Ice forces on an isolated circular pile

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Ice forces on an isolated circular pile

Frederking, R. M. W.; Gold, L. W.

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Reprinter wifhorrt change of pagination from the Proceedings f r o m r l ~ e Firrt I~rternational Conference on Port and Ocean Engineering under Arctic Conditions, Volume I . pp. 73 - 92. 1971. PORT AND OCEAN ENGINEERING UNDER ARCTIC CONDITIONS

TECHNICAL UNIVERSITY OF NORWAY

ICE FORCES ON AN ISOLATED CIRCULAR P I L E

R . F r e d e r k i n g & L. W . Gold

Division of Building R e s e a r c h Ottawa R e s e a r c h Officer & Head

National Re s e a r c h Council of Canada Canada Geotechnical Section

T h i s p a p e r p r e s e n t s a m a t h e m a t i c a l m o d e l f o r calculating the f o r c e s that a moving i c e c o v e r will e x e r t against a n isolated pile.

The m o d e l r e q u i r e s knowledge of the r e l a t i o n s h i p between s t r a i n r a t e in the i c e and the r a t e at which the i c e cover i s moving r e l a t i v e t o the pile. An e x p e r i m e n t a l and a t h e o r e t i c a l a p p r o a c h t o the d e r i v a t i o n of this relationship a r e advanced.

It i s a s s u m e d t h a t c o m p r e s s i v e f a i l u r e of the i c e in the zone adjacent to the pile c o n t r o l s t h e t h r u s t on the pile. It i s f u r t h e r a s s u m e d that t h e m o d e of f a i l u r e in t h i s zone i s comparable t o t h a t f o r uniaxial c o m p r e s s i o n t e s t s in the l a b o r a t o r y . The m o d e l t a k e s into account t h e influence of i c e type, s t r a i n r a t e , pile g e o m e t r y and t e m p e r a t u r e on the t h r u s t on t h e pile.

INTRODUCTION

The prediction of the f o r c e that ice can e x e r t on a s t r u c t u r e i s a design p r o b l e m that h a s s t i l l not been solved adequately. T h i s i s due to lack of knowl- edge of the deformation behaviour and s t r e n g t h of i c e , insufficient information concerning the c h a r a c t e r i s t i c s of i c e c o v e r s a s s o c i a t e d with the design condition, and l a c k of appreciation of the i n t e r a c t i o n between i c e and s t r u c t u r e s . The t h e o r e t i c a l investigation of a n u m b e r of r e l a t i v e l y s i m p l e c a s e s would r e s u l t i n useful p r o g r e s s t o w a r d s establishing design c r i t e r i a . If t h e a s s u m p t i o n s a r e p r o p e r l y chosen, the calculations should give the m a x i m u m loads t h a t could be expected. T h e s e studies could a l s o provide a b a s i s f o r f u r t h e r refinement of the design method based on additional f i e l d , l a b o r a t o r y , and t h e o r e t i c a l studies.

One situation that i s of i n t e r e s t i s t h a t of the f o r c e developed on a rigid c i r c u l a r pile by a laterally-moving i c e c o v e r . In t h i s p a p e r a method i s p r e s - ented f o r calculating the load f o r the condition of continuous contact between the i c e and half the c i r c u m f e r e n c e of the pile. The m o d e l t a k e s into account the s t r a i n r a t e dependence of the r e s i s t a n c e of i c e t o deformation, and the effect of t e m p e r a t u r e .

DEFORMATION BEHAVIOUR OF ICE

I t i s n e c e s s a r y t o have s o m e knowledge of the deformation behaviour of i c e t o p r o p e r l y understand the i n t e r a c t i o n between an i c e cover and a s t r u c t u r e . Under c e r t a i n loading conditions i c e behaves in a v i s c o e l a s t i c m a n n e r .

If

i t i s d e f o r m e d a t a constant r a t e of s t r a i n , i t exhibits an upper yield s t r e s s . The s t r a i n a s s o c i a t e d with yield depends on the type of i c e being d e f o r m e d and the r a t e of deformation. F o r the m o r e r e s i s t a n t types of i c e and the r a t e s of d e f o r -

-

3

m a t i o n t h a t should be considered i n d e s i g n , the s t r a i n at yield i s about 2 x 10 (1).

If the s t r e s s o r r a t e of s t r a i n imposed on the i c e exceeds a f a i r l y c r i t i c a l value, c r a c k f o r m a t i o n i s initiated ( 2 ) . T h i s c r a c k i n g activity c a u s e s a d e t e r i o - r a t i o n of the s t r u c t u r e and contributes t o the o c c u r r e n c e of yield o r f a i l u r e . F o r the loads of i n t e r e s t f o r design, the d e t e r i o r a t i o n of the s t r u c t u r e by c r a c k f o r m a t i o n would be s o extensive a t yield t h a t the i c e could not be a s s u m e d t o have the s a m e deformation p r o p e r t i e s a s in the uncracked s t a t e . The c r a c k s would have s o r e l i e v e d i n t e r n a l c o n s t r a i n t s , however, t h a t i t would be r e a s o n a b l e t o a s s u m e that the uncracked portions have a r e s i s t a n c e t o deformation of about the s a m e value a s that o b s e r v e d in an unconfined c o m p r e s s i o n t e s t .

