Chapter 1 To 5

CHAPTER 1 INTRODUCTION 1.1

GENERAL During the last few decades, the exploration and production of offshore petroleum

reserves have progressively moved to deeper water sites around the world. Operations in 1000 to 1200 m water depth have become common and the offshore industry gears up to venture into ultra-deepwaters of 3000 m and beyond for new finds. Traditional shallow water platforms e.g. jack-up and jacket type drilling and production platforms, have given way to floating platforms, which are more economical in deep waters. These platforms, such as tension leg platforms (TLPs) and spars are kept on station using vertical tendons, which are sometimes combined with more conventional spread mooring systems. Unlike the rigid platforms, the motion characteristics of compliant platforms play significant role in their operations. Therefore, the study of structural behavior and dynamic response of these platform concepts in order to optimize their designs are presently being actively pursued in the literature. In ultra-deepwater sites the use of seabed-mounted platforms becomes uneconomic and use of floating platform becomes the only viable option. Numerous small oil fields have been discovered in very deep waters. New concepts of platform construction, exploration, drilling and production are necessary for economic development of these minimal oil fields in deepwater locations in hostile environment. 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

1

deposits. It is being considered as the next generation of deep water offshore structures by many oil companies. A spar platform consists of a vertical cylinder, which floats vertically in the water. Fig.1.1 shows a typical spar platform with the basic arrangements. 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 m 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 at the center well by separate buoyancy cans,

d)

Always stable because center of buoyancy (CB) is above the center of gravity (CG)

e)

It has favorable 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 2

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 are normally exposed to high waves on semi-submersible, drilling units would be protected inside the spar platform. Sea motion inside spar platform center well would be minimum

Figure 1.2 shows three types of Spar configurations: 1. Classic Spar 2. Truss Spar 3. Cell Spar

3

Fig 1.1 A typical spar platform with basic arrangements and terms

Classic Spar

Truss Spar

Cell Spar

Fig 1.2 Various configurations of a spar platform

4

1.2

OBJECTIVE •

The objective of the project is to develop a program which can be used to perform design of spar platform.

•

The program also includes the analysis of motion response of spar, using simplified approach.

5

CHAPTER 2 DEVELOPMENT OF SCHEME 1.3

GENERAL The computer program starts with reading inputs, mainly wave height (H w ), water

depth(d), Time period(T), Topside Weight (WT) from a file. Using these inputs diameter and thickness of the spar are determined based on buoyancy requirements. Designed structure is checked for safety against hydrostatic pressure, hoop buckling, tension and compression. Circumferential stiffening rings are provided based on API recommendations. Total weight of spar is calculated including weight of stiffener and topside weight. Total buoyancy is determined from the draft and diameter of spar. Then, the hydrostatic stability check is performed in order to ensure stable equilibrium. If the required metacentric height is not achieved, the structure is redesigned after providing ballast. Motion response analysis is done using simplified calculation approach. An optimized design can be achieved by running the program for different drafts to find out corresponding response amplitude operator (RAO).The design corresponding to minimum diameter and minimum RAO can be taken as an optimized design. A flow diagram showing the general flow of the program is given below. Theoretical background of each step involved in the flow diagram is explained in Chapter 3.

6

1.4

FLOW DIAGRAM

Fig 2.2 Flow Diagram

7

CHAPTER 3 THEORETICAL BACKGROUND 3.1

GENERAL The procedure followed for the structural design of spar, weight estimation, hydrostatic stability check and response prediction using simplified approach are described here.

3.2

STRUCTURAL DESIGN The given parameters for the structural design are the topside weight (WT), wave height (Hw) and water depth(d). The aim is to obtain the dimensions of spar.

Initial sizing of the hull is determined by the following steps: •

Initial value for the draft (Df) is assumed.

•

Free board is calculated as Hf = 0.6x Hw +1.5 m.

•

Total length of spar, Ls = Df + Hf

•

Ballast Weight (WB) is initially taken as zero.

