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  Three-dimensional behaviour of elastic marine cables in sheared currents M.A. Vaz a, *, M.H. Patel b a  Laboratory for Subsea Technology, Ocean Engineering Program, Coppe, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil b Santa Fe Laboratory for Offshore Engineering, Department of Mechanical Engineering, University College London, London, UK  Received 20 January 1999; received in revised form 25 June 1999; accepted 25 June 1999 Abstract This paper presents the formulation and solution of governing equations that can be used to analyse the three-dimensional (3D) behaviourof either marine cables during installation or the response of segmented elastic mooring line catenaries as used for floating offshore structureswhen both are subjected to arbitrary sheared currents. The methodology used is an extension of one recently developed for analyses of marinecables when being installed on the seabed or being towed. The formulation describes elastic cable geometry in terms of two angles, elevationand azimuth, which are related to Cartesian co-ordinates by geometric compatibility relations. These relations are combined with the cableequilibrium equations to obtain a system of non-linear differential equations, which are numerically integrated by fourth and fifth orderRunga–Kutta methods. The inclusion of cable elasticity and the ability to consider arbitrary stored currents are key features of this analysis.Results for cable tension, angles, geometry and elongation are presented for three example cases—the installation of a fibre optic marinecable, the static analysis of a deep water mooring line and the response of a telecommunications cable to a multi-directional current profile.  2000 Elsevier Science Ltd. All rights reserved. Keywords : Statics of marine cables; Cable installation; Mooring lines 1. Introduction The exploitation of the ocean’s natural resources is conti-nuing to demand intensive use of cable and line structures inthe marine environment. These include mooring lines foroffshore structures together with flexible riser and umbilicalcatenaries between the seabed and surface vessel offloatingoil production structures. A parallel requirement for theunderstanding of suspended cable span is in the installationprocess of the world’s extensive network of subsea opticalfibre telecommunication cables.Now, as offshore suspended cable systems are beingapplied to even deeper waters, the effects of line elasticityand of sheared current profiles become more important.Deeper water suspended cable structures are more suscep-tible to the effects of currents to the extent that accurateanalysis of their geometry becomes an important designrequirement. Such analysis needs to predict the cabletension and displacement profile, its suspended length andthe horizontal and vertical forces at the top end—the latterbeing an important element of the system’s performance.The behaviour of marine cables during their installationphase is a distinctly different problem both in the physics of the laying process and in the objectives of an analysis of theprocess. An important element of the physics of the installa-tion process is the pay out of the cable and its sinking due tosubmerged weight during its transit to the seabed. At thesame time, the analysis of the installation is concerned withthe position of the cable’s touch down point (TDP) just asmuch as with the tension at the top of the cable suspendedlength.The work of many investigators into the cable installationproblem has demonstrated that inaccuracy in cable TDPposition is due to the presence of unknown ocean currents.Such currents are a combination of components includingtide driven, local wind, Stoke’s wave drift, set up in shallowwaters, storm surges and water density variations (see Ref.[1]). Ocean currents are known to have magnitude anddirection varying in space and time although a depth depen-dent two-dimensional (2D) profile is often used as a designassumption. Analysis shows that uniform in-plane currentchanges the cable inclination and offsets the cable TDP,while non-uniform current whose direction changes withdepth leads to a complex curvature of the cable.Burgess [2] has carried out several studies on theeffect of current on marine cable installation and hasconducted sea trials to evaluate the influence of shearedcurrents on cable deployment. A significant variation inthe current profile was detected for different locations Applied Ocean Research 22 (2000) 45–530141-1187/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved.PII: S0141-1187(99)*Corresponding author. Tel.:  55-21-560-8832; fax:  55-21-290-6626.  