## 2 Materials and Methods
\begin{array}{l}\sigma_{x}=\frac{E_{1}}{1-\
u_{1}^{2}}\Big{(}\varepsilon_{x }+\
u_{1}\varepsilon_{y}-\Big{(}\varepsilon_{x}^{*}+\
u_{1}\varepsilon_{y}^{* }\Big{)}+\
u(1+\
u_{1})\big{(}\varepsilon_{z}^{0}+y\chi-\varepsilon_{z}^{*} \big{)}\Big{)}=\\ \qquad\qquad=\frac{E_{1}}{1-\
u_{1}^{2}}\Big{(}\varepsilon_{x}+\
u_{1} \varepsilon_{y}-\Big{(}\varepsilon_{x}^{*}+\
u_{1}\varepsilon_{y}^{*}\Big{)} +\
u_{1}\big{(}\varepsilon_{z}^{0}+y\chi-\varepsilon_{z}^{*}\big{)}\Big{)}; \\ \sigma_{y}=\frac{E_{1}}{1-\
u_{1}^{2}}\Big{(}\varepsilon_{y}+\
u_{1} \varepsilon_{x}-\Big{(}\varepsilon_{y}^{*}+\
u_{1}\varepsilon_{x}^{*}\Big{)} +\
u_{1}\big{(}\varepsilon_{z}^{0}+y\chi-\varepsilon_{z}^{*}\big{)}\Big{)}; \\ \tau_{xy}=\frac{E_{1}}{2(1+\
u_{1})}\Big{(}\gamma_{xy}-\gamma_{xy}^{*}\Big{)}.\end{array}  \tag{8}\]  
Let us write Equality (8) in matrix form  
\[\{\sigma\}=[D](\{\varepsilon\}  
The subscripts \"\(el\)\" in Formula (11) correspond to elastic deformations, which represent the difference between total and forced deformations. Expression (11) can be represented in the form  
\[\prod_{b}=\frac{1}{2}\int\limits_{A}\sigma_{z}\Big{(}\varepsilon_{z}^{0}+y \chi-\varepsilon_{z}^{*}\Big{)}dA+\frac{1}{2}\int\limits_{A}\{\sigma\}^{T}(\{ \varepsilon\}-\{\varepsilon^{*}\})dA. \tag{12}\]  
Let us consider the first integral separately in (12)  
\[\frac{1}{2}\int\limits_{A}\sigma_{z}\Big{(}\varepsilon_{z}^{0}+y\chi- \varepsilon_{z}^{*}\Big{)}dA=\frac{1}{2}\int\limits_{A}E\Big{(}\varepsilon_{z }^{0}+y\chi-\varepsilon_{z}^{*}\Big{)}^{2}dA+\frac{v}{2}\int\limits_{A}\{ \sigma\}^{T}\Bigg{\{} \begin{array}{c}1\\ 1\\ 0\end{array} \Bigg{\}}\Big{  
For the FE of steel shell, the displacements are approximated as  
\[u(s)=u_{1}+\frac{u_{2}-u_{1}}{l}s. \tag{19}\]  
The circumferential deformations of the shell are defined as  
\[\varepsilon_{s\theta}=\frac{du}{ds}=\left[ \begin{array}{cc}-\frac{1}{l}& \frac{1}{l}\end{array} \right]\left\{ \begin{array}{c}u_{1}\\ u_{2}\end{array} \right\}=[B_{s}]\{U_{s}\}. \tag{20}\]  
These formulas calculate the stresses in the steel  
\[ \begin{array}{c}\sigma_{s\theta}=\frac{E_{s}}{1-v_{s}^{2}}(\varepsilon_{s \theta}+v_{s}\varepsilon_{sz})=\frac{E_{s}}{1-v_{s}^{2}}\big{(}\varepsilon_{s \theta}+v_{s}\big{(}\varepsilon_{s\theta}^{0}+y_{s}\chi\big{)}\big{)};\\ \sigma_{sz}=\frac{E_{s}}{1-v_{s}^{2}}(\varepsilon_{sz}+v_{s}\varepsilon_{s \theta})=\frac{E_{s}}{1-v_{s}^{2}}\big{(}\varepsilon_{z}^{0}+y_{s}\chi+v_{s} \varepsilon_{s\theta}\big{)}.\end{array}  \tag{21}\]  
The \(y_{s}\) coordinate within the FE is not constant. For convenience, it can be calculated in the middle of the element. The potential deformation energy of the FE is written in the form  
\[\prod_{s}=\frac{1}{2}\left(\delta\int\limits_{0}^{l}\sigma_{s\theta}e_{s \theta}ds+\delta\int\limits_{0}^{l}\sigma_{sz}e_{sz}ds\right). \tag{22}\]  
The first integral in (22) is defined as  
\[ \begin{array}{c}\delta\int\limits_{0}^{l}\sigma_{s\theta}e_{s\theta}ds=\frac {E_{s}\delta}{1-v_{s}^{2}}\int\limits_{0}^{l}\Big{(}\{U_{s}\}^{T}[B_{s}]^{T}+v _{s}\big{(}\varepsilon_{z}^{0}+y_{s}\chi\big{)}\Big{)}[B_{s}]\{U_{s}\}ds=\\ \qquad=\frac{E_{s}\delta l}{1-v_{s}^{2}}\{U_{s}\}^{T}[B_{s}]^{T}[B_{s}]\{U_{s }\}+\frac{E_{s}v_{s}\delta l}{1-v_{s}^{2}}\big{(}\varepsilon_{z}^{0}+y_{s} \chi\big{)}[B_{s}]\{U_{s}\}=\\ \qquad=\{U_{s}\}^{T}[K_{s}]\{U_{s}\}+\{U_{s}\}^{T}[B_{s}]^{T}\frac{E_{s}v_{s }\delta l}{1-v_{s}^{2}}\big{(}\varepsilon_{z}^{0}+y_{s}\chi\big{)},\end{array}  \tag{23}\]  
where \([K_{s}]=\frac{E_{s}\delta l}{1-v_{s}^{2}}[B_{s}]^{T}[B_{s}]=\frac{E_{s}\delta }{l\left(1-v_{s}^{2}\right)}\left[ \begin{array}{cc}1&-1\\ -1&1\end{array} \right]\)-- Stiffness matrix of the steel cage. The second integral in (22) is calculated as  
\[