Mooring by Marin

8/9/2019 Mooring by Marin

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1. GeneralaNySIM can be used to analyse the dynamic behaviour of a turret or spread moored vesselsubject to wind, waves and current. The program predicts the mooring loads and tankermotions when the system is exposed to survival or operational environmental conditions.Optionally the dynamic behaviour of the mooring system can be computed in conjunction withthe first and second order motions of the vessel.

Two approaches are possible to model a mooring system in aNySim:

- the quasi-static approach: the shape and tension of mooring lines are derived fromcatenary formulations. The catenary formulations account for the weight and the axial

stiffness of the lines, they do not account for the inertia, the current and the waveforces. The mooring lines are described in 2D, in a vertical plane through anchor andfairlead.

- the dynamic approach: each mooring line is modelled as a chain of discrete massesconnected by springs, allowing for inertia, bending stiffness, drag forces and seabedfriction to act on individual masses. The mooring lines are described in 3D.

For every mooring system, the user can choose between the static approach and thedynamic approach. If the dynamic approach is chosen then the initial shapes of the lines,which are input for the dynamic model, are still calculated thanks to the static approach.

The number of mooring systems and the number of lines per mooring system is unlimited,

although only one mooring system per vessel is possible.

It is not possible to define mooring lines between vessels. The mooring module assumes thateach mooring line runs from a fixed point (anchor) to a fairlead of a vessel. It is also assumedthat the mooring lines are not branched.

The mooring module offers four different ways of defining the initial state of a mooringsystem:

1. the line top tension and line azimuth angle are known; the anchor position and linetop angle are calculated;

2. the line top angle and line azimuth angle are known; the anchor position and line toptension are calculated;

3. the anchor position is known; the top tension and line top angle are calculated;4. the anchor radius and line azimuth angle are known; the top tension and line top

angle are calculated.


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2. Static approach

The static approach uses an iteration procedure within another iteration procedure todetermine the mooring line shape and tension for a given anchor and fairlead position:

- an inner iteration procedure: in this procedure the line length or the line angle at theseabed is varied until the height of the line top equals the height of the fairlead for agiven horizontal tension;

- an outer iteration procedure: in this procedure the horizontal tension is varied until thehorizontal line span equals the distance between the anchor and the fairlead.

During the iteration process extra segment ends are generated at the touch-down point onthe seabed and at the intersection with the water surface to derive the correct shape and

tension.

The following tolerances are applied for the iteration procedures:

- line tension: 0.0001 kN;- line top angle: 0.05 0;- vertical line top position: 0.0005 m;- horizontal line span: 0.0005 m.

The catenary formulations are implemented in the inner iteration procedure. A mooring linemay consist of one or more segments. Typically, the division of a mooring line into segmentsis driven by discontinuities such as different line properties (chain, wire, diameter, rope

material, etc.) and the presence of clump weights or buoys.

The analysis starts with the segment which is connected to the anchor. Given the seabedslope, segment length, weight and stiffness and the horizontal tension (set by the outeriteration procedure) it can be determined whether the segment is completely on the seabed,partly lifted from the seabed or completely free from the seabed.

When it rests completely on the seabed the segment top tension and position follow from:

β−

21 0 0

0T L w L sin( )

L = L +EA 2 EA

X = L cos( β)

Z = L sin( β)T2 = T 1 – w L 0 sin( β)

with: w segment weight per meterL0 segment unstretched lengthL segment stretched lengthEA segment axial stiffness, linearized for T 1 T1 tension at segment startT2 tension at segment top


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β seabed slope in vertical plane thru mooring line

X horizontal segment spanZ vertical segment span

When the segment is completely lifted from the seabed its top tension and positon followfrom:

