Dynamic behavior of offshore spar platforms under regular sea waves

Transcription

1 Dynamic behavior of offshore spar platforms under regular sea waves A.K. Agarwal, A.K. Jain * Department of Civil Engineering, Indian Institute of Technology, Hauz Khas, New Delhi , India Received 10 October 2001; accepted 14 January 2002 Abstract Many innovative floating offshore structures have been proposed for cost effectiveness of oil and gas exploration and production in water depths exceeding one thousand meters in recent years. One such type of platform is the offshore floating Spar platform. The Spar platform is modelled as a rigid body with six degrees-of-freedom, connected to the sea floor by multicomponent 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 provide restoring force in heave, roll and pitch, (b) the mooring lines provide the restoring force which are represented here by nonlinear horizontal springs. A unidirectional regular wave model is used for computing the incident wave kinematics by Airy's wave theory and force by Morison's equation. The response analysis is performed in time domain to solve the dynamic behavior of the moored Spar platform as an integrated system using the iterative incremental Newmark's Beta approach. Numerical studies are conducted for sea state conditions with and without coupling of degrees-of-freedom. Keywords: Wave structure interaction; Offshore structural dynamics; Spar platform; Multi-component catenary mooring

2 488 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) Introduction As offshore oil and gas exploration are pushed into deeper and deeper water, many innovative floating offshore structures are being proposed for cost savings. To reduce wave induced motion, the natural frequency of these newly proposed offshore structures 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 water applications for the drilling, production, processing, storage and offloading of ocean deposits. It is being considered as the next generation of deep water offshore structures by many oil companies. It consists of a vertical cylinder, which floats vertically in the water. The structure floats so deep in the water that the wave action at the surface is dampened by the counter balance effect of the structure weight. Fin like structures called strakes, attached in a helical fashion around the exterior of the cylinder, act to break the water flow against the structure, further enhancing the stability. Station keeping is provided by lateral, multi-component catenary anchor lines attached to the hull near its center of pitch for low dynamic loading. The analysis, design and operation of Spar platform turn out to be a difficult job, primarily because of the uncertainties associated with the specification of the environmental loads. 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 production to 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 buoyancy cans, (d) It is always stable because center of buoyancy (CB) is above the center of gravity (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 platform buoy type structures have been built in the ocean before. For example, a floating instrument platform (FLIP) was built in 1961 to perform oceanographic research (Fisher and Spiess, 1963), the Brent Spar platform was built by Royal Dutch shell as 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.

3 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Glanville et al. (1991) gave the details of the concept, construction and installation of a Spar platform. He concluded that a Spar platform allows flexibility in the selection of well systems and drilling strategies, including early production or predrilling 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 inertia coefficient, C m, as in the standard Morison's equation or using a frequency dependent C m coefficient based on the diffraction theory. The drag forces were computed using the nonlinear term of Morison's equation in both cases. The analysis was performed in time domain. The result showed that using frequency dependent or constant inertia coefficient, C m, produces similar results since most of the wave energy is concentrated over the range of frequencies where the value of C m is equal to 2 for the Spar platform size used in the literature. Mekha et al. (1996) used the same model (Mekha et al., 1995) with frequency dependent C m coefficient based on diffraction theory. Different nonlinear modifications to Morison's equation were induced to account diffraction effects. The results obtained for a variety of sea state conditions were compared to the experimental data. Halkyard (1996) reviewed the status of several Spar platform concepts emphasizing the design aspect of these platforms. Cao and Zhang (1996) discussed an efficient methodology to predict slow drift response of slack moored slender offshore structures due to ocean waves using a hybrid wave model. The hybrid wave model considers the wave interactions in an irregular wave field up to second order of wave steepness and is able to accurately predict incident wave kinematics, including the contributions from nonlinear difference frequency interactions. A unique feature of this approach is that measured wave elevation time series can be used as input and the structure responses to measured incident waves can be deterministically obtained. Ran and Kim (1996) studied the nonlinear response characteristics of a tethered/moored Spar platform in regular and irregular waves. A time-domain coupled nonlinear motion analysis computer program was developed to solve both the static and dynamic behaviours of a moored compliant platform as an integrated system. In particular, an efficient global-coordinate based dynamic finite element program was developed to simulate the nonlinear tether/mooring responses. Using this program, the coupled dynamic analysis results were obtained and they were compared with uncoupled analysis results to see the effects of tethers and mooring lines on hull motions and vice versa. Jha et al. (1997) compared the analytical predicted motions of a floating Spar buoy platform with the results of wave tank experiments considering, surge and pitch motions only. Base-case predictions combine nonlinear diffraction loads and a linear, multi-degree-of-freedom model of the Spar platform stiffness and damping characteristics, refined models and the effect of wave-drift damping, and of viscous forces as well. Consistent choices of damping and wave input were considered in some detail. Fischer and Gopalkrishnan (1998) presented the importance of heave characteristics of Spar platforms that have been gleaned from wave basin model tests, numerical simulations and a combination of the two. The heave performance of a small

