Design Selection Analysis for Mooring Positioning System of Deepwater Semi-submersible Platform

Dongsheng Qiao1 , Jinping Ou1 , Fei Wu2

1. Center for Deepwater Engineering, Dalian University of Technology, Dalian, China 2. Luxun Academy of Fine Arts, Shenyang, China

ABSTRACT Aiming at design selection analysis for mooring positioning system of a semi-submersible platform applied in South China Sea, catenary mooring system, semi-taut mooring system and taut mooring system are respectively considered. The three types of mooring positioning system have the similar static restoring force characteristics, the same mooring line number and angle arrangement. The dynamic coupling effects between the semi-submersible platform and its mooring lines are investigated through numerical simulation method. The 3-dimension hydrodynamic finite element model of semi-submersible platform is built firstly. The wave forces are calculated under diffraction theory by boundary element method, and the wind forces are obtained from the wind tunnel test with the model scale 1:100, and the current forces are considered as steady. The platform motions under combined action of wind, wave and current are solved by Runge-Kutta method respectively under working and extreme conditions in South China Sea. The specific numerical results and analysis conclusions would be helpful for selecting the mooring system and the motion performance study in the semi-submersible platform preliminary design. KEY WORDS: Mooring system; semi-submersible platform; dynamic analysis; response; tension. INTRODUCTION In recent years, the application and research of floating platforms are becoming more and more widespread with the exploration of deepwater hydrocarbon resources to deep and ultra-deep water. The types of floating platforms such as Semi-submersible platform, Spar platform and Floating Production Storage and Offloading (FPSO) are all need to be positioned through mooring system when they are working as production platforms. The integrity of production risers depends on the station keeping ability. Now the floating platforms moves beyond 2000m and targets the 3000m range, so the need for efficient station keeping mooring systems increases. The common used mooring system includes three types which are plotted in Fig. 1: catenary, semi-taut and taut. Catenary mooring system

is widely applied in practice during these years and usually made up of chain, wire or combined of them. Catenary mooring system supplies restoring force to platform depending on weights of mooring line. With the increasing water depth, the high weight of chain becomes one of the restrictions. The representative feature of catenary mooring system is that parts of bottom chain keep lying on seabed when the platform moves. Semi-taut mooring system supplies restoring force to platform depending on weights and elastic deformation of mooring line. Taut mooring system supplies restoring force to platform depending on elastic deformation of mooring line (Liu and Huang, 2007). Compared with catenary mooring system, taut mooring system has some advantages: little positioning radius which may reduce occupation area of seabed and interference with other underwater facility; little length of mooring line which may reduce economic costs (Devlin et al., 1999; Christian and Shankar, 2002). Taut mooring system usually has high restoring stiffness which may cause high mooring line tension, so the safety margin is lower than catenary mooring system. The characteristics of semi-taut mooring system are between catenary and taut mooring system.

(a) Catenary (b) Semi-taut (c) Taut

Fig. 1 Configuration of mooring line Because of patent protection of TLP and Spar platform and other reasons, the semi-submersible platforms are the first chosen in South China Sea (Xie et al., 2007). The HYSY-981 semi-submersible drilling platform has been designed and manufactured by China National Offshore Oil Corporation (Zhu et al., 2010). The semi-submersible platform usually has small waterline areas, low initial investment and operation cost. When the semi-submersible platform is positioned through mooring system, the structure may experience large low-frequency (LF) motions, known as slow-drift motions, under nonlinear low frequency wave forces excitation. Meanwhile, the wave frequency

