- Email: [email protected]
Three-dimensional behaviour of elastic marine cables in sheared currents
Applied Ocean Research 22 (2000) 45–53 www.elsevier.com/locate/apor
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) behaviour of either marine cables during installation or the response of segmented elastic mooring line catenaries as used for floating offshore structures when both are subjected to arbitrary sheared currents. The methodology used is an extension of one recently developed for analyses of marine cables when being installed on the seabed or being towed. The formulation describes elastic cable geometry in terms of two angles, elevation and azimuth, which are related to Cartesian co-ordinates by geometric compatibility relations. These relations are combined with the cable equilibrium equations to obtain a system of non-linear differential equations, which are numerically integrated by fourth and fifth order Runga–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 marine cable, the static analysis of a deep water mooring line and the response of a telecommunications cable to a multi-directional current profile. q 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 continuing to demand intensive use of cable and line structures in the marine environment. These include mooring lines for offshore structures together with flexible riser and umbilical catenaries between the seabed and surface vessel of floating oil production structures. A parallel requirement for the understanding of suspended cable span is in the installation process of the world’s extensive network of subsea optical fibre telecommunication cables. Now, as offshore suspended cable systems are being applied to even deeper waters, the effects of line elasticity and of sheared current profiles become more important. Deeper water suspended cable structures are more susceptible to the effects of currents to the extent that accurate analysis of their geometry becomes an important design requirement. Such analysis needs to predict the cable tension and displacement profile, its suspended length and the horizontal and vertical forces at the top end—the latter being an important element of the system’s performance. The behaviour of marine cables during their installation phase is a distinctly different problem both in the physics of * Corresponding author. Tel.: 155-21-560-8832; fax: 155-21-290-6626. E-mail address: [email protected] (M.A. Vaz)
the laying process and in the objectives of an analysis of the process. An important element of the physics of the installation process is the pay out of the cable and its sinking due to submerged weight during its transit to the seabed. At the same time, the analysis of the installation is concerned with the position of the cable’s touch down point (TDP) just as much as with the tension at the top of the cable suspended length. The work of many investigators into the cable installation problem has demonstrated that inaccuracy in cable TDP position is due to the presence of unknown ocean currents. Such currents are a combination of components including tide driven, local wind, Stoke’s wave drift, set up in shallow waters, storm surges and water density variations (see Ref. [1]). Ocean currents are known to have magnitude and direction varying in space and time although a depth dependent two-dimensional (2D) profile is often used as a design assumption. Analysis shows that uniform in-plane current changes the cable inclination and offsets the cable TDP, while non-uniform current whose direction changes with depth leads to a complex curvature of the cable. Burgess [2] has carried out several studies on the effect of current on marine cable installation and has conducted sea trials to evaluate the influence of sheared currents on cable deployment. A significant variation in the current profile was detected for different locations
0141-1187/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0141-118 7(99)00023-1
46
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Fig. 1. Systems of co-ordinates.
and times. Use of measured current profiles on the numerical simulations improved results for the prediction of the TDP track. Many analysts have considered the problem of marine cable installation starting with the pioneering work of Zajac [3] who formulated the equations and developed a solution for the 2D stationary configuration of marine cables being laid. He also approximated the solution for the 3D problem by a perturbation technique. It was assumed that the cables were subjected to surface currents and zero bottom tension. A steady state formulation and solution for a cable-body system towed by a ship describing a circular path were presented by Choo and Casarella [4]. The elasticity of the cable was not included. Casarella and Parsons [5] and Choo and Casarella [6] presented a systematic and comprehensive review of early work on the simulation of cable-body systems. Pedersen [7] proposed a numerical solution using successive integrations to determine the 2D static configuration of cables and pipelines
during laying. Peyrot and Goulois [8] proposed a numerical solution for analyses of 3D assemblages of cables and substructures. 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 static cables suspended by two points and subjected to self-weight and hydrostatic forces only. Bending stiffness and fluid hydrodynamic actions are not considered. The cable elasticity may be approximately included in the formulae. Leonard and Karnoski [9] developed a numerical algorithm to simulate the stationary 3D cable deployment from a ship travelling at a constant speed and direction in sheared currents. The authors were particularly interested in investigating the configuration during passively controlled cable installation. The dynamic analysis of marine cables has also received much attention in the research literature with development of different algorithms to solve the problem—see for example [2,10–13]. A key feature of this technique is its
Fig. 2. Geometric compatibility.
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
47
Fig. 3. Forces on cable infinitesimal element.
