# Nonlinear Hydrodynamic Analysis of Ships Moored in a VLFS Service Basin in the East Mediterranean Sea

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analysis Method

#### 2.2. Conceptual Framework

#### 2.2.1. Model Description

- SCN2500 F2.2
- Length: 2500 mm;
- 100% energy: 6888.6 kN/m (72% deflection);
- 100% reaction force: 4882 kN (35% and 72% deflection);
- Stiffness: see Figure 5 for stiffness curve and its polynomial representation.

- SCN2500 F3.1:
- Length: 2500 mm;
- 100% Energy: 9151 kN/m (72% deflection);
- 100% reaction force: 6871 kN (35% and 72% deflection);
- Stiffness: see Figure 6 for stiffness curve and its polynomial representation.

- Compliant system:
- Length: ~40 m;
- Stiffness: 7420 kN/m;

- Stiff system:
- Length: ~40 m;
- Stiffness: 10,440 kN/m.

- Diameter: 80 mm;
- Stiffness: 931 kN/m;
- MBL: 4473 kN.

- ${\mathrm{H}}_{\mathrm{mo}}=2\text{}\mathrm{m}\text{}\left({\mathrm{T}}_{\mathrm{p}}=7.71\text{}\mathrm{s}\right)$, exceeding probability: 6%
- ${\mathrm{H}}_{\mathrm{mo}}=2.5\text{}\mathrm{m}\text{}\left({\mathrm{T}}_{\mathrm{p}}=8.62\text{}\mathrm{s}\right)\text{})$, exceeding probability: 3%
- ${\mathrm{H}}_{\mathrm{mo}}=3\text{}\mathrm{m}\text{}\left({\mathrm{T}}_{\mathrm{p}}=9.44\text{}\mathrm{s}\right)$, exceeding probability: 2%
- ${\mathrm{H}}_{\mathrm{mo}}=3.5\text{}\mathrm{m}\text{}\left({\mathrm{T}}_{\mathrm{p}}=10.20\text{}\mathrm{s}\right)$, exceeding probability: 0.5%

#### 2.2.2. Evaluation of Operability Criteria

- 55% MBL for wires;
- 50% MBL for synthetic ropes;
- 45% MBL for polyamide.

#### 2.2.3. Load Cases

## 3. Results

- Container vessel
- 100% efficiency: c1, s1;
- 50% efficiency: c1, c2, s1;

- Bulk carriers
- Cranes: c1, c2, s1;
- Elevator/bucket-wheel: -;
- Conveyor belt: c1–c3, s1–s3;

- Oil tanker
- Loading arms: c1–c3, s1, s2;

- Gas tankers
- Loading arms: -.

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Figure A1.**Load case c1 (${\mathrm{H}}_{\mathrm{mo}}=2\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A2.**Load case c2 (${\mathrm{H}}_{\mathrm{mo}}=2.5\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A3.**Load case c3 (${\mathrm{H}}_{\mathrm{mo}}=3\text{}\mathrm{m}$)—wave frequency position. The surge and sway position are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A4.**Load case c4 (${\mathrm{H}}_{\mathrm{mo}}=3.5\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A5.**Load case s1 (${\mathrm{H}}_{\mathrm{mo}}=2\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A6.**Load case s2 (${\mathrm{H}}_{\mathrm{mo}}=2.5\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A7.**Load case s3 (${\mathrm{H}}_{\mathrm{mo}}=3\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

**Figure A8.**Load case s4 (${\mathrm{H}}_{\mathrm{mo}}=3.5\text{}\mathrm{m}$)—wave frequency position. The surge and sway positions are relative to the position of the Delta, the other positions are absolute position of the ship.

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**Figure 2.**The berth mooring system layout. Top: view from above; bottom: view from the back. Note: the figures are slightly distorted for visual purposes.

