AXIAL DYNAMIC RESPONSE OF PILE FOUNDATIONS: ANALYTICAL STUDY

by

Campbell W. Bryden

B.Sc.E. (Civil Engineering), University of New Brunswick, 2015

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Science in Engineering

in the Graduate Academic Unit of Civil Engineering

Supervisors: Arun Valsangkar, PhD, PEng, Deptartment of Civil Engineering Kaveh Arjomandi, PhD, PEng, Department of Civil Engineering Examining Board: Brian Cooke, PhD, PEng, Department of Civil Engineering Edmund Biden, PhD, Department of Mechanical Engineering Alan Lloyd, PhD, Department of Civil Engineering

This thesis is accepted by the Dean of Graduate Studies

UNIVERSITY OF NEW BRUNSWICK

April, 2017

© Campbell W. Bryden, 2017

ii

ABSTRACT

This thesis re-evaluates the theoretical models reported in the literature for

individual piles subject to axial vibration. Analytical procedures have been used to

investigate three independent sub-topics: (1) development of a closed form solution to

Novak’s elastic theory; (2) formulation of a new mathematical model for axial vibration

of tapered piles; and (3) incorporation of modified elastic parameters in Novak’s theory

to account for nonlinear characteristics of driven piles. The general conclusions obtained

from each study were found to be: (1) the proposed explicit expressions are easily

programmed in spreadsheet software, thus allowing one to avoid the approximations and

interpolations associated with classical design charts; (2) the proposed tapered pile

model respects the uniformly tapered geometry, and is shown to be in good agreement

with more rigorous segment-by-segment procedures reported in the literature; and (3)

Novak’s elastic theory can accurately represent multiple sets of experimental data

reported in the literature provided that modified soil shear moduli values are used.

iii

ACKNOWLEDGEMENTS

I would like to thank the following people:

• My supervisors: Dr. Arun Valsangkar and Dr. Kaveh Arjomandi, for their

guidance and encouragement throughout the completion of my program. Their

mentorship and support is greatly appreciated.

• The Civil Engineering administrative staff (Joyce Moore, Angela Stewart, and

Alisha Hanselpacker) for their help during my time at UNB.

• The Natural Science and Engineering Research Council of Canada, the New

Brunswick Innovation Foundation, and the Association of Professional Engineers

and Geoscientists of New Brunswick for providing financial assistance to help

fund my research.

• My parents, Peter and Melissa, my wife-to-be, Lisa, and our furry companions,

Russ and Pumpkin, for their love, support, and encouragement throughout my

studies.

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Table of Contents ABSTRACT ......................................................................................................................... ii

ACKNOWLEDGMENTS ............................................................................................... iii

Table of Contents .............................................................................................................. iv

List of Tables .................................................................................................................... vi

List of Figures .................................................................................................................. vii

1 Introduction ..................................................................................................................... 1

1.1 Overview ................................................................................................................... 1

1.2 Thesis Structure ........................................................................................................ 3

1.3 Contribution of the Candidate ................................................................................... 4

References ....................................................................................................................... 5

2 Explicit Frequency-Dependent Equations for Vertical Vibration of Piles ..................... 6

Abstract ........................................................................................................................... 6

2.1 Introduction ............................................................................................................... 7

2.2 Background ............................................................................................................... 8

2.3 Explicit Expression for the Dynamic Vertical Response of Piles ........................... 10

2.4 Current Practice ...................................................................................................... 12

2.5 Dynamic Response Examples ................................................................................. 14

2.6 Conclusion .............................................................................................................. 22

2.7 Appendix I: Derivation of Explicit Model .............................................................. 23

2.8 Appendix II: Approximations for Bessel Functions ............................................... 25

Notation ......................................................................................................................... 26

References ..................................................................................................................... 27

3 Dynamic Axial Stiffness and Damping Parameters of Tapered Piles .......................... 29

Abstract ......................................................................................................................... 29

3.1 Introduction ............................................................................................................. 30

3.2 Analytical Model .................................................................................................... 32

3.3 Solution by Numerical Integration .......................................................................... 38

3.4 Approximate Solution ............................................................................................. 41

3.5 Conclusion .............................................................................................................. 50

v

3.6 Appendix I: Derivation of Approximate Model ..................................................... 51

Notation ......................................................................................................................... 53

References ..................................................................................................................... 55

4 Modified Elastic Parameters for the Dynamic Axial Impedance of Driven Piles ........ 58

Abstract ......................................................................................................................... 58

List of Notations ........................................................................................................... 59

4.1 Introduction ............................................................................................................. 60

4.2 Background ............................................................................................................. 62

4.3 Published Experimental Data .................................................................................. 63

4.4 Modified Elastic Model .......................................................................................... 68

4.5 Discussion and Summary of Results ....................................................................... 72

4.6 Conclusion .............................................................................................................. 76

References ..................................................................................................................... 78

5 General Conclusions and Recommendations ................................................................ 81

5.1 General Conclusions ............................................................................................... 81

5.2 Recommendations ................................................................................................... 82

References ..................................................................................................................... 83

Appendix A: Background Information for Novak’s Theory ........................................... 84

Appendix B: Derivation of Adjacent Soil Reaction Parameters ...................................... 91

Appendix C: Letter of Permission for Re-Use of Published Material ........................... 101

Curriculum Vitae

vi

List of Tables Table 2.1: Properties and Dimensions of Example Piles ................................................ 15

Table 2.2: Stiffness and Damping Values from Design Charts ...................................... 18

Table 3.1: Sample Pile and Soil Properties ..................................................................... 39

Table 3.2: Additional Soil Pile Configurations for Analysis (Modifications to Example

in Table 3.1) .................................................................................................. 47

Table 4.1: Physical Properties of Experimental Configurations ..................................... 64

Table 4.2: Apparent Shear Moduli and Modification Factors ........................................ 69

vii

List of Figures Figure 2.1: Stiffness and Damping Parameters of Vertical Response for: a) End Bearing

Piles and b) Floating Piles (reprinted from Novak and El Sharnouby 1983, ©

ASCE) ........................................................................................................... 13

Figure 2.2: a) Stiffness and b) Damping Coefficients as a Function of Frequency for

Piles 1 and 2 .................................................................................................. 16

Figure 2.3: a) Stiffness and b) Damping Coefficients as a Function of Frequency for

Piles 3 and 4 .................................................................................................. 17

Figure 2.4: Response Comparison for Pile 1 .................................................................. 19

Figure 2.5: Response Comparison for Pile 2 .................................................................. 19

Figure 2.6: Response Comparison for Pile 3 .................................................................. 20

Figure 2.7: Response Comparison for Pile 4 .................................................................. 20

Figure 2.8: Dynamic Response of Pile 1 for Various Values of Vb/Vs ........................... 21

Figure 3.1: Geometric Properties of Tapered Pile Model ............................................... 33

Figure 3.2: Forces Acting on a Pile Differential Element: a) Elemental Forces, and b)

Vertical Components of Soil Reaction Forces .............................................. 34

Figure 3.3: Dimensionless Amplitude vs. Frequency for End-Bearing Pile Described in

Table 3.1 ........................................................................................................ 40

Figure 3.4: Dimensionless Amplitude vs. Frequency for Floating Pile Described in

Table 3.1 ........................................................................................................ 40

Figure 3.5: Stiffness and Damping Parameters of Sample Pile with δ = 1.5° using the

Approximate Method (Floating Case) ........................................................... 44

Figure 3.6: Stiffness and Damping Parameters of Sample Pile with δ = 1.5° using the

Approximate Method (End-Bearing Case) ................................................... 44

Figure 3.7: Dynamic Response of Sample Pile Using the Approximate Method

(Floating) ....................................................................................................... 45

Figure 3.8: Dynamic Response of Sample Pile Using the Approximate Method (End-

Bearing) ......................................................................................................... 46

Figure 3.9: Response of Pile 1 for the a) Floating, and b) End-Bearing Scenario .......... 47

Figure 3.10: Response of Pile 2 for the a) Floating, and b) End-Bearing Scenario ........ 48

Figure 3.11: Response of Pile 3 for the a) Floating, and b) End-Bearing Scenario ........ 49

viii

Figure 4.1: Conceptual Diagram of the Nonlinear Model .............................................. 61

Figure 4.2: Comparison of Novak and Grigg’s (1976) Experimental Results with the

Elastic Model ................................................................................................. 66

Figure 4.3: Comparison of El Marsafawi et al. (1992) Experimental Results with the

Elastic Model ................................................................................................. 67

Figure 4.4: Comparison of Elkasabgy and El Naggar’s (2013) Experimental Results

with the Elastic Model ................................................................................... 67

Figure 4.5: Comparison of Novak and Grigg’s (1976) Experimental Results with the

Modified Elastic Model ................................................................................. 70

Figure 4.6: Comparison of El Marsafawi et al. (1992) Experimental Results with the

Modified Elastic Model ................................................................................. 70

Figure 4.7: Comparison of Elkasabgy and El Naggar’s (2013) Experimental results with

the Modified Elastic Model ........................................................................... 71

Figure 4.8: Modification Factor α vs. Dynamic Load Factor ......................................... 74

Figure 4.9: Modification Factor β vs. Dynamic Load Factor ......................................... 75

Figure A.1: Schematic Diagram of Soil-Pile Model ....................................................... 85

Figure A.2: Pile Differential Element ............................................................................. 86

Figure B.1: Three-Dimensional Stress Element with Vertical Stresses Indicated .......... 93

Figure B.2: Adjacent Soil Reaction Parameters vs. Dimensionless Frequency ............ 100

1

1 Introduction

1.1 Overview

Pile foundations are often subjected to dynamic loads, such as those produced by

vibrating machines, wind, traffic, and construction practices. Engineers have to

approximate stiffness and damping parameters of the soil-pile system to facilitate

prediction of the foundations dynamic response (Canadian Geotechnical Society, 2006).

The response is highly sensitive to the frequency of vibration, and displacements can

increase by orders of magnitude when the vibration frequency approached the natural

(resonant) frequency (Prakash and Puri, 1988).

The dynamic parameters of a pile group are determined as a form of summation

of the parameters associated with the individual piles in the group (Dobry and Gazetas,

1988). Novak (1974) developed a mathematical formulation to approximate stiffness and

damping parameters of an individual pile based on the theory of elasticity. The

theoretical model for axial vibration was later improved by incorporating the pile-tip

condition (Novak, 1977), and design charts were developed for practical applications

(Novak and El Sharnouby, 1983). Novak’s (1977) model is cited in numerous standards

(Canadian Geotechnical Society, 2006; U. S. Naval Facilities Engineering Command,

1983) and textbooks (Prakash and Sharma, 1990; Arya et al., 1979), and is routinely

used for the dynamic design of pile foundations.

2

The present study re-evaluates the elastic model developed by Novak (1977) for

axial vibration of individual piles. Three independent problems were investigated, which

are briefly summarized below:

1. Various approximations and interpolations are required to use the charts

currently cited in design standards. Novak’s (1977) theory was reformulated and

a closed-form solution for stiffness and damping parameters was obtained. The

explicit expressions are easily implemented in spreadsheet software for design

applications, and can replace the charts used in practice. This topic is addressed

in Chapter 2.

2. Multiple researchers have presented approximate solutions for uniformly tapered

piles by using a step-taper idealization (dividing the pile into a finite number of

uniform sections). Chapter 3 includes the formulation of a new analytical model

for the dynamic axial impedance of tapered piles, which correctly accounts for

the uniformly tapered geometry.

3. Three independent sets of experimental data reported in the literature for axial

vibration of driven piles were reviewed and compared with Novak’s (1977)

theory. It was found that Novak’s (1977) elastic model could accurately represent

the observed experimental data provided that modified soil shear moduli values

were used. The proposed modified elastic model could potentially replace the

more complicated nonlinear model for design applications. This topic is

addressed in Chapter 4.

3

Relevant background information has been provided in Appendix A pertaining to

the formulation of Novak’s (1977) theory. Novak’s (1977) model assumes that the

adjacent soil reaction parameters are equal to those presented by Baranov (1967).

Baranov’s (1967) original publication is in Russian and numerous steps were omitted

from the published derivation. The author has provided a derivation of Baranov’s (1967)

reaction parameters in Appendix B; this derivation provides a clear understanding of the

assumptions embedded in Novak’s (1977) dynamic model, and will be of use to future

researchers using Baranov’s work.

1.2 Thesis Structure

This thesis has been prepared following the articles format and is composed of

three independent journal articles, which have been included as Chapters 2, 3, and 4.

Introductory and concluding chapters have also been included, along with three

appendices.

Chapter 2 presents an explicit derivation of Novak’s (1977) model for the axial

vibration of piles. This article was submitted to the Practice Periodical on Structural

Design and Construction in May 2016, was accepted in October 2016, and was

published online in November 2016. A letter of permission from the publisher has been

included in Appendix C, which indicates that the author may include the published

article in this thesis.

Chapter 3 presents a new mathematical formulation for axial vibration of tapered

piles. This article was submitted to the International Journal of Geomechanics in

December 2016.

4

Chapter 4 defines a modified elastic model that accounts for nonlinear effects.

This article was submitted to Géotechnique Letters in March 2017.

1.3 Contribution of the Candidate

All three of the articles included in this thesis were co-authored by the

candidate’s supervisors: Dr. Kaveh Arjomandi and Dr. Arun Valsangkar. Dr. Arjomandi

and Dr. Valsangkar initially proposed the field of research: pile dynamics, and provided

indispensable guidance and mentorship throughout the duration of the research project.

The candidate identified the research problem addressed in each article, developed the

analytical models, completed the parametric studies, analyzed the data, and prepared the

manuscripts.

5

References

Arya, S.C., O’Neill, M.W., Pincus, G. (1979). Design of Structures and Foundations for

Vibrating Machines. Gulf Publication Co., Houston, TX. Baranov, V.A. (1967). “On the Calculation of an Embedded Foundation.” Voprosy

Dinamiki i Prochnosti, 14: 195-209. Canadian Geotechnical Society. (2006). Canadian Foundation Engineering Manual:

Fourth Edition. BiTech Publishers Ltd., Richmond, BC, Canada. Dobry, R., and Gazetas, G. (1988). “Simple Method for Dynamic Stiffness and Damping

of Floating Pile Groups.” Géotechnique, 38(4): 557-574. Novak, M. (1974). “Dynamic Stiffness and Damping of Piles.” Canadian Geotechnical

Journal, 11(4): 574-498 Novak, M. (1977). “Vertical Vibration of Floating Piles.” Journal of Engineering

Mechanics Division, 103(1): 153-168. Novak, M., and El Sharnouby, B. (1983). “Stiffness Constants of Single Piles.” Journal

of Geotechnical Engineering, 109(7): 961-974. Prakash, S., and Puri, V. K. (1988). Foundations for Machines: Analysis and Design,

Wiley, New York, USA. Prakash, S., Sharma, H.D. (1990). Pile Foundations in Engineering Practice. John

Wiley & Sons. USA. U.S. Naval Facilities Engineering Command. (1983). “Soil Dynamics and Special

Design Aspects.” NAVFAC DM7.3, Alexandria, VA. USA.

