Dynamic behavior of offshore spar platforms under regular ...eprint.iitd.ac.in/bitstream/2074/1143/1/agarwaldyn2003.pdf · Dynamic behavior of offshore spar platforms under regular

Dynamic behavior of offshore spar platformsunder regular sea waves

A.K. Agarwal, A.K. Jain *Department of Civil Engineering, Indian Institute of Technology, Hauz Khas, New Delhi-110016, India

Received 10 October 2001; accepted 14 January 2002

Abstract

Many innovative floating offshore structures have been proposed for cost effectiveness ofoil and gas exploration and production in water depths exceeding one thousand meters in recentyears. One such type of platform is the offshore floating Spar platform. The Spar platform ismodelled as a rigid body with six degrees-of-freedom, connected to the sea floor by multi-component catenary mooring lines, which are attached to the Spar platform at the fairleads.The response dependent stiffness matrix consists of two parts (a) the hydrostatics providerestoring force in heave, roll and pitch, (b) the mooring lines provide the restoring force whichare represented here by nonlinear horizontal springs. A unidirectional regular wave model isused for computing the incident wave kinematics by Airy's wave theory and force by Mori-son's equation. The response analysis is performed in time domain to solve the dynamicbehavior of the moored Spar platform as an integrated system using the iterative incrementalNewmark's Beta approach. Numerical studies are conducted for sea state conditions with andwithout coupling of degrees-of-freedom.

Keywords: Wave structure interaction; Offshore structural dynamics; Spar platform; Multi-componentcatenary mooring

488 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) 487-516

1. Introduction

As offshore oil and gas exploration are pushed into deeper and deeper water, manyinnovative floating offshore structures are being proposed for cost savings. To reducewave induced motion, the natural frequency of these newly proposed offshore struc-tures are designed to be far away from the peak frequency of the force power spectra.Spar platforms are one such compliant offshore floating structure used for deep waterapplications for the drilling, production, processing, storage and offloading of oceandeposits. It is being considered as the next generation of deep water offshore struc-tures by many oil companies. It consists of a vertical cylinder, which floats verticallyin the water. The structure floats so deep in the water that the wave action at thesurface is dampened by the counter balance effect of the structure weight. Fin likestructures called strakes, attached in a helical fashion around the exterior of thecylinder, act to break the water flow against the structure, further enhancing thestability. Station keeping is provided by lateral, multi-component catenary anchorlines attached to the hull near its center of pitch for low dynamic loading. Theanalysis, design and operation of Spar platform turn out to be a difficult job, primarilybecause of the uncertainties associated with the specification of the environmentalloads. The present generation of Spar platform has the following features:

(a) It can be operated till 3000 Mts. depth of water from full drilling and productionto production only,

(b) It can have a large range of topside payloads,(c) Rigid steel production risers are supported in the center well by separate buoy-

ancy cans,(d) It is always stable because center of buoyancy (CB) is above the center of grav-

ity (CG),(e) It has favourable motions compared to other floating structures,(f) It can have a steel or concrete hull,(g) It has minimum hull /deck interface,(h) Oil can be stored at low marginal cost,(i) It has sea keeping characteristics superior to all other mobile drilling units,(j) It can be used as a mobile drilling rig,(k) The mooring system is easy to install, operate and relocate,(l) The risers, which normally take a breathing in the wave zone from high waves

on semi-submersible, drilling units would be protected inside the Spar platform.Sea motion inside the Spar platform center well would be minimal.

The concept of a Spar platform as an offshore structure is not new. Spar platformbuoy type structures have been built in the ocean before. For example, a floatinginstrument platform (FLIP) was built in 1961 to perform oceanographic research(Fisher and Spiess, 1963), the Brent Spar platform was built by Royal Dutch shellas a storage and offloading platform in the North sea at intermediate water depth(Bax and de Werk, 1974, Van Santen and de Werk, 1976; Glanville et al., 1997).The use of Spar platforms as a production platform is relatively recent.

A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) 487-516 489

Glanville et al. (1991) gave the details of the concept, construction and installationof a Spar platform. He concluded that a Spar platform allows flexibility in the selec-tion of well systems and drilling strategies, including early production or predrill-ing programs.

Mekha et al. (1995) modelled the Spar platform with 3 degrees-of-freedom, i.e.surge, heave and pitch. The inertia forces were calculated using a constant inertiacoefficient, Cm, as in the standard Morison's equation or using a frequency dependentCm coefficient based on the diffraction theory. The drag forces were computed usingthe nonlinear term of Morison's equation in both cases. The analysis was performedin time domain. The result showed that using frequency dependent or constant inertiacoefficient, Cm, produces similar results since most of the wave energy is concen-trated over the range of frequencies where the value of Cm is equal to 2 for the Sparplatform size used in the literature.

Mekha et al. (1996) used the same model (Mekha et al., 1995) with frequencydependent Cm coefficient based on diffraction theory. Different nonlinear modifi-cations to Morison's equation were induced to account diffraction effects. The resultsobtained for a variety of sea state conditions were compared to the experimental data.

Halkyard (1996) reviewed the status of several Spar platform concepts emphasiz-ing the design aspect of these platforms.

Cao and Zhang (1996) discussed an efficient methodology to predict slow driftresponse of slack moored slender offshore structures due to ocean waves using ahybrid wave model. The hybrid wave model considers the wave interactions in anirregular wave field up to second order of wave steepness and is able to accuratelypredict incident wave kinematics, including the contributions from nonlinear differ-ence frequency interactions. A unique feature of this approach is that measured waveelevation time series can be used as input and the structure responses to measuredincident waves can be deterministically obtained.

Ran and Kim (1996) studied the nonlinear response characteristics of atethered/moored Spar platform in regular and irregular waves. A time-domaincoupled nonlinear motion analysis computer program was developed to solve boththe static and dynamic behaviours of a moored compliant platform as an integratedsystem. In particular, an efficient global-coordinate based dynamic finite elementprogram was developed to simulate the nonlinear tether/mooring responses. Usingthis program, the coupled dynamic analysis results were obtained and they werecompared with uncoupled analysis results to see the effects of tethers and mooringlines on hull motions and vice versa.

