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Experimental results on the buckling of freshwater ice sheets
Timco, G. W.; Sinha, N. K.
https://publications-cnrc.canada.ca/fra/droits
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Experimental Results on the Buckling
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Freshwater Ice Sheets
by G.W.
Timco and N.K. Sinha
Reprinted from
Proceedings of the Seventh International Conference on
Offshore Mechanics and Arctic Engineering
Houston, Texas, 7-12 February 1988
Vol. IV, p. 31-38
(IRC Paper No. 1559)
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gauchissementEXPERIMENTAL RESULTS ON THE BUCKLING OF FRESHWATER ICE SHEETS G. W. Timco and N. K. Sinha
National Research Council of Canada Ottawa, Ontario. Canada
A t e s t s e r i e s h a s been c a r r i e d o u t
to
i n v e s t i g a t e t h e l o a d and d e f o r m a t i o n behaviour o f t h i n f r e s h w a t e r i c e s h e e t s l o a d e dto
u l t i m a t e f a i l u r e i n a b u c k l i n g - t y p e f a i l u r e mode. The t e s t s were conducted w i t h load- i n g r a t e s g i v i n g f a i l u r e t i m e s of l e s s t h a n one-half second. C o n s i d e r a b l e m i c r o c r a c k i n g was o b s e r v e d i n t h e i c e . Large d e f l e c t i o n s , on t h e o r d e r of one-half t h e i c e t h i c k n e s s , = r e measured a t f a i l u r e . The l o a d and i c e d e f o r m a t i o n p r o f i l e a r e g i v e n f o r t w e l v e d i f f e r e n t b u c k l i n g e v e n t s .The b u c k l i n g o f f l o a t i n g i c e s h e e t s is an impor- t a n t f a i l u r e mode i n many s i t u a t i o n s where a t h i n ice s h e e t impinges on a wide v e r t i c a l - s i d e d s t r u c t u r e . To d a t e t h e r e have been numerous t h e o r i e s developed t o c a l c u l a t e t h e l o a d and i c e d e f o r m a t i o n behaviour d u r i n g t h e b u c k l i n g e v e n t (Sodhi and FIamea, 1977: Kerr, 1978; Wang, 1978; S o d h i , 1979; Nevel, 1980; K e r t , 1980; and S j U l i n d , 1984, 1985). These have been reviewed r e c e n t - l y by Sodhi and N e w 1 (1980) and Sodhi (1986). A l - though t h e r e h a s been numerous t h e o r e t i c a l t r e a t m e n t s t o t h i s problem, t h e r e is a p a u c i t y o f e x p e r i m e n t a l d a t a i n t h i s a r e a . Sodhi (Sodhi, 1983; Sodhi et a l . ,
1983; Sodhi and Adley, 1984) h a s been t h e p r i n c i p a l i n v e s t i g a t o r , measuring b u c k l i n g l o a d s f o r v a r i o u s geo- metr i e s and l o a d i n g r a t e s under c o n t r o l l e d c o n d i t i o n s i n an i c e modelling basin. I n h i s t e s t s no i n f o r m a t i o n
was
c o l l e c t e d on t h e d e f o r m a t i o n behaviour o r damage p r o c e s s e s o f t h e i c e s h e e t i t s e l f . T h i s paper p r e s e n t s t h e r e s u l t s o f a s e t o f t e s t s i n which b o t h t h e buck- l i n g l o a d and i d e d e f o r m a t i o n b e r e measured t h r o u g h o u t t h e whole b u c k l i n g event. Thetests
were performed u s i n g t h i n s h e e t s o f f r e s h w a t e r i c e , r a t h e r t h a n model i c e a s used by Sodhi. The use of f r e s h w a t e r i c e h a s two distinct advantages i ntests
