## 7 Applications
### Elastoplastic Optimization of a Toroidal Shell under Internal Radial Pressure  
A numerical example with an FGM toroidal shell, as represented in Figure 6, was considered. The toroidal shell was modelled using 24 finite elements in a sequence, starting at the inner point and moving in a clockwise direction (where node 1 was located at the inner point of the circular section of the right side of Figure 6 such that they were numbered in sequence from 1 to 24 in the clockwise direction). The geometry was defined by r = 254 mm, R = 381 mm and initial thickness h = 12.7 mm corresponding to a mass of m = 335.263 kg. The constituents were zirconia and stainless steel, i.e., ZrO\({}_{2}\)/SUS304, with the following properties at 300 K: \(E_{m}\) = 207.79 GPa, \(\
u_{m}\) = 0.3176, \(E_{c}\) = 168.0 GPa, \(\
u_{c}\) = 0.2978, \(\rho_{m}\) = 8160 kg/m\({}^{3}\), \(\rho_{c}\) = 5700 kg/m\({}^{3}\) and q\({}^{*}\) = 90 \(\times\) 10\({}^{9}\) Pa. The inner surface was ceramic and the outer surface was metal. The gradient index \(p\) = 1 was considered, and the shell was subjected to an inner radial pressure load. Figure 7 shows the original and deformed shapes in which the displacements were enlarged 100 times. In this analysis, a finite element mesh with 48 elements was used.  
The optimization of the shell was performed with the objective of minimizing the mass of the vessel, with a constraint on the effective stress. It was required for the load level of 11.16 MPa that the structure should not display elastoplastic behaviour. The structure could be modelled using one or more groups of finite elements to allow for different thicknesses. For the optimization problem, a finite element mesh with 24 elements was used. Considering a uniform thickness, the results obtained were as follows: minimum thickness h = 13.8 mm, which corresponded to a mass of m = 364.301 kg, and a maximum effective stress equal to 395 MPa, which equalized the yield stress (395 MPa). The deformed shapes (enlarged 100 times), considering the initial thickness h = 12.7 mm and the optimized h = 13.8 mm, are shown in Figure 8, with the plastic zone in yellow, and the load-displacement paths are presented in Figure 9.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Initial Design**} & \multicolumn{2}{c}{**Final Design**} \\ \hline \(p\) & **Thickness (mm)** & **Mass (kg)** & **Thickness (mm)** & **Mass (kg)** \\ \hline
5.0 & h = 12.5 & 70.175 & h = 17.19 & 96.504 \\
2.0 & h = 12.5 & 60.782 & h = 17.09 & 83.101 \\
2.0 & h = 12.5 & 60.782 & h = 17.08 * & 83.052 \\
1.0 & h = 12.5 & 51.394 & h = 16.83 & 69.198 \\
0.5 & h = 12.5 & 41.983 & h = 16.45 & 55.250 \\
0.2 & h = 12.5 & 32.574 & h = 15.93 & 41.513 \\ \hline \hline \end{tabular}  
*–calculated by using analytical sensitivities.  
\end{table}
Table 2: Optimization of a clamped plate.  
Figure 6: Toroidal shell with thickness h subjected to internal pressure \(p\).  
Figure 8: Deformed and optimized shapes of an FGM toroidal shell (index \(p=1\)).  
Figure 7: Deformed shapes of an FGM toroidal shell for index \(p=1\).  
Figure 9: Load–displacement paths for the original and optimized toroidal shell.  
Considering two groups of elements, with one group formed by four elements, namely, the ones that plasticize (elements 1, 2, 23 and 24), and another group with 20 elements for the remaining structure, the results obtained were as follows: h\({}_{1}\) = 13.92 mm and h\({}_{2}\) =11.21 mm, resulting in a mass of m = 300.396 kg. Considering four groups of elements, each one with six elements, as described in Table 3 (where, as described before, node 1 was located at the inner point of the circular section of the right side of Figure 6), different results in the final mass were obtained, as shown in Table 4.