1OT5206 Offshore Foundations
Department of Civil & Environmental Engineering
National University of Singapore
Pile Driving Analysis
Y K Chow
2
3One-Dimensional Wave Propagation in Pile
Equilibrium equation (compression as positive)






∂
∂
+−=
∂
∂ dx
x
PPP
t
um 2
2
dx
x
P
t
uAdx
∂
∂
−=
∂
∂
ρ 2
2
AP σ=
x
u
∂
∂
−=ε
ρ = 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)
4Axial stress-strain relationship
x
uEE
∂
∂
−=ε⋅=σ
Hence from Eqns (1) to (4)
2
2
2
2
x
uEA
t
uA
∂
∂
=
∂
∂
ρ
2
2
2
2
2
x
uc
t
u
∂
∂
=
∂
∂
ρ
=
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
x
uE
t
u
∂
∂
=
∂
∂
ρ or
where
(4)
(5)
5Typical wave speed:
26 /10207 mkNE ×=
3/83.7 mt=ρ
smc /5100≈
Time taken to travel from pile head to pile toe and back to the pile
head:
c
Lt 2=
st 067.0
5100
1702
=
×
=
Steel :
where L = pile length
For example, take L = 170 m (say pile penetration of 50 m +
mean water depth of 100 m + nominal 20 m from sea level
to top of follower)
Steel pile :
6General solution to 1-D wave equation
f1(x-ct) = wave propagating in (+)ve x-direction (forward / downward)
)()( '2
'
1
ctxfctxf
x
u
++−=
∂
∂
( ) ( )ctxfctxfu ++−= 21
f2(x+ct)= wave propagating in (-)ve x-direction (backward / upward)
Proof: )()( ''2''12
2
ctxfctxf
x
u
++−=
∂
∂
)()( '2
'
1 ctxcfctxcft
u
++−−=
∂
∂
)()( "2
2"
1
2
2
2
ctxfcctxfc
t
u
++−=
∂
∂
Substitute Eqn (7) into Eqn (5),
- (6)
- (7)
[ ])()()()( "2"12"22"12 ctxfctxfcctxfcctxfc ++−=++−
The expressions are identical on both sides of the equation, hence
satisfying the wave equation
7( )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 = c∆t
Hence u = f1(x-ct) , i.e. wave shape remains unchanged, the wave has merely
advanced a distance ∆x = c∆t
Solutions for velocity and stress :
( ) ( )ctxhctxfE
x
uE −=−−=
∂
∂
−=σ 1
'
1)()( 1
'
1 ctxgctxcft
uv −=−−=
∂
∂
=
Obviously, v and σ also propagate with velocity c and do not change in shape in the
absence of material damping
8Solution of 1-D Wave Equation
2
2
2
2
2
t
u
x
uc
∂
∂
=
∂
∂
( ) ( ) ↑++↓−= )(, 21 ctxfctxftxu
ctxyyfctxfLet −==− );()( 11
Wave equation :
General solution :
'f
y
f;c
t
y;
x
y
1
11 =
∂
∂
−=
∂
∂
=
∂
∂
ctxzzfctxfLet +==+ );()( 22
'
2
2;;1 f
z
fc
t
z
x
z
=
∂
∂
+=
∂
∂
=
∂
∂
'
2
'
1: ffx
uStrain −−=
∂
∂
−=ε
'
2
'
1: cfcft
uvvelocityParticle +−=
∂
∂
=
(1)
(2)
(3)
(4)
9No upward propagating wave, i.e. f2(x+ct) = 0
↓ε=−=ε '1f
↓=−= vcfv '1
cvv +=
↓ε
↓
=
ε
No downward propagating wave, i.e. f1(x-ct) = 0
↑ε=−=ε '2f
↑=+= vcfv '2
cvv −=
↑ε
↑
=
ε
(5)
(6)
10
Downward wave : F = EA ε = - EA f1’
Z
c
EA
v
F
==↓ :
Upward wave : F = EA ε = - EA f2’
Z
c
EA
v
F
−=−=↑ :
where Z = pile impedance
(7)
(8)
11
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
22
ZvFFZvFF −↑=+↓=
22
vzFvvzFv −−↑=+↓=
(12)
12
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 conditions. Follower may
separate from main pile. Solution: Control drop height of hammer.
or
Free end
(iii) impedance change
(iv) external soil resistance
13
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
↓−↑= vv
Z
F
Z
F ↓
−=
↑
−
↓↑=FF
or
Fixed end
14
rit FFF +=
c
EAZ =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)
15
Relationship between transmitted and reflected waves with the
incident wave:
rit vvv +=
112 Z
F
Z
F
Z
F rit −=
( )rit FFZ
ZFor −=
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 FF 1
2
+β
β
= (20)
16






−
+β
β
=−= 1
1
2
iitr FFFF
ir FF 1
1
+β
−β
=
Then,
or (21)
Hence, from Eqn (20),
it vZvZ 12 1
2
+β
β
=
it vv 1
2
+β
=or (22)
Similarly from Eqn (21)
ir vZvZ 11 1
1
+β
−β
=−
ir vv β+
β−
=
1
1
(23)
17
Notes :
If an incident wave meets a section with a smaller impedance (β <
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 (β >
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
1.
