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LPB __ ?79 -~5~"'i
1~!L~o~~~is~~~titute of Technology III Department of Civil Engineering Armour College of Engineering
Geotechnical Series 77-201
FOUNDATION RESPONSE TO SOIL
TRANSMITTED LOADS
by
Satya N, Varadhi
Surendra K, Saxena
Eben Vey
December 1977
Prepared for
NATIONAL SCIENCE FOUNDATION
Research Grant GK-41903
REPRODUCED BY
NATIONAL TECHNICAL INFORMATION SERViCe
u. s. DEPARTMENT OF COMMERCE SPRINGFIELD, VA. 22161
FOUNDATION RESPONSE TO SOIL TRANSMITTED LOADS 7. ;\ lit h,' ,)
3 ",.' " , 'I " /.' ': ". I' P ,.}'{<i:~~,IP1.011f l :''','(/L~S''10tr-",d<~
T:.... ~·L ;,.~;.8 i' .!../ ,.\.' j ,'~- ~/.' '~. Eeport Date
Dec 77 6.
8. Performing Org"nizarion Repr. No. Geotech series 77-201 S. N. Varadhi, S. K. Saxena and E. Vev
!9. :\-r!,\;I,'
ATTENTION
AS NOTED IN THE NTIS ANNOUNCEMENT,
PORTIONS OF THIS REPORT ARE NOT LEGIBLE.
HOWEVER, IT IS THE BEST REPRODUCTION
AVAILABLE FROM THE COPY SENT TO NTIS.
Geotechnical Series 77 ~ 201
FOUNDATION RESPONSE TO SOIL
TRANSMITTED LOADS
by
Satya N. Varadhi Surendra K. Saxena
Eben Vey
A Report on Research Sponsored
by
National Science Foundation
Grant No, GK~41903
Illinois Institute of Technology Department of Civil Engineering Armour College of Engineering
December 1977
(i)l~
NATIONAL SCIENCE FOUNDATION Washington, D.C. 20550
SUMMARY OF COMPLETED PROJECT
Please read illstrnctions on reverse carejillly before completing this form.
Form Approved OMB No. 99R0013
1. INSTITUTION AND ADDRESS 2. NSF PROGRAM 3. PRINCIPAL INVESTIGATOR(S)
Illinois Institute of Tech! Civ'~l and Eben Vey Civil Engineering Dept. Environmental Technology Surendra K, Saxena Chi caao III inois 60616
4. AWARD NUMBER 15. DURATION 6. AWARD PERIOD ENG73-03922A02 (MOSl24 from to Dec I ·78 7GK~1~o~ ACCOUNT NUMBER
8. PROJECT TITLE
Foundation Response to Soil Transmitted Loads 9. SUMMARY (ATTACH LIST OF PUBLICATIONS TO FORM)
This research deals with effects on foundation response produced by stress waves propagating from the underlying soils, In this research the circular footings·were located at the surface and at different depths of burial and a transient load was applied to the footings through the underlying soils, Measurements were made of stress-time and strain-time histories using embedded gages throughout the soil mass, These records were correlated with acceleration-time records of the input load. and the foundation response to provide a comprehensive picture of stresses and strains in a soil mass subjected to an earthquake type loading, The data obtained also evaluates to what extent these stresses and strains are influenced by the foundation and the distribution of pressures in the soil beneath it, In the cases studied. it was noticed that the influence of the mass of the footing on the stress distribution is negligible at a depth more than two footing diameters below the footing, It was also noticed that the stress intensities in the vicinity of the footing increases as the distance from the transient load source decreases~ The investigation also includes some experiments, static and dynamic, conducted using a thin layer of compressible material buried at depth of one radius below the footing, Data from these tests has been analyzed to 1) obtain a relationship between the percentage reduction in deceleration of the footing and the subgrade modulus of the compressible material, 2) obtain a relationship between the percentage reduction of the vertical stress in the vicinity of the footing and the subgrade modulus of the compressible material, and 3) obtain a relationship between the percent. age reduction in deceleration of the footing and percentage reduction in the factor of safety against the bearing capacity failure based on allowable settlement,
9. SIGNATURE OF PRINCIPAL INVESTIGATORI PROJECT DIRECTOr
~ It: --&/C'Q.,~ . TYPED OR PRINTED NAME
Surendra K, Saxena NSF Form 98A (10-77) SUPERSEDES ALL PREVIOUS EDITIONS
II
DATE
12/30/77
I
ACKNOWLEDGEMENTS
This report is part of a series devoted to the problem
of foundation response to soil transmitted loads. The
research program was supported by National Science Foundation
under grant GK~41903 awarded to the Illinois Institute
of Technology,
(iii)
ACKNOWLEDGEMENTS
LIST OF TABLES .
LIST OF FIGURES
LIST OF SYMBOLS
CHAPTER I. INTRODUCTION
TABLE OF CONTENTS
I I. REVIEH OF LITERATURE
Footing Behavior Under Dynamic Loads Recent Soil Dynamics Research at lIT The Isolation Studies . . . . . .
Page .iii
v
vi
xv
1
4
4 15 22
III. PURPOSE AND SCOPE OF INVESTIGATION 27
IV. TEST APPARATUS AND EXPERlr~NTAL PROCEDURES. 31
Test Apparatus .... . Embedded Gages .... . Instrumentation . . . . Experimental Procedures
V. PRESENTATION AND ANALYSIS OF RESULTS
VI.
Footing Response. . . . Stress-Wave Propagation Depth of Burial . Footing Isolation
CONCLUSIONS
APPENDICES
A. SOIL PROPERTIES . . . . . .
31 39 51 56
67
67 78
.126
.139
.175
.180
B. WAVE PROPAGATION VELOCITIES AND SOIL MODULI .184
C. STRESS-DEFORMATION RELATIONSHIP FOR COMPRESSIBLE MATERIALS ·188
BIBLIOGRAPHY . . . . . . . . . . . . . . ·194
iv
LIST OF TABLES
Table 1. Sensitivity Values of Piezoresistive Stress
Gages Under Hydrostatic Conditions . . . .
2. Sensitivity Values of Piezoresistive Stress Gages in Confined Specimens of the Soil
v
Page
42
44
Figure I.
2.
3.
4.
5.
6.
7.
8.
9.
10.
II.
12.
13.
14.
15.
LIST OF FIGURES
Page A General View of the Experimental Apparatus 32
Bottom Movable Steel Plate Mounted on Three Hydro-Line Cylinders . . . . . . . . . 33
Various Size Footings Used in This Study 34
Schematic Diagram of Transient Loading System . . . . . . . . . . 35
A View of the Loading System . 37
Apparatus for Statically Loaded Footing 38
Piezoresistive Soil Stress Gage 40
Schematic Diagram of Water Chamber for Gage Calibration . . . . . . . . . . 41
Typical Calibration Curve for Piezoresistive Gage Under Hydrostatic Conditions . . .' 43
Typical Calibration Curve for Piezoresistive Gage Embedded in Sand 45
IITRI Coil Strain Gage . . . . . . . . 47
Calibration of Strain Gage (1 in. dia. coils) (Displacement vs Gage Output) ....... 48
Calibration of Strain Gage (1 in. dia. coils) (Strain vs Gage Output) ...... 49
Calibration of Strain Gage (3/4 in. dia. coils) (Displacement vs Gage Output) 50
Instrumentation Setup 52
16. Schematic Diagram of Electrical Circuit for the Soil-Strain Gage Unit . . . . . . ., 54
17. Data Recording System (Magnetic Tape Recorder Model 5600B, Honeywell) ........... 55
18. Data Printing Equipment (Visicorder Oscillograph Model l508B, Honeywell). . . . . . . . .. 57
19. Raining Device for Plac'ement of 20-40 Ottavla Sand. . . . . ...... . 58
vi
Figure 20.
2l.
22.
23.
24.
25.
26.
27.
28.
29.
LIST OF FIGURES (Cont'd.)
Page Location of Stress and Strain Measurements 60
A Typical Gage Array of the Embedded Gages 61
Schematic Diagram of Placement Methods for Strain Gages . . . . . . . . . . . . . . 62
Footing Before Being Subjected to Transient Load. . . . . . . . . .. .... 64
Schematic Diagram of Footing Apparatus . .. 69
Acceleration-Time Traces for the Bottom Movable Steel Plate and a Footing (6 in. dia. circular). . . . . . . . . . . . . . . .. 70
Velocity vs Time Relationship for a Surface Footing (6 in. dia. circular). . . . . 73
Displacement vs Time Relationship for a Surface Footing (6 in. dia. circular). 74
Acceleration, Displacement-Time Traces for a Footing (6 in. dia. circular). . . . . . 75
Acceleration-Time Traces for the Bottom Movable Steel Plate and a Footing (3 in. dia. circular). . . . . . . . . . . . . . . . 76
30. Acceleration-Time Traces for the Bottom Movable Steel Plate and a Footing (3 in. by 3 in. square) ................ 77
31. Deceleration of Surface Footing vs Weight of Footing . . . . . . . . . . . .. 79
32. Vertical Stress-Time Traces on Axis of Symmetry at Z = 1/2 in.; 6 in.; 12 in.· (6 in. dia. circular footing). . . . . . . . . . . . 81
33. Vertical Stress-Time Trace at R = 0 in.; Z = 1-1/2 in. (6 in. dia. circular footing).. 82
34. Vertical Stress-Time Traces on Axis of Symmetry at Z = 2 in.; 4 in. (6 in. dia. circular footing) 83
vii
LIST OF FIGURES (Cont'd.)
Figure 35.
Page Vertical Stress-Time Traces on Axis of Symmetry
at Z = 3 in.; 9 in. (6 in. dia. circular footing) . . . . . . . . . . . . . . . . " 84
36. Attenuation of Peak Vertical Stress on Axis of Symmetry for a 6 in. dia. Circular Footing (a c = 0.55 psi) .... . . . . . . . 85
37. Vertical Stress-Time Trace at R = 0 in.; Z = 1/2 in. (3 in. dia. circular footing) 87
38. Vertical Stress-~ime Trace at R = 0 in.; Z = 1 in. (3 in. dia. circular footing). . 88
39. Vertical Stress-Time Trace at R = 0 in.; Z = 1-1/2 in. (3 in. dia. circular footing). 89
40. Vertical Stress-Time Traces on Axis of Symmetry at Z = 6 in.; 12 in. (3 in. dia. circular footing) . . . . . . . . . . . . . . . 90
41. Vertical Stress-Time Trace at R = 0 in.; Z = 9 in. (3 in. dia. circular footing) .., 91
42. Attenuation of Peak Vertical Stress on Axis of Symmetry for a 3 in. dia. Circular Footing (ac = 0.88 psi) ............... 92
43. Vertical Stress-Time Traces on Axis of Symmetry at Z = 1/2 in. ; 1-1/2 in. (3 in. by 3 in. square footing) . .
44. Vertical Stress-Time Traces on Axis of Symmetry at Z = 1 in. ; 6 in. ; 12 (3 in. by 3 in. square footing) . .
45. Attenuation of Peak Vertical Stress on Axis of Symmetry for a 3 in. by 3 in. Square Footing
93
94
(a c = 0.20 psi) .... . . . . . . . . . . 95
46. Stress at Any Point from Circular Bearing Area (Boussinesque's Stress Distribution Theory). 97
47. Peak Vertical Stress Distribution With Depth Under Transient Loads (without any footing on the surface) .......... 98
viii
Figure 48.
49.
50.
LIST OF FIGURES (Cont'd.)
Vertical Stress-Time Traces at 6 in. and at R = 0 in.; 6 in.; 12 in. (6 dia. circular footing) . . . .
Vertical Stress-Time Traces at 3 in. and at R = 0 in.; 6 in.; 12 in. (6 dia. circular footing) . . . . . .
Vertical Stress-Time Traces at 1-1/2 and at R = 0 in.; 6 in.; 12 in. (6 circular footing) ........ .
Depth in.
Depth in.
in. Depth in. dia.
51. Attenuation of Peak Vertical Stress vs Offset from Axis of Symmetry for a 6 in. dia.
Page
101
102
103
Circular Footing (a = 0.55 psi) .... , 104 c
52. Vertical Stress-Time Traces at 6 in. Depth and at R = 0 in.; 3 in.; 6 in. (3 in. dia. circular footing) .............. 105
53. Vertical Stress-Time Traces at 6 in. Depth and at R = 9 in. ;12 in. (3 in. dia. circular footing) . . . . . . . . . .. ... 106
54. Vertical Stress-Time Traces at 3 in. Depth and at R = 0 in.; 6 in.; 12 in. (3 in. dia. circular footing) ..... 107
55. Vertical Stress-Time Traces at 1-1/2 in. Depth and at R = 0 in.; 6 in. (3 in. dia. circular footing) . . . . . . . . . . . . . . . . . . 108
56. Vertical Stress-Time Trace at R = 12 in.; Z = 1-1/2 in. (3 in. dia. circular footing) 109
57. Vertical Stress-Time Trace at R = 0 in.; Z = 1/2 in. (3 in. circular footing) . . . .. 110
58. Vertical Stress-Time Traces at 1/2 in. Depth and at R = 6 in.; 12 in. (3 in. dia. circular footing) . . . . . . . . . . . . . . . . . . III
59. Attenuation of Peak Vertical Stress vs Offset from Axis of Symmetry for a 3 in. dia. Circular Footing Cac = 0.88 psi) ...... 112
:bx
Figure 60.
61.
62.
63.
64.
65.
66.
67.
68.
69.
70.
LIST OF FIGURES (Cont'd.)
Vertical Stress-Time Traces at 6 in. Depth and at R = 0 in.; 6 in.; 12 in (3 in. by
Page
3 in. square footing) . . . . . . . . " 113
Vertical Stress-Time Traces at 3 in. Depth and at R = 0 in.; 6 in.; 12 in. (3 in. by 3 in. square footing) . . . . . . . . .. 114
Vertical Stress-Time Traces at 1-1/2 in. Depth and at R = 0 in.; 6 in.; 12 in. (3 in. by 3 in. square footing) . . . . . . . . . . . 115
Vertical Stress-Time Traces at 1/2 in. Depth and at R = 0 in.; 6 in.; 12 in. (3 in. by 3 in. square footing) . . . . . . . . . . . 116
Attenuation of Peak Vertical Stress vs Offset from Axis of Symmetry for a 3 in. by 3 in. Square Footing (a
c = 0.20 psi) 117
Strain-Time Trace at R = 0 in.; Z = 2 in. (6 in. dia. circular footing) . . 119
Strain-Time Traces on Axis of Symmetry at Z = 3 in.; 4 in. (6 in. dia. circular footing). . . . . . . . . . . . . . . . 120
Vertical Strain Distribution With Depth on Axis of Symmetry for 6 in. dia. Circular Footing . . . . . . . . . . . . . . . .. 122
Vertical Strain Distribution vs Offset from Axis of Symmetry for a 6 in. dia. Circular Footing . . . . . . . . . . . . 123
Nodulus vs Strain Relationship at Zir = 1/2; R/r = 2 (Test No. TA-6l) ...
Modulus vs Strain Relationship at Zir = 1; R/r = 1 (Test No. TA-l1l) ...
124
125
71. Vertical Stress-Time Traces on Axis of Symmetry at Various Depths for a 6 in. dia. Circular Footing With 2 in. Surcharge . . . . . . . . 127
72. Vertical Stress-Time Traces on Axis of Syw~etry at Various Depths for a 6 in. dia. Circular Footing on the Surface of a Soil Media . . . 128
x
Figure 73.
74.
75.
76.
LIST OF FIGURES (Cont'd.)
Vertical Stress-Time Traces on Axis of Symmetry at Various Depths for a 6 in. Circular Footing Buried at 2 in. Depth
Vertical Stress-Time Traces on Axis of Symmetry at Various Depths for a 6 in. Circular Footing Buried at 4 in. Depth
Vertical Stress-Time Traces on Axis of Symmetry at Various Depths for a 6 in. Circular Footing Buried at 6 in. Depth
dia.
dia.
dia.
Attenuation of Peak Vertical Stress With Depth on Axis of Symmetry for a 6 in. dia. Circular Footing at Different Depths of
Page
129
130
131
Burial . . . . . . . . . . . . . . . . . . . 132
77. Acceleration-Time Trace for a 6 in. dia. Circular Footing on the Surface of a Soil Media in the Vessel ........ . 134
78. Acceleration-Time Trace for a 6 in. dia. Circular Footing Buried at 2 in. Depth 135
79. Acceleration-Time Trace for a 6 in. dia. Circular Footing Buried at 4 in. Depth . . . 136
80. Acceleration-Time Trace for a 6 in. dia. Circular Footing Buried at 6 in. Depth 137
81. Deceleration of Footing vs Footing Depth from the Surface of the Soil Media for a 6 in. dia. Circular Footing. . . . . . . . . . . . 138
82. Stress vs Deformation for Compressible
83.
Materials Ml , H2 , M3 ............ 140
Stress vs Deformation for Compressible Materials M4 , M5 . . . . . . . . . . .. 141
84. Acceleration-Time Trace for a 6 in. dia. Circular Footing on the Surface ~>Jithout Any Compressible Material in the Soil Media. . . 143
85. A Compressible Material M1 Placed in Position at a Depth Z/r = 1 Before Being Buried in the Sand . . . . . . . . . . . . . . . . . . 144
xi
Figure 86.
LIST OF FIGURES (Cont'd.)
Acceleration-Time Trace for a 6 in. dia. Circular Footing With a Compressible Material Ml Buried at R/r ='0; Z/r = 1
87. Acceleration-Time Trace for a 6 in. dia. Circular Footing With a Compressible
Page
l4S
Material M2 Buried at R/r = 0; Z/r = 1 ... 146
88. Acceleration-Time Trace for a 6 in. dia. Circular Footing With a Compressible Material M3 Buried at R/r = 0; Z/r = 1 ... 147
89. Acceleration-Time Trace for a 6 in. dia. Circular Footing With a Compressible Material M4 Buried at R/r = 0; Z/r = 1 ... 148
90. Acceleration-Time Trace for a 6 in. dia. Circular Footing With a Compressible Material MS Buried at R/r = 0; Z/r = 1 149
91. Deceleration Reduction vs Subgrade Modulus of Compressible Material . . . . . . . .. ISO
92. Deceleration Reduction vs Log of Subgrade Modulus of Compressible Haterial . . .. 151
93. Stress-Time Trace at R = 0 in.; Z = 1-1/2 in. (6 in. dia. circular footing on the surface) . . . . . . . . . . . . . . . .. 153
94. Stress-Time Trace at R/r = 0; Z/r = 1/2 With a Compressible Material Ml Buried at R/r = 0; Z/r = 1 (6 in. dia. circular footing on the surface) ............... lS4
95. Stress-Time Trace at R/r = 0; Z/r = 1/2 With a Compressible Material H2 Buried at R/r 0; Z/r = 1 (6 in. dia. circular footing on the surface) ............... 155
96. Stress-Time Trace at R/r = 0; Z/r = 1/2 With a Compressible Material M3 Buried at R/r = 0; Z/r = 1 (6 in. dia. circular footing on the surface) . . . . . . . . . . . . . . . . 156
97. Stress-Time Trace at R/r = 0; Z/r = 1/2 Hith a Compressible Material M4 Buried at R/r = 0; Z/r = 1 (6 in. dia. circular footing on the surface) . . . . . . . . . . . . . . . . 157
xii
LIST OF FIGURES (Cont'd.)
Figure
98. Stress-Time Trace at R/r = 0; Z/r - 1/2 With a Compressible Material Ms Buried at Rlr = 0; Z/r = 1 (6 in. dia. circular footing
Page
on the surface) . . . . . . . . . . . .. 158
99. Peak Vertical Stress Reduction vs Subgrade Modulus of Compressible Material ... " 160
100. Peak Vertical Stress Reduction vs Log of Subgrade Modulus of Compressible Haterial 161
101. A View of the Apparatus for Statically Loaded Footing . . . . . . . . . . . . . . . . ., 163
102. Footing Pressure vs Settlement (under static conditions) . . . . . . . . . . . . . . . . 165
103. Ultimate Load Criterion Based on Log Load-Log Settlement Plot (De Beer's Method) 166
104. Footing Pressure vs Settlement With a Compressible Material Ml Buried at R/r = 0; Zir == 1 . . . . . . . . . . . . . . . . . . 167
105. Footing Pressure vs Settlement with a Compressible Material M2 Buried at R/r = 0; Zir = 1 . . . . . . . . . . . . . . . . l6B
106. Footing Pressure vs Settlement With a Compressible Material M3 Buried at Rlr = 0; Zir == 1 . . . . . . . . . . . . . . . . 169
107. Footing Pressure vs Settlement With a Compressible Material M4 Buried at Rlr = 0; Zir == 1 . . . . . . . . . . . . . . . . 170
lOB. Footing Pressure vs Settlement vJith A Compressible Material Ms Buried at R/r = 0; Zir = 1 . . . . . . . . . . . . . . . . . . 171
109. Deceleration Reduction vs Factor of Safety Reduction (factor of safety based on O.lB in. allowable settlement) . . . . . . . . .. 173
110. Deceleration Reduction vs Factor of Safety Reduction (factor of safety based on 0.10 in. allowable settlement) . . . . . . . . .. l7L~
xiii
Figure Ill.
112.
113.
114.
115.
116.
117.
118.
LIST OF FIGURES (Cont'd.)
Page Grain Size Distribution of Ottawa Sand 182
Relative Density vs Specimen Density for Dry Ottawa Sand (20·~40 Mesh) . . . . 183
Vertical Stress Arrival Time vs Depth 187
Stress vs Deformation Relationship for Compressible Material No. 1 (loading and unloading conditions). ". . . 189
Stress vs Deformation Relationship for Compressible Material No. 2 (loading and unloading conditions). . . . 190
Stress vs Deformation Relationship for Compressible Material No. 3 (loading and unloading conditions). .. .. 191
Stress vs Deformation Relationship for Compressible Material No.4 (loading and unloading conditions). . . . . . 192
Stress vs Deformation Relationship for Compressible Material No. 5 (loading and unloading conditions). .. . 193
xiv
LIST OF SYMBOLS
A ..... Contact area of the footing
Al ,A2 ,A3 and A4 .Constants
AB ..... Acceleration of the bottom movable steel plate
~ ..... Acceleration of the footing
a ..... Deceleration of footing
B ., ... One half width of footing
D ., ... Diameter of the footing
Df ..... Footing depth from surface
DR ..... Relative density
CD." .. Dilatational wave velocity
CS ..... Coil spacing
E ..... Modulus of elasticity (Young's modulus)
Ec ..... Constrained moudlus
F.S ... Factor of safety
K ..... Subgrade modulus of the compressible material c
G ..... Specific gravity of sand
v ..... Poisson's ratio
p ..... Mass density
Y ..... Dry density of soil
Ymax " . Maximum dry density
Ymin ... Minimum dry density
g ..... Acceleration due to Earth's gravity
R ..... Radial cylindrical coordinate from axis of symmetry
R ..... Percentage reduction of the deceleration of the a f' ootlng
xv
LIST OF SYMBOLS (Cont'd.)
