Analytical solutions for out-of-plane response of curved beams resting on an elastic foundation under a random moving load

In this study, the vibration behaviour of a horizontally curved beam resting on an elastic foundation and subjected to a random moving force is studied. The governing coupled differential equations for the out-of-plane vibration of curved beams are derived based on the mode superposition method, Duhamel's integral, and the Laplace transformation. Note that the out-of-dynamic responses of curved beams induced by a random moving load are composed of the centred random and mean value. Analytical solutions are proposed for the beam’s vertical and torsional deflections and the bending moment, where the rotary inertia, warping constant, and foundation stiffness are incorporated in these formulations. Because analytical studies on the random vibration of a curved beam resting on an elastic foundation are very limited, the aim of this study is to investigate the effect of relative parameters, namely, high-mode vibrations, the radius of curvature, the moving velocity, and foundation vertical and torsional stiffnesses, on the centred random and mean values of the beam dynamic responses. The results show that high-mode vibrations have a greater impact on the beam responses at higher velocities. To obtain precise results, the first 30 modes are considered in the calculation for the dynamic responses of the beams. The critical velocity is an inherent property of bridges, and it increases as the radius of curvature and foundation stiffness increase. In addition, an increase in the velocity and foundation stiffness results in a reduction in the centred random values for responses.