MECHANICS RESEARCH PORT No ROCK D 22

InterAartmental Rock Mechanics Project, (Departments of Civil Engineering, Geology & Mining & Min!ral Technology), . .r Imperial College of Science & Technology, (University of London), South Kensington, S.W.7.

ROCK MECHANICS RESEARCH REPORT,

Number D.22 October 1970

Influence of Confining Pressure on Young's Modulus of Broken Rock and Soil by B. Stimpson.

,

-.

.

=,

%. .....

*kW..

INFLUENCE OF CONFINING PRESSURE ON YOUNG'S MODULUS OF

BROKEN ROCK AND SOIL

by

B. Stimpson, M.Sc., D.I.C., B.Sc., F.G.S.

Both intact and broken rock exhibit an increasing Young's Modulus with confining pressure, though it is most marked for broken rock. A literature survey showed that (a) very little experimental work had been specifically oriented towards the determination of the stress dependence of Young's Modulus for broken rock, and (b) that available data suggested that the equation relating confining pressure and Young's Modulus was a power law of the same form as that for sands and gravels. This property may, perhaps, also extend to

jointed rock. The importance of stress -dependent Young's Moduli is

briefly discussed.

.

'

..

Nie

. 7

.

, - -

1

.1 e

4

4

S

INFLUENCE OF CONFINING PRESSURE ON YOUNG'S MODULUS OF BROKEN

ROCK AND SOIL

1. INTRODUCTION

Linear infinitesimal elasticity is a commonly assumed property in stress and strain analysis of rock structures because of the mathematical' simplifications it allows. Fe4 earth materials are, in reality, linearly elastic but, in some cases, 'an approximation to non-;inear elasticity is applicable. One major cause of now. linearity is the influence of confining pressure on the 'elastic constants'. For example, materials commonly show an increase in Young's Modulus with confining pressure, an effect which appears to be more marked for broken rock and soil at low confining pressures.

With the advent of numerical procedures of stress and strain analysis it is now possible to incorporate such nonlinear properties in stress -strain analyses. In the author's case (STIMPSON 1971) the problem concerned a buttress dam folpdation which contained a broad shear zone of broken and shattered rock. Although no data on the shear zone material were available it seemed worthwhile to compare finite eltment solutions for the cases of a constant Young's Modulus and a stress dependent Young's Modulus in the shear zone providing some data on any broken rock was available. A literature search produced very little data on this topic.

2. STRESS DEPENDENT YOUNG'S MODULUS - A LITERATURE SEARCH

The stress dependence of Young's Modulus has received most attention from soils engineers. In the general case the modulus, E, is a function of the stress invariant, 0, where 0 is the sum of three orthogonal normal stresses at any point and A, where A is the difference between the major and minor principal stresses at a point. Thus:

E - f ( 0, A)

It is believed that under compressive stresses E increases with an increase in 0 but decreases with an increase in A. The relationship has been verified for sands and gravels tested in triaxial cells (SEED et al (1965), BROWN and PELL (1964)) but no published experiments have been carried out to test its veracity for a truly triaxial (three unequal principal stresses) stress field. SEED (1965) demonstrated that if a triaxial specimen is subjected to cycling of the axial load the recoverable axial strain on each unloading cycle finally reaches a constant value, so that it is possible to define a 'resilient modulus', Er, analogous to the elastic modulus E. The following law was subsequently demonstrated:

Er kio3"1 where a3is the confining pressure and kl, nl are constants which can be determined from a series of experiments at different confining pressures.

Similarly, it was further shown:

Er k20112 where() is the stress invariant (al + 203 )

BARKAN (1962) suggested a relationship which would hold for both cohesive and unconsolidated soils: A

E0 (1 + B 0 where E0 and 8 are experimentally determined constants.

-4-4014104111

I V

t

,

4

I

. ,

..

.

. '

.,

- ,e . , .

, .

. N .

.

1,

,

/ .

-

4

. * .

'

i , .

E .m

, .

. - l .

3

-

This relationship has been shown by HUANG (1968) to be applicable to sands and gravels.

Very little work has been conducted on the stress dependence of Young's Modulus of broken rock. However, several authors investigating the strength of broken rock in triaxial compression give data from which it has been possible to determine whether any Young's Modulus v. confining stress laws, such as those referred to above, are also applicable.

