Lecture 7 stress distribution in soil

Soil Mechanics

1. INTERNATIONAL UNIVERSITY
FOR SCIENCE & TECHNOLOGY
INTEREAOLR UVSEYFYRVC&IHR VCNRUGR VCF2I7R
CIVIL ENGINEERING AND
ENVIRONMENTAL DEPARTMENT
303322 -Soil Mechanics
Stress Distribution in Soil
Dr. Abdulmannan Orabi
Lecture
2
Lecture
7

2. Dr. Abdulmannan Orabi IUST
2
Das, B., M. (2014),
“ Principles of geotechnical
Engineering ” Eighth Edition, CENGAGE
Learning, ISBN-13: 978-0-495-41130-7.
Knappett,J. A. and Craig R. F. (2012),
“ Craig’s Soil
Mechanics” Eighth Edition, Spon Press, ISBN: 978-
0-415-56125-9.
References

3. IStress in soil due to self weight
Stress Distribution in Soil
IStress in soil due to surface load
3
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4. Stress due to self weight
The vertical stress on element A can be determined
simply from the mass of the overlying material.
If represents the unit weight of the soil, the
vertical stress is
I
Variation of stresses with depth
A
Ground surface
N
z
z
⋅
=
γ
σ
I TERAOL UAVSO
LYFLCYA&L
H
GTI27
4
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5. ∑
=
⋅ =⋅ + +⋅ +⋅ =
n
i
i i n n z
h h h h
1
2 2 1 1
......
γ γ γ γ σ
Stress due to self weight
Stresses in a Layered Deposit
The stresses in a deposit consisting of layers of
soil having different densities may be determined a s
Vertical stress at depth z 1
Vertical stress at depth z 2
Vertical stress at depth z 3

D
H
GrTI
r∗
r
H
GDTI
r∗
rbI
D∗
D
d
u
d
l
d
m

r

a
7
r
7
a
7
D
I
r∗
r
I
r∗
rbI
D∗
D
H
G
a
T
I
r
∗

r
b
I
D
∗

D
b
I
a
∗

a 5
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6. With uniform surcharge on infinite land surface
Stress due to self weight
Original
land surface
Conversion land surface
nn
H
GTI.ib3L
3 TLILSL

