Citation
The particle motion field generated by the torsional vibration of a circular footing on sand

Material Information

Title:
The particle motion field generated by the torsional vibration of a circular footing on sand
Creator:
Heller, Lyman Wagner, 1928- ( Dissertant )
Schertmenn, John E. ( Thesis advisor )
Self, Morris W. ( Reviewer )
Nevill, Gale E. ( Reviewer )
Richart, Frank E. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1971
Language:
English
Physical Description:
xix, 270 leaves : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Electric potential ( jstor )
Erosional sand landforms ( jstor )
Footings ( jstor )
Oscilloscopes ( jstor )
Particle motion ( jstor )
Shear stress ( jstor )
Sine function ( jstor )
Transducers ( jstor )
Velocity ( jstor )
Vibration ( jstor )
Civil Engineering thesis, Ph. D ( lcsh )
Dissertations, Academic -- Civil Engineering -- UF ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
Over the past few years, it has been demonstrated that the self-excited vibratory motion of a circular footing on various types of soil can be successfully predicted by a mathematical model derived by assuming that the foundation soil is represented by a homogeneous elastic half-space. This finding suggested that the same model, or variations thereof, might be useful for predicting the particle motion generated within a soil foundation by a vibrating footing. The objective of this study was to test the hypothesized utility of the half-space model for predicting the motion field generated in a natural soil deposit by the forced torsional vibration of a circular footing. the test involved the computation of half-space motion, the measurement of soil motion, and a comparison of the computations to the measurements. A 5-ft-diam footing was vibrated at 5 different frequencies on a natural sand deposit with a shear modulus that varied from about 1,800 psi at a depth of 1 ft to about 23,000 psi at a depth of 35 ft. Resultant particle motions were measured on the footing and at radial distances to 90 ft and at depths to 35 ft. Homogeneous half-space particle motions were computed using a shear modulus of 9,480 psi. The average particle displacement, neglecting damping, was between 1/3 and 1/4 of the computed displacement. The measured displacements were 1/10 of the calculated displacements at the deeper locations and 3 times the calculated displacements near the ground surface. Near the footing, the displacements were in good agreement. Recent literature on the stress conditions in a nonhomogeneous elastic half-space suggested that the particle displacements in a homogeneous half-space could be used to determine the particle displacements in a nonhomogeneous half-space. When the sand deposit was considered as a nonhomogeneous half-space and damping was neglected, the displacements were in good agreement near the footing, the average measured displacement was 60 percent of the computed displacement, and the measured displacements were 1/4 to 2-1/2 times the computed displacements. the material damping effect on surface waves at a similar test site. The correlation between the ration of the computed displacement to the measured displacement and the cone bearing capacity of the sand deposit at various depths suggested that a more accurate and detailed determination of the shear modulus of the sand would improve the correspondence between measured and computed results. Because the accuracy of the particle displacement predictions was adequate to classify transmitted vibrations as either undetectable, readily apparent, or intolerable, the elastic half-space model, adjusted for nonhomogeneous site conditions, was considered a potentially useful analytical representation of a natural soil deposit subjected to torsional footing vibration.
Thesis:
Thesis (Ph. D.)--University of Florida, 1971.
Bibliography:
Includes bibliographical references (leaves 265-270).
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by Lyman Wagner Heller.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
003079561 ( ALEPH )
55540230 ( OCLC )

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Full Text















The Particle Motion Field Generated by the Torsional
Vibration of a Circular Footing on Sand














By

LYMAN WAGNER HELLER


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY









UNIVERSITY OF FLORIDA
1971
















ACKNOWLEDGMENTS


The author wishes to express his sincere appreciation and grati-

tude to his Supervisory Committee for their guidance, their faith, and

their infinite patience during the progress of these studies. The con-

tinued encouragement and counsel of Professors J. H. Schmertmann, Chair-

man, M. W. Self, G. E. Nevill, Jr., and F. E. Richart, Jr., are grate-

fully acknowledged. Special thanks are due to Professor Schmertmann

for directing the work, for his contributions, and for his key sugges-

tions. Particular thanks are also expressed to Professor Richart,

University of Michigan, for his many years of instruction and inspira-

tion, for his service on the Supervisory Committee, and for his guidance

and encouragement during this investigation.

Financial support of the investigation, provided by the Office of

the Chief of Research and Development, Department of the Army, through

the Office of the Chief of Engineers and the Administration of the U. S.

Army Engineer Waterways Experiment Station, is gratefully acknowledged.

The personal efforts and interest of Mr. A. A. Maxwell (deceased), who

was instrumental in initiating the general research task of which this

study is a part, are also acknowledged.

The author wishes to express his appreciation to the Commanding

Officer, Eglin Air Force Base, Florida, and his staff and to the Mobile

District Office, U. S. Army Corps of Engineers, Mobile, Alabama, for

the use of facilities and for field support efforts during the experi-

mental aspects of the study. Special thanks are offered to Mr. Leon










Leskowitz, U. S. Army Electronics Command, Fort Monmouti, New Jersey,

for his cooperation and assistance during the computational aspects of

the work.

Appreciation and gratitude is expressed to the many individuals

at the Waterways Experiment Station who assisted and contributed to the

prosecution of this study. Special thanks are extended to Mr. Monroe B.

Savage, Jr., and Mr. Jack Fowler for their capable and cooperative

assistance during the experimental work. Particular thanks are also

expressed to Miss K. Jones and her helpful staff at the Station's

Reproduction and Reports Office for preparing the reproducible copy

and photographs, and for printing the manuscript.

Finally, the author wishes to thank his wife, Elizabeth, and his

children for their patience and sacrifices during the course of this

study.


iii
















TABLE OF CONTENTS

Page

ACKNOWLEDGMENTS-------------------------------------- ii

LIST OF TABLES--------------------------------------------------- viii

LIST OF FIGURES------------------------------------------------- xi

LIST OF SYMBOLS------------------------------------------------- xiv

ABSTRACT-------------------------------------------------------- xviii

INTRODUCTION---------------------------------------------------- 1

Background------------------------------------------------- 1
Previous Work------------------------------------------------ 3
Related Work------------------------------------------------ 3
Approach to the Investigation--------------------------------
Available Theory-----------------------------------------
Available Experimentation-------------------------------
Comparisons-------------------------------------------- 7
Objective and Goals-- --------------------------------- 7

THE THEORETICAL PARTICLE MOTION GENERATED BY THE TORSIONAL OSCIL-
LATION OF A WEIGHTLESS, RIGID, CIRCULAR DISK ON THE SURFACE
OF AN ELASTIC HALF-SPACE--------------------------------------- 9

Homogeneous (Constant E) Elastic Half-Space------------------ 9
Problem Statement and Approach--------------------------- 9
Equations of Elasticity---------------------------------- 10
Solution to the Equilibrium Equation--------------------- 14
Boundary Conditions-------------------------------------- 15
Applied moment and disk rotation--------------------- 18
Particle Displacements----------------------------------- 19
Evaluation of the Infinite Integral---------------------- 20
Example calculation--------------------------------- 24
Computer Program to Evaluate the Integrals--------------- 29
Nonhomogeneous (Linear E) Elastic Half-Space----------------- 30
Literature----------------------------------------------- 32
Results of Gibson's Solutions---------------------------- 33
Solution for stresses-------------------------------- 33
Strain relationships--------------------------------- 34
Half-Space Under To-sion-------------------------------- 335
Torsional Oscillation------------------------------------ 36
Assumption------------------------------------------ 37
Particle motion------------------------------------- 38









Page

THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCIL-
LATING RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT-------- 40

Description of Test Site------------------------------------ 40
Geographical Location and Geological Setting------------- 40
Soil Exploration----------------------------------------- 41
Borings---------------------------------------------- 43
Penetration tests------------------------------------ 43
Laboratory Tests----------------------------------------- 48
Unit weight------------------------------------------ 48
Gradation-------------------------------------------- 49
Seismic Wave Propagation Tests--------------------------- 50
Design of the Experiment------------------------------------- 53
Foundation Design---------------------------------------- 53
Practical considerations----------------------------- 53
Diameter of the test footing------------------------- 56
Stresses at the footing-soil interface--------------- 57
Stresses near the periphery of the footing----------- 60
Footing emplacement operation------------------------ 63
Position of dead load on cured first pour------------ 66
Rigidity of the footing------------------------------ 68
Limiting torsional moment---------------------------- 72
Dynamic response of the foundation------------------- 73
Vibrator Design------------------------------------------ 74
Power requirements----------------------------------- 75
Frequency and moment capacity------------------------ 78
Foundation and Transducer Location----------------------- 80
Location of the test footing------------------------- 80
Location of transducers------------------------------ 81
Isolation of power and recording facilities---------- 84
Construction of Test Facilities------------------------------ 84
Foundation Construction---------------------------------- 84
Fabrication of the footing form---------------------- 84
Placing the form------------------------------------- 87
First pour of concrete------------------------------- 88
Backfilling------------------------------------------ 95
Second pour of concrete------------------------------ 99
Vibrator Construction------------------------------------ 104
Motor------------------------------------------------ 104
Mounting the vibrator-------------------------------- 104
Operating tests-------------------------------------- 106
Transducer Installation---------------------------------- 110
Performance tests------------------------------------ 110
Boreholes-------------------------------------------- 113
Transducer alignment--------------------------------- 113
Installing procedure-------------------------------- 118
Backfilling------------------------------------------ 118
Particle Motion Measuring System----------------------------- 122
Functional Components---------------------------------- 122
Transducers------------------------------------------ 125









Page

Cables----------------------------------------------- 128
Amplifiers------------------------------------------- 128
Oscillographs and galvanometers---------------------- 129
Reference (calibration) voltage---------------------- 129
System accuracy-------------------------------------- 132
Power generators------------------------------------- 133
Arrangement and Utilization------------------------------ 135
Arrangement of components---------------------------- 135
Utilization of system-------------------------------- 136
Typical vibration test data-------------------------- 140
Schedule of Tests-------------------------------------------- 147
Footing Settlement and Tilt------------------------------ 147
Transducer Operation------------------------------------- 147
Torsional Vibration-------------------------------------- 154
Compression Wave Propagation----------------------------- 155
Transducer and Cable Resistance-------------------------- 155
Results of Measurements-------------------------------------- 158
Footing Settlement and Tilt------------------------------ 158
Compression Wave Propagation----------------------------- 158
Footing source--------------------------------------- 158
Surface source--------------------------------------- 159
Propagation velocities------------------------------- 160
Particle Velocities Due to Torsional Vibration----------- 160
Amplitudes------------------------------------------- 160
Wave propagation velocities-------------------------- 166

COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS------------------ 169

Test of the Calculated Results------------------------------- 169
Solutions at the Surface of a Homogeneous (Constant E)
Elastic Half-Space------------------------------------ 169
Geometrical Damping Law---------------------------------- 171
Position of Disk and Footing----------------------------- 173
Test of the Measured Results--------------------------------- 173
Dynamic Footing Response-------------------------------- 173
Footing-Soil Contact Area-------------------------------- 174
Particle Motion Components------------------------------ 176
Properties of the Sand Deposit--------------------------- 179
Basis for Comparing Results---------------------------------- 182
Measured Motion------------------------------------------ 182
Computed Motion------------------------------------------ 184
Comparison of Normalized Displacements------------------- 185
Comparison of Results---------------------------------------- 185
Normalized (Constant E) Half-Space Displacements--------- 185
Normalized Soil Displacements--------------------------- 187
Ratio of Displacements----------------------------------- 189
Displacements in a Nonhomogeneous (Linear E) Half-Space-- 191
Discussion of Results---------------------------------------- 195
Homogeneous (Constant E) and Nonhomogeneous (Linear E)
Half-Space--------------------------------------------- 195
Characteristics of the Test Site------------------------- 197












Particle Motion Predictions ---------------------------- 200

CONCLUSIONS AND RECOMMENDATIONS---------------------------------- 201

Conclusions-------------------------------------------------- 201
Homogeneous (Constant E) Half-Space---------------------- 201
Nonhomogeneous (Linear E) Half-Space--------------------- 201
Experimental Aspects------------------------------------- 202
Test site-------------------------------------------- 202
Test footing and vibrator---------------------------- 202
Particle motion measuring system--------------------- 203
Results of measurements------------------------------ 203
Computations and Measurements---------------------------- 203
Recommendations --------------------------------------------- 205
Analytical Work------------------------------------------ 205
Experimental Work---------------------------------------- 205
Comparisons---------------------------------------------- 206

APPENDIX A CALCULATIONS FOR THE INTEGRAL I(ao,a,b)------------ 207

APPENDIX B SPECIFICATIONS FOR THE PARTICLE MOTION MEASURING AND
RECORDING SYSTEM----------------------------------------------- 256

Transducers-------------------------------------------------- 256
Three-Component Transducers----------------------------- 256
Single-Component Transducers----------------------------- 257
Cables------------------------------------------------------- 257
Amplifiers----------------------------------------- ---------- 258
Galvanometers------------------------------------------------ 259
Oscillographs and Paper-------------------------------------- 259
Paper Processor---------------------------------------------- 260
Reference (Calibration) Voltage Supply----------------------- 260
Voltmeter---------------------------------------------------- 261
Connections------------------------------------------------- 262
Resistance of Transducer Circuits---------------------------- 263

LIST OF REFERENCES----------------------------------------------- 265














LIST OF TABLES


Page

1. Values of the Common Terms in the Integrand of Il and
I2---------------------------------------------------------- 25
2 25

2. Values, f(a) of the Integrand of I ---------------------- 25

3. Values, f(a) of the Integrand of I2---------------------- 26

4. Values of the Terms in the Integrand of I ----------------- 28

5. Values, f(a) of the Integrand of I ---------------------- 28

6. Calculation Parameters for I(a ,a,b)----------------------- 30

7. Well Log at Auxiliary Field 5------------------------------ 42

8. Boring Log for Hole 1--------------------------------------- 45

9. Boring Log for Hole 2--------------------------------------- 46

10. Average Bearing Capacity of Static Cone Penetrometer ------- 47

11. Results of Laboratory Tests on Samples from Hole 3--------- 48

12. Identification Letter and Weight of Eccentric Masses-------- 79

13. Vibrator Moment Capacity at Various Frequencies------------- 80

14. List of Transducers, Locations, and Transduction Values----- 127

15. Transducers, Recorders, and Recording Sequence-------------- 137

16. Schedule of Torsional Vibration Tests----------------------- 155

17. Results of Footing Source Compression Wave Tests------------ 159

18. Results of Surface Source Compression Wave Tests------------ 160

19. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,380 ft-lb Oscillating at 15 Hz----------------- 161


viii









Page

20. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,502 ft-lb Oscillating at 20 Hz------------------ 162

21. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,470 ft-lb Oscillating at 30 Hz------------------ 163

22. Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,614 ft-lb Oscillating at 40 Hz------------------ 164

23. Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,818 ft-lb Oscillating at 50 Hz------------------ 165

24. Measured Arrival Time and Average Wave Propagation Velocity
for Vibration Tests----------------------------------------- 168

25. Computed and Published Values of the Displacement Function-- 171

26. Measured Motion of the Test Footing------------------------- 174

27. Component Displacement Ratios Averaged over 5 Frequencies--- 179

28. Normalized Half-Space Particle Displacements---------------- 186

29. Influence of Frequency on Half-Space Displacements- -------- 187

30. Normalized Soil Particle Displacements -------------------- 188

31. Influence of Frequency on Soil Displacements---------------- 189

32. Ratio of Half-Space to Soil Displacement-------------------- 190

33. Average Displacement Ratios for 5 Frequencies--------------- 191

34. Normalized Nonhomogeneous Half-Space Displacements ---------- 193

35. Ratio of Nonhomogeneous Half-Space Displacements to Soil
Displacements----------------------------------------------- 194

36. Average Nonhomogeneous Half-Space Displacement Ratios for
5 Frequencies---------------------------------------------- 195

37. Subroutines and Computer Program for the Integral
I(ao,a,b)-------------------------------------------------- 207

38. Value of I(0.36,a,b)-------------------------------------- 211

39. Value of I(o.48,a,b)-------------------------------------- 220

40. Value of I(0.72,a,b)-------------------------------------- 229










Page

41. Value of I(0.96,a,b)----------------------------- 238

42. Value of I(l.20,a,b)-------------------------------------- 247

43. Connection of Measuring and Recording Components----------- 262

44. Electrical Resistance of Transducer Components After
Completing Test Program------------------------------------ 264














LIST OF FIGURES


Page

1. Rigid circular disk on the surface of an elastic half-space- 11

2. Location of 3 exploration borings and 20 friction-cone
penetrations------------------------------------------------ 44

3. Grain-size distribution for six sample depths--------------- 51

4. Shear wave velocity versus depth, surface and empirical
methods----------------------------------------------------- 52

5. Distribution of stresses between a rigid disk and an
elastic half-space------------------------------------------ 55

6. Sketch of concrete footing embedded in soil----------------- 65

7. Distribution of vertical soil stress and dead load pressure
on cured first pour----------------------------------------- 67

8. Plan view sketch of torsional vibrator---------------------- 76

9. Elevation view sketch of torsional vibrator----------------- 77

10. Plan view of the field of transducer locations-------------- 85

11. Section view of the field of transducer locations----------- 86

12. Footing form and soil retaining ring------------------------ 89

13. Excavation for footing and transducer----------------------- 90

14. Transducer embedded below edge of footing------------------- 91

15. Excavation ready to receive footing form and retaining
ring-------------------------------------------------------- 92

16. Placing footing form and retaining ring in excavation------- 93

17. Footing form and retaining ring positioned in excavation---- 94

18. Concrete test cylinders, auxiliary form and reinforcing
mesh---------------------------------------------------- 96

19. Second pour reinforcing mesh placed in first pour----------- 97










Page

20. Position of auxiliary form and backfilling operation-------- 98

21. Cone penetration test adjacent to footing------------------- 100

22. Cone penetration test on backfill--------------------------- 101

23. Second pour of cured concrete in the footing form----------- 102

24. Checking depth and continuity of the air gap---------------- 103

25. Power required to drive the torsional vibrator-------------- 105

26. Vibrator bonded to mounting plate with an epoxy compound---- 107

27. Torsional vibrator mounted on the test foundation----------- 108

28. Assembled vibrator, test footing, and switch box------------ 109

29. Carpenter's level used to check tilt on vibrator frame------ 111

30. Method of attaching transducers to the test footing--------- 112

31. Test site topography, vegetation, and borehole markers------ 114

32. Drill rig used to auger uncased boreholes for the
transducers------------------------------------------------- 115

33. Alignment sleeves bonded to transducers with an epoxy
compound--------------------------------------------------- 116

34. Transducers with support cables and electrical leads-------- 117

35. Apparatus for installing and aligning transducers in
boreholes--------------------------------------------------- 119

36. Sighting bar used to align borehole rod and attached
transducer-------------------------------------------------- 120

37. Borehole, borehole rod, and transducer cables--------------- 121

38. Water hose inserted in borehole during backfilling---------- 123

39. Functional components of the particle velocity measuring
system------------------------------------------------------ 124

40. Two rows of amplifiers mounted in a cabinet-------- ------ 130

41. Type 5-119P4 recording oscillographs------------------------ 131

42. Histogram of measured transduction constants---------------- 134









Page

43. Typical calibration record, oscillograph A, before 50-Hz
vibration test---------------------------------------------- 141

44. Typical calibration record, oscillograph B, before 50-Hz
vibration test---------------------------------------------- 142

45. Typical test record at 50 Hz, oscillograph A, with eccentric
weights on vibrator----------------------------------------- 143

46. Typical test record at 50 Hz, oscillograph B, with eccentric
weights on vibrator----------------------------------------- 144

47. Typical test record at 50 Hz, oscillograph A, without
eccentric weights on vibrator------------------------------- 145

48. Typical test record at 50 Hz, oscillograph B, without
eccentric weights on vibrator------------------------------- 146

49. Settlement measurement at the center of the vibrator-------- 148

50. Settlement measurement at the edge of the footing----------- 149

51. Tilt check with level along vibrator frame------------------ 150

52. Tilt check with level across vibrator frame----------------- 151

53. Tilt check with level on top of test footing---------------- 152

54. Results of footing settlement and tilt measurements--------- 153

55. Compression wave initiated by striking footing-------------- 156

56. Compression wave velocities from hammer blows on footing and
on ground surface------------------------------------------- 157

57. Shear vibration propagation velocities versus depth--------- 167

58. Frequency response of test footing-------------------------- 175

59. Ratio of footing displacement to soil displacement under the
footing---------------------------------------------------- 177

60. Average penetration resistance and cone bearing capacity
versus depth------------------------------------------------ 181

61. Shear wave velocities versus depth-------------------------- 183

62. Cone bearing capacity and displacement ratio versus depth--- 199


xiii















LIST OF SYMBOLS


a = rk

a = frequency ratio; r k
o o
b = zk ; radius of uniformly loaded area

B = arbitrary function

CMP = component of motion

d = galvanometer trace deflection during calibration

D = galvanometer trace deflection during test

e = base of natural logarithms; void ratio; eccentricity

E = Young's modulus

E = Young's modulus for concrete

Ei = exponential integral

E = Young's modulus for soil
s
f = frequency; function

f = function

F = function

g = variable, arbitrary parameter, x/k ; function; gravitational

acceleration

G = shear modulus

G(z) = shear modulus that depends on z

h = interval; depth below ground surface

i= V.Il

im = iml = imaginary part of the integral

I = particle displacement integral









I' = (4/3T)I

J = Bessel function of the first kind and nth order
n
k = wave number, w/Vs

K = kinetic energy

K = coefficient of earth pressure at rest = rr/
o r zz
m = rate of shear modulus change with depth, z

M = moment applied to disk or footing

M' = limiting torsional moment

M, = design moment

n = integer

N(c) = normalized computed motion

N(m) = normalized measured motion

p = power loss

P = total vertical load on disk or footing

q = uniformly distributed vertical load on disk or footing

q = uniformly distributed vertical dead load due '-o second pour

of concrete

r = cylindrical coordinate

r = radius of the disk or footing

rl = radial distance to critical stress point

r2 = radial distance to inside edge of second pour of concrete

r = radial distance to center of pressure

R = radial distance from center of disk or footing

R = real I = real part of the integral

s = variable parameter = r/r

S = deformation ratio











S/N = serial number

t = time

T = transduction constant

u = particle displacement in the r direction

v = particle displacement in the 0 direction

vd = design displacement

v = particle velocity in the e direction

vN = particle displacement in a nonhomogeneous half-space

V = volume; voltage

V = shear wave propagation velocity, \G/p
s
w = particle displacement in the z direction

W = strain energy density; weight

x = variable; arbitrary parameter

y = variable; arbitrary parameter xr ; z + G(0)/m

z = cylindrical coordinate; distance below ground surface

a = variable; integer; attenuation coefficient

S= (x2 k2)1/2 ; G(0)/m

7 = shear strain, unit weight

7.. = shear strain in the plane of i and j

7N = shear strain in nonhomogeneous half-space

A = vertical deflection of concrete footing
c

A = vertical deflection of soil on footing-soil contact area
S
E.. = linear strain on the i plane in the j direction
1J
0 = cylindrical coordinate

X = variable; Lame elastic constant

p = integer










v = integer

S= arbitrary parameter

T = 3.14159+

p = mass density = 7/g

a.. = stress on the i plane in the j direction
1J

a = mean effective stress

T = shear stress

TN = shear stress in a nonhomogeneous half-space

S= angular rotation of disk or footing; angle of internal

friction of soil

tan (' = coefficient of friction

= limiting angular rotation of the disk or footing

w = angular frequency


xvii











Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy

THE PARTICLE MOTION FIELD GENERATED BY THE TORSIONAL VIBRATION
OF A CIRCULAR FOOTING ON SAND

By

Lyman Wagner Heller

August 1971

Chairman: Dr. John H. Schmertmann, P.E.
Major Department: Civil Engineering

Over the past few years, it has been demonstrated that the self-

excited vibratory motion of a circular footing on various types of soil

can be successfully predicted by a mathematical model derived by assum-

ing that the foundation soil is represented by a homogeneous elastic

half-space. This finding suggested that the same model, or variations

thereof, might be useful for predicting the particle motion generated

within a soil foundation by a vibrating footing.

The objective of this study was to test the hypothesized utility

of the half-space model for predicting the motion field generated in a

natural soil deposit by the forced torsional vibration of a circular

footing. The test involved the computation of half-space motion, the

measurement of soil motion, and a comparison of the computations to the

measurements.

A 5-ft-diam footing was vibrated at 5 different frequencies on a

natural sand deposit with a shear modulus that varied from about 1,800

psi at a depth of 1 ft to about 23,000 psi at a depth of 35 ft. Re-

sultant particle motions were measured on the footing and at radial

distances to 90 ft and at depths to 35 ft. Homogeneous half-space


xviii










particle motions were computed using a shear modulus of 9,480 psi.

The average measured particle displacement, neglecting damping, was

between 1/3 and 1/4 of the computed displacement. The measured dis-

placements were 1/10 of the calculated displacements at the deeper loca-

tions and 3 times the calculated displacements near the ground surface.

Near the footing, the displacements were in good agreement.

Recent literature on the stress conditions in a nonhomogeneous

elastic half-space suggested that the particle displacements in a homo-

geneous half-space could be used to determine the particle displacements

in a nonhomogeneous half-space. When the sand deposit was considered

as a nonhomogeneous half-space and damping was neglected, the displace-

ments were in good agreement near the footing, the average measured dis-

placement was 60 percent of the computed displacement, and the measured

displacements were 1/4 to 2-1/2 times the computed displacements. The

material damping effect on the propagating body waves agreed with pre-

vious determinations for this effect on surface wave- at a similar

test site. The correlation between the ratio of the computed displace-

ment to the measured displacement and the cone bearing capacity of

the sand deposit at various depths suggested that a more accurate and

detailed determination of the shear modulus of the sand would improve

the correspondence between measured and computed results.

Because the accuracy of the particle displacement predictions was

adequate to classify transmitted vibrations as either undetectable,

readily apparent, or intolerable, the elastic half-space model, adjusted

for nonhomogeneous site conditions, was considered a potentially useful

analytical representation of a natural soil deposit subjected to tor-

sional footing vibrations.


xix















INTRODUCTION


Background


Soil and foundation engineers who specify and design adequate sup-

port systems for buildings and equipment are commonly concerned with

three aspects of the performance of their foundations: (1) the long-

term load carrying capacity, as it relates to the type of facility and

safety of its inhabitants, (2) the immediate or during-construction

settlements, and (3) the rate and amount of postconstruction

settlement.

There are situations, however, when the engineer must provide a

foundation with additional capabilities. Such a situation occurs when

the foundation supports sustained or transient dynamic loads as devel-

oped by punch presses, forging machines, shock testers, and unbalanced

machinery. For these cases, the prescribed foundation not only must

provide support for the imposed static and dynamic loads to assure the

safe operation of the equipment and the facility, but also must mini-

mize the radiation of undesirable vibrations into the surrounding soil

where they can be transmitted to adjacent inhabited or vibration sensi-

tive areas. Thus, one part of the soil engineer's responsibility is to

provide a foundation that will inhibit or diminish the generation and

transmission of dangerous, troublesome, and annoying ground vibrations.

Crockett (1965) has briefly discussed some of these problems; the

writer is aware of a California case in which a titanium forging plant









was moved from Los Angeles to Ventura County because its foundations

generated suit-worthy ground vibrations.

The reciprocal situation arises when inhabitable or motion sensi-

tive facilities must be built in congested areas with their founda-

tions resting on ground that is shaken by industrial machines, subway

and elevated trains, pile driving, blasting, and pavement breaking op-

erations. In this situation, the engineer's task is to effectively

isolate sensitive structures from the ambient movements of the surround-

ing ground. Margason, Barneich, and Babcock (1967) and Blaschke (1964)

have summarized some of the approaches to these problems.

The common denominator necessary for the rational solution of both

of the above situations is a definition of the characteristics of gen-

erated, transmitted, and received ground vibrations. The soil engineer

currently has little empirical evidence or confirmed theory to guide

him in selecting, designing, or improving his foundations to diminish

transmitted ground vibrations or to minimize received ground vibrations.

In a summary review of a Soviet conference on the dynamics of bases and

foundations, D. D. Barkan (1965) wrote:

...primary attention should be devoted to investigating
the elastic properties of bases [supporting soil] based
on measurements of free and forced vibrations of machine
foundations since not only have effective measures to
combat waves propagating in soils not been worked out,
but there are no methods which permit calculating the
parameters of these waves or a theory for calculating
soil vibrations at various distances from the wave
source as a function of its dimensions, depth of occur-
rence, and mechanical properties of soils. Without a
solution to these problems, it is impossible to de-
velop methods of evaluating the effect of a wave source
on structures, equipment, and people.










Previous Work


One of the earlier investigations of wave propagation through

soils from a surface source was reported by Ramspeck (1936) who noted

the influence of interference waves on the observed amplitude of parti-

cle motion at the ground surface. Over the years, Bernhard (1967) has

conducted, extended, and reported on a variety of similar experiments

and observations. Barkan (1962) studied wave propagation through near-

surface soils with particular attention to the surface wave effects.

Lysmer and Kuhlemeyer (1969) have used discrete models for the study

of surface waves.

Available theory for waves propagating in soil has been summarized

by Woods (1968) and is based primarily on representing the soil by a

homogeneous elastic half-space. He concluded that an oscillating ver-

tical pressure on a circular area at the surface of the half-space pro-

duced dilatational and distortional body waves that radiated into the

half-space and surface waves that propagated along the surface of the

half-space. The annularly diverging, slow-moving, surface wave was

the dominant energy carrier. On the other hand, if an oscillating tor-

sional moment is applied to the surface of a half-space, Reissner

(1937), Reissner and Sagoci (1944), and Bycroft (1956) indicate that

only shear waves are radiated into the half-space.


Related Work


The most complete collection and digest of past theoretical and

experimental work related to this study is available in Richart, Hall,











and Woods (1970); contributions to this collection span almost a cen-

tury. Over the past few years, however, it has been demonstrated that

the self-excited vibratory displacement of a circular footing on various

soil foundations can be successfully predicted by a mathematical model

derived by assuming that the soil supporting the footing is represented

by a homogeneous elastic half-space (Richart and Whitman, 1967). This

result suggested that the same model, or variations thereof, might be

useful for predicting the vibrations within the supporting soil due to

an oscillating footing. If the model should prove useful, it might be

utilized to attack some of the ground motion transmission and isolation

problems outlined in preceding paragraphs.


Approach to the Investigation


Based on the confirmed behavior of self-excited foundations de-

scribed above, it was hypothesized that the elastic half-space model

could be used to predict the particle motion in a soil deposit due to

a vibrating foundation. A test of the hypothesis required two parallel

efforts: (1) the solution and evaluation of the particle motion in an

elastic half-space, and (2) the experimental measurement of the particle

motion in a soil deposit. The approach was in consonance with the views

of Odqvist (1968) who wrote:

In his introductory article, written in 1921, [Richard]
von Mises expressed opinions very much like those I have
been propounding here. ..."There should be no distinc-
tion between theoretical and experimental papers. All
theoretical research depends in the end on observational
facts. Experimental work is useless unless it is under-
taken in the view of some theory."








5
Available Theory

Since the purpose of the theoretical work was to calculate the

particle motion in an elastic half-space due to an oscillating surface

source, the simplest mode of source oscillation was chosen for analy-

sis and interpretation. Bycroft developed solutions for the oscilla-

tion of a rigid circular plate on the surface of a half-space for

four modes of motion: (1) vertical, (2) rocking, (3) sliding, and

(4) torsion. Of these four modes, rotation of the plate about a ver-

tical axis, or torsion, was the simplest because this is an uncoupled

motion and because no dilation of the half-space exists (Reissner,

1937; Bycroft, 1956).

Two general types of solutions were found for the torsional oscil-

lation of a surface source. One assumed that the shear stresses on the

circular contact area were zero at the center and increased linearly

along a radius of the area (Reissner, 1937; Miller and Pursey, 1954).

The other assumed that the contact area was rigid (Reissner and Sagoci,

1944; Bycroft, 1956; Collins, 1962; Stallybrass, 1962, 1967; Awojobi

and Grootenhuis, 1965; Thomas, 1968). From an experimental viewpoint,

duplication of the rigid boundary condition case by an oscillating test

footing presented fewer difficulties than controlling the shear stress

on the contact area. For this reason, the rigid contact boundary con-

dition between the half-space and the oscillator was chosen for calcu-

lating half-space particle motions due to torsional source vibrations.

The solution for the oscillation of a rigid disk on the surface

of a homogeneous (constant E) elastic half-space given by Bycroft

(1956) was appropriate for practical foundation dimensions and typical








6

vibration frequencies. No significantly improved theoretical solutions

for the torsion case have been advanced since the work by Reissner and

Sagoci (1944) and Bycroft (1956). All of the later -ethors evaluate

their work, and check the accuracy of their approximations, by compari-

son with the Reissner and Sagoci or the Bycroft solutions.

Investigations of a nonhomogeneous half-space with elastic moduli

that increase linearly with depth (linear E), subjected to static sur-

face loads, have been conducted by Fr'lich (1934), Borowicka (1943),

Hruban (1948), Curtis and Richart (1955), and Gibson (1967). Discrete

analysis systems have been devised by Lysmer and Kuhlemeyer (1969) and

Lysmer and Waas (1970) for assessing the effects of oscillatory surface

loads on irregular and layered elastic media.


Available Experimentation

Previous experiments to determine the particle motion in a soil,

or other material, due to a torsionally vibrating surface source were

not found. Arnold, Bycroft, and Warburton (1955) computed the response

of a self-excited rigid disk and then measured the response of small

plates mounted on a foam rubber half-space and on a foam rubber layer,

but they did not measure the particle motion in the foam rubber. They

used 3/4- to 4-in.-diam plates on the surface of a 3-ft-square by

1-ft-thick block of laminated foam rubber. Useful experimental data

is evidently quite scarce since Thomas (1968) found it necessary to

compare his theoretical work to the 1955 experiments by Arnold, Bycroft,

and Warburton. Surface motion due to a vertically oscillating source

has received extensive attention (Woods, 1968).










Comparisons

Richart and Whitman (1967) examined existing experimental data for

the vibratory behavior of surface footings founded on soil materials

and compared this data to the theoretically predicted vibratory response

of these same footings on an elastic half-space. These comparisons

confirmed the applicability of the elastic half-space model for

predicting the oscillatory motion of circular foundations on soil.

Similar comparisons of calculated and measured results would indicate

the usefulness of the half-space model for predicting the particle mo-

tions generated in a soil deposit by a vibrating footing.


Objective and Goals


The confirmed utility of the elastic half-space model for predict-

ing the oscillatory behavior of circular footings founded on soil sug-

gested that this same model, or variations thereof, could be useful for

predicting the particle motion field generated within a soil deposit by

a vibrating footing. The objective of this study was to test the hy-

pothesized usefulness of the half-space model for predicting the vibra-

tions transmitted into a soil foundation by an oscillating footing.

A test of the hypothesis involved three specific goals: (1) extend

Bycroft's (1956) solution for the torsional oscillation of a rigid disk

on the surface of an elastic half-space to include the motion of the

half-space and evaluate the resulting expression for absolute values of

the motion of the half-space, (2) conduct a field experiment on a

natural soil deposit that physically represents the boundary conditions








8

assumed for the half-space model and measure the particle motion gene-

rated in the soil by the oscillation of a footing, and (3) compare the

particle displacements predicted by the half-space model to the parti-

cle displacements measured in the natural soil deposit and evaluate the

usefulness of the model for motion prediction purposes.















THE THEORETICAL PARTICLE MOTION GENERATED BY THE
TORSIONAL OSCILLATION OF A WEIGHTLESS, RIGID, CIRCULAR DISK ON THE
SURFACE OF AN ELASTIC HALF-SPACE



Homogeneous (Constant E) Elastic Half-Space



Problem Statement and Approach

A weightless, rigid, circular disk rests on the surface of an iso-

tropic, homogeneous, elastic half-space. A torsional moment, which

varies sinusoidally with time, acts on the weightless disk about a ver-

tical axis through the center of the disk and imparts an oscillatory

rigid body rotation to the contact area between the disk and the half-

space. Since the disk is rigid, the displacement on the contact area

is proportional to the radial distance from the center of the disk.

Except for the contact area between the disk and the half-space, the

horizontal surface of the half-space is free of stress. Particle mo-

tion, stress, and strain vanish at infinity on the hemispherical bound-

ary of the half-space. The particle motion generated within the half-

space is to be determined.

The above situation is a mixed, or third type, boundary value prob-

lem in the theory of elasticity. This problem has been treated by

several investigators in order to establish the characteristic behavior

of the disk under forced torsional vibrations (Reissner and Sagoci,

1944; Bycroft, 1956; Collins, 1962; Awojobi and Grootenhuis, 1965), but

no one has apparently attempted to deduce the particle motion developed

in the body of the half-space.









10

The purpose of this section is to quantitatively evaluate the par-

ticle motion generated in an elastic half-space by the torsional oscil-

lation of a massless rigid disk on the surface of the half-space. The

evaluation has been guided by Bycroft's (1956) analysis for the behavior

of a rigid disk; departures from and extensions to his work have been

accomplished to treat the behavior of the half-space rather than the

circular disk. Figure 1 illustrates the system.


Equations of Elasticity


The equations for strain in cylindrical coordinates r 8 z are

(Timoshenko and Goodier, 1951)


Er -u (1)
rr ar

E = + v (2)
90 r rae


E =w (3)
zz az

: 6w + av (4)
ze re a2z

7 6u + 6w (5)
rz az ar

u + v v (6)
re r9e ar r

the stress-strain relationships in cylindrical coordinates are

(Sokolnikoff, 1956)


rr= X(Err + E + zz) + 2G (7)
rrr rr 9e zz rr


e = X(Er + E + z ) + 2GE
Oe k(rr ee zz ee














dr




ELEMENT b 6





i------- U


PLAN VIEW


ELEVATION VIEW


Rigid circular
half-space.


disk on the surface of an elastic


RIGID DISK


ELASTIC
HALF-SPACE


r


- Ozz


J dz



- U


Figure 1.







12

Szz = X(C + E9 + E ) + 2GGzz (9)
zz rr ee zz zz


ze = G7ze (10)


r = G7 (11)


re = G7re (12)


and the equations of equilibrium for an infinitesima element with dimen-

sions of dr rde and dz are

bcrr Ibe rz rr 9 2u
r+ + + = p (13)
r r be z r t2P (13)


bre 91 909 + ze 2r 2v (4)
+ + + p (14)
br r be az r t2
rat

r z 1 ze z z 2z 1 '
+ +_ + Pz (15)
6r r be az r t2

Since the particle motion, u v w with respect to the coordinates

r e z of the half-space is desired, the equilibrium equations are

written as

S (ru) + av L G 1 b(ru) =b
Sr r r r e azJ r 6e r 9ae










( + G) (ru) +v (
SG r p (17)
r -r r Be 2










( + 2G) + + -- r -U r ar
,z [r 6r r ?A ;z r 2r 8z r


G 1i aw av 6w
+ p (18)
r be r Ge az 2



Solutions to the equilibrium equations are sought which satisfy the

boundary conditions described in the problem statement; i.e.


v(r 9 0 t) = 0 reiwt (r ro) (19)


ze(r 9 0 t) = 0 (r > ro) (20)

where 0 is the angular rotation of the rigid disk. For completeness,

it should be noted at this point that the compatibility conditions are

not involved since the equilibrium equations are expressed in terms of

particle displacements (Sternberg, 1960). This mixed or third boundary

value problem can be reduced to a first boundary value problem by eval-

uating the static shear stresses produced on the surface of the half-

space by the rigid circular disk and assuming that these same stresses

occur on the half-space as the disk undergoes forced torsional oscilla-

tions. A similar approach and assumption is common in the literature

and has been used by Miller and Pursey (1954), Reissner (1937), Sung

(1953), and Hsieh (1962), as well as by Bycroft (1956). For the case

of a rigid circular disk in forced torsional oscillation on an elastic

half-space, these same authors agree that v is the only component of

displacement that occurs. Consequently, the axisymmetric problem is

greatly simplified and the strain equations reduce to

S-v (21)
ze az








= (22)
re 2r r

the stress-strain relationships to

aze = GYze (23)

are = GYre (24)

and the equilibrium equation becomes

cre+ cze 2re r 2v
+ t+ p (25)
r z r 2 P


/2v 2v 1 av 2v
Gv L+ 2- vP (26)
S2 2 r ar (26)


Solution to the Equilibrium Equation
According to Bycroft (1956), solutions to the equilibrium equation
were devised by K. Sezawa in 1929. The solution for v(r,z,t) is
-B(x)Jl(xr) iwt
v(r,z,t) = -B(x e e (27)

where x is an arbitrary parameter, B(x) is an arbitrary function of
x to be determined from the boundary conditions, =(x2 k2) /2
and u is the angular frequency of the rigid disk.
The derivatives of v which satisfy the equilibrium equation are

v [B(x)J(xr) B) J(xr)] e- eiWt (28)
=r )2 xr 1


827 v L/\ /\ B(x) -6z int
2 v B(x) x J1(xr) B( J2(xr e e (29)
r 2 r 2









_ __Bx) 2 2\I-Pz int
2 x J (xr)(x k) eB e


82v B(x) Jl(Xr, 2 -Bz int
t2 1


k2 = G W2


(30)


(31)


(32)


Boundary Conditions

The shear stresses in the elastic half-space are

Sv -B() (2 2)12 it
z(e(r,z,t) = G7z x x = k Jl(xr)eB ei (33)

The boundary conditions for stress on the surface of the half-space

when the rigid disk is rotated statically, so that z = w = k = 0 ,

become


B(x)
oze = G x x Jl(xr)
z9 ~ -- ---xr


= 0


(r S ro)

(r > ro)


(34)

(35)


and the static displacement v at the surface of the half-space is


v(r,0) = -B(x) J (xr)
x 1


Since x and B(x) are arbitrary, set


v = B() J1(xr)dx =


ze = G f B(x)J(xr)dx = 0
0


and let r/ro = s and xro = y so that


where


(36)


(r r )


(r > r)


(37)


(38)









v = r F(y)y~-J1(sy)dy = ros
0
e = G O f (y)J y)y

ze = G F(y)J1(sy)dy = 0


(O < s < 1)



(s >1)


where y is another arbitrary parameter.

To evaluate F(y) in this pair of integral equations, Bycroft
(1956) and Awojobi and Grootenhuis (1965) use work by Titchmarsh
(1948), Busbridge (1938), and Tranter (1951). Busbridge gives the
solution of the equations


/ Y(y)J (xy)dy = g(x)
0

f f(y)J (xy)dy- o


(O < x < 1)


(x > 1)


where g(x) is prescribed and f(y) is to be determined, as

2-/ 2 -a 1+a/2 1 +v+l 2 o,2

0
1 /2 1
+ f ul(1- u2 ) du f g(y)(xy)2+/2 J++ /2(xy)dy


which is valid for a > -2 and where
By substituting = -1 v = ,
tion 43


(-v 1) < (a 1/2) < (v + 1).
and g(x) = Oros into Equa-


(39)



(40)


(41)


(42)


(43)









2-1/2y
F(y) / 12 ()


+ fm2( -


= yror


i
sin y


1 1 2-1/2
f2 ( 02) (/)2


2)-1/2 X 1 Me) (ye) 1
m2)/2dm f (Orom) (yz)3/2J3/2 (yA) d
0


f (1 )-/2
0


1
+ fm3(


1
- m2)-/2dm 3/2 (y)-3/2(sin ya
0


- yl cos yj)dI


4
TT


r (sin y- y cosy
o \ y


The expression for shear stress (Equation 40) is then


Coz(r,0,0)


ro) sin y y cos y (sy)dy
0) -- -y ) 1


0G f
0


-40r G sin xr xr cos xr
sn f xr
f xr
~o


J (xr)dx


4or G 1/2
TT f- --Z (ro J3/2(xro) J(xr)dx
0


Restating the boundary conditions in terms of stress requires that


o- rl 1/2
,ze TT 2 0


Sx1/2 J1(xr)J3/2(xro)dx


(r r r)


= 0 (r > r)

The infinite integral appearing in Equation 48 is one of the special

cases of the discontinuous Weber-Schafheitlin integral tabulated by.


Abramowitz and Stegun (1964)


(44)


(45)

(46)



(47)


(48)


(49)










f t-v+ij (at)J (bt)dt = 0
0 2-v+lab(b2 a2)--1

bV vT-(


where v > 4 > -1 Substituting 4 = 1 and v = 3/2

50 and 51, the values of the integrals in Equations 48



f xl/21(xr)J3/2(xro)dx = 0


1/2' (2 2) -1/2
2 1r (r r2

r3/2 (1/2)
o


(0 < b

(b > a > 0)


into Equations

and 49 are


(r > ro)



(r r )


so that the shear stress on the surface of the half-space in contact

with the rigid disk becomes


__ o r r3/2r2_

0 0o

G r (r < r ) (
n 2 2 O
r -r


Applied moment and disk rotation

The moment applied to the disk is

r
o 2TT
M = f zrrdedr
0 0

16OGr3
3
3


54)


55)


(50)


(51)


(52)



(53)









so the angular rotation of the disk becomes


3 3 (56)
16Gr3


Substituting this value of into Equation 54 gives


z (r,O,O) -3M r r (5)
41r7r3 2 2 (
o r r


Reissner and Sagoci (1944), using a system of oblate spheroidal co-

ordinates instead of dual integral equations, found this same shear-

stress distribution on the contact surface between a torsionally

loaded rigid disk and an elastic half-space.


Particle Displacements

It was previously found that any arbitrary parameter x and

function B(x) will satisfy the equilibrium equation and that a spe-

cific form of B(x) will also satisfy the prescribed boundary condi-

tions. Thus, the specific formulation for B(x) that satisfies the

boundary conditions will also satisfy the equilibrium equation

throughout the half-space.

The solutions to the equilibrium equation

v(r,z,t) = -B(x) J1(xr)e-zeiwt (58)


and the shear-stress expression


z(r,z,t) = G B(x) e-z (59)
aze(rzt) = x J1(xr)e eirt (59)









are related by

v(r,z,t) =- 1 z(r,z,t) (60)
G "z 0

and the specific shear-stress formulation (Equation 46) that satisfies

the boundary conditions is

-46r G o sin xr xr cos xr
z (r,z,t) = f -oxr o o J1(xr)e- iWtdx (61)
0

Substituting Equation 61 into Equation 60 gives the particle dis-

placement in the half-space as

4rroeiut sin xr xr cos xr
v(r,z,t) = f xr J(xr)e dx (62)
0

The normalized particle motion in the half-space with respect to the

tangential displacement of the disk is then

-(rzt) 4 ei t f sin xr xr cos xr
v O t -Xroo J (xr)e- Zdx (63)
0r 0TT Jxr 1


Evaluation of the Infinite Integral

In order to calculate the particle motion at a specific point in

the half-space, it is necessary to evaluate the infinite integral for

particular values of four quantities, r0 r z and k .The four

quantities are reduced to three if a = rk a = r k b = zk and
O o
g is a new arbitrary parameter g = x/k Substituting and represent-

ing the infinite integral by

f sin a g a g cos a g b2(agg
I(aa,b) = 0 o 0 e- -1 1(ag)dg (-64)
0 aagg2 1)
0 o









the particle motion expression becomes


int
v(r,z,t) = 3e I(ao,a,b)
4nGr
o

4r oeiwt
=- I(a ,a,b)
SO


(65)



(66)


Inspection of I(ao,a,b) reveals that the integrand will have

imaginary components when g is less than unity because the term

g2 1 appears in the denominator. The integrand will be real when

g is greater than unity. These characteristics suggest that the inte-

gration should be carried out in two ranges; i.e.


1
I=R f
0


Noting that


1
+ im f
0


+ il2


Co
+ R
1


+ 13


(67)



(68)


g =i g2





e ib l-g= co-s b g2 i sin b l g


(69)


(70)


-1 and 12 are developed from the expression

Ssin a g a g cos ag ib 2
I + il2 =f 0 g e g Jl(ag)dg
0 aog i g2


(71)


1sin a g a g cos a g

I1 sin ag g cos a i sin bl g2J (ag)dg (72)
0 ag i g









sin a g a g cos a g 2
i = f sinag a0go cos b 1 g Jl(ag)dg (73)
0 aog ivl g


The integrands of II and 12 become unbounded as g approaches

unity, so a change in variables is appropriate. Let


g = sin a (74)

dg = cos ada (75)

and
g2 = cos a (76)


where 0 < g <1 and 0 a TTr/2


By replacing the variable g with the variable a
r/2 sin (a sin a) (a sin a) cos (a sin a)

1 a sin
0o

x J1(a sin a) sin (b cos a)do (77)


and
r/2 sin (a sin a) (a sin a) cos (a sin a)

2 a sin a
0 o
x Jl(a sin a) cos (b cos ac)dc (78)


The integrand of the integral

J sin (aog) aog cos aog -b 2-
13 = e Jl(ag)dg (79)
1 a0g g 1

also becomes unbounded when g approaches unity and a change of vari-

ables for I3 is indicated. Let






23

g = sec (80)


dg = sec a tan a dc (81)

and

g2 1 = tan a (82)

where 1 g g s and 0 a nr/2 Using the new variable a instead

of g the integral I becomes


T/2 sin (a sec )- (a sec a) cos (a sec a)
3 f a
0
x Jl(a sec a)e-D tan ado (83)

The above three integrals, II 12 13 are expressed in terms of

circular functions, J1 an integer order Bessel function of the first

kind, and e the base of natural logarithms. The integrands are con-

tinuous functions in the interval of integration, and, of particular

note, all the terms of the integrand can be expressed as a series or as

polynomial approximations. Such formulations make the integrand well

suited for evaluation with a digital computer.

Integration of I 12 and I could be accomplished in a

variety of ways, but perhaps the most obvious is by numerical methods.

One numerical integration scheme is based on Simpson's rule for deter-

mining the area of an irregular figure. The integral expression of

Simpson's rule, given by Abramowitz and Stegun (1964), is


f(x) dx = fo+ k4(f+ f3 +- '2n-
X L
+ 2(f2+ 4 + .2) + 2n (84)








x x
2n o
where h = and n is an integer.
2n

Example calculation

The application of Simpson's rule for calculating uhe approximate

value of II 12 and I can be illustrated by an example.

The integration parameters for the particle motion in a half-

space at a distance of 30 ft from the center of the disk and at a depth

of 15 ft, when the shear wave velocity in the half-space is 650 fps and

the 5 ft diameter disk oscillates at 20 Hz are


k = =/V = 2r20/650 = 0.192 (85)


a = kr = 0.192(2.5) = 0.48 (86)
0 0

a = kr = 0.192(30) = 5.76 (87)


b = kz = 0.192(15) = 2.90 (88)


For illustration purposes, divide the integration interval into 4 equal

parts, i.e. h = 1/4(n/2) = nr/8 Values for the common terms in the

integrand of Il and I are listed in Table 1 and values, f(a) of

the integrand of Il are given in Table 2. The calculation for II

by Simpson's rule is


I = -(T/8)/3 [o + 4(0.0027899 0.0199034) + 2(-0.0031265) + 0


= 0.009779 (89)


Values, f(ot) of the integrand of I2 are given in Table 3. The cal-

culation for 12 by Simpson's rule is










Table 1


Values of the


Common Terms in the Integrand of II


a a s a J (a sin a) sin (a sin a) cos (a sin a)
c sin a cos a o a sin a b cos a 1 o o

0 0.00000 1.00000 0.0000000 0.0000000 2.9000000 0.0000 0.00000 1.00000
Tr/8 0.38267 0.92387 0.1836816 2.2041792 2.6792230 0.5553 0.18265 0.98316
Tr/4 0.70709 0.70709 0.3394032 4.0728384 2.0505610 -0.0931 0.33289 0.94295
3n/8 0.92387 0.38267 0.4434576 5.3214912 1.1098010 -0.3458 0.42906 0.90327
T/2 1.00000 0.00000 0.4800000 5.760000 0.0000000 -0.3163 0.46178 0.88699





Table 2
Values, f(a) of the Integrand of Il

sin (a sin a) a sin a cos

a (ao sin y) /a sin a J1(a sin a) sin (b cos ) f()

0 0.00000 0.0000 0.24192 0.0000000
r/8 0.01126 0.5553 0.44620 0.0027899
r/4 0.03786 -0.0931 0.88701 -0.0031265
31/8 0.06426 -0.3458 0.89570 -0.0199034
q/2 0.07506 -0.3163 0.00000 0.0000000


and 12










Table 3
Values, f(a) of the Integrand of 12


shY a sin~)-a sin~cos


sin (a sin a) a sin o cos

(ao sin a) /a0 sin a


J1(a sin a)


cos (b cos a)


-0.97030
-0.89493
-0.46175
0.44466
1.00000


0.0000000
-0.0055957
0.0016275
0.0098808
-0.0237415


rr/8


3n/8
n/2


0.00000
0.01126
0.03786
0.06426
0.07506


0.0000
0.5553
-0.0931
-0.3458
-0.3163






27

1 = -(n/8)/3[o + 4(- 0.0055957 0.0098808) + 2(0.0016275) 0.0237415]

= 0.010785 (90)

Evaluation of the integral I can be accomplished in the same

manner as that illustrated for II and 12 Table 4 lists the values

of the terms in the integrand of I3 and Table 5 gives the values,

f(c) of the integrand of I The computation for 13 is

13 = (n/8)/3[-0.023713 + 4(-0.0063927 + 0.0010616) + 2(0.0028797) + 0]

= -0.0051415 (91)


The real part of the integral I(ao,a,b) is

Real I = II + 3

= 0.0046375 (92)


and the imaginary part of I(ao,a,b) is

iml = I i

= 0.010785i (93)


The particle displacement for this example in terms of the ro-

tation of the rigid disk is


v(30,15,t) =--- e Ot[O.0046375 + i(0.010785)] (94)

and the peak particle displacement is


I v(30,15) || [(0.0046375)2 + (0.010785)21/ (95)

This example problem illustrates that the integrands of II 12 '










Table 4


Values of the Terms in the Integrand of 13


S sec aa sec a sin (a sec ) cos (a sec a) Js (a sec a) a ta
a see a o o o a sec a 1 tan a b tan a

0 1.00000 0.4800 0.46175 0.88701 5.76000 -0.3163 0.0000 0.0000
n/8 1.08240 0.51955 0.49647 0.86805 6.23462 -0.2243 0.41421 1.20121
1/4 1.41425 0.67884 0.62788 0.77831 8.14608 0.2524 1.0000 2.9000
3TT/8 2.61308 1.25428 0.95033 0.31123 15.05134 0.2032 2.4142 7.00118
T/2 : cc -1 to 1 -1 to 1 c 0.0000 o c





Table 5
Values, f(a) of the Integrand of 13

sin (a sec a) a sec a cos

(ao sec a) /aO J (a sec a) e-b tan c a(a)

0 0.074969 -0.3163 1.00000 -0.023713
T/8 0.094740 -0.2243 0.30083 -0.0063927
n/4 0.207358 0.2524 0.055023 0.0028797
37T/8 1.16658 0.2032 0.000910 0.0010616
n/2 Undefined 0.0000 0.00000 0.00000









and I are well behaved functions of the variable a so that a nu-

merical integration scheme, such as Simpson's rule, should give a good

approximation for the value of these integrals.


Computer Program to Evaluate the Integrals

A computer program, based on Simpson's rule and written in ALGOL

language, was used to evaluate the integrals I1 12 and I

Table 37 of Appendix A lists this program and includes the polynomial

routine for calculating J1 Using the variable x instead of a ,

the integrand of I1 is called RELX(X), the integrand of 12 is

IMX(X) and the integrand of 13 is REH(X). The real part of

I(ao,a,b) is called REINT and the imaginary part IMINT.

The computer calculations are carried out in much the same manner

as illustrated in the example calculations above. The integration in-

terval, 0 to rr/2 radians, has been subdivided into three parts:

0 to 0.5 radians, 0.5 to 1.0 radians, and 1.0 to TT/2 radians. Each

part is independently integrated, using Simpson's rule and a geometri-

cally increasing number of intervals, until two sequential integration

agree to five significant digits. When this criterion is satisfied for

each part of the integration interval, the sum of the parts is con-

sidered to be a sufficiently accurate representation of the integral

for the purpose of this investigation.

Appendix A contains tables of computed I(ao,a,b) values for

several combinations of the variables a a and b The real,

imaginary, and absolute values of I are listed for parameters charac-

teristic of the test site at Eglin Field, Florida, and for the measure-

ments planned at this site. The shear wave velocity at the Florida









site is about 650 fps at a depth of 15 ft, the unit weight of the soil

is about 104 pcf, and the diameter of the footing (disk) is 5 ft.

Table 6 gives the calculation parameters used to compute the values of

I(a ,a,b) contained in Appendix A.


Table 6

Calculation Parameters for I(a ,a,b)
ao


Frequency
Hz

15






20






30






40






50


a
0.36
0.36


0.48


0.72


0.96


1.20


r
ft

0-12.5
0-90
0-90
0-90
0-90
0-90

0-12.5
0-90
0-90
0-90
0-90
0-90

0-12.5
0-90
0-90
0-90
0-90
0-90

0-12.5
0-90
0-90
0-90
0-90
0-90

0-8.75
0-90
0-90
0-90
0-90
0-90


a

0-1.80
0-12.96
0-12.96
0-12.96
0-12.96
0-12.96

0-2.40
0-17.28
0-17.28
0-17.28
0-17.28
0-17.28

0-3.60
0-25.92
0-25.92
0-25.92
0-25.92
0-25.92

0-4.80
0-34.56
0-34.56
0-34.56
0-34.56
0-34.56

0-4.20
0-43.20
0-43.20
0-43.20
0-43.20
0-43.20


z
ft

0
1
5
15
25
35

0
1
5
15
25
35

0
1
5
15
25
35

0
1
5
15
25
35

0
1
5
15
25
35


b

0.00
0.144
0.72
2.20
3.62
5.06

0.00
0.192
0.96
2.90
4.84
6.76

0.00
0.288
1.44
4.34
7.24
10.14

0.00
0.384
1.92
5.80
9.64
13.50

0.00
0.480
2.40
7.24
12.06
16.90









Nonhomogeneous (Linear E) Elastic Half-Space


Because the elastic moduli of soils is known to depend on the mean

effective stress applied to the soil (Hardin and Richart, 1963) and the

effective stress in a soil deposit increases with depth below the ground

surface, a nonhomogeneous elastic half-space would be a more realistic

analytical representation of a soil deposit than a homogeneous elastic

half-space. Thus, it was worthwhile to consider the progress that has

been made toward the use of a nonhomogeneous half-space for foundation

problems and some apparent relationships between a homogeneous and a

nonhomogeneous half-space with an elastic modulus that increases lin-

early with depth.

In a review of existing knowledge of the dynamic behavior of soils

and foundations, Jones, Lister, and Thrower (1966) made particular men-

tion of the need for and the apparent lack of attention to the develop-

ment and application of nonhomogeneous theory to these problems. In

summarizing the analysis of machine foundations on soils with a modulus

that changes with depth, they state:

The problem which arises when the elastic properties
vary continuously with depth, rather than in the
discontinuous fashion typified by layered media, has
received less attention, although it is important,
especially in view of the variation of elastic prop-
erties of non-cohesive soils with the mean stress.
Structures [soil stratification] of this type are
probably rather more frequent in practice than the
layered case. No analytical investigations of the
kind described above are known to the authors. Pauw
(1953), however, has analyzed the problem by assum-
ing essentially that the phenomena can be described
by considering the propagation of a cone-shaped bun-
dle of longitudinal-type waves downwards into the
soil. ...Pauw's approach appears rather unsatisfac-
tory from an analytical point of view.









Some progress, as outlined in the following paragraphs, has been

made since 1966, but rigorous solutions to the dynamic boundary value

problem on a nonhomogeneous elastic half-space have not been found or

attempted herein. Discrete methods, however, such as the finite ele-

ment and lumped mass representations, are developing rapidly and may

soon be capable of solving such problems.


Literature

Seismologists have been concerned with the influence of a variable

modulus earth structure on the speed and period of propagating earth-

quake tremors (Byerly, 1942). Their concern stems from a need to locate

the epicenters of earthquakes and to define the gross structure of the

earth's mantle. Ewing, Jardetsky, and Press (1957) devoted an entire

chapter of their book to wave propagation in media with variable veloc-

ity. Again, the primary purpose of the work was to study the disper-

sion characteristics of propagating seismic waves. A recent paper by

Bhattacharya (1970) gives the solution of the wave equations for an in-

homogeneous media. His work is limited to horizontal shear waves prop-

agating in a plane, and his solutions define the variation of density,

shear modulus, and shear wave velocity with depth. The form of these

variations depends on the solution functions.

Solutions for the static displacement and stresses in a nonhomo-

geneous elastic half-space due to a uniformly distributed strip or cir-

cular surface loading have been developed by Gibson (1967); he con-

sidered that the elastic modulus of the half-space varied linearly with

depth (linear E) and that the half-space was incompressible. Half-

space stress solutions for a point or a line load on the surface of a







33

compressible or incompressible nonhomogeneous elastic half-space with

a modulus that varies with depth have been presented by Curtis and

Richart (1955). Earlier investigations of similar cases have been ac-

complished by Hruban (1948), Borowicka (1943), and Fr'lich (1934).


Results of Gibson's Solutions

As mentioned, Gibson (1967) obtained solutions for the static dis-

placements and stresses in a nonhomogeneous elastic half-space due to a

uniform load distributed along an infinitely long strip or over a cir-

cular area. He assumed that the shear modulus, G(z) varied with

depth, z as

G(z) = G(O) + mz (96)


The equilibrium equations resulting from this assumed modulus variation

were intractable, but they were greatly simplified by assuming that

Poisson's ratio was 1/2.

By the use of Fourier transforms, suitable changes in variables,

and a discontinuous integral satisfying the boundary conditions, Gibson

was able to develop closed form expressions for the displacements and

stresses in the nonhomogeneous, incompressible, elastic half-space.


Solution for stresses

A uniform vertical pressure, q acting on the surface of the

nonhomogeneous half-space over a circular area of radius, b produces

shear stresses, a of
rz


z e Z [F(Y F(p)]} d (97)
~TZ 2J^ 0









where r and z are cylindrical coordinates


0 = G(O)/m (98)

y = z + (99)

K = bJo(r )Jl(bt) (100)

A = [IF(BP) + tB loge (ZB) + 1 + J (101)

F(X) = e21Ei(-2X) loge X (102)

and Ei is an exponential integral (Ambramowitz and Stegun, 1964).

When P -> as a limit, the change in shear modulus with depth

approaches zero, and the nonhomogeneous incompressible half-space be-

comes a homogeneous incompressible half-space. When B 0 the shear

modulus at the surface of the nonhomogeneous half-space approaches zero,

but the value of m is not restricted. Gibson found that the expres-

sions for the stresses were the same in both of these cases and he con-

cluded that the stresses were unaffected by this particular type of non-

homogeneity. This conclusion also results from the Curtis and Richart

(1955) work. Gibson also postulated that the stress components in a

nonhomogeneous half-space, with finite values of G(0)/m may not

differ appreciably from the stress components in a homogeneous

half-space.


Strain relationships

The stress and strain in an elastic material are related by the

elastic moduli of the materials, and the stress at any point in an

elastic body is the product of the strain at that point times the elas-

tic moduli at the same point. In a homogeneous half-space, the shear

stress was represented by








35

7 = Gy (103)


and in a nonhomogeneous half-space, the shear stress was represented by


TN = G(z)N (104)

For an incompressible half-space, Gibson showed that if TN = T

then the ratio of the strains becomes

N G
S- = (105)

Half-Space Under Torsion

The stress and strain conditions developed in a half-space due to

a torsional moment applied to a rigid circular disk on the surface of

the half-space are analogous to the half-space conditions that result

from Gibson's solutions.

The stresses developed in the half-space by the disk are indepen-

dent of the value of Poisson's ratio. Because Gibson assumed a

Poisson's ratio of 1/2 before obtaining solutions for the stresses,

his stress solutions are also valid for the same value of Poisson's

ratio.

Dilatational strains are not developed in the half-space by a

torsional moment applied to the rigid disk. Gibson assumed that the

half-space was incompressible, so, again, no dilatational strains were

developed by the surface loads.

A disk in torsion produces shear stresses on a circular area at

the boundary of the half-space. Gibson's solutions are also for a cir-

cular area loaded by a uniform vertical pressure at the surface of the

half-space.










The above similarities between Gibson's case and the torsional

loading situation lead to the hypothesis that the results of Gibson's

investigations were also applicable to a nonhomogeneous half-space

under torsion. Gibson's results, thus, indicate that the stresses de-

veloped in a homogeneous elastic half-space by a torsionally loaded

rigid disk on the surface of the half-space would be the same as the

stresses developed in a nonhomogeneous elastic half-space by the same

torsional load. The variation of the shear modulus with depth in the

nonhomogeneous half-space under torsional loads should be the same as

that assumed by Gibson: G(z) = G(O) + mz .


Torsional Oscillation

As mentioned before, rigorous solutions to the dynamic boundary

value problem of a rigid circular disk in torsional oscillation on the

surface of a nonhomogeneous elastic half-space have not been found and

are not attempted herein.

Engineers, however, are notoriously proficient in rationalizing a

sufficient number of plausible assumptions to circumvent rigorously in-

tractable problems (Zienkiewicz, 1967). Soil engineers, typically

faced with incomplete, inaccurate information and armed with inadequate,

inappropriate, and often untested theory, have been able to resolve a

variety of problems by the simultaneous application of available knowl-

edge and logical assumptions. The results are usually successful, but

sometimes they are not (Peck, 1967). The following paragraphs are

offered to bridge the gap between what is known and what is needed..









Assumption

Gibson's (1967) correlation between the stresses in a homogeneous

half-space and a nonhomogeneous half-space under static loads was con-

sidered adequate evidence to assume that the dynamic stresses in a homo-

geneous half-space and a nonhomogeneous half-space under dynamic loads

have the same correlation. Thus, it was assumed that the stresses

developed in a homogeneous elastic half-space by a torsionally oscil-

lating rigid circular disk on the surface of the half-space are the

same as the stresses developed in a nonhomogeneous elastic half-

space subjected to the same oscillatory loading.

Low frequencies.--When the frequency of the torsional loading is

low, the second time derivative of the particle displacement is small,

and the equations of equilibrium become nearly homogeneous. The stress

conditions in either the homogeneous half-space or the nonhomogeneous

half-space would approach the static loading situation, so, hypothet-

ically, the two half-spaces would have almost identical stress

conditions.

High frequencies.--The wave fronts propagating in a homogeneous

half-space are located on a spherical surface (Woods, 1968). Wave

fronts propagating in a nonhomogeneous half-space are functions of

source distance, surface reflections, and type of nonhomogeneity

(Byerly, 1942; Brown, 1965). Phase relationships are also distorted

complexly in the nonhomogeneous case--leading to frequency dependent

particle motions. So, for high frequency oscillations, there is prob-

ably less correspondence between the stresses in the two half-spaces.









Particle motion

The strain energy per unit volume generated in a torsionally loaded

homogeneous half-space is (Timoshenko and Goodier, 1951)

1 = 2 2 ( lo6)
W = (o9 + ue) (106)
2G re ze

and the kinetic energy of an oscillating particle in the half-space is

1 2
K = pdVv (107)


The strain energy per unit volume developed in a nonhomogeneous

half-space is


WN = 2GT re + (108)

and the kinetic energy of an oscillating particle in the nonhomogeneous

half-space is


K = pdVVN (109)

Equating the strain energy and the kinetic energy in each of the

above cases (Timoshenko and Goodier, 1951) gave

.2 1 2 2110)
v a + a (110)
Gp = re ze


and


.2 1/2 2
N = G( z ( re ze (1

The loading and mass density, p of each half-space was assumed equal,









so the stresses generated in each half-space were equal and the ratio

of the particle displacement in the nonhomogeneous half-space to the

particle displacement in the homogeneous half-space was



rv -G (112)
v VG z)


In summary, Gibson's (1967) approach to the nonhomogeneous half-

space problem implied that the particle velocities in a nonhomogeneous

(linear E) half-space can be determined from the particle velocities

(or displacements) in a homogeneous (constant E) half-space, as given

by Equation 112. The form of this particle velocity relationship is

similar to Equation 105 for the static strains in the two half-spaces.














THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCILLATING
RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT


Description of Test Site


The selected test site was located at an inactive auxiliary field

on the Eglin Air Force Base, Florida, military reservation. The soil

at the site was a homogeneous marine terrace deposit of poorly graded,

fine- to medium-grained sand. The water table was about 100 ft deep in

this thick, free-draining sand deposit and the shear wave velocity in-

creased significantly with depth.


Geographical Location and Geological Setting

The test site chosen for the experimental work was located in

section 14, range 24 west, township 1 north, Okaloosa county, Florida,

at about 86 degrees and 38 minutes west longitude and 30 degrees and

35-1/2 minutes north latitude. The circular test foundation and the

approximately 100-ft-square test area was about 1,050 ft west and 230 ft

north of the south end of the north-south runway at Piccolo field (aux-

iliary field 5), within the boundaries of Eglin Air Force Base and about

15 miles north of the Gulf of Mexico coastline. Piccolo field was

chosen as a test site because it was militarily inactive and the water

table was unusually deep. The elevation of the area was about 175 ft

above mean sea level and the topography was quite flat; elevations

within the test area varied less than 3 in. Native grasses covered the










ground surface and the area was lightly wooded with indigenous scrub

oak and pine.

Sand deposits in this vicinity are of geologically recent origin

(Cooke, 1945; Vernon and Puri, 1965). The Citronelle formation is

dated somewhere between the Pliocene and the Pleistouene epochs and is

no more than ten million years old; the terrace and fluvial terrace

formations laid down during the Pleistocene epoch are less than one

million years old. Stratigraphically, the Citronelle formation lies

unconformably on older formations, and is overlain by Pleistocene

terrace deposits.

The Pleistocene epoch was characterized by many changes in sea

level due to a sequence of glacial accumulation and subsequent melting.

Sea levels during that time were as much as 270 ft above current levels.

The water from melting glaciers carried a variety of soil material to

the sea where currents and wave action developed the sandy terrace de-

posits. Erosion during low sea levels and redeposition during high sea

levels created a generally flat topography with hidden stratigraphic

features. The three specific marine terraces that were associated with

deposits at the test site are the Brandywine formation, the Cohaire for-

mation, and the Sunderland formation.

Table 7 is a well log taken at auxiliary field 5 by the Layne Cen-

tral Co. and provided by the Directorate of Civil Engineering, Eglin Air

Force Base, Florida; it illustrates the general stratigraphic situation

near the test site.


Soil Exploration

The in situ soil exploration program at the test site was









Table 7
Well Log at Auxiliary Field 5

Depth Below
Ground Surface Well Driller's Identification of
ft Material Penetrated

0 to 20 Sand
20 to 84 Sand with white clay balls
84 to 110 Sand and white clay balls
100 Water table
110 to 156 Sand and gravel with white clay balls
156 to 212 Hard blue sandy clay
212 to 235 Sand, shells, and clay
235 to 307 Sand streaked with blue clay
307 to 343 Sand, shells, and clay
343 to 353 Clay
353 to 360 Hard rock
360 to 400 Clay, shells, and sand
400 to 480 Tough clay, shells, and sand
480 to 518 Soapstone and blue clay
518 to 533 Shell rock
533 to 555 Shell rock with soft places
555 to 585 Hard coarse rock and shells
585 to 598 Extra hard shell rock
598 to 620 Hard shell rock with soft places
620 to 643 Lime rock and very fine shells
643 to 650 Brown sand rock
650 to 666 Lime rock and brown sand rock
666 to 763 Lime rock







43

accomplished with a standard split spoon sampler and with a Begemann me-

chanical static friction cone penetrometer. Three holes were bored to a

depth of 60 ft with the standard sampler, and 20 penetrations were made

to an average depth of 70 ft with the cone penetrometer; locations are

shown in Figure 2. The purpose of these exploration efforts was to as-

sess the suitability of the site for conducting ground motion propaga-

tion experiments.


Borings

Holes 1 and 2 were continuously sampled with a standard split spoon

sampler (ASTM, 1969) and Hole 3 was sampled at 10-ft intervals using a

3-in.-diam, 18-in.-long Shelby tube. Tables 8 and 9 list the standard

penetration resistance of the sampled soil, the driller's visual clas-

sification, and his description of the soil retrieved from Holes 1 and

2, respectively. Hole 2 was located 135 ft west of Hole 1 and Hole 3

was located 15 ft north of Hole 2. The Shelby tube samples from Hole

3 were analyzed in the laboratory.


Penetration tests

Although no fine-grained materials were discovered by exploratory

borings at the test site, other boring and well log data taken at Eglin

have often indicated the presence of clay or marl. Such impermeable

layers could support a perched water table or could impede the infiltra-

tion of meteoric water. The occurrence of perched or transient water

within the mass of soil would, of course, cause density contrasts that

would be detrimental to precise and reproducible ground motion measure-

ments. An attempt to locate possible lenses of fine-grained material

within the selected test site was considered necessary.














MAGNETIC


UNLESS
UNLESS


O-- BORING LOCATION

SrQ PENETRATION LOCATION

C NORTH

----- ----4-

5-FT-DIAM
TEST FOOTING


HOLE 3
-E3"-- ,," ,., CD --1"-E
SCALE HOLE 2 HOLE 1
100' _H I
-^ '(

DIMENSIONED



I4I
4--J


Figure 2. Location of 3 exploration borings and 20 friction-cone penetrations.












Table 8
Boring Log for Hole 1


Sample Depth
ft

0.0 to 1.5

1.5 to 3.0

3.0 to 4.5
4.5 to 6.0
6.0 to 7.5
7.5 to 9.0
9.0 to 10.5

10.5 to 12.0
12.0 to 13.5
13.5 to 15.0
15.0 to 16.5

16.5 to 18.0
18.0 to 19.5
19.5 to 21.0
21.0 to 22.5
22.5 to 24.0

24.0 to 25.5

25.5 to 27.0
27.0 to 28.5

28.5 to 30.0

30.0 to 31.5
31.5 to 33.0
33.0 to 34.5
34.5 to 36.0

36.0 to 37.5

37.5 to 39.0

39.0 to 40.5
40.5 to 42.0
42.0 to 43.5
43.5 to 45.0
45.0 to 46.5
46.5 to 48.0
48.0 to 49.5

49.5 to 51.0
51.0 to 52.5
52.5 to 54.0

54.0 to 55.5
55.5 to 57.0
57.0 to 58.5
58.5 to 60.0


Driller's
Classification and Description
Symbol Description

SM Silty sand, fine grained with surface or-
ganic material--hair roots, etc.
Silty sand, fine grained with trace of
surface organic matter
Silty sand, fine grained


Moisture
Content


6

5

5
5
4
4
3

4
3
4
3


5
7
7
8
5

5

5
6

4

5
6
8
6

8

5

10
8
9
9
7
7
8

7
7
7

8
6
6
6


Reddish tan










Light tan SP-SM


Standard
Penetration Resistance
blows/ft Color

3 Medium brown

3 Brown

3 Tan
4
7
7
6 Light tan

8
11
13
18


23 Medium brown
20 Light red
19
17
15 Tan

16

16
15 Reddish tan

15 Tan

10I
18
18 Reddish tan
19 Tan

15

17

14 Light red


Sample Dept


SP-SM








SM
SC



SM







SP-SM




SC
SP-SM





SC


Sand, poorly graded, with silt fines con-
tent. Fine grained and sharp particles



I
Sand, poorly graded, fine grained. Sharp
particles with slightly silty fines

Silty sand, fine grained
Clayey sand, fine grained


1
Silty sand, fine grained with trace of
clay
Silty sand, fine grained with trace of
clay
Silty sand, fine grained
Silty sand, fine grained with trace of
clay
Sand, poorly graded, fine grained, sharp,
with trace of silt fines



Clayey sand, fine grained
Sand, poorly graded, fine grained with
silt fines
Sand, poorly graded, fine grained with
silt fines
Sand, poorly graded, fine sharp grains,
slight silt content
Clayey sand, fine grained







Clayey sand, fine grained with less
clay content


I
Clayey sand, fine grained with slight
clay content



I
Sand, poorly graded, fine, sharp grains,
trace of silt












Table 9
Boring Log for Hole 2


Moisture Standard
Sample Depth Content Penetration Resistance
ft blows/ft

0.0 to 1.5 6 5
1. to 3.0 5 2
3.0 to 4.5 5 1
4.5 to 6.0 4 6
6.0 to 7.5 4 7

7.5 to 9.0 4 7
9.0 to 10.5 5 9
10.5 to 12.0 6 8
12.0 to 13.5 6 12
13.5 to 15.0 4 10
15.0 to 16.5 6 17

16.5 to 18.0 4 14
18.0 to 19.5 5 19
19.5 to 21.0 6 24
21.0 to 22.5 5 22


22.5 to 24.0 4


24.0 to
25.5 to
27.0 to
28.5 to
30.0 to
31.5 to

33.0 to
34.5 to
36.0 to


37.5 to 39.0 7


39.0 to 40.5 5


40.5 to 42.0 4


42.0 to 43.5
43.5 to 45.0
45.0 to 46.5

46.5 to 48.0
48.0 to 49.5
49.5 to 51.0
51.0 to 52.5
52.5 to 54.0
54.0 to 55.5
55.5 to 57.0
57.0 to 58.5
58.5 to 60.0


Color Symbol


Brown
Light brown



Tan





Reddish tan





Whitish tan


I
Tan






T
Tannish
white







Greyish
white










Grey
Grey
Tan



I


SM




.~P-SM





SM
SM
SP-SM













SC








SP-SM








SP


Driller's
Classification and Description


Description


Silty sand, fine grained with surface
organic matter--roots, etc.



Sand, poorly graded, fine sharp
grains with slight silt




Silty sand with trace of clay
Silty sand with trace of clay
Silty sand with trace of clay
Sand, poorly graded, fine sharp
grains with slight silt



I
Sand, as above, with trace of clay
streaks
Sand, poorly graded, fine sharp
grains with slight silt



Clayey sand
Clayey sand
Clayey sand with less clay content
Clayey sand, fine to medium sharp
grains



Sand, poorly graded, fine to medium
sharp grains with slight silt
Sand, poorly graded, fine to medium
sharp grains, trace of silt and
clay
Sand, poorly graded, fine to medium
sharp grains, trace of silt and
clay
Sand, poorly graded, trace of silt



1
Sand, poorly graded, sharp fine
grains, trace of silt




Drill mud used
Drill mud used


SP-SM
SP-SM
SC
SC


____










The Dutch friction-cone penetrometer, a relatively new soil

exploration tool, offered the most practical means of investigating the

possible existence and extent of fine-grained sedimentary material at

the site; this tool can also be used to reveal density variations

within the mass of sand (Schmertmann, 1967; 1969). Eighteen soundings

were made to depths ranging from 60 to 70 ft, one to a depth of 82 ft,

and another to a depth of 102 ft. The 102-ft-deep sounding revealed

that it had nearly reached the elevation of the permanent water table.

The friction-cone penetrometer exploration did not reveal the

presence of cohesive soils within the investigated area that was sev-

eral hundred yards square. In addition, no perched water table condi-

tions were encountered. The cone bearing capacity data indicated that

there was a significant variation in the density of sand with depth;

however, the density variations with depth were quite consistent at

each sounding location. Thus, the depositional environment at the test

site apparently had laterally homogeneous characteristics that produced

a generally uniform horizontal stratification of the sand. The average

cone bearing capacity at various depths is listed in Table 10.

Table 10
Average Bearing Capacity of Static Cone Penetrometer

Bearing Capacity Bearing Capacity
Depth, ft kg/cm2 Depth, ft kg/cm2
1 28 35 92
5 28 40 104
10 59 45 109
15 99 50 115
20 128 55 123
25 103 60 134
30 89 65 152










Laboratory Tests

Laboratory tests were conducted on the Shelby tube samples ex-

tracted from Hole 3 before the friction-cone penetration tests were

performed. Six samples were obtained at depths of 2.5, 13, 19.5, 31,

41, and 51 ft.


Unit weight

The 18-in.-long Shelby tube samples were divided into three

equal increments. The natural unit weight of the sand retained in

each increment was measured and the color of the material was noted.

The sand from all of the increments in a single sample was then com-

bined and the maximum unit weight (minimum void ratio) and the mini-

mum unit weight (maximum void ratio) of the sample were determined.

Because the volume of the sample was less than 0.1 cu ft, standard

methods and apparatus could not be used. The minimum unit weight was

determined by filling a 2-in.-diam, 4-in.-deep mold with sand poured

from a standard 1/2-in.-diam funnel. The maximum unit weight was de-

termined by filling the mold with sand in three equal layers; each

layer was compacted by 25 blows of a 5.5-1b hammer falling 12 in. on a

2-in.-diam steel platen resting on the sand layer. Table 11 gives the

results of these laboratory tests.


Table 11
Results of Laboratory Tests on Samples from Hole 3

Dry Unit Weight, lb/cu ft
Depth, ft Increment Color Minimum Maximum Natural
1.5 to 3.0 Top) 98.8
Tan and light 90.9 113.2 108.8
2 o90.9 113.2 108.8
3 Continu100.2
(Continued)










Table 11 (Concluded)


Depth, ft
12.0 to 13.5


18.5 to 20.0



30.0 to 31.5


40.0 to 41.5



50.0 to 51.5


Increment
Top
2
3
Top
2
3
Top
2
3
Top
2
3
Top
2
3


Color

Tan

Reddish brown
to brownish
red
Brown
Brown
Reddish brown
Brown
Reddish brown
Reddish brown

Light red to
brown


Dry Unit
Minimum

90.2


84.1


84.8



83.9


84.1


Weight,
Maximum

109.9



109.3


105.3


108.5


106.4


Ib/cu ft
Natural
93.6
93.1
95.7
103.4
106.2
106.4


95.7
95.0
94.8
108.4
103.7
92.2
95.0
94.9


The average natural unit weight (dry) of all the sampled material

was about 99 lb/cu ft, the relative density was 62 percent, and, from

Table 8, the moisture content of the sand was approximately 5 percent.

Thus, the unit weight of the in situ sand was taken as 104 lb/cu ft.


Gradation

An indication of the uniformity of the sand deposit at the test

site was obtained from an inspection of the grain-size-distribution

curves for the sand material sampled at various depths. Similar grain

sizes and distributions at various depths indicate that the material

was deposited during the same or similar geological environments. The

sampled material from Shelby tubes extracted from Hole 3 had an ef-

fective grain size of about 0.14 mm, a uniformity coefficient of


__ r


I









about 2.5, and similar grain-size-distribution curves; Figure 3 shows

the grain-size distribution for these six samples. The uniformity of

the material sampled to a depth of 50 ft suggests that this zone of

sand might have been deposited by just one of the terrace formations

previously mentioned.


Seismic Wave Propagation Tests

Wave propagation tests, as described by Maxwell and Fry (1967),

were conducted to assess the shear wave propagation velocity of the in

situ sand deposit at the test site. The method employs a variable

frequency vibrator to generate Rayleigh waves along the surface of the

ground. An interpretation of the measured length of the propagating

Rayleigh wave with respect to the excitation frequency provides an ap-

proximation to the shear wave velocity at various depths.

Figure 4 is a plot of the results of these tests showing the vari-

tion of in situ shear wave velocity with depth. Figure 4 also shows

the shear wave velocity, V obtained by applying the empirical equa-
s
tions (Richart, Hall, and Woods, 1970)


V = (170 78.2e) a 025 (113)

-0.25
V = (159 53.5e) a2 (114)


and assuming a constant void ratio, e of 0.67 and an earth pressure

coefficient, K of 1/2 (Terzaghi, 1943).



















8 A SAMPLE DEPTH 1.5 TO 3.0 FT
80-- 20

S\ 0 SAMPLE DEPTH 12.0 TO 13.5 FT


I- X SAMPLE DEPTH 18.5 TO 20.0 FT
I-

w
S60 A- SAMPLE DEPTH 30.0 TO 31.5 FT 40


S\ 0 SAMPLE DEPTH 40.0 TO 41.5 FT LU


\ +\ SAMPLE DEPTH 50.0 TO 51.5 FT u
I.-




20 \, \ 80





20 --------_ __- -- --- -\3-------___- -- ------0-- -- ------ 8
w




20 ---0





A-
01I 100

1.0 0.5 0.1 0.05 0.01

GRAIN SIZE IN MILLIMETERS


Figure 3. Grain-size distribution for six sample depths.
















250


SHEAR WAVE VELOCITY IN FPS
500 750


1000


Figure 4. Shear wave velocity versus depth, surface and
empirical methods.


1250









Design of the Experiment


Foundation Design

This section discusses the considerations, approach, and calcula-

tions which were exercised to proportion and design a torsionally os-

cillating footing that served as the source of soil excitation during

the experimental phase of the investigation. The design goal for the

circular test footing placed on a natural sand deposit was to physi-

cally duplicate the boundary conditions assumed for a rigid circular

disk on an elastic half-space. Correspondence of experimental and

theoretical boundary conditions was deemed important for a valid com-

parison between experimental and analytical results. Essentially

elastic behavior of a vibrating foundation on soil was attained by

embedding the footing and limiting the torsion induced soil stresses;

a comparatively rigid foundation was simulated by controlling the flex-

ure of the footing.


Practical considerations

A rigid circular disk pressed vertically against the surface of a

smooth elastic half-space produces a hyperbolic distribution of verti-

cal stress along a radius of the disk which becomes infinite at the

edge of the disk (Timoshenko and Goodier, 1951), and development of

Equation 57 has shown that a torsional moment applied about the verti-

cal axis of a rigid disk produces a similar distribution of shear

stress along a radius of the disk in contact with the half-space. Soil,

or any other material, cannot resist infinite surface stresses and a

footing cannot be perfectly rigid, so the problem was to design and










construct a circular footing that would approximate the theoretical

boundary conditions as closely as possible. Limiting vertical stresses

and shear stresses on the footing-soil contact area are shown in Fig-

ure 5. Because only finite shear stresses can be mobilized near the

edge of the footing, the rotational stiffness of a footing on soil is

considerably less than the rotational stiffness of a disk on a half-

space (Richart and Whitman, 1967).

To mobilize large vertical stresses near the periphery of a cir-

cular footing on the surface of a sand material, correspondingly large

horizontal or confining stresses must be provided or the sand will

yield. Two possible methods of confining the sand at the edge of the

footing were: (1) provide a flexible surcharge such as air pressure

on a membrane, and (2) embed the footing in the sand.

Tests on vertically loaded laboratory scale footings on and in a

sand foundation show that the measured vertical pressure distribution

does not correspond to the theoretical distribution and that footing

embedment improves the correspondence (Chae, Hall, and Richart, 1965;

Drnevich and Hall, 1966; Ho and Lopes, 1969). While conducting his ex-

periments, Woods (1967) found that vibration measurements on the sur-

face of a sandy soil were very sensitive to changes in the near-surface

moisture conditions. Assuming that motion transmission from a surface

source would be similarly influenced by the moisture content of the

near-surface sand, consistent transmission was more likely to be at-

tained with an embedded footing than with a surface footing. The ef-

fect of embedment on the response of vertically oscillating footings is

to increase the resonant frequency and decrease the amplitude of










7.0







6.0







5.0







4.0







3.0







2.0







1.0







0o




Figure 5.


Distribution of stresses between a rigid disk and
an elastic half-space.









footing motion. This effect is quite small, however, for shallow

buried footings which have vertical faces isolated from the soil

(Lysmer and Kuhlemeyer, 1969; Richart, Hall, and Woods, 1970; Novak,

1970). It is likely, though yet untested, that the embedment effects

on a similar footing in torsional oscillation are also small. Thus,

embedding the footing in the sand and isolating the vertical face of

the footing appeared to be a practical method of resolving the con-

finement and vertical pressure distribution problem as well as the

soil moisture fluctuation problem.


Diameter of the test footing

The diameter of the footing was established on the basis of the

vibration frequencies that are commonly imposed on actual foundations,

the range of dimensionless frequency ratios that are usually encoun-

tered in the design of prototype foundations, and the average shear

wave velocity of the sand material at the test site.

Steady state foundation vibrations range from about 10 to 60 Hz

and dimensionless frequency ratios range from 0.2 to .5 (Richart,

Hall, and Woods, 1970). Shear wave velocities in the sand deposit in-

crease with depth; however, at the average 15- to 20-ft depth of the

particle velocity measuring stations, the shear wave velocity was ap-

proximately 650 fps. One form of the dimensionless frequency ratio is

defined by

2nfr
o V5)
s

Where V is the shear wave velocity of the soil. Using the above
S










definition of a and average values of the variables for the test

site, the footing radius was

aV
Os
r
o 2Trf


0.85(650)
2TT(35)


= 2.5 ft (116)


or a footing diameter of approximately 5 ft.


Stresses at the footing-soil interface

The desired stress conditions at the contact between the vibrating

footing and the soil were previously mentioned. The test footing

should develop similar oscillatory stress conditions at its contact

with the soil as were assigned in the Bycroft (1956) theory for the

contact area between a rigid circular disk and the horizontal boundary

of an elastic half-space.

Shear stresses.--The shear stress between a rigid disk and a

half-space due to an oscillatory moment applied to the disk about its

vertical axis of symmetry is


-3M r ( / .
z9 3 F2 2
44rrrr r r

For a torsionally loaded circular footing resting on a sand foundation,

the shear stresses in the sand on the footing-soil interface depend on

the friction developed between the bottom of the footing and the sand.

The limiting value of the shear stress, azt is related to the normal









stress, a acting on the plane of contact by


aze = azz tan (118)


where tan is the coefficient of friction between the bottom face

of the footing and the sand. This equation indicates that a vertical

dead load must be applied to the test footing in order to develop the

necessary normal stresses between the footing and the foundation.

If the shear modulus of the material used to construct a solid

cylindrical test footing is much greater than the shear modulus of the

sand on which it rests, negligible distortion of a radius of the foot-

ing in contact with the soil would occur as a torsional moment was

applied to the footing. The shear modulus of concrete was 300 times

the shear modulus of the sand on the contact area between the footing

and the soil, so a concrete footing was considered to be rigid with

respect to the soil. Further, if the shear stresses in the soil at

the footing-soil interface are limited to about 1/3 of the failure

(slip) value, and these stresses are repetitive, laboratory tests on

sand show that these soils will behave elastically (Timmerman and Wu,

1969).

Since a concrete test footing would be rigid with respect to the

soil, and the soil would behave elastically during torsional footing

oscillations, it was reasonable to expect the shear stresses on a large

portion of the footing-soil interface to be similar to those developed

by a rigid disk on a half-space. Figure 5 shows the probable distribu-

tion of torsion induced shear stresses on the contact area.

So, to represent the boundary conditions assumed by the Bycroft










theory, the footing had to be rigid with respect to torsional deforma-

tion, i.e., a radius of the footing in contact with the soil should not

be distorted during rotation by an applied torsional moment, and the

footing had to be rigid with respect to flexure in a vertical plane.

