Geotechnical Engineering

DHARM

362 GEOTECHNICAL ENGINEERING

a state of plane strain; that is, the strains and stresses in all planes normal to the line of the
loading are identical and it is adequate to consider the conditions in one such plane as in Fig.
10.8 (a). Let the y-axis be directed along the line of loading as shown in Fig. 10.8 (b).

Let us consider a small length dy of the line loads as shown; the equivalent point load is
q′. dy and, the vertical stress at A due to this load is given by:

dσz =

3

2

3

2

3

5

3

22252

(. ).

()/

qdyz

r

qzdy

xyz

′ = ′
ππ++
The vertical stress σz at A due to the infinite length of line load may be obtained by
integrating the equation for d σz with respect to the variable y within the limits – ∞ and + ∞.

∴σz =^3

2

2

3

2

3

22252

3

0 22252

qz dy

xyz

qzdy

xyz

′

++

=

′

++

∞

−∞

∞

ππ() ()//zz

or σz =

221

1

3

222 22

qz

xz

q

z xz

′

+

=

′

π()π +

.

[(/)]

...(Eq. 10.17)

–x

xA

z

+x

q

(a) (b)

x/+

dy

–y

• ¥

+y

+¥

q /unit length¢

O

R

O¢ x

r

A(x, y, z)

z

y

Fig. 10.8 Line load acting on the surface of semi-infinite elastic soil medium

Equation 10.17 may be written in either of the two forms:

σz =

q

z

′.Kl ...(Eq. 10.18)

where Kl =

(/)

[(/)]

2

1 22

π

+ xz

...(Eq. 10.19)

Kl being the influence coefficient for line load using Boussinesq’s theory; or

σz =

2 q 4

z

′

π

.cos θ ...(Eq. 10.20)

since the Y-co-ordinate may be taken as zero for any position of the point relative to the line
load, in view of the infinite extension of the latter in either direction.