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CE5107 Pile FoundationDepartment of Civil EngineeringNational University of Singapore
Pile Driving Analysis &Dynamic Pile Testing
Y K Chow
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One-Dimensional Wave Propagation in PileEquilibrium equation (compression as positive)
dx
xPPP
tum 2
2
dxxP
tuAdx
2
2
AP xu
= density of pile material
A = cross-sectional area of pile
For a one-dimensional rod Axial strain is given by
or
where
(2)
(1)
(3)
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Axial stress-strain relationship
xuEE
Hence from Eqns
(1) to (4)
2
2
2
2
xuEA
tuA
2
22
2
2
xuc
tu
Ec
Eqn
(5) is generally known as the one-dimensional wave equation. “c”
is the “celerity”
or speed of sound in the material, or is simply referred to as the wave speed.
For constant E and A, this gives
2
2
2
2
xuE
tu
or
where
(4)
(5)
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Typical wave speeds:
266 /10401030 mkNtoE 3/4.2 mt
smtoc /40003500
26 /10207 mkNE 3/83.7 mt
smc /5100
Time taken to travel from pile head to pile toe and back to the pile head:
cLt 2
Concrete pile :
st 01.04000
202
st 0078.0
5100202
Concrete : Steel :
Where L = pile length
For example, take L = 20 m
Steel pile :
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General solution to 1-D wave equation
f1 (x-ct) = wave propagating in (+)ve
x-direction (forward / downward)
)()( '2
'1
ctxfctxfxu
ctxfctxfu 21
f2 (x+ct)= wave propagating in (-)ve
x-direction (backward / upward)
Proof: )()( ''2
''12
2
ctxfctxfxu
)()( '2
'1 ctxcfctxcf
tu
)()( "2
2"1
22
2
ctxfcctxfctu
Substitute Eqn
(7) into Eqn
(5),
- (6)
- (7)
)()()()( "2
"1
2"2
2"1
2 ctxfctxfcctxfcctxfc
The expressions are identical
on both sides of the equation, hence satisfying the wave equation
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ctxfu 1
ttcxxfu 1
Consider a forward / downward propagating wave at a given time, t
At time t+t , the wave has moved a distance x
But x = ct
Hence u = f1 (x-ct) , i.e. wave shape remains unchanged, the wave has merely advanced a distance x = ct
Solutions for velocity and stress : ctxhctxfE
xuE
1'
1)()( 1'
1 ctxgctxcftuv
Obviously, v and σ
also propagate with velocity c and do not change in shape in the absence of material damping
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Solution of 1-D Wave Equation
2
2
2
22
tu
xuc
)(, 21 ctxfctxftxu
ctxyyfctxfLet );()( 11
Wave equation :
General solution :
'fyf
;cty
;xy
111
ctxzzfctxfLet );()( 22
'2
2;;1 fzfc
tz
xz
'2
'1: ff
xuStrain
'2
'1: cfcf
tuvvelocityParticle
(1)
(2)
(3)
(4)
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No upward propagating wave, i.e. f2
(x+ct) = 0
'1f
vcfv '1
cvv
No downward propagating wave, i.e. f1
(x-ct) = 0
'2f
vcfv '2
cvv
(5)
(6)
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Downward wave
: F = EA ε
= -
EA f1
’
Zc
EAvF
:
Upward wave
: F = EA ε
= -
EA f2
’
Zc
EAvF
:
where Z = pile impedance
(7)
(8)
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Assuming the pile material remains elastic, the net force and net velocity at any location at a given time can be obtained by superposition of the downward and upward waves:
F = F↓
+ F↑
(9)v = v↓
+ v↑
(10)From Eqns
(7) and (8)F↓
= Z v ↓F↑
= -
Zv↑
(11)By combining Eqns
(9) –
(11), we can separate the downward wave from the upward wave if we know the total (net) force and velocity at a particular point along the pile
22ZvFFZvFF
22vzFvvzFv
(12)
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Boundary Conditions
The following boundary conditions are considered:
(i) free end
(ii) fixed end
Stress free boundary condition, i.e. net force at ‘b’, Fb = 0
0 FFFb FF
A downward propagating compressive wave is reflected at the free end as an upward propagating tensile wave.
