On Modelling of ice ridge formation

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On Modelling of ice ridge formation

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On Modelling of Ice Ridge Formation

by M. Sayed and R. Frederking

Appeared in

Proceedings of IAHR Ice Symposium 1986

lowa City, lowa, 18-22 August 1986

Vol. I, p. 603-614

(IRC Paper No. 1426)

Reprinted with permission

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lAHR

Ice Symposium

1986

Iowa City, Iowa

ON MODELLING OF ICE RIDGE FORMATION

M. Sayed

Associate Research Officer

R. Frederking

Senior Research Officer

National Research Ottawa

Council of Canada Ontario, Canada

National Research Ottawa

Council of Canada Ontario, Canada

Abstract

A review and comparison of calculation methods of ice ridging forces is

presented.

An

overview of a new ridging model that predicts the limiting

rubble height and depth is included. This model uses a diffusion equation to describe the normal stress spread in the bulk rubble.

Results of calculation methods are compared to those of laboratory tests and a typical field situation. Continuum models appear to predict reasonable ridging forces while the potential energy method greatly underestimates the forces.

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I n t r o d u c t i o n

Knowledge of t h e f o r c e s a s s o c i a t e d w i t h i c e r i d g i n g and dimensions of t h e s a i l and k e e l a r e needed t o a d d r e s s s e v e r a l p r a c t i c a l problems. These i n c l u d e t h e p r e d i c t i o n of i c e f o r c e s on wide s t r u c t u r e s , t h e u s e of grounded r u b b l e f o r p r o t e c t i o n of s t r u c t u r e s and f o r c o n s t r u c t i o n p u r p o s e s , and t h e e s t i m a t i o n of t h e f o r c e s a c t i n g on pack i c e o r l a r g e m u l t i y e a r i c e f l o e s . The p r o c e s s of r i d g i n g remains p o o r l y u n d e r s t o o d i n s p i t e of t h e a t t e n t i o n i t h a s a t t r a c t e d . The s i m p l e c a l c u l a t i o n method u s e d i n p r a c t i c e a p p e a r s t o be i n a d e q u a t e i n many s i t u a t i o n s . This p a p e r p r e s e n t s a d i s c u s s i o n of a n a l y t i c a l models, i n c l u d i n g t h e r e s u l t s of a new model, t h a t r e l a t e i c e f o r c e s d u r i n g r i d g i n g t o r u b b l e geometry. Emphasis i s r e s t r i c t e d t o a comparison of t h e methods and t h e c o n s t r a i n t s on t h e i r v a l i d i t y .

Review of Ridging Models

E a r l y o b s e r v a t i o n s of i c e r i d g i n g were r e p o r t e d by Zubov (1945). He

d e s c r i b e d t h e geometry of many i c e r u b b l e f e a t u r e s and s u g g e s t e d e q u a t i n g t h e l o s s of t h e k i n e t i c energy of a n i c e f l o e t o t h e i n c r e a s e i n t h e p o t e n t i a l e n e r g y of t h e r u b b l e i n o r d e r t o c a l c u l a t e r i d g e - b u i l d i n g f o r c e . P a r m e r t e r and Coon (1973) developed a n u m e r i c a l k i n e m a t i c a l model t h a t p r e d i c t e d r i d g e g e o m e t r i e s t h a t were i n e x c e l l e n t agreement w i t h f i e l d o b s e r v a t i o n s . An i c e s h e e t was assumed t o p r o g r e s s h o r i z o n t a l l y i n s t e p s u n t i l i t broke i n bending. Broken i c e b l o c k s were p l a c e d i n t h e k e e l and t h e s a i l a t each s t e p . A f t e r a number of s t e p s , t h e l i m i t i n g h e i g h t and d e p t h of t h e r i d g e were reached. S i n c e no d i r e c t e s t i m a t e of t h e f o r c e i n t h e i c e s h e e t c o u l d be made, P a r m e r t e r and Coon (1973) s u g g e s t e d t h a t a lower bound t o t h e f o r c e c a n be c a l c u l a t e d by e q u a t i n g t h e work done by t h e i c e s h e e t t o t h e i n c r e a s e i n t h e p o t e n t i a l e n e r g y of t h e r i d g e . The r e s u l t i n g v a l u e s were i n t h e r a n g e of 100 t o l o 4 N/m, which was c o n s i d e r e d t o be i n agreement w i t h e x p e c t e d a v e r a g e f o r c e s i n a n i c e c o v e r s e v e r a l hundred k i l o m e t e r s long.

