«Eiiivpean Slüpbiiihliiig» No. 6 — 1968
1 0
W A V E LOADS O N THE FORE-SHIP OF A TANKER
by B. P e d e r s e n ")
Ab.stract
This article describes measurements of water pressures, stresses and vertical accelerations i n the fore-ship of a large 'tanker and the analysis of tlie results. Notations d g h P(x) .X yA V E N N l P(X < X l ) V
w
I R M 7 In ) • draught • acceleration of gravity' vertical distance between longitudi-nal frame and base line
length of longitudinal frame
spacing between longitudinal frames s • I
frequency function of variable x peak to peak value of wave induced variable
maximum recorded value of x i n a short term recording
vertical distance from still water line to aneasuring point A
parameter of the Rayieigh distribu-tion = half the root-niean-square of peak to peak values
number of cycles i n a recording corresponding number of emergences (sulsniergences) of a measuring point probability that .\ is smaller than or equal to xx
ship speed
section modulus of longitudinal frame
vertical relative motion between ship and wave
Euler's constant = 0.5772 stress
natural logarithm
Introduction
Our intention w i t h the measurements described below was to acquire better knowledge of the wave loads acting on the foreship of a tanker. Vertical accelerations, water pressures on the h u l l and stres-ses in longitudinal frames were recorded. F r o m these measurements relative motion between ship and wave was also estimated.
The measurements were carried out during the
I^ijt jior.skc Vcrrilas, Roscaich D e p a i U n u n t .
period March to M a y , 1967, while the ship was i n ordinary service.
D u r i n g this period the ship sailed i n ballast f r o m Europe to the Persian Gulf through the Suez Ca-nal, f r o m there on to Japan i n f u l l load returning to the Persian Gulf i n ballast, and then back to Europe around the Cape of Good Hope i n f u l l load.
Particulars of ship and equipnwnt
These measurements were performed on a tanker w i t h the following main dimensions:
Length between perpendiculars 252 m Moulded breadth 39 m Moulded depth 18 m Draught ^ 14 m Displacement ^ 95,000 tons
The ship has a bulbous bow and operating speed in loaded condition is 16.25 knots.
The water pressures were measured by pressure gauges de\ eloped hy Det norske Veritas. The gauge consists of a sensitix'e membrane w i t h an active strain gauge attached to i t . A dummy strain gauge is introduced in tlie circuit i n order to pre\'ent var-iations i l l temperature influencing the recordings.
Stresses in longitudinal frames were measured by strain gauges mounted on the flange, one at the end of the frame as close as possible to the web frame and the other at the midpoint of the span. The strain gauges were connected in such a way as to measure only the stress difference (i.-e. the sum of the absolute stress values) between the end and the midpoint of the frame in order to obtain results which were not influenced by longitudinal stresses or temperature variations.
Vertical accelerations of the h u l l were measured by a Hottinger Messtechnik accelerometer w i t h na-tural frequency 250 H z and range 0 to 5 g's.
Signals f r o m the gauges were led through a 6-channel Hottinger bridge f o r balancing and ampli-fication and finally recorded on an ultra violet re-corder of type A B E M Ultralette 5651.
The positions of the measuring points are shown i n F i g . 1. The accelerometer was mounted in the centre line of the ship.
Test programme
Recordings w i t h a duration of 15 minutes were •taken every eighth hour when the ship was in open
KEiivopcan ShiphuiUling» No. 6 — IDüH F O R E C A S T L E D E C K O S T R A I N G A U G E S I l P R E S S U R E G A U G E S • A C C E L E R O M E T E R F i g . 1. Positions o f m e a s u r i n g p o i n t s i n the f o r e - s h i p .
number of recordings f r o m each point, This was not achieved, however, as some of the gauges broke down after some time. The pressure gauges failed due to insufficient sealing, and we obtained recor-dings of pressure only during the first part of the voyage.
A t the same time observations of the en\'iron-mental conditions were made through visual esti-mates of significant w a \ e height, average wave period, Beaufort force and relati\'e direction be-tween ship and wave system.
As seen f r o m F i g . 2 the ship sailed i n relatively
T O T A L N U M B E R S O F O B S E R V A T I O N S : 2 3 0 F . C A S T I E O K . 2 3 1 5 » B E A U F O R T F i g . 2. T o t a l d i s t r i b u t i o n o f B e a u f o r t . F i g . 1.
good weather most of the time. This makes extra-polations of results rather uncertain.