4 490 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Spar platform, e.g. mini Spar platform, has been examined and found to be potentially problematic. Chitrapu et al. (1998) studied the nonlinear response of a Spar platform under different environmental conditions such as regular, bi-chromatic, random waves and current using a time domain simulation model. The model can consider several nonlinear effects. Hydrodynamic forces and moments were computed using the Morison's equation. It was concluded that Morison's equation combined with accurate prediction of wave particle kinematics and force calculations in the displaced position of the platform gave a reliable prediction of platform response both in wavefrequency and low-frequency range. Nonlinear coupled response of a moored Spar platform in random waves with and without co-linear current were investigated in both time and frequency domain (Ran et al., 1999). The first and second order wave forces, added mass, radiation damping and wave drift damping were calculated from a hydrodynamic software package called WINTCOL. The total wave force time series (or spectra) were then generated in the time (or frequency) domain based on a two-term volterra series method. The mooring dynamics were solved using the software package WINPOST, which is based on a generalised coordinate based FEM. The mooring lines were attached to the platform through linear and rotational springs and dampers. Various boundary conditions can be modelled using proper spring and damping values. In the time domain analysis, the nonlinear drag forces on the hull and mooring lines were applied at the instantaneous position. In the frequency domain analysis, nonlinear drag forces were stochastically linearized and solutions were obtained by an iterative procedure. Ye et al. (1998) studied the Spar platform response in directional wave environment using the unidirectional hybrid wave model and directional hybrid wave model (UHWM and DHWM). Comparison between numerical results from two different wave models indicated that the slow drifting surge and pitch motions based on DHWM are slightly smaller than those based on UHWM. The slow-drifting heave motions from the two wave models were almost the same because the heave motion was mainly excited by the pressure applied on the structure bottom and the predicted bottom pressure from the two methods had almost no differences. Datta et al. (1999) described recent comparisons of numerical predictions of motions and loads of typical truss Spar platform model test results. The purpose of this comparison was to calibrate hydrodynamic coefficients, which were to be used for the design of a new truss Spar platform for Amoco. Chitrapu et al. (1999) discussed the motion response of a large diameter Spar platform in long crested and random directional waves and current using a timedomain simulation model. Several nonlinearities such as the free surface force calculation, displaced position force computation, nonlinearities in the equations of motion and the effect of wave-current interaction were considered in determining the motion response. The effect of wave directionality on the predicted surge and pitch response of the Spar platform had been studied. It was seen that both wave-current interaction and directional spread of wave energy had a significant effect on the predicted response.

5 2. Structural model A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) 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, Y and 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 closed at its keel. The stability and stiffness is provided by a number of mooring lines attached 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 symmetry of the mooring system, the resultant horizontal force will also occur in this plane and 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 analysis of the platforms. It is assumed that the Spar platform is connected to the sea floor by four multi component catenary mooring lines placed perpendicular to each other, which are attached to the Spar platform at the fairleads. The development of Spar platform model for dynamic analysis involves the formulation of a nonlinear stiffness matrix considering mooring line tension fluctuations due to variable buoyancy and other nonlinearities. The model considers the coupled behaviour of a Spar platform for various degrees-of-freedom. Fig. 1 shows a typical offshore Spar platform. WLLINO RIG HEU-POftT- MOORING TQJIPUENT OAT- WATK SURFACE ACCESS \ K M TO TOPSIDES AND BOAr UNHH3 muw ma -JJJIK Fig. 1. Typical offshore Spar platform.