The lower bound method was a minor p a r t of P a r m e r t e r and Coon's s t u d y , which was concerned w i t h l a r g e - s c a l e movements of i c e covers. S t i l l , because of i t s s i m p l i c i t y and t h e need t o p r e d i c t i c e f o r c e s , s e v e r a l a u t h o r s used t h a t method t o d e r i v e a s i m p l e e q u a t i o n f o r r i d g e - b u i l d i n g f o r c e . That e q u a t i o n i s f r e q u e n t l y (and e r r o n e o u s l y ) a t t r i b u t e d t o

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Parmerter and Coon (1973). A review by Kovacs and Sodhi (1980) d e s c r i b e d t h e l i t e r a t u r e on t h i s "energy balance" method ( s e e a l s o Sodhi and

Kovacs, 1984).

Forces observed i n i c e c o v e r s a t l e n g t h s c a l e s r e l e v a n t t o wide

s t r u c t u r e s (10 t o 100 m) a r e much h i g h e r t h a n t h e v a l u e s p r e d i c t e d by t h e above method. Kovacs and Sodhi (1980) t r i e d t o overcome t h i s d i s c r e p a n c y by adding a f r i c t i o n a l f o r c e . They c o n s i d e r e d t h e c a s e of an i c e s h e e t s l i d i n g o v e r a r u b b l e accumulation t o c a l c u l a t e t h e f r i c t i o n f o r c e on t h e s u r f a c e of t h a t s h e e t . The r e s u l t i s r e s t r i c t e d t o t h e assumed mode of i c e b e h a v i o r and does n o t a p p l y when t h e b u l k r u b b l e i s undergoing

deformation. V i s u a l a c c o u n t s of r i d g e s and r u b b l e f e a t u r e s (Zubov, 1945) i n d i c a t e t h a t complex modes of r u b b l e d e f o r m a t i o n may o c c u r .

Guidance t o t h e f o r m u l a t i o n of more a c c u r a t e models came from l a b o r a t o r y experiments on i c e r u b b l e (e.g., T a t i n c l a u x and Cheng, 1978; Prodanovic,

1979; and Hellmann, 1984). These i n v e s t i g a t i o n s showed t h a t deforming bulk r u b b l e approximately f o l l o w s a Mohr-Coulomb y i e l d c o n d i t i o n . Mellor (1980) r e a l i z e d t h a t a continuum approach c a n be used t o model b r a s h i c e and examined some c a s e s r e l a t e d t o i c e r i d g i n g .

The _{p r e s e n t a u t h o r s (Sayed and F r e d e r k i n g , 1984) developed a continuum} model by c o n s i d e r i n g t h e bulk r u b b l e t o be r i g i d - p l a s t i c obeying t h e Mohr-Coulomb y i e l d c r i t e r i o n . The r i d g e was i d e a l i z e d a s a wedge a t t h e p a s s i v e c r i t i c a l s t a t e . The r e s u l t i n g f o r c e s i n t h e i c e s h e e t were h i g h e r t h a n t h o s e c a l c u l a t e d from t h e p o t e n t i a l energy method. T h i s p l a s t i c i t y a n a l y s i s is probably s u i t a b l e f o r t h e e a r l y s t a g e s of r i d g e b u i l d i n g , when most of t h e r u b b l e i s deforming ( a t y i e l d ) . A s a r i d g e approaches i t s l i m i t i n g h e i g h t and depth a d i f f e r e n t s i t u a t i o n may develop, w i t h only p a r t s of t h e r u b b l e deforming.

S t r e s s D i s t r i b u t i o n i n Bulk Rubble

Although l a b o r a t o r y experiments s u g g e s t t h a t deforming bulk r u b b l e may f o l l o w t h e Mohr-Coulomb c r i t e r i o n , s u b s t a n t i a l e f f o r t i s s t i l l needed t o f u l l y c h a r a c t e r i z e t h e m a t e r i a l behavior. Numerical models can be c o n s t r u c t e d once t h e a p p r o p r i a t e c o n s t i t u t i v e e q u a t i o n s a r e e s t a b l i s h e d .

