IHavl'

ll A11105 157030

NBSIR 86-3452

Impact-Echo: A Method for Flaw

Detection in Concrete Using

Transient Stress Waves

Mary Sansalone

Nicholas J. Carino

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Structures Division

September 1986

U.S. DEPARTMENT OF COMMERCE

“ATIONAl BUREAU OF STANDARDS

QC

100

.U56

NO. 86-3452

1986

NBSIR 86-3452

IMPACT-ECHO: A METHOD FOR FLAW

DETECTION IN CONCRETE USING

TRANSIENT STRESS WAVES

Mary Sansalone

Nicholas J. Carino

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Structures Division

Gaithersburg, MD 20899

September 1986

U.S. DEPARTMENT OF COMMERCE. Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director
: i
ABSTRACT

To develop a basis for a nondestructive test method for heterogenous

solids such as concrete, analytical, numerical, and laboratory studies of

transient stress wave propagation in solid plates and in plates containing

flaws are presented. The technique, which is referred to as the impact-echo

method, involves introducing transient stress waves into a test object by

mechanical point impact and monitoring reflections of the waves from internal

defects and external boundaries using a point receiver located close to the

impact point

.

Exact Green's function solutions for a point source on an infinite

plate are used to study the response of a plate to elastic impact by a sphere

in both the time and frequency domains. Displacement and stress fields

produced by a transient point load on the top surface of an elastic plate

are obtained using the finite element method. The finite element solution

is verified by comparison to Green's function solutions.

The finite element method is used to study transient wave propagation

in plates containing a flat-bottom hole or planar disk-shaped flaws. The

finite element solution for the flat-bottom hole is verified by comparison

to an experimental waveform. The numerical studies demonstrate the power

of the finite element method for studying stress wave propagation within

solids having arbitrary shapes, boundary conditions, and applied loads, and

containing defects.

The development (instrumentation and signal processing) of an experimental impact-echo technique for finding flaws within plain and steel

reinforced concrete is described. Laboratory results are presented;

observations drawn from the analytical and numerical studies are used to

aid in understanding experimental results. Successes achieved in the

laboratory suggest that the impact-echo method has the potential to become

a reliable field technique for detecting flaws within concrete structures.

KEYWORDS: concrete; finite element method; flaw detection; Green's function;

nondestructive testing; stress wave propagation
.
ACKNOWLEDGMENTS

The authors are grateful to Dr. Nelson N. Hsu of NBS for providing

insight into problems involving transient stress wave propagation and signal

processing, for performing the experiment that led to Fig. 6.8(a), and for

providing the Green's function solutions for the response of an infinite

plate that were used in Chapters 4 and 5.

The authors also thank Mr. Thomas M. Proctor and Dr. Edward Magrab of

NBS for helpful discussions and for reviewing parts of this report, and Dr.

John 0. Hallquist of Lawrence Livermore National Laboratory for providing

the finite element code which was used for the numerical studies presented

in Chapters 4 and 6

.

TABLE OF CONTENTS

Page

List of Tables x List of Figures x i

CHAPTER 1: INTRODUCTION 1

1 . 1 Introduct ion 1

1.2 Objectives and Scope of Research 2

1.2.1 Objectives 2

1.2.2 Scope 2

CHAPTER 2: BACKGROUND 5

2.1 Basic principles of elastic wave propagation 5 2.1.1 Wave types 5 2.1.2 Wave velocity 6

2.1.3 Reflection and refraction 8

2.1.4 Diffraction at a crack tip 12

2.1.5 Attenuation and divergence 12

2.2 Stress pulses created by elastic impact 14

2.2.1 Hertz theory of elastic impact 14

CHAPTER 3: REVIEW OF PREVIOUS APPLICATIONS OF ECHO

TECHNIQUES TO CONCRETE 21

3.1 Introduction 21

3.2 Test Methods 21

3.2.1 Through-transmission method ................ 22

3.2.2 Echo methods 22

3.2.3 Resonance method 23

3.3 Past applications 24

3.3.1 Pavements and bridge decks 24

3.3.2 Erosion cavities under slabs and behind

walls 26

3.3.3 Dams 27

3.3.4 Piles 28

3.3.5 Reactor structures 29

3.3.6 Refractory concrete 30

3.3.7 Surface opening cracks in submerged

structures 30

3.4 Summary 31

v

CHAPTER 4: A FINITE ELEMENT STUDY OF TRANSIENT WAVE

PROPAGATION IN PLATES 32

4.1 Introduction 32

4.2 Background 33

4.2.1 Transient wave propagation 33

4.2.2 Green's function solutions 34

4.2.3 Finite element method 36

4.3 Plate Response 38

4.3.1 Epicenter response 40

4. 3. 1.1 Green's function solutions 40

4. 3. 1.2 Comparison with finite element

solution 43

4.3.2 Displacement fields within a plate 45

4.3.3 Impact-Echo response 48

4. 3. 3.1 Green's function solutions ........ 49

4. 3. 3. 2 Comparison with finite element

solutions 50

4.4 Summary 51

CHAPTER 5: FREQUENCY SPECTRUM ANALYSIS OF THEORETICAL

IMPACT-ECHO WAVEFORMS 58

5.1 Introduction 58

5.2 Frequency Analysis 58

5.3 Frequency Spectra of Analytical Waveforms 60

5.3.1 Frequency content of the impact 60

5.3.2 Frequency analysis of infinite plate

response 64

5. 3. 2.1 Contact time of the impact 64

5. 3. 2. 2 Test configuration 67

5.3.3 The effect of clipping or removing the

R-wave signal 69

5.4 Conclusions 71

CHAPTER 6: A FINITE ELEMENT STUDY OF THE INTERACTION OF

TRANSIENT STRESS WAVES WITH PLANAR FLAWS 79

6.1

Introduction 79

6.1.1

Brief review of the use of numerical

methods for NDT applications 80

vii
6.2 Background 82

6.3 Plate with a flat-bottom hole 84

6.3.1 Displacement response at the center of

the hole 86

6.3.2 Displacement fields 88

6.3.3 Impact-Echo response 90

6.3.4 Comparison to experimental waveform 92

6.4 Planar disk-shaped flaws in plates 94

6.4.1 Flaw in aluminum 94

6.4.2 Flaws in concrete 96

6. 4. 2.1 Displacement waveforms 97

6. 4. 2. 2 Displacement and stress fields 98

6.5 The effect of test variables on impact-echo

waveforms 101

6.5.1 Contact time of the impact 101

6.5.2. Ratio of flaw diameter to flaw depth 105

6.5.3 Test configuration 109

6.6 Summary and conclusions Ill

CHAPTER 7: IMPLEMENTATION OF THE IMPACT-ECHO METHOD FOR

FLAW DETECTION IN CONCRETE 130

7.1 Introduction 130

7.2 Development of a measurement technique 130

7.2.1 Impact sources 130

7.2.2 Transducer and filter 131

7.2.3 Oscilloscope 133

7.2.4 Concrete specimens 133

7.2.5 Signal acquisition and processing 135

7.2.6 Determining wave speeds through concrete ... 136

7.3 Waveforms and frequency spectra obtained from a solid plate 139

7.3.1 Contact time of the impact 139

7.3.2 Test configuration 141

7.4 Planar disk-shaped flaws in a plate 143

7.4.1 Effects of test variables on impact-echo

waveforms 144

7. 4. 1.1 Contact time of the impact 145

7. 4. 1.2 Flaw diameter 147

7. 4. 1.3 Flaw depth 150

7.4.2 Sensitivity of the impact-echo method for

detecting planar flaws 151

viii
7.5 Flaws in reinforced concrete 154

7.5.1 Wave reflection from a concrete/steel

interface 154

7. 5.

1.1

Wave reflection from a steel bar

embedded in a concrete plate 157

7.5.2 Waveforms and frequency spectra obtained

from a plate containing steel bars 158

7.5.3 Planar disk-shaped flaws in reinforced

concrete 161

7.5.4 A simulated crack 163

7.6 Studies of other types of flaws 166

7.6.1 Depth of a surface opening crack 166

7.6.2 Flaw oriented at an angle to the surfaces

of a plate 168

7.6.3 Grouted and ungrouted metal ducts 170

7.7 Summary and conclusions 173

CHAPTER 8: SUMMARY, CONCLUSIONS, AND FUTURE RESEARCH 206

8.1 Summary 206

8.2 Conclusions 207

8.2.1 Analytical and finite element studies 208

8.2.2 Laboratory studies 209

8.3 Future research 211

BIBLIOGRAPHY 213

IX
LIST OF TABLES

Page

2.1 Specific acoustic impedances 17

2.2 Contact times, forces, and pressures produced by

the impact of steel spheres on a concrete plate ... 17

x
LIST OF FIGURES

Page

2.1 Representative ray paths produced by impact on a plate 18

2.2 a) Spherical wavefronts produced by impact on the

top surface of a plate; and, b) wavefronts after

reflection from the bottom surface 18

2.3 The behavior of a P-wave incident upon an interface

between two dissimilar media; a) reflection and

refraction; and, b) mode conversion 19

2.4 Reflection coefficients at a concrete/air interface

for an incident P-wave as a function of the

incidence angle (Poisson's ratio = 0.2) 20

2.5 Reflection coefficients at a concrete/air interface

for an incident S-wave as a function of the

incidence angle (Poisson's ratio = 0.2) 20

4.1 Schematic representation of the wavefronts produced

by point impact on a semi-infinite solid 52

4.2 Amplitude of particle displacements in the radiation

pattern produced by a harmonic point source 52

4.3 Test configuration for a plate: a) epicenter; and,

b) impact-echo 53

4.4 Epicenter response to a delta function impact 53

4.5 Epicenter response to impact: a) Green's function

solution; and, b) waveform obtained from finite

element analysis 54

4.6 Displacement and stress fields within a 0.5-m thick

plate 125 p.s after the start of the impact; a) vector

plot of displacements and the location of the waves;

and, b) minimum principal stress contour plot 55

4.7 Vector plots of displacements at various times after

the start of the impact: a) 148 ^us; b) 203 jus; and,

c) 250 jjs 56

4.8 Impact-echo response: a) Green's function solution;

and, b) waveform obtained from finite element

analysis 57
5.1 Example of a frequency spectrum obtained from a time

domain waveform using the Fast Fourier Transform

technique: a) digital time domain waveform; and,

b) frequency spectrum 72

5.2 Stress pulse produced by the elastic impact of a sphere on the top surface of a plate, a) force-time

function; and, b) Fourier transform 73

5.3 Half-cycle, sine-squared force-time function 74

5.4 Comparison of surface displacement waveforms for two

force-time functions (H/T = 0.2 and t c /t 2 p

= 0.25):

a) half-cycle, sine-squared function; and,

b) half-cycle, sine function 75

5.5 The effect of contact time of the impact on the

response of an infinite plate for H/T = 0.2:

a) t c /t2p

= 0.15; b) t c /t2P = 0.25; and.

c) t^ /

t

2 p

— 1.0 ......................................... 76

5.6 The effect of impact source - receiver spacing on

the response of an infinite plate for t c

/

2 ?

= 0*25:

a) H/T = 0.05; b) H/T = 0.2; and, c) H/T = 1.0 77

5.7 Effect on the frequency spectrum due to partially

and completely removing the R-wave signal from the

time domain displacement waveform shown in Figure 5a:

a) R-wave signal clipped at an amplitude of -8.; and,

b) R-wave signal completely removed 78

6.1 Diffraction at crack tips: a) P-ray incident on the

edges of a crack; and, b) cylindrical wavefronts P

d P and P^S emitted from edges 113

6.2 The important variables affecting the response of a planar disk-shaped void in a plate 114

6.3 Finite element model of a plate with a flat-bottom

hole 114

6.4 Displacement waveforms obtained at the center of a flat-bottom hole and at the center of the bottom

surface of a solid plate 115

6.5 Vector plots of the displacement fields in a plate

with a flat-bottom hole: a) 6.1 jus after the start

of impact; and b) 10 jus after the start of impact 116

6.6 Close-ups of vector plots of displacement fields

around the flat-bottom hole at various times after

the start of the impact: a) 12 jus; b) 13.5 jus;

and, c) 15 jus 117

xii
6.7 Impact-echo displacement waveforms obtained at the

surface of a plate with a flat-bottom hole and at the

surface of a solid plate 118

6.8 Comparison of analytical response with experimentally

obtained response for a plate with a flat-bottom hole:

a) experimental waveform; and, b) finite element

waveform 119

6.9 Planar disk-shaped flaw in an aluminum plate:

a) Displacement at center of the top surface of the

flaw; and, b) Displacement at a point on the top

surface of the plate 120

6.10 Planar disk-shaped flaw in a concrete plate:

a) Displacement at center of the top surface of the

flaw; and, b) Displacement at a point on the top

surface of the plate 121

6.11 Displacement and stress fields in a plate containing

a disk-shaped flaw 65 ps after the start of the

impact 122

6.12 Displacement and stress fields in a plate containing

a disk-shaped flaw 80 jus after the start of the

impact 123

6.13 Displacement and stress fields in a plate containing

a disk-shaped flaw 95 jus after the start of the

impact 124

6.14 Displacement and stress fields in a plate containing

a disk-shaped flaw 125 jus after the start of the

impact 125

6.15 The effect on displacement waveforms caused by

varying the contact time of the impact: a) 20 jus;

b) 40 jis; and, c) 80 jus 126

6.16 The effects on surface displacement waveforms caused

by varying the diameter of a flaw. Flaw depth is

0.2 m. Flaw diameter is a) 0.05 m; b) 0.1 m;

c ) 0.2 m ; and , d) 0.4 m 127

6.17 Surface displacement waveform obtained from a plate

containing a 0.1 m diameter flaw located 0.4 m below

the top surface of the plate 128

6.18 The effect on displacement waveforms of increasing

the distance, H, between the impact point and the

point where the surface displacement is measured:

a) 0.025 m; b) 0.1 m; and, c) 0.15 m 129

xiii
7.1 Impactor with spring driven mass 177

7.2 NBS conical displacement transducer 177

7.3 0.2-m thick concrete plate containing steel bars:

a) elevation view; and, b) plan view 178

7.4 0.5-m thick concrete plate containing polyurethane

disks: a) plan view; and, b) elevation view 179

7.5 Formwork for 0.5-m thick concrete plate with disks in

place prior to placement of concrete 179

7.6 Plan view of 0.5-m thick reinforced concrete plate

containing artificial flaws 180

7.7 Formwork for 0.5-m thick reinforced concrete slab with

flaws in place prior to placement of concrete 180

7.8 Test set-up: a) experimental set-up; and, b) schematic

representation 181

7.9 Surface displacement waveform obtained from 0.25-m

thick concrete plate showing computed wave arrival

times 182

7.10 Waveforms and frequency spectra obtained from the

0.25-m thick concrete plate using various duration

impacts: a) 34 ^is; b) 48 ^is; and, c) 90 ps 183

7.11 Waveforms and frequency spectra obtained from the

0.25-m thick concrete plate for various distances

between the impact point and the receiver:

a) 0.025 m; b) 0.05 m; c) 0.125 m; and d) 0.25 m 184

7.12 Test configuration for tests carried out over planar

disk-shaped flaws 185

7.13 Relationship between depth and frequency 185

7.14 Waveform and frequency spectrum obtained from 0.5-m

thick concrete plate 186

7.15 Waveforms and frequency spectra obtained over a 0.2-m

diameter disk-shaped flaw located 0.127 m below the

top surface of the 0.5-m thick plate. Duration of the

impacts was: a) 26 ^is; b) 40 ps; and, c) 80 pa 187

7.16 Waveforms and frequency spectra obtained over various

diameter disk-shaped flaws located 0.25 m below the

top surface of the 0.5-m thick concrete plate: a) 0.1-m

diameter; b) 0.2-m diameter; and, c) 0.5-m diameter .... 188

xiv
7.17 Waveforms and frequency spectra obtained from flaws

located at different depths in the 0.5-m thick

concrete plate: a) 0.127 m; b) 0.258 m; c) 0.38 m; and d) bottom surface of the plate (0.5 m) 189

7.18 Summary of results of planar flaw studies based on frequency spectrum analysis 190

7.19 Wave reflection from concrete/air and concrete/steel

interfaces 191

7.20 Waveforms and frequency spectra obtained from various

diameter bars located 0.06 m below the top surface of

the 0.2-m thick concrete plate; contact time of the

impacts was 20 jus: a) 2.54 cm; b) 3.81 cm; and,

c) 4.45 cm 192

7.21 Waveforms and frequency spectra obtained from various

diameter bars located 0.06 m below the top surface of

the 0.2-m thick concrete plate; contact time of the

impact was 54 jus: a) 2.54 cm; b) 3.81 cm; and,

c) 4.45 cm 193

7.22 Wavelength components of input pulses having a) 20 jus

contact time; and, b) 54 jus contact time 194

7.23 Waveform and frequency spectrum obtained from a plain

section of 0.5-m thick concrete plate containing a variety of flaws and reinforcing bars 195

7.24 Response obtained over a 0.35-m diameter flaw located

0.26 m below the top surface of the 0.5-m thick

reinforced concrete plate: a) test configuration; and,

b) waveforms and frequency spectrum 196

7.25 Response obtained over the 0.35-m diameter flaw shown

in Figure 7.24 for a different test configuration:

a) test configuration; and, b) waveform and frequency

spectrum 197

7.26 Response obtained over a simulated crack in plain

concrete: a) test configuration; and, b) waveform and

frequency spectrum 198

7.27 Response obtained over a simulated crack in reinforced

concrete: a) test configuration; and, b) waveform and

frequency spectrum 199

7.28 Response obtained with impact point and receiver

located on opposite sides of a vertical surface

opening crack: a) test configuration; and, b) waveform

and frequency spectrum 200

xv
7.29 Schematic illustration of interaction of waves with

vertical surface opening crack 201

7.30 Results obtained from a flaw oriented at a 30 degree

angle to the surfaces of the 0.5-m thick plate 202

7.31 Waveforms and frequency spectra obtained at various

points over the inclined flaw shown in Figure 7.30 203

7.32 Responses obtained over hollow and grouted thin metal

ducts for a contact time of 70 jus : a) test

configuration and results for hollow duct; and,

b) test configuration and results for grouted duct 204

7.33 Responses obtained over duct for a contact time of

25 psi a) waveform and frequency spectrum for hollow

duct; and, b) waveform and frequency spectrum for

grouted duct 205

xv i
CHAPTER 1

INTRODUCTION

1.1 INTRODUCTION

For more than thirty years, efforts have been made to apply

stress wave propagation to nondestructive testing of concrete.

These efforts have met with limited success, although some progress

has been made in measuring the thickness of plate elements and for

integrity testing of piles. Progress has been limited because of

the heterogeneous nature of concrete, which strongly attenuates

high frequency waves; thus, traditional wave propagation methods

developed for flaw detection in metals cannot be used for evaluation

of concrete.

The research project that led to this report was aimed at

developing a basis for a nondestructive test method for finding

flaws within concrete using transient stress waves. The method

that has been developed is referred to as the impact-echo method.

The technique involves introducing a transient stress pulse into

a test object by mechanical impact and monitoring surface

displacements caused by the arrival of reflections of the pulse

from internal defects and external boundaries.

The impact-echo test is a simple procedure; however,

successful interpretation of the displacement waveforms requires

an understanding of the interaction of transient stress waves with

internal defects. The current state of knowledge about the

propagation of transient stress waves in bounded solids containing

1
defects is very limited. Thus this study has focused on

understanding the nature of transient stress wave propagation in

solids containing defects, as well as on the development of an

impact-echo method for finding flaws within concrete structures .

1.2 OBJECTIVES AND SCOPE OF RESEARCH

Following is a statement of the main objectives and the scope

of the research work that is presented in this report

.

1.2.1 Objectives

The objectives of this report are:

1) to study the displacement and stress fields created by

transient point impact on an elastic solid;

2) to gain an understanding of transient elastic stress

wave propagation in solid plates and in plates containing planar

flaws; and,

3) to develop a method (testing technique and signal

processing) for finding flaws within hardened concrete which is

based on the generation and reception of transient stress waves.

1.2.2 Scope

Chapter 2 presents a brief review of the basic principles

of elastic wave propagation. Values for wave speeds, reflection

and attenuation coefficients are given for concrete. The Hertz

theory of elastic impact is reviewed and applied to the impact of

a steel sphere on a large concrete plate.
3

Chapter 3 presents a review of the previous applications

of stress wave propagation techniques for evaluating concrete

structures

.

Chapter 4 discusses transient stress wave propagation in

solid plates. The finite element method is used to study the

displacement and stress fields generated by a transient point load

on a plate. The validity of the finite element method for modeling

transient wave phenomenon in elastic solids is established by

comparing surface displacement waveforms obtained from finite element

analyses to exact Green's function solutions for impact on an

infinite plate.

In Chapter 5, the Fast Fourier Transform technique is used

to compute the frequency content of 1) stress pulses generated by

impact and 2) theoretical surface displacement waveforms obtained

from the Green's function solution for an infinite plate. The

effects on displacement waveforms and their corresponding frequency

spectra due to changes in the duration of the impact and due to

changing the test configuration (spacing between the impact point

and the point where the displacement is calculated) are studied.

The effect on the frequency spectrum of clipping or removing the

Rayleigh wave signal is shown.

Chapter 6 presents a finite element study of the interaction

of transient stress waves with planar flaws. The primary focus

of this study is to determine the effects on displacement waveforms

caused by waves diffracted from the sharp edges of a flaw.

Displacement and stress fields within plates containing flat-bottom
4

holes and planar disk-shaped flaws are presented. The finite element

solution is verified by comparing a surface displacement waveform

obtained from the analysis of a plate containing a flat-bottom hole

with an experimentally obtained waveform. To show the effect of

important test variables on surface displacement waveforms, the

results of parameter studies of plates containing planar disk-shaped

flaws are presented. The variables studied included the duration

of the impact, the diameter and the depth of a flaw, and the point

where the displacement is monitored.

Chapter 7 discusses the development of an impact-echo

technique for finding flaws within concrete. Results obtained from

a variety of controlled flaw studies (simulated flaws in plain and

reinforced concrete specimens) are presented. Observations and

conclusions drawn from the analytical studies presented in the

previous chapters are used to help interpret experimental waveforms

and frequency spectra.

Chapter 8 summarizes important conclusions and discusses

ongoing and future work and related applications.
CHAPTER 2

BACKGROUND

The purpose of this chapter is to provide background

information on elastic wave propagation in solids and on the Hertz

theory of elastic impact. This chapter is intended to provide basic

background for those unfamiliar with stress wave propagation in

solids

.

2.1 BASIC PRINCIPLES OF ELASTIC WAVE PROPAGATION

2.1.1 Wave Types

There are three primary modes of stress wave propagation

through isotropic, elastic media: dilatational , distortional , and

Rayleigh waves. Dilatational and distortional waves, commonly

referred to as compression and shear waves, or P- and S-waves, are

characterized by the direction of particle motion with respect to

the direction the wavefront is propagating. In a P-wave, motion

is parallel to the direction of propagation; in the S-wave, motion

is perpendicular to the direction of propagation. P-waves can

propagate in all types of media; S-waves can propagate only in media

with shear stiffness, i.e., in solids. Where there is a solid/liquid

or a solid/gas interface, Rayleigh waves (R-waves) can propagate

along the interface. In an R-wave , particle motion is retrograde

elliptical. The amplitude of motion in the R-wave decreases

exponentially with distance away from a free boundary [13]. The

5
6

shape of the P-, S-, and R-wavefronts depends upon the

characteristics of the source that is used to generate the waves.

There are three idealized types of wavefronts: planar, spherical,

and cylindrical. When the stress waves are generated by a point

source applied normal to the top surface of a plate, the resulting

P- and S-wavefronts are spherical and the R-wavefront is circular.

2.1.2 Wave Velocity

The propagation of transient stress waves through a

heterogenous bounded solid, such as a structural concrete member,

is a complex phenomenon. However, a basic understanding of the

relationship between the physical properties of a material and the

velocity of wave propagation can be acquired from the theory of

wave propagation in infinite isotropic elastic media [86].

In infinite elastic solids, the P-wave velocity, Cp , is a

function of Young's modulus of elasticity, E, the mass density,

P, and Poisson's ratio, v :

( 2 . 1 )

In bounded solids, such as thin plates or long rods, P-wave velocity

can vary depending on the dimensions of the solid relative to the

component wave length( s ) of the propagating wave. For rod-like

structures, such as piles, P-wave velocity is independent of

Poisson's ratio if the rod diameter is much less than the component

wavelength(s) of the propagating wave [5]. In this case, Cp is
7

given by the following equation:

C

P

( 2 . 2 )

For a Poisson's ratio of 0.2, a typical value in concrete, the P-wave

velocity is five percent higher in an infinite solid than in a long

thin rod.

The S-wave velocity, C

g

, in an infinite solid is given by

the following equation:

2 p ( 1 + i/ )

G

P (2.3)

where G = shear modulus of elasticity.

A useful parameter is the ratio of the S- to P-wave speeds:

C.. (2.4)

For Poisson's ratio of 0.2, the ratio of the S- to P-wave velocity

is 0.61.

Rayleigh waves propagate at a velocity, C^, which can be

determined from the following approximate formula [92]:

C R

0.87 + 1.12 v

1 + v (2.5)

For Poisson's ratio of 0.2, the R-wave velocity is 92 percent of

the S-wave velocity.

P- and R-wave velocities are related by the following

equation:

0.87 + 1.12 v

1 + v

( 2 . 6 )
8

Thus, if Poisson's ratio is known, the measured R-wave velocity

can be used to estimate the P-wave velocity [83]. For Poisson's

ratio of 0.2, the R-wave velocity is 56 percent of the P-wave

velocity.

Wave velocity, C, frequency, f, and wavelength, X , are

related by the following equation:

C = f X (2.7)

Transient waves generated by a point source are composed of a range

of frequencies; therefore, they contain a number of different

wavelengths. The frequency content of transient waves is discussed

in detail in Chapter 5.

2.1.3 Reflection and Refraction

When a P- or S-wavefront is incident upon an interface between

dissimilar media, "specular" reflection occurs. (The term specular

reflection is used since the reflection of stress waves is similar

to the reflection of light by a mirror.) Stress waves can be

visualized as propagating along ray paths and the geometry of ray

reflection is analogous to that of light rays. Representative ray

paths are shown in Fig. 2.1. The path of the R-wave is denoted

by an R, and the paths of variously reflected P- and S-rays are

denoted by a P or an S. For example, the 2P-ray (or PP-ray)

represents a P-ray incident upon and reflected by an interface.

The shape of the reflected wavefront can be determined by considering

the reflection of individual rays. Fig. 2.2(a) illustrates the

propagation of P-, S-, and R-wavefronts generated by point impact
9

on a plate. In Fig. 2.2(b), incident rays OA and OB are reflected

as rays AC and BD, and it is seen that the reflected rays behave

as though they were radiating from virtual point O'. Thus, the

reflected wavefronts (2P and 2S) are also spherical.

At a boundary between two different media only a portion

of a stress wave is reflected. The remainder of the wave penetrates

into the underlying medium (wave refraction), as shown in Fig. 2.3a.

