FEM Matrices for Problems of Free Vibrations and Buckling of a Truncated Cone Beam

Abstract

Proper use of materials is one of the most important criteria of a rational design and shaping of engineering constructions. It requires such dimensioning of each element of the construction which will ensure that the element is matched to its load – and this condition is fulfilled only for beams with variable cross section. Hence, it is essential to develop possibilities of calculations of beams with cross section varying along the beam longitudinal axis. This study provides relevant matrices (i.e. stiffness, mass and initial stress matrix) applied in the Finite Element Method for calculations of natural frequencies and buckling critical forces. The matrices have been derived for beams shaped as a truncated cone with a linear generatrix, supported in various ways. The results have been compared to those obtained for the stair-shaped beams approximating the conical ones; a good concordance of results has been stated.