# george-wave-mechanics

```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
∂2u/∂t2 = c2 ∂2u/∂x2
WAVE TRAVEL SPEED
E
c

• E – Modulus of Elasticity
• ρ - Mass Density
WAVE TRAVEL IN A PILE
FORCE A FUNCTION OF X
F
at time t
at time t + Δt
x + ct
X
FORCE A FUNCTION OF t
F
t
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%
• vi = (η2gh)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
ΔΨ =FΔδ
F
Δδ = vΔt

Ψ = Fvdt
FORCE (F)
ENERGY CALCULATION
F
Rod

DISPLACEMENT 
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 F2 dt
Assumes No Reflections
Half Kinetic – Half Strain
R
L1
L
Force
EA
v
c
Force
F
EA
c v
R
2
R
2
R
2
F+R
2
F-R
2
Force
EA
v
c
R
Force,
EA
v
c
R
Force,
EA
v
c
t
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
```