Transition wave in a supported heavy beam

TL;DR

This paper models the propagation of a failure wave in a supported elastic beam, analyzing different support configurations and identifying conditions for steady or accelerating failure wave speeds, with a focus on intersonic propagation.

Contribution

It introduces and compares three models of a supported beam with support failure, revealing the conditions for wave propagation and steady-state speeds, especially at intersonic velocities.

Findings

Failure wave can propagate steadily only at intersonic speeds.

Steady-state wave speed decreases as support stiffness drop increases.

Models show strong similarities between continuous and discrete support configurations.

Abstract

We consider a heavy, uniform, elastic beam rested on periodically distributed supports as a simplified model of a bridge. The supports are subjected to a partial destruction propagating as a failure wave along the beam. Three related models are examined and compared: (a) a uniform elastic beam on a distributed elastic foundation, (b) an elastic beam which mass is concentrated at a discrete set of points corresponding to the discrete set of the elastic supports and (c) a uniform elastic beam on a set of discrete elastic supports. Stiffness of the support is assumed to drop when the stress reaches a critical value. In the formulation, it is also assumed that, at the moment of the support damage, the value of the `added mass', which reflects the dynamic response of the support, is dropped too. Strong similarities in the behavior of the continuous and discrete-continuous models are detected. Three speed regimes, subsonic, intersonic and supersonic, where the failure wave is or is not accompanied by elastic waves excited by the moving jump in the support stiffness, are considered and related characteristic speeds are determined. With respect to these continuous and discrete-continuous models, the conditions are found for the failure wave to exists, to propagate uniformly or to accelerate. It is also found that such beam-related transition wave can propagate steadily only at the intersonic speeds. It is remarkable that the steady-state speed appears to decrease as the jump of the stiffness increases.