response of a tension leg platform to stochastic wave forcescoebenaroya/publications/response... ·...

Response of a tension leg platform to stochastic wave forces

Ron Adrezina, Haym Benaroyab

aDepartment of Mechanical Engineering, University of Hartford, West Hartford, CT 06117-1599, USAbDepartment of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, NJ 08855-0909, USA

Abstract

The equations of motion and the response of a Tension Leg Platform with a single tendon undergoing planar motion are presented. The hull

is represented by a rigid cylindrical body and the tendon as a nonlinear elastic beam. Tendons have often been modeled as massless springs,

which neglects contribution to the response by the wave forces on the tendons and its varying stiffness due to changes in its tension. The

structure is subjected to random wave loading. Linear wave theory is applied and the random wave height power spectrum is transformed into

a time history using Borgman's method. The surge and pitch responses for the hull, and the surge response along the tendon are presented for

two cases. Case 1 represents a tendon with a hull mass two orders of magnitude smaller than the tendon's mass. In Case 2, the hull mass is two

orders of magnitude greater than the tendon. Inclusion of tendon forces were found to signi®cantly increase the amplitude of the surge

response for Case 1 but not for Case 2. q 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Tension Leg Platform; Linear wave theory; Random wave loading

1. Introduction

Of the classes of offshore structures, the Tension Leg

Platform (TLP) is particularly well suited for deepwater

operation. Unlike ®xed structures, its cost does not drama-

tically increase with water depth. The TLP is vertically

moored at each corner of the hull minimizing the heave,

pitch and roll of the platform. This small vertical motion

results in less expensive production equipment than would

be required on a semi-submersible [1, 2].

TLPs are complete oil and natural gas production facil-

ities costing $1 billion or more [3]. The supporting structure

of a TLP consists of a hull, tendons and templates as shown

in Fig. 1. The hull is a buoyant structure with a deck at its

top that supports the oil production facility and crew hous-

ing. Pontoons and columns provide suf®cient buoyancy to

maintain the deck above the waves during all sea states.

These columns are moored to the sea ¯oor through tendons,

and ®xed in place with templates. The hull's buoyancy

creates tension in the tendons.

The tallest TLP at the time of its construction, Shell Oil's

Auger TLP in the Gulf of Mexico, began production in 1994

after an investment of 6 years and $1.2 billion. The Auger

TLP with a crew of 112 has two main decks 90 m by 90 m

(300 ft by 300 ft) with a well bay at its center. Four cylind-

rical columns 22.5 m (74 ft) in diameter and pontoons

[8.5 m by 10.7 m (28 ft by 35 ft) cross-section] comprise

the hull. There are three tendons at each column. Each

tendon, also known as a tether or tension leg, was assembled

from 12 steel pipes with a 66 cm (26 in) diameter and a

3.3 cm (1.3 in) wall thickness, connected end to end, and

a total length of 884 m (2900 ft). During severe storms it

may surge 72 m (235 ft) [3, 4].

The surge, sway and yaw resonance frequencies of TLPs

are below that of the wave frequency range as de®ned by a

power spectrum such as the Pierson±Moskowitz. The

heave, pitch and roll resonance frequencies are above this

range. The resulting response is a desirable feature of TLPs.

Wind, waves and current will cause a TLP to oscillate about

an offset position rather than its vertical position. This offset

in the surge direction has a corresponding setdown, the

lowering of the TLP in the heave direction, which increases

the buoyancy forces. This results in a higher tension in the

tendon than if it was in the vertical position. Higher-order

effects due to the nonlinear nature of the waves and

nonlinear structural properties will affect the dynamic

response and may be of interest. Papers that include varying

levels of higher-order effects are discussed in a review paper

by Adrezin et al. [5].

Song and Kareem [6] analyzed a TLP as a platform±

tendon coupled structure. Here the tendon's curvature,

mass and varying tension are included in the discretized

mass, damping, stiffness matrices and the force vector,

rather than approximated as an equivalent spring. The

stochastic response to random wind and wave forces was

determined. The response of the coupled system was then

Probabilistic Engineering Mechanics 14 (1999) 3±17

0266-8920/99/$ - see front matter q 1998 Elsevier Science Ltd. All rights reserved.

