Stress Distribution

INTERNATIONAL UNIVERSITY
FOR SCIENCE & TECHNOLOGY
‫وا‬ ‫م‬ ‫ا‬ ‫و‬ ‫ا‬ ‫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


∑=
⋅=⋅++⋅+⋅=
n
i
iinnz hhhh
1
2211 ...... γγγγσ
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
∗
∗ ∗
∗
∗ ∗
∗ ∗ ∗

With uniform surcharge on infinite land surface
Original
land surface
Conversion land surface
∗


∗
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

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
!"!#$ %#! ∗

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

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.

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.

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,
Stresses in Saturated Soil without Seepage
0
A
Solid particle
Pore water
)*
)&
+
+

0
A Solid particle
Pore water
)*
)&
+
+
+
+
Forces acting at the points of contact of soil
particles at the level of point A
Stresses in Saturated
Soil without Seepage
& ) ,)* − )- %#!
where
+ . +
/+ 0 1
%#! + .+ 2

)* 2 + 3 4 0
1 + 2 + . +4

)
)
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
: & ) ) ∗ %#!
: ,) ) - &
:
'
: − :
:
'
) %;<

Stress at point C,
• Total stress:
= & ) ∗ %#!
> ,) - &
>
'
> − >
>
'
%;<
Total stress
Pore water
Pressure, u Effective stress
DepthDepth Depth

)
)
5
6
7
8
Valve (open)
?
(
@
AB
-
Stresses in Saturated Soil with Upward Seepage
Stress at point A,
• Total stress:
* & )
* & )
*
'
* − * 0

Stress at point B,
• Total stress:
: & ) ) ∗ %#!
: ,) ) - &
:
'
: − :
:
'
) %;< − &

Stress at point C,
• Total stress:
= & ) ∗ %#!
: ,)
)
- &
>
'
> − >
>
'
%;< −
)
&
>
'
%;< − &
Note that h/H2 is the hydraulic gradient i
caused by the flow, and therefore

Total stress
Pore water
Pressure, u Effective stress
DepthDepth Depth

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
%;<
&
>
'

)
)
5
6
7
8
Valve (open)
?
(
@
AB
-
Stresses in Saturated Soil with Downward Seepage
Stress at point A,
• Total stress:
* & )
* & )
*
'
* − * 0

Stress at point B,
• Total stress:
: & ) ) ∗ %#!
: ,) ) − - &
:
'
: − :
:
'
) %;< &

Stress at point C,
• Total stress:
= & ) ∗ %#!
: ,) −
)
- &
>
' > − >
>
'
%;<
)
& >
'
%;< &

Pore water
Pressure, uTotal stress Effective stress
DepthDepth Depth

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
Sandγ = 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

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

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

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

•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

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

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

Stresses in soil due to surface load
Vertical stress due to a concentrated load
• Boussinesq’s Formula
• Wastergaard Formula

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.
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

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.
Assumption:

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.

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°

According to Boussinesq’s analysis, the vertical stress
increase at point A caused by a point load of magnitude P
is given by
D
∆
∆ M
∆ N
O
P
Q
.
P
D
1

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 z
σ
π
=
+
… … . 7 − 1
1
∆
.
Q
or
2z b
P
I
z
σ =

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).
where
2 5/2
3 1
2 [1 ( / ) ]
bI
r zπ
=
+
… … … … . 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.

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

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
Influence Factor 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
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.
Pressure Distribution Diagram

The vertical stress distribution on a horizontal
plane at depth of z below the ground surface
U
5
5
Distribution on a horizontal plane

The vertical stress
distribution on a vertical
plane at a distance of r
from the point load
.
Distribution on a vertical plane O

U
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.

What is the vertical stress at point A of figure below
for the two loads, P1 and P2 ?
P1 = 350 kNP2 = 470 kN
Z=2.5m
2.3 m1.1 m
A
Worked Examples
Example 2

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

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

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-
… . 7 − 3

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
1P
P
… . 7 − 4

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
… . 7 − 5

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.

Vertical Stress caused by a strip load
α
β
6
B
Vertical stress at point A can be determined by equation:
[ sin cos( 2 )]o
z
q
σ α α α β
π
= + +
P
… . 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.

_
6
4
"
+
_
1 2 2[( )( ) ( )]o
z
q a b b
a a
σ 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
" )where:
`4+ a`
) `4+ a`
… . 7 − 7

^ 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

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 due to a uniformly loaded circular area
1 −
1
Q/ 1 /
… . 7 − 8

6
6'
Q
6'
6

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
where and are functions of z/R and r/R.
1' '
1' '
… . 7 − 9

7
7'
.
Q
7
7'.

Variation of with z/R and r/R.1'

Variation of with z/R and r/R.'

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 . 2 0 2 .+
d
+ 2

where :
… . 7 − 10

c
1
4V
2 ` ` 1
` ` 1
` 2
` 1
+ e
2 ` ` 1
` − ` 1
The influence factor
can be expressed as
`
d
+ 2

where :
… . 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.
1 2
34
6'
* f
g c g

4
6'
6'
6'
6'
− h5
h5i − h5
+ h5 h5
4 3
4
2
9
1
1 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

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

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

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

Approximate Method
B
B + z
2
1
z
"
O
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
… . 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
2V:1H method

Newmark Method
• 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)

Newmark Method
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 B Influence
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

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
" c j
Where I is the influence factor of the chart.
" +00 2 0. . +. + 2+ 3 +3 0. .
j `4 . 3 2 , 0+. + +. `+ 2-
… . 7 − 13

Newmark Method

Stress Distribution

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