Exact Solutions for the Analysis of General Elastically Restrained Nonuniform Beams

Abstract

The exact solutions for the problems governed by a general self-adjoint fourth-order nonhomogeneous ordinary differential equation with arbitrarily polynomial varying coefficients and general elastic boundary conditions are derived in Green’s function form. To illustrate the analysis, the static deflection and dynamic analysis of a general eiastically end restrained Bernoulli-Euler beam with polynomial varying bending rigidity, applied axial and force, and elastic foundation modulus along the beam, subjected to an arbitrary transverse force are presented. The Green’s function is concisely expressed in terms of the four normalized fundamental solutions of the system and these fundamental solutions are given in power series forms. The characteristic equations for elastic stability and free vibrational analysis of the beam can be obtained by setting the denominator of the corresponding Green’s function equal to zero. Finally, examples are given to illustrate the accuracy and efficiency of the analysis.