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Proceedings of the Twenty-second (2012) International Offshore and Polar Engineering ConferenceRhodes, Greece, June 17–22, 2012Copyright © 2012 by the International Society of Offshore and Polar Engineers (ISOPE)ISBN 978-1-880653-94–4 (Set); ISSN 1098-6189 (Set)

www.isope.org

forces excitation may cause significant dynamic responses of platform. These excitations are sensitive to different types of mooring system, so analysis the influence of mooring system to semi-submersible platform during the design stage is necessary. Based on the ocean environmental loads of semi-submersible platform and bearing capacity of mooring line materials, the mooring line number is determined mainly to resist the initial mean load. Usually, the angle arrangement of mooring line is symmetric and uniform unless the orientation of ocean environmental loads is fixed. After several adjustments, the final mooring line number and angle arrangement are fixed. Through detail dynamic analysis, the initial pretension and length are obtained. In past years, many scholars have revealed the coupling effects between floating platform and its mooring system should be considered in predicting their motions (Huse, 1986; Wichers and Huijismans, 1990). Coupled dynamic analysis technique has been developed from quasi-static approach (Cao and Zhang, 1997) to fully couple dynamic approach (Ma et al., 2000; koo et al., 2004; Li et al., 2010). Despite this, only little scholars investigate the impact of difference mooring model to motions of floating platform. Chen et al. (2001) use a quasi-static approach (SMACOS) and a coupled dynamic approach (COUPLE) to calculate motions of a spar and its mooring system in three water depths. Shafieefar and Rezvani (2007) present genetic algorithm to optimize the mooring design of floating platforms. Tong et al. (2009) compare the dynamic effect for semi-submerged platform respectively with catenary and taut mooring system. Sun and Wang (2010) study on motion performance of deepwater spar platform under equally distributed mooring method and grouped mooring method. In this work, global responses analysis of a semi-submersible platform respectively using catenary, semi-taut and taut mooring system in 1500m water depth is calculated. Three different types of mooring system have the similar static restoring force characteristics, the same mooring line number and angle arrangement. The dynamic coupling effects between the semi-submersible platform and its mooring lines are investigated through numerical simulation method which has been validated through model tests by Qiao and Ou (2010). Two environmental conditions in South China Sea are considered, respectively are 1-year working conditions and 100-year extreme conditions. DESCRIPTION OF SEMI-SUBMERSIBLE PLATFORM AND ENVIRONMENTAL CONDITIONS Semi-submersible Platform Configurations The main structure of semi-submersible platform consists of two pontoons, four columns, deck and derrick, and the main characteristic parameters are listed in Table 1. Mooring System Configurations The mooring system is a four (4×4) groups as shown in Fig. 2. Each mooring line consists of three segments: upper chain, middle wire and bottom chain. Each group in the mooring system is separated by 90-degree spacing and each line in the same group is separated by 5-degree spacing. Three types of mooring system, respectively are catenary, semi-taut and taut, are calculated and added to the semi-submersible platform. Considering the gravity, tension and mooring line extension, the piecewise extrapolating method is employed to the static analysis of the

multi-component mooring line (Qiao and Ou, 2009). Aiming to ensure the three types of mooring positioning system have the similar static restoring force characteristics, the arrangement of three segments in taut mooring system may not meet the practical application demands of project, but using in this research is still suitable. Through optimization design of the three types of mooring system, the mooring systems configurations in 1500m water depth are specified in Table 2. The mooring line properties are shown in Table 3. The tension-horizontal displacement characteristic curve of single mooring line is plotted in Fig. 3. The total horizontal force-horizontal displacement is plotted in Fig. 4. Table 1. Parameters of semi-submersible platform

Parameters Value Deck (m) 74.42×74.42×8.60

Column (m) 17.385×17.385×21.46Pontoon (m) 114.07×20.12×8.54 Tonnage (t) 48206.8

Center of gravity from water surface (m) 8.9 Roll gyration radius (m) 32.4 Pitch gyration radius (m) 32.1 Yaw gyration radius (m) 34.4

Initial air gap (m) 14 Diameter of brace (m) 1.8

Water depth (m) 1500

Fig. 2 Mooring system layout Table 2. Mooring system configurations

Length (m) Upper

chain Middle

wire Bottom chain

Pretension (kN)

Catenary 300 2000 1500 3000 Semi-taut 300 1600 800 2900

Taut 900 1100 100 2650 Table 3. Mooring line properties

Item Chain (K4 studless)

