AN ANALYTICAL METHOD FOR THE ANALYSIS OF FUNCTIONALLY GRADED BEAMS RESTING ON A WINKLER ELASTIC FOUNDATION

This publication addresses the analysis of so-called functionally graded beams (FGBs), whose material properties vary along their length according to a specified gradient. The study is devoted to the development of an analytical method (AM) for bending analysis of such beams resting on a homogeneous, continuous Winkler elastic foundation. The material's modulus of elasticity and the external load are assumed to be arbitrary, continuously varying functions dependent on the coordinate of the beam’s neutral axis. The proposed method is based on an exact solution to the corresponding fourth-order ordinary differential equation of bending with variable coefficients. The unknown integration constants are expressed in terms of initial parameters, which are determined by applying the prescribed boundary conditions. The fundamental functions and the particular solution of the differential equation are represented as power series in a dimensionless parameter, with variable coefficients obtained through recursive integral relations. For practical convenience, the fundamental functions and the particular solution are transformed into power series format. This reduces the bending analysis of FGBs to a numerical implementation of explicit analytical expressions for stress-strain state parameters. A numerical example demonstrates the practical application of the AM. A prismatic FGB with a parabolically varying modulus of elasticity is considered. The AM results are presented in both numerical and graphical formats for the case where the left end of the beam is simply supported and the right end is fixed. The obtained numerical values are treated as exact, since the analysis is based on an exact solution of the governing differential equation. The availability of such a method enables the assessment of the accuracy of solutions obtained using various approximate methods. For verification purposes, corresponding solutions obtained by the Finite Element Method (FEM) using the LIRA-SAPR software are also provided. The comparison confirms the validity of the proposed analytical method.