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FACYQRS AFFECTING THE SHEAR STRENGTH
0F COHESEGNLESS SQEL

Thesis ‘0: Nu Degree of M. 5.
MECHISAN STATE MEYERSETY

Wifliam Arthur Sack

i960

(1.

0-169

This is to certify that the

thesis entitled

Factors Affecting the Shear Strength‘

of Cohesionless Soil

presented by

W1 1 11am Arthur Sack

has been accepted towards fulfillment
of the requirements for

_M.S.._ degree in Mineering

We

Major professor

Date May 20, 1960

    

L.
L [B R A R Y
Michigan Statac
University

 
     

 

 

 

 

.ro a“

FACTORS AFFECTING THE SHEAR STRENGTH

OF COHESIONLESS SOIL

by

William Arthur Sack

AN ABSTRACT

Submitted to the College of Engineering
Nfichigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE
Department of Civil Engineering

1960

Approved: ZzzbéfiZi£Z1,4,,/’

WILLIAM ARTHUR SACK ABSTRACT

This thesis reports an investigation of the shear strength of
a cohesionless soil by evaluating the frictional and volume change com-
ponents of shear strengtho The effects of initial void ratio, normal
pressure, and particle shape on shearing resistance were also investi-
gated°

Direct shear tests were made to determine the shear strength
and the volume change of the soil during shear. Friction tests were
performed on quartz to determine the variation of the coefficient of
friction with normal pressure, W.

It was found that the shearing resistance increased almost
linearly with increasing relative density. The increase was due
primarily to the increase in the shear force necessary to do work
against dilation.

The angle of shearing resistance, ¢, decreases with increasing
normal load by as much as 12°. The coefficient of friction for quartz
as found from the shear tests and the friction tests decreases with
increasing normal load in a manner similar to ¢. It is believed that
the decrease in ¢'with increasing W is due mainly to the frictional
prOperties of the mineralo

The more angular sands have values of ¢ 2 or 30 higher than the

round sand”

FACTORS AFFECTING THE SHEAR STRENGTH

0F COHESIONLESS SOIL

by
William Arthur Sack

A THESIS

Submitted to the College of Engineering
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

Department of Civil Engineering

1960

fen-.83

 

 

 

ACKNOWLEDGMENT

The author wishes to express his indebtedness and appreciation
to Dr. T. H. vs, Department of Civil Engineering, Michigan State
University, whose generous help and encouragement made this thesis

possible.

11

ACKNOWLEDGMENT . ..... . .
LIST OF FIGURES . . . . . . . . . . . .
LIST OF TABLE 00000000 o e o e e e s 0

LIST OF SYMBOLS

Chapter

I.

WICIES- -F18urea e o o o o o o o e o o 0

TABLE OF CONTENTS

FUNDAMENTAL CONSIDERATIONS ........

FRICTION . . . . . ..... . . . . .

mmmtmosm .....
assumsorrssrs.......

CONC LIE ION O O O O O O O O O O 0

“b 1e 8 O O O O O O O O O

BIBLIOGRAPHY ............... .

111

Page
11
iv
vi

vii

17

31
33

57
63

Figure

15.

16.
17.
18.
19.
20.
21.

22.

LIST OF FIGURES

Element Acted Upon byg- 1 andw"3 . .

Forces Acting Upon a Spherical Element . . . . .

Mohr's EnvelOpe .....

Direction of EXpansion . . . . . . . . .....
Particle Bridging . . ........ . . . . . .
Mode of Failure . . . . . . . . . . . . . . . . .
Surface Irregularities . . . . . . . . . . . . .

Friction TEsts on Quartz and Flint by Hafiz .

Micrograph of Sub-Angular Sand . . . . . . . .....

.Micrograph of Very-Angular Sand . . . . . . .
Schematic Diagram of Triaxial Cell . . . . . .

Friction Test Apparatus . . . . . . . . . . . .

Stress—Strain and Vblume Change-Strain for Round Sand .

Stress-Strain and Vblume Change-Strain for Sub-Angular

sand 0 0 O O O O O O O O O O O O O O O O O 0 0 O O O

Stress-Strain and Volume Change Change-Strain for Very-
Angum Band 0 e e e e e o e e e e e e o e e o e 0

e0 versus ¢, ¢f, fin, and ¢R for Round Sand . .

eo versus d, ¢f, ¢n, and UR for SuheAngular Sand
eo versus ¢, ¢f, ¢n and ¢R for Very-Angular Sand
DR versus ¢, ¢f, ¢n, and ¢R for Sub-Angular Sand

tvversusDR................

t d versus DR 0 O I O O O O O I O O C O I

W versus ¢, ¢c, ¢f, and ¢n for Round Sand . . . .

iv

Page

#—

\O\0(DO\\11

ll

33
33
33
314
3A
35

36

37
38

39

Al

#3
Ah

Figure
23.
21+.
25.

26.

27.

28.

29.

3o.
31.
32.
33.
31+.
35.
36.
37-

W versus ¢, ¢c’ ¢f’ and ¢n for Sub-Angular Sand . . . .
W versus ¢, ¢c, ¢f, and ¢n for Very-Angular Sand . . .
DR versus ¢, ¢f, and ¢n for each sand . . . . . . . . .

Comparison of U versus e“ for Direct and Triaxial Tests
’ROundSGDd........o. 00000000

Comparison ofd versus eb Far Direct and Triaxial Tests
SUD‘AW Sand 0 e o o e e o e e e e 0' 0 0 0 O 0

Comparison of ¢ versus go for Direct and Triaxial Tests

VOry'MSUJAr 833d 0 o o o s Q . . . .' e o e o e 0

Comparison of Mohr EnvelOpe for Direct and Triaxial
Tasts As Suggested by Hill . . . . . . . . . . . .

Stress-Strain and Pore Pressure-Strain . . . . . . . .

Friction Test on Quartz.. . . ... . . . ... ... ... .-..

QP’ ¢, ¢f, ¢n, and ¢R versus W for Round Sand . . . . .
¢ , ¢, ¢f, ¢n, and ¢R versus W for Sub-Angular Sand . .
¢ , ¢, ¢f, ¢n, and ¢R versus W for Very-Angular Sand .
Loose Rectangular Packing . . . . . . . . . . . . . . .
Dense Rhombic Type Packing- . . . . . . . . . . . . . .

qo versus ,1 for Friction Teats and Direct Shear Tests .

Page
AS

#7

“9

27
51
52
53

55
29
29
56

LIST OF TABLES
Table , P389
I. MaximumandMinimumVoidRatio.............. 57
II. Direct Shear Tests on Sand Type A - W Varied . . . . . . . 58
III. Direct Shear Tests on Sand Type A - w Held Constant . . . 60
IV. Direct Shear Tests on Sand Type B - W Varied . . . . . . . 61
v.DryTriaxialTests....................61

VI. Calculated load Per Particle Direct Shear Test . . . . . . 62

vi

p.

8(a)
6(v)

SYMBOLS

area
a constant for a given material depending upon its stress-
deformation behavior

relative density

diameter

modulus of elasticity

void ratio

initial void ratio

maximum attainable void ratio

minimum attainable void ratio

shear force

the ratio of shear strength to yield pressure for a given

material

id‘s:

m

number of grains or spheres

load per particle

yield pressure of a metal

maximum pressure between two bodies in contact
radii of spherical grains

shear strength

normal load

work

the shearing displacement at which the maximum shearing re-
sistance is obtained

displacement in the horizontal direction
volume change per unit of area, (-) for compression, (/) for

expansion

1"! 4

r153," 3‘ q I
:1

In

'Td

unit strain in the direction of the major principal stress
unit strain in the direction of the minor principal stress
the angle whose tangent equals b(v)/6(A\

coefficient of friction

normal stress

normal effective stress

major, intermediate, and minor principal stresses

shear stress

component of shear stress required to overcome the internal
friction of the sample assuming the individual values of 9
are equal to zero

the same a313, but modified by the collapse of bridges (see
dashed curve-figure 6 ).

the additional shear stress required to produce failure because
the plane of sliding is inclined at some angle to the shear
stress (as computed by Newland and Allely's method)

the shear stress required to do work against volume change
(energy method)

residual shear stress

angle of shearing resistance

the experimental values of ¢ corrected for variations in void
ratio

angle of internal friction corrected for volume change

angle of internal friction as found by Newland and Allely's
method

the residual angle of shearing resistance

angle of sliding friction

viii

I. FUNDAMENTAL CONSIDERATIONS

MOVEMENT OF SAND PARTICLES DURING SHEAR

The shear strength of a cohesionless material is dependent upon
the type and magnitude of the inter-particle movements during shear as
well as its frictional resistance. In 1925, Tarzaghi (l)* pointed out
that shear failure along a surface of sliding in sand occurs progress-
ively and not suddenly. As the shearing stress increases, the resulting
displacement increases more rapidly than the stress (incipient slip).
This may be due to a rotational displacement of the sand particles
without the particles changing their partners. The movement is
resisted largely by the frictional resistance at the points of contact
between grains. At constant shearing stress, the rate of increase of
displacement decreases and eventually ceases. The particles on one
side of the surface of sliding now begin to advance with reference
to those on the other side. The length of movement is far more than
a single particle diameter and hence the grains change partners. The
resistance to this last displacement depends on the degree of inter-
locking of the grains and hence will increase with decreasing porosity.
After a slip within the sand mass, the porosity of the sand adjacent
to the interface is higher than that further away, and therefore
the resistance to sliding will be smaller here than elsewhere in the

mass.

 

*Numbers in parentheses indicate reference listed in Bibliography.

1

Mogami (2) found the movement of sand during shear to consist
of two stages. He used a shear box having a light cover plate but did
not apply a vertical load. In the first stage of the shearing motion,
the sand layer near the shearing plane becomes loose and moves in a
manner similar to a viscous liquid. The layer of motion has a finite
breadth and the cover plate is heaved up continuously by the sand. In
the second stage the motion of the sand becomes constant and is con-
fined to a very thin layer. The heaving of the cover plate ceases and
no vertical force is generated. The shearing force becomes constant.

Rowe (3) found that the angle of shearing resistance depends
upon the degree of interlocking of the soil grains, which in turn
depends upon the fractional movement of the shear planes, called slip.
strain. Slip strain does not increase in prOportion to sample thickness.
The decrease in sample thickness during shear is approximately pro-
portional to the slip strain and is independent of thickness. This is
in agreement with Mogami's observation that the sliding movement takes

place within a narrow zone usually defined as the failure surface.

ANALYSIS OF SHEARING RESISTANCE BY ENERGY CONSIDERATIONS

Volume Change Correction

 

It is well known that a granular material undergoes a volume
change during shear. Reynolds (A) was one of the first to study this
relationship in 1886. He termed it dilatancy as the volume change is
usually positive.

In l9h8, Taylor (5) described the shear strength of sand as
consisting of two parts. The first is the frictional resistance between

grains, which is a combination of rolling and sliding friction. The

second factor he called interlocking. The interlocking of the grains
contributes a large part of the strength in dense sands. Taylor outlined
a method for evaluating the effect of interlocking on shear strength

in terms of strain energy.

Hafiz (6) also analyzed the volume change using energy consider-
ations. Consider a sample ofsand in a direct shear apparatus which is
subjected to a normal effective stress 5" , and a shearing stress I. A
small displacement in the horizontal direction 8(A\ will then result
in a positive volume change of 5(V\ per unit area.

Work done by the applied shear stress is therefore T 8(A)
per unit area. The work is expended in causing the sample to dilate
against normal effective stress 5;, and in overcoming the frictional
resistance of the sample.

Denoting that part of the shear stress which is required to.

overcome the frictional resistance of the sample as‘t'

ram .-. 1:' sun + 3— am or
1; I -§; - §IXB (1)
0' 0- MM

¢f will be called the angle of internal friction and represents that
part of the shearing strength designated as‘t'. ¢f is defined by

‘ a.

tan ¢f : IF: g2)
It should be noted here that the values of't' and ¢f do not represent
the actual mineral friction of the material. It is merely the value
obtained after subtracting the force required to do work against
dilation'tv, from the total shear force. A further correction is
necessary to obtain ¢p, the angle of sliding friction.

The angle ¢, measured directly in the shear test, will be called

the angle of shearing resistance and is equal to

tan ¢ : %%7 (3)

Relationshipretween the Angle of Internal Friction, ¢£, and the Coefficient
of Friction, u '

 

(a) Direct Shear Test "

Bishop (7) analyzed the relationship between ¢f and p in terms of
strain energy for the direct shear test. The analysis is made for the
case of no volume change and takes into account the difference in
magnitude of the three principal stresses at failure.

Consider a small element of a sample acted upon by the principal
stresses 0'7 ands-3 (figure la). The unit strain in the direction of a".
is 6. and in the direction offl'g is E; . Under constant volume conditions

5. : -€3

Fbr displacements in the sample equal

to y and x

01
tan 9 : y/x

log tan 9 2 log y - 10g x '
0'3 —> ‘__U"3
or 1 (sec29d9):dy-d_x_
tan 9 y x
I (a)

6‘- (£5) 2'.” 26‘

and d9 _ 2 E.(sin 9 cos 9)

It is assumed that there will be an 9 /d7
equal probability of contacts between grain (T,Y
X
(1:)

surfaces in all directions. Considering a As

01

solid spherical element which has an equal Figure l
projected area in all directions, the pairs of elements making up its
surface will give the average work done per unit volume due to the

combination of stress and strain in all directions.

If in a spherical element (see figure 2), a plane 0A makes an

angle 9 with the 0} axis, the following

 

 

 

relationships may be derived. ‘0;
0'"=G'Tsin2 9 / ficosa 9 C 5/“.

