Finite element analysis of a third generation containership

REPORT No. 182 S

(S C/269, a, b, c)

Faculty WbMT

Dept. of Marine Techfloicgy

Mekel weg 2, 2623

CD Deft

The Netherlands

NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

NETHERLANDS SHIP RESEARCH CENTRE TNO

SHIPBUILDING DEPARTMENT

LEEGHWATERSTRAAT 5, DELFI'

*

FINITE ELEMENT ANALYSIS OF

A THIRD GENERATION CONTAINERSHIP

(DE EINDIGE ELEMENTEN METHODE

TOEGEPAST OP EEN 3e GENERATIE CONTAINERSCHIP)

by

IR. A. W. VAN BEEK

IRO

December 1973

NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

NETHERLANDS SHIP RESEARCH CENTRE TNO

SHIPBU ILDING DEPARTMENT LEEGHWATERSTRAAT 5. DELFT

*

FINITE ELEMENT ANALYSIS OF

A THIRD GENERATION CONTAINERSHIP

(DE EINDIGE ELEMENTEN METHODE

TOEGEPAST OP EEN 3e GENERATIE CONTAINERSCHIP)

by

IR. A. W. VAN BEEK

(Instituut TNO voor Werktuigkundige Constructies)

IFLO

Onder auspicien van het Nederlands Scheepsstudiecentrum TNO is in 1967 een onderzoek begonnen, gericht op de toepassings-mogelijkheden voor de scheepsbouw vari de eindige elementen methode voor konstruktie-berekeningsdoeleinden. In eerste in-stantie gericht op de ,,Offshore" industrie, kwam later bij de introduktie van onkonventionele scheepsontwerpen met grote dekopeningen en zijluiken, de nadruk op meer scheepsachtige konstrukties te liggen.

Aanvankelijk vond toepassing van de eindige elementen methode plaats op enige gestileerde scheepskonstrukties, zo mogelijk begeleid door een experimentele verifikatie. Een deel van dit onderzoek is gerapporteerd in mededeling 27 S.

Omdat de omvang van een specifiek probleem een zeer belang-rijke rol speelt bij toepassing van numerieke methoden. werd eind 1971 besloten het spanningsverloop van een complete scheepsconstructie uit te rekenen. .Dat de keuze hierbij viel op het derde generatie kontainerschip van de Koninklijke Nedlloyd kwam, behalve doordat in dit schip enige bijzondere konstruk-tieve kenmerken voorkomen vooral, omdat van dit schip reeds veel bijzonderheden bekend waren.

Het betreffende type, een kontainerschip, dat binnen het Scandutch konsortium een lijndienst onderhoudt tussen Europa en het Verre Oosten, kan gekarakteriseerd worden als een open dubbelwandige scheepskonstruktie voorzien van tweelingluiken met de daarbij behorende langsscheepse dekdrager op hart schip. Deze konstruktie verdient in het bijzonder om zijn torsiestijf-heid zeer vecI aandacht.

De tot nu toe gebruikte welvingstheorie voor dunwandige open baikkonstrukties schiet tekort, als analysemethode van dit scheepssterkteprobleem bij deze gekompliceerde 3-dimensionale

plaat-balk konstrukties. J

0m deze reden zijn ter ondersteuning van een theoretische benadering bij enige buitenlandse onderzoekinstellingen kom-plete tamelijk grote rompkonstruktiemodel!en in staal of van kunststof gebouwd en uitvoerig beproefd.

Behalve dat experimentele methoden kostbaar zijn en zeer veci tijd veresen, worden de resultaten beïnvloed door grote onzeker-heden als: schaalfaktoren, materiaaleigenschappen, maaton-nauwkeurigheden, vereenvoudigde konstrukties en problemen in relatie met het in rekening brengen van welvingsverhindering ter plaatse vari dekdwarsbalken, scheepseinden en machinekamer-gedeelte.

Aan de toepassing van de cern. kleven een aantal van ge-noemde nadelen niet.

De wijze waarop de scheepskonstruktie gemoduleerd wordt tot een mathematisch rekenmodel opgebouwd uit een beperkt aantal deJen van eindige afmetingen met bekend zijnde eigen-schappen, hangt grotendeels af van de konstruktiewijze en is mede bepalend voor zowel de analysekosten als de tijdsduur. Deze laatste faktor echter wordt ook voor een aanmerkelijk dccl bepaald door de effektieve inzet aan mankracht en het ter be-schikking hebben van de relevante informatie.

Het betrokken projekt werd uitgevoerd door een team van het Instituut TNO voor Werktuigkundige Construkties, bijgestaan door medewerkers van het Nederlands Scheepsstudiecentrum TNO.

An investigation was conducted in i 967 sponsored by the Nether-lands Ship Research Centre to evaluate the applicability of the finite element method for structure calculation purposes in ship-building. Although it was focussed in the first instance to the offshore industry, with the introduction of unconventional ship designs with large deck-openings and side hatches, the emphasis shifted to more shiplike structures. Initially the finite element method was applied to some simplified structures, checked by experimental tests where feasible. Results are partly reported in our Communication no. 27 S. Because the extent of a specific problem figures prominently in numerical methods,

it was

decided, at the end of 1971 to compute the stress behaviour of a complete hull structure. The third generation containership of the 'Royal Nedlloydlines By", was selected for reasons of both some special structural features and its merits as a medium of several research.

The subject type, a containership, which operates within the ScanDutch consortium between Europe and the Far East, may be characterized as an open double shell ship structure with twin hatches, provided with the matching longitudinal centre girder. This structure has to be investigated thoroughly, particularly with regard to its torsional stiffness. To this very point, the warping theory for thin-walled open prismatic beam structures fails to some extent for analysis of these complicated 3-dimen-sional plate beam assemblies.

For reason of support the theoretical approach, at sorne foreign research laboratories complete hull structure models at a rather large model-scale in steel or synthetic material were built and extensively tested experimentally. Apart from the fact that experimental methods are costly and time-consuming, the results arc

liable to great

uncertainties concerning, scaling errors, material properties, measure deviations, simplified structures taking into account problems

of modelling the physical

warping restraint at deck transverse beams, at ship ends and at machinery room hullstntctures.

The finite element method is thought to be able to eliminate a number of these drawbacks in strength-analysis.

The art of modelling, in where the physical structure is replaced by a mathematical model of discrete elements with known prop-erties, depends largely as a logical consequence of the structure and determines both costs and time duration of the analysis too. The latter factor however, to a substantial part is also deter-mined by an effective organization of manpower and the availabil-ity of the relevant data. The operations were executed by a team of the Institute TNO for Mechanical Constructions, assisted by the Netherlands Ship Research Centre TNO.

