# On the Optimal Design of Steel Shells with Technological Constraints
## 6 Results and Discussion  
Performing the structural optimization, the objective function is chosen to be connected with the variables of dimensioning, i.e., the ring thicknesses \(e_{1}\) and \(e_{2}\). It was necessary for optimization to be connected with the determination of the minimum of energy and total power of dissipation energy of rings of the shell that the conditions of optimum, derived in Section 3, provide. Through the limit analysis, the greatest resistance of the shell is used. While the shell is operating at the limits of its resistance, we perform a variation of the variable dimensioning, during the derivation of the condition of optimum, to reach the minimum of the energy of dissipation. Thus, the \"best\" solution that works with the least material under the limit load \(p_{0}\) was determined. In the condition of optimum, the total power of dissipation energy depends only on bending moments \(M_{x}\), \(M_{px}\), normal forces \(N_{\theta}\), \(N_{p}\), and on-field of displacement rates \(\dot{w}_{x}^{(i)}\) which characterize the collapse mechanism. In the optimal plastic design (Section 4), both analytical and numerical approaches are exposed. To solve this nonlinear problem (Equations (33)-(35)), we start from the fact that it is real and practical. For approximate initial values, the thickness \(e\) from the shell of constant thickness was used, as well as the condition \(e_{1}\leq e_{2}\). Thus, the initial value of \(E_{2}\) of the central ring thickness was chosen in the subroutine as very small in relation to the shell of constant thickness \(e\). As the number of steps increases, the interval in which the unknown \(E_{2}\) is located occurs. The determination of the thickness \(E_{1}\) was carried out \begin{table}
\begin{tabular}{c c c c} \hline \hline \(X_{1}\) & \(\dot{w}_{1}\) & \(X_{2}\) & \(\dot{w}_{2}\) \\ \hline
0.0000000 & 0.0000000 & 0.08000000 & \(-17.97580596\) \\
0.0080000 & \(-1.94955594\) & 0.08080000 & \(-19.08173715\) \\
0.0160000 & \(-3.8896472\) & 0.09600000 & \(-20.108173715\) \\
0.0240000 & \(-5.81104077\) & 0.10400000 & \(-21.05072739\) \\
0.0320000 & \(-7.0422548\) & 0.11200000 & \(-21.90557981\) \\
0.0400000 & \(-9.56019208\) & 0.12000000 & \(-22.66913126\) \\
0.0480000 & \(-11.36992999\) & 0.12800000 & \(-23.33819928\) \\
0.0560000 & \(-13.12468814\) & 0.13600000 & \(-23.90999527\) \\
0.0640000 & \(-14.81598132\) & 0.14400000 & \(-24.38213601\) \\
0.07200000 & \(-16.43563122\) & 0.15200000 & \(-24.75265364\) \\
0.08000000 & \(-17.97580596\) & 0.16000000 & \(-25.02000388\) \\ \hline \(X_{1}\) & \(P_{1}\) & \(X_{2}\) & \(P_{2}\) \\ \hline
0.000 & \(-244\) & 0.08 & \(-136\) \\
0.08 & \(-187\) & 0.16 & \(-22\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The strain rate velocities and their slopes. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{\(P_{\rm s}=0.0345\)} \\ \hline \(X_{1}\) & 0 & 0.09 & 0 & 0.085 \\ \(X_{2}\) & 0.18 & 0.18 & 0.17 & 0.17 \\ \(E_{1}\) & - & 0.02861 & - & 0.02755 \\ \(E_{2}\) & 0.0321 & 0.033 & 0.031 & 0.0321 \\ \hline \multicolumn{3}{c}{\(P_{\rm s}=0.035\)} \\ \hline \(X_{1}\) & 0 & 0.07 & 0.095 & 0.11 \\ \(X_{2}\) & 0.19 & 0.19 & 0.19 & 0.19 \\ \(E_{1}\) & - & 0.02638 & 0.02961 & 0.03096 \\ \(E_{2}\) & 0.0332 & 0.034 & 0.0345 & 0.0344 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The calculated thicknesses of optimal shells. in a similar way. As explained in Section 4.1, the main program decreases the interval of parameter \(\xi\) at the point where the difference \(\xi\cdot E_{2}^{2}-E_{1}^{2}\) from (35) change sign and specifies the new interval of \(\xi\). In Example 1, (Table 1), this change in sign would be between \(\xi=0.7423\) and \(\xi=0.7422\). In this particular case, the interval is reduced because accuracy is required; \(\varepsilon=0.0001\) is reached and this procedure is interrupted. Based on the same Example 1 and Table 2, a diagram of the displacement rates \(\dot{w}_{x}^{(i)}\) as well as the slopes, i.e., tangents, on the plastic hinges are shown (see Figure 6).