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DYNAMIC RESPONSE OF FOOTINGS AND PILES

by

Wing Tai Peter To, B.Sc.

A thesis submitted to theUniversity of Manchester

for the degree of,

Doctor of Philosophyin the

Faculty of Science

Fepruary, 1985;.

To Gertrude and my parents

CONTENTS

ABSTRACTACKNOWLEDGEMENTSDECLARATIONNOTATION

CHAPTER 1 INTRODUCTION

CHAPTER 2 - FORMULATION OF NUMERICAL MODEL2.1 Constitutive Relationships2.2 Material Nonlinearity2.3 Implementation of the Initial Stress Method2.4 Integration Order and Element Types2.5 Deep Foundations Axial Capacity of a Single Pile

in Clay2.5.1 Shaft Capacity2.5.2 Bearing Capacity

2.6 Interface Elements2.7 Influence of Mesh Boundaries in Static Analysis

CHAPTER 3 - FINITE ELEMENT SOLUTION TO THE EQUATION OF MOTION3.1 General Solution Procedure3.2 Dynamic Response Analysis as a Wave Propagation

Problem3.3 Considerations for Dynamic Analysis

3.3.1 Spatial Discretisation3.3.2 Mass Formulation

i

iii

ivv

1

4

5

6

8

11

11

121317

19

20

2122

243.3.3 Temporal Operators and Associated Considerations 253.3.4 Effect of Transmitting Boundaries 28

3.43.3.5 SummarySolution Algorithm Wfison(6 • 1.4) Scheme withInitial Stress Method

31

32

CHAPTER 4 - DYNAMIC RESPONSE OF SHALLOW FOOTINGS4.1 Introduction 354.2 Periodic Excitation of a Smooth Massless Circular

Footing upon a Smooth Elastic Stratum 384.3 Response of Dynamically Loaded Foundations 414.4 Response of a Rigid, Circular Surface Footing

Subjected to a Trapezoidal Pulse 464.5 Acceleration Response of a Circular Surface Footing

Subjected to Impact 504.6 Foundation Response to Indirect Impact 51

4.6.1 Introduction4.6.2 Mesh Design

5153

4.6.34.6.4

Stage I : Static Response of 'Target' Foundation 53Stage II : Dynamic Response of 'Target'

4.6.5FoundationStage III : Dynamic Response of the 'Second'Foundation

CHAPTER 55.1

VIBRATORY PILE DRIVINGHistorical Development

5.2 Comparison between Conventional Impact Pile Drivingand Vibratory Pile Driving

5.3 The Principle of Vibratory Pile Driving5.3.1 Introduction

5.3.2 Mechanisms of Penetration

54

55

56

57

60

60

61

5.4 Finite Element Simulation of Vibratory Driving inCohesive Soils 685.4.1 Elastic Analysis 68

5.4.2 Elastoplastic Analysis 71

5.4.3 Parametric Studies 75

5.5 Environmental Impact of Vibratory Pile Driving 78

CHAPTER 6 IMPACT PILE DRIVING6.1 Introduction6.2 One-Dimensional Analysis6.3 Three-Dimensional Analysis6.4 Deformation Pattern due to Impact Pile Driving6.5 Closed-Ended Piles : Effect of the Damping Parameters

Js and Jp6.6 Open-Ended Piles

Comparison of Behaviour in Driving andStatic Loading

6.6.2 Effect of Adhesion Coefficients ai' ao

6.6.1

6.6.3 Effect of Pile Inertia6.7 Comparison of Driving Performance of Open- and

Closed-Ended Piles6.8 Evaluation of Static Pile Capacity

6.8.1 Introduction6.8.2 Field Load Test6.8.3 Dynamic Methods6.8.4 SummaryThe Case Method6.9

80818689

90

90

90

9394

95

96969798

103103

6.9.1 The Development of the Case Method 1036.9.2 Advantages and Limitations of the Case Method 106

6.9.3 Assessment of the Case Method by AxisymmetricFinite Elements

CHAPTER 7 - CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH7.1 Summary and Conclusions7.2 Suggestions for Future Research

REFERENCES

107

112

116

118

i

ABSTRACT

Dynamic response analyses can be regarded as stress wave propagation

problems. The solution of such by the finite element method entails more

consideration than static problems, since sources of inaccuracies such as

dispersion, spurious oscillations due to mesh gradation, w~ve reflection at

transmitting boundaries, as well as instability or inaccuracy due to temporal

operators and discretisation can arise. The criteria for formulating a

finite element model for dynamic response analysis have been investigated.

Using the relatively simple von-Mises soil model (satisfactory for

undrained saturated clay) three categories of problems have been

investigated:-

(i) The dynamic response analyses of surface footings subjected to periodic

and impact loading have been performed in order to evaluate the finite

element model design criteria. An approximate analysis is also

performed in reducing a three-dimensional indirect impact problem to a

two-dimensional analysis.

(ii) Vibratory pile driving is a relatively new but somewhat unreliable

technique of pile installation. Penetration is instantaneous if

conditions are right, but with the high hire charges and uncertainty in

success the technique is unpopular, especially in clays. In the work

presented it is shown that vibratory installation is possible in

cohesive soils at the fundamental frequency for vertical pile

translation, if a high enough dynamic oscillatory force is provided.

Penetration mechanisms have also been exploited.

(iii)On the other hand, impact pile driving is reliable and widely adopted

in terrestrial as well as offshore construction. Experience in one-

dimensional wave equation analysis is discussed, and further numerical

evaluation of the parameters involved has been carried out by a more

elaborate axisymmetric finite element model. In cohesive soils a

ii

closed-ended pile may be driven more easily than an equivalent open-

ended pile, depending on the level of the internal soil column and the

soil properties. In the light of the growing popularity of non-

destructive determination of the axial load-carrying capacity of piles

by dynamic methods, the possibility of correlating the soil resistance

mobilised in dynamic conditions to the ultimate static capacity is

queried. The semi-empirical Case method has been assessed in detail.

iii

ACKNOWLEDGEMENTS

The author wishes to express his sincere gratitude to:

Professor I.M. Smith for his supervision and constant encouragement

throughout the accomplishment of this research, and for the permission to

use the facilities in the department.

Staff members of the University, Dr. W.H. Craig, Dr. D.V. Griffiths,

Mr. D.C. Proctor, Dr. I. Gladwell and Mr. B. Cathers for their constructive

guidance and interesting discussions.

Dr. Y.K. Chow whose initial work in modelling pile-soil systems by finite

elements has paved the way for the present research.

All his colleagues for enlightening discussions through informal meetings

and research seminars.

The Engineering Department of the University for the award of an Engineering

Scholarship from October 1981 to September 1982.

The Committee of Vice-Chancellors and Principals for the Overseas Research

Student Award covering the period October 1981 to September 1984.

The Croucher Foundation for the award of a Scholarship covering the period

October 1982 to 1985.

iv

DECLARATION

No portion of the work referred to in the thesis has been submitted in

support of an application for another degree or qualification of this or any

other university or other institution of learning.

v

NOTATIONS

Symbols not given below are defined in the text.

Soil and Foundation Parameters

E

G

v

P

Cu

4>u

t/J

OCR

ai' (:to

Young's modulus

Shear modulus

Poisson ratio

Density

Undrained cohesive strength of soil

Undrained angle of friction of soil

Angle of dilation of soil

Over-consolidation ratio

Adhesion coefficients of the inner and outerpile shaft

Radius of footing

Static and dynamic bearing capacity of piles

Smith's (1960) damping parameters for the pileshaft and pile tip respectively

Case damping coefficient

Soil resistance mobilised during driving

Velocity of shear (S-) and compression (P-) waves

vi

Stress and Strain Parameters

o, T Normal and shear stresses

Normal and shear strains

J2

fvm

subscripts:-

Second deviatoric stress invariant

Yield function for von-Mises criterion

x, y, z Orientation of Cartesian coordinates

r, 8 Radial and circumferential orientations

P

D

Out-of-balance stress vector

Additional out-of-balance stress vector,

due to correction of drift

Vectors and Matrices

K

C

M

F

BDYLDS

B

D

Stiffness matrix

Damping matrix

Mass matrix

External force vector

Out-of-balance body force vector

Shape function matrix

Property Matrix

vii

Dynamic Analysis Parameters

Amplitude of dynamic and static input force

x, i, x

t , ,1t

Displacement, velocity and acceleration vectors

Time, time step size

wavelength

velocity of stress wave (in general terms)c

c Courant number

a,/3

f, w

Collocation parameters of the Wilson and

Newmark time integration schemes respectively

frequency and angular frequency

T wave period

Symbols for Transmitting Boundaries

Fixed boundary

Roller boundary

Standard viscous boundary

Conventions

For stresses, strains, forces, displacement and its time derivatives:

Compression and downwards are positive;

Tension and upwards are negative.

1

CHAPTER 1

INTRODUCTION

There are essentially two methods of pile driving, namely by impact or

by vibration. Impact pile driving is the more conventional technique, by

which a pile is hammered into the ground by a number of discrete blows (Fig.

1.1). This has profound applications in offshore construction works, where

the sheer size and cost (and at least up to now, risks) are orders of

magnitude greater than in terrestrial operations. As a result, there arises

the need to assess the driveability of piles, as well as much cheaper, non-

destructive techniques to evaluate the subsequent pile capacities.

On the other hand, the concept of vibratory pile driving is relatively

new, especially to the western world. Continuous, periodic load is applied

by special vibratory hammers mounted on the pile top (Fig. 1.2), the

mechanism of which may be either rotary eccentric or linear. The advantage

of vibratory driving is that provided the appropriate operationg frequency

is selected, penetration can be many times faster than impact driving, or

even virtually instantaneous (Engineering News Record, 1961). The earlier

Russian vibrators operate at relatively low frequencies (less than 60 Hz.),

aiming at the resonant frequency of the soil mass. In contrast the more

recent American versions aim at resonance of the pile, which may be over

100 Hz. So far the operating frequency is determined in the field by trial

and error, and thus in-built flexibility in frequency variability is an

important feature in vibratory hammers. The technique is known to work well

in loose sands, but becoming unreliable in clays. On the whole, there is

still much uncertainty regarding penetration mechanisms as well as subsequent

loading performance of vibratory driven piles.

While experimentation of pile driving in the field tends to be expensive

and time consuming, analysis using the finite element method seems to be a

feasible and effective technique. As with any foundation-soil system, the

\2

dynamic response can be described by the basic equation of motion:

M x + C X + K x • F(t) (1.1)

Solution of the above equation in the time domain will allow prediction of

permanant (i.e. plastic) deformations.

Due to the lack of perfect rigidity, the application of a dynamic load

will give· rise to stress wave propagation within the system. In reality

these stress waves propagate in a continuum (which the soil medium is

assumed to be), with an infinite number of natural frequencies. However,

when this is simulated by a finite element model, the continuum is replaced

by a model having only a limited number of degrees of freedom, and hence a

finite number of natural frequencies. The consequence is that signals

propagating at all frequencies will be somewhat distorted, the severity of

which increases with frequency, as well as any mesh gradation. This

phenomenon is known as dispersion. There also exists a certain frequency

above which the waves are so distorted that they are rapidly attenuated,

leaving behind their energy which tends to cause spurious oscillations.

This 'cutoff' frequency is mainly a function of the size of elements. Thus

is clear that a mesh suitable for static analysis (which may be discretised

in a somewhat random fashion) may not be justified for dynamic analysis. In

this thesis, the characteristics of finite element dynamic response analysis

are considered, and subsequently applied to solve problems of footing

vibrations and pile driving. The only soil material considered is undrained,

saturated and frictionless clay, which can be governed by the relatively

simple von-Mises yield criterion. Effects of cyclic degradation can be

taken into account, but is omitted herein for simplicity.

The layout of the thesis is outlined below.

Chapter 2 reviews the formulation of a finite element model leading to

the solution of static bearing capacity problems. Various numerical

parameters, including those for interface elements used to model slippage

between the pile shaft and the soil are assessed.

3

Chapter 3 points out the fact that finite element practices suitable

for static analyses may not be justified for the solution of dynamic

problems. An example is the use of mesh gradation. By treating a dynamic

response problem as a stress wave propagation problem, various criteria

regarding spatial and temporal discretisation, the influence of artificial

truncating boundaries and various other aspects are examined. An implicit

time integraion scheme suitable for nonlinear dynamic response analysis is

stated.

The solution algorithm proposed in Chapter 3 is utilised to solve a

number of footing vibration problems in Chapter 4, including problems of

periodic excitation as well as pulse loading. By comparing the finite

element solutions of benchmark problems with corresponding closed-form

analytical solutions, the mesh design criteria described earlier is assessed.

A further problem examines the feasibility of approximating a three-

dimensional dynamic response problem by a two-dimensional model.

Chapter 5 examines the relatively new technique of vibratory pile

driving. The state-of-art is described, and is further explored by finite

element simulation. The process of installation, loading performance and

environmental considerations are discussed.

A more usual pile driving technique is the use of steam or diesel

hammers, as practised in both on-land and offshore operations. However,

the considerably larger scale of the latter calls for the assessment of

driveability. In Chapter 6, various parameters influencing driveability

is examined. The driving performance of open- and closed-ended piles

are compared. Furthermore, there are various non-destructive methods of

estimating the capacity of piles, usually by means of dynamic measurements.

One of these, the Case method, is assessed in detail.

tJ ~~ ..c::~ QJ 00t'tI ..... ~~ ~ QJtil p.. ~

t.:JZ->--.0::Cl

W.....J-c,I-w<:o,:::E:-

UJ 0::%: 0...... LL..

Z0:::UJl-I--ec,

uiu0:::0LL..

I-::::Jo,z-.....J-eu-o,>-I-

3JClOd lndNI

t:JZ.......>.......~0

w-'.......o,

>-~0~-c~CO.......

~ >I.U ~.....:E: 0en

Al- u,~

+ z~en W~ ~

n ~<:....... o,~- w~u~0u,

~::::>o,z.......-'-cU.......c,>-~

N

t:J.......u,

.00

dJeiOd IndNI

4

CHAPTER 2

FORMULATION OF NUMERICAL MODEL

2.1 Constitutive Relationships

The modelling of soil behaviour is considerably more difficult than

that of structures. Conditions even of plane strain or axisymmetry imply

that one has to manipulate a complicated three-dimensional combination of

stress states. Moreover, being particulate and heterogeneous in nature

soil can rearrange itself upon a change in stress state. A generalised

constitutive model should thus be capable of taking into account features

like dilatancy, sensitivity, degradation, viscosity, as well as the

generation and subsequent dissipation of excess pore pressures. A lot of

research has been devoted to better the predictive ability of numerical

modelling, for both sands and clays. Important conferences on constitutive

laws were held, and proceedings edited by Parry (1972), Palmer (1973),

Murayamo & Schofield (1977), Yong & Ko (1981), Desai & Saxena (1981) and

Desai & Gallagher (1983) contained many novel and state-of-art information

on the subject. The workshop chaired by Yong & Ko (1981) is especially

interesting since many of the important existing soil models were compared

for their predictive ability. The general conclusion was that a complicated,

all-encompassing soil model is unnecessary and undesirable, as it can be

costly and does not guarantee better quality results than a simpler model.

The behaviour of undrained, saturated clay is probably the one most

amenable to numerical modelling. The strength of the soil is independenttTI•• n

of any change inAstress state as a result of loading, and can be described

by a single parameter cu. The yield condition can be satisfactorily

governed by the von-Mises criterion, the yield functions of which are

fvm...{[3{J; -

{JP;for axisymmetry (2.1)

for plane strain (2.2)

5

where J2 = 16" + (0: _ (1) 2]z x +

Throughout this thesis the above soil model will be used to study the

static and dynamic interaction with foundations.

2.2 Material Nonlinearity

When a soil mass is subjected to substantial deviator stress, yield

(i.e. f = 0) may occur resulting in irrecoverable plastic deformation orvm'flow'. If strain is considered to be seperated into an elastic and a

plastic component, the latter is not only a function of the stress state,

but also depends on the stress path taken. Because of such dependence of

plastic strain on the stress path, it is generally necessary to compute the

increments of plastic strains throughout the loading history and then

obtain the total strain by integration. The relationship of the plastic

strain increment to the corresponding stress increment is described by a

'flow rule', which may be 'associated' (if the strain rate vector is normal

to the yield surface, and is only true when ~ ~ ~) or 'non-associated'

(when Cl> = t/J).There exists a variety of numerical solution procedures developed to

cope with elastoplasticity problems. Iterative techniques are generally

employed, the objectives being to ensure that

(i) equilibrium is satisfied between the externally applied and

the internal stresses; and(ii) the stress state of the soil mass does not violate the specified

yield condition(s), or f ~ o.In finite element implementation the methods can generally be divided

into two categories: the constant stiffness methods, which follow the

concept of the Newton-Raphson method in that a constant global stiffness

matrix is employed for all load increments; and the variable stiffness

methods which involve repeated stiffness matrix assembly and tridiagonalisation

for every load increment. These are summarised in Table 2.1. In the

6

following work the initial stress method is adopted for reasons of efficiency

and versatility.

2.3 Implementation of the Initial Stress Method

The theory and programming strategies of the initial stress method are

well documented (see Table 2.1). Essentially, an elastic solution is

obtained for a load increment, and the stress state is computed for each

gauss point. The values of the yield function subsequently computed will

determine whether the material has locally turned plastic or not. Should

plasticity occur an adjustment process is performed in which the flow and

hardening rules are incorporated. A set of 'bodyloads' is obtained on

integrating the out-of-balance stresses, and is redistributed into the

system as psuedo-loads in the next iteration, until convergence is

achieved (Fig. 2.1).

Refinements in the Solution Algorithm

(i) Subdivision of Strain Increments.

Should the subsequent strain increment be too large, the plasticity

matrix may vary considerably across the increment. A possible remedy is to

subdivide the strain increment into a given number of smaller steps, with

the plastic matrix computed repeatedly for each step (Fig. 2.2).

This procedure has been incorporated by Nayak & Zienkiewicz (1972)

and Chin (1979). Although it has also been incorporated in the computer

programs developed for this project, the implementation is found to be

time consuming, and no subdivision is employed in the following analyses.

(ii) Correction of Drift.

Since the plasticity matrix is not strictly constant for any plastic

strain increment, the correction of the out-of-balance forces will not

revert the stress state directly onto the yield surface. Instead, a

slight drift from the yield surface may exist, the magnitude of which

depends on the gradient of the plastic stress-strain curve (i.e. the

7

degree of hardening) and the magnitude of the strain increment.

Such errors tends to cumulate if uncorrected, resulting in a false

indication of gain in strength by the material. Nayak & Zienkiewicz (1972)proposed a correction procedure assuming that the stress change is normal to

the yield surface:f

( 1 )

{!~}T {:~}

(2.3)

where fl is the yield function of the stress state before drift correction,#fand is the gradient vector.

In the solution procedure this stress correction is added to the out-

of-balance stress vector, and their sum is integrated in the form

.. J [B]T d{vol) (2.4)

in order to establish the body force vector. In practice i ~UD}is found

to be of negligible magnitude when compared with I ~(7pt, so that the local

approximation of flow rate will only introduce minimal error.

(iii)Extra Correction for the body force vector.

The above correction procedures should bring the stress state of a

system to the yield surface, even though it may not be exactly on the

correct position, due to the slackness introduced by the tolerance specified

for the convergence criterion. Taking Fig. 2.1 as an example, convergence

may be achieved after 3 iterations, bringing the stress state to Dl• The

discrepancy DIEI will accumulate if left uncorrected, despite the fact that

the convergence criteria should guarantee such discrepancy to be of small

enough magnitude.

Chin (1979) has proposed a novel correction procedure, such that an

extra set of body forces {FEXTRA} is determined:

{FEXTRAl .. 1FBDYLDS}at convergence - {FBDYLDs}one iteration (2.5)before convergence

8

It has been found that this helps to reduce the number of iterations in

t~e next increment when this extra set of body forces is applied.

(iv) The above solution refinement techniques have been incorporated in the

algorithm for the work presented. Apart from these, other modifications and

refinements have been suggested, for example by Nayak (1971) and Thomas (1984).

Since these are not implemented they are not further described in detail here.

2.4 Integration Order and Element Types

Since the displacement formulation of the finite element method can be

regarded as an extension of the Ritz analysis, a lower bound exists on the

'exact' strain energy of the system considered. In other words, a

displacement formulation will result in overestimating the stiffness of the

system. Thus by integrating the stiffness matrix with a reduced order the

error introduced may somewhat compensate such overestimation. Experience

confirms that an appropriately reduced integration order tends to lead to

improved results in many cases.

Another use of reduced integration is to remedy the incompetence of many

finite elements in the prediction of failure loads. Nagtegaal et al (1974)

and Sloan & Randolph (1982) have shown that when an elastic-perfectly plastic

material is stressed in an undrained condition, it becomes nearly

incompressible when impending collapse is approached. Under such conditions

meshes assembled from a number of popular finite elements will become

over-constrained, resulting in substantial over-estimation of limit loads.

In the case of the 8-node quadrilateral, the performance is found to be

satisfactory in plane strain but not in axisymmetric conditions, a verdict

also applicable to a number of other conventional finite elements.

As a result, two possible alternatives emerged:-

(i) Apply reduced integra~ion to the conventional elements.

The first alternative is to perform reduced integration with the

9

conventional elements in the formulation of the stiffness matrix (and

usually in dynamic analyses, the mass matrix as well). This in effect

reduces the number of constraints in the finite element formulation, and

consequently its stiffness (Fig. 2.3), though in a somewhat abitrary

fashion. Furthermore, in the case of 8-node quadrilateral elements

Naylor (1974) has shown that the (2 x 2) gauss points turn out to be the

best possible positions for the sampling of stress information, even in the

case of near incompressibility.

Due to its relative simplicity and economy the reduced integration

technique in conjunction with 8-node elements have been employed for

some time in Manchester (for example recently by Griffiths, 1980; Chow,

1981; and Smith, 1982) and elsewhere (such as Zienkiewicz et aI, 1975;

and Thomas, 1984).

However, the reduced integration technique is only approximate in

nature, and its disadvantages must not be overlooked. The employment of

reduced integration will destroy the bounding property of the finite

element method mentioned in the first paragraph of this section. Also in

the case of very crude meshes, reduced integration may lead to an inferior

solution due to the failure in accurately capturing the onset and spread of

yield (Fig. 2.4). With regard to 8-node quadrilaterals, (2 x 2) reduced

integration will result in conditions of near incompressibility in the

plastic range to be satisfied only at the integration sampling points.

Furthermore, reduced integration in effect reduces the order of the

elements, and is liable to give rise to zero-energy deformation modes, i.e.

deformation patterns in which the strain field is zero at all gauss points.

Examples of such zero-energy deformation modes are given in Fig. 2.5.

These are also known as 'hourglassing' modes. Hourglassing will result in

extra zero eigenvalues other than those of the rigid body modes, rendering the

stiffness matrix singular. Although this may occur for individual elements,

hourglassing is not likely to occur in a finite element mesh made up from a

10

number of elements for the reason of compatibility. (Nevertheless, it is of

interest to note that if reduced integration is employed to formulate a

consistent mass matrix for eigenvalue evaluation, in either axisymmetry or

plane strain, the resulting mass matrix will be singular. Under such

circumstances no result can be obtained if the mass matrix is tridiagonalised,

but a number of the lower harmonics can be obtained using stiffness

factorisation, although the accuracy falls rather rapidly with increasing

harmonics).

(ii) The Use of High Order Elements.

The second approach to improve limit load prediction by finite element

method is to resort to those elements which can perform satisfactorily in

the presence of incompressibility constraints. These elements tend to

possess internal nodes and are consequently high-order elements. Examples

are the 9-node quadrilateral and the IS-node triangle. Computations using

the latter have been demonstrated by Sloan & Randolph (1982) and de Borst

& Vermeer (1984). The usually large bandwidth generated by meshes assembled

from these elements makes the ordinary equation solution algorithms

inefficient, and special techniques like the frontal solver and static

condensation are advisable. In the case of mesh discretisation in terms

of triangular elements, extra care should be taken in order to preserve the

symmetries present in the system (Robinson, 1971).

This approach is more rigorous than the reduced integration technique

described earlier, and the bounding property of the finite element method

can also be upheld by using exact integration, though at the expense of

considerable extra computer effort.

Nevertheless, in practice the solution quality of the reduced integration

approach in conjunction with 8-node rectangles has not been found to be

inferior to those obtained by IS-node triangles. In the following work

presented, the former and more economical approach is adopted.

11

2.5 Deep Foundations: Axial Capacity of a Single Pile in Clay

The axial capacity of a pile comprises two components, namely the shaft

resistance and the bearing capacity at the pile tip. The analytical

treatments of these follow rather different procedures, and have been

comprehensively reviewed by Esrig & Kirby (1979), Chow (1981) and Randolph

& Wroth (1982).

2.5.1 Shaft Capacity

For piles in cohesive soils the majority of the working load is usually

carried by the shaft resistance component. In the last two decades much

research has been devoted to better the prediction of the shaft resistance

of piles, thanks to the offshore boom. The procedures proposed fall into

two categories: the total stress approach and the effective stress approach.

The total stress approach consists mainly of the a-method (API, 1981), which

relates the shaft capacity to the undrained shear strength of the soil, cu'

a quantity comparatively easy to be measured, such that

T = (2.6)

where a is a parameter to be correlated with the soil profile. Esrig &Kirby (1979) reported large scatter of data when attempting to relate T

to Cu for a number of load tests. This may be caused by:

(i) when a pile is driven through a stratified soil, the dragdown of the

overburden materials tends to alter the shaft resistance of the pile

(Tomlinson, 1971);

(ii) the load tests concerned may be performed after different set-up

periods, and are consequently reconsolidated to different extents.

While (i) can be remedied by adjusting a according to experience, (ii)

suggests that the shaft capacity should be related to the effective rather

than the total shear strength of the soil. Hence despite the popularity

enjoyed, the a-method is only empirical in nature.

On the other hand, effective stress methods attempt to relate the shaft

capacity to the radial effective stress u. 'r and the effective shear strength

12

parameters c', ~' of the soil. However, the radial effective stress u ' isr

difficult to determine, which renders the effective stress methods of less

practical use. In general, ur' is a function of OCR, the vertical effective

stress and the earth pressure coefficient K. At the present stage of

development the effective stress methods still rely on some simplifying

assumptions, and is still far from ready to replace the somewhat empirical

design rules.

The predictive ability of a number of different design methods have

been compared upon a field tension test in silty clay (Pelletier & Doyle,

1982). While the a-method yielded results of 25% over-conservative, the

Esrig & Kirby (1979) effective stress method furnished results of 50% over-

predicted. Burland's (1973) simplified effective stress approach (~-method)

gave the best prediction, being 4% over-predicted. However, the success of

simplified effective stress methods may, as Esrig & Kirby (1979) commented,

be dependent upon a compensation of errors.

All in all, there seems to be still much room for improvement towards

the prediction of shaft capacity of piles at the present stage. The

effective stress approach seems to be one step closer to reality, but its

complexity and assumptions involved prevents it from being industrially

acceptable. In the following work presented, the a-method is adhered to

despite its limitations.

2.5.2 Bearing Capacity

The ultimate bearing capacity of a single pile in"clay Qub is given by

(2.7)

where Ab is the area of the pile base, and

Nc is a bearing capacity factor.

The fact that Qub can be satisfactorily related to the undrained shear

strength of the soil has been qualitatively explained by Burland (1973):-

(i) 'Failure usually takes place through the soil some distance beneath the

base and disturbance during installation of the pile will not greatly

13

affect the major part of the clay involved in the shearing process.'

(ii) 'In the long term the soil beneath the pile tip will normally

experience an increase in effective stress and a consequent increase

in strength. Hence the undrained bearing capacity represents a safe

lower limit.'

Based on the results of model tests reported by Skempton (1951, 1959)

and theoretical analyses and model tests by Meyerhof (1951), Nc is generally

taken as 9. Due to the usually small contribution of bearing capacity to

the overall pile capacity as well as its relatively slow mobilisation rate,

the value of 9 is usually considered satisfactory and taken for granted.

Nevertheless, alternative expressions by Meyerhof (1951) and Vesic (1975)

gives Nc as a function of soil stiffness:

Meyerhof Nc = P Ic + 1u u(2.8)

Vesic : = 1 + ~/2 (2.9)

where Pu is given by Bishop et al (1945) as

Pu = 4/3 Cu (In(Glcu) + 1) (2.10)

Furthermore, Butterfield & Ghosh (1980) reported an Nc value of 11.5

measured in stiff, remoulded London clay. Herein, the finite element analysis

presented in Section 6.6.1 show that Nc is also affected by the adhesion

coefficient a of the pile, and Nc values of up to 12.5 have been obtained.

