Yield point in the semi-infinite solid in case of local loading

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Title:

NATIONAL RESEARCH COUNCIL O F CANADA TechnlcaP Translation TT-134

Yield point In the semi-inf f n i t e sol-Id In case of local loadlng.

( P l i e s s g ~ e n z e bef Ortllcher Belastung des Halbraumes ) ,

Author : H , Hruban.

Reference: Publicat Ion of the International A s s n c i a t Ion for Bridge and Structl~ral E(~yineerlnp: .;,

7,:

179- 214, 1943-44.

The N a t i o n a l R e s e a r c h C o u n c i l , t h r o u g h i t s D i v i s i o n o f B u i l d i n g R e s e a r c h , i s p l e a s e d t o h a v e b e e n a b l e t o a r r a n g e f o r t h e p u b l i c a t i o n o f t h i s t r a n s l a t i o n o f a n i m p o r t a n t p a p e r i n t h e f i e l d o f c i v i l e n g i n e e r i n g , The s i g n i f i c a n c e o f t h e p a p e r i s i n d i c a t e d by t h e a t t e n t i o n g i v e n t o i t by P r o f e s s o r

I,

F,

M o r r i s o n , o f t h e U n i v e r s i t y o f A l b e r t a , The p u b l i c a t i o n o f t h i s t r a n s l a t i o n r e p r e - s e n t s one way i n which t h e D i v i s i o n o f B u i l d i n g R e s e a r c h h o p e s t o work w i t h t h e e n g i n e e r i n g and a r c h i t e c t u r a l d e p a r t m e n t s o f C a n a d i a n u n i v e r s i t i e s i n t h e c o - o p e r a t i v e development o f b u i l d i n g r e s e a r c h i n Canada, March, 1950 R , F o L e g g e t , D i r e c t o r

T r a n s l a t i o n of P a r t of Paper on Y i e l d P o i n t i n t h e S e m i - I n f i n f t e S o l f d i n Case of Local Loadfng

by D r ,

KO

Hruban

Vol, 7, 1943-44

Proceedings I n t e r n a t i o n a l A s s o c i a t i o n f o r Bridge and S t r u c t u r a l Engineerfng, Summary

There a r e b u i l d i n g s which, i n t h e c o u r s e of c e n t u r i e s , a r e c o n t i n u a l l y s i n k i n g more a n d more i n t o t h e s o i l , T h i s phenomenon cannot be a t t r i b u t e d s o l e l y t o t h e c o n s o l i d a t i o n of c o h e s i v e l a y e r s ;

i t a r i s e s r a t h e r from a d i s t u r b a n c e of e q u i l f b r i u m f n t h e mass of e a r t h u n d e r n e a t h t h e b u i l d f n g , a d f s t u r b a n c e which must n o t be con- f u s e d wPth r u p t u r e , I n o r d e r t o I n v e s t i g a t e t h i s phenomenon of p l a s t f c deformatf on, t h e s t r e s s c o n d i t f o n s a r e determfned i n t h e e l a s t i c - f s o t r o p f c s e m i - I n f i n i t e s o l f d whose s u r f a c e f s l o a d e d w i t h a r i g i d body, Rigorous s o l u t i o n s can be o b t a i n e d f o r t h e two- dimensional problem wfth c e n t r a l and a l s o e c c e n t r i c t r a n s m f s s i o n of f o r c e , Based on a simple form of t h e y f e l d condf t i on of t h e l f m f t e d s h e a r i n g energy, t h e magnitude of t h e l o a d f u found a t t h e

s t a r t of permanent p l a s t f c d f splacements f o r l o a d i n g on a s t r i p (eq, 1 5 ) and on a c i r c u l a r a r e a ( m 3 ) , T h i s l i m f t e d l o a d i n g f s found t o be much s m a l l e r t h a n t h e r u p t u r f n g l o a d a s c a l c u l a t e d I n accordance wfth t h e asswnptfons h i t h e r t o made, The Paw of deforma- % I o n of e a r t h masses does not a l l o w t h e t h e o r y of e l a s t i c i t y t o be a p p l i e d d i r e c t l y t o f o u n d a t i o n l a y e r s ; n e v e r t h e l e s s , t h e c o n d i t i o n s of compatfbflf t y Pead t o conePusions whi ch make It p o s s f b l e t o

form a judgment of t h e e q u f l f b r i u m i n l o c a l l y l o a d e d l a y e r s of e a r t h o From t h a t , Pfmitfng v a l u e s of t h e p r e s s u r e on t h e s o f h a n be found i n c a s e of a l o a d e d s t r f p (eq, 27, 28) and a l o a d e d c f r e l e (eq, 29,30), where t h e c o n d i t i o n i s f u l f f l l e d t h a t no c o n t f n u a l s e t t l e m e n t s h a l l

o c c u r o The I n f l u e n c e of t h e d e p t h of t h e f o u n d a t i o n , of t h e w i d t h of s l a b s , and of t h e c o h e s i o n a r e f l l u s t r a t e d by some examples a n d numerical t a b l e s , The f f r s t p a r t of t h e paper p r e s e n t s t h e t h e o r y of s t r e s s d i s t r f b u t i o n f n t h e semf- i n f i n i t e s o l i d due t o l o c a l l o a d i n g , S e e t i o n 7 i n t r o d u c e s s 7 , The Flow C o n d i t i o n

Very l i t t l e f s known up t o now r e g a r d f n g t h e behavfour of v a r i o u s m a t e r i a l s a t t h e f l o w - l f m i t ( y f e l d - p o f n t ) w i t h t r f - a i x i a l p r e s s u r e wfth p r i n c f p a l s t r e s s e s of d f f f e r e n t magnftudes, Most of t h e compression a p p a r a t u s e s hf t h e r t o desfgned, can i n v e s t f g a t e

merely e i t h e r a plane deformation condition or an a x i a l symmetrfc s t r e s s condition, The researches of recent y e a r s on t h e mechanics of p l a s t i c behaviour of mild s t e e l , show, however, t h e i n c o r r e c t - ness of t h e Mohr hypothesis according t o which t h e mean p r i n c i p a l

s t r e s s e x e r t s no influence on the yield-point,

The same conclusions a r i s e from researches which Kjellman I has c a r r i e d out with sand i n a s p e c i a l l y constructed compression apparatus,

If one revfews previous work, one a r r f v e s a t t h e idea, f i r s t expressed by Ruber, and l a t e r more e x a c t l y formulated by s e v e r a l

o t h e r r e s e a r c h workers, t h a t the a b i l i t y of m a t e r i a l s t o behave e l a s t i c a l l y i s l i m i t e d by the f a c t t h a t t h e produced work of de- f o m a t f o n cannot exceed a l i m i t f n g value, Thfs ffrnitfng value appears t o be dependent on t h e accupulated energy I n the m a t e r i a l throughout i t s h i s t o r y o The deformation work r e f e r r e d t o f s pro- p o r t i o n a l t o the sum o f t h e squares of t h e p r i n c i p a l s t r e s s e s ; the accumulated energy i s a f u n c t i o n of t h e p r f n c i p a l normal

s t r e s s e s , i n the components of which, however, the workfng s t r e s s e s of t h e whole of t h e s o l i d substance of the mass a r e t o be included

( t h u s taking i n t o account the i n t r i n s i c pressure and secondary s t r e s s e s ) , I f we designate these t o t a l p r i n c i p a l s t r e s s e s by

6

I~

a g ~ .

