Buckling of functionally graded Euler-Bernoulli and Timoshenko beams with edge cracks

In this paper, an analytical approach was proposed for solving the buckling of functionally gradient material (FGM) Euler-Bernoulli and Timoshenko beams with cracks. The discontinuity of rotation caused by the cracks was simulated by means of the rotational spring model. The governing differential equations for buckling of an FGM beam were established and their solutions were found firstly. The recurrence formula of solution using the transfer matrix method was developed in the current research. Then the eigenvalue equations for buckling of an FGM beam can be conveniently obtained from a third-order determinant. A comprehensive parametric study is conducted to investigate the influences of the locations and number of cracks, shear deformation, material properties, slenderness ratio and various end supports on the critical buckling loads of cracked FGM beams. Numerical examples show that the developed method can simply, exactly and effectively solve the buckling of cracked FGM beams with various conditions.