Stresses in first-year ice pressure ridges

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Stresses in first-year ice pressure ridges

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1 STRESSES I N FIRST-YEAR ICE PRESSURE RIDGES

by

M. Sayed and R.M.W. Frederking

ANALYZED

Reprinted from

Proceedings of the Third International Offshore Mechanics

and Arctic Engineering Symposium, Volume 11 1, 1984

p. 1 7 3 - 177

DBR Paper No. 1193

Division of Building Research

Price $1

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OTTAWA

CL3G.

RES.

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Les c r s t e s d e p r e s s i o n d e l a g l a c e d e mer d e p r e r n i k e ann'ee s o n t mod'elis'ees s o u s forme de prismes bidimensionnels s e comportant s e l o n l a l o i de MohrCoulomb. La r 6 p a r t i t i o n d e s c o n t r a i n t e s p a s s i v e s e s t obtenue e n u t i l i s a n t une s o l u t i o n s i m i l a i r e e t d e s polynomes a p p r o x i m a t i f s de l a v a r i a b l e ddpendante. Ces a n a l y s e s p e r m e t t e n t d e c a l c u l e r l e s f o r c e s h o r i z o n t a l e s dans l e s c o u v e r t u r e s d e g l a c e li'ees

a

l a f o r m a t i o n d e c r L e s d e p r e s s i o n f l o t t a n t e s e t dans c e r t a i n s c a s , aux amas de g l a c e l e long d e s c b t e s .

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STRESSES I N FIRST-YEAR ICE PRESSURE RIDGES

M. Sayed and R.M.W. Frederking Division of Building Research National Research Council Canada

Ottawa, Ontario, Canada

ABSTRACT

F i r s t - y e a r s e a i c e p r e s s u r e r i d g e s a r e modelled a s two-dimensional wedges of a Mohr-Coulomb m a t e r i a l a n d t h e p a s s i v e s t r e s s d i s t r i b u t i o n i s o b t a i n e d u s i n g a s i m i l a r i t y s o l u t i o n and a p p r o x i m a t e p o l y n o m i a l forms of t h e dependent v a r i a b l e . A n a l y s i s g i v e s t h e h o r i z o n t a l f o r c e s i n i c e c o v e r s a s s o c i a t e d w i t h f o r m a t i o n of f l o a t i n g p r e s s u r e r i d g e s and some c a s e s of r u b b l e p i l e - up a g a i n s t s h o r e s . INTRODUCTION I c e f l o e s d r i v e n by wind and c u r r e n t o f t e n f a i l on e n c o u n t e r i n g o t h e r i c e f l o e s o r f i x e d o b s t a c l e s , by c r u s h i n g , f l e x u r e o r b u c k l i n g , w i t h r e s u l t i n g acummulation of i c e blocks. These masses form a v a r i e t y of f e a t u r e s s u c h a s p r e s s u r e r i d g e s , p i l e - u p s on s h o r e s , and r u b b l e f i e l d s s u r r o u n d i n g o f f s h o r e s t r u c t u r e s . There have been e x t e n s i v e f i e l d , l a b o r a t o r y , and a n a l y t i c a l i n v e s t i g a t i o n s of t h e f r e q u e n c y of o c c u r r e n c e and t h e c h a r a c t e r i s t i c s of s u c h r i d g e s and p i l e u p s , and of a s s o c i a t e d l o a d s . Kovacs and Sodhi

(1)

p r e s e n t e d a comprehensive review of t h e l i t e r a t u r e on i c e r u b b l e p i l e u p s ; and a number of r e c e n t s t u d i e s were i n c l u d e d i n t h e P r o c e e d i n g s of t h e Workshop on Sea I c e R i d g i n g and P i l e u p

(2).

E a r l y i n v e s t i g a t i o n s of i c e r u b b l e f e a t u r e s and g e o m e t r i c a l models of r i d g e s were r e p o r t e d by Zubov

(1);

r e c e n t i n v e s t i g a t i o n of f i r s t - y e a r p r e s s u r e r i d g e s i n t h e B e a u f o r t Sea have been p r e s e n t e d by Tucker and Govoni

(4).

