Lifting Force and Bearing Capacity of an Ice Sheet

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Lifting Force and Bearing Capacity of an Ice Sheet

Lofquist, B.

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NATIOBiAL RESEZiRCii COUI.ICIL O F CANADA

Technical T r a n s l a t i o n TT-164

L i f t i n g Force and Bearing Capacity of an I c e Sheet, ( ~ ~ f t k r a f t och E a r f i j r d g a hos e t t ~ s ~ c k e ) Teknisk T i d s k r i f t , No, 25, Stockholm 194.4,

b

B e r t i l L 6 f q u i s t

D r , &gaj, The S t a t e Power Board, Sweden

t r a n s l a t e d by

H O B O G O Nathan

Revised by t h e author !

:

T h i s i s t h e t w e l f t h of t h e s e r i e s of t r a n s l a t i a n s p r e p a r t d f o r Ule Division of Building Research.

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The s t r e n g t h of i c e i s a major s c i e n t i f i c protlem of r e a l concern t o t h e Division of Building Research, The term i s one used t o designate i n simple f a s h i o n a knowledbe of t h e complex mechanical and p h y s i c a l p r o p e r t i e s of i c e , This knowledge i s e s s e n t i a l f o r Ein understanding of p r a c t i c a l problems, such a s t h e p r e s s u r e e x e r t e d by i c e a g a i n s t darcs, and t h e bearing c a p a c i t y of s h e e t s of i c e a s , f o r example, those which cover t h e s u r f a c e s of Canadian l a k e s i n winter, I n t h e course of a search f o r i n f o r n a t i o n on t h i s s u b j e c t , with s p e c i a l reference t o t h e problem of i c e p r e s s u r e s on dams, t h e o p p o r t u n i t y presented i t s e l f of v i s i t i n 4 Stockholm, Sweden,and t h e r e c o n s u l t i n g with members of t h e s t a f f of t h e Royal Swedish S t a t e board of , i v a t e r f a l l s amongst them D r . B e r t i l ~ g f ~ u i s t , t h e author of t h i s

paper o

So l i t t l e information i s a v a i l a b l e i n p r i n t e d form on t h e s t r e n ~ t h of i c e , t h a t permission was sought t o have t h i s paper by

Dro

L o f q u i s t t r a n s l a t e d , This permission was r e a d i l y granted and t h e Division i s glad t o make a v a i l a b l e t h i s u s e f u l paper throu,h t h e Council's t r a n s l a t i o n s e r v i c e ,

It

may u s e f u l l y be menticned t h a t another paper by D r ,

~ b l f ~ u i s t w i l l ,

i t

i s hoped, soon be published i n English a s a p a r t of a symposium on t h e problem of i c e p r e s s u r e s a g a i n s t dams wliich was presented t o t h e American S o c i e t y of C i v i l Engineers a t i t s meet- i n g i n Toronto, Ontario, i n J u l y , 1950, The papers t h e n read a r e being submitted formally t o t h e Society with a view t o t h e i r public- a t i o n and subsequent d i s c u s s i o n , October, 1950 Robert F, Legget, D i r e c t o r , Division of Building Re search

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LIFIING FORCE bl\jL) bhARIIJC CAPACITY Or' AIY ICE SHEET

I n winter, hydraulic c o n s t r u c t i o n works a r e a f f e c t e d Ly f o r c e s s e t up by t h e i c e s h e e t , These f o r c e s , which a c t both i n v e r t i c a l and h o r i z o n t a l d i r e c t i o n , may be r a t h e r l a r g e . Smaller

c o n s t r u c t i o n s , such a s p i t r s , caissons and p i l e groups may be

damaged, among other t h i n g s , due t o t h e f a c t t h a t c o n s t r u c t i o n s a r e r a i s e d by t h e i c e s h e e t a s t h e water l e v e l r i s e s .

The l i f t i n g f o r c e of an i c e s h e e t p a r - t i c u l a r l y a f f e c t s smaller, i s o l a t e d o b j e c t s such a s p i l e s and p i l e groups, which f r e q u e n t l y tecome frozen up during t h e winter u n l e s s s p e c i a l pre- c a u t i o n s a r e taken. Observations i n Northern Sweden have shown t h a t even caissons of considerable s i z e can be l i f t e d by t h e i c e ,

A s f a r a s i s known, no d a t a on t h e magnitude of t h e l i f t - i n g f o r c e s a r e a v a i l a b l e i n t h e l i t e r a t u r e , I n view of t h e g r e a t p r a c t i c a l importance of t h e problem, conditions i n a p o r t i n

Northern Sweden were i n v e s t i g a t e d i n t h e winter of 1939-40 by The Royal Board of Roads and Waterviays, The r e s u l t s o f t h i s i n v e s t i g - a t i o n a f f o r d an i n s i g h t i n t o t h e i c e conditions about p i e r s , I t

was found t h a t some oaissons had been r a i s e d by more than 20 centime- t r e s .