**Figure 4.**Mooring configuration setup. (

**1**) Bow node location, upper view; (

**2**) connection point; (

**3**) mooring lines arrangement. Note: actual geometric configuration and dimensions of the Delta are as shown in Figure 3.

**Figure 5.**Stiffness curve of the SCN2500 F2.2 fender. The 5th degree polynomial was approximated via MATLAB.

**Figure 6.**Stiffness curve of the SCN2500 F3.1 fender. The 5th degree polynomial was approximated via MATLAB.

**Figure 7.**The position of the Delta under the incoming waves—load case c2. Note that the position of the Delta in the horizontal degrees of freedom is dominated by the slow-varying and constant drift load. It is characterized by slow oscillations of large amplitude about a point of equilibrium accompanied by the low amplitude, fast wave frequency oscillation.

**Figure 8.**Load case c2 ${\mathrm{H}}_{\mathrm{mo}}=2.5\text{}\mathrm{m}$—wave frequency position. The surge and sway position are relative to the position of the Delta, the rest are absolute position of the ship.

**Figure 9.**Load case s2 ${\mathrm{H}}_{\mathrm{mo}}=2.5\text{}\mathrm{m}$—wave frequency position. The surge and sway position are relative to the position of the Delta, the rest are absolute position of the ship.

**Figure 11.**Sensitivity test—applied tension (N) on the berths’ mooring lines. Incoming wave angle of about 15 degrees from the head-on direction; otherwise, similar wave condition as for the c2 load case.

Length over all (parallel to the x axis in Figure 3) | 611 m |

Breadth over all | 558 m |

Depth | 50 m |

Draft | 20 m |

displacement | ~1,600,000 ton |

Length over all | 230 m |

Breadth | 32.5 m |

Depth | 19.85 m |

Draft | 14 m |

Displacement | 84,000 ton |

DOF | Delta (s) | Ship (s) |
---|---|---|

Heave | 20.1 | 10.1 |

Roll | 13.6 | 12.9 |

Pitch | 13.6 | 9.0 |

Segment 1 (Ground): Chain K4 Studless | Segment 2: Polyester Cable | Segment 3 (Top): Chain K4 Studless | ||
---|---|---|---|---|

Length | m | 50 | 1584 | 250 |

Diameter | mm | 122 | 223 | 122 |

Mass | kg/m | 326.0 | 31.8 | 326.0 |

Weight in water | N/m | 2429.1 | 78.5 | 2429.1 |

Stiffness AE | kN | 1,327,000 | 384,600 | 1,327,000 |

Mean braking load (MBL) | kN | 14,360 | 13,730 | 14,360 |

**Table 5.**Acceptable motion for moored ships [43].

Ship Type | Cargo Handling Equipment | Surge (m) | Sway (m) | Heave (m) | $\mathbf{Yaw}\text{}(\xb0)$ | $\mathbf{Pitch}\text{}(\xb0)$ | $\mathbf{Roll}\text{}(\xb0)$ |
---|---|---|---|---|---|---|---|

Container vessels | 100% efficiency | 1.0 | 0.6 | 0.8 | 1 | 1 | 3 |

50% efficiency | 2 | 1.2 | 1.2 | 1.5 | 2 | 6 | |

Bulk carriers | Cranes | 2.0 | 1.0 | 1.0 | 2 | 2 | 6 |

Elevator/bucket-wheel | 1.0 | 0.5 | 1.0 | 2 | 2 | 2 | |

Conveyor belt | 5.0 | 2.5 | 3 | ||||

Oil tankers | Loading arms | 3.0 | 3.0 | ||||

Gas tankers | Loading arms | 2.0 | 2.0 | 2 | 2 | 2 |

Compliant System (c) | Stiff System (s) | ||
---|---|---|---|

Load Case | ${\mathbf{H}}_{\mathbf{m}\mathbf{o}}\left(\mathbf{m}\right)$ | Load Case | ${{H}}_{\mathbf{m}\mathbf{o}}\left(\mathbf{m}\right)$ |

