# Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomolding

^{*}

## Abstract

**:**

_{1OPT}= 0.189 h.

## 1. Introduction

## 2. Materials and Methods

#### Flexural Rigidity of the Sandwich Structure

_{0}= E

_{1}J

_{R}

_{1}is the modulus of the elasticity of the surface layers, and J

_{R}denotes the quadratic moment of the reduced cross-sectional area, as shown in Figure 1a.

_{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

_{ef}; see Figure 1b. The effective thickness of the sandwich wall is thus given by the equality of the flexural rigidities according to Figure 1b.

_{1}J

_{R}= E

_{1}J

_{ef}

_{ef}is the quadratic moment of the area of a homogeneous (single-layer) wall section of thickness s

_{ef}, given by

_{ef}= b s

_{ef}

^{3}/12

_{ef}= (12 J

_{R}/b)

^{3}

_{ef}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 E

_{2}/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.

^{3}.

## 3. Results

#### 3.1. Optimization of Sandwich Structures

_{ef}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 ρ

_{1}as the density of the non-lightweight bearing layers, and ρ

_{2}as the density of the lightweight core, the mass per unit length of the member is given by

_{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.

_{1OPT}for the considered structures. Naturally, the t

_{1OPT}/h ratio is constant. For a given material composition of the structure, t

_{1OPT}= 0.189 × h.

_{ef}values are given in Table 3.

_{ef}, as

_{ef}with the effective mass of the structures considered are given in Table 3. As can be seen, the most efficient structure is clearly structure 3v_15. Structures 3v_7 and 3v_10 can no longer be recommended due to the small material savings. This is due to the fact that structures 3v_7 and 3v_10 achieve seemingly high s

_{ef}values, but uneconomically, at the cost of considerable thicknesses of the surface layers t

_{1}. This means that they are similar to a homogeneous wall.

#### 3.2. Stability of Shell Structures

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

b | cross-sectional width of the sandwich element |

E_{1} | calculated modulus of elasticity—non-lightweight layers |

E_{2} | calculated modulus of elasticity—lightweight layer (core) |

E_{ef} | effective modulus of elasticity |

E_{ef theoretical} | theoretical effective modulus of elasticity |

E_{ef experimental} | experimental effective modulus of elasticity |

h | cross-sectional height of the sandwich element |

J_{R} | quadratic modulus of the reduced cross-sectional area |

J_{ef} | effective quadratic modulus of the cross-section of the sandwich element |