Rigidity in this plane means that vertical dead loads applied to the

footing cause negligible bending of any footing radius in a vertical

plane that contains that radius.

Vertical stresses.--A flexurally rigid circular disk, pressed

vertically against the horizontal boundary of an elastic half-space,

develops normal stresses, ozz on the contact area of


zz = (119)
o 2 2
2iro ro r


where P is the total load applied to the rigid disk (Timoshenko and

Goodier, 1951). If a vertical load were applied to a rigid circular

footing resting on soil, and the magnitude of this load was limited

such that the normal stresses between the footing and the soil were

about 1/2 of the stresses that would initiate local failure of the

soil, the soil would react in an essentially elastic manner (Timmer-

man and Wu, 1969). These conditions were prescribed for the designed

test footing, so the distribution of normal stresses on most of the

contact area between a lightly loaded rigid footing and an elastic

foundation material was probably similar to that given by Equation

119. Figure 5 shows the probable distribution of vertical stresses

on the contact surface.

Summary.--It was considered that the stress distribution between








60

a rigid disk and an elastic half-space was a reasonable approximation

to the stress distribution between a lightly loaded rigid footing and

the soil on which it rests provided that the soil stresses were less

than about 1/3 to 1/2 of the value necessary to cause local soil fail-

ure on a large part of the contact area between the footing and the

soil. Unfortunately, reliable measurements of the distribution of

normal and shear stresses in this situation were beyond the current

state-of-the-art.


Stresses near the periphery of the footing

Figure 5 shows that the maximum contact stresses occurred near

the periphery of the circular disk and that slippage between a tor-

sionally loaded circular footing and its foundation was most likely

near the periphery of the footing because the torsion induced shear

stresses approach the normal stresses in this region. This section

views the critical stress region in more detail.

The critical stress conditions were considered to be represented

by the limiting equilibrium state of plane stress for an element (see

Figure 1) of cohesionless soil located on the contact plane near the

circumference of the vertically loaded rigid footing. The lateral

pressure confining the sand at the edge of the footing was taken as

the peak passive soil pressure attainable at this point (Terzaghi,

1943).

Si + sin (Th) (120)
Trr r=ro 1 sin


The maximum vertical stress on the sand under the edge of the footing









necessary to mobilize the passive confining pressure, arr at this

point was (Terzaghi, 1943)


+sin ( ) (121)
1 + sin
0zz 1 sin rr (121)

so the limiting vertical stress near the edge of the footing was


( sin Yh (122)
azz = sinP

where 7 is the effective unit weight of the soil, h is the depth of

the element below the surface of the soil, and 0 is the angle of in-

ternal friction of the soil.

The above equation had implications which influenced the design

and placement of the test footing; it indicated the need for lateral

confining stress at the edge of the footing and the footing burial

necessary to attain confinement. Of course, to conform to the geomet-

ric boundary conditions assumed in the theory, the footing must be as

near the ground surface as possible, and, to conform to the stress con-

ditions assumed in the same theory, the footing must be buried as

deeply as possible with the vertical surface of the footing isolated

from the soil. This dilemma was resolved by recalling that the pri-

mary objective of the experimental work was to assess the particle mo-

tion in a large mass of soil extending to nearly 100 ft from the motion

source, so distorting the geometric position of that source should have

little effect on the measurements; burying the source 1 ft, or 20 per-

cent of its diameter, was judged to be an acceptable bias of the geo-

metric boundary conditions at the source. With this depth of burial,

the maximum normal stress developed near the periphery of the contact







62

surface between the buried test footing and the soil due to a vertical

load on the footing was


ti + 0.5 2 (104)(1)
)= 0i2 )
azz 0.5)

= 936 psf


= 6.5 psi (123)


where 7 and are taken as 104 lb/cu ft and 30 degrees, respec-

tively. Figure 5 shows the limiting stress distribution.

The significance of the inelastic vertical stresses developed near

the periphery of the footing was implied by calculating the portion of

the footing-soil contact area on which elastic stresses act to the

total contact area. Assuming that the 5-ft-diam test footing was a

solid cylinder of concrete, 2 ft high and buried 1 ft in the soil,

Equation 119 was used to compute the approximate radial position, rI ,

of the maximum normal stress of 6.5 psi. If the unit weight of con-

crete is taken as 150 lb/cu ft, and the entire footing acts as a rigid

body


r = 2.47 ft (124)


Thus, for the case assumed, nearly 98 percent of the contact area be-

tween the footing and the soil had normal stresses that were less than

6.5 psi and transmission of elastic stresses during torsional oscilla-

tion of the footing occurred over some 92 percent of the footing-soil

contact area (Timmerman and Wu, 1969). Also, because the shear

stresses at the edge of the footing are less than those at the edge








63

of a rigid disk, the rotational stiffness of the footing, at the limit-

ing moment, was about 75 percent of the rotational stiffness for a

rigid disk.


Footing emplacement operation

Previous paragraphs established the size and position of the test

footing and discussed the desired stress conditions on the footing-soil

contact area. This section sets forth the footing design and placement

method.

To transmit torsional oscillations into the soil, the friction

angle between the base of the footing and the soil should be comparable

to the angle of internal friction, 0 of the soil. This objective was

met by using concrete as the footing material in contact with the soil.

Intimate and uniform contact between the base of the footing and the

sand should result by pouring the concrete directly on the prepared

sand surface.

The only contact allowed between the buried footing and the soil

occurred on a horizontal circular area, so the vertical face of the

circular footing had to be isolated from the soil. This was accom-

plished by a thin steel ring placed between the soil and the cylindri-

cal surface of the footing.

The distribution of vertical contact stresses between the base of

the footing and the soil should be similar to that developed by a rigid

circular disk pressed vertically against a half-space. This objective

was realized by considering the soil stress conditions after excava-

tion, after pouring the footing, and after the application of a dead











load to the cured footing. The last condition is discussed in a fol-

lowing section.

Figure 6 is a sketch of the embedded concrete test footing. The

following sequence of placement operations resulted in the desired ver-

tical stress distribution on the contact surface between the footing

and the soil.

1. The natural soil within a 7-ft-diam circle was excavated to a

depth of 1 ft. The stress change in the soil due to the excava-

tion was equivalent to a uniform unloading pressure of 7h act-

ing on the excavated area.

2. The concentric footing form and soil retaining ring were

placed in the excavation, the first pour of concrete was placed

inside the footing form to a depth that produced a uniform pres-

sure of 7h on the soil, and backfill soil was placed around the

retaining ring to the original ground surface and at its original

in situ density. At the end of these operations, the stresses in

the soil on the footing-soil contact surface and in the vicinity

of the footing were approximately the same as the in situ stresses

before excavation because the added loads were equal to the loads

removed during excavation.

3. The first pour of concrete was allowed to cure to a rigid

mass.

4. A second pour of concrete was added inside the form to act as

a dead load on the rigid first pour. It was this dead load, prop-

erly applied, that produced the desired distribution of normal

stresses on the footing-soil contact area.












PLAN VIEW


A


t


A











RING


GROUND


.I '
SAND FOUNDATION
SECTION A-A

Figure 6. Sketch of concrete footing embedded in soil.

Figure 6. Sketch of concrete footing embedded in soil.










Position of dead load on cured first pour

As mentioned previously, the dead load applied to the cured first

pour was positioned so that the plane contact area between the footing

and the soil was not distorted under this load; i.e., the cured first

pour simulated a rigid disk as it was pressed against the underlying

soil by the dead load.

Figure 7 shows a cross section of the cured first pour with an

axisymmetric, uniformly distributed dead load acting on a part of its

upper surface and a footing-soil contact stress distribution, that

would be developed by a rigid footing, acting on its lower surface.

The simple dead load pressure distribution was chosen to minimize

forming and placement problems as the dead load concrete was placed

on the cured first pour.

If the locus of the center of pressure for the dead load is coin-

cident with the center of pressure acting on the contact area, bending

of the cured first pour, due to the dead load, should not be signifi-

cant; the position of the center of pressure in each case was a cir-

cle with its center on the vertical axis of the footing. In essence,

this approach considered that the soil pressure due to the uniformly

distributed dead load, q acting on the cured first pour of the foot-

ing was represented by an equivalent load distributed along a circle

of radius, r (Richart, 1953). The distance, r to the center of pres-

pressure on the bottom of the footing was






















































SOIL PRESSURE

Figure 7.


ON BASE DUE TO DEAD LOAD

Distribution of vertical
soil stress and dead load
pressure on cured first
pour.







68
r
0
Srozz(rdrde)
r (125)

J zz(rdrde)
0


4 o
Coincidence of the center of pressure for the dead load required that

r


Sq(rdrde)
r = r (126)


r2

from which

r = 0.52035ro (127)



Rigidity of the footing

Having chosen the material for the footing and established the di-

mensions, depth of burial, sequence of emplacement operations, and

system for applying the dead load, it was necessary to evaluate the de-

sign to assess the effective rigidity of the test footing with respect

to the soil. A flexurally rigid footing was desirable to assure the

proper vertical stress distribution on the footing-soil contact area.

The method of evaluating the relative rigidity of the test footing

with respect to the soil was to compute and compare the vertical de-

flection of the soil in the footing-soil contact area for two equiva-

lent loading situations. The first case assumed that the footing was







69

completely flexible and was loaded by a uniformly distributed pressure

acting on its entire upper surface; the load-deformation relations ob-

tained illustrate the rigidity or stiffness of the soil. The estimated

elastic properties and design dimensions of the test footing with an

advantageously positioned equivalent dead load were used to calculate a

second load-deformation relationship for the contact area that illus-

trated the rigidity of the footing. The relative rigidity of the foot-

ing with respect to the soil was taken as the ratio of the footing

load-deformation relationship to the soil load-deformation relationship.

In the first case, Young's modulus, Es for the foundation mate-

rial at a depth of 2-1/2 ft was about 7,050 psi, and Poisson's ratio

was about 1/3. The deflection of the soil at the edge of the flexible

footing, due to the uniform load, q acting on its entire surface is

(Timoshenko and Goodier, 1951)

/ \ 4(1 2)qr (
(w) = E (128)


and at the center of the footing

2(1 2)qro
(w = E (129)
5


The deflection, A within the contact area is (w) (w)r= so


a \ b (130)

S

= 0.00275q in.


where q is expressed in pounds per square inch.







70

In the second case, Young's modulus for concrete, Ec was assumed

to be 3 x 10 psi and Poisson's ratio was taken as 0.17 (Dunham, 1953;

Lin, 1955). The cured first pour of the footing had a diameter of 5 ft

and a planned thickness of about 8-1/4 in. Before computing the actual

design situation, the deformation of the contact area estimated by the

center deflection of an edge supported circular plate supporting a

uniform load, q acting over its entire surface was calculated. Using

thin plate theory (Timoshenko and Woinowsky-Krieger, 1959), the deflec-

tion, Acl is

3(5 + p)(1 )r
A q .(131)
Acl 16E t3(131)
c

= 0.000387q in.


Even under these unrealistic support and loading assumptions, the in-

fluence of the stiffness of the cured first pour decreased the deflec-

tion of the footing-soil contact area by a factor, S of

A
S s (132)
cl

0.00275q
0.000387q


S7.1


The deformation of the footing-soil contact area was next approx-

imated by the deflection of the cured first pour simply supported along

a circle of radius r and loaded by a uniformly distributed dead load

with its center of pressure lying on a circle of radius r ; see











Figure 7. These support and loading assumptions were believed realis-

tic for computing the approximate deformation of the contact area; com-

puting deflections for a flexible, finite plate on an elastic founda-

tion were considered unnecessarily complicated and tedious. With

r2 = 16 in., the dead equivalent load, q that acted on only part of

the circular area, was 1.40 q where q is the uniform dead load

pressure previously assumed to act over the entire area of the footing.

Superposition of three loading situations given by Timoshenko and

Woinowsky-Krieger (1959) was accomplished to calculate the maximum

deflection of the center of the first pour with respect to its edge due

to the distribution of the dead load. A similar displacement of the

contact area was assumed, and, to slide rule accuracy, this displace-

ment, Ac2 was

c2
Ac2 = 0.0000074q (133)

= 0.0000103q

The ratio of the deformation of the contact area for a completely

flexible footing to the deformation of the contact area under the

cured first pour supporting a selectively located equivalent dead load

was used to judge the degree of rigidity of the footing with respect to

the foundation soil. For the above situation, this ratio, S was


S = 0.00275q (134)
0.0000103q

= 266

Thus, since the deformation of the contact area was reduced to less

than 0.5 percent of its free deformation by the effective rigidity of









the cured first pour and the soil stresses were limited to elastic

values, it was reasonable to expect the soil pressure distribution on

the contact area to be nearly the same as that for a rigid footing on

an elastic foundation.


Limiting torsional moment

The maximum torsional moment that could be applied to the test

footing without causing slippage on the contact area was limited by the

critical normal stress developed in the soil, by the dead load on the

footing, and by the location of this critical stress on the contact

area. With the assumption that the friction angle between the footing

and the soil was the same as the angle of internal friction for the co-

hesionless sand foundation, the moment, M' to cause impending slippage

on the contact surface at the locus of the critical normal stress was

computed from

r 2 zz2 2 r 2~
oz1 r3drd
M' f f r f f zz tan r2dedr (135)
o o tr r r o


where az is the critical vertical stress on the contact area and r1

is the radial distance to the location of this critical stress. Sub-

stituting ro = 2.5 ft ozz = 936 psf r1 = 2.47 ft and 0 = 30

degrees into the above expression, the limiting moment, M' is


M' = 4,400 ft-lb (136)


The moment capacity of a rigid disk on a half-space with a similar










shear stress distribution on the contact area, except at the edge,

would be about 5,750 ft-lb. With an assumed soil shear modulus of

4,500 psi under the footing, the angular rotation, due to the

limiting moment was about


"=- 3M'
16Gr3

= 0.0001 radians (137)


and the single amplitude displacement, v of the outside edge of the

footing was 0.003 in.

Transmission of predominantly elastic shear stress into the soil

was possible by limiting the applied moment to less than 1/3 of the

value necessary to initiate slippage at the periphery of the footing-

soil contact area. The design moment, Md and the design displace-

ment, vd were


Md 1,400 ft-lb (138)

and

Vd 0.001 in. (139)


Dynamic response of the foundation

The geometry, weight, and operating frequency of the designed

test foundation were previously established, and the elastic proper-

ties of the test site were estimated and measured. These parameters

were used to determine the resonant frequency of the torsionally vibra-

ting foundation and the static amplitude magnification factor at reso-

nance. The frequency at which resonance occurs and the expected











increase in foundation motion were pertinent to the design of a tor-

sional vibrator to drive the test footing and to the expected experi-

mental measurements.

The designed test footing was a hollow concrete cylinder. The

diameter of the cylinder was 60 in. and it was 28 in. high. A 32-in.-

diam, 18-in.-high cylindrical void was formed interior to and sym-

metric with the outer cylinder; the top of the void was 2 in. below

the top of the outer cylinder. The mass ratio of the footing was 2.1,

the resonant frequency was about 40 Hz, and the amplitude magnifica-

tion ratio was approximately 5.2 (Richart, Hall, and Woods, 1970).

Although the resonant frequency occurred within the planned test fre-

quency range, the magnification of the footing motion did not present

apparent experimental difficulties.


Vibrator Design

A special vibrator was designed to drive the test footing in a

torsional mode of oscillation about the vertical axis of the circular

footing. The vibrator design goals were (a) to minimize mechanical

sources of noise and vibration, (b) to utilize a remote source of

power, and (c) to keep all dynamic forces in a horizontal plane.

Goal (a) was attempted by using an electric motor to drive the rota-

ting eccentric masses through a rubber timing belt, goal (b) by a

long electrical power line from the electric motor to a remote

generator, and goal (c) by rotating the eccentric masses in a hor-

izontal plane. The vibrator was designed to provide a twisting moment

of about 3 ft-kip in the 20- to 50-Hz-frequency range.










Figures 8 and 9 are plan and elevation sketches of the vibrator

components; the layout of these components was dictated by the config-

uration of the test footing. The eccentric masses were separated as

far as practical to develop large twisting moments with small centri-

fugal forces, the horizontal plane containing the rotating masses

was kept as low as practical to reduce rocking of the footing by un-

balanced forces, and the structural frame was extremely rigid to raise

sympathetic vibration frequencies well above the torsional frequencies

applied to the foundation.

Design details such as timing belt layouts, sprocket sizes, bear-

ing loads, and shaft bending and whirling are not mentioned in the fol-

lowing paragraphs, but a discussion of the input power requirements

and torque capacity of the vibrator was considered pertinent to the

design and conduct of the experimental work.


Power requirements

A gross approximation of the power expended on the footing-soil

contact area by a torsionally vibrating footing was made by assuming

a simple relationship between the twisting moment, M applied to the

footing and the rotation, 0 of the footing. To find an upper bound

for the power losses through the foundation, the M versus 0 rela-

tionship was assumed rigid-plastic. The work done by the footing on

the soil per cycle was 4M the work done per second at a frequency

of 50 Hz was 200*M and the horsepower expended was 200MO/550

Using the design moment of 1,400 ft-lb and the design rotation of

0.00004 radians, the power loss, p at a frequency of 50 Hz was




















EDGE OF FOOTING


IDLER


Plan view sketch of torsional vibrator.


Figure 8.























MOTOR


FOOTING


Figure 9. Elevation view sketch of torsional vibrator.











550


= 0.02 hp (140)


Power losses to the ground appeared to be negligible; power losses due

to belt friction, windage, and bearing friction controlled the selec-

tion of an electric motor.

Four precision, sealed, permanently-lubricated, self-aligning

ball bearings were chosen to support the two shafts for the rotating

eccentric weights. With a torque capacity of 3 ft-kip and a shaft

center distance of 31 in., the radial loading on each shaft was about

1,200 lb and the maximum radial bearing load was about 800 lb. At a

frequency of 50 Hz, the peripheral velocity of the 1.625-in.-dian

shaft was 1,280 fpm. Using the average tabulated coefficient of fric-

tion for ball bearings (Oberg and Jones, 1949), the horsepower loss

for the four bearings was


p = (2,400)(1,280)(0.0023)/33,000


= 0.21 hp (141)


The belt friction and windage losses were taken equal to the bearing

losses and a 1-hp motor was judged adequate to drive the torsional

vibrator.


Frequency and moment capacity

The eccentricity of the rotating masses on the torsional vibrator

was 4 in. and the masses were varied to change the torque output at











a constant frequency. The mass was changed by bolting different com-

binations of identically matched weights to each vibrator flywheel.

Table 12 gives the stamped identification letter and weight of each

of the paired eccentric masses and the attaching bolts.


Table 12
Identification Letter and Weight of Eccentric Masses


Mass
Identification

A

B

C

D

E


Weight
lb

1.9771

1.9786

0.9777

0.4770

0.2314


The centrifugal force developed

masses is We) /g so the moment, M

brator was


M 4 W
12 )W(2


Bolt
Identification

E

F

G

H

I through N


Weight
lb

0.135

0.135

0.106

0.106

0.084


by each of the rotating eccentric

,generated by the torsional vi-


ft-lb


(142)


where W is the weight (lb) of the eccentric mass and f is the fre-

quency (Hz). The moment output of the vibrator per pound of eccentric

mass was


M = 1.0616f2
W


(143)


This relationship and Table 12 were used to calculate the moment










developed by the torsional vibrator for typical mass combinations at

frequencies of 15, 20, 30, 40, and 50 Hz. The calculated results are

given in Table 13 and show that the vibrator was capable of developing

the footing design moment of nearly 1,400 ft-lb at a frequency of

15 Hz.


Table 13
Vibrator Moment Capacity at Various Frequencies


Total
Frequency Weight Bolt Weight Moment
Hz Identification Identification lb ft-lb

15 ABCDE E 5.7768 1,380
20 ACD G 3.5378 1,502
30 CD I 1.5387 1,470
40 CD I 1.5387 2,614
40 DE I 0.7924 1,346
50 C I 1.0617 2,818
50 D I 0.561 1,489




Foundation and Transducer Location

Location of the test footing

The Piccolo field test area that had been investigated during the

soil exploration program was examined to locate a favorable site for

placing the circular test foundation. Likely locations were probed

with a portable cone penetrometer similar to that described by Poplin

(1969); its use is shown in Figure 21. The soil was probed to a depth

of 6 to 18 in. on a grid spacing of 5 ft to delineate unusually soft

or hard areas. Of several suitable sites, one was chosen which offered










good lines of sight for surveying work and a minimum number of large

trees; wind action on trees can cause undesirable ground movements.


Location of transducers

The location and position of the particle velocity transducers

with respect to the test foundation was based on (a) the planned future

uses of the test area, (b) the sensitivity of commercial transducers

and available recording systems, (c) the expected wavelengths of

vibrator generated seismic waves, and (d) the possibility that the

test foundation was located on a laterally nonhomogeneous sand deposit.

Future uses of the test area include measurements of ground vibra-

tion developed by other types of foundations in various modes of oscil-

lation. Such experiments would produce surface or Rayleigh waves that

have significant motion to a depth of about one wavelength (Woods,

1968). For the site conditions discussed in a previous section, one

Rayleigh wavelength at 20 Hz would be equal to about 33 ft; transducer

locations to a maximum depth of 35 ft were judged adequate for the test

area.

Commercial particle velocity transducers, suitable for borehole

placement, were available with damped transduction sensitivities of

about 1 to 2 volts/in./sec, available amplifiers had a maximum gain of

about 2,000, and the sensitivity of available high performance gal-

vanometers was about 9 in./volt. Thus, if a 1-in. amplitude oscil-

lograph record is desired, the particle velocity of the transducer

must be at least 1/18,000 in./sec. The decrease in particle motion

with distance from the source of motion, or geometrical damping




Full Text
Table 39
Value of l(0.48,a,b)
220
b
0.000
0.192
a Real Imaginary Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
4.091
-
01
-5.997
-
03
4.092
-
01
0.48
7.351
-
01
-1.179
-
02
7.352
-
01
0.72
2.054
-
01
-I.718
-
02
2.06l
-
01
0.96
1.183
-
01
-2.198
-
02
1.204
01
1.20
8.034
_
02
-2.604
_
02
8.446
_
02
1.44
5.832
-
02
-2.923
-
02
6.523
-
02
1.68
4.288
-
02
-3.146
-
02
5.319
-
02
1.92
3.066
-
02
-3.267
-
02
4.481
-
02
2.16
2.042
-
02
-3.286
-
02
3.868
-
02
2.40
1.174
-
02
-3.204
-
02
3.412
-
02
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
2.113
-
01
-5.975
-
03
2.114
-
01
0.48
2.737
-
01
-1.174
-
02
2.74o
-
01
0.72
1.742
-
01
-1.711
-
02
1.750
-
01
0.96
1.111
-
01
-2.189
-
02
1.132
-
01
1.20
7.756
-
02
-2.594
-
02
8.178
-
02
1.44
5*666
-
02
-2.911
-
02
6.370
-
02
1.68
4.169
-
02
-3.133
-
02
5.215
-
02
1.92
2.989
-
02
-3.254
-
02
4.418
-
02
2.16
2.004
-
02
-3.272
-
02
3.836
-
02
2.40
1.158
-
02
-3.189

02
3-393
_
02
2.64
4.276
-
03
-3.014
-
02
3.044
-
02
2.88
-1.966
-
03
-2.755
-
02
2.762
-
02
3.12
-7.164
-
03
-2.425
-
02
2.528
-
02
3.36
-I.131
-
02
-2.040
-
02
2.332
-
02
3.60
-1.441
-
02
-1.616
_
02
2.165
_
02
3.84
-1.646
-
02
-1.172
-
02
2.021
-
02
4.08
-1.750
-
02
-7.258
-
03
1.895
-
02
4.32
-1.760
-
02
-2.947
-
03
1.784
-
02
4.56
-1.682
-
02
1.053
-
03
1.686
-
02
4.80
-1.530
-
02
4.600
-
03
1.598
-
02
5.04
-1.316
-
02
7.580
-
03
1.519
-
02
5.28
-1.055
-
02
9.906
-
03
1.447
-
02
5.52
-7.629
-
03
1.152
-
02
1.382
-
02
5.76
-4.561
-
03
1.242
-
02
1.323
-
02
6.00
-1.503
-
03
1.259
_
02
1.268
-
02
6.24
1.398
-
03
1.210
-
02
1.218
-
02
6.48
4.010
-
03
1.101
-
02
1.172
-
02
6.72
6.225
-
03
9.422
-
03
1.129
-
02
6.96
7.958
-
03
7.442
-
03
1.090
-
02
(Continued)


54
construct a circular footing that would approximate the theoretical
boundary conditions as closely as possible. Limiting vertical stresses
and shear stresses on the footing-soil contact area are shown in Fig
ure 5. Because only finite shear stresses can be mobilized near the
edge of the footing, the rotational stiffness of a footing on soil is
considerably less than the rotational stiffness of a disk on a half
space (Richart and Whitman, 1967)*
To mobilize large vertical stresses near the periphery of a cir
cular footing on the surface of a sand material, correspondingly large
horizontal or confining stresses must be provided or the sand will
yield. Two possible methods of confining the sand at the edge of the
footing were: (l) provide a flexible surcharge such as air pressure
on a membrane, and (2) embed the footing in the sand.
Tests on vertically loaded laboratory scale footings on and in a
sand foundation show that the measured vertical pressure distribution
does not correspond to the theoretical distribution and that footing
embedment improves the correspondence (Chae, Hall, and Richart, 1965;
Drnevich and Hall, 1966; Ho and Lopes, 1969)- While conducting his ex
periments, Woods (1967) found that vibration measurements on the sur
face of a sandy soil were very sensitive to changes in the near-surface
moisture conditions. Assuming that motion transmission from a surface
source would be similarly influenced by the moisture content of the
near-surface sand, consistent transmission was more likely to be at
tained with an embedded footing than with a surface footing. The ef
fect of embedment on the response of vertically oscillating footings is
to increase the resonant frequency and decrease the amplitude of


127
Table 14 lists the serial number of the transducer, its model des
ignation, its location, and its damped transduction constant.
Table 14
List of Transducers, Locations, and Transductpon Values
Location
Transduction in y
olts/ in./sec
Serial
Radial
Depth
Vertical
Radiajl Transverse
Number
ft
ft
Component
Component Component
Model
L-1B-3DS
Three-C omponent
Transducers
1
30.0
1.0
1.80
1.7C
) 1.70
3
30.0
5.0
1.70
1.7a
) 1.80
4
3-5
1.0
1.70
1.75
i 1.70
5
90.0
5.0
1.75
I.73
¡ 1.70
8
90.0
15.0
1.65
1.73
i 1.80
9
2.646
Footing
1.85
1.8 1.80
10
30.0
15.0
1.80
1.73
i 1.75
11
30.0
25.0
1.80
1.65
; 1.80
12
90.0
35.0
1.75
l.6c
) 1.70
13
30.0
35.0
I.65
1.75
¡ 1.75
14
10.0
5.0
1.80
1.8c
) 1.75
17
10.0
25.0
1.70
1.85
; 1.80
18
10.0
35.0
1.70
1.75
¡ 1.80
19
90.0
1.0
1.75
1.75
; 1.75
20
10.0
1.0
1.75
1.75
i 1.75
21
90.0
25.0
1-75
1.75
i 1.75
22
6o.o
35.0
1.75
1.75
1.75
23
6o.o
1.0
2.36
2.36
¡ 2.36
24
60.0
5.0
2.36
2.36
¡ 2.36
25
6o.o
25.0
2.36
2.36
i 2.36
26
6o.o
15.0
2.36
2.3
¡ 2.36
27
10.0
15.0
2.36
2.3
¡ 2.36
Model L-1D Single-Component Transducers
L-1D-BT
2.500
1.5


1.43
L-1D-TT
2.344
Footing
--

1.43


66
Position of dead load on cured, first pour
As mentioned previously, the dead load applied to the cured first
pour was positioned so that the plane contact area between the footing
and the soil was not distorted under this load; i.e., the cured first
pour simulated a rigid disk as it was pressed against the underlying
soil by the dead load.
Figure 7 shows a cross section of the cured first pour with an
axisymmetric, uniformly distributed dead load acting on a part of its
upper surface and a footing-soil contact stress distribution, that
would be developed by a rigid footing, acting on its lower surface.
The simple dead load pressure distribution was chosen to minimize
forming and placement problems as the dead load concrete was placed
on the cured first pour.
If the locus of the center of pressure for the dead load is coin
cident with the center of pressure acting on the contact area, bending
of the cured first pour, due to the dead load, should not be signifi
cant; the position of the center of pressure in each case was a cir
cle with its center on the vertical axis of the footing. In essence,
this approach considered that the soil pressure due to the uniformly
distributed dead load, q acting on the cured first pour of the foot
ing was represented by an equivalent load distributed along a circle
of radius, r (Richart, 1953)- The distance, r to the center of pres-
pressure on the bottom of the footing was


MOTOR
Figure 9. Elevation view sketch of torsional vibrator.


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS 11
LIST OF TABLES viii
LIST OF FIGURES xi
LIST OF SYMBOLS xiv
ABSTRACT xviii
INTRODUCTION 1
Background
Previous Work 3
Related Work 3
Approach to the Investigation 4
Available Theory
Available Experimentation
Comparisons 3
Objective and Goals-- 3
THE THEORETICAL PARTICLE MOTION GENERATED BY THE TORSIONAL OSCIL
LATION OF A WEIGHTLESS, RIGID, CIRCULAR DISK ON THE SURFACE
OF AN ELASTIC HALF-SPACE 9
Homogeneous (Constant E) Elastic Half-Space 9
Problem Statement and Approach 9
Equations of Elasticity 10
Solution to the Equilibrium Equation l4
Boundary Conditions 15
Applied moment and disk rotation 18
Particle Displacements 19
Evaluation of the Infinite Integral 20
Example calculation 24
Computer Program to Evaluate the Integrals 29
Nonhomogeneous (Linear E) Elastic Half-Space 30
Literature 32
Results of Gibson's Solutions 33
Solution for stresses 33
Strain relationships 34
Half-Space Under Torsion 35
Torsional Oscillation 36
Assumption 37
Particle motion 1 38
IV


26o
8' I
to print timing lines on the moving paper at intervals of 0.01, 0.1,
and 1 sec. This oscillograph type uses 12-in.-wide paper, 250 to 1+75
ft long enclosed in a lightproof magazine.
The Lino-Writ 4 photorecording paper was made by the E. I. du Pont
de Nemours Co., Wilmington, Delaware. The rolls were 12 in. wide,
0.0025 in. thick and 400 ft long. The spectral response of this light-
sensitive paper was orthochromatic and the maximum writing speed with
the standard oscillograph light source was 50*000 in./sec.
Paper Processor
A Consolidated Electrodynamics Corporation type 23-109B oscillo
gram processor was used to develop the exposed photorecording paper.
Reference (Calibration) Voltage Supply
A Consolidated Electrodynamics Corporation, model 3-1^0 power
supply was used as the voltage source for the calibration records.
This unit is completely solid state and provided a regulated DC
voltage that was continuously adjustable from 1 to 24 volts at up to
0.200 amperes. Although 60 Hz-power was supplied to the unit, it
would operate on frequencies from 48 to 420 Hz; it required 25 watts
at 90 to 135 volts. Fifteen minutes of warm-up time was needed for
stable operation; the ambient operating temperature range was 0 to
50 degrees C.
The output voltage varied less than 0.03 percent with a supply
voltage change of 10 percent, and the output voltage varied less than
0.05 percent with an output current variation of 0 to 0.200 amperes.


57
definition of aQ and average values of the variables for the test
site, the footing radius was
Vs
ro 2rrf
_ 0.85(650)
2tt( 35)
= 2.5 ft (116)
or a footing diameter of approximately 5 ft*
Stresses at the footing-soil interface
The desired stress conditions at the contact between the vibrating
footing and the soil were previously mentioned. The test footing
should develop similar oscillatory stress conditions at its contact
with the soil as were assigned in the Bycroft (1956) theory for the
contact area between a rigid circular disk and the horizontal boundary
of an elastic half-space.
Shear stresses.The shear stress between a rigid disk and a
half-space due to an oscillatory moment applied to the disk about its
vertical axis of symmetry is
For a torsionally loaded circular footing resting on a sand foundation,
the shear stresses in the sand on the footing-soil interface depend on
the friction developed between the bottom of the footing and the sand.
The limiting value of the shear stress, <7 Q is related to the normal
Z b


LIST OF FIGURES
Page
1. Rigid circular disk on the surface of an elastic half-space- 11
2. Location of 3 exploration borings and 20 friction-cone
penetrations \ 44
3. Grain-size distribution for six sample depths 51
4. Shear wave velocity versus depth, surface and empirical
methods 52
5. Distribution of stresses between a rigid disk and an
elastic half-space 55
6. Sketch of concrete footing embedded in soil 65
7. Distribution of vertical soil stress and dead load pressure
on cured first pour 67
8. Plan view sketch of torsional vibrator 76
9. Elevation view sketch of torsional vibrator 77
10. Plan view of the field of transducer locations 85
11. Section view of the field of transducer locations 86
12. Footing form and soil retaining ring 89
13. Excavation for footing and transducer 90
14. Transducer embedded below edge of footing 91
15 Excavation ready to receive footing form and retaining
ring 92
l6. Placing footing form and retaining ring in excavation 93
17 Footing form and retaining ring positioned in excavation 94
l8. Concrete test cylinders, auxiliary form and reinforcing
mesh 96
19- Second pour reinforcing mesh placed in first pour 97
xi


48
Laboratory Tests
Laboratory tests were conducted on the Shelby tube samples ex
tracted from Hole 3 before the friction-cone penetration tests were
performed. Six samples were obtained at depths of 2.5 13 19*5 31
4l, and 51 ft.
Unit weight
The 18-in.-long Shelby tube samples were divided into three
equal increments. The natural unit weight of the sand retained in
each increment was measured and the color of the material was noted.
The sand from all of the increments in a single sample was then com
bined and the maximum unit weight (minimum void ratio) and the mini
mum unit weight (maximum void ratio) of the sample were determined.
Because the volume of the sample was less than 0.1 cu ft, standard
methods and apparatus could not be used. The minimum unit weight was
determined by filling a 2-in.-diam, 4-in.-deep mold with sand poured
from a standard l/2-in.-diam funnel. The maximum unit weight was de
termined by filling the mold with sand in three equal layers; each
layer was compacted by 25 blows of a 5 5-lb hammer falling 12 in. on a
2-in.-diam steel platen resting on the sand layer. Table 11 gives the
results of these laboratory tests.
Table 11
Results of Laboratory Tests on Samples from Hole 3
Dry Unit Weight, Ib/cu ft
Depth,
ft
Increment
Color
Minimum ]
Maximum
Natural
1.5 to
3.0
?!
Tan and light
brown
(Continued)
90.9
113.2
98.8
108.8
100.2


Table 40 (Concluded)
237
a
Real
Imaginary
Absolute Value
16.20
2.294
_
03
7.027
_
03
7.392
_
03
16.56
4.255
-
03
5.951
-
03
7.316
-
03
16.92
5.794
-
03
4.342
-
03
7.240
-
03
17.28
6.767
-
03
2.351
-
03
7.164
-
03
17.64
7.086
-
03
I.696
-
04
7.088
-
03
18.00
6.723
-
03
-1.993
-
03
7.013
-
03
18.36
5.719
-
03
-3.927
-
03
6.937
-
03
18.72
4.175
-
03
-5.446
-
03
6.862
-
03
19.08
2.248
-
03
-6.405
-
03
6.788
-
03
19.44
1.330
-
o4
-6.713
-
03
6.714
-
03
19.80
-I.958
_
03
-6.346
-
03
6.641
-
03
20.16
-3.814
-
03
-5.347
-
03
6.568
-
03
20.52
-5.251
-
03
-3.826
-
03
6.497
-
03
20.88
-6.125
-
03
-1.942
-
03
6.426
-
03
21.24
-6.355
-
03
1.057
-
o4
6.356
-
03
21.60
-5.924
-
03
2.105
_
03
6.287
-
03
21.96
-4.885
-
03
3.848
-
03
6.218
-
03
22.32
-3.355
-
03
5.155
-
03
6.151
-
03
22.68
-1.501
-
03
5.897
-
03
6.085
-
03
23.04
4.761
-
o4
6.000
-
03
6.019
-
03
23.40
2.366
_
03
5.465
-
03
5-955
-
03
23.76
3.967
-
03
4.356
-
03
5.891
-
03
24.12
5.112
-
03
2.800
-
03
5.829
-
03
24.48
5.684
-
03
9.722
-
o4
5.767
-
03
24.84
5.631
-
03
-9.264
-
o4
5.706
-
03
25.20
4.965
_
03
-2.688
-
03
5-647
-
03
25.56
3.771
-
03
-4.124
-
03
5.588
-
03
25.92
2.185
-
03
-5.080
-
03
5.530
-
03


205
apparent, or intolerable. Thus, the three goals of this study, motion
computations, motion measurements, and motion comparisions, were
attained and the hypothesized utility of the half-space model was
assessed.
Recommendations
Analytical Work
An attempt to utilize the nonhomogeneous elastic half-space as an
analytical model of a soil deposit should be made. Gibson's (1967)
work could be a starting point for the analysis of a vertically oscil
lating flexible footing that applies a uniform loading to the non
homogeneous half-space.
Discrete analysis methods, such as the lumped mass and finite-
element method, should be used to predict the particle motion generated
in the sand deposit.
Predictions of the particle motion generated by a vertically
oscillating rigid footing should be developed for comparison to meas
ured motion.
Experimental Work
A more accurate and detailed determination of the elastic moduli
in the sand deposit at the test area should be attempted. This might
be accomplished by the use of seismic detectors in a borehole.
Particle motion measurements should be made in the sand deposit
as the test footing is subjected to forced vertical oscillations.


6o
a rigid disk and an elastic half-space was a reasonable approximation
to the stress distribution between a lightly loaded rijgid footing and
the soil on -which it rests provided that the soil stresses were less
than about 1/3 to 1/2 of the value necessary to cause local soil fail
ure on a large part of the contact area between the footing and the
soil. Unfortunately, reliable measurements of the distribution of
normal and shear stresses in this situation were beyond the current
state-of-the-art.
Stresses near the periphery of the footing
Figure 5 shows that the maximum contact stresses occurred near
the periphery of the circular disk and that slippage between a tor-
sionally loaded circular footing and its foundation was most likely
near the periphery of the footing because the torsion induced shear
stresses approach the normal stresses in this region. This section
views the critical stress region in more detail.
The critical stress conditions were considered to be represented
by the limiting equilibrium state of plane stress for an element (see
Figure l) of cohesionless soil located on the contact plane near the
circumference of the vertically loaded rigid footing. The lateral
pressure confining the sand at the edge of the footing was taken as
the peak passive soil pressure attainable at this point (Terzaghi,
1943).
Vr t \ (7h) <120>
The maximum vertical stress on the sand under the edge of the footing


21
the particle motion expression becomes
v(r,z,t) =
icut
3146 -g l(aQ,a,b)
4nGr
icut
40r e'
2 l(a ,a,b)
TT O
(65)
(66)
Inspection of l(a ,a,b) reveals that the integrand will have
imaginary components when g is less than unity because the term
n
\Jg 1 appears in the denominator. The integrand will be real when
g is greater than unity. These characteristics suggest that the inte
gration should be carried out in two ranges; i.e.
Noting that
11a
I = R j + im j + R j
0 0 1
= I + il + I
1 2 3
\ls2 i = i\fi-
(67)
(68)
(69)
and
^-ibVl-g- cos b^l g^ i sin b^l g^
(70)
I and I are developed from the expression
r sin
T 4- - T I
a g a g cos a g 1
o& o& o -lb
Jn 2
S J1(ag)dg
\
as
o
1
CVJ
-i r-
H
i1
H
.\L 2
aQg iv1 g
, sin a g -
i r
aog cos aog /
Jl g2jj-L(ag)dg
0 a g i
o&
v/i g2 V
(71)
(72)


I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
John H. Schmertmann, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
0-4sU**
Morris W. Self
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Gale E. Nevill, Jr.
Chairman, Department of Engineerid
Science and Mechanics


Figure 8. Plan view sketch of torsional vibrator.
CA


259
Galvanometers
Consolidated Electrodynamics Corporation, type 7-364, fluid
damped, high performance galvanometers were used. These units were of
the mirror type and had a high sensitivity. The undamped natural fre
quency of the galvanometers was 833 Hz. With an external damping re
sistance of 200 ohms to provide 64 percent critical damping, the useful
frequency range was 0 to 500 Hz. Terminal resistance of the galvanom
eters was 69 ohms, plus or minus 10 percent, and the maximum safe cur
rent was 0.050 amperes.
The frequency response error was less than plus or minus 2 per
cent and the linearity error was less than 1 percent for a full scale
deflection of 2 in. and an optical arm length of 11.5 in. The nominal
sensitivity of the galvanometers was 0.000397 amperes/in.
Oscillographs and Paper
Two type 5-119^4 oscillographs manufactured by the Consolidated
Electrodynamics Corporation were used. Each oscillograph could accom
modate 36 active type 7-300 series galvanometers and 4 inactive galva
nometers Each unit weighed 185 lb and required a 60-Hz power source
of 105 to 132 volts; it consumed 135 watts on standbyJ 350 watts with
timing and light sources activated, and 600 watts maximum during
recording.
Two tungsten-filament lamps provided the light source for the
optical system. An induction motor with a precision speed control and
a 16-speed gear driven transmission provided constant paper speeds
from 0.1 to 160 in./sec. An electronic timer or counter was employed


LIMITING STRESSES ON THE CONTACT PLANE IN PSI
55
r/ro
Figure 5. Distribution of stresses between a
an elastic half-space.
rigid disk and


400' i 400' I 400'
E3
-3
-£3
£3-
-3-
MAGNETIC NORTH
O BORING LOCATION
Q PENETRATION LOCATION
5-FT-DIAM
TEST FOOTING
O
SCALE
100'
4-
UNLESS DIMENSIONED
387'
400'
CD
HOLE 3
o
- e -D-
HOLE 2
I
£3
£3

i
--
HOLE 1
£3-
-GB E3
Figure 2. Location of 3 exploration borings and 20 friction-cone penetrations.


r
q q
' r
'
1
1 1 1
> ' "
CURED FIRST POUR (RIGID)
w -
1
{ H,M 1
1
o t
SOIL PRESSURE ON BASE DUE TO DEAD LOAD
Figure 7* Distribution of vertical
soil stress and dead load
pressure on cured first
pour.


i' = (4/3tt)i
J = Bessel function of the first kind and nth order
n
k = wave number, oj/v^
K = kinetic energy
K = coefficient of earth pressure at rest = a /oi
o rr zz
m = rate of shear modulus change with depth, z
M = moment applied to disk or footing
M' = limiting torsional moment
= design moment
n = integer
N(c) = normalized computed motion
W(m) = normalized measured motion
p = power loss
P = total vertical load on disk or footing
q = uniformly distributed vertical load on disk or footing
q = uniformly distributed vertical dead load due '"o second pour
of concrete
r = cylindrical coordinate
r^ = radius of the disk or footing
r^ = radial distance to critical stress point
r^ = radial distance to inside edge of second pour of concrete
r = radial distance to center of pressure
R = radial distance from center of disk or footing
RJ = real I = real part of the integral
s = variable parameter = r/rQ
S = deformation ratio
xv


Page
20. Position of auxiliary form and backfilling operation 98
21. Cone penetration test adjacent to footing 100
22. Cone penetration test on backfill 101
23. Second pour of cured concrete in the footing form 102
24. Checking depth and continuity of the air gap 103
25. Power required to drive the torsional vibrator 105
26. Vibrator bonded to mounting plate with an epoxy compound 107
27. Torsional vibrator mounted on the test foundation 108
28. Assembled vibrator, test footing, and switch box 109
29. Carpenter's level used to check tilt on vibrator frame 111
30. Method of attaching transducers to the test footing 112
31. Test site topography, vegetation, and borehole markers ll4
32. Drill rig used to auger uncased boreholes for the
transducers 115
33. Alignment sleeves bonded to transducers with an epoxy
compound ll6
34. Transducers with support cables and electrical leads 117
35 Apparatus for installing and aligning transducers in
boreholes 119
36. Sighting bar used to align borehole rod and attached
transducer 120
37. Borehole, borehole rod, and transducer cables 1 121
38. Water hose inserted in borehole during backfilling 123
39* Functional components of the particle velocity measuring
system 124
40. Two rows of amplifiers mounted in a cabinet 130
41. Type 5-119P4 recording oscillographs 131
42. Histogram of measured transduction constants 134
xii


16
oo
v = rQ J F(y)y1J1(sy)dy = 0rQs
0
00
azQ = G J F(y)J1(sy)dy = 0
(0 < s s 1) (39)
(s > l) (4o)
where y is another arbitrary parameter.
To evaluate F(y) in this pair of integral equations, Bycroft
(1956) and Awojobi and Grootenhuis (1965) use work by Titchmarsh
(1948), Busbridge (1938), and Tranter (1951)* Busbridge gives the
solution of the equations
OO
/ yaf(y)Jv(xy)dy = g(x) (0 < x < 1) (4i)
0
00
/ f(y)Jv(xy)dy =0 (x > 1) (42)
0
where g(x) is prescribed and f(y) is to be determined, as
2
r(i + a/2)
1
xl+ dj 2
+ / u^l U2) du J g(yu)(xy)2+Q!^2 J^^gtxyjdy
1
0
(43)
which is valid for a > -2 and where (-v l) < (or 1/2) < (v + l)
By substituting & = -1 v = 1 and g(x) = 0rQS into Equa
tion 43


10
The purpose of this section is to quantitatively evaluate the par
ticle motion generated in an elastic half-space by the torsional oscil
lation of a massless rigid disk on the surface of the half-space. The
evaluation has been guided by Bycroft's (1956) analysis for the behavior
of a rigid disk; departures from and extensions to his work have been
accomplished to treat the behavior of the half-space rather than the
circular disk. Figure 1 illustrates the system.
Equations of Elasticity
The equations for strain in cylindrical coordinates r 9 z are
(Timoshenko and Goodier, 1951)
e
_ 5u
rr dr
(1)
(2)
e
_ dw
ZZ dz
(3)
7 _
Z0 rd0 dz
(4)
7 = Su + dw
rz dz Sr
(5)
7 = JE_ + Sv I
r@ rd0 dr r
(6)
the stress-strain relationships in cylindrical coordinates are
(Sokolnikoff, 1956)
(7)
a
(8)




192
displacement in a nonhomogeneous half-space with a modulus that varied
according to the properties of the sand at the test site. Because the
vertical dead load applied to the footing during the second pour of
concrete increased the average footing-soil contact pressure by 214
psf--equivalent to an overburden depth of about 2 ft--the shear wave
velocity of the sand supporting the footing was taken, as the velocity
at a depth of 3 ft.
Table 34 lists the normalized nonhomogeneous half-space displace
ments, defined as the displacement of the nonhomogeneous half-space
with respect to the edge of a rigid disk oscillating on the surface of
the nonhomogeneous half-space. Table 35 gives the ratios of the nor
malized nonhomogeneous half-space displacement to the normalized soil
displacement, and Table 36 is a tabulation of the average displacement
ratio at each transducer location for the 5 test frequencies.
The ratios of the nonhomogeneous half-space displacement to the
soil displacement given in Tables 35 and 36 lead to several observa
tions. (l) The computed displacements were less than the measured
displacements near the ground surface. (2) The computed displacements
were more than the measured displacements at the deeper transducer lo
cations. (3) The average measured displacements ranged from 1/4 to
2-1/2 times the measured displacements for all 5 test frequencies.
(4) The calculated displacements approach the measured displacements
near the footing. (5) The displacement ratios generally increased
with increasing vibration frequency. (6) The average displacement
ratio for all frequencies and all locations was about 1.6, indicating
that the average calculated nonhomogeneous half-space displacement was
larger than the measured displacement.


APPENDIX A
CALCULATIONS FOR THE INTEGRAL l(a ,a,b)
Table 37
Subroutines and Computer Program for the
Integral l(aQ,a,b)
REAL PROCEDURE SIMPSON (FCT, A, B, TEST) :
$CARD
VALUE A, B, TEST ; REAL A, B, TEST ; REAL PROCEDURE FCT :
BEGIN
REAL PROCEDURE FINITE (S, FCT, A, B, TEST) ;
VALUE S, A, B, TEST ; REAL A, B, TEST ; INTEGER S d
REAL PROCEDURE FCT ;
BEGIN
INTEGER N, NMBR ;
REAL SI, S2, S4, OLD, NEW, X, H ;
LABEL ABC ;
TEST:=10 (- TEST) ;
OLD:=123456789012
SI:=FCT (S X A) + FCT (SXB) ;
S2:=0 ;
NMBR:=2 ;
ABC: H:=(B A) / NMBR ;
S4:=0 j
FOR N:=1 STEP 2 UNTIL NMBR DO
BEGIN
X:=S X (A + N X H) ;
S4:=s4 + FCT(x) ;
END ;
FINITE:=NEW:=(H X (SI + 2 X S2 + 4 X S4)) / 3 5
IF ABS (NEW OLD) > ABS ( NEW X TEST) THEN
BEGIN
OLD:=NEW ;
S2: =S2 + S4 ;
NMBR:=2 x NMBR ;
GO TO ABC ;
END ;
END FINITE ;
(Continued)
207


132
unknown voltage record was determined by comparing it to the magnitude
of the known voltage record. A record of unknown voltages was called a
test record and a record of known voltages was called a calibration
record.
Calibration voltage supply.--A Consolidated Electrodynamics Cor
poration model 3-140 power supply was used to provide a voltage source
for the calibration records. The direct current output voltage was
variable from 1 to 24 volts and the regulation was 3 parts in 1,000.
A resistance circuit across the output of the power supply extended
the measurable voltage range from 0.000005 to 2.4 volts. Figure 40
shows two voltage supply units next to the top row of amplifiers.
Calibration voltmeter.--An AC or DC differential voltmeter with
a null detector circuit was used to measure the calibration voltage
applied to the amplifier input. The solid state model 887AB voltmeter
was manufactured by Fluke Electronics, Seattle, Washington. It had an
accuracy of 0.005 percent of the measured input voltage plus 5 micro
volts. Figure 40 shows this unit resting on top of the amplifier
cabinet.
The maximum calibration voltage used in the experiments was 2.5
and the minimum was 0.00034, so the error in measuring the calibration
voltage varied from 0.005 to 1.5 percent.
System accuracy
The accuracy of the particle velocity measuring and recording
system was estimated from the results of the transducer calibration
measurements and the error or accuracy specifications for each compo
nent of the system.


129
attributed to the amplifiers was 1.25 percent.
Figure 40 shows two rows of these amplifiers, six in the upper
row and seven in the lower row, mounted in a cabinet.
Oscillographs and galvanometers
Consolidated Electrodynamics Corporation oscillographs and gal
vanometers were used. Two type 5-H9P^ oscillographs, equipped with
fluid-damped, high-performance, type 7-364 galvanometers, recorded the
test data on 12-in.-wide, light-sensitive paper. The galvanometers
had a sensitivity of 0.397 milliamps/in., a useful frequency range from
0 to 500 Hz, a linearity error of less than 1 percent at full scale
deflection, and a frequency response error of less than 2 percent.
The paper speed in the oscillograph was 25.6 in./sec or 2.56 in/sec
and timing lines were printed at 0.01- and 0.1-sec intervals on Lino-
Writ 4 photorecording paper made by the E. I. du Pont de Nemours Co.,
Wilmington, Delaware.
Eleven active galvanometers were used in oscillograph A and 12
active galvanometers were used in oscillograph B. Figure 4l shows
oscillograph A without a dust cover and oscillograph B with a dust
cover.
Reference (calibration) voltage
A particle velocity transducer supplied an unknown, time-
dependent voltage to the input of the amplifier and the time history
of that voltage was recorded by an oscillograph on moving, light-
sensitive paper. To determine the magnitude of the unknown transducer
voltage, a known reference voltage was applied to the input of the
amplifier and recorded on light-sensitive paper. The magnitude of the


i4o
continuous time reference between oscillographs was obtained by record
ing the transverse footing motion on each oscillograph and by a manual
step pulse applied simultaneously to one inactive galvanometer in each
oscillograph. The propagation velocity of compression waves was com
puted from the arrival time of the disturbance at each transducer.
Typical vibration test data
Procedures and methods used to obtain particle velocity measure
ments and records were discussed in the preceding paragraphs. This
section illustrates typical data obtained during a torsional vibration
test at a frequency of 50 Hz.
Calibration records.--Typical calibration records for oscillo
graphs A and B are shown in Figures 43 and 44, respectively. The
labeled galvanometer traces give the oscillograph record number, the
oscillograph identification and galvanometer trace (or channel) num
ber, and a group of four symbols that identifies the serial number
(s/n) of the transducer, the component (CMP) of motion recorded, the
distance (r) of the transducer from the center of the footing, and
the depth (z) of the transducer below the ground surface.
Test with eccentric weights.Figures 45 and 46 show the oscillo
graph records that were obtained during a 50-Hz torsional vibration
test with eccentric weights attached to the flywheels of the vibrator.
Test without eccentric weights.--The eccentric weights were re
moved from the vibrator and the vibrator was operated at the same 50-Hz
frequency while the particle velocity perturbations and the ambient
measuring system disturbances were recorded. Figures 47 and 48 illus
trate typical test records obtained when the vibrator was operated
without the eccentric weights.


Figure 34. Transducers with support cables and electrical leads
H
3


131


^_REC. NO.; CHAN. NO.; TRANS(S/N, CMP, R, Z)
152; Bl; 9, V, 2.6, on footing
I52;B2;9,R,2.6,* on footing
3 *-!
0.173"
152; B3; 4, V, 3.5, I
4' -
/.os"
..152; B4; 4, R, 3.5, I
152; B5; 4, T, 3.5, I **r -
i.152; B6; l-D, T, 2.5, 1.5
152 B 7; l-D, T, 2.3, *
ON FOOTING
152; B8; 14, V, 10, 5
T hit
3J52; B9; 3, V, 10, 5
aoooOCCWWOO^OaoOCiOCqpCl52; BIO; 24, V, 60, 5
152; Bl I; 26, V, 60, 15 152; BI2; 8, V, 90, 15
Figure 44. Typical calibration record, oscillograph B, before 50-Hz vibration test.


Page
Cables 128
Amplifiers + 128
Oscillographs and. galvanometers 129
Reference (calibration) voltage -j- 129
System accuracy 132
Power generators 133
Arrangement and Utilization 135
Arrangement of components 135
Utilization of system 136
Typical vibration test data ] i4o
Schedule of Tests i47
Footing Settlement and Tilt 147
Transducer Operation l47
Torsional Vibration 154
Compression Wave Propagation 155
Transducer and Cable Resistance 155
Results of Measurements 158
Footing Settlement and Tilt 158
Compression Wave Propagation 158
Footing source 158
Surface source 159
Propagation velocities l60
Particle Velocities Due to Torsional Vibration l6o
Amplitudes l60
Wave propagation velocities 166
COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS 1 169
Test of the Calculated Results 169
Solutions at the Surface of a Homogeneous (Constant E)
Elastic Half-Space 169
Geometrical Damping Law 171
Position of Disk and Footing 173
Test of the Measured Results 173
Dynamic Footing Response 173
Footing-Soil Contact Area 174
Particle Motion Components 176
Properties of the Sand Deposit 179
Basis for Comparing Results 182
Measured Motion 182
Computed Motion 184
Comparison of Normalized Displacements 185
Comparison of Results 185
Normalized (Constant E) Half-Space Displacements 185
Normalized Soil Displacements 187
Ratio of Displacements 189
Displacements in a Nonhomogeneous (Linear E) Half-Space 191
Discussion of Results 195
Homogeneous (Constant E) and Nonhomogeneous (Linear E)
Half-Space 195
Characteristics of the Test Site 197
vi


Table 4l (Continued)
b
1.920
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000 +
00
0.48
1.872
-
02
-2.98O
-
02
3.519 -
02
O.96
2.674
-
02
-5-504
-
02
6.119 -
02
1.44
2.005
-
02
-7.196
-
02
7.470 -
02
1.92
2.015
-
03
-7.821
-
02
7.824 -
02
2.40
-2.060
_
02
-7.325
_
02
7.609 -
02
2.88
-4.126
-
02
-5.838
-
02
7-149 -
02
3.36
-5.523
-
02
-3.65O
-
02
6.620 -
02
3.84
-5.992
-
02
-I.I54
-
02
6.102 -
02
4.32
-5.49O
-
02
1.228
-
02
5.625 -
02
4.80
-4.I6I
_
02
3.116
_
02
5.198 -
02
5.28
-2.292
-
02
4.24o
-
02
4.820 -
02
5.76
-2.514
-
03
4.478
-
02
4.485 -
02
6.24
1.590
-
02
3.876
-
02
4.189 -
02
6.72
2.922
-
02
2.622
-
02
3.926 -
02
7.20
3.551
_
02
1.011
-
02
3.692 -
02
7.68
3-427
-
02
-6.224
-
03
3-483 -
02
8.16
2.644
-
02
-1.966
-
02
3.295 -
02
8.64
l.4l4
-
02
-2.787
-
02
3.125 -
02
9.12
1.767
-
04
-2.972
-
02
2.972 -
02
9.60
-1.256
-
02
-2.538
-
02
2.832 -
02
10.08
-2.161
-
02
-1.625
-
02
2.704 -
02
10.56
-2.547
-
02
-4.554
-
03
2.587 -
02
11.04
-2.375
-
02
7.138
-
03
2.480 -
02
11.52
-1.723
-
02
1.643
-
02
2.381 -
02
12.00
-7.600
_
03
2.159
-
02
2.289 -
02
12.48
2.920
-
03
2.185
-
02
2.204 -
02
12.96
1.208
-
02
1.748
-
02
2.125 -
02
13-44
1.807
-
02
9.718
-
03
2.052 -
02
13.92
1.982
-
02
4.051
-
04
1.983 -
02
i4.4o
1.725
_
02
-8.397
_
03
1.918 -
02
14.88
1.115
-
02
-1.486
-
02
I.858 -
02
15.36
3.o4o
-
03
-1.775
-
02
1.801 -
02
15.84
-5.250
-
03
-1.667
-
02
1.748 -
02
16.32
-1.194
-
02
-I.207
-
02
1.697 -
02
16.80
-1.568
_
02
-5.121
_
03
1.650 -
02
17.28
-1.584
-
02
2.556
-
03
1.605 -
02
17.76
-1.257
-
02
9-275
-
03
1.562 -
02
18.24
-6.745
-
03
1.364
-
02
1.522 -
02
18.72
2.518
-
04
1.483
-
02
1.483 -
02
19.20
6.852
_
03
1.274
-
02
1.447 -
02
19.68
1.165
-
02
7.984
-
03
1.412 -
02
20.16
1.368
-
02
1.709
-
03
1.379 -
02
20.64
1.264
-
02
-4.652
-
03
1.347 -
02
21.12
8.892
-
03
-9.715
-
03
1.317 -
02
(Continued)


Table 42
Value of l(l.20,a,b)
247
a
Real
0.00
0.000 +
0.60
4.864 -
1.20
8.515 -
1.80
2.348 -
2.40
7.852 -
3.00
-2.007 -
3.60
-7.773 -
4.20
-9.819 -
0.00
0.000 +
o.6o
2.698 -
1.20
3.403 -
l.8o
1.932 -
2.40
6.481 -
3.00
-2.538 -
3.60
-7.949 -
4.20
-9.778 -
4.8o
-8.488 -
5.4o
-5.076 -
6.oo
-8.627 -
6.6o
2.838 -
7.20
5.049 -
7.80
5.351 -
8.40
3.928 -
9.00
1.449 -
9.60
-1.182 -
10.20
-3.121 -
10.80
-3.834 -
11.4o
-3.223 -
12.00
-1.620 -
12.60
3.627 -
13.20
2.045 -
13.80
2.905 -
i4.4o
2.730 -
15.00
1.662 -
15.60
1.229 -
16.20
-1.343 -
16.80
-2.255 -
17.40
-2.349 -
18.00
-1.648 -
18.60
-4.383 -
19.20
8.431 -
19.80
1.763 -
20.4o
2.033 -
Imaginary Absolute Value
0.000
+
00
0.000
+
00
-8.241
-
02
4.933
-
01
-1.477
-
01
8.642
-
01
-1.829
-
01
2.976
-
01
-1.825
-
01
1.987
-
01
-1.496
-
01
1.509
-
01
-9.445
-
02
1.223
-
01
-3.182
-
02
1.032
-
01
0.000
+
00
0.000
+
00
-8.044
_
02
2.816
-
01
-1.440
-
01
3.695
-
01
-1.781
-
01
2.627
-
01
-1.773
-
01
1.887
-
01
-1.445
-
01
1.467
-
01
-9.014
-
02
1.202
-
01
-2.865
-
02
1.019
-
01
2.505
-
02
8.850
-
02
5.958
-
02
7.827
-
02
6.965
_
02
7.018
_
02
5.695
-
02
6.363
-
02
2.894
-
02
5.820
-
02
-3.715
-
03
5.363
-
02
-3.052
-
02
4.974
-
02
-4.405
_
02
4.637
-
02
-4.180
-
02
4.344
-
02
-2.635
-
02
4.085
-
02
-4.094
-
03
3.856
-
02
1.715
-
02
3.651
-
02
3.065
-
02
3-467
-
02
3.281
-
02
3.301
-
02
2.395
-
02
3.150
-
02
7.956
-
03
3.012
-
02
-9.344
-
03
2.886
-
02
-2.215
-
02
2.770
-
02
-2.660
-
02
2.662
-
02
-2.183
-
02
2.563
-
02
-1.011
-
02
2.471
-
02
4.193
-
03
2.386
-
02
1.613
-
02
2.306
-
02
2.188
-
02
2.231
-
02
1.990
-
02
2.161
-
02
1.134
-
02
2.096
-
02
-5.475
-
o4
2.034
-
02
inued)
00
01
01
01
02
02
02
02
00
01
01
01
02
02
02
02
02
02
03
02
02
02
02
02
02
02
02
02
02
03
02
02
02
02
03
02
02
02
02
03
03
02
02
(Cont


BOREHOLE ROD
SUPPORT CABLE
ELECTRICAL LEAD
BOREHOLE 'JM
Figure 37. Borehole, borehole rod, and transducer cables.


62
surface between the buried test footing and the soil due to a vertical
load on the footing was
= (r^f)211041
= 936 psf
= 6.5 psi (123)
where 7 and / are taken as 104 lb/cu ft and 30 deerees, respec
tively. Figure 5 shows the limiting stress distribution.
The significance of the inelastic vertical stresses developed near
the periphery of the footing was implied by calculating the portion of
the footing-soil contact area on which elastic stresses act to the
total contact area. Assuming that the 5-f't-diam test footing was a
solid cylinder of concrete, 2 ft high and buried 1 ft in the soil,
Equation 119 was used to compute the approximate radial position, r^ ,
of the maximum normal stress of 6.5 psi. If the unit weight of con
crete is taken as 150 lb/cu ft, and the entire footing acts as a rigid
body
r = 2.47 ft (124)
Thus, for the case assumed, nearly 98 percent of the contact area be
tween the footing and the soil had normal stresses that were less than
6.5 psi and transmission of elastic stresses during torsional oscilla
tion of the footing occurred over some 92 percent of the footing-soil
contact area (Timmerman and Wu, 1969). Also, because the shear
stresses at the edge of the footing are less than those at the edge


Table 42 (Continued.)
a
Real
3.00
-7.579 -
3.60
-8.418 -
4.20
-7-426 -
4.80
-4.845 -
5.40
-1.397 -
6.00
1.970 -
6.60
4.376 -
7.20
5.263 -
7.80
4.538 -
8.4o
2.572 -
9.00
6.197 -
9.60
-2.195 -
10.20
-3.546 -
10.80
-3.664 -
11.4o
-2.625 -
12.00
-8.534 -
12.60
1.033 -
13.20
2.430 -
13.80
2.932 -
i4.4o
2.447 -
15.00
1.204 -
15.60
-3.396 -
16.20
-1.663 -
16.80
-2.351 -
17.4o
-2.219 -
18.00
-1.362 -
18.60
-1.088 -
19.20
1.105 -
19.80
1.880 -
20.40
1.985 -
21.00
1.421 -
21.60
4.126 -
22.20
-6.807 -
22.80
-1.490 -
23.4o
-1.757 -
24.00
-1.421 -
24.60
-6.226 -
25.20
3-487 -
25.80
1.158 -
26.4o
1.540 -
27.00
1.384 -
27.60
7.668 -
28.20
-8.445 -
Imaginary Absolute Value
-4.969
-
02
9.063
-
02
-1.191
-
02
8.502
-
02
2.590
-
02
7.865
-
02
5.386
-
02
7.244
-
02
6.527
-
02
6.675
-
02
5-843
_
02
6.166
_
02
3.677
-
02
5.716
-
02
7.605
-
03
5.318
-
02
-2.017
-
02
4.966
-
02
-3.879
-
02
4.654
-
02
-4.376
_
02
4.377
_
02
-3.496
-
02
4.128
-
02
-1.637
-
02
3.905
-
02
5.447
-
03
3.704
-
02
2.348
-
02
3.522
-
02
3.245
-
02
3.356
-
02
3.033
-
02
3.204
-
02
1.870
-
02
3.066
-
02
1.995
-
03
2.938
-
02
-i.4o4
-
02
2.821
-
02
-2.430
_
02
2.712
-
02
-2.589
-
02
2.612
-
02
-1.890
-
02
2.518
-
02
-6.186
-
03
2.431
-
02
7.725
-
03
2.349
-
02
1.819
-
02
2.273
-
02
2.199
-
02
2.201
-
02
1.826
-
02
2.134
-
02
8.677
-
03
2.071
-
02
-3.247
-
03
2.011
-
02
-1.342
-
02
1.955
-
02
-1.856
-
02
1.901
-
02
-1.721
-
02
1.851
-
02
-1.016
-
02
1.803
-
02
-4.862
-
05
1.757
-
02
9.583
-
03
1.714
-
02
1.553
-
02
1.673
-
02
1.596
-
02
1.634
-
02
1.099
-
02
1.596
-
02
2.523
-
03
1.560
-
02
-6.423
_
03
1.526
-
02
-1.281
-
02
1.493
-
02
-1.459
-
02
1.462
-
02
inued)
02
02
02
02
02
02
02
02
02
02
04
02
02
02
02
03
02
02
02
02
02
03
02
02
02
02
03
02
02
02
02
03
03
02
02
02
03
03
02
02
02
03
04
(Cont


Figure 39. Functional components of the particle velocity measuring system.


Figure 22. Cone penetration test on backfill
o
H


showing the water supply hose, is given in Figure 38.
Particle Motion Measuring System
122
The system used to measure and record the particle velocities
generated in the sand deposit by a vibrating footing was capable of
sensing a wide range of frequencies--from a few to several hundreds of
cycles per second. Means are available for eliminating or rejecting
unwanted or unexpected frequencies within this range, and it is some
times advantageous to employ electrical filters and convolution methods
to extract the desired signals from the undesired signals. However,
the a priori rejection of unwanted signal frequencies and the elimina
tion of unexpected particle motion were not considered appropriate for
these experiments. Consequently, no filters, squelch circuits, or
other devices were used to limit the measurement capability of the
detecting and recording system.
Functional Components
The essential components of the measuring and recording system
were (l) a transducer that generated a voltage proportional to its ve
locity, (2) a shielded cable that conducted the generated voltage to
an amplifier, (3) an amplifier or attenuator that adjusted the trans
ducer voltage and provided a proportionate current to a galvanometer,
(4) a galvanometer with a current sensitive mirror that reflected a
beam of light onto moving photographic paper, (5) a regulated voltage
source, and (6) a voltmeter that measured a known reference voltage.
The functional relationships between these components is illustrated
in Figure 39- These components, and some auxiliary items, are briefly


Figure 36. Sighting bar used to align borehole rod and attached transducer.
120


29
and are well behaved functions of the variable o so that a nu
merical integration scheme, such as Simpson's rule, should give a good
approximation for the value of these integrals.
Computer Program to Evaluate the Integrals
A computer program, based on Simpson's rule and written in ALGOL
language, was used to evaluate the integrals 1^ and I .
Table 37 of Appendix A lists this program and includes the polynomial
routine for calculating J Using the variable x instead of a ,
the integrand of 1^ is called RELX(X), the integrand of is
IMX(X) and the integrand of is REH(X). The real part of
l(a ,a,b) is called REINT and the imaginary part IMINT.
The computer calculations are carried out in much the same manner
as illustrated in the example calculations above. The integration in
terval, 0 to rr/2 radians, has been subdivided into three parts:
0 to 0.5 radians, 0.5 to 1.0 radians, and 1.0 to tt/2 radians. Each
part is independently integrated, using Simpson's rule and a geometri
cally increasing number of intervals, until two sequential integrations
agree to five significant digits. When this criterion is satisfied for
each part of the integration interval, the sum of the parts is con
sidered to be a sufficiently accurate representation of the integral
for the purpose of this investigation.
Appendix A contains tables of computed l(aQ,a,b) values for
several combinations of the variables a a and b The real,
o
imaginary, and absolute values of I are listed for parameters charac
teristic of the test site at Eglin Field, Florida, and for the measure
ments planned at this site. The shear wave velocity at the Florida


FOOTING DISPLACEMENT/SOIL DISPLACEMENT
FOOTING DISPLACEMENT IN MILS
15 20 30 40 50
VIBRATION FREQUENCY IN HZ
Figure 59* Ratio of footing displacement to soil displacement under the footing.
177


PLAN VIEW
Figure 6. Sketch of concrete footing embedded in soil


130
CABINET
SIX AMPLIFIERS
TWO VOLTAGE SUPPLY UNITS
SEVEN AMPLIFIERS
(f
%
Figure 1+0. Two rows of amplifiers mounted in a cabinet.


195
Table 36
Average Nonhomogeneous Half-Space Displacement
Ratios for 9 Frequencies
Average Ratio of Normalized Nonhomogeneous Half-
Space Displacement to Normalized Soil Displacement
for 5 Frequencies
Depth Below
Radial
Distance,
fit
Surface, ft
3.5
10
30
60
90
1
1.0
0.68
o.4o
0.46
0.77
5
0.76
0.60
0.63
0.55
15
1.2
2.3
2.0
1.3
25
1.9
3.9
3.7
2.6
35
1.3
2.7
2.9
2.8
Discussion of Results
Homogeneous (Constant E) and Nonhomogeneous (Linear E) Half-Space
A comparison of Tables 32 and 35 indicated that the computed
displacements in a nonhomogeneous half-space were in closer agreement
with the measured displacements than were the homogeneous half-space
displacement calculations. The nonhomogeneous half-space displacements,
however, depend on the computed values of the homogeneous half-space
displacements.
The displacement ratios for the 15 Hz test were used to compare
the homogeneous and nonhomogeneous displacement prediction values. For
the homogeneous case, the average ratio was 2.62 and the standard devia
tion was 2.55j for the nonhomogeneous case, the average ratio was 1.1
and the standard deviation was 0.8l. Since an ideal prediction method
would give a ratio of about 1.0 and a standard deviation of zero,


8o
developed by the torsional vibrator for typical mass combinations at
frequencies of 15, 20, 30, 40, and 50 Hz. The calculated results are
given in Table 13 and show that the vibrator was capable of developing
the footing design moment of nearly 1,400 ft-lb at a frequency of
15 Hz.
Table 13
Vibrator Moment Capacity at Various Frequencies
Frequency
Hz
Weight
Identification
Bolt
Identification
Total
Weight
lb
Moment
ft-lb
15
ABODE
E
5j768
1,380
20
ACD
G
3.5378
1,502
30
CD
I
1.5387
1,470
4o
CD
I
1.5387
2,6i4
40
DE
I
0.7924
1,346
50
C
I
1. <4)617
2,818
50
D
I
0.561
1,489
Foundation and Transducer Location
Location of the test footing
The Piccolo field test area that had been investigated during the
soil exploration program was examined to locate a favorable site for
placing the circular test foundation. Likely locations were probed
with a portable cone penetrometer similar to that described by Poplin
(1969); its use is shown in Figure 21. The soil was probed to a depth
of 6 to l8 in. on a grid spacing of 5 ft to delineate unusually soft
or hard areas. Of several suitable sites, one was chosen which offered


170
and the form that Stallybrass (1962) used was
v(r ) = 9Me g (h + ih ) (148)
0 l6Gr 1 2
o
To compare the real and imaginary parts of these two displacement func
tions at the surface of the half-space, let
"l + ih2 = 5? + iIi(ro)] (l49)
= I'(ro) + iI'(ro) (150)
Because stresses, rather than displacements, were used as the boundary
conditions under the rigid disk, there is some distortion of a radius
on the loaded area; Bycroft (1956) devised a complex averaging method
that gave peripheral displacements of the disk in close agreement with
the Reissner and Sagoci (1944) results.
An averaging technique appeared unnecessary for testing the
validity of the developed calculations, so two values of the edge dis
placement function were computed and compared to the published results--
the value of I'(ro) and the value 2I(r /2) Table 25 gives the
real part and imaginary part of the edge displacement function for
various values of the frequency ratio, a^ An inspection of the tab
ulated values indicated that the computed values of I'(r ) are less
than the published values and the computed values of 2I'(r /2) are
greater than the published values. Thus, the calculated displacement
of the edge of a disk on the surface of the half-space was in reason
able agreement with published solutions.


63
of a rigid disk, the rotational stiffness of the footing, at the limit
ing moment, was about 75 percent of the rotational stiffness for a
rigid disk.
Footing emplacement operation
Previous paragraphs established the size and position of the test
footing and discussed the desired stress conditions on the footing-soil
contact area. This section sets forth the footing design and placement
method.
To transmit torsional oscillations into the soil, the friction
angle between the base of the footing and the soil should be comparable
to the angle of internal friction, 0 of the soil. This objective was
met by using concrete as the footing material in contact with the soil.
Intimate and uniform contact between the base of the footing and the
sand should result by pouring the concrete directly on the prepared
sand surface.
The only contact allowed between the buried footing and the soil
occurred on a horizontal circular area, so the vertical face of the
circular footing had to be isolated from the soil. This was accom
plished by a thin steel ring placed between the soil and the cylindri
cal surface of the footing.
The distribution of vertical contact stresses between the base of
the footing and the soil should be similar to that developed by a rigid
circular disk pressed vertically against a half-space. This objective
was realized by considering the soil stress conditions after excava
tion, after pouring the footing, and after the application of a dead


49
Table 11 (Concluded)
Depth,
ft
Increment
Color
Dry Unit Weight, lb/cu ft
Minimum Maximum Natural
12.0
to 13.5
Top}
93-6
2 1
Tan
90.2
109.9
93-1
3 )
95.7
18.5
to
20.0
Top
Reddish brown
103.4
2
to brownish
84.1
109.3
106.2
3
red
106.4
30.0
to
31-5
Top
Brown
--
2
Brown
84.8
105.3
95-7
3
Reddish brown
95-0
40.0
to
41.5
Top
Brown
94.8
2
Reddish brown
83.9
IO8.5
108.4
3
Reddish brown
103.7
o
o
im
to
51.5
Top}
92.2
2 >
Light red to
84.1
106.4
95.0
3(
brown
94.9
The average natural unit weight (dry) of all the sampled material
was about 99 lb/cu ft, the relative density was 62 percent, and, from
Table 8, the moisture content of the sand was approximately 5 percent.
Thus, the unit weight of the in situ sand was taken as 104 lb/cu ft.
Gradation
An indication of the uniformity of the sand deposit at the test
site was obtained from an inspection of the grain-size-distribution
curves for the sand material sampled at various depths. Similar grain
sizes and distributions at various depths indicate that the material
was deposited during the same or similar geological environments. The
sampled material from Shelby tubes extracted from Hole 3 bad an ef
fective grain size of about 0.14 mm, a uniformity coefficient of


36
The above similarities between Gibson's case and the torsional
loading situation lead to the hypothesis that the results of Gibson's
investigations were also applicable to a nonhomogeneous half-space
under torsion. Gibson's results, thus, indicate that the stresses de
veloped in a homogeneous elastic half-space by a torsibnally loaded
rigid disk on the surface of the half-space would be the same as the
stresses developed in a nonhomogeneous elastic half-space by the same
torsional load. The variation of the shear modulus with depth in the
nonhomogeneous half-space under torsional loads should be the same as
that assumed by Gibson: G(z) = G(o) + mz .
Torsional Oscillation
As mentioned before, rigorous solutions to the dynamic boundary
value problem of a rigid circular disk in torsional oscillation on the
surface of a nonhomogeneous elastic half-space have not been found and
are not attempted herein.
Engineers, however, are notoriously proficient in rationalizing a
sufficient number of plausible assumptions to circumvent rigorously in
tractable problems (Zienkiewicz, 1967) Soil engineers, typically
faced with incomplete, inaccurate information and armed with inadequate,
inappropriate, and often untested theory, have been able to resolve a
variety of problems by the simultaneous application of available knowl
edge and logical assumptions. The results are usually successful, but
sometimes they are not (Peck, 1967). The following paragraphs are
offered to bridge the gap between what is known and what is needed..


b
0.000
0.144
211
Table 38
Value of l(0.36,a,b)
a
Real
Imaginary
Absolute
: Value
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.18
3.974
-
01
-2.557
-
03
3-974
- 01
0.36
7.085
-
01
-5.064
-
03
7.085
- 01
0.54
1.932
-
01
-7.474
-
03
1.933
- 01
0.72
1.080
-
01
-9.739
-
03
1.085
- 01
0.90
7.259
-
02
-I.182
-
02
7.355
- 02
1.08
5.349
-
02
-I.367
-
02
5.521
- 02
1.26
4.138
-
02
-I.526
-
02
4.4ii
- 02
1.44
3.273
-
02
-I.657
-
02
3.669
- 02
1.62
2.599
-
02
-1.757
-
02
3.137
- 02
1.80
2.041
-
02
-1.825
-
02
2.738
- 02
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.18
2.039
-
01
-2.552
-
03
2.039
- 01
0.36
2.622
-
01
-5.054
-
03
2.623
- 01
0.54
1.626
-
01
-7.458
-
03
1.628
- 01
0.72
1.012
-
01
-9.719
-
03
1.016
- 01
0.90
7.017
-
02
-1.179
-
02
7.115
- 02
1.08
5.227
-
02
-1.364
-
02
5.402
- 02
1.26
4.058
-
02
-1.523
-
02
4.335
- 02
1.44
3.214
-
02
-1.654
-
02
3.615
- 02
1.62
2.556
-
02
-1.753
-
02
3.100
- 02
1.80
2.013
_
02
-1.821
_
02
2.715
- 02
1.98
1.547
-
02
-1.856
-
02
2.416
- 02
2.16
1.136
-
02
-1.858
-
02
2.178
- 02
2.34
7.700
-
03
-1.828
-
02
1.984
- 02
2.52
4.414
-
03
-1.768
-
02
1.822
- 02
2.70
1.476
-
03
-1.679
-
02
1.685
- 02
2.88
-1.125
-
03
-1.564
-
02
1.568
- 02
3.06
-3.394
-
03
-1.427
-
02
1.467
- 02
3-24
-5.329
-
03
-I.271
-
02
1.378
- 02
3.42
-6.928
-
03
-1.100
-
02
1.300
- 02
3.60
-8.190
-
03
-9.176
-
03
1.230
- 02
3.78
-9.118
-
03
-7.289
-
03
1.167
- 02
3.96
-9.720
-
03
-5.382
-
03
l.lll
- 02
4.l4
-1.001
-
02
-3.495
-
03
1.060
- 02
4.32
-9-997
-
03
-1.670
-
03
1.013
- 02
4.50
-9.710
-
03
5.564
-
05
9.710
- 03
4.68
-9.173
-
03
1.646
-
03
9.320
-.03
4.86
-8.417
-
03
3.073
-
03
8.960
- 03
5.04
-7.474
-
03
4.309
-
03
8.628
- 03
5.22
-6.382
-
03
5.337
-
03
8.319
- 03
(Continued)


87
the green concrete and as a retaining wall for the surrounding soil.
The rings were separated by a removable wire spacer to provide an air
gap between the inner concrete form and the outer soil retaining wall.
The interior ring was 5 ft OD, 28 in. high, and rolled from
0.125-in.-thick steel plate. It was butt welded, ground smooth on the
outside, and held in a cylindrical shape by an internal plywood dia
phragm. Nine-gage steel wire was lightly tack welded to the bottom
of the form and wrapped circumferentially around the lower 16 in. of
the form at a pitch of 2 in.
The outer ring was 16 in. high and rolled from the same stock as
the inner ring. It was fitted over the wire wrapping on the inner
ring, cut to size, and butt welded in place. A plywood exterior flange
was fitted around the outside of the outer ring to hold it in a cylin
drical shape.
An auxiliary form, to provide a cylindrical void at the center
of the footing, was rolled from 0.0625-in.-thick sheet stock and capped
with an 0.0991-in.-thick circular plate. This form, an inverted tub,
was 32 in. OD and 18 in. high. Number 2 gage, 6-in.-mesh reinforcing
was trimmed to 58 in. OD for placement in the first concrete pour; a
46 -in.-diam, 20-in.-high cylinder of the same mesh was fabricated for
the dead load pour. Anchored baseplates for attaching the torsional
vibrator to the test foundation, brackets for mounting motion trans
ducers to the foundation, and other miscellaneous hardware were also
fabricated.
Placing the form
The form for the test footing was transported to the test site


DEPTH BELOW GROUND SURFACE IN FT
PENETRATION RESISTANCE IN BLOWS/FT
CONE BEARING CAPACITY IN KG/CM2
Figure 60.
Average penetration resistance and cone bearing capacity versus depth.
CD


Voltage drift was less than 0.005 percent/degree F.,
in an 8-hour period.
and 0.05 percent
Voltmeter
An AC or DC differential voltmeter with null detector was used to
measure the calibration voltage applied to the amplifier input. This
solid state model 887 AB voltmeter was manufactured by Fluke Electron
ics, Seattle, Washington. It was a solid state differential voltmeter
and had a DC voltage measuring accuracy of 0.005 percent of the measured
voltage plus 5 microvolts. The unit weighed 14 lb and operated from
either a rechargeable battery or from a 50 to 420 Hz, II5/23O volt AC
power source. The ambient operating temperature range was 0 to 50 de
grees C.
The circuit of this instrument is composed of an AC to DC conver
ter, a DC input attenuator, a DC transistorized voltmeter and a 0 to
11-volt comparison or reference voltage. There were seven scales in
the null range from 0 to 11 volts. Meter resolution and voltage reso
lution allowed readings equal to or greater than 1 microvolt. The DC
input resistance in the null range was infinite; above 11 volts, the
input resistance was 10,000 ohms.
The voltage regulation on the reference supply was 0.0002 percent
for a line voltage variation of 10 percent and the stability of the
reference voltage was 0.0005 percent/hr, 0.0007 percent/day, and
0.0013 percent in 60 days. The total instrument stability was less
than 0.002 percent/yr.


158
transducer component was measured at the transducer cable connection to
assure that the transducer had not been damaged and that it had per
formed consistently throughout the test program. Each length of instru
mentation cable was also checked for continuity and resistance. The
measured transducer resistance values appear1 in Appendix B.
Results of Measurements
Footing Settlement and Tilt
The settlement and tilt of the test footing was measured with a
transit, level rod, and carpenter's level. Measurements with the car
penter's level indicated that the footing did not tilt during the entire
program of footing vibration tests. Figure 54 gives the results of the
elevation measurements in a plot of elevation change vtersus sequential
measurement number. The test footing probably settled about l/l6 in.
during the torsional vibration tests.
Compression Wave Propagation
Footing source
Compression waves initiated by a vertical hammer blow on the top
surface of the footing were detected with the transducers listed in
Table 15. Amplifier gains were adjusted to provide a clear arrival
time on the oscillograph record; calibration was not attempted. Table
17 lists the oscillograph record number, the recording oscillograph
and trace identification, the serial number and component of the de
tecting transducer, and the first arrival time of the detected particle
velocity for these tests. The arrival time values are averages of the
data from at least three hammer blows and measurements by two different
oscillogram readers.


12
ct = \(e + e + e ) + 2G£
zz rr 90 zz' zz
/
(9)
o = G7 n
z9 z9
(10)
a = G7
rz rz
(11)
4Q
CD
II
Q
V
4
CD
(12)
and the equations of equilibrium for an infinitesimal
sions of dr rdQ and dz are
dgrr + 1 ^CTrQ + ^CTrz + grr g99 52u
dr r 39 3z r 2
element with dimen-
(13)
5gr9 + 1 5g99 + 5gz9 + 2gr9 v
3r r 39 3z r t2 p
(14)
drz 1 agz6 5CTzz
3r r 39 3z
(15)
Since the particle motion, u
v w with respect to the coordinates
r 9 z of the half-space is desired, the equilibrium equations are
written as
( \ + 2G)
JL
3r
1 3(ru) + 1 3v + 3w
G 3
1 3(rv)
1 3u
r 3r r 39 3z.
r 39
_r 3r
t 60.
(16)
(X + 2G) -|r
r 39
1 3(ru) + 1 3v + 3w
r 3r r 99 9z
_3. /l 3w 3v\
3z \r 39 3zJ
3 1 3(ru) 1 3u
3r r 3r r 39
(17)


128
Cables
The voltage generated by the particle velocity transducers was
conducted from the test area to truck-mounted amplifying and recording
equipment by 1,000-ft-long instrumentation cables produced by the
Belden Manufacturing Company, Chicago, Ill., and designated as trade
number 8777- Eight cables, with pin connectors at each end, were
used; each cable consisted of six conductors, shields, and shield
grounds, and was adequate to transmit the signals generated by one
model L-1B-3DS 3-component transducer. The continuity and resistance
of each cable and conductor were checked by connecting a known resist
ance across each pair of conductors and measuring the circuit resist
ance at the other end of the cable.
Each conductor had a resistance of 0.016 ohms/ft; the amplifier
impedance was 1 megohm, and the transducer circuit had a resistance
of less than 1,200 ohms. The transducer voltage drop due to the cable
length was negligible.
Amplifiers
Model 1-165 amplifiers, manufactured by the Consolidated Electro
dynamics Corporation (CEC), Pasadena, Calif., were used. These dif
ferential amplifiers had solid state circuitry and gain steps from 10
to 1,000, and they had been designed specifically for driving mirror-
type galvanometers used in CEC recording oscillographs. The frequency
response was accurate to one percent, the full scale linearity was
within 0.25 percent, and the drift with a line voltage change of 10
percent was 0.5 percent. Considering the procedures used to measure
and record the particle velocities, the maximum error that could be


Table 39 (Concluded)
228
b
6.760
a
Real
Imaginary
Absolute
: Value
12.00
4.585 -
03
1-374
_
03
4.786
- 03
12.24
4.719 -
03
3.918
-
o4
4.736
- 03
12.48
4.648 -
03
-5.916
-
o4
4.685
- 03
12.72
4.374 -
03
-1.533
-
03
4.635
- 03
12.96
3.913 -
03
-2.390
-
03
4.585
- 03
13.20
3.286 -
03
-3.I27
-
03
4.536
- 03
13.44
2.524 -
03
-3.7IO
-
03
4.487
- 03
13.68
1.66l -
03
-4.116
-
03
4.439
- 03
13.92
7.380 -
04
-4.328
-
03
4.391
- 03
l4.l6
-2.016 -
o4
-4.339
-
03
4.344
- 03
i4.4o
-I.II5 -
03
-4.150
-
03
4.297
- 03
14.64
-I.96O -
03
-3.772
-
03
4.251
- 03
14.88
-2.698 -
03
-3.226
-
03
4.205
- 03
15.12
-3.296 -
03
-2.539
-
03
4.160
- 03
15.36
-3.728 -
03
-1.745
-
03
4.116
- 03
15.60
-3-975 -
03
-8.831
-
o4
4.072
- 03
15.84
-4.029 -
03
4.425
-
06
4.029
- 03
16.08
-3.889 -
03
8.752
-
o4
3.987
- 03
16.32
-3.565 -
03
1.688
-
03
3.945
- 03
16.56
-3.076 -
03
2.4o4
-
03
3.904
- 03
16.80
-2.446 -
03
2.990
-
03
3.863
- 03
17-04
-1.709 -
03
3.420
-
03
3.823
- 03
17.28
-9.028 -
04
3.674
-
03
3.784
- 03


32
Some progress, as outlined in the following paragraphs, has been
made since 1966, but rigorous solutions to the dynamic boundary value
problem on a nonhomogeneous elastic half-space have not been found or
attempted herein. Discrete methods, however, such as (the finite ele
ment and lumped mass representations, are developing rapidly and may
soon be capable of solving such problems.
Literature
Seismologists have been concerned with the influence of a variable
modulus earth structure on the speed and period of propagating earth
quake tremors (Byerly, 19^2). Their concern stems from a need to locate
the epicenters of earthquakes and to define the gross structure of the
earths mantle. Ewing, Jardetsky, and Press (1957) devoted an entire
chapter of their book to wave propagation in media with variable veloc
ity. Again, the primary purpose of the work was to study the disper
sion characteristics of propagating seismic waves. A recent paper by
Bhattacharya (1970) gives the solution of the wave equations for an in
homogeneous media. His work is limited to horizontal shear waves prop
agating in a plane, and his solutions define the variation of density,
shear modulus, and shear wave velocity with depth. The form of these
variations depends on the solution functions.
Solutions for the static displacement and stresses in a nonhomo
geneous elastic half-space due to a uniformly distributed strip or cir
cular surface loading have been developed by Gibson (1967) he con
sidered that the elastic modulus of the half-space varied linearly with
depth (linear E) and that the half-space was incompressible. Half
space stress solutions for a point or a line load on the surface of a


(JO


Figure 19- Second pour reinforcing mesh placed in first pour.
VO
-3


18
J +|jl-v+lT (at)J (bt)dt = 0
V y v'
= 2^~v+V(b2 a2)^^1
bvr(v n)
(0 < b < a) (50)
(p > a > 0) (51)
where v > p > -1 Substituting p = l and v = 3/2 into Equations
50 and 51, the values of the integrals in Equations 48 and 49 are
w
J x1//2J1(xr)jy2(xro)dx = 0
gl/g (rg rg)
r^/2r(l/2)
-1/2
(r > rQ) (52)
(r £ rQ) (53)
so that the shear stress on the surface of the half-space in contact
with the rigid disk becomes
-40r G
0) =
!'o/V
r~" \l r2 r2
_ -40G r
TT
4
2 2
ro r
(r £ rQ) (54)
Applied moment and disk rotation
The moment applied to the disk is
o 2tt
M
/ / CTz0rrd0dr
0 0
160Gi
3
(55)


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In reference to the following dissertation:
AUTHOR: Heller, Lyman
TITLE: The particle motirTTtd generated by the torsional vibration of a circular
footing on sand / (record number 3079561)
PUBLICATION DATE: 1971
I, 2.yM/},a/ lA/, //fL- as copyHjihf holder for the aforementioned dissertation,
hereby grant specific and limited archive and distribution rights to the Board of Trustees of the University of
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37
Assumption
Gibson's (1967) correlation between the stresses in a homogeneous
half-space and a nonhomogeneous half-space under static loads was con
sidered adequate evidence to assume that the dynamic stresses in a homo
geneous half-space and a nonhomogeneous half-space under dynamic loads
have the same correlation. Thus, it was assumed that the stresses
developed in a homogeneous elastic half-space by a torsionally oscil
lating rigid circular disk on the surface of the half-space are the
same as the stresses developed in a nonhomogeneous elastic half
space subjected to the same oscillatory loading.
Low frequencies.--When the frequency of the torsional loading is
low, the second time derivative of the particle displacement is small,
and the equations of equilibrium become nearly homogeneous. The stress
conditions in either the homogeneous half-space or the nonhomogeneous
half-space would approach the static loading situation, so, hypothet
ically, the two half-spaces would have almost identical stress
conditions.
High frequencies.The wave fronts propagating in a homogeneous
half-space are located on a spherical surface (Woods, 1968). Wave
fronts propagating in a nonhomogeneous half-space are functions of
source distance, surface reflections, and type of nonhomogeneity
(Byerly, 19^2; Brown, 1965). Phase relationships are also distorted
complexly in the nonhomogeneous caseleading to frequency dependent
particle motions. So, for high frequency oscillations, there is prob
ably less correspondence between the stresses in the two half-spaces.