c1 | 2 | s1 | 2 |

c2 | 2.5 | s2 | 2.5 |

c3 | 3 | s3 | 3 |

c4 | 3.5 | s4 | 3.5 |

$\mathbf{Load}\text{}\mathbf{Case}\text{}\left({\mathbf{H}}_{\mathbf{m}\mathbf{o}}\right)$ | Surge (m) | Sway (m) | Heave (m) | $\mathbf{Yaw}\text{}(\xb0)$ | $\mathbf{Pitch}\text{}(\xb0)$ | $\mathbf{Roll}\text{}(\xb0)$ |
---|---|---|---|---|---|---|

c1 (2) | 0.58 | 0.47 | 0.38 | 0.38 | 0.43 | 2.45 |

c2 (2.5) | 1.04 | 0.84 | 0.83 | 0.85 | 0.95 | 3.68 |

c3 (3) | 2.94 | 1.18 | 1.25 | 1.78 | 1.73 | 6.60 |

c4 (3.5) | 14.31 | 1.26 | 1.70 | 3.43 | 2.39 | 7.63 |

s1 (2) | 0.79 | 0.35 | 0.38 | 0.33 | 0.42 | 2.07 |

s2 (2.5) | 2.49 | 0.73 | 0.82 | 0.54 | 0.96 | 3.14 |

s3 (3) | 4.187 | 0.71 | 1.25 | 0.87 | 1.74 | 4.07 |

s4 (3.5) | 11.35 | 1.02 | 1.67 | 1.55 | 2.49 | 5.63 |

$\mathbf{Load}\text{}\mathbf{Case}\text{}\left({\mathbf{H}}_{\mathbf{m}\mathbf{o}}\right)$ | Acceptable Tension on Lines (% of MBL) | Max. Tension on Breast Line (kN) | % of MBL | Max. Tension on Spring Line (kN) | % of MBL | Acceptable Load on Fenders (kN) | Max. Compression on Fender (kN) | % of Max. Reaction Force |
---|---|---|---|---|---|---|---|---|

c1 (2) | 50 | 4930 | 16 | 2301 | 7 | 4882 | 2733 | 56 |

c2 (2.5) | 50 | 8859 | 28 | 2726 | 9 | 4882 | 4630 | 95 |

c3 (3) | 50 | 18,455 | 59 | 5542 | 18 | 4882 | 4824 | 99 |

c4 (3.5) | 50 | 29,772 | 95 | 12,515 | 40 | 4882 | 9789 | 200 |

s1 (2) | 50 | 5297 | 12 | 3704 | 8 | 6871 | 2984 | 43 |

s2 (2.5) | 50 | 7969 | 18 | 4737 | 11 | 6871 | 5205 | 76 |

s3 (3) | 50 | 12,442 | 28 | 6348 | 14 | 6871 | 6910 | 101 |

s4 (3.5) | 50 | 19,041 | 43 | 11,992 | 27 | 6871 | 7515 | 109 |

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**MDPI and ACS Style**

Gafter, R.; Drimer, N.
Nonlinear Hydrodynamic Analysis of Ships Moored in a VLFS Service Basin in the East Mediterranean Sea. *J. Mar. Sci. Eng.* **2022**, *10*, 382.
https://doi.org/10.3390/jmse10030382

**AMA Style**

Gafter R, Drimer N.
Nonlinear Hydrodynamic Analysis of Ships Moored in a VLFS Service Basin in the East Mediterranean Sea. *Journal of Marine Science and Engineering*. 2022; 10(3):382.
https://doi.org/10.3390/jmse10030382

**Chicago/Turabian Style**

Gafter, Roy, and Nitai Drimer.
2022. "Nonlinear Hydrodynamic Analysis of Ships Moored in a VLFS Service Basin in the East Mediterranean Sea" *Journal of Marine Science and Engineering* 10, no. 3: 382.
https://doi.org/10.3390/jmse10030382