# Buckling Analysis of an AUV Pressure Vessel with Sliding Stiffeners

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## Abstract

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## 1. Introduction

## 2. Analysis of the Unstiffened Shell

- The Von Mises simplified equation (see References [39,40,41,42]):$${p}_{cr}=\frac{Eh}{R}\frac{1}{{n}^{2}+\frac{1}{2}{\left(\frac{\pi R}{L}\right)}^{2}}\left[\frac{1}{{\left({n}^{2}{\left(\frac{L}{\pi R}\right)}^{2}+1\right)}^{2}}+\frac{{h}^{2}}{12{R}^{2}\left(1-{\nu}^{2}\right)}{\left({n}^{2}+{\left(\frac{\pi R}{L}\right)}^{2}\right)}^{2}\right]$$
- The Von Mises complete equation (see Reference [39]):$${p}_{cr}=\frac{\frac{1}{12}\left[{\left({n}^{2}+{\left(\frac{\pi R}{L}\right)}^{2}\right)}^{2}-2{\mu}_{1}{n}^{2}+{\mu}_{2}\right]\left(\frac{E}{1-{\nu}^{2}}\right){\left(\frac{h}{R}\right)}^{3}+\frac{Eh}{R}{\left({\left(\frac{nL}{\pi R}\right)}^{2}+1\right)}^{-2}}{\left({n}^{2}-1+\frac{1}{2}{\left(\frac{\pi R}{L}\right)}^{2}\right)}$$$${\mu}_{1}=1+\frac{\rho}{2}\left(3+\nu +\left(1-{\nu}^{2}\right)\rho \right)\phantom{\rule{0ex}{0ex}}{\mu}_{2}=1+\rho \left(1+\nu \right)-{\rho}^{2}\left[\nu \left(1+2\nu \right)+\left(1-{\nu}^{2}\right)\left(1-\rho \nu \right)\left(1+\frac{\left(1+\nu \right)}{\left(1-\nu \right)}\rho \right)\right]\phantom{\rule{0ex}{0ex}}\rho =\frac{1}{1+{\left(\frac{nL}{\pi R}\right)}^{2}}$$
- The Tokugawa equation (see Reference [39]):$${p}_{cr}=\frac{\frac{1}{12}\left[{\left({n}^{2}+{\left(\frac{\pi R}{L}\right)}^{2}\right)}^{2}-\frac{{n}^{4}\left(2{n}^{2}-1\right)}{{\left({n}^{2}+{\left(\frac{\pi R}{L}\right)}^{2}\right)}^{2}}\right]\left(\frac{E}{1-{\nu}^{2}}\right){\left(\frac{h}{R}\right)}^{3}+\frac{Eh}{R}{\left({\left(\frac{nL}{\pi R}\right)}^{2}+1\right)}^{-2}}{\left({n}^{2}-1+\frac{1}{2}{\left(\frac{\pi R}{L}\right)}^{2}\right)}$$
- The U.S. Experimental Model Basin (DTMB) equation (see Reference [39]):$${p}_{cr}=\frac{2.42E}{{\left(1-{\nu}^{2}\right)}^{3/4}}{\left(\frac{h}{2R}\right)}^{5/2}{\left(\frac{L}{2R}-0.45{\left(\frac{h}{2R}\right)}^{1/2}\right)}^{-1}$$
- The Kendrick equation (see Reference [30], page 19):$${p}_{cr}=\frac{Eh}{R}\frac{\left[{\left({\left(\frac{nL}{\pi R}\right)}^{2}+1\right)}^{-2}+\frac{{h}^{2}}{12{R}^{2}\left(1-{\nu}^{2}\right)}{\left({n}^{2}-1+{\left(\frac{\pi R}{L}\right)}^{2}\right)}^{2}\right]}{\left[{n}^{2}-1+\frac{1}{2}{\left(\frac{\pi R}{L}\right)}^{2}\right]}$$$${\varphi}_{cr}=\frac{{p}_{cr}\left(1-{\nu}^{2}\right)}{E}\frac{R}{h}$$$$\lambda =\frac{L}{R}$$$$\xi =\frac{1}{12}{\left(\frac{h}{R}\right)}^{2}$$$$Z=\frac{{L}^{2}}{Rh}\sqrt{1-{\nu}^{2}}$$

- The shell is subjected to a uniform hydrostatic pressure, i.e., both lateral pressure and axial load are considered.
- The material of the shell is homogeneous, isotropic, and has a linear-elastic behavior.
- The formulas are valid for shells whose length is shorter than the critical length (L
_{cr}) given by (see, e.g., Reference [39]):$$\frac{{L}_{cr}}{R}=\left(\frac{16\pi \sqrt{3}}{27}\sqrt[4]{1-{\nu}^{2}}\right)\sqrt{R/h}$$ - The boundary conditions are those of a simply supported shell at both ends, which means (see, e.g., Hoff and Soong [43]):$$w={w}_{,xx}=u={N}_{x\theta}=0,\text{}\mathrm{at}\text{}x=0\text{}and\phantom{\rule{0ex}{0ex}}w={w}_{,xx}={N}_{x}={N}_{x\theta}=0,\text{}\mathrm{at}\text{}x=L\phantom{\rule{0ex}{0ex}}{N}_{x0}={\sigma}_{x0}h=-pR/2,\text{}\mathrm{at}\text{}x=L\text{}(\mathrm{pre}-\mathrm{buckling}\text{}\mathrm{state})$$

- (i)
- The most conservative value for the critical buckling pressure is the one given by the Von Mises complete equation (Equation (2)), which provides a result very close to Kendrick’s equation (Equation (5)).
- (ii)
- The least conservative value for the critical buckling pressure is the one given by the Von Mises simplified equation (Equation (1)).

