Nondestructive Evaluation of Pre-stressed Concrete Cylinder Pipe … · 2016. 1. 22. · the modal...

Nondestructive Evaluation of Pre-stressed Concrete

Cylinder Pipe by Resonance Acoustic Spectroscopy:Theoretical and Modelling Considerations

by

Jonathan Lesage

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of Mechanical and Industrial EngineeringUniversity of Toronto

c� Copyright 2015 by Jonathan Lesage

Abstract

Nondestructive Evaluation of Pre-stressed Concrete Cylinder Pipe by Resonance

Acoustic Spectroscopy: Theoretical and Modelling Considerations

Jonathan Lesage

Doctor of Philosophy

Graduate Department of Mechanical and Industrial Engineering

University of Toronto

2015

The theoretical basis for adapting the technique of resonance acoustic spectroscopy (RAS)

for the purposes of inspecting buried, in-service water mains composed of prestressed

concrete cylinder pipe (PCCP, lined-type) is presented in this thesis. The proposed

nondestructive evaluation (NDE) technique is sensitive to degradation of the outermost

layer (a protective mortar coating) of PCCP which occurs over only part of the pipe’s

circumference. The frequency spectrum of healthy pipes is evaluated through a sequence

of experimental measurements and finite element modelling studies (modal analyses).

Several simplifying assumptions about the vibratory response of PCCP are gleaned from

the modal analysis of healthy pipes, facilitating the development of a model of buried,

water filled sections of pipe subject to varying degrees of damage. The model treats

the pipe as a multilayered cylinder in plane strain condition and is solved via a transfer

matrix (T-Matrix) method which has been extended by the author to accommodate

non-axisymmetric elastic parameters (to represent pipe damage) and coupling to media

external to the pipe (to represent coupling to soil and water). The model is then used to

assess the e↵ects of coupling to the surrounding soil, as well as the e↵ects of increasing

levels of mortar damage. From the results of these investigations, a novel damage metric

called the asymmetry index is defined. This metric is appropriate for assessing damage to

ii

the protective mortar coating, and is based on the splitting of degenerate flexural modes

which occurs when the axial symmetry of a pipe section is perturbed. Finally, a procedure

for collecting and processing the resonant spectra from pipes in the field is presented.

This procedure allows for the asymmetry indices associated with di↵erent vibrational

modes to be computed at various axial locations along the pipeline thus providing a

map of mortar damage along the water main. The primary advantage of the proposed

technique is that it does not require precise knowledge of the pipe’s dimensions or elastic

parameters. In addition, the technique is applicable for any type of surrounding soil.

iii

Acknowledgements

First and foremost, I would like to express my sincerest gratitude and appreciation to my

supervisor, Professor Anthony N. Sinclair, for his guidance, support and encouragement

throughout the entire course of this project. It is due to his confidence in me, as well as

his seemingly boundless patience that I have been able to do my best work; for this I am

truly thankful.

The financial support of ANDEC Manufacturing Ltd. as well as the Ontario Centres of

Excellence is greatly appreciated.

I’d like to thank MUNRO Ltd. for access to their facilities. This was essential for the

experimental components of this project.

Finally, I would like to thank all my great friends and colleagues (past and present) at

UNDEL. I can only hope to be able to work amongst such wonderful and talented people

in the future.

iv

Dedication

To my parents - I am, as always, precariously balanced on your shoulders.

v

Contents

1 Introduction 1

1.1 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Background and Literature Review 7

2.1 Prestressed Concrete Cylinder Pipe . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Basic Structure of PCCP . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Failure of PCCP . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.3 Nondestructive Evaluation . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Resonance Acoustic Spectroscopy . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

vi

2.2.2 Fundamentals of Elastic Resonance . . . . . . . . . . . . . . . . . 14

2.2.3 Resonant Modes of Elastic Cylinders . . . . . . . . . . . . . . . . 17

2.2.4 Circumferential Mode Classification . . . . . . . . . . . . . . . . . 21

2.2.5 Axial Wavelength and Dispersion Relations . . . . . . . . . . . . . 23

2.2.6 Forced Vibration and Attenuation . . . . . . . . . . . . . . . . . . 25

2.2.7 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . 28

3 Characterization of Healthy Pipes 32

3.1 Geometry of Tested/Modelled Pipes . . . . . . . . . . . . . . . . . . . . . 33

3.2 Material Properties of Pipe Constituents . . . . . . . . . . . . . . . . . . 34

3.3 Finite Element Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3.1 E↵ect of Prestress . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2 E↵ect of Bell and Spigot . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.2 Coupling Between Pipes . . . . . . . . . . . . . . . . . . . . . . . 48

vii

4 Mathematical Modelling of Damaged Pipes 53

4.1 Transfer Matrix for Asymmetric Layers . . . . . . . . . . . . . . . . . . 55

4.1.1 Structure of the Coupled T-matrix . . . . . . . . . . . . . . . . . 59

4.2 Axisymmetric, Isotropic Layers . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 Coupling to External Media and Loading Considerations . . . . . . . . . 62

4.3.1 Coupling to Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.2 Coupling to Water . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.3 Loading Considerations . . . . . . . . . . . . . . . . . . . . . . . 67

4.3.4 Solution of Global System . . . . . . . . . . . . . . . . . . . . . . 68

4.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 70

5 Modelling Results for Damaged Pipes 73

5.1 Model of Mortar Damage . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Comparison to Finite Element Results . . . . . . . . . . . . . . . . . . . 76

5.3 E↵ect of Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5.4 E↵ect of Mortar Sti↵ness . . . . . . . . . . . . . . . . . . . . . . . . . . 87

viii

5.5 E↵ect of Damage Thickness . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.6 E↵ect of Damage Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.7 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Proposed Nondestructive Testing Procedure 94

6.1 Sampling and Circumferential Harmonics . . . . . . . . . . . . . . . . . 95

6.2 Localization of Symmetry Axis . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Conclusions and Recommendations 102

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

7.2 Recommendations For Future Work . . . . . . . . . . . . . . . . . . . . . 105

7.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

A Mathematical Expressions 109

B MATLAB Functions 111

C Equivalent Properties for the Mortar/Steel Winding Layer 114

ix

References 119

x

List of Figures

2.1 Cross-section of LCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Ruptured segment of LCP . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Reference Coordinates and Dimensions for the Infinite Cylinder . . . . . 18

2.4 Breathing Mode: n = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Bending/Axial Shear Mode: n = 1 . . . . . . . . . . . . . . . . . . . . . 22

2.6 Flexural Mode: n = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Flexural Mode: n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8 E↵ect of Increasing Attenuation . . . . . . . . . . . . . . . . . . . . . . . 28

2.9 Block diagram for swept frequency RAS experimental setup . . . . . . . 29

2.10 Block diagram for impact testing . . . . . . . . . . . . . . . . . . . . . . 30

3.1 Simplified Model of Healthy LCP . . . . . . . . . . . . . . . . . . . . . . 36

xi

3.2 Mesh for 600mm Section of LCP . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Measured Mode Shapes for the 600mm pipe section . . . . . . . . . . . . 45

3.5 Excitation point Frequency Response Function magnitude: 600mm pipe . 46

3.6 Excitation point Frequency Response Function magnitude: 1200mm pipe 47

3.7 Excitation Point Frequency Response Function Magnitude, 2 sections of

600mm pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Cylindrical annulus with dimensions and cylindrical coordinate system

defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.1 Parametrized Model of Damaged PCCP in Soil . . . . . . . . . . . . . . 74

5.2 Excitation point frequency response function, 600 mm pipe, freely sup-

ported, water filled . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.3 E↵ect of increasing soil dimensions on the frequency response of a 600 mm

pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.4 Comparison of normalized excitation point frequency response functions

computed via finite element analysis and the coupled T-matrix method . 80

5.5 Radial Displacement Magnitude Spectra Measured Coincident with Exci-

tation for Various Soil Types . . . . . . . . . . . . . . . . . . . . . . . . . 83

xii

5.6 Normalized Radial Displacement Magnitude Spectra for n=2,3,4; Adrian,

Catlin and Plainfield Soil . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.7 E↵ect of Mortar Sti↵ness on Asymmetry Indices . . . . . . . . . . . . . . 88

5.8 E↵ect of Damage Thickness on Asymmetry Indices . . . . . . . . . . . . 90

5.9 E↵ect of Damage Angle on Asymmetry Indices . . . . . . . . . . . . . . . 91

6.1 Diagram of Inspection Setup . . . . . . . . . . . . . . . . . . . . . . . . . 95

C.1 Mortar/Steel Winding Layer with Unit Cell Geometry as well as Cylindri-

cal and Fibre Aligned Coordinate Systems Defined . . . . . . . . . . . . 115

xiii

List of Tables

3.1 Geometric parameters for 600mm and 1200mm sections of PCCP, dimen-

sions given in millimetres . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.2 Measured Longitudinal and Shear Speeds for Concrete and Mortar . . . . 38

3.3 Computed eigenfrequencies, with and without considering the e↵ect of

prestress; 600mm pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Computed Eigenfrequencies, With and Without Bell and Spigot Features:

600mm pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5 Computed Eigenfrequencies, With and without bell and spigot features:

1200mm pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.6 Experimental and computed eigenfrequencies: 600mm PCCP . . . . . . . 46

3.7 Experimental and computed eigenfrequencies: 1200mm PCCP . . . . . . 51

3.8 Experimental and Computed Eigenfrequencies, 2 sections of 600mm PCCP 51

xiv

5.2 Wave Speeds and Densities for Adrian, Catlin and Plainfield soils . . . . 82

5.1 Comparison between computed eigenfrequencies and peak frequencies for

a fluid filled damage pipe . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.3 Parameters for Damage Cases 1, 2, 3: Decreasing Mortar Sti↵ness . . . . 88

5.4 Parameters for Damage Cases 4, 5, 6: Increasing Damage Thickness . . . 89

5.5 Parameters for Damage Cases 7, 8, 9: Increasing Damage Angle . . . . . 91

C.1 Fibre and Matrix Properties . . . . . . . . . . . . . . . . . . . . . . . . . 118

xv

List of Symbols

a radius at which pressure field is sampled

a(1,2)n

2⇥ 1 vectors containing the nth integration constants for the elastodymamic solu-

tion in soil

Arr

i,o

, Ar✓

i,o

, Arz

i,o

forcing amplitudes for infinite isotropic cylinder

An

, Bn

, Cn

, Dn

, En

, Fn

nth integration constants/modal amplitudes for isotropic cylinder

solution

b transducer diameter

b(1,2)n

pressure field integration constants

cij

(r, ✓) spatially varying sti↵ness matrix components: i, j = 1, 2, 3

cf

longitudinal wave speed in water

cL

longitudinal wave speed

cL

e

longitudinal wave speed in soil

cp

phase velocity of axially propagating waves

cT

shear wave speed

cT

e

shear wave speed in soil

xvi

Cijkl

general sti↵ness tensor

C(r, ✓) spatially varying sti↵ness matrix

f(t) applied force signal

f(✓,!) =

fr

f✓

�

T

load vector applied to pipe/fluid interface

f (1,2)n

Fourier coe�cients of the load vector applied to the pipe/fluid interface

F (!) Fourier transform of applied force signal

F Fourier coe�cients of the loading on the pipe boundaries

gm

(t) pressure signal recorded by the mth transducer

Gm

(!) Fourier transform of pressure signal recorded by the mth transducer

Hij

(!) frequency response function for ith circumferential position and jth axial position

Hm

(!) component of the frequency domain pressure response normal to the mth trans-

ducer, deconvolved with the input frequency response

Hn

(.) Hankel functions of the first kind (outward propagating), order n

Jn

(.) Bessel function of the first kind, order n

k axial wavenumber

kL

longitudinal wavenumber

kL

e

longitudinal wavenumber in soil

kT

shear wavenumber

kT

e

shear wavenumber in soil

l cylinder length

xvii

L number of discrete frequencies

L1

(✓, @

@✓

),L2

(✓, @

@✓

) 4⇥ 4 linear operator matrices

M number of circumferential locations where the pressure field is to be sampled

Mi

2N ⇥ 2N coe�cient matrix relating fluid displacement Fourier coe�cients to the

corresponding integration constants

Mo

4(N+1)⇥4(N+1) coe�cient matrix relating soil displacement Fourier coe�cients

to the corresponding integration constants

n circumferential wavenumber

N number of terms in Fourier expansion

p(r, ✓,!) pressure field in water

P number of sublayers in the layer approximate model

Q number of axial sample locations along the pipeline

r radial coordinate

R annulus/pipe inner radius

se

vector containing the normal and shear components of stress in soil

Se

o

2(N + 1) ⇥ 1 vector containing the Fourier coe�cients of the normal and shear

components of stress in soil evaluated on the pipe’s outer surface

t time

T(!) transfer matrix

T(1,2)n

(!) 4⇥ 4 block for the relating the nth Fourier coe�cient of the state vector evalu-

ated at the layer’s inner surface to the state vector evaluated at the outer surface

xviii

Tp

(!) transfer matrix for the pth sublayer

u radial displacement field component

u(t) measured acceleration signal

u displacement field vector

ue

displacement field vector in soil

uf

displacement field in water

uo,i

(1,2)(!) (N + 1)⇥ 1 vector containing the Fourier expansion coe�cients of the radial

displacement

u(1,2)n

(!) is the magnitude of the nth normalized frequency response function (radial com-

ponent of displacement)

u(1,2)n

L ⇥ 1 vector defining the symmetric and anti-symmetric components of the nth

Fourier coe�cient of the radial displacement evaluated at the pipe/water interface

u0(1,2)n

L ⇥ 1 vector defining the symmetric and anti-symmetric components of the nth

Fourier coe�cient of the radial displacement evaluated at the pipe/water interface,

computed with respect to the primed coordinate system

U(!) Fourier transform of radial displacement signal

U(!) Fourier transform of measured acceleration signal

U Fourier coe�cients of displacements evaluated at the pipe boundaries

Ue

o

2(N+1)⇥1 vector containing the Fourier components of the displacement solution

in soil evaluated on the pipe’s outer surface

Uf

i

=

U(1)f

i

U(2)f

i

�

T

vector containing the Fourier coe�cients of the radial displace-

ment in the fluid evaluated at r = R

xix

v circumferential displacement field component

vo,i

(1,2)(!) (N + 1) ⇥ 1 vector containing the Fourier expansion coe�cients of the cir-

cumferential displacement

w axial displacement field component

Wi

2N ⇥ 2N coe�cient matrix relating fluid pressure Fourier components to the cor-

responding integration constants

Wo

4(N + 1) ⇥ 4(N + 1) coe�cient matrix relating soil stress Fourier coe�cients to

the corresponding integration constants

x horizontal coordinate

x0 pipe’s approximate symmetry axis

xi,o

state vectors evaluated at the annulus’ inner (i) and outer (o) radii

x(1,2) 4(N +1)⇥1 vectors containing the Fourier coe�cients of the state variable vector

for the symmetric (1) and anti-symmetric (2) modes

x(1,2)n

nth Fourier coe�cient of the state variable vector for the symmetric (1) and anti-

symmetric (2) modes

X(r, ✓,!) vector of state variables

y vertical coordinate

Yn

(.) Bessel function of the second kind, order n

z axial coordinate

Z(!) frequency dependent compliance matrix

↵ angle between impact and pipe symmetry axis

xx

� mortar damage angle

� half of the circumferential footprint of the transducer

� mortar damage thickness

✏ij

strain components, i, j = r, ✓, z

✏n

Neumann factor

⌘L

longitudinal loss factor

⌘T

shear loss factor

⇣n

projection of the normalized symmetric frequency response onto the normalized

anti-symmetric response

✓ angular coordinate

⇥(1,2)n

(✓) nth circumferential basis function matrices for the state variable vector

⇥(1,2)n

(✓) nth circumferential basis function matrices for the external and internal media

r gradient operator

r2 Laplace operator

�, µ Lame Parameters

⇤ asymmetry index, summed over N modes

⇤n

nth asymmetry index

⇤n

nth asymmetry index, averaged over Q axial positions

⇢ density

⇢e

density of soil

xxi

⇢f

density of water

�ij

stress components, i, j = r, ✓, z

�rro,i

(1,2)(!) (N+1)⇥1 vector containing the Fourier expansion coe�cients of the radial

normal stress

�r✓o,i

(1,2)(!) (N+1)⇥1 vector containing the Fourier expansion coe�cients of the shear

stress in the r � ✓ plane

� scalar displacement potential

vector displacement potential

! circular frequency

xxii

Chapter 1

Introduction

Over the past decade, the aging North American water utilities infrastructure has become

a major cause for concern; much of the water distribution network in the United States

and Canada was installed over 50 years ago. Water mains currently in service are made

of various types of materials including cast iron, ductile iron, asbestos cement, steel, pvc

and prestressed concrete cylinder pipe (PCCP). PCCP has been used for high pressure

transmission of drinking and waste water since 1942 [12] and roughly 50 000 kilometres of

PCCP is currently in use across North America. Although PCCP has one of the lowest

water main break rates of all water main materials and PCCP is commonly specified

today for new water infrastructure, several municipalities in Canada and the US have

experienced ruptured water mains causing extensive damage. Mitigation costs associated

with a single burst waterline can be as high as $500 000. In addition, the hypothetical

cost of replacing all of these pipes is approaching $40 billion. Since costs of emergency

repairs or total replacement are both prohibitive to any water utility, there is considerable

interest in reliable nondestructive evaluation (NDE) techniques capable of identifying

damaged pipe sections as a means for optimizing rehabilitation e↵orts. Data gleaned

1

Chapter 1. Introduction 2

from a suitable NDE technique could be used to assess the probability of pipe failure,

thereby allowing utilities to schedule the appropriate maintenance.

The manufacture and design of PCCP conform to the American Water Works Association

(AWWA) C-301 and C-304 standards respectively [12], [13]. PCCP is of 2 types: 1. Lined

Cylinder Pipe (LCP), typical diameter: 400 mm - 1500 mm , and 2. Embedded Cylinder

Pipe (ECP), typical diameter: 1050 mm - 3600 mm. The current study focuses exclusively

on Lined Cylinder Pipe. LCP is composed of 4 main constituents: the concrete core, the

steel cylinder, the prestressed steel winding, and the protective outer mortar layer. The

structural integrity of the pipe is provided by the concrete core which resists the high

internal pressure (up to 1, 720 kPa) of the conveying fluid. The concrete core is wrapped

inside a water-tight steel cylinder. Since concrete is considerably weaker in tension than

in compression, a prestressed wire is helically wound over the steel cylinder and anchored

at the pipe ends in order to maintain a state of compressive stress in the concrete. Thus,

the tensile stresses induced by internal pressure are balanced by the compressive stress

induced by the prestressed wire. The steel cylinder acts as a water tight barrier whereas

the bell and spigot rings welded to either end of the cylinder allow pipe sections to fit

together in the field. Finally, the mortar coating (which is very dense and highly alkaline)

protects the prestressed wire from corrosive elements in the soil where the pipe is buried.

[13].

Pipe rupture initiated by external damage is the most severe and prevalent type of damage

to PCCP [15]. The deterioration process begins with the degradation of the mortar layer

by aggressive elements in the soil (such as chlorides) surrounding the pipe [41]. This allows

acidic ground water to penetrate the mortar coating and corrode the prestressed wire in

localized areas creating stress concentration features [66], [27]. The prestressed wire then

fails by brittle fracture in one or more locations along the axis of the pipe [66], [27].

Loss of compressive prestress in the concrete core occurs in the vicinity of wire breaks.


Bending stresses also develop in the core due to the uneven distribution of prestress which

causes axial and/or circumferential cracking of the concrete and de-bonding of the steel

cylinder. Finally cracks propagate through the core thickness, followed by yielding of the

steel cylinder and complete rupture of the pipe section [57].

The later stages of failure, occurring after the wire has broken in several locations have

been studied extensively [57], [66], [56]. A number of NDE techniques have already been

developed for the assessment of PCCP. The most commonly applied techniques are Re-

mote Field Eddie Current/Transformer Coupling (RTEC/TC), and Acoustic Emissions

Monitoring. RTEC/TC involves measuring distortions in an electromagnetic signal aris-

ing from the presence of broken wires [3], [58]. This method can estimate the number of

wire breaks as well as the approximate axial position of the breaks, however it is com-

pletely insensitive to any other form of damage including degradation of the protective

mortar. Acoustic emission monitoring involves inserting microphones at multiple loca-

tions along the pipeline to capture the characteristic sound wave emitted when a wire

snaps [3], [58]. This technique can determine the approximate number of wire breaking

events starting from the time when monitoring begins, though it cannot assess how many

wire breaks have occurred prior to testing. Both of these techniques have been demon-

strated to be e↵ective at sensing broken wires but have not been shown to be capable of

detecting wire thinning or mortar damage [58].