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A

simpler alternative approach is one used to perform stress calculations in particulate media. Considering that forces are transmitted through random contacts between particles, the stress applied at a boundary has been shown to spread according to the one-dimensional diffusion equation. Smoltczyk (1967), Harr (1977), and Chikwendu and Alimba (1979) developed this theory. Detailed derivations can be found in these references; only a simple case, given by Harr, is mentioned below as a justification and to illustrate this approach. Equilibrium of the blocks shown in Fig. 1 gives

where x and y are Cartesian coordinates, and

a

is the normal stress in

X

the x direction. Expanding

a

using Taylor's theorem and assuming that

X

Ax and Ay tend to zero gives,

where

D

= lim

(AX)

A x + O t ~

Based on physical and dimensional arguments, the diffusivity was chosen as

The value of v given by Harr (1977) is

$,

while Chikwendu and Alimba (1979) calculated 0.273 for spherical particles. Ice rubble would

TC

correspond to high value for v, possibly close to 8, owing to the angularity of typical ice blocks.

It should be noted that Equation

(2)

can be derived using a probabilistic approach that is essentially different from the above simple

deterministic method. The aim of this section is not to present a

rigorous proof but rather to propose the use of Equation (2) to calculate stresses in bulk rubble.

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F I G U R E 1

N O R M A L S T R E S S D I S T R I B U T I O N B E T W E E N I C E B L O C K S

Overview of a Ridge Limiting-Height Model

A new model of r i d g e f o r m a t i o n a s i t r e a c h e s t h e l i m i t i n g h e i g h t and d e p t h i s b r i e f l y d e s c r i b e d . The a n a l y s i s i s t o o l o n g t o be a d e q u a t e l y p r e s e n t e d h e r e . T h e r e f o r e , t h e d i s c u s s i o n i s r e s t r i c t e d t o a n o u t l i n e of t h e approach and p a r t of t h e r e s u l t s .

It i s assumed t h a t h a l f a two-dimensional symmetrical wedge ( a s shown i n Fig. 2 ) r e p r e s e n t s a s a i l o r a k e e l . I n t h i s i d e a l i z e d model, t h e advancing i c e s h e e t a p p l i e s a h o r i z o n t a l f o r c e , F , a t t h e edge of a r u b b l e p i l e - u p of h e i g h t H. A v e r t i c a l body f o r c e , y , a c t i n g downwards, i s t a k e n a s t h e b u l k u n i t weight of t h e s a i l o r buoyancy of t h e k e e l . F I G U R E 2 D E F I N I T I O N S K E T C H O F I C E R I D G I N G M O D E L

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Assuming t h a t deformation i s two-dimensional and n e g l e c t i n g i n e r t i a t e r m s , t h e e q u i l i b r i u m e q u a t i o n s become

where a

,

a

,

and z a r e t h e normal and s h e a r s t r e s s components, and y i s

X Y

t h e u n i t weight of t h e i c e r u b b l e i n t h e s a i l o r k e e l . Equations (21, (51, and ( 6 ) , w i t h t h e a p p r o p r i a t e i n i t i a l and boundary c o n d i t i o n s c a n d e t e r m i n e t h e s t r e s s d i s t r i b u t i o n . D e t a i l s of t h e s o l u t i o n of t h e s e e q u a t i o n s a r e t o o l e n g t h y t o be i n c l u d e d h e r e . Our r e s u l t s show t h a t d i f f e r e n t s t r e s s d i s t r i b u t i o n s can be o b t a i n e d by changing t h e F r a t i o 7 . YH I n t h e e a r l y s t a g e s of t h e development of a r i d g e , t h e h o r i z o n t a l f o r c e a p p l i e d by t h e i c e s h e e t w i l l c a u s e f a i l u r e of t h e r u b b l e , which can t h e n move upwards. The h e i g h t c e a s e s t o i n c r e a s e when i t approaches a v a l u e s u f f i c i e n t t o i n h i b i t f u r t h e r f a i l u r e of t h e rubble. Thus, t h e l i m i t i n g h e i g h t i s assumed t o o c c u r when s t r e s s e s r e a c h t h e c r i t i c a l s t a t e o v e r p a r t of a v e r t i c a l plane. The Mohr-Coulomb c r i t e r i o n i s used t o d e t e r m i n e f a i l u r e zones i n t h e rubble. 'Ihis f a i l u r e c o n d i t i o n i s

where i s t h e a n g l e of i n t e r n a l f r i c t i o n . A t y p i c a l f a i l u r e zone i n t h e r u b b l e a t t h e l i m i t i n g h e i g h t i s shown i n Fig. 3.

