## 1 Introduction  
The procedure for estimating the buckling capacity of spherical shells, developed in the past and described in detail in [1; 2; 3], is based on the assumptions contained in standard EN-1993-1-6 [4] while maintaining an engineering approach. It also complements the recommendation of EDR5 th [5] by unifying the provisions contained therein. The procedure is distinguished by its low complexity and ease of application. Using simple mathematical formulas, it is possible to estimate the buckling capacity of spherical shells composed of different materials with different geometries: thickness \(t\), radius \(R\), and a semi-angle of opening contained within \(10^{\circ}\leq\varphi\leq 90^{\circ}\). During the development of the new procedure, the main assumptions accompanying the dimensioning of the spherical shells contained in the standard [4] and recommendations [5] were not changed. The result was an engineering approach to a very complex issue. The algorithm for proceeding with the estimation of the buckling capacity of the spherical shells fixed along the support edge with a semi-angle of opening contained within \(10^{\circ}\leq\varphi\leq 90^{\circ}\) is as follows:  
The B&M algorithm for calculating the buckling resistance of spherical shells:  
1. Taking material and geometrical data: \(E,f_{y\!k},R\), and \(t\). 2. Calculation of critical and plastic resistance. Determination of slenderness \(\lambda\) \[p_{\text{Rcr}(\text{LBA})}=1.303E\big{(}\frac{t}{R}\big{)}^{2}\] \[p_{\text{RpI}(\text{MNA})}=1.986f_{y\!k}\frac{t}{R}\] \[\text{then }\lambda=\sqrt{\frac{p_{\text{RpI}(\text{MNA})}}{p_{ \text{Rcr}(\text{LBA})}}}\]Selection of shell execution class. Determination of imperfection amplitude ClassA \(\rightarrow\)\(Q\)\(=\)\(40\) ClassB \(\rightarrow\)\(Q\)\(=\)\(25\) ClassC \(\rightarrow\)\(Q\)\(=\)\(16\) then \(\Delta w_{k}=\frac{1}{Q}\sqrt{R}t\)
4. Determination of buckling parameters \(\alpha\) and \(\beta\). Determination of slenderness \(\lambda_{\mathrm{P}}\) \[\alpha(\Delta w_{k}/t) =\frac{0.65}{1+1.8(\Delta w_{k}/t)^{0.8}}\] \[\beta(\Delta w_{k}/t) =0.87\left(\frac{\Delta w_{k}}{t}\right)^{0.026}\] then \(\lambda_{p}=\sqrt{\frac{\alpha}{1-\beta}}\)
5. Determination of working range of spherical shell. Reduction factor \(\chi\) \[\chi(\lambda)=1, \mathrm{dla}\lambda<\lambda_{0}\] \[\chi(\lambda)=a\lambda^{2}+b\lambda+c, \mathrm{dla}\lambda_{0}\leq\lambda\leq\lambda_{p}\] \[\chi(\lambda)=\alpha/\lambda^{2}, \mathrm{dla}\lambda>\lambda_{p}\] \[\mathrm{where}\lambda_{0}=0.2\]
6. Determination of polynomial expressions for elastic-plastic range \[a =\frac{\alpha(0.4-3\lambda_{p})+\lambda_{p}^{3}}{\lambda_{p}^{3}( 0.04-0.4\lambda_{p}+\lambda_{p}^{2})},\] \[p_{\text{RpI}(\text{MNA})}=1.986f_{y\!k}\frac{t}{R}\]0 \[p_{\text{RpI}(\text{MNA})}=1.986f_{y\!k}\frac{t}{R}\]1
7. Determination of characteristic buckling capacity of spherical shell \[p_{\text{RpI}(\text{MNA})}=1.986f_{y\!k}\frac{t}{R}\]2  
It is worth remembering that, in the development of the above calculation formula, as many as seven different forms of imperfections were taken into account, counting from the first two forms of buckling, through linear imperfections from welding, and ending with local dimple. This is described in detail in [3]. In total, more than 8000 numerical analyses have been performed. It is also not insignificant that each of the numerically tested spherical shells with imperfections was analyzed for different manufacturing quality classes (A, B, and C), and thus different amplitudes of imperfections were taken into account, according to [4; 5]. It is worth mentioning at this point that the spherical shells were not classified according to their geometric imperfections during the conducted laboratory tests [6; 7; 8; 9; 10; 11; 12]. Such a classification was included in the studies conducted by [PERSON] and [PERSON] and described in detail in [13; 14].