An a n a l y t i c a l model of t h e k i n e m a t i c s of p r e s s u r e r i d g e f o r m a t i o n was developed by P a r m e r t e r and

Coon

(5).

who o b t a i n e d lower bounds f o r t h e a s s o c i a t e d f o r c e s by e q u a t i n g t h e work done by t h e advancing 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 energy of t h e r i d g e . T h i s approach h a s been a d o p t e d i n a l l s u b s e q u e n t s t u d i e s t h a t e s t i m a t e r i d g i n g o r p i l e u p f o r c e s . Some a u t h o r s have used semi-empirical f o r m u l a s of s o i l mechanics t o i n c l u d e f r i c t i o n a l f o r c e s a s w e l l . K r y

(5)

d i s c u s s e d i c e f a i l u r e modes a s s o c i a t e d w i t h r u b b l e f o r m a t i o n and t h e r e s u l t i n g f o r c e s on wide o f f s h o r e s t r u c t u r e s . M e l l o r

(2)

t r e a t e d b r a s h i c e a s a

Mohr-Coulomb m a t e r i a l and c o n s i d e r e d some s i m p l e c a s e s r e l a t e d t o p r e s s u r e , r i d g i n g .

L a b o r a t o r y e x p e r i m e n t s on model i c e r u b b l e performed by Keinonen and Nyman

(8)

and by

Prodanovic

(9)

s u g g e s t t h a t t h e bulk r u b b l e obeys t h e MohrCoulomb y i e l d c r i t e r i o n . I n o t h e r e x p e r i m e n t s by T a t i n c l a u x and Cheng

(10)

t h e r a t e e f f e c t on s h e a r r e s i s t a n c e was examined. The p r e s e n t s t u d y i s

concerned w i t h t h e development of s o l u t i o n s f o r s t r e s s d i s t r i b u t i o n s i n f i r s t - y e a r p r e s s u r e r i d g e s and some c a s e s of r u b b l e p i l e - u p . GOVERNING EQUATIONS Deformation of bulk r u b b l e d u r i n g r i d g e b u i l d i n g o r r u b b l e p i l i n g i s u s u a l l y comprised of r e l a t i v e motion and r e a r r a n g e m e n t of t h e b l o c k s , i n a d d i t i o n t o f a i l u r e of i n d i v i d u a l b l o c k s . T h i s mode of d e f o r m a t i o n i s s i m i l a r t o t h e b e h a v i o u r of g r a n u l a r m a t e r i a l s and s o i l s . Both o b s e r v a t i o n and e x p e r i m e n t a l e v i d e n c e s u g g e s t t h a t i n s i t u a t i o n s of p r e s e n t i n t e r e s t t h e r u b b l e may obey t h e MohrCoulomb y i e l d c r i t e r i o n . I n t h e f o l l o w i n g a n a l y s i s bulk r u b b l e i s c o n s i d e r e d t o b e a r i g i d - p l a s t i c continuum u n d e r g o i n g q u a s i - s t a t i c , two- d i m e n s i o n a l d e f o r m a t i o n ; t h e r e a r e no e x p e r i m e n t a l d a t a o r f i e l d o b s e r v a t i o n s of p o s s i b l e volume changes d u r i n g d e f o r m a t i o n . A p r e s s u r e - v o i d s r e l a t i o n would be r e q u i r e d i f c o m p r e s s i b l i t y were t o b e c o n s i d e r e d . The i c e b l o c k c o n c e n t r a t i o n ( o r v o i d s r a t i o ) is t h e r e f o r e assumed t o be c o n s t a n t . A s r u b b l e p i l i n g e v e n t s u s u a l l y o c c u r o v e r s h o r t p e r i o d s

(L),

t e m p e r a t u r e and i t s e f f e c t on t h e d e f o r m a t i o n p r o c e s s a r e ignored. The e q u a t i o n s of e q u i l i b r i u m a r e

a

u a T r + + 1 + 2 2 = - y s i n

e

ar r a0 r ( 2 )

where y i s t h e u n i t weight ( o r buoyancy of submerged r u b b l e ) and or, og and r r e a r e normal and s h e a r s t r e s s components, a s shown i n Fig. 1. Compressive s t r e s s e s a r e c o n s i d e r e d p o s i t i v e t h r o u g h o u t t h e p r e s e n t a n a l y s i s .

