisation-based maximum response envelope estimation approach.

4. Numerical results and discussion 4.1. Case study I: Continuous three-span bridge from Eurocode 1

The first case study is a continuous three-span high-speed bridge analysed in Eurocode 1 (CEN (2020)) (Fig. 3). The bridge has been modelled in the commercial FEM code SAP2000 with 8 Euler-Bernoulli beam elements per span. Five vibration modes with resonant frequencies below 30 Hz have been found by linear modal analysis of the FEM,