# On the Optimal Design of Steel Shells with Technological Constraints
## 7 Conclusions  
* The necessary and sufficient conditions of the optimum for the full circular cylindrical shell with technological limitations, composed of rings of variable thicknesses have been derived with one behavioral constraint related to a specific constitutive low, for limit load-carrying capacity \(p_{0}\). The paper proves that the necessary optimality conditions are identical to the sufficient conditions of the optimum. The objective function is cost \(C\). The specific cost \(c_{i}\) is the function of thickness \(e_{i}\) which leads to the shell of minimal volume, and also minimal weight. These conditions are derived for a more complex case, i.e., for a full shell with technological constraints, not a sandwich-type shell. Note that in [23], [PERSON] was the first to derive optimal conditions in the plastic region for a sandwich-type shell, whose thickness can vary continuously from zero to infinity. * The optimal structural design of shell (in detail with three rings) with an analytical approach complemented by numerical tools and direct connection with basic equations of limit analysis (Section 2) is given. In the static part, the problem is reduced to three linear Equations (33)-(35) and the procedure for their numerical solution is briefly explained, which leads to the optimal dimensions of the shell. Within the kinematic part, the energy dissipation of rings \(\mathcal{D}_{1}\) and \(\mathcal{D}_{2}\) were analytically obtained, and, using the optimum conditions, the displacement rates of the shell \(\dot{w}_{x}\) were determined. In this way, the real collapse mechanism is defined with a minimum of dissipative shell energy. This numerical procedure was applied, and the dimensions in numerical examples were obtained. Figure 6: The simply supported full shell composed of three rings. * In numerical examples, the optimal dimensions of shells were successfully calculated, applying the conditions of optimum derived in Section 3. Due to the choice of the cost function \(C\), \(c_{i}=e_{i}\), the shells have minimum volume. In Example 1, the collapse mechanism is shown and confirmed (see Figure 6). For shells of the same limit load \(p_{0}\) and the same length \(L\), the length of ring \(X_{1}\) is varied. By comparing these shells, for technological reasons, composed of three rings, with an optimal shell of constant thickness, a gain of 6.4% was achieved. This gain was for the shell of length \(X_{1}=0.07\). By further calculation, the most favorable length \(X_{1}\) can be obtained. * This type of cylindrical shell constituted of rings is common in the mechanical, chemical industry, etc. With the increase in the number of rings, the gains also increase. Due to complexity, there are very few solutions to this problem given in analytical, closed-form, and only for simple cases with the results of strongly simplifying assumptions are obtained. The solutions in this paper, which are mostly analytical and complemented by exposed numerical tools, can be useful for confirming numerical procedures, as well as providing information for creating concepts and ideas for technological solutions to more practical and complex problems. * In further work, this problem will be extended, especially, to the cases of shells with stiffeners, then with end loads for more complex problems of stability and dynamics from practice. Based on the results presented in this paper, the shells, differing in terms of the limit conditions, types of loads, and the number of rings of various thicknesses may be the object of new research and experimental verification to obtain new results. Conceptualization, D.T. and T.I. ; methodology, D.T. ; software, T.I. and A.I. ; validation, D.T., T.I. and S.Z. ; formal analysis, M.S.S. ; investigation, A.I. ; resources, T.I. ; data curation, T.I. ; writing--original draft preparation, D.T. and T.I. ; writing--review and editing, D.T. ; visualization, M.S.S. ; supervision, S.Z. and M.S.S. All authors have read and agreed to the published version of the manuscript.