2.6 Interface Elements

In order for a pile to penetrate, slip must take place at or close to

the pile-soil interface. Using a Mohr-Coulomb model Randolph & Wroth (1982)

pointed out that there are two possible modes of slip, depending on the

effective strength of the soil. It is not proposed to investigate into

these in great detail here, but the fact that the finite element approach

requires no assumption on the failure mechanism makes it a natural approach

to pile-soil analysis.

14

In some earlier finite element analysis, only a narrow column of

reduced strength soil elements was installed at the vertical interface (Esu

& Ottavani, 1975; Ottavani & Marchetti, 1979; and Hobbs, 1979). Although

the load-displacement response of the pile was reported to be satisfactorily

modelled, Chow (1981) showed that the correct value of limiting skin friction

may not be achieved.

As a result, special interface elements were considered to model the

interaction at pile-soil interfaces. A large number of such elements have

been developed, modelling features like slip, separation, contact and

rotation. Some also incorporate dilation, strain-softening and fluid flow.

Most of the proposed interface elements are concisely summarised by Heuze

& Barbour (1982).

In the following work a simple interface element developed by Chow (1981)

is adopted. This is a 6-node isoparametric element, designed to be

compatible with the 8-node elements used to represent the soil and the pile.

This element has a small nominal thickness (say, 1mm.). The effect of aspect

ratio on the performance of this element is assessed in Section 2.6.2. For

simplicity no dilation or strain-softening is assumed at the interface,

thereby allowing the normal and shear components of deformation be uncoupled.

The interface element is used in axisymmetric context here.

Formulation of Chow's (1981) 6-Node Interface Element

(i) Shape Functions Nij

The shape functions of the interface element follow the ordinary

formulation procedures for isoparametric elements. They are shown in Fig.

2.6.

(ii) Property Matrix D

This is formulated with. the object of modelling the following features:-

(a) Slip will occur when the shear stress at the gaussian integration

points of the interface elements exceed the limiting shear strength

specified (in the present case, a cu);

15

(b) The interface element will transmit normal stress perfectly, and

distortion in joint thickness is negligible.

Assuming that no dilatancy or strain-softening occur at the joints, the

normal and shear components of deformation can be uncoupled. The stress-

strain relationship in axisymmetry can thus be expressed as

o o

(2.11)= o o

T o o vsThe formulation is illustrated in Fig. 2.7. It can be noted that the

model will transmit tension in the same way as compression. In most cases

the inclusion of tension transmission at the pile tip will only cause

secondary effects because:

(a)<t-most of the energy is t~smitted during compression in both static

loading and driving; and

(b) for piles in cohesive soils, load transmission across the tip is

usually secondary to shaft adhesion.

The significance of the various parameters in D are described in turn

below.

The normal stiffness Dnn can theoretically be expressed as

(2.12)

Although this quantity may be physically real and measureable for rocks, it

is uncertain what value is representative for the pile-soil interface

considered here. Ghaboussi et al (1973) recommend a large value to minimise

the change in joint thickness due to stresses. In order to assess the

influence of the value of Dnn' a static analysis was performed on the

CDC 7600 computer with a pile-soil model as shown in Fig. 2.8. Reduced

integration was performed except across the thickness of the interface

16

elements. The results as shown in Fig. 2.9 indicate that pile response is

insensitive to the value assigned to Dnn'

In a similar formulation by Ghaboussi et al (1973) D06 is arbitrarily

assigned to zero. Heuze & Barbour (1982) pointed out that a zero value is

justified as the interface element thickness diminishes.

Ghaboussi et al (1973) expressed Dss' the shear stiffness of the joint

as:

Gjoint when T < limiting shear stress at interface

o when T - limiting shear stress at interface

Chow (1981) proposed an empirical relationship for Gjoint such that

(2.13)

where A is a scalar of the order 10-3• A suitable value can be chosen by

backanalysis of static load test data, but the effect of A has been shown to

be slight except near failure.

(iii) Thickness of Interface Elements

Whether an interface element should have a thickness or not is still a

debatable point. For the interface element adopted here, a small thickness

has to be assumed. Pande & Sharma (1979) have demonstrated that an 8-node

isoparametric interface element with relative displacements as degrees of

freedom can tolerate aspect ratios up to 105 without encountering numerical

difficulties on the CDC 7600 machine.

The 6-node interface element proposed is subjected to a similar

assessment here. The pile-soil model as in Fig. 2.8 is again employed, with

varying aspect ratios assigned to the interface elements. The load-

displacement behaviour for the pile is shown in Fig. 2.10. The curve with

aspect ratio 20 is unrealistic since this indicates a joint thickness of

10 cm! It can be seen that the collapse load prediction is satisfactory

with aspect ratios of the order 103 to 105.

17

2.7 Influence of Mesh Boundaries in Static Analysis

The simulation of a continuum using a discrete model generally requires

the existence of a finite domain within well-defined boundaries. In soil-

foundation interaction problems, any bedrock encountered at depth can

represent such a boundary. However, when no such natural boundaries exist,

which is usual in the lateral extent, artificial boundaries must be

incorporated to truncate the model to a size amenable to computation.

The requirements of such artificial boundaries are different for static

and dynamic problems. The latter is described in Section 3.3.3. As for

static analysis, suitable boundaries range from the simple truncated (i.e.

free, rollers of fixed) to static infinite elements. The latter aims at

modelling the stiffness of the infinite domain by modifying the shape

functions of the boundary elements (Chow & Smith, 1981). Nevertheless, with

the possibility of mesh gradation in static analysis, one can obtain

reasonable results, at least for an elastic-perfectly plastic soil, by

simply incorporated the simple truncated boundaries at a remote distance

from the foundation structure. Hoeg et al (1968) and Griffiths (1982) have

shown that the load-displacement relationship is sensitive to the boundary

distance, but the collapse load remains unaffected.

Regarding the performance of different types of simple truncated

boundaries, the closed-ended pile/von-Mises soil model in Fig. 2.11 is used.

Bedrock is assumed to lie at 14.14 m. below ground level. The lateral

boundary is placed arbitrarily at 6 m. from the centreline of the pile, and

is assumed to be rollers or free in turn. Fig. 2.12 to 2.14 shows that the

nature of the boundary, again, influences only the load-displacement

relationship before failure, but not the collapse load or the extent of the

yielded zone at impending collapse. However, on further examining the

displacement patterns at impending collapse (Figs. 2.15 and 2.16), it can be

seen that the effect of lateral constraint is to limit the displacements to

a more localised scale. The pattern corresponding to lateral constraint is

18

unrealistic because displacements do not diminish significantly with distance

from the pile.

In summary, the collapse load of an elastic-perfectly plastic, purely

cohesive soil is insensitive to the distance and nature of any artificially

imposed boundaries. However, if the displacement response rather than the

collapse load is of interest, then the influence of the far field must be

represented by simple truncated boundaries with lateral constraints, or by

the use of static infinite elements.

Cl)Cl)Q)~~ .-4~ co.... '"~ 0 ~

Cl) en...::::: Q)

'" ~0 ~ Q) ~-..c: ~ x: 0.-4~ CO .-4 00Q) U :>'0'1x: Q) CO.-4

en Z-tiltilQ)~~~..-4~ Cl)en Cl)

Q)Q) ~.-4 ~ 00.Cl ~ ~ -CO ..-4 - ..-4 .-4..-4 ~ '" II"l ~ ,....~ en 0 IoC a--CO .c:: a-- ..0 .-4> ~ ~ ..... -~ Q) - .-4 -QJX: CO ,.... ~

bO Q) U \Cl CIl~ e, ~ a-- :>.CIl 0 CO .-4 CIlE-t p.. x: - Z

-0- - ~ 00..0 -e II"l Q) ..0 0'1

U ,.... ,.... .-4 -..-4 N 0'1 a-- N N - N

~ U - - t) t) 00CI)'" ..-4 - - ..-4- ..-4 ~ til a--CIl 0 ) )1I"l ) 0 ..c: -.-4"'::::: Q) ::s ::s Q),.... Q) til ~ -p.~ ..-4 CIl CO ..-4 a-- ..-4 Q)- ..-40 Q) ~ Q) Q) ~- ~.c::"'" ~ ..c:ux: ~ S S ~- ~ p.,.... ~ ~Cl) Q) ~ ~ Q) Q) Sa-- ..-4 ..-4..-4 ..-4 0 0 ..-4 .-4 ..-4 ::s- ~ S> N U U N CO N::t:- o en

til-e0.c::~Q)x: ~ -..-4 II"ltil CIl ~ IoCtil ~ Q) 0'1Q) ~ '" .-4~ en 0 ~--~ .c:: Q)N~ .-4 ~ ..c:1I:J til..-4 CO Q) 000'1 ..-4~ ..-4X: CIl.-4 ~en ~ .-4 _

:>.~ ..-4 .-4 ee~ CO.-4 ~~ 1-4 t!l COCO~til~0U

Cl) ~Cl) Q)Q) - -~ N .-4 N - N~ '" t) ,.... t) ..... 00en 0 ..-4- a-- .... 00 a--

.c:: )0'1 - :. a-- -.... ~ Q)1oC - ..0 Q) - -CIl Q) ..-40'1 ..-4 - -.~ x: ~- ~ ~~ N .c::~ ~- CO CO ~ ,.... ) ~..-4 Q) :>. >'Q)O'I 0 ..-4~ ..-4 .... CO CO..-4 .... ..c: S1-4 N CO Z ZN- U en

STRESS

initial stresscorrect ion atconverqence

in rernent4~

A

STRA1N

FIG. 2.1 Iteration procedure of the Initial Stress M,tehod.

STRESS

STRAIN

FIG. 2.2 Subdivision of strain increment.

eo2 -I VI-VI

~<I{z<I{

0z-~0eIJJu~

Ii IX::>

0 VI....- 0::,41(._'

::;)U0::-U0::e:l:VII.LI2:

c::0....t.....Cl:!,..00Q).....c::....t"'0Q)t);j"'0Q),..-N><N'-"

c::0.r-!.....Cl:!,..00Q,).....c::....t.....t)Cl:!><Q)-~><~'-"

""""",,,,,,,,,,,,,,,,,,,,,~~~~;:..

~~~~~

~~~~

~

~ } U"'II N- oft .. 02

..,! ·o~ 0• • • •III .. • :JU

0N0·0

oo~ooN

- -C'oI r--. .1.0 11"\

II IIt)

Z Z'-" '-"

c:: ....t0 ..c:..... 00Co Cl:!13 NQ,) ,..

..!o:l Q,)en E-4

0 0 0 00 0 0 0"- -0 LI'I ..:r

('W'~S/'N~)SS3~lS lVJll~3A

oo

oLI'Io·oLI'I..:too

o..;to·o

U"'II't1o

eI-ZUJ~UJ Zu< Cl....J ......a. I-en <:-Cl ~

t:)WI-:z......I-U<:Xw0z<:0wu~0W0::

LLCl

ZClc.n......0::-c0-:::E:ClU

to.N.t:)......LL

·ootooo

U"'I-o·ooo·oIIIoo·o

·oo

Thickness· 0.1 cm

~-_\~___,!

p

10cm

E- 6 X 105 N/cm2

Er·O.O1I·0.0a ·6 X102 N/cm2y

M· 1OPN-cm

I· 10cm

(a) Finite element model considered

MMy

2.0

4 X4

1.0 Gauss integration_._._. 2X2_ ••_ ••_ 3X3

0.5------ 4X4----- Beam theory

My, .y are moment and rotation atfirst yield, respectively

2 3 4 5

(b) Calculated response

FIG. 2.4 Effect of integration orderin elastic-plastic analysisof beam section (from Bathe,1982).

3

5(bl

FIG. 2.5 Zero-energy deformation modes in plane elements.(a) The 'hourglass' modes in a linear elementintegrated by a one-point rule. (b) A quadraticelement integrated by a four-point rule.Differential elements at the gauss points rotatebut do not strain.

3 4

2 3 4

S2

1 6 5

I ,

(a) horizontal joint element1 f)

(b) 'vertical' joint element

N1 : .. 0.25 ~ (1- ~)(1-~)

N21: - 0.25 ~ (1- ~)(1+1)

N3: 0.5 (1-~)(1.f)(1"1\)

N4 : 0 .25 ~ (1+J)(l. 'l)

Ns: 0.25 J (1+~)(1--t)N6= O.5(1-~)(l·~)(1"l)

Nl = - 0.'25 'l(1- ~)(1-1l) ,

N2= O.S(1-~)(l-1\)(1+1)

N3= O.251\(l-J)(1+,\)

N4: 0.25 .,'Jl + ~)(1. ,,)Ns= 0.6(1·~)(1- tt)(l·,\)

~= - 0.25,(1·~)(1-')

where ~ and ~ are local coordinates.

FIG. 2.6 The 6-Node Interface Element and Shape Functions.

//

/~

//

//

//

//

presentformulation

Dnn = 0 for tension (reality)

1

FIG.2.7

!:lo'T'1CI.lCI.lCl)~Cl.eou

T

(a) normal stiffness

= Gjoint

1

(b) shear stiffness

Normal and Shear Characteristics of aNon-dilatant joint.

I:r=': pile~.r- pile

radius 0.5 mI .II

...-v/~/

~tV/v~~v

.iV'-'f'VlI:;Vi/~ ~~//1VI l.tVI

"'VVv

~1YvVVII/V '"''fV!,I

~t IvVVvv l..r

?'v I~V

VV~~~ V

!,I!,II/!,IVr\ ~"

,~ ~~ t-In erface El~ments1..10

" fJ~1.11

~~a:'-"

't If"Pile properties:-

2E = 4E7 kN/m~~ v • 0.25

l.¥~ 1-1'"

Soil Properties:-2E - SE4 kN/m

v .. 0.45 2eu - 100 kN/m«>-1/1-0ex • 0.6

~~

7lM" "

;,H- T. ",. -;. '*

so-

FIG. 2.8 Finite Element Mesh for assessment of InterfaceElement Parameters (Section 2.6).

-0....aa

I"p)(NI

"'0-"ccl N .

a s (J')

· l- f-a z Z

UJ ui::J: LWu ur< ....J..J LU5;

ui~C u

<:LLa.:u.Jf-Z-

CD LL0 0a·a (J')(J')LUZLLLL-f-(J')

ui:::c.....LL

~ 0Z ..:t W..:s:N 8 u0'1 • ZLJ") 0 uiN ::J- ...J

LLC Z0 -.....2s 0-."8

N..;; t:I

2 -LL<{ 0

00.

a 0 a a 0a 0 a a aLon 0 Lon a aN N Lon 0

(·N~)OVOllVJll~3A

0NII

0::: 0('oJ

-c 0·0.

~0 UJ0 11\0 Z0 ..... ....IN )C Cl

" N =>%:a:: "« 0::' -0 .....

<C < (J")0· ..... t-O Z Z

UJ UJ%: :EUJu UJ< _J....I UJa.(/")

UJ.-Cl u

<:LL0::

N UJ0 l-

· Z0 -

LLCl

1"0(J")(J")...- UJ.. ZN ~

II U-CD :::c0 I-0·0 UJ:::cl-

LLCl~

Z W.:::t:. UN Z0'\ WloO :::>N ...:to _J

0 LL0c · Z0 0 -:;:::;;:,-0 0III

-a •.~ C\I.....- .2 c...:J-« 0 u..

00·0 0 0 0 0

0 0 0 0 0Lf'I 0 Lf'I 0 0N ('oJ Lf'I 0

('N~)aVO'lVJll~3A

¢.II

I~

E =pilel: ~Hammer & v =pileAccessories Outer

Wall T

~ ~ Penetra~~ Overal~~

I

I r---Pile WallI

I 5 7115 mI

II

II

II

I + +I

I + +I

II

II

II

II

I§~"-....

~

.~-'I'"

~

...

207E6 kN/m20.3

Diam.= 0.457m.

2Esoil= 6E4 kN/mv i1= 0.48so 2C = 117 kN/mu

hickness= 19mm.tion= 9.14m.1 pile length= 12.6m.

LateralBoundary:free orHr for static

analysis,W for driving

analysis.

Base Boundary: ~ for static analysis,~ for driving analysis.

FIG. 2.11 FINITE ELEMENT MESH FOR RIGDEN'S(1979) CLOSED-ENDED PILE

0.0r::~ ....til au ::l

.... Cl)~ Cl)Q) til

'"'O~Q) til.t: 0E-4~

Cl)

'"'Q)~~~

0000·0-,.....a

0-0·~0 r::

Q)aQ)u

Iol:) til0 ~0 Q.· Cl)0 ....

Q

Q.....E-4

U"'I0 Q)0 .-4· ....0 I)..

..;t00·0M00·0

0'\ooo

Cl)r::o....~....~r::ou>.'"'til~r::::lo~

Noo· N'-.N

o

-oo·,o

--'_--~----~ ~ ~~ -L ~ ~oo 0 0 0 0 0o '" 0 U"'I 0 U"'I'" NO""" U"'I N.-4 _ _

(N~) PU01 paltddV

~ yielded zones

FIG 2.13 YIELDED ZONE AT IMPENDING COLLAPSE.ROLLERS LATERAL CONSTRAINT

~Yielded zones

II

FIG 2.14 YIELDED ZONE AT IMPENDING COLLAPSE I

NO LATERAL CONSTRAINT

I I.:

:1 I I1

! I I I

Iil I I

III I I

III I I

III I I

I

~~

\

\ \

.~\1111 \ \. ,"~'I .

FIG. 2.15 Displacement Plot at impending collapse:Rollers Lateral Constraint. Displacementsmagnified by 200 times.

I I

I I II I

I I \

I \

I I \I

I I

\1

\\ \ \

I \ \

!I \ \ \ \

\ \

1\ \ \\ ~

\

\ \ \ \

I~ \ \

1\ \ \ \ \

• I ,I' " I \

, , ,

. ,

FIG. 2.16 Displacement Plot at impending collapse:No Lateral Constraint. Displacementsmagnified by 200 times.

19

CHAPTER 3

FINITE ELEMENT SOLUTION TO THE EQUATION OF MOTION

3.1 General Solution Procedure

The response of a system subjected to dynamic loading, whether in the

form of impact or periodic excitation, is governed by the general equation

of motion:

M x + C x + K x = ret) (3.1)

When the system is spatially discretised for finite element modelling, ~, C

and K represent the mass, damping and stiffness of the numerical model

.respectively. This system of second order, linear differential equations

can then be solved by any of three methods:-

(i) integrate the system of equations directly step-by-step in the time

domain, which involves obtaining equilibrium between the inertial,

damping and elastic forces at regular time intervals;

(ii) solution by modal superposition, also in the time domain, in

which the dynamic response is determined from a limited number of

modes which are considered to contribute significantly to the

response; and

(iii) transform the equation of motion into the frequency domain, and

solve the system of complex simultaneous equations for steady

state response.

The assessment and implementation of these methods are widely documented,

notably Smith (1982) and Bathe (1982). Since the last two methods cannot cope

with truly nonlinear analysis, which is essential for pile driving problems,

they are not further considered here. As for direct integration methods, a

number of these are available, and the choice of a suitable scheme depends on

the type of problem concerned as well as the capacity of the computer

available (Chow, 1981).

20

3.2 Dynamic Response Analysis as a Wave Propagation ProblemDue to the lack of perfect rigidity in materials, the application of a

dynamic load will cause elastic stress waves to propagate from the source ofdisturbance. Thus a dynamic foundation-soil interaction problem can bevi~alised as a wave propagation analysis. If a periodic load is applied asin the case of machine foundations, the elastic waves will.be forced tovibrate at the frequency of excitation. On the other hand, if an impacttype of loading is applied, as in the case of earthquakes, a large amount ofvibration modes will be excited, which can be 'convolved' if desired bybreaking down into a Fourier spectrum.

Richart et al (1970) illustrated mathematically that for an elastic,semi-infinite, homogeneous and isotropic medium - often termed as an elastic'half-space', there exist three types of stress waves (Fig. 3.la):

(i) a P- (compression) wave, the particle motion associated with whichis a push-pull one parallel to the direction of the wavefront;

(ii) an 5- (shear)vvave, the particles associated with which displacetransversely and normal to the direction of the wavefront; and

(iii) an R- (Rayleigh) wave, the particle motion associated with whichcan be split into an horizontal and a vertical component,the magnitudes of which vary with depth.

The three types of stress waves all propagate at different velocities,and are independent of frequency (Fig. 3.1b). The Rayleigh wave is onlysignificant near the free surface, and obeys a different geometric dampinglaw from that of the P- and S- waves (Ewing et aI, 1967). Miller & Pursey(1955) analysed the classical problem of vertical periodic excitation ofa circular surface footing, and found that the Rayleigh wave carries two-thirds of the total input energy. Moreover, it is known to decay much moreslowly with distance. Despite its importance, there are no provisions

available to model geometric damping of the Rayleigh wave in time domain

finite element analysis. Fortunately, Lysmer & Kuhlemeyer (1969) showed

21

that with the incorporation of the standard viscous boundary, errors due to

Rayleigh waves can be kept to a small magnitude.

It should be noted that the propagation of these elastic stress waves

as described is, strictly speaking, applicable to an elastic transmitting

medium only. Should plasticity occur , the propagation velocities expressed

as a simple function of the elastic modulus is probably not justified.

Studies (Kondner, 1962; Nicholas, 1982) have shown that the velocities of

'plastic stress waves' are functions of the constitutive relationship (in

terms of strain level and strain rate, Fig. 3.2). Seed & Idriss (1970) have

also published data on the reduction in shear modulus in terms of effective

shear strain amplitudes for typical sands and clays. In general the

mathematical formulation of plastic stress waves are very involved, and in

the present work it is assumed that the stress waves are elastic even when

nonlinearity is prevalent.

3.3 Considerations for Dynamic Analysis

Although there are many similarities between static and dynamic response

analyses, the latter is more general in nature, so that a finite element

model amenable to static analysis may not be justified for dynamic response

analysis. The considerations required in formulating a dynamic finite

element model include

(i) spatial discretisation:

(a) )../~xratio;

(b) types of elements used;

(c) mesh gradation;

(ii) mass formulation;

(iii) temporal operator and discretisation; and

(iv) transmitting boundaries.

These are discussed individually in detail below.

22

3.3.1 Spatial Discretisation

(a) *AIAx ratio

When a continuum is simulated by a discretised model, finite elements

or finite difference alike, it is obvious that the accuracy in modelling

wave propagation depends on the number of elements used to represent each

cycle of the wave. Barring the influence of mass idealisations and

temporal operators, each cycle cannot be covered by less than 2 elements

(Fig. 3.3), or else no propagation can occur. On the other hand, if AIAX

is greater than 2, waves can propagate but only at a distorted velocity,

known as the 'phase velocity', which is a function of the wavelength. Such

phenomenon is termed dispersion.

Two adverse effects of dispersion are apparent. Firstly, if a wave

pulse made up from a number of Fourier components (typical of impact or

seismic loading) is propagated across a finite element grid, the frequency

components will all be distorted to a different degree, and consequently out

of phase with each other. Exact wave transmission occurs theoretically only

for a wave with zero frequency. Secondly, for any discretised grid a 'cutoff

frequency' exists such that any wave components with frequencies higher than

this will be doomed to rapid attenuation. The cutoff frequency fco can be

expressed as

f = cIA • c/(n Ax)cowhere c is the velocity of wave propagation,

(3.2)

and n is a number whose value is theoretically 2 as discussed in the last

paragraph, but in practice is affected by factors like mass idealisations as

mentioned (see Section 3.3.2). The attenuated high frequency waves are

undesirable because their energy will remain within the discretised grid

causing spurious node-to-node oscillations. These can be prevented or

minimised by (i) the incorporation of internal soil damping or artificial

viscosities; (ii) the use of a time integration scheme with inherent

* AX is defined herein as the element dimension in the direction of wavepropagation.

23

numerical damping; or (iii) postprocessing the solution by digital filters

(Holmes & Belytschko, 1975). The last remedial procedure, however, is

undesirable because more dispersion will be manifested.

The dispersive characteristics of various dynamic problems have been

investigated (Table 3.1). Excessive dispersion always occurs when the

cutoff frequency is approached, resulting in solution errors of sometimes

over 100% (Kuhlemeyer & Lysmer, 1973). As a result, a number of

recommendations for a limiting A/~x ratio have been put forward, as

summarised in Table 3.2.

The performance of 8-node element meshes with different A/~X ratios

will be investigated in Section 4.2.

(b) Types of elements used

The degree of dispersion is sensitive to the manner of spatial

discretisation, and hence on the types of elements employed. Ba!ant &Celep (1982) compared the performance of one-dimensional models consisting

of 2-node and 3-node line elements respectively (Fig. 3.4). It is apparent

that high order elements are less dispersive than low order ones. Moreover,

high order elements tend to cause less spurious wave reflection as a result

of mesh gradation (see below).

(c) Mesh Gradation

One of the advantages of the finite element method is the possibility

of introducing mesh refinements locally around the zones of interest in the

hope of obtaining a more accurate solution. However, for wave propagation

problems variations in element sizes can cause spurious wave reflections even

in a homogeneous medium. So far mathematical analysis in this respect has

been performed for one-dimensional problems due to the complexities involved.

Ba~ant (1978) studied the case of wave propagation across a grid of

2-node line elements with a single size variation (Fig. 3.5). The amplitude

of the input wave is assumed to be unity. The explicit central difference

24

scheme is employed. It was found that:

(i) spurious wave reflection due to mesh gradation is significant for

small AI R values. From Figs. 3.6(a) and (b), it can be seen that

in order to limit the amplitude and energy flux of the reflected

wave to say 10% of the incoming wave, AIR values of 3.5 and 5 are

suitable for consistent and lumped mass formulations respectively;

(ii) the consistent mass formulation is superior in performance to the

lumped mass formulation as far as minimising spurious wave

reflection is concerned (Fig. 3.6); and

(iii) the size of the time step has no apparent influence on spurious

wave reflection, probably because it has been kept small for

explicit integration in time.

It is apparent, therefore, that by keeping AIR greater than, say 4 for

consistent mass formulation, one can expect only secondary influences from

dispersion and mesh gradation. The use of high order elements is beneficial

(Fig. 3.7), as reported by Bazant & Celep (1982). The introduction of

gradual variation in element size through a transition zone can also help to

alleviate the problem of spurious wave reflections to some extent (Ce1ep &Bazant, 1983) (Fig. 3.8).

3.3.2 Mass Formulation

In analysing one-dimensional wave propagation through a discretised

grid, Belytschko & Mullen (1978) have shown that the cutoff frequency of the

grid is dependent upon the mass formulation employed. For example, for the

simple case of 2-node linear elements with no temporal discretisation errors

assumed (i.e. ~t -. 0), the cutoff frequency fco can be expressed as

3 c2 m) ~x

(3.3)

where m is the degree of lumping, such that m • 1 corresponds to consistent

mass formulation (Archer,

formulation. Thus1963), and m • 0 corresponds to lumped massJOHN RYLAND~

UNIVERSITYUBR;\n'l' OFMANi}H£STliR

25

= f([3 /1f) c/dx for consistent mass, and

( 1 l n } c/dx for lumped mass formulation(3.4)

The consistent mass formulation is thus superior to lumped mass formulation

as far as propagation characteristics is concerned. Similar conclusion

applies to the case of one-dimensional 3-node elements, the solution of

which is complicated and not repeated here.

As for those waves propagating at frequencies less than the cutoff value,

dispersion will occur but the characteristics of which is also dependent on

the mass idealisation adopted. Kreig & Key (1972) have found that consistent

mass tends to overestimate the frequency, while lumped mass does the opposite.

This has been reinforced by the studies of Ba~ant (1978) and Ba~ant & Celep

(1982) •

Fig. 3.4 reproduced from Bazant & Celep (1982), shows the dispersive

characteristics of different mass formulations for one-dimensional linear

(2-node) and quadratic (3-node) elements. Again no temporal discretisation

errors have been introduced in the analysis. The opposite dispersion trends

between consistent (m - 1.00) and lumped (m • 0) masses are clearly

exhibited. The figure also illustrates the superiority of consistent mass

over lumped mass when A/dx is small. However, the cutoff effects are not

shown in the figure, and the left most portion of the curves with A/dx

values corresponding to frequencies higher than the cutoff value are, in the

author's opinion, suspicious.

3.3.3 Temporal Operators and Associated Considerations

In order to perform truly nonlinear analysis and predict permanant

deformations, it is essential to perform finite element computations in the

time domain. The characteristics of both explicit and implicit time

integration algorithms are well established for linear problems, in terms

of stability and accuracy. Performance of various schemes have been assessed

experimentally by Gray & Lynch (1977) for the long-wave surface water

26

equation, and by Brook-Hart (1982) for the equation of motion. Discussions

on the choice between explicit and implicit schemes have been presented by

Key (1978), Nelson (1978) and Smith (1982), and are not reiterated here.

