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INTERNATIONAL UNIVERSITY FOR SCIENCE & TECHNOLOGY
�م وا����������� ا������ ا��و��� ا����� �
CIVIL ENGINEERING AND
ENVIRONMENTAL DEPARTMENT
303322 - Soil Mechanics
Stress Distribution in Soil
Dr. Abdulmannan Orabi
Lecture
2
Lecture
7
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
� Stress in soil due to self weight
Stress Distribution in Soil
� Stress in soil due to surface load
3Dr. Abdulmannan Orabi IUST
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
�
Variation of stresses with depth
A
Ground surface
�
zz ⋅= γσ
� � ��������
�� ��
�� � � � �
4Dr. Abdulmannan Orabi IUST
∑=
⋅=⋅++⋅+⋅=n
i
iinnz hhhh1
2211 ...... γγγγσ
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 as
Vertical stress at depth z1
Vertical stress at depth z2
Vertical stress at depth z3
��
��� � �� ∗ ��
��� � �� ∗ �� � �� ∗ ��
��
��
��
��
����
��
��
�� ∗ ��
�� ∗ �� � �� ∗ ��
��� � �� ∗ �� � �� ∗ �� � �� ∗ ��5Dr. Abdulmannan Orabi IUST
With uniform surcharge on infinite land surface
Stress due to self weight
Original land surface
Conversion land surface
� �
�� � � ∗ � �
� ��
�
�� � ��������
�� ��
6Dr. Abdulmannan Orabi IUST
Stress due to self weight
�� � � ∗ �
Vertical stresses due to self weight increase with depth,There are 3 types of geostatic stresses:
a. Total Stress, σtotal
b. Effective Stress, σ'c. Pore Water Pressure, u
Vertical Stresses
7Dr. Abdulmannan Orabi IUST
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
��!"!#$ � �%#! ∗ �
8Dr. Abdulmannan Orabi IUST
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
� � �& ∗ �
If the pores of a soil mass are filled with water and if a pressure induced into the pore water, tries to separate the grains, this pressure is termed as pore water pressure
9Dr. Abdulmannan Orabi IUST
Stress due to self weight
Effective vertical stress due to self weight of soil
The difference between the total stress (��!"!#$) and the pore pressure (u) in a saturated soil has been defined by Terzaghi as the effective stress ( ).��
'
��' � ��!"!#$ − �
The pressure transmitted through grain to grain at the contact points through a soil mass is termed as effective pressure.
10Dr. Abdulmannan Orabi IUST
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.
11Dr. Abdulmannan Orabi IUST
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
0A
Solid particle
Pore water
)*
)&
+
+
12Dr. Abdulmannan Orabi IUST
0A 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
�� ��&) � ,)* − )-�%#!
where �� � � �+���.��+���
�/+�� � �0 ���1
�%#! � �+��.+�2��������
���� ��
)* � 2���+�34��0 ���
1+�2��+�.�+4�
13Dr. Abdulmannan Orabi IUST
Stress due to self weight
Stresses in Saturated Soil without Seepage
)�
)�
5
6
7
8
Valve (closed)
Stress at point A,
• Total stress:• Pore water pressure:• Effective stress:
�* ��&)�
�* ��&)�
�*' ��* − �* � 0
Stress at point B,• Total stress:
• Pore water pressure• Effective stress:
�: ��&)� �)� ∗ �%#!
�: � ,)��)�-�&
�:' � �: − �:
�:' � )� � �%;<
14Dr. Abdulmannan Orabi IUST
Stress due to self weight
Stresses in Saturated Soil without Seepage
Stress at point C,• Total stress:
�= ��&)� � � ∗ �%#!
�> �,)���-�&
�>' ��> − �>
�>' � � � �%;<
• Pore water pressure:
• Effective stress:
Total stress
Pore water Pressure, u Effective stress
DepthDepth Depth
15Dr. Abdulmannan Orabi IUST
)�
)�
5
6
7
8
Valve (open)
?
(@
AB-�
Stress due to self weight
Stresses in Saturated Soil with Upward Seepage
Stress at point A,• Total stress:
• Pore water pressure:
• Effective stress:
�* ��&)�
�* ��&)�
�*' ��* − �* � 0
16Dr. Abdulmannan Orabi IUST
Stresses in Saturated Soil with Upward Seepage
Stress due to self weight
Stress at point B,• Total stress:
• Pore water pressure
• Effective stress:
�: ��&)� � )� ∗ �%#!