6 1 1 1 9 wherein a g a i n 61 s f g n f f f e s the smallest and

6

_{111 t e l a r g e s t pressure s t r e s s , then t h e flow conditf on can}

i n g e n e r a l be wrf t t e n down i n the form

For p r a c t f c a l computations, a simple form of the f u n c t i o n f

w i l l be chosen and t h e constants e n t e r i n g i n t o f t w i l l be so

determined t h a t t h e y correspond a s w e l l a s possfble wfth experience i n t h e chosen s t r e s s range,

We w i l l be content here wfth the sfmplest form of t h i s f u n c t f o n which leads, with plane deformatfon conditions, t o l f n e a r con-

dftf on equatf ons, We assume t h a t i n t h e loaded materfal, b e s i d e s the s t r e s s e s according t o Sectfons 4 and 6,3 only a h y d r o s t a t i c s t r e s s condlti on p r e v a i l s with all-around p r e s ~ u r e 6 ~ , whi ch i s

1

W,

Kjellman, Proc, of I n t e r , Conf, on S o i l Mechanics, 1936, Vol, 11, page 16,

*

_{Equations i n the t r a n s l a t i o n have been renumbered.}

f These s e c t i o n s precede the t r a n s l a t e d s e c t i o n s i n t h e o r i g i n a l t e x t ,

produced by a combfnation of molecular f o r c e s , previous loading, f n t r i n s f c s t r e s s e s and secondary i n f l u e n c e s , The p r i n c i p a l s t r e s s e s of t h e e l a s t i c c o n d i t i o n s caused by the l o a d i n g wf

lP

.,

Then t h e t o t a l p r i n c i p a l s t r e s s e s i n the amount to8

The d f f f e r e n c e s on t h e l e f t s i d e of e q u a t i o n ( 1 ) (26 i n the paper) a r e (

cI

-

o I I ) =

6

-

629

e t c . and t h e right-hand s i d e becomes a f u n c t f o n of

60s

e f s

,52s CY3. We assume t h a t the r o o t of t h f s f u n c t i o n can be developed i n a power s e r i e s accord-

i n g t o t h e average p r i n c i p a l s t r e s s

6

1

+

6 2

+

5 3 9 of which we

3

r e t a f n only t h e f i r s t two terms, The r e s u l t can be w r f t t e n a s followsg

T h i s i s one simple form of t h e flow c o n d i t i o n f o r i s o t r o p i c m a t e r i a l s , which s t a n d s i n agreement with t h e p r i n c i p l e of l f mfted work of deformation, The c o e f f i c i e n t k and t h e s t r e s s 6 s f g n f f y m a t e r i a l c o n s t a n t s ; k i s conditfoned by t h e i n n e r f r i c t i o n and 0

by t h e i n t e r n a l s t r e s s e s , Three groups of m a t e r i a l s can be df s- t f n g u f s h e d according t o & i c h of t h e s e two v a l u e s i n f l u e n c e t h e p r o p e r t i e s of t h e m a t e r i a l t h e most,

Group I M a t e r i a l s without i n t e r n a l f r i c t i o n , w i t h whfch 6 r e l a t i v e t o

6 z S

and 6 3 a r e very l a r g e and k v e r y s m a l l a s f o r example wf t h metals, (For rnfld s t e e l , i f 6 be t a k e n

equal t o 320,000 kg/cm2, k f s t h e n z s 0,0075) I n t h i s group a l s o belong w a t e r - s a t u r a t e d c l a y s s u b j e c t t o qufck foadfng, The r f g h t

s f d e of r e l a t f o n ( 2 ) can t h u s be assumed a s c o n s t a n t and t h f s transforms f t s e l f i n t o t h e Ruber-von

MI

ses-Hencky flow condf ti on f o r s t e e l ,

5, t h u s s i g n i f i e s t h e f l o w - l i m i t ( y i e l d - p o i n t ) i n t h e s i m p l i - f i e d s t r e s s - s t r a i n diagram w i t h pure t e n s i o n s t r e s s ( s e e

Figure

l),

The same numerical value i s o b t a i n e d from ( 3 ) f o r u n i q x f a l oompressive s t r e s s ,

(b) I n t h e case of p l a i n deformation

-

61

+

6 3 d 2

-

m

I n t h e i n i t i a l s t a g e s of t h e flow process w i t h most m a t e r i a l s

m

= 2 and t h e e q u a t i o n ( 3 ) g i v e s t h e n t h e flow c o n d i t i o n

The s t r e s s d i f f e r e n c e can t h e r e f o r e be h i g h e r i n t h e r a t i o of 2 %

6

=

1,15 t h a n w i t h pure t e n s i o n s t r e s s , Thus t h e r e a r i s e s a n upper and a lower y i e l d p o i n t , which a l s o a g r e e s wf t h observation, 2

( c ) We consider a l s o t h e following axial-symmetric s t r e s s con- d i t i o n , which a r i s e s with t h e loading of t h e half-space through a r i g i d c y l i n d r i c a l d i e ; i n t h e d i r e c t i o n of t h e

Z-axis t h e l a r g e s t p r e s s u r e 6 a c t s , t h e two o t h e r p r i n c i p a l s t r e s s e s a r e e q u a l p r e s s u r e s 0 =

a2.

The e q u a t i o n ( 3 )

-

y i e l d s f o r t h i s case 6

-

o 3

-

6' o ~ o ~ ~ o o o o o o ~ o o . Q o (3c) o o o The s t r e s s d i f f e r e n c e i s t h u s t h e same with u n i a x i a l l o a d i n g , Group 11, F i l l s ( o r d e p o s i t s ) with 6

=

O o I n t h e flow c o n d i t i o n

( 2 ) t h e r e remains only t h e c o n s t a n t k, Since here only comDres- i f & s t r e s s e s (with n e g a t i v e s i g n ) can occur, we writelnow t h e flow c o n d f t f o n ( 2 ) a s f o l l o w s s

( a ) I n u n i a x i a l s t r e s s c o n d i t i o n t h e r e r e s u l t s

D 3

=

0

..

...

....

( 5 a ) T h i s m a t e r i a l possesses no compressiVe s t r e n g t h o

2

Refer t o

P o

Bi,jlaardp Theory of Local P l a s t i c Deformation, Proceedings I o A o B o & S , E , , Vole 6 page 27,

( b ) For the case of plane deformation, the flow c o n d i t i o n runs (with m

=

2 ) " 3

-

6l

-

k

. . . .

O O O O .. . b . O O O . O O O O O O (5b) 3

-

0

\f3

6 1

+

6 3

T h i s i s t h e known equation of c l a s s i c a l e a r t h pressure theory,

i f t h e f r i c t i o n angle i s i n d i c a t e d by

4 k

p),

= a r c s i n

-

.

6

( c ) For t h e a x i a l symmetric loading, we must d i s c r i m i n a t e here between two, cases,

1) The a x l a 1 pressure C 3 i s l a r g e r than t h e mantle pressure

2 6 1

+

bg 61 + a 3

G1 3 620 One n o t i c e s t h a t

3

=

z

so one g e t s t h e flow condition i n t h e form

There appears i n t h i s case another f r i c t i o n angle

9

2

3k

=

a r c s i n +

2 ) The a x i a l pressure

el

i s smaller

t

han the mantle pressure

-

G 2

-

a3"

S u b s t i t u t i o n i n (4) l e a d s t o t he flow condition, 6 3

-

61 3k and t h e f r i c t i o n angle

6 1

+

6 3

= -

a t t a i n s t h e value

43

=

a r c s i n 3k o O O O o O O o O O O O o O ( 5 6 ) 6 - k 3 O l + 6 3 = This a r i s e s from O2

=

+

'33

m

2 4

The theory of e a r t h pressure a g a i n s t r e t a i n i n g w a l l s comes under t h i s s t r e s s conditiono

I f t h e assumed general flow condition ( 2 ) corresponds t o t h e a c t u a l behaviour of t h e m a t e r i a l , d i f f e r e n t magnitudes of t h e f r i c t i on an319 must t h e r e f ore appear with d i f f e r e n t s t r e s s conditions, There r e s u l t s , f o r example, f o r k

=

1 t h e values

42

= 25O20', $ 3 = 37O; t h e f r i c t i o n - a n g l e w i t h plane defor- mation

4

= 35O201 l i e s between these. 5

The measurements of Kjellman mentioned have a c t u a l l y ₁ given, w i t h h i s t e s t sand, a smaller f r i c t i o n angle f o r case t h a n f o r case2, 35O h s compared with 43O; t h e d i f f e r e n c e , however, i s not a s l a r g e a s t h a t given by equations ( 5 c ) and

(5b) 0

6

Various research measurements a r e comparable with flow conditions (5b) and ( 5 c ) ; these show t h a t t h e r e l a t i o n between the two values (

o3

-

61) rn d (

a l

+

g 3 ) i s not l i n e a r but can be represented by a s l i g h t l y curved l i n e , We must t h e r e f o r e

l i m i t the usefhlness of t h e s i m p l i f i e d flow condition ( 2 ) t o a s t r e s s region i n which t h e enveiope of t h e Mohr's c i r c l e s can be s u b s t i t u t e d by t h e s t r a i g h t l i n e of Coulomb, (Figure 2,)

5

It would appear from t h i s , t h a t i f the f r i c t i o n angle obtain-d from a t r i a x i a l t e s t i s t o be a p p l i e d t o r e t a i n i n g w a l l e a r t h pressure theory the value of k should f i r s t be obtained from

*2, and then derived from

it.

