DHARM
ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 835
Stress
Elastic deformation
Fig. 20.22 Stress vs elastic deformation from a
repeated plate load test
The spring constant, kz, may be determined from Cu as follows:
pz = CuSe ...(Eq. 20.65)
or
P
A
z = C
uSe
∴
P
S
z
e
= CuA
But, by definition, kz =
P
S
z
e
∴ kz = CuA ...(Eq. 20.66)
In these equation, pz is the vertical stress, Pz is the vertical load, Se is the elastic part of
the settlement, and A is the area of the plate.
Cu is independent of the foundation base contact area only if the distribution of the
pressure on the foundation is uniform. In reality, the normal stresses in the soil under the
plate (or foundation) are distributed in a rather irregular manner. This leads to the fact that
Cu varies with area of the plate or foundation.
Sadovsky (1928) gave a solution to this problem for a circular base contact area of a rigid
plate. After going through a bit of mathematical analysis including integration, the following
expression is obtained for Cu:
Cu =^113
1
1
. () 2.
E
−v A
...(Eq. 20.67)
where E and v are the elastic constants of the soil. Thus it is seen that Cu is inversely propor-
tional to the square root of the area of the plate, and it is not an absolute property of the soil.
CAu 1 1 = CAu 2 2 ...(Eq. 20.68)
or Cru 1. 1 = Cru 2. 2 ...(Eq. 20.69)
r 1 and r 2 being the equivalent radii of the base plates of areas A 1 and A 2 , respectively.
These equations enable one to calculate the Cu-value for a machine foundation-soil sys-
tem as follows:
Cuf = C
A
up A
p
f
...(Eq. 20.70)