i
I TERAOL UAVSO
LYFLCYA&L
6
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7. Stress due to self weight
H
GTI.i
Vertical stresses due to self weight increase
with depth,
There are 3 types of geostatic stresses:
a.
Total Stress,
σ
total
b.
Effective Stress,
σ
'
c.
Pore Water Pressure, u
Vertical Stresses
7
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8. Stress due to self weight
Consider a soil mass having a horizontal
surface and with the water table at surface
level. The
total vertical stress
at depth z is
equal to the weight of all material (solids +
water) per unit area above that depth ,i.e
Total vertical stress
H
G45468
TI
%#!
.i
8
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9. Stress due to self weight
The pore water pressure at any depth will be
hydrostatic since the void space between the solid
particles is continuous, therefore at depth z:
Pore water pressure
ETI
&
.i
If the pores of a soil mass are filled with water
and if a pressure induced into the pore water, trie s
to separate the grains, this pressure is termed as
pore water pressure
9
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10. Stress due to self weight
Effective vertical stress due to self weight of soi l
The difference between the total stress (
H
G45468
) and
the pore pressure (
u
) in a saturated soil has been
defined by Terzaghi as the effective stress ( ) .
H
G
'
H
G
'
TH
G45468L
•E
The pressure transmitted through grain to grain at
the contact points through a soil mass is termed as
effective pressure.
10
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11. Stress due to self weight
Stresses in Saturated Soil If water is seeping, the effective stress at any
point in a soil mass will differ from that in
the static case.
It will
increase
or
decrease
, depending on the
direction of seepage.
The increasing in effective pressure due to the
flow of water through the pores of the soil is
known as
seepage pressure.
11
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12. A column of saturated soil mass with no seepage of
water in any direction.
The total stress at the
elevation of point
A
can be
obtained from the saturated
unit weight of the soil and
the unit weight of water
above it. Thus,
Stress due to self weight
Stresses in Saturated Soil without Seepage
0
A
Solid particle
Pore water
)
*
)
&
+
+
12
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13. 0
A
Solid particle
Pore water
)
*
)
&
+
+
+
+
Forces acting at the points of contact of soil
particles at the level of point A
Stress due to self weight
Stresses in Saturated
Soil without Seepage
H
GTLI
&L:bP:
*•:=I
%#! where H
GT OYOB&LCOkUCCLBOLOSUL
U&U+BOAYRLYFLzYAROL1
I
%#!T CBOEkBOU2LERAOL UAVSOL
YFLOSULCYA&
)
*T 2ACOBR3UL4UO UURLzYARO
L1LBR2LOSUL BOUkLOB4&UL
13
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14. Stress due to self weight
Stresses in Saturated Soil without Seepage
)
D
)
r
5
6
7
8
Valve (closed)
Stress at point A, •
Total stress:
•
Pore water pressure:
•
Effective stress:
H
*TLI
& )
r
E
*TLI
& )
r
H
*
'
TLH
*• E
*T 9
Stress at point B,
•
Total stress
:
•
Pore water pressure
•
Effective stress
:
H
:TLI
& )
rb:
DL.I
%#!
E
:TLP:
rb:
DL=I
&
H
:
'
TLH
:•E
:
H
:
'
TL:
D2I
%;<
14
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15. Stress due to self weight
Stresses in Saturated Soil without Seepage
Stress at point C , •
Total stress:
H
=TLI
& )
rbi.I
%#!
E
>TLP:
rbi=I
&
H
>
'
TLH
>•E
>
H
>
'
T i2I
%;<
•
Pore water pressure: •
Effective stress:
Total stress
Pore water
Pressure, uEffective stress
Depth DepthDepth
15
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16. )
D
)
r
5
6
7
8
Valve (open)
?
(
@
AB
=i
Stress due to self weight
Stresses in Saturated Soil with Upward Seepage
Stress at point A, •
Total stress:
•
Pore water pressure:
•
Effective stress:
H
*
TLI
&
)
r
E
*
TLI
&
)
r
H
*
'
TLH
*
•E
*
T9
16
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17. Stresses in Saturated Soil with Upward Seepage
Stress due to self weight
Stress at point B,
•
Total stress
:
•
Pore water pressure
•
Effective stress
:
H
:
TLI
&
)
r
b:
DL
.I
%#!
E
:
TLP:
r
b:
DL
bS=I
&
H
:
'
TLH
:
•E
:
H
:
'
TL:
D
2I
%;<
•SLI
&
17
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18. Stresses in Saturated Soil with Upward Seepage
Stress due to self weight
Stress at point C , •
Total stress:
•
Pore water pressure:
•
Effective stress:
H
=TLI
& )
rbi.I
%#!
E
:TLP:
rbib

)
D
i=I
&
H
>
'
TLH
>•E
>
H
>
'
T i2I
%;<−

)
D
iLI
&
H
>
'
T i2I
%;<•ALiLI
&
Note that h/H2 is the hydraulic gradient i
caused by the flow, and therefore
18
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19. Total stress
Pore water
Pressure, u
Effective stress
Depth Depth Depth
Stress due to self weight
Stresses in Saturated Soil with Upward Seepage
19
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20. Stress due to self weight
Stresses in Saturated Soil with Upward Seepage
At any depth z, is the pressure of the
submerged soil acting downward and is the
seepage pressure acting upward.
The effective pressure reduces to zero when these two
pressures balance.
This situation generally is referred to as boiling.
H
>
'
Ti2I
%;<
•A
>C
LiLI
&
T9
A
>C
T
I
%;< I
&
SUkULLA
>CT3kAOA3B&LSD2kBE&A3LVkB2AURO
For most soils,the value of
A
>C
varies from 0.9 to 1.1
i2I
%;<
ALiLI
&
H
>
'
20
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21. )
D
)
r
5
6
7
8
Valve (open)
?
(
@
AB
=i
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
Stress at point A, •
Total stress:
•
Pore water pressure:
•
Effective stress:
H
*
TLI
&
)
r
E
*
TLI
&
)
r
H
*
'
TLH
*
•E
*
T9
21
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22. Stress at point B,
•
Total stress
:
•
Pore water pressure
•
Effective stress
:
H
:
TLI
&
)
r
b:
DL
.I
%#!
E
:
TLP:
r
b:
DL
•S=I
&
H
:
'
TLH
:
•E
:
H
:
'
TL:
D
2I
%;<
bSLI
&
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
22
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23. Stress due to self weight
Stress at point C , •
Total stress:
•
Pore water pressure:
•
Effective stress:
H
=TLI
& )
rbi.I
%#!
E
:TLP:
rbi•

)
D
i=I
&
H
>
'
TLH
>•E
>
H
>
'
Ti2I
%;<b

)
D
iLI
&H
>
'
Ti2I
%;<bALiLI
&
Stresses in Saturated Soil with Downward Seepage
23
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24. Pore water
Pressure, u
Total stressEffective stress
Depth Depth Depth
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
24
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25. Worked Examples
Example 1
A soil profile is shown in figure below. Calculate total
stress, pore water pressure, and effective stress a t A,
B, C, and D.
D
C
B
A
Ground surface
G.W.T
Sand
Clay
Sand
γ
= 16.3 kN/m^3
γ
= 15.1 kN/m^3
γ
= 19.8 kN/m^3
1.8 m
1.6 m
2.9 m
25
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26. Stress due to self weight
Total stressEffective stress Pore water pressure
Depth Depth Depth
γ1X H1
γ1X H1 + γ2X H2
γ1X H1 + γ2X H2 + γ3 XH3
γ1X H1 + γ2X H2 + γsub XH3
γwX Hw
26
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27. To analyze problems such as compressibility of
soils, bearing capacity of foundations, stability
of embankments, and lateral pressure on earth
retaining structures, we need to know the
nature of the distribution of stress along a
given cross section of the soil profile.
Stress due to surface load
Introduction
27
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28. When a load is applied to the soil surface, it
increases the vertical stresses within the soil
mass. The increased stresses are greatest
directly under the loaded area, but extend
indefinitely in all directions.
Introduction
28
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Stress due to surface load

29. •Allowable settlement, usually set by building
codes, may control the allowable bearing
capacity.
•The vertical stress increase with depth must
be determined to calculate the amount of
settlement that a foundation may undergo
Introduction
29
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Stress due to surface load

30. Introduction Foundations and structures placed on the
surface of the earth will produce stresses in
the soil
These stresses will decrease with the
distance from the load
How these stresses decrease depends upon
the nature of the soil bearing the load
30
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Stress due to surface load

31. Individual column footings or wheel loads
may be replaced by equivalent point loads
provided that the stresses are to be
calculated at points sufficiently far from
the point of application of the point load.
Stress Due to a Concentrated Load
31
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Stress due to surface load

32. Stresses in soil due to surface load
Verticalstress due to a concentrated load •Boussinesq’s Formula •Wastergaard Formula Stress Due to a Concentrated Load
32
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33. Stress Due to a Concentrated Load
Boussinesq’s Formula for Point Loads
Joseph Valentin Boussinesq
(13 March 1842 –19 February
1929) was aFrench mathematician and physicist who m ade
significant contributions to the theory of hydrodyn amics, vibration,
light, and heat.
33
Dr. Abdulmannan Orabi IUSTStresses in soil due to surface load

34. In 1885,
Boussinesq
developed the
mathematical relationships for determining
the normal and shear stresses at any point
inside a homogenous, elastic and isotropic
mediums due to a concentrated point loads
located at the surface
Vertical Stress in Soil
Stress Due to a Concentrated Load
34
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35. The soil mass is elastic, isotropic (having
identical properties in all direction
throughout), homogeneous (identical elastic
properties) and semi-infinite depth.
The soil is weightless.
Stress Due to a Concentrated Load
Assumption:
35
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Vertical Stress in Soil

