Free vibration analysis of functionally graded beam with transverse crack using Timoshenko and higher order shear deformation theory

In this research, the dynamic analysis of composite functionally graded thick beam despite transverse cracking with the help of Tymoshenko theory and higher - order shear deformation theory has been studied. The governing equations and boundary conditions for composite functionally graded beam and thick using Timoshenko and Reddy theories and also, the principle of system energy minimization has been obtained. It is assumed that the inhomogeneous mechanical properties of the beam as a function of thickness as a function of power in which the thickness of the beam is variable. The boundary condition as clamp-clamp are also considered. According to these conditions, the closed form solution for natural frequencies of composite functionally graded beam despite the crack is obtained. Then, the results were analyzed and by those published in the literature and with the finite element results obtained by ABAQUS are validated. Finally, the results showed that the percentage of improvement of the answer obtained from Reddy's third order shear theory compared to Tymoshenko's theory for thick beam is about 24% and for thin beam the answers are close.