HOBBS (1966) after testing intact cylinders of a calcareous silty mudstone for their strength* in triaxial compression, proceeded to determine their behaviour in their now broken state. The load -deformation curves of broken specimens showed a zone of linearity in the middle region of the curves from which a Young's Modulus, E. could be determined. If the Young's Modulus is plotted against confining pressure on log -log paper a very close straight line relationship is obtained (Fig. 1). The power law relationship is of the form:

E inkia 3 nt

i.e. identical to the relationships for sands and gravels referred to above.

Specifically, for Bilsthorpe siltston::348

E 1.107 x 105 03 ) lbf/in2

JAEGER (1969) reported triaxial compression tests on six inch diameter cores of closely, randomly jointed Panguna Andesite from Bougainville Island. After several cycles of loading and unloading the load -displacement curves again show a nearly linear portion over about 3/4's of each curve,. and hence a Young's Modulus, E, can be calculated. Once again (Fig. 2) a power law is applicable:

039.441 E 0.550 x 105 lbf/in2

The scatter in the plot for the Panguna Andesite is greater than that for Bilsthorpe siltstone. In addition to natural variations between samples, the increased scatter with the andesite may be attributed to disturbances arising from the difficulties of taking six inch diameter cores of a friable, broken rock and transporting them over large distances to the laboratory where they were again inevitably slightly disturbed during test specimen preparation.

ROSENGREN and JAEGER (1968) measured the mechanical properties of a course -grained marble heated to 6000C. The heat treatment separates the calcite grain boundaries so that the marble becomes a loose aggregate of crystals which the authors accepted as.a model for broken and randomly jointed rock. From their triaxial data at various confining pressures, the tangent modulus of elasticity at an axial stress 01 of 0.50 max (a max is the maximum differential stress) has been plotted against the confining pressure 03 in Fig. 3 on log -log paper. (At an axial stress, al of 0.50 the the load -deformation curve is, for all practical purposes, linear). Except at high and low values of 03 the relationship is given by:

E s 1.601 x 105 03 0.394

BRACE (1968) reports that triaxial tests on broken dunite, gabbro and granodiorite show a very gradual, linear increase of Young's Modulus with confining pressure ranging from 0.65, the magnitude of Young's Modulus for the intact rock at 0.7 kb confining pressure, to the same value as that of intact rock at 3 kb confining pressure The lowest value of confining pressure used in this last study corresponds to a depth of about 3 km. below the earth's surface and represents a common problem in present day rock mechanics. Perhaps

'

.

- -74 k

t

- t

.. , . , ...

: .. ,

: ---

.

.

t .

. _ .

4 \ Or

Y ..

.. ,...,.''

: *i.

,

. .. ,

. 0.348 .

. . ,I./ .1 .

'

_ a. 4, t

:r4 1.-.

)

ti

'

// -

'

I

1) 3

4

lv

1.)

Om.

v

4.

0.

Fig. t Confining presOure, *

a.3 v.

effective Young's Modulus, E,

for broken ailsthorpe silty mud -

1

E x 10 lbf/in.aq.

stone ( after date from Hobbs, D.W. (1966))

1.107 x 105 Oeitihn2 3

10 102

ti

o V

. . S S.

,.1=1.

4r4

I

4-1 Q

4

fr" A . 4

e" P 1 0 ixt

.

.

ef

.

Iv

0

3

at

a-

1 .1. 1 . .,, .I .

s. 6. ' Ii

5 .0 I 1

. t 1.

s a

6.*. 3

.

a t,

I

i

a

r,

at

E

"14

A

r

K.

ti

4." ,

...V

A

7.

1.)4

goo

N

*IP

-

O,.0

, o.

t. ft

.0 '

SU

2

O

104 1' 3

112r.. T -

. . - 3.-4.

. r .1

7 i -3 :

4 r.irrs-Po... .. 4

It

.

I - t- :* t I' 31

S

41' ;f4 a 1-41 4 .11,

, ' ' *s. ' : " , . . - 000-10

4 O. ....II l.. 111. . - .14 44440b

-.19

-",-44 '

. .

103 -

r4!7-. 4. I .T !

3

10

I

I

-

-4 004 .4

- , "'t : : : .....

. . . . . 7 - -44- vs/

:

,

g 17T !

