Table Of ContentDEVELOPMENTS IN GEOTECHNICAL ENGINEERING 70
S t r e s s es a nd D i s p l a c e m e n ts
f or S h a l l ow F o u n d a t i o ns
D. Μ Movie
Faculty of Technical Sciences, University of Novi Sad, Serbia, Yugoslavia
For correspondence:
Ustanicka 152, 11000 Beograd, Serbia, Yugoslavia
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ν
Further titles in this series
1. G. SANGLERAT — THE PENETROMETER AND SOIL EXPLORATION
2. Q. SARUBA AND V. MENCL — LANDSLIDES AND THEIR CONTROL
3. Ε. E. WAHLSTROM — TUNNELING IN ROCK
4. R. SILVESTER — COASTAL ENGINEERING, 1 and 2
5. R. N. YONG AND B. P. WARKENTIN — SOIL PROPERTIES AND BEHAVIOUR
6. Ε. E. WAHLSTROM — DAMS, DAM FOUNDATIONS, AND RESERVOIR SITES
7. W. F. CHEN — LIMIT ANALYSIS AND SOIL PLASTICITY
8. L. N. PERSON — ROCK DYNAMICS AND GEOPHYSICAL EXPLORATION
Introduction to Stress Waves in Rocks
9. M. D. GIDIGASU — LATERITE SOIL ENGINEERING
10. Q. ZARUBA AND V. MENCL — ENGINEERING GEOLOGY
11. Η. K. GUPTA AND Β. K. RASTOGI — DAMS AND EARTHQUAKES
12. F. H. CHEN — FOUNDATIONS ON EXPANSIVE SOILS
13. L. HOBST AND J. ZAJIC — ANCHORING IN ROCK
14. B. VOIGHT (Editor) — ROCKSLIDES AND AVALANCHES, 1 and 2
15. G. LOMNITZ AND E. ROSENBLUETH (Editors) — SEISMIC RISK AND ENGINEERING DECI
SIONS
16. C. A. BAAR — APPLIED SALT-ROCK MECHANICS, 1
The In-Situ Behavior of Salt Rocks
17. A. P. S. SELVADURAI — ELASTIC ANALYSIS OF SOIL-FOUNDATION INTERACTION
18. J. FEDA — STRESS IN SUBSOIL AND METHODS OF FINAL SETTLEMENT CALCULATION
19. A. KEZDI — STABILIZED EARTH ROADS
20. E. W. BRAND AND R. P. BRENNER (Editors) — SOFT-CLAY ENGINEERING
21. A. MYSLIVEC AND Z. KYSELA — THE BEARING CAPACITY OF BUILDING FOUNDATIONS
22. R. N. CHOWDHURY — SLOPE ANALYSIS
23. P. BRUUN — STABILITY OF TIDAL INLETS
Theory and Engineering
24. Ζ. BAZANT — METHODS OF FOUNDATION ENGINEERING
25. A. KEZDI — SOIL PHYSICS
Selected Topics
26. H. L. JESSBERGER (Editor) — GROUND FREEZING
27. D. STEPHENSON — ROCKFILL IN HYDRAULIC ENGINEERING
28. P. E. FRIVIK, N. JANBU, R. SAETERSDAL AND L. I. FINBORUD (Editors) — GROUND
FREEZING 1980
29. P. PETER — CANALS AND RIVER LEVEES
30. J. FEDA — MECHANICS OF PARTICULATE MATERIALS
The Principles
31. Q. ZARUBA AND V. MENCL — LANDSLIDES AND THEIR CONTROL
Second, completely revised edition
32. I. W. FARMER (Editor) — STRATA MECHANICS
33. L. HOBST AND J. ZAJIC — ANCHORING IN ROCK AND SOIL
Second, completely revised edition
34. G. SANGLERAT, G. OLIVARI AND B. CAMBOU — PRACTICAL PROBLEMS IN SOIL ME
CHANICS AND FOUNDATION ENGINEERING, 1 and 2
35. L. RETHATI — GROUNDWATER IN CIVIL ENGINEERING
36. S. S. VYALOV — RHEOLOGICAL FUNDAMENTALS OF SOIL MECHANICS
37. P. BRUUN (Editor) — DESIGN AND CONSTRUCTION OF MOUNDS FOR BREAKWATERS AND
COASTAL PROTECTION
vi
38. W. Κ. CHEN AND G. Y. BALADI — SOIL PLASTICITY
Theory and Implementation
39. Ε. T. HANRAHAN — THE GEOTECHNICS OF REAL MATERIALS
The £, £ method
g k
40. J. ALDORF AND K. EXNER — MINE OPENINGS
Stability and Support
41. J. E. GILLOTT — CLAY IN ENGINEERING GEOLOGY
42. A. S. CAKMAK (Editor) — SOIL DYNAMICS AND LIQUEFACTION
43. A. S. CAKMAK (Editor) — SOIL-STRUCTURE INTERACTION
44. A. S. CAKMAK (Editor) — GROUND MOTION AND ENGINEERING SEISMOLOGY
45. A. S. CAKMAK (Editor) — STRUCTURES, UNDERGROUND STRUCTURES, DAMS AND
STOCHASTIC METHODS
46. L. RETHATI — PROBABILISTIC SOLUTIONS IN GEOTECHNICS
47. Β. M. DAS — THEORETICAL FOUNDATION ENGINEERING
48. W. DERSKI, R. IZBICKI, I. KISIEL AND Z. MROZ — ROCK AND SOIL MECHANICS
49. T. ARIMAN, H. HAMADA, A. C. SINGHAL, M. A. HAROUN AND A. S. CAKMAK (Editors) —