For the plane including OA, .l’v. ...r'
f=fisin2vyl B

farisin‘? 9 / 0'3 cos2 9) COS2W hr;

The force acting on an element at D is Figure 2

(cl/2) (aw) (cl/2 was) (d9) (0').“
Then displacement of the element is
E‘sin 2e d/2 cos 2w
The work done against friction is‘
fisin 2€ (d/2 cosy) (p) (9.14? coswdw d9)
[dismew / (0’. sin2 9 /t!"=5cos2 9) cos2wj
Integrating around the slice BADC with respect to W from
-1\‘/2 to 1/2, one obtains
6.2%3 sin 29 d9 [cam/8) / r. sin2 e / 03 .032 e) (yr/8)]
Integrating around the sphere with respect to 9,
Work : w : 6.}1d3W/32 (307/ 3'3/ 212)

The energy per unit volume is

 

w ( 3gx 8 A)
Fx 11 x d5
or spa/16 (3r. / 363/ 2a) (1+)
Now at constant'volume, the work done byr, andT'3 is
_1_ (r. 43) e. ' (5)
2

By equating equations (A) and (5)

01-03 - 3/8 ,1 (3o: / 3r, / 203.) (6)

Assuming T2. 3 a (0'; /0’3) (7)

 

and substituting (7) into (6)

. - - s
3; 3:3; -3/}1(3_/28)

By Mohr's circle (figure 3)

in '1: «'0’ JFK/EF- l

8 4r —7—.-, .33 F‘T—t/bg 1 (8)
Taking a = 1/2

sin¢f:3/8p(3/1)=3/2p (9)

(b) Triaxial Shear Test

The relation between.¢§ and P was analyzed for the case of the

triaxial test using the same general

 

 

 

approach. - 1
In the case of the triaxial test,
at constant volume .t
61/52.)!532031'1‘151263 ¢
... 55"- " 61/2 lye—0'3 —J M
For shearing strains in the body equal b———-———?21§;§ 2
to y and x (see figure 1), I‘F—ed1 ‘. ae:e1~'
tan 9 : y/x
and d e = 3/h €.sin 29 Figure 3

As in the case of plain strain, a solid spherical element will
be used to compute the average work done per unit volume due to the
combination of stress and strain in all directions. Then the force
acting on an element at D (see figure 2) is

d/2 x dwx d/2 cosxyx d9 x u"
A relative displacement of the element will be

3/h E‘sin 29 x d/2

The work done against friction is

(3/h 6. sin 29 d/2) x (P) x (d2/h cosvdxfde)

xLT-S sinew/ (UT sin2 9 / 0'3 cos2 9) cosZW]
Integrating around the sphere with respect to\r gives

3/32 6‘11 d3 sin 29 d9 [2/30‘3/ (“7 sin2 9 l0} cos2 9) V3]
Integrating with respect to 9 yields

'nyz
.w - it 3/32 Sp 6.3 (2/303 sinZG / 8/301 sin3 9 cos 9

118/303; cos3 9 sin 9) d9 - l/h 6,11 d3 (61/203)
Hence energy per unit volume is

31w a}: (r. .1263) (10)
Now at constant volume, the work done by“? and 0’5 is

1/2 (mas-2.63 {-33 and since€3 : -€./2

w = .6_: (r. 413) (11)
By equatiig equations (10) and (11)

LE” (0‘? 21203,) :_€_.(°’t -°':5)
21‘ «3 2
andE'L:_"F l
03 {EN-1 *
Then from equa Thu 8
sin ¢ : 9
f 3p72n (l2)

ANALYSIS OF SHEARING RESISTANCE BY CONSIDERATION
OF INTER-PARTICLE MOVEMENTS
Newland and Allely (8) analyzed inter-particle movement to
explain the effect of dilatancy on shear strength. Figure A shows a
shear stress applied in a horizontal direction, causing particles
a, b, c, etc., to move to the right relative to particles a', b', c',
etc. Excluding grain failure, for particle a to move to the right

relative to particle a', it must initially slide in a direction making

an angle 9a to the direction of the shear stress. Each particle,
therefore, has a component of move-

ment in the vertical direction and the
mass consequently expands against the

normal stress.

 

Forces parallel and perpendicular to the
initial direction of movement of particle a

may be resolved as Figure A

 

Ta Aa cos 9a -5"’ As sin 9a : tan¢
“Ca Aa sin 9a {5'- Aa cos 9a P

Here SP is defined as the angle of sliding friction and tan ¢P is the
coefficient of friction. Simplifying,

'Ca As :5: As (tanEP/Ga)
Similar relationships may be obtained for other surfaces of sliding as

I;:;Aa tan (¢n g/ 9a)/ . . . AatanA(U: / 93) (13)

.7— na ’Ab.........n

Now if particle a moves a distance 8 (A) in the direction of the

shear stress, it raises against the normal stress? a distances (ya)
such that

tan 9 ;8§za)
A.

Then considering the mass as a whole,

t 9 - 85v;
an _ A (11‘)

As sliding begins, the individual values of G are a maximum
except in very loose sands. Hence the shear stress and rate of volume
expansion will attain maximum values oftmax and 6_E_\g_g_max. I max may be
considered to consist of a shear stress ‘C' neceihagy to overcome

the frictional force, assuming the individual values of 9 are equal

to zero (I) ; tan 95"), plus the shear stress‘td required to overcome
the resistance to expansion against 6" because the individual planes
of sliding are inclined at some angle to the shear stress.

As the shear displacement proceeds, the shear stress drops to
a residual value,TiR. If at that point the expansionh v has ceased,
then the average value of O is equal to zero. ThenTR 6sh£uld equal
the computed value of‘C'. The residual angle of shearing resistance
95R is than equal to ¢P.

Equations (13) and (1h) have not taken into account the fact
that the movement of each particle will be restricted by the movement
of its neighboring particles. Conceivably, the particles having the
steepest initial surface of sliding I ,

_—-. 1
may control the expansion with the ..L—n”

 

remaining particles "bridging" over

their former contacts as shown in Figure 5.

 

 

 

 

Due to the normal load, the bridges may Figure 5
continually develop and collapse as shown
in Figure 6.
t Twas
The slope of the steep-rising ITR
portion of the stepped curve in Figure 6 £9. E
is that which should be used with the peak 3
value of the shear stress in equation 13 g
to obtain the true value of 96“. If the > I STRAtN
flatter slope of the experimental dashed Figure 6

curve is used, the value of ¢u obtained will be larger than the true
value. Because of the continuous collapse of the bridges, a measured

6 V; of zero does not always mean that O is zero. Hence, the computed
h A

10
¢P and't' are dependent upon the mode of failure as well as the
coefficient of friction and will be called gin and 1’". gain is called

the angle of internal friction. Equation 13 now becomes

 

Tmax ; tan (Un / emax) (13)
E
and Qmax : tan '1 Séxgmax
8 A (16)
Then ¢n ; tan-11:: : ¢ - 9 (l7)
6'-

The shear stress represented by 9 is calledTId.

Hence by Newland and Allely's analysis, 9 is deducted from ¢
to obtain ¢n. The value of ¢n so obtained will be equal to ¢P only if
complete "bridging" occurs.

The value of ¢n found from equation 17 will be equal to the
residual shear strength ¢R, when.é{§%_: 0, only if the mode of failure
at the end of the test is the sam: 28 it is at the peak point. Since
it has been mentioned that because of the collapse of bridges the
measuredt v is not a reliable indication of s, it is not likely that

6 ti
¢n will be equal to 95F or ¢R’

II. FRICTION

NATURE OF FRICTION

The experimental laws governing friction state that frictional
resistance is directly proportional to the load and is independent of .

the size of the surface in contact.

Metals

Much more is known about the frictional behavior of metals than
of non—metals. F. P. Bowden 69), who is well known for his studies of
frictional resistance, explains the laws of friction in terms of the
surface contour of solid surfaces. The engineers best surfaces have
irregularities which are thousands of angstrom units high (Figure 7).
Electrical conductivity experiments have

found that for flat steel surfaces, the

 

actual area of contact may be only one
ten-thousandth of the apparent area. Thus
the actual area of contact depends mainly
on the load which is applied to the

surfaces and is directly proportional

 

 

 

to it. Therefore, even with lightly

loaded surfaces, the local pressure at Figure 7

these small points of contact is very high and may cause the hardest
metals to flow plastically until their cross sectional area is sufficient
to support the applied load. The two surfaces thus adhere or weld

ll

12
together at points of contact. The actual area of contact is

A : W/P (18)

m
where W is the load and Pm is the yield pressure of the metal.
Bowdeéggtates that there is strong evidence that the friction
of metals is due, in large measure to adhesion at these contact regions
and represents the force necessary to shear these Junctions. The fric-
tion, F, is approximately equal to As, where s is the shear strength
of the;hnctions. For most solids, whether plastic, brittle, or elastic,
the surface adhesion can be strong.
Since the EEEl.§IEE.°f contact is directly proportional to the
load, so is the friction. The value of the coefficient of friction
y, will then be a constant since
: K

P 3 F : As - x

_ __—F_
P

§.
w w w

(19)
The characteristic frictional prOperties of metals are seen
to be due largely to their ability to flow plastically and to weld

together under load.

Non-Metals

 

Some non-metals have frictional characteristics similar to metals
while others are quite different. Extensive studies were made by Bowden
and Young DJ» to investigate the frictional behavior of diamond. The
deformation of diamond was found to be principally elastic rather than
plastic and hence the real area of contact is expected to be preportional
to W2/3 rather than W. The coefficient would no longer have a constant

value, but vary as W'l/3 since
)1 : :l_\__:: W2/3S- KW-l/3
W

W

 

wpm (20)

l3

Experimentally, P varies as KW'O'z for clean degassed diamond
surfaces. This indicates that the deformation is largely elastic.

The adsorbed surface film of oxygen and other gases normally
present has a marked effect on friction. For clean diamond exposed
to air, u is about. 085 at a load of 10 grams. With the adsorbed
gases removed and specimen tested in vacuo, P.1ncreases to almost .hS.

The orientation of the crystallographic axis of the mineral to

sliding has a large effect upon its resistance to sliding.
FRICTION EXPERIMENTS ON MINERALS

ggggg investigated the frictional characteristics of quartz
and flint. A block of a mineral.was cut flat and its face roughened.
Three 1/8 inch diameter particles were then slid over the block under
various normal loads. For both minerals, the value of gpdecreases
with increasing loads as shown in Figure 8. The average sliding value
is given in Figure 8. The value of P for quartz ranges from .380 to

.h927and for flint .27h to .366 depending on the normal load.

Eschebotarioff and Welch (11) conducted a series of friction
tests on quartz, calcite, pagodite, and perphyllite under dry, moist,
and completely submerged conditions. Dry tests were performed immed-
iately after removing the minerals from the desicator. A two inch
polished cube of each mineral was slid over mineral fragments at normal
loads up to about 36 lb.

The value of the friction remained almost constant for all
loads. The value of P.for quartz varied from .11 in the dry condition

to .hS when submerged and the corresponding values of p for calcite

1h
were .11 and .26.

A distinct difference exists between the frictional character-
istics of the hydrophilic minerals, quartz and calcite, which have an
affinity for water; and the hydrOphobic minerals of the talc variety
which are water repellent. Water has a slight lubricating effect on
the hydrophobic minerals and decreases the frictional resistance.

However, water significantly increases the frictional coefficient for

the hydrophilic minerals.

Penman (l2) investigated the coefficient 3: friction for quartz.
Two fairly large quartz crystals were imbedded in plaster and tested
at a constant rate of strain. The surfaces were washed with soap and
water, rinsed with distilled water and submerged during testing. The
measured frictional coefficient is .650 for normal loads ranging from
2.96 lbs. to 151.3 lbs. For quartz crystals dried in an oven at
1050 C and tested while warm, the value of is .195 for the same range
of normal loads. The area of the upper quartz surface is about 1.2 sq.
inches so that the maximum test load is about 126 psi. No damage on
the quartz surfaces was reported.

To produce higher stresses, three freshly broken chips were moved
over the lower quartz surface while saturated with distilled water.
For normal loads increasing from h.l to 1&5 lbs., P decreases from
.555 to .3h5. Crushing of the points was noted at all loads above
100 lbs. The coefficient of friction thus decreased from .650 for a
normal load of 126 psi to .3h5 at a much higher load.

Recently (1959), friction tests were carried out at the

Norwegian Geotechnical Institute (33). Three points of a mineral were

15
slid over a crystal at a constant velocity while submerged under various
liquids. The value of P for quartz varies from about .0625 to .lhl
when submerged in water, and from .156 to .312 when submerged in alcohol.
The load varied from about 5 to 35 gms per point. Both the load and
the direction of sliding influence the coefficient. The value of F
decreases with increasing normal load for some directions of sliding,

while in others, it is constant.

Shear Strength of a Loose Sand. A series of triaxial tests at the

 

Norwegian Geotechnical Institute (NGI) (14) on a fine loose sand
(primarily composed of quartz) produced some unexpected results. USing
the consolidated undrained constant volume test, at initial porosities
near h3 per cent the angle of internal friction is about 35°. However
as the porosity increases from #3 to h7.5 per cent, ¢f drops off
sharply to around 120. High pore pressures were recorded in the tests
on the very loose sands.

It seems probable that the very low values of ¢f obtained
represent mainly the frictional resistance of the mineral grains
(1. e. ¢ffv ¢ P)° If so, this is in agreement with the results
of friction tests at NGI. Assuming that the value of ¢P is equal to
120 (P = .213), the value of ¢f computed from equation 12 is 16.10,
which is rather low. In other words a value Of/J. equal to 120 cannot

account for an angle of internal friction of 350.
SUMMARY

From the work of Bowden and Tabor, it is seen that friction

between two surfaces will depend on the true area of surface contact.

16
For materials that behave plastically, the actual area of contact
increases in direct pr0portion to the load and hence P.is constant.
However, for non-plastic materials, the true area does not increase
in direct proportion to the load and F.1s not a constant. Fer an
elastic material, P.is pr0portional to the -l/3 power of W.

Large differences in the value of p for the same mineral have
been reported in the literature. Penman, Tschebotarioff, and Hafiz
found the value of P.for quartz to range from .3h5 to .650 when moist
or submerged in.water and from .11 to .195 when dry. The lower values
were obtained at high normal stresses. However the friction tests
at NGI (under submerged conditions) resulted in values of p. from
.0625 to .lhl, while triaxial tests gave values of fin as low as
12° (P z .213). The values of u obtained at NGI seem quite low when

compared with the other tests.

III. EXPERIMENTAL PROGRAM

OBJECT

The purpose of the experimental program is to examine the shear
strength of a cohesionless soil by an evaluation of the frictional and
dilatancy components. It is believed that a study of these components
and the factors influencing their relative magnitude would improve the
understanding of shearing resistance, and facilitate in prediction of
the behavior of a soil under various load conditions in the field.

The investigation includes the effects of initial voil ratio, particle

shape, and normal pressure.
PROPERTIES OF THE SANDS INVESTIGATED

Angularity

Three sands with very different particle shapes were used in
the tests. Ottawa sand.was used because of its characteristic roundness.
A typical Michigan glacial sand was used as a sub-angular soil, and a
residual sand from Georgia containing specs of mica was tested because
of its extremely angular grains. Micrographs of the angular sands are
shown in Figures 9 and 10.