The analysis was executed by means of the general software package ASKA (Automatic System for Kinematic Analysis), a large general-purpose digital computer programme using the finite displacement method.

Most of the structural loads were derived from an extensive investigation on seakeeping with a four-segmented model tested in regular waves as reported in our Report S 173.

Both the still water load and much information were kindly made available by the shipowner.

This study yields a thorough insight into the static elastic strain

singsmethode toepast.

Een dee! der toegepaste belastingsgevallen is verkregen uit re-sultaten van een uitvoerig zeegangsonderzoek met een 4-delig schaalmodel, in regelmatige golven, gerapporteerd in rapport 173 S.

Voorts werd de stilwaterbelasting en verder benodigde infor-matie door de rederij ter beschikkin.g gesteld.

De studie levert een grondig inzicht op in het statisch elastisch rek- en spanningsgedrag, vooral wat betreft de welvingsspan-ningen t.g.v. de kombinatie horizontaal buigend moment en - torsiebelasting. Daarbij werd tevens zeer veel ervaring opgedaan

in de wijze waarop het rekenmodel bepaald wordt van gekompli-ceerde 3-dimensionale konstrukties met een minimum aan korn-puterinvoer en met behoud van de nodige nauwkeurigheid.

Daarbij vormt de berekende stijtheidsmatrix een stuk basis gereedschap voor een dynamische analyse van de romp inklusief dekhuis, lading en meetrillende watermassa. De resultaten van de , overall" spanningsberekeningen dienden bovendien als in-voergegevens voor een gedetailleerde berekening van enige speci-fieke, hoog belaste scheepsdelen.

De thans in voorbereiding zijnde metingen aan het werkelijke schip hebben onder meer tot doel de gemaakte veronderstellingen te verifiëren. Daarbij zullen de statische rekmetingen, de aan-namen die gemaakt werden moeten bevestigen. Dynamische metingen zullen worden verricht om overdrachtsfunkties te kun-nen verifiëren met resultaten beschikbaar uit berekeningen en eerder genoemde modelproeven, zowel voor wat betreft bewe-gingen als belastingen.

HET NEDERLANDS SCHEEPSSTUDIECENTRUM TNO

complex 3-dimensional structures, with minimum input and without substantially sacrificing accuracy.

Besides, the calculated stiffness matrix forms the basic tool for dynamic analysis of the ship hull inclusive deckhouse, cargo and added water-mass. Some specific highly stressed sub-structures will be analysed in detail.

At present, preparations are made for an extensive full scale research incorporating static strain measurements in order to verify the critical assumptions and basic theories of the analysis. Dynamic strain measurements will be executed to verify

transferfunctions with results of calculations and the model experiments mentioned above both for motions and loads.

CONTENTS

page

Summary

7

i

Introduction

7

2

Formulation of the problem

7

2.1

Modelling of shipstructures

7

2.1.1 Description of the model

7

2.2

Loading conditions

Il

3

Results of the analysis

15

3.1

Stress distribution for vertical bending

15

3.1.1 Comparison of finite element results with values obtained with

beam theory

15

3.2

Deformations and stresses for torsional loading and horizontal

bending

17

3.2.1 Distribution of torsional angle

17

3.2.2 Distribution of centre of twist

17

3.2.3 Distortion of the hatch openings

19

3.2.4 Normal stress distribution for torsional loading and horizontal

bending

19

3.2.5 Shear stresses in upperdeck

20

3.3

Comparison of stresses and deformations with corresponding

Germanischer Lloyd results for the "Liverpool Bay" Class.

20

4 Conclusions 21

FINITE ELEMENT ANALYSIS OF A THIRD GENERATION CONTAINERSHIP

by

Ir. A. W. VAN BEEK

Summary

An extensive finite element analysis of a third-generation containership, subjected to bending and torsion is presented. Stresses and deformations are derived for five different loading cases, partly based on experimental results from a scale model in several simulated sea-conditions.

The stress distribution and the amount of warping restraint due to torsional loading will show why torsion-bending theory of thin-walled beams cannot provide accurate results.

i

Introduction

In the calculation of stiffness and strength of

container-ships with wide hatch openings, the beam theory for

bending and the general warping theory for torsion of

thin-walled beams is usually applied. In most cases

reasonable overall results are obtained, when bending

of the ship is concerned. It is, however, experienced

that warping theory for thin-walled beams does not

always provide accurate torsional stresses and

deforma-tions, which can be attributed to the considerable

simplification required to replace the ship hull structure

to a beam with equivalent stiffness and strength.

With the development of very large containerships,

having extremely wide hatch openings over almost the

full length of the ship, the need for an accurate stress

analysis is evident. In the design stage of the

third-generation containerships no service experience for

such large containerships existed and, therefore, at

several research institutes model experiments were

carried out in order to derive stiffness and strength

properties as far as possible [3], [4]. The main purpose

of this analysis is to demonstrate the possibilities of

large-scale finite element calculations of such

com-plicated ship structures. One of the major advantages

of the finite element method over an experimental

analysis is that changes in the design can easily and

quickly be analysed, while this is rather timeconsuming

when an experimental method is used.

This report presents the stress analysis for a Far

East containership being built for the "Koninklijke

Nediloyd N.y."

Figure

1

shows the general arrangement of the

Nedlloyd Far East containership.

The main characteristics of the containership are.

length between perpendiculars

273.00 m

breadth

32.24 m

draught even keel

10.85 m

displacement volume

56.097 m3

depth

25.00 m

Stresses and deformations are calculated for five

dif-ferent loading conditions, which are partly based on

the results of model experiments carried out in the

tank of Netherlands Ship Model Basin (NSMB) at

Wageningen [6].

2

Formulation of the problem

This chapter describes the modelling of the

ship-structures and the loading conditions analyzed.

2.1

Modelling of shipstructures

The finite element method requires that the

ship-structure (Fig. 2) be replaced by a model made up of

discrete structural elements of known elastic and

geo-metric properties. The model has to simulate the elastic

behaviour of the shipstructure sufficiently accurately.

The symmetry of the ship with respect to its

centre-plane justifies an analysis that is restricted to the half

structure.

The effect of one half of the vessel on the other half,

is simulated by suppressing certain freedoms at the

nodes of the plane of symmetry.

