This paper presents a novel modeling approach for the mechanics of multisegment, rod-driven continuum robots. This modeling approach utilizes a high-fidelity lumped parameter model that captures the variation in curvature along the robot while simultaneously defined by a discrete set of variables and utilizes the principle of virtual power to formulate the statics and dynamics of the continuum robot as a set of algebraic equations for the static model and as a set of coupled ordinary differential equations (ODEs) in time for the dynamic model. The actuation loading on the robot by the actuation rods is formulated based on the calculation of contact forces that result in rod equilibrium. Numerical optimization calculates the magnitudes of these forces, and an iterative solver simultaneously estimates the robot's friction and contact forces. In addition, modeling considerations including variable elastic loading among segments and mutual segment loading due to rods terminating at different disks are presented. The resulting static and dynamic models have been compared to dynamic finite element analyses and experimental results to validate their accuracy.