6 492 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) Assumptions and structural idealization The platform and the mooring lines are treated as a single system and the analysis is carried out for the six degrees-of-freedom under the environmental loads. The following assumptions have been made in the analysis: 1. Initial pretension in all mooring lines is equal. However, total pretension changes with the motion of the Spar platform, 2. Wave forces are estimated at the instantaneous equilibrium position of the Spar platform by Morison's equation using Airy's linear wave theory. The wave diffraction effects have been neglected, 3. Integration of fluid inertia and drag forces are carried out up to the actual level of submergence according to the stretching modifications considered in the analysis, 4. Wave force coefficients, C d and C m are independent of frequencies as well as constant over the water depth, 5. Current velocity has not been considered and also the interaction of wave and current 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 stiffness 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-directional wave 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 to the 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 treated as negligible, (c) The change in the line geometry and thereby in the line force due to direct fluid loading caused from waves and /or currents is insignificant, (d) Initial length of the mooring line and anchor line is inclusive of elongation due to 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.

7 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) The force-excursion relationship is nonlinear and requires an iterative solution. Equation of a catenary was used for evaluation of force-excursion relationship of a catenary mooring line. The horizontal projection and vertical projection of any segment hanging freely under its own weight w per unit length as shown in Fig. 2 can be expressed (considering horizontal force (H t ), top slope (q t ), length (S) and weight (W)) as: Y = (H,/fF)[cosh{sinh- 1 (tan(0,))}-cosh{sinh- 1 (tan(0 6 ))}] (1) tan(q b ) = (V-WS)IH t (3) When for any segment the bottom slope (q b ) is zero, the above equation reduces to: Y = (H t / HOtcosMsinh-^tanCG,))} -1] (4) (5) (a) Vo Clump weight W c I, 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.

8 494 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) If H t, Y,W,q t are known then, tan(q b ) = sinh[cosh- 1 [cosh{sdnh- 1 (tan(0,))}-(yft//r,)]] (6) S = flxtan(0,)-tan(0 6 ))/ 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 T o when the segment length is S o For increased average line tension T, the stretched length becomes: S = S o [1 + (T-T O )IEA\ (8) where, E and A are the Young's modulus and effective area of the segment respectively. T and T o is the arithmetic means of the line tensions at two ends. The total weight of any segment (W) remains same, W=(S O W O )/S (9) where W is the modified unit weight due to stretching and W o is the unit weight of the unstreched segment. 5. Analysis of mooring line with distributed clump weight for (horizontal excursion) In the present work H o, q o, h, zero bottom slope and the elastic and physical properties of the segments, as shown in Fig. 3(a), are chosen as the known parameters. The unknowns which are to be evaluated are S c, S h and then Y c, Y h,x c 6. Initial configuration The following steps are used to find the unknowns given above. Step 1 Calculate V o from the known values of H o and q o. 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 S h W cl as the bottom slope is equal to zero). Using the known values of the horizontal force, use Eq. (4) to find Y h. Step 3 Find S c = (V o -S h W cl )/W c and then find Y c using Eq. (1). Step 4 Add up Y c and Y h and compare with h. If the difference is less than a specified limit, go to the next step, otherwise change Vj appropriately and repeat the procedure from step 2. For the first iteration the change of Vj can be taken as ± 1% depending upon the sign of error. For the subsequent iterations the following equation is to be used to get a new value of Vj.

9 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) (a) he s h (b) s cl" s hn Fig. 3. Configuration of mooring line for increased horizontal force, H (Condition 1). (a) Initial configuration, (b) Final configuration. where, k is the number of the last iteration, (Vj) k is the vertical force at the junction of the mooring line and clump weight and e k is the difference between the vertical projection of the hanging length of the mooring line calculated in the kth iteration and the mooring level (h). Step 5 Find X c and X h from Eqs. (2) and (5), respectively. Step 6 Find initial total hanging length S i = S c + S h and its horizontal projection X i = X c + X h. 7. Evaluation of force-excursion (horizontal) relationship for a single mooring line The vertical force (V o ) is changed which allows direct checking of the condition regarding the lifting off of the clump weight. The corresponding Horizontal force is

