Vibrations of Damaged Beams Using a Continuous Crack Model Considering Stiffness and Mass Reduction

Crack models are crucial for analyzing the effects of cracks on structures and for damage detection in civil, mechanical, and aerospace engineering. In the literature, a rotational spring is often introduced to simulate the effects of a crack. This spring-crack model is not consistent with physical reality, as it produces a singularity in the bending moment at the cracked cross-section. In addition, it does not adequately account for the stiffness and mass reduction near a crack simultaneously. Moreover, after cracking, the distributions of effective stiffness and mass distribution near a crack are not the same. The stiffness and mass reduction on the two sides of a crack are not addressed simultaneously in the literature. A novel crack model considering stiffness reduction in the vicinity of a crack is proposed. This study reveals that the stiffness reduction regions are located approximately 45 degrees on either side of a crack for aluminum beams with rectangular cross-sections. The transitional length of stiffness is obtained and discretized into sub-segments. At the nodes, the displacement, slope, shear, and moment continuity are ensured. Point masses on beams are incorporated into the model. The characteristic frequency equation is derived by considering the four continuity conditions at the nodes and the boundary conditions. The general forms of mode shape solutions and modal curvatures are also presented. The results obtained by the proposed crack model are validated by an experiment. Applications of the proposed method to beams with single and multiple cracks are demonstrated. It is found that the modal change due to cracking is more than 9% for the first 3 modes, which can be utilized for crack detection. The proposed method can serve as a benchmark for studying cracked beams and crack detection. Machine learning-based crack detection methods can also benefit from the proposed method.