# On the Optimal Design of Steel Shells with Technological Constraints
## 4 Optimal Plastic Design of Shell Composed of Three Rings
### Statically Admissible Solution  
Based on the static boundary conditions, the continuity of the section forces, and the conditions of symmetry, we obtained the following six equations  
\[\begin{split}& x=0;M_{x}^{(1)}=0;\\ & x=x_{1};M_{x}^{(1)}=-\xi Mp_{2};\\ & x=x_{1};\frac{dM_{x}^{(1)}}{dx}=\frac{dM_{x}^{(2)}}{dx};\\ & x=x_{1};M_{x}^{(2)}=-\xi Mp_{2};\\ & x=x_{2}=L;M_{x}^{(2)}=-Mp_{2};\\ & x=x_{2}=L;\frac{dM_{x}^{(2)}}{dx}=0.\end{split} \tag{25}\]  
Using [PERSON]'s yield condition and Table 11-3 in [21] (p. 277) to our case under consideration, we can obtain for bending moment \(M_{x}^{(i)}\), normal force \(N_{\theta}^{(i)}\), and displacement rate \(\dot{w}_{x}^{(i)}\), as follows  
\[M_{x}^{(i)}=\frac{\sigma_{0}e_{i}^{2}}{2}\Big{(}D_{3}^{(i)}\sin\omega^{(i)}x+D _{4}^{(i)}\cos\omega^{(i)}x+\overline{p_{0}}-1\Big{)}, \tag{26}\]  
\[N_{\theta}^{(i)}=\sigma_{0}e_{i}\Big{(}D_{3}^{(i)}\sin\omega^{(i)}x+D_{4}^{(i) }\cos  
The system (30) and (29) contains seven unknown variables \(e_{1},e_{2}\), constants \(D_{3}^{(1)},D_{4}^{(1)}\), \(D_{3}^{(2)}\), \(D_{4}^{(2)}\) and the parameter \(\xi\). After eliminating constants, three nonlinear equations with unknowns \(e_{1},e_{2}\), and \(\xi\) were obtained  
\[\left(\frac{1}{2}-\frac{P_{s}}{E_{2}}\right)\cdot\left\{\cos\left[(X_{2}-X_{1} )\sqrt{\frac{2}{E_{2}}}\right]-1\right\}+\frac{\xi}{2}-\frac{1}{2}=0, \tag{33}\]  
\[\left[1-\frac{P_{s}}{E_{1}}-\frac{\xi}{2}\left(\frac{E_{2}}{E_{1}}\right)^{2} \right]\cos\left(X_{1}\sqrt{\frac{2}{E_{1}}}\right)-\left(1-\frac{P_{s}}{E_{1 }}\right)-\left(\frac{E_{2}}{E_{1}}\right)^{2}\sqrt{\frac{E_{1}}{E_{2}}} \left(0.5-\frac{P_{s}}{E_{2}}\right)\cdot\sin\left(X_{1}\sqrt{\frac{2}{E_{1}} }\right)\sin\left[(X_{2}-X_{1})\sqrt{\frac{2}{E_{2}}}\right]=0, \tag{34}\]  
\[\xi\cdot E_{2}^{2}=E_{1}^{2} \tag{35}\]  
By introducing \(E_{1}=E_{2}\), then \(X_{2}=X_{1}\) and \(\xi=1\) into (34), this Equation (34) becomes the equation of the shell of constant thickness \(E_{1}\), i.e.,  
\[\left(\frac{1}{2}-\frac{P_{s}}{E_{1}}\right)\cos\left(X_{1}\sqrt{\frac{2}{E_{1 }}}\right)+\frac{P_{s}}{E_{1}}-1=0. \tag{36}\]  
To solve this system of nonlinear equations, a numerical procedure was implemented. Knowing that the ratio of moments of plasticity is as in (29) and \(E_{1}\leq E_{2}\), the variable \(\xi\) must be in the interval \(0\leq\xi\leq 1\). In a computer program, created according to that procedure, we reduce \(\xi\) by a fixed step \(h\). The main program contains two subroutines adopted for the numerical calculation of nonlinear equations. In the first subroutine, we determine \(E_{2}\) and with this value, we obtain \(E_{1}\) from (34). By entering the values \(E_{1}\) and \(E_{2}\), calculated in the first step, into the main program is it checked whether the condition (35) is met. If this condition is not met, we reduce the initial interval for \(\xi\) by the initial step \(h=0.1\) and \(n=10\). Then, the procedure is repeated with a new step \(h_{1}=\frac{h}{n}=\frac{1}{n^{2}}\), etc. We interrupt the procedure \(h_{m}=\frac{1}{m^{m+1}}<\varepsilon\). In this way, the statically admissible solutions, \(E_{1}\) and \(E_{2}\), were obtained. In Sections 5 and 6, Example 1, a part of this procedure will be shown.  
Knowing the values of \(E_{1}\), \(E_{2}\), and \(\xi\), we can calculate the constants \(D_{3}^{(i)}\) and \(D_{4}^{(i)}\), (\(i=1,2\)) from (30). After that, we can calculate the bending moments (26) and the normal forces (27).