6

2 Explicit Frequency-Dependent Equations for Vertical Vibration of Piles* Abstract

Practicing engineers generally obtain stiffness and damping coefficients for

dynamic design of pile foundations from design charts. The design charts require

multiple interpolations, and produce approximate frequency independent coefficients.

Explicit expressions consistent with the underlying theory for the vertical vibration of

single piles are presented in this technical note. These expressions are easily

implemented in a computer program, such as a spreadsheet, for design use. The

proposed method preserves the frequency dependent nature of the dynamic coefficients,

and allows one to account for the true condition of the pile tip. The effectiveness of the

proposed method is illustrated with numerous examples.

_______________________________________________________________________

*Bryden, C., Arjomandi, K., and Valsangkar A. (2016). “Explicit Frequency-Dependent Equations for Vertical Vibration of Piles.” Practice Periodical on Structural Design and Construction, doi:10.1061/(ASCE)SC.1943-5576.0000311. Submitted in May 2016, Accepted in October 2016, and published online in November 2016. Reprinted with permission from ASCE.

7

2.1 Introduction

Piles are often subjected to dynamic loads such as those produced by vibrating

machines, wind, traffic, and construction practices. The theoretical response for a single

pile subjected to dynamic vertical loads has been presented by Novak (1977), Novak and

Sheta (1980), and Han and Sabin (1995), with the latter two having accounted for

nonlinearity in the soil response.

Dynamic design of piles is generally completed using the linear elastic approach

presented by Novak (1977). Experimental data has shown relatively good agreement

between the predicted and observed response using the linear elastic model (Novak and

Grigg, 1976, Puri, 1988). Practising engineers rely on simplified design charts,

developed by Novak and El Sharnouby (1983), to evaluate the dynamic response of pile

foundations (Canadian Geotechnical Society, 2006; U. S. Naval Facilities Engineering

Command, 1983). Many textbooks (Prakash and Sharma, 1990; Arya et al., 1979;

Prakash, 1981) also refer to charts developed by Novak and El Sharnouby (1983).

However, using these design charts has various limitations; the designer must select

either end bearing or floating to determine approximate (frequency independent)

stiffness and damping parameters. To account for a scenario between floating and end

bearing, additional charts from Novak (1977) must be used.

The focus of the present study is to represent the underlying theory in a manner

readily accessible to practicing engineers without incorporating simplification errors.

The theory presented by Novak (1977) is expressed in a way that is easily implemented

in computer software (such as a spreadsheet). The expressions presented in this technical

note preserve the frequency dependent nature of the stiffness and damping parameters,

8

and allow one to consider the true pile tip condition, whether it be floating, end bearing,

or any intermediate scenario.

2.2 Background

The theory presented by Novak (1977) assumes that the pile is vertical, elastic,

circular in cross section, and perfectly bonded to the soil. The governing differential

equation for the vertical vibration of a single pile is:

!!!(!)!!! + 1

!" !!! − !!!! − ! !!! + !!!! ! ! = 0 (2.1)

Where E, A, µ, and c0 are the pile modulus of elasticity, cross sectional area, mass per

unit length, and internal damping coefficient, respectively, and i is the imaginary unit. G

is the shear modulus of the adjacent soil and ω is the harmonic vibration frequency

(rad/s). Sw1 and Sw2 are frequency dependent soil reaction parameters, which were

derived by Baranov (1967) and are defined as:

!!! = 2!!!!! !! !! !! + !! !! !! !!

!!! !! + !!! !! (2.2)

!!! =4

!!! !! + !!! !! (2.3)

Where Jn(a) and Yn(a) are Bessel functions of the first and second kind respectively

(order n). The dimensionless frequency, a0, is defined as:

!! =!!!!!

(2.4)

Where r0 and Vs are the pile radius and adjacent soil shear wave velocity, respectively.

The solution to Equation (2.1) is of the form:

! ! = ! cos ! !! + ! sin ! !! (2.5)

9

Where B and D are integration constants, and λ is equal to:

! = ! 1!" !!! − !!!! − ! !!! + !!!! (2.6)

The integration constants are obtained from boundary conditions. A unit

displacement is applied to the pile head, which leads to the first boundary condition:

w(0) = 1. The vertical soil reaction at the pile tip is assumed to be that of a rigid circular

disk on an elastic half space, and is equal to the axial load at the pile tip. The second

boundary condition is therefore equal to:

!! ! = −!!!!!" !!! + !!!! ! ! (2.7)

Where Gb is the shear modulus of the base soil, and Cw1 and Cw2 are frequency

dependent soil reaction parameters. These parameters are dependent on the soil Poisson

ratio, v, and are defined in Equation (2.8) for v = 0.25 (Novak, 1977). Additional

expressions can be found in Novak (1977), but the response is not sensitive to Poisson

ratio.

!!! = 5.33+ 0.364!! − 1.41!!! (2.8a)

!!! = 5.06!! (2.8b)

Where ab is the dimensionless frequency based on the shear wave velocity of the base

soil, Vb.

!! =!!!!!

(2.9)

Upon evaluation of the integration constants (see Appendix I for complete

derivation), the complex stiffness of the system, K*, is calculated as:

!∗ = −!"!! 0

! 0 = ! + !ℎ (2.10)

10

The complex stiffness is composed of real and imaginary components, k and h, which

are real-valued frequency dependent stiffness and hysteretic damping coefficients,

respectively (Novak, 1974). The equivalent viscous damping coefficient, c, is related to

the hysteretic damping coefficient by:

! = ℎ/! (2.11)

The soil-pile system may now be expressed as a simple single degree of freedom

system, and the dimensionless amplitude of steady state vibration at the pile head is

given by:

!! = !!

!! − !!

!+ !"

!! (2.12)

Where M is the pile head mass.

2.3 Explicit Expression for the Dynamic Vertical Response of Piles

Simple analytic expressions for the stiffness and damping coefficients, k and h,

are presented in Equations (2.13) and (2.14). The real and imaginary components of the

complex stiffness were determined using algebraic manipulations, and various

intermediate parameters, expressed in Equations (2.15) through (2.21), have been

defined to obtain the final expressions. Similar to Novak’s (1977) assumption, the model

assumes zero material damping within the soil. Refer to Appendix I for derivation of

Equations (2.13) through (2.21).

! = −!"! !!!! − !!!! (2.13)

ℎ = −!"! !!!! + !!!! (2.14)

11

where

!! =!!!! + !!!!!!! + !!!

(2.15a)

!! =

!!!! − !!!!!!! + !!!

(2.15b)

!! = !! !!!! − !!!! − !!!!! − !!!!! (2.16a)

!! = !! !!!! + !!!! − !!!!! + !!!!! (2.16b)

!! = !! !!!! − !!!! + !!!!! − !!!!! (2.16c)

!! = !! !!!! + !!!! + !!!!! + !!!!! (2.16d)

!! = sin !! cosh !! (2.17a)

!! = cos !! sinh !! (2.17b)

!! = cos !! cosh !! (2.17c)

!! = − sin !! sinh !! (2.17d)

and

!! = ! cos !2 (2.18a)

!! = ! sin !2 (2.18b)

Parameters R, ϕ, and K’ are dependent on the soil and pile properties, along with the

vibration frequency, and are defined as:

! = !!!" !!! − !!!!

!

+ !!!" !!! + !!!!

!

(2.19)

! = tan!! − !!! + !!!!!!! − !!!!

(2.20)

12

!! = !"!!!!!

(2.21)

It should be noted that the four-quadrant tangent must be used when calculating ϕ; i.e.

-π < ϕ < 0.

Equations (2.13) through (2.21) are easily programmed using spreadsheet

software. Soil parameters Sw1, Sw2, Cw1, and Cw2 are calculated for a range of vibration

frequencies, and the frequency dependent stiffness and damping parameters are

determined. Note that the Bessel functions are native to most computer software

packages, and approximations have been provided in Appendix II.

The proposed model is identical to the original theory presented by Novak

(1977), but has been reformulated in a way that can easily be implemented by practicing

engineers. Equations (2.13) through (2.21) preserve the frequency-dependent nature of

the stiffness and damping coefficients, and allow one to account for the true tip

condition.

2.4 Current Practice

Many practicing engineers use the simplified model presented by Novak and El

Sharnouby (1983) to determine the dynamic response of pile foundations. For the

vertical vibration case, stiffness and damping coefficients are calculated as:

! = !"!!!!! (2.22a)

! = !"!!!!! (2.22b)

13

Novak and El Sharnouby (1983) provide design-plots to determine the stiffness

and damping parameters fv1 and fv2, which have been reproduced in Figure 2.1. These

charts represent approximate (frequency independent) values for two cases: floating

piles and end bearing piles. Novak (1977) provides an additional design chart containing

stiffness and damping parameters for various Vb/Vs ratios, which can be used to account

for pile tip conditions between floating and end bearing. Novak and his colleagues used

stiffness and damping values corresponding to a dimensionless frequency of 0.3 to

develop the design charts (Novak and El Sharnouby, 1983).

Figure 2.1: Stiffness and Damping Parameters of Vertical Response for: a) End Bearing Piles and b) Floating Piles (reprinted from Novak and El Sharnouby 1983, © ASCE)

14

The underlying theory is based on the fact that stiffness and damping are

frequency dependent. The design charts by Novak and El Sharnouby (1983) produce

stiffness and damping coefficients that coincide with a dimensionless frequency of 0.3,

which is not representative of resonance. The operating frequency of a machine is often

much greater than the resonant frequency. However, the amplitude at resonance that

occurs during start-up and ramp-down of machines needs to be considered in the design.

The nature of the tip condition plays a key role in the dynamic response, and end

bearing vs. floating represent only the two extreme cases. A floating pile has the same

soil surrounding the shaft and at the pile base, while an end bearing pile has infinitely

stiff material (bedrock) beneath its base. In reality, piles are often in between floating

and end bearing. The practicing engineer must make some form of approximation in

order to use the plots by Novak and El Sharnouby (1983) for pile tip conditions between

floating and end bearing. To obtain a more accurate response for such an intermediate

scenario, the practicing engineer must use the additional charts provided by Novak

(1977).

2.5 Dynamic Response Examples

Consider four individual piles with the properties given in Table 2.1. The

examples include two concrete piles and two steel pipe piles with slenderness ratios of

approximately 50 and 100. As done by Novak (1977), pile internal damping is neglected

for all cases. The soil properties are constant as: ρs = 1900 kg/m3, v = 0.25, Vs = 60 m/s,

and G = 6.8 MPa. The shear wave velocity of the soil beneath the pile tip is not defined,

as multiple cases are examined to study the influence of tip condition on dynamic

response.

15

Table 2.1: Properties and Dimensions of Example Piles

Pile Property Pile 1 Pile 2 Pile 3 Pile 4 Material Concrete Concrete Steel Pipe Steel Pipe Radius, r0 (m) 0.220 0.220 0.162 0.162 Length, L (m) 11.0 22.0 9.0 18.0 Area, A (m2) 0.152 0.152 0.00891 0.00891 Mass / Unit Length: µ (kg/m) 365 365 69.9 69.9 Elastic Modulus, E (GPa) 22 22 200 200 Internal Damping: c0 0 0 0 0 Pile Head Mass, M (kg) 20000 30000 10000 15000

Note that the pile head mass is increased by 50% when the slenderness is

increased by a factor of two. The increased mass is required in order to mobilize skin

friction and end bearing for the longer pile, such that the effect of the pile tip condition

may be demonstrated.

The proposed explicit model (equivalent to the original theory by Novak, 1977)

is used to compute the response under both end bearing and floating conditions. Figures

2.2 and 2.3 contain stiffness and damping coefficients as a function of frequency for the

concrete and steel pipe piles, respectively.

16

(a)

(b)

Figure 2.2: a) Stiffness and b) Damping Coefficients as a Function of Frequency for Piles 1 and 2

17

(a)

(b)

Figure 2.3: a) Stiffness and b) Damping Coefficients as a Function of Frequency for Piles 3 and 4

18

As seen in Figures 2.2 and 2.3, stiffness and damping coefficient are dependent

on the vibration frequency.

The design charts presented by Novak and El Sharnouby (1983) are now used to

compute dynamic coefficients. Stiffness and damping parameters are obtained from

Figure 2.1a for the end bearing case, and from Figure 2.1b for the floating case. Note

that a transformed modulus of elasticity must be used for piles 3 and 4 when obtaining

parameters from the design charts. Stiffness and damping coefficients are then

calculated with Equation (2.22), and the results are summarized in Table 2.2. The

predicted responses using both methods are plotted in Figures 2.4 through 2.7.

Table 2.2: Stiffness and Damping Values from Design Charts

Dynamic Property Pile 1 Pile 2 Pile 3 Pile 4 End Bearing: fv1 0.023 0.017 0.022 0.017 fv2 0.013 0.021 0.014 0.021 k (N/m) 3.50×108 2.58×108 2.42×108 1.87×108 c (Ns/m) 7.25×105 1.17×106 4.16×105 6.23×105 Floating: fv1 0.010 0.015 0.011 0.015 fv2 0.030 0.030 0.030 0.030 k (N/m) 1.52×108 2.28×108 1.21×108 1.65×108 c (Ns/m) 1.67×106 1.67×106 8.91×105 8.91×105

19

Figure 2.4: Response Comparison for Pile 1

Figure 2.5: Response Comparison for Pile 2

20

Figure 2.6: Response Comparison for Pile 3

Figure 2.7: Response Comparison for Pile 4

21

The approximate response is in relatively close agreement with the frequency

dependent model for all four piles, although the response near resonance is notably

different. The stiffness and damping parameters indicated in Table 2.2 are obtained

using multiple interpolations within design charts. As can be expected, obtaining precise

values with the design charts is extremely difficult and user errors are introduced.