•

Diameter is calculated from buoyancy requirements as

Ds =

(W T + WB)

(Π /4 x ρw x Df) - ( Πx ρs x Ls )/(D/t))

•

Thickness (t) is calculated using assumed D/t ratio. 8

•

Check for hoop stress due to hydrostatic pressure.

fh =

pD s Fhc ≤ 2t SFh

Where, fh = hoop stress due to hydrostatic pressure, MPa p = hydrostatic pressure, MPa SFh = safety factor against hydrostatic collapse (API RP 2A) Fhc = critical hoop buckling stress, MPa (API RP 2A) Multiply p with a factor of 1.25 to take into account the pressure due to wave elevation. •

Check for axial tension. The allowable tensile stress, Ft, for cylindrical members subjected to axial tensile loads should be determined from: Ft = 0.6 Fy

Where, Fy = Yield strength (Mpa) •

Check for axial compression. The allowable axial compressive stress shall be determined from the following AISC formulas for the members with a D/t ratio equal to or less than 60. Fa =

[1 - (Kl/r) 2 /2 Cc 2 ]F y 5/3 + (3(Kl/r)/8 Cc ) - ((Kl/r) 3 /8 Cc 3)

9

for

Kl/r < Cc

Fa =

12 Π2 E 23 (Kl/r) 2

for

Kl/r ≥ Cc

Where, E = Young’s Modulus of elasticity, ksi (MPa) K = effective length factor l = unbraced length (m) r = radius of gyration (m) Cc =

2 Π2 E Fy

xe

xc

For D/t ratio greater than 60, substitute the critical local buckling stress (F or F ,

c

y

whichever is smaller) in determining C and F

•

Check for local buckling:

xe

The elastic local buckling stress, F is determined from

Fxe = 2CEt/D

Where,

10

C= Critical elastic buckling coefficient

D= Outside diameter(m)

t = Wall thickness (m)

xc

The inelastic local buckling stress, F is determined from

Fxc = Fy x [1.64 - 0.23 (D/t)

•

1/4

]

Check for hoop buckling stresses:

Elastic hoop buckling is given by

Fhe =2C hEt/ Ds M =Lx (2 Ds /t)

1/2

/ Ds

Ch =0.44 t/ Ds Ch =0.44 t/ Ds +0.21 (D s /t) Ch =0.736/(M

- 0.636)

Ch =0.755/(M

- 0.559)

@ M ≥1.6 Ds /t 3

/ M

4

Ch =0.8

@ 0.825

Ds /t ≤M <1.6 Ds /t

@3.5

≤M <0.825

@1.5

≤M <3.5

@M <1.5

Where, L is the spacing between cylindrical rings M is geometric parameter Critical hoop buckling stress is given by 11

Ds /t

Elastic

buckling

Fhc =Fhe

@ Fhe ≤0.55 Fy

Inelastic buckling Fhc =0.45 Fy +0.18 Fhe

@ 0.55 Fy
Fhc =1.31 Fy /[1.15 +(Fy / Fhe )]

@ 1.6 Fy
Fhc =Fy

@ Fhe >6.2 Fy

When longitudinal tensile stresses and hoop compressive stresses occur simultaneously, the following interaction equation should be satisfied A2+B2+2ν‫׀‬A‫׀‬B≤1.0 A=

Where,

f a + f b - (0.5 f h ) Fy

A reflects maximum tensile stress combination.

B=

fh ( SFh ) Fhc

ν = poisson’s ratio fa =absolute value fof axial stress,(MPa) fb = absolute value of acting resultant bending stress,(MPa) fh= absolute value for hoop compression,(MPa) SFx= Safety factor for axial tension(API RP2A). SFh= Safety factor for hoop compression(API RP2A).

12

When longitudinal compressive stresses and hoop compressive stresses occur simultaneously, the following interaction equation should be satisfied fh ( SF h ) ≤ 1.0 Fhc f a + 0.5 f h f ( SF x ) + b ( SFb ) ≤ 1.0 Fxc Fy

SFx= Safety factor fort axial compression(API RP2A).

