Determination of the dynamic properties and propeller excited vibrations of a special ship stern arrangement

REPORT NO. 57 M

MARCH 1964

STUDIECENTRUM T.N.O. VOOR SCHEEPSBOUW EN NAVIGATIE

AFDELING MACHINEBOUW - DROOGBAK lA - AMSTERDAM

(NETHERLANDS' RESEARCH CENTRE T.N.O. FOR SHIPBUILDING AND NAVIGATION)

ENGINEERING DEPARTMENT - DROOGBAK IA - AMSTERDAM

DETERMINATION OF THE DYNAMIC PROPERTIES AND

PROPELLER EXCITED VIBRATIONS OF

A SPECIAL SHIP STERN ARRANGEMENT

(BEPALING VAN DE DYNAMISCHE EIGENSCHAPPEN EN DE DOOR DE SCHROEF

OPGEWEKTE TRILLINGEN VAN EEN NIET CONVENTIONEEL ACHTERSCHIP)

by

IR. R. WERELDSMA

Netherlands Ship Model Basin

Issued by the Council This report is not to be published unless verbatim and unabridged.

RESEARCH COMMITTEE

Ir. N. DIJKSHOORN

Ir.

W. H. VAN OORDT

Ir.

W. SPUYMAN

Ir. A. VAN DEN TOORN

CONTENTS

page

Summary 5

Introduction 5

Simplification of the stern and propulsion arrangement 5

Qualitative considerations about the dynamic behaviour of the

ath-wartships elastic supporting system of the propeller 7

Transfer functions obtained by the theoretical approach 8 Determination of the vibratory propeller forces and moments 10

Forced vibrations of the system and results 12

LIST OF SYMBOLS

XYZ co-ordinate system

propeller torque

T; T

propeller excited athwartships bending moments

F propeller thrust

F; F

propeller excited transverse forces

; ei,; e components of displacement

; ,; q components of rotation

sterntube stiffness in X-direction CBy sterntube stiffness in Y-direction

rEx; 5By components of bearing displacement Rx; R components of the bearing forces

propeller inertia about the x-axis

I,

propeller inertia about they-axis

D propeller shaft diameter

MB mass of the bearing support

mass of the propeller

ß angular rotation of the propeller amplitude of harmonic components

phase angle of harmonic components

E modulus of elasticity

SUMMARY

The principal approach to a theoretical analysis for the determination of the athwartships propeller

vibrations is

given. The results show that the fluctuating load of the propeller shaft, due to the

vibratory forces of the propeller, cannot be neglected with respect to the shaft load, resulting from the

average propeller thrust and torque.

1.

Introduction

Propeller operation behind a ship gives rise to

variable loads and vibratory motions of the

propel-ler-propeller shaft-thrust block system and

vibra-tions of the stern.

The relatively high frequency of the excitation

equals the blade frequency and multiples thereof.

Due to the variable rpm of the propeller, the

excitation frequency is variable and can

intro-duce inconvenient resonance phenomena resulting

in increased stresses in the structure.

This paper deals with the athwartships vibrations

of the propulsion system. It is assumed that the

axial phenomena of the propeller (i.e. thrust and

torque fluctuations, axial and torsional vibratory

motions) are not affecting the transverse

behav-iour. In that case only the propeller force

fluctua-tions operating in the transverse direction, exciting

the afterbody via the shaftbearings [1], and the

variable pressure field of the propeller, exciting

the afterbody via the hull [2], have to be taken

into account.

The investigation is carried out for a single screw

cargo liner with the following particulars:

The tests were carried out with a model to a scale

of 1: 27.

2.

Simplification of the stern and propulsion

arrangement

Fig. 1 gives an impression of the unconventional

stern arrangement of the investigated ship. Self

propulsion tests showed that this stern

arrange-ment was better from a propulsion point of view

than the conventional construction.

However, due to the increased elasticity of the

support of the stern tube bearing the possibility

excists that inconvenient critical athwartship

vi-brations could render the construction

unaccept-sterntube bearing

Fig. 1. The stern of the investigated ship model.

5

Length between perpendiculars

152,40 m

Breadth moulded

21,03 m

Draft

8,915 m

Propeller diameter

6,000 m

Number of propeller blades

4

Installed power

13,000 BHP

able. In order to study this problem a theoretical

analysis was made of the dynamic behaviour of the

stern and the propeller.

However, some simplifications and assumptions

had to be made in order to obtain a solution.

