PROCEEDINGS PART 1, .A t 0 MGI , RESE A

. l',~ r!' d'fG L SORAl Y -A,' l : ~ 12 Lyme Rb 0IAHR Hanover, ' H 0375

SYMPOSIUM ON

. ICE PROBLEMS

LULE!SWEDEN

August 7-91978

. ""

u.s. ,ll.nMY C!J

IAHR

INTERNATIONAL ASSOCIATION

FOR HYDRAULIC RESEARCH

SYMPOSIUM ON

ICE PROBLEMS

PROCEEDINGS

PART I

ICE ORCESON

STRUCTURES

IAHR - INTERNATIONAL ASSOCIATION FOR

HYDRAULIC RESEARCH

is a worldwide organization of people and institutions working with

hydraulic problems of all kinds. IAHR was founded in 1935 and has

now about 2 ()()() individual and 265 corporate members. It organizes congresses covering the whole hydraulic field every second year. In order to arrange more thorough discussions of specific aspects a number of sections are established. some of which arrange special conferences - symposia - in congress-free years. The Section for Ice Problems has organized symposia in Reykjavik, Iceland 1970, Leningrad, USSR 1972, Budapest, Hungary 1974, Hannover, USA 1976, and now welcomes all interested to the fifth Symposium in LuleA, Sweden, 7-9 August 1978.

SPONSORS The Symposium is organized by The Town of LuieA and The Division of Water Resources Engineering, University of LuleA in co-operation with The IAHR Committee on Ice Problems.

CO-SPONSORS UNESCO

WMO

The IAHS Commission on Snow and Ice

The International Glaciological Society

ORGANIZING COMMITTEE SVENKOHLER

CHAIRMAN

BERTIL KOHLER SECRETARY

LARS BENGTSSON

ROLF BOHLIN

JIM SANDQVIST

GORAN WESTERSTROM

INTERNATIONAL COMMITTEE o STAROSOLSZKY, HUNGARY

CHAIRMAN G D ASHTON, USA

SECRETARY W BALANIN, USSR

L BENGTSSON, SWEDEN

G FRANKENSTEIN, USA

R FREDERKING. CANADA

E KANAVIN. NORWAY

T J REKONEN. FINLAND

J SCHWARZ. WEST GERMANY

J N SOKOLOV. USSR

T TABATA. JAPAN

P TRYDE. DENMARK

PREFACE

In northern areas ice and snow constitue a significant role in the every day life for half the year. Ice problems occur in the oceans, in coastal zones, in lakes and rivers. Costs associated with ice are in the direct form due to accidents such as clogging of water inletes, lost energy heights at water plants and failure of constructions as well as indirect costs such as necessary increased dimension to make constructions withstand the ice.

Ice engineering of today is still very much based on empirical methods. The IAHS Section for Ice Problems aims at a better understanding of the mechanics of lake! river ice and ice in coastal areas.

The Fifth IAHR-Symposium on Ice Problems in Lulea Sweden 7-9 August 1978 have four themes -ICE IN ESTUARIES ON HARBOURS, MECHANICS OF BROKEN ICE MASSES, ICE FORCES ON STRUCTURES, THERMAL REGIME ON ICE COVERED WATERS of which the first subject is of particular interest to the town of Lulea.

In the proceedings about 65 papers are presented in two volumes. It is my hope that these proceedings will supply a source for new ideas and concepts.

Since the intention was to publish the proceedings in advance of the symposium the quality of the printing may not always be of perfect quality.

This symposium is sponsored by the Town of Lulea and the Division of Water Resources Engineering, University of Lulea.

The organizing committee was headed by Mr Sven Kohler, First Mayour of Lulea, to whom I am most grateful.

The complection of this symposium is due to the efforts of many people. Particularly significant in this regard was the support and effect of Mr Bertil Kbhler and Mrs Inger Astrbm of The Town of Lulea and Mr Jim Sandqvisl of Div. Water Resources Engin. whom I thank.

I would also like to thank professor 0 Starosolszky, president of the IAHR Section of Ice Problems, for his everlasting encouragement and for valuable help.

Lulea, Sweden, June 1978

LARS BENGTSSON

prof head Div Water Resources Engineering

LIST OFINVITED LECTURES

J F KENNEDY, USA

General Lecture on Ice Problems

T CARSTENS, NORWAY

Ice in Estuaries and Harbours

G FRANKENSTEIN, USA

Mechanics of Broken Ice Masses

YU N ST ARSHINOV, USSR

Ice Forces on Structures

P LARSEN, SWEDEN

Thermal Regime of Ice Covered Waters

LISTOFPAPERS PAGE

TRENDS IN THE DOMAIN OF ICE HYDRAULICS 1 oStarosoiszky AN ANALYSIS OF ICE SHEET INDENTATION 13 Ternnce D Ralston A STATISTICAL PREDICTION OF EFFECTIVE ICE CRUSHING STRESSES ON WIDE STRUCTURES 33 PRKry SEA ICEFLEXURAL CREEP 49 Rene TiDawi, JeanRobert Murat PROBABILISTIC ICE-STRUCTURE INTERACTON THEORY 77 F G Bercha, J V Danys, J G Rokne EVALUA TlON OF CREEP PROPERTIES OF SEA ICE BY MEANS OF A BOREHOLE DlLA TOMETER 'TI B Ladanyi, R SaintPiern BUCKLING ANALYSIS OF A SEMIINFINITE ICE SHEET MOVING AGAINST CYLINDRICAL STRUCTURES 117 Yung-Shih WaDg EXPERIMENTAL STIJDY ON THE TESTING METHODS OF STRENGTH AND MECHANICAL PROPERTIES FOR SEA ICE 135 Hiroshi Saeki, Tosbio Nomura, Akira Ozaki FIELD ICE STRAIN MEASUREMENTS 151 Anton ProcIaDovic MODELING TIlE INTERACTION BETWEEN PRESSURE RIDGES AND CONICAL SHAPED STRUCTURES 1~ J lltk W Lewis, Kenneth R CroasdaJe

THE FLEXURAL BEHAVIOUR OF ICE FROM IN SITU CANTILEVER BEAM TESTS 1'T1 R Frederking, FU Hausler A PROCEDURE FOR THE DETERMINATION OF ICE FORCES - ILLUST RATED FOR POLYCRYSTALLINE ICE 117 Kurt M Heinicke, Rolf Remer EFECT OF ICE FORMATION ON HYDRAULIC STRUCTURE ELEMENTS 139 S M Aldnlkov, A I Pekhovi ,E L Razgovorova SEA ICE PRESSURE RIDGES IN THE BEAUFORT SEA 149 B D Wright, J Hnatiuk, A Kovacs MECHANICAL PROPERTIES OF IMPURE ICE SINGLE CRYSTALS AT HIGH TEMPERATURES 173 Tsutomu Nakamura A NEW ICEBREAKING CONCEPT 193 A Freilas, J Mhwan: ICE-STRUCTURE INTERACTION STUDIES ON A LIGHTHOUSE IN THE GULF OF BOm NIA USING RESPONSE SPECTRUM AND POWER SPECTRAL DEN SITY FUNCTION ANALYSES 319 M MiittliDen, D V Reddy, M AroclLiasamy, P S Cbeema

MODE OF FAILURE AND THE ANALYSIS OF ICE LOADS ON BRIDGE PIERS 335 RGerard PROBABILITIES OF THERMAL ICE PRESSURES IN FIVE SWEDISH LAKES 349 Lan Bergdahl, Lan Wemetll80n EXISTING ICE CODES AND SUGGESTED CRITERIA 363 SH Iyer

MATBEMATICAL AND PHYSICAL MODELLING OF ICE 379 H R KlvisDd, S H Iyer

INTERNATIONAL ASSOCIATION FOR HYDRAULIC RESEARCH

4th International Symposium on Ice Problems

Trends in the Domain of Ice Hydraulics

By Dr. O.starosolszky, President

Section on Ice Problems, IAHR

(Research Centre for Water Resources Development,

Budapest, Hungary)

Synopsis

The volume of information on river-, lake- and sea ice has increased considerably during the last decade. The dissemination of recent advances in this domain has been greatly promoted by the activities of several international associations and organizat i ons, including the IAHR Section on Ice Problems, further by t he international symposia held at Reykjavik, Leningrad, Budapest and Hanover.

Beyond regular exchange of experiences, the international organizations and associations dealing with ice problems may great ly promote the development of science and technology by est abl ishing international working groups, making more efficient use of the intellectual capacity available through division of labour.

In the interest of coordinating and organizing activities, a general survey is deemed desirable on the present state of knowledge and on the problems requiring to be solved urgently at international level.

A brief review is thus presented on the knowledge related to the formation, movement, effects of ice as well as to the potential methods of influencing these, indicating the domains of research where phenomena can be cleared and more effective procedures can be developed by international cooperation for planningengineering measures.

Introduction

Round 77 per cent of the f reshwater resources of the Globe are s tored in the form of i ce. The development of polar regions receives ever growing a ttention , the northern countries, par ticularly Canada, the Soviet Union and the United States of America extending their economic act ivities to areas under severe climates. In the cOUIltries i n the temperate zone the lakes and rivers are covered with ice in the winter season, or 8 pa,rt thereof only. Since navigation i s hi ndered, or blocked completely by ice , and the potential development of ice jams presents a potentia l f lood hazard along the maj or streams, economic development direc ts attention also ill t he cOUDtri es under temperate c limates to the formation and movement of ice , further to the aversion of the ha zards resulting therefrom.

The general public becomes aware of the problems r ela ted to ice ma inly at times , when the ullexpected appearance of ice causes an unscheduled interruption of navigation, or when running ice is arrested to form ice jams, raising suddenly the water l evel in the rivers,causing floods and inundation, or when diverse hydraulic structures are destroyed, i.e., when life, ' or proper ty are endangered.

Reoognition of the need t o contr ol ice and of the vast potential of international t echnlco-scient ific cooperation has prompted the sc ient i sts and ellgineers dealing with ice to joinforces on an internationa l level, creating thus - among others the Ice Section within the International Associat ion for Hydraulic Research. The need for international cooperation has become increasingly intensive in recent years and the experts on ice problems ha.ve et regularly at the internationa l symposia on ice problems.

