Mechanical analogy of a constant heave rate

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Mechanical analogy of a constant heave rate

Veda, T.; Penner, E.

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MECHANICAL ANALOGY OF A CONSTANT

HEAVE

RATE

?WALYZED

by

Taka0 Ueda and Edward

Penner

Reprinted, from

Proceedings, International Symposium on Frost Action in Soils

held at

U.

of Lulea, Sweden, February 1978

VOL 1, p

57 67

DBR Paper

No.

764

Division of Building Research

SOMMAIRE

La maquette ouverte simule le comportement de l'action du gel d'un

Gchantillon de sol saturE en contrepoids

1 une tempErature de gel

en "escalier" imposGe

P

une extrGmitG. L'analogie

fournit une

bonne comprGhension des rapports entre le taux

de

soulSvement d6

au gel, le taux de pEn6tration de gel, la direction du mouvement

de l'eau et la surcharge de pression.

- -

- -

Taken from Vol. 2, Proc. of International Symposium on Frost Action in Soils. held at U, of Lulea in February 1978,

Takao Ueda, Guest worker with DBR from Takenaka Technical Research Laboratory, Tokyo

1

Edward Penner, Head, Geotechnical Section, Division of Building Research, National Research Council of Canada, Ottawa

One of the typical features of the frost heaving phenomenon described in the authors' paper (Penner and Ueda, 1977) is that the total heave rate is essentially constant for any particular overburden pressure and cold side temperature. The fact that a large change jn rate of frost penetration occurs after the step change in temperature in the cold side has been imposed indicates that the rate of frost penetration does not have a significant effect on the total heave rate.

By way of review, the test conditions of the experiment were as follows. The soil sample was remoulded, water saturated and subjected to an arbitrary pressure inside the test cell until consolidation was essentially complete. The test cell was then placed inside a constant temperature chamber and conditioned thermally to a temperature of about 1 or 2 O C . One end of the soil sample had free access to water, i.e, the heaving conditions were in the open system mode. When thermal and pressure equilibrium had been achieved a constant cold side freezing temperature was imposed on the end opposite to the water supply. The total heave was continuously recorded as was the temperature distribu- tion and the movement of water into or out of the sample. Freezing took place unidirectionally because the sides of the sample were heavily insulated.

To understand and explain the relationship between constant heave rate, movement of water that is taken in or expelled, and the rate of frost penetration, the mechanical analogy shown in Fig. 1 is proposed.

The liquid-containing cylinder is connected at one end to a reservoir from which liquid can be supplied or withdrawn from the cylinder with a pump. The piston contains a hole through which liquid flows at a constant rate because the head H is held constant under all rates of piston movement in the direction of the reservoir.

The types of flow behaviour that can be induced in the apparatus that are of interest occurs vhen:

(a) the piston is moved so slowly to the right that a larger volume of liquid flows out of the hole than is displaced by the piston. To maintain a constant flow rate and head, H,

the pump supplies liquid from the reservoir to the cylinder (b) the piston is moved so rapidly to the right that liquid is

transferred into the reservoir to maintain a constant head, H.

*

Discussion paper following Session 1 of International Symposium on Frost Action in Soils, Univ. of Lulez, Sweden, 16 to 18 February 1977.

58

Takao Ueda and Edward Penner.

where

VT = the total volume of flow through the hole

A2 = the cross sectional area of the water in the cylinder

X = the distance traversed by the piston

VP, = volume of liquid interchange between reservoir and cylinder. V is (+) positive when liquid is moved into

P,

the cylinder and (-) negative when taken out of the cylinder

The frost heave phenomenon is simulated by the mechanical analogy in the following ways:

(1) The flow rate of liquid through the hole of the piston d h ~

(~2)

represents the total heave rate -

dt

(2) Rate of piston movement

( g )

represents the rate of frost penetration.

dV

( 3 ) Flow rate of liquid into or out of cylinder

( 2 )

from the reservoir reuresents the rate of water flow into or out of

dh the sample

By differentiating eq. (1) with respect to time gives

Based upon the assumption that the head of liquid in the cylinder remains the same the flow rate of liquid through the hole in the piston is given by Bernoulli's equation as follows

-

::

- - A, = constant ( 3 )

where

A1 = factor that takes into account the cross-sectional area of the hole in the piston

g = acceleration due to gravity

H = head of liquid

Substituting eq. (3) into eq. (2) and rearranging the terms, the rate of flow of liquid through the connecting pipe between the reservoir and the cylinder is given by