Wire (Sprial strand)

Diameter (m) 0.095 0.095 Weight in water (N/m) 1605.9 356.9

Axial stiffness (N) 6.7681E8 8.3391E8 Breaking stress (N) 9.0444E6 7.8765E6

1100

Environmental Conditions The environmental conditions considered is 1-year return period and 100-year return period in South China Sea as listed in Table 4, respectively represents working and extreme conditions. The 1-minute mean wind speed, Jonswap wave spectrum and uniform current are used in the numerical simulation. The wind, wave and current are assumed collinear. The environmental heading is assumed to be from X-axis as shown in Fig. 2.

0 20 40 60 80 1001.5

2

2.5

3

3.5x 106

Offset (m)

Tens

ion

(N)

CatenarySemi-tautTaut

Fig. 3 Static offset curve of single mooring line

0 20 40 60 80 100-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0x 107

Offset (m)

Hor

izon

tal r

esto

ring

forc

e (N

)

CatenarySemi-tautTaut

Fig. 4 Surge static offset curve of mooring system Table 4. Environmental conditions

Item 1-year 100-year wind speed (m/s) 23.15 55

Significant wave height (m) 6.0 13.3 Wave Peak period (s) 11.2 15.5 Current speed (m/s) 0.93 1.97

GOVERNING EQUATIONS AND FORMULATION Governing equation of mooring line The mooring line is generally presumed to be a completely flexibile component during the motion response analysis and the motion governing equation is proposed by Berteaux (1976).

( )( )a Dn Dt In ItV Um m F F F F T Gt t

→ →→ → → → → →∂ ∂

+ − = + + + + +∂ ∂

(1)

1/ 212Dn w Dn n n nF C D V V Vρ

→ → → →

= Δ Δ ⋅Δ (2)

12Dt w Dt t tF C D V Vρ π

→ → →

= Δ Δ (3)

21 ( )4In w In

V UF D Ct t

ρ π→ →

→ ∂ ∂= −

∂ ∂ (4)

21 ( )4It w It

V UF D Ct t

ρ π→ →

→ ∂ ∂= −

∂ ∂ (5)

where m is mass of mooring line, am is added mass of mooring line,

V→

is velocity vector of the mooring line, U→

is velocity vector of fluid,

DnF→

is mooring line normal drag forces (per unit length), DtF→

is

mooring line tangential drag forces (per unit length), InF→

is mooring

line normal inertia forces (per unit length), ItF→

is mooring line

tangential inertia forces (per unit length), T→

is tension of mooring line

and G→

is net weight of mooring line, 31025 /w kg mρ = is fluid density,

0.05DtC = is tangential drag coefficient, D is wire diameter, tV→

Δ is relative tangential velocity of the fluid, 1.2DnC = is normal drag

coefficient, nV→

Δ is relative normal velocity of the fluid. 0.25ItC = is tangential inertial coefficient and 0.5InC = is normal inertial

coefficient. ( )V Ut t

→ →

∂ ∂−

∂ ∂ is relative acceleration between fluid and

mooring line. As far as formula (1) is concerned, the motion governing equation is a strong complex time-varying non-linear equation which needs to be solved using a numerical method. The non-linear finite element method is used for the solution in this paper. In ABAQUS, the mooring line is simulated as hybrid beam element (Timoshenko, 1956) and the Newton-Raphson iterative method is used to solve non-linear problem directly. Hydrodynamic Model of Semi-submersible Platform The commercial program AQWA is applied here in the calculating procedure for the coupled motion. The wave forces on the semi-submersible platform are calculated under diffraction theory by boundary element method, and the panel model is sketched in Fig. 5.

