DYNAMIC STRESS TRANSDUCERS AND THE USE
OF CONTINUUM MECHANICS IN THE STUDY

OF VARIOUS SOIL STRESS-STRAIN
RELATIONSHIPS

TIIcsls Ior II“: Degree oI DI’I. D.
MICHIGAN STATE UNIVERSITY

Wesley Lamar Harris
1960

 

This is to certify that the

thesis entitled

Dynamic Stress Transducers and the Use
of Continuum Mechanics in the Study
of Various Soil Stress-Strain
Relationships

presented bg
Wesley Lamar Harris

has been accepted towards fulfillment
of the requirements for

Ph.D degree inAggicultux-al Engineering

MKM

 

fliajor professor

Date OCtOber 28 l 60

0469

 

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DYNAMIC STRESS TRANSDUCERS
AND THE USE OF CONTINUUM MECHANICS
IN THE STUDY OF VARIOUS SOIL STRESS-STRAIN RELATIONSHIPS

by

Wesley Lamar Harris

AN ABSTRACT

Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY
Department of Agricultural Engineering

1960

Na/ I
”PM“ “M f:/ f}:- fieM 94}

fl’fr f/y- («4

ABSTRACT

The changes in soil consolidation resulting from
externally applied forces and the effect of these changes
on the physical properties of the soil have been studied
by many individuals. The results of one of the investi-
gations revealed that the concept of continuum mechanics
could be used as a mathematical model for studying the
soil compaction problem. The development of soil stress-
strain relationships which will permit the prediction of
the changes in the state of compaction caused by various
implements and power units will be a major contribution
toward controlling soil compaction. .

The concept of continuum mechanics was used to
determine various stress-strain relationships. A Six
Directional Stress Transducer capable of measuring
sufficient data to determine the components of the general
stress tensor was developed and compared with the method
used by vanden Berg (1958). A W Cell capable of measuring
mean stress directly was developed and the values of mean
stress calculated from the Type A and 6 DST data were
compared.

The data from a series of 27 tests of 5 replications
composed of three depths below the loading surface, three
moisture contents and three rates of loading are presented.

11

The data was analyzed using MISTIC, an electronic digital
computer, to determine the relationships between the
invariants of the stress tensor and bulk density.

The hypothesis that changes in mean normal stress,
an invariant of the stress tensor, are related to changes
in volumetric strain was tested by measuring the stress
tensor and bulk density in the soil while the soil was
subjected to dynamic loads of various magnitudes. Based
on the data presented, the hypothesis could not be accepted
or rejected. The data indicated that of the four invariants
of the stress tensor investigated the maximum shear stress
related best to changes in bulk density.

The relationships between the invariants and bulk
density were affected by the moisture content at the
higher rate of loading and deeper depths. The rates of
loading data was varied; therefore the effects on the
relationships could not be determined.

The values of mean stress obtained directly from the
W Cell compared best with the values calculated from the
Type A data. Comparison of the two methods of measuring
vertical stress with theoretical values determined with
Froehlick's equation showed good agreement with the Type A
values at the two deeper depths. Fbr the 5-inch depth,
both the 6 DST and Type A data were greater than the

theoretical data for a given surface load.

iii

The relationships between the invariants and bulk
density are exponential for the soil studied. The
relationship between mean stress and applied load appears
to be linear for loads greater than 5 pounds per square
inch.

iv

DYNAMIC STRESS TRANSDUCERS
AND THE USE OF CONTINUUM MECHANICS
IN THE STUDY OF VARIOUS SOIL STRESS—STRAIN RELATIONSHIPS

by

Wesley Lamar Harris

A THESIS
Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and

Applied Science in partial fulfillment 6f
the requirements for the degree of

DOCTOR OP PRILO SOPHY

Department of Agricultural Engineering

1960

 

6/5723?
e/zo/w

VITA
Wesley Lamar Harris
candidate for the degree of

Doctor of Philosophy

Final examination: October 28, 1960; 2:00 P.M.; Room 218
Agricultural Engineering Building

Dissertation: Dynamic Stress Transducers and the Use
of Continuum Mbchanics in the Study of

variousLStress—Strain Relationships

Outline of Studies:

major subject: Agricultural Engineering

Minor subjects: Applied Mechanics
mathematics

Biographical items:

Born: November 12, 1931; Taylorsville, Georgia

Undergraduate studies: University of Georgia
BOSOAOE.’ 1953

University of Georgia
M.S., 195 ,
Michigan State University

1958-1960

Gratuate Studies:

Experience:

1953-1956 Commissioned Officer, Corps of Engineers,

Uhited States Army
1956-1958 Instructor, University of Georgia
1958-1960 Graduate Assistant, Michigan State University

Professional and Honorary Affiliations

American Society of Agricultural Engineers

Registered Land Surveyor
Sigma XI, Alpha Zeta, Sphinx, Blue Key, Omicron Delta

Kappa, Aghon

ACKNOWLEDGEMENTS

In the course of planning and conducting an investi-
ation of the nature presented in this thesis, the cooperation
and advice of numerous individuals is necessary. The
author wishes to express his sincere appreciation to all
those people who contributed to make this thesis possible.
A special note of thanks is extended to those individuals

mentioned below.
Dr. Arthur W. Farrall, Head, Department of Agricultural

Engineering, Michigan State University, provided the
financial assistance for the pursuance of the program of
study.

Dr. wesley F. Buchele, the author's major professor
provided valuable assistance, encouragement, guidance, and
inspiration throughout the author's course of study.

The guidance committee, Prof H. F. McColly and Dr. Carl
Hall, Agricultural Engineering, Dr. L. E. Malvern, Applied
Mechanics, Dr. E. A. Nordhaus, Mathematics, and Dr. A. E.

Erickson, Soil Science Departments, provided constructive

criticisms and suggestions during the course of study and

Preparation of this manuscript.
Col. M. G. Bekker of the Land Locomotion Laboratory

and Mr. Alexander Brede of the Motor Wheel Corporation

arranged for the loan of strain gage equipment from their

repective organizations.
vii

Dr. Clement A Tatro of the Applied Mechanics Department
and Dr. T. H. wu of the Civil Engineering Department,

Michigan State university loaned electronic equipment used

during the experimental tests. In addition Dr. Tatro gave

valuable assistance during the construction and calibration

of the strain gage transducers.
. Jim Cawood and Mr. Glenn Shiffer of the Agricultural

If”
AHA

Engineering Department gave valuable aid in the construction

of the equipment. Mr. Cawood demonstrated expert workman—

ship during the construction of the six directional stress

transducer. Mr. Sniffer was a source of many challenges

which were stimulating and educational.

Mr. Jack V. Stong, Graduate Student and Mr. George
Keller, undergraduate Student, Agricultural Engineering,
assisted in the construction of the soil handling equipment.

Mr. Harold Brockbank of the Agricultural Engineering
Department, and Mr. John Morrison, undergraduate Student,
Agricultural Engineering, assisted during the investigation.

The author is especially grateful to his wife, Libby

for her encouragement and help which made the completion of

this report possible. To his elder son, Wes for his

confidence in his father's ability and his younger son,

Jamie, who didn't appreciate the time his father spent

studying.

viii

TABLE OF CONTENTS

Section

INTRODUCTION . . . . . . . . .
REVIEW OF LITERATURE . . . . .
THEORY . . . . . . . . . . . .
APPARATUS AND INSTRUMENTATION

Design and development of
Stress Transducer . . .

w cell 0 O O O O O O I 0
Soil handling equipment .

Force transducer . . . .

PRWEDURE O O O O O O O O O 0

O O O 0

RESULTS AND DISCUSSION . . . . . . . .
Relationship between mean normal stress

and bulk density . . .

Relationship between second invariant and
bukdenSityocooeooooooc

Relationship between maximum normal stress

and bulk density . . . .

Relationship between maximum shear stress

and bulk density . . .

The effect of moisture content on the

relationship between mean normal stress
andbulkdensitycoooooeoooo

The effect of moisture content on the
relationship between second invariant

and b11115 density 0 o c

The effect of moisture content on the
relationship between maximum normal

stress and bulk density . . . . . . . .

ix

Page
0 O O 0 O O O O 1
O O O O O O O O 4
O O O O O 0 O O 12
o} o o o 20
Six Directional-
. O O O O O O O O O O 20
o o o c e c e o 34
O O O O O O O O 34
O O O O 0 O I O 38
O O O O O O O 43
O O O O O O I O 49
O O O O O O O O 53
c c o o o 58
e c c e o o o o c e o 65
o c I c c o o o 72
O o o o 78
O O O O 83
o o o g 87

Section Page

The effect of moisture content on the
relationship between maximum shear stress
c.0091

andbulkdenSj-tyooooeooeooo

The effect of rate of loading on the
relationship between mean normal stress

and bulk density . . . . . . . . . . . . . . . 91

The effect of rate of loading on the
relationship between second invariant

andbUlkdeHSityoeoccooooeooooo99
The effect of rate of loading on the '
relationship between maximum normal
. 99

stress and bulk density . . . . . . . . . . .

The effect of rate of loading on the
relationship between shear stress
and bulk denSity O O O O O O O O O O 0 O O 0 0

Comparison of three methods used to
determine mean normal stress . . . . . .

106

O O O 106

The effect of rate of loading on the
relationship between mean stress and

applied load . . . . . . . . . . . . . . . . . 118

The effect of moisture content on the
relationship between mean normal
Stress and applied load 0 O O O O O O O O O O O 123

o o o o o o o e 136
o o o o o 138
142

CONCLUSIONS 0 O O O O O O O O O O O 0
REFERENCES 0 O O O O O O 0 O O O O O O O O 0
APPENDIX 0 O O O O O O O O O C

Figure
1.

9.
10.

11.

12.

13.

LIST OF FIGURES

Detail drawing of the four directional
stress transducer . . . . . . . . . . . . .

Drawing of the six directional stress
transducer showing the location of
diaphragm pressure cells . . . . . . . . .

Calibration curve for Redshaw strain gage
on 0.010 inch thick diaphragm 3/4 inch
in diameter 0 O O O O O O O O O O O O O O 0

Calibration curve for Redshaw strain gage
mounted on 0.020 inch thick diaphragm
3/4 inch in diameter . . . . . . . . . . .

A typical calibration curve using bridge
arrangement as shown. One complete
Redshaw 2-ED type strain gage used as
sensing element and the other as dummy
gageSeeeoocccoocococoooo

A view of the six directional stress
tranSduCer O O O O O O O O O O 0 O O O O 0

Calibration device used to obtain calibration
dataforthGGDST.............

The W Cell that was used to measure mean
StreSSdireCtly 0000000000...

A view of the soil handling equipment . . . .

Detail drawing of the W Cell used to measure
mean Stress direCtly o o o o c c o o o o 0

Calibration curve for W Cell showing number
of lines deflection versus applied load . .

A view of the loading plate, force transducer
and hydrau1ic cylinder 0 o o o o e o o o o

A view of the pressure transducers and
balloons used to obtain data: . . . . . . .

xi

Page

Figure Page

14. Detail drawing of force transducer showing
location of strain gages and ball and
socket Joint . . . . . . . . . . . . . . . . . 41

15. Calibration curve for force transducer
showing number of lines deflection
versus applied load . . . . . . . . . . . . . . 42

16. mean normal stress versus bulk density
for data obtained with 6 Directional
Stress Transducer . . . . . . . . . . . . . . . 59

17. mean normal stress versus bulk density
for data obtained with Type A pressure cells . 60

18. Second invariant of stress deviator tensor
versus bulk density for data Obtained
with 6 Directional Stress Transducer . . . . . 66

19. Second invariant of stress deviator tensor
versus bulk density for data obtained
with Type A pressure cells . . . . . . . . . . 67

20. Comparison of the values of the second
invariant calculated from the Type A
and 6 DST data 0 O O O O O O O O O O O O O O I 68

21. maximum normal stress versus bulk density
for data obtained with 6 Directional

Stress Transducer . . . . . . . . . . . . . . . 73

22. maximum normal stress versus bulk density
for data obtained with Type A pressure cells . 74

23. Maximum shearing stress versus bulk density
for data obtained with 6 Directional

Stress TranSducer o o o o e o o e o o o o o c o 79

24. maximum shearing stress versus bulk density
for data obtained with Type A pressure cells . 80

25. The effect of moisture content on the
relationship between mean stress and
bulk density. Data Obtained with 6 DST . . . 84

26. The effect of moisture content on the
relationship between mean stress and
bulk density. Data obtained with Type A

Pressure C3118 o o o c o o o o o c o c o o o o 85
xii“

 

Figure

27. The effect of moisture content on the
relationship between the second invariant
and bulk density. Data obtained with 6 DST . . 88

Page

28. The effect of moisture content on the
relationship between the second invariant
and bulk density. Data obtained with
Type A pressure cells . . . . . . . . . . . . . 89

29. The effect of moisture content on the
relationship between the maximum normal
stress and bulk density. Data obtained

with5DST.e...............o92

30. The effect of moisture content on the
relationship between the maximum normal
stress and bulk density. Data obtained
with Type A pressure cells . . . . . . . . . . 93

31. The effect of moisture content on the
relationship between the maximum shear
stress and bulk denstiy. Data obtained
'1th6DSTocccoooocoeooecoeogs

32. The effect of moisture content on the
relationship between maximum shear stress
and bulk density. Data obtained with
Type A pressure cells . . . . . . . . . . . . . 96

33. The effect of rate of loading on the
relationship between.mean stress and
bulk density. Data obtained with 6 DST . . . 100

34. The effect of rate of loading on the
relationship between mean stress and
bulk density. Data obtained with
Type A pressure cells . . . . . . . . . . . . 101

35. The effect of rate of loading on the
relationship between the second invariant
and bulk density. Data obtained with 6 DST . 103

36. The effect of rate of loading on the
relationship between the second invariant
and bulk density. Data Obtained with
TYPEAPresmecellsoocooooeooo.104

xiii

Figure Page
37. The effect of rate of loading on the
relationship between the maximum
normal stress and bulk density.
Data obtained with 6 DST . . . . . . . . . . . 107

38. The effect of rate of loading on the
relationship between the maxrmwm
normal stress and bulk density.
Data obtained with Type A pressure cells . . . 108

39. The effect of rate of loading on the
relationship between the maximum
shear stress and bulk density. Data
Obtained With 6 DST O O O O O O O O O O O O O 110

40. The effect of rate of loading on the
relationship between the maximum shear
stress and bulk density. Data obtained
with Type A pressure cells . . . . . . . . . . 111

41. Comparison of the regression lines for the
three methods used to determine mean stress.
Data obtained at the 5-inch depth,
1.00 in/sec rate of loading and a moisture
content0f17e41%cocoa-00000000113

42. Comparison of the regression lines for the
three methods used to determine mean stress.
Data obtained at the 5-inch depth,
1.00 in/sec rate of loading and a moisture
content0f10079%000000.0000000114

43. Comparison of the regression lines for the
three methods used to determine mean stress.
Data obtained at the 10-inch depth,
1.00 in/sec rate of loading and a moisture

content of 17.41 % . . . . . . . . . . . . . . 115

44. Comparison of the regression lines for the
three methods used to determine mean stress.
Data obtained at the 10-inch depth,
1.00 in/sec rate of loading and a moisture

content0f10095%oooocooocecoo.116

45. Comparison of the regression lines for the
three methods used to determine mean stress.
Data obtained at the 15-inch depth,
1.00 in/sec rate of loading and a moisture
content of 12.31% . . . . . . . . . . . . . . 117

xiv

Figure Page

46. The effect of rate of loading on the
relationship between mean stress and
applied load at 5-inch depth. Data

obtained with 6 DST at a moisture
content ranging from 8.89 to 10.79% . . . . . 119

47. The effect of rate of loading on the
relationship between mean stress and
applied load at 5-inch depth. Data
obtained with Type A pressure cells at a

moisture content ranging from 8.89 to 10.79% 120

48. The effect of rate of loading on the
relationship between mean stress and
applied load at 5-inch depth. Data obtained
with 6 DST at a moisture content ranging .
from16905 t017067% o c c o o o o c o o c c 0121

49. The effect of rate of loading on the
relationship between mean stress and
applied load at S-inch depth. Data
obtained with Type A cells at a moisture
content ranging from 16.05 to 17.67% . . . . . 122

50. The effect of moisture content on the
relationship between mean stress and
applied load at S-inch depth and 0.38 in/sec
rate of loading. Data obtained with 6 DST . . 124

51. The effect of moisture content on the
relationship between mean stress and
applied load at S-inch depth and
0.38 in/sec rate of loading. Data
obtained with Type A cells . . . . . . . . . . 125

52. The effect of moisture content on the
relationship between mean stress and
applied load at 15-inch depth and
0.62 in/sec rate of loading. Data
obtained with 6 DST . . . . . . . . . . . . . 126

53. The effect of moisture content on the
relationship between mean stress and
applied load at 15-inch depth and 0.62 in/sec
rate of loading. Data obtained with
TypeApressurecells ............127

IV

Figure Page
54. The effect of moisture content on the
relationship between mean stress and

applied load at 10-inch depth and
1.00 in/sec rate of loading. Data
obtained with 6 DST . . . . . . . . . . . . . 129

55. The effect of moisture content on the
relationship between mean stress and
applied load at 10-inch depth and
1.00 in/sec rate of loading. Data
obtained with Type A cells . . . . . . . . . . 130

56. Comparison of the two instruments used
to measure the vertical stress with
theoretical values at a depth of 5 inches
below the loading surface. Measured data
obtained from test 17 . . . . . . . . . . . . 132

57. Comparison of the two instruments used
to measure the vertical stress with
theoretical values at a depth of 10 inches
below the loading surface. measured data
obtained from Test 16 . . . . . . . . . . . . 133

58. Comparison of the two instruments used
to measure the vertical stress with
theoretical values at a depth of 15 inches
below the loading surface. Measured data
obtained from test 15 . . . . . . . . . . . . 134

xvi

LIST OF TABLES
Table Page

1. Average output of number one set of gages
in DST calibrated for one line '
deflection equal to one psi . . . . . . . . . . 31

2. Physical description of the Brookston soil
needintest00000000000000.0046

3. Changes in bulk density produced by the
weight of the soil as determined with
the volumetric transducer . . . . . . . . . . . 47

4. Description of the laboratory tests . . . . . . . 50

5. Statistical analysis for mean normal stress
versus bulk density . . . . . . . . . . . . . . 54

6. Comparison of regression coefficients of
the r m versus bulk density relation
for different stress states and same
stress state for the two methods used
to Obtain data 0 O 0 O O O O O O O O O O O 0 g 57

7. Statistical analysis for second invariant
verS'quulkdenSj-tycoo0000000000062

8. Comparison of regression coefficients
of the II versus bulk density relation
for diffegent stress states and same
stress state for the two methods used
tOObtamdataocoocoeeoeeeoo..64

9. Statistical analysis for maximum normal
stress versus bulk density . . . . . . . . . . 69

10. Comparison of regression coefficients of
the WI versus bulk density relation for
different stress states and same stress
state for the two methods used to obtain

data...ooooeoeoooceoceooo71

11. Statistical analysis for maximum shear
stress versus bulk density . . . . . . . . . , 75

xvii

Table Page

12. Comparison of regression coefficients
for 'X max versus bulk density relations
for different stress states and same
stress state for the two methods used
to Obtain data 0 O O O O O O O O O O O O O 0 O 77

13. Comparison of regression coefficients
of the 0" m versus bulk density
relations at different moisture contents
and between methods for a given test . . . . . 82

14. Comparison of regression coefficients
of the II versus bulk density relations
at constafit depth and different moisture
contents and between methods for a given test. 86

15. Comparison of regression coefficients of
the GI versus bulk density relations at
constant rate, constant depth, and
different moisture contents and between
methods for a given test . . . . .‘. . . . . . 9O

16. Comparison of regression coefficients of
the’rmax versus bulk density relations at
constant rate, constant depth and different
moisture contents and between methods for

a given teSt O O O O O O O O O O O O O O O O O 94

17. Comparison of regression coefficients of
the Em versus bulk density relations at
different rates of loading and between
methOds for a given test 0 c o e o o o o o o o 97

18. Comparison of regression coefficients of
the II versus bulk density relations at
differgnt rates of loading and between
methods for a given test . . . . . . . . . . . 102

19. Comparison of regression coefficients of
' the 01 versus bulk density relations at

different rates of loading and between 105

methods for a given test . . . . . . . . . . .
20. Compar son of regression coefficients of

the max versus bulk density relations at

different rates of loading and between 109

methods for a given test . . . . . . . . . . .

xviii

 

Table Page

21. Comparison of the regression coefficients
calculated for the relationship between
mean normal stress and applied load for
a constant rate of loading of 1.00 inch
Per second 0 O O 0 O O O O O O O O O O O O O O 1 12

A Calculated values of mean normal stress,
second invariant of deviator stress tensor,
maximum and minimum principal stresses,
and maximum shear stress for the Type A Cells
and Six Directional Stress Transducer. . . . . 143

xix

 

INTRODUCTION

A major factor in the advancement of civilization
during the twentieth century has been the mechanization
of agriculture. The extensive use of larger power units
and associated equipment has benefited mankind, but it
has been a source of problems too; 1.3. inadequate soil
air movement, reduced infiltration and percolation rates,
mechanical impedance to roots and reduced crop yields are
caused to some degree by excessive compaction. The complete
solution of these problems will require the combined efforts
of many branches of science. Soil compaction resulting
from large externally applied forces has been studied by
agricultural engineers and soil physicists. Unfortunately
their results to date have not produced an adequate
agricultural soil mechanics.

