# Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomodiding
## 2 Materials and Methods  
### Flexural Rigidity of the Sandwich Structure  
It is inappropriate to design bent elements with a full homogeneous cross-section in terms of the efficient use of material. In fact, the area near the neutral surface is only stressed slightly compared to the stresses on the outermost fibers of the cross-section. High stiffness with minimum weight can be achieved by using a three-layer sandwich wall section. The outer layer and rigid layers are separated by a lightweight foam core of low stiffness, shown in Figure 1. For the flexural rigidity of the layered composite cross-section, the engineering theory of bending of composite elements generally applies the following rule:  
\[K_{0}=\text{E}_{1}\,J_{\text{R}} \tag{1}\]  
where \(\text{E}_{1}\) is the modulus of the elasticity of the surface layers, and \(J_{\text{R}}\) denotes the quadratic moment of the reduced cross-sectional area, as shown in Figure 1a.  
For the value of the quadratic moment of the reduced cross-sectional area from Figure 1a, the following relation can be obtained:  
\[J_{R}=2\left[\frac{1}{12}bt_{1}^{3}+t_{1}b\frac{\left((h-t_{1})^{3}\right)}{4} \right]+\left[\frac{1}{12}\frac{\text{E}_{2}}{\text{E}_{1}}b(h-2t_{1})^{3}\right] \tag{2}\]A sandwich structure of given dimensions \(t_{i}\) and \(h\) substitutes a homogeneous layer of a certain thickness for its flexural rigidity value. We call this the effective thickness of the sandwich wall, \(s_{\textit{eff}}\); see Figure 1b. The effective thickness of the sandwich wall is thus given by the equality of the flexural rigidities according to Figure 1b.  
\[\text{E}_{1}\,J_{R}=\text{E}_{1}\,J_{\textit{eff}} \tag{3}\]  
where \(J_{\textit{eff}}\) is the quadratic moment of the area of a homogeneous (single-layer) wall section of thickness \(s_{\textit{eff}}\), given by  
\[J_{\textit{eff}}=b\,s_{\textit{eff}}^{3}/12 \tag{4}\]  
It is, therefore,  
\[s_{\textit{eff}}=(12\,J_{R}/b)^{3} \tag{5}\]  
The second possibility, especially for cases in which the evaluation of experimental results is required, is to introduce the concept of the effective modulus of elasticity of the sandwich structure. We define this as the modulus of elasticity of an ideally homogeneous (single-layer) wall of an identical thickness to that of the sandwich, with the same value of flexural rigidity as the sandwich structure. According to Figure 1c, this condition is  
\[E_{\textit{eff}}=\text{E}_{1}\frac{12J_{R}}{bh^{3}} \tag{6}\]  
The dependences of the effective thickness \(s_{\textit{eff}}\) and the reduced square moment \(J_{R}\) on the dimensions of the sandwich structure are plotted in Figure 2. The diagram is constructed for the limiting case \(\text{E}_{2}/\text{E}_{1}<<1\), i.e., assuming neglect of the effect of the stiffness of the as-cast layer on the resulting flexural rigidity. This assumption is justified not only by the relatively small value of the modulus of elasticity of the lightweight middle layer, but also by the fact that the bending stresses in the region of the neutral surface of the bent element are low.  
Due to the generally low values of flexural rigidity of walls of shell structures made of thermoplastics and their dependence on the loading time and temperature, there is often a real danger of a limit state of loss of stability of the thin-walled shell of the tank, manifested by its buckling and subsequent collapse [15; 16; 17; 18; 19]. As follows from the presented parametric study, it is necessary to pay increased attention to the problem of the stability of thermoplastic shells during their structural design.  
For the study of sandwich structures, the LLDPE btene-type material (a linear low-density copolymer of polyethylene with btene) from Gerbaldo Polimeri s.p.a (Italy), used  
Figure 1: Reduced sandwich structure area (**a**), effective thickness (**b**), and effective modulus (**c**).  
in rotational molding technology, under the trade name MICROLEX RM 1242 WT, was chosen, with an MFI = 4 g/10 min and a density of 0.938 g/cm\({}^{3}\).