This paper describes the quasi-static formulation of frictionless line contact between flexible beams by employing the mortar finite element approach. Contact constraints are enforced in a weak sense along the contact region using Lagrange multipliers. A simple projection appropriate for thin beams with circular cross-sections is proposed for the computation of contact regions. It is combined with the geometrically exact beam formalism on the Lie group $SE(3)$ S E ( 3 ) . Interestingly, this framework leads to a constraint gradient and a tangent stiffness invariant under rigid body transformations. The formulation is tested in some numerical examples.

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.