2.1.1

Description of the model

The description of the model consists in locating the

nodal points, prescribing the degrees of freedom at

each node as well as defining the elastic elements. The

geometry of the discrete finite element model is

de-scribed by connecting elements to the nodal points.

Therefore, selection of the location of the nodal points

is so arranged that all primary structural members of

the vessel can be represented.

The more nodes that can be used, the more details

that can be included in the model and, therefore, the

greater the accuracy of the analysis.

For the description of the element model 44

trans-verse sections are selected, so that all transtrans-verse

bulk-heads,

stiff webframes and discontinuities of the

structure can be represented.

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22

Frames that cannot be represented in the element

model are lumped to the nearest transverse section.

In a transverse cross-section the nodes are selected

at the intersections of decks, stringers and double

bottom with longitudinal bulkheads and the side shell

of the hull form.

The main characteristics of the element model are as

follows

1621

nodal points

247

element groups

3627

elements

4820

degrees of freedom for symmetric case

4598

degrees of freedom for anti-symmetric case.

Figures 3 through 6 (Appendix) show the element

dis-tribution of the longitudinal members of the model; it

can be concluded that a regular element mesh is

obtain-ed. All members which are supposed to resist against

torsional loading or bending of the hull are taken into

account for the analysis of the model. A number of

stiffeners are lumped at the nearest grid lines as bar

elements.

The transverse boxgirders are accounted for by

beam elements, extension, bending, torsion and shear

effects in bending being taken into account. The profiles

mounted on the transverse bulkheads are represented

by eccentric beam elements, extension and bending

normal to the plane of the bulkhead only being taken

into account. All plate elements are basic plane stress

elements with cornern odes only.

Figure 7 shows the element distribution in a

trans-verse bulkhead.

The beam elements at nodal points 870, 875, 879,

894, 909 represent the transverse boxgirder,

eccentric-ally attached to the bdkhead. Beam element between

nodes 893 and 894 couples the torsional angle of the

transverse boxgirder with the displacements of the

longitudinal bulkhead. Its stiffness against rotation

has to be large compared with the torsional stiffness

of the boxgirder.

The remaining beam elements on Figure 7 represent

the eccentric profiles mounted on the bulkhead.

Other typical transverse sections are given in Figures

8a and 9a. The cross-sections of the element model are

given in Figure 10. lt can be seen that the geometry

of the hulisurface is represented quite accurately.

2.2

Loading conditions

The five loading conditions analyzed are shown in

Figures 11 and 12. Loading case i represents the still

water bending moment distribution caused by the

ship's light weight, dead weight and the buoyance load.

The maximum bending moment appeared to be a

hogging condition: M = 2.665 x l0 Nm.

Frame 165

Fig. 7. Transverse bulkhead.

The remaining loading cases are based on the results

of model experiments carried out in the tank of

NSMB at Wageningen [6]. A scale model of the

Nediloyd-containership was divided

into 4

parts

connected to each other in such a way that at three

sections the bending moments, shear forces and the

torsional moment could be measured for various

regular waves. With shear force and bending moment

at three stations, and knowing that both shear force

and bending moment vanish at fore and aft end of the

ship, ten parameters for the moment distribution over

the ship's length are obtained and a good

approxima-tion for the distribuapproxima-tion can be made.

Because the analysis is restricted to the half ship

structure, the loading has to be divided into symmetric

and anti-symmetric cases with respect to the centre

plane of the ship.

Assuming the moment to vary linearly between two

selected frames of the model, the resultant shear forces

are calculated. The vertical bending moment

distribu-tions, being the symmetric loading cases, are introduced

by prescribing the equivalent vertical loads to the nodal

points lying on the intersection of longitudinal

bulk-head 8361 mm off centre line with the bottom shell.

8 ¡1 869 874 892 873

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12 ;7o Frame lB!. 941 91.3 944 91.5 946 958 953 952 951 950 949 948 947 960 957 971 972 973 874 Fig. 8a. 954 959 975 976 981. 983 982 981 980 979 978 977

Figure 11 shows loading cases I, II and III,

represent-ing the stiliwater bendrepresent-ing moment distribution, the

vertical wave bending moment distribution with p =

1800 and the vertical wave bending moment

distribu-tion with p = 45° at service speed. The angle p denotes

the angle between the ship and the direction of the

wave with wave-length of 243 m.

Loading case IV represents the combination of

horizontal wave bending and torsional loading for the

ship sailing with p

450 in regular waves and loading

case V represents the pure torsional moment

distri-bution.

Figure 12 shows both the horizontal wave bending

moment and the torsional moment distribution.

The horizontal bending moment distribution

is

introduced by prescribing the equivalent horizontal

loads on the nodes lying on the base line of the ship.

Because these loads do not pass through the shear

centre of the cross-sections, a contribution to the

SECTION AT HOLD'S LENGTH

SECTION I

Fig. 8b.

torsion of the ship is given.

The corresponding torsional moment is defined with

respect to the base line as well and is introduced by

vertical loads on the nodes of longitudinal bulkhead

13765 mm off centre line and in the forebody on the

nodes lying on the intersection of side shell with

upper-deck.

The ship in floating condition is considered as a free

body, but for the analysis the rigid bodymotion must

be eliminated. For the symmetric loading cases I, II

and III, the rigid body degrees of freedom are

elimi-nated by suppressing the vertical displacements of

frames 51 and 318 at the base line and one

displace-ment of deck 4 in longitudinal direction.

For the anti-symmetric loading cases IV and V the

rigid body degrees of freedom are eliminated by

sup-pressing the horizontal displacement of frame 51 at

the baseline and at upperdeck level, and the horizontal

displacement of frame 318 at the base line.

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14 3.0 2.5 Moment [109Nm ₂₀ 1.5 1,0 0.5 O -0 5 -1 0 -1 5

Fig. IO. Cross-sections element model of Nediloyd containership.

P

Fig. Il. Still water bending moment and wave bending moment distribution.

f \

''

ç

Still water b?nding moment '

Dv

Vertical wave bendinc moment 18«

moment

Verticol wave hendinç

3

Results of the analysis

This chapter describes stress patterns and deformations

calculated with the finite element method for the five

different loading cases.

3.1

Stress distribution for vertical bending

Figure 13 shows the stress distribution for the

sym-metric loading cases. The highest stress values occur

in the upperdeck as was to be expected. It can be seen

that the distribution of the stresses in the upperdeck

is very similar for the three loading cases. The stress

distribution for these loading cases depends mainly on

the section modulus and the bending moment applied.

