DYNAMIC RESPONSE OF COMPLIANT OFFSHORE

STRUCTURES-REvIEW

By R. Adrezin; P. Bar-Avi,z and H. Benaroya3

ABSTRACT: Offshore compliant structures such as guyed platforms, tension leg platforms and articulated towers
are economically attractive for deep-water conditions because of their reduced structural weight compared to
conventional platforms. The foundations of these kinds of structures do not resist lateral environmental loads
forces; instead, restoring moments are provided by a large buoyancy force, a set of guylines or a combination
of both. These structures have a fundamental frequency well below the ocean wave's lower frequency bound.
As a result of the relatively large displacements, geometric nonlinearity is an important consideration in the
analysis of such a structure. This paper presents a literature review on offshore compliant structures. Our purpose
for this paper was to primarily explore the various modeling approaches used by workers worldwide to better
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understand the behavior of such structures. The review focuses on the static and dynamic response of the
structure due to various environmental conditions, such as wind, waves, and current. Emphasis is placed on
modeling techniques and methods of solution. Many modeling and analysis techniques are common to the
aerospace and ocean engineering communities due to the similarities in structural and environmental complex-
ities. For example, aerodynamic loads on offshore platforms are a significant part of any analysis. Also, fluid
loading models bear great similarities to mechanistic wind loading models. It is in this vein that a review paper
with the focus offshore structures is included in a journal of aerospace engineering.

INTRODUCTION to a base. The tower itself may be either a tubular column or

a trussed steel latticework. The structure's fundamental fre-
Since the 1970s, a need for deep water structures that would quency is designed to be well below those wave frequencies
exploit energy resources such as oil and natural gas has arisen. with high amplitudes. Articulated towers are typically de-
When deep water combined with hostile weather conditions signed for water depths of 100-500 m and are used as single
are considered, conventional fixed offshore structures require point mooring or as loading terminals, control towers and early
excessive physical dimension to obtain the stiffness and and/or full production facilities.
strength needed, and therefore are very costly. Thus, special
deep water platforms called compliant offshore structures had
Design and Construction
to be considered. This kind of structure is flexibly linked to
the seafloor and is free to move with the waves. Since the Very few papers discussing the design and construction as-
foundation of the structure cannot resist lateral forces due to pects of articulated towers are found. The first articulated
waves, current and/or wind, the restoring moment is provided tower ever built was designed in response to an industry call,
by a large buoyancy force, a set of guylines or a combination in 1963, for innovative offshore structures (Hays et al. 1979).
of both. The structure's natural frequency is designed to be A full-scale experimental structure, for a water depth of 100
well below the wave lower frequency bound in order to avoid m (330 ft) was designed by EMH. In 1968, it was constructed
resonances. This results in relatively large displacements, and and installed in the Bay of Biscay. The tower remained on-
thus geometric nonlinearity is an important consideration in site for 3 years during which time many measurements were
the analysis of such structures. Three types of platforms fall taken for a wide variety of weather conditions. This experi-
into the category of compliant structures: guyed tower, tension ment demonstrated that the articulated tower concept can be
leg platform and articulated tower. utilized in the offshore industry.
The feasible applications for each of the different types of Burns and D' Amorim (1977) discussed the development,
structures, and their conclusions are summarized in Table 1 design, and construction of two articulated towers that provide
(Fjeld and Flogeland 1980). Our purpose for this paper was mooring facilities and house flow lines from subsea equipment
to primarily explore the various modeling approaches used by to surface facilities. The towers were designed for a water
workers worldwide to better understand the behavior of such depth of 128 m (420 ft). Environmental loads due to waves,
structures. current and wind were considered in the evaluation of the tow-
er's dynamic response (e.g. surge, pitch and yaw) and the re-
ARTICULATED TOWER action forces in the base. The tower is constructed of the fol-
lowing major parts (see Fig. 1): (I) Base-connects the tower
The articulated tower consists of a vertical column to which to the sea-floor and keeps it from lifting or sliding; (2) uni-
a buoyancy chamber is attached near the water surface and to versal joint-has two degrees of freedom, one can tilt 30° and
which a ballast is usually added near the bottom. The tower the other 90°, so that the tower can be constructed horizontally
is connected to the seafloor through a universal joint connected [it is designed to withstand horizontal loads of 4450 kN (1,000
kips), and downward loads of 8,900 kN (2000 kips)]; (3) bal-
'Grad. Asst., Dept. of Mech. and Aerosp. Engrg., Rutgers Univ., Pis-
last chamber-it is located above the joint and has a diameter
cataway, NJ 08855.
'Senior Research Engineer, Rafael, Israel. of 9.75 m (32 ft); (4) during towing it is pressurized to give
'Prof., Dept. of Mech. and Aerosp. Engrg., Rutgers Univ., Piscataway,
NJ. TABLE 1. Platform Concepts, Areas of Feasibility
Note. Discussion open until March I, 1997. To extend the closing date
one month, a written request must be filed with the ASCE Manager of Application Guyed Tension leg Articulated
Journals. The manuscript for this paper was submitted for review and (1 ) (2) (3) (4)
possible publication on April 26, 1996. This paper is part of the Journal Predrilling Possible Possible Possible
of Aerospace Engineering, Vol. 9, No.4, October, 1996. ©ASCE, ISSN Drilling and production Possible Possible Infeasible
0893-1321/96/0004-0114-01311$4.00 + $.50 per page. Paper No. Subsea installation Feasible Feasible Feasible
13156.

114/ JOURNAL OF AEROSPACE ENGINEERING /OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 Deck "Nordsee." The aim of the test program was to demonstrate
the technical feasibility of the CONAT (Concrete Articulated
Tower) concept. The special features of this concept are the
bottle-shaped concrete tower and the ball joint that creates the
Buoyancy articulated connection. The problems of oscillating platforms
chamber and basic design steps were discussed. And finally the sched-
uled program tests were briefly discussed.
/ Naess (1980) presented the results of an extensive scale
Shaft 1:70 model test of an articulated tower. A 20 m long model
was built of steel and aluminium to give the necessary strength
and covered with polyurethane foam to give the correct outer
dimensions. The tests were done both with and without a
tanker moored to it. The tests included the following meas-
Ballast urements: wave elevation, pitch and roll of the tower measured
chamber with accelerometers, axial and lateral (shear) forces at the uni-
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~ versal joint measured by strain gauges, bending moment near

Universal the buoyancy chamber, and tension force at the mooring cable
)oint also measured by strain gauges. The system was tested under
Base a wind of 24 mis, waves with significant wave height of 5.5
V m and frequency 0.12 Hz, and current velocity 0.5 m/s. From
the tests they found the natural period to be 58.6 s and the
damping ratio , = 0.46. The pitch angle was about 10°. An
FIG. 1. Articulated Tower attempt to measure the natural period in the presence of current
failed because the free oscillations were immediately damped
buoyancy, but it is flooded when installed; (5) shaft-its di- out, a phenomenon which is explained in Bar-Avi and Bena-
ameter is 5.5 m (18 ft) (it connects the ballast and the buoy- roya (I 995c).
ancy chamber and it is full of water during service); and (6) An anonymous trade magazine paper (BHP brings 1990)
buoyancy chamber-the chamber is 9.75 m (32 ft) in diameter describes the world's largest single-point-mooring (SPM) ter-
and provides the vertical force that keeps the tower upright. minal. It was constructed in December 1989 in the Timor Sea
Hays et al. (1979) discussed the operation of an articulated off Australia's northwest coast. The tower was designed to
oil loading tower in the North Sea at a water depth of 122 m survive conditions of significant wave height up to 9 m, wind
(400 ft). The reasons for selecting the articulated tower con- velocity of 47 mis, and current velocity of 2 m/s. The oper-
cept were simplicity of design, that it could remain as an un- ational environmental conditions were a significant wave
manned facility during loading operations, its superior under- height of 3 m, wind velocity of 14.5 mis, and current velocity
water reliability and advantageous motion characteristics (e.g. of 1 m/s. The main components of the SPM are the same as
minimal surge displacement). The structure consists of a steel in any other articulated tower; a ballast, a universal joint, a
body that oscillates about a universal joint connected to the tower and a mooring yoke.
seafloor via a concrete ballast. The head of the tower rotates,
so when a tanker is moored to the tower, the orientation of Dynamic Response
the head is determined by the weather conditions. The tower
was built and tested and, according to the authors, it fulfilled Since articulated towers comply with the environmental
their expectations. forces, they can undergo large displacements. Therefore, the
In two papers (Smith 1979) and (Smith and Taylor 1980), dynamic response of these kinds of structures is very impor-
the applicability, function, and performance of an articulated tant. Most of the studies considered the tower as a rigid body
tower were examined. The construction program covered the having a one or two angular degrees of freedom about a uni-
following aspects: hydrodynamics, materials, economic as- versal joint. Structures having multiple articulations in planar
sessments, and interaction between the structure and the fluid. or 3D motion were also analyzed. Very few studies considered
The tower was designed for a water depth of 250 m, and its the tower as a flexible structure and those that did, used
predicted cost was approximately $700,000,000, while a con- lumped mass or finite-element methods and not continuous
ventional fixed structure would have cost in the order of $1.5 analytical models. The external forces considered by most
billion. An analysis of the response due to waves and wind studies were due to waves, current and wind. Linear wave
was performed and the analytical results were compared with theory was applied, and the forces were approximated by Mor-
experimental results from a 1:64 scale model. A fairly good ison's equation (Morison et al. 1950). This subsection sum-
correlation between the analytical model and the experimental marizes the literature on the dynamic response of articulated
one was found. From the analysis and testing, the authors con- towers. It is divided to three sections:
cluded the following:
o Single-degree-of-freedom (SDOF) systems-the tower
o Articulated towers can perform in a range of functions in was assumed rigid and could rotate about one axis at its
offshore production including serving as a low payload base.
service platform or as a mooring column. o Two-degrees-of-freedom system-the tower was assumed
o A good analytical model to predict the tower's response rigid and could rotate about two axes at its base.
is important. o Multiple articulation and flexible systems-rotation was
o A close collaboration with the oil industry, in order to allowed about one or two axes for a multiarticulated tow-
address real problems, is needed. ers or the tower was considered flexible.

Single-Degree-of-Freedom Systems
In Butt et al. (1980), a large-scale test program for a con-
crete articulated tower was presented. The tests were planned Chakrabarti and Cotter (1978) developed a mathematical
to be performed in the vicinity of a research platform called model to analyze the dynamic response of a tower-tanker sys-
JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/115

J. Aerosp. Eng. 1996.9:114-131.

 tem. The tower was assumed rigid, connected to the tanker ity of a buoyant platform. They used a linear SDOF model as
via a spring with stiffness K. Forces due to waves, current, follows:
and wind are considered collinear. First the static equilibrium
state due to only current and wind was found. Then small x+ 2c.i + [1 + G(t)]x = 0 (6)
perturbations about the equilibrium position were assumed in
where x = displacement; e = damping coefficient; and G(t) =
the formulation of the equation of motion. The tanker was
a nonlinear stochastic time-dependent function due to buoy-
assumed to have two degrees of freedom, one linear (surge)
ancy. It is assumed that G(t) is a narrow-band random process
and the other angular (pitch). The equations for the tower and
with zero mean. A criterion for the mean square stability is
the ship were derived and coupled through the spring con-
obtained from which the following results are found: for e >
necting the tower to the tanker
1 the system is always stable, and for < e < I there are
regions of stability and instability.
°
I,i\! + FD , + CsliJ + Frl cos(8 + <1» = M,e'(e,-a,) (1)
Thompson et al. (1984) investigated the motions of an ar-
mx + FDis - Fr cos 8 = F,ei(e,-rrt> (2) ticulated mooring tower. They modeled the structure as a bi-
linear oscillator which consists of two linear oscillators having
Is~ + FD2s + Csf.L - Fr[Hs cos 8 + (L/2) sin 8] = Mse'(e,-rrl)
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different stiffnesses for each half cycle

(3)

where 1iJ, x, f.L = tower's deflection angle, and the ship's surge
mx + ex + ktx = Fo sin wt, for x > 0 (7)

and pitch, respectively; F r = spring force which couples the mx + ex + k2 x = Fo sin wt, for x < 0 (8)
tower to the ship; C s = buoyancy term; F D " F Din F D2s = drag
forces proportional to the square of the relative velocity be-
tween the fluid and the structure; and cr = wave frequency.
°
where k" k2 = stiffnesses for x > and x < 0, respectively.
The equation is solved numerically for different spring ratios
The equations were solved numerically and the solution was and, as expected, harmonic and subharmonic resonances ap-
compared to experimental results obtained from a model that peared in the response. A comparison between the response
was built with a scale of 1:48. Good correlation between the and experimental results of a reduced-scale model showed
test results and theoretical predictions for small displacements good agreement in the main phenomenon.
was found. When soft spring-mass systems were considered, Chantrel and Marol (1987) presented a study on a tanker
irregular waves produced a drift (static) force that the model moored to an SDOF articulated tower. The objective of the
did not predict. study was to identify the relative importance of the different
In a later paper, the motion of an articulated tower fixed by nonlinear terms in the equation of motion, especially the terms
a universal joint having a single degree of freedom was in- that cause subharrnonic response. A few assumptions were
vestigated (Chakrabarti and Cotter 1979). They assumed linear made in deriving the equation of motion:
waves, small perturbations about an equilibrium position, and
that the wind, current and wave are collinear. Their resulting • Restoring moment due to buoyancy quadratic terms were
equation of motion is neglected.
• The drag force due water velocity was neglected.
(4) • Forces/moments were evaluated at the upright position of
where I = total moment of inertia including added mass; the tower.
B(IjJ) = nonlinear drag term; DIjJ = structural damping; CIiJ = The force in the mooring cable was assumed to have a
restoring moment due to buoyancy; and Mo = magnitude of cubic form.
the wave moment. A linear equation was obtained by assuming
a linear drag force and an analytical solution was obtained. Applying these assumptions resulted in the equation of mo-
The solution was then compared to experimental results, show- tion
ing good agreement as long as the system is inertia dominant,
and not drag dominant. Ie + cIa + c aIaI +
2 KH Yf)8 - p(8" - 8f = Mo cos(wt + e)
Kim and Luh (1981) evaluated the response of an articulated (9)
loading platform in regular waves. The objective of the study
was to develop a reliable technique to predict the loads and where 8 = pitch angle; I = mass moment of inertia that in-
motion of the tower. The following assumptions have been cludes the added mass; C t , C2 = linear and quadratic damping
made: rigid SDOF body, linear drag force, small deflection coefficients; KHYD is the hydrodynamic restoring stiffness; p(8 o
angle, and deep water. In the derivation of the equation of - 8)3 = moment due to the mooring cable which is set to zero
motion, the tower was assumed to be in its upright position for 80 ~ 8; and finally M o cos(wt + €) = external wave mo-
so that geometrical nonlinearities were not included. The an- ment that includes only inertia terms. Linearization of the
alytical solution of the linear equation of motion was obtained equation by assuming small perturbations about an equilibrium
position resulted in
Mo.j>
U + 2')'wnu + w,,[1 + f,. cos(wt + e)]u = (Moll) cos(wt + e)
(10)

This equation is actually the Mathieu equation, and a stability

analysis was performed to show an unstable region around the
first natural frequency of the system. This region, as expected,
gets smaller when linear damping is added to the system. Eq.
(9) was then solved numerically for regular and irregular
waves having the Pierson-Moskowitz spectra and the follow-
ing conclusions were drawn:

• The subharmonic response is due to the nonlinear char-

acteristic of the mooring cable's stiffness.

