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P. KFBIDMAT MAKLUMAT AKADEMIK UNIMAS

1000165877

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THE RELATIONSHIP BETWEEN THE APPLIED TORQUE AND STRESSES IN POST-TENSION STRUCTURES

LIEW FUI KIEW

A thesis submitted in fulfilment of the requirements for the degree of Master of Engineering

Faculty of Engineering UNIVERSITI MALAYSIA SARAWAK

2007

Acknowledgement

First and foremost, I would like to thank the UNIMAS, the Faculty of Engineering and the

Center of Graduate Studies for providing me the opportunity, scholarship and facilities to

conduct this applied research.

Secondly, I would like to extend my gratitude and appreciation to my supervisor, Associate

Professor Dr. Sinin Hamdan for his invaluable advice, constant comment, suggestions and

patience during this research. I am very grateful and thankful to Dr. Mohd. Shahril Osman,

who has given me diligent attention and advises through out the process of this research

work.

I also would like to thank my fellow course mates, laboratory technicians and other for their

invaluable discussion and their patience throughout the research work.

In addition, I want to thank my former colleagues in Technical and QC department at Taiyo

Yuden (Sarawak) Sdn. Bhd. for their technical guidance regarding the use of Scanning

Electron Microscope (SEM) and providing the SEM micrograph.

Furthermore, I wish to express my special thank to Mr. Yong Chin Soon and Ms. Chin Suh

Lin, who take their valuable time to correct the grammatical mistake in this work.

Last but not least, I would like to express my sincere appreciation to my beloved family for

their endless support, understanding and ever lasting love, my brothers and sisters in

Christ, for their pray, love and care.

11

Abstract

This research presents the non-destructive testing (NDT) method to determine the

resultant stresses in mild steel bar usually employed in structures. The technique utilized

ultrasonic pulse-echo that determined the wave velocity change due to torque applied

between bolt and nut.

Mild steel bar with nominal diameter of 18 mm and 25mm were used. The specimen

was loaded by means of a torque wrench, which gave the required amount of moment of

about 240 Nm. This was carefully achieved manually. In order to measure the strain, strain

gauges were employed. The direct strain gauge method gives the strain values. This strain

will be used to calculate the stresses caused by the applied load. The experiment had been

carried out in a control environment with constant temperature.

The relationship between torque- velocity, torque-strain, and velocity-strain is

obtained and compared. The test results indicate that ultrasonic wave velocity decrease

with the applied torque. This decrement in velocity is characteristic of dislocation

dampening behavior. This is due to degradation or loss of strength of the material. The

potential of this NDT method to obtain structure quality and strength determination is

discussed.

111

Abstrak

Laporan ini memaparkan kaedah Ujian Tanpa Musnah untuk memperolehi tegasan dalam

spesimen bar keluli yang lazim digunakan dalam struktur. Teknik ini mengunakan

Kaedah Ultrasonik pulse-echo untuk memperolehi perubahan halaju gelombang yang

disebabkan oleh applikasi tork antara bolt dan nut.

Bar keluli dengan purata diameter 18mm dan 25mm digunakan. Sampel dibebani dengan

mengunakan tork wrench yang memberi moment lebih kurang 240 Nm. Ini dilakukan

dengan cermat secara manual. Untuk memperolehi keterikan, tolok keterikan digunakan.

Kaedah keterikan memberikan nilai keterikan secara terus. Nilai keterikan yang

diperolehi kemudian digunakan dalam pengiraan ketegasan yang disebabkan oleh aplikasi

beban. Eksperimen ini dijalankan dalam persekitaran terkawal dimana suhunya adalah

malar.

Hubungan antara tork -halaju, tork-terikan dan halaju-terikan diperolehi dan

dibandingkan. Keputusan menunjukkan bahawa halaju gelombang ultrasonik berkurang

dengan peningkatan tork. Pengurangan halaju ini adalah satu sifat pelembapan dislokasi

bahan. Ini adalah disebabkan oleh degradasi atau kehilangan kekuatan bahan. Pontensi

bagi kaedah Ujian Tanpa Musnah dalam memperolehi maklumat kualiti struktur dan

kekuatan juga dibincangkan.