F a i l u r e o r yield i n a n unconfined c o m p r e s s i o n t e s t at t h e r a t e s of s t r a i n u n d e r c o n s i d e r a t i o n o c c u r s by t h e f o r m a t i o n of a fault zone in which t h e r e i s a

m a r k e d i n c r e a s e in c r a c k i n g a c t i v i t y . T h i s zone i s a p p r o x i m a t e l y p a r a l l e l t o t h e plane of m a x i m u m s h e a r . F o r s t r a i n r a t e s l e s s t h a n t h a t a s s o c i a t e d with t h e d u c t i l e - t o - b r i t t l e t r a n s i t i o n i n b e h a v i o u r , t h e m a t e r i a l i n the zone r e m a i n s i n t a c t , but t h e d e f o r m a t i o n of t h e s p e c i m e n i s c o n c e n t r a t e d i n t h i s r e g i o n ( i . e . , t h e s t r a i n b e c o m e s , n o n - u n i f o r m ) . At r a t e s of s t r a i n g r e a t e r t h a n t h a t a s s o c i a t e d with t h e d u c t i l e - t o - b r i t t l e t r a n s i t i o n i n b e h a v i o u r , t h e f o r m a t i o n of t h e zone i s a b r u p t and r e s u l t s i n c a t a s t r o p h i c f a i l u r e . It i s a s s u m e d t h a t a s i m i l a r d u c t i l e o r b r i t t l e f a i l u r e b e h a v i o u r t a k e s p l a c e f o r t h e s i t u a t i o n u n d e r c o n s i d e r a t i o n ,

and t h a t the m a x i m u m load i s a s s o c i a t e d with c o m p r e s s i v e yield o r f a i l u r e of t h e

i c e i m m e d i a t e l y a d j a c e n t t o t h e p i l e . I t i s n e c e s s a r y to e s t a b l i s h t h e r e l a t i o n s h i p between t h e s t r a i n r a t e i n t h e i c e i m m e d i a t e l y a d j a c e n t t o t h e pile and t h e r a t e of p e n e t r a t i o n of t h e c o v e r i f t h e foregoing a s s u m p t i o n s a r e t o be u s e d in c a l c u l a t i o n s of t h e m a x i m u m load t h a t c a n be e x e r t e d . T h i s r e l a t i o n s h i p could be e s t a b l i s h e d by m e a s u r e m e n t . T h e s m a l l amount of i n f o r m a t i o n a v a i l a b l e f r o m field s t u d i e s i n d i c a t e s t h a t t h e s h a p e of t h e l o a d - p e n e t r a t i o n c u r v e f o r a p p r o x i m a t e l y c o n s t a n t r a t e of p e n e t r a - tion i s s i m i l a r t o t h a t of the s t r e s s - s t r a i n c u r v e f r o m c o n s t a n t s t r a i n r a t e

t e s t s ( 3 ) . Lf yielding i n t h e i m m e d i a t e vicinity of the pile h a s the s a m e s t r a i n dependence a s i n t h e unconfined c o n s t a n t s t r a i n r a t e t e s t s , c o m p a r i s o n of t h e s e

l a b o r a t o r y r e s u l t s with f i e l d o b s e r v a t i o n s on l o a d and p e n e t r a t i o n would give t h e

dependence of t h e s t r a i n r a t e i n t h i s a r e a on t h e r a t e of p e n e t r a t i o n . T h e r e i s , u n f o r t u n a t e l y , i n s u f f i c i e n t published i n f o r m a t i o n to allow t h i s c o m p a r i s o n t o be m a d e . In a l a t e r s e c t i o n of t h i s p a p e r a t h e o r e t i c a l r e l a t i o n s h i p between t h e r a t e of p e n e t r a t i o n and t h e r a t e of s t r a i n i n t h e i c e i s developed, a s s u m i n g e l a s t i c b e h a v i o u r . T h e a s s u m p t i o n of e l a s t i c b e h a v i o u r should be r e a s o n a b l e f o r a

good p r o p o r t i o n of t h e load build-up p r i o r t o yield a t t h e r a t e s of p e n e t r a t i o n of

i n t e r e s t f o r d e s i g n .

SOLUTION FOR P I L E LOADING

T h e s i t u a t i o n u n d e r c o n s i d e r a t i o n i s shown i n F i g u r e 1 . A pile i n c o n t a c t with t h e i c e c o v e r o v e r the r a n g e 0 ₌

+

_q,

_{i s s u b j e c t t o a f o r c e}

_F

o

( a , 8) =

o

( a , 8) 2 ~ - ~ s 8 s q r ( 1 )

o

( a ,

a )

=

o

( a , 8) = 0 q s e s 2 n - q r ( 2 ) T 8 ( a , 8) = T ' ( a , 8)

=

0 0~ e s 2 r r r r 8 ( 3 ) u ( a , 8) = u r Z ( a ,

8 )

2 T T - q s B < r ( ( 4 )

w h e r e the p r i m e d q u a n t i t i e s r e f e r t o t h e p i l e and the u n p r i m e d t o the i c e .

