Asymptotically correct refinements in numerical cross-sectional analysis of composite beams

The cross-sectional stiffness properties are derived, along with the interior solution for prismatic as well as for initially twisted and curved composite beams, allowing for an oblique cross-sectional frame relative to the reference axes. The variational-asymptotic method is used in the determination of the warping field. It is shown that there is a fundamental difference between the solutions for a prismatic beam and a beam with initial twist and curvature, which in turn produces a limitation of the angle of obliqueness: the effect of obliqueness can be treated exactly for a prismatic beam, while for the initially twisted and curved beam the obliqueness has to be regarded as a small parameter.;In the second part, a general treatment of the non-linear strain field problem based on an asymptotic formulation and the Green definition for the strain field is presented. It was possible to prove that the first approximation for the warping field suffices for capturing of nonlinear effects in an asymptotically correct fashion. The resulting method is able to treat general cross-sectional beam geometries of arbitrary anisotropic material while published work to date can only deal with some special, simplified cases like strips or thin-walled beams. Herein, the numerical cross-sectional analysis is developed based on a finite element discretization. The principal effect turns out to be the so-called "trapeze effect" which is a nonlinear extension-twist coupling.;Finally, the third part deals with transverse shear and restrained torsion (Vlasov) effects which are associated with axially varying resultants. The present approach is able to find a second order asymptotically correct expression for the warping field and the corresponding expression for the strain energy. Then, a Timoshenko-like formulation is sought for general anisotropic beams and arbitrary cross-sectional geometry. It is shown that the problem is overdetermined and the solution can only be obtained as the result of a minimization problem. The determination of the Vlasov effect comes naturally from the formulation which shows the power and the rich potential of the present analysis.