Jha et al. (1997) compared the analytical predicted motions of a floating Sparbuoy platform with the results of wave tank experiments considering, surge and pitchmotions only. Base-case predictions combine nonlinear diffraction loads and a linear,multi-degree-of-freedom model of the Spar platform stiffness and damping character-istics, refined models and the effect of wave-drift damping, and of viscous forces aswell. Consistent choices of damping and wave input were considered in some detail.

Fischer and Gopalkrishnan (1998) presented the importance of heave character-istics of Spar platforms that have been gleaned from wave basin model tests, numeri-cal simulations and a combination of the two. The heave performance of a small

490 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) 487-516

Spar platform, e.g. mini Spar platform, has been examined and found to be poten-tially problematic.

Chitrapu et al. (1998) studied the nonlinear response of a Spar platform underdifferent environmental conditions such as regular, bi-chromatic, random waves andcurrent using a time domain simulation model. The model can consider several non-linear effects. Hydrodynamic forces and moments were computed using the Mori-son's equation. It was concluded that Morison's equation combined with accurateprediction of wave particle kinematics and force calculations in the displaced positionof the platform gave a reliable prediction of platform response both in wave-frequency and low-frequency range.

Nonlinear coupled response of a moored Spar platform in random waves with andwithout co-linear current were investigated in both time and frequency domain (Ranet al., 1999). The first and second order wave forces, added mass, radiation dampingand wave drift damping were calculated from a hydrodynamic software packagecalled WINTCOL. The total wave force time series (or spectra) were then generatedin the time (or frequency) domain based on a two-term volterra series method. Themooring dynamics were solved using the software package WINPOST, which isbased on a generalised coordinate based FEM. The mooring lines were attached tothe platform through linear and rotational springs and dampers. Various boundaryconditions can be modelled using proper spring and damping values. In the timedomain analysis, the nonlinear drag forces on the hull and mooring lines were appliedat the instantaneous position. In the frequency domain analysis, nonlinear drag forceswere stochastically linearized and solutions were obtained by an iterative procedure.

Ye et al. (1998) studied the Spar platform response in directional wave environ-ment using the unidirectional hybrid wave model and directional hybrid wave model(UHWM and DHWM). Comparison between numerical results from two differentwave models indicated that the slow drifting surge and pitch motions based onDHWM are slightly smaller than those based on UHWM. The slow-drifting heavemotions from the two wave models were almost the same because the heave motionwas mainly excited by the pressure applied on the structure bottom and the predictedbottom pressure from the two methods had almost no differences.

Datta et al. (1999) described recent comparisons of numerical predictions ofmotions and loads of typical truss Spar platform model test results. The purpose ofthis comparison was to calibrate hydrodynamic coefficients, which were to be usedfor the design of a new truss Spar platform for Amoco.

Chitrapu et al. (1999) discussed the motion response of a large diameter Sparplatform in long crested and random directional waves and current using a time-domain simulation model. Several nonlinearities such as the free surface force calcu-lation, displaced position force computation, nonlinearities in the equations of motionand the effect of wave-current interaction were considered in determining the motionresponse. The effect of wave directionality on the predicted surge and pitch responseof the Spar platform had been studied. It was seen that both wave-current interactionand directional spread of wave energy had a significant effect on the predictedresponse.

A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) 487-516 491

2. Structural model

The Spar platform is modelled as a rigid cylinder with six degrees-of-freedom(i.e. three displacement degrees-of-freedom i.e. Surge, Sway and Heave along X, Yand Z axis and three rotational degree-of-freedom i.e. Roll, Pitch and Yaw about X,Y and Z axis) at its center of gravity, CG. The Spar platform is assumed to be closedat its keel. The stability and stiffness is provided by a number of mooring linesattached near the center of gravity for low dynamic positioning of the Spar platforms.When the platform deflects the movement will take place in a plane of symmetryof the mooring system, the resultant horizontal force will also occur in this planeand the behavior will be 2-dimensional. It is the force and displacement (excursion)at this attachment point that is of fundamental importance for the overall analysisof the platforms. It is assumed that the Spar platform is connected to the sea floorby four multi component catenary mooring lines placed perpendicular to each other,which are attached to the Spar platform at the fairleads. The development of Sparplatform model for dynamic analysis involves the formulation of a nonlinear stiffnessmatrix considering mooring line tension fluctuations due to variable buoyancy andother nonlinearities. The model considers the coupled behaviour of a Spar platformfor various degrees-of-freedom. Fig. 1 shows a typical offshore Spar platform.

WLLINO RIG

HEU-POftT-

MOORING TQJIPUENT O A T -

WATK SURFACE

ACCESS \ K M TO TOPSIDESAND BOAr UNHH3

muw ma -JJJIK

Fig. 1. Typical offshore Spar platform.


3. Assumptions and structural idealization

The platform and the mooring lines are treated as a single system and the analysisis carried out for the six degrees-of-freedom under the environmental loads. Thefollowing assumptions have been made in the analysis:

1. Initial pretension in all mooring lines is equal. However, total pretension changeswith the motion of the Spar platform,

2. Wave forces are estimated at the instantaneous equilibrium position of the Sparplatform by Morison's equation using Airy's linear wave theory. The wave dif-fraction effects have been neglected,

3. Integration of fluid inertia and drag forces are carried out up to the actual level ofsubmergence according to the stretching modifications considered in the analysis,

4. Wave force coefficients, Cd and Cm are independent of frequencies as well asconstant over the water depth,

5. Current velocity has not been considered and also the interaction of wave andcurrent has been ignored,

6. Wind forces have been neglected,7. Change in pretension in mooring line is calculated at each time step, and writing

the equation of equilibrium at that time step modifies the elements of the stiff-ness matrix,

8. The platform is considered as a rigid body having six degrees-of-freedom,9. Platform has been considered symmetrical along surge axis. Directionality of wave

approach to the structure has been ignored in the analysis and only uni-directionalwave train is considered,

10. The damping matrix has been assumed to be mass and stiffness proportional,based on the initial values.

4. Catenary mooring line analysis

Some of the assumptions made for the analysis of catenary mooring line are:

(a) The sea floor (having negligible slope) offers a rigid and frictionless support tothe mooring line, which is lying on it,

(b) All the components of the mooring line move very slowly inside the water,so that the generated drag forces on the line due to the motion can be treatedas negligible,

(c) The change in the line geometry and thereby in the line force due to direct fluidloading caused from waves and /or currents is insignificant,

(d) Initial length of the mooring line and anchor line is inclusive of elongation dueto the initial line force,

(e) The clump weight segment is inextensible,(f) Anchor point does not move in any direction,(g) Only horizontal excursion of the catenary mooring line is considered.