Following Noble and H u s s a i n ( 4 ) , t h e s t r e s s function f o r the plane s t r a i n ,

e l a s t i c c a s e t h a t i s a p p r o p r i a t e f o r t h e b o u n d a r y conditions i s w h e r e D a r e c o n s t a n t s , a i s t h e r a d i u s of t h e p i l e , and

v

i s P o i s s o n ' s r a t i o . n Ln the i c e t h e r a d i a l s t r e s s i s and t h e r a d i a l s t r a i n i s

r

1

w h e r e G i s t h e s h e a r m o d u l u s . Equation (6) i n d i c a t e s t h a t t h e r a d i a l s t r e s s e s a r e c o m p r e s s i v e , and it w a s found t h a t t h e c i r c u m f e r e n t i a l s t r e s s e s a r e t e n s i l e .

E q u a t i o n s

( 6 )

and ( 7 ) f o r m the b a s i s f o r the solution p r e s e n t e d f o r the p i l e loading p r o b l e m . The foregoing d e r i v a t i o n h a s been f o r the plane s t r a i n c a s e . In t h e i c e c o v e r a t s o m e d i s t a n c e f r o m the pile a plane s t r e s s s i t u a t i o n would e x i s t .

In the r e g i o n v e r y n e a r the contact i n t e r f a c e , h o w e v e r , the plane s t r a i n c a s e i s

m o r e a p p r o p r i a t e (i. e . , t h e t h i c k n e s s of the i c e c o v e r i s a s s u m e d t o be m u c h

g r e a t e r than t h e r e g i o n a t the i n t e r f a c e in which f a i l u r e i s i n i t i a t e d ) .

Solving f o r t h e n e t r e s u l t a n t f o r c e which a n i c e c o v e r can e x e r t i n v o l v e s ,

i n i t i a l l y , a c o n s i d e r a t i o n of the s t r e s s d i s t r i b u t i o n a r o u n d the p i l e . Because the loading i s d y n a m i c the s t r e s s d i s t r i b u t i o n will depend on the s t r a i n r a t e in the

i c e a d j a c e n t to the contact i n t e r f a c e . F r o m equation ( 7 ) , the r a d i a l s t r a i n d i s -

t r i b u t i o n a t t h e i n t e r f a c e i s m Fx ( 2

-

3V)

case

Do 2Gz ( a , 6 ) =

-

t - t 2 ( 1

-

2v)

T

( n f 1 ) r 2 n ( l - V ) a 2 n t 2 D n cosn8 ( 8 ) a 2 a Equation ( 8 ) s i m p l i f i e s t o if P o i s s o n ' s r a t i o h a s the value of

$.

T h i s s i m p l i f i c a t i o n i s c o n s i d e r e d to be justified in the p r e s e n t c a s e b e c a u s e of t h e d e t e r i o r a t i o n of the i c e a t yield due t o

c r a c k f o r m a t i o n , and the v i s c o e l a s t i c behaviour of the i n t a c t m a t e r i a l . If i t i s

a s s u m e d t h a t the solution of Noble and H u s s a i n ( 4 ) i s s t i l l valid, t h e p a r a m e t e r D can be obtained in t e r m s of F x , in which c a s e equation ( 9 ) r e d u c e s t o

0

T h i s equation i s talcen a s being r e p r e s e n t a t i v e of the s t r a i n and s t r a i n r a t e d i s -

t r i b u t i o n a r o u n d the contact i n t e r f a c e . T h e r a d i a l s t r a i n r a t e d i s t r i b u t i o n ,

t h e r e f o r e , i s

w h e r e i s the r a d i a l s t r a i n r a t e a t the i n t e r f a c e on t h e l i n e of a c t i o n of the

f o r c e Fx. T h e n o r m a l i z e d s t r a i n r a t e d i s t r i b u t i o n f r o m t h e above function i s

plotted in F i g u r e 2 along with a s i m p l e s i n u s o i d a l d i s t r i b u t i o n f o r c o m p a r i s o n .

T h e unconfined c o m p r e s s i v e yield o r f a i l u r e s t r e n g t h f o r a given i c e type

i s a function of both t e m p e r a t u r e and s t r a i n r a t e . L e t i t be a s s u m e d t h a t t h i s

function i s s e p a r a b l e (i. e . , 0 = ( T )

.

o 2

( E

)). Over the d u c t i l e r a n g e of YP 1

r a t e i s found to have the f o r m

w h e r e (5: and

a

a r e e x p e r i m e n t a l l y d e t e r m i n e d and

2

i s a r b i t r a r i l y s e l e c t e d t o

0

m a i n t a i n the d i m e n s i o n l e s s f o r m of the equation. Combining equations ( 1 1 ) and

( 1 2 ) g i v e s f o r the s t r a i n r a t e dependence of the s t r e s s d i s t r i b u t i o n a r o u n d t h e

pile f o r m a x i m u m load

Gold and K r a u s z ( 1 ) found that f o r c o l u m n a r - g r a i n e d i c e a t - 1 0 ° C 5 = 565

k

- 7 - 3 -1 c m 2 and

u

= 0.25 o v e r the s t r a i n r a t e r a n g e of 2 x 10 t o 10 s e c w h e r e

-

1

= 1 s e c

.

