# Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomodiding
## 3 Results  
### Optimization of Sandwich Structures  
As can be seen from the diagram in Figure 2, a homogeneous wall of a given \(s_{\mathit{eff}}\) thickness can be replaced by structures with different geometries. Moreover, larger heights \(h\) at lower surface layer thicknesses \(t_{1}\) can be chosen, or conversely, thicker surface layers at lower sandwich heights \(h\) can be chosen. According to the diagram, for example, a homogeneous wall with a thickness of 20 mm will correspond to structures from \(t_{1}\) = 5 mm, \(h\) = 22 mm to \(t_{1}\) = 1.5 mm at \(h\) = 32 mm. In general, with increasing layer values \(t_{1}\), the material utilization and thus the effect of the sandwich structure decreases, due to higher material volumes near the neutral axis. The value of the flexural rigidity can be expressed by Equations (1) and (2). If we denote \(\rho_{1}\) as the density of the non-lightweight bearing layers, and \(\rho_{2}\) as the density of the lightweight core, the mass per unit length of the member is given by  
\[m=b[h\rho_{2}+2t_{1}(\rho_{1}-\rho_{2})] \tag{7}\]  
Finding the optimum thickness of the outer layers \(t_{1}\) for a certain height \(h\) in terms of flexural rigidity obviously means finding the extremum of the function \(K_{0}/m\). For the modulus of elasticity of the non-lightweight material of E\({}_{1}\) = 680 MPa, the modulus of elasticity of the lightweight core of E\({}_{2}\) = 170 MPa, and a degree of relative volume lightweighting of the core of 50%, the result of the optimization of the investigated structures is given in the diagram in Figure 3.  
Table 1 shows the resulting optimum wall thickness values of the non-lightweight layers \(t_{\mathrm{1 OPT}}\) for the considered structures. Naturally, the \(t_{\mathrm{1 OPT}}/h\) ratio is constant. For a given material composition of the structure, \(t_{\mathrm{1 OPT}}\) = 0.189 \(\times\)\(h\). For each structure, the theoretical values of the effective modulus of elasticity according to Equation (6) were compared with those determined experimentally. The results are shown in Table 2.  
Figure 2: Dependence of the effective thickness and reduced square moment on the dimensions of the sandwich structure. It is clear from Table 2 that the experimental values are, in all cases, only insignificantly lower than the theoretical values. Thus, as can be seen, for the practical purposes of predicting the replacement of a homogeneous single-layer wall with a three-layer structure, the theoretical values can be used as a qualified estimate of the flexural rigidity of a given sandwich structure, with 7.07% of maximum difference. The value of the effective thickness of the homogeneous single-layer wall was used to assess the suitability of replacing the homogeneous single-layer wall with a layered structure. The effective thicknesses calculated from the experimental E\({}_{\text{ef}}\) values are given in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure** & **\(\mathbf{s_{ef}}\) (mm)** & **\(\mathbf{m_{sf}}\)/m\({}_{\text{sw}}\) (\(\mathbf{\cdot}\))** \\ \hline
3v\_7 & 6.7 & 1.21 \\
3v\_10 & 8.9 & 1.12 \\
3v\_11 & 9.1 & 1.42 \\
3v\_15 & 12.1 & 1.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effective weight of the structures. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Structure** & **\(\mathbf{s_{ef}}\) (mm)** & **\(\mathbf{m_{sf}}\)/m\({}_{\text{sw}}\) (\(\mathbf{\cdot}\))** \\ \hline
3v\_7 & 6.7 & 1.21 \\
3v\_10 & 8.9 & 1.12 \\
3v\_11 & 9.1 & 1.42 \\
3v\_15 & 12.1 & 1.64 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Effective weight of the structures. Figure 3: Resulting optimum thicknesses of the surface layers of the individual structures. The degree of efficiency of replacing the homogeneous wall with a three-layer wall is also evident from the comparison of material consumption.