Large deflection and rotation of Timoshenko beams with frictional end supports under three-point bending

Three-point bending of a beam is studied based on the Timoshenko beam theory. Large deflection and large rotation of a beam resting on simple supports with friction are calculated for a concentrated force acting at the midspan. Using the Lagrangian kinematic relations, a system of non-linear differential equations are obtained for a prismatic shear-deformable Timoshenko beam. Exact solutions for the deflection, horizontal displacement, and rotation of cross-section are derived analytically. Two deflections of small and large scale exist under three-point bending. The solutions corresponding to linearized model coincide with the well-known solutions to the classical Timoshenko beams. Numerical calculations are carried out to show the effect of the important parameters such as shear rigidity of the beam and the coefficient of friction at the contact position between the beam and supports on the deflection. The load-deflection curves are graphically presented. A comparison of large deflections and large rotations with their classical counterparts and with experimental data is made. The obtained results are useful in safety design of linear and non-linear beams subject to three-point bending.