fatigue testing and analysis theory and practice

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Fatigue Testing
and Analysis
(Theory and Practice)
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 21.6.2004 6:19pm page i
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Fatigue Testing
and Analysis
(Theory and Practice)
Yung-Li Lee
DaimlerChrysler
Jwo Pan
University of Michigan
Richard B. Hathaway
Western Michigan University
Mark E. Barkey
University of Alabama
AMSTERDAM
.

BOSTON
.
HEIDELBERG
.
LONDON
.
NEW YORK
OXFORD
.
PARIS
.
SAN DIEGO
.
SAN FRANCISCO
.
SINGAPORE

SYDNEY

.
TOKYO
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 21.6.2004 6:19pm page iii
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LEE: FATIGUE TESTING AND ANALYSIS Final Proof 21.6.2004 6:19pm page iv
Thank our families for their support and patience.
To my parents and my wife Pai-Jen.
- Yung-Li Lee
To my Mom and my wife Michelle.
- Jwo Pan
To my wife Barbara.
- Richard B. Hathaway
To my wife, Tammy, and our daughters Lauren and Anna.
- Mark E. Barkey
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 21.6.2004 6:19pm page v
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Preface
Over the past 20 years there has been a heightened interest in improving
quality, productivity, and reliability of manufactured products in the ground

vehicle industry due to global competition and higher customer demands for
safety, durability and reliability of the products. As a result, these products
must be designed and tested for sufficient fatigue resistance over a large range
of product populations so that the scatters of the product strength and
loading have to be quantified for any reliability analysis.
There have been continuing efforts in developing the analysis techniques for
those who are responsible for product reliability and product design. The
purpose of this book is to present the latest, proven techniques for fatigue
data acquisition, data analysis, test planning and practice. More specifically,
it covers the most comprehensive methods to capture the component load, to
characterize the scatter of product fatigue resistance and loading, to perform
the fatigue damage assessment of a product, and to develop an accelerated
life test plan for reliability target demonstration.
The authors have designed this book to be a useful guideline and reference to
the practicing professional engineers as well as to students in universities who
are working in fatigue testing and design projects. We have placed a primary
focus on an extensive coverage of statistical data analyses, concepts,
methods, practices, and interpretation.
The material in this book is based on our interaction with engineers and
statisticians in the industry as well as based on the courses on fatigue testing
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 21.6.2004 6:19pm page vii
vii
and analysis that were taught at Oakland University, University of Michigan,
Western Michigan University and University of Alabama. Five major con-
tributors from several companies and universities were also invited to help us
enhance the completeness of this book. The name and affiliation of the
authors are identified at the beginning of each chapter.
There are ten chapters in this book. A brief description of these chapters is
given in the following.
Chapter 1 (Transducers and Data Acquisition) is first presented to address

the importance of sufficient knowledge of service loads/stresses and how to
measure these loads/stresses. The service loads have significant effects on the
results of fatigue analyses and therefore accurate measurements of the actual
service loads are necessary. A large portion of the chapter is focused on the
strain gage as a transducer of the accurate measurement of the strain/stress,
which is the most significant predictor of fatigue life analyses. A variety of
methods to identify the high stress areas and hence the strain gage placement
in the test part are also presented. Measurement for temperature, number of
temperature cycles per unit time, and rate of temperature rise is included. The
inclusion is to draw the attention to the fact that fatigue life prediction is
based on both the number of cycles at a given stress level during the service
life and the service environments. The basic data acquisition and analysis
techniques are also presented.
In Chapter 2 (Fatigue Damage Theories), we describe physical fatigue mech-
anisms of products under cyclic mechanical loading conditions, models to
describe the mechanical fatigue damages, and postulations and practical
implementations of these commonly used damage rules. The relations of
crack initiation and crack propagation to final fracture are discussed in this
chapter.
In Chapter 3 (Cycle Counting Techniques), we cover various cycle counting
methods used to reduce a complicated loading time history into a series of
simple constant amplitude loads that can be associated with fatigue
damage. Moreover the technique to reconstruct a load time history with
the equivalent damage from a given cycle counting matrix is introduced in
this chapter.
In Chapter 4 (Stress-Based Fatigue Analysis and Design), we review methods
of determining statistical fatigue properties and methods of estimating the
fatigue resistance curve based on the definition of nominal stress amplitude.
These methods have been widely used in the high cycle fatigue regime for
decades and have shown their applicability in predicting fatigue life of