RS'" .. Percentage reduction of the vertical stress
RF ..... Percentage reduction of the Factor of Safety (based on allowable settlement)
r .. , .. Radius of the footing
S .. , .. Settlement of the footing
Sa ..... Allowable settlement of the footing
St ., .. Sensitivity
Q .. , .. Static load applied on the footing
z .. , .. Vertical cylindrical coordinate below surface
0c'" .. Static contact pressure
0z'" ,.Vertical stress
°2 ... , .Stress gage number 2
°3 .. , .. Stress gage number 3
°4 ", .. Stress gage number 4
€ ....• Strain at any arbitrary point
£z ... , . Vertical strain at depth Z
£2.'.' . Strain gage number 2
E: 3 ... , . Strain gage number 3
£4'" .. Strain gage number 4
MI ..... Haterial designation number I
M2 ..... Material designation number 2
M3 ..... Material designation number 3
114 , .... Material designation number 4
My .... Material designation number 5
)tvi
CHAPTER I
INTRODUCTION
This thesis deals with effects on foundation response
produced by stress waves propagating from the underlying
soils.
The research described in this thesis is a part of
the research program which began at the Illinois Institute
of Technology and the lIT Research Institute in 1960 to
study the transient stress-strain properties of soil by
means of embedded stress gages and strain gages. The
previous research was concerned with one-dimensional, two-
dimensional and three-dimensional stress and strain through
soil media under transient loads applied through a founda-
tion at the surface.
1
In the design of structures which lie within earthquake
regions, designers have been mainly concerned with lateral
forces and very little attention has been given to forces
in the vertical direction. But during the San Fernando
18* earthquake in Los Angeles ,Southern California, in
February, 1971, vertical loads resulting from the earthquake-
initiated motions in the vertical direction, produced
sufficiently large forces in first-floor columns of some
buildings to collapse the entire structure. SeedS7 has
pointed out that the main lessons to be learned from recent
"/'Numerical superscripts refer to references in the bibliography at the end of the thesis.
earthquakes are the dangers resulting from inadequate con
siderations of site conditions for structure.
As discussed and recommended by Varadhi 7l this study
forms an extension of the work where vertical loads were
transmitted to .small size footings enclosed in a 10 in.
diameter vessel. In this study, however, a 3 ft. diameter
vessel was used in which a 24 in. thick layer of 20-40
2
Ottawa sand was used as a soil mass beneath the footing. Two
circular footings, 6 in. and 3 in. diameter, and a square
footing of 3 in. size, having different contact pressures,
were used. Vertical loads were transmitted to the foundation
from mechanically-induced ground motions by means of three
hydro-line cylinders.
The behavior of surface footings as well as buried
footings was investigated in this study. Measurements were
made of vertical acceleration and displacement-time records
by mounting accelerometers on the footing. Stress-time and
strain-time relations using embedded gages were made at
various locations throughout the soil layer, with special
emphasis to the area in the vicinity of the footing.
Experiments were also conducted using a thin layer of
compressible material buried at a depth of one radius below
the footing. Four sponge materials and a rubber material,
having different compressible characteristics, were used in
the investigation. These compressible materials were used
as damping materials which absorbed some of the compressional
waves travelling towards the bottom of the footing. This
technique of screening the waves (generated at a distance
away from the footing) was defined by Woods 82 as Passive
Isolation. A series of five tests were conducted, each
time using one of the different compressible materials.
3
The foundation response (acceleration-time), and the vertical
stress in the vicinity of the footing were recorded under
dynamic load conditions.
The above test series was repeated for static load
conditions where static loads were applied to the soil
through the footing to estimate the bearing capacity of the
footing (surface footing only).
CHAPTER II
REVIEW OF LITERATURE
4
This review is separated into three studies: first,
the footing behavior under dynamic loads is studied; second,
the recent soil dynamics research at lIT is studied; and,
finally, the isolation studies are covered.
Footing Behavior Under Dynamic Loads
Detailed review of literature concerning the behavior
of foundations under dynamic loads was done by Selig58 . A
brief description of the review is presented herein. Lamb 36
was the first to derive a relationship between the vertical
harmonic force, applied at a point on the surface of an
isotropic, homogeneous, semi-infinite, elastic medium, and
the displacement. In Germany, the German research society
for soil mechanics, DEGEBO (Deutsche, Gesellschaft fur
Bodenmechanik) and the University of Gottingen, together,
. d h .. l' .. 39,40,41 A . 1 carr~e out t e emp~r~ca ~nvest~gat~ons . s~mp e
mass-spring analogy was chosen as the mathematical model to
represent the system. This mass included the mass of the
vibrator and the foundation, but the experiments showed that
the portion of the soil beneath the foundation should also
be included in the mass. The system consisted of a founda
tion and the vibrator supported by a massless, linear spring
with a dash-pot. Barkan3 ,4 performed a series of similar,
but independent, foundation tests in Russia at VIOS, the
institute of eng~neering foundation research. From his tests
he concluded, among other things, that the spring constant
was not a constant but varied with the contact area of the
foundation. Furthermore, he concluded that it varied with
5
the frequency or rate at which the load was applied and
removed. In 1936, Reissner53 derived a solution for a system
of vertical, periodic forces distributed over a circular
area on the surface of an elastic body. He assumed that the
dynamic soil pressure developed was uniform. The resonance
frequency and amplitude of oscillation of the vibrator-soil
system were determined as functions of radius of loaded area,
static weight of the vibrator, weight and frequency of the
vibrating mass, and the modulus of elasticity, Poisson's
ratio and density of the soil. Crockett and Hammond12 ,13
suggested a refinement of the mass-spring analogy, in which
the mass of the vibrating soil be distributed as weight for
the spring rather than lumped with mass of foundation and
vibrator. Vibration-compaction experiments on sand were
carried out at the California Institute of Techno1ogy9.
Based on these experiments an empirical natural frequency
formula was derived, which included the following soil
properties: soil density, soil shearing modulus, weight of
foundation and vibrator, dynamic force and radius of the
foundation.
Eastwood17 has carried out a considerable number of
experiments to obtain a quantitative information about the
effect of the various parameters on the natural frequency of
the foundations and the effect of vibrations on the bearing
6
capacity and settlement of foundations.
Quinlan50 used three types of load distributions
(uniform, parabolic and a load for a rigid base) and derived
a solution for vibration of circular and long rectangular
vibrators assuming various contact pressures. He has shown
a method for determining the phase angle, natural frequency
and the amplitude of vibration. An analytical investigation
was conducted by Sung65 to determine the effect of the
amplitude of oscillation on the resonant frequency.
Slade64 has suggested that different models determined
principally by the energy level of the vibration, must be
used to represent soil. At low energy levels, the soil may
be approximately represented as elastic. Since the dis
placements in the soil are principally irreversable. at
medium-energy levels the soil is considered as nonelastic.
At high-energy levels, the characteristics of the soil change
significantly. Winterkorn80 called this state of soil at
high-energy levels, as a macrometritic liquid. When the
surface of a soil medium is subjected to dynamic loads,
similar to those imparted by a vibrator, there will be several
zones in the soil each with a different energy state. The
highest state is in the vicinity of the vibrating footing,
and it decreases as the distance from the vibrating footing
increases. Because of the difference in the nature of the
soil in these different energy states, Slade feels that one
cannot describe the entire soil medium by a single model;
instead, a discontinuous model must be used to represent the
actual conditions. Slade suggested: 1) a coupling-region
consisting of a spring and a dashpot in place of the high
and medium regions; and, 2) a mass which transmits a
concentrated force to the low-energy region. This low-
energy region is treated as an elastic region and the
solution presented by Lamb 36 is used to determine motion of
the soil.
Arnold, Bycroft and Warburton2 determined the forced
7
vibrations of a rigid body resting on a homogeneous, iso
tropic, elastic medium of infinite surface area and of
constant depth, which may be finite or infinite. Four modes
of vibration of a mass on a circular base were considered:
1) vertical translation; 2) torsion; 3) horizontal translation;
and, 4) rocking. Arnold checked his theory experimentally
using an electromagnetic vibrator on a rubber medium and
concluded that the results agreed favorably with the
theoretical predictions.
Richart55 outlined Sung's work and presented extensive
curves which may be used for analysis and design of founda
tions under dynamic loads. He assumed the foundation to be
rigid.
Triandafilidis67 solved analytically the dynamic problem
for a very simplified, rigid, plastic material. He found
that the footing response is greatly influenced by the
inertia of the participating soil mass. In case of dynamic
problems, the inertia forces play an important role, therefore,
the participating soil mass in the motion is of paramount
importance.
8
11 Cox, et al., analyzed the dynamic response of a smooth
circular punch on the surface of a semi-infinite, rigid and
plastic medium by numerical techniques. In their analysis,
they assumed the soil to be rigid and plastic obeying the
Mohr-Coulomb yield criterion.
Carroll8 studied the behavior of square, rigid footings
resting on the surface of a clay medium (Buckshot Clay)
subjected to static and impulsive vertical loadings, both
experimentally and analytically. Load was applied to the
clay layer through the footing. The purpose of his investiga
tion was to establish relations between applied transient
foundation loads and the resulting foundation motions.
Measurements consisted of recording the vertical displacement
and reaction load of the footing when subjected to known
input load.
From the results of his investigation Carroll categorized
the dynamic load-displacement behavior of the footing in to
three inertial regimes: 1) three-dimensional static in
which inertial forces exert only a minor influence on footing
response and a three-dimensional static analysis should give
reliable results; 2) three-dimensional dynamic in which the
footing response should be considered as a three-dimensional
problem; and, 3) one-dimensional dynamic in which one-
dimensional wave theory, with appropriate dynamic stress-
strain curves, should reliably predict footing response.
Hadala23 , Jackson and Hadala30 at Waterways Experiment
Station used the principles of similitude in the analysis of
9
dynamically loaded square footings on clay. Three non-
dimensional relationships were developed to predict the
maximum displacement and the time of its occurrence. The
first relationship was between the resistance parameter and
the displacement parameter. The second relation was that of
a strength parameter to the displacement parameter. The
third relationship included the strength and inertia
parameters.
In 1965, Whitman78 reviewed the theoretical work on
foundation vibrations and compared the results with some
field tests. He concluded that, if the soil properties-
especially the shear modulus' and Poisson's ratio-can be
estimated more accurately, the theoretical and experimental
results would have a better correlation.
Richart and Whitman56 expanded Whitman's78 review work
to include other modes of vibration. From their study of
model footing tests, they concluded that the theory assuming
the soil media as elastic and semi-infinite, gives good
approximation of the amplitudes of motion, when the vertical
oscillations produce a linear acceleration of the footing of
less than O.5g.
Lysmer and Richart42 developed a method for analysis of
the vertical mode of vibration of a rigid circular footing
resting on the surface of an elastic half-space. Their
solution included all ranges of frequencies for steady state
vibration. From the results of their study it was shown that
all vertically loaded footing-soil systems were strongly
10
damped because of wave propagation into the subsoil. There
fore, it was suggested that the material damping could be
neglected in practical calculations. It was also shown that
the damping ratio decreased with increasing mass ratio of the
system.
Drnevich and Hall lS conducted transient loading tests on
a foot diameter, relatively rigid footing resting on the
surface of a bed of sand. Two types of load pulses were
applied to the footing. The magnitude of the loads was such
that the response of the footing was essentially elastic. The
mass of the footing was also varied to produce different
footing response characteristics. From the analysis of their
test results, it was concluded that the theoretical approach
developed by Lysmer leads to reasonable predictions of the
"elastic" dynamic response of the footing.
Experimental and analytical investigations on the
behavior of small boy footings were conducted by Hadala and
Jackson24. They have found that the static preloads of less
than 30% of static bearing capacity, do not influence either
the change in maximum footing displacement or the change in
time to maximum displacement. Although the preload-induced
static settlements were negligible, it was possible that
the preload-induced field beneath the footing, influenced the
magnitude of the dynamic displacements.
Rao and HoegSI developed a numerical procedure for
predicting the dynamic response of strip footings on
saturated clayey soils. The soil was assumed to be linearly
11
elastic-perfectly plastic, yielding according to Tresca's
criterion and flowing plastically according to the normality
rule of plasticity. They used a discrete lumped parameter
mathematical model replacing the clay continuum. The effect
of various variables on the peak displacement of the footing
was studied. They concluded that the peak displacement is
very sensitive to: 1) pulse duration; 2) applied load pUlse;
and, 3) soil properties like, soil modulus and Poisson's
ratio, but is not sensitive to the density and pressure
distribution under the footing.
Larkin38 used perfectly smooth strip and circular
footings resting on a cohesionless soil to obtain the bearing
capacities. He obtained the bearing capacities of very
shallow footings by integration of S~e equations of plastic
equilibrium of soils. It was found that the footing was
very sensitive to depth of burial. In the case of circular
footing located at the surface, an increase in depth of
about 0.09 to 0.13 of the footing diameter is sufficient
to increase the bearing capacity 100%. This indicated that
the small footing settlements which accompany the loading up
to the point of failure, significantly increase the actual
bearing capacity of footings.
Orrje49 developed a new method to evaluate the
deformation and strength properties of cohesionless soils.
Both laboratory and field tests were conducted on four
different soils. Behavior of these soils was investigated
by measuring the retardation of free falling weight when it
strikes the soil surface. Static plate loading tests were
also conducted on the same soil material for comparison
purposes.
12
The load-deformation curves, the failure loads and the
equivalent moduli of elasticity from the dynamic and static
loading tests were compared at various relative densities
of the soil materials. He also investigated the failure
modes of the dynamic static loading tests. From the analysis
of the test results Orrje concluded that: 1) the shape of
load-settlement curves obtained from the dynamic loading
tests with a free falling weight agreed well with the load
settlement curves for the static plate loading tests; 2)
the dynamic failure load and the dynamic equivalent moduli
of elasticity increased with increasing height of free fall;
3) the dynamic failure load increased with increasing plate
diameter; 4) the failure load also increased with increasing
mass of the falling weight and the equivalent modulus of
elasticity was approximately independent of the mass of the
falling weight; 5) the footing response was sensitive to any
changes in the dry unit weight of the sand (The maximum
retardation increased approximately linearly with increasing
dry unit weight of the sand. The dynamic and static failure
loads also increased linearly with increasing dry unit
weight.); and,6) in the dynamic tests with retardations
between 19 and 6g, the failure mode differed from that of
the static tests.
He also concluded that the depth to which the loading
13
tests affected the underlying soil corresponded to approxi
mately 1.5 plate diameters.
Rhines54 proposed a new foundation model for punch
shear failure in soils. An infinitely long rigid footing
resting on an elastic-perfectly plastic soil was considered in
their analysis. The foundation model consisted of two elastic
spring layers interconnected by an elastic shear layer.
The important characteristics of the foundation
response included: 1) yielding began at and was confined to,
points in the foundation layer directly under the edges of
the punch; 2) the load-deflection curve was bilinear and the
peak corresponded to the oneset of yielding in the shear
layer; and, 3) once yielding occured, the maximum contact
pressure was less than it would have been had the foundation,
under the same load, remained elastic.
Duns16 numerically examined the dynamic behavior of a
rigid circular footing on a linear-elastic isotropic ha1f
space by means of the finite element method. For the
reflection, an equivalent viscous damping was used at the
footing. Results showed a good agreement with the analytical
methods. The calculation of the vertical footing acceler
ation for unsteady (shock) excitation by means of a ~t-method
was confirmed by the results of measurements obtained in a
similar test.
Skormin63 conducted some bearing tests using a circular
plate 60 cm.in diameter resting on the surface of a sand
layer three meters thick. The contact pressure at the base
14
of the plate and the normal stresses in the subgrade were
measured at different loading increments. From the investi
gation it was found that the stress under the foundation was
concentrated more than that suggested by the theory of semi
infinite elastic solids.
Naik48 proposed a method to find displacements and
stresses within a foundation on a semi-infinite mass under
dynamic loads. The soil was assumed to be a semi-infinite,
homogeneous, isotropic, elastic solid and the foundation was
treated as a plate supported on the soil. The finite element
method was used to calculate the soil reaction beneath a
foundation subjected to a dynamic load. He then used Lamb's
study of 1904 to describe the foundation displacement. With
the method proposed in his study, the natural frequency of
vibration for a foundation of any shape can be easily
computed. Some of his conclusions were that the displacement,
due to one or more foundations, at any location on the surface
of the semi-infinite mass can be computed by the method
described in the study. The natural frequency of a given
foundation-soil system can be easily computed. For this
foundation-soil system, the displacement gradient between the
mid-section and the edge of the foundation decreases as the
loaded area increases.
Medearis47 of Kenneth Medearis Associates (KMA), Ft.
Collins, Colorado, prepared a report for the ASCE Research
Council on Performance of Structures. KMA has investigated
the response characteristics of fullscale structures to the
15
transient, earthquake-like ground motions resulted from an
underground nuclear detonation. They used the Plowshare
Project (of U.S. AEC) to study the dynamic response of actual
structures.
Project Rio Blanco was the third Plowshare natural gas
stimulation experiment to be conducted in the United States.
It involved the detonation of 3-30 kiloton nuclear devices
more than a mile below ground level in Rio Blanco County,
Western Colorado on May 17, 1973. Five Rio Blanco area
structures were selected for instrumentation and analysis.
These structures were located between 4 and 40 miles from the
detonation point, and experienced peak accelerations of
0.003-1.0g's. All these structures were previously analyzed
by KMA. Seismic recorders were installed at the base of and
on each of these structures to monitor their responses to the
detonation ground motions. The structures were subsequently
reanalyzed and the theoretical and recorded responses compared.
The results of their study indicated a good correlation
between the recorded responses and those obtained using
modern, computer-oriented, structural dynamics models and
analysis.
It was also indicated that the Uniform Building Code
seismic recommendations are inadequate with regard to both
low-rise and high-rise construction. It was suggested to
revise the Uniform Building Code seismic recommendations.
Recent Soil Dynamics Research at lIT
One-dimensional Studies. The stress wave propagation
16
study at lIT began in 1960. In this study, a series of
experiments were conducted in which long horizontal columns
of sand and clay were subjected at one end to a shock
loading6l, 75. Stress gages were buried along the length
of the specimen and information was obtained on wave velocity,
peak stress attenuation and reflected stresses. This study
also included the development of an embedded soil stress
gage60 . A piezoelectric crystal was used as the sensing
element for this gage. The gage possessed the advantages of
high frequency response and high sensitivity.
Gage Development. Concurrently with the program outlined
above, a program of gage development was conducted to develop
a soil strain gage that would minimize interface of the gage
. h f '1 68 Th' . f f W1t ree S01 movement. 1S gage cons1sts 0 two sets 0
two mechanically uncoupled thin coil disks and associated
electronic circuitry. The principle of operation is that of
a null balance differential transformer, i.e., each set of
coils represents transformer primary and secondary windings
and are so arranged in the circuitry that the resulting
signal is the difference of the individual coil outputs. In
operation, one set of coils is embedded in the soil as the
strain sensing element while the other is positioned
externally at a known spacing to serve as a reference. The
gage is well adapted for use in soil as the sensing elements
are mechanically uncoupled.
Januskevicius and Vey3l performed a series of experi
ments to evaluate the performance of strain gage and the
17
previously mentioned stress gage in soil. In these experi-
ments, the gages were used to measure stresses and strains
in triaxial specimens of sand under both slow and rapid rates
of loading. Evaluation of the results led to two important
conclusions relative to the reliability of the gages to
record true free field stresses and strains:
(1) The stress gage must be calibrated in soil.
Moreover in any stress gage measurement the
modular ratio of the soil to the gage must
lie within determined limits of the ratio at
which the gage was calibrated.
(2) The strain gage output can be related to the
true strain up to 6 percent strain if the
strain is due to volume decrease. Where
strains occur without a change in volume or
if the volume increases during deformation,
the presence of the gage begins to have an
appreciable effect on the deformation at
about 3 percent strain.
Later, a series of one-dimensional wave propagation
experiments were conducted in which the gages were employed
. l' . h '174 ,26 S to measure stress-stra~n re at~ons ~n t e so~ . tress
gages and strain gages were embedded at intervals along the
length of long columns of soil (both sand and clay). The
soil columns were loaded at one end by a shock wave with a
peak stress in the range of 5-300 psi and the gages used to
measure stress-time and strain-time as the stress wave
traversed the specimen. Stress-strain curves were con-
structed by interpolating between gage locations. In a
subsequent study soil specimens were subjected to shock
waves up to 1100 psi peak stress.
Further studies to eliminate certain undesirable
18
features of the piezoelectric gage such as the adverse
effects of moisture and the gage's inability to sense quasi
static loads led to the development of a piezoresistive
A number of gage concepts were evaluated and it
was determined that the deflecting diaphragm type gage was
most satisfactory, provided the deflection was kept suffi
ciently small. A prototype gage was constructed which used
a piezoresistive element on a deflecting diaphragm as the
° 1 t Th ° ° 1 f °d d59 ° h senslng e emen. e prlnClp e actors conSl ere ln t e
gage design include:
(1) application, i.e., free field or against
structures
(2) frequency response
(3) geometry
(4) relative stiffness of the gage and the soil
(5) density of gage relative to soil
(6) nature of sensing element, and
(7) the placement of the gage in soil.
Depending upon a particular application, gages must be
selected or designed (if not available in the market). If
there are any major variations in environmental conditions,
such as temperature and moisture, they must be controlled.
19
For measurement of stress under transient loading condi-
tions, frequency response of the gage must be considered. The
geometry of the gage is also an important factor to be con
sidered. Small size gages for laboratory experiments and
larger size gages for stress measurements in the field are
generally used.
Most important factor to be considered in evaluation of
the response is the difference between the stiffness of the
gage and the soil. By burying the stress gage into the soil,
some soil is displaced. The stiffness of the gage is entire
ly different from that of the soil. Therefore, this will
cause a redistribution of stresses around and in the vicinity
of the gage and produce a higher or lower stress reading, de
pending upon whether the gage has a greater or small stiff-
ness than the soil. It has been found that the thickness-to-
diameter ratio of the gage is also an important factor which
effects the magnitude of the redistribution of stresses. It
is generally accepted that this ratio should be as small as
possible. A typical value28 ,29 of this ratio is 0.2.
