Crack Identification in the Complex Beam-type Structures Based on Frequency Data

This paper presented a numerical methodology to locate and estimate depth of crack in a cantilever beam. The local effect of "softening" at crack location can be simulated by an equivalent spring connecting two segments of the beam. Combined with the Bernoulli-Euler theory for beams, the characteristic equation of a stepped cantilever beam model is derived using transfer matrix method. The resulted frequency equation of a cracked beam generally depends on the boundary conditions including the closer (elastic) end supports and can be formulated analytically by using symbolic codes. The procedure proposed simply operates on some fourth order determinants so that the computational efforts for calculating natural frequencies are significantly reduced. Based con this characteristic equation, two identification methods, the contour curve method and minimizing object function one, are developed to identify the location and depth of the crack, which only requires the first three tested natural frequencies. The proposed approaches are verified through some examples with a single-stepped cantilever beam under the different boundary conditions. It is shown that the location and depth of the crack can be worked out as long as the first three natural frequencies are known. It is also shown that making a "zero setting" may offer a tremendous' improvement in identification of crack parameters. Moreover, the minimizing object function method can easily be adapted for complex boundary conditions and for multi-cracks located in any segments of the beam.