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OCR for page 112
Prediction of Vertical-Plane Wave Loading
and Ship Responses in High Seas
Zhaohui Wang~, Jinzhu Xia2, J. Juncher Jensen~ and A ne Braathen3
(~Technical University of Denn ark, 2The University of Westem AustTalia,
3Det Norske Veritas, Norway)
A'dSTF ACT
Tne non-linear ti s m wa~ md shmmi g-md ced ngid-
body moticns md tmct ral ~e pom s of ships such as
heave, pit h md vedical bend g mome ts are consi te t y
mve tigated based m c rational t me d mcm ship medhod
(~, Wmg md Lm n, 199S1 A hyd odynamic mod I fcr
p~edi ti g sectional r e water force i also outlined fcr fne
mve tigati m of th effect of r e wabr loads m th global
h 11 gi d r bend g m ome t Tne c mputstional ~ e nits based
m fne non linmr time domcm ship fneo y a~e compared wih
fhosebased mbuef llynon linmr3-D panelmedhod SWAN-
DNV mdofnerpubbsh d~esults
Fr m fne rath:r e tffnive c mputatbns md c mpari-
sons, t is fommd thst non linear eff ts are sig d mt m heed
mdb w wave mbue motion~e ~eso~mt regim forboh
heave md pi h moLcns, b w accelemticns md ve tisal
bend g m ome ts f cr two conluirvr ships consid ~ cd wh ~ eas
not sig f~mtfcrcVLC TnenonkneartiesnmoLcns md
tmct ;1 loads of conve tional monoh 11 ships em w r
p~edictedbydnep~ese t m-linmrshipfheoy
INTRODUCTION
Linmr ship fnecries md 3D linmr pote tial fnecries
hcve ben wid y accepted md used by naval archit ts as dne
mcm toob fcr e timalmg fne p f~rna~ of c hip m wave
d e to th ~ektively :nall computational effo t md fne gff~er-
alysatifactoyagreme twihexperime ts Thediff'culties
come m higkr md e~eme ~as md wh n try~ng to estsbli h
maxim m kf time loads fcr tmctural d sir
Nonlineartiesmwave- mdskmmi g-md ced tmct cl
~e pcnses of i ips hcve ben ob~d fi m 911-~ale meas-
memeris md m mod I expemme ts Sham meEnreme ts m
ships wih fm f ms such as wmships (Smif 1966 md
con~irvr ships :;dek et ~, 19 2) m modemte md hec y
seas hcve shovm f tt fne wave-md ced saggi g bend g mm-
me ts m be consid r~oly k ger f m fne wave-md ced hog
gi g bend g m m nts Tne non-linmrity m he v rtisal-pkne
bend g mome ts has to be taken i to accom t m tmctural
d ig To ml mise wave m~mg ~e isl~e md erhame
seakepmgp f mar~ect~ektiwlyhigh ped fastvessels
a~e UtDaDy dEriff~ed wifh krge l ngth to beam mtio, krge b w
ftare md I w block cceffcie t Tnese pmp dies put fnem
outsid fne mplicati mrmge fordne mles of fne ckssidcati m
sochi s fcrh llgad r loads calcubtion h~d6 idual c~nsid
ticns based m dsect cal-ubti m pmcedures a~e herefore
qwed to d rive fne d sign loads Zhe g, 1999) M~my ~
paical non linear ship mehod hcve ben pmposed p~edLct-
i g fne non knear wave- md sk ml g-md ced tmaural
loads wifh n~asonably good accmacy ( e fne pr cd gs of
fne kt mational Ship md Offhore Stmctures C ngress
SSC md he ktemational Tow g Tmk C fe~e
~TTCt
Tne importar~e of fne non-linearibes m heave md pit h
moti ms of ships was not ~ecogmsed m til kte 1 9S0's h~ fne
benchmarkseakepi gexpemmeriscaniedoutfcr TC m c
stmdard h r f m designated fne S175 con~irvr hip by
tw ty f e orgmisaticns, c sig f~mt scatter was fommd m
s me of fne hansfer f mction ~esu ts fcr heaw md pit h m -
ticns mh cd seas ([TTC, 19S7 Lster mod I te tsby ODe,
et al (1992) d m mshated c varbti m of fne heave md pit h
hanslbr fmchons wifh wave amplit d, mdLcati g c non
linear moti m behc iour Rwe tly, r~ mseribe g md Brou~r
(199S) hov~d 6~t linear p~edLch m of fneheaw moti m mcy
beim ffci tffcshiphllisdsirjnedtomi misebohi
sisl~e md wave-md ced motbns Mod I te~mg et f is
stage is till esse tial mh 11 f m optimisati m fcr ekepi g
pe f~rna~
h~ crder to p~edLct nomlinmrities c msisb tly m bodh
wave-md ced rigidbody moti ms md tmctural loads, c ~
tional time-domcm ship mehod was d velop dby =, Wmg
md 3ff~sen (199S) fcr fne ve tisal-pkne probkms A higkr-
crd r crdinary d ffere tisl eq~ti m was used to mpro mab
fnehydodynamicm moyeffectd etofnefie nfacewave
mohon Tne hyd odynamic md ~e bri g forces were e ti-
mabd exactly over fne imtantamous wett d miace Tne
momerium skmmi g'fome was mtomati~3y obt~ned m he
f mmktion The fluid force e p~essi m was coupled wifh fne
tmcturerep~esent dascTimosh kobeamtof mchydo-
1
OCR for page 113
elasticity theory. By specifying wave amplitudes, non-linear
frequency response functions were presented for the S175
Containership in head seas, including the heave and pitch ~-
tions, bow acceleration and sagging~ogging bending moments,
see Figure 1. Two different bow geometries of the ship were
considered to demonstrate the relationship between the bow
flare of ships and the non-linearity of the responses, see Fig-
ure 2. The predicted results were compared with available
experimental data from the elastic model test made by Wata- WAS ~,$,j.S
nabe, Ueno and Sawada (1989) and the experimental investi- ¢
gallon by O'Dea, Powers and Zselecsky (1992~. Very good
agreements were obtained between the predictions and the
measurements for wave-induced rigid-body motions and
bending moments.
