# On the Optimal Design of Steel Shells with Technological Constraints
## 2 Basic Equations for the Limit Analysis of Shells  
We observe a simple case of a circular cylindrical shell subjected to pressure \(p(x)\) in the outward radial direction. A homogeneous and orthotropic material with rigid, perfectly plastic behavior is considered. The cylindrical coordinate system (\(r\), \(\theta\), \(x\)) is chosen with the axis \(x\), along with the symmetry. All functions depend on \(x\) only. The only displacement of the median surface of the shell is in the radial direction. For the assumption that the elements normal to the median surface are straight and normal after deformation, see Figure 1.  
The power of dissipation energy \(\mathcal{D}\) across a shell element is given by  
\[\mathcal{D}=\int\limits_{-\epsilon/2}^{+\epsilon/2}\big{(}\sigma_{x}\dot{\epsilon }_{x}+\sigma_{0}\dot{\epsilon}_{\theta}\big{)}dz, \tag{1}\]  
\(\dot{\epsilon}_{\theta}\) and \(\dot{\epsilon}_{x}\) are the strain rates of unit circular and axial lengths. It follows as  
\[\mathcal{D}=\frac{d^{2}\dot{w}}{dx^{2}}M_{x}+\frac{\dot{w}}{R}N_{\theta} \tag{2}\]  
where the stress resultants \(M_{x}\) and \(N_{\theta}\) are introduced (See Figure 2). Introducing the dimensionless values \(m_{x}\) and \(n_{\theta}\), i.e.  
\[m_{x}=\frac{M_{x}}{M_{px}},\;n_{\theta}=\frac{N_{\theta}}{N_{p\theta}} \tag{3}\]  
where  
\[M_{px}=\frac{\sigma_{0}e^{2}}{4},\;N_{p\theta}=\sigma_{0}e. \tag{4}\]  
Figure 1: Circular cylindrical shell.  
Figure 2: Shell differential element.  
The dissipation \(\mathcal{D}\) from (1) is given  
\[\mathcal{D}=\sigma_{0}e\Big{(}\dot{k}_{x}m_{x}+\dot{\lambda}_{\theta}n_{\theta} \Big{)}. \tag{5}\]  
The strain rates \(\dot{k}_{x}\) and \(\dot{\lambda}\) corresponding to the generalized stresses \(m_{x}\) and \(n_{\theta}\) respectively are the generalized strain rates  
\[\dot{k}_{x}=\frac{e}{4}\frac{d^{2}\dot{w}}{dx^{2}},\ \dot{\lambda}_{\theta}= \frac{\dot{w}}{R} \tag{6}\]  
where \(\dot{w}\) is the displacement rate of the median surface of the shell.  
The equilibrium of the shell differential element leads to the differential equation  
\[\frac{d^{2}M_{x}}{dx^{2}}+\frac{N_{\theta}}{R}-p(x)=0. \tag{7}\]  
By denoting \(M_{x}\) and \(N_{\theta}\) from (3), Equation (7) reads  
\[\frac{d^{2}m_{x}}{dx^{2}}+2\omega^{2}(n_{\theta}-\overline{p})=0, \tag{8}\]  
where  
\[\omega^{2}=\frac{2}{eR},\ \overline{p}=p\frac{R}{\sigma_{0}e}. \tag{9}\]  
Hodge's yield condition. The yield condition shows the points of the structure where stresses \(M_{x}\) and \(N_{\theta}\) are in a state of plasticity. Unlike the previous ones, the stresses \(M_{\theta}\) and \(T_{x}\) do not appear in the dissipation and represent a reaction correlating to the work by [PERSON] [7]. Starting from [PERSON]'s material, we obtained a nonlinear interaction curve. On the suggestion of [PERSON], the linear approximation was made [20]. Using dimensionless variables \(n_{\theta}m_{x}\), the result was [PERSON]'s hexagon (see Figure 3).  
For a shell element in a limited state, the stresses are represented by a point on [PERSON]'s hexagon. The sides of the hexagon provide the relationship variables \(m\) and \(n\) given in [PERSON]'s Table 11-1 in [21] (p. 277). For each side of the hexagon, we find \(\dot{w}\) with (6), \(\dot{w}\) is a continuous function.