# Finite Element Simulations of Novel Submersible Design Based on the ASME Design-by-Analysis Approach

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## Abstract

**:**

## 1. Introduction

## 2. Design Requirements of the Acrylic Hull Submersible

#### 2.1. A Computational Procedure for Structural Analysis of a Passenger Submersible Based on the Design-by-Analysis (DBA) Approach by ASME

#### 2.1.1. Protection against Failure from Plastic Collapse

_{m}), local membrane stress (P

_{L}), and bending stress (P

_{b}). If shell FEs are used, the stress categorisation is straightforward since a shell FE can output membrane and bending stress separately [12,23] (unless specified otherwise) when the shell is built out of a single homogenous isotropic material and a linear static behaviour is considered. If a 3D continuum mechanics (“solid”) FE is used, a process called linearisation [12] must be performed to obtain the stresses from a general stress tensor. Nowadays, there are many stress categorisation algorithms that can readily be integrated into commercial FEA programs, although the process is rather cumbersome and is still an active field of research [24].

_{PL}) used to evaluate local membrane and local membrane and bending stress taken together. Thus, the stress limits can be written as follows:

_{PL}are functions of the material’s yield stress and are related in the following manner: ${S}_{PL}=1.5S$.

#### 2.1.2. Protection against Failure from Buckling

_{B}= 2/β

_{cr}. The capacity reduction factor β

_{cr}for unstiffened and ring stiffened cylinders and cones under external pressure reads 0.8, while for spherical shells and spherical, torispherical, and elliptical heads under external pressure β

_{cr}are equal to 0.124 [12]. As the boundary conditions are critical for the buckling analysis, a special attention must be paid to avoid that boundary conditions prevent some realistic buckling modes.

#### 2.1.3. Protection against Failure from Cyclic Loading

_{b}) elaborated in Section 2.1.1. These are the primary local membrane (P

_{L}), the secondary membrane and bending stress (Q), and peak stress (F). These stress components occur due to structural discontinuities (e.g., change in plate thickness), thermal gradients due to temperature changes, stress concentrations (e.g., junctions, welds, sharp edges), etc. Thus, the total stress to be considered in the fatigue analysis is [12]:

_{k}) can be obtained from diagrams provided in [12]. This allows us to calculate the fatigue damage in the current (k-th) cycle:

_{k}is the number of repetitions of the k-th cycle.

_{1}and N

_{1}denote the number of repetitions of the first (and only) cycle/loading condition and the permissible number of cycle repetitions, respectively.

#### 2.2. Submersible Design Details

^{−5}mm/ (mm °C) is a coefficient of linear thermal expansion at 20 °C calculated according to data obtained from [27], L is length of the cylinder (Table 2), and Δθ = −5 °C is the temperature decrease. Both temperature values are selected only to illustrate the procedure and will depend on ambient temperature in areas where the submersible is assembled and operated, respectively.

## 3. FEM Structural Analysis

_{PL}= 1.5S = 225 MPa. As described in Section 2.1, membrane stresses are relevant in most parts of the structure while local primary membrane stresses are evaluated in the vicinity of the structural discontinuities. Steel parts are modelled using shell elements with mesh size t × t (t is the thickness of the structural element), while longitudinal rods are modelled by circular bar elements.

#### 3.1. FEM Analysis of Submersible Structural Components

#### 3.2. FEM Analysis of Submersible Assembly

#### 3.2.1. Yielding Analysis

#### 3.2.2. Radial and Angular Deformation Analysis

_{max}of the window seat at maximum internal or external pressure must be less than 0.002·D

_{i}, where D

_{i}in our case reads:

- Internal diameter of the acrylic cylinder D
_{i}= 2360 mm; r_{max}= 4.72 mm. - Internal diameter of the acrylic fore sphere D
_{i}= 2001 mm; r_{max}= 4.00 mm.

#### 3.2.3. Buckling Analysis

#### 3.2.4. Fatigue Analysis

- Connection of the aft head and aft cupola ring.
- Connection of the forehead and fore cupola ring.

^{5}. Since the number of cycle repetitions is 25,000, the fatigue assessment criterion (accumulated fatigue damage) according to (7) reads:

#### 3.2.5. Lifting Analysis

- Stresses in the structures are acceptable;
- Relative displacements at the ends of the cylindrical acrylic sections are satisfactorily low and will not cause damage.

^{2}.

