Spectral Analysis of Vertical Vibration Response of Footbridges under Crowd Loads Using Lyapunov Equations

Over the years, various methodologies have emerged for assessing the effect of crowds on footbridges. Given the stochastic nature of such excitations and the desired linear behavior of the structure, employing tools rooted in random-vibration theory is encouraged. This paper presents an analytical approach for evaluating the root-mean-square of vertical vibrations. The approach involves solving the Lyapunov equations for a structural system in the state-space, subjected to a filtered white-noise input. A serviceability evaluation procedure based on the presented analysis approach is presented as well. For the proposed procedure, commodifiable filters which articulate the excitation in the state-space, are developed. These filters express the mean spectral densities of crowd excitations, and account for various distributions of a crowd along a footbridge. A single analysis of a footbridge, subjected to such loads, results with the mean of the root-mean-square response of interest. Hence, the approach may appeal to engineers for implementation in design practice. Furthermore, by using the Lyapunov equations, the suggested procedure significantly surpasses the traditional spectral density analysis in terms of computational efficiency. Therefore, this approach may be preferred in cases when many analyses are required.