**Vibration Analysis of Cracked Beams Using the Finite Element Method**

A. S. Bouboulas, S. K. Georgantzinos and N. K. Anifantis

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51173

## **1. Introduction**

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Most of the members of engineering structures operate under loading conditions, which may cause damages or cracks in overstressed zones. The presence of cracks in a structural member, such as a beam, causes local variations in stiffness, the magnitude of which mainly depends on the location and depth of the cracks. These variations, in turn, have a significant effect on the vibrational behavior of the entire structure. To ensure the safe operation of structures, it is extremely important to know whether their members are free of cracks, and should any be present, to assess their extent. The procedures often used for detection are di‐ rect procedures such as ultrasound, X-rays, etc. However, these methods have proven to be inoperative and unsuitable in certain cases, since they require expensive and minutely de‐ tailed inspections [1]. To avoid these disadvantages, in recent decades, researchers have fo‐ cused on more efficient procedures in crack detection using vibration-based methods [2]. Modelling of a crack is an important aspect of these methods.

The majority of published studies assume that the crack in a structural member always re‐ mains open during vibration [3-7]. However, this assumption may not be valid when dy‐ namic loadings are dominant. In this case, the crack breathes (opens and closes) regularly during vibration, inducing variations in the structural stiffness. These variations cause the structure to exhibit non-linear dynamic behavior [8]. The main distinctive feature of this be‐ havior is the presence of higher harmonic components. In particular, a beam with a breath‐ ing crack shows natural frequencies between those of a non-cracked beam and those of a faulty beam with an open crack. Therefore, in these cases, vibration-based methods should employ breathing crack models to provide accurate conclusions regarding the state of dam‐ age. Several researchers [9-11] have developed breathing crack models considering only the

© 2012 Bouboulas et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Bouboulas et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

fully open and fully closed crack states. However, experiments have indicated that the tran‐ sition between these two crack states does not occur instantaneously [12]. In reference [13] represented the interaction forces between two segments of a beam, separated by a crack, using time-varying connection matrices. These matrices were expanded in Fourier series to simulate the alternation of a crack opening and closing. However, the implementation of this study requires excessive computer time. In references [14, 15] considered a simple peri‐ odic function to model the time-varying stiffness of a beam. However, this model is limited to the fundamental mode, and thus, the equation of motion for the beam must be solved.

when the crack is considered as always open. To assess further the validity of this technique, the quasi-static problem of a three-dimensional rotating beam with a breathing crack is also presented. The formulation of this latter problem is similar to the former one. The main differ‐ ences are: the inertia and damping terms are ignored, any possible sliding occurs in two di‐ mensions and the iterative procedure is applied to load instead of time increment. The flexibility of the rotating beam and the crack state over time are presented for both a transverse and slant crack of various depths. The validation of the present study is demonstrated through

Vibration Analysis of Cracked Beams Using the Finite Element Method

http://dx.doi.org/10.5772/51173

183

In the following, both a two and three-dimensional beam models with a non-propagating surface crack are presented. For both models the crack surfaces are assumed to be planar and smooth and the crack thickness negligible. The beam material properties are considered linear elastic and the displacements and strains are assumed to be small. The region around the crack is discretized into conventional finite elements. The breathing crack behavior is si‐ mulated as a full frictional contact problem between the crack surfaces, which is an inherent‐ ly non-linear problem. Any possible sliding is assumed to obey Coulomb's law of friction, and penetration between contacting areas is not allowed. The non-linear dynamic problem is discussed for the two-dimensional model and the corresponding quasi-static for the threedimensional model. Both problems are solved utilizing incremental iterative procedures. For completeness reasons, the contact analysis in three dimensions and the formulation and sol‐

Figure 1 illustrates a two-dimensional straight cantilever beam with a rectangular cross-sec‐ tion *b* ×*h* and length *L* . A breathing crack of depth *a* exists at position *L <sup>c</sup>* . The crack is located at the upper edge of the beam and forms an angle *θ* with respect to the *x* − axis of the global co‐

ordinate system *x*, *y* . An impulsive load is applied transversally at point *A* (Figure 1).

comparisons with results available from the literature.

ution of the non-linear dynamic problem are presented below.

**2. Finite element formulation**

**a.** *Two-dimensional model*

**Figure 1.** Cracked two-dimensional beam model.

A realistic model of a breathing crack is difficult to create due to the lack of fundamental understanding about certain aspects of the breathing mechanism. This involves not only the identification of variables affecting the breathing crack behavior, but also issues for evaluat‐ ing the structural dynamic response of the fractured material. It is also not yet entirely clear how partial closure interacts with key variables of the problem. The actual physical situation requires a model that accounts for the breathing mechanism and for the interaction between external loading and dynamic crack behavior. When crack contact occurs, the unknowns are the field singular behavior, the contact region and the distribution of contact tractions on the closed region of the crack. The latter class of unknowns does not exist in the case without crack closure. This type of complicated deformation of crack surfaces constitutes a non-line‐ ar problem that is too difficult to be treated with classical analytical procedures. Thus, a suit‐ able numerical implementation is required when partial crack closure occurs.

In reference [16] constructed a lumped cracked beam model from the three-dimensional for‐ mulation of the general problem of elasticity with unilateral contact conditions on the crack lips. The problem of a beam with an edge crack subjected to a harmonic load was consid‐ ered in [17]. The breathing crack behavior was simulated as a frictionless contact problem between the crack surfaces. Displacement constraints were applied to prevent penetration of the nodes of one crack surface into the other crack surface. In reference [18] studied the problem of a cantilever beam with an edge crack subjected to a harmonic load. The breath‐ ing crack behavior was represented via a frictionless contact model of the interacting surfa‐ ces. In [19] studied the effect of a helicoidal crack on the dynamic behavior of a rotating shaft. This study used a very accurate and simplified model that assumes linear stress and strain distributions to calculate the breathing mechanism. The determination of open and closed parts of the crack was performed through a non-linear iterative procedure.

This chapter presents the vibrational behavior of a beam with a non-propagating edge crack. To treat this problem, a two-dimensional beam finite element model is employed. The breath‐ ing crack is simulated as a full frictional contact problem between the crack surfaces, while the region around the crack is discretized into a number of conventional finite elements. This nonlinear dynamic problem is solved using an incremental iterative procedure. This study is ap‐ plied for the case of an impulsive loaded cantilever beam. Based on the derived time response, conclusions are extracted for the crack state (i.e. open or closed) over the time. Furthermore, the time response is analyzed by Fourier and continuous wavelet transforms to show the sensitivi‐ ty of the vibrational behavior for both a transverse and slant crack of various depths and posi‐ tions. Comparisons are performed with the corresponding vibrational behavior of the beam when the crack is considered as always open. To assess further the validity of this technique, the quasi-static problem of a three-dimensional rotating beam with a breathing crack is also presented. The formulation of this latter problem is similar to the former one. The main differ‐ ences are: the inertia and damping terms are ignored, any possible sliding occurs in two di‐ mensions and the iterative procedure is applied to load instead of time increment. The flexibility of the rotating beam and the crack state over time are presented for both a transverse and slant crack of various depths. The validation of the present study is demonstrated through comparisons with results available from the literature.