T h e m a x i m u m value f o r the yield s t r e n g t h o c c u r r e d a t t h e d u c t i l e -

0

-

4 t o - b r i t t l e t r a n s i t i o n , which w a s a s s o c i a t e d with a s t r a i n r a t e of about 2 x 10

- 1 s e c

.

Equation ( 1 3 ) i m p l i c i t l y a s s u m e s t h a t the m a x i m u m s t r e s s c o r r e s p o n d i n g

t o the s t r a i n r a t e h a s been m o b i l i z e d s i m u l t a n e o u s l y o v e r the full s u r f a c e of

contact. In r e a l i t y t h i s will probably not o c c u r . When the load on the pile i s

m a x i m u m , p a r t of the c o n t a c t s u r f a c e w i l l have undergone d e f o r m a t i o n i n e x c e s s

of t h a t f o r yield and p a r t m a y not have u n d e r g o n e sufficient d e f o r m a t i o n to a t t a i n

the m a x i m u m s t r e s s condition. The load c a l c u l a t e d using equation (13) should,

t h e r e f o r e , be g r e a t e r than t h e a c t u a l load f o r t h e a s s u m e d conditions. I t should

not be v e r y m u c h g r e a t e r , h o w e v e r , b e c a u s e t h e s t r e s s depends on t h e s t r a i n

r a t e r a i s e d t o t h e 1/4 p o w e r , and the s t r e s s a t yield i s r e l a t i v e l y i n s e n s i t i v e t o

the s t r a i n (i. e . , t h e s t r e s s - s t r a i n c u r v e h a s a m a x i m u m a t yield).

T h e n o r m a l i z e d f o r m of equation ( 1 3) i s plotted i n F i g u r e 3 f o r

a

= 0. 25. Shown a l s o i s t h e n o r m a l i z e d s t r e s s f o r a s i n u s o i d a l s t r e s s d i s t r i b u t i o n , i. e . ,

w h e r e rl i s one-half the angle of contact. Values f o r the angle of c o n t a c t ,

which depend on t h e e l a s t i c p r o p e r t i e s of the p i l e and i c e , and a r e u s u a l l y l e s s

than

n ,

a r e given by Noble and H u s s a i n (4). Substituting equation ( 1 3) into (15) g' lve s

r'l

S a m p l e c a l c u l a t i o n s showed that t h e f o r c e p e r unit t h i c k n e s s of the c o v e r was

r e d u c e d by l e s s than 5 p e r cent u s i n g t h e s i n u s o i d a l s t r e s s d i s t r i b u t i o n given by

L .

equation (16) with

a

= 0 . 2 5 and

o

=

565 k g / c m ; 1 . e . ,

fq

T h i s r e d u c t i o n would b a l a n c e , a t l e a s t p a r t i a l l y , t h e o v e r e s t i m a t i o n of t h e f o r c e

due t o t h e a s s u m p t i o n t h a t the yield s t r e s s i s m o b i l i z e d o v e r the full contact

s u r f a c e . T h e calculation d o e s i n d i c a t e the r e l a t i v e i n s e n s i t i v i t y of the load t o

s m a l l c h a n g e s i n s t r a i n r a t e .

T h e yield s t r e n g t h of i c e i s t e m p e r a t u r e dependent. T h i s dependence i s

w h e r e Q i s the a p p a r e n t activation e n e r g y

-

3

R i s the g a s c o n s t a n t (1.98 x 10 k c a l / m o l e o ~ ) T i s the t e m p e r a t u r e i n d e g r e e s Kelvin and

-

a.

i s a c o n s t a n t f o r t e m p e r a t u r e e q u a l to To.

value a s t h a t found f r o m c r e e p s t u d i e s , i. e . , about 20 k c a l / m o l e ( 5 ) .

Modifying equation (17) t o t a k e i n t o account t e m p e r a t u r e g i v e s

w h e r e f i s now the load p e r unit t h i c k n e s s a t the l e v e l in the i c e c o v e r f o r which

the t e m p e r a t u r e i s T .