4.
3.
2.
18
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),
Z
F
Z
F
Z
F tri =− tri FFF =−or (26)
From Eqns (24) & (26)
2
)( RForFFRFF rriri =−+=+ (27)
19
From Eqn (26)
2
RFF 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)]
Z
R
Z
Fv rr 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 )
20
One-dimensional wave equation model with soil resistance:
Wave Equation Model
( )tP
x
uEAuk
t
uc
t
uA ss =∂
∂
−+
∂
∂
+
∂
∂
ρ 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
21Schematic in GRLWEAP
22
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
0.656
0.164
0.033
0.492
( )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
23
(b) Lee et al. (1988) Model
ss Gk 75.2=
sss Grc ρπ= 02
Shaft (per unit length of pile shaft) :
Pile toe :
s
s
t v
rGk
−
=
1
4 0
s
ss
t v
Gr
c
−
ρ
=
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
ρs = 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.
24
Pile Drivability Analysis
Pile drivability analysis is essential for the selection of appropriate hammer
for the installation of offshore piles. Example of commercial wave equation
program – GRLWEAP.
Static Soil Resistance at time of Driving (SRD)
The soil resistance at time of driving (SRD) will determine the depth to
which a pile can be driven.
Unplugged pile
Soil column (plug) inside pipe pile moves up the pile during driving -
assume that the internal shaft friction is less than end bearing capacity of
soil plug.
SRD = ∑ fsoAso + ∑ fsiAsi + qbAw (1)
where
fso = unit shaft friction during driving (outside); Aso = outside shaft area
fsi = unit shaft friction during driving (inside); Asi = inside shaft area
qb = unit end bearing pressure; Aw = annulus area of pipe pile
25
Plugged pile
Soil column (plug) inside pipe pile moves down together with pile during
driving - assume that the internal shaft friction is greater than end bearing
capacity of soil plug.
SRD = ∑ fsoAso + qbAp (2)
where
fso = unit shaft friction during driving (outside); Aso = outside shaft area
qb = unit end bearing pressure; Ap = gross cross sectional area of pile toe
26
Unplugged pile Plugged pile
External shaft
friction External shaft
friction
Internal shaft
friction
End bearing
pressure on pile
wall
End bearing
pressure on
soil plug &
pile wall
fsofsi
qb
fso
qb
27
SRD (kN)
28
Notes:
1. Based on statics, it is generally assumed that the pile will behave in a
manner which produces least resistance to pile penetration, i.e. at any
depth, the SRD will be given by the lesser of Eqn (1) and (2). In
reality, pile driving is a dynamic phenomenon and the pile generally
behaves as unplugged during driving (especially during continuous
driving) (Chow, 1982)
2. The unplugged mode in an all clay soil profile will generally give a
higher resistance to pile penetration
3. In commercial wave equation program, the 1-D idealisation of the pile
is unable to distinguish between the inside and outside shaft friction
and model them as a soil resistance at a particular level of the pile.
Although there are specially developed wave equation models that
attempt to simulate the behaviour of the soil plug inside the pipe pile,
these are essentially 1-D model and are thus approximate. Proper
modeling of the soil plug behaviour would require a truly 3-D wave
equation model (Chow, 1982; Smith & Chow, 1982)
29
4. It should be noted that in drivability analysis for the selection of
hammer, it is conservative to over-estimate the SRD. An
underestimate of SRD can lead to premature refusal of piles and has
serious cost implications because of the need to remobilize a larger
hammer (pile refusal is often taken to be 150 blows per foot (0.3m)
or 500 blows/m.
5. On the other hand, over-conservative estimate of SRD may result in
piles being driven deeper than the theoretical estimate and may give
the mistaken impression that the piles are under-capacity.
6. In offshore pile design, piles are driven to design penetration depth
and not to “set” unlike onshore pile design.
30
(a) 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) – Limiting value in API applies
For pipe piles without internal driving shoe, it is often assumed that:
fsi = 0.5 fso (Sam & Cheung, 1993)
(b) 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)
(API (limiting value applies) or Brinch Hansen)
PI
PLw
PLLL
PLwLI −=
−
−
=
31
32
Friction Fatigue
Friction fatigue is the phenomenon where shaft resistance decreases as the
pile penetrates deeper into the soil during driving. Heerema (1980)
introduced this concept that the unit shaft resistance decreases as the piles
are driven deeper into the soil. The exponential decrease in soil resistance
postulated by Heerema (1980) can be seen in Figure below.
