The influence of beam kinematic assumptions in a beam contact benchmark

We present a benchmark to help validating simulation codes for contact problems involv-
ing beams and discuss the influence of different kinematic hypotheses on the solution.
Closed form expressions are derived and the comparison is made with a finite element im-
plementation that uses the mortar method for enforcing the contact constraints [1]. The
test case consists in a semi-infinite cantilever beam subjected to a constant distributed
load and experiencing frictionless contact with a straight rigid substrate. It is shown that
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 con-
trary, 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
speed of decay depends on the magnitude of the shear effect. For both models the dis-
tributed contact force is discontinuous. Moreover, whereas in the first case the transverse
shear force is discontinuous, it becomes continuous when allowing for shear deformation.
It is shown that the numerical value of the total contact force converges to the analytic
solution when the finite element mesh is refined. Our observations are consistent with the
results found in [2, 3] in other beam contact problems. Richer beam models, that allow
for transverse normal deformation, would lead to continuous distributed contact forces.