The pile tip condition is often some intermediate stage between floating and end

bearing. If, for example, the shear wave velocity of the base soil were equal to 180 m/s

(3 times the value of the soil along the pile shaft), an engineer using the design charts

would likely consider it a floating pile as the base is not resting on bedrock. The

predicted response of pile 1 using the explicit model for such a scenario is presented in

Figure 2.8; the approximate model significantly underestimates the resonant amplitude.

The response of pile 1 subject to various shear wave velocity ratios are also indicated in

Figure 2.8.

Figure 2.8: Dynamic Response of Pile 1 for Various Values of Vb/Vs

22

The design charts by Novak and El Sharnouby (1983) produce reasonable results

for end bearing and floating piles, but are not reliable for any intermediate scenario.

Additional plots provided by Novak (1977) may be used to approximate stiffness and

damping parameters for various Vb/Vs ratios, but approximations and additional

interpolations are required. The dynamic response is highly sensitive to parameters fv1

and fv2, and user error is a potential issue when obtaining values from charts. Use of the

design charts can significantly underestimate the amplitude at resonance.

2.6 Conclusion

The explicit model presented in this note is fundamentally identical to the

original theory presented by Novak (1977). Accurate solutions can now be obtained very

quickly once the equations presented in this note are incorporated in a spreadsheet.

Various examples have been analyzed to illustrate the errors associated with design

charts currently used in practice. With the use of spreadsheet software, these errors can

be avoided through implementation of the proposed explicit expressions. The proposed

method preserves the frequency-dependent nature of the stiffness and damping

parameters, and the true nature of the pile tip is accounted for. The practicing engineer

can now readily obtain a response in compliance with the underlying theory while

avoiding the simplifying assumptions and interpolations associated with design charts.

23

2.7 Appendix I: Derivation of Explicit Model

The integration constants in Equation (2.5), B and D, are determined from

boundary conditions. The first boundary condition, w(0) = 1, yields B = 1. The second

integration constant is obtained from Equation (2.7) as:

! = !!!"#$! − !!! + !!!! cos !!!!"#$! + !!! + !!!! !"#$

(2.23)

Where K’ is defined in Equation (2.21). The parameter λ, which is defined in Equation

(2.6), is complex-valued and may be expressed as:

! = !! + !!! (2.24)

The real and imaginary components, λ1 and λ2, are obtained using geometric

representation in the complex plane, and are defined in Equation (2.18). Integration

constant D may then be written as:

! = !! !! + !!! sin !! + !!! − !!! + !!!! cos !! + !!!!! !! + !!! cos !! + !!! + !!! + !!!! sin !! + !!!

(2.25)

Using the complex trigonometric identities indicated in Equation (2.26), the integration

constant D is simplified further:

sin !! + !!! = sin !! cosh !! + ! cos !! sinh !! = !! + !!! (2.26a)

cos !! + !!! = cos !! cosh !! − ! sin !! sinh !! = !! + !!! (2.26b)

! = !! !! + !!! !! + !!! − !!! + !!!! !! + !!!!! !! + !!! !! + !!! + !!! + !!!! !! + !!!

(2.27)

Expansion of the complex products leads to:

! = !! !!!! − !!!! − !!!!! − !!!!! + ! !! !!!! + !!!! − !!!!! + !!!!!!! !!!! − !!!! + !!!!! − !!!!! + ! !! !!!! + !!!! + !!!!! + !!!!!

(2.28)

By defining new parameters (σ1, σ2, σ3, and σ4), D may be expressed as:

! = !1 + !!2!3 + !!4 =

!1!3 + !2!4!32 + !42

+ ! !2!3 − !1!4!32 + !42

= !1 + !!2 (2.29)

24

The integration constant D has now been separated into its real and imaginary

components: C1 and C2, respectively. Equation (2.5) may then be written as:

! ! = cos ! !! + !! + !!! sin ! !! (2.30)

The complex stiffness is obtained from Equation (2.10):

!∗ = −!"!! 0

! 0 = − !"1!! !! + !!! (2.31)

Which may be separated into real and imaginary components as:

!∗ = −!"! !!!! − !!!! − ! !"! !!!! + !!!! = ! + !ℎ (2.32)

25

2.8 Appendix II: Approximations for Bessel Functions

Most analysis software packages (such as Microsoft Excel) can evaluate Bessel

functions using simple commands. Polynomial approximations were presented by

Newman (1984), and are defined in Equations (2.33) through (2.36) to the 6th order. The

approximations are valid for 0 ≤ a ≤ 2, which is the range of interest when determining

stiffness and damping coefficients for pile foundations.

!! ! = 0.999999999− 2.249999879 !3

!+ 1.265623060 !

3!

− 0.316394552 !3

!

(2.33)

!! ! = ! 0.500000000− 0.562499992 !3

!+ 0.210937377 !

3!

− 0.039550040 !3

!

(2.34)

!! ! = 2! ln

!2 !! ! + 0.367466907+ 0.605593797 !

3!

− 0.743505078 !3

!+ 0.253005481 !

3!

(2.35)

!! ! = 2! ln !

2 !! ! − 1! + 0.073735531 !3 + 0.722769344 !

3!

− 0.438896337 !3

!

(2.36)

26

Notation

The following symbols are used in this paper:

A = Pile cross sectional area; Aw = Dimensionless amplitude of vibration; a0 = Dimensionless frequency for adjacent soil; ab = Dimensionless frequency for base soil;

B, D = Integration constants; C1, C2 = Intermediate parameters of explicit model;

Cw1, Cw2 = Base soil reaction parameters; c0 = Pile internal damping coefficient; c = Viscous damping coefficient; E = Pile modulus of elasticity;

fv1, fv2 = Stiffness and damping parameters from Novak and El Sharnouby (1983);

G = Shear modulus of soil along pile shaft; Gb = Shear modulus of soil beneath the pile tip;

h = Hysteretic damping coefficient; K* = Complex stiffness coefficient; K’ = Dimensionless constant;

k = Stiffness coefficient; L = Pile length;

M = Pile head mass; R = Radial parameter; r0 = Pile radius;

Sw1, Sw2 = Adjacent soil reaction parameters; Vs = Shear wave velocity of adjacent soil; Vb = Shear wave velocity of base soil; w = Vertical displacement; z = Depth; λ = Parameter;

λ1, λ2 = Real and imaginary components of λ; µ = Pile mass per unit length; ν = Soil Poisson ratio; ρs = Mass density of soil;

σ1, σ2, σ3, σ4 = Intermediate parameters of explicit model; χ1, χ2, χ3, χ4 = Intermediate parameters of explicit model;

ϕ = Phase angle parameter; ω = Harmonic vibration frequency;

27

References

Arya, S.C., O’Neill, M.W., Pincus, G. (1979). Design of Structures and Foundations for Vibrating Machines. Gulf Publication Co., Houston, TX.

Baranov, V.A. (1967). “On the Calculation of an Embedded Foundation.” Voprosy

Dinamiki i Prochnosti, 14: 195-209. Canadian Geotechnical Society. (2006). Canadian Foundation Engineering Manual:

Fourth Edition. BiTech Publishers Ltd., Richmond, BC, Canada. Han, Y.C., and Sabin, G.C.W. (1995). “Impedances for Radially Inhomogeneous

Viscoelastic Soil Media.” Journal of Engineering Mechanics, 121(9): 939-947. doi:10.1061/(ASCE)0733-9399(1995)121:9(939).

Newman, J.N. (1984). “Approximations for the Bessel and Struve Functions.”

Mathematics and Computation, 43(168): 551-556. Novak, M. (1974). “Dynamic Stiffness and Damping of Piles.” Canadian Geotechnical

Journal, 11(4): 574-498 Novak, M. (1977). “Vertical Vibration of Floating Piles.” Journal of Engineering

Mechanics Division, 103(1): 153-168. Novak, M., and Grigg, R.F. (1976). “Dynamic Experiments with Small Pile

Foundations.” Canadian Geotechnical Journal, 13(4): 372-385. Novak, M., and Sheta, M. (1980). “Approximate Approach to Contact Effects of Piles”.

Dynamic Response of Pile Foundations: Analytical Aspects. In Proceedings of the ASCE National Convention, New York, NY. pp. 53-79.

Novak, M., and El Sharnouby, B. (1983). “Stiffness Constants of Single Piles.” Journal

of Geotechnical Engineering, 109(7): 961-974. Prakash, S. (1981). Soil Dynamics. McGraw Hill Book Co., USA. Prakash, S., Sharma, H.D. (1990). Pile Foundations in Engineering Practice. John

Wiley & Sons. USA.

28

Puri, V.K. (1988). “Observed and Predicted Natural Frequency of a Pile Foundation.” In. Proceedings of the Second International Conference on Case Histories in Geotechnical Engineering, St. Louis, Mo., Paper No. 4.41.

U.S. Naval Facilities Engineering Command. (1983). “Soil Dynamics and Special

Design Aspects.” NAVFAC DM7.3, Alexandria, VA. USA.

29

3 Dynamic Axial Stiffness and Damping Parameters of Tapered Piles*

Abstract

Numerous researchers have shown that tapered piles have improved dynamic

properties in comparison to cylindrical piles. The theoretical models currently reported

in the literature approximate the uniformly tapered pile as a step-tapered pile. The step-

taper idealization produces approximate solutions, which become more accurate as the

number of steps is increased. This paper presents a new theoretical model for obtaining

axial stiffness and damping parameters of tapered piles, which accounts for the

uniformly tapered geometry. The underlying theory is consistent with the traditional

elastic model for the vertical vibration of cylindrical piles. The governing differential

equation for vertical vibration of a uniformly tapered pile is solved numerically using

Maple software, and the dynamic response is analyzed; it is observed that the resonant

amplitude can be significantly reduced with an increased taper angle. A simple

approximate solution is also presented, and the results of a parametric study indicate

good agreement with the exact solution obtained by numerical integration. Frequency-

dependent axial stiffness and damping parameters of tapered piles can thus be obtained

quickly and with sufficient accuracy by implementation of the proposed approximate

method.

_______________________________________________________________________

*Bryden, C., Arjomandi, K., and Valsangkar, A. (2016). “Dynamic Axial Stiffness and Damping Parameters of Tapered Piles.” International Journal of Geomechanics. Submitted in December 2016.

30

3.1 Introduction

Deep foundations (piles) are commonly used in modern construction, and are

often subjected to dynamic loads. Stiffness and damping parameters of the soil-pile

system are required to facilitate the prediction of the foundations dynamic response.

Novak (1977) developed an analytical model for the vertical vibration of individual

cylindrical piles based on elastodynamic theory, and Han and Sabin (1995) proposed a

model accounting for soil nonlinearity. Novak’s (1977) linear model produces results

that are in relatively good agreement with experimental data (Novak and Grigg, 1976;

Puri, 1988). Many textbooks (Prakash and Sharma, 1990; Arya et al., 1979; Prakash,

1981) and design standards (Canadian Geotechnical Society, 2006; U.S. Naval Facilities

Engineering Command, 1983) refer to the elastic theory developed by Novak (1977),

and the corresponding charts developed by Novak and El Sharnouby (1983), for the

dynamic design of cylindrical pile foundations.

It has been shown that the static axial capacity of tapered piles is superior in

comparison to cylindrical piles under similar conditions; experimental data has shown

that the use of tapered piles can increase the static axial capacity by up to 80%

(Rybnikov, 1990; El Naggar and Wei, 1999; Ghazavi and Ahmadi, 2008; Khan et al.,

2008). Analytical models have been developed for predicting the static capacity of

tapered piles (Kodikara and Moore, 1993; Liu et al., 2012), and sufficient agreement

with experimental data has been observed (Kodikara and Moore, 1993). Laboratory

testing, including the use of centrifuge analysis (El Naggar and Sakr, 2000) and

pressurized soil chambers (Wei and El Naggar, 1998), has confirmed the benefits of

tapered piles over conventional cylindrical piles for static applications. Tapered piles

have also been shown to yield better drivability performance in comparison to

31

cylindrical piles under equivalent conditions (Sakr et al., 2007; Ghazavi and Tavasoli,

2012).

The performance of tapered piles for dynamic applications has received attention

in recent years. Dehghanpoor and Ghazavi (2012) developed an analytical model for

tapered piles subject to lateral vibration, while Wu et al. (2014) proposed a model for

torsional vibration.

Multiple researchers have presented theoretical models for the vertical vibration

of tapered piles (Saha and Gosh, 1986; Xie and Vaziri, 1991; Ghazavi, 2008; Cai et al.,

2011; Wu et al., 2013). The theoretical models reported in the literature have all

idealized the uniformly tapered pile as a step-tapered pile. The step-taper idealization

produces approximate solutions, which become more accurate as the number of steps is

increased. The various theoretical models are founded on different assumptions of

elasticity, but have all concluded that tapered piles have improved dynamic properties in

comparison to cylindrical piles. The most accepted theoretical model currently reported

in the literature is that proposed by Ghazavi (2008); in this work, material assumptions

are similar to the accepted cylindrical model developed by Novak (1977) and validation

of the theoretical model was performed with finite element analysis.

This paper presents a new analytical model to obtain axial stiffness and damping

parameters of tapered piles that does not require the step-taper idealization. The

governing differential equation is derived using the elastic assumptions proposed by

Novak (1977), while respecting the uniformly tapered pile geometry. The governing

equation is solved numerically, and stiffness and damping parameters of the soil-pile

system are computed.

32

A simple approximate solution is also presented for calculating the axial stiffness

and damping parameters of tapered piles. The imposed assumptions allow one to obtain

a closed form solution to the governing differential equation, while accounting for the

tapered pile geometry. The proposed expressions are simple, and the predicted response

is observed to be in good agreement with the exact solution obtained by numerical

integration. The approximate solution presented in this paper avoids the system of

equations that is produced by the traditional step-taper analysis.

3.2 Analytical Model

Following the theory developed by Novak (1977), it is assumed that the pile is

vertical, elastic, circular in cross section, and perfectly bonded to the soil. It is assumed

that the soil media is composed of two elastic homogeneous layers, which include the

pile’s adjacent soil and base soil as depicted in Figure 3.1.

33

Figure 3.1: Geometric Properties of Tapered Pile Model

The tapered pile can be defined geometrically in terms of length L, radius at the

ground surface r0, and taper angle δ. The radius of the pile as a function of depth z may

be expressed as:

! ! = !! − ! tan ! 0 ≤ ! ≤ ! (3.1)

The pile is subjected to vertical harmonic loading P(t) at the ground surface, and

a pile element dz experiences vertical displacement w(z,t). The pile element encounters a

soil reaction force, which is composed of normal and shear stresses acting along the

elements circumferential area. Figure 3.2a indicates the forces acting on a pile

differential element. Only the vertical components of the reaction forces are of interest,

as depicted in Figure 3.2b.

