This work focuses on the modelling and numerical simulation of unilateral contact in viscoelastic beam-like structures under dynamic loading, with direct relevance to nuclear industry components such as steam generator tubes, fuel assemblies, and control rods. In these systems, vibrations induced by seismic or accidental events can lead to repeated impacts against rigid stops, generating highly non-smooth dynamical responses. Industrial approaches often employ simplified beam models coupled with penalty-based contact formulations; however, these methods suffer from limited accuracy and strong sensitivity to model parameters. The governing equations involve linear differential operators coupled with contact forces defined through Signorini complementarity conditions, resulting in variational inequalities and time-dependent differential inclusions. To address the limitations of penalty methods, this study investigates the application of Nitsche’s method to beam contact problems. While well established in three-dimensional contact mechanics, Nitsche’s formulation applied directly to classical Euler–Bernoulli and Timoshenko beams degenerates into a simple penalty approach due to insufficient kinematic richness. To recover the theoretical benefits of Nitsche’s method, an enriched Timoshenko beam model incorporating through-the-depth pinching kinematics is proposed, allowing the definition of physically meaningful contact stresses consistent with the underlying three-dimensional framework. From a mathematical perspective, the viscoelastic impact problem is analyzed in both continuous and semi-discrete finite element formulations, yielding systems of differential inclusions. Existence, uniqueness, and regularity results are obtained by extending singular mass techniques originally developed for elastic cases. Ongoing numerical investigations examine the system’s dynamical behavior via bifurcation diagrams, probabilistic characterization of chaotic regimes, energy evolution, and sensitivity to initial conditions and parameters, informing the development of efficient shooting and continuation algorithms. Comparisons with classical penalty-based Timoshenko beam models demonstrate the enhanced robustness and physical consistency of the proposed Nitsche-based enriched formulation.