DHARM

854 GEOTECHNICAL ENGINEERING

and k 2 = spring constant of elastic pad.

Also k 1 = Cu′A 1 ...(Eq. 20.99)

and k 2 =

EA

t

pp

p

...(Eq. 20.100)

Ma

Mf

Tup(hammer)

Anvil

k (for pad) 2

Foundation

Z=0 1

k (for soil) 1

Fig. 20.31 Model for analysis of a hammer foundation

where Cu′ = coefficient of elastic uniform compression of soil under impact,

A 1 = contact area of foundation,

Ap = base area of pad,

Ep = Young’s modulus of the material of the pad,

and tp = thickness of pad.

Starting with possible solutions for z 1 and z 2

such as z 1 = C 1 sin ωnt

and z 2 = C 2 sin ωnt,

C 1 and C 2 being constants, and substituting in the differential equations of motion (Eq. 20.98),
and simplifying, it is possible to develop a frequency equation of the fourth degree as follows:

ωn^4 – (1 + λ 1 ) (ωa^2 + ωl^2 )ωn^2 + (1 + λ 1 )ωa^2 ωl^2 = 0 ...(Eq. 20.101)

where λ 1 = Ma/Mf ...(Eq. 20.102)

ωn^2 =

k

M

EA

a tM

pp

pa

(^2) = ...(Eq. 20.103)
and ωl^2 =
k
MM
CA
afMM
u
af
11
()()+

′

• ...(Eq. 20.104)
  ωa is the limiting natural frequency of the anvil, assuming the soil to be infinitely rigid (k 1 = ∞).
  ωl is the limiting natural frequency of the entire system (anvil and foundation), assuming the
  anvil to be infinitely rigid (k 2 = ∞).
  The positive roots of Eq. 20.101 are designated as ωn 1 and ωn 2. These may be expressed as
  ω^2 n1,2 =
  1
  2
  (( 114 ++±++−+λω ω 1 )( al^22 )) (( 1 )( al^222 )) ( 11 ) al^22 λω ω λωω
  ...(Eq. 20.105)