of t h i s kype: f i r s t l y , t h e mechanical and r h e a l o g i c a l p r o p e r t i e s o f f r e s h w a t e rice
a r e much b e t t e r known t h a n t h o s e o f model i c e , T h i s a i d e i n t h e i n t e r p r e t a t i o n and a n a l y s i sof
t h e test r e a u l t s . S e c o n d l y , f r e s h w a t e rice
is o p t i c a l l y t r a n s p a r e n t , a l l o w i n g o b s e r v a t i o nof
t h e cracking Rsmnted 6t the k m t h Internalonat Conlnrenm onOffshore Mechanics and Arctic Eng~neering
Houston, Texas - February 7-12, 1988 31
a c t i v i t y i n t h e i c e . The b i g d i s a d v a n t a g e w i t h f r e s h - water i c e r e l a t e s t o its h i g h e r s t r e n g t h
-
much h i g h e r l o a d s a r e e n c o u n t e r e d i n t h e t e s t s and t h i s r e s t r i c t s t h e range of e x p e r i m e n t a l parameters. I n t h e s e t e s t s , t h e b u c k l i n g l o a d s and i c e d e f o r m a t i o n o f f r e s h w a t e r i c e were measured o v e r a range o f a s p e c t r a t i o s from 30 to 50 f o r one nominal l o a d i n g r a t e andt m
i n t e r f a c e f r i c t i o n c o n d i t i o n s .The t e s t s were performed i n t h e i c e t a n k i n t h e H y d r a u l i c s L a b o r a t o r y o f t h e N a t i o n a l Research C o u n c i l i n Ottawa ( P r a t t e and Timco, 1981). I n t h e t a n k , which is 21 m x 7 m x 1.2 m, t h i n s h e e t s o f f r e s h w a t e r i c e a p p r o x i m a t e l y 1
cm
t h i c k were grown u s i n g a wet-seeding t e c h n i q u e f o r n u c l e a t i o n . T h i s method p r o d u c e s a f i n e - g r a i n e d , c o l u m n a r - s t r u c t u r e d i c e with g r a i n d i a m e t e r s o f t h e o r d e r o f 0.1 cm. The s t r u c t u r e and mechanical p r o p e r t i e s of t h i s i c e h a w been measured and documen- t e d by Timco and F r e d e r k i n g (1982).The model r e p r e s e n t a t i o n o f a s t r u c t u r e f o r t h e s e b u c k l i n g t e s t s was a 1.9 cm t h i c k plywood s h e e t c u t t o
a width o f 30 cm and p a i n t e d t o a high g l o s s f i n i s h
u s i n g Varathane p a i n t . The board was used w i t h t h i s f i n i s h f o r one t e s t s e r i e s . It was c o v e r e d w i t h a t h i n (0.9 c m ) s h e e t of r u b b e r f o r a second t e s t s e r i e s . P r e v i o u s t e s t s by Timco (1984) u s i n g t h e s e f i n i s h e s gave dynamic i c e - s t r u c t u r e f r i c t i o n c o e f f i c i e n t s o f 0.02 and 0.07 f o r t h e Varathane (V) and rubber (R) f i n i s h e s r e s p e c t i v e l y * . The board was mounted o n t o a s t e e l frame and a t t a c h e d t o a s i x component dynamometer p l a t f o r m which, i n t u r n , was a t t a c h e d t o t h e main c a r - r i a g e o f t h e i c e t a n k f a c i l i t y . With t h i s arrangement i n f o r m a t i o n on t h e t h r e e l o a d and moment components on t h e s t r u c t u r e a r e o b t a i n e d ,
*
These v a l u e s a r e based on t e s t s measuring t h e i c e l o a d componehts on a wide i n c l i n e d s t r u c t u r e . S i n c e t h e y were d e t e r m i n e d using model i c e and low c o n f i n i n g p r e s s u r e s , however, t h e s e f r i c t i o n c o e f f i c i e n t s s h o u l d be c o n s i d e r e d o n l yas
