PILE DRIVINGBY

WAVE MECHANICS

George Goble

Consulting Engineer

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 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 DYNAMIIC 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.

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 – Modulus of Elasticity

• ρ - Mass Density

Ec

WAVE TRAVEL IN A PILE

FORCE A FUNCTION OF X

X

F

at time t at time t + Δt

x + ct

FORCE A FUNCTION OF t

t

F

FORCE-VELOCITY PROPORTIONALITY

ε = (1/c) vσ = (E/c) v

F = (EA/c) v

SO IF THE PARTICLE VELOCITY IS KNOWNTHEN STRESS AND FORCE CAN BE CALCULATED

OR THE REVERSE

SO, FOR GRAPHIC REPRESENTATION THE F – v PROPORTIONALITY CAN BE USED

COMPRESSION AND DOWN VELOCITY POSITIVETENSION 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

cv

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

ΔΨ =FΔδ

Δδ = vΔt

Ψ =

FvdtRod

F

F

DISPLACEMENT

F

OR

CE

(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 F2 dt

Assumes No Reflections

Half Kinetic – Half Strain

L1

L

R2

R2

R2

Force

EA c

v

Force

R

F

EA c

v

F - R 2

Force

F+R 2

EA c

v

R

t

REA c

vForce,

EA c

vForce,

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

ANY QUESTIONS

?