Although the motions and elasticity of the hull,

supporting the shaft and the propeller, affect the

overall behaviour, it is assumed that the hull and

the afterbody are not vibrating. For the

deter-mination of the effect of the elastic afterbody, the

knowledge of the exciting forces and the

mechan-ical impedance of the afterbody

is

necessary.

Research in this field has recently started but

results are not yet available and the mentioned

effect could not be considered.

Further it is assumed that the stern tube bearing,

elasticly supported by the tube, cannot transfer a

bending moment to the shaft nor the reverse. The

elasticity of the stern tube bearing support

is

different in the horizontal and the vertical

direc-tion, due to the fact that the stiffness of the hull

in the horizontal direction is less than that in the

vertical direction. The propeller shaft is supposed

Fz

propeller excited forces

propeller motions

to be built in between the second and the third

bearing of the propeller shaft. All these

simplifica-tions lead to the arrangement shown in fig. 2. For

this case the quantitative information is given in

table

1.

Table 1.

Quantitative information of the system under

consideration.

* see paragraph 4.

Fig. 2. The propulsion system and its simplified support with indication of the positive forces and motions.

Propeller mass M = 2530 kgsec2/iis Propeller moment olintertia

I =

= 1650 kgmsec2

Propeller shaft diameter D = 640 mm Distance a

Distance b

a = 1250 mm b = 2750 mm Stiffness of the stern tube

bearing support in

v-direction

GBZ = 4.8 x l0 kg/rn (C'BX= 0.35v l0 kg/m)*

Stiffness of the stern tube bearing support in

p-direction

CBY = 1.47 x l0 kg/rn

(C'By= 0.lOx 10 kg/m)*

Mass of the stern ttils

3.

Qualitative considerations about the

dynamic behaviour of the athwartships

elastic supporting system of the

propeller

The equation describing the motions in transverse

direction (q, q,

and

Ey)

of an elasticly

sup-ported disc with mass and moment of inertia are

given in ref. [3] and [4].

In fact the actual motion of the system is a

com-bination of two principal motions and

deforma-tions illustrated in fig. 3 and both operating in

two mutually perpendicular planes (horizontal

and vertical).

The four possible motions are mutually coupled

by the mechanical properties of the system.

Between e

and

a coupling excists due to the

fact that from the nature of the system the two

indicated defiections are combined (elastic

coup-ling). A force F or a moment T will introduce

deflections e and çcy simultaneously.

The rotation of the disc gives rise to a gyroscopic

effect, which introduces a mutual coupling of the

two deflections q and q, (precession moment).

For the non rotating case the behaviour in the

horizontal and vertical plane is identical. In each

plane two natural frequencies, performing

com-bined motions of fig. 3, are present. However,

when the system rotates, the gyroscopic effect

changes the vibratory behaviour and gives rise to

four different natural frequencies resulting in a

forward whirl and a counter whirl.

z

deflections = O and q > O

Fig. 3. Principal deformations of a simplified propeller shaft.

z

not deflected shaft

defiections ¿. > O and q O z

For the case that a propeller is supported in the

same way, we can distinguish in addition to the

mechanical coupling terms the so called

hydro-dynamical coupling terms, caused by the geometry

of the propeller operating in water [5].

A vibratory motion qx carried out by a stopped

right hand propeller will generate a vibratory

force F and a moment - T. A motion

L will

generate a moment - T and a force Fi (see

fig. 2). When the propeller is rotating the same

conclusions can be drawn. Due to the phase shift

between the motion and reaction force, additional

forces and motions in the y-direction exist.

Such forces will analogously be generated from

vibratory motions

and q,.

In this way we can distinguish a number of

coeffi-cients affecting the equations of motions.

An approximate value of these coefficients can be

obtained by a two-dimensional analysis [6] [7].*)

This analysis shows that the mentioned

coeffi-cients depend on the rpm of the propeller which

phenomenon must be considered if the speed of

the propeller is variable. Four simultaneous

dif-ferential equations are describing the problem.

Additional mechanical systems, for example the

mass MB and the elasticities CB and CB

affect

the mentioned mechanical coefficients and give

rise to more coupled equations.

The solution of the equations for the steady state

conditions with a constant rpm of the propeller

can be represented by means of transfer functions

giving the ratio of

output motions

input forces

and the phase relations between the input forces

and the output motions as a function of the rpm of

the propeller.

With the knowledge of the input forces, the

mo-tions of the individual parts of the system and the

stresses in the material can be obtained and

verified for acceptance.

