THREE-DIMENSIONAL STATIONARY ANALYSIS … STATIONARY ANALYSIS OF ELASTIC MARINE CABLES UNDER SHEARED...

THREE-DIMENSIONAL STATIONARY ANALYSIS OF

ELASTIC MARINE CABLES UNDER SHEARED

CURRENTS

Murilo Augusto VazLaboratory for Subsea TechnologyOcean Engineering ProgramCOPPE/UFRJ

Abstract - This paper describes a three-dimensional formulation of theequations governing the stationary behaviour of segmented marine cablesduring installation. The cable-laying vessel is assumed to move rectilinearlyand pay out cable at constant speed. The statics of mooring lines representsthe particular case of no vessel motion. The analysis considers the elasticityof the cable and the hydrodynamic loading due to sheared currents. The cablestretching capability may be significant in extreme operation conditions. Thepresence of sheared currents renders analysis three-dimensional. A finitedifference scheme - a Runge-Kutta fourth and fifth order method - isemployed to integrate the set of first-order nonlinear differential equations.The cable geometry is described by elevation and azimuth angles which aregeometrically connected to Cartesian coordinates by compatibility relations.Results of numerical simulations are presented for the static analysis of atypical segmented deep water mooring line used in offshore Campos Basin.

INTRODUCTION

Cables, in general, are axially stiff structures capable of withstanding largetensile forces. On the other hand cables may buckle when subjected tocompressive forces because of the low bending stiffness. This behaviourmakes analysis more difficult because of the geometrical nonlinearity. Marinecables are also subjected to large and complex fluid forces resulting from theaction of potential and viscous flows. For slender structures such as marinecables the fluid force is generally modelled by the modified Morison equationwhich uses the cable-water relative velocity and acceleration. Morison's

Transactions on the Built Environment vol 29, © 1997 WIT Press, www.witpress.com, ISSN 1743-3509

114 Offshore Engineering

equation renders analysis nonlinear through the quadratic dependent term onthe drag force.

The exploitation of the ocean's natural resources has demanded the intensiveuse of cable structures in the marine environment. In the offshore industry,for instance, umbilical cables with hydraulic and/or electrical functions areemployed to transmit power to subsea equipment and hence control theopening and closure of valves. The ocean has also been used as a seaway forthe installation of transcontinental telecommunication fibre optic cables.These cables are very expensive and so may be the cost of downtime andrepair: proper and safe installation needs to be to ensured to minimiseeventual intervention. The operation of laying marine cables is very criticalfrom the structural point of view and may impart permanent damage to thecable.

The use of conventional mooring systems for the stationkeeping of largefloating structures has been extended to deep waters. For an optimum designthe precise determination of the line configuration and its restoring forceresponse is important. A mooring line acts as a geometrically nonlinearspring. Work is necessary to raise the suspended line with consequent storageof potential energy. Changes in the spread mooring system are made possibleby repositioning the anchors, so that a relatively small vertical force should beenough to lift these anchors. This means that a mooring line should operate,in all circumstances, with part of the line lying on the seabed. The horizontalrestoring force as a function of the offset (distance from the anchor to the lineupper point) provides the line stiffness. Another feature of mooring systemsis the intensive use of segmented lines which provide a good restoringcapacity with a light weight line.

Ocean currents affect cable configuration. These currents result from acombination of components such as tide driven, local wind, regular waveStokes drift, ocean circulation, set-up (in shallow waters) and storm surgesand water density sudden variation (Faltinsen*). Ocean currents are reportedto have magnitude and direction varying in space and time but a depthdependent profile is often a good design assumption.