E-mail address: (M.A. Vaz)  and times. Use of measured current profiles on thenumerical simulations improved results for the predic-tion of the TDP track.Many analysts have considered the problem of marinecable installation starting with the pioneering work of Zajac [3] who formulated the equations and developed asolution for the 2D stationary configuration of marine cablesbeing laid. He also approximated the solution for the 3Dproblem by a perturbation technique. It was assumed thatthe cables were subjected to surface currents and zerobottom tension. A steady state formulation and solutionfor a cable-body system towed by a ship describing a circu-lar path were presented by Choo and Casarella [4]. Theelasticity of the cable was not included. Casarella andParsons[5] and Choo and Casarella [6] presented a systema-tic and comprehensive review of early work on the simula-tion of cable-body systems. Pedersen [7] proposed anumerical solution using successive integrations to deter-mine the 2D static configuration of cables and pipelinesduring laying. Peyrot and Goulois [8] proposed a numericalsolution for analyses of 3D assemblages of cables and sub-structures. The loading in the cable was restricted to gravity,thermal expansion and fluid drag. In his textbook, Faltinsen[1] presented the classical analytical solution for staticcables suspended by two points and subjected to self-weightand hydrostatic forces only. Bending stiffness and fluidhydrodynamic actions are not considered. The cable elasti-city may be approximately included in the formulae.Leonard and Karnoski [9] developed a numerical algorithmto simulate the stationary 3D cable deployment from a shiptravelling at a constant speed and direction in shearedcurrents. The authors were particularly interested in inves-tigating the configuration during passively controlled cableinstallation.The dynamic analysis of marine cables has also receivedmuch attention in the research literature with developmentof different algorithms to solve the problem—see forexample [2,10–13]. A key feature of this technique is its  M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53 46Fig. 1. Systems of co-ordinates.Fig. 2. Geometric compatibility.  computational efficiency, making it possible to carry outcalculations in real time or on board vessels.Morerecently,Pinto[14]hasproducedastableformulationfor cables with low or nearly zero tension. In parallel withthis work, Vaz et al. [15] have produced an efficient formu-lation and numerical solution for the 3D behaviour of marine cable installation. The analysis presented in thispaper is an extension of the latter work in two areas—thefirst being the inclusion of sheared currents to bringthe efficient formulation to bear on this. At the same time,the methodology has been developed so that segmentedelastic mooring lines or the flexible riser problem can alsobe efficiently solved in a unified technique. 2. Formulation of the governing equations The solution for the cable steady-state configurationencompasses the formulation of geometric and equilibriumrelations. The governing equations are written for a local co-ordinate system where loads can be more easily calculated.This local system of reference varies its orientation withposition in the cable and is related to the moving frame bya rotation matrix. The formulation presented in this paper issteady-state (time independent), or sometimes calledstationary, as it depends on relative velocities and involvesdynamic forces but it is not an arbitrary function of time. 2.1. Systems of co-ordinates and cable’s kinematics Three Cartesian systems of references, shown in Fig. 1,are adopted. The inertial  OXYZ   system has the plane  OXZ   inthe seabed and the axis  OY   oriented vertically upwards. Theship motion and current profile are defined relative to thissystem. Axes system  Oxyz  has the same orientation as the OXYZ   system but with its srcin at the cable TDP. A localframe of reference defines an individual cable element andis represented by the tangent, normal and binormal unitvectors   t    n   b   respectively, of a cable segment. Trans-formation matrices relate these systems to each other. Thefollowing relation is derived from Fig. 2:   I    J   K     i   j  k    cos     cos      sin     cos      sin    sin     cos     0cos     sin      sin     sin     cos      t   n  b   1  where   i   j   k   and    I     J    K   are the unit vectors of   Oxyz  and OXYZ  , respectively. The azimuth and elevation angles are,respectively,           p   and           p   and  p  is the stretchedarc length along the cable. The relationship between thestretched and unstretched arc length  s  isd  p d s   1      2  where     is the uniaxial strain in the cable and the indepen-dent variables  p  and  s  are also called the material co-ordi-nates. Small deformations are assumed and flexure, shearand torsion effects are not considered. The position of acable element,    R c   may be decomposed into (see Fig. 1)   R c   p  t      r  0  t     r  c   p   3a  where   r  0  t     V  0 t    i  and   r  c   p    X    p   i  Y    p    j   Z    p   k   The functions  X    p    Y    p   and  Z    p   describe the cablegeometric configuration viewed from an observer movingwith the TDP,  t   is the time variable and  V  0  gives the constantvelocity of the TDP. The velocity and acceleration of acable element, respectively,   V  c  and    A c  are given by  V  c   p   dd t    R c   p  t      t   V  0  cos     cos     V  po    n  V  0  sin     cos       b  V  0  sin     3b    A c   p   d 2 d t  2   R c   p  t     V  2po d   d  p  n  d   d  p  cos     b     3c  where  V  po  is the cable pay out rate. The concept of totaldifferentiation is needed because the cable is constantly paid  M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53  47Fig. 3. Forces on cable infinitesimal element.  