The roundness and sphericity of the sub-angular and very-angular
sands are 0.390, 0.800; and 0.175, 0.787 respectively. Roundness is
defined by Wadell (15) as ififl and sphericity as dc/Dc where

H number of corners on a grain

R the radius of the maximum inscribed circle

17

18

r : the radius of curvature of a corner

dC ; diameter of a circle equal in area to the area obtained
when the grain rests on one of its larger faces
Dc ; diameter of the smallest circle circumscribing the grain
reproduction
Minerals

The ottawa sand is composed of quartz, while the angular sand
containei a mixture of feldspar, quartz, and other minerals usually
found in sawed. soils. The very-angular sand contains. a significant

amount of mica.

Grain Size-Graduation

 

Two different ranges in particle size were tested.

Sand type A contains only grain sizes from .590 to .297 mm in

 

diameter. This is the size range that passes a #30 U.S. standard
sieve and is retained on a #50 sieve.

Sand type B contains grain sizes from .250 to .lh9 mm.in

 

diameter. These grain sizes pass a #60 U.S. standard sieve and are
retained on a #100.

The majority of the experimenta1.work was performed using
sand type A. Hence the following discussion refers to type A unless

otherwise specified.

Maximum and Minimum Void Ratio

 

In order to compute relative density_DR, the sands were tested
to find their respective maximum and minimum void ratios. Methods

develOped by J. J. Kblbuszewski (15) were used. Tb obtain the loosest

19
possible state (highest void ratio), 250 grams of dry sand'were placed
:h.a one liter graduated cylinder. The cylinder was shaken a few times,
turned upside down, and then very quickly turned over again. The volume
of the sample was read and its void ratio calculated.

The densest state was obtained by using a vibrating table. The
sand was placed in a brass mold, 3 inches in diameter and 3 inches
deep, which was clamped to the table. A tight fitting cap was placed
on the sand and a 100 gram'weight was placed on top of the cap. The
sand.was placed in 3 layers and vibrated 5 minutes for each layer.

The void ratios obtained are shown in TABLE I.
SHEAR TESTS

Drerriaxial Toots

A series of dry triaxial tests were performed at various degrees
of compaction. The specimens were approximately l.h5 inches in diameter
and 3.10 inches long. A schematic diagram of the triaxial apparatus
is shown in Figure 11.

A constant all around effective stressfg , of about .960 Kg/cm2
was used in all tests. A vacuum was applied to the sample through
the burette E (Figure 11), thus utilizing atmospheric pressure to
supplyiFi.

The deviator stress was applied at a rate of approximately O.h

per cent per minute until failure.

Consolidated Uhdrained Triaxial Tests
A series of consolidated undrained (CU) triaxial tests were
carried out in an attempt to study the effect of an extremely high

void ratio. To obtain the highest possible initial void ratio, the

20

sand.was carefully placed in the membrane at a moisture content of
about 11 per cent. At this moisture content the capillary forces
create an adhesion between the grains resulting in.a "honeycombe"
structure. The sample was then saturated at either a very fast or a
very slow rate in an attempt to allow only a minimum of consolidation
as the capillary tensions were destroyed. The time allowed to saturate
the sample varied from.one to as much as 90 minutes.

The sample was then subjected to a very light vacuum of 1 to 3
inches of mercury through the burette (B in Figure 11), the mold removed
, and the dimensions of the sample measured. Specimen sizes were the
same as those used in the dry triaxial tests above.

After consolidation by a hydrostatic pressurelri, the cell
pressure was increased and at the same time a porewater pressure of the
same magnitude was applied. This procedure was followed in an attempt
to completely saturate the sample by compressing and dissolving the air
bubbles in the porewater. As the cell pressure and pore pressure were
increased simultaneously, the effective stress remains unchanged.

P018 pressure measurements were node by balancing the water level in
the capillary tube A as shown in Figure 11.
The specimen was then sheared at a constant 0'3 under constant

volume conditions by the application of a deviator stress.

Egrect Shear Tbsts .

Well over 50 direct shear tests were made under both saturated
and dry conditions. The direct shear apparatus used takes a circular
specimen 2.5 inches in diameter and approximately 0.8 inches high. The

shear stress‘f, was applied at a rate of approximately 1.0 per cent

21
per minute.
Sand types A and B were tested with the majority of tests run

on type A .

Friction Tests

Friction tests were run on quartz crystals to check the variation
in the coefficient of friction with normal load. The direct shear
apparatus was adapted for this purpose. A quartz crystal approxi-
mately 1}} inches long and it inch wide was set into a block of plaster
of paris which was carefully sized to fit into the stationary part of
the shear apparatus. Another crysta1.was set in a similar manner into
the movable (top) half of the shear apparatus so that its point would
hear on the stationary crystal (see Figure 12).

The quartz was not polished or cleaned in any mnner so as to
leave its surface in the same condition as that of the sands tested.

Both dry and saturated tests were made with normal loads from

lto 21} Kg.

IV. RESULTS OF TESTS

SHEAR STRENGTH FROM DIRECT SHEAR TESTS

The results of the direct shear tests are summarized in TABLES
II, III, and IV. The shear strength was divided into frictional and
volume change components by the energy method and by Newland and
Allely's particle movement method. The angle of internal friction ¢f,
as found by the energy method, was calculated using equations 1 and 2.
The angle of internal friction by Rowland and Allely's method, ¢n, was
computed by equations 15, 16, and 17. The calculated ¢f and ¢n are
also given in TABLES II,QIII, and IV.

The value of ¢ varies from 143.90 to 29.30, while the computed
value of ¢f ranges from hl.8 to 26.50. Newland and Allely's analysis
yields values of ¢n from 10.10 to 25.70.

Typical stress-strain curves are shown in Figures 13, 1h, and
15 for the 3 sands. Values of tan ¢, tan ¢f and the maximum.value of
¢n are plotted versus the horizontal displacement. The volume change
is shown below the stress-strain curves in each case.

The ultimate or residual shear strength of the sands is taken
at the part of the stress-strain curve where the volume change has I

ceased such as point "a" in Figures 13, lb, and 15.

Effect of the Initial Vbid Ratio

The increase in the angle of shearing resistance ¢, with de-
creasing void ratio is a well known relation. The greater shear strength
exhibited by a dense material is due mainly to interlocking of the

22

23
particles. It would be expected than, that the shear force required to
cause dilation,‘[v, would account for most of this increase in strength
and that 961. would be almost unaffected by density.

Figures l6, l7, and 18 show the results of the direct shear tests.
values of ¢, ¢f, ¢n7and ¢R are plotted against eo for a normal stress
of .7575 Kg/cme. The value of ¢ is seen to vary as much as 8°. However,
¢f and ¢R remain nearly constant for all values of eo thus confirming
the concept thatTv is mainly responsible for the variation of ¢
with so. However, ¢n increases significantly with increasing so.

The value of'T§ and‘td expressed as the per cent of the total

shearing resistance increases with decreasing e as shown in TABLES II,

0
III, and IV. The average values of'Cv and‘t’d are 16.9 per cent and

23.7 per cent respectively for the 3 sands.