In a pioneer work Zajac formulated the equations and developed a solutionfor the two-dimensional stationary configuration of marine cables. Thesolution for the three-dimensional problem was approximated by aperturbation technique. It was assumed that the cables were subjected tosurface currents and zero bottom tension. A steady state formulation andsolution for a cable-body system towed by a ship describing a circular pathwere presented by Choo and Casarella . The elasticity of the cable was not


Offshore Engineering 115

included. Casarella and Parsons and Choo and Casarella presented asystematic and comprehensive review of early work on the simulation ofcable-body systems. Pinto has updated and included recent work in thisfield. Pedersen^ proposed a numerical solution using successive integrationsto determine the two-dimensional static configuration of cables and pipelinesduring laying. Peyrot and Goulois^ proposed a numerical solution foranalyses of three-dimensional assemblages of cables and sub-strucutres. Theloading on the cable was restricted to gravity, thermal expansion and fluiddrag. In his textbook Faltinsen^ presented the classical analytical solution forstatic cables suspended by two points, subjected to self weight and hydrostaticforces only. Bending stiffness and fluid hydrodynamic actions are notconsidered. The cable elasticity may be approximately included in theformulas. Leonard and Karnoski developed a numerical algorithm tosimulate the stationary three-dimensional cable deployment from a shiptravelling at constant speed and direction in sheared currents. The authorswere particularly interested in investigating the configuration during passivelycontrolled cable installation.

FORMULATION OF THE GOVERNING EQUATIONS

The solution for the cable steady state configuration encompasses theformulation of geometric and equilibrium relations. The governing equationsare written for a local coordinate system where loads can be more easilycalculated. This local system of reference varies its orientation with theposition in the cable and is related to the moving frame by a rotation matrix.The formulation presented in this paper is steady state (time independent), orsometimes called stationary, as it depends on relative velocities and involvesdynamic forces but it is not a function of time.

Systems of Coordinates and Cable's KinematicsThree Cartesian systems of references, shown in Figure 1, are adopted. Thesystem OXYZ is fixed and describes the touch down point TDP (vessel)motion, the system oxyz is solidary to the TDP and the local system is

represented by their tangent, normal and binomial unit vectors F,n, b. Thesesystems are related to each other the following relation, derived from Figure2:

cos0 cosi|/ —sinQ cos\|/ —sin\\f

sinQ cosO 0

cosOszm]/ -sinQsiny cosxj/



where i ,j,k and I, J, K are the unit vectors of oxyz and QATZ,

respectively. The azimuth and elevation angles are respectively \|/ =vj/(/?)

and 9=9(p) and p is the stretched arclength which is related to the

unstretched arclength s by:

f = l + e (2)ds

where 8 is the uniaxial strain in the cable and the independent variables pand s are also called the material coordinates. Small deformations areassumed and flexure, shear and torsion effects are not considered. The

position of a cable element, R<., may be given by (Figure 1):

(p,f)=F,(f)+f,(p) (3a)

where r,(i) = (F, f)7 and (p) = X(pJ + 7(p)J + Z(p)fr. The functions

Jf(p), Y(p) and Z(p) describe the cable configuration as seen from an

observer moving with the touch down point, t is the time variable and ¥„gives the constant velocity of the TDP. The velocity and acceleration of a

cable element - respectively V<. and A^ - are given by:

d =

^V.

cos9 cosvj/ — I - n sinQ cosxj/ - b sin\\i (3b)

(so

where F is the cable pay out rate. The concept of total differentiation is

needed because cable is paid out (at constant speed) and a formulation fixed inspace is adopted. The geometric compatibility equations (Ala) to (Ale) areused in derivation of equations (3b) and (3c).

Relations for Cable EquilibriumThe equilibrium of external and internal forces in the infinitesimal element ofstretched length dp is calculated next. Newton's second law is invoked toaccount for the "centrifugal" force as the added cable travels at constant speedbut changes direction in curved configurations. The bending stiffness isassumed small - for cables subject to high axial forces the momentsassociated with the curvature (i.e., the geometrical stiffness component) are



higher than the internal moments resulting from the cable-section bendingstiffness. The marine cable is assumed fully immersed in water and the cablerotational motions are not considered.