Implications: Tensile stresses will develop during easy driving (e.g. in soft clay) –
potential problems for concrete piles and at the joints if splicing is poor. Solution: Control drop height of hammer.
or
Free end
(iii) impedance change
(iv) external soil resistance
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Force at ‘b’,
FFFFb 2
A downward propagating compressive wave is reflected at the fixed end as an upward compressive wave. At the fixed end, the compressive stress is doubled.
Implications: Potential problems with toe damage when driving piles into very hard stratum (rock), particularly when overburden soil is soft.
Boundary condition, vb = 0
0 vvvb
vvZ
FZ
F
FF
or
Fixed end
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rit FFF
cEAZ Impedance
rit vvv
Let subscripts i denote incident wave
r denote reflected wave
t denote transmitted wave
Impedance change
At interface “b”, the net force and net velocity is given by the superposition of the incident and reflected wave
Fb
= Fi
+ Fr
vb
= vi
+ vr
(18)
This is equal to the transmitted force and velocity:
(19)
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Relationship between transmitted and reflected waves with the incident wave:
rit vvv
112 ZF
ZF
ZF rit
rit FFZZFor
1
2
Let β
= Z2
/Z1
, then Ft
= β
( Fi
– Fr
)
From Eq
(19),
Fr
= Ft
– Fi
Hence, Ft
= β
[ Fi
– (Ft
– Fi
) ]
or
(β
+ 1)Ft
= 2 β
Fi
it FF1
2
(20)
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11
2iitr FFFF
ir FF11
Then,
or (21)
Hence, from Eqn
(20),
it vZvZ 12 12
it vv1
2
or (22)
Similarly from Eqn
(21)
ir vZvZ 11 11
ir vv
11
(23)
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Notes :
If an incident wave meets a section with a smaller impedance ( < 1) , the reflected velocity wave is of the same sign
as the incident wave.
If an incident wave meets a section with a larger impedance ( > 1) , the reflected velocity wave is of the opposite sign
as the incident wave.
The characteristic of the reflected wave and transmitted wave is
entirely a function of the ratio of the impedance of the 2 sections.
The analysis for pile with a change of impedance is useful for :
(a) interpretation of pile integrity
(b) selection of pile follower/dolly
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4.
3.
2.
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trib FRFFF
At the interface “b”, the net force and net velocity is given by
trib vvvv
External Soil Resistance
Consider now the effect of an external soil resistance (R) on the wave propagating in the pile. The soil resistance is usually in the form: R = ku
+ cv
(24)
(25)
From Eqns
(25) & (11),
ZF
ZF
ZF tri tri FFF or (26)
From Eqns
(24) & (26)
2)( RForFFRFF rriri (27)
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From Eqn
(26)
2RFF it (28)
The effect of an external soil resistance (R) on the propagating wave is to create a reflected wave of the same type as R with magnitude R/2 and a transmitted wave (due to soil resistance) of opposite type as R, also with magnitude R/2.
From the relationship between force and velocity [Eqn
(11)]
ZR
ZFv r
r 2 (29)
Note that this reflected velocity has a similar effect compared to when an incident wave meets a section with an increase in impedance ( see Eqn
(23) with β
> 1 )
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The objective of the low strain test is to provide an assessment of the integrity of the pile, i.e. whether there are any changes in sectional properties along the pile.
Low Strain Test –
Pile Integrity Test
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Intact PileDefective
Pile
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Data Analyser
2 inch diameter test hammer Accelerometer
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View showing the full hammer View showing the impact surface of hammer
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• cracks in pile
• pile joints (driven piles)
• changes in pile section
• high skin friction
• overlapping reinforcements (heavily reinforced piles)
Early reflections in integrity tests may be caused by:
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Limitations :
Advantages :
• Many piles can be tested in a day at low cost
• No pre-selection of piles required
• Minimal preparation required –
mainly trimming of pile head
• Major defects can be easily detected
• No information on bearing capacity of pile
• Minor defects may not be easy to detect
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Cannot estimate pile length for long piles –
low energy hammer impact gets damped out
• Debris at pile toe not easily detectable
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One-dimensional wave equation model with soil resistance:
Wave Equation Model
tPxuEAuk
tuc
tuA ss
2
2
2
2
pile inertia
soil damping
soil stiffness
pile stiffness
Conceptually, the soil is represented as a spring and dashpot.