computational efficiency, making it possible to carry out calculations in real time or on board vessels. More recently, Pinto [14] has produced a stable formulation for cables with low or nearly zero tension. In parallel with this work, Vaz et al. [15] have produced an efficient formulation and numerical solution for the 3D behaviour of marine cable installation. The analysis presented in this paper is an extension of the latter work in two areas—the first being the inclusion of sheared currents to bring the efficient formulation to bear on this. At the same time, the methodology has been developed so that segmented elastic mooring lines or the flexible riser problem can also be efficiently solved in a unified technique. 2. Formulation of the governing equations The solution for the cable steady-state configuration encompasses the formulation of geometric and equilibrium relations. The governing equations are written for a local coordinate system where loads can be more easily calculated. This local system of reference varies its orientation with position 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), or sometimes called stationary, as it depends on relative velocities and involves dynamic 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 in the seabed and the axis OY oriented vertically upwards. The ship motion and current profile are defined relative to this system. Axes system Oxyz has the same orientation as the OXYZ system but with its origin at the cable TDP. A local frame of reference defines an individual cable element and is represented by the tangent, normal and binormal unit ~ respectively, of a cable segment. Transvectors ~t; n~ ; b; formation matrices relate these systems to each other. The
following relation is derived from Fig. 2:
8 9 8 9 2 ~I > > ~i > cos u cos c > > = 6 < > = > < > sin u J~ > > ~j > 6 4 > > ; :~> ; > :~> cos u sin c k K
2sin u cos c cos u 2sin u sin c
38 9 2sin c > > > ~t > 7< = 0 7 ~ n 5> > > ; :~> cos c b
1 ~ K ~ are the unit vectors of Oxyz and where ~i; ~j; k~ and ~I; J; OXYZ, respectively. The azimuth and elevation angles are, respectively, c c p and u u p and p is the stretched arc length along the cable. The relationship between the stretched and unstretched arc length s is dp 111 ds
2
where 1 is the uniaxial strain in the cable and the independent variables p and s are also called the material co-ordinates. Small deformations are assumed and flexure, shear and torsion effects are not considered. The position of a ~ c ; may be decomposed into (see Fig. 1) cable element, R ~ c p; t ~r 0 t 1 ~r c p R
3a
~ where ~r0 t V0 t~i and ~rc p X p~i 1 Y p~j 1 Z pk: The functions X p; Y p and Z p describe the cable geometric configuration viewed from an observer moving with the TDP, t is the time variable and V0 gives the constant velocity of the TDP. The velocity and acceleration of a ~ c are given by cable element, respectively, V~ c and A d ~ R p; t ~t V0 cos u cos c 2 Vpo V~ c p dt c ~ 0 sin c 2 n~ V0 sin u cos c 2 b V 3b 2 du dc 2 ~ ~ c p d R ~ c p; t Vpo ~ n 1 cos b u A dp dp dt2
3c
where Vpo is the cable pay out rate. The concept of total differentiation is needed because the cable is constantly paid
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M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Table 1 HA cable characteristics Weight in air/sea (N/m)
Cable diameter (m)
Drag coefficient
Inertia coefficient
26.49/17.76
0.0332
1.649
1.0
out and a formulation fixed in space is adopted. The geometric compatibility equations (Eqs. (A1a)–(A1c)) are used in the derivation of Eqs. (3b) and (3c).
2.2. Relations for cable equilibrium The equilibrium of external and internal forces in the infinitesimal element of stretched length dp is calculated next. Newton’s second law is invoked to account for the “centrifugal” force originated from the fact that the added cable travels at constant speed but it changes direction when in a curved configuration. The bending stiffness is assumed to be small—for cables subjected to high axial forces the moments associated with the curvature (i.e. the geometrical stiffness component) are higher than the internal moments resulting from the cable-section bending stiffness. The marine cable is assumed fully immersed in water and the cable rotational motions are not considered. The following forces act on the cable element of Fig. 3. ^ ~j self weight: 2wdp “effective” tension: 2T~t and T 1 dT ~t 1 d~t ^ rt uVrt udp~t tangential drag force: Dt 2 12 rCf pdV 1 ^ normal drag force: Dn 2 2 rCD dVrn uVrnb udp~n ^ rb uVrnb udpb~ binormal drag force: Db 2 12 rCD dV ~ inertia force: r^ c dpAc pd^ 2 ~ dpAc hydrodynamic added inertia: rCm 4 where w^ is the cable weight in sea water per unit stretched length, r the sea water density, Cf the friction coefficient, CD the drag coefficient, d^ the cable diameter after the cable stretches, Vrn ; Vrb ; Vrt ; uVrnb u are relative velocities defined in Appendix B, r^ c is the cable physical mass per unit ~c stretched length Cm is the added mass coefficient and A is the cable acceleration. It is assumed that both the cable mass and material density remain invariant during its ^ stretching. p Hence r^ c rc ; w^ w= 1 1 1 and d d= 1 1 1 where rc ; w and d are the cable’s properties before stretching. Furthermore, the cable is assumed to be continuous and extensible with a linear elastic stress–strain relationship given by Hooke’s Law, i.e. s E1; where s is the normal stress, E is the Young’s Modulus and 1 is the unit elonga where A is the tion. The axial stress is also given by s T=A cable’s cross-sectional area and T is the “actual” tension. The “effective” tension T is related to T by T T 2 rgAh 2 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 co-ordinate s results in p dT 1 2 w sin u 2 rCf pd 1 1 1Vrt uVrt u 0 ds 2 T2 2 T2
t
4a
n
4b
b
4c
r du 2 Vpo 111 ds p 1 rCD d 1 1 1Vrn uVrnb u 2 w cos u 0 2 r dc 2 cos u Vpo 111 ds
p 1 rCD d 1 1 1Vrb uVrnb u 0 2 pd2 : where r rc 1 rCm 4 2
3. Numerical solution The solution is obtained by solving a system of seven non-linear first-order ordinary non-linear differential equations. Eqs. (2), (4a) and (A1a)–(A1c) are written in the Cauchy 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. elevation and azimuth angles, tension and three Cartesian coordinates. Note that the numerical scheme allows solution for segmented lines, i.e. cables with different geometrical and material properties. This is specifically important, for instance, when analysing mooring lines composed of segments of steel chains and fibre ropes. For segmented lines, the end conditions of the lower cable section are the initial conditions of the upper cable section. In static analyses the sea bottom may have a slope. The convergence is fast and the method is very robust.