B a s i c a l l y , t h e s t r i c t l y t h e o r e t i c a l determination of t h e l i f t i n g f o r c e of an i c e s h e e t i s d i f f i c u l t s i n c e , i n t h e f i r s t p l a c e , t h e laws f o r t h e p l a s t i c i t y o f . i c e are only roughly known, However,

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by c e r t a i n approxinlations,

it

i s p o s s i i l e t o o b t a i n a r e s u l t which a g r e e s w e l l w i t h t h e a c t u h l c o n d i t i o n s observed d u r i n g t h e above in- v e s t i g a t i o n .

The l i f t i n g f o r c e a c t i n g on a l o n g s t r a i g h t w a l l and t h a t a c t i n g on a n i s o l a t e d o b j e c t a r e determined below, The proilem of t h e l i f t i n g f o r c e i s tantamount t o t h a t of t h e b e a r i n g c a p a c i t y of an i c e s h e e t , The nethod used f o r c a l c u l a t i n g t h e l i f t i n g f o r c e can t h e r e f o r e a l s o be a p p l i e d t o t h e c a l c u l a t i o n of t h e b e a r i n g c a p a c i t y of a n i c e s h e e t , a s shown i n t h e l a s t

c h a p t e r ,

L i f t i n p Porce on a Lonn S t r a i ~ h t \ # a l l

The p a r t s of an i c e s h e e t c l o s e t o s o l i d c o n s t r u c t i o n s , which i s prevented from following t h e r i s i n g of t h e water l e v e l , a r e i n f l u e n c e d by i n c r e a s e d u p l i f t f o r c e s , The l i f t i n g f o r c e i s

t h e n given by t h e displacement due t o t h e b e d i n t h e i c e s h e e t , Iience t h e c h i e f problem i s t o d i s c o v e r what form t h e bends t a k e ,

Assuming t h a t t h e i c e i s an e l a s t i c m a t e r i a l , t h e l i f t i n g f o r c e can be c a l c u l a t e d by t h e t h e o r y of beams and s l a b s on e l a s t i c support, s i n c e t h e upward p r e s s u r e on t h e underside o f t h e i c e s h e e t a t a l l p o i n t s i s p r o p o r t i o n a l t o t h e d e f l e c t i o n ,

The s i m p l e s t c a s e i s t h a t of a lon, s t r a i g h t w a l l f r o z e n i n t o a n i c e s h e e t (Fig, l ) , To bedin with, t h e i c e s h e e t i s

assumed t o be l e v e l and t h a t t h e water l e v e l r i s e s t h e d i s t a n c e 11, L e t u s consider a s t r i p of t h e i c e s h e e t a t r i g h t angle t o t h e w a l l , r i g i d l y f i x e d t o t h e l a t t e r and i n f i n i t e l y long, The i c e s t r i p i s

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a c t e d on by i t s own weight, by t h e upward p r e s s u r e and a t t h e f i x e d end by a v e r t i c a l f o r c e and a negative moment. The weight of t h e i c e s h e e t and t h e corresponding component of t h e u p l i f t f o r c e can be disregarded s i n c e t h e s e f o r c e s balance each o t h e r everywhere and cause no deformations,

The following symbols a r e now introduced:

k

=

modulus of, r e a c t i o n (here 1 t o n per cu, metre)

E

=

modulus of e l a s t i c i t y

d

=

t h i c k n e s s of i c e

1 =

moment of i n e r t i a p e r u n i t of width

p

=

l i f t i n g f o r c e per u n i t of width

For t h e d e f l e c t i o n of t h e i c e s t r i p , t h e following equat- i o n holds;

which has t h e same s o l u t i o n here a s i n t h e case of an i n f i n i t e l y long beam on an e l a s t i c support subjected t o a s i n g l e f o r c e

2 P

,

Hence

-

where

The equation c o n t a i n s a simple ..sine f u n c t i o n m u l t i p l i e d by a damping function, Waves of constant wave l e n g t h form i n t h e

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i c e s t r i p , but with g r e a t l y reduced amplitudes. By equating

x

t o zero and p u t t i n g

Y

=

H

,

t h e l i f t i n g f o r c e i s obtained, v i z . ,

For t h e l i f t i n g f o r c e t o occur, t h e i c e must be a b l e t o r e s i s t tending s t ~ e s s e s which reach a

maximum

a t .the f i x e d end, The maximum moment then becomes

=

and t h e corresponding bending s t r e s s

The aLove expressions apply only a t t h e beginning of a

v e r y r a p i d r i s e i n t h e v;ater l e v e l where t h e i c e may be consicered e l a s t i c , Bowever, i c e i s a highly p l a s t i c m a t e r i a l , i . e , t h e deformations i n c r e a s e vritb time, IGoreover, t h e y depend on t h e t e r ~ p e r a t u r e ana a r e p r o p o r t i o n a t e l y g r e a t e r f o r h i b e r s t r e s s e s t h a n f o r 1or;er ones,

