274
8 Elastic Solutions in Geomechanics
or
σ x =
2qx
2 z
π
x 2 + z 2
2
and
τ xz = −σ θ sin θ cos θ + σ r sin θ cos θ + τ r θ
cos
2
θ sin
2
θ
= 0 +
2q
πr
cos θ sin θ cos θ + 0
=
2q
πr
sin θ cos
2
θ
=
2q
πr
x
√
x 2 + z 2
z
2
x 2 + z 2
or
τ xz =
2qxz
2
π
x 2 + z 2
2
But for the plane strain case,
σ y = v(σ x + σ z )
where v = Poisson’s ratio
Substituting the values of σ x and σ z in σ y , we get
σ y =
2qx
2 z
π
x 2 + z 2
2 +
2qz
3
π
x 2 + z 2
2
v
=
2qvz
π
x 2 + z 2
x
2
+ z
2
or σ y =
2qvz
π(x 2 +z 2 )
Therefore, according to Flamant’s (1892) solution, the following are the stresses
due to a vertical line load on the surface of an half-space.
σ x =
2qx
2 z
π
x 2 + z 2
2
σ y =
2qvz
π
x 2 + z 2