# Wave Mechanics

Urban and Civil

Nov 29, 2013 (4 years and 5 months ago)

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PILE DRIVING

BY

WAVE MECHANICS

George Goble

GOBLE PILE TEST

A STUPID QUESTION

WHAT MAKES A PILE PENETRATE?

A FORCE

IF WE PUSH SLOWLY BUT HARD ENOUGH IT WLL
MOVE DOWN AGAINST THE SOIL RESISTANCE

THE MAGNITUDE OF THE PUSH WILL BE THE PILE
CAPACITY (
BUT HOW DO WE DEFINE CAPACITY
)

BUT WHAT IF WE USE A VERY BRIEF PUSH THAT
WILL PENETRATE THE PILE? PERHAPS AN IMPACT

THAT FORCE WILL BE LARGER THAN THE CAPACITY?

THERE IS A DYNAMIC RESISTANCE

WE WANT TO UNDERSTAND THE EFFECT OF
AN IMPACT ON THE PILE IN ORDER TO DEAL
WITH PROBLEMS LIKE THE ABOVE

WAVE

PROPAGATION

Based on the assumption of linear elastic material

1.
If a force is suddenly applied to the end of a pile a
wave (disturbance) is generated that travels along
the pile. When the wave passes a point on the pile
the point
displaces

with some
velocity

and
acceleration
. A force is present in the pile. The
disturbance can be expressed as a wave of any of
these quantities.

2.
A stress wave propagates unchanged in
magnitude at a constant speed, c, in a uniform
cross section pile.

SOME WAVE SPEEDS

Steel

16,800 feet/sec.

Almost 12,000 miles/hour

Concrete

11,000 to 14,000 feet/sec

Both Modulus and Density Vary so Wave
Speed Varies

Wave Speed Is a Material Property

WAVE MECHANICS

The Hammer Impact Generates a
Stress Wave

The Wave Transmits the Driving
Force

BASIC EXPRESSION GOVERNING ONE
DIMENSIONAL WAVE PROPAGATION

2
u/∂t
2

= c
2

2
u/∂x
2

WAVE TRAVEL SPEED

E

Modulus of Elasticity

ρ
-

Mass Density

E
c
WAVE TRAVEL IN A PILE

FORCE A FUNCTION OF X

X

F

at time t

at time t +
Δ
t

x + ct

FORCE A FUNCTION OF t

t

F

FORCE
-
VELOCITY PROPORTIONALITY

ε

= (1/c) v

σ

= (E/c) v

F = (EA/c) v

SO IF THE PARTICLE VELOCITY IS KNOWN

THEN STRESS AND FORCE CAN BE CALCULATED

OR THE REVERSE

SO, FOR GRAPHIC REPRESENTATION

THE F

v PROPORTIONALITY CAN BE USED

COMPRESSION AND DOWN VELOCITY POSITIVE

TENSION AND UP VELOCITY NEGATIVE

STRESS IMPEDANCE

For Steel

E/c = 30,000/16,800

E/c = 1.80 ksi/ft/sec

So

If an Air Hammer Falls 3.0 feet with an
Efficiency of 65%

v
i

= (
η
2gh)
1/2

= 11.2 ft/sec

η

is

the efficiency

σ = (E/c) v = (1.8)(11.2) = 20 ksi

4.

A stress wave is reflected from the free end of a rod
with the opposite sign. Compression reflects tension.

E

c

v

5.

A stress wave reflects from a fixed end with the same sign.
Compression reflects compression.

6.
An increase in cross section will reflect a wave of the same
sign. A decrease in cross section will reflect a wave of the
opposite sign
.

REFLECTIONS FROM PILE
SECTION CHANGES

Section Increases Reflect Compression and
Up Velocity

Section Decreases Reflect Tension and
Down Velocity

The Larger the Section Change the Larger
the Reflection

7.

If a rigid mass impacts a pile the stress is proportional
to the velocity. The stress decays exponentially.

1

ENERGY CALCULATION

Δ
Ψ

=FΔδ

Δδ = vΔt

Ψ

=

Fvdt

Rod
F
F
DISPLACEMENT


FORCE (F)

8.
The Energy Passing a Point in a Pile
During the Passage of a Stress Wave Is:

Ψ

=

Fvdt

The Energy Passing a Point in a Pile During
the Passage of a Stress Wave Is:

Ψ

=

Fvdt

If F = EA/c (v)

Then
Ψ

= c/EA

F
2
dt

Assumes No Reflections

Half Kinetic

Half Strain

L
1

L

R

2

R

2

R

2

Force

EA

c

v

Force

R

F

EA

c

v

F
-

R

2

Force

F+R

2

EA

c

v

R

t

R

EA

c

v

Force,

EA

c

v

Force,

Soil Resistance Effects on Force and Velocity

Force and Velocity Measurements for Various
Soil Conditions.

Energy transfer in easy driving conditions

Energy transfer in hard driving conditions

Effects of diesel hammer pre
-
ignition on energy transfer

Effects of diesel hammer pre
-
ignition on energy transfer cont.

Force and Velocity Measurements Illustrating
Progressive Concrete Pile Damage