The force-excursion relationship is nonlinear and requires an iterative solution.Equation of a catenary was used for evaluation of force-excursion relationship of acatenary mooring line. The horizontal projection and vertical projection of any seg-ment hanging freely under its own weight w per unit length as shown in Fig. 2 canbe expressed (considering horizontal force (Ht), top slope (qt), length (S) and weight(W)) as:

Y = (H,/fF)[cosh{sinh-1(tan(0,))}-cosh{sinh-1(tan(06))}] (1)

tan(qb) = (V-WS)IHt (3)

When for any segment the bottom slope (qb) is zero, the above equation reduces to:

Y = (Ht/ HOtcosMsinh-^tanCG,))} - 1 ] (4)

(5)

(a) Vo

Clump weightWcI , Sti

Anchor line

Fig. 2. Multi component mooring line. (a) Initial configuration with different sectional properties; (b)Free body diagram of uniform mooring line suspended freely between two points not in the same elevation.


If Ht, Y,W ,qt are known then,

tan(qb) = sinh[cosh-1[cosh{sdnh-1(tan(0,))}-(yfT//r,)]] (6)

S = flXtan(0,)-tan(06))/ W (7)

and X can be evaluated by Eq. (2).The extension of any segment under increased line tension can be approximately

evaluated as follows. Let the initial average line tension be To when the segmentlength is So For increased average line tension T, the stretched length becomes:

S = So[1 + (T-TO)IEA\ (8)

where, E and A are the Young's modulus and effective area of the segment respect-ively. T and To is the arithmetic means of the line tensions at two ends. The totalweight of any segment (W) remains same,

W=(SOWO)/S (9)

where W is the modified unit weight due to stretching and Wo is the unit weight ofthe unstreched segment.

5. Analysis of mooring line with distributed clump weight for (horizontalexcursion)

In the present work Ho, qo, h, zero bottom slope and the elastic and physicalproperties of the segments, as shown in Fig. 3(a), are chosen as the known para-meters. The unknowns which are to be evaluated are Sc, Sh and then Yc, Yh ,Xc

6. Initial configuration

The following steps are used to find the unknowns given above.

Step 1 Calculate Vo from the known values of Ho and qo.Step 2 Find the slope at the junction of the clump weight and mooring line, then

find vertical force at the junction Vj (which will be equal to ShWcl as thebottom slope is equal to zero). Using the known values of the horizontalforce, use Eq. (4) to find Yh.

Step 3 Find Sc = (Vo-ShWcl)/Wc and then find Yc using Eq. (1).Step 4 Add up Yc and Yh and compare with h. If the difference is less than a specified

limit, go to the next step, otherwise change Vj appropriately and repeat theprocedure from step 2. For the first iteration the change of Vj can be takenas ± 1% depending upon the sign of error. For the subsequent iterations thefollowing equation is to be used to get a new value of Vj.


(a)

he sh

(b)

scl"shn

Fig. 3. Configuration of mooring line for increased horizontal force, H (Condition 1). (a) Initial con-figuration, (b) Final configuration.

where, k is the number of the last iteration, (Vj)k is the vertical force at thejunction of the mooring line and clump weight and ek is the differencebetween the vertical projection of the hanging length of the mooring linecalculated in the kth iteration and the mooring level (h).

Step 5 Find Xc and Xh from Eqs. (2) and (5), respectively.Step 6 Find initial total hanging length Si = Sc + Sh and its horizontal projection

Xi = Xc + Xh.

7. Evaluation of force-excursion (horizontal) relationship for a singlemooring line

The vertical force (Vo) is changed which allows direct checking of the conditionregarding the lifting off of the clump weight. The corresponding Horizontal force is


found iteratively, ending with the determination of a new configuration includingthe excursion of the attachment point. The procedures for the two alternative statesof lifting off of the clump weight are given below. Wc = ScWc is the weight of themooring line and Wcl = SclWcl is the total weight of the clump weight. Find theinitial tension in the mooring line as follows:

7.1. Condition 1 (when V<WC + Wcl)

Step 1 Increase Vo by AFFig. 3(b).Step 2 Find the vertical and horizontal force at the junction of the clump weight

and mooring line. Find the new average line tension, T in the mooring line.Step 3 Find the stretched length of mooring line, Scn and hence modified Wc using

eqs. (8) and (9), respectively.Step 4 Find the vertical projections of mooring line, (Ycn) and clump weight (Yhn)

with the help of eqs. (1) and (4), respectively.Step 5 Add up Ycn and Yhn to compare with h. If the difference e is less than the

tolerable limit go to the next step. Otherwise, change horizontal force Happropriately and repeat the procedure from step 2. For the first iteration thechange of H can be taken as ± 1% depending upon the sign of error. Forthe subsequent iterations the following equation is to be used to get a newvalue ofH,

Hk+1= Hk-[ek(Hk^-Hk)l(ek^-ek)\ (10)

where k is the number of the last iteration, Hk is the horizontal force and ek

is the difference between the vertical projections of the hanging length ofthe mooring line calculated in the ML iteration and the mooring level (h).

Step 6 Find the new hanging length of the clump weight Shn from the zero bottomslope condition and add it with the stretched mooring line length Scn to getthe new length, Sf.

Step 7 Find Xcn and Xhn from eqs. (2) and (5), respectively and add them to get thehorizontal projection Xf.

Step 8 Find the elastic stretch of anchor line, ea.

ea = Sa(H-Ho)l(EAa) (11)

where Ea is Young's modulus of the anchor line; Aa is effective area of theanchor line; Ho is initial horizontal force at the top; H is final horizontalforce at the top; and, Sa is length of anchor line.