The p l a s t i c deformation of i c e i s caused by t h e p a r t i c l e s s l i p l ~ i n g clue t o t h e i n f l u e n c e of' normal and shearing s t r e s s e s . I n t h e p r e s e n t case, t h e shearing f o r c e s a r e comparatively small, Therefore,

i t

may t e assumed t h a t t h e p l a s t i c i t y i s clue t o tending

s t r e s s e s alone,

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t h e i c e , Ep

,

has been formulated by Uils Rogen

(2),

v i z e ,

where

6'

denotes t h e s t r e s s ,

h

t h e time,

t

t h e temperature i n degrees Centigrade with reversed s i g n and

c

a constant, which f o r t h e t e s t was between

6

x

lod5

and

9

x expressed i n hours, t o n s and metres. The expression i s based on p r e s s u r e t e s t s with moderate s t r e s s e s ,

It

should be p o s s i b l e t o apply t h i s expression t o t e n s i o n a l s o .

Even f o r such m a t e r i a l s a s g l a s s , t i n , p a r a f f i n , brfck- work and ccncrete, t h e p l a s t i c deformation has been found t o be approximately p r o p o r t i o n a l t o t h e cubic r o o t of t h e time,

The e l a s t i c deformations f, and t h e p l a s t i c deformation

E P are t h e n combined t o form t h e modulus of deformation,

F,

which i s defined by

The l i f t i n g f o r c e , wliich i s obtained when t h e water l e v e l r i s e s r a p i d l y , begins t o diminish imnlediately because 19' t h e p l a s t i c - i t y o f t h e i c e , I n o r d e r t o o b t a i n a r o u g h e s t i m a t e o f t h i s d e c r s & s e ,

E

i n equation ( 2 ) i s replaced by

/-

.

Then

,- ---. >- %-,. .>-". -+-

--The decrease of t h e l i f t i n g f o r c e with t h e time, a s expressed by equation ( 6 ) , i s shown i n Fig, 2. The g r e a t e s t decrease o c c u r s during

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t h e f i r s t hour, After t h a t time, t h e l i f t i n g f o r c e remains almost constant

,

I n nature, however, t h e water l e v e l r i s e s a t a c e r t a i n f i n i t e r a t e , f o r which equation (6) cannot be applied d i r e c t l y , The r i s e of t h e water l e v e l i s a f u n c t i c n of time, I n order t o oLtain t h e l i f t i n g f o r c e a t a c e r t a i n time,

h,

,

it

i s assuned

t h a t t h e i c e sheet i s being r a i s e d by i n f i n i t e l y

small

s t a g e s , A

H.

I f such a minute r i s e occurs i n t h e time

h

,

then, according t o equation ( 6 ) , t h e corresponding p a r t i a l f o r c e i n t h e time

A,

i s t

I f vie assulne t h e d i f f e r e n t p a r t i a l f o r c e s according t o t h e formula could be added then

4 -- -- --

--

k

-'d3

_--rz

_->

f i e

f - .-a

0 t i /

I n order t o solve t h e i n t e g r a l , t h e f u n c t i o n Hmus!; te known. If

it,

i s assumed t h a t t h e water l e v e l r i s e s a t a constant r a t e Q , then

H

=

a h

and

I n t e g r a t i o n can then Le c a r r i e d o u t a n a l y t i c a l l y . Thus

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with t h e aLbreviation

Very soon t h e p l a s t i c deformation becomes considerably g r e a t e r than t h e e l a s t i c deformation, e s p e c i a l l y a t t h e temperat- u r e s near zero degrees

C,

We may t h e r e f o r e w r i t e t h e formula

5 =

u-

and cancel t h e l a s t t h r e e terms i n parentheses,

c v *

t h e n

S i m i l a r l y , t h e bending s t r e s s a t t h e f i x e d end can be derived:

These expressions t h u s give approximate values of t h e l i f t i n g f o r c e and t h e corresponding bending s t r e s s when t h e water l e v e l r i s e s a t t h e r a t e of

a

.

Kith r e s p e c t t o t h e i c e temperature,

t

,

a r e s e r v a t i o n should be made. The formulae apply f o r uniform temperature i n t h e i c e s h e e t , I f t h e upper surface of t h e i c e i s cooled, a non- uniform terqeraturia is obtained s i n c e t h e underside, which i s i n c o n t a c t with t h e watel., has a temperature of OOC, Assunling a

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above formulae hold, i f a c o r r e c t e d mean temperature i s i n t r o - duced, The l a t t e r f a l l s somewhat s h o r t of t h e a r i t h m e t i c mean temperature, Hence, f o r a temperature of t h e upper s u r f a c e of t h e i c e o f , say

-loOc.

