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