The efforts directed at a bet ter understand ing of ice phenomena have been triggered by tangible eoonomic interests,thus primarily extension of t he naVigat ion sea son, prevention of ice jam f loods. multiple use of the shores and riparian areas,and recently the exploitation of the petro leum- and minera l resources of the sea bottom. It is t hus obvious that the cost of research coul d be procured from state, or company fund s f or attaining an economic objective alone, and the sums devoted to such r esearch are moetly directly proportionate t o the int erests i t serves. These are the motives underlying the principal fields of interest a t t he schools developed oyer t he World , s uch as those at Leningrad, Hovosybirsk, Quebec and elsewhere in Canada, at Hanover, Iowa, i n Scandinavia and in Central Europe, to mention a few examples only, without any attempt at assigning special pOSitions , or order of ranks to them.

In the present modest comprehensive report it will be endeavoured on the one hand to outline the recent r esulte of ice hydraulics, emphasizing the ma.i n t rend s, thus without making clam a t comple teness, on the other hand to indicate the problemsthe sol ution of which would close a gap recognized and complete the coverage in this part i cular domain of science .

The br oad question arising is concerned with the subjeo t of ice hydrauliCS, which would define at the same time t he scope of

3

tions on the sea is similarly small. ThuB surprisingly few actua l wa tereurface data are available in the vicinity of freezing on long- and short wave radiation, as well as on the heat losses due to evaporation, conduction and convection. Concerning the magnitude of the heat flux from the soil and ground water to the channel, reliance must be made on the r esults of estimates made at a small number of points. The data ava i labl e on the role of snowfall in the thermal budget are also deplorably inadequate. ConS qu nt therefro , although more or l ess accurate numerical solutions are known for the differential equation describing the thermal budget, the values to be substit uted can at the best be f ound by estimation only.

The present situation is thus that though sound methods have been developed at several research stat i ons and even applied numer ically with the data available for a particular Site, the general application of the thermal budget equation is prevented bythe lack of data. The action to be t aken f ollows automatically,namely climatological stations must be set up - where still missing - to yield the data needed for descri bing ( and predicting)the actual thermal budget of the water space.

The theory of ice formation on lakes i s r ather well understood. The only difficulty arises from the fact that actually few water bodies are truly stationary. The conditions are complicatedby currents and wave action to an extent that the treatment based on static cooling and ice formation seldom reflects accuratelythe actual process. Approximate formulae ha ve been suggested for taki ng the affect of wave motion into account, but the data c ol l ected thus far are insufficient to support their general val idity .

Supercooling and the formation of frazil ice in flowing water have also been described rather well t heoretically. Actual measurements have, however. failed thus f ar in demonstrating the position of the thermal boundary layer on the water surface, its dependence on turbulence and consequently the distribution of the supercooled water particles in the water body. Thus the methods suggested for estimating the mass of frazil ice formed cannot be accepted as sufficiently well founded.

It would seem, therefore, desirable t o clear positively ~he role of various currents and supercooling in the formation of frazil ice by laboratory and field measurements.

The nu.erical description of the growth of ice floes aggregated fro frazil ice is still miSSing, but could presumably be t ackled successfully by the theory of stochastic processes.

2. The physical proper ties of ice

Studies into the physical properties of ice are rendered difficult by several factors, thus firstly the dependence of these properties on temperature, age ( the history of ice), t he salt content of tbe water and the variation thereof with depth. the second being the space of subzero temperature needed for ex

5

\

Verification of the theory under a.ctual, field conditions should be the next task, which in view of the difficulties of measurement is by no means an easy one.

Advances in terrestrial and aerial (space) photography offer the possibility of recording on some streams the passage and arrest of ice at discrete points (ground st'ations) continuously and more or less regularly over the entire length by aerial photogr aphy. A comprehensive assessment of the data may be ~seful in checking and improving the methods available for describing the formation and movement of ioe. (The first steps towards establishi ng a similar interna,tional observation network have alreadybeen done on the Danube.)

A similar task is specified in the resolutions, of the UN Water Conference as well, the initiative for which has been submitted by the IAHR Section on Ice Problems (Rec.A~, subpar.C).

4. Interrelationships between flow and ice

Water as the carrying medium is of fundamental importance to the movement of ice. It should be obvious, however, that part of the energy of the water is consumed for transporting the ice, changing also the boundary conditions of flow.

The redueed conveying capacity under an ice cover is generally recognized. The distortion of the velocity profile in the ice-water boundary layer has been confirmed by numerous observations. Reliable observation data are available on the friction (roughness ) coeffic ient of the ice cover.Once the position, shape and dimensions of an ice jam are known, the backwater effect t hereof, thus the probable level of the ice-jam flood can be predicted with a fair degree of accuracy.

Ice jams cause the upstream water level to rise much faster than ordinary flood waves and are thus extremely dangerous.Still, no warning system has been devised yet, and the measurement of j am dimens i ons r ema i ns also to be solved by remote senSing, since di rect measurements are too dangerous.

5. The passage is ice through structures

At bridges crossing a river, barrages and dams restrictingthe natural cross-section and sometimes even at valley dams precautions must be made to ensure the unobstructed passage of ice and to avoid damage to the structure.

The "ice conveying capaCity" of these structures must be planned carefully and the regulating organs must be operated appropriately. Sometimes it is desirable to pass the ice with a minimum loss of water.

Empirical formulae yielding the smallest ice release span in terms of the dimensions of the arriving ice floes have been de

7

7. Protection against the adverse effects of ice, i ce control

Prevention has been recognized for long as the best method of protec t ion aga inst the destructive effects of ice, but cases are conceivable, where actual protective measures must be taken. The fo rmer involves mostly hydrological-hydraulic means, whereas the mechan ica l properties are of greatest interest in the latter.

The und isturbed formation of aD ice cover is promo ted by the ice def lec tors , or booms designed recently as f loa ting structures, the largest of which has beed applied at the out flow from Lake Erie. Booms may be successful to some extent also in deflectingthe ice, e.g., by those applied on the St.Lawrence River.

Maintenance of t he navigable waterway, or the breakup and clearing of the ice cover i s t he purpose of ice brea~er vessels, which may be assisted i n their work by other means,e.g. ice drill ing, or blasting as well . Cont inuous efforts are made at perfecting t he ice breakers, which range in size from huge ships down to launches. :More organized exchange of data and general informa,tion is considered adv isabl e on their methods and ef ficiencies. It would be per haps of inter es t to publish the major pa.rt iculars stripped of commercial int erests - on the types, for which shipyards would accept orders .

Ic e blasting is mostly an "emergency" method of protect ion, on the effectiveness of which opinions diverge widely . For this very reason an unbiassed appraisal of these methods would be worth the effort, in order to identify the problems which can be solved effectively by them.

A number of devices have been introduced for ice control a.t structures . Electr ic hea.t ing of critical elements,bubble curtains before the structure have proved successful except for extreme rates of freezing .

8. Ice as a load beari ng medium

In Hungary clever people are sa id "to make a living even on i ce" . Indeed, for lack of a better solution, or for i t s own erits ice can be used as the substructure, or even t he pa vemeDt of a transport road. In the firs t case the static, in the latter the dyIlamic load bearillg properti es predominate.

In recent t imes import ant ad vances have been recorded in this domain and the behaviour of i ce as a plastic medium and a continuously supported s l ab under concentrated-, edge- , or uniformlydistributed loads has been described by sophisticated aathematical tools. Interesting phenomena associated with dynamic load transfer under vehic les ha ve also been explored and even stresses and deflections measured in the field. On the basis thereof certa i n rules of t humb have been derived relat i ng t he thickness of the ice cover and the magni tude of load, or vehicle which can be allowed to move on it. In the lat ter case the speed of t he vehi

9

I am convinced that the future problems indicated in my report cannot be solved well and effectively, unless we succeed in finding the ways of organized cooperation!

Comprehensive reviews on ice

Michel,E.: Winter regime of rivera and lakes. Corps of Engineers, U.S.Army, Cold Regions Science and Engineering MonographIII-Bla, Hanover, liew Hampshire, April, 1971

starosolszkJ'O.: Ice in hydraulic engineering. Technical University 0 Norway. Trondheim. Report No 70-1. 1970.

Starosolszky,O.: Relationships of fluvial and ice hydraulics.lIBR-PIANO Symposium on River and Ice.Budapest,1974.Hungary.

11

AN ANALYSIS OF ICE SHEET INDENTATION

Terrance D. Ralston

Exxon Production Research Company

Houston, Texas

ABSTRACT

The indentation of an ice sheet by a flat indenter is of engineering interest because of its similarity to the crushing failure of ice moving against a vertical pier. Several experimental studies of ice sheet indentation have recently been discussed in the literature. A relationship between these results and more basic ice properties is established in this paper using the methods of plastic limit analysis.

The strength of columnar-grained ice is rate dependent, temperature dependent, anisotropic, sensitive to confining stress, and differs in tension and compression. An appropriate failure criterion that describes these effects can be used with the methods of plastic limit analysis to address various ice failure processes, such as the initial penetration of an ice sheet by a flat indenter. This problem has been addressed by others using material descriptions that were originally developed to describe plastic deformation of structural metals. However, metals are essentially isotropic, non-pressure sensitive, and do not differ significantly in tension and compression. The present analysis demonstrates how the basic strength properties of columnar-grainedfreshwater ice can be used to predict ice indentation forces.

13

"

AN ANALYSIS OF ICE SHEET INDENTATION

INTRODUCTION

Recent tests of ice sheet penetration by relatively rigid indenters have been motivated by the need to understand the failure process and resulting loads that occur when a moving ice sheet crushes against a vertical pier or other offshore structure. These tests include ice sheets a few centimeters thick tested in a cold room laboratory (Michel and Toussaint, 1976, Frederking and Gold, 1975, Hirayama et al., 1974), intermediate scale tests of lake ice and river ice tens of centimeters thick (Croasdale et al., 1976, Haynes et al., 1975), and large scale tests of arctic sea ice nearly 1.5 meters thick (Croasdale, 1974). Small scale tests, conducted under carefully controlled laboratory conditions, can be used to evaluate theoretical descriptions of ice failure. Once an adequate ice failure theory has been developed on the basis of small scale tests, it may then serve to interpret larger scale tests and to design full scale data collection programs.