SFb= Safety factor for bending(API RP2A). •

Ring Design: Circumferential stiffening ring size may be selected on the following basis Ic = t L Ds 2 Fhe / 8 E

Where,

Ic = Required moment of inertia of ring composite section L =Ring Spacing (m) D = Diameter (m) An effective width (bf) of shell equal to 1.1(Ds t )1/2 is be assumed as the flange for composite ring section.

For flat bar stiffener minimum dimension provided is 10x76 mm. 13

The width (b) to Thickness(t) ratios of stiffening rings are selected in accordance with AISC requirements.

b=tsx65/(fy)1/2 3.3

WEIGHT ESTIMATION An initial estimate of total weight of the structure is based on Topside weight(WT), hull

weight including the weight of circumferential stiffeners and weight of ballast(W B) which is initially taken as zero.

Weight of Hull is given by,

Ws = π Ds t Lsρs

Weight of stiffeners is given by, Wst = (h w t w ρS π (D S - h w ) + b f t f ρSπ (D S - h w ))((H f + D f )/L)

14

Fig 3.3 Stiffener Details

Total Weight, W = WT + Ws +Wst+ WB

3.4

HYDROSTATIC STABILITY Hydrostatic stability is achieved only if there is a balance between the total downward force and buoyancy. The diameter of spar is selected in such a way as to satisfy this condition. Also, for a floating body to be in equilibrium, metacentric height should be greater than zero. Usually for a spar, the required matacentric height is 4-6 m. Metacentric height is determined as follows.

 Distance of centre of buoyancy from keel, KB = Df /2

 Dead Weight of spar, WD = SPAR weight(Ws) + stiffener weight(Wst)

 Distance of centre of gravity from keel,

KG =

WD L s /2 + WT L s + WB ADF/2 WD + WT + WB

 Metacentric radius, BMT=Transverse moment of inertia(I )/Displaced Volume(V)

 Distance of metacentre from keel , KM=KB + BM

 Metacentric height, GM = KM – KG

15

Where, Df= Draft of spar

Ls= Length of spar

ADF=Additional draft due to Ballast

3.5

RESPONSE PREDICTION USING SIMPLIFIED APPROACH Simplified calculation approach is based on linear potential theory and the superposition

principle, i.e. behavior in irregular sea is modeled by linearly superposing results from regular waves. Hydrodynamically, it is therefore sufficient to analyze a spar platform exposed to regular sinusoidal waves. The simplified method is described by Faltinsen (1990). 3.5.1

The Hydrodynamic Problem Assuming linear damping, the linear equations of motion for surge, heave, and pitch can

be solved in frequency domain. The damping represents the non-potential flow effects. Due to symmetry, the waves can be assumed to propagate along the positive x-axis with no roll, sway and yaw-response of the spar. The heave equation of motion is uncoupled while pitch and surge are coupled. The wave elevation and the velocity potential of incoming waves may be written:

ζ = ζ a sin( ωt − kx )

a z = −ω 2ζ a

and

φ=

sinh k (d + z ) sin ωt sinh kd

ζ a g cosh k (d + z ) cos( ωt − kx ) ω cosh kd

;

a x = ω2ζ a

(3.5.1)

cosh k (d + z ) cos ωt sinh kd

For slender structures like spar, the wavelength is much longer than the diameter i.e. D/L<0.2. The consequence of this long wavelength assumption is that no waves are generated by the hull. 16

Then the diffraction problem may be solved in a simplified manner. The excitation forces are obtained from the incoming wave potential and using analytical expressions for the added mass. No internal flow effects are considered as the spar bottom is closed. The equations to solve are the coupled surge/pitch equations of motion;

A15  η1   B11 B15  η 1   C11 C15  η 1   F1 (t )   M + A11  A    +  B B    +  C C  η  =  F (t ) A + I 55 55  η  51  5   51 55  η 5   51 55   5   5 

(3.5.2)

and the heave equation of motion; (3.5.3)