1V

UN--, 1Vý : ixuaemj,. MALAYSIA SARAµ/gy V43(1() Kota Samarahan

Table of Content

Page Acknowledgement ii

Abstract i i i

Abstrak iv Table of Content v List of Figures viii List of Tables x List of Symbol xi List of Appendixes xiii

Chapter 1 Introduction I

1.1 Background of Study 1.1.1 Steel 1.1.2 History of Ultrasonic 2 1.1.3 Type of Ultrasonic Waves 4

1.1.4 The Beginning of Nondestructive Testing (NDT) 5 1.2 Research Objectives 6 1.3 Scope of the Thesis 7

Chapter 2 Literature Review 8

2.1 Introduction 8 2.2 Torque and Stress Theory 8

2.3 Acoustoelasticity I I

2.4 Stress and Strain Relations 13 2.5 Wave Propagation 16

2.5.1 Wave Propagation in Elastically Isotropic Solid 19

2.6 Wave Speed and Material Factors 22 2.7 Constraints Of Stress Measurement By Using Ultrasonic 27

2.8 Ultrasound Attenuation in Polycrystalline Solids 28 2.9 Ultrasonic Velocity and Microstructure relationship 31

2.9.1 Effect of Porosity 33 2.9.2 Mean Grain Size 33 2.9.3 Grain Boundaries 34

2.10 Literature Summary and Research Constrain 34

Chapter 3 Methodology 37

3.1 Introduction 37

V

3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11

Chapter 4

4.1 4.2

4.2.1 4.2.1.1 4.2.1.2 4.2.1.3 4.2.1.4 4.2.1.5

4.2.1.5.1 4.2.1.5.2 4.2.1.5.3

4.2.1.6 4.2.1.7

4.2.2 4.2.2.1 4.2.2.2 4.2.2.3 4.2.2.4 4.2.2.5

4.2.2.5.1 4.2.2.5.2 4.2.2.5.3

4.2.2.6 4.2.2.7

4.3

Chapter 5

5.1 5.2

Specimen's Material Loading Types i. e Experimental Structure and Setup Ultrasonic Testing Instrument and Types of Ultrasonic Waves Calibration Strain Gauge Method Comparative Study of Ultrasonic Instrument's Readings Density Measurement Poisson's Ratio, Young's Modulus and Shear Modulus Microstructure Analysis Summary of Research Methodology

Result and Discussion

Introduction Experimental Results and Discussions

18mm Specimens Results No grease condition BP grease condition DM grease condition KY grease condition

Comparison between Different Grease for 18mm Specimen 18mm specimen I 18mm specimen 2 18mm specimen 3

SEM results for 18mm Specimens Summary for 18mm Specimens

25mm Specimens Results No grease condition BP grease condition DM grease condition KY grease condition

Comparison between Different Grease for 25mm Specimen 25mm specimen l 25mm specimen 2 25mm specimen 3

SEM Results for 25mm Specimens Summary for 25mm Specimens

Conclusion of Results

Conclusion and Recommendation

Conclusion Recommendation

37 39 40 42 46 49 50 50 52 56

58

58 58

59 59 62 65 68 72 72 75 78 81 83

84 84 87 90 93 96 96 99 102 105 107

108

110

iio 112

vi

References Appendix

VII

113 118

List of Figures

Page Figure 2.1 Variation of stress in a circular member in the elastic region. 9 Figure 2.2 Particles movement and wave direction of longitudinal wave. 18 Figure 2.3 Particles movement and wave direction of shear wave. 19 Figure 2.4 Screw clamping sheet metal. 28 Figure 3.1 Experimental setup to conduct the study. 39 Figure 3.2 Initial set up of the ultrasonic instrument. 44 Figure 3.3 Strain gauge location on the specimen. 47 Figure 3.4 Strain Gauge Divider diagram. 47 Figure 3.5 Flow chart for research methodology 56 Figure 4.1 Specimen identification system. 59 Figure 4.2 18mm specimen 1 for no grease condition. 60 Figure 4.3 18mm specimen 2 for no grease condition. 61 Figure 4.4 18mm specimen 3 for no grease condition. 62 Figure 4.5 18mm specimen 1 for BP grease condition. 63 Figure 4.6 18mm specimen 2 for BP grease condition. 64 Figure 4.7 18mm specimen 3 for BP grease condition. 65 Figure 4.8 18mm specimen 1 for DM grease condition. 66 Figure 4.9 18mm specimen 2 for DM grease condition. 67 Figure 4.10 18mm specimen 3 for DM grease condition. 68 Figure 4.11 18mm specimen 1 for KY grease condition. 69 Figure 4.12 18mm specimen 2 for KY grease condition. 70 Figure 4.13 18mm specimen 3 for KY grease condition. 71 Figure 4.14 18mm Specimen 1: Axial strain, wave velocity, torque. 72 Figure 4.15 18mm specimen 1: Lateral strain, wave velocity, torque. 73 Figure 4.16 18mm specimen 1: 45° strain, wave velocity, torque. 74 Figure 4.17 18mm specimen 2: Axial strain, wave velocity, torque. 75 Figure 4.18 18mm specimen 2: Lateral strain, wave velocity, torque. 76 Figure 4.19 18mm specimen 2: 45° strain, wave velocity, torque. 77 Figure 4.20 18mm specimen 3: Axial strain, wave velocity, torque. 78 Figure 4.21 18mm specimen 3: Lateral strain, velocity, torque. 79 Figure 4.22 18mm specimen 3: 45" strain, wave velocity, torque. 80 Figure 4.23 SEM plate (X1000) for 18mm specimen 1, 2 and 3 after Nital 81