notched shafts and tubular components. The emphasis of this chapter is on
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viii Preface
the applications of these methods to the fatigue design processes in the
ground vehicle industry.
In Chapter 5 (Strain-Based Fatigue Analysis and Design), we introduce the
deterministic and statistical methods for determining the fatigue resistance
parameters based on a definition of local strain amplitude. Other accompan-
ied techniques such as the local stress-strain simulation and notch analysis are
also covered. This method has been recommended by the SAE Fatigue
Design & Evaluation Committee for the last two decades for its applicability
in the low and high cycle fatigue regimes. It appears of great value in the
application of notched plate components.
In Chapter 6 (Fracture Mechanics and Fatigue Crack Propagation), the
text is written in a manner to emphasize the basic concepts of stress concen-
tration factor, stress intensity factor and asymptotic crack-tip field for linear
elastic materials. Stress intensity factor solutions for practical cracked
geometries under simple loading conditions are given. Plastic zones and
requirements of linear elastic fracture mechanics are then discussed. Finally,
fatigue crack propagation laws based on linear elastic fracture mechanics are
presented.
In Chapter 7 (Fatigue of Spot Welds), we address sources of variability in the
fatigue life of spot welded structures and to describe techniques for calculat-
ing the fatigue life of spot-welded structures. The load-life approach, struc-
tural stress approach, and fracture mechanics approach are discussed in
details.
In Chapter 8 (Development of Accelerated Life Test Criteria), we provide
methods to account for the scatter of loading spectra for fatigue design
and testing. Obtaining the actual long term loading histories via real time
measurements appears difficult due to technical and economical reasons.

As a consequence, it is important that the field data contain all possible
loading events and the results of measurement be properly extrapolated.
Rainflow cycle counting matrices have been recently, predominately used
for assessing loading variability and cycle extrapolation. The following
three main features are covered: (1) cycle extrapolation from short term
measurement to longer time spans, (2) quantile cycle extrapolation from
median loading spectra to extreme loading, and (3) applications of the
extrapolation techniques to accelerated life test criteria.
In Chapter 9 (Reliability Demonstration Testing), we present various statis-
tical-based test plans for meeting reliability target requirements in the accel-
erated life test laboratories. A few fatigue tests under the test load spectra
should be carried out to ensure that the product would pass life test criteria.
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Preface ix
The statistical procedures for the choice of a test plan including sample size
and life test target are the subject of our discussion.
In Chapter 10 (Fatigue Analysis in the Frequency Domain), we introduce the
fundamentals of random vibrations and existing methods for predicting
fatigue damage from a power spectral density (PSD) plot of stress response.
This type of fatigue analysis in the frequency domain is particularly useful for
the use of the PSD technique in structural dynamics analyses.
The authors greatly thank to our colleagues who cheerfully undertook the
task of checking portions or all of the manuscripts. They are Thomas Cordes
(John Deere), Benda Yan (ISPAT Inland), Steve Tipton (University of
Tulsa), Justin Wu (Applied Research Associates), Gary Halford (NASA-
Glenn Research Center), Zissimos Mourelatos (Oakland University), Keyu
Li (Oakland University), Daqing Zhang (Breed Tech.), Cliff Chen (Boeing),
Philip Kittredge (ArvinMeritor), Yue Chen (Defiance), Hongtae Kang (Uni-
versity of Michigan-Dearborn), Yen-Kai Wang (ArvinMeritor), Paul
Lubinski (ArvinMeritor), and Tana Tjhung (DaimlerChrysler).