Although soil is one of the oldest materials used by
man, accurate stress-strain relationships for all soil
conditions and types of loading have not been developed.
The main reason that this is true is that agricultural
soils vary in density and texture and are non-homogeneous
and inelastic. Vanden Berg(1960) stated that there is
no analytical method for developing a rigorous stress-
strain relationship. The stress and strain developed in
a soil mass must be measured simultaneously and stress-

strain relationships for soil determined empirically.

The study of soil compaction consists of two phases.
The first phase involves determining the distribution of
stresses in a soil mass caused by externally applied forces.
The second phase involves determining the effect that these
stresses or strains have on the soil mass. Since one of
the effects of the stresses developed is to force a
modification of the stress pattern, the two phases must
be studied simultaneously.

In general the stress distribution developed by an
externally applied load will depend upon several factors
that include the following:

1. The magnitude and type of load

2. The size and shape of the contact area where

the force is applied

3. The distribution of the pressure within the

contact area

4. Moisture content of the soil

5. The initial bulk density of the soil mass.

The largest forces applied to the soil are due to
tractor and implement traffic. While these forces are
not the only causes of soil compaction, they are conceded
to be the major cause. Since these forces are dynamic, to
understand the effect of these forces, the volumetric strain
produced by a dynamic load must be determined.

In order to conduct the above study, the model of a

continuous medium for soils as proposed by Vanden Berg (1958)

was used. He defined soil stress as a set of nine quantities
in the form of a stress tensor instead of a single value.
The stress tensor can be separated into two components, the
mean normal stress tensor or spherical stress tensor and
the stress deviator tensor. The spherical stress tensor

is similiar to hydrostatic pressure and is determined by
taking the algebraic mean of the normal stresses acting

in three mutually perpendicular directions at a point. The
stress deviator tensor differs from the stress tensor in
that the mean normal stress is subtracted from each normal
stress component.

Since any stress-strain relationship will be a
complicated function depending on soil type, moisture
content, rate at which the load is applied and others,
many instrumentation problems are involved. Because of
the great need for an instrument to measure the components
of the stress tensor at a point in the soil, the major
portion of the work presented in this thesis was directed
toward the design, construction and development of a six
directional stress transducer.

The primary objective of this study was to determine
the relationship of dynamic forces, mean normal stress and
volumetric strain. When this relationship is determined,
it will be possible to predict soil compaction as caused
by various implements and power units. This information
will be a major contribution toward the development of

means for controlling soil compaction.

REVIEW OF LITERATURE

Numerous studies have been conducted to evaluate the
stress distribution in soils and the relationship between
the stresses and changes in the soil mass. Obviously, if
this relationship was known, the change in the state of
compaction resulting from externally applied forces could
be predicted. The soil stress-strain relationships were
recently reviewed by vanden Berg (1958).

The deveIOpment of strain-gage pressure transducers
led to the first real progress in accurately measuring
stress in a soil mass. They have not only been used within
the soil mass but at the loading surfaces such as at the
soil-tire interface. The performance of the cells reported
by Cooper (1956) makes the Type A Cell preferable to other
types.

Soil-Tire Interface Pressures

The first recorded effort to measure the magnitude of
the forces applied to the soil by a farm tractor tire was
made by Lask (1958, 1959). Small strain-gage pressure
transducers (column and diaphragm cells) were mounted in
the tire so that the surfaces were flush with the tire
surface. The lower inflation pressures gave a more even
pressure distribution across the tire. The lugs of the

tire carried a larger portion of the load than the undertread.

Additional studies were conducted by Trabbic (1959) using
diaphragm-type pressure transducers in lugs as well as
undertread. The results showed that as the drawbar load
and tire inflation pressure were increased the soil-tire
interface pressure generally increased on the undertread
and leading lug side. The pressure decreased on the lug
face and trailing lug side as the drawbar load was increased.

The soil-tire interface pressure was measured in a
different manner on a smooth tire by vanden Berg and Gill
(1959). Larger diaphragm cells were placed flush with the
surface in a densely packed sand and a tractor equipped
with a smooth tire was towed across the instrument area.
Peak pressures occured just as the tire made contact and
broke contact with the soil. The highest pressures occured
at the center of the tire and progressively decreased
toward the outside edge.

Soehne (1958) theoretically calculated the soil-tire
interface pressures and concluded from measurements made
by Kraft for thin-walled tires on firm soil that the
surface pressure over the entire contact area was approxi-
mately equal to the average pressure. In a study of the
Pressure distribution between a smooth tire and soil,
however, Vanden Berg (1959) found that the pressure
distribution within the contact area was not uniform. He
concluded that Soehne's theory of uniform surface pressure

can not be used without considerable error since the

maximum pressures recorded were twice the average pressure
for the contact area.

The stresses produced in a soil mass as a result of
an externally applied force have been.measured using various
physical principles. A U.S. waterways EXperiment Station
report, as reported by Cooper (1956), reviewed and described
various types of soil pressure cells developed for soil
mechanics studies prior to 1956.

COOper 33' al. (1957) described a strain gage transducer
for measuring normal stress pressures developed by the wheels
of a tractor in the soil. Results obtained with the cell
indicated that the stress distribution under a rolling wheel
was similiar to that described by the empirical equation
0; = Pm (1-cos‘() developed by Hoehlich (1934).

Where:
0;'= vertical normal stress

Pb: applied surface load

°( = polar coordinate.

Reaves and Cooper (1959) studied the stress distribution
under a 12-inch tractor track and a 13-38 tractor tire
carrying the same total dynamic load and pulling the same
drawbar load. They found that the stresses under the tire
were in almost every case twice as large as those under
the track for any position. Pressure measurements were

recorded at 3-inch increments from the center line of tire

and track laterally 12 inches and downward to 42 inches in
Congaree silt loam with the Type A Cells. Also in the

same report, results of comparative stress curves at a
depth of 9 inches in Hiwassee sandy loam under a 13-38 inch
tire and a 12-inch track were presented. They found for
the tire a smooth curve of higher magnitude and shorter
duration then for the track. The curve for the track

showed a vibrating stress which was correlated with stresses
applied to the surface of the soil due to the action of the
drive sprocket. ’

In experiments designed to determine the overall
movement and compaction in a soil mass for the simplified
case of piston sinkage, Soehne gt_ El. (1959) found that
"at some distance from the piston, lines of equal principal
stress appeared to coincide fairly well with lines of equal
compaction, but directly under the piston this was not the
case". The movement of the soil was determined by placing
small lead spheres in the soil and thay plates were made
during each test. The method of determining the directions
of the principal stresses from the deformation of a grid as
developed by Haefeli and reported by Bekker (1957) was used.

Willits (1956) studied the stress produced in soils
by traffic and the relationship between the stresses and
compaction in undisturbed soils. He found that a maximum
stress of over one hundred pounds per square inch near the

surface of the soil was produced under the drive wheel of

a Massey-Harris Clipper combine. The stresses developed
by all traffic decreased rapidly with depth. The amount
of compaction was determined by taking soil samples and
determining the bulk density. The variables affecting the
change in compaction were the vehicle, number of passes,
original soil density and soil moisture content. Cores

of undisturbed soil were subjected to various pressures in
the laboratory to obtain the same change in bulk density
as was produced by the passage of a tractor in the field.
The pressures were similiar to those recorded by the
pressure cells during field tests.

§Qil §tress~Strain Relationships

 

A number of different theories have been applied to
soils. One of the oldest, the Coulomb-Mohr formula (an
empirical relationship) discussed by Terzaghi (1959),
defines the stresses acting on a plane through the soil
mass at the moment of failure.

In studies of agricultural implements 47 years ago,
Berstein developed a sinkage equation that relates the
ground pressure and sinkage of a given loading area.
Bekker (1957) modified Berstein's equation and used it in
his theory of land locomotion. Soil deformation was
defined in terms of certain soil constants "practically
independent" of the size and form of the loading area.
Using the soil value system developed by Bekker, Stong (1960)
found that the soil strength was decreased by plowing and

disking. Vehicle traffic increased the soil strength by
compacting the soil. Within the range of 10—24 percent
moisture content, bulk density has a greater effect on
soil strength than the moisture content. Vanden Berg (1960)
stated that neither the Berstein equation or the Coulomb~
Mohr formula is a logical basis for a general soil mechanics
because they do not relate stress and strain.
Using the model of a continuous medium for soil,
Vanden Berg (1958) defined soil stress in terms of a stress
tensor. The stress tensor was divided into two tensors,
the mean normal stress tensor and the deviatoric stress
tensor. Applying theories of elasticity and plasticity,
he proposed that volume strain is controlled by mean
normal stress. Some of the observations made by Vanden Berg
are:
1. The concept of continuum will apply to loose soils.
2. Of the four invariants of the stress tensors
investigated the mean normal stress related best
to bulk density.
3. It could not be concluded that soil compaction
is independent of the deviatoric stress tensor.
Hovanesian (1958, 1959) found that the density of
agricultural soils was related to mean normal stress by

the following general formula:

“Y: 13 + B ln[(q7fg)+ K/(1+K)] (1)

10

Where:

initial bulk density of soil

initial mean stress

mean stress

bulk density

:x: 4953‘“

and B are soil parameters, assumed constant for
a given soil condition.

He also found that for a given value of mean stress,
impact loads will cause less change in bulk density than
that created by a gradually applied and released load.

In static compression tests, Soehne (1958) filled
cylinders of outside diameter 11.2 inches, height 5.2 inches
and volume 610 cubic inches, with undisturbed soil samples
taken from the field or with loose soil. He found that
the amount of compaction and the reduction of porosity
was related to the pressure by a logarithmic law. The
higher the moisture content the more the soil was compacted
by a given pressure. When the kneading compaction test
was compared with the static compaction of loose soil a
steeper slope resulted from the kneading test.

The following equation was derived from an analysis

of the compaction of arable soils.

n = -A log p+c (2)
Where:

n = porosity

11

A = slope of the curve on a logarithmic scale
p = pressure
c = porosity at a pressure of 10 psi.

This relationship between porosity and pressure is similiar
to the formula used in civil engineering soil mechanics
discussed by Hough (1957).

From studies of the resistance to compression of
confined fragmented soils, Reeves and Nichols (1955)

found the relationship between pressure and amount of

compression to be of the general form y = a e bx
where:

y = amount of compression

x = pressure.

Hendrick (1960) found that the tensile strength of
soil briquettes did not change for loading rates of 0.18
to 4.70 kg/cmZ/sec. Less strain energy was required to
cause failure at the higher loading rates because the
briquettes strained less.

The magnitude of volume strain may be less from a
static load than from a dynamic load such as that produced
by a track or a tractor tire. This latter action may cause
an orientation of particles that will result in a greater
volume strain. Terzaghi (1959) found that vibration 0f

sand resulted in a greater compaction than could be caused

by an equivalent static force. The effect of vibration on

clay was much less because the cohesive bond between clay

particles interferes with intergranular slippage.

THEORY

Using the model of a continuous medium proposed by
Vanden Berg (1958) the forces acting on a volume element
are completely specified by the stress tensor and volumetric
strain by the change in bulk density by ignoring the
shearing deformations and rigid body rotation. To define
the state of stress at a point requires that six independent
values be determined. The volumetric strain can be deter-
mined by measuring the change in bulk density.

The stress vector on any arbitrary plane can be
determined by using matrix algebra (murnaghan 1951. For
example problems see Malvern 1957). For the following
conditions,

1. a plane oriented so that its normal lies in the

Y Z plane and bisects the angle formed by the
positive Y and Z axes, .
2. a plane oriented so that its normal lies in the
Y X plane and biSects the angle formed by the
positive Y and X axes,
‘ and a plane oriented so that its normal lies in

the X Z plane and bisects the angle formed by

the positive X and Z axes;

the directions cosines of a normal vector to each of the

Planes are, respectively,

13

1’ 0' [IE/2’ ”/2 ’
2° VIE/29 VIE/2: O 1
3.13/2. 0. 1372.

For the general stress state the stress tensor is

0; ts tn
.13!!! T3} Zyz (3)
(my. my 0;.
If 1, j, k are unit vectors along the pos1tive X, Y
and Z axes respectively, then the components of the stress
vector acting on the three planes described above can be
obtained by matrix multiplication.
For plane one as defined by condition one the stress
A
vector-r} is
U} 'tmy ‘Zkz
-h.
T1 = (0.1272. 1272) fxy 13 232
rm 2’” 0‘7.

= fi/Za’XyJXZfi . (ii/goat”) ‘3 +{5/2(Z§z+03> 3?. (4)

For plane two (condition two)

.4. U} 29y 2&2
T2 = ((5/2, fi/g. 0) 23W 03 232
’Z'xz Z’yz 02

=f2—/2(fi+2’xy) '3 + fi/2(ny+q§) .3 + fi/2(sz+zyz)-E’ (5)

14

For plane three (condition three)

_§. 0; 'txy 2x2

T; = (43/2. 0 .5/2) 23w 07 fyz
‘le Z52 0?:

=V§)2(0§+2&z)'§ +7572(ny +Zyz)'3'+{§7é(2§z+02)'§2 (6)

If the scalar product of a unit vector A; in the‘direction
of the normal to the plane and the stress vector T; acting
on the plane is determined this will be the magnitude of
the normal stress acting on the plane. The normal stress
U11, acting on the plane can be obtained as follows; for

plane one the unit normal vector is,

:3
II

A
0:11 = 3:1"; (733’

- sUSwU'z) + Zyz (7")

for plane two the unit normal is,

—I
I

.- - -"
32:5/2i+fi/2j+0k

then _;§
032 ... 3‘2 .T2 (8a)
UB2 = %(Ul+03) + 23y (8b)

for plane three the unit normal is,

A -.> -.- “
n3 = {572 i + O 3 + 45/2 k

15

then, d.
0:13 2333.1 (98.)
033 = %(Vi+fih) +‘sz . (9b)

The principal stresses can be determined from the
stress tensor for the general stress state, however, it
is easier to determine the principal values of the stress
deviator tensor and then calculate the principal values
for the stress tensor. The stress tensor can be separated
into a spherical stress tensor and a stress deviator tensor

as follows.

0% ny 1&2 Um 0 O Ul-Ud 29y 2&2
fxy Ty zyz = O “—111 0 + xxy 65-07:: tyz (10)
fxz fyz 0‘2 0 0 fix {x2 tyz (Ta-1?;

Stress Tensor Spherical Deviator Tensor
Stress Tensor

Where:

"a = 1 (Ui+03 +02) (11)

If Si denotes the three principal deviator stresses, the

following relationships are known:

67-07:: (12)
52:52-57} (13)
53:93-73 (14)

079 W5: and V? are the principal stresses of the stress

(I)
_5
N

16

tensor.

It is possible to rotate the coordinate axes to such
a position that all of the shear stresses will be zero and
only these principal stresses will act on the plane. The
problem is to determine the direction cosines nx, ny, and
n so that this condition is present. If '3’ is a unit

2

vector in one of the principal directions and S the
magnitude of the stress vectorzfi, the stress vector on
this plane must be parallel to ‘3 since there are no shear
stress component on the plane perpendicular to ‘3.

Therefore,
A

13:33” (15)

The three components of this vector equation can be

determined by matrix algebra as follows

Sx-S ’2’xy fxz

H

(nX, ny, nz) Z’xy Sy-S {yz
fxz Zyz SZ-S

° = 0 1’s
(Sx-s)nX +23x ny +1kx nZ ( o )
fxy nX +(Sy-S)ny +2’xy nz = 0 (16b)
2&2 nZ +2yz my + (SZ-S)nz = O (160)

Where:

S = 111;, 3y = 03-62;}, and SK = 02-0;

This is a set of three homogeneous linear algebraic

equations for the three unknown direction cosines nx, ny’

17

and nz. The directions cosines must also satisfy the

equation

2 2 2

nx + ny + nZ = 1, (17)

and, therefore, all three cannot be zero. A system of
linear homogeneous equations has a solution other than
the trivial solution if and only if the determinant of

the coefficients is equal to zero, that is, if

Sx" S tyx z'zx
‘fxy 8y
1 'sz {yz SZ-S

(I
O

-S 22y (18)

 

 

Expanding the determinant gives a cubic equation in terms

of the unknown magnitude 8;
s3 - II s - III = O (19)
s S

where IIS and IIIS are algebraic invariants of the stress
deviator tensor and are defined as follows;

-: >21

)2 ‘1’ (.3y "Z

2 ,
113 = lusx-sy) + (Sz-sx
6
2 2 2
+7332 + YZX + ny (203)

Substituting

Sx = Vlfflli 8y = F; ' Vtv 5% = V; '5; 3
118 = _1_[(9'§c-03,)2 + (03, 41212 + "'2 ““3023

6 2 2 2 20
+ ny + 72: + ny ( b)

18

53 Zsy‘ 7&2 Uk-Oh. Zly ¢§z

IIIS 2’yz sy fyz t’yz V3; 4,; {yz (21)
I ikx 12y Sz) ‘tsx Zéy V3 —0B

11
H

 

 

 

 

The roots of equation (19) are the three principal
stresses. The solution may be obtained by making the

substitution (malvern 1957):
s = 2(coso0 II8/3. (22)

From this substitution.the following relation is determined:

/
(Us) 2

Then 30(1, 30(1 + 277', and 30(1- 277' all have the same
cosine given in terms of the invariants of the stress

deviator. Thus the three roots of equation (19) are:

I)
II

II

2(COS°(1) 8/3
II

52 = 2(cos°(2) 3/3

II
S = 2(cos°<3) 8/3

Where:

0(2=o§+277" and o<3=0<,- L73:
3

Now the principal stress values can be determined by using

equations (12), (13) and (14). The values that are obtained

 

 

19

are ordered algebraically from largest to smallest and
designated by E, (El, and (TI-II respectively. The maximum
shear stress, which is a function of the stress deviator,
is given by

Z’max = afi-an) (24)

In order to vertify the hypothesis that soil compaction
developed under dynamic conditions is controlled by mean
normal stress two things must be demonstrated:

1. That mean normal stress does correlate with

bulk density and
2. That the deviator stress tensor does not correlate
with bulk density.
The only measure of the spherical stress tensor is mean
normal stress. many expressions can be used as a measure
of the deviator tensor. Since earlier investigations
have indicated a relationship between maximum shear stress,

maximum normal stress and bulk density, these relationships

will be investigated.

APPARATUS AND INSTRUMENTATION

Design and ngelopme t gf‘g Six Directional Stress

Transducer

Two different models were designed during the
development of the six directional stress transducer
(6 DST). The first model designed consisted of a small
octagonal brass box with eight sensing elements as shown
in Figure 1. Each of the sensing elements was to be made
of 0.025 inch thick stainless steel with two Type A-18,
SR-4 electrical resistance strain gages cemented to the
element. The diametrical pairs of sensing elements would
form the four components of a Wheatstone bridge. With this
arrangement maximum sensitivity and temperature compensation
would be obtained.

Construction of several sensing elements revealed
that pieces of stainless steel of this size and shape were
difficult to work. In addition, since the element was
designed to act as a simply-supported beam, the problem
0f protecting the gages mounted on the element without
restricting the action of the beam was not satisfactorily

accomplished.

Due to these difficulties the second model, the

5 DST, was designed and constructed. (Figure 2) A hollow

brass sphere (3 inches outside diameter and 1 7/8 inches

21

 

.8 SIDES MACHINED
TO .025

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3’ _
. CROSS SECTION
TOP VIEW
.6. |§ -4— §
4 --<-—-§--,J ‘1
4 M19 +<-->-I III
*3 I6 i 12 ’2
T— P——f—'L ‘ ‘ I _ u
M0 \ + I ‘ 1‘ 1 T :3"
. 'r 12 ..
\3 i ' 3 "I" '1' T 1’ 5'”
’ T ’r \‘F -Iv EN
52—...) 3 T no
0 DRILL I'e' g-Isuc-z 'e' I“
éloEEP- 4 HOLES } nun-4 HOLES
FRONT VIEW TYPICAL SIDE VIEW
2 A-IB 33-4 NOTES:
-.v . 6/— 37““ ““553 SENSING ELEuEuT MADE or
”NT—l M —_ .025 STANLESS STEEL
| E7 ~‘5 DRILL. Tun“ TAP HOLES FOR A NUMBER
3" 5’ , J oeux 92°10 2 FLAT no sanw
1+— I- .199 mA

 

 

SCALE FULL SIZE

 

Figure 1. Detail drawing of the Four Directional Stress
Transducer.

