Determine vertical soil pressure under footing
Contents:
Assume stresses distributes in a
2 to 1 slope under footing.
Vertical stress at a depth z
under a rectangular footing is
Dp=
P /[(B+Z)(L+Z)
[1.17]
Example 10: Determine vertical soil pressure using 2:1 method
Given:
Requirement:
Determine vertical soil pressure
at 6’ below bottom of footing
Solution:
q =P /[(B+Z)(L+Z)] = 25000 / [(8+6)(4+6)] = 178.6 lbs/ft2.
Boussinesq equation
Newmark’s influence chart is
developed from Boussinesq’s equation. Boussinesq’s
equation considers a point load on a semi-infinite, homogeneous, isotropic,
weightless, elastic half-space as shown below
By integrating Boussineq’s equation over a circular area
of radius R, one obtains a relationship between DP/P
and R/D as
DP/P
= 1-{1/[1+(R/D)2]3/2}
[1.18]
Newmark’s chart is constructed by rewriting Equation 1.18
as
R/D = [(1-DP/P)-2/3-1]1/2
[1.19]
Or
|
DP/P
|
0.1
|
0.2
|
0.3
|
0.4
|
0.5
|
0.6
|
0.7
|
0.8
|
0.9
|
1.0
|
|
R/D
|
0.27
|
0.40
|
0.52
|
0.64
|
0.77
|
0.92
|
1.1
|
1.39
|
1.91
|
¥
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In Newmark’s chart, if the length of one unit equals to
D, then the radius of the ring is equal to R.
The area inside each circle is the ratio of DP/P.
The area inside the first inside circle is 0.1, and the area inside the
second circle is 0.2, etc. To be
able to measure the area inside the circle, the area is divided in to a number
of pieces. Each piece between the circle and radical lines is a unit.
The number of units multiplied by an influence value should be equal to
the ratio of DP/P
of the circle. Therefore,
DP=P
I N
Where, I is an influence of Newmark’s chart and N is the
number of units inside the circle.
For the chart shown below, there are 200 pieces in the
total area. Then, the influence
value of the chart is 1/200= 0.005. Therefore,
the ratio of DP/P
for the first circle is 20 units x 0.005 = 0.1. Although, the chart is developed
based on circular area, it can be extended to other shape of footing by placing
a scaled footing area on top of the chart.
Example 11: Determine vertical soil pressure using
Newmark’s chart
Given:
Requirement:
Determine vertical soil pressure at 6’ below bottom of
footing at center of footing and at a corner of the footing
Solution:
The footing is draw
to scale according to a depth of 6’. Square
#1 is placed on the chart with center of the footing at the center of circle.
Square #2 is placed on the chart with corner of footing at the center of
the circle.
-
Pressure at the
bottom of footing, P =
25000/36=694 lbs/ft2.
-
The influence
value of the chart is 0.005
-
The number of
unit inside Square #1 is about 68.
-
The vertical at
6’ below the center of the footing DP
= 694*0.005*68 =236 lbs/ft2.
-
The number of
unit inside Square #2 is about 35.
-
The vertical at
6’ below the corner of the footing DP
= 694*0.005*35 =121 lbs/ft2.
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