Experimental stress analysis BE notes by mohammed imran

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GHOUSIA COLLEGE OF ENGINEERING
RAMANAGARAM-562159
EXPERIMENTAL STRESS ANALYSIS
[10ME761]
Mr. MOHAMMED IMRAN
ASST PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING

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UNIT-1 Electrical Resistance Strain Gauges
1. INTRODUCTION
The electrical resistance strain gauge is widely used as it has a good measure of all the
characteristics a strain gauge should process. Three important advantages of electrical
resistance strain gauges are:
(i) They are small size and used in situations where other types of gauges
cannot be used.
(ii) As they have negligible mass, their effect on the quantity being measured
is significant. Further, they respond faithfully to rapidly fluctuating strains.
(iii) As the output is electrical, remote observation is possible. Further, the
output can be displayed, recorded or processed as required.
The principle on which the electrical resistance strain gauge operates was
discovered in 1856 by Lord Kelvin. Using a Wheatstone bridge, he measured the
change in resistance in copper and iron wires due to a tensile strain. He established
that the change in resistance is a function of strain and that different materials have
different sensitivities, i.e. the ratios of change in strain are different.
Lord Kelvin noted that the resistance of a wire increases with increasing strain and
decreases with decreasing strain. The question then arises whether this change in
resistance is due to the dimensional change in the wire under strain or to the change in
resistivity of the wire with strain. It is possible to answer this question by performing
a very simple analysis and comparing the results with experimental data which have
been compiled on the characteristics of certain metallic alloys. The analysis proceeds
in the following manner.
The strain gauge can be easily bonded to the test component with a suitable
adhesive as shown in fig (1). Any strain compressive or tensile in the test component
is faithfully transmitted to the strain gauge, after attaching lead wires to the solder
tabs on the gauge by soft soldering, the grid of the gauge, solder tab and base lead
wires are covered with a protective in coating to prevent oxidation, electrical shorting
and mechanical damage. The electrical circuit required for the measurement of the
very small changes in the gauge resistance is a variation of the well-known
Wheatstone bridge.

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The potentiometer circuit is used in some applications where purely dynamic stress
measurements are involved. Temperature sensitivity, i.e. the error in strain
measurement due to temperature variation can be reduced to a minimum either
through the use of suitable compensation circuits or by using self-temperature-
compensated gauges. However, expensive and complex auxiliary equipment is needed
to energize and record the signal from the gauge. With automatic data acquisition and
processing systems, output from hundreds of strain gauges bonded to the structure
under test can be processed and read out in units of strain or its derived quantities.
Typical applications of electrical strain gauges include:
(i) Experimental study of stresses in transport vehicles – aircraft, ships,
automobiles, trucks, etc.;
(ii) Experimental analysis of stresses in structures and machines –apartment
buildings, pressure vessels, bridges, dams, transmission towers, engines,
steam and gas turbines, machine tools, etc.;
(iii) Experimental verification of theoretical analysis;
(iv) Aid design and development of machines and structures;
(v) Assist failure analysis; and
(vi) As a sensing element in transducers for measurement of force, load,
pressure, displacement, torque, etc.

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2. STRAIN SENSITIVITY IN METALLIC ALLOYS
GAUGE FACTOR: The gauge factor or strain sensitivity of a metal is denoted by FA
and is defined as the ratio of the resistance change in a conductor per unit of its initial
resistance to applied axial strain.
Expression for gauge factor
The resistance R of a straight conductor of length L, area of cross section A and
resistivity ρ is given by -------- (1)
If the conductor is stretched its length will increase and area of cross section will
decrease. This result in a change of resistance R
Taking log on both sides of equation (1)
–
Differentiating we get
Now A = cD
2
Where C = a constant
D = some dimensions of conductor like width, diameter etc.
Therefore
The term dA represents the change in cross-sectional area of the conductor due to the
transverse strain, which is equal to - dL / L .
Where ν is the Poisson’s ratio for the conductor material we get
Therefore Eq. (b) becomes



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But Longitudinal strain ε = dL / L
Hence 
Or