For examples suppose a t r i a x i a l t e s t y i e l d s a value of

$

= 31°20V, then we have

6

Actually k works out t o be 1,4 and 1.2 i n t h e s e cases, More experimental work by means of t h e t r i a x i a l t e s t i s d e s i r a b l e t o i n v e s t i g a t e these r e l a t i o n s and t h u s s u b s t a n t i a t e o r

condemn t h e theory.

For example, W. Bernatzik, "Researches regarding the s t r e n g t h p r o p e r t i e s of sand i p t r i a x i a l s t r e s s conditions", Water

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

I n any case, t h e f a c t s can be e x p l a i n e d through t h e assumption of t h e t h e o r y of t h e l i m i t e d work of deformation, t h a t with t h e use of varioub measuring d e v i c e s d i f f e r e n t magnitudes of t h e f r i c t i o n angle can be e s t a b l i s h e d with t h e

same material'; a c c o r d i n g l y t h e angle of i n t e r n a l f r i c t i o n must be a v a r i a b l e , d e ~ e n d e n t on. t h e magnitude of t h e value

of t h e mean p r i n c i p a l s t r e s s , and there-should be d i s c r i m i n a - t i o n between t h e value

6 .

(with plane deformation c o n d i t i o n )

I A -

and

(I

( i n t h e case c ,.

51

= 3 2 ) o

Group 111, M a t e r i a l s with ~ o h e s i o n and i n t e r n a l f r i c t i o n , The flow c o n d i t i o n ( 2 ) remains w i t h b o t h c o n s t a n t s k and

o

i n f o r c e . It follows, f o r t h e u n i a x i a l s t r e s s condition,- t h a t t h e flow l i m i t i n pure p r e s s u r e l i e s h i g h e r t h a n w i t h pure t e n s i o n , For t r i a x i a l p r e s s u r e s t r e s s c o n d i t i o n s , t h e flow c o n d i t i o n , f o r a l l t h r e e of t h e previous m a t e r i a l groups considered, can be brought t o t h e form

O 3

-

"1 _{= s i n}

...

o . o o o o o o ( 6 )

5 - 1 + 5 3 + 2 b T

The f r i c t i o n a n g l e

.$

h a s again, i n each of t h e s e cases, a d i f f e r e n t value

6

l,

6

,

:l3 and indeed t h e same a s i n t h e

' .*

e q u a t i o n s 5b, 5c, 5d. I n o t h e r r e s p e c t s fhe flow c o n d i t i o n ( 6 ) i s i d e n t i c a l w f t h t h e known Coulomb e q u i l i b r i u m c o n d i t i o n f o r cohesive m a t e r i a l ; t h e cohesion, i n t h e uoulomb sense,

_-

i s g i v e n by OIL! = c =

-

2 t a n

4

.

A s w i t h t h e previous m a t e r i a l groups,

one must a g a i n l i m i t t h e v a l i d i t y of c o n d i t i o n ( 6 ) o n l y t o a d e t e r m i n a t i o n r e g i o n , f o r example

MN

i n Figure 2, i n which t h e curved envelope l i n e of t h e p r i n L i p a l s t r e s s c i r c l e s can be r e p l a c e d by a s t r a i g h t l i n e , Tha magnitude 6 i s dependent

on t h e choice of t h i s r e g i o n ; i t s i g n i f i e s a p r e s s u r e a t r e a s a n d i s t o be put i n t o e q u a t i o n ( 6 ) with a n e g a t i v e s i g n , a s with t h e

p r e s s u r e s t r e s s e s . F 1 and 630 The value O, h e r e no longer s i g n i I f e s t h e flow-limft ( y i e l d - p - l i n t ) with pure unfaxfaP p r e s -

s u r e s t r o s s ; If 6

=

0, t h e magnit,~de conforms t o

1

~ s , = 2 s i n

5-

=-

4\

,

i n d t h u s

obviously according t o t h e value of t h e f r i c t i o n angle and t h e r e w i t h a l s o according t o t h e mean p r i n c i p a l s t r e s s

_s

_{2 0}

The p r e s s u r e s t r e n g t h w i l l be, w i t h p a r t i q l l y h i n d e r e d expansi on, l a r g e r t h a n w i t h pure p r e s s u r e s t r e s s ,

80 The Loadin% a t t h e Limit of t h e E l a s t i c S t a t e ,

We w i l l now, w i t h t h e h e l p of t h e r e l a t i o n s h i p o b t a i n e d i n t h e preceding s e c t i o n , seek t o determine t h e i n t e n s f t ; ~ of t h e s t r i p loading which l i e s a t t h e l i m i t between predominantly e l a s t i c and predominantly p l a s t i c behavfour of t h e s t r e s s e d half-spaceo Ip order t o o b t a i n a p r e c i se d e f i n i t f on of t h i s l i m i t loading, consider f i r s t t h e f o l l o w i n g simple c a s e ; t h e m a t e r f a l of t h e half-space i s e l a s t i c - i s o t r o p i c and belongs t o t h e m a t e r i a l of Group I of t h e preceding s e c t i o n , The flow c o n d i t i on i s , t h e r e f o r e , given, f o r t h e plane deformatf on con- d i t i on by e q u a t i o n ( 3 b ) , Thi

s

means t h a t t h e d i f f e r e n c e of

t h e p r i n c i p a l s t r e s s e s s h a l l not exceed a c o n s t a n t value 2

&

-

;

6*1i s h e r e t h e y i e l d p o i n t i n pure u n i a x i a l t e n s i o n s t r e s s t 3

The upper s u r f a c e of t h e h a l f - s p a c e on b o t h s f d e s of t h e r i g i d - p l a t e s t r i p i s not loaded, and t h e s p e c i f i c w e i g h t of t h e

m a t e r f a l w i l l not be taken i n t o c o n s i d e r a t i o n ,

Then t h e p r i n c i p a l s t r e s s e s w i l l be determfned by t h e formula (19) and t h e i r d i f f e r e n c e i s

I f one p l o t s t h e r e s p e c t i v e v a l u e of ( g l

-

5 3) a t each pofnt of t h e XZ-plane p e r p e n d i c u l a r t o t h e plane of t h e diagram

(Figure 31, one o b t a i n s a s u r f a c e whose contour l i n e s a r e r e - p r e s e n t e d a t p r o j e c t i o n by f sochromes, Thf s s u r f a c e c u t s t h e p i c t u r e plane i n the X-axis, ascends along t h e Z-axis t o over t h e p o i n t C, vhich l i e s a t t h e d f s t a n c e a

f i

from t h e l o a d

s u r f a c e and s i n k s a g a i n g r a d u a l l y with i n c r e a s i n g z - o r d i n a t e of t h e p o i n t s consideredo

The formula (19 i n t h e paper) i s t h e u s u a l w e l l known formula f o r t h e p r i n c i p a l s t r e s s e s i n t h e half-apace due t o a s t r i p - l o a d i n g o R i s t h e d i s t a n c e of t h e p o i n t from t h e c e n t r e of t h e loaded s t r i p , z t h e d i s t a n c e below t h e h o r i z o n t a l s u r f a c e ,

r t h e r a d i u s of t h e c i r c l e passing through t h e edges .f t h e s t r i p and t h e p o i n t under consfderatfon, q t h e t o t a l l o a d on t h e s t r i p per u n i t of l e n g t h of s t r i p o