K_{0} | flexural rigidity of the cross-section of the sandwich element |

m | unit mass of the sandwich structure |

m_{ef} | unit mass of the homogeneous wall of effective thickness |

s_{ef} | effective thickness of the sandwich structure |

t_{1} | thickness of surface—non-lightweight layers |

t_{1OPT} | optimum thickness of the surface—non-lightweight layers |

ρ_{1} | density of the non-lightweight layers |

ρ_{2} | density of the lightweight core |

## References

- Černohlávek, V.; Štěrba, J.; Svoboda, M.; Zdráhal, T.; Suszyński, M.; Chalupa, M.; Krobot, Z. Verification of the Safety of Storing a Pair of Pressure Vessels. Manuf. Technol.
**2022**, 21, 762–773. [Google Scholar] [CrossRef] - Daghighi, S.; Weaver, P.M. Three-dimensional effects influencing failure in bend-free, variable stiffness composite pressure vessels. Compos. Struct.
**2021**, 262, 113346. [Google Scholar] [CrossRef] - Margolis, J.M. Engineering Thermoplastics: Properties and Applications; CRC Press: Boca Raton, FL, USA, 2020. [Google Scholar]
- Shaker, R.; Rodrigue, D. Rotomolding of Thermoplastic Elastomers Based on Low-Density Polyethylene and Recycled Natural Rubber. Appl. Sci.
**2019**, 9, 5430. [Google Scholar] [CrossRef] - Crawford, R.J.; Throne, J.L. Rotational Molding Technology; William Andrew Publishing: New York, NY, USA, 2001. [Google Scholar]
- Novo, A.V.; Bayon, J.R.; Castro-Fresno, D.; Rodriguez-Hernandez, J. Review of Seasonal Heat Storage in Large Basins: Water Tanks and Gravel–Water Pits. Appl. Energy
**2010**, 87, 390–397. [Google Scholar] [CrossRef] - Tkac, J.; Samborski, S.; Monkova, K.; Debski, H. Analysis of mechanical properties of a lattice structure produced with the additive technology. Comp. Struct.
**2020**, 242, 112138. [Google Scholar] [CrossRef] - Xue, B.; Peng, Y.X.; Ren, S.F.; Liu, N.N.; Zhang, Q. Investigation of impact resistance performance of pyramid lattice sandwich structure based on SPH-FEM. Comp. Struct.
**2021**, 261, 113561. [Google Scholar] [CrossRef] - Saifullah, A.; Wang, L.; Barouni, A.; Giasin, K.; Lupton, C.; Jiang, C.; Zhang, Z.; Quaratino, A.; Dhakal, H.N. Low Velocity Impact (LVI) and Flexure-after-Impact (FAI) Behaviours of Rotationally Moulded Sandwich Structures. J. Mater. Res. Technol.
**2021**, 15, 3915–3927. [Google Scholar] [CrossRef] - Błachut, J.; Magnucki, K. Strength, stability, and optimization of pressure vessels: Review of selected problems. Appl. Mech. Rev.
**2008**, 61, 060801. [Google Scholar] [CrossRef] - Renhuai, L.; Jianghong, X. Development of nonlinear mechanics for laminated composite plates and shells. Chin. J. Theor. Appl. Mech.
**2017**, 49, 487–506. [Google Scholar] [CrossRef] - Carrera, E.; Soave, M. Use of functionally graded material layers in a two-layered pressure vessel. J. Press. Vessel Technol.
**2011**, 133, 051202. [Google Scholar] [CrossRef] - Magnucki, K.; Stasiewicz, P. Critical sizes of ground and underground horizontal cylindrical tanks. Thin-Walled Struct.
**2003**, 41, 317–327. [Google Scholar] [CrossRef] - Brar, G.S.; Hari, Y.; Williams, D.K. Calculation of working pressure for cylindrical vessel under external pressure. In Proceedings of the ASME 2010 Pressure Vessels and Piping Division/K-PVP Conference, Bellevue, Washington, DC, USA, 18–22 July 2010. [Google Scholar] [CrossRef]
- Subbaiah, T.; Vijetha, P.; Marandi, B.; Sanjay, K.; Minakshi, M. Ionic Mass Transfer at Point Electrodes Located at Cathode Support Plate in an Electrorefining Cell in Presence of Rectangular Turbulent Promoters. Sustainability
**2022**, 14, 880. [Google Scholar] [CrossRef] - Banghai, J.; Zhibin, L.; Fangyun, L. Failure mechanism of sandwich beams subjected to three-point bending. Comp. Struct.
**2015**, 133, 739–745. [Google Scholar] [CrossRef] - Behravan, A.; Dejong, M.M.; Brand, A.S. Laboratory Study on Non-Destructive Evaluation of Polyethylene Liquid Storage Tanks by Thermographic and Ultrasonic Methods. CivilEng
**2021**, 2, 823–851. [Google Scholar] [CrossRef] - Yang, L.; Chen, Z.; Chen, F.; Guo, W.; Cao, G. Buckling of cylindrical shells with general axisymmetric thickness imperfections under external pressure. Eur. J. Mech. A/Solids
**2013**, 38, 90–99. [Google Scholar] [CrossRef] - Yang, L.; Luo, Y.; Qiu, T.; Zheng, H.; Zeng, P. A novel analytical study on the buckling of cylindrical shells subjected to arbitrarily distributed external pressure. Eur. J. Mech. A/Solids
**2022**, 91, 104406. [Google Scholar] [CrossRef] - Jiroutova, D. Methodology of Experimental Analysis of Long-Term Monitoring of Sandwich Composite Structure by Fibre-Optic Strain Gauges. Manuf. Technol.
**2016**, 16, 512–518. [Google Scholar] [CrossRef] - Beall, G.L. Rotational Molding: Design, Materials, Tooling, and Processing; Hanser: Munich, Germany, 1998. [Google Scholar]
- Scott, D. Products and Applications—Composite and Thermoplastic Tanks, Silos and Other Vessels. Adv. Mater. Water Handl.
**2000**, 175–216. [Google Scholar] [CrossRef] - Beňo, P.; Kozak, D.; Konjatić, P. Optimization of thin-walled constructions in CAE system ANSYS. Tech. Gaz.
**2014**, 21, 1051–1055. [Google Scholar] - Mešić, E.; Muratović, E.; Redžepagić-Vražalica, L.; Pervan, N.; Muminović, A.J.; Delić, M.; Glušac, M. Experimental & FEM analysis of orthodontic mini-implant design on primary stability. Appl. Sci.
**2021**, 11, 5461. [Google Scholar] [CrossRef] - Regassa, Y.; Gari, J.; Lemu, H.G. Composite Overwrapped Pressure Vessel Design Optimization Using Numerical Method. J. Compos. Sci.
**2022**, 6, 229. [Google Scholar] [CrossRef]

**Figure 2.**Dependence of the effective thickness and reduced square moment on the dimensions of the sandwich structure.

**Figure 4.**Example of loss of stability of an underground reservoir produced by rotomolding technology.

**Figure 5.**Example of the result of FEM modelling of the stability capacity of an underground tank loaded with soil pressure.

Structure | h (mm) | t_{1} (mm) | t_{1OPT} (mm) |
---|---|---|---|

3v_7 | 7 | 2.4 | 1.32 |

3v_10 | 10 | 2.5 | 1.90 |

3v_11 | 11 | 1.6 | 2.08 |

3v_15 | 15 | 2.0 | 2.84 |

Structure | E_{ef theoretical} (MPa) | E_{ef experimental} (MPa) | Difference δ (%) |
---|---|---|---|

3v_7 | 678 | 640 | 5.94 |

3v_10 | 612 | 576 | 6.25 |

3v_11 | 450 | 422 | 6.64 |

3v_15 | 424 | 396 | 7.07 |

Structure | s_{ef} (mm) | m_{sef}/m_{3v} (-) |
---|---|---|

3v_7 | 6.7 | 1.21 |

3v_10 | 8.9 | 1.12 |

3v_11 | 9.1 | 1.42 |

3v_15 | 12.1 | 1.64 |

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**MDPI and ACS Style**

Šuba, O.; Bílek, O.; Kubišová, M.; Pata, V.; Měřínská, D.
Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomolding. *Appl. Sci.* **2022**, *12*, 9276.
https://doi.org/10.3390/app12189276

**AMA Style**

Šuba O, Bílek O, Kubišová M, Pata V, Měřínská D.
Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomolding. *Applied Sciences*. 2022; 12(18):9276.
https://doi.org/10.3390/app12189276

**Chicago/Turabian Style**

Šuba, Oldřich, Ondřej Bílek, Milena Kubišová, Vladimír Pata, and Dagmar Měřínská.
2022. "Evaluation of the Flexural Rigidity of Underground Tanks Manufactured by Rotomolding" *Applied Sciences* 12, no. 18: 9276.
https://doi.org/10.3390/app12189276