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STRESS-STRAIN RELATIONSHIPS son sou.
WITH VARIABLE LATERAL STRAIN

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
Osamu Kifani

1966

 

 

MlCHlGAN STATE UNNERSETY
UBRAPY

 

~

 

EEVTLIS7I
'HAR171972

MICHIGAN STATE UNIVERSITY
LIBRARY

ABSTRACT

STRESS-STRAIN RELATIONSHIPS FOR SOIL
WITH VARIABLE LATERAL STRAIN

by Osamu Kitani

A two-dimensional stress—strain law for soil is re-
quired for calculations of the general behavior of soil
under load or when deformed.

A study was made to investigate the behavior of a
cylindrical soil sample which was axially compressed and
laterally confined with springs of various spring rates.
This kind of test has more similarity to the actual situ-
ation where soil expands laterally under increasing axial
stress, and the lateral confinement stress is not a con-
stant but a function of the lateral strain. Moreover,
even a very loose soil can be tested by this simple and
rather inexpensive method.

The soil sample was compressed at a constant speed,
and the axial stress, axial strain and lateral strain were
measured. The lateral stress was calculated from the test
data. Shear strain was obtained from X-ray pictures of
lead spheres buried in the soil sample.

A loam with an average moisture content of l2.A%
dry basis was used with an average initial bulk density
of 0.03A6 lb/in3. A friction reducer was applied on the

surface of the sample and the wall friction was made small

Osamu Kitani

enough to satisfy the assumptions of uniform stress-strain
distribution and negligible shear stress along the wall.

Tests were carried out for the four variable spring
rates (lb/in) of 9.6, 56, 26A and a (fixed wall). From
the test results, relationships between the two principal
stress components and the two principal strain components
were determined in graphical form. It was also found that
there was an approximately linear relationship between
axial stress and lateral stress in the test range of
£1 < 0.36 and :2 < 0.029.

The functional forms of the relationships between
principal stresses and strains were derived by applying
the isotropic hardening theory and by modifying it. The

functions were of the following form;
01 = 8(81 - 262)n
02 = H01

The maximum shear stress was not a function of the
maximum shear strain alone, but was linearly related to
the mean normal stress.

The functional relationship between the mean normal
stress and the volume change (or the bulk density) was

also derived.

 

Approvech‘Mém TI W

Major Professor

Approved WM

Department Chaifman

STRESS-STRAIN RELATIONSHIPS FOR SOIL

WITH VARIABLE LATERAL STRAIN

By

Osamu Kitani

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Engineering

1966

ACKNOWLEDGMENTS

The author expresses his sincere gratitude to his
major professor Dr. P. E. Sverker Persson for the guidance
throughout his study.

To the other members of guidance committee, Dr. B. A.
Stout, Agricultural Engineering Department, Dr. G. E. Mase,
Metallurgy, Mechanics and Material Science Department, and
Dr. 0. B. Andersland, Civil Engineering Department, the
author wishes to express his gratitude for their valuable
suggestions and help.

Dr. L. E. Malvern, Metallurgy, Mechanics and Material
Science Department gave many valuable theoretical suggestions
and advice to the author. Dr. S. Serata, Civil Engineering
Department gave important suggestions on the experimental
phase.

Gratitudes are also expressed to the Fulbright Com-
mission in Japan which delegated the author as an exchange
visitor to the United States.

Mr. J. B. Cawood and his staff in the Agricultural
Engineering Research Laboratory assisted in the development
of the experimental apparatus. Dr. U. V. Mostosky and Miss

D. L. Middleton in Veterinary Clinic provided the author

ii

with X-ray machine and gave assistance. Mr. J. D. Wilson
helped in revising the manuscript.

This dissertation is dedicated to my wife Shigeko.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS . . . . . . .

LIST OF FIGURES . . . . . . .

LIST OF TABLES AND APPENDICES

LIST OF SYMBOLS . . . . . . . .
I. INTRODUCTION. . . . . . . .
II. REVIEW OF LITERATURE . . . . .

III.

IV.

VI.

2.1 Force versus Deformation
2.2 Ultimate Strength Study.

THE PROBLEM . . . .

3.1 Purpose of Study . . .
3.2 Assumptions for Study .

EXPERIMENTAL APPARATUS AND PROCEDURE.

A.l Apparatus . .
A.2 Procedure . .

METHOD OF ANALYSIS. . . . . .

5.1 Stress and Strain Compone

nts .

5.2 Mean Normal Stress, Deviatoric

Stress and Volume Change

aphs.

RESULTS . . . . . .
6.1 Recorded Results . . .
6.2 F1 and F2 Functions in Gr
6.3 01 versus‘kl Relationship .
6.“ 01 versus 02 Relationship
6.5 T versus 7 Relationship.
6.6 “V

cm versus V3 Relationship

iv

Page
11
vi

vii
ix

VII. DISCUSSION . . . . . . .

7.1 Strain-hardening Theory and F1

and F2 Functions. .
7.2 G Function. . . .
7.3 H Function. .
70

Conventional Strain System an

Natural Strain System .

VIII. SUGGESTIONS FOR THE APPLICATIONS.

IX. SUMMARY AND CONCLUSIONS.

9.1 Summary. . . . .
9.2 Conclusions . . .

SUGGESTIONS FOR FUTURE STUDY
REFERENCES 0 O O O O O O O 0
APPENDIX 0 O O O O O O O O

Page

53

53
66

67
71
73

73
75

78
79
81

Figure

I
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I I I I I I I I

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o o o o o o o 0 ”HO 0

LIST OF FIGURES

Stres8¢Strain Relations for Soil
Stress-Strain Curves from Soehne's Tests

General View of Test Device. . . . .
Compression Device.

Test Cylinder .

Layout of Strain Pick-ups, Amplifiers, and
Recorders. . .
Packing the Soil in the Cylinder at A5°. .
Soil Specimen after Compression (Covered with
a thin plastic film sandwiched by the friction
reducer) . . . . . .
X-ray Pictures

Dimensions of the Soil Specimen and the Stress-
Strain Components . . . . . . . . . .
Shear Strain.
Calculation of y" from e1 and e2

w

X-Y Records for Three Tests for Each of Four
Spring Rates. . .

01 versus 32 and e1 Relationship

oz versus :2 and 61 Relationship

cl versus e1 Relationship

log 01 versus log e1 Lines

log 01 versus 61 Curves . . . . . .

01 versus oz Relationship . . . .
7 Values from X-ray Test and from el and e2 .
1 versus y Relationship . .
om versus AV/Vo Relationship .

log om versus log AV/Vo Relationship.

log am versus AV/Vo Relationship .

Isotropic Hardening Theory

the Modified Equation (1). .

the Modified Equation (2)n 8 1

oz from the Modified Equation (2)n - 1 . .
log (01 - 02) versus log (:1 - 2:2) relation-
ship . .

01 from the Modified Equation (2)n = 1/2

°m Calculated from AV/Vo. . .

log 01 versus log e1 Relationship.

01 from
01 from
01 from

vi

Page

17
20
22
25

25
27

29
33
33

Table

LIST OF TABLES AND APPENDICES

6—1. H = 02/01 Values . .

Appendix

A-lo

A-2.

A-3o
A-A.

8-1.
B—2.
8-3.
B-A.
B—S.
B—6.
8-7.

B-8.

0.]. 0
C-2.
C-3 0

Calibration Chart for Axial Force Trans-
ducer. . . . .

Calibration of Axial Displacement of
Compression Piston in Terms of Time. .

Spring Constants k -.' . .

Calibration Chart for Lateral Displacement
Transducer . . . . . . . . . . .

01 Values

el and £1 Values

02 Values . . .

e2 and :2 Values

cm, 01', and 02' Values. . . . . .

1“ Values

r

7" Values

w

AV/VO values 0 o o o o o o o o o
u Values Calculated from Individual Test
01 Values from Isotropic Hardening Theory.
01 Values from the Modified Equation (1)

01 Values from the Modified Equation (2),-
n = l. . . . . . . . . . . . .

vii

Page
“5

82

83
83

8A
85
86
87
88
89
90
9O
91
92
93
93

9A

Appendix

C-U.

C~5.

02 Values from the Modified Equation (2),
n = l . . . . . . . . . . . .

01 Values from the Modified Equation (2),
n = 1/2 . . . . . . . .

TW/O Values.

II
__ .AA
1+

Om Values Calculated from AV/Vo

Comparison Between Conventional Strain
System and Natural Strain System

viii

Page

9“

95
96
96

97

e1
82

F1

x :11 III C)

20

in
A2.

A12

LIST OF SYMBOLS

current section area of soil specimen.
initial section area of soil specimen.
soil parameter; 01 - a(el — 252)n
cohesion

rate of deformation

conventional axial strain

conventional lateral strain

01 function; 01 I F1(51, £2)

02 function; 02 = F2(el, 22)

T function; I - C(Y)

function of volume change in terms of mean normal stress
strain-hardening function; '3 - HIG)
spring constant

current length of soil specimen
initial length of soil specimen

axial displacement (change of length of specimen)
tangential displacement

soil parameter; 01 a a(el - 252)n
axial force

lateral force (spring force)

current radius of soil specimen

initial radius of soil specimen

ix

Ar

51

52

GI

increment of radius

shear stress (ultimate value)

time (time of compression)

current volume of soil specimen

initiai volume of soil_specimen

change ofvolume; AV < 0 for compression
shear strain A
natural axial strain

natural lateral strain

contribution factor of :2 to :1

angle of inclination of plane from the maximum
principal stress direction

soil parameter u - 62/01
Poisson's ratio:
bulk density
axial stress
lateral (radial) stress
deviatoric axial stress
deviatoric lateral stress
‘ 01 + 202
mean normal stress; am - ———;———— for cylindrical case
shear stress ‘

angle of internal friction

 

generalized strain; E f JZT'EiJ €13.

 

' generalized stress; 3 ' //%'°'ij 0'13

On p. 6, 7, and 10, 3 is used as the effective stress
in soil.

