On asymptotically correct Timoshenko-like anisotropic beam theory

This paper presents a ®nite element cross-sectional beam analysis capable of capturing transverse shear e€ects. The approach uses the variational-asymptotic method and can handle beams of general cross-sectional shape and arbitrary anisotropic material. The work builds on previous works which deal with development of the classical beam theory, which includes only extension, torsion, and bending. A Timoshenko-like formulation is sought to achieve a re®ned theory with simple boundary conditions. Apart from some simple special cases, it is shown that this problem is overdetermined. However, it is possible to obtain `the best possibleʹ solution using the least-squares minimization technique. The geometrical meaning of the shear variable for this formulation is also derived. Results are found to be in good agreement with published results for the shear sti€ness coecients for both isotropic and anisotropic beam