Free in-plane vibration of general curved beams using finite element method

The governing differential equations for the free in-plane vibration of uniform and non-uniform curved beams with variable curvatures, including the effects of the axis extensibility, shear deformation and the rotary inertia, are derived using the extended-Hamilton principle. These equations were then solved numerically utilizing the Galerkin finite element method and the curvilinear integral taken along the central line of the curvilinear beam. Based on the proposed finite element formulation, one can easily study curved beams having different geometrical and boundary conditions. Furthermore, those curved beams, excluding the effects of the axis extensibility, shear deformation and the rotary inertia, are modeled and then solved utilizing the finite element method using a new non-isoparametric element. The results for the natural frequencies, modal shapes and the deformed configurations are presented for different types of the curved beams with various geometrical properties and boundary conditions, and in order to illustrate the validity and accuracy of the presented methodology they are compared with those published in literature.