Fig. 5 Panel model of semi-submersible platform According to AQWA, the platform structure is treated as a rigid body, the wind loads and wave forces acting on the platform can be described as follows: Wave forces calculation In the numerical simulation, the transient wave forces acting on platform under irregular waves are approximately given as: (Cummins, 1962; Oortmerssen, 1976)

1101

(1) (2)( ) ( ) ( ) ( 1,2,...,6)i i iF t F t F t i= + = (6)

where (1) ( )iF t and (2) ( )iF t are the first and second order wave forces. They are given as:

(1) 1

0

( ) ( ) ( ) ( 1,2,...,6)t

i iF t h t d iτ η τ τ= − =∫ (7)

(2) 21 2 1 2 1 2

0 0

( ) ( , ) ( ) ( ) ( 1,2,...,6)t t

i iF t h t t d d iτ τ η τ η τ τ τ= − − × =∫ ∫ (8)

where 1( )ih t and 2 ( )ih t are the first and second order impulse response functions in the time domain, and ( )tη is the sea surface elevation.

1( )ih t and 2 ( )ih t are given as:

1 (1)

0

1( ) Re ( ) i ti ih t H e dωω ω

π

∞⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭∫ (9)

1 1 2 2( )2 (2)1 2 1 2 1 22

0 0

1( , ) Re ( , )2

i t ti ih t t H e d dω ωω ω ω ω

π

∞ ∞+⎧ ⎫⎪ ⎪= ⎨ ⎬

⎪ ⎪⎩ ⎭∫ ∫ (10)

where 1( )iH ω and 21 2( , )iH ω ω are the first and second square order

transfer functions for wave force (Teng et al., 1999). Wind forces calculation The wind loads acting on the platform are given as:

2, ,

12wind H d H H HF C A Vρ= (11)

where HV is the average wind velocity at the height H above sea level, 31.29 /kg mρ = is density of air, HA is projected area of platform in the

direction of wind. ,d HC is wind pressure coefficient at the height of

H above sea level, which is obtained from wind tunnel experiment with a 1:100 scaled model made up of columns, deck and derrick (Zhu et al.,2009). Coupled analysis of semi-submersible platform and mooring lines The model of coupled analysis of semi-submersible platform and its mooring lines is shown in Fig. 6. The equation of motion for the coupled system in the time domain is given as follows:

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) 1,2,..,6)

t

kj kj j j kj k j kj j

j j

M m t K t d B t C t

F t G t j

ξ ξ τ τ τ ξ ξ−∞

+ + − + +

= + =

∫&& &

(

(12)

where kjM is the mass matrices of the platform, kjC the hydrostatic

restoring stiffness, kB is the viscous damping of the system, ( )jG t is

the mooring force, ( )jF t is the external forces which contain wind loads and wave forces. Assuming the platform motion is related to simple harmonic motion and compared with the motion equation of the platform in frequent domain, any motion of the platform can be described by Eq.12 (Li and Teng, 2002). kjM and ( )kjK t are given as:

( )kj kjm a= ∞ (13)

0

2( ) ( )cos( )kj kjK t b t dtω ωπ

∞

= ∫ (14)

where kja and kjb are, respectively, the added mass and damping of

the platform in frequent domain. Eq.12 can be described as:

1[ , , ] [ ( ) ( ) ( ) ( )

( ) ( )]

t

j j j kjkj kj

k kj j

F t F t G t K t dM m

B t C t

ξ ξ ξ ξ τ τ τ

ξ ξ−∞

= = + − −+

+ +

∫&& &

&

(15)

Eq.15 is solved by Runge-Kutta method in this paper. The displacement and velocity of the platform at the time step t t+ Δ are written as:

1 2 3( ) ( ) ( ) ( ) / 6t t t t t t M M Mξ ξ ξ+ Δ = + Δ + Δ + +& (16)

1 2 3 4( ) ( ) ( 2 2 ) / 6t t t M M M Mξ ξ+ Δ = + + + +& & (17) where tΔ is taken as the time step, and

1 [ , ( ), ( )]M tF t t tξ ξ= Δ & ,

12

( )[ , ( ) , ( ) ]2 2 2t t t MM tF t t tξξ ξΔ Δ

= Δ + + +&

& ,

1 23

( )[ , ( ) , ( ) ]2 2 2 2t t t tM MM tF t t tξξ ξΔ Δ Δ

= Δ + + +&

& ,

24 3[ , ( ) ( ) , ( ) ]

2 2t tMM tF t t t t t Mξ ξ ξΔ Δ

= Δ + + Δ + +& & .