T2,v = T 1,v + L0 w2 2

1 1,v hT = T + T

2 22 2,v hT = T + T

Q =

−

2 2,v 22,v 2 1,v 1 h

1,v 1

0

T + TT T T T + T ln

T + T1 +

2 EA w L

X =

2,v 2h

1,v 1

T + TQ T ln

T + T

w

Z =−2 1Q (T T )

w

with: T 1,v vertical tension at segment startT2,v vertical tension at segment topTh horizontal tension (is constant over segment length)

When the segment is partly lifted from the seabed it is split in two: a segment completelyresting on the seabed and a segment completely lifted from the seabed. The determination ofthe touch-down point is done at each timestep of the simulation.The same holds for segments which run through the water surface. These segments are alsosplit in two: a segment completely submerged and a segment completely in air. In this waythe correct weight (submerged / in air) can be applied. The intersection point is alsodetermined at each timestep.The water surface level is the still water level: wave elevations are not accounted for.

When the vessel moves towards an anchor it may happen that the mooring line becomesslack: it runs vertical from the fairlead to the seabed. The line takes the shape of an “L”instead of a catenary shape. The catenary formulations cannot be used anymore.To avoid numerical problems the static approach applies the catenary formulations until theline top angle is 88 0. Beyond this limit the tension is found by linear interpolation between thetension at an angle of 88 0 and the tension of the vertical line (90 0). The tension in the lattercase follows from the weight summation of the segments between seabed and fairlead.

Clump weights and buoys can be attached to the mooring lines. Clump weights have aspherical shape, while buoys have either a spherical or a cylindrical shape. The cylindricalbuoys may be horizontal or vertical.At each timestep of the simulation the draft of both clump weights and buoys are calculatedto derive the correct nett force which they exert on a mooring line.


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In the static approach five synthetic ropes can be used:

- polypropylene / polyester;- nylon – double braided;- nylon – 3 strand;- nylon – 8 strand;- Dyneema.

The following figures shows the elongation versus the load as percentage of the break load ofthese ropes:

0

25

50

75

100

0 5 10 15 20 25 30 35

elongation [%]

l o a

d [ % ]

nylon - double braided

nylon - 3 strand

nylon - 8 strand

Dyneema

polyprop / polyester


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During a simulation each segment is checked whether the tension exceeds its breakingstrength. If so, the line breaks, unless the user has specified to ignore the breakingstrength. This option may be useful to derive statistics of the tensions.

In the quasi-static approach the following forces on the mooring lines are ignored:The current drag forces.The wave forces.The bottom friction forces.

3. Dynamic approach

3.1. Basic assumptions

The main assumptions and approximations underlying the simulation algorithm may besummarized as follows:All external forces and internal reactions may be lumped to a finite number of nodes.Fluid forces may be described by the given empirical relations using relative motions andconstant shape coefficients. The local wave velocities at every node is not taken into accountin this algorithm.Sea floor contact can be simulated by critically damped springs in vertical direction while thehorizontal forces are simulated by means of a Coulomb type of friction.

3.2. Equations of motion

The mathematical model for the simulation of the three dimensional behaviour of the mooringlines is based on a lumped mass method. The space wise discretization of the line isobtained by lumping the mass and all forces to a finite number of nodes.

To derive the governing equations of motions for the j-th lumped mass, Newtons law iswritten in global co-ordinates.

( ) j j jA a ( ) x ( ) F ( ) + τ τ = τ &&


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where:

jA = inertia matrix for node j

ja = time dependent added inertia matrix for node j

τ = time

jx&& = acceleration vector (x, y, z) for the j-th lumped mass

jF = nodal force vector for node j

The added inertia matrix can be derived from the normal and tangential fluid inertiacoefficients by directional transformations:

j nj nj tj tja ( ) a ( ) a ( ) τ = ∧ τ + ∧ τ

where a nj and a tj represent the normal and tangential added mass:2

nj ln j j

2tj tn j j

a (C 1) / 4 D L

a (C 1) / 4 D L

= ρ − π

= ρ − π

The matrices [A j], [a j], [∧nj] and [ ∧tj] are given in Appendix L.

