An analytical method for calculating the pure ridge resistance encountered by ships in first year ice ridges

HELSINKI UNIVERSITY OF TECHNOLOGY SHIP HYDRODYNAMICS LABORATORY

OTANIEMI FINLAND REPORT NO 17

AN ANALYTICAL ?THOD FOR CALCULATING THE PURE RIDGE RESISTANCE ENCOUNTERED BY SHIPS IN FIRST YEAR ICE RIDGES

by ARNO KEINONEN

Otaniemi 1979

17 OKt i79

_Lab.

_{y. Scheepstwwkund}

ARCHIEF

Technische Hogeschool

HELSINKI UNIVERSITY OF TECHNOLOGY SHIP HYDRODYNAMICS LABORATÒRY

OTANIENI FINLAND REPORT ÑO 17

AN ANALYTICAL METHOD FOR CALCULATING THE PURE RIDGE RESISTANCE ENCOUNTERED BY SHIPS IN FIRST YEAR ICE RIDGES

by

ARNO KEIÑONEN

Thesis for the degree of Doctor of Technology approved after public examinatibn, and criticism in the Auditorium Ko 216 at thé Helsinki University öf Technolàgy on the 27th of April, 1979 at 12 o'clock noon.

Corrections

ADDITION TO PAGE 11 Abbreviations

CRREL Cold Regions Research and Engineering Läboratory

F.sc. Fùll scale

-lAHR International Association for Hydraulic Research

M.sc. Model Scale

POAC Port and Ocean Engineering under Arctic Conditions WADAM Wärtsi1 Arctic Design and Marketing

WIMB Wärtsilä Icebreaking Model Basin

ISSN 0356-1313 ISBN 951-751-589-8

TKK. OFFSET 1979

page line reads shôuld read

3 1 years year's

6 25 . CONCLUSIONS CNCLUSIONS) IN BRIEF

9 h (or 1) ti (or H, HR., hRID.E)

9 RICE RICE (or RRIDGE)

10 Cl (2.lfla . (2.18.)a 10 C3 (2.lflc (2.18)c 13 15 Ref. 38 ef. O 18 3 Ref's. ',6,24,25 efs. I, 2I 25 13 V1 h( v1 =

.Bh

Bh

1J4 _V1 . . V1 =

ACKNOWLEDGEMENTS

I would like to express my gratitude towards the following persons:

- Mr. E. Mkinen, the manager of Wärtsilä Arctic

Design and Marketing Department (WADA!4), for his active supportof my work.

- Prof. V. Kostilainen for Iis warm and encouraging.

support during the work.

- Prof. E. Palosuo for his kind cooperation and.

discussions on matters coñcerning ice ridges.

-, Mr. T. Nyman and Mr. T. Heideman who have performed

model experiments, and varied analysis of experimental data for this work.

- - The personnel of WADAM for good cooperation during

the work.

- Dr. Lorna Sundström for checking the English

language.

- Mrs. Irma Lauks'io who has typed the work. Finally, the author wishes to acknowledge the financial support of the Academy of Finland and that of Wärtsilä Helsinki Shipyard have made the present work

possible. .

-Helsinki,, November 1978

ABSTRACT

- The resistance to shipts motion in first years ice

ridges has been studied in this work. The ice resistance due to ridges is split into three components: level ice resistance, aging resistance and pure ridge resistance. The pure ridge resistance is solved analytically in the case of a large homogeneous ridge field with a constant thickness, based on shear breaking of the ridge.

The results have been compared with model test results for four models in homogeneous ridge fields and with ship test results for three ships in natural ridge fields. The model tests were made in the Wrtsilä Ice-breaker Model Basin and the ship tests were performed by Wärtsil.

The agreement betwêen the experimental results for model tests and those obtaIned theoretically are

generally good.

In ship tests deviatiòns are found from -11 to +102.. percent between experiment and theory. At this stage of

the art it is proposed that the deviations would mainly be due to the following:

ridge characteristics are not known satisfactorily effects of deviations from theoretical idealizations in ship tests in natural ridges are not known.

5 TABLE OF CONTENTS ACKNOWLEDGEMENTS ABSTRACT NOTATION INTRODUCTION 13 THE PROBLEM 16 1.1 General considerations 16 1.2 Detailed considerations 18

ANALYSIS OF THE PURE RIDGE RESISTANCE 23

2.1 Gênerai 23

2.2 Basis for breaking analysis 24

2.3 Initial resistañce, thin ridge 27

2.3.1 Generai 27

2.3.2 Force geometry 27

2.3.3 Force geometry in breaking plane 31

2.3.4 Resistance to y-plane breaking 34

2.3.5 Forces in the end planes 36 2.4 Thick ridges, initiai resistance

2.4.1 General 38

2.4.2 Conditions of the lower breaking Piane 39

2.4.3 Changes in the y1-piané analysis 39

2.4.4 Changes in the end plane analysis 39 2.5 Developed resistance condition 40

2.5.1 General 40

2.5.2 Thickness growth model for a landing 40

craft bow

2.5.3 General solution for ridge growth 43 2.5.4 Resistance due to the lower breäking

plane 50

2.5.5 Area of the end planes 52

2.6 Parallel rniddle body 52

2.6.1 General 5'

2.6.2 Ice profile around the parallel middle

2.6.3 Side resistance 60

2.6.14 Bottom resistance 60

2.6.5 Penetrations for ridge growth 61 2.7 Summary of resistance components 63

COMPARISON WITH MODEL TEST RESUL'PS 614

3.1 General 6k

3.2 Basic tests for theory 66

3.3 Comparisons with ship model tests 70

3.3.1 General 70

3.3.2 Three small models, D, E, F 71

3.3.3 MODEL A 75

3.14 Sensitivity analysis 77

3.14.1 _General 77

- 3.4.2 Analysis 78

COMPARISON WITH. SHIP.TEST RESULTS 82

14.1 General 82

14.2 Characteristics of natural ridges 83

¿4.3 _{On full scale tests} - 85

¿1.4 Comparison between theoryand full scale 89

tests 14.4.1 SHIP A 89 4.14.2 SHIP 91 V 14.4.3 SHIP C V 93 4.5 Combined comparisons V CONCLUSIONS V V V V ₉₉ REFERENCES TABLES

Ridge resistance analysis,

MODEL D V

Ridge resistanée analysis, MODEL E

Ridge resistance analysis,

MODEL?

V

VRidge resistance analysis,

ÑODELA

V V 101 106 107 -109 110

7

5. Sensitivity of the resülts to the

change òf variables ill

6. Ridg resistance analysis, SHIP A 112 7. Ridge resistance analysi, SHIP B 113

8

NOTATION

Symbol Definition

A area

APB cross sectional area of ice under parallel

hull

cross sectioial area of ice at parallel sides òf hull

As contact area between ice and parallel sides

of' hull

area of breaking plane (upper, lower)

A area of end plane

B - breadth f ship

(x,y,z) force (its axial components) F shear friction resistance

buoyancy force, (components normal and parallel to breaking plane, total) FN(x,y,z,xy,xz) normal. forèe (its cOmponents)

FPB buoyant ice force against flat bottom of

ship

ToT(o,1,2) totál hull force (due to upper, lower breaking plane)

force to break end plane

p normal force, shèàr force to breaking plane frictional forte (its axial components)

P cohesive force

T0

2 3) volume of ice for buoyancy force (its

' _{' compOnents)}

length of parallel waterline -length of parallel part of hull effective parallel hull length

9

R resistance

RICE total ice resistance

RPB total ice resistance flat ship's bottom

R5

totalice resistance parallel ship's sides

pure ridge resistance, totally, at o-speed

2

total ice resistance,rPlafle breaking

' (upper, lòwer)

R

2'

resistanCe due to end plane breaking (upper, lower)

T ship's draught

subdivisions of buoyant ice mass

b length of y-breaking plane

g acceleration of gravity

h (or H) ridge thickness

h3 height of ridge at parallel side plane

average pressure height

added ridge thickness due to development direction cosines of a force for x,y,z-axes