The following forces act on the cable element of Figure 3:

# self weight: -vv dp j

• "effective" tension: -T t and (T + df)ft + di

• tangential drag force: £), = -(l/2)p C, n d V« \V,, dp 7

. normal drag force: D,, = -(l/2)p Q, J F^ 16<p M

* binormal drag force: D, = -(l/2)p C» d V* V^ \ dp b

• Inertia force: p^ dp A^

• hydrodynamic added inertia: p C,, md*/4\dp A^

where w is the cable weight in sea water per unit stretched length, p is the

sea water density, C/ is the friction coefficient, C^ is the drag coefficient, d

is the cable diameter after cable stretches, V^, V^, V«, V^ \ are relative

velocities defined in Appendix B, p^ is the cable physical mass per unit

stretched length and €„, is the added mass coefficient and A<. is the cableacceleration. It is assumed mass conservation and invariability of the cable

material density upon stretching. Hence p^ = p^,, w=w/(l + £) and

d = d/Vl + s where p<_, w and d mean the cable's properties beforestretching.

Furthermore, the cable is assumed continuous and extensible with linearelastic stress-strain relationship given by Hooke's Law, i.e., a = Ez, wherea is the normal stress, E is the Young's Modulus and 8 is the unit

elongation. The axial stress is also given by a = T/A where A is the cable's

cross-sectional area and T is the "actual" tension. The "effective" tension T

is related to T by T -T- pgA[h-Y , where g is the acceleration of

gravity and h is the water depth.

Summing forces parallel to the tangential, normal and binormal axes,respectively (t), (n) and (b), and using the material coordinate s results in:



~ ' ' (t) (4a)

0 (n) (4b)

where p = p,. + pC,,, ——

NUMERICAL SOLUTION

The solution is obtained by solving a system of seven first-order ordinarynonlinear differential equations. Equations (2), (4a) to (4c) and (Ala) to(Ale) are written in Cauchy form to allow numerical solution by a Runge-Kutta solver using the computer package Matlab . Seven initial conditions atthe TOP are required, i.e., elevation and azimuth angles, tension and threeCartesian and one material coordinates. Note that the numerical schemeallows solution for segmented lines, i.e., cables with different geometrical andmaterial properties. This is specially important, for instance, when analysingmooring lines composed of segments of steel chains and fibre ropes. Forsegmented lines the end conditions of the lower cable section are the initialconditions of the upper cable section. In static analyses the sea bottom mayhave a slope. The convergence is fast and method is robust.

RESULTS

The simulation shown below deals with the static behaviour of a segmentedmooring line. Results are presented for a typical segmented mooring lineinstalled in deep waters at the offshore Campos Basin, Brazil. This lineconsists of bottom and upper chain segments and an intermediate wire ropesegment. Their material and geometrical properties are described in Table 1.

Also the design pre-tension is 1351 kN with an anchoring radius of 2387 m.The water depth is assumed constant and equal to 861 m. The data was kindlyfurnished by Cenpes/Petrobras.



CableCharacteristics

D(m)w (m)

• EA (MM)Breaking Load (kN)Total Length (m)

Drag Coefficient Q,

Inertia Coefficient €„

Lower SegmentChain

0.0951729.1

793.88693012403.2

1.6

Middle SegmentWire Rope

0.109420.8536.7971601250

1.8

1.0

Upper SegmentChain0.084

1363.8620.6872301503.2

1.6

Table 1 - Properties of Mooring Line

Figure 4a presents the restoring capacity of a single mooring line in terms ofthe offset, defined as the "horizontal" distance between the anchor and theupper end points. The line exhibits its typical progressively hardeningrestoring characteristic. The line design reveals a good degree ofoptimisation: (1) the line breaking point occurs just after the maximumoperating condition which is "initial offset plus 10% water depth". Also theline tends to break before the anchor lifts. Both situations represent loss ofrestoring capacity. (2) The point at which the wire rope touches the seaflooris away from the other operating extreme condition given by "initial offsetminus 10% water depth". Wire ropes are not very resistant to abrasion.

Figure 4b shows the percentual gain or loss of horizontal load for the mooringline in the presence of currents with results compared to the case of nocurrents. In the operating range the line experiences decreasing influence ofthe marine currents as the offset is increased an the line is tensioned. Also forthe same offset the results show a higher loss for negative directions ofcurrent.