r e p r e s e n t a t i v e v a l u e s f o r t h e p r e s e n t c a s e .To measure t h e d e f l e c t i o n o f t h e i c e s h e e t i n were o f t h e o r d e r o f 0.1 t o 0.2 c m , comparable t o t h e f r o n t of t h e board, an a r r a y o f e i g h t d i s p l a c e m e n t g r a i n s i z e o f t h e i c e , and t h e c r a c k s e x t e n d e d t h r o u g h g a u g e s (DCDTs) was used. A b r a c k e t was c o n s t r u c t e d t o the f u l l i c e t h i c k n e s s . T h i s m i c r o c r a c k d e n s i t y was h o l d t h e DCDTs a t r e g u l a r b u t a d j u s t a b l e s p a c i n g s o n h i g h e s t c l o s e t o t h e board. During t h e l o a d i n g e v e n t t h e i c e a l o n g t h e c e n t r e - l i n e of t h e board b u t perpen- t h i s m i c r o c r a c k i n g c o n t i n u e d u n t i l t h e r e was a sudden d i c u l a r t o it. T h i s b r a c k e t was mounted t o t h e s e r v i c e and c a t a s t r o p h i c f a i l u r e of t h e ice when it s h a t t e r e d c a r r i a g e o f t h e i c e t a n k f a c i l i t y t h e r e b y making it i n t o numerous l a r g e p i e c e s . Although t h e d e t a i l e d c o m p l e t e l y independent of t h e main c a r r i a g e . T h i s pro- p a t t e r n o f c r a c k s would vary from t e s t t o t e s t , t h e r e c e d u r e allowed measurement of i c e d e f l e c t i o n r e l a t i v e were some common f e a t u r e s observed i n each case. For t o a f i x e d , non-moving frame o f r e f e r e n c e . A s c h e m a t i c t h e t e s t s u s i n g t h e Varathane-painted board, t h e i c e 1 o f t h e f u l l t e s t s e t u p is shown i n F i g u r e 1. would f a i l w i t h a c u r v e d a r c geometry s i m i l a r t o t h a t
shown i n F i g u r e 2 f o r t e s t V-3. The l e n g t h (L) of t h e
- OY NAUOMETER PLATFORM a r c was t y p i c a l l y one and one-half t o two t i m e s t h e
width o f t h e b u c k l i n g board and it had a maximum w i d t h
- BUCKLING SERVICE CARRIAGE-
BOARD ( w ) a t t h e mid-point o f t h e board. The d e t a i l s o f t h e
,-
BRACKET FM3 OCaT S\
c r a c k i n g p a t t e r n o u t s i d e t h i s r e g i o n d i f f e r e d from t e s t t o t e s t , b u t t h e r e was always a mix o f r a d i a l andc i r c u m f e r e n t i a l c r a c k s , w i t h f r a c t u r e d i c e e x t e n d i n g t y p i c a l l y two to f o u r times t h e width ( w ) o f t h e f i r s t c i r c u l a r a r c . For t h e rubber-faced b o a r d , on t h e o t h e r hand, t h e l a r g e r c r a c k i n g r a d i u s o f t h e i c e d i d not o c c u r ; i n s t e a d t h e i c e f a i l u r e was much more c o n f i n e d t o t h e r e g i o n i n f r o n t o f t h e board. A t y p i c a l f a i l u r e p a t t e r n is shown i n F i g u r e 3. I n t h i s c a s e , t h e i c e p i e c e s were much s m a l l e r and c o n t a i n e d fewer micro- c r a c k s . T h e r e were, however, a number o f " g r a z i n g n c r a c k s e v i d e n t . These milky-coloured c r a c k s meandered t h r o u g h t h e i c e sample; however, t h e r e was no p h y s i c a l s e p a r a t i o n o f t h e i c e p i e c e s along t h e f r a c t u r e plane.