1 penetrations for ridge growth

1, ,,,

p pressure

p6 pressure against hull with 6-inclination

2 moment levers

, , I

half angle of entrance of aterlifle

Z' half angle of entrance_{normal plane tò total hull force}f waterline of

angle of friction bétween ship's hull and ice angle between vertical plane and parallel middle body side

inner angle of friction of ridge mass angle between horizontal plane and ship's buttock

B

B

B'

10

' (ort') angle between horizontal plane and buttock of normai plane to totál hull force

2 anglebetween horizontal plane and breaking

' plane (for upper breaking plane, lower

brea:ing plane)

6 angle between horizontal plane and

- -increasing thickness under. ship's bottom

£ index for end plane

u coefficient o fric'tion betweenrip'-s- hull

and ice.

soliUity of 1dge mass

. Poissorfs ratio for ridge mass

density of water less density of ice,

A buoyant density of ice

G normal stress

normal stress against ship's hull

o normal stress against breaking plane

y(l,2) _{(upper, lower)}

oc .

normal stress against' end plane

r . shear stress

t0

cohesion of ridge mass r

(] 2) shear stress in-breáking plane (upper, lower)

Te sheai stress- in end piane

* angle between horizontal plane and a

section perpendicular to waterline angle between horizontal plane and a

section perpendicular to total hull force.

V volume of displademént

CONSTANTS

cl area of -breakingplanes (Eq. (2.l7)a)

C2 + , geometric coefficient

verticals (= buttocks) outer hull contours of vertical plane cross-section

INTRODUCTION

The resistance encountered by ships in fast, level ice has historically been the most popular object in the field' of icebreaking research, even though it does not present the most severe obstacle to the passage of ships through ice covered waters. In the Baltic, at least, it is the ice ridges which present the greatest resistance to ice navigation. And yet, the penetration of ridged ice conditions has only occasionally been touched upon in availäble literature and no theoretical or empirical approach is known by the author to exist on this problem.

Enkvist, who has done research on level ice

resistance, Ref. 2 (1972) indicated already the importance and difficulty of ridged ice conditions to ice navigation. In 1976 in the Baltic the Importance of ridges was also clearly indicated in a study named "Vintbott" Ref. 38, where ice navigation was carefully registered during a whole winter by researchers on board icebreakers assisting mercha±it traffic. During "Vintbott" the number of ridges was only 3 to 10 percent of the total distance travelled through ice, but they formed the only ice conditions where the icebreakers had to use rams to penetrate them.

An ice ridge is created whentwo ice fields are compressed into each other. Thus, a ridge consists mainly of ice blocks which break off under the compression. The Ice blocks are pushed both above and below the fast ice field: A schematic cross-section of a ridge, shown in Fig. 1.1, consists of three parts, the sail, the keel and a frozen ice layer. In the Baltic both the sail and the keel consist mostly of loosely bound ice blocks. The sail of a free floating Baltic ridge may typically be 1-3 metres high and the keel to 15 metres deep, although reports have been made of ridges of total thickness up to 31 metres, Palosuo Ref. 3?. A more detailed determination

14

of the characteristics of ridges. is given in Ref. 18. Ice ridges may occur as single ridges (distance between successive ridges over 1000 metres), in ridged ice fields (distance between successive ridges 0-1000 metres), or in ridge fields (ridges in immediate contact with each other), Ref. 18.

The structure of ice ridges has been very little known up to

_1970..

The data on Baltic ridges come from the

work of Palosuo Ref s.

_{29-32 (1970-1975)}

_{and from Ref s. 15,}

16, 18 (1976-78).

This last decade has also seen a number

of publications on ice ridges in other parts of the world. We present here a selection o! further references on ridges: 3, 8-10,

20-22, 33, 34, 38, 39, 41, 43.

In Finland, the history of the research into ship's

behaviour in ice. ridges can.. be said .to. have begun in

_1971,.

when for the first time registration of ice ridges was made before. testing a ship in them. This was done by Wärtsilâ with the Baltic icebreaker "Njord", Ref. 11. The first systematic test series in ice ridges was carried Out in

₁₉₇₄

with the Baltic icebreaker "Apu", Ref s.. 23 and

27.

Since then, icegoing ships have been continuously and suecesfully tested in ice ridges.

The model tests in ice ridges in Wärtsil's Ice.Model Basin were started ïn

₁₉₇₀

(Ref. 1) and thereafter most of the models run in the basin have been tested in simula-ted ice ridges in addition, to other, ice conditions.

In

1973

a sparate project.cwas.started by. Wä&i1A: tu study the behaviour of ships in specifically Baltic type ice ridges. There was a lack of environmental data on the ridges, as well as of appropriate éxperimental data on the behaviour of ships in these ridges. The work reported here is a continuation of that project. Our present purpose is: therefore to lay the foundations for the continuing

15

behavioür in ice ridges. Although the ice rsistance problem in ice ridges is approached here, no attempthas been made to develop any resistance optimization method for

ship's design purposes.The results of this work only covèr a limited portion of all the matters to be considered in

hip's design, arid considered alone they would lead to suboptimization for.an icegoing. vessel.

1. TRE PROBLEM

1.1 Generâl conside±atións

Penetration of ridges by a ship is. a complicated and extensive problem, if the whole process. is considered. The most important factors affecting the penetration of ridges may be divided as follows:

- resistance - sticking in ice

- ice effects on propulsion - inertial effects

This study approaches the resistance problem only. The inertial effects (added masses, accelerations in vertical and horizonts]. directions), which could be attached-to the resistance problem, are at this context kept

separate. Due to the complicated interaction between ice and water in a ridge during ship's penetration, the resistance problem for the case of slow passage only is considered, where bydrodynanhic or mass forces should not be of importance.

--This work is limited to the consideration of free floating, first year-Baltic ridgea.._in contrast to the free floating ridges, there exist ridges on the shoals which stand on the seafloor. The contact between the keel of a ridge and the seafloor gives rise to additional supporting effects on the ridge, which willnot be ana-lyzed here.

- On thé basis of their age., ridges- are classified into

first year and multiyear ridges. In the Baltic there are. no multiyear ridges. The structure of multiyear ridges is considerably more rigid than that of first yeár ridges, although the Arctic first year ridges may be slightly more rigid than the majority of thé ridges in the

Baltic:.-'-J 16

WATER LEVEL

Fig. 1.1. Schematic ridge profile and surface view of a Baltic ridge field.

1.2 Detailed coñsiderations

The analysis of full scale tests of ships in Baltic ice ridges Refs.

4,

6, 2'4, 25 shows that the total ice resistance (total resistance total ice resistance + resistance in ice free water) may be logically split into thièe components (see Fig. 1.2):

- 'level ice resistance

-- aging resistañce

-- pure ridge resistancé

Hv CE

Fig. 1.2. Components of total ice. resistance in ridges.

These componeñts are at this stage assumed to be independent of each other. In Fig. 1.1 the corresponding parts of the ridge are roughly sketched.

Lével ice resistance is due to the fast, level ice field around and inside the ridge. In this work no assumptions are made as to this part of the ice resistance. It may be determined through model tests, by full scale tests or by some theoretical or semiempirical method, whichever is most practical.

19

Aging resistance means that part of the resistance which is related to the changes in the structure of a ridge due to its aging ( or weathering). Thesechanges

involve an increase in the thickness of the frozen ice layer inside the ridge and the freezing together of ice-blocks in the sail and in the keel. Especially the freezing of rafted icé fields at the water surface caus a noticeablé aging resistance component. This part of the resistance wIll be disregarded in the present approach to the resistance problem. In Section 4.3 clear evidence of its existence is shown on an empirical basis in the full scale ship's tests.

Thermainder of the ice resistance, after substraction of the level ice resistance and the aging resistance from the total ice resistance, will be called pure ridge

resistance. This is by far the largest component of the ice resistance in large first year ridges. We shàll concentrate on the analysis of this part of the resistance.