'REG. NO.; CHAN. NO.; TRANS(S/N, CMP, R, Z)
152; Al; 9, T, 2.6, *
ON FOOTING
152; A2; 14, R, 10, 5
152; A3; 14, T, 10, 5
152; A4; 3, R, 30, 5
-Siiar.ja-.
152; A 5; 3, T, 30, 5
Urn*
152; A 6; 24, R, 60, 5
1152; A 7; 24, T, 60, 5
-L 152; A8; 26, R, 60, 15
' /o m
ft********************' -
152; A 9; 26, T, 60, 15
152; AIO; 8, R, 90, 15
152; All; 8, T, 90, 15'
Figure 43. Typical calibration record, oscillograph A, before 50-Hz vibration test.


Table 4l (Continued)
242
b
5.800
a
Real
Imaginary
Absolute Val
7.20
1.522
_
03
-2.45I
_
02
2.455 -
02
7.68
-7.363
-
03
-2.295
-
02
2.410 -
02
8.16
-I.506
-
02
-I.817
-
02
2.360 -
02
8.64
-2.O37
-
02
-1.084
-
02
2.308 -
02
9.12
-2.244
-
02
-2.062
-
03
2.253 -
02
9.60
-2.O92
_
02
6.747
_
03
2.198 -
02
10.08
-I.608
-
02
1.415
-
02
2.142 -
02
10.56
-8.788
-
03
1.894
-
02
2.088 -
02
ii.o4
-3.O62
-
04
2.033
-
02
2.034 -
02
11.52
7.872
-
03
1.8l8
-
02
1.981 -
02
12.00
1.432

02
1.294
-
02
1.930 -
02
12.48
1.794
-
02
5.621
-
03
1.880 -
02
12.96
1.817
-
02
-2.386
-
03
1.832 -
02
13.44
1.506
-
02
-9.602
-
03
1.786 -
02
13.92
9-300
-
03
-1.472
-
02
1.741 -
02
i4.4o
2.056
-
03
-1.686
-
02
I.698 -
02
14.88
-5.257
-
03
-1.571
-
02
1.657 -
02
15.36
-I.I25
-
02
-1.162
-
02
1.617 -
02
15.84
-1.481
-
02
-5.463
-
03
1.578 -
02
16.32
-1.534
-
02
1.485
-
03
1.541 -
02
16.80
-1.286
_
02
7.845
-
03
1.506 -
02
17.28
-7.942
-
03
1.239
-
02
1.472 -
02
17.76
-1.661
-
03
1.430
-
02
1.439 -
02
18.24
4.693
-
03
1.327
-
02
i.4o8 -
02
18.72
9.847
-
03
9.629
-
03
1.377 -
02
19.20
1.281
-
02
4.195
-
03
1.348 -
02
19.68
1.307
-
02
-1.874
-
03
1.320 -
02
20.16
1.066
-
02
-7-324
-
03
1.293 -
02
20.64
6.162
-
03
-1.107
-
02
1.267 -
02
21.12
5.763
-
04
-1.240
-
02
1.242 -
02
21.60
-4.918
_
03
-1.114
_
02
1.218 -
02
22.08
-9.195
-
03
-7.619
-
03
1.194 -
02
22.56
-l.i4i
-
02
-2.651
-
03
1.172 -
02
23.04
-1.118
-
02
2.690
-
03
1.150 -
02
23.52
-8.625
-
03
7.281
-
03
1.129 -
02
24.00
-4.359
_
03
1.019
-
02
1.108 -
02
24.48
6.737
-
o4
I.087
-
02
1.089 -
02
24.96
5.394
-
03
9.238
-
03
1.070 -
02
25.44
8.822
-
03
5.720
-
03
1.051 -
02
25.92
1.028
-
02
1.110
-
03
1.034 -
02
26.40
9.511
_
03
-3.584
-
03
1.016 -
02
26.88
6.756
-
03
-7.369
-
03
9-997 -
03
27.36
2.654
-
03
-9.470
-
03
9.835 -
.03
27.84
-1.884
-
03
-9.493
-
03
9.678 -
03
28.32
-5.881
-
03
-7.494
-
03
9.526 -
03
(Continued)


Figure 29
Carpenter's level used to check tilt on vibrator frame.
H
H
H


Table 40 (Continued)
235
b
7.240
a
Real
Imaginary
Absolute Value
10.80
5.257
_
03
9.294
_
03
1.068
- 02
11.16
7.652
-
03
7.237
-
03
1.053
- 02
II.52
9.323
-
03
4.574
-
03
I.O38
- 02
11.88
1.012
-
02
1.548
-
03
1.024
- 02
12.24
9.965
-
03
-I.557
-
03
1.009
- 02
12.60
8.884
-
03
-4.450
-
03
9.936
- 03
12.96
6.986
-
03
-6.855
-
03
9.788
- 03
13.32
4.462
-
03
-8.545
-
03
9.64o
- 03
13.68
1.563
-
03
-9-364
-
03
9.493
- 03
14.04
-1.417
-
03
-9.24i
-
03
9.349
- 03
i4.4o
-4.182
_
03
-8.201
_
03
9.206
- 03
14.76
-6.457
-
03
-6.363
-
03
9.065
- 03
15.12
-8.017
-
03
-3.926
-
03
8.927
- 03
15.48
-8.715
-
03
-1.150
-
03
8.791
- 03
15.84
-8.495
-
03
1.672
-
03
8.658
- 03
16.20
-7.394
_
03
4.246
_
03
8.527
- 03
16.56
-5.547
-
03
6.306
-
03
8.398
- 03
16.92
-3.161
-
03
7.645
-
03
8.273
- 03
17.28
-5.012
-
o4
8.134
-
03
8.150
- 03
17.64
2.142
-
03
7.738
-
03
8.029
- 03
18.00
4.484
_
03
6.518
-
03
7.911
- 03
18.36
6.278
-
03
4.623
-
03
7.796
- 03
18.72
7.340
-
03
2.272
-
03
7.684
- 03
19.08
7.569
-
03
-2.657
-
o4
7.574
- 03
19.44
6.957
-
03
-2.708
-
03
7.466
- 03
19.80
5.591
-
03
-4.788
-
03
7.361
- 03
20.16
3.637
-
03
-6.281
-
03
7.258
- 03
20.52
1.327
-
03
-7.033
-
03
7.157
- 03
20.88
-1.073
-
03
-6.977
-
03
7.059
- 03
21.24
-3.291
-
03
-6.137
-
03
6.963
- 03
21.60
-5.082
-
03
-4.623
-
03
6.870
- 03
21.96
-6.250
-
03
-2.622
-
03
6.778
- 03
22.32
-6.678
-
03
-3.722
-
04
6.689
- 03
22.68
-6.332
-
03
1.865
-
03
6.601
- 03
23.04
-5.269
-
03
3.833
-
03
6.516
- 03
23.40
-3.625
-
03
5.313
-
03
6.432
- 03
23.76
-1.603
-
03
6.145
-
03
6.350
- 03
24.12
5 566
-
o4
6.246
-
03
6.270
- 03
24.48
2.600
-
03
5.620
-
03
6.192
- 03
24.84
4.293
-
03
4.355
-
03
6.116
- 03
25.20
5.447
-
03
2.613
-
03
6.o4i
- 03
25.56
5-937
-
03
6.051
-
o4
5.968
- 03
25.92
5.721
-
03
-1.427
-
03
5.896
- 03
(Continued)


47
The Dutch friction-cone penetrometer, a relatively new soil
exploration tool, offered the most practical means of investigating the
possible existence and extent of fine-grained sedimentary material at
the site; this tool can also be used to reveal density variations
within the mass of sand (Schmertmann, 1967; 1969)* Eighteen soundings
were made to depths ranging from 60 to 70 ft, one to a depth of 82 ft,
and another to a depth of 102 ft. The 102-ft-deep sounding revealed
that it had nearly reached the elevation of the permanent water table.
The friction-cone penetrometer exploration did not reveal the
presence of cohesive soils within the investigated area that was sev
eral hundred yards square. In addition, no perched water table condi
tions were encountered. The cone bearing capacity data indicated that
there was a significant variation in the density of sand with depth;
however, the density variations with depth were quite consistent at
each sounding location. Thus, the depositional environment at the test
site apparently had laterally homogeneous characteristics that produced
a generally uniform horizontal stratification of the sand. The average
cone bearing capacity at various depths is listed in Table 10.
Table 10
Average Bearing Capacity of Static Cone Penetrometer


Table 39 (Continued)
224
a
Real
Imaginary
Absolute Value
4.08
-I.IO5
_
02
5.941
03
1.254 -
02
4.32
-9.519
-
03
7.744
-
03
1.227 -
02
4.56
-7.671
-
03
9.213
-
03
1.199 -
02
4.80
-5.583
_
03
1.028
_
02
1.170 -
02
5.04
-3.342
-
03
1.091
-
02
l.l4l -
02
5.28
-1.046
-
03
1.108
-
02
1.113 -
02
5-52
1.207
-
03
1.078
-
02
1.084 -
02
5.76
3.320
-
03
1.003
-
02
1.057 -
02
6.00
5.203
-
03
8.887
-
03
I.030 -
02
6.24
6.780
-
03
7.400
-
03
i.oo4 -
02
6.48
7.989
-
03
5-647
-
03
9.783 -
03
6.72
8.786
-
03
3.713
-
03
9.538 -
03
6.96
9.146
-
03
1.692
-
03
9.302 -
03
7.20
9.068
_
03
-3.206
_
04
9.074 -
03
7.44
8.568
-
03
-2.232
-
03
8.854 -
03
7.68
7.684
-
03
-3.956
-
03
8.643 -
03
7.92
6.470
-
03
-5.418
-
03
8.439 -
03
8.16
4.995
-
03
-6.557
-
03
8.243 -
03
8.40
3-339
-
03
-7.330
-
03
8.055 -
03
8.64
1.588
-
03
-7.712
-
03
7.874 -
03
8.88
-1.704
-
04
-7.698
-
03
7.700 -
03
9.12
-1.849
-
03
-7.301
-
03
7.532 -
03
9.36
-3.368
-
03
-6.556
-
03
7.370 -
03
9.60
-4.658
-
03
-5.510
-
03
7.215 -
03
9.84
-5.662
-
03
-4.227
-
03
7.065 -
03
10.08
-6.339
-
03
-2.778
-
03
6.921 -
03
10.32
-6.667
-
03
-1.242
-
03
6.782 -
03
10.56
-6.641
-
03
2.996
-
o4
6.648 -
03
10.80
-6.274
-
03
1.769
-
03
6.519 -
03
li.o4
-5.596
-
03
3.093
-
03
6.394 -
03
11.28
-4.652
-
03
4.210
-
03
6.274 -
03
11.52
-3.499
-
03
5.067
-
03
6.158 -
03
11.76
-2.203
-
03
5.630
-
03
6.045 -
03
12.00
-8.370
_
o4
5.878
-
03
5-937 -
03
12.24
5.269
-
04
5.808
-
03
5.832 -
03
12.48
1.817
-
03
5.435
-
03
5.731 -
03
12.72
2.969
-
03
4.786
-
03
5.632 -
03
12.96
3.925
-
03
3.905
-
03
5.537 -
03
13.20
4.643
-
03
2.845
-
03
5.445 -
03
13.44
5.090
-
03
1.667
-
03
5.356 -
03
13.68
5.251
-
03
4.372
-
04
5.269 -
03
13.92
5.127
-
03
-7.782
-
04
5.186 -
03
l4.l6
4.732
-
03
-I.915
-
03
5.104 -
03
l4.4o
4.094
-
03
-2.914
-
03
5.025 -
03
14.64
3.255
-
03
-3.727
-
03
4.949 -
03
(Continued)


Figure 23 Second pour of cured concrete in the footing form.
102


Table 25
Computed and Published Values of the Displacement Function
171
a
0
2I'(ro/2)
Reissner
and Sagoci
Stallybrass
Real Part of the Displacement Functioii
0.20
0.33595
0.33598
0.36
0.3373
0.3007
0.48
0.3473
0.3120
0.50
0.34872
0.34869
0.72
0.3688
0.3318
0.80
O.36769
0.36769
O.96
0.3913
0.3490
1.00
1.20
0.4129
0.3804l
0.38042
0.3614
Imaginary
Part of the Displacement Function
0.20
-0.00037
-0.00037
0.36
-O.OO217O
-0.002149
0.48
-O.OO509O
-0.005004
0.50
-O.OO566
-0.00563
0.72
-0.01666
-0.01603
0.80
-0.02175
-0.02180
0.96
-O.O3785
-0.03529
1.00
-0.04007
-o.o4oo4
1.20
-O.O6995
-0.06269
Geometrical Damping Law
Dilatational waves are not produced by the torsional oscillation of
a rigid disk on an elastic half-space, and no free or surface waves
exist (Bycroft, 1956). Radiation of energy away from ¡the vibrating
disk is due to shear waves (Richart, Hall, and Woods, ¡1970) These
shear or body waves are governed by a geometrical damping law that
relates particle displacement amplitudes on a propagating spherical wave
front. At large distances from the source, the amplitude of particle
motion is inversely proportional to the source distance (Woods, 1968);


Table 38 (Continued)
218
b
5.O6O
a
Real
Imaginary
Absolute
Value
0.90
-1.208
_
03
8.780
_
04
1.493
_
03
1.08
-1.395
-
03
1.086
-
03
1.768
-
03
1.26
-I.552
-
03
1.308
-
03
2.029
-
03
1.44
-I.674
-
03
1.544
-
03
2.278
-
03
1.62
-1.759
-
03
1.793
-
03
2.511
-
03
1.80
-I.8OI
-
03
2.051
-
03
2.730
-
03
1.98
-1.797
-
03
2.316
-
03
2.932
-
03
2.16
-1.746
-
03
2.584
-
03
3.118
-
03
2.34
-1.645
-
03
2.847
-
03
3.288
-
03
2.52
-1.493
-
03
3.102
-
03
3.442
-
03
2.70
-I.291
-
03
3.340
-
03
3.581
-
03
2.88
-1.041
-
03
3.555
-
03
3.704
-
03
3.06
-7.452
-
04
3.740
-
03
3.813
-
03
3-24
-4.077
-
o4
3.887
-
03
3.908
-
03
3-42
-3.388
-
05
3.990
-
03
3.991
-
03
3-60
3.696
-1
04
4.044
-
03
4.061
-
03
3.78
7.952
-
o4
4.042
-
03
4.119
-
03
3.96
1.234
-
03
3.980
-
03
4.167
-
03
4.14
1.678
-
03
3.857
-
03
4.206
-
03
4.32
2.115
-
03
3.669
-
03
4.235
-
03
4.50
2.536
-
03
3-419
-
03
4.257
-
03
4.68
2.931
-
03
3.106
-
03
4.271
-
03
4.86
3.290
-
03
2.735
-
03
4.278
-
03
5.04
3.602
-
03
2.311
-
03
4.279
-
03
5.22
3.859
-
03
1.840
-
03
4.275
-
03
5.40
4.053
-
03
1.330
-
03
4.266
-
03
5.58
4.179
-
03
7.918
-
o4
4.253
-
03
5.76
4.229
-
03
2.347
-
o4
4.236
-
03
5.94
4.202
-
03
-3.297
-
04
4.215
-
03
6.12
4.096
-
03
-8.896
-
o4
4.192
-
03
6.30
3.911
-
03
-1.433
-
03
4.165
-
03
6.48
3.650
-
03
-1.947
-
03
4.137
-
03
6.66
3.317
-
03
-2.421
-
03
4.107
-
03
6.84
2.919
-
03
-2.843
-
03
4.075
-
03
7.02
2.462
-
03
-3.204
-
03
4.o4i
-
03
7.20
1.957
-
03
-3.496
-
03
4.006
-
03
7.38
1.416
-
03
-3.710
-
03
3.971
-
03
7.56
8.486
-
o4
-3.841
-
03
3.934
-
03
7.74
2.692
-
o4
-3.888
-
03
3.897
-
03
7.92
-3.093
-
04
-3-847
-
03
3.859
-
03
8.10
-8.736

o4
-3.720
-
03
3.821
-
03
8.28
-1.410
-
03
-3.510
-
03
3.783
-
03
8.46
-1.907
-
03
-3.222
-
03
3-744
-
03
8.64
-2.352
-
03
-2.863
-
03
3.705
-
03
8.82
-2.735
-
03
-2.442
-
03
3.667
-
03
(Continued)




POWER REQUIRED IN HP
105
/
/
/
/
0
/
/
/
'
/
/
/
/ O
/
/
/
/
/
V
/
<
\\l
N,
/
/
s
o 10 20 30 40 50
FREQUENCY OF TORSIONAL VIBRATION IN HZ
Figure 25. Power required to drive the torsional vibrator.


io6
resin number 4, made by the Minnesota Mining and Manufacturing Company,
was applied to the plates before the vibrator was set in place on the
test footing. One end of the mounted vibrator, the tqst footing, and
the epoxy compound are shown in Figure 26; the test pqd and mounted
vibrator are completely illustrated in Figure 27. When the epoxy had
cured, the vibrator was fastened to the mounting plates with four
5/8 -in.-diam lug nuts so that tensile cracks would not develop in the
epoxy.
Operating tests
Preliminary experiments were performed to assure that the vibrator,
the test footing, and the power source isolation schemes would work
before the particle motion transducers were buried in the soil deposit.
A 750-ft-long reel of power cable was unrolled on top of the ground,
connected to a remote 10-kw motor generator at one end, and connected
to a terminal box and vibrator motor switch at the other end. The
2-hp electric motor, with a sprocket to develop a vibration frequency
of 40 Hz, and motor balancing weights were assembled on the vibrator,
and the belt was tensioned and adjusted to synchronize the rotating
eccentric masses. The assembled vibrator, the test footing, and the
switch box are shown in Figure 28.
Measurements to determine the settlement and tilt of the test
footing were made during the preliminary operating tests and during the
subsequent field tests. A transit and level rod were used to measure
elevations at the center of the vibrator frame and at four equally
spaced points around the top edge of the footing. A carpenter's level
was used to check tilting of the vibrator frame and the top surface of


136
positive vertical movement was upward. The positive radial direction
was away from the center of the footing and the positive transverse di
rection, in plan, was counterclockwise. The sense of the single com
ponent transducers was the same as the transverse component of the 3-
component transducers. Particle velocities in positive directions
caused the galvanometer traces to move toward the top of the record.
Table 15 lists the transducer serial number or model, the compon
ent of motion recorded, the cable number that carried the signal to the
recording truck, the oscillograph that recorded the motion, and the
position of the galvanometer trace on the photorecording paper. These
items are listed for each set of connections used to record one vibra
tion test. Connection details are given in Appendix B.
Utilization of system
This section describes the procedures and methods that were em
ployed to utilize the measuring system for determining the particle ve
locities generated by a torsionally vibrating footing.
Procedures.--The first set of transducers listed in Table 15 was
connected to the instrumentation cables, appropriate eccentric weights
were attached to the torsional vibrator, and the vibrator was started.
The speed or frequency of the vibrator was measured with an electronic
counter that sampled a transducer signal and displayed the period of the
vibration. The speed of the vibrator was adjusted by changing the gover
nor setting on the generator that powered the vibrator motor. When the
desired frequency was attained, the amplifier gains were adjusted and
trimmed until the observed movement of the galvanometer light beams.,
visible on the oscillograph viewing screen, had an amplitude of about


45
Table 8
Boring Log for Hole 1
Moisture
Standard
Driller
s
Sample Depth
Content
Penetration Resistance
Classification and Description
ft
i
blows/ ft
Color
Symbol
Description
0.0
to
1.5
6
3
Medium
brown
SM
Silty sand, fine grained with surface or-
ganic material--
hair roots, etc.
1.5
to
3.0
5
3
Brown
Silty sand, fine grained with trace of
surface organic matter
3-0
to
4.5
5
3
Tan
Silty sand, fine grained
4.5
to
6.0
5
4
6.0
to
7.5
4
7
7-5
to
9.0
4
7
9.0
to
10.5
3
6
Light tan
SP-SM
Sand, poorly graded, with silt fines con-
tent. Fine grained and sharp particles
10.5
to
12.0
4
8
12.0
to
13.5
3
11
13.5
to
15.0
4
13
15.0
to
16.5
3
18
Sand, poorly graded, fine grained. Sharp
particles with slightly silty fines
16.5
to
18.0
5
23
Medium
brown
SM
Silty sand, fine grained
18.0
to
19.5
7
20
Light
red
SC
Clayey sand, fine grained
19.5
to
21.0
7
19
1
21.0
to
22.5
8
17
1
22.5
to
24.0
5
15
Tan
SM
Silty sand, fine grained with trace of
clay
24.0
to
25.5
5
16
Silty sand, fine grained with trace of
clay
25-5
to
27.0
5
16
Silty sand, line grained
27.0
to
28.5
6
15
Reddish tan
Silty sand, fine grained with trace of
clay
28.5
to
30.0
4
15
Tan
SP-
M
Sand, poorly graded, fine grained, sharp,
with trace of silt fines
30.0
to
31.5
10
31.5
to
33.0
6
18
-
33.0
to
34.5
8
18
Reddish tan
SC
Clayey sand, fine grained
34.5
to
36.0
6
19
Tan
SP-
SM
Sand, poorly graded, fine grained with
silt fines
36.0
to
37.5
8
15
Sand, poorly graded, fine grained with
silt fines
37.5
to
39.0
5
17
Sand, poorly graded, fine sharp grains,
slight silt content
39.0
to
40.5
10
14
Light red
S
Clayey sand, fine grained
40.5
to
42.0
8
12
42.0
to
43.5
9
19
43.5
to
45.0
9
23
45.0
to
46.5
7
20
46.5
to
48.0
7
22
48.0
to
49.5
8
26
Reddish tan
Clayey sand, fine grained with less
clay content
49.5
to
51.0
7
21
51.0
to
52.5
7
24
52-5
to
54.0
7
22
Clayey sand, fine grained with slight
clay content
54.0
to
55.5
8
27
55.5
to
57.0
6
28
57.0
to
58.5
6
24
58.5
to
60.0
6
25
Light
tan
SP-
SM
Sand, poorly graded, fine, sharp grains,
trace of silt


263
Resistance of Transducer Circuits
After the last test was concluded, the resistance of each trans
ducer component was measured at the transducer cable connection to
assure that the transducer had not been damaged and that it had per
formed consistently throughout the testing program. Table 44 lists
the electrical resistance measured for the circuit of each transducer
component.


72
the cured first pour and the soil stresses were limited to elastic
values, it was reasonable to expect the soil pressure distribution on
the contact area to be nearly the same as that for a rigid footing on
an elastic foundation.
Limiting torsional moment
The maximum torsional moment that could be applied to the test
footing without causing slippage on the contact area was limited by the
critical normal stress developed in the soil, by the dead load on the
footing, and by the location of this critical stress on the contact
area. With the assumption that the friction angle between the footing
and the soil was the same as the angle of internal friction for the co
hesionless sand foundation, the moment, M' to cause impending slippage
on the contact surface at the locus of the critical normal stress was
computed from
2n
M' =
/ /
ctzz tan 0,
r -
r 3
1 r drdQ
r 2tt
o
V?-
/ /<* tan f> r2d0dr (135)
rl
where a is the critical vertical stress on the contact area and r,
ZZ JL
is the radial distance to the location of this critical stress. Sub
stituting r = 2.5 ft cj = 936 psf r, = 2.47 ft and 0=30
degrees into the above expression, the limiting moment, M' is
M' = 4,400 ft-lb
(136)
The moment capacity of a rigid disk on a half-space with a similar


completely flexible and was loaded by a uniformly distributed pressure
acting on its entire upper surface; the load-deformation relations ob
tained illustrate the rigidity or stiffness of the soil. The estimated
elastic properties and design dimensions of the test footing with an
advantageously positioned equivalent dead load were used to calculate a
second load-deformation relationship for the contact area that illus
trated the rigidity of the footing. The relative rigidity of the foot
ing with respect to the soil was taken as the ratio of the footing
load-deformation relationship to the soil load-deformation relationship.
In the first case, Young's modulus, E for the foundation mate-
s
rial at a depth of 2-l/2 ft was about 7>050 psi, and Poisson's ratio
was about l/3- The deflection of the soil at the edge of the flexible
footing, due to the uniform load, q acting on its entire surface is
(Timoshenko and Goodier, 1951)
/ \
( /r=r uE
o s
(128)
and at the center of the footing
2(1 p2)iro
E
s
(129)
The deflection, A
s
within the contact area is
A
s
- ( 2tT ^)(l U2) r. n
TTW.
(130)
= 0.002751 in.
where q is expressed in pounds per square inch.


64
load, to the cured, footing. The last condition is discussed in a fol
lowing section.
Figure 6 is a sketch of the embedded concrete test footing. The
following sequence of placement operations resulted in the desired ver
tical stress distribution on the contact surface between the footing
and the soil.
1. The natural soil within a 7-f"t-diam circle was excavated to a
depth of 1 ft. The stress change in the soil due to the excava
tion was equivalent to a uniform unloading pressure of 7h act
ing on the excavated area.
2. The concentric footing form and soil retaining ring were
placed in the excavation, the first pour of concrete was placed
inside the footing form to a depth that produced a uniform pres
sure of 7h on the soil, and backfill soil was placed around the
retaining ring to the original ground surface and at its original
in situ density. At the end of these operations, the stresses in
the soil on the footing-soil contact surface and in the vicinity
of the footing were approximately the same as the in situ stresses
before excavation because the added loads were equal to the loads
removed during excavation.
3- The first pour of concrete was allowed to cure to a rigid
mass.
4. A second pour of concrete was added inside the form to act as
a dead load on the rigid first pour. It was this dead load, prop
erly applied, that produced the desired distribution of normal-
stresses on the footing-soil contact area.


24
where h = 2_ and n is an integer.
2n
Example calculation
The application of Simpson's rule for calculating ohe approximate
value of 1^ , and I can be illustrated by an example.
The integration parameters for the particle motion in a half
space at a distance of 30 ft from the center of the disk and at a depth
of 15 ft, when the shear wave velocity in the half-space is 650 fps and
the 5 ft diameter disk oscillates at 20 Hz are
k = ou/v = 2tt20/650 = 0.192
(85)
. = kr = 0.192(2.5) = 0.48
(86)
a = kr = 0.192(30) = 5.76
(87)
b = kz = 0.192(15) = 2.90
(88)
For illustration purposes, divide the integration interval into 4 equal
parts, i.e. h = l/4(ir/2) = rr/8 Values for the common terms in the
integrand of 1^ and 1^ are listed in Table 1 and values, f(a) of
the integrand of I are given in Table 2. The calculation for 1^
by Simpson's rule is
I1 = -(tt/8)/3 [O + 4(0.0027899 0.0199034) + 2(-0.0031265) + 0]
= 0.009779 (89)
Values, /(a) of the integrand of I are given in Table 3. The cal
culation for I by Simpson's rule is


i48


230
Table 40 (Continued)
b
0.288
a Real Imaginary Absolute Value
10.80
-1.517
-
02
-1.587
-
03
1.525
-
02
11.16
-1.429
-
02
3.691
-
03
1.476
-
02
11.52
-1.171
-
02
8.189
-
03
1.429
-
02
11.88
-7.862
-
03
l.l4l
-
02
1.386
-
02
12.24
-3.271
-
03
1.304
-
02
1.345
-
02
12.6o
1.459
-
03
1.298
-
02
1.306
-
02
12.96
5.747
-
03
1.132
-
02
1.269
-
02
13.32
9.094
-
03
8.355
-
03
1.235
-
02
13-68
1.114
-
02
4.515
-
03
1.202
-
02
l4.o4
1.171
-
02
3-144
-
o4
1.171
-
02
14.40
1.080
_
02
-3.715
_
03
1.142
_
02
14.76
8.592
-
03
-7.088
-
03
1.114
-
02
15.12
5.427
-
03
-9-420
-
03
1.087
-
02
15.48
1.735
-
03
-1.048
-
02
1.062
-
02
15.84
-2.004
-
03
-1.018
-
02
1.038
-
02
16.20
-5.328
-
03
-8.632
-
03
l.oi4
-
02
16.56
-7.844
-
03
-6.077
-
03
9.923
-
03
16.92
-9.276
-
03
-2.873
-
03
9.711
-
03
17.28
-9.492
-
03
5.550
-
o4
9.508
-
03
17.64
-8.514
-
03
3.774
-
03
9.313
-
03
18.00
-6.513
-
03
6.393
-
03
9.126
-
03
18.36
-3.776
-
03
8.111
-
03
8.947
-
03
18.72
-6.730
-
04
8.748
-
03
8.774
-
03
19.08
2.397
-
03
8.268
-
03
8.608
-
03
19.44
5.054
-
03
6.770
-
03
8.448
-
03
19.80
6.981
-
03
4.479
-
03
8.294
-
03
20.16
7.965
-
03
1.709
-
03
8.146
-
03
20.52
7.915
-
03
-1.178
-
03
8.002
-
03
20.88
6.875
-
03
-3.818
-
03
7.864
-
03
21.24
5.007
-
03
-5.890
-
03
7.731
-
03
21.60
2.574
_
03
-7.152
03
7.601
-
03
21.96
-1.036
-
o4
-7.476
-
03
7.477
-
03
22.32
-2.681
-
03
-6.850
-
03
7.356
-
03
22.68
-4.840
-
03
-5.382
-
03
7.239
-
03
23.04
-6.323
-
03
-3.284
-
03
7.125
-
03
23.40
-6.965
-
03
-8.378
-
o4
7.016
-
03
23.76
-6.712
-
03
1.640
-
03
6.909
-
03
24.12
-5.621
-
03
3.837
-
03
6.806
-
03
24.48
-3.854
-
03
5.487
-
03
6.705
-
03
24.84
-1.654
-
03
6.398
-
03
6.608
-
03
25.20
6.923
o4
6.477
-
03
6.514
-
03
25.56
2.885
-
03
5.737
-
03
6.422
-
03
25.92
4.653
-
03
4.295
-
03
6.332
-
03
(Continued)


182
Figure 6l; the relationship of these lines to the data points was in
fluenced by the penetration resistance and cone bearing capacity values
shown in Figure 60.
Basis for Comparing Results
Peak particle velocities were measured in the sand deposit and on
the test footing during the vibration experiments. P^ak particle dis
placements in the half-space and on the rigid disk were calculated.
For steady state sinusoidal vibration, the particle velocity is equal
to the angular frequency times the particle displacement.
The calculated particle displacement is directlv proportional to
the moment applied to the weightless rigid disk. The test footing was
not weightless and the torsional moment applied to th^ soil was not
determined; measuring the moment on the footing-soil interface is dif
ficult and usually inaccurate near the resonant frequency of the foot
ing. These difficulties were eliminated by normalizing the calculated
motion with respect to the motion of the rigid disk and normalizing the
measured motion with respect to the motion of the test footing; the nor
malized motions were then compared. The relationship between the dis
placement of the edge of the disk and the displacement of the edge of
the footing was not investigated in this work; it has been previously
confirmed by Richart and Whitman (1967)-
Measured Motion
The torsionally vibrating footing radiated energy into the sand
deposit in the form of shear waves. Since the energy level of any
particle in the sand deposit is proportional to the square of the


Table 4l (Continued)
245
b
13-500
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.48
6.971
-
o4
4.063
-
04
8.O69
-
o4
O.96
1.409
-
03
7.740
-
04
1.607
-
03
1.44
2.145
-
03
1.064
-
03
2.394
-
03
1.92
2.911
-
03
1.237
-
03
3.163
-
03
2.40
3.699
-
03
1.255
-
03
3.906
-
03
2.88
4.492
-
03
1.080
-
03
4.620
-
03
3.36
5.255
-
03
6.825
-
o4
5.300
-
03
3.84
5.942
-
03
4.083
-
05
5.942
-
03
4.32
6.488
-
03
-8.514
-
04
6.544
-
03
4.80
6.823
-
03
-I.98O
-
03
7.104
-
03
5.28
6.867
-
03
-3.305
-
03
7.621
-
03
5.76
6.548
-
03
-4.759
-
03
8.094
-
03
6.24
5.806
-
03
-6.241
-
03
8.524
-
03
6.72
4.608
-
03
-7.627
-
03
8.911
-
03
7.20
2.959
_
03
-8.771
-
03
9.256
-
03
7.68
9.130
-
o4
-9.518
-
03
9.561
-
03
8.16
-1.423
-
03
-9.725
-
03
9.828
-
03
8.64
-3.889
-
03
-9.277
-
03
1.006
-
02
9.12
-6.279
-
03
-8.109
-
03
1.026
-
02
9.60
-8.357
-
03
-6.226
-
03
1.042
-
02
10.08
-9.883
-
03
-3.713
-
03
1.056
-
02
10.56
-1.064
-
02
-7.431
-
o4
1.067
-
02
11.04
-1.047
-
02
2.434
-
03
1.075
-
02
11.52
-9.304
-
03
5.507
-
03
1.081
-
02
12.00
-7.179
-
03
8.139
-
03
1.085
-
02
12.48
-4.255
-
03
1.001
-
02
1.087
-
02
12.96
-8.089
-
o4
1.085
-
02
1.088
-
02
13.44
2.791
-
03
1.051
-
02
1.087
-
02
13.92
6.124
-
03
8.955
-
03
1.085
-
02
i4.4o
8.770
-
03
6.328
-
03
1.081
-
02
14.88
1.037
-
02
2.909
-
03
1.077
-
02
15.36
1.068
-
02
-8.904
-
o4
1.072
-
02
15.84
9.616
-
03
-4.584
-
03
1.065
-
02
16.32
7.288
-
03
-7.675
-
03
1.058
-
02
16.80
3.983
_
03
-9.725
-
03
1.051
-
02
17.28
1.421
-
o4
-1.043
-
02
1.043
-
02
17.76
-3.697
-
03
-9.660
-
03
1.034
-
02
18.24
-6.978
-
03
-7.514
-
03
1.025
-
02
18.72
-9.213
-
03
-4.290
-
03
1.016
-
02
19.20
-1.006
_
02
-4.612
-
o4
1.007
-
02
19.68
-9.375
-
03
3.396
-
03
9.971
-
03
20.16
-7.265
-
03
6.686
-
03
9.873
-
03
20.64
-4.052
-
03
8.894
-
03
9.773
-
03
21.12
-2.445
-
04
9.670
-
03
9.673
-
03
(Continued)


59
theory, the footing had to he rigid with respect to torsional deforma
tion, i.e., a radius of the footing in contact with the soil should not
be distorted during rotation by an applied torsional moment, and the
footing had to be rigid with respect to flexure in a vertical plane.
Rigidity in this plane means that vertical dead loads applied to the
footing cause negligible bending of any footing radius in a vertical
plane that contains that radius.
Vertical stresses.A flexural.1 y rigid circular disk, pressed
vertically against the horizontal boundary of an elastic half-space,
develops normal stresses, azz on the contact area of
(119)
where P is the total load applied to the rigid disk (Timoshenko and
Goodier, 1951). If a vertical load were applied to a rigid circular
footing resting on soil, and the magnitude of this load was limited
such that the normal stresses between the footing and the soil were
about 1/2 of the stresses that would initiate local failure of the
soil, the soil would react in an essentially elastic manner (Timmer
man and Wu, 1969). These conditions were prescribed for the designed
test footing, so the distribution of normal stresses on most of the
contact area between a lightly loaded rigid footing and an elastic
foundation material was probably similar to that given by Equation
119. Figure 5 shows the probable distribution of vertical stresses
on the contact surface.
Summary.It was considered that the stress distribution between


INNER FORM
WIRE WRAPPING
FLANGE
OUTER WALL
TAPE
Figure 12. Footing form and soil retaining ring.


Table 29
Influence of Frequency on Half-Space Displacements
187
Normalized 50-Hz
Half-Space Displacements/
Normalized 15-Hz
Half-Space Displacements
Depth Below
Radial Distance
from Center
of Disk, ft
Surface, ft
3.5 10
30
60
90
1
1.17 2.00
2.34
2.38
2.38
5
2.12
2.35
2.38
2.38
15
2.51
2.42
2.40
2.38
25
2.6l
2.49
2.43
2.4l
35
2.65
2.56
2.47
2.43
in the half-space. The tabulations reveal that, for equal amplitudes
of disk displacement, 50-Hz oscillations generate half-space displace
ments about 2.4 times the displacements generated by 15-Hz oscillations.
Normalized Soil Displacements
The normalized soil displacements, defined as the ratio of the meas
ured resultant particle displacement in the sand deposit to the meas
ured tangential displacement of the edge of the test footing, are given
in Table 30 for the position of each buried transducer and for the 5
test frequencies. Table 31 lists the ratio of the normalized soil dis
placements measured at 50 Hz to the normalized soil displacements meas
ured at 15 Hz.
Changing the excitation frequency of the test footing from 15 to
50 Hz caused erratic changes in the measured soil displacement field.
Soil displacements were halved at some locations and at others the dis
placement was more than tripled. Considering the entire field of trans
ducers, however, the displacement was increased about 5>0 percent due to
increasing the frequency from 15 to 50 Hz.


NUMBER OF TRANSDUCERS
134
Figure 42. Histogram of measured transduction constants


15
where
V =
Bz2
-B(x) / \/ 2 2\
^ j^xrJix k )
e-pzeiujt
B v B(x) T / \ 2-Bz itut
= V1 J-, (xr)o) e p e
3t2 X 1
,2 p 2
k G ^
(30)
(31)
(32)
Boundary Conditions
The shear stresses in the elastic half-space are
ze(r,z,t)
G?ze = G S = G
M(x?
/ \ -Bz iuut
(xr)e K e
(33)
The boundary conditions for stress on the surface of the half-space
when the rigid disk is rotated statically, so that z = ou = k = 0 ,
become
ozQ G x Jx(xr) (r <; rQ) (34)
- 0 (r > rQ) (35)
and the static displacement v at the surface of the half-space is
v(r,0) = Jx(xr) (36)
Since x and B(x) are arbitrary, set
00
v = f Jx(xr)dx = ^r (r <; rQ) (37)
0
00
CTZ0 = G j B(x)J1(xr)dx =0 (r > rQ) (38)
0
and let r/rQ = s and xrQ = y so that


206
An attempt to relate standard penetration test resistance and cone
bearing capacity to in situ soil stresses should be made in order to
relate these values to shear wave velocity.
Comparisons
Comparative work similar to that illustrated herein should con
tinue, but particle motion measurements from actual operating facil
ities (arsenals, power plants, forging mills, etc.) should receive
increased attention and analysis.


70
In the second case, Young's modulus for concrete, Ec was assumed
to be 3 x 10^ psi and Poisson's ratio was taken as 0.17 (Dunham, 1953>
Lin, 1955). The cured first pour of the footing had a diameter of 5 ft
and a planned thickness of about 8-l/4 in. Before computing the actual
design situation, the deformation of the contact area estimated by the
center deflection of an edge supported circular plate supporting a
uniform load, q acting over its entire surface was calculated. Using
thin plate theory (Timoshenko and Woinowsky-Krieger, 1959)? the deflec
tion, A ^ is
Acl
3(5 + M-)(l p)rc
l6E t3
c
= 0.0003871 in.
(131)
Even under these unrealistic support and loading assumptions, the in
fluence of the stiffness of the cured first pour decreased the deflec
tion of the footing-soil contact area by a factor, S of
S = P- (132)
cl
0.002751
0.0003871
= 7.1
The deformation of the footing-soil contact area was next approx
imated by the deflection of the cured first pour simply supported along
a circle of radius r and loaded by a uniformly distributed dead load
with its center of pressure lying on a circle of radius r ; see


transduction of these units was 2.36 volts/in./sec.
257
Single-Component Transducers
The single-component transducers used to measure particle veloc
ities in the horizontal plane were manufactured by Mark Products, Inc.,
Houston, Texas; both transducers were model L-1D. These transducers
were 2-3/8 in. diam, l-l/2 in. high, and weighed about l-l/2 lb.
The maximum case to coil excursion was 0.090 in., the undamped
natural frequency was 4.5 Hz, and the useful frequency range at 72 per
cent of critical damping was 10 to several hundred Hz. The transduc
tion tolerance was plus or minus 10 percent, the frequency change with
tilt was less than 0.25 Hz at 15 degrees of tilt, and the frequency
change was less than 0.1 percent at half of the rated output.
The coil resistance was 1,480 ohms and the shunt resistance was
3,000 ohms.
Cables
The instrumentation cables were made by the Belden Manufacturing
Company, Chicago, Illinois. This particular cable had a designated
trade number of 8777* The outside diameter of the cable was 0.27 in.
and covered with a 0.030-in.-thick chrome vinyl jacket.
Six number 22-gage stranded and tinned copper conductors with
0.010-in.-thick polypropylene insulation were contained in the cable.
The six conductors were separated into three pairs and each pair was
covered with an aluminized polyester shield; each shield was connected
to a separate ground wire. The conductors had a resistance of


(* CENTER LINE OF TEST FOUNDATION
O 10, 15 O 30, 15 O 60, 15 90, 15 O
O 10, 25
O 30, 25
O 60, 25
90, 25 O
O 10, 35 O 30, 35
- r
O 60, 35
90, 35 O
Figure 11. Section view of the field of transducer locations.
00
ON


176
possibility of slippage on the contact area was recognized and means
for detecting and measuring the slippage were provided. A particle
motion transducer was buried 6 in. below the edge of the footing to
measure the soil motion and an identical transducer was mounted on the
footing to measure its motion. Slippage between the footing and the
soil was indicated if the ratio of the footing displacement to the soil
displacement changed appreciably during the experimental program.
Figure 59 shows the footing/soil displacement ratio plotted versus
(l) the vibration frequency of the footing and (2) th displacement of
the edge of the footing. The measured footing/soil displacement ratio
decays smoothly with increasing frequencies--in agreement with the
computed behavior of an elastic half-space. The measured footing/soil
displacement ratio changes erratically with footing displacement; the
ratio has no apparent correlation with footing displacement.
Since the measured footing/soil displacement ratio was unrelated
to footing displacement and the ratio agreed with halff-space calcula
tions, good contact between the footing and the sand was evident and
transmission of elastic stresses into the sand deposit was indicated.
Particle Motion Components
A rigid disk, in torsional oscillation on the surface of a homo
geneous elastic half-space, generates half-space particle displacements
in a direction that is tangent to the circumference of the disk; no
radial or vertical motion is developed. In the experimental program,
particle motion in the sand deposit due to the torsional oscillation
of the footing was measured in the transverse (tangential), radial, and


20
are related by
v(r,z,t) =
11 (
- g b ze(r,z,t)
(6o)
and the specific shear-stress formulation (Equation 46) that satisfies
the boundary conditions is
-40r G sin xr xr cos xr
0z6(r,Z,,t) =
I
xr
J1(xr)epzelu)tdic (6l)
0
Substituting Equation 6l into Equation 60 gives the particle dis
placement in the half-space as
i, a icut 03
40r e sin xr xr cos xr
v(r,z,t) = 2 / -^r 2 J1(xr)e_pzdx (62)
0
\
The normalized particle motion in the half-space with (respect to the
tangential displacement of the disk is then
v(r,z,t) 4 icut
0r tt
r o
I
sin xr xr cos xr
o o
pxr
- J1(xr)e"^zdx (63)
Evaluation of the Infinite Integral
In order to calculate the particle motion at a specific point in
the half-space, it is necessary to evaluate the infinite integral for
particular values of four quantities, r r z and k The four
quantities are reduced to three if a = rk aQ = rQk b = zk and
g is a new arbitrary parameter g = x/k Substituting and represent
ing the infinite integral by
I (aQ, a, b) = f
sm aQg aQg cos aQg
e
0 aQg(g 1
1/2
J1(ag)dg (-64)


250
b
2.400
7.240
Table 42 (Continued)
a
Real
Imaginary
Absolute Value
28.80
-8.721 -
03
-1.135
- 02
1.432 02
29.40
-I.332 -
02
-4.391
- 03
1.403 02
30.00
-I.322 -
02
3.787
- 03
1.375 02
30.6o
-8.624 -
03
1.036
- 021
1.348 02
31.20
-1.274 -
03
1.316
- 02
1.322 02
31.80
6.236 -
03
1.138
- 02
1.298 02
32.40
1.135 -
02
5.788
- 03
1.274 02
33.00
l.24l -
02
-1.575
- 03
1.251 02
33.60
9.203 -
03
-8.l4l
- 03
1.229 02
3^.20
2.970 -
03
-1.170
- 02
1.207 -.02
34.80
-4.064 -
03
-1.115
- 02
1.187 02
35-40
-9.474 -
03
-6.807
- 03
1.167 02
36.00
-1.147 -
02
-2.842
- o4
1.147 02
36.60
-9.480 -
03
6.124
- 03
1.129 02
37.20
-4.313 -
03
1.023
- 02
1.111 02
37.80
2.166 -
03
1.071
- 02
1.093 02
38.40
7.704 -
03
7.512
- 03
1.076 02
39.00
1.043 -
02
1.838
- 03
1.060 02
39.60
9-511 -
03
-4.296
- 03
1.044 02
40.20
5.356 -
03
-8.776
- 03
1.028 02
4o.8o
-5.131 -
o4
-1.012
- 02
1.013 02
4i.4o
-6.037 -
03
-7.952
- 03
9-984 03
42.00
-9.334 -
03
-3.122
- 03
9.842 03
42.60
-9.337 -
03
2.645
- 03
9.704 03
43.20
-6.136 -
03
7.344
- 03
9.570 03
0.00
0.000 +
00
0.000
+ 00
0.000 + 00
0.60
5.009 -
03
1.976
- 03
5.384 03
1.20
i.oo4 -
02
3.170
- 03
1.053 02
1.80
1.496 -
02
2.883
- 03
1.524 02
2.40
1.934 -
02
6.078
- 04
1.935 02
3.00
2.247 -
02
-3.829
- 03
2.279 02
3.60
2.346 -
02
-1.012
- 02
2.555 02
4.20
2.147 -
02
-1.741
- 02
2.764 02
4.80
1.605 -
02
-2.432
- 02
2.914 02
5.4o
7.386 -
03
-2.921
- 02
3.013 02
6.00
-3.488 -
03
-3.049
- 02
3.069 02
6.60
-1.479 -
02
-2.715
- 02
3.091 02
7.20
-2.426 -
02
-1.908
- 02
3.087 02
7.80
-2.972 -
02
-7.341
- 03
3.06l 02
8.4o
-2.962 -
02
5.950
- 03
3.021 02
9.00
-2.358 -
02
1.805
- 02
2.969 02
9.60
-1.266 -
02
2.621
- 02
2.910 02
10.20
8.306 -
04
2.845
- 02
2.846 02
(Continued)


174
footing, that the resonant frequency would be 40 Hz, and that the mag
nification factor at resonance would be about five. Table 26 lists the
moment applied to the footing, the peak displacement at the edge of the
footing, and the frequency of the oscillations. The ^xact resonant
Table 26
Measured Motion of the Test Footing
Vibration
Frequency
Hz
Moment Applied
to the Footing
ft-lb
Displacement at
Edge of the
Footing
in.
15
1,380
0.00201
20
1,502
0.00719
30
1,470
oj00285
40
2,6l4
0.00249
50
2,818
0.00155
frequency of the footing was not determined because the torsional vi
brator was driven by a constant speed motor and the moment applied to
the footing was limited. Figure 58 is a plot of the vibration fre
quency versus the ratio of the footing displacement to the applied
moment. Table 26 and Figure 58 indicate that the footing displacements
were about twice the anticipated values, the resonant frequency was
perhaps half the anticipated value, and the magnification ratio was
more than three.
Footing-Soil Contact Area
A torsionally vibrating footing transmits its motion to the sup
porting soil by contact friction at the footing-soil interface. The


163
Table 21
Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,470 ft-lb Oscillating at 30 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Rooting
0.0115
O.O316
0.649
2.344
Footing
None
None
0.503
2.5
1.5
None
None
0.385
3.5
1.0
0.0135
0.0261
0.297
10.0
1.0
0.00772
0.00598
O.0655
30.0
1.0
0.00262
0.00301
0.0513
60.0
1.0
0.000930
0.00468
0.0210
90.0
1.0
0.00374
0.00412
0.01057
90.0
5.0
0.000861
0.00303
0.0101
10.0
5.0
0.00419
0.0015$
0.0454
30.0
5.0
0.00245
0.00122
O.OI67
60.0
5.0
0.00178
0.00140
0.00922
60.0
15.0
0.000593

0.00151
90.0
15.0
0.000511
0.000476
0.00202
30.0
15.0
0.000500

0.00288
60.0
25.0
0.000126
0.000442
0.000161
60.0
35.0
0.000239
0.000213
0.000411
90.0
25.0
0.000125
0.000267
0.000817
90.0
35-0
0.000170
0.000180
O.OOO668
10.0
15.0
0.000550
0.00115
0.00488
10.0
25.0
0.000576
0.000489
0.000771
10.0
35.0
0.000522
0.000244
0.000632
30.0
25.0
0.000112
0.000131
o.ooo846
30.0
35.0
0.000236
0.000180
0.000419


133
Figure 42 is a histogram of the transduction constants determined
for 39 transducers. The standard deviation of these measurements was
0.06164 and the probable error in any one measurement was 2.4 percent
(Durelli, Phillips, and Tsao, 1958). The probable eror was about 4.1
percent for the L-ID transducers.
The frequency response errors in the amplifiers were assumed to be
normally distributed and the full scale linearity error of 0.25 per
cent was possible, so the probable amplifier error was about 0.5 per
cent. The same assumptions result in a probable galvanometer error
of 1.4 percent. Calibration voltage errors varied from 0.005 to 1.5
percent, depending on the measured voltage.
Amplitude and time measurements were scaled on developed oscillo
graph records. The scaling errors depend on the measured length and
probably averaged from 1 to 3 percent on the calibration records and
about the same value on the test records.
Considering the entire measurement system, from transducer to in
terpreted data, the estimated maximum error was probably between 7 and
15 percent.
Power generators
Electric power for operating the amplifiers, oscillographs, paper
processor, vibrator motor, and other electrical miscellanea was supplied
by three Kohler gasoline-engine-driven, 60-Hz generators. One 3-phase,
240-volt, 15-kw generator and one single phase, 120/p40 volt, 10-kw
generator were mounted on a trailer. The other generator was a small,
air-cooled, single phase unit that developed 3 kw at 120/240 volts.


Figure 28. Assembled vibrator, test footing, and switch box
o
\o


Table 38 (Continued)
217
b
3.620
5.O6O
Real
Imaginary
Absolute Value
5.76
3.328
-
03
4.230
-
03
5.383
-
03
5.94
3.867
-
03
3.634
-
03
5.306
-
03
6.12
4.306
-
03
2.967
-
03
5.229
-
03
6.30
4.638
-
03
2.245

03
5.153
_
03
6.48
4.854
-
03
1.486
-
03
5.076
-
03
6.66
4.949
-
03
7.093
-
04
5.000
-
03
6.84
4.924
-
03
-6.667
-
05
4.924
-
03
7.02
4.780
-
03
-8.226
-
04
4.850
-
03
7.20
4.521
-
03
-1.540
-
03
4.776
-
03
7.38
4.157
-
03
-2.201
-
03
4.704
-
03
7.56
3.697
-
03
-2.791
-
03
4.632
-
03
7.74
3-155
-
03
-3.295
-
03
4.562
-
03
7.92
2.547
-
03
-3.702
-
03
4.493
-
03
8.10
1.888
-
03
-4.003
-
03
4.426
-
03
8.28
1-197
-
03
-4.192
-
03
4.359
-
03
8.46
4.923
-
04
-4.266
-
03
4.294
-
03
8.64
-2.069
-
o4
-4.226
-
03
4.231
-
03
8.82
-8.826
-
o4
-4.074
-
03
4.169
-
03
9.00
-I.517
_
03
-3.817
_
03
4.108
-
03
9.18
-2.094
-
03
-3.464
-
03
4.048
-
03
9.36
-2.600
-
03
-3.026
-
03
3.990
-
03
9.54
-3.022
-
03
-2.517
-
03
3-933
-
03
9.72
-3.350
-
03
-1.952
-
03
3.877
-
03
9.90
-3-577
-
03
-1.348
-
03
3.823
-
03
10.08
-3.700
-
03
-7.218
-
o4
3.769
-
03
10.26
-3.716
-
03
-9.129
-
05
3.717
-
03
10.44
-3.629
-
03
5.259
-
04
3.667
-
03
10.62
-3-441
-
03
1.113
-
03
3.617
-
03
10.80
-3.162
-
03
1.654
-
03
3.568
-
03
10.98
-2.800
-
03
2.135
-
03
3.521
-
03
11.16
-2.367
-
03
2.543
-
03
3.474
-
03
11.34
-1.877
-
03
2.870
-
03
3-429
-
03
11.52
-1.346
03
3.106
-
03
3.385
-
03
11.70
-7.887
-
04
3.247
-
03
3.341
-
03
11.88
-2.220
-
o4
3.292
-
03
3.299
-
03
12.06
3.377
-
o4
3.240
-
03
3.258
-
03
12.24
8.745
-
04
3.096
-
03
3.217
-
03
12.42
1.373
-
03
2.865
-
03
3.178
-
03
12.60
1.821
-
03
2.557
-
03
3.139
-
03
12.78
2.205
-
03
2.180
-
03
3.101
-
03
12.96
2.516
-
03
1.749
-
03
3-064

03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.18
-2.616
-
o4
1.624
-
o4
3.079
-
o4
0.36
-5.182
-
o4
3.283
-
04
6.134
-
o4
0.54
-7.647
-
o4
5.011
-
o4
9-143
-
04
0.72
-9.963
-
o4
6.835
-
04
1.208
-
03
(Continued)


17
F(y) =
-1/2
2 y
i' i)
i
yl/2 Jj/2 (y) /
*2)
-1/2
(0rQi) di
^ / \ -1/ 2
/ m2\l m / dm x J (^romi)(yi)3/2J3^2(y)dje
O O
^\/ sin y / 3(x
2X-1/2
di
y m3(l m2) dm J (yi)3/2(sin yi yi eos yi)di
O O
4 / sin y y eos y
= ^ ¡r
TT r O
y
)
(44)
The expression for shear stress (Equation 4o) is then
a26(r,0,) = G / (£ )fc0) (
0
sin y y cos y
y
^ Jx(sy)dy
-40r G f sin xr xr cos xr
2_ f 2 2 j (xr)dx
tt J xr_ lv '
(45)
(46)
xr
40r G
TT
f yjjs (xrQ)1//2 J3/2(xro) Jx(xr)dx (47)
0
Restating the boundary conditions in terms of stress requires that
40r G r _/0 f n/_
Z0 TT
= 0
0
(r s rQ)
(48)
(r > rQ)
(49)
The infinite integral appearing in Equation 48 is one of the special
cases of the discontinuous Weber-Schafheitlin integral tabulated by-
Abramowitz and Stegun (1964)


l8o
The average compression wave velocity increases continuously with
depth. Figure 60 shows the penetration resistance and average cone
bearing capacity of the sand deposit plotted versus depth. Both re
sistance and capacity increase with depth to about 20 ft, decrease from
20 to 30 ft, then increase from 30 ft to greater depths.
A comparison of Figures 56, 57 3 and 60 shows that the measured
shear wave propagation velocity correlates well with the penetration
and bearing values at various depths; the compression wave velocity
seems unrelated to penetration and bearing values. A change in k or
o
Poisson's ratio with depth in the sand deposit can account for the
observed differences in shear and compression wave velocities (Richart,
Hall, and Woods, 1970)- The 20-ft-thick layer of sand between 10
and 30 ft deep was probably a beach area during the Pleistocene epoch;
vibration from breaking waves and innumerable changes in effective
stress could have created this anomalous sand layer.
The gross distribution of the shear wave velocity with depth in
the sand deposit at the test site was estimated from the results of
the surface wave exploration method (Maxwell and Fry, 1967), the meas
ured compression wave velocities (Poisson's ratio taken as 1/3), and
the measured shear wave velocities. Figure 6l shows the distribution
of the measured shear wave velocities with depth for each method.
Since all of the methods were either empirical, involved an assumed
Poisson's ratio, or tested only a small part of the sand deposit, there
was considerable scatter in the measured velocity distribution with
depth. The shear wave velocities in the gross sand deposit were esti
mated and established by drawing the two straight lines shown in


19
so the angular rotation of the disk becomes
(56)
Substituting this value of 0 into Equation 54 gives
o
2
r
r
2
> rG) (57)
Reissner and Sagoci (1944), using a system of oblate spheroidal co
ordinates instead of dual integral equations, found this same shear-
stress distribution on the contact surface between a torsionally
loaded rigid disk and an elastic half-space.
Particle Displacements
It was previously found that any arbitrary parameter x and
function B(x) will satisfy the equilibrium equation and that a spe
cific form of B(x) will also satisfy the prescribed boundary condi
tions. Thus, the specific formulation for B(x) that satisfies the
boundary conditions will also satisfy the equilibrium equation
throughout the half-space.
The solutions to the equilibrium equation
(58)
and the shear-stress expression
(59)


82
(Richart, Hall, and Woods, 1970), is greatest for body waves, so
minimum motions were estimated from the previously computed displace
ments due to torsion induced shear waves in an elastic half-space. For
steady state sinusoidal oscillation, the particle velocity, v is re
lated to the particle displacement by
v = u)V (l44)
For the test foundation, oscillated at 15 Hz by a 1,400 ft-lb moment
on the test site sand with an average shear wave velocity of 650 fps,
the particle velocity at a depth of 35 ft and. a distance of 170 ft was
v = (2nf) ^ -p l(a ,a,b) (145)
V 4GTir2
o
= 0.00007 in./sec
If material damping losses are neglected, the calculated particle ve
locity in the ground at this point is 25 percent more than the motion
needed to produce an oscillograph record with a single amplitude of
1 in. Material damping losses, or attenuation of motion in excess
of geometrical damping, reduce the expected particle motion signifi
cantly, so particle motion measurements greater than 90 ft from the
test foundation were judged to be beyond the sensitivity and capacity
of available motion recording equipment.
The horizontal and vertical spacing of 20 available transducers
within 90 ft of the footing and 35 ft below the surface of the sand de
posit was selected to facilitate an interpretation of the veloc
ity of the propagating seismic waves. A close spacing is desirable for


LIST OF SYMBOLS
a = rk
a = frequency ratio; r k
o o
b = zk ; radius of uniformly loaded area
B = arbitrary function
CMP = component of motion
d = galvanometer trace deflection during calibration
D = galvanometer trace deflection during test
e = base of natural logarithms; void ratio; eccentricity
E = Young's modulus
Ec = Young's modulus for concrete
Ei = exponential integral
Eg = Young's modulus for soil
f = frequency; function
/ = function
F = function
g = variable, arbitrary parameter, x/k ; function; gravitational
acceleration
G = shear modulus
G(z) = shear modulus that depends on z
h = interval; depth below ground surface
i = vn
imj = iml = imaginary part of the integral
I = particle displacement integral
xiv


Table 40 (Continued)
231
b
i.44o
a
Real
Imaginary
Absolute 1
7alue
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.36
2.340
-
02
-I.58O
-
02
2.823
-
02
0.72
3.858
-
02
-3.O3O
-
02
4.906
-
02
1.08
4.220
-
02
-4.234
-
02
5.978
-
02
1.44
3.607
-
02
-5.O97
-
02
6.244
-
02
1.80
2.401
_
02
-5.559
_
02
6.055
_
02
2.16
9.468
-
03
-5.594
-
02
5.674
-
02
2.52
-5.065
-
03
-5.217
-
02
5.241
-
02
2.88
-1.787
-
02
-4.478
-
02
4.821
_
02
3.24
-2.778
-
02
-3.460
-
02
4.437
-
02
3.60
-3-410
_
02
-2.267
_
02
4.095
_
02
3.96
-3.655
-
02
-1.014
-
02
3-793
-
02
4.32
-3.521
-
02
1.830
-
03
3.526
-
02
4.68
-3.056
-
02
1.220
-
02
3.291
-
02
5.04
-2.334
-
02
2.013
-
02
3.083
-
02
5.40
-1.449
-
02
2.509
-
02
2.897
-
02
5.76
-5.030
-
03
2.685
-
02
2.732
-
02
6.12
4.004
-
03
2.552
-
02
2.583
-
02
6.48
1.171
-
02
2.152
-
02
2.450
-
02
6.84
1.739
-
02
1.549
-
02
2.329
-
02
7.20
2.059
_
02
8.265

03
2.218

02
7.56
2.117
-
02
7.288
-
04
2.118
-
02
7.92
1.927
-
02
-6.258
-
03
2.026
-
02
8.28
1.530
-
02
-I.I96
-
02
1.942
-
02
8.64
9.854
-
03
-I.582
-
02
1.864
-
02
9.00
3.666
-
03
-1.754
-
02
1.792
-
02
9.36
-2.504
-
03
-I.707
-
02
1.725
-
02
9.72
-7.944
-
03
-1.461
-
02
1.663
-
02
10.08
-1.207
-
02
-1.058
-
02
1.605
-
02
10.44
-1.448
-
02
-5.561
-
03
1-551
-
02
10.80
-1.501
_
02
-I.929
-
04
1.501
-
02
11.16
-1.370
-
02
4.865
-
03
1.454
-
02
11.52
-1.082
-
02
9.033
-
03
1.409
-
02
11.88
-6.795
-
03
1.186
-
02
1.367
-
02
12.24
-2.180
-
03
1.310
-
02
1.328
-
02
12.60
2.443
-
03
1.267
-
02
1.291
-
02
12.96
6.520
-
03
1.073
-
02
1.255
-
02
13.32
9.587
-
03
7.575
-
03
1.222
-
02
13.68
1.132
-
02
3.660
-
03
1.190
-
02
l4.04
1.159
-
02
-5.068
-
04
l.l60
-
02
i4.4o
1.042
_
02
-4.408
_
03
1.131
_
02
14.76
8.028
-
03
-7.582
-
03
1.104
-
02
15.12
4.759
-
03
-9.676
-
03
1.078
-
02
15.48
1.053
-
03
-1.048
-
02
1.053
-
02
15.84
-2.617
-
03
-9.960
-
03
1.030
-
02
(Continued)


248
Table 42 (Continued)
b
0.480
2.400
Real
Imaginary
Absolute Value
21.00
1.602
-
02
-1.157
-
02
1.976
-
02
21.60
6.523
-
03
-1.807
-
02
1.921
-
02
22.20
-4.665
-
03
-1.809
-
02
1.869
-
02
22.80
-1.368
-
02
-1.199
-
02
1.819
-
02
23.4o
-1.760
-
02
-2.135
-
03
1.773
-
02
24.00
-1.534
-
02
7.954
-
03
1.728
_
02
24.60
-7.987
-
03
1.485
-
02
1.686
-
02
25.20
1.734
-
03
1.637
-
02
1.646
-
02
25.80
l.04l
-
02
1.225
-
02
1.607
-
02
26.40
1.515
-
02
4.144
-
03
1.571
-
02
27.00
1.452
-
02
-5.009
-
03
1.536
-
02
27.60
8.965
-
03
-1.206
-
02
1.502
-
02
28.20
5.899
-
o4
-1.469
-
02
1.470
-
02
28.80
-7.637
-
03
-1.220
-
02
i.44o
-
02
29.40
-1.292
-
02
-5.652
-
03
i.4io
-
02
30.00
-1.358
-
02
2.570
-
03
1.382
-
02
30.60
-9.571
-
03
9.590
-
03
1.355
-
02
31.20
-2.444
-
03
1.306
-
02
1.329
-
02
31.80
5.249
-
03
1.193
-
02
1.304
-
02
32.40
1.086
-
02
6.764
-
03
1.280
-
02
33-00
1.255
-
02
-5.339
-
o4
1.256
-
02
33.60
9.882
-
03
-7.388
-
03
1.234
-
02
34.20
3.920
-
03
-1.147
-
02
1.212
-
02
34.80
-3.180
-
03
-1.148
-
02
1.191
-
02
35.40
-8.948
-
03
-7.553
-
03
1.171
-
02
36.00
-1.146
-
02
-1.167
-
03
1.151
-
02
36.60
-9.950
-
03
5.410
-
03
1.133
-
02
37.20
-5.078
-
03
9.919
-
03
1.114
-
02
37.80
1.383
-
03
1.088
-
02
1.097
-
02
38.40
7.166
-
03
8.072
-
03
1.079
-
02
39.00
1.031
-
02
2.580
-
03
1.063
-
02
39.60
9.816
-
03
-3.633
-
03
1.047
-
02
40.20
5.963
-
03
-8.411
-
03
1.031
-
02
4o.8o
1.733
-
o4
-1.016
-
02
1.016
-
02
4i.4o
-5.508
-
03
-8.360
-
03
1.001
-
02
42.00
-9.133
-
03
-3.789
-
03
9.869
-
03
42.60
-9.513
-
03
2.039
-
03
9.729
-
03
43.20
-6.609
-
03
6.955
-
03
9.594
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.60
4.4o6
-
03
-4.218
-
02
4.24i
-
02
1.20
-2.969
-
03
-7.347
-
02
7.353
-
02
1.80
-2.462
-
02
-8.599
-
02
8.945
-
02
2.40
-5.281
-
02
-7.703
-
02
9.339
-
02
(Continued)


196
the nonhomogeneous half-space calculations are closer to the ideal than
the homogeneous half-space calculations.
The decrease in displacement amplitude of propagating waves with
distance from their source, in excess of the decrease due to geometri
cal damping, is caused by material damping (Richart, Hall, and Woods,
1970)- The effect of material damping on the amplitude, v of
particle displacements at distances r^ and r^ from the source of
motion is quantified by an attenuation coefficient, d* such that
v^ = v^(geometrical damping law)(material damping) (158)
= v^(geometrical damping law) e-0i^r4-r3' (159)
For body waves
L = h e-cy(r4-r3) (160)
V3 r4
The displacement ratios computed with respect to .the homogeneous
and nonhomogeneous half-spaces consider geometrical damping, but do
not consider material damping effects, consequently, the average dis
placement ratios have been consistently greater than one. A gross
attenuation coefficient for the entire sand deposit was computed using
the frequency averaged nonhomogeneous displacement ratios for the 20
transducers located from 10 to 90 ft from the center of the footing and
buried from 1 to 35 ft deep. The average displacement ratio for the
nonhomogeneous half-space and for all of these transducer locations was
I.67. The center of these 20 transducers was 50 ft from the center of
the footing and 17.5 ft below the ground, so the distance from this


58
stress, zz
acting on the plane of contact by
(118)
where tan 0' is the coefficient of friction between the bottom face
of the footing and the sand. This equation indicates that a vertical-
dead load must be applied to the test footing in order to develop the
necessary normal stresses between the footing and the foundation.
If the shear modulus of the material used to construct a solid
cylindrical test footing is much greater than the shear modulus of the
sand on which it rests, negligible distortion of a radius of the foot
ing in contact with the soil would occur as a torsional moment was
applied to the footing. The shear modulus of concrete was 3^0 times
the shear modulus of the sand on the contact area between the footing
and the soil, so a concrete footing was considered to be rigid with
respect to the soil. Further, if the shear stresses in the soil at
the footing-soil interface are limited to about l/3 of the failure
(slip) value, and these stresses are repetitive, laboratory tests on
sand show that these soils will behave elastically (Timmerman and Wu,
1969).
Since a concrete test footing would be rigid with respect to the
soil, and the soil would behave elastically during torsional footing
oscillations, it was reasonable to expect the shear stresses on a large
portion of the footing-soil interface to be similar to those developed
by a rigid disk on a half-space. Figure 5 shows the probable distribu
tion of torsion induced shear stresses on the contact area.
So, to represent the boundary conditions assumed by the Bycroft


5
Available Theory
Since the purpose of the theoretical work was to calculate the
particle motion in an elastic half-space due to an oscillating surface
source, the simplest mode of source oscillation was chosen for analy
sis and interpretation. Bycroft developed solutions for the oscilla
tion of a rigid circular plate on the surface of a half-space for
four modes of motion: (l) vertical, (2) rocking, (3) sliding, and
(4) torsion. Of these four modes, rotation of the plate about a ver
tical axis, or torsion, was the simplest because this is an uncoupled
motion and because no dilation of the half-space exists (Reissner,
1937; Bycroft, 1956).
Two general types of solutions were found for the torsional oscil
lation of a surface source. One assumed that the shear stresses on the
circular contact area were zero at the center and increased linearly
along a radius of the area (Reissner, 1937; Miller and Pursey, 1954).
The other assumed that the contact area was rigid (Reissner and Sagoci,
1944; Bycroft, 1956; Collins, 1962; Stallybrass, 1962, 1967; Awojobi
and Grootenhuis, 1965; Thomas, 1968). From an experimental viewpoint,
duplication of the rigid boundary condition case by an oscillating test
footing presented fewer difficulties than controlling the shear stress
on the contact area. For this reason, the rigid contact boundary con
dition between the half-space and the oscillator was chosen for calcu
lating half-space particle motions due to torsional source vibrations.
The solution for the oscillation of a rigid disk on the surface
of a homogeneous (constant E) elastic half-space given by Bycroft
(1956) was appropriate for practical foundation dimensions and typical


168
Table 24
Measured Arrival Time and Average Wave Propagation
Velocity for Vibration Tests
Transducer
Location
Radial Depth
ft ft
Arrival Time in Seconds
15-Hz Test 20-Hz Test 30 Hz-Test
Average
Velocity
fps
Depth
ft
10.0
1.0
0.0170
0.0225
0.0245
395
3
30.0
1.0
0.0620
0.0775
0.0823
383
5
30.0
5-0
0.0620
0.0755
0.0833
627
10
10.0
5.0
0.0l60
0.0255
0.0270
524
15
30.0
15.0
0.0600
0.0595
0.0583
963
20
10.0
15.0
0.0255
O.O3O5
0.0300
655
25
30.0
25.0
0.0570
O.O56O
0.0550
848
39
10.0
25.0
0.0375
O.385
0.0395
30.0
35.0
0.0600
0.610
0.0585


Table 39 (Continued)
223
b
a
Real
Imaginary
Absolute
: Value
O.96O
10.80
-6.914
_
03
-4.366
_
o4
6.927
- 03
11.04
-6.675
-
03
1.174
-
03
6.778
- 03
11.28
-6.O8I
-
03
2.653
-
03
6.634
- 03
11.52
-5.I77
-
03
3.925
-
03
6.497
- 03
11.76
-4.025
-
03
4.931
-
03
6.365
- 03
12.00
-2.697
-
03
5.625
-
03
6.238
- 03
12.24
-I.27I
-
03
5.983
-
03
6.117
- 03
12.48
1.706
-
04
5.997
-
03
5-999
- 03
12.72
1.549
-
03
5.679
-
03
5.887
- 03
12.96
2.792
-
03
5.059
-
03
5.778
- 03
13.20
3.835
-
03
4.181
-
03
5.674
- 03
13.44
4.629
-
03
3.103
-
03
5.573
- 03
13.68
5.139
-
03
1.890
-
03
5.475
- 03
13.92
5-346
-
03
6.131
-
o4
5.381
- 03
l4.i6
5.250
-
03
-6.548
-
04
5.290
- 03
l4.4o
4.865
-
03
-1.844
-
03
5.203
- 03
14.64
4.221
-
03
-2.893
-
03
5.118
- 03
14.88
3-364
-
03
-3.747
-
03
5.035
- 03
15.12
2.346
-
03
-4.365
-
03
4.956
- 03
15.36
1.229
-
03
-4.721
-
03
4.878
- 03
15.60
7.690
-
05
-4.803
-
03
4.804
- 03
15-84
-1.044
-
03
-4.614
-
03
4.731
- 03
16.08
-2.074
-
03
-4.173
-
03
4.661
- 03
16.32
-2.958
-
03
-3.512
-
03
4.592
- 03
16.56
-3.651
-
03
-2.674
-
03
4.526
- 03
16.80
-4.121
-
03
-1.710
-
03
4.461
- 03
17.04
-4.346
-
03
-6.771
-
04
4.399
- 03
17.28
-4.322
-
03
3.652
-
04
4.338
- 03
2.900
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.24
-2.856
-
o4
-2.232
-
03
2.250
- 03
0.48
-6.710
-
04
-4.357
-
03
4.4o8
- 03
0.72
-1.239
-
03
-6.272
-
03
6.394
- 03
O.96
-2.042
-
03
-7.887
-
03
8.147
- 03
1.20
-3.095
-
03
-9.122
-
03
9.633
- 03
1.44
-4.375
-
03
-9.919
-
03
1.084
- 02
1.68
-5.825
-
03
-1.024
-
02
1.178
- 02
1.92
-7-364
-
03
-1.007
-
02
1.247
- 02
2.16
-8.898
-
03
-9.410
-
03
1.295
- 02
2.40
-1.032
-
02
-8.299
-
03
1.325
- 02
2.64
-1.155
-
02
-6.787
-
03
1.339
- 02
2.88
-1.248
-
02
-4.943
-
03
1.342
- 02
3.12
-1.304
-
02
-2.854
-
03
1.335
- 02
3.36
-I.320
-
02
-6.173
-
04
1.322
- 02
3.60
-1.292
-
02
1.663
-
03
1.303
- 02
3.84
-I.219
-
02
3.883
-
03
I.280
- 02
(Continued)


252
Jo
7.24o
12.060
Table 42 (Continued)
a
Real
Imaginary
Absolute
: Val
36.60
-3.927
-
03
1.023
-
02
1.096
- 02
37.20
2.377
-
03
1.053
-
02
1.080
- 02
37.80
7.708
-
03
7.327
-
03
1.064
- 02
38.40
1.033
-
02
1.783
-
03
1.048
- 02
39.00
9.436
-
03
-4.196
-
03
1.033
- 02
39.60
5.429
-
03
-8.6IO
-
03
1.018
- 02
40.20
-2.750
-
o4
-I.OO3
-
02
1.003
- 02
4o.8o
-5.731
-
03
-8.066
-
03
9.895
- 03
4i.4o
-9.125
-
03
-3.460
-
03
9.759
- 03
42.00
-9.376
03
2.179
-
03
9.626
- 03
42.60
-6.483
-
03
6.940
-
03
9.497
- 03
43.20
-1.493
-
03
9.252
-
03
9.371
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.60
-6.956
-
04
1.838
-
03
1.966
- 03
1.20
-I.219
-
03
3.703
-
03
3.899
- 03
1.80
-1.402
-
03
5.595
-
03
5.768
- 03
2.40
-1.087
-
03
7.468
-
03
7.547
- 03
3.00
-1.469
-
04
9.211
_
03
9.212
- 03
3.60
1.494
-
03
1.064
-
02
1.074
- 02
4.20
3.827
-
03
1.151
-
02
1.213
- 02
4.80
6.727
-
03
1.156
-
02
1.337
- 02
5.4o
9.931
-
03
1.051
-
02
1.446
- 02
6.00
1.303
-
02
8.191
-
03
1.539
- 02
6.60
1.552
-
02
4.584
-
03
1.619
- 02
7.20
1.684
-
02
-1.177
-
04
1.684
- 02
7.80
1.649
-
02
-5.466
-
03
1.737
- 02
8.4o
1.415
-
02
-1.078
-
02
1.779
- 02
9.00
9.810
-
03
-1.522
-
02
1.810
- 02
9.60
3.816
-
03
-1.793
-
02
1.833
- 02
10.20
-3.081
-
03
-1.821
-
02
1.847
- 02
10.80
-9.824
-
03
-1.573
-
02
1.854
- 02
11.4o
-1.522
-
02
-1.062
-
02
1.856
- 02
12.00
-1.817
-
02
-3.577
-
03
1.852
- 02
12.60
-1.794
-
02
4.255
-
03
1.844
- 02
13.20
-1.439
-
02
1.133
-
02
1.832
- 02
13.80
-8.049
-
03
1.629
-
02
1.817
- 02
i4.4o
-9.630
-
05
1.799
-
02
1.800
- 02
15.00
7.843
-
03
1.598
_
02
1.780
- 02
15.60
1.405
-
02
1.058
-
02
1.759
- 02
16.20
1.712
-
02
2.927
-
03
1.737
- 02
16.80
1.630
-
02
-5.279
-
03
1.714
- 02
17.40
1.175
-
02
-1.214
-
02
1.690
- 02
(Continued)


Table 39 (Continued.)
225
b
2.900
4.840
a
Real
Imaginary
Absolute 1
Value
14.88
2.265
_
03
-4.316
_
03
4.874
03
15.12
1.182
-
03
-4.654
-
03
4.802
-
03
15.36
6.594
-
05
-4.731
-
03
4.732
-
03
15.60
-1.022
-
03
-4.550
-
03
4.663
-
03
15.84
-2.O25
-
03
-4.127
-
03
4.597
-
03
16.08
-2.89O
-
03
-3.492
-
03
4.532
-
03
16.32
-3.574
-
03
-2.684
-
03
4.470
-
03
16.56
-4.046
-
03
-I.751
-
03
4.408
-
03
16.80
-4.284
_
03
-7-480
_
04
4.349
_
03
17.04
-4.282
-
03
2.697
-
o4
4.291
-
03
17.28
-4.046
-
03
1.246
-
03
4.234
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
-7.454
-
o4
2.765
-
o4
7.950
-
o4
0.48
-1.470
-
03
5-737
-
o4
1.578
-
03
0.72
-2.153
-
03
9.105
-
o4
2.338
-
03
0.96
-2.774
-
03
1.302
-
03
3.065
-
03
1.20
-3.310
-
03
1.759
-
03
3.749
-
03
1.44
-3-741
-
03
2.286
-
03
4.384
-
03
1.68
-4.045
-
03
2.879
-
03
4.965
-
03
1.92
-4.204
-
03
3.530
-
03
5.489
-
03
2.16
-4.202
-
03
4.220
-
03
5.955
-
03
2.40
-4.027
-
03
4.926
-
03
6.363
-
03
2.64
-3.674
-
03
5.620
-
03
6.714
-
03
2.88
-3.143
-
03
6.268
-
03
7.012
-
03
3.12
-2.441
-
03
6.836
-
03
7.259
-
03
3.36
-1.585
-
03
7.289
-
03
7.460
-
03
3.60
-5.981
-
o4
7.594
-
03
7.618
-
03
3.84
4.878
-
o4
7.722
-
03
7.738
-
03
4.08
1.635
-
03
7.651
-
03
7.824
-
03
4.32
2.801
-
03
7.365
-
03
7.879
-
03
4.56
3.940
-
03
6.858
-
03
7.909
-
03
4.80
5.002
-
03
6.134
-
03
7.915
-
03
5.04
5.943
-
03
5.209
-
03
7.902
-
03
5.28
6.717
-
03
4.105
-
03
7.872
-
03
5.52
7.288
-
03
2.857
-
03
7.828
-
03
5.76
7.624
-
03
1.507
-
03
7.771
-
03
6.00
7.704
_
03
1.027
_
o4
7.705
_
03
6.24
7.518
-
03
-I.302
-
03
7.630
-
03
6.48
7.067
-
03
-2.654
-
03
7.549
-
03
6.72
6.363
-
03
-3.898
-
03
7.462
-
03
6.96
5.430
-
03
-4.984
-
03
7.371
-
03
7.20
4.304
-
03
-5.867
-
03
7.276
-
03
7.44
3.029
-
03
-6.509
-
03
7.180
-
03
7.68
1.657
-
03
-6.885
-
03
7.081
-
03
(Continued)


The Particle Motion Field Generated by the Torsional
Vibration of a Circular Footing on Sand
By
LYMAN WAGNER HELLER
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1971

ACKNOWLEDGMENTS
The author wishes to express his sincere appreciation and grati
tude to his Supervisory Committee for their guidance, [their faith, and
their infinite patience during the progress of these studies. The con
tinued encouragement and counsel of Professors J. H. Schmertmann, Chair
man, M. W. Self, G. E. Nevill, Jr., and F. E. Richart,| Jr., are grate
fully acknowledged. Special thanks are due to Professor Schmertmann
for directing the work, for his contributions, and for his key sugges
tions. Particular thanks are also expressed to Professor Richart,
University of Michigan, for his many years of instruction and inspira
tion, for his service on the Supervisory Committee, and for his guidance
and encouragement during this investigation.
Financial support of the investigation, provided by the Office of
the Chief of Research and Development, Department of the Army, through
the Office of the Chief of Engineers and the Administration of the U. S.
Army Engineer Waterways Experiment Station, is gratefully acknowledged.
The personal efforts and interest of Mr. A. A. Maxwell (deceased), who
was instrumental in initiating the general research task of which this
study is a part, are also acknowledged.
The author wishes to express his appreciation to the Commanding
Officer, Eglin Air Force Base, Florida, and his staff and to the Mobile
District Office, U. S. Army Corps of Engineers, Mobile, Alabama, for
the use of facilities and for field support efforts during the experi
mental aspects of the study. Special thanks are offered to Mr. Leon
ii
J

Leskowitz, U. S. Army Electronics Command, Fort Monmoutu, New Jersey,
for his cooperation and assistance during the computational aspects of
the work.
Appreciation and gratitude is expressed to the many individuals
at the Waterways Experiment Station who assisted and contributed to the
prosecution of this study. Special thanks are extended to Mr. Monroe B.
Savage, Jr., and Mr. Jack Fowler for their capable and cooperative
assistance during the experimental work. Particular thanks are also
expressed to Miss K. Jones and her helpful staff at the Station's
Reproduction and Reports Office for preparing the reproducible copy
and photographs, and for printing the manuscript.
Finally, the author wishes to thank his wife, Elizabeth, and his
children for their patience and sacrifices during the qourse of this
study.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS 11
LIST OF TABLES viii
LIST OF FIGURES xi
LIST OF SYMBOLS xiv
ABSTRACT xviii
INTRODUCTION 1
Background
Previous Work 3
Related Work 3
Approach to the Investigation 4
Available Theory
Available Experimentation
Comparisons 3
Objective and Goals-- 3
THE THEORETICAL PARTICLE MOTION GENERATED BY THE TORSIONAL OSCIL
LATION OF A WEIGHTLESS, RIGID, CIRCULAR DISK ON THE SURFACE
OF AN ELASTIC HALF-SPACE 9
Homogeneous (Constant E) Elastic Half-Space 9
Problem Statement and Approach 9
Equations of Elasticity 10
Solution to the Equilibrium Equation l4
Boundary Conditions 15
Applied moment and disk rotation 18
Particle Displacements 19
Evaluation of the Infinite Integral 20
Example calculation 24
Computer Program to Evaluate the Integrals 29
Nonhomogeneous (Linear E) Elastic Half-Space 30
Literature 32
Results of Gibson's Solutions 33
Solution for stresses 33
Strain relationships 34
Half-Space Under Torsion 35
Torsional Oscillation 36
Assumption 37
Particle motion 1 38
IV

Page
THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCIL
LATING RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT 40
Description of Test Site 40
Geographical Location and Geological Setting 40
Soil Exploration 4l
Borings 43
Penetration tests 43
Laboratory Tests 48
Unit weight 48
Gradation 49
Seismic Wave Propagation Tests 50
Design of the Experiment 53
Foundation Design 53
Practical considerations 53
Diameter of the test footing 56
Stresses at the footing-soil interface 1 57
Stresses near the periphery of the footing 60
Footing emplacement operation 63
Position of dead load on cured first pour-^ 66
Rigidity of the footing 68
Limiting torsional moment 72
Dynamic response of the foundation 73
Vibrator Design 74
Power requirements 75
Frequency and moment capacity 78
Foundation and Transducer Location + 80
Location of the test footing 80
Location of transducers 8l
Isolation of power and recording facilities 84
Construction of Test Facilities 84
Foundation Construction 84
Fabrication of the footing form 84
Placing the form 87
First pour of concrete 88
Backfilling 95
Second pour of concrete 99
Vibrator Construction 104
Motor 16*+
Mounting the vibrator 16*+
Operating tests 166
Transducer Installation 110
Performance tests 116
Boreholes 113
Transducer alignment ¡ H3
Installing procedure ll8
Backfilling 1 H8
Particle Motion Measuring System 122
Functional Components 122
Transducers 125
v

Page
Cables 128
Amplifiers + 128
Oscillographs and. galvanometers 129
Reference (calibration) voltage -j- 129
System accuracy 132
Power generators 133
Arrangement and Utilization 135
Arrangement of components 135
Utilization of system 136
Typical vibration test data ] i4o
Schedule of Tests i47
Footing Settlement and Tilt 147
Transducer Operation l47
Torsional Vibration 154
Compression Wave Propagation 155
Transducer and Cable Resistance 155
Results of Measurements 158
Footing Settlement and Tilt 158
Compression Wave Propagation 158
Footing source 158
Surface source 159
Propagation velocities l60
Particle Velocities Due to Torsional Vibration l6o
Amplitudes l60
Wave propagation velocities 166
COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS 1 169
Test of the Calculated Results 169
Solutions at the Surface of a Homogeneous (Constant E)
Elastic Half-Space 169
Geometrical Damping Law 171
Position of Disk and Footing 173
Test of the Measured Results 173
Dynamic Footing Response 173
Footing-Soil Contact Area 174
Particle Motion Components 176
Properties of the Sand Deposit 179
Basis for Comparing Results 182
Measured Motion 182
Computed Motion 184
Comparison of Normalized Displacements 185
Comparison of Results 185
Normalized (Constant E) Half-Space Displacements 185
Normalized Soil Displacements 187
Ratio of Displacements 189
Displacements in a Nonhomogeneous (Linear E) Half-Space 191
Discussion of Results 195
Homogeneous (Constant E) and Nonhomogeneous (Linear E)
Half-Space 195
Characteristics of the Test Site 197
vi

Page
Particle Motion Predictions 200
CONCLUSIONS AND RECOMMENDATIONS 201
Conclusions 201
Homogeneous (Constant E) Half-Space 201
Nonhomogeneous (Linear E) Half-Space 201
Experimental Aspects 202
Test site 202
Test footing and vibrator 202
Particle motion measuring system 203
Results of measurements 203
Computations and Measurements 203
Recommendations 205
Analytical Work 205
Experimental Work 205
Comparisons 206
APPENDIX A CALCULATIONS FOR THE INTEGRAL l(aQ,a,b) 207
APPENDIX B SPECIFICATIONS FOR THE PARTICLE MOTION MEASURING AND
RECORDING SYSTEM 256
Transducers 256
Three-Component Transducers 256
Single-Component Transducers 257
Cables 257
Amplifiers r 258
Galvanometers 259
Oscillographs and Paper 259
Paper Processor 260
Reference (Calibration) Voltage Supply 260
Voltmeter 26l
Connections 262
Resistance of Transducer Circuits 263
LIST OF REFERENCES 265
vii

LIST OF TABLES
Page
1.Values of the Common Terms in the Integrand of I^_ and
2. Values, f(a) of the Integrand of 1^ 25
3. Values, f(a) of the Integrand of I 26
4. Values of the Terms in the Integrand of I 28
5. Values, /(a) of the Integrand of I^ 1 28
6. Calculation Parameters for l(a ,a,b) 30
o
7. Well Log at Auxiliary Field 5 42
8. Boring Log for Hole 1 45
9* Boring Log for Hole 2 46
10. Average Bearing Capacity of Static Cone Penetrometer 47
11. Results of Laboratory Tests on Samples from Hole 3 48
12. Identification Letter and Weight of Eccentric Masses 79
13. Vibrator Moment Capacity at Various Frequencies 80
14. List of Transducers, Locations, and Transduction Values 127
15- Transducers, Recorders, and Recording Sequence 137
l6. Schedule of Torsional Vibration Tests 155
17- Results of Footing Source Compression Wave Tests 159
18. Results of Surface Source Compression Wave Tests l60
19* Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,380 ft-lb Oscillating at 15 Hz l6l
viii

Page
20. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,502 ft-lb Oscillating at 20 Hz 162
21. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,470 ft-lb Oscillating at 30 Hz 163
22. Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,6l4 ft-lb Oscillating at 40 Hz l64
23. Particle Velocity Amplitudes Generated by a Torsipnal
Moment of 2,8l8 ft-lb Oscillating at 50 Hz 165
24. Measured Arrival Time and Average Wave Propagation Velocity
for Vibration Tests l68
25. Computed and Published Values of the Displacement Function-- 171
26. Measured Motion of the Test Footing 174
27. Component Displacement Ratios Averaged over 5 Frequencies 179
28. Normalized Half-Space Particle Displacements 186
29- Influence of Frequency on Half-Space Displacements-- 187
30. Normalized Soil Particle Displacements 188
31* Influence of Frequency on Soil Displacements 189
32. Ratio of Half-Space to Soil Displacement 190
33* Average Displacement Ratios for 5 Frequencies 191
34. Normalized Nonhomogeneous Half-Space Displacements 193
35- Ratio of Nonhomogeneous Half-Space Displacements to Soil
Displacements 194
36. Average Nonhomogeneous Half-Space Displacement Ratios for
5 Frequencies 195
37* Subroutines and Computer Program for the Integral
l(a ,a,b) 207
38. Value of 1(0.36,a,b) 211
39. Value of 1(0.48,a,b) 220
40. Value of 1(0.72,a,b) 229

Page
41. Value of 1(0.96,a,b) 238
42. Value of l(l.20,a,b) - 247
43- Connection of Measuring and Recording Components 262
44. Electrical Resistance of Transducer Components After
Completing Test Program 264
x

LIST OF FIGURES
Page
1. Rigid circular disk on the surface of an elastic half-space- 11
2. Location of 3 exploration borings and 20 friction-cone
penetrations \ 44
3. Grain-size distribution for six sample depths 51
4. Shear wave velocity versus depth, surface and empirical
methods 52
5. Distribution of stresses between a rigid disk and an
elastic half-space 55
6. Sketch of concrete footing embedded in soil 65
7. Distribution of vertical soil stress and dead load pressure
on cured first pour 67
8. Plan view sketch of torsional vibrator 76
9. Elevation view sketch of torsional vibrator 77
10. Plan view of the field of transducer locations 85
11. Section view of the field of transducer locations 86
12. Footing form and soil retaining ring 89
13. Excavation for footing and transducer 90
14. Transducer embedded below edge of footing 91
15 Excavation ready to receive footing form and retaining
ring 92
l6. Placing footing form and retaining ring in excavation 93
17 Footing form and retaining ring positioned in excavation 94
l8. Concrete test cylinders, auxiliary form and reinforcing
mesh 96
19- Second pour reinforcing mesh placed in first pour 97
xi

Page
20. Position of auxiliary form and backfilling operation 98
21. Cone penetration test adjacent to footing 100
22. Cone penetration test on backfill 101
23. Second pour of cured concrete in the footing form 102
24. Checking depth and continuity of the air gap 103
25. Power required to drive the torsional vibrator 105
26. Vibrator bonded to mounting plate with an epoxy compound 107
27. Torsional vibrator mounted on the test foundation 108
28. Assembled vibrator, test footing, and switch box 109
29. Carpenter's level used to check tilt on vibrator frame 111
30. Method of attaching transducers to the test footing 112
31. Test site topography, vegetation, and borehole markers ll4
32. Drill rig used to auger uncased boreholes for the
transducers 115
33. Alignment sleeves bonded to transducers with an epoxy
compound ll6
34. Transducers with support cables and electrical leads 117
35 Apparatus for installing and aligning transducers in
boreholes 119
36. Sighting bar used to align borehole rod and attached
transducer 120
37. Borehole, borehole rod, and transducer cables 1 121
38. Water hose inserted in borehole during backfilling 123
39* Functional components of the particle velocity measuring
system 124
40. Two rows of amplifiers mounted in a cabinet 130
41. Type 5-119P4 recording oscillographs 131
42. Histogram of measured transduction constants 134
xii

Page
43. Typical calibration record, oscillograph A, befozfe 50-Hz
vibration test 1 l4l
44. Typical calibration record, oscillograph B, before 50-Hz
vibration test 1 142
45- Typical test record at 50 Hz, oscillograph A, with eccentric
weights on vibrator 143
46. Typical test record at 50 Hz, oscillograph B, with eccentric
weights on vibrator \ l44
47. Typical test record at 50 Hz, oscillograph A, without
eccentric weights on vibrator 145
48. Typical test record at 50 Hz, oscillograph B, without
eccentric weights on vibrator [ l46
49. Settlement measurement at the center of the vibrator l48
50. Settlement measurement at the edge of the footing 149
51. Tilt check with level along vibrator frame + 150
52. Tilt check with level across vibrator frame 151
53- Tilt check with level on top of test footing 152
54. Results of footing settlement and tilt measurements 153
55* Compression wave initiated by striking footing 156
56. Compression wave velocities from hammer blows on footing and
on ground surface 157
57 Shear vibration propagation velocities versus depth 167
58. Frequency response of test footing 175
59* Ratio of footing displacement to soil displacement under the
footing 1 177
60. Average penetration resistance and cone bearing papacity
versus depth 1 l8l
61. Shear wave velocities versus depth 183
62. Cone bearing capacity and displacement ratio versus depth 199
xiii

LIST OF SYMBOLS
a = rk
a = frequency ratio; r k
o o
b = zk ; radius of uniformly loaded area
B = arbitrary function
CMP = component of motion
d = galvanometer trace deflection during calibration
D = galvanometer trace deflection during test
e = base of natural logarithms; void ratio; eccentricity
E = Young's modulus
Ec = Young's modulus for concrete
Ei = exponential integral
Eg = Young's modulus for soil
f = frequency; function
/ = function
F = function
g = variable, arbitrary parameter, x/k ; function; gravitational
acceleration
G = shear modulus
G(z) = shear modulus that depends on z
h = interval; depth below ground surface
i = vn
imj = iml = imaginary part of the integral
I = particle displacement integral
xiv

i' = (4/3tt)i
J = Bessel function of the first kind and nth order
n
k = wave number, oj/v^
K = kinetic energy
K = coefficient of earth pressure at rest = a /oi
o rr zz
m = rate of shear modulus change with depth, z
M = moment applied to disk or footing
M' = limiting torsional moment
= design moment
n = integer
N(c) = normalized computed motion
W(m) = normalized measured motion
p = power loss
P = total vertical load on disk or footing
q = uniformly distributed vertical load on disk or footing
q = uniformly distributed vertical dead load due '"o second pour
of concrete
r = cylindrical coordinate
r^ = radius of the disk or footing
r^ = radial distance to critical stress point
r^ = radial distance to inside edge of second pour of concrete
r = radial distance to center of pressure
R = radial distance from center of disk or footing
RJ = real I = real part of the integral
s = variable parameter = r/rQ
S = deformation ratio
xv

s/N = serial number
t = time
T = transduction constant
u = particle displacement in the r direction
v = particle displacement in the 8 direction
v^ = design displacement
v = particle velocity in the 8 direction
v = particle displacement in a nonhomogeneous half-space
V = volume; voltage
V = shear wave propagation velocity, vG/p
s
w = particle displacement in the z direction
W = strain energy density; weight
x = variable; arbitrary parameter
y = variable; arbitrary parameter xr^ ; z + G(0)/m
z = cylindrical coordinate; distance below ground surface
a = variable; integer; attenuation coefficient
= (x k ) ; G(0)/m
7 = shear strain, unit weight
7.. = shear strain in the plane of i and j
ij
7^ = shear strain in nonhomogeneous half-space
= vertical deflection of concrete footing
A = vertical deflection of soil on footing-soil contact area
s
e.. = linear strain on the i plane in the j direction
ij
8 = cylindrical coordinate
\ = variable; Lame elastic constant
p = integer
xvi

V
= integer
| = arbitrary parameter
tt = 3.14159+
P =
CTij
a =
t =
tN =
0 =
tan ft' =
r =
0) =
mass density = y/g
stress on the i plane in the j direction
mean effective stress
shear stress
shear stress in a nonhomogeneous half-space
angular rotation of disk or footing; angle of internal
friction of soil
coefficient of friction
limiting angular rotation of the disk or footing
angular frequency
xvii

Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
THE PARTICLE MOTION FIELD GENERATED BY THE TORSIONAL VIBRATION
OF A CIRCULAR FOOTING ON SAND
By
Lyman Wagner Heller
August 1971
Chairman: Dr. John H. Schmertmann, P.E.
Major Department: Civil Engineering
Over the past few years, it has been demonstrated that the self-
excited vibratory motion of a circular footing on various types of soil
can be successfully predicted by a mathematical model derived by assum
ing that the foundation soil is represented by a homogeneous elastic
half-space. This finding suggested that the same model, or variations
thereof, might be useful for predicting the particle motion generated
within a soil foundation by a vibrating footing.
The objective of this study was to test the hypothesized utility
of the half-space model for predicting the motion field generated in a
natural soil deposit by the forced torsional vibration of a circular
footing. The test involved the computation of half-space motion, the
measurement of soil motion, and a comparison of the computations to the
measurements.
A 5-f't-diam footing was vibrated at 5 different frequencies on a
natural sand deposit with a shear modulus that varied
psi at a depth of 1 ft to about 23,000 psi at a depth of 35 ft. Re
sultant particle motions were measured on the footing and at radial
distances to 90 ft and at depths to 35 ft. Homogeneous half-space
from about 1,800
xviii

particle motions were computed using a shear modulus of 9j^80 psi.
The average measured particle displacement, neglecting damping, was
between 1/3 and l/4 of the computed displacement. The measured dis
placements were l/lO of the calculated displacements at the deeper loca
tions and 3 times the calculated displacements near the ground surface.
Near the footing, the displacements were in good agreement.
Recent literature on the stress conditions in a nonhomogeneous
elastic half-space suggested that the particle displacements in a homo
geneous half-space could be used to determine the pahticle displacements
in a nonhomogeneous half-space. When the sand deposit was considered
as a nonhomogeneous half-space and damping was neglected, the displace
ments were in good agreement near the footing, the average measured dis
placement was 60 percent of the computed displacement, and the measured
displacements were l/4 to 2-l/2 times the computed displacements. The
material damping effect on the propagating body waves agreed with pre
vious determinations for this effect on surface wave at a similar
test site. The correlation between the ratio of the computed displace
ment to the measured displacement and the cone bearing capacity of
the sand deposit at various depths suggested that a more accurate and
detailed determination of the shear modulus of the sand would improve
the correspondence between measured and computed results.
Because the accuracy of the particle displacement predictions was
adequate to classify transmitted vibrations as eithejr undetectable,
readily apparent, or intolerable, the elastic half-space model, adjusted
for nonhomogeneous site conditions, was considered a potentially useful
analytical representation of a natural soil deposit ¡subjected to tor
sional footing vibrations.
xix

INTRODUCTION
Background
Soil and foundation engineers who specify and design adequate sup
port systems for buildings and equipment are commonly concerned with
three aspects of the performance of their foundations: (l) the long
term load carrying capacity, as it relates to the type of facility and
safety of its inhabitants, (2) the immediate or during-construction
settlements, and (3) the rate and amount of postconstruction
settlement.
There are situations, however, when the engineer must provide a
foundation with additional capabilities. Such a situation occurs when
the foundation supports sustained or transient dynamic loads as devel
oped by punch presses, forging machines, shock testerjs, and unbalanced
machinery. For these cases, the prescribed foundation not only must
provide support for the imposed static and dynamic loads to assure the
safe operation of the equipment and the facility, but also must mini
mize the radiation of undesirable vibrations into the surrounding soil
where they can be transmitted to adjacent inhabited or vibration sensi
tive areas. Thus, one part of the soil engineers responsibility is to
provide a foundation that will inhibit or diminish the generation and
transmission of dangerous, troublesome, and annoying ground vibrations.
Crockett (1965) has briefly discussed some of these problems; the
writer is aware of a California case in which a titanium forging plant
1

was moved from Los Angeles to Ventura County because its foundations
generated suit-worthy ground vibrations.
2
The reciprocal situation arises when inhabitable or motion sensi
tive facilities must be built in congested areas with their founda
tions resting on ground that is shaken by industrial machines, subway
and elevated trains, pile driving, blasting, and pavement breaking op
erations. In this situation, the engineer's task is to effectively
isolate sensitive structures from the ambient movements of the surround
ing ground. Margason, Barneich, and Babcock (1967) and Blaschke (1964)
have summarized some of the approaches to these problems.
The common denominator necessary for the rational solution of both
of the above situations is a definition of the characteristics of gen
erated, transmitted, and received ground vibrations. The soil engineer
currently has little empirical evidence or confirmed theory to guide
him in selecting, designing, or improving his foundations to diminish
transmitted ground vibrations or to minimize received ground vibrations.
In a summary review of a Soviet conference on the dynamics of bases and
foundations, D. D. Barkan (1965) wrote:
...primary attention should be devoted to investigating
the elastic properties of bases [supporting soil] based
on measurements of free and forced vibrations of machine
foundations since not only have effective measures to
combat waves propagating in soils not been worked out,
but there are no methods which permit calculating the
parameters of these waves or a theory for calculating
soil vibrations at various distances from the wave
source as a function of its dimensions, depth of occur
rence, and mechanical properties of soils. Without a
solution to these problems, it is impossible to de
velop methods of evaluating the effect of a wave source
on structures, equipment, and people.

