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.