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;36&@
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4#C&* ,#?& 4#$."353*+5 +D >&*86$* #59 ."#5
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85."#5&*3*$#5.& V
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-400
-200
0
200
400
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@
J#8
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D(
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@$?ka I665%(68 )" B5 $"&+>&E.#6 D+>486#* $+ =& 8*&9@
F+,&?&>G 3$ 3* $++ *34;63O&9 $+ *3486#$& $"& #.$8#6 *$>&** ,#?&* 4*8>&9 35 '() $&*$*@
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#59 ,39&5* $"& ;86*&@
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25 30 35 40
FORCE
TIME
EFFECT OF Z
1
2
5
10
0.5
0.2
0.1
3"6%$(&)"* 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
!"#$ &'()&* +,'-*. #/01 *&' 2+)/(*34'5
63* /#* *&' .&+$' #7 *&' $30.'
3"6%$(&)"* Q4
0
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
!"#$% '#((") * +,)-" +,..,/0
"12,3"34#. 5"-#% -6)$"7
89:'; '#((") * 0',); +,)-" 26.0"7
3"6%$(&)"* 1]
W3< ;36& P $&> 4# ?&>7 *4#66
;36&@
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%
7/23/2019 Understanding the PDA Pile Stress Wave
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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