34

(a)

(b)

Figure 3.2: Forces Acting on a Pile Differential Element: a) Elemental Forces, and b) Vertical Components of Soil Reaction Forces

The vertical component of the shear force Sv is equivalent to the shear resistance of a

‘cylindrical’ pile with radius r(z), and is equal to (Novak, 1974):

!! = !!!(!)! !, ! !" = ! !!!(!)+ !!!!(!) ! !, ! !" (3.2)

35

Where G is the adjacent soil shear modulus, and Sw1 and Sw2 are soil reaction parameters.

The soil reaction parameters Sw1 and Sw2 were derived by Baranov (1967) and are

defined as:

!!! = 2!!!!! !! !! !! + !! !! !! !!

!!! !! + !!! !! (3.3a)

!!! =4

!!! !! + !!! !! (3.3b)

Where Jn(a) and Yn(a) are Bessel functions of the first and second kind respectively

(order n). The dimensionless frequency a0 is a function of pile radius (which is a

function of depth) and is equal to:

!! ! = ! ! !!!

(3.4)

Where Vs is the adjacent soil shear wave velocity and ω is the harmonic vibration

frequency.

The vertical component of the normal reaction force Nv is taken as the normal

stress acting on the annular region produced by the difference in cross sectional areas

from a depth of z to z + dz, as depicted in Figure 3.2b. The resulting force is assumed to

be that of a rigid circular disk on an elastic half space (Novak 1977), which is expressed

in Equation (3.5) for a solid circular disk.

!! ! ! !, ! = !" ! !! ! !(!, !) = !" ! !!! ! + !!!! ! !(!, !) (3.5)

The soil reaction parameters Cw1 and Cw2 are dependent on the dimensionless

frequency and the adjacent soil Poisson ratio, and are defined in Equations (3.6a) and

(3.6b) for a Poisson Ratio of 0.25 (refer to Novak 1977 for additional expressions).

36

!!! = 5.33+ 0.364!! − 1.41!!! (3.6a)

!!! = 5.06!! (3.6b)

The vertical reaction of the annular projection is therefor equal to:

!! = !! ! − !! ! + !" ! !, ! (3.7)

Dynamic equilibrium of the differential element in Figure 3.2b, including the

inertial and damping forces within the pile, is used in conjunction with elastic

compatibility to obtain the differential equation presented in Equation (3.8).

! ! !!! !, !!!! + !!

!" !, !!" − !" ! !!! !, !

!!! − ! !" !!"

!" !, !!"

+ ! !! ! − !!" ! ! !! ! ! !, ! = 0

(3.8)

Where µ, c0, E, and A are the pile mass-per-unit-length, internal damping coefficient,

modulus of elasticity, and cross sectional area, respectively. The right-most term in

Equation (3.8) represents the soil reaction parameter, and can be expressed as:

! !! ! − !!" ! ! !! ! ! ! !, ! = ! !!! + !!!! + !!! + !!!! !(!, !) (3.9)

Where Sw1 and Sw2 have been defined in Equations (3.3a) and (3.3b), respectively.

Parameters Nw1 and Nw2 are defined in Equations (3.10a) and (3.10b) and are obtained by

evaluating the depth derivative in Equation (3.9).

!!! = tan ! !!! + !!!" !!! (3.10a)

!!! = tan ! !!! + !!!" !!! (3.10b)

Parameters r, Cw1, and Cw2 are functions of depth, as defined in Equations (3.1), (3.6a),

and (3.6b), respectively. For a pile that is undergoing complex vertical vibration:

37

! !, ! = ! ! !!"# (3.11)

Equation (3.8) reduces to the ordinary differential equation:

!!!!!! +

1!!"!"

!"!" +

!!! − !!!! − ! !!! + !!!! + !!! + !!!!!" ! = 0 (3.12)

Equation (3.12) is a homogeneous second order linear differential equation, of

which no simple closed-form solution can be obtained due to the variable nature of the

coefficients. Note that parameters A, µ, Sw1, Sw2, Nw1, Nw2, and w are functions of depth.

The equation can be analyzed using numerical integration techniques or through the

introduction of simplifying assumptions.

The complex stiffness of the system is obtained by imposing boundary

conditions. The first boundary condition involves the application of a unit displacement

to the pile head: w(0) = 1. The second boundary condition defines the vertical soil

reaction at the pile tip as that of a rigid circular disk on an elastic-half-space, with

properties equal to those of the base soil. This boundary condition can be expressed as:

!! ! = −!!!!!!!!!!! + !!!!! ! ! (3.13)

Where Gb, is the shear modulus of the base soil, and rb and Ab are the pile radius and

cross sectional area at the tip (z = L), respectively. The soil reaction parameters Cw1b and

Cw2b are obtained from Equations (3.6a) and (3.6b) (for a base soil Poisson ratio of 0.25),

where r(z) is equal to rb and Vs is replaced with Vb (the shear wave velocity of the base

soil). The complex stiffness K* is then computed as:

!∗ = −!!!!! 0

! 0 = ! + !ℎ (3.14)

Where A0 is the cross sectional area at the pile head (z = 0). The complex stiffness is

frequency-dependent and has real and imaginary components k and h, which represent

38

real stiffness and hysteretic damping coefficients, respectively (Novak, 1974). The

hysteretic damping coefficient is related to the equivalent viscous damping coefficient c

by:

! = ℎ/! (3.15)

The dimensionless amplitude of steady state vibration at the pile head is then equal to:

!! = !!

!! − !!

!+ !"

!! (3.16)

Where M is the pile head mass.

The material assumptions used in this model follow from the work of Novak

(1977) for the axial vibration of cylindrical piles. Ghazavi (2008) made similar material

assumptions through a step-taper idealization; the proposed model is mathematically

equivalent to that developed by Ghazavi (2008) with infinitely many step-segments, and

thus overcomes the need for step-taper analysis.

3.3 Solution by Numerical Integration

Equation (3.12), subject to the boundary conditions described previously, is

solved numerically using the fourth-order Runge-Kutta method (with Maple software).

A sample soil-pile system is examined to illustrate the effect that taper angle has on the

dynamic response. Consider the concrete pile and soil media with properties described in

Table 3.1. The internal damping of the pile is neglected, which is typical for the dynamic

analysis of pile foundations (Novak, 1977). This is the same example pile analyzed by

Ghazavi (2008).

39

Table 3.1: Sample Pile and Soil Properties

Pile Properties Length, L (m) 5.0 Radius of equivalent cylindrical pile, req (m) 0.1 Modulus of elasticity, E (GPa) 20 Unit weight, γp (kN/m3) 24 Soil Properties Shear wave velocity of adjacent soil, Vs (m/s) 82.5 Shear modulus of adjacent soil, G (MPa) 12.5 Unit weight, γs (kN/m3) 18 Poisson’s ratio, v 0.25

The radius of an equivalent cylindrical pile req is defined as the cylindrical radius

that produces a pile with length and volume equal to that of the tapered pile. The

equivalent radius is specified to warrant comparison amongst various taper angles, and

may be computed from the expression presented in Equation (3.17).

!!"! =13 !!! + !!!! + !!! (3.17)

The base soil properties are adjusted to investigate both floating and end-bearing

scenarios, corresponding to Vb/Vs = 1 and 10 000, respectively. A pile head mass of 5000

kilograms is applied at the pile head, and the dynamic response is computed using

Equation (3.16). Figures 3.3 and 3.4 indicate the dynamic response for taper angles of 0,

0.5, 1.0, and 1.5 degrees for the end-bearing and floating cases, respectively.

40

Figure 3.3: Dimensionless Amplitude vs. Frequency for End-Bearing Pile Described in Table 3.1

Figure 3.4: Dimensionless Amplitude vs. Frequency for Floating Pile Described in Table 3.1

41

The dynamic responses presented in Figures 3.3 and 3.4 are in agreement with

those reported by Ghazavi (2008), who performed the analysis using a step-taper

idealization (composed of ten step-segments) and finite element analysis. The resonant

frequency remains approximately constant in both floating and end-bearing scenarios,

while the resonant amplitude is observed to decrease (significantly for the end-bearing

case), with an increased taper angle.

3.4 Approximate Solution

The governing differential equation, Equation (3.12), is based on the physical

geometry of the tapered pile and no simple closed form solution exists due to the

variable nature of the coefficients. The depth-dependence of the coefficients originates

within the radial parameter. If one evaluates the coefficients for a variable radius, but

subsequently fixes the radial parameter, then a closed form solution to Equation (3.12)

can be obtained. The parameters (and derivatives) within the coefficients of Equation

(3.12) are evaluated for r(z), but r(z) is then replaced with req prior to solving the

equation. The tapered geometry is accounted for in the model prior to implementation of

this simplifying assumption, and only marginal errors are introduced.

Incorporation of this assumption facilitates the development of a closed form

solution; expressions for the stiffness and damping coefficients, k and h, are presented in

Equations (3.18) and (3.19). Refer to Appendix I for a complete derivation.

! = −!!!! !!!! − !!!! − ! (3.18)

ℎ = −!!!! !!!! + !!!! (3.19)

42

where:

!! =!!!! + !!!!!!! + !!!

(3.20a)

!! =!!!! − !!!!!!! + !!!

(3.20b)

!! = !! !!!! − !!!! + !!!!! − !!!!!! − !!!!!! (3.21a)

!! = !! !!!! + !!!! + !!!!! − !!!!!! + !!!!!! (3.21b)

!! = !! !!!! − !!!! − !!!!! + !!!!!! − !!!!!! (3.21c)

!! = !! !!!! + !!!! − !!!!! + !!!!!! + !!!!!! (3.21d)

!! = sin !! cosh !! (3.22a)

!! = cos !! sinh !! (3.22b)

!! = cos !! cosh !! (3.22c)

!! = − sin !! sinh !! (3.22d)

and:

!! = ! cos !2 (3.23a)

!! = ! sin !2 (3.23b)

Parameters R, ϕ, K’, and ψ are dependent on the soil and pile properties, along with the

vibration frequency, and are defined as:

! = !!!!!"

!!"!! − ! !!! + !!! − !!!+ !!

!!!"!!! + ! !!! + !!!

! (3.24)

! = tan!!− !!!!!" !!! + ! !!! + !!!

!!!!!" !!"!! − ! !!! + !!! − !!

(3.25)

43

!! = !!!!!!!!

(3.26)

! = − !!!"

tan ! (3.27)

Where Aeq and µeq are the area and mass per unit length of the equivalent cylindrical pile,

respectively. To ensure proper sign convention, the four-quadrant tangent must be used

when calculating ϕ (i.e. - π < ϕ < 0).

Note that for a cylindrical pile, the parameters Nw1, Nw2, and ψ are equal to zero;

the approximate model presented in Equations (3.18) through (3.27) reduces to the

explicit expression for a cylindrical pile proposed by Bryden et al. (2016).

Dynamic properties for the sample pile described in Table 3.1 are computed

using the approximate method for a taper angle of 1.5 degrees. Stiffness and damping

parameters as a function of frequency are presented for the floating and end-bearing

cases in Figures 3.5 and 3.6, respectively.

44

Figure 3.5: Stiffness and Damping Parameters of Sample Pile with δ = 1.5° using the Approximate Method (Floating Case)

Figure 3.6: Stiffness and Damping Parameters of Sample Pile with δ = 1.5° using the Approximate Method (End-Bearing Case)

45

The dynamic response of the sample pile described in Table 3.1 with δ = 1.5° for

the floating and end-bearing cases are shown in Figures 3.7 and 3.8, respectively. The

simple approximate method produces a dynamic response that is observed to be in good

agreement with the exact solution obtained by numerical integration. The most

significant difference between the approximate and exact solution is the predicted

stiffness values for the end-bearing scenario, which differ by up to 20%. This is

acceptable for geotechnical applications given the uncertainty of subsurface material

properties. The proposed method is simple; it avoids the system-of-equations imposed

by the traditional step-taper idealization, and is easily programmed in spreadsheet

software for design applications.

Figure 3.7: Dynamic Response of Sample Pile Using the Approximate Method (Floating)

46

Figure 3.8: Dynamic Response of Sample Pile Using the Approximate Method (End-Bearing)

Numerous additional soil-pile systems are examined to confirm the validity of

the approximate model. In addition to the base soil stiffness and pile taper angle, the pile

slenderness ratio (L/req) and the pile-soil stiffness ratio (E/G) significantly impact the

dynamic response. Table 3.2 specifies three additional soil-pile systems, which are

analyzed to confirm the accuracy of the proposed approximate model; each of these

sample piles has properties identical to those specified in Table 3.1, with the

modifications indicated in Table 3.2. The dynamic response of Piles 1, 2, and 3 (as

specified in Table 3.2) are indicated in Figures 3.9, 3.10, and 3.11, respectively, for both

floating and end bearing scenarios.

47

Table 3.2: Additional Soil Pile Configurations for Analysis (Modifications to Example in Table 3.1)

Updated Property Pile 1 Pile 2 Pile 3 Pile length, L (m) 2.5 5.0 2.5 Shear modulus of adjacent soil, G (MPa) 12.5 50 50

(a)

(b)

Figure 3.9: Response of Pile 1 for the a) Floating, and b) End-Bearing Scenario

48

(a)

(b)

Figure 3.10: Response of Pile 2 for the a) Floating, and b) End-Bearing Scenario

49

(a)

(b)

Figure 3.11: Response of Pile 3 for the a) Floating, and b) End-Bearing Scenario

The dynamic response obtained with the approximate method is in good

agreement with the exact response; the calculated stiffness, damping, resonant

50

amplitude, and resonant frequency values are within 20% of the values obtained by

numerical integration.

3.5 Conclusion

This paper presents a new theoretical model for obtaining axial stiffness and

damping parameters of tapered piles. The underlying theory is consistent with the

commonly accepted elastic model presented by Novak (1977) for cylindrical piles. The

founding assumptions of the proposed theoretical model are consistent with the physical

tapered pile geometry, thus avoiding the step taper idealization of traditional tapered pile

models. The predicted dynamic response is in good agreement with the segment-by-

segment method and finite element analysis conducted by Ghazavi (2008). It is observed

that the resonant amplitude of piles subjected to axial vibrations can be significantly

reduced with an increased taper angle; tapered piles thus have tremendous potential for

application in dynamic design of deep foundations. Though the proposed model is

consistent with those currently available in the literature, additional research is required

in the form of experimental testing to confirm the findings of this analytical study.