I. INTRODUCTION

The stress-strain relationship of soil is the most
important basis for all the problems related to the soil
deformation and failure.

The problem of tillage,in which one of the important
objectives is how to reduce the draft and the tillage
energy, can only be solved completely after the stress-
strain law of soil is established and whereby the draft
and the energy for various tillage tools under certain soil
conditions can be calculated.

The traction problems also require the knowledge about
the stress-strain law. To achieve maximum traction, the
soil deformation and the stress under the tractor tire or
shoe must be calculated. The integration of the stress
along the border will yield the traction force which we
want to maximize.

Thus, agricultural engineers need soil stress-strain
relationship which can be applied to the practical problems,
or at least, can be a basis for them. The main effort in
this thesis has been devoted to get a stress-strain reé
lationship of soil which is applied to the practical

problems.

Agricultural engineers have rather special problems
in the study of stress-strain laws of soil, because the
soils on the field surface are soft and highly compressible.
Accordingly they are work-hardened very much. They also
contain complex organic matter. These features create differ-
ent problems from those in foundation engineering in which

the engineers are mainly interested in compacted soils.

II. REVIEW OF LITERATURE

There are two main categories in the study field of
soil stress-strain relations. They are the study of force
versus deformation and the study of ultimate strength which
may be considered as a part of stress-strain relationship

in the form of yield criterion.

2.1 Force versus Deformation Study

 

Soil changes its rheological properties according to
its texture, moisture content and density as well as loading
history. Therefore, there are several ways of approach in
this field. The theoretical studies so far have been based
upon theories of either elasticity, plasticity or viscoe
elasticity.

The theory of linear elasticity has been used quite
extensively in foundation engineering especially as the
means of calculating initial settlement. It is assumed
that the strain is completely recoverable and it follows
the linear stress-strain relationship as shown in Fig. 2-1
(a—l). In a 3-dimensional case'it is written in the follow-

ing form.

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e = E{°x - v(ay + 02)}

} etc.* (2.1)

Txy

QIH

ny '

where E, G and v are constants.

The agricultural soils are mostly so soft that it is
impossible to assume a complete recovery of strain after
the external force has been removed. Soehnef (1956) re-
ported that the surface soil behaved as a non-linear elastic
material (broken line in Fig. 2%1 (a)) only after it was re—
peatedly loaded up to the highest load encountered.

Thus, the surface soils have quite plastic charac-
teristics. Only particular types of clay of certain moisture
content do, however, show nearly ideal plastic flow which is

defined '

d etc. (2.2)

 

where sx is shear stress or deviatoric stress
K is a constant which represents yield value

IId is the second invariant of rate of defor-
mation

dx is rate of deformation.
This relation corresponds to either one of the flat lines in
Fig. 2-l(c).
By assuming rigid-perfect plasticity with the yield

criterion in the form of Coulomb's equation, we can calculate

 

-_..

*etc. means similar equations for y and 2 components.

stress-strain diatribution of hard (incompressible) soil
under Simple boundary conditions as described by Scott
(1963).p. 510.

Berezantsev (1955) analyzed the soil deformation
under a conical head by means of the plastic equilibrium
theory of soil.

However the actual situation is far more complex.

Soil shows work-hardening due to the increase of strain

(and stress) as shown in Fig. 2-l(d). Drucker gt_al. (1957)
showed that soil should be treated as a work-hardening
material. Drucker (1961) also tried to apply limit analysis
torsoil.

In many cases, efforts have been made to get a stress-
strain relationship of soil as a mixture of non-linear
elasticity and some kind of plasticity through simulation
type tests. That is to say, strain remains even after the
load is removed, sometimes no yield point is reached, or
even if it reaches the yield point, stress does not neces-
sarily remain constant.

Actually most of the studies do not consider the
factor of time. They take into consideration only stress
and strain in the same manner as elasticity. The most
frequently used relationship of this kind is the consoli-

dation equation.

Ae = eo - e = c log JL- (2.3)

where e is void ratio, 0C is a constant. If one—
dimensional consolidation is assumed, eq. (2.3) can be

written in the form of the following stress-strain relation.

ex =»c 10g o-+ c' ' (2.“)

Roberts and Souza (1958) reported that this relationship
was valid also for the very high stress under which the
soil particles themselves are crushed.

Another equation of this type is Bekker's sinkage
equation which has been used in traction problems. The
relationship between penetration pressure p and sinkage 2

under a penetration disk is described as

n
P ' KZ (2.5)
where K and n are constants.
If onemdimensional situation is assumed for eq. (2.5), it

becomes

(2.6)

In the lower axial force range, the axial force versus
displacement curve of Soehne's study (1956) plotted by the
author in Fig. 2-2, shows itself in the form of Bekker's
equation, 'Whereas, in the higher range it obeys the consoli-
dation law. ‘Soehne's soil compression tests were carried
out in a rigid wall cylinder.

Another Stress—strain relationship which has recently

been studied extensively is the viscoelastic theory which

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___ __ ________ _

      

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is a combination of (a—l) and (b) in Fig. 2-1 and expressed

in the following form.

dY
= _‘....fl
Sxy G ny + n dt etc. (2.7)

where G and n are constants.

McMurdie (1963) carried out a set of triaxial tests
and creep tests. He reported that the viscoelastic theory
agreed best with the test results.

Kondner and Krizek (196A) have develOped a vibratory
soil compression test device with which they analyzed the
soil behavior and divided the strain into the storage part
and the dissipation part.

Waldron (1964) showed that the strain super-position
law of viscoelastic theory is valid for soils only within a
limited range of moisture content, stress and time. He also
obtained three viscoelastic functions from his tests.

There are some studies which do not deal with the
constitutive laws of soil, but rather check whether or not
some stress-strain relation of continuous media is valid
for soil. VandenBerg et_al, (1958) suggested that the bulk
density change during soil compaction should be caused only'
by mean normal stress. However, VandenBerg (1962) later
reported that the triaxial tests with diametral meaSurement'
showed that the change in bulk density p is not a function
of mean normal stress cm alone but alSo a function of maxi-

mum shear strain Thax in the following exponential form.

10

m max (2.8)

where a, b are constants.

2.2 Ultimate Strength Study

 

Many studies have been carried out from the vieWpoint

of soil strength based upon the famous Coulomb's law.

S=C+Etan¢ (2.9)

where s is ultimate shear strength, 3 is cohesion and e
is angle of internal friction.

Hendrick and VandenBerg (1961) conducted a tensile
strength test of soil at various loading rates and obtained
c values at tensile failure. They concluded that the ulti-
mate strength does not change with a change in the loading
rate, whereas the deformation at failure decreases as
loading rate increases.

Vomocil gt_al. (1961) carried out a tensile test using
centrifugal force.

The triaxial test is considered as the most reliable
-test to measure the two parameters c and e. Normally soil
fails in a progressive manner. In the case of a triaxial
test the stress condition, however, changes rapidly with the
increase of deformation, especially after maximum stress is
reached. To avgid this, other devices such as torsion
shear apparatus have been developed (for example, Waterway

Experiment Station, 1952).

11

Some people tried to combine deformation and ultimate
strength laws together. Taylor and VandenBerg (1965) intro-

duced deformation J in the failure equation.
K (2.10)

where n and K are constants.
They derived this formula from a series of tests with a

ring shear device in laboratory soil.

III. THE PROBLEM

3.1 Purpose of Study

 

As described in the previous chapter, several stress-
strain laws for soil have been proposed for the one—di-
mensional case. However a two-dimensional stress-strain
relationship is required to solve the practical problems
of tillage and traction, because in soils their strength
as well as their stress-strain level depends largely on
the mean normal stress. This indicates that the stress-
strain relationships of soil, unlike those of metals,
could be largely affected by the mean normal stress. The
particular problem of this study may be defined as describ-
ing how the relationship between axial stress and axial
strain in a compression test of a cylindrical soil specimen
is affected by the lateral (radial) stress and lateral strain.

This kind of study with various lateral stresses can be
carried out with a triaxial compression test apparatus with
lateral strain pick-up. The triaxial test is, however,
limited to those soils with a certain natural strength so that
the sample can maintain its shape without lateral support.
This is a great disadvantage for studying stress-strain re-
lationships of a soft surface soil in agriculture. For

these, it is necessary to provide lateral support for the

12

13

soil specimen. This kind of device could be used even
for a very loose sand.

The triaxial test also takes time to be carried out.
It requires rather expensive equipment. A test device with
lateral confinement can be much simpler, because it does
not require handling the soil specimen in liquid.

Moreover some of the practical problems of tillage
and traction have boundary values in terms of strain rather
than stress, for example, boundary by a rigid blade or an
elastic tire. In this situation, a test with various types
of strain confinement may be more suitable than a stress
confinement test. More generally, almost any element of
soil under load in one direction is subjected to a confine-
ment in the other direction, where the confinement stress is
not a constant but a function of the lateral strain. Because
the soil strength is affected by this lateral stress and
strain, a realistic testing should be made with a strain-

defined lateral confinement.

3.2 Assumptions for Study

Thus, a soil compression test in which lateral strain
confinement is controlled was designed. Actually the lateral
confinement was provided by two springs. There is, there-
fore, an approximately linear relationship between lateral
stress and lateral strain as; 02 = kzez. (cf. eq. (5.4)).

The lateral strain level at a certain axial stress or

1A

axial strain can be controlled through the exchange of
springs with various spring constants k.

The test has to be limited to the loading process
and to the loads below the failure point. Within these
limitations, the stress-strain relationship is considered
to be a single valued, monotonically increasing function.
Hence, the relationship between normal stress and normal
strain in this cylindrical case (or in the general two
dimensional case) could be described by the following two

functions.
01 = F1(€19=€2)

02 ‘ F2(€1I 62)
where 01 and c2 are principal stresses

£1 and c2 are principal strains

For the measurement of these principal stresses and principal
strains as described later, it is assumed that the stress
state is uniform in the whole sample and that the principal
axes are parallel and perpendicular respectively to the
axis of the cylinder. In order for this to be true the
assumption that there is no wall friction must be satisfied.
The shear stress can be calculated from the principal
stresses 01 and 0; assuming homogeneity of the soil. The
shear strain is picked up from X—ray pictures of the soil

as the change of the angle of a line in the soil.