I n t e g r a t i n g equation (19) g i v e s f o r the t o t a l load e x e r t e d on the pile by a n i c e c o v e r of t h i c k n e s s h

T h e t e m p e r a t u r e i n the c o v e r will v a r y f r o m the e q u i l i b r i u m f r e e z i n g

t e m p e r a t u r e a t the i c e - w a t e r i n t e r f a c e , t o s o m e value at the u p p e r s u r f a c e d e -

t e r m i n e d by the w e a t h e r and snow c o v e r . If i t i s a s s u m e d f o r d e s i g n t h a t the

t e m p e r a t u r e d e c r e a s e s l i n e a r l y f r o m Tm a t t h e i c e - w a t e r i n t e r f a c e to s o m e z

v a l u e , T s , a t the s u r f a c e ( i . e . , T = T s t- ( T m

-

T s )

g

w h e r e z i s positive downwards f r o m the s u r f a c e and h i s the i c e s h e e t t h i c k n e s s ) , then equation (20)

w h e r e K ( T ) , a t e m p e r a t u r e c o r r e c t i o n function, i s a constant f o r a given t e m - p e r a t u r e of t h e s u r f a c e and the i c e - w a t e r i n t e r f a c e , and

L(@),

a g e o m e t r y

function, i s a constant f o r given r e l a t i v e e l a s t i c p r o p e r t i e s of the pile and the ice.

Both of t h e s e functions c a n only be e v a l u a t e d n u m e r i c a l l y . The t e m p e r a t u r e

c o r r e c t i o n function i s plotted i n F i g u r e 4 and t h e value of the g e o m e t r y function

f o r t h e c a s e of a r i g i d p i l e i s 1.80. Equation (21) i n d i c a t e s t h a t the r a t i o of t h e t o t a l load t o t h e p r o d u c t of t h e pile d i a m e t e r , d , and i c e t h i c k n e s s , h, h a s t h e

following d e p e n d e n c e on s t r a i n r a t e i. e . , t h e n o r m a l i z e d l o a d on t h e p i l e i s p r o p o r t i o n a l t o t h e yield s t r e n g t h of t h e i c e in c o n t a c t w i t h t h e p i l e . T H E O R E T I C A L R E L A T I O N B E T W E E N STRAIN R A T E AND P E N E T R A T I O N R A T E T o g i v e t h e p r o p e r p e r s p e c t i v e t o t h e d i s c u s s i o n p r e s e n t e d i n t h i s s e c t i o n , c o n s i d e r e q u a t i o n ( 2 1 ) w h i c h s h o w s t h a t t h e p i l e loading d e p e n d s on t h e s t r a i n r a t e r a i s e d t o t h e p o w e r % , w h e r e

a

i s a s m a l l n u m b e r ( f o r c o l u m n a r - g r a i n e d r i v e r i c e a = 0. 25). E v e n a n a p p r o x i m a t e r e l a t i o n , t h e r e f o r e , between s t r a i n r a t e and p e n e t r a t i o n r a t e would be u s e f u l f o r p r e d i c t i n g t h e lpad.

I n t e g r a t i n g e q u a t i o n ( 7 ) f o r t h e r a d i a l s t r a i n g i v e s

F

X r

I

2 2Gu ( r , 9 ) =

-

( 3 - 4 ~ ) d n r

-

( 1 - 2v) - c o s e

- -

Do 4 n ( l - V ) 2 r r w h e r e f ( 9 ) i s an odd f u n c t i o n of

8.

T h e n a t u r a l b o u n d a r y condition t o u s e t o d e - t e r m i n e t h i s function would be u = 0 f o r r = w

,

_{but i t c a n n o t be a p p l i e d e a s i l y}

b e c a u s e of t h e & n r t e r m . T h i s d i f f i c u l t y c a n be c i r c u m v e n t e d by c h o o s i n g a n a r b i t r a r y r a d i u s , &, f o r w h i c h t h e d i s p l a c e m e n t i s n e g l i g i b l e . Applying t h i s c o n - d i t i o n t o e q u a t i o n ( 2 3 ) and a s s u m i n g a P o i s s o n ' s r a t i o of

$

g i v e s 2 w h e r e i2

> >

a

.

F o r r = a and 8 = O

F m

x a 2n D n

2Gu ( a , 0) =

- -

Rn - t D

2 Tr R o L . n - 1 n t 1 ' ( 2 5 )

a a

Using t h e solution given by Noble and H u s s a i n ( 4 ) t o d e t e r m i n e D and D i n

o n

t e r m s of F and combining t h e r e s u l t with equation ( 1 0) with B = 0 g i v e s

X I

E a

= 2.22

u r ( a , 0 ) ( 2 6 )

1.22

[t-

+

R n i

-

0 . 6 1 6

T h e dependence of t h e s t r a i n - d i s p l a c e m e n t r a t i o on a/R i s shown in F i g u r e 5.

