Calculation of wall pressures in a smooth rectangular tank due to movement of liquids

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CALCULATION OF WALL PRESSURES IN A SMOOTH RECTANGULAR TANK DUE TO

MOVEMENT OF LIQUIDS

(Program LR 321)

AUTHOR: _{A. Blixell, Civ.ing., C.Eng.}

DEPARTMENT: _{Structures Section, Research and Technical} Advisory Services.

PRINCIPAL: J.I. Mathewson, C.Eng.

CONTENTS PAGE SUMMARY INTRODUCTION THEORY PROGRAM LR 321 MODEL EXPERIMENTS 9

COMPARISON BETWEEN THEORETICAL CALCULATIONS AND _{... 12}

EXPERIMENTAL RESULTS.

ACKNOWLEDGEMENTS .. 13

SUMMARY

As part of the Society's long-term research program concerning the ship and its behaviour and

responses in an ocean environment this report treats the problem of liquid motions in a smooth, rectangular tank. The liquid is assumed to be ideal whilst the tank is assumed to oscillate

with simple harmonic motion as well as being

subjected to a vertical acceleration. Model

experiments have been performed and a theoretical

method has been derived and programmed, LR 321.

Comparisons are made between experimental and computed results and they show good agreement.

Investigations are now being made to test tanks of other shapes to confirm calculation methods which cater for all tank motions.

LLOYD'S REGISTER OF SHIPPING

7!, Fenchurch Street, London, E.C.3

CALCULATION OF WALL PRESSURES IN A SMOOTH RECTANGULAR TANK DUE TO

MOVEMENT OF LIQUIDS

(Program LR 321)

1. _INTRODUCTION

An increased number of requests regarding _arbitrary

tank fillings in bulk, L.N.G., L.P.G. carriers and chemical tankers resulted in Lloyd's _Register

carrying out investigations to study the motion

of liquids in tanks.

These investigations followed an experimental and theoretical course of study. _Laboratory

tests were performed in order to observe the

liquid movement in an oscillating tank and also to measure the pressures on the tank walls. _These

pressures have been compared with the results of a theoretical method which has been developed

within the Society.

The computer program for this theory enables the designer to predict pressures which can

be used at the design stage to avoid the _possibility

of sloshing damage.

This Certificate is issued upon the terms of the Rules and Regulations of the Society, which provide

that:-"The Committees of the Society use their best endeavours to ensure that the functions of the Society are properly executed, but it is_{to be}

understood that neither the Society nor any Member of any of its Committees nor any of its Officers, Servants or Surveyors is under any circumstances whatever to be held responsible or liable for any inaccuracy in any report or certificate issued by the Society or its Surveyors, or in any entry in the Register Book or other publication of the Society, or for any act or omission, default or negligence of any of its Committeesor any Member thereof, or of the Survcyors, or other Officers, Servants or Agents of the Society".

-2

The liquid motions that occur in a smooth tank

are of the non-linear and linear types. The non-linear type, Fig. 1, is predominant at low

degrees of filling and causes heavy impact pressures.

The linear type, Fig. 2, occurs at higher fillings

and produces comparatively lower pressures.

Fig. la

Non-linear motion

's

l"iq. 11)

-3-Fig. 2a Linear Motion

Fig. 2b

Still level

2.1.1 Hydraulic jump or bore.

When calculating the non-linear pressure against

a tank wall it is essential to know the speed

with which the bore hits the wall, as the dynamic

pressure, in this case, is expressed as

2

= PC

where

= dynamic pressure (N/rn2)

p = density (kg/rn3)

c = velocity of the bore (m/s)

In order to obtain the velocity, c, the strength of the bore, the hydraulic jump which is defined

as the difference between the water depth in

front of the bore, h1, and the water depth behind the bore, h2, Fig.3 has to be known.

y £ H1

_h_

The hydraulic jump n = h2 - h1

Fig. 3 n (1) Centre of rotation 4 2. THEORY 2.1 NON-LINEAR MOTIONS

In this report a method described in Ref. (2)

has been utilized to calculate the hydraulic jump, finally given as:

fl=h4A 1-

where h = undisturbed depth of licruid in the tank (m)