.^s the wave to heading angles are so evenly distributed as shown in F i g . 3., approximately the same amount of data are obtained f r o m each head-ing range. Consequently no notable effect is ex-pected i n the f i n a l results because of skew distrib-ution of headings.
Average speeds as a f u n c t i o n of Beaufort f o r the ballast and f u l l load conditions are given i n F i g . 4 which shows the typical trend of speed reduction as the weather worsens. A ratlier small reduction is observed u n t i l Beaufort 4 to 5 due to increasing wave resistance. Thereafter the drop i n speed is considerable and is due to the voluntary reduction ordered b y the officers as a result of subjective judgement.
Results from the measurements
A n important assumption^ i n theoretical calcula-tions of wave induced mocalcula-tions and loads is that the short t e r m distributions of peak to peak values follow the Rayieigh distribiition given as
' ( X Ï / 2 P
P(x < X l ) = 1 — e E (1)
(see list of notations).
A n attempt was made to verify Oiis assiunption. Several recordings, chosen at random, were there-fore analysed i n tlie following way. .\11 double amplitudes of the recordings were taken, classed according to size and the frequency function and the cumulative distribution was calculated. A
Ray-«Eiiropeaii Shipbuilding)) Nu. 6 —• 1968 1 9 5 1 6 5 -N U M B E R O F O B S E R V A T I O -N S D R A F T B E A U F O R T S U M o D R A F T 0 1 2 3 ( 5 6 S U M B A L L A S T 8 13 19 16 1 0 I B F U L L L O A D 6 K 28 4 9 3 0 13 B K 8 TOTAL N U M B E R O F O B S E R V A T I O N S •. 2 0 8 10 1 3 5 7 9 11
F i g . 3. D e f i n i t i o n o f wa\'c' to hi-ailiiig angle a n d d i . s l i i b u t i o n o f licading.s.
leigh distribution when plotted on WeibuU pro-bability paper appears as a straight line w i t h slope equal to 2. E.xamples of plottings on W e i b u l l pro-bability paper are shown i n Figs. 5 — 7 .
1 frequency function corres^Jonding to eq. 1 is
P
W = ^ «
" E ( 2 )
V E which is the only parameter of the Rayieigh distribution was estimated as described below, and the histograms and corresponding theoretical fre-quency functions are shown i n Figs 8—^10.
I t seems reasonable to conclude that these va-riables follow the Rayieigh distribution.
I t should be mentioned that the recordings labelled Nos. 8 and 4 1 were those showing the largest disagreement w i t h theory.
Estimation of \/E
V E (the square root is introduced i n order to give the parameter the same dimension as the v y - ' 'ble) dclermines the distribution of the vari-al, in question completely and the greater part
B E A U F O R T
Fig. 4. Average speed vs. Beaufort.
of the analysis of dynamic measurements like these consists of estimating V E .
This can be performed i n several ways. The simplest method is contained i n the f o l l o w i n g equa-tion
X „ , / 2
vE = -rzz
v ~This equation gives on average a correct estimate of V E . At the preliminary stage of the analysis, however, different methods were employed and i t was concluded that all the methods tried gave practically the same results. I t may for example be seen from eq. 1 that P(x) = 0.63 when x = 2 VE. V E ma)' therefore be estimated f r o m diagrams like those shown i n I'igs 5 — 7 . V E as given i n Figs 8 — 1 0 is estimated according to eq. 3 and i t may be seen that the results f r o m these t w o methods have given practically identical results.
The relati\e motion between ship and wave is defined in Fig. 1 1 . Also V E for this variable could be estimated f r o m the recordings i n cases w h e n a measuring point every now and then emerged or submerged. I t could be clearly seen f r o m the re-cordings whenever this was the case. I f N i de-notes the number of emergences (submergences) and N the total number of -cycles i n a recording then N i / N is an estimate of the probability that the relative motion is larger than the vertical d i -stance between the still water line and the measur-i n g pomeasur-int. W measur-i t h reference to F measur-i g . 1 2 and eq. 1 measur-i t is then easily shown that the f o l l o w i n g equation is valid
y A ^
(4) R M
N ,
.99909 .999913 .99990 .99950 .99900 f r t i f i t r n- r r i i l i i T f r r [ i: i' ' [ i j
' STRAIN GAUGt riO 2 , j M
/ .10000 . 9 9 9 9 9 . 9 9 9 9 5 . 9 9 9 9 0 .99950 .99900 3 4 5 6 .7 a 9 :
WEIBULL PROBABILITY PAPER u : In [ - l n ( l - P(O-))]
3 4 5 6 7. 8 9 10, 20 30. 40. 5 0 .