 

22

 

    

3 DRILL c'soRE 3- CIA

TO 2; DEPTH- 6 HOLES
EQUALLY SPAOED

DRILL 3 HOLES
QUALLY SPAOED

MOI”

 

DRILL c'soRE
FOR-2'- HEX sooner

@193

TOP VIEW

:2 PRESSURE CELLS wn'n
REDSHAw STRAIN GAGE /

 

SENSING ELEMENT

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

SCALE : . 0 Rune
FULL SIZE

 

CROSS SECTIONAL VIEW

 

 

1”Laura 2. Drawing of the Six Directional Stress Transducer
showing the Location of Diaphragm Pressure Cells.

 

23

inside diameter) was cast in two parts. The two halves
were machined to permit the use of an "O“ ring for a
waterproof connection. A 3/8 inch hexagon socket head

bolt was used to clamp the two hemispheres together. Six
diaphragm pressure cells capable of measuring normal stress
were located in each hemisphere. Three of the pressure
cells are mutually perpendicular and the other three are
oriented in the planes that bisect any two of the three
mutually perpendicular directions. When the two half
spheres were connected, the corresponding cells in each
half sphere were oriented diametrically opposite each other.
These pairs of cells were connected to form two legs of a
Wheatstone bridge. Two 120-ohm wire resistors were used
to complete the Wheatstone bridge.

The diaphragm cells which were used for the sensing
elements were constructed in the following manner. A
length of 3/4 inch diameter cold-rolled steel stock was
chucked in a lathe and a 5/8 inch hole was drilled through
the center of the piece. An 11/16 inch drill was used to
enlarge the hole to a depth of 1/16 inch. A 3/8 inch long
cylinder was then cut from the length.

Diaphragms made of 0.010-and 0.020-inch thick stain-
less steel were rough cut to a one inch diameter with a
metal clipper and soldered with stainless steel solder to
the cylinder wall at the end with the 11/16 inch inside
hole. Finally, the cell was chucked in a lathe and the

24

diaphragm was machined flush with the outside diameter of
the cell wall.

After the cells were constructed, a Saunders-Roe foil
strain gage (Redshaw 1/2—2 ED, 25 ohms, gage factor 2.1)
was cemented to the inside surface of the stainless steel
diaphragm. Since the Redshaw strain gage does not have
lead wires attached as the SR-4 gages, a method for attaching
lead wires to the tabs of the gage had to be devised.

After several preliminary tests the best method found was
to attach a piece of copper wire to the tip of a soldering
gun. The tabs were tinned prior to being mounted on the
diaphragm and after the curing process a 2-inch length of
wire (Belden No. 8430) was soldered to each tab.

The gage was waterproofed with a thin layer of wax.
To protect the strain gage from being damaged by a force
applied to the lead wires, a rubber stopper was cut and
placed in the open end of the pressure cell. The lead
wires were conducted through a hole in the center that was
sealed after the stopper was in place. A four-conductor
shielded cable (Belden Strain Gage Cable No. 8434) was
connected to the 2-inch wires to carry the signal to the
amplifier.

The calibration device as reported by Lask (1958)
was used to calibrate the individual cells. The maximum
design value of 60 psi was selected since this was the
maximum pressure recorded in the soil by present agricultural

25

equipment. Calculations indicated that a 0.010 inch thick
stainless steel diaphragm 3/4 inch in diameter could be
used.

A problem was encountered since the Redshaw strain
gage has only approximately 25 ohms resistance. The
amplifiers and associated equipment available for conducting
the experimental tests had a range of 50 to 500 ohms.
Several calibration tests were conducted using different
bridge arrangements to determine if the bridge could be
balanced and the order of magnitude of the gage output.

A calibration curve for the 0.010 thick diaphragm
using a 120 ohm wire resistor in series with the Redshaw
gage is shown in Figure 3. The results of three tests
show a linear relationship up to 25 psi. Within the range
from 0 to 25 psi, one line deflection represents approxi-
mately 2 1/2 psi. As it was proposed to use two active
gages in each bridge the sensitivity of the bridge would
be doubled or one line deflection would represent 1 1/% P81-
Since the relationship was not linear up to 60 psi, it was
concluded that a thicker diaphragm should be used.

A series of tests were made using 100 and 120 ohms
resistors in series with the 25 ohm Redshaw gage mounted
on a 0.020 inch thick diaphragm 3/4 inch in diameter. The
results of three of these tests using a 100 ohm resistor
are shown in Figure 4. A linear relation was obtained up

to the maximum value of 80 psi used during the tests. One

26

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28

line deflection with the calibration data shown corresponds
to approximately 2 1/4 psi. As a result of these tests

it was concluded that the 0.020 inch thick stainless steel
diaphragm 3/4 inch in diameter would be used for the
sensing elements of the 6 DS transducer.

Since it was proposed to use two active gages in each
bridge, tests were conducted using the arrangement in
Figure 5. The calibration data were obtained from only
one pressure cell being subjected to pressure. During the
experimental tests both pressure cells in a bridge would
be subjected to a load, therefore, a device for obtaining
calibration curves with both cells being subjected to a
load had to be designed and constructed. With 6 holes for
lead wires, 12 for the sensing elements and two for the
connecting bolt , the probability of maintaining a
completely air sealed unit was very small. Therefore, a
calibration device with two nozzles similiar to the one
reported by Lask (1958) was constructed and is shown in
Figure 7. Control valves were installed to permit separate
control of each nozzle. Any combination of pressures could
be obtained.

The calibration data shown in Table 1 proved that the
bridge arrangement with two pressure cells would give an
average of the two separate readings. This was expected,
as it can be shown mathematically. Consider the bridge

arrangement as shown in Figure 5. The output voltage of

 

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29

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Figure 6. A View of the Six Directional Stress Transducer.

 

Figure 7. Calibration Device used to obtain Calibration
Data for the 6 DST.

31

TABLE 1

AVERAGE OUTPUT OF NUMBER ONE SET OF GAGES IN
6 DST CALIBRATED FOR ONE LINE DEFLECTION EQUAL TO ONE PSI

-

 

 

 

 

Load (psi) Lines Deflection
Cell 1 Cell 2 Average Avg. of 3 reps.
20 0 10 9.8
20 10 15 14.8
30 1o 20 20.0
40 20 30 30.0
40 40 40 40.1
40 0 20 19.1
40 10 25 24.3
40 20 30 29.0
40 3O 35 34.7
40 40 40 40.0
the bridge is,
30 = E [El_.._f_32__3 . (24)

R1+R4 R2+R3

Where:

E = supply voltage

U

‘ ' .ce values of elements of
R1 — R1a + R1b reSistar

Redshaw gages.

resistance values of elements of

Redshaw gages.

R3 = R4 = resistance values of Wire resistors

Bo = bridge output voltage

The bridge circuit is in balance and zero output voltage

32

results when

R1 = R2

_.._..__.. __.._.__ ° (25)
R1 + R4 R2 + R3

The change in output voltage for a change in
resistances R1 and R2 can be determined by calculating

the total differential for equation (24).

dEo = 3.30.5131 + Pin—(332 (26a)
3 R1 3 R2

 

313° E(R1+R4-R1) 3130 E(R2+Ra-R2)
331 = (31.34)? 332 = (32.423)?-

Substituting into equation (26a)

dEO = E[.A__R dR1 2 " —3___.R ”‘2 1 (26b)

(R1+R4) (R2+R3)2

For R1a= R1b = R23 = R2b = R and all with gage
factor F, and R3 = R4 = KR, then

 

dEo = EX [dR1a +Eglb.‘ 3323.” E§2h_g (26c)
R R

 

(2+X)2 R R
Setting:
dB1a = F£1a
R
dRTb = F511)

 

33

R

dRZb ___ F5

2
R b

dEo now becomes,

E
dEO = m FE £13 + 81b ’ £23. " 82b]. (273.)

Since the Redshaw had a resistance of 25 ohms, and

the resistor R3 and R4 are 120 ohms, X is equal to 4.8.

£111 + 5.1b - £221 + 52b J.

 

 

 

 

 

_, EF
dE° - 4.82 E 2 2 (27b)
Letting
E1e + 6110 = 51
2
82a + 82b = 82
2
_ B F _ .
dEo - 4.82 [51 £2] (28)

Therefore, the prOposed bridge gives the algebraic
difference of the average strain in arm 1 and the average
strain in arm 2. Since the gages in arm one will be
measuring the change in strain due to tension and those
in arm two due to compression, ‘51 will be a positive

value and 82 will be negative. Our equation now becomes

34

 

_ E
dEo 4 8g [l51'+l&2l ] . (29)

W Cell

 

Another instrument that was designed and constructed
during the experimental tests is the W Call. This instrument,
Figure 8, consists of a non-collapsible plastic tubing
connected to a spherical shaped rubber balloon approximately
3 cm in diameter. The other end of the tubing is connected
to a cylinderical shaped housing containing one of the
diaphragm pressure cells. Details of this housing are
shown in Figure 10. The balloon, tubing and part A of
the housing are completely filled with water. When the
balloon is subjected to stress, its volume of water cannot
decrease,therefore, the pressure within the balloon must
change. This change in pressure is the change in mean
normal stress as water cannot transmit shearing stresses.

Calibration tests were made with the device reported
by Hovanesian (1958) and the results are shown in Figure
11.
§g§$ Handling Equipment

The need for controlling the soil parameters and
providing an accurate means of reproducing the initial
soil conditions to obtain replication results required
the construction of special soil handling equipment. The
equipment described in this thesis was built as a joint

Project between Mr. Jack Stong (1960), Graduate Assistant,

35

 

L { "'

Figure 8. The l Cell used to measure Mean Stress directly.

 

Figure 9. A View of the Soil Handling Equipment.

36

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EIDRILL5
c'eone - om
Toa DEBPTH {/ID
5% on
TOP VIEW
TOP VIEW PART A
SOLDERED
.1 WELDED
'4” -N' , E
i _. I 1_ f i . ,_ I l-NPT‘
T7? . T e T
"2|m : | : g - /
I I 3
L . I ' 8 L?
T u, / |
T PART g 6312 /
1.! A :_ g
I /
It l + 2
”Jr '40 CROSS SECTION

 

 

 

 

I \ PART A _
_ 4%..” wELoeo

SIDE ~VIEW

COPPER wants

an»

SCALE= FULL SIZE

 

 

 

Figure 10. Detail Drawing of the w Cell used to Measure
Mean Stress Directly.

37

 

 

 

 

 

 

 

 

 

LINES DEF LEOTION N0

 

 

 

 

 

 

42 7 I T
INSTRUMENT DATA
CALIBRATED IO LINES ATTN 5
OPERATED ATTN 5
36
30 /
AVG VALU?
0F 3 REPS
24 /
I8 /
/ BRIDGE
ARRANGEMENT
/\ GAGE I GAGE 2
Izon WIRE Izonwne
6 / RESISTOR RESISTOR-
I, I I
o 6 I2 I8 24 30

APPLIED LOAD PSI

Figure 11. Calibration Curve for IT Cell showing Number
01’ Lines Deflection versus Applied Load.

Agricultural Engineering Department and the author.

The system as shown in Figure 9 page 35 was designed
and constructed so that it was not limited to one particular
experiment. There are two main soil tanks for experimental
work and one storage tank. One tank is 5 feet in diameter
and 42 inches high, the other is 4 feet in diameter and
3 feet high. Under each tank is located an 18-inch wide
flat conveyor belt that transports the soil from the tank
to the boot Of a bucket elevator. The bucket elevator,
which is 10 feet in height, was constructed of an 18-inch
wide belting with 14~inch by 7-inch buckets placed 9 inches
on center. At the head Of the elevator is a 6-foot,
reversible conveyor belt used for transporting the soil
to a chute which delivers the soil into the tank to be
filled.

A loading frame used in the application of dynamic
vertical loads was constructed above the smaller tank.

A 15-inch stroke, 4-inch bore hydraulic cylinder was
suspended from the frame above the center of the tank.

A 3/4-inch thick steel plate, 20 inches in diameter, was
constructed for the loading plate (Figure 12). A strain-
Sage transducer designed to measure vertical forces only
was connected between the end Of the hydraulic cylinder
piston and loading plate.

Fbrce Transducer

 

A strain-gage force transducer designed and constructed

..:‘

 

‘ . ' ' I I
,. . r349: ,..- "
I. ‘ .17, gig-3:7," ‘4‘" . '

Figure 12. A View of the Leading Plate, Fbrce
Transducer and Hydraulic Cylinder.

 

Figure 13. A View of the Pressure Transducers
and Balloons used to obtain Data.

 

 

40

by Bellinger (1960) was modified to measure the total
vertical force applied to the loading plate. The semi-ball
as shown in Figure 14 was constructed to fit the external
end Of the hydraulic-cylinder piston. A seat to permit

a flexible joint between the piston and force transducer
was constructed and placed in the force transducer (Figure
14). Calibration test results made with a screw-type

loading machine are shown in Figure 15.

41

 

a————3%o

n——-ZDII———o-l

T
I

HOLE FOR I
I
l
|
I
I

 

 

 

I

' BALL AND
: see
I

l

L__.____.__.I i
F“‘ -' “' ""1
4 A-S SR
STRAIN eases
me

 

 

 

 

 

 

 

 

 

 

 

I
a
‘9 2 A. _
“DRILLIB e suc 2
_ (

I<——— 2 D ———r¢
SEAT FOR BALL

 

 

 

 

 

 

 

 

. 3
PISTON BALL .SCALE- 3- SIZE

 

 

Figure“. Detail drawing of Force Transducer showing
location of Strain Gages and Ball and Socket joint.

42

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

INSTRUMENT DATA /
CALIBRATED 19 LINES ATTN 2
OPERATED ATTN 2
7200 /
6000 Z
AVG VALUES
8 OF 3 REPS
_.|
4800
9:
3
3600
8 /
:1
i /
<2400
BRIDGE
ARRANGEMENT
% I20n Izon
1200 GAGEI GAGE 2 _,
I200 1200 I
GAGE 4 GAGE 3
0 L I
0 8 I6 24 32 40

LINES DE FLECTION NO

Figure 15 .‘ Calibration Curve for Force Transducer showing
Number of Lines Deflection versus Applied Load.

PROCEDURE

Original plans were to use a tank 4 feet in diameter.
However, the results Obtained by replication varied because
the loading plate would not remain level during the loading
process, and the contact surface therefore, could not be
considered as a principal plane. This made it difficult
to use the method of vanden Berg's to check the results
Obtained with the six directional stress transducer. The
soil flowed around the contact area and low values were
obtained from the instruments in the soil and the load
transducer.

It was decided to use a smaller tank to avoid these
difficulties. Measurements showed that the loading plate
would fit inside Of a 55 gallon drum with approximately
an inch of clearance between the plate and the inside wall
of the drum. Tests were conducted to determine the distri-
bution Of vertical stress under the loading plate at the
Proposed test depth. Four of the Type A Cells were placed
on the circumference of a 12-inch diameter circle. Results
showed that the vertical stress pattern around the circum-
ference of the circle was uniform.

The following proceduce was used while conducting a
test. The soil was sprayed with water to Obtain the desired

moisture content and then stored in the larger tank one or

44

two days. During this time a plastic sheet was placed
over the soil to prevent moisture changes. Prior to
conducting a series of tests at this level of moisture
content a 3/4 by 2 inch expanded metal screen was placed
over the top of the drum to remove large clods Of soil
that were formed during the wetting process. The soil
was recirculated several times by the soil handling equip-
ment to increase the homogeneity of the mass.

The small tank was filled to the desired level for

the particular test being run and the surface leveled with

a template. A circle 12 inches in diameter was drawn in

the center of the tank and the Type A Cells, the six
directional stress transducer and balloons for measuring
changes in bulk density and in some tests the W Cell were
positioned as shown in Figure 16. The position of the
gages was checked with a hand level. TO prevent movement
Of the instruments in the process of placing the desired
depth of soil on top Of them, loose soil was placed over
all instruments by hand. _After the tank was filled to
the Operating level, the surface was again leveled with
the template. The loading plate was properly positioned
and the recording instruments activated. The control
lever of the portable hydraulic unit which operated the
hydraulic cylinder was held open until the end of the
piston stroke was reached or until a signal from the

instrument operator was given. Upon completion of a test

the soil and instruments were removed from the tank. The
soil was passed through the 3/4 by 2 inch screen to remove
large blocks of soil formed during the compaction process.

The rate Of loading of the loading plate was changed
by controlling the rate Of fluid flow in the hydraulic
lines. A needle valve was placed in the high pressure line
and insured a constant rate of flow during the test run.

Moisture and bulk density samples were taken in the
loose soil at the level of the instruments prior to the
placement of the instruments. A standard core sampler
was used to take the soil samples.

Since the six directional stress transducer was not
completely airtight at high pressures, checks were made
to determine the magnitude of the static pressure developed
during a test. A piece Of non-collapsible plastic tubing
was connected to a static pressure gage and the open end
placed at the same level of the instruments in the soil.
Readings were taken during a test run and the maximum value
Obtained for all tests was 1.2 inches of water. This was
80 small that any effects on the pressure cells could be
neglected.

The test soil was Brookston sandy loam, which was
Obtained from M.S.U. Farm Crops Farm located on Mt. Hope
road. At the time Of placing the soil in the testing tank
the bulk density was approximately 0.95 to 1.05 (dry "Eight)
depending upon the moisture content. The physical properties

of the soil as reported by Stong (1960) are set forth in
Table 2.

TABLE 2

PHYSICAL DESCRIPTION OF THE BROOKSTON
SOIL USED IN TESTS

 

 

mechanical Analysis

Fine Gravel 1.2%
Coarse Sand 3.6%
Medium Sand 6.1%
Fine Sand 25-87g
very Fine Sand 27.7%
50 Micron 13.4%
5 Micron 5.5%
2 Micron 15.6%
Hygroscopic Coefficient 1.6%
Moisture Equivalent 14.3%
Maximum water Holding Capacity 63.8%
Soil Saturated 37.1%
60 cm Tension 25-4%
Permanent Wilting Point 8.7%
Lower Plastic Limit 21.0%
Upper Plastic Limit 25.5%
Piastic Range 4.5%
Density 2.6%

Since the recording instruments were set on zero
reference after the soil tank had been filled and leveled,
the effect of the weight of the soil on the pressure cells
and balloons was not determined during the experimental
tests. To determine the effect Of the soil weight, a
series of tests were run at 5, 10 and 15—inch depths with
moisture contents of 12.21, 15.19 and 17-21 percents, and

the results are presented in Table 3.

TABLE 3

CHANGES IN BULK DENSITY PRODUCED BY THE
WEIGHT OF THE SOIL AS DETERMINED WITH THE
VOLUMETRIC TRANSDUCER

 

II

Percent Moisture Content

 

 

 

12.21 12:12_ 17.21
Depth B.D. Stress B.D. Stress B.D. Stress
(in) (gm/cc) (psi) (gm/cc) (psi) (gm/cc) (psi)
0 1.064 0.0. 0.962 0.0 0.932 0.0
5 1.068 0.9 0.967 1.0 0.947 1.1
10 1.072 1.2 0.969 1.2 0.952 1.3
15 1.075 1.4 0.970 1.7 0.959 1.8

These errors in initial bulk density cause errors in
the measured values of bulk density Obtained during a test.
A 5.5 percent error for moisture contents less than 15.19
and an error of 8.5 percent for a moisture content of 17.21
are caused provided the bulk density change for the
eXperimental test was as large as 0.2 gm/cc. These errors
would change the intercept values of the relationships
presented but would not change the values of the regression
coefficients. The statistical analyses of the various
relationships were based on the regression coefficients
and therefore, are not affected by the error in neglecting
the errors due to the weight Of the soil.

The vertical stresses produced by the soil weight do

not exceed 2.0 pounds per square inch. As mean stress is

48

an average of three normal stresses the maximum error due
to the weight of the soil mass would not exceed 1.0 pounds

per square inch.

RESULTS AND DISCUSSION

To test the relationship between mean stress and

bulk density under dynamic conditions, different stress

states must be applied to a volume element. If different

stress states are not used, all the stress tensor components

will be linearly related to the applied load. In such

cases all invariants of the stress tensor will be linearly

related and will appear to be related to bulk density.

Three different stress states were measured by Varying the

depth of instruments below the loading plates. Five

replications of each stress state were taken and the

calculated results are reported in Table A in the Appendix.

The bulk density readings reported were an average of
three measurements taken on the periphery of the circles.

The laboratory tests were conducted at three depths,

three rates of loading and three moisture contents. A

series of tests were run at a constant rate of loading
and approximately the same moisture content at the three

Then the rate of loading was changed and the

depths.
This

series repeated at the same moisture content.
Procedure was followed until all possible combinations of
the three rates, moisture contents and depths were used.