3 + B33η 3 + C 33η3 = F3 (t ) ( M + A33 )η

but before the equation can be solved, all the coefficients (Aij, Bij, Cij, and Fi) have to be determined. These coefficients are representing hydrodynamic forces and determining these coefficients, “the hydrodynamic problem”, can be divided into two sub-problems: •

“The diffraction problem”: The forces and moments on the body when the body is fixed and there are incoming regular waves. These hydrodynamic forces are again divided into the Froude-Krylov forces (pressure forces and moments due to undisturbed fluid flow) and the diffraction forces (pressure forces occurring since the body changes the pressure field by its presence in the water). Fi=FFK+FDIF

•

i=1,3,5.

“The radiation problem”: The forces and moments on the body when the body is forced to oscillate and there are no incident waves. These hydrodynamic loads are identified as added mass, damping, and restoring terms. (Aij, Bij, Cij i, j=1, 3, 5). Note that due to the

17

long wavelength characteristic, there is no radiation damping, since it is assumed that no waves are generated by the hull. Consequently Bij consist of non potential flow effects only.

3.5.2

Hydrodynamic Forces

The heave excitation is obtained by integrating the dynamic pressure over the wetted hull surface. The pressure is found by using Bernoulli’s equation. Formally the excitation force can be written as:

Fi = −∫ p tot ni ds = ∫ ρ S

S

∂φtot ni ds ∂t

where φtot =φinco min g +φdiffractio

nt

(3.5.4) n = is the vector normal to the body surface defined to be positive into the fluid. But as previously mentioned a simplified approach based on a long wavelength assumption will be applied. a)

Heave excitation force The Froude-Krylov heave force is obtained by integrating the undisturbed fluid pressure from the incoming wave potential over the bottom of the spar. The diffraction force is obtained in a simplified manner as previously described. Due to the long wavelength assumption, the diffraction term may be simplified and the integral over the wetted surface can be replaced by the quantities at the center of the spar(x=0). This means that structure is assumed transparent with respect to waves. Due to the normal vector of 18

the body surface, only the bottom surface of the spar contributes to the heave force (see figure 3.5.2) F3 ≈ −∫ p FK n3 ds + A33 a z = Aw p | z =−Td +A33 a z | z =−Td S

(3.5.5)

where Td = draft of the spar The above equation becomes: H cos shk ( d − Td ) H sinh k ( d − Td )   F3 =  Aw ρg − ω 2 A33  sin ωt 2 cosh kd 2 sinh kd  

The first term is the Froude-Krylov force while the second term is an approximation for the diffraction force. For a spar platform, the Froude-Krylov term is an order of magnitude larger than the diffraction term, due to low added mass. Therefore, a spar platform does not take advantage of the heave cancellation effect. b)

Heave added mass The heave added mass A33 appears both in the expression for the excitation force, Equation (3.5.5), and a mass term in the equation of motion, Equation (3.5.3). In order to solve the equation of heave motion, it is necessary to estimate the added mass A33. Newman (1985) calculated with the help of numerical methods, the axial added mass for a semi infinite cylinder to be A33=2.064ρr3, where r is the radius of cylinder. For a typical spar, the free surface effects have small influence on the heave added mass i.e. the added mass is basically an end effect. It may be noted that 2.064 ρr3 is fairly close to the displaced mass of the hemisphere 2 2 π ρr 3 = 2.09 ρr 3 . So for a bare cylinder, A33 = π ρr 3 is used. See fig. 3.5.3 (a) 3 3

19

The added mass for a bare cylinder is low compared to the total mass of the spar, and has therefore a relatively small effect. As mentioned earlier, when a heave plate is attached at the bottom of the spar, the heave added mass increases and becomes significantly high. The heave added mass for a spar + disc configuration is estimated as shown below: The added mass of a disc oscillating along its axis approximately equal to the mass of a sphere of water enclosing the disk (Sarpkaya, et al., 1981)

ma =

1 ρDd3 3

For the configuration of a cylinder with a disc attached to its base, if the diameter of the disc is greater than that of the cylinder, there is only a part of the disc on the cylinder side producing added mass effect since the presence of the cylinder (see figure 3.5.3 (b)). Thus, the added mass of a cylinder + disc configuration can be estimated by subtracting approximately the mass of the cylindrical volume of water. After calculations, the added mass of a cylinder + disc becomes:

ma = A33 =

c)

