| A rigorous beam modeling approach has been developed based on the variational-asymptotical method. A finite-element-based nonhomogeneous anisotropic initially curved and twisted beam theory was formulated from geometric nonlinear, three-dimensional elasticity. The kinematics were derived for arbitrary warping (which includes out-of-plane as well as in-plane deformations) based upon the concept of decomposition of the rotation tensor. The three-dimensional strain energy based on this strain field is dimensionally reduced via the variational-asymptotical method. The three-dimensional warping is calculated in terms of the one-dimensional strain measures and the functions in the strain energy become independent of the cross-sectional variables. The resulting equations govern both sectional and global deformation, as well as provide the three-dimensional displacement and strain fields in terms of beam deformation quantities. The formulation also naturally leads to geometrically exact, one-dimensional kinematical and intrinsic equilibrium equations for the beam deformation. This theory is not limited to the low-order theory found in classical approaches. Moreover, it is not based on the usual Saint-Venant approach for the interior problem associated with beams. Rather, the asymptotical method allows for the approximation of the cross-sectional behavior in terms of the eigenfunctions of a certain Sturm-Liouville problem associated with the cross section. These eigenfunctions contain all the necessary information about the nonhomogeneities throughout the cross section of the beam and thus possess the appropriate discontinuities in the derivatives of displacement. The approach is based on the identification of small parameters in the structure, and the cross section may have arbitrary geometry (solid or thin-walled, closed or open). The developed theory was implemented numerically in a computer code called VABS (Variational-Asymptotical Beam Sectional Analysis) to obtain the stiffness constants and warping field over the cross section. For several different cross section shapes and material distribution, beam geometries and loading conditions, results from the current work were compared with experimental, analytical and other numerical results whenever available. |