DOUBLE AMPLITUDE STRESS CT- 10 ( k p / c m ^ ) F i g . -5. C i i u i u l a t i x e cli.stiibution o f .slres.ses.
2 3.
t i i i i 1 n ^ J j M j f a t r t t f i i ü j ' i ' i i i m ' i
.15000 M
.10000 3 4 5 ,6
WEIBULL PROBABILITY PAPER
2 3 4 b. 6. 7, a 9 10 20 30 40. 50
DOUBLE AMPLITUDE WATER PRESSURE p l O ( k p / c m ) F i g . 6. C u m u l a t i v e cli.stribution of w a t e r pressures.
((European Shipbuilding)) No. 6 — 1968 .999'^"' .996 . 9 9 9 9 U .99950 .99900 .99500 .99000 .15000 .10000
VVEIBULL PROBADILfTY PAPER In [ - I n d - P ( A ) ) ]
2 . 3 i . E 6 7 8 9 10 i 2 0
DOUBLE AMPLITUDE ACCELERATION A 1 0 2 ( g ) 3 0 m 5 0 .
F i g . 7. C u m u l a t i v e d i . s t r i l n i l i o i i o f acx-elciation.
f r o m simultaneous recordings f r o m different mea-suring points. These recordings each gave estima-tes of V E R M which were reasonably consistent w i t h each other. I t was also possible to calculate the cumulative distribution of relative motion i n order t o check that the assumption of a Rayieigh distribution is ^'alid also for this variable. Examples are own i n F i g . 13. This result might not seem convincing but i t should be remembered that the determination of y ^ ( y ^ ) is probably inaccurate. The VEKJV'I values given applies only t o the sec-tion 0.028L aft of FP.
I t must be emphasised that a distribution deter-mined by a given V E value is valid only so long as there are no changes i n external conditions like draught, heading, wave height, speed, etc. This condition is assumed to be f u l f i l l e d for the record-ing time used i n these measurements.
V1'- cii' a function of Beaufort
From eq. 1 i t is seen that the probability that the value of the variable x does not exceed 2 • V I 2 is 6 3 ' / . , By using VE-values i n the further analy-sis "'0 are therefore working w i t h values having lli(, auie probability of being exceeded.
The \'ariation of V E w i t h Beaufort was investi-gated for all variables. I n this connection i t should be remembered that the Beaufort numbers given are based on \'isual observations and that they are not necessarily comparable w i t h Beaufort numbers based on w i n d measurements. Examples are shown i n Figs 1 4 — 1 6 . Theoretical calculations of V E f o r relative motion and vertical accelerations based on strip theory calculations as described i n / I , 2 / have been plotted i n Figs 1 5 and 1 6 employing the ob-served relations between Beaufort numbers, wave heights and periods.
The variation of V E w i t h Beaufort seems rea-sonable and the measurements are i n fair consist-ency w i t h the theoretical results.
Correlation
I n order t o f i n d i f any simple relationship could be obtained between local variables like stresses and variables like relative motion (which is quite well established theoretically) a correlation analy-sis between VE-values was performed. Examples are shown i n Figs 1 7 and 18.
Because of the small number of simultaneous recordings the only conclusion that can be drawn
«Euwpcan Shiphuilding» No. 6 — 1968 R E C O R D I N G NO. S 12 20 i n 3 210 2 8 0 3 5 0 D O U B L E A M P L I T U D E ( k p / c m M R E C O R D I N G NO. 181 D R A F T S P E E D B E A U F O R T 14.1 m 15,5 k n o t s 5 J F R E Q U E N C Y F U N C T I O N W= 229 k p / c m ' 5 6 0 8 4 0 1120 1400 D O U B L E A M P L I T U D E I k p / c m M F i g . 8. H i s t o g r a m s o f r t ' c o r d c i l stresses f r o m s t r a i n gauge N o . '2. 70 R E C O R D I N G NO. 41 D R A F T S P E E D B E A U F O R T 14 7 m 1 5 . 5 k n o t s 5 L H E p R E T I C A L _ _R A Y L E I GH_ F R E Q U E N C Y F U N C T I O N W = 0 - 0 1 8 g 5.1 6 8 8 , 5 10 2 D O U B L E A M P L I T U D E ( g l O M R E C O R D I N G NO. 181 2.184 2.912 3 , 6 4 4 3 6 8 D O U B L E A M P L I T U D E I g -101 F i g . 10. H i s t o g r a m s o f r e c o r d e d accelerations. R E C O R D I N G NO. 8 D R A F T S P E E D B E A U F O R T T H E O R E T I C A L R A Y L E I G H F R E Q U E N C Y F U N C T I O N 8 . 0 m 1 7 . 0 k n o t s 4 ^1^= 0. 141 k p / c m ' 0 . 3 2 0 . 4 8 0 . 6 4 0 8 0 D O U B L E A M P L I T U D E ( k p / c m ' 1 F i g . 9. H i s t o g r a m o f r e c o r d e d pressTires f r o m pressure gauge N o . 6. H . HEAVE RM : RELATIVE MOTlüH F i g . H . D e f i n i t i o n o f r e l a t i v e m o t i o n .