The function [ , , ]F t ξ ξΔ & can be solved using the displacement ( )tξ

and velocity ( )tξ& of the platform at the time t , and the displacement

( )t tξ + Δ and velocity ( )t tξ + Δ& of the platform at the time step t t+ Δ can be calculated by using Eqs.16 and 17. The process will be repeated until the computation is completed.

Fig. 6 Coupled analysis model of semi-submersible platform NUMERICAL SIMULATION AND RESULTS ANALYSIS Natural Periods The natural periods for three types of mooring system are presented from free decay tests in calm water based on the first six cycles. The initial amplitudes for surge, heave and pitch respectively are 10m, 4m and 10degree. The natural periods derived from free decay simulations in calm water are summarized in Table 5. Figs. 7-9 respectively show the surge, heave and pitch free decay test results. Table 5. Natural periods

Case Surge (s) Heave (s) Pitch (s) Catenary 149 21 53 Semi-taut 166.5 21 54

Taut 184.5 21 58 According to Table 5, the natural periods of surge and pitch for taut mooring system is longer than that for semi-taut and catenary mooring system, and the one for catenary mooring system is smallest. The reason is that horizontal stiffness of taut mooring system is smallest and horizontal stiffness of catenary mooring system is largest, which could

1102

be obtained from Figs. 3-4. There are no significant changes in the natural periods of heave for three cases, which mean the vertical stiffness of three cases is the same. Natural Damping Ratios According to Figs. 7-9, the natural damping ratios derived from free decay simulations in calm water are summarized in Table 6. The natural damping ratios of surge and pitch for catenary mooring system is little larger than (about 10%) that for semi-taut and taut mooring system, and the one for taut mooring system is smallest. The reason is that total length of catenary mooring line is the largest and the one of semi-taut is the second place, so the drag force in catenary mooring line is the largest and the one of semi-taut is the second place too. Therefore, the damping of catenary mooring line is the largest. The natural damping ratios of heave for three cases is much smaller than the one of surge and pitch, which mean damping ratios of heave are negligible in the analysis. Table 6. Natural damping ratios

Case Surge Heave Pitch Catenary 6.70% 0.68% 4.03% Semi-taut 6.42% 0.51% 3.82%

Taut 6.15% 0.44% 3.70%

0 200 400 600 800 1000-10

-5

0

5

10

Time (s)

Sur

ge d

ecay

(m)

CatenarySemi-tautTaut

Fig. 7 Surge decay in calm water

0 20 40 60 80 100 120 140

1

2

3

4

5

Time (s)

Hea

ve d

ecay

(m)

CatenarySemi-tautTaut

Fig. 8 Heave decay in calm water

0 50 100 150 200 250 300 350-10

-5

0

5

10

Time (s)

Pitc

h de

cay

(m)

CatenarySemi-tautTaut

Fig. 9 Pitch decay in calm water

Motion Responses of Semi-submersible Platform According to calculation method on semi-submersible platform above, the duration of 3h under 1-year and 100-year return period environmental conditions are numerically simulated respectively using catenary, semi-taut and taut mooring systems. The statistics of the semi-submersible motions (surge, heave and pitch) respectively using three types of mooring systems are summarized in Table 7-8 (APPENDIXES). As given in Table 7-8, the LF range of the surge is 0-0.2 rad/s and its WF range is 0.2-1.2 rad/s, while the LF range of heave and pitch is 0-0.25 rad/s and related WF range is 0.25-1.2 rad/s. The motions time series and their spectrums under 1-year return period environmental conditions are plotted in Figs. 10-15, and the ones under 100-year return period environmental conditions are omitted for brief. All spectrums are smoothed by a 10-point averaging window.