Fbr convenience a description of the tests is presented in

Table 4. The values for initial bulk density were obtained

50

TABLE 4

DESCRIPTION OF THE LABORATORY TESTS

 

 

 

 

 

 

Test Rate of Mbisture Initial
No. Depth Loadingg Content Bulk Density_'
(in.) (in/sec) (Per Cent) (gm/cc)
1 10 0.62 8.63 1.08
2 5 0.62 9.89 1.08
3 15 0.62 11.59 1.06
4 10 0.38 11.56 1.06
5 5 0.38 8.89 1.07
6 15 0.38 7.97 1.08
7 5 0.38 11.35 1.08
8 15 0.38 11.39 1.09
9 10 0.38 11.38 1.06
10 15 0.62 12.63 1.06
11 10 0.62 12.43 1.08
12 5 0.62 12.46 1.09
13 10 1.00 17.41 0.94
14 5 1.00 17.41 0.94
15 15 1.00 14.85 1.01
16 10 1.00 14.85 1.02
17 5 1.00 14.34 1.03
18 15 1.00 12.31 1.08
19 10 1.00 10.95 1.06
20 5 1.00 10.79 1.06
21 15 0.62 17.93 0.93
22 10 0.62 17.52 0.91
23 5 0.62 17.67 0.95
24 15 0.38 16.16 1.00
25 10 0.38 15.78 0.98
26 5 0.38 16.05 0.98
27 15 1.00 16.54 0.96

 

51

with a hand sampler at the instrument level prior to filling
fluatank.
The values of Wm, IIS, ”I. VII—.1, fill and {max for
the Type A Cells were computed from four measured stress
values using the appropriate formulae as reported by
vanden Berg (1958). The values for the 6 DST were computed
from 6 measured stress values using equations 7b, 8b, 9b,
12, 13, 14, 19 and 24. MISTIC, an electronic digital
computer at Michigan State University was used to make the
lengthy calculations involved in evaluating the above
equations and the statistical analysis. The sum of least
squares method was used to fit a straight line to the data
plotted on semi-logarithmic paper. An estimate of standard
error was determined for each relationship and a test of
significance of the slope was made by using the "t" test.
The confidence limits at the 95 per cent level for each
relationship were also determine.
The results have been presented under the following
headings:
1. The Relationship between Mean Normal Stress and
Bulk Density
2. The Relationship between Second Invariant and
Bulk Density
3. The relationship between maximum Normal Stress

and Bulk Density

10.

11.

12.

13.

14.

15.

16.

The Relationship between Maximum Shear Stress

and Bulk Density

The Effect of Moisture Content on the Relationship
between Mean Normal Stress and Bulk Density

The Effect of Moisture Content on the Relationship
between Second Invariant and Bulk Density

The Effect of Moisture Content on the Relationship
between Maximum Normal Stress and Bulk Density

The Effect of Moisture Content on the Relationship
between Maximum Shear Stress and Bulk Density
The Effect of Rate of Loading on the Relationship
between Mean Normal Stress and Bulk Density

The Effect of Rate of Loading on the Relationship
between Second Invariant and Bulk Density

The Effect of Rate of Loading on the Relationship
between Maximum Normal Stress and Bulk Density

The Effect of Rate of Loading on the Relationship
between Maximum Shear Stress and Bulk Density
Comparison of Three Methods used to Determine

Mean Normal Stress

The Effect of Moisture Content on the Relationship
between Mean Normal Stress and Applied Load

The Effect of Rate of Loading on the Relationship
between Mean Normal Stress and Applied Load
Comparison of Theoretical Values of Vertical

Stress with Measured Values.

‘d‘l
14

Relationship between Mean Normal Stress and Bulk Density

The results obtained indicated an exponential
relationship between mean normal stress and bulk density.
This exponential relationship had been observed by other
investigators (Soehne, 1953; Hovanesian, 1958; Vanden Berg,
1958). Instead of plotting the results using rectangular
coordinates, semi-logarithmic paper was used to obtain a
straight curve.

The variation which can be seen by examining the data
in Table A of the Appendix required that statistical
analysis be used. The sum of least squares method was
used to determine the best predicting relationship. The
natural logarithm of the values of mean stress were used
instead of the quantity itself. In mathematical terms.
bulk density, the dependent variable, would be described
as a function of mean stress, the independent variable.

In biological statistics the term regression is generally
used and the relationship is defined by the regression
equation.

The regression equations, estimates of standard error
(Sxy) and confidence limits for both the Type A Cells and
6 DST data are given in Table 5. The Type A data is
designated by an "A" following the test number and the
6 DST data by only the test number. The calculated values
0f "t" were compared with the distribution of "t" using

the degrees of freedom (D.F.) shown. All calculated values

TABLE 5

STATISTICAL ANALYSIS FOR MEAN NORMAL STRESS
VERSUS BULK DENSITY

 

 

 

Test Confidence
No. Regression Eqpation DJ. t Limits

1 ln Tug-36.824.30.507 26 38.46 28.63-32.27
1A In furs-26. 11+22o101 25068 20.33.2308?
2 lnTm=-37.41+31.28Y 33 27.93 24.60-29.04
23 1n0'm=-25.41+21.65‘( 16.31 18.94-24.36
3 1n Fuss-23 .08+20.26Y 38 15 . 35 18 .03-22 .49
34 inTm=-21.53+19.07Y 18.93 17.37-20.77
4 1n Tun--36 .29+30 .991 18 12 .01 25 .57-36 .41
44 mama-29.9042532Y 12.28 21.31-30.13
5 1n0'me-34.47+28.767 28 . 26.39 26.53-30.99
54 lnTma-25.14+21.27Y 24.00 19.45-23.09
6 1n¢m=-27.59+23.961r 38 21.59 22.09-25.83
7 inS'm--34.21+29.83Y 33 21.31 26.98-32.68
7A lnTma-26.68+23.58Y 21.60 21.36-25.88
8 lnTma-30.10+26.30Y 38 10.08 21.90-30.70
8‘ 1n Vina-23 097+21 014Y 9065 17.445-24.83
9 in€m=-26.84+24.41Y 33 13.16 22.81-31.15
94 anm:—24.17+21.23Y 14.26 18.20-24.26
10 1116' m.25,34+24.41v 38 11.05 20.68-28. 14
10A inWm--2o.19+18.771 12.37 1631-21-33
11 1n0'm=-34.02+29.84Y 33 15.07 25.81-33.87
.114. 1n¢m=-24.66+22.12Y 17.16 19.50-24.74
12 1an=-23,49+22,14Y 33 16.77 19.46-24.82
12A inw'mg-20.02.19.06Y 17.71 16.86-21.26
13 infm=-38.41+37.967 22 15.43 32.86-43.06
13A lnTma-31.44+31.40Y 12.95 26.38-36.42
14 1nfm=.53.52+51.e31r 18 14.44 44.29-59.37
14‘“ lnfm=~41o76+40.72‘( 15046 35.19-4602.)-

55

TABLE 5 (CONTINUED)

 

 

 

Test - Confidence
No. Regression Equation g DJ. t Limits_ g
15 in Gin=-51.75+49.727 0.35 33 14.54 42.76-56.68
15A 1n €m=~38.15+37.137 0.27 14.00 31.74-42.52
16 In G'm:-34.82+34.681 0.17 33 26.05 32.11-37.53
16A in Gin=-31.73+31.757 0.17 22.19 28.84-34.66
17 ln fm=-32.79+32.83Y 0.23 33 19.43 29.39-36.27
17.4 In me-26.36+26.657 0.26 13.74 22.70-30.60
18 In rue-37.77432.16v 0.18 38 31.84 30.46-33.86
183 lnT‘m:-29.06+25.191 0.17 26.41 23.59-26.79
19.4 In m=-33.44+29.05'r 0.17 22.63 26.89-31.21
20 In 6'm=-35.36+30.427 0.22 38 22.53 28.14-32.70
204 In C'm=-27.75+24.18‘f 0.21 18.31 21.95-26.41
21 In €m=~40.48+39.817 0.16 18 18.10 35.19.4443
21A 1n Tm=-33.23+32.881' 0.15 16.34 28.66-37.10
22 In G'm=-36.92+36.751 0.22 13 9.10 28.02-45.48
22A ln¢m=-33.06+32.76Y 0.14 12.10 26.91-38.61
23 111 Tm=-27.91+27.87v 0.32 13 6.26 18.26-37.48
23A In m=-21.16+21.37*I 0.20 _ 7.50 15.21-27.53
24 In Tug-29.0342854Y 0.29 23 13.27 24.09-32.99
242 In 1"m==-22.30+22.2o1r 0.20 14.68 19.08-25.32
25 In Tm=-23.63+23.73v 0.21 23 16.37 20.73-26.73
25A ln o'ma-19.24+19.56‘/ 0. 0 - 14.02 16.68-22.44

26 1n V'm=-3o.92+30.84‘f

9 23 11.34 25.21-36.4-
26A 1n rm3'23 0 564-23 095Y 7

24

0
0
27 infm=-26.19+25.23r o
0 18.59 16.58-20.32

27A ln Tm=-19.32+18 .70\’

____

were highly significant which means that the regression
coefficients or slopes are different than zero. The true
regression coefficient is within the limits presented for
each relationship. Assuming a normal distribution of error,
one standard error (SXy) would include 68.3% of the values
used to determine the regression equation. The data obtained
with the Six Directional Transducer are consistently more
varied than that obtained with the Type A 08115 as the
standard errors are larger except for three tests.

If a quantity is related to bulk density in a general
manner, the regression lines for each different stress
state should not be significantly different for a given
soil condition. The lines should be parallel or the
difference between slopes should not be significant. The
"t" test was used to test for differences among the lines
for different stress states for each method used. The
results of these tests for the Type A and 6 DST lines are
presented in Table 6.

Differences between 1-3, 14-13, 17-16, 20-19, and
19-18 are significant at the 95 per cent level (*) and
between 14-27, 13-27, 17-15. 23-22 and 23-21 are
significant at the 99 per cent level (*f) with data
obtained with the Type A Cells.

The following tests for data obtained with the 6 DST
show a significant difference; 2-3. 1-3, 5-6, 4-6, 7-9,
14-13, 14-27, 13-27, 17-15, 16-15, 20-19. 19-18, 23-21,

57

TABLE 6

COMPARISON OF REGRESSION COEFFICIENTS OF THE CD VERSUS
BULK DENSITY RELATION FOR DIFFERENT STRESS STATES AND
SAME STRESS STATE FOR THE TWO METHODS USED TO OBTAIN DATA

 

 

 

Tests Degrees t t Degrees
Comp of Eree- 6 Type Test Depth of Free- t
pared dom DST A N0. In. dam
2-1 59 0.55 0.28 1 10 52 6.71**
2-3 71 6.44** 1.54 2 5 55 5.50**
1‘3 64 6039** 2 028* 3 15 76 0072
5-4 46 0.80 1.95 4 10 36 1.58
5-6 66 3.09** 0.82 5 5 56 5.32**
4-6 56 2.50* 1.38 6 15 76 0.96
7-9 66 2.18* 1.27 7 5 66 3.52**
7-8 71 1.19 1.00 8 15 76 1.51
9-8 71 0.57 0.57 9 10 66 1.25
12-11 66 0.32 1.82 10 15 76 2.10*
12-10 71 0.88 0.16 11 10 66 3.27**
14-13 40 3.19** 2.61* 13 10 44 1.90
14—27 36 7.04** 7.82** 14 5 36 2.50*
13-27 40 4.67** 4.84** 15 15 66 2.91*
17-16 66 0.86 2012* 16 10 66 1.50
17-15 66 4.43** 3.19** 17 5 66 2.40*
16-15 66 4.10** 1.79 18 15 76 5.03**
20-19 76 3.36** 2.65* 19 10 76 4,094.
20-18 76 1.03 0.62 20 5 76 3.30**
19-18 76 2.80** 2.42* 21 15 36 2.33*
23-22 26 1.48 2.90** 22 10 26 0.82
23-21 31 2.41* 3.30%e 23 5 26 1.23
22-21 31 0.67 0.04 24 15 46 2.41*
26-25 46 2.31* 1.64 25 10 46 2.07*
25-24 46 1.86 1.29 27 15 36 4.22**

 

 

and 26-25.

The significant differences for the 6 DST and Type A
data between 1-3 could be due to the difference in moisture
content of 2.96%. The results vary between the two methods
used except for the highest rate of loading and the higher
moisture contents. 0n the basis of the data obtained the
hypothesis that changes in bulk density are controlled by
mean normal stress cannot be accepted or rejected.

The results of the comparison of the regression
coefficients for the same stress state obtained by the
two types of instruments are given in columm four of
Table 6. There are significant differences for tests 1,
2, 5, 7, 10, 11, 14, 15, 17, 18, 19, 20, 21, 24, 25 and
27.

Approximately fifty percent of the calculated values
of "t" show a significant difference. A pattern appears
between the 10- and 15-inch depths at high moisture contents
and high rates of loading. At the lower moisture contents
and lower rates of loading significant differences appear
between the 5- and 10-inch depths and 5- and 15-inch depths.

A typical set of data showing the relationship between
mean normal stress and bulk density are shown in Figures 16
and 17.
Rglationship between Second Invariant and Bulk Density

The calculated results showing the relationships

between the second invariant of the stress deviator tensor

PSI

MEAN STRESS

£3() -i2ii-_iliimu-_.iiii_T-n_2.mi_----x__uii..._
1- 5'; DEPTH TEST 7
2— 10 DEPTH TEST 9
3-15" DEPTH TEST 8
1o
5
2 L
1
1
l by; I

59

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.15 1.20 1.25

BULK DENSITY (SM/CC

Figure 16. Mean Normal Stress versus Balk Density
for Data Obtained with Six Directional Stress Transducer.

PSH

STRESS

MEAN

60

 

 

 

 

1 1 1 T 1
1 1 1
20 ———--~ 1 - - 1 . -- -- 1-- 5 "WT'WI
1— 5‘; DEPTH TEST 7 1 1
2-10 DEPTH TEST 9 5 ,3 a
3-15" DEPTH TEST 8 a 1
1 .
10—--—-~-
5—-——- ..-_, ~14 -
6+w -
2mg.
1
I5-V/L” 5
1.15
BULK DENSITY GM/CC
Figure 17. Mean Normal Stress versus Bulk Density

for Data Obtained with Type A Pressure Cells.

 

 

61

and bulk density for both sets of data are presented in
Table 7. The values of "t" are highly significant which
means that the regression coefficients or slopes are
different from zero. The estimates of the standard error
for the 6 DST data are larger than the Type A values
except for three tests.

If a quantity is not related to bulk density in a
general manner, the regression line for each different
stress state should be significantly different from one
obtained for a different stress state. As the second
invariant is one of the three functions of the stress
deviator tensor the results of tests between different
stress states are given in Table 8.

The results for the 6 DST show significant differences
between tests 2—1, 2-3, 1~3. 5-6, 4-6, 12-11, 14-27, 13—27.
17-16, 17-15, 16-15, 20-19, 20-18, 23-21, and 26-25. For
the Type A data significant differences are found between
1-3, 12-11, 14-13. 14-27, 13-27, 17-16, 17-15, 16-15.
20-19, 19-18, 23-22, 23-21, and 26-24.

Comparisons of the regression coefficients between
the two methods used to obtain the data show significant
differences for tests 1, 2, 5. 7. 10. 11. 15. 18, 21, 24,
25, 26, and 27.

These results are similiar to those for the relation-
ship between mean normal stress and bulk density. The

values for the 6 DST data show more differences than the

TABLE 7

62

STATISTICAL ANALYSIS FOR SECOND INYARIANT
VERSUS BULK DENSITY

 

 

 

Test Confidence
No. Regression Equation ix DJ. ' t Limits
1 lnIIB=-36.74+31.127 0.13 26 34.20 29.25-32.99
1A lnIIgz-25.14+21.807 0.13 24.12 19.95-23.65
2A inngg-24.40+21.341 0.20 15.69 18.57-24.11
3 1n113=-21.03+19.42Y 0.27 38 14.71 17.19-21.65
3A lnIIsa-20.92+18.99Y 0.22 17.99 17.20-20.78
4 muss-34.73.30.387 0.23 18 12.30 25.19-35.57
4A lnIIa=-28.94+25.38 0.17 13.66 21.47-29.29
5 lnIIs=-32 .65+27.901r 0. 15 28 25 .60 25 .67-30. 13
5A 1111183-24 0794'210521 O 013 22 .77 19.59.23 045
6 lnIIs=-27 .19+24 .311 0 .22 38 18 . 99 22. 15-26 .47
6A lungs-25.88.22.817 0.19 20.47 20.94-24.68
7A lnIIs=-25.91+23.43Y 0.17 22.16 21.27-25.59
8 lnIIaa-28.45+25.65Y 0.49 38 10.02 21.33-29.97
9 lnIIaa-27.14+25.39Y 0.49 33 13.85 24.94-33.52
9A lnIIa=-23.99+21.6OY 0.27 13.81 18.43-24.77

' 10 11111 2-27.14+25.39Y 0.49 38 10.54 21.33-29.45
101 1nII§=-20.13+19.27Y 0.31 - 12.72 16.71-21.93
11 1nIIB=-32 .19429 .047 0.35 33 13 .38 24.63-33 .45
11A 1n113=-24,73+22.58'r 0.23 16.18 19.83-25.53
12 lnIIs="2° , 58.1.20 . 32Y 0.24 33 16 .66 17.84-22 .80
12A lnIIB=-19.24+18.98.r 0.21 17089 16082-21014
13 lnIIBa-36.24+36.67Y 0.26 22 14.38 31.38-41.96
13A lnIIaa-30.32+30.897' 0.24 13.16 26.02-35.75
14 lnIIa=-41.87+41.47f 0.22 18 13.12 34.83-48.11
14A lnIIsz-39.72+39.28" 0.18 15.44 33.94-44.62

TABLE 7 (CONTINUED)

63

 

 

 

Test Confidence
No. Regression Equation fix DJ. 1: Limits

15 lnII a=-50. 394-49. 247 0.32 33 15.73 42.87-55.61
15A lung-38. 551-38. 087 0.38 13.72 32.43-43.73
16 lnII =-32. 60+33. 337' 0.17 33 25.06 30.62-36.04
16A lnII $-30. 594-31. 26‘! 0.20 20.12 28.11-34.41
17 lnII 3"27' 84-128. 88" 0.21 33 518.63 25.73-32.03
17A lungs-24. 84-125. 797 0.25 14.03 22.05-29.53
18 lungs-35.41130. 907 0.17 38 32.53 29.30-32.50
18A lnIIaa-28.45+25.15T 0.16 28.06 23.63-26.67
19 lnII 38-.37 41+33. COT 0.19 38 23.57 30.64-35.36
19A lnII :.=-34 024-29. 99‘! 0.18 22.14 27.71-32.27
20 lnIIs=---29.41+26.28'r 0.21 38 20.37 24.11-28.45
20A lnII 82-27.23+24.16'Y 0.22 17.99 21.90-26.42
21 lnII --42. 99443. OOY 0.19 18 16.48 37.52-48.48
21A lnII a---=--32. 06+32. 407 0.16 15.16 27.90-36.90
22 lnIIaa-36. 68-1-37. 367 0.20 13 10.15 29. 41-45. 31
22A lnIIez-32. 01+32. 327' 0.15 11.44 26 2.3.-38 41
23 ~ lnII s='25‘ 42+26. 25Y 0.28 13 6.75 17.85-34.65
23A lnII§=-20. 29121. 057 0.21 ‘ 7.10 14.66-27.44
24 lnIIss-28.76+29.10Y 0.29 23 13.53 24.65-33.55
24A lnII 3=-21.65+22.24Y 0.19 15.56 19.28-15.20
25 lnII,a 9“"23- g0+24.277 0.22 23 15.97 21.13-27.41
26 lnIIS=-30.13+30.94Y 0.30 23 11.01 25 .13-36 .75
26A lnIIaz-22 .2242} .2ST 0.22 11.10 18 .93-27.57
27 1nI183-24. 88+24094Y 0.19 18 19004 22.19-27069
27A hush-19 054.18 94Y 0.13 21.69 17.11-20.77

#—

64

TABLE 8

COMPARISON OF RECRESSION COEFFICIENTS OF THE II VERSUS
BULK DENSITY RELATION FOR DIFFERENT STRESS STATES AND
SAME STRESS STATE FOR THE T110 METHODS USED TO OBTAIN DATA

 

 

 

Tests Degrees t t Degrees
Com- of Free- 6 Type Test Depth of Free- t
pared dom DST A No. In. dom
2-1 59 4.49** 0.28 1 10 52 7.28**
2-3 71 3053** 1036 2 5 66 2031*
1-3 64 7.30** 2.02* 3 15 76 0.25
5-4 46 0092 1085 4 1o ‘36 1062
5-6 66 2.14* 0.89 5 5 56 4.43**
4-6 56 2.18* 1.19 6 15 76 0.89
7-9 66 1.98 0.97 7 5 66 3.90**
7-8 71 1.63 0.84 8 15 76 1.27
9-8 71 0008 0005 9 1O 66 1044
12-11 66 3.50** 2.11* 10 15 76 2.15*
12-10 71 1088 0016 11 1O 66 2046*
11-10 71 1013 1065 12 5 66 0083
14-13 40 1.18 2.42* 13 10 44 1.67
14-27 36 4.83** 7.58** 14 5 36 0.54
13-27 40 4.09** 4.77** 15 15 66 2.67**
17-16 66 2.18* 2.27* 16 10 66 1.01
17-15 66 5.83** 3.69%! 17 5 66 1.28
16-15 66 4.68** 2.14* 18 15 76 4.39**
20-19 76 3.53** 3.06** 19 10 76 1.55
20-18 76 2088** 0061 20 5 76 1014
19-18 76 1.24 2.98** 21 15 36 3.14**
23-22 26 2.07 2.76** 22 10 26 1.09
23_21 31 3.58** 3.11** 23 5 26 1006
22-21 31 1.25 0.02 24 15 46 2.66**
26-25 46 2.09% .40 25 10 46 2.18*
26-24 46 0.52 .99** 26 5 46 2.20*
25-24 46 1.83 .26 27 15 36 3.82**

 

 

_

Type A data, however, neither set of values support
completely the part of the hypothesis that the second
invariant is not related to changes in bulk density.

