Static Euler–Bernoulli beams with contact forces: Existence, uniqueness, and numerical solutions

In this paper, we study the Euler–Bernoulli fourth-order boundary value problem (BVP) given by w(4)=f(x,w), for x∈[a,b], subject to specified values of w and w′′ at the endpoints. The structure of the right-hand side f is motivated by (bio)mechanical systems involving contact forces. Specifically, we consider the case where f is bounded above and monotonically decreasing with respect to its second argument. We begin by establishing the existence and uniqueness of solutions to the continuous BVP. We then examine numerical approximations using a finite-difference discretization of the spatial domain. We show that, similar to the continuous case, the discrete problem always admits a unique solution. For a piecewise linear instance of f, the discrete problem reduces to an absolute value equation. We demonstrate that this equation can be solved using fixed-point iterations and prove that the solutions to the discrete problem converge to those of the continuous BVP. Finally, we illustrate the effectiveness of the fixed-point method through a numerical example.