: ! -t

4 . -

4. *- - 1 4. ....6 .... ...0,

. : 4... 1,4. 1, .. 46 . - 1.-- -4- 1 4. . ...... . i . .1 ........ - -... . 4...4 4 4 1-4 - ...... ....

it-- 4 -4 . - 4... - .. . , - t 4. 1

i 4 . -. ... 4 . . .... 4.

I 1 .., . ..... ..... . 4 4 - i' .

101, I

i

100

1

* 4$

t, . 1 ;,....

. 1.....: ! . ,- 1.-... -. .

... ' . l; ii 4

1-.

-r-= -I 1- .!

;:-

- ./

r. ,r. z . ;., .

me 4-.44 4144

. 7 . . _ _. . t . ..

4 141 4 4 44 I . .

f .. ...- 04 .... 4

.. : : ' . i.

.4 ; --.-4,... eq, , ...tr., ..

4 r.1 . .4`. . . - 4 1

. 4- 4 .:if' i 4 4 . I '4 . 4 i .

I t -- 4' 4-

441t 1 4

1 ....:

. 1 Cord #sins pressure.

4 1 4 1-7rr

; . 1 ........i.....: 1.

t -1

1 . '

I

....... - . dr.. - . 1 116 i

.

i

1 I

..: .4, 1"7t..1 . '57, 4

......?..., 4 li (.6 - r 14-#4/ I

4 T. 471 i

- : : 7

-ii- -t- A 4 -4. :..

: ..7 ,- 4 o

.

ar

I IN -

4 ..)

...7. .4 --

If . - 'f: ..

.414. - -

-

.: :

. : 44iii : : - "

7"..-. . , . . .. r

414 4. . - t 14-1,.-L4 4 . .1 ..rir-,,1 -

.1 - - 1 4 IL #

0.0 .

04 04 r, 14.4440

Oh; IP,-

T ' . ,

1 - 4

11 4 . m 111 4 1 4 # 4. 4 .7 11, 6.4

Ir.. , r ..Milk ! 4. . 110. 6 , ......' .1 ..." . 4. 4 . 4 14. - -4 4 - t ' '; -"' '

' : t ltd- ' ' et+. -4- - i 1 ' ' . - . 1 i .

i1/.4 . 04 . tt... 4... if, 4:- - 4 t p 4 # . . - -..

I - .-44 # . - , .

t 4. - - -

p + - - . # boir..

. "'1. I 4 4

.

101

E it 1051bf/in2

1 .00.444

4-

S 4 7 11 9 1

es2

a

4

.

's

el

v-.

ri

- _ . 7 -

Z j 1 ! -

: TIT t .

P

-- .44 1"

61

-,..

---___-_,. - : //c..1 - " -.. ... -"; -;-; .1-

' ....- - . t- - -0 ""' ' ; I t

'...

. . 4. ..r.... 4.I., . . 4 V

4... 4: . 4 0

" T ". 4: ' i L ''' I + 1 f . .. .6 4 6 tr.

' . t 7. i 1 4 II` a ; '4 w 4 '

. . 4. 1** 4 4 . -t- .......r - . ...._ ; t

-4 . . ... ..... .. -. .. :.4.11: 4 .i -_- . 4. .' .: 4. 4. .

- . . , - . I 4 67.4

I . "1"," k

i' I ' - " 1 1 -1 t 4

gi 1-. ... . . - . 1. 1 1

..... . :.: '7-4 -1' t 4

r. fai .

.1" I .

7 07

4* 4..er ri Lo..I .41reor -

Par a 4 ;

a

I . . I

. . . .

, I :

a - -.11a

.

I 4.

4 .4 m. I .

1 - .

: .

;Z = -

-; r

. I : i : 111, :

itit *Fr 4.: 4

.

.4

" .: % r .P.

.

.1** r

.1

. . ,.:\;,-.: 44 I .. - 4.. 4. + ...4 4. ow .

,.. ...4.- 4 ...ay. . i 44 ..1.44r4 4. Ti p- - .. , 1,1, ! o 1 t

1. . I, 14.9 6 td, I 2 4

4 . .e...I. . 4 ; -:24 ,.. ......

!ti'l! it i - r 41,

an. I..

....1.1 P

- F k.1

1: ,i-% 1 !... I:" , v

.....

.:11

`".irtnt

!' n: : a VI 4 .1 II .... .'

1- I . Ini ,_ . ......-. . A . ...