,

f

=

4

should be s u b s t i t u t e d i n t h e above formulae

,

Comparison w i t h Conditions Observed i n t h e F i e l d

During t h e winter of 1939-40, t h e movements of t h e i c e s h e e t i n a Northern Swedish p o r t were observed by t h e Royal Board of Roads and 'Aaterways, &vary second day t h e i c e s h e e t i n t h e v i c i n i t y of p i e r s was l e v e l l e d and i t s t h i c k n e s s measured, I n a d d i t i o n , o b s e r v a t i o n s of the water l e v e l were made each day, and t h e m a x i m u m and minimum a i r temperatures were recorded.

Some t y p i c a l deformations of t h e i c e s h e e t , observed between l e v e l surveys, a r e shown i n Fig,

3,

It i s found t h a t t h e deformations agree w e l l with those obtained f o r a beam on an e l a s t i c support,

I n

t h e case under c o n s i d e r a t i o n , t h e i c e s h e e t was f r o z e n t o a p i e r c o n s i s t i n g of c a i s s o n s i n a s i n g l e row, t h u s conforming with t h e long s t r a i g h t w a l l assumed i n t h e p r e v i o u s c h a p t e r ,

The l i f t i n g f o r c e i s obtained from t h e displacement due t o t h e bend i n t h e i c e s h e e t n e a r t h e c o n s t r u c t i o n . By t h e l e v e l survey, t h e r e f o r e ,

it

i s p o s s i b l e t o c a l c u l a t e d i r e c t l y t h e l i f t i n g f o r c e a c t i n g on c a i s s o n s i n t h e v a r i o u s cases. The l i f t i n g f o r c e i s r e p r e s e n t e d by t h e a r e a between t h e curve and i t s

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I n t h e t a b l e bebow, t h e v a l u e s of t h e l i f t i n 6 f o r c e ,

c a l c u l a t e d f r o n equation (i2) with c

=

8

x 10'5 a r e corripared ~ i t h t h o s e of t h e a c t u a l l i f t i n g f o r c e determined by l e v e l l i n g f o r

v a r i o u s conditions. The temperature of t h e i c e was estimated from t h e a i r temperature. The r a t e a t which t h e water l e v e l r i s e s i s assumed t o be a = he

/

A.

Conditions observed i n t h e f i e l d a g r e e s a t i s f a c t o r i l y with t h e v a l u e s obtained from t h e approximation formula (12)

.

From t h e comparison,

i t appears t o be reasonable t o s e t t h e v a l u e f o r t h e c o n s t a n t c a t about 8

x

1 ~ - 5 (hours, t o n s and metres),

Rise i n t h e

The L a x i m u m L i f t i n i : Force

The l i f t i n g f o r c e s shorn i n t h e t a b l e were determined by l e v e l l i n g i n December, 1939, and January, 1940, A f t e r t h a t time, t h e i c e broke on many occasions, Therefore, t h e comparison could not te extended t o g r e a t e r t h i c k n e s s e s of t h e i c e ,

The maximum l i f t i n g f o r c e i s a f u n c t i o n of t h e bending s t r e n g t h of t h e i c e , The l a t t e r i n c r e a s e s a s t h e temperature decreases. A number

of v a l u e s of t h e Lending s t r e n g t h obtained from v a r i o u s sources a r e given below, Time, Thickcess water l e v e l A, i n

H,

,

i n m.

1

h r s , of t h e i c e ,

d

,

i n m. 0.16 O,22 00 235 0.39 0,405 0,31 0.38 i 0 ~ 6 2 ;

O o 1 4

j 0 0 5 3 I Corrected 72 36

48

48 48 L i f t i n g f o r c e i n mean temp, of i c e ,

-

f i n OC, -AOO

- l o o

-0.5

-4.

0

-6,O

t o n s . p e r

m,

--. C a l c u l a t i o n from ea. (12) 0.69 0,84 l a 5 2 0,67 2.83 Getermirled b-; l e v c l l i n a 0.7

l o o

l 0 6 0.6 2,7

J

(13)

T e s t a t t h e Eoyal Illst, of Tecllc. i ' i ? l k ' r i-reiiger ( 1 9 2 l ) iircLger

ii92lj

Brown (1326) Erown ( 1 j26)

Zending s t r e n g t h of Terripe 1.atw.e i c e i n t o n s p e r sq, of t h z i c e i n

13 0 C.

The bending s t r e s s when t h e i c e broke f o r t h e f i r s t time d u r i : i ~ th e o b s e r v a t i o n s was c a l c u l a t e d by means of equation (13) a s 190 t o n s p e r sq, metre, The mean temperature o f t h e i c e was approximately - C o 5 0 ~ o

The f i r s t c r a c k s appear:. i n t h e underside of t h e i c e n e a r c o n s t r u c t i o n s where t h e temperature

i s

zero degrees

C.

