Bi-directional functionally graded thin-walled non-prismatic Euler beams of generic open/closed cross section Part Ⅰ: Theoretical formulations

摘要

This paper presents a detailed treatment of the formulation for buckling and free vibration analysis of Bi-Directional Functionally Graded (BDFG) Thin-Walled non-prismatic beams of generic open/closed cross section. The theory developed is limited to small strains, moderate deflections and small rotations. It is based on the assumption that the shear strain on the mid-surface contour of the cross section of the member is neglected. Using the membrane theory of shells rigorous expressions for strains are obtained by which the effect of non-linear tapering is considered. The material properties are assumed to be graded both in the longitudinal and depth-wise directions of the plate segment of the thin-walled beams. The governing equations are developed defining the displacements with reference to any arbitrary point (say geometric centre) and get away with the usual shear centre and centroid. For computer solutions for classical buckling and vibration analyses, the Finite Element Method is used. Simpson's rule of numerical integration is adopted for the computation of axial, coupled and bending rigidities as well as inertial properties of the plate segment and the Gaussian Quadrature of numerical integration is used for the computation of flexural stiffness, geometric stiffness and mass matrices of an element over the length of the beam. Critical buckling loads and the natural frequencies and their corresponding buckled and mode shapes for various examples are obtained by solving as an eigen-value problem and compared with the published results in the companion paper.