A l l i c e c o v e r s and f l o e s a r e f i n i t e . F i g u r e 5 s h o w s t h a t once the s i z e of t h e f l o e o r s h e e t e x c e e d s a p p r o x i m a t e l y 50 t i m e s the p i l e r a d i u s , t h e r a t i o of t h e

s t r a i n t o t h e d i s p l a c e m e n t e v a l u a t e d a t the p i l e - i c e i n t e r f a c e i s r e l a t i v e l y i n s e n -

-

2

-

3

s i t i v e t o i t (e. g . , a d e c r e a s e in a/R f r o m 5 x 10 t o 10 c a u s e s a c o r r e s p o n d i n g

d e c r e a s e i n er#ur f r o r h 0 . 4 5 to 0.25). F o r a given floe s i z e t h e r a t i o i s a

c o n s t a n t . R e a r r a n g i n g e q u a t i o n (26) and d i f f e r e n t i a t i n g with r e s p e c t t o t i m e g i v e s ;,(a, 0 )

t r

( a , 0) =

-

kd w h e r e d i s t h e d i a m e t e r of t h e pile and

T h e dependence of k on a/R i s shown i n F i g u r e 6. E q u a t i o n ( 2 7 ) i n d i c a t e s that f o r

a given f l o e s i z e , t h e s t r a i n r a t e in t h e i c e a t t h e p i l e v a r i e s i n v e r s e l y with t h e

p i l e d i a m e t e r . Equation ( 2 7 ) a g r e e s with t h a t given by K o r z h a v i n ( 6 ) i f k i s s e t

-

3 e q u a l t o 2 ( i . e . , a/R = 10 ).

Substituting equation ( 2 7 ) i n t o equation ( 2 1 ) g i v e s

F o r a given m a t e r i a l behaviour and t e m p e r a t u r e condition equation ( 2 9 ) r e d u c e s

w h e r e N i s a p r o p o r t i o n a l i t y f a c t o r t h a t t a k e s i n t o a c c o u n t t e m p e r a t u r e e f f e c t s ,

s t r a i n d i s t r i b u t i o n a r o u n d t h e p i l e , s t r e n g t h p r o p e r t i e s of the i c e , r e l a t i v e

e l a s t i c p r o p e r t i e s of t h e i c e 2nd p i l e , and t h e g e o m e t r y f a c t o r r e l a t i n g s t r a i n 10

r a t e and p e n e t r a t i o n r a t e . T h e n o r m a l i z e d f o r c e ,

-

i s plotted a g a i n s t the s t r a i n r a t e in F i g u r e 7 hd

'

_-

2 f o r the c a s e of T = -1O0C,

oo

= 565 kg c m

,

3, = 0.25, i c e s u r f a c e t e m p e r a - 0

t u r e -1O0C, i c e - w a t e r i n t e r f a c e t e m p e r a t u r e 0 ° C and floe d i a m e t e r f o r t y t i m e s

a

-

4

t h e pile d i a m e t e r ( i . e . , - = 40,

k

= 1 . 2 5 ) . Note the s t r a i n r a t e of 2 x 10 a

-

1

s e c

,

which i s the r a t e a s s o c i a t e d with the d u c t i l e - t o - b r i t t l e t r a n s i t i o n f o r - 4 -1 c o l u m n a r - g r a i n e d i c e a t - 1 0 ° C . F o r s t r a i n r a t e s l e s s than 2 x 10 s e c the

i c e would be expected t o behave in a ductile m a n n e r . F o r s t r a i n r a t e s g r e a t e r - 4 - 1

than about 2 x 10 s e c

,

l a b o r a t o r y o b s e r v a t i o n s i n d i c a t e t h a t t h e s t r e n g t h in s i m p l e c o m p r e s s i o n i s c o n s t a n t o r e v e n d e c r e a s e s with i n c r e a s i n g r a t e ( 7 ) . T h e n o r m a l i z e d f o r c e a t the s t r a i n r a t e a s s o c i a t e d with t h e d u c t i l e - t o - b r i t t l e t r a n s i -

tion would, t h e r e f o r e , be the m a x i m u m t h a t could o c c u r f o r the a s s u m e d con-

ditions. B e a r i n m i n d , h o w e v e r , t h a t t h i s t r a n s i t i o n s t r a i n r a t e i s t e m p e r a t u r e

dependent.

DISCUSSION

T h e t h e o r y developed i s f o r cold i c e (i. e . , t e m p e r a t u r e l e s s than 0 " C ) i n

i n t i m a t e contact with a r i g i d c y l i n d r i c a l pile. The equations obtained should g i v e a n u p p e r l i m i t t o the t h r u s t that c a n be induced, and t h e i r f o r m i s sufficiently

s i m p l e that i t should be r e l a t i v e l y e a s y t o d e t e r m i n e t h e i r validity t h r o u g h field

m e a s u r e m e n t s .

F o r m a n y situations significant m o v e m e n t of the i c e r e l a t i v e to a s t r u c t u r e

only o c c u r s d u r i n g s p r i n g b r e a k u p ( e . g.

,

m a n y p i e r s i t e s ) . Values of l o a d s that have been obtained f o r t h i s condition a r e shown in F i g u r e 7. N e i l l t s r e s u l t s ( 8 )

a r e f o r two s i t e s with p i e r d i a m e t e r s of 0.85 and 2.3 m . The b a r s indicate the

r a n g e in v e l o c i t i e s and i c e floe s i z e s o b s e r v e d . It should be noted t h a t the

2 . 3 m pile w a s inclined a t 23' f r o m t h e v e r t i c a l . T h e o b s e r v e d loads a r e not

m u c h s m a l l e r t h a n the m a x i m u m p r e d i c t e d by equation ( 2 1 ) f o r a n i c e c o v e r a t

0 ° C throughout. F i g u r e 7 i n d i c a t e s the l a r g e r a n g e i n s t r a i n r a t e s t h a t m u s t be

t a k e n into c o n s i d e r a t i o n , both i n d e s i g n and r e s e a r c h .