A ship sailing in a heavy sea may experience shipping of
water on the fore deck. The green water load may result in
severe impact loading on the deck, the superstructure and the
equipment mounted on the deck. Prediction of green water
loads is especially important for fast ships and for FPSOs as
shipping of green water may place severe operational restric-
tions on these kinds of vessels.
Recently, a significant research effort has been initiated
to solve the problem. Model tests have been performed on
FPSOs in MAR[N (Maritime Research Institute Nether-
lands), and design guidelines are issued addressing the bow
shape and the necessary freeboard and breakwater. However,
the present numerical methods cannot predict correctly the
green water loads due to the very complicated and non-linear
water flow around the bow and over the deck. Volume-Of-
Fluid (VOF) methods seem to be the most promising, but re-
quire significant improvement (Fekken, Veldmann and
Buchner, 1999~.
The extreme sagging wave bending moments in ships are
usually determined by taking into account the non-linearities
due to momentum slamming and hydrostatic restoring action.
These non-linearities are very important to container ships
with a large flare, yielding extreme sagging moments twice as
high as those obtained by a linear analysis, see Figure 2.
However, the effect of green water on deck is seldom included
in the calculations of the sectional loads but if it is, the associ-
ated vertical forces are often based just on the static water
head by which the relative motion exceeds the freeboard.
~ ~x. t...
i~
~..~.,,~,,..,;~ Be. .
~ ~ } 54~, I. ~~...v. ~
.$ ~~ ~ .
i]~? my. ~~ .
~ :45 '~'." ~ i; Hi
~ 57~ W_
~$ ~
.'4~ _ ~ ~ i' ~'$ ~ .i
: 5~ ..:.:.:
Atop ~¢, ~~ R:$$:
'5'2 .5~4. i,.
.~v,. :'
'I '$:.'
:;4
: i..
'A a,:.:.
:~5 "A
~ ~.r5 ~
4< ~ i,
:~:
~ Art
·~,]
:.~
., ~
~5<
5~:
'I .i
Figure 1: Calculated non-dimensional frequency response
functions (FRF) of heave, pitch, bow acceleration (FP) and
midship bending moment of the original S175 container ship
for different regular wave amplitudes, Fn=0.25 (Xia, Wang
and Jensen, 19981.
~,$~$~ —.
~~ i,\:~,, . ~ :~.
_ 5.i~~,~,~( I- -:: :~ .~,. ::
~ ~~ $'$~$~ 6~
~^ i$'~.4;~~;) jig ~v Y. ~v. ·* 4
: it. : ~ Add, ~ 3 ~ ~ ~ ~ ~ :5i i
it,, my,, ~ ~-~^i<~.,.^, i,6~
~ ~ .
0.,
.,6~:
;- ;,i<-~:
:~
ma_
me, DAYS...
Hi- .x ~~ ~ ~ '<: i^:
~:$$: :'i,~ii:
:-$
Figure 2: Non-linear sagging (positive) and hogging (nega-
tive) bending moments of the original (O) and the modified
(M) S175 container ship, moving in regular waves, ~ = 1.2 L,
a = L/60 and Fn=0.25. Comparison is made of the experiment
(Watanabe et al., 1989) and the numerical calculation (Xia,
Wang and Jensen, 1998).
Buchner (1994, 1995) has shown by measurements that
the actual pressure due to water on deck might be several
times larger than the static water head. A much more accurate
description of this load was obtained by including a term
proportional to the change of the momentum of the water on
deck. Later, Wang, Jensen and Xia (1998) proposed a modi-
fied formula to account for the forward-speed effect of the
ship. The concept of effective relative motion was used and a
Smith correction factor was introduced to account for the
wave pile-up effect during green water.