The angle of refraction, |3 , is a function of the angle of incidence,

d , and the ratio of wave velocities, C

2 /C^, in the different media,

and is given by Snell's Law [79]:

c

2

sin j3 = sin 6 (2.8)

C

1 Unlike light waves, stress waves can change their mode of

propagation when striking the surface of a solid at an oblique angle

[85]. Incident P-waves can be partially reflected as both P- and

S-waves and can be refracted as both P- and S-waves, depending on

the angle of incidence. Since S-waves propagate at a lower velocity

than P-waves, they will reflect and refract at angles (determined

using Snell's Law), 0

g and 0 s

> that are less than the angles

of reflection and refraction for P-waves, as shown in Fig. 2.3(b).

The relative amplitudes of reflected waves depend upon the

mismatch in specific acoustic impedances at an interface, the angle

of incidence, the distance of an interface from the pulse source,

and the attenuation along the wave path. The influence of each

of these factors is considered in the following discussion.

The portion of an incident plane P-wave that is reflected
10

at an interface between two media of different densities or elastic

moduli depends on the specific acoustic impedances of each medium.

The specific acoustic impedance, Z, of a medium is

Z = p C

p

(2.9)

Since C

p

is approximately equal to ^ E / p ;

Z = /IT (2.10)

Specific acoustic impedance values for P-waves in selected materials

are given in Table 2.1. Eq. (2.9) is also valid for S-waves if

the S-wave velocity is used to calculate acoustic impedance.

The amplitude of particle motion in a reflected ray, ,

is maximum when the angle of incidence of the ray is normal to the

interface and is determined from the following equation [ 5 ]

:

( z

i - z2 ) A = Ax (2.11)

( Zj + z

2 )

where Aj is the amplitude of motion in the incident ray.

Coefficients of reflection for the amplitude of a reflected

wave as a function of the angle of incidence can be determined using

the formulas in Ref. [48], which are applicable for plane waves

incident upon plane boundaries. These formulas were used to

calculate the reflection coefficients for a concrete/air interface.

Fig. 2.4 shows reflection coefficients for an incident P-wave , and

Fig. 2.5 shows reflection coefficients for an incident S-wave.

It is assumed that each incident wave has an amplitude equal to

unity. Each figure is composed of three graphs. The graph in the

upper left gives the reflection coefficients for the wave with the

same mode as the incident wave; for example in Fig. 2.4, which is
11

for an incident P-wave, the upper left graph shows the reflection

coefficients for a reflected P-wave. The graph in the lower right

gives the reflection coefficients for the mode-converted wave.

The lower right graph in Fig 2.4 shows the reflection coefficients

for the S-wave produced by the mode conversion of the incident

P-wave. The graph in the upper right gives the angular relationship

between the incident wave and the mode-converted wave, which is

determined by Snell's Law. The drawing in the lower left gives

an illustrative example.

In the previous discussion it was assumed that reflection

and refraction of wavefronts occurred at planar interfaces between

two dissimilar media. This type of analysis is also applicable

to flaws or discontinuities within a medium.

The ability of stress wave propagation methods to detect

flaws or discontinuities (sensitivity) depends on the component

frequencies (or wavelengths) in the propagating wave and on the

size of the flaw or discontinuity. A general rule is that waves

will diffract or bend around the edges of discontinuities if the

size of the discontinuity is on the order of or less than the

component wavelengths in the propagating wave. Therefore, to detect

flaws on the order of 0.1 m, it is necessary to introduce into the

concrete (P-wave velocity of 4000 m/s) a stress pulse that contains

frequencies greater than approximately 20 KHz (wavelengths less

than approximately 0.1 m)

.
12

2.1.4 Diffraction at a Crack Tip

When a stress pulse is incident upon a crack tip (or the

sharp edge of a discontinuity) diffracted waves are produced. These

waves are spherical and they originate at the crack tip. Mode

conversion of the incident wave also occurs at the crack tip,

producing a second diffracted wave. For example, a P-wave incident

upon a crack tip produces a diffracted P-wave and a diffracted

S-wave

.

2.1.5 Attenuation and Divergence

As a wave propagates through a solid the acoustic pressure

(and thus the amplitude of particle motion) decreases with path

length due to attenuation (scattering and absorption) and divergence.

In a heterogeneous solid, scattering is the result of wave

reflection, refraction, diffraction, and mode conversion at each

interface between dissimilar media. In ordinary concrete the density

and the elastic modulus of the coarse aggregates are generally higher

than those of the mortar; thus, from Eq . (2.10), the specific

acoustic impedance of the coarse aggregate is higher than that of

the mortar. If the wavelength of the propagating wave is less than

the size of the aggregate, this mismatch in impedances causes

scattering of the incident wave as the waves undergo reflection

and refraction from each mortar-aggregate interface [83]. For higher

quality concrete the specific acoustic impedance of the mortar

approaches that of the coarse aggregate and scattering is reduced.

(Quality in this context refers to the density and elastic modulus
13

of the mortar phase. Higher quality is synonymous with higher

density and greater elastic modulus or both.) In evaluation of

concrete, lower frequency waves must be used (i.e., the wavelength

to aggregate size ratio must be increased [84]) to reduce the

attenuation of wave energy due to scattering. However, use of lower

frequency waves reduces the sensitivity of the propagating wave

to small flaws. Thus, there is an inherent limitation in the flaw

size that can be detected within concrete.

Although attenuation of wave energy in heterogeneous solids

is primarily due to scattering, part of the wave energy is absorbed

and turned into heat (hysteretic damping) [85]. In solids damping

is mainly caused by internal friction.

Attenuation also affects the frequency content of the pulse

propagating in a heterogeneous medium [75]. Pulses produced by

point impact contain a range of frequencies. In a material such

as concrete, the higher frequency components of the propagating

pulse will be preferentially attenuated with path length. As a

result, the frequency spectrum of the pulse is continuously shifted

to lower frequencies. Thus, both the sensitivity and the acoustic

pressure of a pulse decrease with path length.

For non-planar waves, reduction of the acoustic pressure

also occurs due to spreading of the wavefront as it propagates

through the test medium (divergence) [79]. For a point source producing a spherical wavefront, divergence causes the pressure

to vary as the inverse of the distance from the source [5].

The path of a received signal includes both the incident
14

path (source to flaw) and the reflected path (flaw to receiver).

Both incident and reflected waves undergo attenuation and divergence;

thus, the amplitude of the signal decreases with total path length.

The relative losses caused by attenuation and divergence will depend

upon the attenuation coefficient of both the test medium and the

frequency content of the propagating wave [48]

.

2.2 STRESS PULSES CREATED BY ELASTIC IMPACT

Electro-acoustic transducers and mechanical impact are the

two common methods used to introduce a stress pulse into a test

object. The selection of a pulse source for a specific application

depends upon the size of the flaws to be detected and on the

characteristics of the test medium. In the inspection of metals,

high frequency, short-duration, pulses are introduced into a test

object by an electro-acoustic transducer. No commercial transducers

are satisfactory for pulse-echo testing of concrete; thus, a

mechanical impact source normally is used to generate a stress pulse

with sufficient acoustic energy to overcome the effects of

attenuation and divergence. The type of impact source that is used

determines the energy and frequency content of the stress pulse.

2.2.1 Hertz Theory of Elastic Impact

Impact is defined as the collision of two solids. In this

report, this collision involves the impact of a small diameter steel

sphere dropped onto the surface of a large concrete plate. The

duration of this impact is short, ranging from approximately 20
15

to 90 us. During the impact a portion of the potential energy in

the sphere is transferred to elastic wave energy in the plate.

The force-time function resulting from the elastic impact

of a sphere on a solid can be approximated as a half-cycle sine

curve

.

t F - Fmax

sin ir 0 < t < t c (2.12)

cc where F = the maximum force (N); and max

t c = contact time of the impact Cs).

The amplitude of this force-time function, F

max , affects the

magnitude of stresses and displacements that occur in the stress

waves generated by impact. The contact time affects the frequency

content of the pulse.

The contact time of an impact produced by dropping a steel

sphere on a concrete slab can be approximated by the Hertz elastic

solution for a sphere dropped onto a thick plate [28]:

R

t = 5.97 [ n ( 6 + S D )!

0 ’ 4 (2.13)

P

(h) 0 ' 1

1 * "p

2

s = 1— (2.14)

E

P

1 - " s

2

S

s - (2.15)

where P

g = density of the sphere (kg/nr);

R = radius of the sphere (m);

h = drop height (m)

;

v = Poisson's ratio for the plate;

= Poisson's ratio for the sphere;

E = Young's modulus of elasticity for the plate

( N/m ) ; and

E = Young's modulus of elasticity for the sphere

(N/m2 );
16

Q 9 For a steel sphere dropped onto a concrete slab (E

p = 36 x 10^ N/mz

and v p = 0.2):

R

t = 0.00858 (2.16)

0

(h) 0 - 1

This relationship indicates that the contact time is a linear

function of the sphere radius and is only slightly affected by drop

height. As will be discussed in Chapter 7, the contact time of

the impact of a steel sphere on concrete may be slightly longer

than that predicted by elastic theory due to inelastic behavior

of the concrete.

The maximum deformation, force, and pressure occur at t c /2.

The maximum force is given by the following equation:

1.140 (v ) 2 m

'max

- (2 - 17 > a

m

15 7r v

2

<* = [ ( <5 + <5 ) m ] 0 * 4 (2.18) m

16 (R) 0 - 5

S P 3

where m. = mass of the sphere (kg).

The maximum contact pressure is:

v

2 m

Pmav = 0.2515 [ ] 0 * 2 (2.19)

( 6 S + 5 p

) 4

(R) 3

o where g = acceleration of gravity (m/s ); v

Q

= velocity of the sphere at impact (m/s).

Since the mass of a sphere is a function of the radius cubed, P nnx

is independent of sphere size. Table 2.2 gives contact times and

maximum contact forces resulting from the elastic impact of 4 to

16 mm diameter steel spheres dropped 0.2 m onto a concrete slab.

The maximum contact pressure in this case is 1675 MPa.
17

TABLE 2.1 Specific Acoustic Impedances

Material Density

(kg/m3)

P-Wave

Velocity

(m/ s)

Specific Acoustic

Impedance

(kg/ (m^-s)

)

Ref.

Air 1.205 343 0.413 14

Concrete* 2300 3000 - 4500 6.9-10.4 x 10 6 -

Granite 2750 5500 - 6100 15.1 - 16.8 x 10 6

6

Limestone 2690 2800 - 7000 7.5 - 18.8 x 10 6

6

Marble 2650 3700 - 6900 9.8 - 18.3 x 10 6

6

Quartzite 2620 5600 - 6100 14.7 - 16.0 x 10 6 6

Soils 1400 - 2150 200 - 2000 0.28 - 4.3 x 10 6

9

Structural

Steel 7850 5940 46.6 x 10 6 48

Water 1000 1480 1.48 x 10 6 48

The mass density of concrete depends on the mix proportions and

the specific gravities of the mix ingredients. The given density

is for an average, normalweight concrete.

TABLE 2.2 Impact of a Steel Ball on a Concrete Plate

Sphere Diameter

(m)

Maximum Force

(N)

0.004 86.75

0.006 195.18

0.008 346.98

0.010 542.16

0.012 780.71

0.014 1062.63

0.016 1387.93
source i „ Receiver

Figure 2.1 Representative ray paths produced by impact u. 0 plate.

\ / \ / v0

'

Figure 2.2 a) Spherical wavefronts produced by impact on the top

surface of a plate; and, b) wavefronts after reflection from the

bottom surface.
sin 9 _

sin ft Ci C2

a) Reflection and Refraction

•in 8 sin P _sin0 t _

sin 3,

^pi ^p2 C»i C»2

b) Mode Conversion

Figure 2.3 The behavior of a P -wave incident upon an interfac

between two dissimilar media; a) reflection and refraction; and

b) mode conversion.
Figure 2.4 Reflection coefficients at a concrete/air interface

for an incident P-wave as a function of the incidence angle

(Poisson's ratio = 0.2).

Figure 2.5 Reflection coefficients at a concrete/air interface

for an incident S-wave as a function of the incidence angle

(Poisson's ration = 0.2).
CHAPTER 3

REVIEW OF PAST APPLICATIONS OF ECHO METHODS TO CONCRETE

3.1 INTRODUCTION

With the exception of visual inspection, the use of acoustic

methods is the oldest form of nondestructive testing. Striking

an object with a hammer and listening to the "ringing" sound is

a common way of detecting the presence of internal voids, cracks,

or delaminations.

In 1929, Sokolov of Russia first suggested the use of ultrasonic

waves to find defects in metal objects [85]. However, it was not

until the nearly simultaneous introduction of pulse-echo flaw

detectors in 1942 by Firestone of the University of Michigan and

by Sproule of England that significant progress was made. Since

that time, ultrasonic pulse-echo testing of metals, plastics, and

other homogeneous materials has developed into an efficient,

reliable, and versatile nondestructive test method. The development

of test techniques and equipment for evaluation of less ideal

materials, such as concrete, has been hindered by the difficulties

inherent in obtaining and interpreting a signal record from a

heterogeneous material.

3.2 TEST METHODS

There are a variety of techniques that are based on using stress

wave propagation for nondestructive testing. The primary

21
22

distinctions between the techniques are in the methods used for

generating and receiving the stress waves. Three methods that have

been used for evaluation of concrete are: 1) the through-transmission

(pulse-velocity) method; 2) echo methods; and 3) the resonance

method. These methods are discussed in the following sections.

3.2.1 Through-Transmission Method

In the through-transmission (or pulse-velocity) method, one

measures the time it takes for a stress pulse to travel from a

transmitting transducer to a receiving transducer located on the

opposite side of a test object. From the measured travel time and

the known distance between transducers, an apparent pulse velocity

is calculated. The computed pulse velocity value can be used to

draw inferences about the integrity of the medium. This approach

has its limitations: it requires two accessible surfaces and it

offers no indication as to the cause of any differences among pulse

velocity values in different parts of a structure. Because of these

limitations, applications of this technique will not be discussed.

3.2.2 Echo Methods

In the echo methods a stress pulse is introduced into the

test medium at an accessible surface by a transmitter or by

mechanical impact. If a transmitter is used, the method is referred

to as a pulse-echo technique; if mechanical impact is used, the

method is referred to as an impact-echo technique. The pulse

propagates through the medium and is reflected by material defect:
23

or by interfaces between phases of different densities or elastic

moduli. These reflected waves, or echoes, are monitored by either

the transmitter acting as a receiver (true pulse-echo) or by another

receiving transducer coupled to the surface of the test object near

the pulse source (pitch-catch). The transducer output is displayed

on an oscilloscope. Using the time base of the display, the travel

time of the pulse can be determined. If the wave velocity in the

medium is known, the round-trip travel time of each echo can be

used to determine the location of a defect or an interface.

3.2.3 Resonance Method

Resonance techniques are used to determine the thickness or

length of a test object. Continuous waves are introduced into a

test object by a transmitter or a mechanical oscillator. The

frequency of these waves is systematically varied until a resonance

condition is set up in a test object. Resonance occurs when the

driving frequency is equal to the frequency of the fundamental mode

of vibration of the test object. The response of the test object

is monitored by a receiving transducer which is located close to

the transmitter. When a resonance condition is achieved, there

is a significant increase in the amplitude of the measured response.

The fundamental mode of vibration is identified by noting the

frequency, f, at which this increase occurs. If the P-wave velocity

in the test object is known, the thickness of the test object, t,

which equals one-half the wavelength of the propagating waves, can

be calculated
24

3.3 PAST APPLICATIONS

The three stress wave propagation methods discussed in Section

3.1 have been used for detecting very large structural cracks in

concrete dams, for integrity testing of slender concrete structures,

such as piles, for measuring the thickness of plate elements, such

as highway slabs, and for several other specialized applications.

These applications are reviewed in the following section.

3.3.1 Pavements and Bridge Decks

Resonance and echo techniques have been developed to measure

the thickness of concrete pavements and to detect delaminations

in bridge decks. In the early 1960's, Muenow [57] developed a technique to measure pavement thickness which was based on

determining the frequency of the fundamental mode of vibration of

the slab in the thickness dimension. A transmitter was used to

excite resonances in the slabs.

Resonance methods have also been used to detect delaminations

in concrete bridge decks. A technique developed in 1973 by

researchers at Texas A & M University [56] used an oscillating steel

plunger to excite the characteristic vibrations of a delaminated

area. As the plunger oscillates at 60 Hz, the vibrations of the

bridge deck were monitored by a receiving transducer. The location

and extent of delaminated areas could be determined from the relative

amplitudes of the received signals. The use of this technique has

been limited by the fact that it cannot be used where concrete decks

have asphaltic overlays exceeding approximately 5 cm.
25

In 1964 s Bradfield and Gatfield [11] of England reported

the development of an echo technique for measuring the thickness

of concrete pavements. Using two 100 kHz resonant transducers (16

cm tall, 10 cm wide, and 25 cm long) in a pitch-catch arrangement,

they were able to measure the thickness of a 12-inch concrete

specimen with an accuracy of 2 percent. However, this system could

not be field tested [38] due to the impracticality of the test

set-up. Besides being bulky, the transducers were coupled to the

concrete by a large plastic block which required a smooth flat

concrete surface for good coupling. Difficulties were also reported

in obtaining reflections from the bottom of rough textured pavement

surfaces

.

In 1968 , Howkins, et al., at IIT Research Institute [38]

independently investigated available resonance and echo techniques

in an attempt to identify a feasible method for pavement thickness

measurements. Tests using the resonance technique proposed by Muenow

[57] and the echo technique proposed by Bradfield and Gatfield [11]

were performed. Although it was felt that the resonance technique

was, in principle, a good approach, significant reservations were

stated concerning the reliability of Muenow's resonance method.

Using an echo technique similar to that developed by Bradfield and

Gatfield, the IIT researchers were able to measure the thickness

of 7 and 10-inch thick portland cement concrete slab specimens,

for both simply supported slabs and slabs supported on a 4-inch

thick gravel base course, and 5-inch thick, simply supported

bituminous concrete slab specimens with an accuracy of + 2 percent.
However, it was concluded that the transducer arrangement was not

practical for field use.

26

An echo system was developed at Ohio State university in

the late 1960's [52] to measure pavement thickness by monitoring

the travel time for an ultrasonic pulse to propagate through the

thickness of the concrete and return to the receiving transducer

at the top surface after being reflected by the concrete

pavement-subbase interface. A large transmitter was needed to

introduce sufficient acoustic energy into the test medium to overcome

wave attenuation problems due to coarse-grained aggregates and to

obtain coherent reflections from rough pavement-subbase interfaces.

The transmitter was a hollow cylinder, with a 46 cm outer diameter,

a 15 cm inner diameter, and a 200 kHz resonant frequency. The

receiving transducer was placed in the center of the transmitter.

Accuracies of plus or minus 3 percent at more than 90 percent of

the test locations were obtained. The accuracy and good performance

of the Ohio State thickness gage was confirmed in independent field

tests conducted in 1976 by Weber, Gray, and Cady [88]. However,

Weber, et al., concluded that the Ohio State instrument needed to

be redesigned to better withstand the rigors of field use before

it could be considered as practical nondestructive testing equipment.

3.3.2 Erosion Cavities Below Slabs and Behind Walls

Yamshchikov, Sidorov, Baukov, and Potapov of Russia [93]

used an acoustic resonance technique for finding large erosion

cavities behind concrete canal walls and below reinforced concrete
27

slabs on soil substrates. In the case studies presented, the canal

walls were 18 to 23 cm thick and the reinforced concrete slab was

10 cm thick. The resonance technique used was an adaptation of

a standard technique in the USSR for checking bonds in multilaminate

plastic structures; this technique is referred to as the Method

of Free Vibrations [50]. Flexural modes of a wall or a slab were

excited by the periodic impact of an electromechanical vibrator

operating at a frequency of 16 Hz. The response of the wall or

slab was monitored by an accelerometer, and a spectral analyzer

was used to obtain the frequency spectrum of the received signal.

The presence of a cavity was detected by monitoring the change in

the amplitude of the frequencies corresponding to the thickness

of the slab. When a cavity was present, there was a noticeable

increase in amplitude. Cavities approximately 2 m and larger were

detected

.

3.3.3 Dams

Engineers in India [58] used an impact-echo method to estimate

the depth and extent of large horizontal cracks that developed in

Koyna dam during a major earthquake in 1967. Very low frequency

stress waves (200-600 Hz) were introduced into the concrete by

mechanical impact with a free falling steel hammer. This range

of frequencies allowed detection of cracks on the order of 15 m

and larger. The energy generated by the mechanical impact was sufficient to obtain reflections from cracks located 100 m away

from the point of impact. After existing cracks were grouted, the
28

impact-echo technique was used to assess the degree of grout

penetration. The assessment was made qualitatively by comparing

the echo amplitudes in signal records obtained before grouting to

those obtained after grouting.

3.3.4 Piles

Since the early 1970's, impact-echo and resonance techniques

have been widely used for integrity testing of concrete piles [3,

12, 24, 25, 36, 37, 61, 62, 67, 81, 83, 90]. The behavior of stress

waves in slender, rod-like structures, such as piles, is well known.

If a pulse is generated by mechanical impact at one end of the rod,

the resulting wavefront is initially spherical but quickly becomes

planar as the pulse propagates down the long slender rod. Plane

wave reflection occurs at the bottom surface, and the reflected

wavefront travels back up the length of the rod to be picked up

the receiving transducer. Thus, the rod acts as a waveguide.

Because of the length of piles, long duration impacts can be used.

As a result, signal analysis is relatively simple.

Steinbach and Vey [83] were among the first to apply

impact-echo techniques to field evaluation of piles. A pulse was

introduced into a concrete pile at the top surface by mechanical

impact and returning echoes were monitored by an accelerometer

mounted on the same surface. The signal record could then be used

to detect partial or complete discontinuities, such as voids, abrupt

changes in cross section, very weak concrete, and soil intrusions,

as well as the approximate location where such irregularities
29

existed. In the absence of major imperfections the location of

the bottom of a sound pile could be determined. However, little

specific information could be obtained as to the extent of defects

or the relative soundness of concrete at the location of an

irregularity. The success of the method is dependent upon the

damping characteristics of the surrounding soil; a high degree of

damping can severely weaken returning echoes.

Resonance methods have also been widely used for integrity

testing of piles and to determine pile performance under load and

to gain information about the soils surrounding the pile. The first

resonance method for integrity testing of piles was developed in

France by Paquet in 1968. Since that time, the use of resonance

methods for testing of piles has become widespread. Techniques

for evaluating the integrity of piles, determining pile performance

under load, and gaining information about soils surrounding piles

are discussed in detail in Refs. [37, 62, and 81],

3.3.5 Reactor Structures

In 1976, Sutherland and Kent [84] of Sandia Laboratories

used an ultrasonic pulse-echo method to measure the thickness of

concrete reactor substructures subjected to the thermal energy of

a hypothetical core meltdown. Two transducers were used in the

pitch-catch mode. The relative position of a concrete-gas interface

subjected to a high heat flux from a plasma jet was monitored as

a function of time to determine the erosion rate of the concrete

substructure

.
30

3.3.6 Refractory Concrete

Claytor and Ellingson [16] used an echo method to measure

the thickness of 30.5 cm thick refractory concrete specimens. It

was found that for frequencies below 100 kHz, the use of a single

transducer as both the transmitter and receiver was impractical

because the ringing of the transmitter obscured the echo signal.

Tests were also carried out using two transducers in a pitch-catch

arrangement; however, the transmitting transducer generated strong

R-waves which interfered with the reception of the echo signal by

the receiving transducer. To reduce R-wave interference, large

diameter (17.8 cm) transducers were constructed. As the response

of a transducer is an averaged phenomenon over the contact area,

the sensitivity of a larger diameter transducer to localized surface

disturbances (R-waves) was reduced.

3.3.7 Surface Opening Cracks in Submerged Structures

Smith [80] demonstrated that Rayleigh waves can be used to

detect surface opening cracks in submerged concrete structures,

such as concrete tanks and offshore structures. Two 0.5 MHz, 25-mm

diameter, P-wave transducers were used as transmitter and receiver.

When a transmitted P-wave strikes the surface of a solid at a

critical angle (defined by Snell's Law), mode conversion occurs producing a R-wave which propagates along the solid-liquid

interface. As the R-wave propagates, mode conversion also occurs,

producing a P-wave which radiates into the liquid at the same

critical angle and is picked up by the receiving transducer. The
31

distance between the two transducers can be adjusted to optimize

the amplitude of the received signal. If the path of the propagating

R-wave is crossed by a crack, reflection occurs and no signal will

be picked up by the receiving transducer. If a crack is favorably

oriented (a crack at 90 degrees to the propagating wave is the best

orientation), the P-waves produced by mode conversion of the

reflected R-wave will be picked up by the transmitting transducer.

Analysis of the received signals obtained from a complete scan,

i.e., from moving the transducers parallel to and over the surface

of the test object in a prearranged pattern, allowed the location

of surface opening cracks to be determined.

3.4 SUMMARY

A review of the literature shows that resonance and echo

methods have been used successfully to detect the thickness of

concrete slabs and to detect voids beneath slabs or behind walls.

Delaminations within bridge decks and large cracks within dams have

been detected. Impact-echo and resonance techniques have been widely

used for integrity testing of concrete piles; the technique and

the instrumentation for the evaluation of this type of structure

are fairly well established. Although limited success has been

achieved in other specialized applications, no standardized stress

wave propagation method currently exists for general nondestructive

evaluation of concrete structures
CHAPTER 4

A FINITE ELEMENT STUDY OF

TRANSIENT WAVE PROPAGATION IN PLATES

4.1 INTRODUCTION

The propagation of waves in an infinite elastic plate has been

studied since the work of Lord Rayleigh [70] and Lamb [49] in the

late 1800's. The work of Miklowitz [55] on transient waves in

elastic waveguides formed the first detailed study of transient

wave propagation in elastic plates. Recently, due to the interest

in the propagation of acoustic emission waves in elastic solids,

the problem of a transient, normal point load on the surface of

an infinite elastic plate has been studied. This is the problem

that is of interest in this chapter. Solutions of this problem

[15, 21, 41, 64, 69] have been obtained using generalized ray theory

[63] in conjuntion with Cagniard's method [63] or with the Fourier

inversion technique [89], the method of superposition of normal

modes [87], and classical integral transform methods [91].

In this chapter, the finite element method is used for the

first time to study stress wave propagation in elastic plates

subjected to transient, normal point loads. Surface displacement

time-histories and internal displacement and stress fields are

presented. To verify the analyses, surface displacement time

histories obtained from the finite element method are compared to

exact solutions obtained using generalized ray theory and the Fourier

inversion technique. (In this report, solutions obtained using

ray theory will be referred to as Green's function solutions.)

32
33

4.2 BACKGROUND

4.2.1 Transient Wave Propagation

Point impact on the surface of a solid gives rise to three

types of disturbances: dilatational and distortional waves which

propagate into the solid along spherical fronts, and a Rayleigh

(R) wave which propagates along a circular front over the surface

of the solid. The dilatational and distortional waves are commonly

referred to as P- and S-waves. In addition, there is a low amplitude

wave known as a head wave. The front of the head wave extends from

the intersection of the P-wavefront with the surface of the solid

to a point that is tangent to the S-wavefront. Fig. 4.1 is a

schematic representation of the P-, S-, R- and head wavefronts

generated by elastic impact on a solid.

Explicit equations for the radiation pattern produced by a

transient point source on a semi-infinite solid have not yet been

derived. Ideas about the nature of this radiation pattern come

from knowledge about the far-field radiation patterns produced by

a harmonic point source on a semi-infinite solid [33, 54, 71 ].