Step 9 Evaluate horizontal excursion (d)

S = {Xf-X^-{Sf-S^ + (ea + ec) (12)

where Si, Xi is initial hanging length of the mooring line and its horizontalprojection, respectively; Sf,Xfis final stretched hanging length of the mooringline and its horizontal projection, respectively; ec is elastic stretch in the


mooring line from eq. (8); and, ea is elastic stretch in the anchor line fromeq. (11).

Step 10 Repeat steps 1 to 9 for the increased values of Vo till V = Wc + Wcl

7.2. Condition 2 (when V > Wc + Wc

Step 1 Increase V o by A Vas shown in Fig. 4(b)Step 2 Find the vertical and horizontal force at the junction of the clump weight

and mooring line. For the value of (V = Vo + AV) find the vertical force

(a)

(b)

(c)

Fig. 4. Configuration of mooring line for increased horizontal force, H (Condition 2). (a) Configurationwhen the far end of the clump just lifts off; (b) Final configuration; (c) Free body diagram of theanchor line.


Va at the junction of the anchor line and the clump weight as shown in Fig.4(c). Find the new average line tension in the mooring line and the anchorline and hence the modified Wc and Wa (considering the system configurationwhen the clump weight is just lifted as shown in Fig. 4(a) as the initialconfiguration).

Step 3 Find Ycn and Ycln using eq. (1) and Yah from eq. (4). Add the projections andcompare with h. If the difference is within tolerable limit go to step 4. Other-wise, change H appropriately and repeat the procedure from step 2 (as instep 5 for condition 1).

Step 4 Find the hanging length of anchor line, Sah using eq. (7) with qb = 0. Addup Scn and Scl to get Sf (which includes the mooring line stretch).

Step 5 Find Xcn and Xcln using eq. (2) and Xah from eq. (5). Add Xcn to Xcl to get Xf.Step 6 Find the elastic stretch of anchor line

(13)

where H' = {H + ^(H2 + ^ } / 2 ; Va = V-Wc-Wcl; Ho = initial horizontalforce when V = Wc + Wcl; H' = increased horizontal force due to increasedvertical force at the top. Wc and Wcl being the total weights of mooring lineand clump weight, respectively.

Step 7 Find the excursion from

8 = 8' + QCf-X't)-(Sf-S't) + (Xah-S'ah) + (e'a + e'o) (14)

Corresponding to changed value of the horizontal force H. Where 8' = excur-sion when the clump weight lifts off; £",, X\ = total hanging stretched lengthand its horizontal projection, respectively when the clump weight just liftsoff; X ah—S' ah = horizontal movement of the clump weight-anchor line junc-tion; e'c = elastic stretch in the mooring line after the clump weight lifts off;and, e'a = elastic stretch in the anchor line.

Step 8 Repeat steps 1 to 7 for increasing values of Vo, till the tension equals thepermissible value (approximately half of the breaking strength of mooringline material). In the above Sah and Xah are found by treating the anchor lineas a freely hanging mooring line and making use of the modified unit weightfor the stretched segment. If Sah is found more than the total (stretch) anchorline length calculate tan(qb) = (Va— Wa)/H, Wa being the total weight ofanchor line and recalculate Sah. Xah is to be evaluated using eq. (2).

The behaviour of mooring system will be planer if the tower excursion takes placein a plane of symmetry of the mooring system. For an excursion of d at the attach-ment point the resultant horizontal force is given by:

H(d) = 2 H/Sjlcosfr-Oj) (15)

wherep is the total number of mooring lines, qj is the angle between thejth mooringline and the direction of excursion. dj is the excursion for the jth mooring line and


Hj(dj) is the associated horizontal force, with dj = Scos(n—6j). The vertical force atthe mooring attachment point will be:

V{5)= (16)

In this study the mooring line is modelled as a nonlinear horizontal spring locatedat the fairleads along the Spar platform center with no hydrodynamic forces appliedon them. The stiffness matrix representing the mooring line is calculated based oneqs. (15) and (16). The stiffness matrix is given below:

mooring(horizontal )

hs hs hs hs hs•ft-ll ^M2 ^M3 ^M4 ^M5

hs hs hs hs hsKhs21 A 2 2 hsK23 hsKhs24 K 2 5 hsK

hs hs hs hs hs hsKhs31 A 3 2 hsK33 hsKhs34 K 3 5 h s

hs hs hs hs hs hsKhs41 A 4 2 hsK43 hsKhs44 A 4 5 KL

hs hs hs hsA51 A j 2 A53 Khs

K53

hs

hs54

hs hs hs

(17)

Khs11 is the sum of the horizontal component of force (surge) of mooring lines forunit displacement along surge direction, Khs21 will be zero as the mooring line placedperpendicular to the surge direction (sway) will not contribute any force, becausethe behavior of the mooring system is planer when Spar platform excursion takesplace in the plane of symmetry, Khs31 is the sum of the vertical component of force(heave) of mooring lines, Khs41 (roll) is zero as there is no force in the sway direction,Khs51 (pitch) is the sum of the moment of horizontal component of force (surge) ofmooring lines about centre of gravity, Khs61 (yaw) will be zero, as no mooring linewill contribute to any force in this direction.

Khs12 will be zero for unit displacement in sway direction as the mooring linesplaced perpendicular to the sway direction (surge) will not contribute any force,because the behavior of the mooring system is planer when Spar platform excursiontakes place in the plane of symmetry, Ki^ is the sum of the horizontal componentof force (sway) of mooring lines, Kh

s32 is the sum of the vertical component of force(heave) of mooring lines, AJl (roll) is the sum of the moment of horizontal compo-nent of force (sway) of mooring lines about centre of gravity, Khs52 (pitch) is zero asthere is no force in the surge direction, Khs62 (yaw) will be zero as no mooring linewill contribute any force in this direction.

K13hs (surge), Khs23 (sway), K33 (heave) will be zero for unit displacement along heave

as there is no horizontal movement of mooring line so there will be no force, Khs(roll), Khs53 (pitch) and K%, (yaw) will be zero because there is no force in this direc-tion.

