Geotechnical Engineering

DHARM

844 GEOTECHNICAL ENGINEERING

Soil

Ms Boundary of

vibrating soil

Machine

Foundation

M

Mf

Fig. 20.27 Machine foundation-soil system
The system is taken to be undergoing purely vertical vibrations and thus considered to
be a system with single degree freedom.
The vibration analysis of a machine foundation may be performed based on either one of
the broad approaches, namely, the Elastic Half-space theory and the Mass-spring-dashpot
model. Depending upon the approach selected, the values of the appropriate soil parameters
have to be determined by a suitable method.
However, it may be noted that, unfortunately, there is no rational method to determine
the magnitude of the mass of soil participating in the vibration, as stated in subsection 20.3.2.
A general guideline is to choose this value to be ranging between zero and the magnitude of
the mass of machine and of the foundation. In other words, the total mass, M, is taken to be
varying between Mf and 2 Mf is most cases.
Empirical approaches, based on different criteria such as type, speed or power of the
machine have been advanced by some research workers; however, all such approaches may
now be considered to be obsolete.

20.4.5Elastic Half-Space Theory
In this theory, a rigid body of known mass is taken to rest upon the surface of an ideal soil, i.e.,
elastic, homogeneous, and isotropic material. It is termed ‘half-space’ because the soil is as-
sumed to extend infinitely in all directions including the depth, with a top surface as a bound-
ary. For mathematical convenience, the foundation/footing is taken to be of circular shape.
The basic soil parameters used in the development of the theory are the shear modulus, G, the
mass density, ρ, and the Poisson’s ratio, v.
The elastic half-space theory may be used to predict the resonant frequency and the
peak amplitude of motion of the system from a single field vibration test. Although the theory
does not explicitly take into account the damping effect of the system, the amplitudes obtained
are finite, indicating that the effect is considered, indirectly (In fact, the nature of damping in
this case may be ‘radiation’ and/or ‘internal’). Further, contact pressure distribution is re-
quired in the analysis.

Reissner (1936) presented an analytical solution for vertical vibrations of a circular disc
resting on an elastic half-space; he considered the contact pressure distribution to be uniform.
Reissner was the first to use elastic half-space theory for soil dynamics problems.