Resonance Acoustic Spectroscopy (RAS) is a technique used to characterize elastic ob-

jects based on their frequency response. Defects in engineering components such as

cracks, voids or inclusions often manifest as pronounced changes in geometric and/or

mechanical properties [22],[2]. Consequently, the presence of damage is reflected in the

resonance spectrum of the component. Spectra collected from damaged components can

be compared to theoretical or reference spectra which characterize healthy components

in order to assess their condition. In the case of PCCP, the RAS technique is potentially


sensitive to mortar degradation.

1.1 Thesis Objective

The main objective of this thesis is to develop an NDE technique capable of detecting

the earliest stage of failure in PCCP: deterioration of the protective mortar layer. The

steps required to meet the stated objective are as follows:

• Perform baseline studies to completely characterize the frequency response of healthy

sections of PCCP. This is to be accomplished by means of numerical modelling (via

commercially available finite element software), the results of which are to be veri-

fied by experimental modal analysis.

• Based on the results of the baseline studies, develop an appropriately simplified

pseudo-analytical model capable of predicting the frequency response of damaged,

water filled pipes that are buried in soil.

• Using the newly developed model of the damaged pipe, determine how the frequency

response of a pipe is a↵ected by deterioration of the protective mortar layer in the

presence of various common types of soil.

• Based on the results of modelling damaged pipes buried in soil, identify a robust

damage metric which scales with the level of deterioration of the protective mortar

layer.

• Define an experimental procedure by which field data can be collected, processed

and used to compute the pipe damage metric, which in turn can be used to infer

the level of damage to the pipeline under investigation.


1.2 Thesis Organization

CHAPTER 2 gives an overview of the construction and stages of failure of PCCP, as well

as available inspection technologies and their corresponding strengths and limitations. In

addition, the fundamentals of NDE by Resonance Acoustic Spectroscopy are presented

with a specific focus on resonant modes in cylindrical structures.

CHAPTER 3 gives an investigation of the frequency response of healthy sections of PCCP

via numerical (Finite Element) and experimental modal analysis. In this chapter, the

elastic parameters of pipe constituents are measured and the e↵ects of uncertainties in

these values on the accuracy of numerical modelling results is determined. Additionally,

the e↵ects of the prestressing wire and the coupling between adjacent pipes on the fre-

quency spectrum are examined as a means of justifying simplifying assumptions used in

modelling damaged, buried pipes.

CHAPTER 4 introduces a new mathematical model for computing the frequency response

of buried sections of PCCP, subject to mortar damage. The formulation is first presented

generally for an arbitrarily inhomogeneous cylinder in plane strain condition. Next, the

specific case of PCCP is addressed, including coupling to the surrounding soil (treated

as an infinite elastic medium) and to the conveying water (treated as an acoustic fluid).

CHAPTER 5 examines the e↵ects of mortar damage of varying degrees on two standard

diameters of PCCP. Additionally, the e↵ect of coupling to the surrounding soil is studied.

A new metric for quantifying perturbations to axial symmetry (called the asymmetry in-

dex) is proposed and shown to increase with increasing levels of mortar damage. For each

standard pipe diameter and type of soil, the sensitivity to mortar damage is discussed.

CHAPTER 6 outlines the experimental procedure for identifying damaged, in-service


pipe segments. This includes defining how to excite the vibration modes of interest,

sample the resulting motion and process the data to obtain values of asymmetry indices

which in turn are used to infer the level of pipe damage.

CHAPTER 7 includes the conclusions and presents recommendations for future work in

this area.

Chapter 2

Background and Literature Review

This chapter provides the necessary background information and reviews literature rel-

evant to the topics covered in this thesis. First, the basic structure and failure process

of Prestressed Concrete Cylinder are briefly discussed. This is followed by a review of

available Nondestructive Evaluation techniques used to assess PCCP. Finally, the funda-

mentals of NDE by Resonance Acoustic Spectroscopy are presented with a specific focus

on resonant modes in cylindrical structures.

2.1 Prestressed Concrete Cylinder Pipe

2.1.1 Basic Structure of PCCP

There are two types of commercially available pre-stressed concrete cylinder pipe: (1)

lined cylinder pipe (LCP), composed of a steel cylinder lined with concrete, wrapped by

a steel wire and coated with a protective layer of mortar, (2) embedded cylinder pipe

7

Chapter 2. Background and Literature Review 8

(ECP), composed of a steel cylinder embedded in concrete, wrapped by a steel wire

and coated with a protective layer of mortar. The manufacturing details and design

requirements for PCCP conform to the American Water Works Association (AWWA)

C-301 and C-304 standards respectively [12], [13]. 1. A cross section of LCP is shown in

figure 2.1.

Figure 2.1: Cross-section of LCP

Each component of PCCP serves a specific purpose. The structural integrity of the pipe

is provided by the concrete core which resists the high internal pressure (up to 1, 720

kPa) of the conveying fluid. Since concrete is considerably weaker in tension than in

compression, a pre-stressed wire is helically wound over the core and anchored at the

pipe ends in order to maintain a state of compressive stress in the concrete. Thus,

the tensile stress induced by internal pressure are balanced by the compressive stress

induced by the pre-stressed wire. The steel cylinder acts as a water tight barrier whereas

the bell and spigot rings welded to either end of the cylinder allow pipe sections to be

fitted together in the field. Finally, the mortar coating (which is very dense and highly

alkaline) protects the pre-stressed wire from corrosive elements in the soil where the pipe

is buried. [13].

Individual pipe sections are manufactured to standard nominal diameters (typically be-

1This thesis is concerned specifically with the nondestructive evaluation of LCP. Further detailsconcerning PCCP are focused accordingly


tween 0.6 and 1.5 meters) and standard lengths (typically between 6.1 and 7.3 meters).

Sections are laid end to end (spigot end of one pipe fits into the bell end of the next) in

a hard packed trench. Junctions between pipes are sealed with grout and then the entire

pipeline is covered with packed soil. Complete installation guidelines for pre-stressed

concrete pipelines are provided in [8].

2.1.2 Failure of PCCP

Pipe rupture initiated by external corrosion is the most severe and prevalent type of

damage to PCCP [15]. The deterioration process leading to catastrophic pipe rupture is

outlined below.

1. Protective qualities of the mortar layer are degraded aggressive elements in the soil

(such as chlorides) surrounding the pipe [41]

2. Acidic ground water penetrates the mortar coating and corrodes the pre-stressed

wire in localized areas creating stress concentration features [66], [27].

3. The pre-stressed wire fails by brittle fracture in one or more locations along the

axis of the pipe [66], [27].

4. Loss of compressive pre-stress in the core occurs in the vicinity of wire breaks.

Bending stresses also develop in the core due to the uneven distribution of pre-

stress [57].

5. Combined loading in the core causes axial and/or circumferential cracking of the

concrete and de-bonding of the steel cylinder from the core [57].

6. Cracks propagate through the core thickness, followed by yielding of the steel cylin-

der and complete rupture of the pipe section [57].


A ruptured segment of 42 inch LCP is shown in Figure (2.2) (photo taken from [33]). The

later stages of failure, occurring after the wire has broken in several locations have been

studied extensively [57], [66], [56]. However, from the perspective of NDE the earliest

stage of failure associated with the corrosion of the protective mortar layer is of greatest

interest. According to [41], corrosion of the mortar layer results in a loss of structural

integrity of the mortar coating. This e↵ect is central to the NDE technique developed in

this thesis.

Figure 2.2: Ruptured segment of LCP

2.1.3 Nondestructive Evaluation

A number of NDE techniques have been developed for the evaluation of PCCP. The most

commonly applied techniques are outlined below.

Visual Inspection and Pipe Sounding

Larger diameter pipes can be evacuated allowing human inspectors to walk through

buried waterlines 2. Evidence of internal corrosion or erosion can be seen directly. In

2This approach is not generally suitable for Lined-Cylinder Pipe as these pipes are too narrow forhuman inspectors to fit inside [44]


addition, inspectors can strike the pipe surface at di↵erent locations and with a steel

hammer attempt to observe hollow sounds indicative of de-bonding of the steel cylinder

and concrete core or de-lamination of the protective mortar [3]. The main advantage

of this technique is that it is inexpensive and does not require any special equipment

or expertise. However, internal signs of damage are visible only when crack growth in

the core has reached an advanced stage and thus can not be relied upon exclusively to

predict failure.

Remote Field Eddie Current/Transformer Coupling (RTEC/TC)

This technique involves generating an electromagnetic field in the centre of a pipe section.

A receiver is used to measure electromagnetic energy transmitted through the steel wire.

The received signal is distorted by the presence of broken wire, allowing the approximate

number and location of wire breaks to be determined. Once the number of wire breaks

is known, analysis can be performed to determine the likelihood of failure and the repair

priority for specific pipe segments. RTEC/TC is limited in terms of its accuracy and has

been found to underestimate or overestimate the number of wire breaks in specific cases

[3]. In addition, wire breaks cannot be observed near the pipe ends and the circumferen-

tial location of breaks cannot be determined by RTEC/TC [3]. Despite these drawbacks,

RTEC/TC remains a useful tool for quantifying the level of damage in the pre-stressing

wire.

Acoustic Emissions Testing

When a wire breaks elastic energy is released causing a stress wave to propagate through

the pipe core and into the water inside. The event can be captured by continuously

recording fluctuations of internal water pressure via hydrophones placed at di↵erent ax-

ial locations along the pipe. Recorded signals are then analyzed to determine if the


fluctuations in pressure are in fact due to a wire breaking and if so, the relative travel

time between multiple receivers can be used to find the location of the damage. A major

limitation of acoustic emissions monitoring is that events which occur prior to moni-

toring are obviously not captured and so the total number of wire breaks is unknown

[3]. This limitation can be overcome to a certain degree by applying RTEC/TC prior to

monitoring to obtain a baseline number of broken wires.

Impact Echo Method

This method involves generating stress waves from the inside of PCCP by impacting the

concrete surface with a hammer. The resulting pipe wall displacement is then recorded

by accelerometers fixed to the concrete surface. The accelerometers record waves that are

reflected from the di↵erent interfaces between layers of the pipe. If the steel cylinder has

de-bonded from the outer concrete or mortar, the acoustic impedance di↵erence at the

interface becomes larger than if the layers are perfectly bonded and so the recorded echo

should be of larger amplitude. This method is also capable of determining loss of pipe

wall thickness by comparing travel times of waves from the same interface at di↵erent

axial locations [26]. The impact echo method may be useful in terms of determining

damage to the inner concrete layer, though it cannot easily determine damage to the

outer mortar or pre-stressing wire [3].

Modal Analysis

Alavinasab et. al. [6], [4] have conducted preliminary numerical research via commercial

finite element software to determine the shifts in eigenfrequency caused by the loss of

pre-stress associated with increasing numbers of wire breaks. The authors found that the

loss of pre-stress caused only very slight changes in resonant frequency (< 1Hz) [6]. A

follow up study proposed that wire breaks could be identified by tracking the curvature


of particular mode shapes [4], however the feasibility of such an approach has not yet

been demonstrated.

A common limitation amongst existing NDE methods used to assess PCCP is their

insensitivity to even the most extreme deterioration of the protective mortar layer. Since

the rupture process begins with appreciable corrosion of the mortar layer, a reliable NDE

method capable of quantifying external damage would provide advanced warning of the

earliest stages of pipeline failure. The technique proposed in this thesis attempts to use

RAS to correlate large scale damage in the protective mortar layer to changes in measured

resonance signatures readily measured from inside of the pipeline.

2.2 Resonance Acoustic Spectroscopy

2.2.1 Introduction

Resonance Acoustic Spectroscopy 3 is a technique used to characterize elastic objects

based on their vibratory response. All elastic bodies of finite dimensions tend to oscillate

freely at discrete frequencies defining the object’s natural or resonant modes of vibration.

These resonant frequencies depend exclusively on the object’s geometry (size, shape) and

physical properties (density, elastic and internal damping parameters) [22]. If a harmonic

excitation is applied to a body with a frequency which matches one of its resonance

frequencies, the corresponding resonant mode is excited resulting in large displacement

amplitudes. Thus by measuring the displacement at some point on the body while varying

the excitation frequency, the object’s resonance spectrum can be determined.

3The terms Resonant Ultrasound Spectroscopy (RUS) and Acoustic Resonance Spectroscopy (ARS)are used interchangeably with RAS in the literature


Defects in engineering components such as the formation of cracks, voids or inclusions

often manifest as pronounced changes in geometric and/or mechanical properties [22].

Consequently, the presence of damage is reflected in the resonance spectrum of the com-

ponent. Spectra collected from damaged components can be compared to theoretical or

reference spectra which characterize healthy components in order to assess their condi-

tion. Therefore, a Nondestructive Evaluation (NDE) scheme based on RAS may provide

an estimate of the overall health of a component with only a few simple measurements

[2].

E↵ective application of RAS requires a coherent numerical model capable of describing

the vibrational characteristics of the component under investigation as well as a pro-

ducible method of measuring the resonance spectrum of the component. Both items

pose considerable challenges in adapting the RAS technique to the evaluation of PCCP.

First, modelling of pipe sections is complicated by their heterogeneous composition as

well as the ambiguity of boundary conditions encountered both in the lab and in the

field. Second, measurement of resonances must be performed from inside of the pipe.

These items will be addressed in later chapters. Here, an introduction to the fundamen-

tals of modelling wave propagation in elastic structures is presented with a specific focus

on resonant modes in layered cylinders. Next, a brief introduction to measurement of

resonant spectra is provided.

2.2.2 Fundamentals of Elastic Resonance

The fundamental concept behind RAS is the relationship between an isolated elastic

body’s geometry, its mechanical properties and its resonant frequencies.This relationship

is best understood in qualitative terms by considering how a disturbance (or mechanical

wave) is propagated through an elastic medium. From the theory of elasticity it is known


that any deformable solid will resist displacement from its equilibrium position. If a

region in the body is forcibly deformed and subsequently released, the restorative quality

of the medium will tend to accelerate elements in the deformed region back towards their

equilibrium positions through action of internal stresses. Since these elements have mass

and thus inertia, they will tend to continue moving past their equilibrium positions forcing

the displacement of adjacent elements. In a medium of infinite dimensions this pattern

continues leading to a traveling wave front which propagates at a speed related to the

material’s sti↵ness and density [42]. The phenomenon of resonance arrises in bodies of

finite dimensions due to the multiple internal reflections of traveling waves reinforcing [2].

When a traveling wave encounters a boundary, some or all of the energy of the incident

wave is reflected (depending on the nature of the boundary). If a pair of traveling

waves reflected from di↵erent boundaries meet with the same phase they will interfere

constructively forming a standing wave. This condition is met only at frequencies where

the wavelength of traveling wave components are related to the physical dimensions of

the body. These frequencies are the natural or resonant frequencies which define the

vibratory response of the body if left to oscillate freely from some initially deformed

configuration.

In the case of forced vibration, waves are generated in the solid by some continuous time

harmonic stress applied over a portion of the boundary. If the wave launched from the

boundary has a frequency corresponding to a resonance frequency, it will be reflected

from some opposing boundary and return to the point of excitation in phase with the

applied stress causing the amplitude of the wave to double. This process is repeated

resulting in a build up of energy and large displacement/velocity amplitudes [18].

Wave propagation in elastic solids is governed by the elastic wave equation which, in the

absence of body forces is given in tensor form by:


Cijkl

uk,lj

= ⇢@2u

i

@t2(2.1)

where ui

component of the displacement vector is the ith component of the displacement

vector, Cijkl

and ⇢ are the sti↵ness tensor and density of the medium respectively (sum-

mation is implied over double indices for i, j, k = 1, 2, 3). Equation 2.1, which applies

for a fully anisotropic linearly elastic continuum, is derived in a straightforward manner

by applying conservation of momentum to an arbitrary solid volume and by making use

stress/strain and strain displacement relations [42] [51]. The elastic equation by itself ad-

mits the time harmonic traveling wave solutions for ui

. When stress and/or displacement

conditions are imposed, an infinite number of standing wave solutions corresponding to

individual resonant modes are obtained. Thus, in principle, if the sti↵ness tensor, density

and dimensions of an object are known it is possible to determine its entire resonance

spectrum. The process of computing normal modes for a body of known geometric and

physical parameters is referred to as the forward modelling problem. Conversely, the

inverse problem consists of estimating the object’s characteristic parameters by attempt-

ing to fit computed resonant frequencies to a su�ciently large set of measured ones.

The inverse problem is typically solved via a nonlinear optimization scheme (generally a

weighted least squared error type algorithm) in which the forward problem is solved with

di↵erent parameters until the discrepancy between numerical and measured spectra is

below some acceptable tolerance [2]. Resonance based material characterization is widely

used in geophysics for determining the physical properties of rock samples [10],[34], [47].

For the purposes of nondestructive evaluation however, it is su�cient to use forward

modelling (where possible) to study qualitative changes to resonance spectra resulting

from perturbations of elastic constants associated with known damage types [22].

At this point it should be noted that analytical solutions to Equation 2.1 are only available


for isotropic or transversely isotropic, homogeneous bodies and for a limited number of

geometries e.g. infinitely long cylinders and spheres. For objects of arbitrary shape,

material symmetry and boundary conditions, a numerical solution (usually based on the

Rayleigh-Ritz procedure) is required. As PCCP is a layered cylindrical component, the

following section will focus on solutions to the wave equation for cylindrical geometries.

2.2.3 Resonant Modes of Elastic Cylinders

Wave propagation in elastic cylinders has been studied extensively. A complete treatment

is provided by Hamidzadeh and Jazar [39]. The first general discussion of wave propa-

gation in infinite cylinders based on three-dimensional elastodynamics was presented by

Gazis [19] 4. Gazis developed an analytical solution for time harmonic wave motions for

the case of an arbitrarily thick, isotropic cylinder bounded by stress free surfaces at its

inner and outer radii. Consider an infinite cylinder with inner radius a and outer radius

b as shown in Figure 2.3. For an isotropic medium, the elastic wave equation reduces to

Navier’s equation:

µr2u+ (�+ µ)rr · u = ⇢@2u

@t2(2.2)

where µ and � are the Lame coe�cients for the cylinder, u is the displacement vector

and r2 is the three dimensional Laplace operator. Equation 2.2 simplifies considerably

by applying the Helmholtz decomposition for u,

u = r�+r⇥ (2.3)

4Previous research in this area had been limited to axially symmetric motions of solid cylinders


Figure 2.3: Reference Coordinates and Dimensions for the Infinite Cylinder

where � and are displacement potentials representing dilatational and shear motions

respectively. Substitution of Equation 2.3 in Equation 2.2 yields the following set of wave

equations in terms of � and ,

r2� =1

c2L

@2�

@t2(2.4a)

r2 =1

c2T

@2

@t2(2.4b)

where cL

and cT

are the longitudinal and transverse (shear) wave speeds respectively and

are given in terms of the Lame parameters

cL

=

s

�+ 2µ

⇢(2.5a)

cT

=

r

µ

⇢(2.5b)

Solutions to Equation 2.4 for the scalar wave potential, �, and the z component of are

fully separated and so can be solved by the standard separation of variables technique.

The r and ✓ components of are coupled and are found by first solving for r

� ✓


and r

+ ✓

, from which the solutions for r

and ✓

are obtained. For the details of

this process see [39], [42]. Since the cylinder is of infinite extent, the z dependence for

both displacement potentials takes the form of a time harmonic wave propagating axially.

The radial and circumferential dependencies are required to satisfy Bessel equations and

harmonic oscillator equations respectively. These dependencies taken together form the

solutions for � and the components of ,

� =1X

n=0

[An

Jn

(kL

r) + Bn

Yn

(kL

r)] cos(n✓)ei(kz�!t) (2.6a)

r

=1X

n=0

[Cn

Jn+1(kT r) +D

n

Yn+1(kT r)] sin(n✓)e

i(kz�!t) (2.6b)

✓

=1X

n=0

�[Cn

Jn+1(kT r) +D

n

Yn+1(kT r)] cos(n✓)e

i(kz�!t) (2.6c)

z

=1X

n=0

[En

Jn

(kT

r) + Fn

Yn

(kT

r)] cos(n✓)ei(kz�!t) (2.6d)

where,

kL

=

s

✓

!

cL

◆2

� k2 (2.7a)

kT

=

s

✓

!

cT

◆2

� k2 (2.7b)

! is the circular frequency, k is the axial wavenumber, n is the circumferential wave num-

ber and An

, Bn

, are unknown integration constants for each circumferential wavenumber.