Figure: Shaft friction resistance at varying depths (Heerema, 1980)
33
Alm and Hamre (2001) CPT based method incorporating Friction Fatigue
Alm and Hamre (2001) proposed a CPT based method incorporating the
concept of friction fatigue in sand and clay. General formulation for shaft friction
along a pile during driving:
s = s,res + (s,i − s,res) ∙ k(d-p)
where
s = pile shaft friction (kPa)
s,res = residual pile shaft friction (kPa) (function of CPT sleeve friction/tip
resistance)
s,i = initial pile shaft friction (kPa) (function of CPT sleeve friction/tip
resistance)
k = shape factor for degradation (function of cone tip resistance and
effective overburden pressure)
d = depth to actual layer (m)
p = pile tip penetration (m)
Details given in Alm and Hamre (2001).
34
Hammers
Hydraulic or steam hammers used. For deep water depths, underwater
hydraulic hammers are used.
Southeast Asian waters (typical water depths: 60 – 110 m):
Some hammers:
(a) IHC90
(b) MHU 500 – 1700 (generally do not use pile cushion)
(c) Menck 3000 – 4600 (generally use with pile cushion – bongossi wood)
Pile diameter ranges from: 36 – 60 in (0.914 – 1.524 m)
Pile wall thickness: 1.25 –2.0 in (32 – 50 mm)
In the North Sea, much larger pile sizes are used:
Diameter of 84 in (2.13 m) and 96 in (2.43 m) with wall thickness of 2.5 in
(63 mm) and 3 in (76 mm) are common
35
Cap Block and Pile Cushion Behaviour
Hysteretic behaviour of cap block and pile cushion.
Hysteresis (a measure of energy loss):
inputenergy
outputenergy
ABCArea
BCDAreae ==2
where e = coefficient of restitution
36
Initial Condition for Computer Program
Computer program uses an initial impact velocity assigned to the hammer
as the starting condition. Potential energy of hammer is converted to
kinetic energy:
hgmevm f=
2
2
1
where m = mass of hammer
h = hammer stroke or drop height (or equivalent stroke for
double acting hammer)
ef = efficiency of hammer
v = impact velocity of hammer
fghev 2=
This efficiency, ef, is not to be confused with the measured energy in the
pile.
Impact velocity of hammer
37
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.
38
Driving Stresses
The 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 Depth
The 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.
39
Pile Refusal
• Review hammer performance – particularly helpful if pile installation is
monitored using a Pile Driving Analyser (PDA) which will allow driving
energy to be monitored and compared against rated energy by hammer
manufacturer
• Review SRD used in pile drivability analysis
• Re-evaluate design penetration – reconsideration of design loads and
design soil parameters.
• Last recourse (very expensive operation) – removal of soil plug inside
pile by drilling to reduce driving resistance and continue driving. It may
be necessary to subsequently have a grout plug to enable the
development of the toe bearing capacity. In more serious cases, it may
be necessary to drill past the pile toe to enable driving to continue.
1. Pile refusal is often taken to be 150 blows per foot (per 0.3m) or 500
blows/m
2. When a pile refuses before it reaches design penetration, one or more
of the following actions can be taken:
40
41
42
43
44
45
SRD (kN)
A typical GRLWEAP output on drivability for another project
46
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 1.5 diameter away from the pile head). The force
trace is obtained from the strain measurements. From the
acceleration trace, the velocity trace is obtained by numerical
integration.
47
48
PDA Model PAX Accelerometer and strain
transducer mounted on a
pile (above water)
Underwater strain
transducer and
accelerometer mounted
on pile.
49
Typical force and velocity versus time curves for an offshore pile
(Vel Msd above is actually pile impedance Z x Vel Msd)
50
Case Method
( ) ( )











 +−+











 ++=
c
LtvtvZ
c
LtFtFR 2
2
2
2
1
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 – 0.25
Silty 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)
51
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
52
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.
53
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55
56
57
58
59
60
61
62
63
64
65
66
References
Alm, T. & Hamre L., (2001), Soil model for driveability predictions based on CPT
interpretations, Proceedings of the International Conference on Soil Mechanics and
Geotechnical Engineering, Balkeema, CONF 15; VOL 2, pp. 1297-1302.
Chow, YK (1982) “Dynamic behaviour of piles”, PhD Thesis, University of Manchester,
UK
Heerema, E. P. (1980). Predicting pile driveability: Heather as an illustration of the
friction fatigue theory. Ground Engng 13,15–37.
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.
Sam, MT and Cheung LY (1993) “Installation of a 479 feet water depth platform in South
China Sea”, Proc 3rd International Offshore and Polar Engineering Conference,
Singapore, pp 288-293
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.