However, dynamic analysis taking nonlinearity into account seems to be

less established. Confidence in the performance of temporal operators tends

to be based on the success of linear analyses. While this may be feasible

for slight nonlinearities (Wilson et aI, 1975), a strongly nonlinear system

may require the use of a smaller time step size to ensure stability and

accuracy. This also applies to the so-called 'unconditionally stable'

algorithms. Weeks (1972) pointed out that the overall characteristics of

time integration schemes on a materially nonlinear system with no algorithmic 1

damping appears to be analogous to a materially nonlinear but algorithmically

damped system.

Stability Analysis

The stability analysis of temporal operators for linear systems has been

developed along two parallel trends:

(i) Lax & Richtmyer (1956) defined the amplification matrix of a

scheme and its spectral radius. The integration method is stable

for a given time step size if the corresponding spectral radius is

not greater than 1. The stability characteristics of various time

integration schemes have been analysed using this concept by

Hilber et al (1977), Hilber & Hughes (1978) and Bathe (1982).

(ii) Dahlquist (1963) introduced the concept of A-stability. An

integration scheme is said to be A-stable if the numerical error

due to integration remains uniformly bounded for any time step

size.

Gear (1969) suggested the use of 'stiffly stable' methods, which have

regions of stability up to the 'cutoff frequency' of the model.

27

Stability in Nonlinear Analysis

In dynamic analysis of materially or geometrically nonlinear systems,

the so-called 'unconditionally stable' temporal operators may become only

conditionally stable (Stricklin et aI, 1971; McNamara, 1974). As the time

step size is gradually increased, the quality of the solution gradually

decreases with no noticeable warning, until the solution eventually becomes

unstable.

Park (1975) concluded that the maximum time step size for stability in

nonlinear analysis depends on the interaction of the integration scheme with

the method used to handle nonlinearity. McNamara (1974) pointed out that for

implicit methods in general convergence will be achieved only when the time

step size approaches the stability limit of the explicit central difference

algorithm. This is especially important for problems of cyclic or reversed

loading when errors can accumulate rapidly. Furthermore, the example problem

in Fig. 3.9 shows that the Newmark (~. 1/4) scheme is inferior in stability

to the Wilson (0- 1.5) and Houbolt methods, despite the fact that a highly

damped time integration scheme does not necessarily possess stability

characteristics superior to a less damped one (Park, 1975).

Accuracy Analysis

The performance of a temporal operator will depend on:

(i) the use of a small time step size dt, since temporal discretisation

is dispersive and tends to dIstort the period (Fig. 3.10). For

periodic excitation problems in which the propagating frequencies

of the stress waves are well-defined, dt can be simply selected

as a small fraction of the period T. As for impact or seismic

problems, usually a range of frequencies are agitated. For

linear problems, the Fourier components of the stress waves can

be identified by frequency domain analysis, whereas in the time

domain dt is chosen as a function of dX/C, where dX is selected

based on the rise time of the input load (Hada1a & Taylor, 1972),

28

or simply by experience (Chow, 1981). In the case of nonlinearproblems, the time step size for implcit time integration schemesmust be reduced to around the order of the stability limit forexplicit algorithms. Some recommended time step sizes for accuracyare summarised in Table 3.3.

(ii) the suitable combination of integration scheme, mass formulationand Courant number C (- c ~t/~x). Be1ytschlo & Mullen (1978)have examined the dispersive characteristics of the centraldifference (Newmark ~ - 0) and the trapezoidal (Newmark ~ = 1/4)methods with different mass formulations in solving the equationof motion (Fig. 3.11). Kreig & Key (1972) have previously proventhat the central difference integrator tends to underestimate theperiod, whereas the trapezoidal integrator does the opposite. Asa result, by using the appropriate mass formulation and Courantnumber (a function of both spatial and temporal discretisation)the dispersive effects can be minimised, as illustrated in Fig.3.11.

3.3.4 Effect of Transmitting BoundariesThe development of transmitting boundaries for dynamic wave propagation

analysis follows a different concept from those established for staticproblems. While static boundaries aim to include the influence of stiffnessoffered by the far field, dynamic boundaries attempt to model the propagationof stress waves to infinity by preventing them from reflecting into thesystem. In this respect the simple truncated boundaries and the staticinfinite elements mentioned in Section 2.7 fail to serve the purpose. Somedynamic energy-absorbing boundaries formulated are frequency-dependent, andare only suitable for analysis in the frequency domain. Analysis performed

in the time domain requires the use of frequency-independent transmitting

b~undaries. The commonly available boundaries for time domain analysis are

29

reviewed below.(a) Simple Truncated Boundaries (Lysmer et aI, 1974; Shaw et aI, 1978)

These elementary boundaries do not prevent wave reflections at all, andstray influence on the solution is possible when the reflected waves reachthe region of interest. The assignment of artificial viscosity or internalsoil damping may alleviate this problem (Roesset & Ettouney, 1977), but theamount of damping to be prescribed tends to be subjective and difficult tojustify. Nevertheless, in order to maintain dynamic equlibrium at discretetime intervals, some fixity (i.e. roller or fixed boundary) must beprovided for problems involving non-vanishing input loads. This oftenimplies the use of an elaborate finite element mesh, and dynamic responsecan only be obtained for the first few oscillations. This is illustrated inthe analyses in Chapters_4 and 5.(b) SuperpOSition Boundary (Smith, 1974; Cundall, 1978)

The original superposition boundary, as proposed by Smith (1974), isonly applicable to linear problems. Cundall et al (1978) modified the modelto take into account any nonlinearities existing in the confined mainregion. Further description is available in Chow (1981).(c) Viscous Boundaries

This is a category of boundary conditions such that the energy of anyimpinging waves are damped out by the artificial damping offered by thetransmitting boundary. While a large number of these are frequency-dependent, four are known to be frequency-independent and are suitable fortime domain analysis. These are described in turn below.Standard Viscous Boundary (Lysmer & Kuhlemeyer, 1969)

The frequency-independent version of the standard viscous boundary isonly formulated to absorb incident P- and S- waves. The Rayleigh wave, whichmay carry two-thirds of the total energy as in the case of verticallyoscillating footing upon an elastic half-space, is not catered for. In

general, the boundary is over 95% efficient in absorbing P- and 5- waves

30

o(White et aI, 1977), except when the incident angle is greater than 60. As

a result of such imperfection, the distance of the boundary from the

excitation source will influence the accuracy of solutions. Roesset &Ettouney (1977) concluded from their numerical experiments on strip footings

that a distance of 5 to 10 times the half-width of the footing is suitable,

depending on the amount of internal damping.

Furthermore, the Lysmer & Kuhlemeyer (1969) formulation of standard

viscous boundary is strictly applicable in plane strain conditions only.

Nevertheless White et al (1977) showed that the formulation can be used to

approximate axisymmetric conditions when the boundary is located further

than lAs from the source of excitation.

The implementation of the standard viscous boundary has been described

by Chow (1981). The resulting damping matrix £ can also be lumped or

consistent as the mass matrix M. For the problems considered in the

following chapters, a lumped damping matrix is always employed for reason of

efficiency.

Unified Boundary (White et aI, 1977)

Developed along the same trend as the standard viscous boundary, the

unified boundary further includes optimisation of its parameters according

to the angle of incidence of the impinging wave and the Poisson ratio of

the medium. Despite the formulation being more involved, the performance is

only marginally superior to that of the standard viscous boundary. The

absorption of Rayleigh waves which constitute as a major energy carrier is

again ignored.

Compatible Viscous Boundary (Akiyoshi, 1978)

This is developed by Akiyoshi (1978) as a perfect absorber for shear

waves in a one-dimensional lumped mass model. The formulation is also

potentially suitable for the perfect absorption of P-waves. However, the

method is at present limited to the analysis of one-dimensional lumped mass

models, and further development is required before it can be extended to

31

more general applications.

Generalised Viscous Boundary (Castellini et aI, 1982)

In the formulation of the generalised viscous boundary, the direction

of propagation of both incident and reflected waves must be known. However,

the direction of the latter is generally unknown and difficult to determine,

thus undermining the effectiveness of the proposed boundary model. Moreover,

the proposed formulation is, strictly speaking, only valid for plane strain

conditions. The performance as compared with the other more popular

boundaries has yet to be assessed.

3.3.5 Summary

From the above discussions it can be seen that it is difficult to

establish a general set of design criteria in formulating finite element

models for dynamic analysis, in order to limit the numerical errors to a

small magnitude. The presence of any nonlinearities will further complicate

matters. The choice of mesh parameters in an ad-hoc fashion, as often

practised in static analysis, can result in gross inaccuracies.

Furthermore, the stringency of mesh design depends also on the types of

problem concerned. Roesset & Ettouney (1977) pointed out that impact loading

and seismic excitation problems can tolerate cruder meshes than forced

periodic oscillation problems, while eigenvalue analysis requires an even

mo~refined mesh to guarantee accuracy.

In general, dynamic response analysis in the time domain requires a

mesh which is much more elaborate then necessary for static problems. For

the solution of two-dimensional problems a mainframe computer is essential.

One is then faced with the choice of adopting lumped mass/explicit time

integration formulation, which requires less storage but more time steps, or

the more stable implicit temporal operators, which tax computer storage as

well as effort in manipulating a large number of simultaneous equations.

In the analysis presented in the following chapters, the a-node

32

rectangle is employed along with an implicit time integration scheme. Thisshould result in superior stability and accuracy characteristics to the one-dimensional lumped mass model on which much of the error analysis descibedearlier are based. The general solution algorithm is developed based on theWilson (0 =1.4) scheme, and the initial stress method is employed to handlematerial nonlinearity.

3.4 Solution Algorithm: Wilson (8 - 1.4) Scheme with Initial Stress methodAn anomaly of the Wilson (8 - 1.4) scheme is that dynamic equilibrium is

only satisfied at the collocation point (t + O~t) (which is beyond the timestepping range in question) and not at the time station (t + ~t). Solutionalgorithms employing the scheme in the solution of nonlinear dynamic response

*problems are well-documented, notably Wilson et al (1973) , MaNamara (1974),Chow (1981), Bathe (1982) and Smith (1982). In the following works presented,the algorithm proposed by Chow (1981) is based upon:-A. Initial Calculations.

1. Assemle the elastic stiffness matrix K, the mass matrix M and the- -damping matrix f. (Since K and M are symmetrical and banded, onlyhalf of the bandwidth is stored; C is lumped in the mannerdescribed by Hinton et aI, 1976).

2. Calculate the following constants:

cl - o ~t; c2 - 6 / cl; c3 • c2 / cl;

c4 - ~t2 / 6; c5 - cl / 2; c6 - c2 / 2.

3. Form the effective stiffness matrix:K - K + c3 ! + c6 C

4. '"Triangulise !.5. Initlialise xo' . and Fo.xo' Xo

(3.5)

(3.6)

* It i s believed that in the table on pg. 246, '0 ~ 1.37' should read as'0 ;t 1.37'.

33

B. Time Stepping Recursion For each time step, iterate to achieve

dynamic equilibrium.

1. Calculate the effective load vector at time (t + 8~t):

+ .£ (c6 !o + 2 ~ + Cs &)VISLDSi

+ TOTBODi + BDYLDSi

(3.7)

where TOTBOD is as in equation 3.13,

BDYLDS is as in equation 3.11 ,

VISLDS is only applicable for piling problems, and as

in equation 3.12,

superscript i denotes the 'i'th iteration of the current

time step.

2. Solve for displacements at time (t + 8~t):...!..!(t + 8~t)

A

= !(t + 8~t) (3.8)

3. Check convergence criterion

i.!(t + 8.1t)

i- .!(t + lMt)

< TOL (3.9)

where TOL is a specified tolerance value.

4. Compute the present state of stresses from the constitutive

relationship.

5. Test if yield has occurred at any ~aussian integration point.

6. If yield has occurred, compute the plastic matrix DP based on the

estimat£d stress state on the yield surface. The out-of-balance

stresses can be computed from

• (3.10)

and evaluate the bodyload vector

BDYLDS • f BT d(uP) devol) (3.11)

34

7. For pile driving problems, compute damping forces at the pile-soil

interface:VISLDS = J * (internal forces at interface) *

(3.12)

8. If convergence has not been achieved, repeat steps 1 - 7.9. For the converged solution

(a) Update stresses and strains(b) Update TOTBOD:

TOTBOD(t + 8.:1t) = TOTBOD(t - (1- 8).:1t) + BDYLDS(t + 8.:1t)

(3.13)

(c) Evaluate accelerations at time (t + 8.:1t):

X(t + 8.:1t) ..(3.14)

(d) Evaluate accelerations, velocities and displacements at time

(t + .:1t):.. .. + (x(t + 8.:1t)- Xt) /8 (3.15)~(t + .:1t) .. xt. + (xt + x(t + .:1t».:1t/2 (3.16).!.( t + .:1t) .. xt

~(t + .:1t) - xt + t xt + c4 (2xt + x(t + .:1t» (3.17)

10. Repeat steps 1 - 9 for new time step.

In the above algorithm, displacements and the time derivatives aremarched along at t , t + .:1t,t + 2.:1t,•••• • On the other hand, stresses andstrains are marched along at t + 8.:1t,t + (1+ 8).:1t,t + (2 + 8Mt, •••• • Thealgorithm has been employed successfully by Chow (1981), and will be assessedagain in Chapter 4. However, McNamara (1974) pointed out that afterconvergence, the initial stress redistribution process (steps B1 - 8) should,strictly speaking, be performed again at 'time t+.:1t. Although this may help

in cases of strong nonlinearity or large time step sizes, the extra

complications and storage requirements introduced are not desirable, and arenot implemented in the programs used herein.

=(I) (I) ~ 0a a 0 .~<1l <1l ... ...~ ~ (I) ~,0 ,0 Q ~ ;::I 000 0 I 00 (I) C" = (I)... ... ~ = :>. <1l ~ ~P- P- ~ ~ -e (I)

= ~ ~ ... ~ :>.Q Q 0 ~ = ... 0 ~I I 0 ~ 0 ~ ~N N <1l ~ P- (I) =(I) = (I) P- = ~~ ~ ~ ~ 0 = <1l 0

= = ;::I ~ 'I"'l ~ ... ~ ~~ ~ P- ~ ... ... (I) ... ~

~ ;::I ... ~ (J

a I I ... 0 ~ Q ;::I .~CII ~ ..... ~ (I) 0 (I) I C" ........ ~ ;::I > (I) N <1l <1l,0 (I) (I) ;::I = = ~ ~0 ;::I ;::I 00 .~ 0 a -e ...... 0 0 = (I) (J = <1l =p.. .~ .~ ~ <1l ~ ...... ... ~ Q Q > ~ Q.. ~ ~ ... I I ~ I ~ I

> > E-4 ..... N (I) ..... .....= ... <1l0 .. .. .. .. .. <1l .. (J ..'I"'l P- ~.... = = = = = (I) = ~ =~ 0 0 0 0 0 .~ 0 ... 0;::I 'I"'l .~ ~ ~ ~ ~ .~ ;::I .~C" ... ... ... ... ... ... (I) ...~ 0 0 ~

0 0 <1l~

0a a a a > <1l a

..... ~ >~ ~ ~ ~ ~ ~ ... 1.1-1 ~ 1.1-1~ 0 0 0 0 0 CJ 0 ~

0... <1l

= = = = = = > = 00 =<1l 0 0 0 0 0 = 0 = 0... .~ ~ ~ .~ ~ 0 ~ 0 ~<1l ... ... ... ... ... t,,) ... ~ ...

1.1-1 ~ ell ~ ~ ell ~ ~1.1-1 ;::I ;::I ;::I ;::I ;::I Q ;::I Q ;::I.~ C" C" C" C" C" I C" I C"Q ~ ~ ~ ~ ~ ..... ~ ..... ~

- - -C""I \J:) 00,.... ,.... ,....C7\ - C7\ C7\- ..... 11"1 ..... .....

N - ,.... - -- ,.... C7\ -... ,.... 0\ ... ..... \J:) 0 - =0 \J:) ..... <1l - ,.... ..lIII ,.... <1loJ:: - C7\ - a C7\ oJ:: ,.... ~... ,.... ..... (I) ..... ..... CJ C7\ .....~

\J:) - ... S ~ - (I) ~ fC7\ 0 ... -~ ~ ~ ... ... :>.- ~ :>. 04 <1l <1l ~ oJ:: 04~ '\:I. <1l (J

0 ... E-4 ... 0 = IlCI = 0

= <1l <1l = ~ :>. ..l1li~ 04 :>. .~ p.. 04 ~ oJ::... :>. i ... (J

= <1l ell = 04 (I) 04 (I)

~ ~ ~ ~ <1l ....... Q. ~ ..... ... :>. a :>. :>.(I) ~ ~ oJ:: (I) ~ ~ ~ .....0 oJ:: ~ ;::I 0 ... 0 ... <1lt,,) Cl) := ~ u ~ := ~ IlCI

·

-Cl)Cl) Si+-I aIc:: ~(1) ,Q

Si .. 0(1) c:: ,..~ 0 Q-(1) .....

+-I 0,.. 111 I111 ~ N(1) §c: ,..•.-4 ~ ,.. 0~ 111 0 ~,.. c:: ~ -,.. aI 00 c: ..... Cl) c::

1.1-1 aI +-I Cl) 0.. 00 111 111 .....'Q § ~ lEI ...(1) c:: ;:l 111+-I ..... ..... e +-I ~Cl) 'Q c:: §aI (1) Cl) 0 (1)

OIl lEI Si ~ +-I ,..OIl aI Cl) 0 Cl)

;:l Cl) ~ Cl) ..... ~ ~Cl) ;:l .0 Cl) Cl)0 0 111 c:: Cl) ~

0 (1) ,.. Si 0 Cl) ,Q

..... c:: Q- 0 ~0

+-I aI "'0 ,..111 00 0 aI ,.. Q-,..

~I Q- 0 'QN ~

aI ~~ 0 "'0 OIl 111

~ .c:: ,.. ~ (1) 111 ,..0 Q- ,.. aI

.-.:: ,::::) 0 ~ aI c::I I I > aI

Si ...-4 .... .... ~ 111 00

§ ,.. ,.. ,.. ,.. ,.. ,....... 0 0 0 0 0 0c:: 1.1-1 1.1-1 ~ 1.1-1 ~ ~.....~ ...;t 00 \C 00 "" ...;t

-M.....C"I...-4-,.. -~ '".....Cl) '":>. - ...-4,.. ~ CO -0 ......c:: ~ 0'1 ~ -+-I ~ 111 N

~,.. - 00(1) +-I '":>. ..... (1) ....(1) .c:: -~

Cl) Cl)0 Cl) aI

...-4 :>. ..... .c::.c:: ..... ,.. ...;:l

~'Q 111

i:oIi: 1-4 ~

Time Integra-tion Scheme Author ~t recommended

Suitablefor non-linear~nalysis?

Yes

Implicit Schemes Bathe (1982)

All

CentralDifference

Euler Scheme

Euler-Lagrangian

Houbolt

ModalSuperposition

Newmark(f3 .. 0.25)

Park

Mcnamara (1974) T/200

Wu & Witmer(l973)

Shantaram et al(1976)

Y'tix/c, where Yes0.45 c f' ~ 0.5 (parabolic el.)0.9 <. f' .. l.0 (linear elem.)

J273 ~x/c (consistent mass), Notix/c (lumped mass)

T/80 or ~x/c

0.99 TIn0.80 TIn

0.75 ~(1 + v)F I(Ef), whereof3 = constant,f(t) = fluidity coefficient,F = a ref. value of field fn.o

0.1 tix/c iii tit 111£ 0.5 tix/cdepending on rise time of load

5 TIn

0.012 T

0.05 tix/c

2 T/'ff

l.2 T/Tr

0.016 T

TABLE 3.3 Recommended Time Step Size for Time Domain Analysis.

Key (1980)

Cormeau (1975)

Hartzman (1974)

Wu & Witmer(1973)

Park (1975)

Farrel & Dai(1971 )

Wu & Witmer(1973)

Key (1980)

Park (1975)

Yes

NoYes

Yes

Yes

Yes

Yes

No

Yes

Yes

Yes

Circular Footing

t

Wave Type Per Cenl ofTolol Energy

Rayleigh 67Shear 26Compression 7

FIG. 3.la Distribution of displacement wavesfrom a circular footing on ahomogeneous, isotropic, elastichalf-space (from Woods, 1968).

~C) 4

>

3>1,'"-0'" 2II;)

'0>

o

FIG. 3.lb Relation betweenPoisson ratio, ,and velocitiesof propogation of compression(P),shear(S), and Rayleigh(R) wavesin a semi-infinite elastic medium(from Richart, 1962).

S-Waves

R-Woves

0.4 0.5

Poisson's Ratio, .,

'"'0~00 10'"=~000~'-'

~~ 5

'"'0~00 10'"=~000~'-'

~~ 5

15

oo 0.5

DYNAMIC STRESS uD (psi)

15Moisture Content = 22.7%Angular Frequency = 25 Hz.Wet Density = 127 pcf (2.034 t/m3)

o ol_-----2~.0------4~.0~--~6~.~0----~8~.0~--~10~.~0----~172~.0

WE (0.001 in/in/sec)

FIG 3.2 PROPAGATION VELOCITY OF COMPRESSION WAVES Vs.AMPLITUDE OF DYNAMIC STRESS AND STRAIN RATE(from Kondner, 1962).

-----._- .._-------------------- --

--...................... _.- .........

.i->:..............................

o

lVN~IS ~NlllIWSNY~l

wu Cf)z-e I-~ Z(J") W.... LCJ W_J ....J-e Wu.... ><(J")>- '<lJ: Nc,

0:: I0 I--w ~L

~ W><:~....J-cCl-0Cf):::lZ-Cf)-ct:JZ........J....JWCl0L

f'I")

f'I")

t:J-LL

m-I.51.0

~.O 2-Node el,ments 1.0(0) (b) 0.50

00.25~ 3-Node elem,nt.Q.I> 0.8S U5;:I;:I

m -1.50I:l~~ 0.'I:l0U 1.0-~ 0.4Q.I>Q.ICl.) 0.$CIS 0.2.t::p..

02 4 • 8 10 I 2 4 • 8 10

1.2 1.02 3-Node element.(c) (d)

~ . 1.1Q.I> 1.01

S;:I;:I

1.0I:l.....~I:l 1.000U- 0.'~Q.I> moOQ.I

0.19Cl.) 0.8CIS.t::p..

0.1 0.88I 2 4 • 8 10 I 2 4 6 8 10

"" II II

m = degree of lumping, such thatm = 0 represents lumped mass formulation,m = 1 represents consistent mass formulation.

FIG. 3.4 Variation of relative velocity with relativewavelength (from Bazant & Celep, 1982).

N

~1

---00.....'"..... - t-- .........,...c::tilNtil~>..0

."OJen::J

."0 0..1

'"'o.c:: ...

c::OJElOJ..... .....

I ~OJ...

.c:: 0..1c::0..1~

N .....I tilc::

00..1enc::~OMA

"" II OJ

8..;t 11'\. I0 ""c:: .... oc:: ~OJ ~El

OJ.-4 11'\~ I

-m-I2.5 ---."1.4

0.' -.-0---,-0.50.4

0.3J

I~I J----0.2

D.I

L/H • 2.5

2.5

3

FIG. 3.6a Amplitude I~Iof spurious reflected wave(from Bazant, 1978).

-- m ~ 0 (I~ed mass)

---II· 0.5

0.2

0.1

--II· 1 (consistent mass)--- II· 1.4 ..

"I h (IOC-scal.)FIG. 3.6b Energy flux of spurious reflected wave to

incoming wave (from Bazant, 1978).

I

Io. a .__ .......---&...II------I...

2--JJ....__--"'-----L.J '---...I.L-_._L.i.-~ __ -...L.---!.-- - -~ J

0.7 5 10 0.7 2 J 5 10

1.& --m ..O---m=O.5

L .' H " 2.~ --111-1---111-1.4

1.4

1.3

3u

2 !I-------_.1.1

FIG. 3.6c Amplitude IVI of diffracted wave(from Bazant, 1978).

1.0 r------------T"'T'""T'"""T'"--,

( a) 2-Node tlements

maO

1.50

o 1.0HIli

1.0 r-------__:.~;;.---r~...-,--...,0.5

0.5

""04('0) 2-Node element.

A

o 0.5 1.51.0

1.0 .-- ~H.:.:/....:".;,__...,.....,,.......,-_r_-_.,.

""a,( C) 2-Node element

oHI"

I.O..-----------.""T'"""T'"-r----..l/"·'( C) 2-Nodt element.

A 05

o 2

HI"

1.0r-----------TT1".111 ..,""04 II,I

II,III,

(b) 3-Node element.maO-0.25 -0.50 -0.75 -1.00 -

I1.50 ,

}j)l2o I

HI"

1.0 r-----------mllT"1

IA I 05

II II04 3-No<1eelement.(b) H/"cl-1

maO-0.75 AcO -0mal-1.5 A-O cO

o2.0 HI"

1.0 r------------rI7T'M""a,( d) 3-Node elements

maO-

8:~8:0.751.00-..,

1.50lJ' ~ I

~~)o 2

HIli

l.o,-----------rnrT'1

IA 105

l/"a,(d) 3-Node elemen ••

HI" cl »1 maOmaO 0.75 A»O cO g:~8mal-1.5 ACO »0 0.75

1.00

2oHI"

FIG. 3.7 Energy flux and amplitude of spurious reflectedwave as a function of element size ratio(from Bazant & Celep, 1982).

0

"...on

on

...C41

...E CQ,I 0

s::. II..... e 0IJ.J :r s ...41 UJ

s::.

"C c .....~

0 II 0 :r Q)

:\ "tI :s. 0 '0

= f() Z

'r"i t::Q)

•.-1 r-. ,., >>.

tU

... >. N on )

tU ... -, 0Q)

> tU'0 N

> cQ)'r"i

Cl] V "'""... Cl]

Q) Cl]on CJ

N Q) "0 c 0 Q) ... -•.-1 N s::. II U

"'"",~ S::M

Cl] •.-1 ,E

11-1 Q)OO

Cl] - Q) aC\Q) g", 'It

... Q)~

Cl] Q) N - "tI~

0 Cl] 0 .. N 00 .J:Cl] Q) ..

.c: 0.. '-" :s

0 c .........

)~ - e 0 s:: s::

eX........ tU

Cl] t:: 0 ... N

... 0 Cl] = 0 (I) Cl) " N 0 :s Q) tU

t:: .... ... 0 0 0 0 0 Q. eo IlQ

Q) rJ) t:: .... Cl] s::e rJ) Q) Cl]

tU w:sQ) Q)

~Cl]

eX II-I.c:~ ... Q)

0 CJ Q.

Q) eo ~ ... Q)

0 Q) eoQ)~~

= ... 0'0 tU Q)

0 Q. t:: ... on :s :su•.-1 0 Q.

"''0... CJ •.-1.... tU S

•.-1 .... ... CJ~ ... 0

rJ) ... •.-1 ....on Q.eo ...

= ~rJ) ... ~

11-1

tU = ... ... 1:11-... .c: tU Q) C...... ... e II.... ... 0 e !! 00

'0 ... Q) II 0 c: .~ LV LI1 0.0 UJ

II C""IE

II !! ... "tI UJ ~0

< Z s::. .. ~..... "tI

j::r,.

0 ~ N ::I: 0~ LI1 Z

N

..Q.-. ....J: •

::. E

--

.,plI 1022 51.. 1E • 28.4 l VJ' pSi (1,96 I lOT kN/ml)

,Q.QZ84Ib/in' (7.S!! l VJ-'kQ/cm')90 llO'81llO'

..

~ f-0,1683

..!!io~....4

LeQend GIv.nOn FIQ. 2

....... The half-beam ill up-\ <,\ ""'. proxirnated by 10 finite elements .

\ "., "\ .'\ ".\ "-.

"_._ Experiment

----- F,n,t. D,ff.rence UslnQ CD, 6t 0 1/3 110-'MC-CD, 6t 'lxIO-l"c- H, Le And W, Le (8 ol.!ll, 6t • 2,!I 110-\ee-- H, Le, 6t • !I.O llO",ae-- W, Le (8 ol,!ll, 6t· !I.O 110·'see- N, Le (/301/41, 6t • 2.!I 110-t"c-N, Le (/3.1/4), 6t

CD - Centrol DiU.r_M • New",o,. 0H • HouDOit 0W •• ,1_ 6L. - Linear Solut 1011Le • NoIIIi'_r SolultOfl .,th LoadConeetiOll

Iu

"jo

T imt, .. et I 104

FIG. 3.9 Comparison of solution schemes for anelastic-plastic beam (from McNamara,1974).

....:1.4 ,....._~_....,...--r.....,....,..._......,._.....,_-r-~144(I)

• >"(I)~ r.z)

C (Courant no ,) = CAt / AX 72

';t; i.o 0 -UlU (I)~Ul

(I)

>. C = ~.t:: 0.8 -72 00Il-I

(I)- 2.0 "0. = -Q) 0.6 -144 (I)

> Ul<0

(I) .t::> Il-I

~ 0.4 -216"0 no friction assumed(I)+'" -288::l 0.2~0u 00 -3602 4 6 10 20 60 100

parts per wavelength (A/AX)

FIG. 3.10 Dispersion of long-wave surfacewater equation using Newmark({3- 0.25) scheme (from Gray &

Lynch, 1977).

1.2 C • 1, f3 • 0

0.8 0.5, f3 • 0

exact0.4

C ..2, f3 =

0.00 0.2 0.4 0.6 0.8 1.0

2 t1x/l<a)

t = 0.5773,{3 = 0

t = 0.3, 13 '" 0

0.8

0.4C - 2, f3 - 0.25

0.2 0.4 0.6 0.82 t1x/.A

1.0 1.2

(b)

FIG.' 3.11 Dispersion for (a) Lumped and (b) ConsistentMass formulations in association with theCentral Difference ([3 - 0) and Trapezoidal([3. 0.25) temporal operators (fromBelytschko & Mullen, 1978). The 'exact' curvesimplies no temporal discretisation errors (Le. C-O).

35

CHAPTER 4

DYNAMIC RESPONSE OF SHALLOW FOOTINGS

4.1 Introduction

The evaluation of footing response under dynamic loading conditions is

essential to the design of machine foundations and those subjected to

external transient loads. The two major aspects in foundation design for

such conditions are change in material strength under cyclic loading and

dynamic amplification of response. The former is a combined effect of

fatigue and viscosity, and is more suitable for laboratory evaluation than

numerical computation (Smith & Molenkamp, 1980). The present chapter

concentrates on the second aspect, which is more amenable to prediction, and

of which the experimental, analytical and numerical basis are comparatively

well-founded. Table 4.1, though by no means comprehensive, serves to

illustrate the volume of literature devoted to the study of the topic.

(i) Experimental Approach

Possibly the earliest recorded field tests of extensive scale were

carried out by the German Research Society for Soil Dynamics (D~E80) in the

1930s. Rotating mass oscillators were employed to evaluate the dynamic

properties of a soil in situ hoping to produce more representative results

than the conventional borehole sampling and laboratory testing procedures.

Refined DEGEBO tests led to the conclusion that there is no definite value

for the natural frequency of a 'soil', but rather the natural frequency of a

'foundation-soil system' depends on the physical properties of both the soil

and the foundation, as well as any natural boundaries present in the environ

of the system.

Extensive laboratory and theoretical works have been presented by

Barkan (1962).

Fry (1963) presented results from field vibration tests on circular

concrete footings which varied from 1.5 m. to 5 m. in diameter, while

36

Maksimov (1963) investigated free vibration of similar footings up to 70 m.in diameter. Vibratory tests on small scale footings were also carried outby Bycroft (1959), Kondner (1965), Novak (1970) and Raman (1975).

Sridharan & Nagendra (1981) analysed data from 442 vibratory tests ina statistical manner. Two generalised equations were derived to predict theresonant frequency and amplitude separately.(ii) Analytical Approach

Since Lamb (1904), most of the analytical solutions have been derived bytreating the soil as an elastic half-space. According to the half-spacetheory all the wave energy generated by footing vibration will either betransmitted into the far field or dissipated by internal damping. The elastichalf-space solutions compared well with experiments in some occasions (e.g.Richart & Whitman, 1967), but not in others (e.g. Maksimov, 1963; Novak,1970). Discrepancies were attributed by Novak (1970) to the soil massbehaving as a stratum of definite thickness, and the nonlinearity of responseat high excitation intensity.

It is interesting to note the comment by Warburton (1959) that a limitedstratum with thickness approaching infinity does not constitute the limitcondition for a half-space model. In the former case, the stress waves willreflect at the base of the stratum and will reach the free surface at steadystate, whereas in the latter case no such reflection will take place at all.The difference in resonance characteristics between the two cases can beseen in Fig. 4.1.(iii)Lumped Parameter Model

Probably first suggested by Reissner (1936), the model was not developeduntil Rausch (1959) and Barkan (1962). The one-degree-of-freedom mass-spring-dashpot system has been employed analytically as an equivalent analogin the analysis od foundation-soil systems (e.g. Lysmer & Richart, 1966)(Fig. 4.2a). Extensions into two dimensions have been initiated by Ang &

Harper (1964), in which the continuum is divided into a large but finite

37

number of mass and stress points (Fig. 4.2b). The mathematical model hasbeen described in detail by Ang & Rainier (1964), and has been employed byHoeg et al (1968) and Hoeg & Rao (1970) with the incorporation ofe1astoplastic springs. Subsequent extension to model radiation damping hasbeen suggested by Agabein et al (1968), and applied by Krizek et al (1972).

The main attraction of the lumped model lies in its relative simplicityand economy in implementation. For simple cases, like the interaction of anelastic soil with a circular or strip footing, standard impedance functionsexist and the resulting solution quality is compatible with finite elementsolutions. On the other hand, for three-dimensional interaction problemsproper three-dimensional finite element analysis is often not financiallyviable, and the lumped model offers as a cheaper alternative than anapproximate two-dimensional finite element analysis for a simplifiedevaluation. Thus it is not surprising that the lumped parameter model stillfinds popularity in modern-day design of dynamically loaded foundations.(iv) Finite Element Model

A more generalised and versatile numerical modelling technique is thefinite element method, which proves its superiority in cases with complicatedgeometries or soil stratification, and in its ability to model constitutiverelationship of soils. The application of the method in the frequencydomain on dynamic footing-soil interaction have been reported by Lysmer &Kuhlemeyer (1969), Roesset & Ettouney (1977) and Chow (1981). Extension tothe dynamic response analysis of gravity platforms in the time domain hasbeen performed by Smith & Molenkamp (1977, 1980).

In this chapter a number of footing vibration problems are to be solvedin the time domain with the algorithm described in Section 3.4. It hasbeen mentioned that the discretised approximation of the model to a continuumimposes more restrictions on the design of the model and its associatedparameters. In order to assess this, a few benchmark problems of both

periodic excitation and pulse loading in nature are solved. In all

38

computations it is assumed that the footing always adheres to the soil mass,

and also there exists no internal viscous damping within the soil mass, except

at the transmitting boundaries.

4.2 Periodic Excitation of a Smooth Massless Circular Footing upon a

Smooth Elastic Stratum

The footing-soil system as shown in Fig. 4.3 is considered. The

boundary conditions specified are (i) no stresses exist at the free surface

beyond the realm of the footing; and (ii) there is no friction between the

footing/elastic stratum and the elastic stratum/rigid stratum interfaces.

Since the wave characteristics are better defined in a periodic oscillation

problem than in a pulse loading problem, it will be easier to assess the

influence of the finite element formulation with the former.

In the case of a rigid footing, a closed-form analytical solution has

been furnished by Warburton (1957):

x (f1 cos wt f2 sin wt) (4.1)

where fl and f2 are functions of a (= r w / V ), the depth factor Ro 0 s(= stratum depth / r ) ando (Fig. 4.4). However, in a prescribed load

problem it is difficult to model the rigidity of a massless footing, and in

bere the footing is simply assumed to be flexible. By prescribing a

uniformly distributed load (Griffiths, 1981), there is no necessity to

allocate elements to represent the footing in the numerical model. The

techniques in modelling footing rigidity are further discussed in Section 4.3.

(i) Effect of AS/~X

When spatial discretisation is considered for a two- (or three-)

dimensional wave propagation problem, the shear wave is more critical than

the compression wave because of its shorter wavelength. In this case,

the wavelength of the S-wave corre{onding to a forcing frequency of 10 Hz.

39

is (2~ V I w) or 62.8 m. In order to investigate the effect of meshs

refinement, uniform meshes (apart from the transition in element sizebeneath the footing) of A I~x ratios from 2 to 16 have been formulated.sthe standard viscous boundary (Lysmer & Kuhlemeyer, 1969) is incorporatedin the lateral extent, but is deliberately kept at a distance of 22 ro(i.e. 172.7 m.) from the centreline of the footing (Fig. 4.5), so that anystray reflections resulting from imperfect absorption of impinging energyat the boundary will not return to the footing within (2 x 21 x ro I Vp)or 1.9 seconds, i.e. 3 cycles. Simple rollers are placed at the bottomboundary to simulate a frictionless interface. Following the recommendationof Bathe (1982), the time step size is chosen to be T/80 or 7.854 msecs.Consistent mass formulation and (2 x 2) reduced integration are employed.The initial conditions are simply taken as that at deadstart, i.e.

x - 0, x - 0 and x • 0 (4.2)From Fig. 4.6 it can be seen that provided A I~x ~ 4, resonablys

accurate solutions can be obtained. On the other hand, with A I~x <4, thes

displacement amplitude becomes irregular and the frequency distorted,suggesting strong dispersion as well as spurious effects of mesh gradation(at region of footing). The static responses obtained from the As/~x - 8and A I~x = 16 meshes are also shown. By comparing the static and dynamics

responses it can be seen that the role of inertia is, at this lowfrequency, to amplify the response of the system.

(ii) Effect of Time Step SizeTo study the influence of time step size, the mesh with A I~x - 16 iss

employed. The Wilson (8 - 1.4) algorithm is again adopted using time stepsizes varying from T/80 to T/2. Fig. 4.7 shows that severe dispersion isevident when At is greater than T/7 (i.e. C - c~t/~x >2), and when At isincreased to T/2 rapid attenuation occurs. Furthermore, the displacement

amplitudes exhibit convergence only when ~t < T/14 (i.e. C - 1). Since

the problem considered here is linear in nature the Bathe (1982)

40

recommendation of ~t = T/80 seems to be too stringent.

(iii)Effect of Transmitting Boundaries

Roesset & Ettouney (1977) have shown that for low frequency excitation

the steady state response of an elastic system is rather insensitive to the

nature of the transmitting boundaries incorporated. However, due to the

often formidaL!e cost and resource required for time domain analysis,

computation to steady state may not be practical, but instead only the first

few cycles are usually evaluated.

Fig. 4.8 - 4.10 shows the footing response for the first 5 cycles with

different lateral boundary conditions and distances incorporated in the mesh,

which has a A /AX ratio of 16, and a time step size of T/20. The initials

responses of all curves are more or less the same, but the curves in

Figs. 4.8 - 4.9 gradually diverge9 progressing from the mesh with the closest

lateral boundary to the one with lateral boundary farthest apart. The

influence of the trapped spurious energy as a result of the lack of lateral

radiation damping can be clearly seen. On the other hand9 Fig. 4.10 shows

that provided the distance of the lateral viscous boundary is greater than

A /2 (i.e. 2 r from the centreline in the present case), the boundarys 0

absorbs incident energy waves satisfactorily and the response becomes

insensitive to the boundary distance. This reinforces the finding of White

et al (1977) as mentioned earlier in Section 3.3.4 (c).

(iv) Effect of Mass Formulation and Integration Order

So far the problem has been analysed using consistent mass formulation

and (2 x 2) reduced integration. Fig. 3.4 shows that the influence of mass

idealisation is only significant when A /~x is small. The same applies tos

the effect of integration order used, as shown by Chow (1981)~

When the mass formulation and integration order are varied for the"present problem with A /~x a 4 and A/AX - 16 meshes, it has been found thats s

the response is insensitive to the mass formulation, but is sensitive to the

41

integration order when A I~x = 4 (Fig. 4.11). The reduced integration, again,s

relaxes the system and amplifies the response. However, the difference inresponse is minimal when A /~x = 16. The above applies to T/80 ~ ~t , T/14.s

4.3 Resonance of Dynamically Loaded FoundationsThe consideration of resonance, in terms of frequency and amplitude, is

important in the design of dynamically loaded foundations (Whitman & Richart,1967). Following the discussions in Section 4.1, the finite element methodcan be regarded as a feasible technique in the prediction of resonance.Three approaches can be considered:(i) determination of the eigensolutions of the system, for example by QR

or Lanczos algorithms (Heshmati, 1983);(ii) frequency domain analysis to evaluate steady state response over a

frequency range; and(iii)time domain analysis to evaluate the transient response (of the first

few cycles) at a number of frequencies.While (i) seems to be the most direct method to pin-point the natural

frequencies, it is difficult to identify the one that corresponds to theappropriate mode of deformation. This will be illustrated in the examplethat follows. Furthermore, both (i) and (ii) are suitable for linearanalysis only. While repeating the time domain analysis (as in Section 4.2)over a number of frequency values is a somewhat laborious process, it isthe only finite element approach for truly nonlinear analysis. In thefollowing example, both (i) and (iii) will be illustrated.

The problem to be considered is shown in Fig. 4.12. The analyticalsolution furnished by Warburton (1957), reproduced herein as Fig. 4.1,predicts the fundamental frequency for vertical translation as 10.4 Hz.(corresponding to a • 0.653). In order to test the performance of theo

finite element method, the eigensolutions are to be sought over the

frequency range of, say, 4 to 15 Hz. On the other hand, forced oscillation

42

computations in the time domain are also to be performed over this frequencyrange.

A mathematical analysis on the accuracy of finite element eigensolutionsis given by Fried (1971). In the present example, eigenvalue analysis isperformed on the same mesh as discretised for forced vibration analysis (seebelow). Since only a portion at the lower end of the frequency spectrum isof interest, the Lanczos algorithm with stiffness factorisation isappropriate.

As for mesh design, the finite elements must be small enough to transmita frequency as high as 15 Hz. Following the results of Section 4.2, a As/~x

ratio of 4 is adopted, limiting the element size to a maximum of 1.667 m.The distance of the lateral viscous boundary is kept at 10 r from theo

footing realm, thus reuiring a total of 7 columns of elements in the model(Fig. 4.13). In the forced vibration computations, 20 time steps are usedto cover every load cycle, i.e. ~t = T/20, and following Section 4.2, thisshould be adequate for linear analysis.

Modelling of Footing RigidityIn finite element analysis, there are four usual techniques for modelling

a ridid footing:(i) by prescribing equal footing displacements rather than prescribing

external loads as input, in conjunction with the 'large spring technique'(Smith, 1982). However, in doing so it is impossible to recover theinternal reactive forces by the usual J BT ~ d(vol) computation (Chin,1979).

(ii) The vertical degrees of freedom at the footing base are tied by stifftruss elements which are then connected to a stiff beam element.External loads are then prescribed on the beam. This technique isproposed by Chow (1981), but has not been implemented. In any case,

the magnitude of stiffness and the masses of the additional elements

remains to be justified, and the implementation is relatively difficult.

43

(iii)ln the case of vertical oscillation of surface footing on an elastic

half-space, Sung (1953) has established dimensionless charts for

estimating the dynamic response characteristics of footings subjected

to different load distributions: uniform, rigid base and parabolic.

Richart (1953) further exploited these to point out the inter-

convertibility between different load distributions by an appropriate

modification in the loading area. The concept of 'effective radius'

is defined, such that each of the pressure distributions can be

converted to an equivalent uniformly distributed load (Fig. 4.14).

For example, a rigid footing of radius 1 m. and dimensionless mass3ratio B (= m/(pr »)of 15, installed upon a half-space with shear waveo

velocity of 1 m/sec and Poisson ratio 0.25, can be 'converted' to a

flexible footing (i.e. uniformly distributed load) with effective

radius of 1.273 m. and dimensionless mass ratio of 7.27. As long as

the effective radius remains constant, the various systems will yield

similar dynamic response characteristics.

(iv) A simple and more general procedure is to assign the footing a

stiffness much higher than that of the soil. The usual stiffness

matrix assembly procedure will then lead to a rough footing/soil

formulation. In order to simulate a smooth rigid footing, Chow (1981)

has suggested neglecting the horizontal stiffness terms contributed·by

the footing at its base during the assembly of the global stiffness

matrix. The implementation is tested here by subjecting the footing in

Fig. 4.12 to sinusoidal oscillation with w ~ 60 rad/sec. The steady

state amplitude is given analytically by Warburton (1957) as

_ Fdyn (f12 + f22)G ro (1 + ba02fl)2 + (ba02f2)2

(4.3)

or in this case, 3.83 mm.

The computed amplitudes after 10 cycles with different Ef ti /e i100 ng sovalues are plotted in Fig. 4.15. Virtually constant values are obtained

44

for both the 'rough' and 'smooth' footings with 105 Esoil ~

109 Esoil (i.e. 5 x 109 kN/m2 ~ Efooting ~ 5 x 1013 kN/m2).Efooting E

Fig. 4.16also confirms that the footing is rigid enough to establish a uniform

4> 10 Esoil (i.e. Efooting >displacement beneath it for Ef ti00 ng

a 25 x 10 kN/m). However, for higher E values the amplitudefootingcomputed rises slightly, and then enters into a region of instability,

and ultimately stabilises to a negilgibly small value. In practice, a

value of Ef ti ~ 105 E il should guarantee footing rigidity and avoid00 ng sonumerical ill-conditioning. It should be noted that the procedure to

simulate the smoothness of the footing is only an approximate one, and

consequently the finite element bounding property is destroyed, resulting

in an under-stiffened response.

Eigenproblem Analysis

The evaluation of eigensolutions of the equation M x + C X + K x -

o is difficult as the solutions may not be real numbers (Gupta, 1974). ihe

problem can be much simplified by ignoring the damping term C and determing

the approximate solution instead. As discussed earlier the Lanczos method

with stiffness factorisation is adopted to extract the eigensolutions at

the lower end of the frequency spectrum. The lateral boundary is assumed to

be free. The use of exact integration is required for consistent mass

formulation of the a-node element to avoid matrix singularity as discussed

in Section 2.4.2. Some of the eigenmodes obtained are plotted in Figs. 4.17 -4.19. The effect of fixing the lateral boundary is to stiffen the system and

consequently increases the eigenvalues, while lumped mass formulation has

been found to induce the opposite effect. These results are not presented

any further herein because it can already be seen from Figs. 4.17 - 4.19that it is impossible to identify the eigenmode corresponding to vertical

footing translation. The exercise of forced oscillation computations may be

more fruitful.

45

Forced Oscillation Analysis

The transient response of the footing-soil system at various frequencies

(including all corresponding to the eigenvalues) are computed using the

algorithm in Section 3.4. The response of the first few cycles at four

particular forcing frequencies are presented in Figs. 4.20 - 4.23. Fig. 4.24

shows the amplitude of footing displacement at the 4th, 7th and 10th cycles

over the frequency spectrum of interest. it can be seen that resonance

marked by an ever-growing displacement amplitude occurs at around the second

eigenvalue, i.e. 10.5 Hz. Slight shift of response peaks for different

cycles are likely to be caused by initial transients. The computed finite

element result compares favourably with Warburton's (1957) solution of

approximately 10.4 Hz. Of course the resonant amplitude in reality is finite,

due to the internal viscosity of soils.

Amplification of response also occurs at frequencies other than at

resonance. A dynamic magnification factor can be defined as

DMF (of nth cycle) - (4.4)

As static analysis reveals that the static displacement corresponding to Fdynis 0.609 mm., the DMF plot as in Fig. 4.25 can be obtained. Dynamic

magnification of over 10 times of the static value can be obtained at

resonance after 10 cycles. At frequencies lower than the resonant frequency

the reponse is relatively steady (e.g. Fig. 4.20), and the DMF is always

greater than 1, suggesting amplification effects of the inertia. In

contrast at frequencies higher than the resonant frequency the oscillations

are typically irregular and unpredictable (e.g. Fig. 4.24).

Such finite element procedures can be easily extended to study other

problems such as footing embedment, the effect of nonlinearity and so on.

46

4.4 Response of a Rigid, Circular Surface Footing Subjected to a TrapezoidalPulseThe case of a footing mounted on the surface of an elastic half-space

has been analysed by Lysmer & Richart (1966) using both the half-space theoryand a simplified analog model. Compatible results have been achieved. Onthe other hand, Duns & Butterfield (1968) tackled similar cases with thefinite element method in the time domain, whereby the soil was treated as afinite medium in both lateral and vertical extents. A graded mesh oftriangular elements have been employed, with no apparent considerations formesh and time step design shown in the work, and the finite element resultshave not been compared with solutions in closed-form or obtained byalternative approaches.

Herein the Lysmer & Richart (1966) problem of a rigid, circular surfacefooting subjected to a trapezoidal pulse is repeated using the finite elementformulation in the time domain. The required data is reproduced in metricunits in Fig. 4.26. The influence of the far field in the lateral extent canbe taken into account as before by the incorporation of standard viscousboundaries. However, since the load function is of a quasi-static ratherthan a vanishing type, some fixity must be incorporated at the base boundaryof the finite element mesh in order to preserve vertical equilibrium atdiscrete time steps, because the standard viscous boundary cannot support anon-vanishing load. In other words, a half-space cannot be properly modelledby a finite element formulation in the time domain for quasi-static analysis,and the model has to be arbitrarily truncated at a certain depth. Ofcourse it is desirable to place the artificial base boundary as deep as

,possible in order to approach the vertical infinity of the half-space, insofaras the available computer resources allow.Mesh Discretisation

In contrast to the two previous problems dealing with forced periodic

excitation for which the propagation frequencies of the stress waves are well

47

defined, an impact type of loading will agitate not only a single frequency

but a width of the frequency spectrum. Thus it is essential that the

element size is small enough so that the majority of the agitated frequency

components can propagate with minimal dispersion. Hadala & Taylor (1972)

recommended design of the element size to be based on the rise time of the

load function. This rise time is considered as 1/4 of the period in a load

cycle, while the corresponding frequency to this period is taken as 1/4 of

the cutoff frequency. In other words,

rise time tr = T/4 .. 1/(4f)

and f = fco/4

where fco is the cutoff frequency.

Thus tr • l/fco

(4.5)

(4.6)

(4.7)

In the case of one-dimensional linear elements with lumped mass formulation,

equation (3.4) gives

fco = c/(n:t1x) (3.4)

On the other hand, in the context of two-dimensional wave propagation through

a mesh of 8-node rectangular elements, similar determination of the cutoff

frequency fco is complicated, and from previous experience it is recommended

herein to take

fco - c/(4t1x)

where t1x is the limiting element dimension.

Substituting (4.8) into (4.7),

.:\x .. ctr/4

In the present problem,

Vs - 91.45 m/sec. and

tr .. 0.05 sec.

Thus equation (4.9) gives a limiting element size

.:\x .. 1.143 m.

Overall Mesh Size

As the half-space extends to infinity both laterally and vertically,

(4.8)

(4.9)

48

some decision has to be made on the positions where artificial truncatingboundaries are to be placed.

Since radiation damping cannot be modelled in the vertical direction,stress waves impinging at the base boundary will be reflected into the meshand eventually return to the footing in due time, thereby masking up theresponse. The crucial stress wave in this case is the faster travelling P-wave, the return time of which can be expressed as

RT 2 x Depth of Base Boundary / Vp (4.5)so that in order to secure a minimum return time of say 0.5 sec. in thiscase, the depth of the base boundary must be place at 45.72 m. (or 30 ro)beneath the footing.

As for the lateral boundary, it has been shown in Section 4.2(iii) thatin general the standard viscous boundary performs satisfactorily in theabsorption of impinging energy waves in axisymmetry provided that it isinstalled at a reasonable distance from the excitation source. However, atthe presence of a quasi-static load as in this case, stress waves will tendto decay in due time, and the condition approaches one of static loading. Asthe viscous boundary will not offer any static stiffness, it will graduallydegenerate into a free boundary. Thus it is important to provide enoughelements in the lateral extent to furnish the static stiffness. In view ofthis, the lateral boundary for the present problem is placed at a distance of10 ro from the centreline of the footing, so that under the maximum imposedload the finite element model gives a static footing settlement of 6.976 mm.,which compares well with the static settlement upon a half-space of 7.061 mm.

The resulting mesh, as shown in Fig. 4.27, has 1833 nodes, 3454 degreesof freedom and a half-bandwidth of 91. It is implemented on the CYBER 205machine which is capable of vector processing (for arrays of less than 65Kin length).

Effect of Temporal Operator and Discretisation

For an impact type of problem in which a range of frequencies are

49

excited, the selection of a suitable time step size can be conveniently based

on the Fourier stability limit, i.e. ~x/c. Although in the application of

an implicit operator to linear problems there is no limit on the time step size

for stability reasons, it is in the interest of solution accuracy that a

reasonably small ~t is employed.

For the mesh in Fig. 4.27 the smallest element dimension is 0.762 m.,

which is the width of the elements beneath the footing. Thus the Fourier

stability limit is ~t = 0.762/Vp = 4.167 msecs, although the limit for other

elements is 1.143/Vp or 6.25 msecs.

The effect of time step size on the solution of the present problem using

Wilson (0 = 1.4) and Newmark (~ e 0.25) methods is shown in Figs. 4.28 - 4.33.

The mass ratio of the footing B (- (1 - v)m/(4 r03» is taken as 1 for

comparison purposes. It can be seen that for both algorithms the quality of

solution begins to deteriorate at a time step size of-around 12.5 msecs. (or

3 times the Fourier stability limit), but on the whole reasonably accurate

results have been obtained, compared with the solutions of Lysmer & Richart

(1966) as reproduced in Fig. 4.34. Nevertheless, when ~t ~ 12.5 msecs. the

numerically undamped Newmark (~ - 0.25) solution begins to display significant

spurious oscillations, while the response of the overdamped Wilson (0 - 1.4)

scheme remains stable. Thus it is not surprising that in the presence of

nonlinearity a large time step size will render the Newmark scheme unstable.

This will be further illustrated in Section 6.3. In the present case it is

interesting to note that even when the external load vanishes after 1 sec.,

the computed displacements do not oscillate about the time-axis as expected

for a quasi-static response. This is possibly due to the stray energies

trapped in the mesh because of (i) the attenuation of high-frequency wave

components, (ii) unwanted wave reflection at the base boundary, and

(iii) the imperfection of the lateral viscous boundary.

Effect of Mass Ratio B

The effect of footing inertia on the dynamic response is analysed in

50

Figs. 4.35 - 4.37. The Wilson (6 = 1.4) algorithm is employed in this case.

It can be seen from Fig. 4.35 that the displacement responses for B = 1 and

5 are essentially similar to the Lysmer & Richart (1966) computation (Fig.

4.34), with the latter achieving a more amplified peak than the former, whose

response approaches that of quasi-static in nature. On the other hand, Figs.

4.36 - 4.37 indicate that it is the lighter footing which gives rise to

higher velocity and acceleration peaks.

4.5 Acceleration Response of a Circular Surface Footing subjected to Impact

It has been shown that the finite element analysis in the time domain is

well suited to the computation of displacement response as a result of dynamic

loading. However, in some categories of dynamic problems the object is the

determination of acceleration responses, for example in convolution analysis

of seismic problems. In general, accurate computation of acceleration by

the finite element method is not easy, because the derivatives of displacements

are very sensitive to the damping and dispersive characteristics of the

time integration scheme adopted.

A footing impact test performed by Drnevich et al (1965) and reported by

Lysmer & Richart (1966) is considered herein. The 0.305 m. (1 ft.) diameter

footing is seated on dense Ottawa sand in a 1.45 m. (4.75 ft.) square sand

bin bounded by 1.22 m. (4 ft.) high concrete walls. The footing was subjected

to controlled impact pulses and the transient response was recorded by an

accelerometer mounted on the footing.

The actual recorded load history is not available in digital form, and

has to be approximated from Lysmer & Richart (1966) (Fig. 4.38) and curve-

fitted by spline functions (Fig. 4.39). The critical rise time of the load

is difficult to estimate. In order to ascertain the effect of spatial

discretisation, 4 meshes of different degrees in refinement (Fig. 4.40) are

assessed. The time step size adopted for all meshes is that corresponding to

the Fourier statbility limit of the most refined mesh.

51

The computed responses using the Wilson (8 - 1.4) scheme and the

Newmark (fo = 0.25) scheme are shown in Figs. 4.41 - 4.43 and 4.44 - 4.46

respectively. In both cases convergence occurs for Meshes C and D, the

accelerations of which agree well with the experimental results (Fig. 4.38)

showing that the critical rise time is about half of the overall rise time

from zero to peak load (equation (4.9». The acceleration peaks computed

from both time integration schemes are similar but smaller in magnitude than

reality, probably due to the smoothening effect of the spline curve-fitting

on the load function.