PII: S0266-8920(98)00012-5

compared to a system in which the 16 tendons were modeled

as equivalent springs. For surge, sway and heave motions

the difference between results of the mean value was

between 3 and 8%. The differences for the heave, roll and

pitch motions were between 30 and 58%. These large differ-

ences were attributed to the heave, roll and pitch motions

being controlled by the elastic force in the tendons and

hence more sensitive to its characteristics. Use of an equiva-

lent spring does not include the effect of roll and pitch on the

tendon's stiffness. It was also shown that the importance of

including the nonlinear effects of the tendon increases with

greater water depth.

Kim et al. [7] studied the nonlinear response of a TLP

subjected to random waves, steady winds and currents. In

the model, the hull and tendons are coupled and large displa-

cements are allowed. The ®rst and second-order wave forces

were determined using a higher-order boundary element

method. The response was studied in the time domain utiliz-

ing a three-dimensional hybrid element method. The hull

was treated as a rigid body and the tendons were modeled

with three-dimensional beam elements connected at the top

and bottom with spring±damper elements. Responses were

analyzed for surge, heave and pitch motion, offset and

setdown. The mean tension in the tendons was found to be

about 15% higher than the pre-tension due to the large mean

displacements. For a TLP model with 415 m tendons, the

mean offset and setdown were found to be approximately 20

and 0.5 m respectively.

Wave-induced forces for a TLP in an offset position may

differ from those at the undisplaced position. Li and Kareem

[8] analyzed this effect by adding displacement-induced

feedback forces to the wave-induced loads calculated at

the TLP's undisplaced position. Applying this method to a

TLP model resulted in a surge response due to the feedback

forces that were of the same magnitude as the wave

frequency response. Drift induced by second-order wave

potential was not included.

Patel and Lynch [9] modeled the hull as a rigid body with

six degrees of freedom subject to wave forces. It was

coupled to the ®nite element model of the tendon. Differ-

ences between the quasi-static and dynamic tendon models

were minimal except for tendons of lengths of the order of

1500 m or greater. Therefore, the effect of the tendon's

dynamics on the hull's response was only signi®cant when

the tendons were long, had a large mass per unit length and

the hull's displacements were small. The bending stresses in

the tendons were also found to be small. The highest values

were found for short tendons with large outer diameters and

thin wall thicknesses.

Patel and Park [10] studied the combined axial and lateral

response of tendons. The tendon was modeled as a straight

simply supported column subject to both horizontal and

vertical motions at its top. This motion represented the

response of the hull. At the tendon's top, the lateral force

was assumed sinusoidal. Results from studies of three differ-

ent tendon lengths were presented. The analysis showed that

a greater amplitude of vibration will result from the

combined excitation than the individual axial and lateral

excitations. This was particularly signi®cant in even

numbers of the instability region of the Mathieu stability

chart. The frequency of oscillation was also found to be

dependent on the relative magnitudes of the individual exci-

tations for the combined response.

Park et al. [11] performed a reliability analysis on 4- and

6-column TLPs. Two failure criteria were considered for the

tendons at the four corners, ultimate tensile strength and

negative tension. The results indicated that negative tension,

or slacking in the tendon, may result in high impulsive

stresses at the return of positive tendon tension. This was

determined to be the governing failure criterion.

Mekha et al. [12] created three different tendon models.

The ®rst was as a massless spring to provide a constant

lateral stiffness with TLP setdown disregarded. The second

allowed for time varying axial forces in the spring with one-

third of the tendon mass lumped at the attachment points.

The third tendon model consisted of ¯exural beam elements

and allowed for the time varying axial forces. Hydrody-

namic loads on the tendons were included in the third

model only. The hull was modeled as a rigid body subject

to regular waves and analyzed at different wave depths,

wave heights and wave periods. The amplitude of the

surge response was found to be independent of the model

used. For a constant wave period it increased linearly with

wave height, and decreased linearly with increasing wave

frequency for a constant wave height. The surge mean offset

may more than double for the third model where hydrody-

namic loads on the tendons were included. The maximum

tendon forces were not affected by the hydrodynamic loads.

However, the minimum decreased by 10%. Neglecting the

R. Adrezin, H. Benaroya / Probabilistic Engineering Mechanics 14 (1999) 3±174

Fig. 1. Schematic of a Tension Leg Platform.

inclusion of the hull setdown in the ®rst model resulted in a

maximum tendon force that was about 15% higher.