J. Aerosp. Eng. 1996.9:114-131.

 • The subharmonic response occurs for very specific envi- function using a least-squares method to get the following
ronmental conditions. Duffing equation
• A region of parametric instability that depends on the sys-
tem's damping was found. Ie + cO + k,ll + k2 11 2 + k3 11 3 = M o cos wt (12)

where k], k2 , k3 = spring constants depending on buoyancy,

Jain and Datta (1987) and Datta and Jain (1990) investigated gravity and the mooring lines. The equation of motion is
the response of an articulated tower to random wave and wind solved approximately in closed form and numerically, and sta-
forces. In the numerical solution of the SDOF equation of bility analysis is performed. The closed-form solution agrees
motion, the tower is discretized into n elements having appro- very well with the numerical solution. The main results of their
priate masses, volumes and areas lumped at the nodes, and analyses are that as damping decreases, jump phenomena and
there is viscous damping. The equation of motion is higher subharmonics occur, and chaotic motion occurs only
for large waves and near the first subharmonic (excitation fre-
1[1 + l3(t)]e + cO + R[I + v(t)]ll = F(t) (11) quency equals twice the fundamental frequency); the system
where 113(t) = time varying added mass term; Rv(t) time is very sensitive to initial conditions.
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varying buoyancy moment; and F(t) = random force due to Gottlieb et al. (1992) analyzed the nonlinear response of an
wave and wind. The Pierson-Moskowitz spectrum is assumed SDOF articulated tower. In the derivation of the equation, the
for the wave height and Davenport's spectrum is assumed for expressions for the buoyancy moment arm, added mass term,
the wind velocity. The equation is solved in the frequency and drag and inertia moments are evaluated along the station-
domain using an iterative method, which requires that the de- ary upright tower position and not at the instantaneous position
flection angle ll(t) and the forcing function F(t) be decomposed of the tower. The governing equation is of the form
into a Fourier series. The coefficients of the sin and cos are
then found iteratively. From their parametric study, the follow-
e + "yO + R(S) = M(O, t) (13)
ing was concluded: where R(6) = a sin 6 and a = linear function of buoyancy and
gravity; M(O, t) = drag moment; and "Y = normalized structural
• Nonlinearities such as large displacements and drag force damping. Approximate harmonic and subharmonic solutions
do not influence the response when only wind force is are derived using a finite Fourier series expansion, and stabil-
considered. ity analysis is performed by a Lyapunov function approach.
• The random wind forces result in higher responses than The solution shows a jump phenomenon in the primary and
do only wave forces. secondary resonances.
• The root-mean-square response due only to wind forces Gerber and Engelbrecht (1993) investigated the response of
varies in a linear fashion with the mean wind velocity. an articulated mooring tower to irregular seas. It is an exten-
sion of earlier work done by Thompson et al. The tower is
In a later paper, (Jain and Datta 1991) used the same equa- modeled as a bilinear oscillator
tion and the same method of solution to investigate the re-
sponse due to random wave and current loading. The wave
mx + ex + k,x = F(t), for x> 0 (14)
loadings (drag, inertia and buoyancy) are evaluated via nu- mx + ci + k2 x = F(t), for x < 0 (15)
merical integration. The following results were obtained from
the parametric study: where k, and k 2 represent stiffnesses for positive and negative
displacements of the articulated tower. The random forcing
• The dynamic response is very small since its fundamental function F(t) is assumed to be the sum of a large number of
frequency is well below the wave's fundamental fre- harmonic components, each at different frequencies, a proce-
quency. dure similar to that proposed by Borgman (1969). The equa-
• Nonlinear effects (drag force, variable buoyancy) have tion is then solved analytically since it is linear for each half
considerable influence on the response. cycle. The solution is obtained for different cases; linear os-
• Current velocity has a large influence on the response. cillator (both stiffnesses are the same), bilinear oscillator, and
for the case of impact oscillator (a rigid cable) in which os-
cillation can occur only in one half of the cycle. For future
Virgin and Bishop (1990) studied the domains of attraction
study they suggest inclusion of nonlinear stiffness and/or using
(catchment regions) for an SDOF articulated tower connected
a different spectrum to describe the wave height.
to a tanker. This was done using numerical techniques based
Bar-Avi and Benaroya (1996) investigated the nonlinear re-
on Poincare mapping ideas. A basic bilinear oscillator model
sponse of an SDOF articulated tower. The equation of motion
was assumed, the equation of motion was the same as (14)
was derived via Lagrange's equation. Nonlinearities due to
and (15). This equation can exhibit multiple solutions, but in
geometry and wave drag force are considered. A combined
the example solved, the coefficients (stiffness and mass) were
wave and current field, coulomb friction force, and vortex
chosen so that only two solutions may coexist, depending on
shedding force are included in the analysis. The governing
initial conditions; harmonic and four-order subharmonic. The
equation of motion is
equation was solved numerically and it was shown that a do-
main of attraction could be found.
Choi and Lou (1991) have investigated the behavior of an
J(ll)e + ce + Mgh(ll, t) = Mf/(ll, t) - Mj,(ll) (16)
articulated offshore platform. They modeled it as an upright where J(ll) = position-dependent moment of inertia that in-
pendulum having one degree of freedom (DOF), with linear cludes added mass terms; C = structural damping coefficient;
springs at the top having different stiffnesses for positive and M gb (6, t) = a time- and position-dependent moment due to
negative displacements (bilinear oscillator). The equation of gravity and buoyancy; Mf/(ll, t) = fluid moment due to inertia,
motion is simplified by expanding nonlinear terms into a drag, and vortex shedding force; and M j ,(6) = friction moment.
power series and retaining only the first two terms. They as- The influence on the response of current velocity and direc-
sumed that the combined drag and inertia moment is a har- tion, significant wave height and frequency, and damping
monic function. The discontinuity in the stiffness is assumed mechanism was analyzed. The following observations were
to be small, and thus replaced by an equivalent continuous made:
JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/117

J. Aerosp. Eng. 1996.9:114-131.

 • The equilibrium position is proportional to the product of where [M], [C], [K] = mass (including added mass terms),
the current velocity squared and the drag coefficient. damping and stiffness matrices of the tanker; [F] = force vec-
• The highest response is when the current direction is per- tor acting on the tanker; and {x}, {x}, {xl = acceleration,
pendicular to the wave. velocity, and position vectors of the tanker. The tanker's and
• The response to sub/superharmonics and harmonic exci- tower's equations are coupled by adding a constant mooring
tation demonstrate beating. force [FM] to (17) and (18) at each time step of the integration.
• For most excitation frequencies, the response is quasiper- The solution gives the low frequency motion of the tanker as
iodic, but for certain frequencies chaotic behavior was well as the tower's response. The high-frequency motion of
observed. the tanker was assumed to be unaffected by the fact that the
• Damping (friction, structural) has a stabilizing effect. tanker is moored. Therefore, the tanker's high-frequency am-
plitudes were calculated independently from the low fre-
A simplified equation for an SDOF articulated tower was quency, and then the responses were added together. The equa-
presented in Bar-Avi and Benaroya (1995d). The equation was tions were solved numerically and compared to test results to
derived using the Taylor expansion of the fully nonlinear equa- show a reasonably good correlation, but according to the au-
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tion derived in a later paper (Bar-Avi and Benaroya 1996). thors, a more accurate model should be developed. The effect
Terms of second power or less were kept and the solutions of of the tanker on the surrounding wave field was also investi-
both the fully nonlinear and the simplified equations were gated to find a 0-10% change in the wave velocity and ac-
compared. From the comparison it was found that the simpli- celeration. The effect of these changes on the response was
fied equation predicts the tower's response very well over a not investigated.
broad regime of behavior. Analytical expressions for the nat- Chakrabarti and Cotter (1980) investigated transverse mo-
ural frequency and equilibrium position due to current were tion, the motion perpendicular to the horizontal velocity. The
presented. It was also shown that current causes an additional tower pivot is assumed to have two angular DOF and is taken
damping mechanism in the system that can be expressed as to be frictionless. It is also assumed that the motion is not
(1I3)CD pDd 3 U/J, where CD is the drag coefficient, p is the coupled, so the in-line solution is found (the same as in the
water density, D is the tower diameter, d is the water depth, previous paper), from which the relative velocity between the
Uc is the current velocity, and e is the angular displacement. tower and the wave is obtained. The lift force (in the transverse
This result agrees with the experimental results presented by direction) can then be determined and the linear equation of
Naess (1980). motion is solved analytically and compared to experimental
results. The comparison shows good agreement, especially
Two OOF Systems when the drag terms are small.
Schellin and Koch (1985) calculated the dynamic response
Kirk and Jain (1977) investigated the dynamic response of due to waves and compared results with model tests. The cal-
a two DOF articulated tower to noncolinear waves and current. culation of the response was done for three different sets of
The two equations of motion were obtained via Lagrange's fluid coefficients; coefficients that depend on the wave period,
equation. The waves were assumed linear with the current coefficients selected from experimental data and coefficients
modifying the frequency and amplitude. Forces due to buoy- that are calculated using diffraction theory. The tower was as-
ancy, wave drag and inertia, and added mass were considered. sumed rigid and connected to the sea-floor via a two DOF
The equations were solved numerically, and the influence of universal joint. Forces due to wind, wave and current as well
drag coefficient and wave direction was analyzed. From the as nonlinearities due to geometry and wave drag force are
solution they concluded that: considered. The tower was divided into N elements for which
the following force was found:
• Higher drag coefficients result in lower response.
• The maximum deflection occurs when the current and the Fi = CMiUwi + cn;u wi lu",1 + Fn; + Fc; + F w, - mi r, - Cv;r,Ju",1
waves are in the same direction. (19)

where F i = vector of the total force acting on the element due

When vortex shedding forces are included the last conclu- to fluid inertia force CMiU w '; fluid drag force CDiUWi1ur<l1. buoy-
sion is not correct as shown in Bar-Avi and Benaroya (1996). ancy, gravity and wind forces F Hi, F Git F Wi, inertia force due
Olsen et al. (1978) evaluated the motion and loads acting to tower's acceleration ID,r" and drag force due to tower ve-
on a single-point mooring system. The tower was modeled as locity cn;rijureJI. Summing all forces on each element and
a rigid body connected to the seafloor via a universal joint. multiplying by the moment arm leads to the equations of mo-
The equation for the tower and the tanker were derived sep- tion
arately. To derive the equations of motion for the tower, it was
N
divided into N elements having two DOF each; a horizontal
and vertical displacement and the forces due to wave, current L [(r, -
;... 1
HZ) X Fa = 0 (20)
and wind were evaluated at each element. Hence, 2N nonlinear
differential equations were found where H = distance in the z-direction between the universal
N joint and the mean water level. The equations of motion were
L [(f
;al
iV - zo'k) X FiV + f iH X FiH ] =0 (17) solved numerically for an idealized tower that consists of a
series of circular cylinders. The numerical solutions were com-
pared to experimental results of a model which has been built
where f iV• f iH = displacements of element i in the vertical and to a scale of 1:32.75 and the following conclusions have been
horizontal directions; FiV, FiH = environmental loads acting on
drawn:
element i in the vertical and horizontal direction; and Zo' k =
motion of the universal joint in the z-direction. The tanker was
modeled as a rigid body having three transverse DOE The • Proper choices of the drag and the added mass coefficients
equations are derived in the tanker coordinate system and then result in good correlation between the theoretical and test
transformed into the tower's coordinate system to yield results for the tower's deflection and horizontal reaction
force on the universal joint.
[M]{x} + [CHi} + [K]{x} = [F] (18) • The correlation of the vertical reaction force is not good.
118/ JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 • The added mass coefficient has a predominant effect on • Structural viscous damping
the dynamic response and the drag coefficient almost • Drag fluid force due to waves and current, and wind cou-
none. pled to the structure
• The theoretical results for waves having small period • Inertia, buoyancy, and added mass fluid forces
(high frequency) do not correlate as well as results for • Vortex shedding loads due to waves and wind
wave with large period. • Wave slamming that was modeled as a periodic impulsive
force
Liaw studied the nonlinear dynamic response of articulated • Gyroscopic moments due to the rotation of the earth (Cor-
towers subjected to regular waves (Liaw 1988). The tower is iolis acceleration).
defined by a second order differential equation of a simple
oscillator, All fluid forces due to waves, current, and wind are deter-
mined at the instantaneous position of the tower, resulting in
x+ 2
w x = aU + a131U - xl(U - x) (21) two, highly nonlinear, coupled, ordinary differential equations
where U = normalized water particle velocity; x = normalized with time-dependent coefficients, with rotation angle <1>, and
structural displacement; and a, 13 = constants depending on deflection angle e
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the fluid coefficients and tower dimensions. The nonlinearities

are due to large displacement and velocity of the tower that J:ff(e)e + ce + I s(6)<V + M: (6) = MJ,C6, t) + M~m(6, t)
h