15

As the first step, the author assumed that there is
a functional relationship between shear stress 1 and shear

strain 7 independent of normal stress and strain components.

T ' G(Y)

Thus the stress-strain relationship in any two—di-
mensional case (even when principal components are not
known) is fully described by the three functions F1, F2
and 0. These functions make it possible to calculate the
three stress components if the three strain components are
given.

It might also be possible to calculate the stress-
strain distribution under certain boundary conditions using
the equations of equilibrium, compatibility and boundary
values.

The above mentioned two-dimensional approach might
be applied to some of the practical three-dimensional pro-
blems by further breaking down the problem into numbers of

two-dimensional problems.

IV. EXPERIMENTAL APPARATUS AND PROCEDURE
A.1 Apparatus

The general view of the test equipment is given in
Fig. A—l. It is composed of a soil compression device, test
cylinder, amplifiers, strip-chart recorder, X-Y recorder

and X-ray machine.

Compression Device

The soil compression device compresses the cylindrical
soil specimen at a constant speed. Three speeds can be
selected. The picture of the compression device is given
in Fig. “—2. The designed maximum compression force is
1000 lbs. and the maximum piston stroke is A.5-inches. The
upper and the lower position of the piston is controlled by
two limit switches. A 1/4 H.P. induction motor drives the
piston screw through two worm gears, a V-belt and a chain.

The axial force transducer is made as’a ring of high
strength aluminum alloy. Four SR-A strain gages are at-
tached to the inside wall of the ring. The calibration
(App. A-l) shows good linearity.

The axial displacement of the piston could be picked
up by a strain gage equipped cantilever connected to the

driving screw. However, since the axial displacement has

16

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.01....
Ono-q.‘.

- -\\oe\eeao ...1 .a
. teacnahoeoeut.oI. s
. sheesm~eao.~.e;:
“\soseueuec.‘ . A I .\
w‘sga~.e.e a...~ -.. an
“s..t.0...b urc.b 0A

23.....13...
.. .2.....:...._...

m *M.m.VCWIro»rW»W

,.'-.0Oc--.

.. .7.”th

a. z. is...
4.. 1.6 . :3.
I .asfifzz,

A
-

      

   

a-
c'
a.
~ .
Cu.‘
0»

‘ - l \
-*‘f
"“1er

18

moa>em coammenQEoo
caeasm owed .m

.mIe .mem
eon pee: mefisasm .m

ampaon Edam mmnIx .m neoscmcmnp meson Hmax< .m

mews“ mcficmoq .:

I.» -
I-IVII’I‘nIc.iCOC
' a fill.
.I'I'DO

\ a
I 0. ..-.IF ' . '

O D
I ’ D I.’ . C ‘ ‘

' -‘

neccaazo ewes .H

In.
I
r I
c
O»
.1.

. IOII ..
I ' o".~
I o v f
. ‘ 'I
a.-. .

 

19

complete linearity in terms of driving time as shown in
App. A-2, it is simpler and more accurate to get the

axial displacement from the driving time.

Test Cylinder and Stress-Strain Pickup Device

As shown in Fig. 4-3, the test cylinder has an over-
lapping split side wall. The inside diameter is 3-inches.
The height is 7-inches. The cylinder is made of 0.0156-
inch thick steel plate. The friction in the overlap is
reduced by means of a friction reducer which is mentioned
later.

The lateral confinement of the soil in the cylinder
is provided by two springs which deform tangentially along
the guide rods. Several springs with various spring con-
stants were made so that the condition of lateral confine-
ment can be controlled from an almost free expansion to an
almost rigid wall. The spring constants are given in App.
A-3.

The expansion of the cylinder is measured by the two
cantilever type strain gage transducers made of spring steel
with a Teflon tip on the end of the beam. The calibration
curve is given in App. A-A. The relationship between radial
expansion (tangential displacement) and the output reading
on the X-Y recorder chart shows fairly good linearity. From
the radial expansion the lateral strain of the soil can be

calculated.

U1 :WNH

 

 

 

Cylinder wall Springs
Upper piston Strain gages
Lower piston Soil

Lateral displacement
transducer 1
Spring guide rod

Lead spheres
Friction reducer

oxoooxiox

Fig. n-3, Test Cylinder

21

The strain gages on the axial force transducer and
lateral displacement transducers are hooked up with Brush
RD5612 and BL310 amplifiers and a Brush recorder (Fig. A-A).
In addition a Moseley 135 X-Y recorder was used for record-
ing the axial force and the lateral displacement with better
accuracy than the strip-chart recorder. With time marks on
the X-Y curve produced by feeding time signals into the X-Y
recorder, a complete set of data is obtained on the X-Y re-
corder chart, because axial displacement is obtained from

the time marks.

Sgil Preparation Device

The main part of the soil preparation device consists
ofqa pneumatic vibrator and a test cylinder holder. When
lead spheres are buried in a A5° line a special holder base

is used as shown in Fig. A-S.

§:331»Machine

A General Electric Model 11DB6, 150KV Medical X-ray
machine in the Veterinary Clinic of Michigan State Univer-
sity was used. Kodak Industrial AA Type X-ray film was
chosen to get an accurate picture of the lead spheres in
the soil specimen. The film attached with magnets to a
plate of steel and a lead shield were placed just behind
the test cylinder (Fig. A-2). To secure an exact distance
between the X—ray source and the film a wooden frame was

used.

22

 

 

 

 

 

 

 

 

oaeonooom use .oaoaufiaas< .mqsnxOAm seesaw mo packed .aI: .mfim

 

 

nonpooem

 

 

usenoIanApm _
.

 

 

NIIIIVII

 

nonnooom wa

 

 

 

 

 

.11.

w

 

 

 

_. xnms mafia.

 

 

 

 

 

 

 

 

 

msmacaaaea

 

 

 

 

 

. 0Q vcm o<

 

 

 

 

 

 

 

 

 

II IIIIIIIIIIIIII III! I

23

4.2 Procedure

 

Sgil

A loam with average moisture content of 12.A% dry
basis (range ll.5-l3.0%) was used. The soil was sieved
with a l/A-inch mesh sieve. Then it was kept in a can to
keep a constant moisture content and uniform distribution

of soil water during a series of tests.

Prepgration of Soil Specimen

 

The test cylinder was placed in the cylinder holder
with the piston (and a thin disk plate on t0p of it) at
the bottom of the cylinder and clamped between two holders.
The friction reducer was applied on the inside of the
cylinder wall with a painting roller. The friction re-
ducer is a mixture of graphite and grease with a weight
ratio of 1:3. The friction reducer was also applied on
one side of a sheet of plastic wrapping film. The total
amount of the friction reducer turned out to be quite im-
portant for this test. Therefore, the amount of the friction
reducer applied to a specimen was controlled within the
range of 0.1A3-O.176 lbs. (65-80 grams), corresponding to
an average total thickness of 0.075-inch.

With a small amount of soil in it to prevent the film
sticking together, the friction reducer-coated plastic film
was applied tightly inside the cylinder. A small cup of
soil corresponding to a layer of approximately l-inch was

then put into the cylinder. The pneumatic vibrator vibrated

24

the cylinder for two minutes for each additional cup of
soil. When the soil surface came up near the middle of
the cylinder height, the cylinder holder was placed on the
45° holder base so that a 45° surface was created (Fig. 4-5).
The lead spheres were placed 1/4-inch apart on this surface
by means of an aluminum plate with holes. Then, another cup
of soil was added and vibrated. After the cylinder was set
in the initial vertical position again, the same process
(adding a cup of soil and vibrating for two minutes) was
repeated until a desired amount of soil was packed in the
cylinder. After covering the top of the packed soil with
the upper part of the plastic film, the upper disk plate
and the piston rod (total weight is 1.9 lbs.) was put on
top of the wrapped soil and the vibration was carried out
for three minutes. This completed the preparation of a
soil specimen with an average bulk density of 0.0346 lb/in3
(0.96 g/cm3), (range; 0.0337 - 0.0352 lb/in3).

Sometimes a compression test was carried out without
a X—ray test, because the transportation to the Veterinary
Clinic for X-ray introduced some error in the initial con-
dition of soil. In these cases no lead spheres were needed
and the soil was packed just layer after layer without the

45° holder base.

Test Procedure

 

The compression test was carried out as soon as the

soil preparation was finished. Before starting the

 

06 civil-o.
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ll
09.
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ot4

25

Packing the Soil in the Cylinder at 45°

Soil Specimen after Compression

4-6.

(Covered with a thin plastic film
sandwitched by the friction reducer)

      

n...
O
t
a
r
Pb
8.1
d V
n
.l C
1.1
yt
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m o
_ +uu a,
........ .. 4. won» .U—I
........_..... TP

.1.“te.. I c I

. e I. r

in . I . .!.\ e
LE.DD|I-'| .I J. ..'--.IIH\.. 90;

 

1.
3.
Fig.

Fig.

26

compression the initial height of the specimen was measured.
The initial diameter was kept constant by the holder. The
transducers were connected to the amplifiers and the re-
corders (Fig. 4-4) and the instruments were balanced.

The compression of the soil was made at the rate of
0.092-in/min. The X-Y recorder plotted the axial force
versus tangential displacement curve and the strip-chart
recorder recorded every separate component versus time. An
X-ray picture was taken every 2.5 minutes without stopping
the compression and a time mark was put down on the X-Y
curve at the time of exposure. The X-ray machine was set
for 125 KV, 300mA intensity with 40-inch focal distance and
1.5 second shutter speed. Fig. 4-7 shows two X-ray pictures.

At the end of each test before unloading, the test
cylinder was pushed back toward the upper piston and the
friction at the maximum lateral stress was measured. The
coefficient of the wall friction fell in the range of
0.037 - 0.053.

