Application of genetic algorithm in crack detection of beam-like structures using a new cracked Euler–Bernoulli beam element
Graphical abstract
Highlights
► A crack detection approach for detecting crack depth and location in a beam. ► The stiffness matrix for the cracked beam element is derived. ► The stiffness matrix is computed using conjugate beam concept and Betti's theorem. ► The proposed beam element is employed for solving forward problem. ► The inverse problem is solved using the new finite element and genetic algorithms.
Introduction
Existing cracks in structures present serious threats to suitable performance of structures. The material fatigue may be considered as the primary reason for the failure of structures. Therefore, the methods that may detect and localize the cracks in structures have been the theme of many researches available in the literature.
Computational and experimental studies have proved that the presence of cracks results in some changes in the vibration properties of these structures, mainly as the structures lose their original stiffness due to the presence of crack [1], [2]. As a result, the monitoring of vibration properties changes over time may provide widely used nondestructive approaches for evaluating damage severities and computing the remaining structure life. Therefore, representation of reliable models for the mechanical behavior of cracked elements may play an important role in this relation.
A suitable mesh of finite elements may provide a detailed model of the crack and its surrounding zone. Many finite element analyses have been presented to simulate crack's initiation and propagation under applied loading (see for example [3], [4]). The use of such methods usually requires re-meshing the crack tip region. In other words, these methods need considerable computational efforts to precisely model the stress singularity at the crack tip. Furthermore, the crack extension procedures are computationally time-consuming and difficult. For some applications on the other hand, the global behavior of a cracked structure is of principal importance, while the local behavior of the structure in the vicinity of the cracks’ tip may be ignored. Inverse problems are samples of these applications, in which cracks’ location and depth are needed when searching for potential cracks. In such applications, it is required to simulate the crack effects without actually modeling the crack itself. In this relation, different analytical methods have been applied to illustrate the dynamic behavior of cracked beams. One of the methods has modeled the crack by reducing the corresponding section modulus [5], while another one has reported a model, in which the cracked sections were replaced by a rotational spring [6]. Chondros et al. [7] have presented a complete vibration behavior of a cracked beam faced with single- or double-edge open cracks, in which the theory of fracture mechanics was used to describe the local flexibility of cracked section.
Various methods for solving crack identification problem were usually based on the variation of natural frequencies [8], [9], [10], measuring of dynamic flexibility [11], and comparison of mode shapes [12]. Among these crack identification methods, change of natural frequencies is more effective, inexpensive, and fast tool for non-destructive testing [8]. Comprehensive literature reviews are available in the field of structural damage detection [8], [13], [14] in which the vibratory behavior of beams with multiple cracks for forward [15], [16], [17], [18], [19] and inverse [20], [21], [22], [23], [24] problems are among the most important problems in the last two decades.
An inverse problem may be considered as the location and size detection of the beam's cracks. Actually, the conventional mathematically based methods (i.e., hard-computing methods) are not so efficient in solving inverse problems. Biologically inspired soft-computing methods are based on nature's problem-solving strategies. Soft-computing methods have capabilities which are appropriate for solving inverse problems in engineering [25]. Currently, these methods consist of evolutionary computational models (such as genetic algorithms) [26], [27], [28] and various neural networks [29], [30], [31] among others.
In this paper, an approach for crack detection of beam-like structures, based on a new macro beam element which includes single embedded transverse crack with any depth, is developed. In this element, the crack's effect is introduced by modifying the stiffness matrix of the element. The components of the stiffness matrix represented in closed-forms expressions are developed using the conjugate beam concept and Betti's theorem. The proposed macro element is used for obtaining natural frequencies and eigenvectors of the beam-like structures knowing the cracks’ characteristics. For modeling beam-like structures with an arbitrary number of cracks, the required number of new macro element may be embedded easily. Afterwards, with knowing the natural frequencies and eigenvectors, an inverse problem is established, in which the cracks’ location and depth are calculated. In the inverse approach, an optimization problem according to the new macro element and GAs is solved to search the solution. In this approach, the inverse problem is addressed as a minimization problem with the fitness function being the differences between the actual outputs and the relevant computed outputs from a candidate beam crack configuration. Using the available results of 2D finite element analysis as well as experimental data, some numerical examples are tested to verify the performance of the proposed algorithm.
Section snippets
New cracked finite element
Consider an Euler–Bernoulli beam element with a single- or double-sided open crack as depicted in Fig. 1. As shown in this figure, the presence of the crack is introduced by modifying the stiffness matrix of the element. In this method, the entire beam element is divided into two beam segments, and the crack position is considered as the separation point. It is also assumed that the cracks remain open during the analysis. Furthermore, we assume that the entire beam element has a uniform
Verification by 2D finite element results
The proposed cracked beam element is verified considering various loadings, and boundary conditions of cracked beams. Results from a two-dimensional (2D) finite element model are employed to confirm the accuracy of the present cracked beam element.
All the cracked beam models are made of a material with the following properties: the modulus of elasticity E = 200 GPa, the material mass density ρ = 7800 kg/m3, and the Poisson ratio v = 0.3. The considered geometric data are: the beam length L = 4 m, the
Crack detection in beams using GAs
The solution of inverse problems, such as crack detection in beams, may be basically considered as an optimization problem. The GA is an optimization method that simulates the natural evolution phenomena based on Darwin's theory. The GA operates on an initial population of randomly generated candidate solutions, encoded as chromosomes. Applying the principle of survival of the fittest to hopefully produce better and better approximations, the GA may gradually find the best individual, achieved
Numerical examples
In order to assess the feasibility and accuracy of the proposed algorithm, a simple-supported beam is selected and analyzed in this section. The beam's geometry is defined by the following parameters: the beam length L = 4 m, the cross-section of the beam is a rectangle with depth h = 200 mm and width w = 100 mm. The material properties of the beam are as follows: the modulus of elasticity E = 200 GPa, the material mass density ρ = 7800 kg/m3, and the Poisson ratio ν = 0.3.
Conclusions
In the present research, a crack detection methodology was proposed for detecting cracks in beam-like structures. To this end, a new beam element with single embedded crack was derived. The derivation is based on a simplified model, where each crack is replaced by a linear rotational spring, connecting two neighbor elastic segments. The closed-form expressions for the stiffness matrix of the cracked element are computed using the conjugate beam concept and Betti's theorem. The proposed beam
Acknowledgements
The authors would like to thank anonymous reviewers for their valuable suggestions and comments which helped us to improve the quality of the paper.
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