A beam contact benchmark with analytic solution

This paper presents a test case to help validate simulation codes for contact problems involving beams. A closed form solution is derived and the comparison is made with a finite element (FE) implementation that uses the mortar method for enforcing the contact constraints. The test case consists of a semi-infinite cantilever beam subjected to a constant distributed load and experiencing frictionless contact with a straight rigid substrate. Both an Euler-Bernoulli and a Timoshenko beam model are considered and the influence of the differing kinematic hypotheses is analyzed. In the case of the Euler-Bernoulli beam the distributed contact force is equal to the load along the contact region except at the boundary where a point load appears. On the contrary, the rigid substrate exerts a fully distributed load on the Timoshenko beam which decays exponentially from the first contact point and tends towards the applied load. The rate of decay depends on the relative shear deformability. Moreover, whereas in the first case the transverse shear force is discontinuous, it becomes continuous when allowing for shear deformation. An example of benchmarking is given for a particular FE code. The error with respect to the exact solution can be computed and it is shown that the numerical solution converges to the analytic solution when the FE mesh is refined.