Khs14 (surge) will be zero for unit rotation in roll as the behaviour of the mooringsystem is planer, Khs24 is the horizontal component of force (sway) of the mooringline present in the sway direction, Khs34 is the vertical component of force (heave) of


the mooring line present in the sway direction, K\\ (roll) is the sum of the momentof horizontal component of force in sway direction about centre of gravity, Khs54

(pitch) will be zero because there is no force in surge direction, Khs64 (yaw) will bezero, as no mooring line will contribute any force in this direction.

s15 will be the horizontal component of force (surge) of the mooring line presentin the surge direction for unit rotation in pitch, Afl (sway) will be zero as the behav-iour of the mooring system is planer, Khs35 is the vertical component of force (heave)of the mooring line present in the surge direction, Khs45 (roll) will be zero as there isno force in sway direction, A«j| (pitch) is the sum of the moment of horizontal compo-nent of force in surge direction about centre of gravity, Khs65 (yaw) will be zero, asno mooring line will contribute any force in this direction.

Khs16 (surge) and Khs26 (sway) will be zero because opposite direction force willnullify for unit rotation in yaw, Khs36 is the sum of vertical component of the forces(heave) of the mooring lines, Khs46 (roll) and K56hs (pitch) will be zero because thereis no force in this direction, Khs66 (yaw) is the sum of the moment of the horizontalcomponent of force about centre of gravity.

8. Equation of motion of spar platform

The equation of motion of the Spar platform under regular wave is given below:

[M\{X} + [Q{X} + [K]{X} = {F(t)} (18)

where {X} = structural displacement vector; {X} = structural velocity vector; {X} =structural acceleration vector; M = MSparplatform + Maddedmass; K = Khydrostatic +Kmooring(horizontal); C = s t r u c t u r e damping matrix; F(t) is the hydrodynamic forcing

vector.The mass matrix represents the total mass of the Spar platform including the mass

of the soft tanks, hard tanks, deck, ballast and the entrapped water. The added massmatrix is obtained by integrating the added mass term of Morison's equation alongthe submerged draft of the Spar platform. Mass is taken as constant and it is assumedthat the masses are lumped at the center of gravity. The structural damping matrixis taken to be constant and is dependent on mass and initial stiffness of the structure.The elements of [C] are determined by the equation given below, using the orthog-onal properties of [M] and [K], where, x is the structural damping ratio, O is modalmatrix, wi is natural frequency and mi is the generalized mass.

O r [ q O = [2xiwimi] (19)

The stiffness matrix consists of two parts: the restoring hydrostatic force and thestiffness due to mooring lines. The coefficients, Kij of the stiffness matrix of Sparplatform are derived as the force in degree-of-freedom i due to unit displacement inthe degree-of-freedom j, keeping all other degrees-of-freedom restrained. The coef-ficients of the stiffness matrix have nonlinear terms. Further, the mooring line tensionchanges due to the motion of the Spar platform in different degrees-of-freedom which


makes the stiffness matrix response dependent. Fig. 5 shows the degrees-of-freedomof the Spar platform at its center of gravity.

The hydrostatic stiffness is calculated based on the initial configuration of the Sparplatform and is given by

0 0 0 0 0 0

0 0 0 0 0 0

0 0 ^ 0 0 0

0 0 0 Khy44 0 0

0 0 0 0 Khy55 0

0 0 0 0 0 0

hydrostatic — (20)

where,

ss = 'xuf

K

(21)

(22)

n (23)

h1 = Scb—Scg; D is the diameter of the Spar platform; Scg, Scb are the distance fromthe keel of the Spar platform to its center of gravity and center of buoyancy respect-ively; Hd is the draft of the Spar platform; and gw is the weight density of water.

Heave (3)

Surge(1)

Roll (4)

Fig. 5. Degrees-of-freedom at CG of Spar platform.


9. Hydrodynamic forcing vector

Ocean surface waves refer generally to the moving succession of irregular humpsand hollows of ocean surface. They are generated primarily by the drag of the windon the water surface and hence are the greatest at any offshore site, when stormconditions exist there. For analyzing the offshore structures, it is customary to ana-lyze the effects of the surface waves on the structures either by use of a single designwave chosen to represent the extreme storm conditions in the area of interest or byusing the statistical representation of waves during extreme conditions. In either case,it is necessary to relate the surface wave data to the water velocity, acceleration andpressure beneath the waves. A unidirectional wave model is used for computing theincident wave kinematics. The kinematics of the water particles has been calculatedby Airy's wave theory. The sea surface elevation, h(x,t) is given as:

h(x,t) = 1Hcos(kx - wt) (24)

The horizontal and vertical water particle velocities are given as:

u = . ;/'cos{kx - wt) (25)2 sinh(kh)

v = . w Hsinh(ky)sin(kx - wt) (26)

2 sinh(kh)

k and w denotes the wave number and the wave frequency, respectively.

k = 2p/L and w = 2p/Pwhere, P = wave period; x = point of evaluation of water particle kinematics from theorigin in the horizontal direction; t = time instant at which water particle kinematics isevaluated; L = wave length; H = wave height; and, h = water depth

The acceleration of the water particle in horizontal and vertical directions aregiven as:

w2Hcosh(ky) ." sin(hc-cot) (27)

)2 sinh(kh)

w2Hsinh(ky)— 0 • u n i [

2 sinh(kh)cos(kx-wt) (28)

where y = height of the point of evaluation of water particle kinematicsA simplified alternative proposed in this study is to predict the response of a deep-

drafted offshore structure based on the slender body approximation, that is, withoutexplicitly considering the diffraction and radiation potential due to the presence ofthe structure. For typical deep-water offshore structures such as Spar platform, theratio of the structure dimension to spectrum-peak wave length is small. Hence, it isassumed that the wave field is virtually undisturbed by the structure and that theMorison's equation is adequate to calculate the wave exiting forces. The wave loads


on a structure are computed by integrating forces along the free surface centrelinefrom the bottom to the instant free surface at the displaced position. Use of Morison'sequation with modification has proved to be capable of capturing the trend of theresponse as well as most of the nonlinearity associated with it. The added mass isbased on the initial configuration of the Spar platform and is added to the massmatrix. The added mass force per unit of length is given by

^ (29)

where rw = mass density of the fluid; D = diameter of Spar platform; Cm = inertiacoefficient; and, X = acceleration of Spar platform

The drag force, which includes the relative motion between the structure and thewave, per unit of length is given by

FD = ^CJXu-JQlu-Xl (30)

where Cd = drag coefficient; u = velocity of the fluid; and, X = velocity of Spar plat-form

The inertia force by Morison's equation per unit of length is

PD2

FI = -j-pXnfi (31)

ii= acceleration of the fluid.