[

(

1 1 ρDd3 − ρ Dd3 − Dd2 − Ds2 3 6

)

Dd2 − Ds2

]

(3.5.6)

Horizontal excitation forces The excitation forces and total added mass for lateral motions are estimated using strip theory and the two-dimensional added mass for a cylinder in infinite fluid. For a two dimensional cylinder section in infinite fluid, the excitation force can be written f strip = ( A11( 2 D ) + ρ πr 2 ) a . The first term is the diffraction force and the second term is the Froude-

20

(2 D) Krylov force. A11 is the two dimensional added mass of the section, r = radius of the cylinder,

and a is the undisturbed water particle acceleration. It can be noted that this expression for the force corresponds to the inertia term in Morrison’s equation with inertia coefficient Cm=2. The total surge and pitch excitation forces are obtained by integrating the unit length force on each horizontal strip along the wetted hull surface. The pitch excitation moment is taken about VCG, see figure 3.5.1 H 2 cosh k ( d + z )  π f strip =  D 2 ρC M ω  cos ωt 2 sinh kd 4 F1 = ∫ f strip dz

H 2 1 π =  D 2 ρC M ω 2 sinh kd 4



∫ cosh k (d + z )dz  cos ωt

(3.5.7)

F5 = ∫ f strip Z strip dz

H 2 1 π = − D 2 ρC M ω 2 sinh kd 4

∫ cosh k (d + z )( z

vcg

 − z )dz  cos ωt 

(3.5.8)

where z vcg =z coordinate of center of gravity as shown in figure 3.5.1

The integration is to be done over the wetted length of the cylinder. d)

Horizontal added mass The added mass coefficients Aij, i,j = 1,5 are determined by considering forced surge and pitch oscillation of the spar, see figure 3.5.1. Under combined surge/pitch oscillations, every strip   along the hull has the acceleration a strip =η1 + Z strip η5 . Z strip is again the vertical distance from

21

the strip to the vertical centre of gravity, VCG. Water cannot penetrate the spar hull, so when the strip is accelerated by astrip, a pressure field is set up on the hull’s surface to displace the water. (2D) The strip will “feel” a counteracting inertia force, a strip A11 .

The global reaction forces due to the forced oscillations (F1,

RAD

and F5,

) are obtained by

RAD

integrating the reaction force on each strip. The added mass coefficients Akj are then found based on the definition of added mass:  j Fk = −Akj η

(3.5.9)

   πD 2  πD 2 1      F1, RAD = −∫ A11( 2 D ) alocal dz = −η ρ dz − η ρ − ( z vcg − z ) dz  5 ∫ ∫    4 4    

[

]

(3.5.10)

5 ) is A15. 1 ) in the above equation is A11 and of (-η Coefficient of (- η

Also,

   πD 2 πD 2 1  5  F5, RAD = −∫ A11( 2 D ) alocal Z strip dz = −η  ∫ ρ 4 − ( z vcg − z ) dz   −η ∫ ρ 4   

[

]

[( z

vcg

]

 2 − z ) dz   

(3.5.11) 5 ) is A55. 1 ) in the above equation is A51 and of (-η Coefficient of (- η

22

Fig. 3.5.1

Spar geometry

23

Fig. 3.5.2

Forces on a spar platform

Cylinder of diameter Ds

B Hemispherical fluid mass acting as heave added mass Fig. 3.5.3 (a) Heave added mass for cylinder

24

Fig 3.5.3 (b) 3.5.3

Added mass of a disc attached to a cylinder

Hydrostatic restoring forces

Since the spar is free floating, only hydrostatic terms are contributing to the restoring matrices: C11 = 0