36. The distribution of
σ
zin the elastic medium
is apparently radially symmetrical.
The stress is infinite at the surface directly
beneath the point load and decreases with the
square of the depth.
Vertical Stress in Soil
Stress Due to a Concentrated Load
36
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37. At any given non-zero radius, r, from the point
of load application, the vertical stress is zero
at the surface, increases to a maximum value at
a depth where , approximately, and
then decreases with depth .
E TFGHIJ
°
Vertical Stress in Soil
Stress Due to a Concentrated Load
37
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38. Vertical Stress in Soil
According to Boussinesq’s analysis, the vertical st ress
increase at point A caused by a point load of magni tude P
is given by
Stress Due to a Concentrated Load
D
LH
G
LH
M
LH
N
O
i
P
Q
.
P
D
i
1
38
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39. Vertical Stress in Soil
Stress Due to a Concentrated Load
According to Boussinesq’s analysis, the vertical st ress
increase at point A caused by a point load of magni tude P
is given by
2 2 5/2
3 1
2 [1 ( / ) ]
z
P
z r z
σ
π
=
+
39
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…….
7−1
1
LH
G
.Q
i
or
2
z b
P
I
z
σ
=

40. Equation shows that the vertical stress is I
Directly proportional to the load
I
Inversely proportional to the depth squared, and
I
Proportional to some function of the ratio ( r/z).
Vertical Stress in Soil
Stress Due to a Concentrated Load
where
2 5/2
3 1
2 [1 ( / ) ]
b
I
r z
π
=
+
40
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………….
7−2

41. It should be noted that the expression for z is
independent of elastic modulus (E) and
Poisson’s ratio (
µ
), i.e. stress increase with depth
is a function of geometry only. Vertical Stress in Soil
Stress Due to a Concentrated Load
41
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42. r/Z I
Br/Z I
Br/Z I
Br/Z I
B
0.00
0.4775
0.18
0.4409
0.36
0.3521
0.55
0.2466
0.01
0.4773
0.19
0.4370
0.37
0.3465
0.56
0.2414
0.02
0.477
0.20
0.4329
0.38
0.3408
0.57
0.2363
0.03
0.4764
0.21
0.4286
0.39
0.3351
0.58
0.2313
0.04
0.4756
0.22
0.4242
0.40
0.3294
0.59
0.2263
0.05
0.4745
0.23
0.4197
0.41
0.3238
0.60
0.2214
0.06
0.472
0.24
0.4151
0.42
0.3181
0.61
0.2165
0.07
0.4717
0.25
0.4103
0.43
0.3124
0.62
0.2117
0.08
0.4699
0.26
0.4054
0.44
0.3068
0.63
0.2070
0.09
0.4679
0.27
0.4004
0.45
0.3011
0.64
0.2024
0.1
0.4657
0.28
0.3954
0.46
0.2955
0.65
0.1978
0.11
0.4633
0.29
0.3902
0.47
0.2899
0.66
0.1934
0.12
0.4607
0.30
0.3849
0.48
0.2843
0.67
0.1889
0.13
0.4579
0.31
0.3796
0.49
0.2788
0.68
0.1846
0.14
0.4548
0.32
0.3742
0.50
0.2733
0.69
0.1804
0.15
0.4516
0.33
0.3687
0.51
0.2679
0.70
0.1762
0.16
0.4482
0.34
0.3632
0.52
0.2625
0.71
0.1721
0.17
0.4446
0.35
0.3577
0.53
0.2571
0.72
0.1681
0.54
0.2518
0.73
0.1641
Influence Factor I b
42
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43. r/Z I
Br/Z I
Br/Z I
Br/Z I
B
0.74
0.1603
0.94
0.0981
1.14
0.0595
1.34
0.0365
0.75
0.1565
0.95
0.0956
1.15
0.0581
1.35
0.0357
0.76
0.1527
0.96
0.0933
1.16
0.0567
1.36
0.0348
0.77
0.1491
0.97
0.0910
1.17
0.0553
1.37
0.0340
0.78
0.1455
0.98
0.0887
1.18
0.0539
1.38
0.0332
0.79
0.1420
0.99
0.0865
1.19
0.0526
1.39
0.0324
0.80
0.1386
1.0
0.0844
1.20
0.0513
1.40
0.0317
0.81
0.1353
1.01
0.0823
1.21
0.0501
1.41
0.0309
0.82
0.1320
1.02
0.0803
1.22
0.0489
1.42
0.0302
0.83
0.1288
1.03
0.0783
1.23
0.0477
1.43
0.0295
0.84
0.1257
1.04
0.0764
1.24
0.0466
1.44
0.0283
0.85
0.1226
1.05
0.0744
1.25
0.0454
1.45
0.0282
0.86
0.1196
1.06
0.0727
1.26
0.0443
1.46
0.0275
0.87
0.1166
1.07
0.0709
1.27
0.0433
1.47
0.0269
0.88
0.1138
1.08
0.0691
1.28
0.0422
1.48
0.0263
0.89
0.1110
1.09
0.0674
1.29
0.0412
1.49
0.0257
0.90
0.1083
1.10
0.0658
1.30
0.0402
1.50
0.0251
0.91
0.1057
1.11
0.0641
1.31
0.0393
1.51
0.0245
0.92
0.1031
1.12
0.0626
1.32
0.0384
1.52
0.0240
0.93
0.1005
1.13
0.0610
1.33
0.0374
1.53
0.0234
43
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Influence Factor I b

