## 2 Materials and Methods  
The calculation scheme of the structure is shown in Figure 1. The three-dimensional problem can be reduced to the two-dimensional one by introducing the hypothesis of plane sections. Plane triangular finite elements (FE) for concrete (Figure 2) and one-dimensional two-node finite elements for the steel shell (Figure 3) will be used. Plane triangular finite elements have 2 degrees of freedom at each node (displacements \(u_{i}\) and \(v_{i}\)). Linear functions perform displacement approximation  
\[u =N_{1}u_{1}+N_{2}u_{2}+N_{3}u_{3} \tag{1}\] \[v =N_{1}v_{1}+N_{2}v_{2}+N_{3}v_{3}\]  
where \(N_{i}=\frac{1}{2A}(a_{i}+b_{i}x+c_{i}y)\), \(A\) is the area of the finite element, \(a_{1}=x_{2}y_{3}-x_{3}y_{2}\), \(b_{1}=y_{2}-y_{3}\), \(c_{1}=x_{3}-x_{2}\). Figure 1: Calculation scheme. Coefficients \(a_{2}\), \(b_{2}\), \(c_{2}\), \(a_{3}\), \(b_{3}\), \(c_{3}\) are determined by cyclically changing the indices \(1\to 2\to 3\to 1\). The strain vector in the \(xy\) plane is written as  
\[\{\varepsilon\}=\left\{ \begin{array}{c}\varepsilon_{x}\\ \varepsilon_{y}\\ \gamma_{xy}\end{array} \right\}=\left\{ \begin{array}{c}\frac{\partial u}{ \partial x}\\ \frac{\partial v}{\partial y}\\ \frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\end{array} \right\} =[B]\{U\}, \tag{2}\]  
where \([B]=\frac{1}{2A}\left[ \begin{array}{cccccccc}b_{1}&0&b_{2}&0&b_{3}&0\\ 0&c_{1}&0&c_{2}&0&c_{3}\\ c_{1}&b_{1}&c_{2}&b_{2}&c_{3}&b_{3}\end{array} \right]\), \(\{U\}=\left\{ \begin{array}{cccccccc}u_{1}&v_{1}&u_{2}&v_{2}&u_{3}&v_{3}\end{array}  \right\}^{T}\) is the vector of nodal displacements. The deformation \(\varepsilon_{z}\) is represented as a sum of the axial deformation \(\varepsilon_{z}^{0}\) and the deformation caused by the change in curvature \(\chi\)  
\[\varepsilon_{z}=\varepsilon_{z}^{0}+y\chi. \tag{3}\]  
When deriving the resolving equations, the presence of forced deformations \(\varepsilon_{x}^{*}\), \(\varepsilon_{y}^{*}\), \(\varepsilon_{z}^{*}\), \(\gamma_{xy}^{*}\)--which may include creep deformations, shrinkage, temperature deformations, dilatations, and the deformation caused by the change in curvature \(\chi\). Figure 3: One-dimensional two-node FE for modeling the steel shell. Figure 2: Plane triangular FE for concrete modeling. tion effects, etc.--is considered. The relationship between stresses and deformations of concrete, taking into account forced deformations, is written in the form  
\[ \begin{array}{l}\varepsilon_{x}=\frac{1}{E}\big{(}\sigma_{x}-\
u(\sigma_{y}+ \sigma_{z})\big{)}+\varepsilon_{x}^{*};\\ \varepsilon_{y}=\frac{1}{E}\big{(}\sigma_{y}-\
u(\sigma_{x}+\sigma_{z})\big{)} +\varepsilon_{y}^{*};\\ \varepsilon_{z}=\frac{1}{E}\big{(}\sigma_{z}-\
u(\sigma_{y}+\sigma_{x})\big{)} +\varepsilon_{z}^{*};\\ \gamma_{xy}=\frac{2(1+\
u)}{E}\tau_{xy}+\gamma_{xy}^{*}.\end{array}  \tag{4}\]  
From (4) the stress \(\sigma_{z}\) can be excluded  
\[\sigma_{z}=E(\varepsilon_{z}-\varepsilon_{z}^{*})+\
u\big{(}\sigma_{y}+\sigma_ {x}\big{)}=E\Big{(}\varepsilon_{z}^{0}+y\chi-\varepsilon_{z}^{*}\Big{)}+\
u \big{(}\sigma_{y}+\sigma_{x}\big{)}. \tag{5}\]  
Substituting (5) into the first two equations in Equation (4), following equations can be obtained  
\[ \begin{array}{l}\varepsilon_{x}=\frac{1}{E_{1}}\big{(}\sigma_{x}-\
u_{1} \sigma_{y}\big{)}+\varepsilon_{x}^{*}-\
u\big{(}\varepsilon_{z}^{0}+y\chi- \varepsilon_{z}^{*}\big{)};\\ \varepsilon_{y}=\frac{1}{E_{1}}\big{(}\sigma_{y}-\
u_{1}\sigma_{x}\big{)}+ \varepsilon_{y}^{*}-\
u\big{(}\varepsilon_{z}^{0}+y\chi-\varepsilon_{z}^{*} \big{)},\end{array}  \tag{6}\]  
where \(E_{1}=E/(1-\
u^{2})\), \(\
u_{1}=\
u/(1-\
u)\). The last equation in (4) can be written as  
\[\gamma_{xy}=\frac{2(1+\
u_{1})}{E_{1}}\tau_{xy}+\gamma_{xy}^{*}. \tag{7}\]  
Let us express in (6) and (7) stresses through deformations  
\[