0 2000 4000 6000 8000 100000

10

20

30

40

50

60

Time (s)

Sur

ge (m

)

Catenary Semi-taut Taut

Fig. 10 Time series of surge motions

0 0.05 0.1 0.15 0.20

1000

2000

3000

4000

5000

6000

Fre (rad/s)

Sur

ge re

spon

se p

ower

den

sity

(m2*

s)

CatenarySemi-tautTaut

0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

Fre (rad/s)

Sur

ge re

spon

se p

ower

den

sity

(m2*

s)

CatenarySemi-tautTaut

Fig. 11 Surge motions spectrums Based on the results of surge motions of semi-submersible platform under 1-year and 100-year return period environmental conditions, the average surge motion for three types of mooring system is catenary < semi-taut < taut. In the LF range, the change laws of standard deviations of surge motion are the same as average surge motion. The decrease is because damping contribution for semi-submersible platform increases with efficient mooring line length increases. The efficient mooring line length of catenary mooring system is larger than that for semi-taut and taut mooring system, and the one for taut mooring system is smallest. In the WF range, the standard deviations of surge motion show insignificant changes in three cases. This is because

1103

when the natural frequencies of the semi-submersible platform are far smaller than the WF range, the inertia forces in the WF range are dominant. Thus the damping of the mooring lines does not make significant contributions to the reduction of the surge in the WF range. Further, the total mooring lines tensions applied on the semi-submersible platform are much smaller than the wave forces on the semi-submersible platform in the WF range. Based on the results of heave and pitch motions of semi-submersible platform, the average and standard deviations in LF range for three types of mooring system is still catenary < semi-taut < taut, and the results under 100-year return period environmental conditions enlarge the change laws. The reason is the mooring damping increases with the excitation motion of mooring line increases, and the excitation motion under 100-year conditions is significant larger than 1-year conditions. In the WF range, the standard deviations also show insignificant changes in three cases.

0 2000 4000 6000 8000 10000-4

-2

0

2

4

Time (s)

Hea

ve (m

)

CatenarySemi-tautTaut

Fig. 12 Time series of heave motions

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Fre (rad/s)

Hea

ve re

spon

se p

ower

den

sity

(m2*

s)

CatenarySemi-tautTaut

0.4 0.6 0.8 1 1.20

1

2

3

4

5

6

7

Fre (rad/s)

Hea

ve re

spon

se p

ower

den

sity

(m2*

s)

CatenarySemi-tautTaut

Fig. 13 Heave motions spectrums Mooring Line Tensions With the same as motions responses of semi-submersible platform, two mooring lines are chosen to analyze, #1 is in downstream and the most unloaded mooring line, and #8 is in upstream and the most loaded mooring line. The statistics of two mooring line tensions respectively using three types of mooring systems are summarized in Table 9 (APPENDIXES). The mooring line tensions time series and their spectrums under 1-year return period environmental conditions are

plotted in Figs. 16-17, and the ones under 100-year return period environmental conditions are omitted for brief. All spectrums are smoothed by a 10-point averaging window too.

0 2000 4000 6000 8000 10000-6

-4

-2

0

2

4

6

Time (s)

Pitc

h (d

eg)

Catenary Semi-taut Taut

Fig. 14 Time series of pitch motions For the most unloaded mooring line #1, the changes laws of average mooring line tension respectively in catenary, semi-taut and taut mooring system are not fixed. The reason is the initial pretension of mooring line is not uniform. But the change laws of dynamic average mooring line tension both is catenary > semi-taut > taut, which is calculated through average mooring line tension minus their initial pretension. The standard deviations in LF and WF range for three types of mooring system is still catenary > semi-taut > taut and the results under 100-year return period environmental conditions enlarge the change laws too. The reason is that transfers of mooring line tension are related to the length and shape of mooring line. To unloaded mooring line #1, the efficient length in catenary mooring system is larger than the one in semi-taut and taut mooring system, so the standard deviations for catenary system is the largest. The phenomenon may cause more severe fatigue problem in catenary mooring system. For the most loaded mooring line #8, the change laws of average and standard deviations are just the opposite of #1. The reason is the motion responses of semi-submersible using taut mooring system are the largest, and this phenomenon is the most obvious in #8 under 100-year conditions.