etching. Figure 4.24 SEM plate for 18MS1 : Side A, Centre and Side B. 81 Figure 4.25 SEM plate for 18MS2 : Side A, Centre and Side B. 82 Figure 4.26 SEM plate for 18MS3 : Side A, Centre and Side B 82 Figure 5.1 25mm specimen 1 for no grease condition. 84 Figure 5.2 25mm specimen 2 for no grease condition. 85 Figure 5.3 25mm specimen 3 for no grease condition. 86 Figure 5.4 25mm specimen 1 for BP grease condition. 87 Figure 5.5 25mm specimen 2 for BP grease condition. 88 Figure 5.6 25mm specimen 3 for BP grease condition. 89

viii

Figure 5.7 Figure 5.8 Figure 5.9 Figure 5.10 Figure 5.11 Figure 5.12 Figure 5.13 Figure 5.14 Figure 5.15 Figure 5.16 Figure 5.17 Figure 5.18 Figure 5.19 Figure 5.20 Figure 5.21 Figure 5.22 Figure 5.23 Figure 5.24 Figure 5.25

25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen 25mm specimen25mm 25mm 25mm 25mm

1 for DM grease condition. 2 for DM grease condition. 3 for DM grease condition. 1 for KY grease condition. 2 for KY grease condition. 3 for KY grease condition. 1: Axial strain, wave velocity, torque. 1: Lateral strain, wave velocity, torque. 1: 45° strain, wave velocity, torque. 2: Axial strain, wave velocity, torque. 2: Lateral strain, wave velocity, torque.

specimen 2: 45° strain, wave velocity, torque. specimen 3: Axial strain, wave velocity, torque. specimen 3: Lateral strain, wave velocity, torque. specimen 3: 45° strain, wave velocity, torque.

SEM plate (X1000) for 25mm specimen 1, 2 and 3. SEM plate for 25mm specimen 1 : Side A, Centre and Side B. SEM photo for 25mm specimen 2: Side A, Centre and Side B. SEM photo for 25mm specimen 3: Side A, Centre and Side B.

90 91 92 93 94 95 97 98 99 100 101 102 103 104 105 106 106 106 107

IX

List of Tables

Page Table 1. 1

Table 4.1

Modes of wave and its particle vibration

Specimen identification and test condition

5

58

X

List of Symbols

t = Maximum shearing stress

r = Shearing stress

I,, = Polar moment of inertia of a circular area

v 1 � = Longitudinal wave velocities in unstressed portion

VI0 = Transverse ( Shear) wave velocities in unstressed portion

a 1, = Accoustoelastic constants of the longitudinal waves

aT = Accoustoelastic constants of the transverse waves

ß = Fraction of uniformly stressed length

Le = Uniformly stressed length

L = Total Length

a = Uniform tensile stress

C = Linear Elastic Moduli or stiffness constants

A, fl = Lame' constants,

a, = Stress tensor

G = Modulus rigidity,

K = Bulk modulus

P = Acoustic pressure

= Strain

V/, = Longitudinal thin-bar velocity

E = Young's modulus of elasticity

XI

p = Material Density

f = Ultrasonic frequency.

A = Wavelength,

D = Grain size

a = Linear attenuation coefficient

S = Scattering factor

T = Curie temperature

L = Longitudinal Moduli,

G = Shear Moduli. / Elastic moduli,

a = Stress

F = Applied force,

A = Cross sectional area

Al = Changes in length

1 = Length of specimen.

dR, = Changes in electrical resistance

F = Strain gauge factor.