I n o r d e r t o be on t h e s a f e site, t h e bending s t r e n g t h of t h e i c e ,

$

,

i s s e t a t 200 t o n s p e r sq,

m,

a s f a r as t h e l i f t i n g f o r c e i s concerned,

The maximum l i f t i n g f o r c e i s obtained f r o m e q u a t i o n s (12) and (13). The t o t a l r i s e i n t h e water l e v e l ,

u,

=oX., i s introduced i n t o t h e s e equations. A f t e r e l i m i n a t i n g

h,

i n both e q u a t i o n s , a s i l i ~ p l e e x p r e s s i o n i s f i n a l l y obtained, v i z ,

,

It seems s u r p r i s i n g a t f i r s t t h a t t h e maximurr~ l i f t i n g f o r c e should be indepencient o f the c o e f f i c i e n t of p l a s t i c i t y and t h e temperature of the i c e , This can be explained by t h e f a c t t h a t a t a c e r t a i n v a l u e o f t h e r i s e i n t h e water l e v e l ,

yo

,

t h e

(14)

l i f t i 3 g force may t e the same f o r Loth low and big;, p l a s t i c i t i t s , i f t h e water l e v e l r i s e s a t a slower r a t e i n t h e f i r s t csse then i n t h e l a t t e r ,

k l ~ o s t e x a c t l y t h e same expression a s (14) can be OL- t a i n e d d i r e c t l y from t h e i n i t i a l formulae ( 2 ) and

( 3 )

by elirnin- a t i n g

E-

,

Then

To o ~ t a i n information on t h e a p p l i c a b i l i t y of t h i s ex- p r e s s i o n ,

i t

i s necessary t o introduce t h e p l a s t i c i t y i n t o t h e c a l c u l a t i o n .

The maximum l i f t i n g f o r c e , according t o equation (14) has been p l o t t e d i n Fig,

4

f o r various thicknesses of t h e i c e , I n order t o o b t a i n the maximun l i f t i n g f o r c e ,

i t

may be necessary i n some c a s e s f o r t h e water l e v e l t o r i s e a t a very slow r a t e ,

It t h e time i s limited, s a y t o f o u r days

(96

hours), t h e curves i n t e r s e c t a s can Le seen i n t h e Figure.

For small r i s e s i n t h e water l e v e l , t h e g r e a t e s t l i f t i n g f o r c e s a r e obtained when t h e r i s e s occur r a p i d l y , when t h e deformations a r e q u i t e e l a s t i c . Then, according t o equation (21, t h e l i f t i n g f o r c e i s p r o p o r t i o n a l t o the i n c r e a s e

4

and

(15)

L i f t i n 2 Force on I s o l a t e d Construction

The above expression cannot be a p p l i e d t o c o n s t r u c t i o n s o f s h o r t l e n g t h , The l i f t i n g f o r c e p e r u n i t of l e n g t h becomes g r e a t e r here due t o t h e f a c t t h a t t h e i c e slieet a c t s l i k e a s l a b on an e l a s t i c support,

A s i m i l a r problem was d e a l t with a s e a r l y a s 1863 by

H , ~ e r t z ( 3 1 , who d e r i v e d a n expression f o r an e l a s t i c i c e s h e e t

with t h e l o a d a p p l i e d a t t h e c e n t r e , The l i f t i n g f o r c e on i s o l a t e d , c i r c u l a r c o n s t r u c t i o n s can be c a l c u l a t e d with t h e a i d of a work by

H, S c h l e i c h e r

(41,

i n wilich a l a r g e numLer of c a s e s of a x i a l l y symmetrical l o a d s a p p l i e d t o a c i r c u l a r s l a b on a n e l a s t i c support a r e thoroughly s t u d i e d , The s o l u t i o n s obtained a r e i n the form of v s r i o u s e y l i n d e r f u n c t i o n s , which have been taLulated i n h i s book, On account of t h e i r complicated s t r u c t u r e , t h e s e f u n c t i o n s a r e n o t open t o treatment of t h e problem i n a manner s i m i l a r t o t h a t f o r t h e long s t r a i g h t wall. However, a conception of t h e c o n d i t i o n s can Le obtained i n t h e csse of i s o l a t e d o b j e c t s i f t h e i c e i s assumed t o be e l a s t i c and i f t h e modulus of e l a s t i c i t y i s replaced by a mean value of t h e modulus of deformation

F ,

I f t h i s method i s a p p l i e d t o t h e l i f t i n g f o r c e on a l o n g

w a l l ,

i t

i s found t h a t t h e maximum l i f t i n g f o r c e f o r a time l i m i t

of f o u r days can be obtained from equations (2) and (3) by i n t r o d - ucing

F

=

3,900 t o n s p e r sq, metre i n t o t h e s e equations, T h i s may Le considered a convenient mean value, Moreover, i t w i l l make h a r d l y any d i f f e r e n c e whether t h e time l i l i i i t i s t h r e e o r f i v e days,

A s b e f o r e , t h e tending s t r e n g t h i s assumed t o be 200 t o n s p e r sq, metre.