RECENT ADVANCES IN LIFELINE EARTHQUAKE ENGINEERING
50. Β. M. DAS — EARTH ANCHORS
51. K. THIEL — ROCK MECHANICS IN HYDROENGINEERING
52. W. F. CHEN AND X. L. LIU — LIMIT ANALYSIS IN SOIL MECHANICS
Second, completely revised edition
53. W. F. CHEN AND E. MIZUNO — NONLINEAR ANALYSIS IN SOIL MECHANICS
54. F. H. CHEN — FOUNDATIONS ON EXPANSIVE SOILS
Second, completely revised edition
55. J. VERFEL — ROCK GROUTING AND DIAPHRAGM WALL CONSTRUCTION
56. Β. N. WHITTAKER AND D. J. REDDISH — SUBSIDENCE
Occurrence, Prediction and Control
57. E. NONVEILLER — GROUTING
Theory and Practice
58. V. KOLAR AND I. NEMEC — MODELLING OF SOIL-STRUCTURE INTERACTION
59A. R. S. SINHA (Editor) — UNDERGROUND STRUCTURES
Design and Instrumentation
59B. R. S. SINHA (Editor) — UNDERGROUND STRUCTURES
Instrumentation and Constructions
60. R. L. HARLAN, Κ. E. KOLM AND E. D. GUTENTAG — WATER-WELL DESIGN AND CONSTRUC
TION
61. I. KAZDA — FINITE ELEMENT TECHNIQUES IN GROUNDWATER FLOW STUDIES
with applications in hydraulic and geotechnical engineering
62. L. FIALOVSZKY — SURVEYING INSTRUMENTS AND THEIR OPERATIONAL PRINCIPLES
63. H. GIL — THE THEORY OF STRATA MECHANICS
64. Η. K. GUPTA — RESERVOIR-INDUCED EARTHQUAKES
65. V. J. LUNARDINI — HEAT TRANSFER WITH FREEZING AND THAWING
67. E. JUHASOVA — SEISMIC EFFECTS ON STRUCTURES
To my wife
Milica
and to my sons
Uros and Strahinja
ix
P R E F A CE
The analysis of the interaction between a structural foundation and the
supporting soil media is of primary importance to structural and geotechnical
engineering* Theoretical results of these analyses provide information which
can be used in foundation design. In order to achieve more precise solutions
for the stresses and deformations produced by various types of loading, more
complex soil modulus have been used.
Chapter 1 provides the solutions for stresses and displacements produced
by a strip foundation of any rigidity resting on an elastic half-space,
subjected to vertical and horizontal uniform loads, triangular and trapezoidal
loads. In Chapter 2, the solutions for stresses and displacements produced by
a strip foundation of any rigidity resting in a layered half-space are given.
In Chapters 3 and 4, the same problems related to the strip foundation resting
on a homogeneous soil of finite thickness and in a layered soil of finite
thickness are analysed. Chapter 5 is devoted to the interaction analysis of a
circular foundation of any rigidity resting on an elastic half-space. The
application of numerical methods such as finite difference and finite element
methods is also discussed. In Chapters 6, 7 and 8, soil-foundation interaction
problems related to a circular foundation of any rigidity resting on a layered
half-space, on a homogeneous layer of finite thickness and on a layered soil
of finite thickness are analysed. In Chapters 9 and 10, soil-foundation
interaction problems related to a rectangular foundation of any rigidity,
resting on an elastic half-space and on a homogeneous layer of finite
thickness, are considered. The results of stresses and displacements for ring
foundations are given in Chapter 11. The methods for determining the soil
deformation parameters and the results of the geotechnical investigations
related to laboratory, model and field tests are presented in Chapter 12. In
Chapter 13, theoretical settlement values are compared with the observed
values. In this analysis special attention was pade to the effect of
mechanical soil disturbance.