59 Takao Ueda and Edward Penner.

A p p l i c a t i o n of t h e mechanical analogy t o t h e f r o s t heaving s i t u a t i o n a s g i v e n above e q u a t i o n ( 4 ) can be r e w r i t t e n a s f o l l o w s where A = c r o s s - s e c t i o n a l a r e a of t h e c o n n e c t i n g p i p e t o t h e 3 d V ~ r e s e r v o i r , and t h e t o t a l flow r a t e , r e p r e s e n t s d h ~ d h ~ - hence = d t and t h e r e f o r e by d i v i d i n g e q u a t i o n by A3 g i v e s I f i t i s assumed t h a t t h e r e l a t i o n s h i p between f r o s t p e n e t r a t i o n and time can be r e p r e s e n t e d by t h e S t e f a n e q u a t i o n , t h a t i s ,

t h e f o l l o w i n g f r o s t heaving e q u a t i o n i s i n t h e same form a s eq. (3) (Penner and Ueda, 1977) and can be d e r i v e d by i n t e g r a t i n g eq. (6) w i t h r e s p e c t t o t i m e

where

It h a s been shown, t h e r e f o r e , t h a t t h e analogy model h e r e can s i m u l a t e t h e a c t u a l behaviour of water movement i n t o o r o u t of t h e sample d u r i n g t h e f r e e z i n g p r o c e s s . The l i q u i d i n t h e c y l i n d e r i s , of c o u r s e , c o n s i d e r e d t h e i n s i t u w a t e r . I t f o l l o w s , t h e r e f o r e , t h a t i f t h e p i s t o n i s moved i n such a way s o t h a t t h e flow o u t of t h e h o l e i n t h e p i s t o n e x a c t l y e q u a l s t h e volume swept by t h e movement of t h e p i s t o n t h e r e i s no p a s s a g e of l i q u i d between r e s e r v o i r and t h e c y l i n d e r . T h i s i s t h e c a s e where o n l y i n s i t u water f r e e z e s .

Other c o n s i d e r a t i o n s - t h e i n f l u e n c e of c o l d s i d e t e m p e r a t u r e and overburden p r e s s u r e

d h ~

As was shown by Line11 and Kaplar (1959) t h e t o t a l heave r a t e can

be e x p r e s s e d by an e x p o n e n t i a l f u n c t i o n . F i g u r e 2 shows a l i n e a r r e l a t i o n s h i p between l o g a r i t h m of t h e t o t a l heave r a t e and overburden p r e s s u r e f o r f i v e s o i l t y p e s t h a t have been used i n o u r e x p e r i m e n t s , t h a t i s

60 Takao Ueda and Edward Penner.

theref ore

and A, B and C are constants for a given type of soil and P is the

overburden pressure.

Now consider the relationship between cold side temperature and total heave rate. Figure 3 shows total heave rate versus the square root of the temperature of the cold side for Leda clay. The temper- ature changes were in the range -0.30 to -2.95O~. There is a linear dependance of the heave rate on the square root of temperature, T from -0.3 to -1 . ~ O C . Below about-1.5'~ the rate of increase with temperature decrease is much less. In the first straight-line portion

where

T is the cold side temperature

T* critical cold side temperature for a given soil type.

Thus from eq. (9) and (10)

Integrating eq. (11) with respect to time yields -C2 P

hT = Cl e

fi

t

Inserting the above expression for total heave into eq. (7) which is the integrated form of the analogy eq. (6) gives

where

C1, C and C are constants depending on 2

soil type.

Equation (13), based upon the experimental results, represents the general relationship between pressure, cold side temperature and heave by water intake.

Differentiating eq. (13) with respect to time gives the rate equation

61 Takao Ueda and Edward Penner.

L e t t i n g hQ t h e heave due t o water i n t a k e e q u a l 0 and s o l v i n g

f o r t i m e i n eq. (13) g i v e s

I t w i l l be remembered from Penner and Ueda (1977) t h a t t l i s t h e time

( i f water e x p u l s i o n has o c c u r r e d and where i n t a k e h a s followed) when t h e o r i g i n a l e x p u l s i o n h a s j u s t been r e c o v e r e d . dhll

.

S i m i l a r l y l e t t i n g - I n eq. (14) e q u a l 0 and s o l v i n g f o r t g i v e s d t t o t h e minimum i n t h e water e x p u l s i o n c u r v e , i . e . , t h e p o i n t of t u r n - around.

Equations (15) and (16) show t h a t b o t h t l and t o d e c r e a s e a s t h e c o l d s i d e t e m p e r a t u r e becomes c o l d e r and i n c r e a s e a s t h e overburden p r e s s u r e becomes l a r g e r .

C a l c u l a t e d c u r v e s f o r s o i l No. 2 (Penner and Ueda, 1977) a r e

shown i n F i g . 4 . The c o n s t a n t s C1, C 2 and C3 were determined d i r e c t l y

from t h e heaving c u r v e s observed. As shown i n F i g . 4 t h e dependency

of heave by w a t e r i n t a k e on t h e overburden p r e s s u r e i s r e a d i l y s i m u l a t e d , i . e . , i n c r e a s i n g t h e overburden p r e s s u r e i n c r e a s e s b o t h t o and t l .

