Bayesian estimation of instantaneous frequency reduction on cracked concrete railway bridges under high-speed train passage

Abstract

When a train passes along prestressed concrete (PC) bridges, crack opening reduces the girder stiffness, providing important indicators for the practical evaluation of bridge performance. However, this phenomenon is difficult to detect in the free vibrations induced by trains or via hammer tests as these vibrations have very small amplitudes. Moreover, although the bridge under a passing train vibrates with large amplitude, the vibrational state is forced, and the girder stiffness induced by crack opening occurs only in the downward displacement. To estimate this complex but valuable system, this study establishes a time-varying autoregressive with eXogenous (TV-ARX) model and a hierarchical Bayesian estimation (Bayesian TV-ARX) method. The exogenous variables in the TV-ARX model are the modal-transformed moving loads, which constitute the main excitation frequency of the running train. The variation of the autoregressive (AR) coefficient includes the modal characteristic as a random walk process. When a train passes, the Bayesian TV-ARX estimates the instantaneous amplitude-dependent drop of the bridge frequency from the displacement response. After formulating the Bayesian TV-ARX, the effects of the vehicle–bridge interaction (VBI), train speed, and track irregularity on the accuracy of the instantaneous stiffness decrease estimated via the Bayesian TV-ARX were verified using VBI simulations of a nonlinear beam, in which the bending stiffness decreased only during a downward displacement. Even when the VBI effects overlapped, the model accurately estimated the reduced bridge stiffness due to crack opening. Next, the Bayesian TV-ARX was applied to two PC bridges on a real high-speed railway. On one of the bridges, the estimated stiffness decreased by ~ 12% when the bridge was downward-displaced by crack opening during the train passage. Such results on a real bridge under operation have not been previously reported. Four months later, the amplitude-dependent decrease in the bridge frequency was amplified by crack propagation. These results are important evidences of a nonlinear and nonstationary system. Besides solving real problems, the proposed method is expected to significantly contribute to condition-based maintenance and structure-health monitoring of PC bridges. In particular, it enables early detection of damage and deterioration in addition to bridge performance evaluation over time.