Geotechnical Engineering

DHARM

ELEMENTS OF SOIL DYNAMICS AND MACHINE FOUNDATIONS 855

The differential equations of motion (Eq. 20.98) may be solved for the known initial
conditions:

when t = 0, z 1 = z 2 = 0, z& 1 = 0; and &z 2 = va

The solution is

z 1 =

()()

()

ωωωω sin sin

ωω ω

ω

ω

ω

ω

anan

an n

a

n

n

n

n

v tt

2

2

22

1

2

2

1

2

2

2

1

1

2

2

−−

−

−

F

HG

I

KJ

...(Eq. 20.106 (a))

z 2 =

va tt

nn

an

n

n

an

n

() n

()sin ()sin

ωω

ωω

ω

ω

ωω

ω

ω

1

2

2

2

2

2

2

1

1

2

1

2

2

−^2

−

−

F −

HG

I

KJ

...(Eq. 20.106 (b))

Barkan (1962) observed from his experiments, that vibrations occurred at the lower
principal frequency only, and as such, it may be assumed that the amplitude of vibrations for
sin ωn 1 t (where ωn 1 > ωn 2 ) equals zero.

Then the approximate expressions for z 1 and z 2 are as follows:

z 1 = –

()()

()

ωωωω sin

ωω ω ω

anan ω

an n n

vtan

2

1

22

2

2

2

1

2

2

2

2

2

−−

−

...(Eq. 20.107 (a))

z 2 = −

−

−

()

()

sin

ωω

ωωω

an ω

nnn

vtan

2

1

2

1

2

2

2

2

2 ...(Eq. 20.107 (b))

The maximum amplitudes of motion occur when sin ωn 2 t = 1.

These are—

for foundation-soil-system

A 1 (= z1max) = −

−−

−

()()

()

ωωωω

ωω ω ω

anan

an n n

va

2

2

22

1

2

2

1

2

2

2

2

...(Eq. 20.108 (a))

for anvil

Aa(= z2 max) = −

−

−

()

()

ωω

ωωω

an

nnn

va

2 12

1

2

2

2

2

...(Eq. 20.108 (b))

The stress in the elastic pad, σp, is given by

σp =

k

A

zz

p

2

() 21 − or

k

A

AA

p

a

2

()− 1 ...(Eq. 20.109)

The basic model is applicable if there is a uniform contact between the elastic pad and
the anvil as well as between the pad and the top surface of the foundation block. However, it
has been generally observed that the contacts are not uniform by virtue of the fact that the
bottom surface of anvil and the top surface of the foundation are relatively rough.

Also, the hammer foundation-soil system is the case of free vibration with damping.
Satisfactory solutions are not available to date to analyse the system as Mass-spring-dashpot
model with two-degree freedom.
Further, hammer foundations are generally embedded in the soil either partially or
completely. Embedment makes the analysis rather complex.
Therefore, one has to resort to the use of empirical correlations of Barkan (1962), based
on his experimental investigations, to take care of the influence of damping of the system, non-
uniform contact of the elastic pad, and depth of embedment.