Figure 1 Schematic showing the experimental set-up Although t h e l o a d i n g t i m e t o f a i l u r e was w r y
for the buckling tests. s h o r t (<0.5 s) and t h e b u c k l i n g e v e n t o c c u r r e d very r a p i d l y , it was o b s e r v e d t h a t s e v e r a l o f t h e l a r g e i c e For t h e i n t e r a c t i o n e v e n t , a c l e a n i n t e r f a c e was f r a g m e n t s were very h i g h l y deformed with a n e t perman- c u t i n t h e i c e s h e e t p a r a l l e l t o t h e plywood b o a r d e n t p l a s t i c deformation. T h i s was e s p e c i a l l y e v i d e n t a c r o s s a 5 m width o f t h e tank. The main c a r r i a g e , f o r t h e i c e i n t h e Varathane t e s t s e r i e s . At f i r s t which c o n t a i n e d t h e b u c k l i n g board, was t h e n d r i v e n a t g l a n c e t h i s r e s u l t a p p e a r s s u r p r i s i n g s i n c e it c o u l d be a c o n s t a n t speed o f 2 cm-s-' t o w a r d s t h e i c e edge. a r g u e d t h a t t h e l o a d i n g t i m e was so s h o r t t h a t t h e Although t h e c a r r i a g e is c a p a b l e o f much h i g h e r s p e e d s , whole e v e n t c o u l d be c o n s i d e r e d a s an e l a s t i c , b r i t t l e t h e t e s t s were c o n f i n e d t o t h i s lower speed range. f a i l u r e . However, a l l of t h e s e p i e c e s were "damaged", O t h e r w i s e , damage t o t h e d i s p l a c e m e n t g a u g e s c o u l d s i n c e t h e y c o n t a i n e d a l a r g e number o f microcracks. r e s u l t due t o t h e l o n g e r s t o p p i n g d i s t a n c e of t h e c a r - The l a r g e d e f o r m a t i o n of t h e i c e p i e c e s was c e r t a i n l y r i a g e a t h i g h e r speeds. The d a t a a c q u i s i t i o n system c a u s e d by a mechanism o f c r a c k enhanced creep.
was s t a r t e d j u s t b e f o r e t h e board made c o n t a c t with t h e i c e s h e e t . A l l d a t a c h a n n e l s were sampled a t a r a t e o f 100 Hz, d i g i t i z e d and s t o r e d for l a t e r a n a l y s i s . The d a t a were a n a l y s e d such t h a t l o a d and r e l a t i v e d i s - placement o f t h e i c e c o u l d be d e t e r m i n e d a t any time d u r i n g t h e l o a d i n g event. From t h i s i n f o r m a t i o n it was p o s s i b l e t o g e t load-time c u r v e s , d e f l e c t i o n - t i m e c u r v e s , and l o a d - d e f l e c t i o n c u r v e s t h r o u g h o u t t h e whole p r o c e s s . I n t h i s i n i t i a l t e s t s e r i e s , s i x t e s t s were performed f o r e a c h o f t h e two s t r u c t u r e s u r f a c e f i n i s h e s (i. e. M r a t h a n e and r u b b e r ) . MICROCRACKS
RESULTS AND DISCUSSION
Observations of t h e I c e Behaviour
Because o f t h e r e l a t i v e l y s h o r t t i m e t h a t t h e b o a r d was pushing a g a i n s t t h e i c e s h e e t b e f o r e f a i l u r e of t h e i c e , it was d i f f i c u l t t o d e t e r m i n e v i s u a l l y a l l t h e d e t a i l s o f t h e b u c k l i n g process. There were, how- e v e r , s e v e r a l noteworthy e v e n t s . A s t h e board was p r e s s i n g a g a i n s t t h e i c e s h e e t , t h e s h e e t m u l d s t a r t t o deform and b u c k l e w i t h a wave which had a maximum a m p l i t u d e e i t h e r down i n t o t h e water o r up i n t o t h e a i r . The d i r e c t i o n v a r i e d from t e s t t o t e s t b u t t h e r e was a p r o p e n s i t y f o r it t o buckle down i n t o t h e water. With f u r t h e r i n c r e a s e i n t h e l o a d , numerous m i c r o c r a c k s
were formed i n t h e i c e i n t h e r e g i o n d i r e c t l y i n f r o n t Figure 2 Schematic showing a typical f a i l u r e pattern
o f t h e i n d e n t o r , i n a manner s i m i l a r t o t h a t p r e v i o u s l y i n t h e ice using the 7Wcathane (71) covered
LONG 'GRAZING' CRACKS
I
I
1
-BUCKLINGLRUBBER
BOARD F i g u r e 3 S c h e m a t i c showing a t y p i c a l f a i l u r e p a t t e r ni n the ice u s i n g the r u b b e r (R) w v e r e d 0.6
b u c k l i n g board.