(16)

The e q u a t i o n s f o r t h e l i f t i n g f o r c e and s t r e s s e s a r e q u i t e l o n c and t a L l e v a l u e s a r e requirkd f o r t h t ; i r numerical a y j d i c a t i o n , For t h i s reason, t h e r e s u l t of the c a l c u l a t i o n i s given h e r e only f o r some c a s e s which a r e of i n t e r e s t ,

An i c e sheet of tilickness

d

i s frozen t o a c y l i n d r i c a l o b j e c t of diameter

D

( ~ i g , 5 ) , The bending s t r e s s e s reach t h e breaking p o i n t when t h e water l e v e l r i s e s a d i s t a n c e

H o

The g r e a t e s t l i f t i n g f o r c e t h u s obtained has been c a l c u l a t e d f o r v a r i o u s v a l u e s of

d

and

2 .

Consider an i s o l a t e d p i l e of 8 i n , diameter w a t e r l i n e . This corresponds t o

=

0 , 2 metres. Then t h e l i f t i n g f o r c e s

shown i n Fid,

6

a r e obtained, O ~ v i o u s l y , th e f o r c e s a r e v e r y g r e a t , For exar~lple, with a n i c e s h e e t

0.75

m, t h i c k , a l i f t i n g f o r c e of

30

t o n s i s o l t a i L l e d f o r a n i n c r e a s e i n t h e water l e v e l of 0,35 n ~ e t r e s , Calculated p e r u n i t l e n g t h of circumference t h e l i f t i n g ' f o r c e on t h e p i l e i s

48

t o n s p e r metre whereas, accordin6 t o F i g ,

4,

t h e l i f t i n g f o r c e on a long s t r a i g h t w e l l only amounts t o 3.5 t o n s p e r metre where

4 = 0.35,

Great l i f t i n g f o r c e s may occur even i f t h e water l e v e l only r i s e s a fey; c e n t i n ~ e t r e s , h p e r i e n c e shows, moreover, t h a t i n Northern Swedish r i v e r s i s o l a t e d p i l e s o r groups of p i l e s a r e u s u a l l y l i f t e d by t h e i c e during t h e w i n t e r u n l e s s proper p r e c a u t i o n s a r e taken. Sometimes they can even be l i f t e d s e v e r a l metres s i n c e they p a r t i c i p a t e i n t h e r i s i n g , but not i n t h e sub- s i d i n g

of

t h e i c e , Fig, 70

(17)

C a l c ~ l a t e d p e r n e t r e of c i r c u r ~ f e r e n c e , t h e l i f t i n g f o r c e & c r e a s e s a s t h e diameter i n c r e a s e s , and approaches t h e v a l u e s oLtaii~ed f o r a long s t r a i g h t wall, I n Fig, .S t h e max-

ixurn l i f t i n g f o r c e i s shown as a f u n c t i o n of t h e diameter of c o n s t i v c t i o n s of c y l i n d r i c a l form,

For a c o n s t r u c t i o n having another form, t h e Lehav- i o u r of t h e s t r e s s e s i n t h e i c e changes. I n such cases,

it

i s ciifi'icult t o d e r i v e the s t r e s s e s t h e o r e t i c u l l y , For i n s t a n c e , f o r c o n s t r u c t i c r : ~ of square form i t would be s a f e t o c a l c u l a t e a s f o r t h e i n s c r i b e d c i r c l e because a n g l e s r e s u l t i n s t r e s s concent- r a t i o n s ~ h i c h cause t h e i c e t o Lreak more e a s i l y , S i m i l a r l y , a r e c t a n k u l a r form may be replaced by a form with s t r a i g h t s i d e s and semi-circular ends. Then t h e l i f t i n g f o r c e on the s t r a i g h t s i d e s can t e c a l c u l a t e d i n t h e same manner a s t h a t on a long s t r a i g h t wall,

For the above c a l c u l a t i o n , it i s assurned t h a t t h e i c e i s of uniform thickness and i s homogeneous throughout, This i s p r a c t i c a l l y never the case when t h e i c e a t t a i n s a g r e a t thickness, Eecause of v a r i a t i o n s of t h e water l e v e l , t h e i c e s h e e t breaks time a f t e r time and t h e water f o r c e s ' i t s way up and

f r e e z e s on t o p of t h e i c e , Therefore, t h i s i c e i s t h i c k e r c l o s e t o c o n s t r u c t i o n s , The newly formed i c e o f t e n c o n s i s t s of ltslush i c e " , i , e , a frozen mixture of snow and water, Slush i c e i s not

a s s t r o n g a s black i c e and s i n c e

i t

i s f r e q u e n t l y cracked i n t h e v i c i n i t y of constructions, i t i s doubtful whether t h e thickening