3
Previous Work
One of the earlier investigations of wave propagation through
soils from a surface source was reported by Ramspeck (1936) who noted
the influence of interference waves on the observed amplitude of parti
cle motion at the ground surface. Over the years, Bernhard (1967) has
conducted, extended, and reported on a variety of similar experiments
and observations. Barkan (1962) studied wave propagation through near
surface soils with particular attention to the surface wave effects.
Lysmer and Kuhlemeyer (1969) have used discrete models for the study
of surface waves.
Available theory for waves propagating in soil has been summarized
by Woods (1968) and is based primarily on representing the soil by a
homogeneous elastic half-space. He concluded that an oscillating ver
tical pressure on a circular area at the surface of the half-space pro
duced dilatational and distortional body waves that radiated into the
half-space and surface waves that propagated along the surface of the
half-space. The annularly diverging, slow-moving, surface wave was
the dominant energy carrier. On the other hand, if an oscillating tor
sional moment is applied to the surface of a half-space, Reissner
(1937), Reissner and Sagoci (1944), and Bycroft (1956) indicate that
only shear waves are radiated into the half-space.
Related Work
The most complete collection and digest of past theoretical and
experimental work related to this study is available in Richart, Hall,

4
and. Woods (1970); contributions to this collection span almost a cen
tury. Over the past few years, however, it has been demonstrated that
the self-excited vibratory displacement of a circular footing on various
soil foundations can be successfully predicted by a mathematical model
derived by assuming that the soil supporting the footing is represented
by a homogeneous elastic half-space (Richart and Whitman, 1967). This
result suggested that the same model, or variations thereof, might be
useful for predicting the vibrations within the supporting soil due to
an oscillating footing. If the model should prove useful, it might be
utilized to attack some of the ground motion transmission and isolation
problems outlined in preceding paragraphs.
Approach to the Investigation
Based on the confirmed behavior of self-excited foundations de
scribed above, it was hypothesized that the elastic half-space model
could be used to predict the particle motion in a soil deposit due to
a vibrating foundation. A test of the hypothesis required two parallel
efforts: (l) the solution and evaluation of the particle motion in an
elastic half-space, and (2) the experimental measurement of the particle
motion in a soil deposit. The approach was in consonance with the views
of Odqvist (1968) who wrote:
In his introductory article, written in 1921, [Richard]
von Mises expressed opinions very much like those I have
been propounding here. ..."There should be no distinc
tion between theoretical and experimental papers. All
theoretical research depends in the end on observational
facts. Experimental work is useless unless it is under
taken in the view of some theory."

5
Available Theory
Since the purpose of the theoretical work was to calculate the
particle motion in an elastic half-space due to an oscillating surface
source, the simplest mode of source oscillation was chosen for analy
sis and interpretation. Bycroft developed solutions for the oscilla
tion of a rigid circular plate on the surface of a half-space for
four modes of motion: (l) vertical, (2) rocking, (3) sliding, and
(4) torsion. Of these four modes, rotation of the plate about a ver
tical axis, or torsion, was the simplest because this is an uncoupled
motion and because no dilation of the half-space exists (Reissner,
1937; Bycroft, 1956).
Two general types of solutions were found for the torsional oscil
lation of a surface source. One assumed that the shear stresses on the
circular contact area were zero at the center and increased linearly
along a radius of the area (Reissner, 1937; Miller and Pursey, 1954).
The other assumed that the contact area was rigid (Reissner and Sagoci,
1944; Bycroft, 1956; Collins, 1962; Stallybrass, 1962, 1967; Awojobi
and Grootenhuis, 1965; Thomas, 1968). From an experimental viewpoint,
duplication of the rigid boundary condition case by an oscillating test
footing presented fewer difficulties than controlling the shear stress
on the contact area. For this reason, the rigid contact boundary con
dition between the half-space and the oscillator was chosen for calcu
lating half-space particle motions due to torsional source vibrations.
The solution for the oscillation of a rigid disk on the surface
of a homogeneous (constant E) elastic half-space given by Bycroft
(1956) was appropriate for practical foundation dimensions and typical

6
vibration frequencies. No significantly improved theoretical solutions
for the torsion case have been advanced since the work by Reissner and
Sagoci (19^4) and Bycroft (1956). All of the later authors evaluate
their work, and check the accuracy of their approximations, by compari
son with the Reissner and Sagoci or the Bycroft solutions.
Investigations of a nonhomogeneous half-space with elastic moduli
that increase linearly with depth (linear E), subjected to static sur
face loads, have been conducted by Frolich (1934), Borowicka (1943),
Hruban (1948), Curtis and Richart (l955)} and Gibson (1967)* Discrete
analysis systems have been devised by Lysmer and Kuhlemeyer (1969) and
Lysmer and Waas (1970) for assessing the effects of oscillatory surface
loads on irregular and layered elastic media.
Available Experimentation
Previous experiments to determine the particle motion in a soil,
or other material, due to a torsionally vibrating surface source were
not found. Arnold, Bycroft, and Warburton (1955) computed the response
of a self-excited rigid disk and then measured the response of small
plates mounted on a foam rubber half-space and on a foam rubber layer,
but they did not measure the particle motion in the foam rubber. They
used 3/4- to 4-in.-diam plates on the surface of a 3-ft-square by
1-ft-thick block of laminated foam rubber. Useful experimental data
is evidently quite scarce since Thomas (1968) found it necessary to
compare his theoretical work to the 1955 experiments by Arnold, Bycroft,
and Warburton. Surface motion due to a vertically oscillating source
has received extensive attention (Woods, 1968).

7
Comparisons
Richart and Whitman (1967) examined existing experimental data for
the vibratory behavior of surface footings founded on soil materials
and compared this data to the theoretically predicted vibratory response
of these same footings on an elastic half-space. These comparisons
confirmed the applicability of the elastic half-space model for
predicting the oscillatory motion of circular foundations on soil.
Similar comparisons of calculated and measured results would indicate
the usefulness of the half-space model for predicting the particle mo
tions generated in a soil deposit by a vibrating footing.
Objective and Goals
The confirmed utility of the elastic half-space model for predict
ing the oscillatory behavior of circular footings founded on soil sug
gested that this same model, or variations thereof, could be useful for
predicting the particle motion field generated within a soil deposit by
a vibrating footing. The objective of this study was to test the hy
pothesized usefulness of the half-space model for predicting the vibra
tions transmitted into a soil foundation by an oscillating footing.
A test of the hypothesis involved three specific goals: (l) extend
Bycroft's (1956) solution for the torsional oscillation of a rigid disk
on the surface of an elastic half-space to include the motion of the
half-space and evaluate the resulting expression for absolute values of
the motion of the half-space, (2) conduct a field experiment on a
natural soil deposit that physically represents the boundary conditions

8
assumed, for the half-space model and measure the particle motion gene
rated in the soil by the oscillation of a footing, and (3) compare the
particle displacements predicted by the half-space model to the parti
cle displacements measured in the natural soil deposiip and evaluate the
usefulness of the model for motion prediction purpose^.

THE THEORETICAL PARTICLE MOTION GENERATED BY THE
TORSIONAL OSCILLATION OF A WEIGHTLESS, RIGID, CIRCULAR DISK ON THE
SURFACE OF AN ELASTIC HALF-SPACE
Homogeneous (Constant E) Elastic Half-Space
Problem Statement and. Approach
A weightless, rigid, circular disk rests on the surface of an iso
tropic, homogeneous, elastic half-space. A torsional moment, which
varies sinusoidally with time, acts on the weightless disk about a ver
tical axis through the center of the disk and imparts an oscillatory
rigid body rotation to the contact area between the disk and the half
space. Since the disk is rigid, the displacement on the contact area
is proportional to the radial distance from the center of the disk.
Except for the contact area between the disk and the half-space, the
horizontal surface of the half-space is free of stress. Particle mo
tion, stress, and strain vanish at infinity on the hemispherical bound
ary of the half-space. The particle motion generated within the half
space is to be determined.
The above situation is a mixed, or third type, boundary value prob
lem in the theory of elasticity. This problem has been treated by
several investigators in order to establish the characteristic behavior
of the disk under forced torsional vibrations (Reissner and Sagoci,
1944; Bycroft, 1956; Collins, 1962; Awojobi and Grootenhuis, 1965), but
no one has apparently attempted to deduce the partida motion developed
in the body of the half-space.
9

10
The purpose of this section is to quantitatively evaluate the par
ticle motion generated in an elastic half-space by the torsional oscil
lation of a massless rigid disk on the surface of the half-space. The
evaluation has been guided by Bycroft's (1956) analysis for the behavior
of a rigid disk; departures from and extensions to his work have been
accomplished to treat the behavior of the half-space rather than the
circular disk. Figure 1 illustrates the system.
Equations of Elasticity
The equations for strain in cylindrical coordinates r 9 z are
(Timoshenko and Goodier, 1951)
e
_ 5u
rr dr
(1)
(2)
e
_ dw
ZZ dz
(3)
7 _
Z0 rd0 dz
(4)
7 = Su + dw
rz dz Sr
(5)
7 = JE_ + Sv I
r@ rd0 dr r
(6)
the stress-strain relationships in cylindrical coordinates are
(Sokolnikoff, 1956)
(7)
a
(8)

11
RIGID DISK
'/////////
ELASTIC
HALF-SPACE
-a
rr
~zz
_L
ELEMENT
dz
Z
U

w
ELEVATION VIEW
Figure 1. Rigid circular disk on the surface of an elastic
half-space.

12
ct = \(e + e + e ) + 2G£
zz rr 90 zz' zz
/
(9)
o = G7 n
z9 z9
(10)
a = G7
rz rz
(11)
4Q
CD
II
Q
V
4
CD
(12)
and the equations of equilibrium for an infinitesimal
sions of dr rdQ and dz are
dgrr + 1 ^CTrQ + ^CTrz + grr g99 52u
dr r 39 3z r 2
element with dimen-
(13)
5gr9 + 1 5g99 + 5gz9 + 2gr9 v
3r r 39 3z r t2 p
(14)
drz 1 agz6 5CTzz
3r r 39 3z
(15)
Since the particle motion, u
v w with respect to the coordinates
r 9 z of the half-space is desired, the equilibrium equations are
written as
( \ + 2G)
JL
3r
1 3(ru) + 1 3v + 3w
G 3
1 3(rv)
1 3u
r 3r r 39 3z.
r 39
_r 3r
t 60.
(16)
(X + 2G) -|r
r 39
1 3(ru) + 1 3v + 3w
r 3r r 99 9z
_3. /l 3w 3v\
3z \r 39 3zJ
3 1 3(ru) 1 3u
3r r 3r r 39
(17)

13
3u 3w
3z ar
)
g a_ /1 aw 3v
r ae \r 30 3z
(
(18)
Solutions to the equilibrium equations are sought which satisfy the
boundary conditions described in the problem statement; i.e.
v(r 0 0 t) = 0 re^U)^
(r <; rQ) (19)
(r > rQ) (20)
az0(r 0 0 t) = 0
where is the angular rotation of the rigid disk. For completeness,
it should be noted at this point that the compatibility conditions are
not involved since the equilibrium equations are expressed in terms of
particle displacements (Sternberg, i960). This mixed or third boundary
value problem can be reduced to a first boundary value problem by eval
uating the static shear stresses produced on the surface of the half
space by the rigid circular disk and assuming that these same stresses
occur on the half-space as the disk undergoes forced torsional oscilla
tions. A similar approach and assumption is common in the literature
and has been used by Miller and Pursey (195*0, Reissner (1937), Sung
(1953), and Hsieh (1962), as well as by Bycroft (1956). For the case
of a rigid circular disk in forced torsional oscillation on an elastic
half-space, these same authors agree that v is the only component of
displacement that occurs. Consequently, the axisymmetric problem is
greatly simplified and the strain equations reduce to
(21)

7 = 5X I
r0 dr r
the stress-strain relationships to
CTz0 G7z0
ct n = G7 n
r0 r0
and the equilibrium equation becomes
da.
r0
5 a,
z0
2a.
Sr
dz
re 5 V
c 2 P
Bt
G
1 dv
r dr
14
(22)
(23)
(24)
(25)
(26)
Solution to the Equilibrium Equation
According to Bycroft (1956), solutions to the equilibrium equation
were devised by K. Sezawa in 1929- The solution for v(r,z,t) is
v(r,z,t) =
-B(x) v lx pz l(l)t
x
(27)
where x is an arbitrary parameter, B(x) is an arbitrary function of
x to be determined from the boundary conditions, ¡3 = (x2 k2
:)Vi
and tu is the angular frequency of the rigid disk.
The derivatives of v which satisfy the equilibrium equation are
dv
dr
= [B(x)J2(xr) J-j_(xr)J ePze1U)t
(28)
d2v
dr2
B(x) x Jx(xr) J2(xr)
e-pzeia)t
(29)

15
where
V =
Bz2
-B(x) / \/ 2 2\
^ j^xrJix k )
e-pzeiujt
B v B(x) T / \ 2-Bz itut
= V1 J-, (xr)o) e p e
3t2 X 1
,2 p 2
k G ^
(30)
(31)
(32)
Boundary Conditions
The shear stresses in the elastic half-space are
ze(r,z,t)
G?ze = G S = G
M(x?
/ \ -Bz iuut
(xr)e K e
(33)
The boundary conditions for stress on the surface of the half-space
when the rigid disk is rotated statically, so that z = ou = k = 0 ,
become
ozQ G x Jx(xr) (r <; rQ) (34)
- 0 (r > rQ) (35)
and the static displacement v at the surface of the half-space is
v(r,0) = Jx(xr) (36)
Since x and B(x) are arbitrary, set
00
v = f Jx(xr)dx = ^r (r <; rQ) (37)
0
00
CTZ0 = G j B(x)J1(xr)dx =0 (r > rQ) (38)
0
and let r/rQ = s and xrQ = y so that

16
oo
v = rQ J F(y)y1J1(sy)dy = 0rQs
0
00
azQ = G J F(y)J1(sy)dy = 0
(0 < s s 1) (39)
(s > l) (4o)
where y is another arbitrary parameter.
To evaluate F(y) in this pair of integral equations, Bycroft
(1956) and Awojobi and Grootenhuis (1965) use work by Titchmarsh
(1948), Busbridge (1938), and Tranter (1951)* Busbridge gives the
solution of the equations
OO
/ yaf(y)Jv(xy)dy = g(x) (0 < x < 1) (4i)
0
00
/ f(y)Jv(xy)dy =0 (x > 1) (42)
0
where g(x) is prescribed and f(y) is to be determined, as
2
r(i + a/2)
1
xl+ dj 2
+ / u^l U2) du J g(yu)(xy)2+Q!^2 J^^gtxyjdy
1
0
(43)
which is valid for a > -2 and where (-v l) < (or 1/2) < (v + l)
By substituting & = -1 v = 1 and g(x) = 0rQS into Equa
tion 43

17
F(y) =
-1/2
2 y
i' i)
i
yl/2 Jj/2 (y) /
*2)
-1/2
(0rQi) di
^ / \ -1/ 2
/ m2\l m / dm x J (^romi)(yi)3/2J3^2(y)dje
O O
^\/ sin y / 3(x
2X-1/2
di
y m3(l m2) dm J (yi)3/2(sin yi yi eos yi)di
O O
4 / sin y y eos y
= ^ ¡r
TT r O
y
)
(44)
The expression for shear stress (Equation 4o) is then
a26(r,0,) = G / (£ )fc0) (
0
sin y y cos y
y
^ Jx(sy)dy
-40r G f sin xr xr cos xr
2_ f 2 2 j (xr)dx
tt J xr_ lv '
(45)
(46)
xr
40r G
TT
f yjjs (xrQ)1//2 J3/2(xro) Jx(xr)dx (47)
0
Restating the boundary conditions in terms of stress requires that
40r G r _/0 f n/_
Z0 TT
= 0
0
(r s rQ)
(48)
(r > rQ)
(49)
The infinite integral appearing in Equation 48 is one of the special
cases of the discontinuous Weber-Schafheitlin integral tabulated by-
Abramowitz and Stegun (1964)

18
J +|jl-v+lT (at)J (bt)dt = 0
V y v'
= 2^~v+V(b2 a2)^^1
bvr(v n)
(0 < b < a) (50)
(p > a > 0) (51)
where v > p > -1 Substituting p = l and v = 3/2 into Equations
50 and 51, the values of the integrals in Equations 48 and 49 are
w
J x1//2J1(xr)jy2(xro)dx = 0
gl/g (rg rg)
r^/2r(l/2)
-1/2
(r > rQ) (52)
(r £ rQ) (53)
so that the shear stress on the surface of the half-space in contact
with the rigid disk becomes
-40r G
0) =
!'o/V
r~" \l r2 r2
_ -40G r
TT
4
2 2
ro r
(r £ rQ) (54)
Applied moment and disk rotation
The moment applied to the disk is
o 2tt
M
/ / CTz0rrd0dr
0 0
160Gi
3
(55)

19
so the angular rotation of the disk becomes
(56)
Substituting this value of 0 into Equation 54 gives
o
2
r
r
2
> rG) (57)
Reissner and Sagoci (1944), using a system of oblate spheroidal co
ordinates instead of dual integral equations, found this same shear-
stress distribution on the contact surface between a torsionally
loaded rigid disk and an elastic half-space.
Particle Displacements
It was previously found that any arbitrary parameter x and
function B(x) will satisfy the equilibrium equation and that a spe
cific form of B(x) will also satisfy the prescribed boundary condi
tions. Thus, the specific formulation for B(x) that satisfies the
boundary conditions will also satisfy the equilibrium equation
throughout the half-space.
The solutions to the equilibrium equation
(58)
and the shear-stress expression
(59)

20
are related by
v(r,z,t) =
11 (
- g b ze(r,z,t)
(6o)
and the specific shear-stress formulation (Equation 46) that satisfies
the boundary conditions is
-40r G sin xr xr cos xr
0z6(r,Z,,t) =
I
xr
J1(xr)epzelu)tdic (6l)
0
Substituting Equation 6l into Equation 60 gives the particle dis
placement in the half-space as
i, a icut 03
40r e sin xr xr cos xr
v(r,z,t) = 2 / -^r 2 J1(xr)e_pzdx (62)
0
\
The normalized particle motion in the half-space with (respect to the
tangential displacement of the disk is then
v(r,z,t) 4 icut
0r tt
r o
I
sin xr xr cos xr
o o
pxr
- J1(xr)e"^zdx (63)
Evaluation of the Infinite Integral
In order to calculate the particle motion at a specific point in
the half-space, it is necessary to evaluate the infinite integral for
particular values of four quantities, r r z and k The four
quantities are reduced to three if a = rk aQ = rQk b = zk and
g is a new arbitrary parameter g = x/k Substituting and represent
ing the infinite integral by
I (aQ, a, b) = f
sm aQg aQg cos aQg
e
0 aQg(g 1
1/2
J1(ag)dg (-64)

21
the particle motion expression becomes
v(r,z,t) =
icut
3146 -g l(aQ,a,b)
4nGr
icut
40r e'
2 l(a ,a,b)
TT O
(65)
(66)
Inspection of l(a ,a,b) reveals that the integrand will have
imaginary components when g is less than unity because the term
n
\Jg 1 appears in the denominator. The integrand will be real when
g is greater than unity. These characteristics suggest that the inte
gration should be carried out in two ranges; i.e.
Noting that
11a
I = R j + im j + R j
0 0 1
= I + il + I
1 2 3
\ls2 i = i\fi-
(67)
(68)
(69)
and
^-ibVl-g- cos b^l g^ i sin b^l g^
(70)
I and I are developed from the expression
r sin
T 4- - T I
a g a g cos a g 1
o& o& o -lb
Jn 2
S J1(ag)dg
\
as
o
1
CVJ
-i r-
H
i1
H
.\L 2
aQg iv1 g
, sin a g -
i r
aog cos aog /
Jl g2jj-L(ag)dg
0 a g i
o&
v/i g2 V
(71)
(72)

22
1 .
sin a g a g cos a
il2 = / 2 COS b yjl gc Jn(ag)dg
/
0 aQg i v1 g
i \ :
X
(73)
The integrands of I and I become unbounded as g approaches
unity, so a change in variables is appropriate. Let
g = sin a
dg = cos oda
(74)
(75)
and
v77
g = COS cy
(76)
where 0 g 1 and 0 s a tt/2
By replacing the variable g with the variable &
J11/2 sin (a sin cy) (a sin a) cos (a sin o o o
0
a sm n
o
x J-^a sin O') sin (b cos cy)dcy (77)
and
X2= "
TT/
/
0
/2
sin (a sin ¡y) (aQ sin ¡y) cos (aQ sin a sin a
o
X J^(a sin cy) cos (b cos cy)djy (78)
The integrand of the integral
/ sin (a g) a g cos a g / 2 ,
v o& o& e-b^g -1 j
vTlT n
aQgvg 1
(ag)dg (79)
also becomes unbounded when g approaches unity and a[ change of vari
ables for is indicated. Let

g = sec a
23
(80)
cLg = sec & tan a cLq1
(81)
and
1 = tan o/
(82)
where 1 ^ g ^ < and 0 <. o¡ <. w/2 Using the new variable a instead
of g the integral becomes
sin (aQ sec a) (aQ sec a) cos (aQ sec a)
0
circular functions, J an integer order Bessel function of the first
kind, and e the base of natural logarithms. The integrands are con
tinuous functions in the interval of integration, and, of particular
note, all the terms of the integrand can be expressed as a series or as
polynomial approximations. Such formulations make the integrand well
suited for evaluation with a digital computer.
Integration of 1^,1^, and I could be accomplished in a
variety of ways, but perhaps the most obvious is by numerical methods.
One numerical integration scheme is based on Simpsons rule for deter
mining the area of an irregular figure. The integral expression of
Simpson's rule, given by Abramowitz and Stegun (1964), is
o
+ *h*h +---W + An <84>

24
where h = 2_ and n is an integer.
2n
Example calculation
The application of Simpson's rule for calculating ohe approximate
value of 1^ , and I can be illustrated by an example.
The integration parameters for the particle motion in a half
space at a distance of 30 ft from the center of the disk and at a depth
of 15 ft, when the shear wave velocity in the half-space is 650 fps and
the 5 ft diameter disk oscillates at 20 Hz are
k = ou/v = 2tt20/650 = 0.192
(85)
. = kr = 0.192(2.5) = 0.48
(86)
a = kr = 0.192(30) = 5.76
(87)
b = kz = 0.192(15) = 2.90
(88)
For illustration purposes, divide the integration interval into 4 equal
parts, i.e. h = l/4(ir/2) = rr/8 Values for the common terms in the
integrand of 1^ and 1^ are listed in Table 1 and values, f(a) of
the integrand of I are given in Table 2. The calculation for 1^
by Simpson's rule is
I1 = -(tt/8)/3 [O + 4(0.0027899 0.0199034) + 2(-0.0031265) + 0]
= 0.009779 (89)
Values, /(a) of the integrand of I are given in Table 3. The cal
culation for I by Simpson's rule is

Table 1
Values of the Common Terms in the Integrand of I]_ and Iq
O'
sin a
cos a
aQ sin o'
a sin a
b cos a
J^a sin a)
sin (a sin a)
cos (aQ sin a)
0
0.00000
1.00000
0.0000000
0.0000000
2.9000000
0.0000
0.00000
1.00000
tt/8
O.38267
O.92387
O.I836816
2.2041792
2.6792230
0.5553
0.18265
O.98316
tt/4
0.70709
0.70709
0.3394032
4.0728384
2.0505610
-O.093I
0.33289
0.94295
3n/8
0.92387
O.38267
0.4434576
5.3214912
I.IO98OIO
-O.3458
0.42906
0.90327
tt/ 2
1.00000
0.00000
0.4800000
5.76OOOO
0.0000000
-O.3163
0.46178
0.88699
Table 2
Values, f(a) of the Integrand of Ii
sin (aQ sin ot) aQ sin a cos
O'
(aQ sin a) /a sin a
J^a sin a)
sin (b cos a)
f()
0
0.00000
0.0000
0.24192
0.0000000
tt/8
0.01126
0.5553
0.44620
0.0027899
tt/4
0.03786
-O.O93I
0.88701
-0.0031265
3tt/8
0.06426
-O.3458
0.89570
-0.0199034
n/2
0.07506
-O.3163
0.00000
0.0000000

Table 3
Values, /(a) of the Integrand of I2
sin (aQ sin a) a sin a cos
a
(a sin cv) /a sin c
J^(a sin o')
cos (b cos oi)
f(a)
0
0.00000
0.0000
-O.97O3O
0.0000000
tt/ 8
0.01126
0.5553
-O.89493
-0.0055957
n/4
0.03786
-O.O93I
-0.46175
0.0016275
3tt/8
0.06426
-O.3458
0.44466
0.0098808
tt/ 2
0.07506
-O.3163
1.00000
-0.0237415

27
I = -(n/8)/3[o + 4(- 0.0055957 0.0098808) + 2(0.0016275) 0.0237415]
- O.OIO785 (90)
Evaluation of the integral I can be accomplished in the same
manner as that illustrated for 1^ and Table 4 lists the values
of the terms in the integrand of I and Table 5 gives the values,
/(a) of the integrand of I The computation for is
I = (n/8)/3 [-O.023713 + 4(-O.OO63927 + O.OOIO616) + 2(0.0028797) + 0]
= -0.0051415 (91)
The real part of the integral l(a ,a,b) is
Real I = 1^ + I3
= 0.0046375 (92)
and the imaginary part of l(a ,a,b) is
iml = I i
= 0.OIO785i
(93)
The particle displacement for this example in terms of the ro
tation of the rigid disk is
40r .
v(30,15,t) = ^ elu)t[0.0046375 + i(0.010785)]
(94)
and the peak particle displacement is
40r
v(30,15)
TT
(0.0046375) + (0.010785)'
1/2
(95)
This example problem illustrates that the integrands of 1^ I ,'

Table 4
Values of the Terms in the Integrand of I3
a
sec a
a sec 01
0
sin (a sec a)
0
cos (aQ sec q?)
a sec oi
J^(a sec o)
tan o'
b tan oi
0
1.00000
0.4800
0.46175
0.88701
5.76000
-O.3163
0.0000
0.0000
rr/8
1.08240
0.51955
0.49647
0.86805
6.23462
-0.2243
0.41421
1.20121
tt/4
1.41425
0.67884
0.62788
0.77831
8.14608
0.2524
1.0000
2.9000
3tt/8
2.61308
1.25428
0.95033
0.31123
15-05134
0.2032
2.4142
7.00118
tt/ 2
cc
oc
-1 to 1
-1 to 1
CC
0.0000
CC
OC
Table 5
Values, f( sin (a sec n) a sec q; cos
o o
(aQ sec a) /a.Q J^(a sec a) g-b tan a
0
0.074969
-O.3163
1.00000
-0.023713
tt/ 8
0.094740
-0.2243
0.30083
-0.0063927
tt/4
0.207358
0.2524
0.055023
0.0028797
3rr/8
I.I6658
0.2032
0.000910
0.0010616
tt/2
Undefined
0.0000
0.00000
0.00000

29
and are well behaved functions of the variable o so that a nu
merical integration scheme, such as Simpson's rule, should give a good
approximation for the value of these integrals.
Computer Program to Evaluate the Integrals
A computer program, based on Simpson's rule and written in ALGOL
language, was used to evaluate the integrals 1^ and I .
Table 37 of Appendix A lists this program and includes the polynomial
routine for calculating J Using the variable x instead of a ,
the integrand of 1^ is called RELX(X), the integrand of is
IMX(X) and the integrand of is REH(X). The real part of
l(a ,a,b) is called REINT and the imaginary part IMINT.
The computer calculations are carried out in much the same manner
as illustrated in the example calculations above. The integration in
terval, 0 to rr/2 radians, has been subdivided into three parts:
0 to 0.5 radians, 0.5 to 1.0 radians, and 1.0 to tt/2 radians. Each
part is independently integrated, using Simpson's rule and a geometri
cally increasing number of intervals, until two sequential integrations
agree to five significant digits. When this criterion is satisfied for
each part of the integration interval, the sum of the parts is con
sidered to be a sufficiently accurate representation of the integral
for the purpose of this investigation.
Appendix A contains tables of computed l(aQ,a,b) values for
several combinations of the variables a a and b The real,
o
imaginary, and absolute values of I are listed for parameters charac
teristic of the test site at Eglin Field, Florida, and for the measure
ments planned at this site. The shear wave velocity at the Florida

30
site is about 650 fps at a depth of 15 ft, the unit weight of the soil
is about 104 pcf, and the diameter of the footing (disk) is 5 ft.
Table 6 gives the calculation parameters used to compute the values of
l(a ,a,b) contained in Appendix A.
Table 6
Calculation Parameters for l(aQ,a,b)
Frequency
Hz
a
0
r
ft
a
z
ft
b
15
0.36
0-12.5
0-1.80
0
0.00
0-90
0-12.96
1
o.i44
0-90
0-12.96
5
0.72
0-90
0-12.96
15
2.20
0-90
0-12.96
25
3.62
0-90
0-12.96
35
5.06
20
0.48
0-12.5
0-2.40
0
0.00
0-90
0-17.28
1
0.192
0-90
0-17.28
5
0.96
0-90
0-17.28
15
2.90
0-90
0-17.28
25
4.84
0-90
0-17.28
35
6.76
30
0.72
0-12.5
0-3.60
0
0.00
0-90
0-25.92
1
0.288
0-90
O-25.92
5
1.44
0-90
0-25.92
15
4.34
0-90
0-25.92
25
7.24
0-90
0-25.92
35
io.i4
40
O.96
0-12.5
0-4.80
0
0.00
0-90
0-34.56
1
0.384
0-90
0-34.56
5
1.92
0-90
0-34.56
15
5.80
0-90
0-34.56
25
9.64
0-90
0-34.56
35
13.50
50
1.20
0-8.75
0-4.20
0
0.00
0-90
0-43.20
l
0.480
0-90
0-43.20
5
2.40
0-90
0-43.20
15
7.24
0-90
0-43.20
25
12.06
0-90
0-43.20
35
16.90

Nonhomogeneous (Linear E) Elastic Half-Space
Because the elastic moduli of soils is known to depend on the mean
effective stress applied to the soil (Hardin and Richart, 1963) and the
effective stress in a soil deposit increases with depth below the ground
surface, a nonhomogeneous elastic half-space would be a more realistic
analytical representation of a soil deposit than a homogeneous elastic
half-space. Thus, it was worthwhile to consider the progress that has
been made toward the use of a nonhomogeneous half-spaco for foundation
problems and some apparent relationships between a homogeneous and a
nonhomogeneous half-space with an elastic modulus that increases lin
early with depth.
In a review of existing knowledge of the dynamic behavior of soils
and foundations, Jones, Lister, and Thrower (1966) made particular men
tion of the need for and the apparent lack of attention to the develop
ment and application of nonhomogeneous theory to these problems. In
summarizing the analysis of machine foundations on soils with a modulus
that changes with depth, they state:
The problem which arises when the elastic properties
vary continuously with depth, rather than in the
discontinuous fashion typified by layered media, has
received less attention, although it is important,
especially in view of the variation of elastic prop
erties of non-cohesive soils with the mean stress.
Structures [ soil stratification] of this type are
probably rather more frequent in practice than the
layered case. No analytical investigations of the
kind described above are known to the authors. Pauw
(1953), however, has analyzed the problem by assum
ing essentially that the phenomena can be described
by considering the propagation of a cone-shaped bun
dle of longitudinal-type waves downwards into the
soil. ...Pauw's approach appears rather unsatisfac
tory from an analytical point of view.

32
Some progress, as outlined in the following paragraphs, has been
made since 1966, but rigorous solutions to the dynamic boundary value
problem on a nonhomogeneous elastic half-space have not been found or
attempted herein. Discrete methods, however, such as (the finite ele
ment and lumped mass representations, are developing rapidly and may
soon be capable of solving such problems.
Literature
Seismologists have been concerned with the influence of a variable
modulus earth structure on the speed and period of propagating earth
quake tremors (Byerly, 19^2). Their concern stems from a need to locate
the epicenters of earthquakes and to define the gross structure of the
earths mantle. Ewing, Jardetsky, and Press (1957) devoted an entire
chapter of their book to wave propagation in media with variable veloc
ity. Again, the primary purpose of the work was to study the disper
sion characteristics of propagating seismic waves. A recent paper by
Bhattacharya (1970) gives the solution of the wave equations for an in
homogeneous media. His work is limited to horizontal shear waves prop
agating in a plane, and his solutions define the variation of density,
shear modulus, and shear wave velocity with depth. The form of these
variations depends on the solution functions.
Solutions for the static displacement and stresses in a nonhomo
geneous elastic half-space due to a uniformly distributed strip or cir
cular surface loading have been developed by Gibson (1967) he con
sidered that the elastic modulus of the half-space varied linearly with
depth (linear E) and that the half-space was incompressible. Half
space stress solutions for a point or a line load on the surface of a

33
compressible or incompressible nonhomogeneous elastic ljialf-space with
a modulus that varies with depth have been presented by Curtis and
Richart (1955). Earlier investigations of similar cases have been ac
complished by Hruban (1948), Borowicka (1943), and Frolich (1934).
Results of Gibson's Solutions
As mentioned, Gibson (1967) obtained solutions for the static dis
placements and stresses in a nonhomogeneous elastic half-space due to a
uniform load distributed along an infinitely long strip or over a cir
cular area. He assumed that the shear modulus, G(z) varied with
depth, z as
G(z) = G(o) + mz (96)
The equilibrium equations resulting from this assumed modulus variation
were intractable, but they were greatly simplified by assuming that
Poisson's ratio was l/2.
By the use of Fourier transforms, suitable changes in variables,
and a discontinuous integral satisfying the boundary conditions, Gibson
was able to develop closed form expressions for the displacements and
stresses in the nonhomogeneous, incompressible, elastic half-space.
Solution for stresses
A uniform vertical pressure, q acting on the surface of the
nonhomogeneous half-space over a circular area of radius, b produces
shear stresses, ct of
rz
00
CTrz = m if e'5z It &[F(5y) \ 1? (97)
'o 1 ;

where r and z are cylindrical coordinates
3 = G(0)/m
(98)
y = z + 3
(99)
K = bJ^r^Jjbl)
(100)
A = [5PF(§3) + 53 ige (§3) + 1 + § 5p]
(101)
F(l) = e2XEi(-2\) logg \
(102)
and Ei is an exponential integral (Ambramowitz and Stegun, 1964).
When 3 - oo as a limit, the change in shear modulus with depth
approaches zero, and the nonhomogeneous incompressible half-space be
comes a homogeneous incompressible half-space. When 3 ->0 the shear
modulus at the surface of the nonhomogeneous half-space approaches zero,
but the value of m is not restricted. Gibson found that the expres
sions for the stresses were the same in both of these J^ases and he con
cluded that the stresses were unaffected by this particular type of non
homogeneity. This conclusion also results from the Curtis and Richart
(1955) work. Gibson also postulated that the stress components in a
nonhomogeneous half-space, with finite values of G(0)/m may not
differ appreciably from the stress components in a homogeneous
half-space.
Strain relationships
The stress and strain in an elastic material are related by the
elastic moduli of the materials, and the stress at any point in an
elastic body is the product of the strain at that point times the elas
tic moduli at the same point. In a homogeneous half-space, the shear
stress was represented by

35
t = G7
(103)
and in a nonhomogeneous half-space, the shear stress was represented by
tn = g(z)7b
(io4)
For an incompressible half-space, Gibson showed that if = T ,
then the ratio of the strains becomes
7-
_N G
7 G(z)
(105)
Half-Space Under Torsion
The stress and strain conditions developed in a half-space due to
a torsional moment applied to a rigid circular disk on the surface of
the half-space are analogous to the half-space conditions that result
from Gibsons solutions.
The stresses developed in the half-space by the disk are indepen
dent of the value of Poisson's ratio. Because Gibson assumed a
Poisson's ratio of l/2 before obtaining solutions for the stresses,
his stress solutions are also valid for the same value of Poisson's
ratio.
Dilatational strains are not developed in the half-space by a
torsional moment applied to the rigid disk. Gibson assumed that the
half-space was incompressible, so, again, no dilatational strains were
developed by the surface loads.
A disk in torsion produces shear stresses on a circular area at
the boundary of the half-space. Gibson's solutions are also for a cir
cular area loaded by a uniform vertical pressure at the surface of the
half-space.

36
The above similarities between Gibson's case and the torsional
loading situation lead to the hypothesis that the results of Gibson's
investigations were also applicable to a nonhomogeneous half-space
under torsion. Gibson's results, thus, indicate that the stresses de
veloped in a homogeneous elastic half-space by a torsibnally loaded
rigid disk on the surface of the half-space would be the same as the
stresses developed in a nonhomogeneous elastic half-space by the same
torsional load. The variation of the shear modulus with depth in the
nonhomogeneous half-space under torsional loads should be the same as
that assumed by Gibson: G(z) = G(o) + mz .
Torsional Oscillation
As mentioned before, rigorous solutions to the dynamic boundary
value problem of a rigid circular disk in torsional oscillation on the
surface of a nonhomogeneous elastic half-space have not been found and
are not attempted herein.
Engineers, however, are notoriously proficient in rationalizing a
sufficient number of plausible assumptions to circumvent rigorously in
tractable problems (Zienkiewicz, 1967) Soil engineers, typically
faced with incomplete, inaccurate information and armed with inadequate,
inappropriate, and often untested theory, have been able to resolve a
variety of problems by the simultaneous application of available knowl
edge and logical assumptions. The results are usually successful, but
sometimes they are not (Peck, 1967). The following paragraphs are
offered to bridge the gap between what is known and what is needed..

37
Assumption
Gibson's (1967) correlation between the stresses in a homogeneous
half-space and a nonhomogeneous half-space under static loads was con
sidered adequate evidence to assume that the dynamic stresses in a homo
geneous half-space and a nonhomogeneous half-space under dynamic loads
have the same correlation. Thus, it was assumed that the stresses
developed in a homogeneous elastic half-space by a torsionally oscil
lating rigid circular disk on the surface of the half-space are the
same as the stresses developed in a nonhomogeneous elastic half
space subjected to the same oscillatory loading.
Low frequencies.--When the frequency of the torsional loading is
low, the second time derivative of the particle displacement is small,
and the equations of equilibrium become nearly homogeneous. The stress
conditions in either the homogeneous half-space or the nonhomogeneous
half-space would approach the static loading situation, so, hypothet
ically, the two half-spaces would have almost identical stress
conditions.
High frequencies.The wave fronts propagating in a homogeneous
half-space are located on a spherical surface (Woods, 1968). Wave
fronts propagating in a nonhomogeneous half-space are functions of
source distance, surface reflections, and type of nonhomogeneity
(Byerly, 19^2; Brown, 1965). Phase relationships are also distorted
complexly in the nonhomogeneous caseleading to frequency dependent
particle motions. So, for high frequency oscillations, there is prob
ably less correspondence between the stresses in the two half-spaces.

38
Particle motion
The strain energy per unit volume generated in a torsionally loaded
homogeneous half-space is (Timoshenko and Goodier, 1951)
w = m (4 + aze)
(106)
and the kinetic energy of an oscillating particle in the half-space is
(107)
1 2
K = i pdVv
The strain energy per unit volume developed in a nonhomogeneous
half-space is
WN 2G(z) (CTr0 + ze)
(108)
N
and the kinetic energy of an oscillating particle in the nonhomogeneous
half-space is
% = I dWN
(109)
Equating the strain energy and the kinetic energy in each of the
above cases (Timoshenko and Goodier, 1951) gave
.2 1
V Gp
(4 + 4)
(no)
and
2 1 ( 2 2 \
N G(z)p \CTr0 az0/
(HD
N
The loading and mass density, p of each half-space was assumed equal,

39
so the stresses generated in each half-space were equal and the ratio
of the particle displacement in the nonhomogeneous half-space to the
particle displacement in the homogeneous half-space was
(112)
In summary, Gibson's (1967) approach to the nonhomogeneous half
space problem implied that the particle velocities in a nonhomogeneous
(linear E) half-space can be determined from the particle velocities
(or displacements) in a homogeneous (constant E) half-space, as given
by Equation 112. The form of this particle velocity relationship is
similar to Equation 105 for the static strains in the two half-spaces.

THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCILLATING
RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT
Description of Test Site
The selected test site was located at an inactive auxiliary field
on the Eglin Air Force Base, Florida, military reservation. The soil
at the site was a homogeneous marine terrace deposit of poorly graded,
fine- to medium-grained sand. The water table was about 100 ft deep in
this thick, free-draining sand deposit and the shear wave velocity in
creased significantly with depth.
Geographical Location and Geological Setting
The test site chosen for the experimental work was located in
section 14, range 24 west, township 1 north, Okaloosa county, Florida,
at about 86 degrees and 38 minutes west longitude and 30 degrees and
35-1/2 minutes north latitude. The circular test foundation and the
approximately 100-ft-square test area was about 1,050 ft west and 230 ft
north of the south end of the north-south runway at Piccolo field (aux
iliary field 5)? within the boundaries of Eglin Air Force Base and about
15 miles north of the Gulf of Mexico coastline. Piccolo field was
chosen as a test site because it was militarily inactive and the water
table was unusually deep. The elevation of the area was about 175 ft
above mean sea level and the topography was quite flat; elevations
within the test area varied less than 3 in. Native grasses covered the
40

4i
ground surface and the area was lightly wooded with indigenous scrub
oak and pine.
Sand deposits in this vicinity are of geologically recent origin
(Cooke, 1945j Vernon and Puri, 1965)- The Citronelle formation is
dated somewhere between the Pliocene and the Pleistocene epochs and is
no more than ten million years old; the terrace and fluvial terrace
formations laid down during the Pleistocene epoch are iless than one
million years old. Stratigraphically, the Citronelle formation lies
unconformably on older formations, and is overlain by Pleistocene
terrace deposits.
The Pleistocene epoch was characterized by many changes in sea
level due to a sequence of glacial accummulation and subsequent melting.
Sea levels during that time were as much as 270 ft above current levels.
The water from melting glaciers carried a variety of soil material to
the sea where currents and wave action developed the sandy terrace de
posits Erosion during low sea levels and redeposition during high sea
levels created a generally flat topography with hidden stratigraphic
features. The three specific marine terraces that were associated with
deposits at the test site are the Brandywine formation, the Cohaire for
mation, and the Sunderland formation.
Table 7 is a well log taken at auxiliary field 5 by the Layne Cen
tral. Co. and provided by the Directorate of Civil Engineering, Eglin Air
Force Base, Florida; it illustrates the general stratigraphic sibilation
near the test site.
Soil Exploration
The in situ soil exploration program at the test site was

42
Table 7
Well Log at Auxiliary Field j?
Depth Below
Ground Surface
Well Driller's Identification of
ft
Material Penetrated
0 to
20
Sand
20 to
84
Sand with white clay
balls
84 to
110
Sand and white clay
balls
100
Water table
110 to
156
Sand and gravel with
. white clay balls
156 to
212
Hard blue sandy clay
212 to
235
Sand, shells, and clay
235 to
307
Sand streaked with blue clay
307 to
343
Sand, shells, and clay
343 to
353
Clay
353 to
360
Hard rock
360 to
400
Clay, shells, and sand
4oo to
480
Tough clay, shells,
and sand
480 to
518
Soapstone and blue c
lay
0
-p
3
Lf\
533
Shell rock
533 to
555
Shell rock with soft
places
555 to
585
Hard coarse rock and
. shells
585 to
598
Extra hard shell rock
598 to
620
Hard shell rock with
soft places
620 to
643
Lime rock and very f
'ine shells
643 to
650
Brown sand rock
650 to
666
Lime rock and brown
sand rock
666 to
763
Lime rock

43
accomplished with a standard split spoon sampler and with a Begemann me
chanical static friction cone penetrometer. Three holes were bored to a
depth of 60 ft with the standard sampler, and 20 penetrations were made
to an average depth of JO ft with the cone penetrometer; locations are
shown in Figure 2. The purpose of these exploration efforts was to as
sess the suitability of the site for conducting ground motion propaga
tion experiments.
Borings
Holes 1 and 2 were continuously sampled with a standard split spoon
sampler (ASTM, 1969) and Hole 3 was sampled at 10-ft intervals using a
3-in.-diam, l8-in.-long Shelby tube. Tables 8 and 9 list the standard
penetration resistance of the sampled soil, the drillers visual clas
sification, and his description of the soil retrieved from Holes 1 and
2, respectively. Hole 2 was located 135 ft west of Hole 1 and Hole 3
was located 15 ft north of Hole 2. The Shelby tube samples from Hole
3 were analyzed in the laboratory.
Penetration tests
Although no fine-grained materials were discovered by exploratory
borings at the test site, other boring and well log data taken at Eglin
have often indicated the presence of clay or marl. Such impermeable
layers could support a perched water table or could impede the infiltra
tion of meteoric water. The occurrence of perched or transient water
within the mass of soil would, of course, cause density contrasts that
would be detrimental to precise and reproducible ground motion measure
ments. An attempt to locate possible lenses of fine-grained material
within the selected test site was considered necessary.

400' i 400' I 400'
E3
-3
-£3
£3-
-3-
MAGNETIC NORTH
O BORING LOCATION
Q PENETRATION LOCATION
5-FT-DIAM
TEST FOOTING
O
SCALE
100'
4-
UNLESS DIMENSIONED
387'
400'
CD
HOLE 3
o
- e -D-
HOLE 2
I
£3
£3

i
--
HOLE 1
£3-
-GB E3
Figure 2. Location of 3 exploration borings and 20 friction-cone penetrations.

45
Table 8
Boring Log for Hole 1
Moisture
Standard
Driller
s
Sample Depth
Content
Penetration Resistance
Classification and Description
ft
i
blows/ ft
Color
Symbol
Description
0.0
to
1.5
6
3
Medium
brown
SM
Silty sand, fine grained with surface or-
ganic material--
hair roots, etc.
1.5
to
3.0
5
3
Brown
Silty sand, fine grained with trace of
surface organic matter
3-0
to
4.5
5
3
Tan
Silty sand, fine grained
4.5
to
6.0
5
4
6.0
to
7.5
4
7
7-5
to
9.0
4
7
9.0
to
10.5
3
6
Light tan
SP-SM
Sand, poorly graded, with silt fines con-
tent. Fine grained and sharp particles
10.5
to
12.0
4
8
12.0
to
13.5
3
11
13.5
to
15.0
4
13
15.0
to
16.5
3
18
Sand, poorly graded, fine grained. Sharp
particles with slightly silty fines
16.5
to
18.0
5
23
Medium
brown
SM
Silty sand, fine grained
18.0
to
19.5
7
20
Light
red
SC
Clayey sand, fine grained
19.5
to
21.0
7
19
1
21.0
to
22.5
8
17
1
22.5
to
24.0
5
15
Tan
SM
Silty sand, fine grained with trace of
clay
24.0
to
25.5
5
16
Silty sand, fine grained with trace of
clay
25-5
to
27.0
5
16
Silty sand, line grained
27.0
to
28.5
6
15
Reddish tan
Silty sand, fine grained with trace of
clay
28.5
to
30.0
4
15
Tan
SP-
M
Sand, poorly graded, fine grained, sharp,
with trace of silt fines
30.0
to
31.5
10
31.5
to
33.0
6
18
-
33.0
to
34.5
8
18
Reddish tan
SC
Clayey sand, fine grained
34.5
to
36.0
6
19
Tan
SP-
SM
Sand, poorly graded, fine grained with
silt fines
36.0
to
37.5
8
15
Sand, poorly graded, fine grained with
silt fines
37.5
to
39.0
5
17
Sand, poorly graded, fine sharp grains,
slight silt content
39.0
to
40.5
10
14
Light red
S
Clayey sand, fine grained
40.5
to
42.0
8
12
42.0
to
43.5
9
19
43.5
to
45.0
9
23
45.0
to
46.5
7
20
46.5
to
48.0
7
22
48.0
to
49.5
8
26
Reddish tan
Clayey sand, fine grained with less
clay content
49.5
to
51.0
7
21
51.0
to
52.5
7
24
52-5
to
54.0
7
22
Clayey sand, fine grained with slight
clay content
54.0
to
55.5
8
27
55.5
to
57.0
6
28
57.0
to
58.5
6
24
58.5
to
60.0
6
25
Light
tan
SP-
SM
Sand, poorly graded, fine, sharp grains,
trace of silt

Table 9
Boring Log for Hole 2
46
Moisture Standard Driller's
Sample Depth Content Penetration Resistance Classification and Description
ft
i
blows/ft
Color
Symbol
Description
0.0 to 1.5
6
5
Brown
SM
Silty san
d, fine grained with surface
l. to 3.0
5
2
Light brown
organic
matter--
roots, etc.
3.0 to it.5
5
1
4.5 to 6.0
4
6
6.0 to 7.5
4
7
Tan
SP-
SM
Sand, poorly graded, fine sharp
grains
with slight silt
7-5 to 9.0
4
7
9.0 to 10.5
5
9
10.5 to 12.0
6
8
12.0 to 13-5
6
12
Reddish tan
SM
Silty sand with tr
ace of clay
13.5 to 15.0
4
10
SM
Silty sand with trace of clay
15.0 to 16.5
6
17
SP-
SM
Sand, poorly graded, fine sharp
grains
with slight silt
16.5 to 18.0
4
14
18.0 to 19.5
5
19
Whitish tan
19.5 to 21.0
6
24
21.0 to 22.5
5
22
Sand, as
above, with trace of clay
streaks
22.5 to 24.0
4
24
Sand, poorly graded, fine sharp
grains
with slight silt
24.0 to 25.5
5
22
25.5 to 27.0
4
20
)
27-0 to 28.5
12
10
Ta
n
sc
Clayey sand
28.5 to 30.0
8
11
Clayey sand
30.0 to 31.5
5
16
Clayey sand with less clay content
31.5 to 33.O
9
20
Clayey sand, fine
to medium sharp
grains
33.0 to 34.5
8
28
34.5 to 36.0
10
21
36.0 to 37-5
7
18
Tannish
SP-
SM
Sand, poorly graded, fine to medium
white
sharp g
rains with slight silt
37-5 to 39.O
7
23
Sand, poorly graded, fine to medium
sharp grains, trace of silt and
clay
39-0 to 40.5
5
33
Sand, poorly graded, fine to medium
sharp grains, trace of silt and
clay
40.5 to 42.0
4
26
Greyish
SP
Sand, poorly graded, trace of silt
white
42.0 to 43.5
5
29
43.5 to 45.0
5
32
45.0 to 46.5
33
Sand, poorly graded, sharp fine
grains,
trace of
silt
46.5 to 48.0
4
24
48.0 to 49.5
7
27
49.5 to 51.0
4
29
51.0 to 52.5

52
Gre
y
Drill mud
used
52.5 to 54.0

47
Grey
54.0 to 55.5

35
Tan
SP-
SM
55.5 to 57.O

34
SP-
SM
57.0 to 58.5

35
sc

47
The Dutch friction-cone penetrometer, a relatively new soil
exploration tool, offered the most practical means of investigating the
possible existence and extent of fine-grained sedimentary material at
the site; this tool can also be used to reveal density variations
within the mass of sand (Schmertmann, 1967; 1969)* Eighteen soundings
were made to depths ranging from 60 to 70 ft, one to a depth of 82 ft,
and another to a depth of 102 ft. The 102-ft-deep sounding revealed
that it had nearly reached the elevation of the permanent water table.
The friction-cone penetrometer exploration did not reveal the
presence of cohesive soils within the investigated area that was sev
eral hundred yards square. In addition, no perched water table condi
tions were encountered. The cone bearing capacity data indicated that
there was a significant variation in the density of sand with depth;
however, the density variations with depth were quite consistent at
each sounding location. Thus, the depositional environment at the test
site apparently had laterally homogeneous characteristics that produced
a generally uniform horizontal stratification of the sand. The average
cone bearing capacity at various depths is listed in Table 10.
Table 10
Average Bearing Capacity of Static Cone Penetrometer

48
Laboratory Tests
Laboratory tests were conducted on the Shelby tube samples ex
tracted from Hole 3 before the friction-cone penetration tests were
performed. Six samples were obtained at depths of 2.5 13 19*5 31
4l, and 51 ft.
Unit weight
The 18-in.-long Shelby tube samples were divided into three
equal increments. The natural unit weight of the sand retained in
each increment was measured and the color of the material was noted.
The sand from all of the increments in a single sample was then com
bined and the maximum unit weight (minimum void ratio) and the mini
mum unit weight (maximum void ratio) of the sample were determined.
Because the volume of the sample was less than 0.1 cu ft, standard
methods and apparatus could not be used. The minimum unit weight was
determined by filling a 2-in.-diam, 4-in.-deep mold with sand poured
from a standard l/2-in.-diam funnel. The maximum unit weight was de
termined by filling the mold with sand in three equal layers; each
layer was compacted by 25 blows of a 5 5-lb hammer falling 12 in. on a
2-in.-diam steel platen resting on the sand layer. Table 11 gives the
results of these laboratory tests.
Table 11
Results of Laboratory Tests on Samples from Hole 3
Dry Unit Weight, Ib/cu ft
Depth,
ft
Increment
Color
Minimum ]
Maximum
Natural
1.5 to
3.0
?!
Tan and light
brown
(Continued)
90.9
113.2
98.8
108.8
100.2

49
Table 11 (Concluded)
Depth,
ft
Increment
Color
Dry Unit Weight, lb/cu ft
Minimum Maximum Natural
12.0
to 13.5
Top}
93-6
2 1
Tan
90.2
109.9
93-1
3 )
95.7
18.5
to
20.0
Top
Reddish brown
103.4
2
to brownish
84.1
109.3
106.2
3
red
106.4
30.0
to
31-5
Top
Brown
--
2
Brown
84.8
105.3
95-7
3
Reddish brown
95-0
40.0
to
41.5
Top
Brown
94.8
2
Reddish brown
83.9
IO8.5
108.4
3
Reddish brown
103.7
o
o
im
to
51.5
Top}
92.2
2 >
Light red to
84.1
106.4
95.0
3(
brown
94.9
The average natural unit weight (dry) of all the sampled material
was about 99 lb/cu ft, the relative density was 62 percent, and, from
Table 8, the moisture content of the sand was approximately 5 percent.
Thus, the unit weight of the in situ sand was taken as 104 lb/cu ft.
Gradation
An indication of the uniformity of the sand deposit at the test
site was obtained from an inspection of the grain-size-distribution
curves for the sand material sampled at various depths. Similar grain
sizes and distributions at various depths indicate that the material
was deposited during the same or similar geological environments. The
sampled material from Shelby tubes extracted from Hole 3 bad an ef
fective grain size of about 0.14 mm, a uniformity coefficient of

50
about 2.5, and similar grain-size-distribution curves; Figure 3 shows
the grain-size distribution for these six samples. The uniformity of
the material sampled to a depth of 50 ft suggests that this zone of
sand might have been deposited by just one of the terrace formations
previously mentioned.
Seismic Wave Propagation Tests
Wave propagation tests, as described by Maxwell and Fry (1967)?
were conducted to assess the shear wave propagation velocity of the in
situ sand deposit at the test site. The method employs a variable
frequency vibrator to generate Rayleigh waves along the surface of the
ground. An interpretation of the measured length of the propagating
Rayleigh wave with respect to the excitation frequency provides an ap
proximation to the shear wave velocity at various depths.
Figure 4 is a plot of the results of these tests showing the vari-
tion of in situ shear wave velocity with depth. Figure 4 also shows
the shear -wave velocity, V obtained by applying the empirical equa-
s
tions (Richart, Hall, and Woods, 1970)
V = (170 78.2e) o25
s
(113)
vq = (159 53-5e) a0,25
(H4)
and assuming a constant void ratio, e of O.67 and an earth pressure
coefficient, Kq of 1/2 (Terzaghi, 1943).

PERCENT FINER BY WEIGHT
100
's
80
60
40
\
\*
\
A
SAMPLE DEPTH 1.5 TO 3.0 FT
O SAMPLE DEPTH 12.0 TO 13.5 FT
X SAMPLE DEPTH 18.5 TO 20.0 FT
A SAMPLE DEPTH 30.0 TO 31.5 FT
O SAMPLE DEPTH 40.0 TO 41.5 FT
+ SAMPLE DEPTH 50.0 TO 51.5 FT
20
40
60
\
rr
\ \
\
20
I
80
\
\
-Nr
*
A" A
m c
5 2
o it
!!
100
1.0
0.5
0.1
0.05
0.01
GRAIN SIZE IN MILLIMETERS
Figure 3- Grain-size distribution for six sample depths.
vn
H
PERCENT COARSER BY WEIGHT

DEPTH BELOW GROUND SURFACE IN FT
52
SHEAR WAVE VELOCITY IN FPS
Figure 4. Shear wave velocity versus depth, surface and
empirical methods.

Design of the Experiment
Foundation Design
This section discusses the considerations, approach, and calcula
tions which were exercised to proportion and design a torsionally os
cillating footing that served as the source of soil excitation during
the experimental phase of the investigation. The design goal for the
circular test footing placed on a natural sand deposit was to physi
cally duplicate the boundary conditions assumed for a rigid circular
disk on an elastic half-space. Correspondence of experimental and
theoretical boundary conditions was deemed important for a valid com
parison between experimental and analytical results. Essentially
elastic behavior of a vibrating foundation on soil was attained by
embedding the footing and limiting the torsion induced soil stresses;
a comparatively rigid foundation was simulated by controlling the flex
ure of the footing.
Practical considerations
A rigid circular disk pressed vertically against the surface of a
smooth elastic half-space produces a hyperbolic distribution of verti
cal stress along a radius of the disk which becomes infinite at the
edge of the disk (Timoshenko and Goodier, 1951)j and development of
Equation 57 has shown that a torsional moment applied about the verti
cal axis of a rigid disk produces a similar distribution of shear
stress along a radius of the disk in contact with the half-space. Soil,
or any other material, cannot resist infinite surface stresses and a
footing cannot be perfectly rigid, so the problem was to design and

54
construct a circular footing that would approximate the theoretical
boundary conditions as closely as possible. Limiting vertical stresses
and shear stresses on the footing-soil contact area are shown in Fig
ure 5. Because only finite shear stresses can be mobilized near the
edge of the footing, the rotational stiffness of a footing on soil is
considerably less than the rotational stiffness of a disk on a half
space (Richart and Whitman, 1967)*
To mobilize large vertical stresses near the periphery of a cir
cular footing on the surface of a sand material, correspondingly large
horizontal or confining stresses must be provided or the sand will
yield. Two possible methods of confining the sand at the edge of the
footing were: (l) provide a flexible surcharge such as air pressure
on a membrane, and (2) embed the footing in the sand.
Tests on vertically loaded laboratory scale footings on and in a
sand foundation show that the measured vertical pressure distribution
does not correspond to the theoretical distribution and that footing
embedment improves the correspondence (Chae, Hall, and Richart, 1965;
Drnevich and Hall, 1966; Ho and Lopes, 1969)- While conducting his ex
periments, Woods (1967) found that vibration measurements on the sur
face of a sandy soil were very sensitive to changes in the near-surface
moisture conditions. Assuming that motion transmission from a surface
source would be similarly influenced by the moisture content of the
near-surface sand, consistent transmission was more likely to be at
tained with an embedded footing than with a surface footing. The ef
fect of embedment on the response of vertically oscillating footings is
to increase the resonant frequency and decrease the amplitude of

LIMITING STRESSES ON THE CONTACT PLANE IN PSI
55
r/ro
Figure 5. Distribution of stresses between a
an elastic half-space.
rigid disk and

56
footing motion. This effect is quite small, however, for shallow
buried, footings which have vertical faces isolated from the soil
(Lysmer and Kuhlemeyer, 1969? Richart, Hall, and Woods, 1970; Novak,
1970). It is likely, though yet untested, that the embedment effects
on a similar footing in torsional oscillation are alscj small. Thus,
embedding the footing in the sand and isolating the vertical face of
the footing appeared to be a practical method of resolving the con
finement and vertical pressure distribution problem as well as the
soil moisture fluctuation problem.
Diameter of the test footing
The diameter of the footing was established on the basis of the
vibration frequencies that are commonly imposed on actual foundations,
the range of dimensionless frequency ratios that are usually encoun
tered in the design of prototype foundations, and the average shear
wave velocity of the sand material at the test site.
Steady state foundation vibrations range from about 10 to 60 Hz
and dimensionless frequency ratios range from 0.2 to jj.,5 (Richart,
Hall, and Woods, 1970). Shear wave velocities in the sand deposit in
crease with depth; however, at the average 15- to 20-ft depth of the
particle velocity measuring stations, the shear wave velocity was ap
proximately 65O fps. One form of the dimensionless frequency ratio is
defined by
Where V
s
a
o
2nfro
V
s
is the shear wave velocity of the soil.
(115)
Using the above

57
definition of aQ and average values of the variables for the test
site, the footing radius was
Vs
ro 2rrf
_ 0.85(650)
2tt( 35)
= 2.5 ft (116)
or a footing diameter of approximately 5 ft*
Stresses at the footing-soil interface
The desired stress conditions at the contact between the vibrating
footing and the soil were previously mentioned. The test footing
should develop similar oscillatory stress conditions at its contact
with the soil as were assigned in the Bycroft (1956) theory for the
contact area between a rigid circular disk and the horizontal boundary
of an elastic half-space.
Shear stresses.The shear stress between a rigid disk and a
half-space due to an oscillatory moment applied to the disk about its
vertical axis of symmetry is
For a torsionally loaded circular footing resting on a sand foundation,
the shear stresses in the sand on the footing-soil interface depend on
the friction developed between the bottom of the footing and the sand.
The limiting value of the shear stress, <7 Q is related to the normal
Z b

58
stress, zz
acting on the plane of contact by
(118)
where tan 0' is the coefficient of friction between the bottom face
of the footing and the sand. This equation indicates that a vertical-
dead load must be applied to the test footing in order to develop the
necessary normal stresses between the footing and the foundation.
If the shear modulus of the material used to construct a solid
cylindrical test footing is much greater than the shear modulus of the
sand on which it rests, negligible distortion of a radius of the foot
ing in contact with the soil would occur as a torsional moment was
applied to the footing. The shear modulus of concrete was 3^0 times
the shear modulus of the sand on the contact area between the footing
and the soil, so a concrete footing was considered to be rigid with
respect to the soil. Further, if the shear stresses in the soil at
the footing-soil interface are limited to about l/3 of the failure
(slip) value, and these stresses are repetitive, laboratory tests on
sand show that these soils will behave elastically (Timmerman and Wu,
1969).
Since a concrete test footing would be rigid with respect to the
soil, and the soil would behave elastically during torsional footing
oscillations, it was reasonable to expect the shear stresses on a large
portion of the footing-soil interface to be similar to those developed
by a rigid disk on a half-space. Figure 5 shows the probable distribu
tion of torsion induced shear stresses on the contact area.
So, to represent the boundary conditions assumed by the Bycroft

59
theory, the footing had to he rigid with respect to torsional deforma
tion, i.e., a radius of the footing in contact with the soil should not
be distorted during rotation by an applied torsional moment, and the
footing had to be rigid with respect to flexure in a vertical plane.
Rigidity in this plane means that vertical dead loads applied to the
footing cause negligible bending of any footing radius in a vertical
plane that contains that radius.
Vertical stresses.A flexural.1 y rigid circular disk, pressed
vertically against the horizontal boundary of an elastic half-space,
develops normal stresses, azz on the contact area of
(119)
where P is the total load applied to the rigid disk (Timoshenko and
Goodier, 1951). If a vertical load were applied to a rigid circular
footing resting on soil, and the magnitude of this load was limited
such that the normal stresses between the footing and the soil were
about 1/2 of the stresses that would initiate local failure of the
soil, the soil would react in an essentially elastic manner (Timmer
man and Wu, 1969). These conditions were prescribed for the designed
test footing, so the distribution of normal stresses on most of the
contact area between a lightly loaded rigid footing and an elastic
foundation material was probably similar to that given by Equation
119. Figure 5 shows the probable distribution of vertical stresses
on the contact surface.
Summary.It was considered that the stress distribution between

6o
a rigid disk and an elastic half-space was a reasonable approximation
to the stress distribution between a lightly loaded rijgid footing and
the soil on -which it rests provided that the soil stresses were less
than about 1/3 to 1/2 of the value necessary to cause local soil fail
ure on a large part of the contact area between the footing and the
soil. Unfortunately, reliable measurements of the distribution of
normal and shear stresses in this situation were beyond the current
state-of-the-art.
Stresses near the periphery of the footing
Figure 5 shows that the maximum contact stresses occurred near
the periphery of the circular disk and that slippage between a tor-
sionally loaded circular footing and its foundation was most likely
near the periphery of the footing because the torsion induced shear
stresses approach the normal stresses in this region. This section
views the critical stress region in more detail.
The critical stress conditions were considered to be represented
by the limiting equilibrium state of plane stress for an element (see
Figure l) of cohesionless soil located on the contact plane near the
circumference of the vertically loaded rigid footing. The lateral
pressure confining the sand at the edge of the footing was taken as
the peak passive soil pressure attainable at this point (Terzaghi,
1943).
Vr t \ (7h) <120>
The maximum vertical stress on the sand under the edge of the footing

6l
necessary to mobilize the passive confining pressure, o' > at this
point was (Terzaghi, 19^+3)
_ 1 + sin t> \
azz 1 sin j# 'rr'
so the limiting vertical stress near the edge of the footing was
(121)
(122)
where 7 is the effective unit weight of the soil, h is the depth of
the element below the surface of the soil, and ft is the angle of in
ternal friction of the soil.
The above equation had implications which influenced the design
and placement of the test footing; it indicated the need for lateral
confining stress at the edge of the footing and the footing burial
necessary to attain confinement. Of course, to conform to the geomet
ric boundary conditions assumed in the theory, the footing must be as
near the ground surface as possible, and, to conform to the stress con
ditions assumed in the same theory, the footing must be buried as
deeply as possible with the vertical surface of the footing isolated
from the soil. This dilemma was resolved by recalling that the pri
mary objective of the experimental work was to assess the particle mo
tion in a large mass of soil extending to nearly 100 ft from the motion
source, so distorting the geometric position of that source should have
little effect on the measurements; burying the source 1 ft, or 20 per
cent of its diameter, was judged to be an acceptable bias of the geo
metric boundary conditions at the source. With this depth of burial,
the maximum normal stress developed near the periphery of the contact

62
surface between the buried test footing and the soil due to a vertical
load on the footing was
= (r^f)211041
= 936 psf
= 6.5 psi (123)
where 7 and / are taken as 104 lb/cu ft and 30 deerees, respec
tively. Figure 5 shows the limiting stress distribution.
The significance of the inelastic vertical stresses developed near
the periphery of the footing was implied by calculating the portion of
the footing-soil contact area on which elastic stresses act to the
total contact area. Assuming that the 5-f't-diam test footing was a
solid cylinder of concrete, 2 ft high and buried 1 ft in the soil,
Equation 119 was used to compute the approximate radial position, r^ ,
of the maximum normal stress of 6.5 psi. If the unit weight of con
crete is taken as 150 lb/cu ft, and the entire footing acts as a rigid
body
r = 2.47 ft (124)
Thus, for the case assumed, nearly 98 percent of the contact area be
tween the footing and the soil had normal stresses that were less than
6.5 psi and transmission of elastic stresses during torsional oscilla
tion of the footing occurred over some 92 percent of the footing-soil
contact area (Timmerman and Wu, 1969). Also, because the shear
stresses at the edge of the footing are less than those at the edge

63
of a rigid disk, the rotational stiffness of the footing, at the limit
ing moment, was about 75 percent of the rotational stiffness for a
rigid disk.
Footing emplacement operation
Previous paragraphs established the size and position of the test
footing and discussed the desired stress conditions on the footing-soil
contact area. This section sets forth the footing design and placement
method.
To transmit torsional oscillations into the soil, the friction
angle between the base of the footing and the soil should be comparable
to the angle of internal friction, 0 of the soil. This objective was
met by using concrete as the footing material in contact with the soil.
Intimate and uniform contact between the base of the footing and the
sand should result by pouring the concrete directly on the prepared
sand surface.
The only contact allowed between the buried footing and the soil
occurred on a horizontal circular area, so the vertical face of the
circular footing had to be isolated from the soil. This was accom
plished by a thin steel ring placed between the soil and the cylindri
cal surface of the footing.
The distribution of vertical contact stresses between the base of
the footing and the soil should be similar to that developed by a rigid
circular disk pressed vertically against a half-space. This objective
was realized by considering the soil stress conditions after excava
tion, after pouring the footing, and after the application of a dead

64
load, to the cured, footing. The last condition is discussed in a fol
lowing section.
Figure 6 is a sketch of the embedded concrete test footing. The
following sequence of placement operations resulted in the desired ver
tical stress distribution on the contact surface between the footing
and the soil.
1. The natural soil within a 7-f"t-diam circle was excavated to a
depth of 1 ft. The stress change in the soil due to the excava
tion was equivalent to a uniform unloading pressure of 7h act
ing on the excavated area.
2. The concentric footing form and soil retaining ring were
placed in the excavation, the first pour of concrete was placed
inside the footing form to a depth that produced a uniform pres
sure of 7h on the soil, and backfill soil was placed around the
retaining ring to the original ground surface and at its original
in situ density. At the end of these operations, the stresses in
the soil on the footing-soil contact surface and in the vicinity
of the footing were approximately the same as the in situ stresses
before excavation because the added loads were equal to the loads
removed during excavation.
3- The first pour of concrete was allowed to cure to a rigid
mass.
4. A second pour of concrete was added inside the form to act as
a dead load on the rigid first pour. It was this dead load, prop
erly applied, that produced the desired distribution of normal-
stresses on the footing-soil contact area.

PLAN VIEW
Figure 6. Sketch of concrete footing embedded in soil

66
Position of dead load on cured, first pour
As mentioned previously, the dead load applied to the cured first
pour was positioned so that the plane contact area between the footing
and the soil was not distorted under this load; i.e., the cured first
pour simulated a rigid disk as it was pressed against the underlying
soil by the dead load.
Figure 7 shows a cross section of the cured first pour with an
axisymmetric, uniformly distributed dead load acting on a part of its
upper surface and a footing-soil contact stress distribution, that
would be developed by a rigid footing, acting on its lower surface.
The simple dead load pressure distribution was chosen to minimize
forming and placement problems as the dead load concrete was placed
on the cured first pour.
If the locus of the center of pressure for the dead load is coin
cident with the center of pressure acting on the contact area, bending
of the cured first pour, due to the dead load, should not be signifi
cant; the position of the center of pressure in each case was a cir
cle with its center on the vertical axis of the footing. In essence,
this approach considered that the soil pressure due to the uniformly
distributed dead load, q acting on the cured first pour of the foot
ing was represented by an equivalent load distributed along a circle
of radius, r (Richart, 1953)- The distance, r to the center of pres-
pressure on the bottom of the footing was

r
q q
' r
'
1
1 1 1
> ' "
CURED FIRST POUR (RIGID)
w -
1
{ H,M 1
1
o t
SOIL PRESSURE ON BASE DUE TO DEAD LOAD
Figure 7* Distribution of vertical
soil stress and dead load
pressure on cured first
pour.

68
U
f r0zz( rdrd.0)
o
r =
(125)
J azz^rdrd0)
o
TT
Coincidence of the center of pressure for the dead load required that
U
J rq(rdrd0)
^2
r
o
j q(rdrd@)
(126)
from which
r2 = 0.52035rQ
(127)
Rigidity of the footing
Having chosen the material for the footing and established the di
mensions, depth of burial, sequence of emplacement operations, and
system for applying the dead load, it was necessary to evaluate the de
sign to assess the effective rigidity of the test footing with respect
to the soil. A flexurally rigid footing was desirable to assure the
proper vertical stress distribution on the footing-soil contact area.
The method of evaluating the relative rigidity of the test footing
with respect to the soil was to compute and compare the vertical de
flection of the soil in the footing-soil contact area for two equiva
lent loading situations. The first case assumed that the footing was

completely flexible and was loaded by a uniformly distributed pressure
acting on its entire upper surface; the load-deformation relations ob
tained illustrate the rigidity or stiffness of the soil. The estimated
elastic properties and design dimensions of the test footing with an
advantageously positioned equivalent dead load were used to calculate a
second load-deformation relationship for the contact area that illus
trated the rigidity of the footing. The relative rigidity of the foot
ing with respect to the soil was taken as the ratio of the footing
load-deformation relationship to the soil load-deformation relationship.
In the first case, Young's modulus, E for the foundation mate-
s
rial at a depth of 2-l/2 ft was about 7>050 psi, and Poisson's ratio
was about l/3- The deflection of the soil at the edge of the flexible
footing, due to the uniform load, q acting on its entire surface is
(Timoshenko and Goodier, 1951)
/ \
( /r=r uE
o s
(128)
and at the center of the footing
2(1 p2)iro
E
s
(129)
The deflection, A
s
within the contact area is
A
s
- ( 2tT ^)(l U2) r. n
TTW.
(130)
= 0.002751 in.
where q is expressed in pounds per square inch.

70
In the second case, Young's modulus for concrete, Ec was assumed
to be 3 x 10^ psi and Poisson's ratio was taken as 0.17 (Dunham, 1953>
Lin, 1955). The cured first pour of the footing had a diameter of 5 ft
and a planned thickness of about 8-l/4 in. Before computing the actual
design situation, the deformation of the contact area estimated by the
center deflection of an edge supported circular plate supporting a
uniform load, q acting over its entire surface was calculated. Using
thin plate theory (Timoshenko and Woinowsky-Krieger, 1959)? the deflec
tion, A ^ is
Acl
3(5 + M-)(l p)rc
l6E t3
c
= 0.0003871 in.
(131)
Even under these unrealistic support and loading assumptions, the in
fluence of the stiffness of the cured first pour decreased the deflec
tion of the footing-soil contact area by a factor, S of
S = P- (132)
cl
0.002751
0.0003871
= 7.1
The deformation of the footing-soil contact area was next approx
imated by the deflection of the cured first pour simply supported along
a circle of radius r and loaded by a uniformly distributed dead load
with its center of pressure lying on a circle of radius r ; see

71
Figure 7. These support and loading assumptions were believed realis
tic for computing the approximate deformation of the contact area; com
puting deflections for a flexible, finite plate on an elastic founda
tion were considered unnecessarily complicated and tedious. With
r^ = l6 in., the dead equivalent load, q that acted on only part of
the circular area, was 1.40 q where q is the uniform dead, load
pressure previously assumed to act over the entire area of the footing.
Superposition of three loading situations given by Timoshenko and
Woinowsky-Krieger (1959) was accomplished to calculate the maximum
deflection of the center of the first pour with respect to its edge due
to the distribution of the dead load. A similar displacement of the
contact area was assumed, and, to slide rule accuracy, this displace
ment, A was
c2
Aq2 = 0.0000074q (133)
= 0.0000103q
The ratio of the deformation of the contact area for a completely
flexible footing to the deformation of the contact area under the
cured first pour supporting a selectively located equivalent dead load
was used to judge the degree of rigidity of the footing with respect to
the foundation soil. For the above situation, this ratio, S was
0.002751 (134)
0.0000103q
= 266
Thus, since the deformation of the contact area was reduced to less
than 0.5 percent of its free deformation by the effective rigidity of

72
the cured first pour and the soil stresses were limited to elastic
values, it was reasonable to expect the soil pressure distribution on
the contact area to be nearly the same as that for a rigid footing on
an elastic foundation.
Limiting torsional moment
The maximum torsional moment that could be applied to the test
footing without causing slippage on the contact area was limited by the
critical normal stress developed in the soil, by the dead load on the
footing, and by the location of this critical stress on the contact
area. With the assumption that the friction angle between the footing
and the soil was the same as the angle of internal friction for the co
hesionless sand foundation, the moment, M' to cause impending slippage
on the contact surface at the locus of the critical normal stress was
computed from
2n
M' =
/ /
ctzz tan 0,
r -
r 3
1 r drdQ
r 2tt
o
V?-
/ /<* tan f> r2d0dr (135)
rl
where a is the critical vertical stress on the contact area and r,
ZZ JL
is the radial distance to the location of this critical stress. Sub
stituting r = 2.5 ft cj = 936 psf r, = 2.47 ft and 0=30
degrees into the above expression, the limiting moment, M' is
M' = 4,400 ft-lb
(136)
The moment capacity of a rigid disk on a half-space with a similar

73
shear stress distribution on the contact area, except at the edge,
would be about 5>750 ft-lb. With an assumed soil shear modulus of
4,500 psi under the footing, the angular rotation, f" due to the
limiting moment was about
" = _2L
l6Gr3
o
= 0.0001 radians (137)
and the single amplitude displacement, v of the outside edge of the
footing was 0.003 in.
Transmission of predominantly elastic shear stress into the soil
was possible by limiting the applied moment to less than 1/3 of the
value necessary to initiate slippage at the periphery of the footing-
soil contact area. The design moment, and the design displace
ment, v^ were
Md <; 1,400 ft-lb (138)
and
0.001 in. (139)
Dynamic response of the foundation
The geometry, weight, and operating frequency of the designed
test foundation were previously established, and the elastic proper
ties of the test site were estimated and measured. These parameters
were used to determine the resonant frequency of the torsionally vibra
ting foundation and the static amplitude magnification factor at reso
nance. The frequency at which resonance occurs and the expected

74
increase in foundation motion were pertinent to the design of a tor
sional vibrator to drive the test footing and to the expected experi
mental measurements.
The designed test footing was a hollow concrete cylinder. The
diameter of the cylinder was 60 in. and it was 28 in. high. A 32-in.-
diam, 18-in.-high cylindrical void was formed interior to and sym
metric with the outer cylinder; the top of the void was 2 in. below
the top of the outer cylinder. The mass ratio of the footing was 2.1,
the resonant frequency was about 40 Hz, and the amplitude magnifica
tion ratio was approximately 5*2 (Richart, Hall, and Woods, 1970).
Although the resonant frequency occurred within the planned test fre
quency range, the magnification of the footing motion did not present
apparent experimental difficulties.
Vibrator Design
A special vibrator was designed to drive the test footing in a
torsional mode of oscillation about the vertical axis of the circular
footing. The vibrator design goals were (a) to minimize mechanical
sources of noise and vibration, (b) to utilize a remote source of
power, and (c) to keep all dynamic forces in a horizontal plane.
Goal (a) was attempted by using an electric motor to drive the rota
ting eccentric masses through a rubber timing belt, goal (b) by a
long electrical power line from the electric motor to a remote
generator, and goal (c) by rotating the eccentric masses in a hor
izontal plane. The vibrator was designed to provide a twisting moment
of about 3 ft-kip in the 20- to 50-Hz-frequency range.

75
Figures 8 and 9 are plan and elevation sketches of the vibrator
components; the layout of these components was dictated by the config
uration of the test footing. The eccentric masses were separated as
far as practical to develop large twisting moments with small centri
fugal forces, the horizontal plane containing the rotating masses
was kept as low as practical to reduce rocking of the footing by un
balanced forces, and the structural frame was extremely rigid to raise
sympathetic vibration frequencies well above the torsional frequencies
applied to the foundation.
Design details such as timing belt layouts, sprocket sizes, bear
ing loads, and shaft bending and whirling are not mentioned in the fol
lowing paragraphs, but a discussion of the input power requirements
and torque capacity of the vibrator was considered pertinent to the
design and conduct of the experimental work.
Power requirements
A gross approximation of the power expended on the footing-soil
contact area by a torsionally vibrating footing was made by assuming
a simple relationship between the twisting moment, M applied to the
footing and the rotation, f> of the footing. To find an upper bound
for the power losses through the foundation, the M versus f> rela
tionship was assumed rigid-plastic. The work done by the footing on
the soil per cycle was 4M0 the work done per second at a frequency
of 50 Hz was 2OOM0 and the horsepower expended was 2OOM0/550 .
Using the design moment of 1,400 ft-lb and the design rotation of
0.00004 radians, the power loss, p at a frequency of 50 Hz was

Figure 8. Plan view sketch of torsional vibrator.
CA

MOTOR
Figure 9. Elevation view sketch of torsional vibrator.

