Semi-analytical Approach for Free Vibration Analysis of Variable Cross-Section Beams Resting on Elastic Foundation and under Axial Force

Abstract

In this paper, free vibration of an Euler-Bernoulli beam with variable cross-section resting on elastic foundation and under axial tensile force is considered. Beam’s constant height and exponentially varying width yields variable cross-section. The problem is handled for three different boundary conditions: clamped-clamped, simply supported-simply supported and clamp-free beams. First, the equation of motion that governs the free vibration is derived and then dimensionless frequencies are determined using differential transform method (DTM). DTM is a semi-analytical approach based on Taylor expansion series that is a powerful tool in solution ordinary and partial differential equations. The effects of axial force, elastic foundation coefficient and non-uniformity parameter on dimensionless frequencies are investigated. Wherever possible, comparisons are made with the studies in open literature. Results show, the DTM yields rapid convergence without any frequency missing although convergence rate depends on boundary conditions. Also, dimensionless frequencies are sensitive to axial force rather than other parameters.