Geotechnical Engineering

DHARM

PILE FOUNDATIONS 663

Wave equation method

As the top of the pile is struck with the pile hammer, the impact energy of the blow causes a
stress wave to be transmitted through the length of the pile. Some of the force is absorbed by
the surrounding soil, while some is imparted to the soil at the pile tip. What has come to be
known as the ‘wave equation method’ involves the application of the wave transmission theory
to determine the load-carrying capacity developed by the pile and also to determine the maxi-
mum stresses that occur within the pile during driving.

The dynamic process of pile driving is analogous to the impact of a concentrated mass
upon an elastic rod. The rod is restrained partially by skin friction and by end bearing at the
tip. The system can be approximated by a lumped mass elastic model (Ramot, 1967), as shown
in Fig. 16.5. The distributed mass of the pile is represented by a series of small concentrated
masses, linked by springs that simulate the longitudinal resistance of the pile. The skin fric-
tion can be represented by a rheological model of damping or surface restraint that includes
friction and elastic distortion also.

Pile

Ram

Cap

Wr

Wc

W 1

W 2

W 3

W 4

W 5

W 6

W 7

kcb

kc

kp1

kp2

kp3

kp4

kp5

kp6

ks

Qeb

f 1

f 2

f 3

f 4

f 5

f 6

f 7

Fig. 16.5 Wave equation method for pile capacity
When the hammer strikes the pile cap, a force is generated that accelerates the cap (Wc)
and compresses it. This transfers a certain force to the top segment of the pile (W 1 ), and causes
it to accelerate, slightly after the acceleration of the pile cap. The compressive force induced in
the top segment accelerates the next segment of the pile (W 2 ). Thus, a wave of compression
moves down the pile. The vertical force at any instant is equal to the compression of the spring.
The force wave is partially dissipated in overcoming skin friction on the way down; at the
bottom, the remaining force overcomes end-bearing. In order that the pile penetrate deeper,
the force of the wave must exceed the summation of ultimate values of skin friction and end-