V. = Voltage across the strain gauge.

W, = Weight of sample in air

W = Weight of sample in water

p, N- = Water density, 1 g cm 3

U = Poisson's Ratio

N = Number of Grains / Pores

Xil

List of Appendixes

Page Appendix A Acoustical Properties of Solids 118

Appendix B Detail results of 18mm specimen 1, specimen 2 and specimen 119 3.

Appendix C Detail results of 25mm specimen 1, specimen 2 and specimen 3 122

Appendix D 18mm specimen 1, specimen 2 and specimen 3 micrograph 125

Appendix E 25mm specimen 1, specimen 2 and specimen 3 micrograph 128

Appendix F 18 mm specimen grain size, pore size and quantity 131

Appendix G 25mm specimen grain size, pore size and quantity 132

Appendix H Summary of Poisson Ratio, Young Modulus and Shear 133 Modulus

XIII

CHAPTER 1

INTRODUCTION

1.1 Background of Study

1.1.1 Steel

What is steel? Steel is iron added with carbon with a rate close to 0%, corresponding

to very slight traces, up to 2%. Steels in general have lower carbon content than cast irons,

and lower amounts of impurities like phosphorus and sulfur. Other alloying elements such

as Boron, Chromium, Cobalt, Columbium (Niobium), Copper, Molybdenum, Nickel,

Nitrogen, Selenium, Tantalum, Titanium, Tungsten, and Vanadium are added to improve

corrosion, high temperature, and mechanical properties of steel.

There are two major families of steel: alloy steels and non-alloy steels. Alloy refers to

chemical elements other than carbon added to the iron in accordance with a minimum variable

content for each. Non-alloy steel containing by weight less than 0.25 percent of carbon.

Non-alloy steel is categorized as ferrous metal and its application as structural members

such as I-beams in construction, bar products, axles and railroad rails are considered the

most important technological development (Kalpakjian. S., 2001).

Residual stress are developed in steel during the forming and shaping of

reinforcement steel at the stage of production; however, these stresses are released during

the end of production by stress-relief annealing and yield insignificant effect. However,

when steel reinforcement are utilized as structural members, various kinds of load are

applied causing bending, tensioning or compressing stresses developed within these

I

structural members. Resultant loading develops during stages of structure fabrication and

their service periods. Thus, steel reinforcement experiences stresses within the structural

members, which will give significant and crucial effect to the structure. For instance,

brittle fracture will occur when structural members are overloaded with substantial tensile

stress, which exists in the form of nominal, residual or both stresses. By knowing the

resultant loading of structural members, the level of stresses in steel reinforcement can be

obtained indirectly. Therefore, immediate action could be taken to avoid fatal accidents and

deterioration of structure caused by failure of steel reinforcement when stresses in

structural members reach beyond the acceptable stress level.

By determining the resultant loading on the structural members, the stress

experienced will be known. In order to determine and monitor the loading and stress

experienced by structural members of steel reinforcement during the servicing period and

fabrication stages, on site determination method must be used. Non-destructive testing

(NDT) method is favorable to accomplish in -situ determination of resultant loading, where,

after the test the structural members do not require any repair of deterioration and

structural members could be used safely without causing harms to users.

1.1.2 History of Ultrasonic

Prior to World War II, sonar, the technique of sending sound waves through

water and observing the returning echoes to characterize submerged objects, inspired early

ultrasound investigators to explore ways to apply the concept to medical diagnosis. In 1929

and 1935, Sokolov studied the use of ultrasonic waves in detecting metal objects.

2

Mulhauser, in 1931, obtained a patent for using ultrasonic waves, using two transducers to

detect flaws in solids. Firestone (1940) and Simons (1945) developed pulsed ultrasonic

testing using a pulse-echo technique.

Ultrasonic is a branch of acoustics dealing with frequencies generally beyond the audible

limit (Mclntire et al, 1991). According to Blitz et. al. (1996), ultrasonic is the name given to

study and application of ultrasound. Ultrasound is a type of sound, where the sound is produced

when a pressure wave that is generated by mechanical disturbances propagates through a

medium. The sound above the audible range of human beings is called ultrasound, where the

vibration of ultrasound has high frequency above the limit of human ear.

Ultrasound needs medium to allow the propagation of waves from one point to

another such as air, liquid and solid. In contrast, ultrasound does not travel through the

vacuum. The elastic property of medium is responsible for sustaining the vibration for

propagation of waves; thus, ultrasonic wave is also termed as elastic waves. Although the

researches and studies of ultrasonic as a form of energy had been started since World War I,

the utilization capability of such energy is bestowed to the animals such as bats, dolphins,

moths etc. for purpose of locating foods, navigation and detecting danger(Ensminger, 1988).