A r e l a t i o n s h i p between t h e f o r c e F and t h e l i m i t i n g s a i l h e i g h t o r k e e l d e p t h H was found u s i n g t h e above approach. For t h e v a l u e s v = 0.4 and ( = 50°, which a r e l i k e l y t o r e p r e s e n t t h e p r o p e r t i e s of i c e r u b b l e , t h e r e s u l t is

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D I M E N S I O N L E S S H O R I Z O N T A L D I S T A N C E , x l H F I G U R E 3 C O N T O U R S OF T H E R A T I O O F M A X I M U M S H E A R S T R E S S TO A V E R A G E N O R M A L S T R E S S S H O W I N G F A I L U R E Z O N E I N A R U B B L E P I L E - U P A T T H E C R I T I C A L H E I G H T I + = 5 0 ° , v = 0 . 4 )

45'. Details of this analysis will be presented elsewhere (M. Sayed and

R.M.W. Frederking.

A

model of ice rubble pile-up. Manuscript in

preparation.).

Comparison of Calculation Methods

The force predicted from the limiting-height model (Equation (8)) is

proportional to the square of the height.

A

similar relationship was

obtained from the plasticity analysis (Sayed and Frederking, 1984)

where C depends on the side slopes of the rubble.

A

wedge as shown in

Fig.

2

with one side at a 45O slope, and the other side horizontal

corresponds to

C

= 1.08.

The force calculated using the potential energy method (Kovacs and Sodhi, 1980) would be

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F r i c t i o n a l o n g an i c e s h e e t s l i d i n g o v e r t h e s l o p e i s

Ff = pyitH c o t

p

( 1 1 )

where p i s a c o e f f i c i e n t of f r i c t i o n and 3 ,! i s t h e s l o p e . T h i s would i n c r e a s e t h e f o r c e g i v e n by Equation ( 1 0 ) by approximately 20% f o r t h e c a s e of wedge w i t h s l o p e = 45' and u s i n g p = 0.1.

For n e u t r a l l y buoyant r i d g e s , t h e r a t i o of k e e l d e p t h t o s a i l h e i g h t depends on t h e u n i t weight and buoyancy of rubble. It i s s i m p l e t o show t h a t t h e l i m i t i n g - h e i g h t and p l a s t i c i t y models (Equations ( 8 ) and ( 9 ) ) g i v e

where F s , Fk, ys, and yk a r e t h e f o r c e s a s s o c i a t e d w i t h s a i l and k e e l f o r m a t i o n , t h e u n i t weight of t h e s a i l , and t h e buoyancy of t h e k e e l r e s p e c t i v e l y . The p o t e n t i a l energy method p r e d i c t s f o r c e s l i n e a r l y p r o p o r t i o n a l t o r u b b l e h e i g h t ; consequently f o r c e s a s s o c i a t e d w i t h t h e s a i l and k e e l would be e q u a l .

Forces p r e d i c t e d by t h e v a r i o u s method f o r a t y p i c a l r i d g e ( i c e c o n c e n t r a t i o n = 0.75, i c e s p e c i f i c g r a v i t y = 0.9 and t = 1.5 m) a r e compared i n Fig. 4. Although no f i e l d measurements d i r e c t l y r e l a t i n g f o r c e s t o r i d g e g e o m e t r i e s a r e a v a i l a b l e , c o n s i d e r as an example observed r i d g e s i n t h e s o u t h e r n Beaufort Sea. Geometry of f l o a t i n g r i d g e s and r u b b l e accumulations a g a i n s t s t r u c t u r e s can be uniform over l e n g t h s of t h e o r d e r of 100 m. S a i l h e i g h t s c a n r e a c h 5 m and k e e l d e p t h s do n o t exceed 20 m. Forces i n i c e c o v e r s of commensurate l e n g t h s ( s e e

Johnson e t a l . , 1985; and Sayed e t a l . , 1986) a r e of t h e o r d e r of 100 kN/m. It can be s e e n from Fig. 4 t h a t t h e l i m i t i n g h e i g h t and p l a s t i c i t y models a p p a r e n t l y g i v e v a l u e s of t h e c o r r e c t o r d e r , whereas t h e p o t e n t i a l energy method g r o s s l y u n d e r e s t i m a t e s i c e f o r c e s .