(5)

(6)

A t t h e s i d e s l o p e S = 0 and 9 =

-

6. Equation (13) g i v e s two v a l u e s of correspondfng t o t h e a c t i v e and p a s s i v e s t a t e s . For t h e p a s s i v e s t a t e

Approximate S o l u t i o n

Analysis i s s i m p l i f i e d by c o n s i d e r i n g t h e s t r e s s a n g l e , 6, t o be a l i n e a r f u n c t i o n of 8. T h i s

assumption was used by Nadai

(14)

f o r the a c t i v e s t a t e , and is i n r e a s o n a b l e agreement w i t h t h e numerical s o l u t i o n s of Sokolovski

(11)

and Marais

(13).

The p a s s i v e s t a t e may be somewhat d i f f e r e n t , but t h e l i n e a r d i s t r i b u t i o n s t i l l a g r e e s w i t h t h e e x a c t s o l u t i o n f o r t h e l i m i t i n g c a s e of h o r i z o n t a l s i d e s (apex a n g l e = n)

o r an i n f i n i t e s l o p e of any i n c l i n a t i o n . Thus, t h e s t r e s s a n g l e i s given by

where and $; a r e t h e v a l u e s of J, a t t h e s i d e s l o p e s . These could be determined from Eq. (12) and (141, and a1 and a;! a r e t h e a n g l e s between t h e s i d e s l o p e s and t h e v e r t i c a l d i r e c t i o n . For symmetrical wedges, only h a l f t h e wedge need be s t u d i e d and t h e v a l u e of $ a t t h e c e n t r e l i n e w i l l be R

.

Equation (15) t h e n reduces

2 t 0

The s t r e s s f u n c t i o n , S, i s approximated by a polynomial form

The c o e f f i c i e n t s So, S1,

. . .

a r e determined by s a t i s f y i n g t h e boundary c o n d i t i o n s and t h e e q u i l i b r i u m of f o r c e s on s e c t o r s of t h e wedge. Terms of o r d e r g r e a t e r t h a n

e2

a r e n e g l e c t e d . A s S = 0 a t t h e boundaries, Eq. (17) may be w r i t t e n a s

For symmetrical wedges ( a l = a 2 ) , S i s an even f u n c t i o n of 8 and Eq. (18) reduces t o

The unknown c o e f f i c i e n t , So, may be o b t a i n e d from t h e balance of v e r t i c a l f o r c e s a c t i n g on a s e c t o r of t h e wedge of r a d i u s r. T h i s c o n d i t i o n i s

S u b s t i t u t i n g Eq. (15) and (18) i n ( 8 ) and ( 3 ) , t h e c o n d i t i o n g i v e n by Eq. ( 2 0 ) , g i v e s So. The i n t e g r a t i o n i n Eq. (20) can be o b t a i n e d i n a c l o s e d form, a l b e i t a

very l e n g t h y one. It i s s i m p l e r and more p r a c t i c a l t o e v a l u a t e it numerically.

The a d d i t i o n a l term

s3e3

was i n c l u d e d i n a n e x t e n s i o n of Eq. (18) t o t e s t t h e convergence of t h e p r e s e n t s o l u t i o n f o r asymmetrical wedges. The

a d d i t i o n a l c o n d i t i o n was t h e e q u i l i b r i u m of h o r i z o n t a l f o r c e s on a s e c t o r of t h e wedge. The r e s u l t s , i n t h i s case, were alm s t i d e n t i c a l t o those o b t a i n e d by

₉

n e g l e c t i n g S38

.

The p r e s e n t s o l u t i o n approximates t h e e x a c t s o l u t i o n by power s e r i e s expansion. It i m p l i e s t h a t t h e y i e l d c r i t e r i o n and s i m i l a r i t y c o n d i t i o n a r e s a t i s f i e d everywhere. The e q u i l i b r i u m of f o r c e s , however, i s only s a t i s f i e d i n a n a v e r a g e s e n s e over t h e wedge. It could be s a t i s f i e d over any a r b i t r a r y number of s m a l l e r s e c t o r s of t h e wedge by i n c l u d i n g more terms i n t h e power s e r i e s of Eq. (17).