The inclusion of the soil increases the complexity of the problem. Hence, the above equation is generally solved using numerical methods:
• finite difference method• finite element method• method of characteristics
Modelling
of the pile is relatively straight forward. The main difficulty is modelling
the soil behaviour.
Note: More sophisticated 3-D wave equation model (Chow, 1982) is available that can simulate the pile and soil (especially) in a more rational manner but commercially 1-D wave equation computer program continues to be used
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Typical “quake”
value,
Qu
JvRRD 1
Soil Models
)5.2(1.0 mminQu
Soil resistance during driving
mmtointoQu 105.24.01.0
Typical damping coefficient,
(a) Smith (1960) Model
Shaft :
Toe :
Soil type Jshaft Jtoe
Clay Sand
0.6560.164
0.0330.492
msJ /
Parameters to define curve:
• Ru
= max static resistance of soil spring
• Qu
= “quake”
value –
limiting elastic displacement
• J = damping coefficient
• R = static soil resistance
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(b) Lee et al. (1988) Model
ss Gk 75.2
sss Grc 02
Shaft (per unit length of pile shaft) :
Pile toe :
s
st v
rGk
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s
sst v
Grc
1
4.3 20
Developed at the National University of Singapore. Theory based on vibrating pile in an elastic continuum.
where
Gs = soil shear modulus
s = soil density
vs = soil Poisson’s ratio
r0 = pile radius
The expressions above have physical representations (stiffness and radiation damping) and are characterized by parameters that can be determined in the laboratory.
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Pile Drivability AnalysisPile drivability analysis is essential for the selection of appropriate hammer for the installation of piles.
Static Soil Resistance at time of Driving (SRD or Ru
)The soil resistance at time of driving will determine the depth to which a pile can be driven.
Ru
= ∑
fs
As
+ qb
Ab
where
fs
= unit shaft friction during driving
As
= shaft area
qb
= unit end bearing pressure
Ab
= gross cross sectional area of pile toe
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Unit shaft friction (fs
)Clay: fs
= cr
Remoulded
undrained
shear strength (cr
) –
generally estimated from liquidity index based on Skempton
& Northey
(1952) or using following formula from Wood (1990):
cr
= 2 x 100(1-LI)
kPa
where liquidity index , LL = liquid limit, PL is the plastic limit,
PI is the plasticity index, and w is the water content. Alternatively, cr
= cu
/S where S
is sensitivity of clay –
as a rule of thumb a value of 3 is sometimes used.
Sand: K σv
’ tan δ
(similar to static value)
Unit end bearing pressure (qb
)Generally assumed to be similar to static bearing capacity theory:
Clay: qb
= 9 cu
Sand: qb
= Nq
σv
’
where Nq
= f(Φ)(Brinch
Hansen)
PIPLw
PLLLPLwLI
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Cap Block and Pile Cushion Behaviour
Hysteretic behaviour
of cap block and pile cushion.
Hysteresis (a measure of energy loss):
inputenergyoutputenergy
ABCAreaBCDAreae 2
where e = coefficient of restitution
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Initial Condition for Computer Program
Most computer programs use an initial velocity
assigned to the ram as the starting condition. Potential energy of ram is converted to kinetic energy:
hgmevm f2
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= efficiency of hammer
fghev 2
This efficiency, ef
, is not
to be confused with the measured energy in the pile
Definition of Pile Penetration per blow (Set)
Smith (1960)’s soil model:
Pile penetration per blow = δmax
– Qu
Most computer programs stop computation when the pile toe velocity becomes zero.
NUS computer program (and soil model) compute the true set, i.e.
gives the final penetration of the pile toe when it comes to rest.
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Driving StressesThe wave equation program also gives the driving stresses in the
pile. The maximum driving stresses should be kept within reasonable limits.
Drivability Curves: Blow count versus DepthThe blow count versus depth curves should be produced for various hammers to determine suitable hammers to be used for the pile installation
Set-up or Relaxation•
The driving of piles in clay (particularly soft clay) results in the generation of excess pore water pressure. Subsequent consolidation will result in gain in soil strength. Thus if the driving process is interrupted, the soil will exhibit set-up effects, hence driving will be more difficult.•
Driving in dense sand may give rise to an opposite phenomenon –
“relaxation”. A decrease in driving resistance is possible.