�: � ,)��)� � �-�&
�:' ��: − �:
�:' �)� � �%;< − ��&
17Dr. Abdulmannan Orabi IUST
Stresses in Saturated Soil with Upward Seepage
Stress due to self weight
Stress at point C,
• Total stress:• Pore water pressure:• Effective stress:
�= ��&)� � � ∗ �%#!
�: � ,)��� ��
)�
�-�&
�>' ��> − �>
�>' � � � �%;< −
�
)�
��&
�>' � � � �%;< − ���&
Note that h/H2 is the hydraulic gradient icaused by the flow, and therefore
18Dr. Abdulmannan Orabi IUST
Total stressPore water Pressure, u
Effective stress
DepthDepth Depth
Stress due to self weight
Stresses in Saturated Soil with Upward Seepage
19Dr. Abdulmannan Orabi IUST
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.
�>' � � � �%;< − �>C ��& � 0
�>C ��%;<
�&�.�>C � 3.���3+��D2.+���3�.+2���
For most soils, the value of �>C varies from 0.9 to 1.1
� � �%;<���&
�>'
20Dr. Abdulmannan Orabi IUST
)�
)�
5
6
7
8
Valve (open)
?
(@
AB-�
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
Stress at point A,• Total stress:
• Pore water pressure:
• Effective stress:
�* ��&)�
�* ��&)�
�*' ��* − �* � 0
21Dr. Abdulmannan Orabi IUST
Stress at point B,• Total stress:
• Pore water pressure
• Effective stress:
�: ��&)� � )� ∗ �%#!
�: � ,)��)� − �-�&
�:' ��: − �:
�:' �)� � �%;< � ��&
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
22Dr. Abdulmannan Orabi IUST
Stress due to self weight
Stress at point C,• Total stress:
• Pore water pressure:
• Effective stress:
�= ��&)� � � ∗ �%#!
�: � ,)��� −�
)�
�-�&
�>' ��> − �>
�>' � � � �%;< �
�
)�
��& �>' � � � �%;< � ���&
Stresses in Saturated Soil with Downward Seepage
23Dr. Abdulmannan Orabi IUST
Pore water Pressure, uTotal stress Effective stress
DepthDepth Depth
Stress due to self weight
Stresses in Saturated Soil with Downward Seepage
24Dr. Abdulmannan Orabi IUST
Worked Examples
Example 1
A soil profile is shown in figure below. Calculate total stress, pore water pressure, and effective stress at 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
25Dr. Abdulmannan Orabi IUST
Stress due to self weight
Total stress Effective stress Pore water pressure
DepthDepthDepth
γ1 X H1
γ1 X H1 + γ2 X H2
γ1 X H1 + γ2 X H2 + γ3 X H3
γ1 X H1 + γ2 X H2 + γsub X H3
γw X Hw
26Dr. Abdulmannan Orabi IUST
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
27Dr. Abdulmannan Orabi IUST
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
28Dr. Abdulmannan Orabi IUST
Stress due to surface load
•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
29Dr. Abdulmannan Orabi IUST
Stress due to surface load
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
30Dr. Abdulmannan Orabi IUST
Stress due to surface load
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
31Dr. Abdulmannan Orabi IUST
Stress due to surface load
Stresses in soil due to surface load
Vertical stress due to a concentrated load • Boussinesq’s Formula • Wastergaard Formula
Stress Due to a Concentrated Load
32Dr. Abdulmannan Orabi IUST
Stress Due to a Concentrated Load
Boussinesq’s Formula for Point Loads
Joseph Valentin Boussinesq (13 March 1842 – 19 February
1929) was a French mathematician and physicist who made significant contributions to the theory of hydrodynamics, vibration, light, and heat.