The density of a gage is another important factor to be
considered in dynamic measurements. For instance, if the
density of the gage is higher than that of the soil replaced
by the gage, the free motion in the soil is interfered. As a
result, the gage inertia produces a stress which either in-
creases or decreases the stress in free soil conditions. This
effect becomes more significant at higher rates of loading. By
selecting a proper gage material, one which weighs same as the
soil replaced by the gage, density matching can be achieved.
Most of the gages are constructed using a flexible
diaphragm supported at the perimeter of the gage by a
rigid ring. The deflection of the diaphragm is sensed by
piezoresistive strain gage bonded to the diaphragm. Due
to the less rigid diaphragm and more rigid case, the soil
arches the stress onto the gage perimeter causing a higher
than free-field value on the perimeter (ring) and a lower
than free-field value on the diaphragm.
20
Selig59 in his review suggested that the gage design
must also take into consideration the methods of placement.
Gage placement method and calibration procedures are
discussed in detail in Chapter IV of this thesis.
Footings on Sand. The experimental programs using
surface applied time dependent loads were undertaken to
measure the behavior of sand beneath a rigid circular
footing76 ,77. Two types of loads were applied to the
footing; an impulsive-type dynamic load and a quasi-static
load. Embedded soil stress gages and soil strain gages were
employed to obtain the complete stress-strain-time relation
ship in the soil at various depths below the surface on the
centerline of the loaded area. Data was also obtained off
the centerline. Stress-time and strain-time measurements
were obtained at the same or symmetrical locations.
These studies led to the following conclusions:
(1) The soil response to the impacting load
exhibited both elastic and plastic
characteristics. During the initial
loading phase plastic behavior was noted
primarily in the stress measurements.
Plastic behavior was subsequently noted
in the displacement and strain measurements
as the inertia effects decreased.
(2) The attenuation of the peak vertical stress
with depth on the axis of symmetry was
essentially spatial as predicted from the
theory of elasticity even though the peak
surface pressure was considerably greater
than the bearing capacity of the soil. The
absence of evidence of attenuation due to
inelastic effects was ascribed to the signi
ficant inertia forces acting during the initial
loading phase.
(3) An initial decrease in volume was followed
by a volume increase as yielding occured for
tests on both medium and dense Ottawa sand.
(4) Relating to a foundation subjected to an
impulsive load, the experimental data
indicated that the problem may be divided into
three inertial regions----one-dimensiona1
dynamic, three-dimensional dynamic, and three
dimensional static----depending on the size
of the footing, the rise time of the load
pulse, and the soil stiffness.
21
22
Footings on Clay. An experimental program similar to
that conducted for sand was also carried out for clay (EPK
Clay)37, In addition to the experimental work a computer
program was developed6 to predict wave developments in clay
using an axisymmetric model, and assuming the clay to be an
elastic-perfectly plastic material subjected to a known
impulsive loading applied through a circular footing at the
surface. A finite difference technique was used to integrate
the differential equations of motion. In order to get
comparative results, assuming the elastic-plastic behavior
of clay, several trail values were given to the modulus of
elasticity, Poisson's ratio, and the yield limit in simple
shear using three types of boundaries-rigid and flexible, for
various sizes of footings.
A study of the computed stresses and strains in the soil
as a function of time indicated the sensitivity of the results
to changes in individual soil parameters and a comparison of
the theoretical values with those from the experiments yielded
approximate values of the parameters for the clay. The
latter were well within the limits of what had been pre
viously reported for clay with similar properties.
The Isolation Studies
The isolation technique has been applied to the earth
quake problem to reduce the effect of transient loadings on
the structure. In 1935, Green2l proposed that buildings be
constructed with a flexible first story.
Barkan5 reviewed many applications of trenches and sheet
pile barriers in isolation problems and presented some
empirical rules for design.
23
Matsushita and Izumi44 proposed a double basement design
which incorporates a flexible first story.
Woods 82 conducted some field tests to evaluate the
effectiveness of open trenches in reducing the amplitude of
vertical ground motion. He used annular trenches surrounding
the source of vibration to reduce the disturbance of the
adjacent ground, and defined this isolation as 'active'
isolation (isolation at the source). He defined 'passive'
isolation as employing some barriers (straight trenches) at
remote points from the source of vibrations but near a site,
to reduce the amplitude of vibration. From the active
isolation tests, he concluded that a maximum depth of 0.6
wavelengths was required for a trench to satisfy the
amplitude reduction criteria of 0.25. Circular trenches were
effective in reducing motions outside the trench to some
distance (~10 wavelengths) away. From the passive isolation
tests, he concluded that larger trenches were required at
greater distances from the source to accomplish a given
amplitude reduction. He also concluded that sheet-wall
barriers were not as effective as open trenches in screening
surface waves.
Matsushita and Izumi45 conducted fundamental studies on
the 'Earthquake Free Mechanisms' to decrease earthquake forces
applied to the buildings. They defined 'Earthquake Free
Mechanism' as the mechanism to be installed in a structure,
24
to decrease the input earthquake energy. The effective
application of the mechanisms and the media is to install
them near the boundary between the structure and the soil
since the earthquake forces are fed to the structures through
the boundary. They proposed a concept of isolating buildings
with rollers which would cause the building to rise when
displaced laterally. The four types of mechanisms satisfying
the necessary conditions were: 1) Bearing Mechanism; 2)
Column and Rod Mechanism; 3) Wall and Brace Mechanism; and,
4) Boundary Mechanism. The 'bearing mechanism' consisted of
using bearings including ball bearings. The 'column and rod
mechanism' consisted of flexible but strong columns and rods.
The 'wall and brace mechansim' produced 'restoring force
deflection' relations against lateral displacement. And the
'boundary mechanism' provided shock absorbing materials at
the boundary between the structure and the ground. Mechanisms
shown in their study were designed for about and less than
twenty storied R.C. buildings built on good ground.
From the results of earthquake response calculations and
recorded data they concluded that the acceleration at the top
of the building of medium height was three times the input
acceleration at the base and, in case of structures with
'Earthquake Free ~1echanism', the acceleration values at the
top of the building were small. Some of the calculated
results showed more than a 50% decrease of input earthquake
forces.
Gupta and Chandrasekaran22 considered an absorber system
25
having both linear and elasto-plastic restoring force
characteristics and viscous type of damping in their analysis.
They considered a parent system represented by a mathematical
model of idealized single degree of freedom. Some of their
conclusions were that the response reduction factor (8) de
creased as the mass ratio (~a: the ratio of sum of masses of
all absorbers to that of the parent system mass) was increas
ed. The response of the parent system was increased as the
damping of absorbers was increased. The response of ab
sorbers themselves decreased with the increase in numbers.
Wirsching and Yao8l studied some techniques for
improving the safety of seismic structures. In particular,
the effect of passive motion reducing device on seismic
structures was investigated. They used a Monte Carlo
technique for analog simulations of the random earthquake
and corresponding response processes. The seismic response
of a single-story system having a nonlinear isolator was
studied. They also considered the same nonlinear isolator
with 5 and 10 story models.
From the results of the single-story tests, two general
conclusions were drawn. They found that it is desirable to
have significant damping in the isolator (perfect isolation
occurs when the isolator stiffness approaches zero). And
the isolator must maintain sufficient stiffness to prevent
excessive displacement of a structure in a strong wind.
They also conducted two test series with the absorber
(using the optimum value of damping) in the system of the
26
five and ten story models. From the analysis of the results,
they concluded that: 1) a considerable reduction in structural
response could be realized with the isolator; 2) there was
little dispersion in the peak response data with the isolator
in the system; and, 3) the absorber, used with the isolator,
caused an increase in the response. They also reported that
the use of the abosrbers did not help the isolated system
because the top floor motion was relatively small and little
energy was absorbed.
Finally, on the basis of safety considerations and
without regard to the cost, they concluded that the design
modifications were ranked in the order of their efficiency
as: 1) the isolator system; 2) the isolator-absorber system;
3) the absorber-damper system; 4) the absorber system; and,
5) the damper system. The ranking was based on only limited
studies of each of the systems.
CHAPTER III
PURPOSE AND SCOPE OF INVESTIGATION
Based upon the review presented in Chapter II the
present state of knowledge in the area of footing response
under dynamic loads may be summarized briefly as follows.
27
Earlier theoretical investigations were based on the
assumption that soil behaved like an homogeneous, isotropic
and a linear elastic material. But the general behavior
of soil is nonhomogeneous, anisotropic, inelastic and
nonlinear. However, for small strains the behavior may be
considered as that of a linear elastic material. Some 51 recent studies also included complex constitutive relations
into the analysis, in which soil was assumed as linearly
elastic-perfectly plastic material.
All of the studies to date, both analytical and
experimental, have dealt with stress-time and strain-time
measurements throughout the soil mass under the effect of a
surface applied time-dependent load. Measurements were made
in the soil as a function of direction and distance from the
foundation applied load. In these studies the foundation
was initially in motion and the soil was at rest.
The February 9, 1971 San Fernando earthquake in
California demonstrated that ground accelerations can occur
in excess of those previously considered in the design of
dams10 . It is recognized, of course, that in most earth
quakes, horizontal loads are more damaging to structures
and actual foundation-soil structure interaction results
28
from loads that are inclined. These inclined loads could be
resolved into two components vertical and horizontal and
considered independently. Of the two, only the vertical
component involves the support of the structure under both
static and dynamic loads. If the foundation is designed with
a safety factor of 2 or 3 for example, as is normally the
case, against bearing failure by shear in the soil under
static load, the capacity of the soil to transmit superimposed
vertical dynamic loads before yielding, is likely to be far
in excess of the strength of the columns or piers. In the
San Fernando earthquake in Los Angeles18 the failure in the
columns of bridges and buildings was caused by the motions
transmitted to the foundation soil from the base rock which
propagated upward through the soil to the structure. They
were found to be quite severe in localized areas with
accelerations estimated near the 1.Og level.
Examples of other than natural occuring phenomena that
transient vertical dynamic loads to structure are under-
ground explosions, large drop hammer forges, demolition
operations, pile driving and even heavy truck traffic for
some soil conditions, e.g., soft saturated clays underlying
a thin dessicated surface stratum. Design of foundations,
considering possible structural failure due to vertical
dynamic loads, has received much less attention than has the
effects of horizontal loads. In many of these cases it is
possible to establish the input load spectrum with reasonable
accuracy. The problem lies in evaluating the soil trans-
29
missibility and the soil-foundation interaction that one
also needs for an evaluation of the force-deflection response
of the structure.
Thus, it is evident that a need for a basic understanding
of this phenomenon exists. To accomplish this, an experi
mental program was undertaken in which the foundation was
situated at the surface and at different depths of burial,
and vertical transient loads were applied to the foundation
through the underlying soil. Therefore, in this study the
foundation was initially at rest and the soil at the inter
face between the soil and the foundation was in motion. In
the previous studies, the foundation was initially in motion
and the soil was at rest, thus inertia effects in the present
study were reversed. Initial stress conditions in the soil
mass were also varied by varying the foundation size and mass.
Measurements were made of stress-time and strain-time
histories using embedded gages throughout the soil mass and
particularly in the vicinity of the foundation. From the
results of the test data an attempt was made to obtain a
comprehensive picture of stresses and strains in a soil
mass,and to evaluate the extent of the influence of founda
tion and distribution of pressures in the soil beneath it,
on the stresses and strains. Measurement of acceleration
time histories were also made. These records were integrated
to provide velocity-time and displacement-time relations for
the foundation.
In addition, the study included some efforts to reduce
30
the effect of input transient loads on the foundation
response. This was accomplished by providing a layer of
compressible material at a depth of one radius. The measure
ments included the acceleration-time histories and subgrade
modulus of the compressible material at the static contact
pressure level. An attempt was also made to establish a
relation between the reduction factor of the footing response
and the reduction of the safety factor against bearing
capacity failure based on an allowable settlement.
31
CHAPTER IV
TEST APPARATUS AND EXPERIMENTAL PROCEDURES
Test Apparatus
A general view of the experimental apparatus used in
this study is shown in Fig. 1. A pressure vessel 3 ft. dia.
by 3 ft. deep with both ends open was used. A movable steel
plate 1/2 in. thick, supported by three I-sections welded
(to improve the rigidity of the plate) to the plate at 120 0
apart, as sho~vn in Fig. 2, was positioned at the bottom
inside the vessel. The vertical movements of this plate
were maintained vertically by three hydro-line cylinders.
A 24 in. thick layer of dry Ottawa sand was used as a soil
media beneath the footing. The properties of this sand are
given in Appendix A. Rigid footings, circular and square
shapes 6 in. dia., 3 in. dia. and 3 in. by 3 in., as shown
in Fig. 3, were used. Ground motions induced mechanically
using a loading apparatus specially designed for this purpose
were transmitted to these footings through the underlying
layer of sand bed by moving the bottom plate. The behavior
of surface as well as buried footings was studied.
A schematic diagram of the loading apparatus is shown
in Fig. 4. As shown, the apparatus consisted of a nitrogen
cylinder, accumulator, solenoid-dump valve, flow control
valve, relief valve, distribution chamber, three hydro-line
cylinders, oil storage and oil dum9. Oil under pressure in
the system was released to the hydro-line cylinders and hence
the transient loads to the bottom movable plate attached to
32
Figure 1. A General View of the Experimental Apparatus
a) Top View
b) Bottom View
Figure 2. Bottom Hovable Steel Plate Mounted on Three Hydro-Line Cylinders
33
34
Figure 3. Various Size Footings Used in This Study
2 to
n l
oad
su
pp
ort
ed
by t
hre
e
Hy
dro
-lin
e cy
lin
ders
Swi t
ch
+
•• -
,So
len
oid
F
low
co
ntr
ol
I'
cf1
dum
p v
alv
e
-D
istr
ibu
tio
n'
cham
ber
~
Oil
du
mp
f -
Acc
umul
ator
Oil
IN
Gas~
\ 2 ,
Oil
N
2 G
as
Sup
ply
Fig
ure
4
. S
chem
atic
Dia
gra
m o
f T
ran
sien
t L
oad
ing
S
yst
em
W
\JI
36
the cylinders. A view of this system is shown in Fig. 5.
Embedded stress gages and strain gages were positioned
at various locations throughout the soil layer and in the
vicinity of the footing as the sand was being placed in the
vessel. Two accelerometers (High Resonance Frequency, Shock
Accelerometer Model 2225, ENDEVCO Dynamic Instrument Division),
one on the bottom movable plate of the vessel and the other
one on the top of the footing, and a DCDT were used to record
the acceleration-time and displacement-time relationships.
Stress-time and strain-time histories in the vicinity
of the footing were recorded by using the embedded gages.
For this purpose a l4-Channel magnetic tape recorder
(Magnetic Tape Recorder, Model 5600B, Honeywell) and a visi
corder (Model l508B, Hone~~ell) were used. A bank of four
oscilloscopes (Dual Beam Oscilloscopes Type 502, Tektronix
Inc.) were also used to record the response (for some tests)
of the gages and accelerometers, and traces recorded on a
polaroid film simultaneously, for comparison purposes. An
internal triggering system was used in the first oscilloscope
of this bank and the remaining oscilloscopes were simultane
ously triggered by means of a trigger wire.
A series of static load-displacement tests were also
conducted for a 6 in. dia. surface footing. These tests
were conducted in the same vessel as the dynamic tests,
using the apparatus shown in Fig. 6. The load was applied to
the footing at a constant strain rate through a gear box
arrangement. The footing pressure was measured by means of
37
Figure 5. A View of the Loading Svste8
C-C
lam
p E
lectr
ic
Mot
or
Cha
in
11
t. ~
~
~ ,
2'x
2")
1 i"
-An
gle
Iro
n S
up
po
rt'
Def
orm
atio
n d
ial
gage
rin
g
ves
sel
6"
Dia
. F
oo
tin
g
San
d su
rfac
e
::S":'=
;TC;' .:,:,
.; ',' c':;;
.. ::;-"
.:':' -
'T,:·.
',':::;
: :::., ,
" : ,'.
;'.:':
;;,,
~:.:
:')~i
i{;' ;\'
;;):
',. ': (
: ',:,:,
~ .' i: :
:,:: '.: N
:.
re" '...
18"
... ,.
....
: , .•.. "
.. °.
0"
' .. r:.-
'_1- ~I-,
18
" ~~\ .. )
.ii
.' . . : :.' ::.':
.. ".
: ::::
:: . "
.. : ~
Fie
ure
6
. Ap~aratus
for
Sta
ticall
y L
oad
ed
Fo
oti
ng
C-C
lam
p
w
00
a proving ring dial gage, and the penetration was measured
by a deformation dial gage.
Embedded Stress Gages
39
Stress-time measurements were recorded by using the
piezoresistive gages developed at lIT Research Institute62 .
This gage is schematically shown in Fig. 7. The overall
dimensions of this gage are 1.5 in. in dia. by 0.255 in.
thick and with a frequency response of 60 KHz. The gage
consists of a disk-shaped aluminum case with stiff walls and
a thin diaphragm machined in the bottom surface. The gage
operates on the deflected diaphragm principle and the de
flection of the diaphragm is sensed by a piezoresistive strain
gage. These gages were constructed with four active arms
forming a complete ~Vheatstone Bridge and a balance unit to
provide an initial balancing of the bridge circuits. This
balance unit was also provided with four different calibration
resistors to allow for variations in excitation voltages.
The excitation voltage used throughout the experimental
program was 5.0 volts d.c. These stress gages were cali
brated both under hydrostatic conditions as well as in
confined specimens of sand.
Calibration of Stress Gages. The schematic diagram
of the device used for hydrostatic calibrations is shown
in Fig. 8. The device consisted of two steel cups filled
with water, each covered by a thin rubber membrane. These
cups were then clamped together face-to-face with the gage
between them. A known air pressure was applied to the water
Drill & Tap 4-40
train Gage
==. =~ rf-------- 1.5" d 0 I~ 0.75 .. ---' ....
~tJ~~======~~ ·0 ! o
Figure 7. Piezoresistive Soil Stress Gaee
40
~ / ~ --
~2.5" 2.5" ~
g v
.... I .. ~---- 6.5"
(a)
(b)
~I
Ve nts for Air Pressure
R ubber Membrane
Clamps to hold the Rubber Membrane
Pressure Chambers (water inside)
Stress Gage
Pressure Gage
Figure 8. Schematic Diagram of ~~Jater ChaHber for Gage Calibration
41
42
in both the cups and hence to both sides of the gage uniform-
ly. This pressure was increased in increments of 10 psi to
a maximum of 80 psi, while the gage output was being recorded
for each increment on an X-Y plotter. The pressure was then
decreased to zero in 10 psi decrements with the gage output
again being recorded. This procedure was repeated several
times to check reproducibility. A typical calibration curve
under hydrostatic conditions is shown in Fig. 9. For all
gages the curves obtained for the ambient normal pressure
calibrations were straight lines passing through the origin.
The sensitivity values of these gages under hydrostatic
conditions are tabulated in Table 1.
Table 1. Sensitivity Values of Piezoresistive Stress Gages Under Hydrostatic Conditions
Stress Gage Number 2 3 4
Sensitivity (mv/v/psi) 0.0648 0.0603 0.0441
These stress gages were also calibrated in confined
specimens to determine the effect of confinement on soil
calibration. These gages were placed at 1/3, 2/3 and mid
depth of a 7 in. long by 5 in. diameter confined specimens
of the same soil used in the test vessel. The calibration
results of the gages placed at mid-depth were used in the
analysis, because the stress and strain at mid-depth of the
sample are least effected by the end plates. The experiments
-CI)
CL -.. t/) t/) Q) ...
80
60
1;;40
"i --Q. CL «
20
o 100
Stress Gage No. 4 Excitation Voltage: Output Reference Output of Recorder: Gage Sensitivity :
5 volts R3 5 mv/l0Q% Output 0.0441 mv/v/psi
200 300 400 Gage output, (0/0)
Fir,ure 9. Typical Calibration Curve for Piezoresistive Gage Under Hydrostatic Conditions
44
conducted by Januskevicius and Vey31 have shown that the
measurement of stresses at this depth are true representation
of the actual stresses. Thus it was assumed that the
vertical stress acting at these gage locations was equal to
the applied pressure. The sensitivity values of these gages
in confined specimens of the soil used in this program are
tabulated in Table 2., and a typical calibration curve for
these piezoresistive stress gages is shown in Fig. 10.
Table 2. Sensitivity Values of Piezoresistive Stress Gages in Confined Specimens of the Soil
Stress Gage Number 2 3 4
Sensitivity (mv/v/psi): 0.0747 0.0728 0.0558
Embedded Strain Gages. Strain-time measurements were
recorded by using the mechanically uncoupled strain sensing
elements operating on the transformer principle. These gages
were developed at IITRI and have been extensively evaluated
in the laboratory68,69,27,33 These gages consist of two
identical sets, each set with two coils. Some strain gage
units consisted of 3/4 in. dia. by 1/16 in. t.hick coils
and the other units consisted of 1 in. dia. by 1/8 in. thick
(Bison Instruments) coils. Each set of soils represents
primary and secondary windings of a transformer. Soil de
formations are measured by the resulting changes in the
25
20
-.. 0- .- .. :
15
-= -a • . - 'ii. ~ 1
0 5 o 2
0
Str
ess
Gag
e N
o.4
E
xci
tati
on
Vo
ltag
e:
Out
put
Ref
eren
ce
Out
put
of
Rec
ord
er:
Gag
e S
en
siti
vit
y
60
8
0
5 v
olt
s R
) 5
mv/
100%
Out
put
0.05
58 m
v/v
/psi
100
120
140
Gag
e o
utp
\l (%
) F
igu
re
10
. T
yp
ical
Cali
bra
tio
n C
urv
e fo
r P
iezo
resi
stiv
e G
age
Em
bedd
ed
in S
and
.~
I.J1
46
mutual inductance of the coils which are directly pro-
portiona1 to the changes in spacing of the embedded coils.
d h 1 · f . .. 31,68,69 h Base on t e eva uat~on 0 past ~nvest~gat~ons , t ese
gages can be reliably used up to 6% compressive strains. The
reliable limit of strain measurements in case of tensile
strains is 3%, and the frequency response is 10 KHz.
Calibration of Strain Gages. In operation, one set of
coils (in-situ coils) is embedded in the soil, and the other
set (reference coils) is mounted on an adjustable micrometer
mount as shown in Fig. 11. For calibration purposes the in
situ coils set is mounted on another adjustable micrometer
mount. This calibration test consisted of applying known
displacements in 0.005 in. increments to one set of coils,
while the other set of coils is at a certain fixed spacing,
and recording each instrumentation-system response. A
typical calibration curve for one soil strain gage is sho\vu
in Fig. 12. The gage output values are again plotted against
the compressional and tensional strains as shown in Fig. 13.