78
2OOM0
P = 550
= 0.02 hp (l40)
Power losses to the ground appeared to be negligible; power losses due
to belt friction, windage, and bearing friction controlled the selec
tion of an electric motor.
Four precision, sealed, permanently-lubricated, self-aligning
ball bearings were chosen to support the two shafts for the rotating
eccentric weights. With a torque capacity of 3 ft-kip and a shaft
center distance of 31 in., the radial loading on each shaft was about
1,200 lb and the maximum radial bearing load was about 800 lb. At a
frequency of 50 Hz, the peripheral velocity of the 1.625-in.-diam
shaft was 1,280 fpm. Using the average tabulated coefficient of fric
tion for ball bearings (Oberg and Jones, 194-9), the horsepower loss
for the four bearings was
p = (2,400)(l,280)(0.0023)/33,000
= 0.21 hp
The belt friction and windage losses were taken equal to the bearing
losses and a 1-hp motor was judged adequate to drive the torsional
vibrator.
Frequency and moment capacity
The eccentricity of the rotating masses on the torsional vibrator
was 4 in. and the masses were varied to change the torque output at

79
a constant frequency. The mass was changed by bolting different com
binations of identically matched weights to each vibrator flywheel.
Table 12 gives the stamped identification letter and weight of each
of the paired eccentric masses and the attaching bolts.
Table 12
Identification Letter and Weight of Eccentric Masses
Mass
Identification
Weight
lb
Bolt
Identification
Weight
lb
A
1.9771
E
0.135
B
1.9786
F
0.135
C
0.9777
G
0.106
D
0.4770
H
0.106
E
0.2314
I through N
0.084
The centrifugal force developed by each of the rotating eccentric
2
masses is Weu) /g so the moment, M generated by the torsional vi
brator was
M = (i)w () ir ft-lb (142)
where W is the weight (lb) of the eccentric mass and f is the fre
quency (Hz). The moment output of the vibrator per pound of eccentric
mass was
= 1.06l6f2 (143)
w
This relationship and Table 12 were used to calculate the moment

8o
developed by the torsional vibrator for typical mass combinations at
frequencies of 15, 20, 30, 40, and 50 Hz. The calculated results are
given in Table 13 and show that the vibrator was capable of developing
the footing design moment of nearly 1,400 ft-lb at a frequency of
15 Hz.
Table 13
Vibrator Moment Capacity at Various Frequencies
Frequency
Hz
Weight
Identification
Bolt
Identification
Total
Weight
lb
Moment
ft-lb
15
ABODE
E
5j768
1,380
20
ACD
G
3.5378
1,502
30
CD
I
1.5387
1,470
4o
CD
I
1.5387
2,6i4
40
DE
I
0.7924
1,346
50
C
I
1. <4)617
2,818
50
D
I
0.561
1,489
Foundation and Transducer Location
Location of the test footing
The Piccolo field test area that had been investigated during the
soil exploration program was examined to locate a favorable site for
placing the circular test foundation. Likely locations were probed
with a portable cone penetrometer similar to that described by Poplin
(1969); its use is shown in Figure 21. The soil was probed to a depth
of 6 to l8 in. on a grid spacing of 5 ft to delineate unusually soft
or hard areas. Of several suitable sites, one was chosen which offered

81
good, lines of sight for surveying work and a minimum number of large
trees; wind action on trees can cause undesirable ground movements.
Location of transducers
The location and position of the particle velocity transducers
with respect to the test foundation was based on (a) the planned future
uses of the test area, (b) the sensitivity of commercial transducers
and available recording systems, (c) the expected wavelengths of
vibrator generated seismic waves, and (d) the possibility that the
test foundation was located on a laterally nonhomogeneous sand deposit.
Future uses of the test area include measurements of ground vibra
tion developed by other types of foundations in various modes of oscil
lation. Such experiments would produce surface or Rayleigh waves that
have significant motion to a depth of about one wavelength (Woods,
1968). For the site conditions discussed in a previous section, one
Rayleigh wavelength at 20 Hz would be equal to about 33 ft; transducer
locations to a maximum depth of 35 ft were judged adequate for the test
area.
Commercial particle velocity transducers, suitable for borehole
placement, were available with damped transduction sensitivities of
about 1 to 2 volts/in./sec, available amplifiers had a maximum gain of
about 2,000, and the sensitivity of available high performance gal
vanometers was about 9 in./volt. Thus, if a 1-in. amplitude oscil
lograph record is desired, the particle velocity of the transducer
must be at least l/l8,000 in./sec. The decrease in particle motion
with distance from the source of motion, or geometrical damping

82
(Richart, Hall, and Woods, 1970), is greatest for body waves, so
minimum motions were estimated from the previously computed displace
ments due to torsion induced shear waves in an elastic half-space. For
steady state sinusoidal oscillation, the particle velocity, v is re
lated to the particle displacement by
v = u)V (l44)
For the test foundation, oscillated at 15 Hz by a 1,400 ft-lb moment
on the test site sand with an average shear wave velocity of 650 fps,
the particle velocity at a depth of 35 ft and. a distance of 170 ft was
v = (2nf) ^ -p l(a ,a,b) (145)
V 4GTir2
o
= 0.00007 in./sec
If material damping losses are neglected, the calculated particle ve
locity in the ground at this point is 25 percent more than the motion
needed to produce an oscillograph record with a single amplitude of
1 in. Material damping losses, or attenuation of motion in excess
of geometrical damping, reduce the expected particle motion signifi
cantly, so particle motion measurements greater than 90 ft from the
test foundation were judged to be beyond the sensitivity and capacity
of available motion recording equipment.
The horizontal and vertical spacing of 20 available transducers
within 90 ft of the footing and 35 ft below the surface of the sand de
posit was selected to facilitate an interpretation of the veloc
ity of the propagating seismic waves. A close spacing is desirable for

high frequency waves with short wavelengths and a larger spacing is
desirable for lower frequency waves with long wavelengths. The short
est expected wavelength, for a 50 Hz shear wave, was 13 ft; the long
est expected wavelength, for a 15 Hz compression wave, was 87 ft.
Thus, about seven high frequency shear wavelengths and one low
frequency compression wavelength would be contained in the zone of
particle motion measurements. Transducer spacing in planes parallel
to the ground surface was varied from about half of the shortest ex
pected wavelength to half of the longest expected wavelength, a range
from 6.5 to 30 ft; transducer spacing in the vertical direction was
varied from 4 ft near the ground surface to 10 ft at greater depths.
The natural processes that deposited the sand formation at the
test site include the effects of ocean currents and wave action. These
effects were undoubtedly related to the direction of the shoreline and
the direction of the prevailing winds during the Pleistocene epoch.
Therefore, there was a possibility that the sand deposit in the im
mediate vicinity of the test foundation had undetectable seismic wave
propagation characteristics that were direction dependent. Previous
sections have established the gross properties of the test site, so,
for consistency, it was necessary that the particle motion measurements
reflect the gross seismic characteristics of the test site. One means
of relating the measurements to gross properties is by a random sam
pling of the motion in several different directions from the test
foundation. Practical geometrical field constraints (trees, access
roads, borehole drilling operations, etc.) precluded locating the trans
ducers in random directions from the test footing. The depth of the

84
five transducers at each of 4 radial distances from the footing was
selected by drawing five numbered slips of paper, in sequence, from a
transit case.
Figure 10 shows a plan view of the location of each particle
motion transducer with respect to the test foundation, and Figure 11
is a section view that shows the transducer positions within the soil
deposit; each location is identified by its radial distance in feet
from the center of the test foundation, r and its depth in feet
below the ground surface, z Footing locations are indicated by an
asterisk.
Isolation of power and recording facilities
The power source for the electric motor on the torsional vibrator
and for the amplifiers and recording system was located about 6¡?0 ft
from the test foundation in order to isolate the noise and vibration
developed by this power equipment from the field of motion transducers
located in the sand near the test foundation. An electric power line
connected the motor to the generator and 1,000-ft-long cables connected
the motion transducers to the recording instruments. Number 8 gage
conductors were adequate to supply power to the 220-volt vibrator
motor.
Construction of Test Facilities
Foundation Construction
Fabrication of the footing form
Two concentric steel rings were fabricated to serve as a form for

O 90, 1
O 90, 15
O 90, 25
O 90, 5
O 90, 35
O 60,35
O 60- 1
O 60, 5
O 60, 25
O 60, 15
30, 1 O
30, 5 O
SCALE
20- ^ | 30, 15 O
30, 25 O
30, 35 O
\
Figure 10. Plan view of the field of transducer locations.
oo
vn

(* CENTER LINE OF TEST FOUNDATION
O 10, 15 O 30, 15 O 60, 15 90, 15 O
O 10, 25
O 30, 25
O 60, 25
90, 25 O
O 10, 35 O 30, 35
- r
O 60, 35
90, 35 O
Figure 11. Section view of the field of transducer locations.
00
ON

87
the green concrete and as a retaining wall for the surrounding soil.
The rings were separated by a removable wire spacer to provide an air
gap between the inner concrete form and the outer soil retaining wall.
The interior ring was 5 ft OD, 28 in. high, and rolled from
0.125-in.-thick steel plate. It was butt welded, ground smooth on the
outside, and held in a cylindrical shape by an internal plywood dia
phragm. Nine-gage steel wire was lightly tack welded to the bottom
of the form and wrapped circumferentially around the lower 16 in. of
the form at a pitch of 2 in.
The outer ring was 16 in. high and rolled from the same stock as
the inner ring. It was fitted over the wire wrapping on the inner
ring, cut to size, and butt welded in place. A plywood exterior flange
was fitted around the outside of the outer ring to hold it in a cylin
drical shape.
An auxiliary form, to provide a cylindrical void at the center
of the footing, was rolled from 0.0625-in.-thick sheet stock and capped
with an 0.0991-in.-thick circular plate. This form, an inverted tub,
was 32 in. OD and 18 in. high. Number 2 gage, 6-in.-mesh reinforcing
was trimmed to 58 in. OD for placement in the first concrete pour; a
46 -in.-diam, 20-in.-high cylinder of the same mesh was fabricated for
the dead load pour. Anchored baseplates for attaching the torsional
vibrator to the test foundation, brackets for mounting motion trans
ducers to the foundation, and other miscellaneous hardware were also
fabricated.
Placing the form
The form for the test footing was transported to the test site

88
and. prepared for embedment at the selected footing location. Adhesive
tape was placed along the bottom of the form to prevent sand from in
truding into the air gap between the inner form and outer wall when the
form was placed on the soil. Tape was also placed along the upper
edge of the outer wall to keep debris and water out of the air gap.
Figure 12 shows the footing form attached to a vehicle hoist, the inner
form, the outer wall, the plywood flange, one end of the wire wrapping,
and the tape on the bottom edge of the form.
The footing form was placed in a prepared 7-ft-diam, 1-ft-deep
excavation. Care was exercised to insure that the horizontal bottom
surface of the excavation was exposed by cutting away the natural soil;
filling slight overcuts was not allowed. An additional excavation was
made to place a Mark Products, Inc., model L-1D transducer precisely
6 in. below the edge of the footing with minimum disturbance of the
natural soil. Figure 13 shows the footing excavation, the transducer,
and the cutting operation before planting the transducer in the posi
tion illustrated in Figure l4. Sand was tamped around the transducer,
the connecting cable was brought to the ground surface, and the excava
tion shown in Figure 15 was prepared to receive the footing form. Fig
ure 16 is a photograph of the form as it was aligned and lowered into
the prepared excavation; Figure 17 shows the form positioned in the
excavation and a view of the internal plywood diaphragm. As measured
on various diameters with a carpenter's level, the plane of the upper
edge of the emplaced form was exactly horizontal.
First pour of concrete
Before the first pour of concrete could be placed in the footing

INNER FORM
WIRE WRAPPING
FLANGE
OUTER WALL
TAPE
Figure 12. Footing form and soil retaining ring.


Figure 14. Transducer embedded below edge of footing

Figure 15. Excavation ready to receive footing form and retaining ring
MD
ro

VO
U)
Figure 16. Placing footing form and retaining ring in excavation

Figure 17. Footing form and retaining ring positioned in excavation
vo
-p-

95
form, the internal plywood diaphragm had to be removed]. A band of
stiff concrete was placed along the bottom edge of the interior form
to hold the form in a circular shape while the first pour was cast and
hardening. A few bags of mortar mix sackcrete were mixed, hand placed,
and reinforced with a single loop of l6-gage steel wife. After the
mortar mix hardened, the wooden diaphragm was removed, the steel mesh
reinforcing was positioned 2 in. above the sand surface, the first pour
of concrete was placed in the form to a depth of 9 in., and three test
cylinders of concrete were prepared. The auxiliary form, the second
pour reinforcing mesh, the test cylinders, and the footing form after
the first pour are shown in Figure 18. Before initial set of the first
concrete pour occurred, the second pour reinforcing mesh was positioned
in the footing form with its top strand 1 in. below the top edge of the
form, as shown in Figure 19. The auxiliary form was centered within
the footing form and its leveled top surface was positioned 2 in. below
the top edge of the footing form; Figure 20 shows the location of the
auxiliary form.
Ultimate strength and static modulus tests were conducted on the
three 5,000-psi concrete test specimens taken from the first pour. The
12-day strength of specimens 1, 2, and 3 was 3>700 psi, 3>710 psi, and
3,480 psi, respectively, and the static modulus, measured during these
tests, was 4.31 X 10^ psi, 4.26 x 10^ psi, and 3-99 X 10^ psi,
respectively.
Backfilling
In order to preserve the cylindrical shape of the
outer soil re
taining ring when the wire spacer was removed from the
air gap, a band

TEST CYLINDERS
FORM
REINFORCING MESH
AUXILIARY FORM
Figure 18. Concrete test cylinders, auxiliary form and reinforcing mesh
vo
ON

Figure 19- Second pour reinforcing mesh placed in first pour.
VO
-3

Figure 20
Position of auxiliary form and backfilling operation
M3
co

99
of stiff mortar mix concrete was placed around the bottom of the re
taining ring and reinforced with a single loop of l6-gage wire. After
the band of concrete had cured the excavation was manually backfilled
with lifts of sand, as illustrated in Figure 20. The lifts were com
pacted with a wooden tamper until they attained cone penetration values
similar to the adjacent soil deposit; this procedure jls depicted in
Figures 21 and 22.
Second pour of concrete
Transducer mounting brackets were bolted to the inner form above
the first pour of concrete. The second concrete pour was placed in
the footing form 6 days after the first pour, rodded, and struck level
with the top of the form. Two test cylinders were also cast. An
chored steel plates for mounting a torsional vibrator on the footing
were placed in the green concrete using a predrilled plywood template
to hold the plates until the concrete hardened. Figure 23 shows the
test footing after the second pour of concrete had hardened and the
plywood template had been removed. The concentrically-wound wire
spacer was pulled out of the air gap and a feeler strip and the steel
tape shown in Figure 24 were used to check the continuity and depth of
the gap.
Ultimate strength and static modulus tests were tonducted on two
5,000-psi concrete specimens taken from the second pour. The 6-day
strength of specimens 4 and 5 was 3*150 and 3,260 psi, respectively,
and the static modulus was 3*H X 10^ 'and 3-18 X 10^ psi,
respectively.


Figure 22. Cone penetration test on backfill
o
H

Figure 23 Second pour of cured concrete in the footing form.
102

Figure 2b. Checking depth and continuity of the air gap.
103

io4
Vibrator Construction
Motor
The torsional vibrator was designed to accommodate one or two
electric motors, depending on its power requirements One variable
speed electric motor was obtained and mounted on the fabricated vi
brator for testing. The motor had an advertised rating of 1 hp at
5,000 rpm with a constant torque output to 500 ppm. Speed control was
accomplished with a silicon controlled rectifier; the motor weighed
only 16 lb, and it appeared to be ideally suited for driving the vibra
tor. Tests quickly showed that one motor would not drive the vibrator
at its rated speed and that the actual output of the motor was about
0.6 hp. A second variable speed motor for tandem operation could not
be obtained.
A constant speed, 1,725-ppm, single phase electric motor, rated at
1 hp, was used to run in the vibrator. A variety of timing belt
sprockets and a 2-hp, 3,450-rpm, electric motor were obtained so that
the vibrator could be operated at different frequencies. Motor rpm
was measured under load and no-load conditions, using different sprock
ets, in order to determine the power requirements of 1¡;he torsional vi
brator. Figure 25 is a plot of the results of these measurements.
Mounting the vibrator
The vibrator, drive motors, and sprocket combinations to produce
torsional oscillations at 15, 20, 30, 40, and 50 Hz were transported to
the test site and the vibrator was mounted on the test foundation. The
surfaces of the vibrator mounting plates on the concrete footing were
cleaned and sanded. An epoxy compound, Scotchcast electrical insulating

POWER REQUIRED IN HP
105
/
/
/
/
0
/
/
/
'
/
/
/
/ O
/
/
/
/
/
V
/
<
\\l
N,
/
/
s
o 10 20 30 40 50
FREQUENCY OF TORSIONAL VIBRATION IN HZ
Figure 25. Power required to drive the torsional vibrator.

io6
resin number 4, made by the Minnesota Mining and Manufacturing Company,
was applied to the plates before the vibrator was set in place on the
test footing. One end of the mounted vibrator, the tqst footing, and
the epoxy compound are shown in Figure 26; the test pqd and mounted
vibrator are completely illustrated in Figure 27. When the epoxy had
cured, the vibrator was fastened to the mounting plates with four
5/8 -in.-diam lug nuts so that tensile cracks would not develop in the
epoxy.
Operating tests
Preliminary experiments were performed to assure that the vibrator,
the test footing, and the power source isolation schemes would work
before the particle motion transducers were buried in the soil deposit.
A 750-ft-long reel of power cable was unrolled on top of the ground,
connected to a remote 10-kw motor generator at one end, and connected
to a terminal box and vibrator motor switch at the other end. The
2-hp electric motor, with a sprocket to develop a vibration frequency
of 40 Hz, and motor balancing weights were assembled on the vibrator,
and the belt was tensioned and adjusted to synchronize the rotating
eccentric masses. The assembled vibrator, the test footing, and the
switch box are shown in Figure 28.
Measurements to determine the settlement and tilt of the test
footing were made during the preliminary operating tests and during the
subsequent field tests. A transit and level rod were used to measure
elevations at the center of the vibrator frame and at four equally
spaced points around the top edge of the footing. A carpenter's level
was used to check tilting of the vibrator frame and the top surface of

Figure 2.6. Vibrator bonded to mounting plate with an epoxy compound.
01

Figure 27. Torsional vibrator mounted on the test foundation.
H
o
00

Figure 28. Assembled vibrator, test footing, and switch box
o
\o

110
the footing. The torsional vibrator was run for half-hour periods
using a sequence of increasing eccentric masses; settlement and tilt
observations were made before and after each half-hour interval. Dur
ing this period, no measurable tilt or settlement was observed for ec
centric masses less than or equal to the design moment mass. Figure 29
shows the operating vibrator and the carpenters level used to check
tilting of the vibrator frame.
Measurements were made to determine the motion of the test footing
due to the effects of the torsional vibrator. A three component par
ticle velocity transducer was attached to one of four mounting brackets
located around the periphery of the footing and a single component
transducer, identical to the one buried under the edge of the footing,
was bonded to the top surface of the footing with an epoxy compound.
These preliminary measurements indicated that the vertical and radial
components of footing motion were less than 10 percent of the tangen
tial motion. Figure 30 illustrates the methods that were used to at
tach the particle motion transducers to the test foundation.
Transducer Installation
Performance tests
An operating test on each transducer was conducted before plac
ing it in the ground. The magnitude and phase of the transverse (T)
and radial (R) components of motion, and the sense of this motion, were
determined by mounting the transducer on the test footing, operating
the vibrator, and recording the results on an oscillograph. The po
larity of the electrical cable connection was changed on some of the
transducers so that all recorded components had the same phase. The

Figure 29
Carpenter's level used to check tilt on vibrator frame.
H
H
H

Figure 30. Method of attaching transducers to the test footing.
112

113
radial component of transducer 19 (Serial Number 19) did not work prop
erly, and the radial component of transducer 11 produced about half the
signal amplitude of the other transducers. Since replacement trans
ducers were not available and the transverse component of motion was of
primary interest, these two transducers were used to detect transverse
and vertical (v) components of particle motion.
Boreholes
Figures 10 and 11 give the location and depth of the field of
transducers installed at the test site. These locations were surveyed,
center hubs were driven, and marking stakes were placed to indicate the
position and depth of each borehole. A view of the test site topog
raphy, the vegetation, and some of the marking stakes is shown in
Figure 31* Six-in.-diam vertical holes were augered with the drill
rig pictured in Figure 32; the holes were uncased.
Transducer alignment
The orientation and depth of the buried three component particle
velocity transducers were controlled by an indexed, square, borehole
rod. One end of the rod was telescoped into an aligning sleeve at
tached to the transducer case and the other end of the rod was attached
to a sighting bar. The elevation and plumb of the rod were measured
with a transit and carpenter's level. The method of attaching the
square alignment sleeve to the transducers using a wooden jig and an
epoxy compound is illustrated in Figure 33- A group of transducers
prepared for installation in a borehole is shown in Figure 3^; each
transducer shown has one flexible support cable and one electrical
cable attached to it.

Figure 31. Test site topography, vegetation, and borehole markers
H
H
-p-

Figure 32. Drill rig used to auger uncased boreholes for the transducers
115

Figure 33 Alignment sleeves bonded to transducers with an epoxy compound.
ELECTRICAL AND SUPPORT CABLES
TRANSDUCERS
ALIGNMENT SLEEVES
WOODEN JIGS
9TT

Figure 34. Transducers with support cables and electrical leads
H
3

118
Installing procedure
One end of the indexed borehole rod was slipped into the trans
ducer alignment sleeve. The transducer and rod were lowered into the
hole by the support cable. If the hole was deep, additional 10-ft-
long sections of indexed rod were attached to the first section. When
the desired depth was reached, the support cable and borehole rod were
clamped to a movable wooden cross member supported by a wooden frame
work; this arrangement is shown in Figure 35* A sighting bar was at
tached to the borehole rod and the rod was rotated by the wooden cross
member until the bar was aimed at the center of the test footing; this
step is illustrated in Figure 36.
Backfilling
A method for placing the borehole backfill sand in a manner simi
lar to the natural depositional processes that formed the test site was
devised. The boreholes were filled with screened sand dropped down the
borehole from the ground surface into a slurry of sand and water at the
bottom of the hole. Quarter-in.-mesh hardware cloth was used as a sieve
for the backfill sand; a barrel of water and a long hose provided water
for the slurry in the bottom of the borehole. When enough backfill sand
had been added to the hole to cover and support the transducer, the
borehole rod was slipped upward and out of the transducer alignment
sleeve and removed from the borehole. The transducer support cable
was relaxed and allowed to hang loosely in the hole while the back
filling operation continued to the ground surface. A detail view of
the top of the borehole, the borehole rod, the support cable, and the
electrical cable is presented in Figure 37 and a comprehensive view,

Figure 35 Apparatus for installing and aligning transducers in boreholes.
119

Figure 36. Sighting bar used to align borehole rod and attached transducer.
120

BOREHOLE ROD
SUPPORT CABLE
ELECTRICAL LEAD
BOREHOLE 'JM
Figure 37. Borehole, borehole rod, and transducer cables.

showing the water supply hose, is given in Figure 38.
Particle Motion Measuring System
122
The system used to measure and record the particle velocities
generated in the sand deposit by a vibrating footing was capable of
sensing a wide range of frequencies--from a few to several hundreds of
cycles per second. Means are available for eliminating or rejecting
unwanted or unexpected frequencies within this range, and it is some
times advantageous to employ electrical filters and convolution methods
to extract the desired signals from the undesired signals. However,
the a priori rejection of unwanted signal frequencies and the elimina
tion of unexpected particle motion were not considered appropriate for
these experiments. Consequently, no filters, squelch circuits, or
other devices were used to limit the measurement capability of the
detecting and recording system.
Functional Components
The essential components of the measuring and recording system
were (l) a transducer that generated a voltage proportional to its ve
locity, (2) a shielded cable that conducted the generated voltage to
an amplifier, (3) an amplifier or attenuator that adjusted the trans
ducer voltage and provided a proportionate current to a galvanometer,
(4) a galvanometer with a current sensitive mirror that reflected a
beam of light onto moving photographic paper, (5) a regulated voltage
source, and (6) a voltmeter that measured a known reference voltage.
The functional relationships between these components is illustrated
in Figure 39- These components, and some auxiliary items, are briefly

Figure 38. Water hose inserted in borehole during backfilling
H
ro
00

Figure 39. Functional components of the particle velocity measuring system.

125
discussed in this section; detailed specifications and electrical con
nections are given in Appendix B.
Transducers
All of the particle velocity transducers used in the experimental
work were obtained from Mark Products, Inc., Houston, Texas. One trans
ducer component consisted of a spring attached to a free mass at one
end and to the transducer case at the other end; a coil of wire was
fastened to the mass and a permanent magnet was attached to the case.
When the case was moved, an induced voltage was generated in the coil
of wire that was proportional to the relative velocity between the coil
and the magnetic field developed by the magnet. A 3-'Component trans
ducer sensed particle velocity in the vertical (v), radial (R), and
tangential (T) directions and was composed of 3 sensing units in a
single housing.
Twenty-two transducers of the 3-component type and 2 transducers
of the single component type were used in the experimental work. The
3-component transducers were model L-1B-3DS and the single component
transducers were model L-1D. Figure 30 shows these two types of trans
ducers mounted on the test foundation.
Identification.--A serial number was stamped on the case of each
3-component transducer; the orientation of the radial and tangential
components of motion was indicated by an arrow engraved on the top of
the transducer case. The two single component transducers were not
identified by number, but were referred to as L-1D-BT (for the buried
transducer) and L-1D-TT (located on top of the test footing).
Brief specifications.--The 3-component transducers had a diameter

126
of 3 in. and a length of 12 in.; each transducer weighed 6 lb (a unit
weight of 122 lb/cu ft). The aluminum case and 150-ft-long conductor
cable were waterproof to a pressure of 1,000 psi and the maximum coil
movement was l/l6 in. The undamped natural frequency was 4.5 Hz, the
damping was 65 percent of critical, and the transduction was constant
from 10 to several hundred Hz.
The single component transducers were 2-3/8 in. in diameter, 1-1/2
in. deep, and weighed about l-l/2 lb. The maximum coil movement was
0.09 in., the undamped natural frequency was 4.5 Hz, the damping was
about 72 percent of critical, and the transduction was constant from
10 to several hundred Hz.
Location.The location of the transducers that were buried in
the sand deposit was specified by a radial distance, r from the
center of the footing and by a depth, z below the surface of the
ground; all distances and depths were measured in feet. Footing lo
cations were indicated by an asterisk.
Calibration.The ratio of the voltage generated by a transducer
to the velocity of the transducer case was called the transduction con
stant and was expressed in volts per in. per second. For the L-1B-3DS
transducers, the transduction tolerance given by the manufacturer was
plus or minus 5 percent. The validity of the transduction tolerance was
checked by calibrating 13 model L-1B-3DS transducers that had been in
field service. The 39 calibrations gave transduction values from 1.60 to
I.85 volts/in./sec and an arithmetic average transduction value of 1.74
volts/in./sec. The measured tolerance was about plus 6 percent and
minus 8 percent. The transduction tolerance of the L-1D transducers
was plus or minus 10 percent, as given by the manufacturer.

127
Table 14 lists the serial number of the transducer, its model des
ignation, its location, and its damped transduction constant.
Table 14
List of Transducers, Locations, and Transductpon Values
Location
Transduction in y
olts/ in./sec
Serial
Radial
Depth
Vertical
Radiajl Transverse
Number
ft
ft
Component
Component Component
Model
L-1B-3DS
Three-C omponent
Transducers
1
30.0
1.0
1.80
1.7C
) 1.70
3
30.0
5.0
1.70
1.7a
) 1.80
4
3-5
1.0
1.70
1.75
i 1.70
5
90.0
5.0
1.75
I.73
¡ 1.70
8
90.0
15.0
1.65
1.73
i 1.80
9
2.646
Footing
1.85
1.8 1.80
10
30.0
15.0
1.80
1.73
i 1.75
11
30.0
25.0
1.80
1.65
; 1.80
12
90.0
35.0
1.75
l.6c
) 1.70
13
30.0
35.0
I.65
1.75
¡ 1.75
14
10.0
5.0
1.80
1.8c
) 1.75
17
10.0
25.0
1.70
1.85
; 1.80
18
10.0
35.0
1.70
1.75
¡ 1.80
19
90.0
1.0
1.75
1.75
; 1.75
20
10.0
1.0
1.75
1.75
i 1.75
21
90.0
25.0
1-75
1.75
i 1.75
22
6o.o
35.0
1.75
1.75
1.75
23
6o.o
1.0
2.36
2.36
¡ 2.36
24
60.0
5.0
2.36
2.36
¡ 2.36
25
6o.o
25.0
2.36
2.36
i 2.36
26
6o.o
15.0
2.36
2.3
¡ 2.36
27
10.0
15.0
2.36
2.3
¡ 2.36
Model L-1D Single-Component Transducers
L-1D-BT
2.500
1.5


1.43
L-1D-TT
2.344
Footing
--

1.43

128
Cables
The voltage generated by the particle velocity transducers was
conducted from the test area to truck-mounted amplifying and recording
equipment by 1,000-ft-long instrumentation cables produced by the
Belden Manufacturing Company, Chicago, Ill., and designated as trade
number 8777- Eight cables, with pin connectors at each end, were
used; each cable consisted of six conductors, shields, and shield
grounds, and was adequate to transmit the signals generated by one
model L-1B-3DS 3-component transducer. The continuity and resistance
of each cable and conductor were checked by connecting a known resist
ance across each pair of conductors and measuring the circuit resist
ance at the other end of the cable.
Each conductor had a resistance of 0.016 ohms/ft; the amplifier
impedance was 1 megohm, and the transducer circuit had a resistance
of less than 1,200 ohms. The transducer voltage drop due to the cable
length was negligible.
Amplifiers
Model 1-165 amplifiers, manufactured by the Consolidated Electro
dynamics Corporation (CEC), Pasadena, Calif., were used. These dif
ferential amplifiers had solid state circuitry and gain steps from 10
to 1,000, and they had been designed specifically for driving mirror-
type galvanometers used in CEC recording oscillographs. The frequency
response was accurate to one percent, the full scale linearity was
within 0.25 percent, and the drift with a line voltage change of 10
percent was 0.5 percent. Considering the procedures used to measure
and record the particle velocities, the maximum error that could be

129
attributed to the amplifiers was 1.25 percent.
Figure 40 shows two rows of these amplifiers, six in the upper
row and seven in the lower row, mounted in a cabinet.
Oscillographs and galvanometers
Consolidated Electrodynamics Corporation oscillographs and gal
vanometers were used. Two type 5-H9P^ oscillographs, equipped with
fluid-damped, high-performance, type 7-364 galvanometers, recorded the
test data on 12-in.-wide, light-sensitive paper. The galvanometers
had a sensitivity of 0.397 milliamps/in., a useful frequency range from
0 to 500 Hz, a linearity error of less than 1 percent at full scale
deflection, and a frequency response error of less than 2 percent.
The paper speed in the oscillograph was 25.6 in./sec or 2.56 in/sec
and timing lines were printed at 0.01- and 0.1-sec intervals on Lino-
Writ 4 photorecording paper made by the E. I. du Pont de Nemours Co.,
Wilmington, Delaware.
Eleven active galvanometers were used in oscillograph A and 12
active galvanometers were used in oscillograph B. Figure 4l shows
oscillograph A without a dust cover and oscillograph B with a dust
cover.
Reference (calibration) voltage
A particle velocity transducer supplied an unknown, time-
dependent voltage to the input of the amplifier and the time history
of that voltage was recorded by an oscillograph on moving, light-
sensitive paper. To determine the magnitude of the unknown transducer
voltage, a known reference voltage was applied to the input of the
amplifier and recorded on light-sensitive paper. The magnitude of the

130
CABINET
SIX AMPLIFIERS
TWO VOLTAGE SUPPLY UNITS
SEVEN AMPLIFIERS
(f
%
Figure 1+0. Two rows of amplifiers mounted in a cabinet.

131

132
unknown voltage record was determined by comparing it to the magnitude
of the known voltage record. A record of unknown voltages was called a
test record and a record of known voltages was called a calibration
record.
Calibration voltage supply.--A Consolidated Electrodynamics Cor
poration model 3-140 power supply was used to provide a voltage source
for the calibration records. The direct current output voltage was
variable from 1 to 24 volts and the regulation was 3 parts in 1,000.
A resistance circuit across the output of the power supply extended
the measurable voltage range from 0.000005 to 2.4 volts. Figure 40
shows two voltage supply units next to the top row of amplifiers.
Calibration voltmeter.--An AC or DC differential voltmeter with
a null detector circuit was used to measure the calibration voltage
applied to the amplifier input. The solid state model 887AB voltmeter
was manufactured by Fluke Electronics, Seattle, Washington. It had an
accuracy of 0.005 percent of the measured input voltage plus 5 micro
volts. Figure 40 shows this unit resting on top of the amplifier
cabinet.
The maximum calibration voltage used in the experiments was 2.5
and the minimum was 0.00034, so the error in measuring the calibration
voltage varied from 0.005 to 1.5 percent.
System accuracy
The accuracy of the particle velocity measuring and recording
system was estimated from the results of the transducer calibration
measurements and the error or accuracy specifications for each compo
nent of the system.

133
Figure 42 is a histogram of the transduction constants determined
for 39 transducers. The standard deviation of these measurements was
0.06164 and the probable error in any one measurement was 2.4 percent
(Durelli, Phillips, and Tsao, 1958). The probable eror was about 4.1
percent for the L-ID transducers.
The frequency response errors in the amplifiers were assumed to be
normally distributed and the full scale linearity error of 0.25 per
cent was possible, so the probable amplifier error was about 0.5 per
cent. The same assumptions result in a probable galvanometer error
of 1.4 percent. Calibration voltage errors varied from 0.005 to 1.5
percent, depending on the measured voltage.
Amplitude and time measurements were scaled on developed oscillo
graph records. The scaling errors depend on the measured length and
probably averaged from 1 to 3 percent on the calibration records and
about the same value on the test records.
Considering the entire measurement system, from transducer to in
terpreted data, the estimated maximum error was probably between 7 and
15 percent.
Power generators
Electric power for operating the amplifiers, oscillographs, paper
processor, vibrator motor, and other electrical miscellanea was supplied
by three Kohler gasoline-engine-driven, 60-Hz generators. One 3-phase,
240-volt, 15-kw generator and one single phase, 120/p40 volt, 10-kw
generator were mounted on a trailer. The other generator was a small,
air-cooled, single phase unit that developed 3 kw at 120/240 volts.

NUMBER OF TRANSDUCERS
134
Figure 42. Histogram of measured transduction constants

Arrangement and Utilization
Arrangement of components
135
For a given steady state footing vibration test of constant fre
quency, the 22, 3-component transducers and the two single component
transducers generated a total of 68 time-dependent voltage signals.
Since only 24 amplifiers and calibration circuits were available, all
of the generated signals could not be recorded simultaneously. To
measure all of the signal voltages, four sets of transducer signals
were recorded in sequential order during one steady state, constant
frequency, footing vibration test; the footing motion was included in
al 1 of the records. Switching from one set of transducers to another
was accomplisned by changing transducer leads at the pin connections
between the transducer cables and the 1,000-ft-long instrumentation
cables. The transducers selected for simultaneous recording were lo
cated along a radial from the footing to facilitate an interpretation
of wave propagation velocities.
The active galvanometer traces appearing on the recording paper
were identified with respect to the oscillograph that recorded the data
and to the order of appearance on the recording paper. The numerical
order of appearance was from the top of the record to the bottom of the
record, considering the paper moved from left to right as the records
were made. The oscillograph printed its identification code and a se
quence number on each record produced.
The direction of the transducer movement with respect to the di
rection of the galvanometer trace movement was consistently related.
Positive movement of a 3-cpnent transducer was in the direction in
dicated by the radial and transverse arrows engraved on the housing;

136
positive vertical movement was upward. The positive radial direction
was away from the center of the footing and the positive transverse di
rection, in plan, was counterclockwise. The sense of the single com
ponent transducers was the same as the transverse component of the 3-
component transducers. Particle velocities in positive directions
caused the galvanometer traces to move toward the top of the record.
Table 15 lists the transducer serial number or model, the compon
ent of motion recorded, the cable number that carried the signal to the
recording truck, the oscillograph that recorded the motion, and the
position of the galvanometer trace on the photorecording paper. These
items are listed for each set of connections used to record one vibra
tion test. Connection details are given in Appendix B.
Utilization of system
This section describes the procedures and methods that were em
ployed to utilize the measuring system for determining the particle ve
locities generated by a torsionally vibrating footing.
Procedures.--The first set of transducers listed in Table 15 was
connected to the instrumentation cables, appropriate eccentric weights
were attached to the torsional vibrator, and the vibrator was started.
The speed or frequency of the vibrator was measured with an electronic
counter that sampled a transducer signal and displayed the period of the
vibration. The speed of the vibrator was adjusted by changing the gover
nor setting on the generator that powered the vibrator motor. When the
desired frequency was attained, the amplifier gains were adjusted and
trimmed until the observed movement of the galvanometer light beams.,
visible on the oscillograph viewing screen, had an amplitude of about

Table 15
Transducers, Recorders, and Recording Sequence
Oscillograph
A
B
Galvanometer
Cable
Transducer Serial/Model
No. and Motion
Component
Trace No.
No.
First Set
Second Set
Third Set
Fourth Set
1
1
9-T
9-T
9-T
9-T
2
4
20-R
14-R
10-R
27-R
3
4
20-T
14-t
10-T
27-T
4
5
1-R
3-R
25-R
17-R
5
5
1-T
3-T
25-T
17-T
6
6
23-R
24-R
22-R
18-R
7
6
23-T
24-T
22-T
18-T
8
7
19-R
26-R
21-R
11-R
9
7
19-T
26-T
21-T
11-T
10
8
5-R
8-R
12-R
13-R
li
8
5-T
8-T
12-T
13-T
l
1
9-v
9-v
9-v
9-v
2
1
9-R
9-R
9-R
9-R
3
2
4-v
4-v
4-v
4-v
4
2
4-R
4-R
4-r
4-R
5
2
4-t
4-T
4-t
4-t
6
3
L-1D-BT-T
L-1D-BT-T
L-1D-BT-T
L-1D-BT-T
7
3
l-id-tt-t
L-ID-TT-T
L-UD-TT-T
L-ID-TT-T
Q
14-V
0
V
-tu-V
27-V
9
5
1-V
3-v
25-V
17-v
10
6
23-V
24-V
22-V
18-V
li
7
19-v
26-V
21-V
11-V
12
8
5-V
8-V
12-V
13-v
Note: V is vertical, R is radial, and T is transverse.
H
(JO
^3

138
1 in. When the 11 light beam adjustments were completed on oscillograph
A, and 12 on oscillograph B, the vibrator was stopped.
A DC reference or calibration voltage for each transducer channel
was applied to the amplifier input and adjusted to deflect each light
beam about 1 in.; each transducer channel had a separate calibration
voltage circuit. The calibration voltage for each transducer was
measured with a precision voltmeter, read, and manually recorded on a
data sheet. The oscillograph was started and run at a paper speed of
2.56 in./sec while the calibration voltages were sequentially applied
to the input of the amplifiers. The oscillograph record produced by
these operations was called a calibration record.
After the calibration records were recorded, the vibrator was
started, the vibration frequency was checked, and two test records of
the transducer signals were recorded at a paper speed of 25-6 and
2.56 in/sec. The vibrator was then stopped, the eccentric weights
were removed from the vibrator, and the vibrator was started. Another
test record of the transducer signals was recorded at a paper speed of
25.6 in./sec. The vibrator was stopped, the second set of transducers
was connected to the instrumentation cables listed in Table 15, the
eccentric weights were attached to the vibrator, and the entire se
quence of events outlined above was repeated.
The transducer array and the measurement and recording system were
also used to determine the propagation velocity of compression waves
through the soil deposit. This was accomplished by tapping the test
foundation or the ground surface with a hammer and recording the re
sulting transducer motion with the oscillographs.
Methods.The magnitude of the peak particle velocity detected by

139
each transducer was determined by comparing the calibration record to
the test record. Wave propagation velocities between adjacent trans
ducers were determined by the arrival time of the same phase of motion
measured on the oscillograph record.
Exposed, light-sensitive, photorecording paper was developed and
dried in a chemical bath processor and the displacement of the galva
nometer trace on the calibration record was measured with a ruler.
The test record with eccentric weights on the vibrator was compared to
the test record without eccentric weights on the vibrator; the differ
ence between these two test records was attributed to the particle ve
locity generated by -the torsional vibration of the test footing. The
peak-to-peak (double) amplitude of the galvanometer trace due to the
eccentric weights on the vibrator was measured with a ruler on the test
record. Transducer detected particle velocity was determined from the
expression
V D 1
d 2 T
(in./sec) (146)
where V is the impressed calibration voltage, d is the trace dis
placement measured on the calibration record, D is the peak-to-peak
amplitude of the trace movement measured on the test record, and T is
the transduction constant of the transducer.
The velocity of the seismic wave that propagated away from a
source of motion was determined from the travel time measured on the
test record and the known position of the transducers. Timing lines
were printed on the test records by the oscillographs at 0.01- and
0.1-sec intervals; these lines were used to reference and measure the
time lapse between events on a single oscillograph test record. A

i4o
continuous time reference between oscillographs was obtained by record
ing the transverse footing motion on each oscillograph and by a manual
step pulse applied simultaneously to one inactive galvanometer in each
oscillograph. The propagation velocity of compression waves was com
puted from the arrival time of the disturbance at each transducer.
Typical vibration test data
Procedures and methods used to obtain particle velocity measure
ments and records were discussed in the preceding paragraphs. This
section illustrates typical data obtained during a torsional vibration
test at a frequency of 50 Hz.
Calibration records.--Typical calibration records for oscillo
graphs A and B are shown in Figures 43 and 44, respectively. The
labeled galvanometer traces give the oscillograph record number, the
oscillograph identification and galvanometer trace (or channel) num
ber, and a group of four symbols that identifies the serial number
(s/n) of the transducer, the component (CMP) of motion recorded, the
distance (r) of the transducer from the center of the footing, and
the depth (z) of the transducer below the ground surface.
Test with eccentric weights.Figures 45 and 46 show the oscillo
graph records that were obtained during a 50-Hz torsional vibration
test with eccentric weights attached to the flywheels of the vibrator.
Test without eccentric weights.--The eccentric weights were re
moved from the vibrator and the vibrator was operated at the same 50-Hz
frequency while the particle velocity perturbations and the ambient
measuring system disturbances were recorded. Figures 47 and 48 illus
trate typical test records obtained when the vibrator was operated
without the eccentric weights.

'REG. NO.; CHAN. NO.; TRANS(S/N, CMP, R, Z)
152; Al; 9, T, 2.6, *
ON FOOTING
152; A2; 14, R, 10, 5
152; A3; 14, T, 10, 5
152; A4; 3, R, 30, 5
-Siiar.ja-.
152; A 5; 3, T, 30, 5
Urn*
152; A 6; 24, R, 60, 5
1152; A 7; 24, T, 60, 5
-L 152; A8; 26, R, 60, 15
' /o m
ft********************' -
152; A 9; 26, T, 60, 15
152; AIO; 8, R, 90, 15
152; All; 8, T, 90, 15'
Figure 43. Typical calibration record, oscillograph A, before 50-Hz vibration test.

^_REC. NO.; CHAN. NO.; TRANS(S/N, CMP, R, Z)
152; Bl; 9, V, 2.6, on footing
I52;B2;9,R,2.6,* on footing
3 *-!
0.173"
152; B3; 4, V, 3.5, I
4' -
/.os"
..152; B4; 4, R, 3.5, I
152; B5; 4, T, 3.5, I **r -
i.152; B6; l-D, T, 2.5, 1.5
152 B 7; l-D, T, 2.3, *
ON FOOTING
152; B8; 14, V, 10, 5
T hit
3J52; B9; 3, V, 10, 5
aoooOCCWWOO^OaoOCiOCqpCl52; BIO; 24, V, 60, 5
152; Bl I; 26, V, 60, 15 152; BI2; 8, V, 90, 15
Figure 44. Typical calibration record, oscillograph B, before 50-Hz vibration test.

(JO

JREC. NO.; CHAN. NO.; TRANS(S/N, CMP, R, Z)
I I A I LI .1
,153; Bl; 9, V. 2.6,**,
W'l'l'IMflJIl i s
153; B2; 9, R, 2.6, *i / f
fUNNIS \
MMJiMl
153; B3 ; 4, V, 3.5, I
hmiiiMi
Ll 5 3 ; B 4 ; 4, R, 3.5, I ^
153; B5 ; 4, T, 3
, VJ VJ VJ VJ YJ YJ YJ'
153; B6 ; l-D, T, 2.5, 1.5
mimmww
mimiiiMU
^ 153; BIO; 24, V, 60, 5
' I53 Bl I; 2 6, V,6 0 15A T /y]M
a! IJ. \J. \I L£LULU \j JU
, 153; BI2; 8, V, 90, I5U^,
to mmimriP
9tft.fci
Figure 46. Typical test record at 50 Hz, oscillograph B, with eccentric weights on vibrator.
-p-

Uj
REC. NO.; CHAN. NO.; TRANS (S N, CMP, R, Z)
i i i i i i i i i i i i i i
155; AI; 9, T, 2.6, *
155; A2 ; 14, R, 10, 5
^155; A3; 14, T, 10, 5
155; A4; 3, R, 30,
I I I I I I I I I I I II I I
A/W 155; A5; 3, T, 30, 5
U4^4-41J41
155; A 6; 24, R, 60, 5
w
VA
Ml!
155; A 7; 24, T, 60, 15
155; A 8; 26, R, 60, 15
155; A9; 26, T, 60, 15
.1 : 1
'
¡ r 1
A1
1 1
; 8,
NOTE: TEST RECORD SHOWS WAVE FORMS,
NOT AMPLITUDE COMPARISONS.
jyM"\ \ps
Figure 47. Typical test record at 50 Hz, oscillograph A, without eccentric weights on vibrator.
-p-
VJI

-155; B 5; 4, R, 3.5, I.
FT i i i i n i iir tiiir~rir~irii r r~rT t t L J I T F
REC. NO.; CHAN. NO.; TRANS (S/N, CMP, R, Z)
H I *J IA I L I J Ul LI I I f,
'>55; B I; 9, V, 2.6, *
:i i! j in i r 11 m i
>55; B2; 9, R, 2.6, *
II
155; B 3; 4, V, 3.5,
155; B4; 4, R, 3.5, h
/W'
'-155; B6 ; l-D, T, 2.5, 1.5'
'-155; B 7; l-D, T, 2.3, *1
i/|l55|b81_I41 JO, 5A(\(\|\|/|/I
jV155; B9 ; 3, V, 10, 5 y/\A\^V^//\A
^155; BIO; 24, V, 60, 5
.11II 1.1.111II 1.11.1
v
AI55; BI I; 26, V, 60, I5VN/J\A\
I I I I II I I
fJI55; BI2; 8, V, 90, 15lU^vKyKjMv/W/\Ak>>/1w,MjA^
iiiiiim
f.
V',
yy\rwVr
vV,/
M/v/X^ ajv
M
AA\j VJAfil\\\)\JT
'/7.
aUVM
M
A /W>A
w
/VVvW/M^
lA
Iv/^/'YW /'A
/'M^WWW^A^W^v./hW,
A A A A
/M
\v#
//W
A
NOTE: TEST RECORD SHOWS WAVE FORMS,
, NOT AMPLITUDE COMPARISONS.
/\ A ^ ^ ^ A A \ ^ V k/^ A AAA/NAr|/yY\/\^M/W\A>pv|Vluy\IA/\/MVA/K
A A a
ON FOOTING
M Af
-V
n."
ON FOOTING1
/M
\A/V\aA/WW
A/ifV
ON FOOTING-
/u
\W
A/\ ImI A/*
aIaIaLa/
i/AA\
K/ H
Figure 48. Typical test record at 50 Hz, oscillograph B, without eccentric weights on vibrator.
F-*
-P"
ON

147
Schedule of Tests
Footing Settlement and Tilt
When sand materials with relative densities less than 70 percent
are subjected to vibration, densification of the sand and surface set
tlement are expected (D'Appolonia, Whitman, and D'Appolonia, 1969).
The relative density of the sand deposit at the Piccolo field test site
was about 62 percent, so settlement of the vibrating test footing was a
distinct possibility.
Settlement and tilt measurements were made periodically during
the entire six-week testing program. Figure 49 shows a level rod po
sitioned for measuring the elevation of the center of the vibrator
frame and Figure 50 shows an elevation measurement at one of four loca
tions around the top edge of the footing. Twenty-six elevation and
tilt measurements were conducted during the six-week test period.
A carpenter's level was used to check the tilt of the footing.
Figures 51 and 52 show this level on the frame of the vibrator and
Figure 53 shows it on the top surface of the footing. Results of the
settlement measurements are plotted in Figure 54.
Transducer Operation
The operation of the 3-conrponent transducers was tested by mount
ing each transducer on the test foundation, operating the torsional
vibrator, and recording the transducer signals with an oscillograph.
The oscillograph record was examined to insure that the amplitude and
phase of the transducer signal compared favorably with the L-1D-TT
transducer bonded to the top of the test foundation.
After testing the transverse component, the transducer was

i48


Figure 51. Tilt check with level along vibrator frame.
150


Figure 53. Tilt check with level on top of test footing.
VJ1
ro

CHANGE IN ELEVATION
1 6 11 16 21 26
ELEVATION READING NUMBER
Figure 54. Results of footing settlement and tilt measurements.

154
rotated. 90 degrees in its mounting bracket to locate the radial com
ponent of motion tangent to the circular footing. The vibrator was
started and a signal from the radial component of the transducer was
recorded. No satisfactory means of testing the vertical component was
found.
While connected to the recording oscillograph, the transducers
were tapped lightly with a wooden survey stake to relate the direction
of movement of the transducer to the movement of the galvanometer
trace.
The quantitative results of the transducer operation tests and
the directionality tests are not included in this study.
Torsional Vibration
The torsional vibrator was operated at five different frequencies
and with various eccentric weights during the conduct of the particle
velocity measurement program. Table 16 gives the frequency of the tor
sional oscillation, the horsepower rating of the motor driving the vi
brator, the eccentric weight used in each flywheel of the vibrator, the
maximum time-dependent moment applied to the footing, and the oscillo
graph record numbers obtained for each test situation. Table 16 indi
cates that torsional moments greater than the footing design moment, but
less than the critical moment, were applied to the test footing during
the high frequency vibration tests. A greater moment was needed to
produce a useable oscillograph record of the particle velocities gen
erated at frequencies of 40 and 50 Hz.

155
Table 16
Schedule of Torsional Vibration Tests
Frequency
Hz
Motor
hp
Eccentric
Weight
lb
Moment
ft-lb
Oscillograph
Record
No.
15
1
5.7768
1,380
188-195,
257-264
20
1
3.5378
1,502
62-78, 85-
88, 82
30
1
1.5387
1,470
27-29, 35-
4l, 89-96
4o
2
1.5387
2,6i4
118-135,
166-174,
249-256
50
2
1.0617
2,818
148-163
Compression Wave Propagation
The propagation velocity of compression waves through the buried
transducer array was measured by recording the arrival- time of an im
pulsive disturbance at known transducer locations. A hammer blow on
the test foundation was detected by each set of transducers and re
corded by an oscillograph; Figure 55 shows the technique that was used.
A hammer blow on a 3-in.-wide, 10-in.-long, l/4-in.-thick steel plate
located on the ground surface directly above each vertical set of trans
ducers was also detected and recorded. The results of these tests
are shown in Figure 56.
Transducer and Cable Resistance
At the conclusion of the test program, the resistance of each

Figure 55. Compression wave initiated by striking footing
vn
ON

o
5
10
15
20
25
30
35
6.
157
COMPRESSION WAVE VELOCITY IN FPS
500 1,000 1,500 2,000 2,500
\
\
s
\
\
\\
\

\ \
\
\
V
o HAMMER BLOW ON FOOTING
Q HAMMER BLOWS ON GROUND
SURFACE (AVERAGE VELOCITY)
S
\
\
\
\
\\
v*,
Compression wave velocities from hammer blows on footing and
on ground surface.

158
transducer component was measured at the transducer cable connection to
assure that the transducer had not been damaged and that it had per
formed consistently throughout the test program. Each length of instru
mentation cable was also checked for continuity and resistance. The
measured transducer resistance values appear1 in Appendix B.
Results of Measurements
Footing Settlement and Tilt
The settlement and tilt of the test footing was measured with a
transit, level rod, and carpenter's level. Measurements with the car
penter's level indicated that the footing did not tilt during the entire
program of footing vibration tests. Figure 54 gives the results of the
elevation measurements in a plot of elevation change vtersus sequential
measurement number. The test footing probably settled about l/l6 in.
during the torsional vibration tests.
Compression Wave Propagation
Footing source
Compression waves initiated by a vertical hammer blow on the top
surface of the footing were detected with the transducers listed in
Table 15. Amplifier gains were adjusted to provide a clear arrival
time on the oscillograph record; calibration was not attempted. Table
17 lists the oscillograph record number, the recording oscillograph
and trace identification, the serial number and component of the de
tecting transducer, and the first arrival time of the detected particle
velocity for these tests. The arrival time values are averages of the
data from at least three hammer blows and measurements by two different
oscillogram readers.

Table 17
Results of Footing Source Compression Wave Tests
159
Oscillograph
Record No.
247
239
238
237
Oscillograph
and Trace
Identification
B-l
B-8
B-9
B-10
B-ll
B-12
A-l;
B-l,2
A-2
A-4
A-6
B-ll
A-10
A-l;
B-l,2
b-8
A-4
A-6
A-8
A-10
A-l;
B-l,2
B-8
A-4
B-10
A-8
A-10
Transducer No.
and
Component
9-v
20-V
1-V
23-V
19-V
5-V
9-V,R,T
14-R
3-R
24-R
2 6-V
8-R
9-V,R,T
10-V
25-R
22-R
21-R
12-R
9-V,R,T
27-V
17-R
18-V
11-R
13-R
Time of
First Arrival
sec
0.0000
0.0081
0.0280
0.0550
O.O83O
0.0810
0.0000
0.0135
O.O36I
0.0612
0.0547
0.0745
0.0000
0.0344
0.0524
0.0555
0.0723
0.0728
0.0000
0.0160
0.0210
0.0243
0.0320
0.0342
Surface source
Compression waves, initiated by vertical hammer blows on a steel
plate located on the surface of the ground and just above the uppermost
buried transducer, were detected and recorded at radial distances of
10, 30j 80, and 99 ft from the footing. The oscillograph record number,
the recording oscillograph and trace identification, the serial number
and component of the detecting transducer, and the first arrival time

i6o
of the detected particle velocity for these tests are given in table 18.
Propagation velocities
Figure 56 is a plot of the compression wave propagation velocity
versus depth as determined by the footing source test and the surface
source tests.
Table 18
Results of Surface Source Compression Wave Tests
Oscillograph
Record No.
283
281
280
279
Oscillograph
and Trace
Identification
A-2,3; B-8
A-4,5; b-9
A-6; B-10
A-8
A-10
A-2,3; b-8
a-4,5; b-9
a-6,7; B-10
a-8,9; B-11
A-10,11;B-12
b-8
B-9
B-10
B-11
B-12
A-2,3; B-8
a-4,5; b-9
a-6,7; B-10
a-8,9; B-11
A-10,11; B-12
Transducer No.
and
Component
20-V,R,T
l4-V,R,T
27-V,R
17-R
18-R
1-V,R,T
3-V,R,T
10-V,R,T
11-V,R,T
13-V,R,T
23-V
24-V
2 6-V
25-V
22-V
19-V,R,T
5-V,T,T
8-v,r,t
21-V,R,T
12-V,R,T
Time of
First Arrival
sec
0.0000
0.030
0.022
0.021
0.025
0.0000
0.0115
0.0200
0.0250
0.0290
0.0000
0.0115
0.0240
0.0225
0.0240
0.0000
0.0280
0.0150
0.0210
0.0285
Particle Velocities Due to Torsional Vibration
Amplitudes
The results of the particle velocity measurements on the foot
ing and in the soil deposit are given in Tables 19 through 23.

i6i
Table 19
Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,380 ft-lb Oscillating at 19 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
0.00723
0.0119
0.206
2.344
Footing
None
None
0.177
2.5
1.5
None
None
0.124
3.5
1.0
0.00373
0.00463
0.0827
10.0
1.0
0.00174
0.00105
0.0190
30.0
1.0
0.00152
0.00268
0.00949
60.0
1.0
0.00111
0.000542
O.OO663
90.0
1.0
O.OOO565
0.000513
0.00430
90.0
5.0
0.000344
0.000628
0.00335
10.0
5.0
0.00102
0.00129
0.0139
30.0
5.0
0.000679
0.000377
0.00588
60.0
5.0
O.OOO56O
0.000417
O.OO356
60.0
15.0
o.oooo846
0.0000627
0.000523
90.0
15.0
0.000111
0.0000473
O.OOO513
30.0
15.0
0.000194

O.OOO688
60.0
25.0
0.0000226
0.0000540
0.000154
60.0
35.0


0.000112
90.0
25.0
0.0000463
0.0000397
0.000138
90.0
35.0
0.0000293
0.0000160
0.000130
10.0
15.0
--
0.000421
0.00168
10.0
25.0


o.ooo466
10.0
35.0


0.000219
30.0
25.0
0.0000832
o.oooo485
0.000318
30.0
35.0


0.000182

162
Table 20
Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,502 ft-lb Oscillating at 20 Hz
Peak Amplitude in Inches per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
0.0320
0.0286
1.043
2.344
Footing
None
None
0.8470
2.5
1.5
None
None
0.6084
3-5
1.0
0.0128
0.0207
0.472
10.0
1.0
0.0171
0.00777
f
0.102
30.0
1.0
0.00832
0.0121
0.0592
60.0
1.0
0.00724
0.00353
1
0.0303
90.0
1.0
0.00191
0.00226
0.0189
90.0
5.0
0.000770
0.00482
>
0.0109
10.0
5.0
0.00829
0.00931
0.0597
30.0
5.0
0.00523
0.00818
0.0227
60.0
5.0
0.00272
0.00386
0.0134
60.0
15.0
0.000765
0.00111
0.000932
90.0
15.0
0.000832
O.OOO858
0.00137
30.0
15.0
O.OOO663
o.ooi48
0.000768
60.0
25.0
0.000284
0.000280
0.000679
60.0
35.0
0.000557
0.000437
0.000393
90.0
25.0
0.000552
0.000313
0.000312
90.0
35.0
0.000428
0.00018
2
O.OOO365
10.0
15.0
0.00192
0.00254
0.00611
10.0
25.0
0.00117
0.00097
8
0.00106
10.0
35.0
0.000782
0
00
0
0
0
0
'2
0.000620
30.0
25.0
0.000319
0.000255
O.OOO677
30.0
35.0
O.OOO386
O.OOO38O
0.000734

163
Table 21
Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,470 ft-lb Oscillating at 30 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Rooting
0.0115
O.O316
0.649
2.344
Footing
None
None
0.503
2.5
1.5
None
None
0.385
3.5
1.0
0.0135
0.0261
0.297
10.0
1.0
0.00772
0.00598
O.0655
30.0
1.0
0.00262
0.00301
0.0513
60.0
1.0
0.000930
0.00468
0.0210
90.0
1.0
0.00374
0.00412
0.01057
90.0
5.0
0.000861
0.00303
0.0101
10.0
5.0
0.00419
0.0015$
0.0454
30.0
5.0
0.00245
0.00122
O.OI67
60.0
5.0
0.00178
0.00140
0.00922
60.0
15.0
0.000593

0.00151
90.0
15.0
0.000511
0.000476
0.00202
30.0
15.0
0.000500

0.00288
60.0
25.0
0.000126
0.000442
0.000161
60.0
35.0
0.000239
0.000213
0.000411
90.0
25.0
0.000125
0.000267
0.000817
90.0
35-0
0.000170
0.000180
O.OOO668
10.0
15.0
0.000550
0.00115
0.00488
10.0
25.0
0.000576
0.000489
0.000771
10.0
35.0
0.000522
0.000244
0.000632
30.0
25.0
0.000112
0.000131
o.ooo846
30.0
35.0
0.000236
0.000180
0.000419

164
Table 22
Particle Velocity Amplitudes Generated, by a Torsional
Moment of 2,6l4 ft-lb Oscillating at 40 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
O.OI98
0.0249
O.895
2.344
Footing
None
None
O.588
2.5
1.5
None
None
0.457
3.5
1.0
0.0119
0.0286
0.3509
10.0
1.0
0.0111
0.0109
O.O986
30.0
1.0
0.00594
0.0217
0.0355
60.0
1.0
0.00330
0.00497
0.0259
90.0
1.0
0.00168
0.0186
0.00323
90.0
5.0
0.00168
0.00160
0.00712
10.0
5.0
0.00952
0.00860
0.0380
30.0
5.0
0.00557
0.00328
0.0231
60.0
5.0
0.00197
0.00173
0.00708
60.0
15.0
0.00102
0.000524
0.00078
90.0
15.0
0.000762
0.000323
O.OOI98
30.0
15.0
0.000732
0.000828
0.00173
60.0
25.0
0.000337
0.000236
0.000266
60.0
35.0
O.OOO386
0.000272
O.OOO517
90.0
25.0
0.000172
0.0000968
0.000391
90.0
35.0
0.000319
0.000213
0.000236
10.0
15.0
0.000959
0.0022$
0.00405
10.0
25.0
O.OOO563
O.OOO76O
0.000787
10.0
35.0
0.000517
o.ooo484
0.000375
30.0
25.0
O.OOO3O3
0.000201
0.000785
30.0
35.0
O.OOO356
0.000443
0.000673

165
Table 23
Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,8l8 ft-lb Oscillating at 50
Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
0.0168
0.0268
O.87O
2.344
Footing
None
None
0.456
2.5
1.5
None
None
0.393
3-5
1.0
0.00793
0.0156
0.324
10.0
1.0
0.0120
O.OI69
0.0948
30.0
1.0
0.00554
0.0210
0.0395
60.0
1.0

0.00351
0.0101
90.0
1.0
0.00107
O.OO354
0.00584
90.0
5.0
O.OOO87O
O.OOO333
0.00402
10.0
5-0

0.00462
0.0291
30.0
5.0
--
0.00379
0.0162
60.0
5.0
0.00193
0.00123
0.00822
60.0
15.0
0.000233
0.000420
0.00180
90.0
15.0
0.000467
0.000527
0.000801
30.0
15.0
0.000426
--
0.00388
60.0
25.0
0.000175
0.000181
0.000745
60.0
35.0
0.000304

0.000991
90.0
25.0
0.000716
0.000186
0.000453
90.0
35.0
0.000198
0.000209
0.000451
10.0
15.0
0.00154

0.00710
10.0
25.0
0.000630

0.000594
10.0
35.0
0.000766

0.000686
30.0
25.0

0.000221
0.000583
30.0
35.0

0.000422
0.00159

166
The frequency of the oscillatory torsional moment applied to the test
footing, the peak value of the applied moment, and the peak amplitude
of the particle velocity generated by the applied torsional moment
are listed for each table, as well as the location of the transducer
and the component of motion that was measured. A dash (--) entry in
these tables indicates that there was no significant difference be
tween test records taken with and without eccentric weights on the
vibrator.
Wave propagation velocities
The propagation velocity of waves developed by the steady state
torsional vibration of the test footing was determined by comparing
the phase of the transverse component of footing motion to the phase
of the transverse component of transducer motion. The time interval
to establish the same phase angle at the buried transducers was meas
ured on an oscillograph record and the time for the wave to propagate
from one transducer location to another was determined.
The arrival time for the same phase of motion was used to estab
lish the propagation velocity between transducers located at a radial
distance of 10 and 30 ft from the center of the footing. Table 24
lists the arrival time and calculated propagation velocities between
adjacent transducers for vibration tests at 15, 20, and 30 Hz. The
transducer spacing was too large to interpret the phase relationships
for higher frequency vibration tests. Figure 57 is a plot of the
average propagation velocity versus depth compared, again, with the
results of empirical shear wave velocity equations (Richart, Hall, .
and Woods, 1970)-

5
10
15
20
25
30
35
167
SHEAR VIBRATION PROPAGATION
VELOCITY IN FPS
250 500 750 1,000
57. Shear vibration propagation velocities
versus depth.

168
Table 24
Measured Arrival Time and Average Wave Propagation
Velocity for Vibration Tests
Transducer
Location
Radial Depth
ft ft
Arrival Time in Seconds
15-Hz Test 20-Hz Test 30 Hz-Test
Average
Velocity
fps
Depth
ft
10.0
1.0
0.0170
0.0225
0.0245
395
3
30.0
1.0
0.0620
0.0775
0.0823
383
5
30.0
5-0
0.0620
0.0755
0.0833
627
10
10.0
5.0
0.0l60
0.0255
0.0270
524
15
30.0
15.0
0.0600
0.0595
0.0583
963
20
10.0
15.0
0.0255
O.O3O5
0.0300
655
25
30.0
25.0
0.0570
O.O56O
0.0550
848
39
10.0
25.0
0.0375
O.385
0.0395
30.0
35.0
0.0600
0.610
0.0585

COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS
Test of the Calculated Results
Computed particle displacements, generated by the: torsional oscil
lation of a rigid circular disk on the surface of an elastic half-space
were discussed in a previous section. The validity of| the computed
results was tested by comparing them with some accepted solutions and
known laws governing wave propagation. The motion of the surface of
the half-space under the rigid disk was compared to published work,
and the geometrical damping law for body waves was used to test the
calculated motion within the half-space.
Solutions at the Surface of a
Homogeneous (Constant E) Elastic Half-Space
As mentioned in a previous section, solutions for the torsional
oscillation of a weightless rigid disk on the surface if a homogeneous
elastic half-space are not lacking (Reissner and Sagocp, 1944; Bycroft,
1956; Stallybrass, 1962, 1967; Thomas, 1968). Since Stallybrass (1962)
tabulated the results of his work and that of Reissner and Sagoci
(1944), a comparison with these results was convenient]
The expression for the particle displacement developed from
Equation 65 was
^l(real) + il(imaginary)J
(147)

170
and the form that Stallybrass (1962) used was
v(r ) = 9Me g (h + ih ) (148)
0 l6Gr 1 2
o
To compare the real and imaginary parts of these two displacement func
tions at the surface of the half-space, let
"l + ih2 = 5? + iIi(ro)] (l49)
= I'(ro) + iI'(ro) (150)
Because stresses, rather than displacements, were used as the boundary
conditions under the rigid disk, there is some distortion of a radius
on the loaded area; Bycroft (1956) devised a complex averaging method
that gave peripheral displacements of the disk in close agreement with
the Reissner and Sagoci (1944) results.
An averaging technique appeared unnecessary for testing the
validity of the developed calculations, so two values of the edge dis
placement function were computed and compared to the published results--
the value of I'(ro) and the value 2I(r /2) Table 25 gives the
real part and imaginary part of the edge displacement function for
various values of the frequency ratio, a^ An inspection of the tab
ulated values indicated that the computed values of I'(r ) are less
than the published values and the computed values of 2I'(r /2) are
greater than the published values. Thus, the calculated displacement
of the edge of a disk on the surface of the half-space was in reason
able agreement with published solutions.

Table 25
Computed and Published Values of the Displacement Function
171
a
0
2I'(ro/2)
Reissner
and Sagoci
Stallybrass
Real Part of the Displacement Functioii
0.20
0.33595
0.33598
0.36
0.3373
0.3007
0.48
0.3473
0.3120
0.50
0.34872
0.34869
0.72
0.3688
0.3318
0.80
O.36769
0.36769
O.96
0.3913
0.3490
1.00
1.20
0.4129
0.3804l
0.38042
0.3614
Imaginary
Part of the Displacement Function
0.20
-0.00037
-0.00037
0.36
-O.OO217O
-0.002149
0.48
-O.OO509O
-0.005004
0.50
-O.OO566
-0.00563
0.72
-0.01666
-0.01603
0.80
-0.02175
-0.02180
0.96
-O.O3785
-0.03529
1.00
-0.04007
-o.o4oo4
1.20
-O.O6995
-0.06269
Geometrical Damping Law
Dilatational waves are not produced by the torsional oscillation of
a rigid disk on an elastic half-space, and no free or surface waves
exist (Bycroft, 1956). Radiation of energy away from ¡the vibrating
disk is due to shear waves (Richart, Hall, and Woods, ¡1970) These
shear or body waves are governed by a geometrical damping law that
relates particle displacement amplitudes on a propagating spherical wave
front. At large distances from the source, the amplitude of particle
motion is inversely proportional to the source distance (Woods, 1968);

172
this damping law stems from the conservation of energy along a spheri
cal wave front.
The validity of the computed particle displacements was tested by
applying the geometrical damping law. At a vibration frequency of
30 Hz, the displacement at a distance of 60 ft and a depth of 25 ft was
v(60,25) = (0.008150) (151)
4nGr
o
At the same frequency, and at a distance of 84 ft and a depth of 35 ft,
the displacement was
v(84,35) -^-p (0.005817) (152)
4nGr
o
The damping law for body waves requires that these two displacements be
related by
v(84,35) ---6 + v(60,25) (153)
^842 + 352
Cancelling common terms and substituting in the above equation
0.005817 si || (0.00815)
Cl 0.005821 (154)
The calculated displacement amplitudes agreed with the geometrical
damping lawthe difference was less than 0.1 percent.
Since the calculated surface displacements agreed with published
results and the body wave particle displacements conformed to known
geometrical damping laws, the computed field of half-space motion was
considered valid; calculated motion could be confidently compared to
experimentally measured motion.

173
Position of Disk and Footing
The calculated displacement results are for a rigid disk on the
surface of a half-space. The footing, however, was buried 1 ft below
the surface of the ground. The calculated particle displacement at a
radial distance of 3-5 ft and a depth of 1 ft was 11.7 to 15.7 percent
less than the surface displacement at the same radial distance. The cal
culated displacement at a radial distance of 10 ft and a depth of 1 ft
was 1.0 to 1.4 percent less than the surface displacement at the same
radial distance. Less error occurs at greater distances and depths.
Since the elevation of the bottom of the footing was the same as
the transducer located at a radial distance of 3-5 ft, the calculated
surface motion at a radial distance of 3-5 ft was used to predict the
motion of this transducer. The difference between the location of the
disk and the location of the footing had a small to negligible influ
ence on the calculated particle displacements in the half-space; the
error involved in a comparison between the calculated motion and the
measured motion was considered insignificant.
Test of the Measured Results
This section interprets some of the test measurements that were
related to the dynamic behavior of the footing, to the assumed footing-
soil interface conditions, to the components of particle motion in the
sand deposit, and to a more complete assessment of the soil properties.
Dynamic Footing Response
The design of the footing anticipated that a torsional moment of
1,400 ft-lb would produce a 0.001-in. displacement at the edge of the

174
footing, that the resonant frequency would be 40 Hz, and that the mag
nification factor at resonance would be about five. Table 26 lists the
moment applied to the footing, the peak displacement at the edge of the
footing, and the frequency of the oscillations. The ^xact resonant
Table 26
Measured Motion of the Test Footing
Vibration
Frequency
Hz
Moment Applied
to the Footing
ft-lb
Displacement at
Edge of the
Footing
in.
15
1,380
0.00201
20
1,502
0.00719
30
1,470
oj00285
40
2,6l4
0.00249
50
2,818
0.00155
frequency of the footing was not determined because the torsional vi
brator was driven by a constant speed motor and the moment applied to
the footing was limited. Figure 58 is a plot of the vibration fre
quency versus the ratio of the footing displacement to the applied
moment. Table 26 and Figure 58 indicate that the footing displacements
were about twice the anticipated values, the resonant frequency was
perhaps half the anticipated value, and the magnification ratio was
more than three.
Footing-Soil Contact Area
A torsionally vibrating footing transmits its motion to the sup
porting soil by contact friction at the footing-soil interface. The

175
FREQUENCY IN HZ
Figure 58. Frequency response of test footing.

176
possibility of slippage on the contact area was recognized and means
for detecting and measuring the slippage were provided. A particle
motion transducer was buried 6 in. below the edge of the footing to
measure the soil motion and an identical transducer was mounted on the
footing to measure its motion. Slippage between the footing and the
soil was indicated if the ratio of the footing displacement to the soil
displacement changed appreciably during the experimental program.
Figure 59 shows the footing/soil displacement ratio plotted versus
(l) the vibration frequency of the footing and (2) th displacement of
the edge of the footing. The measured footing/soil displacement ratio
decays smoothly with increasing frequencies--in agreement with the
computed behavior of an elastic half-space. The measured footing/soil
displacement ratio changes erratically with footing displacement; the
ratio has no apparent correlation with footing displacement.
Since the measured footing/soil displacement ratio was unrelated
to footing displacement and the ratio agreed with halff-space calcula
tions, good contact between the footing and the sand was evident and
transmission of elastic stresses into the sand deposit was indicated.
Particle Motion Components
A rigid disk, in torsional oscillation on the surface of a homo
geneous elastic half-space, generates half-space particle displacements
in a direction that is tangent to the circumference of the disk; no
radial or vertical motion is developed. In the experimental program,
particle motion in the sand deposit due to the torsional oscillation
of the footing was measured in the transverse (tangential), radial, and

FOOTING DISPLACEMENT/SOIL DISPLACEMENT
FOOTING DISPLACEMENT IN MILS
15 20 30 40 50
VIBRATION FREQUENCY IN HZ
Figure 59* Ratio of footing displacement to soil displacement under the footing.
177

178
vertical directions. An indication of the agreement between the direc
tion of the half-space particle motion and the measured particle motion
was taken as the ratio of the measured transverse component of motion
to the resultant motion, where the resultant motion wdis the square root
of the sum of the squares of each motion component. The measured
component/resuitant displacement ratios for each transducer location
were averaged over the test frequencies of 15, 20, 30J 40, and 50 Hz.
Table 27 gives the component/resultant displacement ratios for the
footing and for each transducer location. The average value of the
transverse/resultant displacement ratio for the 21 transducers buried
in the sand deposit and vibrated at five different frequencies was
0.872; the average ratio on the footing was 0.999* The average
vertical/resultant ratio was 0.262 for locations in the soil and 0.0250
on the footing. Radial/resultant displacement ratios averaged 0.248 in
the sand deposit and O.O385 on the footing. Motion imparted to the
soil by the torsional oscillation of the footing was aflmost entirely
transverse. Soil motion near the footing was also essentially trans
verse, but vertical and radial components of motion became more signi
ficant at increasing distances from the footing and at increasing
depths below the ground surface.
Since the dominant component of particle motion measured in the
sand deposit was in the transverse direction, and the half-space model
predicts only transverse particle motion, reasonable agreement between
the measured particle motion direction and the half-space particle
motion direction was apparent.

Table 27
Component Displacement Ratios Averaged over 5 Frequencies
179
Depth Below On Radial Distance from Center of Footing, ft
Ground.
, ft
Footing 3.5
10
30
60
90
Ratio
of Transverse Displacement
to Resultant Displacement
0
0.999
1
0.997
O.986
0.928
O.969
0.899
5
O.981
0.967
0.959
0.954
15
0.931
0.843
0.802
0.869
25
0.722
0.927
0.724
0.743
35
0.672
0.870
0.794
0.774
Ratio
of Vertical Displacement
to Resultant Displacement
0
0.0250
l
0.0351
0.122
0.121
0.l4i
0.222
5
0.107
0.l4l
0.204
0.136
15
0.l60
0.255
0.370
0.34o
25
0.470
0.227
0.333
0.490
35
0.522
0.257
0.398
0.452
Ratio of Radial Displacement to Resultant Displacement
0
0.0385
l
O.O636
0.101
0.297
0.161
0.309
5
0.130
0.166
0.181
0.233
15
0.264
0.246
0.280
0.286
25
0.319
0.242
o.46i
0.290
35
0.303
0.305
0.268
0.309
Properties of the Sand Deposit
A better understanding of the properties of the sand deposit was
obtained from the test measurements. Figure 56 shows the measured com
pression wave propagation velocity versus depth for the footing source
and surface source tests, and Figure 57 shows the measured shear wave
propagation velocity versus depth for steady state vibration tests.
The shear wave velocity near the footing increases wiith depth to 20 ft,
decreases from 20 to 25 ft, then increases from 25 ft to greater depths.

l8o
The average compression wave velocity increases continuously with
depth. Figure 60 shows the penetration resistance and average cone
bearing capacity of the sand deposit plotted versus depth. Both re
sistance and capacity increase with depth to about 20 ft, decrease from
20 to 30 ft, then increase from 30 ft to greater depths.
A comparison of Figures 56, 57 3 and 60 shows that the measured
shear wave propagation velocity correlates well with the penetration
and bearing values at various depths; the compression wave velocity
seems unrelated to penetration and bearing values. A change in k or
o
Poisson's ratio with depth in the sand deposit can account for the
observed differences in shear and compression wave velocities (Richart,
Hall, and Woods, 1970)- The 20-ft-thick layer of sand between 10
and 30 ft deep was probably a beach area during the Pleistocene epoch;
vibration from breaking waves and innumerable changes in effective
stress could have created this anomalous sand layer.
The gross distribution of the shear wave velocity with depth in
the sand deposit at the test site was estimated from the results of
the surface wave exploration method (Maxwell and Fry, 1967), the meas
ured compression wave velocities (Poisson's ratio taken as 1/3), and
the measured shear wave velocities. Figure 6l shows the distribution
of the measured shear wave velocities with depth for each method.
Since all of the methods were either empirical, involved an assumed
Poisson's ratio, or tested only a small part of the sand deposit, there
was considerable scatter in the measured velocity distribution with
depth. The shear wave velocities in the gross sand deposit were esti
mated and established by drawing the two straight lines shown in

DEPTH BELOW GROUND SURFACE IN FT
PENETRATION RESISTANCE IN BLOWS/FT
CONE BEARING CAPACITY IN KG/CM2
Figure 60.
Average penetration resistance and cone bearing capacity versus depth.
CD

182
Figure 6l; the relationship of these lines to the data points was in
fluenced by the penetration resistance and cone bearing capacity values
shown in Figure 60.
Basis for Comparing Results
Peak particle velocities were measured in the sand deposit and on
the test footing during the vibration experiments. P^ak particle dis
placements in the half-space and on the rigid disk were calculated.
For steady state sinusoidal vibration, the particle velocity is equal
to the angular frequency times the particle displacement.
The calculated particle displacement is directlv proportional to
the moment applied to the weightless rigid disk. The test footing was
not weightless and the torsional moment applied to th^ soil was not
determined; measuring the moment on the footing-soil interface is dif
ficult and usually inaccurate near the resonant frequency of the foot
ing. These difficulties were eliminated by normalizing the calculated
motion with respect to the motion of the rigid disk and normalizing the
measured motion with respect to the motion of the test footing; the nor
malized motions were then compared. The relationship between the dis
placement of the edge of the disk and the displacement of the edge of
the footing was not investigated in this work; it has been previously
confirmed by Richart and Whitman (1967)-
Measured Motion
The torsionally vibrating footing radiated energy into the sand
deposit in the form of shear waves. Since the energy level of any
particle in the sand deposit is proportional to the square of the

5
10
15
20
25
30
35
40
183
SHEAR WAVE VELOCITY IN FPS
250 500 750 1,000 1,250
o
S
\
\
l&\
\
COMPUTI
ON A VE
:D MOTION WAS
iLOCITY OF 650
BASED
-PS
u\
\
o
\
o
lOltJ
\
\
\
O \
o
\
\
\
c* n
/O
o
o
n
t
a U
|
Ky
o SURFA
CE WAVE METHO
LJ
D
i
i
i
i
to1 a
Q VIBRATION METHOD
9 COMPRESSION WAVE FROM SURFACE
ol
\a QP
IS COMPRESSION WAVE FROM FOOTING
I I

c
i
i
Figure 6l. Shear wave velocities versus depth.

184
velocity of that particle, three component transducers were necessary
to detect and measure the resultant particle velocity in the sand de
posit. (The resultant particle velocity was taken as the square root
of the sum of the squares of each component of particle velocity.) The
measured particle displacement at each transducer location, normalized
with respect to the measured displacement of the edge of the test foot
ing was
N(m)
cu(Resultant particle velocity in the sand)
uu(Particle velocity at the edge of the footing)
(155)
where w(m) was the dimensionless, normalized, particle displacement
of the buried transducers with respect to the test footing.
Computed Motion
The calculated half-space particle displacement, normalized with
respect to the calculated particle displacement at the edge of the rigid
disk, was
N(c)
l(aQ,a,b)
l(a ,a
o o
,0)
l(aQ,a,b)
I(a ,a ,0)
o o
(156)
where N(c) was the dimensionless, normalized particle displacement,
calculated for the half-space with respect to the edge of the rigid
disk, and I was the calculated value of the displacement integral.

185
Comparison of Normalized Displacements
The agreement between the calculated particle displacements in
the half-space and the measured particle displacements in the sand
deposit was tested by computing the ratio of the calculated normalized
displacement to the measured normalized displacement. If the ratio
was near one, good agreement was indicated; if the ratio was less than
one, the calculated displacement was less than the measured displace
ment; if the ratio was more than one, the calculated displacement was
more than the measured displacement.
Comparison of Results
Normalized (Constant E) Half-Space Displacements
The normalized half-space displacements, defined as the ratio of
the particle displacement in the half-space to the displacement of the
edge of the rigid disk, are given in Table 28 for various radial dis
tances from the center of the disk, for various depths below the sur
face of the half-space, and for frequencies of 15, 20, 30, 40, and 50
Hz. A shear wave velocity of 650 fps and a unit weight of 104 lb/cuft
were assumed for the half-space with a 5-ft-diam disk on its surface.
An examination of Table 28 shows that the normalized half-space
displacements depend on the oscillation frequency of the rigid disk
and on the position of the particle in the half-space. The magnitude
and distribution of the frequency influence was illustrated by the tab
ulations listed in Table 29. This table gives the ratio of the normal
ized half-space displacements at a frequency of 50 Hz to the normalized
half-space displacements at a frequency of 15 Hz for various locations

Table 28
Normalized. Half-Space Particle Displacements
186
Half-Space Displacement/^Displacement of Edge of Disk
Depth Below Radial Distance from Center of Disk, ft
Surface, ft
3-5 10
30
60
90
Vibration Frequency, 15 Hz
1
0.418 0.0510
0.0143
0.00701
0.00466
5
0.0395
0.0139
0.00697
0.00465
15
0.0134
0.01133
0.00659
0.00453
25
0.00593
0.00835
0.00597
0.00432
35
0.00322
0.00598
o.c
10523
0.00405
Vibration Frequency, 20 Hz
1
0.424 0.0601
0.0180
0.00889
0.00592
5
0.0471
0.0175
0.00884
0.00590
15
0.0170
0.0144
0.00838
0.00576
25
0.00747
0.0106
0.00758
0.00550
35
0.00408
0.00759
0.00665
0.00515
Vibration Frequency, 30 Hz
1
o.44i 0.0772
0.0244
0.0121
0.00809
5
O.O616
0.0238
0.0121
0.00807
15
0.0233
0.0197
0.0115
0.00788
25
0.0104
0.0146
o.q
1104
0.00753
35
0.00571
0.0105
0.00915
0.00707
Vibration Frequency, 40 Hz
1
o.46o 0.0914
0.0295
0.0147
0.00980
5
0.0738
0.0288
0.0146
0.00977
15
0.0287
0.0240
0.0139
0.00956
25
0.0131
0.0180
0.0127
0.00916
35
0.00719
0.0131
0.0112
0.00861
Vibration Frequency, 50 Hz
1
0.475 0.102
0.0334
0.0167
0.0111
5
O.O838
0.0326
0.0166
0.0111
15
0.0337
0.0274
0.0158
0.0108
25
0.0155
0.0208
0.0145
0.0104
35
0.00853
0.0153
0.0129
0.00982

Table 29
Influence of Frequency on Half-Space Displacements
187
Normalized 50-Hz
Half-Space Displacements/
Normalized 15-Hz
Half-Space Displacements
Depth Below
Radial Distance
from Center
of Disk, ft
Surface, ft
3.5 10
30
60
90
1
1.17 2.00
2.34
2.38
2.38
5
2.12
2.35
2.38
2.38
15
2.51
2.42
2.40
2.38
25
2.6l
2.49
2.43
2.4l
35
2.65
2.56
2.47
2.43
in the half-space. The tabulations reveal that, for equal amplitudes
of disk displacement, 50-Hz oscillations generate half-space displace
ments about 2.4 times the displacements generated by 15-Hz oscillations.
Normalized Soil Displacements
The normalized soil displacements, defined as the ratio of the meas
ured resultant particle displacement in the sand deposit to the meas
ured tangential displacement of the edge of the test footing, are given
in Table 30 for the position of each buried transducer and for the 5
test frequencies. Table 31 lists the ratio of the normalized soil dis
placements measured at 50 Hz to the normalized soil displacements meas
ured at 15 Hz.
Changing the excitation frequency of the test footing from 15 to
50 Hz caused erratic changes in the measured soil displacement field.
Soil displacements were halved at some locations and at others the dis
placement was more than tripled. Considering the entire field of trans
ducers, however, the displacement was increased about 5>0 percent due to
increasing the frequency from 15 to 50 Hz.

Table 30
Normalized Soil Particle Displacements
188
Depth Below
Soil
Displacement/Displacement of Edge of Footing
Radial Distance from Center of Footing, ft
Ground, ft
3.5
10
30
60
90
Vibration Frequency, 15 Hz
1
0.439
0.101
0.0528
0.0357
0.0231
5
0.0739
0.0314
0.0192
0.0181
15
0.00915
0.00378
0.00282
0.00279
25
0.00247
0.00176
0.000873
0.000799
35
0.00116
O.OOO963
0.000593
0.000709
Vibration Frequency, 20 Hz
1
0.523
0.114
0.0674
0.0347
0.0212
5
0.0675
0.0273
0.0157
0.0133
15
0.00762
0.00198
0.00181
0.00201
25
0.00206
0.000874
0.000871
0.000782
35
0.00142
0.00101
0.00090
O.OOO65
Vibration Frequency, 30 Hz
1
0.555
0.123
0.0957
o.o4oi
0.0222
5
0.0850
0.0314
0.0177
0.0196
15
0.00940
0.00546
0.00303
0.00398
25
0.00201
0.00161
0.000908
0.00162
35
0.00159
0.000956
0.000971
0.00133
Vibration Frequency, 40 Hz
1
0.562
0.159
0.0670
0.0425
0.00651
5
0.0640
0.0383
0.0120
0.0119
15
0.00756
0.00328
0.00221
0.00343
25
0.00196
0.00138
0.000780
O.OOO699
35
0.00128
o.ooi4o
0.00112
0.000718
Vibration Frequency, 50 Hz
1
0.666
0.199
0.0926
0.0219
0.0142
5
0.0605
0.0342
0.0175
0.00848
15
0.0149
0.00801
0.00384
0.00219
25
0.00178
0.00128
0.00162
0.00178
35
0.00211
0.00338
0.00213
0.00110

Table 31
Influence of Frequency on Soil Displacements
189
Normalized 50-Hz Soil Displacements/
Normalized 15-Hz Soil Displacements
Depth Below Radial Distance from Center of Footing, ft
aid, ft
3.5
10
30
60
90
1
1.5
2.0
1.8
0.6l
0.6l
5
0.82
1.1
0.91
0.47
15
1.6
2.1
1.4
0.78
25
0.72
0.73
1.9
2.2
35
1.8
3.5
3.6
1.6
Ratio of Displacements
The comparison between calculated half-space displacements and
measured soil displacements is presented in Table 32. This table lists
the ratio of the normalized half-space displacements 1po the normalized
soil displacements. A ratio of unity signifies agreement between the
calculations and the measurements, a ratio of less than one means that
the calculated displacement was less than the measurement, and a ratio
of more than one indicates that the calculated displacement was greater
than the measured displacement. Ratios are given at various radial
distances and depths for frequencies of 15, 20, 30, 40, and 50 Hz.
Table 33 lists average displacement ratios for the 5 tjest frequencies
at various distances and depths.
The ratios of the homogeneous half-space displacement to the soil
displacement given in Tables 32 and 33 lead to several observations.
(l) The computed displacements were less than the measured displacements
near the ground surface. (2) The computed displacements were more than
the measured displacements at the deeper transducer locations. (3) The

Table 32
Ratio of Half-Space to Soil Displacement
190
Normalized Half-Space Displacement/
Normalized Soil Displacement
Depth Below Radial Distance, fjt
Surface, ft
3.5 10
30
60
90
Vibration Frequency
, 15 Hz
1
O.95 0.51
0.27
0.20
0.20
5
0.54
0.44
0.36
0.26
15
1.5
3.0
2.3
1.6
25
2.4
4.7
6.8
5.4
35
2.8
6.2
8.8
5-7
Vibration Frequency
, 20 Hz
1
0.8l 0.53
0.27
0.26
0.28
5
0.70
0.64
0.56
0.44
15
2.2
7.3
4.6
2.9
25
3-6
12
8.7
7.0
35
2.9
7-5
7.4
7.9
Vibration Frequency
, 30 Hz
1
0.79 0.63
0.26
0.30
0.36
5
0.73
0.76
0.68
0.4i
15
2.5
3.6
3.8
2.0
25
5.2
9.1
11
4.7
35
3.6
11
9-4
5.3
Vibration Frequency
, 40 Hz
l
0.82 0.57
0.44
0.35
1.5
5
1.1
0.75
1.2
0.82
15
3.8
7-3
6.3
2.8
25
6.7
13
16
13
35
5.6
9-3
10
12
Vibration Frequency
, 50 Hz
1
0.71 0.51
0.36
0.76
0.78
5
1.4
0.95
0.95
1.3
15
2.3
3.4
4.1
4.9
25
8.7
16
9.0
5.8
35
4.0
4.5
6.0
8.9

Table 33
Average Displacement Ratios for j? Frequencies
191
Average Ratio of
Normalized Half-Space Displacement
Depth Below
to ;
Normalized Soil Displacement for 5 Frequencies
Radial Distance, ft
Surface, ft
3.5
10
30
60
90
1
0.82
0.55
0.32
0.37
0.62
5
O.89
0.71
0.75
O.65
15
2.5
4.9
4.2
2.8
25
5.3
11
6.1
7.2
35
3.8
7.7
8.3
8.0
average measured displacements ranged from 1/10 to 3 times the calcu
lated displacements. (4) The calculated displacements approach the
measured displacements in the vicinity of the footing. (5) The average
displacement ratio for each frequency generally increased with increas
ing vibration frequency. (6) The average displacement ratio for all
frequencies was about 3-75 so the half-space calculations predicted
considerably larger average displacements than were measured.
Displacements in a Nonhomogeneous (Linear E) Half-Space)
The relationship between the particle displacement, v in a homo
geneous half-space with a shear modulus, G to the particle displace
ment, v in a nonhomogeneous half-space with shear modulus, G(z) ,
that is a function of depth, z was discussed in a prior section.
This relationship was
v = v
N
' G '
GR
1/2
(157)
The shear wave velocity data shown in Figure 6l was u^ed to compute
G(z) and the above expression was used to calculate 1phe particle

192
displacement in a nonhomogeneous half-space with a modulus that varied
according to the properties of the sand at the test site. Because the
vertical dead load applied to the footing during the second pour of
concrete increased the average footing-soil contact pressure by 214
psf--equivalent to an overburden depth of about 2 ft--the shear wave
velocity of the sand supporting the footing was taken, as the velocity
at a depth of 3 ft.
Table 34 lists the normalized nonhomogeneous half-space displace
ments, defined as the displacement of the nonhomogeneous half-space
with respect to the edge of a rigid disk oscillating on the surface of
the nonhomogeneous half-space. Table 35 gives the ratios of the nor
malized nonhomogeneous half-space displacement to the normalized soil
displacement, and Table 36 is a tabulation of the average displacement
ratio at each transducer location for the 5 test frequencies.
The ratios of the nonhomogeneous half-space displacement to the
soil displacement given in Tables 35 and 36 lead to several observa
tions. (l) The computed displacements were less than the measured
displacements near the ground surface. (2) The computed displacements
were more than the measured displacements at the deeper transducer lo
cations. (3) The average measured displacements ranged from 1/4 to
2-1/2 times the measured displacements for all 5 test frequencies.
(4) The calculated displacements approach the measured displacements
near the footing. (5) The displacement ratios generally increased
with increasing vibration frequency. (6) The average displacement
ratio for all frequencies and all locations was about 1.6, indicating
that the average calculated nonhomogeneous half-space displacement was
larger than the measured displacement.

