Existence and uniqueness of nonlinear deflections of an infinite beam resting on a non-uniform nonlinear elastic foundation

Abstract

We consider the static deflection of an infinite beam resting on a nonlinear and nonuniform elastic foundation. The governing equation is a fourth-order nonlinear ordinary differential equation. Using the Green's function for the well-analyzed linear version of the equation, we formulate a new integral equation which is equivalent to the original nonlinear equation. We find a function space on which the corresponding nonlinear integral operator is a contraction, and prove the existence and the uniqueness of the deflection in this function space by using Banach fixed point theorem.

Existence and uniqueness of finite beam deflection on nonlinear non-uniform elastic foundation with arbitrary well-posed boundary condition

“…Here, w(x) is a vertical downward load density on the beam, and -f (u(x), x) is the nonlinear and non-uniform elastic force density by the elastic foundation, which can depend on both the location x on the beam and the deflection u(x) at x. Beam deflection is one of the basic and important problems in structural mechanics and mechanical engineering, and it has a lot of applications [1,3,4,[7][8][9][10][11][12][13][15][16][17][18].…”

“…There have been various attempts [1,3,4,[7][8][9][10][11][12][13]17] to generalize the classical linear uniform problem LDE(w) to nonlinear or non-uniform settings. For infinitely long beam, Choi and Jang [7] obtained an existence and uniqueness result for the following infinite version: (4)…”

“…They showed the existence and the uniqueness of the solution to the boundary value problem consisting of (1.4) and (1.5) in some regions of C 0 (R) around the zero function. Following the framework of [7], we will construct a nonlinear operator Ψ :…”

An efficient method for the static deflection analysis of an infinite beam on a nonlinear elastic foundation of one-way spring model

“…It is always fascinating to find the closed form solution, some authors constructed closed-form solutions for the static and dynamic response of a uniform beam resting on a linear elastic foundation [1][2][3] and others proposed closed-form solution under the linearity assumption by using Green's function technique [4][5][6][7]. The static, dynamic and elastic stability analysis of a beam resting on a nonlinear elastic foundation was discussed by Beaufait and Hoadley [8], Massalas [9], Lakshmanan [10], and Hui [11], and by the same author and co-authors [12][13][14][15].…”

A Numerical Approach to Static Deflection Analysis of an Infinite Beam on a Nonlinear Elastic Foundation: One-Way Spring Model

“…Jang and Sung [22] proposed a new functional iterative method for static beam deflection, which has a variable cross-section. Choi and Jang [23] proved the existence and uniqueness of the nonlinear deflections of an infinite beam resting on a nonlinear elastic foundation using the Banach fixed point theorem. Jang [24] also proposed a new iterative method for the large deflection of an infinite beam resting on an elastic foundation based on the v. Karman approximation of geometrical nonlinearity.…”