A set Of typical curves showing the relationship
between the second invariant and bulk density are shown
in Figures 18 and 19. A comparison of the second invariants
for test 18 is shown in Figure 20. The large difference
in the curves at high loads is due to the neglected values
of the shearing stresses Of equation (20b) in calculating
the second invariant with the Type A data.
Relation§h22_between Maximum Normal Stress and Bulk Density

The regression equations, standard errors and confi-
dence limits for maximum normal stress versus bulk density
are given in Table 9. Comparing the calculated values of
"t" with those in the "t" table shows that all are highly
significant. Therefore, all slopes are significantly
different from zero.

Results of the "t" tests for the maximum normal stress-
bulk density relationship are presented in Table 10.
Differences between 2-1, 2-3. 1-3, 7’8, 12'119 12‘109
14-27, 13-27, 17-16, 17-15, 16-15. 20-19. 18-20. 23-22.
23-21, 26-25, and 25-24 of the 6 DST data are significant.
There are significant differences_between 12-11, 14-27,
13-27, 17-16, 17-15, 16-15, 20-19. 19-18. 23-22. and

23-21 for the data Obtained with the Type A Cells.

Tests 1, 2, 5, 7, 10, 15, 18, 21, 24, 26 and 27 show

PSI

INVARIANT

SECOND

66

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

 

T T T I
i a l I
20 . -—— A ~T~ “-le in
l- 5'uDEPTH TEST 7 i E
2-IO DEPTH TEST 9 ‘
3-15"DEPTH TEST 8
I ?
l0—-—-——-—-—-~-——- WT« w
A - 1 _ __.;_._ __._
i a E ‘
i ‘T E
2 L._.__..- :1 ._ 11.. ,: r. r-l-._-'+___._.___..__.7
. E i l
; i 2
| H¥— [ I [
LIZ LI? l.22
BULK DENSITY GM/CC
Figure 18 . Second Invariant Of Stress Deviator Tensor

versus Bulk

Density for Data Obtained with Six Directional

Stre ss Transducer .

PS1

INVARmmH'

SECOND

67

 

 
  
   
  

 

20

 

 

I— 5":DEPT TEST
2-10 DEPTH TEST 9
3-l5"DEPTH TEST 8

‘—

l0

 

 

 

 

 

 

 

 

 

 

 

hD

 

 

 

 

U7 L22

BULK DENSITY GM/CC

Figure 19. Second Invariant Of Stress Deviator Tensor
versus Bulk Density for Data Obtained with Type A

Pressure Cells.

68

 

84

72 LA .L f

0 TYPE A CELLS

 

 

 

 

 

 

A SDST

N

g 60

'5.

s. 48 /
0:

§ /
a

36 -

C3

2!

O

8 /
m 24 /

 

.2 /;/
o we”

0 5 IO I5 20 25
APPLIED LOAD P51

Figure 20 . Comparison of the Values of the Second Invariant
Calculated from the Type A and 6 DST Data.

 

 

 

 

 

 

 

69

TABLE 9

STATISTICAL ANALYSIS FOR MAXIMUM NORMAL STRESS
VERSUS BULK DENSITY

Confidence

 

‘ Test
N0. Regression Emzation SIX D.P. , t limits
1 In fia—36.53+30.4Ov 0.16 26 33.41 28.53-32.27
1A In Gib-25.774.21.477 0.26 39.51 19.25-23.59
2 In Wis-27.30.24.011! 0.18 33 23.54 21.94-26.08
2A In n=-23.97+2O.18Y 0.20 30.74 17.41-22.95
3 ln g=-21.48+19.23v 0.27 38 14.64 16.55-21.91
3A In =-21.25+18.31Y 0.24 32.56 16.35-20.27
4 In f=-34.39+29.451' 0.21 18 13.09 24.72-34.18
4A In =—30.09+25.46Y 0.16 28.17 21.72-29.20
5 In 0"=-32.31.26.97r 0.16 28 23.25 24.60-29.34
5A In =-25.61+21.38Y 0.15 39.45 19.13-23.63
6 In fis—27.76+24.13Y . 0.25 38 16.53 21.67-26.59
6A In fi=-25.97+22.93Y 0.19 40.34 21.01-24.85
7 In Gi=- 4.34 29.81Y 0.25 33 _19.48 26.70-32.92
7A anl=-36.57:23.10Y 0.17 44.09 20.96-25.24
8 In =-26.97+23.791r 0.47 38 9.67 19.64-27.94
9 I G's--2820 25.641 0.49 33 14.73 25.27-33.35.
9A 1: ==--25041:21096Y 0030 25033 18024-25050
10 In I=-28.20 25.647 0.49 38 10.64 21.58-29.70
10A In .-.-21.37:19.471 0.31 25.73 16.87-22.07
11 l r... 2.14 28.42Y 0.35 33 10.49 22.91-33.93
12 e- .52 13.301! 0.23 33 16.07 16.42-21.18
1 _.__. . .37 0.31 22 11.42 28.52-41.18
.3. 131;.-333333934 0.2. 23... 25.14.35...
14 z— . .27 0.21 18 11.70 28.91-41.55
14A 33‘ gig-35.591.332.391! 0.18 2.5-59 28132-3935

TABLE 9 (CONTINUED)

70

 

 

 

Test _ Confidence
N0. Regression Egation _§yx D.F. t Limit

15 In FI=-49.35+47.53Y 0.29 33 16.80 41.77-53.29
15A In 0"I=-4O.33+38.77r 0.29 27.43 32.95-44.59
16 In =-31.48+31.52r 0.15 33 26.93 29.14-33.90
16A 1n I=-30.88+30.58Y 0.22 36.20 26.12-34.04
17 In P=~24.09+24.56Y 0.21 33 15.85 21.41-27.71
17A In I-.--24.42+24.38Y 0.25 27.27 20.70-28.06
18 In G‘I--33.49+28.671r 0.21 38 24.30 26.68-30.66
18A In 1"1=-28.20+24.O7i 0.17 50.94 22.42-25.72
19 In S's-34.39.29.877 0.18 38 22.46 27.63-32.11
19A In I=-35.88+30.167 0.24 35.46 27.66-33.56
20 In 6" --25.74+22.661 0.22 38 16.79 20.38-24.94
20A 1n I--28.34+24.14Y 0.24 32.92 21.66-26.62
21 In r--45.77+44.8SY 0.25 18 13.08 37.68-52.08
21A In I=-32.68+32.067 0.17 - 27.58 27.21-36.91
22 In FI=-37.10+37.07Y 0.19 13 10.62 29.54-44.60
22A an'I=-31.40+30.75‘( 0.19 17.63 23.08-38.42
23 In 614-24. 12+24.27Y 0.24 13 7.29 17.08-31.46
23A In I=-21.16+20.81‘1’ 0.24 12.44 13.68-27.94
24 In n=-29.04+28.66v 0.22 23 14.93 24.69-32.63
24A Inn=-23.OO+22.67'f 0.19 31.90 19.77-25.57
25 In 11"=-23.11+23.46~r 0.22 23 15.43 20.32-26.60
25A In I=-19.66+19.451r 0.22 26.25 16.35-22.55
26 In r--3o.39.3o.48v 0.31 23 10.51 24.48-36.48
26A In I=-22.08+22.13Y 0.22 21.87 17.87-26.39
27 In g‘;=-24.92+24.46Y 0.21 18 16.87 21.41-27.51
27A In I=-20.16+19.01Y 0.14 38.27 16.91-21.11

71

TABLE 10

COMPARISON OF REGRESSION COEFFICIENTS FOR U} VERSUS
BULK DENSITY RELATION FOR DIFFERENT STRESS STATES AND
SAME STRESS STATE FOR THE Two METHODS USED TO OBTAIN DATA

 

Degrees

 

Tests Degrees t t
Comp of Free- 6 Type Test Depth of Free- t
pared dam DST A No. In. dam
2-1 59 4.67** 0.74 1 10 52 6.32**
2-3 71 2.87** 1.05 2 5 66 2.25*
1-3 64 6.97** 1.99 3 15 76 0.52
7-9 66 1.66 0.56 7 5 66 3.62**
7-8 71 2.08* 0.67 8 15 76 0.70
9-8 71 0.58 0.15 9 1O 66 1.39
12-11 66 3.26** 2.31* 10 15 76 2.16*
11-10 71 0.77 1.61 12 5 66 0.13
14-13 40 0.09 1.08 13 10 44 1.11
14-27 36 3.22** 5.43** 14 5 36 0.21
13-27 40 3.08** 4.17** 15 15 66 2.18*
17-16 66 3.58** 2.50* 16 10 66 0.46
17—15 66 7.12** 4.25** 17 5 66 0.08
16-15 66 5.23** 2.46% 18 15 76 3.88**
20-19 76 3.80** 2.83** 19 10 76 0.34
20-18 75 3.35%! 0.05 20 5 76 0.74
19-18 76 0.67 3.73** 21 15 36 3.10**
23-22 26 2.65* 2.05* 22 10 26 1.27
23-21 31 4.31** 2.79** 23 5 26 0.74
22-21 31 1.60 0.30 24 15 46 2.52*
26-25 46 2.14% 1.05 25 10 46 1.88
26-24 46 0.52 0.22 26 5 46 2.35*
25-24 46 2.12% 1.57 27 15 36 3.09**

 

 

72

a significant difference between the two sets of data
obtained with the Type A Cells and 6 DST.

The significant differences again appear around the
highest rate of loading and high moisture contents. The
Type A values are not as varied as the 6 DST data but the
conclusion must be drawn that the changes in bulk density
are not independent of the maximum normal stress. This is
particularly true at the lower moisture contents and lower
rates of loading.

The set of typical curves for this relationship are
shown in Figures 21 and 22.

Relationship between Maximum Shear Stress and Bulk Density

 

The third function used to represent the stress
deviator tensor was the maximum shear stress. The regression
equations, standard errors and confidence limits are listed
in Table 11. Again all the calculated values of "t" are
highly significant.

The results of comparisons of the stress states are
given in Table 12. For the data Obtained with the 6 DST,
comparisons between 2-3, 1-3, 7-8. 12-11. 12-10, 14-27,
13-27, 17-16, 17-15, 16-15, 20-19, 20-18, 23—22, 23-21,
26-25 and 25—24 show significant differences. Results
of the Type A data Show the differences between 12-11,
17—16, 16-15, and 23-22 to be significant at the 95% level
and 14-27, 13-27, 17-15, 20-19. 19-18, 23-22. and 23-21
to be significant at the 99% level. The differences

I
5

PS

MAX NORMAL STRESS

73

 

 

—-.. -.....~. ‘ -.- .,-._4p—v.—_..—--—.-—-—

 

20 --
1- 5" DEPTH TEST 7
2-10"DEPTH TEST 9
3—15“OEPTH TEST 8

    

 

 

 

[ I
I
«E.-- -1!+__+.:.I_—__.

 

I
I
r
i E;

 

 

 

 

 

 

 

_.__.__..- t
_____._ .. - - _. ____ _
I
I
5 r——-—-- «I —- --- "“31"“ ..—-_.~_ .. I
I
;
..____-...II_W._- .. 5.-.. - -
. I
I ;
I
I-————~—- L 1 —- s A“ ———-—-1
z
2 -———-- - é ~ ~-
I i t
I i
Q I ;
I I
I
I ..,- _ _ I
1.15 1.20 1.25

BULK DENSITY GM/CC

Figure 21. Maximum Normal Stress versus Bulk Density
for Data Obtained with Six Directional Stress Transducer.

PSI

MAX NORMAL STRESS

74

 

I

I
I

1- 5: DEPTH TEST 7
2-10 DEPTH TEST 9

3— I 5" DEPTH TEST 8

I

I

 

 

 

 

 

 

 

 

I I I
I
10 __..._-_I I I
I—......-.._._ _ f I - I . _- ___.._ -- .._____ /r
——--~ I - I - - III—I
..___._...-._._;I “Elf In-.. - I I i _._.._.+.__.___I
I ' 1 I
s ... — I- ~ - ~ I -~- I
I I I I
5 I I
,r____..:_ i _. ”I .. I i I”. .22.... -I_.____._I
I I
' I
i- 2..."...

 

 

 

I.16 1.21 l.26
BULK DENSITY GM/CC

Figure 22 . Maximum Normal Stress versus Bulk Density
for Data Obtained with. Type A Pressure Cells.

75

TABLE 11
STATISTICAL ANALYSIS FOR MAXIMUM SHEAR STRESS

VERSUS BULK DENSITY

 

 

 

Test Confidence
N0. Regression Eguation fig D.F. 1: Limits
1 lnl’max:-74.29+61.85Y 0.26 26 34.17 58.13-65.57
1A lnYmax=-51.30+42.91"' 0.29 21.13 38.74-47.08
2 luggr-58JS-1-52J6T 0.39 33 19.78 47.30-57.02
2A 1n 8-49 O37+41o65r 0041 . 14084 35 093-4703?
3 lnzmaxe-74.29+61.85T 0.26 38 14.39 33.53-42.43
3A ln’Zmaxs-43.49+37.627 0.48 15.86 33.62-41.62
4 m7 =-69.27 59.437 0.44 18 12.59 49.51-69.35
5 I hem-64.7154n17 0.32 28 23.32 49.36-58.86
5A 13%...-5133143661 0.30 19.77 37.99-49.01
6 I =- 6.14 48.897 0.50 38 16.74 43.97-53.81
7 In’lmax=-7O.84+61.58Y 0.51 33 19.80 55.26-67.90
7A Inznnx--53.23+46.47Y 0.35 21.51 42.08-50.86
8 I Imam-56.12 49.541 0.97 38 9.77 40.99-58.09
8A Igtmnx=-5O.43:43.621f 0.45 11.18 37.04-50.20
zmaxs- 00 10991 1001 33 14044 52 034-69050
3A Igfmax=-50,.76:24.01Y 0.59 12.77 36.99-51.03
10 .-.-- .O 1. 91’ 1.01 38 10.48 43.63-60.35
2::
11 1' =-64.40 57.02? 0.73 33 12.62 47.83-66.21
11A Ig’gmax=-53.22:46.76Y 0.53 14.23 40.07-53.45
a- , .31 0.46 33 16.17 33.07-42.59
73A Ignaz-.3 372133.677 0.42 17.53 33.26-41.96
gnu..- , 0. 3y 0.58 22 12.38 58.78-82.39
{max-l. . 3 5 .997 0.41 18 11.34 57.03-82.95

TABLE 11 (CONTINUED)

75

 

 

 

Test . Confidénce
N0. Reg-6331011 Equation 1; D.F. 1: Limit

15 InTmaxe-100.09+96.53Y 0.58 33 17.02 85.00-108.06
15A lnYmax=-81.34+78.387 0.61 13.17 66.28-80.48
16 In’tmax=-63.47+63.64Y 0.32 33 25.46 58.56-68.72
16A ln‘lmaxa-61.91+61.49Y 0.45 17.58 54.37-68.61
17 In? =-49.85+50.84Y 0.43 33 16.04 44.39-57.29
17A Indigc-49.28+49.437 0.49 13.71 42.09-56.77
18 In’gnaxa-68.51+58.71Y 0.41 38 25.53 54.83-62.59
18A 1nmax=-58.12+49.71Y 0.34 26.44 46.54-52.88
19 In =—69.11+60.15Y 0.37 38 22.03 55.55-64.75
19A 1n max=-72.70+62.17Y 0.48 17.69 56.25-68.09
20 In’! =-50.02+46.73Y 0.43 38 18.25 42.26-51.20
20A ln’Zmax=-56.49+48.ZGY 0.47 16.62 43.37-53.15
21 ' litmus-923561.287 0.48 18 13.85 77.44-105.12
21A ln'tmax=-54.76+53.75Y 0033 13.86 54009-73041
22 lnfmax=-74.92+74.07Y 0.38 13 10.73 59.87-90.07
22.4 ln‘tmax--63.84+62.64Y 0.38 8.96 47.54-77.174
23 In’l'maxg-49.00+49.38‘r 0.50 13 7.11 34.38-64.38
24 In =-58.87+58.20Y 0.55 23 14.30 49.78-66.62
24.4 In =-45.16+44.68Y 0.37 16.08 38.93-50.43
25 In =-47.21+47.92Y 0.44 23 15.82 41.65-54.19
25A In =-39.70+39.48‘/ 0.46 12.57 32.98-45.98
26 11,1 =-61.46+61.77‘Y 0.63 23 10.47 49.57-73.97
26A minus-44.704.44.967 0.45 10.62 36.21-53.71
27 ln?m.x=-50.05+49.14Y 0.43 18‘ 16.55 42.90-55.38
273 In max--40.60+38.35Y 0.29 19.29 34.17-42.53

 

77

TABLE 12

COMPARISON OF REGRESSION COEFFICIENTS FOR/rm VERSUS
BULK DENSITY RELATION FOR DIFFERENT STRESS STATES AND
SAME STRESS STATE FOR THE TWO METHODS USED TO OBTAIN DATA

 

 

Tests Degrees t t Degrees
Com- of Rree- 6 Type Test Depth of Free- t
Egged dom DST A No. In. dom
2-3 71 6.63** 1.10 2 66 5.40**
5-4 46 1.01 1.63 4 10 36 1.58
5-6 66 1.40 0.76 5 56 3.32**
4—6 56 1.90 1.03 6 15 76 .80
7-9 66 1.83 0.60 7 66 3.99**
7-8 71 2.02* 0.64 8 15 76 0.93
9-8 71 0.37 0.07 9 10 66 1.46
12-11 66 3.77** 2.33* 10 15 76 2.13*
12-10 71 2.5 * 0.52 11 10 66 1. 4
11-10 71 0.75 1.60 12 5 66 0.07
14-13 40 0.07 1.36 13 10 44 1.31
14-27 36 3.04** 5.63** 14 5 36 0.08
13-27 40 3.34** 4.15** 15 15 66 2.21*
17-16 66 3.17** 2.40* 16 10 66 0.50
17—15 66 7.03** 4.16** 17 66 0.29
16-15 66 5031** 2045* 18 15 76 3003**
20-19 76 3.53** 3.06** 19 10 76 '0.45
19-18 76 0.40 3.13** 21 15 36 3.43**
23-22 26 2.60* 2.26* 22 10 26 1.25
23-21 31 4.37** 2.84** 23 5 26 0.87
22-21 31 1.70 0.13 24 15 46 2.74**
26-25 46 2.09* 1.04 25 10 46 1.93
26-24 46 . 0 0.06 26 5 46 2.31
25-24 46 2.03* 1.24 ‘ 27 15 36 3.02*

 

*

 

78

between tests 13-27 and 14-27 are probably partly due to
the differences in moisture content, the moisture content
of tests 13 and 14 is 17.41 percent and only 16.54 percent
for test 27. To a smaller degree, this could also be true
for the differences between test 15-17 and 16-17 because
there is a 0.51% moisture content difference. The
difference in moisture content for tests 18 and 19 is
1.36 percent and 1.52 percent for tests 18 and 20. The
estimates of standard error for tests 2, 6, 9, 13, and 23
are larger than the other values of standard errors
indicating more error in obtaining these test data. Based
on this information and data, the conclusion that the
maximum shear stress is best related to changes in bulk
density might be made.

A typical set of curves showing the relationship
between maximum shear stress and bulk density is shown
in Figures 23 and 24.
The Effect g£_Moisture Content on the Relationship between

 

{can Normal Stress and Bulk Density

 

Since the moisture content of the soil is one of the
variables in agricultural soils which has been found to be
a factor in soil compaction, three different moisture
contents were used during the experimental tests. The
range used was from 7.97 percent to 17-93 percent (dry
weight basis). Within the five replications of a test the

moisture content remained constant. However, in tests

79

 

 

 

 

 

~ * . . N
. . . _
_ m a u .
_ . .
21.1 11.7. 1&1! 1 1 1+ -1 I 1 1111...
. _. W
. m .
m m . ..

1.22

1.17

DENSITY
Maximum Shearing Stress versus 13qu Densit

1.12

 

I

 

.mm

mmmmhm adulm x<2

_.17981 ........ .1 1
TTT .
. SSS W
. EEE . .
. TTT _ .