2. Problem description

In this paper, the response of a TLP with a single tendon

undergoing planar motion will be presented. The hull is

represented by a rigid cylindrical body and the tendon as a

nonlinear elastic beam. Tendons have often been modeled

as massless springs, which neglects the contribution to the

response by the wave forces on the tendon and its varying

stiffness due to changes in its tension. The TLP is subjected

to random wave loading.

2.1. Assumptions

² Tendon length is much greater than its diameter

² Tendon is inextensible and shear effects are ignored

² Planar motion

² Hull is treated as a rigid body

² Hull pitch angle is small due to excessive buoyancy

forces, therefore f is small

² The top of the hull always remains above the wave eleva-

tion

² The bottom of the hull is always submerged (otherwise

unstable)

² The hull is represented by a hollow cylinder

² Center of gravity of the hull is located at the geometrical

center of the hull

² Water level is assumed constant and horizontal across the

hull's column at a given instant of time

The assumption that the tendon length is much greater

than its diameter allows shear-deformation effects to be

disregarded because these will be considerably less than

the effects due to the bending energy. In addition, high

frequency motions are not of interest in the present study

because they are well above the range of loading. These two

assumptions allow for the use of Euler±Bernoulli beam

theory [13].

The natural frequencies for transverse vibration are

signi®cantly lower than those for longitudinal vibration.

The ratio of the transverse to longitudinal frequency is of

the order of the tendon diameter to its length. Because the

tendon length is much greater than its diameter, the long-

itudinal frequencies are well above the range of the envir-

onmental loading and will be disregarded in this study.

Therefore, the assumption that the tendon is inextensible

is valid.

The assumption of planar motion is reasonable if vortex

shedding is disregarded. Vortex shedding results in an out-

of-plane response which needs to be explored further

[14±21].

Treatment of the hull as a rigid body is a reasonable

assumption. Its stiffness results in natural frequencies well

above the range for wave, wind and current loading. The

requirement that the top of the hull always remains above

the waves is necessary to allow it to safely support oil and

gas production activities. Due to the mass of the hull and the

¯exibility of the tendons, the bottom of the hull will always

remain submerged for the system to remain stable.

The representation of the hull as a hollow cylinder with a

center of gravity located at its center (together with the

planar motion restriction) may affect the pitch response.

An actual hull is top heavy with the production facility

and crew housing at its top and buoyant pontoons and

columns at its bottom. The derived equations allow for the

center of gravity to be at any position.

The water level is assumed constant and horizontal across

the hull's column at a given instant of time. This simpli®es

the calculation of the center of buoyancy. The validity of

this assumption is dependent on the wave height, wave

frequency and the column's diameter. The free water

surface will also remain below the top of the hull.

A typical TLP has one or more tendons at each corner.

This model has a single tendon pinned to the hull and the sea

¯oor. The pitch response from a single tendon model may

differ signi®cantly from that of a multiple tendon model.

The additional tendons can restrict the pitch response.

3. Derivation of the Lagrangian

The Lagrangian for the tendon, L, is the sum of the ener-

gies:

L � Ek 2 Es 2 Ep (1)

where Ek represents the kinetic energy, Es represents the

strain energy and Ep includes all potential energy other

than due to strain. The tendon is modeled as a beam in

this paper. Due to the length of a tendon compared to its

diameter, modeling the tendon as a cable may be a reason-

able assumption. However, in order to maintain generality

and retain higher-order effects, a beam model is used which

retains the tendon's stiffness [10, 12, 22].

3.1. Strain energy

A body subject to forces may undergo rigid body motions

and deformations. The rigid body motions do not induce

stress and therefore do not produce strain. The study of a

deformable body is required to determine the strain energy

in a system. A point on the undeformed body has the coor-

dinate (X,Y) and an attached in®nitesimal line segment has

the length dS. After the body is deformed the coordinate for

the point (X,Y) is now (x,y) and the new length is ds. The

vertical and horizontal displacement of a point due to defor-

mation is represented by u and v respectively. They are

de®ned as:

u � x 2 X (2)

v � y 2 Y (3)

R. Adrezin, H. Benaroya / Probabilistic Engineering Mechanics 14 (1999) 3±17 5

The transverse and shear strain are substantially smaller

than the axial strain; therefore only this strain will be

considered. Strain induced by torsion and temperature

changes will also be disregarded.

It is assumed that deformations are within a material's

linearly elastic range with a modulus of elasticity E. It is

also assumed that a cross-sectional plane will remain plane

after deformation; therefore a neutral axis will exist.