Fig. 4.2 shows the angular variation of the amplitude of

displacements within the P- and S-waves for a material with a

Poisson's ratio equal to 0.2. In the P-wave , the amplitude of the

displacements is maximum along the centerline of the plate and

decreases to zero at the surface. In the S-wave, the amplitude

of displacements is zero at the centerline of the plate and at the

surface and is maximum along a ray located approximately 37 degrees

from the centerline. There is a discontinuity in the S-wave
34

displacements at an angle, 6C , given by the following equation:

# c = arcsin (C

s /C

p

) (4.1)

where C = S-wave speed; and

s Cp = P-wave speed.

In this chapter, the finite element method is used to study

the internal displacement and stress fields produced by a transient

point load on a plate. The displacement fields produced by a

transient point load are compared to those shown in Fig. 4.2.

4.2.2 Green's Function Solutions

Theoretical solutions for transient wave propagation in solids

are available for a limited number of problems; these solutions

can be used to obtain the displacement response at a single point

in a solid. The displacement, u(r,t), at a point due to an impact

at some other point, a distance r away on an elastic body, can be

obtained by knowing the impulse response function of the body,

G(r,t), and the force-time function of the impact, F(t). The impulse

response is called the dynamic Green's function of the elastic body.

This Green's function is defined as the impulse (dirac delta

function) response of the b.ody for a particular impact configuration

(impact at one location and the receiver at a different location).

The response of the body for an arbitrary force-time function can

be found by solving the following convolution integral:

oo

u(r,t) = J'

G(r, t - t )F( r )d r (4.2) — oo Since the delta function is causal, from the condition of

causality, the impulse response is also causal [65]. Thus,
35

G(r,t-r)=0 r >

t

so that Eq. (4.2) becomes:

u(r , t)

tG(r,t - r )F( r )dr — oo The force-time function, F( r ), is also causal; thus,

F( r ) = 0 r < 0

and Eq. (4.2) reduces to Duhamel's Integral [65]:

(4.3)

(4.4)

(4.5)

u(r,t)

= J

G(r, t - t )F( r )dr (4.6)

0

Thus, Eq. 4.2 is reduced to a definite integral which can be solved

numerically using a digitized form of G(r,t).

The Green's function solution is the exact solution to the

partial differential equation and associated boundary conditions

governing transient elastic wave propagation. The solution is in

the form of an infinite series expansion which is referred to as

a generalized ray expansion. Stress waves can be visualized as

propagating along ray paths. Each term in the series corresponds

to the arrival of successive stress waves which propagate along

the various ray paths that connect the impact source to the

receiver. For a given time duration, a finite number of rays (terms

in the series expansion) contribute to the total displacement

response at the receiver.

Explicit formulae for Green's function solutions which are

amenable to numerical computations have been derived only for simple

geometries, such as a semi-infinite space or an infinite plate.

(For computation of the Green's function, see Refs. [15, 42, 64].)
36

The Green's function solution for an infinite plate is applicable

to a finite plate for the duration of time prior to the arrival

of the first wave reflected from the sides of the plate. To obtain

displacement and stress fields in bounded plates, the finite element

method was used.

4.2.3 Finite Element Method

The finite element method is a numerical technique for obtaining

approximate solutions to the partial differential equations that

arise from boundary value problems. The method involves dividing

a continuum into a finite number of discrete parts - the finite

elements. The discretized representation of the continuum is

referred to as the finite element model. For stress analysis, the

behavior of each element is described by a set of assumed functions

which represent the variation of displacements or stresses within

that element. Variational principles are used to formulate

force-displacement element equations. These element equations are

then used to construct the global equations which describe the

behavior of the entire continuum. Solution of these global equations

gives the displacements or stresses at points in the element [26].

An explicit, two-dimensional (axisymmetric or plane strain),

finite element code (DYNA2D), developed at Lawrence Livermore

Laboratory for solving finite-deformation, dynamic contact-impact

problems [29, 32, 33], was used to perform the studies discussed

in this report. An input generator (MAZE) [34] was used to create

the finite element model. A mini-computer with a virtual operating
system, 8 MBytes of memory, and a floating point processor was used

to carry out the analyses.

37

In DYNA2D, a continuum is divided into elements using constant

strain (or linear displacement) triangles and quadrilaterals [32].

Higher order elements (e.g., linear strain, quadratic strain) are

not available in DYNA2D because they are computationally expensive

in wave propagation applications compared to the use of constant

strain elements. For a particular element type, element size

determines the accuracy of the finite element solution. In wave

propagation problems, the optimum element size depends on the

geometry of the continuum and on the characteristics of the dynamic

loading. For the constant strain quadrilaterals and the dynamic

loading functions used in the plate analyses presented in this

chapter, studies were carried out to determine the optimum element

size. Comparisons were made between finite element displacement

time-histories obtained at points on the top and bottom surfaces

of a plate and the waveforms obtained at the same points by the

Green's function solution for an infinite plate. For 0.25- to 0.5-m

thick plates subjected to a force-time function which simulated

impact by a steel sphere (contact time of impact was 25 to 31 /as),

rectangular elements with dimensions on the order of 0.02 times

the plate thickness were found to give accurate results. The elastic

material properties used in the analyses were representative of

concrete

.

In dynamic finite element analyses, numerical integration of

the equations of motion must be carried out; DYNA2D uses the central
difference method [32] to accomplish this integration. The central

difference method requires a small time step for numerical

stability. This is not a drawback since wave propagation

applications require very small time steps to obtain an accurate

solution and to capture the true dynamic response. Numerical

stability requires that the time step, h, meets the following

criterion:

h < hmov = L / C

max p

(4.7)

where L = shortest dimension of the element, m; and

Cp = P-wave speed in the material, m/ s

.

In DYNA2D, the time step is taken as 0.67hmax unless the user

specifies some other value. During an analysis, data are stored

in data files at intervals specified by the user. In the analyses

discussed in this chapter, data were stored every 2 ps . An

interactive graphic post-processor (ORION) [35] was used to process

the results of the analyses.

Before the finite element code could be used with confidence

to study transient wave propagation in bounded solids containing

internal flaws, solutions obtained by the method were verified using

Green's function solutions for infinite plates.

4.3 PLATE RESPONSE

The successful implementation of the impact-echo technique

as a method for flaw detection in heterogenous materials, such as

concrete, requires an understanding of the reflection of transient

stress waves by the free boundaries of a solid and the interaction

of waves with internal defects A first step is understanding the
39

response of an infinite plate to impact. In the following discussion

of the elastic response of a plate to point impact on the top

surface, the following analytical results are presented: 1) the

displacement time-history obtained at the bottom surface of the

plate directly under the impact point; 2) displacement fields

recorded at successive times to show transient stress waves

propagating within the plate; and 3) the displacement time-history

of a point on the top surface of the plate near the point of impact.

For the case of a sphere impacting on a plate, Eq. (4.6) can

be used to predict the surface displacement that will be detected

by a receiving transducer located on either the top or bottom surface

of the plate. Two test configurations are considered in this study;

these are shown in Fig. 4.3. Fig. 4.3(a) shows the receiver located

at the epicenter, that is, on the bottom surface of the plate

directly under the point of impact. Fig. 4.3(b) shows the

impact-echo configuration - the receiver is located on the top

surface of the plate near the point of impact. For this

configuration, the separation between impact point and receiver

is denoted by H.

The time-history of the contact force generated by the elastic

impact of a sphere dropped on the surface of a plate can be

approximated by a half-cycle sine curve (see Fig. 4.1). If the

appropriate Green's function, G(r,t), is known, then the

displacement, u(r,t), can be computed by numerical solution of the

convolution integral given by Eq. (4.6) [10].

In this study, the Green's function for a unit step, force-time
40

function on an infinite plate was obtained using a computer code

recently developed at NBS [21, 39]. This program computes the

response for a unit step force-time function input. To obtain the

Green's function (impulse response), the derivative with respect

to time of the step function solution is calculated. The step

function response is calculated using nondimensionalized variables

so that the solutions are applicable to a plate of any thickness.

Values of the step function response are calculated at prescribed

time steps so that the computer solution is a discretized

representation of the true solution. The only input parameters

required are the source-receiver geometry and the ratio of S- to

P-wave speeds. In the analyses presented in this paper, the ratio

was 0.61 (Poisson's ratio equal to 0.2).

4.3.2 Epicenter Response

4. 3. 2.1 Green's Function Solutions

Before considering the response of a plate to impact by a

sphere, the impulse response is shown. In the impulse response,

wave arrivals correspond to abrupt discontinuities in the waveform.

It is therefore easier to determine the displacements caused by

each individual wave arrival.

Since the numerical solution used in this study results in

a discrete representation of the step function response, the

derivative of this solution (the impulse response) also has a

discrete representation.

The impulse response for a 0.25-m thick plate is shown in Fig.
41

4.4. The P- and S-wave speeds are 4000 and 2440 m/ s , respectively.

A time step of 1 jis was used in the calculations. This response

consists of normal surface displacements caused by the arrival of

direct P- and S-waves, multiply reflected waves (3P, 3S, 5P , etc)

and mode-converted waves (2PS, P2S, etc.). The arrival times of

these waves are indicated on the waveform.

The P-wave generated by impact on the top surface of the plate

is the first wave to arrive at the epicenter; it is a compression

wave (a wave causing compressive stress at the wavefront) and it

causes a large downward displacement of the surface. This

compression wave will be reflected at the bottom surface of the

plate (free boundary) as a tension wave. The tension wave will

propagate back up through the plate to be reflected at the top

surface as a compression wave. (The multiply-reflected P-wave is

now called the 3P-wave because when it arrives at the bottom of

the plate it will have traveled through the thickness of the plate

three times.) When the 3P-wave arrives at the bottom surface it

pushes the surface downward. This cycle is repeated so that every

multiply-reflected P-wave arriving at the bottom surface of the

plate (5P, 7P, etc.) is a compression wave and causes a downward

displacement of the plate surface.

Notice that the amplitude of the surface displacements caused

by successive P-wave arrivals decreases. This is due to divergence

(spherical beam spreading).

Theoretically, an S-wave arriving at the epicenter has no

vertical displacement component (see Fig. 4.2). However, the arrival
42

of the S-wavefront is still easy to identify because the arrival

of the wavefront corresponds to a rapid upward movement of the

downwardly displaced bottom plate surface.

The waveform obtained from the Green's function solution for

a point located a distance r from an impulse point source in an

infinite solid shows displacements produced by the arrivals of the

P- and S-wavefronts . No other displacements occur. However, in

the impulse response of the infinite plate (Fig. 4.4), notice that

in addition to the displacements caused by P-, S-, and mode-converted

waves, there are displacements that occur between the arrivals of

each of these waves. These intermediate displacements are referred

to in this report as "wakes"; they result from the transient point

source being applied normal to a stress-free boundary.

To obtain the epicenter response caused by a sphere impacting

the top surface of the plate, the waveform shown in Fig. 4.4 must

be convolved with the force-time function shown in Fig. 4.1 (Eq.

(2.12)). Using the identity for the derivative of convolution,

a mathematically equivalent approach is to convolve the response

computed for a unit step function, H(r,t), with the derivative of

the force-time function, d[F(t)]/dt [10]. This approach is

numerically more accurate in this case. Thus Eq . (4.6) can be

written in the following form:

dF( t- )

dt

(4.8)

o

The waveform obtained by this convolution is shown in Fig. 4.5(a).

The time step used in these calculations was 2 us.
43

In this analysis, the contact time of the impact was 31 p.s ,

which is equal to one-half the time required for a P-wave to travel

from the impact point to the epicenter. The waveform generated

by a 31 ;us point impact is much smoother than the impulse response

that was shown in Fig. 4.4. (Chapter 5 will present a more detailed

discussion of the effect of contact time on surface displacement

waveforms .

)

The arrivals of P-, S-, and mode-converted wavefronts are

indicated on the calculated waveform. The displacements caused

by the large amplitude P-wave arrivals dominate the waveform. Notice

that there is a second dip in the waveform after the end of the

direct P-wave and before the arrival of the S-wavefront. The steady

change in displacement (wake) between these two waves in the impulse

function response (see Fig. 4.4) gives rise to this second dip.

4. 3. 1.2 Comparison with Finite Element Solution

The impact response of the same plate was also calculated using

an axisymmetric finite element model. In both the Green's function

solution and the finite element analysis the plate was unsupported.

Impact on the top surface of the plate was simulated by applying

a uniform stress over an element at the center of the plate. The

time history of the applied stress was a half-cycle sine curve with

a duration of 31 ;us. The values of the material properties used

in the analysis were: a modulus of elasticity of 33100 MPa, a

Poisson' ratio of 0.2, and a density of 2300 kg/m^. These values

result in P-, S-, and R-wave speeds of 4000, 2440, and 2240 m/ s

,
respectively. Fig. 4.5(b) shows the normal displacement at the

epicenter of the plate. The computed arrival times of P-, S-, and

mode-converted waves are indicated on the waveform.

44

The response obtained by the finite element analysis can be

compared with the Green's function solution for an infinite plate

for the period of time before wave reflections return from the sides

of the bounded plate used in the finite element analysis. If the

shape and magnitude of the perturbations in the waveform obtained

from the Green's function solution (Fig. 4.5(a)) are compared with

those in the waveform obtained from the finite element analysis,

it is seen that there is good agreement between the two waveforms.

In the waveform obtained from the finite element analysis,

there is a series of low amplitude, extraneous oscillations (ringing)

between the back of the S-wave (128 ps) and the arrival of the

3P-wavefront. This ringing is due to the excitation of spurious

modes of vibration in the constant strain finite elements used in

DYNA2D. These modes are referred to as "zero energy" or "hourglass

modes" [29] and they are due to distortions of the elements. A

decrease in the contact time of the impact causes more rapid changes

in displacement; this causes greater distortion of elements and

tends to increase ringing. Artificial viscosities are introduced

in DYNA2D to damp out the ringing [23], but it generally cannot

be completely suppressed. The ringing is particularly evident in

this case because there is a relatively quiescent period between

the rapid, large amplitude change in displacements which occurs

prior to 128 ps and the arrival of the 3P-wave.
45

4.3.2 Displacement Fields within a Plate

A single finite element analysis solves for displacements and

stresses over the entire domain (the collection of finite elements)

at each time step during the specified time of analysis. These

results can be used to study the dynamic displacement and stress

fields that are produced within a solid.

An axisymme tr ic , finite element analysis was performed for

a 25-;js duration impact on a 0.5-m thick, 1.5-m diameter,

unsupported, plate. Material properties were identical to those

used in the previous analysis. A 0.5-m thick plate was used in

this analysis so that the P- and S-waves generated by the 25 ps

impact became separated as they propagated through the plate; this

makes the displacement fields created by the waves easier to study.

The right hand side of Fig. 4.6(a) shows the displacement field

in the plate 125 ^s after the start of the impact. (Since the

displacement field is axisymmetric , only half of the plate is shown.)

At 125 ps the P-wavefront arrives at the epicenter of the plate.

The position of the P- and S-wavefronts are indicated the left hand

side of the figure. The magnitude and direction of the average

nodal displacement of each element is indicated by a vector. The

relative lengths of the vectors depend on the magnitude of the

largest displacement that occurs within the plate at a particular

time. The vector lengths are also adjusted -by a scale factor which

is not under the user's control. Therefore, the vector plots shown

in Figs. 4.6(a) and 4.7 are not drawn to the same scale; this must

be remembered when comparing the figures.
46

As mentioned, particle motion in a P-wave is parallel to the

direction of wave propagation. In Fig. 4.6(a), the vectors within

the P-wave are oriented along rays emanating from the impact point.

This orientation is consistent with the direction of motion. The

magnitude of the displacements in the P-wave are not uniform along

the spherical wave. Displacements are maximum near the centerline

of the plate (ray connecting the impact point to the epicenter)

and they diminish to almost zero at the top surface of the plate.

This pattern of displacements is in agreement with that shown in

Fig. 4.2.

The motion in an S-wave is perpendicular to the direction of

wave propagation. In Fig. 4.6, the S-wave is easy to identify

because of the orientation and large amplitude of the vectors within

the wave. As expected, vectors are perpendicular to rays emanating

from the impact point. Displacements along a spherical surface

within the plate were studied to determine the effects caused by

the S-wave. The displacements in the S-wave are approximately zero

at the center of the plate and become larger along rays located

at increasing angles from the centerline. A study of displacement

time-histories obtained for various elements along a spherical front

inside the plate showed that, near the critical angle (approximately

37 degrees from the centerline of the plate), there is a

discontinuity in the displacements in the spherical S-wavefront.

This discontinuity agrees with that predicted by the radiation

pattern shown in Fig. 4.2. Near the surface, it is difficult to

determine the amplitude of the displacements in the S-wave because-
47

of interference due to the displacements caused by the R-wave.

Note that in the vector displacement field all effects are

superimposed; each vector represents the total displacement of any

given element.

Fig. 4.6(b) shows a contour plot of minimum principal

(compression) stress. The stresses in the P-wave are greatest at

the centerline of the plate and decrease toward the surface. Since

a state of pure shear stress is equivalent to a state of equal

biaxial tension and compression, the plot of minimum principal stress

also shows the stress variation in the S-wave. The stresses in

the S-wave are lowest at the centerline and increase toward the

surface. In the region near the surface of the plate, the stresses

caused by the R-wave are superimposed upon those produced by the

S-wave making it is difficult to separate the stresses caused by

each wave.

The observed patterns of displacements and stresses in the

P- and the S-waves is similar to those expected based on the

displacement fields produced by a harmonic point source (Fig. 4.2).

However, in addition to the these P- and S-wave radiation patterns,

Fig. 4.6 shows that in the region between the P- and S-waves there

are displacement and stresses that resemble those that occur in

a P-wave; this is the "P-wake." There is also a region of nonzero

displacements and stresses trailing the S-wave (the "S-wake") that

resemble the patterns in the S-wave. Thus, the disturbances

generated by impact on a plate are not confined in the P- and

S-waves
Figs. 4.7(a) through 4.7(c) show displacement fields obtained

from the finite element analysis, along with corresponding

representations of the positions of the P-, S-, and mode-converted

waves, at 148, 203, and 250 ^is after the start of the impact.

The displacement field at 148 ^is (Fig. 4.7(a)), shows reflection

of the P-wave at the bottom surface of the plate. The S-wave created

by mode-conversion of the incident P-wave (referred to as the

PS-wave) is not yet discernible as it is masked by the displacements

caused by the reflected P-wave.

At 203 }is (Fig. 4.7(b)), the S-wavefront arrives at the

epicenter. The bottom surface of the plate is displaced downward

at this time because of the effect of the preceding P-wave and P-wave

wake. The S-wave wake is clearly visible.

At 250 ;us (Fig. 4.7(c)), the front of the reflected P-wave

arrives at the top surface of the plate. The PS-wave is now easily

discernible. Reflection of the S-wave is occurring and the

mode-converted P-wave (referred to as SP) that was generated by

the reflection of the incident S-wave is seen emerging from the

front of the reflected S-wave.

Once multiple reflections of the P-, S-, and mode-converted

waves begin to occur, the disturbances created by individual waves

become more difficult to distinguish in the displacement fields.

4.3.3 Impact-Echo Response

The use of the impact-echo method for nondestructive testing

involves interpretation of displacement waveforms obtained near
the point of impact. In this section, a surface displacement

waveform obtained from the Green's function solution for an infinite

49

plate is compared with results obtained by the finite element method.

4. 3. 3.1 Green's Function Solution

The normal displacement calculated at a point on the top surface

of a infinite plate due to impact at another point on the same

surface is shown in Fig. 4.8(a). The thickness, T, of the plate

was 0.25 m. The spacing, H, between the impact point and the point

where the displacement was calculated (the point where a receiving

transducer would be located) was 0.05 m (see Fig. 4.3(b)). The

ratio of the S- to the P-wave speed was 0.61 and the contact time

of the impact was 31 ps as in the epicenter analysis.

The waveform shown in Fig. 4.8(a) consists of displacements

caused by the arrival of multiply reflected P- and S-waves and

mode-converted waves. In addition, there is an initial large

amplitude displacement caused by the R-wave propagating along the

top surface of the plate. In the figure, the R-wave arrival is

denoted by an R, and multiple P- and S-wave arrivals and

mode-converted wave arrivals are indicated. For this particular

test configuration (H/T = 0.2), the normal displacements caused

by the S-wave are very small.

As discussed previously, the P-waves which arrive at the top

surface are tension waves; the arrival of each tension wave pulls

the surface downward. Thus the perturbations in the impact-echo

response have a pattern similar to the epicenter response.
4. 3. 3. 2 Comparison with Finite Element Solution

The displacement waveform obtained from the finite element

50

analysis of the 0.25-m thick plate subjected to a 31 p.s duration

impact is shown in Fig. 4.8(b). The spacing between the impact

and the point where the waveform was recorded is 0.05 m as in the

Green's function solution. Material properties are the same as

those used in the previous analyses.

If the shape and relative magnitudes of the perturbations in

the waveform obtained from the finite element analysis (Fig. 4.8(a))

are compared with those in the waveform obtained from the Green's

function solution (Fig. 4.8(b)), it is seen that there is good

agreement between the two waveforms.

As in the epicenter response obtained from the finite element

analysis, low amplitude oscillations due to excitation of the zero

energy modes of the finite elements occur in Fig. 4.8(b). After

the R-wave has passed the receiver, the surface displacement should

go to zero, as shown in Fig. 4.8(a), until reflections arrive from

the bottom of the plate. However, the oscillations due to excitation

of the zero energy modes cause the computed surface displacement

to oscillate about zero for a short time. In this case, the zero

energy modes are excited by the element distortion caused by the

rapid, large amplitude changes in displacement that occur in the

R-wave. This numerical ringing does not affect the displacement

pattern due to the multiply reflected waves.
51

4.4 SUMMARY

The internal displacement and stress fields produced by a

transient point load on the top surface of an elastic plate were

studied using the finite element method. It was shown that in

addition to P- and S-waves, intermediate displacement and stress

fields (wakes) are produced by a transient point load applied normal

to a stress-free boundary. Surface displacement waveforms computed

by the finite element method showed good agreement with those

obtained from the Green's function solution.

The study presented in this chapter has demonstrated the

potential of the finite element method for becoming a powerful tool

for understanding the interaction of stress waves with defects within

solids. Such knowledge is essential for successful implementation

of nondestructive testing techniques based on stress wave

propagation, such as the impact-echo method. The power of the finite

element method lies in its ability to analyze solids having arbitrary

shapes, boundary conditions, and applied loads, and to generate

complete pictures of displacement and stress fields in a

computationally efficient manner.

Chapter 6 will present finite element studies of the diffraction

of transient stress waves by flat-bottom holes and circular disks

within plates - problems for which no Green's function solutions

currently exist.
52

Figure 4.1 Schematic representation of the spherical wavefronts

produced by point impact on a semi-infinite solid.

Figure 4.2 Amplitude of particle displacements in the radiation

pattern produced by a harmonic point source.
53

Figure 4.3 Test configuration for a plate: a) Epicenter; and

b) Impact-Echo.

Figure 4.4 Epicenter response to a delta function impact.
54

P 3P 6P

P 3P

Figure 4.5 Epicenter response to impact: a) Green's function

solution; and b) Waveform obtained from the finite element analysis.
55

F(t)

Figure 4.6 Displacement and stress fields within a 0.5-m thick

plate 125 p s after the start of the impact: a) vector plot of

displacements and the location of the wavefronts; and, b) minimum

principal stress contour plot.
56

Figure 4.7 Vector plots of displacements at various times after

the start of the impact: a) 148 /as; b) 203 jlis; and, c) 250 ^is.
R

TIME (s)

Figure 4.8 Impact-Echo response: a) Green's function solution;

and, b) waveform obtained from finite element analysis.
CHAPTER 5

FREQUENCY SPECTRUM ANALYSIS OF

THEORETICAL IMPACT-ECHO WAVEFORMS

5.1 INTRODUCTION

In Chapter 4, displacement waveforms obtained from an infinite

elastic plate were studied in the time domain. In this chapter,

the results of analytical studies carried out in the frequency domain

are presented. The Fast Fourier Transform technique is used to

determine the frequency content of stress pulses generated by elastic

impact on a thick plate. Frequency spectra obtained by

transformation of theoretical displacement waveforms are studied

as a function of the contact time of the impact and the spacing

between impact source and receiver. The effects on frequency spectra

of clipping or removing the R-wave signal in time domain waveforms

are discussed.

5.2 FREQUENCY ANALYSIS

The multiple reflections from the top and bottom surfaces

of a plate give the displacement response a periodic character.

In bounded solids containing flaws, reflections occur from a variety

of interfaces and free boundaries. As a result, time domain

waveforms become complex and difficult to interpret. However, if

the waveforms are transformed into the frequency domain, multiple

reflections from each interface become dominant peaks in the

frequency spectrum - at values corresponding to the frequency of

arrival of reflections from each interface. These frequencies can

58
59

be used to calculate the location of each interface. It will be

shown that data interpretation is generally simpler in the frequency

domain than in the time domain.

The transformation from the time to the frequency domain

is based on the idea that any waveform can be represented as a sum

of sine curves, each with a particular amplitude, frequency, and

phase shift. This transformation is carried out using Fourier

transforms. As an example. Fig. 5.1(a) shows the digital time

domain waveform, g(t), given by the function:

g(t) = sin 27r(20)t + 2 sin 27r(40)t + 3 sin 27r(60)t (5.1)

where t = time (s).

This function is composed of three sine curves of different

amplitudes having frequencies of 20, 40, and 60 Hz. The time

interval between points in the digital signal is 0.001 seconds;

this is equivalent to a sampling frequency of 1000 Hz. The frequency

spectrum was obtained from a digital time domain waveform containing

256 points using the Fast Fourier Transform (FFT) technique. A

personal computer and an FFT program given in Appendix [ B ] of Ref.

[82] were used to perform the calculations. The frequency spectrum

obtained by the FFT technique contains half as many points as the

time domain waveform and the maximum frequency in the spectrum is

one-half the sampling rate, which for this example is 500 Hz. Fig.

5.1(b) shows the initial portion of the frequency spectrum obtained

by the FFT technique; the peaks occur at 20, 40, and 60 Hz. Each

of the peaks corresponds to one of the component sine curves in

Eq. (5.1). The frequency interval in the spectrum is equal to the
maximum frequency divided by the number of points in the spectrum.

For this example, the interval is equal to 500 Hz / 128 points.

60

that is, 3.9 Hz. Since the frequency interval is inversely

proportional to the sampling rate, a slower sampling rate enhances

resolution in the frequency domain. However, to avoid errors in

frequency analysis, the sampling rate should be greater than twice

the maximum frequency that appears in the time domain waveform [82].

5.3 FREQUENCY SPECTRA OF ANALYTICAL WAVEFORMS

Frequency analyses were carried out for force-time functions

generated by the elastic impact of a sphere on a plate and for the

surface displacement waveforms obtained from the Green's function

solution of point impact on an infinite plate. The frequency content

of impact force-time curves and displacement waveforms was studied

as a function of the contact time of the impact and the spacing

between source and receiver. The results are used to gain an

understanding of how experimental test parameters affect the shape

of the frequency spectrum.

5.3.1 Frequency Content of the Impact

As has been discussed, the force-time function, F(t), produced

by the elastic impact of a sphere on a plate can be approximated

as a half-cycle sine curve (Eq. (2.12). This force-time function

is shown in Fig. 5.2(a).

