The finite element method is applied by various researchers for identification cracks of the faulty dynamic structures, which are described in this section:
Xu et al. [39] have proposed a crack identification method using curvatures and continuous wavelet transforms finite element analysis of beams. A uniform acrylonitrile butadiene styrene cantilever beam with crack has been used in the experimental analysis to know the fidelity of the proposed method. They have found a good agreement between the finite element analysis and experimental analysis results. Wang et al. [40] have performed experimental and numerical analysis of a laminated T700/BA9912 composite under low-velocity impact to analyze the damage behavior. The 3D Hashin damage criterion and the cohesive zone model are used in the finite-element investigation, and the numerical simulation displays the de-lamination, matrix damage, and fiber breakage present in the composite material. A Non-Damage Inspection (NDI) method is performed to evaluate the effectiveness of numerical models. They have found that the predicted size and shape of the de-lamination area well agreed with the NDI result and the fiber breakage in the bottom layers of the laminates are in good agreement with the experimental analysis. An analytical discrete element technique and finite element method to determine the dynamic response of the un-damped Euler Bernoulli beam with breathing cracks under a point moving mass has been presented by Ariaei et al. [41]. The
effect of the moving mass, velocity, location and size of the crack on beam deflection is investigated.
A 3D finite element method to find out plane normal and shearing stresses over critical surfaces of adhesively bonded single lap joints with FRP composite adherends has been analysed by Panigrahi [42]. The stress distribution over mid surface of the adhesive layer of damaged and non-damaged model has been compared with the literature. The parameters of vibration and stress distribution obtained in [41-42] can be used to designed soft computing AI models for recognition of damage. Potirniche et al. [43] have applied finite element method using ABAQUS software for fatigue and fracture application. Two dimensional elements with edge crack are considered in the analysis. They have determined the stiffness matrix of the crack element from the Castiglione‟s first principle and compared the results of the proposed method with results obtained from the physical method and they have found good agreement between the results. A non-destructive damage detection method to study the influence of a crack on the dynamic response of a cantilever beam subjected to bending has been described by Hearndon et al. [44]. They have created a cracked finite element model to compute the influence of damage location and severity on the stiffness of structures. Due to presence of crack, the change in beam deflection is observed, which leads to the reduction of the global stiffness of the beam, upset the vibration responses. This property has been used by authors in structural health monitoring purpose.
Al-said [45] has proposed a mathematical model to identify crack location and depth in stepped cantilever Euler Bernoulli beam carrying a rigid disk at its tip. He has described lateral vibration of the beam using a simple mathematical model combined with mode method and long-range equation. The proposed FEA based method is capable of recognition of crack location and depth. Andreausa et al. [46] have studied the non-linear response of a cantilever cracked beam subjected to harmonic loading with two dimensional finite element formulations which is capable to display a breathing crack behavior via a frictionless contact model of the interacting surfaces. A numerical technique to analyze the free vibration analysis of uniform and stepped cracked beam with circular cross section has been presented by Kisa and Gurel [47]. The beam was assumed to be detached in two parts from crack section, the finite element and component mode synthesis method used in analysis to achieve the goal. The modal parameters of a cracked beam obtained from the free vibration analysis, which can be used in the crack identification process. A finite element method for crack identification of the beam for
free and forced response analysis has been proposed Karthikeyan et al. [48]. A transverse surface crack is considered in the Timoshenko beam model. Chasalevris and Papadopoulos [49] have presented an inverse crack identification method using the dynamic behavior of a shaft with two transverse surface cracks. They have proposed that the identification method gives not only the depth and the location of the crack, but also the regular orientation of crack around axes of shafts.
Nahvi and Jabbari [50] have presented analytical and experimental approach for identification of crack by vibration measurement. An experimental investigation is performed of the cracked cantilever beam excited by hammer and vibration responses recorded by accelerometer moving along the length of the beam. To recognize the crack contours of normalized frequency was plotted in term of relative crack depth and crack location. They have recognized crack depth and crack location by the intersection of contours with constant natural frequency planes. Law and Lu [51] have proposed crack detection method of a beam structure with Dirac delta function and based dynamic measurement in the time domain. The proposed method was based on model superposition and optimization technique with regularization on the solution. They have compared proposed identification damage algorithm with experimental analysis.
Vibration parameters (natural frequencies and mode shapes) of cracked beam using finite element method have been calculated by Zheng and Kessissoglou [52]. They added overall flexibility matrix to a flexibility matrix of the heavy beam instead of local flexibility to obtain total flexibility matrix. They have found that the overall flexibility matrix gives more accurate vibration parameters than local flexibility matrix. It is reported that overall flexibility of system can be used for identification of damage present in the structures.
A Single Damage Indicator (SDI) factor to localize and quantify a crack in beam like structure by relating with fractional change in natural frequency has been proposed Kim and stubb [53]. They have proposed two models one for crack location and another for crack depth by using the parameter fractional change in modal energy to change in natural frequencies due to presence of crack. In the first model, (crack location model) fractional change in measured the eigenvalue (Zi) and FEM based theoretical modal sensivity of the ith modal stiffness corresponding to jth element Fij is described respectively. The theoretical modal curvature is determined from a third order interpolation function of displacement modeshape. This can be defined as: