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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

14 0

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.