The approximate method presented in this paper is observed to be in good

agreement with the exact solution obtained by numerical integration; stiffness, damping,

resonant amplitude, and resonant frequency values are observed to be within 20% of the

exact values for the soil-pile models analyzed in this paper. The approximate method is

simple, and can easily be programmed in spreadsheet software for design applications.

Reasonably accurate frequency-dependent axial stiffness and damping coefficients can

thus be obtained with minimal computational effort for tapered piles.

51

3.6 Appendix I: Derivation of Approximate Model

If one assumes that the pile’s cross sectional area consists of a solid circular disc

with varying radius as expressed in Equation (3.1), the coefficients in Equation (3.12)

reduce to:

!!!!!! + − 2 tan !!(!)

!"!"

+!!! ! ! − !!!! − ! !!! ! ! + !!!! ! ! + !!!(! ! ) + !!!! ! !

!" ! ! ! = 0(3.28)

The founding assumption of the approximate model is through the substitution r(z) = req

in Equation (3.28), which eliminates the variable nature of the coefficients. The

governing differential equation then takes the form:

!!!!!! + − 2 tan !!!"

!"!"

+!!!!" − !!!! − ! !!! !!" + !!!! !!" + !!!(!!") + !!!! !!"

!!!"! = 0

(3.29)

The general solution to Equation (3.29) is:

! = !!!!! ! cos ! !

! + ! sin ! !! (3.30)

Where B and D are integration constants, ψ is defined in Equation (3.27), and λ is equal

to:

! = !!!" !!"!! − !!!! − ! !!! + !!!! + !!! + !!!! − !! (3.31)

The parameter λ is composed of real and imaginary components, λ1 and λ2, which have

been defined in Equations (3.23a) and (3.23b), respectively.

52

The integration constants in Equation (3.30) are evaluated from boundary

conditions. The first boundary condition, w(0) = 1, produces B = 1. The second

boundary condition, defined in Equation (3.13), produces:

! = !!! sin ! + !!! cos ! − !!!! + !!!!! cos !!!! cos ! − !!! sin ! + !!!! + !!!!! sin ! (3.32)

Which is equivalent to:

! = !! !! + !!! (!! + !!!)+ !!!(!! + !!!)− !!!! + !!!!! !! + !!!!! !! + !!! !! + !!! − !!!(!! + !!!)+ !!!! + !!!!! !! + !!!

(3.33)

Where χ1, χ2, χ3, and χ4 are defined in Equations (3.22a), (3.22b), (3.22c), and (3.22d),

respectively. Expansion of the complex products in the numerator and denominator of

Equation (3.33) leads to:

! = !! !!!! − !!!! + !!!!! − !!!!!! − !!!!!! + ! !! !!!! + !!!! + !!!!! − !!!!!! + !!!!!!!! !!!! − !!!! − !!!!! + !!!!!! − !!!!!! + ! !! !!!! + !!!! − !!!!! + !!!!!! + !!!!!!

(3.34)

Defining new parameters, σ1, σ2, σ3, and σ4, allows for the development of simple

expressions for the real and imaginary components of the integration constant D.

! = !! + !!!!! + !!!

= !!!! + !!!!!!! + !!!

+ ! !!!! − !!!!!!! + !!!

= !! + !!! (3.35)

The solution to Equation (3.29) with the imposed boundary conditions is thus equal to:

! = !!!!! cos ! !

! + (!! + !!!) sin ! !! (3.36)

And the complex stiffness is calculated from Equation (3.14) as:

!∗ = −!!!!! 0

! 0 = −!!!1!! !! + !!! − !! (3.37)

Which may be separated into real and imaginary components, corresponding to stiffness

and damping coefficients, respectively, as shown in Equation (3.38).

!∗ = −!!!! !!!! − !!!! − ! − ! !!!! !!!! + !!!! = ! + !ℎ (3.38)

53

Notation

The following symbols are used in this paper:

A = Pile cross sectional area as a function of depth; A0 = Cross sectional area of the pile at the pile head; Ab = Cross sectional area of the pile at the pile tip:

Aeq = Cross sectional area of the equivalent cylindrical pile; Aw = Dimensionless amplitude of vibration; a0 = Dimensionless frequency;

B, D = Integration constants; C1, C2 = Intermediate parameters of approximate model;

Cw = Adjacent soil reaction parameter (normal); Cw1, Cw2 = Real and imaginary components of Cw;

Cw1b, Cw2b = Base soil reaction parameters; c0 = Pile internal damping coefficient; c = Equivalent viscous damping coefficient; E = Pile modulus of elasticity; G = Shear modulus of adjacent soil;

Gb = Shear modulus of base soil; h = Hysteretic damping coefficient; i = Imaginary unit;

K* = Complex stiffness; K’ = Dimensionless constant;

k = Stiffness coefficient; L = Pile length;

M = Pile head mass; Nv = Vertical component of the adjacent soil normal force;

Nw1, Nw2 = Adjacent soil annular reaction parameters; P = Harmonic load at the pile head; R = Radial parameter;

R0 = Reaction parameter for a circular disk on elastic half-space; r = Pile radius as a function of depth;

r0 = Radius of the pile at the pile head; rb = Radius of the pile at the pile tip;

req = Radius of the equivalent cylindrical pile; Sv = Vertical component of the adjacent soil shear force; Sw = Adjacent soil reaction parameter (shear);

Sw1, Sw2 = Real and imaginary components of Sw; t = Time;

Vs = Shear wave velocity of adjacent soil;

54

Vb = Shear wave velocity of base soil; w = Vertical displacement; z = Depth; δ = Pile taper angle; γp = Pile unit weight; γs = Soil unit weight; λ = Parameter;

λ1, λ2 = Real and imaginary components of λ; µ = Mass per unit length of tapered pile (function of depth);

µeq = Mass per unit length of the equivalent cylindrical pile; ν = Soil Poisson ratio;

σ1, σ2, σ3, σ4 = Intermediate parameters of approximate model; χ1, χ2, χ3, χ4 = Intermediate parameters of approximate model;

ϕ = Phase angle parameter; ψ = Dimensionless parameter; ω = Harmonic driving frequency;

55

References

Arya, S.C., O’Neill, M.W., Pincus, G. (1979). Design of Structures and Foundations for

Vibrating Machines. Gulf Publication Co., Houston, TX. Baranov, V.A. (1967). “On the Calculation of an Embedded Foundation.” Voprosy

Dinamiki i Prochnosti, 14: 195-209. (In Russian). Bryden, C., Arjomandi, K., and Valsangkar, A. (2016) “Explicit Frequency-Dependent

Equations for Vertical Vibration of Piles.” Practice Periodical on Structural Design and Construction, 10.1061/(ASCE)SC.1943-5576.0000311.

Cai, Y. Y., Yu, J., Zheng, C., Qi, Z., and Song, B. (2011). “Analytical Solution for

Longitudinal Dynamic Complex Impedance of Tapered Pile.” Chinese Journal of Geotechnical Engineering, 33(2): 392-398. (In Chinese).

Canadian Geotechnical Society. (2006). Canadian Foundation Engineering Manual:

Fourth Edition. BiTech Publishers Ltd., Richmond, BC, Canada. Dehghanpoor, A., and Ghazavi, M. (2012). “Response of Tapered Piles Under Lateral

Harmonic Vibrations.” International Journal of GEOMATE, 2(2): 261-265. El Naggar, M. H., and Wei, J. Q. (1999). “Axial Capacity of Tapered Piles Established

from Model Tests.” Canadian Geotechnical Journal, 36: 1185-1194. El Naggar, M. H, and Sakr, M. (2000). “Evaluation of Axial Performance of Tapered

Piles from Centrifuge Tests.” Canadian Geotechnical Journal, 37: 1295-1308. Ghazavi, M. (2008). “Response of Tapered Piles to Axial Harmonic Loading.

Canadian Geotechnical Journal, 45(11): 1622-1628. 10.1139/T08-073. Ghazavi, M., and Ahmadi, H. A. (2008). “Long-Term Capacity of Driven Non-Uniform

Piles in Cohesive Soil – Field Load Tests.” In Proceedings of the 8th International Conference on the Application of Stress-Wave Theory to Piles, 139-142. Lisbon, Portugal.

Ghazavi, M., and Tavasoli, O. (2012). “Characteristics of Non-Uniform Cross-Section

Piles in Drivability.” Soil Dynamics and Earthquake Engineering, 43: 287-299.

56

Han, Y.C., and Sabin, G.C.W. (1995). “Impedances for Radially Inhomogeneous Viscoelastic Soil Media.” Journal of Engineering Mechanics, 121(9): 939-947. 10.1061/(ASCE)0733-9399(1995)121:9(939).

Khan, M. K., El Naggar, M. H., and Elkasabgy, M. (2008). “Compression Testing and

Analysis of Drilled Concrete Tapered Piles in Cohesive-Frictional Soil.” Canadian Geotechnical Journal, 45: 372-392.

Kodikara, J. K., and Moore, I. D. (1993). “Axial Response of Tapered Piles in Cohesive

Frictional Ground.” Journal of Geotechnical Engineering, 119(4): 675-693. Liu, J., He, J., Wu, Y. P., and Yang, Q. G. (2012). “Load Transfer Behaviour of a

Tapered Rigid Pile.” Geotechnique, 62(7): 649-652. Novak, M. (1974). “Dynamic Stiffness and Damping of Piles.” Canadian Geotechnical

Journal, 11(4): 574-498 Novak, M. (1977). “Vertical Vibration of Floating Piles.” Journal of Engineering

Mechanics Division, 103(1): 153-168. Novak, M., and El Sharnouby, B. (1983). “Stiffness Constants of Single Piles.” Journal

of Geotechnical Engineering, 109(7): 961-974. Novak, M., and Grirr, R. F. (1976). “Dynamic Experiments with Small Pile

Foundations.” Canadian Geotechnical Journal, 13(4): 372-385. Prakash, S. (1981). Soil Dynamics. McGraw Hill Book Co., USA. Prakash, S., Sharma, H.D. (1990). Pile Foundations in Engineering Practice. John

Wiley & Sons. USA. Puri, V. K. (1988). “Observed and Predicted Natural Frequency of a Pile Foundation.”

In. Proceedings of the Second International Conference on Case Histories in Geotechnical Engineering, St. Louis, Mo., Paper No. 4.41.

Rybnikov, A. M. (1990). “Experimental Investigations of Bearing Capacity of Bored-

Cast-in-Place Tapered Piles.” Soil Mechanics and Foundation Engineering, 27(2): 48-52.

57

Saha, S., Gosh, D.P. (1986). “Vertical Vibration of Tapered Piles.” Journal of Geotechnical Engineering, 112(3): 290-302.

Sakr, M., El Naggar, M. H., and Nehdi, M. (2007). “Wave Equation Analyses of

Tapered FRP-concrete Piles in Dense Sand.” Soil Dynamics and Earthquake Engineering, 27: 166-182.

U.S. Naval Facilities Engineering Command. (1983). “Soil Dynamics and Special

Design Aspects.” NAVFAC DM7.3, Alexandria, VA. USA. Wei, J., and El Naggar, M. H. (1998). “Experimental Study of Axial Behaviour of

Tapered Piles.” Canadian Geotechnical Journal, 35: 641-654. Wu, W., Jiang., G., Dou, B., and Leo, C. J. (2013). “Vertical Dynamic Impedance of

Tapered Pile Considering Compacting Effect.” Mathematical Problems in Engineering, Article ID: 304856. http://dx.doi.org/10.1155/2013/304856.

Wu, W., Jiang, G., Huang, S., and Xie, B. (2014). “Torsional Dynamic Impedance of

Tapered Pile Embedded in Layered Viscoelastic Soil.” Electronic Journal of Geotechnical Engineering, 19: 4585-4600.

Xie, J., and Vaziri, H.H. (1991). “Vertical Vibration of Nonuniform Piles.” Journal of

Engineering Mechanics, 117(5): 1105-1118.

58

4 Modified Elastic Parameters for the Dynamic Axial Impedance of Driven Piles*

Abstract

A limited number of researchers have reported experimental data for the axial

dynamic response of individual pile foundations in the literature. When the elastic model

cannot accurately represent the observed response, the nonlinear model is employed.

The present study illustrates that the elastic model can produce a response comparable to

the nonlinear model provided that modified shear moduli values are used. The concept

has practical applications, as the nonlinear model is computationally more complex and

introduces numerous additional parameters pertaining to the inner weak soil. It is shown

that a reduction in adjacent soil shear modulus and an increase in base soil shear

modulus can produce an elastic response in good agreement with the limited sets of

experimental data reported in the literature for driven piles. A total of nine experimental

response curves originating from three test configurations were analyzed for the

completion of the present study. It is found that the shear moduli modification factors

are dependent on numerous parameters, including: the soil-pile stiffness ratio, the

magnitude of dynamic loading, the axial capacity utilization ratio, and the degree of soil

disturbance during installation. Based on the limited data analyzed in the present study,

ranges of shear-moduli modification factors are presented for design purposes.

Keywords: Geotechnical Engineering, Piles & Piling, Dynamics.

_______________________________________________________________________

*Bryden, C., Arjomandi, K., and Valsangkar, A. (2017). “Modified Elastic Parameters for the Dynamic Axial Impedance of Driven Piles.” Géotechnique Letters. Submitted in March 2017.

59

List of Notations

A = Pile cross-sectional area; A0 = Amplitude of steady-state vibration; Ab = Area of pile base; c0 = Pile internal damping coefficient;

DLF = Dynamic load factor; E = Pile modulus of elasticity;

Gb = Shear modulus of base soil; Gb(app) = Apparent shear modulus of base soil;

Gs = Shear modulus of adjacent soil; Gs(app) = Apparent shear modulus of adjacent soil;

h = Hysteretic damping coefficient; i = Imaginary unit;

K* = Complex Stiffness; k = Stiffness coefficient; L = Pile length;

M = Pile head mass; me = Eccentric mass-moment of dynamic load;

r = Pile radius; Sw1, Sw2 = Adjacent soil reaction parameters;

Vs = Shear wave velocity of adjacent soil; Vb = Shear wave velocity of base soil; Ws = Static axial load; w = Vertical deformation; z = Depth; α = Adjacent soil shear modulus modification factor; β = Base soil shear modulus modification factor; µ = Pile mass-per-unit-length; ν = Soil Poisson ratio; ρ = Pile material density; ρs = Soil density; ω = Harmonic vibration frequency; ωr = Resonant frequency;

60

4.1 Introduction

Pile foundations often experience dynamic loads, which can occur in the lateral,

axial, and/or torsional direction(s). An appropriate theoretical model must be employed

for one to predict the foundations dynamic response. The present study is focused on

axial vibration of an individual pile, for which multiple theoretical models have been

reported in the literature.