11-- 1...? 1.11 .1 1. 11.
. mmm L m
. PPP. m
. EEE . _ . .

11--60...... 1..-...» - .. - iii.
~ _ l .1 _ W P. H a I w
m. m 5 2 ..

GM/CC

BULK

)1
1‘.

for Data Obtained with Six Directional Stress Transduce

Figure 23.

PSI

MAX SHEAR STRESS

80

 

 

 

 

 

  
   

 

 

 

 

 

 

 

 

1 1
i .1
20 ‘1 --.... -.._-. ---..-.....-..1_- --......._..
1— 5" DEPTH TEST 1
2-10: DEPTH TEST 9
3-15 DEPTH TEST 8 w
I , :
10.....-1-_-.-_,-., —— -—+—————.——.
»—-—-~1 ._ 1
1--.} - _. 1....
NWT...“ fi-_ - - '
5 _.. i. 2.... ..
- . 1--....
1
i
1—-°—-- - “--t —----—- --1
2 —-—-—— ~ 1 ~-
i ‘ . i i
I .,L _ 1 J A 1
1.13 1.18 1.28

BULK DENSITY GM/CC

Figure 24 . Maximum Shearing Stress versus Bulk Density
for Data Obtained with Type A Pressure Cells.

81

conducted a day or more apart the moisture content was not
the same. The exact percent moisture was not known until
after the tests were run, the soil samples dried, and the
percentages calculated.

To determine the effect of moisture content on the
relationship between mean normal stress and bulk density
the "t" test for the regression coefficients was used.

The coefficients from two relationships determined at the

same depth and rate of loading but with different moisture
contents were compared. The results of these comparisons

are presented in Table 13. I

For the 0.38 inch per second rate of loading
significant differences were obtained only for the 6 DST
data between tests 4-9, and 4-25. The moisture contents
for tests 4 and 9 are 11.56- and 11.38-percent which
indicates that an error must have been made in obtaining
the data for test 4. The estimate of standard error,
however, does not indicate a large variation in the data
used to determine the regression coefficient. Under these
conditions, the conclusion that changes in bulk density are
independent of moisture content could be made.

Significant differences were found between tests
2-12, 3-21, and 10-21 for the 6 DST data and 1-72, 11-22,
3-21 and 10-21 for the Type A data under the 0.62 inch
Per second loading condition. The values for the Type A

data were obtained for the 10— and 15-inch depths between

TABLE
COMPARISON OF REGRESSION COEFFICIENTS OF THE Ga

82

13

VERSUS BULK DENSITY RELATIONS AT DIFFERENT
MOISTURE CONTENTS AND BETWEEN METHODS FOR A GIVEN TEST

 

 

 

 

 

 

 

t t
Type 6 Test M.C.
Tests v.3. A DST No. f v.5. t
Rate of Loading 0.38 Inch per Second
5-7 61 1.64 0.60 5 8.89 56 5.32**
5-26 .51 1.09 0.71 7 11.35 66 3.52**
7-26 56 0.15 0.33 26 16.05 .46 1.94
4-9 51 1074 3020** 4 11056 36 2049*
4-25 41 2.45 5.01** 9 11.38 66 1.26
9-25 56 0.82 0.32 25 15.78 46 2.02*
6-24 61 0.13 1.89 8 11.39 76 1.51
8-24 61 0.40 0.66 24 16.16 46 2.41*
Rate of Loading 0.62 Inch per Second
2-12 66 1.51 5.34** 2 9.89 66 5.60**
2-23 46 0.09 0.87 12 12.46 66 1.80
12-23 46 0.76 1.10 23 17.67 26 1.12
1-11 59 0.01 0.30 1 8.63 52 6.7142
1-22 39 3.74** 1.50 11 12.43 66 3.27**
11-22 46 3.55** 1.54 22 17.52 26 0.82
3-10 76 0.16 1.61 ,3 11.59 76 0.72
3-21 56 6.14** 7.62** 10 12.53 76 2.10%
10-21 56 5.60** 4.94** 21 17.93 36 2.33*
Rate of Loading 1.00 Inch per Second
20-17 71 1.05 1.11 20 10.79 76 3.30**
20-14 56 5.62** 5.58** 17 14.34 . 66 2.40*
17-14 51 4.31** 4.79** 14 17.41 36 2.50*
19-16 71 1.41 1.34 19 10.95 76 4.09**
19-13 60 0.86 0.16 16 14.85 66 1.50
16-13 55 0.12 1.17 13 17.41 44 1.90
18-15 71 4.24** 4.92** 18 12.31 76 5.03**
18-27 56 4.68** 4.48** 15 14.85 66 2.91**
15-27 51 6.50** 6.78** 27 16.54 36 4.22**

 

moisture contents of 11.5- and 17.9—percent. This
difference may not have been evident at the 5-inch depth
due to instrument effects on the stress pattern so near
the loading surface. It is not clear why the effect of
moisture at the 10—inch depth was not found with the 6 DST
unless the larger size of the instrument has a greater
effect on the stress pattern for a deeper depth. The
conclusion that moisture content does affect the relation-
ship between mean normal stress and bulk density for 0.62
inch per second rate of loading and depths of 1C and 15
inches could be made.

For the 1.00 inch per second rate of loading condition,
differences between tests 20-14, 17-14, 18-15, 18-27, and
15-27 were significant. These results are consistent
except for the difference between tests at the 10-inch
depth. Therefore, the conclusion could be made that the
moisture content affects the relationship between mean
stress and bulk density at the 5- and 15-inch depths at
the 1.00 inch per second rate of loading.

The results from Tests 1,11, and 22 are shown in
Figures 25 and 26.

The Effect g£_Mgisture Content 22 the Relationship between

 

§§cond Invariant and Bulk Density
Results of the comparison between regression coeffi-
cients for the relationship between the second invariant

and bulk density are presented in Table 14. The significant

MEAN STRESS PSI

20

 

84

 

 

 

 

  
   

 

 

 

 

T T 1 V 1
1- 8.637. TEST 1 1 3 1
-2—12437. TEST 11 ..,_. 1
317.52% TEST 22 1 2
5
" i
- M...
..
1 ..
1
.‘ 1'
5 1 1
1 L 1
1.15 1.30

BULK DENSITY GM/CC

Figure 25 . The Effect of Moisture Content on the
Relationship between the Mean Stress and Bulk Density.
Data obtained with 6 DST.

MEAN STRESS .PSI

20

1
10 ......_.-_ ,
~ '.~--

85

 

‘ 1 1 1
1-8.63% TESTI 1 1
-2—124351. TEST 11 -1-.-

 

 

317.52% TEST 22

1
1
1 1
1 1
1
1
5.

   

1
1
1
1
1
1
1
1

   

  

 

 

 

 

1.00 1.16 1.30

BULK DENSITY GM/CC
Figure 26 . The Effect 01’ Hoisture Content on the

Relationship between lean Stress and Bulk Density.
Data obtained with Type A Pressure Cells.

COMPARISON OF REGRESSION COEFFICIENTS OF THE
VERSUS BULK DENSITY RELATIONS AT CONSTANT RATE,

86

TABDE 14-

11a

CONSTANT DEPTH, AND DIFEERENT.MOISTURE CONTENTS AND
BETWEEN METHODS FOR A GIVEN TEST

 

 

 

t t
Type 6 Test M.C.
Tests D.F. A DST No. D.F. t
Rate of Loading 0.38 Inch per Second
5-7 61 1.35 1.41 5 8.89 56 4.43**
5-26 51 0.75 1.01 7 11.35 66 3.90**
4-9 51 1.56 1.54 4 11.56 36 1. 62
4-25 41 2.41* 2.10* 9 11.38 66 1. 44
9—25 56 0.90 0.43 25 15.78 46 2.18
6—8 76 0.57 0.47 6 7.97 76 0.89
6-24 61 0.31 1.91 8 11.39 76 1. 27
8-24 61 0.31 1.03 24 16.16 46 2.65**
Rate of Loading 0 62 Inch per Second
2-12 66 1.37 3.13** 2 9.89 66 2.31*
2-23 46 0 09 0.27 12 12.46 66 0.83
12-23 46 o 66 1.45 23 17.67 26 1.06
1-11' 59 0.53 0.88 1 8.63 52 7.28**
1-22 39 3.55** 1.65 11 12.43 66 2.4611
11-22 46 3.06** 1,95 22 17.52 26 1.09
3-21 56 5. 62** 7. 72** 10 12.63 76 2.14*
10-21 56 5. 00** 4.68** 21 17.93 36 2.84*
Rate of Loading 1.00 Inch per Second
20-1 1 0. 72 1. 29 20 10.79 76 1.14
20-1Z 56 5. 26** 4. 45** 17 14.34 66 1928
17-14 51 4. 30** 3 58** 14 17.41 36 0.54
19-16 71 1.10 0.17 19 10.95 76 1.54
19-13 60 0. 33 1.26 16 14.85 66 0.52
16-13 55 0. 49 1.16 13 17.41 44 1. 67
18-15 71 4. 42*} 5. 61** 18 12.31 76 4039**
18-27 56 4. 96** 3. 68** 15 14.85 66 2.66**
15-27 51 6. 57** 7.16** 27 16.54 36 3.82**

 

87

differences follow the same pattern as found for the
relationship between mean normal stress and bulk density.
In general, at the higher rates of loading the moisture
content of the soil does affect the relationship between
the second invariant and bulk density. The relationships
for Tests 1, 11 and 22 have been plotted and are shown

in Figures 27 and 28. There are no significant differences
for the 6 DST lines. However, for the Type A lines signifi-
cant differences are found between Test 1-22 and 11-22, as
can be seen in Figure 28. The slopes of Tests 1 and 11

are significantly different from the slope of Test 22.

The Effect of Moisture Content on the Relationship between

 

Maximum Normal Stress and Bulk Density

The results of the "t" tests between slopes of the
regression equations for the maximum normal stress-bulk
density relationships are listed in Table 15. The
difference between Tests 4 and 25 for the lowest rate of
loading is significant for both the Type A and 6 DST data.
For the 0.62 inch per second rate, significant differences
are found between Tests 2-12, 3-10, 3-21, and 10-21 for
the 6 DST lines and 1-22, 3-21, and 10-21 for the Type A.
Significant differences are found between Tests 18-15,

18-27, and 15-27 for both sets of data under the 1.00 inch

per second rate of loading. In addition differences

between the Type A data for Tests 20-14, and 17-14 are

Significant. Based on this data the conclusion that the

INVARIANT 13512

SECOND A

20

5

88

 

 

     
  

 

 

 

 

 

 

 

 

 

 

l I '
I-8.63% TEST I 1 1 1
.2243; TEST ll ____..:-.______.“ 1
317.52% TEST 22 1 1 1
1 1 1
1 1 : '
1 i 1
1 1 1
1 : 1
____..._-4.-i__.__.-- -_-_-.- 4 ._ . m 4. ._..-
b—-—-——-.———————..T-.—---. ..-. _ i. ___a _- -.._-;., -.
.____.- _. -- ,. 4.. H- .
___.__ 4 __ ---__ 4
1 1 ‘
._._......--.-_4___ - _ ,. . . Q, ,
1 ;
1 : ’
. g 1
M..- 1 _. 4.” . h i
1 1
1 1 1
1,00 1.15
’ BULK DENSITY GM/CC
Figure 27 . The Effect of Moisture Content on the

Relationship between the Second Invariant and Bulk
Data obtained with 6 DST.

Density .

 

 

INVARTANT p312

SECOND

T T T
1-8.63%. TEST 1 4
317.52% TEST 22
1 1 f:
¥ 1
10 -—- 1 .
.4- i- -
__ -4 4.- _-
5221M,
' 1
2...... .. -4 - - ..
__, fl 4.
1
1
1
2 ._..____ 1-
1
I "(IL ' 1 1 . .
0.97 ”2 I.27

89

 

 

 

 

   

 

 

 

 

 

 

BULK DENSITY GM/CC

Figure 28 . The Effect of Moisture Content on the
Relationship between the Second Invariant and Bulk
Density. Data obtained with Type A Pressure Cells.

90

TABLE 15

COMPARISON OF REGRESSION COEFFICIENTS OF THE “E
VERSUS BULK DENSITY RELATIONS AT CONSTANT RATE,
CONSTANT DEPTH AND DIFEERENT MOISTURE CONTENTS AND

BETWEEN METHODS FOR A GIVEN TEST

 

 

 

t t
Type 6 Test M.C.
Tests D.F. A DST No. 1% D.F. t
Rate of Loading 0.38 Inch per Second
5-7 61 1.63 1.48 5 8. 89 56 4.80**
5-26 51 0.36 1.12 7 11.35 66 3.62“
7-26 56 0.42 0.20 26 16. 05 46 2.35*
4-25 41 2.58* 2.21* 9 11.38 66 1.39
6-8 76 0.59 0.14 6 7.97 76 1.04
6-24 61 0.14 2.35* 8 11.39 76 0.70
8-24 61 0.45 1.56 24 16.16 46 2.52*
Rate of Loading 0.62 Inch per Second
2-12 66 0.93 3.36** 2 9 89 66 2.25*
2-23 46 0.18 0.07 12 12. 46 66 0.13
12-23 46 0.64 1.55 23 17. 67 26 0.74
3-21 56 5. 32** 6. 98** 10 12.63 76 2.16*
10-21 56 4.53** 4. 59** 21 17.93 36 3.10**
Rate of Loading 1.00 Inch per Second
20-17 71 0.10 0.92 20 10.79 76 0.74
17-14 51 3.12** 0.20 14 17.41 36 2.28*
19-16 71 0.01 0.93 19 10.95 76 0.34
16-13 55 0.05 1.02 13 17.41 44 1.11
18-27 56 3.61114 2.2511 15 14. 85 66 2.18*
5-27 51 6.52** 7.26** 27 16. 54 36 3.09**

 

91

moisture content affects the relationship at the higher
rates of loading and 15-inch depth could be made.

A set of typical curves showing the relationship
between maximum normal stress and bulk density are pre-
sented in Figures 29 and 30.

The Effect 3: Moisture gpntent 23 the Relationship between

 

 

maximum_§hear Stress and Bulk Density

 

The effect of moisture content on the relationship
between maximum shear stress and bulk density follows the
same pattern as the other relationships as shown in Table 16.
The conclusion that moisture content affects the relationship
at the higher rates of loading and the 15-inch depth could
be made.

The values for Tests 1, 11 and 22 were used to plot
the set of typical curves shown in Figure 31 and 32.

223 Effect 2; §§£g_2£ Loading g§_the Relationship between
E§§g_Normal Stress and Bulk Density

Three rates of loading were used as described in
Table 4. In order to determine the effect of the rate
of loading, the regression coefficients of two tests at
the same depth and approximately the same moisture contents
but with different rates were compared by means of the
"t" test. The results of these comparisons are given in
Table 17.

Difference between Tests 14—23 and 14-26, 5 inches

below the loading surface are significant for both sets of

PSI

MAX NORMAL STRESS

92

 

I T
l-8.63% TEST l ,T 1
20 _.. 2424370 TEST H l L
347.52% TEST 22 g 73

 

 

 

 

 

 

 

 

 

 

 

 

i
1

 

 

 

 

 

Figure 29.
Relationship
Bulk Density.

 

GM/CC

BULK DENSITY

The Effect of Moisture Content on the

between the Maximum Normal Stress and
Data Obtained with 6 DST.

PSI

MAX NORMAL STRESS

r I T I
1-853‘7. TEST! 1
20 _..2-12.43°/. TEST 11 WiWWl W
347.52% TEST 22 1 1
1 .
10 W-_
1
ill a}-
i
2 .m- "19““
l w ' h
1.00 1.15 1.30

93

 

 

 

 

 

 

 

 

 

BULK DENSITY GM/CC

Figure 30. The Effect of Moisture Content on the
Relationship between the maximum Normal Stress and
Bulk Density. Data obtained with Type A Pressure Cells.

, TABLE 16

COMPARISON OF REGRESSION COEFFICIENTS OF THE ’I.’ max

VERSUS BULK DENSITY RELATIONS AT CONSTANT RATE,

CONSTANT DEPTH AND DIFFERENT MOISTURE CONTENTS AND
BETWEEN METHODS FOR A GIVEN TEST

 

 

 

 

 

 

t t
Tyge 6 Test .M.C.
Tests D.F. DST NO. D.P. t
Rate of Loading 0.38 Inch per Second
5-7 61 0.96 1.93 5 8.89 56 3.32**
5-26 51 0.31 1.21 7 11.35 66 3.99**
4-25 41 2.29* 2.10* 9 11.38 66 1.46
9-25 56 0.97 0.80 25 15.78 46 1.91
6-8 76 0.51 0.11 6 7.97 76 0.80
6-24 61 0.34 1.86 8 11.39 76 0.93
8-24 61 0.22 1.33 24 16.16 46 2.74**
Rate Of Loading 0.62 Inch per Second
2- 2 66 1.14 7.10** 2 9.89 66 5.40**
12-23 46 0.501.57 23 17.67 26 0.87
1-11 59 1. 00 0. 99 1 8.63 52 6.96**
1-22 39 2. 71** 1. 82 11 12.43 66 1.84
11-22 46 2. 06* 2.16* 22 17.52 26 1.25
3-10 76 0.50 2.49* 3 11.59 76 0.10
3-21 56 5.05** 7.51 10 12.63 76 2.13%
10-21 56 4.37** 4.76** 21 17.93 26 3.43**
Rate of Loading 1.00 Inch per Second
0 2 0.99 20 10.79 76 0.39
38:12 g; 3 52** 3. 46** 17 14.34 66 0.29
17-14 51 3 27*! 2 76** 14 17.41 36 0008
19-16 71 0.14 0.94 19 10.95 22 8.;3
19-13 60 0. 25 1.65 16 14.85 . 1
16-13 55 0.14 1.12 13 17.41 44 1.3 **
18_15 71 4. 59*} 6 18** 18 12031 76 3003*
13-27 55 4.15** 2 55* 15 14.85 66 2.21**
15-27 51 6.38** 7:40** 27 15.54 35 3.02

 

 

95

 

 

 

 

 

 

 

‘ . 1 1 1T 1
l-8.63/o TEST I § 1 !
20..2—12.43°/. TEST 11 ..._-.;-+ 1..
317.52% TEST 22 1 1
I I ; 1
_ IO e»~—4+» i
(D WW-..-__W§. W , __
“-
a; "H“ 1“ T
(D PW JW - ..1
DJ § '
E ;
co 5 ”W "I “ ' j
a: ~-—-~ 1
<1 1
DJ 1
I 1
U) W- W]
x 1 .
4 1 :
2 L t
2 ----*~ 1 T f
1 E
‘ 1 1
I _/, 2 I I '
I 0.97 ”2

BULK DENSITY GM/CC

Pi e 31. The Effect of Moisture Content on the
Refiionship between the Maximum Shear Stress and
Bulk Density. Data obtained with 6 DST.

 

 

 

 

 

 

 

 

 

 

l-8.63°/. TEST! 1 s 1
20 ._ 242.43% TEST II W-21-.-_-W_-. ; .- 1
347.52% TEST 22 f
1 :
I. 1 '
_ i 1
3’- '° ::":1: g i 1
a “Hm-i :
Lu ‘
E T”
‘2 5mg-.-
i
a: 2 y
< 1.... “T“
w
:1: 1 _
m M" D1?“ 21
x 1
4 z 3
2 2 WW ;
' #er L L
093 1.08
BULK DENSITY GM/CC
Figure 32. The Effect of Moisture Content on the

Relationship between the maximum Shear stress and Bulk

Density.

Data obtained with Type A Pressure Cells.

COMPARISON OF REGRESSION COEFFICIENTS OF THE 6" m

97

TABLE 17

VERSUS BULK DENSITY RELATIONS AT DIFFERENT
RATES OF LOADING AND BETWEEN METHODS FOR A GIVEN TEST

 

 

 

 

 

t t
Type 6 [ Test Rate
Tests D.F. A DST No. (inlsec ) D.F. t
. 5 Inches below Loading Surface
2-20 71 1.35 0.50 5 0.38 56 5.32**
2-5 61 0.24 1.63 2 0.62 66 5.60**
5-20 66 - 1.83 0.96 20 1.00 76 3.30**
14-23 31 4.99** 4.19** 14 1.00 36 2.50*
14-26 41 4.82** 4.66** 23 0.62 26 1.23
10 Inches below Loading Surface
4-19 56 1.35 ' 2.14* 19 1.00 76 4.09**
1-4 44 1.56 0.18 1 0.62 52 6.71**
1-19 64 4.51** 3.77** 4 0.38 36 1.58
13-22 35 0.37 0.26 13 1.00 44 1.90
13-25 45 4,24** 4.98** 22 0.62 26 0.82
22-25 36 4.33** 3.03** 25 0.38 46 2.08*
15 Inches below Loading Surface
27-21 36 6.30** 5.85** 27 1.00 36 4,22**
27-24 41 1.93 1.35 21 0.62 36 2.33*
21-24 41 4.25** 3.66** 24 0.38 46 2.41*

 

 

98

data. The differences for the other tests compared could
have been affected by the differences in moisture contents
and therefore, were not significant. Based on the data
of the three tests compared with approximately the same
moisture contents the conclusion that rate of loading
affects the relationship between mean normal stress and
bulk density at the 5-inch depth and high moisture contents
could be made.