## 4. Discussion about Non-Metallic Parts of the Pressure Hull

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- How Far Does Light Travel in the Ocean? Available online: https://oceanservice.noaa.gov/facts/light_travel.html (accessed on 26 June 2022).
- Stachiw, J.D. Handbook of Acrylics for Submersibles, Hyperbaric Chambers and Aquaria; Best Publishing Company: North Plam Beach, FL, USA, 2003. [Google Scholar]
- Du, Q.; Hu, Y.; Cui, W. Safety assessment of the acrylic conical frustum viewport structure for a deep-sea manned submersible. Ships Offshore Struct.
**2017**, 12(S1), 221–229. [Google Scholar] [CrossRef] - Pranesh, S.B.; Kumar, D.; Subramanian, V.A.; Sathianarayanan, D.; Ramadass, G.A. Numerical and experimental study on the safety of viewport window in a deep sea manned submersible. Ships Offshore Struct.
**2020**, 15, 769–779. [Google Scholar] [CrossRef] - Pope, G.T. Touring the Deep. Pop. Mech.
**1988**, 10, 67–70, 102. [Google Scholar] - Forman, W.R. Submersible Observation Vessel. U.S. Patent US4928614, 29 May 1990. [Google Scholar]
- Kohnen, W. Manned Underwater Vehicles 2017–2018 Global Industry Overview; Marine Technology Society (MTS): Washington, DC, USA, 2018. [Google Scholar]
- Rules for Building and Classing Underwater Vehicles, Systems and Hyperbaric Facilities; ABS: Spring, TX, USA, 2021.
- Xing, Y.; Santoso, T.A.D.; Ma, Y. Technical-Economic Feasibility Analysis of Subsea Shuttle Tanker. J. Mar. Sci. Eng.
**2022**, 10, 20. [Google Scholar] [CrossRef] - Hughes, O.W.; Paik, J.K. Ship Structural Analysis and Design; SNAME: New York, NY, USA, 2010. [Google Scholar]
- Payer, H.G.; Schellin, T.E. A class society’s view on rationally based ship structural design. Ships Offshore Struc.
**2013**, 8, 319–336. [Google Scholar] [CrossRef] - ASME Boiler & Pressure Vessel Code, Section VIII, Division 2; ASME: New York, NY, USA, 2019.
- The Design-by-Analysis Manual; European Commission: Petten, The Netherlands, 1999.
- Diamantoudis, A.T.; Kermanidis, T. Design by analysis versus design by formula of high strength pressure vessels: A comparative study. Int. J. Press. Vessel. Pip.
**2005**, 82, 43–50. [Google Scholar] [CrossRef] - Karthikeyan, K.M.B.; Balasubramanian, T.; Bruce, R.A.; Premkumar, P. Pressure Vessel Design by Design by Analysis Route. In Proceedings of the IOP Conference Series: Materials Science and Engineering, Chennai, India, 20 February 2020. [Google Scholar]
- Cho, S.-R.; Muttaqie, T.; Lee, S.H.; Paek, J.; Sohn, J.M. Ultimate Strength Assessment of Steel-Welded Hemispheres under External Hydrostatic Pressure. J. Mar. Sci. Appl.
**2020**, 19, 615–633. [Google Scholar] [CrossRef] - Cho, S.-R.; Muttaqie, T.; Do, Q.T.; Kim, S.; Kim, S.M.; Han, D.-H. Experimental investigations on the failure modes of ring-stiffened cylinders under external hydrostatic pressure. Int. J. Nav. Archit.
**2018**, 10, 711–729. [Google Scholar] [CrossRef] - Pranesh, S.B.; Sathianarayanan, D.; Ramadass, G.A. Design standards for steel spherical pressure hull for a manned submersible. J. Ocean Eng. Mar. Energy
**2022**, 8, 137–151. [Google Scholar] [CrossRef] - Ma, Y.; Xing, Y.; Ong, M.C.; Hemmingsen, T.H. Baseline design of a subsea shuttle tanker system for liquid carbon dioxide transportation. Ocean Eng.
**2021**, 240, 109891. [Google Scholar] [CrossRef] - Sohn, J.M.; Hirdaris, S.; Romanoff, J.; Kim, S.J. Development of Numerical Modelling Techniques for Composite Cylindrical Structures under External Pressure. J. Mar. Sci. Eng.
**2022**, 10, 466. [Google Scholar] [CrossRef] - Zhou, M.; Patel, A.; Wang, B.; Jin, W. Design Optimization of Pressure Vessel in Compliance With Elastic Stress Analysis Criteria for Plastic Collapse Using an Integrated Approach. J. Press. Vessel. Technol.
**2021**, 143, 011303. [Google Scholar] [CrossRef] - Rules for Conditions of Classification; ABS: Spring, TX, USA, 2022.
- Prebeg, P.; Palaversa, M.; Andric, J.; Tomicic, M. Adaptation of FEM-based open-source software for ship structural analysis. Ships Offshore Struct.
**2022**, 1–12. [Google Scholar] [CrossRef] - Duan, C.; Ding, L.; Lu, M. Discussion on the Implementation of the Primary Structure Method in Design by Analysis. Appl. Mech. Mater.
**2017**, 853, 341–345. [Google Scholar] [CrossRef] - Saghi, H.; Parunov, A.; Mikulic, A. Resistance Coefficient Estimation for a Submarine’s Bare Hull Moving in Forward and Transverse Directions. Appl. Sci.
**2022**, 12, 10953. [Google Scholar] [CrossRef] - Gatin, I.; Čokić, J.; Romić, D.; Parunov, J. CFD Study on the Influence of Exostructure Elements on the Resistance of a Submarine. J. Mar. Sci. Eng.
**2022**, 10, 1542. [Google Scholar] [CrossRef] - ASME PVHO-1 Safety Standard for Pressure Vessels for Human Occupancy; ASME: New York, NY, USA, 2019.
- Bergstrom, J. Mechanics of Solid Polymers: Theory and Computational Modeling; William Andrew-Elsevier: San Diego, CA, USA, 2015. [Google Scholar]
- Palaversa, M.; Parunov, J. FEA of Hyperelastic Structures: A Case from the Submarine Design. In Proceedings of the Sorta 2022, Malinska, Croatia, 7 September 2022. [Google Scholar]
- Kim, B.; Lee, S.B.; Lee, J.; Cho, S.; Park, H.; Yeom, S.; Park, S.H. A Comparison Among Neo-Hookean Model, Mooney-Rivlin Model and Ogden Model for Chloroprene Rubber. Int. J. Precis. Eng. Manuf.
**2012**, 13, 759–764. [Google Scholar] [CrossRef] - Det Norke Veritaas. Fatigue Assessment of Ship Structures, Classification Notes No. 30.7; Det Norke Veritaas: Høvik, Norway, June 2010. [Google Scholar]
- Ćorak, M.; Šperanda, Z.; Čokić, J.; Parunov, J. Structural analysis of tourist submarine with acrylic hull. In Sustainable Development and Innovations in Marine Technologies, 1st ed.; Ergin, S., Soares, C.G., Eds.; CRC Press: London, UK, 2022. [Google Scholar] [CrossRef]
- Young, W.C.; Budynas, R.G. Roark’s Formulas for Stress and Strain, 7th ed.; McGraw-Hill: New York, NY, USA, 2002. [Google Scholar]