From p o i n t C i t forms two r i s i n g r i d g e s mounting towards t h e edges of t h e l o a d s u r f a c e , which become i n f i n i t e l y high a t an i n f i n i t e s i m a l d i s t a n c e from t h e edges, With t h e s m a l l e s t loadfng,

the'=

a u s t , t h e r e f o r e , a l r e a d y be p l a s t i c d e f o r n a t i o n a t t h e edge r e g i o n , and indeed n o t o n l y f n t h e s t r e s s e d h a l f -

apace b u t a l s o i n t h a t of t h e l o a d - t r a n s m i t t i n g body, I n Figure 3, t h a t r e g i o n i s d e s i g n a t e d by c r o s s h a t c h i n g , i n whfch t h e p r i n c i p a l s t r e s s d i f f e r e n c e exceeds t h e y i e l d p o i n t 2

6-

T;T 2 6, if t h e c r o s s - s e c t i o n loading amounts t o = o 2a 0070 q 6

Thereby a change i n t h e s t r e s s c o n d i t i o n w i l l be induced, The a n a l y s f s of t h i s c o n d i t i o n belongs t o t h e s t a t i c a l l y indetepsafnate task of p l a s t i c i t y mechanics, f o r i t f s dependent on t h e

p l a s t i c displacement a t t h e edge of t h e body and i n t h e h a l f - space, and d e f i e s a t p r e s e n t a r i g o r o u s t h e o r e t i c a l t r e a t m e n t , One can, n e v e r t h e l e s s , o b t a i n a n approximate value f o r t h e edge p r e s s u r e

_{e A ,}

by assuming a s known t h e p a t h s of t h e f s o -

s t a t i c c u r v e s ( and t h e r e w i t h t h e s l i p l i n e s ) i n t h e immediate neighbourhood of t h e edge p o i n t ,

( a ) The Rankine e q u i l i b r i u m c o n d i t i o n ,

The d i r e c t i o n s of t h e p r i n c i p a l s t r e s s e s a r e p a r a l l e l t o t h e X and

Z

axes, To t h e r i g h t from edge p o i n t A t h e flow c o n d i t i o n w i t h 6

=

0 p r e v a i l s ; t o t h e l e f t from i t t h e flow c o n d i t i o n w i t h t h e l a r g e s t p r e s s u r e s t r e s s

a d ,

The e q u i l i b r i u m immediately below t h e edge p o i n t r e q u f r e s t h a t on both s i d e s t h e same

a,

occurs; t h e flow c o n d i t i o n (36)

(b) The Boussinesq-Resal e q u i l i b r i u m c o n d i t i o n ,

It w i l l be assumed t h a t a sheaf of s l i p l i n e s w i l l be formed by r a y s whfch go o u t from t h e edge p o i n t , T h i s assumption

"

g i v e s w i t h t h e flow c o n d i t i o n (3b)

9

S , A , Caquot, "Equilib d e s Massif a Frottement I n t e r n w , P a r i s 193a0 Po 5g0

The a c t u a l edge s t r e s s probably w l l l be n e a r the value ( 8 a ) and t h e r e s u l t a n t change i n p r e s s u r e d f s t r f b u t f o n on t h e loaded sur-

f a c e can be r e p r e s e n t e d perhaps by t h e broken l f ne i n Ffgure 3, The s t r e s s pofnt f s rounded o f f and d i s p l a c e d towards t h e middle of t h e l o a d s t r i p , That t h e s t r e s s c o n d i t f o n of t h e h a l f - s p a c e f s changed and t h e r e b y a l s o t h e l i m f t of the p l a s t f c r e g i o n i a evf dent from a comparf son of t h e s t r e s s o p t f e photograph i n Ffgure 4 w i t h t h e t h e o r e t i c a l course of t h e fsochromes i n t h e edge region, T h i s s t r e s s rearrangement remains confined, however, according t o t h e p r i n c i p l e of S t , Venant, i n the fmmedfate

nefghbourhood of t h e d l sturbanee c e n t r e , I n t h e p l a s t f c region, t h e cross-extenaion number, f o e o P o f s s o n ~ a number, s i n k s

t o

t h e value m = 2 and t h e d e r i v e d c u b i c compression becomes n i l , A correspondfng p o r t f o n of t h e m a t e r f a l whfch has become p l a s t f c must t h e r e f o r e be squeezed out sfdewfse from t h e edge, The edge

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

t h i s body f s h a r d e r than t h a t of t h e h a l f -space, One can c l e a r l y observe t h i s , f o r example, wfth t h e t e s t loadfng of cohesfve s o i l ,

The displacement of t h e r f g f d loading-body f s gfven, t h e r e - f o r e , i n t h f s s t a g e of loadfng process, by t h e e l a s t f c eompression I n t h e Z-axis; t h e r e f o r e continued s i n k i n g can not occur a s long a s t h e flow r e g i o n remafns l f m f t e d t o t h e edge region.

If t h e l o a d now be Increased, t h e n t h e two r e g i o n s f n whfch t h e c r i t i c a l s t r e s s d i f f e r e n c e w i l l o v e r s t e p the e l a s t f c con- d f t i o n grow u n t i l f i n a l l y , wfth t h e value of t h e p r i n c f p a l s h e a r s t r e s s given by (18) 1 0

,

t h e y reach t h e pofnt C a t t h e 2-axfs, The mean loadfng a t t h f s i n s t a n t amounts t o

2 60 3

*&

%

= 8 = 2,356 6, 0 0 0 0 0 0 0 0 . 0 0 G 0 ~ 0 (91

6

$ 1 The course of t h e isochrome (

c l

-

83)

=

0.490

-*

gofng through C I s r e p r e s e n t e d i n F f g w e 30 I n t h e e l a s t f c

c o n d f t i o n t h e c r f t f c a l shear s t r e s s was exceeded i n t h e whole regf on e n c l o s e d between t h e two branches of t h i s curveo The formatfon of t h e p l a s t f c r e g i o n caused I n t h f s way, whose depth now a t t a i n s t h e o r d e r of magnftude of h a l f t h e s t r f p wf dth, c a l l s f o r t h a f u r t h e r rearrangement of t h e p r e s s u r e df s t r f b u t f on,

I U

Expression (18) 5. s f o r t h e maximum mean s t n e s s d i f f e r e n c e on t h e Z-axis under a s t r i p loading, It I s

max ( O X

-

= z ) = 2

If t h e l o a d be i n c r e a s e d f u r t h e r , t h e n a connected p l a s t f c r e g i o n forms, somewhat according t o Ffgure 5 (12) which envblops t h e remaining e l a s t i c core under t h e l o a d s u r f a c e and s e p a r a t e s

i t

from t h e o u t e r remaining e l a s t i c regfon of t h e half-space, A t t h i s s t a g e of t h e loading process t h e m a t e r f a l can, however,

s t i l l not s l i d e s i n c e t h e s l i p l i n e s G cannot y e t develop, The m a t e r i a l which h a s become p l a s t f c with t h e Pofsson number rn = 2 must, however, due t o i t s constancy of volume, be squeezed out

near t h e p l a t e edges and t h e l o a d body s i n k s w i t h t h e e l a s t i c core, whereby the d i s t a n c e

ON

becomes s m a l l e r , Consequently, t h e s t r e s s a t N i n c r e a s e s , a f u r t h e r t h f n l a y e r of t h e m a t e r f a l becomes a f f e c t e d through the flow p r o c e s s and t h e s i n k i n g pro-

ceeds with s m a l l e r v e l o c i t y , The continued s e t t l e m e n t can t h e n f f r s t cease f f t h e wefght of t h e d i s p l a c e d m a t e r f a l a g a i n e s - t a b l i s h e s t h e e l a s t f c c o n d i t i o n under t h e middle of t h e p l a t e , On t h e b a s f s of t h e s e c o n s f d e r a t f o n s , t h e r e q u i r e d l i m i t i n g l o a d can be d e f i n e d a s t h a t l o a d magnitude w i t h vh i c h t h e c r i t i c a l s t r e s s r e l a t i o n i s reached a t any p o i n t on t h e l i n e of a c t i o n of t h e a c t i n g f o r c e , The c r f t f c a l r e l a t i o n 1s understood t o be t h e f u n c t i o n of t h e p r l n c i p a l s t r e s s e s l i m f t e d by t h e flow condf t f on,

