Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions

Abstract

A unified method is presented for the analysis of vibration and stability of axially loaded non-uniform beams with abrupt changes of cross-section. The beam may also be supported on Winkler elastic foundation, and both the axial force and the foundation stiffness can be varied arbitrarily. The method is based on the Euler-Lagrangian approach using a family of C1 admissible functions as the assumed modes. The assumed modes comprise essentially the vibration modes of a single span hypothetical prismatic beam with the same end supports but without the intermediate supports, modified by piecewise C1 cubic polynomials. The chosen admissible functions therefore possess both the advantages of fast convergence of the eigenfunctions and the appropriate order of continuity at the location of abrupt change of cross-section. The method allows extensive use of matrix notations and programming is rather straightforward. Numerical results also show that the method is versatile, efficient and accurate.