Finally, we would like to thank our wives and children for their love,
patience, and understanding during the past years when we worked most of
evenings and weekends to complete this project.
Yung-Li Lee, DaimlerChrysler
Jwo Pan, University of Michigan
Richard B. Hathaway, Western Michigan University
Mark E. Barkey, University of Alabama
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x Preface
Table of Contents
1 Transducers and Data Acquisition 1
Richard B. Hathaway, Western Michigan University
Kah Wah Long, DaimlerChrysler
2 Fatigue Damage Theories 57
Yung-Li Lee, DaimlerChrysler
3 Cycle Counting Techniques 77
Yung-Li Lee, DaimlerChrysler
Darryl Taylor, DaimlerChyrysler
4 Stress-Based Fatigue Analysis
and Design 103
Yung-Li Lee, DaimlerChrysler
Darryl Taylor, DaimlerChyrysler
5 Strain-Based Fatigue Analysis
and Design 181
Yung-Li Lee, DaimlerChrysler
Darryl Taylor, DaimlerChyrysler
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xi
6 Fracture Mechanics and Fatigue Crack
Propagation 237

Jwo Pan, Uni versity of Michigan
Shih-Huang Lin, University of Michigan
7 Fatigue of Spot Welds 285
Mark E. Barkey, University of Alabama
Shicheng Zhang, DaimlerChrysler AG
8 Development of Accelerated Life
Test Criteria 313
Yung-Li Lee, DaimlerChrysler
Mark E. Barkey, University of Alabama
9 Reliability Demonstration Testing 337
Ming-Wei Lu, DaimlerChrysler
10 Fatigue Analysis in the Frequency
Domain 369
Yung-Li Lee, DaimlerChrysler
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xii Table of Contents
About the Authors
Dr. Yung-Li Lee is a senior member of the technical staff of the Stress Lab &
Durability Development at DaimlerChrysler, where he has conducted re-
search in multiaxial fatigue, plasticity theories, durability testing for automo-
tive components, fatigue of spot welds, and probabilistic fatigue and fracture
design. He is also an adjunct faculty in Department of Mechanical Engineer-
ing at Oakland University, Rochester, Michigan.
Dr. Jwo Pan is a Professor in Department of Mechanical Engineering of
University of Michigan, Ann Arbor, Michigan. He has worked in the area of
yielding and fracture of plastics and rubber, sheet metal forming, weld
residual stress and failure, fracture, fatigue, plasticity theories and material
modeling for crash simulations. He has served as Director of Center for
Automotive Structural Durability Simulation funded by Ford Motor Com-
pany and Director for Center for Advanced Polymer Engineering Research

at University of Michigan. He is a Fellow of American Society of Mechanical
Engineers (ASME). He is on the editorial boards of International Journal of
Fatigue and International Journal of Damage Mechanics.
Dr. Richard B. Hathaway is a professor of Mechanical and Aeronautical
engineering and Director of the Applied Optics Laboratory at Western
Michigan University. His research involves applications of optical measure-
ment techniques to engineering problems including automotive structures
and powertrains. His teaching involves Automotive structures, vehicle sus-
pension, and instrumentation. He is a 30-year member of the Society of
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xiii
Automotive Engineers (SAE) and a member of the Society of Photo-Instru-
mentation Engineers (SPIE).
Dr. Mark E. Barkey is an Associate Professor in the Aerospace Engineering
and Mechanics Department at the University of Alabama. He has conducted
research in the areas of spot weld fatigue testing and analysis, multiaxial
fatigue and cyclic plasticity of metals, and multiaxial notch analysis. Prior to
his current position, he was a Senior Engineering in the Fatigue Synthesis and
Analysis group at General Motors Mid-Size Car Division.
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xiv About the Authors
1
Transducers and Data
Acquisition
Richard Hathaway
Western Michigan University
Kah Wah Long
DaimlerChrysler
1.1 introduction
This chapter addresses the sensors, sensing methods, measurement