C a l c u l a t e d f o r c e s a r e compared t o t h e e x p e r i m e n t a l r e s u l t s of Timco and

Sayed (1986) i n Fig. 5. The experiments produced two-dimensional

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10 20 30 40 5 0 D E P T H , rn F I G U R E 4 P R E D I C T E D T O T A L I C E F O R C E S A N D T H E C O R R E S P O N D I N G K E E L A N D S A I L D E P T H S C A L C U L A T I O N S : 1 - P L A S T I C I T Y M E T H O D 2 - L I M I T I N G H E I G H T A N D D E P T H M E T H O D 3 - P O T E N T I A L E N E R G Y M E T H O D , F R I C T I O N A D D E D

concentration of 0.75, ice thickness of 0.05 m, and ice specific gravity of 0.95. Forces corresponding to both keel and sail formation are added to give the total force in the ice sheet. The limiting-height and plasticity models give forces of the same order though lower than the experimental results. The potential energy method predicts very low forces.

It appears from the above examples that continuum models predict

reasonable ice ridging forces. The plasticity model gives forces higher than the limiting-height model by approximately 40%. This is expected because the plasticity analysis considers all rubble to be yielding, which may be suitable for the early stage of ridge development. A

similar trend was observed in model tests by Timco and Sayed (1986).

The potential energy method greatly underestimates ridging forces. It seems to be more suitable for situations where an ice sheet slides over a shore. For example, the forces calculated by Kovacs and Sodhi (1980) approached reasonable values for cases where ice would slide for

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0 200 400 6 0 0 800 1 0 0 0 T O T A L F O R C E , N l m F I G U R E 5 C O M P A R I S O N OF P R E D I C T E D T O T A L F O R C E - R U B B L E H E I G H T A N D D E P T H W I T H THE E X P E R I M E N T A L R E S U L T S O F T l M C O A N D S A Y E D ( 1 9 8 6 1 C A L C U L A T I O N S : 1 - P L A S T I C I T Y M E T H O D 2 - L I M I T I N G H E I G H T A N D D E P T H M E T H O D 3 - P O T E N T I A L E N E R G Y M E T H O D . F R I C T I O N A D D E D r e l a t i v e l y l o n g h o r i z o n t a l d i s t a n c e s (20 m t o 90 m) b e f o r e r u b b l e p i l e - u p s 3 m t o 6 m h i g h were formed. Summary

A review and comparison of c a l c u l a t i o n methods of i c e r i d g i n g f o r c e s was p r e s e n t e d . A d i f f u s i o n e q u a t i o n was proposed a s a r e p r e s e n t a t i o n of normal s t r e s s s p r e a d i n bulk r u b b l e . A model of r i d g e s a t t h e i r l i m i t i n g h e i g h t and d e p t h t h a t u s e s t h a t approach was o u t l i n e d .

C a l c u l a t i o n s i n d i c a t e t h a t t h e p l a s t i c i t y and l i m i t i n g - h e i g h t models p r e d i c t f o r c e s of t h e o r d e r e x p e c t e d when deformation i s uniform. This c o r r e s p o n d s t o r i d g e s and f l o a t i n g r u b b l e p i l e - u p a g a i n s t o r n e a r wide

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s t r u c t u r e s . The p o t e n t i a l energy method g r e a t l y u n d e r - e s t i m a t e s i c e f o r c e s i n t h e s e c a s e s . It seems more s u i t e d f o r s i t u a t i o n s where a n i c e s h e e t s l i d e s over a s h o r e f o r r e l a t i v e l y l o n g d i s t a n c e s .