10. Solution of equation of motion

In time domain using numerical integration technique the equation of motion canbe solved, incorporating all the time dependent nonlinearities, stiffness coefficientchanges due to mooring line tension with time, added mass from Morison's equation,and with evaluation of wave forces at the instantaneous displaced position of thestructure. Wave loading constitutes the primary loading on offshore structures.Dynamic behaviour of these structures is, therefore, of design interest. When thedynamic response predominates, the behaviour under wave loading becomes nonlin-ear because the drag component of the wave load, according to Morison's equation,varies with the square of the velocity of the water particle relative to the structure.At each step, the force vector is updated to take into account the change in themooring line tension. The equation of motion has been solved by an iterative pro-cedure using unconditionally stable Newmark's Beta method. The algorithm basedon Newmark's method for solving the equation of motion is given below:

Step 1 The stiffness matrix [K], the damping matrix [C], the mass matrix [M], theinitial displacement vector {X0}, the initial velocity vector {Xo} are given asthe known input data.


Step 2 The force vector {F(t)} is calculated.Step 3 The initial acceleration vector is then calculated as below:

{Xo} = ^F0-CX0-KX0)

Step 4 d = 0.5, a = 0.25*(0.5 + d)2 and with the integration constants as:

a0 = \l{aAf), a1 = 8/aAt, a2 = 1/aAt, a3 = (1/2a)—1, a4 = (d/a)

- 1 , a5 = (Af/2)[(<5/a)-2], a6 = At(l-8), a7 = dAt

where, At is taken as the time step,Step 5 K = K+ aoM+ a1CStep 6 For each time step, the following are calculated:

Ki* = Ft+ht + M{aJCt + aX + ajtt) + C{axXt + aJC, +

Xt+At = Xt + ctgXf + a-jXt+At

Step 7 The values of X, X, X, which are calculated at the time step t + At are usedto evaluate Ft + At such that convergence is achieved to the accuracy of0.01%, before going to the next time step, otherwise iteration is carried out.Since the [K] of the Spar platform is response dependent, the new [K] isgenerated and the difference from the old [K] is used from step No. 5 onwardsby taking it to Ft + At-

11. Numerical results and discussions

Fig. 6 shows the plan and schematic elevation of the Spar platform with differentenvironmental loading. Wind and current have not been studied herein.

The particulars for the multi component catenary mooring line are given inTable 1.

Two cases are taken for initial horizontal force of 2500 kN (Case A) and 2000kN (Case B). Fig. 7 shows the force-excursion relationship of a single mooring line.

Increase in initial horizontal force at the attachment point makes the system tautas it decreases the length of the mooring line. The horizontal force-excursion relationof a single mooring line depends mainly on the initial configuration. The initialconfiguration is greatly influenced by the initial horizontal force, inclination at thefairlead point and its level from the sea floor and also to some extent by the sub-merged unit weight of the clump weight. The horizontal force in the mooring linebecomes higher, after the response of the Spar platform goes beyond 12 m, for initialhorizontal force of 2000 kN than for the 2500 kN. The Spar platform dimensionsand wave data are given in Table 2


Plan shoving Hie position of mooring line

Wave

^Anchor

Fig. 6. Schematic elevation of Spar platform.

11.1. Effect of initial horizontal force, Ho at the top of mooring line on theresponse of Spar platform

Two cases are taken for the initial horizontal force, Ho at the top of mooring linefor the evaluation of response of Spar platform: (A) 2500 kN and (B) 2000 kN.Table 3 shows the natural time period for both the cases at the time, t = 0 and atsteady state response.

In the calculation of natural time period, only diagonal term of the stiffness matrixis effective, as the mass matrix is diagonal due to lumped mass assumption.

It is observed from Table 3 that at time t = 0 for surge degree-of-freedom, naturaltime period of Spar platform for case B is more than case A, since at time t = 0,


Table 1Data for multi component catenary mooring line

Diameter of mooring line and anchor lineDiameter of clump weightEffective area of mooring line and anchor lineEffective area of clump weightWeight of mooring line and anchor lineWeight of clump weightLength of clump weightLength of anchor lineMean sea levelHeight of fairlead pointAngle of inclination at the fairlead point

0.0889 m1.055 m0.0032 m2

0.8742 m2

293.2 N/m25000 N/m40 m800 m914.4 m808.8 m30 degree

6000 -,

5000 -

4000 -

3000 -

2000^

i 1 1 0 -

hor force ^/y*

-"""^ ver force

-60 -40 -20 0 20

Excursion(meter)

40 60

hor force(2500kN)hor force(2000kN)

-verforce(2500kN)

-verforce(2000kN)

Fig. 7. Multi component force-excursion relationship of single mooring line.

Table 2Dimensions of Spar platform and wave data

Weight of the structureHeight of the Spar platformRadius of the Spar platformDistance of center of gravity to buoyancyDistance of center of gravity from keelDistance of center of gravity to fairleadsStructural damping ratioWave periodWave heightDrag coefficient (Cd)Inertia coefficient (Cm)

2.6 * 106 kN216.4 m20.26 m6.67 m92.4 m0.2 m0.05 & 0.0312.5 sec7 m1.0 & 0.02.0 &1.8


Table 3Natural time period for Spar platform with different initial horizontal force (sec)

Time Instant Case Ho (kN) Surge Sway Heave Roll Pitch Yaw

Response at t

Steady stateResponse

= 0 ABAB

2500200025002000

215.43254.60392.23360.41

215.43254.60215.43254.60

28.0428.0439.7939.79

50.8450.8450.8450.84

50.8450.8450.8450.84

102.97115.13102.97115.13

the response is nearly zero and from Fig. 7, the horizontal mooring force is morein case A than case B, so case B is providing less stiffness than case A. Whereasit is observed from steady state response values in surge degree-of-freedom that thehorizontal mooring force for case B (for nearly 12 mts surge response) is more incomparison to case A. Accordingly the natural time period for case B decreases fromcase A in surge degree-of-freedom.

