# Numerical Modelling for Synthetic Fibre Mooring Lines Taking Elongation and Contraction into Account

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

#### 2.1. Loads on Mooring Line

**r**in the global coordinate system Oxyz (see Figure 1). A point on the curve is defined by an arc length $\tilde{s}$ of the extended mooring line. Within the dynamic analysis, the position vector

**r**is dependent on the time variable t. An equivalent segment of the mooring line is set according to the approach on the basis of the effective tension and the effective load [51,52]. This approach is developed in order to include the influence of the outer hydrostatic pressure due to the seawater on the mooring line. The segment of the mooring line is shown in Figure 1.

**q**

_{E}contains buoyancy

**q**

_{B}, weight of the mooring line

**q**

_{W}(i.e., dry weight), and hydrodynamic loads

**q**

_{H}

**g**is the gravity acceleration vector pointing in the opposite direction of the z-axis of the global coordinate system. Cross-sectional area of the segment is given by $\tilde{A}$.

**q**

_{H}from Equation (1) are formulated by the Morison equation:

_{A}, C

_{M}, and C

_{D}are the added mass, the inertial coefficient, and drag coefficient, respectively. The cross-section area of the extended segment is denoted by $\tilde{A}$ (as in the expression for the buoyancy). The diameter, which is elongation dependent, is given as $\tilde{D}$. By the same conservation principle used in Equation (7) for $\tilde{A}$, the relation for $\tilde{D}$ can be formulated as

**r**with respect to the extended segment length as $\mathrm{d}r/\mathrm{d}\tilde{s}$. This is in contrast to current practice where the unit tangent is evaluated on the basis of the non-extended segment length in the form $\mathrm{d}r/\mathrm{d}s$, which is a poor approximation for synthetic fibre mooring lines. This is because, as already mentioned, synthetic mooring lines usually undergo a relatively large axial deformation ε and thus great differences in the extended $\mathrm{d}\tilde{s}$ and the non-extended length $\mathrm{d}s$ exist, as given by Equation (5). Therefore, the model developed here has an advantage in calculating hydrodynamic loads since it incorporates a more accurate expression for tangential and normal components of velocities and accelerations.

#### 2.2. Equations of Motion

**F**

_{E}is the cross-section internal force (see Figure 1). The balance of moments on the same segment can be expressed as

_{E}is a scalar value and it represents the effective tension force [51,52]. By combining Equations (11) and (13), one obtains the equation of motion of the mooring line:

#### 2.3. Discretisation of the Motion Equation

_{E}is given in a simplified manner as

_{ijk}is the Levi–Civita symbol. All indices used here are in the range from 1 to 3. The unknown variables T

_{E}and r

_{i}are approximated as

_{l}and P

_{m}are shape functions (i.e., polynomials) defined in the interval 0 ≤ s ≤ L, where L is the length of the finite element (see [48]). Unknown node displacements U

_{il}and node effective tensions λ

_{m}are expressed as

_{br}and W

_{jq}are nodal velocities and accelerations of the surrounding seawater, respectively. It should be noted that all integrals in the above set of equations can be evaluated analytically except for the integrals for the drag forces, i.e., the Equations (37)–(39).

^{2}of the Taylor series in Equation (20). The quadratic influence on the inertial forces can be noticed in Equation (30) where the term is multiplied by ${\lambda}_{n}{\lambda}_{m}$ besides multiplication by ${\ddot{U}}_{jk}$.

_{il}and the effective tension λ

_{m}. Consequently, an additional equation is needed, and it is obtained using the axial elongation condition.

## 3. Case Studies

#### 3.1. Straight Segment

^{2}.

#### 3.1.1. Inertial Forces on the Straight Segment

#### 3.1.2. Drag Forces on the Straight Segment

#### 3.2. Curved Segment

#### 3.2.1. Inertial Forces on the Curved Segment

_{3}is a third-order polynomial (see Equation (25)). The integration was carried out over the half-length of the segment, since it was assumed again that a single node receives only half of the total finite element load. In an analogue way, inertial forces of the surrounding seawater can be approximated as

#### 3.2.2. Drag Forces on the Curved Segment

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The segment of the mooring line [48].

Designation | Quantity | Unit |
---|---|---|

Length, L | 10.0 | m |

Diameter, D | 160 | mm |

Distributed mass, m | 17.197 | kg/m |

Wet weight, q_{w} | 44.102 | N/m |

Linearised axial stiffness, AE | 1.6812 · 10^{8} | N |

Added mass coefficient, C_{A} | 1.15 | |

Inertial coefficient, C_{M} | 2.15 | |

Drag coefficient, C_{D} | 2.2 |

$\mathit{\epsilon}$ | ${\mathit{f}}^{\mathit{A}},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{A}0},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{A}},\mathbf{N}$ | $\left({\mathit{F}}_{11}^{\mathit{A}0}-{\mathit{f}}^{\mathit{A}}\right)/{\mathit{f}}^{\mathit{A}},\mathbf{\%}$ | $\left({\mathit{F}}_{11}^{\mathit{A}}-{\mathit{f}}^{\mathit{A}}\right)/{\mathit{f}}^{\mathit{A}},\mathbf{\%}$ |
---|---|---|---|---|---|