This calibration procedure was repeated for coil spacings of
0.4 in., 0.5 in. and 0.6 in. and the gage output values versus
coil movements are plotted in Fig. 14 for. compressional
strains.
In operation, the actual spacing of these in-situ coils
in sand, however, was not the same as they were initially
placed, due to the displacement of the coils as the sand was
being poured. The actual spacing of the in-situ coils is
determined by adjusting the reference coils until a null
1. Driver Coils Circui t
2. Tuned Amplifier and Detector Circuit
). Output Device
Adjustable Micrometer
Figure 11. IITRI Coil Strain Gage
L:·7
"
600
-> e -"'400 &. -::s o
Displacement (Ten)
0.02'1
0.01" .' 0.01"
Displacement (Comp)
-> 400.§ -a. -::s o
600 Note '. Out t 1 pu va ues are
800 Strain gage.,4
amplified ten times
Figure 12. Calibration of Strain Gage (1 in. dia. coils) (Displacement vs Gage Output)
48
E (%) - Tension 3 2 I
600
-> e -'5 400 Q. -:s o
200
2 E (%) - Compression
200
-E -400 -;
Q. -6
600 Note: Output values are
amplified ten times
800 Strain gag"4
Figure 13. Calibration of Strain Gage (1 in. dia. cOils) (Strain vs Gage Output)
1~9
2400
2000
1600
-> e --&. 1200 -<5
800
400
o
Note: Output values are amplified ten times
0.005" 0.015" Displacement (Compression)
Strain gage 13
50
0.020"
Figure 14. Calibration of Strain ~age (3/4 in. dia. coils) (Dis~lacement vs Gage Output)
balance is obtained.
Instrumentation
51
A view of the instrumentation system is shown in Fig. 15.
The acceleration-time records of the bottom movable steel
plate in the vessel and the footing were recorded by means
of accelerometers. For this purpose two high-resonance
frequency (80 KHz) shock accelerometers (The ENDEVCO, Model
2225 Accelerometer) were used. The model 2225 incorporates
Piezite Element Type P-8 in a shear design that is capable of
measurements to 20,000 g and is not affected by transverse
components of acceleration. These accelerometers were of
piezoelectric-type transducers which generated an electric
charge. This output charge from the accelerometers was
carried to the electrostatic charge amplifiers (Charge
amplifier Model 566, Kistler Instrument Corp.) by means of
low noise cables, and was converted to a voltage signal. This
converted voltage signal was then recorded on a recording
equipment.
Stress-time relations were recorded by using piezo
resistive gages. These gages as described earlier were
constructed with two active arms having a nominal resistance
of 350 ohms, and two dummy resistors forming a complete
bridge. Only a strain gage element was mounted in the gage
body at its center. This system provided the gage with only
one active diaphragm. These gages were then connected to a
bridge balance unit which provided initial balancing of the
bridge circuits and an excitation voltage of 5 volts d.c.
52
Figure 15. Instrumentation Setup
53
Strain-time relations were recorded by using the
mechanically uncoupled strain sensing elements operating on
the transformer princip1e~ In this, air-core true differ
ential transformer was used instead of a conventional linear
variable differential transformer with iron core. Fig. 16
is a block diagram of the basic components of this differential
transformer for application as a soil strain sensor. As
shown in Fig. 11, only one driver- and one sensor-coil are
embedded in the soil. The other driver and sensor coils are
mounted on an adjustable micrometer mount external to the
soil sample under test. The driver coils (connected in
series) are excited with an a.c. current. The pick-up coils
are also connected in series with the phase of the individually
induced signals in opposition so that the total resulting
signal is the difference of the outputs of the two pick-up
coils. ~l'hen the coils in each set are equally spaced, the
resulting differential output is zero or nulled.
Acceleration-time, stress-time and strain-time responses
were recorded and stored on a l4-channel magnetic tape re
corder (Magnetic Tape Recorder Model 5600B, Honeywell). This
model magnetic tape system is a full 14-channe1 instrumenta
tion grade recorder which handles virtually any tape record
ing requirement. and is capable of recording/reproducing up
to l4-channels of data at seven servo controlled tape speeds.
A view of the data recording system is shown in Fig. 17. The
frequency response of direct electronics is up to 300 KHz and
F.M. electronics to 40 KHz. The entire experimental data was ' .. ' ,",,.-.,""
Mic
rom
eter
Mou
nt
-+
-n
oo
~
y D
riv
er C
oi 1
"1
"1'"l
0
~Secondary
Sen
sor
Co
il
Pri
mar
A.
. E
xci t
ati
on
S
ourc
e
50 K
C O
scil
lato
r /' "
-S
ign
al
/ A
mp
lifi
er
and
Ind
icato
r D
et.
Cir
eui t
''4
' ,-
.'<
"
,-,
1""''
\'' ,~
,,'"
/'
Pri
mar y
Dri
ver
Co
il
..tl
ll
n \01
2 S
econ
dary
Sen
sor
Co
il
So
il
-------
Fig
ure
1
6.
Sch
emat
ic
Dia
gra
m o
f E
lectr
ical
Cir
cu
it fo
r th
e S
oil
-Str
ain
Gag
e U
nit
L1
1 -po
..
Figure 17. Data Recording SYstem (Nagnetic Tape Recorder Hodel 5600B, Honeywell)
55
56
recorded at a high speed of 60 ips (inch per second) and
was reproduced at a low speed 7-1/2 ips. The output data
from the tape recorder was amplified in a multi-channel d.c.
amplifier (Multi-Channel DC Amplifier, Model Accudata 117,
Honeywell) and was printed on a sensitive paper. A visi-
corder oscillograph (Visicorder Oscillograph Model lS08B,
Honeywell) was used to print this data from the multi
channel d.c. amplifier. A view of this data printing equip
ment is shown in Fig. 18. A bank of four oscilloscopes (Dual
Beam Oscilloscopes, Type 502, Tektronix Inc.) was also used
to check the response of the accelerometers and the embedded
gages, before it was recorded by the magnetic tape recording
system. The oscilloscopes were also used for calibration
purposes and to record the response on a polaroid film. An
internal triggering system was provided in the first oscillo
scope of the bank, and the remaining oscilloscopes were
simultaneously triggered by connecting them in series.
Experimental Procedures
In the preparation of the sand bed in the vessel, the
most uniform placement was obtained by a slow rain of sand34
(termed here Pluvial Compaction). This was the method used
at IITRI and later by Wetzel at IIT77. This "raining
technique" was also used by the Waterways Experiment Station7
(WES) , where they employed three devices. In their paper,
Bieganousky and Marcuson7 have referred to other experimentors
who have also used this raining technique. A schematic dia-
gram of the equipment used for this purpose is shown in Fig. 19.
57
Figure 18. Data Printing Equip~ent (Visicorder Oscillograph Model l508B, Honeywell)
I •
Io+---Crane
Sand Rain
r
'.' . . ,., .: ... " . ~ . ~~:~ . ." . , ,-.' ..
,<t'\ ' .... --'100..--- Variable-length
pully system
t~ _______ ~1 I~ ______ ~l
I
n
Figure 19. Raining Device for Placement of 20-40 Ottawa Sand
58
J
59
As shmm in this figure, a free fall dis tance of 12 in.
produced medium densities of the order 104.5 + 1.0% pcf. The
properties of the sand used in this experimental program are
given in Appendix A. Stress gages and strain gages were
oriented at desired locations as shown in Fig. 20, while
placing the sand in the vessel. A typical gage array of these
stress gages and strain gages is shown in Fig. 21.
The stress gages were placed by the set-on-surface
method. Hadala25 in his conclusions suggested that this
set-on-surface method was the best gage placement method for
sands. Ingram28 ,29 also reported that the tests in his
experimental program have indicated a minimum mean registra-
tion by the set-on-surface method. The procedures for placing
the stress gages consisted of the following steps:
1. The sand surface was levelled,
2. The stress gage was placed on the levelled
surface with its active face perpendicular
to the direction of loading,
3. The gage was slightly pressed into the
surface of the soil,
4. Sand filling was resumed by placing sand
on top and around the gage.
For strain gages to sense vertical strains, the gage
placing method is shown in Fig. 22. The procedure was as
follows:
1. The sand surface was levelled and a coil of
one set (in-situ coil set) was placed on the
i I
I~~
area
3
®
3 e
-~
c - - .... .. ;6 $
0 0
-lD
0
9 ..
12 ~
6 J 0 e e e
Ra
diu
s. ~
(in
)
9 12
i
I 6
) e
0 s
e •
• •
0-
stre
ss
poin
t
e -
stra
in
poin
t •
-st
ress
, stra
in p
oint
I~
I
Fig
ure
2
0.
Lo
cati
on
of
Str
ess
an
d S
train
l1
easu
rem
ents
i8
I
0\ o
61
Figure 21. A Typical Gage Arrav of the Embedded Gages
Alignment Rod (tapered)
Approximately 0.6 in.
a) Vertical Strain
rApproximatelY 0.6 in.
~ Alignment Rod (tapered)
b) Radial and Tangential Strain
Figure 22. Schematic Diagram of Placement Hethods for Strain Gages
62
levelled surface,
2. A tapered rod was placed through a hole in
the center of the coil. The taper of the
rod served to hold the coil flat on the
surface as the sand was being placed above
it. Additional soil was then compacted
around the rod to obtain the cover necessary
to give the desired gage spacing,
3. The second coil was slid over the rod and
placed on the soil surface,
4. Additional soil was compacted to a height of
2 to 3 in. above the top of the coil and
the alignment rod carefully removed,
5. The second set of coils (reference coils)
was positioned on a micrometer mount which
served as the reference.
63
After the vessel was filled with sand to the desired
height, the surface was levelled and the .footing was placed
on the surface at the center as shown in Fig. 23. Transient
loads were then applied to the bottom movable steel plate
inside the vessel which transmitted the ground motions
through the underlying soil. A schematic diagram and a view
of the loading apparatus is shown in Fig. 4 and Fig. 5,
respectively. The procedure in applying the transient loads
consisted of the following steps (see Fig. 4):
1. The nitrogen gas (N2-gas) under pressure was
supplied to the accumulator from the nitrogen
Figure 23. Footing Before Being Subjected to Transient Load
64
cylinder,
2. The accumulator consisted of a rubber bladder
inside, separating N2-gas from the hydraulic
oil. The nitrogen gas under pressure was
compressed by using a hand pump. Therefore,
by Newtons Third Law (For every action, there
is an equal but opposite reaction), the N2-gas
and the hydraulic oil were under pressure up
to a certain location in t~e pipeline system,
where the solenoid dump valve is located.
Under normal conditions this solenoid valve
will be closed, and will be open when the
electrical circuit is closed by pressing the
switch,
3. When the switch was pressed, the hydraulic
oil under pressure was delivered through a
flow control valve to the distribution chamber,
which finally distributed an equal quantity
of oil to each of the three hydro-line cylinders.
The piston rods of these hydro-line cylinders
were connected to the bottom of the movable
65
steel plate at 120 0 apart, which supported the
soil-footing system inside the vessel. Therefore,
the sudden release of pressure to the hydro-
line cylinders caused a dynamic movement of
the bottom movable steel plate along with the
soil and the footing inside the vessel, hence
transmitting the ground motions through
underlying soil.
66
67
CHAPTER V
PRESENTATION AND ANALYSIS OF RESULTS
The experimental program consisted of three test series.
Each test series represented a set of tests on various size
and shape of rigid footings having different contact pres
sures. The test series on a 6 in. dia. circular footing
having a contact pressure 0c = 0.55 psi was designated as
TA-No. A total of 130 dynamic tests were conducted on this
footing. Similarly, TB-No. represented a test series with
46 tests conducted on 3 in. dia. circular footing with a
contact pressure 0c = 0.88 psi. The third series, TD-No.,
represented a set of experiments (a total of 48 tests) con
ducted using 3 in. by 3 in. square footing having a contact
pressure 0c = 0.20 psi. Test series TA also included some
static bearing capacity and dynamic response tests with a
compressible material (6 in. dia. circular) beneath the sur
face footing (6 in. dia. circular), at a depth one radius
below the footing.
The test data was analyzed with respect to: (1) the
footing response; (2) the characteristics of the one
dimensional stress-wave propagated upwards towards the bottom
of the footing through the sand mass; (3) the effect of depth
of burial on the footing response and stress distribution
beneath the footing; and, (4) the footing isolation.
Footing Response
Transient loads were transmitted to the bottom of the
footing through the underlying soil mass. The acceleration-
time response of the bottom movable steel plate (AB) and
the footing (AT) were recorded by means of accelerometers
mounted as shown in Fig. 24. As shown in this figure, the
accelerometer mounted on top of the footing is identified
as AT and the one mounted upside-down on bottom of the
68
movable steel plate as AB. Therefore, the acceleration-time
traces corresponding to the footing and the bottom movable
steel plate are identified as AT and AB, respectively,
throughout this experimental program. A typical record of
the acceleration-time traces for the bottom movable steel
plate and a footing is shown in Fig. 25. As shown in this
figure, the first trace on the top identified as AB,
corresponds to the input source, that is the bottom movable
steel plate. The second trace on the bottom is identified
by ~, corresponds to the response of the footing, resting
on the surface of a sand layer 24 in. thick, inside the
vessel (3 ft. diameter). As shown in Fig. 24, the accelero
meter on the bottom movable steel plate is mounted upside
down. Therefore, the acceleration-time trace identified as
AB throughout this thesis, is upside down. In other words,
the trace is opposite to that of the trace corresponding to
the footing (~). In this trace AT' the portion of the curve
above the zero line corresponds to deceleration of the footing,
whereas in trace AB the portion of trace below the zero line
represents deceleration of the bottom movable steel plate.
Similarly, the portion of the trace ~ below zero line
corresponds to acceleration towards gravity of the footing;
Figure 24.
Sand Layer
(24 in. thick)
Accelerometer Lead
Accelerometer (AT)
Sand Surface
Bottom Movable Steel Plate
Accelerometer (AB)
Accelerometer Lead
Schematic Diagrao of Footing Apparatus
69
~~~~~
~-........;-,-.~-:-"--.
-:r
.---
...-,
;..
,,;
.,;-
..--~.-.-, -->
, '-'-'~ -
---' . '~-~-. -,,
--~-.-.
..~.----~-,.--.~~.
---;
--"-
."'T
'--"
'-~
I r .-
~~
!~
1~~~J'
~fV<\"
--<~-.
~".-.
..:~ ~
Ao#;;t~~
.,-~ ~
',. ~
--#"
~"'J~'
~~
,;:,
st
.. 6.
.954
.g i
n.
A.
. ~
~'''''
I>i<
1~""
i I ~v~~ ,
,"
.. :~
<~
I :
: f~
~!1'0~'
... ~~~
._
~ 1
._i~_~,·. "
i,~.~.'. N0
~;_*'.
' ,,,,
,,,_'~
-...t:
.:::r:
:::;:~
I.""It
"_\",,
,"'''_
I!'''-
''~_ .,
,_ ,~ .. ~
"'I\"'
" ....... '
,_ .•
Fig
ure
2
5.
Acc
eler
ati
on
-Tim
e T
race
s fo
r th
e B
otto
m ~1o
vabl
e S
teel
Pla
te
and
a
Fo
oti
ng
(6
in
. d
ia.
cir
cu
lar)
-.)
o
71
and, in trace AB, the portion above the zero line represents
acceleration towards gravity of the bottom movable steel
plate. As shown in Fig. 25, the input response spectra
consisted of two peaks. The first peak occured in 3.5 msec
and attenuated in less than 40 rnsec. This corresponds to
the response of the input source. The second peak, which
has a higher amplitude than the first peak, occured in about
200 msec. This peak attenuated in about 65 msec and is
attributed to the bouncing and ringing effect of the bottom
movable steel plate. Even though the amplitude of this peak
is much higher than the first peak, the time in which it
attenuates (65 msec) is much less than the time in which it
occurs (200 msec). The footing acceleration-time response
(~), shown in Fig. 25, also consisted of two peaks corres
ponding to the input response trace. The first pair of
peaks attenuated in about 40 msec. After 200 msec, the
footing started moving in the direction of gravity with an
acceleration 19 and contirtue.d to ,move for 50 msae, then it
decelerated. The average value of peak deceleration
amplitude for a 6 in. dia. footing (ac = 0.55 psi), measured
from zero line is 4.87g. This peak attenuated in about 12.5
msec to 50 msec. The pattern of the latter portion of the
footing response trace is almost similar to the response
traces of the actual structures under explosion-induced ground
motions47 . They differ only in their amplitudes and time.
The footing response (acceleration-time) trace (AT)'
shown in Fig. 25, is integrated once to obtain the velocity-
72
time relationship as shown in Fig. 26, which is again inte
grated to obtain d.isplacement-time relationship for the
footing as shown in Fig. 27. The acceleration-time trace
from the same test (TA-125) was also recorded on a ?olaroi~
film, as shown in Fig. 28. A direct current differential
transformer (DCDT) was also used to record the displacement
time response of the footing. The displacement versus time
relationship, shown in Fig. 27, is exactly the same as the
trace shown in Fig. 28 recorded directly by using a DCDT.
As shown in Fig. 27, the maximum displacement of the
footing is 0.158 ft. (1.89 in.). A sudden release of
pressure to three hydro-line cylinders causes displacement of
the bottom movable steel plate, the sand bed and the footing
with a momentum. After the piston rods travel to the maximum
stroke length, the sand bed with the footing (which is in a
state of motion), is stopped suddenly. This sudden re-
striction provided to a body in motion causes it to move
backwards with a negative velocity as shown in Fig. 26. The
footing then moves forward and finally comes to a complete
rest, as shown by the displacement-time curve in Fig. 27 and
28. A typical acceleration-time trace for a 3 in. dia.
circular footing (0 = 0.88 psi) is shown in Fig. 29. The c
average value of peak deceleration amplitude for this footing
is 4.20g. The average value of peak deceleration amplitude
for a 3 in. by 3 in. square footing (oc = 0.20 psi) is
7.84g, and the acceleration-time trace for this footing is
shown in Fig. 30. In all these experiments the same input
pressure was maintained. These peak deceleration values were
1.2
0.8
- CJ • ~ 0
.4
41
- - ~ -'(j o - CD
Tes
t N
o.
TA
-125
P
hoto
No.
PA
-86
> 0
.0 r'
5
0
100
150
20
0
\ 2
'so I
30
0
Tim
e (m
sec)
0.4
Fig
ure
26
. V
elo
cit
y v
s T
ime
Rela
tio
nsh
ip
for
a S
urf
ace
Fo
oti
ng
(6
in
. d
ia.
cir
cu
lar)
-..
.J
W
16
......
:: 1
2 - C'I 2 ~ .. - I 8
u 0 Ci.
til
2i
4 o
Tes
t N
o.
TA
-125
P
ho
to N
o.
PA
-86
50
10
0·
150
Tlm
e,(
m ••
c)
20
0
25
0
30
0
Fig
ure
27.
D
isp
lace
men
t v
s Ti
P'!e
R
ela
tio
nsh
ip
for
a S
urf
ace
Fo
oti
ng
(6
in
. d
ia.
cir
cu
lar)
'-J
+:-
-1'
r-50
mse
c T
A-1
25
AT
1.
398
0.4
5
in.
Fig
ure
2
8.
Accele
rati
on
, D
isp
lace
nen
t-T
ime
Tra
ces
for
a S
urf
ace
Fo
oti
ng
(6
in
. d
ia.
cir
cu
lar)
-.J
U1
I ~r 'I' "',
/ ,,',
~I!'
'''t
l ,
. ; r' ,'-,
' ",
f'"
,
t j,
Fig
ure
2
9.
Accele
rati
on
-Tin
e
Tra
ces
for
the
Bo
ttO
R 1~ovable
Ste
el
Pla
te
and
a
Fo
oti
ng
(3
in
. d
ia.
cir
cu
lar)
-..
.J
0"'\
,/ ,.,;,.,.
.~~.
-.--,.~-.--~-:-:-:.
'"
i n
; \
':, .,_
~. _J
{ i
. ""'l
~ I ~
, ,II'
:~.
'.
"
'.:,:.,
1/ f 'w
lt_O_
. m_se.
..,. ..... c,.
·~ .. 1 .'
__
(
, "-!"
~ ..
L-"J
' """
:"'i'.'
. ,1 .... ·
"..'~1""'-:
l·.
! i , -tl
~'\
A
.~ ....
,--
t~~· .
:~.-
1 '-.
--
.
!~ .•
L~-
-;l
I·; ~
!~_ .
. ~~~
"".:.l.
. .
1 7.:
•
"
i : \
<~.
~. , -t
"f ~
.~
Fig
ure
3
0.
(3
in.
by Acceleration-Ti~e
Tra
ces
for
the
Bo
tto
r ?1
ov
able
S
teel
Pla
te
and
a
Fo
oti
ng
')
.
) .J
ll
1.
squ
are
-...
..J
-.....J
plotted against their weights and a linear relationship is
obtained as shown in Fig. 31. This linear relationship
between deceleration of the footing and weight of the
footing is justified because the deceleration of a body is
inversely proportional to its mass.
Stress-Wave Propagation
78
In the experiments transient load was applied to the
bottom of the footing through the underlying soil mass. \~en
the hydraulic oil under pressure in the loading apparatus was
released to the hydro-line cylinders, the bottom movable
steel plate attached to the piston rods was moved upwards.
This generates a stress-wave which propagated up towards
the bottom of the footing through the sand mass. To study
the characteristics of this one-dimensional stress-wave
propagated up towards the bottom of the footing through the
sand mass, embedded stress gages and strain gages were placed
at various locations in the sand. Stress-time and strain
time histories, at the respective gage stations as shown in
Fig. 20, were recorded as the stress wave passed. Wave
propagation velocities and sand moduli are given in Appendix
B.
Stress Measurements. Stress-time histories at various
gage locations were recorded using embedded stress gages.
The location of the stress measurements is shown in Fig. 20.
The description of the gages, calibration procedures and
recording instrumentation are described in Chapter IV. Three
to four measurements were made at each gage location and
12 8 -0- -
6
o
4 2~,----__
__ ~ __
____
__ ~ __
____
__ ~ __
____
__ ~ __
____
__ ~
o 4
8 12
16
2
0
W O
bs).
Fir
-ure
3
1.
Decele
rati
on
of
Su
rface F
oo
tin
g v
s H
eig
ht
of
Fo
oti
ng
.......