9, The Lfrnftfng Load w i t h Plane Deformation Conditfons,

Let u s c o n s i d e r f i r s t of a l l t h e g e n e r a l case, t h a t t h e upper s u r f a c e of t h e h a l f - s p a c e s u p p o r t s a uniformly d i s t r f - buted l o a d Po on b o t h s i d e s of t h e p l a t e - s t r i p a s i n Ffgure 6, The m a t e r i a l i s d e s f g n a t e d according t o ( 7 ) by t h e f r f c t f o n angle and t h e c o n s t a n t 6 = - c l c o t

8

10 We w i l l n o t , f o r t h e p r e s e n t , t a k e t h e s p e c f f f c wefght of t h e half-space i n t o account, The p r e s s u r e d i s t r i b u t i o n under t h e r i g i d - p l a t e s t r i p i s gf,ven by (25) l2 _{and t h e}

p r i n c i p a l s t r e s s e s under t h e c e n t r e of t h e p l a t e according t o (24) and (17)

s

'Po

=

- -

_

9

-

. -

a2

m - 1

n-

_{s 3}

These t h e o r e t i c a l formulae f o p t h e s t r e s s e s a r e worked o u t i n t h e f f r s t p a r t of t h e paper and a r e here r e f e r r e d t o by t h e i r numbers a s given i n t h a t p a r t ,

We work o u t t h e r e l a t i o n vhfch t h e l e f t s i d e of t h e f l o w c p n d f t i o n ( 6 ) forms w i t h v, One s u b s t i t u t e s t h e v a l u e s (10) and

$

= a r c

2

,

a n d o b t a i n s : 9 m-2 -2 a

?,

s i n Jcos2 13 v

(8)

=

x P . + ~

%a . O o O o O O O O O ( l l ~ m -2a

R

m e + '

$a

.

s i n 6

+

2c, c o t v l T h i s r e l a t i o n r e a c h e s a maximum w i t h

S m

of t h e a n g l e

2

.

which i s c o n d i t i o n e d by t h e e q u a t i o n a s i n Jm c o s 0 0 D O o O O O D 0 0 0 0 0 0 0 0 0 ( f 2 ) The maximum v a l u e of v must s a t i s f y t h e flow c o n d i t i o n ( 6 1

~ ( 6 , )

=

s i n

gl

.

.. ..

. .

...

.

..

(13) B y a l i m i n a t i o n of

d,

from e q u a t i o n s (12) and

(131,

one o b t a i n s t h e l i m i t i n g l o a d q a s a f u n c t i o n of t h e g i v e n value s o With t h e a b b r e v i a t i o n t h e r e r e s u l t s 13

The c o e f f i c i e n t s K1 a r e g i v e n i n Table I f o r v a r i o u s v a l u e s of t h e f r i c t i o n a n g l e o Table I P e r m i s s i b l e S o i l P r e s s u r e s f o r S t r f p Loadings C o e f f f c f e n t s of t h e Equations (15) and (27) From (12) and (13) f o l l o w s f u r t h e r P

-

s i n

4

1 s i n

8

=Ij

The o r d i n a t e

zm

of t h e p o i n t C, a t which t h e r e l a t i o n r e a c h e s f t s maximum, amountst-0 <.,.--..,-, , 2

+

s i n $J P P

-

s f n Q , l

The s m a l l e s t value of t h i s magnitude (with

9

=

0 ) P s

min

,

z

= a&

Example 1, A load

f

s t r a n s m i t t e d through a small r a i l t o a broad concrete block f a r from i t s edges, I f one s u b s t f t u t e s

t h e envelope curve according t o Figure 2, f o r whfch the c o n c r e t e of t h e block was determfned f o r plane deformation c o n d i t i o n s i n t h e p r e s s u r e r e g i o n by a s t r a i h t l i n e , t h e n t h i s f s d e t e r -

d

mined by t h e values Cl = 26 kg/cm = 4 l o O

The l i m i t i n g l o a d under t h e s t e e l r a i l f s, when t h e block i s otherwise unloaded, according t o (P5),

31r2)3 cos

@

1 2

.,-_ o1

=

397 kg/cm

The compressive s t r e n g t h of t h i s c o n c r e t e amounts t o about 1x0 kg/cm2 and t h e b r e a k i n g load, computed a c c o r d i n g t o Caquot,

5

2

9770 kg/crn

,

a c c o r d i n g t o R i t t e r 600 kglcm With a l o a d of 2

400 kg/cm

,

however, a c o n t i n u o u s g r a d u a l s i n k i n g of t h e r a i l i n t o t h e c o n c r e t e can r e s u l t *

10, The Limit Loading w i t h A x i a l Sgmmetric S t r e s s Condition,

The s t r e s s d l s t r i b u t i o n under a r i g i d c i r c u l a r d i e , whfch P i e s on t h e otherwi s e unloaded t o p s u r f a c e o f t h e h a l f - s p a c e ,

was determined b y Boussfnesq i n t h e y e a r 1885, I f a uniform l o a d i n g 'o i s a p p l f e d around t h e d i e a s i n F i g u r e 7 t h e r e r e s u l t s , wfth

t h e use of h i s s o l u t i o n and w i t h r e f e r e n c e t o t h e convenient c o n s i d e r a t i o n i n t h e d e r i v a t i o n of t h e e q u a t i o n (251, t h e f u n c t i on f o r t h e p r e s s u r e d i s t r i b u t f on a t t h e s o l e w i t h

( N o t a t i o n a c c o r d i n g t o Fig, 7 , )

The p r e s s u r e d i s t r f b u t f o n h a s a s i m f l a r c o u r s e a s w i t h t h e p l a n e deforrnati on condi ti ona ( s e e F i g u r e 31,

On t h e b a s i s of t h e assumed d e f i n i t i o n of t h e l i m i t

l o a d i n g , we a a t f s f y o u r s e l v e s w i t h t h e knowledge of t h e p r f n - efpaP s t r e s s e s i n t h e a x i s of symmetry, We i n t r o d u c e t h e e y l f n d r i c a l e o - o r d i n a t e s

r

and

z b

The w e l l known r e l a t i o n a from t h e e l a s t i e t h e o r y g i v e t h e f o l l o w i n g c o n t r i b u t i o n ' of

t h e l o a d f n g p'dk, of t h e whole c i r c l e r a d i u s

r,

t o t h e p r f n c f p a l s t ~ e s s e s a t p o i n t M(o,z)s

( N o t a t i o n a c c o r d i n g t o Fig, 7 ;

m

s i g n i f i e s P o f s s o n Q s number), Then we have t o c a r r y through t h e i n t e g r a t i o n from

I n a n e l e m e n t a r y f a s h i o n one f i n d s f r (- r d r ?,!a2- P 2 The s u b s t f t u t i o n of t h e l i m i t s , and t h e i n t e g r a t i o n r e s u l t s I n e q u a t i o n s (18) g f v e t h e s t r e s s components I n o r d e r t o o b t a i n t h e p r i n c i p a l s t r e s s e s

3'

_{a n d}

₆

_of 1

t h e complete l o a d i n g of 7, one h a s s t i l l t o add t h e 3 components a c c o r d f n g t o e q u a t f o n (24) 1 4

.,

If one f n d f c a t e s t h e h a l f opening a n g l e by c and t h e % e f t s i d e of t h e f l o w c o n d i t i o n ( 6 ) by v, one f f n d s 3.4 _Po These a r e

o

-

x

-

6 y = E ~ ~ 9

This r e l a t i o n s h i p reaches i t s maximum value i f t h e numerator of the expression (20) i s d i r e c t l y m u l t i p l i e d by the number

2 2

3m +

+

2 cos

J

-

2 s i n

J

m

The value of the opening angle thus conditioned w i l l be

rm.