The axial strain is derived due to bending stresses only for

the reasons discussed above. The tendon is assumed to be in

its undeformed position when vertical. Therefore, it is desir-

able to express the orientation of an in®nitesimal segment ds

with respect to the x-axis. As shown in Fig. 2, this angle is u .

Assuming in®nitesimal strain, the strain energy may be

written as

Es �Xs�l

s�0

1

2EI�s� 2u

2s

� �2

ds (4)

I(s) is the area moment of inertia of the tendon and the

relationship 2u2s

is equal to the inverse of the radius of curva-

ture R. This is derived in Cartesian coordinates as

R�x� �1 1

2v

2x

� �2 !3=2

22v

2x2

(5)

The coordinate s is a path variable along the neutral axis.

An incremental length, ds, may be derived as follows:

ds2 � dx2 1 dv2 (6)

ds ��1 1 vx�x; t�2

qdx (7)

3.1.1. Constraint equation between u and v

The assumption of an inextensible tendon leads to a

constraint relationship between coordinates u and v. Recall

that u � x 2 X. From Fig. 2, we can conclude that if the

beam is vertical and straight in its undeformed position then

coordinate X � S. Furthermore, if inextensible, S � s. This

leads to the relationship

X � s (8)

which will be of use in ®nding the constraint relationship.

Assigning j as a dummy variable for x, the coordinate x may

be found as follows:

s �Zj�x

j�0

��1 1

2v�j; t�2j

� �2s

dj (9)

If the slope 2v2j is assumed small, which is a reasonable

assumption based on the observed motion of a TLP, then

applying a Taylor Expansion yields:

s � x 11

2v2v

2j

��j�x

j�02

1

2

Zj�x

j�0v22v

2j2dj (10)

The boundary condition that v� 0 at the base, j � 0, will

be applied. Solving for x and substituting into Eq. (2) along

with Eq. (8) results in the following holonomic constraint

relationship:

u � 21

2v2v

2j

��j�x

11

2

Zj�x

j�0v22v

2j2dj (11)

The second term of the above equation must be solved

numerically if no further assumptions are made. However,

for a TLP, the second term will be much smaller than the

®rst. This is due to the high tension in the tendons minimiz-

ing the variation of the slope along its length. The vertical

velocity is derived from the above equation as

2u

2t� 2

1

2

2v

2t

2v

2j

��j�x

21

2v22v

2j2t

��j�x

11

2

Zj�x

j�0

2v

2t

22v

2j21 v

23v

2j22t

( )dj (12)

3.2. Kinetic energy

For an incremental length of tendon in the perturbed state,

ds, the corresponding incremental mass, dm, is as follows:

dm � 1

4p rTp�s� D2

o 2 D2i

� �1 CArflp�s�D2

o

h ids (13)

where Do and Di are the outer and inner diameters of the

tendon. The tendon and the seawater densities are repre-

sented by rTp(s) and r¯p(s) respectively, in the perturbed

con®guration. If the tendon is assumed inextensible, this

results in a constant volume and lengths dX and ds are

equal. Therefore the tendon density is assumed constant:

rTp(s) � rTp. Water density will also be assumed constant:

r¯p(s)� r¯p. When the perturbed length ds is projected onto

the x-axis, it results in a length dx # ds (see Fig. 2). We can

see that the upper bound dx � ds occurs when the tendon is

in its undeformed (vertical) position.


Fig. 2. Incremental element of mass dm shown in its undeformed position

dS, its perturbed position ds and its projection dx.

The incremental mass in the x±y frame may be written

with a constant density and varying diameters to represent

an elliptical horizontal cross-section of the deformed

tendon. A second alternative assumes that the constant

diameters along the path variable s will be used, and the

density will vary. For instance, to account for dx , ds the

density will increase, rT(x) . rTp, where the tendon density

in the projected con®guration is represented by rT(x).

Applying the second alternative with these assumptions

and satisfying the continuity equation results in the follow-

ing relationships between the densities for the perturbed and

present Cartesian con®gurations:

rT�x� � rTp

��1 1 vx�x; t�2

q(14)

rfl�x� � rflp

��1 1 vx�x; t�2

q(15)

The second term of Eq. (13) is the added mass, where CA

is the non-dimensional added mass coef®cient that is related

to the inertia coef®cient CM by

CA � CM 2 1 (16)

For a very large tendon length to diameter ratio, as is the

case for these structures, CM approaches its theoretical limit

of 2 and CA approaches 1. This added mass is due to the

entraining of ¯uid particles by the motion of the tendon.