In this way the stresses in the stern tube, the

reaction forces of the stern tube bearing and the

maximum stresses in the propeller shaft can be

predicted.

*) The analyses, given in the references, are only valid for

the axial direction. For the application in this case an

extension of the existing theory must be made for the trans-verse direction.

4. The transfer functions obtained by the

theoretical approach

From the schematic arrangement, as given in

fig. 2,

the

transfer functions

are

determined.

Every motion in the transverse direction is

in-fluenced by all athwartship components. This

means that for every considered motion four

trans-fer characteristics exist.

As we are interested in four propeller motions

(ex,

, e,

and q) and two bearing motions CBX

and e, twenty four transfer functions must be

determined.

For an arbitrary rpm the corresponding

hydro-dynamic propeller coefficients are approximately

analysed and the steady state solution in terms of

amplitude ratio and phase relation can be

ob-tained from the equations and represents one

point of the presented curves.

The transfer characteristic concerning the

har-monic excitation with blade frequency is indicated

with ml.

For the same rpm, however, higher harmonics,

6 4 3 2 8 4 3 2 2 3 4 5 102 rpm of the propeller

Fig. 4. Theoretically obtained transfer function of Eßx/Fx.

(non-critical case) E i0 00 s

t

o

equal to multiples of the blade frequency, are also

generated.

Additional curves for these excitation frequencies

are determined (indicated with m2, m3 and m4).

Some typical transfer functions are given in fig. 4

and 5. As can be concluded from fig. 4 a resonance

phenomenon at blade frequency occurs for about

114 rpm.

Higher harmonics of the excitation give rise to

corresponding critical values.

Fig. 5 shows that more natural frequencies are

present according to the above mentioned

be-haviour of shaft whirling.

Fortunately this additional critical phenomenon

at maximum propeller speed is only excited by the

relatively small higher components of the

propel-ler

excitation and does not introduce serious

dynamic loads.

Other critical frequencies, that must be present in

accordance with the description of paragraph 3,

are not in the range of operational propeller rpm.

From the character of the transfer functions as

given in fig. 4 and 5 can be seen that for the

nom-6 4 3 2 10.8 = o-E 3 2 10-e IT 102 rpm of the propeller

Fig. 5. Theoretically obtained transfer function of cp/T. (non_critical case) q', T, _m4 m3 ¡ + m2

T''

rol

_.. j

f\

I t

¡

I,

_-

₄ k k m4

1m3

IHIR'

\\

\\

m4\ \m3 r_. _Th m2 mi

L,

m4m3k

m2 1m1 + 2 4 5 102 4 5 102 00 . 120° 00 C 0) E 240° -C o--3 60° 0° 00 C E _240° o-360°

2 5 102

rpm of the propeller

Fig. 6. Theoretically obtained transfer function of EBX/Fx.

(critical case)

jable 2. Review of propeller excited forces and moments.

E .° 10.0 I o 6 4 3 2 10-e o 0_r o-E 4 2 1 .1O 3 4 5 101 rpm of the propeller

Fig. 7. Theoretically obtained transfer function of (critical case) 9 Es, F, m4 m3

Ii

\

I I1

I

\

_/'

/1

_/

_.t___ L -

\

ç,-, T, m4 m3 m2

!

R

i

ml

I

I

/

i!I

A

/

g

4

\

\

lm4.m34

m2 Torque (m.ton)

-

= 76.2±A5 sin(4nß+'n)

n =k

A1 = 7.58

A2 = 1.53

A:i = 0.37

A4 = 0.22

= ±114°

0112 = ± 105

= + 72°

= ± 12°

A1 = 10.94 V'i = + 1000 Thrust (ton)

_{A2 = 1.66}

V'I = + 950

F = 83.0±L40 sin (4nß+9')

A3 = 0.27

=

90 n 1 A4 = 0.17 ± 170°

A1 = 2.59

= +124°

Horizontal transverse force (ton)

_{A, = 0.15}

= + 122°

F = -3.86+Asin(4nß+11'n)

A3 = 0.21 13 = + 118°

A4= (LII

1114= +124°

A1 1.99 V'i = -104°

\rtica1 transverse force (ton)

_A.