6- , : 4 6..14 1

I I

I4

-.4. ..,.: : .1

4.4-. . I

' ''- 46. ow; ;! 1! ;. ....

' . I

77?!; II I 4 - "- 7' 47777P '

4 ... i . . ....k. ,i . ' . .. .. . . .,. ., a .1 t. ; .. . .

, . - , . ........s.i. G.

.. .1

.

. V't; . 4, ..-. , 1.- . r. -it '' v,.... , . . ' . 01

'

....,.......trii.... ! I. L 1... I t *A". . I ag

r. r 4 .. 4%1

1

'f , Li. Ok i 1 4.6 .... .., .4 b . el .. : , i

.? i 4 ' ' 4 . I .4 lo . . e, ... . . ., 4

, - - I 1 . -.4 .,. f. 3 . t ,. t I

. t i . I !.- . 1 i t . . , . 1. 's

. 4 4. _N .., - 4 't.'I.' --"N ' ., -: et- 1 f: -t 1 t rrf Nit. t"

-1:t.s...-.7

a :÷:A11:'.1-:!4- , ,:: 11 : .

, . ; ;:1 - --::.= L ....- - . .....-

: 1' ' 4 ..? , ' : " ! : . :. i : .... :

7 : v- . 1 . 4

- 2 rt-:- 7:t , . . .. -t .. -r.-. +' ., ' 1 : . - .. . :7. ;:.: ;

-4:2 --.. 4 '77 -..---........,.., . ,..7.7-

ft 7: .:- L ; ... ii:.1.: -I -.;... ..1 ..'

- I I

44 r " I , ' 7

irr- I

P.Tri rt.l'J i+4f.

II

q";.. : I

4.1 .1 31..r -

t . ; sr.

-44 .

. . ! i ; ;. t'Y -..

,tirtrt .

., m-

- . ; : . , : ' ; . *. r.::'2.:11...-1 . - r . 71 , Gam , 4. 4 . 1 41 . 41.

. . .. a. . ,. r. wait , 1. - 4 IP 7 . ;

. ..... .4 ,. 10". IRO

f 4. ; . ? ... , 4 go . 11 4/4 4. -.4 a 7 ...- ' 1 OP. - 4.... 4 - t re - ,- e .... 4

. Iay . ,I1L-; . - 4 -

r f % v.- 3 3, 1. 0.-1 .

p .3 ..: 10 If an I a

4" I 77:.-17.:.1.

; x-: . t : " :efie : ; . .

4 t I .: pomp,. .

.t.1...: 1.;;- ,t .1 1 i .,' . :t at _ .13 -

.? - .-t r.r,.. .....

1.3. i ' '. .1 : : t ; r -

..... ..1 . . a. -' t. I ... ,. ... ..,... .r?

,... .4 . ..:....r.%.1 If

In. - -6 04 1 e.

i - 4 ... 4

. .... 1 40. .

...1 .a. . ..... T.. :3 ' ....,, ......

4 ... - .4 t s I . - ' 1- cit`i 1 -.: r-- 97

al.., r- .4

.....4.- 4 . , ; f44, l'; ' It

.

' I 4 - ef I . I ... i a a . .4 , 4 . 4.,

I '

4 II 4 2

.4, so :qv

"4"1.4

4./0.... .0 ' 0.44i1 14 -t0.441. . - 4

1.

1

IMP

4 '

mi

-

.1 t , :!1"oe. "We. 4,- . b. , . 4 .z...,1-. ...I 1.:

o 't - i 4 ...4 4..

I W I

1 . I' 1

- i

r. 1_-..

t=.1 :

.

"1 .. Ln

4 11.;.'

- - .

.40

-

4

I

4

s

. , - - ;

: Tye 4-4 4:-4 . .

t I : .4 . .

"+1.49 4 .

. . 1

r '

la Clair, +40466ii

. UV :UPI

- .

W

61.

...."14.7.

I

7 3

4

.. 41

- 1

7

4

1

)

-Tsang" Ifodulus..t

" jointsd

data (1969r)

a

S 3

.41

10

--4 1 I

t 1

1: ;4- .1-444 : riF 1 i 4 4 d

ra- T.. - Ir f

. 4

4 4 . 11- i

t- ; I

.

i ..4.-r --;

10 ,

: 4 6 .

