Mathematical communities which works on various one-dimensional contact problems with linear beams such as Euler–Bernoulli or Timoshenko beams have developed their mathematical theorems and numerical schemes.
In this talk, we consider mathematical and numerical approaches to dynamic contact problems with a nonlinear beam, the so-called Gao beam, which is a more interesting topic to them nowadays. Contact conditions are Signorini’s types and from a numerical point of view they would be better to be interpreted as complementarity conditions (CCs). We formulate time discretizations based on a truncated variational formulation. We prove the convergence of numerical trajectories and also derive a new form of energy balance. A fully discrete numerical scheme is implemented to present numerical results.