10 496 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) found iteratively, ending with the determination of a new configuration including the excursion of the attachment point. The procedures for the two alternative states of lifting off of the clump weight are given below. W c = S c W c is the weight of the mooring line and W cl = S cl W cl is the total weight of the clump weight. Find the initial tension in the mooring line as follows: 7.1. Condition 1 (when V<W C + W cl ) Step 1 Increase V o 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, S cn and hence modified W c using eqs. (8) and (9), respectively. Step 4 Find the vertical projections of mooring line, (Y cn ) and clump weight (Y hn ) with the help of eqs. (1) and (4), respectively. Step 5 Add up Y cn and Y hn to compare with h. If the difference e is less than the tolerable limit go to the next step. Otherwise, change horizontal force H appropriately and repeat the procedure from step 2. For the first iteration the change of H can be taken as ± 1% depending upon the sign of error. For the subsequent iterations the following equation is to be used to get a new value ofh, H k+1 = H k -[e k (H k^-h k )l(e k^-e k )\ (10) where k is the number of the last iteration, H k is the horizontal force and e k is the difference between the vertical projections of the hanging length of the mooring line calculated in the ML iteration and the mooring level (h). Step 6 Find the new hanging length of the clump weight S hn from the zero bottom slope condition and add it with the stretched mooring line length S cn to get the new length, S f. Step 7 Find X cn and X hn from eqs. (2) and (5), respectively and add them to get the horizontal projection X f. Step 8 Find the elastic stretch of anchor line, e a. e a = S a (H-H o )l(eaa) (11) where E a is Young's modulus of the anchor line; A a is effective area of the anchor line; H o is initial horizontal force at the top; H is final horizontal force at the top; and, S a is length of anchor line. Step 9 Evaluate horizontal excursion (d) S = {X f -X^-{S f -S^ + (e a + e c ) (12) where S i, X i is initial hanging length of the mooring line and its horizontal projection, respectively; S f,x f is final stretched hanging length of the mooring line and its horizontal projection, respectively; e c is elastic stretch in the

11 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) mooring line from eq. (8); and, e a is elastic stretch in the anchor line from eq. (11). Step 10 Repeat steps 1 to 9 for the increased values of V o till V = W c + W cl 7.2. Condition 2 (when V > W c + 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 = V o + AV) find the vertical force (a) (b) (c) Fig. 4. Configuration of mooring line for increased horizontal force, H (Condition 2). (a) Configuration when the far end of the clump just lifts off; (b) Final configuration; (c) Free body diagram of the anchor line.

12 498 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) V a 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 anchor line and hence the modified W c and W a (considering the system configuration when the clump weight is just lifted as shown in Fig. 4(a) as the initial configuration). Step 3 Find Y cn and Y cln using eq. (1) and Y ah from eq. (4). Add the projections and compare with h. If the difference is within tolerable limit go to step 4. Otherwise, change H appropriately and repeat the procedure from step 2 (as in step 5 for condition 1). Step 4 Find the hanging length of anchor line, S ah using eq. (7) with q b = 0. Add up S cn and S cl to get S f (which includes the mooring line stretch). Step 5 Find X cn and X cln using eq. (2) and X ah from eq. (5). Add X cn to X cl to get X f. Step 6 Find the elastic stretch of anchor line where H' = {H + ^(H 2 + ^}/2; V a = V-W c -W cl ; H o = initial horizontal force when V = W c + W cl ; H' = increased horizontal force due to increased vertical force at the top. W c and W cl being the total weights of mooring line and clump weight, respectively. Step 7 Find the excursion from (13) 8 = 8' + QC f -X' t )-(S f -S' t ) + (X ah -S' ah ) + (e' a + e' o ) (14) Corresponding to changed value of the horizontal force H. Where 8' = excursion when the clump weight lifts off; ",, X\ = total hanging stretched length and its horizontal projection, respectively when the clump weight just lifts off; X ah S' ah = horizontal movement of the clump weight-anchor line junction; 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 V o, till the tension equals the permissible value (approximately half of the breaking strength of mooring line material). In the above S ah and X ah are found by treating the anchor line as a freely hanging mooring line and making use of the modified unit weight for the stretched segment. If S ah is found more than the total (stretch) anchor line length calculate tan(q b ) = (V a W a )/H, W a being the total weight of anchor line and recalculate S ah. X ah is to be evaluated using eq. (2). The behaviour of mooring system will be planer if the tower excursion takes place in a plane of symmetry of the mooring system. For an excursion of d at the attachment 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 mooring line and the direction of excursion. dj is the excursion for the jth mooring line and