\D
80
average values were used to determine the stress-distribution
in the soil under the various size footings. Attenuation of
peak vertical stress on the axis of symmetry and offset from
the axis of symmetry for each footing were determined. The
general form of stress-time traces recorded in the vicinity
of the footing followed the same pattern as the acceleration-
time trace.
Typical stress-time histories recorded at different
depths on the axis of symmetry for a 6 in. dia. footing
resting on the surface of sand are shown in Fig. 32 through
35. A total of three stress gages were used, each having a
different sensitivity. To obtain data at different depths,
tests were repeated under identical test conditions. This
data was used to obtain peak vertical stress values at
various depths on the axis of symmetry. The distribution of
peak vertical stress, with depth on the axis of symmetry for
a 6 in. dia. circular footing resting on the surface, is
shown in Fig. 36. .. 51 Based on a recent study by Rao and Hoeg
wherein the effect of footing rigidity was examined, a uni
form stress distribution on the base of the footing is
considered. Therefore, assuming a uniform distribution, a
static contact pressure of 0.55 psi for this footing was
obtained. From Fig. 36, the peak vertical stress on the
bottom of the footing under dynamic loading conditions is
10.9 times its static contact pressure (0 = 0.55 psi). c
A similar type of dynamic tests were also conducted
with a surface footing 3 in. dia. Cac = 0.88 psi). Again,
,.A
Ii I \
,l
' t I
,t. ~
,I, \'
il
l ii'
II i
i _~~,"~1\~
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\
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,
. I"l
orl
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,
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T~st N
o. rrA~83.'
H 1\ i ~
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f I
, i
i
c , ~4~
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iii
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. Ii I
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St.
2
.68
,pei
pln
R
O
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lit
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Fig
ure
3
2.
Vert
ical
Str
ess
-TiI
'1e
Tra
ces
on
A
xis
o
f Symr.~etry at
Z =
1/2
in
; 6
in.
(6
in.
dia
, cir
cu
lar
foo
tin
v)
12
in
.
CJ
r'
Fip
;ure
3
3.
foo
tin
r;)
~,
, '. : ~I , . U\ ~ i i)
i,
\
't I
., ~o m
5eC
-.j
.r-! CI.l Pi
$ C\l
R=Q~t:Zf;~i~',
"\"'1
\'
I ..
...
;
i ,I i
Vert
ical
Str
ess
-Tin
e
Tra
ce at ~
= Q
in
. ; ~{
. i ~
'7
1-:
i\ .i
" ;
! '
,! \
'/ '
; Tea
t Nt>
. T
k-s
j .
--'
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k~~~
i "('f~
."
j;, :i;t
:F~:,
, -Ir~-'#
-J
J ui:
'
I .
-j
,;~
1-1
/2
in.
(6
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dia
. cir
cu
lar
G~
N
Fip
,ure
d
ia.
Tes
t N
o.f
A-1
0
1.,.50
msec~
. ,::::
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ri at
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si/
in
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ert
ical
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ess-T
ine
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ces
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o
f SV
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'1et
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at
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cu
lar
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C)
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d iii
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i11';·
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h~:f't
tf'~:;
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Fig
ure
3
5.
Vert
ical
Str
ess
-Th
1e
Tra
ces
on
!\x
is
of
Sv
nm
etr
v at
Z
= 3
in
. 9
in.
dia
. cir
cu
lar
footin~)
. (6
in
.
c;; ~
85
CTZ tpsil
O __ ----~2~----~4------~------~8
2
4
~ .s 2r=S' - "- ~ ..,6
R
8
0 (f)
10 z
12
Figure 36. Attenuation of PeaY. Vertical Stress on Axis of Syrnmet~y for a 6 in. dia. Circular ~ootin8 (oc = 0.55 ~si)
86
three to four tests were conducted to obtain representative
values at each gage station and the average value was used
to plot the stress distribution. Typical stress-time traces,
recorded at various depths on the axis of symmetry, for this
footing are shown in Figs. 37 through 41. The response of
these traces is compatible to that of the acceleration-time
response of the footing. Data from other tests conducted
under identical test conditions was used to obtain peak
vertical stress values at various depths on the axis of
symmetry. Attenuation of peak vertical stress on the axis
of symmetry, for a 3 in. dia. circular footing resting on
the surface, is shown in Fig. 42. As shown in this Fig. 42,
the peak vertical stress on the bottom of the footing under
dynamic loading conditions is 10.7 times its contact pressure
(0 = 0.88 psi). c A set of experiments similar to those conducted on 6 in.
dia. and 3 in. dia. circular footings were also conducted on
a 3 in. by 3 in. square footing having a very low contact
pressure (oc = 0.20 psi). Again, in this case, three to four
tests were conducted at each gage station and data from
different tests conducted under identical test conditions was
used to obtain peak vertical stress values at various depths
on the axis of symmetry. Typical stress-time traces are
shown in Figs. 43 and 44. From average values of these test
data, a stress distribution curve with depth on the axis of
symmetry is plotted as shown in Fig. 45. In this case, the
peak vertical stress on the bottom of the footing is 22 times
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Fig
ure
3
7.
[oo
tin
?;)
Vert
ical
Str
ess
-Tim
e
Tra
ce at ~
= 0
in
. Z
= 1
/2
in.
(3
in.
dia
. cir
cu
lar
CC
)
-..j
\ i\. :1
1
.t
co
\.Q
. C\
l
Tes
t N
o.
']$.
-12
~. I
I
j I
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Fir
,ure
3
8.
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tin
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I ""'
I;<'~f
i·'
Vert
ical
Str
ess
-Tim
e
Tra
ce at
R
= 0
in
. Z
==
1 in
. (3
in
. d
ia.
cir
cu
lar
CO
C)
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~~ '~=
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;
'"
.~
'J"
I ;
(':
' ';
, '
;
. _i
, i~
T~ \.
\~
" ,I I I'f"
'Q
=O'!,
• I N
-1d
."
'."~
.-' '
,' ti
=
2 ."
.
Fi~ure
39
. V
ert
ical
Stress-~iTIe
Tra
ce at
R =
0
in.
foo
tin
g)
.;--,,~ ..
.. "\~
';'f ... .
tW(:~'H"
':' .~',
•..
. ..k
~ .. ;-_
.. +~:
. '""
, .. ,"
V' ':k
;' ";-
~';-' ..
" T
est
HD,' .~
'1" i
...
';"
~.,,-.
'.
,'"
...
;"-' -'
'. ; :
.
""
....
I' '"
"
, I
I, '.
,i
, I
Ii I
i I
I ! I
~ "
!
, i
Ii
" i I
, ,I,
l" i
)'-."
:I
!' ~
, '''' __
.t.
.
Ii'!
Ii i
, \ J , I" ,:;;
of"
'
Z =
1-1
/2
in.
;' --
--:t
-
. 1
'i' ,-
t' .I
L . ...
\...__ ~
. -" ~-
~ -'I;'; ,~
<. '
,c fi1
"~"i
-,i
V 't
' ";
,;
'., 'r
'fr~
./'
-...
.. -r
'""""""
... '~
, I
+" ',
:i .-
I., '
!'-:" j "
'i
'I (3
in
. d
ia.
cir
cu
lar
co
\0
. , ... '
50 D
!Se~
f ..
+ •
. X.
,tNC
).~.
:tJ
.t~i
.".
! f·
i',·
~·t,
,(..
, .
;'~
i. Ii"
. ;.
'.'.,.J.'.
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....
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,~ .'
~"tf
';.
~~.r
'_,_
.!._
. j'"lr
_ "'.:'
R=O
" Z
=6"
.
,.
St.
,2
.'l4
psi
!in
~:'~
.. ' il
pi . 'i\
,~~\.
~ .i~,1
Jf
'f '"t
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[-
. I=l
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j1 "
'B=O
'~
Z=
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t "
,o1;\\~~/';;·V·i~li~'l
~ •. t·.~
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',~(
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:' •
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1
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.61'
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. '.\!4
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, .. v.''t~\''''f.¥w.t,l~1;Ny·,,''It;'i#, ...
. :~V*
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\;;.
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....
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,.'
: : ..
.... ~.
. '--"
'~'~.~~
.. ::~ .
. '.~.
:'".
,
," \
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:'\(~l.~
.~.'~.f.
--~·"
,·'~"I
~~~-::".~;, '
.' ~""'
1l.' ...
~'1-~
;:~1l.f!
'!''<~:
'"
Fis
,;u
re
L~·O
. V
ert
ical
Str
ess-T
ine T
races
on
A
xis
o
f S
vrru
:let
rv at
Z
dia
. cir
cu
lar
foo
tin
g)
6 in
. 12
in
. (3
in
.
\D
o
~.
_ .
• t45
0~!p
!c .. ·-
; .'
I
,", I.'.'
,Jj,,
'[ ",'
i .. _
....
T
R=O
" . JE
.'. j:?~ :";."
j;,; ~
J~'"
J " \,;~/
.1
1;:
. ",.,'
.~".
"" ""
"'*'~.
:;';":
J .. :~~
,".
Fig
ure
4
1.
Vert
ical
Str
ess
-Tim
e
Tra
ce at
P
foo
tin
g)
I) in
.,
Z
') in
. (3
in
. d
ia.
cir
cu
lar
',0
t-
'
92
or-____ ~2T_----~4~-----6T-----~8~----~IO I
2
12
Figure 42. Synmetry
A .. ttenuation of Peak Vertical Stress on Axis of for a 3 in. dia. Circular Foot~.ng (oc = O.GG psi)
Fig
ure
L;
.3.
Vert
ical
Str
ess
-Tim
e
Tra
ces
on
Ax
is
of
SVII
1f.1
etry
at
Z
= 1
/2 in
.;
1-1
/2 in
. (3
in
. b
y
3 in
. sq
uare
fo
oti
ne)
'.C'
L-:
fig
ure
t:,
.L~.
Vert
ical
Str
ess
-Tim
e T
races
on
A
xis
o
f SY
lTI.m
etry
at
Z
(3
in.
by
3
in.
squ
are
fo
oti
ng
)
lin
.;
6 in
. 1
2
in.
\0
-P'-
95
O'z (psi)
o 2 4 6 8 r-~----T-------~------~------~
2
4
28=3" - "'" .. , .5 -..,6
R
8 z
10
12 <:> .. "
Figure 1~5. Attenuation of Peak Vertical Stress on Axis of S~~etry for a 3 in. by 3 i~. Souare Footin~ (oc = 0.20 psi)
96
its static contact pressure (ac = 0.20 psi). From the above
results, it appears that the magnification of the vertical
dynamic stress on the bottom of the footing is related to
the deceleration of the footing.
To compare the pattern of these stress distribution
curves, shown in Figs. 36, 42 and 45, with the theoretical
stress distribution curves under static loading conditions,
a set of curves for three footings were drawn using32 ,l,
Boussinesque's stress distribution theory, as shown in Fig.
46.
Tests were also conducted on the same soil mass inside
the vessel under identical test conditions, but without any
footing on the surface. To obtain a representative stress
value at various depths on the axis of symmetry, three to
four tests (sometimes even five tests) were conducted at
each gage position. Using these representative values, a
stress distribution curve in the soil mass under dynamic
loading conditions was drawn, as shown in Fig. 47. On the
same graph, overburden pressure at various depths is plotted
for comparison purposes. The data indicated that the stress
distribution under dynamic loading conditions almost follows
the same pattern as the overburden pressure, except for a
sudden decrease near the surface. This may be due to the
fact that the soil particles near the surface will not have
any confinement and when a transient load is transmitted from
the bottom through the soil, the soil particles near the
surface will be in a state of near liquefaction.
o
2
~ 3 . N
6 "
Q 6" dio. footing (A)
• 3" dia. footing (8)
m 3"x3" footl"9(D) (equivalent dlo.=3.39")
97
Figure 46.· Stress at Any Point from Circular Bearing Area (Boussinesque's Stress Distribution Theory)
98
rTz (psi)
O~ ____ ~~ ____ ~1.~0 ______ ~1.~5 ____ ~2~.0 ______ ~a5
2
4
-c :::;6 ...
8
10
12
\ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .\ 'j.\ ~\ ~
~ ~\ ~\ ~\
\ \
00
o 0
Fip'ure 47. Peak Vertical Stress Distribution T-Jith Denth Under Transient Loads ('I;-1ithout any footing on the surface)
99
The stress distribution shown in Fig. 47 was plotted
also, to compare with the stress distribution curves (with
a footing on the surface) shown. in Figs. 36, 42 and 45, and
thus to investigate the influence of the mass of the footing,
on the vertical stress distribution in the soil. In all
the three cases of surface footings subjected to transient
load, it was noticed that the influence of the mass of the
footing on the vertical stress distribution is negligible
at a depth of two footing diameters below the footing. This
. b" t . h 0 . , l' 49 ~s reasona Ly ~n agreemen w~t rrJe s conc us~on . Orrje
conducted both laboratory and field tests on four different
soils. The behavior of these soils was investigated by
measuring the retardation of free falling weight when it
strikes the soil surface. One of his conclusions was that
the depth to which the loading tests affected the underlying
soil corresponded to approximately 1.5 plate diameters.
Landsman37 conducted dynamic tests on Kaoline clay. Behavior
of this soil was investigated by measuring the deceleration
of a free falling weight when it strikes the soil surface. He
also used embedded stress gages at various depths in the soil
to study the stress distribution. From the stress dis
tribution relationships it was noticed that, at a depth equal
to twice the footing diameter, the stress had reduced to 10
percent of the applied value.
Tests were also conducted with a surface footing and
with stress gages embedded in the soil at various offset
distances from the axis of symmetry, to determine the stress
100
distribution along the gage locations. These stress
distributions were also investigated at various depths. A
typical set of stress-time traces for a 6 in. dia. circular
footing are shown in Figs. 48 through 50. These traces follow
ed the same pattern as the acceleration-time traces of the
footing. Attenuation of peak vertical stress versus offset
from the axis of symmetry for this footing is shown in Fig.
5l.
A similar set of stress-time traces in the case of a
3 in. dia. circular footing on the surface are shown in
Figs. 52 through 58. These traces also responded accordingly
as the acceleration-time response of the footing. Attenuation
of peak vertical stress versus offset from the axis of
symmetry for this footing is shown in Fig. 59.
Stress-time histories were also recorded at various
offset distances from the axis of symmetry and at various
depths with a 3 in. by 3 in. square footing on the surface.
A typical set of these traces are shown in Figs. 60 through
63. The data was used to obtain peak vertical stress
distribution with offset from the axis of symmetry as shown
in Fig. 64.
Comparing the stress distribution curves of the three
footings, it was noticed that the pattern of stress distribu
tion, with offset distance for a 6 in. dia. circular footing
on the surface differed from that of the 3 in. dia. circular
and 3 in. by 3 in. square footing on the surface. The
pattern of stress distribution shown in Figs. 59 and 64 for
Fig
ure
4
8.
Vert
ical
Str
ess
-Tin
e T
race
s at
6 in
. Je
pth
an
d at
R
(6
in.
dia
. cir
cu
lar
foo
tin
8)
o in
.;
6 in
. 12
in
. I-
' o I-
'
Fig
ure
4
9.
Vert
ical Stress-Ti~e
Tra
ces
at
3 in
. D
enth
an
d at ~
(6
in.
dia
. cir
cu
lar
footin~)
o in
.;
6 in
.;
12
in
. r.
l (.)
N
Fig
ure
5
0.
Vert
ical
Str
ess-T
ine ~races
at
1-1
/2
in. ne~th
and
at
R
(6
in.
dia
. cir
cu
lar
foo
tin
r.)
o in
.;
6 in
.;
12
in
. r-
' o u..
'
- fI) Q. -
R (
in)
o 3
6 9
12
I Iii
"
2L
z/r=
2 ~
=:::::==
_
1" ~
6i.
8
-0--
--0
-Z
= I. 5
in.
~
IJ:r z
= 3
.0 in
.
-E--
-0-
z =
6.0
in.
I... 2
r= 6
" ~
~R
z
Fic
ure
5
1.
Att
en
uati
on
o
f P
eak
Vert
ical
Str
ess
vs
Off
set
fro
m A
xis
0::
SY
Dnn
etry
fo
r a
6 in
. d
ia.
Cir
cu
lar
Fo
oti
ng
(a
c =
0.5
5 p
si)
I-
' .,
:)
~
Fig
ure
5
2.
Vert
ical
Str
ess
-Tim
e T
races
at
6 in
. ~enth
and
at
P (3
in
. d
ia.
cir
cu
lar
foo
tin
g)
o in
. 3
in.
6 in
. .....
o V
1
... :i~r"
;!:L
l'
r.~]~~'
, ~ .. ~':. ,-
: " ~("": +~ ( .-~"
, -! ,
'" '~'~
I i"
j:;"
Fig
ure
5
3.
Vert
ical
Str
ess
-Tin
e
Tra
ces
at
6 in
. D
epth
an
d at
R
dia
. cir
cu
lar
foo
tin
g)
9 in
.;
12
in.
(3
in.
1--' o (J'\
Fi8
ure
5
4.
Vert
ical
Str
ess
-Tim
e
Tra
ces
at
3 in
. D
en
th
an
d at ~
= 0
in
. (3
in
. d
ia.
cir
cu
lar
foo
tin
g)
6 in
. 12
in
.
r-'
o -.....J
Fie
ure
5
5.
Vert
ical
Str
ess
-Tim
e T
race
s at
1-1
/2
in.
Dep
th
and
at
R =
0 in
.;
6 in
. d
ia.
cir
cu
lar
foo
tin
g)
(3
in.
t--' o co
Of _
_ ~t N
.o.j
T"~. -,
:'I' .
.
-,
; -.
-i
t'~"';i;
-'/ -~
X' .
---+
~ ~~·r!
"'.--
.t_ -+
----
1..-
}.4"'-~
, ''-'
c;.
,.:.
,
' ,
1 "
-'"
-I '.
d~dl
iti~
t;.r~~ ..
.
i '
'I ..•.... I
~
'Ii .. '
'F".
'.
". ,
~,.
" R
=l2r
t•
z=i.1
n'
, ,
'..
'~.~"""
,..
. ~~
'"
" *."
'. ','
" ""
1i:i
:j,C
~!"'
" ".
'.' ~
.6,.1
," I'
~;'
'" ~l'"
'., ';
*~; ... ,
• ,
r":'
~"""" :~
,',Jl"':
:;::;:"'
~:';:·:'
i ,.'"
) I
) I
Sl.
"
' I
: \
ft'
I -t
I \
i ,
,!.;,),~IJ
.J.t#J.w~_
.. 1'J~;J
,\~I\>I:,,~w
..~vJ ri1
NN~NJN
~y.;~:
,lN~~~
~\_~:!
f\!\\'
\'1' ",
. "
'!!k!i~¥
,·WN~Ni.
~Y'{~~ir
l~NfH~V~
WJi~:~~
, '
. '·.
(1'1
1 •.
Fig
ure
5
6.
Verti
ca
l S
tres
s-T
ime
Tra
ce a
t ~
foo
tin
g)
12
in.
2 =
1-1
/2
in.
(3
in.
dia
. cir
cu
lar
t-'
o '.0
Fig
ure
5
7.
Vert
ical
Str
ess
-Tim
e T
race at
R =
0
in.;
Z
=
1/2
in
. (3
in
. d
ia.
cir
cu
lar
foo
tin
g)
t-'
t-'
:.::>
Fig
ure
SD
. V
ert
ical
Str
ess
-Tim
e
Tra
ces
at
1/2
in
. D
eryt
h an
d at
R =
6 in
.;
12
in.
dia
. cir
cu
lar
foo
tin
g)
(3
in.
I-'
t-'
t-'
R (I
n)
Ct
:3 6
9 12
I
I I
I
2t z/r
=4
-!-J7~
~ m
_4
Ow
.,g
. N
b
6 ~z=O.5In.
• •
z=1.
5 in
.
-G--
--8
-z =
3 .0
in.
~z=6.0in.
2 II
10
-. r=
3.,
----~--_r--~---~~R
z
Fi~ure
59
. A
tten
uati
on
o
f P
eal:
V
ert
ical
Str
ess
vs
Off
set
fro
m A
xis
o
f S~
TITl
.lTl
etry
fo
r a
3 in
. d
ia.
Cir
cu
lar
Fo
oti
ng
CO
c =
0.3
3 p
si)
t-
' t-
' tv
-'<
' •.
,-. -,
"I -, I
-f
I -
, -'l
Fig
ure
6
0.
Vert
ical
Str
ess
-Tim
e
Tra
ces
at
6 in
. D
ep
th
an
d at
R =
a i
n.
C3
in.
by
3 in
. sq
uare
fo
oti
ng
) 6
in.
12
in.
I--'
I-
-'
W
Fig
ure
6
1.
Vert
ical
Str
ess
-Tim
e T
race
s at
3 in
. D
enth
an
d at
R =
a i
n.,
6
in.;
12
in
. (3
in
. b
y
3 in
. sq
uare
fo
oti
ng
) r-
' r-
' .p
'
r _,'.~~
::~C~~
~; ,,~
:?,~,~
ry~)~
,-,..
;.,"t1
.,"1l
· .. ;
' .. ~ 1'''
'"1''
Ji]~f~if~~· •
..••. :;r
L~J
.j "".'
I,'"
i '
:::~
t"N !
it
.q~ .•
'-.',
"'[
j.: \"
1#\ • \
i~)\''
'' ,
.1'~1'noJ"i~~ i
':cl~;
4~ :~~
--,--' "
" I"
. '''
'-/t
c+"-
,, I
f'''
" '
, ,
" I '.
~ :, 1
'1' ;
" . ,'
, j
I ' -'I
!.
j; ....
I
R=O
" T
Z=tt'
,
. ~
.r-!
-rl
t,
i
:j
j;
''''i\'''-
.:.'+.
,-' ,
'".= r7 '
i_,~
' ,c
l#'
! 2
.68
psi/
in
!:~:d~~*"~,~",,f'"
~"1' vr'
1 \
El
1lF"
'J'
I ~
. _.-11
. j
'T~': !~
' t
¥\<!"
'St'
. 3.
56 p
si/i
n:
Jt.~>'f!
f-,;t
.. ~, ' '
~_~;
: ,~frijJili~J;'i~""F~1\
Fig
ure
6
2.
Vert
ical
Str
ess
-Tim
e T
races
at
1-1
/2
in.
Den
th
an
d at
!" (3
in
. b
y
3 in
. sq
uare
footin~)
o in
. 6
in.
12
in
. I-
-'
I--'
V
l
i,
\".
," "
~"
". -
. ..
;...
.., .
....
