## 2 Materials and Methods  
### Static-Strength Analysis  
The choice method of static-strength analysis depends on the type of structure. The linear analysis LA is based on linear geometrical and physical relations between displacements, strains, and stresses. The linear analysis of the finite element method (FEM) is reduced to the solution of a linear set of algebraic Equation (1):  
\[\mathbf{K_{L}}\cdot\mathbf{q}=\mathbf{P} \tag{1}\]  
where: \(\mathbf{K_{L}}\)--linear stiffness matrix of the structure, \(\mathbf{q}\)--vector of nodal displacements, \(\mathbf{P}\)--vector of nodal load. The FEM software is additionally equipped with a linear LBA bifurcation analysis, the main purpose of which is to solve its own problem (2):  
\[[\mathbf{K_{L}}+\mu\cdot\mathbf{K_{G}}]\cdot\mathbf{q}=0 \tag{2}\]  
where: \(\mathbf{K_{G}}\)--geometrical stiffness matrix, \(\mu\)--load multiplier. Linear analysis with linear bifurcation analysis works well for the design of single-layer steel domes. Shallow, single-layer domes are subject to large displacement gradients and should be designed according to geometrically nonlinear analysis. The mathematical model of nonlinear FEM analysis corresponds to a set of nonlinear algebraic equations. The system of equations can be formulated as a sum or an increment. In the first case, the equations have the form (3):  
\[\mathbf{K_{s}}(\mathbf{q})\mathbf{q}=\mathbf{P} \tag{3}\]  
where: \(\mathbf{K_{s}}\)--secant stiffness matrix of the structure. In the second case, the equations have the form (4):  
\[\mathbf{K_{T}}(\mathbf{q})\cdot\Delta\mathbf{q}=\Delta\mu\cdot\mathbf{P}+ \mathbf{R} \tag{4}\]  
where: \(\mathbf{K_{T}}(\mathbf{q})\)--tangent stiffness matrix of the structure, \(\mathbf{R}=\mathbf{P}-\mathbf{F}\)--vector of residual forces, \(\mathbf{F}\)--vector of internal forces. In the equilibrium state \(\mathbf{R}=0\), while in the iterative process, the \(\mathbf{R}\) standard defines the distance from the equilibrium state. The iterative process converges if \(\mathbf{R}\)-\(>0\). The tangent stiffness matrix \(\mathbf{K_{T}}\) of the structure is obtained by aggregating the stiffness matrix of the elements \(\mathbf{K_{T}^{e}}\):  
\[\mathbf{K_{T}}=\sum_{i=1}^{\mathrm{e}}\mathbf{K_{T}^{e}}=\sum_{i=1}^{\mathrm{ e}}(\mathbf{K_{L}^{e}}+\mathbf{K_{CN}^{e}}+\mathbf{K_{u1}^{e}}+\mathbf{K_{u2}^{e}}) \tag{5}\]  
where \(\mathbf{K_{T}^{e}}\) is the tangent stiffness matrix of the element composed of linear stiffness matrix \(\mathbf{K_{L}^{e}}\), stress stiffness matrix \(\mathbf{K_{GN}^{e}}\) and nonlinear stiffness matrices: \(\mathbf{K_{u1}^{e}}\) and \(\mathbf{K_{u2}^{e}}\). In geometrically nonlinear problems, we are dealing with a quadratic form of the eigenproblem:  
\[[\mathbf{K_{L}}+\mu\cdot[\mathbf{K_{GN}}+\mathbf{K_{u1}}+\mathbf{K_{u2}}]] \cdot\mathbf{q}=0 \tag{6}\]  
The constant arc-length method developed by [PERSON] [30] was applied to compute nonlinear equations using Autodesk Robot Structure Professional 2022 software (finite element method code). In this paper, the Schwedler dome is subjected to large displacement gradients and susceptible to stability loss from the condition of node snapping. Nonlinear geometrical relations are defined in the Lagrangian description. The constitutive relations are linear. The geometrical nonlinear analysis GNA is performed using the \"Autodesk Robot Structural Analysis\" software (finite element method code). In this paper, the single-layer steel domes are modeled using frame elements, which have 6 degrees of freedom at the node (Figure 1) (u\({}_{\mathrm{x}}\), u\({}_{\mathrm{y}}\), u\({}_{\mathrm{z}}\), R\({}_{\mathrm{x}}\), R\({}_{\mathrm{y}}\), R\({}_{\mathrm{z}}\), where u denotes displacements in the X\({}_{\mathrm{G}}\), Y\({}_{\mathrm{G}}\), or Z\({}_{\mathrm{G}}\) direction in a global coordinate system, and R denotes rotations around the axes). The local coordinate system of an element is a dextrorotary Cartesian system with the following orientation of axes.