13 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Hj(dj) is the associated horizontal force, with dj = Scos(n 6j). The vertical force at the mooring attachment point will be: V{5)= (16) In this study the mooring line is modelled as a nonlinear horizontal spring located at the fairleads along the Spar platform center with no hydrodynamic forces applied on them. The stiffness matrix representing the mooring line is calculated based on eqs. (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 hs Khs 2 1 A 22 hsk 23 hskhs 24 K 2 5 hsk hs hs hs hs hs hs Khs 31 A 32 hsk 33 hskhs 34 K 3 5 h s hs hs hs hs hs hs Khs41 A 42 hsk 43 hskhs 44 A 45 K L hs hs hs hs hs A51 Aj 2 A53 K53 54 Kh s hs hs hs hs (17) K h s11 is the sum of the horizontal component of force (surge) of mooring lines for unit displacement along surge direction, Khs21 will be zero as the mooring line placed perpendicular to the surge direction (sway) will not contribute any force, because the behavior of the mooring system is planer when Spar platform excursion takes place in the plane of symmetry, Khs 3 1 is the sum of the vertical component of force (heave) of mooring lines, K h s41 (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) of mooring lines about centre of gravity, Khs61 (yaw) will be zero, as no mooring line will contribute to any force in this direction. K hs 1 2 will be zero for unit displacement in sway direction as the mooring lines placed perpendicular to the sway direction (surge) will not contribute any force, because the behavior of the mooring system is planer when Spar platform excursion takes place in the plane of symmetry, Ki^ is the sum of the horizontal component of force (sway) of mooring lines, K h s 3 2 is the sum of the vertical component of force (heave) of mooring lines, AJl (roll) is the sum of the moment of horizontal component of force (sway) of mooring lines about centre of gravity, Khs5 2 (pitch) is zero as there is no force in the surge direction, Khs62 (yaw) will be zero as no mooring line will contribute any force in this direction. K 1 3hs (surge), Khs 23 (sway), K 33 (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), Khs 53 (pitch) and K%, (yaw) will be zero because there is no force in this direction. K h s14 (surge) will be zero for unit rotation in roll as the behaviour of the mooring system is planer, Khs 2 4 is the horizontal component of force (sway) of the mooring line present in the sway direction, Khs 3 4 is the vertical component of force (heave) of

14 500 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) the mooring line present in the sway direction, K\\ (roll) is the sum of the moment of horizontal component of force in sway direction about centre of gravity, Khs5 4 (pitch) will be zero because there is no force in surge direction, Khs64 (yaw) will be zero, as no mooring line will contribute any force in this direction. s15 will be the horizontal component of force (surge) of the mooring line present in the surge direction for unit rotation in pitch, Afl (sway) will be zero as the behaviour of the mooring system is planer, Khs3 5 is the vertical component of force (heave) of the mooring line present in the surge direction, Khs45 (roll) will be zero as there is no force in sway direction, A«j (pitch) is the sum of the moment of horizontal component of force in surge direction about centre of gravity, Khs65 (yaw) will be zero, as no mooring line will contribute any force in this direction. Kh s 1 6 (surge) and Khs2 6 (sway) will be zero because opposite direction force will nullify for unit rotation in yaw, Khs3 6 is the sum of vertical component of the forces (heave) of the mooring lines, Khs46 (roll) and K5 6 hs (pitch) will be zero because there is no force in this direction, Kh s 6 6 (yaw) is the sum of the moment of the horizontal component 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 = M Sparplatform + M addedmass ; K = K hydrostat i c + Kmooring(horizontal); C = structure 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 mass matrix is obtained by integrating the added mass term of Morison's equation along the submerged draft of the Spar platform. Mass is taken as constant and it is assumed that the masses are lumped at the center of gravity. The structural damping matrix is 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 orthogonal properties of [M] and [K], where, x is the structural damping ratio, O is modal matrix, wi is natural frequency and m i is the generalized mass. O r [qo = [2xiw i mi] (19) The stiffness matrix consists of two parts: the restoring hydrostatic force and the stiffness due to mooring lines. The coefficients, K ij of the stiffness matrix of Spar platform are derived as the force in degree-of-freedom i due to unit displacement in the degree-of-freedom j, keeping all other degrees-of-freedom restrained. The coefficients of the stiffness matrix have nonlinear terms. Further, the mooring line tension changes due to the motion of the Spar platform in different degrees-of-freedom which

15 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) makes the stiffness matrix response dependent. Fig. 5 shows the degrees-of-freedom of the Spar platform at its center of gravity. The hydrostatic stiffness is calculated based on the initial configuration of the Spar platform and is given by hydrostatic ^ Khy Khy (20) where, ss = 'xuf (21) K (22) n (23) h1 = S cb S cg ; D is the diameter of the Spar platform; S cg, S cb are the distance from the keel of the Spar platform to its center of gravity and center of buoyancy respectively; H d is the draft of the Spar platform; and g w is the weight density of water. Heave (3) Roll (4) Surge(1) Fig. 5. Degrees-of-freedom at CG of Spar platform.