The present paper outlines several of the recent valida-
tions and applications of the non-linear hydroelasticity
method for heave and pitch motions, vertical bending nD-
ments and other wave-induced responses of ships. A short
introduction of the non-linear time-domain strip theory model
OCR for page 114
wi3 be giwn m 5 ti m 2 7h mod 3i g of fne lo git dinal
dish ~buh m of g e water losUs wi3 be inh od ced m 5 h m
3 h~ S :cti m 4, th non kneariti s of wave losUs md ship n~
ponses mhed seasw31be discussedfcr th 5175 conluirvr
ships Comp ms m wi3 be mcd of fne p~ese t p~edicticns
wifh of ner mm~erical md experimental ~esultg particuk Iy,
fne f lly 3-D non linmr simubti m by SWAN-DNV l\d~
gest, Brsa~en md Vcds, 1995) 5 ti m 5 of f is paper wi3
be d voted to fne p~edich m md validsti m of wave losUs md
ship ~e ponses m all hecdmgs fcr c psrnmax con~i~r ship
md c V CG 7his will also d momiosb th ~ektiomhip b~
twendnenon lineariti s mdfneh llf ms
7PHE 7~3ME DOMAIN 57RII' THEORY
Aecrd g to ~s, Wmg md Irnsen (1995), fne non-linmr
time d mamhyd odynamic force F. t) A fne lo git dinal
positionz mbueh llmaybeecp~essedby
F (z, t) = Dt
i ( i i Dt )
(1)
wh ~eI m rep~ese ts bodh he impulsive md m mo y effects
m fne hyd odynsmic m m~ium; DlDt is fne total d rivAive
wifh ~esp t to time t, D—= aa U aa, wifh U bei g th
forward ped of fne ship; ()~) = a;; Aj(z,z) md
B j (z, z) are fne socclled fiequency-md pend t hyd ody-
namic cceffcie ts d rived by c mtionsl mproxmuti m fi m
fne fiequency d p nd t cdd d~ss md dsmpi g cceffl
cie ts Fmth mmore, fne ~ektive moti m
7(z,t) =w(z,t) ;(z,t),wh ~e wz,9 i heve tisalmotim
offneh 3 md ;(z,t) isth wawelevAimwihSmihccr-
rection
f fl~e fiequency-md p nd nt hyd odynamic cceffcie ts
A j (z, z) md B j (z, z) a'e tskff~ as flm ticns of only z, i e
fl~e chmge of w tt d body suriNce is neglect d B - ti m (1)
rep~esent ctime-domcmcowtemart of fl~e linmr ship fl~eo-
ri s, fcr example, Salvesen, Tuck md Falti en (19701 Gem
erally, J=3 suffces fcr mo t sectionsl shapes for mmehic
ship motionprobl ms
By i teg ati m of fl~e high r crd r d ffererbal equati m m
B uAi m (1) md by mcomomti m of fl~e hyd oststic buoy-
ancy force f mmd r fl~e imtardamous wave miace md fl~e
g e wAer fome fO,, fl~e total non linear exbmal fluid force
Z(z,t)acti g maships:ctim mbeep~essedas
Z(z,t)= iii—+U8m Dz am6D ~
Dt ~ az Dt az ~ Dt J
D +f6 +few
(2)
wh ~e mW(z,Z) is he cdded mass of fl~e ship sech m wh n
fl~eoscilhti gfiequencytend toird~mity; D accowtsfcr
fl~e m morial'hyd odynamic effect wiflh q; go~r~ d by
fl~e foil wi g set of d ffe~e tial eqwticns
i8~(~)=1, t(~,:) Bj Iqs(~J) (i Bj ,+Aj t)D
j=1,2, ,} (3)
Thefl dtmmofZ(z,t)mB—tim(2)isth momerium
sk.rnmi g f~ It i assumed to be zero whff~ fl~e ship s~
ti m e it w3ter Th till-w3ter ~e ponse of flhe ship d e to
fl~e d ffe~e of fl~e dish ~buti m of fl~e w ight md fl~e buoy-
ancyforcesi ig oredmth cal-ubticns
MODELING OF GRb~N WATER LOADS
A h ief inh oducti m to fl~e f mmkti m of fl~e g e w+
ter sectionsl force f~ (mB - tim2) is gi~enbelow,wheress
z d t3iLd d ri ati m m be fommd m Wang, Jff~mn md Xis
(1995) mdWmg(2000)
Th w rtisal 1 3df~ p r m it le gth d e to g e w3ter m
d k m z lo git dinsl posih m z md 3t z time t i taken to be
f - (z,t) = me,(~,t) Dt [ ew( ' Dt ]
d ected pos tively upward Here z~(z,t) is defm d as he f-
fechve ~ektiw moti m md mO, d notes fl~e imtardam 3s
mass p r m it le 3th of g e w3ter:
Th effectiw ~ ektive m oh m z~ (z,t) is taken to be z flm~
tim of fl~e minal ~ektive mohm
z~(z,t)=w(z,t) ;(z,t) based mth mmdisbrbedwave
cl w3ti m ;(z, t),
t(2 I C~)z~(z,t); Deck merged
~ l(3 2C~)z~(z,t); Deck 3bmerged
H ~e C~ is fl~e Smiflh conection factcr fcr fl~e imtardam 3s
wetted body secticns
The g en w3ter mass is taken to be propodiorul to fl~e
eff tivew3t:rhighth~(z,t)mhed k:
3
OCR for page 115
mgW(x, t) = p Be (x) he (x, t)
(6)
where he(x~t)=-ze(x~t)-Df (x) with Df (x), the free-
board; Be(X) is an effective breadth of the green water . In the
present study Be(X) is taken to be half the sectional breadth
B4(x) of the deck, i.e., Be (x) = 0.5 By (x) .
Due to the dominant positive component in the second
term of Equation 4, the force few will be directed upward just
at the moment when the water enters or leaves the deck. This
unphysical behaviour is excluded by assuming fgw = 0 if fgw >
O.
Equation 4 is of the same form as suggested by Buchner
(1994, 1995), except that the forward speed effect is included
in the definition of velocities and accelerations. The present
approach simply treats the green water load in the same way
as the added mass of water for a submerged section. The
change in wave profile due to the bow and the flow of water
on the deck is accounted for by using ze~x,t), instead of zn~x,t).
The Smith correction is introduced because it has been suc-
cessfully used in the strip theories to account for the diffrac-
tion effect of the incident waves, and it gives a plausible varia-
tion of Ze with the geometry of the submerged part of the sec-
tion, the wave elevation and the frequency. For instance, an
increasing bow flare will increase Cs and thus decrease Ze
When the wavelength is long or the wave frequency is small,
the effective relative motion is close to the nominal relative
motion, which indicates a physically rational asymptotic be-
haviour of the dynamic wave deformation, see Figure 3. The
relative motion amplitude Za in Figure 3 is shown as a func-
tion of the wave frequency ce for the S175 ship sailing in a
head sea with a Froude number of 0.25. Hereafter, subscript
a represents the amplitude of the full and a is defined as wave
amplitude. The relative motion amplitude Za is defined as half
the peak-to-peak value). The location considered is FP and
the wave amplitude a of the regular waves is L/6O, where L is
the length (175 m) of the ship. At this wave amplitude sig-
nificant non-linearities are present in the responses, see Figure
2. Figure 3 illustrates that Equation 5 yields a rather good
agreement with the measurements over the whole frequency
range, whereas the nominal relative motions Zn deviate quite
strongly from the measurements.