Novak (1974) developed a mathematical formulation to predict stiffness and

damping constants of the soil-pile system based on elastic theory. Novak later improved

the model to account for the pile-tip condition (Novak, 1977), and presented simplified

charts for design applications (Novak and El Sharnouby, 1983). Novak’s elastic model is

cited in multiple design standards (Canadian Geotechnical Society, 2006; U.S. Naval

Facilities Engineering Command, 1983) and textbooks (Prakash and Sharma, 1990;

Arya et al., 1979) and is considered common practice for dynamic design of deep

foundations.

Novak and Sheta (1980) and Han and Sabin (1995) developed theoretical models

that account for soil nonlinearity by incorporating an inner annular region of weaker soil

surrounding the pile. The nonlinear model requires the prediction of numerous

additional parameters, including: the weak zone thickness, weak zone shear modulus,

weak zone Poisson ratio, mass participation factor, and pile separation length (Elkasabgy

and El Naggar, 2013). Figure 4.1 contains a schematic diagram of the nonlinear model.

61

Figure 4.1: Conceptual Diagram of the Nonlinear Model

A limited number of experimental results have been reported in the literature.

Novak and Grigg (1976) and El Marsafawi et al. (1992) performed dynamic experiments

on small pipe-piles driven in native soil at the University of Western Ontario. Elkasabgy

and El Naggar (2013) reported dynamic results for full-scale driven pipe-piles located in

Ponoka, Alberta, Canada. Puri (1988) analyzed a full-scale driven concrete pile, and Han

and Novak (1988) reported results for a small pipe-pile placed in an excavation and

subsequently backfilled. Manna and Baidya (2009) analyzed full-scale bored cast-in-situ

concrete piles, while Sinha et al. (2015) reported results for a pipe-pile driven in an

undersized borehole with bentonite slurry placed beneath the toe. Each experimental

publications compares the observed response with that obtained by theoretical analysis.

It is reported that the nonlinear model is superior to the elastic model, as a closer

62

agreement with experimental data can be obtained provided that the numerous additional

parameters are appropriately selected (Elkasabgy and El Naggar, 2013).

The present study shows that the elastic model developed by Novak (1977) can

produce a response in good agreement with experimental data for driven piles provided

that modified material properties are used. The modified elastic model can produce a

response with accuracy comparable to that of the nonlinear model, while avoiding the

computational rigor and numerous parameter approximations associated with the

nonlinear model. It is shown that experimental data reported in the literature can be well

defined by the elastic model by simply modifying the shear modulus of the founding soil

media. Ranges of shear-moduli modification factors are presented for design

applications based on the experimental configurations analyzed in the present study.

4.2 Background

Novak’s (1977) theory assumes that the pile is elastic, oriented vertically,

circular in cross section, and perfectly bonded to the soil. The soil surrounding the pile is

modelled as a series of infinitesimally thin elastic horizontal layers, thus wave

propagation occurs only in the radial direction, and the pile tip is assumed to rest on an

elastic half-space. When the pile is subject to axial vibration at the pile head, the

governing differential equation for vertical deformation w at depth z is:

!!!(!)!!! + 1

!" !!! − !!!!! − ! !!! + !!!! ! ! = 0 (4.1)

Where A, E, µ, and c0 represent the pile cross sectional area, modulus of elasticity, mass

per unit length, and internal damping coefficient, respectively; Gs is the shear modulus

of the adjacent soil, ω is the harmonic vibration frequency, and Sw1 and Sw2 are soil

63

reaction parameters (defined by Baranov, 1967). The particular solution of Equation 4.1

produces the complex stiffness of the soil-pile system K*, as defined in Equation 4.2.

!∗ = −!"!! 0

! 0 = ! + !ℎ (4.2)

Where k and h are frequency-dependent stiffness and hysteretic damping coefficients,

respectively. Refer to Bryden et al. (2016) for an explicit derivation of the stiffness and

damping coefficients. The soil-pile system may then be defined as a simple single

degree of freedom system with pile head mass M; the amplitude of steady state vibration

A0 is equal to:

!! =!" !!

! −!!! ! + ℎ ! (4.3)

Where me is the eccentric-mass-moment of the dynamic load.

4.3 Published Experimental Data

Three independent sets of experimental data have been analyzed extensively for

the completion of the present study, which include those reported by: Novak and Grigg

(1976), El Marsafawi et al. (1992), and Elkasabgy and El Naggar (2013). The

experimental configuration and site conditions for each scenario have been summarized

in Table 4.1. The driven pile results reported by Puri (1988) could not be included in the

analysis due to a lack of available data, and the results presented by Sinha et al. (2015)

were omitted due to the additional complexities introduced by the undersized borehole

and bentonite slurry.

Tab

le 4

.1: P

hysi

cal P

rope

rties

of E

xper

imen

tal C

onfig

urat

ions

Phys

ical

/Mat

eria

l Pro

pert

y of

Exp

erim

ent

Nov

ak

and

Gri

gg

1976

(1

)

El

Mar

safa

wi

et a

l 199

2 (2

)

Elk

asab

gy

and

El

Nag

ger

2013

(3

) Pi

le P

rope

rties

:

R

adiu

s, r (

m)

0.04

5 0.

0508

0.

162

A

rea,

A (m

2 ) 0.

0014

4 0.

0019

0.

0094

1

B

ase

Are

a, A

b (m

2 ) 0.

0014

4 0.

0081

1 0.

0824

Le

ngth

, L (m

) 2.

25

2.75

8.

4

D

ensi

ty, ρ

(kg/

m3 )

7850

80

00

8000

El

astic

Mod

ulus

, E (G

PA)

200

200

210

Soil

Prop

ertie

s:

Pois

son

Rat

io, v

0.

25

0.25

0.

5

Sh

ear W

ave

Vel

ocity

of A

djac

ent S

oil,

V s (m

/s)

116.

5 12

5 20

0

Sh

ear W

ave

Vel

ocity

of B

ase

Soil,

Vb (

m/s

) 23

3 19

3 23

5

D

ensi

ty, ρ

s (kg

/m3 )

1796

17

80

1820

Lo

ad P

rope

rties

:

Pi

le H

ead

Mas

s, M

(kg)

12

31.9

94

1.0

4849

.5

Ec

cent

ric-M

ass-

Mom

ent,

me

(kg.

mm

) 9.

8 –

39.2

2.

45 –

9.8

4 91

– 2

10

64

65

Each experimental configuration was analyzed under three separate dynamic

axial loads within the eccentric-mass-moment range indicated in Table 4.1. The shear

wave velocity of the adjacent soil and base soil are defined in Equations 4.4a and 4.4b,

respectively.

!! =!!!!

(4.4a)

!! =!!!!

(4.4b)

Where Gs and Gb represent the shear modulus of the adjacent soil and base soil,

respectively.

Novak’s (1977) model assumes that the adjacent soil properties are uniform

across the entire pile length. Weighted average values for the adjacent soil shear wave

velocity and density have thus been reported in Table 4.1. The procedure developed by

Novak and Aboul-Ella (1978) accounts for piles penetrating layered soil media, and may

be used if a more thorough analysis is desired or if significant differences in material

properties are observed amongst layers. The soil Poisson ratio was assumed to be either

0.25 or 0.5 based on the dominating conditions of the particular test site, as indicated in

Table 4.1. Note that pile and soil material damping are neglected for the analysis.

The observed experimental results for configurations 1, 2, and 3 (for three

dynamic load conditions) are presented in Figures 4.2, 4.3, and 4.4, respectively. The

experimental values were obtained by graphical interpretation from the original

reference. The theoretical response for each experimental configuration, as defined in

Table 4.1, has been computed following Novak’s (1977) model and is presented in the

66

corresponding figure. As seen in Figures 4.2 through 4.4, the linear model does not

accurately represent the experimental results. This was also the conclusion attained in

the original publications, and thus the nonlinear model was employed to obtain a

theoretical response in agreement with the experimental data (Elkasabgy and El Naggar,

2013). Note that Novak and Grigg (1976) do in fact report a theoretical response in

agreement with the experimental data using the elastic model; this was because

frequency independent dynamic coefficients were used, which are not representative of

the underlying theory.

Figure 4.2: Comparison of Novak and Grigg’s (1976) Experimental Results with the Elastic Model

67

Figure 4.3: Comparison of El Marsafawi et al. (1992) Experimental Results with the Elastic Model

Figure 4.4: Comparison of Elkasabgy and El Naggar’s (2013) Experimental Results with the Elastic Model

68

4.4 Modified Elastic Model

A parametric study was conducted to fit the theoretical elastic response curves

with the experimental data. The soil-shear modulus is the material property with the

most influence on dynamic behavior, and was therefor chosen as the parameter of

interest. The shear moduli of the adjacent soil and base soil were modified to achieve

sufficient agreement between the theoretical and experimental response. The updated

values are termed the apparent shear modulus of the adjacent soil and base soil: Gs(app)

and Gb(app), respectively.

Figures 4.5, 4.6, and 4.7 contain the updated dynamic response for each

configuration based on the apparent shear moduli values specified in Table 4.2. Note

that all other physical parameters remain unchanged and are equal to those indicated in

Table 4.1. The shear moduli modification factors, α and β, are defined in Equations 4.5a

and 4.5b and have been indicated in Table 4.2 for each configuration.

! = !! !""!!

(4.5a)

! = !! !""!!

(4.5b)

As shown in Figures 4.5 through 4.7, the elastic model can closely represent the

experimental data for all three configurations provided that modified soil-shear moduli

values are used.

Tab

le 4

.2: A

ppar

ent S

hear

Mod

uli a

nd M

odifi

catio

n Fa

ctor

s

Exp

erim

enta

l Con

figur

atio

n O

rigi

nal S

hear

M

odul

us (M

Pa)

App

aren

t She

ar

Mod

ulus

(MPa

) M

odifi

catio

n Fa

ctor

G

s G

b G

s(ap

p)

Gb(

app)

α

β N

ovak

and

Grig

g 19

76

m

e =

9.8

kg.m

m

24.4

97

.4

17.9

9 47

3.1

0.73

80

4.85

3

m

e =

19.7

kg.

mm

24

.4

97.4

11

.48

610.

7 0.

4708

6.

263

m

e =

39.2

kg.

mm

24

.4

97.4

10

.64

398.

0 0.

4368

4.

082

El M

arsa

faw

i et a

l. 19

93

m

e =

2.45

kg.

mm

27

.8

66.3

6.

10

2642

.9

0.21

95

39.8

60

m

e =

4.92

kg.

mm

27

.8

66.3

3.

43

2608

.7

0.12

34

39.3

45

m

e =

9.84

kg.

mm

27

.8

66.3

3.

24

2137

.8

0.11

67

32.2

43

Elka

sabg

y an

d El

Nag

gar 2

013

m

e =

91 k

g.m

m

72.8

10

0.5

0.33

38

68.4

0.

0045

38

.488

m

e =

160

kg.m

m

72.8

10

0.5

0.48

49

65.5

0.

0066

49

.403

me

= 21

0 kg

.mm

72

.8

100.

5 0.

56

4085

.9

0.00

77

40.6

52

69

70

Figure 4.5: Comparison of Novak and Grigg’s (1976) Experimental Results with the Modified Elastic Model

Figure 4.6: Comparison of El Marsafawi et al. (1992) Experimental Results with the Modified Elastic Model

71

Figure 4.7: Comparison of Elkasabgy and El Naggar’s (2013) Experimental Results with the Modified Elastic Model

It is observed that the modification factors are dependent on the magnitude of the

applied dynamic load. A dimensionless quantity termed the dynamic load factor DLF

has been defined in Equation 4.6, which facilitates comparison amongst the

experimental configurations analysed.

!"# = !" !!!!!

(4.6)

Where the numerator represents the amplitude of dynamic loading at resonant frequency

ωr and Ws is the static axial load.

72

4.5 Discussion and Summary of Results

The present study shows that close agreement between Novak’s (1977) elastic

model and experimental data (for driven piles) can be attained by implementing a

reduction in adjacent soil shear modulus and an increase in base soil shear modulus. The

modified elastic model is observed to produce a response with accuracy equivalent to

that of the nonlinear model (refer to the nonlinear response plots presented by Elkasabgy

and El Naggar, 2013), while requiring sufficiently fewer parameter approximations.

The apparent reduction in adjacent soil shear modulus is hypothesized to result

from numerous nonlinear characteristics, including: the strain dependence of the soil

shear modulus, pile-soil separation, and lack of resistance mobilization. It has been

shown that the shear modulus of soil decreases as the state of strain increases

(Likitlersuang et al., 2013). The state of strain within the soil media surrounding a pile

subjected to axial vibration is dependent on the magnitude of static and dynamic loading

and varies both radially and axially, thus producing a complex nonlinear system.

When the pile loading is less than a critical value, there exists a point of fixity at

some depth along the pile shaft; only the shear resistance of the portion of pile above

such point of fixity is mobilized. Novak’s (1977) model produces stiffness and damping

values that are representative of the full pile length. It therefore seems logical that a

reduced adjacent soil shear modulus be used in Novak’s (1977) model to account for

regions where resistance is not utilized and for the reduction in shear modulus that

occurs in regions of larger strain.

The increase in apparent base soil shear modulus can be attributed to lack of

mobilization; the apparent shear modulus of the base soil may be more representative of

the pile material itself if there exists a point of fixity along the pile shaft. The observed

73

increase in base soil shear modulus could also be attributed to the installation method;

the experimental configurations analyzed in the present study were installed by driving,

thus increasing the compacted state of soil immediately surrounding the pile tip. Notably

lower β factors are reported for Novak and Grigg’s (1976) experiment in comparison to

El Marsafawi et al. (1992) and Elkasabgy and El Naggar (2013); this is likely attributed

to the fact that Novak and Grigg (1976) used open-ended pipe piles, while El Marsafawi

et al. (1992) and Elkasabgy and El Naggar (2013) used closed-ended pipe piles thus

imposing a higher degree of compaction.