44. r/Z I
Br/Z I
Br/Z I
Br/Z I
B
1.54
0.0229
1.66
0.0175
1.86
0.0114
2.5
0.0034
1.55
0.0224
1.67
0.0171
1.88
0.0109
2.6
0.0029
1.56
0.0219
1.68
0.0167
1.90
0.0105
2.7
0.0024
1.57
0.0214
1.69
0.0163
1.92
0.0101
2.8
0.0021
1.58
0.0209
1.70
0.0160
1.94
0.0097
2.9
0.0017
1.59
0.0204
1.72
0.0153
1.96
0.0093
3.0
0.0015
1.60
0.0200
1.74
0.0147
1.98
0.0089
3.5
0.0007
1.61
0.0195
1.76
0.0141
2.0
0.0085
4.0
0.0004
1.62
0.0191
1.78
0.0135
2.1
0.0070
4.5
0.0002
1.63
0.0187
1.80
0.0129
2.2
0.0058
5.0
0.0001
1.64
0.0183
1.82
0.0124
2.3
0.0048
1.65
0.0179
1.84
0.0119
2.4
0.0040
44
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Influence Factor I b

45. Equation may be used to draw three types of pressur e
distribution diagram. They are:
I
The vertical stress distribution on a horizontal
plane at depth of z below the ground surface
I
The vertical stress distribution on a vertical plan e
at a distance of r from the load point, and
I
The stress isobar.
Vertical Stress in Soil
Pressure Distribution Diagram
45
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46. I
The vertical stress distribution on a horizontal
plane at depth of z below the ground surface
U
5
u
5
l
Vertical Stress in Soil
Distribution on a horizontal plane
46
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47. I
The vertical stress
distribution on a vertical
plane at a distance of
r
from the point load
.
H
G
7
Vertical Stress in Soil
Distribution on a vertical plane
O
47
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48. H
Gr
H
Ga
H
GD
U
Vertical Stress in Soil
Stress isobars
An
isobar
is a line which
connects all
points of equal
stress
below the ground
surface. In other words, an
isobar is a
stress contour
.
48
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49. What is the vertical stress at point A of figure be low
for the two loads, P 1 and P2 ?
P1= 350 kN P2= 470 kN
Z= 2.5 m
2.3 m 1.1 m
A
Worked Examples
Example 2
49
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50. A four concentrated forces are located at corners o f
a rectangular area with dimensions 8 m by 6 m as
shown in figure in the next slide. Compute the
vertical stress at points A and B, which are locate d
on the lines A –A’ , B –B’ at depth of 4 m below
the ground surface.
Worked Examples
Example 3
50
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51. 700 kN
700 kN
700 kN 700 kN
4 m
4 m
8 m
B
A’
A
B’
Worked Examples
Example 3
Dr. Abdulmannan Orabi IUST
51

52. Vertical Stress in Soil
Westergaard Formula
Westergaard proposed a formula for the
computation of vertical stress H
G
by a point load,
P at the surface as
H
GT
O +
IVi
D
+
D
bL
.
i
D
aWD
In which
µ
is Poisson’s ratio
BT
1−2X /,2−2X-
52
Dr. Abdulmannan Orabi IUST
….
7−3

53. Vertical Stress in Soil
Stress below a Line Load
The vertical stress increase due to line load , ,
inside the soil mass can be determined by using the
principles of the theory of elasticity, or
H
G
H
G
T
IL3Li
a
V
P
D
bi
D D
This equation can be rewritten as
H
G
3Wi
T
2
V
1
b
Pi
D
D
1
i
P
P
53
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….
7−4