In a stochastic sea an average value Cs is applied with the
individual wave amplitudes as weight factors. Numerical c-
sults for the relative motion in stochastic seaways and com-
parison with experimental results can be found in (Wang et
al., 1998~.
:.
~~-
~ !--
.............................................................................................................................
. ~.
z—. f.
e 5.
.A'...V: ~ ~ :.~..V. ..
(2.1(s~ ~ ~
,. ..
}: ~
· )` ~
it..
:~: :~5
.
:~.
· : ~
.~
~ ~ Is , ~ .: . ~ ~ ~ ;. ,,:
..~:
Figure 3: Relative motion amplitude za at the FP for the
S175 container ship sailing in regular head waves. Wave an-
plitude a = L/6O, En = 0.25 (Wang et al, 1998~. Measured re-
sults from Watanabe et al. (19891.
THE S175 CONTAINER SHIP IN HEAD WAVES
Prediction of the non-linear motions and wave loads of
the S175 container ship in head waves is presented below
with comparisons with 3D non-linear results by SWAN-
DNV (Adegeest, Braathen and Vada, 1998) and other avail-
able publications. The body-plan and the main particulars of
the ship are identical to those used in Watanabe et al. (1989~.
Figures 4 and 5 compare the non-linear calculation by the
present method with the experimental data for the first har-
monics of heave and pitch of the original S175 container ship.
The presentation is made for three wavelengths, AL =1.O, 1.2,
1.4, and two Froude numbers, En = 0.2, 0.275, for both ~n-
plitude and phases (defined as leads relative to the wave ele-
vation at the LCG, with wave and heave defined as positive
upward and pitch defined as positive bow down). The pre-
dieted non-dimensional amplitudes of the heave and pitch as
functions of the wave steepness ka seem to agree very well
with the experimental results. It is seen that as the wave ~n-
plitude increases, the non-dimensional heave and pitch nD-
notonously decrease. Also presented in Figures 4 and 5 are
the numerical results obtained by the 3-D non-linear time-
domain codes SWAN-DNV and LAMP-4 (tin et al., 1994),
the partly non-linear simulation (Xia and Wang, 1997) where
only non-linearity in hydrostatic and 'momentum slamming'
forcing is considered. Other results in Fig. 4 and 5 are ~-
tained by various non-linear strip theory formulations (ISSC,
2000~. The discrepancy between the partly non-linear simula-
tion and the non-linear and experimental results indicates that
non-linearities in the hydrodynamic forcing due to variation of
added mass and damping are important. The present non-
linear theory prediction seems to agree very well with the ~-
perimental results, particularly for the variation of heave and
pitch amplitudes with wave sloop. It is noted that green wa-
ter on deck is not detected in the computations for the non-
linear strip theory results in Figures 4 to 6.
4
OCR for page 116
The bow acceleration is a sensitive indication of higher
harmonics since in regular waves acceleration corresponds to
multiplying the displacement by m2, 4~327 9~2 for the first,
second and third harmonics, respectively. If the response is
properly represented by a Volterra functional series, these
harmonics would be expected to vary as the square of the
wave amplitude for the second harmonic, cube for the third
harmonic, etc. The comparison of the bow acceleration of the
second and the third harmonics (non-dimensionalised by the
... : I
I..? L
.
if-)
:~.: ..
At.
:~:
~ .~
........
....... ~! :
~ .~s, 0.~ ~ ~ ~ ~
my:
acceleration of the gravity g) for the wavelength ILL and the
Froude numbers En= 0.2 is presented Figure 6. The present
results agree well for the third harmonics, whereas they u~-
der-predict the second harmonics when the wave steepness
ka > 0.08, i.e. a > 2.2m. The second harmonics results (dot
line) obtained by the quadratic strip theory (Jensen and
Pedersen, 1979) and the results by various non-linear strip
theory formulations (ISSC, 2000) are also included in Figure
6.
D.)
~:
~-
:~
b~ ~4 ~~,~ ID ~~ ~ ~ ~6 tI~
=~ ~ ,~ *~ ~
A.
.~
Figure 4: Comparison of the calculated and the experimental (hollow square points, O'Dea et al., 1992) magnitudes and phases
of the heave (left) and pitch (right) of the original S175 container ship with respect to wave steepness, Fn=0.2. Solid lines for the
present method, solid circle points for SWAN-DNV, dash lines for the partly non-linear simulation (Xia and Wang, 1997) and
other strip theories (ISSC, 20001.
5
OCR for page 117
t w ~
:~: ~
j' ::
(:~
~ .$
DO
I.
Figure 5: Comparison of the calculated and the experimental (hollow square points, O'Dea et al., 1992) magnitudes and phases
of the heave (left) and pitch (right) of the original S175 container ship with respect to wave steepness, Fn=0.275. Solid lines for
the present method, solid circle points for SWAN-DNV, solid square points for LAMP-4, dash lines for the partly non-linear
simulation (Xia and Wang, 1997) and other strip theories (ISSC, 20001.
The load due to green water on deck might influence the
maximum wave-induced sagging bending moment if the two
events are in phase. Figure 7 illustrates the predicted force
and response time histories of the S175 container ship in
regular waves with two different wavelengths VL and
W1.2L. The time histories include the input wave elevation (,
the effective relative ze7 the buoyancy force Jib, the momentum
slamming force f l, the green water force fgW and the hydro-
static part of the green water force fS, all at the FP and the
midship wave bending moment M/. The incident waves are
regular head-sea waves with the wave amplitude a = L/60 and
Fn = 0.25. The result for the bending moment on the assump-
tion of no green water is included for comparison. Here the
hull is taken to be fairly rigid to suppress hydroelastic effects.