½

1+

[3 r

H1 = Y- co-ordinate of undisturbed surface of liquid (rn)

L = length of tank (m)

(-)

= amplitude of oscillation angle (-)

w- w0

-1

(s

w = actual rotational frequency of the tank (s)

(A)

= natural frequency of the liquid

(s)

(m/s) The strength of the hydraulic jump is thus a function of the tank length, the location of the centre of rotation, the amplitude of the rotation angle, the depth of the liquid and the frequency of the rotation.

2.1.1 Non-linear pressure

Once the hydraulic jump is calculated, the

velocity of the bore is calculated using the expression:

2 h.H L2 Pt = d + PS (4)

c=

and using this value of c in ea. (1) the dynamic pressure is given.

The total pressure is finally calculated as:

]½

= O when y = -H

Dy

where = velocity potential

level

-6-where

PS = static pressure (N/m2)

2.2 LINEAR MOTIONS

The expressions for calculating the linear motions of a liquid are obtained using a number of boundary

conditions and the features of potential theory.

2.2.1 Natural Period of the Liquid

The formula for natural period of the liquid is derived from a system of four differential

equations. D2

+ WT

=Owheny=b

Eq. (5) is the equation of continuity for an inviscid

and incompressible liquid. The following equation

says that the pressure on the free surface is

constant. The last two equations describe the

boundary conditions, which in this case are that the velocity at the tank walls and tank bottom

is equal to zero.

-The solution of this system of equations gives the

formula for the natural period:

T = "EA

-y

where A = 1,2 denotes the first, second... natural periods.

2.2.2 Linear Pressure

Using eqs. (5) and (6) and substituting the two boundary conditions (7) and (8) with:

--H

-

xy-which represent a sinusoidal oscillation of the

tank:

0 =0

o

L

X =

±-where O = amplitude of rotation

o

w = the forced rotation frequency t = time

(s) (9)

A solution to the system of equations (5),

(6),

(10) and (11) is given in Ref.

(3) as:

= ®WCOS

wt

[Xy

(_1)k4 L2

sin(2k+1) rr/L

k =0

_{(2k+1)3 jr3 sinh(2k+l)Trh/L}

(2k+l)ir (b-y)

(2k+l)ir

(2k+1)rr h

2 cosh

+ I(g-bw

sinh

L

cosh

(2k+1)iî (y+H)/L

ac u _ax

V =-

-8

(w2 -w2

cosh

(2k+l) irh/L

2k+1

-1

where w

=-

(s

The velocity potential function

(x,y,t)

describes the flow of the liquid in the tank completely.

It is then possible to calculate pressure against

the tank walls using the general expression:

p =p

-

½(u2 +

(13)

Ea

where

= height of the liquid at the tank

side from the tank bottom.

2

(m /s)

= velocity of the liquid in the

x-direction.

(m/s)

= velocity of the liquid in the

y-direction.

(m/s)

The height of the liquid at the tank side above

the still level is calculated using the familiar

expression:

'

E'=-

(14)

(12)

-9--3. PROGRAM LR 321

A computer program, LR 321, has been developed

to perform the pressure calculations for non-linear and non-linear pressures, using eqs. (4)

and (13). The program chooses for low degrees

of filling non-linear maximum pressures and for

higher degrees of filling linear maximum pressures, during one cycle of oscillation. The choice is

based on the relation between still water depth

and tank length.

3.1 INPUT FOR PROGRAM LR 321

The program requires for its execution data

about the tank size, the height of the centre of rotation above the tank bottom, the percentage of filling, the amplitude of the rotation angle, the density of the liquid and finally a value of the vertical acceleration of the tank, which caters for the location of the tank in the ship.

The numerical data is preceded by a title card

which will label the output.

