Exact solution for the severity of transverse cracks in prismatic beams

Abstract

We propose herein a mathematical relation to calculate the severity of transverse cracks which affect a prismatic beam. It is known that the damage severity depends on the depth of the crack located at the slice on which the biggest bending moment is achieved. The frequency drop due to this crack is proportional to the severity associated with it. For all other crack locations, the effect is diminished in relation to the curvature registered at the affected slice. It is essential to obtain a relationship to correctly express the severity with respect to the crack depth. To find it, we designed a model of a cantilever beam with cracks located near the fixed end. We observed that the deflection of the beam’s free end increases the closer the crack to the fixed end is, but after a certain limit, the deflection decreases. We concluded that the deflection decrease occurs because the deformation around the crack is restricted by the fixing condition. To find the true severity, we derived the pseudo-severity for six crack positions and estimate the severity using the linear and second-order polynomial regression curves. Next, we used twelve points for interpolation and found a very similar severity. Finally, we performed modal analysis for the beam with cracks at different positions and found the mathematical relation developed to predict frequency changes that involves the severity provides accurate results.