'

. .. 4

. A:4 . ,

4 . 1 .4 't . , 0014 ., , :

' f -

.11..... ll .41It

y .... lib.

-"!-- r--- T--1

100

it ...... 44 i 0,44011 I 44.41. l.4.4.4446 .4 4

4

I 1 II e 2

- irt-Tri"Trri- 11'1" ti. : ; --t *- --err !- 4T- - .

. . . ,

""! 'I 7 1-1 1 I 1-"01-!"17 . ,r- .- . ... . . 4h

.: -....se - e ...: ., ....... :.......: - ......... ' -:. ...... . ' : ; i : ; : : . ; . .: .

. , . : ' ' i . : . '. :' ' 1, '.. :" -"r"-r T11-1. - t'''rtrt'll fr. - rr, 1'

! ; t I I 1}

. , i.titi i . ; f .'. ' ... "' !

. - t 4 ' - ' :+4" :4-f":"''L ..

1 1 .

. . , . .

101 Tangential Young's Modulus, Et, lbf/in2

f -

, . .

1

'

.

: : 2 4'4 i; . .... .

4 It 4

I '

. . . . ,

4 - - .14

. 4 . . .

3 1 P I a

1 1

I

.0.

S

S

a

a . 1

,

-I.

'

'

7.

a.

- f-- 4- 1 r-; : . /7 - 'T -T. rt i

'' L.+ , .71 1:17.741: 7, :.::: . . 4, ... i . . . i ti

. .7.:"--t...:. 11:: .. ' .1 - ..1 f'

. -.- .. . .,.,.....i. -4.... . -i, *

.' ,......

I 'It. -1 1 i i : ...... L...,...4--,.:,,..i., ; , , .. I'. I ..... 1- f I ...+ 4.4 ± -.4,

- e 6-6 r It', . rrerir 447 .1 1 `.11-4.L...41 , , t 4. 4 .; 4..,..,.... r - . . 4 - & .. f , .....1..14. T.. t ;144 14.et:

tI ..; . -.4..... . 10 44 ...1 I 1 . itsti.it k !..4S.

,- t!. 441-I:. . .4tr - -- 4 p,.., . , . .

; , t -t - 11 I -4 4 6

-404 .; 6 4 /.. .41 14.,, . T . ; 1 6

. 44.. a.. .i .144111" 4- .

. --, . it- .f t......,, 4441....- ..

i. ' 2_

7:, '

I "

' - :

3 , ; " I. 10 ,

4 "-

. . . : : " :

' . .

. g 1 . -.t.- ; :- : : 1'

11 . ...-

21. j . . . : ; 4 ,

4. .

I. 1 : : .. ..

.

°IS I 1 6 O.

.i . - , 6 i , . M ip I. .

. .., ' .4 .1. 6 .. ... ...1 1 .

i 4. S. , .4 4 1 4

, ..7 .. . .. . I . , r r

1 / 1 -0 a

6' t

1 1., -r-. - s- , . .

7 I

61., 4.4.11.44

i!=:::41:.1 4144....664... I

1144;61'.: ; . 614 .4. g

IA: .1, .

64i I

11;0:1444t. :44&" ` 41-'1

.414 ill, !II firL:40: .14414 3 .4

Ii.14.14! 4,;" .4 1; f..;

4,1

.. 4 .. . 11. ; i,- . .

Af.t.t..1 ...,.II i i. .t.....:.+_..

.., _....... ....,:_.,.........._?....

.c. '''-' 1..: - " a pr,:.... . , 6

-.. -4 . . 7. , ,I. , _ . e . ...?' , ren- ! .. ... ,o., t 19! .....

O . a n Vria . , isi . i ; . . t

- If, .1. , . . 4 . .

. . L I, N 4 .

.....1 ... ..

2 -.-.10. . 1 - .. ef 1 .1 - ' . .1 re t.i. ......+ 411..., ' 1111.1..-..........

. 1 le . .., 16

1. ;- a 1. ;.! 1

.. i - aa,

I . a . ...I

' 1

;*..1.1 f ,i Tri".7 -il-ri--1 ii T.1-7

....p...4,.t. i I p . ti i

.2. . .1.4.44+ ./ ,....1 ..1 2i .. -A-.... 1 - .:

:" i i

4 0.... .6 416. 4 - 4:.

i :LI' , .....

f . - 1

- 7.

1

4 i 7'. . s'r )/ :

.' ' . ' . .