The ice used in small scale tests must be carefully grown to provide the desired structure with crys tal sizes sufficiently small to eliminate crystal size related effec ts on the test data. Ice sheet temperature grad ients pose an added complication in most naturally occurring situat i ons. Several methods to minimize these effects have been used in laboratory tests. Hirayama et al. (1974) conducted tests on floating ice sheets that had been flooded to guarantee a uniform ice sheet temperature of OC . Mic hel and Toussaint (1976) conducted tests on ice sheets that were removed from the ice growth tank and supported in a cold room test machine . This ice was tested at the uniform temperature of the cold room (-100 C) . Basic ice property data (tensile strength, uniaxial and biaxial compressive strengths) are usually measured with the ice specimens maintai ned at a uniform temperature. These data can readily be used to develop failure criteria applicable to uniform temperature identation tests conducted on ice sheets of the same structure. The analys is presented in this paper is an example of a failure criterion developed from basic ice properties and applied to the indentation strength of uniform t emperature sheet ice.

Analytical expressions for the indentation strength of sheet ice have been proposed by various authors on the basis of both empirical models derived from exper imental data and theoretical considerations. The variables that are generally thought to be important include the indenter width, ice sheet thickness, indentation velocity, contact conditions between the ice and indenter, and the mechanical properties of the ice. Universal agreement does not presently exist for either the appropriate algebraic equations relating these variables or the relative importance of the variables.

15

Theoretica l descriptions of ice indentat i on requ ire an ice fai lure criterion. Pl as t ic ity methods have been applied to ice crushing failure by Michel and Toussai nt (1976), Croasdale et al. (1976), Reinicke and Ralston (1977), and Ralston (1977). Both Michel and Toussaint and Croasdale et al. used fa i lure criteria that have been developed for applications in metal plasticity. These criteria describe isotropic materials tha t have equal tensile and compressive strengths and are not sensitive to hydrostatic pressure. Michel and Toussaint used the plane strain solution for the flat punch indentation of a half space, with ice failure described by a von Mises material model, to interpret experimental data. This known so lution gives an average i ndenter pressure p = FfDt (where F = total force, D = indenter width, and t = ice thickness) of nearly three times C , the unconfined compressive strength; i.e.,

x

p = 2.97 C (1)x

Since ice strength is rate dependent, a definition of strain rate for ice indentation is needed to compare indentation and unconfined strength data. Michel and Toussaint empirically defined the effective indentation strain rate by

i =Vf4D , (2) where V is the punch velocity, and presented a comparison of indentation and uniaxial ice strength data as a function of strain rate. Thi s comparison included dat a f rom several authors and is reproduced in Fig. 1. All of these tests used laboratory grown ice and, with the exception of t he Hirayama et al . data, were conducted at a test t emperature of ~ooC. Mi chel and Toussai nt ad justed the Hi rayama et al. data f rom OC to -10C for this compari son. Michel and Toussaint concluded that the i ndentation strengt h of laboratory grown S2 ice is approximately 3 times the in-plane un iaxial compressive strength over the entire range of strain rates at the -10C temperature.

Michel and Toussaint al so conducted tests at a constant effective strain rate (given by equation (2)), to investigate the effect of the Dft aspect ratio on ice indentation pressure. These data, normalized by the unconfined compres sive strength , are plotted as indivi dual data points in Fig. 2. The i n-plane, unconfined compress ive strength was not explicitly measured in these tests. However, following Mi che l and Toussaint, a reasonable approxi mation can be obtai ned by us i ng one third of the average of t he i ndentation strengths measured with i ndenters greater than 50 mm wide. Th i s val ue is 2244 KPa and was taken to be C

x in Fig. 2. Michel and Toussaint concluded that for constant st rain rate tests, the indentati on pressure did not depend significantly on the Dft ratio and that the Von Mises material model adequately represented the data. The potent ial increase in indentation pressure indicated by tests 13-17 was not considered to be conclusive because of possible effects related to t he size of the small indenters 25 mm ) and the ice crystal diameter (7.5 mm ) for these tests. The excellent agreement of tests 15 (D = 6.3 mm, t = 25.4 mm) and test 17 (D = 25 .4 mm , t = 101.6 mm), however, suggests that even the low aspect rati o tests may have produced meaningful data.

Croasdale et al. (1976) presented an analysis of ice indentation with ice strength described by a Tresca plasticity model . The Tresca

17

and von Mises materi al models are very similar and the analysis presented by Croasdale et al. can be readily adapted to the von Mises yield function. The curves shown in Fig. 2 are a straightforward extension of that analysis to the von Mises material. Croasdale et al . used the methods of plastic limit analysis to derive upper and lower bounds for t he indentation pressure on rough and smooth indenters. The ir results imply t hat the pressure should decrease as the st ructure becomes wide relative to the ice sheet thickness, i . e., as the aspect ratio increases. The results further imply that the condi t ion of plane strain, which Mi chel and Toussaint assumed to apply for all Oft , is only applicab le to a von Mises material when O/t < 0. 25. The apparent disagreement between this theory and the experimental data presented by Mi chel and Toussaint is a consequence of the von Mises materi al assumpt i on. The strength of 52 ice is not isotropic, pressure insensi t ive , or equal in tension and compression. In the following, we show that satisfacory agreement can be obtained when these factors are taken into account by a general i zation of the von Mises yield cri t er i on.

ICE PROPERTIES

The strength of columnar-grained freshwater ice is both ani sotropic and pressure sensitive. The strength tests indicated in Fig. 3 have been di scussed in the l iterature (Ralston , 1977) and shown t o be cons i stent with the behavior expected for a transversely isotropic ma t er ial with different strengths in tension and compression. The unconfined strength data of Carter and Michel (1971) ind icate vertical to horizontal compressive strength ratios of as much as a factor of 2. Frederking's biaxial tests provide an indi cation of 52 ice strength behavior in other than uniaxial stress states . These data are reproduced in Fig. 4. IILateral" confinement (with respect t o t he crystal growth direction), as indicated by test type A in Fi g. 3, produces s ign ifi cant increases in ice strength. This type of test leads to ice st rengths that are at least twice the uniaxial st rength at rap id strain rat es and as much as five times the uniaxial strength at s l ow strai n rates. The "vertical" confinement of tes t type B has negli gib l e effect on i ce strength in the absence of latera l stresses.

These data clearly demonstrate that ice is not a von Mi ses material; however, the generalization of the von Mises criterion descr'i bed by Ralston (1977) can be used to describe these effects. Thi s generalizat ion is given by

f(2) = al (a - a )2 + a2 (a - a }2 + a 3 (ax - a )2 + a4 L 2 y z z x y YZ (3) + as L 2 + as t 2 + a7 a + a8 a + ag a -I,

zx xy x y z

where the stress components refer to the coordinate syst em i n Fi g. 3 and the coeffi cients al ... ,ag. are determined by the materi al strengt h propert ies. This;s a yield criterion in the sense of plas t ici ty t heory. Stress states for which f(a) < 0 are taken to be elastic , while those for which f(o) =0 are said to lie on the yiel d surface. The materi al will not support stress states for whi ch f (2) > O.

19

Fig. 5. Lower bound stress field . (After Reinicke and Ralston, 1977) .

~y

Fig. 6. Upper bound 'velocity field. (After Reinicke and Ralston, 1977) .

21

TABLE 1

Regional Stress Components in the Lower Bound

Stress Field (After Reinicke and Ralston, 1977)

-0 -0 tRegion X Y xy

1 PI 0 0 2 PI + 1 sin262 1 cos26l 1 sin6 l cos6 l 3 PI + 2 sin262 2 cos262 2 sin62 cos62 4 PI P2 0 5 PI + 1 sin26l 1 cos26l 1 sin6 l cos6 l

+ 2 sin262 + 2 cos262 + 02 sin62 cos62 6 PI + 2 sin262 P2 + 2 cos262 2 sin62 cos62 7 PI + 1 sin26l P2 + 1 cos26l 1 si n6 l cos6 l

+ 2 sin262 + 02 cos262 + 02 sin62 cos62 8 PI + 22 sin262 P2 + 202 cos262 0

. 269 PI +01 sln 1 P2 + 01 cos26l S i n6 l cos6 l1 + 202 sin262 + 202 cos262

10 PI + 201 sin26l P2 + 201 cos26l 0

+ 202 sin262 + 202 cos262

of the constraints determines a lower bound for the indentation pressure. The best lower bound can be determined by numerically maximizing the expression for olb subject to the eleven constraints.

The velocity fi~ld shown in Fig. 6 can be used to construct an upper bound for the failure load. It is described by fourteen parameters that can be selected to determine the least upper bound. The seven angles ~l' ~2' ~3' ~4' ~2' ~3, ~4 specify the geometry of the deforming region, the four angles 61, 62, 63 , 64 specify the direction of the velocity in the four rigidly moving regions, and the three magnification factors a, ~, y specify the changes in velocity magnitude across the inner surfaces of velocity discontinuity, i.e. V2 = aV 1 , Va =~V2' V4 =yV3 . An upper bound for the indentation force is obtained by setting the rate of external work equal to the rate of internal dissipation of energy, i.e.

where F is an upper bound for the indentation pressure, V is the relative velocity between the indenter and the ice sheet, and DT is the total rate of energy dissipation from the seven surfaces of velocity discontinuity on each side of the indenter. An upper bound for the

23

PLANE ST. RESSfr-UY

-6 -5

,

I I

: ~ j~ u:s=1.01

/i/i -2

-3

-5

/'L PLANE STRAIN CONFINING~Uy

PLATES ~

U x

Fig.7. Plane stress and plane strain ice yield surfaces .

/ /

1 UX

uy/C xA., /-2 _

STRAI.