The applications of ultrasonic by introducing ultrasonic waves into medium are

based on the intensity of ultrasonic. Applications of high intensity ultrasonic are intend to

cause an effect on medium or its contents i. e. medical therapy or welding of plastics and

metals while application of low intensity ultrasonic are purposely to pass information via

3

medium or learn something about the medium without changing the state of medium i. e.

non-destructive testing of materials, medical diagnose and depth sounding.

Ultrasonic that is used in non-destructive testing (NDT) involves applications

such as distance gauging, flaw detection and measuring parameters related to material

structure i. e. elastic moduli and grain size ( Blitz et al., 1996). By using low intensity

ultrasonic in NDT, object under examination will not experiences changes in dimension and

structure. As long as the applied stress of ultrasound waves does not exceed the elastic

limit, the object could be used even after the examination.

1.1.3 Type of Ultrasonic Waves

Ultrasonic waves can be differential and divided by modes of ultrasonic waves

propagate in the medium such as longitudinal wave, shear wave, Lamb wave, Love waves

and Rayleigh waves. It also serves as a basis for different applications of ultrasonic waves.

Longitudinal and shear waves are useful in ultrasonic inspection. On the other hand,

Rayleigh waves are useful because of its sensitivity in detecting surface defects.

Modes of wave traveling within medium are influenced by types of medium.

Medium of gases are only capable of transmitting longitudinal waves, whereas longitudinal

and shear waves can travel in liquids and large variety of waves in solids (Ensminger,

1988). The particle vibration in the wave differentiates each wave modes as listed (Table

1.1).

4

Modes Of Wave Particle Vibration

Longitudinal Parallel to wave direction

Shear (transverse) Perpendicular to wave direction

Component perpendicular to surface Plate wave - Lamb

(extensional wave)

Parallel to plane layer, perpendicular to wave Plate wave - Love

direction

Surface (Rayleigh) Elliptical orbit - symmetrical mode

Table 1.1: Modes of wave and its particle vibration.

1.1.4 The Beginnings of Nondestructive Testing (NDT)

Nondestructive testing had been practiced for many decades now. Initial rapid

developments in instrumentation spurred by the technological advances from the 1950's

continue today. During the earlier days, the primary purpose was the detection of defects.

As a part of "safe life" design, it was intended that a structure should not develop

macroscopic defects during its life, with the detection of such defects being a cause for

removal of the component from service. In response to this need, increasingly sophisticated

techniques using ultrasonic, eddy currents, x-rays, dye penetrates, magnetic particles, and

other forms of interrogating energy had emerged.

In the early 1970's, the continuous improvement of the technology, in particular its

ability to detect small flaws, lead to the unsatisfactory situation that more and more parts

5

had to be rejected, even though the probability of failure had not changed. However, the

discipline of fracture mechanics emerged, which enabled one to predict whether a crack of a

given size would fail under a particular load if a material property, fracture toughness, was

known. Other laws were developed to predict the rate of growth of cracks under cyclic

loading (fatigue). With the advent of these tools, it became possible to accept structures

containing defects if the sizes of those defects were known. This formed the basis for new

philosophy of "fail safe" or "damage tolerant" design. Components having known defects

could continue in service as long as it could be established that those defects would not

grow to a critical, failure producing size.

The challenge was thus detection was not enough, one needed to also obtain

quantitative information about flaw size to serve as an input to fracture mechanics based

predictions of remaining life. These concerns, which were felt particularly strong in the

defense and nuclear power industries, led to the creation of a number of research programs

and the emergence of quantitative nondestructive evaluation (QNDE) as a new discipline.

1.2 Research Objectives

The objective of this research is as follows:

I. Determine the change of ultrasonic wave velocity in various diameter of mild

steel bar due to the torque applied.

II. Determine the relationship between the applied torque and the resultant

strains in mild steel bar.

III. Compare the resultant strain experience by the different diameter of mild steel bar.

IV. Obtain the correlation between ultrasonic wave velocity with direct

measurement strains.

6

V. Analyzed the mild steel microstructure using SEM to understand the

underlying reason for result obtained.