The area moment of inertia for the tendon's cross-section

is

I1�s� � 1

64p D4

o 2 D4i

� �(17)

and its mass moment of inertia for a thin hollow disk

dJ1 � dmI1�s�A�s�

� 1

64p rT D2

o 2 D2i

� �1 CArflD

2o

h i� D2

o 1 D2i

� � ��1 1 vx�x; t�2

qdx (18)

where subscript 1 denotes the tendon and 2 the hull in the

following sections.

The kinetic energy due to the tendon's linear and angular

velocities may be represented by

Ek �Zs�l

s�0

1

2dm V2

x 1 V2y

h i1Zs�l

s�0

1

2dJ1

_u 2 (19)

Substituting Eqs. (7), (13) and (18) into Eq. (19) with the

velocities in the x-direction, Vx, and the y-direction, Vy,

represented by ut(x,t) and vt(x,t) respectively, results in the

kinetic energy expressed in the present Cartesian frame,

Ek �Zx�Lx

x�0

1

8p rTp D2

o 2 D2i

� �1 CArflpD2

o

h i�

��1 1 vx�x; t�2

qut�x; t�2 1 vt�x; t�2h i

11

128p rT D2

o 2 D2i

� �1 CArflD

2o

h iD2

o 1 D2i

� ��

��1 1 vx�x; t�2

q_u �x; t�2

�dx (20)

where the angular velocity, _u �x; t�, is found from the geome-

try as follows:

_u �x; t� � vxt�x; t�1 1 vx�x; t�2

(21)

The top of the tendon, coordinate s� l, projected onto the

x-axis, is represented by Lx.

3.3. Potential energy due to gravity and buoyancy

The potential energy due to gravity and buoyancy is

expressed as

Ep �Zx�Lx

x�0Fgbdsx

�Zx�Lx

x�02

1

4rT�x�p D2

o 2 D2i

� ��l 2 s�g

�

21

4rfl�x�pD2

o�l 2 s�g�

dsx (22)

where Fgb is the force due to gravity and buoyancy of the

tendon. Eq. (22) is an approximation to the inclusion of the

membrane strain in the Lagrangian and the work due to

vertical forces represented as 2R

Fgbudx. The change in

the vertical projection of an element ds is

dsx � ds 2 dx ��1 1 vx�x; t�2

q2 1

� �dx (23)

Substituting Eqs. (4), (20) and (22) into Eq. (1) yields the

Lagrangian, which is then substituted into Hamilton's

Principle.

4. Coupling the tendon with the hull

In Fig. 3 the bottom beam represents the tendon with a

horizontal displacement of v(x,t). The top mass represents

the entire hull and its payload. It is modeled as a rigid body

with length LH and diameter DH and rotates about the hinge

connecting it to the tendon with angle f (t).

4.1. Equation of motion for hull pinned to tendon

The kinematics and kinetics of the hull are now

developed.


4.1.1. Kinematics

The displacement vector from the origin to the hull's

center of mass, ~r�t�, is written as

~r�t� � �Lx�t�1LH

2cos f�i 1 �v�Lx�t�; t�1

LH

2sin f�j (24)

The velocity and the acceleration of the hull are

~_r�t� � � _Lx�t�2LH

2_f sin f�i 1 � _v�Lx�t�; t�1

LH

2_f cos f�j

~�r�t� � � �Lx�t�2LH

2�f sin f 2

LH

2_f 2 cos f�i 1 � �v�Lx�t�; t�

1LH

2�f cos f 2

LH

2_f 2 sin f�j (25)

The moment arm, lb, between the hull's buoyancy forces

and the pivot in the inertial coordinate system can be shown

to be

lb � D2H

32LHS�v; t� 2 1 tan2f� �

1LHS�v; t�

2

" #sin f (26)

4.1.2. Kinetics

The sum of the inertia, weight and buoyancy forces of the

hull are resisted by the tendon (see Fig. 4). These are given

by

F2b � rflgVHS�v; t� (27)

F2g � Mg (28)

and the vertical and horizontal reaction forces at the hinge

(point c) are respectively Fcx and Fcy.

The relationship between the mass moment of inertia for

the hull about its center of mass and the pivot is

JH � Jcm 1 �MLH

2

� �2

(29)

where �M is the mass of the hull plus its added mass.