CONFININGc$-"U y /o

PLATES /

-3 U x o _/

Fig. 8. Plane stress and plane strain yield functions for the von Mises criterion.

25

DATA FROM MICHEL AND TOUSSAINT (1976)

D~50 mm

o t~----~----~----~----~--~~--~',7{r---~o 1 2 J 4 5 6

ASPECT RATIO - D/t

Fig. 9. Comparison of computed bounds for indentation pressure with test data.

Fig. 10. Comparison of plane stress ice yield function with plane strain von Mises yield function.

27

consequence of the ice crys tal s i ze . Although the s i ze of these. indenters was comparable to the crys t al di ameter, the size of the plastic deforming region may have been sufficiently large to obscure individual crystal effects.

The plane stress analysis with the generalized von Mises yield function (equation (3a leads to an indentation pressure that is approximately equal to that given by the plane strain analysis with the standard von Mi ses yie l d criterion. This result can perhaps be explained by the compari son of t he two criteria i n Fig. 10. The plane strain von Mises cri t er i a approx imates the plane stress ice behavior in much of the compress ion-compress ion quadrant. Th i s approximation extends to suff i cient ly hi gh compressive stresses to adequately approximate the solution to the i ndentation problem.

The choi ce of velocity fi elds used to construct upper bounds for the failure pressure can be gui ded by observations of actual ice failure. The present construct ion is similar to that observed by Michel and Toussaint (1976) . If the yiel d cr iterion changes as a result of different ice or t est condit ions, one would expect the failure mode to also change. Croasda le et a l . (1976) observed a wedge-type failure in their lake ice t ests . This failure mode was used in their plastic limit analysis with the standard von Mises criterion, and a similar mechanism could be developed for t he present material description.

LIMITATIONS

The present analysis provi des an adequate explanation of carefully controlled l aboratory tests of a particular type of ice loaded at a uniform -10C temperature . The plast i c limit analysis procedure, should be applicab le to other situati ons as well. This approach transforms appropri ate ice property data i nto ice indentation pressure. Largescale property data , on a scale comparable to the structure width or ice thickness di mensions, may be needed for the direct use of this approach in practical appl ications , such as the design of offshore structures. This data may be de ri ved by ext rapolation of small scale data, large scale measurements, or a comb i nati on of both. The extrapolation of small scale data shou l d account for scale effects , struct ural inhomogeniety through t he i ce thickness , thermal gradients, etc. Even in the absence of l arge- scale data, however , this theory provides a framework for the qual itati ve interpret at ion of field observations of ice failure against exist i ng st ructures.

Failure mechanisms other than plastic deformat i on occur in many situations and of ten lead t o reduct ions in the ice fai lure load. Following the ini tial fail ure at t he edge of the ice plate, Michel and Toussaint repor t ed t hat the i ndenter continued to crush through the ice at substanti ally reduced loads. This continuous crushing action is not directly descr ibed by the present analysis. Ins t abilities, in the form of elastic or plasti c buckli ng, also occur in nature. The type of material desc r i pti on, or yi eld criterion, discussed here wou l d apply to instabili t i es; however , t he plastic limit analysis procedure does not. It applies on ly to failure by plastic collapse and does not describe the load reductions tha t result from instabilities.

29

REFERENCES

1. Carter, D., and Michel, B., 1971. lois et mecanismes de

l'apparente fracture fragi l e de 1a glace de riviere et de lac.

Universite laval. Faculte des Sc iences. Departement de Genie

Civil. Section Mecanique des Glaces. Rapport S-22.

2. Chen, W. F., 1975. limit analysis and soil plasticity,

Elsevier Press, Amsterdam.

3. Croasdale, K. R., 1974. Crushing strength of Arctic ice. (in

Reed, J. C., and Sater, J. E. , ed. The coast and she l f of the

Beaufort Sea. Proceedi ngs of ~ sympOSlum on Beauf~ea coast

and shelf research. Arlington, Virginia, Arctic Institute of

North America, p. 377-99).

4. Croasdale, K. R., Mo rgenstern, N. R., and Nuttall, J . B., 1976. Indentation Tests to Investigate Ice Pressure on Vertical Piers (Preprint), Symposium on Applied Glaciology, International Glaciological Society, Cambridge, England.

5. Frederking, R. M. W., and Gold, l. W., 1975. Experimental study of edge loadi ng of ice plates. Canadian Geotechnical Journal, Vol. 12, No.4, p. 456-63.

6. Haynes, F. D., Nevel, D. E., and Farrell, D. R., 1975. Ice Force Measurements on the Pembina River, Alberta, Canada, Tech. Report 269, Cold Regions Research and Engineering laboratory.

7. Hirayama K., Schwarz, J., and Wu, H. C., 1974. An investigation of ice forces on vert i cal structures, Iowa Institute of Hydraulic Research, University of Iowa (IIHR Report No. 158).

8. Michel, B., and Paradis, M., 1976. Analyse statistique des lois du fluage secondai re de la glace de riviere et de lac. Universite laval. Faculte des Sciences et de Genie. Departement de Genie Civil. Section Mecanique des Glaces. Rapport GCS-76-02.

9. Michel, B., and Toussai nt, N. , 1976 . Mechanisms and theory of

indentation of ice plates (Preprint), Symposium on Applied Gla

ciology, Inte rnational Glac iol ogical Society, Cambridge, England.

10. Prodanovic, A. , 1976. Plane stress limit analysis with quadratic yield criteria, Int ernal communication .

11. Ralston, T. D., 1977. Yield and plastic deformation in ice crushing failure (Preprint), ICSI/AIDJEX Symposium on Sea Ice--Processes and Models, Seat tle, Washington.

12. Reinicke, K. M., and Ralston, T. D., 1977. Plastic limit analysis with an anisotropic, parabolic yield function, Int. J. Rock Mech. and Mini ng Sciences, Vol. 14, pp 147-154.

31

A STATISTICAL PREDICTION OF

EFFECTIVE ICE CRUSHING STRESSES

ON WIDE STRUCTURES

by

P.R. KRY

Imperial Oil Limited, Production Research Division

Calgary, Alberta, Canada T2G 2B3

When an ice sheet moves past a wide structure, the ice failure mechanism is characterized by an irregular contact zone and non-simultaneous failure across the width of the structure. Where sufficient ice motion occurs to prevent formation of a frozen bond between a structure and an ice sheet, this failure mode becomes the design condition for ice crushing against the structure. Due to the nature of the failure, the effective stress across any local region of the failure interface is an irregular function of the ice sheet motion. This observation is developed into the concept of independent zones of crushing in front of a wi de structure. It is assumed that as the ice sheet moves past the structure, effective stresses for each zone are developed inderendently of those in adjacent zones. In each zone, stresses are characterized statistically by the probability of exceedance of a stress level and the mean continuous duration of that exceedance. A statistical summation is used to calculate risk of stress exceedance as a function of stress level and extent of ice movement over the anticipated life of the structure. The use of ice sheet thickness to scale ice sheet disrlacement is advocated. A design stress can be defined by the level of risk of stress exceedance which is acceptable in a particular instance. Published data is used in example calculations to demonstrate the reduction in design stress derived from a conventional extrapolation of the data.

33

INTRODUCTIOr~

Imrerial Oil Limited has constructed fourteen islands in the landfast ice zone of the Southern Beaufort Sea as offshore exrloration drilling platforms (Garratt and Kry, 1977). They have withstood both summer storms and movements of 2 m thick ice sheets during the winter. To safely and economically design an island to withstand th i s latter condition requires a knowledge of the effective stress at which the ice will fail against a wide structure for relatively limited ice movements.

Ice failure stress depends on failure mode, movement rate, ice type, temperature and width of the structure. A design ice failure stress would be that obtained for the most severe expected combination of these conditions. Since sufficient small-scale motions, including tidal effects, appear to prevent a frozen-in condition (Kry, 1977), the anticipated failure mode is continuous crushing (Niell, 1976) which is strongly influenced by local contact effects. l1acroscopic cracking, in particular cleavage cracks, and a significant variation of the effective failure stress about its mean value as ice penetration occurs characterize the failure mode (Nevel et al, 1972; Hirayama et al, 1974; Blenkarn, 1970; Uiell, 1972). As an i ce sheet continuously crushes against a very wide structure, the in f l uence of local contact effects imp l ies that the opportunity for non-simultaneous ice failure across the structure is enhanced. This leads to the intuitive expectation that design ice failure stresses for a wide structure are less than for a narrow structure. Non-simultaneous failure implies that at anyone point in time, different local areas of the fai l ure region are in different stages of failure. Local stresses have significant variation about their mean whereas across the entire width, the average stress should remain closer to the mean.

The expected reduction in ice stress on a structure depends on the extent of ice movement during the life of the structure. The greater the movement at a particular location, the greater the chance for a high level of stress to occur. HO\Olever, if movements of the thickest ice sheets are suffi ciently limited, there is a significant statistical reduction in the expected peak stress for a wide structure compared to that for a narrow structure.

This paper presents a particular statistical approach to estimating the influence of str ucture width on design stress. It is an approach which extrapolates emp i rical data for continuous crushing against a relatively narrow indentor to a very wide "indentor. A limited amount of published data is used solely for the purposes of demonstration. For purposes of design a rigorous examination of the appropriateness of the empirical data is required.

ASSUMPT IONS

The particular statistical approach is based on the assumrtion of the existence of independent zones of continuous crushing. It is assumed that the interaction width along which the ice is fa i ling may be divided into a number of independent equivalent zones.

The assumption of independent zones is that stress levels in anyone zone are not influenced by stresses in adjacent zones. As an

,

35

given by the Exponential distribution as exp (-L/r1) (Feller, 1950). The length L is the extent of ice movement expected over the life of the structure and can be written as a fraction r of the recurrence interval M as

L = rt~ (2) If r 1, then the probability of havin9 one or more exceedances

of aD in the length L is given by

1 - exp (-LIM) = 1 - exp (-r) '" r (3)

That is, r is the risk of exceedance of aD over the life of the structure. Given r "and the expected ice movement over the life of the structure L, the recurrence interval M necessary to ensure the risk r is given by Equation (2).

