A Rankine–Timonshenko–Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation

Abstract

Abstract A refined shear deformable beam theory is developed to analyze and design anisotropic beams via the mixed variational theorem (MVT). The developed theory is referred to as a Rankine–Timoshenko–Vlasov (RTV) beam theory since it is able to account for the torsional warping restraint as well as the shear deformation. The MVT is employed to systematically blend the asymptotic stress field and the intuitive displacement field, so that one can derive a simple yet accurate beam model. The asymptotic stress field is adopted from the finite element-based asymptotic expansion method, whereas the displacement field is intuitively assumed to have the form of a traditional shear deformable and torsional warping restrained beam model. The two fields are synthesized by a strain energy transformation via the MVT, which yields the RTV beam model whose degrees of freedom are the same as the traditional refined beam theories. The resulting generalized beam stiffness matrix is different from those traditional theories qualitatively as well as quantitatively. This leads to the accurate prediction of torsional rigidity, bimoment stiffness, transverse shear stiffness, and composite coupling stiffness. The thin-walled composite beams with closed and open cross-sections are taken as illustrative examples to demonstrate the accuracy and validity of the developed RTV theory.