Solving boundary value problems of deformable systems

Abstract

Solving the non-linear boundary value problem that governs the equilibrium state of a deformable system is reduced to solving a sequence of linear boundary value problems for partial derivatives of unknown functions with respect to the load parameter These linear problems describe a change of the solution of the non-linear problem when the load parameter is subject to a small increment. The increment of each unknown function is a product of the load parameter increment by the mentioned partial derivative of the corresponding function. It is assumed that values of the unknown functions are known in the initial state when the numerical parameter has the initial value. The differential equations and boundary conditions of the linear boundary value problem are derived by a linearisation of differential equations and boundary conditions of the source non-linear problem. The required solution of the non-linear boundary value problem is a result of summing over all increments of the unknown functions. Use of the present method is demonstrated by solving the two-point boundary value problem on large deflections of a flexible cable freely fixed at a horizontally sliding end support. A pilot computer program has been developed for solving this problem. Numerical realisation of the present method is described in detail. Computer programs for numerical solution of the discussed non-linear boundary value problems can be simply realised by using the described algorithm.