Three replications were made for each test. Three
kinds of springs were chosen so that it was possible to
get the stress-strain curves at various levels of lateral
confinement. A test with fixed wall (kzerw) was also con-
ducted by replacing the springs and the guide rods with

bracing bolts and nuts.

27

Swans:

negrooflm zmst
coammenano nepm<

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an: .wE

scammmnoEoo onomom

 

V. METHOD OF ANALYSIS

5.1 Stress and Strain Components

 

The stress components 01, 02 and T, and the strain
components 61, £2 and y (natural strain) e1, e2 and y(con-

ventional strain) are obtained as follows:

Axial Stress 01

 

Due to the low friction on the pistons and on the
cylinder wall,* the stress distribution in the soil specimen
is considered to be uniform. Since the shear stress on the
surface of the soil specimen is negligible due to the low
friction, the axial and the radial directions are considered
as principal directions.

From the axial force transducer output which is re-
corded on the Y-coordinate of a X—Y recorder chart, the
axial force P1 is obtained by means of the calibration

chart App. A—l. Then,

01:21
A

where A is the current area of the soil cylinder, and

= 2= £112: 2
A "(r0 + Ar) nr02(l + r0) A0 (1 + e2)

 

* This is also confirmed by the uniform movement of the
lead spheres buried near the pistons and the cylinder wall.

28

 

 

 

 

I
“)1
AI

4\

 

 

 

 

 

 

Ir—---——-_—-

 

 

 

 

 

 

Dimensions of the soil specimen and
the stress-strain components.

Fig. 5.1.

30

Therefore, P

= Ao(l +162)2 (5'1)

 

0l

where e2 is calculated from eq. (5.5).

Axial Strain el and 51

fi—

 

Since the piston displacement shows complete linearity
in terms of compression time t as shown in App. A-2, the

axial displacement A2 is obtained from the compression time.

Then,
2.
e1 = %3 (5.2)
e] = -1n(l — e1) (5.3)

This strain is assumed the same for the whole sample for

the same reasons given for the axial stress.

Lateral Stress 02

 

The lateral stress is assumed constant along the wall
of the cylinder. The equilibrium equation for one-half of
the cylindrical shell is considered. The spring force (for
one spring = P2) is in balance with the total force due to
02 in the direction of P2. Considering the symmetry, the

equation of equilibrium becomes

TI
2(2P2) = 2£f% (rde)ozsin0 = 2r262

Therefore,

31

Since P2 = kAiz where A22 is spring deflection,

 

 

 

o a 2kA£2 = 2kA£2
2 r2, (1‘0 'I’ Ar)l
ZWPO + A22 All
Using the relation Ar = r - r0 = 2" - r0 = 75?,
the above equation becomes
02 = uflkAr LITI'kez (5.14)

 

 

(r0 + ArIR 3 (l + €2)(l - e1)£0
where el and e2 are obtained from eq. (5.2) and eq. (5.5)

respectively.

Lateral Strain e2 and 52

The lateral strain is also considered to be constant.
From Fig. 5-1, eq. (5.5) is obtained. The tangential displace-
ment A22 is obtained from the output records of the lateral

displacement transducer by using the calibration chart in App.

 

A-4. .
_ Ar = A22

e2 ‘ FF 2nro (5.5)

62 = in (l + e2) (5.6)

Shear Stress T

 

Since it is assumed that there is no wall friction,
both the axial and the radial directions are principal
stress directions. The shear stress on the plane of incli-
nation 6 from the radial direction is

01 - O2
sin20 (5.7)

U1 - 02

at e (5.8)

u
f'
I-I
q:
u
l\)

32

Shear Strain y

 

Shear strain is obtained from the change of the angle
of line of lead spheres buried in the soil. As shown in

Fig. 5-2,

Y = 26 (5.9)

“It
1.,-

Assuming uniform distribution of the strain, it is also
possible to calculate shear strain from the normal strain

components in the E plane as shown in Fig. 5-3.

A2
1 _ __
y" = 26 = 2 E - arctanI————K%Q} (5.10)
1: 1+—
I'0
y" appr. = 9 - 2 arctanIl — e1 - e2}

IT

5.2 Mean Normal Stress, Deviatoric Stress
’ and Volume Change

 

 

Mean Normal Component and Deviatoric Components
It is common to divide the normal stress into a mean
normal component and deviatoric components.

Mean normal component is

01 + 02 + 03

 

em = 3 in the general case,
01 + 202 f th 11 d 1 1 (5 11‘
am "_—T§‘_—— or e cy n r ca case. . )

Deviatoric components for the cylindrical case are

 

 

 

 

 

 

O)

.11.

‘4

= tan a + tan B 6 c + B

 

 

 

 

 

Fig. 5-2. Shear Strain

 

r—-' - -"I—-—- _—I

 

 

 

 

>4
0

Tij—

 

E»

 

 

 

 

Fig. 5-3.,

Calculation of y“ from e1
1:

6=0I=B
. ny = 26
x0 = r0
xgxigri
02,0 02.0

It 2r
v arctan(33%)

II
:I
I
ID
’1
0
CT
I!)
:3
A
H
I
(D
N
eq.:

and e2

34

01' =‘ai -“om 35%(01 - 02) (5.12)

02' - 02 - om . -%(01 - 6,) (5.13)

Deviatoric stress is considered to cause the change of
shape. Since both of the components of the deviatoric
stress contain (01 - 02), the shear strain 7 will be a

function of (01 — 02).
Y ' G1<°1 ' 02) = G2(T)

Volumetrip_Changg

The volumetric change 16.

AI _ V - V0 . ;L 2 _ '
V0 V0 Vo{2"(r° + Ar) (20 - A2) V0}
therefore,
9% .. (1 + e2)’-(1 -. e1) - 1 (5.14)

Eq. (5.14) implies

In(1 + %%I a 2In(1 + 02) + In(1 . e1)

'Therefore, £n(lf+'%%)g- 262 -'e1 (5.15)

The mean normal stress is generally considered to
contribute to the vdlume change. Therefore, the following

function H may be expected.

AV
V6- ' H(O‘m)

VI. RESULTS

6.1 Recorded Results

 

Fig. 6-1 shows the results of 12 tests superimposed
on a chart. The three replications for each spring rate
show considerable deviation, yet the general tendency due
to the change of lateral confinement can be clearly seen.
The triangles indicate the same axial displacement (time).

From this record, together with the calibration
chart of the transducers in App. A-l, App. A-2 and App.
A-4, each component of the stress and strain as well as the
volume change was calculated using the equations in Chapter
V. The calculated results are tabulated in App. B. The
main results are presented graphically in the following

sections.

6.2 F1 and F2 Functions in Graphs

 

Fig. 6-2 shows the relationship between 01 and 22
with :1 as contours. Since 01 shows a certain unique (single
value) functional relationship with el and £2, a 01 value can
be obtained graphically if 51 and £2 values are given.

02 versus :2 and £1 relationship is shown in Fig.
6-3. The general tendency is similar to 01 versus £2 and

a, relationship with smaller scale factor for 02. Therefore,

35

mouse mafiaam snow mo Some soa wpmmp means 90% menooop wa

.HIQ .mas

Acfiv psopzo amusemcmsp pamEeomHQmHQ amnepmq

 

 

cH\nH m.m u x
open weanam

    

sd\na mm u x
some wsflnom

Acfievp use» cosmmoposoo
ma oawcwfiso some you shaman one

m
m

H

 

 

l
\,
\xx
m.“ \\\
\
\\ \\
mH \ \
\
\ x
T:
\ T:
.02 pmme .II IIIII \ .,
\ \
.oz pmoe IIIIIIIIII \\
. \ \. . \
.oz pmme \\ \ \
\ x
\ \ \
om \
\
_ \
QA\AH sz I a \\ om
was.“ 5.3m \ 0Q:
\ \. Hams omens

    

 

 

(uT) qndqno Jeonpsusaq some; Terxv

37
”I? Spring rate (lb/in)

u0_ ----o—---k = 9.6

\\D

X k=56

 

01

——A——- k = 264

 

 

 

 

 

Fig. 6—2.~ 01 versus 62 and £1 relationship

38

8-— Spring rate
(lb/in)
A ----o--- k = 9.6
/
/ 56

 

 

 

 

 

Fig. 6-3. 02 versus 82 and el.relationship

39

if 61 and 52 are known, 02 is directly obtained from_this
figure. £1 and a, values have to be determined for a
practical use of these graphs. This can be done by X-ray
method burying lead spheres in the soil at every node of
an imaginary network.

Since formulation of F1 and F2 introduces a certain
error, this graphital method may be the most accurate way
to get the stress components if the £1 and :2 network is

dense enough.

 

6.3 01 versus 21 Relationship

Fig. 6-4 shows the relationship between 01 and c1. In
the higher compression range the effectof spring rateis
clear: ,higher spring rate creates higher axial stress at the
same level of axial strain. In the lower compression range,
the effect of spring rate is, however, inconsistent.) This
may partly come from the variation of initial density rang-
ing from 0.0337 to 0.0352 lb/in3. Thisvariation corresponds
to a variation in axial displacement amounting up to 0.26-
inch (:1 = 0.044) which is large enough to create the in-
consistency. The-low initial density might be.another reason,
because in lower-density,'soil particles and aggregates have
more freedomtof sliding or rolling. This might imply a more
unstable.mechanical condition whichmakes it possible for“.
the soil to take more than one stress path in the compression
' process.) I
The smaller slope of the initial compression lines

of log algyersusglogsepficurves inTFig. 6-5 indicates that

40

01

psi

30

2O

10

 

40

Spring rate (lb/in)

 

----o-——— k = 9.6
x-—-— k = 56

-——-A———-— k = 264
:I kw ..

 

 

,/
/ ,o
X/ /,

/' //
/EJ //°'
.////$3§7//0’/’
./ ”'

’

/ ‘T
I,“

1 I l I I I
.10 L .20 .30

Fig. 6—4. 01 versus 61 relationship

 

41

there may be some critical point where the soil becomes
more stable and begins to show a higher rate of work-
hardening.

There are two commonly used soil compression
equations which may fit to the test results. They are a
modified form of Bekker's sinkage equation and of the consoli—
dation equation (eq.(2.4) and eq.(2.6)). Fig. 6—5 shows log
01 versus log 61 relationship, where eq. (2.6) should be a
straight line. Fig. 6-6 presents log 01 versus 51 relation-
ship, where eq. (2.4) should be a straight line. The log
01 versus 61 curves in Fig. 6—6 are not linear. This means
that the equation similar to the consolidation equation is
not valid here. The log 01 versus log 61 relation in Fig.
6—5 can be represented by two straight lines which are des-
cribed by the following equation with different values for

the constants a and n.

01: aeln ((3.1)

This means that the equation similar to Bekker's sinkage

equation is valid.

6.4 01 versus 02 Relationship

 

The relationship between 01 and 02 is shown in Fig.
6-7. From this figure, there seems to exist a linear re-

lationship between 01 and 02 that can be expressed by

02 = pol. (6.2)

01

psi

 

42

 

Spring rate (lb/in)

 

 

 

—-40
----o---- k = 9.6
_30
--‘X-—- k = 56
---20 —--—A—-— k = 264
D km 0°
—-10
r- 5
/
x /
— 2
/
/
/
/
— 1
e1
.10 O .30 . O
L l l l L f l I 1 L.
Fig. 6-5.- log 01 versus log 61 lines

01

psi

--40

......30

-'20

 

 

43
a

Spring rate (lb/in)
//3
....-..o ---- k = 9.6 /
/
/’

 

 

A x
k = 6 U/
x 5 /
/
__A__ k = 2614 / X
0’ /
Cl 4 R300 x
A
//
/Dx
/
/
//x
/
Io
/
/
/
E:1

.20 .30

 

Fig. 6-6. log 01 versus 51 curves

44

and Ho

QfiQmQOHpmHmp No mSmam> Ho ..m|m .mHm

om OH

 

:mm
mm
m.m
:mm
mm
m.m

Acfi\navx
bump mcanom

)<

msHm> HMSUH>H©QH .+

msam> ommhm><

 

 

 

«ma

No

45

The n values which are calculated from App. B—1 and

App. B-3 are tabulated in Table 6-1.

TABLE 6-l.-qu = 02/01 values.

 

Axial displacement Spring rate k(lb/in)

A£(in)

 

 

k I 9.6 k = 56 k = 264
0.23 0.151 0.108 0.202
0.46 0.156 0.115 0.130
0.69 0.139 0.130 0.103
0.92 0.122 0.152 0.123
1.15 0.105 0.172 0.125
1.38 0.179 0.152
1.61 0.181 0.174
1.84 0.181 0.194

 

Considering the large variation of n values calculated for
the individual test (App. B—9), it will be possible to
assume that u is a constant for various lateral strain con-
finement within this test range of £1 < .36 and £2 < .029
and with a possible exception for the very low compression
range. The idea of u - constant is similar to the idea of
the coefficient of earth pressure in soil mechanics. The
coefficient of earth pressure is defined as the ratio of the
lateral stress to the vertical stress. According to

Scott (1963) p. 403, this coefficient is defined for both

the soil conditions at failure and below failure (soil at

»46

rest). The earth pressure coefficient is considered

as a constant for a soil of semi—infinite mass in which a
soil element under a vertical pressure can expand laterally
with certain lateral confinement, which is similar situation
to this test. The average value of u here in this test is

u = 0.141.

6.5 T versus 7 Relationship

 

The 7 values measured from x—ray pictures show good
coincidence with y calculated from el and e2 (eq. (5.10))
as shown in Fig. 6—8. This fact implies that the effect of
the wall friction is small. Since the latter calculation
method has higher accuracy so far as the wall friction is
negligible, the values calculated from e1 and e2 are used
for the following discussion. The r values were calculated
from 01 and 02 using eq. (5.8).

Fig. 6~9 shows the relationship between T and y.
Spring rate apparently affect the T value. This indicates

that T is not a function of,y alone.

A1

6.6 cm versus V0

Relationship

 

Fig. 6—10 gives the relationship between the mean
normal stress and the volume change. The figure shows
approximately the same tendency for all spring rates.

The'om versus %% relationship on a log-log chart
is shown in Fig. 6-11. The same relationship is shown on

a semi-log chart in Fig. 6—12.

47

mm cam Hm Scam paw pump zmnlx Eon mmsHm> > _.mnm .wfim

Aeflvaa mm.H mH.H mm. mm.

_ _ d _

 

om i-II- msam>
webmasoHMo
m.m loxl
mm .--x--
mzam>

 

m.m -!-nO-u- omnzmmms

Aca\navx
mums wcflnqm

 

48

QHQmQOHpmHmp > msmhm> p .mlm .wfim

 

O_m .

:mm H x
sm-x
smug

Asfixnav mums mausam

 

 

OH-I

 

 

49

Spring rate (lb/in)

 

 

o k = 9 6
x k = 56
-~—-A-— k = 264

\X

 

A
)<
X A
/
XA
/O
23>"
AV
/0 - 10 - 20 v0

Fig. 6-10. cm versus AV/Vo relationship

50

 

 

20 -
V A
o F Spring rate (lb/in) //
m
psi .___°___ k = 9.6 ’
10 _.
' -——x-— k = 56 X
5 -- x/
_ X/
0
_ XA
O
X
1.... O
6 - x
~ .A
O
2 L, 414 1 l L - 11 J 1 I l I l l
— 02 - 05 - 10 -.20 -.30 —.40
fl
V0

Fig. 6—ll.-—log om versus log %% relationship.

20

psi
10

 

51

 

 

X’
Spring rate (lb/in)
____O___ k=9.6
- —x— k=56
_ .—_4A__. k = 264
O
l I I l l J
— 05 - 10 - 15 — 20 — 25 — 30
AK _
V0

Fig. 6-12.--1og om versus %% relationship.

52

Fig. 6-12 shows approximately a straight line in the
higher compression range of 51 > 0.15. This straight line

is expressed by eq. (6.3).

log Om = k0 - —— for 51 > 0.15 (6.13)

where k0 is a constant.

This is a similar form to the original consolidation equation.
Since T takes various values at the same level of y

and T is linearly related to am as discussed later in Chapter

VII. 7.2, the bulk density (or volume) calculated from Vanden-

Berg's equation (2.7) will take various values for the same

value of am. This conflicts with the fact that Om versus

%% curve shows approximately the same tendency for various

lateral confinements. VandenBerg's equation does not, there—

fore, seem to be valid here.

VII. DISCUSSION

7.1 Strain—hardening Theory_and
F1 and F2 Functions

 

The principal stress components 01 and 02 are ob-
tained graphically from Fig. 6—2 and Fig. 6e3 if two
principal strain components 61 and 82 are known. This is,
however, not suitable for calculations of a general case.
For the development of a general theory, the F1 and F2

functions have to be formulated.

Isotropic Hardening Formula

 

The theory of strainehardening developed in metal
plasticity may be useful to explain the increase in soil
strength with compaction and to formulate a stress—strain
relationship for soil. The basic idea of isotropic harden—
ing theory is quoted from Hill (1950) p. 23.

Assuming that a material is work-hardened isotropi-
cally, the one-dimensional stress-strain relationship
ox = HI(epx) (p implies plastic component of strain) can
be extended to the two- or three—dimensional case by using
a generalized stress 6 and a generalized strain 2;. The

one-dimensional relationship.can be easily obtained from

a tensile test of a metal piece for example, and the

53

54

function form HI is determined. Then, the same function

can be assumed for E and Id? .

HI p

E = HI(IdEp) (7.1)

Since a soft surface soil has only negligible elastic
strain, it can be assumed that the total strain*equals
plastic strain for soil, that is, Ep = E. It is also as-
3. Thus eq. (7.1) becomes

sumed here that Id:

 

6‘ = BIG) (7.2)
where E = [’goijoij (7.3)
E = ,/:’23—Eij Eij (7.4)

For the cylindrical case,

2(01- 0'2) 0 U
01': and02'=03'—-(132)

3

 

 

Therefore,

 

E @401- 02)2{<-§->2 + ($2 + <--§>2} = 01- 02

} (7.5)

 

/-:23-(€12 + 2822)

mI

The soil test corresponding to the tensile test of
metals will be a compression test without lateral confine-
ment. It is, however, almost impossible to do this test for
a soft soil. Instead, the necessary relationship can be

W

*Total strain includes volumetric strain, because it con—
tributes to the strain—hardening.

55

obtained from a compression test of soil in a fixed wall
cylinder with suitable friction reducer on the wall. In

this case 02 is no longer zero. Extending the test result

of eq. (6-2) to the fixed wall case, 02 can be calculated
from 02 = uol. And, for this case 01 — 02 = 3 and e- = //§ 3
(as a Special case of eq. (7.5)).

From the fixed wall test result presented in Chapter

VI. 6.3, the same equation as eq. (6.1) can be used for H

I
function.
01: aeln fOI‘ 81> 0.15 (7.6)
where a = 493 (psi)
n = 2.414

Since in the lower compression range of 81 i 0.15, the effect
of lateral confinement is not consistent as discussed in
Chapter VI. 6.3, the theoretical consideration here is limited
to the higher compression range of 51 > 0.15.

Since eq. (7.6) is introduced, the prototype equation

becomes
n
01 - 02 = (1 - u)ael (7.7)
Therefore the general form is

3‘ = (1 - wax/g E)“ (7.8)

Substituting eq. (7.5) into eq. (7.8),

56

 

01 - 02 = (1 - u)a{//%- %(512 + 2522)}n

and using eq. (6.2), eq. (7.9) is obtained.