The term on the left hand side of equation (2) which represents the change in
resistance / unit, resistance / unit strain is defined as the gauge factor and is denoted
by FA or SA.
Hence FA = SA = 
Where SA or FA is the sensitivity of the metallic alloy used in the conductor and is
defined as the resistance change per unit of initial resistance divided by the applied
strain.Examination of the above Eq (2) Shows that the strain sensitivity of any alloy is
due to two factors, namely,
 The change in the dimensions of the conductor, as expressed by the 1 + 2 term.
 The change in specific resistance with respect to ( dρ / ρ) / ϵ.
Experimental results show that F A varies from about 2 to 4 for most metallic alloys.
If strain sensitivity approaches to 2 when the gauge experiences plastic deformation,
this specify that specific resistance is zero (0) and Poisson’s ratio approaches to 0.5
ie., if have strain gauge which has strain sensitivity close to 2 from elastic (E.R) to
plastic region (P.R) it do not need any modification it becomes linear as shown in
figure.
The strain sensitivity is modified slightly by
the form of construction and the pattern of
the strain gauge then sensitivity of strain
gauge is termed the gauge factor F and is
given by
The manufacturer gauge factor ‘F’ for strain gauges is determined normally through a
calibration test in a uniaxial stress field, Ex. The tensile test it is to be note that the
strain gauge bonded to the calibration test is a destructive test
2
E.R
P.R
R/R
% of Strain 
Fig(2) R/R V.S % of Strain 

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3. GAUGE CONSTRUCTION :
It is theoretically possible to measure strain with a single length of wire as the sensing
element of the strain gauge; however circuit requirements needed to prevent
overloading of the power supply and minimum resistance required from
instrumentation point of view is 100Ω (i.e. lower limit of resistance)
If for example, diameter of the conductor is 0.025mm (0.001in) and resistance per
meter is 1000 calculate to have a minimum resistance of 100Ω, what is the length
of wire is required.
As a result a 100Ω strain gauge fabrication from wire having a diameter of 0.025mm
(0.001in) & have resistance of 25 Ω / in (1000 Ω) requires a single length of wire 4 in
(100mm) long, to make a measurement. But it is too long, obviously one cannot
measure strain at a point using a long wire! Hence, the gauge is formed by folded grid
etched on metal foil & wire grid.
Resistance strain gauges with a metallic-sensing element may be broadly classified
into four groups:
(i) Un-bonded- wire strain gauges,
(ii) Bonded-wire strain gauges,
(iii) Foil strain gauges, and
(iv) Wieldable strain gauges.
Construction of bonded- wire strain gauges:
Two methods of construction are generally employed for constructing bonded-wire
strain gauges. In the flat-grid type gauge illustrated shown in fig (3), the sensing
element or grid is formed by winding the wire around pins on a jig. The grid is then
lowered on a backing material and cemented to it before withdrawing the pins. The
grid is then covered with a suitable protective material. In the wrap-around type gauge
(fig 4), the wire is wound in the form of a helix around a thin walled cylinder of
insulating material. This cylinder is then flattened and bonded between two sheets of a
suitable insulating material. Alternatively, the wire may be wound on a thin card of
appropriate size and bonded between two sheets of insulating material.
As the wrap-around type gauge has two layers of wire and three layers of insulating
material it is considerably thicker than the flat-grid type gauge. The wrap-around

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gauge is not suitable for use on thin sections subjected to bending as the strain
indicated by it is inaccurate due to thickness effects. The performance of the wrap-
around gauge is unsatisfactory when transient or rapidly varying strain and/or
temperature are involved. Generally, flat-grid gauges are preferred as they are
superior to wrap-around gauges in terms of hysteresis, creep, elevated-temperature,
performance, stability, especially under hydrostatic pressure fluctuations, and current
carrying capacity. Formerly, only wrap-around gauges were available in shorter (less
than 6mm) gauge lengths. Today, mainly through improvements in production
methods, flat-grid gauges are also available in shorter gauge lengths.
Construction of foil gauges:
In the foil gauge, the foil grid is made by etching the desired grid pattern in a metal
foil only a few microns in thickness. The grid pattern can also be cut from the foil
using high-precision dies. The foil grid is carefully bonded to a thin flexible carrier or
backing as shown in fig (5). Any conceivable grid configuration can be produced by
these processes accurately.