4.6 Foundation Response to Indirect Impact

4.6.1 Introduction

The previous sections of this chapter are concerned with the dynamic

response of a surface footing subjected to direct vertical loading. Except

for some standard cases for which closed-form analytical solutions are

available, various numerical approaches are often resorted to in practical

studies of soil-structure interaction problems. The potential of the finite

element method has been demonstrated in axisymmetric contexts in the previous

sections. Although in theory the method can be extended to 'proper' three-

dimensional analysis, such practice is bound to be extremely expensive and

laborious, although an explicit temporal operator in conjunction with a

lumped mass formulation can help to improve the efficiency to a certain

extent. This point has been demonstrated by the computation of Nelson (1978)

using his 'soil island' model.

In order to enhance the competitiveness of the finite element approach,

the idealisation of a dynamic three-dimensional soil-structure interaction

problem to an 'equivalent' two-dimensional plane strain formulation is often

considered.

A number of such idealisation procedures have been attempted in the

past. For example, Isenberg & Adham (1972) simply replaced an axisymmetric

52

foundation by a plane strain one with the same width and density. Luco &Hadjian (1974) adjusted both the width and density of the idealisation in

order to preserve as far as possible the dynamic stiffness characteristics

(and consequently the natural frequencies) of the three-dimensional prototype

over the frequency range of interest (Fig. 4.47a); however in doing so they

discovered that radiation damping is always over-estimated by the plane

strain model, resulting in an under-predicted response which can lead to

unsafe design.

Another example is the implementation of FLUSH (Lysmer et aI, 1975), in

which the original footing width and density are retained in the 'equivalent'

plane strain model, and furthermore viscous dashpots are incorporated on the

lateral truncating planes to absorb the SH and SV waves generated due to

shearing at the boundary planes (Fig. 4.47b). The concept of FLUSH has

recently been challenged by Gazetas & Dobry (1985), who contend that a plane

strain idealisation will always overestimate the amount of radiation damping

in the original three-dimensional conditions, due to destructive interference

of stress waves, and extra dampers incorporated in an already over-damped

representation will only aggravate the problem further. However, Gazetas &Dobry (1985) based their arguments on the results of analyses using lumped

parameter models, and conclusions are drawn from the comparison of lumped

damping coefficients rather than on the actual response magnitudes. A

problem is presented below which seem to draw the opposite conclusion to

that of Gazetas & Dobry (1985).

It is desired to estimate the response of a circular surface footing

when another one nearby is subjected to impact. This problem has practical

implications on the design of nuclear installations against indirect impacts,

or in the dynamic response analysis of multi-base offshore gravity platforms.

The data of the problem is presented in Fig. 4.48. Both the 'target' and

'second' footings are effectively rigid so that differential movements are

minimal. Due to the complexity of the problem a step-by-step process with

53

intermediate verification stages is employed:-

(i) static analysis of the 'target' foundation only in both axisymmetry and

plane strain, in order to calibrate the response of the latter to the

former;

(ii) transient response analysis of the 'target' foundation only, subjected

to the full design impact pulse in axisymmetry and the calibrated impact

pulse in plane strain; and finally

(iii)transient plane strain analysis including both the 'target' and 'second'

foundations under the calibrated impact pulse.

4.6.2 Mesh Design

The influence of mesh design on static analysis is much smaller than in

dynamic analysis, but it is appropriate to employ the same mesh in both

stages (i) and (ii). From Fig. 4.48 the critical rise time of the load

function can be taken as 0.01 sec. If only vertical impact is considered, a

further assumption can be made in the interest of storage economy, that only

shear waves will propagate horizontally and only compression waves will

travel vertically. Thus using equation (4.9),

~x - trVs/4 = 0.25tr jE/(2p(1 + v) - 0.4626 m.

~y - trVp/4 - 0.25tr jE(l - v)/(p(l + v)(l - 2v» - 1.70 m.

The mesh used for stages (i) and (ii) is shown in Fig. 4.49 and that for

stage (iii), assuming a 2 m. gap between the foundations, shown in Fig. 4.50.

In the present problem, mesh gradation is inevitable in order to

accommodate the geometry of the foundations. Since the implicit Wilson

(8 - 1.4) operator is employed herein, it is reasonable to employ a time step

size conforming to the Fourier stability limit of the general element size,

i.e. 0.4626/Vs or 1.70/Vp' i.e. 1.5 msecs.

4.6.3 Stage I : Static Response of 'Target' Foundation

Fig. 4.51 shows the static displacement response of the 'target' footing

in the soil stratum. The response in axisymmetry is not as substantial as

54

that in plane strain, because a greater volume of soil has to be mobilisedfor a given displacement magnitude. Under the maximum load applied thecircular footing settles about 4.9 mm.

In order to calibrate the plane strain model the stiffness of such maybe altered, but this will destroy the idea of modelling the soil propertiesin a realistic manner. Instead, no allowance is made for the underestimationof stiffness in plane strain, but rather the maximum stress level is factoredto a magnitude such that the plane strain model will yield an equal staticdisplacement to the axisymmetric model. From Fig. 4.51 a maximum stress levelof 340 kN/m2 is applicable to the idealised plane strain model.

4.6.4 Stage II : Dynamic Response of the 'Target' FoundationThe algorithm described in Section 3.4 is employed to perform both

axisymmetric and plane strain analyses of the 'target' foundation at theappropriate stress levels (Fig. 4.52). The footing is modelled as effectivelyrigid by assigning its Young's modulus to be 105 times that of the soil, asdiscussed in Section 4.3. The width of the footing is taken as the same inboth analyses, but it is not clear whether the footing mass or densityshould be kept equal to the axisymmetric case for a better approximation.Fortunately, with the relatively small size of the footing the difference inresponse between these inertial formulations is found to be less than 1%.

The computed displacement response of the footing and of the groundsurface some distance away are as shown in Fig. 4.53. The decay of peakdisplacements with distance is shown in Fig. 4.54. It can be seen that thedisplacement response directly beneath the footing is very similar betweenaxisymmetry and plane strain. The correlation deteriorates with increasingdistance, due to the larger damping in the axisymmetric formulation(contrary to Gazetas & Dobry, 1985). Nevertheless, the plane strain modelwill furnish an upper bound as far as displacement response is concerned,only the margin of safety become extremely large at remote distances fromthe excitation source.

55

Similar comparisons between the acceleration responses are presented in

Figs. 4.55 - 4.56. Again, the agreement between response characteristics of

the axisymmetric and plane strain formulations is remarkable, but the plane

strain model does not serve as an upper bound solution at close distances to

the 'target' foundation. However, with the larger damping possessed by the

axisymmetric formulation, the peak response correlates reasonably well with

the plane strain formulation at distances greater than 4.5 m.

4.6.5 Stage III : Dynamic Response of the 'Second' Foundation

It is only at this final stage that both the 'target' and the 'second'

foundations are modelled. The plane strain idealisation subjected to the

calibrated load as in Stage II is considered. The vertical displacement and

acceleration responses at the edges and centreline of the 'second' footing

are shown in Figs. 4.57 - 4.58. Rocking motion is clearly exhibited. Also,

comparing with the responses obtained for Stage II (Figs. 4.53 - 4.56) the

stiffening effect of the 'second' footing is apparent.

The resulting responses computed at this stage also serve as convenient

(though approximate) input data for subsequent vibration analysis on the

superstructure of the 'second' foundation.

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FIG. 4.47b Plane Strain Approximation of a 3-D Footingwith viscous dashpots incorporated on thelateral truncating planes to absorb SH andSV waves, as applied in FLUSH (Lysmer etaI, 1975).

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56

CHAPTER 5

VIBRATORY PILE DRIVING

5.1 Historical Development

The concept of vibratory pile driving was probably born in 1930 and found

the first industrial application in 1932 when timber piles were vibrodriven

by Hertwig in Berlin-Charlottenburg, Germany, as reported by Lorenz (1960).

Research into the vibratory technique started at almost the same time in

Russia as a by-product of soil dynamics research by Pavlyuk (1931). Small-

model experimentation on driving and extracting from soil was conducted in

1934. In 1948-49, a milestone was laid when vibrators were employed on a

large scale in the construction of the first Russian cellular cofferdam at

the Gorky Hydroelectric Development. A total of 3,700 sheet piles weighing

5,000 tons altogether was driven into saturated sand to 9 - 12 m. penetration,

the average driving time being 2 - 3 minutes (Barkan, 1957). The Russians

had found the vibratory technique so satisfactory that during the post-war

years, sheet piles were mainly driven and extracted by vibrators rather than

conventional hammers at all hydrotechnica1 construction sites. Tsap1in (1953)

investigated the mechanism of vibratory hammers in cohesive soils. Other

important papers depicting the Russian developments include, chronologically,

Medvedev (1953), Neumark (1953), B1ekhman (1954), Barkan (1967), Smorodinov

et a1 (1967) and Ganichev (1973).

The reported achievements of the Russian engineers have led to further

exploitation in vibrodriving in other countries in the 1950s. In China,

Mao (1958) reported the success in vibrodriving open-ended precast concrete

columns, although the operation in clay was found to be an order of magnitude

slower than in sand. Engineers in France and Germany have also attempted to

design vibratory equipment to work at frequencies higher than the Russian

machines (up to 50 Hz.), although wear on machine parts eventually rendered

the operating frequency to 25 Hz. maximum.

57

In the United States a major breakthrough occurred in the 1960s when

Bodine launched a high frequency vibratory pile driver that can operate above

100 Hz. Numerous successes were recorded, as reviewed by Ross Esson (1963).

The high frequency driver, however, is believed to work on a different

mechanism from the low frequency version, and will be discussed in detail in

Section 5.3.

The vibratory driving equipment described so far generates the simple

harmonic motion by the mechanism of rotary eccentrics (Fig. 5.1). A more

recent and different vibrator is the Christiani-Shand hydraulic vibrator, as•reported in the Ground Engineering magazine (1972). SuchAvibrator works on

the principle of linear motion and hence it is possible to introduce

frequency variation over an extremely wide range (la - 120 Hz.) Pile

driving trial$ using this vibrator have been reported by Pearson (1974).

5.2 Comparison between Conventional Impact Pile Driving and Vibratory

Pile Driving

The conventional impact type of pile driving technique can be compared

with vibratory pile driving in the following aspects:-

(a) Form of Force Input

With regard to the conventional impact type of pile driving, the ram

may simply be raised by a crane and dropped onto the pile head. The lifting

can be achieved by using steam pressure, air pressure, hydraulic pressure or

combustion of diesel fuel. The standard of merit depends on the blow rate

and the amount of energy delivered to the pile per blow (or 'enthru' energy

after Housel, 1965).

In general impact hammers have blow rates varying from 40 - 200 blows

per minute. At these blow rates the pile motion due to the preceeding blow

will have ceased before the next impact is encountered (Fig. 1.1).

On the other hand, in vibratory driving a continuous sinusoidal

oscillating force acts in conjunction with a static force (bias) (Fig. 1.2).

58

The magnitude of the force amplitude is relatively smaller than the peak in

impact driving, but the continuous transmission of energy is adequate to

cause much more rapid penetration of piles.

(b) Consumption of Energy

The impact pile driver delivers a discrete, finite amount of energy per

blow. On the contrary, vibratory drivers are variable-energy devices that

coonsume whatever power is required in order to maintain the prescribed

oscillatory mode. Sieffert & Levacher (1982) compared the amount of

transmitted energy required to obtain a given pile penetration in dry sand

between impact pile driving, vibratory driving and static loading. They

concluded that vibratory driving requires the least amount of transmitted

energy, while impact driving requires most (Fig. 5.2).

(c) Mechanism of Penetration

In impact driving it is usually the dynamic imp~lse rather than the

relatively small hammer weight that causes pile penetration. As for

vibratory driving the mechanisms of penetration are still under discussion,

but are thought to be due to one of the two reasons:-

(i) change in soil properties upon cyclic loading, for example

fluidisation in loose sand;

(ii) resonance of pile or pile-soil system causing plastic deformation,

which is possible in dense sand and clay.

In (i), the presence of the relatively small static bias is important to

cause the pile to sink. In (ii), it is the oscillatory motion that causes

large displacement and plastic deformation as a result of resonance. This

will be discussed in greater detail in Section 5.3.

(d) Load-Settlement Behaviour

Hunter & Davisson (1969) concluded for piles in sand that there is less

densification below the pile tip for vibratory driving than by the more

conventional impact type of driving. From load transfer tests conducted

they also found that the tip capacity for the conventionally impact driven

59

pipe pile is I! times the tip capacity for an equivalent vibrodriven pile.

The tests conducted by Smart (1969) also arrived at about the same proportion

for tip capacity, but found that the overall pile capacity is more or less

the same for both types of driven piles, owing to a higher skin friction

developed at the end of vibratory driving. Smart (1969) also found that

vibratory driving tends to cause less settlement for a given loading than

impact driving. Furthermore, field observations by Paunescu & Mateescu (1971)

have led to findings that vibrodriven piles are 20% - 50% higher in load-

carrying capacity than equivalent impact driven piles, depending on the

initial compaction of the sand.

Nevertheless, while the load-carrying capacity of impact driven piles

is relatively easy to be controlled, the capacity of vibratory driven ones

is rather uncertain. While local ground improvement can be achieved by

vibratory driving in loose sands, weakening of the soil skeleton may result

for the same operation in dense sands.

As for cohesive soils, the amount of data on vibratory driving operations

is relatively limited. Smart (1969) states that there is some evidence that

vibratory driving leads to better shaft adhesion and thus more superior pile

capacity than impact driving. However, in the case of floating piles

penetrating saturated clays, Paunescu & Mateescu (1971) found that the

vibratory technique engenders a decrease in the load-carrying capacity of

piles, and is thus not recommended.

On the whole, until the frontier of knowledge in soil dynamics is

further advanced, it is understandable that engineers remain sceptical to

the performance of vibratory driven piles.

60

5.3 The Principle of Vibratory Pile Driving

5.3.1 Introduction

The basic phenomenal evidence of vibratory pile driving has been

summarised by Blekhman (1954):-

(a) 'In the presence of vibrations, a relatively small weight of the

unit becomes adequate to assure the rapid penetration of piles

into the soil to a considerable depth.

(b) Within fixed parameters for the pile, soil and vibrator, the

driven depth is limited.

(c) The driving of a pile is possible only if the pressure on the

soil (in the simplest case - the weight of the installation)

exceeds a certain value; with a further increase in pressure,

the rate at first increases sharply but then remains practically

constant.

(d) With sufficiently small amplitudes of vibration, the penetration

of piles does not occur, regardless of the magnitude of the

acceleration amplitude.

(e) In some cases the phenomenon of resonance is clearly observed;

the character of the resonance curve is typical of nonlinear

systems.'

Although vibratory driving has been practised and studied for some

time now, there is still no unified and comprehensive theory which can

adequately explain the mechanism(s) of penetration, and to provide guidelines

on the optimum values of various parameters (operating frequency, dynamic

force amplitude, etc.) to be used in different conditions. On the academic

front, proposed solutions (Table 5.1) are often derived from limited

variations of pile-soil conditions, and expressed in terms of all forms of

dimensionless parameters. The validity of extrapolating any solution to

61

an untried situation is dubious. On the industrial front, the technique,

though powerful at times, is often discredited for its lack of reliability.

Operations in the sites are generally performed in a trial-and-error fashion,

and seldom applied in cohesive soils. With the hire charges of vibratory

hammers much higher than conventional hammers, more knowledge on vibratory

pile driving must be gathered before the technique can become widely

accepted.

5.3.2 Mechanisms of Penetration

It is believed that piles penetrate upon vibratory driving because of

the following mechanisms:-

(i) penetration by modification of soil properties; and/or

(ii) penetration by resonance.

Both mechanisms are supported by numerous literatures (Table 5.1). Whichever

mechanism is more likely to occur (or even a combination of both) depends on

the vibration parameters as well as the type of soil (Table 5.2). These are

discussed in more detail below.

5.2.3.1 Penetration by Modification of Soil Properties

Studies in Sand

Under favourable conditions (e.g. light piles in loose sands, or low

end-bearing piles in dense sands), pile penetration may occur due to soil

fluidisation.Slade (1953) defined three states of soil response:-

62

(i) Sub-threshold (Elastic Response) state.

In this state, the typical acceleration is less than 0.6 g. The

excitation energy in this state is low, so that variations in the particulate

configuration of the soil are periodic. Normally this implies the existence

of dynamic stability, unless the soil density is less than a certain critical

value when compaction will occur. Rodger & Littlejohn (1980) claimed that

the shear strength of soils has not been found to decrease by more than 5%

in this state.

(ii) Trans-threshold (Compaction Response) state.

The applied acceleration is typically 0.7 g - 1.5 g, and the soil tends

to undergo irrecoverable local changes without resulting in drastic changes

in the statistical characteristics of the soil. Rodger & Littlejohn (1980)

remarked that the decrease in shear strength of the soil is governed by an

exponential function of the acceleration, the parameters of which depends on

the grain size and shape, as well as the overburden pressure.

(iii) Fluidisation (Liquefaction) state.

In theory, fluidisation can occur when the acceleration is greater than

1 g. However, in practice the acceleration amplitude has to be around 1.5 g

before the excitation energy is high enough for the soil medium to change its

statistical characteristics and acquire the properties of a viscous fluid and

thereby allowing the pile to sink.

The displacement amplitude is also important. Winterkorn (1953)

postulated that the threshold value of displacement amplitude to cause

fluidisation should be of the order of magnitude of the sand particles. This

has been confirmed by the vibratory driving tests on a model pile performed

by Littlejohn et al (1974). In loose sands, the fluidised zone is generally

believed to extend beneath the pile, whereas in dense sands any fluidisation

must be confined to the vicinity of the pile shaft.

Effect of Frequency

Barkan (1957) reckoned that frequency variation has practically no

63

effect on the penetration speed, and hypothesised that end-bearing resistance

increases with frequency. Consequently the idea of high frequency vibratory

driving is discouraged. However, using rotary eccentric machines (such that

dynamic force 0< w2) Littlejohn et al (1974) demonstrated a linear relationship

between frequency and penetration velocity (Fig. 5.3). In the figure a

frequency threshold is evident.

Effect of Dynamic Force

The magnitude of the dynamic force is responsible for the amplitude of

vibration. Both Barkan (1957) and Littlejohn et al (1974) found a linear

relationship between the dynamic force amplitude and penetration velocity

(Fig. 5.4). The corresponding relationship between displacement amplitude

and penetration velocity exhibits a threshold amplitude of 0.3 mm. (Fig. 5.5),

which is close to the mean grain diameter of the sand used in the experiment.

Effect of Static Surcharge (Bias)

The effect of fluidisation is to reduce the soil resistance against

penetration. The experience of Barkan (1962b) indicated that penetration

occurs when the static surcharge is greater than a threshold value (Fig. 5.6).

On the other hand, the model pile of Littlejohn et al (1974) penetrated under

its own weight upon vibration (Fig. 5.7).

Judging from the experimental data mentioned above, fluidisation can be

justified to constitute one of the possible penetration mechanisms in sandy

soils. However, fluidisation never occurs in clays. Other important

studies on fluidisation include Wilheim & Valentine (1951), Winterkorn (1953),

Barkan (1962a, 1965), and Lee & Seed (1967).

Studies in Clay

Although successful reports of vibratory pile driving in clays are

numerous (e.g. Tsaplin, 1953; Barkan, 1957; Mao, 1958; Pearson, 1974;

and Byles, 1981), the performance tends to be inconsistent and unpredictable.

The high hire charges of the plant and the relatively low rate of success

64

further inhibits the accumulation of knowledge through field experimentation.

Arguments for pile penetration in cohesive soils by modification of

soil properties include:-

(i) If sufficient moisture content is present, thixotropic transformation

can occur.

(ii) From a series of unconfined uniaxial vibratory compression element

tests, Kondner (1962) concluded that:

(a) The strength of clay under vibratory loading is considerably less

(by 50% - 80%) than under static loading;

(b) The elastic dynamic response of the material is nonlinear, and is

a function of the stress level (Fig. 3.2); and

(c) The response of the material is frequency dependent.

5.3.2.2 Penetration by Resonance

In sites underlain by clay or dense sands, the theories of reduction in

shaft resistance and fluidisation tend to fall short of a true account of

pile penetration upon vibration. Many classic publications (Table 5.1)

advocate the phenomenon of resonance, at which the amplified motion is large

enough to overcome the elastoplastic and viscous components of soil resistance.

It is believed that the low frequency (5 - 60 Hz.) Russian version and

the high frequency (over 100 Hz.) American version of vibrators operate to

different characteristics:-

(a) Low Frequency Vibrators (5 - 60 Hz.)

(i) Severe oscillations are induced in the soil mass rather than in the

pile;

(ii) Using vibrators with rotary eccentric mechanism the response curve has

a broad crest over the resonance range, so that sharp tuning is not

required;

(iii)The pile vibrates more or less as a rigid body. No nodes (i.e.

stationary points in vibration) appear along the pile, but one node

65

may exist beneath the pile within the soil.

(b) High Frequency Vibrators (over 100 Hz.)

(i) The vibrator induces severe oscillations in the pile, around its

natural frequency corresponding to longitudinal vibration;

(ii) Using rotary eccentric machines the response curve has a narrow peak in

the critical range, so that sharp tuning is required;

(iii)The pil~ vibrates with different amplitudes along its length;

(iv) Several harmonics may exist in the pile at the same time.

As a result of (b)(i), a number of formulae have been put forward

attempting to predict the frequency corresponding to high frequency vibratory

pile driving:-

(i) Based on vibrodriving tests of model piles Bernhard (1968) suggested the

expression

f - j(E/p)/(3,) (5.1)where, is the length of the pile.

The expression neglects the influence of the soil, and has been found to

underestimate the true resonant frequency consistently.

(ii) Kovacs & Michitti (1970) established another expression assuming that a

soil mass equal to 1/3 of the pile mass actually contributes to the

vibration. While this effective soil mass is lumped at the pile tip, an

additional pile length and an empirical constant is incorporated to

compensate the effect of vibrator attachments. However, the damping and

stiffness effects of the soil are still not taken into account.

(iii)An attempt to take into account the full soil effects is presented by

Satter (1976):

w 21 • K + EA(~/I)Z(I/2)

M1 + M2 + (pAI/Z)(5.2)

where K is the soil stiffness, and

Ml, MZ are the concentrated masses at the top and tip of the pile

respectively.

66

However, Satter (1976) chose to ignore the soil stiffness term in his

numerical illustration.

(iv) Smart et al (1977) later observed discrepancies in Satter's calculations

and suggested a different expression assuming the pile tip to be free:

w /W = wc cot(wc)p m (5.3)such that wp = pile weight,

wm = weight of lumped mass, and

c = JE/p = velocity of sound in the pile.

Smart et al (1977) also remarked that the contribution of the soil

stiffness is more important than the contribution of its inertia in

most cases of high frequency vibratory driving.

(v) A more sophisticated analytical expression is given by Srinivasulu et

al (1978), with the possibility of accounting for full soil effects.

The resonant frequency, the amplitude of the pile tip and the position

of any node can be determined. As before, material linearity is

inherently assumed.

Studies in Sand

Effect of Frequency

Nassar (1965) vibrated a 19mm (3/4") brass model pile through a wide

range of frequencies. The force input at the pile top and the force output

at the pile tip were measured. It was found that maximum penetration rate,

which corresponded to maximum force input/output amplitudes, occured at

specific resonance frequencies (Fig. 5.8). Furthermore, by attaching an

acelerometer at different positions on the pile, the shape of the standing

wave can be determined. It can be seen from Fig. 5.9 that the pile vibrates

more or less as a rigid body at the first resonant frequency. At the second

resonant frequency, the A/I value corresponds well with Bernhard's (1968)

prediction (equation 5.1).

The above results were found to be insenstive to the depth of penetration,

67

suggesting that both fluidisation and resonance apparently occur at the same

time.

Effect of Dynamic Force

By varying only the dynamic force input, Ghahramani (1967) categorised

the force response into three domains (Fig. 5.10):

(i) Sinusoidal Resistance Domain: When the dynamic force is small and the

soil remains elastic, no permanant deformation will occur, and the

force input and output are in phase with each other (Fig. s.lOa). The

limiting force that allows such linear response is termed 'resistance

threshold'.

(ii) Impact Domain: When the dynamic input force is further increased to a

value greater than the total static weight of the pile assembly, the

pile tip will tend to separate from the soil during a portion of the

load cycle. As a consequence, the response pattern degenerates from a

sinusoid to one with discrete spikes, signifying impact when the pile

tip makes contact with the soil again after separation (Fig. S.lOb-d).

Separation between the pile tip and the soil can be confirmed by a null

tip resistance measured.

(iii)Phase Instablity Domain: The magnitude of the impact spike increases

with the dynamic input force only up to a certain extent. Further

increase in the input sinusoid will result in an increase in the

kinetic energy of the pile, but the amount of soil resistance mobilised

(SRD) will not be increased. This point is known as the 'impact

threshold'. However. at this point the phase between force input and

output begins to increase (Fig. S.10e). first abruptly and then steadies

off as the dynamic force is further increased (Fig. 5.11). The impact

threshold, representing the point when maximum soil resistance can be

mobilised, is thus the optimum dynamic input force for vibratory driving.

68

Effect of Static Surcharge (Bias)

On further experimentation, Ghahramani (1967) found that the impact

threshold value varies linearly with the static surcharge (Fig. 5.12). Heavy

piles thus act as better energy transmitters and penetrate faster than light

piles provided that the power supplied to the vibrator is adequate.

Studies in Clay

Although vibratory driving is less popular in cohesive soils, studies

(Bernhard, 1968; Nassar, 1965; and Guyton, 1968) have indicated directly

or indirectly its feaibility. However, only limited results have been

reported and parametric analysis is scarce. In view of this, the finite

element model described in Chapters 2 and 3 seems to be suitable for studying

the performance of vibratory driving in cohesive soils.

5.4 Finite Element Simulation of Vibratory Driving in Cohesive Soils

So far the majority of convincing evidence of pile penetration as a

result of vibratory driving occur in sands. Postulated mechanisms of

fluidisation and resonance have been supported by experimental facts.

However, clays are also known to lose strength under cyclic loading, and

will obviously resonate at the appropriate frequencies. Thus, apart from

the higher viscous damping which is unfavourable to penetration, vibratory

driving should also be theoretically feasible in cohesive soils to some

extent.

A convenient and relatively cheap method of investigation is the finite

element approach. The algorithm developed in Section 3.4 will be used herein

to examine the vibratory response of the pile-soil system, first for an

elastic case, and then an elastoplastic analysis.

5.4.1 Elastic Analysis

In order to assess the performance of the pile-soil finite element

representaion in dynamic response analysis, the resonance experiment performed

69

by Novak & Grigg (1976) is modelled as a benchmark problem. The experiment

consisted of a steel pipe pile of 889 mm. in diameter installed in a 2 m.

layer of fine silty sand, and with the tip bearing on a bed of gravel and

till. The pile top is firmly connected to a surface footing/oscillator

arrangement and the oscillating force produced by the latter is proportional

to the square of the frequency (i.e. rotary eccentric mechanism). The

properties of the pile-soil system are as shown in Fig. 5.13.

In formulating the finite element model, the following assumptions have

to be made:-

(i) the soil medium is assumed to be linearly elastic, due to the small

displacements involved;

(ii) the surface footing/oscillator arrangement is idealised as two concentric

rings having the same diameter and wall thickness as the pile. In order

to preserve their masses in the simulation the densities are augmented

accordingly;

(iii)since no information about the internal soil column is available, it is

assumed that it does not exist. The effect of the internal soil column

on dynamic response will be assessed in the elastoplastic analysis in

Section 5.4.2;

(iv) the shear modulus of the interface element is assumed to be 0.5% of

that of the soil;

(v) for the sake of economy the mesh is so discretised that ~x = As/4 and

~y • Ap/4 respectively;

(vi) complete absorption of impinging energy is assumed at the lateral

standard viscous boundaries;

(vii)in the continuum model solutions presented by Novak & Grigg (1976) the

fixity of the pile tip is assumed. In the finite element analysis

presented herein no such assumption on pile fixity is required, but the

physical properties of the gravel and till stratum is assumed as stated

in Fig. 5.13.

70

The usual spatial discretisation calculations are presented in Table

5.2. A suitable frequency range to be considered is 12 - 60 Hz. Since the

element size requirements are different for the three soil strata by virtue

of their difference in physical properties, a terraced lateral boundary is

employed such that its distance from the excitation source is not less than

half of the shear wavelength at each layer, in order to enable the plane

strain formulation of the standard viscous boundary to perform satisfactorily

in axisymmetry (fig. 5.14). The extent of the 'gravel and till' stratum is

not known, but in the finite element representation vertical fixity by means

of rollers must be placed at the base of the mesh, which is to be kept as

remote as possible. As in Section 4.4, the lack of geometric damping at the

mesh base will give rise to stray wave reflections which will mask the pile

response in due time. The return time of the stray reflections is especially

critical for low frequency oscillations as less cycles will be completed for

a given return time. In other words, when wave reflections occur at

artificial truncations, the lowest frequency of the range governs the overall

mesh size required, which has to be compromised with the limiting element

size governed by the highest frequency of interest.

