CALCULATION OF WAVE LOADS
ON SENISUBMERSIELE PLATFORMS
S. SPASSOV
SU WMAR Y
Today, for the exploration and mining of
oil,
gas and all Sorts of mineral resources in the
seabed and the substrata, various types of
semi-sub-mersible drilling platforms are being
constructed.
These marine structures should be
operated
stably around the fixed position. Their
construc-tion and shape are therefore planned to keep
themfirm against wave forces in general.
Promotion
of
basic studies on the calculation method
of the wave
exciting force and structural strength,etc.,
for
these marine structures is now under
consideration,
and from this point of view
wave loads on
marine
structures should be studied.
1. INTRODUCTION
Semi-submersible type platforms are mostly
de-veloping in the last years. In designing
such marine
structures, special attention should be
paid to the
performance in wind and waves:
to minimize the motion in waves and to
main-tain the position during operation;
to have sufficient structural strength
against
wave loads.
Accordingly, estimation of hydrodynamic forces
on the structure is the most essential
problem.Bas-ed on the hydrodynamic forces, the motion
in waves
and the wave loads can be estimated, thus enabling
the optimum design of a structure. It is hoped that
such is made with good accuracy by the
theoretical
calculation at the stage of initial
design, without
conducting experiments for each
case. In the
pre-sent investigation an attempt
was made to apply the
hydrodynamic treatment of wave loads
estimation.
Greatest attention should be
paid to the
general
12 APR. 1988
formulation of the problem and to wave exciting
for-ces calculation. Results from calculations and
ex-periments should be compared.
2. LINEAR POTENTIAL THEORY DESCRIPTION
The following assumptions are considered
to
describe the equations of motion of a marine
struc-ture in waves (2), (7), (8), (14), etc.:
- Waves and motions are of small amplitude.
- The floating structure is considered as a
rigid
body.
- Oscillations are sinusoidal about a state of rest.
- The fluid is assumed to be ideal and irrotational.
- The water depth
is infinite.
- The influence of viscosity on the
wave-exciting
forces is neglected.
Let us introduce the coordinate system
01fixed in the space and the coordinate system
Oxyzfixed in the body. In calm water the OI
plane
coincides with the
Oxy plane (Fig. 1). It is
as-sumed that the direction of wave propagation
con-cludes an angle of
degrees relative to the
x-axis of the floating body. The
system of
coordina-tes of water particles motion is defined by
(.1,3).
From this it follows that
= X cosp + y smp
= x smp + y cosp
Given the above conditions and assumptions,
the
problem is reduced to the following boundary -value
problem of potential theory. It is required to
find
a velocity potential
(x,y,z,t) =(x,y,z)e'.
The potential flow field can be characterized
by avelocity potential divided into contributions from
all modes of motion and from the incident and
dif-fracted wave fields
6
=
o
do +
j =1
where
s = s eX0050 +
ysina) - ict
So
- amplitude of the wave
- wave number
Xa
- angle of
incidence
Lab.
y. Scheepsbouwkundt
S. Spa ssovARCHEF
Technische Hogeschool
yv
- the boundary condition on the sea floor
(5) 0
Dz
- the boundarj' condition on the body surface. Due to the linearization, this boundary condition may be applied to the surface L in its equilibrium po-sition. Thus
(6) O
an
Dn(7)=n
IL
j = i...6
an
'Here n is the normal vector, pointing outside the body, and n1 through n6 are the generalized direction cosines on L. The potentials cd and
(j=1.,.6) should satisfy moreover the boundary con-dition in infinity, the so-called radiation con-dition, which states:
(8) lim ( - ik = O
R- R
The incident wave potential is given by
(9)
ol
kh
1 cosk(z+c) ec05ys
in which
g - acceleration due to gravity c - distance from origin to sea bed h - water depth.
Supposing that the unknown potentialsd and can be determined, the pressure on the surface
L can be found from Bernoulli's equation in line-arized form 2 (10) p(x,y,z,t) = = -p
{()S
o+ 6 Joi
=1 z=
-in which p - soecific density of water.
The wave exciting forces (k=1,2,3) and
S.Spassov
2
ments (k=4,5,6) in k-direction are:
X
k
«
=
2Sltf()dL
w
Using (7), (11) can he reformulated:
Xwk =
2Sitjf() Dk dL.
For these boundary conditions we may apply Green's theorem, and find
dL
=ffk
dLL an
The latter results make it possible to elimi
-nate the diffraction potential fromthe wave excit-ing force. So it is possible to avoid the diffrac-tion problem altogether. The wave loads can be com-puted from the potentials of the incoming waves and the oscillation potentials
Xwk
25-itff(24k
4k-- dL.
It is most convenient here to separate the ex-citing force into two parts: the incident wave part
X' and the diffraction part , so that
wk wk Xwk
Xk+Xk
with X' = dL wk Lan
Xk
-pw2S0e ittff1)52
dL L Dnwhere Xk is the so-called Froude-Kriloff force,and
Xk - diffraction force.
