## 2 Materials and Methods  
### Subject of the Analysis  
The structural elements considered in this research are characterised by their unique geometry. Figure 2 presents the structural scheme of the elements that were imitated in the FEA software used. The element has a rectangular shape in a plane view, and its dimensions are described by \(L_{x}\) and \(L_{y}\). The whole element is divided into identical periodic cells. In the X and Y axes' directions, the element has \(n_{x}\) and \(n_{y}\) periodic cells, respectively. The dimensions of the one periodic cell are as follows: \(L_{x}/n_{x}\) and \(L_{y}/n_{y}\). The whole part has a constant thickness \(t\). The corrugation amplitude of the sinusoidal waves is also constant and described as \(f\). The properties of the material are described as follows: \(E_{s}\) as Young'smodulus, \(\
u_{s}\) as Poisson's ratio, and \(\rho_{s}\) as density. The assumed shell geometry of the analysed element can be described using Formula (1). \[z(x,y)\ =\ -f\cdot\sin\biggl{(}\frac{2\pi\cdot n_{x}}{L_{x}}\,x\biggr{)}\cdot \sin\biggl{(}\frac{2\pi\cdot n_{y}}{L_{y}}\cdot y\biggr{)} \tag{1}\]  
Three different types of analyses were conducted. Boundary conditions, which refer to the constraints assumed in the numerical models, were identical for each of these analyses. The displacements of each edge of the element in each direction (u\({}_{x}\), u\({}_{y}\), and u\({}_{z}\)) were blocked, whereas the rotations (\(\varphi_{x}\), \(\varphi_{y}\), and \(\varphi_{z}\)) were not. The scheme of the numerically analysed element is shown in Figure 3.  
For these straightforward support conditions, it is reasonable to determine the deformation of the mid-plane of the plate utilising a differential equation, which is represented as:  
\[D_{b}\biggl{(}\frac{\partial^{4}w(x,y)}{\partial x^{4}}+2\frac{\partial^{4}w( x,y)}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w(x,y)}{\partial y^{4}} \biggr{)}\ =\ q(x,y). \tag{2}\]  
Figure 3: The scheme of the elements analysed in this study with marked support conditions and applied force. Figure 2: The scheme of the elements analysed in this study with marked geometrical and material parameters. This equation provides a mathematical framework for determining the deflection of the mid-plane under various loading conditions. For flat, non-corrugated, plates, the bending stiffness is defined as:  
\[D_{b}\ =\ \frac{Eh^{3}}{12(1-\
u^{2})}, \tag{3}\]  
where h denotes the thickness of the plane element. Similarly, for the analysis of natural frequencies and dynamic responses, despite the availability of known differential equations that can resolve such problems for flat plates (Formulas (4) and (5), respectively), determining a single bending stiffness value for a corrugated plate is impractical. Consequently, accurately capturing the complex behaviour of corrugated elements necessitates the use of numerical models that replicate the precise geometry of the structures. \[D_{b}\left(\frac{\partial^{4}w(x,y)}{\partial x^{4}}+2\frac{\partial^{4}w(x, y)}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w(x,y)}{\partial y^{4}} \right)+\rho h\omega^{2}w(x,y)\ =\ 0, \tag{4}\]  
\[D_{b}\left(\frac{\partial^{4}w(x,y)}{\partial x^{4}}+2\frac{\partial^{4}w(x,y) }{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}w(x,y)}{\partial y^{4}} \right)+\rho h\omega^{2}\frac{\partial^{2}w(x,y,t)}{\partial y^{2}}\ =\ q(x,y,t). \tag{5}\]  
The first analysis was the static analysis. It was performed in both the ABAQUS and RFEM software. In this analysis, a uniform load \(F\) was applied (Figure 4), opposite to the Z axis direction. Based on these results, the maximum deflection of the whole model and the global maximum value of von Mises stresses were obtained and then compared. A list of the parameters adopted for this analysis is presented in Table 1. The assumed number of various parameters resulted in a total of 360 combinations. The second analysis concerned the natural vibrations of such steel shells. It was performed in both FEM software as well.