C 33 = ρgA w

and

C 55 = ∆GM

(3.5.12)

Here Aw is the waterplane area, GM is the metacentric height, and Δ is the displaced weight of the spar. 3.5.4

Damping Effects In general, both generation of waves (radiation damping) and viscous forces (non

potential flow effects) are contributing to the total damping of a floating body. In the simplified analysis it is assumed that the wave generation by the body is negligible, i.e. there is no radiation damping. This approximation is relevant for survival conditions (long wave periods). For shorter wave periods on the other hand, where radiation effects are more important, damping effects have a small influence on the linear wave frequency response. Viscous damping, which plays a crucial role in the resonant response, is an empirical input to the analysis, and is not explicitly calculated. In the region around resonance, which is important in this study, the radiation damping is small. It is therefore assumed that the important damping effects are caused by viscous forces on the platform hull, on mooring lines, on risers, and other appendices. It is believed that these drag forces have a quadratic behavior. However, only linear damping forces are included in this

25

simplified linear frequency domain analysis. For simplicity, the linear damping coefficients Bii are here calculated as ratios of the critical damping (ξ=B/Bcritical): B55 = 2ξ5

( A55 + I 55 )C 55

and

B33 = 2ξ3 ( A33 + M )C 33

(3.5.13)

The values of ζ5 and ζ3 used for the calculation of B55 and B33 are obtained from experiments. 3.5.5

Response Amplitude Operators (RAOs)

When all the coefficients (Aij, Bij, Cij and Fi) are established, the equations of motions are solved by assuming steady state solutions oscillating with the same frequency as the excitation. The

(

)

iω t assumed solutions η i = η i e are substituted into the equations of motion (3.5.2) and (3.5.3).

The motion response amplitude ηic is complex. Motion transfer function or response amplitude operators (RAO) are defined as the frequency dependent steady state motion response amplitude divided by the wave elevation amplitude: RAO 1 (ω) = η1 / ζa

[m/m]

RAO 3 (ω) = η3 / ζ a

[m/m]

RAO 5 (ω) = η5 / ζ a

[rad/m]

(3.5.14)

Phase angles describing the phase shift between the wave elevation, at x=0, and the motion response are defined as:

 Re(η ic )   θ i = tan −1  Im ( η ) ic   3.5.6

(3.5.15)

Solution of Equation of Motion

26

The floating structure dynamics can be considered as the case where the structure floating in water is free to move in six directions when subjected to waves. The EOM will be of the form same as for the SDOF spring mass system with damping except that instead of single direction, the system is free to move in all six directions (Bhattacharyya, 1978). Thus the EOM is:

(M

+ A) x + Bx + Cx = F ( t )

Where (M+A) is a 6x6 mass matrix, B is 6x6 damping matrix, C is a 6x6 stiffness matrix, and  is acceleration vector, η  is velocity vector and η is the F(t) is a 3x1 force vector. η

displacement vector for structure oscillatory motion. Out of the 6 degrees of freedom, sway, roll, yaw are restrained in the present case. Thus only i=1,3,5 (i=mode of motion) are the remaining degrees of freedom. Heave (i=3) is uncoupled from surge and pitch. Surge (i=1) and pitch (i=5) are coupled. Thus, the EOM are equations 3.5.2 and 3.5.3. Following the same approach as for a SDOF spring-mass system, according to linear theory, the responses of the vessel will be directly proportional to wave amplitude –‘LINEAR’- and occurs at the same frequency as that of excitation, ω. Excitation is sinusoidal and so is the response also. We know F3 ( t ) = F3o Sin ωt So η 3 = η 3o Sin ( ωt − θ 3 ) where θ3 is the phase shift between wave elevation and motion response. Writing in complex form: 27

η3 ( t ) = η3 e iωt

where η3 is complex amplitude of vessel response in heave direction. Also, η 3 = iωη3 e iωt 3 = −ω 2η3 e iωt η

