## 4 Numerical Examples
### The Case for an FGM Ring Reinforced in a Homogeneous Plate
It is clear evidence from the moment distribution in Figure 6b that the bending moment changes continuously in the ring and the plate with no jump for both linear FGM ring and exponential FGM ring. Meanwhile, both of them significantly reduce the moment concentrations near the hole. The only difference of the results between the linear and exponential FGM ring is the distribution shapes of the moment in the ring. The linear FGM ring causes the moment to vary along a linear way in the ring, while the exponential FGM ring causes a moment with a similarly exponential change. Figure 7 shows the influence of the width of the reinforcing ring on the distribution of bending moment. It is supposed here the ratio of Young's modulus \(E_{h}/E_{p}=0.5\). Figure 7a,b show that the bending moments in the ring and the plate increase steadily as the width of the homogeneous ring increases. The difference of moments at the interface almost keeps constant. For the linear and exponential FGM rings in Figure 7c-f, the bending moments at the edge of the hole change very slightly as the width of the ring increases, but the maximum moment in the ring decreases gradually. Therefore, the increase of the widths of the ring is beneficial to reduce the overall moments in the structure. In particular, if the width of the ring is taken large enough in Figure 7f, the result will be reduced to the case of an infinite FGM plate with a circular hole as shown in Figure 3a. Figure 6: Distribution of moments \(M_{\theta}/M_{0}\) along the radial direction (\(\theta=\pm 90^{0}\)) for different types of ring: (a) \(E_{h}=1.5E_{p}\); (b) \(E_{h}=0.5E_{p}\). Finally, the variations of SCF with the ratio of Young's modulus \(E_{h}/E_{p}\) and radius ratio \(r_{N}/r_{0}\) are analyzed, and the results are shown in Figure 8 where the linear FGM ring is taken as an example. The SCF \(K_{r_{0}}\) at the edge of the hole and \(K_{r_{N}}\) in plate at the edge of the ring are defined as  
\[K_{r_{0}}=\left(M_{r_{0}}\right)_{\text{max}}/M_{0}, \tag{22}\]  
Figure 7: Distribution of moments \(M_{\theta}/M_{0}\) along the radial direction (\(\theta=\pm 90^{0}\)) for different widths of ring: (**a**) variation of [PERSON]’s modulus for homogeneous ring; (**b**) distribution of moments for homogeneous ring; (**c**) variation of [PERSON]’s modulus for linear FGM ring; (**d**) distribution of moments for linear FGM ring; (**e**) variation of [PERSON]’s modulus for exponential FGM ring; (**f**) distribution of moments for exponential FGM ring  
\[K_{r_{N}}=\left(M_{r_{s}}\right)_{\max}/M_{0}, \tag{23}\]  
where \(\left(M_{r_{0}}\right)_{\max}\) and \(\left(M_{r_{s}}\right)_{\max}\) are the maximum bending moments at the edge of the hole and in plate at the edge of the ring, respectively. It is found from Figure 8a that the SCF \(K_{r_{0}}\) grows quickly while \(K_{r_{N}}\) decreases slowly as the abscissa \(E_{h}/E_{p}\) increases. When \(E_{h}/E_{p}>1\), that is the Young's modulus at the edge of the hole is larger than that of the plate, the SCF \(K_{r_{0}}\) is always higher than that of the homogeneous perforated plate. When \(E_{h}/E_{p}<1\), the SCF in the structure attains the minimum value at the intersections of the curves of \(K_{r_{0}}\) and \(K_{r_{0}}\). The corresponding abscissas of the intersections are the optimal ratio of Young's modulus to realize the minimum stress concentration. It can be seen from Figure 8b that there is a special case as \(E_{h}/E_{p}=\)1.0, i.e., the Young's modulus in the ring equal to that in the plate. The SCF \(K_{r_{0}}\) remains constant and equal to 1.79, which is the result of a homogeneous plate with a circular hole. For other ratios of \(E_{h}/E_{p}\), the SCF \(K_{r_{0}}\) changes slowly with the abscissa \(r_{N}/r_{0}\) increasing, while the SCF \(K_{r_{N}}\) decreases rapidly in the approximate range \(r_{N}/r_{0}=1\sim 2\) and then changes slightly as the abscissa \(r_{N}/r_{0}\) increases.