The values of n are restricted to integers by the condition of continuity of circumfer-

ential dependence for the displacement field [42], thus by superposition, the complete

solutions are obtained by summation of the displacement potentials over n. Substi-

tution of the displacement potentials into Equation 2.3 gives the displacement solu-

tion up to the arbitrary constants. To determine the resonant frequencies, the stress

free conditions at the inner and outer radii of the cylinder must be enforced, that is:

�rr

(r = a, b) = �r✓

(r = a, b) = �rz

(r = a, b) = 0 5. This leads to a system of homoge-

neous equations of the form

Mn

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

An

Bn

Cn

Dn

En

Fn

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

= 0 (2.8)

where Mn

is a 6 ⇥ 6 matrix with Bessel function entries. Non-trivial solutions for the

vector of integration constants is obtained by setting the determinant of Mn

equal to

zero, which yields the frequency equation for the cylinder. For each value of axial and

circumferential wave number, an infinite number of frequency solutions exists [19], [1].

For each frequency, the vector

An

Bn

Cn

Dn

En

Fn

�

T

gives the corresponding

eigenvector, which can be used to define the resonant displacement and stress fields [39].

The solution procedure outlined above is referred to as normal mode expansion as the

solution for the field variables is expanded in terms of orthogonal circumferential modes.

5The derivation of displacement potentials follows the same procedure outlined above except theBessel functions of the second kind, Yn(.), are excluded so that displacement is finite at the origin [42]


2.2.4 Circumferential Mode Classification

Armenakas et al. [1] showed that wave motions in cylinders could be categorized accord-

ing to circumferential wave number as follows:

1. Breathing mode for n = 0, characterized by harmonic vibration of the entire cross

section in the radial direction

2. Torsional mode for n = 0, associated with circumferential motion independent of ✓

3. Axial modes for n = 0, involving both radial and axial displacements independent

of ✓

4. Bending and axial shear modes for n = 1, in which the cross section is undeformed

and bending occurs normal to z

5. Flexural Modes for n > 1, involving all three components of displacement coupled

together

It has been shown that the spectrum of a cylinder is mostly dependent on the lowest

four circumferential modes n = 0, 1, 2, 3 [39]. The lowest cross-sectional mode shapes

are shown in Figures 2.4, 2.5, 2.6, 2.7. The flexural modes are of greatest interest for

the purposes of RAS as they are easily excited through the application of a radial stress

on the inner (or outer) surface of the cylinder. Flexural modes are also the easiest

modes to identify experimentally as their associated mode shapes form distinctive lobar

patterns having 2n circumferential nodes. Furthermore, flexural modes occur in pairs

called doublets, both of which have the same frequency [11]. These are degenerate modes

(corresponding to di↵erent polarizations of the same basic mode shape) and will appear as

a single spikes of modal amplitude in experimental spectra so long as the cylinder under


investigation is symmetric about its axis. If the symmetry is compromised by some

form of axially asymmetric damage, the members of a doublet will occur at di↵erent

frequencies, an e↵ect known as peak splitting [2], [10]. Depending on the degree to which

symmetry is destroyed, formerly degenerate peaks may be su�ciently separated as to

resolve the split in the measured spectrum, thus serving to identify defects along the

circumference of the cylinder.

The modes discussed above represent the lowest frequency, structural modes associated

with vibration of the cross section as a whole. However, at higher frequencies the wave-

length becomes comparable to the thickness dimension of the cylinder leading to the

development of thickness modes which feature nodes distributed along the radial dimen-

sion [39]. As the RAS technique presented in this thesis is restricted to the low frequency

range, these modes will not be described in detail here. The reader is referred to Chapter

4 of reference [39].

Figure 2.4: Breathing Mode: n = 0

Figure 2.5: Bending/Axial Shear Mode: n = 1


Figure 2.6: Flexural Mode: n = 2

Figure 2.7: Flexural Mode: n = 3

2.2.5 Axial Wavelength and Dispersion Relations

Thus far the value of axial wavenumber has been left undefined. In fact k is a continuous

parameter for the infinite cylinder since no boundary conditions are imposed in the axial

direction [42]. An alternative interpretation of the frequency equation presented in the

previous section comes from the relationship between axial wavenumber, frequency and

phase velocity given by

k =!

cp

(2.9)

where, cp

is the phase velocity of axially propagating waves. By employing this definition

of axial wavenumber, the frequency equation implicitly relates cp

to ! which in turn

defines the dispersion relation for a particular circumferential mode. The dispersion

relation describes how quickly particular wave motions (longitudinal, flexural, etc.) travel

along the axis of the cylinder as a function of frequency. Dispersion curves for each mode

can be generated by numerically searching for zeros of the frequency equation. In addition


to relating phase speed to frequency, the dispersion curves also define cut-o↵ frequencies

for each mode below which the mode will not propagate [43].

In order to determine standing wave solutions, the values of k have to be restricted by

enforcing boundary conditions in the axial direction. For free vibration of an infinite

cylinder, the only such condition corresponds to k = 0 [1]. Gazis [19] showed that for

this special case, there are two types of motion (both independent of z) :

1. Plane strain vibration: characterized by coupled radial and circumferential dis-

placements with w = 0. These are flexural modes where the cross section deforms

uniformly over the entire length of the cylinder

2. Axial shear vibration: characterized by axial displacement only generated through

r � z shearing action

These motions are uncoupled from each other and so they can be analyzed independently

[19].

In the case of finite cylinders, stress and/or displacement conditions on the cylinder

ends cause waves propagating axially to reflect and reinforce for discrete values of axial

wavenumber. Finite cylinders can be studied directly only for simply supported end

conditions [39]. Simply supported end conditions do not adequately describe physically

realizable situations (i.e. stress free ends or displacement fixed ends), though they are

useful in terms of understanding the general character of three dimensional standing wave

solutions in cylinders. Consider a cylinder of length 2l with the origin of z located at the

midpoint. The end conditions for which axial wavenumber can be written in closed form

are as follows,


u(z = ±l) = v(z = ±l) = 0 ! k =m⇡

l(2.10a)

w(z = ±l) = 0 ! k =(2m+ 1)⇡

2l(2.10b)

where m = 1, 2, 3, . . .. By linearity, the displacement potentials are then given by a

summation over m and n with each (m,n) pair defining a particular resonant mode.

These modes are characterized by cross sectional patterns similar to those in Figures

2.4, 2.5, 2.6, 2.7, whose amplitude is modulated along the cylinder axis according to the

value of m. The limiting case where the length of the pipe approaches infinity gives

k = 0 for both sets of end conditions. This corresponds to the special case of plane strain

vibration discussed above. Clearly any real structure will be of finite length, though it has

been shown experimentally [40] that for cylinders with l > 8b, the resonance spectrum

is accurately represented by the plane strain vibration assumption. For cylinders with

l < 8b or when the vibration response is measured near the cylinder ends, the plane strain

approximation is no longer appropriate and the actual end conditions must be satisfied

to determine the full three dimensional spectrum.

2.2.6 Forced Vibration and Attenuation

Resonant frequencies of cylinders are determined by solving the frequency equation, which

itself is a complicated transcendental function involving parametric Bessel functions of the

first and second kind. Solutions can be found by using standard root finding algorithms.

An alternative approach proposed by Hamidzadeh et al. [39] involves applying time

harmonic boundary stresses to the inner and/or outer surface of the cylinder of the

following form:


�rr

i

= Arr

i

cos(n✓)ei(kz�!t)

�r✓

i

= Ar✓

i

sin(n✓)ei(kz�!t)

�rz

i

= Arz

i


(2.11a)

�rr

o

= Arr

o


�r✓

o

= Ar✓

o

sin(n✓)ei(kz�!t)

�rz

o

= Arz

o


(2.11b)

where �rr

i,o

, �r✓

i,o

and �rz

i,o

are the stresses on the inner (subscript i) and outer (sub-

script o). The constants Arr

i,o

, Ar✓

i,o

and Arz

i,o

represent the relative magnitudes of the

stress components applied to the cylinder boundaries. For any circumferential and axial

wavenumber combination, continuity of stresses at the boundary results in an inhomo-

geneous linear system:

Mn

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

An

Bn

Cn

Dn

En

Fn

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

=

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

Arr

i

Ar✓

i

Arz

i

Arr

o

Ar✓

o

Arz

o

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(2.12)

This system can be solved for the modal amplitudes,

An

Bn

Cn

Dn

En

Fn

�

T

,

by inverting matrix Mn

for each n, k,!:


2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

An

Bn

Cn

Dn

En

Fn

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

= M�1n

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

Arr

i

Ar✓

i

Arz

i

Arr

o

Ar✓

o

Arz

o

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(2.13)

Then the resonance spectrum of the cylinder is obtained by plotting the magnitude of

the displacement field at an antinode.

This model approximates the forced vibration response of a cylinder and therefore re-

quires some energy dissipation mechanism so that displacements remain finite. This is

accomplished through the introduction of attenuation and/or coupling to an external

unbounded medium. In accordance with the equations of viscoelasticity, attenuation in

solid elastic materials can be approximated numerically by using complex valued shear

and longitudinal moduli [42],

E = E 0(1 + i⌘L

) (2.14a)

µ = µ0(1 + i⌘T

) (2.14b)

where E 0 and µ0 are the Young’s and shear (same as second Lame parameter) moduli

and ⌘L

and ⌘T

are the associated longitudinal and shear loss factors respectively. The

loss factors take values less than unity and quantify the degree of internal frictional

losses (higher values denote higher levels of attenuation). In the absence of attenuation,


resonances have infinite magnitude and are infinitely narrow. Increasing attenuation has

the e↵ect of broadening resonances while decreasing their amplitude as shown in figure

2.8.

520 530 540 550 560 570 580 5900

0.2

0.4

0.6

0.8

1

1.2x 10

−4

Frequency (Hz)

Am

plit

ude

ηL = 0.02

ηL = 0.01

ηL = 0.005

Figure 2.8: E↵ect of Increasing Attenuation

This broadening e↵ect can make it di�cult to resolve closely spaced resonances in exper-

imental spectra [2]. Accordingly, it is common to test in a region of the spectrum where

the resonant frequencies are well separated, typically at low frequencies.

2.2.7 Experimental Procedures

Experimental determination of the resonance frequencies of infrastructure components

and other concrete structures is commonly done via swept frequency testing (for smaller

components) or via impact-echo testing (for larger components). Both procedures are

outlined here.


Swept Frequency Testing

Swept frequency testing involves exciting sinusoidal waves in the structure with a piezo-

electric transducer fixed to some part of its boundary. The response is measured with

another transducer fixed at another location. If the excitation is sinusoidal than the

steady state of the response is also sinusoidal. Should the frequency of the excitation

correspond to a resonance frequency, the response of the system will be of much higher

amplitude than for non-resonant frequencies. By sweeping through a number of exci-

tation frequencies and recording the amplitude of displacement, the spectrum can be

completely determined. A block diagram for a typical swept frequency, contact RAS

setup is shown in Figure 2.9.

Figure 2.9: Block diagram for swept frequency RAS experimental setup

A signal generator is used to supply a voltage to the excitation transducer at a particular

frequency. Once a suitable time has past and the transient e↵ects have died down, the

amplitude of the recieved sinusoid is recorded by an oscilloscope or personal computer.

This process is repeated at each frequency until the spectrum has been determined over

the desired range with the desired resolution. Typically the received signal is averaged

over a number of samples in order to reduce the e↵ects of noise. Both the receiver and

transmitter should be weakly coupled to the sample to avoid excessively loading it [17],

[18]. This technique is used almost exclusively for the determination of elastic parmeters


of geophysical media [2], [11], [10]. In terms of concrete testing, published results have

been limited to a few laboratory studies involving smaller rectcircumferential articles

[17], [47]. This is due to the fact that concrete components of interest are usually quite

large (e.g. bridge decks, foundation of buildings, etc.) and so their resonant frequencies

can be as low as a few hertz [68]. Consequently, specialized low frequency transducers

are required [2].

Impact Testing

Impact testing involves applying a broadband excitation to the surface of the sample

with a small hammer and analyzing the transient response in the frequency domain.

Application of a short-duration impulse causes the structure to respond at all of its

natural frequencies within the band of the excitation simultaneously. A block diagram

of a typical impact test is shown in Figure 2.10.

Figure 2.10: Block diagram for impact testing

The hammer strike generates an impulsive stress wave in the sample composed of an

infinite number of individual frequency components within the range dictated by the du-

ration of the strike. The sample will then oscillate freely at resonant frequencies within

the bandwidth of the excitation. This response is picked up by the receiver (either a piezo-


electric transducer or accelerometer), digitized by an oscilloscope and then saved to a PC.

The spectrum is then determined by applying a Fast-Fourier Transform (FFT) to the

time domain response. Generally, multiple strikes are averaged to obtain a more repeat-

able spectrum. More sophisticated setups allow the input waveform to be recorded via a

load cell fixed to the hammer so that the response can be deconvolved with the excita-

tion. This technique is considerably less time consuming than the swept frequency testing

procedure as it does not require sweeping through many frequencies, rather the whole

spectrum is determined from a single measurement. On the other hand, the impulse-echo

method is more sensitive to noise [18]. For a more comprehensive modal analysis, where

the mode shapes are also desired in addition to the frequency response, the response is

recorded at multiple locations on the specimen under investigation. Experimental modal

analysis based on impact testing is discussed in greater detail in Chapter 3.

A subset of impact testing is the impact-echo (IE) method. This technique was developed

in the mid 1980s by Sansalone et. al. [52] and has since become widely used to detect

flaws in large concrete structures [68], [53], [60]. The focus of the impact-echo method

di↵ers slightly from traditional RAS studies in so far as the IE method is concerned with

studying standing waves between parallel (or near parallel) interfaces as a means for

estimating changes in sonic velocity or layer thickness, whereas RAS is concerned with

observing frequency shifts or mode splitting for lower frequency structural modes.

Chapter 3

Characterization of Healthy Pipes

Prior to determining the e↵ect of mortar damage on the resonance spectrum of buried

pipelines, it was first necessary to evaluate the frequency spectrum of healthy pipes

through a sequence of experimental measurements and finite element modelling studies.

The purpose of these investigations was to determine the accuracy of modelling PCCP in

the presence of uncertainties/inaccuracies in the elastic properties as well as the e↵ects

of certain structural features, such as the bell/spigot interface and wire prestress. In this

chapter, the focus is placed on free vibrations of a single segment of newly-manufactured

PCCP. The following specific points are addressed:

• What are the primary resonant modes that are visible and identifiable? Which are

candidates for a NDE technique to characterize PCCP damage?

• For a finite element model of PCCP, what is an appropriate way to handle the

prestressed spirally wound wire? What e↵ect does the prestress have on the pipes

resonant behaviour?

• To what extent do the bell and spigot on the ends of each PCCP segment a↵ect

32

Chapter 3. Characterization of Healthy Pipes 33

the finite element model of its free vibrations?

• What values should be used in the finite element model for material properties of

each PCCP layer? How well does the spectrum of such a model match the spectrum

measured experimentally?

• How is the resonant spectrum of PCCP a↵ected by the joining of multiple sections

into one long pipe?

The results of these investigations will be used to justify certain simplifications used to

model the more complicated case of damaged, buried pipelines detailed in Chapter 4.

Experimental studies were conducted in collaboration with MUNRO LTD 1. Much of the

material presented in this chapter, including but not limited to all figures and tables,

were published in the ASCE Journal of Pipeline Systems [48] and is reprinted here with

permission from the ASCE. This material may be downloaded for personal use only. Any

other use requires prior permission of the American Society of Civil Engineers.

3.1 Geometry of Tested/Modelled Pipes

The structure of lined type PCCP is seen in Figure 2.1. Specific geometric parameters in-

cluding layer thicknesses, wire spacing etc., vary depending on the pipe diameter, working

pressure and depth to which the pipe is buried. Appropriate values for these parameters

as well as material properties of the pipe constituents are codified in the AWWA C-304

design standard. Two segments of pipe were considered, having nominal diameters of 600

mm and 1200 mm. The geometric parameters associated with these particular segments

are listed in Table 3.1.

1MUNRO LTD, 8807 Simcoe Road 56, Utopia, Ontario, Canada, L0M 1T0


For both pipes the steel cylinder thickness, wire diameter and mortar thickness are 1.54

mm, 4.88 mm and 19 mm respectively. The length of the bell, spigot and pipe are 114

mm, 127 mm and 6096 mm respectively.

3.2 Material Properties of Pipe Constituents

Wave speeds (both longitudinal and shear) were measured for each pipe constituent

by applying the ultrasonic pulse velocity method (see [60] for details) with samples of

mortar and concrete. The samples tested were taken from market-ready sections of LCP,

provided by MUNRO LTD. Wave speeds were calculated at 4 di↵erent positions on each

sample. The wave speed at each location was computed to 3 significant figures using

1MHz contact probes. Each measurement was repeated 10 times. The resulting mean

values for longitudinal and shear wave speeds (cL

and cT

respectively) and associated

uncertainties are summarized in Table 3.2.

For the purposes of modelling the mortar and concrete are considered to be macroscopi-

cally homogeneous/isotropic, taking the wave speeds as the mean of the values measured

at the di↵erent locations. Densities (measured by the pipe manufacturer) are: 2390 kg

m�3 for concrete and 2242 kg m�3 for mortar. The longitudinal speed, shear speed and

density of the steel constituents are taken to be the standard published values, 5900 m

s�1, 3200 m s�1, 7800 kg m�3 respectively. With the assumption of isotropy, any other

relevant elastic parameters such as Young’s Modulus, Poisson’s ratio, Lame constants,

etc., are readily determined.


3.3 Finite Element Modelling

In computing the natural frequencies of PCCP, the following assumptions were made to

simplify the analysis:

• Damping in all pipe constituents (for frequencies < 750Hz) is su�ciently low as to

not e↵ect the natural frequencies.

• All pipe constituents are macroscopically isotropic/homogeneous

• The steel winding and surrounding mortar behave as a fibre reinforced composite,

with ✓ being the reinforced direction 2.

The third assumption allows the steel winding and surrounding mortar to be treated as

a single orthotropic layer; thus the di�culty in meshing the steel wire is avoided. The

e↵ective sti↵ness matrix and density can be computed by the standard rule of mixtures

formulae given in [30]. The procedure for obtaining the homogenized properties of the

mortar/steel winding layer is outlined in Appendix C. The resulting e↵ective sti↵ness

matrices for the 600 mm and 1200 mm pipes are found in Equations C.9a and C.9b (of

Appendix C), respectively. With the aforementioned assumptions in mind, the complex

geometry of LCP (see Figure 2.1) is reduced to the simplified geometry is shown in Figure

3.1. Using the properties of mortar and steel quoted above and following the procedure

outline in Appendix C, the sti↵ness matrix for the steel/mortar composite layer is found

to be:

2The winding angle for LCP is typically very small (< 2 degrees from the ✓ direction)


Spigot Pipe Bell

Concrete

Steel

ConcreteSteelSteel Wire / Mortar CompositeMortar

MortarSteel

Figure 3.1: Simplified Model of Healthy LCP

Figure 3.2: Mesh for 600mm Section of LCP

Each layer in each component was meshed with mapped, 8 node solid elements. Conti-

nuity of displacements between layers and components was ensured by subdividing each

layer such that nodes were common at the component boundaries. All finite element

analysis was performed using the commercial finite element software, COMSOL. In all

cases, convergence was established by uniformly increasing the mesh density in all pipe

constituents until the di↵erence between all eigenfrequencies below 750 Hz for successive

runs was below 0.1 Hz. The mesh used for the 600 mm pipe is shown in Figure 3.2.


3.3.1 E↵ect of Prestress

In modelling the undamaged pipe sections, the influence of the prestressed wire on the

spectrum was investigated numerically. According to the AWWA standard, the wire

should be stressed to 75% of its ultimate tensile strength, which for 4.88mm, ASTM

A648 Class III wire is �ut

=1,740MPa. This corresponds to an initial strain in the

wire of 0.0068. Since the wire and surrounding mortar are treated as a fibre reinforced

composite, the dynamic strain in the wire and mortar are modelled as equal in the

reinforced direction, however the mortar is formed around the wire after it has already

been pre-tensioned, thus the true initial strains in the wire and surrounding mortar are

not the same. In this study, the influence of the pre-stress is approximated by specifying a

uniform initial circumferential strain of ✏✓✓0 = 0.0068 in the steel wire / mortar composite

layer; this will tend to over estimate the pre-stressing e↵ect on resonant frequencies.

Vibration of prestressed components can be handled in COMSOL using the prestressed

eigenvalue analysis option. This type of analysis solves the stationary problem to obtain

the initial displacement field imposed by pre-stress (or pre-strain) and then performs

an eigenvalue analysis assuming finite strains to incorporate the e↵ects of geometric

nonlinearity associated with the initial strain field. Both prestressed eigenvalue analysis

and linear eigenvalue analysis (not including the e↵ects of pre-stress) were performed on

the 600 mm section of PCCP to quantify the perturbation in eigenfrequency caused by

the prestressed wire. The results are summarized in Table 3.3.

