We investigate the load transmission along an elastic rod of finite cross-section in contact with a rigid cylinder, as system often referred to as the generalized capstan problem. In the presence of friction, the idealized classic capstan equation predicts that the tension along a perfectly thin and flexible filament increases exponentially along the contact region. In practical applications, however, the validity of the idealized capstan equation is compromised due to the interplay between finite rod thickness, bending stiffness, and the forces applied at the rod extremities. Here, we combine precision model experiments, finite element simulations, and theoretical modeling to investigate the contact mechanics and the force transmission along an elastic rod in frictional contact with a rigid cylinder. We study two cases when the rod is either static or sliding. First, we focus on the static case, in the absence of friction, by considering equal loads at both extremities of the rod. We show that as the loading force is increased, the nature of contact transitions from a localized region to an extended band at the surface of the cylinder. The latter is characterized by double-peaked contact force distribution. In the sliding case, friction is activated by inducing a relative motion between the rod and the cylinder. We applied a fixed loading force at one rod extremity while pulling the other extremity at a constant velocity. The driving force is monitored during sliding. For increasing loading forces, we find that the force ratio is non-monotonic and displays a local minimum, in contradiction with the constant ideal capstan prediction. This minimum force ratio coincides with the transition from a single contact point to an extended contact region. A theoretical analysis based on Euler's elastica serves to rationalize the results from the physical and numerical experiments. In addition to predicting the nature of the contact region (single point versus extended line), our model provides quantitative predictions for the wrapping angle and the driving-to-loading force ratio. Finally, we leverage our mechanics-based framework to predictively understand the force ratio at the ends of two commercially available engineering belts (spring-steel and polyurethane) in sliding contact with a steel cylinder.