01 = a(512 + 2.22)? (7 9)
However, this formula looks unreasonable because 01 in—
creases as 52 increases even if 22 is in the direction
of expansion. Strain-hardening of metals occurs equally
under compression or tension. For soils, the situation is
different, soil is work-hardened by compression but softened
by expansion. Therefore, the plus sign in front of 2522
could be changed into a minus sign in the case of expansion.
If 22 is in the direction of compression the sign should re-
main plus. Then, the following equations are obtained.

2
2

a(e12 — 2522) (7.10)

C’1

n
ua(512 - 2e22)? (7.11)

02

The 01 values calculated from eq. (7.10) are listed
in App. C-l. The comparison between the calculated values
and the measured values is shown in Fig. 7—1. It shows al-
most no effect of 22. It also shows a large deviation from
the measured value when the lateral confinement is weak.
This might mainly be due to the fact that in soil only a
small strain is produced in the lateral direction even if
a large axial strain is put in; in other words, 62/61

values for soils are comparatively smaller than for metals.

40 ——

01

psi

20 ~—

10 *—

 

.10

57

  
 
  
 
 
  
  
  
  

Calculated from

n
01 = a(612 - 2822)?

Spring rate (lb/in)

 

.— k 3 9.6
_——+—— k = 56
————‘h—- k a 264
-———I_—- k Ash
Measured value

———-o—-— k = 9.6

l '29 I '10 1 61

7-1. 01 from Isotropic hardening theory.

58

The reason will be that most of the input energy is dissi—
pated in the form of Volume change and only a small part is
used for lateral expansion. Probably soil is not work-

hardened isotrOpically as is metal.

Modification ovasotropic Hardening Equation

1. 'One of the main reasons for the discrepancy of
eq. (7.10) with the test results will be the assumption that
61 and 22 contribute equally to a]. If it is assumed that a
part of 01 is contributed from oz (that is, a; - 01(1) + 01(2))
and also 02 = aezn as well as 02 - pol, then 01(1) - aeln and
01(2) = s %ac2n. Also modifying eq. (7.10) into a simpler

form, eq. (7.12) is obtained.
-n l n ‘ ,
01' 3(81 - E 82 ) (7.12)

However eq. (7.12) has the same tendency as eq. (7.10) as
shown in Fig. 7—2 and App. C-2.

2. The discrepancy of-eq. (7.12) mainly comes from
the assumption oz - aezn based upon some idea of isotropy."
This assumption together with 02 I u01 leads us to the re-
lationship.§%: a u. The test result apparently shows that
this ratio is not a censtant. That means there is a non-
linear interaction between :1 and’ez. This may be described
in the form of 01 = f1(el) + f2(ez) + f3(e1,ez). If the
interaction f3(e1,e2) is expressed by a form of pblynomial
n- . .
izl
(7.12) in the following form.

bieln'1(n252)1, then it may be reasonable to modify eq.

  
 
  
  
   
  

59
40 r
Calculated from
01 01 = a(€1n - %€2n)
Spring rate (lb/in)
psi
__.._. k = 9.6
30 _ “—+— k = 56
———q‘——-
____.__.
Measured
-._-O.._
_._..x-_
—--A——
20 —-
.._-c|__
lO —
.10 .20 .30 El
0 l I I I I

 

 

Fig. 7—2. 01 from the modified equation (1).

6O

01 = an:1 - n(262)}n (7.13)
where n is a contribution factor of £2 to :1.

If the contribution factor n

1, that is, the inter-
action term is contributed equally from :1 and e2, the

following form is derived.
n
01" a(81- 262) (7.14)

The calculated value of 01 according to eq. (7.14) is given
in Fig. 7-3 and App. C-3. The same a and n values from the
fixed wall test as in the calculation of eq. (7.10) were
used for this calculation. The calculated results show
fairly good agreement with the test results for various
lateral confinements.

Using the equation 02 = uol, eq. (7.15) is obtained.
n
02 = 118(81 - 262) (7.15)

The calculated values of 02 from eq. (7.15) are given in
Fig. 7—4 and App. C—4. Considering the fairly large vari-
ation in u value, the calculated values are considered to
be approximately in agreement with the measured values.

Since (01 - 02) is the generalized stress in the
cylindrical case, the general form corresponding to eq. (7.8)
is 01 — 02 = (1 — u)a(el — 282)n (7.16)
log (01 - 02) versus log (31 - 252) relationship should be
a straight line. This is shown in Fig. 7—5 in which measured

values are plotted.

40

01

psi

30

20

10

 

,10

Calculated from

01 = a(el - 2ez)n

Measured value

Spring rate (lb/in)

k

NP? K

61

 
  
  
  
  
  
   
 
 
 
  

= 9.6
= 56
= 264

k‘”

.20 030 £1

 

Fig. 7-3.

01 from the modified equation ( 2)n

1

psi

 

62

 

A
/
ll
,Calculated from /
_ f /
02 =-Lla(€1 - 282)n ‘ //
I /
Spring rate (lb/in) /
O— k. = 9.6 // 1

  
    
   
   

“-'4"" ‘k‘* 56
—-A—‘ k- 264
Measured value

--—-o--- k .- 9.6
—--—x-¥-f— k = 56
_.......A..- k - 264

1 I I 1 I 1

.20 _ ' .30 e-

. 744..“62 from the modified ecuation (12) n ='1

 

63

 

40—- Spring rate (lb/in)
.___o..__ k = 9.6 9/
30r-
Z‘I —-X-— k = 56
Q 51
,\ -—~A-—- k = 264 ,x
D __0_ km °° (02 was
' ax
H calculated from
0
v 02 = “01) /
X
AM
10,— /
,0
-X
x’7/
5-
A.
X
0
D
X.
61 '> .15
2__ A 0 range
L J I 1 I I l I l I
.06 .08 .10 .20 .30 .40
(81 - 262)
Fig. 7-5.- log (01 — 02) versus log (51 - 252)

relationship

64

Other values for n, for example n = %, gives larger
deviation from the measured values as shown in Fig. 7—6 and
App. C-5.

Thus, eq. (7.14) and eq. (7.15) can be used as F1
and F2 functions for the range of 61 > 0.15. If the initial
bulk density is high, these equations could probably be used
in the lower range of 61 also with different values for

parameters a and n.

7.2 G Function

 

As shown in Fig. 6-9, I is not a function of y alone,
because I takes various values at the same level of 7.
Actually T seems to be affected by cm, which gives a reason

’1'"
to study 3E. The values are given in App. C-6. They are

m

fairly constant for all the spring rates with an average
In

value 35 a 1.00. ig'and Om are calculated from eq. (5.8)
m

and eq. (5.11) in which only measured values of 01 and 02

are used. (The relationship 02 = pol is not used.)
The same conclusion is also derived from the re—

lationship 02 = pol, because

T. = -(01 r 02) = l — u 01 (7'17)

 

qa

-Therefore,

Tn
i - 3” ‘ ‘4) (7.18)

am 2(1 + 2p)

65

 
  
  
  
  
   
   
  
 
  
 
  
  
   

4O -
Calculated from
01
c1 = 3(81 - €2)n
psi
Spring rate (lb/in)
‘—".“—' k = 9.6
30 " '_"+'— k = 56
-——43-—- k = 264
—D—— k A: on
Measured value
--- O-- k = 9.6
---K-—- k = 56
20 - --—A--
_-...r_1__
lO -
.10 . 0 . 0 e
O I 2| I a l 1

 

 

Fig. 7-6. 01 from the modified equation (2) n = %

66

where u is the average value for all the test range and

u = 0.141.
Tn
Hence, ;E = 1.01 from eq. (7.18). This agrees
m

naturally with the above calculation. So far as the re—
lationship 02 = 401 is valid, T" has a linear relationship
with cm in the form of eq. (7.18).

Since y is a function of el and e2 in the form of
eq. (5.10), it is possible to express a G function in terms
of £1 and 52. Using eq. (7.14) and eq. (7.17), a G function

in the following form is obtained.

T. = Sig—“l am - 2.:2)r1 (7.19)

7.3 H Function

 

An experimental equation similar to the original

consolidation equation is derived in Chapter VI, 6.6.

However, if the relationship between cm and %%

should be consistent with eq. (7.14), the following

equation (7.20) is derived.

Since £n(l + %%)= 2e2 — a, (from eq. (5.15))
1 + 2 n
and Om = ——-3——E' 3(51 " 252) 9
a 1 + 2p _ A! n ~
cm 3 a{ £n(l + V0)} (7.20)

where AV < 0 for compression.

.67

In Fig. 7:7, the am values calculated from eq. (7.20)
are plotted against the cm values which are directly obtained
from the measured value of 01 and 02. They show a fairly
good coincidence considering the large deviation of measured
values and of the u value in the calculation.

cm can be expressed in terms of bulk density p.

9
Since p = g = ———EKV (accordingly %% I -%§), eq. (7.20) be—
1+W
comes
=l+2u Ln
Om -—§——— a(2np0) (7.21)

7.4 Conventional Strain System and
Natural Strain System

In Fig. 7—8, 01 is expressed in terms of the con-
ventional strain e1, although the strain is expressed as
natural strain in all other diagrams in this thesis. A
comparison of this figure with Fig. 6-5 shows that Fig. 6.5
may give a slightly clearer idea in dividing the whole
function into two straight lines. The difference is, how—
ever, almost negligible. Both systems fit to the form of
01 = aeln or 01 - aeln, although the values of a and n are
different for each system. Therefore, from the viewpoint
of formation of stress-strain relationships,there will be
no appreciable difference between the two systems.

Although the conventional strain system has simpler
. equations in most cases as listed in App. C-8, theoretically

speaking, natural strain is better for handling large strain.

68

Calculated from

Om " om = l_i§3E a{—2n(l + %%)}n
psi - Spring rate (lb/in)
15 _. ._——o-——- k = 9.6
_ ———+———- k = 56
-—-‘v-—- k = 264

am from measured 01 and 02

 

---0—-“' k = 9.6
’ ——-x-—- k = 56
10 _. ———A-—-— k -'-' 264

 

 

 

5 —
x/
//
_. /
//+
X/A/
.. //
O
,// :28 AV
' XV V°
O
/
// / I I I _I
l I

—.10 —.2o --30

Fig. 7-7. am calculated from AK.

