Nontrivial solution for buckling of thin beam with discountinues axial quadratic material gradation function subjected to axial span-load

In materials science Functionally Graded Materials (FGMs) can be described by continuous variation in structure and composition over volume, resulting in corresponding changes in the properties of the material. The materials can be designed to achieve specific gradation function and applications. The power series can be used to approximate the transition between two materials. Many methods have been developed for modeling the mechanical behavior of FGMs. The buckling analysis of thin beam with axial gradation required more computational efforts than transverse gradation. This research considers stepwise quadratic material gradation in the axial direction and axial compressive load applied between restraints. The insertion point of concentrated axial load is coincided with discontinuity point of material gradation in the axial direction. The various boundary conditions at beam ends are considered. The equations of shear force and bending moment equilibrium, deflection continuity as well as boundary conditions are written in the matrix form. The nontrivial solution is employed to obtain characteristic equation and first buckling loads. The matrix operations are used to calculate corresponding first mode shapes.

Keywords:

Buckling , Discontinues quadratic axial gradation , Euler-Bernoulli beam , Axial span-load , Nontrivial solution