For W1.2L it is seen that the green water force fgW is
larger than the momentum slamming force f l at the FP, but
contrary to the momentum slamming force f/, fgW appears
slightly after the peak in the midship bending moment. Hence,
the green water load only marginally influences the peaks of
the midship bending moment. The magnitude of the green wa-
ter force is seen to be about twice the hydrostatic value fS
given by the first term in Equation 4. If the nominal relative
motion Zn is used instead of ze7 no green water on deck ~-
pears, see Figure 3, for this wave amplitude. In Buchner
(1994) the average value of the ratio between the local pres-
6
OCR for page 118
sure in the centre line at FP and the static water head is found
to be about 3.5 from measurements on a frigate at 20 knots.
As the pressure must be expected to vary both in magnitude
and phase over the breadth of the deck and with the highest
value in He centre line, He presently predicted value of fig,, is _ '° ~~ ,
. . ~ .
believed to be of the correct magnitude.
The results for AL show only marginally water on deck.
Hence, fgw is small but the phase nearly coincides with the
peak of the midship bending moment. A small reduction in
the peak sagging moment is therefore seen in Figure 7.
.~
do:
:~
:;~,r~,£ ~,~.~l~.i ~, ,, , ~ At.:
~.:.~.6
Figure 6: Comparison of the calculated and the experimental
(cross and square points, O'Dea et al., 1992) bow acceleration
(15%L aft of FP) of the second and third harmonics of the
original S175 container ship with respect to wave steepness,
AIL =1.O, Fn=0.2. Solid line for the present method, dot line
for the quadratic strip theory and other lines, ISSC, 2000.
SHIP RESPONSES IN ALL HEADINGS
For short-term and long-term predictions, it is important
to investigate the motions and loads in all wave directions.
This is because the highest stresses may be expected in 30
and 60 degrees wave directions, either approaching from the
bow or the stern due to the superposition of the vertical and
lateral bending moments and torsional moments. In this sec-
tion comparison with the model experiments of a panamax
container ship (Tan, 1972) and a VLCC (very large crude car-
rier) (Tanizawa et al., 1993) will be made for all headings.
7
~ ~ ~ ~ 0 ~ ~ ~
15 ~ .'..............
~ U ~ ~ ~ ~
Z7~ ~ A loot
o
--60~ 11 ~ 40~,i
n. , ~ . I:, I O
-25t
0.5
n ~
-^ -30
1 ~
*103 Exc! tw =
0.S
-0.5
it' :1
* 103 Excl it =
A. -
3 sees Time , 3 sees, Time
Figure 7: Time histories of the wave elevation A, the effec-
tive relative motion ze7 the buoyancy force fb, the momentum
slamming force fat, the green water force fgW and the hydro-
static part of the green water force fS, all at the FP, together
with the midship bending moment Me for the S175 container
ship in regular head waves. Wave amplitude a = L/6O, En =
0.25 (Xia, Wang and Jensen, 19981.
A Panamax Container Ship
Towards the end of 1969, collaborative research was
performed in the Netherlands on ship behaviour at sea, par-
ticularly for third generation container ships, designed for
trade on the Far East with low block coefficient. Model ~-
periments were carried out to investigate the effect of wave
direction, length, height, and ship speed on the bending and
torsional moments and shear forces. A detailed description of
the experiments including the body plan and the main par-
ticulars of the ship can be found in Tan (1972~.
The model tests were conducted in seven wave direc-
tions: ~ = 25, 45, 65, 18O, 205, 225 and 245 degrees, (180
degrees denote head waves). Seven wavelength to ship length
ratios are used: NIL = 0.35, 0.5, 0.6, 0.7, 0.9, 1.1 and 1.4. The
wave height was kept constant at L/60. A range of ship
OCR for page 119
speeds between Fn=0.22 and Fn=0.27 was investigated. Here
comparisons are only made for Fn=0.245, since the responses
very slightly depend on the speed in the considered speed
range (Flokstra, 1974~.
The comparison of the frequency response functions of
amplitude of heave zala, pitch Ga/(ka), vertical acceleration
Lwa /( ga ~ at the FP, midship bending moment Ma/(pgaBL2)
at the seven headings is shown in Figures 8 to 11. More de-
tailed comparisons including phase and structural responses
at other sections are given in Wang (20003.
The agreement is generally good for the frequency c-
sponse functions, defined as half to peak-to-peak value. It is
seen that the non-linear calculation of motions (Figures 8 to
10) agrees well with the experiment for head, bow and beam
~ A;
,.s ~
) ~
o.s .
waves, whereas the linear calculation over-predicts the nD-
tions as also found by Flokstra (1974) and Wahab and Vink
(1975~.
The non-linearity of motions is obvious for head and
bow waves in the motion-wave resonant region, but small for
quartering waves. The difference between the amplitudes of
linear and non-linear calculations of vertical bending moment
is small, but the small difference does not mean small non-
linear effect of the structural responses. As seen in Figures
11, the increase due to non-linear effect of the sagging lro-
ments is about the same as the corresponding decrease of the
hogging moments.
29.
: ~-5'*.~:
.5~,~:
,V5:~5 $;
..~ ~ .5] .
. ~ 55 ~ ;~ :.
: :. ' ''
,~ ~,.;s .. .. .'2
. ~,2~ .~
~ > j _ : ........ V ~ A ~ ~ . ~ . a _
. i.,, f~ ~',~ ",.