Performing a force balance on the hull yields the follow-

ing equations for the reaction forces and the equation of

motion of the hull in terms of generalized coordinate f :

Fcx � 2 �M� �Lx�t�2LH

2�f sin f 2

LH

2_f 2cos f�

1 rflgVHS�v; t�2 Mg (30)

Fcy � 2 �M� �v�Lx�t�; t�1LH

2�f cos f 2

LH

2_f 2 sin f� (31)

JH�f 2

1

2MgLH sin f 1 rflgVHS

� �v; t� D2H

32LHS�t� 2 1 tan2f� �

1LHS�v; t�

2

" #sin f

1 �MLH

2� �v�Lx�t�; t� cos f 2 �Lx�t� sin f� � Qf (32)

where

LHS�v; t� � d 1 h�v; t�2 Lx�t�cos f

(33)

The tension due to the hull equals Fcx, where the hull's

inertia is �M �Lx�t�, the weight of the hull is Mg and its buoy-

ancy is rflgVHS�v; t�. The volume of the submerged portion

of the hull, VHS(v,t), varies with the displacement of the hull

in time. If the hull is assumed to pivot about its attachment

to the tendon, then for a constant cross-sectional area the

submerged volume is:

VHS�v; t� � pD2

H

4

d 1 h�v; t�2 Lx�t�cos f

(34)

4.2. Equations of motion for a tendon with a pinned hull

The total work W may be expressed as

W �Zx�Lx

x�0Fcx 1 Fx

Hfl

ÿ �dsx 1 Fcyv�x; t� x�Lx

�� (35)

and 2FyHfl is added to the boundary condition at the top of

the tendon. The forces due to the wave forces will be

included in Qf¯, Qy¯, FxHfl, F

yHfl and M¯, which represent

the generalized moment on the hull about the pin between


Fig. 4. Free body diagram of the hull.Fig. 3. Model of tendon with hull.

the hull and tendon, the generalized force on the tendon, the

horizontal and vertical forces due to the ¯uid forces on the

hull, and the moment at the ends of the tendon, respectively.

Applying the extended Hamilton's Principle will result in

the equations of motion and the boundary conditions for the

tendon, the top of which corresponds to the surge and heave

response of the hull. The following equation is for the pitch

response of the hull:

Jcm�f 2

1

2MgLH sin f 1 rflgVHS

� �v; t� D2H

32LHS�t� �2 1 tan2f�1LHS�v; t�

2

" #sin f

2 �MLH

2� �v�Lx�t�; t�cos f 2 �Lx�t�sin f 1

LH

2�f � � Qffl

(36)

5. Stochastic wave forces

5.1. Wave kinematics

In the analysis presented in this paper, linear wave theory

is applied. The following are the assumptions required for

its proper application [23]:

² The wave amplitude is small compared to the wave

length and the water depth.

² The water depth is constant.

² The water is non-viscous and irrotational.

² The water is incompressible and homogenous.

² Coriolis force due to the earth's rotation is negligible.

² Surface tension is negligible.

² The sea ¯oor is smooth and impermeable.

² The sea level atmospheric pressure is uniform.

The random wave height power spectrum is transformed

into a time history using the method by Borgman [24] and

applied in the paper by Bar-Avi and Benaroya [25]. The

wave surface pro®le is given by

h�y; t� � H

2cos�ky 2 vt 1 1� (37)

The Pierson±Moskowitz spectrum for the signi®cant

wave height Hs is of the form

Sh�v� � A0

v5e2B=v4

(38)

where the constants A0 and B are de®ned as

A0 � 8:1e 2 3g2 (39)

B � 3:11

H2s

(40)

The wave elevation is approximated as

�h�y; t� ��

A0

4BN

r XNn�1

cos kny 2 vnt 1 1n

ÿ �where the partition frequencies v n are

vn � B

ln�N=n�1 B=F4

� � 14; n � 1; 2;¼N (42)

and the relationship between the wave numbers, kn, and the

frequencies is de®ned by

v2n � gkn tanh knd (43)

F is the terminal frequency, where the wave energy spec-

trum is truncated at Sh (F). The horizontal wave velocity is

uw �XNn�1

��A0

4BN

rvn

cosh kny

sinh kndcos kny 2 vnt 1 1n

ÿ �(44)

and the vertical wave velocity is

ww �XNn�1

��A0

4BN

rvn

sinh kny

sinh kndsin kny 2 vnt 1 1n

ÿ �(45)

From the literature, linear wave theory appears to be an

excellent approximation for the forces on the tendon. On the

hull, however, nonlinear effects may be of importance. For

example, the variable wetting of the hull's columns may

lead to such higher-order effects as ringing [26±28].

