## 7 Applications
### Elastoplastic Analysis of a Pressure Vessel with a Torispherical End  
To compare the capability of the present model, the elastoplastic analysis was carried out by considering the proposed application of [PERSON] and [PERSON] [PERSON] and [PERSON] (1981): a pressure vessel with torispherical ends made of isotropic material, subjected to a radial pressure load. The dimensions were as follows: radius of curvature of the spherical crown R = 100 in, \(\phi_{0}\) = 27.89\({}^{\circ}\), cylinder diameter D = 100 in, radius of the toroidal part r = 6 in and thickness t = 0.8 in. The material properties were as follows: E = 30 \(\times\) 10\({}^{6}\) lb/in\({}^{2}\) and \(\sigma_{Y}\) = 30 \(\times\) 10\({}^{3}\) lb/in\({}^{2}\). In Figure 10, the results obtained with the present model and the results obtained by the authors mentioned before are shown. A very good agreement was observed. The static elastoplastic analysis of an FGM cylindrical pressure vessel with a torispherical end was performed. The constituents were zirconia and stainless steel, i.e., ZrO\({}_{2}\)/SUS304, at a temperature of 300 K. This structure is represented in Figure 11, where R = D\({}_{1}\) = 135 mm, t = 1.27 mm, r\({}_{1}\) = 10.1 mm, and is subjected to an inner pressure load. The material properties of the constituents were as follows: \(E_{m}\) = 207.7877 GPa, \(v_{nt}\) = 0.3175, \(\rho_{m}\) = 8160 kg/m\({}^{3}\), \(E_{c}\) = 168.0629 GPa, \(v_{c}\) = 0.2978 and \(\rho_{c}\) = 5700 kg/m\({}^{3}\). The yield stress for SUS304 was \(\sigma_{Ym}\) = 400 MPa and the stress-strain transfer parameter was q* = 90 \(\times\) 10\({}^{9}\) Pa. The gradient index considered was \(p\) = 0.5. The initial mass was m = 0.379 kg. The deformed shape (with u\({}_{z}\) enlarged 10 times) of the toroidal shell, considering an internal pressure of 2 MPa, is presented in Figure 12. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Initial Design**} & \multicolumn{2}{c}{**Final Design**} \\ \hline
**Number of** & \(\mathbf{p}\) **Index and** & **Mass (kg)** & **Thickness (mm)** & **Mass (kg)** \\ \hline
1 & \(p\) = 1; h = 12.70 & 335.263 & h = 13.80 & 364.301 \\ \hline
1 & \(p\) = 1; h = 12.70 & 335.263 & h = 13.88 * & 366.402 \\ \hline
2 & \(p\) = 1; h = 12.70 & 335.263 & h\({}_{1}\) = 13.92 & 300.396 \\ \hline
4 & \(p\) = 1; h = 12.70 & 335.263 & h\({}_{1}\) = 13.86 & \\  & & & h\({}_{2}\) = 9.37 & 222.418 \\  & & & h\({}_{3}\) = 7.51 & \\  & & & h\({}_{4}\) = 7.32 & \\ \hline \hline \end{tabular}  
*–calculated by using analytical sensitivities.  
\end{table} 
Table 4: Optimization of a toroidal shell. \begin{table}
\begin{tabular}{c c} \hline \hline
**Group Number** & **Elements in the Group** \\ \hline
1 & 1 2 3 22 23 24 \\
2 & 4 5 6 19 20 21 \\
3 & 7 8 9 16 17 18 \\
4 & 10 11 12 13 14 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Four groups of elements. 395.8 MPa and a yield stress equal to 395.8 MPa. For index \(p\) = 0.5, the deformed shapes (enlarged 10 times), considering h = 1.27 mm and optimized h = 2.0 mm, are presented in Figure 13, with the plastic zone in yellow. The respective load-displacement curves for the apex are shown in Figure 14.  
with exponent indexes \(p=2.0\) and \(5.0\), this load level was not reached because they collapsed before the uniform radial external pressure of 20 MPa could be achieved, as summarized in Table 6. The spherical cap for \(p=2.0\) collapsed at 18.90 MPa and \(\upmu_{z}=30.2\) mm, and for \(p=5.0\), it collapsed at 17.20 MPa and \(\upmu_{z}=30.5\) mm. In addition, we considered the case of the spherical cap divided into two groups of elements to allow for different thicknesses in each group. Group 1 had the elements 1, 2 and