systems, data acquisition, and data interpretation used in the experimental
work that leads to fatigue life prediction. A large portion of the chapter is
focused on the strain gage as a transducer. Accurate measurement of strain,
from which the stress can be determined, is one of the most significant
predictors of fatigue life. Prediction of fatigue life often requires the experi-
mental measurement of localized loads, the frequency of the load occurrence,
the statistical variability of the load, and the number of cycles a part will
experience at any given load. A variety of methods may be used to predict the
fatigue life by applying either a linear or weighted response to the measured
parameters.
Experimental measurements are made to determine the minimum and
maximum values of the load over a time period adequate to establish the
repetition rate. If the part is of complex shape, such that the strain levels
cannot be easily or accurately predicted from the loads, strain gages will need
to be applied to the component in critical areas. Measurements for tempera-
ture, number of temperature cycles per unit time, and rate of temperature rise
may be included. Fatigue life prediction is based on knowledge of both the
number of cycles the part will experience at any given stress level during that
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1
life cycle and other influential environmental and use factors. Section 1.2
begins with a review of surface strain measurement, which can be used to
predict stresses and ultimately lead to accurate fatigue life prediction. One of
the most commonly accepted methods of measuring strain is the resisti ve
strain gage.
1.2 strain gage fundamentals
Modern strain gages are resistive devices that experimentally evaluate the
load or the strain an object experiences. In any resistance transducer,
the resistance (R) measur ed in ohms is material and geometry dependent.
Resistivity of the material (r) is expressed as resistance per unit length  area,

with cross-sectional area (A) along the length of the material (L) making up
the geometry. Resistance increases with length and decreases with cross-
sectional area for a material of constant resistivity. Some sample resistivities
(mohms-cm
2
=cm) at 208C are as follows:
Aluminum: 2.828
Copper: 1.724
Constantan: 4.9
In Figure 1.1, a simple wire of a given length (L), resistivity (r), and cross-
sectional area (A) has a resistance (R) as shown in Equation 1.2.1:
R ¼ r
L
p
4
D
2
0
B
@
1
C
A
¼ r
L
A

If the wire experiences a mechanical load (P) along its length, as shown
in Figure 1.2, all three parameters (L, r, A) change, and, as a result, the end-
to-end resistance of the wire changes:

L
D
R = Resistance (ohms) L = Length r = Resistivity [(ohms ϫ area) / length)]
A = Cross-sectional Area:
Area is sometimes presented in circular mils, which is the area of a 0.001-inch diameter.
r
figure 1.1
A simple resistance wire.
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2
Transducers and Data Acquisitio n
DR ¼ r
L
Â
L
L
p
4
D
2
L
0
B
@
1
C
A
À r Â
L
p

4
D
2
0
B
@
1
C
A
(1:2:2)
The resistance change that occurs in a wire under mechanical load makes it
possible to use a wire to measure small dimensional changes that occur
because of a change in component loading. The concept of strain (e), as it
relates to the mechanical behavior of loaded components, is the change in
length (DL) the component experiences divided by the original component
length (L), as shown in Figure 1.3:
e ¼ strain ¼
DL
L
(1:2:3)
It is possible, with proper bonding of a wire to a structure, to accurately
measure the change in length that occurs in the bonded length of the wire.
This is the underlying principle of the strain gage. In a strain gage, as shown
in Figure 1.4, the gage grid physically changes length when the material to
which it is bonded changes length. In a strain gage, the change in resistance
occurs when the conductor is stretched or compressed. The change in resist-
ance (DR) is due to the change in length of the conductor, the change in cross-
sectional area of the conductor, and the change in resistivity (Dr) due to
mechanical strain. If the unstrained resistivity of the material is defined as r
us

and the resi stivity of the strained material is r
s
, then r
us
À r
s
¼ Dr.
L
L
D
L
r
L
P
P
figure 1.2
A resistance wire under mechanical load.
P
P
DL/2
DL/2
L
e = DL/L
Where
e = strain; L = original length;
DL = increment due to force P
figure 1.3
A simple wire as a strain sensor.
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Strain G age Fundamen tals