16 502 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) Hydrodynamic forcing vector Ocean surface waves refer generally to the moving succession of irregular humps and hollows of ocean surface. They are generated primarily by the drag of the wind on the water surface and hence are the greatest at any offshore site, when storm conditions exist there. For analyzing the offshore structures, it is customary to analyze the effects of the surface waves on the structures either by use of a single design wave chosen to represent the extreme storm conditions in the area of interest or by using 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 and pressure beneath the waves. A unidirectional wave model is used for computing the incident wave kinematics. The kinematics of the water particles has been calculated by 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 Hs i nh(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/P where, P = wave period; x = point of evaluation of water particle kinematics from the origin in the horizontal direction; t = time instant at which water particle kinematics is evaluated; L = wave length; H = wave height; and, h = water depth The acceleration of the water particle in horizontal and vertical directions are given as: w 2 Hcosh(ky). " 2 sinh(kh) sin(hc-cot) (27) ) w 2 Hsinh(ky) 0 uni[ 2 sinh(kh) cos(kx-wt) (28) where y = height of the point of evaluation of water particle kinematics A simplified alternative proposed in this study is to predict the response of a deepdrafted offshore structure based on the slender body approximation, that is, without explicitly considering the diffraction and radiation potential due to the presence of the structure. For typical deep-water offshore structures such as Spar platform, the ratio of the structure dimension to spectrum-peak wave length is small. Hence, it is assumed that the wave field is virtually undisturbed by the structure and that the Morison's equation is adequate to calculate the wave exiting forces. The wave loads

17 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) on a structure are computed by integrating forces along the free surface centreline from the bottom to the instant free surface at the displaced position. Use of Morison's equation with modification has proved to be capable of capturing the trend of the response as well as most of the nonlinearity associated with it. The added mass is based on the initial configuration of the Spar platform and is added to the mass matrix. The added mass force per unit of length is given by ^ (29) where r w = mass density of the fluid; D = diameter of Spar platform; C m = inertia coefficient; and, X = acceleration of Spar platform The drag force, which includes the relative motion between the structure and the wave, per unit of length is given by F D = ^CJXu-JQlu-Xl (30) where C d = drag coefficient; u = velocity of the fluid; and, X = velocity of Spar platform The inertia force by Morison's equation per unit of length is PD 2 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 can be solved, incorporating all the time dependent nonlinearities, stiffness coefficient changes 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 the structure. Wave loading constitutes the primary loading on offshore structures. Dynamic behaviour of these structures is, therefore, of design interest. When the dynamic response predominates, the behaviour under wave loading becomes nonlinear 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 the mooring line tension. The equation of motion has been solved by an iterative procedure using unconditionally stable Newmark's Beta method. The algorithm based on 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], the initial displacement vector {X 0 }, the initial velocity vector {X o } are given as the known input data.

18 504 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) Step 2 The force vector {F(t)} is calculated. Step 3 The initial acceleration vector is then calculated as below: {X o } = ^F 0 -CX 0 -KX 0 ) Step 4 d = 0.5, a = 0.25*(0.5 + d) 2 and with the integration constants as: a 0 = \l{aaf), a 1 = 8/aAt, a 2 = 1/aAt, a 3 = (1/2a) 1, a 4 = (d/a) -1, a 5 = (Af/2)[(<5/a)-2], a 6 = At(l-8), a 7 = dat where, At is taken as the time step, Step 5 K = K+ a o M+ a 1 C Step 6 For each time step, the following are calculated: Ki* = F t+ht + M{aJC t + ax + ajt t ) + C{a x X t + ajc, + X t+ A t = X t + ctgxf + a-jx t+ At Step 7 The values of X, X, X, which are calculated at the time step t + At are used to evaluate F t + At such that convergence is achieved to the accuracy of 0.01%, 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] is generated and the difference from the old [K] is used from step No. 5 onwards by taking it to F t + At Numerical results and discussions Fig. 6 shows the plan and schematic elevation of the Spar platform with different environmental loading. Wind and current have not been studied herein. The particulars for the multi component catenary mooring line are given in Table 1. Two cases are taken for initial horizontal force of 2500 kn (Case A) and 2000 kn (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 taut as it decreases the length of the mooring line. The horizontal force-excursion relation of a single mooring line depends mainly on the initial configuration. The initial configuration is greatly influenced by the initial horizontal force, inclination at the fairlead point and its level from the sea floor and also to some extent by the submerged unit weight of the clump weight. The horizontal force in the mooring line becomes higher, after the response of the Spar platform goes beyond 12 m, for initial horizontal force of 2000 kn than for the 2500 kn. The Spar platform dimensions and wave data are given in Table 2

