A unilateral contact problem with the heavy elastica

A heavy inextensible elastic beam of infinite length and lying on a rigid foundation, is loaded by a concentrated force directed opposite to the gravity field. As a result a region of non-contact develops. This contact problem, in which finite deflections are considered, leads to a free boundary value problem for a system of non-linear ordinary differential equations. This system is discretized by means of finite-differences, and a Newton-Raphson method is applied to solve the nonlinear equations. In this type of problems the matrix has not a band structure and this hampers the use of available fast numerical procedures. However, through the use of an artificial devise this difficulty could be circumvented. Numerical results are given and their accuracy is discussed, in particular for large values of the external force.