It is readily observed that the prestressed eigenfrequencies di↵er by less than 0.1 %

in all cases, from which we conclude that the pre-stress induced by the wire does not

significantly impact the spectrum of PCCP. Consequently, prestress is ignored in all

further computation of resonant frequencies.


Table 3.1: Geometric parameters for 600mm and 1200mm sections of PCCP, dimensionsgiven in millimetres

NOMINAL PIPE CONCRETE WIRE BELL BELLPIPE INSIDE CORE PITCH INSIDE OUTSIDE

DIAMETER DIAMETER THICKNESS DIAMETER DIAMETER600 610 38 32.5 699 8001200 1219 76 16.5 1372 1473

Table 3.2: Measured Longitudinal and Shear Speeds for Concrete and Mortar

1 2 3 4c

L

(ms�1) c

T

(ms�1) c

L

(ms�1) c

T

(ms�1) c

L

(ms�1) c

T

(ms�1) c

L

(ms�1) c

T

(ms�1)Concrete 4850 ±40 2860 ±50 4770 ±20 2880 ±10 4810 ±10 2920 ±20 4700 ±30 2790 ±90Mortar 4140 ±50 2660 ±10 4200 ±10 2680 ±10 4220 ±30 2690 ±10 4260 ±30 2790 ±20

Table 3.3: Computed eigenfrequencies, with and without considering the e↵ect of pre-stress; 600mm pipe

prestressedFrequency Frequency Di↵erence

(Hz) (Hz) (%)89.3 89.3 0.00220.3 220.1 -0.09280.3 280.1 -0.07287.3 287.2 -0.03296.7 296.6 -0.03312.5 312.4 -0.03346.0 345.9 -0.03350.4 350.5 0.03380.0 379.7 -0.08404.1 403.9 -0.05483.1 482.9 -0.04549.4 549.0 -0.07577.1 576.9 -0.03680.7 680.5 -0.03699.5 699.5 0.00718.6 718.0 -0.08


3.3.2 E↵ect of Bell and Spigot

In order to determine the e↵ect of the bell and spigot features on the spectrum of a single

segment of PCCP, eigenfrequencies were computed by modelling the pipe without the bell

and spigot such that the pipe component spanned the entire length. These frequencies

are compared to those computed using the simplified model of PCCP (including the bell

and spigot) shown in Figure 3.1. The results for 600mm and 1200mm pipe are compared

in Tables 3.4 and 3.5 respectively.

It is observed that the di↵erence between the two cases is generally slight < 1.5%). One

outlier is the highest mode for 600mm pipe which shows a relative error of 3.24%. This

is likely due to the fact that this mode is localized in the spigot end and should therefore

be more sensitive to the specific geometry of the spigot. Consequently, in order to best

approximate the actual pipe for all modes, the bell and spigot features should be included

in the model.

3.4 Experimental Modal Analysis

3.4.1 Experimental Setup

Experimental resonant frequencies and mode shapes for each pipe section were obtained

via impact testing. Impact testing involves exciting a structure with a short duration

impulse, f(t), and recording the resulting vibratory response of the structure (normal

acceleration in this case), u(t), at one or more locations. The impact is delivered with an

instrumented source so that applied force, f(t), can be measured. Performing a Fourier

transform on the applied force and measured acceleration signals:


Table 3.4: Computed Eigenfrequencies, With and Without Bell and Spigot Features:600mm pipe

Frequency FrequencyWith Bell and Without Bell and

Spigot Spigot Di↵erence(Hz) (Hz) (%)89.3 89.6 0.34220.1 220.9 0.36280.1 283.5 1.21287.2 284.8 0.84296.6 291.2 1.82312.4 308.5 1.25345.9 344.6 0.38350.5 351.9 0.40379.7 381.2 0.40403.9 403.2 0.17482.9 482.2 0.14549.0 551.5 0.46576.9 576.1 0.14680.5 679.9 0.09699.5 702.5 0.43718.0 721.8 0.53759.2 783.8 3.24


Table 3.5: Computed Eigenfrequencies, With and without bell and spigot features:1200mm pipe

Frequency FrequencyWith Bell and Without Bell and

Spigot Spigot Di↵erence(Hz) (Hz) (%)124.5 123.4 0.88127.7 125.9 1.41147.4 145.9 1.02154.2 152.8 0.91209.2 206.6 1.24305.7 301.7 1.31321.1 319.5 0.50345.2 342.9 0.67350.6 346.0 1.31354.8 353.2 0.45361.3 357.9 0.94384.6 382.4 0.57416.0 410.8 1.25425.7 423.5 0.52485.3 482.3 0.62487.5 486.0 0.31527.9 521.8 1.16560.3 556.4 0.70611.4 612.0 0.10635.7 629.0 1.05645.2 642.5 0.42646.3 641.6 0.73650.6 645.6 0.77660.6 657.3 0.50679.6 677.8 0.26699.7 697.5 0.31709.1 708.3 0.11721.4 722.0 0.08


F {f(t)} (!) = F (!) (3.1a)

F {u(t)} (!) = U(!) (3.1b)

where F (!) and U(!) are the Fourier transforms of the force and measured acceleration

respectively. The Fourier transform of radial displacement, U(!), can be obtained in

terms of the Fourier transform of acceleration as follows:

U(!) = � U(!)

!2(3.2)

The frequency response of the structure (assuming the response is linear) is then obtained

by the following operation:

H(!) =U(!)F ⇤(!)

F (!)F ⇤(!)= � 1

!2

U(!)F ⇤(!)

F (!)F ⇤(!)(3.3)

where,H(!) is the (compliance) frequency response function in units of displacement/force.

Equation 3.3 is essentially a deconvolution operation which minimizes the e↵ect of recorded

noise (noise summing on the output), where the numerator is the spectrum of the cross-

correlation of u(t) and f(t) and the denominator is the auto-correlation of f(t). Since

the modes under consideration vary in both the circumferential and axial directions, it

is necessary to vary the relative position of the impact and receiver.

Each pipe was laid on foam supports (to approximately simulate free boundaries) and an

accelerometer was mounted near the spigot end of the pipe, at a circumferential position


located 180 degrees from the supports. The pipe was then impacted with an instru-

mented hammer (Kistler 3 Model 9724A) on the pipe’s inner surface at 18 equally spaced

circumferential positions (indexed by i = 0, 1, ..., 17). Each circumferential position was

impacted 5 times for the purpose of averaging. This process was repeated with the ac-

celerometer mounted at 18 equally spaced locations along the axis of the pipe (indexed

by j = 0, 1, ..., 17). The impact and response signals at each circumferential and axial

location, fi

(t) and uj

(t) respectively, were digitized and stored on a laptop computer.

Applying Equation (3.3) to each ij pair and averaging over the repeated strikes gives the

frequency response function of the pipe Hij

(!), corresponding to the ith circumferential

and jth axial position.

Hij

(!) =1

5

5X

k=1

= � 1

!2

Ujk

(!)F ⇤ik

(!)

Fik

(!)F ⇤ik

(!)(3.4)

Here the Fast Fourier Transform (FFT) operation was used to obtain an approximation

of the continuous Fourier transform of the digitized signals. Experimental resonant fre-

quencies can be obtained by identifying peaks in the magnitude of Hij

(!). Approximate

mode shapes are then obtained by plotting the imaginary part of Hij

(!) as a function of

position. The useful frequency range for the impact hammer used was roughly 0 to 750

Hz. Accordingly, we will restrict our analysis to that range in the following sections. For

the case where the two 600mm pipe sections fit together end to end, the above procedure

was the same except the pipes were laid on wooden supports spaced 0.5 meters apart.

Experimental modal analysis was performed on market ready pipe sections in order to

compare with the modelled results. Tests were performed on sections of 600mm and

1200mm pipe. The spectrum of each pipe is obtained by plotting the excitation point

3Kistler Instrument Corp. Amherst, NY


frequency response function, |H00(!)|. The spectra are annotated with the (n,m) values

to indicate the circumferential and axial wave numbers determined by examining the

experimental mode shapes. The designations of A and S accompanying the value of m

identify whether the axial dependence of the mode shape is anti-symmetric or symmetric

about the centre line of the pipe. In cases where a degenerate pair has split, the value of n

is accompanied by C or S to denote either the cos(n✓) or sin(n✓) dependence of the mode.

Spectra for the 600mm pipe and 1200mm pipes are shown in Figures 3.5 and 3.6 respec-

tively. Experimental mode shapes are obtained by plotting the imaginary part of the

frequency response function at axial and circumferential positions. To demonstrate, the

mode shapes with circumferential and axial wavenumber combinations: (1, 1A), (2, 1S)

and (2, 1A) are shown in Figures 3.4a, 3.4b and 3.4c, respectively.

An eigenfrequency analysis was performed in COMSOL for both pipes examined ex-

perimentally (as described in the previous section). The computed and experimental

eigenfrequencies are compared in Tables 3.6,3.7 4.

The discrepancy between the computed and measured eigenfrequencies is partially at-

tributable to errors in measuring the e↵ective wave speeds in the concrete and mortar

constituents. Specifically, it is noted that the wave speeds used for the purposes of mod-

elling represent an average of values measured at di↵erent locations on small samples

taken from a di↵erent pipe section. Therefore, these values may deviate appreciably

from true bulk averaged properties. Additionally, it is known that the Young’s modu-

lus and density of the concrete and mortar decrease with age as the samples dry out

over time [13]. Since the samples used to measure wave speed and density were not the

same age as the pipes tested, the spectra based on the sample properties are expected

to deviate slightly from the experimental spectra. Here it is noted that better agreement

with experimental results could likely be achieved by setting up a minimization prob-

4Certain modes that were predicted by the eigenfrequency analysis were not observed experimentally


DATA ACQUISITION

SYSTEMLAPTOP

INSTRUMENTEDHAMMER

ACCELEROMETER

FOAM SUPPORTS

i

i

..

Figure 3.3: Experimental Setup

206 Hz

(a) (1, 1A)

302 Hz

(b) (2, 1S)

321 Hz

(c) (2, 1A)

Figure 3.4: Measured Mode Shapes for the 600mm pipe section


Figure 3.5: Excitation point Frequency Response Function magnitude: 600mm pipe

Table 3.6: Experimental and computed eigenfrequencies: 600mm PCCP

Experimental Computedn m Frequency Frequency Di↵erence

(Hz) (Hz) (%)1 1A 206 220 6.802 0 291 280 -3.782 1S 302 297 -1.672 2A 321 312 -2.802S 2S 351 351 0.002C 2S 355 351 -1.272 3A 409 404 -1.222 3S 480 483 0.631 2A 519 549 5.782 4A 564 577 2.302 4S 664 681 2.563 0 697 759 8.90


(2,0)

(1,1S)

(2,1S) (2,2A)(2S,2S)

(2C,2S)

(3,0)

(3,1S)(2,3A)

(3,2S)

(1,2A)

(3,3A) (2,3S)

(4,0)

(2,4A) (3,4A)

Figure 3.6: Excitation point Frequency Response Function magnitude: 1200mm pipe


lem in which the norm of the di↵erences between computed and measured frequencies is

minimized with respect to the mortar and concrete wave speeds, i.e. solving the inverse

problem [2].

Another source of experimental error comes from the fact that the protective mortar

layer was considered to be of a uniform 19mm thickness for the purposes of modelling,

whereas the actual mortar thickness on the sections tested was not uniform and varied

by as much as 4mm.

It is observed that the (2,2S) mode for both pipes is split. This is due to the fact that

this mode has anti-nodal points coincident with the pipe supports, thus the asymmetry

in the boundary conditions will have a greater influence on this mode. Otherwise, the

supports had no discernible influence the experimental spectra.

The mean values of the di↵erence between computed and experimental frequencies are

3.13% for the 600mm section and 4.31% for the 1200mm section. These di↵erences are

consistent with the sources of experimental error discussed previously. It is therefore

concluded that the simplified model proposed in 3.3, including the homogenized mor-

tar/steel winding layer is su�ciently accurate for the purposes of computing resonant

frequencies of PCCP.

3.4.2 Coupling Between Pipes

Coupling between adjacent pipes was studied experimentally by performing experimental

modal analysis on 2 sections of 600mm pipe fit together in a manner similar to how they

are assembled in the field. The experimental spectrum for the joined pipes is shown in

Figure 3.7.


(1,S)

(1,1A)

(1,2S)

(1,2A)(1C,3S)

(1S,3S)

(2C,0) (2S,0)

(2,3A)(2,3S)(2,4A)(2,4S)

(2,5S) (2,6S)

(3,0)

Figure 3.7: Excitation Point Frequency Response Function Magnitude, 2 sections of600mm pipe


From examining the mode shapes associated with each resonant peak, several distinct

modes were identified and compared to eigenfrequencies computed by assuming the two

pipes to be identical with displacements being continuous across the bell/spigot interface.

The results are compared in Table 3.8.

From Table 3.8 it is observed that the experimental frequencies are all below the computed

frequencies. This is likely due to the fact that displacements across the bell/spigot

interface were assumed to be continuous in modelling which may not have been the case

in the experimental setup. Continuity of displacement across the bell/spigot interface

is a reasonable assumption if there exists a su�ciently high static force on the joint to

keep the components in normal contact and the maximum static frictional force is not

exceeded. For practical reasons, the joint in the experimental setup could not be sealed

with grout as it would normally be in the field, thus it is likely that separation of the two

pipes occurred during testing. The fact that the experimental frequencies are lower in

all cases tends to suggest that the joint was less than perfectly rigid in the experimental

setup. Additionally, it is observed that for a particular m value, the symmetric modes

deviate more from the numerical results than the anti-symmetric ones. This is due to the

fact that the symmetric modes will have large modal displacements at the bell/spigot

interface and are more e↵ected by the joint sti↵ness. Despite the fact that the interface

is not perfectly bonded, the experimental mode shapes indicate that the individual pipes

are in fact well coupled; that is the contact sti↵ness at the joint is high enough such that

standing waves do not develop in each individual section. In the field these pipes will be

coupled even more strongly due to the grout placed between pipe sections.

From the perspective of an NDE method, it is clear that directly comparing computed

and experimental spectra cannot identify the condition of individual pipe sections, even

for the simple case where the pipe is freely supported due to the uncertainty in the mortar

thickness, constituent material properties / age of the pipe. It is therefore necessary to


Table 3.7: Experimental and computed eigenfrequencies: 1200mm PCCP


(Hz) (Hz) (%)2 0 130 125 -4.231 1S 143 147 3.082 1S 152 154 1.452 2A 207 209 1.062S 2S 295 306 3.632C 2S 297 306 2.933 0 326 345 5.893 1S 376 361 -3.912 3A 398 416 4.523 2S 436 426 -2.361 2A 451 488 8.093 3A 490 485 -0.962 3S 504 528 4.744 0 553 645 16.672 4A 606 636 4.903 4A 643 646 0.51

Table 3.8: Experimental and Computed Eigenfrequencies, 2 sections of 600mm PCCP


(Hz) (Hz) (%)1 1S 20 24 18.911 1A 56 64 13.511 2S 83 118 42.081 2A 154 186 20.501C 3S 198 258 30.541S 3S 205 258 26.082C 0 269 289 7.342S 0 275 289 5.002 3A 299 313 4.812 3S 305 328 7.382 4A 326 342 4.892 4S 336 367 9.292 5S 390 430 10.262 6S 464 513 10.573 0 690 782 13.36


determine how individual modes will be influenced by the presence of mortar damage.

From Figures 3.5, 3.6 it is clear that the plane strain modes (modes withm = 0) dominate

the spectrum, though several additional, axially dependent modes are also clearly defined.

However, it is reasonable to assume that the visibility of these axially dependent modes

(m � 1) will be further diminished in an assembled pipeline as the ends of each pipe will

no longer be freely supported, rather they will be coupled to an adjacent pipe. Without

a strong reflection of guided waves from the pipe ends, strong standing wave modes

are unlikely to develop, thus only the plane strain modes should be observable in the

spectrum of the pipe. For this reason, only the plane strain modes will be considered in

modelling the e↵ects of mortar damage in the following chapter.

Chapter 4

Mathematical Modelling of

Damaged Pipes

In conducting baseline experiments on healthy, unburied pipelines (see Chapter 3) it was

observed that the coupling between adjacent pipe sections was su�ciently strong as to

prevent axially dependent standing wave modes from developing in individual sections

of PCCP. Consequently, in modelling a joined pipeline it was assumed that the pipeline

behaves like an infinitely long cylinder where the frequency response of a cross-section

in the vicinity of an excitation is determined by considering the 2 dimensional, plane

strain response of the local cross-section. In the current chapter, a novel approach for

computing the frequency response of cylindrically orthotropic, inhomogeneous cylinders

is developed as a means of predicting the e↵ect of asymmetric mortar damage on the

frequency spectrum of PCCP. The method combines the laminate approximate model

proposed by Chen [31], with a modified Galerkin/spectral method (similar to the one

detailed in [65]) to obtain a coupled transfer matrix which relates the modal components

of stress and displacement on the inner surface of the pipe to the the corresponding

53

Chapter 4. Mathematical Modelling of Damaged Pipes 54

components on the the outer surface of the pipe. This approach allows the wave motion

on the pipe’s outer surface to be coupled to the external soil, which is modelled as an

isotropic, elastic solid of infinite extent. The wave motion on the pipe’s inner surface is

then coupled to the water which is modelled as a simple acoustic fluid 1.

The general layout of this chapter is as follows:

• Section 4.1 outlines the derivation of a coupled Transfer Matrix (T-Matrix). This is

an original development of the author which extends the existing Transfer Matrix

method for functionally graded, axially symmetric cylindrical layers [14], [31] to

accommodate coupling between modes and splitting of degenerate modes which

occurs when axial symmetry is removed i.e. material properties become a function

of the circumferential coordinate, ✓. In modelling damaged PCCP, this method is

used to handle the damaged mortar layer.

• Section 4.2 adapts the exact T-Matrix method for axially symmetric, isotropic

layers as derived in [62], to be compatible with the coupled T-Matrix defined in

Section 4.1.

• Section 4.3 describes how the global T-matrix is coupled to the external soil (mod-

elled as a semi-infinite elastic medium) and the internal fluid. The result is a

frequency dependent compliance matrix which relates the Fourier coe�cients of

the displacement on the pipe’s inner and outer surfaces to the Fourier coe�cients

of the loading on the pipe’s inner and outer surfaces.

• Section 4.4 outlines the steps required to implement the algorithm numerically.

