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8 Elastic Solutions in Geomechanics
Mindlin involved with point load (either vertical or horizontal) acting in the interior
of an elastic half-space. This solution has found many applications to determine the
stresses and displacement fields surrounding an axially loaded pile and in the study
of interactions between foundations and ground anchors. The following sections
describe the selected fundamental solutions in geomechanics.
8.2 Kelvin’s Problem
The problem is to determine the stresses in an infinite elastic body when a point load
is acting in the interior of a body (as shown in Fig. 8.1).
A point load of magnitude 2P is considered to be acting at a point in the interior of an infinite elastic body. In the cylindrical co-ordinate system, the following
displacements can be obtained by Kelvin’s solution (Kelvin 1848).
Displacement in radial direction = u r =
P rz
8π G(1−v)R 3
Tangential displacement = u θ =0
Vertical displacement = u z =
P
8π G(1 − v)
2(1 − 2v)
R
+
1
R
+
z
2
R 3
(8.1)
Similarly, the stresses are given by
σ r = −
P
4π(1 − v)
(1 − 2v)z
R 3
−
3r
2 z
R 5
σ θ =
P(1 − 2v)z
4π(1 − v)R 3
σ z =
P
4π(1 − v)
(1 − 2v)z
R 3
+
3z
3
R 5
(8.2)
τ r z =
P
4π(1 − v)
(1 − 2v)r
R 3
+
3r z
2
R 5
Fig. 8.1 Kelvin’s Problem
Infinite elastic
body
2P
Z
r