are coupled to the wave force. The equation was solved nu- + M~,(6) - M~C6, t)
merically and the following conclusions were drawn:
J~(6)<t> + C<!> + I.(6)<!>e = Mj,C6, t) + M~mC6, t) + M:(6)
• In addition to subharmonic resonances of order 1/2, there
were also subharmonics of orders 1/3, 1/5, 1/7 and so on. - M;C6, t) (22)
• Chaotic motion was found in certain frequency regions.
where M sb ' Mj " M jl , M w , and M sm = moments due to gravity,
• Before and within the chaotic regions, bifurcation behav-
friction, fluid forces, wind, and wave slamming, respectively.
ior was identified.
The superscripts <I> and e denote about which axis.
The equations of motion were numerically solved and the
In a later paper (Liaw et al. 1989), formulated the equations following observations were made:
of motion for a two DOF articulated tower using Lagrange's
equations, and then solved and analyzed the large motion of
• An analytical expression for the equilibrium position due
the structure. This was done using Euler's theorem, which
to current and wind was found.
states that "if a body has one point 0 fixed, then any dis-
• The response due to wave slamming is very small since
placement of the body from one given position to another is
an impulsive force is attenuated when the pulse duration
equivalent to a rotation about a unique axis through 0." The
is shorter than the system's fundamental period; this is
equation was solved for three cases. First, the static equilib-
the case here.
rium inclination of the tower due to current was obtained.
• Wind loads and current loads affect the equilibrium po-
Next, the response due to linear waves with height of 3 m and
sition of the tower.
period of 17 s was evaluated. Finally, the previous waves
• The Coriolis acceleration force has a small but important
along with orthogonal current were applied and the solution
influence on the response, since it causes a coupling so
was found. All three cases were compared to the solution ob-
that planar motion is not possible under real conditions.
tained by Leonard and Young (l985a), who used a finite-ele-
• The regions in which the beating phenomenon occurs are
ment method, and the results matched quite well.
very small and not as pronounced as in an SDOF system.
Liaw et al. (1992) showed that the subharmonic phenome-
non, which occurs in articulated towers, is due to the coupling • Due to the system's nonlinear behavior, chaotic regions
exist.
between the wave force and the structure. They used the equa-
tions that were developed in their previous paper (Liaw et al.
1989), but reduced them from two DOF to an SDOF system. Later, Bar-Avi and Benaroya (l995a, 1995e) analyzed the
The equation was solved numerically and harmonic and sub- response of a two DOF tower, where key parameters were
harmonic responses were obtained. The following observations taken to be random variables. The wave height, drag, inertia
were made: and lift coefficients, and coulomb friction coefficient were as-
sumed to be random uniformly distributed variables. The non-
linear differential equations of motion, (22), were numerically
• The amplitude of the response in the subharmonic region
can be as high as the one in the harmonic region. solved and Monte-Carlo simulations were performed to eval-
uate the average response and the standard deviation. It was
• The initial conditions determine the final steady state re-
sponse. found that the standard deviation for the rotation angle is larger
than that of the deflection angle. The value of the friction
coefficient has a very small influence on the average response,
Similar results for a single degree of freedom model were
unlike the wave height and the drag coefficient.
obtained in a study presented by Bar-Avi and Benaroya (1996),
although, for a two DOF system [see Bar-Avi and Benaroya
(l995b)] it was found that the subharmonic response is not as Multiple Articulations and Flexible Systems
pronounced as in the SDOF model. In Jain and Kirk C1981) a double articulated offshore struc-
Bar-Avi and Benaroya (l995c, 1995b) investigated the re- ture subjected to waves and current loading was analyzed.
sponse of a two DOF articulated tower to deterministic load- They assumed four DOF, two angular degrees for each link.
ing. The nonlinear differential equations of motion were de- The equations of motion were derived using Lagrange's equa-
rived using Lagrange's equations. The tower was assumed to tions. In deriving the equations of motion the following as-
have the same dynamic properties as an upright spherical pen- sumptions were made: drag and inertia forces tangent to the
dulum with additional effects and forces: tower are negligible, and the wave and current velocities are
evaluated at the upright position (small angles assumption).
• Coulomb friction in the pivot (hinge) The linearized equations were solved to find the natural fre-
JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/119

J. Aerosp. Eng. 1996.9:114-131.

 quencies of the system and then numerically solved to find the displacement vectors; and {F(t)} is the generalized force vec-
response due to colinear and noncolinear current and wave tor due to wave and current. The response was evaluated nu-
velocities. They found that when the wave and the current merically for steady current only and then for waves. The re-
velocities are colinear, the response of the top is sinusoidal, sults were compared to those presented by Jain and Kirk
while for non-colinear velocities the response is a complex (1981). From this comparison it was concluded that the 3D
three dimensional (3D) whirling oscillation. finite-element method is adequate. For linear analysis it re-
Seller and Niedzwecki (1992) investigated the response of quires more time than other linear computer schemes, but
a multi-articulated tower in planar motion (upright multipen- when nonlinearities are included the method actually requires
dulum) to account for the tower flexibility. The restoring mo- less time than others.
ments (buoyancy and gravity) were taken as linear rotational Sebastiani et al. (1984) presented the design and dynamic
springs between each link, although the authors state that non- analysis of a 1,000 m single-point mooring tower in the Med-
linear springs are more adequate for this model. Each link is iterranean Sea. The tower consists of four slender columns,
assumed to have a different cross section and density. The about 3 m in diameter each, connected via a universal joint.
equations of motion are derived using Lagrange's equations, A buoyancy chamber is welded to the upper column, just un-
in which the generalized coordinates are the angular deflec- derneath the deck. The tower is so flexible that several natural
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tions of each link. The equations in matrix form are frequencies are within the range of the wave frequencies.
Therefore, it operates in a resonance condition. The structure
[M]{S} + [K]{EI} = {Q} (23) was modeled using finite elements, and forces due to wave,
current, and wind were considered. In the dynamic analysis
where [M] = a mass matrix that includes the actual mass of that was performed for survival and station keeping conditons,
the link and added mass terms; {Q} = generalized force ma- irregular seas characterized by the Pierson-Moskowitz spec-
trix; {e} contains the angular displacements at each of the trum were considered. To gain a better understanding of the
three hinges; and the stiffness matrix [K] includes buoyancy dynamic behavior, a model test program was launched. A
and gravity effects. Damping and drag forces are not included model having a of scale of 1: 107.5 was built and tested. The
in the model. The homogeneous equations for a triarticulated tests were performed for the structure alone in wind, current,
tower are numerically solved to study the effects of different and waves, and then with a tanker moored to the structure.
parameters, such as link length, material density and spring The test results showed reasonable agreement with the simu-
stiffness, on the natural frequency of the system. lations for the maximum forces. As for the dynamic behavior,
In Haverty et al. (1982) and McNamara and Lane (1984), the theoretical predictions did not agree with the tests.
a finite-element method is used to calculate the response of a Hanna et al. (1988) investigated the dynamic response of
planar flexible multiarticulated tower. Examples of the re- triarticulated towers subjected to wave, wind, and current. The
sponse of single-point mooring, biarticulated and multiarticu- tower geometry and dynamic characteristics were optimized
lated towers were presented. To derive the equation of motion, such that tower periods fall outside the 5-20 s range, and
the displacement was decomposed into a rigid body motion reaction forces and weight are minimized. The model con-
and a deformed motion. Two coordinate systems were used: sisted of three rigid segments with different lengths and
one fixed and the other attached to the tower's rigid body masses, and a total length of 914.4 m (3,000 ft). Each segment
motion. The deformation was first expressed in the rotating had a single DOF and they were connected via a rotational
system and then transformed into the fixed coordinate system spring. Thus, three linear ordinary differential equations were
in which the equation of motion was expressed for each ele- obtained for small angles
ment, to find
[M]{x} + [K]{x} = {F} (26)
Mw + Kw = Kwrb + F, (24)
where [M], [K] = 3 X 3 mass and stiffness matrices, respec-
where M = mass matrix; K = stiffness matrix; w, w rb = total tively; {F} = forcing vector due to wave, wind, and current;
and rigid body displacements, respectively; and F = force due and {x} and {x} = displacement and acceleration vectors, re-
to wave and current, calculated by Morison's equation. For spectively. Eq. (26) was used to determine the static stability
the random wave, the Pierson-Moskowitz spectrum was trans- due to offsets of the deck weight. Values for segment length,
formed into the time domain using Borgman's method (Borg- weight, and joint stiffness were found for the highest critical
man 1969). The equations were solved numerically using a load. To analyze the dynamic response and the stresses, large
finite difference method in which artificial damping was intro- angular deflections were considered. The tower was divided
duced which, according to the authors, does not significantly into N elements each having a single DOE Nonlinearities due
influence the response. It was found that the finite element to geometry and drag forces were included, resulting in
solution using 21 elements was stable up to a time step of 0.7
s. A solution for the same problem, based on numerical inte- [M]{ii} + [C]{ti} + [K]{u} = {pet, u, ti)} (27)
gration of the Lagrange's equations (not presented), was com- where [M], [C], [K], {u}, {ti}, and {ii} = mass, damping,
pared to the finite-element solution and the results agreed ex- stiffness, displacement, velocity, and acceleration matrices and
actly except for a few initial cycles. The method presented can {pet, u, u)} = vector of the forces due to waves and colinear
be extended to more realistic problems such as two DOF uni- current approximated by Morison's equation, and due to static
versal joints. wind loads. Numerical solutions were obtained for determin-
The objective of the paper by Leonard and Young (1985a) istic and irregular waves (i.e. not sinusoidal) having the Pier-
was to develop a solution method to evaluate the dynamic son-Moskowitz spectra. From the analysis it was concluded
response of an articulated tower. The method is based on 3D that compliant towers with multiple articulations provide an
finite elements. The tower was subjected to wave, current and attractive concept to optimize the dynamic response without
nonlinearities due to geometry and drag force were included. penalizing the structure's weight. Furthermore, the method of
The equations of motion are analysis can be utilized for 3D structures and also other similar
[M]{ij} + [C]{q} + [K]{q} = (F(t)}, (25) compliant towers with multiple articulations.
Helvacioglu and Incecik (1988) described analytical models
where [M], [C], [K] are the mass, damping and stiffness ma- to predict the dynamic response of a single and biarticulated
trices; {q}, {q}, {q} = generalized acceleration, velocity, and tower subjected to waves and wind. The analytical solutions
120/ JOURNAL OF AEROSPACE ENGINEERING I OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 were compared to test measurements. The effects of changes moment occurs at the position of the chamber, while for
in the buoyancy position, joint location, and deck weight on deep water (300 m), the maximum bending moment can
the bi-articulated tower response were studied. In both models, occur at the midspan or at the buoyancy chamber, de-
planar motion was assumed, and although it wasn't mentioned pending on the chamber's position.
in the paper, fluid drag forces were not included, and therefore • Larger buoyancy forces cause a decrease in the tower's
simple equations were derived that resulted in simple analyt- deflection and an increase in the bending moment.
ical solutions. In both models the equations were simple os- • Current load has a significant effect on the deflection and
cillators with damping subjected to harmonic forces. From the the bending moment.
parametric study it was found that the buoyancy tank position • There is a nonlinear relation between the total force and
has a significant effect on the natural frequencies. According the deflection.
to the authors, the mathematical model for the biarticulated
tower correlated reasonably with the test results. Mathisen and Bergan (1992) outline a general approach to
Active control of offshore articulated towers was discussed large displacement static and dynamic analysis of an intercon-
in a paper by Yoshida et al. (1988). A preliminary attempt was nected rigid and deformable multibody system submerged or
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to control the dynamic response of an articulated tower sub- floating in water. The system's equations of motion were gen-
jected to regular waves. Two models were used; one was a erated by combining the equation of motion derived for each
rigid body having a single DOF, and the other was a flexible subsystem, which can be either rigid or deformable. The in-
tower fixed at the bottom. The control scheme was expressed vestigation is based on the Lagrangian description of motion
as a combination of feedforward control based on the distur- in which the current coordinate of a material point is described
bance and a feedback control. The feedback control copes with in terms of its initial material point and time. The equation of
the higher-order noise remaining after the compensation of the motion of each part was derived using variational methods,
feedforward control. Two feedforward control schemes were and then combined with a nonlinear finite-element displace-
discussed. One is to compensate for the whole wave force ment formulation. The formulation was applied to a biarticu-
acting on the structure, while the other was on-off control to lated tower, and the purpose was to find the response of the
compensate for the principle Fourier components of the wave top of the platform, as well as to evaluate the distribution of
force. The simulation results for both models showed that the the axial force and bending moment along the tower. The
response of the controlled structure was reduced to about 30% equations were solved for a deterministic wave height of 30
of those of the uncontrolled system. m with a period of 30 s, and irregular (random) waves having
In a later paper, Yoshida et al. (1989) discussed the experi- the Jonswap spectrum.
mental results of the response of an actively controlled tri-
articulated tower. The application of active control to offshore TENSIONED LEG PLATFORMS
structures is advantageous, increasing strength (stiffness), and
reducing weight. The structure can be artificially stiffened and Of the classes of offshore structures, the tension leg plat-
damped by means of active control according to the environ- form (TLP) is particularly well suited for deepwater operation.
mental conditions. Ultrasonic sensing systems were used to Unlike fixed structures, its cost does not dramatically increase
measure the deflection of each segment of the tower. The data with water depth. The TLP is vertically moored at each corner
from the measuring system was processed and the signals for of the hull minimizing the heave, pitch and roll of the plat-
the controllers were obtained. The control force was generated form. The resulting small vertical motion results in less ex-
by thrusters which were built into each segment. Optimal con- pensive production equipment than would be required on a
trol was applied to several cases: semisubmersible (Natvig and Teigen 1993; Chakrabarti 1987).
This structure, as opposed to the guyed tower, cannot be as-
• A neutral model, in which the buoyancy and gravity sumed to be a rigid body, and continuous elastic models have
forces are equal, was controlled. The response of the to be considered.
model against an imposed displacement was controlled.
The thrusters had a phase delay, and therefore vibrations Components
in high frequency could not be controlled.
• An unstable model, in which the buoyancy force was less TLPs are complete oil and natural gas production facilities
than the gravity force was controlled. In this case the costing $1 billion or more (Salpukas 1994). The supporting
structure was stabilized but again high frequency vibra- structure of a TLP consists of a hull, tendons and templates,
tions could not be controlled. shown in Fig. 2. The hull is a buoyant structure with a deck
• Static deflection due to current was controlled success- at its top that supports the oil production facility and crew
fully, but with large deflection angle the high gain nec- housing. Pontoons and columns provide sufficient buoyancy
essary to control the structure caused some instability. to maintain the deck above the waves during all sea states.
These columns are moored to the seafloor through tendons,
Ganapathy et al. (1990) developed a general finite-element and fixed in place with templates. The hull's buoyancy creates
program for the analysis of the nonlinear statics and dynamics tension in the tendons.
of articulated towers. The tower was modeled as a 3D beam The tallest TLP at the time of its construction, Shell Oil's
element, which includes axial shear and bending deformations. Auger TLP in the Gulf of Mexico, began production in 1994
The equations of motion have the standard finite-element for- after an investment of six years and $1.2 billion. The Auger
mulation 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 cylindrical
[M]{ii} + [C]{u} + [K]{u} = {F(t)} (28)
columns [22.5 m (74 ft) diameter] and pontoons [8.5 m by
Linear wave theory was assumed and the wave force was eval- 10.7 m (28 ft by 35 ft) cross section] comprise the hull. There
uated via Morison's equation. The equations were numerically are three tendons at each column. Each tendon, also known as
solved and the effects of the water depth, buoyancy force mag- tether or tension leg, was assembled from 12 steel pipes con-
nitude and position, and wave and current loads were inves- nected end to end with a 66 cm (26 in.) diameter and a 3.3
tigated and the following conclusions were drawn: cm (1.3 in.) wall thickness, and a total length of 884 m (2,900
ft). During severe storms it may surge 72 m (235 ft) (Robison
• For moderate water depth (100 m), the maximum bending 1995; Salpukas 1994).
JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/121

