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Tendons are a widely used actuation strategy for continuum robots that enable forces and moments to be transmitted along the robot from base-mounted actuators. Most prior robots have used tendons routed in straight paths along the robot. However, routing tendons through general curved paths within the robot offers potential advantages in reshaping the workspace and enabling a single section of the robot to achieve a wider variety of desired shapes. In this paper, we provide a new model for the statics and dynamics of robots with general tendon routing paths that is derived by coupling the classical Cosserat-rod and Cosserat-string models. This model also accounts for general external loading conditions and includes traditional axially routed tendons as a special case. The advantage of the usage of this coupled model for straight-tendon robots is that it accounts for the distributed wrenches that tendons apply along the robot. We show that these are necessary to consider when the robot is subjected to out-of-plane external loads. Our experimental results demonstrate that the coupled model matches experimental tip positions with an error of 1.7% of the robot length, in a set of experiments that include both straight and nonstraight routing cases, with both point and distributed external loads.