54. Vertical Stress in Soil
Vertical Stress caused by a horizontal line load
The vertical stress increase ( H
G
) at point A in
the soil mass caused by a horizontal line load
can be given as :
H
G
T
IL3LLPLi
D
V
P
D
bi
D D
1
i
3LWERAOL&URVSOL
P
54
Dr. Abdulmannan Orabi IUST
….
7−5

55. Vertical Stress in Soil
Vertical Stress caused by a strip load
The fundamental equation for the vertical stress
increase at a point in a soil mass as the result of
a line load can be used to determine the vertical
stress at a point caused by a flexible strip load o f
width B.
The term strip loading will be used to indicate a
loading that has a finite width along the x axis
but an infinite length along the y axis.
55
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56. Vertical Stress in Soil
Vertical Stress caused by a strip load
α
β
6
i
n
B
Vertical stress at point A can be determined by equation:
[ sin cos( 2 )]
o
z
q
σ α α α β
π
= + +
P
56
Dr. Abdulmannan Orabi IUST
….
7−6

57. B
[
i T 9HIJL\
n
i T 9HJL\
i T \
]
^
0.5 \
0.25 \
Worked Examples
Example 4
Refer to figure below, The magnitude of the strip
load is 120 kPa. Calculate the vertical stress at
points, a , b, and c.
57
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58. _
r
6
4

"
+
_
D
1 2 2
[( )( ) ( )]
o
z
qa b b
a a
σ α α α
π
+
= + −Vertical Stress Due to Embankment Loading
The vertical stress increase in the soil mass due t o
an embankment of height H may be expressed as
Vertical Stress in Soil

"
TI2L:
where:
ITERAOL UAVSOLYFLU`4BRa`UROLLCYA& )
T

UAV

O
YF
O


U`4BRa`URO
58
Dr. Abdulmannan Orabi IUST
….
7−7

59. ^
7
6
2 `
120 aO+
3 `
2 `
Refer to figure below. The magnitude of the load is
120 kPa. Calculate the vertical stress at points,
A , B, and C.
Worked Examples
Example 4
59
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60. 1-Under the center
:
The increase in the vertical
stress (H
G) at depth z ( point A)under the center
of a circular area of diameter D = 2R carrying
a uniform pressure q is given by
Vertical Stress in Soil
Vertical Stress due to a uniformly loaded circular area
H
G
T3
1−
1
QWi
D
bT
aWD
60
Dr. Abdulmannan Orabi IUST
….
7−8

61. Vertical Stress in Soil
Vertical Stress due to a uniformly loaded circular area
6
6
'
i
n
Q
6
'
6
i

61
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62. Vertical Stress in Soil
2-At any point:
The increase in the vertical
stress (H
G
)at any point located at a depth z at
any distance r from the center of the loaded
area can be given
Vertical Stress due to a uniformly loaded circular area
where
and are functions of z/R and r/R.
H
G
T3
1
'
b\
'
1
'
\
'
62
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….
7−9

63. Vertical Stress in Soil
Vertical Stress due to a uniformly loaded circular area
i
7
7
'
.
i
n
Q
7
7
'
.


63
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64. Vertical Stress in Soil
Variation of with z/Rand r/R.
1
'
64
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65. Vertical Stress in Soil
Variation of with z/Rand r/R. 1
'
65
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66. Variation of with z/Rand r/R. \
'
Vertical Stress in Soil
66
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67. Vertical Stress in Soil
Variation of with z/Rand r/R. \
'
67
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68. Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
The increase in the vertical stress ( H
G
)at depth z under a
corner of a rectangular area of dimensions
B = m z
and
L = n z
carrying a uniform pressure q is given by:
z o z
q I
σ
=
c
GTARF&EUR3ULFB3OYkL2UzUR2ARVLYRLOSULkBOAYL
d
i
LBR2L
\
i

where :
68
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….
7−10

69. Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
c
GT
1
4V
IL`LR
`
D
b R
D
b TL
`
D
b R
D
b `
D
R
D
b T
`
D
b R
D
b I
`
D
b R
D
b T
b OBR
er
IL`LR
`
D
b R
D
b T
`
D
b R
D
− `
D
R
D
b T
The influence factor
can be expressed as
` T
d
i
LLLBR2LR T
\
i
where :
69
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….
7−11