out and a formulation fixed in space is adopted. Thegeometric compatibility equations (Eqs. (A1a)–(A1c)) areused in the derivation of Eqs. (3b) and (3c). 2.2. Relations for cable equilibrium The equilibrium of external and internal forces in theinfinitesimal element of stretched length  dp  is calculatednext. Newton’s second law is invoked to account for the“centrifugal” force srcinated from the fact that the addedcable travels at constant speed but it changes direction whenin a curved configuration. The bending stiffness is assumedto be small—for cables subjected to high axial forces themoments associated with the curvature (i.e. the geometricalstiffness component) are higher than the internal momentsresulting from the cable-section bending stiffness. Themarine cable is assumed fully immersed in water and thecable rotational motions are not considered.The following forces act on the cable element of Fig. 3. ã  self weight:   wdp   j ã  “effective” tension:  T   t   and   T   dT    t   d   t   ã  tangential drag force:  D t    12   C  f    dV  rt  V  rt  dp  t  ã  normal drag force:  D n    12   C  D   dV  rn  V  rnb  dp  n ã  binormal drag force:  D b    12   C  D   dV  rb  V  rnb  dp  b ã  inertia force:     c dp   A c ã  hydrodynamic added inertia:    C  m   d  2 4  dp   A c where   w  is the cable weight in sea water per unit stretchedlength,     the sea water density,  C  f   the friction coefficient, C  D  the drag coefficient,   d   the cable diameter after the cablestretches,  V  rn   V  rb   V  rt    V  rnb   are relative velocities definedin Appendix B,     c  is the cable physical mass per unitstretched length  C  m  is the added mass coefficient and    A c is the cable acceleration. It is assumed that both the cablemass and material density remain invariant during itsstretching. Hence     c      c    w    w     1      and   d    d       1       where    c   w  and  d   are the cable’s properties beforestretching.Furthermore, the cable is assumed to be continuous andextensible with a linear elastic stress–strain relationshipgiven by Hooke’s Law, i.e.       E      where     is the normalstress,  E   is the Young’s Modulus and     is the unit elonga-tion. The axial stress is also given by        T      A  where  A  is thecable’s cross-sectional area and   T   is the “actual” tension.The “effective” tension  T   is related to   T   by   T    T     gA  h  Y     where  g  is the acceleration of gravityand  h  is the water depth.Summing forces parallel to the tangential, normal andbinormal axes, respectively (t), (n) and (b), and using thematerial co-ordinate  s  results ind T  d s   w  sin     12    C  f   d    1       V  rt  V  rt     0   t   4a  T      1    V  2po   d   d s  12    C  D d    1       V  rn  V  rnb   w  cos       0   n   4b  T      1    V  2po   d   d s cos     12    C  D d    1       V  rb  V  rnb     0   b   4c  where          c    C  m  d  2 4   3. Numerical solution The solution is obtained by solving a system of sevennon-linear first-order ordinary non-linear differential equa-tions. Eqs. (2), (4a) and (A1a)–(A1c) are written in theCauchy form to allow numerical solution by a Runge–Kutta solver using the computer package Matlab (1991).Seven initial conditions at the TDP are required, i.e. eleva-tion and azimuth angles, tension and three Cartesian co-ordinates. Note that the numerical scheme allows solutionfor segmented lines, i.e. cables with different geometricaland material properties. This is specifically important, forinstance, when analysing mooring lines composed of segments of steel chains and fibre ropes. For segmentedlines, the end conditions of the lower cable section are theinitial conditions of the upper cable section. In staticanalyses the sea bottom may have a slope. The convergenceis fast and the method is very robust. 4. Results Three computations are presented to illustrate the solu-tion method and the main type of analyses. The first dealswith the installation of a telecommunication cable under atransverse sheared current, whereas the second examplesimulates the static behaviour of a segmented mooringline. A third example considers the effect on a telecommu-nication cable of a sheared current in mutually perpendicu-lar vertical planes.  M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53 48Table 1HA cable characteristicsWeight in air/sea (N/m) Cable diameter (m) Drag coefficient Inertia coefficient26.49/17.76 0.0332 1.649 1.0
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