Figures 4.8 and 4.9 show the shear modulus modification factors α and β as

functions of the dynamic load factor for each experimental configuration. The adjacent

stiffness ratios and base stiffness ratios have also been indicated in Figures 4.8 and 4.9,

respectively, for each configuration. Note that a transformed pile modulus of elasticity is

used when calculating the adjacent and base stiffness ratios indicated in Figures 4.8 and

4.9.

74

Figure 4.8: Modification Factor α vs. Dynamic Load Factor

Upon examination of Figure 4.8, one may conclude that the modification factor α

increases as the adjacent stiffness ratio increases. Significantly lower α values have been

reported for Elkasabgy and El Naggar’s (2013) results in comparison to the other two

configurations. This may also be attributed to the fact that, in addition to the lower

adjacent stiffness ratio, the portion of axial capacity utilized by Elkasabgy and El

Naggar (2013) was substantially less. The experiments conducted by Elkasabgy and El

Naggar (2013) utilized approximately 5% of the piles static axial capacity (48 kN static

load; 900 kN capacity), whereas El Marsafawi et al. (1992) utilized 53% of the static

axial capacity (9 kN static load; 17 kN capacity), and Novak and Grigg (1976) utilized

67% of the static axial capacity (12 kN static load; 18 kN capacity). The ultimate axial

capacities were approximated following the procedures outlines by Das (2011).

75

Figure 4.9: Modification Factor β vs. Dynamic Load Factor

There appears to be two distinct regions of curves in Figure 4.9: those

representative of close-ended driven pipe-piles, and that representative of an open-ended

driven pipe-pile. The modification factor β is observed to decrease as the axial capacity

utilization ratio increases amongst experimental configurations analyzed in this research.

The modification factor β may also be related to the base stiffness ratio, but a generic

correlation may not be concluded from the limited data.

Based on the limited data available, a range of modification factors has been

developed for design purposes. It is recommended that for low-displacement driven piles

(open-ended pipe piles): α be selected between 0.40 and 0.75, where a lower value

corresponds to a larger dynamic load, and β be selected between 4.0 and 6.5. It is

recommended that for high-displacement piles (closed-ended piles): α be selected

between 0.005 and 0.25, where a lower value corresponds to a lower adjacent soil

76

stiffness ratio, and β be selected between 38 and 50. The adjacent soil stiffness

modification factor α is observed to span a broad range; the upper and lower bounds

differ by a factor of 50 for high-displacement piles. Additional experimental

investigations are required to facilitate convergence of the specified modification factor

ranges. The experimental configurations analyzed for the present study involved piles

with a slenderness ratio (L/r) of approximately 50; further investigations are required to

investigate the impact of slenderness ratio on the modification factors.

4.6 Conclusion

Three independent sets of experimental data pertaining to the axial vibration of

driven piles have been analyzed in the present study. It is shown that Novak’s (1977)

elastic model can accurately represent the observed response for all three experimental

configurations provided that modified soil shear moduli values are used. A reduction in

adjacent soil shear modulus and an increase in base soil shear modulus can produce a

response with accuracy comparable to that of the nonlinear model. The shear modulus

modification factors are shown to depend on: the soil-pile stiffness ratio, the magnitude

of dynamic loading, the axial capacity utilization ratio, and the degree of soil disturbance

during installation. It is hypothesized that the modification factors are also dependent on

the pile slenderness ratio, but further investigation is required.

Based on the limited data analyzed in the present study, it is concluded that (for a

slenderness ratio of 50) shear moduli modification factors be selected in the range of: α

= 0.40 to 0.75 and β = 4.0 to 6.5 for low-displacement driven piles, and α = 0.005 to

0.25 and β = 38 to 50 for high-displacement driven piles.

77

The fact that modified shear moduli incorporated in the elastic model can

produce a response comparable to that of the nonlinear model has practical applications.

The elastic model is computationally simpler and avoids the approximation of weak

zone material properties. Nonlinear characteristics are present in most physical

scenarios, and the proposed modification factor concept introduces a simple method to

approximate such nonlinear properties. It is recommended that further experimental

investigations be completed to determine quantitative empirical correlations between the

shear moduli modification factors and the physical properties of experimental

configurations.

78

References

Arya, S.C., O’Neill, M.W., Pincus, G. (1979). Design of Structures and Foundations for

Vibrating Machines. Gulf Publication Co., Houston, TX.

Baranov, V.A. (1967). On the Calculation of an Embedded Foundation. Voprosy

Dinamiki i Prochnosti, 14: 195-209. (In Russian).

Bryden, C., Arjomandi, K., and Valsangkar, A. (2016). Explicit Frequency-Dependent

Equations for Vertical Vibration of Piles. Practice Periodical on Structural Design

and Construction, 10.1061/(ASCE)SC.1943-5576.0000311.

Canadian Geotechnical Society. (2006). Canadian Foundation Engineering Manual:

Fourth Edition. BiTech Publishers Ltd., Richmond, BC, Canada.

Das, B. (2011). Principles of Foundation Engineering, Cengage Learning, Stamford, CT

El Marsafawi, H., Han, Y.C., and Novak, M. (1992). Dynamic Experiments on Two Pile

Groups. Journal of Geotechnical Engineering, 118(4): 576-592.

Elkasabgy, M. and El Naggar, M.H. (2013). Dynamic Response of Vertically Loaded

Helical and Driven Steel Piles. Canadian Geotechnical Journal, 50: 521-535.

Han, Y., and Novak, M. (1988). Dynamic Behaviour of Single Piles Under Strong

Harmonic Excitation. Canadian Geotechnical Journal, 25: 523-534.

Han, Y.C., and Sabin, G.C.W. (1995). Impedances for Radially Inhomogeneous

Viscoelastic Soil Media. Journal of Engineering Mechanics, 121(9): 939-947.

10.1061/(ASCE)0733-9399(1995)121:9(939).

Likitlersuang, S. et al. (2013). Small Strain Stiffness and Stiffness Degradation Curve of

Bankok Clays. Soils and Foundations, 53(4): 498-509.

Manna, B., and Baidya, D.K. (2009). Vertical Vibration of Full-Scale Pile – Analytical

79

and Experimental Study. Journal of Geotechnical and Geoenvironmental

Engineering, 135(10): 1452-1461.

Novak, M. (1974). Dynamic Stiffness and Damping of Piles. Canadian Geotechnical

Journal, 11(4): 574-498.

Novak, M. (1977). Vertical Vibration of Floating Piles. Journal of Engineering

Mechanics Division, 103(1): 153-168.

Novak, M., and Aboul-Ella, F. (1978). Impedance Functions of Piles in Layered Media.

Journal of Engineering Mechanics Division, 104(3): 643-661.

Novak, M., and El Sharnouby, B. (1983). Stiffness Constants of Single Piles. Journal of

Geotechnical Engineering, 109(7): 961-974.

Novak, M., and Grigg, R. F. (1976). Dynamic Experiments with Small Pile Foundations.

Canadian Geotechnical Journal, 13(4): 372-385.

Novak, M., and Sheta, M. (1980). Approximate Approach to Contact Effects of Piles.

Dynamic Response of Pile Foundations: Analytical Aspects. Proceedings of the

ASCE National Convention, New York, NY. pp. 53-79.

Prakash, S., Sharma, H.D. (1990). Pile Foundations in Engineering Practice, John

Wiley & Sons. Hoboken, NJ.

Puri, V. K. (1988). Observed and Predicted Natural Frequency of a Pile Foundation.

Proceedings of the Second International Conference on Case Histories in

Geotechnical Engineering, St. Louis, Mo., Paper No. 4.41.

Sinha, S.K., Biswas, S., and Manna, B. (2015). Nonlinear Characteristics of Floating

Piles Under Rotating Machine Induced Vertical Vibration. Geotechnical and

Geological Engineering, 33:1031-1046.

80

U.S. Naval Facilities Engineering Command. (1983). Soil Dynamics and Special Design

Aspects. NAVFAC DM7.3, Alexandria, VA. USA.

81

5 General Conclusions and Recommendations

5.1 General Conclusions

This thesis addresses three topics pertaining to the analytical study of individual

piles subject to axial vibration, which include: (1) development of a closed form solution

to Novak’s (1977) elastic theory; (2) formulation of a new mathematical model for axial

vibration of tapered piles; and (3) incorporation of modified elastic parameters in

Novak’s (1977) theory to account for nonlinear characteristics of driven piles. The

specific conclusions for each topic are indicated in Chapters 2 through 4, respectively,

and have been summarized below:

1. The proposed explicit expressions are identical to the original theory developed

by Novak (1977), and are easily programmed in spreadsheet software for design

applications. The practicing engineer can obtain a response in compliance with

the underlying theory by implementing the proposed explicit expressions, thus

avoiding the various assumptions and interpolations associated with classical

design charts.

2. A new theoretical model for the axial vibration of tapered piles has been

developed following the material assumptions defined by Novak (1977) for

cylindrical piles. It is shown that the resonant amplitude of piles subject to axial

vibration can be significantly reduced with an increased taper angle. A simple

approximate solution is also presented, which is observed to be in good

agreement with the exact solution obtained by numerical integration.

82

3. It is shown that Novak’s (1977) elastic model can reasonably represent numerous

sets of experimental data reported in the literature provided that modified soil

shear moduli values are used. A reduction in adjacent soil shear modulus and an

increase in base soil shear modulus can produce a response in agreement with

experimental data. Based on the limited data available, ranges of shear moduli

modification factors have been provided for design applications. The proposed

modification factor concept introduces a simple method to account for nonlinear

characteristics.

5.2 Recommendations

The author wishes to make the following recommendations for future research:

• The explicit expressions developed in Chapter 2 are for the axial vibration case;

it is recommended that closed form solutions be developed for lateral and

torsional vibration following a similar methodology.

• It is recommended that experimental research be conducted to validate the

analytical model proposed in Chapter 3 for the axial vibration of tapered piles.

• It is recommended that additional experimental data be collected such that

empirical correlations between experimental configurations and the shear moduli

modification factors identified in Chapter 4 may be determined.

• It is recommended that additional research be conducted to investigate non-linear

effects.

83

References

Novak, M. (1977). “Vertical Vibration of Floating Piles.” Journal of Engineering Mechanics Division, 103(1): 153-168.

84

Appendix A: Background Information for Novak’s Theory

85

This section proceeds to derive the governing differential equation for the axial

vibration of a cylindrical pile based on Novak’s (1977) elastic model. Equation A.9d is

the starting point of Chapters 2 and 4 of this thesis (i.e. Equations 2.1 and 4.1).

Consider the soil-pile system depicted in Figure A.1, which is subjected to axial

harmonic vibration P(t) at the pile head. The pile has length L, radius r0, and cross-

sectional area A.

Figure A.1: Schematic Diagram of Soil-Pile Model

The pile head motion may be expressed as a simple single degree of freedom

system with equation of motion as shown in Equation A.1.

86

! !!!!!! + ! !"!" + !" = !! sin!" (A.1)

Where the right hand side of Equation A.1 is the harmonic forcing function with

amplitude P0 and frequency ω that produces vertical deformation W at time t; M is the

pile head mass, and k and c are the equivalent stiffness and damping parameters of the

soil-pile system, respectively.

Novak’s (1977) theory is used to determine the equivalent stiffness and damping

parameters, k and c, of the soil pile system. This is accomplished by assuming the soil

media is composed of two regions: the adjacent soil, and the base soil, as shown in

Figure A.1. To determine the equivalent stiffness and damping parameters of the soil-

pile system, a pile differential element of length dz is analyzed under harmonic

vibration. Consider the differential element shown in Figure A.2, where P + dP is the

axial load at depth z, P is the axial load at depth z + dz, and S is the adjacent soil reaction

force acting on the circumferential area.

Figure A.2: Pile Differential Element

87

Dynamic vertical equilibrium of the differential element in Figure A.2, including the

inertial and damping forces, leads to the following:

! − ! + !" + !"# + !"# !!!!!! + !!!"

!"!" = 0 (A.2a)

!"!" = ! !

!!!!! + !!

!"!" + ! (A.2b)

Where w is the axial deformation as a function of depth z, and c0 and µ represent the

internal damping and mass per unit length, respectively. The soil reaction force S was

defined by Baranov (1967), and is equal to:

! = ! !!! + !!!! ! (A.3)

Where G is the adjacent soil shear modulus, and Sw1 and Sw2 are soil reaction parameters.

Refer to Appendix B for a complete derivation of Baranov’s (1967) reaction parameters.

Combining Equations A.2b and A.3 leads to:

!"!" = ! !

!!!!! + !!

!"!" + ! !!! + !!!! ! (A.4)

Assuming that the pile deformation follows a linear-elastic response, the stress-

strain relationship of the pile material may be expressed as:

! = !" → !! = ! !"!" (A.5)

Where σ, ε, and E represent the pile stress, strain, and modulus of elasticity, respectively.

Differentiating Equation A.5 with respect to depth z produces:

!"!" = !" !

!!!!! (A.6)

88

Substituting Equations A.6 in A.4 leads to:

! !!!!!! + !!

!"!" + ! !!! + !!!! ! − !" !

!!!!! = 0 (A.7)

Equation A.7 is a homogeneous second order partial differential equation with constant

coefficients. The deformation w may be expressed as:

! = ! ! !!"# (A.8)

Where ω is the vibration frequency. Substitution of Equation A.8 in A.7 produces:

! !!

!!! ! ! !!"# + !!!!" ! ! !!"# + ! !!! + !!!! ! ! !!"#

− !" !!!!! ! ! !!"# = 0

(A.9a)

−!!!! ! !!"# + !!!!" ! !!"# + ! !!! + !!!! ! ! !!"# − !"!!"# !!! !!!!

= 0

(A.9b)

−!! !!! !!!! + −!!! + !!!! + ! !!! + !!!! !(!) = 0 (A.9c)

!!! !!!! + 1

!" !!! − !!!! − ! !!! + !!!! !(!) = 0 (A.9d)

Equation A.9d is the governing differential equation for the axial deformation of

a cylindrical pile based on Novak’s (1977) elastic model. This is the starting point of

Chapters 2 and 4 of this thesis. Refer to Section 2.2 for the remaining steps in deriving

the equivalent stiffness and damping parameters k and c of the soil-pile system.