The influence of the decks in the engine room on the

stress distribution is clearly shown in sections 5 and 6.

Figure 14 shows the longitudinal stress distribution

over the half cross-section at frame 130/131 for

still-water bending. It is seen here that bending of the

double shell structure mainly occurs near the

connec-tion with the upperdeck of the engine room.

Table 1 and 2 compare the stresses in the centre-line

girder with the stresses in the deck plating for

still-water bending and vertical wave bending respectively.

The centre-line girder, coamings are neglected,

appear-ed to be quite effective for vertical bending of the hull,

which is attributed to the fact that the bending moment

varies slowly in longitudinal direction. Because the

normal forces in the centre-line girder can only be

introduced by bending of the transverse boxgirders

and the columns in the centre plane, it is to be expected

that for short waves the centre-line girder will be less

effective.

Just before and behind the engine room stresses in

the centre-line girder increase due to local bending.

This local bending effect will be inaccurate because

in the model the centre-line girder is eccentrically

connected with the nodes of the upperdeck only.

3.1.1

Comparison of finite element results

with values obtained with beam

theory

The stresses in the main longitudinal bulkhead show

that the deviations from a linear stress distribution are

Fig. 12. Horizontal wave bending and torsional moment distribution.

-

-Torsional niornent 4(45 DH _MT Horizontal bending

,4

45 moment 130 150 2 0

-

FI 0.30 0.25 I'lomnt 1109Nm) ₀₂₀

t

_0.15 0.10 0.05 O -0.05 -0.10 -0.15

16

Table 2. Stresses in centre-line girder and deck plating for loading case II, vertical wave bending.

Table 3.

Comparison of beam theory with

finite element stresses in centre-line stresses in deck 1

UPPER DECK

LOADING CASE I -STILL WATER BENDINO

LOADING CASED; VERTICAL WAVE BENDING%(.1BO

LOADING CASE - - VERTICAL WAVE BENDINS.j 45 MAIN LONGITUDINAL BULKHEAD

12

small except near large discontinuities of the structure.

[t is therefore to be expected that the beam formula

will provide quite accurate results for this type of

containership, when bending is concerned. For

cross-section at frame 189 with neutral axis 11.24 m above

base-level, moment of inertia 398 m4, and bending

moment 2.27 x lO Nm, the stresses according to the

beam formula are compared with the F.E.M. results.

Fr0rn 130-121

Fig. 14. Stress distribution for still water bending moment.

3 53.5 40.9 23.5 23.5 5 60.3 97.9 95.2 64.2 6 90.2 135.8 107.3 79.8 8 85.5 84.2 86.6 89.1 10 74.0 72.6 80.6 81.6 12 60.4 55.1 56.5 54.8 14 37.1 32.1 29.4 30.8 16 28.9 35.5 19.1 17.3 position in figure 13 girder o [N/mm2] [N/mm] centre of gravity upperdeck level 13765 mm off C.L. at side shell 3 12.8 10.0 9.7 10.6 5 14.7 20.0 22.1 13.8 6 36.7 48.2 36.0 27.7 8 36.2 39.6 42.2 43.1 lO 36.2 33.4 46.7 47.4 12 30.0 23.8 28.5 27.7 14 13.6 8.8 6.7 6.9 16 7.6 9.1 2.4 2.2 method frame 189 stresses [N/mm2] duff.

F.E.M. beam formula

deck i bottom 83.5

-65.5

78.5 -64.1 6 T Table 1. Stresses in centre-line girder and deck plating for

loading case 1, still water bending

position in figure 13

centre line girder [N/mm0J stress in deck I o [N/mm2] centre of upperdeck gravity level 13765 mm off C.L. at side shell 'O 12 14 16

0 80 NORMAL STRESS DISTRIBUTION IN UPPEROECK ANO LONGITUDINAL BULKHEAD FOR STILL WATER

STRESS SCALE CN Vmm2 BENDING AND VERTICAL WAVE BENDINO

Fig. 13. Normal stress distribution in upperdeck and longitudinal bulkhead for still water bending and vertical wave bending.

The results given in table 3 justify the conclusion

that beam theory will give satisfactory overall results

when bending of the ship is considered.

3.2

Deformations and stresses for torsional loading

and horizontal bending

This chapter describes the distribution of torsional

angle, centre of twist, hatch diagonal deformation

and the stress distribution.

Because the differences between loading case IV,

the combination of torsional loading and horizontal

bending and the pure torsional loading case V are

small, the discussion is mainly restricted to the results

from pure torsional loading.

3.2.1

Distribution of torsional angle

For the pure torsional loading (loading case V), the

torsional angle curve is derived. Neglecting

deforma-tion of the cross-secdeforma-tion in its plane, the torsional

angle, '/, is expressed as:

u[i]

z[i] o Torstonot Ongte (103rad] -5 5 7 -8 -g -10

Fig. 15. Distribution of torsional angle. where

rotation of cross-section about longitudinal axis

u[i]: vertical displacement of node i

z{i]

: horizontal co-ordinate of node i.

For the calculation of

t,

the displacements of the nodes

on the main longitudinal bulkhead are used.

Figure 15 shows the torsional angle as a function of

the length coordinate of the ship. It may be seen that

the engine room has a considerable influence on the

distribution of the torsional angle.

3.2.2

Distribution of centre of twist

Several authors have paid attention to the distribution

of the centre of twist, being the collection of points

that do not translate when the ship is loaded in torsion.

En [1] and [3], theoretical values were compared

with experimentally obtained results; differences of

I 00%Ç and more were found.

It is felt that the importance of the centre of twist

is overestimated in [I] and [3], since with modern

numerical methods such as the finite element method

one need not know the centre of twist or the shear

centre.

AP 50 100 50 200 250 ER

18

Moreover, the centre of twist of a floating body is

not fully determined since a rigid body motion will

influence the centre of twist. The assumption (made by

many authors [I], [3}, [5), that the centre of twist must

pass through the shear centre, only holds for a prismatic

beam when its rigid body modes are suppressed.

With the same assumptions as used for the

deriva-tion of the torsional angle, the centre of twist, Yt, is

expressed as:

-

i/J(xxo)u_b

yt

where

Yt

= distance from base line to centre of twist of a

cross-section

= torsional angle of a cross-section

çti = a rigid body rotation about the vertical axis,

where = O

x0 = length co-ordinate of section where

= O

Ub

horizontal displacement of point on base line.