-0.25

= - 65°

F = 4.026+A sin (4nß+ V'e)

A, = 0.33

V'a = - 90°

= i

A.1 0.07 _{= + 154}

A1 = 12.35

V'i = - 65°

Horizontal bending moment due to thrust eccentricity (m.ton)

_{A = 0 45}

₌

3

T

_{2l.46+Asifl(4nß+V'n)}

A: = 0188 V'a = -170°

n =

A4 = 0.87

= - 38°

A1= 4.97

- 62°

Vertical bending moment due to thrust eccentricity (m.ton)

_{A. = 0.87}

_{12 = - 96°}

= 6.34+EASifl(4flß+V'n)

A3 = 0.68

= +179

=

A4 = 0.43

ií' = + 149° 0° 2 102 0° -120° m2 ml -120° 00 00 C C) -240° -240° o--C o--360° -360° 3 4 102 ml E 1O bO 6 4 3 2 o 10.2 0_r o-E

04

3 2 i

o-mal speed of the propeller (114 rpm) the condition

is more or less critical for the blade frequency.

In order to get an idea about the effects of exact

critical operation at 114 rpm the point of support

of the stern tube in the hull is shifted ahead to

such an extent that

became maximum for 14

FI

rpm of the propeller.

This new condition with new elasticities of the

stern tube bearing support

(C'BZ

and

C'BY, see

table 1) gives rise to new transfer characteristics.

For comparison purposes some of the new functions

are indicated in fig. 6 and 7.

With the knowledge of the propeller excited

vibra-10 o

5

o 15 -10 5

-

lo

15

00 \

vertical bending moment due to the thrust eccentricity

900

vertical transverse forte

(Fx) -thrust variations (Fz) _torque variations (-Tz) o ₀₀

_/

9O \ fi

/

900

tory forces and the transfer functions the stern

tube bearing forces, the shaft bending moment and

the motions of the system can be calculated.

5. Determination of the vibratory propeller

forces and moments

The variable propeller forces are experimentally

determined described in ref. [1] and [8].

For this application the thrust eccentricity is

re-presented as a bending moment. The results of

the measurements are given in fig. 8. For

com-pleteness the thrust and torque variations

gene-rated by the propeller are also given.

The harmonic analyses are given in table 2.

fi

30 15

horizontal transverse force 200 100 o 30 15

30 15

0 0,

of average propeller torque

/0 _{propeller radius}

fi

1800 o

i 8O

00 _0°

Fig. 8. Experimentally obtained propeller excited vibratory forces.

+5

n

10°

+ 10

R=3rn

horizontal bending

moment due to the

thrust eccentricity 0 +5 +10 +15 °/ of average thrust x propeller radius X ₁₀ 90° 90° 10 5

11 e

...

_UUUUUU.

...uu..u..

_u

...

U.!

... ...

_{0210.2__}

u..

_uuu

+ 0.4 1 m

...0

!fl

uuuuruuuu - __

_{.1il!ì..c...u.}

muiuuu

achwartship

iuuuuruuuiiuuu

...

Q.410

propeller rpm non critical Fig. 9. Resulting vibrator-displacement m =114 critical condition condition of the propeller. U0.2 .102

0 nUUu

e

U

u

u t .u....u...u...

propeller rpm non - critical = 114 condition athwartship propeller.

uuuui

u.u....

--uiiiuuu

uuuutiiuuu

450

uuutiu

UU1!1U

critical condition Fig. to. Resulting vibratory rotation of the

u

...

UUUUI1UUU

_ituuuu

...

uuuuuauu

uuuuuruuu

uuuuiuuuuu

0.2

uuuuuuiu

0 + 0.2 1O 0,4 0.4 102m

rad.RU

UUU

±0.3103m

I

j,

0

\

/

'

s'

,

/

.-

/

/

-.

-_0 3 10 m 450 ₉₀₀ 90° s' propeller rpm = 114 non - critical condition critical condition

Fig. ti.

Resulting athwartship motion of the stern tube.

,

-/

/

45°( s' I

s-t'-

_s' t

i

s'

0.3

0 +O.3lOam

...u...u

24ton

UUUI2

4t

-R

UUUWAUUUuU 1Y

_!AM

uuiuuiuuu iiiuu

uuuu ° uurnuuuuuuuu

propeller rpm = 114 non - critical condition critical condition

Fig. ta.

Reaction force on the stern propellerload. to the total

.uu..uuu...

.

UUUUUU

_'u.

R y

uuuuuui

uuuuuuiiiuu

uuuaimu

uuuuuiirnuu

uuuuurnuu

_UUUULU

uuuuuiiiuu

6. Forced vibrations of the system and

results

With the assumption that the principle of

super-position is valid, the sum of the individual effects

of each propeller induced harmonic fluctuating

force or moment gives the overall effect, resulting

in vibratory displacements and forces.

