The refined theory of beams for a transversely isotropic body

The refined theory of transversely isotropic beams is proposed on the basis of the classical elasticity theory. By using E-L solution and Lur'e method, the refined theory provides the solutions of transversely isotropic beams without ad hoc assumptions. Exact solutions, including a fourth-order part and a transcendental part, are obtained for beams with homogeneous boundary conditions, whereas approximate solutions are derived for beams under transverse surface loadings by dropping terms of high order. It is shown that the displacements and stresses of the beam can be represented by the angle of rotation and the deflection of the neutral surface. In this paper, separate discussions are given to the cases in which characteristic roots are distinct or equal to each other. To the authors' knowledge, the latter has not been covered in the literature. To illustrate the application of the beam theory developed, three examples are examined: a cantilever beam under end loading, a simply supported beam under uniform loading, and a cantilever beam under linear loading. Results show that the refined theory of transversely isotropic beams can be degenerated into that of isotropic beams by omitting anisotropic terms.