19 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Plan shoving Hie position of mooring line Wave ^Anchor Fig. 6. Schematic elevation of Spar platform Effect of initial horizontal force, H o at the top of mooring line on the response of Spar platform Two cases are taken for the initial horizontal force, H o at the top of mooring line for 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 at steady state response. In the calculation of natural time period, only diagonal term of the stiffness matrix is 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, natural time period of Spar platform for case B is more than case A, since at time t = 0,

20 506 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Table 1 Data for multi component catenary mooring line Diameter of mooring line and anchor line Diameter of clump weight Effective area of mooring line and anchor line Effective area of clump weight Weight of mooring line and anchor line Weight of clump weight Length of clump weight Length of anchor line Mean sea level Height of fairlead point Angle of inclination at the fairlead point m m m m N/m N/m 40 m 800 m m m 30 degree , hor force ^ -"""^ ver force ^/y* i Excursion(meter) hor force(2500kn) hor force(2000kn) -verforce(2500kn) -verforce(2000kn) Fig. 7. Multi component force-excursion relationship of single mooring line. Table 2 Dimensions of Spar platform and wave data Weight of the structure Height of the Spar platform Radius of the Spar platform Distance of center of gravity to buoyancy Distance of center of gravity from keel Distance of center of gravity to fairleads Structural damping ratio Wave period Wave height Drag coefficient (C d ) Inertia coefficient (C m ) 2.6 * 10 6 kn m m 6.67 m 92.4 m 0.2 m 0.05 & sec 7 m 1.0 & &1.8

21 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) Table 3 Natural time period for Spar platform with different initial horizontal force (sec) Time Instant Case H o (kn) Surge Sway Heave Roll Pitch Yaw Response at t Steady state Response = 0 A B A B the response is nearly zero and from Fig. 7, the horizontal mooring force is more in case A than case B, so case B is providing less stiffness than case A. Whereas it is observed from steady state response values in surge degree-of-freedom that the horizontal mooring force for case B (for nearly 12 mts surge response) is more in comparison to case A. Accordingly the natural time period for case B decreases from case A in surge degree-of-freedom. For sway degree-of-freedom at time t = 0 it is observed that natural time period for case B is more than case A, since at time t = 0, the response is nearly zero and from Fig. 7, the horizontal mooring force is more in case A than case B, so case B is providing less stiffness than case A. There is no change in the natural time period at steady state response, because a unidirectional wave train has been considered causing zero response in sway degree-of-freedom. For heave degree-of-freedom only hydrostatic force influences the stiffness matrix for cases A and B and as there is no change in the hydrostatic force so there is no change in the time period for both cases A and B. Time period increases in both cases 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 change in mooring stiffness because a unidirectional wave train has been considered resulting in nearly zero response in roll and little change in response of pitch degree-of-freedom. For yaw degree-of-freedom at time t = 0 it is observed that natural time period for case B is more than case A, since at time t = 0, the response is nearly zero and from Fig. 7, the horizontal mooring force is more in case A than case B, so case B is providing less stiffness than case A. There is no change in the time periods at steady state response, because a unidirectional wave train is considered causing nearly zero response in yaw degree-of-freedom. Table 4 gives the comparison between the maximum value of the steady state response for cases A and B for different initial horizontal force, H o at the top of mooring 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 in comparison to case A. Figs. 8 and 9 show the comparison between the steady state response time histories of the coupled surge and heave response for different initial horizontal force, H o at the top of mooring line. From Figs. 8 and 9 it is observed

22 508 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Table 4 Maximum response for different initial horizontal force Case H o (kn) Max. displacement (m) Max. rotation (radian) Surge Heave Pitch A B time (sec) Wave period 12.5 sec Wave heigth 7 m Fig. 8. Effect of initial horizontal force at the top of mooring line in coupled surge response of a Spar platform. 2.5 time (sec) Wave period 12.5 sec Wave heigth 7 m kn -D kn Fig. 9. Effect of initial horizontal force at the top of mooring line in coupled heave response of a Spar platform.