From the environmental point of view,oUr considerations will limited to a structurally homogeneous, large ridge field with a constant thickness. In natural conditions the ridges may have a considerably varied cross-sectional pro-file., Nevertheless the most difficult ridge conditions for a ship are a long ridge field with continuously large keel depth profilé. This approaches the case of constant thick-ness, because then the variations of the profile become unimportant còmpared with the continuously large amount of

ce around the huÏl of the ship, especially when the ship at any moment interacts with a rather large part of the ridge field. The penetration of ridged ice fields or single ridges presents a less severe obstacle to ice navigation, but not without some importance. Their penetration equations wil1 determinable by interpolation after analysis of the pure ridge resistance. Equations of motion for ridge

penetration are given, in Ref. 19.

the following scheme:

Initial resistance. The initial resistance is deter-mined to be the resistance of a ridge field surrounding the forward body of a.ship.arid having initially constant thickness everywhere around the forward body of the ship. In Fig. 1.3 a schematic illustration of the case of initial resistance is- shown.

Deve1oed resistance. The -developed resistance is determined to correspond to a developed -ridge profile around the hull of a ship. The devej.oped ridge profile is a profile that stays constant under further penetra-tiori, once it has been established after a certain distance of penetration. The developed resistance may itself be divided further into two parts, that due to the forward body and that due to the. parallel middle body of -a ship. The decreasing aft body of the ship also comes into contact with ice, but as its effect on the resistance is minor and interacts with the propul-sion, it will be ignored at this stage. In Fig. l» a schematic -presentation of the developed resistance condition is shown.

The initial resistance -will be -used for the direct comparison with theory and with the corresponding basic experiments, due -to the geometric simplicity of

the condition. The developed resistance is directly comparable with scale model results in large

homoge-neous ridge fields,. and. for full sca-le test results it

is an upper bound approach.

-For the purposes of analysis, a simple geometry is chosen for the ship. This geometry is the combination of planes -shown in Pig. 1.5. The bow angles a,

(and ID) may be compared with the integrated -forward

-body angles determined by -Erkvist, -Ref. 2.- The

following planes can be- distinguished in the hull: - side planes of forward body (ABCD, BA'D'-C) - bottom plane (DTD'J'C'J)

SI-np's MOVEMENT SHIP RIDGE 21

.

\-/

L' / L

-'- \_. -

\.,

L-/

-L- . I L'

-L'

,

-7

'-

.- '- - - L' -

L'

r

'L- V

- I V /\

L

-V \_'-

\

L' L'

7

-'-y' L-I

Fig. 1.3. Initial resistance condition, submerged part of ship hull. For ship geometry, see Fig. 1.5.'

SHIP SHIP'S

MO VE M E NT

RIDGE

XY - PLANE

22

- parallel side planes (ADJI, A'D'J'I')

(- side planes of aft body symmetrical to the forward

body)

Assumptions are made such that the parallel middle body of a ship will be GFMN and such that the whole area of the parallel side planes and the bottom plane belong to the parallél middle body. Then the

resis-tance düe to the forward body may be ättributed to its side planés (ABCD and A'BCD'). Otherwise, a con ventiönal ship's deéign is assumed with no forward screws nor any other departures from a smooth hull.

XZ-PLANE

Fig.l.5. Ship's geometry.

23

2. ANALYSIS OF THE PURE RIDGE RESISTANCE

2.1 Genéral

In this analysis the aim is to produce a

quanti-tative, comparative presentation of the different resistance components of the pure ridge resistance as a function of ship and environmental parameters. The corresponding mathe-matical models representing each of the components will be kept at a simple level, because this is a first theoretical approach to the problem. The simplicity allows bettr

control of the results. To preserve this control throughout, a sensitivity analysis will be presented for each resistance component (Section 3.11). The sensitivities of the results to changes in the basic variables or in the governing models indicate the possible sensitive points of the theory which should first be questioned when problems arise in the inter-pretation or comparisons of the results.

If óne ignores the level ice and the weathering, a ridge field-consists of loose, broken ice blocks both in the sail and the keèl, Ref. 18. Its response to ship move-ment may at .any momove-ment be determined s the force för breaking the mass consisting of the loose ice blocks and removing it from the passage of the ship. In Section 2.2 a solution for this kind of' breaking problem of masses is presented. The basis for the method is found in soil mechanical problems (Coulomb's method).

In Section 2. 3 the initial resistance is analyzed fox' ridges shallower than ship's draught. .Here the basic method

for the determination of pure ridge resistance is présented. In Section 2.11 the initial resistance is touched upon for ridges deeper than ship's draught. The changes in the

conditions for the basic method are presented.

In Section 2.5 the developed resistance condition is analyzed. A kinematic model for development of the ridge

2k

around the hull of a ship is presented, and the corres-ponding changes in the constants of the basic method are derived.

In Section 2i6 the resistance due to the parallel middle body-óf a ship is dealt with.

2.2 Basis for breaking analysis

In the area of soil mechanics, problems of the carry-ing capacity of soil are solved in connection with loads caused by different types of structures. There exist a multitude of methods for these purposes (see for example

Rets. 5 and 7).

-A major part of an ice ridge is subjected during ship penetration to similar types of loads as soi1.The diere-rence between the earth and ice ridge probleci is that the soil must not break, whereas an ice ridge must break.

Thus, en using approximate methods, the desired direction of approach changes from the non-breaking 'condition to a breaking condition. At. the same time, the purely static nature of the soil problem changes to a dynamic problem. Thus in the ship problem there is a need to analyze the problem beyond the stage of the breaking of the masa to gain knowledge of the development of the breaking condition. :In addition,the complicated geometry of a ship, even when

simplified, leads to a three-dimensional handling of the problem or at least to a need for boundary conditions to

solve the end effects. . - - -

-When choosing a method for analyzing- the breaking forces in ship-ice interaction in ice ridges, it is consi-dered most important that the analysis can be computerized

so as to albi systematic calculations. -It is also desirable to be able freely to vary the initial environmental condi-tions (friction, cohesion etc...). The analysis should be such thàt a practical kinematic model may be devèboped

-FTOT

Fig. 2.1. Coulomb's method

25

These conditions have led the author to Coulomb's method, wliich is fully presented in Refs. 5 and 7. A short presentation is given here:

The method of Coulomb is based on an assumption of the breaking of the mass in a plane, when the mass is loaded with a wàll. The breaking condition is soled by using a force balance calculation im possible breaking planes. The actual breaking plane where the breaking conditions are first reached is found by derivation. The method for a vertical frictionless wall is shown in Fig. 2.1. BC is thè breaking plane with an inclination y from the horizon.

WATER LEVEL

F0=cohesiOn resistance

:ear friction

resistance

FG :buoyancy force of ice FTQT=force against the wall

' ;inner friction angle

Tri the figure it can be seen that the action lines of the forces FTO., F , FG and F pass through a single point, which lies im the beaking plane. In the condition of Fig. 2.1 the method of Coulomb is accurate. The equation for projections of the forces onto the L-L-lime normal to F

iS

sin(y+B')+F cosß'FTOT cos(y+ ß') O

T0

where B' is the inner friction angle of the mass. When substituting the condition for breaking,

(2.1) r r+ o tanß'

where r0 is the cohesion (the shear strength at zero normal load) of the mass and o is the normal stress.

26

-And for a wall with a height b, détermiñing

F

:rh/siny

-to O

we get aibrmula for FTOT. per unit width pgh2tan(y+ B')

h coaß' FTOT - 2tany + siny cos(ß'+y) where is the solidity oÍ the mass (O no ice

i = only ice, no, water between ice blocks).

When a minimum for this is searched for- by varying y the breaking condition is determined. The minimum is found from the first derivative. For the vertical frictionlèB wallthe solution is y = (1i12-B'/2).

The method overestimates the results when the coeffi-cient of friction against the wall departs from zero, according to Refs.5and 7.The method also becomes inaccurate when the wall is inclined and when the bottom of the ridge is not horizontál. All these changes cause one or several of the action lines of the forces to change. This causes a rotating moment for which a correction should be made if theoretically fully analyzed. In any event, the actual

changes or these are not known and the momentum equation is at this stage disregarded. The theory will be developed as Coulomb's method, and the results will be compared with

appropriate test results. A departure from Coulomb's method will be made such that the friction between the ice and the wall is added to the analysis.

