Flexible endoscopy and catheterization typically involve inserting a flexible shaft into a curved channel. Understanding the mechanics involved in the insertion process is crucial for the structural design, actuation, sensing, control, and navigation of these flexible medical tools. However, the ever-changing contacts and friction between the insertion shaft and the pathway make the mechanics complicated. Existing analytical models simplify the problem by neglecting the friction and assuming specific boundary conditions that are valid only in a few specific instances. In the meantime, FEM models have trade-offs between computation speed, accuracy, and stability. We developed an efficient theoretical framework to model the insertion process with friction, promoting fast and accurate computation of the mechanics involved. The inserting shaft is segmented based on the evolving contacts; system equations are formulated with friction-included force equilibrium and boundary conditions. The model is verified through experiments; channels with different shapes/curvatures were considered. The root-mean-square errors between the model and measured insertion forces are less than 0.055N (average percentage error less than 9.62%). This model will enhance the fundamental understanding of the insertion process's mechanics and benefit the engineering (design, actuation, and control) and medical practices of related medical tools (e.g., endoscopic instruments and catheters).