Table 34
Normalized Nonhomogeneous Half-Space Displacements
193
Depth Below
Surface, ft
1
5
15
25
35
l
5
15
25
35
1
5
15
25
35
1
5
15
25
35
l
5
15
25
35
Nonhomogeneous Half-Space Displacements/
Displacement of Edge of]
Radial Distance from Center c
Disk
3f Disk,
ft
3-5 10
30
J
w~
90
Vibration Frequency, 15 Hz
0.517 0.0631
0.0177
0.(
30867
0.00576
0.0333
0.0117
04
30588
0.00392
0.00625
0.00529
0.(
30307
0.00211
0.00211
0.00297
0.(
30213
0.00154
0.00113
0.00209
0.(
30183
0.00142
Vibration Frequency, 20 Hz
0.524 0.0743
0.0223
0.0110
0.00732
0.0397
0.0148
0.00746
0.00498
0.00793
0.00672
0.00391
0.00269
0.00266
0.00377
0.00270
O.OOI96
0.00143
0.00266
0.(
30233
0.00180
Vibration Frequency, 30 Hz
0.546 0.0955
0.0302
0.(
3150
0.0100
0.0520
0.0201
0.0102
0.00681
0.0109
0.00919
0.00537
0.00368
0.00370
0.00520
0.00370
0.00268
0.00200
0.00367
0.(
30320
0.00247
Vibration Frequency, 40 Hz
O.569 0.113
0.0365
0.0182
0.0121
0.0622
0.0243
o.<
3123
0.00824
0.0134
0.0112
0.00649
0.00446
0.00466
0.00641
0.00452
0.00326
0.00252
0.00458
0.00392
0.00301
Vibration Frequency, 50 Hz
O.588 0.126
0.0413
0.0207
0.0137
0.0707
0.0275
o.oi4o
0.00936
0.0157
0.0128
0.00737
0.00504
0.00552
0.00741
0.00516
0.00370
0.00299
0.00535
0.00451
0.00344

194
Table 35
Ratio of Norihomogeneous Half-Space Displacements
to Soil Displacements
Depth Below
Normalized Nonhomogeneous Half-Space Displacements/
Normalized Soil Displacements
Radial Distance, fit
Surface, ft
¡O
r1
ir\
CO
30
60
90
Vibration Frequency
, 15 Hz
1
1.2 0.63
0.34
0.24
0.25
5
0.45
0.37
0.31
0.22
15
0.68
1.4
1.1
0.76
25
0.85
1.7
2.4
1.9
35
0.97
2.2
3.1
2.0
Vibration Frequency
, 20 Hz
1
1.0 0.65
0.33
0.32
0.35
5
0.59
0.54
0.48
0.37
15
1.0
3.4
2.2
1.3
25
1.3
4.3
3.1
2.5
35
1.0
2.6
2.6
2.8
Vibration Frequency
, 30 Hz
1
O.98 0.78
0.32
0.37
0.45
5
0.6l
0.64
O.58
0.35
15
1.2
1.7
1.8
0.92
25
1.8
3.2
4.1
1.7
35
1.3
3.8
3.3
1.9
Vibration Frequency
, 40 Hz
1
1.0 0.71
0.54
0.43
1.86
5
0.97
0.63
1.0
0.69
15
1.8
3.4
2.9
1.3
25
2.4
4.6
5.8
4.7
35
2.0
3.3
3.5
4.2
Vibration Frequency
, 50 Hz
1
0.88 O.63
0.45
0.95
0.96
5
1.2
0.80
0.80
l.l
15
1.1
1.6
1.9
2.3
25
3.1
5.8
3.2
2.1
35
1.4
1.6
2.1
3.1

195
Table 36
Average Nonhomogeneous Half-Space Displacement
Ratios for 9 Frequencies
Average Ratio of Normalized Nonhomogeneous Half-
Space Displacement to Normalized Soil Displacement
for 5 Frequencies
Depth Below
Radial
Distance,
fit
Surface, ft
3.5
10
30
60
90
1
1.0
0.68
o.4o
0.46
0.77
5
0.76
0.60
0.63
0.55
15
1.2
2.3
2.0
1.3
25
1.9
3.9
3.7
2.6
35
1.3
2.7
2.9
2.8
Discussion of Results
Homogeneous (Constant E) and Nonhomogeneous (Linear E) Half-Space
A comparison of Tables 32 and 35 indicated that the computed
displacements in a nonhomogeneous half-space were in closer agreement
with the measured displacements than were the homogeneous half-space
displacement calculations. The nonhomogeneous half-space displacements,
however, depend on the computed values of the homogeneous half-space
displacements.
The displacement ratios for the 15 Hz test were used to compare
the homogeneous and nonhomogeneous displacement prediction values. For
the homogeneous case, the average ratio was 2.62 and the standard devia
tion was 2.55j for the nonhomogeneous case, the average ratio was 1.1
and the standard deviation was 0.8l. Since an ideal prediction method
would give a ratio of about 1.0 and a standard deviation of zero,

196
the nonhomogeneous half-space calculations are closer to the ideal than
the homogeneous half-space calculations.
The decrease in displacement amplitude of propagating waves with
distance from their source, in excess of the decrease due to geometri
cal damping, is caused by material damping (Richart, Hall, and Woods,
1970)- The effect of material damping on the amplitude, v of
particle displacements at distances r^ and r^ from the source of
motion is quantified by an attenuation coefficient, d* such that
v^ = v^(geometrical damping law)(material damping) (158)
= v^(geometrical damping law) e-0i^r4-r3' (159)
For body waves
L = h e-cy(r4-r3) (160)
V3 r4
The displacement ratios computed with respect to .the homogeneous
and nonhomogeneous half-spaces consider geometrical damping, but do
not consider material damping effects, consequently, the average dis
placement ratios have been consistently greater than one. A gross
attenuation coefficient for the entire sand deposit was computed using
the frequency averaged nonhomogeneous displacement ratios for the 20
transducers located from 10 to 90 ft from the center of the footing and
buried from 1 to 35 ft deep. The average displacement ratio for the
nonhomogeneous half-space and for all of these transducer locations was
I.67. The center of these 20 transducers was 50 ft from the center of
the footing and 17.5 ft below the ground, so the distance from this

197
point to the edge of the footing was 50.28 ft. Substituting and can
celling appropriate terms gave
(l6l)
and
(162)
a = 0.0102
This coefficient of attenuation agrees well with the value of 0.009
given by Richart, Hall, and Woods (1970) for Rayleigh waves at a simi
lar Eglin Field, Florida, test area.
Characteristics of the Test Site
Since the nonhomogeneous half-space calculations, averaged over the
5 test frequencies and over the entire field of buried transducers, were
in reasonable agreement with the average measured motion, but there was
a wide variation from this average at various transducer locations and
for various excitation frequencies, some explanation for the observed
variation was sought. Recalling that a much improved correlation be
tween computed and measured displacements resulted from the use of more
realistic soil moduli, and that the soil modulus appeared to vary con
siderably with depth, suggested that the variation in the ratio of
computed to measured displacements might be due to local variations in
the soil modulus. Figure 6l shows the range and variation in the soil
shear modulus with depth at the test site, as determined by conventional
seismic methods; modulus changes with distance and direction from the
footing were not determined. Because seismic methods did not reveal de
tailed soil modulus changes within the soil mass and ¡because a more

198
detailed determination was needed, other indicators of the variation of
the in situ soil stresses, and the corresponding modulus variations,
had to be used.
Gibbs and Holtz (1957) found that the in situ stress conditions in
a sand correlate with the penetration resistance of a;standard split-
spoon sampler; Figure 60 shows the relationship between penetration
resistance and bearing capacity of a cone penetrometer. Figure 62 is
a plot of the average cone bearing capacity of the sand deposit versus
depth and a plot of the average nonhomogeneous half-space displacement
ratio, as given in Table 36, versus the depth of the transducer in the
sand deposit. The two plots in Figure 62 showed that the average dis
placement ratio was related to the average cone bearihg capacity at
various depths in the sand deposit.
An interpretation of the observed correlation between the nonhomo
geneous half-space displacement ratios and the cone bearing capacity
shown in Figure 62 was considered worthwhile. Where the cone bearing
capacity was unusually high, lateral soil stresses werje high and the
shear modulus of the sand was greater than the value used in the dis
placement calculations; a calculation modulus less than the actual
modulus would result in larger computed displacements and in larger
displacement ratios. Conversely, where the cone bearing capacity was
unusually low, the lateral soil stresses were low and the shear modulus
of the sand was less than the value used in the displacement calcula
tions; the computed displacements were low and the computed displacement
ratios were also low. A quantitative relationship between cone bearing
capacity and dynamic shear modulus, however, was not found.

5
10
15
20
25
30
35
199
AVERAGE CONE BEARING CAPACITY IN KG/CM2
25 50 75 100 125
0.6 1.2 1.8 2.4 3.0
AVERAGE DISPLACEMENT RATIO, N(c)/N(m)
Figure 62. Cone bearing capacity and displacement ratio
versus depth.

200
Particle Motion Predictions
The magnitude of the predicted particle motion inj a nonhomogeneous
half-space depends on the accuracy and detail of the shear modulus de
terminations throughout the sand deposit, on the material damping in
the soil, and on the validity of the conversion from homogeneous half
space displacements to nonhomogeneous half-space displacements. The
degree to which the calculated motion agrees with the measured motion
also depends on the consonance between the assumed boundary conditions
for the calculations and the physical boundary conditions that existed
during the experiments.
Because the homogeneous half-space calculations Were based on one
shear modulus for the entire sand deposit and because the nonhomogeneous
half-space calculations were based on an average variation in shear
modulus with depth in the sand deposit, the accuracy lof the particle
motion predictions were also based on the average displacement ratios,
as given in Table 36. The maximum displacement ratio in this table
is 3.9s the minimum is 0.40, and the average is 1.64. Neglecting damp
ing, the measured displacements ranged from 1/4 to 2-[L/2 times the pre
dicted displacement. This range was rather broad, however, the predic
tion accuracy was adequate to classify the generated vibrations as
(l) undetectable, (2) quite apparent, or (3) intolerable (Richart, Hall,
and Woods, 1970). To this degree of accuracy, then, the half-space
particle motion predictions were judged to be useful.

CONCLUSIONS AND RECOMMENDATIONS
Conelusions
Homogeneous (Constant E) Half-Space
Bycroft's (1956) solutions for the vibration of a rigid disk on
the surface of an elastic half-space can be extended ^o calculate the
particle motion generated in the half-space by the vibrating disk. The
resulting infinite integral expression for the displacements can be
satisfactorily evaluated by numerical integration methods such as
Simpson's rule. The calculated displacements in the half-space agree
with the geometrical damping law for body waves and the surface dis
placements under the disk agree with the exact solution given by
Reissner and Sagoci (1944) and with the variational solution given by
Stallybrass (1962).
Nonhomogeneous (Linear E) Half-Space
Static solutions for the stresses, strains, and settlement devel
oped in and on a nonhomogeneous elastic half-space subjected to surface
loads have been developed and presented by Frolich (1934), Borowicka
(1943), Hruban (1948), Curtis and Richart (1955), and Gibson (1967).
For a uniformly loaded, incompressible half-space with a shear modulus
that varies linearly with depth, Gibson showed that the stresses in this
half-space were the same as those in a homogeneous half-space. Rigorous
solutions for oscillating loads on a nonhomogeneous half-space are .
201

202
lacking, but discrete methods for solving such problems are developing
rapidly (Lysmer and Waas, 1970).
Experimental Aspects
Test site
Soil conditions.--The soil at the test site was & laterally homo
geneous marine terrace deposit of poorly graded, fine to medium grained
sand. The water table was about 100 ft deep in this thick, free-
draining sand deposit and the shear wave velocity increased signifi
cantly with depth. The average dry unit weight of the sand was 99
lb/cu ft, the relative density was about 62 percent, and the water con
tent was about 5 percent. The penetration resistance and cone bearing
capacity generally increased with depth to 20 ft, decreased from 20 to
30 ft, then increased again at greater depths.
Shear wave velocities.--A range of shear wave velocities was ob
tained at various depths in the sand deposit by three different in situ
seismic methods. The velocities obtained by measuring the phase of
waves emanating from the footing seemed to correlate with the penetra
tion resistance of the sand deposit.
Test footing and vibrator
The 5-ft-diam test footing had to be embedded in the sand to a
depth of 1 ft to create a favorable state of stress at its contact with
the soil. The footing was essentially rigid with respect to the sup
porting soil and it was large enough to simulate full-scale foundation
conditions.

203
Mechanical losses accounted for most of the power needed to drive
the torsional vibrator. Power losses to the ground were less than 1/10
hp.
Particle motion measuring system
The particle motion measuring system used to detect, amplify, and
record the particle velocities generated in the sand deposit had ade
quate sensitivity and resolution to fulfill the purpose of the experi
mental program. The maximum probable error in the particle velocity
determinations, due to the measuring system, was between 7 and 15
percent.
Results of measurements
Torsional vibration of the test footing during the experimental
program did not cause it to tilt, but it probably settled about l/l6 in.
The maximum oscillatory displacement of the edge of the footing was
0.0072 in. and the minimum was 0.0015 in. Torsional oscillation of the
footing did not develop slippage between the footing and the soil on
the footing-soil contact area.
The maximum particle velocity generated in the soil deposit near
the footing was 0.47 in./sec and the minimum at a distant location was
0.00011 in./sec. Wear the footing, the direction of particle motion in
the sand deposit was essentially tangential to the circular footing; the
dominant direction of motion throughout the sand deposit was also
tangential to the footing.
Computations and Measurements
When the soil deposit was considered as a homogeneous elastic

20k
half-space, and material damping was neglected, the average measured
particle displacement was about l/3 to l/k of the calculated displace
ment. The frequency averaged particle displacements measured at 21 lo
cations in the sand deposit were l/lO to 3 times the calculated particle
displacements. Near the footing, the measured and computed displace
ments were in good agreement.
When the soil deposit was considered as a nonhomogeneous elastic
half-space and material damping was neglected, the average measured
particle displacement was about 60 percent of the calculated displace
ment. The frequency averaged particle displacements measured at 21 lo
cations in the sand deposit were l/k to 2-l/2 times the calculated par
ticle displacements. The calculated and measured displacements were in
good agreement near the footing. The ratio of the computed displace
ments to the measured displacements correlated with the cone bearing
capacity of the sand deposit at various depths. A mojre accurate and
detailed determination of the shear modulus of the sand deposit could
improve the correspondence between calculated and measured particle
displacements.
Results of the computed nonhomogeneous half-space particle dis
placements and the measured particle displacements indicated that the
nonhomogeneous elastic half-space model was a reasonable and potentially
useful analytical representation of a natural soil deposit subjected to
forced foundation vibrations. The material damping effect on body waves
agreed with previous determinations for this same effect on surface
waves. The accuracy of the displacement predictions was adequate to
classify the transmitted vibration as either undetectable, readily

205
apparent, or intolerable. Thus, the three goals of this study, motion
computations, motion measurements, and motion comparisions, were
attained and the hypothesized utility of the half-space model was
assessed.
Recommendations
Analytical Work
An attempt to utilize the nonhomogeneous elastic half-space as an
analytical model of a soil deposit should be made. Gibson's (1967)
work could be a starting point for the analysis of a vertically oscil
lating flexible footing that applies a uniform loading to the non
homogeneous half-space.
Discrete analysis methods, such as the lumped mass and finite-
element method, should be used to predict the particle motion generated
in the sand deposit.
Predictions of the particle motion generated by a vertically
oscillating rigid footing should be developed for comparison to meas
ured motion.
Experimental Work
A more accurate and detailed determination of the elastic moduli
in the sand deposit at the test area should be attempted. This might
be accomplished by the use of seismic detectors in a borehole.
Particle motion measurements should be made in the sand deposit
as the test footing is subjected to forced vertical oscillations.

206
An attempt to relate standard penetration test resistance and cone
bearing capacity to in situ soil stresses should be made in order to
relate these values to shear wave velocity.
Comparisons
Comparative work similar to that illustrated herein should con
tinue, but particle motion measurements from actual operating facil
ities (arsenals, power plants, forging mills, etc.) should receive
increased attention and analysis.

APPENDIX A
CALCULATIONS FOR THE INTEGRAL l(a ,a,b)
Table 37
Subroutines and Computer Program for the
Integral l(aQ,a,b)
REAL PROCEDURE SIMPSON (FCT, A, B, TEST) :
$CARD
VALUE A, B, TEST ; REAL A, B, TEST ; REAL PROCEDURE FCT :
BEGIN
REAL PROCEDURE FINITE (S, FCT, A, B, TEST) ;
VALUE S, A, B, TEST ; REAL A, B, TEST ; INTEGER S d
REAL PROCEDURE FCT ;
BEGIN
INTEGER N, NMBR ;
REAL SI, S2, S4, OLD, NEW, X, H ;
LABEL ABC ;
TEST:=10 (- TEST) ;
OLD:=123456789012
SI:=FCT (S X A) + FCT (SXB) ;
S2:=0 ;
NMBR:=2 ;
ABC: H:=(B A) / NMBR ;
S4:=0 j
FOR N:=1 STEP 2 UNTIL NMBR DO
BEGIN
X:=S X (A + N X H) ;
S4:=s4 + FCT(x) ;
END ;
FINITE:=NEW:=(H X (SI + 2 X S2 + 4 X S4)) / 3 5
IF ABS (NEW OLD) > ABS ( NEW X TEST) THEN
BEGIN
OLD:=NEW ;
S2: =S2 + S4 ;
NMBR:=2 x NMBR ;
GO TO ABC ;
END ;
END FINITE ;
(Continued)
207

208
Table 37 (Continued)
REAL PROCEDURE INFINITE (S, FCT, A, TEST) ;
VALUE S, A, TEST ; REAL A, TEST ; INTEGER S ;
REAL PROCEDURE FCT ;
BEGIN
REAL WHOLE, PART, TEST1 ;
WHOLE:=0 ;
TEST1:=10 (-TEST-1) ; DO
BEGIN
PARTINFINITE (S, FCT, A, A+l, TEST) ;
INFINITE:=WHOLE:=WHOLE + PART ;
A: =A + 1 ;
END UNTIL ABS (PART) < ABS (WHOLE X TEST1) ;
END INFINITE ;
IF A / -@68 AND B -f @68 THEN SIMPSON: =FINITE ( 1,FCT,A,B,TEST)ELSE
IF A / -@>68 AND B = @68 THEN SIMPSON: INFINITE ( 1,FCT, A,TEST)ELSE
IF A = -@68 AND B = @68 THEN SIMPSON:=INFINITE (-1,FCT, 0,TEST)
+INFINITE ( 1,FCT, 0,TEST)ELSE
SIMPSON:INFINITE (-1,FCT,-B,TEST) ;
END SIMPSON ;
$ CARD LIST
REAL PROCEDURE JONE(x);
$ CARD
COMMENT SEE MATH TABLES & OTHER AIDS TO COMPUTATION, 1957,PAGE 86 ;
VALUE X; REAL X;
BEGIN
REAL T,PONE,QONE,Y,ONE;
IF X <0 THEN ONE:=-l ELSE ONE:=1;
X:=ABS(x);
IF X<4.0 THEN BEGIN Y:=T:=x/4.0;T:=TXT;
JONE :=(((((((-.0001289769XT+.OO2206915 5)XT-.0236616773)XT+.1777582922)XT
-.8888839649)XT+2.6666660544)XT-3999999971)XT+1.9999999998) XYXONE
END
ELSE BEGIN
Y:=T:=4.0/x;T:=TXT;
PONE:= (((((.0000042414XT-.000020092)>T+.0000580759) XT -.0002232030)XT
+.0029218256) XT+.3989422819)X2.50662827463;
QONE:=(((((-0000036594XT+.00001622) XT -.0000398708)XT+.0001064741) XT
-.00063904) xr+.0374oo8364)xyx2.50662827463;
JONE:-SORT(2/(3.l4l59265358xX))x(PONExCOS(X-2.35619449018)-QONEXSIN
(X-2.35619449018))X0NE
END 5
END OF PROCEDURE JONE ;
$CARD LIST
(Continued)

209
Table 37 (Continued)
COMMENT INTEGRATION PROGRAM
BEGIN
$$A START
$ CARD
$$A JONE
$ CARD
$$A SIMPSON
$ CARD
$$A INVTAN
$ CARD
REAL AO,A,B,REINT,IMINT;
INTEGER OLDT,NEWT,ELAPT;
FORMAT Fl("AO =",F6.2,/," B =" ,P7.3,////,"A/A0" ,X5,"A" ,X8,
"REAL,X7,"IMAGINARY",X4,"ABS VALUE",X5,"ARGUMENT",//),
F2(p4.1,f8.2,3E13.3,F11.3,I3,i4);
REAL PROCEDURE RELX(x); VALUE X; REAL X;
BEGIN
IF X NEQ 0 THEN
BEGIN REAL SX,AOSX;
SX:=SIN(x);
AOSX: =^OXSX;
RELX: =-(SIN(A0SX)-A0SXXC0S(A0SX) )xJONE(AXSX)xSIN(BXCOS(x) )/AOSX;
END ELSE
RELX:=0;
END OF THE PROCEDURE RELX;
REAL PROCEDURE REH(x); VALUE X; REAL X;
BEGIN
IF X NEQ Pl/2 THEN
BEGIN REAL SX,AOSX;
SX: =l/C0S(x);
AOSX: =AOXSX;
REHX: =(SIN(AOSX) -AOSXXCOS(AOSX) )XJ0NE(AXSX)X
(IF B NEQ 0 THEN EXP(-BXSIN(X)XSX) ELSE l)/AO;
END ELSE
REHX:=0;
END OF THE PROCEDURE REHX;
REAL PROCEDURE IMX(x); VALUE X; REAL X;
BEGIN
IF X NEQ 0 THEN
BEGIN REAL SX,AOSX;
SX:=SIN(x);
AOSX: =AOXSX;
IMX: =-(SIN(AOSX) -AOSXXCOS(AOSX) )xJ0NE(AXSX)x
(if B NEQ 0 THEN COS(BXCOS(x)) ELSE 1/A0SX;
END ELSE
IMX:=0;
END OF THE PROCEDURE IMX;
(Continued)

210
Table 37 (Concluded)
OLDT:=0;
READ(CARD,AO,B);
EM) OF DATA;
WRITE(PRINT,FI,AO,B);
FOR A:=0 STEP -5XA0 UNTIL 40.1XA0 DO
BEGIN
IF A NEQ 0 THEN
BEGIN
REINT:=0;
IF B NEQ 0 THEN
REINT:=SIMPSON(RELX,0,. 5,5)+SIMPSON(RELX,.5,1,5)+
SIMPSON(RELX,1,PI/2,5);
REINT:=REINT+SIMPSON(REHX,0,.5,5)+SIMPSON(REHX,.5,1,5)+
S IMPSON(REHX,1,Pl/2,5);
IMINT:-SIMPSON(IMX,0,.5,5)+SIMPSON( IMX,.5,1,5)+SIMPSON(IMX,1,Pl/2,5);
ELAPT:=((NEWT:-TIME(2))-OLDT)/60; OLDT:-NEWT;
WRITE(PRINT,F2,A/AO,A,REINT, IMINT,SQRT(REINT*2+3MINT*2),
INVTAN(IMINT,REINT),ELAPT,NEWT/60);
END ELSE WRITE(PRINT,F2,A/AO,A,0,0,0,0);
END;
$$A FINISH
END.

b
0.000
0.144
211
Table 38
Value of l(0.36,a,b)
a
Real
Imaginary
Absolute
: Value
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.18
3.974
-
01
-2.557
-
03
3-974
- 01
0.36
7.085
-
01
-5.064
-
03
7.085
- 01
0.54
1.932
-
01
-7.474
-
03
1.933
- 01
0.72
1.080
-
01
-9.739
-
03
1.085
- 01
0.90
7.259
-
02
-I.182
-
02
7.355
- 02
1.08
5.349
-
02
-I.367
-
02
5.521
- 02
1.26
4.138
-
02
-I.526
-
02
4.4ii
- 02
1.44
3.273
-
02
-I.657
-
02
3.669
- 02
1.62
2.599
-
02
-1.757
-
02
3.137
- 02
1.80
2.041
-
02
-1.825
-
02
2.738
- 02
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.18
2.039
-
01
-2.552
-
03
2.039
- 01
0.36
2.622
-
01
-5.054
-
03
2.623
- 01
0.54
1.626
-
01
-7.458
-
03
1.628
- 01
0.72
1.012
-
01
-9.719
-
03
1.016
- 01
0.90
7.017
-
02
-1.179
-
02
7.115
- 02
1.08
5.227
-
02
-1.364
-
02
5.402
- 02
1.26
4.058
-
02
-1.523
-
02
4.335
- 02
1.44
3.214
-
02
-1.654
-
02
3.615
- 02
1.62
2.556
-
02
-1.753
-
02
3.100
- 02
1.80
2.013
_
02
-1.821
_
02
2.715
- 02
1.98
1.547
-
02
-1.856
-
02
2.416
- 02
2.16
1.136
-
02
-1.858
-
02
2.178
- 02
2.34
7.700
-
03
-1.828
-
02
1.984
- 02
2.52
4.414
-
03
-1.768
-
02
1.822
- 02
2.70
1.476
-
03
-1.679
-
02
1.685
- 02
2.88
-1.125
-
03
-1.564
-
02
1.568
- 02
3.06
-3.394
-
03
-1.427
-
02
1.467
- 02
3-24
-5.329
-
03
-I.271
-
02
1.378
- 02
3.42
-6.928
-
03
-1.100
-
02
1.300
- 02
3.60
-8.190
-
03
-9.176
-
03
1.230
- 02
3.78
-9.118
-
03
-7.289
-
03
1.167
- 02
3.96
-9.720
-
03
-5.382
-
03
l.lll
- 02
4.l4
-1.001
-
02
-3.495
-
03
1.060
- 02
4.32
-9-997
-
03
-1.670
-
03
1.013
- 02
4.50
-9.710
-
03
5.564
-
05
9.710
- 03
4.68
-9.173
-
03
1.646
-
03
9.320
-.03
4.86
-8.417
-
03
3.073
-
03
8.960
- 03
5.04
-7.474
-
03
4.309
-
03
8.628
- 03
5.22
-6.382
-
03
5.337
-
03
8.319
- 03
(Continued)

Table 38 (Continued)
212
b
0.144
a Real Imaginary Absolute Value
5.40
-5.178
03
6.141
03
8.033
03
5.58
-3.901
-
03
6.714
-
03
7.765
-
03
5.76
-2.589
-
03
7.056
-
03
7.516
-
03
5-94
-1.281
-
03
7.168
-
03
7.281
-
03
6.12
-1.154
-
05
7.061
-
03
7.061
-
03
6.30
1.185
-
03
6.751
_
03
6.854
-
03
6.48
2.281
-
03
6.257
-
03
6.659
-
03
6.66
3.248
-
03
5.602
-
03
6.475
-
03
6.84
4.067
-
03
4.813
-
03
6.301
-
03
7.02
4.722
-
03
3.920
-
03
6.136
-
03
7.20
5.200
-
03
2.952
-
03
5.980
-
03
7.38
5.498
-
03
1.943
-
03
5.832
-
03
7.56
5.615
-
03
9.223
-
o4
5.690
-
03
7.74
5.555
-
03
-7.873
-
05
5.556
-
03
7.92
5.328
-
03
-1.032
-
03
5.427
-
03
8.10
4.949
-
03
-1.911
-
03
5.305
-
03
8.28
4.434
-
03
-2.693
-
03
5.188
-
03
8.46
3.805
-
03
-3.360
-
03
5.076
-
03
8.64
3.085
-
03
-3.895
-
03
4.969
-
03
8.82
2.299
-
03
-4.289
-
03
4.866
-
03
9.00
1.474
_
03
-4.534
_
03
4.768
_
03
9.18
6.361
-
o4
-4.629
-
03
4.673
-
03
9.36
-1.895
-
04
-4.578
-
03
4.582
-
03
9.54
-9.782
-
04
-4.387
-
03
4.495
-
03
9.72
-1.707
-
03
-4.067
-
03
4.4ll
-
03
9.90
-2.357
-
03
-3.632
-
03
4.330
-
03
10.08
-2.910
-
03
-3.100
-
03
4.251
-
03
10.26
-3.353
-
03
-2.490
-
03
4.176
-
03
10.44
-3.676
-
03
-1.823
-
03
4.103
-
03
10.62
-3.874
-
03
-1.122
-
03
4.033
-
03
10.80
-3.944
-
03
-4.096
-
o4
3.965
-
03
10.98
-3.889
-
03
2.923
-
o4
3.900
-
03
11.16
-3.714
-
03
9.624
-
04
3.837
-
03
11.34
-3.428
-
03
1.581
-
03
3.775
-
03
11.52
-3.044
-
03
2.131
-
03
3.716
-
03
11.70
-2.576
-
03
2.597
-
03
3.658
-
03
11.88
-2.042
-
03
2.968
-
03
3.602
-
03
12.06
-1.459
-
03
3.234
-
03
3.548
-
03
12.24
-8.481
-
o4
3.391
-
03
3.496
-
03
12.42
-2.280
-
04
3.437
-
03
3.445
-
03
12.60
3.816
-
04
3.374
_
03
3.395
-
03
12.78
9.619
-
04
3.206
-
03
3.347
-
03
12.96
1.496
-
03
2.942
-
03
3.300
-
03
(Continued)

Table 38 (Continued)
213
b
0.720
a Real Imaginary Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.18
1.902
-
02
-2.426
-
03
1.917
-
02
0.36
3.248
-
02
-4.805
-
03
3.283
-
02
0.54
3.847
-
02
-7.089
-
03
3.912
-
02
0.72
3.869
-
02
-9.235
-
03
3.978
-
02
0.90
3.582
-
02
-1.120
-
02
3.753
-
02
1.08
3.174
-
02
-I.295
-
02
3.428
-
02
1.26
2.739
-
02
-1.445
-
02
3.097
-
02
1.44
2.315
-
02
-I.567
-
02
2.796
-
02
1.62
1.913
-
02
-I.66O
-
02
2.533
-
02
1.80
1.536
-
02
-I.722
-
02
2.307
_
02
1.98
1.183
-
02
-1.753
-
02
2.114
-
02
2.16
8.533
-
03
-I.752
-
02
1.948
-
02
2.34
5.474
-
03
-I.72O
-
02
1.805
-
02
2.52
2.660
-
03
-1.659
-
02
1.680
-
02
2.70
9.957
-
05
-I.57I
_
02
1.571
-
02
2.88
-2.192
-
03
-I.458
-
02
1.475
-
02
3.06
-4.201
-
03
-1.324
-
02
1.389
-
02
3-24
-5.916
-
03
-I.I72
-
02
1.313
-
02
3-42
-7.327
-
03
-I.OO6
-
02
1.245
-
02
3.60
-8.428
_
03
-8.3OI
-
03
1.183
-
02
3.78
-9.219
-
03
-6.482
-
03
1.127
-
02
3.96
-9.705
-
03
-4.646
-
03
1.076
-
02
4.14
-9.896
-
03
-2.836
-
03
1.029
-
02
4.32
-9.806
-
03
-1.091
-
03
9.867
-
03
4.50
-9.457
-
03
5.536
-
04
9.473
-
03
4.68
-8.873
-
03
2.063
-
03
9.110
-
03
4.86
-8.083
-
03
3.4io
-
03
8.773
-
03
5.04
-7.120
-
03
4.569
-
03
8.460
-
03
5.22
-6.019
-
03
5.523
-
03
8.169
-
03
5.40
-4.816
-
03
6.258
-
03
7.897
-
03
5.58
-3.550
-
03
6.768
-
03
7.642
-
03
5.76
-2.257
-
03
7.051
-
03
7.404
-
03
5.94
-9.739
-
04
7.113
-
03
7.179
-
03
6.12
2.646
-
04
6.963
-
03
6.968
-
03
6.30
1.427
-
03
6.617
-
03
6.769
-
03
6.48
2.484
-
03
6.094
-
03
6.581
-
03
6.66
3.412
-
03
5.419
-
03
6.403
-
03
6.84
4.191
-
03
4.616
-
03
6.235
-
03
7.02
4.806
-
03
3.716
-
03
6.075

03
7.20
5.247
-
03
2.748
-
03
5.923
-
03
7.38
5.509
-
03
1.744
-
03
5.779
-
03
(Continued)

Table 38 (Continued)
214
b
0.720
2.200
a
Real
Imaginary
Absolute Value
7.56
5-593
_
03
7.338
_
o4
5.641
- 03
7.74
5.504
-
03
-2.523
-
o4
5.510
- 03
7.92
5.252
-
03
-I.I87
-
03
5.385
- 03
8.10
4.852
-
03
-2.045
-
03
5.265
- 03
8.28
4.321
-
03
-2.803
-
03
5.151
- 03
8.46
3.680
-
03
-3.445
-
03
5.041
- 03
8.64
2.953
-
03
-3.955
-
03
4.936
- 03
8.82
2.165
-
03
-4.323
-
03
4.835
- 03
9.00
1-342
-
03
-4.545
_
03
4.739
- 03
9.18
5.089
-
04
-4.618
-
03
4.646
- 03
9.36
-3.O8I
-
o4
-4.546
-
03
4.556
- 03
9.54
-I.O85
-
03
-4.337
-
03
4.470
- 03
9-72
-I.8OO
-
0.3
-4.001
-
03
4.388
- 03
9.90
-2.434
-
03
-3.554
-
03
4.308
- 03
10.08
-2.97O
-
03
-3.013
-
03
4.231
- 03
10.26
-3.396
-
03
-2.397
-
03
4.157
- 03
10.44
-3.702
-
03
-1.728
-
03
4.085
- 03
10.62
-3.882
-
03
-1.027
-
03
4.016
- 03
10.80
-3.936
-
03
-3.181
04
3.949
- 03
10.98
-3866
-
03
3.781
-
04
3.884
- 03
11.16
-3.677
-
03
i.o4o
-
03
3.821
- 03
11.34
-3.380
-
03
1.649
-
03
3.761
- 03
11.52
-2.986
-
03
2.188
-
03
3-702
- 03
11.70
-2.512
-
03
2.64i
-
03
3.645
- 03
11.88
-1.973
-
03
2.999
-
03
3.590
- 03
12.06
-1.388
-
03
3.252
-
03
3.536
- 03
12.24
-7.769
-
o4
3.396
-
03
3.484
- 03
12.42
-1.590
-
04
3.430
-
03
3.434
- 03
12.60
4.463
-
o4
3.355
-
03
3.385
- 03
12.78
1.021
-
03
3.177
-
03
3-337
- 03
12.96
1.547
-
03
2.904
-
03
3.291
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.18
8.055
-
o4
-1.511
-
03
1.713
- 03
0.36
1.526
-
03
-2.988
-
03
3-355
- 03
0.54
2.085
-
03
-4.397
-
03
4.867
- 03
0.72
2*429
-
03
-5.706
-
03
6.201
- 03
0.90
2.523
-
03
-6.885
-
03
7.333
- 03
1.08
2.357
-
03
-7.909
-
03
8.253
- 03
1.26
1.943
-
03
-8.754
-
03
8.967
- 03
1.44
1.306
-
03
-9.404
-
03
9.495
- 03
1.62
4.847
-
o4
-9.845
-
03
9.857
- 03
1.80
-4.772
-
04
-1.007
-
02
1.008
- 02
1.98
-1.533
-
03
-1.007
-
02
1.019
- 02
2.16
-2.635
-
03
-9.863
-
03
1.021
- 02
2.34
-3.739
-
03
-9.444
-
03
1.016
- 02
2.52
-4.803
-
03
-8.829
-
03
1.005
- 02
(Continued)

Table 38 (Continued)
215
b
2.200
a
Real
Imaginary
Absolute Value
2.70
-5.789
_
03
-8.037
_
03
9.905 03
2.88
-6.665
-
03
-7.089
-
03
9.730 03
3.06
-7-402
-
03
-6.010
-
03
9.534 03
3-24
-7.979
-
03
-4.829
-
03
9.326 03
3-42
-8.379
-
03
-3.576
-
03
9.110 03
3.60
-8.592
-
03
-2.282
-
03
8.890 03
3.78
-8.614
-
03
-9.798
-
o4
8.670 03
3.96
-8.445
-
03
2.995
-
o4
8.451 03
4.14
-8.093
-
03
1.525
-
03
8.235 03
4.32
-7.568
-
03
2.669
-
03
8.024 03
4.50
-6.886
-
03
3.704
_
03
7.819 03
4.68
-6.067
-
03
4.609
-
03
7.620 03
4.86
-5.135
-
03
5.365
-
03
7.426 03
5.04
-4.115
-
03
5-957
-
03
7.240 03
5.22
-3.035
-
03
6.374
-
03
7.059 03
5.40
-1.923
_
03
6.612
_
03
6.886 03
5.58
-8.086
-
04
6.670
-
03
6.719 03
5.76
2.799
-
o4
6.552
-
03
6.558 03
5.94
1.315
-
03
6.266
-
03
6.403 03
6.12
2.273
-
03
5.826
-
03
6.254 03
6.30
3.130
-
03
5.248
-
03
6.111 03
6.48
3.867
-
03
4.552
-
03
5.973 03
6.66
4.469
-
03
3.759
-
03
5.84o 03
6.84
4.925
-
03
2.896
-
03
5.713 03
7.02
5.225
-
03
1.986
-
03
5.590 03
7.20
5.369
-
03
1.056
-
03
5.472 03
7.38
5-357
-
03
1.334
-
o4
5.358 03
7.56
5.194
-
03
-7.582
-
o4
5.249 03
7.74
4.890
-
03
-1.595
-
03
5.143 03
7.92
4.458
-
03
-2.355
-
03
5.o4i 03
8.10
3-914
-
03
-3.019
-
03
4.943 03
8.28
3-277
-
03
-3.573
-
03
4.848 03
8.46
2.569
-
03
-4.004
-
03
4.757 03
8.64
1.811
-
03
-4.303
-
03
4.669 03
8.82
1.027
-
03
-4.467
-
03
4.584 03
9.00
2.397
_
04
-4.495
-
03
4.501 03
9.18
-5.280
-
o4
-4.390
-
03
4.422 03
9.36
-1.255
-
03
-4.159
-
03
4.344 03
9.54
-1.920
-
03
-3.814
-
03
4.270 03
9.72
-2.507
-
03
-3.367
-
03
4.198 03
9.90
-3.001
_
03
-2.834
-
03
4.128 03
10.08
-3.390
-
03 .
-2.234
-
03
4.060 03
10.26
-3.666
-
03
-1.586
-
03
3.995 03
10.44
-3.824
-
03
-9.107
-
o4
3.931 03
10.62
-3.862
-
03
-2.285
-
o4
3.869 03
(Continued)

Table 38 (Continued)
216
b
2.20G
3.620
a Real Imaginary Absolute Value
10.80
-3.784
-
03
4.4oo
-
04
3.809
_
03
10.98
-3.594
-
03
1.075
-
03
3.751
-
03
11.16
-3.301
-
03
1.660
-
03
3.695
-
03
11.34
-2.917
-
03
2.177
-
03
3.64o
-
03
11.52
-2.455
-
03
2.6i4
-
03
3.587
-
03
11.70
-I.932
-
03
2.960
-
03
3.535
_
03
11.88
-1.366
-
03
3.206
-
03
3-484
-
03
12.06
-7.726
-
04
3.347
-
03
3.435
-
03
12.24
-1.722
-
04
3.383
-
03
3.388
-
03
12.42
4.173
-
04
3.315
-
03
3-341
-
03
12.60
9.785
-
o4
3-147
-
03
3.296
_
03
12.78
1.495
-
05
2.888
-
03
3.252
-
03
12.96
1-953
-
03
2.547
-
03
3.209
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.18
-4.219
-
04
-4.392
-
o4
6.091
-
o4
0.36
-8.481
-
o4
-8.615
-
o4
1.209
-
03
0.54
-1.282
-
03
-1.250
-
03
1.791
-
03
0.72
-1.726
-
03
-1.590
-
03
2.347
-
03
0.90
-2.180
-
03
-1.867
_
03
2.870
_
03
1.08
-2.642
-
03
-2.069
-
03
3.356
-
03
1.26
-3.107
-
03
-2.187
-
03
3.800
-
03
1.44
-3.569
-
03
-2.212
-
03
4.199
-
03
1.62
-4.019
-
03
-2.142
-
03
4.554
-
03
1.80
-4.446
-
03
-I.975
-
03
4.865
-
03
1.98
-4.838
-
03
-I.7II
-
03
5.131
-
03
2.16
-5.182
-
03
-I.357
-
03
5.357
-
03
2.34
-5.467
-
03
-9.202
-
o4
5.544
-
03
2.52
-5.680
-
03
-4.100
-
04
5.695
-
03
2.70
-5.812
_
03
1.605
_
04
5.814
_
03
2.88
-5.852
-
03
7.765
-
o4
5.904
-
03
3.06
-5.795
-
03
1.422
-
03
5.967
-
03
3-24
-5.637
-
03
2.079
-
03
6.008
-
03
3-42
-5-375
-
03
2.730
-
03
6.029
-
03
3.60
-5.012
-
03
3.356
-
03
6.032
-
03
3.78
-4.552
-
03
3.940
-
03
6.020
-
03
3.96
-4.001
-
03
4.465
-
03
5.996
-
03
4.14
-3.371
-
03
4.915
-
03
5.960
-
03
4.32
-2.674
-
03
5.277
-
03
5.916
-
03
4.50
-1.924
-
03
5.540
_
03
5.864
_
03
4.68
-1.138
-
03
5.694
-
03
5.806
-
03
4.86
-3.328
-
04
5-733
-
03
5.743
-
03
5.04
4.725
-
o4
5 656
-
03
5.676
-
03
5.22
1.259
-
03
5.462
-
03
5.606
-
03
5.40
2.009
_
03
5.155
-
03
5-533
_
03
5.58
2.704
-
03
4.74i
-
03
5.458
-
03
(Continued)

Table 38 (Continued)
217
b
3.620
5.O6O
Real
Imaginary
Absolute Value
5.76
3.328
-
03
4.230
-
03
5.383
-
03
5.94
3.867
-
03
3.634
-
03
5.306
-
03
6.12
4.306
-
03
2.967
-
03
5.229
-
03
6.30
4.638
-
03
2.245

03
5.153
_
03
6.48
4.854
-
03
1.486
-
03
5.076
-
03
6.66
4.949
-
03
7.093
-
04
5.000
-
03
6.84
4.924
-
03
-6.667
-
05
4.924
-
03
7.02
4.780
-
03
-8.226
-
04
4.850
-
03
7.20
4.521
-
03
-1.540
-
03
4.776
-
03
7.38
4.157
-
03
-2.201
-
03
4.704
-
03
7.56
3.697
-
03
-2.791
-
03
4.632
-
03
7.74
3-155
-
03
-3.295
-
03
4.562
-
03
7.92
2.547
-
03
-3.702
-
03
4.493
-
03
8.10
1.888
-
03
-4.003
-
03
4.426
-
03
8.28
1-197
-
03
-4.192
-
03
4.359
-
03
8.46
4.923
-
04
-4.266
-
03
4.294
-
03
8.64
-2.069
-
o4
-4.226
-
03
4.231
-
03
8.82
-8.826
-
o4
-4.074
-
03
4.169
-
03
9.00
-I.517
_
03
-3.817
_
03
4.108
-
03
9.18
-2.094
-
03
-3.464
-
03
4.048
-
03
9.36
-2.600
-
03
-3.026
-
03
3.990
-
03
9.54
-3.022
-
03
-2.517
-
03
3-933
-
03
9.72
-3.350
-
03
-1.952
-
03
3.877
-
03
9.90
-3-577
-
03
-1.348
-
03
3.823
-
03
10.08
-3.700
-
03
-7.218
-
o4
3.769
-
03
10.26
-3.716
-
03
-9.129
-
05
3.717
-
03
10.44
-3.629
-
03
5.259
-
04
3.667
-
03
10.62
-3-441
-
03
1.113
-
03
3.617
-
03
10.80
-3.162
-
03
1.654
-
03
3.568
-
03
10.98
-2.800
-
03
2.135
-
03
3.521
-
03
11.16
-2.367
-
03
2.543
-
03
3.474
-
03
11.34
-1.877
-
03
2.870
-
03
3-429
-
03
11.52
-1.346
03
3.106
-
03
3.385
-
03
11.70
-7.887
-
04
3.247
-
03
3.341
-
03
11.88
-2.220
-
o4
3.292
-
03
3.299
-
03
12.06
3.377
-
o4
3.240
-
03
3.258
-
03
12.24
8.745
-
04
3.096
-
03
3.217
-
03
12.42
1.373
-
03
2.865
-
03
3.178
-
03
12.60
1.821
-
03
2.557
-
03
3.139
-
03
12.78
2.205
-
03
2.180
-
03
3.101
-
03
12.96
2.516
-
03
1.749
-
03
3-064

03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.18
-2.616
-
o4
1.624
-
o4
3.079
-
o4
0.36
-5.182
-
o4
3.283
-
04
6.134
-
o4
0.54
-7.647
-
o4
5.011
-
o4
9-143
-
04
0.72
-9.963
-
o4
6.835
-
04
1.208
-
03
(Continued)

Table 38 (Continued)
218
b
5.O6O
a
Real
Imaginary
Absolute
Value
0.90
-1.208
_
03
8.780
_
04
1.493
_
03
1.08
-1.395
-
03
1.086
-
03
1.768
-
03
1.26
-I.552
-
03
1.308
-
03
2.029
-
03
1.44
-I.674
-
03
1.544
-
03
2.278
-
03
1.62
-1.759
-
03
1.793
-
03
2.511
-
03
1.80
-I.8OI
-
03
2.051
-
03
2.730
-
03
1.98
-1.797
-
03
2.316
-
03
2.932
-
03
2.16
-1.746
-
03
2.584
-
03
3.118
-
03
2.34
-1.645
-
03
2.847
-
03
3.288
-
03
2.52
-1.493
-
03
3.102
-
03
3.442
-
03
2.70
-I.291
-
03
3.340
-
03
3.581
-
03
2.88
-1.041
-
03
3.555
-
03
3.704
-
03
3.06
-7.452
-
04
3.740
-
03
3.813
-
03
3-24
-4.077
-
o4
3.887
-
03
3.908
-
03
3-42
-3.388
-
05
3.990
-
03
3.991
-
03
3-60
3.696
-1
04
4.044
-
03
4.061
-
03
3.78
7.952
-
o4
4.042
-
03
4.119
-
03
3.96
1.234
-
03
3.980
-
03
4.167
-
03
4.14
1.678
-
03
3.857
-
03
4.206
-
03
4.32
2.115
-
03
3.669
-
03
4.235
-
03
4.50
2.536
-
03
3-419
-
03
4.257
-
03
4.68
2.931
-
03
3.106
-
03
4.271
-
03
4.86
3.290
-
03
2.735
-
03
4.278
-
03
5.04
3.602
-
03
2.311
-
03
4.279
-
03
5.22
3.859
-
03
1.840
-
03
4.275
-
03
5.40
4.053
-
03
1.330
-
03
4.266
-
03
5.58
4.179
-
03
7.918
-
o4
4.253
-
03
5.76
4.229
-
03
2.347
-
o4
4.236
-
03
5.94
4.202
-
03
-3.297
-
04
4.215
-
03
6.12
4.096
-
03
-8.896
-
o4
4.192
-
03
6.30
3.911
-
03
-1.433
-
03
4.165
-
03
6.48
3.650
-
03
-1.947
-
03
4.137
-
03
6.66
3.317
-
03
-2.421
-
03
4.107
-
03
6.84
2.919
-
03
-2.843
-
03
4.075
-
03
7.02
2.462
-
03
-3.204
-
03
4.o4i
-
03
7.20
1.957
-
03
-3.496
-
03
4.006
-
03
7.38
1.416
-
03
-3.710
-
03
3.971
-
03
7.56
8.486
-
o4
-3.841
-
03
3.934
-
03
7.74
2.692
-
o4
-3.888
-
03
3.897
-
03
7.92
-3.093
-
04
-3-847
-
03
3.859
-
03
8.10
-8.736

o4
-3.720
-
03
3.821
-
03
8.28
-1.410
-
03
-3.510
-
03
3.783
-
03
8.46
-1.907
-
03
-3.222
-
03
3-744
-
03
8.64
-2.352
-
03
-2.863
-
03
3.705
-
03
8.82
-2.735
-
03
-2.442
-
03
3.667
-
03
(Continued)

Table 38 (Concluded)
219
b
5.O6O
a
Real
Imaginary
Absolute Value
9.00
-3-047
_
03
-I.97O
_
03
3.628 03
9.18
-3.280
-
03
-1.457
-
03
3.590 03
9.36
-3.431
-
03
-9.183
-
04
3.551 03
9.54
-3.494
-
03
-3.657
-
o4
3.513 03
9.72
-3.471
-
03
I.865
-
o4
3.476 03
9.90
-3.36I
-
03
7.246
-
o4
3.438 03
10.08
-3.I69
-
03
1.235
-
03
3.401 03
10.26
-2.9OO
-
03
1.705
-
03
3-364 03
10.44
-2.563
-
03
2.123
-
03
3.328 03
10.62
-2.I65
-
03
2.480
-
03
3.292 03
10.80
-I.72O
-
03
2.766
-
03
3.257 03
10.98
-I.237
-
03
2.975
-
03
3-222 03
11.16
-7.314
-
04
3.102
-
03
3.187 03
11.34
-2.158
-
o4
3.146
-
03
3.153 03
11.52
2.960
-
o4
3.105
-
03
3.119 03
11.70
7.904
-
04
2.983
-
03
3.086 03
11.88
1.255
-
03
2.784
-
03
3.054 03
12.06
1.676
-
03
2.514
-
03
3.021 03
12.24
2.045
-
03
2.181
-
03
2.990 03
12.42
2.352
-
03
1.795
-
03
2.958 03
12.60
2.589
-
03
1.367
-
03
2.928 03
12.78
2.751
-
03
9.103
-
o4
2.897 03
12.96
2.834
-
03
4.369
-
o4
2.868 03

Table 39
Value of l(0.48,a,b)
220
b
0.000
0.192
a Real Imaginary Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
4.091
-
01
-5.997
-
03
4.092
-
01
0.48
7.351
-
01
-1.179
-
02
7.352
-
01
0.72
2.054
-
01
-I.718
-
02
2.06l
-
01
0.96
1.183
-
01
-2.198
-
02
1.204
01
1.20
8.034
_
02
-2.604
_
02
8.446
_
02
1.44
5.832
-
02
-2.923
-
02
6.523
-
02
1.68
4.288
-
02
-3.146
-
02
5.319
-
02
1.92
3.066
-
02
-3.267
-
02
4.481
-
02
2.16
2.042
-
02
-3.286
-
02
3.868
-
02
2.40
1.174
-
02
-3.204
-
02
3.412
-
02
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
2.113
-
01
-5.975
-
03
2.114
-
01
0.48
2.737
-
01
-1.174
-
02
2.74o
-
01
0.72
1.742
-
01
-1.711
-
02
1.750
-
01
0.96
1.111
-
01
-2.189
-
02
1.132
-
01
1.20
7.756
-
02
-2.594
-
02
8.178
-
02
1.44
5*666
-
02
-2.911
-
02
6.370
-
02
1.68
4.169
-
02
-3.133
-
02
5.215
-
02
1.92
2.989
-
02
-3.254
-
02
4.418
-
02
2.16
2.004
-
02
-3.272
-
02
3.836
-
02
2.40
1.158
-
02
-3.189

02
3-393
_
02
2.64
4.276
-
03
-3.014
-
02
3.044
-
02
2.88
-1.966
-
03
-2.755
-
02
2.762
-
02
3.12
-7.164
-
03
-2.425
-
02
2.528
-
02
3.36
-I.131
-
02
-2.040
-
02
2.332
-
02
3.60
-1.441
-
02
-1.616
_
02
2.165
_
02
3.84
-1.646
-
02
-1.172
-
02
2.021
-
02
4.08
-1.750
-
02
-7.258
-
03
1.895
-
02
4.32
-1.760
-
02
-2.947
-
03
1.784
-
02
4.56
-1.682
-
02
1.053
-
03
1.686
-
02
4.80
-1.530
-
02
4.600
-
03
1.598
-
02
5.04
-1.316
-
02
7.580
-
03
1.519
-
02
5.28
-1.055
-
02
9.906
-
03
1.447
-
02
5.52
-7.629
-
03
1.152
-
02
1.382
-
02
5.76
-4.561
-
03
1.242
-
02
1.323
-
02
6.00
-1.503
-
03
1.259
_
02
1.268
-
02
6.24
1.398
-
03
1.210
-
02
1.218
-
02
6.48
4.010
-
03
1.101
-
02
1.172
-
02
6.72
6.225
-
03
9.422
-
03
1.129
-
02
6.96
7.958
-
03
7.442
-
03
1.090
-
02
(Continued)

Table 39 (Continued)
221
b
0.192
a
Real
Imaginary
Absolute
: Value
7.20
9.151
_
03
5.199
_
03
1.053
- 02
7.44
9.780
-
03
2.823
-
03
1.018
- 02
7.68
9.846
-
03
4.444
-
04
9.856
- 03
7.92
0.378
-
03
-1.814
-
03
9.552
- 03
8.16
8.433
-
03
-3.841
-
03
9.267
- 03
8.4o
7.087
-
03
-5.545
-
03
8.998
- 03
8.64
5.431
-
03
-6.854
-
03
8.745
- 03
8.88
3.570
-
03
-7.720
-
03
8.506
- 03
9.12
1.613
-
03
-8.120
-
03
8.279
- 03
9.36
-3.318
-
04
-8.057
-
03
8.064
- 03
9.60
-2.161
-
03
-7.557
-
03
7.860
- 03
9.84
-3.783
-
03
-6.668
-
03
7.667
- 03
10.08
-5.120
-
03
-5.456
-
03
7.482
- 03
IO.32
-6.114
-
03
-4.001
-
03
7.307
- 03
IO.56
-6.726
-
03
-2.392
-
03
7.139
- 03
10.80
-6.941
-
03
-7.223
-
04
6.979
- 03
11.04
-6.764
-
03
9.142
-
04
6.826
- 03
11.28
-6.222
-
03
2.431
-
03
6.680
- 03
11.52
-5.358
-
03
3.749
-
03
6.539
- 03
11.76
-4.233
-
03
4.807
-
03
6.405
- 03
12.00
-2.918
-
03
5.556
-
03
6.276
- 03
12.24
-1.494
-
03
5.968
-
03
6.152
- 03
12.48
-4.096
-
05
6.033
-
03
6.033
- 03
12.72
1.359
-
03
5.760
-
03
5.918
- 03
12.96
2.632
-
03
5.178
-
03
5.808
- 03
13.20
3.712
-
03
4.328
-
03
5.702
- 03
13.44
4.547
-
03
3.268
-
03
5-599
- 03
13.68
5.099
-
03
2.063
-
03
5.501
- 03
13.92
5-348
-
03
7.853
-
04
5.405
- 03
14.16
5.290
-
03
-4.934
-
o4
5.313
- 03
i4.4o
4.939
-
03
-1.702
-
03
5.224
- 03
14.64
4.324
-
03
-2.776
-
03
5.138
- 03
14.88
3.487
-
03
-3.660
-
03
5.055
- 03
15.12
2.481
-
03
-4.311
-
03
4.974
- 03
15.36
1.368
-
03
-4.701
-
03
4.896
- 03
15.60
2.133
-
o4
-4.816
_
03
4.821
- 03
15.84
-9.189
-
o4
-4.658
-
03
4.747
- 03
16.08
-1.966
-
03
-4.243
-
03
4.676
- 03
16.32
-2.872
-
03
-3.603
-
03
4.607
- 03
16.56
-3.590
-
03
-2.779
-
03
4.540
- 03
16.80
-4.087
-
03
-1.823
-
03
4.475
- 03
17.04
-4.34o
-
03
-7.917
-
o4
4.412
- 03
17.28
-4.343
-
03
2.555
-
o4
4.350
- 03
(Continued)

Table 39 (Continued.)
222
O.96O
a
Real
Imaginary
Absolute
Value
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.24
2.120
-
02
-5.458
-
03
2.189
- 02
0.48
3.621
-
02
-I.072
-
02
3.776
- 02
0.72
4.274
-
02
-I.56I
-
02
4.550
- 02
0.96
4.245
-
02
-1.995
-
02
4.690
- 02
1.20
3.8l8
-
02
-2.36O
_
02
4.488
- 02
1.44
3.205
-
02
-2.644
-
02
4.155
- 02
1.68
2.525
-
02
-2.838
-
02
3-799
- 02
I.92
1.837
-
02
-2.938
-
02
3-465
- 02
2.16
1.172
-
02
-2.942
-
02
3.167
- 02
2.40
5-487
_
03
-2.854
_
02
2.906
- 02
2.64
-1.743
-
04
-2.679
-
02
2.679
- 02
2.88
-5.148
-
03
-2.427
-
02
2.481
- 02
3.12
-9.336
-
03
-2.111
-
02
2.309
- 02
3.36
-1.267
-
02
-1.746
-
02
2.157
- 02
3.60
-I.510
_
02
-1.346
-
02
2.023
- 02
3-84
-1.661
-
02
-9.305
-
03
1.904
- 02
4.08
-I.722
-
02
-5.151
-
03
1.798
- 02
4.32
-1.698
-
02
-1.167
-
03
1.702
- 02
4.56
-1.597
-
02
2.498
-
03
I.616
- 02
4.80
-1.428
-
02
5.713
-
03
1.538
- 02
5-04
-I.205
-
02
8.371
-
03
1.467
- 02
5.28
-9-4i4
-
03
i.o4o
-
02
1.402
- 02
5.52
-6.523
-
03
1.174
-
02
1.343
- 02
5.76
-3.529
-
03
1.239
-
02
1.288
- 02
6.00
-5.819
-
o4
1.237
-
02
1.238
- 02
6.24
2.181
-
03
1.171
-
02
1.191
- 02
6.48
4.639
-
03
1.050
-
02
1.148
- 02
6.72
6.689
-
03
8.830
-
03
1.108
- 02
6.96
8.256
-
03
6.809
-
03
1.070
- 02
7.20
9.291
-
03
4.561
-
03
1.035
- 02
7.44
9-774
-
03
2.212
-
03
1.002
- 02
7.68
9.711
-
03
-1.122
-
04
9.712
- 03
7.92
9.137
-
03
-2.294
-
03
9.421
- 03
8.16
8.110
-
03
-4.230
-
03
9-147
- 03
8.4o
6.706
-
03
-5.833
_
03
8.888
- 03
8.64
5.019
-
03
-7.037
-
03
8.644
- 03
8.88
3.151
-
03
-7.800
-
03
8.412
- 03
9.12
1.209
-
03
-8.103
-
03
8.193
- 03
9.36
-7.009
-
o4
-7.954
-
03
7.985
- 03
9.60
-2.48o
_
03
-7.381
_
03
7.787
- 03
9.84
-4.039
-
03
-6.436
-
03
7.598
- 03
10.08
-5.307
-
03
-5.184
-
03
7.419
- 03
10.32
-6.227
-
03
-3.707
-
03
7.247
- 03
10.56
-6.767
-
03
-2.094
-
03
7.084
- 03
(Continued)

Table 39 (Continued)
223
b
a
Real
Imaginary
Absolute
: Value
O.96O
10.80
-6.914
_
03
-4.366
_
o4
6.927
- 03
11.04
-6.675
-
03
1.174
-
03
6.778
- 03
11.28
-6.O8I
-
03
2.653
-
03
6.634
- 03
11.52
-5.I77
-
03
3.925
-
03
6.497
- 03
11.76
-4.025
-
03
4.931
-
03
6.365
- 03
12.00
-2.697
-
03
5.625
-
03
6.238
- 03
12.24
-I.27I
-
03
5.983
-
03
6.117
- 03
12.48
1.706
-
04
5.997
-
03
5-999
- 03
12.72
1.549
-
03
5.679
-
03
5.887
- 03
12.96
2.792
-
03
5.059
-
03
5.778
- 03
13.20
3.835
-
03
4.181
-
03
5.674
- 03
13.44
4.629
-
03
3.103
-
03
5.573
- 03
13.68
5.139
-
03
1.890
-
03
5.475
- 03
13.92
5-346
-
03
6.131
-
o4
5.381
- 03
l4.i6
5.250
-
03
-6.548
-
04
5.290
- 03
l4.4o
4.865
-
03
-1.844
-
03
5.203
- 03
14.64
4.221
-
03
-2.893
-
03
5.118
- 03
14.88
3-364
-
03
-3.747
-
03
5.035
- 03
15.12
2.346
-
03
-4.365
-
03
4.956
- 03
15.36
1.229
-
03
-4.721
-
03
4.878
- 03
15.60
7.690
-
05
-4.803
-
03
4.804
- 03
15-84
-1.044
-
03
-4.614
-
03
4.731
- 03
16.08
-2.074
-
03
-4.173
-
03
4.661
- 03
16.32
-2.958
-
03
-3.512
-
03
4.592
- 03
16.56
-3.651
-
03
-2.674
-
03
4.526
- 03
16.80
-4.121
-
03
-1.710
-
03
4.461
- 03
17.04
-4.346
-
03
-6.771
-
04
4.399
- 03
17.28
-4.322
-
03
3.652
-
04
4.338
- 03
2.900
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.24
-2.856
-
o4
-2.232
-
03
2.250
- 03
0.48
-6.710
-
04
-4.357
-
03
4.4o8
- 03
0.72
-1.239
-
03
-6.272
-
03
6.394
- 03
O.96
-2.042
-
03
-7.887
-
03
8.147
- 03
1.20
-3.095
-
03
-9.122
-
03
9.633
- 03
1.44
-4.375
-
03
-9.919
-
03
1.084
- 02
1.68
-5.825
-
03
-1.024
-
02
1.178
- 02
1.92
-7-364
-
03
-1.007
-
02
1.247
- 02
2.16
-8.898
-
03
-9.410
-
03
1.295
- 02
2.40
-1.032
-
02
-8.299
-
03
1.325
- 02
2.64
-1.155
-
02
-6.787
-
03
1.339
- 02
2.88
-1.248
-
02
-4.943
-
03
1.342
- 02
3.12
-1.304
-
02
-2.854
-
03
1.335
- 02
3.36
-I.320
-
02
-6.173
-
04
1.322
- 02
3.60
-1.292
-
02
1.663
-
03
1.303
- 02
3.84
-I.219
-
02
3.883
-
03
I.280
- 02
(Continued)

Table 39 (Continued)
224
a
Real
Imaginary
Absolute Value
4.08
-I.IO5
_
02
5.941
03
1.254 -
02
4.32
-9.519
-
03
7.744
-
03
1.227 -
02
4.56
-7.671
-
03
9.213
-
03
1.199 -
02
4.80
-5.583
_
03
1.028
_
02
1.170 -
02
5.04
-3.342
-
03
1.091
-
02
l.l4l -
02
5.28
-1.046
-
03
1.108
-
02
1.113 -
02
5-52
1.207
-
03
1.078
-
02
1.084 -
02
5.76
3.320
-
03
1.003
-
02
1.057 -
02
6.00
5.203
-
03
8.887
-
03
I.030 -
02
6.24
6.780
-
03
7.400
-
03
i.oo4 -
02
6.48
7.989
-
03
5-647
-
03
9.783 -
03
6.72
8.786
-
03
3.713
-
03
9.538 -
03
6.96
9.146
-
03
1.692
-
03
9.302 -
03
7.20
9.068
_
03
-3.206
_
04
9.074 -
03
7.44
8.568
-
03
-2.232
-
03
8.854 -
03
7.68
7.684
-
03
-3.956
-
03
8.643 -
03
7.92
6.470
-
03
-5.418
-
03
8.439 -
03
8.16
4.995
-
03
-6.557
-
03
8.243 -
03
8.40
3-339
-
03
-7.330
-
03
8.055 -
03
8.64
1.588
-
03
-7.712
-
03
7.874 -
03
8.88
-1.704
-
04
-7.698
-
03
7.700 -
03
9.12
-1.849
-
03
-7.301
-
03
7.532 -
03
9.36
-3.368
-
03
-6.556
-
03
7.370 -
03
9.60
-4.658
-
03
-5.510
-
03
7.215 -
03
9.84
-5.662
-
03
-4.227
-
03
7.065 -
03
10.08
-6.339
-
03
-2.778
-
03
6.921 -
03
10.32
-6.667
-
03
-1.242
-
03
6.782 -
03
10.56
-6.641
-
03
2.996
-
o4
6.648 -
03
10.80
-6.274
-
03
1.769
-
03
6.519 -
03
li.o4
-5.596
-
03
3.093
-
03
6.394 -
03
11.28
-4.652
-
03
4.210
-
03
6.274 -
03
11.52
-3.499
-
03
5.067
-
03
6.158 -
03
11.76
-2.203
-
03
5.630
-
03
6.045 -
03
12.00
-8.370
_
o4
5.878
-
03
5-937 -
03
12.24
5.269
-
04
5.808
-
03
5.832 -
03
12.48
1.817
-
03
5.435
-
03
5.731 -
03
12.72
2.969
-
03
4.786
-
03
5.632 -
03
12.96
3.925
-
03
3.905
-
03
5.537 -
03
13.20
4.643
-
03
2.845
-
03
5.445 -
03
13.44
5.090
-
03
1.667
-
03
5.356 -
03
13.68
5.251
-
03
4.372
-
04
5.269 -
03
13.92
5.127
-
03
-7.782
-
04
5.186 -
03
l4.l6
4.732
-
03
-I.915
-
03
5.104 -
03
l4.4o
4.094
-
03
-2.914
-
03
5.025 -
03
14.64
3.255
-
03
-3.727
-
03
4.949 -
03
(Continued)

Table 39 (Continued.)
225
b
2.900
4.840
a
Real
Imaginary
Absolute 1
Value
14.88
2.265
_
03
-4.316
_
03
4.874
03
15.12
1.182
-
03
-4.654
-
03
4.802
-
03
15.36
6.594
-
05
-4.731
-
03
4.732
-
03
15.60
-1.022
-
03
-4.550
-
03
4.663
-
03
15.84
-2.O25
-
03
-4.127
-
03
4.597
-
03
16.08
-2.89O
-
03
-3.492
-
03
4.532
-
03
16.32
-3.574
-
03
-2.684
-
03
4.470
-
03
16.56
-4.046
-
03
-I.751
-
03
4.408
-
03
16.80
-4.284
_
03
-7-480
_
04
4.349
_
03
17.04
-4.282
-
03
2.697
-
o4
4.291
-
03
17.28
-4.046
-
03
1.246
-
03
4.234
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
-7.454
-
o4
2.765
-
o4
7.950
-
o4
0.48
-1.470
-
03
5-737
-
o4
1.578
-
03
0.72
-2.153
-
03
9.105
-
o4
2.338
-
03
0.96
-2.774
-
03
1.302
-
03
3.065
-
03
1.20
-3.310
-
03
1.759
-
03
3.749
-
03
1.44
-3-741
-
03
2.286
-
03
4.384
-
03
1.68
-4.045
-
03
2.879
-
03
4.965
-
03
1.92
-4.204
-
03
3.530
-
03
5.489
-
03
2.16
-4.202
-
03
4.220
-
03
5.955
-
03
2.40
-4.027
-
03
4.926
-
03
6.363
-
03
2.64
-3.674
-
03
5.620
-
03
6.714
-
03
2.88
-3.143
-
03
6.268
-
03
7.012
-
03
3.12
-2.441
-
03
6.836
-
03
7.259
-
03
3.36
-1.585
-
03
7.289
-
03
7.460
-
03
3.60
-5.981
-
o4
7.594
-
03
7.618
-
03
3.84
4.878
-
o4
7.722
-
03
7.738
-
03
4.08
1.635
-
03
7.651
-
03
7.824
-
03
4.32
2.801
-
03
7.365
-
03
7.879
-
03
4.56
3.940
-
03
6.858
-
03
7.909
-
03
4.80
5.002
-
03
6.134
-
03
7.915
-
03
5.04
5.943
-
03
5.209
-
03
7.902
-
03
5.28
6.717
-
03
4.105
-
03
7.872
-
03
5.52
7.288
-
03
2.857
-
03
7.828
-
03
5.76
7.624
-
03
1.507
-
03
7.771
-
03
6.00
7.704
_
03
1.027
_
o4
7.705
_
03
6.24
7.518
-
03
-I.302
-
03
7.630
-
03
6.48
7.067
-
03
-2.654
-
03
7.549
-
03
6.72
6.363
-
03
-3.898
-
03
7.462
-
03
6.96
5.430
-
03
-4.984
-
03
7.371
-
03
7.20
4.304
-
03
-5.867
-
03
7.276
-
03
7.44
3.029
-
03
-6.509
-
03
7.180
-
03
7.68
1.657
-
03
-6.885
-
03
7.081
-
03
(Continued)

Table 39 (Continued)
226
b
4.84C
6.760
a Real Imaginary Absolute Value
7.92
2.427
-
04
-6.977
-
03
6.981
-
03
8.16
-1.153
-
03
-6.784
-
03
6.881
-
03
8.4o
-2.472
-
03
-6.314
_
03
6.781
-
03
8.64
-3.659
-
03
-5.590
-
03
6.681
-
03
8.88
-4.664
-
03
-4.644
-
03
6.582
-
03
9.12
-5.444
-
03
-3.521
-
03
6.483
-
03
9.36
-5.968
-
03
-2.271
-
03
6.386
-
03
9.60
-6.217
-
03
-9.506
-
04
6.289
-
03
9.84
-6.183
-
03
3.789
-
o4
6.194
-
03
10.08
-5.871
-
03
1.658
-
03
6.101
-
03
10.32
-5.301
-
03
2.830
-
03
6.009
-
03
10.56
-4.502
-
03
3.842
-
03
5.919
-
03
10.80
-3.516
-
03
4.650
-
03
5.830
-
03
11.04
-2.392
-
03
5.221
-
03
5.743
-
03
11.28
-1.185
-
03
5.532
-
03
5.658
-
03
11.52
4.716
-
05
5.574
-
03
5.574
-
03
11.76
1.245
-
03
5.350
-
03
5.493
-
03
12.00
2.352
-
03
4.875
-
03
5.413
-
03
12.24
3.317
-
03
4.178
-
03
5.335
-
03
12.48
4.097
-
03
3.296
-
03
5.258
-
03
12.72
4.656
-
03
2.277
-
03
5.183
-
03
12.96
4.974
-
03
1.172
-
03
5.110
-
03
13.20
5.038
-
03
3.765
-
05
5.038
-
03
13.44
4.852
-
03
-1.070
-
03
4.968
-
03
13.68
4.429
-
03
-2.096
-
03
4.900
-
03
13.92
3-797
-
03
-2.991
-
03
4.833
-
03
l4.l6
2.989
-
03
-3.715
-
03
4.768
-
03
i4.4o
2.052
-
03
-4.231
_
03
4.702
-
03
14.64
1.034
-
03
-4.525
-
03
4.642
-
03
14.88
-1.122
-
05
-4.581
-
03
4.581
-
03
15.12
-I.O3O
-
03
-4.402
-
03
4.521
-
03
15.36
-I.972
-
03
-4.003
-
03
4.463
-
03
15.60
-2.79I
-
03
-3.409
-
03
4.406
-
03
15.84
-3.446
-
03
-2.654
-
03
4.350
-
03
16.08
-3.909
-
03
-1.781
-
03
4.296
-
03
16.32
-4.159
-
03
-8.358
-
04
4.242
-
03
16.56
-4.188
-
03
1.304
-
o4
4.190
-
03
16.80
-3.999
_
03
1.067
-
03
4.139
-
03
17.04
-3.606
-
03
1.928
-
03
4.089
-
03
17.28
-3.035
-
03
2.668
-
03
4.o4o
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.24
2.426
-
o4
3.250
-
04
4.056
-
o4
0.48
4.913
-
o4
6.415
-
04
8.080
-
.04
0.72
7.517
-
o4
9.408
-
o4
1.204
-
03
0.96
1.029
-
03
1.214
-
03
1.591
-
03
(Continued)

Table 39 (Continued)
227
b
6.760
a
Real
Imaginary
Absolute Val
1.20
1.326
_
03
1.453
_
03
1.967 -
03
1.44
1.645
-
03
1.648
-
03
2.328 -
03
1.68
1.985
-
03
1.790
-
03
2.673 -
03
1.92
2.345
-
03
1.872
-
03
3.001 -
03
2.16
2.719
-
03
1.885
-
03
3.308 -
03
2.40
3.101
-
03
1.822
_
03
3.596 -
03
2.64
3.479
-
03
1.678
-
03
3.863 -
03
2.88
3.844
-
03
1.451
-
03
4.108 -
03
3.12
4.l8l
-
03
1.138
-
03
4.333 -
03
3.36
4.475
-
03
7.4l8
-
04
4.536 -
03
3.60
4.711
_
03
2.677
_
o4
4.719 -
03
3.84
4.874
-
03
-2.760
-
o4
4.881 -
03
4.08
4.948
-
03
-8.773
-
04
5.025 -
03
4.32
4.921
-
03
-1.521
-
03
5.151 -
03
4.56
4.783
-
03
-2.188
-
03
5.259 -
03
4.80
4.525
-
03
-2.858
_
03
5.352 -
03
5.o4
4.145
-
03
-3.506
-
03
5.429 -
03
5.28
3-645
-
03
-4.no
-
03
5.493 -
03
5.52
3.029
-
03
-4.643
-
03
5-544 -
03
5.76
2.310
-
03
-5.083
-
03
5.583 -
03
6.00
1.504
_
03
-5.406
_
03
5.612 -
03
6.24
6.324
-
o4
-5.595
-
03
5.630 -
03
6.48
-2.802
-
o4
-5.633
-
03
5.640 -
03
6.72
-1.205
-
03
-5.512
-
03
5.642 -
03
6.96
-2.111
-
03
-5.227
-
03
5.637 -
03
7.20
-2.966

03
-4.780

03
5.625 -
03
7.44
-3.738
-
03
-4.180
-
03
5.608 -
03
7.68
-4.398
-
03
-3.443
-
03
5.585 -
03
7.92
-4.918
-
03
-2.590
-
03
5.558 -
03
8.16
-5.276
-
03
-1.649
-
03
5.528 -
03
8.4o
-5.455
_
03
-6.509
_
o4
5.493 -
03
8.64
-5.444
-
03
3.679
-
o4
5.456 -
03
8.88
-5.240
-
03
1.370
-
03
5.416 -
03
9.12
-4.848
-
03
2.318
-
03
5.374 -
03
9.36
-4.281
-
03
3.175
-
03
5.330 -
03
9.60
-3.559
_
03
3.906
_
03
5.284 -
03
9.84
-2.708
-
03
4.483
-
03
5.237 -
03
10.08
-1.761
-
03
4.881
-
03
5.189 -
03
10.32
-7.561
-
o4
5.084
-
03
5.140 -
03
10.56
2.667
-
o4
5.083
-
03
5.090 -
03
10.80
1.266
_
03
4.879
-
03
5.o4o -
03
11.04
2.200
-
03
4.478
-
03
4.990 -
03
11.28
3.030
-
03
3.900
-
03
4.939 -
03
11.52
3.723
-
03
3.168
-
03
4.888 -
03
11.76
4.248
-
03
2.313
-
03
4.837 -
03
(Continued)

Table 39 (Concluded)
228
b
6.760
a
Real
Imaginary
Absolute
: Value
12.00
4.585 -
03
1-374
_
03
4.786
- 03
12.24
4.719 -
03
3.918
-
o4
4.736
- 03
12.48
4.648 -
03
-5.916
-
o4
4.685
- 03
12.72
4.374 -
03
-1.533
-
03
4.635
- 03
12.96
3.913 -
03
-2.390
-
03
4.585
- 03
13.20
3.286 -
03
-3.I27
-
03
4.536
- 03
13.44
2.524 -
03
-3.7IO
-
03
4.487
- 03
13.68
1.66l -
03
-4.116
-
03
4.439
- 03
13.92
7.380 -
04
-4.328
-
03
4.391
- 03
l4.l6
-2.016 -
o4
-4.339
-
03
4.344
- 03
i4.4o
-I.II5 -
03
-4.150
-
03
4.297
- 03
14.64
-I.96O -
03
-3.772
-
03
4.251
- 03
14.88
-2.698 -
03
-3.226
-
03
4.205
- 03
15.12
-3.296 -
03
-2.539
-
03
4.160
- 03
15.36
-3.728 -
03
-1.745
-
03
4.116
- 03
15.60
-3-975 -
03
-8.831
-
o4
4.072
- 03
15.84
-4.029 -
03
4.425
-
06
4.029
- 03
16.08
-3.889 -
03
8.752
-
o4
3.987
- 03
16.32
-3.565 -
03
1.688
-
03
3.945
- 03
16.56
-3.076 -
03
2.4o4
-
03
3.904
- 03
16.80
-2.446 -
03
2.990
-
03
3.863
- 03
17-04
-1.709 -
03
3.420
-
03
3.823
- 03
17.28
-9.028 -
04
3.674
-
03
3.784
- 03

Table 40
229
Value of l(0.72,a,b)
b
0.000
0.288
a
Real
Imaginary
Absolute Val
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.36
4.345
-
01
-I.963
-
02
4.350
-
01
0.72
7.815
-
01
-3.776
-
02
7.825
-
01
1.08
2.300
-
01
-5.3OI
-
02
2.360
-
01
1.44
1.331
-
01
-6.428
-
02
1.478
-
01
1.80
8.231
_
02
-7.O82
_
02
1.086
_
01
2.16
4.625
-
02
-7.23I
-
02
8.583
-
02
2.52
1.828
-
02
-6.885
-
02
7.124
-
02
2.88
-3.386
-
03
-6.101
-
02
6.110
-
02
3-24
-I.986
-
02
-4.966
-
02
5.349
-
02
3.60
-3.I33
-
02
-3.599
-
02
4.772
-
02
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.36
2.303
-
01
-1.947
-
02
2.311
-
01
0.72
3.010
-
01
-3.744
-
02
3.033
-
01
1.08
1.972
-
01
-5.255
-
02
2.040
-
01
1.44
1.238
-
01
-6.371
-
02
1.393
-
01
1.80
7.781
_
02
-7.016
_
02
1.048
_
01
2.16
4.399
-
02
-7.160
-
02
8.403
-
02
2.52
1.717
-
02
-6.813
-
02
7.026
-
02
2.88
-4.210
-
03
-6.030
-
02
6.045
-
02
3.24
-2.043
-
02
-4.900
-
02
5.309
-
02
3.60
-3.147
-
02
-3.54o
-
02
4.737
-
02
3.96
-3.739
-
02
-2.079
-
02
4.278
-
02
4.32
-3.847
-
02
-6.494
-
03
3.902
-
02
4.68
-3.532
-
02
6.281
-
03
3.588
-
02
5.04
-2.879
-
02
1.654
-
02
3.321
-
02
5.40
-1.996
-
02
2.360
-
02
3.091
-
02
5.76
-1.000
-
02
2.714
-
02
2.892
-
02
6.12
-7.971
-
05
2.717
-
02
2.717
-
02
6.48
8.746
-
03
2.409
-
02
2.562
-
02
6.84
1.563
-
02
1.854
-
02
2.425
-
02
7.20
2.000
_
02
1.138

02
2.301
-
02
7.56
2.160
-
02
3.572
-
03
2.189
-
02
7.92
2.050
-
02
-3.949
-
03
2.088
-
02
8.28
1.707
-
02
-1.035
-
02
1.996
-
02
8.64
1.188
-
02
-1.497
-
02
1.912
-
02
9.00
5.688
_
03
-1.744
_
02
1.834
-
02
9.36
-7.136
-
o4
-1.761
-
02
1.763
-
02
9.72
-6.554
-
03
-1.565
-
02
1.697
-
02
10.08
-1.118
-
02
-1.193
-
02
1.635
-
02
10.44
-l.4i4
-
02
-7.024
-
03
1.578
-
02
(Continued)

230
Table 40 (Continued)
b
0.288
a Real Imaginary Absolute Value
10.80
-1.517
-
02
-1.587
-
03
1.525
-
02
11.16
-1.429
-
02
3.691
-
03
1.476
-
02
11.52
-1.171
-
02
8.189
-
03
1.429
-
02
11.88
-7.862
-
03
l.l4l
-
02
1.386
-
02
12.24
-3.271
-
03
1.304
-
02
1.345
-
02
12.6o
1.459
-
03
1.298
-
02
1.306
-
02
12.96
5.747
-
03
1.132
-
02
1.269
-
02
13.32
9.094
-
03
8.355
-
03
1.235
-
02
13-68
1.114
-
02
4.515
-
03
1.202
-
02
l4.o4
1.171
-
02
3-144
-
o4
1.171
-
02
14.40
1.080
_
02
-3.715
_
03
1.142
_
02
14.76
8.592
-
03
-7.088
-
03
1.114
-
02
15.12
5.427
-
03
-9-420
-
03
1.087
-
02
15.48
1.735
-
03
-1.048
-
02
1.062
-
02
15.84
-2.004
-
03
-1.018
-
02
1.038
-
02
16.20
-5.328
-
03
-8.632
-
03
l.oi4
-
02
16.56
-7.844
-
03
-6.077
-
03
9.923
-
03
16.92
-9.276
-
03
-2.873
-
03
9.711
-
03
17.28
-9.492
-
03
5.550
-
o4
9.508
-
03
17.64
-8.514
-
03
3.774
-
03
9.313
-
03
18.00
-6.513
-
03
6.393
-
03
9.126
-
03
18.36
-3.776
-
03
8.111
-
03
8.947
-
03
18.72
-6.730
-
04
8.748
-
03
8.774
-
03
19.08
2.397
-
03
8.268
-
03
8.608
-
03
19.44
5.054
-
03
6.770
-
03
8.448
-
03
19.80
6.981
-
03
4.479
-
03
8.294
-
03
20.16
7.965
-
03
1.709
-
03
8.146
-
03
20.52
7.915
-
03
-1.178
-
03
8.002
-
03
20.88
6.875
-
03
-3.818
-
03
7.864
-
03
21.24
5.007
-
03
-5.890
-
03
7.731
-
03
21.60
2.574
_
03
-7.152
03
7.601
-
03
21.96
-1.036
-
o4
-7.476
-
03
7.477
-
03
22.32
-2.681
-
03
-6.850
-
03
7.356
-
03
22.68
-4.840
-
03
-5.382
-
03
7.239
-
03
23.04
-6.323
-
03
-3.284
-
03
7.125
-
03
23.40
-6.965
-
03
-8.378
-
o4
7.016
-
03
23.76
-6.712
-
03
1.640
-
03
6.909
-
03
24.12
-5.621
-
03
3.837
-
03
6.806
-
03
24.48
-3.854
-
03
5.487
-
03
6.705
-
03
24.84
-1.654
-
03
6.398
-
03
6.608
-
03
25.20
6.923
o4
6.477
-
03
6.514
-
03
25.56
2.885
-
03
5.737
-
03
6.422
-
03
25.92
4.653
-
03
4.295
-
03
6.332
-
03
(Continued)

Table 40 (Continued)
231
b
i.44o
a
Real
Imaginary
Absolute 1
7alue
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.36
2.340
-
02
-I.58O
-
02
2.823
-
02
0.72
3.858
-
02
-3.O3O
-
02
4.906
-
02
1.08
4.220
-
02
-4.234
-
02
5.978
-
02
1.44
3.607
-
02
-5.O97
-
02
6.244
-
02
1.80
2.401
_
02
-5.559
_
02
6.055
_
02
2.16
9.468
-
03
-5.594
-
02
5.674
-
02
2.52
-5.065
-
03
-5.217
-
02
5.241
-
02
2.88
-1.787
-
02
-4.478
-
02
4.821
_
02
3.24
-2.778
-
02
-3.460
-
02
4.437
-
02
3.60
-3-410
_
02
-2.267
_
02
4.095
_
02
3.96
-3.655
-
02
-1.014
-
02
3-793
-
02
4.32
-3.521
-
02
1.830
-
03
3.526
-
02
4.68
-3.056
-
02
1.220
-
02
3.291
-
02
5.04
-2.334
-
02
2.013
-
02
3.083
-
02
5.40
-1.449
-
02
2.509
-
02
2.897
-
02
5.76
-5.030
-
03
2.685
-
02
2.732
-
02
6.12
4.004
-
03
2.552
-
02
2.583
-
02
6.48
1.171
-
02
2.152
-
02
2.450
-
02
6.84
1.739
-
02
1.549
-
02
2.329
-
02
7.20
2.059
_
02
8.265

03
2.218

02
7.56
2.117
-
02
7.288
-
04
2.118
-
02
7.92
1.927
-
02
-6.258
-
03
2.026
-
02
8.28
1.530
-
02
-I.I96
-
02
1.942
-
02
8.64
9.854
-
03
-I.582
-
02
1.864
-
02
9.00
3.666
-
03
-1.754
-
02
1.792
-
02
9.36
-2.504
-
03
-I.707
-
02
1.725
-
02
9.72
-7.944
-
03
-1.461
-
02
1.663
-
02
10.08
-1.207
-
02
-1.058
-
02
1.605
-
02
10.44
-1.448
-
02
-5.561
-
03
1-551
-
02
10.80
-1.501
_
02
-I.929
-
04
1.501
-
02
11.16
-1.370
-
02
4.865
-
03
1.454
-
02
11.52
-1.082
-
02
9.033
-
03
1.409
-
02
11.88
-6.795
-
03
1.186
-
02
1.367
-
02
12.24
-2.180
-
03
1.310
-
02
1.328
-
02
12.60
2.443
-
03
1.267
-
02
1.291
-
02
12.96
6.520
-
03
1.073
-
02
1.255
-
02
13.32
9.587
-
03
7.575
-
03
1.222
-
02
13.68
1.132
-
02
3.660
-
03
1.190
-
02
l4.04
1.159
-
02
-5.068
-
04
l.l60
-
02
i4.4o
1.042
_
02
-4.408
_
03
1.131
_
02
14.76
8.028
-
03
-7.582
-
03
1.104
-
02
15.12
4.759
-
03
-9.676
-
03
1.078
-
02
15.48
1.053
-
03
-1.048
-
02
1.053
-
02
15.84
-2.617
-
03
-9.960
-
03
1.030
-
02
(Continued)

b
1.44o
4.34o
Table 40 (Continued)
a
Real
Imaginary
Absolute
i Value
16.20
-5.804
_
03
-8.231
_
03
1.007
- 02
16.56
-8.138
-
03
-5.558
-
03
9.855
- 03
16.92
-9.366
-
03
-2.3IO
-
03
9-647
- 03
17.28
-9.385
-
03
1.092
-
03
9.448
- 03
17.64
-8.238
-
03
4.221
-
03
9.257
- 03
18.00
-6.115
-
03
6.703
-
03
9.073
- 03
18.36
-3.314
-
03
8.257
-
03
8.897
- 03
18.72
-2.O73
-
o4
8.725
-
03
8.727
- 03
19.08
2.809
-
03
8.090
-
03
8.564
- 03
19.44
5.366
-
03
6.471
-
03
8.4o6
- 03
19.80
7.162
-
03
4.104
-
03
8.254
- 03
20.16
8.002
-
03
1.308
-
03
8.108
- 03
20.52
7.814
-
03
-1.553
-
03
7.967
- 03
20.88
6.657
-
03
-4.123
-
03
7.830
- 03
21.24
4.707
-
03
-6.092
-
03
7.698
- 03
21.60
2.233
-
03
-7.234
-
03
7.571
- 03
21.96
-4.398
-
o4
-7.434
-
03
7.447
- 03
22.32
-2.972
-
03
-6.698
-
03
7.328
- 03
22.68
-5.053
-
03
-5.146
-
03
7.212
- 03
23.04
-6.436
-
03
-2.998
-
03
7.100
- 03
23.40
-6.971
_
03
-5.396
_
o4
6.991
- 03
23.76
-6.615
-
03
1.913
-
03
6.886
- 03
24.12
-5.440
-
03
4.052
-
03
6.784
- 03
24.48
-3.617
-
03
5.621
-
03
6.684
- 03
24.84
-1.392
-
03
6.439
-
03
6.588
- 03
25.20
9.443
_
o4
6.425
-
03
6.494
- 03
25.56
3.097
-
03
5.604
-
03
6.403
- 03
25.92
4.801
-
03
4.102
-
03
6.315
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.36
-3.279
-
03
-2.867
-
o4
3.291
- 03
0.72
-6.439
-
03
-2.996
-
04
6.446
- 03
1.08
-9-346
-
03
1.935
-
o4
9-348
- 03
1.44
-1.183
-
02
1.347
-
03
1.191
- 02
1.80
-1.370
_
02
3.212
_
03
1.4o8
- 02
2.16
-1.477
-
02
5.722
-
03
1.584
- 02
2.52
-1.484
-
02
8.699
-
03
1.720
- 02
2.88
-1.381
-
02
1.187
-
02
1.821
- 02
3.24
-1.165
-
02
1.489
-
02
1.890
- 02
3.60
-8.432
-
03
1.739
-
02
1.933
- 02
3.96
-4.352
-
03
1.905
-
02
1.954
- 02
4.32
2.887
-
o4
1.957
-
02
1.958
- 02
4.68
5.106
-
03
1.879
-
02
1.948
- 02
5.04
9.673
-
03
1.667
-
02
1.927
- 02
(Continued)