0 0.05 0.1 0.15 0.2 0.250

10

20

30

40

50

60

70

Fre (rad/s)

Pitc

h re

spon

se p

ower

den

sity

(deg

2*s)

CatenarySemi-tautTaut

0.4 0.6 0.8 1 1.20

5

10

15

20

Fre (rad/s)

Pitc

h re

spon

se p

ower

den

sity

(deg

2*s)

CatenarySemi-tautTaut

Fig. 15 Pitch motions spectrums

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CONCLUSIONS Through comparison of global responses of a semi-submersible platform and mooring line tensions respectively using catenary, semi-taut and taut mooring system in South China Sea. Meanwhile, the three types of mooring systems have the similar static restoring force characteristics, the same mooring line number and angle arrangement. The following preliminary findings are made: (1) Under the 1-year and 100-year return periods environmental conditions in South China Sea, the global responses of the semi-submersible are logical. The influences of using catenary, semi-taut and taut mooring system to the motions and mooring line tension respectively are similar.

0 2000 4000 6000 8000 100002.4

2.6

2.8

3

3.2

3.4

3.6

3.8x 106

Time (s)

Tens

ion

(N)

Catenary (#1) Semi-taut (#1) Taut (#1) Catenary (#8) Semi-taut (#8) Taut (#8)

Fig. 16 Time series of mooring line tension

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2x 1012

Fre (rad/s)

Tens

ion

resp

onse

pow

er d

ensi

ty (N

2*s)

Catenary (#1)Semi-taut (#1)Taut (#1)Catenary (#8)Semi-taut (#8)Taut (#8)

0.2 0.4 0.6 0.8 1 1.20

0.5

1

1.5

2

2.5

3

3.5x 109

Fre (rad/s)

Tens

ion

resp

onse

pow

er d

ensi

ty (N

2*s)

Catenary (#1)Semi-taut (#1)Taut (#1)Catenary (#8)Semi-taut (#8)Taut (#8)

Fig. 17 Mooring line tension spectrums (2) The natural periods of surge and pitch for taut mooring system is longer than that for semi-taut and catenary mooring system, and the one for catenary mooring system is smallest. (3) The natural damping ratios of surge and pitch for catenary mooring system is little larger (about 10%) than that for semi-taut and taut mooring system, and the one for taut mooring system is smallest. The damping ratios of heave are negligible in the analysis.

(4) Only according to motion responses of semi-submersible platform, the catenary mooring system is the most suitable because of the smallest motions, and the semi-taut mooring system is the second place. (5) Only according to mooring line tensions, the numerical simulation results are controversial. For the most unloaded mooring line, the taut mooring system is the most suitable because of its little amplitude changes. For the most loaded mooring line, the catenary mooring system is the most suitable because of its small extreme value. The semi-taut mooring system is the second place too. ACKNOWLEDGEMENTS This paper is funded in part by National Basic Research Program of China (Grant NO. 2011CB013702; 2011CB013703), Science Fund for Creative Research Groups of the National Natural Science Foundation of China (Grant NO. 50921001) and Fundamental Research Funds for the Central Universities. REFERENCES Berteaux, HO (1976). Buoy Engineering, Wiley Interscience

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APPENDIXES Table 7. Statistics of motions of semi-submersible platform under 1-year return period environmental conditions

Surge (m) Heave (m) Pitch (deg) Motion Catenary Semi-taut Taut Catenary Semi-taut Taut Catenary Semi-taut Taut Average 22.82 29.52 35.04 -0.01 -0.05 -0.39 0.40 0.72 0.78