4. Results Three computations are presented to illustrate the solution method and the main type of analyses. The first deals with the installation of a telecommunication cable under a transverse sheared current, whereas the second example simulates the static behaviour of a segmented mooring line. A third example considers the effect on a telecommunication cable of a sheared current in mutually perpendicular vertical planes.
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
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Fig. 4. (a) XY projection; (b) ZY projection; (c) elevaton angle; (d) Azimuth angle; (e) effective tension distribution; (f) distribution of cable elongation; and (g) suspended length and TDP radius.
4.1. Installation of telecommunication cable in transverse current The installation of a heavy armoured (HA) optical fibre cable is used in this case. The cable properties are chosen to
be the same as those used by Hopland [16] and are described in Table 1. An axial stiffness EA 1:0 × 108 N is assumed. The equation of the laying ship’s track is taken as ~r0 t S0 1 V0 t~i
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M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Fig. 4 (continued).
where V0 is the ship speed, t time and S0 is the initial distance between the axes systems OXYZ and oxyz. S0 may be taken as zero without losing the generality of the solution. The idea behind this simulation is to evaluate the effect of transverse marine current (relative to the ship track) on the configuration of a fibre-optic cable during installation. Tangential drag forces are neglected and the cable is assumed to be uniform. The marine current profile is ~ 1:0 Y=h 2 0:5K ~ where h 1000 m is taken to be U ~ is the unit vector in the Z direction. the water depth and K This current profile is a function of depth only, which makes the problem solution stationary. In other words, for any point in a plane parallel to OXZ, the current profile remains the same function of Y. As the ship moves, the cable encounters the same depth dependent current although this current may have components in both the OX and OZ directions. The laying vessel is assumed to move at a constant speed of 1.5 m/s, which is also the pay out rate. The current profile varies linearly from 0.5 m/s at the sea level to 20.5 m/s at the seafloor. There is no current velocity component in the direction of the vessel motion. Results are presented for four levels of pre-tension at the cable TDP. Fig. 4a–g presents the results of such an analysis. In each of these figures, t0 is the cable pre-tension at the lower end.
These figures are obtained by integrating the governing equation from the TDP upwards so that the initial (boundary) conditions at the TDP need to be known. Fig. 4a and b presents the cable configuration (which is constant in time) as an observer would view it from the TDP. Fig. 4a, the XY projection, shows the in-plane frame moving from a straight line configuration for low tension—as predicted by Zajac [3]—to a catenary-like shape for higher pre-tensions. Fig. 4b, the ZY projection, shows the out-of-phase projection of the cable configuration. Lower pre-tension increases the cable curvature. If there was no transverse current there would be no ZY projection. Fig. 4c and d, respectively, show the distribution of elevation and azimuth angles as a function of depth. Higher the pre-tension, the higher is the variation of elevation and smaller is the variation of azimuth. For low pre-tension, the solution approaches Zajac’s result of a constant angle (straight line). Near the TDP, the cable develops large elevation gradients. The azimuth distribution also indicates a progressive shift from the straight-line cable laying with a constant angle to a nearly constant azimuth as the pre-tension (and stiffness) increases. Fig. 4e shows the linear distribution of tension in the cable with tangential drag forces neglected. Maximum and minimum tensions occur at the cable’s upper and lower ends, respectively. The actual cable tension is lower than the effective tension except at the sea surface where they are equal. Finally Fig. 4f presents the cable elongation, i.e. the amount of cable stretched when axial stiffness is included. The cable elasticity is only noticeable when there is a significant pre-tension. The larger elongations are found at the sea surface, as expected. Fig. 4g shows the cable suspended length and the horizontal distance from the TDP to the upper point (TDP radius). TDP radius is the horizontal distance between the TPD and the cable upper end. As in a mooring line, the higher the pre-tension, the higher is the offset and the suspended length as well as the stretched length and the TDP radius. 4.2. Mooring line Results are presented for a typical segmented mooring line installed in deep waters at offshore Campos Basin,
Table 2 Properties of line segments
D (m) w (m) EA (MN) Breaking load (kN) Total length (m) Drag coefficient CD Inertia coefficient Cm
Lower segment chain
Intermediate segment wire rope
Upper segment chain
0.095 1729.1 793.88 6930 1240 3.2 1.6
0.109 420.8 536.79 7160 1250 1.8 1.0
0.084 1363.8 620.68 7230 150 3.2 1.6
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
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Fig. 5b shows the percentage gain or loss of horizontal load for the mooring line in the presence of currents. Results are compared with the case of no currents. In the operating range, the line experiences decreasing influence of the marine currents as the offset is increased and the line is stretched. For deeper water, the role of marine currents on the mooring line is expected to be even greater. 4.3. Telecommunication cable in 3D current profile This example uses the same cable as in Section 4.1, but imposes a more complex 3D current profile given by the equation ~ u cos 2pY ~I 1 u sin 2pY K ~ U h h
Fig. 5. (a) Mooring line stiffness; and (b) reduced stiffness due to current.