It has been observed i n a c t u a l l a b o r a t o r y experiments t h a t a t t h e

same overburden p r e s s u r e a lower c o l d s i d e t e m p e r a t u r e r e d u c e s t l and t o ( F i g . 5) a s p r e d i c t e d by e q s . (15) and ( 1 6 ) .

Next c o n s i d e r t h e r a t i o of t h e r a t e of i n s i t u heave and r a t e of

heave by water i n t a k e t o t h e t o t a l heave r a t e based on e q . (14). The

r a t i o of t h e r a t e of w a t e r i n t a k e t o t o t a l heave i s a s f o l l o w s

I f t h e o b s e r v a t i o n can be continued f o r a long time t h e f o l l o w i n g r e s u l t i s o b t a i n e d

Takao Ueda and Edward Penner. 62 C a l i m dhll/dt l i m 3 - -

z

t + - d i d t - t+- (l - -C2 P ) = l (18) C 1 e

JT

t h a t i s a f t e r long p e r i o d s of time when f r o s t p e n e t r a t i o n h a s stopped

I

a l l t h e heave ( t o t a l heave) i s by w a t e r i n t a k e . S i m i l a r l y , t h e r a t i o of t h e r a t e of i n s i t u heave t o t h e t o t a l heave r a t e can be g i v e n a s f o l l o w s C a 3 dhi/dt

5-K

--

= d h T / d t -C2 P C e 1

fi

l i m d h i / d t -- - - 0 t+m d h T / d t (20)

which means no i n s i t u heave t a k e s p l a c e a f t e r t h e f r o s t p e n e t r a t i o n h a s s t o p p e d .

Summarizing t h e r e s u l t s a s g i v e n by e q s . (18) and (20) means t h a t a f t e r s u f f i c i e n t time h a s e l a p s e d and t h e p e n e t r a t i o n of t h e f r o s t l i n e h a s h a l t e d t h e t o t a l heave would t a k e p l a c e completely by w a t e r i n t a k e from t h e o u t s i d e r e s e r v o i r .

ACKNOWLEDGEMENT

The analogy which forms t h e b a s i s of t h e paper was s u g g e s t e d by M r . Tom Walton o f t h e N a t i o n a l Energy Board and i s g r a t e f u l l y acknowledged by t h e a u t h o r s . This p a p e r i s a c o n t r i b u t i o n from t h e D i v i s i o n of B u i l d i n g Research, N a t i o n a l Research Council of Canada, and i s p u b l i s h e d w i t h t h e approval of t h e D i r e c t o r of t h e D i v i s i o n . R3FERENCES

Penner, Edward and Ueda, Takao, 1977. The dependence of f r o s t heaving on l o a d a p p l i c a t i o n -- p r e l i m i n a r y r e s u l t s . I n t e r n a t . Synp. F r o s t A c t i o n i n S o i l s , Univ. o f ~ u l e s , Sweden (Feb. 1977).

L i n n e l l , K . A . and K a p l a r , C.W., 1959. The f a c t o r of s o i l and m a t e r i a l type i n f r o s t a c t i o n . Highway Res. Board B u l l . 225, 1959, p. 81-128.

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FROST P E N E T R A T I O N RATE ( 5 ) L l Q U l D I N C Y L I N D E R REPRESENTS I N S I T U W A T E R ( 6 ) R E S E R V O I R L l Q U l D REPRESENTS " I N F L O W " O R " E X P U L S I O N " WATER N O T E S - ( 1 ) I F P I S T O N M O V E D Q U I C K L Y T O R I G H T , L l Q U l D I S P U S H E D O U T O F C Y L I N D E R T O R E S E R V O I R ( 2 ) I F P I S T O N M O V E D S L O W L Y T O R I G H T , RESERVOIR S U P P L I E S L l Q U l D T O C Y L I N D E R

m

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T I M E F I G U R E 1 M E C H A N I C A L A N A L O G Y O F C O N S T A N T H E A V I N G R A T E E R 5 6 1 5 - I

P R E S S U R E , k g l c r n L

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T O T A L H E A V E R A T E V S P R E S S U R E , PENNER 8 U E D A ( 1 9 7 7 )

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B A S E D O N S E P A R A T E E X P E R I M E N T S TESTS N O S . 1 , 3 , 5 , 7 , 8 , 9 B A S E D O N O N E E X P E R I M E N T W I T H C O L D S l D E T R E D U C E D D A I L Y

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I I F I G U R E 3 T O T A L H E A V E R A T E V S

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I N F L U E N C E O F C O L D S I D E T E M P E R A T U R E O N t i A N D t o F O R L E D A C L A Y

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