33Dr. Abdulmannan Orabi IUST
Stresses in soil due to surface load
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
34Dr. Abdulmannan Orabi IUST
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:
35Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
The distribution of σz in 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
36Dr. Abdulmannan Orabi IUST
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 � 39.25°
Vertical Stress in Soil
Stress Due to a Concentrated Load
37Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by
Stress Due to a Concentrated Load
D
∆��
∆�M ∆�N
O
�
P
Q
.P
D
�
1
38Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Stress Due to a Concentrated Load
According to Boussinesq’s analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by
2 2 5/2
3 1
2 [1 ( / ) ]z
P
z r zσ
π=
+
39Dr. Abdulmannan Orabi IUST
…… . 7 − 1
1
∆��.
Q
�
or
2z b
PI
zσ =
Equation shows that the vertical stress is
� Directly proportional to the load
� Inversely proportional to the depth squared, and
� 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 ( / ) ]bI
r zπ=
+
40Dr. Abdulmannan Orabi IUST
………… . 7 − 2
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
41Dr. Abdulmannan Orabi IUST
r/Z IB
r/Z IB
r/Z IB
r/Z IB
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 Ib
42Dr. Abdulmannan Orabi IUST
r/Z IB r/Z IB r/Z IB r/Z IB
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
43Dr. Abdulmannan Orabi IUST
Influence Factor Ib
r/Z IB r/Z IB r/Z IB r/Z IB
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
44Dr. Abdulmannan Orabi IUST
Influence Factor Ib
Equation may be used to draw three types of pressure distribution diagram. They are:
� The vertical stress distribution on a horizontal plane at depth of z below the ground surface
� The vertical stress distribution on a vertical plane at a distance of r from the load point, and
� The stress isobar.
Vertical Stress in Soil
Pressure Distribution Diagram
45Dr. Abdulmannan Orabi IUST
� The vertical stress distribution on a horizontal plane at depth of z below the ground surface
U
5�
5�
Vertical Stress in Soil
Distribution on a horizontal plane
46Dr. Abdulmannan Orabi IUST
�The vertical stress distribution on a vertical plane at a distance of rfrom the point load
.��
�
Vertical Stress in Soil
Distribution on a vertical plane O
47Dr. Abdulmannan Orabi IUST
���
���
���
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.
48Dr. Abdulmannan Orabi IUST
What is the vertical stress at point A of figure below for the two loads, P1 and P2 ?
P1 = 350 kNP2 = 470 kNZ
= 2
.5 m
2.3 m 1.1 m
A
Worked Examples
Example 2
49Dr. Abdulmannan Orabi IUST
A four concentrated forces are located at corners of 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 located on the lines A – A’ , B – B’ at depth of 4 m below the ground surface.
Worked Examples
Example 3
50Dr. Abdulmannan Orabi IUST
700 kN700 kN
700 kN700 kN
4 m
4 m
8 m
B
A’
A
B’
Worked Examples
Example 3
Dr. Abdulmannan Orabi IUST 51
Vertical Stress in Soil
Westergaard Formula
Westergaard proposed a formula for the computation of vertical stress �� by a point load, P at the surface as
�� �O+
2V�� +� �.�
� �/�
In which µ is Poisson’s ratio
+ � 1 − 2X /,2 − 2X-
52Dr. Abdulmannan Orabi IUST
… . 7 − 3
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
��
�� �2 ��
V P� � �� �
This equation can be rewritten as��
/��
2
V 1 �P�
� � 1
�
P
P
53Dr. Abdulmannan Orabi IUST
… . 7 − 4
Vertical Stress in Soil
Vertical Stress caused by a horizontal line load
The vertical stress increase (��) at point A in the soil mass caused by a horizontal line load can be given as :
�� �2 P��
V P� � �� �
1
�
/���������
P
54Dr. Abdulmannan Orabi IUST
… . 7 − 5
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 of 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.
55Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Vertical Stress caused by a strip load
αβ
6
�
�
B
Vertical stress at point A can be determined by equation:
[ sin cos ( 2 ) ]oz
qσ α α α β
π= + +
P
56Dr. Abdulmannan Orabi IUST
… . 7 − 6
B
[
� � 0.25\
���0.5\
� � \
]
^
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.