V0

69

 

 

 

50 ~—
40 Spring rate (lb/in)
//
30 __ —---o--— k = 9.6
X k = 56
20 '_ ——A—— k = 264
01 D kay 00
psi
10 —-
a
5 .—
I
l —
.8 —
.6 t
I l l I I

 

 

 

.20 .30
e1(in/in)

Fig. 7-8.--log 01 versus log e1 relationship.

70

This is especially true when the volume change or bulk
density is handled. The author has, therefore, used

natural strain for the theoretical calculations.

VIII. SUGGESTIONS FOR THE APPLICATIONS

1. Method of obtaining parameters

The simplest method to get the three parameters a,
n and u will be a compression test of soil in a thin-walled
cylinder on which strain gages are attached to pick up 02
under the condition of :2 I 0. The wall must be lubricated
with friction reducer.

By plotting 01 and 61 on a log-log chart, a and n

are obtained. u can be calculated from eq. (6.2).

2. Calculation of stress components

If the principal strain components 61 and £2 are
given, 01: 02 and r can be calculated from equations (7.14),
(7.15) and (7.17).

The X—ray technique could be used for measuring the
strain in the soil. Ex’ ey and ny are obtained from the
deformation of the grid of lead spheres buried in the soil.

:1 and 52 are easily calculated from them.

3. Boundary value problems

Simple force-deformation problems of soil might be
solved by a numerical method using the stress-strain re-
lationship together with the equation of stress equilibrium

and strain compatibility as well as boundary values. It

71

72

might be possible to compute stress-strain distribution by
selecting a network over the stress field and calculate the

stress-strain values at each node step by step.

4. Simulation type study

A simple approximation method might be possible for
those practical problems in which the situation could be
simulated by a compression test of soil under certain lateral
strain confinement. For example, the problem of soil de-
formation under the tractor tires might be simulated by this

laboratory test of variable lateral confinement.

ix.‘ SUMMARY AND CONCLUSIONS

9.1 Summary

A two-dimensional stress-strain law for soil is
required to solve the problems of tillage and traction.

The author studied the behavior of a cylindrical
'soil Sample which was laterally confined with.springs or
various spring rates. 'The sample was compressed axially
'at the rate of 0:092 in/min and the axial stress ansttrain
were measured. The lateral strain was also measured with
strain'gage'transducers. Since the sample was laterally
confined by the springs; the lateral stress could be ob-
{tained'from the known spring rate. The shear'streSSVWas
calbulated from the two normal stresses. 'The shear strain
‘was meaSured'fromsX-ray photographs of soil sample in.which
small lead spheres were buried. The change in the angle of
the lines between the spheres corresponds to the shear
strain. “The shear strain was also calculated from '
twoflprincipal Strain components assuming that there was no
wall friction.

A loam with an average moisture content of 12.4%
dry”basis was uSed. 'The soil was packed in the cylinder

with a pneumatic vibrator.¥ The average initial bulk

'73

74

density was 0.0346 lb/in3. A friction reducer which was

a mixture of grease and graphite was applied inside of the
cylinder wall and on the pistons. The wall friction was
small enough to satisfy the assumption that the stress
distribution was uniform in the sample and that the shear
stress on the surface of the sample was negligible.

Four tests were carried out for the various spring
rates (lb/in) of 9.6, 56, 264 and w (fixed wall). Three
replications were made for each test.

It was first assumed that the functions which de—
scribed the relationship between principal stresses and
principal strains are unique for the loading (compression)
process below the failure point. A series of tests gave

the following relationships

01 F1(€1,€2)

F2(51:€2)

G2

The functional form of F1 and F2 was derived by applying
'the isotropic hardening theory and by modifying it. They
were in the form of power functions.

The maximum shear stress was also formulated in
terms of 51 and 52.

The functional relationship between mean normal
stress cm and volume change as well as between cm and bulk

density change was derived by using F1 and F2 functions.

study.

75

9.2 Conclusions

 

The following conclusions have been made from this

1.

A soil compression test under various lateral

strain confinement has several advantages over a
compression test under stress confinement (such as
triaxial test) for obtaining stress-strain law
below failure point. The advantages are (a)
easiness of handling, (b) being able to handle
with any kind of soil, (c) more similarity to

the actual situation where soil expands laterally
under increasing vertical stress, and (d) simpler
and lower cost test equipment. The test gives us
useful information on stress-strain relationship
of soil.

The effect of wall friction is negligible when a
friction reducer is applied properly. The shear
strain obtained from X—ray pictures shows good
agreement with those values calculated from
principal strains on the assumption that there

is no wall friction. This implies that expensive
X-ray test can be eliminated.

From a set of tests in which lateral confinement
spring rate is changed from almost free expansion
to a rigid wall, two graphs showing the relation-

ships 01 = F1(€1,€2) and 02 = F2(el,ez) have been

76

obtained. Therefore, principal stresses can be
obtained graphically, if principal strains are
known (e.g. frome—ray test).

02/01 I constant (u) is noticed throughout
various lateral strain confinements within this
test range of e1 < 0.36 and £2 < 0.029.

A power equation 01 = aeln is valid for the 01
versus 81 relationship in this test range of

01 < 40 psi. TWO straight lines are noticed on
the log 01 - log el-graph.

By starting-from the isotropic hardening theory
of metal plasticityand modifying it, the follow-
ing functional relationships which show fairly
good agreement with test results have been de-
rived for the range of 51" 0.15 (bulk density
> 0.04 lb/in3).

01 = a(el -.2€2)n
02 = ua(e1 - 262)n

The three soil parameters a, n and u can be ob-
,tained'frOm a simple fixed wall compressidn test
'Iwith lateral pressure-pick-up.

The maximum shear stress 1; takes various values
at the same level of maximum shear strain. 7%
seems to be affected by mean normal stress a

m
in the form of_

77

. 3(1 - u) 0
2(1 + 211) m2

Tn
7;-

1" could also be formulated in terms of £1 and
1+-

82,

T." '3 LEI—L). 3(61 - 262)n for 61 > 0.15.

I:

The functional relationship between mean normal
stress and volume change seems to be in the

following form from the test results.

AV
108 O‘m'ko°-fi for €1>0.15
However, the following function is obtained by

using F1 and F2 functions.

= 1 - 2n AX n

A similar equation is obtained for the relation—
ship between am and bulk density.

Theoretically, a natural strain system is better
- 'than-a conventional strain system for dealing
with a large strain. However,_no.practical
difference for the formulation of a stress-

Istrain law is noticed between these systems.

SUGGESTIONS FOR FUTURE STUDY

1. Improvement of the test equipment in the follow-
ing points are desired.
(a) Greater e2 range
(b) More rigid cylinder wall
(c) Better lubrication method to get smaller,
more uniform and constant effect of wall
friction.
2. Similar tests should be carried out for other
soils of various texture, moisture content and density.
Another soil parameter n in the following form might have

to be introduced.

n
01 3(81 - 2n62)

n
au(el - 2062)

O2

_ 3. After part two is studied, a field test equip-
ment similar to the fixed wall test cylinder with u (and n)
pick-up transducers should be developed.
Development of a soil strain transducer is desired.
4. The applications of the stress—strain law should
be studied extensively. One of the first examples should be
the penetration test which has been used so much as a method

of measuring soil strength and deformation.

78

REFERENCES

Ahlvin,,R. G. and D. N. Brown
1960- Duplication of prototype stress-strain relation-
ships in soil masses by laboratory tests. Proc.

ASTM 60, 1137—1150.

Berezant sev , V. G.
1955 Limiting loads in the indentation of a cohesive
material by spherical and conical punches.
Translated from Russian. Sci. Inf. Dept. NIAE.

Cooper, A. w. and M. L. Nichols

1959 Some observations on soil compaction tests,
Agr. Engr. 40-5, 264-267.

Drucker, D. C. , et a1. '
1957 Soil mechanics and work- hardening theories of
plasticity. Trans. ASCE 122: 338- 346.

Drucker, D. C.
1961 On stress- strain relations for soils and loading
capacity. Proc. International Conf. on the
Mechanics of $0114Vehicle.System No. 1, 15-24.

Hendrick, J. G. and G. E. VandenBerg
1961 Strength and energy relations of a dynamically
loaded clay soil. Trans. ASAE 4-1, 31-32, 36.

Hill, R.
1950 The mathematical theory of plasticity. Oxford
Univ. Press.

Kondner, R. L. and R. J. Krizek
1964 A vibratory uniaxial compression device for
‘ coh_esive soil. Proc. ASTM 64, 934-943.

Lotspeich, F. B.
1964:. Strength and bulk density of compacted mixture
of kaolinite and glass beads. Proc. Soil Sci.
Soc. 28:737-743.

McMurdie, J. L.

1963 Some characteristics of the soil deformation
process.. Proc. Soil Sci. Soc. 27:251-254.

80

Reaves, C. A., and M. L. Nichols

1955 Surface soil reaction to pressure. Agr. Engr.
36-12, 813-816.

Roberts, J. E., and J. M. Souza

1958 The compressibility of sand. Proc. ASTM 58:
1269—1277.

Scott, R. F.

1963 Principles of soil mechanics. Addison-Wesley
Publ. Co.

Soehne, W. n
1956 Einige Grundlagen fur eine Landtechnische
Bodenmechanik. Grundlagen der Landtechnik Hf.
7, 11-27.

Soehne, W. H., W. J. Chancellor and R. H. Schmidt
1959 Soil deformation and compaction during piston
sinkage. ASAE paper 59-100.

Taylor, J. H., and G. E. VandenBerg
1965 The role of displacement in a simple traction
system. ASAE paper 65-122.

VandenBerg, G. E., W. F. Buchele, and L. E. Malvern
1958 Application of continuum mechanics to soil com-
paction. Mich. Agr. Exp. Sta. Jour. No. 2251.