The Fourier transform, F(s), of the continuous force-time

function given by Eq. (2.12) can be obtained in closed form solution
61

[10]. The amplitude spectrum of this transform is:

t sin 7r ( s - 0.5) sin ir(s + 0.5)

F(s) = [ + ] (5.2)

2 7t(s-0.5) tt(s + 0.5)

where s = f • t c

; and

f = frequency (Hz)

.

A plot of Eq. (5.2) is shown in Fig. 5.2(b). The X- and Y-axes

are normalized with respect to the contact time. To obtain the

actual frequency, X-axis values must be divided by the contact time

of a specific impact.

The frequency content of the input pulse is directly related

to the contact time of the impact. The shorter the contact time

the broader the range of frequencies contained in the pulse, but

the lower the amplitude of each component frequency. For a long

contact time, the pulse is made up primarily of large amplitude,

low frequency components. The range of frequencies contained in

a stress pulse determines the thickness of a plate that can be

measured and the size of the flaw that can be detected using the

impact-echo method. Higher frequency pulses are needed to measure

the thickness of thin plates, and, as will be discussed in subsequent

chapters, higher frequency pulses are needed to detect small flaws.

Notice that in the spectrum of the input pulse (Fig. 5.2(b)),

there are frequency values which have zero amplitude. These "zeroes"

occur at frequencies equal to 1.5/t

c

, 2.5/t , 3.5/t c

, etc. The

frequency difference between two successive zeroes is equal to 1 /

t

c

.

For example, if the first and second zeroes occur at 30 and 50 kHz,

respectively, then, 1 / t

c = 20 kHz, or t c = 50 ps.

As has been mentioned, the impact of a sphere on a plate
62

generates an R-wave which propagates along the surface of a test

object. This R-wave produces an initial, large amplitude response

in the displacement waveform, which resembles a half-cycle sine

curve with a half-period approximately equal to the contact time

of the impact (see Fig. 4.8(a)). Therefore, the frequency spectrum

of the plate response is composed of the spectrum due to P- and

S- wave reflections and the spectrum due to the R-wave.

Since the R-wave gives information about the contact time

of the impact, the frequency spectrum of experimental waveforms

can be used to determine the duration of the input pulse. In this

study, the contact time produced by the impact of steel spheres

on a plate was estimated by finding the difference between the

"zeroes" in the frequency spectrum.

In the theoretical and finite element studies presented in

this report, the half-cycle sine curve (Eq. (2.12)) is used as the

force-time function for the impact of a sphere on a thick plate.

This is in accordance with the Hertz theory of elastic impact.

In work carried out by Chang [17], it was shown that for impacts

that do not involve large amounts of energy, the Hertz theory is

very good. However, in his comparisons of Hertz theory to

experiments involving the impacts of small diameter steel balls

on a glass plate, it can be seen that the initial and final slopes

of the experimental curves are not as steep as the initial slopes

predicted by theory. An alternative approximation to the

experimentally obtained force-time function is the half-cycle sine

squared function, used by Lange and Ustinov [50, 51]. This function
63

is given by the following equation:

t F(t) = F

max

sin 2

(it — ) 0 < t < t c (5.3)

fc c

Fig. 5.3 shows a comparison of the force-time functions described

by Eqs. (2.12) and (5.3). The maximum amplitude of both curves

is the same. However, the initial and final slopes of the

sine-squared curve (Eq. (5.3)) are much flatter than the slopes

in the half-cycle sine curve (Eq. (2.12)).

The amplitude spectrum of the Fourier transform of the

sine-squared curve [51] is very similar to that shown in Fig.

5.2(b), except that the frequencies having zero amplitude occur

at 2/t

c

, 3/t

c

, 4/t

c

, etc. Thus, a pulse produced by an impact with

a half cycle sine-squared force-time function contains a slightly

broader range of frequencies than a pulse produced by an impact

with a half cycle sine force-time function. The force-time curves

obtained by Chang [17] appear to fall somewhere in between the sine

and the sine-squared approximations.

Fig. 5.4 shows a comparison between a Green's function

solution calculated for a half-cycle sine-squared force-time function

(Fig. 5.4(a)) and the solution for a half-cycle sine force-time

function (Fig. 5.4(b)). In both solutions, the value of H/T was

equal to 0.2, and the value of t c /t2P was ecl ua l to 0.25. The surface

displacement waveform obtained for an impact having a sine-square

force-time function is very similar to that obtained for a sine

curve. There are two minor differences between the two waveforms.

1) There is less area under a sine-squared curve (Eq. (5.3));
64

therefore, less total energy is applied during the impact. As a

consequence, the amplitudes of the displacements in Fig. 5.4(a)

are slightly less than those in Fig. 5.4(b); 2) The initial and

final slopes of the sine-squared curve are flatter than those in

the sine-curve. Therefore the displacements caused by the arrivals

of wavefronts are not as sharp in the waveform generated using the

sine-squared curve (Fig. 5.4(a)). Although the following discussion

is restricted to surface displacement responses of plates generated

by impacts having a half-cycle sine force-time function, the trends

and conclusions are equally applicable to responses generated with

impacts having half-cycle sine-squared force-time functions.

5.3.2 Frequency Content of Infinite Plate Response

5. 3. 2.1 Contact Time of the Impact

The normal surface displacement waveforms obtained from an

infinite plate subjected to impact are affected by the contact time

of the impact. The left side of Fig. 5.5 shows a series of

displacement waveforms calculated for a test configuration that

corresponds to a ratio of impact source - receiver spacing (H) to

plate thickness (T) of 0.2 (see Fig. 4.3(b)). The only variable

in Fig. 5.5 is the contact time of the impact, which is also given

in terms of a ratio: the contact time of the impact (t

c

) divided

by the arrival of the first P-wave reflection returning from the

bottom surface of the plate (t 2p)* Waveforms are shown for t c /top

ratios of 0.15, 0.25, and 1.0.

For H/T values less than about 0.25, the path length of the*
2P-wave is approximately twice the plate thickness; therefore, the

travel time, t, between the successive arrivals of P-wave echoes

from the bottom surface of the plate is approximately

65

2T

t = —

The corresponding frequency, f, of wave arrivals is

t 2T

(5.5)

(5.6)

In the spectra shown in the right hand side of Fig. 5.5, the X-axis

represents a dimensionless, normalized frequency (fn

), which is

obtained using the following formula:

Tf

fn- - (5.7)

C

P

Combining Eqs. (5.6) and (5.7) gives fQ equal to 0.5. This is

the normalized frequency corresponding to P-waves arriving from

the bottom surface of the plate. Each of the frequency spectra

shown in Fig. 5.5 has a peak at f equal to 0.5.

For an impact produced by a short contact time and for the

given test configuration (an H/T value of 0.2), the sudden downward

displacements in the waveform (Fig. 5.5(a)) correspond to the arrival

of the R-wave followed by waves returning from the bottom surface

of the plate. The arrival of the first P-wave (2P) corresponds

to the downward displacement beginning at t R equal to 2. In using

the impact-echo technique, the arrival time of this first P-wave

can be used to calculate the plate thickness, if the P-wave velocity

is known. However, as the contact time increases (Fig. 5.5(c)),
66

the response becomes smoothed out due to the convolution of the

impulse response function of the plate with the force-time function.

Wave arrival times no longer agree with the beginning of the downward

displacements in the time domain waveform. Thus, in time domain

analysis, it is essential to have a short contact time in order

to accurately determine wave arrival times.

In frequency analysis, however, it is not essential that

the impact have a short contact time. As was shown in Fig. 5.2(b),

a short contact time results in a force-time pulse which has energy

distributed over a broader frequency range. Thus, a pulse produced

by a short contact time has a smaller concentration of energy at

the frequency which corresponds to the multiple P-wave reflections

as compared to a pulse produced by a longer contact time. Therefore,

as is apparent in Fig. 5.5, the peak in the frequency spectrum

obtained from a waveform produced by an impact having a short contact

time has a smaller amplitude.

The contact time of the impact affects the R-wave signal

in both the time and the frequency domains. The Fourier transform

of the plate response is composed of the transform of displacements

caused by waves reflected from the bottom surface of the plate and

the transform of displacements caused by the R-wave. Therefore,

the relative magnitude and duration of the initial displacement

produced by the R-wave as compared to the displacements produced

by internal reflections is important. The R-wave in the time domain

waveform generated by the impact with the shortest contact time

(Fig. 5.5(a), t c

/ t 2P equal to 0.15) causes displacements that an
67

approximately eight times the amplitude of the displacements produced

by reflections from the bottom surface of the plate. Thus the

frequency components due to the R-wave are a large portion of the

spectrum in Fig. 5.5(a). Notice the similarity between the spectrum

shown in Fig. 5.5(a) and the spectrum of the impact force-time

function shown in Fig. 5.2(b). In contrast, a waveform generated

by an impact having a long contact time (Fig. 5.5(c), t c

/ t 2P ecl ua l

to 1.0) produces a long duration R-wave signal that causes

displacements that are only approximately twice that caused by

internal reflections. The frequency spectrum is dominated by a

single peak which corresponds to waves arriving from the bottom

surface of the plate, since the frequency components due to the

R-wave are relatively minor.

In practice, the peak in the frequency spectrum can be used

to calculate the plate thickness, T, using the formula

C

p T = —- (5.8)

2 f

P

where f = peak frequency.

5. 3. 2. 2 Test Configuration

In addition to the contact time of the impact, the surface

displacement waveform is affected by the distance between the impact

point and the point where the displacement is calculated. A

convenient parameter for describing the test configuration is the

ratio of the distance between the impact point and point where the

displacement is calculated (H) to the plate thickness (T). Fig.
68

5.6 shows a series of displacement waveforms and their corresponding

frequency spectra for H/T values of 0.05, 0.2, and 1.0. The ratio

of contact time of the impact to the 2P-arrival time is the same

in all the traces and is equal to 0.25.

For a low H/T value (Fig. 5.6(a)), the displacement due to

the R-wave is very large relative to displacements due to internal

reflections, and as discussed previously the Fourier transform of

the R-wave signal dominates the frequency spectrum. Notice that

the frequency spectrum shown in Fig. 5.6(a) is nearly identical

to that shown in Fig. 5.2, except for the addition of a small peak

corresponding to the frequency of P-wave reflections from the bottom

surface of the plate.

The plate response obtained at a point corresponding to a

large H/T value (Fig. 5.6(c)) produces results that are difficult

to interpret in both the time and the frequency domains. In this

case, the R-wave and the 2P-wave arrive at the receiver at the same

time, and the 2P-wave arrival is masked in the transducer response

to the very large amplitude R-wave. In addition, there are

perturbations in the displacement waveform due to relatively large

amplitude S-wave reflections, making the arrival times of individual

P- and S-waves difficult to distinguish. There is no peak in the

frequency spectrum which corresponds to the frequency of multiple

P-wave reflections from the bottom of the plate. These observations

can be explained by the physics of wave reflection. The amplitude

of P- and S-wave reflections depends, in part, upon the angle that

an incident wave strikes a free boundary or an interface. For an
69

H/T value of 0.05, the amplitudes of reflected P-waves are near

their maximum value, while the amplitudes of reflected S-waves are

very small. In contrast, for an H/T value of 1.0 the incident angles

are large, and the amplitudes of reflected S-waves are near maximum,

while the amplitudes of reflected P-waves have decreased to

approximately half their value at an H/T value of 0.05 (Section

2.1.3). In addition, for large H/T values, S-wave reflections have

a larger displacement component normal to the surface. Thus, for

large H/T values, S-wave reflections make a significant contribution

to the normal surface displacement. This complicates both the time

domain waveform and the frequency spectrum, and makes it impossible

to determine the thickness of a plate from the dominant peak in

the frequency spectrum.

For the H/T values studied, those in the range of 0.2 to

0.5 give the best results by minimizing the complications caused

by large amplitude R-waves and those caused by S-wave reflections.

5.3.3 Clipping and Removing the R-Wave Signal

When taking measurements in the laboratory, it is preferable

to amplify the signal produced by the reflections returning from

internal flaws or the bottom surface of the specimen. This can

be achieved by changing the gain setting on the digital oscilloscope

so that echo signals occupy the full scale of the display. As a

result, the R-wave signal will be cut-off (clipped). Clipping the

R-wave in the time domain waveform changes the nature of the R-wave

representation in the frequency spectrum, since the clipped R-wave
now resembles a square wave with a duration approximately equal

to the contact time of the initial impact. Compare the frequency

70

spectrum shown in Fig. 5.6(a) to that shown in Fig. 5.7(a). Both

were obtained from the displacement waveform shown in Fig. 5.6(a).

The frequency spectrum in Fig. 5.6(a) is obtained from the original

unaltered waveform, while that in Fig. 5.7(a) was obtained after

clipping the R-wave signal at an amplitude of -8. In the spectrum

obtained from the trace with the clipped R-wave, the frequency peak

corresponding to the thickness of the plate is much easier to

identify. Thus experimental frequency spectra are made simpler

to interpret by amplifying the displacement pattern produced by

internal reflections and allowing the R-wave signal to be

automatically clipped. The contact time of the impact can still

be determined from the zeroes in the frequency spectrum. The only

difference is that the zeroes in the Fourier transform of a square

wave of duration t

£ occur at 1.0/tc

, 2.0/t

c

, 3.0/t

c

, etc [10].

If the R-wave is removed completely from the waveform, only

frequencies obtained by transforming the displacements produced

by internal reflections are present in the spectrum. For example,

if the R-wave is completely removed from the waveform shown in Fig.

5.6(a), the spectrum in Fig. 5.7(b) is obtained. The spectrum

contains no information about the contact time of the impact.

Thus clipping the R-wave signal, as is' automatically done in

experimental work, appears to be the optimum approach. The peak

corresponding to the thickness of the plate is easier to identify,

and information about the contact time of the impact is preserved.
71

5.4 CONCLUSIONS

The Fast Fourier Transform technique was used to study the

frequency content of pulses and theoretical surface displacement

waveforms produced by the impact of a sphere on a thick plate.

Analytical results show that frequency spectrum analysis

can be used to determine the frequency content of a pulse generated

by impact, the contact time of the impact, and the thickness of

plate specimens.

To facilitate interpretation of frequency spectra it is

advantageous to: (a) clip the R-wave signal in the displacement

waveform; (b) avoid using impacts having ? very short contact times

relative to the time of arrival of the first P-wave; and (c) keep

the distance between the impact point and the point where

displacement is monitored within 0.2 to 0.5 of the plate thickness.

Frequency spectrum analysis is shown to be simpler than time

domain analysis. In the time domain, the arrival time of P-waves

must be determined. Accurate interpretation of time domain

displacement waveforms must be carried out by an experienced person

and is feasible only for simple geometries. However, in the

frequency domain, the depth of reflecting interfaces is easy to

determine from the dominant peaks in the spectra. In Chapter 7,

the ideas presented in this chapter are extended to the study of

displacement waveforms and frequency spectra obtained during

impact-echo testing of solid concrete plates and concrete plates

containing flaws.
72

uta

3

Ia.2<

Figure 5.1 Example of a frequency spectrum obtained from a time

domain waveform using the Fast Fourier Transform technique. a)

digital time domain waveform; and, b) frequency spectrum.
AMPLITUDE/CONTACT

TIME

73

0 ® 2 4

FREQUENCY x CONTACT TIME

Figure 5.2 Stress pulse produced by the elastic impact of a sphere

on the top surface of a plate: a) force-time function; and, b)

Fourier transform.
AMPLITUDE

74

*

Figure 5.3

functions

.

Comparison of half-cycle sine and 6ine-squared force-time
75

Figure 5.4 Comparison of surface displacement waveforms for two

force-time functions (H/T = 0.2 and t £ /t

2 p e 0.25): a ) half-cycle

sine-squared function; and, b) half-cycle sine function.
76

Figure 5.5 The effect of contact time of the impact of the response

of an infinite plate for H/T » 0.2: a) t _/to P - 0.15; b) t /t 7p - 0.25; and, c) t c /t 2p = 1.0.
77

Figure 5.6 The effect of impact source - response of an infinite plate for t /t 2p =

b) H/T = 0.2; and, c) H/T = 1.0.

receiver spacing on the

0.25: a) H/T = 0.05;
0.4

78

u

o2

3

a.2

<

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

0 2 4 6 8 10

NORMALIZED FREQUENCY

NORMAUZED FREQUENCY

Figure 5.7 Effect on the frequency spectrum due to partially and

completely removing the R-wave signal from the time domain

displacement waveform shown in Figure 5.5(a): a) R-wave signal

clipped at an amplitude of —8; and, b) R-wave signal completely

removed

.
CHAPTER 6

A FINITE ELEMENT STUDY

of the

INTERACTION OF TRANSIENT STRESS WAVES WITH PLANAR FLAWS

6.1 INTRODUCTION

This chapter presents finite element studies of the

interaction of transient stress waves with planar discontinuities

in an elastic solid. Emphasis is placed on the effects on

displacement waveforms caused by waves diffracted from the edges

of a flaw.

The finite element results had to be verified by comparison

with experimentally obtained waveforms; thus an aluminum plate

containing a flat-bottom hole was chosen for the initial finite

element studies because the physical specimen was easy to fabricate.

Displacement waveforms obtained from analyses of planar,

disk-shaped flaws in aluminum and concrete plates are presented.

Displacement and stress fields produced by the interaction of

transient waves and a planar disk-shaped flaw in a concrete plate

are studied. Finally, parameter studies of concrete plates

containing disk-shaped flaws were carried out to determine the effect

of important test variables on surface displacement waveforms.

The variables that were studied included: the diameter and depth

of a planar flaw; the contact time of the impact (frequency content

of the waves); and, the distance between the impact point and the

point where the displacement is recorded. The results of these

studies are used in Chapter 7 to help interpret experimental

impact-echo displacement waveforms.

79
80

6.1.1 The Use of Numerical Methods for the Study of Stress Wave

Propagation in Solids

In the past 15 years, advances in computer technology and the

availability of computer power has enabled the use of numerical

models to study the stress wave propagation in solids. The use

of approximate numerical methods to study nondestructive testing

(NDT) related problems has been motivated by the limited number

of analytical solutions that are available for these types of

problems, and because of the difficulties involved in obtaining

quantitative information from experimental techniques. With the

emphasis on the development of reliable NDT methods, there was and

is a need for alternate methods. Numerical methods provide the

ability to model arbitrary geometries and boundary conditions

(loading and support conditions) and to obtain full field solutions.

The early work in the use of numerical models to study stress wave

propagation in solids was carried out by researchers in geophysics

and seismology. Application of numerical models to NDT related

problems soon followed.

In 1982, Bond [7] wrote a comprehensive summary of methods

available for the computer modelling of ultrasonic waves in solids.

He reviews Ray Tracing methods, Finite Difference methods (explicit

schemes and the Method of Characteristics), Finite Element methods,

and several other lesser known modelling techniques. Most of the

work that has been done in the past has involved the use of ray

tracing methods and finite difference methods; Bond's summary

provides an excellent review of the progress that has been achieved

using these methods.
Boundary Element (Boundary Integral Equation) methods have

also been applied to steady-state and transient stress wave

propagation problems. Ref. [59] presents a summary of the different

boundary element methods that have been used to study stress wave

propagation. Ref. [44] discusses the types of problems that have

been studied using these methods. One class of problems that has

been studied extensively is dynamic stress concentrations created

by the interactions of stress waves with cavities and elastic

inclusions in infinite or semi-infinite solids.

In 1982, Bond, in his review of numerical models, stated

that finite element techniques had yet to be applied to pulsed

elastic wave problems related to NDT [7]. In other fields, such

as seismology and electromagnetism, finite element methods have

been used to study wave problems. Many of these applications are

summarized by Bond [7].

Finite element methods have been applied to the study of

elastic stress wave propagation in rods [77] and layered waveguides

[46], and to flexural wave propagation in beams [94]. The method

was used to determine the plane strain, steady-state response of

an elastic half-space to a surface pressure wave (Cole-Huth problem)

and the axisymmetric response of an elastic half-space to a

concentrated, downward step force applied at a given depth beneath

the free surface of the half-space (Pekeris-Lif son problem) [77].

Much work has been done on the use of finite elements for

problems involving stress wave propagation produced by blast [18]

or impact [95]. Much of this work is classified, but some of the
82

finite element codes developed for these applications (such as DYNA,

the code used to carry out the finite element analyses presented

in this report) are suitable for NDT related studies. The

capabilities of available codes are discussed in Ref. [95].

A review of the applications of the finite element method

to stress wave propagation in solids revealed only one series of

studies that used the method for NDT related problems. In these

studies, a combined finite element and analytical method was used

for analyzing scattering of plane, longitudinal waves and vertically

polarized shear waves by cavities in an infinite elastic medium

[76], and for scattering of plane horizontally polarized shear waves

by surface breaking cracks [1, 20]. The defects were contained

in an interior bounded region which was represented by finite

elements. Wavefunction expansions are used to represent the field

in the exterior region. A nodal force-displacement relation was

developed to satisfy the continuity conditions along the boundaries

separating the two regions. Results obtained by the method for

scattering of plane, longitudinal and shear waves by a circular,

cylindrical cavity are shown to agree with the exact solution [ 76 ],

The work in this report represents the first application

of the finite element method to the study of transient wave

propagation in solids containing flaws.

6.2 BACKGROUND

When a transient stress wave is incident upon a crack within

a solid, the following phenomena occur: specular reflection from
83

the crack face; diffraction at the crack tip; and, mode conversion

of waves on the crack face and at the crack tip. Fig. 6.1(a) shows

the specularly reflected P-wavefront (2P) and the mode-converted

PS-wavefront produced when a spherical P-wave is incident upon a

circular crack. Rays OA, which intersect the crack tips, will be

diffracted as shown in Fig. 6.1(b), producing two diffracted waves,

P^P and P^S. The designation used for diffracted waves includes

the incident wave and the wave produced by diffraction. For example,

P^S is the designation for the diffracted S-wave that is produced

by an incident P-wave. The diffracted waves propagate along

cylindrical wavefronts; the crack tip is the center of the cylinder.

A similar set of specular (2S), mode-converted (SP), and diffracted

waves (S^P and S^S) are produced when the S-wave is incident upon

a crack. The amplitude of particle motion in each of the specularly

reflected, mode-converted, and diffracted waves varies with direction

and depends upon the size, geometry, and depth of the crack, the

frequency content of the waves incident upon the crack, and the

orientation of the crack with respect to the propagating waves.

As shown in Fig. 6.1(b), there is a region beneath the crack called

the shadow zone, where direct P- and S-waves cannot penetrate.

Fig. 6.2 shows a schematic representation of point impact

on a plate containing a planar disk-shaped flaw. The center of

the flaw is located directly under the point of impact and the flaw

is parallel to the top surface of the plate. This is the case that

will be discussed in this report. For this condition, the important

variables affecting the response are the diameter, D, and depth,
T, of the flaw, and the frequency content of the stress waves

generated by impact. The frequency content is determined by the

84

contact time, t c

, of the impact (Section 5.3.1). A long contact

time produces stress waves made up primarily of large amplitude,

low frequency components. Stress waves produced by a short duration

impact contain a broader range of frequencies; however, the amplitude

of each component frequency is lower. This is important because

it is the frequency content of the waves that determines the size

of a flaw that can be detected by monitoring specular reflections

using the impact-echo technique. Waves containing higher frequency

components are needed to detect smaller flaws.

6.3 PLATE WITH A FLAT-BOTTOM HOLE

In the impact-echo method, a surface displacement waveform

recorded near the point of impact is used to gain information about

the interior of the solid. When sharp discontinuities exist, the

displacements caused by the arrival of diffracted waves are added

to the displacements caused by the arrival of reflected waves.

Correct interpretation of waveforms requires an understanding of

the displacements caused by diffracted waves. In this study, the

finite element method is used to gain this understanding.

A plate containing a flat-bottom hole is the simplest test

spec imen that can be fabricated to model a solid containing an

internal flaw with sharp edges. A physical specimen was needed

to verify the results from the finite element analyses.

In the discussion of the elastic response of a plate with
a flat-bottom hole to point impact on the top surface, the following

analytical results are presented: 1) the displacement time-history

obtained on the surface of the hole, directly under the point of

impact; 2) displacement fields recorded at successive times to

show transient stress waves propagating within the plate; and 3)

the displacement time-history of a point on the top surface of the

plate, near the point of impact. To clearly identify the effects

caused by diffraction, the displacement time-histories obtained

from the plate with a flat-bottom hole are compared to time-histories

obtained from a solid plate which had the same thickness as the

distance to the flat-bottom hole. Finally, a displacement

time-history obtained from a finite element analysis is compared

to an experimentally obtained impact-echo displacement waveform.

The dimensions of the plate with the flat-bottom hole are

shown in Fig. 6.3. The diameter, D, is 0.038 m and the depth of

the hole, T, is 0.038 m; this geometry corresponds to a D/T value

equal to 1. The axisymmetric , finite element model is shown on

the right hand side of the figure. The plate is made of aluminum

and the values of the material properties used in the analysis were:

a modulus of elasticity of 7.1 x 10^ kPa, a Poisson's ratio of 0.33,

and a density of 2700 kg/m . These values resulted in P-, S-, and

R-wave speeds of 6242, 3144, and 2930 m/s, respectively. In the

region above and near the edge of the hole, 0.75-mm square elements

were used to construct the finite element model. The time history

of the impact loading was a half-sine curve with a 2 ;us contact

time (t

c ). The arrival time of the 2P-wave reflected from the
86

surface of the hole is 12.2 jus; this results in a t

c

/ t£p value of

0.16. The transient load was applied as a uniform pressure over

one element at the centerline of the plate. In the analyses of

aluminum plates discussed in this chapter, values of displacement

and stress were stored in data files every 0.2 us.

6.3.1 Displacement Response at the Center of the Hole

The vertical displacement waveform obtained at the center

of the flat-bottom hole (point A in Fig. 6.3) is shown as a solid

line in Fig. 6.4. This response consists of vertical surface

displacements caused by the arrival of the direct waves, multiply

reflected waves (3P), mode-converted waves (2PS), and diffracted

waves (P^P, P^S, S^P, S^S, SP^P, etc). The computed arrival times

of the various waves are indicated on the waveform. The analysis

was terminated before waves reflected from the sides and the bottom

of the plate arrived at the hole.

For comparison, the dashed line in Fig. 6.4 shows the response

of a point at the center of the bottom surface of a solid aluminum

plate. The plate was 0.038 m thick, and was subjected to a 2 ys

duration impact, as in the analysis of the plate containing the

flat-bottom hole. The finite element mesh for the plate was the

same as the mesh for the top 0.038-m thickness of the flat-bottom

hole model (Fig. 6.3), that is, the elements below the depth of

the hole were eliminated. The vertical surface displacement for

the plate consists of displacements caused by direct, multiply

reflected, and mode-converted waves; diffracted waves are absent.
87

The waveform obtained from the flat-bottom hole is the

superposition of this plate response and the displacements caused

by diffracted waves. Thus, the differences between the waveforms

shown in Fig. 6.4 are due to the displacements caused by diffracted

waves from the edges of the flat-bottom hole.

Using the waveform obtained from the solid plate as a

baseline response, that is, the component of the response that is

common to both waveforms, the effect of the diffracted waves on

the displacement pattern in the waveform obtained from the surface

of the flat-bottom hole can be determined. The arrival of diffracted

wave P^P produces a noticeable discontinuity in the displacement

pattern, but causes only a slight increase in the magnitude of the

displacement compared with the baseline response. The P^S-wave

causes an upward displacement of the surface which pushes the surface

well above its original undisturbed position. The S^P-wave also

produces an upward displacement of the hole; however, the magnitude

of this displacement is less than that produced by P^S. The S^S-wave

causes an increase in the magnitude of the upward displacement that

occurs after the 3P-wave. Arrivals of subsequent diffracted waves

(3P^P and 3P^S) produce noticeable displacements in the waveform

that are absent in the waveform obtained from the solid plate.