Substituting in EOM for heave 3 + B33η 3 + C 33η3 = F3 (t ) ( M + A33 )η

We get,

[− ω

2

( M + A33 ) + iω B33 + C33 ]η 3 = F30

Linear Transfer Function H (ω) is ratio of output amplitude to input amplitude. So: η3 = H (ω) F3o

Thus,

H (ω ) =

[− ω

1

2

( M + A33 ) + iωB33 + C33 ]

Thus, η3 =η3o = H (ω) F3o

and,

θ 3 = arg ( H ( ω ) )

28

So the time series for heave response can be found out for various frequencies. Similar procedure is to be followed for surge and pitch motion response. The final EOM become,

[− ω

2

( M + A11 ) + iωB11 + C11 ]η 1 + [ − ω 2 A15 + iωB15 + C15 ]η 5 = F1o

[− ω

2

A51 + iωB51 + C51 η1 + − ω 2 ( I 55 + A55 ) + iωB55 + C55 η 5 = F5o

]

[

]

The above simultaneous linear equations are then solved for unknowns η1 and η5 . 3.5.7

Natural Periods

1. Heave Natural period

TN ,3 = 2π

M + A33 C 33

2. Pitch Natural Period

TN ,5 = 2π

29

I 55 + A55 C 55

CHAPTER 4 RESULTS & DISCUSSIONS 4.1

GENERAL The outputs for three different trial runs for the program are presented and discussed in this

chapter. Predicted heave responses for all three trials are also plotted with respect to wave period. A parametric study of heave response with diameter and draft as parameters is also done in an attempt to optimize the response of the spar platform.

4.2

TRIAL 1

4.2.1

Inputs

Water Depth= 1000 m Wave Height = 15 m Topside Weight = 50000 kN 4.2.2

Outputs

SPAR DATA Diameter of spar (m)

:

21.4

Thickness of spar (mm)

:

71.2

Draft of spar (m)

:

49.0

Free board of spar (m)

:

10.5

Displacement (KN)

:

1.815e+05

30

Dead weight of spar (kN)

:

22475.02

Mass of spar (kg)

:

2.312e+06

Ballast weight (kN)

:

1.090e+05

Mass moment of inertia (kg/m2)

:

5.349e+05

Spacing between stiffeners (m)

:

4.30

Moment of inertia required (mm4)

:

2.54e+10

Moment of inertia provided (m4)

:

2.54e+10

Effective flange width (mm)

:

1357.0

Outstand length of stiffener (mm)

:

643.0

Thickness of the ring stiffener (mm)

:

70.0

Flange width of stiffener (mm)

:

1283.0

Weight of stiffeners (kN)

:

9653.0

Vertical centre of gravity of spar (m)

:

20.6825

Vertical centre of buoyancy of spar (m)

:

24.6680

Transverse BM (m)

:

0.0153

Metacentre of the spar (m)

:

24.6833

STIFFENER DETAILS

STABILITY CHECK

31

Metacentric height, GM (m)

:

4.5643

Fig 4.2.1 Heave RAO plot for Trial 1

4.3

TRIAL 2

4.2.1

Inputs

Water Depth= 300 m Wave Height = 23 m Topside Weight = 10000 kN

32

4.2.2

Outputs

SPAR DATA Diameter of spar (m)

:

12.0

Thickness of spar (mm)

:

40.0

Draft of spar (m)

:

40.0

Free board of spar (m)

:

15.3

Displacement (kN)

:

4.764e+04

Dead weight of spar (kN)

:

6738.03

Mass of spar (kg)

:

6.069e+05

Ballast weight (kN)

:

3.090e+04

Mass moment of inertia (kg/m2)

:

8.546e+04

Spacing between stiffeners (m)

:

3.0

Moment of inertia required (mm4)

:

2.1e+09

Moment of inertia provided (mm4)

:

2.1e+09

Effective flange width (mm)

:

775.0

Outstand length of stiffener (mm)

:

368.0

Thickness of the ring stiffener (mm)

:

40.0

Flange width of stiffener (mm)