3
R ¼
rL
A
DR
R
%
DL
L
þ
Dr
r
À
DA
A
(1:2:4)
The resistance strain gage is convenient because the change in resistance
that occurs is direct ly proportional to the change in length per unit length
that the transducer undergoes. Two fundamental types of strain gages are
available, the wire gage and the etched foil gage, as shown in Figure 1.5. Both
gages have similar basic designs; however, the etched foil gage introduces
some additional flexibility in the gage design process, providing additional
control, such as temperature compensation. The etched foil gage can typically
be produced at lower cost.
The product of gage wi dth and length defines the active gage area, as shown
in Figure 1.6. The active gage area charact erizes the measurement surface and
the power dissipation of the gage. The backing length and width define the
required mounting space. The gage backing material is designed such that high
Backing Material
Strain Wire

Lead Wires
figure 1.4
A typical uniaxial strain gage.
Foil Gage
gage base
Resistance
wire
gage leads
Wire Gage
Etched
resistance
foil
figure 1.5
Resistance wire and etched resistance foil gages.
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4
Transducers and Data Acquisitio n
transfer efficiency is obtained between the test material and the gage, allowi ng
the gage to accurately indicate the component loading conditions.
1.2.1 GAGE RESISTANCE AND EXCITATION VOLTAGE
Nominal gage resistance is most commonly either 120 or 350 ohms.
Higher-resistance gages are available if the application requires either a
higher excitation voltage or the material to which it is attached has low
heat conductivity. Increasing the gage resistance (R) allows increased excita-
tion levels (V) with an equivalent power dissipation (P) requirement as shown
in Equation 1.2.5.
Testing in high electrical noise environments necessitates the need for
higher excitation voltages (V). With analog-to-digital (A–D) conversion for
processing in computers, a commonly used excitation voltage is 10 volts. At
10 volts of excitation, each gage of the bridge would have a voltage drop

of approximately 5 V. The power to be dissipated in a 350-ohm gage is thereby
approximately 71 mW and that in a 120-ohm gage is approximately 208 mW:
P
350
¼
V
2
R
¼
5
2
350
¼ 0:071W P
120
¼
V
2
R
¼
5
2
120
¼ 0:208W (1:2:5)
At a 15-volt excitation with the 350-ohm gage, the power to be dissipated
in each arm goes up to 161 mW. High excitation voltage leads to higher
signal-to-noise ratios and increases the power dissipation requirement. Ex-
cessively high excitation voltages, especially on smaller grid sizes, can lead to
drift due to grid heating.
1.2.2 GAGE LENGTH
The gage averages the strain field over the length (L) of the grid. If the gage

is mounted on a nonuniform stress field the average strain to which the active
gage length
backing
length
gage
width
backing
width
gage lead
gird
length
grid
area
figure 1.6
Gage dimensional nomenclature.
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Strain G age Fundamen tals
5
gage area is exposed is proportional to the resistance change. If a strain field
is known to be nonuniform, proper location of the smallest gage is frequently
the best option as shown in Figure 1.7.
1.2.3 GAGE MATERIAL
Gage material from which the grid is made is usually constantan. The
material used depends on the application, the material to which it is bonded,
and the control required. If the gage material is perfectly matched to the
mechanical characteristics of the material to which it is bonded, the gage can
have pseudo temperature compensation with the gage dimensional changes
offsetting the temperature-related component changes. The gage itself will be
temperature compensated if the gage material selected has a thermal coeffi-
cient of resistivity of zero over the temperature range anticipated. If the gage

has both mechanical and thermal compensation, the system will not produce
apparent strain as a result of ambient temperature variations in the testing
environment. Select ion of the proper gage material that has minimal tem-
perature-dependant resistivity and some temperature-dependent mechanical
characteristics can result in a gage system with minimum sensitivity to
temperature changes in the test environment. Strain gage manufacturers
broadly group their foil gages based on their application to either aluminum
or steel, which then provides acceptable temperature compensation for
ambient temperature variations.
The major function of the strain gage is to produce a resistance change
proportional to the mechanical strain (e) the object to which it is mounted
experiences. The gage proportionality factor, commonly called the gage
factor (GF), which makes the two equations of 1.2.6 equivalent, is defined
A gage length that is one-tenth of the corresponding dimension of any
stress raiser where the measurement is made is usually acceptable.
Peak Strain
Indicated Strain
Strain
Position
X
figure 1.7
Peak and indicated strain comparisons.
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6
Transducers and Data Acquisition
in Equation 1.2.7. Most common strain gages have a nominal gage factor of
2, although special gages are available with higher gage factors.
e ¼ Strain ¼
DL
L

