High-fidelity sectional analysis of warping functions, stiffness values and wave properties of beams

• Present a systematic formulation of geometrically exact beam theories.
• Present an advanced 2D sectional analysis method for obtaining warpings and stiffness values.
• Point out errors in the literature about sectional analysis.
• Present detailed examinations of boundary and central solutions.
• Present a 2D dynamic sectional analysis method.

A systematic formulation for geometrically exact deformation analysis and 2D cross-sectional analysis for obtaining structural stiffness values and warping functions of general isotropic/anisotropic beams is presented here. The formulation is developed using easy-to-understand mathematics and new concepts of local displacements, orthogonal virtual rotations and Jaumann stress and strain measures, and provides straightforward explanations for and clear insight into the physical meanings of all structural terms. The derived beam theory accounts for initial curvatures, geometric non-linearities, in-plane and out-of-plane warpings, elastic couplings, and 3D stresses. For sectional analysis, the six constraints for preventing rigid-body motions and singularity in analysis are obtained naturally and uniquely in this formulation. Central solutions from sectional analysis of different beam cross-sections are presented to reveal the problems of other methods in the literature, especially the popular but erroneous method using flexural deformation to derive shear stiffness values and shear correction factors instead of the correct method using pure shearing. Decaying boundary solutions from sectional analysis are not well examined and understood in the literature, and non-decaying boundary solutions have never been studied at all. This paper presents methods to use decaying boundary solutions for analysis of problems involving warping restraints, free edges, and local self-balanced load distributions. Furthermore, the influences of different dynamic warpings on different wave speeds are also demonstrated.