J. Aerosp. Eng. 1996.9:114-131.

 YAW
allowed for the reduction in the number and size of the tendons

t
HEAVE
used, but with a high initial capital cost (Oppenheim and Flet-
ter 1991; Dove and Lohr 1993; Robison 1995). Yoshida et al.
(1994) discussed the active control of a TLP using a thruster
~ SURGE ~ ROLL
system.
~ea et al. (1994) have developed a methodology for com-
SWAY
parmg offshore production systems. It is based on analyzing
PITCH
/ the alternative designs through the project's complete life cy-
cle. It is an interdisciplinary approach that results in a set of
PLATFORM risk costs for each design. These values are an indication of
z z where the greatest risk and what cost-effective measures can
::2 ::2 be implemented to reduce the level of risk.
::J ::J
....J ....J
o() 0
()
On-Site Assembly
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The construction of a TLP at sea presents many engineering

PONTOON challenges. The hull of the Auger TLP was towed 11,000 km
(6,800 miles) before being mated with its 21,700 t (24,000
ton) deck during 5-7.7 mls (l0-15 knot) winds (Robison
1995). The hull may be constructed as one unit and towed to
the site or built in modules and assembled onsite (Sato et al.
TENDONS 1989; Robison 1995). Each tendon consists of several steel
pipes that may be welded onshore and towed to or assembled
on-site. The former reduces the construction time at sea but
r-TEMPLATES the forces on the tendon during towing and up-ending must
be considered. The Hutton and Snorre TLPs used threaded
/ tendon couplings whereas the Auger TLP used snap together
FIG. 2. Tension Leg Platform couplings, and single piece tendons were used in the construc-
tion of the Jolliet and Heidrun TLPs (Pallini and Yu 1993;
Robison 1995). Once all tendon connections are complete, the
Sato et al. (1989) developed a basic planning method to h~ll is deballasted and the tendons are pretensioned. Wybro
design TLPs based on a customer's requirements. It begins discusses the current methods for tendon installation (Wybro
with the design of the deck dimensions and layout and the ~995). A new method utilizing the Platform Arrestor Concept
determination of the weight of the onboard equipment. An IS presented to allow for the simultaneous lock-off of all the
initial hull displacement is then assumed and the hull partic- tendons. This offers reduced resonant behavior during instal-
ulars such as column displacement, pontoon diameter and air lation and lower cost for deepwater operation.
gap are designed. These designs are based on numerical anal-
ysis and tank testing. The actual displacement is then com- Dynamic Response
puted and compared with the initial assumed displacement.
Adjustment to the hull displacement is made and the process The surge, sway, and yaw resonant frequencies of TLPs are
is repeated until all design criteria are satisfied. below that of the wave frequency range as defined by a power
The number of tendons and its dimensions are determined spectrum such as the Pierson-Moskowitz (Chakrabarti 1987).
from the hull buoyancy, water depth, and oceanic meteorolog- To heave, pitch, and roll resonant frequencies are above this
ical conditions. A high tendon outer diameter to wall thickness range. The resulting response is a desirable feature of TLPs.
ratio is used to reduce the loss of payload due to tendon Wind, waves and current will cause a TLP to oscillate about
weight. The template design is dependent on the hull displace- an offset position rather than its vertical position. This hori-
ment and type of soil on the seafloor. Piles are used to anchor zontal offset results in a greater submergence of the hull
each template in place. A simple analysis using linear theory (known as set down) due to the geometrical constraint of the
was performed to aid in this initial TLP design cycle. The effectively inextensible tendon length. This lowering of the
tendon's top and bottom connectors use a flex element that hull increases the buoyancy forces due to the greater volume
allows approximately a :t 10° rotation with minimal bending of displaced water. This results in a higher tension in the ten-
moment (Pallini and Yu 1993). The connector may be a com- dons than if tendons were in the vertical position. Higher-order
plex mechanism and its selection is based on the desired in- effects (e.g. ringing) due to the nonlinear nature of the waves
stallation method. As TLPs are designed for greater depths, and nonlinear structural properties will affect the dynamic re-
the effect of environmental loads on the longer tendons and sponse and may be of interest. Papers that include varying
risers greatly increase and must be thoroughly analyzed and levels of higher-order effect are included in this review.
carefully designed (Lim and Hatton 1991). Rainey presented a dynamic analysis of vertically moored
The Auger TLP uses a spread well system (several individ- tethered buoyant platforms such as articulated towers and
ual wellheads) with a fixed drilling derrick. This design, in TLPs (Rainey 1977). The structure was modeled as a spherical
which the risers are spread over a larger area at the seafloor mass representing the platform moored to the seafloor with a
as compared to the deck, was implemented due to concerns of cable in zero viscosity water. The platform is treated as sub-
vortex shedding (Abott et al. 1994). Rather than using a merged in order to eliminate the effects at the free water sur-
thruster system, a multileg catenary mooring system is used face. The following governing equation was used for a sub-
to meet the additional station keeping restoring force required merged platform in surge motion:

~ PV] x + +[(PV - ~ PVV] x == ~ pVfI

to position the derrick over each wellhead. An eight-point cat-
enary system is used on the Auger TLP. The components in- [M + M)g + (29)
clude mooring lines [12.7 em (5 in.) diameter wire and 12.7
em (5 in.) chain], submersible buoys, pile anchors, and linear where x == surge displacement of the platform; M == platform
winches for positioning the hull. The addition of this system mass; p == water density; l == cable length; g == gravity; V ==
122/ JOURNAL OF AEROSPACE ENGINEERING I OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 platform volume; V and H = vertical and horizontal water ve- irregular wave forces, with and without current, and compared
locities, respectively. The coefficient of x represents the mass to Stokes' second-order wave theory. A 2D TLP mathematical
of the platform and the horizontal added mass. The second- model consisting of four columns and four pontoons was cre-
term represents the effect due to excess buoyancy and the ver- ated. The hull was modeled as a three DOF rigid body un-
tical load on the TLP. The right-hand side of (29) is the hor- dergoing surge, heave, and pitch. The tendons were treated as
izontal load on the platform in the x-direction. Similar massless springs providing axial and lateral stiffness at their
equations represented the motion due to sway and yaw. The connection with the hull. The stiffness matrix is calculated
right-hand side of (29) was zero for the yaw equation. Heave, from the initial pretensioning of the tendons. The force vector
roll, and pitch motions were considered to be of the second is calculated using the Morison equation to determine the hy-
order and therefore not considered. Enhancements to (29) were drodynamic forces and also include effects due to variations
made to account for viscosity and free water surface effects. of tendon tension. Linear wave theory was used to evaluate
Small-wave theory was applied at the free water surface and the wave kinematics up to the mean water level. The following
added to the vertical load exerted on the platform by the approximate methods were used to evaluate the wave kine-
waves. This results in the following governing equation for matics from the mean water level to the wave free surface:
surge motion only: hyperbolic extrapolation, linear extrapolation, stretching meth-
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ods, and uniform extrapolation. For a TLP subject to regular

oX + 2c.x + [1 + g(t)]x = f(t) (30)
waves, the surge amplitude turns out to be not affected by the
where f(t) and get) represent the horizontal and vertical force method chosen. However, the surge mean drift was very sen-
histories, respectively. The units for the platform mass, added sitive to the method use. Heave amplitude and mean offset
mass, cable length and platform excess buoyancy were se- were both affected by the method selected but were not sig-
lected to yield values of unity. In (30), the damping coefficient nificantly different from calculating the response to the mean
c provides a measure of the damping effect due to the viscosity water level only. The pitch response at its natural frequency
of the water. This differential equation was then solved and a was amplified at the free water surface, particularly for irreg-
stability analysis was performed and compared to empirical ular waves and was affected by the approximation selected.
results from a physical TLP model immersed in water of a Song and Kareem (1994) analyzed a TLP as a platform-
depth of approximately 1.2 m. Tests were performed with both tendon coupled structure. Here, the tendon's curvature, mass
regular and irregular waves. Dynamic instability was found and varying tension are modeled rather than approximated as
due to regular waves at the predicted wave periods and am- an equivalent spring. The stochastic response to random wind
plitudes. The irregular waves required a significantly higher and wave forces was determined. The response of the coupled
amplitude to cause instability in the model. system was then compared to a system in which the tendon
Armenis et al. (1991) performed a time domain dynamic was modeled as an equivalent spring. For surge, sway, and
analysis on a TLP. Nonlinearities modeled included large ten- heave motions the difference between results of the mean
don displacements, coupling between the hull and tendons, value was between 3% and 8%. The differences for the heave,
coupling between horizontal and vertical motions of the plat- roll and pitch motions were between 30% and 58%. These
form, and the set-down effect. Diffraction effects were in- large differences were attributed to the heave, roll, and pitch
cluded, but second-order wave forces due to potential effects motions being controlled by the elastic force in the tendons,
were neglected. A finite-element analysis was performed on a and hence, more sensitive to its characteristics. Use of an
TLP model consisting of four columns and four tendons. The equivalent spring does not include the effect of roll and pitch
structure was modeled with beam elements and the tendons on the tendon's stiffness. It was also shown that the impor-
with truss elements. The platform was modeled as a rigid body tance of including the nonlinear effects of the tendon increases
with six degrees of freedom. The nonlinear governing equation with greater water depth. A parallel computation scheme to
for the N-degree-of-freedom system is reduce computational time over that of a single microprocessor
was also presented. The set of parallel microprocessors are
[M]{r} + {Ri(r)} = {R e} + {F(t)} + {F(t, r, t, f)} (31)
divided into two groups, each solving one substructure in the
where [M] represents the N X N mass matrix; r = acceleration time domain. At certain intervals data is shared between the
vector with respect to a fixed coordinate system; t and f are two groups to couple the substructures.
relative velocity and acceleration vectors between the platform Johnson (1994) created a finite-element analysis based soft-
and wave; {Ri(r)} includes elastic forces; and {R e } includes ware package to aid in the design of TLPs. A 2D model with
the mean static loading due to wave, wind, and current forces coupled hull and tendons was subjected to wave, wind, and
and tenson weight. {F(t)} represents the time-dependent load- current loads. The hull consisted of a platform, four cylinders
ing and {F(t, r, t, f)} is the wave and current loading vector. and four pontoons and was treated as a rigid body. The tendons
The latter is a function of platform displacement, and the rel- were constructed of several beam-column elements and pro-
ative velocity and acceleration between the platform and the vide resistance to the hull's surge, heave, and pitch motions.
waves. Linear wave theory was used to determine the wave Linear wave theory and the Morison equation were used to
pressure field velocity and acceleration on each submerged el- determine the hydrodynamic loading on the structure. The fi-
ement of the model. The Morison equation and MacCarny and nite-element analysis may be performed with either the trap-
Fuchs formula were used to calculate the inertia forces on the ezoidal or Houbolt's method. The TLP response is presented
cylinders. (The hull's columns and the tendons may be mod- graphically in real time.
eled as cylinders). The MacCarny and Fuchs formula accounts Kim et al. (1994) studied the nonlinear response of a TLP
for diffraction effects and can be applied to any ratio of wave- subjected to random waves, steady winds, and currents. In the
length to cylinder diameter. Wave loads on the pontoons were model, the hull and tendons are coupled, and large displace-
calculated with the Morison equation and two-dimensional ments are allowed. The first- and second-order wave forces
(2D) strip theory. A finite-element analysis was then per- were determined using a higher-order boundary-element
formed. The results compared favorably with an experimental method. The response was studied in the time domain utilizing
model subjected to regular and irregular waves. a 3D hybrid element method. The hull is treated as a rigid
Mekha et al. (1994) studied the nonlinear effect of evalu- body, and the tendons were modeled with 3D beam elements
ating the wave forces on a TLP up to the wave free surface. connected at its top and bottom with spring-damper elements.
Several approximate methods were evaluated for regular and Responses were analyzed for surge, heave, and pitch motion;
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J. Aerosp. Eng. 1996.9:114-131.

 offset; and setdown. The mean tension in the tendons was resulting response closely matched that obtained from a time
found to be about 15% higher than the pretension due to the domain analysis with a reduction in CPU time.
large mean displacements. For a TLP model with 415 m ten- Kareem and Zhao (1994b), the surge response to wind,
dons, the mean offset and setdown were found to be approx- wave and current was found using a standard one DaF gov-
imately 20 m and 0.5 m, respectively. Liu et al. (1995) used erning equation. Wind and wave processes were assumed
the higher-order boundary element method to model a fixed Gaussian, but the structural velocity was not. This led to
and compliant TLP subject to second-order mean and double- highly nonlinear terms when expanded as an equivalent pol-
frequency wave loads. The second spatial derivatives of the ynomial. In this study, terms above quadratic were neglected.
first-order velocity potential were included for the body and A frequency domain analysis was performed with the response
free surfaces. The tendons were modeled as massless springs. cumulants based on Volterra theory.
The free-surface component was found to dominate the heave The response where only wind forces were included was
force and pitch moment. Large differences between the fixed determined in Kareem and Zhao (1994a). The response of a
and compliant TLPs were observed in the calculation of the one DaF TLP was presented in order to demonstrate the meth-
yaw moment. odology. A nonlinear wind gust loading factor was also de-
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Kareem (1983) presented time histories of wind velocity veloped.