1. 3 Scope of the Thesis

The thesis consists of five Chapters. Chapter one is the introduction, which describe

the fundamental, purpose and general overview of the project to the reader. This is followed

by second chapter, which mainly review the previous works and researches that had been

carried out in related field. Chapter three presents the methodology to carry out the

experiment in laboratory. A major portion of this chapter is dealing with the usage of

Ultrasonic as nondestructive test technique. This followed by sample preparation methods

for SEM analysis. All the information gathers through journal, article, handbooks,

technical manual and other related materials aimed to provide essential ingredient for

completing this project. Chapter four consists of results and discussion gathered by

experiment performed in laboratory. Chapter five is the conclusion and the

recommendation for further work.

CHAPTER 2

THEORY AND LITERATURE REVIEW

2.1 Introduction

Structures in general can be strengthens using different mechanism such as direct

reinforcement, pre-stressed reinforcement, post-tensioning and etc. In pre-stress structures,

a known stress in being applied to the steel reinforcement members and thus the resulting

stress in the structure is known.

In the case of the post-tension structures, the tension is only applied after the beam

or structure has been produced. This method is used to increase the load bearing capacity of

structures. The load is being applied to the tendons inside the structure via bolts and nut

arrangement and thus the resulting stress in the beam depends on the level of torque

applied to the nut. There has been little investigation on the relationship between the

applied torque and the induced stresses in such bolt and nut arrangement.

2.2 Torque and Stress Theory

In order to establish a relation between the internal torque and the stresses it set up

in members with circular cross-section, it is necessary to assume that stress is proportional

to strain in the elastic case. Strain varies linearly from the center, and therefore stresses

vary linearly from the central axis of a circular member. The stresses induced by the

8

assumed distortions are shearing stresses and lie in the plane parallel to the section normal

to the axis of a rod. The variation of shearing stress is illustrated in Figure 2.1.

Figure 2.1: Variation of stress in a circular member in the elastic region

Where r = the shearing stress

= the maximum shearing stress

p = distance from center, o

dA =2npdp, circular cross section area

c= maximum distance from center, o

Unlike the case of an axially loaded rod, the stress intensity is not uniform. The

maximum shearing stress occurs at points most remote from the center 0 and is

designated These points, such as point C' in figure 2.1 lie at the periphery of a section

at a distance c from the center. While, by virtue of a linear stress variation, at any arbitrary

9

point at a distance p from 0, the shearing stress is r = P Once the stress distribution c

at a section is established, the resistance to torque in terms of stress can be express. The

resistance to the torque so develop must be equivalent to the internal torque. Hence, an

equation can be formulated as the following:

Torque, T = fPt,,,

a\ dAp

c(2.2.1)

The integral sums up all the torque developed on the cut by the infinitesimal forces acting

at a distance p from a member's axis, 0 in figure 2. 1, over the whole area A of the cross

section, and T is the resisting torque. At any given section im,,, and c are constant; hence

the above relation can be written as:

f pr "'al

f c1A p= T , C I

(2.2.2)

However, f p'dt1, the polar moment of inertia of a cross sectional area, is also a constant for

a particular cross sectional area and is designated I,,. For a circular section dA=27rpdp,

where 27Tp is the circumference of an annulus with a radius p of width dp. Hence

fp'dA = f2zp'dp = 27s

, i o

4 P 4

IITc' zd'

2 32(2.2.3)

where d is the diameter of a solid circular shaft.

By using the symbol I,, for the polar moment of inertia of a circular area, the above equation

may be written more compactly as:

10

TcZmax (2.2.4)

This equation is the well-known torsion formula for circular shaft that expresses the

maximum shearing stress in terms of the resisting torque and the dimension of a member.

A more general relation for a shearing stress t at any point at distance p from the center of

a section is

r_Pr TP

C 1

2.3 Acoustoelasticity

i,

(2.2.5)

The ultrasonic stress measurement technique is based on the acoustoelastic effect

(Bray. D. E, 1996). The acoustoelasticity concept originated from the interest in the

measurement of third-order elastic constants (TOECs) in crystals. Hughes and Kelly (1953)

developed the theory of acoustoelasticity and used ultrasonic waves to determine these

constants in crystals by varying the applied stress. Ultrasonic wave propagating through an

elastic material under stressed conditions change speed a little due to the stress. In other

word, acoustoelasticity is the term applied to changes in velocity or attenuation wrought by

applied or residual stress. This change is called an acoustoelastic effect, and the

acoustoelastic technique can be applied to stress analyses of materials (Benson and

Raelson , 1959; Bergman and Shahbender, 1958; Crecraft, 1967; Fukuoka 1993 and

Iwashimizu, Y. 1994). In practice, it is easier to measure velocity changes although velocity

is a weak function of stress (Mclntire et. al, 1991)

I I