Geotechnical Engineering

DHARM

828 GEOTECHNICAL ENGINEERING

η 2 = A M

Mee D

.

. ()

=

−+

ξ

ξξ

2

1422 2 2 ...(Eq. 20.57)

Then means that η 2 = η 1 .ξ^2 ...(Eq. 20.58)

It can also be shown that

η2 max =

1

21 DD−^2

...(Eq. 20.59)

The relationship between ξ and η 2 (or A) for different values of D is shown in Fig. 20.17.

0.4

0.6

1.00

0.8

123

x

5

4

3

2

1

h^2

D=0

D = 0.1

D = 0.2

Fig. 20.17 η 2 Versus ξ
It can be seen from this figure that for quadratic excitation the maximum value of η 2 (or
A) occurs at a value of ξ greater than unity when damping is present. As the value of D in-
creases, the peak η 2 (or A) deviates more from ξ = 1. Thus resonant conditions tend to occur at
a frequency ratio more than unity.

In this case also, the effect of damping is to make the peak amplitude finite and to make
the ξ-value corresponding to the peak amplitude deviate from unity. It can also be seen that
the influence of damping is large when resonance condition occurs and it decreases when the
amplitude of motion is different from the peak amplitude; the greater the difference the smaller
the influence of damping ratio.

20.3 FUNDAMENTALS OF SOIL DYNAMICS

‘Soil Dynamics’ has all ready been defined as that discipline which deals with the behaviour of
soil under dynamic loading. The sources of dynamic loading have also been enumerated earlier.