(18)

a c t u a l l y r e s u l t s i n a s t r e n ~ t h e n i n g , Conditions i n t h e p o r t i n v e s t i g a t e d i n d i c a t e t h a t t h i s i s not t h e case, althouzh i n t h e v i c i n i t y of p i e r s t h e i c e mas sometimes found t o be two metres t h i c k ,

I n view of t h i s a bending-strength value of l e s s than 200 t o n s p e r sq, metre might be s u f f i c i e n t l y s a f e . P a r t i c u l a r l y i n t h e case of i s o l a t e d smaller c o n s t r u c t i o n s , a reduction of t h e assurned bending-strength seems t o be reasonable since here t h e i c e breaks more f r e q u e n t l y than i n t h e case of a long wall. This problem r e q u i r e s f ~ r t h e r i n v e s t i g a t i o n ,

Bearing Capacity of Ice

The thickness of i c e required i n order t o support a person o r a horse, e t c . , i s known from p r a c t i c a l experience, However, f o r more unusual loads

i t

may t e d i f f i c u l t t o estirnate

t h e t e a r i n g c a p a c i t y from empirical knowledge alone. An in- v e s t i g a t i o n i n t o t h e bearing s t r e n g t h of i c e s h e e t s a t v a r i o u s t h i c k n e s s e s t h u s seems appropriate,

L i f t i n g f o r c e and bearing c a p a c i t y a r e b a s i c a l l y t h e same p r o p e r t y of an i c e s h e e t , but seen from d i f f e r e n t p o i n t s o f view, The time fac%or has t h e r e f o r e t h e same i n f l u e n c e on t h e bearing capacity a s on t h e l i f t i n g force. The s m a l l e s t bearing c a p a c i t y i s obtained f o r a n instantaneous load rhen t h e deformation i s e l a s t i c . The smallest bearing s t r e n g t h t h u s depends on t h e modulus of e l a s t i c i t y of i c e , Some of t h e v a l u e s

(19)

f o r t h e modulus of e l a s t i c i t y of i c e ]lave t e e n t a k e n from t h e l i t e r a t u r e and a r e l i s t e d Lelow. hiodulus of e l a s t i c i t y f o r i c e i n t o n s p e r sq.

m.

Bevan 1826 Frankenheim 1858 I,.osley 1871 Reuch 1880 Less 1902 Loch

1713

'I 1914 iireiiger 1922 tt tt II n

Reucll obtained h i s value by determining t h e v i b r a t i o n frequency f o r p r i s m a t i c i c e bars. The o t h e r v a l u e s were obtained from p r e s s u r e and bending t e s t s .

For c a l c u l a t i o n s of t h e bearing c a p a c i t y a high v a l u e should be assumed f o r t h e modulus of e l a s t i c i t y , say

lo6

t o n s p e r sq. metre, I n t h i s connection, t h e tending s t r e n g t h of black i c e which h a s no cracks should not be chosen higher than 100 t o n s p e r sq. metre,

I f

H

i s eliminated i n equations ( 2 ) and(3), then t h e bearing s t r e n g t h f o r instantaneous l o a d s applied on a s t r a i g h t l i n e i s obtained:

The f a c t o r 2 e n t e r s i n t o t h e c a l c u l a t i o n s i n c e t h e i c e s h e e t s c o n t r i b u t e s t o both s i d e s of t h e load.

(20)

I f the load i s applied gradually, a consideraLily g r e a t e r bearing s t r e n g t h i s obtained, This becones evident i f t h e r a t e a t aliicll t h e viater l s v c l r i s e s , 0

,

i s eliminated i n

equations (12) and ( 1 3 > , The11

This expression approximately gives t h e influence of t h e time f a c t o r on t h e bearing s t r e n g t h , Fig,

9

shows t h e bearing

6

s t r e n g t h f o r an instantaneous load with

E =

10 t o n s p e r sq, m e and f o r a load appJied gradually f o r 24 hours (

A,

=

24,

f

=

0

5

and C

=

8

x 10-

i,

The two curves may be considered a s l i m i t cases, Fig, 1 0 gives t h e bearing s t r e n g t h f o r a concentrated l o a d with