Thus, for this particular flat-bottom hole geometry and for the

2 us duration impact used in the analysis, large amplitude,

diffracted waves are produced which significantly alter the

displacement pattern obtained at the center of the hole compared

with that obtained at the bottom of a solid plate.
88

6.3.2 Displacement Fields

The right sides of Figs. 6.5(a) and (b) show vector plots

of the displacement fields in the plate with the flat-bottom hole

at 6.1 and 10 ;us after the start of the impact. (Since the

displacement field is axisymmetric , the field for only half of the

specimen is shown.) The corresponding positions of the P-, S-,

mode-converted, and diffracted wavefronts are indicated on the left

side of each figure.

The displacement field at 6.1 ps (Fig. 6.5(a)) shows the

P-wavefront arriving at the hole. The S-wavefront has traveled

approximately half the distance to the hole. The pattern of

displacements between the P- and S-waves, the P-wake, is evident.

Fig. 6.5(b) shows the displacement field at 10 ^is. The

reflected P-wave (2P) is evident. The mode-converted PS-wave,

produced by the incident P-wave, is difficult to distinguish as

it overlaps the S-wave. The surface of the hole is displaced

downward due to the displacements caused by the preceeding P-wave

and the P-wave wake, which at 10 ps is incident upon the hole (see

Fig. 6.4). The pattern of displacements trailing the S-wave, the

S-wake, is clearly evident. The low amplitude head wave can also

be distinguished. Diffracted waves P^P and P^S have been produced

by the P-wave incident at the edge of the hole. The Pj P-wave is

clearly evident in the shadow zone behind the hole (Fig. 6.5(b))

where the direct P-wave cannot penetrate. The circular pattern

of displacements caused by P^S is larger in amplitude than that

caused by P^P , and is easily identified around the edge of the hole.
89

To show the effects of the diffracted waves on the

displacement response at the hole, the vector plots shown in the

right side of Figs. 6.6(a), (b), and (c) are close-ups of the region

near the hole. The plots represent the displacement fields at 12,

13.5, and 15 ^us after the start of the impact. The positions of

the wavefronts are shown on the left side of each figure.

In the displacement field at 12 ^us, (Fig. 6.6(a)), the S-wave

is incident on the center of the hole. The P^P-waves are overlapping

in the region above the hole.

Fig. 6.6(b), the displacement field at 13.5 ;us, shows the

S-wave incident upon the edge of the hole. In the radiation pattern

of the S-wave (Fig. 4.2), vertical displacements in the center of

the plate are very small; therefore, the S-wave does not cause

significant downward movement of the center region of the hole.

Thus, at 13.5 jjs, the center region of the hole is recovering and

moving upward. At larger angles in the radiation pattern, the

vertical component of the displacement in the S-wave increases.

Thus, the outer region of the hole is depressed by the incident

S-wave. The fronts of diffracted waves P^S (one from each edge)

have just arrived at the center of the hole. F-ecall that in the

displacement waveform obtained from the surface of the flat-bottom

hole (solid line in Fig. 6.4), it was P^S that produced the large

upward displacement that pushed the center of the hole well above

its original undisturbed position. In Fig. 6.4, at 13.5 ps, the

center of the hole is moving rapidly upward, but it is still

displaced below its undisturbed position.
90

At approximately 15 ps , the upward displacement of the center

of the hole has reached its maximum (see Fig. 6.4). The center

of the hole is displaced upward, while the edge of the hole is pushed

down by the incident S-wave. The diffraction pattern has become

complicated; at 15 ;us four diffracted wavefronts have been produced

by the incident P- and S-waves. Each of these diffracted waves

will give rise to new diffracted waves when they reach the opposite

edge of the hole [2]. These doubly diffracted wavefronts have not

been discussed as they appear to be of secondary importance. The

arrival of multiply reflected P- and S-waves from the top surface

of the solid, such as the 3P-wave (seen at the top of Fig. 6.6(c)),

will also produce diffracted waves. In addition, diffracted waves

will be reflected from the top surface of the plate, giving rise

to new diffracted waves when they strike the edge of the hole.

6.3.3 Impact-Echo Response

Displacement responses recorded on the top surface of a plate

are referred to as impact-echo waveforms. Impact-echo waveforms

obtained from both the plate with the flat-bottom hole and the solid

plate can be used to determine the effects caused by diffracted

waves on the displacement response at the top surface. The effects

caused by diffracted waves can be more difficult to determine at

the top surface of the plate than at the center of the surface of

the hole, because surface waves often mask the effects caused by

the specularly reflected and diffracted waves. However, since the

relative wave arrival times of surface waves, reflected waves, and
diffracted waves differ at increasing distances from the impact

point, the problem can be overcome by studying waveforms obtained

at different points along the surface.

A study of impact-echo waveforms obtained from the plate

with the flat-bottom hole showed that P^P, P^S, S^P, and S^S all

cause upward movement of the top surface of the specimen. This

is in contrast to the downward displacements caused by specularly

reflected P-waves. Displacement waveforms represent the

superposition of displacements caused by specularly reflected and

diffracted waves. Thus, depending upon the arrival times of the

diffracted waves relative to the specularly reflected waves, the

effect of a diffracted wave may only appear to lessen or increase

the displacements caused by specularly reflected waves.

As an example, the solid line in Fig. 6.7 shows a displacement

waveform obtained on the top surface of the plate with the

flat-bottom hole. The separation, H, between the impact point and

the point where the response was recorded was 0.025 m (point B in

Fig. 6.3). The response consists of displacements caused by the

arrivals of direct P-, S- and R-waves propagating along the surface,

reflected P- and S-waves (2P,2S, 4P) , mode-converted waves (PS),

and diffracted waves (P^P, P^S, S^P , S^S, etc.). The arrival times

of these various waves are indicated on the waveform. In the figure,

arrival times of diffracted waves correspond to the arrival of waves

diffracted from the point on the edge of the hole nearest the

receiver; these diffracted waves produce the most significant effects

in the displacement response.
92

For comparison, the dashed line in Fig. 6.7 represents an

impact-echo response that was obtained from the analysis of the

0.038-m thick plate subjected to a 2 ps duration impact. As for

the plate containing the flat-bottom hole, the separation between

the impact point and point where the waveform was recorded was 0.025

m. The response is dominated by the arrival of the P-waves which

cause downward displacements; the top surface is never displaced

above its undisturbed position.

From a comparison of the waveforms in Fig. 6.7, it is

concluded that the diffracted waves cause the following effects:

P^P noticeably reduces the downward displacement caused by the

2P-wave; P^S appears to have little effect, although it slightly

reduces the downward displacement caused by the PS-wave; S^P and

S^S both cause upward displacements.

If another point on the top surface of the plates had been

chosen for this comparison, the displacement response caused by

superposition of the specular and diffracted waves would be

different, as both the arrival times and the amplitudes of the waves

change relative to one another.

6.3.4 Comparison to Experimental Waveform

Since no exact solutions are available to compare with the

finite element displacement waveforms, the finite element solution

was verified by comparison to an experimentally obtained waveform.

An impact-echo test was carried out on an aluminum plate containing

a flat-bottom hole with dimensions as shown in Fig. 6.3. The impact
93

point was located at the center of the specimen; the spacing between

the impact and receiving transducer was 0.029 m. The impact source

was a 1.6-mm diameter steel ball, dropped from a height of 3 cm.

A high fidelity, conical, broadband displacement transducer,

developed at NBS for acoustic emission testing [68], was used as

the receiver. The output of this transducer is proportional to

normal surface displacement. The small diameter of the conical

tip of the transducer (1.5 mm) approximated a point receiver. (This

transducer will be discussed in more detail in Chapter 7.) Fig.

6.8(a) shows the experimental waveform up to a time of 60 jus. Using

the measured response, the contact time of the impact was determined

to be 10 jus. The waveform includes reflections from the sides and

bottom of the plate. At 40 jus, the P-wave reflection from the bottom

of the plate arrives at the receiver causing a large downward

displacement; at 52 jus, the R-wave reflected from the side of the

plate arrives at the receiver also causing a large downward

displacement

.

Fig. 6.8(b) shows a displacement waveform obtained from a

finite element analysis of the plate containing the flat-bottom

hole subjected to an impact having a force-time history in the shape

of a half-sine curve with a contact time of 10 jus. The spacing

between impact point and the point where the displacement was

recorded was 0.029 m, as in the experiment.

There is good agreement between the experimental response

and the response predicted by the finite element analysis, that

is, there is agreement in the pattern of displacements caused by
94

the surface wave, specular reflections, and diffracted waves. The

10 ;us duration impact results in a t c /t2P value of approximately

0.8. For this contact time and for the spacing between the impact

point and the receiver, effects due to individual wave arrivals

are difficult to discern. The superposition of effects causes the

arrival of the initial internal reflections to be hidden in the

response of the surface to the large amplitude R-wave. However,

the important feature in both waveforms is that effects caused by

diffracted waves move the surface above its undisturbed position.

This comparison verifies that the finite element method can

be used to model transient wave propagation in elastic solids

containing flaws. Thus, the method can be used to study the

interaction of transient waves with flaw geometries more likely

to be encountered in actual impact-echo testing. In the following

section, the discussion of a plate containing a flat-bottom hole

is extended to study a similar, but more realistic, geometry - a

plate containing a planar, disk-shaped flaw.

6.4 PLANAR DISK-SHAPED FLAWS IN PLATES

6.4.1 Flaw in Aluminum

The dimensions of an aluminum plate containing a disk-shaped

flaw are the same as those shown in Fig. 6.3 for the plate containing

the flat-bottom hole. The depth and diameter of the circular flaw

are identical to the same dimensions for the flat-bottom hole.

The finite element model is identical to that shown in Fig. 6.3,

with the addition of the elements required to form a disk-shaped
void from the hole. The void is 0.001 m thick. The analysis was

carried out using the same material properties and loading conditions

as used in the analysis of the plate with the flat-bottom hole.

95

Displacement waveforms obtained at the center of the top

surface of the flaw (point A in Fig. 6.3) and at a point on the

top surface of the plate, 0.025 m from the point of impact (point

B in Fig. 6.3), are shown in Figs. 6.9(a) and (b), respectively.

Wave arrival times are indicated on both waveforms. These waveforms

do not need detailed explanation as they are very similar to those

obtained from the plate containing the flat-bottom hole (Figs. 6.4

and 6.7). There are minor differences in the waveforms obtained

from the surface of the hole (Fig. 6.4) and the disk-shaped flaw

(Fig. 6.9(a)) which occur after the arrival of the S^S- and

2PS-waves. The waveforms obtained at the top surface of the plates

are almost identical, except that the displacements in the waveform

obtained from the plate containing the disk-shaped flaw (Fig. 6.9(b))

appear shifted slightly above those obtained from the plate with

the flat-bottom hole after the arrival of the S^P-wave . Thus, it

appears that the presence of the material below the flaw does not

significantly change the overall response. (This conclusion is

only valid for the period of time prior to wave reflections from

the bottom of the plate.)

The displacement patterns previously discussed were obtained

for planar flaws in aluminum, a material with a Poisson's ratio

of 0.33 (ratio of S- to P-wave speeds equal to 0.50). As explained,

aluminum was used so that waveforms obtained from the finite element
96

analysis could be compared to waveforms obtained from an easily

fabricated test specimen. However, the primary focus of the research

program is in applying the impact-echo method to concrete, a material

with a Poisson's ratio of approximately 0.2 in the elastic range.

For a Poisson's value of 0.2, the S-wave speed is 61% of the P-wave

speed; this ratio is significantly different than that in aluminum.

This difference affects the relative arrival times of the

wavefronts. For comparison, the displacement patterns obtained

from planar disk-shaped flaws in plates having elastic properties

representative of concrete are presented.

6.4.2 Flaws in Concrete

In this discussion, the following results are presented:

displacement time-histories obtained on the top surface of the flaw,

directly under the point of impact, and at a point on the top surface

of the plate, near the point of impact; and, displacement and stress

fields recorded at successive times to show the interaction of

transient stress waves with a planar flaw.

In the analyses presented in this section, the concrete was

modeled as a linearly-elastic , homogeneous solid, with the following

material properties: a modulus of elasticity of 3.31 x 10^ kPa;

a Poisson's ratio of 0.2; and, a density of 2300 kg/m . These values

resulted in P-, S-, and R-wave -.speeds of 4000, 2440, and 2240 m/s,

respectively. In the region above and near the edge of the flaw,

5-mm square elements were used in the finite element model.
6. 4. 2.1 Displacement Waveforms

An analysis was performed for a planar, disk-shaped void

97

in a 0.5-m thick concrete plate. The diameter and the depth of

the flaw were both 0.2 m for a D/T value of 1, as in the analyses

of the flaws in alumimum. The time history of the impact load was

a half-sine curve with a 16 us contact time. The arrival time of

the 2P-wave reflected from the flaw was 100 ps, which resulted in

a t c /t 2 p

value equal to 0.16, as in the analyses of flaws in

aluminum.

Displacement waveforms obtained at the center of the top surface

of the flaw and at a point on the top surface of the plate, 0.05

m from the impact point, are shown in Figs. 6.10(a) and (b),

respectively. Wave arrival times are indicated on the waveforms.

The displacement caused by each wave is the same as has been

discussed; however, since there is less difference between the Pand S-wave speeds, wave arrivals in concrete are more closely spaced

than in aluminum. Thus, the displacement waveforms obtained from

the concrete specimen are different from those previously shown

for the same flaw in aluminum. For example, compare the displacement

pattern in Fig. 6.9(a) with that in Fig. 6.10(a). In the concrete

specimen, the P^S-wave arrives somewhat later than the S-wave, rather

than at approximately the same time as in the aluminum plate; thus,

the large upward displacement caused by the P^S-wave is easily

identified in Fig. 6.10(a). As in aluminum, if the response obtained

from the concrete plate containing the disk-shaped flaw is compared

to the response obtained from a solid concrete plate (such as in
98

Fig. 4.5), the effects caused by diffracted waves are clearly

evident. Diffracted waves produce more frequent fluctuations in

the displacement and they move the surface above its undisturbed

position. For the top surface displacement shown in Fig. 6.10(b),

the most noticeable effect due to diffracted waves is the large

upward surface displacement caused by the arrivals of the P^S- and

S^P-waves

.

In Fig. 6.10(b), there is a period after the arrival of the

R-wave when the displacement appears to oscillate about its

undisturbed position. These oscillations are due to the numerical

ringing which occurs due to excitation of the zero energy modes

of the constant strain finite elements.

6. 4. 2. 2 Displacement and Stress Fields

An analysis was performed for impact on a 0.5-m thick, 1.5-m

diameter, concrete plate containing a disk-shaped void. The void

was Q.Ol-m thick and 0.1 m in diameter; it was located 0.25 m below

the top surface of the plate, resulting in a D/T value of 0.4.

The contact time of the impact was 20 us. The arrival time of the

2P-wave reflected from the top surface of the void was 125 us for

a t c /t2 pvalue of 0.16.

The right sides of part (a) in Figs. 6.11 through 6.14 show

vector plots of the displacement fields in the plate 65, 80, 95,

and 125 /is after the start of the impact. The corresponding

positions of the P-, S-, mode-converted, and diffracted wavefronts

are shown on the left side of part (a) of each figure. Part (b)
99

of each figure shows a contour plot of minimum principal

(compression) stress.

The P-wavefront arrives at the flaw at 62.5 jus, and is

incident upon the edge of the flaw at 64 ^is. At 65 jus, the stress

contours in Fig. 6.11(b) show that diffraction of the P-wave is

occurring; the P^P-wave front is beginning to emerge from the edge

of the flaw. The S-wavefront has traveled approximately two-thirds

of the distance to the flaw. Rays emanating from the impact point

and intersecting the edges of the flaw delineate the shadow zone

beneath the flaw. Only diffracted waves can penetrate this zone.

Fig. 6.12 shows the displacement and stress fields at 80

jus. The front of the reflected P-wave (2P) is just overlapping

the front of the direct S-wave. The top surface of the flaw is

displaced downward because of the effects caused by the direct

P-wave. The diffracted P^P-waves have overlapped in the center

region of the plate, above and below the flaw; thus, the entire

shadow zone is stressed. Close inspection of Fig. 6.12(a) shows

that the displacement vectors in the diffracted P^P-waves are

oriented in spherical patterns emanating from the edges of the flaw.

The direction of displacements of the vectors in the P^P-waves is

outward, away from the edges of the flaw. The stress contours in

Fig. 12(b) show that the P^P-wave is a compression wave.

. At 95 jus (Fig. 6.13), the P-wave has completely passed the

flaw. This wavefront is now formed by the P^P-wavefront in the

shadow zone and the original P-wavefront outside the shadow zone.

Note that the part of the wavefront formed by the diffracted waves
100

is no longer spherical. Reflection and diffraction of the P-wake

are occurring

.

Fig. 6.14 shows the displacement and stress fields at 125

us. The 2P-wave reflected from the surface of the flaw arrives

at the top surface of the plate. The PS-wave produced by

mode-conversion of the P-wave incident upon the flaw has traveled

approximately 60 percent of the distance from the flaw to the top

surface of the plate. Reflection and diffraction of the S-wave

by the flaw has occurred. The P-wake has reformed below the flaw.

The P-wavefront is nearing the bottom surface of the plate.

Compare Fig. 6.14 to Fig. 4.6 which shows the displacement

and stress fields in a solid, 0.5-m thick, plate at 125 us. In

the plate containing the flaw (Fig. 6.14) diffracted waves allowed

the P- and S-wavefronts to penetrate the shadow zone. Thus, the

displacement and stress fields produced by the P- and S-waves in

the plate containing the flaw are similar to those in the solid

plate (Fig. 4.6). However, in the fields obtained from the plate

containing the flaw, the displacement and stresses caused by the

waves produced by the interaction of the P- and S-waves with the

flaw are also present. These waves will reflect from the surfaces

of the plate and interact with the flaw producing new reflected

and diffracted waves. In addition, subsequent interactions of the

reflected P- and S-waves with the top and bottom surfaces and the

edges of the flaw will result in new reflected and diffracted waves.

Thus, the presence of the flaw greatly complicates the displacement

and stress fields.
101

A displacement waveform recorded near the point of impact

at the top surface of a plate containing a flaw consists of

displacement caused by waves reflected and diffracted from the flaw

and waves reflected from the bottom surface of the plate and

subsequently diffracted by the flaw. Multiple reflections and

mode-conversions of all these waves occur, complicating the

displacement waveform. The relative importance of the effects caused

by each of these phenomena on surface displacement waveforms depends

on the flaw geomtry and on the test conditions.

6.5 THE EFFECT OF TEST VARIABLES ON IMPACT-ECHO WAVEFORMS

To better understand how surface displacement waveforms are

affected by flaw geometry and test conditions, parameter studies

were carried out. The variables examined are shown in Fig. 6.2;

they include: the contact time of the impact; the diameter and the

depth of a flaw; and, the test configuration, that is, the point

where the displacement is recorded.

6.5.1 Contact Time of the Impact

As discussed in Chapter 5 (Section 5.4.1), the frequency

content of the stress waves produced by impact depends on the contact

time of the impact. As was shown in Fig. 5.4, a shorter contact

time produces a broader range of frequencies in the waves; however,

the amplitude of each component frequency is lower. Waves produced

by a long contact time are made up primarily of large amplitude,

low frequency components. This is important because it is the
frequency content of the waves produced by impact that determines

the size of the flaw that can be detected. For a flaw to be

102

detected, it is generally stated that the dimensions of the flaw

must be on the order of, or larger than, the component wavelengths

in the propagating waves. (This general rule will be referred to

as the "one-wavelength" criterion.) Higher frequency components

have shorter wavelengths; thus, waves containing higher frequencies

are reflected by smaller flaws. However, high frequency waves have

less penetrating ability, as they are attenuated more quickly than

low frequency waves in a heterogenous material such as concrete.

Three analyses of a 0.5-m thick concrete plate containing

a disk-shaped flaw were carried out. The variable in the analyses

was the contact time of the impact. The test configuration is shown

in Fig. 6.2. The flaw diameter was 0.2 m and it was located 0.13

m below the top surface of the plate (D/T value of 1.54). The

separation between the impact point and the point where the waveform

was recorded was 0.02 m (H/T value of 0.15). The contact times

of the impacts were 20, 40, and 80 ^lis.

Using the "one-wavelength" criterion, a 0.2-m diameter flaw

will reflect frequencies having wavelengths on the order of, or

greater than, approximately 0.2 m; therefore, for the flaw to be

detected, sufficient energy must be contained in the frequencies

approximately equal to and greater than 40 kHz. The normalized

amplitude spectrum of the force-time history of a half-cycle sine

curve (Fig. 5.4) can be used to determine the frequency content

of the various impacts. By multiplying the horizontal axis by each
103

contact time, it is found that the 20, 40, and 80 ps duration impacts

have a first zero amplitude at 75 , 37.5 , and 18.75 kHz,

respectively. Figs. 6.15(a)-(c) show the surface displacement

waveforms obtained from the plate for each of these contact times.

Wave arrival times are indicated on the waveforms. Notice that

in all three waveforms, regardless of the contact time, effects

caused by the diffracted waves cause the surface to be displaced

above its undisturbed position, indicating that a flaw is present.

A 20-jis duration impact results in a combination of conditions

which allow displacements caused by specularly reflected,

mode-converted, and diffracted waves to be identified in the waveform

shown in Fig. 6.15(a). These conditions are as follows: A large

amount of the total energy is contained in frequencies in the range

of 40 kHz and higher; thus, a large portion of the energy in the

pulse is reflected by the flaw. The arrival of the first reflection

from the flaw (2P-wave) occurs at 100 us. This results in a t c /t 2 p

value of 0.32. As discussed in Chapter 5, a small t c /t2P value

means that the contact time is short enough relative to the time

of the arrival of the first internal reflection (2P-wave) so that

the displacement pattern produced by internal reflection is not

masked by the R-wave arrival. In this case, the perturbations in

the waveform can be used to accurately determine the depth of the

flaw.

In the surface displacement produced by a 40-ps duration

impact (Fig. 6.15(b)), P-wave arrivals from the flaw dominate the

response. The waveform is smoother, as the displacements caused
104

by diffracted waves are masked by the general smoothing effect of

a longer duration impact (see Section 5.3.2). A contact time of

40 jus results in a t c /t 2 p

value of 0.64. This is twice as large

as in the previous case; however, the R-wave does not interfere

with the displacement pattern produced by internal reflections.

There is still good agreement between calculated P-wave arrival

times and the large downward displacements in the waveform; thus,

the depth of the flaw can be accurately determined from a time domain

analysis. The diameter of the flaw cannot be determined in this

case because the arrival of diffracted waves from the edges of the

flaw cannot be identified.

Fig. 6.15(c) shows the surface displacement waveform produced

by an 80-jus duration impact. In this case, t c /t2P is equal to 1.3;

the duration of the impact is longer that the arrival time of the

2P-wave. The initial part of the displacement pattern produced

by internal reflections from the flaw is lost in the surface response

to the large amplitude R-wave, and there is no agreement between

the perturbations in the waveforms and calculated wave arrival

times. However, because of effects caused by diffraction which

push the surface displacement above its undisturbed position, the

presence of the flaw can still be detected, even though the depth

of the flaw cannot be determined from a time domain analysis.

In summary, as the contact time of the impact increases and

more of the energy produced by impact is contained in the low

frequency components of the input pulse, the following trends arc

observed: 1) The displacement response becomes smoother; 2) Surface
105

displacements caused by the R-wave begins to mask the displacements

caused by internal reflections from the flaw; and, 3) Calculated

wave arrival times do not agree with the perturbations in the

waveforms

.

6.5.2 Ratio of Flaw Diameter to Flaw Depth

Figs. 6.16(a)-(d) show surface displacement waveforms obtained

from 0.05, 0.1, 0.2, and 0.4 m diameter, disk-shaped flaws which

were located 0.2 m below the top surface of a 0.5 m thick concrete

plate. Flaw geometries are shown schematically on the left of each

figure. These flaw geometries resulted in D/T values of 0.25, 0.5,

1, and 2, respectively.

In each analysis, the spacing between the impact point and

the receiver was 0.05 m; that is for each flaw geometry, the waveform

was recorded at the same point on the top surface of the plate.

Thus, the position of the receiver relative to the edge of the flaw

was different in each case: for the 0.05 m diameter flaw (Fig.

6.16(a)), the waveform was recorded off the edge of the flaw; for

the 0.1 m diameter flaw (Fig. 6.16(b)), the waveform was recorded

directly over the edge of the flaw; for the larger flaws (Figs.

6.15(c) and (d)), the waveforms were recorded over the flaw. The

point where the waveform was recorded relative to the location of

the flaw is important, as it will affect the amplitude and arrival

times of the various echoes produced by the flaw. This parameter

will be studied in more detail in the following section.

The contact time of the impact was 16 ps (t

c /t 2 p

value of
106

0.16). Wave arrival times are indicated on the waveforms. The

following paragraphs discuss the effects on the surface displacement

waveforms caused by increasing the flaw diameter.

As flaw diameter increases at a given depth (Fig. 6.16(a)

to (d)), the arrival times of diffracted waves change relative to

the arrival times of specularly reflected and mode-converted waves.

The larger the diameter of the flaw, the greater the time lag between

the arrival of the specular reflection produced by an incident wave

and the diffracted waves produced by the same wave. Therefore,

as the diameter increases, the superposition of displacements caused

by specular, mode-converted, and diffracted waves is different.

The amplitude of specularly reflected waves increases as

the size of the flaw increases. This increase in amplitude results

from the fact that a larger flaw reflects a greater range of

frequencies. For flaws having diameters of 0.05, 0.1, 0.2, and

0.4 m, frequencies in the range of and greater than 80, 40, 20,

and 10 kHz, respectively, will be reflected. For a contact time

of 16 jus , the first zero amplitude in the spectrum occurs at a

frequency of 93.7 kHz. Thus, as the diameter of the flaw decreases

from 0.4 m to 0.05 m, the amount of energy reflected by the flaw

decreases signf icantly , until, for the smallest flaw (Fig. 6.16(a)),

displacement caused by wave arrivals from the flaw are difficult

to distinguish from the noise (numerical in this case). Reflections

from this small flaw are also difficult to discern because the

waveform was recorded at a point off the edge of the flaw. For

this particular test configuration (impact point and the point when
the displacement was recorded are equally distant from the edge

of the flaw) , only diffracted waves from the flaw contribute to

the surface displacement waveform.

107

The amplitude of diffracted waves depends, in part, on the

position of the edges of the flaw with respect to the radiation

pattern of the incident waves. As was mentioned in Chapter 4, the

radiation pattern for P- and S-waves (Fig. 4.2) shows that the

amplitude of the displacements in the P-wave is greatest at the

center of the wavefront and gradually decrease along rays located

at increasing angles from the centerline of the plate; displacements

in an S-wave are zero at the center of the wavefront and increase

to a maximum along rays located at approximately 40 degrees from

the centerline of the plate (Poisson's ratio equal to 0.2).