: : . , : .,

;:-1 rfiVr'' ; I - 7..; 4.÷;...;....1.): e.1 41 1 ': 4 I .r.... ti. .....,, L. l',.. k

.. i : . ' ,' . . I: . .. :, 2.+.1.:. .. ,.. l: -I.:. '

' '

. 1 1.... . ...

.1 7-..-..... pf... ..... 7r../ .4.: 1.1 .. . ....v. .. c 1 .i.i

,

i ; ! . 4;:ii, * *0 %-;* -I' *

. . 4 1 4

. t ,i .

f

.

. . t . n1 .

: a .! I : ..: ' : 0 i :

4 ......!...i. ' --6:::.....,'....1..: .;...-;.......1 i......

' 7 1 . t . -..-..... ---..... ....... !. ....., . - ........... - : : ' : :

. r . . . 4; .:r I

4 1 f)". li 1!1 4.1 i :* z

. 4:17 ; t. ' r ;I:

, - 14.!..dt t.* I .11. -

rl t- 1.-t- + , - - 4 , i vt 1 ..1/- ., . -4. 1.4 4 1 1.... 1 ti ti .

.....e44 474- I. ;- 4 ... 4: . .1 4 ."."r1SP.'.

; ,

TT : j. I .4 . .... .1. 4,- 4. 1 . i , . 1 ..." : I 4 - -, .1: T

I 1 ,. - .' i 4 4 t i 4... 0

''' 1' i'...-. -4 -1 - 14. 4.4 .. ...0.14. 101 . . . ___. . t .. .. ' ) I

1 2 i

- 4

4aN .1., 1 ..t,

. . :ig 1.1 : t .

IC i , .. - - i 1.

4....-.7 L, 1, . ........ a 4.1. ID.. .. 196, '

11 ; t :, I i , ,.1 - i. . ... 1 * . 11 . t . 4 4 ,.1 i. ...11 ... A.. .. 4.1.; . .., k

,p. . i ,-,...4 wir.- 4 ;-:r! ; r 1 . i' 110 I

r i 1 . 4 1 \-1' 1 4

i . . % f .4 4 . i g IC g

4. g ... a ... .. . .. . .. k.

1 k .- . 1 , , I ... I

4 ' S i e : . . i le. . . I

''.7-irr.:- ...1_41.. : I. N ?:! -.{ ; .'l .. :...:.' 44.4:-.t! 4. 4 ..4...t

t - ....a t 4 : . :, .01,- :

*14...t....1-4. ...t.1 . - mom. I. : i : T. : . . : .

I 6. , . a, . . ' . . . . . t. .. 4

."1 : 1-1-'::- e . r-:-.----:-. . . . ..

b : : .: ....: .f . r . .., r-1 -'. . i ..

. :

1'1". ' ' t Tr.-tr.:L....I"' !:-1:44:::T-.4 , 7 I I_ i.- 1 tt i i:'./

s., . :. - ,

" 3 " : it': P '.' . L

.4. '....

--,......t......-.... .1.-........... ... ....,.....:-... 4 4 '

\ 1 : : .... .jo! . . . so :1 iT I ; .- . . . -4: ; . -- !..7.. . e : t 4..4. ' :I : -

... ! . :a .

t.. ., f;.: I

-.. . . r , .. 0. ,:. , . ..; ..... - ...1 ..... ..- .i ., .... , . . .g. . . 4 . . .. .. .

10-1 ' .2. 1 .. '... : - ' " 1.:

: l' ll 7.1:: 6.16 . :' r: . 71; .. 1 -1 r I

, t ti..

4..

16. a.

.... i .

1

1 A tit* ri

1

4

- ."4

1 I

. . .4 s

f 4, .t .

,..;

,.

.

f

tar

' " *1 ' 3 '4,4ittlij itreitt,

!--_:, t --.. Ir.

i -.; T.-:' . . - t -: it - r- !-. rFrrp : -":"`r ..., i': . rl: , ' ti _11 ..4....4- :...-...c.. ..7.1.r... rt _ .-

-/-'

......_ .......... . t.. ,...,........ ........ 4.. N.- ...a ! : g t .I ja* ..

: . . ( '... ter delta frost.

i ;1- = i: t -,f ,.:.

t .4 .:!. - 111:.':11.i Ii:'..1 1i.l.:! - :milt, I... :. t

-trintttr ... ..i. I- 4.