E

=

t o n s per sq. n o a s well a s with

F

=

6,400 t o n s p e r sq, m, The l a t t e r case corresponds approximately t o a time

l i m i t

of 24 hcurs wilere

f

equals zero,

It

i s assuned t h a t t h e l o a d i s

spread over a c i r c u l a r a r e a with diameter equal t o t h e tllicltness of t h e i c e , For concentrated loads, the time apparently has n o t a s g r e a t an influence a s f o r a load applied on a s t r a i g h t l i n e . The c a l c u l a t i o n s a r e based on Schleicller s book /cf

,

(4)

/

Liany o t h e r l o a d cases can be d e a l t with s i m i l a r l y ,

R i t h regard t o t h e bending s t r e n g t h of t h e i c e , which i s here assumed t o be 100 t o n s per sq, m,, this v ~ i l l vary con- s i d e r a b l y with d i f f e r e n t kinds of i c e , The p a r t i c u l a r s cont- ained i n l i t e r a t u r e a r e very sparse, D i f f e r e n t i n v e s t i g a t i o n s show, however, t h a t t h e s t r e n g t h of t h e i c e i s l e s s with l o a d s

(21)

of long d u r a t i o n than with momentary l o a d s , Therefore t h e d i f f e r e n c e Letween t h e curves i n Figs.

9

and 1 0 i s sornev;hat u n r e l i a b l e on t h e a c t u a l c o n d i t i o n s .

The author i s s u e s a warning a g a i n s t t h e e m p l o p e n t of l o a d s , having t h e magnitude of those shown i n Figs.

9

and 10. Very high f a c t o r s of s a f e t y silould t e adopted, Loacls of l o n g d u r a t i o n on a s h e e t of i c e should be avoided a l t o g e t h e r , as a r e s u l t of t h e downward d e f l e c t i o n , melting of t h e i c e nay t a k e p l a c e from beneath,

I f t h e r e a r e h o l e s and c r a c k s i n t h e i c e i n t h e v i c i n i t y of t h e l o a d , high v a l u e s f o r t h e b e a r i n g c a p a c i t y

cannot be expected of course, s i n c e t h e water forced o u t f u r t h e r i n c r e a s e s t h e weight on t h e i c e sheet.

Conclusion

An i c e s h e e t can e x e r t a considerable f o r c e on con- s t r u c t i o n s which a r e f r o z e n i n , A s has been shown, t h e l i f t i n g f o r c e and t h e Lending-strength of i c e on a l o n g s t r a i g h t w a l l can Le derived with c e r t a i n approximations, t a k i n g i n t o account t h e t e ~ p e r a t u r e of t h e i c e and i t s p l a s t i c i t y a s w e l l a s a slowly r i s i n g water l e v e l , Then expressions (12) and (13) a r e obtained and t h e s e agree w e l l with c o n d i t i o n s observed i n t h e f i e l d ( c f , T a b l e ) ,

From equations (12) and. (13) a simple formula (14) f o r maximum l i f t i n g f o r c e i s obtained, This i s found t o Lie l a r g e l y independent of t h e modulus of e l a s t i c i s y of t h e i c e

(22)

and i t s c o e f S i c i e n t of p l a s t i c i t y . This formula i s shown p l o t t e d i n Fig,

4,

For i s o l a t e d c y l i n d r i c a l o b j e c t s , t h e l i f t i n g f o r c e may be c a l c u l a t e d apy;roxin;ately, i f t h e i c e i s assumed t o Le e l a s t i c and i f a low value i s introduced f o r t h e modulus of e l a s t i c i t y , The l i f t i n g f o r c e on an i s o l a t e d p i l e i s shown i n Fig,

6

and t h e l i f t i n g f o r c e a s a f u n c t i o n of t h e diameter f o r c o n s t r u c t i o n s of c y l i n d r i c a l form i n F i g , 8, For prisma- t i c bodies of r e c t a n g u l a r form, t h e l i f t i n g f o r c e may be c a l - c u l a t e d i n t h e same manner a s f o r bodies with s t r a i g h t s i d e s and semi-circular ends,

The bearing c a p a c i t y of i c e i s shown i n Fig,

9

f o r l o a d s a p p l i e d on a s t r a i g h t l i n e and i n Fig, 10 f o r l o a d s spread over a c i r c u l a r a r e a with diameter equal t o t h e t h i c k n e s s of t h e i c e ,

(23)

References

1. Barnes, H O T , "Ice Engineerin&", l,on-treal 192b,

2, Eoyen, 1 J i l s . "Ice P r e s s u r e due t o Temperature Rise", S-tocl:holm 1922,

11

3.

kiertz,H, '"E ber u a s i;leicl,,c;-,;icht sch-tlimrnenuer e l a s t i s c i ~ e r

Y l a L t ~ n ~ ~ , ='m.l, I h ~ s , 22 ( 1 8 ~ 4 ) 9.

445"

4.

S c h l e i c h e r , P, M i ~ r e i s p l a t t s n auf e l a s t i s c h e r U n t s r l a g e H , L e r l i n , 1926,