Autoparametric Resonance in a Pedestrian Steel Arch Bridge: Solferino Bridge, Paris

Autoparametric resonance is treated as the reason of arising excessive lateral vibrations in the steel arch bridge (the Solferino Bridge). To explain this phenomenon, a physical model (a double pendulum) is proposed. Its behavior, as a rule, depends on dynamic characteristics of a bridge rather than on its type. The response of a two degree-of-freedom system with quadratic nonlinearities in the presence of two-to-one autoparametric resonance is investigated. The perturbation method of multiple time scales is used to construct first-order nonlinear differential equations and to determine steady state solutions and their stability. Bifurcation analysis is performed to determine a critical (threshold) value in the external load (control) parameter. The autoparametric resonance becomes possible if an excited torsional mode is near a primary resonance and an external load parameter caused by pedestrians is equal or higher than its critical value. When the increasing load parameter passes through the critical value (because a quantity of pedestrians on the bridge is increased), a jump phenomenon (or fast growth) is observed for the lateral mode, the torsional mode is saturated and has much smaller amplitudes. Field tests were held to understand a phenomenon of an excessive lateral movement, and to enforce the Solferino Bridge. Theoretical results of the present paper are compared with those experimental measurements. Swaying of pedestrian bridges can be treated as a two-step process. The first step (achievement of parametric resonance), described in this paper, is the condition for the beginning of the second step—the process of possible synchronization of applied forces and the interactions between them and the lateral and torsional modes of vibration.