fluctuations as single and multiple-point Gaussian random pro- Li and Kareem (1993) presented parametric models to gen-
cesses. For single-point loading, an equation relating drag erate time histories of waves. Autoregressive and moving av-
force as a function of fluctuating wind velocity was given for erages (ARMA), convolution, and interpolation techniques
the surge direction. When multiple-point loading is desired, were used in these models. These methods were demonstrated
equations for force in the surge direction, and moments in the with an application to a simple TLP modeled as a six DaF
yaw and pitch directions were presented. Here, the wind ve- rigid body subject to a random wave field. The dynamic re-
locity field is a function of time and its position in the vertical sponse of the TLP was calculated in the time domain. In this
(sway-heave) plane. These equations were transformed into the paper, loading due to wave drift forces was not included, but
frequency domain (Kareem 1985). Both frequency and time was addressed in a subsequent report (Kareem and Li 1994).
domain analyses for a TLP subject only to wind were imple- Li and Kareem (1995) presented a stochastic decomposition
mented to determine the response in the surge, yaw, and pitch technique to improve the efficiency of conventional frequency
directions. The hull was modeled as a rigid body, and the domain analysis and apply it to a TLP subject to random wind
tendons as massless springs. The motions were assumed to be and wave fields.
uncoupled and solved numerically. Results showed that the The numerical response of a TLP subject to wind and wave
surge mode is sensitive to the static and dynamic effects of loading was compared to the empirical results obtained by
wind forces. Pitch was minimally affected but may be of con- scaled tank testing by Vickery (1995). The numerical model
cern for fatigue analysis due to the low structural and hydro- was analyzed in both the time and frequency domains. The
dynamic damping. Kareem recommended a frequency domain TLP's hull was modeled as a six DaF rigid body moored to
analysis at preliminary design stages and a time domain anal- the sea floor with tendons that are assumed to remain straight.
ysis at later stages. A boundary element model was created of the submerged por-
To develop a better understanding of aerodynamic loading, tion of the TLP. As the hull translates in the surge direction,
mean aerodynamic force and moment coefficients were deter- the tendon stretches, resulting in a higher tension. The total
external force included first- and second-order wave forces,
mined from wind tunnel tests (Kareem et al. 1986). These tests
wind forces, and the drag term from the Morison equation. A
were conducted on a I: 128 scale model TLP hull with major
fourth-order Runge-Kutta differential equation solver was used
components (i.e., derricks, cranes, living quarters) in various
to solve for the time-domain response. In the frequency do-
configurations. main analysis, nonlinearities were neglected. A I :200 scale
Wave-induced forces for a TLP in an offset position may
experimental TLP model was tested in a wind-wave flume by
differ from those at the undisplaced position. Li and Kareem Vickery. The model consisted of a square deck, four columns,
(1992) analyzed this effect by adding displacement induced and four pontoons. Each column was restrained with one cable
feedback forces to the wave induced loads calculated at the representing the tendons. Additional modules were mounted
TLP's undisplaced position. Applying this method to a TLP on the platform including a scaled derrick and crew quarters.
model resulted in a surge response due to the feedback forces All six rigid body displacements of the hull, and one forward
that were of the same magnitude as the wave frequency re- and one aft tendon tension were measured. The response was
sponse. Drift induced by second order wave potential was not measured in a total of 35 wind-wave conditions (seven wave
included. heights each at five different wind speeds). The comparison
Kareem and Li (1990, 1993) have studied the response of between experimental results and computational analyses fo-
TLPs in the frequency domain. A procedure was described to cused on the surge response. Vickery (1995) concluded that
analyze the coupled six DaF motion of a TLP subjected to the effect of wind on the surge response was highly dependent
current random wind and wave loads. Conventional frequency on the size of the incident waves. The importance of the re-
domain analysis was applied to linear and linearized systems, sponse due to the wind diminished with an increase in wave
whereas nonlinear systems are typically solved utilizing a time height. For realistic combinations of wind and wave loads, the
domain approach. In this paper, the nonlinear terms are ex- surge response showed an increase of approximately 30-
panded into multivariate orthogonal polynomials. The zeroth- 100% as compared to a wave load only case. The heave re-
order terms in the polynomials represent the mean forces, the sponse was also significantly influenced by the wind loads.
first-order corresponds to the wave, wind, and damping forces, The mean tension in the tendons was dominated by wind
and the second-order accounts for the slowly varying drift forces, but its dynamic tension was not affected.
forces. This may be solved with perturbation or iterative tech- Jefferys and Patel (1982) created a linear analytical tendon
niques with spectral convolution and quadratic transformation model, and both linear and nonlinear finite-element models of
methods, but with significant increases in central-processing- a tendon. Assumptions included constant tension along the
unit (CPU) requirements. A new spectral decomposition ap- length of the tendon, pin joints at both ends, planar motion,
proach was developed to reduce computational requirements. and negligible bending forces. Longitudinal motions were also
It is a stochastic approach that more efficiently provides the neglected. This resulted in the following governing equation
input to be used by the perturbation or iterative methods. The for a tendon:
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J. Aerosp. Eng. 1996.9:114-131.

 il scattering and radiation problems. The surge response com-
my + w(y, y, r) = T-{
az
(32) puted analytically compared favorably with the results of a
numerical study that used the boundary element method.
where m = mass per unit length of the tendon; y = lateral Mullarkey and McNamara (1994) studied a TLP subject to
displacement; r = radius; T = constant tension; z = vertical first-order wave forces. The effects of added mass and radia-
coordinate; and w(y, y, r) include the added mass and the drag tion damping were considered. Semianalytical solutions for the
force. In addition to solving (32) for y, a modal analysis was hydrodynamics of the columns and pontoons were presented
performed and the results were compared with those from the and compared to the results of a radiation-diffraction panel
finite-element models. This nonlinear drag force was included program for wave-body interactions.
as an equivalent linear damping constant. Two approaches Park et al. (1991) performed a reliability analysis on four-
were presented to calculate this constant. The whole-tether ap- and six-column TLPs. Two failure criteria were considered for
proach assumes that the response is dominated by a solid body the tendons at the four corners: one, ultimate tensile strength
rocking mode and that higher modes are negligible. The ele- and the other, negative tension. Negative tension, or slacking
ment-by-element approach is an iterative technique that begins in the tendon, may result in high impulsive stresses at the
with an initial guess calculated using the whole-tether ap- return of positive tendon tension. The TLP was modeled as a
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proach to determine the tendon's velocity and then includes six DOF system with the hull treated as a rigid body and the
this in subsequent iterations. The whole-tether method was tendons as massless springs. It was subject to a severe storm
found to be more computationally efficient and the differences whose intensity was represented by a maximum wave height
between the two methods were negligible. treated as a random variable. Only first-order wave effects
The finite-element model where nonlinear drag forces were were considered. Two angles of wave attack were considered:
included was used in a study of the coupled dynamics of the 0° and approximately 45°. The failure probabilities were de-
hull with the tendons by Patel and Lynch (1983). The hull was termined for a 20-yr service life with the 45° case being more
modeled as a rigid body with six DOF subject to wave forces. critical. The results indicated that negative tension was the
It was coupled with the finite-element model of the tendon. governing failure criterion. Although not modeled, a sensitiv-
This was accomplished by first calculating the response of the ity analysis was performed to indirectly study the effect of
hull while assuming a quasi-static tendon stiffness. The hull's wind, current, and second-order wave forces. These forces
displacements were then used in the numerical analysis of the may cause horizontal mean drifts. Inclusion of drift in the
tendon's stiffness. The hull's response was then reevaluated. model was found to increase the tension in the tendons,
Only one iteration was performed since the hull's response thereby improving reliability by reducing the chance of tendon
was predominantly due to inertia, and the effect of the ten- slacking.
don's stiffness was secondary. Several perturbations were per-
formed with varying physical properties. Patel and Lynch con- Seismic Response
cluded that differences between the quasi-static and dynamic
tendon models were minimal except for tendons of lengths on Liou et al. (1988) studied the response of TLPs to vertical
the order of 1,500 m or greater. Therefore, the effect of the seismic excitation. The system was modeled as two masses,
tendon's dynamics on the hull's response was only significant representing the foundation and platform, connected with a
when the tendons were long, had a large mass per unit length, single tension element representing all the tendons. Both ver-
and the hull's displacements were small. The bending stresses tical and oblique stress waves travelling through the soil were
in the tendons were also determined to be small. The greatest used to solve for the response. Two accelerograms from actual
values were found for short tendons with large outer diameters earthquakes, along with two modified accelerograms, were se-
and thin wall thicknesses. lected as inputs to the system. These modifications were made
Patel and Patel (1995) studied the combined axial and lateral to create accelerograms that are closer to the American Petro-
response of tendons. The tendon was modeled as a straight leum Institute's recommended design response spectrum for
simply supported column subject to both horizontal and ver- offshore structures. The responses of the normalized tendon
tical motions at its top. This motion represented the response force at its top and bottom were presented. It was concluded
of the hull. The governing equation for the lateral motion of that the seismic response should be included in TLP design.
the tendon is The maximum seismic response was due to the vertical earth-
quake ground motions in a narrow band about the fundamental
2
ay
M at2
a'y
+ EI ax'
fly
- (To - S cos wt) ax2 + B"
Iayat Iayat = 0 (33)
frequency of the TLP; therefore selection of the accelerogram
is important.
The numerical response of a TLP subject to an earthquake
where y(x, t) = lateral displacement; M = structural and added was compared to the empirical results obtained by water-tank
mass; EI = structural flexural rigidity; To = constant axial ten- testing mounted on a shaker table by Kawanishi et al. (1991).
sion; S cos wt = axial force; and B v = drag-induced viscous The TLP was studied with the hull vertically moored and with
damping. At the tendon's top, the lateral force was assumed different horizontal offsets. The analytical model consisted of
sinusoidal. Assuming a solution for y(x, t), the resulting equa- a hull represented by a six DOF rigid body. The weight, in-
tion of motion was solved numerically. The initial position of ertia, and hydrodynamic forces of the tendon were neglected
the top end was at the midpoint of the hull's surge motion. with the mooring force on the hull treated as an axial force
Results from studies of three different tendon lengths were due to the tendon's elasticity. Wind, current, and wave drift
presented. The analysis showed that a greater amplitude of forces were constant. The structure was vertically excited to
vibration will result from the combined excitation than the simulate seismic forces. The nonlinear governing equations
individual axial and lateral excitations. This was particularly were numerically integrated. The experimental model con-
significant in even numbers of the instability region of the sisted of six columns connected by pontoons moored with one
Mathieu stability chart. The frequency of oscillation was also rod at each corner. To simulate offset conditions, a constant
found to be dependent on the relative magnitudes of the in- horizontal force is exerted by a weight through a pulley at-
dividual excitations for the combined response. tached to the hull. Accelerometers were used to measure the
Lee and Lee (1993) presented an analytical solution for a hull's heave response. Ring gauges measured the tension in
two-dimensional TLP subject to wave-induced surge motion. the tendons. Analysis of the TLP when vertically moored com-
It was approached as a boundary value problem divided into pared favorably with the empirical results. As the hull's am-
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J. Aerosp. Eng. 1996.9:114-131.

 plitude increased at resonance, nonlinearities resulted from the dons resulting in a springing effect. Springing could lead to
slack in the tendons. Response of the TLP in the offset con- earlier fatigue in the tendons.
dition showed higher tensions for the upstream tendons than Kim et al. (1991) studied the TLP responses under random
for the downstream tendons. Greater offset corresponded to a sea conditions. A digital second-order polyspectral analysis
larger tension and a lower response amplitude in the upstream was used to compute a power transfer function. The results of
tendons, the reverse was true for the downstream tendons. It an experimental 1:54 scale TLP model was used to quantify
was concluded that the tension in the tendons due to earth- the amount of power extracted from the incident wave spec-
quakes was greater than that due to waves. trum and transferred to each spectral component of the sum-
Kawanishi et al. (1994) studied the response of a TLP sub- and difference-frequency response. In the linear spectral
ject to tsunami waves (waves often caused by earthquakes). model, the input wave only affected the surge response at the
In the analytical model, the hull was modeled as a rigid body same frequency. For the second-order spectral model, the sum
and Morison's equations were used. Tendon tension was found and difference frequencies were important and considered.
to become 146% of the initial tension due to the tsunami Surge response and wave height were measured from the
waves. scaled model simulating a TLP moored in 450 m (1,500 ft) of
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Venkataramana (1994) studied the seismic response of a water. The TLP was subject to 45° unidirectional irregular
TLP in steady currents. The horizontal earthquake excitation waves generated according to the Pierson-Moskowitz spec-
was modeled stochastically, and the system response was an- trum. The results showed that the low-frequency surge re-
alyzed in the frequency domain. The governing equation was sponse is dependent on the difference frequencies with a far
obtained by dividing the system into two substructures. These smaller contribution from the sum frequencies.
were the tether and the pile-soil foundation. To determine the Marthinsen and Muren (1994) analyzed the measured re-
static deflected shape of the tendon subject to a steady current, sponse from the Snorre TLP. This TLP operates in the North
the tendon was modeled as a cantilever beam. This deflected Sea at a water depth of 310 m and produces 30,000 m'
shape is then discretized into 51 lumped masses with the hull (190,000 barrels) of oil each day. The springing standard de-
treated as a concentrated mass at its top. The pretension in the viations were presented for the measured roll and pitch re-
tendon is included in the geometric stiffness term. It was found sponse and compared to the calculated pitch response. These
that the horizontal response of the tendon increased with input calculations were performed with the second-order potential
ground acceleration and was independent of the level of steady theory and were found to underestimate the response.
current. The vertical response was zero with no current and Natvig and Vogel (1991) explored the effect of sum fre-
increased with increasing current and seismic forces. If suffi- quencies on tendon fatigue and extreme response. They stud-
ciently large, the vertical displacements may cause slackening ied a TLP subject to direct wave, slowly varying wave drift,
in the tendons. Horizontal displacements were on the order of wind gust, and sum-frequency excitations. A linearized fre-
102 times greater than the vertical displacements. The hori- quency domain procedure was implemented and the effect of
zontal and vertical displacements were reduced when the pile- various types of damping was presented. The second-order
soil foundation was included in the dynamic model as com- wave sum-frequency response analysis was based on quadratic
pared to a fixed base. transfer functions (QTF) obtained from three different insti-
tutions. The QTF represents the total wave force normalized
Springing and Ringing with respect to the incoming wave amplitudes. Only sum fre-
quencies near the natural frequencies for motion in the vertical
Springing is a steady-state response due to first- and second- plane were considered. In cases where the QTF data did not
order wave diffraction sum-frequency effects. Ringing is a include frequencies near these resonant frequencies a linear
transient response due to various nonlinear free surface effects. interpolation was performed. The greatest force spectral den-
Springing and ringing may contribute to fatigue and cause sity was found where a sum frequency had a corresponding
jerks in the platform that may affect the comfort of the crew. difference frequency that was smallest. The influence of sum-
Natvig and Teigen (1993), in their review of hydrodynamic frequency forcing was highly dependent on the damping in the
challenges in TLP design, included the need for further re- system. Potential damping due to wave radiation, viscous
search of springing and ringing. The designers of the Heidrun damping, column footing damping from model tests, soil
concrete TLP paid special attention to the effect of ringing damping, and structural damping were discussed. Structural
(Munkejord 1995). The following papers discuss current in- damping, though normally neglected, should be considered.
vestigations into these effects. For typical welded structures 0.5-0.8% can be used, and for
Second-order wave forces are quadratically nonlinear and the TLP components a value of 0.2% was recommended. A
may cause low-frequency drift oscillations and affect the high- comparison was made between the three QTFs for one sum
frequency vertical modes. This nonlinearity is due to the wave frequency and a single wave heading. The vertical sum-fre-
forces varying with the square of the incident wave heights, quency forcing was found to be less important than the mo-
the variable wetting of the TLP columns, and the effect of the ment forcing. Problems can also result from using a single
velocity squared term in the Bernoulli equation (Kim et al. heading since the QTF data set may contain a lower or higher
1991; Natvig and Vogel 1991; Chen et al. 1995). Solving the energy level at that heading. This will greatly affect tendon
quadratic velocity potential yields pairs of frequencies. Adding fatigue analysis. The second-order sum-frequency forcing also
or subtracting one frequency from another within a pair results had a significant impact on the extreme tendon response un-
in a frequency that is referred to as a sum or difference fre- dergoing a large 100-yr sea state.
quency. Neglecting these second-order wave forces, the TLP's Natvig (1994) studied the ringing response observed in
surge, sway, and yaw resonance frequencies were below that TLPs. Springing, due to first- and second-order diffraction
of the wave frequency range. The heave, pitch, and roll res- sum-frequency effects, has a slowly varying amplitude with
onance frequencies were above this range. If the second-order only moderate variation. A TLP's response due to ringing re-
wave forces were included, the difference frequencies may be sembles that of a struck bell. Its amplitude builds quickly and
in a bandwidth to increase the TLP's horizontal response, and then decays slowly. It is a transient state that can only occur
the sum frequencies may increase its vertical response. The when the steady-state springing is present; however, springing
additional horizontal response could affect drilling operations can occur without ringing. It is believed that ringing-type re-
and the sum frequencies would create a stretching of the ten- sponses are due to the various nonlinear effects of free surface
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J. Aerosp. Eng. 1996.9:114-131.