Then t h e flow condition ( 6 ) i s

2 2

-

₊

_{~ ( C O S}

_{d m}

_-

_{s i n}

_S,)

2 * m 2

=

s i n

8

(20a) 2 3m +

+

2 ( c o s

Sm

-

s i n 6,)

m

(The s u b s c r i p t of t h e f r i c t i o n angle shows here t h a t t h i s angle

i s

t o be d i s t i n g u i s h e d from t h a t of t h e plane p r ~ b l e m ) ~ Out of (20a) follows

2 4 ( m

+

1)

+

(m + 2) s i n 5 2

cos

6

,

=

0 0 0 0 0 0 (21)

4m(3

-

s i n g 2 )

and t h e o r d i n a t e of t h e point of the Z-axis a t & i c h t h e flow l i m i t f s reached f s

.---

-...---

1)

+

(m

+

2) s i n s 2

'

m

₁₎

-

_(5m

₊

_{2) s i n} (21a)

9 2

I f one s u b s t f t u t e s i n Equation (20) the angle

d'

given by equation ( 2 1 ) , one o b t a i n s t h e maximum value ~ ( 8 , )

=

s i n * a s a f u n c t i o n of the loading and of t h e f r i c t i o n angle * 2 0

I n thf s way t h e l i m i t i n g load f s determinedo W t h the abbrevf a t i on f 4 ( 3

-

s i n

5

2 )

- -

-

'2

_(ern

_-1

-

_{s m}

_{+ 2} ~ 2

m

4m

I n s o i l mechanics t h e P o i s s o n number

rn

i s commonly t a k e n a s 2,

T h i s o r d i n a t e i s , w i t h a l l v a l u e s of 2, l a r g e r t h a n a , Example 2, A c i r c u l a r d i e i s p r e s s e d i n t o a m e t a l body of" c o n s i d e r a b l y l a r g e r s i z e ; t h e f l o w l i m i t of t h e m a t e r i a l f n pure t e n s i o n s t r e s s i s 6,

,

and t h e P o i s s o n number amounts t o 1 0

.

The upper s u r f a c e of t h e b l o c k 1s o t h e r w i s e n o t loaded,

Y 3

Po

=

00 Since

5

=

0 , i t f o l l o w s from (22) 2

,=I,($$)

= 4 0 1 6

The c r o s s - s e c t i on loading, w i t h which t h e overcoming of t h e continuous p e n e t r a t i o n of t h e df e i n t o t h e b l o c k b e g i n s , i s g i v e n a c c o r d i n g t o (23) w i t h

-

6 v 1 5

C 2 - - 0 2 *

With t h e v a l u e

m

= 2 one o b t a i n s , a c c o r d i n g t o (24) and ( 2 3 ) , 1 6

l=

% = T s

q a 2

=

2,676* a s compared with 2,36

6,,

f o r s t r i p l o a d a c c o r d i n g t o ( 9 ) o

l5

The p r i n c i p a l s h e a r s t r e s s i s one-half t h e d i f f e r e n c e between t h e p r i n c i p a l s t r e s s e s , I n pure t e n s f on one p r i n c i p a l s t r e s s i s 6,

,

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

11, A p p l i c a t i o n t o Foundation Ground T e s t i n g , i , e , S o i l Mechanicso

If we wish t o u t i l i z e t h e r e s u l t s of t h e preceding con- s i d e r a t i o n s i n foundation problems, then we must f i r s t of a l l i n v e s t i g a t e how f a r and i n which d i r e c t i o n t h e s t r e s s c o n d i t i o n of t h e pseudo-solid e a r t h mass d e p a r t s from t h a t of t h e e l a s t i c - i s o t r o p f c h a l f -space with c o n s t a n t e l a s t f c modulio We aim a % f i n d i n g a g a i n t h e loading a t the l i m i t of t h e q u a s i - e l a s t f e behavfour of t h e loaded s o f l stratum, t h a t i s , t h e h i g h e s t l o a d with rvhich t h e s e t t l e m e n t , a f t e r t h e e q u a l f z a t i o n of t h e hydro- dynamfc s t r e s s , s t i l l remains a t a c o n s t a n t f i n a l value, T h i s l i m f t loading i s a g a i n t o be d i s t i n g u i s h e d from t h e r u p t u r a l o a d whfch w i l l be s i g n i f i c a n t l y h i g h e r , because before t h e r u p t u r e , a n e x t e n s i v e r e g i o n around t h e l o a d s u r f a c e must g e t i n t o t h e p l a s t i c s t a t e , The quoted knowledge i n S e c t i o n I

shows, however, t h a t w i t h c o n s t r u c t i o n ground, t h e permf s s f b l e s o l e p r e s s u r e must not exceed t h e r e q u i r e d l i m i t l o a d i n g ;

continuous s e t t l e m e n t , which l e a d s t o c r a c k i n g of t h e construc- t i o n , a f t e r t h e course of a long time can a l s o then occur even

i f t h e s a f e t y f a c t o r r e l a t i v e t o t h e ground r u p t u r e i s s u f -

f i c i e n t l y l a r g e , To a b b r e v i a t e , we w i l l a l s o desfgnate a s

e l a s t i c t h a t c o n d i t i o n of t h e e a r t h mass w i t h which flow r e g i o n s have not yet reached t h e l i n e of a c t i o n of t h e r e s u l t a n t of t h e a c t i n g sole-pressure, although t o be sure t h e b u f l d f n g ground may not behave e l a s t i c a l l y ( f n t h e t r u e sense o f t h e word),

Nf t h t h e I n v e s t i g a t i o n of t h e e q u i l i b r i u m condf t i o n s i n cohesfve s o l 1 types, one h a s t o b e a r i n mind the circumstance t h a t , f o r t h a t purpose only, t h e s t r e s s e s a c t i n g on t h e s o l i d substance a r e e f f e c t i v e ; t h a t p a r t of t h e s t r e s s whfch i s t r a n s m i t t e d through t h e pore water i s t h u s t o be l e f t o u t of c o n s i d e r a t i o n ,

An e a r t h stratum, which i s s t r e s s e d merely through i t s s p e c i f i c weight and a uniformly d l s t r i b u t e d loadfng

?

over f t s e n t f r e upper s u r f a c e , f i n d s i t s e l f i n t h e equflibr?um con- dfition which 1 s desfgnated by t h e v e r t f c a l and h o r i z o n t a l

p r i n c i p a l s t r e s s e a 61=

-

Po

-

y z ,

6"

=

6?

=

xb,,

Here

y

( 2 5 )

s t a n d s f o r t h e s p e c i f i c welght, z t h e depth below t h e upper s u r f a c e , and 3c t h e e a r t h p r e s s u r e a t r e s t , which here r e p l a c e s t h e value 1 of t h e e l a s t f c m a t e r i a l , The Poisson number

m - 1

m i s v a r f a b l e w i t h e a r t h masses; i t i s , t o a l l appearances, a f u n c t i o n of t h e s t r e s s r a t i o v which forms t h e l e f t s i d e of t h e f l o w c o n d i t i o n (61, I f v approaches t h e v a l u e s i n $ t h e n m s i n k s t o 2 ; i n t h e f l o w c o n d i t i o n a volume i n c r e a s e even

occurs w i t h dense packed s o i l t y p e s ; t h a t i s t o s a y thereby

The c o e f f i c f e n t of e a r t h p r e s s u r e a t r e s t 'X must be dependent on t h e valuo of the f r i c t f on a n g l e , Thf s f o l l o w s from t h e c o n s i d e r a t i o n t h a t a n i n f i n 1 te l y extended h o r i z o n t a l e a r t h s t r a t u m cannot f a i l by s l i d i n g , i f

i t

be s t r e s s e d o n l y by I t s own weigFt, I n t h i s case, f o r a c o h e s i o n l e s s m a t e r i a l

G z

<

s i n

*.

.

_>

I.

-

s i n 4

6 2 + ijz 1

+

s i n

9

The v a l u e s of %

,

e s t a b l i s h e d s o f a r by measurement, a g r e e q u i t e w e l l w i t h t h e r e l a t i o n

% = I .

-

s i n

5

It g i v e s , f o r example, f o r sand w i t h

4

=

35O t h e coef- f i c i e n t of e a r t h - p r e s s u r e a t r e s t 0043, f o r f a t c l a y w i t h

9

= 16O t h e c o e f f i c i e n t 0.72, which a g r e e s w i t h e x p e r i e n c e , Thus, e q u a t i o n s (25) g i v e

-

s i n

dp

1

Now we have s t i l l t h e i n f l u e n c e of t h e s o l e p r e s s u r e t o determine, I n t h a t , i t i s n o t p e r m i s s i b l e t o proceed from t h e r e l a t i o n s h i p of t h e s t r e s s t o t h e deformation of an element of t h e mass, Boussinesq has a l r e a d y engaged himself w i t h t h i s q u e s t i o n o 16 Under t h e assumption of a completely loose p u l v e r f z e d mass he a r r i v e d a t a r e l a t i o n of t h e form

-

-

6,. g ' y

+

6 - 2 B x 3 (1

-

2 h L ' X )

...