Therefore, a P-wave incident upon the edges of a small flaw, such

as the 0.05 m and 0.1 m diameter flaws shown in Figs. 6.16(a) and

(b), will produce large amplitude diffracted waves, while an S-wave

incident upon the same flaws will produce diffracted waves of much

lower amplitude. The edges of these small flaws (Figs. 6.16(a)

and (b)) intersect rays located at 7.1 and 14 degrees from the

centerline of the plate. In contrast, the 0.2 and 0.4 m diameter

flaws (Fig. 6.16(c) and (d)) have edges intersecting rays located

at 26.6 and 45 degrees from the center of the wavefront. In these

cases, the amplitude of displacements in the diffracted waves

produced by an incident S-wave will be near maximum; however, the

amplitude of the displacements in the diffracted waves produced

by an incident P-wave will be less than for a smaller diameter flaw.
108

In addition, each diffracted wave also has its own radiation

pattern. (Radiation patterns have not been calculated for transient

waves, but it can be assumed that they are similar to those for

harmonic waves, just as the radiation patterns for P- and S-waves

produced by a harmonic source were similar to those produced by

a transient point source.) Thus the amplitude of the surface

displacement caused by a diffracted wave will also depend upon where

the displacement is being monitored relative to the edge of a flaw.

To show the effect of increasing depth on surface displacement

waveforms, an analysis was carried out for a 0.1 m diameter flaw

located 0.4 m below the top surface of the plate (D/T value of

0.25). All other parameters were the same as those shown in Fig.

16(b). The value of the t c /t2P ratio and the magnitude of the impact

force-time history were also the same as in the previous analyses.

Fig. 6.17 shows the displacement waveform obtained from this

analysis; for this flaw geometry the waveform was recorded directly

over the edge of the flaw. Wave arrivals are difficult to identify

in this waveform. Divergence due to beam spreading causes spherical

waves to undergo attenuation at the rate inversely proportional

to the distance traveled (Section 2.1.6). Compare this waveform

to the waveform obtained from the 0.1 m flaw located at a depth

of 0.2 m (Fig. 6.16(b)). By increasing the depth of the 0.1 m

diameter flaw by a factor of two, the amplitude of reflected waves

arriving at the top surface of the plate is reduced by a factor

of two, making it difficult to detect a flaw that was easily detected

at the shallower depth
109

The flaw located at a depth of 0.4 m has a D/T value of 0.25,

which is equal to the D/T value of the 0.05 m flaw shown in Fig.

6.16(a); both are difficult to detect. Thus, for a given duration

impact, the D/T value is a useful parameter for determining whether

a flaw can be detected. Additional studies showed that for a given

depth, T, the critical value of the diameter, D, increases as the

contact time of the impact increases. Thus, these three variables

are integrally related.

6.5.3 Test Configuration

As mentioned in the previous section and in Section 5. 3. 2. 2,

surface displacement waveforms are affected by the test

configuration; that is, they are affected by the locations of the

impact point and the point where the displacement is being recorded

relative to the location of a flaw. To study the effect of test

configuration on displacement patterns, surface displacement

waveforms were obtained at various distances (H) from the impact

point by recording the time-history of the displacements at nodes

along the surface of the finite element model. Flaw diameter was

0.2 m and flaw depth was 0.2 m, giving a D/T value of 1. The contact

time of the impact was 16 ps.

Figs. 6.18(a)-(c) show displacement waveforms obtained for

the following H values: 0.025, 0.10, and 0.15 m, respectively.

The waveform shown in Fig. 6.18(a) was obtained at a point located

over the flaw near the impact point; Fig. 6.18(b) shows a waveform

recorded directly over the edge of the 0.2 m diameter flaw; and
Fig. 6.18(c) shows a waveform obtained at a point that was not

located over the flaw. Wave arrival times are indicated on each

110

waveform. The displacement patterns in this series of waveforms

are discussed in the following paragraphs.

As H increases, the time lag between specular reflections

and diffracted waves decreases. Each waveform represents the

superposition of the displacements caused by both sets of waves;

therefore, as H increases, the displacement pattern in the waveforms

gradually changes as the arrival times of specularly reflected and

diffracted waves shift relative to each other. In Fig. 6.18(a),

specularly reflected and diffracted wave arrivals are distinct,

and the displacements caused by each individual arrival can be

discerned in the waveform. In Fig. 6.18(c), however, effects caused

by specular reflections and the arrival of the corresponding

diffracted waves are nearly exactly superimposed in time. For

example, the downward displacement caused by the 2P-wave is reduced

by the simultaneous arrival of the P^P-wave which causes an upward

displacement of the plate surface. Similarly, the PS- and P^S-waves,

and the 2S- and S^S-waves arrive at about the same times, and the

displacements that occur due to these simultaneous arrivals represent

the superposition of the displacements caused by each wave.

The waveform obtained over the edge of the flaw (Fig. 6.16(b))

shows more perturbations than the waveform obtained over the flaw

(Fig. 6.18(a)) and the waveform obtained at the point on the surface

that was not over the flaw (Fig. 6.18(c)). This characteristic

can be used to help locate the edges of a flaw.
As H increases, the amplitudes of the displacements caused

by arrivals of specularly reflected, mode-converted, and diffracted

111

waves change signficantly (see Section 5. 3. 2. 2).

6.6 SUMMARY AND CONCLUSIONS

This chapter has focused on the interaction of transient

stress waves with planar flaws in plates. The elastic responses

produced by surface impact on plates containing flat-bottom holes

and planar disk-shaped flaws were studied. Surface displacement

waveforms obtained from plates containing flaws were compared to

waveforms obtained from solid plates to determine how surface

displacement patterns are affected by the waves created by

diffraction at the sharp edges of a discontinuity. It was shown

that the displacement waveforms obtained from a plate containing

a flat-bottom hole are very similar to those obtained from a plate

containing a planar disk-shaped flaw.

Parameter studies of plates containing planar disk-shaped

flaws were carried out to study the effect of test variables on

surface displacement waveforms. The following conclusions were

made

:

1) The ratio of flaw diameter to flaw depth (D/T) is a useful

parameter for characterizing planar flaws. The larger the D/T value

of a flaw, the easier it is to detect the flaw. The critical D/T

value increases as the contact time of the impact increases.

2) If the diameter of a flaw is sufficiently large, accurate

determination of the depth of a flaw from time domain analysis can
112

be made using surface displacement waveforms, if the contact time

of the impact is less than approximately two/thirds of the time

it takes for the first P-wave reflection from the flaw to arrive

at the top surface of the plate.

3) A waveform recorded close to the impact is simpler to

interpret and allows the most accurate determination of flaw depth.

In Chapters 4 and 6, comparisons of displacement waveforms

obtained from finite element analyses to exact and experimentally

obtained displacement waveforms established the validity of using

finite element analysis for the study of transient wave propagation

in elastic solids. This study lays the groundwork for finite element

studies of elastic solids of arbitrary geometry and studies of solids

containing a variety of types of flaws.

In the following chapter, these numerical studies will be

used to help interpret experimental displacement waveforms obtained

from controlled flaw studies
113

(b)

Figure 6.1 Diffraction at the edges of a crack: a) P-ray

upon the edges of a crack; and, b) cylindrical wavefronts

P^S emitted from tips.

inc ident

PjP and
114

t

H

Figure 6.2 The important variables affecting the response of a planar disk-shaped void in a plate.

T = 0.038 m

0.089 m

0.070 m

Figure 6.3 Finite element model of a plate with a flat-bottom hole.
DISPLACEMENT

115

Figure 6.4 Displacement waveforms at the center of the flat-bottom

hole and at the epicenter of a solid plate.
116

R

Zone

Figure 6.5 Vector plots of the displacement fields in a plate with

a flat-bottom hole: a) 6.1 ;js after the start of impact; and b)

10 ^us after the start of impact.
117

(a)

Figure 6.6 Close-ups

the flat bottom hole

a) 12 ;us; b) 13.5 jus;

of vector plots of displacement fields

at various times after the start of

and, c) 15 jus.

around

impac t

:
DISPLACEMENT

118

TIME (ps)

Figure 6.7 Impact-echo displacement waveforms from a plate with

a flat-bottom hole and from a solid plate.
DISPLACEMENT

119

0 20 40 60

TIME (ms)

Figure 6.8 Comparison of analytical response with experimentally obtained response for a plate with a flat-bottom hole: a) experimental waveform; and, b) finite element waveform.
0 10 20 30

TIME (ps)

Figure 6.9 Planar disk-shaped flaw in an aluminum plate: a)

displacement at the center of the top surface of the flaw; and,

b) displacement at a point on the top surface of the plate.
!

Figure 6.10 Planar disk-shaped flaw in a concrete plate: a)

displacement at center of the top surface of the flaw; and, b)

displacement at a point on the top surface of the plate.
122

R

(a)

Figure 6.11 Displacement and stress fields in a plate containing

a disk-shaped flaw 65 ps after the start of the impact.
123

R

Figure 6.12 Displacement and stress fields in a plate containing

a disk-shaped flaw 80 jjs after the start of the impact.
124

Figure 6.13 Displacement and stress fields in a plate containing

a disk-shaped flaw 95 /js after the start of the impact.
125

Figure 6.14 Displacement and stress fields in a plate containing

a disk-shaped flaw 125 ^is after the start of the impact.
4P 6P

126

Figure 6ol5

the contact

The effect on displacement waveforms caused by varying

time of the impact: a) 20 us; b) 40 ps; and, c) 80 ps.
DISPLACEMENT

DISPLACEMENT

127

Figure 6.16 The effects on surface displacement waveforms caused

by varying the diameter of the flaw. Flaw depth is 0.2 m. Flaw

diameter is: a) 0.05 m; b) 0.1 m; c) 0.2 m; and, d) 0.4 m.
128

Figure 6.17 Surface displacement waveforms obtained from a plate

containing a 0.1-m diameter flaw located 0.4 m below the top surface

of the plate.
129

F igure

the dis

surface

and , c

)

6.18 The effect on displacement waveforms of increasing

tance, H, between the impact point and the point where the displacement is measured: a) H = 0.025 m; b) H = 0 1 H = 0.15 m.

’
CHAPTER 7

IMPLEMENTATION OF THE IMPACT-ECHO METHOD FOR

FLAW DETECTION IN CONCRETE

7.1 INTRODUCTION

The experimental phase of this work had two primary

objectives: 1) to develop an impact-echo measurement technique

(instrumentation and signal acquisition and processing) for finding

flaws within plain and reinforced concrete; and 2) to confirm the

observations and conclusions drawn from the analytical and finite

element studies of solid plates and plates containing planar flaws.

This chapter describes the development of an impact-echo

technique for concrete. Results obtained from controlled laboratory

flaw studies in plain and reinforced concrete plates are presented.

7.2 DEVELOPMENT OF A MEASUREMENT TECHNIQUE

7.2.1 Impact Sources

Initially, small steel ball bearings, dropped from a height

of 0.2 m, were used as the impact source. The contact times of

4- to 16-mm diameter, steel ball bearings on concrete ranged from

approximately 20 to 90 us. The impact of these steel balls produced

sufficient energy to penetrate 0.5-m thick concrete plate elements.

The contact time produced by the impact of steel balls on

concrete was estimated by finding the difference between successive

zeroes in the frequency spectrum (Section 5.3.1). Experimental

contact times for a particular steel ball varied depending upon

the condition of the concrete surface under the point of impact.

130
131

A hard, smooth surface produced contact times that were in almost

exact agreement with Hertz elastic theory (Section 2.2.1). A rough

textured surface tended to produce longer contact times than those

predicted by elastic theory probably because local crushing of the

paste phase of the concrete under the point of impact.

The contact time could also be estimated from the duration

of the R-wave signal in the time domain waveform. Analytical and

finite element studies showed that the time difference between the

front and the back of the R-wave signal is approximately equal to

the difference obtained from successive zeroes in the frequency

domain; this time difference is therefore also an accurate estimate

of the contact time of the impact.

Small diameter ball bearings are convenient to use in

laboratory studies of horizontal concrete specimens. However, a

more practical impact source would be needed for testing vertical

surfaces and for field testing. Thus spring-loaded impactors

(Equotip impactors, manufactured by Proceq SA, Zurich, Switzerland)

designed for hardness testing of metals were adapted for use as

impact sources. Two impactors were used: one had a 3-mm diameter,

spherical steel tip attached to a 5.5-g spring-driven mass; the

other had a 5-mm diameter tip attached to a 20 g mass. The contact

times produced by the impactors ranged from approximately 40 to

80 yus , depending on the condition of the concrete surface. Fig.

7.1 shows an impactor containing the spring-driven mass.
132

7.2.2 Transducer and Filter

The receiving transducer was a conical, broadband,

displacement transducer which was recently developed at NBS for

acoustic emission testing of metals [68]. This transducer, which

is shown in Fig. 7.2, consists of a small conically shaped

piezoelectric element cemented to a large brass backing. The output

of the transducer is proportional to normal surface displacement.

The brass backing damps out the ringing of the transducer element

so that the recorded response is a faithful measurement of the

surface displacement. Because of its small contact area the

transducer closely approximates a point receiver.

As the transducer was developed for testing of metals, a

thin sheet of metal foil had to be used between the conical element

and the concrete surface to complete the transducer circuit. In

the initial phase of the work, a sheet of aluminum foil was used.

The foil was coupled to the concrete surface using water soluble

gel. The transducer was designed for use of a polished metal

surface; thus, grinding of the concrete surface was generally

required to achieve proper coupling between the conical element

and the surface when the aluminum foil was used. As grinding of

the surface was tedious as well as impractical, the foil was replaced

with a thin sheet of lead. Use of the lead, which is malleable

and conforms to the concrete surface, eliminated the need for using

the gel and for surface preparation, greatly simplifying the

experimental procedure.

The transducer was protected from electromagnetic noise by
133

a metal shield. The shield was also used to house a unity gain

amplifier. The transducer was used with a high pass filter which

strongly attenuated frequencies less than 2 kHz. The filter was

used to reduced the effects caused by a low frequency resonance

of the transducer assembly (the large brass backing on the small

conical element acting as a mass on a spring).

7.2.3 Oscilloscope

A digital storage oscilloscope operating at a 500 kHz sampling

frequency (2 jus sampling rate) was used to store and process

received signals. Time domain signals contained 1024 sampling

points; therefore, a 2 p.s sampling rate corresponds to a resolution

of 0.488 kHz in the frequency domain (Section 5.2).

The oscilloscope was programmable which facilitated signal

processing, and it was equipped with two disk drives so that data

could be permanently stored. The oscilloscope was interfaced with

a plotter so that waveforms, frequency spectra, etc., could also

be plotted directly. Data could also be transferred to a personal

computer for further analysis and/or plotting.

7.2.4 Concrete Specimens

Experimental displacement waveforms were obtained from large

concrete plates, in which artificial flaws were embedded at known

locations. Four plates were cast; these included:

1) a solid, 0.25-m thick plate, which was 1.125 x 1.125

m in plan.
134

2) a 0.2-m thick plate, 0.5 x 1 m in plan, which contained

steel bars placed 0.06 m below the top surface of the

plate. Plan and elevation views are shown in Fig. 7.3.

Bar diameters ranged from 19.1 to 44.5 mm.

3) a 0.5-ro thick plate, 2 x 4.5 m in plan, in which were

embedded at various depths 25-mm thick disks cut from

rigid, polyurethane foam insulation board. Disk diameters

ranged from 0.05 to 0.5 m. Plan and elevation views

of this plate are shown in Fig. 7.4. Fig. 7.5 shows

the formwork for the plate with the flaws in place prior

to casting of the concrete.

4) a 0.5-m thick plate, 2 x 2.25 m in plan, which contained

a variety of artificial flaws, including: a thin metal

duct partially filled with grout; 0.075-mm thick plastic

sheets to simulate real cracks; and circular and

rectangular flaws cut from the polyurethane foam board.

A mat of steel reinforcing bars was placed over half

the plate at a depth of approximately 5 cm below the

top surface of the plate; bar diameters were 22.2 and

25.4 mm. Plan and elevation views of this plate are

shown in Fig. 7.6. Fig. 7.7 shows the formwork for the

plate with artificial flaws in place prior to casting

of the concrete.

In plates Nos. 1 and 3, the coarse aggregate was crushed

limestone with a nominal maximum size of 12 mm, and the average

28-day cylinder compressive strength was 25.9 MPa. In plate No. 2,
135

the coarse aggregate was rounded gravel with a maximum size of 12

mm; the 28-day cylinder compressive strength was 34 MPa. In plate

No. 4, the coarse aggregate was crushed limestone with a maximum

size of 19 mm; 28-day cylinder compressive strength was 28.4 MPa.

7.2.5 Signal Acquisition and Processing

The test set-up is shown in Fig. 7.8. Data acquisition by

the oscilloscope was triggered by the voltage produced by the

transducer response to the arrival of the large amplitude R-wave.

Once a waveform was acquired, it could be studied in the time domain

or transformed into the frequency domain by use of the Fast Fourier

Transform technique.

In time domain analysis, the exact time of the start of the

impact (the beginning of the waveform) must be known so that the

time of wave arrivals can be determined from the perturbations in

the waveform. The time of the start of the impact was determined

indirectly. Knowing the velocity of the R-wave in a concrete

specimen and the distance between the impact point and the receiving

transducer, the time required for the R-wave to travel from the

impact point to the receiver can be calculated. (Recall from Section

2.1.2 that the R-wave speed is approximately equal to 0.56 times

the P-wave speed.) As the arrival of the R-wave was easily

identified in waveforms, the time of the impact can be determined.

The initial studies of displacement waveforms were carried

out in the time domain. Later it was found that interpretation

of displacement waveforms was simpler in the frequency domain.
In the following sections, both the displacement waveforms and

frequency spectra are shown; however, most of the numerical results

are based on frequency spectrum analyses.

136

7.2.6 Determining Wave Speeds Through Concrete

To determine the location of a reflecting interface from

the dominant peak in the frequency spectrum, the P-wave velocity

in the test object must be known. Two methods were used for

determining P-wave velocity in concrete: 1) the through

transmission, pulse-velocity method (ASTM C 597 [4]); and 2) the

impact-echo method. As discussed in Chapter 3, in the pulse-velocity

method, the transmitter and the receiver are located on opposite

sides of the test object and the time for an ultrasonic P-wave pulse

to travel from transmitter to receiver through the test object is

measured. The ultrasonic pulse-velocity is obtained by dividing

the distance between the transducers by the travel time. The

commercial test apparatus uses lightly damped transducers; therefore

the pulse that is generated contains multiple cycles of P-waves.

The transit time is based on detecting the arrival of the leading

edge of the pulse. In the impact-echo method, the transient response

of a test object of known thickness subjected to surface impact

is recorded. Then the peak frequency in the amplitude spectrum

obtained from transforming the time domain waveform is used to

calculate the wave velocity using Eq . (5.7). Thus the P-wave

velocity is calculated indirectly.

Previously, it was thought that the pulse-velocity method
137

could be used to independently measure P-wave velocities in

concrete. However, a discrepancy was found when pulse velocities

determined from ultrasonic pulse-velocity measurements (obtained

using 54 kHz transducers, 5 cm in diameter) were used in frequency

spectrum analyses. When the ultrasonic pulse velocity was used

to calculate the depths of reflecting interfaces using the peak

frequency in the amplitude spectra, the calculated depths exceeded

the actual depths in all test situations. When direct comparisons

were made betw-een ultrasonic pulse velocities measured through the

thickness of solid concrete plates and the velocity calculated from

peak frequencies obtained from impact-echo tests on the same slabs,

it was found that the P-wave velocities calculated by the impact-echo

method were approximately 10 percent slower than the velocities

determined by the pulse velocity method.

A time domain study of waveforms obtained from interfaces

of known depth showed that the onset of the downward displacements

in the waveforms agreed with wave arrival times computed using the

P-wave velocity obtained from the impact-echo method. Results of

tests on the 0.25-m thick concrete plate illustrate the difference

between the two velocities. Pulse-velocity measurements, taken

at the center of the plate, indicated a wave speed of 4020 m/s.

Fig. 7.9 shows the waveform obtained from an impact-echo test (H/T

value of 0.2) carried out at the same point. The peak in the

corresponding frequency spectrum was 7.32 kHz, which gives a computed

P-wave velocity of 3660 m/ s . For a velocity of 3660 m/s, the arrival

times of the first (2P) , second (4P) , and third (6P) P-waves are
138

137, 274, and 410 ps

,

respectively, and for a velocity of 4020 m/s,

the arrival times are 124, 249, and 374 ,us. Arrows indicating the

arrival of P-waves corresponding to these velocities are superimposed

on the waveform. Notice that the arrival times corresponding to

the velocity of 3660 m/ s have better overall agreement with the

beginning of the sudden downward displacements that characterize

P-wave arrivals.

Similar results indicating a difference between P-wave

velocities obtained using pulse-velocity and resonance measurements

were obtained by others [66] when dynamic modulus of elasticity

values obtained from through-transmis s ion , pulse-velocity tests

were compared to values obtained by resonant frequency testing (ASTM

C 215 [4]). It was found that the elastic modulus values based

on resonance measurements were 8 to 23 percent less than those based

on pulse-velocity measurements. Since P-wave velocity is

proportional to the square root of the elastic modulus, this

difference in modulus values corresponds to a 4 to 12 percent

difference in P-wave velocities obtained using the two methods.

To the author's knowledge there is no verifiable explanation for

this difference.

Thus, for accurate determination of flaw depth, the P-wave

velocity must be determined from an impact-echo test on a part of

the structure being evaluated where the thickness is known. This

velocity can then be used to accurately determine the depth of

internal interfaces in other parts of the structure based on the

peak frequencies obtained from impact-echo tests.
139

7.3 WAVEFORMS AND FREQUENCY SPECTRA OBTAINED FROM A SOLID PLATE

The impact-receiver configuration for tests carried out on

the solid plate (Plate No. 1) is shown in Fig. 4.3(b). For the

results presented in this section, the thickness of the plate (T)

is equal to 0.25 m; thus, the only two variables in this test

configuration are the contact time of the impact (t

c

) and the

distance between the impact point and the receiving transducer (H).

The effect on experimental waveforms and frequency spectra caused

by varying these two parameters are shown in the following sections.

Experimental observations are compared to observations drawn from

the studies of theoretical waveforms and spectra (Chapter 5).

The P-wave speed in the plate was 3660 m/ s . Note that in

the experimental waveforms shown in this chapter, R-wave signals

are clipped so that displacement patterns produced by internal

reflections are accentuated and interpretation of frequency spectra

is simplified (see Section 5.4.3). The frequency spectra shown

in this chapter are arbitarily limited to a range of 0 to 50 kHz;

this was done to clearly show peak frequencies which, for the

specimens tested, are in the range of 0 to 25 kHz.

7.3.1 Effect of Contact Time of the Impact

In Section 5. 3. 2.1, it was shown that as the contact time

of the impact increases, the perturbations in theoretical

displacement waveforms, computed using Green's function solutions,

become smoothed out. For impacts having contact times approaching

the arrival time of the 2P-wave reflected from the bottom of the
140

plate, wave arrival times do not agree with the beginning of the

downward displacements in the waveform. However, it was shown that

in the frequency domain, a short contact time is not essential.

In fact, an impact having a longer contact times produces a higher

amplitude peak in the frequency spectrum at a value corresponding

to the frequency of P-wave arrivals from the bottom surface of the

plate, because more energy is confined in the frequency which has

a wavelength corresponding twice the thickness of the plate (2T

= ^ , see Section 3.2.3). These observations were based on studies

of theoretical displacement waveforms and their frequency spectra.

To verify these observations, experimental results are presented.

Fig. 7.10 shows a series of displacement waveforms and the

corresponding frequency spectra obtained for various contact times.

Contact time was varied by using spheres 6.35, 8, and 16 mm in

diameter (contact times of 34, 48, and 90 ^is). The distance between

the impact point and the receiver was 0.05 m (H/T value of 0.2).

The 2P-wave arrival time was 137 ^is; thus, the contact times

correspond to t

c /t2P values of 0.25, 0.35, and 0.66, respectively.

Calculated wave arrival times are indicated on the waveforms.

For the P-wave speed of 3660 m/s , the 0.25-m thickness of

the plate corresponds to a resonant frequency of 7.06 kHz. For

the 2 jus sampling rate used in capturing experimental waveforms,

the possible values in the frequency spectrum occur at 6.84 and

7.32 kHz. The value of 7.06 kHz falls almost exactly between these

two possible values. Therefore, the peak in spectra obtained for

a thickness of 0.25 m, can occur at either 6.84 or 7.32 kHz.
141

There is a noticeable difference in the experimental waveforms

obtained from the concrete plate as compared to the theoretical

waveforms computed using Green's function solutions (Chapter 5).

In the theoretical solutions, except for the initial upward

displacements caused by the surface waves, the surface of a solid

plate is never displaced above its undisturbed position (e.g., see

Figs. 5.5 and 5.6). However, in experimental waveforms, the large

amplitude R-wave appears to cause an overshoot response of the

transducer, which occurs just after the R-wave passes by the

transducer. This overshoot causes the displacement pattern produced

by internal reflections to be shifted upward.

As the contact time of the impact increases (Figs.

7 .10(a)-(c) ) , the displacement response becomes smoother. For the

longest contact time (Fig .7 .10(c) ; t c^ t 2P va l ue of 0.66) the

calculated wave arrivals times no longer agree with the start of

the downward dips in the waveform; however, the 6.84 kHz peak in

the frequency spectrum can still be used to determine the thickness

of the plate regardless of the contact time. This is because the

periodicity of the major perturbations in the waveform is determined

by the thickness of the plate. As expected, the amplitude of the

peak in the frequency spectrum increases as the contact time

increases. These experimental observations verify those drawn from

the theoretical study.

7.3.2 Test Configuration

In Section 5. 3. 2. 2, it was shown that interpretation of
142

waveforms and frequency spectra is simpler for test configurations

where the distance between the impact point and the receiver is

small relative to the thickness of the plate (small H/T value).

This was because a small H/T value minimizes the complications caused

by large amplitude R-wave signals and those caused by S-wave

reflections. H/T values less than approximately 0.5 were shown

to give interpretable results. These theoretical observations are

compared to experimental results in the following discussion.

Figs. 7.11(a)-(d) show a series of displacement waveforms

and their corresponding frequency spectra for distances of 0.025,

0.05, 0.125, and 0.25 m between the impact point and the receiving

transducer. These distances correspond to H/T values of 0.1, 0.2,

0.5, and 1, respectively. An 8-mm diameter sphere (contact time

approximately 50 ;us) was used for all the test configurations.

Note that 0.025 m is the smallest practical spacing between the

impact point and the receiver because of the 0.02 m radius of the

transducer backing.

Waveforms and frequency spectra obtained for H/T values of

0.1 (Fig. 7.11(a)) and 0.2 (Fig. 7.11(b)) are similar. However,

as the H/T value increases from 0.2 to 0.5 (Fig. 7.11(b) and (c)),

both the displacement waveforms and the frequency spectra become

more complicated. For an H/T value of 1 (Fig. 7.11(d)), the peak

frequency in the spectrum is not as pronounced and its value shifts

from 7.32 to 7.81 kHz; a depth determination based on this higher

value would be incorrect. As discussed in Section 5. 3. 2. 2, this

trend is observed because the amplitude of P-wave reflection:.
143

decreases relative to the amplitude of S-wave reflections.