The U m b D e f l e c t i o n o f the Ice
Using t h e l o a d c e l l a t t a c h e d to t h e b u c k l i n g b o a r d and t h e a r r a y o f e i g h t EDTs r e s t i n g on t h e s u r f a c e o f t h e i c e , it is p o s s i b l e t o q u a n t i f y t h e behaviour o f t h e b u c k l i n g event. F i g u r e 4 shows t h e load-time behaviour and d e f l e c t i o n - t i m e f o r each DCDT f o r t h e b u c k l i n g e v e n t shown i n F i g u r e 2. Note t h a t t h e p s i - t i o n ( x ) of each d i s p l a c e m e n t t r a n s d u c e r is g i v e n r e l a - t i v e t o t h e edge o f t h e o r i g i n a l i c e i n t e r f a c e [ s e e F i g u r e 11. I n a l l c a s e s , t h e l o a d i n c r e a s e d monotoni- c a l l y t o f a i l u r e , and t h e r e l a t i v e amount o f d e f l e c t i o n v a r i e d a l o n g t h e measuring a x i s . From t h e s e d a t a it is p o s s i b l e to c o n s t r u c t a p i c t u r e of t h e p r o g r e s s i o n o f t h e i c e d e f o r m a t i o n d u r i n g t h e b u c k l i n g event. F i g u r e 5 shows t h e d e f l e c t i o n o f t h e i c e i n f r o n t o f t h e board f o r f o u r d i f f e r e n t t i m e s d u r i n g one l o a d i n g e v e n t . Note t h a t t h e a m p l i t u d e o f t h e d e f o r m a t i o n i n c r e a s e s w i t h i n c r e a s i n g t i m e ( o r l o a d ) . The " k i n k s " i n t h e c u r v e a r e due t o t h e d i s c r e t e measuring p i n t s on t h e i c e s u r f a c e ; i n a n a l y s i s , each p o i n t was c o n n e c t e d l i n e a r l y with t h e neighbour, r e s u l t i n g i n t h e s h a r p edges. I n r e a l i t y , t h e c u r v a t u r e o f t h e i c e would l i k e l y be c o n t i n u o u s and smooth. Knowing t h e i c e t h i c k n e s s and t h e d e f l e c t i o n a t f a i l u r e , it is p o s s i b l e t o r e c o n s t r u c t t h e deformed s h a p e of t h e i c e s h e e t a t t h e time when t h e i c e s h a t - t e r e d , ( i . e . a t f a i l u r e ) . T h i s r e c o n s t r u c t i o n is shown f o r a l l s i x i c e s h e e t s i n F i g u r e 6 f o r t h e Varathane f i n i s h and i n F i g u r e 7 f o r t h e rubber f i n i s h . As p r e v i o u s l y d i s c u s s e d , t h e d e f l e c t i o n o f t h e i c e c a n be up o r down. F u r t h e r , t h e maximum a m p l i t u d e o f t h e b u c k l i n g wave c a n be q u i t e l a r g e and, i n t h e s e t e s t s , up t o one-half of t h e i c e t h i c k n e s s . The p o s i t i o n o f I t h e f r a c t u r e of t h e f i r s t c i r c u m f e r e n t i a l a r c is shown by arrows i n t h e F i g u r e s . It a p p e a r s t h a t t h i s b r e a k
k
o c c u r s i n a d i f f e r e n t r e g i o n depending upon t h e s u r f a c e f i n i s h o f t h e b u c k l i n g board. With t h e v a r a t h a n e b o a r d , t h e f i r s t c i r c u m f e r e n t i a l c r a c k o c c u r r e d between t h e board and t h e peak d i s p l a c e m e n t p o s i t i o n whereas f o r t h e rubber-faced board, it o c c u r r e d on t h e f a r s i d e o f t h e peak d i s p l a c e m e n t . TlME(sec)
0.20 TlME(sec)
F i g u r e 4 L o a d - t h e and d e f l e c t i o n - t i m e b e h a v i o u r for one o fthe
b u c k l i n g tests (Test P 3 ) .,
,
I
l
l
TEST V - 3
0 10 20 30 40 50
HORIZONTAL POSITION !CMl
A summary of a l l t e s t r e s u l t s is g i v e n i n T a b l e 1. F i g u r e 5 D e f l e c t i o n p r o f i l e o f the ice for f o u r d i f f e r e n t times d u r i n g the b u c k l i n g event.