Card i Cols. i - 80 Title,format 20A4 Card 2-(N-i) Cois. 1 - 10 Length of the tank in

metres, format F.lO

Cols.11 - 20 Location of the centre

of rotation above the tank

bottom in metres, format F.lO. Cols.21 - 30 Depth of liquid in per cent

of total tank depth, format

F. 10 Cols.31 - 40 Total tank depth in metres,

format F.10.

Cols.41 - 50 Amplitude of roll or pitch

angle in degrees, format F.i0. Cols.51 - 60 Density of liquid in kg/rn3.

format F.1O.

Cols.6l - 70 Vertical acceleration of tank in fraction of g, the

constant of gravity, format F. 10

lo

-Card N Blank

The figure N might be of any value, which means

as many cases as required might be run at the same

time. _{The program terminates when a blank card}

is read. An input data sheet is shown in Fig. 19.

The calculation time for one case on the Society's

I.B.M. 360/40 computer is approximately 3 minutes.

The choice of the numerical values of the amplitude

of rotation angle and vertical acceleration of the tank is at present under investigation _on long-term probability of occurence basis. These results will be reported separately.

3.2 OUTPUT FROM LR 321

The program first prints the title of the run,

see Fig. 20, and after that the input data is

listed.

The calculated natural frequency and period is then printed.

Pressure values are given by the program on the

vertical tank side at 10 equally spaced positions. These pressures are calculated for 8 periods at

each position based on the natural period of that

filling. _{The pressure is in N/rn2 and then divided}

by pg which gives an equivalent head for each period and position.

The position _{of the points on the ta.nk wall is} expressed in a y coordinate where y = O is the centre of rotation specified in the input.

An error message might appear for very low

degrees of filling which states 'No Calculations for this filling, negative squareroot'. _{This is} due to the fact that the theory is not valid when the liquid does not cover the whole bottom surface.

MN

A number of systematic variations of angle of

rotation, tank size and filling have been run and the results are represented in Figs. 21 - 26. In the figures, the numbers l,2...1O, represent

the 10 equally spaced positions along the bulkhead

where i is bottom and lO is top.

4.

Two plastic models were built to study the phenomenon of liquid motions in an oscillating

tank. The models represent an actual tank and

the scales are 1/20 and 1/32 respectively. r

.756 m

Dimensions of model tank in 1/20 of full scale tank

Fig. 5

12 housings were fitted at the tank side and 2 housings were fitted at the tank top in order to

have the 4 available pressure transducers plugged

into the required positions. The transducers are of the piezo-electrical type, which induce a charge when the pressure is changed. The

4

-j

-I

-c

RESULTS

12

-charge is then amplified and recorded using a Uy- recorder. The natural frequency of these

pressure transducers is very high and therefore no resonance phenomenon between the transducer

and the liquid is likely to occur.

All tests have been performed at Technische

Iloge-school, Deift, Holland, using their oscillating rig and electronic equipment.

5. _{COMPARISON BETWEEN THEORETICAL CALCULATIONS AND EXPERIMENTAL}

An extensive number of experimental tests have been run and for the corresponding cases the

theo-retical values have been calculated using program

LR 321. The results for a number of significant cases are illustrated graphically in Figs. 6 - 18,

where the total head in metres of fresh water is

plotted versus the period T. _{The dimensions of the} small tanks in the figures are in metres and the position of the transducer is signified by an

arrow.

In general it can be seen that the test results and the calculated va1ues. agree within acceptable

limits.

Comparisons have also been made between actual damages and calculated values, and in these cases

the theory has shown that structural collapse was possible.

13

-6.

The author wishes to thank Professor Ir J.

Gerritsma and his staff for help and useful discussions, and the Society would like to

express their apnreciation to The Shipbuilding Laboratory, Deift, for permission to use their

experiment equipment.

Further thanks are due to Mr.

A. C.

of the Society's laboratory, Crawley, for the design and construction of the models and for

performing the major part of experiments.

A. Blixell,

Lloyd's Register of Shipping

LL

. Mathewson,

P ncipal Surveyor to

14

-7. REFERENCES

Lamb, H.: "Hydrodynamics", Dover Publication, New York.