Therefore, S-wave arrivals have a significant influence on the

displacement response, complicating both the waveform and the

frequency spectrum. In addition, as observed in the theoretical

frequency spectra in Fig. 5.6, the amplitude of the peak in the

frequency spectrum decreases as H increases, because the amplitude

of the displacements caused by the arrival of P-waves decreases

as the receiver gets farther from the impact point.

In summary, experimental results obtained from solid plates

are in agreement with observations drawn from Green's function

solutions for an infinite plate.

7.4 PLANAR DISK-SHAPED FLAWS IN A PLATE

Studies of planar disk-shaped flaws in a 0.5-m thick plate

(plate 3) were carried out. The diameters and locations of these

flaws were shown in Fig. 7.4. The impact-receiver configuration

for tests carried out over the flaws is shown in Fig. 7.12. The

distance between the impact point and the receiver was 0.05 m.

The P-wave speed in the concrete was 3530 m/ s . This speed

was an average speed which was determined from impact-echo tests

carried out over interfaces located 0.13, 0.25, 0.38, and 0.5 m

below the top surface of the plate. Fig. 7.13 shows the known depths

plotted versus the inverse of twice the peak frequencies in the

corresponding spectra. A linear regression analysis was carried

to determine the best fit line through these points. The slope

of this line is the P-wave speed (Eq. 5.8).
144

To determine the reliability and the sensitivity of the

impact-echo technique for locating planar flaws in concrete, the

effects on surface displacement waveforms and their corresponding

frequency spectra of the following test variables were studied:

1) contact time of the impact; 2) flaw diameter; and, 3) flaw depth.

To establish the baseline response of the solid 0.5-m thick

plate for comparison with responses obtained from portions of the

plate containing internal flaws. Fig. 7.14 shows a waveform and

the corresponding frequency spectrum obtained over a solid portion

of the plate for impact produced by a 12.7-mm diameter sphere

(contact time approximately 60 ps ) . The peak amplitude in the

frequency spectrum which corresponds to the frequency of P-wave

reflections from the bottom surface of the plate occurs at a value

of 3.42 kHz.

7.4.1 Effect of Test Variables on Impact-Echo Waveforms

In Section 6.5, the finite element method was used to study

the effects on impact-echo waveforms caused by changing the contact

time of the impact, the diameter and the depth of a flaw, and the

spacing between the impact point and the point where the displacement

was monitored. These studies showed that for a given impact, the

diameter to depth ratio of a flaw (D/T) is a useful parameter for

determining whether or not a flaw will be detected. For flaws at

a given depth, the critical value of flaw diameter increases as

the contact time of the impact increases. If the diameter of a

flaw was sufficiently large, it was found that the depth of a flaw
145

could be determined accurately from time domain analysis of

displacement waveforms if the contact time of the impact was less

than or equal to two-thirds of the arrival time of the 2P-wave

reflected from the surface of the flaw.

In the following sections, the observations drawn from the

finite element analyses are verified by observations drawn from

experimental results. In addition, interpretation of experimentally

obtained waveforms by frequency spectrum analysis is emphasized.

7. 4. 1.1 Contact Time of the Impact

Fig. 7.15 shows a series of displacement waveforms and the

corresponding frequency spectra which were recorded over a 0.2-m

diameter flaw located 0.127 m below the top surface of the concrete

plate. The H/T value for this test configuration was 0.38. The

contact times of the impacts in Figs. 7.15(a)-(c) were varied by

dropping 4-, 8-, and 16 -mm diameter spheres (contact times of 26,

40, and 80 ps) . The arrival time of the 2P-wave reflected from

the flaw was 74 ^s, which resulted in t

c /t 2 p

values of 0.35, 0.54,

and 1.08.

In the spectrum obtained for the shortest contact time (Fig.

7.15(a)), the peak amplitude at 14.2 kHz gives a calculated flaw

depth of 0.124 m (Eq. (5.8). Notice that in Fig. 7.15(a) there

is also a lower amplitude peak at 3.42 kHz which is the frequency

of P-wave reflections from the bottom surface of the plate (see

Fig. 7.14). Recall from the displacement and stress fields obtained

from the finite element analyses (Section 6. 4. 2. 2) that waves
146

diffract around the edges of a small flaw, reform below the flaw,

and are subsequently reflected by the bottom surface of the plate.

Thus, the frequency spectrum would be expected to contain a high

amplitude peak at a frequency value corresponding to the frequency

of wave reflections from the bottom surface of the plate.

As observed in the finite element study, the smoothing effect

on waveforms produced by increasing the contact time is evident

in the waveforms shown in Fig. 7.15(b) and (c). In the waveforms

shown in Figs. 7.15(a) and (b), P-wave arrivals from the surface

of the flaw dominate the displacement pattern.

Notice that in the frequency spectra shown in Fig. 7.15,

the amplitude of the 14.2 kHz peak decreases, and the amplitude

of the 3.42 kHz peak increases as the contact time increases, until

in Fig. 7.13(c) there is only a very low amplitude remnant of a

peak at 14.2 kHz. For the 0.2-m diameter flaw to be detected using

the "one-wavelength" criterion (Section 6.5.1), the stress waves

produced by impact must contain frequencies of 18 kHz or higher

(wavelengths of 0.2 m or less). For the 20. 40, and 80 ^s duration

impacts, the first zero in the frequency spectrum of the input pulse

occurs at 58, 38, and 19 kHz, respectively (see Fig. 5.2(b)). Thus,

the pulse produced by the 80 ^is duration impact contains very little

energy in the frequency range of 18 kHz and higher; therefore, there

is no large amplitude peak at 14.2 kHz - the frequency of P-wave

reflections from the surface of the flaw.

An accurate determination of the depth of the flaw can be

made using the frequency spectra shown in Figs. 7.15(a) and (b).
The flaw would not be detected using the spectrum shown in Fig.

7.15(c). However, the effects on the waveform in Fig. 7.15(c) caused

by diffraction indicate that a flaw or some other irregularity is

present. As shown in waveforms obtained from finite element

analyses, diffraction effects make the waveform appear very different

from a waveform obtained from a solid plate subjected to an impact

having approximately the same contact time. In the waveform in

Fig. 7.15(c), there are higher frequency osillations superimposed

upon the lower frequency oscillations caused by P-wave reflections

from the bottom surface of the plate. In this waveform, the effects

produced by reflections from the bottom surface of the plate and

the effects caused by the presence of the flaw are both clearly

evident.

7. 4. 1.2 Flaw Diameter

Figs. 7.16(a)-(c) show waveforms and their corresponding

frequency spectra obtained over 0.1, 0.2, and 0.5-m diameter flaws

located 0.25 m below the top surface of the plate. These flaw

geometries resulted in D/T values of 0.4, 0.8, and 2, respectively.

Impacts were produced by a 8—mm diameter sphere (contact times of

approximately 40 ^is). The arrival time of the 2P-wave reflection

from the surface of the flaw was 142 ^is, for a t c /t 2 p

value of 0.28.

The waveform and the frequency spectrum obtained from the

solid 0.25-m thick plate subjected to approximately the same duration

impact (t

c /t2p va l ue of 0.25) was shown in Fig. 7.10(b). (The P-wave

speed in this plate was not significantly different than the P-wave
148

speed measured in the 0.5-m thick plate which contained the flaws.

Both plates were made from the same batch of concrete.) This

waveform and spectrum represent the baseline response of a plate

of the same thickness as the depth of the flaws in this study; that

is, it is the response caused by specularly reflected and

mode-converted waves.

The waveform and spectrum obtained from the 0.5-m diameter

flaw (Fig. 7.16(c)) are very similar to those obtained from the

solid plate (Fig. 7.10(b)) because the diameter of the planar flaw

is large relative to the depth of the flaw (D/T equal to 2). Notice

that the displacement waveform in Fig. 7.16(c) exhibits a series

of low amplitude oscillations that are not present in Fig. 7.10(b).

These oscillations are due to the effects caused by waves diffracted

from the edges of the disk-shaped flaw. In the frequency spectrum

these oscillations produce several lower amplitude peaks in the

range of 20 to 25 kHz that were not present in the frequency spectrum

obtained from the solid plate (Fig. 7.10(b)).

In the spectrum obtained from the 0.2-m diameter flaw (Fig.

7.16(b)) there is a single dominant peak in the spectrum at a

frequency which corresponds to the depth of the flaw. However,

the amplitude of this peak is significantly less than the amplitude

of the peak obtained from the 0.5-m diameter flaw, which in turn

is less than the amplitude of the peak in the spectrum obtained

from a solid plate. This indicates that the specular reflections

produced by smaller flaws are lower in amplitude from those produced

by larger flaws or the bottom surface of a plate of the same
149

thickness as the depth of the flaw. The decrease in the amplitude

of specular reflections is directly related to the amount of energy

contained in the frequencies having wavelengths equal to or less

than the diameter of the flaw (one-wavelength criterion). As flaw

diameter decreases, less energy is reflected.

An accurate determination of the depth of both the 0.2 and

0.5-m diameter flaws (D/T values of 0.8 and 2) can be made from

the spectra in Figs. 7.16(b) and (c). This is not the case for

the 0.1-m diameter flaw (D/T value of 0.4).

The amplitude of the peak in the frequency spectrum produced

by specular reflection from the surface of the 0.1-m diameter flaw

(Fig. 7.16(a)) is much less than that obtained from the 0.2-m

diameter flaw. Therefore, the other frequencies, which are also

present in the spectra obtained from the 0.2- amd 0.5-m diameter

flaws, appear much larger relative to the 7.32 kHz peak. For

example, in Fig. 7.16(a), a peak at 9.76 kHz has become larger than

the peak corresponding to the depth of the flaw. Using this

spectrum, the presence of the flaw would be detected, but an accurate

determination of the depth of the flaw could not be made.

Just as in the waveforms obtained from finite element

analyses, it is observed that as the flaw diameter decreases, the

amplitude of specular reflections from the flaw decreases and the

displacements caused by diffracted waves have a more significant

effect on the waveform. Therefore, effects produced by diffracted

waves in displacement waveforms can be used to gain information

about the relative sizes of flaws.
7. 4. 1.3 Flaw Depth

A series of waveforms obtained over flaws located at different

150

depths, and the corresponding frequency spectra, are shown in Fig.

7.17. Fig. 7.17(a) was obtained over a 0.2-m diameter flaw located

0.127 m below the top surface of the plate (D/T value of 1.6).

Figs. 7.17(b) and (c) were obtained over 0.5-m diameter flaws located

0.25 and 0.38 m below the top surface of the plate (D/T values of

1.9 and 1.3, respectively). Fig. 7.17(d) was obtained from a solid

section of the 0.5-m thick plate. The contact time of the impact

was approximately 40 ^us for the 0.127- and 0.25-m deep flaws, and

60 }is for the 0.38-m deep flaw and the 0.5-m thick plate. A longer

duration impact was used in the tests carried out over the deeper

flaw so that the input pulse had greater penetrating ability. These

contact times corresponded to values of 0.5, 0.3, 0.3, and

0.2, respectively.

As expected, the peak in the frequency spectra in Figs.

7.17(a)-(d) shifts to a lower frequency value as the depth of the

reflecting interface increases. Using Eq . (5.8), the frequency

values of 14.2, 7.08, 4.39, and 3.42 kHz correspond to depths of

0.124, 0.25, 0.40, and 0.52 m, respectively. These calculated values

are very close to the depths that were measured prior to casting

the concrete. As depth increases, the 0.488 kHz resolution in the

frequency spectrum (2 ps sampling rate) results in calculated depths

that are not as precise. For example, at a depth of 0.127 m,

adjacent points in the frequency spectrum correspond to a 0.004

m difference in depth; at a depth of 0.5 m, adjacent points in the
151

frequency spectrum correspond to a 0.048 m difference in depth.

Thus, the peak frequency values obtained from the spectra in Fig.

7.17 are the closest frequencies to the values that correspond to

the measured depths of the flaws. When probing thicker elements,

more precise depth measurements can be made by using a slower

sampling rate in the time domain (see Section 5.2).

For the 0.2-m diameter flaw located close to the surface

(Fig. 7.17(a)) two peaks are present at frequencies corresponding

to P-wave reflections from the surface of the flaw (14.2 kHz) and

from the bottom of the plate (3.42 kHz). Compare this spectrum

to that shown in Fig. 7.17(d); the 0.5-m thick plate produces a

single peak at the 3.42 kHz frequency value.

7.4.2 Sensitivity of the Impact-Echo Method for Detecting

Planar Flaws

In the previous section, the effects of various test

parameters on impact-echo waveforms have been shown. In this

section, the results of a frequency domain study of waveforms

obtained over all the various planar flaws in the concrete plate

will be summarized and the sensitivity of the impact-echo method

for detecting planar flaws will be discussed.

The sensitivity study involved determining the depths of

flaws from the peak amplitude in frequency spectra. Flaw diameters

ranged from 0.05 to 0.5 m and were located 0.127 to 0.38 m below

the surface of the plate (see Fig. 7.4). The contact times of the

impacts ranged from 20 to 90 ^is. The spacing between the impact

point and the receiving transducer was 0.05 m in every test.
152

Three criteria were established for categorizing the results

of each impact-echo test. These criteria were as follows:

1) If a flaw could be detected and its depth accurately

determined, then the test result was labeled with a "YES";

2) If a flaw could be detected, but an accurate determination

of its depth could not be made, then the test results was labeled

with a "?"; and,

3) If a flaw could not be detected from a frequency domain

analysis, then the test result was labeled with a "NO".

The results of all tests performed are shown in Fig. 7.18.

Each test is plotted in terms of the t c /t2P value of the impact

and the D/T value corresponding to the flaw geometry and the type

of symbol identifying the results of each test*. The results fall

into three distinct regions.

1) If the D/T value is greater than approximately 0.6 and

the t c

/ t 2P va l ue is approximately less than 0.75, then all the

results fell into the "YES" category - the flaw could be detected

and its depth accurately determined. Examples of results which

fall into this category are the frequency spectra shown in Figs.

7.17(a)-(d).

2) If, however, the D/T value becomes less than 0.6, then

the results fell into the category labeled with a "?"

,

in which

a flaw could still be detected, but an accurate determination of

square corresponds to a YES; a diamond corresponds to a and, an X corresponds to a NO.
its depth could not be made. Fig. 7.16(a) is an example of

frequency spectrum that falls into this category.

a

153

3) If the t c

/ fc 2p value becomes greater than approximately

0.75 for any value of D/T, then the results fell into the "NO"

category - the flaw could not be detected from a frequency domain

analysis. Fig. 7.15(c) is an example of a spectrum that falls into

this category.

These results are valid for the range of D/T values considered

in these tests (D/T values less than or equal to 2).

In concluding this discussion, it should be mentioned that

the depth of a flaw or the bottom surface of a plate can be

accurately determined from the peak in the frequency spectrum even

for tests carried out near the edges of the plate. In the time

domain, displacements due to reflections of P-, S-, and R-waves

from the edges of a plate are superimposed upon the displacements

due to reflections from internal flaws or the bottom surface of

the plate, making a surface displacement waveform difficult to

interpret. However, the frequency domain analysis is a simple

technique for separating effects that are superimposed in the time

domain. Generally, in a frequency spectrum obtained from a waveform

that contains multiple R-wave reflections, the multiple reflections

from an internal flaw or from the bottom surface of the plate will

produce a peak at the frequency corresponding to the depth of the

reflecting interface that is distinct from the frequencies

corresponding to multiple R-wave reflections.
154

7.5 FLAWS IN REINFORCED CONCRETE

The results presented in the previous sections have

demonstrated that the impact-echo technique can be used successfully

to locate flaws within plain concrete. However, before the test

technique could be considered as a possible field-worthy technique,

it also had to be demonstrated that the technique can be used to

detect flaws within reinforced concrete. Tests were carried out

on a 0.2-m thick plate containing solid steel bars of various

diameters (Fig. 7.3), and the effects caused by the bars on

waveforms and frequency spectra were determined. Subsequently,

a 0.5-m thick, reinforced concrete plate was cast in which were

embedded a variety of artificial flaws (Figs. 7.6 and 7.7). Results

obtained from impact-echo tests on this plate were used to determine

whether flaws could be detected under layers of reinforcing steel.

Before presenting experimental results, the wave reflection

from a concrete/steel interface is discussed.

7.5.1 Wave Reflection from a Concrete/Steel Interface

As discussed in Section 2.1.3, both reflection and refraction

occur at an interface between two solids. The portions of the

incident energy that are reflected and refracted depend upon the

angle of the incident wave and the acoustic impedance of the two

solids. Previously, reflection from concrete/air interfaces (stress

free boundaries) has been discussed; for normal incidence, almost

total reflection occurred because the acoustic impedance of air

was negligible compared to that of concrete. However, the acoustic
155

impedance of steel is higher than that of concrete; therefore, the

type of wave reflection that occurs from a concrete/steel interface

is different from the type of reflection that occurs from a

concrete/air interface. This difference is explained qualitatively

in Fig. 7.19. Fig. 7.19(a) shows an infinite concrete plate bounded

by air; Fig. 7.19(b) shows an infinite concrete plate bounded on

the bottom by steel and on the top by air.

First consider the case where impact occurs on a plate with

stress free boundaries (Fig. 7.19(a)). The P-wave generated by

the impact is a compression wave. When this wave is incident upon

the bottom surface of the plate, it is reflected as a tension wave.

This tension wave propagates back up through the plate to be

reflected at the top surface as a compression wave, and the cycle

begins again. Every P-wave arriving at the top of the plate is

a tension wave which causes a downward displacement of the surface.

The period between successive downward dips in the waveform is the

time between successive P-wave arrivals, that is, the time for a

P-wave to travel twice the thickness of the plate. Thus, if a

frequency spectrum were computed, the value of the peak in the

frequency spectrum would correspond to twice the plate thickness

(Eq. (5.8)).

Next, consider impact on the top surface of a concrete plate

which is in contact with a thick layer of steel at its bottom surface

(Fig. 7.19(b)). When the initial compression wave generated by

the impact is incident upon the concrete/steel interface it does

not change sign; that is, it is reflected as a compression wave.
156

When this compression wave reaches the top surface of the plate

it causes an upward displacement of the surface. However, since

the top surface of the plate is a stress free boundary, the incident

compression wave is reflected as a tension wave. This wave

propagates down through the plate and is reflected at the

concrete/steel interface unchanged. When this tension wave reaches

the top surface of the plate it causes a downward displacement of

the surface. The incident tension wave is reflected as a compression

wave and the cycle begins again. Therefore, in the surface

displacement waveform (Fig. 7.19(b)), the periodicity of

displacements (time between successive upward or downward

displacements) is twice as long as in waveform obtained from the

plate with free boundaries (Fig. 7.19(a)). This explanation was

verified by a finite element analysis of a layered (concrete/steel)

plate

.

The period in the waveform shown in Fig. 7.19(b) is equal

to the time it takes for a wave to travel four thicknesses of the

plate. Therefore, the value of the peak in the frequency spectrum

obtained by tranforming this waveform would correspond to four plate

thicknesses; the peak frequency value will be half that obtained

from the waveform shown in Fig. 7.19(a).

Based on this qualitative analysis, the peak amplitude

produced by reflections from steel bars would be expected to occur

at a frequency value that is approximately one-half the value that

would be obtained from a concrete/air interface. Therefore, to

determine the depth of a concrete/steel interface from the peak
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a cylindrical inclusion, the waves will also bend around the

inclusion; this phenomenon is similar to the process that occurred

for a wave incident upon a planar flaw. Thus, the displacement

and stress fields produced by transient waves incident upon a steel

bar embedded in a concrete plate are more complicated than those

produced by the interaction of transient waves with a planar

disk-shaped flaw (concrete/air interface).

To show the effect of steel bars on waveforms and frequency

spectra, selected results obtained from a 0.2-m thick plate in which

were embedded various diameter steel bars (Fig. 7.3) are discussed

in the following section.

7.5.2 Waveforms and Frequency Spectra Obtained from a Concrete

Plate Containing Steel Bars

Impact-echo tests were carried out over 25.4-, 38. 1-, and

44.5-mm diameter bars. Fig. 7.20 shows waveforms and frequency

spectra generated by impacts having contact times of approximately

20 ^us, and Fig. 7.21 shows waveforms and spectra generated by impact

having contact times of approximately 54 ^is. The distance from

the top surface of the plate to the top of the bars was 0.06 m.

The arrival time of the 2P-wave reflected from the top of the bar

was 29 ^s.

The distance between the impact point and the receiving

transducer was 0.05 m. Both the impact point and the receiver were

located over the centerline of the bar. The peak frequency obtained

from an impact-echo test over a solid portion of the 0.2-m thick

plate was 10.25 KHz, which results in a computed P-wave speed of
4100 m/s. Using Eq. (7.1) a the frequency corresponding to P-wave

reflections from the top of the bars would be expected to be about

159

17 KHz.

In the waveforms shown in Fig. 7.20, the effect of the bars

appears as higher frequency oscillations superimposed on the lower

frequency oscillations caused by waves reflected from the bottom

surface of the plate. In the frequency spectra, it is seen that

the reflections from the bars give rise to multiple peaks. These

frequency peaks occur at values ranging from 17 to 23 kHz.

Reflections from the bottom of the plate produce a frequency peak

at a value slightly less than the 10.25 kHz obtained from a solid

part of the plate. In Fig. 7.20(c) (4.45-cm bar), there are a group

of lower frequency peaks. These peaks could be produced by the

interaction of P-and S-waves with a larger diameter bar. For small

diameter bars, P-wave reflections dominate the waveform. For a

larger diameter bar, the S-waves that are incident upon the bar

are larger in amplitude (see Fig. 4.2); therefore, diffracted and

reflected S-waves have a more significant effect on the displacement

waveform and thus the frequency spectrum.

Notice that in Fig. 7.20, the amplitudes of the peak (or

peaks) corresponding to the frequency of waves reflected from the

bottom surface of the plate decreases and the amplitudes of the

peaks corresponding to the frequency of waves reflected by the bar

increases with increasing bar diameter. The group of peaks produced

by the bar changes from being a secondary effect in Fig. 7.20(a),

to being the dominant effect in Fig. 7.20(c), because the portion
160

of the incident energy that is reflected by the bar increases as

the bar diameter increases. A plot of the amplitude versus the

wavelength of each component wavelength in the pulse produced by

a 20 ^us duration impact is shown in Fig. 7.22(a). It is seen that

the pulse contains sufficient energy in the range of frequencies

having wavelengths that are approximately equal to or smaller than

the diameter of all three bars.

Notice that in Fig. 7.21, the displacement waveforms are

characterized by the broad, large amplitude oscillations which are

produced by a longer duration impact. Waves produced by 54-^s

duration impacts contain a much narrower range of frequencies. As

shown in Fig. 7.22(b), there is little energy in the range of

frequencies having wavelengths that are approximately equal to or

smaller than the diameters of the bars. Therefore in the frequency

spectrum shown in Fig. 7.21, the amplitudes of the frequencies

associated with reflections from the bars are very small.

In Fig. 7.21(c), the effects produced by the bar are more

apparent because of the multiple peaks that appear which correspond

to frequencies of waves reflected from the bottom of the plate.

If the bar were not present there would only be a single peak.

This phenomenon may be useful as an indication of bar size.

These results show that by increasing the contact time of

the impact, the effects produced by bars on waveforms and frequency

spectra can be significantly reduced. This is important because

the primary purpose of impact-echo testing will be to detect flaws

located beneath reinforcing bars, not to locate the bars themselves.
161

Conversely! if one wishes to interrogate for the existence of bars,

then short duration impacts must be used. Note that using longer

contact times to reduce the effects caused by bars, limits the size

of the flaws that can be detected within reinforced concrete; flaws

on the order of the size of the bars will not be detected. However,

flaws of this size are usually not a concern in concrete stuctures.

In the following section waveforms and the corresponding

frequency spectra generated by impacts on a reinforced concrete

plate containing planar disk-shaped flaws are presented. Longer

duration impacts are used to reduce the effects caused by the steel

reinforcing bars on measured signals.

7.5.3 Planar Disk-Shaped Flaws in Reinforced Concrete

Results discussed in this section were obtained from the

0.5-m thick concrete plate shown in Figs. 7.6 and 7.7. Steel

reinforcing bars were placed over one-half of the plate, parallel

to both sides of the plate, and spaced 20 cm on center. Bars, 22.2

mm in diameter, were placed 73 mm below the top surface of the plate;

on top of these bars was placed a layer of 25.4-mm diameter bars.

The depth of the 25.4 mm bars was 48 mm.

The P-wave speed was obtained from an impact-echo test over

the 0.5-m thickness of the plate, using a sampling rate of 5 ^us.

The peak frequency was 4.10 kHz and the computed P-wave speed was

4100 m/s (Eq. 5.8). To establish the baseline response of the plate.

Fig. 7.23 shows a waveform and frequency spectrum obtained over

a solid portion of the 0.5-m thick plate. The sampling rate in
162

this case was 2 yus, as will be the case for all of the other results

that will be presented in this and the following section. For a

2 yus sampling rate, the peak amplitude occurs at a frequency of

3.91 kHz.

Figs. 7.24 and 7.25 show waveforms and frequency spectra

obtained over a 0.35-m diameter flaw which was located 0.26 m below

the top surface of the plate (H/T value of 0.19). The 5.5 g impactor

was used as the impact source (contact times approximately 70 yjs).

The arrival time of the 2P-wave reflected from the surface of the

flaw was 125 yus. In part (a) of each figure, the test configuration

is shown; that is, the locations of the impact point and the

receiving transducer relative to the reinforcing bars and to the

flaw are shown.

As shown in Fig. 7.24(a), the receiver was located over a

25.4 bar and the impact point was 0.05 m away from the centerline

of the bar. In the spectrum shown in Fig. 7.24(b), there is a peak

at 2.44 kHz which corresponds to the frequency of P-waves diffracted

around the flaw and reflected from the bottom of the plate, and

a peak at 7.81 kHz which is the frequency of P-waves reflections

from the surface of the 0.26-m deep flaw (Eq. (5.8)). There are

also minor peaks at 14.6 and 21 kHz. These are the frequencies

of reflections from the top of the bars. Using Eq . (7.1), these

frequencies give depths of 71 and 49 mm, respectively; these depths

are in good agreement with the known depths of 73 and 48 mm.

The test configuration in Fig. 7.25(a) was similar to that

in Fig. 7.24(a), except that the impact point was also located over
163

a 25.4-mm diameter reinforcing bar. The frequency spectrum obtained

for this test configuration is similar to that shown in Fig. 7.24(b),

except that the secondary effects caused by the presence of the

bars are different; the frequencies near 21 kHz are larger than

the 14.6 kHz peak.

The key point is that the effects caused by the reinforcing

steel are secondary and the disk-shaped flaw embedded in the

reinforced concrete plate was easily detected. Similar results

were obtained for the 0.205-m diameter flaws which were located

0.26 and 0.13 m below the top surface of the reinforced concrete

plate (see Fig. 7.6). In the following section results obtained

from a part of the reinforced concrete plate that contained a planar

crack-like flaw are discussed.

7.5.4 A Simulated Crack in Plain and Reinforced Concrete

Wave reflection from an actual crack in concrete is more

complicated than the reflection that occurs from planar disk-shaped

flaws. For the case of a disk-shaped flaw, the reflecting interface

was a relatively smooth concrete/air interface, which reflects almost

100 percent of an incident wave's energy. The surfaces of a crack

are rough and may be partially in contact with each other because

of interlocking of aggregate particles. If contact between crack

surfaces exists or is caused by the displacements produced by

incident waves, then only a portion of the energy will be reflected;

the remainder will be transmitted into the underlying concrete.

