## 2 Case Study of Cracked Rotor Using a Three-Dimensional Finite Element Model
The final mesh for the shaft bulk and immediate crack region can be seen in Figure 4.  
The model was simulated by varying the shaft rotation angle from 0\({}^{\circ}\) to 345\({}^{\circ}\) in an increment of 15\({}^{\circ}\). As the shaft rotates, the position of the crack changes relative to the neutral plane of bending. Any portion of the crack above and below the neutral axis experiences compressive and tensile stresses, respectively. Compressive stresses mean crack surfaces are under contact for crack closure while tensile stresses mean the crack surfaces are under separation for crack opening. The neutral axis orientation angle at the crack location was identified by determining the coordinates of several nodes which have zero normal stress. Since the evaluated stress was rarely exactly zero at nodes, pairs of nodes with opposite stress signs that also shared an element edge were probed for their axial stress magnitude and spatial coordinates. According to the intermediate value theorem, there must be a point of zero stress, i.e., the neutral plane, and by using a linear trend line the location can be inferred. This idea is visualised in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mesh** & **Number of Elements in Model** & **Closed Area of Crack** & **Shaft Deflection (mm)** \\ \hline
1 & 63,000 & 0.28628 & 0.06517 \\
2 & 143,000 & 0.29116 & 0.06316 \\
3 & 325,000 & 0.28268 & 0.06292 \\
4 & 860,000 & 0.28095 & 0.06272 \\
5 & 79,000 & 0.28592 & 0.06321 \\
6 & 165,000 & 0.28083 & 0.06321 \\
7 & 239,000 & 0.27662 & 0.06322 \\
8 & 289,000 & 0.28019 & 0.06322 \\
9 & 177,000 & 0.27284 & 0.06323 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of convergence test for various meshes. Figure 3: Boundary conditions of the 3-D finite element shaft model. Figure 6 shows the neutral axis location and inclination for various shaft rotation angles (0\({}^{\circ}\)-180\({}^{\circ}\)). In this figure, the shaded region represents the open portion of the crack and the remaining region is the non-cracked area. It is assumed that the non-cracked area is the effective cross-sectional area at the location of the crack and the bending moment vector \(M\) acts horizontally about the centroid of the non-cracked area. The inclination angle of the neutral plane (from the horizontal) is non-zero when the shaft rotates to 45\({}^{\circ}\) (when the crack is fully open but the crack front is not horizontal) and to 90\({}^{\circ}\) and 135\({}^{\circ}\) (crack is partially open). According to beam theory [26], if the direction of the bending moment is not in a line with the planes of symmetry of a cross-section, then the neutral plane should be inclined relative to the symmetry planes. It is clear that the FEA probed values and trend line enforce this idea whenever the crack is partially open and therefore disagrees with the use of a horizontal neutral axis [12]. At the 0\({}^{\circ}\) and 180\({}^{\circ}\) cases, the non-cracked area is symmetrical relative to the direction of the bending moment and thus the FEA extracts that the trend line is horizontal. It should be noted that the numerically predicted trend line is included to reduce the semi-elliptical crack front to a straight crack front. Figure 4: Mesh 9—the chosen mesh showing (**a**) radial and (**b**) axial refinement, particularly local to the crack. Figure 5: Neutral plane inference based on pairs of nodes on the same element edge with oppositely signed normal stress.