Linear and Nonlinear Vibration of Axially Loaded Timoshenko Beam with Elastic Supports: Effects of Transition Parameter

Abstract

Axially loaded beam-like structures, such as beam segments and piers of cable-stayed bridges, are widely used in civil engineering. This paper presents a study on the linear and nonlinear vibration of an axially loaded Timoshenko beam with elastic supports. The governing equations of the system are derived using Timoshenko beam theory and von Kármán geometric nonlinearity within the framework of Hamilton’s principle. A transition parameter is introduced to describe the direction of the axial internal force during the beam’s deformation. Galerkin’s method is employed to reduce the nonlinear partial differential equations, from which linear solutions can be obtained. Nonlinear discretized equations are solved using the incremental harmonic balance (IHB) method. Frequency-response curves of the beam system are tracked by pseudo arc-length continuation, and Floquet’s theory is used to determine the system’s stability. Numerical examples are presented to validate the proposed solutions, comparing the results with the literature and the Runge-Kutta method. The effects of the transition parameter, elastic support’s stiffness, and supported position are investigated for both linear and nonlinear vibration of the axially loaded Timoshenko beam. Of particular interest is the significance of the transition parameter, especially for a Timoshenko beam subjected to extreme axial compression with a small slenderness ratio. The present analysis provides critical insights into the design and analysis of such axially loaded structures in various civil engineering applications.