. ~\ljd
; •. -
-. .L
:$:3
.t
.\~
' .. ">
_'_ .
. ....
' "
';.~ ,.
,', ',,'
, •.•
• '~.~,
'" :...,."" ..•. ' •
. ".' •.
... " .. ~ ... ' •
•. '.,', "'
.'~.
~~ d~;
;'"-"'.
, :
1 ...,
.• -
n'
·'-c·
n"
•• .',.
• ,-If
•
~, ".,
... n
n'
.l, ..
'~:I~';¥".
_ .. ",_·"
,-....
·')'~.'{"
"".'
""
.. '
r,
~
·1
1 ,
' R
=6n ,:z
:=t"
f~:T
':'-
--'
St .•
2
.. ,71i
.·:PS
I/.in
\ r ~~.:;
, .
I ,"y
..
"-1'"
. ' .';
'. '~'"
.'1~"·
: '.
I '
: . ..
. I"
, !
I
',"i
~fil
if.J
.tfY
W· ;
:"'il-~'f;l/(II'IfJ ,
::;.'JIJ#I~#lJI##WPI,*,~',~lf>!.iJI,t#i#tv.\!~M##.M,ii.fi'?;JJW;!iff'(~' ",
,,\,.1
',,, '
. (''',
''ti", .
. '''')'
','
",'
'"
..
" ".. .j",
.>."
.,.:
;,--
:,.
~:4~~ii."
~;'<
. ' ..
"-
.' _
::/tf
-' ,,..
I
Fig
ure
6
3.
Vert
ical
Str
ess-T
ine
Tra
ces
at
1/2
in
. D
en
th
an
d at
R
(3
in.
by
3
in.
squ
are
fo
oti
ng
) -
'
o in
. 6
in.
12
in
. ~,..J
r-'
ry.
Fi~ure
64
. b
y
3 in
.
RU
n)
o 3
6 ¥
If
I I
-u; 0. - ... 4
b
6 8
• •
z=O
.5 in
. -0
----
Q.-
z = I.
5 in
. -I
ll
IJ:>.
z = 3
. 0 in
. -Q
---.
.-.G
. z =
6. 0
in.
~ 28
=3" .,
z
/d:)
·R
Att
en
uati
on
o
f P
eak
Vert
ical
Str
ess
v
s O
ffse
t fr
om
Ax
is
of
Svv
.met
ry fo
r a
3 in
. S
qu
are
Fo
oti
ng
(a
c =
0
.20
~s
i)
. .
I-'
I-'
---J
118
a 3 in. dia. circular and 3 in. by 3 in. square footings
respectively, agreed well except for their stress intensity
values on the axis of symnletry. The data shown in Figs. 51,
59 and 64 indicated that the size of the footing affects the
stress distribution in the soil. Attenuation was very rapid
in a distance equal to the diameter of the footing. At a
distance equal to 1.5 times the diameter of the footing from
the axis of symmetry, the effect of footing on the vertical
stress distribution is negligible.
Strain Measurements. Strain-time histories at various
gage locations were recorded by using the embedded strain
gages. Location of these strain measurements is shown in Fig.
20. Details of these strain gages, the calibration procedures
and the method of placement in the soil are described in
Chapter IV. Typical visicorder traces of the output from the
embedded strain gages at various positions on the axis of
symmetry with a 6 in. dia. circular footing on the surface
are shown in Figs. 65 and 66. As shown in these figures,
strain-time trace consisted of positive (trace above zero
line) or compressive strain, and negative (trace below zero
line) or tensile strain. These strain-time records are
consistent with the acceleration-time and stress-time records.
Initial compressional strain is followed by a tensile strain,
which occurs in about 200 msec, followed by a compressional
strain. Data from these strain-time records was used to
plot vertical strain distribution with depth on the axis of
symmetry for a 6 in. dia. circular footing on the surface,
Fig
ure
6
5.
'
,.",~,
;~~&?"
;.¥i4~
",,,,;
¥.i~
. . ,~. \
,,~:
"
. L.
. . ~.
'
,!:"
.
; Z
=2"
-' ,;
,.v~
Str
ain
-Tim
e T
race
a
t :!1
=
o in
. '7
<~
".,- = 2
in
.
T.'
.,;~.
\:": \ (6
l. in
.
Tes
t N
o.
-'\
1
, TA
-18
V""
.. ' ."".'
2'
, . 'J
t-i
dia
. cir
cu
lar
foo
tin
g)
~ ~
Fig
ure
6
6.
cir
cu
lar
-
Str
ain
-Tin
e T
races
on
A
xis
o
f SY~netry
at
Z
= 3
fo
oti
ng
) .
-in
.
TA
-13
!~-in
.;
(6
in.
dia
.
t-'
N
.:::>
121
and is shown in Fig. 67. Vertical strain distribution versus
offset from the axis of symmetry for this footing was also
drawn at various depths as shown in Fig. 68.
Stress-Strain Relations. Data was obtained by the
stress and strain gages embedded at the same depth and same
offset distance from the axis of the footing, but on opposite
sides of the footing. Typical stress-strain and modulus-
strain relationships constructed at two locations are shown
in Figs. 69 and 70. These relationships corresponded to the
peak values of the stress-time and strain-time traces which
generally occured in about 250 msec. It should be noted that
these curves represent a one-dimensional stress-strain
relationship. General characteristics of stress-time traces
indicate that the time in which peak vertical stress occurs
is very small, and in these experiments was found to be about
1 msec to 1.5 msec. During this period the stresses increased
very rapidly from zero to their maximum values. Strains were
relatively small with little yielding of the soil and there
was no evidence of failure zones and volume contraction.
Based on these observations, the soil behavior during this
phase may be approximated as one-dimensional. The constrained
modulus of the soil at various strain levels and at various
locations can be computed by constructing similar graphs,as
shown in Figs. 69 and 70. Methods for constructing these
stress-strain relationships is well described by Landsman in
h · k37 ~s wor .
122
o 0.5 1.5 2.0
2
-c:
N 3"
4
5
6
Fip."ure 67. Vertical Strain Distribution :'Jith ~enth on Axis ~f Symnetry for a 6 in. dia. Circular Footing -
-0
~ - N
W
R (
in)
o 3
6 9
12
i I
I I
I
0.8
1.2
1.6
Fig
ure
6
8.
Vert
ical
Str
ain
Dis
trib
uti
on
vs
Off
set
from
Ax
is
of
Sym
met
ry
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
,....
... N
W
~.O
- 2.0 -iii ~ -b
N
1.0
o
1500
_1000 Ci· ~ -o
W
I 1 I
: ",/" I
1/ ;:'"
/:/ I //1 114/ 1 1/ l' 1 // //1 E3 1 1
II' I ~~~ II " I ...... 1
/1' /" I 1 .,.,~... I I'~I'/ I ~ ....... r
II' / • .,.,-, I II ",,,I' ...... ' I 1 I
~/ ",,,I' _"'~"" I I I 1-------r4' / ~ ... "" I I ~------- I I
124
,,'" ... ~... J-----'-----. I I ...... _---- • I. -------------T---- . ~ _---.:-...::. _____ t-____ -------------
0.1 0.2 0.3 0.4 .,
--~------------. --E2---~-------1 ---, 1 I I I
E ,I ·--..!:3.----------f----i-----
I I , I I 1 I I J 1 1 1 J 1 I
. ____ ~----------1----J _____ L ________ . 1 • I 1 J I I I I I 1
E I I I I --~--'E , _____ l. ____ l ____ J ________ L___________ .
___ ------H-----~-----l-----l-------- I -------------J-----
Figure 69 .. Modulus vs Strain Re1ationshin at Z/r = 1/2; ~/r = 2 (Test No. TA-6l)
3.0
- 2.0 "iii ~ -b
N
. 1.0
0.0
600
~ 400 (I) ~ -
CJ I&J
200
o
125
0.2 0.4 0.6 0.8 Ej E2~3 E4
EZ (%)
-~~-------- ----_. -~~------- ----_.
Figure 70. Modulusvs Strain Relationship at Z/r = 1; R/r = 1 (Test No. TA-lll)
126
Depth of Burial
Stress-time measurements were made at various depths
beneath the footing on the axis of symmetry, with the footing
buried at various depths. Measurements were also made with
a surcharge equivalent to the height of the footing. Tests
were conducted with a 6 in. dia. circular footing (a c
0.55 psi); and, other size footings were not included in
this investigation. Typical stress-time traces on the axis
of symmetry at various depths beneath the footing with a 2 in.
thick surcharge around it are shown in Fig. 71. A similar
set of stress-time traces for a surface footing are shown in
Fig. 72. The traces, when the footing was buried at 2 in.
depth, are shown in Fig. 73. Similarly Fig. 74 shows stress
time traces at various depths beneath the footing buried 4 in.
from the surface. In the case of a footing buried at 6 in.
depth, stress-time measurements were made at only two
locations on the axis of symmetry as shown by the two traces
in Fig. 75. General behavior of all these stress-time traces
were similar to that of the acceleration-time traces of the
footing. The data from these traces, shown in Figs. 71
through 75, was used to draw vertical stress distribution with
depth on the axis of symmetry beneath the footing at various
depth of burial. This relationship is shown in Fig. 76.
From the data it was noticed that the vertical stress
intensities in the vicinity of the footing increase as the
depth of burial (depth of embedment) increases (as the
distance from the transient load source decreases).
Fig
ure
7
1.
Vert
ical Stress-Ti~e
Tra
ces
on
A
xis
o
f Sv~etry at
Vari
ou
s D
ep
ths
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
W
ith
2
in.
Su
rch
are
e
I-'
N
'-J
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'7 -
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t 2'
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St.
J ."
56 p
si/i
n
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<'
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mse
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, ' ,'
I ,,,.,,,,,,~~·.,,,,,,,,,~,,,,,,,,,,,,,,,..,,,,,,,~,,,,,,~,~I./'~7<Ifi,~i>.,:,,,,,,,,~,,.",",,'"
, t..
.'
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Fir
;ure
7
2.
Vert
ical
Str
ess-T
ine
Tra
ces
on
A
xis
o
f S
ym
netr
y at
Vari
ou
s D
en
ths
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
o
n
the
Su
rface o
f a
So
il
Med
ia
I-'
N
C8
'"
~:.7t1'!f""e;--,~---- i"
II.
""
\ "
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e", 'T~'"
._;.:
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~~
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,>,: .~ ...
~j'
;r.
;, II Ii T
est
No
. TA
-86
, D
= 2
" f
I
~.
a 2
. I",
• "
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JI
I' if
', ,,;
.lttJlo.
7~::iIIV
: "'Y
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,
St.
2
.68
psi
in
,1
I
~t~
~ R
=O"
2=1.
. ..
t-,,'
2 1"
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.. 1
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,;U
R=
0"
Z=
4"
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,. ',i
ii',;
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'{"iff:(
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f.~'~
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' ,'~.
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R-O
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=1
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1 50
mse
c -,',~:~*, ..
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t.
.3. 5
6 p
si/
in
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T<'
ip;u
re
73
. V
ert
ical
Str
ess
-TiT
ne
Tra
ces
on
A
xis
o
f SV
T'JI
'1et
rv
at
Vari
ou
s D
ep
ths
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
n8
B
uri
ed
at
2 in
. D
en
th
f-'
N
'-D
FiG
ure
d
ia.
f: ,
,'I ~r:; , . ~ \ t \
~. \'
~.
.r-!
..-I
Te
st
No.
TA
..,91
...
D
==
4,"
. ,.
'£
, ,I 'I i
.. ~ 1:1
i ;
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:....-..
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"111
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f!\ \!#;
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't ii'''!
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II \..
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t.
2. 7
4 p
si li
n
R=O
" Z
=4"
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. q'
,;/""M
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St •
. 3.
56 p
si/
in
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.... :;
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" :'?~..,~~~,,,<
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.. f.
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-I
~ '~r~'
_-.!
~
, 50
mse
c I
I"
,.
71:..
V
ert
ical
Str
ess
-Tim
e
Tra
ces
on
A
xis
o
f S
yn
Iaetr
y at
Vari
ou
s D
ep
ths
for
a 6
in.
Cir
cu
lar
Fo
oti
ng
B
uri
ed
at
& i
n.
Den
th
t-'
W
. .,:)
~~.
'
l~
, ., t~j
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. 'I
.~
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J;.s..
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'llui
t"1t
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>
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j' ,=
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1,<
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;O"
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i,,-
' ....
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III
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ll \~\'~:'i\
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+
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.44
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3.5
6 p
Sl
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=O"
; Z=
6"
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eC
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,·", .. _~
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Fig
ure
7
5.
Vert
ical
Str
ess
-Tim
e
Tra
ces
on
A
xis
o
f S
yn
Fle
try
at
Vari
ou
s D
ep
ths
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
B
uri
ed
at
6 in
. D
en
th
r'
W
t-'
o 2.0
2.0
4.0
-c ::; 6.0 N
8.0 --------
10.0
12.0
IT z (psI) 4J.0 E). 0 8.0
I . 0
,---------------I I
,@ I I
~ Footing with 2" surcharge
-A- Surface footlno
~ Footing buried 2" deep
~@&- Footing burled 4' deep
A- Footing buried 61 deep
132
Figure 7'6. Attenuation of Peak Vertical Stress ~7ith Denth on Axis of Synmetry for a 6 in. dia. Circular Footing' at Different Depths of Burial
133
Acceleration-time traces for the footing were also
recorded at the same time as the stress-time traces were
recorded, in the tests described in the preceding paragraph.
Typical acceleration-time traces for this footing at various
depths of burial (embedment) are shown in Figs. 77 through
80. The data from these figures was combined to obtain a
relationship between the deceleration of the footing and
depth of burial (embedment). This relationship is shown in
Fig. 81. As shown in Fig. 81, deceleration amplitude of the
footing decreased linearly with depth of burial (embedment)
of the footing, thus indicating greater damping. This is
in agreement with other investigations. Whitman and Richart79
have included some references on tests of footings partially
embedded as well as for footings at the surface. They stated
that the partial embedment reduces the magnitude of motions
at the resonant peaks.
The relationship shown in Fig. 81 is also justified by
the fact that embedment increases the stiffness of the
system. The increase in stiffness comes from three sources46 :
(1) resistance to deformation due to the embedment; (2) side
friction between the soil and the foundation; and, (3)
increased soil stiffness due to the confinement (overburden).
Lysmer and Kuhlemeyer43 used a finite element technique
to study the effect of embedment of a circular footing on
the dynamic soil response. Their results indicated that the
effect of embedment is to increase the resonant frequency
and to decrease the dynamic amplification at resonance.
I?:
;\ I.
!
~
I'
i
• ,.
,;r
"if
" .
.':.;
'.
. N
..
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! \
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It
·· .... >~
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(\
:TJt~~· r
it~·
>i>;
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l .. ,.
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t"''
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Fi~u
re
77
. A
ccele
rati
on
-Tim
e
Tra
ce
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
o
n
the
Su
rface
of
a S
oil
H
ed
ia in
th
e
Vess
el
t-'
w
-:>
0-'
"
(1)
(1)
"'"
0
:i
<c:
~~
III
Fig
ure
7
f3.
Den
th
'i~_4';'
:"i.;
'.'''"\
-y
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-7~ ~,,
*:'
......... -~,:~-4~ .. -
"-r
' :.
-.. ""
'--, ..
........
-~--.;.
:'"
:;
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':
~-''
7 •
.... '.
ll···':·;~··
..
t -"
--.~
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Acc
eler
ati
on
-Tim
e T
race
fo
r a
6 in
. d
ia.
Cir
cula
r F
oo
tin
g
Bu
ried
a
t 2
in.
r-'
u,.)
lJI
Fig
ure
7
9.
Dep
th
,;."
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rati
on
-Tim
e
Tra
ce
for
a 6
in.
dia
. C
ircu
lar ~ootin~
Bu
ried
at
4 in
.
r"
W
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;ure
8
0.
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rati
on
-Tim
e ~r
ace
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
B
uri
ed
at
6 in
. D
ep
th
/-'
W
---i
o
2
-c: --~ 3
o
4
6
138
a (g)
2 6
Figure 81. Deceleration of Footinf, vs :f<'ootinp; :1eryth fro!"} the Surface of the Soil Hedia for a 6 in. dia. Circular Footing
139
Valliappan, Favaloro and White70 considered a rigid
strip footing resting on an isotropic and homogeneous soil
mass vertically excited by a harmonic force in the axis of
symmetry. They used a finite element mesh for this model.
They concluded that, for increasing embedment of footing,
there is a reduction in the vibration amplitude because of
the apparent increase in soil stiffness, and not the damping
characteristics of the system.
Footing Isolation
In the past, various isolation techniques have been
used for the earthquake problem to reduce the effect of
transient loadings on the structure. Review of related
studies on this subject is presented in Chapter II. A
passive isolation technique was used in this investigation.
For this purpose four sponge materials and a rubber material
having different compressible characteristics were used.
These compressible materials were of the same size as the
footing (6 in. dia. circular) and their thicknesses varied
from 1/4 in. to 1/2 in. Static compression tests were
conducted on these materials, and stress versus deformation
relationships were obtained, as shown in Figs. 82 and 83. The
same relationships under loading and unloading conditions are
shown in Figs. 114 through 118 (see Appendix C). Using these
relationships, subgrade modulus (Kc) of each compressible
material at static contact stress (ac = 0.55 psi) level was
computed. For example:
Stress (psI)
o 0.5 1.0 1.5 e---------r-------~~------~
--b 0.2 )( .. c o -o
~ 0.3 -4D C
0.4
0.5
Figure 82. Stress vs Deformation for COITmress::i_ble ~1aterials l1 1<1' M . l' "2' - ~3
-•• ~ --'0 )(
c o .: o
Stress (psi) 0~ ____ ~Q~5 ______ ~I.~O ______ ~15
1.0
2.0
3.0
! 4.0 .2 CD o
5.0
6.0
7.0
141
Figure 83. Stress vs Defort:l.ation for Comnressib1e !~ateri?.ls 114 , MS
142
KC = [static contact stress/deformation corresponding
to the static contact stress], (psi/in).
Acceleration-time response of the footing was recorded
with a footing (6 in. dia. circular) on the surface, but
without any compressible material buried in the soil. The
acceleration-time trace corresponding to this is shown in
Fig. 84, and, is used as a reference for the other tests
conducted under identical test conditions, but with a
compressible material buried at one radius depth. A series
of dynamic tests were conducted with a compressible material
buried beneath the footing at a depth of one radius. A
compressible material Ml placed in position at this depth is
shown in Fig. 85. Acceleration-time traces corresponding
to the five compressible materials are shown in Figs. 86
through 90. Percentage reduction in deceleration (Ra) of the
footing was computed for each case. For example:
R = 100 [deceleration of the footing without a
any compressible material buried in
the soil - deceleration of the footing
with a compressible material buried in
the soil]/[deceleration of the footing
without any compressible material buried
in the soil].
These Ra values were plotted against the Kc values and
a relationship was obtained as shown in Fig. 91. These
values were again used to draw Ra versus log Kc plot and a
straight line relationship was obtained, as shown in Fig. 92.
~ .',
,I Ii ,
1 \
(I
! 1I;
I"
( 11
I
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. C
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. ' ....
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'=
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III!
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X_" M
lb,.!
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t >.
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Fi~ure
84
. A
ccele
rati
on
-Tim
e
Tra
ce
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
on
th
e
Su
rface
Hit
ho
ut
Any
C
om
pre
ssib
le '
l<-1
ater
ial
in
the
So
il ~
~edia
r'
.j:>
v,
.)
144
Figure 85. A Compressible Material Ml Placed in Position at a Depth Zir = 1 Before Being Buried in the Sand
;;'
. ~:;'~}
. '7&
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,-,
,. ..
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f;-)
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Fig
ure
8
6.
Accele
rati
on
-Tim
e
Tra
ce
for
a 6
in.
dia
. C
ircu
lar
i"o
oti
ng
H
ith
a
Co
mp
ress
ible
M
ate
rial
Ml
Bu
ried
at
R/r
=
0;
Z/r
=
1
l-~
.f-'
lr
Fig
ure
3
7.
Accele
rati
on
-Tin
e T
race
fo
r a
6 in
. d
ia.
Cir
cu
lar
Fo
oti
ng
Uit
h
a C
om
pre
ssib
le
Nate
rial1
12
Bu
ried
at
R/r
=
0;
Zir
=
1 I-
' .p
"-
0-
e.,"
.•. "
.. r.,\"
1 .. 1
;
lit.·
_, ..
' .• ,
r.
, ~
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-~.~
: ~ i \
f' \ \ \
,~
!"ig
ure
3
8.
Acc
eler
ati
on
-Tim
e T
race
fo
r
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
Hit
h
a C
om
pre
ssib
le
Ha
teria
l ~1
3 B
uri
ed a
t fl
/r
= 0;
Z
ir
= 1
r'
.~
'-.J
::0':"
"..1- .,'1
",
tv
!I
:":~! ..
....
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'. lb
. ~;U.
'$l1lJ_.
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-"
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'
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Fig
ure
8
9.
Accele
rati
on
-Tim
e
Tra
ce
for
a 6
in.
dia
. C
ircu
lar
Fo
oti
ng
\v
i th
a
Co
mp
ress
ible
~1aterial
H[
Bu
ried
at
R/r
=
0
; 7.
./r
=
1 }
t-'
.P'
C)
-::'
~.
'.
1>'"
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~ ~
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•
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r·· I"
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Fir
,ure
9
0.
Accele
rati
on
-Lim
e
Tra
ce fo
r a
6 in
. d
ia.
Cir
cu
lar
Fo
oti
ng
\A
Tith
a
Co
mp
ress
ible
}~terial
MS
Bu
ried
at
R/r
=
0;
Z/r
=
1
t-'
.~
1..0
'0
~ - o
sow
!-is
60
cr::
40'.
.20 o
10
Ro=
7S
.4-4
2.7
1o
gk
c
01-1
2
HI
20
3
0
40
5
0
·60
k ca
t CT
C st
ress
lev
el (
ps
i/In
)
Fig
ure
9
1.
Decele
rati
on
Red
ucti
on
v
s S
ub
gra
de ~1odulus
of
CO
P1
pre
ssib
le M
ate
rial
r-'
\Jt
o
80
60
RQ
= 7
8.4
-42
.710
9 kc
- ~ - D
0::
40
.eM
2
20
o_,~~ __
____
____
____
~ __
__ ~ __
~~~ __
~~~ __
____
__ ~ __
__ ~ __
~ __
__ ~
.1 5
10
50
70
k c
at C
7'cs
tr •• 'e
ve'
(psi
/in)
Fig
ure
9
2.