1Here the term acoustic fluid denotes a fluid which is Newtonian, irrotational and inviscid


4.1 Transfer Matrix for Asymmetric Layers

Consider the infinitely long cylindrical layer seen in Figure 4.1, composed of linearly

elastic material characterized by density, ⇢(r, ✓), and elastic tensor, C(r, ✓), both of which

can vary as an arbitrary function of the radial coordinate, r, and the circumferential

coordinate, ✓.

x

y

✓

r

C(r, ✓), ⇢(r, ✓)

Rh

Figure 4.1: Cylindrical annulus with dimensions and cylindrical coordinate system de-fined

The layer is assumed to be orthotropic and in plane strain condition (the axial (z) com-

ponent of strain is zero) and the relevant, non-zero stress components, �rr

, �✓✓

, �r✓

, are

given in terms of the non-zero strains, ✏rr

, ✏✓✓

, ✏r✓

, and spatially varying tensor compo-

nents, cij

(r, ✓), i, j = 1, 2, 3 as follows[42]:

2

6

6

6

6

4

�rr

(r, ✓)

�✓✓

(r, ✓)

�r✓

(r, ✓)

3

7

7

7

7

5

=

2

6

6

6

6

4

c11(r, ✓) c12(r, ✓) 0

c22(r, ✓) 0

SYM c33(r, ✓)

3

7

7

7

7

5

2

6

6

6

6

4

✏rr

(r, ✓)

✏✓✓

(r, ✓)

2✏r✓

(r, ✓)

3

7

7

7

7

5

(4.1)

Where the normal and in-plane shear strains are related to radial displacement, u, and

transverse displacement, v, by the strain-displacement relations [42]:


✏rr

=@u

@r(4.2a)

✏✓✓

=1

r

✓

u+@v

@✓

◆

(4.2b)

✏r✓

=1

2

✓

1

r

@u

@✓+@v

@r� v

r

◆

(4.2c)

Furthermore, the wave motion in the layer is governed by the 2-dimensional, frequency

domain equations of elasticity in polar coordinates [42]:

@�rr

@r+

1

r

@�r✓

@✓+

1

r(�

rr

� �r✓

) + ⇢!2u = 0 (4.3a)

@�r✓

@r+

1

r

@�✓✓

@✓+

2

r�r✓

+ ⇢!2v = 0 (4.3b)

By substituting Equation 4.2 into Equation 4.1 and then substituting the resulting ex-

pressions for the stress components into Equation 4.3, a matrix equation of the following

form is obtained:

L1

@X

@r= L

2

X (4.4)

where, X(r, ✓,!) =

u v �rr

�r✓

�

T

is the vector of state variables and L1

and L2

are linear operators defined in matrix form in Equations A.1a and A.1b respectively. The

solution for X must be 2⇡ periodic in ✓ and as such can be expanded as a Fourier series:


X(r, ✓,!) =1X

n=0

⇥

⇥(1)n

x(1)n

(r,!) +⇥(2)n

x(2)n

(r,!)⇤

(4.5)

Here the superscripts 1, 2 refer to modes which are symmetric and anti-symmetric about

the x (horizontal) axis respectively. The matrices ⇥(1,2)n

(✓) are 4 ⇥ 4 diagonal matrices

defined in Equation A.2a, A.2b; these represent the basis functions for the ✓ dependence

of X. The vectors, x(1,2)n

are the nth components of the Fourier expansion of X. By

substituting Equation 4.5 into Equation 4.4 and enforcing orthogonality with respect to

the ✓ basis function matrices, ⇥(1,2)m

(✓):

"

Z 2⇡

0

⇥(1)m

1X

n=0

L1

⇥(1)n

d✓

#

dx(1)n

dr+

"

Z 2⇡

0

⇥(1)m

1X

n=0

L1

⇥(2)n

d✓

#

dx(2)n

dr

=

"

Z 2⇡

0

⇥(1)m

1X

n=0

L2

⇥(1)n

d✓

#

x(1)n

+

"

Z 2⇡

0

⇥(1)m

1X

n=0

L2

⇥(2)n

d✓

#

x(2)n

(4.6a)

"

Z 2⇡

0

⇥(2)m

1X

n=0

L1

⇥(1)n

d✓

#

dx(1)n

dr+

"

Z 2⇡

0

⇥(2)m

1X

n=0

L1

⇥(2)n

d✓

#

dx(2)n

dr

=

"

Z 2⇡

0

⇥(2)m

1X

n=0

L2

⇥(1)n

d✓

#

x(1)n

+

"

Z 2⇡

0

⇥(2)m

1X

n=0

L2

⇥(2)n

d✓

#

x(2)n

(4.6b)

Equations 4.6a and 4.6b together give an infinite by infinite system of ordinary di↵erential

equations in r. The Galerkin approximation involves enforcing orthogonality for a finite

number of terms in the summations in Equations 4.6a and 4.6b. Taking the summation

over n up to N terms, Equations 4.6a and 4.6b yield the following 8(N + 1)⇥ 8(N + 1)

system after integration over ✓:


2

6

4

A(11) A(12)

A(21) A(22)

3

7

5

d

dr

2

6

4

x(1)

x(2)

3

7

5

=

2

6

4

B(11) B(12)

B(21) B(22)

3

7

5

2

6

4

x(1)

x(2)

3

7

5

A(r,!)dx

dr= B(r,!)x

(4.7)

where, x(1) =

x(1)0 x(1)

1 . . . x(1)N

�

T

and x(2) =

x(2)0 x(2)

1 . . . x(2)N

�

T

are

vectors of Fourier coe�cients associated with the symmetric and anti-symmetric com-

ponents respectively. The blocks, A(ij) and B(ij), i = 1, 2, j = 1, 2, are each composed

of (N + 1) ⇥ (N + 1) sub-blocks of size 4 ⇥ 4 obtained by imposing orthogonality for

each n,m combination. The equations for each sub-block of A(ij) and B(ij) are defined

in Equations A.3a, A.3b respectively. Equation 4.7 is a system of ordinary di↵erential

equations with variable coe�cients, which in general cannot be solved analytically. A

solution is sought using the layer approximate model proposed by Chen [31], wherein

the annulus is subdivided into several layers over which A and B are independent of

r. If an annulus with inner radius, R, and thickness, h, is subdivided into P layers,

then the inner, mean and outer radii of the pth sublayer (p = 1, 2, ..., P ) will be given

by ri

p

= R + h(p�1)P

, rm

p

= R + h(2p�1)2P and r

o

p

= R + hp

P

respectively. For a su�ciently

large P , it is assumed that A and B are roughly constant in each sublayer, taking their

respective values at rm

p

. This allows Equation (4.7) to written in standard linear form:

dx

dr= A�1(r

m

p

,!)B(rm

p

,!)x (4.8)

If A�1(rm

p

,!) is invertible, Equation (4.8) is solved via matrix exponentiation on the

interval ri

p

r ro

p

(for a given value of !):

x(r,!) = exp⇥

(r � ri

p

)A�1(rm

p

,!)B(rm

p

,!)⇤

x(ri

p

,!) (4.9)


Evaluating Equation (4.9) at the sublayer’s outer surface:

x(ro

p

,!) = exp

h

PA�1(r

m

p

,!)B(rm

p

,!)

�

x(ri

p

,!)

xo

p

= Tp

(!)xi

p

(4.10)

Equation 4.10, states that the state variables evaluated at the outer surface of the pth

sublayer are related to the state variables evaluated at the inner surface by a linear

transformation, Tp

(!), which is the transfer matrix for the pth sublayer [14], [31]. Since

the state variables must be continuous between adjacent sublayers, one can obtain the

transfer matrix for the entire annulus by multiplying the individual sublayer matrices:

x(R + h,!) =P

Y

p=1

exp

h

PA�1(r

m

p

,!)B(rm

p

,!)

�

x(R,!)

xo

=P

Y

p=1

Tp

(!)xi

(4.11)

4.1.1 Structure of the Coupled T-matrix

The coupled T-matrix developed in the previous section assumes the following block

form:

2

6

4

x(1)o

x(2)o

3

7

5

=

2

6

4

T(11) T(12)

T(21) T(22)

3

7

5

2

6

4

x(1)i

x(2)i

3

7

5

(4.12)

where, x(1)i

,x(1)o

are the state variable vectors at the inner and outer surfaces of the

cylinder associated with the symmetric modes, and x(2)i

,x(2)o

are the state variable vectors

at the inner and outer surfaces of the cylinder associated with the anti-symmetric modes.


Blocks T(ij) represent how the symmetric and anti-symmetric modes are coupled. In

the case where the elastic tensor and density are even functions of the circumferential

coordinate, it can be readily shown that no coupling between the symmetric and anti-

symmetric modes exist, thus: T(12) = T(21) = 0. This implies that for cylinders with one

axis of axial symmetry, the symmetric and anti-symmetric modes decouple and can be

solved independently.

Each block, T(ij), is itself composed of 4⇥ 4 sub-blocks:

T(ij) =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

T(ij)00 T(ij)

01 . . . T(ij)0N

T(ij)10 T(ij)

11 . . . T(ij)1N

. . . . . .

. . . . . .

. . . . . .

T(ij)N0 T(ij)

N1 . . . T(ij)NN

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(4.13)

Each sub-block, Tmn

, represents how each Fourier component of the state vector evalu-

ated on the outer surface, xo

n

, is coupled to the corresponding components of the state

vector evaluated on the inner surface, xi

n

. For the limiting case where the elastic tensor

and density are purely functions of the radial coordinate (axially symmetric cylinders),

it is readily shown that T(12) = T(21) = 0 and blocks T(11) and T(22) become block diago-

nal; this indicates that the circumferential harmonics become decoupled. For n = 0, the

symmetric and anti-symmetric modes give the longitudinal and torsional modes series

respectively. For n � 1, each block gives a separate flexural mode series. The symmet-

ric and anti-symmetric modes have identical flexural mode spectra, di↵ering only by a

rotation of ⇡

2n in the mode shape’s circumferential dependence. The flexural modes for

symmetric cylinders are said to be degenerate [2]. When symmetry is removed, the flexu-

ral modes become distinct, an e↵ect known as mode or peak splitting [2]. Therefore, if a


cylindrical component’s frequency response di↵ers for the symmetric and anti-symmetric

modes, cylindrical asymmetry can be inferred. Furthermore, circumferential dependence

on material properties results in coupling between circumferential harmonics, as the mode

shapes for each resonance can no longer be described by a single value of n .

In the case of the damaged mortar layer, the full procedure outlined in this section will

be used to obtain a coupled T-matrix. The same procedure is also applied to obtain the

T-matrix for the mortar/steel wire layer which as discussed in Chapter 3, is modelled as a

cylindrically orthotropic composite medium. Since there is no circumferential dependence

on the density or the sti↵ness tensor for the mortar/steel wire layer, the resulting T-

matrix becomes block diagonal, with a 4⇥ 4 block corresponding to each value of n for

n = 0, 1, 2, ..., N . This can be shown to be equivalent to taking the T-matrices derived in

[32] for each n and cascading them diagonally to form a 2(N+1)⇥2(N+1), block diagonal

matrix - the first N +1 rows being associated with the symmetric modes and the second

N +1 rows being associated with the anti-symmetric modes. Since the mortar/steel wire

layer is axially symmetric, only blocks associated with the n = 0 will di↵er - representing

the breathing modes for the symmetric and the torsional modes for the anti-symmetric.

4.2 Axisymmetric, Isotropic Layers

As discussed in the previous section, T-matrices for axially symmetric layers assume block

diagonal form. In the special case of isotropic layers, the T-matrix associated with each

circumferential harmonic, T(1,2)n

, can be formulated analytically using Bessel functions of

the first and second kinds [62]. To form a T-matrix for an axially symmetric, isotropic

layer, the individual T-matrices for each value of n are placed along the diagonal of a

2(N + 1)⇥ 2(N + 1) matrix of zeros:


T =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

T(1)0 0 . . . 0

0 T(1)1 . . . 0

. . . .

. . . .

. . . .

T(1)N

T(2)0

T(2)1

.

.

.

0 0 . . . T(2)N

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(4.14)

The procedure outlined in [62] involves evaluating the analytic expressions for the state

variables at the inner and outer surfaces of a cylindrical annulus (x(1,2)i

,x(1,2)o

), solving

for the integration constants in terms of x(1,2)i

, and then substituting the result into the

expression for x(1,2)o

. The resulting coe�cient matrix for each value of n gives the blocks

T(1,2)n

. The procedure is best implemented by using a computer algebra system to ensure

the correctness of the lengthy entries which are omitted here. The full expressions for

the entries in T(1,2)n

can be found in [62].

4.3 Coupling to External Media and Loading Con-

siderations

Pipe sections are connected to soil on their outer surface and water on their inner surface.

Additionally, it is assumed that there is an arbitrarily distributed load on the pipe’s


inner surface. In order to accommodate these features, the coupled T-matrix developed

in Section 4.1 must be re-arranged such that the Fourier coe�cients for the displacement

and stress components are grouped together:

2

6

4

Uo

So

3

7

5

=

2

6

4

T1

T2

T3

T4

3

7

5

2

6

4

Ui

Si

3

7

5

(4.15)

where, Uo,i

=

uo,i

(1) uo,i

(2) vo,i

(1) vo,i

(2)

�

T

and

So,i

=

�rro,i

(1) �rro,i

(2) �r✓o,i

(1) �r✓o,i

(2)

�

T

. The vectors,

uo,i

(1,2),vo,i

(1,2),�rro,i

(1,2),�r✓o,i

(1,2) are column vectors of length (N + 1) containing the

Fourier expansion coe�cients of the displacement and stress components evaluated on

the inner (subscript i) and outer (subscript o) surfaces of the pipe. Matrix blocks,

T1

,T2

,T3

,T4

, are of dimensions 2(N+1)⇥2(N+1), and are obtained by partitioning the

global T-matrix in Equation 4.11. The T-matrix is then re-arranged by swapping rows

and columns to group stress and displacement components as they appear in Equation

4.15.

4.3.1 Coupling to Soil

The soil surrounding the pipe is assumed to behave as a linearly elastic, isotropic material

characterized by longitudinal and shear wave speeds, cL

e

and cT

e

, and density, ⇢e

2. The

soil layer is assumed to be infinite in extent and as such the relevant elastodynamic

solution is one which propagates radially outward, which for an isotropic medium is as

follows [20]:

2Damping in the form of isotropic loss factors can be incorporated by using complex valued wavespeeds


ue

(r, ✓,!) =1X

n=0

8

>

<

>

:

⇥(1)n

2

6

4

H 0n

(kL

e

r) �n

r

Hn

(kT

e

r)

�n

r

Hn

(kL

e

r) H 0n

(kT

e

r)

3

7

5

a(1)n

+⇥(2)n

2

6

4

H 0n

(kL

e

r) n

r

Hn

(kT

e

r)

n

r

Hn

(kL

e

r) H 0n

(kT

e

r)

3

7

5

a(2)n

9

>

=

>

;

(4.16)

where, ue

(r, ✓) =

ue

ve

�

T

is the displacement vector in the soil layer , ⇥(1)n

=

diag[cos(n✓) sin(n✓)], ⇥(2)n

= diag[sin(n✓) cos(n✓)], kL

e

= !

c

L

e

, kT

e

= !

c

T

e

and Hn

(.) are

Hankel functions of the first kind 3. Vectors, a(1)n

and a(2)n

contain integration constants as-

sociated with the symmetric and anti-symmetric modes respectively. The corresponding

stress components are obtained using the isotropic stress-strain and strain displacement

relations with the displacement solution of Equation 4.16.

se

(r, ✓,!) =1X

n=0

n

⇥(1)n

S(1)n

(r)a(1)n

+ ⇥(2)n

S(2)n

(r)a(2)n

o

(4.17)

where, se

(r, ✓,!) =

�rr

e

�r✓

e

�

T

; S(1)n

and S(2)n

are 2 ⇥ 2 matrices whose entries are

lengthy expressions containing Hankel functions of the first kind of order n. The first N

Fourier coe�cients of the displacement components evaluated at the outer pipe radius

are obtained by evaluating Equation 4.16 at r = R+ h (expanded to N terms) and then

enforcing orthogonality with respect to ⇥(1)m

(✓), ⇥(2)m

(✓). This process results in a system

of equations, which can be re-arranged to give the following:

3Hankel functions of the first kind (outward propagating) are typically written as H(1)n (.). Here the

superscript is suppressed as all Hankel functions used are of the first kind and the superscript (1) hasbeen used to denote quantities associated with the anti-symmetric modes.


Ue

o

= Mo

a (4.18)

where, Ue

o

=

U(1)e

(R + h) U(2)e

(R + h) V(1)e

(R + h) V(2)e

(R + h)

�

T

contains the

Fourier coe�cients of displacement components evaluated at r = R+h, a =

a(1) a(2)

�

T

is the vector of undetermined integration constants and Mo

is a 4N ⇥ 4N coe�cient ma-

trix.

The same procedure, when applied to the stress component solution given in 4.17 pro-

duces the following system:

Se

o

= Wo

a (4.19)

where, Se

o

=

�(1)rr

e

(R + h) �(2)rr

e

(R + h) �(1)r✓

e

(R + h) �(2)r✓

e

(R + h)

�

T

contains the

Fourier coe�cients of stress components evaluated at r = R+h and Wo

is a 4N⇥4N co-

e�cient matrix. By solving Equation 4.18 for a and substituting the result into Equation

4.19, the vector, Se

o

, is then given in terms of the vector, Ue

o

:

Se

o

= Wo

M�1o

Ue

o

(4.20)

4.3.2 Coupling to Water

The water inside of the pipe is well approximated as a simple acoustic fluid, characterized

by a speed of sound, cf

and a density, ⇢f

. The pressure field, p(r, ✓,!), in the fluid core


must then satisfy the frequency domain acoustic wave equation in polar coordinates:

@2p

@r2+

1

r

@p

@r+

1

r2@2p

@✓2= �!

2

c2f

p (4.21)

Equation 4.21 admits a solution of the following form:

p(r, ✓,!) =1X

n=0

⇢

✏n

b(1)n

(!) cos(n✓)Jn

✓

!

cf

r

◆

+ b(2)n

(!) sin(n✓)Jn

✓

!

cf

r

◆�

(4.22)

where, b(1)n

and b(2)n

are integration constants associated with the symmetric and anti-

symmetric modes respectively and Jn

(.) are Bessel functions of the first kind 4. It can be

shown that the frequency domain displacement field can be related to the pressure field

as follows [61]:

uf

(r, ✓,!) =

2

6

4

uf

vf

3

7

5

=1

⇢f

!2rp(r, ✓,!)

(4.23)

where, uf

, is the displacement field in the fluid core. By enforcing orthogonality of

uf

(R, ✓,!) expanded toN terms, with cos(m✓) and sin(m✓),m = 0, 1, ..., N , the following

system of equations is obtained:

Uf

i

= Mi

b (4.24)

4Bessel functions of the second kind are suppressed so that the pressure field is finite at the origin


where, Uf

i

=

U(1)f

i

U(2)f

i

�

T

is a 2N ⇥ 1 vector containing the Fourier coe�cients of

the radial displacement evaluated at r = R, Mi

is a 2N ⇥ 2N coe�cient matrix, and

b =

b(1) b(2)

�

T

is a vector of pressure field integration constants (n = 0, 1, ..., N).

Similarly, by enforcing orthogonality of p(R, ✓,!) expanded to N terms, with cos(m✓)

and sin(m✓), m = 0, 1, ..., N , the following system of equations is obtained:

Pf

i

= Wi

b (4.25)

where, Pf

i

=

P(1)f

i

P(2)f

i

�

T

is a 2N ⇥1 vector containing the Fourier coe�cients of the

fluid pressure evaluated at r = R and Wi

is a 2N ⇥ 2N matrix. Solving Equation 4.24

for b and substituting the result into Equation 4.25 gives:

Pf

i

= Wi

M�1i

Uf

i

(4.26)

Here the integration constants have been eliminated so that the undetermined quantities

to be solved for are the Fourier coe�cients of the water’s radial displacement field at the

pipe/fluid interface.

4.3.3 Loading Considerations

Vibration of the pipe is induced by a harmonic load applied at r = R and is assumed to

be arbitrarily distributed in ✓:


f(✓,!) =

2

6

4

fr

f✓

3

7

5

=1X

n=0

n

⇥(1)n

f (1)n

(!) + ⇥(2)n

f (2)n

(!)o

(4.27)

where the Fourier coe�cients of f , f (1)n

, f (2)n

, are determined as follows:

f (1)n

(!) =✏n

⇡

Z 2⇡

0

⇥(1)n

f(✓,!)d✓ (4.28a)

f (2)n

(!) =✏n

⇡

Z 2⇡

0

⇥(2)n

f(✓,!)d✓ (4.28b)

where, ✏n

= 1/2 for n = 0 and ✏n

= 1 for n � 1, is called the Neumann factor.

4.3.4 Solution of Global System

Displacement and stress must be continuous across the the soil/pipe interface, therefore,

the Fourier components of the pipe displacement and stress fields must be equal to the

Fourier components of the soil displacement and stress fields evaluated at r = R + h, so

that the left hand side of Equation 4.15 becomes:

2

6

4

Uo

So

3

7

5

=

2

6

4

Ue

o

Se

o

3

7

5

=

2

6

4

I 0

Wo

M�1o

0

3

7

5

2

6

4

Ue

o

0

3

7

5

(4.29)

where, I is a 2N⇥2N identity matrix. On the pipe’s inner surface, the radial displacement

in the fluid must be equal to the radial displacement in the pipe. Additionally, conti-

nuity of stress requires: �rr

(R, ✓,!) = �p(R, ✓,!) + fr

(✓,!) and �r✓

(R, ✓,!) = f✓

(✓,!).


These conditions imply that the Fourier coe�cients of displacement and stress satisfy

the following:

2

6

4

Ui

Si

3

7

5

=

2

6

4

Uf

i

Sf

i

3

7

5

+

2

6

4

0

f

3

7

5

=

2

6

4

Mi

0

Wi

0

3

7

5

2

6

4

Uf

i

0

3

7

5

+

2

6

4

0

f

3

7

5

(4.30)

where, Uf

i

=

Uf

i

vi

�

T

, Sf

i

=

�Pf

i

0

�

T

, f =

f (1) f (2)�

T

, Mi

=

2

6

4

I 0

D 0

3

7

5

and Wi

=

2

6

4

Wi

M�1i

0

0 0

3

7

5

. D is a diagonal matrix whose entries are defined in A.4.