I am grateful to several people for their assistance in preparing this
manuscript. Mrs. Ruzica Vidovic has gone for beyond the usual duties of a
χ
typist* Her assistance is greatly appreciated. My thanks are due to Mr. Randy
Major for his conscientious proof-reading of the English text. Likewise, I am
grateful to my assistent Mr. Mitar Djogo for his help in the preparation of
the necessary computer programs.
Finally, it is with great pleasure that I acknowledge the help and
encouragement of my wife, Milica, throughout the preparation of the
manuscript.
November, 1991.
D. M. Milovic
1
Chapter 1
STRIP FOUNDATION IN ELASTIC HALF-SPACE
1.1 FLEXIBLE STRIP FOUNDATION; VERTICAL UNIFORM LOAD;
HOMOGENEOUS SOIL
The problem of determination of stresses and displacements in an elastic and
isotropic half-space, produced by a uniformly distributed vertical load over a
flexible strip foundation has been studied by several investigators (Steinbren-
ner, 1934; Kolosov, 1935; Newmark, 1935).
The componental stresses produced by vertical concentrated force acting on
the surface of the elastic and isotropic half-space are given by the following
expressions:
°°^ (l.D
z
z 2TTR5
l_J_3zxi. .,J . . _ R_ >i_2i2Rzl .
σ { ( 12) + ± } (12)
X 2π R2 R3 R (R+z) R(R+z)2
(1.3)
y 2π R2 R3 R (R+z) R(R+z)2
. J-_L jxyz _ (1-2y)(2R+z)xy
Τ (1 4)
Xy 2π R2 R3 R(R+z)
τ = — 1.5
XZ 2TTR5
- 3pyz2
τ (1.6)
where Ρ = the concentrated force; R2 = x2+y2+z2; y = Poisson's ratio.
In Fig. 1.1 are shown the componental stresses produced by vertical concentra
ted force P.
2
Ρ
χ
υ
ζ
Γ2=χ2*γ2 y N(x,y.z)
W /
Fig. 1.1 Stresses at point Ν acted upon by point load P.
These equations are known as Boussinesq's equations.
The componental displacements in the x, y and ζ directions are:
xz _ (1-2y)x
u - Pil+H) 1 (1.7)
2πΕ R R2 R+z
_ P(1+y) i| yz (1-2y)y
(1.8)
2πΕ R R2 R+z
- ΡΟ+μ) ι
w (1.9)
2πΕ R
The line load p' acting on the surface of a half-space, can be divided in an
infinite number of elementary forces p'dy. By replacing in the Boussinesq equa
tion (1.1) the load Ρ with dP = p'dy and σ with do Mel an (1932) has obtained
ζ 2>
by integration the following expression for the vertical stress σ_:
+0O
3pVdy _ 2p'z
(1.10)
2u(/x2+y2+z2]
and:
2p'zx*
(1.11)
TTR"
(1.12)
xz *R<
3
Let us consider now a flexible strip foundation of width B = 2b, loaded by
uniform load p, as shown in Fig. 1.2.
Β
t i lt
* » * »
N(x.z)
Fig. 1.2 Flexible strip foundation; vertical uniform load p.
By replacing in the equations (1.10-1.12) χ by (χ-ξ) dp" by ράξ and integra
ting these expressions between the limits -b and +b, one obtains:
2ράζζ:
da =
z
π[(χ-ξ)2+ζ2]
+b
iP_J
A ^>< arctg-^)
(arctg +
z π -b [(χ-ξ)2+ζ2]2
_ 2bpz(x2-z2-b2)
= Pi, (1.13)
7r[(x2+z2-b2)2+4b2z2]
+b
= 2p_ / (χ-ξ)2ζ JL J>I>L arctg-^-) +
σ άζ = (arctg +
X π -b [(χ-ξ)2+ζ2]2 π ζ ζ
2bpz(x2-z2-b2)
= Ρίχ (1.14)
Tr[(x2+z2-b2)2+4b2z2]
+b
- 2 J (x-ξ) 4bpxz2
τ P = Pi (1.15)
xz
π -b [(χ-ξ)|+ζ2]2 Tr[(x2+z2-b2)2+4b2z2]