((European Shiphuilding» No, 6 — 1968
F i g . 12. Short t e r m cli.stribution o f r e l a t i v e m o t i o n . Shaded are epresent the p r o b a b i l i t y f o r a r e l a t i v e m o t i o n larger
t h a n Y A . respectively y B
is that the results indicate that a simple relation seems to e.xist between looal stresses and relative motion and that the local jiressure height may be estimated f r o m F i g . 19. The line in Fig. 17 labelled
is based on Fig. 19 and gives the stress difference between the end and the m i d p o i n t of the longitu-dinal frame when a symmetrical bending moment curve is assumed. The factor 0.1 is introduced i n order t o obtain the stress i n k p / c m ^ when R M is given i n metres.
The comparatively good results of this correla-tion analysis suggests that further investigacorrela-tion i n this f i e l d might lead to valuable results.
The good correlation between stresses and accele-ration is a little surprising and may be quite acci-dental.
The tank at the section where the m a j o r i t y of the measuring points were situated was empty.
.99999 .99995 .99990 .99950 .99900 .99500 .99000 .15000 m .10000 2 3 M .5 I .6 .7 .8 .9
WEIBULL PROBABILITY PAPER
3 4 5 6 7 a 9 10 20 - 30.
DISTANCE FROM WATER SURFACE
u = In [ - I n ( 1 - P (RM ))] TO MEASURING POINT ( m e t e r )
«Enwpcdii Sl>i)>huihling» No. 6 — 1968 NUMBER OF V ï - V A L U E S B E A U F O R T 0 1 2 3 i. 5 6 S U M B A L L A S T 3 6 5 5 8
— —
2 7 F U L L L O A D 3 6 22 3fl 26 13 112 S U M 6 12 2 7 13 3 t 13 i 139 B E A U F O R T 0 1 2 3 L 5 6 S U M D A L L A S ! 3 9 10 18 16 1 0 57 F U L L L O A D 3 B 25 : l 26 13 C 120SUM 6 17 3 5 59 L2 K L 177 F i g . 15. As'evage -^/E f o r acceleration vs. B e a u f o r t .
F i g . 14. Average - \ / E f o r stress r e c o r i l c d at m e a s T i r i n g p o i n t N o . 3 VS. B e a u f o r t . too 200 100 C O R R E L A T I O N C O E F F I C I E N T 0 9 0 7 E Q U A T I O N F O R R E G R E S S I O N OF I T V E R S U S R M ^ - 1 3 5 . N U M B E R O F S I M U L T A N E O U S R E C O R D I N G S 5 3 E Q U A T I O N F O R R E G R E S S I O N O F R M V F R S I I S C7 6 6 l ^ j ^ - 1 5 7 1 + 1 . 5 7 . 0 0 1 0 7 ( \ / E J _ - I 3 5 I
w
/
,
/ / /
w
/
M E A N O F \/F(p 4 + / *. • ^ ^ E ^ ^ O l 1^ 1 0 ' • 2 ^ ' ^ P M 4 4 + / *. • • » 7 / • Th
I I 1,0 2 0 3,0 4,0 \ / E ™ l m l F i g . 17. C o r r e l a t i o n b e t w e e n - y / E f o r stresses r e c o r d e d at measuring p o i n t N o . 2 a n d r e l a t i v e m o t i o n . F u l l l o a d .«Europcan SJiipbtiildiiig» No. 6 — 1968
Long term distributions
I n our theoretical calculations of long term distributions of wave induced motions and loads i t is assumed that i n the long run both V E and the variable itself f o l l o w W e i b u l l distributions, ge-nerally given as
- ( I f
P(X < X l ) 1 — e N U M B E R O F l / Ë " - V A L U E S B E A U F O R T 0 1 1 3 i 5 6 S U M B A L L A S T F U L L L O A D -1 3 3 22 B 15 1 S 12 S3 S U M 0 1 3 2 5 23 9 1. 6 5The Rayieigh distribution is thus a special case of the W e i b u l l distribution w i t h k = 2.