For sway degree-of-freedom at time t = 0 it is observed that natural time periodfor case B is more than case A, since at time t = 0, the response is nearly zero andfrom Fig. 7, the horizontal mooring force is more in case A than case B, so case Bis providing less stiffness than case A. There is no change in the natural time periodat steady state response, because a unidirectional wave train has been consideredcausing zero response in sway degree-of-freedom.

For heave degree-of-freedom only hydrostatic force influences the stiffness matrixfor cases A and B and as there is no change in the hydrostatic force so there is nochange in the time period for both cases A and B. Time period increases in bothcases A and B at steady state response, because mass is increased by added mass,which makes the system more flexible in heave degree-of-freedom.

For roll and pitch degree-of-freedom it is observed that there is much less changein mooring stiffness because a unidirectional wave train has been considered resultingin nearly zero response in roll and little change in response of pitch degree-of-free-dom.

For yaw degree-of-freedom at time t = 0 it is observed that natural time periodfor case B is more than case A, since at time t = 0, the response is nearly zero andfrom Fig. 7, the horizontal mooring force is more in case A than case B, so case Bis providing less stiffness than case A. There is no change in the time periods atsteady state response, because a unidirectional wave train is considered causingnearly zero response in yaw degree-of-freedom.

Table 4 gives the comparison between the maximum value of the steady stateresponse for cases A and B for different initial horizontal force, Ho at the top ofmooring line.

Table 4 shows that there is a decrease of 14.55% in surge, decrease of 23.38%in heave and decrease of 0.52% in pitch responses when case B is considered incomparison to case A. Figs. 8 and 9 show the comparison between the steady stateresponse time histories of the coupled surge and heave response for different initialhorizontal force, Ho at the top of mooring line. From Figs. 8 and 9 it is observed


Table 4Maximum response for different initial horizontal force

Case Ho (kN) Max. displacement (m) Max. rotation (radian)

AB

25002000

Surge

-15.769-13.474

Heave

-1.779-1.363

Pitch

-0.0385-0.0383

-12

time (sec)5 7.5 10 12.5

Wave period 12.5 secWave heigth 7 m

Fig. 8. Effect of initial horizontal force at the top of mooring line in coupled surge response of aSpar platform.

-2 -

-2.5

2.5time (sec)

5 7.5

Wave period 12.5 secWave heigth 7 m - 2500 kN - D - 2000 kN

Fig. 9. Effect of initial horizontal force at the top of mooring line in coupled heave response of aSpar platform.


that both surge and heave responses have negative offset and are oscillating at nega-tive mean. It is also observed that variation in initial horizontal force affects surgeand heave responses significantly.

Decrease in initial horizontal force makes the system slack as it decreases thelength of the mooring line. The effect of initial horizontal force causes horizontaland vertical excursions, which changes nonlinearly with the change in initial horizon-tal force. For having lesser mooring system stiffness in B, the structure is moreflexible and gives lower dynamic response, although the static contribution ofresponse being more for lower stiffness of the structure. Case A, on the other hand,gives higher response as the structure is stiff and produces comparatively moredynamic response whereas the static response is lower than case B. This is due tononlinear behavior of the cable force where for case B the net force in the entirecable system is more than for case A. So for lower initial horizontal force it becomesstiffer in comparison to higher initial horizontal force. This indicates that the betterperformance of Spar platforms can be achieved with lesser stiffness of mooring sys-tem.

11.2. Effect of coupling of stiffness matrix on the response of Spar platform

Two cases are taken: (A) for the coupled stiffness matrix and (B) for the uncoupledstiffness matrix with the initial horizontal force of 2500 kN. Table 5 gives the com-parison between the maximum values of the steady state response for coupled anduncoupled stiffness matrix.

Table 5 shows that there is a decrease of 0.16% in surge response, 98.56% inheave response and 2.08% in pitch response, when uncoupled stiffness matrix isconsidered in comparison to coupled stiffness matrix. Figs. 10 and 11 show thecomparison between the time histories of the coupled surge and heave response forcoupled and uncoupled stiffness matrix. From Fig. 10 it is observed that surgeresponse for both the cases has negative offset and is oscillating at negative mean.From Fig. 11 it is observed that heave response for the coupled case has negativeoffset and is oscillating at negative mean, whereas for uncoupled case it has both+ve and —ve values. The stiffness matrix plays the most important role on the overallresponse analysis because it is response dependent.

The sway, roll and yaw response is zero for the uncoupled case as an unidirectionalwave is taken, while for the coupled case the responses are almost zero that means

Table 5Maximum response for different stiffness assumption

Stiffness

CoupledUncoupled

Max. displacement (m)

Surge

-15.769-15.744

Heave

-1.779-0.0256

Max. rotation (radian)

Pitch

-0.0385-0.0377


2.5time (sec)

5 7.5 12.5


Fig. 10. Effect of coupling of stiffness matrix in coupled surge response of a Spar platform.

2.5time (sec)

5 7.5 10 12.5

0.5

0

1 -0.5

1I "'•3ft

•I -1.5-2 -

p—H—H—a—D-

-2.5

-•—•—H—B—B—B—B—B

Wave period 12.5 secWave heigth 7 m - coupled —•— uncoupled

Fig. 11. Effect of coupling of stiffness matrix in coupled heave response of a Spar platform.

in these degrees-of-freedom there is no displacement and rotation. The result showsthat coupling of degrees-of-freedom has a significant effect on the response behaviorin heave and pitch degrees-of-freedom. In all further studies, coupled stiffness matrixhas been considered.

11.3. Effect of structural damping on the response of spar platform

Two cases are taken for structural damping ratio of 3% and 5% with the initialhorizontal force of 2500 kN. Table 6 gives the response for 5% and 3% structuraldamping ratio.

