Dynamic stiffness for piecewise non-uniform Timoshenko column by power series—part I: Conservative axial force

Abstract

The dynamic stiffness method uses the solutions of the governing equations as shape functions in a harmonic vibration analysis. One element can predict many modes exactly in the classical sense. The disadvantages lie in the transcendental nature and in the need to solve a non-linear eigenproblem for the natural modes, which can be solved by the Wittrick–William algorithm and the Leung theorem. Another practical problem is to solve the governing equations exactly for the shape functions, non-uniform members in particular. It is proposed to use power series for the purpose. Dynamic stiffness matrices for non-uniform Timoshenko column are taken as examples. The shape functions can be found easily by symbolic programming. Step beam structures can be treated without difficulty. The new contributions of the paper include a general formulation, an extended Leung's theorem and its application to parametric study.