Verhagen,J.II.C. and van Wijngaarden, L.

"Non-linear Oscillations of Fluid in a Container", J. Fluid Mech. (1965) , vol. 22, pp. 737-751.

Hajiwata, K. and Yamamoto, Y. : " A Theory of

Sloshing in Cargo Oil Tanks", JSNAJ, 112 (1962)

Head

1.

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Exreri.rnent

Theory

0.

1.36

i

Fig. 6

Head

1.2

1.1

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Exne riment

Theory

0.75E

1.06

Secs.

Fig. 7

Experiment

Theory

2

Fig. 8

COMPARISON BETWEEN }EASURED AND CALCULATED PRESSURE HEADS

i

ecs

Head

Experiment

Theory

0. .56

1.06

\

\

\

\

i

Fig. 9

COMPARISON BETWEEN YEASURED AND CALCULATED PRESSURE HEADS

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

secs

Experirent

Theory

p..'

1.06

-o

secs

Fig. 10

0.5

0.4

0.3

0.2

0.1

Head

Experiment

Theory

0.

756j

I

y

1.06

i

Fig. 11

Head

0.3

0.2

0.1

f

Experiment

Theory

Fig. 12

COMPARISON BETWEEN MEASURED AND CALCULATED PRESSURE HEADS

0.756

1.06

-o

a,

Secs

Experiment

Theory

0.756

Fig. 13

COMPARISON BETWEEN MEASURED AND CALCULATED PRESSURE HEADS

1.06

T

Experiment

Theory

secs

i

Fig. 14

COMPARISON BETWEEN MEASURED AND CALCULATED PP½ESSURE HEADS

y

e

0.756

_-,

p

1.06

Experiment Theory

Fig. 15

0.75 6

COMPARISON BETWEEN MEASURED AND CALCULATED PRESSURE HEPDS

1.06 Head In L

03

0.2

0.1

j

Experiment

Theory

1 ₂ 3 4

Fig. 16

COMPARISON BETWEEN MEASURED AND CALCULATED "ESSURE HEADS

L m y

1.

Secs

-I

Experirent

0.755

Theory

1.06

sec s

Fig. 17

Head

ExneriIent

Theory

0.756

1.0

Fig. 18

COMPARISON BETWEEN MEASURED AND CALCULATED PRESSURE HEADS

y

LR 321

i

TITLE

ROTATION FILLING TANK ANGLE OF

10 ABOVE BOT.20 % 30 DEPTH 40 ROTATION 50 DENSITY 60 ACCELERATION₇₀

CALCULATION OF PRESSURE ON TANK WAILS

CALCULATICNS TO 8E REPRESENTED IN FIGURES AND DIAGRAMS FOR REPORT PRESENTATION.

TANK LENGTH 20.000 METERS CENTER 0F ROTATION - 7.500 METRES, POSITIVE ABOVE TANK BOTTOM

DEPTH OF LIQUIO - 50.0 PERCENT TANK DEPTH - 16.000 METRES ROLLPITCH ANGLE lo_o DEGREES

DENSITY OF 110010- 1000.0 KG/M3 VETICAL ACCELERATION OF TANK- 0.20 TIMES 6 (9.81 M/52)

CALCULATED NATLAL FREQUENCY - 1.145 RAD/SEC NATURAL PERIOD - 5.490 SECS

RUN ON aol i/72.

BULKHEAD PRESSURES V-COORDINATE.

V-COORDINATE.