In principal, once a value of Mis known, a desi9n stress can be directly determined since M is a function of stress level. In practice, for extreme values of a, evaluation of M(a) is less certain than evaluation of P(a) and D(a).

Since P(a) is a single valued function of a it may be inverted to allow D to be expressed as a function of P. Once the functional relationship is determined for the particular structure and ice conditions, it may be substituted into Equation 1 and with the appropriate val ue of Mfrom Equation 2, the specific value of P which corresponds to a risk r, is determined. The appropriate design stress aD is then that stress which has that instantaneous probability of exceedance. Simultaneously there is a risk r that this design stress will be exceeded over the life of the structure.

The statistical functions P and D are not known directly for the case of a very wide structure. For relatively narrow structures, data necessary to evaluate P and D are more readily available. By assuming the statistical functions for a narrow structure are appropriate for a single zone, and assuming the interaction across a wide structure occurs via a number of statistically independent equivalent zones, then P and D can be evaluated for the wide structure and used to estimate a design stress.

The probability distribution p(o) for the total stress over a number of zones is a function of the distribution parameters for the stress in each zone. When the zones are equivalent and independent there are several statistical theorems which establish the functional dependence. Assuming indepengence of a number of distributions. the mean ang variances of the sum of the distributions is equal to the sum of the means and variances of the individual distributions (Feller, 1950). Furthermore the skewness of the sum of the distributions is equal to the skewness of the individual distributions divided by the square root of their number (Shaw, 1976). These theorems apply directly to the distributions describing the force in each zone, summed for all zones to give the total force on the structure. When the forces are expressed as stresses for equivalent independent zones the theorems imply

37

Y (7) = 9 where y is related to the probability P as indicated in Table 1. The relation between y and P follows directly from the definition of the log-normal distribution and an evaluation of the Probability Integral.

Using Table 1 and Equation (7) it may be verified that for extreme stresses, an order of magnitude change in P causes a relative change in the corresponding stress level of aprroximately (Og - 1)/2.

By using a particular distribution, an analytical form can be obtained for the variation of the rrobability distribution with the number of zones. Taking the one zone distribution to be log-normal, two of Equations (4), (5) or (6) may be used to define a new log-normal distribution representing the distribution of stresses for the sum of n zones. As only two parameters define a log-normal distribution, the distribution of the total stress cannot be exactly log-normal. However, the inconsistency is not large. For example if Equations (4) and (5) are used to define a new log-normal distribution for 16 zones, its skewness is within 7% for 0gl = 1.6 and within 1% for 0gl = 1.2 of the skewness given by Equation l6).

Using Equations (4) and (5) algebraic manirulation yields

(8)exp / ln (1 - *+ *exp (ln2 9 1 ))

and

= (9 ) on / 1 _ 1 + 1 exp (ln2 )

n n 91

as the relationship between the n-zone median stress and geometric standard deviation (on and 0gn) and the l-zone values (01 and 0gl).

P 10- 1 10- 2 10- 3 10- 4 10- 6

y 1.28 2.33 3.09 3.72 4.265 4.753

Table 1. Probability P and associated exponent of geometric standard deviation, y for a log-normal distribution.

39

The decrease in variance of the stress with increasing number of zones is evident in Figure 4. This decrease is quantitatively presented in Table 2 in terms of the geometric standard deviation of the fitted log-normal distributions for data from both the A and B cases of Toussaint of Figure 2. Also presented in Table 2 are the respective median stresses for the distributions and predicted values from Equations (8) and (9). The results indicate the conservative nature of the predictions. The geometric standard deviations for the simulations are less than predicted since the simulations do not consider the possibility of any stress higher than what was observed, while Equations (8) and (9) are based on stresses which are not bounded.

The functional dependence of the duration distribution 0 on the probability distribution P was investigated by fitting a power law function as

(10) where Do and a are parameters determined from the data. Since the primary interest centers on extreme events, only data for P < 0.10 was used to determine Do and a. Table 3 presents the two parameters and the coefficient of detennination (r2). The results show that the fit is reasonable lr2 > 0.9) and that the relationship is not strongly dependent on the number of zones. Variation between the two cases is as great as between two zones for the same case, and there is no apparent systematic variation.

The results of Table 3, if averaged indicate

Do = 0.04 0.01

and

a = 0.35 0.13

If these values are used in Equation 10 which is then combined with Equation (1), the probability of exceedance P may be written as a

iiumber r2Case of Zones Do a

1 0.044 0.286 0.89 2 0.046 0.393 1.00A 4 0.034 0.350 0.90 8 0.059 0.576 1.00 1 0.045 0.418 0.99 2 0.037 0.246 0.90B 4 0.057 0.419 0.95 8 0.017 0.141 0.68

Table 3. Parameters Do and a and coefficient of determination r2 for power law fit to measured duration and probability distributions from simulated traces.

41

relati vely narrow structures. The part i cular assumption of independent zones of fai lure i s cons i s tent with the concept of design stresses fo r wide structures be ing cl oser t o t he average stress than the extreme stress observed for a narrow structure. Primarily the examrl e appl i cation of the method i s intended to illustrate a reduction of effec t i ve stress when a structure i s sufficientl y wide that failur e occurs nonsimultaneously along t he inte racti on width. This failure mode i s characterized by la rge scale crac king activity, resulting i n a st ress which is best described in a s tatistical sense.

The relation between number of zones and actual struc ture width is related to the assumpti on of independence and the part i cular emp ir i cal data used to represent the stati stics for a single zone. Ideally , t he indentor used to generate one zone statistics should be at least as wide as the largest pieces observed to fail as independent units agains t a very wide structure. A width of four or five ice thicknesses may be an appropriate lower limi t since i t is at that aspect ratio tha t end effects for indentors experimen ta ll y tend to become insignifican t (Ni ell, 1975).

Schwarz (1 970 ) obtained data from ice crushing against a pil e with an aspect ratio of about 4. The pile was instrumented wi th an array of load cells and t he resu l ts demonstrated that lower mean effect ive stresses occurred across the ent ire width than across any part ial wi dth. Were the assumption of independent zones to hold, it would predi ct the same mean s t ress (Equati on (4 ) ) for part of the width as for the ent i re width. In terms of the concepts rresented in this paper, that i s , gi ven zones of fai lure , t here would have to be an anti-correlati on between stresses in any two zones to account for a reduction in the mean. Shou l d this type of anti-correl ati on continue to larger zone sizes, the present approach would be conserva tive . Considerably more experi mental and theoretical effort mus t be expended t o properly addres s thi s question.

It must be emphasized that the appropriate empirical dat a i s essential for a reali sti c applicat ion of the methods presen t ed. In particular, the proper combination of the parameters, ice temperature, crystallography, salinity and renetration rate, t hat l ead to t he hi9hest secondary stress l evels must be established.

AC KNmJLEDGEMErn

The permi ss ion of Imperial Oil Limited to present the concerts in this paper i s gra t efull y acknow l ed~ed. The author i s indebted to Prof. B. r1ichel fo r permission to use the experimental data to ill us trate the method. Tha nks are due particularly to r~r. J.D. ~'Jheeler and Mr. A.H. Shaw for discussions of the statistical techniques which considerably clarified the author' s apprecia t ion of the concept s . The author thanks r~rs . R. Hume for her tllOUghtful preparation of the manuscri pt.

43 -~

----------

~ U r-~~~~-4----~~--~~ '"DC ~

'"

ICE MOVEMENT

Figure 1. Schematic stress - displacement trace illustrating basis for probability and duration distribution calculations

AFTER TOUSSAINT ( 1975) A

I'- =1.34 MPa S2 5 ex =0.36 MPa T TJ =0051

0 . 5

o . 0oL---'----LL 1 1 2 3 4

PENETRATION ( I CE THICKNESSES )

R E2 . O ~ S S

~ A

AFTER TOUSSAINT (1975 ) B

I'- = 1.94 MPa ex = 103 MPa TJ = 0 475

/~ ~ Vi 2 -1.____-1-___ .1....-_ _ _ ---1._11

I 2 PENETRATION ( I CE THICKNESSES )

Figure 2. Sample data from Toussaint (1975). A: 10 Crl wide indentor, 5 cm thick ice sheet, 1 cm S-l penetration rate. B: 5 cm wide indentor, 5 cm thick ice sheet, 0.5 cm S-l penetration

rate.

5 .0

D

0" TOUSSAINT B :Ii

J . O on on w

, 0'"

'"

~

w > ;:: U I 0 w ....

....

w

~_ . __ ---L...o 5 o 0 0 0 "00 o SO 010 0 . 01 0 . 001

PROBABILITY OF EXCEEDANCE

Figure 3. Probability distributions for sample data from Toussaint (1975) plotted with log-normal distribution axes.

45

3.0r-,--------,--------------.-------------.--------------,

o 2.0 Q..

~

p

----10-1 ------------------~

TOUSSAINT A

0~---------7--------------~------------~------------~1 5 10 15 20 NUMBER OF ZONES

6.0r-,-----,-~--------------~------------~------------~

p

4 .0 10 -5

10 -3

10 -1 2.0

TOUSSAINT B O~________~______________L_____________~____________~

0 Q..

~ V)

V)

w a:: l-V)

1 5 10 15 20

NUMBER OF ZONES

Fi9ure 5. Stress exceeded with probability P as a function of number of zones for two cases of sample data.

47

SEA ICE - FLEXURAL CREEP

RENE TINAWI JEAN-ROBERT MURAT

Associate Professor Research Associate Department of Civil Engineering Department of Civil Engineering Ecole Polytechnique Ecole Polytechnique Box 6079, Station "A" Box 6079, Station "A" Montreal, Canada Montreal Canada

ABSTRACT

This paper describes a theoretical (Finite Element) and experimental investigation of the flexural behaviour of sea ice beams and plates. About one hundred beams were tested elastically and fifteen beams in creep. The elastic behaviour for a brine volume of 26%0 compares favourably with the findings of others. As for the secondary creep, a law of the form : = 2.305 X 10- 12 0 2 32 is suggested. Elastic tests on seven simply supported circular plates show a reduction in the elastic modulus (compared to beams ). Creep tests on plates show that for secondary creep, a law of the

3form t = 1.63 X 10- 12 0 governs.