e /
DR
R
(1:2:6)
GF ¼
DR
R

e
¼
DR
R

DL
L

therefore DR ¼ GF  e  R (1:2:7)
The gage factor results from the mechanical deformation of the gage grid
and the change in resistivity of the material (r) due to the mechanical strain.
Deformation is the change in length of the gage material and the change in
cross-sectional area due to Poisson’s ratio. The change in the resistivity, called
piezoresistance, occurs at a molecular level and is dependent on gage material.
In fatigue life prediction, cyclic loads may only be a fraction of the loads
required to cause yielding. The measured output from the instrumentation
will depend on the gage resistance change, which is proportional to the strain.
If the loads are relatively low, Equation 1.2.7 indicates the highest output and
the highest signal-to-noise ratio is obtained with high-resistance gages and a
high gage factor.
Example 1.1. A 350-ohm gage is to be used in measuring the strain magni-
tude of an automotive component under load. The strain gage has a gage

factor of 2. If the component is subjected to a strain field of 200 microstrain,
what is the change in resistance in the gage? If a high gage factor 120-ohm
strain gage is used instead of the 350-ohm gage, what is the gage factor if the
change in resistance is 0.096 ohms?
Solution. By using Equation 1.2.7, the change in resistance that occurs with
the 350-ohm gage is calculated as
DR ¼ GF  e  R ¼ 2  200  10
À6
 350 ¼ 0: 14 ohms
By using Equation 1.2.7, the gage factor of the 120-ohms gage is
GF ¼
DR
R
e
¼
0:096
120
200 Â 10
À6
¼ 4
1.2.4 STRAIN GAGE ARRANGEMENTS
Strain gages may be purchased in a variety of arrangements to make
application easier, measurement more precise, and the information gained
more comprehensive. A common arrangement is the 908 rosette, as shown
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Strain G age Fundamen tals 7
in Figure 1.8. This arrangement is popular if the direction of loading is
unknown or varies. This gage arrangement provides all the information
required for a Mohr’s circle strain analysis for identification of principle
strains. Determination of the principle strains is straightforward when a

three-element 908 rosette is used, as shown in Figure 1.9.
Mohr’s circle for strain would indicate that with two gages at 908 to each
other and the third bisecting the angle at 458, the princi ple strains can
be identified as given in Equation 1.2.8. The orientation angle (f) of principle
strain (e
1
), with respect to the X-axis is as shown in Equation 1.2.9, with the
shear strain (g
xy
) as given in Equation 1.2.10:
e
1,2
¼
e
x
þ e
y
ÀÁ
2
þ = À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
x
À e
y
ÀÁ
2
þg
2
xy

q
2
(1:2:8)
tan 2f ¼
g
xy
e
x
À e
y
(1:2:9)
g
xy
¼ 2e
45
À e
x
À e
y
(1:2:10)
The principle strains are then given by Equations 1.2.11 and 1.2.12:
figure 1.8
Three-element rectangular and stacked rectangular strain rosettes.
XՆ0
YՆ90
45
figure 1.9
Rectangular three-element strain rosette.
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8

Transducers and Data Acquisition
e
1
¼
e
x
þ e
y
ÀÁ
2
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
x
À e
y
ÀÁ
2
þ 2e
45
À e
x
À e
y
ÀÁ
2
q
2
(1:2:11)
e

2
¼
e
x
þ e
y
ÀÁ
2
À
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
x
À e
y
ÀÁ
2
þ 2e
45
À e
x
À e
y
ÀÁ
2
q
2
(1:2:12)
Correspondingly, the principle angle (f) is as shown in Equation 1.2.13:
tan 2f ¼
2e