Y-C OUED I NATE V-COU AD NA TE Y-COOPDIUATE. Y_CUr)ROI NATE-Y-COc10 I Y-COORO INATE. Y-COUTOINATE- V-COORDINATE- V-COORDINAIE- V-COORDINATV-C3ORD INST E- V-COORDINATE-Y- COO 10 1 NATE V-COORDINATE. V-COORDINATE. V-COORDINATY-C 00KO INAT

Y- COO RO I NAT E- Y-COJROINATY-C OLIROINAT E-V-C 00 RO INAT Y-CUORO IN AT E. Y-COURt) INATE V-COORDINATE-V-COORDINATE. V-COORDINATE. V-COORDINATE. Y-COORDIT)ATE v-COORDINATE. Y-COUROINATE-V-COORDINATE. Y-COJROINATE-V-COORDINATE. y-COORDINArE-V-COORDINATE. V-COORDINATE-Y-COORDINArE. V-COORDINATE. Y-CO)ITOINATF. ,COOc)INATE. Y-C OUR D INA ET

V-COORDINATE-Y-C DORt) ¡TIAr E. Y-CUORDINAIF. Y-C (t)tt O INRI E-V-COORDINATE. V-COURT) INATF. Y-COJRD INSTE-V-COOED INRI E Y-CO) AT) I NATE. Y-CCJOROINATE.

Y-C O)) '(T) ¡NA T

E--6.700 METRES PERIOD. 2.745 SECS PRESSURE. 0_IO8AAF 06 N/M2 HEAD- 11.066 METRES -6.700 METTES PERIOD. 3.843 SECS PRFSSIJRF. 0.97IRRE 05 N/M2 TIRAD- 9.904 METRES

-6700 METRES PERIOD. 4.941 SECS T'RTSSUAF- 0.10004E 04 N/P? fI'A0. 10.198 METPES

-6.700 METAL-S PERIOD. 6.039 SECS PRESSUAE 0.14412E '76 NI"? HEAT)- 14.E91 METRES -6.700 METRES PERIOD- 7.137 SECS PRESSURc. 0.1IR?4F 06 '11M? ITAfl. 12.053 NFTRRS -6.700 METRES PERIOD. 8.234 SECS PRESSURE- 0.11283E 06 N/M2 UPAD. 11.502 METRFS -6.700 METRES PERIOD- 9.332 SECS PRESSURE- 0.11045E 06 './M2 HfAD 11.759 METRES -6.700 METRES PERIOD- 10.4" SECS PRTSSUPE- 0.109171' 06 IT HrAD. 11.123 METR1'S

-6.700 MrTTTS PERIOD. 11.52(1 SECs PRESSURE- 0.10827E '16 N/M2 HEAD- 11.flR MErDES

--5.100 METRES PERIOD. 2.745 SECS PRESSURE- 0.682051' 05 NIP? HEAD 9.00) Mc'TRES -5.100 METRES PERIOD- 3.043 SECS PRESSURE- 0.77558E 75 N/M2 HEAD- 7.906 METRES

-5.10) METRES PERIOD. 4.941 SECS PRESSURE- 0.82107E 05 N/M2 TITAD.- 8.310 METRES

-5.100 METRES PERIOD- 6.039 SECS PRESSURF. 0.12557E 0E, E4/M2 H1'A'7 12.800 MEPS

-5.100 METRES PTRIOD. 7.137 SECS PRESSI)RR. 0.99396(1 05 N/M2 HEAD- 10.132 METRFS -5.100 METRES PERIOD- 8.234 SECS PRESSURE- 0.93960(1 05 PIIM2 HEAD. 9.578 METRES -5.100 METRES PERIOD. 9.332 SECS PRESSURE. 0.915811' 05 N/M2 HFAD 9.315 METRES -5.100 METRES PERIOD- 10.430 SECS PRESSURE. 0.902501' 05 NIP? HEAD- 9.70') METRES -5.100 METDES PERIOD- 11.528 SECS PRSSURE- O.6?4lAE 05 N/P2 HEA') 9.114 ETRES

-3.500 hEERES PERIOD. 2.745 SECS PRESSURE- 0.718051' OS N/P? 11(140. 7.320 METRES

-3.500 METRES PERIOD- 3.843 SECS PRESSURE- 0.59616(1 05 TI/N? HEAD. 6.079 METRES -3.50') METRES PERIOD. 4.941 SECS PRESSURE. 0.63672E 05 N/M2 HEAO 6.',RA PRIER5