Table 40 (Continued)
233
b
4.340
a
Real
Imaginary
Absolute Value
5.40
1.356
_
02
1.330
_
02
1.899
- 02
5.76
1.639
-
02
8.921
-
03
1.866
- 02
6.12
1.787
-
02
3.888
-
03
1.829
- 02
6.48
1.784
-
02
-I.362
-
03
1.789
- 02
6.84
1.628
-
02
-6.356
-
03
1.748
- 02
7.20
1.333
_
02
-1.064
_
02
1.706
- 02
7.56
9.283
-
03
-1.380
-
02
1.663
- 02
7.92
4.520
-
03
-1.557
-
02
1.622
- 02
8.28
-4.842
-
o4
-1.580
-
02
1.580
- 02
8.64
-5.238
-
03
-1.448
-
02
1.540
- 02
9.00
-9.278
_
03
-I.180
-
02
1.501
- 02
9.36
-1.222
-
02
-8.044
-
03
1.463
- 02
9.72
-1.379
-
02
-3.636
-
03
1.426
- 02
10.08
-I.387
-
02
9-559
-
o4
1.390
- 02
10.44
-I.25O
-
02
5.248
-
03
1.355
- 02
10.80
-9.869
-
03
8.799
-
03
1.322
- 02
11.16
-6.303
-
03
1.126
-
02
1.290
- 02
11.52
-2.212
-
03
1.240
-
02
1.259
- 02
11.88
1.944
-
03
1.214
-
02
1.230
- 02
12.24
5.709
-
03
1.057
-
02
1.201
- 02
12.60
8.686
_
03
7.894
-
03
1.174
- 02
12.96
1.057
-
02
4.454
-
03
1.147
- 02
13.32
1.120
-
02
6.577
-
04
1.122
- 02
13.68
1.054
-
02
-3.057
-
03
1.097
- 02
i4.o4
8.714
-
03
-6.277
-
03
1.074
- 02
i4.4o
5.969
-
03
-8.654
-
03
1.051
- 02
14.76
2.649
-
03
-9.948
-
03
1.029
- 02
15.12
-8.460
-
o4
-1.005
-
02
1.008
- 02
15.48
-4.110
-
03
-8.987
-
03
9.882
- 03
15.84
-6.775
-
03
-6.923
-
03
9.687
- 03
16.20
-8.553
-
03
-4.132
-
03
9.498
- 03
16.56
-9.267
-
03
-9.601
-
o4
9.317
- 03
16.92
-8.870
-
03
2.209
-
03
9.i4i
- 03
17.28
-7.446
-
03
5.006
-
03
8.972
- 03
17.64
-5.194
-
03
7.114
-
03
8.808
- 03
18.00
-2.4o6
_
03
8.309
_
03
8.650
- 03
18.36
5.717
-
o4
8.478
-
03
8.497
- 03
18.72
3.381
-
03
7.634
-
03
8.350
- 03
19.08
5.694
-
03
5.909
-
03
8.206
- 03
19.44
7.252
-
03
3.536
-
03
8.068
- 03
19.80
7.892
-
03
8.157
-
o4
7.934
- 03
20.16
7.565
-
03
-1.915
-
03
7.804
- 03
20.52
6.340
-
03
-4.330
-
03
7.678
- 03
20.88
4.390
-
03
-6.149
-
03
7.556
- 03
21.24
1.970
-
03
-7.171
-
03
7.437
- 03
(Continued)

Table 40 (Continued)
234
b
4.340
7.240
a
Real
Imaginary
Absolut Val
21.60
-6.142
_
04
-7.296
_
03
7.322
_
03
21.96
-3.047
-
03
-6.535
-
03
7.210
-
03
22.32
- 5.040
-
03
-5-004
-
03
7.102
-
03
22.68
-6.362
-
03
-2.910
-
03
6.997
-
03
23.04
-6.874
-
03
-5.233
-
04
6.894
-
03
23.40
-6.535
-
03
1.860
-
03
6.795
-
03
23.76
-5.409
-
03
3.950
-
03
6.698
-
03
24.12
-3.654
-
03
5.500
-
03
6.6o4
-
03
24.48
-I.5OI
-
03
6.337
-
03
6.512
-
03
24.84
7.779
-
o4
6.376
-
03
6.423
-
03
25.20
2.901
-
03
5.633
-
03
6.336
-
03
25.56
4.613
-
03
4.219
-
03
6.251
-
03
25.92
5.714
-
03
2.325
-
03
6.169
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.36
1.073
-
03
5.050
-
04
1.185
-
03
0.72
2.154
-
03
9-467
-
04
2.353
-
03
1.08
3.247
-
03
1.264
-
03
3-485
-
03
1.44
4.346
-
03
1.398
-
03
4.566
-
03
1.80
5.429
-
03
1.300
-
03
5.583
-
03
2.16
6.458
-
03
9.291
-
04
6.525
-
03
2.52
7.381
-
03
2.6o4
-
o4
7.385
-
03
2.88
8.129
-
03
-7.101
-
04
8.160
-
03
3.24
8.625
-
03
-1.962
-
03
8.846
-
03
3.60
8.793
-
03
-3.448
-
03
9.444
-
03
3.96
8.560
-
03
-5.090
-
03
9-959
-
03
4.32
7.872
-
03
-6.784
-
03
1.039
-
02
4.68
6.704
-
03
-8.404
-
03
1.075
-
02
5.04
5.065
-
03
-9.809
-
03
i.io4
-
02
5.40
3.004
_
03
-I.086
-
02
1.127
-
02
5.76
6.164
-
o4
-1.142
-
02
1.143
-
02
6.12
-1.963
-
03
-1.139
-
02
1.155
-
02
6.48
-4.566
-
03
-1.069
-
02
1.163
-
02
6.84
-7.002
-
03
-9.330
-
03
1.166
-
02
7.20
-9.074
_
03
-7.334
-
03
1.167
-
02
7.56
-1.060
-
02
-4.809
-
03
1.164
-
02
7.92
-1.143
-
02
-1.916
-
03
1.159
-
02
8.28
-1.146
-
02
1.144
-
03
1.152
-
02
8.64
-1.066
-
02
4.135
-
03
1.143
-
02
9.00
-9.047
_
03
6.817
-
03
1.133
-
02
9.36
-6.741
-
03
8.961
-
03
1.121
-
02
9.72
-3.913
-
03
1.038
-
02
1.109
-
02
10.08
-7.931
-
o4
1.093
-
02
1.096
-
02
10.44
2.355
-
03
1.056
-
02
1.082
-
02
(Continued)

Table 40 (Continued)
235
b
7.240
a
Real
Imaginary
Absolute Value
10.80
5.257
_
03
9.294
_
03
1.068
- 02
11.16
7.652
-
03
7.237
-
03
1.053
- 02
II.52
9.323
-
03
4.574
-
03
I.O38
- 02
11.88
1.012
-
02
1.548
-
03
1.024
- 02
12.24
9.965
-
03
-I.557
-
03
1.009
- 02
12.60
8.884
-
03
-4.450
-
03
9.936
- 03
12.96
6.986
-
03
-6.855
-
03
9.788
- 03
13.32
4.462
-
03
-8.545
-
03
9.64o
- 03
13.68
1.563
-
03
-9-364
-
03
9.493
- 03
14.04
-1.417
-
03
-9.24i
-
03
9.349
- 03
i4.4o
-4.182
_
03
-8.201
_
03
9.206
- 03
14.76
-6.457
-
03
-6.363
-
03
9.065
- 03
15.12
-8.017
-
03
-3.926
-
03
8.927
- 03
15.48
-8.715
-
03
-1.150
-
03
8.791
- 03
15.84
-8.495
-
03
1.672
-
03
8.658
- 03
16.20
-7.394
_
03
4.246
_
03
8.527
- 03
16.56
-5.547
-
03
6.306
-
03
8.398
- 03
16.92
-3.161
-
03
7.645
-
03
8.273
- 03
17.28
-5.012
-
o4
8.134
-
03
8.150
- 03
17.64
2.142
-
03
7.738
-
03
8.029
- 03
18.00
4.484
_
03
6.518
-
03
7.911
- 03
18.36
6.278
-
03
4.623
-
03
7.796
- 03
18.72
7.340
-
03
2.272
-
03
7.684
- 03
19.08
7.569
-
03
-2.657
-
o4
7.574
- 03
19.44
6.957
-
03
-2.708
-
03
7.466
- 03
19.80
5.591
-
03
-4.788
-
03
7.361
- 03
20.16
3.637
-
03
-6.281
-
03
7.258
- 03
20.52
1.327
-
03
-7.033
-
03
7.157
- 03
20.88
-1.073
-
03
-6.977
-
03
7.059
- 03
21.24
-3.291
-
03
-6.137
-
03
6.963
- 03
21.60
-5.082
-
03
-4.623
-
03
6.870
- 03
21.96
-6.250
-
03
-2.622
-
03
6.778
- 03
22.32
-6.678
-
03
-3.722
-
04
6.689
- 03
22.68
-6.332
-
03
1.865
-
03
6.601
- 03
23.04
-5.269
-
03
3.833
-
03
6.516
- 03
23.40
-3.625
-
03
5.313
-
03
6.432
- 03
23.76
-1.603
-
03
6.145
-
03
6.350
- 03
24.12
5 566
-
o4
6.246
-
03
6.270
- 03
24.48
2.600
-
03
5.620
-
03
6.192
- 03
24.84
4.293
-
03
4.355
-
03
6.116
- 03
25.20
5.447
-
03
2.613
-
03
6.o4i
- 03
25.56
5-937
-
03
6.051
-
o4
5.968
- 03
25.92
5.721
-
03
-1.427
-
03
5.896
- 03
(Continued)

Table 40 (Continued)
236
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000 +
00
O.36
-4.503
-
o4
-4.036
-
o4
6.047 -
o4
0.72
-9.I2O
-
o4
-7.870
-
o4
1.205 -
03
1.08
-1.395
-
03
-I.I30
-
03
1.795 -
03
1.44
-I.907
-
03
-1.411
-
03
2.372 -
03
1.80
-2.45O
-
03
-I.609
-
03
2.931 -
03
2.16
-3.O2I
-
03
-I.703
-
03
3.468 -
03
2.52
-3.613
-
03
-I.672
-
03
3.981 -
03
2.88
-4.207
-
03
-1.499
-
03
4.466 -
03
3-24
-4.782
-
03
-I.I69
-
03
4.923 -
03
3.60
-5.306
_
03
-6.742
_
04
5-348 -
03
3.96
-5.742
-
03
-1.374
-
05
5-742 -
03
4.32
-6.051
-
03
8.026
-
04
6.104 -
03
4.68
-6.191
-
03
1.753
-
03
6.434 -
03
5.04
-6.121
-
03
2.8o4
-
03
6.733 -
03
5.40
-5.807
-
03
3.909
-
03
7.000 -
03
5.76
-5.226
-
03
5.007
-
03
7.238 -
03
6.12
-4.368
-
03
6.031
-
03
7.447 -
03
6.48
-3.242
-
03
6.906
-
03
7.629 -
03
6.84
-1.875
-
03
7-557
-
03
7.786 -
03
7.20
-3.190
-
o4
7.913
-
03
7.919 -
03
7.56
1.355
-
03
7.914
-
03
8.030 -
03
7.92
3.058
-
03
7.522
-
03
8.120 -
03
8.28
4.685
-
03
6.719
-
03
8.191 -
03
8.64
6.126
-
03
5.518
-
03
8.245 -
03
9.00
7.274
-
03
3.962
-
03
8.283 -
03
9.36
8.030
-
03
2.127
-
03
8.307 -
03
9-72
8.317
-
03
1.159
-
o4
8.318 -
03
10.08
8.087
-
03
-1.944
-
03
8.317 -
03
10.44
7.328
-
03
-3.911
-
03
8.306 -
03
10.80
6.068
_
03
-5.642
-
03
8.286 -
03
11.16
4.378
-
03
-7.001
-
03
8.257 -
03
11.52
2.366
-
03
-7.873
-
03
8.221 -
03
11.88
1.747
-
o4
-8.176
-
03
8.178 -
03
12.24
-2.033
-
03
-7.872
-
03
8.130 -
03
12.60
-4.085
-
03
-6.. 968
-
03
8.077 -
03
12.96
-5.814
-
03
-5.524
-
03
8.020 -
03
13.32
-7.074
-
03
-3.648
-
03
7.959 -
03
13.68
-7.753
-
03
-1.487
-
03
7.894 -
03
i4.o4
-7.789
-
03
7.802
-
04
7.828 -
03
i4.4o
-7.171
_
03
2.962
_
03
7.758 -
03
14.76
-5.948
-
03
4.871
-
03
7.688 -
03
15.12
-4.223
-
03
6.337
-
03
7.615 -
03
15.48
-2.148
-
03
7.229
-
03
7-541 -
03
15.84
9.234
-
05
7.466
-
03
7-467 -
03
(Continued)

Table 40 (Concluded)
237
a
Real
Imaginary
Absolute Value
16.20
2.294
_
03
7.027
_
03
7.392
_
03
16.56
4.255
-
03
5.951
-
03
7.316
-
03
16.92
5.794
-
03
4.342
-
03
7.240
-
03
17.28
6.767
-
03
2.351
-
03
7.164
-
03
17.64
7.086
-
03
I.696
-
04
7.088
-
03
18.00
6.723
-
03
-1.993
-
03
7.013
-
03
18.36
5.719
-
03
-3.927
-
03
6.937
-
03
18.72
4.175
-
03
-5.446
-
03
6.862
-
03
19.08
2.248
-
03
-6.405
-
03
6.788
-
03
19.44
1.330
-
o4
-6.713
-
03
6.714
-
03
19.80
-I.958
_
03
-6.346
-
03
6.641
-
03
20.16
-3.814
-
03
-5.347
-
03
6.568
-
03
20.52
-5.251
-
03
-3.826
-
03
6.497
-
03
20.88
-6.125
-
03
-1.942
-
03
6.426
-
03
21.24
-6.355
-
03
1.057
-
o4
6.356
-
03
21.60
-5.924
-
03
2.105
_
03
6.287
-
03
21.96
-4.885
-
03
3.848
-
03
6.218
-
03
22.32
-3.355
-
03
5.155
-
03
6.151
-
03
22.68
-1.501
-
03
5.897
-
03
6.085
-
03
23.04
4.761
-
o4
6.000
-
03
6.019
-
03
23.40
2.366
_
03
5.465
-
03
5-955
-
03
23.76
3.967
-
03
4.356
-
03
5.891
-
03
24.12
5.112
-
03
2.800
-
03
5.829
-
03
24.48
5.684
-
03
9.722
-
o4
5.767
-
03
24.84
5.631
-
03
-9.264
-
o4
5.706
-
03
25.20
4.965
_
03
-2.688
-
03
5-647
-
03
25.56
3.771
-
03
-4.124
-
03
5.588
-
03
25.92
2.185
-
03
-5.080
-
03
5.530
-
03

Table 4l
238
Value of l(0.96,a,b)
b
0.000
0.384
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000 +
00
0.48
4.610
-
01
-4.459
-
02
4.632 -
01
0.96
8.222
-
01
-8.316
-
02
8.264 -
01
1.44
2.446
-
01
-I.IO7
-
01
2.685 -
01
1.92
1.235
-
01
-I.238
-
01
1.749 -
01
2.40
4.888
_
02
-1.216
_
01
1.310 -
01
2.88
-4.490
-
03
-1.053
-
01
1.054 -
01
3.36
-4.087
-
02
-7.839
-
02
8.840 -
02
3.84
-6.117
-
02
-4.564
-
02
7.632 -
02
4.32
-6.602
-
02
-1.243
-
02
6.718 -
02
4.8o
-5.779
-
02
1.624
-
02
6.002 -
02
0.00
0.000
+
00
0.000
+
00
0.000 +
00
0.48
2.514
-
01
-4.392
-
02
2.552 -
01
0.96
3.263
-
01
-8.188
-
02
3-364 -
01
1.44
2.077
-
01
-1.089
-
01
2.345 -
01
1.92
1.119
-
01
-1.217
-
01
1.654 -
01
2.40
4.370
_
02
-1.193
_
01
1.271 -
01
2.88
-7.018
-
03
-1.031
-
01
1.034 -
01
3.36
-4.206
-
02
-7.642
-
02
8.723 -
02
3.84
-6.139
-
02
-4.401
-
02
7-553 -
02
4.32
-6.570
-
02
-1.121
-
02
6.665 -
02
4.8o
-5.719
-
02
1.702
-
02
5.967 -
02
5.28
-3.949
-
02
3.688
-
02
5.403 -
02
5.76
-1.714
-
02
4.630
-
02
4.938 -
02
6.24
5.110
-
03
4.518
-
02
4.547 -
02
6.72
2.315
-
02
3.521
-
02
4.214 -
02
7.20
3.411
_
02
1.947
-
02
3.927 -
02
7.68
3.673
-
02
1.731
-
03
3.677 -
02
8.16
3.149
-
02
-1.427
-
02
3.457 -
02
8.64
2.031
-
02
-2.553
-
02
3.262 -
02
9.12
6.068
-
03
-3.028
-
02
3.088 -
02
9.60
-8.015
_
03
-2.820
-
02
2.932 -
02
10.08
-1.907
-
02
-2.038
-
02
2.791 -
02
10.56
-2.508
-
02
-8.957
-
03
2.663 -
02
11.04
-2.523
-
02
3.374
-
03
2.546 -
02
11.52
-2.000
-
02
1.396
-
02
2.439 -
02
12.00
-1.091
_
02
2.071
-
02
2.341 -
02
12.48
-1.813
-
04
2.250
-
02
2.250 -
02
12.96
9.791
-
03
1.932
-
02
2.166 -
02
13.44
1.694
-
02
1.221
-
02
2.088 -
02
13.92
1.994
-
02
2.951
-
03
2.016 -
02
(Continued)

Table 4l (Continued)
239
b
0.384
a
Real
Imaginary
Absolute Value
l4.4o
1.843
_
02
-6.327
_
03
1.948
_
02
14.88
1.302
-
02
-I.363
-
02
1.885
-
02
15.36
5.121
-
03
-1.753
-
02
1.826
-
02
15.84
-3.409
-
03
-1.737
-
02
1.770
-
02
16.32
-I.07O
-
02
-1.345
-
02
1.718
-
02
16.80
-I.523
_
02
-6.813
_
03
I.669
-
02
17.28
-1.620
-
02
9-379
-
o4
1.622
-
02
17.76
-1.357
-
02
8.064
-
03
1.578
-
02
18.24
-8.IO9
-
03
1.305
-
02
1.537
-
02
18.72
-I.I56
-
03
1.493
-
02
1.497
-
02
19.20
5.702
-
03
1.343
-
02
1.459
-
02
19.68
1.098
-
02
9.070
-
03
1.424
-
02
20.16
1.359
-
02
2.922
-
03
1.390
-
02
20.64
1.309
-
02
-3.582
-
03
1.357
-
02
21.12
9.742
-
03
-9.OO3
-
03
1.326
-
02
21.60
4.397
_
03
-1.220
-
02
1.297
-
02
22.08
-1.684
-
03
-1.257
-
02
1.269
-
02
22.56
-7.135
-
03
-I.OI6
-
02
1.242
-
02
23.04
-1.079
-
02
-5.608
-
03
1.216
-
02
23.52
-1.191
-
02
-4.047
-
06
1.191
-
02
24.00
-1.036
_
02
5-377
-
03
1.167
-
02
24.48
-6.580
-
03
9-359
-
03
1.144
-
02
24.96
-1.494
-
03
1.112
-
02
1.122
-
02
25.44
3.730
-
03
1.036
-
02
1.101
-
02 ,
25.92
7.936
-
03
7.331
-
03
1.080
-
02
26.4o
1.023
_
02
2.793
_
03
1.061
-
02
26.88
1.018
-
02
-2.201
-
03
1.042
-
02
27.36
7.879
-
03
-6.532
-
03
1.023
-
02
27.84
3.909
-
03
-9.267
-
03
1.006
-
02
28.32
-7.923
-
o4
-9.855
-
03
9.887
-
03
28.80
-5.161
_
03
-8.239
-
03
9.722
-
03
29.28
-8.242
-
03
-4.848
-
03
9.562
-
03
29.76
-9.395
-
03
-4.913
-
o4
9.408
-
03
30.24
-8.426
-
03
3.836
-
03
9.258
-
03
30.72
-5.617
-
03
7.177
-
03
9.114
-
03
31.20
-1.647
_
03
8.821
-
03
8.973
-
03
31.68
2.569
-
03
8.455
-
03
8.837
-
03
32.16
6.088
-
03
6.222
-
03
8.705
-
03
32.64
8.150
-
03
2.671
-
03
8.577
-
03
33.12
8.341
-
03
-1.370
-
03
8.452
-
03
33.60
6.671
-
03
-4.992
-
03
8.332
-
03
34.08
3-564
-
03
-7.401
-
03
8.214
-
03
34.56
-2.495
-
o4
-8.096
-
03
8.100
-
03
(Continued)

Table 4l (Continued)
b
1.920
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000 +
00
0.48
1.872
-
02
-2.98O
-
02
3.519 -
02
O.96
2.674
-
02
-5-504
-
02
6.119 -
02
1.44
2.005
-
02
-7.196
-
02
7.470 -
02
1.92
2.015
-
03
-7.821
-
02
7.824 -
02
2.40
-2.060
_
02
-7.325
_
02
7.609 -
02
2.88
-4.126
-
02
-5.838
-
02
7-149 -
02
3.36
-5.523
-
02
-3.65O
-
02
6.620 -
02
3.84
-5.992
-
02
-I.I54
-
02
6.102 -
02
4.32
-5.49O
-
02
1.228
-
02
5.625 -
02
4.80
-4.I6I
_
02
3.116
_
02
5.198 -
02
5.28
-2.292
-
02
4.24o
-
02
4.820 -
02
5.76
-2.514
-
03
4.478
-
02
4.485 -
02
6.24
1.590
-
02
3.876
-
02
4.189 -
02
6.72
2.922
-
02
2.622
-
02
3.926 -
02
7.20
3.551
_
02
1.011
-
02
3.692 -
02
7.68
3-427
-
02
-6.224
-
03
3-483 -
02
8.16
2.644
-
02
-1.966
-
02
3.295 -
02
8.64
l.4l4
-
02
-2.787
-
02
3.125 -
02
9.12
1.767
-
04
-2.972
-
02
2.972 -
02
9.60
-1.256
-
02
-2.538
-
02
2.832 -
02
10.08
-2.161
-
02
-1.625
-
02
2.704 -
02
10.56
-2.547
-
02
-4.554
-
03
2.587 -
02
11.04
-2.375
-
02
7.138
-
03
2.480 -
02
11.52
-1.723
-
02
1.643
-
02
2.381 -
02
12.00
-7.600
_
03
2.159
-
02
2.289 -
02
12.48
2.920
-
03
2.185
-
02
2.204 -
02
12.96
1.208
-
02
1.748
-
02
2.125 -
02
13-44
1.807
-
02
9.718
-
03
2.052 -
02
13.92
1.982
-
02
4.051
-
04
1.983 -
02
i4.4o
1.725
_
02
-8.397
_
03
1.918 -
02
14.88
1.115
-
02
-1.486
-
02
I.858 -
02
15.36
3.o4o
-
03
-1.775
-
02
1.801 -
02
15.84
-5.250
-
03
-1.667
-
02
1.748 -
02
16.32
-1.194
-
02
-I.207
-
02
1.697 -
02
16.80
-1.568
_
02
-5.121
_
03
1.650 -
02
17.28
-1.584
-
02
2.556
-
03
1.605 -
02
17.76
-1.257
-
02
9-275
-
03
1.562 -
02
18.24
-6.745
-
03
1.364
-
02
1.522 -
02
18.72
2.518
-
04
1.483
-
02
1.483 -
02
19.20
6.852
_
03
1.274
-
02
1.447 -
02
19.68
1.165
-
02
7.984
-
03
1.412 -
02
20.16
1.368
-
02
1.709
-
03
1.379 -
02
20.64
1.264
-
02
-4.652
-
03
1.347 -
02
21.12
8.892
-
03
-9.715
-
03
1.317 -
02
(Continued)

Table 4l (Continued)
24l
b
1.920
5.800
a
Real
Imaginary
Absolute
: Value
21.60
3.362
_
03
-1.243
_
02
1.288
- 02
22.80
-2.666
-
03
-I.232
-
02
1.260
- 02
22.56
-7.859
-
03
-9.512
-
03
1.234
- 02
23.03
-1.112
-
02
-4.737
-
03
1.208
- 02
23.52 :
-I.I8I
-
02
8.844
-
o4
1.184
- 02
24.00
-9.879
-
03
6.090
_
03
1.160
- 02
24.48
-5.856
-
03
9-757
-
03
1.138
- 02
24.96
-7.000
-
04
1.114
-
02
1.116
- 02
25.44
4.418
-
03
1.002
-
02
1.095
- 02
25.92
8.377
-
03
6.741
-
03
1.075
- 02
26.40
1.035
-
02
2.093
-
03
1.056
- 02
26.88
9.971
-
03
-2.852
-
03
1.037
- 02
27.36
7.409
-
03
-6.997
-
03
1.019
- 02
27.84
3.300
-
03
-9.457
-
03
1.002
- 02
28.32
-1.400
-
03
-9.747
-
03
9.847
- 03
28.80
-5.635
-
03
-7.876
-
03
9.684
- 03
29.28
-8.488
-
03
-4.326
-
03
9.527
- 03
29.76
-9.374
-
03
6.705
-
05
9.374
- 03
30.24
-8.159
-
03
4.307
-
03
9.226
- 03
30.72
-5.177
-
03
7.463
-
03
9.083
- 03
31.20
-l.l4l
-
03
8.871
-
03
8.944
- 03
31.68
3.027
-
03
8.273
-
03
8.809
- 03
32.16
6.4oi
-
03
5.860
-
03
8.678
- 03
32.64
8.258
-
03
2.220
-
03
8.551
- 03
33.12
8.232
-
03
-1.808
-
03
8.428
- 03
33.60
6.381
-
03
-5.320
-
03
8.308
- 03
34.08
3.167
-
03
-7.555
-
03
8.191
- 03
34.56
-6.614
-
o4
-8.051
-
03
8.078
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.48
-9.957
-
o4
4.203
-
03
4.320
- 03
0.96
-1.491
-
03
8.324
-
03
8.457
- 03
1.44
-1.052
-
03
1.221
-
02
1.225
- 02
1.92
6.222
-
04
1.558
-
02
1.559
- 02
2.40
3.630
_
03
1.805
_
02
l.84i
- 02
2.88
7.818
-
03
1.915
-
02
2.069
- 02
3.36
1.276
-
02
1.846
-
02
2.244
- 02
3.84
1.779
-
02
1.570
-
02
2.373
- 02
4.32
2.207
-
02
1.085
-
02
2.46o
- 02
4.80
2.476
_
02
4.241
_
03
2.512
- 02
5.28
2.512
-
02
-3.445
-
03
2.536
- 02
5.76
2.275
-
02
-1.124
-
02
2.538
- 02
6.24
1.764
-
02
-1.803
-
02
2.522
- 02
6.72
1.026
-
02
-2.273
-
02
2.494
- 02
(Continued)

Table 4l (Continued)
242
b
5.800
a
Real
Imaginary
Absolute Val
7.20
1.522
_
03
-2.45I
_
02
2.455 -
02
7.68
-7.363
-
03
-2.295
-
02
2.410 -
02
8.16
-I.506
-
02
-I.817
-
02
2.360 -
02
8.64
-2.O37
-
02
-1.084
-
02
2.308 -
02
9.12
-2.244
-
02
-2.062
-
03
2.253 -
02
9.60
-2.O92
_
02
6.747
_
03
2.198 -
02
10.08
-I.608
-
02
1.415
-
02
2.142 -
02
10.56
-8.788
-
03
1.894
-
02
2.088 -
02
ii.o4
-3.O62
-
04
2.033
-
02
2.034 -
02
11.52
7.872
-
03
1.8l8
-
02
1.981 -
02
12.00
1.432

02
1.294
-
02
1.930 -
02
12.48
1.794
-
02
5.621
-
03
1.880 -
02
12.96
1.817
-
02
-2.386
-
03
1.832 -
02
13.44
1.506
-
02
-9.602
-
03
1.786 -
02
13.92
9-300
-
03
-1.472
-
02
1.741 -
02
i4.4o
2.056
-
03
-1.686
-
02
I.698 -
02
14.88
-5.257
-
03
-1.571
-
02
1.657 -
02
15.36
-I.I25
-
02
-1.162
-
02
1.617 -
02
15.84
-1.481
-
02
-5.463
-
03
1.578 -
02
16.32
-1.534
-
02
1.485
-
03
1.541 -
02
16.80
-1.286
_
02
7.845
-
03
1.506 -
02
17.28
-7.942
-
03
1.239
-
02
1.472 -
02
17.76
-1.661
-
03
1.430
-
02
1.439 -
02
18.24
4.693
-
03
1.327
-
02
i.4o8 -
02
18.72
9.847
-
03
9.629
-
03
1.377 -
02
19.20
1.281
-
02
4.195
-
03
1.348 -
02
19.68
1.307
-
02
-1.874
-
03
1.320 -
02
20.16
1.066
-
02
-7-324
-
03
1.293 -
02
20.64
6.162
-
03
-1.107
-
02
1.267 -
02
21.12
5.763
-
04
-1.240
-
02
1.242 -
02
21.60
-4.918
_
03
-1.114
_
02
1.218 -
02
22.08
-9.195
-
03
-7.619
-
03
1.194 -
02
22.56
-l.i4i
-
02
-2.651
-
03
1.172 -
02
23.04
-1.118
-
02
2.690
-
03
1.150 -
02
23.52
-8.625
-
03
7.281
-
03
1.129 -
02
24.00
-4.359
_
03
1.019
-
02
1.108 -
02
24.48
6.737
-
o4
I.087
-
02
1.089 -
02
24.96
5.394
-
03
9.238
-
03
1.070 -
02
25.44
8.822
-
03
5.720
-
03
1.051 -
02
25.92
1.028
-
02
1.110
-
03
1.034 -
02
26.40
9.511
_
03
-3.584
-
03
1.016 -
02
26.88
6.756
-
03
-7.369
-
03
9-997 -
03
27.36
2.654
-
03
-9.470
-
03
9.835 -
.03
27.84
-1.884
-
03
-9.493
-
03
9.678 -
03
28.32
-5.881
-
03
-7.494
-
03
9.526 -
03
(Continued)

Table 4l (Continued)
243
b
5.800
9.64o
a
Real
Imaginary
Absolute Val
28.80
-8.503
_
03
-3.956
_
03
9.378
_
03
29.28
-9.229
-
03
3.204
-
04
9.235
-
03
29.76
-7.96O
-
03
4.402
-
03
9.096
-
03
30.24
-5.O2I
-
03
7.422
-
03
8.961
-
03
30.72
-I.090
-
03
8.762
-
03
8.829
-
03
31.20
2.964
-
03
8.181
_
03
8.702
_
03
31.68
6.267
-
03
5.857
-
03
8.577
-
03
32.16
8.128
-
03
2.335
-
03
8.457
-
03
32.64
8.185
-
03
-1.594
-
03
8.339
-
03
33.12
6.474
-
03
-5.073
-
03
8.225
-
03
33.60
3.409
-
03
-7.362
-
03
8.113
-
03
34.08
-3.126
-
o4
-7-999
-
03
8.005
-
03
34.56
-3.870
-
03
-6.886
-
03
7.899
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.48
-5.678
-
o4
-1.473
-
03
1.578
-
03
0.96
-1.228
-
03
-2.881
-
03
3.132
-
03
1.44
-2.061
-
03
-4.153
-
03
4.636
-
03
1.92
-3.127
-
03
-5.203
-
03
6.070
-
03
2.40
-4.452
_
03
-5.932
-
03
7.417
-
03
2.88
-6.015
-
03
-6.231
-
03
a. 661
-
03
3.36
-7.746
-
03
-5.991
-
03
9.793
-
03
3.84
-9.516
-
03
-5.121
-
03
1.081
-
02
4.32
-1.114
-
02
-3-572
-
03
1.170
-
02
4.80
-1.241
_
02
-I.352
_
03
1.248
_
02
5.28
-1.306
-
02
1.443
-
03
1.314
-
02
5.76
-1.289
-
02
4.629
-
03
1.370
-
02
6.24
-1.172
-
02
7.928
-
03
1.415
-
02
6.72
-9.482
-
03
1.099
-
02
1.451
-
02
7.20
-6.228

03
1.342
-
02
1.479
-
02
7.68
-2.165
-
03
1.484
-
02
1.500
-
02
8.16
2.357
-
03
1.495
-
02
1.513
-
02
8.64
6.870
-
03
1.358
-
02
1.521
-
02
9.12
1.084
-
02
1.073
-
02
1.525
-
02
9.60
1.372
_
02
6.611
-
03
1.523
-
02
10.08
1.509
-
02
1.651
-
03
1.518
-
02
10.56
1.468
-
02
-3.578
-
03
1.511
-
02
11.04
1.242
-
02
-8.402
-
03
1.500
-
02
11.52
8.565
-
03
-1.216
-
02
1.487
-
02
12.00
3.571
_
03
-1.429
_
02
1.473
-
02
12.48
-1.887
-
03
-1.444
-
02
1.457
-
02
12.96
-7.032
-
03
-1.256
-
02
1.439
-
02
13.44
-1.110
-
02
-8.880
-
03
1.421
-
02
13.92
-1.346
-
02
-3.935
-
03
1.403
-
02
(Continued)

Table 4l (Continued)
244
b
a
Real
Imaginary
Absolute Value
i4.4o
-1.375
-
02
1.526
-
03
1.383
-
02
14.88
-1.190
-
02
6.654
-
03
1.364
-
02
15.36
-8.219
-
03
1.063
-
02
1.344
-
02
15-84
-3.292
-
03
1.282
-
02
1.324
-
02
16.32
2.064
-
03
1.288
-
02
1.304
-
02
16.80
6.955
_
03
1.080
_
02
1.284
_
02
17.28
1.056
-
02
6.951
-
03
1.264
-
02
17.76
1.228
-
02
2.017
-
03
1.245
-
02
18.24
1.185
-
02
-3.140
-
03
1.225
-
02
18.72
9.354
-
03
-7.619
-
03
1.206
-
02
19.20
5.278
_
03
-1.064
_
02
1.188
02
19.68
3.691
-
04
-1.168
-
02
1.169
-
02
20.16
-4.476
-
03
-1.060
-
02
1.151
-
02
20.64
-8.380
-
03
-7.626
-
03
1.133
-
02
21.12
-1.065
-
02
-3.334
-
03
1.116
-
02
21.60
-1.089
_
02
1.458
_
03
1.098
_
02
22.08
-9.098
-
03
5.849
-
03
1.082
-
02
22.56
-5.655
-
03
9.027
-
03
1.065
-
02
23.04
-1.239
-
03
1.042
-
02
1.049
-
02
23.52
3.294
-
03
9-795
-
03
1.033
-
02
24.00
7-075
-
03
7.320
-
03
1.018
-
02
24.48
9-397
-
03
3.506
-
03
1.003
-
02
24.96
9.844
-
03
-8.826
-
04
9.883
-
03
25.44
8.369
-
03
-4.983
-
03
9-740
-
03
25.92
5.301
-
03
-8.004
-
03
9.600
-
03
26.40
1.276
-
03
-9-377
-
03
9-463
-
03
26.88
-2.895
-
03
-8.869
-
03
9.330
-
03
27.36
-6.386
-
03
-6.622
-
03
9.199
-
03
27.84
-8.519
-
03
-3.119
-
03
9.072
-
03
28.32
-8.901
-
03
9.153
-
o4
8.947
-
03
28.80
-7.493
_
03
4.664
-
03
8.826
-
03
29.28
-4.618
-
03
7.382
-
03
8.707
-
03
29.76
-8.851
-
o4
8.546
-
03
8.591
-
03
30.24
2.933
-
03
7.954
-
03
8.478
-
03
30.72
6.063
-
03
5.766
-
03
8.367
-
03
31.20
7.884
_
03
2.460
-
03
8.259
-
03
31.68
8.054
-
03
-1.267
-
03
8.153
-
03
32.16
6.575
-
03
-4.645
-
03
8.050
-
03
32.64
3.783
-
03
-6.991
-
03
7.949
-
03
33.12
2.812
-
o4
-7.845
-
03
7.850
-
03
33.60
-3.196
_
03
-7.064
-
03
7.754
-
03
34.08
-5.933
-
03
-4.844
-
03
7.659
-
03
34.56
-7.380
-
03
-1.673
-
03
7.567
-
03
(Continued)

Table 4l (Continued)
245
b
13-500
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.48
6.971
-
o4
4.063
-
04
8.O69
-
o4
O.96
1.409
-
03
7.740
-
04
1.607
-
03
1.44
2.145
-
03
1.064
-
03
2.394
-
03
1.92
2.911
-
03
1.237
-
03
3.163
-
03
2.40
3.699
-
03
1.255
-
03
3.906
-
03
2.88
4.492
-
03
1.080
-
03
4.620
-
03
3.36
5.255
-
03
6.825
-
o4
5.300
-
03
3.84
5.942
-
03
4.083
-
05
5.942
-
03
4.32
6.488
-
03
-8.514
-
04
6.544
-
03
4.80
6.823
-
03
-I.98O
-
03
7.104
-
03
5.28
6.867
-
03
-3.305
-
03
7.621
-
03
5.76
6.548
-
03
-4.759
-
03
8.094
-
03
6.24
5.806
-
03
-6.241
-
03
8.524
-
03
6.72
4.608
-
03
-7.627
-
03
8.911
-
03
7.20
2.959
_
03
-8.771
-
03
9.256
-
03
7.68
9.130
-
o4
-9.518
-
03
9.561
-
03
8.16
-1.423
-
03
-9.725
-
03
9.828
-
03
8.64
-3.889
-
03
-9.277
-
03
1.006
-
02
9.12
-6.279
-
03
-8.109
-
03
1.026
-
02
9.60
-8.357
-
03
-6.226
-
03
1.042
-
02
10.08
-9.883
-
03
-3.713
-
03
1.056
-
02
10.56
-1.064
-
02
-7.431
-
o4
1.067
-
02
11.04
-1.047
-
02
2.434
-
03
1.075
-
02
11.52
-9.304
-
03
5.507
-
03
1.081
-
02
12.00
-7.179
-
03
8.139
-
03
1.085
-
02
12.48
-4.255
-
03
1.001
-
02
1.087
-
02
12.96
-8.089
-
o4
1.085
-
02
1.088
-
02
13.44
2.791
-
03
1.051
-
02
1.087
-
02
13.92
6.124
-
03
8.955
-
03
1.085
-
02
i4.4o
8.770
-
03
6.328
-
03
1.081
-
02
14.88
1.037
-
02
2.909
-
03
1.077
-
02
15.36
1.068
-
02
-8.904
-
o4
1.072
-
02
15.84
9.616
-
03
-4.584
-
03
1.065
-
02
16.32
7.288
-
03
-7.675
-
03
1.058
-
02
16.80
3.983
_
03
-9.725
-
03
1.051
-
02
17.28
1.421
-
o4
-1.043
-
02
1.043
-
02
17.76
-3.697
-
03
-9.660
-
03
1.034
-
02
18.24
-6.978
-
03
-7.514
-
03
1.025
-
02
18.72
-9.213
-
03
-4.290
-
03
1.016
-
02
19.20
-1.006
_
02
-4.612
-
o4
1.007
-
02
19.68
-9.375
-
03
3.396
-
03
9.971
-
03
20.16
-7.265
-
03
6.686
-
03
9.873
-
03
20.64
-4.052
-
03
8.894
-
03
9.773
-
03
21.12
-2.445
-
04
9.670
-
03
9.673
-
03
(Continued)

Table 4l (Concluded)
246
a
Real
Imaginary
Absolute 1
CD
21.60
3.547
_
03
8.89O
_
03
9.572
_
03
22.08
6.708
-
03
6.685
-
03
9.470
-
03
22.56
8.722
-
03
3.420
-
03
9.369
-
03
23.04
9.260
-
03
-3.572
-
o4
9.267
-
03
23.52
8.241
-
03
-4.012
-
03
9.166
-
03
24.00
5.848
_
03
-6.927
-
03
9.065
-
03
24.48
2.498
-
03
-8.6IO
-
03
8.965
-
03
24.96
-1.221
-
03
-8.78I
-
03
8.866
-
03
25.44
-4.662
-
03
-7.425
-
03
8.767
-
03
25.92
-7-224
-
03
-4.793
-
03
8.669
-
03
26.40
-8.463
-
03
-1.365
-
03
8.573
-
03
26.88
-8.175
-
03
2.242
-
03
8.477
-
03
27.36
-6.428
-
03
5.380
-
03
8.383
-
03
27.84
-3.555
-
03
7.488
-
03
8.289
-
03
28.32
-9.204
-
05
8.197
-
03
8.197
-
03
28.80
3.322
-
03
7.394
-
03
8.106
-
03
29.28
6.060
-
03
5.248
-
03
8.017
-
03
29.76
7.624
-
03
2.174
-
03
7.928
-
03
30.24
7.742
-
03
-1.245
-
03
7.841
-
03
30.72
6.4io
-
03
-4.366
-
03
7-755
-
03
31.20
3.900
-
03
-6.606
-
03
7.671
-
03
31.68
7.022
-
o4
-7-555
-
03
7.588
-
03
32.16
-2.567
-
03
-7.053
-
03
7.506
-
03
32.64
-5.283
-
03
-5.217
-
03
7.425
-
03
33.12
-6.936
-
03
-2.418
-
03
7-346
-
03
33.60
-7.225
-
03
7.913
-
04
7.268
-
03
34.08
-6.114
-
03
3.787
-
03
7.191
-
03
34.56
-3.839
-
03
5.991
-
03
7.116
-
03

Table 42
Value of l(l.20,a,b)
247
a
Real
0.00
0.000 +
0.60
4.864 -
1.20
8.515 -
1.80
2.348 -
2.40
7.852 -
3.00
-2.007 -
3.60
-7.773 -
4.20
-9.819 -
0.00
0.000 +
o.6o
2.698 -
1.20
3.403 -
l.8o
1.932 -
2.40
6.481 -
3.00
-2.538 -
3.60
-7.949 -
4.20
-9.778 -
4.8o
-8.488 -
5.4o
-5.076 -
6.oo
-8.627 -
6.6o
2.838 -
7.20
5.049 -
7.80
5.351 -
8.40
3.928 -
9.00
1.449 -
9.60
-1.182 -
10.20
-3.121 -
10.80
-3.834 -
11.4o
-3.223 -
12.00
-1.620 -
12.60
3.627 -
13.20
2.045 -
13.80
2.905 -
i4.4o
2.730 -
15.00
1.662 -
15.60
1.229 -
16.20
-1.343 -
16.80
-2.255 -
17.40
-2.349 -
18.00
-1.648 -
18.60
-4.383 -
19.20
8.431 -
19.80
1.763 -
20.4o
2.033 -
Imaginary Absolute Value
0.000
+
00
0.000
+
00
-8.241
-
02
4.933
-
01
-1.477
-
01
8.642
-
01
-1.829
-
01
2.976
-
01
-1.825
-
01
1.987
-
01
-1.496
-
01
1.509
-
01
-9.445
-
02
1.223
-
01
-3.182
-
02
1.032
-
01
0.000
+
00
0.000
+
00
-8.044
_
02
2.816
-
01
-1.440
-
01
3.695
-
01
-1.781
-
01
2.627
-
01
-1.773
-
01
1.887
-
01
-1.445
-
01
1.467
-
01
-9.014
-
02
1.202
-
01
-2.865
-
02
1.019
-
01
2.505
-
02
8.850
-
02
5.958
-
02
7.827
-
02
6.965
_
02
7.018
_
02
5.695
-
02
6.363
-
02
2.894
-
02
5.820
-
02
-3.715
-
03
5.363
-
02
-3.052
-
02
4.974
-
02
-4.405
_
02
4.637
-
02
-4.180
-
02
4.344
-
02
-2.635
-
02
4.085
-
02
-4.094
-
03
3.856
-
02
1.715
-
02
3.651
-
02
3.065
-
02
3-467
-
02
3.281
-
02
3.301
-
02
2.395
-
02
3.150
-
02
7.956
-
03
3.012
-
02
-9.344
-
03
2.886
-
02
-2.215
-
02
2.770
-
02
-2.660
-
02
2.662
-
02
-2.183
-
02
2.563
-
02
-1.011
-
02
2.471
-
02
4.193
-
03
2.386
-
02
1.613
-
02
2.306
-
02
2.188
-
02
2.231
-
02
1.990
-
02
2.161
-
02
1.134
-
02
2.096
-
02
-5.475
-
o4
2.034
-
02
inued)
00
01
01
01
02
02
02
02
00
01
01
01
02
02
02
02
02
02
03
02
02
02
02
02
02
02
02
02
02
03
02
02
02
02
03
02
02
02
02
03
03
02
02
(Cont

248
Table 42 (Continued)
b
0.480
2.400
Real
Imaginary
Absolute Value
21.00
1.602
-
02
-1.157
-
02
1.976
-
02
21.60
6.523
-
03
-1.807
-
02
1.921
-
02
22.20
-4.665
-
03
-1.809
-
02
1.869
-
02
22.80
-1.368
-
02
-1.199
-
02
1.819
-
02
23.4o
-1.760
-
02
-2.135
-
03
1.773
-
02
24.00
-1.534
-
02
7.954
-
03
1.728
_
02
24.60
-7.987
-
03
1.485
-
02
1.686
-
02
25.20
1.734
-
03
1.637
-
02
1.646
-
02
25.80
l.04l
-
02
1.225
-
02
1.607
-
02
26.40
1.515
-
02
4.144
-
03
1.571
-
02
27.00
1.452
-
02
-5.009
-
03
1.536
-
02
27.60
8.965
-
03
-1.206
-
02
1.502
-
02
28.20
5.899
-
o4
-1.469
-
02
1.470
-
02
28.80
-7.637
-
03
-1.220
-
02
i.44o
-
02
29.40
-1.292
-
02
-5.652
-
03
i.4io
-
02
30.00
-1.358
-
02
2.570
-
03
1.382
-
02
30.60
-9.571
-
03
9.590
-
03
1.355
-
02
31.20
-2.444
-
03
1.306
-
02
1.329
-
02
31.80
5.249
-
03
1.193
-
02
1.304
-
02
32.40
1.086
-
02
6.764
-
03
1.280
-
02
33-00
1.255
-
02
-5.339
-
o4
1.256
-
02
33.60
9.882
-
03
-7.388
-
03
1.234
-
02
34.20
3.920
-
03
-1.147
-
02
1.212
-
02
34.80
-3.180
-
03
-1.148
-
02
1.191
-
02
35.40
-8.948
-
03
-7.553
-
03
1.171
-
02
36.00
-1.146
-
02
-1.167
-
03
1.151
-
02
36.60
-9.950
-
03
5.410
-
03
1.133
-
02
37.20
-5.078
-
03
9.919
-
03
1.114
-
02
37.80
1.383
-
03
1.088
-
02
1.097
-
02
38.40
7.166
-
03
8.072
-
03
1.079
-
02
39.00
1.031
-
02
2.580
-
03
1.063
-
02
39.60
9.816
-
03
-3.633
-
03
1.047
-
02
40.20
5.963
-
03
-8.411
-
03
1.031
-
02
4o.8o
1.733
-
o4
-1.016
-
02
1.016
-
02
4i.4o
-5.508
-
03
-8.360
-
03
1.001
-
02
42.00
-9.133
-
03
-3.789
-
03
9.869
-
03
42.60
-9.513
-
03
2.039
-
03
9.729
-
03
43.20
-6.609
-
03
6.955
-
03
9.594
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.60
4.4o6
-
03
-4.218
-
02
4.24i
-
02
1.20
-2.969
-
03
-7.347
-
02
7.353
-
02
1.80
-2.462
-
02
-8.599
-
02
8.945
-
02
2.40
-5.281
-
02
-7.703
-
02
9.339
-
02
(Continued)

Table 42 (Continued.)
a
Real
3.00
-7.579 -
3.60
-8.418 -
4.20
-7-426 -
4.80
-4.845 -
5.40
-1.397 -
6.00
1.970 -
6.60
4.376 -
7.20
5.263 -
7.80
4.538 -
8.4o
2.572 -
9.00
6.197 -
9.60
-2.195 -
10.20
-3.546 -
10.80
-3.664 -
11.4o
-2.625 -
12.00
-8.534 -
12.60
1.033 -
13.20
2.430 -
13.80
2.932 -
i4.4o
2.447 -
15.00
1.204 -
15.60
-3.396 -
16.20
-1.663 -
16.80
-2.351 -
17.4o
-2.219 -
18.00
-1.362 -
18.60
-1.088 -
19.20
1.105 -
19.80
1.880 -
20.40
1.985 -
21.00
1.421 -
21.60
4.126 -
22.20
-6.807 -
22.80
-1.490 -
23.4o
-1.757 -
24.00
-1.421 -
24.60
-6.226 -
25.20
3-487 -
25.80
1.158 -
26.4o
1.540 -
27.00
1.384 -
27.60
7.668 -
28.20
-8.445 -
Imaginary Absolute Value
-4.969
-
02
9.063
-
02
-1.191
-
02
8.502
-
02
2.590
-
02
7.865
-
02
5.386
-
02
7.244
-
02
6.527
-
02
6.675
-
02
5-843
_
02
6.166
_
02
3.677
-
02
5.716
-
02
7.605
-
03
5.318
-
02
-2.017
-
02
4.966
-
02
-3.879
-
02
4.654
-
02
-4.376
_
02
4.377
_
02
-3.496
-
02
4.128
-
02
-1.637
-
02
3.905
-
02
5.447
-
03
3.704
-
02
2.348
-
02
3.522
-
02
3.245
-
02
3.356
-
02
3.033
-
02
3.204
-
02
1.870
-
02
3.066
-
02
1.995
-
03
2.938
-
02
-i.4o4
-
02
2.821
-
02
-2.430
_
02
2.712
-
02
-2.589
-
02
2.612
-
02
-1.890
-
02
2.518
-
02
-6.186
-
03
2.431
-
02
7.725
-
03
2.349
-
02
1.819
-
02
2.273
-
02
2.199
-
02
2.201
-
02
1.826
-
02
2.134
-
02
8.677
-
03
2.071
-
02
-3.247
-
03
2.011
-
02
-1.342
-
02
1.955
-
02
-1.856
-
02
1.901
-
02
-1.721
-
02
1.851
-
02
-1.016
-
02
1.803
-
02
-4.862
-
05
1.757
-
02
9.583
-
03
1.714
-
02
1.553
-
02
1.673
-
02
1.596
-
02
1.634
-
02
1.099
-
02
1.596
-
02
2.523
-
03
1.560
-
02
-6.423
_
03
1.526
-
02
-1.281
-
02
1.493
-
02
-1.459
-
02
1.462
-
02
inued)
02
02
02
02
02
02
02
02
02
02
04
02
02
02
02
03
02
02
02
02
02
03
02
02
02
02
03
02
02
02
02
03
03
02
02
02
03
03
02
02
02
03
04
(Cont

250
b
2.400
7.240
Table 42 (Continued)
a
Real
Imaginary
Absolute Value
28.80
-8.721 -
03
-1.135
- 02
1.432 02
29.40
-I.332 -
02
-4.391
- 03
1.403 02
30.00
-I.322 -
02
3.787
- 03
1.375 02
30.6o
-8.624 -
03
1.036
- 021
1.348 02
31.20
-1.274 -
03
1.316
- 02
1.322 02
31.80
6.236 -
03
1.138
- 02
1.298 02
32.40
1.135 -
02
5.788
- 03
1.274 02
33.00
l.24l -
02
-1.575
- 03
1.251 02
33.60
9.203 -
03
-8.l4l
- 03
1.229 02
3^.20
2.970 -
03
-1.170
- 02
1.207 -.02
34.80
-4.064 -
03
-1.115
- 02
1.187 02
35-40
-9.474 -
03
-6.807
- 03
1.167 02
36.00
-1.147 -
02
-2.842
- o4
1.147 02
36.60
-9.480 -
03
6.124
- 03
1.129 02
37.20
-4.313 -
03
1.023
- 02
1.111 02
37.80
2.166 -
03
1.071
- 02
1.093 02
38.40
7.704 -
03
7.512
- 03
1.076 02
39.00
1.043 -
02
1.838
- 03
1.060 02
39.60
9-511 -
03
-4.296
- 03
1.044 02
40.20
5.356 -
03
-8.776
- 03
1.028 02
4o.8o
-5.131 -
o4
-1.012
- 02
1.013 02
4i.4o
-6.037 -
03
-7.952
- 03
9-984 03
42.00
-9.334 -
03
-3.122
- 03
9.842 03
42.60
-9.337 -
03
2.645
- 03
9.704 03
43.20
-6.136 -
03
7.344
- 03
9.570 03
0.00
0.000 +
00
0.000
+ 00
0.000 + 00
0.60
5.009 -
03
1.976
- 03
5.384 03
1.20
i.oo4 -
02
3.170
- 03
1.053 02
1.80
1.496 -
02
2.883
- 03
1.524 02
2.40
1.934 -
02
6.078
- 04
1.935 02
3.00
2.247 -
02
-3.829
- 03
2.279 02
3.60
2.346 -
02
-1.012
- 02
2.555 02
4.20
2.147 -
02
-1.741
- 02
2.764 02
4.80
1.605 -
02
-2.432
- 02
2.914 02
5.4o
7.386 -
03
-2.921
- 02
3.013 02
6.00
-3.488 -
03
-3.049
- 02
3.069 02
6.60
-1.479 -
02
-2.715
- 02
3.091 02
7.20
-2.426 -
02
-1.908
- 02
3.087 02
7.80
-2.972 -
02
-7.341
- 03
3.06l 02
8.4o
-2.962 -
02
5.950
- 03
3.021 02
9.00
-2.358 -
02
1.805
- 02
2.969 02
9.60
-1.266 -
02
2.621
- 02
2.910 02
10.20
8.306 -
04
2.845
- 02
2.846 02
(Continued)

251
b
7.240
Table 42 (Continued)
a Real Imaginary Absolute Value
10.80
1.377
_
02
2.4l4
_
02
2.779
_
02
11.4o
2.304
-
02
1.426
-
02
2.710
-
02
12.00
2.637
-
02
1.264
-
03
2.640
-
02
12.60
2.298
-
02
-1.154
-
02
2.571
-
02
13-20
1.383
-
02
-2.087
-
02
2.503
-
02
13-80
1.450
-
03
-2.432
-
02
2.437
-
02
i4.4o
-1.077
-
02
-2.113
-
02
2.372
-
02
15.00
-1.952
-
02
-1.234
-
02
2.309
-
02
15.60
-2.248
-
02
-5.334
-
04
2.248
-
02
16.20
-1.900
-
02
1.088
-
02
2.190
-
02
16.80
-1.029
-
02
1.869
-
02
2.133
-
02
17.4o
9.837
-
04
2.076
-
02
2.079
-
02
18.00
1.146
-
02
1.671
-
02
2.026
-
02
18.60
1.809
-
02
7.943
-
03
1.976
-
02
19.20
1.908
-
02
-2.788
-
03
1.928
-
02
19.80
1.432
_
02
-1.221
_
02
1.881
_
02
20.40
5.455
-
03
-1.754
-
02
1.837
-
02
21.00
-4.674
-
03
-I.732
-
02
1.794
-
02
21.60
-1.294
-
02
-I.I83
-
02
1.753
-
02
22.20
-1.688
-
02
-2.93O
-
03
1.714
-
02
22.80
-1.545
-
02
6.498
_
03
1.676
-
02
23.40
-9.269
-
03
1.352
-
02
1.639
-
02
24.00
-4.521
-
o4
1.603
-
02
i.6o4
-
02
24.60
8.153
-
03
1.342
-
02
1.570
-
02
25.20
1.385
-
02
6.684
-
03
1.538
-
02
25.80
1.495
-
02
-1.904
-
03
1.507
-
02
26.40
1.125
-
02
-9.558
-
03
1.476
-
02
27.00
4.116
-
03
-1.388
-
02
1.447
-
02
27.60
-4.071
-
03
-1.360
-
02
1.419
-
02
28.20
-1.065
-
02
-8.969
-
03
1.392
-
02
28.80
-1.357
_
02
-1.622
-
03
1.366
-
02
29.40
-1.200
-
02
5.986
-
03
1.341
-
02
30.00
-6.6IO
-
03
1.139
-
02
1.317
-
02
30.60
7.388
-
04
1.291
-
02
1.293
-
02
31.20
7-597
-
03
1.019
-
02
1.271
-
02
31.80
1.175
-
02
4.233
-
03
1.249
-
02
32.40
1.193
-
02
-2.906
-
03
1.228
-
02
33.00
8.196
-
03
-8.860
-
03
1.207
-
02
33.60
1.899
-
03
-1.172
-
02
1.187
-
02
34.20
-4.821
-
03
-1.064
-
02
1.168
-
02
34.80
-9.745
-
03
-6.088
-
03
1.149
-
02
35.40
-1.130
-
02
3.254
-
o4
1.131
-
02
36.00
-9.084
-
03 6.436
(Continued)
-
03
1.113
-
02

252
Jo
7.24o
12.060
Table 42 (Continued)
a
Real
Imaginary
Absolute
: Val
36.60
-3.927
-
03
1.023
-
02
1.096
- 02
37.20
2.377
-
03
1.053
-
02
1.080
- 02
37.80
7.708
-
03
7.327
-
03
1.064
- 02
38.40
1.033
-
02
1.783
-
03
1.048
- 02
39.00
9.436
-
03
-4.196
-
03
1.033
- 02
39.60
5.429
-
03
-8.6IO
-
03
1.018
- 02
40.20
-2.750
-
o4
-I.OO3
-
02
1.003
- 02
4o.8o
-5.731
-
03
-8.066
-
03
9.895
- 03
4i.4o
-9.125
-
03
-3.460
-
03
9.759
- 03
42.00
-9.376
03
2.179
-
03
9.626
- 03
42.60
-6.483
-
03
6.940
-
03
9.497
- 03
43.20
-1.493
-
03
9.252
-
03
9.371
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.60
-6.956
-
04
1.838
-
03
1.966
- 03
1.20
-I.219
-
03
3.703
-
03
3.899
- 03
1.80
-1.402
-
03
5.595
-
03
5.768
- 03
2.40
-1.087
-
03
7.468
-
03
7.547
- 03
3.00
-1.469
-
04
9.211
_
03
9.212
- 03
3.60
1.494
-
03
1.064
-
02
1.074
- 02
4.20
3.827
-
03
1.151
-
02
1.213
- 02
4.80
6.727
-
03
1.156
-
02
1.337
- 02
5.4o
9.931
-
03
1.051
-
02
1.446
- 02
6.00
1.303
-
02
8.191
-
03
1.539
- 02
6.60
1.552
-
02
4.584
-
03
1.619
- 02
7.20
1.684
-
02
-1.177
-
04
1.684
- 02
7.80
1.649
-
02
-5.466
-
03
1.737
- 02
8.4o
1.415
-
02
-1.078
-
02
1.779
- 02
9.00
9.810
-
03
-1.522
-
02
1.810
- 02
9.60
3.816
-
03
-1.793
-
02
1.833
- 02
10.20
-3.081
-
03
-1.821
-
02
1.847
- 02
10.80
-9.824
-
03
-1.573
-
02
1.854
- 02
11.4o
-1.522
-
02
-1.062
-
02
1.856
- 02
12.00
-1.817
-
02
-3.577
-
03
1.852
- 02
12.60
-1.794
-
02
4.255
-
03
1.844
- 02
13.20
-1.439
-
02
1.133
-
02
1.832
- 02
13.80
-8.049
-
03
1.629
-
02
1.817
- 02
i4.4o
-9.630
-
05
1.799
-
02
1.800
- 02
15.00
7.843
-
03
1.598
_
02
1.780
- 02
15.60
1.405
-
02
1.058
-
02
1.759
- 02
16.20
1.712
-
02
2.927
-
03
1.737
- 02
16.80
1.630
-
02
-5.279
-
03
1.714
- 02
17.40
1.175
-
02
-1.214
-
02
1.690
- 02
(Continued)

Table 42 (Continued)
253
b
a
Real
Imaginary
Absolute Value
12.060
18.00
4.521
-
03
-I.603
-
02
1.665
-
02
18.60
-3.652
-
03
-1.599
-
02
l.64i
-
02
19.20
-1.076
-
02
-I.205
-
02
1.616
-
02
19.80
-1.503
-
02
-5.202
-
03
1.591
-
02
20.4o
-1.54o
-
02
2.803
-
03
1.566
-
02
21.00
-1.181
-
02
9.894
-
03
1.541
-
02
21.60
-5.239
-
03
1.423
-
02
1.516
-
02
22.20
2.550
-
03
1.470
-
02
1.492
-
02
22.80
9-453
-
03
1.124
-
02
1.468
-
02
23.40
1.362
-
02
4.825
-
03
1.445
-
02
24.00
1.395
-
02
-2.730
-
03
1.422
02
24.60
1.043
-
02
-9.326
-
03
1.399
-
02
25.20
4.094
-
03
-1.314
-
02
1.377
-
02
25.80
-3.215
-
03
-1.316
-
02
1.355
-
02
26.40
-9.410
-
03
-9.447
-
03
1.333
-
02
27.00
-1.274
-
02
-3.137
-
03
1.312
-
02
27.60
-1.232
-
02
3.902
-
03
1.292
-
02
28.20
-8.330
-
03
9.613
-
03
1.272
-
02
28.80
-2.022
-
03
1.236
-
02
1.252
-
02
29.40
4.706
-
03
i.i4o
-
02
1.233
-
02
30.00
9.858
-
03
7.098
-
03
1.215
-
02
30.60
1.194
-
02
8.028
-
o4
1.197
-
02
31.20
1.039
-
02
-5.560
-
03
1.179
-
02
31.80
5.769
-
03
-1.008
-
02
1.161
-
02
32.40
-4.729
-
o4
-1.144
-
02
1.144
-
02
33.00
-6.4o4
_
03
-9.286
-
03
1.128
_
02
33.60
-1.023
-
02
-4.365
-
03
1.112
-
02
34.20
-1.082
-
02
1.762
-
03
1.096
-
02
34.80
-8.072
-
03
7.186
-
03
1.081
-
02
35.40
-2.909
-
03
1.025
-
02
1.066
-
02
36.00
3.021
_
03
1.007
-
02
1.051
-
02
36.60
7.881
-
03
6.760
-
03
1.037
-
02
37.20
1.013
-
02
1.431
-
03
1.023
-
02
37.80
9.171
-
03
-4.211
-
03
1.009
-
02
38.40
5.363
-
03
-8.391
-
03
9-959
-
03
39.00
-3.692
-
05
-9.829
-
03
9.829
-
03
39.60
-5.292
-
03
-8.131
-
03
9.702
-
03
40.20
-8.746
-
03
-3-904
-
03
9.578
-
03
4o.8o
-9.343
-
03
1.459
-
03
9.456
-
03
4l.4o
-6.956
-
03
6.229
-
03
9.338
-
03
42.00
-2.410
-
03
8.901
-
03
9.222
-
03
42.6o
2.799
-
03
8.668
-
03
9.109
-
03
43.20
6.990
-
03
5.666
-
03
8.998
-
03
(Continued)

Table 42 (Continued)
b
16.900
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.60
-9.668
-
o4
-2.730
-
04
1.005
-
03
1.20
-1.942
-
03
-4.826
-
o4
2.001
-
03
1.80
-2.925
-
03
-5.66I
-
o4
2.980
-
03
2.40
-3.907
-
03
-4.628
-
o4
3.934
-
03
3.00
-4.855
-
03
-1.177
-
o4
4.856
_
03
3.60
-5.717
-
03
5.136
-
o4
5.740
-
03
4.20
-6.417
-
03
1.457
-
03
6.580
-
03
4.80
-6.857
-
03
2.708
-
03
7.372
-
03
5.40
-6.925
-
03
4.225
-
03
8.112
-
03
6.00
-6.510
-
03
5.919
-
03
8.798
-
03
6.6 0
-5.518
-
03
7.646
-
03
9.429
-
03
7.20
-3.897
-
03
9.214
-
03
1.000
-
02
7.80
-1.663
-
03
1.039
-
02
1.052
-
02
8.4o
1.083
-
03
1.094
-
02
1.099
-
02
9.00
4.138
-
03
1.063
-
02
i.i4o
-
02
9.60
7.201
-
03
9.305
-
03
1.177
-
02
10.20
9.896
-
03
6.929
-
03
1.208
-
02
10.80
1.181
-
02
3.605
-
03
1.235
-
02
11.4o
1.257
-
02
-3.935
-
o4
1.258
-
02
12.00
1.190
_
02
-4.628
-
03
1.277
_
02
12.60
9.695
-
03
-8.54o
-
03
1.292
-
02
13.20
6.095
-
03
-1.153
-
02
1.304
-
02
13.80
1.490
-
03
-1.304
-
02
1.313
-
02
i4.4o
-3.505
-
03
-1.272
-
02
1.319
-
02
15.00
-8.121
_
03
-1.044
-
02
1.323
-
02
15.60
-1.157
-
02
-6.448
-
03
1.324
-
02
16.20
-1.317
-
02
-1.309
-
03
1.324
-
02
16.80
-1.255
-
02
4.139
-
03
1.321
-
02
17.4o
-9.693
-
03
8.921
-
03
1.317
-
02
18.00
-5.o4o
_
03
1.211
_
02
1.312
_
02
18.60
5.896
-
o4
1.304
-
02
1.306
-
02
19.20
6.122
-
03
1.145
-
02
1.298
-
02
19.80
i.o44
-
02
7.570
-
03
1.290
-
02
20.40
1.262
-
02
2.158
-
03
1.280
-
02
21.00
1.216
-
02
-3.684
-
03
1.270
-
02
21.60
9.102
-
03
-8.707
-
03
1.260
-
02
22.20
4.089
-
03
-1.180
-
02
1.249
-
02
22.80
-1.789
-
03
-1.224
-
02
1.237
-
02
23.40
-7.203
-
03
-9.912
-
03
1.225
-
02
24.00
-1.090
_
02
-5.329
-
03
1.213
-
02
24.60
-1.200
-
02
4.431
-
o4
1.201
-
02
25.20
-1.024
-
02 6.033
(Continued)
-
03
1.188
-
02

255
Table 42 (Concluded)
16.900
a
Real
Imaginary
Absolute
rH
>
25.80
-6.035
_
03
1.009
_
02
1.176
- 02
26.40
-4.200
-
o4
1.162
-
02
1.163
- 02
27.00
5.211
-
03
1.026
-
02
1.150
- 02
27.60
9.446
-
03
6.342
-
03
1.138
- 02
28.20
1.122
-
02
8.852
-
o4
1.125
- 02
28.80
1.008
-
02
-4.704
_
03
1.100
- 02
29.40
6.354
-
03
-8.977
-
03
1.110
- 02
30.00
1.032
-
03
-I.O82
-
02
1.087
- 02
30.6o
-4.465
-
03
-9.778
-
03
1.075
- 02
31.20
-8.669
-
03
-6.145
-
03
1.063
- 02
31.80
-1.046
-
02
-9.282
_
o4
1.050
- 02
32.40
-9.387
-
03
4.441
-
03
1.038
- 02
33.00
-5.765
-
03
8.493
-
03
1.027
- 02
33.60
-6.273
-
o4
1.013
-
02
1.015
- 02
3^.20
4.585
-
03
8.923
-
03
1.003
- 02
34.80
8.4i4
-
03
5.249
-
03
9.917
- 03
35.40
9.803
-
03
1.731
-
o4
9.804
- 03
36.00
8.391
-
03
-4.852
-
03
9.693
- 03
36.60
4.620
-
03
-8.396
-
03
9.583
- 03
37.20
-3.982
-
o4
-9.467
-
03
9.476
- 03
37.80
-5.202
-
03
-7.792
-
03
9.369
- 03
38.40
-8.4o6
-
03
-3.896
-
03
9.265
- 03
39-00
-9.101
-
03
1.055
-
03
9.162
- 03
39.60
-7.123
-
03
5.600
-
03
9.061
- 03
40.20
-3.093
-
03
8.4io
-
03
8.961
- 03
4o.8o
1.769
-
03
8.685
_
03
8.863
- 03
4i.4o
6.012
-
03
6.381
-
03
8.767
- 03
42.00
8.382
-
03
2.225
-
03
8.672
- 03
42.60
8.203
-
03
-2.514
-
03
8.579
- 03
43.20
5.568
-
03
-6.407
-
03
8.488
- 03

APPENDIX B
SPECIFICATIONS FOR THE PARTICLE MOTION
MEASURING AND RECORDING SYSTEM
Transducers
Three-Component Transducers
The 3-component particle velocity transducers were manufactured by
Mark Products, Inc., Houston, Texas; all transducers were model L-1B-3DS.
The 3-in.-diam, 12-in.-long aluminum case housed three single component
transducers oriented in the vertical, radial, and transverse directions.
One unit weighed 6 lb, had 150-ft-long support and conductor cables,
and had a waterproof pressure rating of 1,000 psi. The shielded con
ductor cable contained six number 22 gage stranded wires and a ground
wire.
The maximum case to coil excursion was l/l6 in., the undamped
natural frequency was 4.5 Hz, and the useful frequency range at 65 per
cent of critical damping was 10 to several hundred Hz. The transduc
tion tolerance was plus or minus 5 percent, the frequency change with
tilt was less than 0.25 Hz at 15 degrees of tilt, and the frequency
change was less than 0.25 Hz at maximum coil excursion.
Transducers numbered 1 through 21 had a coil resistance of 87O
ohms and a damping shunt resistance of about 6,000 ohms. The nominal
transduction of these units was 1.7 volts/in./sec.
Transducers numbered 22 through 27 had a coil resistance of 1,480
ohms and a damping shunt resistance of 12,000 ohms. The nominal
256

transduction of these units was 2.36 volts/in./sec.
257
Single-Component Transducers
The single-component transducers used to measure particle veloc
ities in the horizontal plane were manufactured by Mark Products, Inc.,
Houston, Texas; both transducers were model L-1D. These transducers
were 2-3/8 in. diam, l-l/2 in. high, and weighed about l-l/2 lb.
The maximum case to coil excursion was 0.090 in., the undamped
natural frequency was 4.5 Hz, and the useful frequency range at 72 per
cent of critical damping was 10 to several hundred Hz. The transduc
tion tolerance was plus or minus 10 percent, the frequency change with
tilt was less than 0.25 Hz at 15 degrees of tilt, and the frequency
change was less than 0.1 percent at half of the rated output.
The coil resistance was 1,480 ohms and the shunt resistance was
3,000 ohms.
Cables
The instrumentation cables were made by the Belden Manufacturing
Company, Chicago, Illinois. This particular cable had a designated
trade number of 8777* The outside diameter of the cable was 0.27 in.
and covered with a 0.030-in.-thick chrome vinyl jacket.
Six number 22-gage stranded and tinned copper conductors with
0.010-in.-thick polypropylene insulation were contained in the cable.
The six conductors were separated into three pairs and each pair was
covered with an aluminized polyester shield; each shield was connected
to a separate ground wire. The conductors had a resistance of

258
0.016 ohms/ft, the suggested working voltage was 300 volts, and the
maximum operating temperature was 105 degrees C.
The working voltage between adjacent shields was 50 volts and the
breakdown voltage was 1,500 volts. The nominal capacitance between
conductors was 30 micro-microfarads/ft. The capacitance between
shields was 115 micro-microfarads/ft and the capacitance between a
conductor and a shield was 30 micro-microfarads/ft.
Amplifiers
The model 1-165 amplifiers were manufactured by the Consolidated
Electrodynamics Corporation, Pasadena, California. These differential
amplifiers had solid state components and were designed specifically
to drive high frequency recording galvanometers. Gain steps of 10,
20, 50, 100, 200, 500, and 1,000 were available to an accuracy of plus
or minus 2 percent and a vernier gain adjustment provided gains from
10 to 2,000. The amplifiers could operate from a 117 volt AC, 48- to
420-Hz power supply of 10 watts; the power supplied for these experi
ments was 60 Hz. The output frequency response error was less than
1 percent between 0 and 5>000 Hz and the linearity error was less than
0.25 percent at full scale amplification. Drift was less than 0.3 per
cent in 8 hours and 0.5 percent with a 10 percent change in line
voltage.
The input sensitivity was 0.010 to 1 volt to provide a full scale
output of 10 volts and the input impedance was 1 million ohms.
The output capability was 0.100 amperes at 10 volts and the out
put impedance was 1 ohm.

259
Galvanometers
Consolidated Electrodynamics Corporation, type 7-364, fluid
damped, high performance galvanometers were used. These units were of
the mirror type and had a high sensitivity. The undamped natural fre
quency of the galvanometers was 833 Hz. With an external damping re
sistance of 200 ohms to provide 64 percent critical damping, the useful
frequency range was 0 to 500 Hz. Terminal resistance of the galvanom
eters was 69 ohms, plus or minus 10 percent, and the maximum safe cur
rent was 0.050 amperes.
The frequency response error was less than plus or minus 2 per
cent and the linearity error was less than 1 percent for a full scale
deflection of 2 in. and an optical arm length of 11.5 in. The nominal
sensitivity of the galvanometers was 0.000397 amperes/in.
Oscillographs and Paper
Two type 5-119^4 oscillographs manufactured by the Consolidated
Electrodynamics Corporation were used. Each oscillograph could accom
modate 36 active type 7-300 series galvanometers and 4 inactive galva
nometers Each unit weighed 185 lb and required a 60-Hz power source
of 105 to 132 volts; it consumed 135 watts on standbyJ 350 watts with
timing and light sources activated, and 600 watts maximum during
recording.
Two tungsten-filament lamps provided the light source for the
optical system. An induction motor with a precision speed control and
a 16-speed gear driven transmission provided constant paper speeds
from 0.1 to 160 in./sec. An electronic timer or counter was employed

26o
8' I
to print timing lines on the moving paper at intervals of 0.01, 0.1,
and 1 sec. This oscillograph type uses 12-in.-wide paper, 250 to 1+75
ft long enclosed in a lightproof magazine.
The Lino-Writ 4 photorecording paper was made by the E. I. du Pont
de Nemours Co., Wilmington, Delaware. The rolls were 12 in. wide,
0.0025 in. thick and 400 ft long. The spectral response of this light-
sensitive paper was orthochromatic and the maximum writing speed with
the standard oscillograph light source was 50*000 in./sec.
Paper Processor
A Consolidated Electrodynamics Corporation type 23-109B oscillo
gram processor was used to develop the exposed photorecording paper.
Reference (Calibration) Voltage Supply
A Consolidated Electrodynamics Corporation, model 3-1^0 power
supply was used as the voltage source for the calibration records.
This unit is completely solid state and provided a regulated DC
voltage that was continuously adjustable from 1 to 24 volts at up to
0.200 amperes. Although 60 Hz-power was supplied to the unit, it
would operate on frequencies from 48 to 420 Hz; it required 25 watts
at 90 to 135 volts. Fifteen minutes of warm-up time was needed for
stable operation; the ambient operating temperature range was 0 to
50 degrees C.
The output voltage varied less than 0.03 percent with a supply
voltage change of 10 percent, and the output voltage varied less than
0.05 percent with an output current variation of 0 to 0.200 amperes.

Voltage drift was less than 0.005 percent/degree F.,
in an 8-hour period.
and 0.05 percent
Voltmeter
An AC or DC differential voltmeter with null detector was used to
measure the calibration voltage applied to the amplifier input. This
solid state model 887 AB voltmeter was manufactured by Fluke Electron
ics, Seattle, Washington. It was a solid state differential voltmeter
and had a DC voltage measuring accuracy of 0.005 percent of the measured
voltage plus 5 microvolts. The unit weighed 14 lb and operated from
either a rechargeable battery or from a 50 to 420 Hz, II5/23O volt AC
power source. The ambient operating temperature range was 0 to 50 de
grees C.
The circuit of this instrument is composed of an AC to DC conver
ter, a DC input attenuator, a DC transistorized voltmeter and a 0 to
11-volt comparison or reference voltage. There were seven scales in
the null range from 0 to 11 volts. Meter resolution and voltage reso
lution allowed readings equal to or greater than 1 microvolt. The DC
input resistance in the null range was infinite; above 11 volts, the
input resistance was 10,000 ohms.
The voltage regulation on the reference supply was 0.0002 percent
for a line voltage variation of 10 percent and the stability of the
reference voltage was 0.0005 percent/hr, 0.0007 percent/day, and
0.0013 percent in 60 days. The total instrument stability was less
than 0.002 percent/yr.

262
Connections
Seven pin connectors were used to join the transducer cable to the
instrumentation cable at the test site and at the truck that housed the
amplifiers and recording system. The connection of the transducers to
the cable was discussed in the text. The connections between the
cables, amplifiers, and oscillographs is given in Table 43.
Table 43
Connection of Measuring and Recording Components
Cable
No.
Transducer
Component
Amplifier
Serial No.
Galvanometer
Oscillograph A
Position No.
Oscillograph B
1
Vertical
8270
3
Radial
6152
6
Transverse
8120
3
2
Vertical
6279
9
Radial
3092
12
Transverse
8222
15
3
L-1D-BT
8258
18
L-ID-TT
6329
20
4
Vertical
6406
23
Radial
6305
6
Transverse
8271
9
5
Vertical
8208
26
Radial
8201
12
Transverse
4ll6
15
6
Vertical
4134
29
Radial
8214
18
Transverse
6326
20
7
Vertical
3121
32
Radial
8273
23
Transverse
6415
26
8
Vertical
8276
35
Radial
8142
29
Transverse
8256
32

263
Resistance of Transducer Circuits
After the last test was concluded, the resistance of each trans
ducer component was measured at the transducer cable connection to
assure that the transducer had not been damaged and that it had per
formed consistently throughout the testing program. Table 44 lists
the electrical resistance measured for the circuit of each transducer
component.

264
Table 44
Electrical Resistance of Transducer Components
After Completing Test Program
Location
Circuit
Resistance
in Ohms
Serial
Number
Radial
ft
Depth
ft
Vertical
Component
Radial
Component
Transverse
Component
Model L-1B-3DS Three Component Transducers
1
30.0
1.0
750
^50
750
3
30.0
5.0
780
780
760
4
3-5
1.0
760
800
760
5
90.0
5.0
740
760
770
8
90.0
15.0
750
760
780
9
2.646
Footing
780
810
800
10
30.0
15.0
780
780
780
li
30.0
25.0
810
44o
810
12
90.0
35.0
790
770
800
13
30.0
35.0
800
790
800
l4
10.0
5.0
760
790
780
17
10.0
25.0
810
810
810
18
10.0
35.0
800
790
830
19
90.0
1.0
760
790
800
20
10.0
1.0
760
760
750
21
90.0
25.0
800
800
790
22
6o.o
35.0
1,400
1,350
1,390
23
6o.o
1.0
1,310
1,300
1,320
24
6o.o
5.0
1,320
1,350
1,350
25
6o.o
25.0
1,380
1,320
1,380
26
6o.o
15.0
1,380
1,390
1,350
27
10.0
15.0
1,380
1,380
1,360
Model
L-1D Single Component Transducers
L-1D-BT
2.500
1.5

--
1,110
L-1D-TT
2.344
Footing


1,100

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BIOGRAPHICAL SKETCH
Lyman Wagner Heller, fourth son of Earl Wagner and Kathryn
Goembel Heller, was born July 22, 1928, at Geneseo, Illinois. He
graduated from Geneseo High School in 1946, entered the University
of Illinois in 1947? and received a B.S. in Agricultural Engineering
(machine design) in 1950. Following employment with the John Deere
Ottumwa Works, he was entered into the U. S. Army in 1951 and served
at the Office, Chief of Engineers until 1953- Employment with Ameri
can Air Filter, Bendix Aviation, and the consulting fjirm of Clark,
Daily, and Dietz preceded his B.S. in Civil Engineering from the
University of Illinois in 1957- He enrolled in the Graduate School
of the University of Florida in 1957, was a teaching assistant in the
Department of Civil Engineering, and received an M.S. in Engineering
in 1959- From 1959 to 1965, he was employed by the Naval Civil Engi
neering Laboratory, Port Hueneme, California, and was awarded a Navy
fellowship for studies in soil dynamics at the Graduate School of the
University of Florida. He is presently employed by the U. S. Army
Engineer Waterways Experiment Station, Vicksburg, Mississippi.
Lyman Wagner Heller is married to the former Ida Elizabeth Dean
and is the father of three sons, Earl, Alan, and Charles. He is a
registered Professional Engineer in the State of Florida and a member
of the American Society of Civil Engineers, the National Society of
Professional Engineers, the Scientific Research Society of America,.
Sigma Tau, Chi Epsilon, and Phi Kappa Phi.

I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
John H. Schmertmann, Chairman
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
0-4sU**
Morris W. Self
Professor of Civil Engineering
I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Gale E. Nevill, Jr.
Chairman, Department of Engineerid
Science and Mechanics

I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Frank E.
Professor of Civil Engineering,
University of Michigan
Formerly on Graduate Staff of the
University of Flotrida
This dissertation was submitted to the Dean of the Cpllege of Engineer
ing and to the Graduate Council, and was accepted as partial fulfill
ment of the requirements for the degree of Doctor of Philosophy.
August, 1971
Dean, Graduate School

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TITLE: The particle motirTTtd generated by the torsional vibration of a circular
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203
Mechanical losses accounted for most of the power needed to drive
the torsional vibrator. Power losses to the ground were less than 1/10
hp.
Particle motion measuring system
The particle motion measuring system used to detect, amplify, and
record the particle velocities generated in the sand deposit had ade
quate sensitivity and resolution to fulfill the purpose of the experi
mental program. The maximum probable error in the particle velocity
determinations, due to the measuring system, was between 7 and 15
percent.
Results of measurements
Torsional vibration of the test footing during the experimental
program did not cause it to tilt, but it probably settled about l/l6 in.
The maximum oscillatory displacement of the edge of the footing was
0.0072 in. and the minimum was 0.0015 in. Torsional oscillation of the
footing did not develop slippage between the footing and the soil on
the footing-soil contact area.
The maximum particle velocity generated in the soil deposit near
the footing was 0.47 in./sec and the minimum at a distant location was
0.00011 in./sec. Wear the footing, the direction of particle motion in
the sand deposit was essentially tangential to the circular footing; the
dominant direction of motion throughout the sand deposit was also
tangential to the footing.
Computations and Measurements
When the soil deposit was considered as a homogeneous elastic


Table 39 (Continued)
221
b
0.192
a
Real
Imaginary
Absolute
: Value
7.20
9.151
_
03
5.199
_
03
1.053
- 02
7.44
9.780
-
03
2.823
-
03
1.018
- 02
7.68
9.846
-
03
4.444
-
04
9.856
- 03
7.92
0.378
-
03
-1.814
-
03
9.552
- 03
8.16
8.433
-
03
-3.841
-
03
9.267
- 03
8.4o
7.087
-
03
-5.545
-
03
8.998
- 03
8.64
5.431
-
03
-6.854
-
03
8.745
- 03
8.88
3.570
-
03
-7.720
-
03
8.506
- 03
9.12
1.613
-
03
-8.120
-
03
8.279
- 03
9.36
-3.318
-
04
-8.057
-
03
8.064
- 03
9.60
-2.161
-
03
-7.557
-
03
7.860
- 03
9.84
-3.783
-
03
-6.668
-
03
7.667
- 03
10.08
-5.120
-
03
-5.456
-
03
7.482
- 03
IO.32
-6.114
-
03
-4.001
-
03
7.307
- 03
IO.56
-6.726
-
03
-2.392
-
03
7.139
- 03
10.80
-6.941
-
03
-7.223
-
04
6.979
- 03
11.04
-6.764
-
03
9.142
-
04
6.826
- 03
11.28
-6.222
-
03
2.431
-
03
6.680
- 03
11.52
-5.358
-
03
3.749
-
03
6.539
- 03
11.76
-4.233
-
03
4.807
-
03
6.405
- 03
12.00
-2.918
-
03
5.556
-
03
6.276
- 03
12.24
-1.494
-
03
5.968
-
03
6.152
- 03
12.48
-4.096
-
05
6.033
-
03
6.033
- 03
12.72
1.359
-
03
5.760
-
03
5.918
- 03
12.96
2.632
-
03
5.178
-
03
5.808
- 03
13.20
3.712
-
03
4.328
-
03
5.702
- 03
13.44
4.547
-
03
3.268
-
03
5-599
- 03
13.68
5.099
-
03
2.063
-
03
5.501
- 03
13.92
5-348
-
03
7.853
-
04
5.405
- 03
14.16
5.290
-
03
-4.934
-
o4
5.313
- 03
i4.4o
4.939
-
03
-1.702
-
03
5.224
- 03
14.64
4.324
-
03
-2.776
-
03
5.138
- 03
14.88
3.487
-
03
-3.660
-
03
5.055
- 03
15.12
2.481
-
03
-4.311
-
03
4.974
- 03
15.36
1.368
-
03
-4.701
-
03
4.896
- 03
15.60
2.133
-
o4
-4.816
_
03
4.821
- 03
15.84
-9.189
-
o4
-4.658
-
03
4.747
- 03
16.08
-1.966
-
03
-4.243
-
03
4.676
- 03
16.32
-2.872
-
03
-3.603
-
03
4.607
- 03
16.56
-3.590
-
03
-2.779
-
03
4.540
- 03
16.80
-4.087
-
03
-1.823
-
03
4.475
- 03
17.04
-4.34o
-
03
-7.917
-
o4
4.412
- 03
17.28
-4.343
-
03
2.555
-
o4
4.350
- 03
(Continued)


172
this damping law stems from the conservation of energy along a spheri
cal wave front.
The validity of the computed particle displacements was tested by
applying the geometrical damping law. At a vibration frequency of
30 Hz, the displacement at a distance of 60 ft and a depth of 25 ft was
v(60,25) = (0.008150) (151)
4nGr
o
At the same frequency, and at a distance of 84 ft and a depth of 35 ft,
the displacement was
v(84,35) -^-p (0.005817) (152)
4nGr
o
The damping law for body waves requires that these two displacements be
related by
v(84,35) ---6 + v(60,25) (153)
^842 + 352
Cancelling common terms and substituting in the above equation
0.005817 si || (0.00815)
Cl 0.005821 (154)
The calculated displacement amplitudes agreed with the geometrical
damping lawthe difference was less than 0.1 percent.
Since the calculated surface displacements agreed with published
results and the body wave particle displacements conformed to known
geometrical damping laws, the computed field of half-space motion was
considered valid; calculated motion could be confidently compared to
experimentally measured motion.


Table 4l (Continued)
243
b
5.800
9.64o
a
Real
Imaginary
Absolute Val
28.80
-8.503
_
03
-3.956
_
03
9.378
_
03
29.28
-9.229
-
03
3.204
-
04
9.235
-
03
29.76
-7.96O
-
03
4.402
-
03
9.096
-
03
30.24
-5.O2I
-
03
7.422
-
03
8.961
-
03
30.72
-I.090
-
03
8.762
-
03
8.829
-
03
31.20
2.964
-
03
8.181
_
03
8.702
_
03
31.68
6.267
-
03
5.857
-
03
8.577
-
03
32.16
8.128
-
03
2.335
-
03
8.457
-
03
32.64
8.185
-
03
-1.594
-
03
8.339
-
03
33.12
6.474
-
03
-5.073
-
03
8.225
-
03
33.60
3.409
-
03
-7.362
-
03
8.113
-
03
34.08
-3.126
-
o4
-7-999
-
03
8.005
-
03
34.56
-3.870
-
03
-6.886
-
03
7.899
-
03
0.00
0.000
+
00
0.000
+
00
0.000
+
00
0.48
-5.678
-
o4
-1.473
-
03
1.578
-
03
0.96
-1.228
-
03
-2.881
-
03
3.132
-
03
1.44
-2.061
-
03
-4.153
-
03
4.636
-
03
1.92
-3.127
-
03
-5.203
-
03
6.070
-
03
2.40
-4.452
_
03
-5.932
-
03
7.417
-
03
2.88
-6.015
-
03
-6.231
-
03
a. 661
-
03
3.36
-7.746
-
03
-5.991
-
03
9.793
-
03
3.84
-9.516
-
03
-5.121
-
03
1.081
-
02
4.32
-1.114
-
02
-3-572
-
03
1.170
-
02
4.80
-1.241
_
02
-I.352
_
03
1.248
_
02
5.28
-1.306
-
02
1.443
-
03
1.314
-
02
5.76
-1.289
-
02
4.629
-
03
1.370
-
02
6.24
-1.172
-
02
7.928
-
03
1.415
-
02
6.72
-9.482
-
03
1.099
-
02
1.451
-
02
7.20
-6.228

03
1.342
-
02
1.479
-
02
7.68
-2.165
-
03
1.484
-
02
1.500
-
02
8.16
2.357
-
03
1.495
-
02
1.513
-
02
8.64
6.870
-
03
1.358
-
02
1.521
-
02
9.12
1.084
-
02
1.073
-
02
1.525
-
02
9.60
1.372
_
02
6.611
-
03
1.523
-
02
10.08
1.509
-
02
1.651
-
03
1.518
-
02
10.56
1.468
-
02
-3.578
-
03
1.511
-
02
11.04
1.242
-
02
-8.402
-
03
1.500
-
02
11.52
8.565
-
03
-1.216
-
02
1.487
-
02
12.00
3.571
_
03
-1.429
_
02
1.473
-
02
12.48
-1.887
-
03
-1.444
-
02
1.457
-
02
12.96
-7.032
-
03
-1.256
-
02
1.439
-
02
13.44
-1.110
-
02
-8.880
-
03
1.421
-
02
13.92
-1.346
-
02
-3.935
-
03
1.403
-
02
(Continued)


208
Table 37 (Continued)
REAL PROCEDURE INFINITE (S, FCT, A, TEST) ;
VALUE S, A, TEST ; REAL A, TEST ; INTEGER S ;
REAL PROCEDURE FCT ;
BEGIN
REAL WHOLE, PART, TEST1 ;
WHOLE:=0 ;
TEST1:=10 (-TEST-1) ; DO
BEGIN
PARTINFINITE (S, FCT, A, A+l, TEST) ;
INFINITE:=WHOLE:=WHOLE + PART ;
A: =A + 1 ;
END UNTIL ABS (PART) < ABS (WHOLE X TEST1) ;
END INFINITE ;
IF A / -@68 AND B -f @68 THEN SIMPSON: =FINITE ( 1,FCT,A,B,TEST)ELSE
IF A / -@>68 AND B = @68 THEN SIMPSON: INFINITE ( 1,FCT, A,TEST)ELSE
IF A = -@68 AND B = @68 THEN SIMPSON:=INFINITE (-1,FCT, 0,TEST)
+INFINITE ( 1,FCT, 0,TEST)ELSE
SIMPSON:INFINITE (-1,FCT,-B,TEST) ;
END SIMPSON ;
$ CARD LIST
REAL PROCEDURE JONE(x);
$ CARD
COMMENT SEE MATH TABLES & OTHER AIDS TO COMPUTATION, 1957,PAGE 86 ;
VALUE X; REAL X;
BEGIN
REAL T,PONE,QONE,Y,ONE;
IF X <0 THEN ONE:=-l ELSE ONE:=1;
X:=ABS(x);
IF X<4.0 THEN BEGIN Y:=T:=x/4.0;T:=TXT;
JONE :=(((((((-.0001289769XT+.OO2206915 5)XT-.0236616773)XT+.1777582922)XT
-.8888839649)XT+2.6666660544)XT-3999999971)XT+1.9999999998) XYXONE
END
ELSE BEGIN
Y:=T:=4.0/x;T:=TXT;
PONE:= (((((.0000042414XT-.000020092)>T+.0000580759) XT -.0002232030)XT
+.0029218256) XT+.3989422819)X2.50662827463;
QONE:=(((((-0000036594XT+.00001622) XT -.0000398708)XT+.0001064741) XT
-.00063904) xr+.0374oo8364)xyx2.50662827463;
JONE:-SORT(2/(3.l4l59265358xX))x(PONExCOS(X-2.35619449018)-QONEXSIN
(X-2.35619449018))X0NE
END 5
END OF PROCEDURE JONE ;
$CARD LIST
(Continued)


Figure 30. Method of attaching transducers to the test footing.
112


35
t = G7
(103)
and in a nonhomogeneous half-space, the shear stress was represented by
tn = g(z)7b
(io4)
For an incompressible half-space, Gibson showed that if = T ,
then the ratio of the strains becomes
7-
_N G
7 G(z)
(105)
Half-Space Under Torsion
The stress and strain conditions developed in a half-space due to
a torsional moment applied to a rigid circular disk on the surface of
the half-space are analogous to the half-space conditions that result
from Gibsons solutions.
The stresses developed in the half-space by the disk are indepen
dent of the value of Poisson's ratio. Because Gibson assumed a
Poisson's ratio of l/2 before obtaining solutions for the stresses,
his stress solutions are also valid for the same value of Poisson's
ratio.
Dilatational strains are not developed in the half-space by a
torsional moment applied to the rigid disk. Gibson assumed that the
half-space was incompressible, so, again, no dilatational strains were
developed by the surface loads.
A disk in torsion produces shear stresses on a circular area at
the boundary of the half-space. Gibson's solutions are also for a cir
cular area loaded by a uniform vertical pressure at the surface of the
half-space.