I n order to check i f this assumption is reasonable the cumulative distribution of V E was calculated and plotted on W e i b u l l probability paper as shown i n Figs 20—22. The distribution of stresses shows the largest deviation f r o m the required straight line but, at least i n the most interesting upper part of the curve, a W e i b u l l distribution seems ade-quate. The number of measurements is, however, too small to allow any definite conclusions to be drawn regarding the hypothesis of longterm d i -stributions of V E following W e i b u l l di-stributions.
F i g . 16. Average y E f o r r e l a t i v e m o t i o n vs. B e a u f o r t . 4 0 0 300 2 0 0 100 C O R R E L A T I O N C O E F F I C I E N T 0 9 2 t R E G R E S S I O N E Q U A T I O N = "32 * - 0 0 2 0 9 ) ^ N U M B R E C O ER O F S I N R D I N & S ' 1 I U L T A N E O U 2 \
\
\
/
/ +—
«•—
* * / * / M E A N O F + • /1
4-\ * Pl * * 1 ^ m l > j Z l"n 1-0.02 0.04 0.06 0 0 8 \ ^ A l g )«Etiiüj)ctin S}iii>J)iiil(liug» No. 6 — 1968 STILJ^VVATEJ? LINE F i g . 19. D y n a m i c pressure h e i g h t h p distance f r o m b o t t o m . IS a f u n c t i o n o f Conclusions
T h e most important resuUs of this investigation may be summarised as follows:
1. The Rayieigh distribution is w e l l suited f o r describing short term variations of water pres-sure, local stress, vertical acceleration and rela-tive motion.
2. The residts indicate no significant departure f r o m the Weibull distribution in tlu; description of the long-term distributions of V E .
3. A simple relationship between local stresses and relative motion is indicated but further investi-gation is necessary.
4. Measured acceleration and estimated relative motion show fair consistency w i t h theoretical,
results. '
Ackno wledgement
Instrumentation and installations were carried out at the Electronic Laboratory of Det norske Veritas and the measurements by M r . A. Larsen. The analysis was performed by the group for Sea Load Investigations under guidance of M r . N . N o r d e n s t r ö m .
R E F E R E Z V C E S ;
[ 1 ] . B . Pedersen: C o m p u t e r p r o g r a m s p e c i f i c a t i o n N V 410. T r a n s f e r f u n c t i o n s f o r w a v e i n d u c e d ship motions and loads.
D e t norske Veritas, Research D e p a r t m e n t , R e p o r t N o . 68—13—S.
[ 2 ] , N . N o r d e n s t r o m : Calculations of w a v e i n d u c e d m o -tions and loads. Progress r e p o r t N o . 4. A p i l o t s t u d y w i t h the c o m p u t e r p r o g r a m N V 4 ( ) 3 .
D e t norske Veritas, I^esearch D e p a r t m e n t , Report N o . 66—11—S. .99999 .99995 .99990 .99950 i .99900 ,99500 .99000 - f f .95000 .15000 .10000 1 .2 .3 .1 .5 .6 .7 .8 .9 1. W E I B U L L P R O B A B I L I T Y P A P E R , u = I n [ - I n ( I - P( V¥c-))1 4 . 5 . 6 . 7 . B . 9 , 1 0 30. 40. 5 0 . \ / Ë < r x l O - ' ( - Ï E ^ F i g . 20. C u m u l a t i v e d i s t r i b u t i o n o f \ / E f o r stress.
.99999 .99995 .99990 .99950 .99900 .99500 .99000 .30000 .25000 .20000 .10000 .99999 .99995 . 9 9 9 9 0 .99950 .99900 .99500 .99000 f f .95000 .1 .2 ,3 A 5 .6 .7 .8 9 1. W E I B U L L P R O B A B I L I T Y P A P E R , u = I n [ - I n (1 - P ( \ /E ' J ) ) ] 2 . 3. 4. 5. 6. 7 e. 9. 10 F i g . 2 1 . CLimulative d i s t r i b u t i o n o f - y / E f o r acceleration. -1. O 1 20. 3Ü. 40. 5 0 . X 1 0 " ( g ) .25000 .20000 ,15000 4 , .4 .5 .6 .7 B .9 1. V / E I B U L L P R O B A B I L I T Y P A P E R . u = I n [ - In ( 1 - P ( v / Ë f a ) ) ] 30 40 5 0 .
