## 2 Materials and Methods
\begin{array}{c}\epsilon_{x}=\frac{1}{E(\Gamma)}\big{(}\sigma_{x}-\
u\big{(} \sigma_{y}+\sigma_{z}\big{)}\big{)}+\epsilon_{d};\\ \epsilon_{y}=\frac{1}{E(\Gamma)}\big{(}\sigma_{y}-\
u(\sigma_{x}+\sigma_{z}) \big{)}+\epsilon_{d};\\ \epsilon_{z}=\frac{1}{E(\Gamma)}\big{(}\sigma_{z}-\
u\big{(}\sigma_{x}+\sigma_ {y}\big{)}\big{)}+\epsilon_{d};\\ \quad\quad\quad\quad\gamma_{xy}=\frac{2(1+\
u)}{E(\Gamma)}\tau_{xy},\end{array}  \tag{32}\]  
where \(\epsilon_{d}=-g_{0}\Gamma^{2}/3\) are the dilatation deformations, \(g_{0}\) is the dilatation module, \(\Gamma=\sqrt{\frac{2}{3}\sqrt{\left(\epsilon_{1}-\epsilon_{2}\right)^{2}+\left( \epsilon_{2}-\epsilon_{3}\right)^{2}+\left(\epsilon_{1}-\epsilon_{3}\right)^{2}}}\) is the intensity of shear deformations. Dilatation deformations can be considered as a special case of forced deformations \(\epsilon_{xx}^{*}\), \(\epsilon_{y}^{*}\), \(\epsilon_{z}^{*}\). The dilatation modulus \(g_{0}\) in [PERSON]'s theory is defined as  
\[g_{0}=-\frac{\theta_{c}}{\Gamma_{c}^{2}}, \tag{33}\]where \(\theta_{c}\) is the ultimate volumetric deformation of concrete at pure shear, \(\Gamma_{c}\) is the limiting intensity of shear strains at pure shear, calculated as  
\[\Gamma_{c}=\frac{2T_{c}}{G_{0}}. \tag{34}\]  
In Formula (33) \(T_{c}=\sqrt{R_{b}R_{bt}/3}\) is the limiting intensity of shear stresses, \(R_{b}\) and \(R_{bt}\) are the compressive and tensile strength of concrete, respectively, \(G_{0}=E_{0}/(2\left(1+\
u_{b}\right))\) is the initial shear modulus of concrete, \(E_{0}\) is the initial modulus of elasticity of concrete. The formula determines the secant module \(\mathrm{E}(\Gamma)\)  
\[E(\Gamma)=E_{0}\bigg{(}1-\frac{\Gamma}{2\Gamma_{s}}\bigg{)}, \tag{35}\]  
where \(\Gamma_{s}\) is the limiting intensity of shear deformations, depending on the nature of the stress state  
\[\Gamma_{s}=\Gamma_{c}k(\lambda,\omega),\ k(\lambda,\omega)=\frac{\lambda(1+ \omega)}{2}+\sqrt{\frac{\lambda^{2}(1+\omega)^{2}}{4}+(1+\omega)}. \tag{36}\]  
The parameter \(\omega\) in (35) is defined as  
\[\omega=e\bigg{(}\frac{S}{T}\bigg{)}^{3}, \tag{37}\]  
where \(T=\frac{1}{\sqrt{6}}\sqrt{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma _{2}-\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}}\) is the shear stress intensity, \(e=\frac{R_{b}R_{bt}}{3T_{c}^{2}}-1\), \(S=\sqrt{3}\left[\frac{1}{2}(\sigma_{1}-\sigma)(\sigma_{2}-\sigma)(\sigma_{3} -\sigma)\right]^{\frac{1}{3}}\), \(\sigma=(\sigma_{1}+\sigma_{2}+\sigma_{3})/3\). The parameter \(\lambda\) in (35) is calculated by the formula  
\[\lambda=\frac{f\sigma}{T}, \tag{38}\]  
where \(f=\frac{3T_{c}\left(R_{b}-R_{bt}\right)}{R_{b}R_{bt}}\). The calculation is performed by a stepwise method. The load is increased in small portions \(\Delta F\). The dilatation strain increment is defined as  
\[ \begin{split}&\Delta\varepsilon_{d}=\varepsilon_{d}(\Gamma+ \Delta\Gamma)-\varepsilon_{d}(\Gamma)=-\frac{g_{0}}{3}\big{(}\Gamma^{2}+2 \Gamma\Delta\Gamma+\Delta\Gamma^{2}\big{)}-\frac{g_{0}}{3}\Gamma^{2}=\\ &=-\frac{g_{0}}{3}\big{(}2\Gamma\Delta\Gamma+\Delta\Gamma^{2} \big{)}.\end{split}  \tag{39}\]  
The value of \(\Delta\Gamma^{2}\) in comparison with the first term in the parenthesis can be neglected, since it is a quantity of a higher order of smallness, and then Formula (38) takes the form  
\[\Delta\varepsilon_{d}=-\frac{2g_{0}\Gamma\Delta\Gamma}{3}. \tag{40}\]  
The material of the shell is assumed to be ideal elastoplastic; the Huber-Mises-Genka plasticity condition is used  
\[\frac{1}{\sqrt{2}}\sqrt{\sigma_{sz}^{2}-\sigma_{sz}\sigma_{g\phi}+\sigma_{s \theta}^{2}}=R_{y}, \tag{41}\]  
where \(R_{y}\) is the yield strength of steel. For calculations, the authors have developed the program in the MATLAB environment.