For the relationships obtained from the 10-inch data,
differences between Tests 1-19, 13-25, and 22-25 for both
sets of data are significant. In addition, the difference
between Tests 4 and 19 is significant for the 6 DST data.
These differences are probably due to the difference in
moisture contents of the tests compared. The conclusion
that the rate of loading does not affect the relationship
between mean stress and bulk density at the 10-inch depth
could be made.

The results at the 15-inch depth show a significant
difference between the 1.00 and 0.62 inch per second rates
of loading and between the 0.62 and 0.38 rates. The
difference between the 1.00 and 0.38 rates is not signifi-
cant. The significant differences between the 1.00 - 0.62
and 0.62 - 0.38 rates could be due to the differences in
moisture contents. Based on this data the conclusion that
the rate of loading at the 15-inch depth affects the

relationship between mean normal stress and bulk density

99

could not be made.

A set of typical curves showing the results of Tests
5, 2, and 20 are shown in Figures 33 and 34.
The Effect 2£_§§tg.g£_Loading 2n the Relationship between

 

 

Second Invariant and Bulk Density

 

The results of the "t" tests between regression
coefficients for the second invariant-bulk density relation-
ships are presented in Table 18.

The comparisons for the 5- and 10-inch depths revealed
results similiar to those for the mean stress relationships.
The results for the 10-inch depth show significant differ-
ences between Tests 13-25 and 22-25 for both the Type A
and 6 DST data. Differences between Test 4-19 and 1-19
are significant for the Type A data.

A set of typical curves are shown in Figures 35 and 36.
The Effect 2£.§§£2.2£ Loading 2n the Belationship between
Maximum Normal Stress and REIT Density

 

Comparisons between the regression coefficients
Obtained from the regression equations of the maximum
normal stress-bulk density relationships are shown in
Table 19. The results for the 5-inch depth show significant
differences between the Type A data for tests 14-23, and
14-26. Difference between Tests 5-20 and 14-23 are signifi-
cant for the 6 DST data.

For the 10-inch depth, the Type A data show that the

rate of loading affects the relationship between maximum

100

 

 

 

 

 

 

 

 

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BULK DENSITY

Figure 33 .

The Effect of Rate of Loading on the

Relationship between Mean Stress and Bulk Density.

Data obtained with 6 DST.

PSI

MEAN STRESS

 

101

 

 

T ‘- 3. 1. I
i i =-

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BULK DENSITY (SM/CC

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i
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Figure 34.. The Effect of Rate of Loading on the

Relationship between mean Stress and Bulk
Data obtained with Type A Pressure Cells.

Density.

COMPARISON OF.REGRESSION COEFFICIENT
VERSUS BULK DENSITY RELATIONS AT

102

TABEE 18

s or THE 113
DIFFERENT

RATES OF LOADING AND BETWEEN METHODS FOR A GIVEN TEST

 

 

 

 

 

‘JPW t t
Type 6 Test Rate
Tests D.F. A DST No. (in/sec) v.3. t
5 Inches below Loading Surface
5-20 66 1.61 0.96 20 1.00 76 1.14
2-5 61 0.11 1.87 2 0.62 66 2.31*
2-20 71 1.48 0.68 S 0.38 56 4.43**
14'23 31 4067** 3004** 14 1000 36 0054
14-26 41 4.87** 2.49* 23 0.62 26 1.06
23-26 36 0.61 0.98 26 0.38 46 2.20*
10 Inches below Loading Surface
1-49 44 1.73 0.28 1 0.62 52 7.28**
13-22 35 0.40 0.15 13 1.00 44 1.67
13-25 45 4006** 4018** 22 0.62 26 1.09
22-25 36‘ 3.98** 3.29** 25 0.38 46 2.18*
15 Inches below Loading Surface
2 -2 5 ,8 &* 6,18** 27 1.00 36 3.82**
23-21 21 3.9% 1.65 21 0.62 36 3.14**
21-24 41 3.95** 4.11** 24 0.38 46 2.66**

 

 

INVARIANT p312

SECOND

20

IO

103

 

  

 

  
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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LI LIZ 1.22 l.32
BULK DENSITY GM/CC

Figure 35 . The Effect of Rate of Loading on the
Relationship between the Second Invariant and Bulk
Density. Data obtained with 6 DST.

104

 

20

l-03
2-062
3-I.OO

EC TEST 2

7SEC TEST 20

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INVARIANT 9an

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I.I8

BULK DENSITY GM/CC

Pi e 35 . The Effect of Rate of Loading on the
Reggiionship between the Second Invariant and Bulk
Density. Data obtained with Type A Pressure Cells.

BULK DENSITY RELATIONS AT DIFFERENT RATES OF
LOADING AND BETWEEN METHODS FOR A GIVEN TEST

105

TABLE19
COMPARISON or REGRESSION COEFFICIENTS or THE 03 VERSUS

 

 

 

 

 

t t
' Lype 6 Test Rate
Tests D.F. A DST NO. (in/sec) D.F. t
5 Inches below Loading Surface
5-20 66 1.50 2.42* 20 1.00 76 0.74
2-20 71 1.98 0.80 5 0.38 56 3.50**
14-23 31 3021** 2044* 14 1.00 36 0.21
14-26 41 3.65** 1.14 23 0.62 26 0.74
23-26 36 0.34 1.41 26 0.38 46 2.35*
10 Inches below Loading Surface
1-4 44 1.92 0.39 1 0.62 52 6.32**
1-19 64 4.44** 0.33 4 0.38 35 1.39
13-22 35 0.07 0.48 13 1.00 44 1.11
13-22 45 3.71** 3.34** 22 0.62 26 1.27
22-25 36 2.93** 3.58** 25 0.38 46 1.88
15 Inches below Loading Surface
27-21 36 5.18** 5.48** 27 1.00 36 3.09**
27-24 41 2 013* 1075 21 0062 36 3 .10**

106

normal stress and bulk density. The 6 DST data supports
the Type A data for the higher moisture contents but not
at the lower moisture contents.

The results for the 15-inch depth show agreement
between the two methods except for Tests 27-24. The Type A
data is significant, however, the 6 DST is not.

The results of Tests 1, 11, and 22 are shown in
Figure 37 and 38.

Eggthfect g£_§§tg'2£_Loading on the Relationship between
Maximum Shear Stress and Bulk Density

The results of the "t" tests are presented in Table 20.
They are similiar to the results obtained for the relation-
ship between maximum normal stress and bulk density. In
general, the conclusion that the rate of loading affects
the relationship between maximum shear stress and bulk
density at the higher moisture contents and lower depths
could be made.

A typical set of curves for this relation are shown
in Figures 39 and 40.

ggmparison 2£_Three Methods Used to Determine Mean Normal

 

Stress

The regression equations for the relationship between
mean normal stress and applied surface loads for the
calculated values Of mean normal stress Obtained from the
Type A and 6 DST data and the values measured directly

with the waell were determined. The results are shown

 

 

 

 

 

20

MAX NORMAL STRESS PSI

N

107

 

 

 

E5

0'

 

I-osejzssc TEST 5
2-O.62./'SEC TEST 2
3-l.00/SEC TEST 20

 

 

 

“Thwarl. --

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l.2|

BULK DENSITY GM/CC

Figure 37 . The Effect of Rate of Loading on the
Relationship between the Maximum Normal Stress and
'Bulk Density. Data obtained with 6 DST.

l.3l

 

MAX NORMAL STRESS PSI

108

 

I
I

l- 0.38/SEC TEST 5
2-0.62/SEC TEST 2
3- LOO/SEC TEST 20

 

 

 

 

  
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

L33

BULK DENSITY GM/CC

Figure 38 . The Effect of Rate of Loading on the
Relationship between the Maximum Normal Stress and
Bulk Density. Data obtained with Type A Pressure Cells.

AND BETWEEN METHODS FOR A GIVEN TEST

109

TABBE 20

COMPARISON OF REGRESSION COEFFICIENTS OF THE Tmax VERSUS
BULK DENSITY RELATIONS AT DIFFERENT RATES OF LOADING

 

 

 

 

 

 

t t
Type 6 Test Rate
Tests D.F. A DST No. (infieec) 12.3. 1:
5 Inches below Loading Surface
5-20 66 1.31 2.10* 20 1.00 76 0.39
2-20 71 1.64 4.16** 5 0.38 56 3.32**
14-23 31 3.49** 2.22* 14 1.00 36 0.08
14-26 41 3.75** 0.96 23 0.62 26 0.87
23-26 36 0.50 1.36 36 0.38 46 2.32*
10 Inches below Loading Surface '
1-19 64 4.75** 0.52 4 0.38 36 1.58
13-22 35 0.23 0.49 13 1.00 44 1.31
13-25 45 3.59** 3.51** 22 0.62 26 1.25
22-25 36 3.02** 3055** 25 0038 46 1093
15 Inches below Loading Surface
- .08** 5.83** 27 1.00 36 3.02**
S;—§4 i? g.85 1.80 21 0.62 36 3.4?**
21-24 41 3.54** 4.27** 24 0.38 46 2.74**

 

PSI

MAX SHEAR STRESS

110

 

 

 

 

 

 

 

 

 

 

 

 

 

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1.06 “6 I26

BULK DENSITY GM/CC

Figure 39 . The Effect of Rate of Loading on the
Relationship between the Maximum Shear Stress and
Bulk Density. Data obtained with 6 DST.

 

 

111

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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BULK DENSITY

(NM/CC

The Effect of Rate of Loading on the
Data obtained with Type A Pressure

40.

Figure

Relationship between the maximum Shear Stress and

Bulk Density.

Cells.

 

 

in Figures 41, 42, 43, 44, and 45. Theoretically, the
relationships should go through the origin, however, as
the curves show this was not true. The sum of least
squares method of obtaining the best predicting straight
line for the points was used to permit the use of a
statistical method for comparing the three methods.

The "t" test was applied to the regression coefficients
to check for significant differences and the results are
presented in Table 21. The most consistent results were
obtained between the Type A Cells and the W Cell. The
differences between the Type A and 6 DST data were the

most varied.

TABLE 21

COMPARISON OF THE REGLESSION COEFFICIENTS CALCULATED FOR THE
RELATIONSHIP BETWEEN MEAN NORMAL STRESS AND APPLIED ICLI
FOR A CONSTANT RATE OF LOADING OF 1.00 INCH PER SECOND.

 

.—__m

‘—

 

t t t
Test Depth M.C. Type A Type A 6 DST
No. In. a 6 DST W Cell W Cell D.F.
14 5 17.41 4.07** 1.22 0.33 36
20 5 10.79 3.57** 1.36 1.26 76
13 10 17.41 0.06 0.65 2.75** 44
19 10 10.95 2.72** 2.82** 0.81 76
18 15 12.31 0.44 1.32 3.90** 76

 

 

 

 

113

 

 

 

 

     

 

 

 

 

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I2 ~--— I- TYPE A CELLS I “MM—""1
2‘ GUST ‘
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APPLIED LOAD PSI

Figure 41. Comparison of the Regression Lines for
the Three Methods used to Determine Mean Stress.
Data obtained at the 5-inch Depth, 1.00 in/sec Rate
of Loading and a Moisture Content of 17.41%.

114

 

   
       

 

 

 

'4 I I I I0 9 e
I I 3 i
TEST 20 i I
I2 ~———- I- TYPE A CELLS -—-—~f-—~-~—- —-—-—---+
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APPLIED LOAD PSI

Figure 42 . Comparison of the Regression Lines

for the Three Methods used to Determine Mean Stress.
Data obtained at the S-inch Depth, 1.00 in/eec Rate
of Loading and a Moisture Content of 10.79%.

115

 

 

 

 

 

 

I4 I I I 1‘
I I I I
TEST I3 I
I2 ~— I—TYPE A CELLS --» ”+- w
2- GDST I
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APPLIED LOAD PSI

31 e 43 .
Data obtained at the 10-1

of Loading and a Moisture Content of 17.41%.

Comparison of the Regression Lines for

gur termine mean Stress.
the Three l9th°d8 used ‘°n2§ Depth, 1.00 in/sec Rate

116

 

 

 

 

 

 

 

l4 * I I
I I ‘ e2
TEST I9 I ;
I2 I- TYPE A CELLS ~—~—--I_.--_--.-_-.--.--_I ___._
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APPLIED LOAD PSI

Figure 44. Comparison of the Regression Lines for
the Three Methods Used to Determine Mean Stress.
Data obtained at the 10-inch Depth, 1.00 in/sec Rate
of Loading and a Moisture Content of 10.95%.

117

 

 

 

 

'4 I I I

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TEST I8 ; I

I2 A— l— TYPE A CELLS MIMW..-

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APPLIED LOAD p3.

Figure 45. Comparison of the Regression Lines for
the Three Methods used to Determine Mean Stress.
Data obtained at the 15-1nch Depth, 1.00 in/seo Rate
of Loading and a Moisture Content of 12.31 %.

 

 

I
e
i
I
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118

On the basis of this data the measured values of
mean stress obtained with the W Cell will be similiar
to those calculated with the Type A method.
The Effect 2£_gggg_2f_Loadins 22 the Relationship between

Mean Stress and Applied Load

 

To determine the effect of the rate of loading on the
relationship between mean normal stress and applied surface
load the average values of mean stress for the S replications
were plotted versus applied load. The curves shown in
Figures 46 through 49 show, in general the results obtained.

The curves persented in Figures 46 and 47 are for 5
inches below the surface and moisture content ranging from
8.89 to 10.79 percent. This difference in moisture content
could contribute to part of the difference obtained. The
differences between the 1.00 inch per second and 0.62 inch
per second are similiar for the two sets of curves. There
is only 0.90 percent difference in moisture content. The
difference between the 0.38 inch per second rate and 1.00
inch per second rates is not as consistent, which could be
due to the difference in moisture of 1.90 percent. In
general, these results are the same at the 10— and 15—inch
depths within the range of moisture content from 8.89 to
12.31 percent.

The results at a moisture content ranging from 16.05
to 17.67 percent and a depth of 5 inChes are shown in

Figures 48 and 49. The two sets of curves are in very

 

 

 

 

 

119

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u 0.38':/SEo TEST 5

o O.62'{SEC TEST 2

:2 e—Alod/SEC TEST 20 fi/

 

 

 

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MEAN STRESS
(D
T
I

 

 

 

 

 

 

 

 

 

 

 

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O 5 I0 I5 20 25
APPLIED LOAD PSI

Figure 45 . The Effect of Rate of Loading on the
Relationship between lean Stress and Applied Load
at 5-inch Depth. Data obtained with 6 DST at a
Moisture Content Ranging from 8.89 to 10.79%.

120

 

 

 

 

 

 

 

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n deer/SEO TEST 5 f
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I2 ~—A nod/SEC TEST 20 /
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l0 IS 20 25
APPLIED LOAD PSI

Figure 47. The Effect of Rate of Loading on the
Relationship between mean Stress and Applied Load at
S-inch Depth. Data obtained with Type A Cells at a
moisture Content Ranging from 8.89 to 10.79%.

121

 

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0.83 8'/SEC TEST 26
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APPLIED LOAD PSI

Figure 48. The Effect of Rate of Loading on the
Relationship between Mean Stress and Applied Load at
S-inch Depth. Data obtained with 6 DST at a Boisture
Content Ranging from 16. 05 to 17. 67%

122

 

 

 

 

 

 

 

 

 

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IO I5 20 25
APPLIED LOAD PSI

Figure 49. The Effect of Rate of Loading on the Relationship
between Mean Stress and Applied Load at 5 inches Depth. Data
obtained with Type A Cells. At a Moisture Content Ranging
from 16.05 to 17.67%.

123

good agreement. In general, these results are similiar
to those at the 10- and 15-inch depth for 15.78 to 17.93
.percent moisture contents.

Based on this data, the conclusion is that for a
given applied load, the lowest values of mean stress will
be obtained for the 1.00 inch per second rate of loading.

The Effect 2: Moisture Content 2n Relationship between

 

 

Mean Normal Stress and Applied Lgad

 

To determine the effect of moisture content on the
relationship between mean normal stress gand applied
surface loads, the averages of 5 replications of data
from the 6 DST and Type A cells were plotted. The results
at the 5-inch depth for the 3 rates of loading are shown
in Figures 50 through 53.

For the 0.38 inch per second rate (Figures 50 and 51)
the results are approximately the same for the two methods.
The curves show a difference between the highest and lowest
moisture content. The "t" tests between the highest and
lowest moisture contents for the relationship involving
bulk density were also significant. These results are
representative of results for the 10- and 15-inch depths.
The conclusion that for a given load the highest values
of mean stress would be obtained for the high moisture
content could be made.

The results at the 0.62 inch per second rate (Figures

52 and 53) are similiar for both sets of curves. The

124

 

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- /5

'2 ____0 use TEST 7
A I605 TEST 2e

 

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. I I
o _ I I l
O 5 IO I5 20 25
APPLIED LOAD PSI
Figure 50. The Effect 1’ M i tur 1:
Relationship between Idea: Stgegs azdcxgplrfzdogozlde

at S-inch Depth and 0.38 in/sec Rate of Load
Data obtained with 6 DST. ing.

 

125

 

 

    

 

  

 

 

 

 

 

 

  
 

 

 

 

 

 

 

I4 + +
I I I
a see TEST 5 f
'2 o ”.35 TEST 7 I ___.__.__
A I605 TEST 26
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O 5 IO I5 20 25
APPLIED LOAD PSI
Figure 51 . The Effect of Moisture Content on the

Relationship betwee
S"inch Depth and 0.
obtained with Type

n Mean Stress and Applied Load at
38 in/sec Rate of Loading. Data

A Cells.

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I- ' ¢ -
(D 6——————---.—--+~ nuw-w ”-4;- —— «g --—«+
I I
z I i° I
<1 I i
In . I
E .. a. _ m- 4 ,_.___.....__
I I
O 5 IO I5 20 25
APPLIED LOAD PSI
Figure 52 . The Effect of MOisture Content on the

 

126

 

 

 

 

 

 

 

 

 

 

Relationship between mean Stress and Applied Load
at 15 Inches Depth and 0.62 in/sec Rate of Loading.
Data obtained with 6 DST.

127

 

CI ”.59 TEST 3
'2 ___0 I263 TEST IO 1
A|7.93 TEST 2| I

I I
I I
I() I .WHI.-_
I
I
I
I

I 4 I I T I
I
i I

 

 

   

 

PSI

I i
£3___.lim-u-wla_l-““w__n,,

 

I

 

MEAN STRESS

 

 

 

 

 

 

 

 

o 5 IO I5 20 25
APPLIED LOAD PSI ,

Pi e 53 . The Effect of Moisture Content on the
Refgzionship between mean Stress and Applied Load
at 15 Inches Depth and 0.62 in/sec Rate of Loading.

Data obtained with Type A Pressure Cells.

128

results, in general, are the same for the 10- and 15-inch
depths. The maximum value of mean stress that could be
developed for the 17.93 percent moisture content was less
than 5 pounds per square inch. The differences between the
highest and lowest moisture contents appear to be signifi—
cant.

Figures 54 and 55 show the curves for the 1.00 inch
per second rate. The 6 DST and Type A curves are in good
agreement. The results indicate little or no effect of .
moisture content on the mean stress-applied load relation.
In addition, they represent the pattern at the 10- and 15-
inch depths. The conclusion that moisture content within
the range used for these tests has little or no effect on
the relationship between mean stress and applied load at
the 1.00 inch-per—second rate of loading could be made.

Comparison 2£_Theoretical_values 2£_Vertical Stress with

 

Measured values

 

The vertical normal stress produced in the soil by
a loading plate can be determined with a semi-empirical

equation developed by Froehlick as reviewed by Soehne (1957).

The equation is as follows:

0. = —2-—g§—-cos(c+2) O (30)
21T'z
where:
C = concentration factor

Z = distance below loading plate

 

 

 

 

l4

l2

IO

PSI

MEAN STRESS
m

49%“ .. a a ___.._...I
I :
2 IMMIMW—I
I I
I I I
o I I I
o 5 IO I5 20 25

129

 

 

 

I ' I
I I I
a loss TEST I9 I
___0I4.85 TEST I6 I
AI7.4| TEST l3 I
I I
I I
- I
I
__....._- HIM m I--- _. m
I
I... i. WI awn”; - M. ”howls

 

 

 

 

 

 

 

 

 

APPLIED LOAD PSI

Figure 54 . The Effect of Moisture Content on the
Relationship between Mean Stress and Applied Load
at 10 Inches Depth and 1.00 in/sec Rate of Loading.
Data obtained with 6 DST.