In figs. 9 and 10 the athwartship displacements

and rotations of the propeller due to the vibratory

excitation is given. The non-critical and the

crit-ical case are indicated with full and dotted lines

respectively.

The total transverse displacements of the stern

tube bearing is given in fig. 11.

In addition to excitations by the variable forces

and moments the system is

excited by static

athwartship forces and moments i.e. the average

transverse force and the moment due the average

thrust eccentricity.

For the maximum loads in the system these

com-ponents have to be taken into account.

Fig. 12 gives the reaction forces on the stern tube

bearing resulting from the dynamic and the static

part of the transverse force and the thrust

eccen-tricity. The propeller weight however is exicuded.

Due to the eccentricity of the thrust the bending

moment in the shaft becomes maximum just in

front of the propeller. The pattern of this moment

is given in fig. 13.

The average bearing forces as exerted by the

shaft on the tube are orientated upwards (see fig.

12). The propeller weight is more or less reducing

this force by its opposite direction.

The maximum value of the bending moment due

to the propeller weight occurs on another shaft

location than that introduced by the thrust

eccen-tricity and cannot be compared.

The bending moment in the shaft due to the static

propeller forces is negligible as compared with

the variable bending moment.

For the determination of the shaft diameter, the

load due to the bending moment has to be taken

into account in addition to the normal thrust and

torque loads.

12

7.

Conclusions and recommendations

It can be concluded from the above described

investigation that the instationary hydrodynamic

load of the propeller due to the ships wake field

gives rise to dynamic shaft loads that are

com-parable with the loads duc to the propeller thrust

and torque and cannot be neglected in special

ranges of propeller speeds.

As this analysis is based on various assumptions

it

is recommended to complete this type of

in-vestigation with measurements on the full scale

ship. A comparison of the predicted and measured

phenomena gives an impression about the

ad-missibility of these assumptions.

7; + lOtm

----s

,

.. / ..

j'

0

/

I _'S 110

'

.5- -

/

-/05

450 900 1

-l2Otm

propeller non rpm = 114 - crìtical condition condition bending moment in

shaft due to the

load. 9Q0 1 .5-. I - -- critical Fig. 13. Macimum the propeller total propeller I J

.,

450

I

_.., 5

T

-\

S-.--120

0 +8Otm

List of References

I. R. WERELDSMA, Experimental determination of thrust eccentricity and transverse forces, generated by a

screw propeller. mt. Shipbuilding Progr. Vol 9,

1962.

P. C. PIEN and N. L. FICKEN, The measurement of prop-eller induced vibratory forces on scale ship models.

D.T.M.B. Report July 1959.

R. N. ARNOLD and L. MAUNDER. Gyrodynamics.

Aca-demic Press New York.

N. H. JASPER, A theoretical approach to the problem of

critical whirling speeds of shaftdisk systems. D.T.M.B. Report 827; Dec. 1954.

R. WERELDSMA, Some investigations into the dynamic

behaviour of a ship propeller (to be published). H. SCHWANECKE, Gedanken über die hydrodynamischen

Kraftwirkungen an Schwingenden Schiffspropel-1cm.

Mitteilung Nr. 240 der Versuchsanstalt für Wasserbau und Schiffbau, Berlin. Schiffstechnik Band 7 (1960) Heft 38, p.p. 170-176.

P. D. RITGER and J. P. BRESLIN, A theory for the

quasi-steady and unquasi-steady thrust and torque of a

propeller in a ship wake. Stevens Institute of

Tech-nology N.J. Experimental towing tank. Report

No. 686, July 1958.

J. D. VAN MANEN and R. WERELDSMA, Propeller excited vibratory forces in the shaft of a single screw tanker. Tnt. Shipbuildg. Progr. Vol. 7, No. 73,

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No. 37 M Propel 1er excited vibratory forces in the shaft of a single screw tanker.

By dr ir J. D. van Manen and ir R. Wereldsrna. June 1960. No. 38 S Beamknees and other bracketed connections.

By prof. ir H. E. Jaeger and ir J. .7. W. Nibbering. January 1961.

No. 39 M Crankshaft coupled free torsional-axial vibrations of a ship's propulsion system. By ir D. van Don and N. J. Visser. September 1963.

No. 40 S On the longitudinal reduction factor for the added mass of vibrating ships with rectangular cross-section.

