Dynamic analysis of an arch due to a moving load

The radial (in-plane) bending-vibration responses of a uniform circular arch under the action of a moving load were investigated by means of the arch (curved beam) elements. Instead of the complex explicit-form shape functions given by the existing literature, the simple implicit-form shape functions associated with the radial (normal), tangential and rotational displacements of the arch element were derived. Based on the relationships between the nodal forces and nodal displacements of an arch element the elemental stiffness matrix was obtained, and based on the equation relating the kinetic energy and nodal velocities the elemental consistent mass matrix was determined. Assembly of the elemental property matrices yields the overall stiffness and mass matrices of the complete circular arch. The analytical free vibration analysis results were used to confirm the reliability of the presented stiffness and mass matrices for the arch element. Then the dynamic responses of a typical segmental circular arch, with constant curvature, due to a concentrated load moving along the circumferential direction were discussed. In addition to the circular arch, a hybrid (curved) beam composed of a circular-arch segment and two identical straight-beam segments was also studied. All numerical results were compared with the finite element solutions based on the conventional straight-beam elements and reasonable agreement was achieved. Influence of the moving speed, centrifugal force and frictional force on the dynamic behaviors of the circular arch and the hybrid beam was investigated.