Finite element model of circularly curved Timoshenko beam for in-plane vibration analysis

Curved beams are used so much in the arches and railway bridges and equipments for amusement parks. There are few reports about the curved beam with the effects of both the shear deformation and rotary inertias. In this paper, a new finite element model investigates to analyze In-Plane vibration of a curved Timoshenko beam. The Stiffness and mass matrices of the curved beam element was obtained from the force-displacement relations and the kinetic energy equations, respectively. Assembly of the elemental property matrices is simple and without need to transformation matrix because of using the local polar coordinate system. The natural frequencies of curved Euler-Bernoulli beam with large thickness are not sufficiently accurate. In this case, using the curved Timoshenko beam element is necessary. Moreover, the influence of vibration absorber is discussed on the natural frequencies of the curved beam.

Exact dynamic stiffness method for planar natural frequencies of curved Timoshenko beams

An exact dynamic stiffness matrix, which defines the planar motion of a circularly curved Timoshenko beam member, is developed from the closed-form solution to the governing differential equations. This matrix and a variation of the Wittrick-Williams algorithm are combined in a stiffness formulation in such a way that the required natural frequencies, which correspond to the solutions of a transcendental eigenvalue problem, are converged upon unambiguously, to any desired accuracy, for any plane structure composed of such members. The effects of rotary inertia and shear deflection, uniquely described by the parameters r and s respectively, can be accounted for in any combination. Any particular effect can be neglected by setting the relevant parameter to zero. An example is included that highlights the effects of every possible combination of rotary inertia and shear deflection on the natural frequencies of a simple two-span arch structure, and comparisons are made with published work to confirm the accuracy of the method.