Parametric instability of functionally graded Timoshenko beams with an open edge crack

This paper studies the parametric instability of functionally graded beams with open edge cracks subjected to an axial excitation. The beam is clamped at both ends and movable in the longitudinal direction with materials properties varying exponentially through the thickness direction. Theoretical formulations are based on Timoshenko beam theory and the linear rotational spring model. The governing equations of motion are derived by using the Hamilton's principle and solved by using Galerkin's technique and the Least Squares method. The boundary points on the unstable regions are determined by Bolotin's method. Numerical results are presented to show the influences of crack location, crack depth, material property gradient, and beam slenderness ratio on the principal unstable region of the cracked functionally graded beams