The distribution of the centre of twist depends on how

the shipstructure is fixed in space. With parameter i'

12 10 -8 -lo -12 o A.P = -0.09216 10 rod -0.204l 1O3rd

Fig. 16. Distribution of centre of twist for '.

it is possible to analyze the centre of twist distribution

for various conditions.

When the engine room part would restrain the

warping of the

cross-section completely,

a good

criterion for fixation of the ship would be to require

the longitudinal displacements for the cross-section in

the engine room part to be zero. The numerical

analysis indicated that the engine room part, although

considerably rigid, cannot be assumed to give complete

warping restraint.

Here the requirement is made, that in the cross-section

just before the engine room, the longitudinal

displace-ment in the point of longitudinal bulkhead, where the

warping stress vanishes, becomes zero. The

corres-ponding value for the rigid body rotation yielded:

= 009216 x iO

rad

and the distribution of the centre of twist is shown in

Figure 16. It may be seen that the centre of twist

distribution does not follow the shear centre.

Figure 16 shows the distribution of the centre of

twist for a larger value of

iIi,

so that the centre of

twist at mid-length of the ship comes close to the shear

centre. Also in this case, in particular in the engine

room part and in the forebody. the centre of twist

deviates considerably from the shear centre.

3.2.3

Distortion of the hatch openings

An important deformation occurring when the

con-tainership is loaded in torsion, is the distortion of the

hatch openings. The diagonal deformations,

corres-ponding to the pure torsional loading case V, with its

maximum torque of 0.194 x l0 Nm at mid-length of

the ship, are given in Table 4.

From figure 16 it can be seen that the largest

de-formation occurs in hatch No. 6, being just at

mid-length of the open part between engine room and

fore-deck.

3.2.4

Normal stress distribution for

torsional loading and horizontal

bending

Figure 17 shows the normal stress distribution for the

anti-symmetric loading cases IV and V. It may be

seen that the so called warping stresses reach their

maximum values at the intersection of main

longi-tudinal bulkhead and upperdeck near the engine room.

Figure 18 shows the warping stress distribution in

the most heavily stressed section, being just before the

engine room. The influence of the decks, which results

15 14 13 12 10

Fig. 7a. Hatch numbers for NEDLL0YD-containership.

UPPER DECK

S s

'I

in shear lag effects in the longitudinal bulkhead is

clearly shown. Moreover, it can be seen that a

pronoun-ced bending effect of the double shell structure exists,

which is attributed to the differences of torsional

stiff-ness between engine room part and container hold part.

The bending stresses in the transverse boxgirders are

quite large for loading cases IV and V. If such a

box-girder is eccentrically attached to a bulkhead, then the

stresses increase due to its larger bending stiffness.

The largest bending stresses occur near mid-length

of the open part between engine-room and fore-deck

where

the

hatch

diagonal

deformation

becomes

maximum.

Table 4. Distortion of the hatch openings for loading case V.

7 6 deformation of direction [mm] 5

\deformation

of direction [mm]

-12.2

-19.1

-21.9

-24.3

-25.3

-25.1

-21.9

-20.6

-18.3

-14.2

- 6.3

- 6.6

- 5.1

- 3.8

Fig. 17. Normal stress distribution in upperdeck and longitudinal bulkhead for horizontal wave bending and torsional loading.

2 13.2 3 20.2 4 23.7 5 25.5 6 25.7 7 25.0 8 21.8 9 20.2 lo 18.0 11 12.1 12 7.0 13 6.2 14 4.8 15 3.7 6 B 10 12 14 16

LOADING CASE ] HORIZONTAL WAVE RENDING AND TORSIONAL LOADING

LOADING CASE SI:---- TORSIONAL LOADING

MAIN LONGITUDINAL BULKHEAD

B 10 12 14 16

NORMAL STRESS DISTRIBUTION IN UPPERDECK ANO

STRESS SCLE N

LONGITUDINAL BULKHEAD FOR HORIZGNTAL WAVE BENDING AND TORSIONAL LOADING

20

Fig. 18. Torsional stress distribution in the most heavily stressed section.

The bending stresses in the centre-line girder are small

for the anti-symmetric loading cases, due to its large

flexibility.

In the torsional strength of containerships with wide

hatch openings, usually the warping theory of

thin-walled beams has been applied [5]. The numerical

o

results of this finite element analysis indicated that the

ship's ends and the engine room part do not give

complete warping restraint and this is felt to be one

of the major difficulties in application of the beam

theory on this type of structures. Moreover, it can be

seen that the bending effect of the double shell structure

and the transverse boxgirders is very important.

3.2.5

Shear stresses in upperdeck

Figure 19 shows the shear stresses in the upperdeck,

calculated at the centres of gravity of the plate elements.

The shear-stress distribution will be discontinuous

because of the bending moments from the transverse

boxgirders and the discontinuities in the plate thickness.

3.3

Comparison of stresses and deformations with

corresponding Germanischer Lloyd results for

the "Liverpool Bay" Class

Table 5 compares still-water bending stresses of the

Nedlloyd-containership with corresponding

Germa-nischer-Lloyd results for the "Liverpool Bay" Class,

given in [3].

It may be seen that the stress gradient in the deck

plating just before and after the engine room is not

given by the Germanischer Lloyd results.

Table 6 shows the bending stresses in the transverse

vu".

u'.

Fig. 19. Shear stresses in upperdeck for loading case V: Pure torsional loading.

r

-4

j; E E _j".

I'

' -j ,----1 AP, F.P 12 11 10 9 8 7 6 5 4 3 2

E

29 30 IS 13 17 31 32 21 25

stresses for still water bending [N/mm2]

stresses for pure torsional loading [N/mm2]

hatch diagonal deformation [mm]

boxgirders of the Nedlloyd containership and the

"Liverpool Bay" Class ship for torsional loading.

When the same torsional moment is applied for both

ships, the stresses in the boxgirders of the Nediloyd

are much higher than in those of the "Liverpool Bay"

Class. The comparison of the hatch diagonal

deforma-tion is shown in Table 7.

Due to the smaller side box breadth of the Nediloyd

(2.38 m compared with 2.78 m) and the longer open

part, the torsional stiffness will be smaller and in

consequence the hatch diagonal deformation will be

greater.

Moreover, the transverse boxgirders have a beam of

1.4 m compared with 0.8 m in the "Liverpool Bay"

Class.

16

Fig. 20. Stress comparison diagram.