3. CALCULATION 0F WAVE EXCITING FORCES
It is common practice (2), (8), (15), (17) to
separate
2
j
=)akJ
ibk.The real part ak are the so-called added mass coefficients, the imaginary part bkj are the
so-called damping coefficients.
Then we can find (1)
(16) Xk
= -
a bmo- i=1 1=1
d
Xwk + Xw - incident wave potential
- potential of the diffracted waves
cP - potential due to the motions of
the body in j-th mode.
The individual potentials are all solutions
of the Laplace equation
(3)
=0
while the following conditions must be satisfied:
- the linearized free surface condition
and since on L
=
-an
an we findddL
= L an (4) z = Od
From (16) it follows that X is the product of the added mass aik of the structure and the
acce-leration of water particles in the undisturbed wave and X is the product of the damping bik of the structure and velocity S. of water particles in the undisturbed wave.
Now
(17) Xk = Xk + xdi + xd2w wk
Later in i = 1,2... 6 corresponds to different mode motions.
The semisubsiersible structures are built out of well-shaped elements like spheres, plane areas, elements whose dimensions are small relative to the wave length, horizontal and vertical cylinders,etc.
In our calculation we consider that the inter-action between the platform elements is neglected, which means that the hydrodynamic properties of one element of the construction are not affected by the existence of neighbouring elements. With this assumption we can solve the problem by calculation of wave loads on every element determinatively (4),
(6), (8) (13).
3.1. Calculation of Wave Exciting Forces on Some Semisubmersible Elements
For a submerged sphere with diameter D in waves, which are long compared to the diameter, the vertical force is
= _(pV+azz)c2e_kz1
S0coswt+be1S0sinat
where z1 is the depth of the body and h - depth of the water, and i = 3.
For a horizontal cylinder with diameter D
and length L5, the vertical force is
112 L5
Xwk= t
f
(pA+azz)w2sink(h-z1)
sin(wt-kx)dxj
-112L5 sinkh
For a square barge with length L5, beam B5 and draft T5, the horizontal force, neglecting damping effects, is
TÍJ
Xk
pgs0f45inskc05ssk5in
=Ix
k2sin x tanhkh sinhk(h-T5) coshkh 4cos(L5cosi.i)sin(--B5ksinii) +Sa
x o xx ksinp cos sinkh S. Spa ssovwhich is illustrated in Fig. 2.
In Fig. 3 is illustrated the wave exciting for-ce on a submerged hull in beam seas.
3.2. Calculation of Wave Exciting Forces on a Semi su bmersible
When we solve the problem of wave exciting for-ces on the whole semisubmersible structure, it is
necessary to create an accurate mathematical model which should include all force components important from physical viewpoint. Hooft (7), Kim and Chou(9), Kokinowrahos(10), Tasai (16) and others have deve-loped approximative methods for calculating sea loads and semisubmersible motions with different pe-culiarities.
The method which is used in this paper is based on the strip theory (1), (7), (9), (14), (16) in
deep water. According to these basic ideas, the wave exciting forces on the semisubmersible for surging should be presented approximately as
Xk = Xwk(F.K) + Xwk(Y)
Xwk(FK) = Xwk(F.K)l + Xwk(FK)2 Xwk(Y) = Xwk(Yw)1 + Xwk(y
where
Xwk(FK)l - Froude-Kriloff force acting upon col umns
XWk(F.K)2 - Froude-Kriloff force acting upon caissons
xwk(yw( - diffraction force acting upon
co-1 umns
- diffraction force acting upon caissons.
The transformation of results from local sys-tems for each element to Oxyz system is made fol-lowing Hooft's instruction (7).
In Fig. 4 are presented Xk, xwk(FK) Xk(9w) for a platform (16).
In Fig. 5 are presented the Froude-Kriloff
mo-ment and the momo-ment from diffraction waves for yaw-ing actyaw-ing on caissons. The accurate determination of wave loads is reflected on the precision calcula-tions of mocalcula-tions. In Fig. 6 the added mass and
damping coefficients are taken from (19), and the experiment verification is taken from (20).
3
3.3. Calculation of Wave Exciting Forces on a
Semisubmersible with Conditions of Connection
be-tween the members
We have examined the values of wave excited fon ces individually for the separate members. But for
accurate estimation of the values of the
floating
structure as a whole, it is necessary to pay
at-tention to effects provided by the
connections
be-tween these members. Following (9), (13) and others,
it is seen that three-dimensional calculation
me-thods give a better prognosis for wave loads
than
two-dimensional ones (see i. 3.2), although
the
difference is small and the results have
qualita-tive agreement. There is not a sufficient number of
works about correction of the two-dimensional model
for agreement of computer and model tests
results.
In Fig. 7 is shown the comparison from
experiment
and two calculations for sway exciting force -
onewithout correction for connection between submerged
hulls and columns, and the other with this
correc-tion. It is well seen that the dashed curve has
abetter agreement with model tests carried out in the
BSHC deep water tank with half part of a
semisub-mersible for escaping from partial connection
ef-fects. This attempt to make the well known
strip
theory more precise for this purpose is based on the
consideration that Froude-Kriloff force acts on the
connecting surface in opposite directions
onthe
two members. Using some geometrical
correlations
and aforementioned ideas, a result is obtained which
is illustrated in Fig. 7, where L, B, T are
lower
hull dimensions.