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Methods used to estimate the pile bearing capacity :
Dynamic Pile Testing (High Strain Test)
(a) Efficiency of piling hammer in driven piles(b) Driving stresses in driven piles(c) Assessment of pile integrity(d) Bearing capacity and load-settlement response of pile
(a) Case Method(b) Stress-Wave Matching Technique
Objectives –
To obtain:
Test method: During the impact of the hammer, the stress waves are measured using strain transducers and accelerometers mounted on the pile (at least 1 diameter away from the pile head –
not an issue with offshore piles as driving is above water during the testing). The force trace is obtained from the strain measurements. From the acceleration trace, the velocity trace is obtained by numerical integration.
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Case Method
cLtvtvZ
cLtFtFR 2
22
21
1111
RtFJRR cs 12
Assuming that all the soil damping
is concentrated at the pile toe, the static component or bearing capacity of pile under static load is given by
Suggested damping factor, Jc
From the force and velocity versus time curves, the total soil resistance (includes both static and dynamic components) is given by
Sand :
0.1 –
0.15 ;Silt :
0.25 –
0.4 ;Clay : 0.7 –
1.0
“Correct”
Jc value obtain from correlation with static load test or stress wave matching analysis.
Silty
Sand :
0.15 –
0.25Silty
Clay :
0.4 –
0.7
where t1
is generally taken as the time when F(t1
) is maximum and Z is the pile impedance (= EA/c)
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Available computer programs :
Stress-Wave Matching Technique
The force-time history or velocity-time history is used as a boundary condition in a wave equation computer program. For instance, if the velocity-time history is used as the input, the wave equation program computes the force-time history and this is compared with the measured values. The soil resistance, soil stiffness and damping values are adjusted iteratively until the computed and measured values agree closely or until no further improvements can be made. When this stage is reached, the soil parameters used in the wave equation model are assumed to be representative of those in the field. The bearing capacity of the pile and the load-settlement response are then determined.
• CAPWAPC• TNOWAVE• NUSWAP
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Notes:
1.
The test results are representative of the conditions at the time of testing. For instance in the case of driven piles tested at the
end of driving in clay soils, the capacity obtained is generally a lower bound. Pile should be retested a few days after pile installation to allow set-
up to occur.
2.
If the impact energy used during testing is insufficient to move
the pile adequately, the pile capacity obtained may be a lower bound. The capacity obtained is actually the mobilised
static resistance.
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ReferencesChow, YK (1982) “Dynamic behaviour
of piles”, PhD Thesis, University of Manchester, UK
Chow, YK, Radhakrishnan, R, Wong, KY, Karunaratne
and Lee, SL (1988) “Estimation of pile capacity from stress-wave measurements”, Proc 3rd
International Conference on the Application of Stress-Wave Theory on Piles, Ottawa, pp 626-634.
Chow, YK, Yong, KY, Wong, KY and Lee, SL (1990) “Installation of long piles through soft clay”, Proc 10th
Southeast Asian Geotechnical Conference, Taipei, pp 333-338.
Lee, SL, Chow, YK, Karunaratne, GP and Wong, KY (1988) “Rational wave equation model for pile driving analysis”, Journal of Geotechnical Engineering, ASCE, 114, No 3, pp 306-325.
Lee, SL, Chow, YK, Somehsa, P, Kog, YC, Chan, SF and Lee, PCS (1990) “Dynamic testing of bored piles for Suntec
City Development”, Prof Conference on Deep Foundation Practice in Singapore,.
Skempton, AW and Northey, RD (1952) “The sensitivity of clay”, Geotechnique, Vol
3, No 1.
Smith, EAL (1960) “Pile driving analysis by the wave equation”, Journal for Soil Mechanics and Foundations Division, ASCE, 86, SM4, pp 35-61.
Smith, IM and Chow, YK (1982) “Three-dimensional analysis of pile drivability”, Proc 2nd
International Conference on Numerical Methods in Offshore Piling, Texas, Austin, pp 1-19.
Wong, KY (1988) “A rational wave equation model for pile driving analysis”, PhD Thesis, National University of Singapore.
Wood, DM (1990) “Soil behaviour
and critical state soil mechanics”, Cambridge University Press.