23 A.K. Agarwal, A.K. Jain/Ocean Engineering 30 (2003) that both surge and heave responses have negative offset and are oscillating at negative mean. It is also observed that variation in initial horizontal force affects surge and heave responses significantly. Decrease in initial horizontal force makes the system slack as it decreases the length of the mooring line. The effect of initial horizontal force causes horizontal and vertical excursions, which changes nonlinearly with the change in initial horizontal force. For having lesser mooring system stiffness in B, the structure is more flexible and gives lower dynamic response, although the static contribution of response 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 more dynamic response whereas the static response is lower than case B. This is due to nonlinear behavior of the cable force where for case B the net force in the entire cable system is more than for case A. So for lower initial horizontal force it becomes stiffer in comparison to higher initial horizontal force. This indicates that the better performance of Spar platforms can be achieved with lesser stiffness of mooring system 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 uncoupled stiffness matrix with the initial horizontal force of 2500 kn. Table 5 gives the comparison between the maximum values of the steady state response for coupled and uncoupled stiffness matrix. Table 5 shows that there is a decrease of 0.16% in surge response, 98.56% in heave response and 2.08% in pitch response, when uncoupled stiffness matrix is considered in comparison to coupled stiffness matrix. Figs. 10 and 11 show the comparison between the time histories of the coupled surge and heave response for coupled and uncoupled stiffness matrix. From Fig. 10 it is observed that surge response 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 negative offset 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 overall response analysis because it is response dependent. The sway, roll and yaw response is zero for the uncoupled case as an unidirectional wave is taken, while for the coupled case the responses are almost zero that means Table 5 Maximum response for different stiffness assumption Stiffness Max. displacement (m) Max. rotation (radian) Surge Heave Pitch Coupled Uncoupled

24 510 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) time (sec) Wave period 12.5 sec Wave heigth 7 m Fig. 10. Effect of coupling of stiffness matrix in coupled surge response of a Spar platform time (sec) I "' 3 ft I p H H a D- Wave period 12.5 sec Wave heigth 7 m - H B B B B B - 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 shows that coupling of degrees-of-freedom has a significant effect on the response behavior in heave and pitch degrees-of-freedom. In all further studies, coupled stiffness matrix has been considered Effect of structural damping on the response of spar platform Two cases are taken for structural damping ratio of 3% and 5% with the initial horizontal force of 2500 kn. Table 6 gives the response for 5% and 3% structural damping ratio. Table 6 shows that there is an increase of 0.23% in surge, decrease of 3.58% in heave and decrease of 0.26% in pitch direction when 5% structural damping is considered in comparison to 3% structural damping. Figs. 12 and 13 show the comparison between the time histories of the coupled surge and heave response for structural damping on the responses have of the Spar platform. From Figs. 12 and 13 it is

25 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Table 6 Maximum response for different structural damping ratio Damping Max. displacement (m) Max. rotation (radian) Surge Heave Pitch 5% 3% time (sec) 12.5 Wave period 12.5 sec Wave heigth 7 m structural damping 5% -D- structural damping 3% Fig. 12. Effect of structural damping in coupled surge response of a Spar platform. 2.5 time (sec) Wave period 12.5 sec Wave 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 oscillating at negative mean. It is observed that for higher structural damping ratio there is no effect in surge and pitch response, whereas it affects heave responses significantly.

26 512 A.K. Agarwal, A.K. Jain / Ocean Engineering 30 (2003) Table 7 Maximum response for variation in C m Inertia coefficient Max. displacement (m) Max. rotation (radian) Surge Heave Pitch C m = 2.0 C m = Effect of inertia coefficient C m on the response of spar platford Two cases are taken for C m equal to 2 and 1.8 with the initial horizontal force of 2500 kn. Surge force, heave force and pitch moment reduces when C m reduces from 2 to 1.8 while calculating force using Morison's equation. Table 7 gives the response for C m equal to 2 and 1.8. Table 7 shows that there is a decrease of 15.79% in surge, decrease of 7.25% in heave and decrease of 10.39% in pitch response when C m equal to 1.8 is considered in comparison to C m equal to 2. Figs show the comparison between the time histories of the coupled surge, heave and pitch responses for different values of inertia coefficient, C m on the response of Spar platform. From Figs. 14 and 15 it is observed that both surge and heave responses have negative offset and is oscillating at 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 responses are significantly affected with the decrease in inertia coefficient C m Effect of drag coefficient C d on the response of Spar platform Two cases are taken for C d equal to 1 and 0 with initial horizontal force of 2500 kn. Surge and heave force reduces as the total force decreases when drag coefficient 2.5 time (sec) Wave period 12.5 sec Wave heigth 7 m Fig. 14. Effect of inertia coefficient in coupled surge response of a Spar platform.