Comparative study of direct and inverse problems of cracked beams

In recent decades, the analysis and evaluation of the cracked structures were hot spots in several engineering fields and has been the subject of great interest with important and comprehensive surveys covering various methodologies and applications, in order to obtain reliable and effective methods to maintain the safety and performance of structures on a proactive basis. The presence of a crack, not only causes a local variation in the structural parameters (e.g., the stiffness of a beam) at its location, but it also has a global effect which affects the overall dynamic behavior of the structure (such as the natural frequencies). For this reason, the dynamic characterization of the cracked structures can be used to detect damage from non-destructive testing. The objective of this paper is to compare the accuracy and ability of two methods to correctly predict the results for both direct problem to find natural frequencies and inverse problem to find crack’s locations and depths of a cracked simply supported beam. Several cases of crack depths and crack locations are investigated. The crack is supposed to remain open. The Euler–Bernoulli beam theory is employed to model the cracked beam and the crack is represented as a rotational spring with a sectional flexibility. In the first method, the transfer matrix method is used; the cracked beam is modeled as two uniform sub-segments connected by a rotational spring located at the cracked section. In the second method which is based on the Rayleigh’s method, the mode shape of the cracked beam is constructed by adding a cubic polynomial function to that of the undamaged beam. By applying the compatibility conditions at crack’s location and the corresponding boundary conditions, the general forms of characteristic equations for this cracked system are obtained. The two methods are then utilized to determine the locations and depths by using any two natural frequencies of a cracked simply supported beam obtained from measurements as inputs. The two approaches are compared with a number of numerical examples for simply supported beams including one crack. The theoretical results show that the accuracy of the Rayleigh’s method to predict natural frequencies decreases for higher modes when crack depth increases. It is also found that for the inverse problem, the transfer matrix method show a good agreement with those obtained from previous works done in this field.