Identification of Cracking in Beam Structures Using Timoshenko and Euler Formulations

Timoshenko and Euler beam formulations, using energy approach, have been used to estimate the influence of crack size and location on the natural frequencies of cracked beams. Fracture mechanics approach has been used to consider the effect of cracking on the dynamic response of the beam. Galerkin’s approach has been used to solve the problem numerically. It is shown that for slender beams the deep beam influence is felt only when the $\left[\right(\mathrm{basic}\mathrm{bending}\mathrm{length})/h]$ ratio of the fundamental sinusoid of a beam becomes very small for higher modes. When the $(l/h)$ ratio becomes small $\left(<10\right)$, the influence of shear rotation and rotary inertia effects become dominant; the inclusion of these effects makes the beam less stiff than a Euler beam. The crack influence on Euler and Timoshenko beams are similar for beams with $l/h>10$; but when $l/h<10$, the results of cracked Euler and Timoshenko beams slowly become different and diverge. The frequency contour method identifies the crack size and location properly, using the lower order frequencies. When structural symmetry gives an ambiguity regarding the crack location, the vibration behavior of the same beam with an asymmetrically placed mass, in conjunction with the frequency contour method, would uniquely identify the crack size and location.