CONCLUSIONS

A three-dimensional steady state formulation for elastic segmented marinecables during installation in sheared currents is presented. The timeindependent set of first-order nonlinear ordinary differential equations aresolved by a Runge-Kutta integrator. The convergence of the numericalalgorithm is fast and method is robust. A case study is presented for the staticanalysis of a segmented deep water mooring line employed offshore Brazil.



ACKNOWLEDGEMENTS

The support of the Brazilian Council of Research (CNPq) is acknowledged.

REFERENCES

[1] Faltinsen, O. M: "Sea Loads on Ships and Offshore Structures",Cambridge University Press, 1990.

[2] Zajac, E. E.: "Dynamics and Kinematics of the Laying and Recovery ofSubmarine Cable", Bell System Technical Journal, Vol. 36, No. 5,September 1957, pp. 1129-1207.

[3] Choo, Y.-I1 and Casarella, M. J.: "Configuration of a Towline Attached toa Vehicle Moving in a Circular Path", Journal of Hydronautics, Vol. 6,No 1, Jan. 1972, pp. 51-57.

[4] Casarella, M. J. and Parsons, M.: "Cable Systems under HydrodynamicLoading", Marine Technology Society Journal, Vol. 4, No 4, July-Aug.1970, pp. 27-44.

[5] Choo, Y.-I1 and Casarella, M. J.: "A Survey of Analytical Methods forDynamic Simulation of Cable-Body Systems", Journal of Hydronautics,Vol. 7, No 4, Oct. 1973, pp. 137-144.

[6] Pinto, W. T.: "On the Dynamics of Low Tension Marine Cables", Ph DDissertation, Department of Mechanical Engineering, University CollegeLondon, 1995.

[7] Pedersen, P. T.: "Installation of Cables and Pipelines", InternationalShipbuilding Progress, Vol. 22, 1975, pp. 399-408.

[8] Peyrot, A. H. and Goulois, A. M.: "Analysis of Cable Structures",Computers and Structures, Vol. 10, 1979, pp. 805-813.

[9] Leonard, J. W. and Karnoski, S. R.: "Simulation of Tension ControlledCable Deployment", Applied Ocean Research, Vol. 12, No 1, 1990, pp.34-42.

[10]Pro-matlab for VAX/VMS Computers, The Math Works, Inc., Jan 1 1991.

Appendix A - Equations of Geometric CompatibilityThe following relationships are derived from Figure 2.

dY _ 1 df ^^— = cos\|/ cos9 = (Ala)dp 1 + s as


</y . a i <#— = smv =dp 1 + s ds


(Alb)

dZ . ^ 1 dZ— = sm\y cos 8 =dp 1 + s ds

(Ale)

_ , , ^ ,, ^Then —- + — + — =1 and —- + — + — =l + e.

Appendix B - Relative Velocity Fluid/Structure

The sheared current velocity is a function of depth only and is given by:

(Bl)

There is no vertical component of current. Hence the relative velocity

V, - V^ - U is given by:

V = +n

-b

cosi|/ - -

- Ux (Y)] sinQ cos \|/ + U (Y) sinQ siny ]+ (B2)

And the magnitude of the radial relative velocity is



A Y

O

Figure 1 - Systems of Coordinates

yA

dv

cable infinitesimal element

ds s ds

dp

dz

dx

Figure 2 - Geometric Compatibility



dp

A 0 and VB (6 + d9) and (H/

1: -Tl

2: (T + dT

3: - w dp j

6: -

7: P C,,

Figure 3 - Forces on Cable Infinitesimal Element

8000 _,

6000 -

4000 -

2000 -

cable onseafloor

2.20Offset (km)

2.50 2.6

Figure 4a - Mooring Line Stiffness


3OQ*

s.y

i

(T)

Percentage Differencein Horizontal Force (%)

i i i i I i i i i I i i i i i I I l I l l l i I

I I I

r

w

I

OQ


THREE-DIMENSIONAL STATIONARY ANALYSIS … STATIONARY ANALYSIS OF ELASTIC MARINE CABLES UNDER SHEARED...

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