Table 6 shows that there is an increase of 0.23% in surge, decrease of 3.58% inheave and decrease of 0.26% in pitch direction when 5% structural damping is con-sidered in comparison to 3% structural damping. Figs. 12 and 13 show the compari-son between the time histories of the coupled surge and heave response for structuraldamping on the responses have of the Spar platform. From Figs. 12 and 13 it is


Table 6Maximum response for different structural damping ratio

Damping Max. displacement (m) Max. rotation (radian)

5%3%

Surge Heave

-15.769-15.733

-1.779-1.845

Pitch

-0.0385-0.0386

2.5time (sec)

12.5


structural damping 5%- D - structural damping 3%

Fig. 12. Effect of structural damping in coupled surge response of a Spar platform.

2.5time (sec)

5 7.5 10 12.5

-1.5

-2 -

-2.5 -Wave period 12.5 secWave heigth 7 in

- structural damping 5%- structural damping 3%

Fig. 13. Effect of structural damping in coupled heave response of a Spar platform.

observed that both surge and heave response has negative offset and is oscillatingat negative mean. It is observed that for higher structural damping ratio there is noeffect in surge and pitch response, whereas it affects heave responses significantly.


Table 7Maximum response

Inertia coefficient

Cm = 2.0Cm= 1.8

for variation in Cm

Max. displacement (m)

Surge

-15.769-13.279

Heave

-1.779-1.650

Max. rotation (radian)

Pitch

-0.0385-0.0345

11.4. Effect of inertia coefficient Cm on the response of spar platford

Two cases are taken for Cm equal to 2 and 1.8 with the initial horizontal force of2500 kN. Surge force, heave force and pitch moment reduces when Cm reduces from2 to 1.8 while calculating force using Morison's equation. Table 7 gives the responsefor Cm equal to 2 and 1.8.

Table 7 shows that there is a decrease of 15.79% in surge, decrease of 7.25% inheave and decrease of 10.39% in pitch response when Cm equal to 1.8 is consideredin comparison to Cm equal to 2. Figs. 14-16 show the comparison between the timehistories of the coupled surge, heave and pitch responses for different values ofinertia coefficient, Cm on the response of Spar platform. From Figs. 14 and 15 it isobserved that both surge and heave responses have negative offset and is oscillatingat negative mean. From Fig. 16 it is observed that pitch response oscillates both on+ve and —ve side of the axis. It is observed that surge, heave and pitch responsesare significantly affected with the decrease in inertia coefficient Cm.

11.5. Effect of drag coefficient Cd on the response of Spar platform

Two cases are taken for Cd equal to 1 and 0 with initial horizontal force of 2500kN. Surge and heave force reduces as the total force decreases when drag coefficient

2.5time (sec)

10 12.5


Fig. 14. Effect of inertia coefficient in coupled surge response of a Spar platform.


2.5time (sec)

5 7.5 10 12.5

-1.5

-2.5

n—•—•—•—•—n—•—•—•—n-I A A A A A A -A—A—A-

Wavc period 12.5 secWave heigth 7 in -A-Cm=2.0 -d-Cm =1.8

Fig. 15. Effect of inertia coefficient in coupled heave response of a Spar platform.

2.5time (sec)

5 7.5 10 12.5

0.02

-0.06

Wave period 12.5 secWave heigth 7 m -Cm = 2.0 -n-Cm=1.8

Fig. 16. Effect of inertia coefficient in coupled pitch response of a Spar platform

is zero, while calculating force using Morison's equation. Table 8 gives the responsefor Cd equal to 1 and 0.

Table 8 shows that there is a decrease of 24.47% in surge, decrease of 6.30% inheave and increase of 0.52% in pitch response when Cd equal to 0 is considered in

Table 8Maximum response for variation in Cd

Drag coefficient Max. displacement (m) Max. rotation (radian)

Surge Heave Pitch

Cd=1 15.769-11.910

-1.779-1.667

-0.0385-0.0387


time (sec)

Fig. 17. Effect of drag coefficient in coupled surge response of a Spar platform.

comparison to Cd equal to 1. Figs. 17 and 18 shows the comparison between thetime histories of the coupled surge and heave responses for different values of dragcoefficient, Cd on the response of the Spar platform. From Figs. 17 and 18 it isobserved that both surge and heave responses have negative offset and are oscillatingat negative mean.

Although the Spar platform exhibits inertia dominated force regime but the influ-ence of coefficient of drag is appreciable in surge response and little in heaveresponse.

12. Conclusions

Based on the numerical study conducted on the Spar platform, the following con-clusions can be drawn:

-2 -

-2.5

2.5time (sec)

5 7.5 10 12.5


-Cd=1.0 -D-Cd = 0.0

Fig. 18. Effect of drag coefficient in coupled heave response of a Spar platform.


1. The force-excursion relation of a single mooring line depends mainly on the initialhorizontal force at the top of the mooring line. The initial structural time periodof Spar platform in surge degree-of-freedom is higher with lower value of initialhorizontal force than with higher value of initial horizontal force, due to higherstiffness caused by higher initial horizontal force. Whereas structural time periodof Spar platform at steady state response, in surge degree-of-freedom with lowerinitial horizontal force is less than the value with higher initial horizontal force.This is due to nonlinear behaviour of the cable force where for lower initial hori-zontal force the net force in the entire cable system is more with lower responsevalue than for higher initial horizontal force, thus causing higher stiffness withlower initial horizontal force.

2. With lower initial horizontal force, the structure is more flexible and gives lowerdynamic response, although the static contribution of response being more due tolower stiffness of the structure. While the higher initial horizontal force giveshigher response as the structure is relatively more stiffer and produces compara-tively more dynamic response although the static response contribution is lowerin this case. Variation in initial horizontal force affects surge and heaveresponses significantly.

3. The coupling of degrees-of-freedom in stiffness matrix of Spar platform plays animportant role in the dynamic behaviour of offshore Spar platform as the responseis significantly affected by considering coupled stiffness matrix. Heave responseis affected most while considering coupled stiffness matrix.

4. Due to change in structural damping ratio there is no effect in surge and pitchresponse, whereas it affects heave response significantly.

5. It is necessary to evaluate the proper value of inertia coefficient so that waveforce can be accurately estimated as it has significant effect on the response ofthe Spar platform. Surge, heave and pitch responses proportionately vary with thevalue of inertia coefficient.

6. Although the Spar platform exhibits inertia dominated force regime but the influ-ence of coefficient of drag is appreciable in surge response and slightly affectingthe heave response.

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