Buckling Analysis of Discontinues Fractional Axially Graded Thin Beam with Piecewise Axial Load Function Rested on Rotational Spring Hinges

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. These kinds of materials are designed to achieve specific properties for specific applications. For the first time, the effects of stepwise fractional axial material gradation pattern and axial compressive load with piecewise function on buckling behaviour of Euler-Bernoulli beam rested on semi-rigid restraints are studied. It is worth mentioning that the more computational efforts are required to solve current problem with respect to the buckling problem of transversely graded beam due to discontinues material gradation especially in the axial direction, axial span-load with piecewise function and natural conditions of rotational spring hinges. The deflection continuity, natural equations as well as boundary conditions are written in the matrix form. The beam discretizing and nontrivial solution are employed to obtain buckling characteristic equation and matrix operations are used to calculate corresponding first mode shapes. Compatibility with various conditions and eliminating convergence drawbacks of conventional numerical tools are advantages of the proposed method. It is observed that the buckling load is decreased by increasing lengths of beam parts and increased by increasing rotational stiffness at semi-rigid supports. In the case of homogeneous beam, the result validity is proved by observing an excellent agreement between results of current work and well-known data in literature.