Stress distribution through soil When a footing is loaded, a certain stress will be applied on the soil immediately below the footing. The applied stress on the soil will decrease away from the footing location as the stress is distributed over a larger area. Consider a small soil element as shown below and the stresses applied on the element under plain strain condition. Figure – Stress on a soil element under a strip footing. Due to the stress applied on the soil from the foundation, the increase in the normal and the shear stress in the soil mass under the foundation may lead to failure or cause deformation of the soil causing settlement of the foundation. Therefore, it is very important to study the stress distribution under a loaded area. Even though the soil is neither elastic nor homogeneous, most of the stress distributions are obtained assuming linear elastic and homogeneous soil medium below the foundation. Under the working loads, the soil is stress well below the ultimate stress, as a factor of safety of about 3 against ultimate stress in generally used. Therefore, under the working conditions, the assumption of linear elastic behaviour may be somewhat accurate. The most important stress component in the design of foundation is the vertical stress and hence the distribution of vertical stress with depth is considered here. F X Z ZZ XX ZX XZ
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Stress distribution through soil
When a footing is loaded, a certain stress will be applied on the soil immediately below the footing.
The applied stress on the soil will decrease away from the footing location as the stress is distributed
over a larger area. Consider a small soil element as shown below and the stresses applied on the
element under plain strain condition.
Figure – Stress on a soil element under a strip footing.
Due to the stress applied on the soil from the foundation, the increase in the normal and the shear
stress in the soil mass under the foundation may lead to failure or cause deformation of the soil
causing settlement of the foundation. Therefore, it is very important to study the stress distribution
under a loaded area.
Even though the soil is neither elastic nor homogeneous, most of the stress distributions are obtained
assuming linear elastic and homogeneous soil medium below the foundation. Under the working
loads, the soil is stress well below the ultimate stress, as a factor of safety of about 3 against ultimate
stress in generally used. Therefore, under the working conditions, the assumption of linear elastic
behaviour may be somewhat accurate.
The most important stress component in the design of foundation is the vertical stress and hence the
distribution of vertical stress with depth is considered here.
F
X
Z
ZZ
XX
ZX
XZ
Vertical stress distribution in homogeneous isotropic soil medium
Approximate stress distribution
It is assumed that the stress within the soil mass is distributed under the footing so that a larger area is
involved in carrying the applied load with the depth and the boundary of the area involved in carrying
the applied load is linearly varying with the depth as shown in the following Figure.
As the concentrated force applied on the foundation is resisted by the soil pressure developed over a
(L + 2Z) x (B + 2Z) the stress developed at a depth Z, Z is given by the following relationship.
)2)(2( ZLZB
FZ
Generally, is taken as 0.5 with the tan = 0.5/1, = 26.5o
Boussinesq (1883) solved the problem of stress inside a semi-infinite mass due to a point load acting
on the surface.
L
Z 1
F = F/(BL)
z
B
L
B + 2Z
L + 2Z
The stress at point (x, y, z) due to a point load, Q, acting on the ground surface at the originate of a
rectangular coordinate system is given by Boussinesq as given by the following equations:
2
2
2
3
R
Qzz
[1]
323
2
5
2
)(
2
)(
1
3
21
2
3
R
z
zRR
xzR
zRRR
zxQx
[2]
323
2
5
2
)(
2
)(
1
3
21
2
3
R
z
zRR
xzR
zRRR
zyQy
[3]
235 )(
2
3
21
2
3
zRR
xyzR
R
xyzQxy
[4]
Where
222 zyxR
The practical application of the above equations is a difficult task. Therefore, other charts and Tables
are prepared to estimate of the stress increased due to a loaded area.
Considering a circular loaded area with radius r is loaded with a surface stress of qo, the stress
variation along the vertical axis through the centre may be estimated using the Equation proposed by
Boussinesq for the vertical stress due to a concentrated surface load, as shown in the following Figure.
Z
Y
X
y
x
z
Q
The loaded area is considered with small areas dA and the vertical stress at point D due to the
concentrated force qodA acting over the small area. Integrating the stress due to force acting over
small areas, the vertical stress increment is estimated, as given below as Equation [5].
2
32)/(1
10.1
zr
qo [5]
Newmarks Charts
The Boussinesq Equation for the vertical stress given in Equation [5] is re-arranged to develop the
Newmark’s charts. The Equation [5] is rearranged to give Equation [6].
113
2
o
v
q
q
z
r [6]
As the stress ratio (qv/qo) varies from 0.1 to 0.9, the ratio (r/z) varies as given in the following table.