' ~ 35: ~ ~ I:
. ~~ ~ it, ,.
. ~ 5~& .~
'N ~ o ~
.~ :.:.:.:.:.: , ::.: : ::::.\., ~ I . : :.: . ~ : : :
'.: ~~-"~=}:,,.
5~.: .~ ::::::::: 5::5:
~5 :~t ~) :'~ jt'ik}
~ 55:i a: ~ ~ ~ ~~
~~6'lr''~'
Figure 8: Frequency response functions of the heave Gala of the container ship. Fn=0.245.
AS
0.5
~ O ~
~ ~L/s
Figure 9: Frequency response functions of the pitch ~J(ka) of the container ship. Fn=0.245.
)s
0
_ no
~-
o
T 2 3 4
Ws~ Wit
_ ~
5
I I
1 ~ 3 4
8
s
OCR for page 120
to .
6() -
40 _ ...
20 _..
o C
! ! ! 60 _
180 HLILnle6aor ~
, N Exp 0
40 _ ~ _ 40
2() _. i ~ _ 2()
() //, to
A.,
65
I ! T
, \
205
_ ~
_ ~ ~
~ '.~
1 1 1 —
45
! T
[,t,
40
20
o ,°~-~^ id
60
40
2()
o .
25
I T I
225 ,"
...................
_ .. .. :.1. : a ........... _
f ',~
1 2 3 4 5
_ 60
.._ 40
20
o
I I T
245 ,,"
_ ~ ~,
_ , ~ ..
to
j i
2 3
i ~
4 5
Figure 10: Frequency response functions of the vertical acceleration Lwa l(ga) at the FP of the container ship. Fn=0.245.
0.04
0.03
2
0.01
() ,
.. . .
0.03
(1 ()2
() (11
, , ~
IXO
............
Sagging
Hogging - - - - --
Linear - - - - - - - -
Exp. O
0.04
0.03
0.02
0.01
l
205
~~
_ ... ~.~ ~ ........
...... /~ '` ..~
lo,/ , '\ ,.;'-'^~-1
O.( ! ! ! ! I ! 1
65 1 1 45 1
............... .' ~ L A
1 1 9 ~
_ ~
... .
(' _
0.04
0.03
.02
0.01
1 225
V6 - ~
1 7 ' !
1 25
L------ -------------------- ------- ------ ----------r'-~-
1 . . . ..
1 2 3 4
0.04
0.03
0.02
0.01
- o
5
,: .~ ]
L ~ ".--.~;M
1 2 3 4 5
Figure 11: Frequency response functions of the midship sagging and hogging bending momentMa/(pgaBL2) of the container
ship. Fn=0.245.
A Very Large Crude Carrier
Here a VLCC is used to validate the program and to in-
vestigate the nonlinearities of ship motion and responses as
another kind of ship different from container ships. VLCCs
have more vertical sidewalls and larger block coefficients than
container ships. The non-linear effect of the VLCCs may be
expected to be small.
A free-run experiment of a VLCC model ship was carried
out in the Ship Research Institute of Japan ~anizawa et al.,
1993). The model tests were conducted in seven wave direc-
tions: p=0, 30, 60, 90, 120, 150 and 180 degrees. (180 degrees
denoted head waves). Ten wavelength to ship length ratios
were used: ~L=0.2, 0.3, 0.4, 0.5, 0.625, 0.75, 0.875, 1.0, 1.25
and 1.5. The wave height H was kept constant at L/64. The
Froude number was 0.131. The ship motion, the vertical and
lateral bending moments amidships, the relative water level
and the wave pressure were measured.
The comparison of the frequency response functions of
heave za/a, pitch (3al(ka) and midship bending moment
Ma/(pgaBL2) amidships at the seven headings is shown in
Figures 12 to 14. Generally, the calculated results agree rea-
sonably well with the experiments and, as expected, the dif-
ferences between the linear and the non-linear results are very
small. Calculations have also been performed for greater wave
heights, but the frequency response functions change little.
It is seen from Figures 12 that the ship experiences larger
heave motion in bow waves than in head and quartering
waves. The predicted amplification of the heave motion for
bow waves in the heave-wave resonant region is not as large
as measured in the experiment. The vertical bending moment
curves in head and bow waves in Figures 14 show multiple
peaks. The occurrence of the two larger peaks for the head sea
9
OCR for page 121
case can be explained by considering the bending moment as
the sum of moments due to the hydrodynamic forces and to
I.5
Llnear = . IXO
Exp. 0
5 _ , _
()5 _ j _
O O^~1
2 ! ! !
6()
. ....... . .. . . ...
_ ... ........... ...... ..
_....................................................
r.~ _ ........................ , .... , _ 1.5
_ ' ~ 1
_ ............ .>f ...... _ 0.5
9~ I i O
, ! !
l _..................................
0.5 ~ 0.5
O O
A/L A/L
l
150
the inertial forces
! ! I
120
_ 0~=::=
_ ~ ....... _
_ 2 _ , , _
_ 0.5 _ 5;:C=
o
0 0.5 1 1.5
A/L
. 2 _
1.5 _........
0.5 _ ..
Figure 12: Frequency response functions of the heave Za/a of the VLCC. Fn=O. 131.
1.5 Llnear ~ 18() 1.5
l Exp. O
0.5 _ .
O _
· o
................ ~
~,
^~ I . u
1.5 ! ! !
6()
1 _ _
0.5 _ ...... ' ~.~=~===
<~~
O ~ I I
0.03 1 ! ~
[,.\,.,
0.021
H=L/64 180
L~near ~ : :
Exp O /~\N,
' /~; O\
. /00.
[, _..~...........................
60
_...........................................................
