# On the Optimal Design of Steel Shells with Technological Constraints
## 4 Optimal Plastic Design of Shell Composed of Three Rings
### Kinematically Admissible Solution and Optimality Conditions  
In this part of the paper, the boundary conditions of the shell (37) and the condition of the compatibility of displacements rates (38) will be used (see Figure 5)  
\[x=0;\dot{w}_{x}^{(1)}=0; \tag{37}\]  
\[x=x_{1};\dot{w}_{x}^{(1)}=\dot{w}_{x}^{(2)}. \tag{38}\]  
The conditions (37) and (38) reads  
\[D_{1}^{(1)}\sin\omega^{(1)}x+D_{2}^{(1)}\cos\omega^{(1)}x=0,\ (x=0), \tag{39}\]  
\[D_{1}^{(1)}\sin\omega^{(1)}x_{1}+D_{2}^{(1)}\cos\omega^{(1)}x_{1}=D_{1}^{(2)} \sin\omega^{(2)}x_{1}+D_{2}^{(2)}\cos\omega^{(2)}x_{1}. \tag{40}\]  
Two equations with four unknown variables \(D_{1}^{(1)}\), \(D_{2}^{(1)}\), \(D_{1}^{(2)}\) and \(D_{2}^{(2)}\) are obtained, and using (31) and (32) it follows as  
\[D_{2}^{(1)}=0; \tag{41}\]\[D_{1}^{(1)}\sin\biggl{(}X_{1}\sqrt{\frac{2}{E_{1}}}\biggr{)}=D_{1}^{(2)}\sin \biggl{(}X_{1}\sqrt{\frac{2}{E_{2}}}\biggr{)}+D_{2}^{(2)}\cos\biggl{(}X_{1}\sqrt {\frac{2}{E_{2}}}\biggr{)}. \tag{42}\]  
Application of Optimum Conditions  
The conditions of optimum (18), derived in Section 3. for this case, will take the form  
\[\frac{\partial\mathcal{D}_{1}}{\partial e_{1}}=A_{1}\frac{dc_{1}}{de_{1}}, \tag{43}\]  
\[\frac{\partial\mathcal{D}_{2}}{\partial e_{2}}=A_{2}\frac{dc_{2}}{de_{2}}. \tag{44}\]  
For a specific cost, we choose \(c_{1}=e_{1}\) and \(c_{2}=e_{2}\), which leads to the shell of minimal volume and minimal weight. From Equations (43) and (44) it follows  
\[\frac{\partial\mathcal{D}_{1}}{\partial e_{1}}=2\pi Rx_{1}. \tag{45}\]  
\[\frac{\partial\mathcal{D}_{2}}{\partial e_{2}}=2\pi R(x_{2}-x_{1}). \tag{46}\]  
By substituting \(m_{x}\) and \(n_{\theta}\) from (3) and (4) and \(\dot{k}_{x}\), \(\dot{\lambda}_{\theta}\) and \(\dot{w}_{x}\) from (6) and (28) into the expression (5) of the power of dissipation energy of the ring of thickness \(e_{1}\), \(\mathcal{D}_{1}\) is obtained as  
\[ \begin{array}{l}\mathcal{D}_{1}=\Bigl{(}\frac{\pi Rx_{0}e_{1}^{2}\omega^{(1 )}}{2}+\frac{2\pi Rx_{0}e_{1}}{\omega^{(1)}}\Bigr{)}\Bigl{[}D_{1}^{(1)}\cos \Bigl{(}\omega^{(1)}x_{1}\Bigr{)}-D_{2}^{(1)}\sin\Bigl{(}\omega^{(1)}x_{1} \Bigr{)}\Bigr{]}-\\ \frac{\pi Rx_{0}e_{1}^{2}}{2}\Bigl{[}D_{1}^{(2)}\omega^{(2)}\cos\Bigl{(}\omega ^{(2)}x_{1}\Bigr{)}-D_{2}^{(2)}\omega^{(2)}\sin\Bigl{(}\omega^{(2)}x_{1} \Bigr{)}\Bigr{]}-\frac{2\pi Rx_{0}e_{1}^{2}}{\omega^{(1)}}D_{1}^{(1)}.\end{array}  \tag{47}\]  
The optimum condition (45) with the dissipation \(\mathcal{D}_{1}\) from (47) becomes  
\[ \begin{array}{l}\overline{D}_{1}^{(1)}\Bigl{[}4.5\sqrt{\frac{E_{1}}{2}}\cos \Bigl{(}X_{1}\sqrt{\frac{2}{E_{1}}}\Bigr{)}+1.5X_{1}\sin\Bigl{(}X_{1}\sqrt{ \frac{2}{E_{1}}}\Bigr{)}-3\sqrt{\frac{E_{1}}{2}}\Bigr{]}+\overline{D}_{2}^{(1 )}\Bigl{[}1.5X_{1}\cos\Bigl{(}X_{1}\sqrt{\frac{2}{E_{1}}}\Bigr{)}-4.5\sqrt{ \frac{E_{1}}{2}}\sin\Bigl{(}X_{1}\sqrt{\frac{2}{E_{1}}}\Bigr{)}\Bigr{]}-\\ \overline{D}_{1}^{(2)}E_{1}\sqrt{\frac{2}{E_{1}}}\cos\Bigl{(}X_{1}\sqrt{\frac {2}{E_{1}}}\Bigr{)}+\overline{D}_{2}^{(2)}E_{1}\sqrt{\frac{2}{E_{1}}}\sin \Bigl{(}X_{1}\sqrt{\frac{2}{E_{1}}}\Bigr{)}-2X_{1}=0,\end{array}  \tag{48}\]  
where the notation for constants \(\overline{D}_{i}^{(i)}\) was introduced  
\[\overline{D}_{i}^{(i)}=\frac{\sigma_{0}}{R}D_{i}^{(i)},\;i=1,2. \tag{49}\]  
Due to the symmetry of the shell, only half of it was considered, as well as half of the dissipation \(\mathcal{D}_{2}\) of the central ring. By analogous procedure as for \(\mathcal{D}_{1}\), a dissipation \(\mathcal{D}_{2}\) was obtained  
\[