5.2. Forces

Morison's equation is generally applied to slender

offshore structures. For a cylindrical element, the equation

for the force per unit length normal to the cylinder is given

by

Ffl � CDrD

2Vn

fl 2 VnTj j Vn

fl 2 VnT

ÿ �1 CMrp

D2

4_Vn

fl (46)

where the viscous, frictional drag coef®cient CD and the

inertia coef®cient CM are functions of the Reynolds number,

the wave characteristics, the cylinder diameter and its

roughness. The ®rst term of Eq. (46) is the drag force,

which is the force per unit length required to hold the cylin-

der in place subject to a stream of relative velocity

Vnfl 2 Vn

T

ÿ �. The second term is the force per unit length

required to hold the cylinder in place subject to a constant

free stream acceleration of _Vnfl. The acceleration of the

cylinder _VnT was already accounted for in the added mass

term of Eq. (13).

5.2.1. Fluid forces on the tendon

The generalized force per unit length on the tendon will

be determined from Eq. (46), with the following relation-

ships for the velocities and accelerations:

VnT � vt��

1 1 v2x

p (47)


Vnfl � uw 2 wwvx��

1 1 v2x

p (48)

_Vnfl � _uw 2 _wwvx��

1 1 v2x

p (49)

resulting in the following equation:

Qyfl � CDrfl

Do

2

uw 2 vt 2 wwvx��1 1 v2

x

p�� uw 2 vt 2 wwvx��

1 1 v2x

p !

1 CMrflpD2

o

4

_uw 2 _wwvx��1 1 v2

x

p !(50)

5.2.2. Fluid forces on the hull

The displacement vector, rH, from the origin of the iner-

tial frame to any point along the centerline of the hull is as

follows:

rH � Lx�t�1 x cos f�t�� i 1 v Lx�t�; t

ÿ �1 x sin f�t��

j (51)

and its velocity is

_rH � _Lx�t�2 x _f �t� sin f�t�� i

1 _v Lx�t�; tÿ �

1 x _f �t� cos f�t�� j (52)

which provides the value of VnT. The force per unit length on

the hull due to the wave forces in the inertial frame is

�FHfl ��Fx

Hfl

�FyHfl

( )(53)

resulting in a moment about the pivot between the hull and

the tendon of

Qffl �Zx�Lx 1 LHS cos f�t�

x�Lx

�FyHflx 2 �Fx

Hflx tan f�t�� dx (54)

and resultant forces on the tendon of

FxHfl �

Zx�Lx 1 LHS cos f�t�

x�Lx

�FxHfldx (55)

and

FyHfl �

Zx�Lx 1 LHS cos f�t�

x�Lx

�FyHfldx (56)

6. Numerical solution

The partial differential equations are discretized spatially

using the ®nite difference method. The resulting ordinary

differential equations are solved using the Runge±Kutta

method. This is implemented with Matlab [29]. The hull's

mass includes both structural and non-structural (e.g. crew

housing, storage tanks) mass. The submerged hull volume

as expressed by Eq. (34) was reduced by a factor of 4. These

adjustments to the mass and volume were made to raise the

TLP's ®rst natural frequency into the range of the Pierson±

Moskowitz spectrum for a hull mass of 4.53e6 kg (see Table

1). It is desirable in an actual design for this frequency

corresponding to surge to be below this range. Here, we

are looking to study extreme conditions. Figs 5 and 11 exhi-

bit a growing high frequency component in the response of

the surge at 282 m along the length. This results from using

a maximum time step in the Runge±Kutta method that is too

large. Reducing this parameter eliminates these high

frequencies.

7. Responses

The surge and pitch responses for the hull, and the surge

response along the tendon, are presented for two cases. Case

1 represents a tendon with a hull mass two orders of magni-

tude smaller than the tendon's mass. In Case 2, the hull mass

is two orders of magnitude greater than the tendon. The

mass moment of inertia of the hull varies with the change

in mass but all other parameters are constant, as shown in

Table 1. Figs 5±15 show the responses and the power spec-

tral densities for the two cases. For each case, a damped free

vibration response, TLP subjected to random wave forces on

the hull only, and wave forces on the hull and tendon are

presented. Fig. 10 shows the response with a signi®cant

wave height of 15 m, all other forced responses use a signif-

icant wave height of 9 m.