By ir W. P. A. Jansen and dr J. A. Sparenberg. April 1961.

No. 41 S Stresses in flat propeller blade models determined by the moiré-method.

By ir F. K. Ligienberg. June 1962.

No. 42 S Application of modern digital computers in naval-architecture.

By ir H. J. Zunderdorp. June 1962.

No. 43 C Raft trials and ships' trials with some underwater paint systems.

By drs P. de Wolf and A. M. van Landen. July 1962.

No. 44 S Some acoustical properties of ships with respect to noise-control. Part I. By ir J. H. Janssen. August 1962.

No. 45 5 Some acoustical properties of ships with respect to noise-control. Part II. By ir J. H. Janssen. August 1962.

No. 46 C An investigation into the influence of the method of application on the behaviour of anti-corrosive paint systems in seawater.

By A. M. van Landen. August 1962.

No. 47 C Results of an inquiry into the condition of ships' hulls in relation to fouling and corrosion.

By ir H. C. Ekama, A. M. van Landen and dis. P. de Wolf. December 1962.

No. 48 C Investigations into the use of the wheel-abrator for removing rust and miliscale from shipbuilding steel (Dutch)

Interim report.

By ir J. Remmelts and L. D. B. van den Burg. December 1962.

No. 49 S Distribution of damping and added mass along the length of a shipmodel.

By prof. ir J. Gernitona and W. Beukelrnan. March 1963.

No. 50 S The influence of a bulbous bow on the motions and the propu!sion in longitudinal waves.

By prof. ir J. Gerritsrna and W. Beukelrnin. April 1963.

No. 51 M Streas measurements on a propeller blade of a 42.000 ton tanker on full scale.

By ir H. R. Weretdsnja. January 1964.

No. 52 C _{Cimparative investigations on the surface preparation of shipbuilding steel by using wheel-abrators and the} application of shop-coats.

By ir H. C. Ekama, A. M. van Londen and irJ. Reininelts. July 1963.

No. 53 S The braking of large vessels.

By prof ir H. E. Jaeger. August 1963.

No. 54 C _{A study of ship bottom paints in particular pertaining to the behaviour and action of anti-fouling paints.}

By A. M. van Londen. September 1963.

No. 55 S Fatigue of ship structures.

By ir J. J. W. Nibbering. September 1963.

No. 56 C The possibilities of exposure of anti-fouling paints in Curaçao, Dutch Lesser Antilles.

By drs P. de Wolf and Mrs M. Meuter-Schriel. November 1963.

No. 57 M _{Determination of the dynamic properties and propeller excited vibrations of a special ship sternarrangement.}

By ir R. Wereldcnna. Maart 1964.

No. 58 S Numerical calculation of vertical hull vibrations of ships by discretizing the vibration system. By J. de Vries. April 1964.

C'ommunications

No. 1 M Report on the use of heavy fuel oil in the tanker "Auricula" of the Anglo-Saxon Petroleum Company (Dutch).

August 1950.

No. 2 S

Ship speeds over the measured mile (Dutch).

By ir W. H. C. E. Rösingh. February 1951.

No. 3 S

On voyage logs of sea-going ships and their analysis (Dutch).

By prof ir J. W. Bonebakker and ir J. Cerrilsrna. November 1952.

No. 4 S

Analysis of model experiments, trial and service performance data ofa single-screw tanker.

By prof in J. W. Bonebakker. October 1954.

No. 5 S _{Determination of the dimensions of panels subjected to water pressure only or to a combination of water pressure}

and edge compression (Dutch).

By prof ir H. E. Jaeger. November 1954.

No. 6 S

Approximative calculation of the effect of free surfaces on transverse stability (Dutch). By ir L. P. Herfst. April 1956.

No. 7 S On the calculation ofstresses in a stayed mast.

By ir B. Burghgraef August 1956.

No. 8 S _{Simply supported rectangular plates subjected to the combined action of a uniformly distributed lateral load and} compressive forces in the middle plane.

By ir B. Burghgraef February 1958.

No. 9 C

Review of the investigations into the prevention of corrosion and lòuling of ships' hulls (Dutch). By ir H. C. Ekama. October 1962.

No. 10 S/M Condensed report of a design study for a 53,000 dwt-class nuclear powered tanker.

By the Dutch International Team (D.I.T.) directed by ir A. M. Fabery de Jonge. October 1963. M = engineering department

S = shipbuilding department