; . - 7 - : ; ',' :. I 1 " 1; : 11:...:1ti,,T.:2 ti

- - "' I . ..r- : .1,

, ;: .7:-.3. f 7-1- .-

.......... 1 e .. - ... ....4 ..

, .17 , _..f.'.. '

.4 , . II

4 .... I. g

0 4.1 ! r.1 it .ar.:.-t&;.' i ... . 0...... . ... T . i

i ..- . . . .....- . 1_,

6......1/4 rt "- t' '..: -1..

............- -....4-t.."7 111; : ;

1 .1 .... ....

....

1 11 r in . I 8111 .

..6 , ... 4. i t 11 . . 41C VI ... 7 ..: .

...k.t. 1..., 1 ,, ,. .... I. i i i-. . :4 1 ...i .

. .r....4,4, . i .. f ... ....4 ..

.....,).; I. . : t ---...,,,L..... . - ! i .11.1 .

S Oi. 2 II 1 1 - 3 1

. . ,, - 102 .6 .. .;

. -

1 a

31

111111111MI

4 - . I . .4;az ti41.

:

. :

i Ste 11, .4 14.4, . r . 4 . : . " . , . . . ,,.. . . ,

. - - i TI ri: - . is.

, .4 , .; . .. , 4 i a :IN

: : 1 1 i i e cr.:::i.

. f 60.. I,

t ,.. I :. I lir:10a* t..0 : " : ! .. . . : t :::!::1 ..,:: . L . . .0 '

, ..; qt :.: : 1;7:: 4 i : 4 . . : ' ,...1 .?

'I I

I .

4 1

I.

-

4 5

? !or-M.; ;

,

. . .

1 . 4

. r 0.

- 06 .0. ti 1. 1

I

I1,

.00..0

-

v.- T

- 4 4

4

r

.0 0- ..0,r f 4 r

I , .

44

, .

' ,

. 4

4

..,

1 II .

II

4

.

I $ '

I' 11 11

for

and Jaeger MOO) t

,5

because of the success of rock mechanics techniques in deep level mines,

many tests are carried out at high stress levels totally outside the range

of most engineering problems. Reference to Figs. 1 and 2 shows that the

minimum confining pressure to which both the Bilsthorpe siltstone and the Panguna Andesite were subjected was 50 lbf/in2, and the highest confining

pressures used were 4000 lbf/in2 and 6000 lbf/in2 onthese two materials respectively. For most surface and near -surface structures a range of confining

pressures for triaxial testing from 0 to 100 lbf/in2 should be sufficient. As

Figs. 1 and 2 demonstrate, this range of stresses is poorly covered by both

Hobbit and Jaeger's data. The maximum confining pressures ̀ are also totally

unrealistic for ordinary engineering rock structures except the very deepest

mines and in very localised zones of high stress concentrations. It seems

probable, however, film the analogl with sands and gravels, which have been

tested at low stress (SEED et al (1965)) that a power -law type

relationship is valid for broken... rock at very low levels also.

No experiments have yet Oen conducted to determine the relationship between Young's Modulus, E, and the stress invariant, e, and the difference

between the major and minor principal stresses A for broken rock.

It is concluded from the limited evidence available the Young's Modulus -

confining pressure relationships foi,broken rock in triaxial compression is

identical in form to that of uncon dated soils, viz.

E =

3. THE IMPORTANCE OF A STRESS -DEPENDENT YOUNG'S MODULUS IN DESIGN

Most of the studies to investigate this topic have been directed towards soil foundations. HUANG (1968) studied the problem of a pavement resting on sand in which Young's Modulus varied _with 0, the sum of the principal stresses, and found only a sliOit effect of this constitutive equation on the vertical stresses and displagements, as cowered to Young's Modulus. His work corroborates the results of experimental model studies with sands subjected tc uniform circular loads (KOLBUSZEWSKI an HU (1961), 1.LLWD6D (1956) HOLDEN (1967) WES(1954))

In tne author's study of the buttress dam foundation most marked effect self incorporating a stress dependent Young's Modulus was on the upstream tensile stresses which decreased in extent and magnitude over those in the linear elastic case.