The nodal force vector F j contains the following contributions:a. segment tension T( τ)b. buoyancy and weight F w c. fluid forces F f(τ)d. sea floor reactive forces F r(τ)e. buoy forces F b(τ)

Since the tangential stiffness of the line, represented by its modulus of elasticity EA, is anorder of magnitude higher than the stiffness in normal direction, the tension is taken intoaccount in the solution procedure directly.

The tension vector on the j-th node results from the tension and orientation of the adjacentline segments:

T j j 1 j j j 1 j 1F ( ) T ( ) x ( ) /L T ( ) x ( )/L−− −τ = τ ∆ τ − τ ∆ τ

where:

( ) j j 1 jx x x+∆ = − j

j oj j

T ( )L ins tan taneous length of segment j L 1

EA

τ= = +


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3.3. Constitutive stress-strain

In order to derive consistent segment tensions and displacements, a Newton-Raphsoniteration using the additional constraint equation for the constitutive stress-strain relation isapplied:

22 j2

j j oj j

T ( )( ) x ( ) L 1

EA

τ σ τ = ∆ τ − +

1k 1 k k kT ( ) T ( ) ( ) ( )−+ τ = τ − ∆σ τ σ τ

where:

σ = segment length error vector ( σ1, …, σ j, …, σN)kT = tentative segment tension vector at the k-th iteration

(T1, …, T j, …, T N) with Tk(τ+∆τ) = T( τ)

[ ]∆σ = length error derivative matrix [ ]T

∂ σ∂

Rewriting the constitutive stress-strain relation results in:

( ) ( ) ( ) ( ) ( )2 2kk k k j j n n n n2

oj 1 oj j j j oj j j j oj

EA T( ) L {1 } 2 x x x 1 x x 1 x

EA T + +

σ τ = − + − ⋅ δ + − δ + δ + − δ +

Where ( )k

n jxδ is the change in position of node j at time n τ at iteration index k.

This formulation is used to prevent loss of accuracy in the calculations.

For each time step the above given system of equations should be solved until acceptableconvergence of T k(τ+∆τ) is obtained. The initial tentative tension can be taken equal to thetension in the previous step.

Each node j is connected to the adjacent nodes j-1 and j+1, hence the above given equationrepresents a tridiagonal (Nx3) system. Such equations may be efficiently solved by theso-called Thomas algorithm.

If no acceptable convergence of T k(τ+∆τ) is obtained the algorithm reduces the time step andproceeds from the last time step for which acceptable convergence was obtained. If the timesteps becomes to small (0.1 / 2 5) before convergence, the solution at the previous time stepis accepted. This is acceptable because this problem generally only occurs at the slack (lessloaded) lines.


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The added mass term can be moved to the left hand side of the equation of motions (reactionforces - see Section 5.10.2), while the other two terms remain at the right hand side of theequation (excitation forces).

Inclined membersBoth the normal and the tangential drag and inertia loads can be determined in a similarmanner using C I and C D coefficients. The forces are calculated in a local system of axes (seethe figure on the next page), using the following equations.

Normal inertia + Froude-Krylov force 2In In nF C D L u4π= ρ ⋅ ⋅ ⋅ ⋅ ⋅&

Tangential inertia + Froude-Krylov force 2It It tF C D L u4π= ρ ⋅ ⋅ ⋅ ⋅ ⋅&

Normal drag force ( )Dn Dn n n n n1

F C D L u x u x2

= ⋅ρ ⋅ ⋅ ⋅ ⋅ − ⋅ −& &

Tangential drag force ( )Dt Dt t t t t1

F C D L u x u x2

= ⋅ρ ⋅ ⋅ ⋅ ⋅ − ⋅ −& &

Normal added mass ( ) 2n Ina C 1 D L4π= ρ ⋅ − ⋅ ⋅ ⋅

Tangential added mass ( ) 2t Ita C 1 D L4π= ρ ⋅ − ⋅ ⋅ ⋅

Both for tangential and normal fluid forces, as well as for the added mass, the samereference area (D.L) and volume ( π /4.D².L) are used.