49

SEA ICE - FLEXURAL CREEP

I NTRODUCTI ON There is no doubt about the growing interest in the research field

of sea-ice. This increase in activity is mainly due to oil companies investing in the exploration of oil and natural gas. Research on sea-ice at Ecole Polytechnique was initiated because of the Polar Gas Project reported by McGovern & all. This project includes the transportation and storage, on the floating ice sheet, of heavy equipment not only on a short term basis but also on a long term basis.

While the short term static loading presents no major difficul ty, the long term creep analysis is certainly a very challenging problem with many consequences. It must also be pointed out that from the theoretical point of vue a number of authors such as Neve1 2 , Williams 3 , have made attempts at the classical solution of infinite or semi-infinite plates under long term loadings. Others'+' 5,6,7,8,9, 10 have attempted sol utions for the creep of ci rcular plates. While most of the classical or numerical solutions would be based on a creep law of the power from, the actual value of such a law, specially under bi-axial or tri-axial states of stress is still a long way away at least for sea-ice. In fact this is now the most important information yet to be found simply because of the existance of the Finite Element technique 11. The geometry, for any floating ice sheet under any loading or boundary condi tions can be simulated wldcr in s tant or long term loading with remarquable accuracy. The speci:ll purposc J.'.inite Elemcnt progr,lJns developed by Garbaccio 12 , Katona l3 , Ilrudl.' yIO, ~luratl'+' 15 and the availability of commercial programs

51

Coldroom wall Adjustable overflow pipe

Polye t hylene

Insulat ion

Sealed cell neoprene foam

heating wire:s~____~~~t-____~~I

drain

Ring support

Salt solution

o 50cm , I I , J

figure 1: Freez j ng tank

Ma nomet er Speed reducer Electric engine

Air release Output=1700 kPa ~

J.ac k By pass for manual calibration

, Ai r ( 14,000 k Pa)

120 1201201cm Fi gure 2 : Fle xural testing apparatus

53

0.1 ~~~~-+________~________+-______~________~______~

o o 200 400 600 800 1000 1200

Time 1 minutes

Figure 3: Deflection (wh/L2) versus time for flexural creep of beams

55

FLEXURAL CREEP OF BEAMS

Fifteen sea-ice beams have been subjected to a constant load P for 17 to 48 hours. The deflection curves for five different stress levels, ranging from 276 kPa to 552 kPa, have been transformed i nto a non-dimensional form wh/L 2 and plotted in figure 3, It is clear from the experimental findings that a stationary condition is always obtained. This suggests that secondary creep is well achieved and hence a strain rate of the power form is not an unusual choice. Hence,

( 3)

where 0 is the uniaxial stress and a is a numerical factor.

The non-dimensional deflection rates for each test have been computed by least square fit and reported in table 1. The obtained deflection rates are then plotted against stress on a log-log scale as shown in figure 4. Using least square fit, the equation of the straight line is

0.269 X 10- 12 (0) 2 . 32 (4)

where 0 lS in kPa and the time is in minutes.

Now, from the theoretical point of vue, Hult 25 obtains,using the elastic analogy, a differential equation of the form

( 5)

where

1 2nb (~) 2+ -;; (6 )1 + 2n 2

w is the central deflection and M the bending moment.

Integrating over the beam length and using the boundary conditions yields

. n 2L2 1) (1 + 2n)n (Sn + 13 1 w = Ct 0 9Fl (l+""I1 3n 8 ~) ( 7)

substituing for n = 2.32 into (7) gives L2

0 2.32O.117 ( 8)h a

or

wh 32 V- = 0.117 a 0

2. (9 )

57

Comparing (4) and (9) yields 2 . 30 5 x 1a- 1 2 a 2. 3 2 (10)

where a is still in kPa and the time in minutes.

ELASTIC TESTS ON CIRCULAR PLATES

Tests without water

Tests on a circular plate were started by lowering the water level after formation of an ice thickness of 95 mm (plate number 6). When the overall temperature reached -lOC a central load, acting over an area 178 mm in diameter, was applied. Displacements were recorded on four X-Y plotters: one for the central, two for quarter radius and one for the half radius deflections as shown in photograph 2. It was then decided to use the Finite Element "AXI" program 14 with 9 elements to see which value of the elastic modulus fitted best the experimental results. Also, three different boundary conditions were tried to see the effect of the ring support. All these values are reproduced in figure s. Since it was difficult to break the plate, because the pneumatic jack reached its maximum capacity, it was therefore decided to use the plate again at -13C and -SoC.

It is clear from figure 5 that the boundary condition that fits best the experiment is the simple support and the corresponding values of the clastic modulus are:

for SoC E 0.997 X 10 6 kPa for -lOC E = 1.09 X 10 6 kPa (11) for -13C E 1.15 x 10 6 kPa

I f the val ue of E at -lOC is now considered as a reference with Vb .026, and using the proposed formula of Weeks &Assur24 :

E = Eo (l - Vb) 4 ( 12)

the foIl owing results are obtained:

for - SoC Vb 0.049, E 0.991 x 10 6 kPa

for -13 DC Vb = 0.020, E = 1.12 X 10 6 kPa (13)

(ompari;;on between the resul ts obtained experimentally and by USlng c'luutlon ll.2i confirms the validity of such an equation at least for the small r:lIlgc of tl'mperatures considcreJ. Ilm"ever, it is distrubing to note thGt the value of E = 1.09 X 10 6 kPG i;; about three times smaller than the cla;;tic moJulus obtaineJ from beam tests (E = 3.89 X 10 6 kPa).

TIlis is explained by realising that the loading rate on the plate of 500 N/s gives rise to stress rates that are for the majority of the plate much less than the corresponding rate for the beams. The stress rate distribution is shown in figure 6 for both the plate and the beam tests. It is obvious that only 15% of the plate was subjected to the

59

same stress rate as the beam. Vaudrey &Katona 23 report a value of 5.53 x 10 6 kPa for stress rates varying between 50 to 100 kPa/s on circular plates 53 cm in radius. However, that value of E was measured using strain gages not deflections. Kerr 26 , analysed Bernstein 27 results using plate theory for a temperature range beh'een _7( and _100( and finds a value for E = 1.03 X 10 6 kPa which is quite comparable to the findings reported here but probably quite accidental too. In fact little is available in the litterature on the elastic testing of plates because of the great difficulties in achieving high loading rates and also because of the required automation for recording of data.

Tests with water

Table 2 shows all the available data on plates 1, 2, 3, 4, 5 and 8 that were tested with water underneath. Plate 7 is not reported due to technical difficulties encountered. Since the bottom of the plate was at _2( and the air temperature at -10(, a linear variation in temperature was assumed. If the salinity is assumed parabolic 18 it can be easily shown using the formula of Frankenstein &Garner 28 that

8Z 2 - 8Z + 6 ( 49.185 Vb = I l + 0.532) (14)1000 -8Z - 2

where 0 ,,;;;; Z ,,;;;; 1 is measured from the bottom of the plate. Hence a parabolic approximation in E yields:

E = Eo (0.572 + 0.931 Z - 0.65S Z2) (IS)

where for Vb = .026 and E = 1.09 X 10 6 kPa, equation (12) yields Eo = 1.21 X 10 6 kPa. This parabolic variation in the elastic modulus (figure 7a) shifts the neutral axis at .528h from the bottom of the plate. More important still, is the variation in tensile flexural strength. Using the formula proposed by Weeks & Assur 21

(16 )

yields after substitution of (14) into (16) the curve shown in figure 7b. From (}t, the octahedral stress Toet lS also plotted using

(17)

Since the applied load on the plate of 2.2 kN develops octahedral stresses in excess of the limit value, cracks occur at the bottom of the plate and continue to propagate as shown in figure 8 for plate number 3. Hence for all practical purposes it can be considered that the element directly under the applied load is cracked and a linear elastic analysis is no longer val id. Von Mises yield cri terion is used to evaluate the evolution of cracks across the thickness. Away from the centrally loaded portion of the plate the behaviour is more duct ile due to the lower stress rates.

1110 other problem to be solveuis the bet that uSlng the elastic mouulus of the plate obt:llned without \v:lter shmvs little correlation to

61

---

---

----

------

---

--

--

----

o

0 .5

1.0

1.5

E E

2 .0

c o o u C1l

....

C1l "0

o u

-....

C1l >

0 .5

1.0

1.5

o

0.5

I I

.---'-. ...-

/" -- --. .,,-

---

--

PLATE 4 , h = 92 mm

,/

Hi A .~ --

.~ .,,-"'- 2SO , t ,tVt , as: ~:-- .,,- -~ ./.'

I'0t-~--1.51

PLATE 3 , h = 95 mm

Experiment o

-

-

--

---

-..-:

PLATE 5 , h = 99 mm

o~--------~~--.~~_~~.~_~~~_____-

0.5

1.0 PLATE 8 , h = 137mm

Figure 9: Corolation between experimental deflections and different Finite Element idealisations

63

!O x 10-3r

9

8

7

6

.r.IN~o:: 5

4

3

2

________~--------~._--------~----------~----------._--------~ Plate 4

Plate I

Plate 2

Plate 5

Pia te 3

o o 5 10 15 20 25 30

Time , hours

Figure 10: Central deflection (wh/R2) versus time for flexural creep of simply supported circular plate (-lODC)

65

1/10 1/6 1/3 1/2 2/ 5/6 R r3 ~ 0 I 0 a.. .:.t:

~ I 0

2

o

Experiment

3

Figure 12: Function aCr): Comparison between experimental and analytical results

Figure 13: Evolution of the deflected shape of plate no 3 with time

67

o

E E 2 c: o

o 4 E

~

o ...

Q) "0

o u .....