For convenience the balance equation is split into two separate equations-: the shear equation and --the equation for the force normal to the breaking plane. Thus the physically meaningful normal and shear stresses in the breaking plane

are obtained directly from the analysis, if required.

27

2.3 Initial resistance, thin ridges

2.3.1

In Fig. 2.2 an initial resistance condition is shown with the notation that will be used in the analysis. The thickness of the ridge may not reach be].ow'the ship's draught in the caseof a thin ridge.

In Sub-section 2.3.2 the relationship between the direction cosines of the forces and the hull angles are, derived.

The force geometry in the basic condition is formula-ted In Sub-section 2.3.3. The method for the analysis of the breaking plane (plane ABPL in Fig. 2.1), and the corres-ponding breaking force is derived in Sub-section 2.3.I.

The third dimension is introduced in Sub-section 2.3.5, where the breaking conditions are formulated for the so-called end planes. These are the planes HAL and OBP in Fig. 2..

2.3.2 Porse geometri

In connectiñ with initial resistance the analysis of force direction cosines will be done. Several versions of such analysis exist, but to date all of these have been done with assumptions appropriate to level ice resistance ana-lysis(Refs. 2, 12, 26, 35, 36). The earlier analyses were either done in order to determine the relationships between the net vertical force and the total resistive force, or donotncIude friction in a convenient manner.

For the purposes of this work the direction of the frictional force will be given two possibilities, the direction of the waterline and the direction of ship's verticals. The direction of the relative motion between ice

28

and depends in a complicated way on the shape of the

for-ward body' of the ship, on the côefficient of frictiori,..

ridge thickness and ship's speed. The direction of friction

for a landing craft bow shape is in the direction of the

verticals in the flat part of the bow and that for a bow

with' a vertical stern and vertical sides is in the direction

of the. waterlines. The more the stem angle decreases,.the

more the direction of friction departs from..this. The more

the half-angle of the waterline decreases from 9O.degrees,

the more the direction of friction departs from the

direction of verticals.

In this work both limiting cases are kept in the

analysis throughout.

The notation of the. hull angles is used according to

Enkvistef. 2) as in Fig. 2.2a. From Fig. 2.2b-d the

directiOn cosines for the normal force can be written

F FN ' FN

sin4' sins,

sin4 coss,

= cos*

FNXY

_-

Nxz

cosb

s invii sins

-. s nip, -ç- - cose -

51fl$

From the last of these the relationship between the hull

anglés cz,p,and v

can be determined and any one of these

may be eliminated bY expressing it in terms of the

remaining two..

Using the expression F

FN tanB, where ß is the

friction angle between the hull of the ship and the ice,.

thedirection cosines are developed for the total force as

follows (see Pigs. 2.2 and 2.3):

a) for friction in the direction of the waterlines

P:F

_{X C}

+F

_jx

FMX

coordinates xy-plané FNy Nxy 29 CL Nx FNxyslna 'Ny

FNXYc0S

Fig. 2.2,. Components of ñormal force.

FNXZ _cOse

'Nxy FNsin*

30

-F1, cosa FN tanß cosa

F F sini siria + F tanß cOsa

_-x

N N

- ç;; -

FN/co8ß

cosß (sirn, sins + tanß cosa)

F :

+F

y

Ny uy

FN sinI cosa

F siñ* cosa - F tañß sins

.Y_ N

- N.

- F - F /cosB

TOT N

(sint cosa '- tanß sins)

+ Fz

'Nz = FN cosl!I F 0 liz - F F costi, (2.) n'

FTOT NIcOSB - cose cosß

a) xy-plàne, friction in direction of waterlines

F F, cosa

Fuy=Fu

sins

F O

b) xz-plane, friction in direction of-verticals

Fig. 2.3 Components of frictional force. WL

ux F1 cose Fuy O

31

b) for friction in the direction of verticals1 we correspondinglY obtain

The apparent hull angles corresponding. to the plane normal to the total force are next determined by means of these direction cosines. The apparent angles are labelled cx', , S,.

(2.8) a' = arctan(l'/m') (2.9) 5' arctan(l'/n') (2.10) 5' arctan(tan5'/Sin')

It can be seen at once that, for example, j 1' the dif fe-rence between the two directions of friction is the

substi-tution of cos5 for cosa. Thus if the a and 5 angles are aboutme, there is no difference between these two direction cosines (Eqs. (2.2),(2.5).

2.3.3 Force geQTner! jn

In Fig. 2. the waterline of the ship is ABQ.., h is the ridge thickness below the waterline. Only a submerged ridge is considered in this model, which means that the whole ridge will be considered, but as if it were completely

submerged. The effects of this assumption on the results may be considered by a convenient choice of the coefficient of

friction in those cases where it may not be assumed to be constant everywhere. This is discussed in Section 4.2 in connection with full-scale test results.

-The plane ABCD is the side plane of the forward body of the ship. The part of the ridge that will be considered is ABPLHO, where the angle y is a variable expressing the

(2.5)

lt =

cosß (sin5 Sins + CO55 tanS) (2.6) mt = cosß sin5 cosa

32

possible direction of breaking planes in the ridge-.

Pig. 2.i. Stress analysis geometry for the side plane of, the forward body of a ship (ABCD).

A further assumption n the model is tha thé1

inoring of the plane AND of the ship's hüÏi will be :CÖm pensated for by the addition of BOC. The breaking, in addition to the yplane (ABPL), will be assumed to take -place in ALH and BPO-planes which will be éalled end planes. These planes are in the direction of the A-A-cut

in Fig. 2.14,(for A-A see Fig. 2.2).

-In addition,sliding öf- the ice against the ship's hull in the ABCD-plane occurs, and this requires us to consider a fully developed force of friction between

ship's hull, and the ice. Also, the ridge mass is assumed -to

be incompressible. -.

The forces acting in the model are; Fig. 2.5: As input, the. total ship's force,

TOT' transmitted

from the hull of the ship to ice. lXie to friction, it is inclinéd away from the normal to the hull plane by the amount of the frictional angle.

The buoyancy force of ice, FG, vertically upwards. A force in ari arbitrary, so called y-plane. This force

.33

is presented through its normal and tangential components, Faya

Forces in the ènd planes (ALH, BFO), presented through their normal and tangential compònents, the tangential component being F (not shown iñ Fig. 2.5) The analysis is first done by determining the breaking condition, when the forces acting in the end planes are neglected.

FIg. 2.5. Components of total ship's force in y-plane,

In Fig. 2.5 the components of the total, ship's force are presented in the A-A plane. In this figure. the positive directions for the tangential (shear) and normal forces in the y-plane are shown. It should be noted that i', the

apparent hull angle', is used instead of p. Thus friction is automatically included. The expressions for the compo-nents of the total force in the y-plane are

F = F ' sin(y+,'-9O°) -F cos(y+4,')

TOTlt FTOT1 cos(y+4i'-90°) FTOT1 sin(y+p')

In Fig. 2.6 the components of the buoyancy forée in they-plane are shown. ]n this figure the real î4i angle is used and the A-A plane is shown. The expressions for the

components are 'thus

34

(2.12)b F F siny

Gty G

is calculated according to

= G,

where expresses the buoyant density of ice and G is the. volume of ice corresponding to HAL.. Furthermore,

where b is the 1ength of the third dimension, (Fig. 2.4), and A is the area of ari end plane, given by -

-.2 .2 h2 1 (2.13) A _{= 2tany + 2tan* - T} + b B - 2slns Thu s l G - 4sina +

and considering both sides of the ship, gBh

(2.14) P0

2sin tany tan4i

Pig. 2.6. Components-of buoyancyforce in y-plané, -and the end plane.force.

2.3.4 Resistance .to-gØg

First Summing the forces in the y-plane Eqs. (2.11),

(2.1 2). and (2.14) (2.1.5)a. F _{T 1} 0y

OT

2

5L

-t)cosy

p,ghB

35

b. F.