Brazil. This line consists of bottom and upper chain segments and an intermediate wire rope segment. Their material and geometrical properties are described in Table 2. 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. Fig. 5a presents the restoring capacity of a single mooring line in terms of the offset, this being defined as the “horizontal” distance between the anchor and the upper end point. Results are plotted for in-line constant profile marine currents of ^1.0 m/s and for no current. The line exhibits its typical progressively hardening restoring characteristic. The line design optimisation reveals two issues: (1) the line breaking point occurs just after the maximum operating condition which is “initial offset plus 10% water depth”. Also, the line tends to break before the anchor lifts. Both situations represent loss of restoring capacity. (2) The point at which the wire rope touches the seafloor is away from the other operating extreme condition given by “initial offset minus 10% water depth”. Wire ropes are not very resistant to abrasion and contact with seabed soil should be avoided.
where h 1000 m is the water depth and u is taken to have values of 1.5, 0.75 m/s and zero (no current). The Z coordinate of the top end is set to zero and only one level of pretension at the cable TDP is computed. Fig. 6a–f presents the results of this analysis. The projections of the cable configuration are shown in Fig. 6a and b. In the absence of currents, the cable geometrical configuration entirely lies in the XY plane with a perfect catenary shape. Sheared currents displace the cable laterally and this is more noticeable with the higher magnitude of the current. The elevation and azimuth angles are shown in Fig. 6c and d, respectively, as functions of the vertical co-ordinate. These figures illustrate how difficult it is, for example, to extrapolate the cable configuration underwater with the knowledge of the top angles only. Fig. 6e shows the linear variation of cable effective tension with water depth. Fluid velocity could only affect tension distribution if tangential drag forces were considered (Eq. (4a)). Fig. 6f presents the distribution of cable elongation. The higher the magnitude of the current, the longer are the suspended lengths, as are the cable elongations. The stretched suspended cable lengths for u 1:5; 0.75 m/s and no current are equal to 4782.7, 4156.8 and 3840.9 m, respectively. 5. Conclusions A 3D steady-state formulation for elastic segmented marine cables during installation in sheared currents is presented. The time independent set of first-order non-linear ordinary differential equations are solved by a Runge–Kutta integrator. Case studies are presented for three typical situations: the installation of a heavy armoured fibre optical telecommunication cable in the presence of transverse and 3D currents, and the static analysis of a segmented deep water mooring line subjected to in-line current and employed offshore Brazil. The results presented demonstrate the importance of
52
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Fig. 6. (a) XY projection; (b) ZY projection; (c) elevaton angle; (d) Azimuth angle; (e) effective tension distribution and (f) distribution of cable elongation.
current profile, both in terms of magnitude and direction, for prediction of suspended line behaviour. In particular, the analyses demonstrate the uncertainty of predicting line configurations by using only the top angles and also the difficulty of accurately predicting TDP position without good current information.
The 3D steady-state model developed in this paper is comprehensive as it includes the effects of sheared currents, cable elasticity, segmented cables and laying vessel speed, allowing an accurate analysis of the installation of marine cables subjected to time independent environmental actions. The numerical model is also robust and computationally
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
efficient demanding little CPU time. It can, therefore, be used in real time or on board installation or offshore vessels requiring only reliable current data for operationally effective prediction of underwater cable or line behaviour. Acknowledgements
Appendix A. Equations of geometric compatibility The following relationships are derived from Fig. 2: dX 1 dX cos c cos u dp 1 1 1 ds
A1a
dY 1 dY sin u dp 1 1 1 ds
A1b
dZ 1 dZ sin c cos u dp 1 1 1 ds
A1c
Then dX 2 dY 2 dZ 2 1 1 1 dp dp dp
and dX 2 dY 2 dZ 2 1 1 1 1 12 : ds ds ds
Appendix B. Relative velocity fluid/structure The sheared current velocity is a function of depth only and is given by ~ ~ UX Y~I 1 UZ YK U
~ is given by Hence the relative velocity V~ r V~ c 2 U 8 ~tV0 2 Ux Y cos u cos c 2 Vpo 2 Uz Y cos u sin c1 > > > < ~ V~ r 1n2V : 0 2 Ux Y sin u cos c 1 Uz Y sin u sin c1 > > > : ~ 2bV0 2 Ux Y sin c 1 Uz Y cos c
B3
This work was carried out with the support of the facilities of The Laboratory for Subsea Technology, Federal University of Rio de Janeiro and The Santa Fe Laboratory for Offshore Engineering, University College London. The first author acknowledges the support of the Brazilian Council of Research (CNPq). Data on the mooring line case considered here were kindly supplied by CENPES/Petrobras, Brazil.
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B1
There is no vertical component of current. In terms of the local system, the current profile is 8 ~tUx Y cos u cos c 1 Uz Y cos u sin c1 > > < ~ 2~nUx Y sin u cos c 1 Uz Y sin u sin c1 : B2 U > > : ~ x Y sin c 1 Uz Y cos c 2bU
And the magnitude of the radial relative velocity is uVrnb u p 2: Vrn2 1 Vrb
References [1] Faltinsen OM. Sea loads on ships and offshore structures, Cambridge, MA: Cambridge University Press, 1990. [2] Burgess JJ. The deployment of an undersea cable system in sheared current, Proceedings of BOSS’94, 1994. p. 327–34. [3] Zajac EE. Dynamics and kinematics of the laying and recovery of submarine cable. Bell System Technical Journal 1957;36(5):1129– 207. [4] Choo Y-Il, Casarella MJ. Configuration of a towline attached to a vehicle moving in a circular path. Journal of Hydronautics 1972;6(1):51–7. [5] Casarella MJ, Parsons M. Cable systems under hydrodynamic loading. Marine Technology Society Journal 1970;4(4):27–44. [6] Choo Y-Il, Casarella MJ. A survey of analytical methods for dynamic simulation of cable-body systems. Journal of Hydronautics 1973;7(4):137–44. [7] Pedersen PT. Installation of cables and pipelines. International Shipbuilding Progress 1975;22:399–408. [8] Peyrot AH, Goulois AM. Analysis of cable structures. Computers and Structures 1979;10:805–13. [9] Leonard JW, Karnoski SR. Simulation of tension controlled cable deployment. Applied Ocean Research 1990;12(1):34–42. [10] Burgess JJ. Modelling of undersea cable installation with a finite difference method, Proceedings of the First International Offshore and Polar Engineering Conference, ISOPE’91, vol. II, 1991. p. 222–7. [11] Burgess JJ. Equations of motion of a submerged cable with bending stiffness, International Conference on Offshore Mechanics and Arctic Engineering, OMAE’92, vol. I-A, 1992. p. 283–9. [12] Triantafyllou MS, Chryssostomidis C. The dynamics of towed arrays. Journal of Offshore and Arctic Engineering 1989;111:208–13. [13] Triantafyllou MS, Hover FS. Cable dynamics for tethered underwater vehicles. MIT Sea Grant College Program, Report No. 90-4, Massachusetts Institute of Technology, Cambridge, MA, 1990. [14] Pinto WT. On the dynamics of low tension marine cables, PhD dissertation, Department of Mechanical Engineering, University College London, 1995. [15] Vaz MA, Patel MH, Witz JA. Three dimensional transient analysis of the intallation of marine cables. Acta Mechanica 1997;124(1–4):1– 26. [16] Hopland S. Investigation of cable behaviour in water during laying of fiberoptic submarine cables. International Wire & Cable Symposium Proceedings. 1993. p. 734–9.