Substituting Equations 4.29 and 4.30 into Equation 4.15 gives:

2

6

4

I 0

Wo

M�1o

0

3

7

5

2

6

4

Ue

o

0

3

7

5

=

2

6

4

T1

T2

T3

T4

3

7

5

8

>

<

>

:

2

6

4

Mi

0

Wi

0

3

7

5

2

6

4

Uf

i

0

3

7

5

+

2

6

4

0

f

3

7

5

9

>

=

>

;

(4.31)

Equation 4.31 can then be re-arranged so that the quantities to be determined, Ue

o

and

Uf

i

are grouped together on the left hand side of the equation:

2

6

4

�⇣

T1

Mi

+T2

Wi

⌘

I

�⇣

T3

Mi

+T4

Wi

⌘

Wo

M�1o

3

7

5

2

6

4

Uf

i

Ue

o

3

7

5

=

2

6

4

T1

T2

T3

T4

3

7

5

2

6

4

0

f

3

7

5

K1U = K2F

(4.32)

From Equation 4.32, the Fourier coe�cients for the displacement field at the pipe bound-


aries, U, are found by matrix inversion:

U = K�11 K2F

U = ZF(4.33)

Here, the matrix Z(!) can be thought of as a compliance matrix, relating the Fourier

coe�cients of displacement to the Fourier coe�cients of applied load, F(!). One can

then obtain the compliance matrix for a range of frequencies, from which U(!) can be

readily computed for any form of excitation with expansion coe�cients F.

4.4 Numerical Implementation

The process of obtaining the frequency response for a damaged pipe is as follows:

1. For a particular frequency, !, choose the number terms in the circumferential

Fourier expansion, N , to include in the approximation.

2. Form the T-matrices for the axially symmetric, isotropic layers (concrete, steel and

undamaged mortar) according to the procedure outlined in Section 4.2.

3. Form the T-matrices for the asymmetric and orthotropic layers (damaged mortar

and steel/mortar composite respectively) according to the procedure outlined in

4.1. The number of sublayers used should be steadily increased until all entries in

the approximate T-matrix converge below a specified tolerance.

4. Obtain the global T-matrix for the pipe by multiplying the T-matrices for the


individual layers together (left to right) in order of outermost to innermost .

5. Form the coe�cient matrices associated with the soil: Mo

,Wo

, as described in

Subsection 4.3.1.

6. Form the coe�cient matrices associated with the water: Mi

,Wi

, as described in

Subsection 4.3.2.

7. Using the global T-matrix and coe�cient matrices, compute the compliance matrix,

Z(!) from Equations 4.32 and 4.33.

8. Obtain the vector of Fourier coe�cients of the load applied to the pipe/water

interface, f =

f (1) f (2)�

T

, as described in Subsection 4.3.3. Form the global

load vector, F =

0 f

�

T

.

9. Solve for the interface displacement vector, U(!), by multiplying Z(!) by F(!).

This procedure is repeated for all ! in the frequency range of interest. The number

of terms, N , should be su�ciently large so that the elements of U are not a↵ected

by increasing N further. This algorithm was implemented for the specific loading and

damage cases detailed in Chapter 5 using MATLAB, a commercially available scientific

computing language. MATLAB is well suited to this purpose as it has built-in libraries

to compute the values of Bessel and Hankel functions as well as extensive linear algebra

libraries to carry out the required matrix operations. A brief description of the MATLAB

functions used to compute the frequency response of damaged PCCP are included in

Appendix B. The technique outlined in this section has the primary advantage of using

the exact series solutions for the water, soil and axially symmetric, isotropic layers. Only

the orthotropic and non-axially symmetric layers need to be treated in an approximate

manner. This results in an e�cient pseudo-analytical method which does not require

discretization of the entire domain, which becomes particularly impractical when trying


to model the unbounded soil. In addition, the present method allows for more general

insights about the frequency response of non-axially symmetric cylinders. By observing

the structure of the coupled T-matrix one can readily see that for non-axially symmetric

cylinders, the o↵-diagonal blocks are non-zero, leading to the mode coupling and mode

splitting phenomena.

Chapter 5

Modelling Results for Damaged

Pipes

The e↵ects of mortar damage are investigated numerically in this Chapter based on the

mathematical model outlined in Chapter 4. Computations were carried out using the

MATLAB functions described in Appendix B. A simplified model of mortar damage is

presented and the relevant damage parameters to be investigated defined. The e↵ects of

the surrounding soil, mortar sti↵ness, thickness to which the mortar damage penetrates

and angle over which the mortar damage extends are studied individually.

5.1 Model of Mortar Damage

The precise form of mortar damage encountered in the field is not known, however,

post-mortem investigations on burst sections of PCCP have revealed that the mortar

degenerates in localized patches in the vicinity of the wire breaks. Accordingly, the

73

Chapter 5. Modelling Results for Damaged Pipes 74

simplified model shown in Figure 5.1 was used for the purposes of predicting the frequency

response of a damaged section of PCCP buried in soil. The damaged portion of the

mortar is assumed to be characterized by a simultaneous reduction in bulk and shear

sti↵ness (moduli). This area of reduced sti↵ness is of thickness, � and extends over an

angle �. Therefore, the sti↵ness matrix in the damaged mortar layer can be written as

a piecewise constant function of ✓, allowing the integrals in Equations A.3a and A.3b

to be determined in closed form. Furthermore, if the x axis is chosen to be along the

axis of symmetry of the pipe, the sti↵ness matrix becomes an even function of ✓, and

the symmetric and anti-symmetric modes can be shown to decouple, allowing them to

be solved separately.

�

Soil

Damaged MortarUndamaged Mortar

f(✓,!)

↵

Water

�

R

Concrete

Steel

Steel/Mortar

Figure 5.1: Parametrized Model of Damaged PCCP in Soil

The excitation, f(✓,!), is a point force applied to the pipe/fluid interface, with unit

magnitude, radially directed at an angle ✓ = ↵ from the pipe’s axis of symmetry. More

precisely:

f(✓,!) = �(✓ � ↵)r (5.1)


where, �(.) is the Dirac-Delta function. The functional form of the load is chosen to

excite all frequencies and n components with unit amplitude. The Fourier coe�cients

of the forcing function are determined from Equations 4.28a and 4.28b. Additionally, it

is assumed that material damping in all of the pipe constituents as well as the soil and

water is su�ciently small in the frequency range of interest (< 2000 Hz) such that it

may be safely neglected; however, significant loss of vibrational energy and consequent

broadening of spectral peaks will be seen to occur due to coupling between the pipe and

the semi-infinite soil. Material damping has previously been found to be unimportant

compared to the radiation of vibrational energy into the surrounding soil [46]. Finally, it

is assumed that the soil behaves as a simple linearly elastic, isotropic medium, as detailed

in the previous chapter. Here the poro-elastic nature of the soil is neglected and the soil

is simply characterized by a shear wave speed, cT

e

, and fast compressional wave speed,

cL

e

. For evaluating the e↵ects of varying degrees of mortar damage, the pipe is assumed

to be surrounded by a relatively soft soil known as Adrian soil with longitudinal wave

speed cL

e

=373 m/s, shear wave speed cT

e

=152 m/s, and density, ⇢e

=920 kg/m3 [55]. The

influence of sti↵er soil types on the pipe’s spectrum is also evaluated.

In the subsequent sections, both small (600mm) and large (1200mm) diameter LCP sec-

tions are considered. The geometric parameters for both pipes as provided by Munro

LTD. are listed in Table 3.1 of Chapter 3. The three lowest vibration modes, correspond-

ing to n = 2, 3, 4 are of greatest practical interest as they can be readily excited by an

impact source with a usable bandwidth of 1500 Hz for the 600mm pipe and 1000 Hz

for the 1200mm pipe. The mode shapes for the symmetric and anti-symmetric modes

di↵er by an angle of ⇡

2n radians, therefore for each n, the symmetric and anti-symmetric

modes are optimally excited by an impact directed at an angle ↵ = ⇡

2n (halfway between

the mode shapes). Accordingly, we have chosen to perform computations with ↵ = ⇡

12

so that both members of the n = 3 doublet are equally excited and the n = 2 and n = 4


base modes are not impacted at an anti-node.

5.2 Comparison to Finite Element Results

In order to establish the accuracy of the proposed method, the results generated using the

new coupled T-matrix approach are compared results obtained using the commercially

available finite element software COMSOL. The following 2 cases are considered:

1. A freely supported, fluid filled pipe section subject to mortar damage

2. A soil loaded, fluid filled pipe section subject to mortar damage

For the first case, a simple eigenfrequency analysis was performed in COMSOL and

the computed frequencies were compared to those obtained by generating the frequency

response to a radially directed point load using the coupled T-matrix method; where

the resonant frequencies are taken to be the peak frequencies in magnitude response.

This is done primarily to establish the accuracy of the technique in the absence of soil

loading which cannot be modelled exactly with finite element analysis. In the second

case, a swept frequency analysis is performed in COMSOL and the radiation boundary

conditions are approximated by implementing low reflecting boundary conditions where

the soil domain is truncated in the simulation. Although the finite element model cannot

exactly treat the unbounded surrounding soil, the results are used to show that as the

size of soil domain increases, the (excitation point) frequency response calculated using

finite element analysis approaches the frequency response obtained via the proposed

method. In both cases, a significantly damaged 600 mm section of PCCP with the

following damage parameters was considered: damaged mortar wave speeds at 25 % of


their nominal values, damage angle � = 20� and damage thickness, � = 0.75 ⇥ 19mm.

The density and speed of sound for water were taken to be ⇢w

= 1000 kg/m3 and cw

= 1500 ms�1, respectively. For the soil loaded case, Adrian soil was used (see in the

following section for the relevant properties). In both cases, a free mesh composed of

2nd order triangular elements was used. Initially, the mesh density was controlled such

that there was a minimum of 5 elements per (smallest) acoustic wavelength (� = c

f

), as

suggested in the COMSOL user’s reference [16]. Further mesh refinement was found not

to be necessary in either case.

For the freely supported case, the T-matrix method was employed to generate the fre-

quency response to a harmonic point load directed at an angle of ↵ = ⇡

12 , as to ensure

that both members of split degenerate modes are excited with at least some energy. The

number of modes used in the approximation, as well as the number of sublayers used

for the mortar/steel winding and damaged mortar layers were increased until the peak

frequencies identified in the excitation point frequency response function converged (to

within 1 Hz). Since the radial wavelength of these modes is large, only was sublayer was

required for the mortar/steel winding and damaged mortar layers. The number of terms

in the approximation, N , required for convergence was 30. The normalized excitation

point frequency response with modes identified is seen in Figure 5.2. Here the modes

are identified by the primary circumferential wave number, n, followed by either: (1) to

indicate a symmetric mode or (2) to indicate an anti-symmetric mode.


Frequency (Hz)200 400 600 800 1000 1200

Nor

mal

ized

Am

plitu

de

0

0.1

0.2

0.3

0.4

0.5

0.6

2 (2)

2 (1)

3 (1) 3 (2)

4 (1)

4 (2)

Figure 5.2: Excitation point frequency response function, 600 mm pipe, freely supported,water filled

The resonant frequencies identified by a finite element eigenfrequency analysis are com-

pared with those identified by peaks in the excitation point frequency response, computed

using the proposed coupled T-matrix approach are compared in Table 5.1.

For each mode considered, the di↵erence between the resonant frequencies computed

using the two methods is less than 0.5 %. This indicates that the proposed method is in

excellent agreement with the standard finite element modelling approach. The T-matrix

approach seems to give slightly higher resonant frequencies; this may be due to the fact

that the sharp transition in damaged mortar modulus is di�cult to represent using the

relatively smooth harmonic basis functions, however, given the fact that the finite element

method is also only an approximate solution, it is not known which method is closest to

the true result.


For the second case, a swept frequency, finite element analysis was performed using

COMSOL. Since the finite element model cannot directly handle the unbounded soil

medium, the surrounding soil had to be truncated and non-reflecting boundaries im-

posed. Referring to the simplified model of PCCP shown in Figure 5.1, the rectangle

of soil surrounding the damaged pipe is taken to be of dimensions 2H ⇥ 2H and the

non-reflecting boundary conditions are specified along the dashed lines. Non-reflecting

boundary conditions are commonly used in wave propagation problems where only outgo-

ing waves are to be considered. Essentially, an impedance boundary condition is imposed,

where the impedance on the boundary is computed to match the impedance of the wave

motions (longitudinal and shear) incident upon it [16]. This technique cannot completely

prevent reflections from the boundaries, only significantly damp them. Additionally, the

behaviour of the pipe coupled to a finite soil domain will only approach the behaviour

of the pipe coupled to an unbounded soil domain as the dimensions of the soil approach

infinity (i.e. H ! 1); this trend is shown in Figure 5.3.

Frequency (Hz)200 400 600 800 1000 1200 1400 1600 1800 2000

Nor

mal

ized

Am

plitu

de

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4H = 10 6H = 15 6H = 20 6

Figure 5.3: E↵ect of increasing soil dimensions on the frequency response of a 600 mmpipe


Here it is observed that the frequency response converges as the soil dimension increases.

Having limited computational resources, a soil thickness of 20 times the shear wavelength

was the largest soil domain which could be feasibly computed. The normalized excitation

point frequency response for this case, computed via finite element analysis, is compared

to the response computed using the coupled T-matrix approach in Figure 5.4.

Frequency (Hz)0 200 400 600 800 1000 1200 1400 1600 1800 2000

Nor

mal

ized

Am

plitd

ue

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4T - MatrixFEA

Figure 5.4: Comparison of normalized excitation point frequency response functions com-puted via finite element analysis and the coupled T-matrix method

Here, it is observed that the frequency responses computed using both methods are very

similar; the maximum di↵erence between them is 7.4 %. The curves are expected to

converge if the dimensions of the soil in the finite element model could be extended

further. This comparison clearly shows that the proposed method gives similar results

to the closest available standard modelling method (finite element analysis). In terms of

relative e�ciency, the finite element model took nearly 1.5 hours to compute whereas the

coupled T-matrix method took less than 2 minutes. It is therefore concluded that the

coupled T-matrix method is especially useful when computing the frequency response

of circumferentially inhomogeneous materials connected to unbounded domains, which


cannot be modelled directly using finite element analysis.

5.3 E↵ect of Soil

As discussed in Chapter 4, a reduction in the elastic properties of the mortar layer

occurring over only part of the pipe’s circumference results in fully coupled T-matrices

for both the symmetric and anti-symmetric modes. This leads to 2 observable changes

in the frequency response of damaged as compared to undamaged PCCP:

1. Flexural mode splitting - the symmetric and anti-symmetric flexural modes have

distinct frequency responses.

2. Mode coupling - the frequency responses exhibit coupling between circumferential

harmonics (n components).

The former is a consequence of the well-studied phenomenon of eigenvalue splitting,

whereby the resonant frequencies of flexural mode doublets split into distinct values [45],

[2]. The latter e↵ect, which is less well discussed in the literature, is due to the fact

that when a structure deviates from perfect axial symmetry, its resonant modes can no

longer be described by a circumferential basis functions (cos(n✓) and sin(n✓)) with a

single circumferential wave number, n, and are instead contaminated by additional wave

numbers [45].

Typically, Resonance Acoustic Spectroscopy is used in cases where damping (material

and/or radiative) is relatively low, where closely spaced resonant peaks do not overlap;

this allows the resonant frequencies to be determined easily by identifying peaks in the

magnitude spectrum as measured at some point on the structure under investigation


[2]. Evidence of damage is then inferred by locating closely spaced peaks where only a

single peak should be present, indicating that the degenerate members of a doublet have

split. The degree to which the peaks split will scale with the level of deviation from axial

symmetry. This approach to assessing asymmetry proves ine↵ective in the presence of

significant radiative damping. Accordingly, a new method of quantifying deviations from

axial symmetry (a measure of mortar damage in the present study) is developed in what

follows.

The spectral broadening e↵ect of coupling to the surrounding soil is demonstrating by

simulating the spectra obtained using three common soil types: Adrian soil, Catlin soil

and Plainfield soil. The wave speeds and densities for each soil type are taken from [55]

and summarized in Table 5.2. Here the mortar wave speeds, damage angle and damage

thickness are all held constant with: c0L

m

= 0.25⇥4331 = 1083 m/s, c0T

m

= 0.25⇥2514 m/s

= 629 m/s, � = 20�, � = 0.75 ⇥ 19mm = 14.25mm . Only 600mm pipes are discussed

in this section, since the influence of soil type is independent of pipe diameter.

Table 5.2: Wave Speeds and Densities for Adrian, Catlin and Plainfield soils

Soil Type cL

e

(m/s) cT

e

(m/s) ⇢e

(kg/m3)

Adrian 373 152 920

Catlin 463 188 1270

Plainfield 634 259 1510

Figure 5.5 shows the radial displacement magnitude spectra for each soil type, measured

coincident with the excitation, i.e. |u(R, ✓ = ⇡

12 ,!)|.

From Figure 5.5, the lowest three modes: I, II and III are identified by locating peaks

in the magnitude spectra. It is observed that as the soil becomes denser and more sti↵,

the spectral peaks become broader. This is due to the fact that the acoustic impedance


Table 5.1: Comparison between computed eigenfrequencies and peak frequencies for afluid filled damage pipe

Finite Element T-matrixMode Eigenfrequency Peak Frequency Di↵erence

(Hz) (Hz) (%)2 (1) 200 201 0.4992 (2) 207 207 0.0003 (1) 588 589 0.1703 (2) 602 603 0.16604 (1) 1141 1142 0.0884 (2) 1159 1161 0.172

200 400 600 800 1000 1200 14000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

Nor

mal

ized

Rad

ial D

ispl

acem

ent M

agni

tude

AdrianCatlinPlainfieldI

I

II

III

mode

mode

mode

Figure 5.5: Radial Displacement Magnitude Spectra Measured Coincident with Excita-tion for Various Soil Types


of the soil more closely matches that of the pipe constituents, allowing greater coupling

between the pipe and soil, thus higher radiative damping. For the Catlin and Plainfield

soil types, the spectral peaks become so broad as to make them di�cult to locate. In

fact, all three soil types provide su�cient radiative damping as to obscure the individual

members of the split doublet, despite the fact that the level of damage considered here

is significant. This is due to the fact that the spectrum evaluated at a particular point

is composed of the sum of contributions of the symmetric and anti-symmetric modes for

each value of n. Since both the peaks associated with the symmetric and anti-symmetric

modes are significantly broadened due to radiative damping, their sum produces a single

broad peak, despite the fact that the symmetric and anti-symmetric peak frequencies are

distinct from one another. The di↵erence between the symmetric and anti-symmetric

modes becomes clearer when observing the corresponding n components of frequency

response. Figures 5.6 a, 5.6 b and 5.6 c show |u(1)n

(!)| (symmetric) and |u(2)n

(!)| (anti-

symmetric) for n =2,3 and 4, respectively; where, |u(1,2)n

(!)| is the magnitude of nth

normalized frequency response function (radial component of displacement), defined as

follows:

u(1,2)n

(!) =u(1,2)n

(!)

||u(1,2)n

(!)||(5.2)

where, ||.|| denotes the L2 Euclidean norm.


0 500 1000 15000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Frequency (Hz)

Nor

mal

ized

Mag

nitu

deAdrianCatlinPlainfield

n = 2 component of mode I

(a) |u(1)2 (!)| (solid lines), |u(2)

2 (!)| (dashed lines)

0 500 1000 15000

0.01

0.02

0.03

0.04

0.05

0.06

Frequency (Hz)

Nor

mal

ized

Mag

nitu

de

AdrianCatlinPlainfield

n = 3 componentof mode II

n = 3 component of mode I

(b) |u(1)3 (!)| (solid lines), |u(2)


0 500 1000 15000.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

Frequency (Hz)

Nor

mal

ized

Mag

nitu

de

AdrianCatlinPlainfield

n = 4 componentof mode III

n = 4 component of mode

n = 4 componentof mode

II

I

(c) |u(1)4 (!)| (solid lines), |u(2)


Figure 5.6: Normalized Radial Displacement Magnitude Spectra for n=2,3,4; Adrian,Catlin and Plainfield Soil

From 5.6 it can be seen that mode I is primarily composed of the n = 2 component, mode

II is primarily composed of the n = 3 component and mode III is primarily composed

of the n = 4 component. In each case the primary peaks associated with the symmetric

components have maxima that are slightly lower in frequency than the anti-symmetric


component. This is the result of the mode splitting phenomenon discussed earlier. Mode

coupling is also evident; in Figure 5.6 b secondary peaks in the symmetric frequency re-

sponse indicate that the n = 3 component contributes significantly to mode I. Similarly,

in Figure 5.6 c, secondary peaks in the symmetric frequency response indicate that the

n = 4 component contributes significantly to modes I and II. Both the lower peak frequen-

cies and the increased mode coupling seen in the symmetric response can be explained by

the fact that the form of the mortar damage represents a greater change to the symmet-

ric modes as compared to the anti-symmetric modes. For all soils considered, the peaks

are very broad; as a result, the resonant frequency of the pipe is not well approximated

by the location of the maximum spectral response. Consequently, a simple comparison

of the peak frequencies of each mode, as is done in traditional RAS studies, is not an

appropriate measure of asymmetry in the presence of high radiative damping. However,

the e↵ect of mode splitting is to shift the anti-symmetric frequency response with respect

to the symmetric response, while the e↵ect of greater mode coupling experienced by the

symmetric modes is to introduce secondary peaks in only the symmetric response. The

combination of these two factors leads to the symmetric and anti-symmetric frequency

responses diverging. Therefore, another more robust measure of axial symmetry comes

from directly comparing the symmetric and anti-symmetric frequency responses by way

of the Euclidean inner product:

⇣n

=�

�

⌦

u(1)n

(!), u(2)n

(!)↵

�

� =

�

�

�

�

Z

u(1)n

(!)u⇤(2)n

(!)d!

