The document discusses three approximate methods for calculating stresses: [1] Equivalent point-load method, [2] Two-to-one load distribution method, and [3] Sixty degree distribution method. It then provides examples of using the equivalent point-load method to calculate vertical stress at a point below a loaded foundation. Specifically, it calculates the vertical stress at a point 3m below the center of a 3x1.5m foundation carrying a uniform load of 40kN/m^2.

1. Government College Of Engineering,
Aurangabad.
Created By-
Nikhil Pakwannne
1

2. Approximate Methods for
calculation of stresses

3. 1) Equivalent Point-Load Method
2) Two-to-one Load Distribution Method
3) Sixty Degree Distribution Method

4.  The vertical stress at a point under a
loaded area of any shape can be
determined by dividing the loaded
area into small area & replacing the
distributed load on each small area by
an equivalent load acting at the
centroid of the area shown in fig.

6.  E.g. in fig. shown Q=qa² for each area. The
total load is thus converted into no. of point
loads. The vertical stress at any point below
or outside the loaded area is equal to the sum
of the vertical stresses due to these
equivalent point loads using equation,
Q1( IB )1 Q 2 ( IB ) 2 ... Qn ( IB ) n
z
z²
Or
n
1
z Qi ( IB )i
z² i i

7.  this equation gives fairly accurate
results if the side a of the small area
unit is equal to or less than one-third
of the depth z of point P at which the
vertical stress is required.

8. 1) A rectangular foundation 3 x 1.5m carries a
uniform load of 40 kN/m². Determine the
vertical stress at P which is 3m below ground
surface. Use equivalent point load method

9.  Solution:-
Let us divide the loaded area into 9 small
areas of size 0.5 x 1m.
Load on each area = 40 x (1x0.5) = 20kN
the stresses at point P are determined due to
9 point loads, using Boussinesq’s solution,
for load (1) & (4),
2 2
r 1.5 (0.25)
r
r 1.521, 0.507
z

10.  For load 2,3,5,6,
r 0.52 (0.25) 2
r
r 0.559, 0.186
z
 For load 7,
r 0.752 (0.5) 2
r
r 0.901, 0.300
z

11.  In this case,
3 20 2 4
z
2 (3) 2 [(1 (0.507) 2 ]2.5 [(1 (0.186) 2 ]2.5
2 1
[(1 (0.30) 2 ]2.5 [(1 (0.559) 2 ]2.5
z 1.061 (1.129 3.674 1.612 0.507 )
z 7.34 kN / m 2

12.  The actual distribution of load with the depth
is complex. However, it can be assumed to
spread approximately at a slope of two
vertical & one horizontal.
 Thus the vertical pressure at any depth z
below the soil surface can be determined
approximately by constructing a frustum of
pyramid of depth z & side slope 2:1.The
pressure distribution is assumed to be
uniform on horizontal plane at that depth.

14.  The average vertical stress Z depends upon
the shape of the loaded area,
1) Square Area (B x B):-
qB ²
z
(B z)²
2) Rectangular Area (B x L):-
q (B L)
z
(B z)(L z)

15. 3) Strip Area (width B, unit length):-
q (B 1)
z
(B z) 1
4) Circular Area (diameter D):-
qD²
z
(D z)²

16.  The above method gives fairly accurate
values of the average vertical stress if the
depth z is less than 2.5 times the width of
the loaded area. The max. stress is
generally taken as 1.5 times the average
stress determined above.

17. 1)A long strip footing of width 2m carries a
load of 400kN/m. Calculate the max. stress
at a depth of 5m below the centre line of the
footing. Compare the result with 2:1
distribution method.
Solution:-
q
z (2 sin 2 )
In this case, b=1m & z=5m.
tan =1/5=0.2 & 2 = 0.395 radians

18. taking q=400/2=200 kN/m²,
200
z ( 0.395 0.385) 49.6kN / m ²
2:1 distribution method:
q B 200 2
z 57.1kN/m ²
B z 2 5
57.1 49.6
percentageerror 100 15.2%
49.6

19.  This method is similar to the preceding
method. In this case, the pressure
distribution is assumed along line making an
angle of 60º with the horizontal instead of
63.5º(2:1) method gives approximately the
same results.

20.  Soil
mechanics and
foundation engg by
K.R.ARORA
 http://en.wikipedia.org

21. Appreciate
your time and attention!