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) behaviour of either marine cables during installation or the response of segmented elastic mooring line catenaries as used for floating offshore structures when both are subjected to arbitrary sheared currents. The methodology used is an extension of one recently developed for analyses of marine cables when being installed on the seabed or being towed. The formulation describes elastic cable geometry in terms of two angles, elevation and azimuth, which are related to Cartesian co-ordinates by geometric compatibility relations. These relations are combined with the cable equilibrium equations to obtain a system of non-linear differential equations, which are numerically integrated by fourth and fifth order Runga–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 marine cable, the static analysis of a deep water mooring line and the response of a telecommunications cable to a multi-directional current profile. q 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 continuing to demand intensive use of cable and line structures in the marine environment. These include mooring lines for offshore structures together with flexible riser and umbilical catenaries between the seabed and surface vessel of floating oil production structures. A parallel requirement for the understanding of suspended cable span is in the installation process of the world’s extensive network of subsea optical fibre telecommunication cables. Now, as offshore suspended cable systems are being applied to even deeper waters, the effects of line elasticity and of sheared current profiles become more important. Deeper water suspended cable structures are more susceptible to the effects of currents to the extent that accurate analysis of their geometry becomes an important design requirement. Such analysis needs to predict the cable tension and displacement profile, its suspended length and the horizontal and vertical forces at the top end—the latter being an important element of the system’s performance. The behaviour of marine cables during their installation phase is a distinctly different problem both in the physics of * Corresponding author. Tel.: 155-21-560-8832; fax: 155-21-290-6626. E-mail address: [email protected] (M.A. Vaz)
the laying process and in the objectives of an analysis of the process. An important element of the physics of the installation process is the pay out of the cable and its sinking due to submerged weight during its transit to the seabed. At the same time, the analysis of the installation is concerned with the position of the cable’s touch down point (TDP) just as much as with the tension at the top of the cable suspended length. The work of many investigators into the cable installation problem has demonstrated that inaccuracy in cable TDP position is due to the presence of unknown ocean currents. Such currents are a combination of components including tide driven, local wind, Stoke’s wave drift, set up in shallow waters, storm surges and water density variations (see Ref. [1]). Ocean currents are known to have magnitude and direction varying in space and time although a depth dependent two-dimensional (2D) profile is often used as a design assumption. Analysis shows that uniform in-plane current changes the cable inclination and offsets the cable TDP, while non-uniform current whose direction changes with depth leads to a complex curvature of the cable. Burgess [2] has carried out several studies on the effect of current on marine cable installation and has conducted sea trials to evaluate the influence of sheared currents on cable deployment. A significant variation in the current profile was detected for different locations
0141-1187/00/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S0141-118 7(99)00023-1
46
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Fig. 1. Systems of co-ordinates.
and times. Use of measured current profiles on the numerical simulations improved results for the prediction of the TDP track. Many analysts have considered the problem of marine cable installation starting with the pioneering work of Zajac [3] who formulated the equations and developed a solution for the 2D stationary configuration of marine cables being laid. He also approximated the solution for the 3D problem by a perturbation technique. It was assumed that the cables were subjected to surface currents and zero bottom tension. A steady state formulation and solution for a cable-body system towed by a ship describing a circular path were presented by Choo and Casarella [4]. The elasticity of the cable was not included. Casarella and Parsons [5] and Choo and Casarella [6] presented a systematic and comprehensive review of early work on the simulation of cable-body systems. Pedersen [7] proposed a numerical solution using successive integrations to determine the 2D static configuration of cables and pipelines
during laying. Peyrot and Goulois [8] proposed a numerical solution for analyses of 3D assemblages of cables and substructures. 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 static cables suspended by two points and subjected to self-weight and hydrostatic forces only. Bending stiffness and fluid hydrodynamic actions are not considered. The cable elasticity may be approximately included in the formulae. Leonard and Karnoski [9] developed a numerical algorithm to simulate the stationary 3D cable deployment from a ship travelling at a constant speed and direction in sheared currents. The authors were particularly interested in investigating the configuration during passively controlled cable installation. The dynamic analysis of marine cables has also received much attention in the research literature with development of different algorithms to solve the problem—see for example [2,10–13]. A key feature of this technique is its
Fig. 2. Geometric compatibility.