-3.500 METRES PFRIOD- 6.039 SECS PRESSURE- 0.10813E 06 N/M2 TIFA')- 11.D'3 METFS

-3.500 METRES PERlaI). 1.137 SECS PRESSURE- 0.81272(1 05 N/P? HEAD- 8.285 METRES -3.500 METRES PERIOD- 8.234 SECS PRESSURE. 0.75596C 05 R/M? HFAD. 7.706 MCIRRS -3.500 METRES PERIOD- 9.332 SECS PRESSIJ'E- 0.73O91T 05 1/II2 HEAD. 7.451 METRES

-3.10) METRES PERIOD. 10.430 SECS PRESSURE- 0.716111E 05 N/M2 HEAD- 7.107 METRES

-3.500 METRES PERIOD- 11.25 SECS PR1'5SUPE. 0.70787E 05 RIM? HAD 7.716 MrTRFS

-1.900 METRES PERlOn. 2.745 SECS PRESSURE. 0.58111E 05 RPM? HEAD- 5.924 METRES -1.900 METRES PERIOD. 3.843 SECS PRESSURE. 0.42976(1 05 T4/M2 HEAD- 4.381 METQFS -1.900 METRES PERIOD- 4.941 SECS PRESSTJRF. 0.44897E 05 N/P? HEAD- 4577 METRES -1.900 METRES PERIOD- 6.059 SECS PRESSURE- O.923?IE 05 N/N? HEAD. 9.411 METRES -1.900 METTES PERICO. 1.13? SECS PRIS5tjAE- 0.63771(1 05 N/P? HES'). 6.507 METRES -1.900' METRES PEIOfl- 8.234 SECS PRESSURE. 057663(1 05 N/N? MEAD. 5.578 P510(15

-1.900 METRES PERIOD- 9.332 SECS PRESSURE= 0.54926E 05 N/H? HEAD 5509 METRES -1.900 METRES PERIOD- 10.430 SECS PRESSURE- 0.53310(1 05 N/H? HEAD- 5.4'0 METRES

-1.900 METFES PERIOD. 11.528 SECS ORESSURE. 0.52373E 05 N/N? HEAD. 5.130 HErDES

-O.roO METRES PERIOD- 2.745 SECS PATASUPE. 0.46611E 05 NIM2 1I"AD- 4.7't MFTES

-0.300 METRES PERIOD- 3.843 SECS PRESSURE- '7.27242E 05 N/P? H(1A0 2.777 METRES -0.300 METRES PERlER). 4.941 SECS PRESSURE. 0.26224E 05 R/M? HEAD. 2.673 ME1ES

-0.300 METRES PERIOD. 6.039 SECS PRESSURE- 0.77571(1 05 N/M2 HEAD- 7.902 M(1T°(1S -0.300 METRES PERIOD- 1.137 SECS PArSSO!(1'. _{0.461146(1 05 NFM2} lEAD. 4.778 MRT°ES -0.300 METRES PERIOD- 8.234 SECS PRESSURE. 0.40117E 05 N/N? MEA'). 4.089 METRES -0.300 METRES PERIOD. 9.332 SECS PRES$'JPE 0.37049(1 05 N/N? HEAD- 3.777 METR1'S -0.300 METRES PERlaI). 10.430 SECS PRESSURE. 0.3578EE OS N/M2 HEAD- 3.o9E. METRES -0.300 METRES PERIOD- 11.528 SECS °RESSUAE- 0.341380 05 N/M2 HEAD. .48D METRES 1.300 METRES PERIOD- 2.145 SECS PRESSURE- 0.36840E 05 H/M2 HEAD. 3.755 METRFS 1.300 METRES PEPTDT)- 3.843 SECS PRESSURE- 0.0 P41M2 HEAD- 0.0 METRES