'TOT1 + FGT

HPgh2B

TOTl'

2sin

tany

The corresponding stresses are nèeded for determination of the breaking condition. The area of the y-plane for two

ship's sides is (2.16) A 2 y siny 2sino - -Then (2.17)a. a = y y Ft b t y Ay

Thése formulae will be simplified by use of the following notation: hB -(2.18)a. Cl A siny sin C2 1 1 - tany tan

PDgh2B

C3

Then the stresses,Eqs. (2.l7a,b), are expressed as-(2.l9)a. ay = [FToTl cos(y+ip')+C2C3coSy]/Cl

b. ty z

_{sin(y+*')C2C3sinyl/Cl}

Next, the breaking condition, Eq. (2.1),

Ty z + ay tanß' is ransfomed to give ty - t0

which is then substituted into the first of Eqs. (2.19)

cos(y+ji')+C2C3cosy]/C1 2sina

.36

so that t1 can be solved:

t1 = tanBt[_FTOTl.eos(Y+tpt)+C2C3cOsY]/Cl+t This is equated against Eq. (2.19)b

ToTl

sin(y+')-C2C3siny]/Cl

T0T1 cos(y+')+C2C3cosy] tanß'/Cl+t0 and for

T0Tl we ohtain

r0Cl+C2 C3 (siny+ cosy tanß')

-(2.20) _FTOT1

sin(y+')+tanB'cos(y+*')

This expression will be used to determine the y-plane for shear breaking of the ridge mass. The desired y-plane is found by varying the value of y. The actual breaking

.flane, dénoted by

l' is the one

which

gives a miniwwu

value for _T0T1 The resistive force is then obtained by using the direction cosine 1', Eqs. (2.2) and (2.5)

(2.21) R1

2.3.5 Forces in the

In Fig. 2.4 have been drawn the planes HAL and OBP limiting thàt part of the ridge

which

is moving when breaking oécurs in the i-plane (ABPL). These two vertical planes present the simplest kinematically possible

conditions which allow for end effects. Also, after having assumed the y-plane breaking, the breaking of these two pianes represents the minimum energy solution. The planes.

(HAL, OPB) will be-called endplanesu.

When the ridge mass breaks along the three planes (ABPL, HAL, OBP), it may be examined as if it were a rigid body. Then in each end plane there will be a normal stress which is found by using the ridge Poisson's coefficient

(see Rets. 17, 28), proportional to the stress against the hull of the ship. The normal stresS (as) in an end plane

37

where is the stress against the hull.

FTOT1 cosß FTOT1 coeB sins sing,

hull area of Fwd ship hB

substitution gives

V COSB

_{SimS Slfl*}

hB(l-v)

The next step is to calculate the force needed for shear breaking in the end planes. The breaking condition is again _{TE= Oc} tanß'+ Then

y

TOTl cosß sins sin* tanß'

TE = hB(l-v) I T0

Tò exceed this stress in the end planes, a force is required which is found by multiplying the stress r by the area of all corresponding end planes. The area for

one end plane is(Eq. (2.13))

_h2,l

+ 1

2 'tany tanis

In the simple presentation of ship's geometry there are four end planes. For the-twO fôremobt of them the end plane force for breaking is

F

2A .

t

C C C

vF cosß sins sintp tanB' -2, l

,l

., TOT1 -

--- tan4, tan'r'' hB(1v) O

This force acts parallel to the r-plane and its resistive compOnent is expressed as

F.cosy

sins'

sin(y '*')

FOr the remaining two end planes a- pure cohesive breaking will be assumed, because in those planes the hull causes tension instead of compression. This includes the

V a =

E

v

38

assumptión that cohesion may be physically interpreted as the- -force needed-to cause continuous internaì ice --biöck

structure deformation. The cohesion itself then not only includes -the bonding of ice blocks by freezing or adhesion but also the force needed to cause a change in the order. of the ice. block structure. The resistance due to these two end planes is

R

2A

cosy

81fl

C to -sin(y+p')

The total ènd plane reSistance is

(2.22) -R

=R +R

ci c2

2.11 Thick ridges, initial resistance

2.11.1 General

-The -eondition of a thick- ridge does not present -any

experimentally interesting advantages over or -additions to these presented in Séction 2.3-. The condition is briefly

touched upon, becausé in the developed condition the lower

breaking plane is important.

-The basic analysis would be as presented in Section 2.3 but the tact that the ridge- has a portiön above ship's draught causes some changes in the constants and

introduces :anadd±tionai breaking plane y2.. The geomet.rj-: of the condition in. the A-A plane is shown in Fig. 2.7.

-A-A-PLANE

-r s - L

Fig. 2.7. Initial geômetry for a ridge, thicker than.

-39

In the following $ub-sectionsthe matters are dis-cussed which are characteristic of this condition.

2.11.2 Conditions of the lower breaking glane

When h > T, the breaking must occur in two planes, Yj and y2-planes, to allow the kinematic conditions to be fulfil-led. These planes must for kinematic reasons be parallel in the direction given by y1. Henceforth, the notation y2 will in some cases be used for the lówer breaking plane to show clearly the local difference between the two breaking planes.

In the initial resistance conditiön the lower breaking plane does not have an equivalent plane on the experimental level and thus no analysis is needed.

2.11.3 Changes in the y1-plane analysis

a dynamic case the buoyancy forces due to the ice mass corresponding to

RSE of Fig.

2.7 will not have any

effect on the breaking in the y1-plane. Thus the mass corresponding to ALSR will alone give rise to force components in the y1-plane. The resulting change in the analysis is such that C3 will be given by

- T2).

(2.23) C3 = 2sina

-2.11.11 Changes in, the end Liane ana1siS

End plane breaking takes place in the ALSR plane instead of in the original ALH-plàne. The change in the añalysiS is correspondingly given by substitution of the

original A Eq. (2.13) by

2.5. Developed resistance condition

2.5.1 Gener1

The principle for analyzing the resistance of the ice mass against breaking, presented in Section 2.3, is also applicable to the developed resistance condition. The main difference lies in the amount of ice that has to be taken into account in determining the buoyancy force. This means that the constant corresponding tô the amount of ie--(C3) must be analyzed for the developed condition separately. Before this can be done, a model must be devised for the devèlopment of the ridge profile around the hull of a ship during penetration. Such a model is presented in

Sub-section 2.5.2 for a landing craft bow o show the basic principles for the growth. In Sub-section 2.5.3 ageneral model for ridge development is presented based on the principles contained in Sub-section 2.5.2. The results of

this model are given in terms of the constant C3, appli-cable to the original calculation method, Eq. (2.20).

-In Sub-section 2.5.ZI the conditions and resistance due to anothér, lower breaking plane are analyzed.

2.5.2

To illustrate the logic of the analysis, a kinematic ridge growth model for a landing craft bow will be deve-loped. Only the case of a ridge with h < T will be consi-dered.

According to the breaking force analysis in Section 2.3, a straight rupturé liné is assumed for the kinematic model. When the y-planes àre used for sliding, the deve-lopment of the ridge mass profile will take place as shown in Fig. 2.8, for a ianding.craft bow.

In this eìample a developed resistance condition in the forward body area is reached in Fig. 2.8(7).

0

Lu

Fig. 2.8. Development of ridge profile arouñd a landing craft bow.

M IS ERE SHIP RIDGE A £

A'7

--i

4?

not important for the constants in the breaking analysi&j but the amount of moving ice mass is. .Neverthe].ess the

geometry is important as a basis for determination 'of the development of the profile 'in different ridge thicknesses and for different bow shapes.

The geometry 'of completed ridge growth around the forward body is shown in Fig. 2.9. The initial analysis is changed such -that tbe buoyancy force ls-increased-:by an

amount corresponding to FEt.T of Fig. 2.9. Thus Thu p gB sin(+y) p (:BEIIN)

OTOT sine amy

Pig. 2.9. Developed geometry, landing. craft bow.

XZ- PLANE

Fig. 2.ìO. Penetration for ridge growth, landing craft bow.

The distance of penetration, starting from the initiai resistance condition, that is needed for the completed ridge gròwth in the bow area is seen in Fig. 2.10 as NN'.

