Extended Finite Element Method for Statics and Vibration Analyses on Cracked Bars and Beams

In this paper, the extended finite element method (XFEM) is employed to investigate the statics and vibration problems of cracked isotropic bars and beams. Three kinds of elements namely the standard, the blended and the enriched elements are utilized to discretize the structure and model cracks. Two techniques referred as the increase of the number of Gauss integration points and the rectangle sub-grid are applied to refine the integration within the blended and enriched elements of the beam in which the priority of the developed rectangle sub-grid technique is identified. The stiffness and the mass matrices of the beam are extended by considering the Heaviside and the crack tip functions. In a plane stress analysis, the effects of various crack positions and depths, different boundary conditions and other geometric parameters on the displacement and the stress contours are detected. Moreover, in a free vibration analysis, changes of the natural frequencies and the mode shapes due to the aforementioned effects are determined.