VandenBerg, G. E. I
1962 Triaxial measurements of shearing strain and
compaction in unsaturated soil. ASAE paper 62—648.

Vomocil, J. A., L. J. Waldron, and W. J. Chancellor
1961 Soil tensile strength by centrifugation. Proc.
Soil Sci. Soc. 25-3, 176-180.

Waldron, L. J.
1964 Soil viscoelasticity; superposition tests.
Proc. Soil Sci. Soc. 28, 323-328.

1964 Viscoelastic function for soils. Proc. Soil
Sci. Soc. 28, 329-333.

Waterways Experiment Station
1952 Torsion shear apparatus and testing procedures.
WES Bulletin 38, 75.

1961 Physical components of the shear strength of

saturated clays. WES miscellaneous paper No.
3—428.

APPENDIX

82

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85

APPENDIX B-l

01 Values (psi) From eq. (5.1)

 

Spring rate k(lb/in)

 

 

A2.
(in) k = 9.6 k = 56 k - 264 k «aw
0 0 0 0 0
0.23 .60 1.45 .85 .85
0.46 1.35 3.05 2.30 1.98
0.69 2.43 4.90 4.85 3.68
0.92 3.98 7.24 7.94 6.55
1.15 6.37 10.35 12.53 10.99
1.38 14.18 18.85 17.99
1.61 19.38 27.66 26.50
1.84 26.13 39.51 41.27

 

86

APPENDIX B—2

 

 

 

 

 

 

 

(a) el Values (in/in) From eq. (5.2)
02 Spring rate k(lb/in)

(in) k - 9.6 k a 56 k a 264 e co
0 0 0 0 0
0.23 .0379 .0379 .0384 .0370
0.46 .0758 .0758 .0768 .0741
0.69 .1137 .1137 .1152 1111
0.92 .1516 .1516 .1536 1481
1.15 .1895 .1895 .1920 .1852
1.38 .2273 .2304 2222
1.61 .2652 .2688 2593
1.84 .3031 .3072 2963

(b) 81 Values From eq. (5.3)
A2 Spring rate k(lb/in)

(in) k=9.6 k=56 k=264 kmoo
0 0 0 0 0
0.23 .0387 .0387 .0391 .0377
0.46 .0788 .0788 .0798 .0770
0.69 .1207 .1207 .1224 .1177
0.92 .1644 .1644 .1668 .1603
1.15 .2101 .2101 .2132 .2048
1.38 .2579 .2619 .2513
1.61 .3082 .3131 .3002
1.84 .3611 .3670 .3514.

 

02 Values (psi)

87

APPENDIX B-3

From eq. (5.4)

 

 

 

Spring rate k(lb/in)

(i:) k = 9.6 k = 56 k = 264
0 0 0 0
0.23 .09 .16 .17
0.46 .21 .35 .30
0.69 .34 .64 .50
0.92 .48 1.10 .98
1.15 .67 1.78 1.57
1.38 2.54 2.87
1.61 3.52 4.82
1.84 4.74 7.68

 

APPENDIX B—4

88

 

 

 

 

 

 

 

(a) e2 Values (in/in) From eq. (5.5)

A1 Spring rate k(lb/in)

(in) = 9. = 56 k = 264 km on
0 0 0 0 0
0.23 .0044 0013 .0003 0
0.46 .0099 0028 .0005 0
0.69 .0153 0049 .0008 0
0.92 .0211 0081 .0015 0
1.15 .0280 0126 .0023 .0001
1.38 0172 .0040 .0004
1.61 0228 .0064 .0010
1.84 0293 .0097 .0017

(b) :2 From eq. (5.6)

A2 Spring rate k(lb/in)

(in) = 9. k = 56 k = 264 a...
0 0 0 0 0
0.23 .0044 .0013 .0003 0
0.46 .0099 .0028 .0005 0
0.69 .0152 .0049 .0008 0
0.92 .0209 .0081 .0015 0
1.15 .0276 .0125 .0023 .0001
1.38 .0171 .0040 .0004
1.61 .0226 .0064 .0010
1.84 .0289 .0097 .0017

 

89
APPENDIX B-5

(a) om Values (psi) From eq. (5.11)

 

Spring rate k(lb/in)

 

 

 

 

 

 

A52.
(in) k = 9.6 k = 56 k = 264
0 0 0 0
0.23 .26 .59 .40
0.46 .59 1.25 .97
0.69 1.04 2.06 1.95
0.92 1.65 3.15 3.30
1.15 2.57 4.64 5.22
1.38 6.42 8.20
1.61 9.14 12.43
1.84 11.87 18.29
(b) 61', 62' Values (psi) From eq. (5.12) and eq. (5.13)
Spring rate k(lb/in)
(ii) k = 9.6 k = 56 k = 264
01' 02' 01‘ 02' 01' 02'
0 0 0 0 0 0 0
0.23 .34 -.17 .86 -.43 .45 -.23
0.46 .89 -.45 1.80 -.90 1.33 -.67
0.69 1.39 -.70 2.84 -1.42 2.90 -1.45
0.92 2.33 —1.17 4.09 -2.05 4.64 —2.32
1.15 3.80 —1.90 5.71 -2.86 7.31 -3.65
1.38 7.76 -3.88 10.65 -5.33
1.61 10.57 -5.29 15.23 -7.61
1.84 14.26 -7.13 21.22 -10.28

 

90

APPENDIX B—6

 

 

 

 

 

 

 

I" Values (psi) From eq. (5.8)
tr
A2 Spring rate k(lb/in)
(in) k = 9.6 k = 56 k = 264
0 0 0 0
0.23 .26 .65 .34
0.46 .67 1.35 1.00
0.69 1.05 2.13 2.18
0.92 1.75 3.07 3.48
1.15 2.85 4.29 5.48
1.38 5.82 7.99
1.61 7.93 11.42
1.84 10.70 15.92
APPENDIX B-7
y" Values From eq. (5.10)
1;-
A2 Spring rate k(lb/in)

(in) k = 9.6 k = 56 k = 264
0 0 0 0
0.23 .0430 .0400 .0394
0.46 .0882 .0814 .0802
0.69 .1358 .1228 .1228
0.92 .1840 .1714 .1676
1.15 .2356 .2202 .2140
1.38 .2716 .2628
1.61 .3255 .3194
1.84 .3804 .3674

 

91

APPENDIX B-8

AV/Vo Values From eq. (5.14)

 

 

 

A2 Spring rate k(lb/in)

(in) = 9.6 k = 56 = 264
O 0 0 0
0.23 .0294 -.0354 .0378
0.46 .0575 -.0706 .0759
0.69 .0864 —.1050 .1138
0.92 .1155 -.1378 .1511
1.15 .1435 -.1689 .1883
1.38 —.2005 .2243
1.61 —.2313 .2594
1.84 -.2616 .2937

 

92

 

 

 

 

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93

APPENDIX C-1

01 Values from Isotropic Hardening Theory

n
o = a(612 - 2522)? (psi)

 

Spring rate k(lb/in)

 

 

 

 

 

 

AR.
(1“) k = 9.6 k = 56 k = 264 k e
0.92 6.19 6.29 6.54 5.94
1.15 11.17 11.36 11.82 10.73
1.38 18.61 19.41 17.58
1.61 28.58 29.87 27.00
1.84 41.84 43.81 39.48
APPENDIX C-2
01 Values from the Modified Equation (1)
01 = a(e1n - %Ezn) (psi)
At Spring rate k(lb/in)
(in) k = 9.6 k = 56 k = 264 k e
0.92 6.01 5.96 6.54 5.94
1.15 10.81 11.31 11.82 10.73
1.38 18.54 19.41 17.58
1.61 28.40 29.87 27.00
1.84 41.54 43.81 39.48

 

 

.II 1‘ l'.' A .

94
APPENDIX C-3

01 Values from the Modified Equation (2),, n = l

01 3 8(61 — 282)n (psi)

 

Spring rate k(lb/in)

 

 

09.

(in) k = 9.6 k = 56 k = 264 k:v m
0.92 3.11 4.91 6.26 5.94
1.15 5.47 8.40 11.21 10.70
1.38 13.27 18.02 17.44
1.61 19.62 27.02 26.56
1.84 27.67 38.46 38.57

APPENDIX C—4

02 Values from the Modified Equation (2), n = l

02 = ua(€1 - 262)n (p81)

 

Spring rate k(lb/in)

 

 

All.

(in) k = 9.6 k = 56 k = 264 k5»
0.92 0.46 0.73 0.93 0.89
1.15 0.82 1.25 1.67 1.59
1.38 1.98 2.68 2.60
1.61 2.92 4.03 3.96

1.84 4.12 5.73 5.75

 

 

 

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95

APPENDIX C-5

01 Values from the Modified Equation (2), n

01: a(51- €2)n (1381)

NIH

 

Spring rate k(lb/in)

 

 

1
(in) = 9.6 k = 56 k = 264 key a
0.92 4.54 5.59 6.39 5.94
1.15 8.13 9.85 11.51 10.72
1.38 15.85 18.71 17.51
1.61 23.94 28.43 26.78
1.84 34.48 41.11 39.03

 

96

APPENDIX 046

Tn/c Values
V m

 

r?

 

 

 

 

 

 

A2 Spring rate k(lb/in)
(in) k = 9.6 k = 56 k = 264
o -_ __ __
0 23 1.00 1.10 0.85
0.46 1.14 1.08 1.03
0.69 1.01 1.03 1.12
0 92 1.06 0.97 1.05
1 15 1.11 0.92 1.05
1.38 0.91 0.97
1.61 0.87 0.92
1.84 0.90 0.87
APPENDIX C-7
Om Values calculated from %%, (psi) From eq. (7.20)
Al Spring rate k(lb/in)

(in) k = 9.5 k = 56 k = 264
0.92 1.35 2.13 2.71
1.15 2.36 3.63 4.85
1 38 5.76 7.79
1.61 8.50 11.69
1 84 11.98 16.65

 

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