Multiple-crack damage detection in multi-step beams by a novel local flexibility-based damage index

In this article we address a simple method for detecting, localizing, and quantifying multiple cracks occurred in Euler–Bernoulli multi-stepped beams, using the measurement of natural frequencies and estimating the uncracked mode shapes. The analysis is based on energy method and Euler–Bernoulli beam theory. We model cracks as rotational springs and demonstrate a relationship among natural frequencies, crack locations and depths. This method can be applied to both forward and inverse problems. The main advantage of this method is that it has the power of detecting the unknown number of cracks. The concise and simple calculations, good accuracy and elimination of complicated matrix calculations are the other advantages. In addition, for reducing numerical complexity, we use global interpolation functions as is common in Rayleigh–Ritz method, instead of using piecewise continuous mode shapes for a multi-step beam. We present numerical examples for a two-step cantilever beam, including one, two, and three cracks to validate the method.