8.4 Flamant’s Problem
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8.4 Flamant’s Problem
Consider a case of a line load of intensity “q” per unit length acting on the surface
of a homogeneous, elastic and isotropic half-space as shown in Fig. 8.6.
The stresses at a point P (r, ) can be determined by using the stress function
φ =
q
π
r θ sin θ
(8.12)
In the polar co-ordinate system, the expressions for the stresses are as follows:
σ r =
1
r
∂φ
∂r
+
1
r 2
∂
2
φ
∂θ 2
(8.13)
and
σ θ =
∂
2
φ
∂r 2
(8.14)
τ r θ = −
∂
∂r
1
r
∂φ
∂θ
(8.15)
Now, differentiating Eq. (8.12) with respect to r, we get
∂φ
∂r
=
q
π
θ sin θ
Similarly
∂
2 φ
∂r 2 = 0 and σ θ =0
Also, differentiating Eq. (8.12) with respect to θ, we get
Fig. 8.6 Vertical line load
on surface of an half-space