45
À e
x
À e
y
e
x
À e
y
(1:2:13)
With principle strains and principle angles known, principle stresses can be
obtained from stress–strain relationships. Linear stress–strain relationships
are given in Equations 1.2.14–1.2.25. In high-strain environments, these
linear equations may not hold true.
The linear stress–strain relationships in a three-dimensional state of stress
are shown in Equations 1.2.14–1.2. 16 for the normal stresses. The stresses and
strains are related through the elastic modulus (E) and Poisson’s ratio (m):
e
x
¼
1
E
s
x
À ms
y
þ s
z
ÀÁÂÃ
(1:2:14)

e
y
¼
1
E
s
y
À ms
x
þ s
z
ðÞ
ÂÃ
(1:2:15)
e
z
¼
1
E
s
z
À ms
x
þ s
y
ÀÁÂÃ
(1:2:16)
The relationship between shear strains and shear stresses are given in
Equation 1.2.17. Shear strains and shear stresses are related through the
shear modulus (G):

g
xy
ÂÃ
¼
1
G
s
xy
g
xz
½¼
1
G
s
xz
g
yz
ÂÃ
¼
1
G
s
yz
(1:2:17)
Equations 1.2.18–1.2.20 can be used to obtain the normal stresses given
the normal strains, with a three-dimensional linear strain field:
s
x
¼
E

1 þ mðÞ1 À 2mðÞ
¼ 1 ÀmðÞe
x
þ me
y
þ e
z
ÀÁÂÃ
(1:2:18)
s
y
¼
E
1 þ mðÞ1 À2mðÞ
¼ 1 À mðÞe
y
þ me
x
þ e
z
ðÞ
ÂÃ
(1:2:19)
s
z
¼
E
1 þ mðÞ1 À 2mðÞ
¼ 1 ÀmðÞe
z

þ me
x
þ e
y
ÀÁÂÃ
(1:2:20)
Shear stresses are directly obtaine d from shear strains as shown in Equa-
tion 1.2.21:
s
xy
¼ Gbg
xy
c s
xz
¼ Gbg
xz
c s
yz
¼ Gbg
yz
c (1:2:21)
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 22.6.2004 2:54pm page 9
Strain G age Fundamen tals 9
Equations 1.2.22 and 1.2.23 can be used to obtain principle stresses from
principle strains:
s
1
¼
E
1 þ m

2
ðÞ
(e
1
þ me
2
)(1:2:22)
s
2
¼
E
1 þ m
2
ðÞ
(e
2
þ me
1
)(1:2:23)
Principle stresses for the three-element rectangular rosette can also be
obtained directly from the measur ed strains, as shown in Equat ions 1.2.24
and 1.2.25:
s
1
¼
E
2
e
x
þ e

y
ÀÁ
1 À mðÞ
þ
1
1 þ mðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
x
À e
y
ÀÁ
2
þ 2e
45
À e
x
À e
y
ÀÁ
2
q
!
(1:2:24)
s
2
¼
E
2
e

x
þ e
y
ÀÁ
1 À mðÞ
À
1
1 þ mðÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
e
x
À e
y
ÀÁ
2
þ 2e
45
À e
x
À e
y
ÀÁ
2
q
!
(1:2:25)
1.3 understanding wheatstone bridges
The change in resistance that occurs in a typical strain gage is quite small,
as indicated in Example 1.1. Because resistance change is not easily measured,
voltage change as a result of resistance change is always preferred. A Wheat-

stone bridge is used to provide the voltage output due to a resistance change
at the gage. The strain gage bridge is simply a Wheatstone bridge with the
added requirement that either gages of equal resistance or precision resistors
be in each arm of the bridge, as shown in Figure 1.10.
1.3.1 THE BALANCED BRIDGE
The bridge circuit can be viewed as a voltage divider circuit, as shown in
Figure 1.11. As a voltage divider, each leg of the circuit is exposed to the same
e
0
R
2
D
R
1
R
4
B
R
3
AC
E
e
x
figure 1.10
A Wheatstone bridge circuit.
LEE: FATIGUE TESTING AND ANALYSIS Final Proof 22.6.2004 2:54pm page 10
10
Transducers and Data Acquisitio n