0... Q) >

6

8

10

AXI 15

assumed that the deformation rate wis proportional to the elastic deformation and sol ved equation (22) by use of GalerKin' s approach. More recently Hrudey 10, 31 proposed a finite element formulation and solved the non-linear equations by a Newton-Raphson technique.

Both developments yield a deformation rate of the form:

w(r) h p n a (r) (h 2) (24)R2

where w(r) is the deflection rate at a point of radius r, h is the thickness of the plate, R its radius and P the applied central load.

The deflection rate w has been computed by least square fit for each test at five different locations along the radius of the plate. The results in the non-dimensional form wh/R2 are reported in table III. The corresponding regression coefficients are also indicated in parenthesis.

The obtained deflection rates are plotted against P/h 2 on a loglog scale in figure 11. The values of parameters nand a(r) of equation (24) have been obtained by least square fit for each line and are reported as well as the regression coefficient (r.c.) in table IV. The consistancy in the numerical value of n (approximatly 3) confirms a solution of the form of equation (24).

The finite element model used for elastic analysis has been used for a creep analysis with a law of the form:

(25)

The parameter a adjusted to match the experimental central deflection rate was found to be 1.625 x 10- 12 For the same conditions, Hrudey's solution will give a = 1.72 x 10- 12 which is quite close, while for Malinin's, a = 2.67 x 10- 12

Using these values, the function a(r) has been evaluatcd accordingly to each approximation and the resulting values arc also reported in table IV. These values, as well as the experimental ones, have been plotted on figure 12. The agreement is best between AXI and Hrudey even if the mathematical model is slightly different. However, both di ffer from the experiment.

It is also interesting to study the evolution of the deflections across the plate as shown in figure 13. The results of AX I , using the creep 1 a\,

(26)

arc also plotted after ~l time of 24 hours. 111e discrepancy bebreen theoretical and experiment;! 1 central deflection can be explained by the influence of ]lrilll~II'Y crecjl, which is not taken into consjderation by the program. It C;lJl ~llso cont rioutc towurds :111)' di fference in the actual deflected sh;lpcs. Comp;lrison with other boundary conditions ,:md in particular Hith an infln.itc plate on clastic foundation (Neve1 2 ), seems to

69

indicate that the simply supported circular plate, possibly because of its high stress gradient, over-emphasizes the effect of primary creep.

Photograph 3 shows the plate deflected unde r a constant static load. Photograph 4 shows the same plate cut along a diameter in order to illustrate the large deflections obtained after 70 hours of loading.

A comparison between (10) and (26) shows that the creep law obtained from plate tests is different from the beam tests for the same brine volume. This is not surprising because the plate behaviour is very different from a statically determinate beam where internal forces can easily redistribute themselves within the plate. As for the literature search, the work of Frederking &Gold 32 sug~ests a value for n ranging between 1.5 to 2 and the work of St-Pierre 3 and Ladanyi 34 , an average value of 2.1.

All other figures in the literature reported by Glen 35 , Jellinek &Bril1 36 , Krauz 37 , Mellor &Testa 38 , Nadreau 39 , refer to fresh water ice and average a val ue for n around 3.

CONCLUSION

The testing in flexure of sea-ice beams and plates with an average brine volume of about 26ho was reported. About one hundred beams were tested elastically and the flexural strength of simply supported beams average about 570 kPa. However, when the beams are tested upside down the value is twice as high which means that crystallography is as important as the brine volume for determination of flexural strength. The elastic modulus averaged a value of 3.89 x 10 6 kPa. The f i fteen beams tested in flexural creep for a range of stress varying from 276 kPa to 552 kPa indicate that secondary creep is always obtained and a Power law of the form 2.305 x 10- 12 (0)2.3 2 is suggested.

Elastic tests on simply supported plates l-vi th and wi thout water suggest that the elastic modulus obtained from deflection measurement on th e plate is always much smaller than the corresponding value for beams. furthermore, when water is present underneath, a pressure must be applied (because of the high loading rate) in order to ge t an acceptable correlation between theoretical and experimental resul t s . llowever, the main reason as to why the elastic modulus is reduceu is becaus e of th e large difference in the stress rate across the plate. lucally, the plate should be analy zed with a variable elastic mouulus across it s rauiu5 dependin g on the stress rate and the finite clement technique is ideally suited for that purpose. Work is now in progress to investigate this effect.

As for the creep flexure of plates, a power law of the form E = 1.6::'5 x 10- 12 (0) 3 is suggested. This is different from the value obt ained for beams under the same brine volume. However, since the theoretical resul ts based on a simple power law are not in total agreement wi th the experiments, this probl em des erves further attention. An attempt to a summation of two laws, as suggest ed by Nixon 4o , might be useful since it would cover a wider spectrum of effective stress across the plate.

71

Poly technique de Montreal, (Canada), to be pub li:;hed. [20] Tabata,T. ,Fujino K. ,Masaaki A. ,1966."StuLli,-':' of the Mechanical Pro

perties of Sea Ice,XI",International Conference on Low Temperature Science,Sapporo,Japan,(Aug.1966).Physics of Snow and Ice,vol.I.

[21] Weeks W.F.,Assur A.,1966."The Mechanical PropertLes of Sea Ice".Ice Pressure against Structures,Proceedings of ;1 Confl'H'Ill'l' held at Laval University,Quebec,Nov.1966.

[22] Dykins J .E. ,1971."Ice-Engineering.Material PropeniL's of Saline Ice for a Limited Range of Conditions".Naval Civil Engineering Laboratory, Technical Report R720,Port Hueneme,Cal.

[23] Vaudrey K.D.,Katona M.G.,1975."An Elastic Structural Analysis of Floating Ice Sheets by the Finite Element Method". Third International Conference on Port and Ocean Engineering under Arctic Conditions, University of Alaska(Aug.1975).

[24J Schwarz J. ,Weeks W.S.,1976."Engineering Properties of Sea Ice", Symposium on Applied Glaciology, Cambridge,England,(Sept.1976).

[25] HuH J .A. ,1966. "Creep in Engineering Structures" .Blaisdell Publishing company.

[26] Kerr A.D. ,1975. "The Bearing Capacity of Floating Ice Plates Subjected to Static or Quasi-Static Loads.A Critical Survey",CRREL Research Report ,RR333.

[27] Bernstein S. ,1929. "The Railway Ice Crossing"(Text in Russian), Trudy Nauchno.Teckhnicheskogo Komiteta Narodnogo Komissariata Putei Soobshcheniia,vol.84.

[28] Frankenstein G. ,Garner R. ,1967. "Equations for Determining the Brine Volume of Sea Ice from -0.5C to -22.9C",Journal of Glaciology, vOl.6,n048,pp943-944.

[29] Odqvist F.K.G.,1974."Mathematical Theory of Creep and Creep Rupture", Oxford ~1athematical Monographs ,second edi tion.

[30] Timoshenko S.,Woinowsky-Krieyer S.,1961."Theorie des Plaques et Coq ues" , Deuxi erne edi ti on, Li brai rie Pol ytechniq ue , ch . Berange r , Paris. Dunod ed.

[31] Hrudey T .M. ,1971. "A Creep Bending Analysis of Plates by the Fini te Element Method" .National Research Council of Canada, Aeronautical Report LR 552.

[32] Fredcrking R.M.W. ,Gold L.W. ,1975. "A Des i gn ApproJch for the TLmedependant Bearing Capacity of Icc C:overs".NI~C:,Div. of Hllilding I

- --

TABLE III: Flexural creep of plates

1 2 54Plate no 3

92 89 92114 102h (mrn) 2.54 2.00 1. 57 3.49 1. 70P (kN)

P /h 2 (kPa) 299 412 165253 120

0.91510.70 3.27 0.413 11.80r - = 0 (0.9973)(0.9995) (0.9989) (0.9986) (0.9948)R

0.7390.392 10.689.32 2.93r 1 (0.9941) (0.9994)(0.9998) (0.9995)(0.9996)R 10

wh 10 6 0.7249.62r 1 8.84 2.33 0.337"R2x = (0.9999) (0.9981)(0.9995) (0.9986) (0.9974)R 6mn- 1

5.51 0.430r 1 1.58 0.1985.04 R="3 (0.9994) (0.9986)(0.9998) (0.9999) (0.9983)

r 1 2.63 0.782 0.099 2.65 0.2l9 - = (0.9931)(0.9997) (0.9999) (0.9987) (0.9998)R 2

TABLE IV: Values of a(r) x 10l3. (0 in kPa ,mJ t LI1 mn)

r R=O

r 1 "R-TO

r R

1 6

r 1 R-3"

r R

1 2

r 2r 5 "R-3 R-6

Experiment r.c.

n aCr)

0.975 2.97 2.779

0.975 2.96 2.530

0.971 2.97 2.215

0.975 2.95 1.422

0.970 2.94 0.792

.-\\1 15 a(r) 2.779 2.564 2.301 1.737 1.222 0.770 0.364

I-IRUOCY 10 a(r) 2.779 2.568 2.357 1.834 1.326 0.847 0.423

aCr) 2.779 2.699 2.583 2.179 1.671 1.114 0.548

75

PROBABILISTIC ICE-STRUCTURE INTERACTION THEORY

by

F.G. Bercha

President

F.G. Bercha and Associates Limited

339 - 6 Avenue S.W.

Calgary, Alberta T2P OR8

J.V. Danys Superintendent, Civil Engineering

Marine Aids Division, Canadian Coast Guard Ministry of Transport

Place de Ville, Tower A

Ottawa, Ontario KIA ON7

J.G. Rokne

Senior Analyst

F.G. Bercha and Associates Limited

ABSTRACT

The development of a theory for selecting design ice forces for conical and cylindrical bottom-founded offshore structures was carried out utilizing probabilistic analytical methods and associated computational techniques. Flexural and crushing ice sheet failure phenomena were described utilizing extensions of the customary deterministic theories. Methods of probabilistic analysis, involving random variable techniques, were utilized to generate a set of probabilistic equations describing each of the pure mode interactions. These equations yielded values of the mean force and the associated standard deviation for any given set of principal input random variables, permitting the specification of a unique ice load spectrum. The magnitude of ice load associated with any given probability of occurrence was then obtainable directly from the spectrum. Numerical sensitivity analysis was util i zed to identify salient combinations of random variables likely to occur , such as those associated with transitional or mixed ice sheet failur e modes. Several relevant ice-structure interaction spectra were cons tructed and discussed in the light of practical design implications.

