Geotechnical Engineering

DHARM

STRESS DISTRIBUTION IN SOIL 377

Q 2 Q 3

Q 1 Q 4

B

R 1

R 4

R 2

R 3

sA

z

z

10 .8 Approximate Methods

Approximate methods are used to determine the stress distribution in soil under the influence

of complex loadings and/or shapes of loaded areas, saving time and labour without sacrificing

accuracy to any significant degree.

Two commonly used approximate methods are given in the following subsections.

10.8.1Equivalent Point Load Method

In this approach, the given loaded area is divided into a

convenient number of smaller units and the total load

from each unit is assumed to act at its centroid as a point

load. The principle of superposition is then applied and

the required stress at a specified point is obtained by

summing up the contributions of the individual point

loads from each of the units by applying the appropriate

Point Load formula, such as that of Boussinesq.

Referring to Fig. 10.20, if the influence values are

KKBB 12 ,,... for the point loads Q 1 ,Q 2 , ..., for σz at A, we

have:

σz = diQK 12 BB 12 ++Q K ... ...(Eq. 10.55)
If a square area of size B is acted on by a uniform
load q, the stress obtained by Newmark’s influence value
differs from the approximate value obtained by treating
the total load of q.B^2 to be acting at the centre. It has been established that this difference is
negligible for engineering purposes if z/B ≥ 3. This give a hint to us that, in dividing the loaded
area into smaller units, we have to remember to do it such that z/B ≥ 3; that is to say, in
relation to the specified depth, the size of any unit area should not be greater than one-third of
the depth.

10.8.2 Two is to One Method

This method involves the assumption that the stresses get distributed uniformly on to areas

the edges of which are obtained by taking the angle of distribution at 2 vertical to 1 horizontal

(tan θ = 1/2), where θ is the angle made by the line of distribution with the vertical, as shown

in Fig. 10.21.

The average vertical stress at depth z is obtained as:

σzav= qBL

BzLz

..

()()++

...(Eq. 10.56)

The discrepancy between this and the accurate value of the maximum vertical stress is

maximum at a value of z/B = 0.5, while there is no discrepancy at all at a value of z/B ≈ 2.

Fig. 10.20 Equivalent point

load method