57Dr. Abdulmannan Orabi IUST
_�
6
4
"
+
_�
1 2 2[( ) ( ) ( )]oz
q a b b
a aσ α α α
π
+= + −
Vertical Stress Due to Embankment Loading
The vertical stress increase in the soil mass due to an embankment of height H may be expressed as
Vertical Stress in Soil
" � � � )where:
� � �������� �`4+�a`��� ��
) � ����� ���`4+�a`��
58Dr. Abdulmannan Orabi IUST
… . 7 − 7
^ 7 6
2`
120aO+
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
59Dr. Abdulmannan Orabi IUST
1- Under the center: The increase in the vertical
stress (��) 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
�� � 1 −1
Q/� � � 1 �/�
60Dr. Abdulmannan Orabi IUST
… . 7 − 8
Vertical Stress in Soil
Vertical Stress due to a uniformly loaded circular area
6
6'
�
�
Q6'
6
�
61Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
2- At any point: The increase in the vertical stress (��) 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.
�� � 1' � \'
1' \'
62Dr. Abdulmannan Orabi IUST
… . 7 − 9
Vertical Stress in Soil
Vertical Stress due to a uniformly loaded circular area
�
7
7'
.
�
�
Q
7
7'.
63Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Variation of with z/R and r/R. 1'
64Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Variation of with z/R and r/R. 1'
65Dr. Abdulmannan Orabi IUST
Variation of with z/R and r/R. \'
Vertical Stress in Soil
66Dr. Abdulmannan Orabi IUST
Vertical Stress in SoilVariation of with z/R and r/R. \'
67Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
The increase in the vertical stress (��) 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 zq Iσ =
c� � ������3�+3� .20�2��� ���.+�� d
�+�2
\
�
where :
68Dr. Abdulmannan Orabi IUST
… . 7 − 10
Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
c� �1
4V
2`� `� � �� � 1
`� � �� �`��� � 1
`� � �� � 2
`� � �� � 1� �+�e�
2`� `� � �� � 1
`� � �� −`��� � 1
The influence factor
can be expressed as
` �d
�+�2� �
\
�where :
69Dr. Abdulmannan Orabi IUST
… . 7 − 11
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
34
6'��* � ��� � ��� � ��� � ��f
��g � c�g
70Dr. Abdulmannan Orabi IUST
Vertical Stress in Soil
Vertical Stress Caused by a Rectangular loaded area
71Dr. Abdulmannan Orabi IUST
4
6'
6'
6'
6'
−h5�
�h5i−h5�
+ h5� h5
4 3
4
2
9
11 2
3
5
5
1
8
7 9
87
4
7
3
��* � ��� − ��� − ��� � ��f
Variation of with m and nc�
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
72Dr. Abdulmannan Orabi IUST
Variation of with m and nc�
nm
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
73Dr. Abdulmannan Orabi IUST
Variation of with m and nc�
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
74Dr. Abdulmannan Orabi IUST
Variation of with m and nc�
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
75Dr. Abdulmannan Orabi IUST
Approximate Method
B
B + z
2
1
z��
"
O
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2V:1H method
A simple but approximate method is sometimes used for calculating the stress change at various depths as a result of the application of a pressure at the ground surface.
The transmission of stress is assumed to follow outward fanning lines at a slope of 1 horizontal to 2 vertical.
Approximate Method
For uniform footing (B x L) we can estimate the change in vertical stress with depth using the Boston Rule. Assumes stress at depth is constant below foundation influence area
B
B + z
2
1
z
�� � "d\
,d � �-,\ � �-
��
"
O
" �O
d � \
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… . 7 − 12
2V:1H method
Approximate Method
B + z
L
B
z
Stress on this plane " �j
d ∗ \
Stress on this plane at depth z, �� � "d\
,d � �-,\ � �-
Rectangular footing
B
B + z
2
1
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2V:1H method
Newmark Method
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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.
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)
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Newmark Method
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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
Newmark Method
A BInfluence value = 0.005
Total number of block on chart = 200 and influence value = 1/200
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
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The footing plan will be placed on the influence chart with the point for which the stress is desired 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
�� � "cj
Where Iistheinfluencefactorofthechart. " � +00��20.���. ���+.+ � ��2+�� �3 ��+3�0.���.
j � ��`4. ������3 ���2,0+.��+������+.���`+�2-
84Dr. Abdulmannan Orabi IUST
… . 7 − 13
Newmark Method
85Dr. Abdulmannan Orabi IUST