A l s o shown in F i g u r e 7 a r e r e s u l t s obtained by Nuttall and Gold ( 9 ) f r o m l a b o r a t o r y t e s t s simulating pile loading by c o l u m n a r - g r a i n e d i c e c a r r i e d out a t

-10°C. T h e t h e o r e t i c a l c u r v e f o r a t e m p e r a t u r e of - 1 0 ° C throughout the i c e i s

shown f o r c o m p a r i s o n . The s t r a i n r a t e s w e r e evaluated a t the point w h e r e c r e e p s t r a i n w a s e q u a l t o the a s s u m e d s t r a i n a t yield ( 2 x l o - ) ) . In a l l c a s e s t h i s s t r a i n o c c u r r e d i n the r e g i o n of p r i m a r y c r e e p . Note that the o b s e r v e d

l o a d s a r e s n l a l l e r t h a n t h e t h e o r e t i c a l l y p r e d i c t e d o n e s . If t h e s t r a i n r a t e s

a s s o c i a t e d with t h e s e c o n d a r y c r e e p s t a g e a r e u s e d i n t h e c a l c u l a t i o n s , t h e

p r e d i c t e d l o a d s f o r given conditions a r e l e s s t h a n t h e ones a p p l i e d .

I t would be e x p e c t e d t h a t a n i c e c o v e r would be i n i n t i m a t e contact w i t h a

p i l e only d u r i n g t h e v e r y i n i t i a l p e r i o d of i t s m o v e m e n t r e l a t i v e to i t . Once t h e f a i l u r e p r o c e s s had been i n d u c e d , c o n t a c t between t h e i c e and pile would p r o b a b l y

no l o n g e r be continuous. I t would be e x p e c t e d t h a t s t r e s s c o n c e n t r a t i o n s would

e x i s t a t the points of c o n t a c t f r o m which f a i l u r e could continue with r e l a t i v e e a s e .

T h e load r e s u l t i n g f r o m s u c h p a r t i a l c o n t a c t conditions would be l e s s t h a n t h a t

p r e d i c t e d a s s u m i n g i n t i m a t e contact.

D u r i n g s p r i n g b r e a k u p t h e i c e c o v e r c a n be e x p e c t e d to be a t O°C t h r o u g h -

out, and the width of t h e f l o e s m u s t be e q u a l t o o r l e s s t h a n t h a t of t h e channel.

L a b o r a t o r y s t u d i e s i n d i c a t e t h a t t h e s t r e n g t h of m e l t i n g i c e i s l e s s t h a n t h a t which

would be p r e d i c t e d by equation ( 1 8 ) ,

( 6 ) .

T h e f o r e g o i n g f a c t o r s m u s t be t a k e n into c o n s i d e r a t i o n i n field i n v e s t i g a t i o n s and when e s t a b l i s h i n g d e s i g n c r i t e r i a .

They could a c c o u n t , along with t h e p o s s i b l e d e c r e a s e in s t r e n g t h with i n c r e a s e in

s t r a i n r a t e i n t h e b r i t t l e r e g i o n , f o r t h e d i f f e r e n c e between the p r e d i c t e d and

o b s e r v e d f o r c e s shown i n F i g u r e 7 . E q u a t i o n ( 3 0 ) i n d i c a t e s t h a t t h e n o r m a l i z e d load d e c r e a s e s a s t h e p i l e d i a m e t e r i n c r e a s e s . I t m i g h t b e c o n s i d e r e d t h a t t h i s i s a g e o m e t r y e f f e c t , but i t i s i n r e a l i t y a s t r a i n r a t e e f f e c t . T h i s c a n b e a p p r e c i a t e d by r e f e r r i n g t o equation ( 2 7 ) which i n d i c a t e s t h a t t h e s t r a i n r a t e in t h e i c e a t t h e pile v a r i e s i n v e r s e l y a s t h e p i l e d i a m e t e r ' . T h i s p o s s i b i l i t y m u s t be t a k e n into c o n s i d e r a t i o n if t h e r e l a -

t i o n between r a t e of p e n e t r a t i o n and s t r a i n r a t e i n t h e i c e a t t h e pile i s t o be

d e t e r m i n e d by c o m p a r i s o n of f i e l d and l a b o r a t o r y o b s e r v a t i o n s .

