Natural modes of Bernoulli-Euler beams with symmetric cracks

Abstract

An approximate Galerkin solution to the one-dimensional cracked beam theory developed by Christides and Barr for the free bending motion of beams with pairs of symmetric open cracks is suggested. The series of comparison functions considered in the Galerkin procedure consists of the mode shapes of corresponding uncracked beam. The number of terms in the expansion is determined by the covnergence of the natural frequencies and confirmed by studying the stress concentration profile near the crack. This approach allows the determination of the higher natural frequencies and mode shapes of the cracked beam. It is found that the Christides and Barr original solution was not fully converged and that cracks render the convergence of the Galerkin's procedure very slow by affecting the continuity characteristics of the solution of the boundary value problem. To validate the theoretical results, a two-dimensional finite element approach is proposed, which also allows one to determine the parameter that controls the stress concentration profile near the crack tip in the theoretical formulation without requiring the use of experimental results. Very good agreement between the theoretical and finite element results is observed.