 variable wetting. The maximum ringing force may be as high wave height. Ratios of wave runup between the aft and for-
as the forces due to other dynamic effects. However, since ward columns were presented for different column spacing as
ringing rarely occurs (in general very few sea states cause a function of wave period. Wave runup was found to be greater
ringing), it may not have a significant effect on fatigue. The for closer column spacing. Wave upwelling, the local ampli-
response from model tests was presented. Ringing was found fication of the incident wave field, can damage the underside
for all headings with diagonal waves being the most signifi- of the platform if sufficiently large. Raising the platform in-
cant. Ringing was considerably reduced with slight changes in creases cost and decreases stability. Maximum upwelling val-
heading from the diagonal. Single high waves with steep fronts ues as a function of the incident wave period were presented.
and/or backs, but not breaking waves, resulted in ringing. As the column spacing increased, the magnitude of the wave
Ringing was not sensitive to increases in damping, but was upwelling decreased and the wave period with the peak am-
affected by vertical shifts in the TLP's center of gravity. An plitude shifted to lower wave periods.
analytical model was also created where the tendons were Niedzwecki and Rijken (1994) conducted experimental
treated as massless springs. Certain nonlinear effects (those studies to analyze the effect of nonlinear wave runup on TLPs
found to be most important are mentioned in the foIlowing and the nonlinear response of risers. The hull was modeled as
paragraph) were included and the response for one loading a single truncated cylinder piercing the water surface. A ran-
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case was compared to that of a model test. The importance of dom wave field was used and statistical moments and proba-
each effect in this loading case was presented. The frequency- bility plots were presented. The surface elevation of the waves
dependent wave-force coefficients from diffraction, the fre- was found to be non-Gaussian with higher crest amplitudes
quency-dependent wave-crest amplification around a column, and lower trough amplitudes. Increased wave steepness led to
the time derivative of the added mass, and the wave slap on a more pronounced non-Gaussian behavior. Testing performed
the columns were found to be of major importance. with the truncated cylinder in place resulted in wave runup
Gurley and Kareem (1995) simulated the ringing response amplitudes that significantly exceeded the significant wave
using a nonlinear Duffing osciIlator. The pitch response of a height.
lO-m-Iong immersed column subjected to linear and nonlinear Druggal and Niedzwecki (1995) modeled a TLP tendon or
waves was studied. The Morison equation and stretching the- riser as a single flexible cylinder to simulate a depth of 1,000
ory were used to compute the wave force up to the instanta- m. The cylinder was mounted in a wave basin and preten-
neous water level. The effect of the change in added mass due sioned. A different curvature response pattern was measured
to the varying water level was included. Ringing was only just below the still water level as compared to the middle level
present when nonlinear wave effects were included. Increasing of the cylinder. This was attributed to the depth dependency
the system center of mass eliminated the ringing response and of the Keulegan-Carpenter number. The probability density
the effect due to changes in the level of damping was insig- function of the measured curvature was found to be non-Gaus-
nificant. The response due to the grouping of waves was also sian and would result in a larger curvature than a Gaussian
studied. Ringing was observed for a single group of large assumption would predict.
waves, with the effect lessened if multiple groups were ap- Faltinsen et al. (1995) analyzed a vertical circular cylinder
plied. These results compared favorably with that of Natvig subject to nonlinear wave loads. The authors found that the
(1994). second- and third-order wave forces near the free surface were
of comparable magnitude. Kim (1993) presented equations in
Effect of Columns closed form to determine the interaction of waves between
multiple vertical circular cylinders. Wavelength, column spac-
Columns were often incorporated into the single rigid body ing, and wave heading were determined to be important fac-
representing the hull in the foregoing papers. The columns are tors. Zhu et al. (1995) studied the vibration of a circular cyl-
the structural elements that pierce the water surface. Therefore,
inder in the wake of another circular cylinder. Two cylinders
in addition to wind and current forces, they are subject to wave were positioned at six different distances apart. Unsteady-flow
loads that include the wave-free surface effects. The wake cre-
theory and direct measurements were used. It was found that
ated by one column may affect another making the positioning
the position of the cylinder in the wake significantly affected
of the columns important. The columns are also subject to its stability. Mizutani et al. (1994) analyzed wave forces on
variable added mass and damping.
two and three large-diameter cylinders. They found that wave
Chung (1994) studied the effect on the added mass and
diffraction affected the lead cylinder more in the two-cylinder
wave damping coefficients due to the free surface effect. Ex-
experiment and the center cylinder in the three-cylinder study.
periments showed a larger effect on the added mass coefficient
for the vertical oscillation than for the horizontal oscillation.
Tendons
Others have studied hydrodynamic forces on vertical cylinders
either numerically (Weggel and Roesset 1994) or experimen- In many of the reviewed papers, tendons were treated as
tally (Gopalkrishnan et al. 1991; Hogedal, et aI. and Burcharth massless springs. In the more advanced structural models, the
1994; Hoshino and Sato 1994). tendons themselves were modeled. Tendons can be modeled
Vortex-shedding and vortex-induced vibration can be of sig- as beams, cables, or strings with varying degrees of complex-
nificance in the study of TLPs since it can result in large am- ity. Tendons and risers have many similarities in that they are
plitudes of vibration (Billah 1989). This effect has been in- both very long submerged hollow pipes subject to hydrody-
vestigated numerically (Lee et al. 1994) and experimentally namic loads. As the water depth increases, the need to consider
(Moe et al. 1994a; Sibetheros et al. 1994; Sumer and Kozak- higher-order effects in both the loading and the structure in-
iewicz 1994; Sunahara and Kinoshita 1994). Chung et al. creases. Two fundamental differences between the two is the
(1994) discussed means to reduce the effect of vortex shedding extremely high tension experienced by the tendons and the
that included the use of helical strakes (cables wrapped around fluid flowing within the riser (Moe et al. 1994b). Analysis
the cylinder along a helical path). techniques for risers may be applied to tendons. Patel and Se-
Niedzwecki and Huston (1992) conducted 1:250 scaled yed's (1995) review paper includes 74 references on modeling
wave tank tests. A four-column TLP model with and without and analysis techniques for flexible risers.
pontoons was studied with varying column spacing. Regres- As with the columns discussed earlier, vortex-shedding and
sion formulas were developed for the wave runup on a forward vortex-induced vibration may need to be considered. Although
leg as a function of horizontal wave velocity and incident the tendon diameters are significantly smaller than the column
JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/127

J. Aerosp. Eng. 1996.9:114-131.

 diameters, there are generally multiple tendons at each comer Deck
of the hull. This may lead to interactions between the tendons
at one comer as well as between the comers. Bokaian (1994)
modeled a tendon or riser subject to vortex shedding in a ver-
ticaIly sheared flow. For a tendon of uniform mass and cross
section, a closed-form solution was presented. It showed that
multiple structural modes are usually excited by the shear flow.
The numerical results were in agreement with reported ex-
perimental observations. Mooring Line
Several papers are cited here that present methods to aid in
modeling tendons as beams (Kim and Lee 1990; Saevik and Shaft
Berge 1993; Chucheepsakul et al.; 1994; Yoo et al. 1995; Iura
and Atluri 1995; Chucheepsakul et al. 1995). The dynamic
response of cables may also be useful in the modeling of ten-
Downloaded from ascelibrary.org by Cambridge University on 06/24/15. Copyright ASCE. For personal use only; all rights reserved.

dons (Benedettini and Rega 1987; Takahashi and Konishi

1987a,b; Benedettini and Rega 1989; Cheng and Perkins
1992b; Perkins 1991; Howell 1991; Irvine 1992; Cheng and
Perkins 1992a; Cheng and Perkins 1994; Benedettini et al.
1995) including the case of extreme tension (Shin 1991) and Base
negative tension (Triantafyllou and HoweIl 1994).
Seyed and Patel (1991) investigated tendon response subject
to short duration tension loss. The governing equation of mo-
tion for the lateral motion of the tendon was solved analyti-
caIly to study the dynamic pulse buckling behavior. Since
higher modes are affected by pulse bucking, bending stiffness
was included. Compressive stress as a function of time was
presented. It was found that TLPs can be designed to allow
FIG. 3. Guyed Tower
for a tension loss of a few seconds.
Mekha et al. (1996) created three different tendon models.
The first was as a massless spring to provide a constant lateral the root-mean-square tower rotation. The problem is solved
stiffness with the TLP setdown neglected. The second allowed for the case where the tower motion is limited to planar mo-
for time-varying axial forces in the spring with one-third of tion. One general conclusion from this study is that a stiff
the tendon mass lumped at the attachment points. The third cable array is desirable when the tower is subject to normal
tendon model consisted of flexural beam elements and allowed operating loads, but that a softer system with more compliance
for the time-varying axial forces. Hydrodynamic loads on the is of advantage when the tower is subjected to storm loads.
tendons were included in the third model only. Wheeler's ap- Such additional compliance will permit storm energies to be
proximation (Chakrabarti 1987) was used to determine the dissipated by the structural dynamics rather than only because
wave forces up to the wave-free surface. This stretching of structural strain energy. The structure is modeled as a rigid
method extends the wave kinematics in the same form as cal- inverted pendulum with a point mass at the top, subjected to
culated to the mean water level, a hyperbolic cosine function, environmental forces along with the cable forces.
up to the free surface. The hull was modeled as a rigid body In a finite-element approach, Leonard and Young (1985b)
subject to regular waves and analyzed at different wave depths, prepared a three-dimensional model to simulate the static and
wave heights, and wave periods. The amplitude of the surge dynamic response of a compliant tower to the ocean environ-
response was found to be independent of the model used. For ment. The aforementioned nonlinearities are included here as
a constant wave period it increased linearly with wave height, well as the spatial variation of the fluid loading. Five valida-
and decreased linearly with increasing wave frequency for a tion problems are presented, as follows: (1) the case of a
constant wave height. The surge mean offset may more than steady tow of a cable with a sphere at the end; (2) the response
double for the third model where hydrodynamic loads on the of the articulated tower to steady current; (3) the response of
tendons were included. The maximum tendon forces were not the articulated tower to waves; (4) the response to two-dimen-
affected by the hydrodynamic loads. However the minimum sional waves; and (5) the response of a tension leg structure
decreased by 10%. Neglecting the inclusion of the huIl set- to waves. Owen and Shi (1991) also used a finite-element
down in the first model resulted in a maximum tendon force analysis to perform parametric studies of guyed-tower dy-
that was about 15% higher. namic response. Numerical results were compared to model
tests, showing the results to be good. The key conclusion was
that guyline stiffness is the most important characteristic de-
GUYED TOWERS
termining guyed-tower response.
Guyed tower platforms are a slender uniform jacket, sup- Ryu and Yun (1992) studied the reliability of guyed-tower
ported by piles which are closely clustered around the center structures against anchor pile failure. This failure will occur
of the structure, as shown in Fig. 3. Lateral support is through generally during a large storm event and may be either a sin-
an array of mooring lines configured for designs that optimize gle-exceedance failure or due to high-cycle fatigue. As with
the tensions in the various lines. Hanna et al. (1983) discuss other work on this type of structure, the equations of motion
how such a mooring system is inherently nonlinear, and pro- contain nonlinearities due to the cable restoring forces. Nu-
ceed to model the system for a dynamic analysis. The structure merical results are provided for a hypothetical 300 m (1,000
is modeled as a lumped mass system. Nonlinearities are intro- ft) structure situated in the Gulf of Mexico. In a further work
duced into the system due to the force deflection of a typical (Ryu and Yun 1994), the nonstationary response of a guyed
mooring array, and due to nonlinear soil effects. tower to strong earthquakes and moderate waves was ana-
Wilson and Orgill (1984) studied the optimal cable config- lyzed. The solution is based on a stochastic linearization that
urations for passive dynamic control of compliant towers. In results in the solution of a linear equation of motion. The
particular, an optimization technique is developed to minimize earthquake is modeled as a filtered Kanai-Tajimi spectrum.
128/ JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 The nonstationarity results in time-dependent response statis- to loads due to wave slamming, wind and coriolis acceleration." Proc.•
tics. A key conclusion is that the moderate wave loading con- 36th A/AAIASMEIASCEIAHSIASC Struc.• Struct. Dyn., and Mat. Conf,
257-265.
tributes comparably to the structural response as does the Bar-Avi, P., and Benaroya, H. (1995d). "Simplified equation for a SDOF
earthquake. articulated tower." J. Engrg. Mech.
A characteristic of guyed towers is that they have a very Bar-Avi, P., and Benaroya, H. (1995e). "Stochastic response of an artic-
low natural frequency, making them susceptible to low-fre- ulated tower." Int. J. of Nonlinear Mech.
quency excitation such as second-order drift forces in random Bar-Avi, P., and Benaroya, H. (1996). "Nonlinear dynamics of an artic-
seas. Bisht and Jain (1994) have studied such low-frequency ulated tower submerged in the ocean." J. of Sound and Vibration,
190(1), 77 -103.
drift oscillations of guyed towers. A numerical technique is Bea, R., Cornell, C., Vinnem, J., Geyer, J., Shoup, G., and Stahl, B.
used to solve the nonlinear equations of motion. A key result (1994). "Comparative risk assessment of alternative TLP systems:
is that the response due to second-order wave drift force alone structure and foundation aspects." J. Offshore Mech. and Arctic Engrg.,
shows a resonating effect at the structure's natural frequency 116, 86-96.
that is comparable to the first-order wave response. However, Benedettini, E, and Rega, G. (1987). "Non-Iinear dynamics of an elastic
when both drift and wave forces are combined, hydrodynamic cable under planar excitation." Int. J. Non-Linear Mech., 22(6), 497-
509.
damping attenuates the drift effects.
Downloaded from ascelibrary.org by Cambridge University on 06/24/15. Copyright ASCE. For personal use only; all rights reserved.