In the present case, the limiting size of soil elements in the vertical

direction is given by

~y - Vp / (4 x 60)

If there exist n rows of elements beneath the pile tip, then the return time

RT of the reflected wave to the pile tip is

RT - 2n~y / Vp (5.5)

For any frequency f, the number of load cycles LC free from stray wave

reflections is

Le • 2nf~y / vp (5.6)In the present problem, 8 rows of elements are installed beneath the pile tip.

Substituting n - 8 and equation (5.4) into equation (5.6), LC can be found

equal to 0.8 for f - 12 Hz., and 4 for £ = 60 Hz.

71

Again, since no definite conclusions can be drawn from the eigenmodes

of the system (e.g. Figs. 5.15 - 5.17), computation of transient responses

at discrete frequencies are carried out. The eccentric moment adopted

corresponds to the middle curve in Fig. 5.23, namely 1.708 lb-in. All the

natural frequencies within the range are covered, and otherwise forced

oscillations are performed at 3 Hz. intervals. As in Section 4.3, the time

step size used is 1/20 of the period. This has been checked as adequate,

with results similar to Fig. 4.6, despite the non-homogenity of the system.

The initial conditions are x = 0, x = 0 and x = o.Typical transient responses computed are exhibited in Figs. 5.18 - 5.21.

The peak response at various cycles over the frequency spectrum considered is

shown in Fig. 5.22. The dotted portions of each curve indicate the possibility

of stray effects as a result of wave reflections from the artificial base

boundary. Maximum response is found to occur at around 45.4 Hz., which is

within the range obtained by Novak & Grigg (1976) (Fig. 5.23). However,

direct comparison of response magnitude is not possible because the Novak &Grigg solution represents response at steady state, which has not been

reached in the finite element results. By further computing the steady

response (Fig. 5.24) the dynamic magnification factors can be determined over

the spectrum (Fig. S.2S). Maximum amplification can be seen to correlate

with the fifth eigenvalue, namely at 44.2 Hz.

It can be concluded that although resonance will occur at an

eigensolution, it is not possible to predict which eigensolution is the

one of interest. The only value of eigensolution analysis seems to be

giving an idea on the frequency range at which forced oscillation analysis

should be performed.

5.4.2 Elastoplastic Analysis

The elastic analysis above is generally satisfactory for small

amplitude vibration analYSis, for example of machine foundations. However,

· .

On hindsight, Mesh A can be seen to yield an unrealisticallyhigh static pile capacity. This is due to the coarseness of themesh around the pile tip, so that the onset of bearing failure isnot properly captured. When Mesh A is replaced by a graded mesh(more refined around the pile tip) with the same overall size,the static pile capacity has been found to be the same as thatyielded using Mesh B, although a stiffer load-deformation responsewas observed due to the mesh coarseness. The subsequent dynamicanalysis involving the 'coarse' Mesh A (i.e. Figs. 5.28-5.34,5.39-5.41, 5.49-5.55) are consistent with the overestimated staticpile capacity, so that maximum load levels up to 1330 kN. (97.4%of static pile capacity) has been sustained, as shown in Fig. 5.51.Although the qualitative conclusions are un~ffected, more realisticresults would be obtained by repeating these analyses using agraded mesh with the same limiting element dimensions but suitablymodified values of the static and dynamic forces.

72

when the load level is more substantial as in the case of vibratory pile driving,

extension to elastoplastic analysis is required. A nominal pile-soil system

is employed herein to study vibratory driving in undrained saturated clay

(von-Mises material). The occurrence of cyclic degradation in clays is

relatively unpredictable and difficult to model numerically, and is ignored

here. In other words, any penetration of the pile must occur through

overcoming the shear strength of the soil, as in the case of impact driving.

The vibrator data is based on the linear motion electro-hydraulic

vibrator, as reported by Pearson (1974). The data of the complete set-up

is given in Table 5.4. However, the use of numerical approach allows

experimentation on frequencies beyond the actual operating frequency range

of the machine. In the present example, it is considered desirable to cover

a frequency range from 3.5 to 220 Hz. in order to explore both high and low

frequency vibrodriving. As in Section 5.4.1, it is not possible to model

radiation damping at the mesh base. Following equation (5.4),

~y = Vp / (4 x 220),

and from equation (5.6),2n (3.5) Vp / (4 x 220 x Vp)

for a minimum frequency of 3.5 Hz. Hence in order to achieve one cycle

LC z

without the influence of stray wave reflection at this frequency, the value

n must not be less than 1261 It is thus obviously not practical to cover the

analysis within the proposed frequency range with a single mesh size.

Rather, three meshes of different overall sizes (named A, B and C) are

employed (Fig. 5.26), with details as summarised in Table 5.5. The static

~ile tip displacements for the three meshes are plotted in Fig. 5.27.As the role of viscous damping in the soil and at the pile-soil

interface is only to diminish the severity of response, it is ignored in the

foregoing computations of this section for the sake of clarity and simplicity.

In the light of previous experience indicating that only limited information

can be retrieved from eigensolutions, the determination of them is dispensed

73

with herein.

Results and Interpretations

Forced oscillations are performed with the three meshes over their

designated frequency range, initially at 5 Hz. intervals and then at 2.5 Hz.

intervals in ranges where amplification is detected. The peak displacement

responses at individual frequencies over the considered spectrum are shown

in Fig. 5.28, and their corresponding amplitudes shown in Fig. 5.29. The

results are not shown in terms of DMF as before, because the maximum load

applied in this case exceeds the static pile capacity in meshes Band e, as

seen from Fig. 5.27. The maximum soil resistance mobilised during driving

(SRD) also follows a similar trend, and is presented in Fig. 5.30. From

these figures two resonant frequencies can be identified. The first of

these occurs in Mesh A at around 18.5 Hz. (Figs. 5.31 - 5.34) and also in

Mesh B at around 12.5 Hz. (Figs. 5.35 - 5.3%). The difference in resonance

position reflects the differences in mesh characteristics and the level of

nonlinearity encountered. While the resonant response of Mesh A exhibits

little nonlinearity and hence achieves only limited penetration, plasticity

is evident in Mesh B as reflected by the flattened SRD peaks in Fig. 5.35,

and some pile penetration is recorded (Fig. 5.36).

At this first resonance the responses of the pile top and the pile tip

are in concert with each other. The minor difference between them indicates

the elasticity of the pile. Profiles of longitudinal displacement (Fig. 5.39)

and acceleration (Fig. 5.40) along the pile confirm that the pile tends to

respond as a rigid body and suffers only minimal deformation at this resonant

frequency, similar to the observation by Nassar (1965) (Fig. 5.9). Regarding

the response in the soil, Fig. 5.41 shows that the soil in the lateral

vicinity of the pile displaces much more than the soil at depth. Attenuation

of response with distance is apparently slight, and this can be undesirable

for construction in tight areas (see Section 5.5).

74

Apart from the fundamental pile-soil frequency just described, a second

harmonic can be detected in Mesh C at around 210 Hz. (Figs. 5.29 - 5.30).

Although the maximum load level far exceeds the ultimate static capacity of

the pile, the dynamic amplitude obtained is very much diminished compared

with the first peak. This is characteristic for vibrations above the

fundamental frequency under a constant load, which is different from that

obtained for an eccentric load mechanism as in Fig. 5.8.

The response of the system at this second harmonic as shown in Figs.

5.42 - 5.45 is rather intriguing, because instead of penetration the pile

seems to lift itself up from the equilibrium position. In fact, it is

likely that such momentary lift-up of the pile is the reason behind the

impact mechanism observed by Ghahramani (1967) in Fig. 5.10. The impact

spike is the consequence of the rebonding after the pile tip has been

lifted up and separated from the soil for a part of each cycle. However,

such impact mechanism is beyond the modelling ability of the simple

interface element employed here, which does not simulate separation. Of

course, should the analysis be performed in a reasonably loose sand

fluidisation may also occur in conjunction with the impact mechanism. Such

analysis definitely requires a soil model with a much more sophisticated

constitutive relationship.

Another interesting feature apparent in the response curves at this

second harmonic is that the response at the pile top and the pile tip are

exactly out of phase with each other. If the displacement (Fig. 5.46) and

acceleration (Fig. 5.47) along the length of the pile is plotted at a number

of instances, it can be seen that a node tends to occur very near the pile

top. This is in contrast with Nassar's (1965) result (Fig. 5.9) in which

the node occurs at about 1/3 of the pile (and hammer) length from the tip.

This may be due to the fact that Nassar's observation is made in a

fluidising soil in which the restraint at the pile shaft is minimal, whereas

in the present case a non-fluidising, elastoplastic soil is considered,

75

so that the severe restraint on the pile from the mudline downwards pushes

the node above the mud line.

Fig. 5.48 shows a typical displacement plot of the system vibrating at

the second harmonic. Again the vibration attenuates more rapidly with depth

than with lateral distance. However, if fluidisation and pile-soil interface

damping are present, lateral attenuation will be much more severe than

shown.

Summary

From the analysis above it can be seen that a pile will penetrate on

vibration:-

(a) at the fundamental frequency of the pile-soil system (when the

predominant oscillations are within the soil mass) if the stress level

is high enough to cause cumulative elastoplastic deformation; or

(b) at the second harmonic (when the predominant oscillations are due to

longitudinal deformation of the pile) 'possibly' if

(i) the dynamic force is substantial enough to cause separation between

the pile tip and the soil mass during part of a cycle, and

subsequently rebonding with an impact stress level high enough to

cause elastoplastic deformation; and/or

(ii) the dynamic force is large enough to cause the soil to fluidise,

subject to the conditions described in Section 5.3.2.1.

However, these two penetration conditions at the second harmonic are

only qualitative inferences and have not been numerically verified in

the present work.

5.4.3 Parametric Studies

Since the pile-soil model employed herein has been found to be unable

to account satisfactorily for vibratory penetration above the fundamental

frequency in a quantitative manner, further parametric studies using the

present model will be sensible and reliable if they are confined to analyses

76

around the fundamental frequency of the system. This is to be performed in

Sections 5.4.3.1 - 5.4.3.4. Nevertheless, since the position of the second

harmonic has been claimed to be influenced mainly by the pile and only weakly

by the soil, the position of resonance in different soils can be compared in

the hope of obtaining at least qualitative conclusions. This is also studied

in Section 5.4.3.4.

5.4.3.1 Effect of Static Surcharge

The role of the static surcharge is to determine the initial level of

the applied load. The displacement response in Fig. 5.32 corresponds to

Mesh A being subjected to a static surcharge of 26.9775 kN. and a dynamic

force of 529.74 kN~ Only a small amount of net penetration can be obtained.

However, as the static surcharge is raised, the net penetration as computed

from the same mesh is also found to increase, and a trend resembling that in

Fig. 5.36 results. The maximum pile tip responses for various static

surcharge magnitudes are shown in Fig. 5.49. It can be seen that the response

tend to shift to the lower frequencies as the level of nonlinearity increases.

Some measure of the penetration rate can be obtained by considering the

maximum pile tip displacement after three (response) cycles. Fig. 5.50

shows that the peak displacement increases more or less linearly with static

surcharge for the early part of the curve, resembling Fig. 5.7, and then

turns more steeply at high surcharge levels, resembling Fig. 5.6. Apparently

whether the graph intercepts the x-axis or the y-axis depends mainly on the

degree of nonlinearity caused by the dynamic force.

5.4.3.2 Effect of Dynamic Force

The role of the dynamic force is to produce oscillations about a certain

mean level which somewhat depends on the static surcharge. Fig. 5.51 shows

that as the magnitude of the dynamic force is increased, the resonance peak

shifts to a frequency substantially lower than that corresponding to an equal

increase in static surcharge. However, by comparing Fig. 5.52 with Fig. 5.50

77

it can be seen that the peak response is dependent on the total load level

rather than on the relative proportion of static and dynamic forces.

?4.3.3 Effect of Internal Soil Column

The presence of the internal soil column in open-ended pile complicates

the pattern of soil resistance mobilisation. While under static loading the

internal shaft resistance is always in static equilibrium with the bearing

pressure, in dynamic conditions the soil column tends to plug only

intermittently depending on the instantaneous relative magnitudes of the two

soil resistance components aforementioned. A full discussion is presented

in Section 6.6.

Fig. 5.53 - 5.54 shows the effect of the internal soil column on the

vibratory response for Mesh A. On the whole, the effect of the internal soil

column is relatively small, but it can be seen that a long internal soil

column is actually advantageous to vibratory pile driving (at the fundamental

frequency). This is possibly due to the amplification effect of the inertia

of the soil column, which dominates over the relatively slight restraining

effect of the higher internal shaft resistance (see Section 6.6). This

trend is in complete contrast to the behaviour in impact driving, as depicted

in Fig. 6.30.

5.4.3.4 Effect of Soil Properties

If a pile penetrates by causing fluidisation in the soil, the influence

of the soil on penetraion will only be minimal, as has been observed by

Schmid (1969). However, if pile penetration occurs by overcoming the

elastoplastic soil resistance at resonance, then the properties of the soil

are likely to exert some influence on the position and magnitude of the

resonant response.

In order to investigate this, frictionless soils with Cu = 50, 100 and

200 kN/m2 are considered. The E/cu ratio is kept as 50rr, and interface

damping is, again, assumed to be always zero. Since the critical element

78

size corresponds to the weakest soil (assuming the same spatial discretisation

for the analysis of all three soil types), meshes have to be re-designed in

a similar manner as before. The resulting mesh employed in determining the

responses around the fundamental frequencies of the three soils has a

limiting element width and height of 0.58 m. and 2.96 m. respectively. On

the other hand, the mesh used for analysis around the second harmonics has

limiting element dimensions of only 0.0625 m. by 0.3186 m.

The results in Figs. 5.55 - 5.56 show that for a pile penetrating a non-

fluidising soil the influence of the soil is very significant at both first

and second harmonics. In general, the stronger, stiffer soil resonates at a

comparatively high frequency, while more penetration may be achieved in the

weaker, softer soil at a lower resonant frequency.

In the case of a fluidising soil, the frequency of the second harmonic

can be estimated by the various formulae proposed as summarised in Section

5.3.2.2(b). Using the simplest expression of Bernhard (1968), the second

harmonic of the present pile, irrespective of the soil effects, is given by

f - j(E/P)/(3&) • 156 or 190 Hz., depending on whether & includes the hammer

or not. It can be seen that with the influence of the non-fluidising soil

as in the present case, a stiffer response generally results and the second

harmonic tends to occur at a higher frequency.

5.5 Environmental Impact of Vibratory Pile Driving

The vibratory pile driver often operates at the resonant frequencies of

pile-soil systems, and it may be worrying that excessive ground vibrations

may result in detrimental effects to nearby structures.

Numerous reports (DOE/CIRIA, 1980; Palmer, 1982) exist blaming

vibratory driving operations for causing undue nuisance, but on the other

hand there are case histories in which the high frequency vibratory driver

has been used successfully in preferance to the conventional impact hammers

in order to cope with stringent restrictions on noise and vibrations

79

(Reseigh, 1962; Green Mountaineer, 1963; Davisson, 1971). Bernhard (1968)

has contended that there are reasons to be confident in minimising the

possibility of causing excessive vibrations or nuisance to the neighbourhood:

(i) Frequencies of over 100 Hz. are much higher than most natural frequencies

of buildings and soils;

(ii) Waves propagating at high frequencies suffer more damping than those at

low frequencies;

(iii)The displacement amplitudes required to generate the same power are

smaller at high than at low frequencies.

With regard to the analysis performed in Section 5.4.2, attenuation

patterns can be determined for vibratory driving at both the first and

second harmonics. Typical results are shown in Figs. 5.57 - 5.58. It can

be seen that without fluidisation high frequency vibration does not seem to

attenuate any more rapidly than low frequency waves. Comparing with Fig. 6.19

the levels of soil vibration due to impact and vibratory driving seem to be of

the same order of magnitude. On the other hand, if fluidisation occurs (in

sandy soils due to vibration) damping is likely to increase causing rapid

attenuation. It is useful to confirm this by further research using a more

sophisticated soil model.

In conclusion, the present sceptical attitude of the authorities

(DOE/CIRIA, 1980) towards vibratory driving is justified, because of the

relatively high degree of uncertainties involved, and the lack of an

established track record. The practice of vibratory pile driving in densely

populated areas is not recommended. Useful guidelines have been suggested

by O'Neill (1971) and Littlejohn et al (1974) on the acceptable vibration

levels regarding human sensitivity (nuisance), risk of damage to structures,

interference to fine-tolerance machines and dynamic ground settlement. Some

alleviating procedures have also been proposed.

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"" Q) CIl 13Q) Q) e I CII"I Cl.

"'"' +oJ <1.1 Cl. ..-1 • QlN.... = "'"' 00 ~e <1.1Ajl., H ~ ~- ~-

........"'"'..;t

LI"\

=0....+oJc.JQICIl-13<1.1

"'"'.J:J0

""jl.,=0....+oJCIl

"'"'"'"'.~t)ell0Q)

"'"'....jl.,

t).~+oJellCIl

"'"'~""0

N .....::x=

ell0 =1.0 0.~ell +oJco co

""~ Q)ell "Cl<1.1 ...."" ellQj =+oJ 0

= U....=..... 00

0 ....ell

>-. QIc.J ~= ..c::<1.1=' ell0" Q)Qj :E,........

~M.LI'\.... ~>clco ~

13 ~<1.1 E-tod~

~Cl)

~ole

Vibrator DataDynamic Force = 54 t. (529.7 kN.)Operating Frequency Range = 10 - 240 Hz.Static Weight = 2.75 - 6.5 t.Assumed Dimensions : 2 m. high, with same cross-section

as pile

Pile DataMaterial : SteelOuter Radius = 0.2 m.Wall Thickness - 0.02 m.Total Length - 9 m.Initial Penetration = 5 m.Level of Internal Soil Column - 3 m. above pile tip

Soil DataC - 100 kN/m2u

0(.: CM!

t/J,I/J = 0

E/C - 500uPoisson ratio • 0.48Density 3- 2.0 tIm

J , J - 0s p

TABLE 5.4 Hammer, Pile and Soil Data for VibratoryDriving Analysis (Section 5.4.2)

Shear Wave Vel. Poisson Ratio Comp. Wave Vel.V (m/sec) V (m/sec)s p

Pile 3189 0.3 5966Interface Element 6.5 - -Soil 91.9 0.48 468.6

Mesh A Mesh B Mesh C

Min. Freq. (Hz.) 3.5 10.0 110.0

Max. Freq. (Hz.) 28.0 120.0 220.0

.1x = V /4f (m) 0.82 0.191 0.104s max

.1y = V /4f (m) 4.184 0.976 0.5325p maxNo. of columns 12 12 12of elementsNo. of rows 24 24 23of elementsOverall Width (m) 6.76 1.728 1.032of meshOverall Depth (m) 77.944 26.616 16.325of mesh

TABLE 5.5 Mesh Design Considerations for Vibratory PileDriving Problem (Section 5.4.2).

-~c "-"0Cu-o0.-eg.o :2o~OCul>~u..->0

--,.-IIIII

--,-- --,.- -,-- -,-- -.--I Position of Excentric Masses II ' I I

I : I II I II I II I I

-r--r-I II II I

III

Angle of Rotation

Operating PrincipleTwo disks rotate in opposite directions with the disturbing frequency, each bearing an excentric mass. In the upper

part 01 the figure, the parr 01 disks is to be seen lor several positions 01 one revolution. The resultant 01 the centrifugallorces is always directed vertically. It. size is plotted axainst the an,le 01 rotation. One gets a single harmonic disturbingIorce, the amplitude growing with the square 01 the frequency.

Engine

Mudline

Engine

Air spring

FIG. 5.1

pile resistanceShaft

point resistance

Vibratory pile driver with rotary eccentric mechanism.

II

400 I/I

V/

,..; I"~: ,. "~ li;j /'. l,t:

200 fl~:·1 /Pi 1. model /Length- 1.76 m ~/• Ext- 3S.3 mm ~Area • 518 mm"

lOO ,/,/

~,/.;'

500 Energy (J)

0.4 0.6 Emdeddeddepth (m

I Driving: hamme~ drop O.S m. mass of hammers 4.18 ka2 Static piling3 High-frequency vibro-piling (2966 HI).

FIG. 5.2 Transmitted energies on top ofpile versus embedded depth(from Sieffert & Levacher, 1982)

-CJQ)fJl-~'-"

200

100

oIII

static surcharge - 0.15 kN.pile diameter = 21mm.

o SOFrequency (Hz.)

FIG. 5.3

10 20 30 40 60

Effect of Frequency (from Littlejohn et aI, 1974).

.z,:.!

M..... ......~.

0

II...... -N ZN ,:.!:::c Cl) II -00

N ~ ~ Cl)M (d Cl) to).J:: .... ~II to) Cl) M 0~ S ~>- =' t1Sto) (I) or1 to)c:: '1j or1 -Cl) to) S ...:t'=' .~ Cl) (d ......0" .... ..... c:: 0\Cl) (d .~ >- .....~ .... c, '1j~ (I)

.....(d

....Cl)

c::.J::0~Cl)

N .............or1toJ

~~~-Cl)to)~0~to)or1et1Sc::..... !l~0....to)QI........~

...:t'.Lf'\.o1-4~

-~ -..:1'- ......., 0\. Cl -C""IZ Q,I ..,.!>o! e -Q,I CISM U-- CIS ..,-- - Q,I. 1:100 Cl)

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Q,I >. Q,ICO ~ -N ~ 0 ..,

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1:10 U

~CIS-1:10Cl)....Q

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1601~~

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o 0 0 0 0CIO ,..... I.C LI"I -e(~as/mm) AllJOlaA uOlle~lauad

oN

o-JOHN RnJ~r;s

UNIVERSITYLIBRARY OFMP-.NCtlESTER

-en-~ 200->-!: 160uo....Qj

> 120c::o.......co 80,.....~~ 40

-1-0.c:4-s-

o

pile diameter - 450 mm.

4 6Static Surcharge

FIG. 5.6 Effect of Static Surcharge (Bias)(from Barkan, 1962b).

frequency - 29 Hz •dynamic force - 0.7 kN •pile diameter - 21 mm.

o SO 100 150Static Surcharge (~

FIG. 5.7 Effect of Static Surcharge (Bias)(from Littlejohn et aI, 1974).

-II) 14oZ:;:)

oGo

18

10

6

2

o 80 1001000 4000 eeoc 60002000

FIG. 5.8 Vibratory frequency versus force amplitudes(from Nassar, 1965).

1.00'

F F F F Fj

RESOHANC£fREQUENCY

N (Hz)110 1250 2150 41110 52110

HUMHR 01" HOO£8

m 0 , I 2 I

>"A(APPROXIMATE )

1.00 1.&0 1.00 0.71

FIG. 5.9 Wave form of model piles at various resonancefrequencies as observed by Nassar (1965).

Upper trace: sinusoidal driving forceLower trace: dynamic point resistanceParameters: N = lOO Hz. bias = 5 Ib (constant)

I cm = 5 lb. I cm = 2 ms (horizontal scale)(a) Driving force is less than resistance threshold.(b) Driving force is greater than resistance threshold but smaller than the impact

threshold.(c) Driving force still smaller than impact threshold.(d) Driving force equals impact threshold.(e) Driving force larger than impact threshold.

FIG. 5.10 Dynamic point resistance versus sinusoisdaldriving force for constant bias force of5 1bs. (from Ghahramani, 1967).

1&1UZ

~(I)

iii1&1Cl:

¥ laoc1&11&1Q.U

150~~211. (/)C 1&1 120Z .... 1&1)-:::l Cl:0Q. (!IZ 1&1 90z- 0

1&10I&IZ~o( 60.... PHASE INSTABILITY NEAR1&1

IM~CT THRESHOL.D FORCECD 301&1(I)C 0%Cl.

-30

-606 8 10 12 14 16 18 20

SINUSOIDAL FORCE INPUTIMPACT ( POUNDS)

THRESHOLD

FIG. 5.11 Phase between tip resistance and drivingforce (from Ghahramani, 1967).

100 Hz

O(/)...JO

~~ S(/)0I&IQ.~~ 4....t-U

i3

2

2 4 6 7 8

BIAS-SURCHARGE (POUNDS)

FIG. 5.12 Variation of impact threshold with biasforce (from Ghahramani, 1967).

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Gravel& Till

¢_

~/

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f:

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f

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1:

lI'

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viscous boundary ~

FIG. 5.14 F.E. MESH LAYOUT FOR PILE-SOILRESONANCE ANALYSIS (SECTION 5.4)

FIG. 5.15

r--- _

t-I---_+__~

t-t----+__~

1"1_ - _

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MODE SHAPE FOR EIGENVALUE 1 (FREQUENCY = 12.6 HZ.)

~~~~

II

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--- I--

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FIG. 5.16 MODE SHAPE FOR EIGENVALUE 5 (FREQUENCY = 44.3 HZ.)

I\

,~~~~~~

FIG. 5.17 MODE SHAPE FOR EIGENVALUE 10 (FREQUENCY = 56.7 HZ.)

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(0)

20 6030 40FREQUENCY (CPS)

~O

SYMBOL LB-IN.C 3.403A 1.70B<> QSS!!

p

.

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o Cl. • o. e»,

20 30 40

FREQUENCY (CPS)

60

Steady-state response of a vertically loadedpile, observed experimentally by Novak &Grigg (1976), expressed in Ca) dimensional,and (b) dimensionless forms.

.o

CD.o

.o

.o

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MESH A

- pile top __

t!I -_ pile top _

tjII

I-mudline - -- mudline

- pile tip-- - pile tip - __

MESH C

MESH B

N.B. (i) Boundary conditions for MeshesB & C are similar to that ofMesh A, but omitted in figuresfor clarity sake •

(ii) Details of spatial discretisationare presented in Table 5.5.

FIG. 5.26 Layout of Meshes A, Band C(Section 5.4.2).

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80

CHAPTER 6IMPACT PILE DRIVING

6.1 IntroductionWith regard to the installation, control and design procedures of

offshore piles there has been only limited technological advances since thelate sixties. Although experience has been accumulated to better the designand construction methods, the problems likely to be encountered today arestill very much the same as those stated in the state-of-art reports byMcClelland et al (1969), Toolan & Fox (1977) and Agarwal et al (1978). In anutshell, problems tend to occur in offshore construction unfamiliar interrestrial operations because of the sheer scale of piles and penetrationdepths required.

In the process of installation, difficulties may arise as the size ofthe piles tend to out-scale the hammer capacities available. The assessmentof pile driveability thus become\an important procedure in offshore piledesign. The one-dimensional wave equation analysis proposed by E.A.L. Smith(1960) (and its variations) is still the most popular technique employednowadays, although the axisymmetric finite element analysis advocated bySmith & Chow (1982) is starting to gain popularity. Both of thesecomputation techniques allow easy determination of driving stresses withinthe pile, the maximum of which is important in the selection of suitablepile sizes.

Due to the limited maximum power delivered by the hammer in every blow,any stoppage in driving (e.g. for splicing) can cause large set-ups whichmay be detrimental to the driving operation. Remedial measures as summarisedby McClelland et al (1969) can be implemented if required.

Having installed a pile to the required penetration successfully, the

next task is to ascertain that the required axial load-carrying capacity has

been achieved. While pile load testing is feasible though expensive for

81

onshore construction, it is practically and economically very difficult forlarge offshore piles, except perhaps on small diameter conductors. Theproblem is even more difficult in construction upon unfamiliar deposits likecalcareous and cemented soils which may provide extremely low shaftresistance. Accordingly a variety of non-destructive pile testing techniquesbegin to bud, based on the measurement of longitudinally propagating stresswave signals. These are also attractive to be used in onshore jobs, forreasons of economy, convenience and speed. An important sideline in thestress wave measurement techniques is the testing of pile integrity, whichis especially important for concrete piles.

6.2 One-Dimensional AnalysisThe realisation that pile driving cannot be adequately accounted for by

Newtonian rigid body mechanics has led to the development of the stress wavepropagation theory. Due to the elasticity of piles, stress waves aregenerated as a result of hammer blows. These waves propagate at the velocityof sound along the pile, so that the entire length of the pile is notstressed uniformly. This is a definite improvement over all pile drivingformulae, in which pile rigidity is always assumed.

The one-dimensional wave equation theory was first pursued by Isaacs(1931) and Fox (1932), although solutions for all but some very simpleboundary conditions were not possible until the dawning of the computer era.The classic publication by E.A.L. Smith (1960) represented a breakthrough,in which a finite difference approach was advocated. Implementation of thisapproach was subsequently illustrated by Samson et al (1963) and Forehand &Reese (1964), and today it still forms the framework of many commercial piledriveability programs. Recently, Smith (1982) adopted a finite elementapproach to the same analysis, and it is gradually gaining popularity. The

advantages of both the finite difference and finite element formulations are

outlined below:-

82

(a) Finite Difference Scheme.

