Understanding the PDA Pile Stress Wave

7/23/2019 Understanding the PDA Pile Stress Wave

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-400

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0

200

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600

800

0 1 2 3 4 5 6

Force&

Vel*Z(Tons)

t/(L/c)

SWA plot

Piletop force

Force

Vel*Z

Wave up

Wave down

PDA Force

PDA Vel*Z

PDA Wup

PDA Wdn

I"35 ;36& *&.E+5 *8=*E$8$&9@

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0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 5 10 15 20 25 30 35 40

FORCE

TIME

EFFECT OF Z

1

2

5

10

0.5

0.2

0.1

3"6%$(&)"* U

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 5 10 15 20 25 30 35 40

FORCE

TIME

EFFECT OF Vo

H = 0.6

0.8

1

1.2

1.5

2

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0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20 25 30 35 40

FORCE

TIME

EFFECT OF W1

KeepingZ=1,K=,

vary W

2

5

10

0.5

0.2

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1 E. A.

L.

Smith, Pile driving analysis by the wave equation,

J.

Soil Mech. Found. ASCE, 86 35-61, 1960).

2

G. G. Goble and F. Rausche,

WEAP8 6 program documentation in

4

Vols.

Federal Highway Administration,Office of

,-$4%$76(/ P4%


36/37

Implementation, Washington DC, 1986.

3

F. Rausche, G. G. Goble and G. E Likins, Recent WEAP developments,

Proc. 3rd Int. Con on Application of

Sires-Wave Theory

to

Piles ,

Ottawa, 1988, pp. 164-173.

4. P. Midde ndorp and

A

F. van Weele, Application of characteristic tress wave method in offshore practice,

Proc. 3rd

Int.

Con

on Numerical Methods in Ofshore Piling,

Nantes, Supplement, 1986, pp. 6-18.

5 M. F. Randolph, Analysisof the dynamics of pile driving, n P.

K

Banerjee and R Buttefield eds.),

Developments

in

oil Mechanics V: Advanced Geotechnical Analyses, Elsevier Applied Science, also Univers ity

o

Western Aus tralia

Research Repor t No. G1001,

1991.

APPENDIX I: SOLUTION OF RAM/CUSHION/ANVIL MODEL

The Laplace transform for the anvil velocity in the ram/cushion/anvil model is

r

Naming th e coefficients of the cubic denominator

298 A J

DEEKS

A N D M F R A N D O L P H

equation 32) can be written as

a

L g

a + a s + u2s2 + s3

Letting

and

if p is greater than zero, the cubic denominator of equation (58) has one real root and two

imaginary roots. The inverse transform can be found readily if the following substitutions are

made:

This allows the cubic denominator to be factorized, and equation (58) becomes

a0

(S

bi ) [ ( s

+ b2 2+ 01

lif

which can be expanded to

Llif a {

s + b 2 +

w2

+ (b2 1 ) 2

s +

bl S + b2)2+ w

S + b2) + w2

Performing the inverse Laplace transform, the anvil velocity can be found.

af a

e -b if -bz f (coswt* + b,--b,sinut*)]

m + b2

l)

This solution can be written in a simpler, more convenient form by making

c a

c1 bl

c2

b2

l, 4 arctan-, F, ___

w + cz

The anvil velocity and the force on the pile head are then

The spring force can be found by using the equilibrium equation

f, n:

+ if

HAMMER IMPACT FOR PI LE DRIVING

299

Differentiating equation 70), substituting into equation 71), and simplifying, he spring force can

be expressed as

where

For most combinations of

rn

and

kr

p2

is greater than zero, and the solution presented above

applies. For a small range of combinations,

p2

is less than or equal to zero, and a different

solution must be used. When this

is

the case, the cubic denominator has three real roots, and

equation

58)

can be written in the following way:

0

+ 271

b 2 = ~ + 2 J Q c o s -

3

a 0

f

471

b3 ~ J Q c o s

3

0

arccos

6)

If bl, b2 and b3 are all distinct, equation 74) becomes

1

1

Lti:

=

a

+

s

+ bI) b2 l) b3 l) S + b2) bl 2) b3 2)

+

S f b3) bl 3) b2 3)

and the solution

is

given by

f ,*

Fp(e-blr*

, ~ - b 2 t , ~ - b t )

with

300 A

J DEEKS AND

M. F.

RANDOLPH

The force in the spring is

with

86)

a2 2 b3 z

a2 l b3 l

a2

3 bz l

a2

, b2 3

As

______

Bs=--

o a2

l)

2 l ) b 3 l

Fs b

However, iftwo of the roots are the same, the solution is different again. When is zero and

b3 is

equal to b Z , he solution becomes

f t

= F,e-clr* [l -c2r l b , t * ) ]

f

FSe-lr*[l -c2* 1

b z A s t * ) ]

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Understanding the PDA Pile Stress Wave

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