Exact dynamic stiffness method for non-uniform Timoshenko beam vibrations and Bernoulli–Euler column buckling

The exact dynamic stiffness method is further extended, using a recently developed approach for vibration of Bernoulli–Euler members, to flexural free vibration of non-uniform Timoshenko beams with gradual or stepwise non-uniformity of geometric and/or material properties and to Euler buckling of similarly non-uniform columns. Two key strategies are emphasized: (i) formulation of the governing ordinary differential equations (ODE) for dynamic stiffnesses and their derivatives and the solution of the ODE problem by standard ODE solvers; and (ii) establishment of mesh generation rules for the two problems. Extension of the method to three-dimensional frames with non-uniform members poses no major theoretical hurdles. Numerical examples, including challenging problems, are given to show the effectiveness, efficiency, accuracy and reliability of the proposed method which, unlike the finite element method, is exact and so can be iterated until any preset accuracy is achieved.