Bending solutions of FGM Timoshenko beams from those of the homogenous Euler–Bernoulli beams

By using mathematical similarity and load equivalence between the governing equations, bending solutions of FGM Timoshenko beams are derived analytically in terms of the homogenous Euler–Bernoulli beams. The deflection, rotational angle, bending moment and shear force of FGM Timoshenko beams are expressed in terms of the deflection of the corresponding homogenous Euler–Bernoulli beams with the same geometry, the same loadings and end constraints. Consequently, solutions of bending of the FGM Timoshenko beams are simplified as the calculation of the transition coefficients which can be easily determined by the variation law of the gradient of the material properties and the geometry of the beams if the solutions of corresponding Euler–Bernoulli beam are known. As examples, solutions are given for the FGM Timoshenko beams under S–S, C–C, C–F and C–S end constraints and arbitrary transverse loadings to illustrate the use of this approach. These analytical solutions can be as benchmarks in the further investigations of behaviors of FGM beams.