89

Once expressions for the equivalent stiffness and damping parameters of the soil-

pile system have been defined, the equation of motion for vertical deformation at the pile

head (Equation A.1) may be analyzed. The solution to Equation A.1 is of the form:

! ! =!! ! +!!(!) (A.10)

Where Wp(t) is the particular solution, and Wc(t) is the complementary solution. It is

assumed that start-up and ramp-down of machines occurs over sufficient time such that

transient motion is negligible. For steady state vibration of a damped system, the

complementary component vanishes. The steady state solution of Equation A.1 is

therefore equal to:

! ! !"#$%&!!"#"$ =!! ! = !! sin!" + !! cos!" (A.11)

Where C1 and C2 are constants of integration, which may be evaluated using the method

of undetermined coefficient:

!! =!! ! −!!!

!!! − ! ! + !" ! (A.12a)

!! = − !!!"!!! − ! ! + !" ! (A.12b)

The steady state solution of Equation A.1 is therefore:

!(!) = !! ! −!!!

!!! − ! ! + !" ! sin!" −!!!"

!!! − ! ! + !" ! !! cos!" (A.13)

The amplitude of steady state vibration is then equal to:

!! =!!

! −!!! ! + !" ! (A.14)

90

Since the entire model is based on elastic material assumptions, it is convenient

to normalize the amplitude of steady state vibration. For dynamic loads originating from

an eccentric rotating mass, the amplitude of the forcing function is equal to:

!! = !"!! (A.15)

Where me represents the mass-moment of the rotating body (i.e. the mass of the rotating

body multiplied by its eccentricity). The dimensionless amplitude of steady state

vibration is then defined as:

!! = !!!!" → !! =

!!

!! − !!

!+ !"

!! (A.16)

The amplitude of vibration approaches me/M at high frequencies; the

dimensionless amplitude therefore approaches a value of 1 at high frequencies. The

dimensionless amplitude is used throughout this thesis to develop dynamic response

plots.

91

Appendix B: Derivation of Adjacent Soil Reaction Parameters

92

The adjacent soil reaction parameters, referred to by Novak (1977) as Sw1 and Sw2,

were derived by Baranov (1967). The original derivation was published in Russian, and

multiple steps were omitted from the publication. The following section proceeds to re-

derive the adjacent soil reaction parameters from basic principles. The detailed

derivation is provided for clear understanding of the fundamental assumptions

incorporated in Novak’s (1977) dynamic model.

Consider a circular pile that is embedded in an elastic half-space and oriented

perpendicular to the free surface. When the pile is subject to vertical vibration, the

dynamic motion is transferred to the soil media along the piles circumferential area. It is

assumed that the pile is perfectly connected to the soil and that shear waves propagate

radially from the piles longitudinal axis. The concept of shear waves propagating only in

the radial direction is equivalent to dividing the elastic half-space into infinitesimally

thin non-interacting horizontal layers. The model is axisymmetric about the pile’s

longitudinal axis therefore cylindrical coordinates are desirable. The problem will be

formulated in Cartesian coordinates for simplicity, and a change of coordinates will

subsequently be imposed.

A soil-media differential element has been presented in Figure B.1. Only the

vertical stresses are of interest, as indicated in Figure B.1. The stresses on opposing

faces differ by an amount equal to the stress-gradient multiplied by the elemental

thickness.

93

Figure B.1: Three-Dimensional Stress Element with Vertical Stresses Indicated

Dynamic vertical equilibrium of the element leads to the expression in Equation B.1.

!! = !" (B.1a)

!! +!!!!" !" − !! !"!# + !!" +

!!!"!" !" − !!" !"!#

+ !!" +!!!"!" !" − !!" !"!# = !"#"$"% !

!!!!!

(B.1b)

Where the differential element of dimensions dx, dy, and dz has vertical normal stress σz,

vertical shear stresses τxz and τyz, density ρ, and is subject to vertical displacement w at

time t.

! !

!

!"!"

!"τ!"

τ!" +!τ!"!! !"

τ!! +!τ!!!! !!

τ!!

σ! +!σ!!! !!

σ!

¥ This is the approximate expression for volumetric strain, which is valid for the small strain condition of a differential element.

94

Simplification of Equation B.1b produces:

!!!!" +

!!!"!" + !!!"!! = ! !

!!!!! (B.2)

The three-dimensional elemental stress-strain relationships have been defined in

Equation B.3.

!! = !! + 2!!! (B.3a)

!!" = !!!" (B.3b)

!!" = !!!" (B.3c)

Where λ is the Lamé constant, G is the shear modulus, ! is the volumetric strain, εz is the

axial strain in the z direction, and γ is the shear strain in the corresponding plane. Recall

the following additional definitions:

! = !! + !! + !! ¥ (B.4a)

!! =!"!" , !! =

!"!" , !! =

!"!" (B.4b)

!!" =!"!" +

!"!" (B.4c)

!!" =!"!" +

!"!" (B.4d)

Where u represents x-deformation, v represents y-deformation, and w represents z-

deformation. Substitution of the expressions in Equation B.4 into Equation B.2 leads to:

!!" !! + 2!!! + !

!" !!!" + !!" !!!" = ! !

!!!!! (B.5a)

! !!"!"!" +

!"!" +

!"!" + 2! !

!"!"!" + ! !

!"!"!" +

!"!" + ! !

!"!"!" +

!"!"

= ! !!!!!!

(B.5b)

95

Using Young’s Theorem (the fact that the order of differentiation does not matter)

Equation B.5b can be simplified as follows:

! !!"!"!" +

!"!" +

!"!" + 2! !

!!!!! + !

!!!!"!# +

!!!!!! +

!!!!"!# +

!!!!!! = ! !

!!!!! (B.6a)

! !!"!"!" +

!"!" +

!"!" + 2! !

!!!!! + !

!!"

!"!" +

!"!" + ! !!!

!!! +!!!!!!

= ! !!!!!!

(B.6b)

! !!"!"!" +

!"!" +

!"!" + ! !

!"!"!" +

!"!" +

!"!" + ! !!!

!!! +!!!!!! +

!!!!!!

= ! !!!!!!

(B.6c)

! + ! !!!" + !

!!!!!! +

!!!!!! +

!!!!!! = ! !

!!!!! (B.6d)

Note that the second term in Equation B.6d contains the Laplacian operator in Cartesian

coordinates; Equation B.6d can therefore be expressed as:

(! + !) !!!" + !∇!! = ! !

!!!!! (B.7)

Equation B.7 is the governing equation of motion. Due to the axisymmetric

condition of the problem, it is convenient to use cylindrical coordinates. The volumetric

strain and Laplacian operator have been defined in cylindrical coordinates in Equation

B.8.

!!!" = !! + !! + !! =

!!!!" + 1

!!!!!" + !!! + !"!" (B.8a)

∇!! = !!!!!! +

!"!"! +

!!!!!!!! +

!!!!!! (B.8b)

96

For axisymmetric vertical vibration of an infinitesimally thin horizontal layer, the

following conditions apply:

!! = !! =!"!" =

!"!" = 0 (B.9)

With the conditions indicated in Equation B.9, Equation B.7 may be expressed in

cylindrical coordinates as:

! !!!!!! +

!"!"! = ! !

!!!!! (B.10a)

!!!!!! +

!"!"! −

!!!!!!!! = 0 (B.10b)

For vertical vibration of the form:

! !, ! = !!"#!(!) (B.11)

Equation B.10b may be expressed as:

!!!!! !!"#! ! + 1!

!!" ! ! !!"# − !

!!!!!! ! ! !!"# = 0 (B.12a)

!!"#!′′+ 1! !!"#!′− !

!! !!!!!!"# = 0 (B.12b)

!!! + 1!!! + !

!!! ! = 0 (B.12c)

Where w’ represents a radial derivative (with respect to r). Equation B.12c is a variation

of Bessel’s equation of order zero, and has general solution of the form:

! ! = !!!!(!)!!!

! ! + !!!!(!)!!!

! ! (B.13)

97

Where c1 and c2 are constants of integration, and Hn(1)(x) and Hn

(2)(x) are Hankel

functions of the first and second kind (order n), respectively, which are defined as:

!!! ! = !! ! + !!!(!) (B.14a)

!!! ! = !! ! − !!!(!) (B.14b)

Where Jn(x) and Yn(x) are Bessel functions of the first and second kind, respectively

(order n).

Based on the time dependence defined in Equation B.11: the H0(1)(x) term is

representative of converging (incoming) waves, while the H0(2)(x) term is representative

of diverging (outgoing) waves. Since the only source of vibration is located at the origin

and the elastic medium extends to infinity, only divergent waves will exist; the

coefficient for the H0(1)(x) term c1 must therefore equal zero.

The second integration constant may be defined in terms of the pile’s axial

deformation. At the pile edge, the vertical deformation w(r0) may be defined as:

!(!!) = !!!!(!)!!!

! !! (B.15a)

!! =! !!

!!(!) !!!

! !!

(B.15b)

The particular solution to Equation B.12c is thus:

! ! = ! !!!!!

!!!! !

!!(!) !!!

! !! (B.16)

98

The shear stress τrz along a cylindrical fronts circumferential area is equal to:

!!" = !!!" = ! !"!" = ! !!" ! !!

!!!!!!! !

!!!!!!! !!

= !" !!!!!

!!!! !!

!!" !!!

!!!

! !

(B.17a)

!!" =!" !!

!!!!!!! !!

− !!!

! !!!!!!

! ! (B.17b)

Evaluated at the pile edge, the shear stress along the piles circumferential area is equal

to:

!!" = −!" !!!!!

!!!!

!!!! !!

!!!!!!! !!

(B.18)

Integrating the stress along the pile circumferential area for a unit length, the reaction

force S is equal to:

! = − !!" 1 !! !"!!

!= −2!!!!!" = 2!"# !!

!!!

! !!!!!

!!!! !!

!!!!!!! !!

(B.19)

Defining the dimensionless frequency a0 as:

!! =!!!

! !! =!!!!! =

!!!!! (B.20)

99

The soil reaction force may be expressed as:

! = 2!"# !! !!!!! !!!!! !!

(B.21)

Which may be separated into real and imaginary components using the definition of the

Hankel functions (defined in Equation B.14):

! = 2!"# !! !!!! !! − !!! !!!! !! − !!!(!!)

(B.22a)

! = 2!"# !! !!!! !! − !!! !!!! !! − !!! !!

!! !! + !!! !!!! !! + !!! !!

(B.22b)

! = 2!"# !! !!!! !! !! !! + !! !! !! !! + ! !! !! !! !! − !! !! !! !!

!!! !! + !!! !! (B.22c)

! = 2!"# !! !!!! !! !! !! + !! !! !! !!

!!! !! + !!! !!+ ! !! !! !! !! − !! !! !! !!

!!! !! + !!! !! (B.22d)

Using the following property:

!! !! !! !! − !! !! !! !! = 2!!!

(B.23)

The soil reaction force S may be expressed as shown in Equation B.24.

! = 2!"# !! !!!! !! !! !! + !! !! !! !!

!!! !! + !!! !!+ !

2!!!

!!! !! + !!! !! (B.24a)

! = !" !! 2!!!!! !! !! !! + !! !! !! !!

!!! !! + !!! !!+ ! 4

!!! !! + !!! !! (B.24b)

! = ! !!! + !!!! ! !! (B.24c)

100

Where Sw1 and Sw2 are the adjacent soil reaction parameters defined as:

!!! = 2!!!!! !! !! !! + !! !! !! !!

!!! !! + !!! !! (B.25a)

!!! =4

!!! !! + !!! !! (B.25b)

For harmonic axial vibration, the vertical deformation at the soil-pile interface varies

with time and depth as shown in Equation B.26.

!(!!) = ! !, ! = ! ! !!"# (B.26)

The soil reaction force along the pile shaft (per unit length) is therefore equal to:

! = ! !!! + !!!! !(!, !) (B.27)

The expression in Equation B.27 is the soil reaction model presented by Baranov

(1967), which is also the adjacent soil-reaction model used by Novak (1977). Figure B.2

shows the values of reaction parameters Sw1 and Sw2 for the small values of dimensionless

frequency commonly encountered during the dynamic analysis of pile foundations.

Figure B.2: Adjacent Soil Reaction Parameters vs. Dimensionless Frequency

101

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RE: Authors Permission for Reuse of Own Material

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P PERMISSIONS <permissions@asce.org> #Fri 1/6, 2:30 PM

Campbell William Bryden $

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2017-02-22, 9:21 AMRE: Authors Permission for Reuse of Own Material

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From:CampbellWilliamBryden[mailto:c.bryden@unb.ca]Sent:Friday,January06,20171:23PMTo:PERMISSIONS<permissions@asce.org>Subject:AuthorsPermissionforReuseofOwnMaterial

Hello,

Iamintheprocessofcomple>ngmygraduateprogramattheUniversityofNewBrunswick,andhaveauthoredmaterialpublishedinASCE'sPrac,cePeriodicalonStructuralDesignandConstruc,on.Iamlookingtoobtainale6erofpermissionfromASCEsta>ngthatIhavetherighttoincludethemanuscriptinmythesis(i.e.thefinalversionpriortoASCEcopyedi>ng).Thear>cleofinterestislistedbelow:

Bryden,C.,Arjomandi,K.,andValsangkar,A.(2016).ExplicitFrequency-DependentEqua>onsforVer>calVibra>onofPiles.Prac,cePeriodicalonStructuralDesignandConstruc,on,DOI:10.1061/(ASCE)SC.1943-5576.0000311

Pleaseletmeknowifyourequireanyaddi>onalinforma>on.Thankyou,

CampbellW.Bryden,B.Sc.Eng.ResearchAssistant,DepartmentofCivilEngineeringUniversityofNewBrunswickoffice:H229,HeadHalltel:506-447-0334

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Curriculum Vitae

Candidate’s full name: Campbell William Bryden Universities attended: University of New Brunswick, M.Sc.E. (Civil Engineering), 2015-2017 University of New Brunswick, B.Sc.E. (Civil Engineering; Minor in Mathematics), 2015 Publications: Bryden, C., Arjomandi, K., and Valsangkar, A. (2017). “Modified Elastic Parameters for

the Dynamic Axial Impedance of Driven Piles.” Submitted to Géotechnique Letters. Bryden, C., Arjomandi, K., and Valsangkar, A. (2016). “Dynamic Axial Stiffness and

Damping Parameters of Tapered Piles.” Submitted to the International Journal of Geomechanics.

Bryden, C., Arjomandi, K., and Valsangkar, A. (2016). “Explicit Frequency-Dependent

Equations for the Vertical Vibration of Piles.” Practice Periodical on Structural Design and Construction. doi:10.1061/(ASCE)SC.1943-5576.0000311.