Flexible endoscopes are widely used in minimally invasive surgical robot systems. Various kinematic models have been developed for describing the deformation of such endoscopes. For joint-type flexible endoscopes, most existing models neglect the effect of internal friction and cannot precisely show the shape. In this paper, we propose a new nonlinear bending model. The rubber tube and metal net at each joint are approximated as a tube under elastic deformation and are assigned an equivalent bending stiffness. The internal friction force is also taken into account to build the moment balance equation at each joint. Groups of experiments were performed to validate the nonlinear model. The results closely confirm the model's predictions. The model's tip position error during the bending and unbending phases are 1.48 +/- 0.99 mm and 1.68 +/- 0.91 mm respectively; the bending angle errors are -5.50 +/- 2.54 degrees and 1.68 +/- 3.66 degrees, respectively The model can also take account of the hysteresis effect of the bending, which is quite common for cable-driven flexible robots. Moreover, the model has good computational efficiency, making it suitable for real-time control.