Effect of Relative Density

 

The relative density of a soil is defined as

DR 3 emax ‘ e

e '8

max min (21)

Hence, a soil in its densest possible state would have a DR of 100
per cent and in its loosest possible state a DR of 0 per cent.

The values of ¢, ¢f, ¢n, and ¢R are plotted against DB in
Figure 19 for the sub-angular sand. It may be seen that ¢ increases
almost linearly with increasing DR as would be eXpected. ¢f and ¢R
however, which have very similar values, are almost independent of DR'
The value of ¢n decreases with increasing DR.

Tb study the effect of DR on the shear strength due to volume

change, the percentages of‘C§ and'td are related to DR in Figures 20

2h
and 21. The values of 'Cv and 13d as a percentage of the total shear
strength increase with increasing DR. It is interesting to note that

the curves for the 3 sends are quite similar.

Effect of Normal Load

Taylor and others have reported that ¢ is also dependent on
normal pressure. A series of tests were performed varying the normal
load from 8 to 32 Kg. An attempt was made to keep the variation in
eO to a minimum.

The angle of shearing resistance decreases up to 10.60 with in-
creasing load as is shown in Figures 22, 23, and 2h. The angles of
internal friction,¢f and ¢n, decrease with increasing W by an amount
similar to ¢. This indicates that the components of shear strength
attributed to'l:v and 11d (which had already been deducted from 95 to
obtain ¢f and ¢n respectively) are not responsible for the decrease.
The decrease is believed to be primarily a function of the frictional
characteristics of the mineral grains as discussed in a subsequent part
of this paper.

In order to compensate for changes in ¢ which may have been due
to variations in so, the relationships between ¢ and eo (Figures l6,
l7, and 18) were used to correct the experimental values. The corrected
¢ versus eo curves are also plotted in Figures 22, 23, and 2L and are

designated as ¢c'

Effect of Particle Shape

 

A sand containing angular grains is expected to have a higher

shear strength than a sand with predominantly round grains. Figure 25

25
shows the results of the direct shear tests on the round (ottawa),
sub-angular (glacial), and very-angular (residual) sands. The values
of ¢, ¢f and ¢n are plotted against DR for each sand.

There is no significant difference in the shearing resistance
of the sub-angular and very-angular sands. However, the round sand is
seen to have values of ¢, ¢f, and ¢n which are 2 to 3 degrees lower
than the other sands. One reason for this difference may be that it
requires more work to roll and slide a random arrangement of cubes over
one another than spheres.

It must be recognized here that the difference between the shearing
resistance of the round and angular sands shown in Figure 25 is also

influenced by the mineral composition of the sands.

Strain at Maximum Shear

 

To obtain a better insight into the various factors contributing
to the volume change component of shear strength, an analysis was made
of the shear displacement at which the maximum shear occurred, Am.

The tests indicate that.Am (see TABLES II, III, and IV) is
influenced mainly by 3 factors. These factors are initial void ratio,
grain shape, and particle size. A low initial void ratio causes the
maximum shear resistance to occur at a lower strain than a high initial
void ratio. The maximum shear occurs at a smaller strain for the round
than for the angular sands. The value of Am is smaller for type B
sand (which contained smaller grains) than for type A sand.

At maximum shear strength all sands show expansion even though
they had at first contracted. The particle diameter of the sand ranges
from 0232 to .00586 inches while the strain at maximum shear ranges

from .035 to .150 inches.

26

The observed relationship between Am and e0 may be eXplained as
follows. When a normal load is applied to a very loose sand the whole
mass undergoes consolidation. As the sand is subjected to a shearing
strain (as in the direct shear test) the particles in and around a
relatively narrow shearing zone will consolidate without changing
partners until they attain a certain critical density or void ratio.
At this point the grains in the shear zone begin to rise up on one
another causing expansion of the mass against W. This results in a
maximum value of the shear stress. In a very dense sand however, as
shearing takes place there is little or no consolidation of the particles
in the shearing zone. Therefore at a relatively small strain the
particles begin to slide and roll over one another producing a maximum
value of I.

As noted above, the round sand reaches its maximum shear strength
at a lower strain than the more angular sands. A possible explanation
for this might be that the round sand consolidates more readily during
shear and hence reaches its critical density very quickly. Expansion
thus begins at a lower strain.

It was found that Am is smaller for sands with smaller grains.
Since maximum shearing resistance occurs as the particles rise upon
one another expanding against‘w, it would be expected that expansion

would begin sooner in a finer sand, thus causing Am to occur at a

relatively low strain.
DRY TRIAXIAL TESTS

A series of dry triaxial tests were performed for comparison

with the direct shear tests. Pertinent data from the tests are shown

27
in TABLE V. The effective normal stress on the shear plane 0." , was
maintained at about 1.52 Kg/cm2 for all tests. The values of ¢ obtained
are plotted against co in Figures 26, 27, and 28 (line B).

Line A in Figures 26, 27, and 28 represents the results of direct
shear tests at at? of .7575 Kg/cme. Using equation 23 (see page28 ),
and the respective values of K and B as found for each sand, the values
of ¢ were corrected to a normal pressure of 1.52 Kg/cm2 to obtain a
better comparison with the triaxial tests. The computed values are shown
as line C.

The values of ¢ as obtained from the direct and triaxial tests
are seen to agree within 1 or 2° except for the sub-angular sand where
there is a ho difference. The agreement is believed to be good con-
sidering the limited number of triaxial tests performed.

It has been suggested by Hill(%hat in the direct shear test, the
deformation is so constrained as to be effectively a simple shear in a
narrow zone. The direct shear tests thus 15 b
give a Mbhr stress envelope such as a in a

Figure 29. The triaxial test, however,

gives a tangent to the stress circle such IA? 4.x

 

as b. Thus the relation between the shear-

 

\.
q:

ing resistance as obtained by the triaxial
and the direct shear test would be Figure 29

sinve tan x (22)
The values of ¢ from the triaxial tests were reduced by equation 22
in order to give a comparison with envelOpe a. The values obtained are
plotted as line D in Figures 26, 27, and 28. The agreement between

the two is not satisfactory.

28
CONSOLIDATED UNDRAINED TRIAXIAL TESTS

Figure 30 shows a typical plot of deviator stress and pore :
pressure versus per cent strain for a typical test. The porosity is
hl per cent and the value of ¢f is 32.80. Since the tests are not
successful in producing extremely high porosities with exceptionally

low values of ¢f, the series was discontinued.
RELATIONSHIP BETWEEN SHEAR STRENGTH AND FRICTION

Friction Tests

 

The results of the friction tests on quartz are shown in Figure 31.
It may be seen that the coefficient of friction is not a constant, but
is a function of W. The data was found to follow the general equation

11 : NE (23)

K and B are constants depending upon the stress-deformation character-
istics of the material.

The values of K and B are equal to 0.629 and 0.138. No signi-
ficant change innF.was observed when the quartz surfaces were saturated
with water. When the normal load exceeded 16 Kg., pieces of the point

broke off and the lower quartz surface was damaged.

Friction From Direct Shear Tests

The coefficient of friction was computed for each direct shear
test using equation 9. The computed coefficients for the 3 sands
.decrease with W according to equation 23. For the round, sub-angular,
and very-angular sands the respective values of K and B are 0.h9l,
0.139; 0.h89, 0.0965; and 0.610, 0.171. These values fall in the same

range as the values obtained from the friction test.