Observed r i d g e k e e l d e p t h and s a i l h e i g h t s , and measured i c e p r e s s u r e s i n t h e s o u t h e r n Beaufort Sea a r e b e s t e x p l a i n e d by t h e p l a s t i c i t y and

l i m i t i n g h e i g h t models t h a t p r e d i c t a n H~ dependence of F r a t h e r t h a n t h e l i n e a r H dependence of t h e energy models.

R e f e r e n c e s

Chikwendu, S.C., and Alimba, M. 1979. D i f f u s i o n analogy f o r some s t r e s s computations. J o u r n a l of t h e G e o t e c h n i c a l Engineering D i v i s i o n , P r o c e e d i n g s , American S o c i e t y of C i v i l E n g i n e e r s , Vol. 105, No. GT11, Proceedings Paper 1111, p. 1337-1342.

H a r r , M.E. 1977. Mechanics of p a r t i c u l a t e media. McGraw-Hill, New York, N.Y.

Hellmann, J.-H. 1984. B a s i c i n v e s t i g a t i o n s on mush i c e . P r o c e e d i n g s , I n t e r n a t i o n a l A s s o c i a t i o n f o r H y d r a u l i c Research, I c e Symposium, Hamburg, Vol. 3, p. 37-55.

Johnson, J.B., Cox, G.F.N., and Tucker, W.B. 1985. Kadluk i c e s t r e s s measurement program. 8 t h I n t e r n a t i o n a l Conference on P o r t and Ocean E n g i n e e r i n g under A r c t i c C o n d i t i o n s (POAC), N a r s s a r s s u a q , Greenland, 1985, Vol. 1, p. 88-100.

Kovacs, A. and Sodhi, D.S. 1980. Shore i c e p i l e - u p and r i d e - u p : F i e l d o b s e r v a t i o n s , models, t h e o r e t i c a l a n a l y s i s . Cold Regions S c i e n c e and Technology, Vol. 2, p. 209-298.

M e l l o r , M. 1980. Ship r e s i s t a n c e i n t h i c k b r a s h i c e . Cold Regions S c i e n c e and Technology, Vol. 3, p. 305-321.

P a r m e r t e r , R.R. and Coon, M. 1973. Model of p r e s s u r e r i d g e f o r m a t i o n i n s e a i c e . J o u r n a l of Geophysical Research, Vol. 77(33),

p. 6565-6575.

Prodanovic, A. 1979. Model t e s t s of i c e r u b b l e s t r e n g t h . P r o c e e d i n g s , 5 t h I n t e r n a t i o n a l Conference on P o r t and Ocean E n g i n e e r i n g u n d e r A r c t i c C o n d i t i o n s , Trondheim, Norway, Vol. 1, p. 89-105.

Sayed, M. and F r e d e r k i n g , R.M.W. 1984. S t r e s s e s i n f i r s t - y e a r i c e p r e s s u r e r i d g e s . P r o c e e d i n g s , 3rd I n t e r n a t i o n a l O f f s h o r e Mechanics and A r c t i c E n g i n e e r i n g Symposium, New O r l e a n s , Vol. 3, p. 173-177.

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Sayed, M., Croasdale, K.R., and Frederking, K.M.W. 1986. Ice stress measurements in a rubble field surrounding a caisson retained island. Proceedings, International Conference on Ice Technology, MIT, Cambridge, Mass., June 10-12, 1986.

Smoltczyk, H.V. 1967. Stress computation in soil media. Journal of the Soil Mechanics and Foundations Division, Proceedings, American Society of Civil Engineers, Vol. 93 (No. SM~), p. 101-124.

Sodhi, D.S. and Kovacs, A. 1984. Forces associated with ice pile-up and ride-up. Proceedings, International Association for Hydraulic Research, Ice Symposium, Hamburg, Vol. 4, p. 239-262.

Tatinclaux, J.C. and Cheng, S.T. 1978. Characteristics of river ice jams. Proceedings, International Association for Hydraulic Research, Lulea, Sweden, Vol. 2, p. 461-475.

Timco,

G.W.

and Sayed, M. 1986. Model tests of the ridge-building process. Proceedings, International Association for Hydraulic Research, Symposium on Ice, Iowa City, Iowa, August 18-22, 1986. Zubov, N.N. 1945. Arctic Ice. Izdatel'stvo Glavsevmorputi, Moscow,

1945, (Translation AD42697 2, National Technical Information Service, Springfield, Va.).