1

Government College Of Engineering, Aurangabad.

Approximate Methods for calculation of stresses

Created By-

Nikhil Pakwannne

Approximate Methods for calculation of

stresses

1) Equivalent Point-Load Method

2) Two-to-one Load Distribution Method

3) Sixty Degree Distribution Method

Approximate Methods for calculation of stresses

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.

Equivalent Point-Load Method

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,

Or

²

)(...2)()( 211

z

IQIQIQz

nBnBB

n

ii

iIBQiz

z )(²

1

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.

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

Problem on Equivalent Point-Load Method

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),

507.0,521.1

)25.0(5.1 22

z

rr

r

For load 2,3,5,6,

For load 7,

186.0,559.0

)25.0(5.0 22

z

rr

r

300.0,901.0

)5.0(75.0 22

z

rr

r

In this case,

2/34.7

)507.0612.1674.3129.1(061.1

mkNz

z

5.2]2)559.0(1[(

15.2]2)30.0(1[(

2

5.2]2)186.0(1[(

45.2]2)507.0(1[(

22)3(2

203

z

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.

Two-to-one Load Distribution Method

The average vertical stress Z depends upon the shape of the loaded area,

1) Square Area (B x B):-

2) Rectangular Area (B x L):-

z)²(B

²

qB

z

z)z)(L(B

L)(B

q

z

3) Strip Area (width B, unit length):-

4) Circular Area (diameter D):-

1z)(B

1)(B

q

z

z)²(D

D²

q

z

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.

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:-

In this case, b=1m & z=5m.tan =1/5=0.2 & 2 = 0.395 radians

Problem on Two-to-one Load Distribution Method

)2sin2( q

z

taking q=400/2=200 kN/m²,

2:1 distribution method:

²/6.49)385.0395.0(200

mkNz

%2.151006.49

6.491.57error percentage

57.1kN/m²52

2200

zB

Bqz

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.

Sixty Degree Distribution Method

Soil mechanics and foundation engg by K.R.ARORA

http://en.wikipedia.org

REFERENCES

Thank You

Appreciate your time and attention!