(i) It allows easy incorporation of restitution of hammer accessories,no-tension condition at the pile base, as well as separationbetween the ram and the pile during rebound.

(b) Finite Element Scheme.(i) For a modest increase in storage requirements the finite element

method warrants the use of implicit time integration algorithms,thereby increasing the stability characteristics of the system andallowing a somewhat larger time step size.

(ii) Apart from lumping the masses at the ends of the springs one mayopt to distribute the mass evenly along the pile length by theformulation of the consistent mass matrix.

(iii)The program can be written in a modular manner for bettersoftware adaptability.

The one-dimensional model has subsequently been refined (Matlock & Foo,1980; Heerema & de Jong, 1980), reviewed (Holloway et aI, 1978; Goble etaI, 1980; Poulos & Davis, 1980; Saxena, 1981), assessed qualitatively(Rigden & Poskitt, 1977; Smith, 1980) as well as quantitatively byparametric analysis (Coyle et aI, 1977; Dover et aI, 1982). However, itshould be pointed out that the values of some parameters used seem to bebased upon experience rather than derived from appropriate physical propertiesof the system. Thus much care and expertise is required when therecommendations for one particular site are extended to another.(i) Ram Velocity and Cushion Properties

These are determined from standard information in the manuals providedby hammer manufacturers and used as input initial conditions. In fact thesevalues depend on the actual set-up, and deviations from the 'recommended'.values can cause substantial (over 20%) difference in driveability predictions(Dover et aI, 1982). Goble (1982) argued that such input conditions are

still superior to the manufacturer-supplied 'force-time records' which are

83

usually too generalised to be representative. The treatment of differentkinds of diesel hammers are more complicated, and detailed numericalmodelling has been reported by Goble & Rausche (1976).(ii) Damping Parameters Js' Jp

In order to correlate the dynamic and static resistances offered by thesoil, viscous dashpots are incorporated at the nodes of the one-dimensionalmodel. Theoretically this is a device used to simulate in a gross manner(a) the viscous damping at the pile-soil interface; (ii) internal viscousdamping within the soil mass; and (iii) radiation damping into the farfield. A survey of different damping laws and corresponding valuesrecommended and employed by various researchers is exhibited in Tables 6.1 -6.3. The lack of consensus is quite apparent, suggesting that in practicethe damping parameters are possibly not unique physical properties of thesoil, but merely correlation parameters between reality and whatever

.numerical idealisation employed, and within which uncertainties andinaccuracies are concealed. Holloway et al (1978) even reported that withaccurate simulation of the impact stress wave the Case Western researchershave managed to reduce the damping parameters towards zero.

In any event, the application of any textbook values to an unfamiliarsite can result in gross inaccuracies, since driveability predictions aresensitive to the damping values assumed. This will be further discussed inSection 6.3. A sensible procedure is to calibrate the wave equationparameters based on dynamic measurements locally in the field (Hirsch et aI,1975; Santoyo & Goble, 1981).(iii)Soil Resistance Ru

In order to model the elastoplastic resistance offered by the soil,massless springs are incorporated at the nodal points of the one-dimensionalmodel. The assumed resistance may be simply elastic-perfectly plastic, or

follows complex t - Z curves as suggested by Vijayvergiya (1977) and Kraftet al (1981). It is usual to obtain the soil resistance values from static

84

design methods, or by backfiguring from static field tests. However, the

different mobilisation patterns between static loading and pile driving cast

doubt on such correlation.

For closed-ended piles in clays (Fig. 6.1) the shaft resistance can be

simply expressed as 2nro,acu' where , is the segment length and a is the

adhesion coeficient as specified by the API-method. As for the tip

resistance, it is often attractive to represent it as ~r02(NDcU)' where ND

is the dynamic bearing capacity factor, and is assumed to be equal to its

static counterpart, Nc or 9. However, Smith & Chow (1982) have shown that

while this is valid for very stiff clays (cu ~ 400 kN/m2), the dynamic

bearing capacity for soft clays (cu ~ 25 kN/m2) can be as high as 40.Even more uncertainties exist in the treatment of open-ended piles.

Most computer codes require at the time of input to specify whether the pile

.plugs or not. A plugged pile is then considered in a similar manner as a

closed-ended pile (Fig. 6.2), while an unplugged pile requires computation

of shaft resistances at both inner and outer pile walls (Fig. 6.3).

Although these may be mobilised at different rates the one-dimensional model

may not be able to take this fact into account. The Dynpac program proposed

by Heerema & de Jong (1979) alleviates this problem to a certain extent,

although assumptions are still required for the values of limiting soil

resistances and quakes of the internal soil column. More refinements of the

code are necessary.

The determination of input values for ultimate tip resistances follows

the previous discussion for closed-ended piles, except in this case the tipresistance assumed is

2 2~ (ro - rl ) NDI Cu for unplugged piles, and(6.1)

2~ ro ND2 Cu for plugges piles

where NDI = ND2 = Nc • 9.

This is again in gross error as Chow (1981) showed that NDI ranges from 50

85

(for soft clays) to 120 (for stiff clays), while ND2 ranges from 9 to 20.(iv) Soil 'quake'

The 'quake' is a numerical parameter specially devised for the one-dimensional model. It is supposed to represent the vertical displacementof the pile at the point concerned just before plastic deformation'occurs.A constant value of 2.5 mm. was originally recommended by Smith (1960),based on his experience. Smith & Chow (1982) found that good correlation tothis value can be obtained by identifying

(a) the tip quake as the pile tip displacement when ND attains itsmaximum value; and

(b) the side quake as the maximum pile top displacement attained duringdriving.

Subsequent parametric analysis by Forehand & Reese (1964) and Ramey & Hudgins(1977) confirmed that variations of the quake value between 1.3 mm. to 7.6 mm.caused little variations in driveability predictions. Using axisymmetricfinite element analysis Smith & Chow (1982) also showed that although thedynamic response of the pile is not properly followed, the one-dimensionalmodel can predict pile set rather accurately by subtracting the quake fromthe first displacement peak at the pile tip (which is the point when mostwave equation codes cease computing).

Nevertheless, Authier & Fe1lenius (1980) have furnished two cases whenthe driving of moderately-sized piles (0.3 m. in diameter) into glacial tillmaterial encountered quakes of the order of 20 mm.! The reasons for such'unusually large' quakes are still unknown, but have been postulated as dueto pore pressure set-up effects. Large quakes can only be detected bydynamic measurements in the field.(v) Spatial and Temporal Discretisation

As an impact type of wave propagation problem, the accuracy of piledriving analysis depends on the ability of the numerical model to propagate

the agitated frequency components 'faithfully' along the pile. Based on

86

his experience and the propagation velovity of the pile material JE/p

Smith (1960) recommended a pile segment length of 0.3 m. (10 ft.) and a time

step size of 0.25 msec. Parametric analysis by Chow (1981) and Dover et al

(1982) confirmed that such recommendation is satisfactory, at least for the

hammer-pile systems available today. In fact any spurious energy remaining

due to discretisation may by damped out by the damping parameters Js and Jp'

It is of interest to note that the shape of the pile (e.g. closed-ended,

open-ended, H-pile or sheet pile) is only represented by a gross cross-

sectional area term in the formulation. While this can be accused of

oversimplification by neglect of phenomena like plugging, this may sometimes

help to circumvent the necessity of pursuing an elaborate three-dimensional

analysis for the sake of representing the proper pile geometry, e.g. in the

case of H-piles.

Summary

The one-dimensional wave equation code is well established today and has

the advantage of relatively small core storage requirements so that mounting

onto small computers is possible. Provided suitable values for various

parameters are selected, reasonably accurate results can be obtained.

However, the justification of these input values is often not easy, and

experience and expertise is often indispensible. In designing for pile

driving operations on an unfamiliar site, it is advisable to perform

dynamic measurements (e.g. obtain force-time and velocity-time records)

on the piling arrangement in order to calibrate the driveability program on

a local basis. In this manner, uncertainties and inaccuracies concealed in

the numerical code can be minimised. On the whole, the wave equation

approach remains undeniably as a prodigious stride from the conventional pile

driving formulae.

6.3 Three-Dimensional (Axisymmetric) Model

While the one-dimensional wave equation model attempts to simulate

87

piling behaviour through somewhat fictitious parameters, the full three-

dimensional (or more precisely, axisymmetric) finite element model deals with

everyday soil parameters and furnishes a more genuine picture on the driving

process. Suitable constitutive soil models can be incorporated, and no

assumptions have to be made regarding plugging conditions or distribution of

soil resistance. Both radiation and viscous damping can be catered for,

alongside with cyclic degradation relationships if desired. The only major

disadvantage, however, is that such a model requires considerably more core

storage than the less elaborate one-dimensional code so that at least 'mini'-

computers in current terms must be resorted to.

The three-dimensional model has been programmed by Chow (1981) to solve

driveability problems. The input initial condition used is the same as for

the one-dimensional model, namely the ram velocity. Chow (1981) adopted the

implicit Wilson (8 = 1.4) algorithm for time stepping, and computation to

steady state gives the pile set per blow. Reasonable results have been

obtained.

It has been mentioned in Section 6.2 that by inputting the ram velocity

rather than the force-time curve, mesh design is difficult to justify. This

also applies to the axisymmetric model, except in this case it is the soil

elements rather than the pile elements that govern discretisation, due to

their lower wave velocities.

In the numerical simulation of the pile driving tests performed by

Rigden et al (1979) at the Building Research Establishment site at Cowden

(thereafter called the 'Rigden piles'), Chow (1981) adopted a maximum pile

segment length of 1.36 m. and a maximum transverse element dimension of

2.0 m. (Fig. 2.11). The use of a uniform mesh is not possible due to the

actual geometry repre~tation of the pile. Mesh gradation will also allowed

more accurate modelling of the elastoplastic reponse of the soil in proximity

to the pile. The time step size employed is the same as the recommended

value by Smith (1960), namely 0.25 msec. Apparently all these figures have

88

chosen based on experience, and somewhat on the recommendations for one-

dimensional models. Thus before proceeding to further driveability analysis

utilising the three-dimensional model, the spatial and temporal discretisation

employed by Chow (1981) is reassessed.

(i) Spatial Discretisation.

Using exactly the same discretisation scheme as Chow (Fig. 2.11),

results for Rigden's closed-ended pile are obtained. The standard viscous

boundary has been incorporated at both the lateral and base mesh boundaries,

since the load function vanishes with time. The computed responses are

shown as the solid lines in Figs. 6.4 - 6.7.

From Fig. 6.4, the rise time of the SRD response is approximately 3.85

msec. The mesh can thus be refined by designing to this rise time using

equation (4.9), giving a maximum soil element size of 0.124 m. transversely

and 0.629 m. longitudinally, as shown in Fig. 6.8 (assuming that only P-

waves propagate vertically).

The responses computed using the 'refined' mesh are shown as dotted

lines in Figs. 6.4 - 6.7. Essentially the same penetration is predicted by

both meshes (i.e. approximately 10.1 mm., compared with the actual set of

10.2 mm.), although the refined mesh shows a less oscillatory response. The

SRD and velocity responses are essentially similar up to around 0.015 secs.,

or 4.5 llc after the peak response is encountered. This shows that the mesh

adopted by Chow (1981), as in Fig. 2.11, should be adeq~ate in both the

prediction of driveability (Section 6.2) and pile capacity using the Case

method (section 6.9.3, since the responses up to 2 llc only are of interest).

(ii) Temporal Discretisation.

Because of the non-homogenity of the system and the actual geometry of

the pile, a three-dimensional pile-soil representation often involves a

large variation in element sizes, as illustrated by the two meshes considered

in (i). As a result the use of a time step size conforming to the Fourier

89

stability limit (i.e. ~t - ~x/c) of the smallest element may not be toopractical at all, especially when an implicit time integration algorithm isemployed.

Using Chow's mesh (Fig. 2.11) as an example, the critical time step sizeis governed by the pile element at the tip, with a thickness of only 0.025 m.Since the P-wave velocity of the pile is 5966 m/sec., the Fourier stabilitylimit is only 4.2 x 10-6 sec., or 60 times smaller than Smith's (1960)recommendation of 0.25 msec.

Nevertheless, Figs. 6.9 - 6.16 confirm that the use of such a small timestep size is impractical and unnecessary. In fact, for both Wilson (8 - 1.4)and Newmark (~= 0.25) methods, convergence can be obtained up to, incidentally,a ~t value compatible with Smith's recommendation. Although the predictionsof displacement response by both implicit methods are more or less the same,the predicted acceleration responses are markedly different. The peakacceleration obtained by Wilson (8 = 1.4) scheme (Fig. 6.13) is about 50% ofthat obtained by Newmark (~ - 0.25) method (Fig. 6.16), and the former isvirtually damped out by 50 msecs., by virtue of the inherent numericaldamping of the algorithm. This conforms well with the acceleration recordsreported by Rigden et al (1979), showing that the additional numericaldamping happens to reflect the internal viscosity of the soil. On the otherhand, the lack of numerical damping in the Newmark (~ - 0.25) algorithm leadsto only minimal decay in the computed acceleration response.

6.4 Deformation Pattern due to Impact Pile DrivingFigs. 6.17 and 6.18 shows the extent of the yielded zone and the

displacement vectors of the pile-soil system during driving. While the soilyields with much the same pattern as in static loading, displacements aremore localised especially around the pile tip. Attenuation (Fig. 6.19) ismore rapid than vibratory pile driving (Figs. 5.56 - 5.57), probably

because of the additional interface damping modelled in the impact case.

90

6.5 Closed-Ended Piles: Effect of the Damping Parameters Js and Jp

The driveability analysis presented in Section 6.3 and 6.4 utilisedSmith's (1960) linear damping law, and Rigden's (1979) recommended value ofJs = 0.328 sec/m. and Jp = 0.00328 sec/m. for this particular Cowden clay.However, these tend to be merely subjective values unsupported by anyappropriate experimentation. It would be illustrative to find out howsensitive the blow count is towards variations in these values.

Figs. 6.20 - 6.21 shows the variations in blow count for Rigden's c10sed-ended pile with Js and Jp varying within their commonly recommended andemployed values (Tables 6.1 - 6.3). While Fig. 6.21 shows that the predictedblow count changes by less than 10% for a change in value of Jp from 0 to1.0 sec/m., it is much more sensitive to Js (100% variation when Js increasesfrom 0 to 1), as seen in Fig. 6.20. This is logical because Js is associatedwith a much larger area of influence than Jp in deep foundations.Furthermore, Fig. 6.20 also shows that it is possible to achieve the sameblow count prediction with different combinations of Js and Jp' This latterconclusion has also been reached by Holloway et al (1979) in the context ofone-dimensional wave equation analysis using the program DUKFOR.

In summary, the modelling of pile-soil interface damping in terms ofJs and Jp can have significant influence on driveabi1ity predictions. Asmentioned in Section 6.2, field calibration of these parameters with thenumerical models employed is definitely an indispensible procedure forobtaining accurate results.

6.6 Open-Ended Piles6.6.1 Comparison of Behaviour in Driving and Static Loading

The analysis of open-ended piles is more complicated than of closed-ended piles, because additional consideration is required on the behaviour

of the internal soil column. The conditions of plugging are illustrated inFigs. 6.2 - 6.3, namely:

PLUGGING occurs if NO PLUGGING occurs if

Ti ~ Rp Ti <: Rp

If PLUGGING occurs then If NO PLUGGING occurs then

Rud = T + Rb, ~d - T + Ti + Rb,0 0

Rb ..,R + Rp Rb - Rs s

91

It has been mentioned in Section 6.2 that a one-dimensional model cannot

simulate the behaviour of the internal soil column effectively. Often the

plugging status has to be designated as an input condition. On the contrary,

the more elaborate axisymmetric finite element model requires no assumptions

about spring constants or plugging status, but rather the latter can be

determined by evaluating Ti and Rp at every load increment or time step.

Thus it is employed herein to investigate the behaviour of Rigden's open-

ended pile in static loading and impact driving. The mesh in Fig. 6.22 is

employed, which is similar to Fig. 2.11 except that an internal soil column

rises to 3.69 m. above the pile tip, at which no steel cap has been

installed. The value of a taken is 0.64, which is the backfigured value

reported by Rigden et al (1979).

Static Loading

The mobilisation of the various soil resistance components in static

loading is shown in Fig. 6.23. The external shaft resistance is mobilised

relatively quickly to the full extent, while the internal shaft resistance

is only mobilised to a maximum of 63.4 kN. at failure, which is about 17% of

the full capacity considering ai ..,0.64.

A second observation is that the internal soil column is always in

static equilibrium, with the internal shaft resistance mobilised equal to

the bearing pressure beneath the soil column. Thus an open-ended pile always

plugs in static loading.

Furthermore, the distribution of bearing resistance across the base of

92

the pile is not uniform, in the sense that the pressure beneath the pileannulus being 2.5 times that beneath the soil plug. If the value of Nc isdetermined from the expression

Nc = (Rs + Rp)/(nr02cu), (6.2)a value of 11.3 can be obtained.

In the light of the above observations, it is interesting to furtherinvestigate the behaviour of a pile with very low shaft resistance, forexample a bitumen-coated pile, to minimise negative skin friction. Will theinternal shaft resistance mobilise to a greater extent? And since Ti • Rp'will the value of Nc be affected by the value of a?

In order to answer these questions the analysis is repeated with thesame pile-soil model but different values of a. The inner and outer adhesioncoefficients are assumed to be equal all the time. Fig. 6.24 shows thatas the value of a decreases, the maximum internal shaft friction mobilisedalso decreases, so that full mobilisation of the internal shaft friction isnever attained in static loading (in a uniform soil, at least). SimilarlyFig. 6.25 shows that the value of N also decreases from about 12.4 (whenc

ao = ai = 1.0) to 7.9 (when Qo = ai - 0), possibly due to the reduction ofrestraint along the pile shaft. The same trend also applies to closed-endedpiles. It is interesting to find that the value of Nc conforms more toVesic's expression (equation 2.9) rather than the conventional Skemptonvalue of 9.

Impact DrivingIn the case of impact driving, the soil resistance components are at

dynamic equilibrium with the externally applied momentum. This means thatthe internal shaft resistance is not necessarily of the same magnitude asthe bearing pressure of the internal soil column. Field observations(Kindel, 1977; Vijayvergiya, 1980; and Motherwell & Husak, 1982) and

numerical evaluations (Heerema & de Jong, 1980; Chow, 1981) have shown thatopen-ended piles tend to behave as unplugged or partially plugged during

93

driving. Heerema & de Jong and Poskitt (1980) have offered the following

qualitative explanations:

(i) Stress waves propagate much faster in the (steel) pile than in the

internal soil column during driving. Simultaneous response of the pile

and soil column is not likely;

(ii) As the typical pile acceleration during driving amounts to several

hundred 'g's, the inertia of the internal soil column will be so large

that the available shear stress at the pile-soil interface will not be

adequate to bring the soil column into coalesced motion.

When the pile-soil system of Fig. 6.22 is subjected to impact driving

(ram velocity = 4.11 m/sec. from Rigden et aI, 1979), the various soil

resistance components mobilised behave as transient functions, as shown in

Fig. 6.26a. Unlike the previous static loading case, both the internal and

external shaft resistance components are fully mobilised at an early stage

after impact. The transient responses of the internal shaft resistance Ti

and the soil column bearing pressure Rp are shown again in Fig. 6.26b for

greater clarity, and it can be seen that the two quantities will only

approach equilibrium when the impact dies down. Such typical driving record

shows that an open-ended pile will plug intermittently during driving, i.e.

at those durations when Ti > Rp. As a result a one-dimensional wave equation

analysis which requires clear-cut assumption of the plugging status of the

pile will fall short of a proper simulation of impact driving of open-ended

piles.

6.6.2 Effect of Adhesion Coefficients ai' ao

In performing driveability analyses in clays the values of the API

adhesion coefficients ai and ao are usually backfigured from static load

tests (sometimes of small diameter conductors). Since the physical

properties of the soil inside and around the pile can be considered as the

the same, it seems intuitive that ao - ai' as employed by Mizikos &

94

Fournier (1982). On the other hand, Hill (1966) and Heerema & de Jong (1980)suggested that 'arching' of the internal soil column during driving will

result in ai being smaller than ao• Vijayvergiya (1980) suggested the

following values for normally consolidated to slightly over-consolidated

clays: 0.1, ai ' 0.3, and 0.5 ~ ao ~ 1.0.It is apparent that the ai < ao concept is more in line with the static

loading results of Fig. 6.23, whereby the inner and outer shaft resistance

components are mobilised at different rates and to different extents. On the

contrary, complete mobilisation of both components at some stage during

impact driving tends to substantiate ai K ao• In any case, what is the

effect on driveability upon a variation of ao and ai?

Fig. 6.27 shows the predicted blow count of Rigden's open-ended pile

modelled in axisymmetry as before, but with varying values of ao and ai• It

can be seen that pile driveability only decreases mildly with increasing

value of ai assumed, but is much more dependent on the ao value. This is

because the value of ai only governs the limiting soil resistance mobilised

along the internal pile-soil interface, T~lt, which mayor may not contribute

to the overall SRD of the set-up. When the inner shaft resistance is fully

mobilised, however, T~lt will be greater than Rp' and the internal soil

column plugs, as can be observed in the driving record of Fig. 6.26. Theultvalue of Ti ,and consequently ai' is thus immaterial as far as the maximum

overall SRD is concerned (Fig. 6.28), and hence is only secondary in

importance to driveabi1ity.

6.6.3 Effect of Pile Inertia

So far only the influence of the soil on driveabi1ity has been

discussed. While the one-dimensional wave equation model may not be able to

furnish a realistic simulation of soil effects, the influence of pile

properties can be easily assessed. Coyle et al (1977) found that at low soil

resistance a light pile is easier to drive than a heavy pile. The phenomenon

95

is reversed in the case of high soil resistance (Fig. 6.29). It is explained

that when the soil resistance is low, a large amount of the impact energy is

absorbed by the inertia of the heavy pile, and consequently penetration is

little. On the other hand in strong soils the larger capacity of the heavy

pile to transmit the stress waves to the soil exceeds the inertial effects,

thus making it easier to drive.

This implies that apart from changing to a larger hammer to improve

driveability, one can also resort to a more substantial pile in stiff soils.

The choice seems to be a balance between the availability and hire charges

of the hammers, and the cost of material and transportation of heavier piles.

6.7 Comparison of Driving Performance of Open- and Closed-Ended Piles

Conventionally the open-ended pile has been adopted by the offshore

construction industry in preference to closed-ended piles. Two reasons for

this can be conceived:

(i) It is intuitive that open-ended piles are easier to drive than closed-

ended piles because a smaller amount of soil has to be displaced in the

course of penetration;

(ii) Remedial measures (e.g. installation of insert piles) are easier to

implement in open-ended piles when difficulty in driving or unexpectedly

low pile capacities are encountered.

While the first point may be true for end-bearing piles, the differences

in driving performance between open- and closed-ended piles in cohesive soils

have been shown by Rigden et al (1979) to be small, with the closed-ended

pile having the slight edge. If driveability ceases to be the inhibiting

criterion for the adoption of closed-ended piles, the more extensive

remoulding of the soil due to its full-displacement manner can be taken

advantage of, so that a greater long-term pile capacity can be achieved

(Carter et aI, 1980).

The main difference in open- and closed-ended piles is the existence of

96

the internal soil column in the former. This will give rise to an additionalshaft adhesion component when unplugged, or base bearing plus inertialeffects when plugged. These are elements resisting pile driveability. Onthe other hand, in order for a closed-ended pile to penetrate, a much higherbase bearing resistance (about twice as that for an open-ended pile,according to Chow, 1981) must be mobilised.

The influence of the level of the internal soil column (which in practicecan be measured by means of a plug follower after driving) on the driveabilityof open-ended pile is assessed, and compared with an equivalent closed-endedpile (Figs. 2.11 and 6.22). The external and internal shaft adhesioncoefficients are assumed to be equal in the former. It can be seen inFig. 6.30 that the open-ended pile is superior only when a is relativelysmall and the soil column level very low. Thus it makes sense to drill outthe soil column in the case of hard driving, in order to improve driveability.Despite lower soil resistances encountered (Fig. 6.31) the open-ended pile issomewhat inferior to the closed-ended pile regarding driveability in cohesivesoils. This must therefore be attributed to the high inertial effects of theinternal soil column.

6.8 Evaluation of Static Pile Capacity6.8.1 Introduction

Having installed the piles successfully to their specified penetrationsit is important to confirm by testing that the required load-carryingcapacity has been attained. Evaluation of pile capacity also furnishesinformation regarding the adequacy of design, the effects of installation,as well as the effects of consolidation in the case of cohesive soils.

The axial capacity of single piles can be determined by using one ormore of the following methods s

(i) theoretical design methods (already discussed in Section 2.5);

(ii) static field loading tests; and

97

(iii)dynamic methods.

6.8.2 Field Load Test

Fuller & Hoy (1970) stated that the results of a load test on a single

pile can be extrapolated to other piles provided that:

(i) the other piles are of the same design, material and similar size as the

test pile;

(ii) subsoil conditions are comparable to those and the test pile locations;

(iii)installation methods and equipment used are similar to those used for

the test piles; and

(iv) piles are to be driven to similar penetration depth or resistance or

both as the test piles to take account of soil stratification effects.

Although static pile load tests are the most direct and realistic method

in obtaining the load-carrying capacity of piles, the following limitations

are imminent:

(i) such tests are inevitably expensive and time-consuming. As a result,

in offshore construction design parameters tend to be based upon load

tests of much shorter piles (so that length reduction and soil

stratification effects cannot be assessed) or on small diameter

conductors (which may be driven only to a limited depth and then

predrilled, the conditions are not strictly comparable to that for a driven

pile), both of which entails much uncertainties in the design procedure;

(ii) it is sometimes more difficult to determine the tip resistance of end-

bearing piles under static than under driving conditions. For example,

when a pile is being driven through soft clay to reach a bearing sand

stratum, the mobilised shaft resistance is comparatively small through

constant remoulding. On the other hand, the shaft resistance can be

rather substantial if the static loading test is carried out after

allowing for consolidation.

98

6.8.3 Dynamic Methods

6.8.3.1 Introduction

An alternative to the previous methods is to evaluate the pile capacity

by making use of the information recorded during driving. Dynamic methods

are generally non-destructive in nature, and no robust equipment is required

on site. While pile driving records can furnish information on assessing

the immediate pile capacity, long-term effects like soil remoulding,

reconsolidationand cyclic degradation can be taken into account by subjecting

the pile to a few further blows after a set-up period.

Nevertheless, dynamic methods are not problem free. Apart from

assumptions involved in each particular dynamic method, the all-important

question is whether the soil resistance predicted from driving is the same as,

or related to, the static capacity or not. Gersevanov (1948) was probably

the first to reckon that it was possible to determine the pile capacity by

dynamic testing, with sufficient accuracy for practical purposes. On the

contrary, Kezdi (1957) believed that in many cases there is no correlation

between static and dynamic soil resistances. Today the uncertainty has

been clarified to some degree by the following concepts:-

(i) All dynamic testing methods measure the soil resistance mobilised

during driving rather than the static soil resistance. These are

definitely different if the hammer is not substantial enough to

mobilise the full capacity, or if soil freezing or reconsolidation

changes the bearing conditions of the soil. The former can be remedied

by using a large enough hammer to drive the pile, and the latter can

be accounted for by restriking the pile some time after the freezing

or reconsolidation has occurred.

(ii) In general the dynamic soil resistance R can be represented as the sum

of two components (Fig. 6.32):

R = S + D (6.3)

where S is the soil resistance mobilised during driving (SRD), and

99

D is the damping force, usually considered as an empirical function

of the penetration velocity v.

Up to the present the frontier of knowledge is not very well advanced on

the so-called viscosity arising from relative movement between the pile

and the soil. The uncertainty how D should be related to the pile

velocity has been discussed in Sections 6.2(ii) and 6.5. It is

believed that D is generally small in sands, but can be significant in

clays. Since one cannot have an accurate grasp on D the dynamic methods

are generally less satisfactory in clays than in sands.

(iii)Smith & Chow (1982) have shown that in general the dynamic bearing

capacity of undrained clay ND is considerably larger than the

corresponding static Nc value. Furthermore, the results of Section 6.6

show that in the context of open-ended piles the mobilisation pattern

of static and dynamic resistances are different. These two pieces of

evidence suggest that even if the SRD can be correctly evaluated, it

may not be justified to equate it to the static resistance.

(iv) The magnitude of excess pore water pressure generated during driving can

be significantly higher than that generated during static loading.