Benedettini, E, and Rega, G. (1989). "Planar non-linear oscillations of

In a series of papers (Kahla 1994, Kahla 1995a,b) numerical elastic cables under superharmonic resonance conditions." J. Sound
dynamic analyses of guyed towers are carried out showing and Vibration, 132(3), 353-366.
circumstances where large amplitude oscillations of the guy Benedettini, E, Rega, G., and Alaggio, R. (1995). "Non-linear oscilla-
cables are possible, causing tensions large enough to fail the tions of a four-degree-of-freedom model of a suspended cable under
cables leading to structural collapse. multiple internal resonance conditions." J. Sound and Vibration,
182(5), 775 - 798.
"BHP brings Challis onstream with world's largest SALRAM." (1990).
Concluding Remarks Oc. Industry, 53-56.
Billah, K. (1989). "A study of vortex-induced vibration," PhD thesis,
The important class of offshore structures known as com- Princeton Univ., Princeton, N.J.
pliant structures have been studied over the past two decades. Bisht. R., and Jain, A. (1994). "Low frequency drift oscillation of off-
Such structures have found primary offshore application in the shore guyed towers." Proc.. 4th Int. Offshore and Polar Engrg. Conf,
oil industry, but also in cases where a stable ocean platform 1,122-129.
is needed for communication and mooring. Bokaian, A. (1994). "Lock-in prediction of marine risers and tethers."
J. Sound and Vibration, 175(5), 607 -623.
This paper summarizes the literature focused on articulated Borgman, L. (1969). "Ocean wave simulation for engineering design."
towers, tension leg platforms, and guyed towers. Forces on J. Wtrwy. and Harb. Diy., ASCE, 95, 557-583.
these structures due to the ocean and atmosphere include ocean Burns, G., and D' Amorim, G. (1977). "Buoyant tower for phase one
waves and current, wind, buoyancy, and friction at the base. development of Garoupa field." Proc.. 9th Annu. Offshore Technol.
Various models have been developed for these forces, with a Conf,I77-184.
spectrum of sophistication. Similarly, the complex equations Butt, H., Salewski, J., and Wagner, P. (1980). "A large-scale test with
the concrete articulated tower conat in the vicinity of the research plat-
of motion governing structural response find numerous meth- form 'nordsee'." Proc., Int. Conf on Marine Sci. and Oc. Engrg., 31
ods for their solution. -47.
The interested worker will find here the necessary back- Chakrabarti, S. (1987). Hydrodynamics of offshore structures. Computa-
ground on this problem, and then will be able to proceed with tional Mechanics Publications, Inc.
the research literature. Our objective here was to primarily Chakrabarti, S., and Cotter, D. (1978). "Analysis of a tower-tanker sys-
explore the various modeling approaches used by workers tem." Proc.• 10th Annu. Offshore Techno!' Con!, 1301-1310.
Chakrabarti, S.. and Cotter, D. (1979). "Motion analysis of articulated
worldwide to better understand the behavior of such structures.
tower." J. Wtrwy., Port. Coast., and Oc. Diy., ASCE, 105,281-292.
What we tended to see in the literature was either very so- Chakrabarti, S., and Cotter, D. (1980). "Transverse motion of articulated
phisticated large-scale computational models or very simple tower." J. Wtrwy., Port. Coast.. and Oc. Diy., ASCE, 107,65-77.
analytical models with complex loading functions. A general Chantrel, J., and Marol, P. (1987). "Subharmonic response of articulated
conclusion is that much can yet be learned using nonlinear loading platform." Proc.• 6th Con! on Offshore Mech. and Arctic
differential equations of motion and realistic loading models. Engrg., 35-45.
The benefit to the computational modelers would be signifi- Chen, X., Molin, B., and Petitjean, E (1995). "Numerical evaluations of
the springing loads on tension leg platforms." Marine Struct., 8, 501-
cant. 524.
Cheng, S., and Perkins, N. (l992a). "Closed-form vibration analysis of
Acknowledgment sagged cable/mass suspensions." J. Appl. Mech., 59, 923-928.
Cheng, S., and Perkins, N. (1992b). "Free vibration of a sagged cable
This work is supported by the Office of Naval Research Grant No. supporting a discrete mass." J. Acoustic Soc. Am., 91(5), 2654-2662.
NOOOI4-93-1-0763. The first two writers are grateful for their ONR-spon- Cheng, S., and Perkins. N. (1994). "Theoretical and experimental anal-
sored Fellowship, and the third writer for additional support. Our program ysis of the forced response of sagged cable/mass suspensions." J. Appl.
manager, Dr. T. Swean, is sincerely thanked for this support as well as Mech., 61, 944-948.
for his interest in our work. Choi, H., and Lou, J. (1991). "Nonlinear behavior of an articulated off-
shore loading platform." Appl. Oc. Res.. 12(2), 63 - 74.
APPENDIX. REFERENCES Chucheepsakul, S., Buncharoen, S., and Huang, T. (1995). "Elastica of
simple variable-are-length beam subjected to end moment." J. Engrg.
Abbott, P., D'Souza, R., Solberg, I., and Eriksen, K. (1994). "Evaluating Mech., 121(7), 767 -772.
deepwater development concepts." Proc., SPE Int. Petro Conf and Ex- Chucheepsakul, S., Buncharoen, S., and Wang, C. (1994). "Large de-
hibition of Mexico, 111-127. flection of beams under moment gradient." J. Engrg. Mech., 120(9),
Armenis, D., Angelopoulos, T., and Papanikas, D. (1991). "Time domain 1848-1860.
simulation of the dynamic behavior of a tension leg platform." Proc.• Chung, J. (1994). "Added mass and damping on an oscillating surface-
1st Int. Offshore and Polar Engrg. Conf, I, 100-107. piercing circular column with circular footing." Proc.• 4th Int. Offshore
Bar-Avi, P., and Benaroya, H. (1995a). "Dynamic response of an artic- and Polar Engrg. Conj, 3, 182-189.
ulated tower to random waves and current loads." Proc.• 3rd Int. Conj Chung, J., Whitney, A., Lezius, D., and Conti, R. (1994). "Flow-induced
on Stochastic Struct. Dyn. torsional moment and vortex suppression for a circular cylinder with
Bar-Avi, P., and Benaroya, H. (l995b). "Response of a two DOF artic- cables." Proc., 4th Int. Offshore and Polar Engrg. Con!, 3, 447 -459.
ulated tower to different environmental conditions." Int. J. of Nonlin- Datta, T., and Jain, A. (1990). "Response of articulated tower platforms
ear Mech. to random wind and wave forces." Compo and Struct., 34(1), 137-
Bar-Avi, P., and Benaroya, H. (1995c). "Response of an articulated tower 144.

JOURNAL OF AEROSPACE ENGINEERING / OCTOBER 1996/129

J. Aerosp. Eng. 1996.9:114-131.

 Dove, P., and Lohr, C. (1993). "Lateral mooring systems for tension leg Kareem, A., and Li, Y. (1994). "Stochastic response of a tension leg
platforms." Offshore and Arctic Operations, 51, 129-138. platform to viscous and potential drift forces." Probabilistic Engrg.
Duggal, A., and Niedzwecki, J. (1995). "Dynamic response of a single Mech.,9,1-14.
flexible cylinders in waves." J. Offshore Mech. and Arctic Engrg., 117, Kareem, A., and Zhao, J. (I 994a). "Analysis of non-Gaussian surge re-
99-104. sponse of tension leg platforms under wind loads." J. Offshore Mech.
Faltinsen, 0., Newman, J., and Vinje, T. (1995). "Nonlinear wave loads and Arctic Engrg., 116, 137 -144.
on a slender vertical cylinder." J. Fluid Mech., 289, 179-198. Kareem, A., and Zhao, J. (1994b). "Response statistics of tension leg
Fjeld, S., and Flogeland, S. (1980). "Deepwater platforms problem ar- platforms under wind and wave loads: a statistical quadratization ap-
eas." Proc., Eur. Offshore Petro Conf. and Exhibition, 223-234. proach." Struct. Safety and Reliability, 497-503.
Ganapathy, C., Samantaray, B., and Nagamani, K. (1990). "Truss type Kareem, A., Lu, P., Finnigan, T., and Liu, S. (1986). "A wind tunnel
articulated towers for deep water applications." Proc., 9th Int. Conf. investigation of aerodynamic loads on a typical tension leg platform."
on Offshore Mech. and Arctic Engrg., 477-484. Proc., Offshore Technol. Conf., 187 - 197.
Gerber, M., and Engelbrecht, L. (1993). "The bilinear oscillator: the re- Kawanishi, T., Kato, T., Takamura, H., and Kobayashi, H. (1991). "Earth-
sponse of an articulated mooring tower driven by irregular seas." Oc. quake response of the tension leg platform under offset condition."
Engrg., 20(2), 113 -133. Proc., 1st Int. Offshore and Polar Engrg. Conf., 1,80-86.
Gopalkrishnan, R., Triantafyllou, M., and Grosenbaug, M. (1991). "In- Kawanishi, T., Ohashi, S., Han-Ya, H., and Kobayashi, H. (1994). "Tsu-
fluence of amplitude modulation of the fluid forces acting on a vibrat- nami response of the tension leg platform under unbalanced initial ten-
Downloaded from ascelibrary.org by Cambridge University on 06/24/15. Copyright ASCE. For personal use only; all rights reserved.

ing cylinder in cross-flow." Proc., 1st Int. Offshore and Polar Engrg. sion." Oc. Conf. Rec., 2, 42-47.
Conf., 2, 132-139. Kim, C., and Luh, P. (1981). "Motions and loads of an articulated loading
Gottlieb, 0., Yim, C., and Hudspeth, R. (1992). "Analysis of nonlinear platform in waves." Oc. Engrg., 956-961.
response of an articulated tower." Int. J. Offshore and Polar Engrg., Kim, C., Kim, M., Liu, Y., and Zhao, C. (1994). "Time domain simu-
2(1), 61-66. lation of nonlinear response of a coupled TLP system in random seas."
Hanna, S., Karsen, D., and Yeung, J. (1988). "Dynamic response of a Proc., 4th Int. Offshore and Polar Engrg. Conf., 1, 68-77.
compliant tower with multiple articulations." Proc., 7th Int. Conf. on Kim, M.-H. (1993). "Interaction of waves with n vertical circular cyl-
Offshore Mech. and Arctic Engrg., 257-269. inders." J. Wtrny., Port, Coast., and Oc. Engrg., ASCE, 119(6),671-
Hanna, S., Mangiavacchi, A., and Suhendra, R. (1983). "Nonlinear dy- 689.
namic analysis of guyed tower platforms." J. Energy Resour. Technol., Kim, S., Powers, E., Miksad, R., and Fischer, E (1991). "Quantification
105, 205-211. of nonlinear energy transfer to sum and difference frequency responses
Haverty, K., McNamara, J., and Moran, B. (1982). "Finite dynamics of TLP's." Proc., 1st Int. Offshore and Polar Engrg. Conf., 1,87-92.
motions of articulated offshore loading towers." Proc., Int. Conj. of Kim, Y., and Lee, P. (1990). "Nonlinear motion characteristics of a long
Marine Res. Ship Technol. and Oc. Engrg. vertical cylinder." Proc., ist PacifidAsia Offshore Mech. Symp., 247-
Hays, D., McSwiggan, M., and Vilain, R. (1979). "Operation of an ar- 256.
ticulated oil loading column at the Beryl field in the North Sea." Proc., Kirk, C., and Jain, R. (1977). "Response of articulated tower to waves
II th Annu. Offshore Technol. Conf., 1805 - 1815. and current." Proc., 9th Annu. Offshore Technol. Conf., 545-552.
Hevacioglu, I., and Incecik, A. (1988). "Dynamic analysis of coupled Gurley, K. R., and Kareem, A. (1995). "Numerical experiments in ringing
articulated tower and floating production system." Proc., 7th Int. Conf. and springing of offshore platforms." Proc., 10th ASCE Engrg. Mech.
on Offshore Mech. and Arctic Engrg., 279-287. Conf., ASCE, New York, N.Y., 1-4.
Hogedal, M., Skourup, J., and Burchart, H. (1994). "Wave forces on a Lee, C., and Lee, J. (1993). "Wave-induced surge motion of a tension
vertical smooth cylinder in directional waves." Proc., 4th Int. Offshore leg structure." Oc. Engrg., 20(2), 171 - 186.
and Polar Engrg., Conf., 3, 218-224. Lee, Y., Hong, S., and Kang, K. (1994). "A numerical simulation of
Hoshino, K., and Sato, H. (1994). "Experimental study on hydrodynamic vortex motion behind a circular cylinder above a horizontal plane
forces acting on an oscillating column with circular footing." Proc., boundary." Proc., 4th Int. Offshore and Polar Engrg. Conf., 3, 427-
4th Int. Offshore and Polar Engrg., Conj., 3,173-181. 433.
Howell, C. (1991). "Numerical analysis of 2-d nonlinear cable equations Leonard, J., and Young, R. (l985a). "Coupled response of compliant
with applications to low-tension problems." Proc., 1st Int. Offshore offshore platforms." Engrg. Struct., 7, 74-84.
and Polar Engrg. Conf., 2, 203-209. Leonard, J., and Young, R. (l985b). "Coupled response of compliant
Irvine, M. (1992). Cable structures. Dover Publications, Inc., New York, offshore platforms." Engrg. Struct., 7, 74-84.
N.Y. Li, Y., and Kareem, A. (1992). "Computation of wind-induced drift
Iura, M., and Atluri, S. (1995). "Dynamic analysis of planar flexible forces introduced by displaced position of compliant offshore plat-
beams with finite rotations by using inertial and rotating frames." forms." J. Offshore Mech. and Arctic Engrg., 114, 175-184.
Compo and Struct., 55(3), 453-462. Li, Y., and Kareem, A. (1993). "Parametric modelling of stochastic wave
Jain, A., and Datta, T. (1991). "Nonlinear behavior of articulated tower effects on offshore platforms." App/. Oc. Res., 15,63-83.
in random sea." J. Engrg. for Industry, 113, 238-240. Li, Y., and Kareem, A. (1995). "Stochastic decomposition and application
Jain, K., and Datta, T. (1987). "Stochastic response of articulated tow- to probabilistic dynamics." J. Engrg. Mech., ASCE, 121(1), 162-174.
ers." Proc., Deep Offshore Technol. 4th Int. Conf. and Exhibit, 191- Liaw, C. (1988). "Bifurcations of subharmonics and chaotic motions of
208. articulated towers." Engrg. Struct.. 117 -124.
Jain, R., and Kirk, C. (1981). "Dynamic response of a double articulated Liaw, C., Shankar, N., and Chua, K. (1989). "Large motion analysis of
offshore loading structure to noncollinear waves and current." J. En- compliant structures using Euler parameters." Oc. Engrg., 16(6), 545-
ergy Resour. Techno/., 103,41-47. 557.
Jefferys, E., and Patel, M. (1982). "On the dynamics of taut mooring Liaw, C., Shankar, N., and Chua, K. (1992). "Subharmonic motions and
systems." Engrg. Struct., 4, 37-43. wave force-structure interaction." Marine Struct., 5, 281-295.
Johnson, C. (1994). "A computer aided design approach for deep water Lim, E, and Hatton, S. (1991). "Design considerations for TLP risers in
tension leg platforms." Offshore and Arctic Operations, 58, 77-87. harsh environments." Proc., 1st Int. Offshore and Polar Engrg. Conf.,
Kahla, N. (1994). "Dynamic analysis of guyed towers." Engrg. Struct., 2, 182-189.
16(4),293-301. Liou, G., Penzien, J., and Yeung, R. (1988). "Response of tension-leg
Kahla, N. (1995a). "Equivalent beam-column analysis of guyed towers." platforms to vertical seismic excitations." Earthquake Engrg. Struct.
Compo and Struct., 55(4), 631-645. Dyn., 16, 157-182.
Kahla, N. (1995b). "Influence of star mounts on guyed towers." Compo Liu, Y., Kim, M., and Kim, C. (1995). "The computation of second-order
and Struct., 54(5), 989-995. mean and double-frequency wave loads on a compliant TLP by
Kareem, A. (1983). "Nonlinear dynamic analysis of compliant offshore HOBEM." Int. J. Offshore and Polar Engrg., 5(2), 111-119.
platforms subjected to fluctuating wind." J. Wing Engrg. and Industrial Marthinsen, T., and Muren, J. (1994). "Snorre TLP-analysis of mea-
Aerodyn., 14, 345-356. sured response." Proc., 4th Int. Offshore and Polar Engrg. Conf., I,
Kareem, A. (1985). "Wind-induced response analysis of tension leg plat- 32-39.
forms." J. Struct. Engrg., ASCE, 111(1),37-55. Mathisen, K., and Bergan, P. (1992). "Large displacement analysis of
Kareem, A., and Li, Y. (1990). "Response of tension leg platforms to submerged multibody systems." Engrg. Computations, 9, 609-634.
wind, wave, and currents: a frequency domain approach." Proc., Off- McNamara, J., and Lane, M. (1984). "Practical modeling for articulated
shore Technol. Conf., 437 -442. risers and loading columns." J. Energy Resour. Technol., 444-450.
Kareem, A., and Li, Y. (1993). "Wind-excited surge response of tension- Mekha, B., Johnson, c., and Roesset, J. (1994). "Effects of different
leg platform: frequency-domain approach." J. Engrg. Mech., ASCE, wave free surface approximations on the response of a TLP in deep
119(1),161-173. water." Proc., 4th Int. Offshore and Polar Engrg. Conf., I, 105-115.