Decele
rati
on
"!:
ledu
ctio
n v
s L
og
o
f S
ub
p;r
ade
Ho
du
lus
of
Co
mp
ress
ible
tfa
teri
al
t--'
lJ
1 t-
-'
152
This straight line relationship is also represented by the
equation Ra = 78.4 - 42.7 log Kc. In a generalized form this
can be expressed as Ra = Al - A2 log Kc' where Al and A2 are
constants.
As shown in Figs. 91 and 92 there is a slight deviation
of a value corresponding to the compressible material de
signated as M2 . This may be due to the fact that the size
of voids in this sponge material M2 were slightly bigger than
the size of the sand particles. Therefore, when a transient
load was applied, a stress wave was generated and travelled
in a direction perpendicular to the plane of the compressible
material buried in the sand. Some sand particles were
penetrated into the voids of the sponge material. Thus the
actual compressible characteristics of this material were
not fully mobilized.
Stress-time trace of a stress gage embedded at R/r = 0;
Zir = 1/2 and without any compressible material buried in
the soil is shown in Fig. 93. Stress-time measurements were
also made with the stress gage embedded in the soil at mid
depth between the bottom of the footing and the compressible
material (i.e., at R/r = 0; Z/r = 1/2). Stress-time traces
were recorded each time a test was run with a compressible
material buried in the soil. These stress-time traces are
shown in Figs. 94 through 98. The stress-time histories
indicated that there is a considerable amount of peak
vertical stress reduction by the presence of a compressible
material buried in the soil, which damped out some of the
Fig
ure
9
3.
Str
ess
-Tim
e T
race at
R
on
th
e
su
rface)
o in
.;
Z
1-1
/2
in
(6
in.
dia
. cir
cu
lar
foo
tin
g
t-....
Ln
W
Fig
ure
9L
.l.
Str
ess
-Tim
e T
race at
?../
r =
0
; Z
/r
=
1/2
;J
ith
a
Co
mp
ress
ible
H
ate
rial
111
Bu
ried
at
R/r
=
0
; Z
/r
=
1 (6
in
. d
ia.
cir
cu
lar
foo
tin
g
on
fue
su
rface)
~,
l.f1
-~
.' ..• f:
.:::¥>
:-:~..
.;....
.-
"~ ~-
--7~
::~!
jt:1
~'~~
={~-
,.T
-.. "---~ ..
--
;, -l--~
~-; ,L
"
'~," '"
~M!'t
,J1
, .~
",,,
, ';
, '~
" :'
fIl'
'--"
";-~
~r '~-
A
' '-~r
-,
".Ji
:~,-
':,j
~:.:;f
,-------,
, .. --~,,-
/:'- .
,-~
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y--
::--
-.--
' ~~.~,-
-"--,.
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j '" '
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_ "i,L
.~'W~i i_
.t'iL*_-
.. : . .J:!~ . .J.
i:~~~L...~L~"'w'!#I~~,J~
Fig
ure
9
5.
Str
ess
-Tim
e
Tra
ce at
'R./
r =
0;
Z/r
=
1/2
~Ji
t11
a C
om
pre
ssib
le H
ate
rial
H2
Bu
ried
at
R/r
=
0
; Z
/r
=
1 (6
in
. d
ia.
cir
cu
lar
foo
tin
g
on
th
e
su
rface)
-
"
t-'
U1
U1
FiR
ure
9
6.
Str
ess
-Tim
e T
race at
:l/r
=
0;
7./
r =
1/2
T
-Jit
h a
CO
I!1!
:,re
ssib
le ~taterial
113
Bu
ried
at
R/r
=
0;
Z/r
~ 1
(6
in
. d
ia.
cir
cu
lar
foo
tin
g
on
th
e su
rface)
t-'
lJ1
0'0
'i.
:;,
-~---fi\
. i ,
!
r-
lL'
t 1
, l·f
___ ..
~ .. ,-+
~\
, I
~'
.. ~!'4
~.
+. ",
. f ..
, ..
f~-"
~' j-
' iT
c~~~
~l~
I
.~. j
A
-~
--.........~
'" ,.
"
.~f.
tI
~
::-/
~:'-
:' '
".~"
. : ...
. ln
l., -
.t~:.r
r.;...
1..!."
.1
01..
fi2. £
r.-
2 ;;., ...
; .-~
.<fi
~~:.rt
~~·'·'
:
--.. ~:
.:.
:..:
....
.---
-:-~
--.-
j:'~::::J1
'. I
J' .
I ~
.....•.
... ~
..
tiM
!I\"
T.'"
:: III
~~ "..
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_.,\""
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1M' ~'N"~""""~' .
. ~_,..
. H
J
UA
, ; 4
!?
Q
..."
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. ~
.. n
hP
J.
' ..
'l'n
,tw
",
.,..
". T
V".
. .
Fig
ure
9
7.
Str
ess
-Tim
e
Tra
ce at
R/r
=
0
; Z
/r
=
1/2
H
ith
a
Co
mp
ress
ible
'1
ate
rial
HA
B
uri
ed
at ~/r
= 0
; Z
/r
= 1
(6
in
. d
ia.
cir
cu
lar
foo
tin
g
on
th
e
su
rface)
. t-
-' V
I -J
'--'-
""..f
'.-'-*
<_.
C
':,: "l'.
... _
' __ '. > .~
;.
.§f
.
,.
' .. 'f.
·' .
..... -
.-. .
...
..•.
';~;
-/.'
-.;"
'v.!
.
. '-
;--;
~-+.
-...
. '.
'.'
.', !{LI
i •• ;; __
_ ~.\;
. ... J,! .
. -,~L~.£.Jf"i_.;...J.:"f.,~iii.~~;~
Fig
ure
9
0.
Str
ess
-Tim
e
Tra
ce at
R/r
=
0
; Z
/r
=
1/2
TJi
th
a C
OT
'lp
ress
ible
H
ate
rial
J;~5
Bu
ried
at ~/r
= 0
; 7
./r
= 1
(6
in
. d
ia.
cir
cu
lar
foo
tin
g
on
th
e
su
rface)
f-'
In
00
159
stress waves. Peak vertical stress values from Figs. 94
through 98 were compared with that of Fig. 93, and percentage
reduction of the peak vertical stress (Rs) at R/r = 0;
Zlr = 1/2 corresponding to each compressible material were
computed. For example:
R = 100 {peak vertical stress without any compressible s material buried in the soil - peak vertical
stress with a compressible material buried
in the soil]/[peak vertical stress without
any compressible material buried in the
soilJ .
These Rs values were plotted against the Kc values and a
relationship shown in Fig. 99 was obtained. These Rs values
were again plotted against log Kc values and a straight line
relationship (similar to that shown in Fig. 92) was obtained,
and is shown in Fig. 100. This straight line relationship
is represented by the equation Rs = 89.3 - 30.7 log Kc'
This equation is also expressed in a generalized form as
Rs = A3 - A4 log Kc ' where A3 and A4 are constants. The
relationship as derived above may prove to be very useful
in the design of earthquake resistant structures.
A series of static load-displacement tests were also
conducted for a 6 in. dia. surface footing with and without
a compressible material buried at a depth equal to the radius
of the footing. These tests were conducted under identical
test conditions in the same vessel as the dynamic tests. A
schematic diagram of the loading apparatus is shown in Fig. 6,
if 8
0
-.. o II ~ c\i 6
0
:::::
II .::::
N -o a:,fh
40
@M
--
-2
l--1
@
~l
I I
I I
I «
20
10
2
0
30
5
0
60
4
0
k c
at
CTc
stre
ss l
evel
, (p
si li
n )
Fig
ure
99
. P
eak
Vert
ical
Str
ess
R
edu
ctio
n v
s S
ub
gra
de
Ho
du
lus
of
Co
mp
ress
ible
Uate
rial
I-'
0\
o
100 .. ·~80 .. o II
~
0:: ."
~60
II
~
fool -a rn
0:
: 4
0
Rs=
89
3-3
0.7
Iog
kc
20L.
~~WL _
__
_ ~ _
_ ~~~~~~ _
__
_ ~ _
_ ~~~~
5 10
5
0
70
kc
o
tCT c
s
tre
ss
le
ve
l, (
psi
/in
)
Fig
ure
1
00
. P
eal:
V
ert
ical
Str
ess
Red
ucti
on
vs
Lo
g o
f S
ub
gra
de
~todu1us
of
Co
mp
ress
ible
H
ate
ria1
I-
' 0
'\
I-'
162
and, a view of this apparatus is shown in Fig. 101. As
shown in these figures, load was applied to the soil through
the footing at a constant strain rate through a gear box
arrangement. The footing pressure and penetration measure-
ments were made using a proving ring dial gage, to measure
pressure, and a deformation dial gage, to measure penetration
(settlement) of the footing.
Ultimate Load Criterion. Several ultimate load criteria
were evaluated for use in this study. The first versatile
ultimate load criterion, recommended by Vesic72 ,73, defines
the ultimate load as the point at which the slope of the
load-displacement curve first reaches zero or a steady,
minimum value. Another (De Beer's) criterion14 defines the
ultimate load at the point of break of the load-settlement
curve in a log-log plot. Both criteria require that the
loading test be carried to very large displacements. There-
fore, from the practical point of view, another criterion
must be selected based on critical settlement (allowable
settlement). One criterion used in practice73 , where a
peak load cannot be established, is to adopt a limit of
critical settlement as 5% to 15% of the footing width.
Critical settlement (allowable settlement) criterion was
adopted in this study. Considering 0.5 in. settlement of
the load area (1 ft. square) under a given load q per unit
area, the allowable settlement at the same load per unit area
of a 6 in. dia. circular footing (using its equivalent width)
was computed using the formula suggested by Terzaghi and
163
Figure 101. A Vie,,, of the Apparatus for Statically Loaded Footing
164
Peck66 . Thus a critical settlement of 0.18 in. was obtained
for a 6 in. dia. footing.
A footing pressure versus settlement relationship, for a
6 in. dia. circular footing on the surface (without any com
pressible material buried in the sand) is shown in Fig. 102.
These values, footing pressure and settlement were again
plotted on a log-log plot as shown in Fig. 103. Using De
Beer's14 method, ultimate load was obtained from this log-log
plot. Surprisingly, this ultimate load corresponded to a
critical settlement of 0.18 in. (3% of the footing diameter).
Factor of Safety (F.S.). Factor of safety against
bearing capacity failure, based on allowable settlement, was
computed for a 6 in. diameter surface footing, with and with
out any compressible material buried 'in the sand. For this
purpose two levels of allowable settlement were considered.
Therefore, factor of safety values based on 0.18 in. and
0.10 in. allowable settlements were computed from the footing
pressure versus settlement relationships. Footing pressure
versus settlement relationships for this footing, with a
different compressible material buried each time in the soil
at Zlr = 1, are shown in Figs. 104 through 108. These re-
lationships were used to compute the factor of safety. Per
centage reduction, in the factor of safety values (RF) with
respect to the factor of safety computed from Fig. 102, was
calculated. For example, for material Ml (F.S. values based
on 0.18 in. settlement);
RF = 100 [F.S. from Fig. 102 - F.S. from Fig. 104]1
[F.S. from Fig. 102]
o 1.
0 2.
0
0.1
C 0
.2
- j - - ~ 0.3
0.4
0.5
Foot
ing
pres
sure
(p
si)
3.0
4.0
5.0
6.0
Q !
6"
dio
. fo
otl
nO
--...
. T
-c:' ~:
.i
;:
.. cu
III'
Fig
ure
10
2.
Fo
oti
ng
Pre
ssu
re v
s S
ett
1en
en
t (u
nd
er sta
tic co
nd
itio
ns)
7.0
..,
I-' ~
VI
0.1
0.0
7'
0.10
0.2
0.2
0
.4
0.6
1.
0 Q
/AyD
(0
/.)
2.0
4.0
6.0
10
20
40
6
0
100
qu
It
3.30
psi
F.S
. 3.
30/0
.55
= 5.
99
~O.4
S
ettl
emen
t:
3% o
f th
e d
iam
eter
c 'U) 0
.6
1.0
2.0
4.0
6.0
J J I I I I I ~l
-_
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
__
.!ia.
.failu
re
Fig
ure
1
03
. U
ltim
ate
Loa
d C
rite
rio
n B
ased
on
Log
L
oad-
Log
S
ett
leM
en
t P
lot
(De
Beer'
s M
etho
d)
I-' '" '"
o
0.1
~ 0
.2
- -c ~ GJ =
Q)
0.3
U
)
0.4
0.5
Foot
InQ
pre
ss
".
(psi
) 1.
0 2.
0 3.
0 4
.0
5.0
6.
0 7.
0 ...,
Fig
ure
lO
LL
Fo
oti
ng
P
ressu
re v
s S
ett
lem
en
t TT
5_th
a
Com~ressible n
ate
rial
~'fl
Bu
ried
at
R/r
= 0
; Z
/r =
1
l-' '" '-I
o 0.1
:s 0
.2
- - i e .!? - - eX 0
.3
0.4
0.5
2.0
Fo
otin
g pr
essu
re
( psi
)
3.0
4.0
6.
0 -,
Fig
ure
1
05
. F
oo
tin
g P
ress
ure
v
s S
ett
lem
en
t T
:Jit
h a
Co
mp
ress
ible
Mate
rial
H2
Bu
ried
at
TJ../
r =
0;
Z/r
= 1
I-
' 0
'\
00
Foo1
ing
pres
sure
(ps
i) o
1.0
2.0
3.0
4.0
5.
0 6
.0
7.0
L:
i i
•
0.1
";;
0.2
.- - -c Q
) E
;!! -~ 0.
3
en 0
.4
0.5
F
ieu
re
10
6.
Fo
oti
ng
P
ress
ure
vs
Sett
lem
en
t tn
th
a C
om
pre
ssib
le n
ate
ria1
113
B
uri
ed
at
R/r
= 0
; Z
/r =
1
I-'
0\
'-::>
Foot
inop
ress
ure
(psi
) o
1.0
2.0
'3_0
4
.0
5.0
6.0
C 0
.2
.- - -c ~ E
~ - - ~ 0
.3
0.4
0.5
SL--
----
---r
----
----
--r-
----
----
-r--
----
----
r---
----
---T
----
----
--,
\ o @
r;J
r;J
@
@
r;J
r;J
r;J
@
Fir
,ure
1
07
. F
oo
tin
e P
ress
ure
v
s S
ettl
eMen
t lJ
ith
a
Co
mp
ress
ible
Hate
rial
M4
Bu
ried
at
R/r
= O
J Z
/r =
1
t-'
-....J o
Foot
ino
pres
sure
(p
sI)
o 1.
0 2.
0 3.
0 4.
0 5.
0 6.
0
C 0
.2
- c • j - -; 0.3
C
I)
0.4
0.5
F
igu
re
10
8.
Fo
oti
nr.
; P
ress
ure
v
s S
ett
lem
en
t l-
lith
a
Co
mp
ress
ible
M
ate
rial
MS
Bu
ried
at
R/r
=
0;
Z/r
= 1
I-
' ~
I-'
172
These RF values were plotted against Ra values (from Fig.
92) and a straight line relationship was obtained, as shown
in Fig. 109. Similarly RF values computed with F.S. values
based on 0.10 in. allowable settlement were also plotted
against the same Ra values, as before, and again a straight
line relationship was obtained. This relationship is shown
in Fig. 110. The relationships as derived above may prove to
be very useful in design of earthquake resistant structures.
These relationships are applicable to a selected criterion
of settlement. For any other criterion a new relationship
has to be established. However, these types of relation
ships provide a useful mechanism for reducing the foundation
deceleration based on the settlement criterion.
- ~ ....., o 0:
:
90
60
30
<:>
o
<:>
30
\..'~
,'b
~o· R
F (%
)
<:>
60
9
0
Pig
ure
1
09
. D
ecele
rati
on
Red
uct
ion
vs
Facto
r o
f S
afe
ty ~eduction
(facto
r o
f sa
fety
base
d
on
0
.18
in
. al
low
·ab
le
sett
lem
en
t)
~
-....J
W
- ';I. - a a:: 9
0
60
30
o 3
0
.~
~-
,0
o·
+
R F
{%
}
60
9
0
Fig
ure
1
10
. D
ecele
rati
on
Red
uct
ion
vs
Facto
r o
f S
afe
ty R
edu
ctio
n
(facto
r o
f sa
fety
bas
ed
on 0
.1
in.
all
ow
ab
le
sett
lem
en
t)
I-'
'-J
~
CHAPTER VI
CONCLUSIONS
175
Based on the results of the experimental stress-wave
propagation investigation the following conclusions were
drawn:
1. The general stress-time response in the
vicinity of the footing followed the same
pattern as the acceleration-time response.
The peak vertical stress intensities near
the surface on the axis of symmetry were
in the order of 11 times the static contact
pressure of the footing with an exception of
lighter footings, in which case these
intensities were much higher.
2. The data indicates that the size of the
footing affected the stress distribution
in the soil. In all the three cases in
vestigated, the attenuation of peak
vertical stress was very rapid in a distance
equal to the diameter of the footing. The
data also indicates that at a distance equal
to 1.5 times the diameter of the footing from
the axis of symmetry, the effect of footing
on the vertical stress distribution is
negligible.
3. In all the three cases of surface footings
subjected to transient load, it was
noticed that, at a depth equal to twice
the footing diameter, the influence of
footing mass on the vertical stress
distribution is negligible.
4. Stress intensities in the vicinity of
the footing increased with depth of burial
(depth of embedment) of the footing (as
the distance from the transient load source
decreased).
According to the results of the footing response
experiments it was concluded that:
1. There exists a linear relationship between
deceleration of the footing and weight of
the footing. Deceleration of the footing was
inversely proportional to its weight.
2. Deceleration amplitude of the footing decreased
with the depth of burial (embedment) of the
footing. thus indicating a greater damping
and increased stiffness (since the maximum
depth of burial was only one footing diameter,
the soil above the footing had a negligible
contribution to the weight of the footing).
Based on the analyses of results from the footing
isolation experiments it was concluded that:
1. A linear relationship can be generated
between the percentage reduction in
176
deceleration amplitude of the footing
(Ra) and log of sub grade modulus (Kc)
of the compressible material computed
at (a c ) stress level. This straight line
equation is represented by the equation
Ra = 78.4 - 42.7 log Kc' In a generalized
form this can be expressed as Ra = Al -
A2 log Kc ' where Al and A2 are constants.
177
2. Similar linear relationship can be established
between the percentage reduction of the vertical
stress (Rs) in the vicinity of the footing and
log of subgrade modulus (Kc) of the compressible
material at (a c ) stress level. This linear
plot, on semi-log graph can be represented
by the equation Rs = 89.3 - 30.7 log Kc ' and
in a generalized form, Rs A3 A4 log Kc '
where A3 and A4 are constants.
3. The data from static load-displacement tests
(with and without a compressible material
buried in the soil) indicated that a linear
relationship can be generated between the
percentage reduction in the factor of safety
(based on allowable settlement) of a surface
footing (RF) and percentage reduction in
deceleration of the footing (Ra ). For an
allowable settlement of 3% of the footing
diameter (0.18 in.), the deceleration
amplitude of the footing is reduced by
30%, however, the factor of safety
reduced to 65%. Similarly, for an allow
able settlement of 1.67% of the footing
diameter (0.10 in.), the factor of safety
reduced to 68% for the same reduction in
deceleration.
178
The above conclusions were drawn based on small scale
model tests. A comparison of small scale model test results
with large size prototype structures may prove helpful in
developing some similitude criterion and may also provide
better understanding of the influence of the vertical
component of earthquake induced forces. It must be
recognized that information from large size structures in
a meaningful form may not be available now and obtaining the
information may involve considerable time. Furthermore, it
may not be possible to establish one-to-one scale factor
between the small scale models and prototype structures,
but relationships similar to those presented in this
investigation can be developed and established.
Future research should include:
(a) Extensive analysis of the experiments
conducted, perhaps by the finite element
technique to provide a better understanding
of the foundation response, and also to
predict the stress intensities in the
vicinity of the foundation for a given
input spectra.
(b) Use of cohesive material (clay) as a soil
media beneath the footing to transmit
transient loads to the bottom of the footing.
(c) More number of compressible materials in
footing isolation studies.
(d) Evaluation of the constants AI' A2 , A3 and
A4 in various soil types.
179
APPENDIX A
SOIL PROPERTIES
180
APPENDIX A
SOIL PROPERTIES
181
In the experimental program of this thesis, the soil
used was dry, 20-40 Ottawa sand. The sand was uniformly
graded, and consisted of subrounded particles. This soil
is classified as (SP) according to the Unified Soil
Classification System (USeS). The grain size distribution
of this soil is shown in Fig. Ill. The specific gravity
test was conducted for this soil material, and a value of
2.66 is obtained.
Maximum and minimum dry density tests were conducted
for this soil, according to the ASTM standards. The maximum
and minimum dry densities obtained are 110 pcf and 92 pcf,
respectively. A graph, as shown in Fig. 112, is drawn using
the maximum and minimum dry densities, in order to obtain
the relative density at any known initial dry density.
-c CI) u ...
100
80
tf40
20
o 1.0
~
'" \ ~ \
I , I I
0.8 0.6
I I 20. 30-
G:2.66
\
\ \ ~ I Ib .~
I I I I ~ 0.4 0.3 0.2
Partlcte dia meter (m m)
I I I I I 40 50 60 70 100
U.S. std. sieve number
Figure 111. Grain Size Distribution of Ottawa Sand
182
0.1
-~
80
90
~IOO -110
120
130
.............
--
o
.............. ........ ........
t'...... ...... ...............
-- --- --- ---
20 40
r---......... r---........
---1---~ I ............. I
T I
60 80
80
90
-~ u 100 ~
~ 110
120
130
100
S E
)...
Figure 112. Relative Density vs Specimen Density for Dry Ottawa Sand (20-40 Hesh)
183
184
APPENDIX B
WAVE PROPAGATION VELOCITIES AND SOIL MODULI
185
APPENDIX B
WAVE PROPAGATION VELOCITIES AND SOIL MODULI
According to the elastic half-space theory, there are
two basic types of waves, body waves and surface waves. Again
the body waves are subdivided into two important waves, the
compression or dilatational-also called P-wave, and the shear
wave or S-wave. The only surface wave is the Rayleigh wave
also called R-wave. Lord Rayleigh52 was first to study the
motion of a wave confined to a zone near the boundary of the
half-space. Lamb36 described in detail the surface motion
that occurs at distances far away from the point source, at
the surface of the half-space. These Rayleigh waves propagate
radially outward on a cylindrical wave front. The particle
motion, associated with this wave at the surface, is a
retrograde ellipse.