Consequently, the effective shear strength of the soil in proximity to

the pile will be lower during driving than under static loading

conditions. The correlation between static and dynamic capacities are

thus dubious.

The dynamic methods available today can be categorised as follows:-

(i) pile driving formulae;

(ii) one-dimensional stress wave propagation analysis;

(iii)optimisation method; and

(iv) the finite element method.

These are discussed in turn below.

100

6.8.3.2 Pile Driving FormulaeThe earliest available dynamic method, they relate the ultimate capacity

of driven piles to the set. There exists a vast number of pile drivingformulae, all of which contain two inherent and somewhat empirical assumptions:(i) the pile is regarded as a rigid body during driving, and(ii) static and dynamic soil resistances are taken to be equal.

Some of the formulae (e.g. Hiley's) are still widely acclaimed (Rigden &Poskitt, 1977) and practised (Piling Construction Ltd., 1984) nowadays.

The existing formulae are of different degrees in reliability. Theyhave been assessed by Sorensen & Hansen (1957), Agerchov (1962), Housel (1966),Olsen & Flaate (1967), Rausche et al (1972), Hansen & Schroeder (1977) andRamey & Hudgins (1977). On the whole the formulae are oversimplified andunreliable. They cannot take into account effects like soil stratification,pile elasticity as well as any non-uniformity in cross-section of the hammer-pile arrangement. Moreover, any safety factor applied tends to be subjectiveand cannot be fully justified.

6.8.3.3 One-Dimensional Wave-Equation AnalysisThe application of one-dimensional wave equation analysis to predict

pile capacity can be divided into three different approaches:(i) the production of bearing graphs;(ii) semi-empirical methods with stress wave measurements; and(iii)CAPWAP analysis.(i) Bearing Graph Analysis.

The procedure has been demonstrated by Coyle et al (1977). By modellingpile driving at various penetrations, a series of corresponding pile sets aredetermined. A plot of blow count versus soil resistance (e.g. as in Fig.6.32) or soil resistance versus penetration can then be produced. With suchgraphs available, the pile capacity (and also driveability, which is the

inverse of the problem) can be estimated for a given hammer-pi Ie-soil system.

101

(ii) Semi-Empirical Methods with Stress Wave Measurements.

In common wave equation pile driving analysis, as performed in (i)

above, the input condition is the ram velocity or the impact load history.

The actual energy transferred to the pile, or ENTHRU, is dependent on the

input condition as well as the accurate modelling of the hammer and cushion

properties, which can be rather difficult. In order to improve the quality

of solutions, one can measure the driving response at the pile top to

evaluate the performance of the hammer. Alternatively, one may utilise the

dynamic measurements at the pile top as the input conditions, and exclude

the hammer and accessories from the numerical model altogether. This forms

the basis of a number of semi-empirical methods, for example the Case method

and the TNO method (Gable et aI, 1980). These methods are only semi-empirical

in nature because a number of idealisations are involved in their formulation.

Their major common advantage is that answers can be obtained in real time in

the field, with the help of some handy electronic equipment. The Case method

will be further discussed and assessed in Section 6.9.

(iii)CAPWAP Analysis.

Besides developing the Case method, researchers at Case Western Reserve

University have also put forward the CAPWAP analysis in the early 1970s, in

which the force and acceleration at the pile top are monitored using strain

gauge and accelerometer. Either the force signal or the valocity signal

(integrated from the acceleration signal) is then used as the input condition

to a wave equation pile driving model, the output of which is to be matched

against the remaining unused signal by trial and error. In practice, the

velocity record is always used as the input condition, and the output is to

be matched with the force record. Otherwise, the reverse operation will

involve comparing the derivatives of displacement responses, which have been

shown in Section 6.2 to be very sensitive to the time integration scheme

adopted. Correlation of the records over a duration of 4l/C is recommended,

after which the numerical model should be completely calibrated. Again, the

102

often inaccurate modelling of the hammer and cushion can be bypassed. The

method has been comprehensively illustrated by Goble & Rausche (1980), and is

not further described herein.

Instead, certain points of interest are discussed below regarding dynamic

measurements in the offshore context:-

(a) Near-perfect matching between pile top force and velocity records is only

possible for piles with constant impedance (i.e. EA/c or pile mass * cIa).

The condition, however, is usually not met in offshore piles, the section

of which near the mudline is often thickened to resist bending moments.

Change of cross-section will also occur if a driving shoe or an end-

plate (in the case of closed-ended piles) is installed. This sudden

change of impedance will disrupt the force and velocity records due to

partial reflection of stress waves at the transition. Subjective

adjustments to the measured signals to smoothen the consequent stray

oscillations may be necessary.

(b) In the usual situation of an open-ended pile being driven by an under-

water hammer, it has been found that the force-time measurements are

likely to be influenced by pressure surges of the internal water column.

It is virtually impossible to determine the energy transmitted to

produce the 'water hammer', and once again subjective corrections on

the force record may be required to eliminate the stray effects.

6.8.3.4 Optimisation Method

The optimisation technique developed by Dolwin & Poskitt (1982) is based

on a 'least square' method used in conjunction with a nonlinear optimisation

procedure. It aims at the assessment of the hammer and cushion parameters,

soil resistances and wave equation parameters that produces the best fit

between observed and computed stresses at the pile top. The optimised

acceleration-time record can then be compared with the measured acceleration

curve to provide an independent check on the validity of the desired

parameters.

103

The method has been applied by Fugro (1983) and found to be time

consuming, and results obtained are sensitive to the input for hammer and

accessories. Further refinements of the method seems necessary.

6.8.3.5 The Finite Element Method

In theory if the approaches discussed in Section 6.8.3.3 are performed

with axisymmetric finite elements, better quality solutions should be

obtained. However, the much more elaborate nature of the 'three-dimensional'

analysis defeats the object of the semi-empirical methods because on-site

solutions will then not be possible with current hardware. The application

of the finite element technique in performing CAPWAP in axisymmetric context

is definitely undesirable especially for long offshore piles, as exhaustive

trials and errors at low turnarounds severely tax the effectiveness of the

method.

6.8.4 Summary

Dynamic testing methods undoubtedly serve to provide a quick and economical

indication of the static axial capacity of driven piles. Effects of soil

remoulding and set-up can also be taken into account. However, at the

present stage of progress dynamic testing does not constitute on its own as

a satisfactory method for estimating pile capacity. Besides, such information

is usually available only after most of the design decisions have been taken.

Thus the role of dynamic testing is mainly one of control.

6.9 The Case Method

6.9.1 The Development of the Case Method

In the sixties and seventies, considerable effort has been devoted by

the research group in Case Western Reserve University to the development of

a simple and cheap method using electronic measurements taken during pile

driving to predict the static axial capacity of piles. Such method is

especially useful because it can be applied in the field. This so-called

104

'Case' method has actually been developed in three stages, as reported by

Goble et al (1968, 1970), Rausche et al (1972), Ferahian (1972) and Gravere

et al (1980).

(i) Phase I - Rigid Body Model.

The pile is assumed to be a rigid mody of mass m. In order to eliminate

the effects of viscous damping, Newton's second law of motion is applied at

the instant when the pile velocity is zero, denoted as to:

Ro = F(to) m x(to) (6.4)

where Ro is the mobilised soil resistance (SRD),

F(to) is the driving force on the pile at time to' andx(to) is the pile acceleration at time to'

Usually Ro is identified with the static axial capacity of the pile. Since

equation (6.4) is derived by treating the pile as a rigid body, the force and

acceleration records are assumed to be uniform throughout the length of the

pile. In practice, the responses at the pile top are recorded and employed

as the representative values.

The results of Phase I was found to exhibit considerable scatter, and a

slight shift in to (possibly due to measurement inaccuracies) often caused

substantial differences in the predictions.

(ii) Phase II - Averaging of Accelerations.

In order to eliminate the scatter experienced in Phase I, Goble et al

(1968) considered the average slope of the velocity curve instead of x(to)

in equation (6.4), giving:

= (6.5)

where to is the time of zero velocity, and

tl is the time of maximum velocity.

A further proposal was to average the force record as well, but the

results were little affected because of the relatively smooth nature offorce-time records in general.

105

(iii)Phase IIa - Consideration of Wave Propagation.By considering the propagation of stress waves Phase II can be further

improved. The force term is calculated as an average, and the time intervalconsidered is strategically restricted to 2lfc (i.e. the period of thestress wave):

F(to) + F(to + 2lfc)2= (6.6)

When Phase IIa was first derived, to was taken as the time of zero piletop velocity, presumably in order to minimise the uncertainties of viscousdamping. The formulation was eventually branched into two versions(Ferahian, 1977):(a) the time-delay approach, and(b) the damping approach.(a) Time-Delay Approach.

In this version of the Case method, the first sampling time to is set at

= + ~(2Ifc) (6.7)where tmax is the time at which the pile top velocity first reaches a maximum,

and ~ is an empirical time-delay coefficient, the value of which depends onthe type of soil, and is based on the backanalysis from 60 pile loading testsin Ohio. The recommended values of 8 for steel and timber piles invarious soils are: 0 for sand, 0.25 for non-plastic silt, 1.20 for stiffclay, and 1.40 for weak plastic silt and soft clay. The static pile capacityis then predicted by Ro as from equation (6.6).(b) Damping Approach.

In this version to is taken to be the time at which the maximum piletop velocity is attained. The value of Ro determined from equation (6.6) isthen split into a static soil resistance component S and a damping forcecomponent D, i.e.

= s + D (6.8)By further assuming

(6.9)

106

where jc is an empirical, dimensionless Case damping coefficient,(EA/c) is the impedance of the pile, andvtip is the pile tip velocity at time to' which is given by

vtip = 2(EA/c)(2vo - Ro/(EA/c» (6.10)The recommended value of jc is derived from 71 pile loading tests in

Ohio. Fig. 6.33 shows that despite the small scale of the data base, thefluctuations of jc values are rather large. The general guidance given byFerahian (1977) is: 0.05 for sand, 0.15 for silty sand, 0.2 for sandy silt,0.3 for silt, 0.55 for silty clays and clayey silt, and 1.1 for clay. Thelarge but uncertain jc value for clay renders accurate prediction of pilecapacity difficult, and disappointing results are often obtained.

6.9.2 Advantages and Limitations of the Case Method6.9.2.1 Advantages(i) As for any other dynamic methods, testing can be performed at the time

of driving or afterwards as desired, thus allowing effects of soilremoulding and set-up to be appraised.

(ii) By placing accelerometers and strain gauges at the pile top, inputconditions are obtained for the numerical model without bothering aboutthe often inaccurate simulation of the hammer and accessories.

(iii)Since the Case method is computationally simple, special 'pile drivinganalysers' (Gravere & Hermansson, 1980; Byles, 1984) have been developedto record and display results in the field.

6.9.2.2 Limitations(i) The Case method shares the characteristic of other dynamic methods that

the soil resistance mobilised during driving rather than static loadingis predicted. Moreover, the estimation of damping is somewhat

empirical. The small databases and large deviations in ~ and jc furtherlimit the reliability of the method.

(ii) Although wave mechanics is involved in the selection of the sampling

107

times, the Case method is essentially developed from a rigid bodyformulation. Thus it is not surprising when Garbrecht (1978) comparedthe performance of the wave equation analysis (by Tavenas & Audibert,1977) and the Case method (by Rausche et aI, 1972) and found that thestandard deviation of the former is considerably smaller than that ofthe latter.

6.9.2.3 ApplicationsDespite the assumptions and simplifications involved, the Case method

can be used successfully as a cheap, convenient and quick means to complementstatic loading tests. Santoyo & Goble (1981) utilised the results of staticload tests to calibrate the local value of jc for the same site. Thesubsequent application of the Case damping expression led to consistentresults. It is believed that the use of the Case method will be effectiveas long as the criteria of Fuller & Hoy (1972), as stated at the beginningof Section 6.8.2, are obeyed.

6.9.3 Assessment of the Case Method by Axisymmetric Finite Elements6.9.3.1 Evaluation of Risden's Closed-Ended Pile

Using the same data as in Section 6.3, the force-time relationships atthe pile top as derived from strain and from velocity are computed andpresented in Fig. 6.34. Slight secondary oscillations superimposed on themain trend of the curves reflect the somewhat oversimplified meshdiscretisation, as discussed in Section 6.3.Case prediction of Pile Capacity

Applying equations (6.6) and (6.9) on Fig. 6.34,Ro = 1488 kN. andD = 5837jc kN.

..Since static load test on the same mesh (Fig. 2.11) reveals that

S = 1476 kN.,

substituting this into equation (6.8) gives

108

0.002.

This is substantially smaller than the usually large value of jc

associated with cohesive soils, but rather it seems to echo the comment of

Holloway et al (1978) that accurate simulation and measurement of the

stress waves tend to reduce the damping parameter in the wave equation

analysis to zero. From this damping version of the Case method, the

damping resistance predicted is given by (R - S) or only 12 kN., which

constitutes only 0.8% of the total soil resistance R.

Comparison between Case and Finite Element Predictions

The soil resistance and interface damping mobilised as computed from the

finite element model is shown in Fig. 6.35. A maximum damping force of

1137 kN. has been attained, which is much larger than that predicted by the

Case method. The time at which the maximum SRD occurs (namely 3.71 msec.) is

also different from the time at which the peak pile top force is recorded

(namely 2.31 msec.).

Another dubious point is that the Case method is derived based on the

assumption that the pile behaves as a rigid body. This will require the

pile to respond with the same displacement, velocity and acceleration along

its length. Fig. 6.36 shows that while the velocity distribution is

rather uniform at the time (to + 2lfc) after impact, this is not the case at

time to. Thus the employment of the pile top velocity at time to in the

Case expression (6.6) seems to be unjustified.

6.9.3.2 Effects of the Soil

While the 'hammer domain' in the driving record can be taken as extending

from the instant of impact to the instant when the pile top displacement

reaches a maximum, the record beyond this peak is mainly influenced by the

properties of the soil (Fig. 6.37). The damping version of the Case method

utilises the portion of the driving record of 21fc duration from the response

peak onwards in order to determine the pile capacity, or more precisely, the

SRD of the hammer-pile-soil system.

109

The performance of the Case method in purely cohesive soils of differentstrengths have been assessed by Smith & Chow (1982). Driving records wereinitially obtained by modelling hammer impact upon a closed-ended pile usingan axisymmetric finite element model, as in Section 6.9.3.1. Moreover, forsimplicity Js and Jp were assumed to be zero. As a result, it is natural toassume jc = 0 as well, giving Ro - S in equation (6.8). It turned out fromthe analysis that the Case method overpredicted the capacity in weak clays(cu = 25 kN/m2) by as much as 56%, while underpredicted the capacity in stiffclays (cu = 500 kN/m2) by 38%.

From this numerical experiment, the following points can be raised:(i) In their analysis Smith & Chow (1982) assumed no interface damping in

the production of the driving records, and thereby putting Js - Jp - O.However, the damping version of the Case method requires jc ~ 1.1 forcohesive soils in order to correlate between static and dynamic soilresistances. Is there any correlation at all between JSt Jp and jc?

(ii) It is obvious that the amount of dynamic resistance mobilised depends onthe energy transferred from the hammer to the pile (ENTHRU). While theoverprediction for weak clays may result from a misrepresentation of jc(i.e. Js = Jp - 0 does not imply jc - 0), the underestimation of pilecapacity in strong soils may be due to the inadequacy of the hammer.

(iii)Smith & Chow (1982) have also shown that the maximum tip resistancemobilised during driving is a function of both the strength (i.e. cu)and the stiffness (i.e. ~ - E/cu) of the soil.In order to further investigate the effects of the above points on the

predictive ability of the Case method, a simple case like the one adopted bySmith & Chow (1982) is considered, except that the shaft adhesion coefficienta is also assumed to be zero. Soils of various strengths (cu • 25, 100 and500 kN/m2) and stiffness (250"~ ~ 1250) are considered. Responses at thepile top are computed using the mesh as in Fig. 2.11 as before, and the totalsoil resistance Ro can be determined by applying the Case damping formula.

110

On the other hand,S is supposed to be equal to the ultimate static axialpile capacity, and can thus be determined from static finite element analysisperformed with the same mesh. The damping resistance D, and subsequently theCase damping coefficient jc' can thus be backfigured.

The value of jc obtained for different soil properties are shown inFig. 6.38. Although Js and Jp have been designated as zero, the correspondingjc'S are definitely non-zero, although they are substantially smaller thanthe recommended value for clay, namely 1.1, as stated in Section 6.9.1. Thefigure also suggests that the value of jc does not follow any rational trendwith the soil strength. This point is further investigated.Weak Soils

Fig. 6.39 shows the proportion of damping backfigured from the Caseanalysis. This proportion increases as the strength of the soil decreases,but is virtually insensitive to soil stiffness (in contrast to the commentsof Novak, 1982). In the case of a weak soil (cu - 25 kN/m2), over 80% of thepredicted soil resistance Ro is due to viscosity. The margin of errorinvolved in estimating S in this case is thus significant. The limitedpredicitive ability of the Case method in weak soils has also been admittedrecently by Goble (1982).Stiff Soils

As the strength of the soil increases, the proportion of damping amongthe overall soil resistance Ro decreases (Fig. 6.39), and the reliability inthe prediction of S should improve. One possible dubious point is whetherthe hammer employed is adequate to fully mobilise the (dynamic) soilresistance in this case or not. Accordingly the influence of the hammercapacity on the Case prediction is investigated with a soil of stiffness~ - 500. Figs. 6.40 - 6.41 indicate that the damping coefficient jc tendsto stabilise for high ram velocities. On the other hand, when the hammer is

inadequate so that the predicted overall soil resistance Ro is less than the

static axial pile capacity 5, 'negative damping' is exhibited. The under-

III

prediction of the Case method in stiff clays as claimed by Smith & Chow (1982)is likely to be a reflection of their analysis using an inadequate hammerrather than an indication of the predicitve ability of the Case method.

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0 0 0 0 0 0 0

-- " -N " CO

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...... U ...... to I'll~ to tI.l ~ to

+.I +J= +.I ~ 1:1 +.I OJ OJto Cl) 0 OJ> = +.I 1:1 1:1..-I OJ Cl) .c be Cl) 0...... ...... CIl be = 'Cl +.I...... >-- = ::l ::I be +.I::l 0 I'll to 0 ..-I ::ltI.l U ::c Z ~ c:¥l til

:JI'll...:lbec::..-I0.

~~...toOJc::oM...:l-0..00'1......-CIl...c+.IoMEltI.l

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= CIl..-I ...

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0bI) oM= ...oM to

~>>--~ ot:l

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r.t.J...:l

~

FIG. 6.1 Soil resistance during drivingof closed-ended pile:Rud = To + Rb

FIG. 6.2 So11 resistanceof an unpluggedR - T + T +ud 0 iTi < Rp' Rb - Rs

during drivingopen-ended pile:Rs,

FIG. 6.3 Soil resistance during drivingof a plugged open-ended pile:Rud ..To + \'Ti ...Rp' \ - Rs + Rp

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0. =~ ....0 ~..c:: Q)

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I

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Overall vertical dimensionand boundary conditionsare the same as in fig. 2.11

FIG. 6.8 'REFINED' MESH FOR CLOSED-ENDED PILE ANALYSIS

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FIG. 6.17 YIELDED ZONE UPON DRIVING RIGDEN'S (1979)CLOSED-ENDED PILE

I"

I'

II

II

II

II

~: '

FIG. 6.18 Displacement Plot for pile driving after 0.00825 sec.Displacements magnified by 200 times.

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o Z0 LU

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Plug length• 3.69 m.

~

Data i~

- ~~~::~o!iesexceptcase i

~ i--length

p.3 10.

IIII--Pile WallI

I,It

It

II

I + +t

I + +

I

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T

s the same as in fig. 2.11,that the pile in the present

s open-ended, with a plugof 3.69 m.

LateralBoundary:free for static

analysis,~ for driving

analysis.

Base Boundary: ~ for static analysis,~ for driving analysis.

FIG. 6.22 MESH FOR OPEN-ENDED PILE ANALYSIS

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~(j)

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WALL THICKNESS' 1.5 It.

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o 50 100 150 200 250 300 _ 350

BLOWS PER FOOT

FIG. 6.29 Effect of pile wall thickness :Vulcan 020 hammer, I-D wave equationanalysis (from Coyle et aI, 1977).

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R

Static Capacit S

I Difference possibly due to:-(i) inadequate hammer capacity;(ii) soil remoulding/set up/

degradation;(iii)difference in mobilisation

of into shaft res. in thecase of open-ended piles.

Blow Count

FIG. 6.32 Difference between dynamic and static pile capacities.

------ ---------------

-

---=--_-f---

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112

CHAPTER 7CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

7.1 Summary and ConclusionsFor soil-foundation interaction problems full-sized field experimentation

is often regarded as a non-viable method of data collection. Within thelimits of various resources available, physical or numerical modelling offermore feasible alternatives. However, models are almost always imperfect,and it is important to ensure that the limitations of whatever model appliedwill only cause secondary side effects (if any at all) upon the resultsobtained.

In general the emphases of static and dynamic numerical modelling arerather different. For example, the former aims at the formulation ofaccurate constitutive soil models, and such investigation may end up with oneso complicated that it will be too costly (computer timewise or storage-wise)to implement in nonlinear transient analysis, a solution of which may requireas many as hundreds or thousands of time steps. Instead many dynamicanalysts have to be content with either a one-dimensional spring-and-dashpottype of nonlinear analysis, or simply a materially linear analysis which canbe performed in the frequency domain.

Herein some compromise has been sought by performing truly nonlinearsoil-foundation interaction analysis in two-dimensions (i.e. in plane strainor axisymmetry) using the relatively simple von-Mises model, which is rathersatisfactory for total stress analysis in undrained, saturated, purelycohesive soil. In effect the soil has been treated as a continuum, therebyignoring particulate effects like cyclic degradation.

Despite its conservative nature the API-method remains as a populartechnique for estimating the limiting shaft adhesion on the pile. Chow's(1981) interface element has been incorporated to model slip and

incompressibility (but not separation) of pile-soil interfaces. The nature

113

of the mesh boundaries will influence the load-displacement relationshipalthough the ultimate failure resistance is not affected. Ideally staticinfinite elements should be employed.

Due to the elasticity of the soil and foundation, the application of adynamic load can be regarded as the instigation of stress wave propagation.This gives rise to such undesirable effects as dispersion, spuriousoscillations due to mesh gradation, instability and accuracy problems oftemporal operators in time domain analysis, as well as wave reflection fromthe transmitting boundaries. Such effects do not exist for static problems.Hence a mesh justified for analysing a static situation may not be justifiedin the analysis of a dynamic counterpart.

The criteria for spatial and temporal discretisation have been examinedby solving a number of dynamic response problems, and the finite elementresults are compared with closed-form solutions or experimental observations.Only time domain analysis using implicit algorithms has been investigated.In general, mesh gradation in dynamic analysis is not as beneficial as instatic situations, because elements are confined to a limiting size dependingon the frequency content of the load function. Thus a mesh designed foranalysing wave action (typical frequency of which is less than I Hz.) on afoundation-soil system may be inadequate for earthquake analysis (typicalfrequency range up to 25 Hz.) of the same system. Furthermore, if nonlinearityexists, the time step size of implicit algorithms has to be reduced even tothe order of the Fourier stability limit.

Due to stringent discretisation requirements and the relativelylarge number of time steps involved, dynamic analyses can be rather expensiveespecially for three-dimensional situations. Approximation of three-dimensional problems by plane strain counterparts has often been contemplated.The dynamic response analysis of a foundation subjected to indirect impacthas been considered in a rational step-by-step manner. Reasonable planestrain responses have been obtained. The radiation damping 'paradox'

114

suggested by Gazetas & Dobry (1985) is queried.The Wilson (8 = 1.4) scheme in conjunction with the initial stress

method have been applied to the study of vibratory pile driving. Threeelaborate meshes of different overall sizes have been adopted to span afrequency range of 3.5 - 220 Hz. The undrained saturated clay considered issupposed to constitute about the most adverse situation for vibratorypenetration, due to the absence of fluidisation effects. However, theanalysis has shown that provided the vibratory system is substantial enough(usually in terms of a large oscillatory force) to overcome the elastoplastic(and viscous, which has been ignored herein) soil resistances, penetration atresonance is possible, at least at the fundamental frequency corresponding tothe operating range of the 'low-frequency' Russian and European vibrators.Qualitative agreement with experimental results of other researchers for thisvibration mode has been obtained. However, the pile-soil model employed isnot sophisticated enough to predict the response at the second harmonic,corresponding to the operating frequency range of the American vibrators,

.when pile-soil separation at the pile tip (and possibly fluidisation in thecase of loose sands) will occur. Nowadays vibratory pile driving is onlypractical in terrestrial operations. Extension to offshore piling is notgenerally feasible due to the enormous power required, not to mention theuncertainties and high hire charges involved, and the possible need toredrive to a final 'set'.

On the other hand, the more widely employed pile driving technique isby means of hammer impacts. In the case of offshore piles, sheer sizenecessitates the evaluation of pile driveability. A one-dimensional t - zanalysis may yield satisfactory results, although the dynamic response of thesoil is not properly followed. Nevertheless, an axisymmetric finite elementanalysis may be more realistic but requires a much more elaborate mesh and asubstantially smaller Fourier stability limit (corresponding to that of the

soil rather than the pile). The latter is relaxed herein by using an

115

implicit temporal operator, at the expense of a larger computer storage. On

average the cost of a two-dimensional axisymmetric analysis is about two

orders of magnitude more than an one-dimensional analysis. Hence despite the

axisymmetric model appearing to be a good research tool, its practical use

may not be justified yet.

Much diversity exists in the proposal of damping laws and corresponding

values for modelling the dynamic behaviour of pile-soil interfaces. These,

however, seem to serve a role as correlation between prediction and

obeservation, rather than as physical properties of the soil. Regarding

Smith's (1960) linear damping law, driveability has been found to be more

sensitive to the value assumed for Js than for Jp. The axisymmetric finite

element analysis also permits detailed analysis of plugging in open-ended

piles. It has been found that while an open-ended pile always plugs in

static loading, it only plugs intermittently upon dynamic loading. The

mobilised base bearing in both static loading and impact driving conditions

have been shown to depend on values of the adhesion coefficients. Uncertainty

exists on whether ai should be smaller than ao' but the influence of the

former on driveability is small. In cohesive soils, a closed-ended pile can

be driven more easily than an open-ended pile, depending on the level of

the internal soil column.

By means of instrumentation, non-destructive testing of pile capacity

is possible. However, such tests tend to measure the dynamic pile capacity,

which is difficult to correlate strictly to the static capacity. A number of

dynamic testing methods have been reviewed, and the semi-empirical Case

method has been numerically assessed. Since the damping or time-delay

parameters are derived from a small database, extension to general use is

only possible by correlation on a local scale. Care and expertise are

indispensible at this stage of development.

116

7.2 Suggestions for Future ResearchThe work presented herein has shed some light on performing nonlinear

transient soil-foundation interaction analysis using the finite elementmethod. As the emphasis is placed on the dynamic behaviour of finite elementsystems, the relatively simple von-Mises soil model has been used to avoidextra complications in constitutive relationships. This model may well besatisfactory in assuming undrained conditions for saturated clay, but forgenerality more sophisticated soil models are necessary to take account ofthe change in pore pressures during installation and subsequent consolidation,as well as deformations due to both shear and compressibility. Such a modelis under development (Griffiths et aI, 1982), and can considerably widen thescope of the present work.

It has been mentioned that the evaluation of dynamic response of surfacefootings has found two contexts of application: impact response of nuclearinstallations and wave loading of offshore gravity structures. More reliableestimation of cyclic degradation, material and radiation damping isdesirable. In the numerical aspects, more efficient and economic three-dimensional dynamic analysis algorithms should be invented to take advantageof the rapid rate of development of computer hardware. On the design aspects,engineers may be called upon to devise means to augment the dampingcharacteristics of foundations in order to minimise dynamic amplification:this is especially important when offshore construction is moved to deeperwaters and regions with higher waves.

Regarding pile foundations, future development of the effective stressapproach of analysis should be encouraged, in order to gain a betterunderstanding of the build-up and dissipation of excess pore pressures dueto static or dynamic loading, and subsequently improve the current simplifiedmethods of pile design. Better quantification of interface viscous dampingis required, especially in cohesive soils, to improve the prediction of

dynamic behaviour of piles, and the dynamic pile testing methods.

117

The pile driving analyses presented in Chapters 5 and 6 can be improved

in the following ways:

(i) the use of a more elaborate soil model to take into account effects like

fluidisation, pore pressure responses and cyclic degradation;

(ii) the application of a more sophisticated interface element to allow

separation and rebonding between the pile tip and the soil; and

(iii)extension to the analysis of c-~soils.

ll8

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