Piecewise linear warping theory for multilayered elastic beams

Warping due to transverse shear in multilayered elastic beams is studied in this paper. The Bernoulli-Kirchhoff hypothesis that plane sections remain plane after deformations, with independent rotations, is assumed for each lamina to account for the out-of-plane deformation of the composite cross section. The effects of shear are included by taking the rotations independent of the transverse deflection, as in Timoshenko beam theory. The result is a simple piecewise linear warping theory for layered composite beams. The solution to the governing equations is presented in terms of the eigenvalues and eigenvectors of a generalized matrix eigenvalue problem associated with the coefficient matrices that appear in the governing equations. The problem of a two-layered cantilever beam subjected to a uniformly distributed loading is solved in detail to show the effects of different elastic moduli on the interfacial shear stress. Compared with a finite-element solution, the current theory yields significant improvement over elementary beam theory (excluding warping) in predicting the interface shear stress.