Buckling of functionally graded and elastically restrained non-uniform columns

Columns with non-uniform distribution of geometrical or material parameters i.e. functionally graded material distribution, varying cross-sectional area and flexural stiffness provide an economical solution to carry the desired higher compressive loads in engineering structures. In this paper, a low-dimensional mathematical model is presented, which is capable of computing the buckling loads of uniform and non-uniform functionally graded columns in the axial direction. The columns with spatial variation of flexural stiffness, due to material grading and/or non-uniform shape, are approximated by an equivalent column with piecewise constant geometrical and material properties. Such a formulation leads to certain transcendental eigenvalue problems where the matrix elements are transcendental functions. This model is further extended in analyzing some uniform and non-uniform elastically restrained or braced axially graded columns with equal or unequal spans. The mathematical modeling, closed-form transcendental functions and numerical solution technique are described and several examples of estimating buckling loads for various boundary configurations are presented. Some of the results are also validated against available solutions, representing the convergence, effectiveness, accuracy and versatility of the proposed modeling and numerical method. Formulation of such low-dimensional eigenvalue problems can also be extended for analyzing, designing and optimizing the static and dynamic behavior of structural components that are made of functionally graded materials.