An analytical solution for out-of-plane deflection of a curved Timoshenko beam with strong nonlinear boundary conditions

This paper presents a simple method for finding the analytical solution for nonlinear boundary problems. The shifting function method is applied to developing the static deflection of an out-of-plane curved Timoshenko beam with nonlinear boundary conditions. Considering the out-of-plane motion of a uniform curved Timoshenko beam of constant radius R and a doubly symmetric cross section, three coupled governing differential equations are derived via Hamilton's principle. After some simple arithmetic operations, the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. An example is given to illustrate the analysis and show that the proposed method performs very well for problems with strong nonlinearity.