Table 5. Comparison of still water bending stresses with Germanischer Lloyd results for "Liverpool Bay", given in [3]

Table 6. Comparison of torsional stresses with corresponding Germanischer Lloyd results for "Liverpool Bay", given in [3]

Table 7. Comparison of hatch diagonal deformation with corresponding Germanischer Lloyd results for the "Liverpool Bay" [3]

4

Conclusions

/

The results of the analysis show that finite element

analysis, if properly carried out, may be used to

obtain accurate information about stress and

displacement response of complex shipstructures.

It will be clear that the art of modelling is of

major importance both with respect to overall

problem simplification as well as the modelling

of detail members.

It may be concluded, that a containership as

modelled in this analysis will give an accuracy,

which is

fully satisfactory with respect to the

overall response.

In order to obtain a detailed picture of the stresses

position on Fig. 20

Nediloyd loading case V Liverpool Bay

MT = MTmax (1 -cos 2rx/L) Mrmax 1.94x 108 Nm MTmax=4.OX 108 Nm MTmax4.O>(lO8 Nm

29 12.7 26 34 30

_-16.5

_{- 34}

_{- 40}

13 64.2 132 86 17

-58.8

_-121

_{- 86}

31 70.4 145 63 32

-63.6

-131

- 63

21 63.1 130 103 25

_-51.5

_-106

_-103

7 95.2 71 8 64.2 71 Il 107.3 80 12 _{79.8 80} 15 80.6 63 16 81.6 63 7 25.0 51.6 32.5 11 12.1 25.0 17.7 13 6.2 12.6 7.4

Nediloyd

_{i ,.}

-\Liverpoo1 Bay

position on Fig. 20 M11max = 2.665X10 Nm = 2.0X10 Nm

hatch _{Nediloyd loading case V Liverpool Bay}

number on _{MT = MTmax (1 -cos 27tx/L)}

22

at a hatch corner, a local piece of the structure

has to be isolated and remodelled a much finer

element grid being used. Displacements or stresses

have to be prescribed, where the local piece of

structure is taken from the major structure.

Local analysis will be necessary for the hatch

corner just before the engine room and for the

connection of a few transverse boxgirders with

the double shell structure.

The stress distribution for vertical bending is

mainly determined by cross-sectional properties

of the ship, such as moment of inertia, and

significant

discontinuities

of the shipstructure

such as change of container hold part to engine

room.

Beam theory will give satisfactory overall results,

when bending of the ship is considered.

Since horizontal forces influence the torque of the

ships, one should analyze the combination of

horizontal bending and torsional loading.

The engine room has a considerable influence on

the distribution of the torsional angle.

With application of the finite element method to

the torsional analysis of open ships the shear

centre position, which is difficult to determine

accurately, is not necessary.

The hatch diihal deformation due to torsional

loading becomes maximum at mid-length of the

open part between engine room and fore-deck.

The maximum hatch diagonal extension is 26 mm

for pure torsional loading with the maximum

torque of 0.194 x l0 Nm amid-ships, and a

dis-tribution according to results of model

experi-ments.

Warping stresses due to torsional loading reach

their maximum values at the intersection of main

longitudinal bulkhead and upperdeck near the

engine room.

If a transverse boxgirder is eccentrically attached

to a bulkhead, the bending stresses due to

torsion-al loading are higher because of the larger bending

stiffness of the boxgirder with effective plate width

of the bulkhead.

Il. The ship's ends and the engine room part,

although considerably rigid compared with

con-tainer hold parts, do not give complete warping

restraint: this is

felt to be one of the major

difficulties in application of beam theory to this

type of ship structure. Moreover, the bending

stiffness of the double shell structure and the

transverse boxgirders play an important role in

the torsional stiffness and strength analysis.

12.

Shear stresses due to torsion remain quite low

compared with bending stresses and warping

stresses and they become maximum in the

upper-deck.

Acknowledgement

The Netherlands Ship Research

Centre TNO is

greatly indebted to Mr. E. Vossnack, Chief Naval

Architect at the Technical Advicebureau of the

Nether-lands Shipping Union Liner Group. His interest and

excellent cooperation has considerably contributed to

the performance and the execution of the project.

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PRICE PER COPY DFL. 10.- (POSTAGE NOT INCLUDED)

M = engineering department S = shipbuilding department C = corrosion and antifouling department

Reports

90 5 Computation ofpitch and heave motions for arbitrary ship forms. w. E. Smith, 1967.

9 1 M Corrosion in exhaust driven turbochargers on marine diesel engines using heavy fuels. R. W. Stuart Mitchell. A. J. M. S. van Montfoort and V. A. Ogale, I 967.

92 M Residual fuel treatment on board ship. Part II. Comparative cylinder wear measurements on a laboratory diesel engine using filtered or centrifuged residual fuel. A de Mooy, M. Verwoest and G. G. van der Meulen, 1967.

93 C Cost relations of the treatments of ship hulls and the fuel con-sumption of ships. H. J. Lageveen-van Kuijk, 1967.

94 C Optimum conditions for blast cleaning of steel plate. J. Rem-melts, 1967.

95 M Residual fuel treatment on board ship. Part I. The effect of cen-trifuging, filtering and homogenizing on the unsolubles in residual fuel. M. Verwoest and F. J. Colon, 1967.

96 5 Analysis of the modified strip theory for the calculation of ship motions and wave bending moments. J. Gerritsma and W. Beu-kelman, 1967.

97 S On the efficacy of two different roll-damping tanks. J. Bootsma and J. J. van den Bosch, 1967.

98 S Equation of motion coefficients for a pitching and heaving des-troyer model. W. E. Smith, 1967.

99 S The manoeuvrability of ships on a straight course. J. P. Hooft, 1967.

100 5 Amidships forces and moments on a CB = 0.80 "Series 60" model in waves from various directions. R. Wahab, 1967. 101 C Optimum conditions for blast cleaning of steel plate. Conclusion.

J. Remmelts, 1967.

102 M The axial stiffness of marine diesel engine crankshafts. Part I. Comparison between the results of full scale measurements and those of calculations according to published formulae. N. J. Visser, 1967.

103 M The axial stiffness of marine diesel engine crankshafts. Part II. Theory and results of scale model measurements and comparison with published formulae. C. A. M. van der Linden, 1967. 104 M Marine diesel engine exhaust noise. Part I. A mathematical model.

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105 M Marine diesel engine exhaust noise. Part II. Scale models of exhaust systems. J. Buiten and J. H. Janssen, 1968.

106 M Marine diesel engine exhaust noise. Part Ill. Exhaust sound criteria for bridge wings. J. H. Janssen en J. Buiten, 1967. 107 5 Ship vibration analysis by finite element technique. Part I.