Other analytical work on strip loadings on soil and rock foundations (HOEG,

CHRISTIAN and WHITMAN (1968), MORGENSTERN and TAMULY PHUKAN (1968)) has demonstrated that vertical stresses appear, for all practical purposes, to be

little dependent on the constitutive equation of the material. These studies support the observation that methods of predicting settlements in soils which assume a linear elastic stress distribution and a suitable non-linear stress - strain relation, have been used with success for a long time, and suggest that

a similar approach for rock foundations should be applicable.

4. CONCLUSIONS

It has been shown that the equation of confining stress dependence of Young's Modulus for broken rock (and perhaps jointed rock) is of the same form as that for unconsolidated soils. The simple power law is easily accommodated in finite

programmes.

Finally, it is apparent that the confining pressures used in triaxial r-rk

testing have in the past often been at magnitudes far beyond those likely to ite

met in most rock structures.

$: .1

: .

,

V . .1. 4 %

4 6 6 l r

-.1 e'

. ..,

I , . , , ':, 1',1 . . .

I .. . . ,. .. . , .

.. .

. ,. A.

,

. . .

t

..

,

'

.

-

.,

.

4

'

A

1r,

1$.1o3n1

4 . .

-

11

.

-

\

"A orb..

.

,

- - q

' .

"Mu

7*

REFERENCES

ALLWOOD. R.J. (1956) An experimental investigation of the distribution of pressures in sand. Ph.D. Thesis, University of Birmingham.

BARKAN, D.D. (1962) Dynamics of Bases and Foundations. McGraw-Hill Book Co. Inc., p. 9, 13.

BRACE, W.F. (1969) Micromechanics in rock systems. Int. Conf. Structure, Solid Mechanics and Engrg. Design in Civ. Eng. Materials, Southampton Univemity.

BROWN, S.F. and PELL, P.S. (1964) An experimental investigation of the stresses, strains and deflections in a layered pavement structure subjected to dynamic loads. 2nd Int. Conf. on the Structural Design of Asphalt Pavements, University of Michigan, Ann Arbor, Michigan, pp.384 -403.

HOBBS, D.W. (1966) A study of the behaviour of broken rock tinder triaxial compression. Int. J. Rock Mech.Min. Sci., Vol.3, No.l. pp.11 -43.

HO1G, K., CHRISTIAN, J.T. and WHITMAN, R.V. (1968) Settlement of strip load on elastic -plastic soil. Proc. /6CE, J. Soil Mechanics i Foundations Div. Vol. 94, No. SH2, pp.431 -446.

HOLDEN, J.C. (1967) Stresses and strains in a sand mass subjected to a uniform circular load. Departmental Report No. 13, Civ. Eng. Dept., Univ. of Melbourne, Australia.

JAEGER, J.C. (1969) The behaviour of closely jointed rock. Dept. Geophys. and Geochemistry, Australian National University, Canberra.

KOLBUSZEWSKI, J. and HU, G.C.Y. (1961) An interim report on pressure distributions and measurements in sands. Proc. Midland Society Soil Mech. and Found. Engrg., Vol. 4, p.73.

N.R. and TAHULY, PHUKAN, A.L. (1968) Stresses and displacements in a homogeneous non-linear foundation. Int. Symp. Rock Mechs., Madrid.

ADSENGREN, K. and JAEGER, J.C. (1968) The mechanical properties of an interlocked low porosity aggregate. Geotechnique, Vol. 18, No.3, pp.317-326.

SEED, H.B., MITRE, F.G., MONISMITH, C.L. and CHAR, C.K. (1965) Fr fiction of pavement deflections from laboratory repeated load test . Report No. TE-65-6, Soil Mechanics and Bituminous Materials Resea h Laboratory, University of California, Berkeley.

STIMPSON, B. (1971) Stress and strain analysis in layered rock dam foundations with particular reference to Farahnaz Pahlavi Dam, Iran, Ph.D. London Univ.

--WES. (1954) U.S. Corps of Engineers. Investigations of pressures and deflections for flexible pavements. Report No. 4, Homogeneous Sand Test Sections, Waterways Experiment Station (WES), Tech. Memo., No. 3-323, Vicksburg, Miss.

t ""116

!1-- "

. 4

1.4 .

;

. $

-

to

4/.

,.

4- 4 IV' .

/ 64'

. .

' . a-

4

A I

f WT.

1 --L..'

.

,

-

a

ft .., ..

' p

.

+L. -No. - - ".

W.