Figure 27. Torsional vibrator mounted on the test foundation.
H
o
00


209
Table 37 (Continued)
COMMENT INTEGRATION PROGRAM
BEGIN
$$A START
$ CARD
$$A JONE
$ CARD
$$A SIMPSON
$ CARD
$$A INVTAN
$ CARD
REAL AO,A,B,REINT,IMINT;
INTEGER OLDT,NEWT,ELAPT;
FORMAT Fl("AO =",F6.2,/," B =" ,P7.3,////,"A/A0" ,X5,"A" ,X8,
"REAL,X7,"IMAGINARY",X4,"ABS VALUE",X5,"ARGUMENT",//),
F2(p4.1,f8.2,3E13.3,F11.3,I3,i4);
REAL PROCEDURE RELX(x); VALUE X; REAL X;
BEGIN
IF X NEQ 0 THEN
BEGIN REAL SX,AOSX;
SX:=SIN(x);
AOSX: =^OXSX;
RELX: =-(SIN(A0SX)-A0SXXC0S(A0SX) )xJONE(AXSX)xSIN(BXCOS(x) )/AOSX;
END ELSE
RELX:=0;
END OF THE PROCEDURE RELX;
REAL PROCEDURE REH(x); VALUE X; REAL X;
BEGIN
IF X NEQ Pl/2 THEN
BEGIN REAL SX,AOSX;
SX: =l/C0S(x);
AOSX: =AOXSX;
REHX: =(SIN(AOSX) -AOSXXCOS(AOSX) )XJ0NE(AXSX)X
(IF B NEQ 0 THEN EXP(-BXSIN(X)XSX) ELSE l)/AO;
END ELSE
REHX:=0;
END OF THE PROCEDURE REHX;
REAL PROCEDURE IMX(x); VALUE X; REAL X;
BEGIN
IF X NEQ 0 THEN
BEGIN REAL SX,AOSX;
SX:=SIN(x);
AOSX: =AOXSX;
IMX: =-(SIN(AOSX) -AOSXXCOS(AOSX) )xJ0NE(AXSX)x
(if B NEQ 0 THEN COS(BXCOS(x)) ELSE 1/A0SX;
END ELSE
IMX:=0;
END OF THE PROCEDURE IMX;
(Continued)


79
a constant frequency. The mass was changed by bolting different com
binations of identically matched weights to each vibrator flywheel.
Table 12 gives the stamped identification letter and weight of each
of the paired eccentric masses and the attaching bolts.
Table 12
Identification Letter and Weight of Eccentric Masses
Mass
Identification
Weight
lb
Bolt
Identification
Weight
lb
A
1.9771
E
0.135
B
1.9786
F
0.135
C
0.9777
G
0.106
D
0.4770
H
0.106
E
0.2314
I through N
0.084
The centrifugal force developed by each of the rotating eccentric
2
masses is Weu) /g so the moment, M generated by the torsional vi
brator was
M = (i)w () ir ft-lb (142)
where W is the weight (lb) of the eccentric mass and f is the fre
quency (Hz). The moment output of the vibrator per pound of eccentric
mass was
= 1.06l6f2 (143)
w
This relationship and Table 12 were used to calculate the moment


CHANGE IN ELEVATION
1 6 11 16 21 26
ELEVATION READING NUMBER
Figure 54. Results of footing settlement and tilt measurements.


Leskowitz, U. S. Army Electronics Command, Fort Monmoutu, New Jersey,
for his cooperation and assistance during the computational aspects of
the work.
Appreciation and gratitude is expressed to the many individuals
at the Waterways Experiment Station who assisted and contributed to the
prosecution of this study. Special thanks are extended to Mr. Monroe B.
Savage, Jr., and Mr. Jack Fowler for their capable and cooperative
assistance during the experimental work. Particular thanks are also
expressed to Miss K. Jones and her helpful staff at the Station's
Reproduction and Reports Office for preparing the reproducible copy
and photographs, and for printing the manuscript.
Finally, the author wishes to thank his wife, Elizabeth, and his
children for their patience and sacrifices during the qourse of this
study.
iii


258
0.016 ohms/ft, the suggested working voltage was 300 volts, and the
maximum operating temperature was 105 degrees C.
The working voltage between adjacent shields was 50 volts and the
breakdown voltage was 1,500 volts. The nominal capacitance between
conductors was 30 micro-microfarads/ft. The capacitance between
shields was 115 micro-microfarads/ft and the capacitance between a
conductor and a shield was 30 micro-microfarads/ft.
Amplifiers
The model 1-165 amplifiers were manufactured by the Consolidated
Electrodynamics Corporation, Pasadena, California. These differential
amplifiers had solid state components and were designed specifically
to drive high frequency recording galvanometers. Gain steps of 10,
20, 50, 100, 200, 500, and 1,000 were available to an accuracy of plus
or minus 2 percent and a vernier gain adjustment provided gains from
10 to 2,000. The amplifiers could operate from a 117 volt AC, 48- to
420-Hz power supply of 10 watts; the power supplied for these experi
ments was 60 Hz. The output frequency response error was less than
1 percent between 0 and 5>000 Hz and the linearity error was less than
0.25 percent at full scale amplification. Drift was less than 0.3 per
cent in 8 hours and 0.5 percent with a 10 percent change in line
voltage.
The input sensitivity was 0.010 to 1 volt to provide a full scale
output of 10 volts and the input impedance was 1 million ohms.
The output capability was 0.100 amperes at 10 volts and the out
put impedance was 1 ohm.


Table 4
Values of the Terms in the Integrand of I3
a
sec a
a sec 01
0
sin (a sec a)
0
cos (aQ sec q?)
a sec oi
J^(a sec o)
tan o'
b tan oi
0
1.00000
0.4800
0.46175
0.88701
5.76000
-O.3163
0.0000
0.0000
rr/8
1.08240
0.51955
0.49647
0.86805
6.23462
-0.2243
0.41421
1.20121
tt/4
1.41425
0.67884
0.62788
0.77831
8.14608
0.2524
1.0000
2.9000
3tt/8
2.61308
1.25428
0.95033
0.31123
15-05134
0.2032
2.4142
7.00118
tt/ 2
cc
oc
-1 to 1
-1 to 1
CC
0.0000
CC
OC
Table 5
Values, f( sin (a sec n) a sec q; cos
o o
(aQ sec a) /a.Q J^(a sec a) g-b tan a
0
0.074969
-O.3163
1.00000
-0.023713
tt/ 8
0.094740
-0.2243
0.30083
-0.0063927
tt/4
0.207358
0.2524
0.055023
0.0028797
3rr/8
I.I6658
0.2032
0.000910
0.0010616
tt/2
Undefined
0.0000
0.00000
0.00000


251
b
7.240
Table 42 (Continued)
a Real Imaginary Absolute Value
10.80
1.377
_
02
2.4l4
_
02
2.779
_
02
11.4o
2.304
-
02
1.426
-
02
2.710
-
02
12.00
2.637
-
02
1.264
-
03
2.640
-
02
12.60
2.298
-
02
-1.154
-
02
2.571
-
02
13-20
1.383
-
02
-2.087
-
02
2.503
-
02
13-80
1.450
-
03
-2.432
-
02
2.437
-
02
i4.4o
-1.077
-
02
-2.113
-
02
2.372
-
02
15.00
-1.952
-
02
-1.234
-
02
2.309
-
02
15.60
-2.248
-
02
-5.334
-
04
2.248
-
02
16.20
-1.900
-
02
1.088
-
02
2.190
-
02
16.80
-1.029
-
02
1.869
-
02
2.133
-
02
17.4o
9.837
-
04
2.076
-
02
2.079
-
02
18.00
1.146
-
02
1.671
-
02
2.026
-
02
18.60
1.809
-
02
7.943
-
03
1.976
-
02
19.20
1.908
-
02
-2.788
-
03
1.928
-
02
19.80
1.432
_
02
-1.221
_
02
1.881
_
02
20.40
5.455
-
03
-1.754
-
02
1.837
-
02
21.00
-4.674
-
03
-I.732
-
02
1.794
-
02
21.60
-1.294
-
02
-I.I83
-
02
1.753
-
02
22.20
-1.688
-
02
-2.93O
-
03
1.714
-
02
22.80
-1.545
-
02
6.498
_
03
1.676
-
02
23.40
-9.269
-
03
1.352
-
02
1.639
-
02
24.00
-4.521
-
o4
1.603
-
02
i.6o4
-
02
24.60
8.153
-
03
1.342
-
02
1.570
-
02
25.20
1.385
-
02
6.684
-
03
1.538
-
02
25.80
1.495
-
02
-1.904
-
03
1.507
-
02
26.40
1.125
-
02
-9.558
-
03
1.476
-
02
27.00
4.116
-
03
-1.388
-
02
1.447
-
02
27.60
-4.071
-
03
-1.360
-
02
1.419
-
02
28.20
-1.065
-
02
-8.969
-
03
1.392
-
02
28.80
-1.357
_
02
-1.622
-
03
1.366
-
02
29.40
-1.200
-
02
5.986
-
03
1.341
-
02
30.00
-6.6IO
-
03
1.139
-
02
1.317
-
02
30.60
7.388
-
04
1.291
-
02
1.293
-
02
31.20
7-597
-
03
1.019
-
02
1.271
-
02
31.80
1.175
-
02
4.233
-
03
1.249
-
02
32.40
1.193
-
02
-2.906
-
03
1.228
-
02
33.00
8.196
-
03
-8.860
-
03
1.207
-
02
33.60
1.899
-
03
-1.172
-
02
1.187
-
02
34.20
-4.821
-
03
-1.064
-
02
1.168
-
02
34.80
-9.745
-
03
-6.088
-
03
1.149
-
02
35.40
-1.130
-
02
3.254
-
o4
1.131
-
02
36.00
-9.084
-
03 6.436
(Continued)
-
03
1.113
-
02


Table 4l (Continued)
239
b
0.384
a
Real
Imaginary
Absolute Value
l4.4o
1.843
_
02
-6.327
_
03
1.948
_
02
14.88
1.302
-
02
-I.363
-
02
1.885
-
02
15.36
5.121
-
03
-1.753
-
02
1.826
-
02
15.84
-3.409
-
03
-1.737
-
02
1.770
-
02
16.32
-I.07O
-
02
-1.345
-
02
1.718
-
02
16.80
-I.523
_
02
-6.813
_
03
I.669
-
02
17.28
-1.620
-
02
9-379
-
o4
1.622
-
02
17.76
-1.357
-
02
8.064
-
03
1.578
-
02
18.24
-8.IO9
-
03
1.305
-
02
1.537
-
02
18.72
-I.I56
-
03
1.493
-
02
1.497
-
02
19.20
5.702
-
03
1.343
-
02
1.459
-
02
19.68
1.098
-
02
9.070
-
03
1.424
-
02
20.16
1.359
-
02
2.922
-
03
1.390
-
02
20.64
1.309
-
02
-3.582
-
03
1.357
-
02
21.12
9.742
-
03
-9.OO3
-
03
1.326
-
02
21.60
4.397
_
03
-1.220
-
02
1.297
-
02
22.08
-1.684
-
03
-1.257
-
02
1.269
-
02
22.56
-7.135
-
03
-I.OI6
-
02
1.242
-
02
23.04
-1.079
-
02
-5.608
-
03
1.216
-
02
23.52
-1.191
-
02
-4.047
-
06
1.191
-
02
24.00
-1.036
_
02
5-377
-
03
1.167
-
02
24.48
-6.580
-
03
9-359
-
03
1.144
-
02
24.96
-1.494
-
03
1.112
-
02
1.122
-
02
25.44
3.730
-
03
1.036
-
02
1.101
-
02 ,
25.92
7.936
-
03
7.331
-
03
1.080
-
02
26.4o
1.023
_
02
2.793
_
03
1.061
-
02
26.88
1.018
-
02
-2.201
-
03
1.042
-
02
27.36
7.879
-
03
-6.532
-
03
1.023
-
02
27.84
3.909
-
03
-9.267
-
03
1.006
-
02
28.32
-7.923
-
o4
-9.855
-
03
9.887
-
03
28.80
-5.161
_
03
-8.239
-
03
9.722
-
03
29.28
-8.242
-
03
-4.848
-
03
9.562
-
03
29.76
-9.395
-
03
-4.913
-
o4
9.408
-
03
30.24
-8.426
-
03
3.836
-
03
9.258
-
03
30.72
-5.617
-
03
7.177
-
03
9.114
-
03
31.20
-1.647
_
03
8.821
-
03
8.973
-
03
31.68
2.569
-
03
8.455
-
03
8.837
-
03
32.16
6.088
-
03
6.222
-
03
8.705
-
03
32.64
8.150
-
03
2.671
-
03
8.577
-
03
33.12
8.341
-
03
-1.370
-
03
8.452
-
03
33.60
6.671
-
03
-4.992
-
03
8.332
-
03
34.08
3-564
-
03
-7.401
-
03
8.214
-
03
34.56
-2.495
-
o4
-8.096
-
03
8.100
-
03
(Continued)


Page
41. Value of 1(0.96,a,b) 238
42. Value of l(l.20,a,b) - 247
43- Connection of Measuring and Recording Components 262
44. Electrical Resistance of Transducer Components After
Completing Test Program 264
x


22
1 .
sin a g a g cos a
il2 = / 2 COS b yjl gc Jn(ag)dg
/
0 aQg i v1 g
i \ :
X
(73)
The integrands of I and I become unbounded as g approaches
unity, so a change in variables is appropriate. Let
g = sin a
dg = cos oda
(74)
(75)
and
v77
g = COS cy
(76)
where 0 g 1 and 0 s a tt/2
By replacing the variable g with the variable &
J11/2 sin (a sin cy) (a sin a) cos (a sin o o o
0
a sm n
o
x J-^a sin O') sin (b cos cy)dcy (77)
and
X2= "
TT/
/
0
/2
sin (a sin ¡y) (aQ sin ¡y) cos (aQ sin a sin a
o
X J^(a sin cy) cos (b cos cy)djy (78)
The integrand of the integral
/ sin (a g) a g cos a g / 2 ,
v o& o& e-b^g -1 j
vTlT n
aQgvg 1
(ag)dg (79)
also becomes unbounded when g approaches unity and a[ change of vari
ables for is indicated. Let


74
increase in foundation motion were pertinent to the design of a tor
sional vibrator to drive the test footing and to the expected experi
mental measurements.
The designed test footing was a hollow concrete cylinder. The
diameter of the cylinder was 60 in. and it was 28 in. high. A 32-in.-
diam, 18-in.-high cylindrical void was formed interior to and sym
metric with the outer cylinder; the top of the void was 2 in. below
the top of the outer cylinder. The mass ratio of the footing was 2.1,
the resonant frequency was about 40 Hz, and the amplitude magnifica
tion ratio was approximately 5*2 (Richart, Hall, and Woods, 1970).
Although the resonant frequency occurred within the planned test fre
quency range, the magnification of the footing motion did not present
apparent experimental difficulties.
Vibrator Design
A special vibrator was designed to drive the test footing in a
torsional mode of oscillation about the vertical axis of the circular
footing. The vibrator design goals were (a) to minimize mechanical
sources of noise and vibration, (b) to utilize a remote source of
power, and (c) to keep all dynamic forces in a horizontal plane.
Goal (a) was attempted by using an electric motor to drive the rota
ting eccentric masses through a rubber timing belt, goal (b) by a
long electrical power line from the electric motor to a remote
generator, and goal (c) by rotating the eccentric masses in a hor
izontal plane. The vibrator was designed to provide a twisting moment
of about 3 ft-kip in the 20- to 50-Hz-frequency range.


78
2OOM0
P = 550
= 0.02 hp (l40)
Power losses to the ground appeared to be negligible; power losses due
to belt friction, windage, and bearing friction controlled the selec
tion of an electric motor.
Four precision, sealed, permanently-lubricated, self-aligning
ball bearings were chosen to support the two shafts for the rotating
eccentric weights. With a torque capacity of 3 ft-kip and a shaft
center distance of 31 in., the radial loading on each shaft was about
1,200 lb and the maximum radial bearing load was about 800 lb. At a
frequency of 50 Hz, the peripheral velocity of the 1.625-in.-diam
shaft was 1,280 fpm. Using the average tabulated coefficient of fric
tion for ball bearings (Oberg and Jones, 194-9), the horsepower loss
for the four bearings was
p = (2,400)(l,280)(0.0023)/33,000
= 0.21 hp
The belt friction and windage losses were taken equal to the bearing
losses and a 1-hp motor was judged adequate to drive the torsional
vibrator.
Frequency and moment capacity
The eccentricity of the rotating masses on the torsional vibrator
was 4 in. and the masses were varied to change the torque output at


THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCILLATING
RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT
Description of Test Site
The selected test site was located at an inactive auxiliary field
on the Eglin Air Force Base, Florida, military reservation. The soil
at the site was a homogeneous marine terrace deposit of poorly graded,
fine- to medium-grained sand. The water table was about 100 ft deep in
this thick, free-draining sand deposit and the shear wave velocity in
creased significantly with depth.
Geographical Location and Geological Setting
The test site chosen for the experimental work was located in
section 14, range 24 west, township 1 north, Okaloosa county, Florida,
at about 86 degrees and 38 minutes west longitude and 30 degrees and
35-1/2 minutes north latitude. The circular test foundation and the
approximately 100-ft-square test area was about 1,050 ft west and 230 ft
north of the south end of the north-south runway at Piccolo field (aux
iliary field 5)? within the boundaries of Eglin Air Force Base and about
15 miles north of the Gulf of Mexico coastline. Piccolo field was
chosen as a test site because it was militarily inactive and the water
table was unusually deep. The elevation of the area was about 175 ft
above mean sea level and the topography was quite flat; elevations
within the test area varied less than 3 in. Native grasses covered the
40


Nonhomogeneous (Linear E) Elastic Half-Space
Because the elastic moduli of soils is known to depend on the mean
effective stress applied to the soil (Hardin and Richart, 1963) and the
effective stress in a soil deposit increases with depth below the ground
surface, a nonhomogeneous elastic half-space would be a more realistic
analytical representation of a soil deposit than a homogeneous elastic
half-space. Thus, it was worthwhile to consider the progress that has
been made toward the use of a nonhomogeneous half-spaco for foundation
problems and some apparent relationships between a homogeneous and a
nonhomogeneous half-space with an elastic modulus that increases lin
early with depth.
In a review of existing knowledge of the dynamic behavior of soils
and foundations, Jones, Lister, and Thrower (1966) made particular men
tion of the need for and the apparent lack of attention to the develop
ment and application of nonhomogeneous theory to these problems. In
summarizing the analysis of machine foundations on soils with a modulus
that changes with depth, they state:
The problem which arises when the elastic properties
vary continuously with depth, rather than in the
discontinuous fashion typified by layered media, has
received less attention, although it is important,
especially in view of the variation of elastic prop
erties of non-cohesive soils with the mean stress.
Structures [ soil stratification] of this type are
probably rather more frequent in practice than the
layered case. No analytical investigations of the
kind described above are known to the authors. Pauw
(1953), however, has analyzed the problem by assum
ing essentially that the phenomena can be described
by considering the propagation of a cone-shaped bun
dle of longitudinal-type waves downwards into the
soil. ...Pauw's approach appears rather unsatisfac
tory from an analytical point of view.


11
RIGID DISK
'/////////
ELASTIC
HALF-SPACE
-a
rr
~zz
_L
ELEMENT
dz
Z
U

w
ELEVATION VIEW
Figure 1. Rigid circular disk on the surface of an elastic
half-space.


Table 40 (Continued)
236
a
Real
Imaginary
Absolute Value
0.00
0.000
+
00
0.000
+
00
0.000 +
00
O.36
-4.503
-
o4
-4.036
-
o4
6.047 -
o4
0.72
-9.I2O
-
o4
-7.870
-
o4
1.205 -
03
1.08
-1.395
-
03
-I.I30
-
03
1.795 -
03
1.44
-I.907
-
03
-1.411
-
03
2.372 -
03
1.80
-2.45O
-
03
-I.609
-
03
2.931 -
03
2.16
-3.O2I
-
03
-I.703
-
03
3.468 -
03
2.52
-3.613
-
03
-I.672
-
03
3.981 -
03
2.88
-4.207
-
03
-1.499
-
03
4.466 -
03
3-24
-4.782
-
03
-I.I69
-
03
4.923 -
03
3.60
-5.306
_
03
-6.742
_
04
5-348 -
03
3.96
-5.742
-
03
-1.374
-
05
5-742 -
03
4.32
-6.051
-
03
8.026
-
04
6.104 -
03
4.68
-6.191
-
03
1.753
-
03
6.434 -
03
5.04
-6.121
-
03
2.8o4
-
03
6.733 -
03
5.40
-5.807
-
03
3.909
-
03
7.000 -
03
5.76
-5.226
-
03
5.007
-
03
7.238 -
03
6.12
-4.368
-
03
6.031
-
03
7.447 -
03
6.48
-3.242
-
03
6.906
-
03
7.629 -
03
6.84
-1.875
-
03
7-557
-
03
7.786 -
03
7.20
-3.190
-
o4
7.913
-
03
7.919 -
03
7.56
1.355
-
03
7.914
-
03
8.030 -
03
7.92
3.058
-
03
7.522
-
03
8.120 -
03
8.28
4.685
-
03
6.719
-
03
8.191 -
03
8.64
6.126
-
03
5.518
-
03
8.245 -
03
9.00
7.274
-
03
3.962
-
03
8.283 -
03
9.36
8.030
-
03
2.127
-
03
8.307 -
03
9-72
8.317
-
03
1.159
-
o4
8.318 -
03
10.08
8.087
-
03
-1.944
-
03
8.317 -
03
10.44
7.328
-
03
-3.911
-
03
8.306 -
03
10.80
6.068
_
03
-5.642
-
03
8.286 -
03
11.16
4.378
-
03
-7.001
-
03
8.257 -
03
11.52
2.366
-
03
-7.873
-
03
8.221 -
03
11.88
1.747
-
o4
-8.176
-
03
8.178 -
03
12.24
-2.033
-
03
-7.872
-
03
8.130 -
03
12.60
-4.085
-
03
-6.. 968
-
03
8.077 -
03
12.96
-5.814
-
03
-5.524
-
03
8.020 -
03
13.32
-7.074
-
03
-3.648
-
03
7.959 -
03
13.68
-7.753
-
03
-1.487
-
03
7.894 -
03
i4.o4
-7.789
-
03
7.802
-
04
7.828 -
03
i4.4o
-7.171
_
03
2.962
_
03
7.758 -
03
14.76
-5.948
-
03
4.871
-
03
7.688 -
03
15.12
-4.223
-
03
6.337
-
03
7.615 -
03
15.48
-2.148
-
03
7.229
-
03
7-541 -
03
15.84
9.234
-
05
7.466
-
03
7-467 -
03
(Continued)


38
Particle motion
The strain energy per unit volume generated in a torsionally loaded
homogeneous half-space is (Timoshenko and Goodier, 1951)
w = m (4 + aze)
(106)
and the kinetic energy of an oscillating particle in the half-space is
(107)
1 2
K = i pdVv
The strain energy per unit volume developed in a nonhomogeneous
half-space is
WN 2G(z) (CTr0 + ze)
(108)
N
and the kinetic energy of an oscillating particle in the nonhomogeneous
half-space is
% = I dWN
(109)
Equating the strain energy and the kinetic energy in each of the
above cases (Timoshenko and Goodier, 1951) gave
.2 1
V Gp
(4 + 4)
(no)
and
2 1 ( 2 2 \
N G(z)p \CTr0 az0/
(HD
N
The loading and mass density, p of each half-space was assumed equal,


Table 32
Ratio of Half-Space to Soil Displacement
190
Normalized Half-Space Displacement/
Normalized Soil Displacement
Depth Below Radial Distance, fjt
Surface, ft
3.5 10
30
60
90
Vibration Frequency
, 15 Hz
1
O.95 0.51
0.27
0.20
0.20
5
0.54
0.44
0.36
0.26
15
1.5
3.0
2.3
1.6
25
2.4
4.7
6.8
5.4
35
2.8
6.2
8.8
5-7
Vibration Frequency
, 20 Hz
1
0.8l 0.53
0.27
0.26
0.28
5
0.70
0.64
0.56
0.44
15
2.2
7.3
4.6
2.9
25
3-6
12
8.7
7.0
35
2.9
7-5
7.4
7.9
Vibration Frequency
, 30 Hz
1
0.79 0.63
0.26
0.30
0.36
5
0.73
0.76
0.68
0.4i
15
2.5
3.6
3.8
2.0
25
5.2
9.1
11
4.7
35
3.6
11
9-4
5.3
Vibration Frequency
, 40 Hz
l
0.82 0.57
0.44
0.35
1.5
5
1.1
0.75
1.2
0.82
15
3.8
7-3
6.3
2.8
25
6.7
13
16
13
35
5.6
9-3
10
12
Vibration Frequency
, 50 Hz
1
0.71 0.51
0.36
0.76
0.78
5
1.4
0.95
0.95
1.3
15
2.3
3.4
4.1
4.9
25
8.7
16
9.0
5.8
35
4.0
4.5
6.0
8.9


INTRODUCTION
Background
Soil and foundation engineers who specify and design adequate sup
port systems for buildings and equipment are commonly concerned with
three aspects of the performance of their foundations: (l) the long
term load carrying capacity, as it relates to the type of facility and
safety of its inhabitants, (2) the immediate or during-construction
settlements, and (3) the rate and amount of postconstruction
settlement.
There are situations, however, when the engineer must provide a
foundation with additional capabilities. Such a situation occurs when
the foundation supports sustained or transient dynamic loads as devel
oped by punch presses, forging machines, shock testerjs, and unbalanced
machinery. For these cases, the prescribed foundation not only must
provide support for the imposed static and dynamic loads to assure the
safe operation of the equipment and the facility, but also must mini
mize the radiation of undesirable vibrations into the surrounding soil
where they can be transmitted to adjacent inhabited or vibration sensi
tive areas. Thus, one part of the soil engineers responsibility is to
provide a foundation that will inhibit or diminish the generation and
transmission of dangerous, troublesome, and annoying ground vibrations.
Crockett (1965) has briefly discussed some of these problems; the
writer is aware of a California case in which a titanium forging plant
1


Table 39 (Continued.)
222
O.96O
a
Real
Imaginary
Absolute
Value
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.24
2.120
-
02
-5.458
-
03
2.189
- 02
0.48
3.621
-
02
-I.072
-
02
3.776
- 02
0.72
4.274
-
02
-I.56I
-
02
4.550
- 02
0.96
4.245
-
02
-1.995
-
02
4.690
- 02
1.20
3.8l8
-
02
-2.36O
_
02
4.488
- 02
1.44
3.205
-
02
-2.644
-
02
4.155
- 02
1.68
2.525
-
02
-2.838
-
02
3-799
- 02
I.92
1.837
-
02
-2.938
-
02
3-465
- 02
2.16
1.172
-
02
-2.942
-
02
3.167
- 02
2.40
5-487
_
03
-2.854
_
02
2.906
- 02
2.64
-1.743
-
04
-2.679
-
02
2.679
- 02
2.88
-5.148
-
03
-2.427
-
02
2.481
- 02
3.12
-9.336
-
03
-2.111
-
02
2.309
- 02
3.36
-1.267
-
02
-1.746
-
02
2.157
- 02
3.60
-I.510
_
02
-1.346
-
02
2.023
- 02
3-84
-1.661
-
02
-9.305
-
03
1.904
- 02
4.08
-I.722
-
02
-5.151
-
03
1.798
- 02
4.32
-1.698
-
02
-1.167
-
03
1.702
- 02
4.56
-1.597
-
02
2.498
-
03
I.616
- 02
4.80
-1.428
-
02
5.713
-
03
1.538
- 02
5-04
-I.205
-
02
8.371
-
03
1.467
- 02
5.28
-9-4i4
-
03
i.o4o
-
02
1.402
- 02
5.52
-6.523
-
03
1.174
-
02
1.343
- 02
5.76
-3.529
-
03
1.239
-
02
1.288
- 02
6.00
-5.819
-
o4
1.237
-
02
1.238
- 02
6.24
2.181
-
03
1.171
-
02
1.191
- 02
6.48
4.639
-
03
1.050
-
02
1.148
- 02
6.72
6.689
-
03
8.830
-
03
1.108
- 02
6.96
8.256
-
03
6.809
-
03
1.070
- 02
7.20
9.291
-
03
4.561
-
03
1.035
- 02
7.44
9-774
-
03
2.212
-
03
1.002
- 02
7.68
9.711
-
03
-1.122
-
04
9.712
- 03
7.92
9.137
-
03
-2.294
-
03
9.421
- 03
8.16
8.110
-
03
-4.230
-
03
9-147
- 03
8.4o
6.706
-
03
-5.833
_
03
8.888
- 03
8.64
5.019
-
03
-7.037
-
03
8.644
- 03
8.88
3.151
-
03
-7.800
-
03
8.412
- 03
9.12
1.209
-
03
-8.103
-
03
8.193
- 03
9.36
-7.009
-
o4
-7.954
-
03
7.985
- 03
9.60
-2.48o
_
03
-7.381
_
03
7.787
- 03
9.84
-4.039
-
03
-6.436
-
03
7.598
- 03
10.08
-5.307
-
03
-5.184
-
03
7.419
- 03
10.32
-6.227
-
03
-3.707
-
03
7.247
- 03
10.56
-6.767
-
03
-2.094
-
03
7.084
- 03
(Continued)


-155; B 5; 4, R, 3.5, I.
FT i i i i n i iir tiiir~rir~irii r r~rT t t L J I T F
REC. NO.; CHAN. NO.; TRANS (S/N, CMP, R, Z)
H I *J IA I L I J Ul LI I I f,
'>55; B I; 9, V, 2.6, *
:i i! j in i r 11 m i
>55; B2; 9, R, 2.6, *
II
155; B 3; 4, V, 3.5,
155; B4; 4, R, 3.5, h
/W'
'-155; B6 ; l-D, T, 2.5, 1.5'
'-155; B 7; l-D, T, 2.3, *1
i/|l55|b81_I41 JO, 5A(\(\|\|/|/I
jV155; B9 ; 3, V, 10, 5 y/\A\^V^//\A
^155; BIO; 24, V, 60, 5
.11II 1.1.111II 1.11.1
v
AI55; BI I; 26, V, 60, I5VN/J\A\
I I I I II I I
fJI55; BI2; 8, V, 90, 15lU^vKyKjMv/W/\Ak>>/1w,MjA^
iiiiiim
f.
V',
yy\rwVr
vV,/
M/v/X^ ajv
M
AA\j VJAfil\\\)\JT
'/7.
aUVM
M
A /W>A
w
/VVvW/M^
lA
Iv/^/'YW /'A
/'M^WWW^A^W^v./hW,
A A A A
/M
\v#
//W
A
NOTE: TEST RECORD SHOWS WAVE FORMS,
, NOT AMPLITUDE COMPARISONS.
/\ A ^ ^ ^ A A \ ^ V k/^ A AAA/NAr|/yY\/\^M/W\A>pv|Vluy\IA/\/MVA/K
A A a
ON FOOTING
M Af
-V
n."
ON FOOTING1
/M
\A/V\aA/WW
A/ifV
ON FOOTING-
/u
\W
A/\ ImI A/*
aIaIaLa/
i/AA\
K/ H
Figure 48. Typical test record at 50 Hz, oscillograph B, without eccentric weights on vibrator.
F-*
-P"
ON


BIOGRAPHICAL SKETCH
Lyman Wagner Heller, fourth son of Earl Wagner and Kathryn
Goembel Heller, was born July 22, 1928, at Geneseo, Illinois. He
graduated from Geneseo High School in 1946, entered the University
of Illinois in 1947? and received a B.S. in Agricultural Engineering
(machine design) in 1950. Following employment with the John Deere
Ottumwa Works, he was entered into the U. S. Army in 1951 and served
at the Office, Chief of Engineers until 1953- Employment with Ameri
can Air Filter, Bendix Aviation, and the consulting fjirm of Clark,
Daily, and Dietz preceded his B.S. in Civil Engineering from the
University of Illinois in 1957- He enrolled in the Graduate School
of the University of Florida in 1957, was a teaching assistant in the
Department of Civil Engineering, and received an M.S. in Engineering
in 1959- From 1959 to 1965, he was employed by the Naval Civil Engi
neering Laboratory, Port Hueneme, California, and was awarded a Navy
fellowship for studies in soil dynamics at the Graduate School of the
University of Florida. He is presently employed by the U. S. Army
Engineer Waterways Experiment Station, Vicksburg, Mississippi.
Lyman Wagner Heller is married to the former Ida Elizabeth Dean
and is the father of three sons, Earl, Alan, and Charles. He is a
registered Professional Engineer in the State of Florida and a member
of the American Society of Civil Engineers, the National Society of
Professional Engineers, the Scientific Research Society of America,.
Sigma Tau, Chi Epsilon, and Phi Kappa Phi.


5
10
15
20
25
30
35
40
183
SHEAR WAVE VELOCITY IN FPS
250 500 750 1,000 1,250
o
S
\
\
l&\
\
COMPUTI
ON A VE
:D MOTION WAS
iLOCITY OF 650
BASED
-PS
u\
\
o
\
o
lOltJ
\
\
\
O \
o
\
\
\
c* n
/O
o
o
n
t
a U
|
Ky
o SURFA
CE WAVE METHO
LJ
D
i
i
i
i
to1 a
Q VIBRATION METHOD
9 COMPRESSION WAVE FROM SURFACE
ol
\a QP
IS COMPRESSION WAVE FROM FOOTING
I I

c
i
i
Figure 6l. Shear wave velocities versus depth.


Figure 31. Test site topography, vegetation, and borehole markers
H
H
-p-


198
detailed determination was needed, other indicators of the variation of
the in situ soil stresses, and the corresponding modulus variations,
had to be used.
Gibbs and Holtz (1957) found that the in situ stress conditions in
a sand correlate with the penetration resistance of a;standard split-
spoon sampler; Figure 60 shows the relationship between penetration
resistance and bearing capacity of a cone penetrometer. Figure 62 is
a plot of the average cone bearing capacity of the sand deposit versus
depth and a plot of the average nonhomogeneous half-space displacement
ratio, as given in Table 36, versus the depth of the transducer in the
sand deposit. The two plots in Figure 62 showed that the average dis
placement ratio was related to the average cone bearihg capacity at
various depths in the sand deposit.
An interpretation of the observed correlation between the nonhomo
geneous half-space displacement ratios and the cone bearing capacity
shown in Figure 62 was considered worthwhile. Where the cone bearing
capacity was unusually high, lateral soil stresses werje high and the
shear modulus of the sand was greater than the value used in the dis
placement calculations; a calculation modulus less than the actual
modulus would result in larger computed displacements and in larger
displacement ratios. Conversely, where the cone bearing capacity was
unusually low, the lateral soil stresses were low and the shear modulus
of the sand was less than the value used in the displacement calcula
tions; the computed displacements were low and the computed displacement
ratios were also low. A quantitative relationship between cone bearing
capacity and dynamic shear modulus, however, was not found.


262
Connections
Seven pin connectors were used to join the transducer cable to the
instrumentation cable at the test site and at the truck that housed the
amplifiers and recording system. The connection of the transducers to
the cable was discussed in the text. The connections between the
cables, amplifiers, and oscillographs is given in Table 43.
Table 43
Connection of Measuring and Recording Components
Cable
No.
Transducer
Component
Amplifier
Serial No.
Galvanometer
Oscillograph A
Position No.
Oscillograph B
1
Vertical
8270
3
Radial
6152
6
Transverse
8120
3
2
Vertical
6279
9
Radial
3092
12
Transverse
8222
15
3
L-1D-BT
8258
18
L-ID-TT
6329
20
4
Vertical
6406
23
Radial
6305
6
Transverse
8271
9
5
Vertical
8208
26
Radial
8201
12
Transverse
4ll6
15
6
Vertical
4134
29
Radial
8214
18
Transverse
6326
20
7
Vertical
3121
32
Radial
8273
23
Transverse
6415
26
8
Vertical
8276
35
Radial
8142
29
Transverse
8256
32


7
Comparisons
Richart and Whitman (1967) examined existing experimental data for
the vibratory behavior of surface footings founded on soil materials
and compared this data to the theoretically predicted vibratory response
of these same footings on an elastic half-space. These comparisons
confirmed the applicability of the elastic half-space model for
predicting the oscillatory motion of circular foundations on soil.
Similar comparisons of calculated and measured results would indicate
the usefulness of the half-space model for predicting the particle mo
tions generated in a soil deposit by a vibrating footing.
Objective and Goals
The confirmed utility of the elastic half-space model for predict
ing the oscillatory behavior of circular footings founded on soil sug
gested that this same model, or variations thereof, could be useful for
predicting the particle motion field generated within a soil deposit by
a vibrating footing. The objective of this study was to test the hy
pothesized usefulness of the half-space model for predicting the vibra
tions transmitted into a soil foundation by an oscillating footing.
A test of the hypothesis involved three specific goals: (l) extend
Bycroft's (1956) solution for the torsional oscillation of a rigid disk
on the surface of an elastic half-space to include the motion of the
half-space and evaluate the resulting expression for absolute values of
the motion of the half-space, (2) conduct a field experiment on a
natural soil deposit that physically represents the boundary conditions


high frequency waves with short wavelengths and a larger spacing is
desirable for lower frequency waves with long wavelengths. The short
est expected wavelength, for a 50 Hz shear wave, was 13 ft; the long
est expected wavelength, for a 15 Hz compression wave, was 87 ft.
Thus, about seven high frequency shear wavelengths and one low
frequency compression wavelength would be contained in the zone of
particle motion measurements. Transducer spacing in planes parallel
to the ground surface was varied from about half of the shortest ex
pected wavelength to half of the longest expected wavelength, a range
from 6.5 to 30 ft; transducer spacing in the vertical direction was
varied from 4 ft near the ground surface to 10 ft at greater depths.
The natural processes that deposited the sand formation at the
test site include the effects of ocean currents and wave action. These
effects were undoubtedly related to the direction of the shoreline and
the direction of the prevailing winds during the Pleistocene epoch.
Therefore, there was a possibility that the sand deposit in the im
mediate vicinity of the test foundation had undetectable seismic wave
propagation characteristics that were direction dependent. Previous
sections have established the gross properties of the test site, so,
for consistency, it was necessary that the particle motion measurements
reflect the gross seismic characteristics of the test site. One means
of relating the measurements to gross properties is by a random sam
pling of the motion in several different directions from the test
foundation. Practical geometrical field constraints (trees, access
roads, borehole drilling operations, etc.) precluded locating the trans
ducers in random directions from the test footing. The depth of the


i6i
Table 19
Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,380 ft-lb Oscillating at 19 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
0.00723
0.0119
0.206
2.344
Footing
None
None
0.177
2.5
1.5
None
None
0.124
3.5
1.0
0.00373
0.00463
0.0827
10.0
1.0
0.00174
0.00105
0.0190
30.0
1.0
0.00152
0.00268
0.00949
60.0
1.0
0.00111
0.000542
O.OO663
90.0
1.0
O.OOO565
0.000513
0.00430
90.0
5.0
0.000344
0.000628
0.00335
10.0
5.0
0.00102
0.00129
0.0139
30.0
5.0
0.000679
0.000377
0.00588
60.0
5.0
O.OOO56O
0.000417
O.OO356
60.0
15.0
o.oooo846
0.0000627
0.000523
90.0
15.0
0.000111
0.0000473
O.OOO513
30.0
15.0
0.000194

O.OOO688
60.0
25.0
0.0000226
0.0000540
0.000154
60.0
35.0


0.000112
90.0
25.0
0.0000463
0.0000397
0.000138
90.0
35.0
0.0000293
0.0000160
0.000130
10.0
15.0
--
0.000421
0.00168
10.0
25.0


o.ooo466
10.0
35.0


0.000219
30.0
25.0
0.0000832
o.oooo485
0.000318
30.0
35.0


0.000182


V
= integer
| = arbitrary parameter
tt = 3.14159+
P =
CTij
a =
t =
tN =
0 =
tan ft' =
r =
0) =
mass density = y/g
stress on the i plane in the j direction
mean effective stress
shear stress
shear stress in a nonhomogeneous half-space
angular rotation of disk or footing; angle of internal
friction of soil
coefficient of friction
limiting angular rotation of the disk or footing
angular frequency
xvii


Page
THE MEASURED PARTICLE MOTION GENERATED BY A TORSIONALLY OSCIL
LATING RIGID CIRCULAR FOOTING ON A NATURAL SAND DEPOSIT 40
Description of Test Site 40
Geographical Location and Geological Setting 40
Soil Exploration 4l
Borings 43
Penetration tests 43
Laboratory Tests 48
Unit weight 48
Gradation 49
Seismic Wave Propagation Tests 50
Design of the Experiment 53
Foundation Design 53
Practical considerations 53
Diameter of the test footing 56
Stresses at the footing-soil interface 1 57
Stresses near the periphery of the footing 60
Footing emplacement operation 63
Position of dead load on cured first pour-^ 66
Rigidity of the footing 68
Limiting torsional moment 72
Dynamic response of the foundation 73
Vibrator Design 74
Power requirements 75
Frequency and moment capacity 78
Foundation and Transducer Location + 80
Location of the test footing 80
Location of transducers 8l
Isolation of power and recording facilities 84
Construction of Test Facilities 84
Foundation Construction 84
Fabrication of the footing form 84
Placing the form 87
First pour of concrete 88
Backfilling 95
Second pour of concrete 99
Vibrator Construction 104
Motor 16*+
Mounting the vibrator 16*+
Operating tests 166
Transducer Installation 110
Performance tests 116
Boreholes 113
Transducer alignment ¡ H3
Installing procedure ll8
Backfilling 1 H8
Particle Motion Measuring System 122
Functional Components 122
Transducers 125
v


147
Schedule of Tests
Footing Settlement and Tilt
When sand materials with relative densities less than 70 percent
are subjected to vibration, densification of the sand and surface set
tlement are expected (D'Appolonia, Whitman, and D'Appolonia, 1969).
The relative density of the sand deposit at the Piccolo field test site
was about 62 percent, so settlement of the vibrating test footing was a
distinct possibility.
Settlement and tilt measurements were made periodically during
the entire six-week testing program. Figure 49 shows a level rod po
sitioned for measuring the elevation of the center of the vibrator
frame and Figure 50 shows an elevation measurement at one of four loca
tions around the top edge of the footing. Twenty-six elevation and
tilt measurements were conducted during the six-week test period.
A carpenter's level was used to check the tilt of the footing.
Figures 51 and 52 show this level on the frame of the vibrator and
Figure 53 shows it on the top surface of the footing. Results of the
settlement measurements are plotted in Figure 54.
Transducer Operation
The operation of the 3-conrponent transducers was tested by mount
ing each transducer on the test foundation, operating the torsional
vibrator, and recording the transducer signals with an oscillograph.
The oscillograph record was examined to insure that the amplitude and
phase of the transducer signal compared favorably with the L-1D-TT
transducer bonded to the top of the test foundation.
After testing the transverse component, the transducer was


Table 4l (Continued)
24l
b
1.920
5.800
a
Real
Imaginary
Absolute
: Value
21.60
3.362
_
03
-1.243
_
02
1.288
- 02
22.80
-2.666
-
03
-I.232
-
02
1.260
- 02
22.56
-7.859
-
03
-9.512
-
03
1.234
- 02
23.03
-1.112
-
02
-4.737
-
03
1.208
- 02
23.52 :
-I.I8I
-
02
8.844
-
o4
1.184
- 02
24.00
-9.879
-
03
6.090
_
03
1.160
- 02
24.48
-5.856
-
03
9-757
-
03
1.138
- 02
24.96
-7.000
-
04
1.114
-
02
1.116
- 02
25.44
4.418
-
03
1.002
-
02
1.095
- 02
25.92
8.377
-
03
6.741
-
03
1.075
- 02
26.40
1.035
-
02
2.093
-
03
1.056
- 02
26.88
9.971
-
03
-2.852
-
03
1.037
- 02
27.36
7.409
-
03
-6.997
-
03
1.019
- 02
27.84
3.300
-
03
-9.457
-
03
1.002
- 02
28.32
-1.400
-
03
-9.747
-
03
9.847
- 03
28.80
-5.635
-
03
-7.876
-
03
9.684
- 03
29.28
-8.488
-
03
-4.326
-
03
9.527
- 03
29.76
-9.374
-
03
6.705
-
05
9.374
- 03
30.24
-8.159
-
03
4.307
-
03
9.226
- 03
30.72
-5.177
-
03
7.463
-
03
9.083
- 03
31.20
-l.l4l
-
03
8.871
-
03
8.944
- 03
31.68
3.027
-
03
8.273
-
03
8.809
- 03
32.16
6.4oi
-
03
5.860
-
03
8.678
- 03
32.64
8.258
-
03
2.220
-
03
8.551
- 03
33.12
8.232
-
03
-1.808
-
03
8.428
- 03
33.60
6.381
-
03
-5.320
-
03
8.308
- 03
34.08
3.167
-
03
-7.555
-
03
8.191
- 03
34.56
-6.614
-
o4
-8.051
-
03
8.078
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.48
-9.957
-
o4
4.203
-
03
4.320
- 03
0.96
-1.491
-
03
8.324
-
03
8.457
- 03
1.44
-1.052
-
03
1.221
-
02
1.225
- 02
1.92
6.222
-
04
1.558
-
02
1.559
- 02
2.40
3.630
_
03
1.805
_
02
l.84i
- 02
2.88
7.818
-
03
1.915
-
02
2.069
- 02
3.36
1.276
-
02
1.846
-
02
2.244
- 02
3.84
1.779
-
02
1.570
-
02
2.373
- 02
4.32
2.207
-
02
1.085
-
02
2.46o
- 02
4.80
2.476
_
02
4.241
_
03
2.512
- 02
5.28
2.512
-
02
-3.445
-
03
2.536
- 02
5.76
2.275
-
02
-1.124
-
02
2.538
- 02
6.24
1.764
-
02
-1.803
-
02
2.522
- 02
6.72
1.026
-
02
-2.273
-
02
2.494
- 02
(Continued)


Table 30
Normalized Soil Particle Displacements
188
Depth Below
Soil
Displacement/Displacement of Edge of Footing
Radial Distance from Center of Footing, ft
Ground, ft
3.5
10
30
60
90
Vibration Frequency, 15 Hz
1
0.439
0.101
0.0528
0.0357
0.0231
5
0.0739
0.0314
0.0192
0.0181
15
0.00915
0.00378
0.00282
0.00279
25
0.00247
0.00176
0.000873
0.000799
35
0.00116
O.OOO963
0.000593
0.000709
Vibration Frequency, 20 Hz
1
0.523
0.114
0.0674
0.0347
0.0212
5
0.0675
0.0273
0.0157
0.0133
15
0.00762
0.00198
0.00181
0.00201
25
0.00206
0.000874
0.000871
0.000782
35
0.00142
0.00101
0.00090
O.OOO65
Vibration Frequency, 30 Hz
1
0.555
0.123
0.0957
o.o4oi
0.0222
5
0.0850
0.0314
0.0177
0.0196
15
0.00940
0.00546
0.00303
0.00398
25
0.00201
0.00161
0.000908
0.00162
35
0.00159
0.000956
0.000971
0.00133
Vibration Frequency, 40 Hz
1
0.562
0.159
0.0670
0.0425
0.00651
5
0.0640
0.0383
0.0120
0.0119
15
0.00756
0.00328
0.00221
0.00343
25
0.00196
0.00138
0.000780
O.OOO699
35
0.00128
o.ooi4o
0.00112
0.000718
Vibration Frequency, 50 Hz
1
0.666
0.199
0.0926
0.0219
0.0142
5
0.0605
0.0342
0.0175
0.00848
15
0.0149
0.00801
0.00384
0.00219
25
0.00178
0.00128
0.00162
0.00178
35
0.00211
0.00338
0.00213
0.00110


Uj
REC. NO.; CHAN. NO.; TRANS (S N, CMP, R, Z)
i i i i i i i i i i i i i i
155; AI; 9, T, 2.6, *
155; A2 ; 14, R, 10, 5
^155; A3; 14, T, 10, 5
155; A4; 3, R, 30,
I I I I I I I I I I I II I I
A/W 155; A5; 3, T, 30, 5
U4^4-41J41
155; A 6; 24, R, 60, 5
w
VA
Ml!
155; A 7; 24, T, 60, 15
155; A 8; 26, R, 60, 15
155; A9; 26, T, 60, 15
.1 : 1
'
¡ r 1
A1
1 1
; 8,
NOTE: TEST RECORD SHOWS WAVE FORMS,
NOT AMPLITUDE COMPARISONS.
jyM"\ \ps
Figure 47. Typical test record at 50 Hz, oscillograph A, without eccentric weights on vibrator.
-p-
VJI


LIST OF REFERENCES
Abramowitz, M., and Stegun, I. A. (1964), Handbook of Mathematical Func
tions, Applied Mathematics Series AMS 55, National Bureau of Stan
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Arnold, R. N., Bycroft, G. N., and Warburton, G. B. (1955), "Forced
Vibrations of a Body on an Infinite Elastic Solid," Journal of
Applied Mechanics, Transactions of the American Society of
Mechanical Engineers, Vol. 22, Sept., pp. 391-400.
ASTM (1969), "Standard Method for Penetration Test and Split-Barrel
Sampling of Soils," ASTM Standards, American Society for Testing
and Materials, Part II, March, pp 492-494.
Awojobi, A. 0., and Grootenhuis, P. (1965), "Vibration of Rigid Bodies
on Semi-Infinite Elastic Media," Proceedings of the Royal Society of
London, Series A, Vol. 287, pp. 27-63.
Barkan, D. D. (1962), Dynamics of Bases and Foundations, McGraw-Hill
Book Co., New York, 434 pp.
Barkan, D. D. (1965), "Basic Problems of the Dynamics of Bases and
Foundations," Soil Mechanics and Foundation Engineering, UDC
624.131.0063 (translated from Osnovaniya, Fundamently i Mekhanika
Gruntov), No. 6, Nov.-Dec., pp. 29-30.
Bernhard, R. K. (1967), "Stress and Wave Patterns in Soils Subjected to
Dynamic Loads," Research Report No. 120, U. S. Army Cold Regions
Research and Engineering Laboratory, Hanover, N. H., 57 pp.
Bhattacharya, S. N. (1970), "Exact Solutions of SH Wave Equation for
Inhomogeneous Media," Bulletin of the Seismological Society of
America, Vol. 60, No. 6, Dec., pp. 1847-1859*
Blaschke, T. 0. (1964), "Underground Command Center," Civil Engineering,
Vol. 34, No. 5, May, pp. 36-39*
Borowicka, H. (1943), "Die Druckausbreitung im Halbraum bei linear
zunehmenden Elastizitatsmodul," Ingenieur-Archiv, Vol. l4, No. 2,
p. 75-
Brown, J. W. (1965), "Ray Theory for Energy Transmission in Snow,"
Bulletin of the Seismological Society of America, Vol. 55, No. 6,
Dec., pp. 1039-1047.
265


268
Maxwell, A. A., and Fry, Z. B. (1967), "A Procedure for Determining
Elastic Moduli of In Situ Soils by Dynamic Techniques," Proceedings
of the International Symposium on Wave Propagatiqn and Dynamic
Properties of Earth Materials, University of New Mexico Press,
Albuquerque, pp. 913-919
Miller, G. F., and Pursey, H. (195*0, "The Field and Radiation Imped
ance of Mechanical Radiators on the Surface of a Semi-Infinite
Isotropic Solid," Proceedings of the Royal Society of London,
Series A, Vol. 223, pp. 521-541.
Novak, M. (1970), "Prediction of Footing Vibrations," Journal of the
Soil Mechanics and Foundations Division, Proceedings of the Ameri-
can Society of Civil Engineers, Vol. 96, No. SM3, May, pp. 837-861.
Oberg, E., and Jones, F. D. (1949), Machinery's Handbook, l4th ed.,
Industrial Press, New York, pp. 519-533
Odqvist, F. K. G. (1968), "Nonlinear Mechanics, Past, Present and
Future," Applied Mechanics Reviews, Transactions of the American
Society of Mechanical Engineers, Vol. 21, No. 12, Dec., pp. 1213-
1222.
Pauw, A. (1953), "A Dynamic Analogy for Foundation-Soil Systems," Sym
posium on Dynamic Testing of Soils, Special Technical Publication
No. 156, American Society for Testing Materials, pp. 90-112.
Peck, R. B. (1967), "Stability of Natural Slopes," Journal of the Soil
Mechanics and Foundations Division, Proceedings of the American
Society of Civil Engineers, Vol. 93, No SM4, July, pp. 403-417
Poplin, J. K. (1969)5 "Soils Engineering in the Design and Performance
of Artillery Foundations," Miscellaneous Paper S-69-7, U. S. Army
Engineer Waterways Experiment Station, CE, Vicksburg, Miss.
Ramspeck, A. (1936), "Die Interferenz elastischer Wellens im Unter-
grund," Deutschen Forschungsgesellschaft fr Bodenmechanik
(DEGEBO), Heft 4, Springer, Berlin, pp. 17-29
Reissner, E. (1937), "Freie und erzwungene Torsionschwingungen des
elastischen Halbraumes," Ingenieur-Archiv, Vol. 8, No. 4,
pp. 229-245
Reissner, E., and Sagoci, H. F. (1944), "Forced Torsional Oscillations
of an Elastic Half-Space," Journal of Applied Physics, Vol. 15,
Sept., pp. 652-662.
Richart, F. E., Jr. (1953), Discussion of "Vibrations in Semi-Infinite
Solids Due to Periodic Surface Loading," by T. Y. Sung, Symposium
on Dynamic Testing of Soils, Special Technical Publication No.
156, American Society for Testing Materials, pp. 64-68.


50
about 2.5, and similar grain-size-distribution curves; Figure 3 shows
the grain-size distribution for these six samples. The uniformity of
the material sampled to a depth of 50 ft suggests that this zone of
sand might have been deposited by just one of the terrace formations
previously mentioned.
Seismic Wave Propagation Tests
Wave propagation tests, as described by Maxwell and Fry (1967)?
were conducted to assess the shear wave propagation velocity of the in
situ sand deposit at the test site. The method employs a variable
frequency vibrator to generate Rayleigh waves along the surface of the
ground. An interpretation of the measured length of the propagating
Rayleigh wave with respect to the excitation frequency provides an ap
proximation to the shear wave velocity at various depths.
Figure 4 is a plot of the results of these tests showing the vari-
tion of in situ shear wave velocity with depth. Figure 4 also shows
the shear -wave velocity, V obtained by applying the empirical equa-
s
tions (Richart, Hall, and Woods, 1970)
V = (170 78.2e) o25
s
(113)
vq = (159 53-5e) a0,25
(H4)
and assuming a constant void ratio, e of O.67 and an earth pressure
coefficient, Kq of 1/2 (Terzaghi, 1943).


Table 42 (Continued)
253
b
a
Real
Imaginary
Absolute Value
12.060
18.00
4.521
-
03
-I.603
-
02
1.665
-
02
18.60
-3.652
-
03
-1.599
-
02
l.64i
-
02
19.20
-1.076
-
02
-I.205
-
02
1.616
-
02
19.80
-1.503
-
02
-5.202
-
03
1.591
-
02
20.4o
-1.54o
-
02
2.803
-
03
1.566
-
02
21.00
-1.181
-
02
9.894
-
03
1.541
-
02
21.60
-5.239
-
03
1.423
-
02
1.516
-
02
22.20
2.550
-
03
1.470
-
02
1.492
-
02
22.80
9-453
-
03
1.124
-
02
1.468
-
02
23.40
1.362
-
02
4.825
-
03
1.445
-
02
24.00
1.395
-
02
-2.730
-
03
1.422
02
24.60
1.043
-
02
-9.326
-
03
1.399
-
02
25.20
4.094
-
03
-1.314
-
02
1.377
-
02
25.80
-3.215
-
03
-1.316
-
02
1.355
-
02
26.40
-9.410
-
03
-9.447
-
03
1.333
-
02
27.00
-1.274
-
02
-3.137
-
03
1.312
-
02
27.60
-1.232
-
02
3.902
-
03
1.292
-
02
28.20
-8.330
-
03
9.613
-
03
1.272
-
02
28.80
-2.022
-
03
1.236
-
02
1.252
-
02
29.40
4.706
-
03
i.i4o
-
02
1.233
-
02
30.00
9.858
-
03
7.098
-
03
1.215
-
02
30.60
1.194
-
02
8.028
-
o4
1.197
-
02
31.20
1.039
-
02
-5.560
-
03
1.179
-
02
31.80
5.769
-
03
-1.008
-
02
1.161
-
02
32.40
-4.729
-
o4
-1.144
-
02
1.144
-
02
33.00
-6.4o4
_
03
-9.286
-
03
1.128
_
02
33.60
-1.023
-
02
-4.365
-
03
1.112
-
02
34.20
-1.082
-
02
1.762
-
03
1.096
-
02
34.80
-8.072
-
03
7.186
-
03
1.081
-
02
35.40
-2.909
-
03
1.025
-
02
1.066
-
02
36.00
3.021
_
03
1.007
-
02
1.051
-
02
36.60
7.881
-
03
6.760
-
03
1.037
-
02
37.20
1.013
-
02
1.431
-
03
1.023
-
02
37.80
9.171
-
03
-4.211
-
03
1.009
-
02
38.40
5.363
-
03
-8.391
-
03
9-959
-
03
39.00
-3.692
-
05
-9.829
-
03
9.829
-
03
39.60
-5.292
-
03
-8.131
-
03
9.702
-
03
40.20
-8.746
-
03
-3-904
-
03
9.578
-
03
4o.8o
-9.343
-
03
1.459
-
03
9.456
-
03
4l.4o
-6.956
-
03
6.229
-
03
9.338
-
03
42.00
-2.410
-
03
8.901
-
03
9.222
-
03
42.6o
2.799
-
03
8.668
-
03
9.109
-
03
43.20
6.990
-
03
5.666
-
03
8.998
-
03
(Continued)


Page
Particle Motion Predictions 200
CONCLUSIONS AND RECOMMENDATIONS 201
Conclusions 201
Homogeneous (Constant E) Half-Space 201
Nonhomogeneous (Linear E) Half-Space 201
Experimental Aspects 202
Test site 202
Test footing and vibrator 202
Particle motion measuring system 203
Results of measurements 203
Computations and Measurements 203
Recommendations 205
Analytical Work 205
Experimental Work 205
Comparisons 206
APPENDIX A CALCULATIONS FOR THE INTEGRAL l(aQ,a,b) 207
APPENDIX B SPECIFICATIONS FOR THE PARTICLE MOTION MEASURING AND
RECORDING SYSTEM 256
Transducers 256
Three-Component Transducers 256
Single-Component Transducers 257
Cables 257
Amplifiers r 258
Galvanometers 259
Oscillographs and Paper 259
Paper Processor 260
Reference (Calibration) Voltage Supply 260
Voltmeter 26l
Connections 262
Resistance of Transducer Circuits 263
LIST OF REFERENCES 265
vii


165
Table 23
Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,8l8 ft-lb Oscillating at 50
Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
0.0168
0.0268
O.87O
2.344
Footing
None
None
0.456
2.5
1.5
None
None
0.393
3-5
1.0
0.00793
0.0156
0.324
10.0
1.0
0.0120
O.OI69
0.0948
30.0
1.0
0.00554
0.0210
0.0395
60.0
1.0

0.00351
0.0101
90.0
1.0
0.00107
O.OO354
0.00584
90.0
5.0
O.OOO87O
O.OOO333
0.00402
10.0
5-0

0.00462
0.0291
30.0
5.0
--
0.00379
0.0162
60.0
5.0
0.00193
0.00123
0.00822
60.0
15.0
0.000233
0.000420
0.00180
90.0
15.0
0.000467
0.000527
0.000801
30.0
15.0
0.000426
--
0.00388
60.0
25.0
0.000175
0.000181
0.000745
60.0
35.0
0.000304

0.000991
90.0
25.0
0.000716
0.000186
0.000453
90.0
35.0
0.000198
0.000209
0.000451
10.0
15.0
0.00154

0.00710
10.0
25.0
0.000630

0.000594
10.0
35.0
0.000766

0.000686
30.0
25.0

0.000221
0.000583
30.0
35.0

0.000422
0.00159


DEPTH BELOW GROUND SURFACE IN FT
52
SHEAR WAVE VELOCITY IN FPS
Figure 4. Shear wave velocity versus depth, surface and
empirical methods.


Figure 51. Tilt check with level along vibrator frame.
150


Figure 32. Drill rig used to auger uncased boreholes for the transducers
115


75
Figures 8 and 9 are plan and elevation sketches of the vibrator
components; the layout of these components was dictated by the config
uration of the test footing. The eccentric masses were separated as
far as practical to develop large twisting moments with small centri
fugal forces, the horizontal plane containing the rotating masses
was kept as low as practical to reduce rocking of the footing by un
balanced forces, and the structural frame was extremely rigid to raise
sympathetic vibration frequencies well above the torsional frequencies
applied to the foundation.
Design details such as timing belt layouts, sprocket sizes, bear
ing loads, and shaft bending and whirling are not mentioned in the fol
lowing paragraphs, but a discussion of the input power requirements
and torque capacity of the vibrator was considered pertinent to the
design and conduct of the experimental work.
Power requirements
A gross approximation of the power expended on the footing-soil
contact area by a torsionally vibrating footing was made by assuming
a simple relationship between the twisting moment, M applied to the
footing and the rotation, f> of the footing. To find an upper bound
for the power losses through the foundation, the M versus f> rela
tionship was assumed rigid-plastic. The work done by the footing on
the soil per cycle was 4M0 the work done per second at a frequency
of 50 Hz was 2OOM0 and the horsepower expended was 2OOM0/550 .
Using the design moment of 1,400 ft-lb and the design rotation of
0.00004 radians, the power loss, p at a frequency of 50 Hz was


VO
U)
Figure 16. Placing footing form and retaining ring in excavation


164
Table 22
Particle Velocity Amplitudes Generated, by a Torsional
Moment of 2,6l4 ft-lb Oscillating at 40 Hz
Peak Amplitude in Inches
per Second
Transducer
Location
Vertical
Radial
Transverse
Radial, ft
Depth, ft
Component
Component
Component
2.646
Footing
O.OI98
0.0249
O.895
2.344
Footing
None
None
O.588
2.5
1.5
None
None
0.457
3.5
1.0
0.0119
0.0286
0.3509
10.0
1.0
0.0111
0.0109
O.O986
30.0
1.0
0.00594
0.0217
0.0355
60.0
1.0
0.00330
0.00497
0.0259
90.0
1.0
0.00168
0.0186
0.00323
90.0
5.0
0.00168
0.00160
0.00712
10.0
5.0
0.00952
0.00860
0.0380
30.0
5.0
0.00557
0.00328
0.0231
60.0
5.0
0.00197
0.00173
0.00708
60.0
15.0
0.00102
0.000524
0.00078
90.0
15.0
0.000762
0.000323
O.OOI98
30.0
15.0
0.000732
0.000828
0.00173
60.0
25.0
0.000337
0.000236
0.000266
60.0
35.0
O.OOO386
0.000272
O.OOO517
90.0
25.0
0.000172
0.0000968
0.000391
90.0
35.0
0.000319
0.000213
0.000236
10.0
15.0
0.000959
0.0022$
0.00405
10.0
25.0
O.OOO563
O.OOO76O
0.000787
10.0
35.0
0.000517
o.ooo484
0.000375
30.0
25.0
O.OOO3O3
0.000201
0.000785
30.0
35.0
O.OOO356
0.000443
0.000673


o
5
10
15
20
25
30
35
6.
157
COMPRESSION WAVE VELOCITY IN FPS
500 1,000 1,500 2,000 2,500
\
\
s
\
\
\\
\

\ \
\
\
V
o HAMMER BLOW ON FOOTING
Q HAMMER BLOWS ON GROUND
SURFACE (AVERAGE VELOCITY)
S
\
\
\
\
\\
v*,
Compression wave velocities from hammer blows on footing and
on ground surface.


I certify that I have read this study and that in my opinion it con
forms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of
Doctor of Philosophy.
Frank E.
Professor of Civil Engineering,
University of Michigan
Formerly on Graduate Staff of the
University of Flotrida
This dissertation was submitted to the Dean of the Cpllege of Engineer
ing and to the Graduate Council, and was accepted as partial fulfill
ment of the requirements for the degree of Doctor of Philosophy.
August, 1971
Dean, Graduate School


4i
ground surface and the area was lightly wooded with indigenous scrub
oak and pine.
Sand deposits in this vicinity are of geologically recent origin
(Cooke, 1945j Vernon and Puri, 1965)- The Citronelle formation is
dated somewhere between the Pliocene and the Pleistocene epochs and is
no more than ten million years old; the terrace and fluvial terrace
formations laid down during the Pleistocene epoch are iless than one
million years old. Stratigraphically, the Citronelle formation lies
unconformably on older formations, and is overlain by Pleistocene
terrace deposits.
The Pleistocene epoch was characterized by many changes in sea
level due to a sequence of glacial accummulation and subsequent melting.
Sea levels during that time were as much as 270 ft above current levels.
The water from melting glaciers carried a variety of soil material to
the sea where currents and wave action developed the sandy terrace de
posits Erosion during low sea levels and redeposition during high sea
levels created a generally flat topography with hidden stratigraphic
features. The three specific marine terraces that were associated with
deposits at the test site are the Brandywine formation, the Cohaire for
mation, and the Sunderland formation.
Table 7 is a well log taken at auxiliary field 5 by the Layne Cen
tral. Co. and provided by the Directorate of Civil Engineering, Eglin Air
Force Base, Florida; it illustrates the general stratigraphic sibilation
near the test site.
Soil Exploration
The in situ soil exploration program at the test site was


125
discussed in this section; detailed specifications and electrical con
nections are given in Appendix B.
Transducers
All of the particle velocity transducers used in the experimental
work were obtained from Mark Products, Inc., Houston, Texas. One trans
ducer component consisted of a spring attached to a free mass at one
end and to the transducer case at the other end; a coil of wire was
fastened to the mass and a permanent magnet was attached to the case.
When the case was moved, an induced voltage was generated in the coil
of wire that was proportional to the relative velocity between the coil
and the magnetic field developed by the magnet. A 3-'Component trans
ducer sensed particle velocity in the vertical (v), radial (R), and
tangential (T) directions and was composed of 3 sensing units in a
single housing.
Twenty-two transducers of the 3-component type and 2 transducers
of the single component type were used in the experimental work. The
3-component transducers were model L-1B-3DS and the single component
transducers were model L-1D. Figure 30 shows these two types of trans
ducers mounted on the test foundation.
Identification.--A serial number was stamped on the case of each
3-component transducer; the orientation of the radial and tangential
components of motion was indicated by an arrow engraved on the top of
the transducer case. The two single component transducers were not
identified by number, but were referred to as L-1D-BT (for the buried
transducer) and L-1D-TT (located on top of the test footing).
Brief specifications.--The 3-component transducers had a diameter


b
1.44o
4.34o
Table 40 (Continued)
a
Real
Imaginary
Absolute
i Value
16.20
-5.804
_
03
-8.231
_
03
1.007
- 02
16.56
-8.138
-
03
-5.558
-
03
9.855
- 03
16.92
-9.366
-
03
-2.3IO
-
03
9-647
- 03
17.28
-9.385
-
03
1.092
-
03
9.448
- 03
17.64
-8.238
-
03
4.221
-
03
9.257
- 03
18.00
-6.115
-
03
6.703
-
03
9.073
- 03
18.36
-3.314
-
03
8.257
-
03
8.897
- 03
18.72
-2.O73
-
o4
8.725
-
03
8.727
- 03
19.08
2.809
-
03
8.090
-
03
8.564
- 03
19.44
5.366
-
03
6.471
-
03
8.4o6
- 03
19.80
7.162
-
03
4.104
-
03
8.254
- 03
20.16
8.002
-
03
1.308
-
03
8.108
- 03
20.52
7.814
-
03
-1.553
-
03
7.967
- 03
20.88
6.657
-
03
-4.123
-
03
7.830
- 03
21.24
4.707
-
03
-6.092
-
03
7.698
- 03
21.60
2.233
-
03
-7.234
-
03
7.571
- 03
21.96
-4.398
-
o4
-7.434
-
03
7.447
- 03
22.32
-2.972
-
03
-6.698
-
03
7.328
- 03
22.68
-5.053
-
03
-5.146
-
03
7.212
- 03
23.04
-6.436
-
03
-2.998
-
03
7.100
- 03
23.40
-6.971
_
03
-5.396
_
o4
6.991
- 03
23.76
-6.615
-
03
1.913
-
03
6.886
- 03
24.12
-5.440
-
03
4.052
-
03
6.784
- 03
24.48
-3.617
-
03
5.621
-
03
6.684
- 03
24.84
-1.392
-
03
6.439
-
03
6.588
- 03
25.20
9.443
_
o4
6.425
-
03
6.494
- 03
25.56
3.097
-
03
5.604
-
03
6.403
- 03
25.92
4.801
-
03
4.102
-
03
6.315
- 03
0.00
0.000
+
00
0.000
+
00
0.000
+ 00
0.36
-3.279
-
03
-2.867
-
o4
3.291
- 03
0.72
-6.439
-
03
-2.996
-
04
6.446
- 03
1.08
-9-346
-
03
1.935
-
o4
9-348
- 03
1.44
-1.183
-
02
1.347
-
03
1.191
- 02
1.80
-1.370
_
02
3.212
_
03
1.4o8
- 02
2.16
-1.477
-
02
5.722
-
03
1.584
- 02
2.52
-1.484
-
02
8.699
-
03
1.720
- 02
2.88
-1.381
-
02
1.187
-
02
1.821
- 02
3.24
-1.165
-
02
1.489
-
02
1.890
- 02
3.60
-8.432
-
03
1.739
-
02
1.933
- 02
3.96
-4.352
-
03
1.905
-
02
1.954
- 02
4.32
2.887
-
o4
1.957
-
02
1.958
- 02
4.68
5.106
-
03
1.879
-
02
1.948
- 02
5.04
9.673
-
03
1.667
-
02
1.927
- 02
(Continued)


g = sec a
23
(80)
cLg = sec & tan a cLq1
(81)
and
1 = tan o/
(82)
where 1 ^ g ^ < and 0 <. o¡ <. w/2 Using the new variable a instead
of g the integral becomes
sin (aQ sec a) (aQ sec a) cos (aQ sec a)
0
circular functions, J an integer order Bessel function of the first
kind, and e the base of natural logarithms. The integrands are con
tinuous functions in the interval of integration, and, of particular
note, all the terms of the integrand can be expressed as a series or as
polynomial approximations. Such formulations make the integrand well
suited for evaluation with a digital computer.
Integration of 1^,1^, and I could be accomplished in a
variety of ways, but perhaps the most obvious is by numerical methods.
One numerical integration scheme is based on Simpsons rule for deter
mining the area of an irregular figure. The integral expression of
Simpson's rule, given by Abramowitz and Stegun (1964), is
o
+ *h*h +---W + An <84>


Page
20. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,502 ft-lb Oscillating at 20 Hz 162
21. Particle Velocity Amplitudes Generated by a Torsional
Moment of 1,470 ft-lb Oscillating at 30 Hz 163
22. Particle Velocity Amplitudes Generated by a Torsional
Moment of 2,6l4 ft-lb Oscillating at 40 Hz l64
23. Particle Velocity Amplitudes Generated by a Torsipnal
Moment of 2,8l8 ft-lb Oscillating at 50 Hz 165
24. Measured Arrival Time and Average Wave Propagation Velocity
for Vibration Tests l68
25. Computed and Published Values of the Displacement Function-- 171
26. Measured Motion of the Test Footing 174
27. Component Displacement Ratios Averaged over 5 Frequencies 179
28. Normalized Half-Space Particle Displacements 186
29- Influence of Frequency on Half-Space Displacements-- 187
30. Normalized Soil Particle Displacements 188
31* Influence of Frequency on Soil Displacements 189
32. Ratio of Half-Space to Soil Displacement 190
33* Average Displacement Ratios for 5 Frequencies 191
34. Normalized Nonhomogeneous Half-Space Displacements 193
35- Ratio of Nonhomogeneous Half-Space Displacements to Soil
Displacements 194
36. Average Nonhomogeneous Half-Space Displacement Ratios for
5 Frequencies 195
37* Subroutines and Computer Program for the Integral
l(a ,a,b) 207
38. Value of 1(0.36,a,b) 211
39. Value of 1(0.48,a,b) 220
40. Value of 1(0.72,a,b) 229


APPENDIX B
SPECIFICATIONS FOR THE PARTICLE MOTION
MEASURING AND RECORDING SYSTEM
Transducers
Three-Component Transducers
The 3-component particle velocity transducers were manufactured by
Mark Products, Inc., Houston, Texas; all transducers were model L-1B-3DS.
The 3-in.-diam, 12-in.-long aluminum case housed three single component
transducers oriented in the vertical, radial, and transverse directions.
One unit weighed 6 lb, had 150-ft-long support and conductor cables,
and had a waterproof pressure rating of 1,000 psi. The shielded con
ductor cable contained six number 22 gage stranded wires and a ground
wire.
The maximum case to coil excursion was l/l6 in., the undamped
natural frequency was 4.5 Hz, and the useful frequency range at 65 per
cent of critical damping was 10 to several hundred Hz. The transduc
tion tolerance was plus or minus 5 percent, the frequency change with
tilt was less than 0.25 Hz at 15 degrees of tilt, and the frequency
change was less than 0.25 Hz at maximum coil excursion.
Transducers numbered 1 through 21 had a coil resistance of 87O
ohms and a damping shunt resistance of about 6,000 ohms. The nominal
transduction of these units was 1.7 volts/in./sec.
Transducers numbered 22 through 27 had a coil resistance of 1,480
ohms and a damping shunt resistance of 12,000 ohms. The nominal
256


71
Figure 7. These support and loading assumptions were believed realis
tic for computing the approximate deformation of the contact area; com
puting deflections for a flexible, finite plate on an elastic founda
tion were considered unnecessarily complicated and tedious. With
r^ = l6 in., the dead equivalent load, q that acted on only part of
the circular area, was 1.40 q where q is the uniform dead, load
pressure previously assumed to act over the entire area of the footing.
Superposition of three loading situations given by Timoshenko and
Woinowsky-Krieger (1959) was accomplished to calculate the maximum
deflection of the center of the first pour with respect to its edge due
to the distribution of the dead load. A similar displacement of the
contact area was assumed, and, to slide rule accuracy, this displace
ment, A was
c2
Aq2 = 0.0000074q (133)
= 0.0000103q
The ratio of the deformation of the contact area for a completely
flexible footing to the deformation of the contact area under the
cured first pour supporting a selectively located equivalent dead load
was used to judge the degree of rigidity of the footing with respect to
the foundation soil. For the above situation, this ratio, S was
0.002751 (134)
0.0000103q
= 266
Thus, since the deformation of the contact area was reduced to less
than 0.5 percent of its free deformation by the effective rigidity of


113
radial component of transducer 19 (Serial Number 19) did not work prop
erly, and the radial component of transducer 11 produced about half the
signal amplitude of the other transducers. Since replacement trans
ducers were not available and the transverse component of motion was of
primary interest, these two transducers were used to detect transverse
and vertical (v) components of particle motion.
Boreholes
Figures 10 and 11 give the location and depth of the field of
transducers installed at the test site. These locations were surveyed,
center hubs were driven, and marking stakes were placed to indicate the
position and depth of each borehole. A view of the test site topog
raphy, the vegetation, and some of the marking stakes is shown in
Figure 31* Six-in.-diam vertical holes were augered with the drill
rig pictured in Figure 32; the holes were uncased.
Transducer alignment
The orientation and depth of the buried three component particle
velocity transducers were controlled by an indexed, square, borehole
rod. One end of the rod was telescoped into an aligning sleeve at
tached to the transducer case and the other end of the rod was attached
to a sighting bar. The elevation and plumb of the rod were measured
with a transit and carpenter's level. The method of attaching the
square alignment sleeve to the transducers using a wooden jig and an
epoxy compound is illustrated in Figure 33- A group of transducers
prepared for installation in a borehole is shown in Figure 3^; each
transducer shown has one flexible support cable and one electrical
cable attached to it.


Figure 20
Position of auxiliary form and backfilling operation
M3
co


194
Table 35
Ratio of Norihomogeneous Half-Space Displacements
to Soil Displacements
Depth Below
Normalized Nonhomogeneous Half-Space Displacements/
Normalized Soil Displacements
Radial Distance, fit
Surface, ft
¡O
r1
ir\
CO
30
60
90
Vibration Frequency
, 15 Hz
1
1.2 0.63
0.34
0.24
0.25
5
0.45
0.37
0.31
0.22
15
0.68
1.4
1.1
0.76
25
0.85
1.7
2.4
1.9
35
0.97
2.2
3.1
2.0
Vibration Frequency
, 20 Hz
1
1.0 0.65
0.33
0.32
0.35
5
0.59
0.54
0.48
0.37
15
1.0
3.4
2.2
1.3
25
1.3
4.3
3.1
2.5
35
1.0
2.6
2.6
2.8
Vibration Frequency
, 30 Hz
1
O.98 0.78
0.32
0.37
0.45
5
0.6l
0.64
O.58
0.35
15
1.2
1.7
1.8
0.92
25
1.8
3.2
4.1
1.7
35
1.3
3.8
3.3
1.9
Vibration Frequency
, 40 Hz
1
1.0 0.71
0.54
0.43
1.86
5
0.97
0.63
1.0
0.69
15
1.8
3.4
2.9
1.3
25
2.4
4.6
5.8
4.7
35
2.0
3.3
3.5
4.2
Vibration Frequency
, 50 Hz
1
0.88 O.63
0.45
0.95
0.96
5
1.2
0.80
0.80
l.l
15
1.1
1.6
1.9
2.3
25
3.1
5.8
3.2
2.1
35
1.4
1.6
2.1
3.1


269
Richart, F. E., Jr., Hall, J. R., Jr., and Woods, R. D. (1970), Vibra
tions of Soils and Foundations, Prentice-Hall, Englewood Cliffs,
N. J., 360 pp.
Richart, F. E., Jr., and Whitman, R. V. (1967), "Comparison of Footing
Vibration Tests with Theory," Journal of the Soil Mechanics and
Foundations Division, Proceedings of the American Society of Civil
Engineers, Vol. 93, No. SM6, Nov., pp. 143-168.
Schmertmann, J. H. (1967)5 "Static Cone Penetrometers for Soil Explora
tion," Civil Engineering, Vol. 37, No. 6, June, pp. 71-73.
Schmertmann, J. H. (1969)5 "Dutch Friction Cone Penetrometer Explora
tion of Research Area at Field 5, Eglin AFB, Florida," Contract
Report S-69-4, U. S. Army Engineer Waterways Experiment Station,
CE, Vicksburg, Miss.
Sokolnikoff, I. S. (1956), Mathematical Theory of Elasticity, 2nd ed.,
McGraw-Hill Book Co., New York, 476 pp.
Stallybrass, M. P. (1962), "A Variational Approach to a Class of Mixed
Boundary-Value Problems in the Forced Oscillations of an Elastic
Medium," Proceedings of the 4th U. S. National Congress of Applied
Mechanics7 American Society of Mechanical Engineers, pp. 391-400.
Stallybrass, M. P. (1967)5 "On the Reissner-Sagoci Problem at High
Frequencies," International Journal of Engineering Science, Vol. 5,
No. 9, Sept., pp. 689-703.
Sternberg, E. (i960), "On Some Recent Developments in the Linear Theory
of Elasticity," Proceedings of the 1st Symposium on Naval Struc
tural Mechanics, Pergamon Press, London, pp. 48-72.
Sung, T. Y. (1953)5 "Vibrations in Semi-Infinite Solids Due to Periodic
Surface Loadings," Symposium on Dynamic Testing of Soils, Special
Technical Publication No. 156, American Society for Testing Mate
rials, pp. 35-64
Terzaghi, K. (1943), Theoretical Soil Mechanics, John Wiley and Sons,
Inc., New York, pp. 22-45.
Thomas, D. P. (1968), "Torsional Oscillations of an Elastic Half-Space,"
Quarterly Journal of Mechanics and Applied Mathematics, Vol. 21,
No. 1, Feb., pp. 51-65.
Timmerman, D. H., and Wu, T. H. (1969), "Behavior of Dry Sands Under
Cyclic Loading," Journal of the Soil Mechanics and Foundations
Division, Proceedings of the American Society of Civil Engineers,
Vol. 95, No. SM4, July, pp. 1097-1112.


Table 27
Component Displacement Ratios Averaged over 5 Frequencies
179
Depth Below On Radial Distance from Center of Footing, ft
Ground.
, ft
Footing 3.5
10
30
60
90
Ratio
of Transverse Displacement
to Resultant Displacement
0
0.999
1
0.997
O.986
0.928
O.969
0.899
5
O.981
0.967
0.959
0.954
15
0.931
0.843
0.802
0.869
25
0.722
0.927
0.724
0.743
35
0.672
0.870
0.794
0.774
Ratio
of Vertical Displacement
to Resultant Displacement
0
0.0250
l
0.0351
0.122
0.121
0.l4i
0.222
5
0.107
0.l4l
0.204
0.136
15
0.l60
0.255
0.370
0.34o
25
0.470
0.227
0.333
0.490
35
0.522
0.257
0.398
0.452
Ratio of Radial Displacement to Resultant Displacement
0
0.0385
l
O.O636
0.101
0.297
0.161
0.309
5
0.130
0.166
0.181
0.233
15
0.264
0.246
0.280
0.286
25
0.319
0.242
o.46i
0.290
35
0.303
0.305
0.268
0.309
Properties of the Sand Deposit
A better understanding of the properties of the sand deposit was
obtained from the test measurements. Figure 56 shows the measured com
pression wave propagation velocity versus depth for the footing source
and surface source tests, and Figure 57 shows the measured shear wave
propagation velocity versus depth for steady state vibration tests.
The shear wave velocity near the footing increases wiith depth to 20 ft,
decreases from 20 to 25 ft, then increases from 25 ft to greater depths.


13
3u 3w
3z ar
)
g a_ /1 aw 3v
r ae \r 30 3z
(
(18)
Solutions to the equilibrium equations are sought which satisfy the
boundary conditions described in the problem statement; i.e.
v(r 0 0 t) = 0 re^U)^
(r <; rQ) (19)
(r > rQ) (20)
az0(r 0 0 t) = 0
where is the angular rotation of the rigid disk. For completeness,
it should be noted at this point that the compatibility conditions are
not involved since the equilibrium equations are expressed in terms of
particle displacements (Sternberg, i960). This mixed or third boundary
value problem can be reduced to a first boundary value problem by eval
uating the static shear stresses produced on the surface of the half
space by the rigid circular disk and assuming that these same stresses
occur on the half-space as the disk undergoes forced torsional oscilla
tions. A similar approach and assumption is common in the literature
and has been used by Miller and Pursey (195*0, Reissner (1937), Sung
(1953), and Hsieh (1962), as well as by Bycroft (1956). For the case
of a rigid circular disk in forced torsional oscillation on an elastic
half-space, these same authors agree that v is the only component of
displacement that occurs. Consequently, the axisymmetric problem is
greatly simplified and the strain equations reduce to
(21)


5
10
15
20
25
30
35
199
AVERAGE CONE BEARING CAPACITY IN KG/CM2
25 50 75 100 125
0.6 1.2 1.8 2.4 3.0
AVERAGE DISPLACEMENT RATIO, N(c)/N(m)
Figure 62. Cone bearing capacity and displacement ratio
versus depth.


COMPARISON OF COMPUTED AND EXPERIMENTAL RESULTS
Test of the Calculated Results
Computed particle displacements, generated by the: torsional oscil
lation of a rigid circular disk on the surface of an elastic half-space
were discussed in a previous section. The validity of| the computed
results was tested by comparing them with some accepted solutions and
known laws governing wave propagation. The motion of the surface of
the half-space under the rigid disk was compared to published work,
and the geometrical damping law for body waves was used to test the
calculated motion within the half-space.
Solutions at the Surface of a
Homogeneous (Constant E) Elastic Half-Space
As mentioned in a previous section, solutions for the torsional
oscillation of a weightless rigid disk on the surface if a homogeneous
elastic half-space are not lacking (Reissner and Sagocp, 1944; Bycroft,
1956; Stallybrass, 1962, 1967; Thomas, 1968). Since Stallybrass (1962)
tabulated the results of his work and that of Reissner and Sagoci
(1944), a comparison with these results was convenient]
The expression for the particle displacement developed from
Equation 65 was
^l(real) + il(imaginary)J
(147)


30
site is about 650 fps at a depth of 15 ft, the unit weight of the soil
is about 104 pcf, and the diameter of the footing (disk) is 5 ft.
Table 6 gives the calculation parameters used to compute the values of
l(a ,a,b) contained in Appendix A.
Table 6
Calculation Parameters for l(aQ,a,b)
Frequency
Hz
a
0
r
ft
a
z
ft
b
15
0.36
0-12.5
0-1.80
0
0.00
0-90
0-12.96
1
o.i44
0-90
0-12.96
5
0.72
0-90
0-12.96
15
2.20
0-90
0-12.96
25
3.62
0-90
0-12.96
35
5.06
20
0.48
0-12.5
0-2.40
0
0.00
0-90
0-17.28
1
0.192
0-90
0-17.28
5
0.96
0-90
0-17.28
15
2.90
0-90
0-17.28
25
4.84
0-90
0-17.28
35
6.76
30
0.72
0-12.5
0-3.60
0
0.00
0-90
0-25.92
1
0.288
0-90
O-25.92
5
1.44
0-90
0-25.92
15
4.34
0-90
0-25.92
25
7.24
0-90
0-25.92
35
io.i4
40
O.96
0-12.5
0-4.80
0
0.00
0-90
0-34.56
1
0.384
0-90
0-34.56
5
1.92
0-90
0-34.56
15
5.80
0-90
0-34.56
25
9.64
0-90
0-34.56
35
13.50
50
1.20
0-8.75
0-4.20
0
0.00
0-90
0-43.20
l
0.480
0-90
0-43.20
5
2.40
0-90
0-43.20
15
7.24
0-90
0-43.20
25
12.06
0-90
0-43.20
35
16.90


Table 38 (Continued)
214
b
0.720
2.200
a
Real
Imaginary
Absolute Value
7.56
5-593
_
03
7.338
_
o4
5.641
- 03
7.74
5.504
-
03
-2.523
-
o4
5.510
- 03
7.92
5.252
-
03
-I.I87
-
03
5.385
- 03
8.10
4.852
-
03
-2.045
-
03
5.265
- 03
8.28
4.321
-
03
-2.803
-
03
5.151
- 03
8.46
3.680
-
03
-3.445
-
03
5.041
- 03
8.64
2.953
-
03
-3.955
-
03
4.936
- 03
8.82
2.165
-
03
-4.323
-
03
4.835
- 03
9.00
1-342
-
03
-4.545
_
03
4.739
- 03
9.18
5.089
-
04
-4.618
-
03
4.646
- 03
9.36
-3.O8I
-
o4
-4.546
-
03
4.556
- 03
9.54
-I.O85
-
03
-4.337
-
03
4.470
- 03
9-72
-I.8OO
-
0.3
-4.001
-
03
4.388
- 03
9.90
-2.434
-
03
-3.554
-
03
4.308
- 03
10.08
-2.97O
-
03
-3.013
-
03
4.231
- 03
10.26
-3.396
-
03
-2.397
-
03
4.157
- 03
10.44
-3.702
-
03
-1.728
-
03
4.085
- 03
10.62
-3.882
-
03
-1.027
-
03
4.016
- 03
10.80
-3.936
-
03
-3.181
04
3.949
- 03
10.98
-3866
-
03
3.781
-
04
3.884
- 03
11.16
-3.677
-
03
i.o4o
-
03
3.821
- 03
11.34
-3.380
-
03
1.649
-
03
3.761
- 03
11.52
-2.986
-
03
2.188
-
03
3-702
- 03
11.70
-2.512
-
03
2.64i
-
03
3.645
- 03
11.88
-1.973
-
03
2.999
-
03
3.590
- 03
12.06
-1.388
-
03
3.252
-
03
3.536
- 03
12.24
-7.769
-
o4
3.396
-
03
3.484
- 03
12.42
-1.590
-
04
3.430
-
03
3.434
- 03
12.60
4.463
-
o4
3.355
-
03
3.385
- 03
12.78
1.021
-
03
3.177