70. The increase in the stress at any point below a
rectangular loaded area can be found by dividing
the area into four rectangles. The point A’ is the
corner common to all four rectangles.
Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
1 2
3 4
6
'
H
G(
TH
Gr
bH
GD
bH
Ga
bH
Gf
H
Gg
T3c
Gg
70
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71. Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
71
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4
6
'
6
' 6
'
6
'
− h
5l
bLLh
5i − h
5m
+ h
5u
h
5
43
4
2
9
1
12
3
5
5
1
8
79
8 7
4
7
3
H
G(
T
H
Gr
−
H
GD
−
H
Ga
b
H
Gf

72. Variation of with m and n
c
G
n
m
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
0.1 0.0047 0.0092 0.0132 0.0168 0.0198 0.0222 0.0242 0.0258
0.2 0.0092 0.0179 0.0259 0.0328 0.0387 0.0435 0.0474 0.0504
0.3 0.0132 0.0259 0.0374 0.0474 0.0559 0.0629 0.0686 0.0731
0.4 0.0168 0.0328 0.0474 0.0602 0.0711 0.0801 0.0873 0.0931
0.5 0.0198 0.0387 0.0559 0.0711 0.0840 0.0947 0.1034 0.1104
0.6 0.0222 0.0435 0.0629 0.0801 0.0947 0.1069 0.1168 0.1247
0.7 0.0242 0.0474 0.0686 0.0873 0.1034 0.1169 0.1277 0.1365
0.8 0.0258 0.0504 0.0731 0.0931 0.1104 0.1247 0.1365 0.1461
0.9 0.0270 0.0528 0.0766 0.0977 0.1158 0.1311 0.1436 0.1537
1.0 0.0279 0.0547 0.0794 0.1013 0.1202 0.1361 0.1491 0.1598
72
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73. Variation of with m and n
c
G
n
m
0.9 1 1.2 1.4 1.6 1.8 2.0 2.5
0.1 0.0270 0.0279 0.0293 0.0301 0.0306 0.0309 0.0311 0.0314
0.2 0.0528 0.0547 0.0573 0.0589 0.0599 0.0606 0.0610 0.0616
0.3 0.0766 0.0794 0.0832 0.0856 0.0871 0.0880 0.0887 0.0895
0.4 0.0977 0.1013 0.1063 0.1094 0.1114 0.1126 0.1134 0.1145
0.5 0.1158 0.1202 0.1263 0.1300 0.1324 0.1340 0.1350 0.1363
0.6 0.1311 0.1361 0.1431 0.1475 0.1503 0.1521 0.1533 0.1548
0.7 0.1436 0.1491 0.1570 0.1620 0.1652 0.1672 0.1686 0.1704
0.8 0.1537 0.1598 0.1684 0.1739 0.1774 0.1797 0.1812 0.1832
0.9 0.1619 0.1684 0.1777 0.1836 0.1875 0.1899 0.1915 0.1938
1.0 0.1684 0.1752 0.1851 0.1914 0.1955 0.1981 0.1999 0.2024
73
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74. Variation of with m and n
c
G
n
m
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
1.2 0.0293 0.0573 0.0832 0.1063 0.1263 0.1431 0.1570 0.1684
1.4 0.0301 0.0589 0.0856 0.1094 0.1300 0.1475 0.1620 0.1739
1.6 0.0306 0.0599 0.0871 0.1114 0.1324 0.1503 0.1652 0.1774
1.8 0.0309 0.0606 0.0880 0.1126 0.1340 0.1521 0.1672 0.1797
2.0 0.0311 0.0610 0.0887 0.1134 0.1350 0.1533 0.1686 0.1812
2.5 0.0314 0.0616 0.0895 0.1145 0.1363 0.1548 0.1704 0.1832
3.0 0.0315 0.0618 0.0898 0.1150 0.1368 0.1555 0.1711 0.1841
4.0 0.0316 0.0619 0.0901 0.1153 0.1372 0.1560 0.1717 0.1847
5.0 0.0316 0.0620 0.0901 0.1154 0.1374 0.1561 0.1719 0.1849
6.0 0.0316 0.0620 0.0902 0.1154 0.1374 0.1562 0.1719 0.1850
74
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75. Variation of with m and n
c
G
n
m
0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5
1.2 0.1777 0.1851 0.1958 0.2028 0.2073 0.2103 0.2124 0.2151
1.4 0.1836 0.1914 0.2028 0.2102 0.2151 0.2184 0.2206 0.2236
1.6 0.1874 0.1955 0.2073 0.2151 0.2203 0.2237 0.2261 0.2294
1.8 0.1899 0.1981 0.2103 0.2183 0.2237 0.2274 0.2299 0.2333
2.0 0.1915 0.1999 0.2124 0.2206 0.2261 0.2299 0.2325 0.2361
2.5 0.1938 0.2024 0.2151 0.2236 0.2294 0.2333 0.2361 0.2401
3.0 0.1947 0.2034 0.2163 0.2250 0.2309 0.2350 0.2378 0.2420
4.0 0.1954 0.2042 0.2172 0.2260 0.2320 0.2362 0.2391 0.2434
5.0 0.1956 0.2044 0.2175 0.2263 0.2324 0.2366 0.2395 0.2439
6.0 0.1957 0.2045 0.2176 0.2264 0.2325 0.2367 0.2397 0.2441
75
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76. Approximate Method
B
B + z
2
1
z
H
G