�

�

�

�

(5.3)

where the superscript ⇤ represents complex conjugation and the integrations are over the

frequency range of interest. Here ⇣n

represents the projection of u(1)n

(!) onto u(2)n

(!).

Since both u(1)n

(!) and u(2)n

(!) have been normalized to have L2 norms equal to unity,

0 ⇣n

1. In the limiting case where there is no damage, the symmetric and anti-


symmetric frequency responses will be identical and therefore ⇣n

= 1. On the other hand,

if the damage is significant, the symmetric and anti-symmetric frequency responses will

di↵er greatly, producing a small value for ⇣n

. It is preferable to have a damage metric

which scales with the level of asymmetry. Accordingly, we define the asymmetry index,

⇤n

, as the complement of ⇣n

, that is:

⇤n

= 1� ⇣n

(5.4)

where, 0 ⇤n

1. Here as damage increases causing a greater perturbation to the

axial symmetry of the pipe, ⇣n

will decrease and thus ⇤n

will increase. In subsequent

sections, the e↵ects of mortar sti↵ness, mortar damage thickness and damage angle will

be assessed by computing the asymmetry indices for n = 2, 3, 4.

5.4 E↵ect of Mortar Sti↵ness

In this section, the e↵ect of a reduction of mortar sti↵ness (an approximation of the

influence of mortar damage) is quantified. Here we consider a simultaneous reduction in

both bulk and shear moduli (assuming density remains constant) such that the longitu-

dinal and shear wave speeds (c0L

m

and c0T

m

, respectively) take on 75 %, 50 % and 25 % of

their undamaged values. For each case the damage angle, �, and thickness, �, are held

constant. The parameters associated with the three damage cases studied in this section

are listed in Table 5.3 below.


Table 5.3: Parameters for Damage Cases 1, 2, 3: Decreasing Mortar Sti↵ness

Case Soil Type � (deg) � (mm) c0L

m

(m/s) c0T

m

(m/s)

1 Adrian 20 0.75 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

2 Adrian 20 0.75 ⇥ 19 0.50 ⇥ 4331 0.50 ⇥ 2514

3 Adrian 20 0.75 ⇥ 19 0.25 ⇥ 4331 0.25 ⇥ 2514

The values of the asymmetry indices, ⇤n

; n = 2, 3, 4, for the 600mm and 1200mm pipes

are shown in Figures 5.7a and 5.7b respectively.

n2 3 4

$n

#10-3

0

1

2

3

4

5

6

7

8Case 1: c0

Lm

= 3248 m/s, c0T

m

= 1886 m/s

Case 2: c0L

m

= 2166 m/s, c0T

m

= 1257 m/s

Case 3: c0L

m

= 1083 m/s, c0T

m

= 629 m/s

(a) Asymmetry Indices, 600mm Pipe

n2 3 4

$n

#10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Case 1: c0

Lm

= 3248 m/s, c0T

m

= 1886 m/s

Case 2: c0L

m

= 2166 m/s, c0T

m

= 1257 m/s

Case 3: c0L

m

= 1083 m/s, c0T

m

= 629 m/s

(b) Asymmetry Indices, 1200mm Pipe

Figure 5.7: E↵ect of Mortar Sti↵ness on Asymmetry Indices

From Figure 5.7 it is observed that the asymmetry indices for each value of n increase

as the damaged mortar sti↵ness decreases. This is the expected result as decreasing the

mortar sti↵ness leads to a greater deviation from perfect axial symmetry and therefore

a greater di↵erence between the symmetric and anti-symmetric frequency responses for

each value of n. Another observable feature is that, the values of ⇤n

increase with n.

Damage to the mortar perturbs the symmetric modes to a greater extent as n increases,


whereas the anti-symmetric modes are relatively una↵ected by the damage. This is due

to the fact that the damage is centered on the anti-node of all the symmetric modes

whereas the anti-symmetric modes are centered on an axis rotated ⇡

2n radians from the

symmetry axis of damage. The result is that the di↵erence between the symmetric and

anti-symmetric frequency response functions, and thus the asymmetry index, ⇤n

, increase

with n. Finally, by comparing Figures 5.7a and 5.7b, it is observed that the values of

asymmetry index for the 600mm pipe are much greater than the corresponding values

of asymmetry index for the 1200mm pipe. The larger pipe has a greater wall thickness

than the smaller pipe, thus the reduction of mortar sti↵ness over the same damage angle

and to the same damage thickness represents a much smaller perturbation to the axial

symmetry for the 1200mm pipe than for the 600mm pipe.

5.5 E↵ect of Damage Thickness

In this section, the e↵ect of a reduction of damage thickness is quantified. Here we

consider a reduction of the thickness to which the mortar damage extends; such that �

is 25 %, 50 % and 75 % of the total mortar thickness of 19mm. For each case, the damage

angle and mortar sti↵ness values are held constant. The parameters associated with the

three damage cases studied in this section are listed in Table 5.4 below.

Table 5.4: Parameters for Damage Cases 4, 5, 6: Increasing Damage Thickness


m

(m/s) c0T

m

(m/s)

5 Adrian 20 0.25 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

6 Adrian 20 0.50 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

7 Adrian 20 0.75 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514


The values of the asymmetry index, ⇤n



n2 3 4

$n

#10-3

0

1

2

3

4

5

6

7

8Case 4: " = 4.75 mmCase 5: " = 9.5 mmCase 6: " = 14.25 mm

(a) Asymmetry Indices, 600mm Pipe

n2 3 4

$n

#10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Case 4: " = 4.75 mmCase 5: " = 9.5 mmCase 6: " = 14.25 mm

(b) Asymmetry Indices, 1200mm Pipe

Figure 5.8: E↵ect of Damage Thickness on Asymmetry Indices

From Figure 5.8 it is observed that the asymmetry indices, for both diameters of pipe

and for each value of n, increase as � increases. The trend is similar to that observed for

a reduction in mortar sti↵ness, as increasing the thickness to which the mortar damage

penetrates represents a progressive disruption to the axial symmetry of the pipe and thus

a corresponding increase in the asymmetry index.

5.6 E↵ect of Damage Angle

In this section, the e↵ect of increasing the damage angle, �, is quantified. Here we

consider mortar damage extending over increasing angles such that the damage extends

over progressively greater shares of the pipe’s outer circumference. For each case, the


mortar sti↵ness and thickness to which the mortar damage has penetrated are held

constant. The parameters associated with the three damage cases studied in this section

are listed in Table 5.4 below.

Table 5.5: Parameters for Damage Cases 7, 8, 9: Increasing Damage Angle


m

(m/s) c0T

m

(m/s)

7 Adrian 10 0.75 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

8 Adrian 15 0.75 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

9 Adrian 20 0.75 ⇥ 19 0.75 ⇥ 4331 0.75 ⇥ 2514

The values of the asymmetry index, ⇤n



n2 3 4

$n

#10-3

0

1

2

3

4

5

6

7

8Case 7: - = 10°

Case 8: - = 15°

Case 9: - = 20-

(a) Asymmetry Index, 600mm Pipe

n2 3 4

$n

#10-3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Case 7: - = 10°

Case 8: - = 15°

Case 9: - = 20°

(b) Asymmetry Index, 1200mm Pipe

Figure 5.9: E↵ect of Damage Angle on Asymmetry Indices

From Figure 5.9 it is observed that the asymmetry indices, for both diameters of pipe

and for each value of n, increase as � increases. This is again the expected e↵ect since


increasing the damage angle causes the pipe to become progressively less axially symmet-

ric, leading to a more pronounced di↵erence between the frequency responses associated

with the symmetric and anti-symmetric modes; the asymmetry index then increases as a

consequence of this di↵erence. Here it is noted that positive correlation between damage

angle and asymmetry index holds only for �

2 < ⇡

2n . In the case where �

2 > ⇡

2n , the rela-

tionship between damage angle and asymmetry index becomes more complicated as the

damage then extends over an anti-node for the anti-symmetric modes as well. The re-

sult is that the symmetric and anti-symmetric frequency responses are both significantly

a↵ected by the damage; consequently, the corresponding values of ⇤n

are lower.

5.7 Summary of Results

In the current chapter, the e↵ects of mortar damage were investigated numerically. First,

a parametrized model of the damaged pipe buried in soil was presented. Next, the e↵ects

of the surrounding soil on the frequency response of the pipe were investigated; from

which it was determined that for common soil types the associated radiative damping

was so great as to obscure split degenerate modes, even in the presence of significant

mortar damage. To address this issue, a new method for assessing damage called the

asymmetry index was defined. The asymmetry index quantifies deviations from perfect

axial symmetry by comparing the di↵erence between the symmetric and anti-symmetric

frequency responses via the Euclidean inner product. The asymmetry index was then

computed for the following cases:

1. Decreasing mortar sti↵ness, keeping damage thickness and angle constant

2. Increasing the thickness to which mortar damage penetrates, keeping mortar sti↵-

ness (waves speeds) and damage angle constant


3. Increasing the angle over which mortar damage extends, keeping mortar sti↵ness

and mortar damage thickness constant

In each case, the asymmetry index was found to increase as the level of damage was

increased (for all circumferential wave numbers). This indicates that the asymmetry

index is in fact a suitable indicator of mortar damage and structural asymmetry more

generally. It was also determined that the values of asymmetry index were much greater

for the 600mm diameter pipe than for the 1200mm diameter pipe, which implies that

nondestructive testing of PCCP based on the method presented in this Chapter might be

more appropriate for smaller diameter pipes, though in principle, the method will work for

larger pipes as well. Finally, it was determined that the asymmetry index increased with

circumferential wave number, n, which implies that the asymmetry index becomes more

e↵ective as a damage metric when used with higher modes. On the other hand, higher

order modes su↵er greater radiative damping and are of lower amplitude and so may

be di�cult to observe in the field. Unfortunately, the results obtained from numerical

modelling could not be experimentally verified as access to buried and damaged pipelines

could not be secured.

Chapter 6

Proposed Nondestructive Testing

Procedure

In this chapter a method for processing and interpreting nondestructive testing data

from pipes in the field is presented. In Chapter 5 the e↵ects of mortar damage on the

frequency response of PCCP were found to be most clearly identifiable by computing the

asymmetry index, ⇤n

, which requires the n components of the pipe’s radial displacement,

u(1,2)n

(R,!), evaluated at the pipe’s inner surface. A technique is presented which allows

the n components of displacement field to be computed from sampling the pressure field in

the water inside of the pipeline at a finite number of points using conventional immersion

transducers. The n components of the symmetric and anti-symmetric frequency responses

are then used to estimate the axis of symmetry for the pipe section.

94

Chapter 6. Proposed Nondestructive Testing Procedure 95

6.1 Sampling and Circumferential Harmonics

Consider a buried section of PCCP to be inspected at regular intervals along the pipe’s z

axis by exciting the local cross section with a short duration impact, f(t), directed radially

outward (along the x axis in Figure 6.1). The functional form of the impact, f(t), is to

be recorded directly via a load cell at each inspection location. The resulting transient

pressure field is sampled at M equally spaced points around the pipe’s circumference,

at a distance, a, from the centre of the pipe via ordinary piston type transducers of

diameter, b. The impact is directed at some unknown angle, ↵, to the axis of symmetry

of the pipe, x0. A diagram of the inspection setup is shown in Figure 6.1.

f(t)

Soil

Watera

PCCP

z

x

y

Transducerm

(a) Plan View

x0

x↵

f(t)r ✓

✓m

am

P

(b) Cross-Sectional View

Figure 6.1: Diagram of Inspection Setup

If the response measured at point m(a, ✓m

), m = 0, 1, 2, ...,M � 1 is denoted by gm

(t),

then the corresponding deconvolved frequency response, Hm

(!) is given by:

Hm

(!) =Sgf

(!)

Sff

(!)=

Gm

(!)F ⇤(!)

F (!)F ⇤(!)(6.1)


where Gm

(!) and F (!) are the Fourier transforms gm

(t) and f(t). The function Hm

(!)

represents the component of the frequency domain pressure response normal to the trans-

ducer face, integrated over the surface of the transducer. If the transducer diameter, d,

is small in comparison to the variation of the pressure field over the transducer face then

it can be shown that Hm

(!) is proportional to the pressure field evaluated at r = a,

integrated over the circumferential footprint of the transducer 1:

Hm

(!) ⇡ c

Z

✓

m

+�

✓

m

��

p(a, ✓,!)d✓ (6.2)

where c is an arbitrary constant and � = tan�1�

d

2a

�

is half of the circumferential footprint

of the transducer. If, in the frequency range of interest, the pressure field is adequately

represented by N circumferential harmonics, then the righthand side of Equation 6.2 can

be expressed by integrating the pressure solution in Equation 4.22, expanded to N terms:

Hm

(!) =

Z

✓

m

+�

✓

m

��

N

X

n=0

✏n

b(1)n

(!) cos(n✓)Jn

✓

!

cf

a

◆

+ b(2)n

(!) sin(n✓)Jn

✓

!

cf

a

◆�

d✓ (6.3)

By measuring the frequency response at M = 2N + 1 points, spaced �✓ = 2⇡2N+1 radians

apart, Equation 6.3 gives a (2N + 1)⇥ (2N + 1) system of equations at each frequency:

H = Pb

b = P�1H(6.4)

1In cases where the transducer diameter is larger, the surface integration becomes more complicatedbut can still be computed numerically.


where H =

H0(!) H1(!) H2(!) . . . H2N+1(!)

�

T

,

b =

b(1)0 (!) b

(1)1 (!) b

(2)2 (!). . . b

(1)N

| b(2)1 (!) b

(2)2 (!) . . . b

(2)N

(!)

�

T

and P

is a (2N + 1) ⇥ (2N + 1) matrix whose entries can be inferred from the ordering of H

and b with Equation 6.3. Once b has been determined, Equation 4.23 can be used to

compute the n components of radial displacement evaluated at the pipe/water interface:

u(1,2)n

(R,!) =1

⇢f

!2

d

dR

Jn

✓

!

cf

R

◆�

b(1,2)n

(!) (6.5)

In practice, the frequency response will only be known at L discrete frequencies as com-

puted by the standard Fast Fourier Transform (FFT). Equation 6.5 can then be used to

obtain matrices, u(1) =

u(1)0 u(1)

1 . . . u(1)N

�

T

, u(2) =

u(2)1 u(2)

2 . . . u(2)N

�

T

,

where the column vectors u(1,2)n

(of length L) are the nth Fourier components of the pipe’s

inner surface displacement evaluated at L discrete frequencies.

6.2 Localization of Symmetry Axis

It is assumed that in the field, mortar damage will occur in localized patches such that

there exists some axis, x0, about which the pipe’s elastic parameters are nearly symmetric.

More technically, the axis of symmetry will be an axis whereby the coupling between

the symmetric and anti-symmetric modes is minimized 2. The n components of the

pipe/water interface displacement as computed by Equation 6.5 are with respect to a

polar coordinate system with ✓ measured from the line of application of the impact (see

2In practice the mortar damage is unlikely to be exactly symmetric about any axis, rather onlyapproximately such that the anti-symmetric components of the pipe’s elastic parameters are much smallerthan the symmetric components


Figure 6.1). The Fourier coe�cients can be transformed to ones computed with respect

to a new system with the circumferential coordinate, ✓0 = ✓+↵, by applying the Fourier

shift theorem:

u0(1)n

= cos(n↵)u(1)n

+ sin(n↵)u(2)n

(6.6a)

u0(2)n

= � sin(n↵)u(1)n

+ cos(n↵)u(2)n

(6.6b)

where, u0(1)n

and u0(2)n

are the symmetric and anti-symmetric Fourier coe�cients computed

with respect to the primed coordinate system. Recall that the level of asymmetry for

the nth Fourier coe�cient is quantified by the asymmetry index, ⇤n

, defined in Equation

5.4. Substituting Equation 6.6 into Equation 5.4 gives ⇤n

as a function of ↵:

⇤n

(↵) = 1�

�

�

�

�

�

*

u0(1)n

||u0(1)n

||,

u0(2)n

||u0(2)n

||

+

�

�

�

�

�

(6.7)

Here the inner product operation is a simple dot product since the quantities involved

are vectors instead of continuous functions of frequency. When ↵ is chosen such that x0

is aligned with the symmetry axis of the pipe, the frequency response associated with the

symmetric and anti-symmetric modes will be least similar, as the coupling between the

two modes is minimized. Thus ⇤n

will be closest to 1 since the normalized dot product

in Equation 6.7 will be closest to 0. Accordingly, the sum of ⇤n

over n will be maximized

when the correct value of ↵ is chosen, that is:

maximize↵

N

X

n=0

⇤n

(↵)

!

(6.8)


The optimum value of ↵ can be readily found by a direct search, with 0 ↵ 2⇡.

Once ↵ has been determined, the corresponding values of ⇤n

evaluated at ↵ give the

asymmetry indices required to gage pipe damage.

6.3 Test Procedure

The general test procedure for detecting mortar damage in PCCP as proposed in this

thesis is presented in this section, however the practical design aspects associated with

the development of an automated system are beyond the scope of the present work. The

basic premise of such an automated procedure would be to construct a testing apparatus

featuring an impact source outfitted with a load cell and an array of transducers. This

apparatus would travel along the interior of the pipeline under investigation, inspecting

the local cross-section at regular intervals by means of computing the asymmetry indices

associated with each value of n. The axial distance between test locations would depend

on the desired axial resolution.

The steps involved in the proposed NDE procedure are as follows:

1. Move testing apparatus to the starting axial position, z0

2. Apply impact and record the impact signal, f(t), and the pressure responses, gm

(t)

3. Compute the transfer functions, Hm

(!) according to Equation 6.1, with respect to

unprimed coordinate system

4. Repeat steps 2 and 3 at least 3 times and average the resulting transfer functions

to reduce the e↵ects of noise


5. Use the averaged transfer functions to compute the matrices u(1,2) as outlined in

section 6.1

6. Use u(1,2) to compute the location of the symmetry axis as detailed in section 6.2

7. Use Fourier coe�cients of the symmetric and anti-symmetric modes as computed

with respect to the primed coordinate system (u(1,2)n

) to compute the asymmetry

indices for the current axial position, z0, i.e. compute ⇤n

(z0)

8. Moving the test apparatus along the pipe, repeat steps 1-7 to obtain the asymmetric

indices at Q distinct axial locations, i.e. compute ⇤n

(z0),⇤n

(z1), ...,⇤n

(zQ�1).

Knowing the value of ⇤n

at Q distinct axial locations along the pipe, ⇤n

(zj

) for j =

0, 1, ..., Q � 1, it is then possible to obtain an average value of of the asymmetry index

for the pipeline under investigation as:

⇤n

=1

Q

Q�1X

j=0

⇤n

(zj

) (6.9)

Theoretically, the asymmetry indices will be identically zero at any axial location where

the pipe is undamaged, however in practice small deviations in wall thickness, soil wave

speeds/density, etc, will lead to small, non-zero values of ⇤n

, even at locations where the

pipe has not su↵ered significant mortar damage. If the pipe is assumed to be undamaged

in most test locations and the soil type exterior to the pipe is the same over the entire

pipeline, then ⇤n

will represent baseline level of asymmetry which quantifies the deviation

from perfect axial symmetry for the actual pipeline. The level of damage at any particular

axial position can then be assessed by comparing the ⇤n

to ⇤n

. If ⇤n

is significantly

higher than ⇤n

for all n considered, then the pipeline has likely undergone appreciable

mortar damage at that location. The exact threshold by which ⇤n

must exceed ⇤n

for


a pipe to be identified as damaged will have to be determined through testing many

pipelines in the field. As a first step, the asymmetry indices for pipelines inspected by

more established technologies, such as Remote Field Eddie Current/Transform Coupling,

should be collected and damage locations identified by both methods compared.