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47
Fig. 3. Forces on cable infinitesimal element.
computational efficiency, making it possible to carry out calculations in real time or on board vessels. More recently, Pinto [14] has produced a stable formulation for cables with low or nearly zero tension. In parallel with this work, Vaz et al. [15] have produced an efficient formulation and numerical solution for the 3D behaviour of marine cable installation. The analysis presented in this paper is an extension of the latter work in two areas—the first being the inclusion of sheared currents to bring the efficient formulation to bear on this. At the same time, the methodology has been developed so that segmented elastic mooring lines or the flexible riser problem can also be efficiently solved in a unified technique. 2. Formulation of the governing equations The solution for the cable steady-state configuration encompasses the formulation of geometric and equilibrium relations. The governing equations are written for a local coordinate system where loads can be more easily calculated. This local system of reference varies its orientation with position 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), or sometimes called stationary, as it depends on relative velocities and involves dynamic 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 in the seabed and the axis OY oriented vertically upwards. The ship motion and current profile are defined relative to this system. Axes system Oxyz has the same orientation as the OXYZ system but with its origin at the cable TDP. A local frame of reference defines an individual cable element and is represented by the tangent, normal and binormal unit ~ respectively, of a cable segment. Transvectors ~t; n~ ; b; formation matrices relate these systems to each other. The
following relation is derived from Fig. 2:
8 9 8 9 2 ~I > > ~i > cos u cos c > > = 6 < > = > < > sin u J~ > > ~j > 6 4 > > ; :~> ; > :~> cos u sin c k K
2sin u cos c cos u 2sin u sin c
38 9 2sin c > > > ~t > 7< = 0 7 ~ n 5> > > ; :~> cos c b
1 ~ K ~ are the unit vectors of Oxyz and where ~i; ~j; k~ and ~I; J; OXYZ, respectively. The azimuth and elevation angles are, respectively, c c p and u u p and p is the stretched arc length along the cable. The relationship between the stretched and unstretched arc length s is dp 111 ds
2
where 1 is the uniaxial strain in the cable and the independent variables p and s are also called the material co-ordinates. Small deformations are assumed and flexure, shear and torsion effects are not considered. The position of a ~ c ; may be decomposed into (see Fig. 1) cable element, R ~ c p; t ~r 0 t 1 ~r c p R
3a
~ where ~r0 t V0 t~i and ~rc p X p~i 1 Y p~j 1 Z pk: The functions X p; Y p and Z p describe the cable geometric configuration viewed from an observer moving with the TDP, t is the time variable and V0 gives the constant velocity of the TDP. The velocity and acceleration of a ~ c are given by cable element, respectively, V~ c and A d ~ R p; t ~t V0 cos u cos c 2 Vpo V~ c p dt c ~ 0 sin c 2 n~ V0 sin u cos c 2 b V 3b 2 du dc 2 ~ ~ c p d R ~ c p; t Vpo ~ n 1 cos b u A dp dp dt2
3c
where Vpo is the cable pay out rate. The concept of total differentiation is needed because the cable is constantly paid
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Table 1 HA cable characteristics Weight in air/sea (N/m)
Cable diameter (m)
Drag coefficient
Inertia coefficient
26.49/17.76
0.0332
1.649
1.0
out and a formulation fixed in space is adopted. The geometric compatibility equations (Eqs. (A1a)–(A1c)) are used in the derivation of Eqs. (3b) and (3c).
2.2. Relations for cable equilibrium The equilibrium of external and internal forces in the infinitesimal element of stretched length dp is calculated next. Newton’s second law is invoked to account for the “centrifugal” force originated from the fact that the added cable travels at constant speed but it changes direction when in a curved configuration. The bending stiffness is assumed to be small—for cables subjected to high axial forces the moments associated with the curvature (i.e. the geometrical stiffness component) are higher than the internal moments resulting from the cable-section bending stiffness. The marine cable is assumed fully immersed in water and the cable rotational motions are not considered. The following forces act on the cable element of Fig. 3. ^ ~j self weight: 2wdp “effective” tension: 2T~t and T 1 dT ~t 1 d~t ^ rt uVrt udp~t tangential drag force: Dt 2 12 rCf pdV 1 ^ normal drag force: Dn 2 2 rCD dVrn uVrnb udp~n ^ rb uVrnb udpb~ binormal drag force: Db 2 12 rCD dV ~ inertia force: r^ c dpAc pd^ 2 ~ dpAc hydrodynamic added inertia: rCm 4 where w^ is the cable weight in sea water per unit stretched length, r the sea water density, Cf the friction coefficient, CD the drag coefficient, d^ the cable diameter after the cable stretches, Vrn ; Vrb ; Vrt ; uVrnb u are relative velocities defined in Appendix B, r^ c is the cable physical mass per unit ~c stretched length Cm is the added mass coefficient and A is the cable acceleration. It is assumed that both the cable mass and material density remain invariant during its ^ stretching. p Hence r^ c rc ; w^ w= 1 1 1 and d d= 1 1 1 where rc ; w and d are the cable’s properties before stretching. Furthermore, the cable is assumed to be continuous and extensible with a linear elastic stress–strain relationship given by Hooke’s Law, i.e. s E1; where s is the normal stress, E is the Young’s Modulus and 1 is the unit elonga where A is the tion. The axial stress is also given by s T=A cable’s cross-sectional area and T is the “actual” tension. The “effective” tension T is related to T by T T 2 rgAh 2 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 co-ordinate s results in p dT 1 2 w sin u 2 rCf pd 1 1 1Vrt uVrt u 0 ds 2 T2 2 T2
t
4a
n
4b
b
4c
r du 2 Vpo 111 ds p 1 rCD d 1 1 1Vrn uVrnb u 2 w cos u 0 2 r dc 2 cos u Vpo 111 ds
p 1 rCD d 1 1 1Vrb uVrnb u 0 2 pd2 : where r rc 1 rCm 4 2
3. Numerical solution The solution is obtained by solving a system of seven non-linear first-order ordinary non-linear differential equations. Eqs. (2), (4a) and (A1a)–(A1c) are written in the Cauchy 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. elevation and azimuth angles, tension and three Cartesian coordinates. Note that the numerical scheme allows solution for segmented lines, i.e. cables with different geometrical and material properties. This is specifically important, for instance, when analysing mooring lines composed of segments of steel chains and fibre ropes. For segmented lines, the end conditions of the lower cable section are the initial conditions of the upper cable section. In static analyses the sea bottom may have a slope. The convergence is fast and the method is very robust.
