Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 831

20.3.4Methods of Analysis
The following are the two broad approach to analyse a machine foundation system undergoing
vibrations:
(a) Mass-spring-dashpot model
(b) Elastic half-space theory in which the soil on which the machine foundation rests is
considered to be as an elastic half-space.

Detailed consideration of different types of vibration, involving the use of the classical
mass-spring-dashpot model has been already given in the previous section.

Further consideration of the elastic half-space approach will be taken up in the next
section.

The objective of the design procedure is the determination of a foundation soil system
which supports the machine satisfactorily. The supported machine may itself be the source of
dynamic loads or it may require isolation from external sources of excitation.

The soil parameters required in the first approach are the ‘spring constant’ and the
‘damping ratio’. Those required in the second are the ‘shear modulus’ and the ‘Poisson’s ratio’
of the soil.

The main difficulty in soil dynamics and machine foundations consists in the determi-
nation of the appropriate soil parameters to the desired degree of accuracy. Detailed treat-
ment of the determination of dynamic soil parameters is given in a later sub-section.

20.3.5 Wave Propagation through Soil

For several reasons, the applicability of Hooke’s law soil is limited. The elastic constants of soil

depend on normal stresses and elastic deformations may affect the initial internal stresses

which always exist in soil. It should also be noted that the propagation of waves may be greatly

influenced by dissipative properties of soil which govern the absorption of wave energy.

There are two basic types of elastic wave: ‘body waves’ which travel through the interior

of the soil mass and ‘surface waves’ which travel at or near the surface of the material. Body

waves are further subdivided into two modes. ‘Dilatational’, ‘compression’ or ‘P-wave’ and ‘shear’

or ‘S-wave’. Surface waves are subdivided into four modes-‘Rayleigh’ or ‘R-wave’, ‘Compres-

sion bar’ wave, ‘Hydrodynamic’ or ‘H-wave’, and ‘coupled’ or ‘C-wave’. (The last two are of

somewhat dubious origin).

Dilatational or P-Wave

This wave induces motion of the soil particles in the direction of the propagation of the wave.

The velocity of the wave, Cp, may be expressed as

Cp =

E()

()( )

1

112

−

+−

υ

ρυ υ ...(Eq. 20.61)

where E and υ = Modulus of elasticity and Poisson’s ratio of the soil,

and ρ = density of the soil (mass per unit volume).
P-Waves can be propagated in any direction within the soil mass.