I I I ~ I I I I I ~ ~
ICE PROnLE AT WURE
0 0.2
A
1 1 1 1 l 1 1 1 ! 1 1
0.4
t
r\
ICE PRdFllE AT F N U E-I
HOAlZOMAL POSITION (CM) HORlZOMAL POSITION ICMI
Pigure 6 Deformation of the ice sheet a t f a i l u r e for each o f the six tests
I l l l l l l l l l l ICE PROFILE AT WUlRE
0 9
-
HOWZOMPL FtHm( M I 0.4 a2-
5
%
1
mm
OF UPR3EPORtED ICE SKETy
0.6 0.8 1.0 I 1 I I I I I I I I I a 10 20 ao 4a 50Figure 7 Beformatien of the ice eheet at f a i l u r e for each of the six teats m i * the %ubber finisb p h l b g board.
The l o a d i n g t i m e t o f a i l u r e ( t f ) was o b t a i n e d from t h e load-time c u r v e , and t h e a v e r a g e s t r e s s - r a t e i n t h e ice was d e f i n e d a s F i g u r e 8 shows t h e peak p r e s s u r e v e r s u s a v e r a g e s t r e s s r a t e f o r a l l o f t h e t e s t s . A r a t e e f f e c t i n t h e buck- l i n g p r e s s u r e c a n be s e e n o n l y f o r t h e r u b b e r - f a c e d board. For t h e M r a t h a n e c o a t e d b o a r d , no r a t e e f f e c t is e v i d e n t o v e r t h e l i m i t e d r a n g e o f s t r e s s - r a t e s o f t h e t e s t s . However, b e c a u s e t h e r e was some v a r i a t i o n i n t h e a s p e c t r a t i o o f t h e s e t e s t s , t h e s i t u a t i o n is more complex. F i g u r e 9 shows t h e b u c k l i n g p r e s s u r e v e r s u s a s p e c t r a t i o (B/h) f o r a l l o f t h e t e s t s . I t c a n be s e e n t h a t t h e lower b u c k l i n g p r e s s u r e s t e n d e d t o c o i n c i d e w i t h h i g h e r a s p e c t r a t i o s ( i . e . t h e t h i n n e r
i c e s h e e t s ) . These c o r r e s p o n d t o t h e lower s t r e s s - r a t e s . Examination o f F i g u r e s 6 and 7 show t h a t t h e r e l a t i v e d e f o r m a t i o n a t f a i l u r e was g r e a t e r f o r t h e t h i c k e r i c e s h e e t s (compare e.g. F i g u r e 7-R2 f o r t h e t h i n n e r i c e t o F i g u r e 7-R5 f o r t h e t h i c k e r i c e ) . A t
comparable a s p e c t r a t i o s and l o a d i n g s t r e s s - r a t e s , t h e p r e s e n t d a t a do not show any s i g n i f i c a n t e f f e c t o f s u r f a c e f r i c t i o n on t h e b u c k l i n g load.