To simulate a crack, a 0.075-mm thick plastic sheet was
164

embedded in the plain and reinforced sections of the concrete plate

(see Fig. 7.6). The sheet was 0.3 by 1 m in dimension. It was

placed on the fresh concrete during casting of the plate after the

concrete had been placed in the form to a level of approximately

0.18 m (0.32 m from the top of the form). Concrete was then placed

on top of the sheet and casting of the plate was completed. (The

depth of the crack was not known as precisely as for the planar

disk-shaped flaws which were fixed in place prior to placement of

the concrete.) The plastic conformed to the surfaces of the concrete

between which it was placed.

Figs. 7.26 and 7.27 show results obtained from the simulated

crack in a plain section of the plate and in a reinforced section

of the plate. Fig. 7.26(a) shows the test configuration for an

impact-echo test carried out over the crack-like flaw in plain

concrete; Fig. 7.26(b) shows the waveform and frequency spectrum

that were obtained. The impact source was the 20-g impactor (contact

time approximately 80 ps ) . A single peak occurs in the frequency

spectrum at a value of 6.84 kHz. This frequency value corresponds

to a depth of 0.30 m, which is close to the approximate depth of

the crack that was measured during casting of the plate. There

is also a lower amplitude peak at a frequency value of 3.42 kHz.

This peak is related to the frequency of P-waves diffracted around

the edges of the crack and reflected from the bottom surface of -

the plate. The fact that this lower frequency peak is shifted to

a value less than the 3.91 kHz value corresponding to P-vave

reflected directly from the bottom surface of a solid plate (Fig.
7.23), indicated that waves were diffracted around the edges of

the crack and not transmitted across the crack.

165

Fig. 7.27(a) shows the test configuration for an impact-echo

test carried out over the crack-like flaw in reinforced concrete;

Fig. 7.27(b) shows the waveform and frequency spectrum that were

obtained. As in the previous case, the impact source was the 20-g

impactor (contact time approximately 80 us). The highest amplitude

peak in Fig. 7.27(b) occurs at a value of 7.32 kHz instead of 6.84

kHz as in Fig. 7.26(b). A frequency of 7.32 kHz corresponds to

a depth of 0.28 cm which is still close to the approximate depth

of the crack measured during placement of the concrete. The lower

frequency peak occurs at 3.91 kHz, the frequency of P-waves reflected

from the bottom surface of the plate. There are also additional

peaks that occur at values slightly higher than the 3.91 and 7.32

kHz peaks. These multiple peaks are thought to be caused by the

presence of the reinforcing bars, as they are similar to the multiple

peaks that were observed in the frequency spectra obtained from

the plate containing bars (Figs. 7.20 and 7.21).

This result is encouraging as it appears to show that the

impact-echo method will be able to locate cracks in plain and

reinforced concrete. To extend the scope of the flaw studies, prior

to field testing the method, other types of flaws were simulated.

The results of these studies are discussed in the following section.

7.6 STUDIES OF OTHER TYPES OF FLAWS

In the 0.5-m thick concrete plates, several other types of
166

flaws that may be encountered in concrete structures were simulated.

Results are presented for a simulated, vertical surface opening

crack, a disk-shaped flaw inclined at an angle to the surface, and

a thin metal duct, partially filled with mortar grout.

7.6.1 Depth of a Surface Opening Crack

To simulate a vertical crack in concrete, a 0.075-mm thick

plastic sheet was hung between two vertical metal supports which

were spaced 0.4 m apart. The supports were fixed to the bottom

of the form prior to casting the 0.5-m thick concrete plate (Fig.

7.7). The plastic sheet was 0.21 m deep. The objective was to

locate the depth of a visible surface opening crack.

The impact point was located on one side of the crack and

the receiving transducer was placed on the opposite side. Both

the impact point and the receiver were located 0.06 m away from

the crack. The test configuration is shown in Fig. 7.28(a). The

impact source was the 5.5-g impactor (contact time approximately

72 us).

Diffraction of waves incident upon the bottom edge of the

crack causes the edge to act as a source of cylindrical waves.

Fig. 7.29(a) shows a P-wave incident upon the bottom edge of a

vertical surface opening crack. Fig. 7.29(b) shows the diffracted

P- and S-waves that are produced by the incident P-wave. The region

behind the crack is a shadow zone - only diffracted waves can

penetrate this zone. These diffracted waves are the first waves

to arrive at the receiving transducer. Reflection of these
diffracted waves occurs at the top surface of the plate. As these

reflected waves propagate down through the plate they are incident

upon the bottom edge of the crack giving rise to new diffracted

167

waves. Thus, the cycle of diffraction at the bottom edge of the

crack and reflection at the top surface of the plate begins again;

this cycle gives rise to a resonance condition with a frequency

of wave arrivals at the receiver which corresponds approximately

to twice the depth of the crack. In addition, diffracted waves

are reflected by the bottom surface of the plate, as are the direct

P- and S-waves that propagate past the crack. As these reflected

waves propagate back up through the plate, they are incident upon

the bottom edge of the crack giving rise to additional diffracted

waves. Therefore there are additional resonances that are set-up

between the bottom surface of the plate and the bottom edge of the

crack, the top and bottom surfaces of the plate. Thus, in a

frequency spectrum, it would be expected to that these different

resonances would give rise to multiple high amplitude peaks.

The waveforms and frequency spectrum obtained from the

impact-echo test are shown in Fig. 7.28(b). As expected, the

spectrum looks different from those presented in previous sections.

There are three closely spaced peaks at 3.42, 4.88, and 6.35 kHz

and a lower amplitude peak at 9.27 kHz. Each of these peaks

corresponds to various periodicities of waves reflected between

the surfaces of the plate and the bottom edge of the crack.

The peak occuring at the highest frequency value (9.27 kHz

in this case) is the frequency that the cycle of P-wave reflection
168

from the top surface of the plate and diffraction from the bottom

edge of the crack occurs. For a relatively close spacing between

the impact point and the receiver, this frequency corresponds

approximately to twice the depth of the crack (Eq. (5.8)). For

this example, a frequency of 9.27 kHz corresponds to a depth of

0.22 m, which is close to the measured depth of 0.21 m.

7.6.2 Flaw Oriented at an Angle to the Surfaces of a Plate

A 0.5-m diameter disk-shaped flaw was embedded in the 0.5-m

thick plain concrete plate at an angle of 30 degrees to the top

and bottom surfaces of the plate (see Figs. 7.4 and 7.5). Fig. 7.30

shows an elevation view of the cross-section of the plate containing

the inclined flaw. The location of the disk prior to placement

of the concrete is shown by the dashed line. During concrete

placement, the disk broke, creating a discontinuity in the surface

of the flaw. Thus the inclined flaw actually consisted of two

discontinous pieces.

Impact-echo tests were carried out at points A through E

shown in Fig. 7.30; these points were spaced 0.1 tn apart on the

surface of the plate over the centerline of the flaw. The impact

sources for the tests were 8- or 9.5-mm diameter spheres (contact

times of 44 to 52 ^s). The receiver was located at points A through

E and the impact points was located 0.03 m to the right of the

receiver, as shown in Fig. 7.30.

Fig. 7.31 shows the waveform and the corresponding frequency

spectrum obtained at each test point. The frequency value of the
169

peak amplitude in each spectrum was used to calculate the depth

of the reflecting interface at each point (Eq. 5.8). The fact that

the frequency peak shifts to higher and higher values in Figs.

7.31(a) through (e) indicates that the flaw is not parallel to the

top surface. Therefore, the calculated depths of the reflecting

interfaces are not vertical depths, but are perpendicular distances

from the impact point to the plane of the flaw. At each test point,

an arc with a radius equal to the calculated depth at that point

is drawn. The location of the interface is found by drawing a

tangent the arcs, as shown in Fig. 7.30. Note that at point C there

are two peaks in the frequency spectrum (Fig. 7.31(c)); each of

these peaks was used to calculate two radii to be drawn from point

C. The arc calculated from the frequency of 7.81 kHz was consistent

with the tangent drawn through points D and E, while the arc

calculated from the frequency of 6.84 kHz was consistent with the

tangent drawn through points A and B. Therefore, it appears that

the disk broke near point C.

In this study, tests were also carried out for the impact

point located to the left of the receiver and adjacent to the

receiver on a line perpendicular to the cross-section shown in

Fig. 7.30. In some cases, the value of the peak frequency shifts

to a slightly lower value; this is expected since the point of ray

reflection is at a lower point on the flaw. In all cases, the

reflecting interface was easily detected and it could be determined

that the interface was inclined.
170

7.6.3 Grouted and Ungrouted Metal Duct

In post-tensioned concrete structures, a common concern is

whether the ducts containing the post-tensioning strands have been

fully filled with mortar grout after the post-tensioning operation

is completed. A nondestructive test method is needed to determine

the extend of grout penetration in these ducts. As a first look

at this problem, a thin galvanized sheet metal duct, 0.1 m in

diameter and 1.52 m long, was partially filled over a length of

0.58 m with mortar grout. After the mortar hardened, the partially

filled duct was embedded in the 0.5-m thick reinforced concrete

plate. The distance from the top surface of the plate to the top

of the duct was 0.15 m (Fig. 7.6 and 7.7).

A brief discussion of the nature of wave reflection from

a thin steel interface between concrete and air and between concrete

and mortar is followed by a discussion of the experimental results.

When an interface is very thin (an interface is considered

thin when its thickness is small compared to the component

wavelengths in the propagating wave [43]), as is the case for the

0.5-mm thick steel duct, then the nature of wave reflection is

different than that discussed in Section 7.5.1. Consider a concrete

plate bonded to a thin steel layer which has a stress-free boundary

at its lower surface (a hollow duct). Since the stresses are zero

at the steel/air interface, and the steel is thin, then the stresses

are also approximately zero at the concrete/steel interface. Thus

the presence of the thin steel layer will not affect the propagating
171

wave, and wave reflection occurs as from an air interface; incident

waves are reflected at the air interface as waves of opposite sign.

If the thin layer of greater acoustic impedance (rigidity)

is between two solids of like acoustic impedances (steel duct between

concrete and mortar in the case of a fully grouted duct) , then the

thin layer is in effect transparent to the incident waves. The

interface does not interfere with stress wave transmission [43];

thus, the waves propagate directly through the fully grouted duct.

Note that the thin sheet of plastic placed in the concrete reflected

incident waves. Although the layer was thin, it was composed of

a material that had a lower acoustic impedance (lower rigidity)

than the concrete on either side of it; therefore, the layer of

plastic was not transparent to the propagating waves.

Figs. 7.32 and 7.33 show waveforms and frequency spectra

obtained by tranforming waveforms recorded over the grouted and

hollow sections of the duct. The results shown in Fig. 7.32 were

generated by the 20-g impactor (contact times approximately 70 ^s).

The results shown in Fig. 7.33 were generated by a 7-mm diameter

sphere (contact times approximately 44 ^lis). Both the impact point

and the receiver were located over the centerline of the duct. The

distance between the impact point and the receiver was 0.05 m.

The waves generated by the 70-^is duration impact contained

very little energy in the frequency range that would be reflected

by the duct. However, effects caused by the presence of the duct

can be seen in the frequency spectra. In the spectrum obtained

from the hollow duct (Fig. 7.32(b)), there is a single peak at 3.42
172

kHz; this value is related to the frequency of P-waves diffracted

around the hollow duct and reflected from the bottom of the plate.

If Figs. 7.32(a) and (b) are compared, two differences are apparent:

1) the relative amplitude of the 3.42 kHz peak as compared to the

other frequencies that are present in the spectrum in Fig. 7.32(b)

is much higher than the relative amplitude of the 3.92 kHz peak

as compared to the other frequencies that are present in the spectrum

in Fig. 7.32(a). In addition, the amplitude of the 3.42 kHz peak

is approximately three times greater than the amplitude of the peak

in the spectrum obtained from the grouted duct. These differences

in amplitude indicate that losses occur as the waves propagate

through the grouted duct. Losses probably result from the fact

that the actual test situation - thin steel layer between mortar

and the concrete - is not an ideal case of a thin, higher acoustic

impedance layer between two materials with the same impedance.

The acoustic impedance of the mortar was somewhat less than that

of the concrete; thus, wave traveling through the duct also undergo

multiple reflections within the duct, scattering the wave energy.

2) The highest amplitude occurs at a frequency of 3.91 kHz, the

value obtained from a solid plate, rather than 3.42 kHz. This

indicates that the waves traveled directly through the grouted duct

rather than around the duct as was the case for a hollow duct.

The 44-^is duration impact generated waves containing a broader

range of frequencies than the 70-yas duration impact; thus, the

effects caused by waves reflected from the surface of the duct could

be studied. In the spectrum obtained from the hollow duct (Fig.
173

7.33(b)) , in addition to the 3.42 kHz peak produced by waves

reflected from the bottom surface of the plate, there is a high

amplitude peak at 13.7 kHz which corresponds to a depth of 0.15

m (Eq. (5.8)), the measured distance to the top of the duct.

However, in the spectrum obtained from the grouted duct (Fig.

7.33(a)), the only peak occurs at 3.91 kHz. Because of the

relatively low amplitude of this peak, the frequencies produced

by the R-wave (Chapter 5) are apparent in the spectrum. Thus, the

hollow duct acts like a cylindrical air void and produces high

amplitude reflections, while the grouted duct cannot be detected

because the thin interface is transparent to the propagating waves.

Thus, it appears possible to distinguish ducts that are hollow

or only partially grouted (air voids exist) from ducts that are

fully grouted using frequency spectra obtained from impact-echo

testing carried out directly over the ducts. For impacts having

longer contact times, it seems that the absence of the grout can

be inferred from the difference in the relative amplitudes of the

peaks that are related to the plate thickness. For impacts having

shorter contact times, the absence of the grout can be inferred

by a peak at a frequency corresponding to the depth of the duct.

7.7 SUMMARY and CONCLUSIONS

This chapter has described the development and implementation

of an impact-echo technique for flaw detection in plain and

reinforced concrete.

To confirm the conclusions drawn from the studies of
174

theoretical displacement waveforms and their corresponding frequency

spectra (Chapters 4 and 5), experimental studies on solid plates

were carried out. The effects caused by changing the test

configuration (impact source - receiver spacing) and by varying

the contact time of the impact on waveforms and frequency spectra

were studied. To determine the thickness of plate elements, it

it was found that it was necessary to keep the distance between

the impact point and the receiving transducer less than approximately

0.5 times the thickness of the plate. The contact time of the impact

is of minor importance as long as the stress pulse produced by the

impact contains sufficient energy in the range of frequencies that

have wavelengths corresponding to twice the plate thickness. These

results were as expected based on the results presented in Chapters

4 and 5

.

To confirm conclusions drawn from time domain studies of

plates containing planar flaws (Chapter 6), experimental studies

were carried out on a plate containing planar disk-shaped flaws.

The effects on waveforms caused by varying the contact time of the

impact, and the diameter and depth of planar, disk-shaped flaws

verified observations drawn from the finite element analyses

presented in Chapter 6. In the experimental studies, frequency

spectrum analysis was emphasized, and the effects of the

above-mentioned variables on frequency spectra were also discussed.

From frequency spectrum analysis, it was found that a flaw will

be detected and accurately located using impacts having contact

times ranging from 20 to 80 ^is and D/T values less than or equal
175

to 2 if the following two conditions are satisfied: 1) the ratio

of the diameter of the flaw to the depth of the flaw (D/T) must

be greater than approximately 0.6; and, 2) the ratio of the contact

time of the impact to the arrival time of the 2P-wave reflected

from the surface of the flaw (t

c /t2 p) must be less than approximately

0.75.

Wave reflection from a concrete/steel interface was discussed

and the effects on waveforms and frequency spectrum produced by

steel bars embedded in concrete were determined. It was shown that

flaws within reinforced concrete could be detected by using longer

contact times to minimize the effects caused by the reinforcing

bars

.

The results of several other types of flaw studies were

presented. It was shown that the depths of simulated horizontal

cracks in plain and reinforced concrete, the depth of a vertical

surface opening crack, and the orientation of a flaw placed at an

angle to the surfaces of a plate could be determined, and a hollow

metal duct could be distinguished from a fully grouted duct.

In all of the studies presented in this chapter, frequency

domain analysis of displacement waveforms was emphasized. Both

time domain and frequency domain analyses require knowledge of the

P-wave velocity in a test object. It was shown that P-wave velocity

is affected by the measurement method. Velocities determined using

the through- transmis s ion , ultrasonic pulse-velocity method were higher than velocities determined using the impact-echo method.

Thus velocities obtained from an impact-echo test on a part of a
test object of known thickness must be used for accurate

interpretation of impact-echo waveforms and frequency spectra.

176

The results presented in this chapter show that the

impact-echo technique works very well in laboratory studies. The

successes that have been achieved in the laboratory strongly suggest

that the method has the potential to become a practical field

technique for detecting flaws within concrete structures.
177

7.1 Impac tor' with spring driven mass.

Output

to preamp

4

7.2 NBS conical displacement transducer
178

(a)

(b)

7.3

view;

_L

0.06m

T

0.15 | 0.2 0.2 0.2 0.15 0.1

* Diameter of steel bar (cm)

t © • • • • .2m

J_

4.45* 2.54 3.18 3.81 1.91

0.2-m thick concrete plate containing steel bars: a) elevation

and, b) plan view.
175

(b)

t 0.5

1

7.4 0.5-m thick concrete plate containing polyurethane disks: a)

plan view; and, b) elevation view.

7.5 Formwork for 0.5-m thick concrete plate with disks in place

prior to placement of concrete.
180

2.2 5m

0.21

0.10 o

0.16

Inclined

at 46*

0.30*

-

0.30 8

)U26 — r) r

0.26 “iM 2

2.0m

0.2m

* Numbers Indicate depth of flaw

2.22cm bars 2.54cm bars

7.6 Plan view of 0.5-m thick reinforced concrete plate containing

artificial flaws.

7.7 Formwork for 0.5-m thick reinforced concrete slab with flaws

in place prior to placement of concrete.
181

7.8 Test set-up

representation

.

a) experimental set-up and , b ) schema tic
VOLTAGE

182

6

7.9 Surface displacement waveform obtained from 0.25-m thick

concrete plate showing computed wave arrival times.
7.10 Waveforms and frequency spectra obtained from the 0.25-m thic

concrete plate using various duration impacts: a) 34 ps; b) 48 ^is

and, c) 90 ^is.
DEPTH

(m)

0.05 m

165

Source

: w .

£ n

Receiver

t - 4 ,

• %

• Polyurethane

L disk »;?

E

io

o

7.12 Test configuration for tests carried out over planar

disk-shaped flaws.

0.6

0.5 -

0.4 -

0.3 -

0.2 -

0

0

l 0.02 —i

1 1 1 1 1 1 1 1 1 1 r 0.04 0.06 0.08 0.1 0.12 0.14

1/(2*FREQUENCY) x 1000 (»)

0.1 6

7.13 Relationship between depth and frequency.
187

o.oi

MJO3

0

TIME (pa)

500

i

FREQUENCY (KHi)

7.15 Waveforms and frequency spectra obtained over a 0.2-m diameter

disk-shaped flaw located 0.127 m below the top surface of the 0.5-m

thick plate. Duration of the impacts was: a) 26 us; b) 40 us: and.

c) 80 fis . r r
189

7.17 Waveforms and frequency spectra obtained from flaws located

at various depths in the 0.5-m thick concrete plate: a) 0.127 m; b) 0.258 m; c) 0.38 m; and d) bottom surface of the plate (0.5 m)

.
191

EFFECT OF INTERFACE

Air i> 2P 4P 6 P

* * • •

«

• t

- i

%

.

* * • • •

• .

• *. • . . *

Concrete’

,

i

1

1 lU

I

l

1 U-J

! : -

•••

1. .

* • *

1

‘ •

1 1• •

• % Air

Air v 2

>

• 0 0

. ' •

• • * •

Concrete •*. . . st*«i xxxx>o<><x><Xxx

| Compression

t Tension

i

(b)

7.19 Wave reflection from concrete/air

interfaces

.

and c one r e t e / s t ee

1
193

f

. 2.54 cm

1

{

/|

3.81 cm

1

\

A 4.45 cm

I 1 I 1 0 1000 0 50

TIME (ps) FREQUENCY (KHz)

7.21 Waveforms and frequency spectra obtained from different

diameter bars located 0.06 m below the top surface of the 0.2-m

thick concrete plate; contact time of the impact was 54 pa: a) 2. 54

cm; b) 3.81 cm; and, c) 4.45 cm.
195

(b)

7.23 Waveform and frequency spectrum obtained from a plain section

of 0.5-m thick concrete plate containing a variety of flaws and

reinforcing bars.
19 ?

(a)

Impact

4.8cm

* kL.

&——nr-—

o

-

* j 2.22cm * * a' •

• ^ * »

* —i i » ^

. h-3 5cm H ••*

6 .

. k

7.3cm

l

f

T 26cm

_L

(b)

7.25 Response obtained over the 0.35 -td diameter flaw shown in

Fig. 7.24 for a different test configuration: a) test configuration;

and, b) waveform and frequency spectrum.
199

(a)

(b)

7.27 Response obtained over a simulated crack in reinforced

concrete: a) test configuration; and, b) waveform and frequency

spec trum.
7.29 Schematic illustration of interaction of waves with vertica

surface opening crack.
>

5,86

\

6.35

L

6.84

J 1 i

M

7,81

ill

A 9.28

TIME (ps) FREQUENCY (KHz)

7.31 Waveforms and frequency spectra obtained at various poin

over the inclined flaw shown in Fig. 7.30.
VOLTS

VOLTS

205

7.33 Responses obtained over duct for a contact time of 25 ^ia: a)

waveform and frequency spectrum for hollow duct; and, b) waveform

and frequency spectrum for grouted duct.
207

propagation in plates containing flat-bottom holes and planar

disk-shaped flaws. The analyses presented in this report were

limited to axisymmetric problems. The capability of the finite

element code to accurately model displacements caused by the

diffraction effects produced by the interaction of transient waves

with a planar flaw was verified by comparison to an experimentally

obtained surface displacement waveform. Parameter studies were

carried out to determine the effects of test variables, such as

flaw geometry and the contact time of the impact, on surface

displacement waveforms.

The development of an experimental impact-echo technique

for finding flaws within plain and reinforced concrete was

described. The key features of the technique are:

1)

a point source - impact of steel sphere or mechanical

impactor on concrete;

2) a point receiver - a broadband, displacement transducer,

with a small conical tip; and,

3) frequency spectrum analysis of displacement waveforms.

Laboratory results obtained from impact-echo tests on solid

plates and plates containing flaws were presented. Observations

and conclusions drawn from the analytical and finite element studies

were used to aid in interpreting experimental results.

8.2 CONCLUSIONS

The significant conclusions reached as a result of the studies

presented in this report are stated in the following sections.
209

of surface displacement waveforms obtained from plates containing

disk-shaped flaws:

1) The larger the ratio of the diameter of the flaw to

the depth of the flaw (D/T), the easier it is to detect

the flaw.

2) The critical D/T value increases as the contact time

of the impact increases.

3) As was found for solid plates, effects produced by

a flaw on surface displacement waveforms are most pronounced

when the waveforms are generated using impacts having

shorter contact times and are recorded close to the point

of impact; these waveforms are the simplest to interpret.

The numerical flaw studies presented in this report

demonstrate that the finite element method is a powerful tool for

studying stress wave propagation within solids having arbitrary

shapes, boundary conditions, and applied loads, and containing

defects

.

8.2.1 Laboratory Studies

To accurately interpret waveforms and frequency spectra,

the P-wave velocity must be determined from an impact-echo test

on a part of the test object of known thickness. P-wave velocities

obtained using the ultrasonic, through-transmission method are approximately 10 percent higher than those calculated from

impact-echo results.

Frequency spectrum analysis simplifies signal interpretation.
211

8.3 FUTURE RESEARCH

The availability of two- and three-dimensional dynamic finite

element computer codes, such as the DYNA code used in this report,

and the availability of supercomputers, now make it possible to

study a wide variety of nondestructive testing related problems

which could not be studied using other existing analytical or

numerical methods . The two-dimensional studies of planar flaws

in plates that were presented in this report are the first of an

almost unlimited number of stress wave propagation problems that

could be studied using the finite element method. For example,

the standard flat-bottom hole specimens that are used routinely

in ultrasonic flaw calibration studies could be analyzed. A

classical three dimensional problem that could be solved is the

interaction of transient waves with a cylindrical inclusion.

The studies in this report have been restricted to plate-like

structures. Before the field potential of the method can be

realized, analytical and experimental studies of other structural

elements, such as rectangular beams and round columns must be carried

out. In addition, the ability of the impact-echo method to detect

other types of flaws needs to be investigated. For example, a common

type of flaw in concrete structures is "honeycombing" - concrete

that is not fully consolidated.

Field testing of the proposed impact-echo method is

essential. A practical and efficient field system must be assembled

and evaluated under actual field conditions. The key component

of this field system will be the development of a rugged displacement
BIBLIOGRAPHY

1. Abduljabbar, Z., Datta, S., and Shah, A., "Diffraction of

Horizontally Polarized Shear Waves by Normal Edge Cracks in a Plate,"

Journal of Applied Physics, Vol. 54, No. 2, February, 1983, pp

.

461-472.

2. Achenbach, J., Gautesen, A., and McMaken, H., "Diffraction of

Elastic Waves by Cracks - Analytical Results," Elastic Waves and

Non-Destructive Testing of Materials, ASME, AMD, Vol. 29, Y.H. Pao,

ed . , 1978, pp. 33-52.

3. Alexander, A. M., "Feasibility of Sonic Pulse-Echo Technique,"

Misc. Paper C-77-11, U.S. Army Engineer Waterway Experiment Station,

Vicksburg, Miss., April, 1980, 25 pp

.

4. American Society for Testing and Materials, 1985 Annual Book

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NBS-114A irev. 2 -ac) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 86-3452

2. Performing Organ. Report NoJ 3. Publication Date

SEPTEMBER 1986

4. TITLE AND SUBTITLE

Impact-Echo: A Method for

Flaw Detection in Concrete Using Transient Stress Waves

5. AUTHOR(S)

Mary Sansalone and Nicholas J. Carino

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8 . Type of Report & Period Covered

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City, State, ZIP)

National Bureau of Standards

Department of Commerce

Gaithersburg, Maryland 20899

10.

SUPPLEMENTARY NOTES

1 Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a si gn ificant

bi bliography or literature survey, mention it here)

To develop a basis for a nondestructive test method for heterogenous

solids, analytical, numerical, and laboratory studies of transient stress

wave propagation in plain plates and in plates containing flaws are presented. The test method involves introducing transient stress waves into

a test object by mechanical point impact and monitoring reflections of

the waves from internal defects and external boundaries using a point receiver. Exact Green's function solutions for a point source on an infinite plate are used to study the response of a plate to elastic impact by

a sphere in both the time and frequency domains. Displacement and stress

fields produced by a transient point load on the top surface of a plate

are obtained using the finite element method. The finite element method

is also used to study transient wave propagation in plates containing

planar flaws. The development of an experimental impact-echo technique

for finding flaws within plain and steel reinforced concrete is described

Laboratory results are presented. Successes achieved in the laboratory

suggest that the impact-echo method has the potential to become a reliable field technique for detecting flaws within concrete structures.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) concrete; finite element method; flaw detection; Green's function;

nondestructive testing; stress wave propagation

13.

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