RESULTS

Values of So a r e p r e s e n t e d i n Fig. 2 f o r symmetrical wedges and i n Fig. 3 f o r asymmetrical wedges w i t h one h o r i z o n t a l s i d e , 62 = 0 , S t r e s s

d i s t r i b u t i o n s i n wedges of any s i d e s l o p e can be o b t a i n e d , a s w e l l , from t h e p r e s e n t a n a l y s i s . A s expected, s t r e s s e s i n c r e a s e w i t h i n c r e a s i n g e q u i v a l e n t a n g l e e of i n t e r n a l f r i c t i o n , $ ' , and w i t h d e c r e a s i n g s i d e s l o p e , 6. The c a l c u l a t e d v a l u e s of S o a r e less, by 3 x , than

Lp

exact v a l u e s f o r a h o r i z o n t a l s u r f a c e

(11

-

s i n + ' ]

)

owing t o approximation of t h e e x a c t s o l u t i o n c o s i n e f u n c t i o n by a parabola. The e r r o r i s expected t o d e c r e a s e f o r s m a l l e r apex a n g l e s ( l a r g e r 6). When t h e a n a l y s i s was extended t o t h e a c t i v e s t a t e t o examine i t s accuracy, t h e r e s u l t s were i n good agreement w i t h Marais' numerical v a l u e s

(13).

The maximum e r r o r was 3% f o r h o r i z o n t a l s u r f a c e s and decreased f o r l a r g e r v a l u e s of 6.

The h o r i z o n t a l f o r c e i n t h e i c e s h e e t a d j a c e n t t o a symmetrical r u b b l e wedge ( s a i l o r k e e l ) may be determined by i n t e g r a t i n g t h e normal s t r e s s a l o n g t h e v e r t i c a l a x i s ( 8 = O),

From Eq. ( 3 t o 8, 1 9 )

where H is h e i g h t of wedge (from apex t o base, s e e F i g u r e 4a) and t h e a n g l e ,

to,

i s

1 .

The o r d e r of

2

t h e s e f o r c e s may be i l l u s t r a t e d by c o n s i d e r i n g a n example of t y p i c a l v a l u e s f o r a f i r s t - y e a r p r e s s u z e r i d g e of a symmetrical t

₅

i a n g u l a r s a i l ( 6 ' = 60°, 6 = 25" and y = 6200 N/m ). The f o r c e a s s o c i a t e d w i t h k e e l formation may be t a k e n a s approximately e q u a l t o t h a t a s s o c i a t e d w i t h t h e s a i l ( s i n c e weight of t h e s a i l e q u a l s buoyancy of t h e k e e l ) . Thus, f o r a s a i l h e i g h t of 3 m t h e f o r c e i t h e i c e s h e e t d u r i n g r i d g e b u i l d i n g would be 2.34 x 10' N/m.

ksymmetrical r u b b l e f e a t u r e s , a s shown i n Fig. 4b, may have s a i l and k e e l apexes a t d i f f e r e n t v e r t i c a l

*This i s t h e v a l u e of $' f o r an i n t e r n a l f r i c t i o n a n g l e , 6 = 50" and a cohesion c o e f f i c i e n t , k = 0.15.

(7)

0 10 20 30 40 50 60 S I D E S L O P E . 8 ( d e g r e e s ) F i g u r e 2 S t r e s s c o e f f i c i e n t , So, f o r symmetrical wedges p l a n e s . The h o r i z o n t a l f o r c e i n t h e i c e s h e e t s h o u l d b e e s t i m a t e d by i n c l u d i n g t h e s h e a r s t r e s s e s on p a r t of t h e h o r i z o n t a l b a s e of t h e wedges i n a d d i t i o n t o t h e normal s t r e s s e s on t h e v e r t i c a l p l a n e s t h r o u g h t h e apexes. Note t h a t

JI

can be determined from t h e boundary v a l u e s and f i n e a r d i s t r i b u t i o n of J, g i v e n by Eq. (14) and (15). Normal and s h e a r s t r e s s e s on t h e h o r i z o n t a l b a s e of t h e wedge a r e g i v e n by 'Jx = _{c o s e} [ l 2 s i n + ' c o s 2 ( $

+

e ) ] (23) T =

YSH

s i n $ ' s i n Z ( ~ , + 8 ) c o s e (24) The s t r e s s e s on t h e b a s e of t h e s a i l of a r i d g e would n o t b e i n e q u i l i b r i u m w i t h t h o s e on t h e b a s e of t h e k e e l i f b o t h s a i l and k e e l had a common p l a n e boundary a t t h e w a t e r l e v e l . I n s t e a d , a t w a t e r l e v e l a l a y e r of r u b b l e o r an i c e s h e e t must e x i s t t h a t h a s s t r e s s e s from t h e s a i l and k e e l a c t i n g on d i f f e r e n t s i d e s . I n t e g r a t i o n of t h e s h e a r s t r e s s e s a l o n g t h i s l a y e r g i v e s t h e same t o t a l h o r i z o n t a l f o r c e i n t h e i c e c o v e r a s t h a t p r e d i c t e d by Eq. (22).

