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8 Elastic Solutions in Geomechanics
Fig. 8.9 Pressure bulb on
which the principal stresses
are constant
z
b 2 =
πσ 1
2q
=
1
2h
where h is a constant.
But b
2
= x
2
+ z
2
Therefore,
z
(x
2 +z 2 )
=
1
2h
or b
2
=
x
2
+ z
2
= 2 h z
which is the equation of a circle with radius h centred on the z-axis at a depth h
beneath the origin, as shown in Fig. 8.9.
At every point on the circle, the major principal stress is the same. It points directly
at the origin. If a larger circle is considered, the value of σ 1 would be smaller. This
result gives us the idea of a “pressure bulb” in the soil beneath a foundation.
8.5 Cerrutti’s Problem
This is a more complicated problem than Boussinesq’s or Kelvin due to the absence
of radial symmetry. Figure 8.10 shows a horizontal point load P acting on the surface
of a semi-infinite soil mass.
The point load represented by P acts at the origin of co-ordinates, pointing in
the x-direction. Due to the absence of symmetry, a rectangular co-ordinate system is
used in the solution.
According to Cerrutti’s solution (Cerrutti 1884), the displacements are given by
u x =
P
4π G R
1 +
x
2
R 2 + (1 − 2v)
R
R + z
−
x
2
(R + z) 2
(8.19)