29

The angle of sliding friction ¢ is plotted against W in

}1
Figures 32, 33, and 3h. The values of ¢c’ ¢f, ¢n, and ¢R are shown for
comparison.

It may be seen that the decrease in ¢R’ ¢, ¢f, and ¢n, with in-
creasing W‘ésimilar to that of (JP. Hence it seems that the decrease in
shearing resistance with increasing normal load is primarily a function
of the frictional properties of the minerals involved.

(a) Comparison of Normal Stresses During Shear

In order to better compare the values of P measured in the
friction tests with those calculated from the shear tests,.it is necessary
to estimate the contact stresses between the sand particles in the shear
tests.

Assuming the particles are spherical, Hafiz computed the contact
load per particle for extremely loose and dense configurations.. For
particles in a loose rectangular pattern (Figure 35), each sphere
touches six others. Then in a cross section of area A, the number of
spheres is N, the diameter of each sphere is d, and
N : A/d2

For a normal load of W, the load per particle,

 

p, will be
P : HE?
A

In a dense rhombic type_packing,
each particle touches twelve others. Figure 35
It may be seen from Figure 36 that

B:1+5°, Ly3h5°andW/n:1+pcos\{z 351%

then p : w ; Wd2
2.83N 2.83A

 

The loose packing has a void ratio

of 0.92 and the dense 0.35. Figure 36

30
Assuming that the average void ratio of the sands is midway

between the dense and loose packs, the load per particle p would be

p : m2 (21+)
A

Using equation 24 , the approximate load per particle was computed for
each normal load and is shown in TABLE VI.

The contact pressure between particles was computed with the
Hertz Equations (18) for an elastic material. For two spherical

bodies in contact, the maximum pressure, qo, is

 

S’WE2
RlZS/ R2: . (25)

 

and in the case of a ball pressed into a plane surface,

' 3
qo : 0.388 WE2
F2; (26)

In these equations,

W : Normal Load
E : Modulus of Elasticity
R1’ R2 : radius

An average modulus for quartz may be taken as 9.25 x 108 gms/cm? (19).

The radius of the quartz point was estimated to be between 1/16
inch and 1/8 inch. The coefficient of friction is plotted against qo
for the direct shear tests and the friction tests in Figure 37. The
contact pressure computed from Hafiz's friction tests is also plotted
-in Figure 37.

The curve for friction tests is not in agreement with those for
the direct shear and the friction tests by Hafiz. At least part of the

difficulty lies in the uncertainty of the radius of the quartz point.

V. CONCLUSION

VOID RATIO AND RELATIVE DENSITY

The well known increase in shearing resistance with decreasing
eo or increasing DH is primarily due to the increased shear force
necessary to cause dilation against W. The angle of shearing resistance
as well as the part of the shear strength necessary to overcome volume
change increase almost linearly with increasing DR’

¢f and ¢R, which were found to be quite similar, are essentially
independent of eo and DR. The value of ¢n, however, increases with

increasing void ratio.
NORMAL LOAD

The angle of shearing resistance was found to decrease as much as
12° with increasing normal load, W. The values of ¢f and ¢n decrease
with increasing normal load by an amount similar to {25. Since the
dilatancy components have already been deducted from ¢ to obtain ¢f
and ¢n, the decrease in shearing resistance with increasing W cannot be

due to dilatancy.
PARTICLE SHAPE

The values‘Ev.and'[d as a percentage of the total shear strength
for the 3 sands are quite similar when plotted against DR‘ The more
angular sands have values of ¢, ¢f, ¢n, and ¢R 2 or 3° higher than the

round sand.

31

32
COMPARISON OF THE ENERGY AND PARTICLE DISPLACEMENT
METHODS OF ANALXSIS
The analysis of the shear strength by the energy method
leading to ¢f and NP seems to yield quite consistent results. However,
the analysis of particle displacements gives values of ¢n which
increase with the initial void ratio. The uncertainties concerning

the mode of failure make the physical significance of On rather dubious.
FRICTION

The coefficient of friction was found to be a function of W in
both the friction and the direct shear tests. The data was found to
fit the general equation

FWB
The values of K and B for the friction tests are 0.629 and 0.138. In
the direct shear tests, the respective values of K and B for the round,
sub-angular, and very-angular sands are 0.h9l, 0.139; 0.h89, 0.0965; and
0.610, 0.171.

As W increases, 0P decreases in a manner similar to the decrease
in ¢R and ¢. It is therefore believed that the decrease in shearing
resistance with increasing normal load is mainly due to the frictional
pr0perties of the mineral.

The frictional resistance between two minerals will increase as
the true area of contact between them. For both the friction test and
the direct shear test on the quartz sand, the value of B was approxi-
mately 0.139. The true area of contact A is pr0portiona1 to WO'861'

Hence, the deformation behavior of the quartz lies between the elastic

(A proportional to W2/3) and the plastic (A pr0portional to W) states.

33

 

o HAF’VZ FR\CT\ON TESTS

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0216 0217240

.0. was;

 

 

 

 

 

 

tV(°/o GIT)

FIGURE 2.0
W814 KG
- 7.8.0
- 24.0
SUB-AME:-
-zo.o
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535.0I .030 .1 730 .zoo

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FIGUREZ I

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‘6‘ FIGURE 22

ROUND SAND
¢¢ -A
:58 O - x
A’s-o
q’n-n

-36

 

 

I: \ \o

-26 \
\ n
3 W— '9 Z? 13‘ fi'\ -3?-

 

NORMAL LOAD ( K63

 

 

MI

 

 

 

FIGURE 2‘!)
SUE’ANG. SAND
«Ia-A

4; .- x

Q+~e

Ia-s

552° \ o O
a
\f‘b"
-30 \ \ E.
a ......
S I}. I? ZLO 234 2.3 3.7..

 

NORMAL. LOAD (Km

 

 

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D
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VERY-ANS SAND

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NORMAL LOAD (KG)

 

 

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DEVIATOR STRESS (KG/cm‘)

PORE PRESSURE( KG/sz)

3.7.0.

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FIGURE '50

ROUND SAND

 

POROSITY = 4 I “I.

q, = 32.8.

 

 

4 8 I2. I16 210

L l I

°/c ST RAIN

 

 

 

51

 

 

FIGURE BI

FRICTION TEST ON
QUARTZ POINTS

 

 

 

NORMAL LOAD (KGI

 

 

52

53

 

 

 

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FIGURE 37
TESTS ON

‘_. .300 QUARTZ

.{00 .450 .sbo ' .150 l I 1.50 L'SO 2.1.30 2.150

MAXIMUM PRESSURE; qo
( x m" qm/cmz')

 

 

56

 

 

5SAND BIEVE era; .°max 3min
Round 50-50 .7575 .488
Subnlngular 30-50 .854 .559
Veryehngular 30-50 1.204 .763
TABLE I MAXIMUM AND MINIMUM VOID RATIO

 

 

57

W—m 4-4 -—

 

 

 

 

 

