<p>A novel framework for modelling shear-deformable rods undergoing large deformations, with a focus on contact problems, is introduced in this paper. By treating the shear strain as an independent variable rather than the rotation angle customary in Timoshenko-type formulations we benefit from a two-stage solution strategy. A computationally efficient Bernoulli–Euler solution is first obtained and subsequently used as a reference state, from which the release of the unshearability constraint leads to rapid convergence toward a neighbouring shear-deformable configuration. Exploiting the smallness of the resulting deformation change further enables an incremental formulation of the rod theory that delivers accurate results at minimal additional cost. In weakly nonlinear settings, a single linear correction step proves sufficient, whereas problems involving contact require a Newton-type iterative scheme. The proposed framework is implemented using an isogeometric discretization with freely adjustable continuity of the approximation. Numerical examples with available analytical solutions demonstrate improved efficiency compared to the classical Timoshenko formulation, fast convergence with respect to mesh refinement, and reliable resolution of pronounced contact pressure peaks. The final example of a belt-pulley contact problem shows the potential of the proposed approach in a practically relevant setting.</p>