43

In Fig. 2.10 the distance in front of the bow where the ridge starts to get thicker in the developed con-dition is AH. The expressions for these quantities are

NN' :T(!__+

tany h AE

tany

For large values of -angle, which is at the same time the angle for ridge growth in front of the bow

(see Fig. 2.9), the ridge mass cannot stay piled up with a wall steeper than the natural slope of a ridge keel. However, in a dynamic condition this is possible and also, as it is now set up, the geometry itself does not have any relevance in the analysis of the forces.

2.5.3 General solution for ridge growth

Up to thispoint we have considered the geometric mass shown in Fig. 2.11 as DE'C'A. For a kinematic model it is of importance to use the kinematic mass, which is given in Fig. 2.11 as DECA. The difference arises from consideration of the direction of the frictional force in the breaking geometry.

When the half angle of the waterline, a, is less than

900, _{and so departs from the landing craft bow type, the}

ridge grows according to a different geometry. In a

landing craft bow the ice moves in the xz-plane only, with all ice passing under the hull. In the case of a normal bow shape, the ice moves also in the xy- and yz-planes, accor-ding to the same principles. In addition the penetrations needed for the developmèñt of the ridge resistance have a different expression. Fig. 2.11 shows the x-y image of one basic geometry of the ridge growth. In this figure it can be seen that a completed ridge growth around the forward body (:corresponding to the case of a landing craft bow type) will be achieved in the area AFEB, only. The corres-ponding bottom surface image in an Xy-plane is shOwn

jfl

Fig....2.12.. The bottom profile, of the developed ridge is

-such -that the ridge reaches its greatest depth in the line 3K (Fig. 2.12), and behind that when penetration is continued. At the points E, D and C no ridge growth will take place.

a) HD'BI2

b) HD> B12

F-1g. 2.11. Geometry for growth model.

For determination of the buoyáncy force the volume of the. ice will be divided into thxeè separate. parts, which are shown for thin (h T) and thick ridges (h T) sepa-rately in Fig. 2.13 a and b.

Fig. 2.12. Bottom of developed ridge. o) h T

b) hT

145 DEF V1 FEBA V2 ABC V3

Fig. 2.13. Division of ice masses for the developed resistance condition.

h2B1iP

(2.25) _{FG1=' 2sins}

or

Zi6

Determi:flation O1 Fj,:jS straightforWard' and for--the.

wboIe shIp we obtain.

+ when h T

tarn' tanb

T Bppg

('2.26) Ff

_2si + ,) when. h - T... correspondingly, the expressions for F2 in the two

cases- will be, for tÌíe whole ship

hB(T-h). 3. 1 C (2.27) F02 sins + when h T arid TB(h-T) uo g

G2 sins tan'r tanii

The determination of F Ì more complicated and, in

addition, to the division due' to ridge thickness, another divisionI will be made into two subcases which are

shown-in Fig. 2.11 -a and. b shown-in xy-piane views.

The difference between- the cases shown in Fig. 2.11 a.

and b is that in the case (a) the ridge growth in the line. JK will reach the maximum corresponding to complete growth for the case of a landing craft bow, whereas in the case (b) the thickness. growth will not reach the same depth in-the- JK-llne, Fig. 2.12.

-For determination of the total ice vòlume' G3, the volume will be divided into three: parts V1,-Y2

anV as

shown in, Fig-. 2.12. V1 -and -V3 are equál, by symmetry.. The general. formixlae' or F3 i-s thus, for-the' whole ship,

(2.29) F03 (V2 + 2v1) 2

-and the generai expressions for V1 and V2 will be,. for

-FÊ.h1.DH

6

FE.h.HG

2

where Figs. 2.11 and2.13 show FE,DH, HG and h for the different eases. In the following we shall derive

expressions for these. The cases of Fig. 2.11 a and b are considered separately -md eacI öf theth is dividéd still further into two subcases for h < T and h > T. respectively.

he case of Fig. 2.11 a. This case is characterized by the fact that DH < B/2 and V2 is thus limited by the breadth of the ship. First the case h < T is considered

(Fig. 2.13a).

In this case h h

FE FH + HE

DE' ama' h sifla' J. l HE DE

co5(a'-a) - cos(a'-a) tan4' tany

DE DE cosa' -. h cosa' 1 1

- tana

tan

tana cos(a'-a)tafl* tany PE - 1 -1--)h (aj' tana+cosa')

tanip tany tana cos(ci'-a)

h cosa' J. - l

coS(a') tarni, tany

2 hh2cosat(sinattana+coßa) Vi=(tan* tä 6cos2(a'-a) tana

HG - Dli

B.hcosa'

I (2.30) HG 2 cos(a'-a)tan4i taziy Dli 147 h h(sina' tana+cosa') y

(1

1

)X.

2 - tant tany 2tana cos(a'-a) B.h cosa' 1 1

[rcos('(t

(for oneS ship 's side) (for one ship's side)

hus fo the whole ship, substitution in Eq. 2.29) gves 1 1. h h tana+cosa' F03- _tan,tany X taflacos(OE'-a) COSU' 1 1

3 eos(a-a) tan* tany

8h òosa'

i i

+ r cos(a'-u)

tan1p

2upg

Eventually, with the notaton C2

(2_t

-C2 hh(sina' tan+cosa')Ii8Pg

(2.31)

_{tana cos('a)}

b cosOE' C2).

2 3 coB('-cz)

Next, for h . T (Fig. 2.13b1, the equation will be changed.

Then h= T and

T ama' HE - cos('-a) H T cosa' C2 - tana cos(a!-a) T(sina' tana+cosa') C2 tans cos(a1-a) T cosa' C2 DB _{- cos(a'-a)}

C?2 hT2 cosOE' (sins' tana+cosa')

-V1 6 cos'(OE'-rz) tans

_B_Tcosa' C2

(2.32) HO - _cos(a'-s) (for oñe. ship's side)

C.2 hT(sinu' tana4cosa!)

_{B T co' C)}

V2 =

2 tans cos(a'-a) 2 cos(a'-s) and by substitution in Eq. (2.29),

C2hT(sim'tam+cosa')Ii8Pg

_{B T cosa'C2}

(2.33) F03

The case of Fig. 2.11b. This case is characterized by the fact that DG > B/2 Then DR is constant (=312) and

< h. First, for h < T (Fig. 2.13a), FE.: AB = 3H + AH (2;3) B tana' BH DR tana' 2 FE + tana')

For h a linear interpolation is made such that approaches zero when HG approaches DE'. Thus

HG = h(l - T)

hcosa'

,l

₊ i

cos(a!-cs) tan* tany

h cosa' 1 1 B

(2.35) h hO. cos(a'-) tanil, tany h(t11,

1)

By substitution,

B2h

_h cosa'

.1 1

- T(tanatt1')(l

2 cosa'-a)tan, tany )B

AH : DHltana and ¿ . 'tarn tany cosa' C2 B h(. 1 , tana' )(.l 2 cos V tarn h C2

For h T (Fig; 2.13b) we correspondingly obtain B 2tana

B hcosa'

B h :.i , 1cos(a!_«)c2 B h cosa' C2

V2__fl(tantt9X1 (1 h C2 'cos(a'-cz) Thus B hcosu'C2 )( B hcos'C2 (2.36)

_{FG3-_(Ej+tanat)(1}

2

t.J

'50 .T ccsa'C2 (2.37)

h :-T(1

cos(a'--u) - T cosa.'C2 B. (2.38) HG -'T cosa.' C2 B _____ cos(3'-a) 2)B Tecsa' B

_BT,I

,., V2_4....tan1:tan T C2' os(a'-a) 2 Tcosa'C2 B ' -_BT 1 -cos('-c*) 2 B TcOscz'C2

(2.39) FG3_(ttana!X].

_{T C2}

The correct case oüt of, these four is chosen by the

division between .h .< 'T and h > 'T. By' calculating ..h qs:. (2.35)(2.37) the additional subdivision may be done. If exceeds h whén h< T, Eq. (2.35),' the correct 'G3 value is

found

from

Eq. (2.31) and if hx exceeds T when h > T, Eq.. (2.37), the correct F03 value is found from Eq. (2.33).