In Fig. 8 is illustrated the surge
exciting
force for the same half of semisubmersible
model.
It is seen that model test and computer results are
in good agreement for x/L
1.31.5 which may
de-pend on wavemaker capabilities
and on the
goodmeasurement of short waves.
4. CONCLUSIONS
A practical method has been developed
for
predicting the wave loads on a semisubmersible-type
platform by using the strip theory and Hooft's
me-thod (8).
The following important conclusions
can bedrawn from the analysis:
Using the Haskind relation, diffraction and
Froude-Kriloff forces are determined.
Wave-excited forces for semi- or
submergedcylindrical elements are determined for an
arbi-trary heading angle.
Investigation is made on the
conditions
of connection between the members.
It is seen
that
the correction of swaying force wave shows a better
agreement between calculations and model tests. For
surging force this is not so well seen.
An attempt is made to take into account the
geometrical conditions in the determination
of
Froude-Kriloff forces correction. The initial
re-sults give us grounds to hope that the direction of
investigation is right.
It is necessary to continue the work
for
finding correcting coefficients for all modes
of
motions and wave heading angles.
The development of a two-dimensional method
for prediction of wave loads and motions of a
semi-submersible platform can be used for design
tools
as an effort to obtain a hydrodynamically
optimumconfiguration.
REFERENCES
Blagoveshchenskiy S., Holodiniyj A.,
Seine-nov-Tyanshanskly V.
Ship Motion', Leningrad,
1969(in Russian):
Boroday I., Necvetaev Vu.,
Ship
Seakeep-ing", Sudostroenie Publ., Leningrad, 1982 (in
Rus-sian).
Galahov I., Litonov O., Aliseychik
A."Float-ing Oil-DrillA."Float-ing Platforms
(in Russian).
Flokstra C.
'Wave Forces on a
Vertical
Cy-linder in Finite Water Depth".
Fujii H., Takahashi T."Estimation of
Hydro-dynamical Forces Acting on a Marine
Structure",Tech-nical review, vol. 7, No. 2, 1970.
Hogben N., Osborne 3., Standing B.
"WaveLoading on Offshore Structures - Theory and
Experi-ment", RINA, 1974.
Hooft 3. "A Mathematical Method of
Determi-ning Hydrodynamically Induced Forces on a
Semisub-mersible", SNAME, 1971.
Hooft J.
"Advanced Dynamics of Marine
Struc-tures', John Wiley & Sons, 1982.
Kim C., Chou F. 'Motions of a
Semisubmer-sible Drilling Platform in Head Seas", TSNAME,1974.
Kokinowrachos K. "Hydrodynaniic
Analysis
of Large Offshore Structures", 5th Int. Ocean
Deve-lopment Conference, Tokyo, 1978.
Maeda H. "Wave Excitation Forces
on TwoDimensional Ship of Arbitrary Sections',JSNAJ, vol.
126, 1969.
Morison J., O'Brien N., Johnson J.,
ShaafS. "The Force Exerted by Surface Waves on Piles"
AIME, 189, 1950.
Nojiri N.
"A Study of Hydrodynamic
Pres-sures and Wave Loads on Three-Dimensional
Floating
Bodies", IHI Engineering Review, 1981.
Oortmerssen G.
"Calculation of Wave
Loadsand Motions with Strip Method and Diffraction
The-ory".
Salvesen N., Tuck E., Faltinsen O.
"Ship
Motions and Sea Loads", TSNAME, 1971.
Tasei F., Arakawa H., Karihara M. A Stu-dy on the Motions of a Semi-Submersible Catamaran Hull in Regular Waves", Research Institute for
Ap-plied Mechanics, 1970.
Takagi M., Arai S., Takezawa S., Tanaka K. A Comparison of Methods for Calculating the Motion of a Semisubmersible, Ocean Engineering, 1985.
Vugts J. The Hydrodynamic Coefficients for Swaying, Heaving and Rolling Cylinders in a
Free Surface", Deift, 1970.
Spassov S. Application of Two Linear Me-thods on Slender Bodies Hydrodynamic Coefficients Sofia, 1MM, 1981.
Spassov S., Vassilev P., Chernev I.Experi-mental Investigation on Heading Angle Influence on
Linear Motions of Semisubmersibles", Varna, Naval Architect Symposium of Young Scientists, 1985.
Spassov S., M.Sc., Research Scientist Bulgarian Ship Hydrodynamics Centre Varna 9003 Bulgaria 1.0 0.5 io7 2.0
-
Kim [9]
o
Coto 3.2 15 S.Spassov 5 0' 5Fig.1
Fig.2
Fig. 3
f
10 15 w [rod/s J1u3O°
0
2.50
05 1.0 15 20XIL
O