20Oo'
_ ~ ..........
~ ~ I ~
A/L
! 1
150
_ " ~
.
30
_ ~
~5
. . ~
_..;.._~,?...............
! ~ !
_ ... _
_ ' ,'.~
/f
~o .
~, 0.5 1 1.5
A/L A/L A/L
Figure 13: Frequency response functions of the pitch °a/(ka) of the VLCC. Fn=O. 131.
1 °.°31 ! ! ! i ° °3
150 1 1 . 120
no E~ n L~
0.03 1 ~ ~ ~ 1 0.03 1 ~ ~ ! 1
30 1 1 0 1
100:~o —
() 0.5 1 1.5 2
A/L A/L
10
~ l.5r
1 0.5~
1, ' . 90 1
..... , , , 2
~1 1
-
0 0.5 1 1.5 2
1 'Oo 1
0 0.5 1 1.5 2
! ! !
....................................................
0.03 1 1 ~ ~ I
1 . 90 1
0.02 F---------~- A
(),o~..~4
0 0.5 1 1.5 2
OCR for page 122
Figure 14: Frequcacy resi ocs e fcac tiocs of the midshti
CONCLUSION
h~ f is pep r, w hcve outh~ed sever ti rece t v tlid 4hns md
mph thons of a t linear time-dom tm sh ip heo y medhod
fm heave md p~tch moLcns, ve ti ti bend g mome ts md
of uer wave-mdhced t ponses of ships Fmm he e tffnive
c mput thom md comparisons, it is fommd fi gt fue t linear
fueo y s~gm ~ mdy mprm s fue over til accmacy of pt di~
t'm of bodh heave, pit h mothns md stmct ml loads of
monoh 11 sh ps m h gh mas Mme speci ~ ti y, fue foDmwi g
conchsmns maybe d a~rt:
-7.1~et lit~reff tsa~esignif'mtmfuemothns md
stmctm ti t pom s of fue two contamer ships consid t d,
wh teas not s~gm ~ mt fm fue V G.
- Fm fue two contamer ships, fue t m-hme ity of fue
motmns ems sho ger m h td md bmw wave m motion
wave t sot mt regmn, but less importmt m q~arbrmg wave
- h~ fue shp~e t sot mt regi m fm h td md bmw
wa~s fm fue two contamer ships mve tigabd, heave, pit h
bmwaccelemhm mdhoggmgbe dmgmome tmonotmN~sly
de~ee w i fue tggi g bend g mome t inct~ases wifh
m re mg wave tepness
RRbEgENGbd
Ad ge t,L,Brsathen,A mdVcda,T, 1998,'dvti 4imof
medhod fm e tmt th m of exheme t m-linear ship t ponses
based m mmt ~ ti simuitthns md mod I t ts", F cd-
mgs: 2 d Sympos m m Ncvti Hydodynamics, Voll,
pp 70-84
Buchmer, B. 1 994, "On he effect of g e t wabr impacts m
sh~p safety (c p~lot shrdy)", F oc NAV' 94, Vol. I, 5 ssi m
Buchner, B. 1995, "Onfue impact of g etwlter loadhg m
Shp md Offhore Umt L sig ", fo Si th tetnbot ti
Sympos tm m F acti ti L sig of Ships md Mobik Umts,
5 tl. 1 430-1 443
Fekken, G. Veldmnm, AEF md Buchner, B. 1999,
Smmukhon fGtertWtt:rL dmgUsmgfo Nc i:r-Stoks
E~hom Froc 7 t Grd: Nmm:ri ti Shp Hydody-
Floksha, G. 1 974, "Comparis m of ship moti m fo mies wifh
e rmme t fo ont r hip", tet 4iot ti Shiptuildi g
Frogress,Vo 21,No238, 168-lS3
..~ becdhg momectMJ(pg 3L2)
of the V CC Fc=0 131
ISSC, 2000, "G mmite Vi 1: m E=eme Ehg Gsd r L td-
m, F oc of h i4h tetTntiot ti Ship & Offshme Stm~
ITTC,1987,'Repotoffo 5 kepmgGmmitte",Foc of
fo 18h ktemahot ti Tmwi g Tank G fe~e, Vol. 1,
pp 401-468
Jff~sert, J. J. md Federsen, F T. 1979, "Waw-kdhced E nd
mgM merismShps cQuadrdicThoy",Ttms RNA,
Vol. 121, pp 151-165
Kmseriberg, G K md Brouwer, R 1998, "Hyd odynamic
d velopme t r c fiigtte f r fue 2 cent y, F oc of
Lm, W-M, M i hold M, Stivesen. N. md Ytt, D KF,
1994 1arge~mpliud moLcns md wave loads fm ship d
s~g , F cd gs of 2i h Symposimm m Ncv ti Hyd ody-
Mek,M t ti, 1972,' mctmlid ig o ft OCL
o binerships",Trans RNA,Volll4,pp 41-292
O Dea, J. Fmw s, E md Zselecsky, J. I 9 92, "Experimenb4
d temmmsh m of nonlimariti s m ve ti ti pkne ship mo-
h~ms, F cd gs: 19h Symposimm m Nw4 Hydody-
Sal N,Tt k E G mdF O. i370 Shipm -
tiom: md sea loads ', Tr ms SNA d, 78, pp 25 -287
Smi6 GS, 1966,'Mes eme t of set ice shes~es mwar-
sh ,ms, G f e~e m Stt sses m 5 t ice, h~st of Civ: E~s,
Tan, SG, 1972,'Wavel tdm~uemeris mcmod lofc
krge corda~r h p, Nefo rkmd Ship Research G ter, R
potl735
Tamzawa, K, Tagmchi, H. Samta, T. mdWatana~, 1, 1993,
~pe mmentai shrdy of wave pt ssure 0 V CC mnnmg m
Wahah, R md V'rlc, J H, 1 975, 'Wave mdhced mothns md
i on shps m hi e w s L tet 4io Shiptuiidi g
F og ess,V L2 N 249, 151-18 L
Wang, Z, 20L10, "Hyd oekstic anaiy is of high ped shp;',
FhD E me; g ~, ~ ~ `~. b >F :~ ~mL
11
OCR for page 123
Wang, Z. Jem n, JJ mdXia, J. 1998, "On fne ffect of
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Xia, J. md Wang, Z. 1997, "Time-D mam Hyd ceksticity
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Xia, J. Wang, Z. md Jff~sen, JJ, 199d, Nonlimar Waw
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729
12
OCR for page 124
L J Doctors
University of New South Wholes
Aushalic
The review r would like to con mmlc~e the four
mthors on c mo t prscti 91 Ed interesting piece of
research it is particularly heartenmg to see the
m my plotted examples of nonlinearity, exemplified
by c comparison of response curves cone pondmg
to differing wave amplitudes
Could the mfhors fir t clarify precisely what
nonlinear effects are included in their theory? The
text in the section "Th Time-D mom Ship
Theory" seems to suggest that She nonlinearity is
included with respect to She buoyancy force, green-
water force, Ed She slamming torce On the other
h Ed, has the Imearized free-smfcce condition been
used for the other components of force?