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Fluid Forces in the local co-ordinate system.

Note on coefficient definitionsIn the calculation of the inertia and Froude-Krylov loads (contribution 2.), as well as in the

calculation of the added mass term (contribution 3.) the coefficient C I is used. In the inertia +Froude-Krylov term “C I.D

2” is used, while in the added mass term “(C I - 1).D2” is used. This

difference is a result of the Froude-Krylov contribution, which is present only in the excitationforce (contribution 2.) and not in the added mass (contribution 3.). The Froude-Krylov force isby definition proportional to the element volume. Because the C I coefficient is also based onthe element volume, this results in the “-1” difference.

The use of both a “C I.D2”-term and a “(C I - 1).D

2”-term in the same equation implies that theC I coefficients in this equation may only be based on the element volume (or a volumetricequivalent diameter D). Using a C I coefficient based on a different diameter than thevolumetric equivalent diameter would lead to a different ratio between “added mass” and“inertia + Froude-Krylov” contributions.

For riser, pipe and wire elements the volumetric equivalent diameter is (almost) equal to thenominal diameter. For chain elements, however, the volumetric equivalent diameter is not equal to the nominal (link) diameter.

ApplicationThe orbital velocities and accelerations due to waves are not taken into account in thecalculation of fluid loads on the mooring lines. The following formulas are used for thecalculation of the drag loads and the added mass.


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For chain a typical value D volumetric / Dnominal ~ 1.88 can be used.

The drag and inertia coefficients specified in the input have to be based on the nominaldiameter D for all element types (including chain). Since the coefficients (C I - 1) are used inthe calculation of the added mass, the C I-coefficient should be larger than 1.0.

The default values for C I and C D are based on model tests for mooring chains, steel wirepipes and riser sections, see Reference [21] and on values from the available literature. Forchain, the values based on the equivalent volumetric diameter and based on the nominaldiameter have both been included in the table below. In calculations use the values based onthe nominal (link) diameter!

The following values are recommended :

CIn C It C Dn C Dt

Chain - Nominal diameter based 3.1 1.7 2.4 0.8

( - Volumetric diameter based 1.6 1.2 1.3 0.4 )

Steel wire 1.6 1.2 1.3 0.4

Pipe / riser 2.0 1.2 1.3 0.2


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3.7. Newmark

Assuming that all nodal force contributions are formulated in terms of node positions,velocities and accelerations, the motion equations are solved by a Newmark algorithm whichhas been adapted for variable time steps.

2 j j j j j

1x ( ) x ( ) x ( ) x ( ) x ( )

2

τ + ∆τ = τ − τ ⋅∆τ + − α τ + α ⋅ τ + ∆τ ⋅∆τ

& && &&

{ } j j jx ( ) x( ) (1 )x ( ) x ( )τ + ∆τ = τ + − δ τ + δ τ + ∆τ ∆τ& & && &&

3.8. Discretization aspects

The choice of nodes and elements along the line is of importance from the point of view ofaccuracy and computational efficiency.

The line discretization should be such that:The geometry is represented accurately, the first check can be made on the staticconfiguration.Mass and weight lumping is acceptable.The angles between two successive elements remain within approximately 20 degrees.The fluid forces may be assumed as constant over the element length.

Reference is made to the discussion in Reference [2].

Displacements

• Keel point:1/2 L PP , midship at keel level.

• Anchor position:Position of the anchor location given in global coordinate system.

• Turret location:Position of the origin of the turret coordinate system relative to the vessel keel coordinatesystem. Per extension, this point is also called mooring system origin (MSO) which canbe used for a spread mnooring system as well.