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The width of the section at each turn of the straight filament portions as shown in
figure is increased to reduce the resistance in the transverse direction to a relatively
low value. This configuration makes the foil gauge quite insensitive to strains in the
transverse direction. The large surface area near the ends of the straight filaments also
ensures that linear conditions prevail over the complete active length of the grid.
As foil gauges have a greater bonding (surface) area to cross-sectional area
ratio than wire-gauges, they have enhanced (avoid) heat-dissipation properties. As this
permits use of higher voltage levels for gauge excitation, higher sensitivity can be
achieved. As the foil gauge has a larger contact area for bonding onto the test
component, the stress in the adhesive is lower. Consequently, the stress relaxation and
hysteresis are significantly less in foil gauges. For these reasons the performance of
the foil gauge is superior to that of the wire gauge. Currently, foil gauges are used
extensively. The use of wire gauges is mostly limited to applications such as stress
analysis at elevated temperatures, where it still possesses an edge over the foil gauges.
Gauge materials:
The strain sensitive alloy used in the wire or foil grid determines to a great
extent the operating characteristics of a strain gauge. Other factors which influence
significantly the performance of a strain gauge are the properties of the backing
material and the bonding material.
The desirable features or properties in a grid material are:
(i) High gauge factor, gauge factor constant over a wide range of strain,
(ii) High specific resistance
(iii) Low temperature coefficient of resistance
(iv) High elastic limit,
(v) High fatigue strength,
(vi) Good workability, soldering and weldability,
(vii) Low mechanical hysteresis,
(viii) Low thermal emf when joined with other materials, and
(ix) Good corrosion resistance.

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Probably the most serious problem in strain measurement is sensitivity to temperature
variations. This effect is minimized through the use of self-temperature-compensated
gauges and/or the bridge-compensation method.
Thermal emf superimposed on the gauge output must be avoided if dc
circuitry is employed. This factor presents no problem in the case of ac circuitry.
Corrosion at a junction between the grid and lead wire could possibly result in a
miniature rectifier; this would be more serious in an ac circuit.
In some applications the influence of strong magnetic fields on gauge performance is
of great importance. Grid materials of high nickel content are susceptible to the effect
of magnetostriction and magnetoresistivity. The apparent strain caused by these two
effects may be significant enough to preclude use of such materials as gauge grid
material.
Some of the important alloys or its equivalent that ore commonly used as
gauge grid material are
(i) Constantan or Advance,
(ii) annealed Constantan,
(iii) iso-elastic,
(iv) Nichrome V
(v) Karma (Ni-Cr-Al alloy with iron),
(vi) iron-chromium aluminium alloy (Armor D), and
(vii) Platinum-tungsten alloys.
Backing materials or carrier materials:
A strain gauge backing material has several functions to perform. It provides support
to the grid and ensures dimensional stability of the grid. It also provides mechanical
protection to the grid during handling and mounting. The backing material transmits
the strain from the test material to the grid and provides electrical insulation between
the grid and the test material or component.
The backing should be stiffer than the grid to support it and also to ensure that
the grid is not disturbed when it is strained. However, the backing should be flexible
enough to faithfully follow strain changes in the test component without in any way

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altering it. There are several backing materials which satisfy these requirements. They
are
i) Paper ii) Polymide iii) Epoxy plastics iv) Epoxy-phenolic resin
v) Glass fibre-reinforced epoxy-phenolic vi) Bakelite
A Good Carrier Materials should have the following characteristics
1. Minimum thickness
2. High Mechanical strength
3. High dielectric strength
4. Minimum Temperature restrictions
5. Good Adherence to cement used
4. ADHESIVES AND MOUNTING TECHNIQUE
Adhesives or bonding cement:
The bonded wire or foil gauge should be bonded to the test component with a
suitable adhesive. The strain gauge adhesive should be sufficiently elastic to faithfully
transfer strain in the test component to the gauge – sensing element or grid. For
optimum performance, the adhesive prescribed by the gauge manufacturer should be
used and the recommended procedure for mounting the gauge should be followed.
Several important factors have to be considered while selecting the adhesive
for a particular strain gauge and test component combination. It is very important to
ensure that the adhesive is compatible with both the gauge-backing material and test
material. The adhesive should not damage either of them. The adhesive should also
have long-term stability and high creep resistance at the maximum strain level over
the expected temperature range of operation. The adhesive should also have high
insulation resistance. The main types of adhesives which are commonly used are
i) Nitro-cellulose cement
ii) Epoxy cements
iii) Cyanoacrylate cement
iv) Phenolic adhesives
v) Ceramic cements