A s t h e value of the exponent,

a

,

in equation ( 3 0 ) i s p r o b a b l y l e s s t h a n 0. 5, t h e unit load developed on t h e p i l e will not be s t r o n g l y dependent on r a t e of p e n e t r a t i o n and pile d i a m e t e r in t h e d u c t i l e r e g i o n . T h e r a n g e of d i a m e t e r a n d

r a t e of p e n e t r a t i o n will have t o c o v e r s e v e r a l d e c a d e s , t h e r e f o r e , t o d e t e r m i n e p r o p e r l y the v a l i d i t y of the e q u a t i o n f r o m f i e l d i n v e s t i g a t i o n s .

CONCLUSIONS

I t h a s b e e n shown t h a t t h e d e p e n d e n c e of t h e t h r u s t e x e r t e d on a c i r c u l a r

w h e r e

6

i s t h e r e l a t i v e r a t e of m o v e m e n t between t h e p i l e and t h e i c e ,

a

i s a n exponent t h a t d e p e n d s on t h e type of i c e , and N i s a f a c t o r taking into a c c o u n t

t e m p e r a t u r e e f f e c t s , s t r a i n d i s t r i b u t i o n about t h e p i l e , s t r e n g t h of t h e i c e , e l a s t i c

p r o p e r t i e s of the i c e and pile and g e o m e t r i c f a c t o r s . It g i v e s a n u p p e r l i m i t t o

t h e p r e d i c t e d load, i s s u f f i c i e n t l y s i m p l e t o be r e a d i l y c h e c k e d by f i e l d m e a s u r e - m e n t s , and p r o v i d e s a b a s i s f o r f u t u r e d e v e l o p m e n t s and r e f i n e m e n t s of d e s i g n c r i t e r i a f o r i c e t h r u s t on p i l e s . T h i s p a p e r i s a c o n t r i b u t i o n f r o m t h e D i v i s i o n of Building R e s e a r c h , N a t i o n a l R e s e a r c h Council of C a n a d a , and i s p u b l i s h e d w i t h t h e a p p r o v a l of t h e D i r e c t o r of t h e D i v i s i o n . R E F E R E N C E S ( 1 ) Gold, L . W . and K r a u s z , A . S. 1971. I n v e s t i g a t i o n of t h e M e c h a n i c a l P r o p e r t i e s of S t . L a w r e n c e R i v e r I c e . C a n . G e o t e c h . J o u r . , Vol. 8 , No. 2, p. 163-169. ( 2 ) G o l d , L . W . 1970. T h e F a i l u r e P r o c e s s i n C o l u m n a r - G r a i n e d I c e . P h . D. T h e s i s , M c G i l l U n i v e r s i t y , M o n t r e a l , Q u e b e c , C a n a d a . ( 3 ) P e y t o n , H. R. 1966. S e a I c e S t r e n g t h . G e o p h y s i c a l I n s t i t u t e , U n i v e r s i t y of A l a s k a , R e p o r t No. UAGR-182.

( 4 ) Noble, B . and H u s s a i n , M . A . 1969. E x a c t Solution of C e r t a i n D u a l

S e r i e s f o r Indentation and I n c l u s i o n P r o b l e m s . Int. J . Engng. S c i . , Vol. 7 , p. 1149-1161. (5) R a m s e i e r , R. 0 . 197 1. P r i v a t e C o m m u n i c a t i o n . (6) K o r z h a v i n , K. N. 1962. I c e P r e s s u r e a g a i n s t S t r u c t u r e s . Akad. Nauk SSSR., S i b i r . O t d . , N o v o s i b i r s k . (In R u s s i a n ) ( 7 ) C a r t e r , D. 1968. P r e l i m i n a r y A n a l y s i s of C o n t r o l l e d F o r m a t i o n and of s o m e M e c h a n i c a l P r o p e r t i e s of C o l u m n a r I c e with H o r i z o n t a l c - a x i s . M . S c . T h e s i s , L a v a 1 U n i v e r s i t y , Q u e b e c , Q u e b e c , C a n a d a . (In F r e n c h ) ( 8 ) N e i l l , C.R. 1970. S t u d i e s of I c e P r e s s u r e on B r i d g e P i e r s i n A l b e r t a , C a n a d a . LAHR I c e S y m p o s i u m , Reykjavik.

(9) Nuttall, J . and Gold, L . W. 1968. M o d e l Study of Ice P r e s s u r e s . N a t i o n a l R e s e a r c h C o u n c i l of C a n a d a , T e c h n i c a l M e m o r a n d u m

m

z

0

- I- 3 m E l- m _- n LL! +- Q oi

z

- Q oi +- m

Q = 20 KCAL/MOLE (COLUMNAR GRAINED RIVER ICE) R = 1.98 x 10-3 K C A L / M O L E O K

T = ICE SURFACE TEMPERATURE -DEGREE KELVIN

-

-

-

- -

-

- -

-

-

-

- 0

-

5 - 1 0 - 1 5

-

2 0 - 2 5 - 3 0

-

3 5 I C E S U R F A C E T E M P E R A T U R E

-

D E G R E E S C E L C I U S F I G U R E 4 T E M P E R A T U R E C O R R E C T I O N F A C T O R F O R C O L U M N A R G R A I N E D I C E C O V E R S a ~ + 7 8 9 - 7