0.00 | −118.5 | −118.5 | −118.5 | 0.0 | 0.0 |

0.05 | −118.5 | −130.6 | −118.6 | 10.2 | 0.1 |

0.10 | −118.5 | −143.4 | −119.0 | 21.0 | 0.4 |

0.15 | −118.5 | −156.7 | −120.2 | 32.2 | 1.5 |

0.20 | −118.5 | −170.6 | −122.9 | 44.0 | 3.7 |

0.25 | −118.5 | −185.2 | −127.3 | 56.2 | 7.4 |

0.30 | −118.5 | −200.3 | −134.2 | 69.0 | 13.2 |

$\mathit{\epsilon}$ | ${\mathit{f}}^{\mathit{M}},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{M}0},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{M}},\mathbf{N}$ | $\left({\mathit{F}}_{11}^{\mathit{M}0}-{\mathit{f}}^{\mathit{M}}\right)/{\mathit{f}}^{\mathit{M}},\mathbf{\%}$ | $\left({\mathit{F}}_{11}^{\mathit{M}}-{\mathit{f}}^{\mathit{M}}\right)/{\mathit{f}}^{\mathit{M}},\mathbf{\%}$ |
---|---|---|---|---|---|

0.00 | 221.6 | 221.5 | 221.6 | 0.0 | 0.0 |

0.05 | 221.6 | 244.3 | 221.7 | 10.2 | 0.1 |

0.10 | 221.6 | 268.1 | 222.5 | 21.0 | 0.4 |

0.15 | 221.6 | 293.0 | 224.9 | 32.2 | 1.5 |

0.20 | 221.6 | 319.0 | 229.7 | 44.0 | 3.7 |

0.25 | 221.6 | 346.2 | 238.0 | 56.2 | 7.4 |

0.30 | 221.6 | 374.4 | 250.9 | 69.0 | 13.2 |

$\mathit{\epsilon}$ | ${\mathit{f}}^{\mathit{D}},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{D}0},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{D}},\mathbf{N}$ | $\left({\mathit{F}}_{11}^{\mathit{D}0}-{\mathit{f}}^{\mathit{D}}\right)/{\mathit{f}}^{\mathit{D}},\mathbf{\%}$ | $\left({\mathit{F}}_{11}^{\mathit{D}}-{\mathit{f}}^{\mathit{D}}\right)/{\mathit{f}}^{\mathit{D}},\mathbf{\%}$ |
---|---|---|---|---|---|

0.00 | −902.0 | −902.0 | −902.0 | 0.0 | 0.0 |

0.05 | −924.3 | −1044.2 | −925.1 | 13.0 | 0.1 |

0.10 | −946.0 | −1200.6 | −953.0 | 26.9 | 0.7 |

0.15 | −967.3 | −1371.8 | −992.4 | 41.8 | 2.6 |

0.20 | −988.1 | −1558.7 | −1052.1 | 57.7 | 6.5 |

0.25 | −1008.4 | −1761.7 | −1142.4 | 74.7 | 13.3 |

0.30 | −1028.4 | −1981.7 | −1275.7 | 92.7 | 24.0 |

$\mathit{\epsilon}$ | ${\mathit{f}}^{\mathit{D}},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{D}0},\mathbf{N}$ | ${\mathit{F}}_{11}^{\mathit{D}},\mathbf{N}$ | $\left({\mathit{F}}_{11}^{\mathit{D}0}-{\mathit{f}}^{\mathit{D}}\right)/{\mathit{f}}^{\mathit{D}},\mathbf{\%}$ | $\left({\mathit{F}}_{11}^{\mathit{D}}-{\mathit{f}}^{\mathit{D}}\right)/{\mathit{f}}^{\mathit{D}},\mathbf{\%}$ |
---|---|---|---|---|---|

0.00 | 902.0 | 902.0 | 902.0 | 0.0 | 0.0 |

0.05 | 924.3 | 1044.2 | 925.1 | 13.0 | 0.1 |

0.10 | 946.0 | 1200.6 | 953.0 | 26.9 | 0.7 |

0.15 | 967.3 | 1371.8 | 992.4 | 41.8 | 2.6 |

0.20 | 988.1 | 1558.7 | 1052.1 | 57.7 | 6.5 |

0.25 | 1008.4 | 1761.7 | 1142.4 | 74.7 | 13.3 |

0.30 | 1028.4 | 1981.7 | 1275.7 | 92.7 | 24.0 |

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**MDPI and ACS Style**

Ćatipović, I.; Alujević, N.; Rudan, S.; Slapničar, V. Numerical Modelling for Synthetic Fibre Mooring Lines Taking Elongation and Contraction into Account. *J. Mar. Sci. Eng.* **2021**, *9*, 417.
https://doi.org/10.3390/jmse9040417

**AMA Style**

Ćatipović I, Alujević N, Rudan S, Slapničar V. Numerical Modelling for Synthetic Fibre Mooring Lines Taking Elongation and Contraction into Account. *Journal of Marine Science and Engineering*. 2021; 9(4):417.
https://doi.org/10.3390/jmse9040417

**Chicago/Turabian Style**

Ćatipović, Ivan, Neven Alujević, Smiljko Rudan, and Vedran Slapničar. 2021. "Numerical Modelling for Synthetic Fibre Mooring Lines Taking Elongation and Contraction into Account" *Journal of Marine Science and Engineering* 9, no. 4: 417.
https://doi.org/10.3390/jmse9040417