Nonlinear Vibration Model for Initially Stressed Beam-Foundation System

Abstract

An analytical solution for nonlinear vibration of an initially stressed beam with elastic end restraints resting on a nonlinear elastic foundation is obtained. As a first step in solving nonlinear vibration equation, the linear vibration mode functions for a beam with elastic end restraints resting on a linear elastic foundation are obtained. Then, the nonlinear vibration equation is solved by employing the linear mode functions to obtain frequency equation and nonlinear response using Jacobi elliptic integral. The nonlinearity due to lateral vibrations, the nonlinearity of foundations and lateral displacement due to lateral elastic restraints at beam ends not included in previous analytical work are considered in the present work. The effects of spring stiffness at the beam ends, foundation stiffness, axial load and vibration amplitude on the frequency parameter are studied. The present solution can be used to measure the accuracy of approximate methods. In the present work, the effects of above mentioned parameters are taken into account. The nonlinear vibration of an initially stressed beam with elastic end restraints resting on a nonlinear elastic foundation is solved using elliptic integrals. The obtained solutions are verified against those obtained from numerical methods and found in close agreement. Parametric study to investigate the influences of foundation stiffness, elastic end restraints stiffnesses, initial axial load and vibration amplitude are conducted and results are depicted in graphs for a wide range of the different practical characteristics. 2. ANALYSIS 2.1. Vibration Equation