TEST # 60 “KS???“ bf" ID; ¢R° ,1 A mom...)
DRY TESTS
Round Sand
5 .615 8 57.5 55.1 54.0 54.0 .584 .060
16 .564 12 55.0 51.6 50.2 50.2 .549 .055
1 .504 16 52.5 28.8 27.5 29.9 .522 .080
-10 .578 20 54.6 28.9 26.6 28.8 .522 .090
;19 .521 28 54.0 27.6 25.5 28.1 .509 .050
5 .508 52 29.5 26.5 25.7 25.4 .297 .045
. . Sub-Angular Sand
9 7 .724 8 40.8 57.8 55.8 59.1 .410 .080
i 17 .685 12 59.8 55.2 52.5 55.6 .585 .050
‘ 8 .770 16 55.9 55.2 51.9 54.2 .566 .070
.11 .699 20 56.6 51.8 29.6 55.2 .552 .070
15 .645 24 58.0 55.6 51.5 52.8 .570 .050
20 .665 28 58.1 52.0 29.1 52.6 .554 .070
9 .756 52 55.6 52.6 '52.2 51.1 .560 .070
TABLE II DIRECT SHEAR TESTS ON SAND TYPE A—W VARIED

 

 

TETI 0°

Veg-Angular Sand

 

6 1.03 8
18 .939 12

2 1.01 16
12 1.04 20
15 .923 24
4 0.892 32
Round.Band
3-3 .525 8
1-8 .598 16
2-5 .550 32

W(K8)

fie

43.9
40.1
36.8
36.2
36.4

33.3

bro

41.8
37.2
32.8
33.4
32.1

30.3

An“

40.1
35.4
30.9
31.9
30.1

29.2

SATURATED'TESTS

41.2

34.5

33.5

33.8
29.7

27.1

29.7

27.9

25.0

152

42.0
36.3
35.0
34.8
34.2

30.1

36.0
30.3

28.6

’1

.445
.403
.362
.368
.355

.337

.372
.330

.304

Am (NU-0

.150
.080
.080
.100
.080

.110

.035

.045 4

.040

TABLE II Continued DIRECT SHEAR TESTS 0N SAND'TYPE.A4W VARIED

 

 

 

 

 

TEST I 00
Round Sand
26A .590
27 .557
28 .525
29B .462

31.8

34.2

38.5

40.1

Sub-Angler SaE‘d

 

 

25 .640 37.6
22 .619 37.8
23A. .586 39.6
24A .562 42.4
Very-Angular Sand

30 .969 35.6
31 .974 36.6
32 .814 40.6
33 .764 41.0

TABLE III

4’

30.9
28.1
30.8

28.4

33.0
32.8
28.8

32.8

32.5
34.8
32.6

32.7

bn

1113065

25.9

27.2

23.4

31.0

30.4

27.5

31.0

34.0

28.2

28.0

25’

DRY TETS - w = 24 Kg

29.3

29.3

29.3

30.5

33.4

33.4

32.0

32.5

33.5

33.5

33.2

)1

.343
.314
.342

.318

.364
.362
O 322

.362

.359

..381

.359

.361

DIRECT SHEAR TESTS ON SAND TYPE A?" HELD

A m(|NCHE$)

.100
.045
.065

.035

.065
.055
.050

.040

.090

.110

.060

.050

CONSTANT

 

 

 

TM} .0 Hits) 0° 4° 41°

 

DRY TESTS
Sub-Angplar Sand
50 0.672 24 35.3 29.1 26.7
51 0.672 12 37.8 32.3 29.7

”R0

34.7

35.2

P Aunbumfif
.369 .045
.411 .045

TABLE IV DIRECT SHEAR TESTS 0N SAND TYPE B - I'VARIED

 

 

 

 

TEST i so 6'3 (Kg/omz)
Round.Sand

l .553 .960

3 .570 .960

5 1.350 .960

Subntngulqp Sand

 

7 .706 .960

VeryhAngglar Sand

2 .848 .925
4 .870 .960
6 1.00 .960

62-5.

2.21
2.10

1.74

2.80

3.26

3.10

2.42

TABLE'V DRY'TRIAXIAL TESTS

(Kg/0mg) 93°

32.3

31.4

28.4

36.2

39.6

38.1

33.8

 

 

61

 

 

SAND TYPE.A

“1(8) p (aim/grain)
8 .301 gms

12 .452

16 .605

20 .755

24 .904

28 1.052

52 1.200

TABLE VI CALCULATED LOAD PER PARTICLE-DIRECT SHEAR TEST

 

 

62

10.

11.

12.

13.

11}.

BIBLImRAPHY

Terzaghi, K. Erdbaumechanik auf Bodenphysikalischen Grundlage, p. 399,
1925.

ngami, T. "On the law of Friction In Sand," Proc. 2nd Int. Conf.
on Soil Mach. and Found. Eng. Vol. 1, p. 51, 1948.

Rowe, P. W. "A Stress-Strain Theory for Cohesionless 8011 with
Applications to Earth Pressures at Best and braving Walls,"

Reynolds, 0. "Experiments Showing Dilatancy, a Pr0perty of Granular
Material, Possibly Connected with Gravitation," Scientific
Papers by 0. Reynolds, Vol. 2, pp. 217-227, 1885

Taylor, D. w. Fundamentals of Soil Mechanics, John Wiley and Sons,
Inc., pp- 335-351. 19h8-

Hafiz, M. A. A. "Strength Characteristics of Sands and Gravels,"
A Thesis presented for the Degree of Doctor of PhilosoPhy,
University of Ipndon, 1950.

BishOp, A. W. Appendix , Thesis by 'M. A. A. Hafiz, see reference

Newland, P. L. and Allely, B. B. "Volume Changes In Drained Triaxial
Tests on Granular Lhterials," Geotechnique, Vol. 7, p. 17, 1957.

Bowden, F. P. "Adhesion and Friction," Endeavor, Vol. 16, pp. 5-18,
1957-

Bowden, F. P., and Young, J E. "Friction of Diamond Graphite, and
Carbon, and the Influence of Surface Films," Proc Royal Soc.
of London, Vol. 288, p. 2444,1951.

Tschebotarioff, G. P. Soil Mechanics , Foundations L and mrth
Structures, McGraw-Hill, Inc. , pp. 120-125, 1951.

Penman, A D M "Shear Characteristics of a Saturated Silt,
Measured in Triaxial Compression," Geotechnique, Vol. 3,
pp 312- -328 1952- -3

Norwegian Geotechnical Institute, Literature Concerning Friction
In Sand--The Friction of Minerals, An Internal Report, F 152,

19 59-

BJerrum, L., Kringstand, S., and KummeneJe, . The Shear Strength
of A Fine Loose Sand Norwegian Geotecfnical Institute, F. 9, 1958.

63

15.

l6.

17.

18 .

l9.

Wadell, H. "Sphericity and Roundness of Rock Particles ," Jour.
Geology, Vol. 41, p. 310, 1933.

Kolbuszewski, J. J. "General Investigation of the Fundamental
Factors Controlling loose Packing of Sands," Proc. 2nd Int.
Conf. on Soil Mech. and Found. Eng. Vol. 7, pp.TL7-7+9, 1918.

Hill, R. fI'Lhe Mathematical Theory of Plasticity, Oxford at the

Hertz, H. Theory of Elasticity, by Timoshenko, McGraw-Hill Book
00., Inc., p. 343, 19?}.

Birch, F., Handbook of Phygical Constants, Geological Society of
America, paper #36, p. 1+5, 1942.

 

1“. {'71. g “a C 6-3 Y
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Rum

 

 

 

MICHllGAN STATE UNIVERSITY LIBRAR REI

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