 

 

 

130

I4 ‘ ' I
I I I I
u l0.95 TEST l9 I '
OI4.85 TEST l6

IZWA I7.4l TEST I3
I

 

 

 

I
I

PSI

 

 

MEAN STRESS

 

 

 

 

 

\DOIC
I'—“_""'~ ._ __

I
0 5 IO IS 20 25
APPLIED LOAD PSI

Figure 55 . The Effect of Moisture Content on the
Relationship between Mean Stress and Applied Load
at 10 Inches Depth and 1.00 in/sec Rate of Loading.
Data obtained with Type A Cells.

 

 

 

131

6

polar coordinate

Q = force applied.

The average measured values of the vertical stress
for the 5 replications at the 5, 10, and 15-inch depths
at a moisture content of 14.34, 14.85 and 14.85 percent
and the 1.00 inch per second rate of loading were plotted
versus applied surface loads. Theoretical values were
calculated from equation (30) using concentration factors
to obtain curves corresponding to the measured values.
The results are shown in Figures 56, 57 and 58.

The curves in Figure 56 are for the S-inch depth.
The theoretical curve was the maximum that could be
obtained with any concentration factor. The value used
was two. Both the measured values are greater than the
theoretical. The differences may be due to the influence
of the instruments on the stress pattern near the surface.

Results for the 10-inch depth are shown in Figure 57.
With a concentration factor of six the theoretical and
Type A values are in good agreement at the lower loads
and only a 10.35 percent difference at the highest load.
The 6 DST values are lower than the other two for loads
less than 7.5 pounds per square inch and are increasingly
greater with increasing load. The highest difference
between the Type A and 6 DST values is 31.78 percent and
19.34 percent between the theoretical values and 6 DST

values.

 

 

 

132

 

28 I I I

 

 

D THEORETICAL
24 A TYPE A CELL L l
' O 603T I I
I I . A
5 . I I
CI20”“ ~ ,5 y * ~ -- Ins-mm- ------,--—--—---4
I A I

63

VERTICAL NORMAL STRESS
co F6

 

 

 

 

 

I

j . I

o . I I I

O 5 IO I5 20 25
APPLIED LOAD PSI

Figure 56 . Comparison of the Two Instruments Used
to measure the Vertical Stress with Theoretical values
at a Depth of 5 Inches below the Loading airface.
Measured Data obtained from Test 17.

 

 

 

133

 

 

 

 

 

 

 

 

 

 

 

28 I I I
D THEORETICAL I
24 ATYPE A CELL I
0 COST . I” /
I ' I I o
.. I ‘ I A
<0 I I I
Cl-20+---~~-- —- .»--- ~ I « I — ~+- ~—-«-~«~
I I I II
CD ' ' I o I
an s I A I
5 I6I-— , i I - {WWW--
i I O _ n I
5 I I I
012W. A I u ”MI
2 I I ' I
.J I 0? I L;
a I r I
-,; 8L» . e- .._.I
a: I I I I I
3’ . 3 I I
+. , I --- -m- --I.- --...._..._.I.
o I I
O 5 IO I5 20 ' 25

APPLIED LOAD PSI

Figure 57 . Comparison of the Two Instruments Used
to measure the vertical Stress with Theoretical
values at a Depth of‘KDInches below the Loading

Surface. Measured Data obtained from Test 16.

134

 

 

 

 

 

 

28 I I I
E] THEORETICAL
24 ____,.,“__A TYPE A CELL ~ I
O 6DST ”"‘*T”“"‘“‘”““ I
I I 3 I
I T '
<75 I I I
0'20 _._- I I “ " ‘I
(n o
(O
I?
:7) I6 “““““ I —- ~ 4
.J o A
q B
2 .
5 I2 ~—— /i? I - -I
2 'o
_, /3
8 o/
I: 8*“ A“ ‘ I * ““
m I ;
Lu ’ I
> , 5 I
I -o , _
I
A . I
I I g
IO IS 20 25

 

APPLIED LOAD PSI

Figure 58 . Comparison of the Two Instruments Used
to Measure the Vertical Stress with Theoretical Values
at a Depth of 15Inches below the Loading mrface.
Measured Data obtained from Test 15.

The curves for the 15-inch depth are shown in Figure
58. The theoretical and Type A curves again are in good
agreement for a concentration factor of four. The maximum
percent difference is only 7.75, which occured at the
highest load of 21.6 pounds per square inch. The values
obtained with the 6 DST are lower than the theoretical
and Type A values for loads less than 12.0 pounds per
square inch. Above 12.0 pounds per square inch the
differences are greater with increased loads. The maximum
difference between the Type A and 6 DST values is 27.55
percent as compared with 36.68 percent between the 6 DST
and theoretical values.

On the basis of this data the conclusion that both
instruments gave higher values than the theoretical values
for loads greater than 12 pounds per square inch could be
made. In addition the 6 DST gave measured values greater
than the Type A values for loads greater than 12 pounds
per square inch.

The differences between the theoretical values and
measured values could be due to the differences in their

volumes which may interfere with the stress distribution

in the soil mass.

CONCLUSIONS

In the loose soil used for the experimental tests,

data presented indicate the following.

1.

3.

The data obtained with the Six Directional Stress
Transducer were more varied than that obtained
with the Type A Cells.

The hypothesis that changes in bulk density are
controlled by mean normal stress cannot be accepted
or rejected.

The data obtained for both methods do not support
completely the part of the hypothesis that the
second invariant, maximum normal stress and‘
maximum shear stress are not related to changes

in bulk density.

The maximum shear stress was best related to
changes in bulk density.

For the 0.38 inch per.second rate of loading,

a range of moisture content from 7.97 percent to
16.16 percent had no effect on the relationship
between mean stress and bulk density.

The moisture content had an affect on the,-
relationship between mean normal stress and bulk
density for the 0.62 inch per second rate of
loading at depths of 10 and 15 inches below the

10.

11.

12.

137

loading surface and the 1.00 inch per second
rate at the 15 inch depth.

The relationships between the invariants and
bulk density at the higher rates of loading

at the 15 inch depth were affected by the
moisture content of the soil.

The mean stress-bulk density relationship was
not affected by the rate of loading at the three
depths below the loading surface.

The values of mean stress obtained directly with
the w Cell compared best with the values calculated
from the Type A data.

The lowest values of mean stress produced in the
soil mass for a given applied load occured under
the 1.00 inch per second rate of loading.

The vertical stresses measured with the Type A
Cell were in good agreement with values calculated
with Froehlick's equation at the 10- and_15-inch
depths. .

The relationships between the invariants and
bulk density were exponential.

REFERENCES

Anonymous (1958). Soil compaction committee report. Joint
American Society of Agricultural Engineers and Soil Science
Society of American Committee Report. Agricultural

Engineerin539z173-176.

Bekker, M. G.(1960). mechanical properties of soil and
'compaction problems. Unpublished A.S.A.E. Paper No. 60-126.
St. Joseph, Michigan.

 

Bekker M; G.(1956). Theo of Land Locomotion university
of‘Michigan Press, Ann Iffior.‘520fipp. ’

Cooper, Arthur W.(1956). Investigations of an instrumentation
for measuring pressure distributions in soils. Thesis for
degree of Ph.D., Michigan State Univ., East Lansing,
(Unpublished).

Cooper, A. W., G. E. vanden Berg and H. F. McColly (1957).
Strain gage cell measure soil pressure. Agricultural
Engineering 38:232-235, 246.

 

Gill, William R.(1959) Soil compaction by traffic.
Agricultural Engineering 40:392-394. 400.

Hendrick, James G.(1960). Strain energy and tensile strength
of a dynamically loaded clay soil. Thesis for degree of
M.S., Auburn Univ., Auburn, Alabama (unpublished).

Hough, B. K.(1957). Basic Soils Engineering. The Ronald
Press 00., New York. 513 pp.

Hovanesian, J. F.(1958). Development and use of a volumetric
transducer for studies of parameters upon soil compaction.
Thesis for degree of Ph.D., Michigan State Univ., East Lansing
(Unpublished).

Hovanesian, J. D., and W. F. Buchele (1959). Development
of a recording volumetric transducer for studying effects
of soil parameters on compaction. Transactions, A.s.A.E.
2:78-81.

Lask, Kay V.(1958). Instrumentation and measurement of
soil-tire contact pressures. Thesis for degree of M.S.,
Michigan State Dniv., East Lansing (Unpublished).

139

Lee, G. H.(1950). Ag Introduction to E1 erimental Stress
Analysis. John Wiley andfgbns, Inc., ew ork.

Malvern, L. E.(1957). Introduction to the mechanics of a
continuous medium. unpublished notes for graduate course
in Applied mechanics Dept., Michigan State Univ., East
Lansing. ‘ .

murnaghan, J}(1951). Finite Deformation g£_§g Elastic
Solid. John Wiley and Sons, Few YorE.

Perry, C. C. and H. R. Lissner (1955). The Strain Gage
Primer. The maple Press 00., York, Pa. 281 pp.

Popov, E. P.(1958). mechanics 2; materials. Prentice—
Hall Civil Engineering and EEgineeFIfiE'flEEEanics Series,
Englewood Cliffs, N. J.. 441 pp.

Reaves, C. A. and A. W. Cooper (1960). Stress distribution
in soils under tractor loads. Agricultural Engineering
41:20-21, 31.

Reavcs, C. A. and M. L. Nichols (1955). Surface soil
geaction to pressure. Agricultural Engineering 36:813-816.

Redshaw, S. C. (1954). A sensitive miniature pressure.
JOur. 3: Sci. Inst. 31:467-469.

Reed, I. F., A. W. Cooper and C. A. Reeves (1959). Effects
of two-wheel and tandem drives on traction and soil
compacting stresses. Transactions, A.S.A.E. 2:22-25.

Rosenbach, Joseph B., Edwin A. Whitman and David Moskovitz
$1243). 21223 Trigonometry. Ginn and Company, New Ybrk.
PP.

Snedecor, George W. (1956). Statistical Methods. 5th ed.
The Iowa State College Press, Ames. 534 pp.

Soehne, W.(1958). Fundamentals of pressure distribution
and soil compaction under tractor tires. _Agricultural
Engineering 39:276-281, 290. . .

Soehne, w. H., w..J. Chancellor and R. H. Schmidt (1959).
Soil deformation and compaction during piston sinkage.
unpublished A.S.A.E. Paper NO. 59-100. St. Joseph, Michigan

140

Stong, Jack V.(1960). Basic factors affecting the strength
and sinkage of tillable soils. Thesis for degree of m.S.,
Michigan State Univ., East Lansing (Unpublished).

Terzaghi, x.(1943). Theoretical Soil mechanics. Jehn Wiley
and Sons, Inc., New YorE. 515 pp. ‘—

 

Timoshenko S.(1936). Theo§y of Elastic Stability. MCGraw
Hill Book 60., new YOrk. p57

Timoshenko, s. and J. N. Goodier (1951).' Theor of
Elasticity. McGraw Hill Book Co. 2nd ed. New York?

Trabbic, Gerald W.(1959). The effect of drawbar load
and tire inflation on soil-tire interface pressure.
Thesis for degree of M.S., Michigan State Univ., East
Lansing (Unpublished).

Trabbic, G. W., K. V. Lask and W. R. Buchele (1959).
measurement of soil-tire interface pressures. Agricultural
Engineering 40:678-681.

Vanden Berg, Glen E.(1958). Application of continuum
mechanics to compaction in tillable soils. Thesis for
degree of Ph.D., Michigan State Univ., East Lansing
(Unpublished).

Vanden Berg, G. E.(1960). Requirements for a soil mechanics.
Unpublished A.S.A.E. Paper No. 60-127. St. Joseph, Michigan.

vanden Berg, G. E. and W. R. Gill (1959). Pressure
distribution between a smooth tire and soil. unpublished
A.S.A,E. Paper No. 59-108. St. Joseph, Michigan.

Weaver, H. A. and V. C. Jamison (1951). Effects of
moistuge on tractor tire compaction of soil. Soil Science,
1:15- 3. -

Willits, Nathan A. (1956). measurement of pressures in
soils produced by traffic and the relationship between
those pressure and compaction in undisturbed soils.
Thesis for degree of Ph.D., Michigan State Univ., East
Lansing (unpublished).

141

Wilson, S. D.(1952). Effect of compaction on soil
properties. Proc. of the m.I.T. Conference on Soil
Stabilization, 84-161 pp.

APPENDIX

TABEE A.

CALCULATED VALUES OF MEAN NORMAL STRESS, SECOND INVARIANT OF DEVIATOR

MAXIMUM AND MINIMUM PRINCIPAL STRESSES, AND MAXIMUM SHEAR STRESS

ROR THE TYPE A CELLS AND THE S

IX DIRECTIONAL STRESS TRANSDUCER

STRESS TENSOR,

 

 

Bulk

Load Density

6 DST
0?.

(psi) (psi)‘(psi) (psi)-(psi) (psi)

Type A Cell
V3:

12::

(psi) (em/cc) (psi) (psi)’(psi) (psi) (psi) (psi)

“Z

 

93

3

V7»

 

 

0:...

.22:

 

TEST 1

NONmeP
s s s s s s

NOOWF‘G‘WO
FFNNN

d'mkol‘COPM
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v-s-NMO‘PN

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£11

TABLE A (CONTINUED)

6Dfl

 

 

 

 

fink
Load Density

 

 

 

(psi) (gm7cc3 (psi) Tpsi

 

Sufi“) OWN ““0 ‘-

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TEST 2

144

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P

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1229 A Cell

59

 

fink
Load Density

 
 

(psi) (psi) psi

(psi)‘?5éi)

 

(£51) (sm7éc) (psi) (psi ‘ psi

 

  

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TABLE A (CONTINUED)

Wk

1%

 

a;

Mfié

iifi

 

6 DST
_:I__£E___!Eé_

(psi) (p91) (psi) (psi) (psi) (psi)

 

T'

Tyge A Cell

a.

Wk

 

Rflk

TEST 3

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TABLE A (CONTINUED)

 

 

T e A Cell

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O'OOPPFNM

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o PNMMVVWO

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o r-q-comwrom
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FPNNNNNN

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.

(psi) (psi)‘(psi) (psi) (psi) (psi)

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aoonv-oomm

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(psi) (sm/bc) (psi) (psi)‘(psi) (psi) (psi) (psi)

 

aux
Load Benoit

MMMNON ooxotmnund-m mMNd'mb-M
0.000 0000000 '0000000
OPvamm OPNMQ'UMO (>vaQO
mMNNNOa) bmmv-bnnn PO‘PKDK‘CDN
0.00000 00000.0 .000...
OPNMd-u‘un OPNMMvm OOPNNmm
OCOIANNMV' ObmmNNO meNOOF
0...... 0.00.. 0......
PPNMQ‘MKD PPNMQ'IAKD OPNMd'mm
nmNb-moxo M\OU\N€DMM \OMOMMNQ
.0... 0.00000 ~0000000
Nd'h-O‘Nmb: Nd-b-ONtnw wav-d'l‘v'
PFP PPv-I- PPv-N
O‘b-OPOV'O" PONtov-O‘b: PNFPMQQ
0...... 000 '0000000
d'M M OMQIDH‘3K3M PWNMPd'
oanNfid’ PNMIn v-qun
P
MmOVro‘tO‘ MonOO‘MO‘ GNOOFr-bq-
000.000 0 0.. 000.000
PN‘nmbwm Fmémwwm v-Nvunb-aog
mPNNMMM Inks-(uncut mmopmm‘.
PNNNNNN v-PNNNNN PFNNNNN
000000 .0000 0.00.00
PPFPPPF PFPv-I-v-v: PFPPPPP
«wwwnvms Ntasetea ”twaetoa

O
“éd&&fiéw pmmmwag nnmmNmmg

TABLE A (CONTINUED)

7

Y6DST

r

(ps1) (ps1)’(ps1) (ps1) (ps1) (ps1)

.un.

 

'3

2 (em/so) (ps1) (ps1)‘(ps1) (ps1) (ps1) (ps1)

A

re ‘A Cell

I

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(psi

 

 

 

711111:
Load Benoit

POiOOan-P

ONMU‘QO"
PF

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OOPFPNM

OCVNNMO‘

PNNMMQQ

orbvth

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PPNN

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Od'lnomd'w
PQ'CONIA
PF

NV'NPOQP

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22

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PP

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P

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o "NMMV’M

CDCDNNONK~

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PFC-P

TEST 10

#oonmmmr

00......

OPNMVWOQON

NQNN P‘OW"

OOOFFPFN

MO‘U‘OW PG) N

00......

OOFNNMMQ'

PQQQPWNV’

PN #FOQ‘Q"

PPPN

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o o o o o o o o
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PNMQ"

P

mv-Q-l‘ko #0101

or-NMVWDI‘O‘

QKMOOV'MMG'

QPNM‘WQWP

WOMNMK‘PV’

OPFNNNMM

PNOQ‘ONd'OQ

OPPNNMd'Q'

PMWPPMmN

N #00 :MU‘Q

CDMWMQMF’P
o o o o o o o o
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r—

 

0"

fat-Q1:

(psi) (gm/6c) (psi) (ps1)’(pai) (psi) (psi) (psi)

(psi) (psi)‘ (psi) (psi) (psi) (psi)

 

 

 

.MK‘WOO‘MNWQ

OPNMIDC‘QO‘

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00......

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PFPN

MQC’NOV‘OO‘M

ONP-QMNQK‘
meom"
P

MMWFOU‘CQM

OPNMlnmPO‘

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00......

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F

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m

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if

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T2.

(psi) (psi)’(psi) (psi) (psi) (p81)

 

SE

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max

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'qu;

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mommwemw

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a:

TABLE A (CONTINUED)

 

 

 

6 DST

 

“Zn-‘1

’(p81) (p81) (psi) (psi)

 

e A Cell
::

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Load Dena

 

(psi) (gm/be) (psi) (p81)

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P

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TABLE 11(CONTINUED)

I:
r

 

 

6 DST

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T

 

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3

TEST

Reg1 1

NPNPFPP

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PNQ'IOQO

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160

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NIn MOPMOIA

md-Q'PNMM
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v-rmnb-mv-m
1-1-

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PNMMQQQ

\OMMPOQU‘

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{-meth

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ONIOONFQN

NIOOMIOOQ
PPFNN

o o o O O O 0
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NOb-FOPN

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v-v-v-

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O‘QOMNPP

0......

MQMQIOQO
FFNNM

OQIfiIfiMIfiC

o o o o o o e
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o FNMV’U‘Q

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FPFFP
O O O O O

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1.08

thNMd’mQ

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TABLEJA (CONTINUED)

 

 

6 DST

’V33"iEZZII:IEEZIIJEEIIZEEEZIIZEEEt

eACflJ

Bulk

Load Deneit

-:_

 

r

(p81) (em/be) (psi) (pai)‘(p81) (psi) (psi) (P81)

 

 

(psi) (p81)’(pai) (psi) (p81) (psi)

PPFNV’NN
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P‘-

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QQMMPWN

0......

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1-1-

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o o e o O O O
v-r-r-q-v-PP

d-PNNMQ'IDIO

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gd'flID O‘NIOQ ‘-
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161

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«unborn-PO)

PNMIAIOP'F

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o

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0 O O O C C C
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TEST 13

unnamed-tn

ONMVQD-

\OOQNQ’V

onmmw

QCMIOOIn

v-mhomoo
PF!-

QOQPNP

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Fount-

(Dov-mood)

.00...

O NMV'IO'Q

TABLE A.(CONTINUED)

J
-_‘

 

 

 

 

_:
f

6 DST

e A Cell

psi) (psi)‘(psi) (psi) (psi) (psi)

Bulk

Load Density

1: V—- 9

(psi) (psi)‘(p81) (psi) (psi) (psi)

E3.

 

 

 

(psi) (em/be) (

man-meow

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MMNQII‘O

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PP!-

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o o o o o o
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mm MMMV

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162

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vwnnnqmnv

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QPMMQI‘

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oommb-b-oo.

o PNMV'IO

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mmbkoko

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r-mmb-mm

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F

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xo¢mxomm

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ou'u-Ofl‘o'

chommm
v-v-

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O O O O O O
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PPNM

TEST 14

MID MQW

FNQ’MP

Q‘NFNQ

sséss

NIhPOM

0....

OPVmF

ummo

«macs-u)
P1-

InMQV'N

NPQ d’Q
v-rmn

QPNPO

ONV'InQ

OPIOWIn

PPNM¢

v-r-v-mb

OFNNM

MQ I‘QIO

FNMV’M

MQO‘NU‘
q) Pv-
0:3

TABLE A (common)

___._‘-l

 

 

—;
_:f

6 DST

M

Bulk

Load Density

T??—

 

 

11!

V31:

 

gzpe A Cell
02..

2’3

Vi

I

 

 

 

A

1'! \OPNIOO
m o o o o o
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v

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m o o e o 0
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A

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m o o e o o
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v

A

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v FF
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TEST 15

Qb-QQQQQ

OPMIAFQO‘

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passes:

VQV‘NIOV‘In

OOPNMQ'd'

v-(novor-ur

v-Mb-Oéb-O
FPPN

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meowo
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TmmEA(mmmmmm)

 

 

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TEST 21

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181