• Fairlead position:Position of the fairlead attachments points given in either the vessel keel coordinatesystem (spread mooring) or the turret coordinate system (turret moorings).

• COG:Position of the vessel Centre of Gravity in the vessel keel coordinate system.


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4. References

Remove all unquoted references at the end

[1] MARIN Report No. 12619-1-BT, DYNFLOAT oscillation tests.

[2] MARIN Report No. 12619-2-OE, DYNFLOAT validation study.

[3] Lahey Compiler.

[4] “Prediction of Wind and Current Loads on VLCC's”, OCIMF Publication, 1994.

[5] Wichers, J.E.W., “Simulation Model for a Single Point Moored Tanker”, PhD-Thesis,

Delft University of Technology, 1988.[6] “Low Frequency Motions of a Semi-submersible in Waves”, J.A. Pinkster and R.H.M.

Huijsmans, BOSS '82 Conference, MIT Cambridge, Massachusetts.

[7] Pinkster, J.A., “Drift Forces in Directional Seas”, MARINTEK Conference, Shanghai,1985.

[8] Oortmerssen, G. van, Pinkster, J.A. and Boom, H.J.J. van den, “Computer Simulationof Moored Ship Behaviour”, Journal of Waterway, Port, Coastal and OceanEngineering, Vol. 112, No. 2, 1986.

[9] “Algorithm for Computing the Mixed Radix Fast Fourier Transform”, R.C. Singleton,IEEE Trans., Audio Electro acoust., Vol. AU-17, June 1969.

[10] “Static 2-D Solution of a Mooring Line of Arbitrary Composition in Vertical andHorizontal Operating Modes”, B.W. Oppenheim and P.A. Wilson.

[11] Oortmerssen, G. van, “The Motions of a Moored Ship in Waves”, PhD-thesis, DelftUniversity of Technology, 1976.

[12] Vugts, J.H., “The Hydrodynamic Forces and Ship Motions in Waves”, PhD-thesis , DelftUniversity of Technology, 1970.

[13] Chain catalogue, Ramnäs.

[14] Chain catalogue, Vicinzy, 1993.

[15] Steel wire catalogue, Den Haan Wire Ropes.

[16] Catalogue of synthetic ropes, Hawkins & Tipson Ropemakers Ltd., 1976.

[17] Huijsmans, R.H.M., “Mathematical Modelling of the Mean Wave Drift Force in Current -A Numerical and Experimental Study”, PhD-thesis, Delft University of Technology,1996.


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[18] Van den Boom, H.J.J.: “Dynamic Behaviour of Mooring Lines”, BOSS Conference,Delft, 1985.

[19] Huijsmans, R.H.M. and Wichers, J.E.W., “A Computation Model on a Chain-TurretMoored Tanker in Irregular Seas”, OTC paper 6594, Houston, May 1991.

[20] Dercksen, A. and Wichers, J.E.W., “A Discrete Element Method on a Chain TurretTanker Exposed to Survival Conditions”, BOSS Conference, March 1992.

[21] MARIN report No. 45064-2-RD, “Mooring Line Dynamics - Phase I : Development of aMathematical Model”, November 1984.

[22] O.M. Faltinsen, “Sea Loads on Ships and Offshore Structures”, Cambridge UniversityPress, 1990.

[23] Prins, H.J., “Time Domain Calculations of Wave Drift Forces and Moments”, PhD-thesis, Delft University of Technology, 1995.

[24] Prins, Henk J. and Hermans, Aad J., “Wave Drift Damping of a 200 kDWT Tanker”,Journal of Ship Research, Volume 40, Number 2, June 1996.

[25] Aranha, J.A.P., “A Formula for Wave Drift Damping in the Drift of a Floating Body”,Journal of Fluid Mechanics, 275, 1994.

[26] Pinkster, J.A., “Low Frequency Wave Exciting Forces on Floating Structures”, PhD-thesis, Delft University of Technology, 1980.

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