## 3. Analysis of the Conventional Stiffened Shell

_{B}, which is the total length of the AUV shell), and ${P}_{c5}$ represents the externally applied pressure that makes the stress measured in the circumferential direction, at the mid surface of the plating and midway between stiffeners, reach the yield stress (${\sigma}_{y}$) of the material, and is given by the expression (see MacKay [30], page 23):

- ${R}_{f}={R}_{o}-h-{h}_{f}$ = $R-\left(h/2\right)-{h}_{f}$ (radius to the extreme fiber of frame flange),
- ${A}_{f}={h}_{f}{h}_{w}$ (cross-sectional area of frame).

## 4. Analysis of the Shell with Sliding Stiffeners

## 5. Geometrically Nonlinear Analysis of the Shell Considering Small Imperfections

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 10.**Schematic view of the internal structure of the autonomous underwater vehicle (AUV) and its sliding stiffeners: (

**a**) overall structural arrangement; (

**b**) ring stiffener detail.

**Figure 11.**Geometries considered in the analyses: (

**a**) four conventional stiffeners rigidly attached to the shell (Section 3), (

**b**) shell with sliding stiffeners regularly spaced along the shell.

**Figure 16.**Most refined mesh used for the shell with sliding stiffeners: front view of the stiffening members.

**Figure 17.**Critical pressures obtained by the finite element method for the three alternative shell designs.

**Figure 18.**Maximum shell imperfections at the mid-section: (

**a**) 5% and (

**b**) 10% of the shell thickness.

**Figure 19.**Critical buckling pressure for the perfect shell (linear bifurcation analysis) and geometrically nonlinear analysis curves for shells with imperfections.

Equation | ${\mathit{\varphi}}_{\mathit{c}\mathit{r}}$ | ${\mathit{p}}_{\mathit{c}\mathit{r}}\text{}\left(\mathbf{MPa}\right)$ |
---|---|---|

(1) Von Mises (simplified) | 1.7182 × 10^{−3} | 8.86 |

(2) Von Mises (complete) | 1.4212 × 10^{−3} | 7.33 |

(3) Tokugawa | 1.5173 × 10^{−3} | 7.82 |

(4) DTMB | 1.6372 × 10^{−3} | 8.44 |

(5) Kendrick | 1.4355 × 10^{−3} | 7.40 |

Mean Value: | 1.5459 × 10^{−3} | 7.97 |

Parameter | Value |
---|---|

L_{B} (length between bulkhead supports) | 840 mm |

L_{f} (length between stiffeners) | 168 mm |

L = L_{f} − h_{w} (unsupported length) | 161.65 mm |

h_{w} (stiffeners’ thickness) | 6.35 mm |

h_{f} (stiffeners’ height) | 6.35 mm |

N (number of ring stiffeners) | 4 |

**Table 3.**Calculated critical pressures for the conventional stiffened shell using analytical formulations.

$\mathbf{Equation}\text{}\mathbf{to}\text{}\mathbf{Estimate}\text{}{\mathit{p}}_{\mathit{s}}$ | ${\mathit{p}}_{\mathit{s}}\text{}\left(\mathbf{MPa}\right)$ | ${\mathit{p}}_{\mathit{c}\mathit{r}}\text{}\left(\mathbf{MPa}\right)$ |
---|---|---|

(1) Von Mises (simplified) | 8.86 | 12.31 |

(2) Von Mises (complete) | 7.33 | 10.78 |

(3) Tokugawa | 7.82 | 11.27 |

(4) DTMB | 8.44 | 11.89 |

(5) Kendrick | 7.40 | 10.85 |

Mean Value: | 7.97 | 11.42 |

Eigenmode | Buckling Pressure (MPa) | ||
---|---|---|---|

Unstiffened Shell | Conventional Stiffened Shell | Sliding Stiffeners | |

1º | 9.093 | 11.295 | 12.853 |

2º | 15.852 | 19.273 | 19.651 |

3º | 19.474 | 22.979 | 27.728 |

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**MDPI and ACS Style**

Siqueira Nóbrega de Freitas, A.; Alfonso Alvarez, A.; Ramos, R., Jr.; de Barros, E.A. Buckling Analysis of an AUV Pressure Vessel with Sliding Stiffeners. *J. Mar. Sci. Eng.* **2020**, *8*, 515.
https://doi.org/10.3390/jmse8070515

**AMA Style**

Siqueira Nóbrega de Freitas A, Alfonso Alvarez A, Ramos R Jr., de Barros EA. Buckling Analysis of an AUV Pressure Vessel with Sliding Stiffeners. *Journal of Marine Science and Engineering*. 2020; 8(7):515.
https://doi.org/10.3390/jmse8070515

**Chicago/Turabian Style**

Siqueira Nóbrega de Freitas, Artur, Alexander Alfonso Alvarez, Roberto Ramos, Jr., and Ettore Apolonio de Barros. 2020. "Buckling Analysis of an AUV Pressure Vessel with Sliding Stiffeners" *Journal of Marine Science and Engineering* 8, no. 7: 515.
https://doi.org/10.3390/jmse8070515