77

INTRODUCTION

It is known that many designs arrived at using conventional safety factors result in excess strength with which is associated extra mass, weight, components, and other aspects, all of which ultimately result in extra costs. Questions such as "How safe is a design?" or "How can designs be made to specific levels of accuracy?" point directly to probabilistic and statistical methods.

Also, ice loads are particularly amenable to a probabilistic treatment ment because of the relatively large scatter of ice property data. Through the genera t ion 'and ava ilability of ce l oa '-spectra basedO'ii statistical knowledge of ice properties, a more realistic and reliable design for offshore structures should be possible, ultimately resulting in better economy by facilitating less conservative design in some cases "and red.ucing- -the_ number of failures in other cases.

In this paper, following a brief review of some fundamental concepts, a probabilistic form for the flexural and crushing equations for the inte ractions 1Jetween level i ce covers" and coni~a l s t r c res wi be developed. First, the pertinent deterministic equations [lJ* will be summarized; next, a probabilistic form of the equations will be generated, followed by presentation of selected results for both the deterministic and probabilistic analyses.

REVIEW OF SOME PROBABILISTIC FUNDAMENTALS

Make the assumption that certain variables are known as probability distributions. These variables may be termed random variables [3J.

Take the case of the ice thickness, h. The distribution of ice thickness over an area could take on the form shown in Figure 1. The exact form of this curve is not of major importance. What is of importance is that there is a certain maximum thickness, h = hmax ' for each sample area which is investigated. The distribution of such h is what is important in the calculation of forces on the structure. Clearly, the curve for h will be bell-shaped, with almost no sample area having very small or very large maximum ice thicknesses (ignoring open water). A curve for the maximum h would probably look like that shown in Figure l(~.

If the shape of the curve is known on the basis of experiments, it can be used directly in the calculations. To make use of such a curve, one either uses a Monte Carlo method, which is very costly in terms of computing effort, or one approximates the curve with a distribution that is easier to handle analytically.

The most analytically convenient and physically realistic distribution is the normal distribution. It is a symmetric distribution about the mean h for h with' a standard deviation sh and may be written as

-

SSF (1) s

*Numbers in square brackets correspond to publications listed under References.

79

3000

PURE CRUSHING

2500 .-~- 68 % CCNFIDENCE

LIMITS

95 % CONFIDENCE

LI MIT S

2000

95 % CONFIDENCE LI MITS

)( 1500 u...

1000

~-;It-- PURE FLEXURE

500

--68 % CONFIDE NCE LIMITS

o 20 30 40 50 60 70 80 90 a (DEGREES)

FIrURI: l PROBABILISTIC PREDICTIONS OVFR ENTIRE RANCE 0:,: CONE ANGLE

81

10

3[Eh10 (4a)12p

BO = 2 sin (t] (4b) II bllO (4c)

x 10 [1. 83 + .916\l + .073n 2 ] (4d)Jl

CT = [1 + ~ 2~Jlk] (4e) k = 1.1 + 1.2n - .O1211 3 (4f)

h2F a k (4g)z b

1 BO f CT

2F ~ F x (4h)x z if

~ (]J cosa + sina)/(cosa - sinaJl) (4i)

Jl = c + c v . (4j)1 2where Fz and Fx are the vertical and horizontal interaction force components, respectively, 10 is the characteristic length, y represents the angle sub tended by the ice sheet, and the other variables were either defined earlier, or represent auxiliary functions which need not be defined for the purposes of this paper.

For crushing, the following simplified equations from [2] are relevant:

N -if 4 sina a c h b (Sa)

F - N x sina (Sb)x

F N x cosu (Sc)z

For compound or bimodal interactions, the two sets of equations can be combined with the use of a set of mix coefficients which determine the force proportions that each mode is likely to contribute to the interaction. Such a mix theory has been developed by the writer, but will be described elsewhere.

83

s x

]J ~ [[~]'

And, again, using the method of partial derivatives, one can

obtain,

b 2+ X .916[~Jl,,4 h-U- 3~ .073[~J-U h-\} (15)l2p l2p Sh{ (1. 38 + .69v) (l~pJ % h_l;4 - .054 b2 [l~pJ-% h-%}.

In fact, from Equation (9), it can be shown,

1 + .55 ~Xh + .6 X~(l~pr~ h+~ b - .006b 3 X~(l~pr~ h-% (16) ]J]J ]J

and it is the expansion for CT that takes into account the effect of random fragmentation at the interface. The standard deviation for this effect is then,

[['~~r s 2 + [aCrr ' [acrr ,y'Sc '-='=' an sh + ax Sxt;T ]J ]J {[;~]' [. 55h + .6(1~p J % h% b - .006b3 [1~pJ% h-% 12 (17a) + [s ~]J ~]'r + .1Sb l~P + . 0075b 3 (l~P J % h_9~r[ r ~ h-~3.5S + [sx" T- (.SSh + .6b[1~pr'" h~ - .006b3(1~pJ% h-14]]}% .x 2 [

]J

It is possible to further vary the fragmentation effect by introducing a factor, rf' which modulates the spread of the CT distribution in accordance with the integrity weighting of the ice interface. Thus, an auxiliary equation to (17a) is

(17b)

Then, develop Equation (8) in distributed form and get, 1. 26

k = Yo s 0 + - .012h' [1.1 Eh3r4 [Ehb3

3 %~1 0 f [l2p l2p (18) From this one gets at once,

85

(1=75 0 (585,246) (2532,595 )

UJ 0 0 ~ 0 UJ ><

~ u...

FXf

,.....u... 1\ coII0 ...... I

>l-

--'

co

From the above development, it is possible to express each of the two sets of resultant forces in the form of a couple, specifying a unique force spectrum.

RESULTS OF PROBABILISTIC ANALYSIS

Probabilistic analysis was carried out using computer programs which evaluate both the mean value and the associated variance for the vertical and horizontal resultant forces for different interaction cases. Table 2 shows a sample printout of the probabilistic analysis, for the case of h = 3 ft., o f = 100 psi, 0c = 400 psi, ~O = .4, and v = 1.5 fps, for b = 20 ft. for the random variable standard deviations suggested in Table 1. In the printout, the quantities designated by FXBAR AND FZBAR correspond to the horizontal and vertical flexural forces while their crushing counterparts are designated by SXBAR and SZBAR. The quantity adjacent to each of the force columns, SFX, SSX, SFZ, AND SSZ, represent the calculated value of the standard deviation corresponding to each of the left adjacent force predictions.

From elementary statistical theory, it is also possible to determine the various confidence limits for the random variable distributions from a knowledge of the standard deviation. For a normal distribution, it is known that the 95% confidence limits are given by the mean plus or minus twice the standard deviation, while the 68% confidence limits are given by the mean plus or minus the standard deviation itself. Figure 2 shows a plot for the selected exemplary set of parameters, showing the pure modes and the 95% and 68% confidence limit ranges about the pure modes based on the probabilistic analysis. In addition, the mean value graph of a linear mode mix prediction is shown. The mixed mode predictions range generally within the overlap area corresponding to the 95% confidence limits. Probabilistic mixed mode range can be seen to occur between 75 and 80, suggesting that a mix over 5 may be an accurate way of describing the transitional failure mode region.

Figure 3 shows the normal distributions for the flexural and crushing failure modes for a specific cone angle. The overlap area represents the probability of failure or action in the combined mode. Finally, as the cone angle increases from 75 to 80, the two mode distributions would be drawn closer and closer together giving an ever increasing probability of combined failure load. At a = 75, as shown in Figure 4(a), the probability does not exceed that corresponding to 95% design criterion. However, as the angle is increased to 78 and more, the probability of a mixed mode failure approaches certainty--and goes well over the 95% confidence limits corresponding to the distributions, as illustrated in Figures 4(b) and 4(c).

Exact values of the different load probabilities can be obtained easily with the use of the standard normal distribution curve. Normalization of the x-axis values in a force distribution such as those shown in Figure 4 results in the standard form of the curves, which permits direct interpretation of the areas under the curve as probability of occurrence of the normalized quantities.

89

----

FLEXUR E

J-I.-:l'r------=....::,__--

0

8 0

IV 0 0 0

>-;:J H CJ c:: ;>;:I t'l ." )( W .I"

.--...

.--...(')

0 0 0

CJ '-' ~ 0 z ~ H V) H Z c:: t'l 0 '-'

~ 0 0 0

lrI o o o

0o o o

f ( F )

I

I 95% FXF \ \

\

\

-95% Fxc

68% Fxc

'---.... 68% FXf "-..

~ 5,10,20 MIX~

CRUSHING (3331,766 )

\ 95% Sx (6000,1000)

68% Fxc

F ( 4780,2007)Xf

.... ---

--

-- .........

.........

=----------~-95% Sx (6000,250) 68% (6000,500) " -- - " \ ~ )-" - - __ \ --68% Sx (6000,250'- ............ ---------------~\

- - ---STRUCTURAl STRENGTH - - -A Sx.

_-----~~s, (6000,1000)

---- _..- ~ 5 (6000,500)/' - -x _--- (6000,250)

__ / x_/ ~--'~--------------7r"--- -68-/0 FXf

/ -- ...-:--

"

91

(3330)

CONCLUSIONS

A probabilistic analysis of the loads generated during interactions of circular or conical offshore structures with moving level ice was carried out. The analysis yields continuous ice load spectra corresponding to any set of principal input random variables, which can be used as an aid to more realistic design of offshore facilities.

The the best of the writer's knowledge, the above treatment of ice loads has not been presented previously in the literature.

Not only can the load spectra be used directly to design structures, but also they can lead to a better understanding of the relationship among and probable proportion of different ice sheet failure modes in a given interaction.

Certain recommendations for further work can logically be made on the basis of the above investigation. These recommendations include the following:

(