The f o r c e o b t a i n e d from t h e p o t e n t i a l energy approach f o r a symmetrical wedge is (Kovacs and

Sodhi

(1)).

where yi and t a r e t h e u n i t weight and t h i c k n e s s of i c e b l o c k s , r e s p e c t i v e l y . The r a t i o of t h e f o r c e s

p r e d i c t e d from t h e p r e s e n t a n a l y s i s t o t h o s e from t h e p o t e n t i a l energy approach i s

T y p i c a l v a l u e s of t h e v a r i a b l e s i n Eq. ( 2 6 ) g i v e r a t i o s of t h e o r d e r of 20.

The p r e c e d i n g a n a l y s i s a p p l i e s t o newly formed, f l o a t i n g r u b b l e w i t h a wedge-shaped s a i l o r k e e l and two p l a n e , s t r e s s - f r e e s i d e s . Grounded r u b b l e , on t h e o t h e r hand, would have d i f f e r e n t boundary c o n d i t i o n s .

0 10 20 30 4 0 50 60 S I D E S L O P E , 6 ( d e g r e e s ) F i g u r e 3 S t r e s s c o e f f i c i e n t , So, f o r a s y m e t r i c a l wedges w i t h one h o r i z o n t a l s i d e ( 6 2 = 0 ) The s i m i l a r i t y s o l u t i o n c o u l d be e x t e n d e d t o t r e a t s u c h c a s e s o n l y i f t h e g r o u n d i n g s t r e s s on t h e k e e l s i d e were p r o p o r t i o n a l t o t h e d i s t a n c e from t h e apex.

Shore p i l e - u p s o c c u r r i n g above w a t e r l e v e l may c o r r e s p o n d t o t h e above a n a l y s i s i f t h e r u b b l e i s p r e s s e d a g a i n s t a n o b s t a c l e i n s u c h a way t h a t a p a s s i v e stress s t a t e e x i s t s . An i c e s h e e t o r a l a y e r of r u b b l e a d v a n c i n g on t h e s h o r e would be s u b j e c t e d t o r e s i s t a n c e due t o r u b b l e on i t s t o p s u r f a c e ( o r t h e b a s e of r u b b l e wedge) and t o t h e r e s i s t a n c e of

grounding on i t s bottom s u r f a c e . Rubble f o r c e s can be e s t i m a t e d a s f o r f l o a t i n g i c e . Ground r e s i s t a n c e may be assumed t o be e q u a l t o t h e normal f o r c e from t h e r u b b l e wedge m l t i p l i e d by a f r i c t i o n c o e f f i c i e n t between t h e s o i l and i c e . Note t h a t t h e s h e a r s t r e s s e s from t h e ground and t h e r u b b l e wedge a r e n o t i n

e q u i l i b r i u m and a c t i n a d i r e c t i o n o p p o s i t e t o t h e movement of t h e advancing i c e s h e e t . O t h e r p i l e - u p s i t u a t i o n s ( n o t c o n s i d e r e d h e r e ) may o c c u r where t h e r e a r e no v e r t i c a l o b s t a c l e s e x e r t i n g h o r i z o n t a l f o r c e on

ICE SHEET SAIL

a ) S Y M M E T R I C A L R I D G E ICE SHEET f - ---,--- !

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'

.

KEEL C _ _ _ - - b l A S Y M M E T R I C A L R U B B L E F i g u r e 4 I c e r u b b l e f e a t u r e s

(8)

(9)

reprinted from

Proceedings of the Third International Offshore M a n i c s and

Arctic Engineering Symposium

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Volume II I

Editor: V. J. Lunardini (Book No. 100173) published by

THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 East 47th Street, New York, N. Y. 10017