4. Results Three computations are presented to illustrate the solution method and the main type of analyses. The first deals with the installation of a telecommunication cable under a transverse sheared current, whereas the second example simulates the static behaviour of a segmented mooring line. A third example considers the effect on a telecommunication cable of a sheared current in mutually perpendicular vertical planes.
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Fig. 4. (a) XY projection; (b) ZY projection; (c) elevaton angle; (d) Azimuth angle; (e) effective tension distribution; (f) distribution of cable elongation; and (g) suspended length and TDP radius.
4.1. Installation of telecommunication cable in transverse current The installation of a heavy armoured (HA) optical fibre cable is used in this case. The cable properties are chosen to
be the same as those used by Hopland [16] and are described in Table 1. An axial stiffness EA 1:0 × 108 N is assumed. The equation of the laying ship’s track is taken as ~r0 t S0 1 V0 t~i
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Fig. 4 (continued).
where V0 is the ship speed, t time and S0 is the initial distance between the axes systems OXYZ and oxyz. S0 may be taken as zero without losing the generality of the solution. The idea behind this simulation is to evaluate the effect of transverse marine current (relative to the ship track) on the configuration of a fibre-optic cable during installation. Tangential drag forces are neglected and the cable is assumed to be uniform. The marine current profile is ~ 1:0 Y=h 2 0:5K ~ where h 1000 m is taken to be U ~ is the unit vector in the Z direction. the water depth and K This current profile is a function of depth only, which makes the problem solution stationary. In other words, for any point in a plane parallel to OXZ, the current profile remains the same function of Y. As the ship moves, the cable encounters the same depth dependent current although this current may have components in both the OX and OZ directions. The laying vessel is assumed to move at a constant speed of 1.5 m/s, which is also the pay out rate. The current profile varies linearly from 0.5 m/s at the sea level to 20.5 m/s at the seafloor. There is no current velocity component in the direction of the vessel motion. Results are presented for four levels of pre-tension at the cable TDP. Fig. 4a–g presents the results of such an analysis. In each of these figures, t0 is the cable pre-tension at the lower end.
These figures are obtained by integrating the governing equation from the TDP upwards so that the initial (boundary) conditions at the TDP need to be known. Fig. 4a and b presents the cable configuration (which is constant in time) as an observer would view it from the TDP. Fig. 4a, the XY projection, shows the in-plane frame moving from a straight line configuration for low tension—as predicted by Zajac [3]—to a catenary-like shape for higher pre-tensions. Fig. 4b, the ZY projection, shows the out-of-phase projection of the cable configuration. Lower pre-tension increases the cable curvature. If there was no transverse current there would be no ZY projection. Fig. 4c and d, respectively, show the distribution of elevation and azimuth angles as a function of depth. Higher the pre-tension, the higher is the variation of elevation and smaller is the variation of azimuth. For low pre-tension, the solution approaches Zajac’s result of a constant angle (straight line). Near the TDP, the cable develops large elevation gradients. The azimuth distribution also indicates a progressive shift from the straight-line cable laying with a constant angle to a nearly constant azimuth as the pre-tension (and stiffness) increases. Fig. 4e shows the linear distribution of tension in the cable with tangential drag forces neglected. Maximum and minimum tensions occur at the cable’s upper and lower ends, respectively. The actual cable tension is lower than the effective tension except at the sea surface where they are equal. Finally Fig. 4f presents the cable elongation, i.e. the amount of cable stretched when axial stiffness is included. The cable elasticity is only noticeable when there is a significant pre-tension. The larger elongations are found at the sea surface, as expected. Fig. 4g shows the cable suspended length and the horizontal distance from the TDP to the upper point (TDP radius). TDP radius is the horizontal distance between the TPD and the cable upper end. As in a mooring line, the higher the pre-tension, the higher is the offset and the suspended length as well as the stretched length and the TDP radius. 4.2. Mooring line Results are presented for a typical segmented mooring line installed in deep waters at offshore Campos Basin,
Table 2 Properties of line segments
D (m) w (m) EA (MN) Breaking load (kN) Total length (m) Drag coefficient CD Inertia coefficient Cm
Lower segment chain
Intermediate segment wire rope
Upper segment chain
0.095 1729.1 793.88 6930 1240 3.2 1.6
0.109 420.8 536.79 7160 1250 1.8 1.0
0.084 1363.8 620.68 7230 150 3.2 1.6
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Fig. 5b shows the percentage gain or loss of horizontal load for the mooring line in the presence of currents. Results are compared with the case of no currents. In the operating range, the line experiences decreasing influence of the marine currents as the offset is increased and the line is stretched. For deeper water, the role of marine currents on the mooring line is expected to be even greater. 4.3. Telecommunication cable in 3D current profile This example uses the same cable as in Section 4.1, but imposes a more complex 3D current profile given by the equation ~ u cos 2pY ~I 1 u sin 2pY K ~ U h h
Fig. 5. (a) Mooring line stiffness; and (b) reduced stiffness due to current.