The body waves propagate radially outward from the source
along hemispherical wave fronts. The particle motion assoc
iated with the compression wave is a push-pull motion in the
direction of the wave front; and, the particle motion
associated with the shear wave is a transverse displacement
normal to the direction of the wave front. The distribution
of displacement waves from a circular footing on a homo
geneous, isotropic, elastic half-space is schematically shown
by Woods 82 , in his work. There is also a considerable amount
of information relative to the propagation of seismic waves
in homogeneous, isotropic, elastic half_space35 ,19,20.
In the experimental work of this thesis, compressional
186
waves were transmitted to the bottom of the footing,through
the soil media. Stress-time relationships were recorded at
each gage station along the vertical axis of symmetry. From
these records the arrival times of the compressi~Qna.l wave
were computed at each gage station. These arrival times
were plotted against the distance of the gage from the loading
source, as shown in Fig. 113, to represent the average
dilatational (compressional) wave velocity. The average
dilatational wave velocity observed in these experiments
was 557 ft./sec.
This dilatational wave velocity varies from soil to soil.
It is primarily a function of the constrained modulus (Ec)
and the mass density (p), and is represented by the equation:
C =~ D ~-t
where, E is defined as the modulus of deformation for tests, c
in which no lateral movement of the material is allowed, and
is given by the equation:
= E (1 -v) (1 + v)(l - 2v)
24
IS
- .5 - s:. 0.1
2
~
6 o 2
'--
velo
city
of
peak
str
ess
55
7ft
/sec 3
Arr
ivol
tim
e (m
sec)
Fig
ure
11
3.
Vert
ical
Str
ess
Arr
ival
Tim
e v
s D
epth
4
,.....
<Xl
-....J
188
APPENDIX C
STRESS-DEFORMATION RELATIONSHIP FOR COHPRESSIBLE MATERIALS
-c: .--(\J '0 )(
0.2
0.4
g 0.6 ;
I ~ -CD
o 0.8
1.0
1.2
, , , I I , \ . , \
Stress (psI) 0.5 1.0 1.5
\ . \ \ . \ \ " .
L' ..... .:. .. · u-...I~~ unloadlnQ °c:urv;
Figure 114. Stress vs Deformation Relationship for Compressible Material No. 1 (loading and unloading conditions)
189
Stress (psi) o 0.5 1.0 I 5 e-------~----~~----~.
0.5
- 1.0 c \ -- \
C\J \ I 0 \
\ )( \ .. 1.5 \ c \ 0 - \ - \ 0 E \ ... \ 0 \ - \ • o 2.0 \
® \ \ \ \ \ \
2.5 \ \ \ \ \ \ \ \ \
3.0 \
UnIOadingJ " ..... -- . curve
Fieure 115. Stress vs Deformation Relationship for Compressible l1aterial No. 2 (loading and unloading conditions)
190
Stress (psi) 0~ _____ 0~.~5 ____ ~1~.0~ ____ ~1.5
1.0 '
-.5 - , ' ~ 2.0 " 0\. x \ .. \ .~ \ '0 \ E 3.0 \ o , ~ \ Q \
'\ , , , 4.0
~::~~------~ 5.0
Figure 116. Stress vs Deformation Relationship for Compressible Material No. 3 (loading and unloading conditions)
191
Stress (psI) 0~ _____ 0~.~5 ______ ~LO ______ ~1.5
1.0
-.5 -";2 2.0
4.0
5.0
\ \ I I I I , \ I \ \ \ \ \ \ \ \ , , >--- .
~Oading curv;-----
Figure 117. Stress vs Deformation Relationship for Compressible 11aterial No. 4 (loading and unloading conditions)
192
1.0
2.0
3.0 -.5 --10 )(
.. 4.0 g ;:
~ .2 ., o 5.0
6.0
7.0
I , I \ I I I \ I I \ I I I , I I \ I I , \ \ , I \ I , I I , \ \ \ \ \ \ \ \ \ \
Stress (psi)
0.5 1.0 1.5
, , , ' .... " .... ,
' .... unloading curveJ -------· .
Figure 118. Stress vs Defornation Relationship for Compressible 1faterial No. 5 (loading and unloading conditions)
193
194
BIBLIOGRAPHY
1. Ahlvin, R.G., and Ulery, H.H., "Tabulated Values for Determining the Complete Pattern of Stresses, Strains, and Deflections Beneath a Uniform Circular Load on a Homogeneous Half Space", Highway Research Board Bulletin 342, Jan., 1962, pp. 1-13.
2. Arnold, R.N., Bycroft, G.N., and Warburton, G.B., "Forced Vibrations of a Body on an Infinite Elastic Solid", Journal of Applied Mechanics, September, 1955, pp. 39l~OO.
3. Barkan, D.D., "Field Investigation of the Theory of Vibration of Massive Foundations Under Machines", Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Vol. II, 1936.
4. Barkan, D.D., "Experimental Study of Vibrations of Mat Foundations Resting on Cohesive Saturated Soils", Transactions, Institute for Engineering Foundation Research (VIOS); Moscow, U.S.S.R., 1934.
5. Barkan, D.D., Dynamics of Bases and Foundations, HcGraw-Hill Book Co., Inc., New York, New York, 1962.
6. Bassol, Jean Luc., "Numerical Analysis of Stress Wave Propagation in an Elastic-Plastic Medium", Unpublished M.S. Thesis, Illinois Institute of Technology, May, 1971.
7. Bieganousky, Wayne A., and Marcuson, William F., III, "Uniform Placement of Sand", Journal of the Geotechnical Engineering Div., ASCE, Vol~02, No. GT3, March 1976, pp. 229-233.
8. Carroll, W.F., "Dynamic Bearing Capacity of Soils, Report No.5, Vertical Displacements of Spread Footings on Clay: Static and Impulsive Loadings", U.S. Army Engineer Waterways Experiment Station Technical Report No. 3-599, Vicksburg, Mississippi, September, 1963, pp. 1-154.
9. Converse, F. J., Housner, G. W., Hausman, D. A., Quinlan, R.M., and Paul, A., "An Investigation of the Compaction of Soils by Vibration", California Institute of Technology, Research Project, U.S. Army Engineering and Development Labs., The Engineer Center, Fort Belvoir, Virginia, March, 1950.
195
BIBLIOGRAPHY (Cont'd.)
10. Cortright, Clifford J., "A New Look at California Dams and Earthquakes", Meeting Preprint 2011, ASCE National Structural Engineering Meeting, San Francisco, California, April, 1973, pp. 1-10.
11. Cox, A.D., Eason, G., and Hopkins, H.G., "Axially Symmetric Plastic Deformations in Soils", Philosophical Transactions, The Royal Society of London, Series A, No. 1036, Vol. 254, London, England, August, 1961, pp. 1-45.
12. Crockett, J.H.A., and Hammond, R.E.R., "The Natural Oscillation of Ground and Industrial Foundation", Proceedings, Second International Conference on Soil Mechanics and Foundation Engineering, Vol. III, June, 1948, pp. 88-93.
13. Crockett, J. H. A., and Hammond, R. E. R., "Dynamic Principles of Machine Foundations and Ground", Proceedings, Institution of Mechanical Engineers, Vol. 160, pp. 512-523, discussion, 1949. pp. 523-553.
14. De Beer, E.E., "Experimental Determination of the Shape Factors and the Bearing Capacity Factors of Sand", Geotechnique 20, No.4, 1970, pp. 387-411.
15. Drnevich, Vincent, P., and Hall, J.R, .Jr .. , "Transient Loading Tests on a Circular Footing", Journal of the Soil Mechanics and Foundations Division, ASCE, Vo1.~ No. SM6, Nov., 1966, pp. 153-167.
16. Duns, C.S., "Vertical Response of a Rigid Base to Dynamic Loading", Civil Engineering and Public Works Rev. 64, No. 760, 1969, pp. 1081-1095.
17. Eastwood, W., "The Factors which Affect the Natural Frequency of Foundations and the Effect of Vibrations on the Bearing Power and Settlement of Foundations on Sand", Structural Engineering, Vol. 31, March, 1953, pp. 82-98.
18. Elliot) A. L., "Lessons from a Major Earthquake", American Highways, Vol. 50, No.2, April, 1971.
19. Ewing, W. M., Jardetzky. ~J. S., and Press, F., Elastic Waves in La~ered Media. McGraw-Hill Book Co., Inc., New York, 1 57.
20. Grant, F.S., and West, G.F., Interpretation Theory in
BIBLIOGRAPHY (Cont'd.)
Applied Geophysics. International Series in The Earth Sciences, McGraw~Hi1l Book Co., Inc., New York, 1965. . .
21. Green, N.B., "Flexible First Story Construction for Earthquake Resistance", Transactions, ASCE, Vo1. 100, 1935, pp. 645-674.
196
22. Gupta, Y. P., and Chandrasekaran, A. R., "Absorber System for Earthquake Excitations", 4WCEE. Vol. II, Santiago De Chile, January, 1969, pp. 139-148.
23. Hadala, P.F., "Dynamic Bearing Capacity of Soils, Report No.4, Investigation of a Dimensionless Load Displacement Relation for Footings on Clay", U.S. Army Engineer Waterways Experiment Station Technical Report No. 3-599, Vicksburg, Mississippi, June, 1965. -
24. Hada1a, P. F., and Jackson, J. G. Jr., "A Model Study of the Small Boy Footing Behavior", U.S. Army Engineer Waterways Experiment Station Technical Report No. 3-793, Vicksburg, Mississippi, August, 1967, pp. 1-266.
25. Hada1a, P.F., "The Effect of Placement Method on the Response of Soil Stress Gages ll
, Proceedings, International Symposium on Wave Propagation and Dynamic Properties of Earth Materials, Albuquerque, New Mexico, Aug., 1967, pp. 255-263.
26. Hampton, D., and Wetzel, R.A., IIStress Wave Propagation in Confined Soils", Proceedings, Symposium on Wave Propagation and Dynamic Properties of Earth Materials, Albuquerque, New Hexico, August, 1967, pp. 433-442.
27. Hicks, C.~V., Janza, F.J., and Reid, W.G., "Evaluation of Soil Strain Gage Instrumentation", Air Force Weapons Laboratory Technical Report No. AFWL-TR-67-50, Kirtland AFB, New Mexico, Aug., 1967, pp. 1-280.
28. Ingram, J.K., "The Development of a Free-Field Soil Stress Gage for Static and Dynamic Measurements", Instruments and Apparatus for Soil and Rock Mechanics, American Society-Ior Testing-orMaterials Special Publication, ASTM STP 392, Washington, D.C., 1965, pp. 20-36.
197
BIBLIOGRAPHY (Cont'd.)
29. Ingram, J.K., "Development of a Free-Field Soil Stress Gage for Static and Dynamic Measurements", U.S. Army Engineer Waterways Experiment Station TeC1inical Report No. 1-184, Vicksburg, Mississippi, Feb., 1968, pp. 1=rI2.
30. Jackson, J. G. Jr., and Hadala, P., "Dynamic Bearing Capacity of Soils, Report No.3, The Application of Similitude to Small-Scale Footing Tests", U.S. Arm! Engineer Waterways Experiment Station Technica Report No. 3-599, Vicksburg, Mississippi, December, 1964. --
31. Januskevicius, C.K., and Vey, E., "Stresses and Strains in Triaxial Specimens Using Embedded Gages", American Society for Testin~ of Materials, STP No. 392, Instrumentation an Apparatus for Soil and Rock Mechanics, 1965, pp. 37-54.
32. Jumikis, A.R., "Vertical Stress Tables for Uniformly Distributed Loads on Soil", Engineering Research Publication No. 52, College of Engineering, Bureau of Engineering Research, Rutgers University, New Brunswick, New Jersey, 1971.
33. Keller, R.W., and Anderson, H.E., "Development of a Soil Strain Gage for Laboratory Dynamic Test", Air Force Weapons Laboratory Technical Report No. AFWL-TDR-64-7, Kirtland AFB, New Mexico, 1964.
34. Kolbuszewski, J., and Jones, R. H., "The Preparation of Sand Samples for Laboratory Testing", Proceedings, Hidland Soil Mechanics and Foundations Engineering Society, Vol. 4, 1961, p.l07.
35. Kolsky, H., Stress Waves in Solids. Dover Publications, New York, 1963, pp. 4-98; also Clarendon Press, Oxford, 1953.
36. Lamb, H., "On the Propagation of Tremors over the Surface of an Elastic Solid", Philosophical Transactions of the Royal Society of London, Series A., Vol. 203, 1904, pp. 1-42.
37. Landsman, R.B., "Stress Wave Propagation in a Clay Soil Under an Impulsively Loaded Circular Footing", Unpublished Ph.D. Dissertation, Illinois Institute of Technology, May, 1973.
BIBLIOGRAPHY (Cont'd.)
38. Larkin, L.A., "Theoretical Bearing Capacity of Very Shallow Footings", Journal of the Soil Hechanics and Foundations Division, ASCE-;-VoI:"94, No. SM6, November, 1968, pp. 1347-1347.
198
39. Lorenz, H., Hertwig, A., and Frusk, G., "Die Ermittlung der fur das Bauwesen wishtigsten Eigenschaften durch erzwungene Schwingungen", Veroffentlichungen des Instituts der deutschen Forschungsgesellschaft fur Bodenmechanik (DEGEBO) an der Technischen Hochschule Berlin, Heft I, Berlin, Deutschland, Verlag von Julius Springer, 1933.
40. Lorenz, H., "Neue Ergebnisie der dynamischen Baugrunduntersuchung", Zeitschrift des Vereins deutscher Ingenieure, Band 78, Nr. 12, 1934 S. 379-385.
41. Lorenz, H., "Zeichnerische Auswertung von Resonanzkurven" , Ingenieur - Archiv, Band 5, 1934, S. 376-394.
42. Lysmer, J., and Richart, F. E., Jr.. "Dynamic Response of Footings to Vertical Loading", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. ~No. SM1, January, 1966, pp. 65-91.
43. Lysmer, J., and Kuhlemeyer, R.L., "Finite Dynamic Model for Infinite Media", Journal of the Engineering Mechanics Division, ASC~ Vol. 95, No. EM4, August, 1969, pp. 859-877.
44. Matsushita, K., and Izumi, M., "Some Analysis on Mechanisms to Decrease Seismic Force Applied to Buildings", 3WCEE, Vol. III, February, 1965, pp. 342-359. .
45. Matsushita, K., and Izumi, M., "Studies on Mechanisms to Decrease Earthquake Forces Applied to Buildings", 4WCEE, Vol. II, Santiago De Chile, January, 1969, pp. 117-125.
46. McNeill, R.L., "Machine Foundations-The State-of-theArt", Soil Dynamics Speciality Conference, Mexico City, Hexico, August, 1969.
47. Medearis, K. G., "A Comparative Study of" "Structural Response to Explosion-Induced Ground Motions", A Report Prepared for the ASCE Research Council on Performance of Structures, ASCE Project
BIBLIOGRAPHY (Cont'd.)
210.03/17-2.72-3, American Society of Civil Engineers, New York, 1975, pp. 1-138.
48. Naik, T.R., "Analysis of Foundations on a SemiInfinite Elastic Mass Under Dynamic Loads", Unpublished Ph.D. Thesis, University of Wisconsin, January, 1972.
49. Orrje, 0., "Dynamic Loading Tests on Cohesionless Soils With Free Falling Weights, Compared With Static Plate Loading Tests", Division of Soil Mechanics, Royal Institute of Technology, Stockholm, 1968, pp. 1-42.
199
50. Quinlan, P.M., "The Elastic Theory of Soil Dynamics", Symposium on Dynamic Testing of Soils, ASTM Special Technical Publication No. 156, July, 1953, pp. 3-34.
51. Rao, H.A.B., and Hoeg, K., "Two-Dimensional Analysis of Stress and Strain in Soils, Report No.4, Dynamic Response of Strip Footing on ElasticPlastic Soils", U.S. Army Engineer waterwats Experiment Station Technical Report No. 3- 29, Vicksburg, Mississippi, November, 19b'7,pp. 1-72.
52. Rayleigh, Lord, "On Waves Propagated Along the Plane Surface of an Elastic Solid", London Mathematical Society Proc., 17, pp. 4-11.
53. Reissner, E., "Stationare, axialsymmetrische, durch eine schiittelnde Masse erregte Schwingungen eines homogenen elastischen Halbraumes", IngenieurArchiv, Band 7, 1936, S. 381-396.
54. Rhines, W.J., "Elastic Plastic Foundation Model for Punch-Shear Failure", Journal of the Soil Hechanics and Foundations Division, ASCE-,-V~ ~No. SM3, May, 1969, pp. 819-828.
55. Richart, F.E., "Foundation Vibrations", Journal of the Soil Mechanics and Foundations Division, ASC~ --Vol. 86, No. ,SM~August, 1960, pp. 1-34.
56. Richart, F. E., and Whitman, R. V., "Comparison of Footing Vibration Tests with Theory", Journal of the Soil Mechanics and Foundations Division, ASCE,VOl~, No. SMll, November, 1967, pp. 143-168.
57. Seed, H. B. J "Some Lessons from Recent Earthquakes in Relation to Seismic Design in North Carolina Adjacent
200
BIBLIOGRAPHY (Cont'd.)
States", Sixth Annual H.N. Show Lecture Series in Civil Engineering, Nor~arorrna State UniversITy, Raleigh, North Carolina, November 1970.
58. Selig, E.T., "Response of Foundations to Dynamic Loads", Unpublished M.S. Thesis, Illinois Institute of Technology, January, 1960.
59. Selig, E.T., "A Review of Stress and Strain Measurement in Soil", Proceedings, Soil-Structure Interaction Symposium, Tucson, Arizona, September, 1964, pp. 172-186.
60. Selig, E.T., and Wetzel, R.A., "A l'1iniature Piezoelectric Gage for Static and Dynamic Soil Stress Measurement", Contract Report No. 1-105, U.S. Armk Engineer Waterwa¥s Experiment Station, Vic sburg, Mississipp~, November, 1964.
61. Selig, E.T., and Vey, Eben, "Shock-Induced Stress Wave Propagation in Sand", Journal of the Soil Mechanics and Foundations Division, ASCE-,-Vo1. ~No. SM3, May, 1965, pp. 19-49.
62. Selig, E.T., and Tobin. H., "Fiezoresistive Soil Stress Gages", Air Force Weapons Laboratory Report No. AFWL-TR-66-51, lIT Research Institute, May, 1966.
63. Skormin, G.A., and Malysev, M.B., "Investigation of Stress Distribution in Sand Under a Circular Footing During Loading Increase'!, Osnov. Fund. Mech. Grunt. 12, No.5, 1970.
64. Slade, J.J., "A Discontinuous Model for the Problmes of Soil Dynamics", Symposium on Dynamic Testing of Soils, ASTM Special Technical Publication No. 156, July, 1953, pp. 69-76.
65. Sung, Tse Yung, "Vibrations in Semi-Infinite Solids Due to Periodic Surface Loading", Symposium on Dynamic Testing of Soils, ASTM Special Technical Publication No. 156, July, 1953, pp. 35-68.
66. Terzaghi, K., and Peck, R.B., Soil Mechanics in Engineering Practice. New York, John ~iley & Sons, Inc., New York, 1967.
67. Triandafilidis, G.E., "Analytical Study of Dynamic Bearing Capacity of Foundations", Defense Atomic Support Agency Technical Report No. 1189,
201
BIBLIOGRAPHY (Cont'd.)
Washington, D.C., January, 1961.
68. Truesdale, W.B., "Development of a Small Strain Gage", Air Force Weapons Laboratory Technical Report No. AFWL-TDR-63-3, Kirtland AFB, New Mexico, 1963.
69. Truesdale, W.B., and Schwab, R.B., "Soil Strain Gage Instrumentation", Air Force Weapons Laboratory Technical Report No. AFWL-TDR-65-207, Kirtland AFB, New Mexico, 19b5, pp. 1-92.
70. Valliappan, S., Favaloro, J.J., and White, W., "Dynamic Analysis of Embedded Footings", Technical notes, Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT2, February, 1977, pp. 129-133.
71. Varadhi, S.N., "Foundation Response Due to Disturbances In Underlying SoilH
, Unpublished M.S. Thesis, Illinois Institute of Technology, Chicago, Illinois, December, 1972.
72. Vesic, A.S., I!Bearing Capacity of Deep Foundation in Sand", National Academy of Sciences, National Research Council, Highway Research Record, No. 39, 1963, pp. 112-153.
73. Vesic, A.S., "Analysis of Ultimate Loads of Shallow Foundations", Journal of the Soil Mechanics and Foundations Division, ASC~V~99, No. SMl-,-January, 1973, pp. 45-73.
74. Vey, E., and Strauss, L., "Stress-Strain Relationships in Clay Due to Propagating Stress Waves", Proceedings, Symposium on Wave Propagation and Dynamic Properties of Earth Materials, Albuquerque, New Mexico, August 1967, pp. 575-586.
75. Wetzel, R.A., "Shock Induced Stress Wave Propagation in a Cohesive Soil", Unpublished M.S. Thesis, Illinois Institute of Technology, Chicago, Illinois, January, 1965.
76. Wetzel, R.A., "Axisymmetric Stress Wave Propagation in Sand", Unpublished Ph.D. Dissertation, Illinois Institute of Technology, June, 1969.
77. Wetzel, R.A., and Vey, E., "Axisymmetric Stress Wave Propagation in Sand", Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. SM5, September, 1970, pp. 1763-1786.
202
BIBLIOGRAPHY (Cont'd.)
78. Whitman, R. V., "Analysis of Foundation Vibrations", Presented to Symposium on Man-made Vibrations in Civil Engineering, British National Section, International Association for Earthquake Engineering, London, April, 1965.
79. Whitman, R. V., and Richart, F .E., "Design Procedures for Dynamically Loaded Foundations", Journal of the Soil Mechanics and Foundations DiVision, ASCE, Vol.~ No. SM6, November, 1967, pp. 169-193.
80. ~Vinterkorn, H. F., "Macromeritic Liquids", Symposium on Dynamic Testing of Soils, ASTI1 Special Technical Publication No. 156, July, 1953, pp. 77-89.
81. Wirsching, Paul H., and Yao, James, T.P., "A Statistical Study of Some Design Concepts in Earthquake Engineering", Bureau of Engineering Research. The University of New Mexico Technical Report CE-2l(70) NSF-065, Albuquerque, New Mexico, May, 1970, pp. 1-169.
82. Woods, R.D., "Screening of Surface Waves in Soils", Journal of the Soil Mechanics and Foundations Division--, ASCE,-voT. 94, No. SM~ July, 1968, pp. 951-979.
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