General review and application to simple structures, statically loaded. S. Hylarides, 1967.

108 M Marine refrigeration engineering. Part I. Testing of a decentraI-ised refrigerating installation. J. A. Knobbout and R. W. J. Kouffeld, 1967.

109 S A comparative study on four different passive roll damping tanks. Part I. J. H. Vugts, 1968.

110 S Strain, stress and flexure of two corrugated and one plane bulk-head subjected to a lateral, distributed load. H. E. Jaeger and P. A. van Katwijk, 1968.

111 M Experimental evaluation of heat transfer in a dry-cargo ships' tank, using thermal oil as a heat transfer medium. D. J. van der Heeden, 1968.

112 S The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. J. H. Vugts, 1968.

113 M Marine refrigeration engineering. Part II. Some results of testing a decentralised marine refrigerating unit with R 502. J. A. Knob-bout and C. B. Colenbrander, 1968.

114 S The steering of a ship during the stopping manoeuvre. J. P. Hooft, 1969.

115S Cylinder motions in beam waves. J. H. Vugts, 1968.

116 M Torsional-axial vibrations of a ship's propulsion system. Part I. Comparative investigation of calculated and measured torsional-.

axial vibrations in the shafting of a dry cargo motorship.

C. A. M. van der Linden, H. H. 't Hart and E. R. Doffin, 1968. 117 S A comparative study on four different passive roll damping

tanks. Part II. J. H. Vugts, 1969.

118 M Stern gear arrangement and electric power generation in ships propelled by controllable pitch propellers. C. Kapsenberg, 1968.

i I 9 M Marine diesel engine exhaust noise. Part IV. Transferdamping data of 40 modelvariants of a compound resonator silencer. J. Buiten, M. J. A. M. de Regt and W. P. Hanen, 1968. 120 C Durability tests with prefabrication primers in use steel of plates.

A. M. van Londen and W. Mulder, 1970.

121 S Proposal for the testing of weld metal from the viewpoint of brittle fracture initiation. W. P. van den Blink and J. J. W. Nib-bering, 1968.

122 M The corrosion behaviour of cunifer 10 ailoys in seawaterpiping-systems on board ship. Part 1. W. J. J. Goetzee and F. J. Kievits, 1968.

123 M Marine refrigeration engineering. Part III. Proposal for a specifi-cation of a marine refrigerating unit and test procedures. J. A. Knobbout and R. W. J. Kouffeld, 1968.

124 S The design of U-tanks for roll damping of ships. J. D. van den Bunt, 1969.

I 25 S A proposal on noise criteria for sea-going ships. J. Buiten, 1969. I 26 S A proposal for standardized measurements and annoyance rating

of simultaneous noise and vibration in ships. J. H. Janssen, 1969. 127 S The braking of large vessels II. H. E. Jaeger in collaboration with

M. Jourdain, 1969.

I 28 M Guide for the calculation of heating capacity and heating coils for double bottom fuel oil tanks in dry cargo ships. D. J. van der Heeden, 1969.

129 M Residual fuel treatment on board ship. Part III. A. de Mooy, P. J. Brandenburg and G. G. van der Meulen, 1969.

I 30 M Marine diesel engine exhaust noise. Part V. Investigation of a double resonatorsilencer. J. Buiten, 1969.

131 S Model and full scale motions of a twin-hull vessel. M. F. van Sluijs, 1969.

132 M Torsional-axial vibrations of a ship's propulsion system. Part II. W. van Gent and S. Hylarides, 1969.

133 S A model study on the noise reduction effect of damping layers aboard ships. F. H. van ToI, 1970.

134 M The corrosion behaviour of cunifer-lO alloys in seawaterpiping-systems on board ship. Part II. P. J. Berg and R. G. de Lange,

1969.

135 S Boundary layer control on a ship's rudder. J. H. G. Verhagen, 1970.

136 S Observations on waves and ship's behaviour made on board of Dutch ships. M. F. van Sluijs and J. J. Stijnman, 1971. 137 M Torsional-axial vibrations of a ship's propulsion system. Part III.

C. A. M. 'ian der Linden, 1969.

138 S The manoeuvrability of ships at low speed. J. P. Hooft and M. W. C. Oosterveld, 1970.

139 S Prevention of noise and vibration annoyance aboard a sea-going passenger and carferry equipped with diesel engines. Part I.

Line of thoughts and predictions. J. Buiten, J. H. Janssen,

H. F. Steenhoek and L. A. S. Hageman, 1971.

140 S Prevention of noise and vibration annoyance aboard a sea-going passenger and carferry equipped with diesel engines. Part II. Measures applied and comparison of computed values with measurements. J. Buiten, 1971.

141 S Resistance and propulsion of a high-speed single-screw cargo liner design. J. J. Muntjewerf, 1970.

142 S Optimal meteorological ship routeing. C. de Wit, 1970. 143 S Hull vibrations of the cargo-liner "Koudekerk". H. H. 't Hart,

1970.

144 S Critical consideration of present hull vibration analysis. S. Hyla-rides, 1970.

145 S Computation of the hydrodynamic coefficients of oscillating cylinders. B. de Jong, 1973.

146 M Marine refrigeration engineering. Part IV. A Comparative stuyd on single and two stage compression. A. H. van der Tak, 1970. 147 M Fire detection in machinery spaces. P. J. Brandenburg, 1971. 148 S A reduced method for the calculation of the shear stiffness of a

ship hull. W. van Horssen, 1971.

149 M Maritime transportation of containerized cargo. Part II. Experi-mental investigation concerning the carriage of green coffee from Colombia to Europe in sealed containers. J. A. Knobbout, 1971. 150 S The hydrodynamic forces and ship motions in oblique waves.

151 M Maritime transportation of containerized cargo. Part I. Theoretical and experimental evaluation of the condensation risk when transporting containers loaded with tins in cardboard boxes. J. A. Knobbout, 1971.

152 S Acoustical investigations of asphaltic floating floors applied on a steel deck. J. Buiten, 1971.

i 53 S Ship vibration analysis by finite element technique. Part II. Vibra-tion analysis. S. Hylarides, 1971.

155 M Marine diesel engine exhaust noise. Part VI. Model experiments on the influence of the shape of funnel and superstructure on the radiated exhaust sound. J. Buiten and M. J. A. M. de Regt, 1971. 156 S The behaviour of a five-column floating drilling unit in waves.