Chapter 7

Conclusions and Recommendations

7.1 Conclusions

A series of numerical and experimental studies was undertaken in this thesis in order

to establish a new Nondestructive Evaluation technique capable of detecting the earliest

stages of deterioration of Prestressed Concrete Cylinder Pipe associated with degrada-

tion of the protective mortar layer. As a first step, a comprehensive modal analysis of

undamaged pipe sections was performed. The wave speeds for each pipe constituent were

measured from samples of these materials provided by the manufacturer, using the ultra-

sonic pulse velocity method. Using the measured wave speeds, a three dimensional finite

element model of the pipe was used to compute the lowest 16 eigenfrequencies of 600 mm

and 1200 mm sections of PCCP. The analysis was done with and without including the

e↵ects of the pre-stressing wire, from which it was determined that the pre-stressing wire

had a negligible e↵ect on the lowest 16 resonant modes of PCCP (< 0.1%). The same

finite element analyses were also performed with the bell and spigot features included

102

Chapter 7. Conclusions and Recommendations 103

and the results compared to a simplified model of PCCP where the pipe section was

modelled as a simple multilayered cylinder. The comparison revealed that the bell and

spigot features did not greatly a↵ect the values of resonant frequencies for most modes

but needed to be included in the model in order to obtain accurate mode shapes for

comparing to the experimental mode shapes.

Experimental modal analysis of market-ready sections of 600 mm and 1200 mm diameter

PCCP was performed from which the resonant frequencies (< 750 Hz) were determined

and compared to the eigenfrequencies computed using finite element analysis. The aver-

age di↵erence between the experimental and computed resonant frequencies was found

to be less than 5%. It was determined that one reason for the discrepancy was likely

due to the fact that the mortar thickness varied significantly along the axis of the pipe.

Additionally, the elastic parameters of the actual pipe constituents are expected to di↵er

from the values measured from the samples as these samples came from di↵erent batches

of concrete and mortar than the pipe constituents.

The e↵ect of coupling between pipes was also investigated experimentally by fitting 2

sections of 600 mm pipe together and performing modal analysis on the joined pipes. By

examining the measured spectra it was determined that the coupling between the two

individual sections was su�ciently strong as to prevent axially dependent standing waves

from developing in the individual pipe sections. From this observation it was concluded

that in actual buried pipelines, where each pipe is strongly coupled to another, the

plain strain flexural modes, which are already dominant in the spectra of individual pipe

segments with free ends, will be the only modes discernible in the measured frequency

response.

Based on the modal analysis of healthy pipes it was assumed that a joined pipeline

behaves like an infinitely long, multilayered cylinder where the frequency response of


a cross-section in the vicinity of an excitation is determined by considering the 2 di-

mensional, plane strain response of the local cross-section. A novel pseudo-analytical

technique was developed to solve the elastodynamic problem (the 2 dimensional, fre-

quency domain version), for multilayered cylinders where one or more layers have ✓

dependent material properties. The approach involves extending the standard transfer

matrix (T-matrix) formulation to non-axially symmetric cylindrical layers, which allows

mortar damage to be treated as a perturbation to the axial symmetry of a mortar layer.

The surrounding soil (treated as a semi-infinite isotropic continuum) and the water inside

the pipe (treated as an acoustic fluid) both admit analytical solutions which are coupled

to the global transfer matrix of the pipe. The result is a matrix equation which relates

the frequency dependent Fourier coe�cients of pipe wall displacements to the frequency

dependent Fourier coe�cients of loads applied to the fluid/pipe and soil/pipe interfaces.

Using the pseudo-analytical solution method presented in Chapter 4, a series of para-

metric studies was performed to determine the e↵ects of various soil types as well as

varying degrees of mortar damage. By examining the frequency spectra (evaluated co-

incident with the simulated excitation) from damaged pipes coupled to various common

soil types, it was determined that the splitting of degenerate flexural modes, which oc-

curs as a result of deviation from axial symmetry, could not be directly observed, even

for cases of extreme mortar damage. This was due to the spectral broadening e↵ect

caused by coupling to the surrounding soil. To address this issue, a new damage met-

ric, called the asymmetry index, was proposed which directly compares the symmetric

and anti-symmetric frequency responses and scales with the level of pipe damage. The

asymmetry index was then computed for increasing levels of pipe damage caused by: 1)

a loss of mortar sti↵ness (shear and longitudinal), 2) an increase in thickness to which

the damage occurs and 3) an increase in angle over which the damage extends. In all

cases, the computed values of asymmetry index were found to increase as the level of


pipe damage increased. Accordingly, it was concluded that the asymmetry index is a

suitable metric for assessing the condition of PCCP, at least in principle.

A procedure for inspecting buried water mains, based on computing the asymmetry

indices at various axial locations along the pipeline, was presented. This requires the

symmetric and anti-symmetric responses to be determined at each location so that the

current values for asymmetry indices can be calculated. It was shown that the symmetric

and anti-symmetric frequency dependent Fourier coe�cients of the pipe/water interface

can be evaluated by sampling the pressure field in the water at at least 2N + 1 discrete

points, where N is the largest circumferential wave number present in the response. In

order to obtain the correct asymmetry index values for comparison, the Fourier shift

theorem is employed to locate the approximate symmetry axis of the local pipe cross

section. The approximate symmetry axis is one for which the sum of asymmetry index

values for all modes is maximized. Damaged cross sections are identified by comparing

the values of asymmetry index for a particular axial location to the average value for the

entire pipeline.

7.2 Recommendations For Future Work

Several additional lines of inquiry have been identified by the author that could not

be fully investigated during the course of this thesis work. These areas of research are

outlined below should future researchers choose to pursue them.

• The numerical model presented for computing frequency spectra for water filled,

damaged pipes buried in soil should be validated experimentally. Unfortunately

this key step in establishing a Nondestructive Evaluation Method for PCCP could


not be carried out by the author as access to buried pipelines was not available.

Experimental validation of the method is required in order to be able to rely on

the computed values of asymmetry indices for the various levels of mortar damage

and soil types presented in Chapter 5.

• The feasibility of using higher order modes (n > 4) as a means for assessing mortar

damage should be assessed. In Chapter 5 it was determined that the asymmetry

index values increased as n increased, suggesting that higher circumferential har-

monics are more sensitive to mortar damage. It remains to be seen whether or

not these modes can be readily excited or observed due to the increased levels of

radiative and material damping at higher frequencies.

• The e↵ects of later stages of damage on the asymmetry indices should be deter-

mined. The expressed focus of the current research was to find a way to identify

mortar damage. The method developed for this purpose relies on the fact that

increasing mortar damage manifests by degrading the axial symmetry of the pipe

(asymmetry index increases with mortar damage). However, the later stages of

damage, namely: wire breaks, de-bonding of the steel cylinder and cracking of

the concrete core, will also occur over only part of the pipe’s circumference. Con-

sequently, the e↵ects of these types of damage are also expected to significantly

degrade the axial symmetry of the pipe, leading to high values of measured asym-

metry indices. This could be accomplished by measuring asymmetry indices ex-

perimentally from pipes that have been identified as damaged by existing NDE

methods.

• Extensive field research is required to establish the appropriate threshold for the

asymmetry index values (above baseline asymmetry). As mentioned in Chapter

6, the process of identifying damaged sections of PCCP will essentially involve

computing the asymmetry indices at several, regularly spaced axial locations along


the pipe. The pipe can be considered to have undergone damage at a particular

location if the asymmetry indices computed for that location are above the average

of the asymmetry indices for the entire pipeline by some threshold value. In order

to establish that threshold value, the proposed technique should be used on several

waterlines and the ones that show anomalously high asymmetry values should be

excavated to verify that significant mortar damage has in fact occurred.

• The coupled T-matrix formulation should be extended to 3 dimensions. The ex-

tension to the typical T-matrix formulation presented in Chapter 4 can be readily

extended to three dimensional cylindrical geometries, so long as the variation in

elastic parameters is not a function of the axial coordinate, z. Such a T-matrix

formulation could prove useful in guided wave and scattering studies involving com-

ponents with elastic parameters that are not perfectly axially symmetric.

7.3 Contributions

Major contributions of this thesis are outlined as follows:

• Formulation of a coupled transfer matrix method for computing the frequency re-

sponse of infinitely long cylinders with elastic parameters and/or density which

are arbitrary functions of the radial and circumferential coordinates. The result-

ing structure of the coupled T-matrix and allows for insight into the splitting of

degenerate flexural modes as well as coupling between circumferential harmonics

which occurs when structures deviate from axial symmetry. This represents a gen-

eralization of the concept of transfer matrices for cylindrical geometries which has

not yet been seen in the literature. The coupled T-matrix can be readily coupled


to isotropic layers as well as solid or fluid media described by exact, series solu-

tions; allowing for an e�cient pseudo-analytical solution to problems having only

1 non-axially symmetric layer as was demonstrated in the current study.

• Definition of the asymmetry index as a new way to quantify mode splitting caused

by damage in the presence of significant damping. In traditional NDE applications

of resonance acoustic spectroscopy, mode splitting is quantified by looking at the

di↵erence in frequency between the members of a split degenerate mode. This

process is not possible for highly damped structures as spectral broadening obscures

the individual resonant peaks. The asymmetry index, on the other hand, compares

the symmetric and anti-symmetric frequency responses via the normalized inner

product and was shown to scale with damage. The asymmetry index may be useful

in other RAS studies where significant levels of damping occur.

• Characterization of the frequency spectrum of healthy sections of PCCP; including

the development of a simplified finite element model of PCCP which predicts the

eigenfrequencies (below 750 Hz) from the elastic properties of the pipe constituents

as estimated from measured values of longitudinal and shear wave speeds. The

model of healthy pipe sections was also validated experimentally.

• Demonstration of the possibility of using resonance acoustic spectroscopy for the

purposes inspection of in-service water mains composed of sections of lined type

PCCP.

Appendix A

Mathematical Expressions

L1

=

2

6

6

6

6

6

6

6

4

c11 0 0 0

0 c33 0 0

� c12r

0 1 0

�1r

@

@✓

c12 0 0 1

3

7

7

7

7

7

7

7

5

(A.1a)

L2

=

2

6

6

6

6

6

6

6

4

� c12r

� c12r

@

@✓

1 0

� c33r

@

@✓

c33r

0 1

c22r

2 � ⇢!2 c22r

2@

@✓

�1r

�1r

@

@✓

� 1r

2@

@✓

c22 �⇢!2 � 1r

2@

@✓

c22@

@✓

0 �2r

3

7

7

7

7

7

7

7

5

(A.1b)

⇥(1)n

=

2

6

6

6

6

6

6

6

4

cos(n✓) 0 0 0

0 sin(n✓) 0 0

0 0 cos(n✓) 0

0 0 0 sin(n✓)

3

7

7

7

7

7

7

7

5

(A.2a)

109

Appendix A. Mathematical Expressions 110

⇥(2)n

=

2

6

6

6

6

6

6

6

4

sin(n✓) 0 0 0

0 cos(n✓) 0 0

0 0 sin(n✓) 0

0 0 0 cos(n✓)

3

7

7

7

7

7

7

7

5

(A.2b)

A(ij)mn

=

Z 2⇡

0

⇥(i)m

L1

⇥(j)n

d✓, i, j = 1, 2 (A.3a)

B(ij)mn

=

Z 2⇡

0

⇥(i)m

L2

⇥(j)n

d✓, i, j = 1, 2 (A.3b)

Dnn

=nc

f

!

Jn

⇣

!

c

f

R⌘

J 0n

⇣

!

c

f

R⌘ (A.4)

Appendix B

MATLAB Functions

The following is a brief description of the main computer codes developed for numerical

calculations performed in MATLAB.

[Z1,Z2]=Z_asym_soil_fluid(N,K,f,p) – this function computes the symmetric and

anti-symmetric dynamic compliance matrices at a specified number of frequencies (Z1

and Z2 respectively) for a multilayered cylinder with an arbitrary number of layers having

elastic parameters and/or density defined as a piecewise functions of ✓ of the following

form:

cij

(✓) =

8

>

<

>

:

cij1

: ��

2 ✓ �

2

cij2

: otherwisei, j = 1, 2, 3 (B.1a)

⇢(✓) =

8

>

<

>

:

⇢1 : ��

2 ✓ �

2

⇢2 : otherwise(B.1b)

111

Appendix B. MATLAB Functions 112

The cylinder is coupled to an elastic solid on its outer surface and an acoustic fluid on

its inner surface. The input parameters are as follows:

• N: The number of modes used in the approximation

• K: A vector where the jth element is the number of sublayers used to approximate

the jth layer. For axisymmetric, isotropic layers the number of sublayers is 1.

• A vector of L frequencies: f (in MHz).

• A structure of material and geometric parameters, p, containing the fields:

– p.R: Internal radius (mm)

– p.h: Vector where the jth element specifies the thickness of the jth layer (mm)

– p.rho: A cell array where jth element is a vector defining the densities (in

g/cm3) for the jth layer. The elements are of the form [⇢1 ⇢2] for non-axisymmetric

layers and [⇢] for axisymmetric layers.

– p.c: A cell array where the jth element containing elastic parameters for the

jth layers. For non-axisymmetric layers the elements are sti↵ness coe�cient

matrices is composed of column vectors and are of the form [c1 c2] (GPa),

ci

=

c11i

c12i

c22i

c33i

�

T

, i = 1, 2. For axisymmetric, isotropic layers,

the elements are vectors of the form [cL

cT

] (km/s).

– p.beta: A cell array where the jth element is the angle � (degrees) for the jth

layer. For axisymmetric layers, � is an empty vector.

– p.s: A vector which defines the wave speeds (km/s) and density (g/cm3) of

the external solid medium of the form

cL

e

cT

e

⇢e

�

.

– p.w: A vector which defines the wave speed (km/s) and density (g/cm3) of

the internal acoustic fluid medium of the form

cf

⇢f

�

.

Appendix B. MATLAB Functions 113

The output arrays, Z1 and Z2, are of size (4N + 2)⇥ (4N + 2)⇥ L.

[U1,U2]=Z_to_U(Z1,Z2,F) – this function takes the dynamic compliance arrays output

from Z_asym_soil_fluid (Z1, Z2), the Fourier coe�cients of the load applied to the

cylinder’s inner and outer surfaces, F and returns the Fourier coe�cients of displacement

evaluated on the inner and outer surfaces for the symmetric (U1) and anti-symmetric

(U2) modes. The loading array, F, is of the form

f (1) f (2)�

,

f (1,2) =

f (1,2)i0

f (1,2)i1

. . . f (1,2)i

N

f (1,2)o0 f (1,2)

o1 . . . f (1,2)o

N

�

T

. U1 and U2 are arrays,

each of dimension L⇥ (4N +2), representing the symmetric and anti-symmetric Fourier

coe�cients of the cylinder wall displacements for L discrete frequencies.

an=asym_index(U1,U2) – this function takes the symmetric and anti-symmetric Fourier

coe�cients of cylinder wall displacement (U1, U2) and computes the asymmetry index

using Equation 5.4. The result is the vector an, of dimension (N + 1)⇥ 1, where the jth

element represents the asymmetry index for mode n = j � 1, ⇤j�1.

u=u_theta(U1,U2,theta) – this function takes the symmetric and anti-symmetric Fourier

coe�cients of cylinder wall displacement (U1, U2), and computes the radial component

of the cylinder displacement at the fluid/cylinder interface, u, evaluated at an angle

✓ =theta (radians).

Appendix C

Equivalent Properties for the

Mortar/Steel Winding Layer

For the purposes of modelling, the steel winding and surrounding mortar are treated as

a cylindrical, fibre reinforced composite layer as seen in C.1.

114

Appendix C. Equivalent Properties for the Mortar/Steel Winding Layer115

ᵠz

r

zᶰ

z

r , x3

x1

x2Unit Cell

l

w d

Figure C.1: Mortar/Steel Winding Layer with Unit Cell Geometry as well as Cylindricaland Fibre Aligned Coordinate Systems Defined

The x1 axis is aligned with the axis of the fibre at an angle � to the layer’s axis, z. In

these types of composites, the strain can be related to stress in the x1 � x2 � x3, fibre

aligned system, by a compliance matrix as follows [30]:

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

✏11

✏22

✏33

2✏23

2✏31

2✏12

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

=

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

1E1

�⌫12E1

�⌫13E1

0 0 0

1E2

�⌫23E2

0 0 0

1E3

0 0 0

1G23

0 0

1G13

0

SYM 1G12

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

�11

�22

�33

�23

�31

�12

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

✏ = S�

(C.1)

where, Ei

, ⌫ij

and Gij

(i, j=1,2,3) are the longitudinal moduli, Poisson’s ratios and shear


moduli respectively. For a unidirectional material, the fibre distributions are similar in

the x1 and x3 directions so that it can approximated as transversely isotropic in the

x1 � x3 plane, leading to the following simplifications [30]:

E1 = E3 (C.2a)

G23 = G13 (C.2b)

⌫12 = ⌫23 (C.2c)

The sti↵ness matrix, C with respect to the polar coordinate system can be obtained

by inverting the compliance matrix in the fibre aligned system and then pre and post

multiplying the result by transformation matrices:

C = R1(�)SR2(�) (C.3)

where, R1(�) and R1(�) both feature elements involving sin(�) and cos(�) terms. For

PCCP, the fibre direction is almost coincident with the ✓ direction, that is � ⇡ ⇡

2 and the

transformation matrices become the identity matrix: R1(�) ⇡ R1(�) ⇡ I [30]. Accord-

ingly, from equation C.3, the sti↵ness matrix in the polar coordinate system is simply

the inverse of the compliance matrix in the fibre aligned system (for � ⇡ ⇡

2 ):

C = S�1 (C.4)

The independent elements of S can be estimated by standard rule of mixtures type


micromechanical equations for fibre reinforced composites. With reference to the unit

cell seen in Figure C.1, the fraction of the unit volume associated with the fibre, ⌘f

, and

the volume fraction associated with the surrounding matrix, ⌘m

, can be computed using

the fibre diameter, d, layer thickness, w, and fibre spacing, l, as follows [30]:

⌘f

=⇡d2

4wl(C.5a)

⌘m

= 1� ⌘f

(C.5b)

The macroscopic density is determined by a direct rule of mixtures [30]:

⇢ = ⌘f

⇢f

+ ⌘m

⇢m

(C.6)

The required independent elastic moduli and Poisson’s ratios are determined from the

following [30]:

E2 = ⌘f

Ef

+ ⌘m

Em

(C.7a)

E1 = E3 =E

f

Em

⌘f

Em

+ ⌘m

Ef

(C.7b)

G13 = ⌘f

Gf

+ ⌘m

Gm

(C.7c)

G23 = G12 =G

f

Gm

⌘f

Gm

+ ⌘m

Gf

(C.7d)

⌫12 = ⌫23 = ⌘f

⌫f

+ ⌘m

⌫m

(C.7e)

⌫13 =E2

2G13� 1 (C.7f)


In the case of the mortar/steel winding layer, the fibre constituent is the steel wire and

the matrix material is the surrounding mortar. The relevant elastic moduli are readily

computed from the measured wave speeds (cL

, cT

) and density (⇢); they are as follows:

Table C.1: Fibre and Matrix Properties

Material ⇢ (kg/m3) cL

(m/s) cT

(m/s) E (GPa) G (GPa) ⌫

Steel (fibre) 7800 5900 3200 206 80 0.29

Mortar (matrix) 2242 4330 2510 35 14 0.25

The layer thickness for both the 600 and 1200 mm pipe sections is taken to be the same

as the wire diameter, that is, d = w = 4.88 mm. The wire spacing is 32.5 mm for the 600

mm pipe and 16.5 mmm for the 1200 mm pipe. Using these dimensions and the values

in Table C.1, Equation C.6 can be used to find the combined density for the layer; given

in Equations C.8a (600 mm pipe) and C.8b (1200 mm pipe):

⇢ = 2900 kg/m3 (C.8a)

⇢ = 3530 kg/m3 (C.8b)

Similarly, the sti↵ness matrices computed using the elastic properties in Table C.1 and

Equations C.1 and C.4 together with Equations C.7a - C.7f are found in Equations C.9a

(600 mm pipe) and C.9b (1200 mm pipe):


C =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

43.0 15.0 �0.837 0 0 0

66.2 15.0 0 0 0

43.0 0 0 0

15.7 0 0

21.9 0

SYM 15.7

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(GPa) (C.9a)

C =

2

6

6

6

6

6

6

6

6

6

6

6

6

6

6

4

50.6 18.7 �8.23 0 0 0

91.5 18.7 0 0 0

50.7 0 0 0

17.5 0 0

29.4 0

SYM 17.5

3

7

7

7

7

7

7

7

7

7

7

7

7

7

7

5

(GPa) (C.9b)

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