**Figure 3.**Scheme of a joint between the acrylic cylinder and the supporting steel ring (dimensions in mm).

**Figure 6.**Loads and boundary conditions on the submarine structural assembly (force in N, pressure in MPa).

**Figure 11.**Plate bottom von Mises stresses for design depth and detail of the connection of the aft head and the first ring (stresses in MPa).

**Figure 12.**Fatigue life of the analysed detail [12].

**Figure 14.**The von Mises stresses in MPa of the vessel for the lifting condition; (

**a**) plate elements, (

**b**) solid elements.

**Figure 16.**Loads and vertical displacements of the assembled structure for the emergency lifting in damaged condition (force in N, pressure in MPa).

**Figure 18.**High-stress area in lugs for launching/recovering in rough seas: (

**a**) plate elements with mid-plate von Mises stress above 192 MPa; and (

**b**) solid elements with von Mises stresses above 230 MPa.

Feature | Value |
---|---|

Test depth | 65 m |

Length overall | 25.09 m |

Beam | 4.750 m |

Draft | 3.420 m |

Dry weight | 155 tons |

Hull acrylic outer diameter | 2.64 m |

Forward speed (max) | 2.5 knots |

Passengers | 48 |

Dimension | Value |
---|---|

Length of the cylinder (L) | 2.4 m |

Inner diameter of the cylinder (d) | 2.34 m |

Thickness of the cylinder (h) | 140 mm |

Thickness of the side gasket (t) | 10 mm |

Component | Von Mises Membrane Stresses (MPa) | Buckling Factor |
---|---|---|

Battery cylinder pod | 13 | 8.5 |

Fore cupola | 164 | 31.5 |

Aft cupola | 163 | 31.5 |

Transverse ring | 106 | 4.5 |

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**MDPI and ACS Style**

Ćorak, M.; Šperanda, Z.; Čokić, J.; Palaversa, M.; Parunov, J.
Finite Element Simulations of Novel Submersible Design Based on the ASME Design-by-Analysis Approach. *J. Mar. Sci. Eng.* **2023**, *11*, 275.
https://doi.org/10.3390/jmse11020275

**AMA Style**

Ćorak M, Šperanda Z, Čokić J, Palaversa M, Parunov J.
Finite Element Simulations of Novel Submersible Design Based on the ASME Design-by-Analysis Approach. *Journal of Marine Science and Engineering*. 2023; 11(2):275.
https://doi.org/10.3390/jmse11020275

**Chicago/Turabian Style**

Ćorak, Maro, Zdenko Šperanda, Juvel Čokić, Marin Palaversa, and Joško Parunov.
2023. "Finite Element Simulations of Novel Submersible Design Based on the ASME Design-by-Analysis Approach" *Journal of Marine Science and Engineering* 11, no. 2: 275.
https://doi.org/10.3390/jmse11020275