J. P. Hooft, 1971.

157 S Computer programs for the design and analysis of general cargo ships. J. Holtrop, 1971.

158 S Prediction of ship manoeuvrability. G. van Leeuwen and

J. M. J. Journée, 1972.

159 S DASH computer program for Dynamic Analysis of Ship Hulls. S. Hylarides, 1971.

160 M Marine refrigeration engineering. Part VII. Predicting the con-trol properties of water valves in marine refrigerating installations A. H. van der Tak, 1971.

161 5 Full-scale measurements of stresses in the bulkcarrier m.v. 'Ossendrecht'. 1st Progress Report: General introduction and information. Verification of the gaussian law for stress-response to waves. F. X. P. Soejadi, 1971.

162 S Motions and mooring forces of twin-hulled ship configurations. M. F. van Sluijs, 1971.

163 S Performance and propeller load fluctuations of a ship in waves. M. F. van Sluijs, 1972.

164 5 The efficiency of rope sheaves. F. L. Noordegraaf and C. Spaans, 1972.

165 S Stress-analysis of a plane bulkhead subjected to a lateral load. P. Meijers, 1972.

166 M Contrarotating propeller propulsion, Part I, Stern gear, line shalt system and engine room arrangement for driving contra-rotating propellers. A. de Vos, 1972.

167 M Contrarotating propeller propulsion. Part II. Theory of the dynamic behaviour of a line shaft system for driving contra-rotating propellers. A. W. van Beek, 1972.

169 S Analysis of the resistance increase in waves of a fast cargo ship. J. Gerritsma and W. Beukelman, 1972.

170 5 Simulation of the steering- and manoeuvring characteristics of a second generation container ship. G. M. A. Brummer, C. B. van de Voorde, W. R. van Wijk and C. C. Glansdorp, 1972. 172 M Reliability analysis of piston rings of slow speed two-stroke

marine diesel engines from field data. P. J. Brandenburg. 1972. 173 S Wave load measurements on a model of a large container ship.

Tan Seng Gie, 1972.

174 M Guide for the calculation of heating capacity and heating coils for deep tanks. D. J. van der Heeden and A. D. Koppenol, 1972. 175 S Some aspects of ship motions in irregular beam and following

waves. B. de Jong, 1973.

176 S Bow flare induced springing. F. F. van Gunsteren, 1973. 177 M Maritime transportation of containerized cargo. Part III. Fire

tests in closed containers. H. J. Souer, 1973. 178 S Fracture mechanics and fracture control for ships.

J. J. W. Nibbering, 1973.

179 5 Effect of forward draught variation on performance of full ships. M. F. van Sluijs and C. Flokstra, 1973.

184 S Numerical and experimental vibration analysis of a deckhouse. P. Meijers, W. ten Cate, L. J. Wevers and J. H. Vink, 1973. 185 S Full scale measurements and predicted seakeeping performance

of the containership "Atlantic Crown". W. Beukelman and M. Buitenhek, 1973.

182 S Finite element analysis of a third generation containership. A. W. van Beek, 1973.

Communications

15 M Refrigerated containerized transport (Dutch). J. A. Knobbout, 1967.

16 S Measures to prevent sound and vibration annoyance aboard a seagoing passenger and carferry, fitted out with dieselengines (Dutch). J. Buiten, J. H. Janssen, H. F. Steenhoek and L. A. S. Hageman, 1968.

17 S Guide for the specification, testing and inspection of glass

reinforced polyester structures

in shipbuilding (Dutch). G.

Hamm, 1968.

18 S An experimental simulator for the manoeuvring of surface ships. J. B. van den Brug and W. A. Wagenaar, 1969.

19 S The computer programmes system and the NALS language for numerical control for shipbuilding. H. le Grand, 1969.

20 S A case study on networkplanning in shipbuilding (Dutch). J. S. Folkers, H. J. de Ruiter, A. W. Ruys, 1970.

21 5 The effect of a contracted time-scale on the learning ability for manoeuvring of large ships (Dutch). C. L. Truijens, W. A. Wage-naar, W. R. van Wijk, 1970.

22 M An improved stern gear arrangement. C. Kapsenberg, 1970. 23 M Marine refrigeration engineering. Part V (Dutch). A. H. van der

Tak, 1970.

24 M Marine refrigeration engineering. Part VI (Dutch). P. J. G. Goris and A. H. van der Tak, l70.

25 S A second case study on the application of networks for pro-ductionplanning in shipbuilding (Dutch). H. J. de Ruiter, H. Aartsen, W. G. Stapper and W. F. V. Vrisou van Eck, 1971. 26 S On optimum propellers with a duct of finite length. Part 11.

C. A. Slijper and J. A. Sparenberg, 1971.

27 5 Finite element and experimental stress analysis of models of shipdecks, provided with large openings (Dutch). A. W. van

Beek and J. Stapel, 1972.

28 S Auxiliary equipment as a compensation for the effect of course instability on the performance of helmsmen. W. A. Wagenaar, P. J. Paymans, G. M. A. Brummer, W. R. van Wijk and C. C. Glansdorp, 1972.

29 5 The equilibrium drift and rudder angles of a hopper dredger with a single suction pipe. C. B. van de Voorde, 1972.

30 5 A third case study on the application of networks for production-planning in shipbuilding (Dutch). H. J. de Ruiter and C. F. Heu-nen, 1973.

31 5 Some experiments on one-side welding with various backing materials. Part I. Manual metal arc welding with coated

electro-des and semi-automatic gas shielded arc welding (Dutch).

J. M. Vink, 1973.

32 S The application of computers aboard ships. Review of the state of the art and possible future developments (Dutch). G. J. Hoge-wind and R. Wahab, 1973.

33 S FRODO, a computerprogram for resource allocation in network-planning (Dutch). H. E. I. Bodewes, 1973.

34 S Bridge design on dutch merchant vessels; an ergonomic study.

Part I: A summary of ergonomic points of view (Dutch).

A. Lazet, H. Schuffel, J. Moraal, H. J. Leebeek and H. van Dam, 1973.

35 S Bridge design on dutch merchant vessels; an ergonomic study. Part Il: First results of a questionnaire completed by captains, navigating officers and pilots. J. Moraal. H. Schuffel and A. Lazet, 1973.

36 S Bridge design on dutch merchant vessels; an ergonomic study. Part III: Observations and preliminary recommendations. A. Lazet, H. Schuffel, J. Moraal, H. J. Leebeek and H. van Dam, 1973.

37 S Application of finite element method for the detailed analysis of hatch corner stresses (Dutch), J. H. Vink, 1973.

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