, (26) Herefn

E

s i g n i f i e s t h e r e s p e c t i v e e x t e n s i o n and A a m a t e r f a l X

constant, The Boussfnesq mass i s volume c o n s t a n t , t h e r e f o r e only one c o n s t a n t appears i n t h e s t r a i n law, The a c t u a l

behavfour of t h e s o i l t y p e s correspond, however, i n no way t o t h e p r o p e r t i e s of t h i s i d e a l mass, a s r e c e n t measurements c l e a r l y show,

Figure 8 g i v e s t h e s t r e s s - s t r a i n diagram f o r a sand c y l i n d e r which was p r e s s u r e s t r e s s e d under a c o n s t a n t l y h e l d mantle p r e s s u r e

v o

in

t h e a x i a l d i r e c t i o n

Z,

If t h e law (26)

2.6

Boussfnesq°, " ~ s s a i Theorique sur 1 f E q u i l i b r e d e s Massifs ~ u l v e r u l e n t s " , Bruxelles, 1870, S o 270

were v a l i d , then t h e l i n e

K

woyld have t o show an o p p o s i t e c u r v a t u r e , Clay t e s t samples have given i n t h e t r i a x i a l com- p r e s s i o n a p p a r a t u s s i m i l a r s t r e s s - s t r a i n diagrams b o t h w i t h 17 A I d r a i n e d c y l i n d e r s and w i t h u n d r a i n e d c y l i n d g r s , The l i n e s h o l d t h e same c h a r a c t e r f o r c o n c r e t e and n a t u r a l s t o n e , a l - though w i t h d i f f e r e n t d e g r e e s o f c u r v a t u r e , On t h e b a s i s o f t h e r e s u l t s o f t h e s t r a i n measurements i n t r i a x i a l com- p r e s s i o n , t h e s u p p o s i t i o n . . can be e x p r e s s e d t h a t a l l b u i l d i n g m a t e r i a l s f o l l o w a s i m i l a r law, whereby t h e curve K d e p a r t s more o r l e s s from t h e broken l i n e OBC, ~w, t h a t t h e compression modulus i s more o r l e s s v a r i a b l e By 6, and 6 = f s t o be understood

t h e t o t a l a c t i n g p r i n c i p a l s t r e s s e s i n t h e s o l i d mass ( t o t a l cohesion, i n t e r n a l s t r e s s e s , e t c

.

) ,

As l o n g a s t h e p r o p o r t i o n

,%

does n o t exceed perhaps

6 9

h a l f o f t h e c r i t i c a l s t r e s s r a t i o

,

t h e curve K c a n a l s o

6-

be s u b s t i t u t e d i n t h i s r e g i o n by a s t r a i g h t l i n e OA f o r e a r t h masses, However, l o c a l i z e d l o a d i n g of t h e h a l f - s p a c e p l a i n l y d e a l s w i t h s t r e s s e s f o r which t h e behaviour o f t h e mass l i e s i n t h e f i n a l r e g i o n , which c o r r e s p o n d s t o t h e curved s e c t o r AD. (Thfs r e g i o n o c c u p i e s approximately t h e s u r f a c e which i n F i g u r e

4

i s l i m i t e d below by isochrome 2 and above by t h e a r e of isochrome 3 r u n n i n g immediately under t h e l o a d s u r f a c e ) , I n t h i s r e g i o n , much l a r g e r deformation3 a r e broqght about f o r a n a p p l i c a b l e s t r e s s c o n d i t i o n w i t h a c o n s t a n t e l a s t i c modulus,

( t h a t i s i n s t e a d of

OM)

t h a n t h e c o n t i n u i t y of t h e mass p e r m i t s ; i n t h i s r e g i o n a o r o s s e x t e n s i o n d e v e l o p s

i n

approxi- m a t e l y a h o r i z o n t a l d i r e c t i o n , and i s h i n d e r e d by t h e envelop- i n g mass, and t h u s t h e compression o f t h e o u t e r l y i n g p o r t i o n o f the h a l f - s p a c e must be t h e same,

It f o l l o w s from t h i s t h a t t h e p r e s s u r e d i s t r i b u t i o n i n t h e loaded e a r t h mass must be d i f f e r e n t from t h a t i n e l a s t i e m a t e r i a l s , It w f l l be s i m i l a r t o the s t r e s s c o n d i t i o n fh a

loaded s t r a t u m which i s h e a t e d a l o n g t h e Z-axis,

We w i l l n o t go f u r t h e r i n t o the computation of t h i s p r e s s u r e d i s t r i b u t i o n , la The t h e o r e t i c a l t r e a t m e n t w i l l , i n my o p i n i o n , be a b l e t o e s t a b l i s h t h e a c t u a l s t r e s s e s i n a n e a r t h mass o n l y when i n v e s t i g a t i o n s of the deformation under

17

L o

Rendulic

-

Der Bauing,

1936

S o

559

and 1937 S O

4590

LU

Ko Hruban, Der Spannungszustand d e s i m I n n e r n b e a n s p r u c h t e n Halbraumes, I n g e n i e u r - a r c h i o

1943,

H. 1 S

9,

t r f a x f a l s t r e s s c o n d i t i o n s have accumulated f u r t h e r knowledge, However, t h e y y f e l d t h e c r i t e r i a from whfch can be drawn t h e following requirementso

(a) I n t h e considered r e g i o n of l a r g e s t r e s s d i f f e r e n c e s , s t r o n g e r compression appears i n e a r t h masses t h a n i n e l a s t i c m a t e r i a l s ; t h e predominant h o r i z o n t a l p r i n c i p a l s t r e s s

61

shows, however, a p r o p o r t i o n a t e l y l a r g e r I n c r e a s e t h a n t h e predominant v e r t i c a l p r i n c i p a l s t r e s s

6 3 0

(b) Thereby t h e p r i n c i p a l s t r e s s r a t i o i n t h i s r e g i o n becomes more favourable t h a n i n t h e e l a s t i c half-space,

( c ) Wfth I n c r e a s i n g d i s t a n c e from t h i s region, t h e p r e s s u r e d i s t r i b u t i o n i n t h e e a r t h mass approaches t h e s t r e s s c o n d i t i o n i n t h e e l a s t i c h a l f - s p a c e ,

These t h e o r e t i c a l r e s u l t s a r e confirmed through t h e f o l - lowing experience :

The v e r t f c a l s t r e s s i n s a n d and loam was measured l9 with numerous r e s e a r c h e s and a c t u a l l y found t o be g r e a t e r t h a n t h e t h e o r y of e l a s t i c m a t e r i a l s I n d i c a t e s , Lines, which r e p r e s e n t t h e s e t t l e m e n t a s a f u n c t i o n of t h e loading, run i n t h e r e g f o n of t h e p e v f s s i b l e magnitude of l o a d i n g i n f o u n d a t i o n con-

20

d f t i o n s e i t h e r l i n e a r l y ( a s i f t h e mass followed Hookeos Law] o r a r e more s l i g h t l y curved t h a n t h e s t r e s s - s t r a i n curve of

Figure 8, With l o c a l i z e d loading of a c l a y block, approxfmatefy h a l f t h e s i n k i n g was found a s compared with t h a t corresponding

t o a p r e s s u r a t e s t on t h e same m a t e r i a l and comparative f n v e s t f - 2 1

g a t 1 ons with e l a s t i c masses A l l of t h e s e phenomena can be t r a c e d back t o the deformation law, i n t h e same way a s t h e

known d i f f e r e n c e between t h e bending and t h e t e n s i o n s t r e n g t h s of concrete i s explainedo

F i n a l l y , we come t h u s t o t h e view t h a t f t i s not p o s s i b l e wfth t h e p r e s e n t s t a t e of i n v e s t i g a t i o n s t o a s s i g n s t r e s s e s I n t h e e a r t h masses w i t h t h e same d e p e n d a b i l i t y a s f n a n e l a s t f c 19

