# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 4 Convergence Studies  
In the following convergence studies, a single-edge cracked square plate with \(c/a=0.4\) is considered (\(c\) and \(a\) are as shown in Figure 3). The Poisson's ratio is 0.3. There are no geometric boundary conditions applied to the stress calculation. The plate is subjected  
Figure 5: The finite element model discretization in ANSYS. to a uniform tension with the intensity being \(\sigma\). The tension is applied symmetrically at the top edge \(y=b\) and bottom edge \(y=0\). Table 1 lists the dimensionless stress intensity factors \(K_{I}/\left(\sigma\sqrt{\pi c}\right)\) that are obtained using different numbers of elements ranging from \(79\times 79\) to \(159\times 159\). The present results are compared with the published results obtained using the moving least square (MLS)-Ritz method [40] and a modified mapping collocation approach [41], respectively. It can be seen from Table 1 that the present results converge monotonically as the number of elements increases, and the results are very close to the published results. The stress distribution obtained using \(119\times 119\) elements is shown in Figure 6. The stress concentration near the crack tip can be observed. As expected, the normal stresses are symmetric about the centreline, while the shear stress is antisymmetric. Overall, the stress distribution of a cracked plate is significantly different from that of an intact plate, where \(\sigma_{yy}=\sigma\) and \(\sigma_{xx}=\sigma_{xy}=0\). Therefore, for a loaded cracked plate, it is necessary to determine the stress distribution before commencing natural frequency calculation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**XFEM**} & \multirow{2}{*}{**[PERSON] et al. [40]**} & \multirow{2}{*}{**[PERSON] [41]**} \\ \hline
**79 \(\times\) 79** & **99 \(\times\) 99** & **119 \(\times\) 119** & **139 \(\times\) 139** & **159 \(\times\) 159** \\ \hline
2.338 & 2.335 & 2.333 & 2.332 & 2.331 & 2.333 & 2.320 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The convergence of the dimensionless stress intensity \(K_{I}/\left(\sigma\sqrt{\pi c}\right)\) for a single-edge cracked square plate with \(c/a=0.4\) under uniform uniaxial tension. Figure 6: The stress distribution in the cracked square plate when \(\sigma=1.3255\times 10^{6}\) Pa.  
Based on the stress distribution, the natural frequency of the loaded cracked plate is computed in the next step. To compare with the published results, the cracked plate is simply supported on all edges. Table 2 shows the convergence of dimensionless frequency parameter \(\Omega=\omega a^{2}\sqrt{\rho h/D}\) of the first five modes with respect to the number of terms in Equations (5) and (6). Results from [40] are listed as a comparison. In addition, results obtained using ANSYS are presented as a reference. It can be seen that the natural frequencies converge fast as the number of terms increases. With \(I=J=8\) and \(N=3\), current results are almost identical to those in [40]. The FE model in ANSYS is discretised into 45,750 shell elements (SHELL281). SHELL281 is capable of analysing thin to moderately-thick shell structures. The small variation between current results and ANSYS results is because SHELL281 is developed based on the Reissner-Mindlin shell theory. Overall, current results agree well with those obtained from ANSYS. The first six mode shapes when \(I=J=8\) and \(N=3\) are presented in Figure 7. The discontinuity in mode shapes caused by the crack can be observed. Mode 2 and mode 3 (or mode 5 and mode 6) are identical modes when the plate is intact. After introducing the crack, their mode shapes become different. The above convergence studies indicate that the natural frequencies of a loaded single-edge cracked rectangular plate can be calculated with high accuracy. Additional numerical results of the natural frequencies and mode shapes of unloaded cracked rectangular plates when the side crack has varying location, length, and ori-entatio