# Strength Analysis of Cylindrical Shells with Tangential Nozzles under Internal Pressure
## 3 Limit-Load Analysis
### Results of Limit-Load Analysis  
The limit load for a cylindrical shell without a nozzle is primarily determined by the maximum pressure exerted on the shell. This critical load depends not only on the material strength but also on the diameter and thickness of the shell. It can be derived from analyzing the von Mises stress or the circumferential stress induced by internal pressure. Through the finite element calculation, the limit load of four groups of cylindrical shell models without nozzles is obtained as listed in Table 3, and the von Mises stress value in the limit state is also listed in the table [18]. It can be seen that the von Mises stresses are very close to the yield strength of the model with a relative error of less than 2%, indicating that the limit-load calculation is accurate.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(D_{i}/T\) & **Limit Load/MPa** & **von Mises Stress/MPa** \\ \hline
71.43 & 9.040 & 287.47 \\
83.33 & 7.765 & 286.94 \\
100 & 6.480 & 286.22 \\
125 & 5.190 & 285.42 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Limit-load value of the cylindrical shell without nozzles.  
Figure 4: Schematic diagram of the load and constraint application (\(d_{o}/D_{i}=0.3\), \(t/T=1\), \(D_{i}/T=100\)).  
As an example, Figure 5 is the displacement-load curve of a given structure in the limit-load analysis process. The displacement-load curve serves as a numerical treatment method for determining the attainment of ultimate load, showcasing the relationship between load and displacement in the finite element model as load steps increase. If the slope of the curve during the horizontal stage is less than \(1/100\)th of the elastic slope, it can be inferred that the limit load has been reached. Figure 5 demonstrates convergence in the finite element calculation results, indicating the achievement of the limit state.  
The von Mises stress contours of the structure corresponding to Figure 5 under the limit load are depicted in Figure 6. It is evident that the region where yield first occurs and the limit load is controlled primarily lies on the inner surface of the cylinder in the connection area between the tangential nozzle and cylindrical shell.  
Figure 7 shows the variation in the limit load of the cylindrical shell having a tangential nozzle with the structural parameters. Obviously, in the range of the designed parameters, with the increase in \(D_{i}/T\), the limit-load value decreases significantly. Keeping \(D_{i}/T\) unchanged, with the increase in \(t/T\) and the decrease in \(d_{o}/D_{i}\), or in other words, the larger the wall thickness and the smaller the outer diameter of the tangential nozzle, the greater the limit load of the cylindrical shell with the tangential nozzle or the stronger the bearing  
Figure 5: Displacement–load curve of the cylindrical shell with a tangential nozzle for limit–load analysis (\(d_{o}/D_{i}=0.3\), \(t/T=1\), \(D_{i}/T=100\)).  
Figure 6: Contours of the von Mises stress at the cylindrical shell with a tangential nozzle under the limit load (\(d_{o}/D_{i}=0.3\), \(t/T=1\), \(D_{i}/T=100\)).  
capacity of the structure is, as a result of a less weakening degree of the tangential nozzle on the strength of the cylindrical shell.