## 5 Conclusions  
Based on [PERSON]'s complex variable methods, the bending stress concentration at a circular hole in a functionally graded thin plate is analyzed. The method of piece-wise homogeneous layers is used to deal with the arbitrary variation of elastic properties along the radial direction in the plate. The effects of different variations of elastic parameters on the distribution of bending moments are discussed for two different cases: a whole FGM plate with a circular hole and an FGM ring reinforced in a homogeneous perforated plate. The following conclusions can be drawn. 1. For the case of a whole FGM plate, it is found that the bending stress concentration, caused by the geometric discontinuity of the hole in the traditional homogeneous plate, can be well relieved or even eliminated by choosing an appropriate radially increasing Young's modulus in the plate. The radial change of Poisson's ratio has little effect on reducing the bending stress concentration. 2. For the case of an FGM ring reinforced in a homogeneous plate, it is found the FGM reinforcing ring with radially increasing Young's modulus can not only relieve the bending stress concentration  
Figure 8: Stress concentration factor (SCF) for a homogeneous plate with a circular hole reinforced with a linear FGM ring: (**a**) variation of SCF with \(E_{h}/E_{p}\); (**b**) variation of SCF with \(r_{s}/r_{0}\). around the hole, but also avoid the stress concentration at the interface between the ring and the plate compared with that of the homogeneous reinforcing ring. 3. The linear and exponential FGM rings can both reduce the stress concentration near the hole, and their effects are almost equally good. The only difference of their results are the distribution shapes of the bending moments. The linear FGM ring makes the moment vary along a linear way in the ring, while the exponential FGM ring brings the moment with a similarly exponential change. 4. The increase of the width of the reinforcing ring is beneficial to relieve the bending stress concentration, but the effect will not be obvious after the width reaches a certain value. Conceptualization, Q.Y. and B.Y. ; software, H.C.; numerical analysis, Y.T. All authors have read and agreed to the published version of the manuscript. The work was supported by the Natural Science Research Project of Jiangsu Higher Education Institutions (No. 19 KJA150007), the Project of \"Six Talent Peaks\" in Jiangsu Province (No. GDZB-014) and the Natural Science Foundation of Zhejiang Province (No. LY18A020009). Q.Y. is grateful to the support of Funding of Jiangsu Overseas Visiting Scholar Program for University Prominent Young & Middle-aged Teachers and Presidents. The authors declare no conflict of interest. The following set of \(2m+3\) linear equations are derived by comparing the coefficients of the same power of \(-2\), \(-1\), \(0\)\(k\) and \(-k\) of \(e^{i\theta}\) in Equation (12)  
\[(1-v_{1})\bigg{[}a^{(1)}+b_{1}^{(1)}\bigg{]}-(3+v_{1})b_{1}^{(1)}=0,\] (A1)  
\[(1-v_{1})\bigg{[}a^{(1)}+2b_{2}^{(1)}\bigg{]}-(3+v_{1})b_{0}^{(1)}=0,\] (A2)  
\[(1-v_{1})\overline{a_{-1}^{(1)}}+(3+v_{1})b_{2}^{(1)}=0,\] (A3)  
\[(1-v_{1})\bigg{[}ka_{k}^{(1)}+(k+1)b_{k+1}^{(1)}\bigg{]}-(3+v_{1})\overline{b _{-(k-1)}^{(1)}}=0,\] (A4)  
\[(1-v_{1})\bigg{[}\overline{ka_{-k}^{(1)}}+(k-1)\overline{b_{-(k-1)}^{(1)}} \bigg{]}+(3+v_{1})b_{k+1}^{(1)}=0,\] (A5)  
where \(k=2,3,\cdots,m+1\).