When h. does no.t exceed these limits, the correct 'ira-lue

for FG3' is found for h < T from. Eq. (2.36) and for h > T from Eq. (2.39)', respectively.

The proper combination of -FG1 Eqs. (2.25-), (2.26),

G2 Eqs. (2.27), (2.28) and F Eqs. (2.31), (2-.33'), (2.36),

(2.39) gives C3.fox the original method Eqs.. (2.20), (2.1k)

(2.110) C3 _{(F01 + F02 + FG3)iC2}

a.5.1I esistance:due to the 'lower breakin.2ìne

In the developed resistance. ..carfdition

'the

-lomr breaking plane also 'assumes 'importance. When the ship to ridge interaction -is not sta-tic, even though only the slow motion case is studied, it is 'evident that forces are

direc.tèd' from

the

hull to the lower breaking plane 'which do not cause compression but rather tension 'in, the lower. breaking p]ane. When dynamic conáidèrat'ïons enter', it

is-51

alsò clear that the ice below the lower bréaking plane does not provide any buoyancy forces in that plane. It is evident rrom the kinematic point of view and from model test observations that thé ice mass does not behave

differently in the front of the, bow of a ship at different speeds, except for at high velocities, where the ice is thrown down so that it submerges telow the depth assumed

in the kinematic, model. . .

Let us now interpret the breaking of the ice mass in the lower breaking plane as a cohesive shear break

(compare Sub-section 2.3.5).

The area of the lower breaking plane, for the deter-mination of resistance, when h T, is given by the

quadri-lateral with parallel sides AD and JK (Fig. 2.12). The formula for JK is

-. JK = HG/sin

HG is found from the appropriate one of Eqs. (2.30), (2.32).

The height of' the quadrilateral, is JL JL = h/siny2 .

'Thus.the area for one plane is (Ay2)

--' B ,HG

- 2sin ama

For the whole ship

(B+2HG)h

(?.Il) Ay _2sinasifly2 (h < T)

When h > T,Ay2 takes on another form. In addition to the area given, in Eq. (2.kl) the area of the 'lower breaking plane consists of the area between A and B shown in Fig.

2.l3b . This area is expressed as

Ay2 = Ay2 (for h.T) + (h > T)

The expressions for HG must now be chosen from Eqs. (2.31),

(2.38)

52

The resistance' due tò cohési've breakieg of the -plán'è

isthen

(2.3) Ry2 = Ay2 COSY2 cosa'

2.5.5 Area of the end 2lanes

The end planes can be seen: from Fig. .2.13 to be rhombusès. The expression for one end piane is, instead :o.,.Eq. (2.13'), ror all values of h

A

_{E -}

Th.sinçY+*-)

amy

s].rllp

2.6 Parallel middle body

2.6.1 enera1

In ships which have a parallel middle body, this part of the hull will cause a resistance component, which

is connected with the amount -of ice mass around the

-parallel part of the ship, at the sides and beiowthe ship's bottom. This resistance component is a linear function of the coefficient of friction between ship's hull and ice with the assumption that no change-:inthe:-ice

profile will ocmur due to change. in. this coefficient.

-Even a ship with -no parallel mjdd± body has approxi-mately flat bottom and sides, as can be seen- in the

repre-sentation of a ship's hull considered 'in this work (Pig. 1.51. The length of this apparently parallel body is-shown in Pig. 1.5 -as DF + MJ-and is expressed as

-(2.lIli.) L' = -T/tan$ -. .

In Sub-section 2.6.2 the pröfilè of thé-ice around the-middle body will be analyzed.

- In Sub-sectioñ 2.6.3 the frictional resistance due'

to the ship's sides i-s analyzedusing the profile-from Sub-section 2.6.2 as input data.

-53

In Sub-section 2.6.4 the frictional resistance due to the ice beneath the ship is analyzed.

It is important to get data on the length of a ridge field at which a developed resistance condition will be reached and when the icé starts to accumulate under the ship. The penetration lengths at which the different stepS of the penetration are reached are analyzed in Sub-section 2.6.5.

2.6.2 Ice profile around the arallel middle body

The development of an ice ridge in the forward body portion of a ship determines what kind of ice mass cross

section enters the parallel part of the ship. This kind of cross section, when it once has reached its maximum

dimensions, will be assumed to pass unchanged along the whole, parallel middle body during penetration. Underwater observations in the ice model basin support this kind of

assumption in long homogeneous ridges.

The input data for determination of the ice profile is obtained from the analysis in Sub-section 2.5.3 and there are correspondingly four cases to be considered separately, two subcases for each of h < T and h > T. Figs. 2.14 a-d present these cases. For clarity, the xy-planes are also drawn, as in Fig. 2.12. A basic requirement for the cross-section DH''G''CQ'H' is that its area has to be

equal to that of the corresponding equivalent cross-section of the submerged part. of the main beam of thé ship. In addition, for symmetry reaSons, DH''H' E CG'G'' in each

case.

The dimensions determined in Figs. 2.14 a-d are good for the idealised conditions of a vertical sided Ship only. Also, the vertical slope of the ridge mass shown in Figs.

2.l a and b is purely theoretical with no corresponding equivalent in natura.

The vertical wall of the ridge mass is eliminated by allowing the masS to break, as shown in Fig. 2.15. The amount of ice that breaks and moves away from its original

54

h <T

DG

h'csc'C2

- cos(u'-) HG

-HD

HI) H'H'' OtGit: h: h h < T. DG ND HG r >3,-2 B -.

hos'C2

B

- cos('-)

H'H' G'G'': h: DE': hC2

Fig 2' 1' ¡

i

:profilè. around

55

h >T

DG T cosa'C2 - ìos('-a) -Dl! HG DG - Dl! H'H'': G'G'' = T

h >T

DG > - -HD

T COS'C2

B HG - cos(a'-a) 2

= GG

= h DE': T.C2 (Pig. 2.1k cont.)

b)

Where

Fig. 2.15. EliminatIon of the vertical ridge Wall.

position,üflder the hull to the sides is given iñ Fig. 2.15 a by

ABCE. This

area is calculated by substracting

DEC from ABP

h2 (2.k5) ABb

_-

_2tanß' DEC DC DE

smB'

DC :DB-CB

with

h

DB

-

tanß' 56

AB:h

CB HO ofFig. 2.l4a KLMN ABCE AB = H'B'' of Fig.2.14b .KLNN = ABE

and

CB

_{- 2 -}

B

h COS'

C2 cos(a'-a)

Thus

h

B + h COSOE' - C2 DC

tanS'

2

cos(a )

Further,

DCsin atri

cos(s'-)

DE

C2cosci'

Thus

h2

1 ,h

B+hcosa'

2

(2.146)

ABCE-

_2tanB'

2

'tanß'

- cos(s'-a) C2)

sin atri

cos(a'-a)

C2cosa'

_sins

sinElBö - atn°

B']

If DB

CB there does not exist any DEC triangle and

instead of ABCE the breaking mass isABD.

Eqs. (2.45) and (2.6) apply for a single ship's side.

The additional amount of ice now at the sides of the

hull is the sane as has bròken away from below. When this

ice is assumed to spread out evenly,, the additional depth

of the ridge at the sides is

ABCE

B/2

'CB

Thus the limit where this kind of total breaking to.the

side takes place is at ridge thickness

sinEl8O -

atn

_{- coscz'.C2}

cos('-cl)

2 b+

ABCE

_{- T}

- CB

-57

When h

T, the other limit is reached where no, vertical

wall exists and no breaking takes Place.. Between these two

extremes a linear interpolation of the amount of broken.

ice can be made.

-

-In the case of Fig

2 15b (corresponding to 2 114 b)

the same type of determination' is done ror the breaking

ice mass.

ABE ABt) - BDE

BDEBD

DE.

ITIB'

h coso'. B BD h1 h -- tanß' -- tanß' C2 h DE = BD siñ atn 58

sin(l8- atn

-h'

(?7)

ABD - 2tanß' Urns (2.148) ABE = 2tanß' tanB'

(h.:,)c2

-sin atn _B/2 .