AVERAGE LOADING STRESS RATE (MPa-s-I)
F i g u r e 8 B u c k l i n g p r e s s u r e s ( a t f a i l u r e ) p e r s u s a v e r a g e stress rate for
a l l
of the tests.I n comparing t h e t e s t r e s u l t s to c u r r e n t t h e o r y , t h e a n a l y s i s by Sodhi and Hamza (1977) seems to be t h e most a p p r o p r i a t e . They used a f i n i t e e l e m e n t a n a l y s i s to s t u d y t h e b u c k l i n g o f a s e m i - i n f i n i t e ice s h e e t l o a d e d by a u n i f o r m l y d i s t r i b u t e d l o a d o v e r a f i n i t e
l e n g t h o f t h e s t r a i g h t boundary. The s t u d y assumed a 1.0 f r i c t i o n l e s s boundary c o n d i t i o n on t h e s t r a i g h t e d g e
and
i t y . f i x e d These boundary a s s u m p t i o n s c o n d i t i o n s s h o u l d g i v e o n t h e t h e e d g e s lower a t l i m i t i n f i n - o f " 28 30 35 ASPECT RATIO O 40 (D/h) 45 O 50 t h e b u c k l i n g load. The r e s u l t s were p r e s e n t e d i n
terms
2
o f non-dimensional b u c k l i n g p r e s s u r e (Pf/BkE ) v e r s u s
B/E where Pf is t h e b u c k l i n g l o a d , B t h e s t r u c t u r e
w i d t h , k t h e - f o u n d a t i o n m d u l u s and t h e c h a r a c t e r i s - Buckling pressure (at failure) mrsus aspect t i c l e n g t h ( s e e F i g u r e 1 0 ) . The l a t t e r is r e l a t e d to ratio h r a l l o f t h e tests.
t h e e f f e c t i v e m d u l u s (E) o f t h e s h e e t by
where
u
is P o i s s o n ' s r a t i o .Sodhi and Hamza's (1977) a n a l y s i s assumes a f r i c - t i o n l e s s boundary c o n d i t i o n and a n i n c r e a s i n g u n i f o r m l o a d a t t h e l o a d i n g p o i n t a l o n g t h e f r e e edge. I n com- p a r i n g t h e p r e s e n t t e s t r e s u l t s w i t h t h e i r a n a l y s i s it s h o u l d be k e p t i n mind t h a t n e i t h e r o f t h e s e c o n d i t i o n s a r e met. I n t h e e x p e r i m e n t s a uniform d i s p l a c e m e n t was a p p l i e d t o t h e f r e e edge o f t h e i c e s h e e t and not a uniform load. Second, t h e c o n t a c t r e g i o n o f t h e s t r u c - t u r e is not f r i c t i o n l e s s a s i d e a l i z e d i n t h e t h e o r e t i - c a l t r e a t m e n t .
I n o r d e r t o compare t h e r e s u l t s o f t h e p r e s e n t tests w i t h Sodhi and Hamza's a n a l y s i s , it is n e c e s s a r y
to d e t e r m i n e e i t h e r t h e c h a r a c t e r i s t i c l e n g t h o r t h e e f f e c t i v e m d u l u s o f t h e ice. T e s t s t o measure t h e s e were performed u s i n g t h e " p l a t e method" w i t h a n
a p p r o a c h s i m i l a r to t h a t d e s c r i b e d by Sodhi e t a l . (1982). Values o f t h e c h a r a c t e r i s t i c l e n g t h were d e t e r m i n e d and t h e e f f e c t i v e modulus was c a l c u l a t e d u s i n g E q u a t i o n ( 3 1 , assuming a v a l u e f o r - P o i s s o n ' s r a t i o o f 0.3. The c h a r a c t e r i s t i c l e n g t h was a f u n c t i o n o f t h e ice t h i c k n e s s . The a v e r a g e v a l u e f o r t h e e f f e c - t i v e modulus was 9.0 GPa.