"
O
76
Dr. Abdulmannan Orabi IUST
2V:1H method
A simple but approximate method is sometimes used f or
calculating the stress change at various depths as a
result of the application of a pressure at the grou nd
surface.
The transmission of stress is
assumed to follow outward
fanning lines at a slope of
1
horizontal to 2 vertical.

77. Approximate Method
For uniform footing (B x L) we can estimate the
change in vertical stress with depth using the Bost on
Rule. Assumes stress at depth is constant below
foundation influence area
B
B + z
2
1
z
H
GT

" d \
Pdbi=LP\bi=L
H
G

"
O

"T
O
d2L\L
77
Dr. Abdulmannan Orabi IUST
….
7−12
2V:1H method

78. Approximate Method
B + z
L
B
z
Stress on this plane

"T
j
d∗\
Stress on this plane at depth z,
H
GT

" d \
Pd b i=LP\ b i=L
Rectangular footing
B
B + z
2
1
78
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2V:1H method

79. Newmark Method
79
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•Stresses due to foundation loads of
arbitrary
shape
applied at the ground surface
•Newmark’s chart provides a graphical
method for calculating the stress increase due
to a uniformly loaded region, of arbitrary
shape resting on a deep homogeneous
isotropic elastic region.

80. Newmark Method
•The Newmark’s Influence Chart method
consists of concentric circles drawn to scale,
each square contributes a fraction of the
stress.
•In most charts each square contributes
1/200 (or 0.005) units of stress. (influence
value, I)
80
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81. Newmark Method
81
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The use of the chart is
based on a factor
termed the influence
value, determined from
the number of units
into which the chart is
subdivided.
Influence value 0.005
A
B
1 unit

82. Newmark Method
A B
Influence
value = 0.005
Total number of block on chart = 200 and influence
value = 1/200

83. The influence chart may be used to compute
the pressure on an element of soil beneath a
footing, or from pattern of footings, and for
any depth z below the footing. It is only
necessary to draw the footing pattern to a
scale of z = length AB of the chart. (If z=
6m
and AB =
30mm
, the scale is
1/200
).
Newmark Method
83
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84. The footing plan will be placed on the influence
chart with the point for which the stress is desire d at
the center of the circles.
Newmark Method
The units (segments or partial segments) enclosed
by the footing are counted, and the increase in
stress at the depth z is computed as
H
GT3
" c j
Where
I
is the influence factor of the chart
.

"T Bzz&AU2LzkCCEkULYRLOSULBkUB
LFYER2BOAYRL3YROB3OLzkUCCEkUL L
j T RE`4UkLYFLERAOCL3YEROU2LPLzBkOAB&LERAOCLBkULUCOA` BOU2=
84
Dr. Abdulmannan Orabi IUST
….
7−13

85. Newmark Method
85
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