It is noted Nat She i fluency of nonlinearity is
generally Nat increcsmg see wave amplitude leads
to c reduction in the ~esponse~mplit de operator
RAO), et least es far es heave, pitch, Ed bow
acceleration me concerned How ver, the opposite
appears to be She case for midship bending moment
This is seen in Figure 2, for example, most clearly
near She resonance is this c generally true result for
of her vessels as w 11?
The much improved correction when utli ing the
nonlinear ah or. is seen m Figme 5 th ough Figure
11 On the other hmd, She case of c 45-degree
heeding m Figme 5 shows poorer conelation
betw en the experiments Ed both the Imear Ed
nonlinear theo y This seems to suggest c difficulty
with the experiments rather oh m with the theory
Could this point be elaborated upon?
Once again, I would like to express my appreciation
to the mthors for c most i fommative paper
AUTHOR'S REPLY
As expressed in Equation 2, the non-lmearities are
included m the theo y with re pect to She buoy m.
truce, Been water loads, summing action Ed the
other hyd odynamic effects such es inertia Ed
damping temms calculated to the in d mt meous w t
surface of the body On She other h Ed, the
Imearized fiee-surface condition has been used for
the hyd odynamic calculations et the in t mtaneous
submersion
Based on our calculations, the trend of non-linearity
show m Figme I is generally co finned for
container ships Ed frigates No opposite bend has
yet been found for other ship types
The poorer correction of heave m the 45-degree
hecdmg in Figure 5 might be due to the difficulties
in the hecdmg control of the self-propelled model
tests in this heeding A other reason might be the
non-linear coupling betw en the heave Ed roll
motions, which is not Included m the calculation
but could be signffic mt in the m odel testing for the
45-degree heeding
DISCUSSION
H Kcgemoto
University of Tokyo, Jcp m
The mfhors ckim that the nonlinear characteri tics
of motion responses observed et reso mt region,
where the motion responses per unit wave height
are reduced es She wave height becomes larger, c m
be accounted for by the potenticlbased nonlinear
theo y However, I thi k Ed I under t Ed that it is
generally ~ reed that such nonlmearities m She
reso mt r mge are c msed due to the nonlinear
characteristics of viscous damping forces
AUTHOR'S REPLY
We agree with Professor Kcgemoto that viscosity
con have m effect on vertical ship motions For
example, Beukelm m (I 953) experimentally st died
c ship-like mod I with rect mgmlar cross-sectiom in
regular heed see waves of different wavelengths
Ed found that viscous effect, particularly flow
separation around shalom corners, may signffi Fitly
contribute to heave Ed pitch damping However,
viscous effect is not likely to be import mt for hea ve
Ed pitch motions of fine ships such es frigates Ed
container ships Fcltinsen Ed Svensen, 1990) A d
it has been common practice to include viscous
effect in the prediction of roll motion but neglect
the effect of viscosity on heave Ed pitch motions
of ships The characteristics of motion responses
observed in She present crurlysis may be intemoeted
es c non-linear potential flow effect, including the
non-lmearity in the potenticl-flowbased
hyd odynamic damping Similar motion
characteristics have been observed based on use of
non-linear potential flow ah or. Ed 3-D Rmkme
source method es reviewed by Beck Ed Reed
(2000) et this symposium A recent discussion on
viscous effects on wave-induced ship motions may
be found in the proceedings of ISSC (2000, pp
26g-269)
Refetem es:
Beck, R. F. Ed Reed, A M, 2000, Modern ses-
keeping computations, 23 Svmunsium on Naval
Hvd ndy tmms, France
OCR for page 125
Beukekmcn, W. 1983, Verticcl motions md cdded
resist mce of c ~ect mgukr md tri mgukr cylinder in
wa~es, Report No 594, Ship Hyd odynamics
Lctorcto y, De ft University of Technology
Fcltinsen, O. md Svensen, T. 1990, h~corporction
of seckeepmg fheories m C D, Intl S mnosmm on
CFD md C D m Ship Design, Wcgeningen, The
Netherl mds
ISSC, 2000, "Committee V 1: on Ext~eme Hull
Girder Loading", Froc of the 14th Interm~tiorurl
Shiu & Offshme Stmctures Con~ress, Vol. 2
Representative terms from entire chapter:
container ship
