<p>A newly developed contact detection method designed for the interaction between flexible slender structures and rigid sheaves is proposed. The former are discretized by means of two-dimensional beam finite elements based on the absolute nodal coordinate formulation. The contact detection relies on the geometrically exact intersection points of the third-order polynomial representing a beam element and the circular boundary of the sheave. The contact force is obtained from a spring–damper penalty formulation. The proposed method is applied to a numerical model of a pulley system, which is analyzed under static and dynamic conditions. Comparative simulations are performed employing a relevant, existing method based on an approximation of the polynomial by piecewise linear segments. In addition, a semi-analytical model is derived to serve as a reference. In the static case, convergence studies reveal a similar order of convergence for the exact as well as the segment method, however, with a significantly higher coarse-mesh accuracy of the exact one. This behavior is qualitatively reproduced by the dynamic results leading to a significant improvement in computational efficiency of the proposed method.</p>