DHARM

364 GEOTECHNICAL ENGINEERING

The previous case of line load may be applied and integration used to obtain the stresses
at any point such as A due to the strip load.

Considering the effect of a small width dx of the strip load and taking it as a line load of
intensity (q. dx) per unit length in the Y-direction, the value of d σz at A is given by:

dσz =

(^24)
π

. θ

(. )

cos

qdx

z

Since x = z tan θ and dx = z sec^2 θ.dθ,

dσz =

(^22242)
π
θθθ
π
.qz. sec. cos d cos θθ.
z
= q d
Integrating within the limits, θ 1 and θ 2 for θ, we get
σz =
q
π
[sincos]θθθ+ θ 1 θ^2 ...(Eq. 10.24)
or σz =
q
π
[(θθ 21 −+) (sinθ θ2 2cos −sinθ θ1 1cos )]

q
π
()(sinsin)θθ 211 θ 2 θ 1
2
L −+ 22 −
NM
O
QP
∴ σz =
q
π
[(θθ 21 −+) cos(θθ 21 +). sin(θθ 21 −)] ...(Eq. 10.25)
Similarly, starting from Eqs. 10.21 and 10.22, and adopting precisely the same proce-
dure, one arrives at the following:
σx =
q
π
[(θθθ−sin cos )]θ 1 θ^2 ...(Eq. 10.26)
or σx =
q
π
[(θθ 21 −−) (sinθ θ2 2cos −sinθ θ1 1cos )]

q
π
()(sinsin)θθ 211 θ 2 θ 1
2
L −− 22 −
NM
O
QP
∴ σx =
q
π
[()cos().sin()]θθ 21 −−θθ 21 + θθ 21 − ...(Eq. 10.27)
τxz =
q
π
[sin^2 θ]θ 1 θ^2 ...(Eq. 10.28)

q
π
(sin^2 θθ 2 −sin^21 )
\ τxz =
q
π
[sin (θθ 21 +−) .sin (θθ 21 )] ...(Eq. 10.29)
The corresponding principal stresses may be established as:
s 1 =
q
π
(sin)θθ 00 + ...(Eq. 10.30)
and s 3 =
q
π
(sin)θθ 00 − ...(Eq. 10.31)
where θ 0 = θ 2 – θ 1 [See Fig. 10.9 (a)].