Brazil. This line consists of bottom and upper chain segments and an intermediate wire rope segment. Their material and geometrical properties are described in Table 2. 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. Fig. 5a presents the restoring capacity of a single mooring line in terms of the offset, this being defined as the “horizontal” distance between the anchor and the upper end point. Results are plotted for in-line constant profile marine currents of ^1.0 m/s and for no current. The line exhibits its typical progressively hardening restoring characteristic. The line design optimisation reveals two issues: (1) the line breaking point occurs just after the maximum operating condition which is “initial offset plus 10% water depth”. Also, the line tends to break before the anchor lifts. Both situations represent loss of restoring capacity. (2) The point at which the wire rope touches the seafloor is away from the other operating extreme condition given by “initial offset minus 10% water depth”. Wire ropes are not very resistant to abrasion and contact with seabed soil should be avoided.
where h 1000 m is the water depth and u is taken to have values of 1.5, 0.75 m/s and zero (no current). The Z coordinate of the top end is set to zero and only one level of pretension at the cable TDP is computed. Fig. 6a–f presents the results of this analysis. The projections of the cable configuration are shown in Fig. 6a and b. In the absence of currents, the cable geometrical configuration entirely lies in the XY plane with a perfect catenary shape. Sheared currents displace the cable laterally and this is more noticeable with the higher magnitude of the current. The elevation and azimuth angles are shown in Fig. 6c and d, respectively, as functions of the vertical co-ordinate. These figures illustrate how difficult it is, for example, to extrapolate the cable configuration underwater with the knowledge of the top angles only. Fig. 6e shows the linear variation of cable effective tension with water depth. Fluid velocity could only affect tension distribution if tangential drag forces were considered (Eq. (4a)). Fig. 6f presents the distribution of cable elongation. The higher the magnitude of the current, the longer are the suspended lengths, as are the cable elongations. The stretched suspended cable lengths for u 1:5; 0.75 m/s and no current are equal to 4782.7, 4156.8 and 3840.9 m, respectively. 5. Conclusions A 3D steady-state formulation for elastic segmented marine cables during installation in sheared currents is presented. The time independent set of first-order non-linear ordinary differential equations are solved by a Runge–Kutta integrator. Case studies are presented for three typical situations: the installation of a heavy armoured fibre optical telecommunication cable in the presence of transverse and 3D currents, and the static analysis of a segmented deep water mooring line subjected to in-line current and employed offshore Brazil. The results presented demonstrate the importance of
52
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
Fig. 6. (a) XY projection; (b) ZY projection; (c) elevaton angle; (d) Azimuth angle; (e) effective tension distribution and (f) distribution of cable elongation.
current profile, both in terms of magnitude and direction, for prediction of suspended line behaviour. In particular, the analyses demonstrate the uncertainty of predicting line configurations by using only the top angles and also the difficulty of accurately predicting TDP position without good current information.
The 3D steady-state model developed in this paper is comprehensive as it includes the effects of sheared currents, cable elasticity, segmented cables and laying vessel speed, allowing an accurate analysis of the installation of marine cables subjected to time independent environmental actions. The numerical model is also robust and computationally
M.A. Vaz, M.H. Patel / Applied Ocean Research 22 (2000) 45–53
efficient demanding little CPU time. It can, therefore, be used in real time or on board installation or offshore vessels requiring only reliable current data for operationally effective prediction of underwater cable or line behaviour. Acknowledgements
Appendix A. Equations of geometric compatibility The following relationships are derived from Fig. 2: dX 1 dX cos c cos u dp 1 1 1 ds
A1a
dY 1 dY sin u dp 1 1 1 ds
A1b
dZ 1 dZ sin c cos u dp 1 1 1 ds
A1c
Then dX 2 dY 2 dZ 2 1 1 1 dp dp dp
and dX 2 dY 2 dZ 2 1 1 1 1 12 : ds ds ds
Appendix B. Relative velocity fluid/structure The sheared current velocity is a function of depth only and is given by ~ ~ UX Y~I 1 UZ YK U
~ is given by Hence the relative velocity V~ r V~ c 2 U 8 ~tV0 2 Ux Y cos u cos c 2 Vpo 2 Uz Y cos u sin c1 > > > < ~ V~ r 1n2V : 0 2 Ux Y sin u cos c 1 Uz Y sin u sin c1 > > > : ~ 2bV0 2 Ux Y sin c 1 Uz Y cos c
B3
This work was carried out with the support of the facilities of The Laboratory for Subsea Technology, Federal University of Rio de Janeiro and The Santa Fe Laboratory for Offshore Engineering, University College London. The first author acknowledges the support of the Brazilian Council of Research (CNPq). Data on the mooring line case considered here were kindly supplied by CENPES/Petrobras, Brazil.
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B1
There is no vertical component of current. In terms of the local system, the current profile is 8 ~tUx Y cos u cos c 1 Uz Y cos u sin c1 > > < ~ 2~nUx Y sin u cos c 1 Uz Y sin u sin c1 : B2 U > > : ~ x Y sin c 1 Uz Y cos c 2bU
And the magnitude of the radial relative velocity is uVrnb u p 2: Vrn2 1 Vrb
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