1.3)0 METRES PElInO- 4.941 SECS PRESSURE- 0.79021(1 04 N/M2 HEAD. 0.906 M(1TR(15 1.300 METRES PERIOD. 6.039 SECS PRESSURE. 0.63996(1 05 N/M2 HT40 6.524 METRES 1.300 METRES PERIOD- 7.137 SECS DRESS)JRF. 0.30476(1 05 N/N? HEAD- 3.107 METRES 1.300 METRES PERIOO- 8.234 SECS PRESSURE. 0.22936(1 05 N/M2 HEAD. 2.338 METRES 1.300 METRES PERIOD 9.332 SECS PRESSURE- 0.104365 05 N/M2 HFAD 1.981 METRES 1.300 METRES PERIOD. 10.430 SECS PTESStJRE. 0.171911E 05 N/M2 HEAD. 1.773 METRES 1.300 METRES PERIOD 11.528 SECS PEE$S)ITE- 0.16068E 03 N/M2 HTAO. 1.638 METRES

2.900 MITRES PFRIOO- 2.745 SECS PRESSURE. 0.0 N/N? HEAD. 0.0 METRES

2.900 METRES PERIOD- 3.843 SECS PRESSURE. 0.0 N/P? HEAD. 0.)) METRES

2.900 METRES PERIOD. 4.941 SECS P0E55)JSE- 0.0 N/M2 HEAD- 0.0 METR1'S

2.900 METRES PERIOD- 6.039 SECS PRESSURE- 0.51891(1 05 N/P? HEAD. 5.290 METRES 2.900 METRES PERIOD. 7.137 SECS PRESSURE. 0.14680E 05 N/M2 HEAD. 1.496 METPES 2.900 METRES PERIOD- 8.234 SECS PRESSORE- 0.0 11/112 HEAD- 0.0 METRES

2.900 MEERES PERIOD. 9.332 SECS PRESSURE. 9.0 N/P? HEAD- 0.0 METRES

2.900 METRES PERIOD 10.430 SECS PRESSURE- 0.0 N/N? HEAD- 0.0 METRES

2.900 METRES PERIOD. 1I.S28 SECS PRES'JflE- 0.0 NIP? HEAD. 0.') METRES

4.500 METRES PERIOD- 2.745 SECS PRESSURE- 0.0 N/M2 HERD. 0.0 METRES

4.500 METRES PERIOD. 3.843 SECS PRESSURE- 0.0 NIP? HEAD. 0.0 METRES

4.500 METRES PERIOD- 4.941 SECS PRESSURE. 0.0 N/M2 HEAD 0.0 METRE$

4.500 METRES PERIOD- 6.039 SECS PRESSURE- 0.41442E 05 N/P? HEAD. 4.224 METRES 4.500 METRES PERIOD- 7137 SECS PRESSURE. 0.0 N/M2 HEAD- 0.0 METRES

4.500 METRES PERIOD- 8.234 SECS PRESSURE- 0.0 N/M2 HEAD- 0.0 METRES

4.500 METRES PERIOD- 9.312 SECS PRESSURE. _0.0 N/P? HEAD. 0.0 METRES

4.500 METRES PERIOD. 10.430 SECS PRFSSURE. 0.0 N/P? :t1'AD. (7.0 MEÎR1'S

4.500 METRES PERIOD. 11.528 SECS PRESSURC- 0.0 RIM? HEAD- 0.0 METRES

6. 100 MF TRE S PEP I 01)- 2. 745 SECS PAF 55'J.RE. 0.0 N/P? HEAT)'. O .0 METIFS 6.100 METRES PERIOD- 3.843 SECS PRtSOAE- 0.0 NIP? HEAD- 0-0 METRES

6.100 METRES PERIOD- 4.941 SECS PPESÇUR(1. 0.0 N/Mi HEAD- 0.0 METRES

6.100 METRES PERIOD. 6.039 SECS PRESSURE. 0.0 RUM? HEAD. 0.0 MFTRF5

6.100 METRES PERIOD- 7.137 SECS PRESSURE. '7.0 N/u? HEAD. 0.0 M5TPFS 6.100 METRES PERIOD- 8.234 SECS PPTSS:JAE. 0.0 NIP? HEAD. 0.0 METRES