Ho P r e s s , Die Bautechnik, 1934, So 569, 20

Ringlfng und Biemond, Proc, I n t o Conf, on S o i l Mech, 1936, Volo 1 So 1060

21

A , B , Mason, Proc, I n t o Conf, on S o i l Mechanics, 1936,

m a t e r i a l if one wfshes t o evade a l l assumptions n o t s u p p o r t e d by c o n c l u s i v e e x p e r i e n c e , \Ye a r e , t h e r e f o r e , n o t i n a

p o s i t i o n t o determine t h e l i m i t l o a d i n g a t t h e f l o w l i m i t

w i t h o u t a s i m i l a r h y p o t h e s i s , N e v e r t h e l e s s we can, w i t h t h e use of s t r e s s c o n d i t i o n s of t h e e l a s t i c h a l f - s p a c e , determine t h e l a r g e s t l o a d i n g , (which perhaps w i l l be a minimum) t h a t w i l l produce no c o n t i n u o u s f l o w phenomenon, T h i s h i g h e s t

l o a d i n g remaf n s t h u s on t h e s i d e of s a f e t y , and we w i l l d e s f g - n a t e i t a s t h e p e r m i s s i b l e s o i l p r e s s u r e ; p e r m i s s i b l e , wf t h c o n s i d e r a t i o n of t h e e q u i l i b r i u m c o n d i t i o n s i n t h e e a r t h mass, The magnitude of t h e e x p e c t e d s e t t l e m e n t due t o t h e consol-

i d a t i o n of t h e c o h e s i v e s o i l s t r a t u m i s n a t u r a l l y a l s o t o be t a k e n i n t o c o n s i a e r a t i o n , w i t h t h i s c h o i c e of t h e s o l e p r e s - s u r e l i m i t , 12, P e r m i s s i b l e S o i l P r e s s u r e w i t h Loaded S t r i p s On t h e b a s i s of t h e above d i s c u s s i o n , l e t u s t r a n s f o r m t h e e q u a t i o n (15) w i t h c o n s i d e r a t i o n of (25a), I n t h e d e r i v a t i o n , t h e s t r e s s parameter d i s c o n s i d e r e d c o n s t a n t ; t h e i n f l u e n c e of t h e s p e c i f i c weight depends however, accord- i n g t o (25a), on t h e d e p t h , We a l s o assume, t h e r e f o r e , t h a t t h e s t r e s s (25a) remains c o n s t a n t a t t h e c r i t i c a l d e p t i n s e r t i n (25a) t h e s m a l l e s t p o s s i b l e v a l u e of

zm

= a

Fnd

a c c o r d i n g t o (16a), Thus, t h e p r i n c i p a l s t r e s s e s i n t h e c r i t i c a l r e g i o n a r e ( N o t a t i o n a c c o r d i n g t o F i g o 6 , ) For t h e e q u i l i b r i u m of s t r a t u m I, t h e s t r e s s r a t i o a t t h e p o i n t C whose o r d i n a t e i s g i v e n by e q u a t i o n ( 1 6 ) i s a g a i n deef s f ve, The p e r m i s s i b l e e a r t h p r e s s u r e amounts t o

Thf s can be m i t t e n i n t h e form -fa

=

AICL + Bppo + Clya

i n which t h e c o n t r i b u t i o n of t h e cohesion C1 of t h e upper

s u r f a c e loading

Po

and t h e h a l f p l a t e width a appear s e p a r a t e d from one another, Herein 3/ s f g n i f f e s the s p e c i f f c weight of t h e s o i l below t h e f o o t i n g s o l e , The c o e f f f c f e n t s A1, B1, C1 a r e given i n Table I f o r v a r i o u s magnitudes of t h e f r i c t i o n angle

+

l o

Example 3 0

For a completely c o h e s i o n l e s s sand wlth = 40'

0

3/

= 1800 kg/m3 t h e perm1 s s i b l e s o i l p r e s s u r e g i v e s , from

Table I wfth foundation depth of 2 meters, t h e follovving valueso ( a ) No ground water present..

P o

= 2 x 0,18 = 0036 kg/cm2

2 p l a t e width ' 2 m

Pa

= 3.20

x

0,36

+

3.10 x 0,18 x 1 , O = 1.7 kg/cm

2 1 0 m

pa

= 3.20 x 0.36

+

3.10 x 0,18 x 5,O = 3,9

( b ) The s o l e l e v e l

P

meter below groundwater t a b l e s

3 (pore volume 32$, s p e c i f f c wefght under buoyancy 1120 kg/m )

2 p l a t e width 2 m

pa

= 3,20 x 0.29

+

3,10 x 0.112 x 1 , O = 1,31g/cm

9

Example 40

The s o l 1 under a p a r t of t h e p a r t y w a l l mentioned i n S e c t f on 1 showed i n t e s t ng with the s h e a r a p p a r a t u s a t r u e

_k

cohesion C1 = 0.10 kg/cm and a f r i c t i o n a n g l e

P l

= 22O; t h e width of t h e foundation amounted t o 1,20 meters, t h e s p e c i f i c

0

weight of t h e s o i l under t h e foundation so

_2!

6 y = 2000 kg/mao The foundation depth gave

Po

= 1,04 kg/cm ,

Equation (27) giveso

pa

= 1,04

+

8,45

@ , l o

x 0,927

+

0,117 (1,04

a

0,20 x 0.60 x

f i g

On t h e o t h e r hand, t h e r u p t u r e l o a d amounts (with C1 c o t

2

2 a c c o r d i n g t o Caquot: 1 , 0 4 x V05

+

0 0 2 5

x

6 , 5 = 9,4 kg/cm a c c o r d i n g t o R f t t e r t 1,04

x

5,6

+

0 0 2 5 x 4,15

+

0 , 2 x 0,6 Continuous s e t t l e m e n t , whfch made n e c e s s a r y a r e c o n s t r u c t f ~ n of t h e f o u n d a t i o n s , took p l a c e wfth a n a v e r a g e l o a d of 3 , 2 kg/cm2, t h a t i s a f a c t o r of s a f e t y a g a i n g t r u p t u r e of more t h a n 2, Layered B u i l d i ng Ground, The p e r m i s s i b l e s o i l p r e s s u r e a c c o r d i n g t o ( 2 7 ) was d e r i v e d under t h e assumption t h a t t h e f o u n d a t i o n was p l a c e d on a v e r y t h i c k s o i l stratum, f o r which

y

,

C1, c o u l d be assumod a s c o n s t a n t s o I n r e a l i t y , however, b u i l d i n g ground c o n s i s t s of l a y e r s of d i f f e r e n t c h a r a c t e r i s t i c s ; i f , t h e r e f o r e , one of t h e deeper l y i n g s t r a t a i s n o t a s s t r o n g a s t h e m a t e r i a l immediately under t h e f o u n d a t i o n s o l e , t h e n t h e e q u i l i b r i u m c o n d i t f on of t h i s s t r a t u m s h o u l d a l s o be t e s t e d o A s m a l l e r p e r m i s s i b l e s o i l p r e s s u r e

&

w i l l r e s u l t t h a n t h a t vh f c h cor- responds w f t h t h e upper s t r a t u m ,

lRJe c o n s i d e r a n y p o i n t M(o, z ) i n t h e Z-axis i n Ffgure 6

and d o s f g n a t e b y

P p

t h e v e r t i c a l p r e s s u r e s t r e s s which e x i s t s a t t h i s p o i n t i n t h e s o l f d s u b s t a n c e of t h e e a r t h due t o t h e weight of t h e e a r t h column of h e f g h t z l y i n g above i t , (Thf s p r e s s u r e f s t o be e s t a b l i s h e d w i t h c o n s i d e r a t i o n of t h e even- t u a l water buoyancy,) The flow c o n d i t f on ( 6 ) , f o r t h e p o i n t

M c o n s i d e r e d , i s , when CV1 and P1 a r e t h e m a t e r i a l c o n s t a n t s a t

Mo,

I Y s i n

$',

+ q

( P o +

PI)

-Zap, z T ' y3

(Po

+P,

)

-+:

$2'

-

2aP. + C , I = s i n

$ ,

Tr s c o t

q:

It g f v e s t h e s o i l p r e s s u r e which f s permf s s f b l e a t t h e founda- t f o n s o l e w i t h c o n s i d e r a t i o n of t h e f l o w l i m i t a t t h e p o i n t Mo