Free Vibration Analysis of Beams with an Open Crack at Clamped End by the Sub-Domain Chebyshev–Ritz Method

Free vibration of beams with an open crack at a clamped end, based on the exact, small-strain and linear two-dimensional (2-D) elasticity theory, is studied. The crack is described by a very narrow gap which is considered to have free boundaries. In the analysis, the cracked beam with mixed boundary conditions is divided from the tip of crack along the beam length into two sub-beams with pure boundary conditions. The 2-D displacement field in each sub-beam is individually developed by the Ritz method. The admissible functions of the displacements in each sub-beam are taken as the Chebyshev polynomials multiplied by suitable boundary characteristic functions which satisfy the geometric boundary conditions of the sub-beam. The total eigenvalue equation of the cracked beams can be established by using the displacement continuity conditions at the interface of two sub-beams to integrate the sub-eigenvalue equations. The method is called as the sub-domain Chebyshev–Ritz method. Three kinds of boundary conditions (clamped–simply supported, clamped–clamped and cantilevered) are considered in numerical examples. The convergence studies show that the first eight natural frequencies of the cracked beams have an accuracy of at least three significant figures. Compared with finite-element solutions and available results in literature, the present solutions exhibit excellent agreement. The effects of size parameters such as the height–span ratio and crack depth on the first several natural frequencies and mode shapes of cracked beams are performed in detail.