## 2 Materials and Methods
\begin{array}{c}\delta\int\limits_{0}^{l}\sigma_{sz}e_{sz}ds=\frac{E_{s} \delta}{1-v_{s}^{2}}\int\limits_{0}^{l}\Big{(}\varepsilon_{z}^{0}+y_{s}\chi+ v_{s}\{U_{s}\}^{T}[B_{s}]^{T}\Big{)}\big{(}\varepsilon_{z}^{0}+y_{s}\chi\big{)} ds=\\ \qquad=\frac{E_{s}\delta}{1-v_{s}^{2}}\left(\int\limits_{0}^{l}\big{(} \varepsilon_{z}^{0}+y_{s}\chi\big{)}^{2}ds+v_{s}\int\limits_{0}^{l}( \varepsilon_{0}+y_{s}\chi)[B_{s}]\{U_{s}\}ds\right).\end{array}  \tag{24}\]  
The first integral in (24) vanishes after minimization over the vector of nodal displacements. The second integral is written as  
\[\frac{E_{s}\delta v_{s}}{1-v_{s}^{2}}\int\limits_{0}^{l}(\epsilon_{0}+y_{s} \chi)[B_{s}]\{U_{s}\}ds=\{U_{s}\}^{T}[B_{s}]^{T}\frac{E_{s}\delta v_{s}l}{1-v_ {s}^{2}}\Big{(}\varepsilon_{z}^{0}+y_{s}\chi\Big{)}. \tag{25}\]  
External forces on the displacements of the CFST column in the \(xy\) plane do not perform work; therefore, the total energy functional for the problem under consideration is equal to the potential deformation energy. By minimizing the potential energy of deformation along the vector of nodal displacements, a system of linear algebraic equations can be obtained  
\[[K]\{U\}+\{F_{b}\}+\{F_{s}\}-\{F^{*}\}=0, \tag{26}\]where \([K]=[K_{b}]+[K_{s}],\{F_{b}\}=[B]^{T}\{\sigma_{1}\}A=[B]^{T}A\frac{E_{1}v_{1}}{1-v_ {1}^{2}}\left(\epsilon_{z}^{0}+y_{c}\chi-\epsilon_{z}^{*}\right)\left\{ \begin{array} []{c}1\\ 1\\ 0\end{array} \right\}\),  
\[\{F_{s}\}=[B_{s}]^{T}\frac{E_{s}\delta v_{s}l}{1-v_{s}^{2}}\left(\epsilon_{z} ^{0}+y_{s}\chi\right). \tag{27}\]  
The vector \(\{F_{s}\}\) in (27), as well as the matrix \([K_{s}]\) in (23) are written in the local coordinate system of the element. When compiling the system of equations of the FEM, the coordinates are transformed according to the formulas  
\[\{\overline{U}\}=[L]\{U\};\,[K]=[L]^{T}[\overline{K}][L];\,\{F\}=[L]^{T}\{ \overline{F}\};\,[L]=\left[ \begin{array}{ccc}\cos\alpha&\sin\alpha&0&0\\ 0&0&\cos\alpha&\sin\alpha\end{array} \right]. \tag{28}\]  
where \(\{\overline{U}\},\{\overline{F}\},\{\overline{K}\}\) are the nodal displacement vector, load vector, and stiffness matrix in the local coordinate system; \(\{U\},\{F\},[K]\) are the same for the global coordinate system. System (26), in addition to the vector of nodal displacements, contains two more unknowns: the quantities \(\epsilon_{z}^{0}\) and \(\chi\). Equilibrium conditions act as two additional equations  
\[ \begin{array}{c}F=-\sum\sigma_{z,i}A_{i}-\delta\sum\sigma_{sz,i}l_{i};\\ M=F\epsilon=-\sum\sigma_{z,i}y_{c,i}A_{i}-\delta\sum\sigma_{sz,i}y_{s,i}l_{i}.\end{array}  \tag{29}\]  
The stress \(\sigma_{z}\) in the \(i\)-th triangular FE of concrete is defined as  
\[ \begin{array}{c}\sigma_{z}=E\big{(}\epsilon_{z}^{0}+y_{c}\chi-\epsilon_{z}^{ *}\big{)}+\
u\big{(}\sigma_{x}+\sigma_{y}\big{)}=E\big{(}\epsilon_{z}^{0}+y_{ c}\chi-\epsilon_{z}^{*}\big{)}+\
u\big{\{}\begin{array}{ccc}1&1&0\end{array}\big{\}} \{\sigma\}=\\ \quad=E\big{(}\epsilon_{z}^{0}+y_{c}\chi-\epsilon_{z}^{*}\big{)}+\
u\big{\{} \begin{array}{ccc}1&1&0\end{array}\big{\}}\,([D][\{D\}\{U\}-\{\epsilon^{*} \})+\{\sigma_{1}\})=\\ \quad=\bigg{(}E+\frac{2E_{1}v_{1}}{1-v_{1}^{2}}\big{)}\big{(}\epsilon_{z}^{0}+ y_{c}\chi-\epsilon_{z}^{*}\big{)}+\
u\big{\{}\begin{array}{ccc}1&1&0\end{array} \big{\}}\,[D]([B]\{U\}-\{\epsilon^{*}\}).\end{array}  \tag{30}\]  
The stress \(\sigma_{sz}\) in the \(i\)-th one-dimensional FE of the steel shell is defined as  
\[\sigma_{sz}=\frac{E_{s}}{1-v_{s}^{2}}\Big{(}\epsilon_{z}^{0}+y_{s}\chi+\
u_{ s}\epsilon_{s0}\Big{)}=\frac{E_{s}}{1-v_{s}^{2}}\Big{(}\epsilon_{z}^{0}+y_{s} \chi+\
u_{s}[B_{s}][L]\{U\}\Big{)}. \tag{31}\]  
Thus, for the total number of nodes \(n\), the problem is reduced to \(2n+2\) equations with \(2n+2\) unknowns. As equations establishing the relationship between stresses and deformations in concrete, the dependences of the deformational theory of plasticity of concrete by [PERSON] [29] can be used as  
\[