:

453.0

STIFFENER DETIALS

33

Weight of stiffeners (kN)

:

1727.58

Vertical centre of gravity of spar (m)

:

15.8646

Vertical centre of bouyancy of spar (m)

:

19.8628

Transverse BM (m)

:

0.0062

Metacentre of the spar (m)

:

19.8690

Metacentric height, GM (m)

:

4.2325

STABILITY CHECK

4.3.1 Heave RAO plot for Trial 2

34

4.4 TRIAL 3 4.4.1

Inputs

Water Depth= 500 m Wave Height = 10 m Topside Weight = 40000 kN 4.4.2

Outputs

SPAR DATA Diameter of spar (m)

:

16.0

Thickness of spar (mm)

:

64.0

Draft of spar (m)

:

62.0

Free board of spar (m)

:

7.50

Displacement (kN)

:

1.275e+05

:

17506.0

Dead weight of spar

(kN)

Mass of spar

(kg)

:

1.624e+06

Ballast weight

(kN)

:

70000.0

:

5.506e+05

Spacing between stiffeners (m)

:

3.1898

Moment of inertia required (mm4)

:

1.13e+10

Mass moment of inertia (kg/m2) STIFFENER DETIALS

35

Moment of inertia provided (mm4)

:

1.14e+10

Effective flange width (mm)

:

1110.0

Outstand length of stiffener (mm)

:

552.0

Thickness of the ring stiffener (mm)

:

60.0

Flange width of stiffener (mm)

:

752.0

Weight of stiffeners (kN)

:

6493.3

Vertical centre of gravity of spar (m)

:

27.07

Vertical centre of buoyancy of spar(m)

:

31.13

Transverse BM (m)

:

0.0081

Metacentre of spar (m)

:

31.1403

Metacentric height GM (m)

:

4.3148

STABILITY CHECK

36

4.4.1

4.4

Heave RAO plot for Trial 3

DISCUSSIONS The table 4.5.1 shows the results obtained after various trial runs for the following input data. Water Depth= 500 m Wave Height = 10 m Topside Weight = 40000 kN Table 4.5.1 Output for different trial runs

Diameter(m)

Draft(m)

Heave RAO(m/m)

Dead wt spar(KN)

Wave Period(s)

15.9492

62.2644

4.975

17506.166

5.8298

16.1279

61.1333

4.986

17610.3452

5.7834

16.3375

60.0256

4.997

17779.4454

5.7381

16.5526

58.9157

5.01

17950.831

5.6925

16.7473

57.7804

5.023

18061.3342

5.645

16.9742

56.6667

5.038

18237.5735

5.598

17.2075

55.5509

5.055

18416.5486

5.5517

17.4477

54.4329

5.072

18598.4381

5.5045

17.695

53.3128

5.091

18783.4367

5.4569

17.9771

52.2107

5.111

19035.8789

5.4103

18.2402

51.0858

5.131

19227.9381

5.3618

18.5395

49.9778

5.154

19488.3131

5.3143

18.8481

48.8666

5.177

19753.1892

5.2664

37

Fig 4.5.1 Heave RAO vs Diameter plot

Fig 4.5.2 Heave RAO vs Draft plot 38

Fig 4.5.3 Dead Weight of spar vs Diameter plot The data given in table 4.5.1 is represented gaphically in figures 4.5.1, 4.5.2, 4.5.3. Figure 4.5.1 shows that heave RAO also increases as diameter increases.From the figure 4.5.3 it is also evident that as diameter increases, dead weight of the spar also increases.So inorder to achieve an optimised design of spar, it is better to select the minimum possible diameter.

39

CHAPTER 5 CONCLUSION A computer program which can be used to perform design of spar platform has been developed. The program also determines the response amplitude operators (RAO) for the heave, surge and pitch motions using simplified calculation approach. Thus, the program can be effectively used to get an optimized design of a spar platform. The functionality of the program can be improved by adding a module to automate the optimization part of the design process, which is presently done manually.

40