σ 4.84 5.30 5.50 0.34 0.34 0.34 0.91 0.91 1.15 Max. 38.59 46.68 52.82 1.41 1.28 0.92 3.69 4.08 5.09 Min. 11.56 17.28 22.60 -1.12 -1.25 -1.65 -2.07 -2.41 -2.72 LF σ 4.83 5.30 5.51 0.02 0.02 0.03 0.67 0.66 0.98

LF Max. 38.39 46.60 52.62 0.65 0.53 0.20 2.98 3.25 4.27 LF Min. 12.43 18.06 23.23 -0.13 -0.27 -0.68 -0.52 -0.60 -1.19 WF σ 0.54 0.58 0.62 0.34 0.34 0.34 0.62 0.62 0.62

WF Max. 12.87 16.48 19.15 1.22 1.26 1.33 2.37 2.40 2.96 WF Min. -3.52 -4.21 -4.72 -1.19 -1.19 -1.16 -2.39 -2.34 -2.19

Table 8. Statistics of motions of semi-submersible platform under 100-year return period environmental conditions

Surge (m) Heave (m) Pitch (deg) Motion Catenary Semi-taut Taut Catenary Semi-taut Taut Catenary Semi-taut Taut Average 88.99 91.42 93.94 -0.04 -0.34 -1.09 1.13 2.07 2.20

σ 6.60 7.34 7.67 1.09 1.11 1.14 1.94 2.12 2.62 Max. 120.77 125.85 133.60 3.58 3.41 2.97 6.63 7.68 8.32 Min. 68.96 73.03 75.52 -4.19 -4.63 -5.67 -6.45 -6.62 -6.82 LF σ 8.52 7.20 6.50 0.05 0.10 0.20 1.17 1.58 2.22

LF Max. 122.96 120.48 119.44 1.00 0.84 0.38 3.81 4.15 5.02 LF Min. 49.53 52.23 54.08 -0.33 -0.86 -2.08 -2.33 -3.69 -4.33 WF σ 1.92 1.95 1.98 1.09 1.10 1.12 1.43 1.43 1.42

WF Max. 39.49 41.57 42.95 3.61 3.62 3.90 6.16 7.11 9.86 WF Min. -8.82 -9.12 -9.36 -3.79 -3.88 -3.95 -4.70 -4.78 -4.80

Table 9. Statistics of mooring line tensions

1-year return period environmental conditions (kN) 100-year return period environmental conditions (kN) #1 #8 #1 #8 Tension

Catenary Semi-taut Taut Catenary Semi-

taut Taut Catenary Semi-taut Taut Catenary Semi-

taut Taut

Average 2774.53 2735.01 2556.18 3408.66 3366.80 3241.15 2123.60 2260.48 2337.68 4689.69 5086.54 5690.83σ 57.46 47.64 26.88 76.17 78.86 99.23 65.09 46.66 18.00 198.15 411.97 707.00

Max. 2907.40 2846.34 2628.06 3669.15 3642.97 3632.97 2289.85 2383.01 2394.81 5644.75 7377.02 7677.25Min. 2585.85 2581.07 2464.66 3237.07 3212.38 3048.43 1870.19 2077.18 2274.50 4256.86 4289.56 4372.36LF σ 57.39 47.62 26.88 76.13 78.01 99.36 65.11 46.11 17.80 196.04 401.63 651.96

LF Max.

2900.48 2839.95 2630.41 3665.83 3638.21 3611.84 2548.89 2595.34 2544.50 5532.51 7351.26 7657.39

LF Min. 2593.43 2586.79 2462.74 3248.76 3220.05 3055.53 1890.38 2087.86 2277.77 3980.15 4211.20 4332.51WF σ 5.81 5.20 4.14 7.54 8.98 10.29 14.49 12.91 6.93 42.65 106.01 289.12

WF Max.

34.51 30.92 21.10 187.84 196.68 265.65 80.73 65.45 40.66 689.72 914.68 1713.10

WF Min.

-162.34 -158.01 -128.17 -56.47 -60.27 -83.30 -420.51 -345.34 -212.72 -181.33 -517.81 -1623.7

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