Civil Engineering Reference
In-Depth Information
3.5.2 Stresses induced by point load
The simplest case of applied loading has been illustrated in Example
3.3.
However, most loads are applied
to soil through foundations of finite area so that the stresses induced within the soil directly below a
particular foundation are different from those induced within the soil at the same depth but at some radial
distance away from the centre of the foundation.
The determination of the stress distributions created by various applied loads has occupied researchers
for many years. The basic assumption used in all their analyses is that the soil mass acts as a continuous,
homogeneous and elastic medium. The assumption of elasticity obviously introduces errors but it leads
to stress values that are of the right order and are suitable for most routine design work.
In most foundation problems, however, it is only necessary to be acquainted with the
increase
in vertical
stresses (for settlement analysis) and the
increase
in shear stresses (for shear strength analysis).
Boussinesq (
1885
) evolved equations that can be used to determine the six stress components that act
at a point in a semi-infinite elastic medium due to the action of a vertical point load applied on the hori-
zontal surface of the medium.
His expression for the increase in vertical stress is:
3
Pz
3
∆
σ
=
z
5
2
2
π
(
r
2
+
z
2
)
where
P
=
concentrated load
r
x
2
y
2
(see Fig.
3.9,
inset).
=
+
The expression has been simplified to:
K
P
z
∆
σ
z
=
2
where K is an influence factor.
Values of K against values of r/z are shown in Fig.
3.9.
Fig. 3.9
Influence coefficients for vertical stress from a concentrated load (after Boussinesq,
1885
).