The p r e s e n t t e s t r e s u l t s were a n a l y z e d u s i n g t h e non-dimensional a p p r o a c h d i s c u s s e d above. The r e s u l t s a r e shown i n F i g u r e 10. Because o f t h e l i m i t e d r a n g e o f t h e t e s t p a r a m e t e r s , most o f t h e d a t a is c l u s t e r e d i n o n e a r e a . I n a l l c a s e s t h e measured v a l u e s were h i g h e r t h a n t h e c a l c u l a t e d values. On a v e r a g e , t h e measured b u c k l i n g l o a d was a b o u t t w i c e a s h i g h a s S o d h i and Hamza' s (1 977) lower-bound value.
T h i s paper d e s c r i b e s a t e s t s e r i e s i n v e s t i g a - t i n g t h e b u c k l i n g b e h a v i o u r o f t h i n s h e e t s o f f r e s h -
S o d h i , D. S.
,
1979. "Buckling A n a l y s i s o f Wedge-shaped F l o a t i n g I c e S h e e t s " . Proc. F i f t h I n t e r n a t i o n a l C o n f e r e n c e on P o r t and Ocean E n g i n e e r i n g under40 I 1 1 1 1 1 1 1 1 I 1 1 1 1 1 1 1 1 I A r c t i c C o n d i t i o n s (POAC)
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lyl>t--+-
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1 D e t e r m i n a t i o n o f Buckling Loads o f Cracked I c e
0.1 1 10 20 30 Sheets". Proc. T h i r d I n t e r n a t i o n a l O f f s h o r e
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m1.
111, pp. 183-186.-
R5
;@;;
-
-
\"'
R4R1 T H E ~ E T I C A L SOLUTION-
-
FOR FRICTIONLESS BUCKLING\ -
\R2 (SODHI AND HAMZA. 1977)
-
- -
\
\ \
J
\ 4
S o d h i , D.S. and Hamza, H.E., 1977. "Buckling A n a l y s i s o f a S e m i - i n f i n i t e I c e Sheet". Proc. F o u r t h F i g u r e 10 Non-dimensionalixed b u c k l i n g p r e s s u r e I n t e r n a t i o n a l C o n f e r e n c e o n P o r t and Ocean a n a l y s i s of the p r e s e n t tests. me t h e o r e - 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)
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1986. " F l e x u r a l and Buckling F a i l u r e o f F l o a t i n g I c e S h e e t s A g a i n s t S t r u c t u r e s " . Proc. IAHR Symposium on I c e ,ml.
11, pp. 339-359, Iowa C i t y , Iowa, U.S.A.water ice. The f u l l r e s u l t s a r e p r e s e n t e d i n t a b u l a r and g r a p h i c a l form. The r e s u l t s c l e a r l y show t h a t t h e i c e c a n deform s i g n i f i c a n t l y b e f o r e f a i l u r e . Micro- c r a c k i n g i n t h e i c e c a n l e a d t o l a r g e permanent d e f o r - m a t i o n o f t h e broken i c e p i e c e s through a mechanism o f c r a c k enhanced c r e e p . Buckling p r e s s u r e s ranged from 1.0 t o 3.5 MPa depending upon l o a d i n g s t r e s s - r a t e and/or a s p e c t r a t i o .
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11, pp. 857-866, Quebec C i t y , S t r u c t u r e s n . Proc. IAHR Symposium o n I c e , P a r t I,Canada. pp. 117-1 33, L u l e a , Sweden.
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,
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1985. " V i s c o - e l a s t i c B u c k l i n g A n a l y s i s o f F l o a t i n g I c e Sheets". Cold Regions S c i e n c e and Technology, 11:241-246.S t r u c t u r e w i d t h B = 30 cm; speed = 2 an-s-'
reprinted from
published by
Seventh International Conference on Offshore Mechanics and Arctic Engineerng
-
Volume I VEditors: D.S. Sodhi, C.H. Luk, and N.K. Sinha (Book No. 102500)
THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS 345 East 47th Street, New York, N.Y. 10017