130 I JOURNAL OF AEROSPACE ENGINEERING I OCTOBER 1996

J. Aerosp. Eng. 1996.9:114-131.

 Mekha, B., Johnson, C., and Roesset, J. (1996). "Implications of tendon (1989). "NKK TLP (tension leg platform)." NKK Tech. Rev., 55,
modeling on nonlinear response of TLP." J. Struct. Engrg., ASCE, 103-114.
122(2), 142-149. Schellin, T., and Koch, T. (1985). "Calculated response of an articulated
Mizutani, N., Kim, C., Iwata, K., Matsudaira, R., Miyaike, Y., and Yu, tower in eaves comparison with model tests." Proc., 4th Offshore
H. (1994). "Wave forces acting on multiple cylindrical structures with Mech. And Arctic Engrg. Symp., Vol. I, ASME, New York, N.Y.
large diameter." Proc., 4th Int. Offshore and Polar Engrg. Conf, 3, Sebastiani, G., Brandi, R., Lena, F. D., and Nista, A. (1984). "Highly
244-251. compliant column for tanker mooring and oil production in 1000 m
Moe, G., Holden, K., and Yttervoll, P. (1994a). "Motion of spring sup- water depth." Proc., 16th Annu. Offshore Technol. Conf, 379-388.
ported cylinders in subcritical and critical water flows." Proc., 4th Int. Seller, L., and Niedzwecki, J. (1992). "Response characteristics of multi-
Offshore and Polar Engrg. Conf, 3,468-475. articulated offshore towers." Oc. Engrg., 19(1), 1-20.
Moe, G., Stromsem, K., and Fylling, I. (1994b). "Behavior of risers with Seyed, F., and Patel, M. (1991). "Parametric studies of flexible risers."
internal flow under various boundary conditions." Proc., 4th Int. Off- Proc., 1st Int. Offshore and Polar Engrg. Conj.
shore and Polar Engrg. Conf, 2, 258-262. Shin, H. (1991). "Analysis of extreme tensions in a snapping cable."
Morison, J., O'Brien, M., Johnson, J., and Schaaf, S. (1950). "The force Proc., 1st Int. Offshore and Polar Engrg. Conf, 2, 216-221.
by surface waves on piles." Petro Trans., 189, 149-154. Sibetheros, I., Miksad, R., Ventre, A., and Lambrakos, K. (1994). "Flow
Muhuri, P., and Gupta, A. (1983). "Stochastic stability of tethered buoy- mapping on the reversing vortex wake of a cylinder in planar harmonic
ant platforms." Oc. Engrg., 10(6),471-479. flow." Proc., 4th Int. Offshore and Polar Engrg. Conf, 3, 406-412.
Mullarkey, T., and McNamara, J. (1994). "Evaluation of the three-di- Smith, J. (1979). "The application of a concrete articulating buoyant
Downloaded from ascelibrary.org by Cambridge University on 06/24/15. Copyright ASCE. For personal use only; all rights reserved.

mensional linear hydrodynamics of the ISSC TLP using a semiana- column to offshore drilling and production systems." Proc., Symp. on
Iytical method." Proc., 4th Int. Offshore and Polar Engrg. Conj., I, New Technol. for Exploration and Exploitation of Oil and Gas Resour.,
61-67. 377-389.
Munkejord, T. (1995). "Heidrun concrete TLP: update." Offshore and Smith, J., and Taylor, R. (1980). "The development of articulated buoyant
Arctic Operations, 68, 189-196. system column system as an aid to economic offshore production."
Naess, A. (1980). "Loads and motions of an articulated loading platform Proc., Eur. Offshore Petro Conf and Exhibition, 545-557.
with moored tanker." Proc., 12th Annu. Offshore Technol. Conf, 409- Song, X., and Kareem, A. (1994). "Combined system analysis of tension
417. leg platforms: a parallel computational scheme." OMAE, I, 123-134.
Natvig, B. (1994). "A proposed ringing analysis model for higher order Sumer, B., and Kozakiewicz, A. (1994). "Visualization of flow around
tether response." Proc., 4th Int. Offshore and Polar Engrg. Conf, I, cylinders in irregular waves." Proc., 4th Int. Offshore and Polar Engrg.
40-51. Conj., 3,413-420.
Natvig, B., and Teigen, P. (1993). "Review of hydrodynamic challenges Sunahara, S., and Kinoshita, T. (1994). "Flow around circular cylinder
in TLP design." Int. J. Offshore and Polar Engrs., 3(4), 241-249. oscillating at low keulegan-carpenter number." Proc., 4th Int. Offshore
Natvig, B., and Vogel, H. (1991). "Sum-frequency excitations in TLP and Polar Engrg. Conf, 3,476-483.
design." Proc., 1st Int. Offshore and Polar Engrg. Conf, I, 93-99. Takahashi, K., and Konishi, Y. (1987a). "Non-linear vibrations of cables
Niedzwecki, J., and Huston, J. (1992). "Wave interaction with tension in three dimensions. Part I: Non-linear free vibrations." J. Sound and
leg platforms." Oc. Engrg., 19(1), 21- 37. Vibration, 118(1), 69-84.
Niedzwecki, J., and Rijken, O. (1994). "Characterizing some aspects of Takahashi, K., and Konishi, Y. (l987a). Non-linear vibrations of cables
stochastic wave-structure interactions." Stochastic Dyn. and Reliability in three dimensions. Part II: Out-of-plane vibrations under in-plane
of Nonlinear Oc. Sys., 77, 109-118. sinusoidally time-varying load." J. Sound and Vibration, 118(1),85-
Olsen, 0., Braathen, A., and Loken, A. (1978). "Slow and high frequency 97.
motions and loads of articulated single point mooring system for large Thompson, J., Bokaian, A., and Ghaffai, R. (1984). "Stochastic and cha-
tanker." Norwegian Marine Res., 14-28. otic motions of compliant offshore structures and articulated mooring
Oppenheim, B., and Fletter, J. (1991). "Design notes on spread mooring towers." J. Energy Resour. Technol., 106, 191 -198.
systems." Proc., 1st Int. Offshore and Polar Engrg. Conf, 2, 259- Triantafyllou, M., and Howell, C. (1994). "Dynamic response of cables
265. under negative tension: an ill-posed problem." J. Sound and Vibration,
Owen, D., and Shi, Y. (1991). "Parametric studies of guyed tower plat- 173(4),433-446.
forms." J. Offshore Mech. and Arctic Engrg., 113, 228-234. Venkataramana, K. (1994). "Earthquake response of tension-leg-plat-
Pallini, Jr., J., and Yu, A. (1993). "TLP tendon connections top and bot- forms in steady currents." Earthquake Engrg. and Struct. Dyn., 23,
tom terminations and tendon pipe couplings." Offshore and Arctic Op- 63-74.
erations, 51, 111-118. Vickery, P. (1995). "Wind-induced response of tension leg platform: the-
Park, W., Yun, c., and Yu, B. (1991). "Reliability analysis of tension leg ory and experiment." J. Struet. Engrg., ASCE, 121(4),651-663.
platforms by domain crossing approach." Proc. 1st Int. Offshore and Virgin, L., and Bishop, S. (1990). "Catchment regions of multiple dy-
Polar Engrg. Conf, I, 108-116. namic responses in nonlinear problems of offshore mechanics." J. Off-
Patel, M., and Lynch, E. (1983). "Coupled dynamics of tensioned buoy- shore Mech. and Arctic Engrg., 127 -133.
ant platforms and mooring tethers." Engrg. Struct., 5, 299-308. Weggel, D., and Roesset, J. (1994). "Vertical hydrodynamic forces on
truncated cylinders." Proc., 4th Int. Offshore and Polar Engrg. Conf,
Patel, M., and Sayed, F. (1995). "Combined axial and lateral responses
3,210-217.
of tensioned buoyant platform tethers." Engrg. Struct., 17(10), 687-
Wilson, J., and Orgill, G. (1984). "Optimal cable configurations for pas-
695.
sive dynamic control of compliant towers." J. Dyn. Sys., Measurement,
Patel, M., and Seyed, F. (1995). "Review of flexible riser modelling and
and Control, 106, 311 - 318.
analysis techniques." Engrg. Struct., 17(4),293-304.
Wybro, P. (1995). "Advances in methods for deepwater TLP installa-
Perkins, N. (1991). "Planar and non-planar response of a suspended cable
tions." Offshore and Arctic Operations, 68, 201-212.
dri ven by small support oscillations." Proc., 1 st Int. Offshore and Po- Yoo, H., Ryan, R., and Scott, R. (1995). "Dynamics of flexible beams
lar Engrg. Conf, 2, 210-215. undergoing overall motions." J. Sound and Vibration, 181 (2), 261-
Rainey, R. (1977). "The dynamics of tethered platforms." Proc., Meeting 278.
of the Royal Inst. of Naval Architects, 59-80. Yoshida, K., Suzuki, H., and Mishima, S. (1989). "Response control of
Robison, R. (1995). "Bullwinkle's big brother." Civ. Engrg., 44-47. articulated compliant tower." Proc.. 8th Int. Conf on Offshore Mech.
Ryu, c., and Yun, C. (1992). "Reliability of offshore guyed tower against and Arctic Engrg., 199-207.
anchor pile failure." Engrg. Struct., 14(2), 112-120. Yoshida, K., Suzuki, H., and Ona, N. (1988). "Control of dynamic re-
Ryu, c., and Yun, C. (1994). "Nonstationary response analysis of off- sponse of tower-like offshore structures in waves." Proc., 7th Int. Conf
shore guyed tower for strong earthquake under moderate random on Offshore Mech. and Arctic Engrg., Vol. I, ASME, New York, N.Y.,
waves." Proc., 4th Int. Offshore and Polar Engrg. Conf, I, 116-121. 249-256.
Saevik, S., and Berge, S. (1993). "Correlation between theoretical pre- Yoshida, K., Suzuki, H., Nam, D., Hineno, M., and Ishida, S. (1994).
dictions and testing of two 4-inch flexible pipes." Offshore and Arctic "Active control of coupled dynamic response of TLP hull and tendon."
Operations, 51, 63-78. Proc.. 4th Int. Offshore and Polar Engrg. Conf, I, 98-104.
Salpukas, A. (1994). "Oil companies drawn to the deep." The New York Zhu, S., Cai, Y, and Chen, S. (1995). "Experimental fluid-force coeffi-
Times, DI, D5. cients for wake-induced cylinder vibration." J. Engrg. Mech., 1003-
Sato, M., Natsume, T., Kodan, N., Ishikawa, K., Ito, S., and Tono, K. 1015.

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