6.100 METRES PERITO. 9.332 SECS PR1'SStJRE- 0.0 NIP? H1'A0 0.0 METRES

6.100 MFT1'ES PERIOD. 10.410 SECS PMTSSURE. 0.0 N/M2 MEAD 0.0 METRES

6.100 METRES PERIOO- 11.528 SECS PRESS)JRE. 0.0 NIP'? HEAD- 0.0 METRES

7.700 METRES PERIOD- 2.745 SECS PRESSUAF- 0.0 19/1(2 HFET)' flI) METRES

7.100 METRES PERIOD- 3.843 SECS PRESÇIJR(1- 0.0 N/M2 HEAD. 0.)) METRES

7.700 METRES PERIOD- 4.9'.l SECS PRESSURE. 0.0 N/N? HEAD- (1.0 METRES

7.700 METRES PERIOD- 6.039 SECS PES5UR(1- 0.0 N/P? H1'R1). 0.0 METAFS

7.700 PITEES DIRIOD. 7.137 SECS PRESSURE- 0.0 N/N? HEAD- 0.0 MFTSFS

1.700 METRES PERI(tD 8.234 SECS PRESSuRE. 0.0 N/H? HSO 0.0 MFTRRS

7.700 PETItES PERIÍ)9 9.332 SFCS PRESSURT. 0.0 N/M2 HEAD. 0.0 METRES

7.700 MITRES PERlEN)- 10.430 SECS PRESSURE- 0.0 N/P? HE4I). 0.1) METRES

7.100 MElEES PERIOD- 1i.2B SECS PRFSSIIRE 0.0 N/H? HEAD. 0.0 PURRS

Y-COURD I '.\t E-Y-COQ RD t NAT V-COORDINATE. V-COORDINATE. V-COORDINATE. Y-COOROIN AT E. YCOORDINAR -Y-COORDINATE. Y-CUOPDIN.TE-V-COORDINATE. V_COORDINATE. V-COOROINAÍE' V-COORDiNATE. Y-C DOPO INST E. V-COORDINATE-Y-CUOTDINAT F. Y-COOPDI NATE V-COORDINATE. V-COORDINATE. V-COORDINATE. Y-0007n INAT E V-COORDINATE. Y-C 00 RO IN! E YC OU RD ¡N T. V-COORDV-CCORO INATE-Y-COOAOINAT E V-COORDINATE. VCOORDINATE -V-COORDINATE. V-COORDINATE- V-COOROIMAIE- Y-COOSOINATE-V-COORDINATE. Y-CDCRD I NATE.

V-COORDINATE-5

lo

Fiq. 21

15

PRESSURE HEAD AS A FUNCTION OF TANK SIZE, CALCULJ\TED VALUES, NON-LINEAR MOTION

20

loo

He 12 11 10 9 8 7 6 5 4 3 2

i

PRESSURE AS A FUNCTION 0F TANK SIZE, CALCULATED VALUES, LINEAR MOTIONS

20

I

Fig. 22

28 26 24 22 20 18 16 14 12 lo 8 5

Vertical acceleration 0.3'i

lo

Fi9. 23

PRESSURE AS A FUNCTION OF AMPLITUDE 0F OSCILLATION,

CALCULATED VALUES, NCLINEAR MOTIONS

15 i im 20 20.0 Amplitude degrees

28

20

lo

rn

_{hits the roof}

_jj4

:

r

21.

Amplitude

degrees

Fig. 24

PRESSURE AS A FUNCTION OF A1!P1ITUDE OF OSCILLATION,

He ad

Vertical acceleration 0.3ci

22

20

AND LINEAR MOTIONS.

80

90

100

23.0

filling

70

lo

Fig. 25

Equivalent total head due to tank motion

Static head when liquid hits the roof

Head

e e i e

m

Vertical acceleration

O.2a

Static hea. for still level

e e i

27 24 21 18 15 12 9 6 3

Fig. 26

CALCULATED PRESSURE DISTRIBUTION ALONG A BuLKHEAD, LINEAR MOTIONS

V

4