Free Vibrations of Beams With a Single-Edge Crack

The equation of motion and associated boundary conditions are derived for a uniform Bernoulli-Euler beam containing one single-edge crack. The generalized variational principle used allows for modified stress, strain and displacement fields that satisfy the compatibility requirements in the vicinity of the crack. The concentration in stress is represented by introducing a crack function into the beam's compatibility relations. A displacement function is also introduced to modify the in-plane displacement and its slope near the crack. Both functions are chosen to have their maximum value at the cracked section and to decay exponentially along the beam's longitudinal direction. The rate of exponential decay is evaluated from finite element calculations. The resulting equation of motion is solved for simply supported and cantilevered beams with single-edge cracks by a Galerkin and a local Ritz procedure, respectively. These theoretical natural frequencies and mode shapes match closely with experimental and finite element results. The possibility of determining the damage properties of cracked beams from changes in dynamic behavior is discussed.