- --- -

sinß' Bifl(180° atn - B')

wicìi are ror one ship's side,Eqa. (2.1461,(2.148). The additional depth of.the ridge at the side Is

ABE h COSO'

cos(o'-aÌ -

-and the Limits for total breaking -and no breaking are

h+h

ABE

x h cosa'

C2-and

reapeet;ively. Again, interpolatión between these limits is doné ilBearly.

PiBl1y, the tota]. amouñt of -ice below the hull is

for the whole ship for APB (under the flat bottom) and for one ship's side for A3 ( at the parallel side). Case a)(Figs. (2.144)a, (2.15)a)

(2.19) APB h(B _- C2) - ABCE 2 (B - h tanB)i - APB 2 (2.50) hs 2h ASCE Case b)(Figs. (2.11)b, (2.15)b) -hcoss' B (2.51)

A5 = (l

cos(a'-s) :2 ABE 2 A (B - h tan8)h. - APB PS.-(2.52) 59 ABE f-h + h

COS'

2 cos(cz Case c) (Fig.-(2.114)C) (2.53)

A5

(h - TB + T(B -

_CosOE'-

C2) A (B - T tan)T PS 2 (?.5) h5 h + T Case d) (Fig. (2.1'4)d)

T cos2

-(2.55) APB - (h - T)B BT(1 cos(;'-a) 2 (B - T tanB)T_A5 Ap5

₂

(2.56) hs = h +

In. these equations h5 is the pressure height of the ridge and B the slope angle of the parallel side.

a.6.3 Side resistance

-6-o

The resistance of ice against the parallel part of the ship's aldea. is indisputably a purely frictional resistance due to friction between the ship's hull and the: ice.

In Sub-section 2.6.2 the geometri of the ice sur-rounding. the parallel middle body was analyzed so that the pressure. against the hull can be expressed as a function of the ice geometry.

The pressure in the' vertical direction is

pgh and

in the horizontal direction, 5p6t'(1-'j)1. When the side of the ship Is inclined at an- anglè B, an interpolation of the pressure for any desired B-direction will be made as

(2.57) P = + sinS-),

hs expresses the pressure height, P5 is the pressure-. When h3 < T, the average pressure height against the side of the ship = hsl2 (h3 Eqs. (2.50), (2.52)) and when hs > T this is = (2h3 -- T)/2 (h5 Eqs. (2.5-1k) and

(2.56)). The resistance is then

(2.58)

R3 = A5

ßhS '

-where A3. is the contact area between the parallel ship's sides and the ice, is the average pressure against the ship's sides and i is the coefficient of. friction between hull and ice. Thus

A3 = 2.L'h3/cosß when hs < T

or

= 2L'T/cosB

when b3 T, for the whole ship.

2.6.l ttom resistance

The maximum resistance due to ice under the bottom is

considered in the situation where the whole length of the bottom is coveréd by ice. In. this case the resistance is a

61

purely frictional force corresponding to the buoyancy force of thé ice under the bottom. The buoyancy force is

F

-A

PB - PB B"A p

with APB from Eqs.

(2.49),(2.51),(2.53)

or (2.55). The resistance is then

(2.59)

RPB

UPBP'UBP

The resistance is a linear function of the length of the parallel middle body covered by ice , stàrting from zero at the bow end.

2.6.5

Penetrations for ridge growth

Fig.

2.16

shows an A-A-section of a ship with the notations used here. for ridge growth. The different penetrations are markèd, starting from the initial

-resistance condition. Their identification Is

11 the distance of penetration required to reach

the developed resistance thickness ( maximum thickness)

12 = penetration required before (additional) ice starts to pass under the bottom

13 = distance of increasing ice thickness under the bottom

14 = (= 12+13) penetration required to reach the maximum ridge thickness below the bottom of the parallel middle body (= the minimum length of ridge to achieve developed resistance condition) l, the distance from the front of the stem at

which the ridge growth starts

li = penetration into the ridge before the initial condition is achieved.

The expressions for these for h < T are given in the direction of the ship's movement, by. dividing the pene-trations in the A-A-plane by siria, as follows:

'2 6Ò'

i

T

i

+

i

A-A.- SECTION

INITIAL STEM: SITJON.

Fig. 2.16. Penetrations for ridge growth.

For 12

the volume- ( or areas) of ship penetration

and' that of ice growth are set equal, when ice starts tò pass under the bottom.

h( + (T - h)

_{= 12 sinh}

thus

- (T. - h) ._.i.__ ₊

(2.ol) _{12. - sins 'tany tan4,}

13 = I _{+ l} - 12 w here = tarfl' sins (2.62.) h + (2.63) 14 = 11 + 1,. + tany'sins (2.64) tani sins STEM SITIQN: IN DEVELOPED RIDGE NDITION _WL

- When the thickness of the ridge is greater than. the

63

for 1, i, ana i. The other expressiáns become

(2.65) ₁₂ lY (2.66) 13 - 1]

'26"

' T

i - tanijí sins

From Fig. 2.19 the ahgle 6 is déteìminèd äs

h

(2.68)a. 6 atn

13 for h < T and h < h

b. 6 atn

.13 for h > T and h < T

2.7 Summary of resistance components

In the preceding sections(2.3 through 2.6) the pure ridge resistance (RTOTO) has been analyzed into its components. Thus RTOT0 is expressed as the sum

(2.69) _RTOT0 R11 + R12 + R _{+ RFB + RPS} where the components are obtained as follows:

R11(2.21),

_R2(2.43),

R - (2.22)

6

3. COMPARISON WITH MODEL TEST RESULTS

3.]. General

In Wärtsilä's Icebreakiñg Model Basin, the first year ice ridges have normally been modelled by means of a homogeneous ice block structure with constant thickness, Ref. 6.. Fig. 3.1 shows a, schematic picture and photographs of this kind of ridge. The structure has been slightly

frozen on its top. These, kinds of ridges are called large. homogeneous ridge fields. As was established in Section .2.1, when accepting the homogeneous large. ridge fields for theoretical consideration, this kind of large homogeneous

field is the most severe that a ship may encounter.

For the comparative purposes of this work, the follo-wing sets of model ship test. results- are used:

-In addition, the characteristics of an ice block structure similar to that in model ice ridges have been studied separately, Rets. 18, 28 (1978).' The data on the ship models are given in Tables 1

-In Section 3.2 the characteristics of the broken, ice block structure are presented as they are determined in Refs. 17, 28. Also,, comparisons are nade between the theory and the basic experiments on the ice block structure of Refs. lÎ 28.

-In Section 3.3 the theoretical and experimental results of pure ridge resistance for ship models are compared. In Section 3.14 a sensitivity analysis is presented for the theory.

Test series Originally reported

- MODEL A Ref. 6

- MODEL D Ref. 13

- MODEL E Réf. 13

65

a) overview

b side view

Fig. 3.1. _{Modelled large homogeneous ridge field (a) from} above and (b) from the side.

1.5

1.0

0.5

66

3.2 Basic tests for theory

Shear test results on the broken, ice block structure of the model ridges (Refs. 17, 28) are shown in Pig. 3.2. The cohesion of the ridge block structure is il N/rn2 and

the inner friction coefficient 1.078. Thus the breaking equátion is

(3.1) = 1]. + 1.078 a (N/rn2)

SHEAR. STRENGTH.

tkN /m4]

AVERAGE ICE BLOCK

THICKNESS 20mm

I t=0.Oi1,1.078xS (kN/m2

0.5

10

NORMAI.. STRESS. ¿

IkNlmZ)

FIG.3.2COHESIVE AND SHEAR STRENGTH

0F RIDGE MASS

. _aq

30

RESISTANCE

[N i

9-_____.-_

---20

AVERAGE ICE BLOCK THICKNESS 22mri

#

7.

/

I

/

lis., -9

/

FROM EXPERIMENTS REF. 17

HULL

0.16

-FROM THEORY

/ç.

,C"ALCULATED

FROM THE THEORY

OF PLASTICITY

INCLINATION ANGLE

10°

20°

300

LO°

50°

60°

700

800 900