Theoretical approach for triply coupled spatial torsional-flexural vibration of asymmetric thin-walled beam bridge under moving train loads

Currently, asymmetric thin-walled U-shaped girders in urban rail transit are widely applied, and triply coupled spatial torsional-flexural forced vibrations of asymmetric thin-walled beam bridges can be induced by moving train loads, which further endangers structural or driving safety. However, no such work on the corresponding theoretical analytical solutions is reported. Thus, taking the simply supported asymmetric thin-walled U-shaped girders in the urban rail transit line 1 project of Chongqing City as an example, this study focuses on the theoretical approach for triply coupled spatial torsional-flexural vibration of asymmetric thin-walled beam bridge under moving train loads. The moving trains can be modeled as concentrated moving loads, and the triply coupled spatial dynamic governing equations under concentrated moving loads are firstly developed considering the effects of rotary inertia and warping resistance. The equations are then decoupled and resolved using the mode superposition method, Laplace integral transformation, and the relationship between the three roots and constant coefficients of a cubic equation. The analytical vibration response expressions in vertical, lateral and torsional directions are derived from this approach, and the resonance conditions are further explored. The results conclude that the vertical, lateral and torsional responses will be simultaneously set in resonance if the parameter related to velocity Ωcn approaches the natural frequency ωni. Subsequently, a novel numerical approach that combines the Galerkin approach and a precise integration method is employed to verify the validity of the above analytical solutions. On this basis, an extensive parametric study is performed. As per the results, the asymmetry, warping effect and eccentricity of the concentrated moving load to the shear center along the Y axis all have a significant impact on the vibration responses in lateral and torsional directions. Specifically, for this asymmetric thin-walled U-shaped girder in urban rail transit bridges, the peak lateral and torsional vibration responses are approximately 3.12 and 2.13 times that of the corresponding monosymmetric one, respectively. The lateral and torsional vibration responses with the warping effect are 28.46% and 14.83% higher than those without the warping effect, respectively.