8. Discussion and summary

The responses exhibit both transient and steady-state

behavior. For the lower hull mass, the ®rst damped natural

frequency was found to be approximately 0.44 Hz for surge

and 0.83 Hz for pitch. For the larger hull mass, the pitch and

surge responses exhibited a frequency of 0.06 Hz, which is

within the range of the Pierson±Moskowitz spectrum. The

response of the tendon at a node 282 m from its bottom also

shows less dominant frequencies at 0.12, 0.17 and 0.22 Hz.

This was typical of the nodes away from the boundaries.


Table 1

TLP properties

Property Value

Tendon length 470 m

Tendon outer diameter 0.6096 m

Tendon inner diameter 0.5890 m

Tendon density 7800 kg/m3

Hull length 50 m

Hull outer diameter 30 m

Hull inner diameter 29 m

Hull inertia 171 and 1.71e6 kg/m2

Hull mass 453 and 4.53e6 kg

Modulus of elasticity 204e9 N/m2


Fig. 6. PSD of the damped free vibration of the TLP with a hull mass of 453 kg.

Fig. 5. Damped free vibration of the TLP with a hull mass of 453 kg.

8.1.1. Case 1

Inclusion of tendon force signi®cantly increases the

amplitude of the surge response. This supports the conclu-

sions by Patel and Lynch [9]. However, because the funda-

mental frequency is well above the range of the Pierson±

Moskowitz spectrum, the amplitudes were small.

8.1.2. Case 2

Inclusion of the tendon force does not signi®cantly affect

the amplitude of the surge response as supported by the

literature [6, 12]. However, researchers have also shown

that at greater tendon lengths, the importance of including

a more sophisticated tendon model increases [6, 9]. The

amplitudes of the response for Case 2 were greater than

for Case 1 because the TLP has a fundamental frequency

which corresponds to the wave frequencies.

8.1.3. Case 2

The hull pitch response shows a non-zero equilibrium

point. This is due to the combination of insuf®cient hull

buoyancy and a buoyant tendon. The tendon's buoyancy

is greater than its weight resulting in an upward force at

the attachment point with the hull. This causes the hull to

tip in order to increase its submerged volume rather than

sinking vertically. When a third case was studied (not

presented here) with an increased hull volume, the equili-

brium con®guration became the vertical upright position

(f � 0).

The analytical model presented in this paper would be

useful to the designers of TLPs in determining the overall

dynamic response characteristics. The structural and ¯uid

parameters listed in Tables 1 and 2 may be varied to deter-

mine their effects. Additionally, deterministic wave forces,

current and wind may be applied [30]. This analysis should

be performed prior to the creation of a large ®nite element

model, scale model tank testing or other design-speci®c

analysis. The usefulness of an analysis such as this is in

specifying the design space.

Acknowledgements

This work was supported by the Of®ce of Naval Research

Grant no. N00014-94-1-0753. The ®rst author is grateful for

his ONR-sponsored Fellowship, and the second author for

additional support. Our program manager, Dr. T. Swean, is

sincerely thanked for this support as well as his interest in

our work.


Fig. 7. Response of the TLP with a hull mass of 453 kg and waves on the hull only.

Table 2

Fluid properties

Property Value

Mean water level 500 m

Water density 1025 kg/m3

Signi®cant wave height 9 and 15 m

Drag coef®cient 1.0

Inertia coef®cient 2.0


Fig. 9. PSD of the TLP with a hull mass of 453 kg and waves on the tendon and the hull.

Fig. 8. Response of the TLP with a hull mass of 453 kg and waves on the tendon and the hull.


Fig. 10. Response of the TLP with a hull mass of 453 kg and waves on the tendon and the hull, signi®cant wave height of 15.

Fig. 11. Damped free vibration of the TLP with a hull mass of 4.53e6 kg.


Fig. 13. Response of the TLP with a hull mass of 4.53e6 kg and waves on the hull only.

Fig. 12. PSD of the damped free vibration of the TLP with a hull mass of 4.53e6 kg.


Fig. 15. Response of the TLP with a hull mass of 4.53e6 kg and waves on the tendon and the hull.

Fig. 14. PSD of the TLP with a hull mass of 4.53e6 kg and waves on the hull only.

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