A boundary shape function method for analyzing nonlinear composite beams, subjecting to nonlinear boundary moment conditions

Abstract

For a boundary value problem of nonlinear composite beam, the boundary conditions are fulfilled automatically by a boundary shape function (BSF), which includes exact solution as a member. For any given free function, we develop a method to construct the BSF and derive two iterative algorithms to approximate the exact solution. Let the solution be a BSF and then the free function be a new variable. Consequently, we can exactly convert the nonlinear-nonuniform composite beam equation, subjecting to nonlinear boundary conditions, to a corresponding initial value problem for the new variable with its terminal values deemed as unknown parameters in the ordinary differential equations (ODEs), whereas the initial conditions are given. In doing so, we can develop fast convergence iterative algorithms with the errors between 10-10 and 10-11, upon comparing numerical solutions to exact ones.