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Desirable characteristics of the Bonding cement or Adhesive
1. High mechanical strength
2. High creep resistance
3. High dielectric strength
4. Minimum temperature restrictions
5. Good adherence giving shear strength 10.5 to 14 Mpa
6. Minimum moisture absorption
7. Ease of application
8. Low setting time
A Typical Method for Bonding Strain Gauges:
A strain gauge can only give best results if it is bonded to the test piece in such
a manner so that the strain experienced by the gauge grid is precisely the same as the
strain of the test specimen. To achieve this the proper installation of gauge is very
important. In fact the bonding technique depends upon the type of gauge and the
cement to be used, which is supplied by the manufacturer. Hence the instructions as
per manufacturers catalogue should be followed. However, the following steps, in
general, may be followed while applying the strain gauge to the test specimens:
1. Surface preparation
2. Gauge preparation
3. Adhesive preparation
4. Gauge installation
5. Lead wire connection
6. Environmental protection.
A typical method of bonding a strain gauge is described here.
1. Surface preparation
The surface to which the gauge is to be bonded should be properly clean,
smooth and have the proper chemical affinity to the adhesive. Using emery paper or
cloth, any rust or paint on the surface is removed to obtain a smooth but not highly
polished surface. The prepared surface is then washed with a solvent to remove metal
or dirt particles and grease. Some of the cleaning fluids suitable for this purpose are
acetone, trichloroethylene, methyl-ethyl-ketone, chlorethene NU and Freon TF. To
achieve the degree of cleanliness required, the surface may have to be washed several
times and the washed surface wiped with clean paper towel or lint-free cloth until the
wiping cloth or paper no longer picks up dirt. The cleaned surface may be treated with
a basic solution to give the surface the proper chemical affinity to the adhesive.

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2. Gauge preparation
Fig (6) Gauge Location
The strain gauge with its backing (bonding surface) also degreased is bonded to the
test surface immediately after it is cleaned. The location of the gauge on the test
surface is marked with a pencil of hardness 4H to 8H. The lines are scribed outside
the area where the gauge is to be bonded. The gauge is then laid in position.
3. Adhesive preparation
Fig (7) Adhesive preparation
The supply leads on the gauge are taped so that the gauge can be raised up for
applying the adhesive without altering its location. Alternatively, the gauge can be
positioned by using a rigid transparent tape shown in figure (7a). After the adhesive is
applied to the area marked on the test surface, the gauge is brought into position and
is pressed down gently with a thumb as shown in figure. This gentle pressure with the
thumb is to squeeze out excessive adhesive and to ensure that there are no air bubbles
between the test surface and the backing Figure (7b). Terminal tabs may be cemented
along with the gauges.

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4. Gauge installation
The gauge installation is then subjected to a proper combination of pressure
and temperature for a recommended length of time to ensure complete cure of the
adhesive. During the curing process, the adhesive may expand because of heat,
undergo a volume reduction during polymerization, and contract upon cooling. Any
residual stresses in the adhesive will deform the relatively flexible gauge grid element
and influence the output of the strain gauge. Therefore the curing is critical and
should be complete particularly when the gauge installation is for long-term strain
measurements.
5. Lead wire connection
Fig (8) Lead – wire layout
Lead wires are needed to transmit the electrical signals from strain gauges to
the strain measuring instrument. The lead-wire system connected to the gauges must
perform satisfactorily under all environmental conditions. The lead wires should have
low resistance and low temperature coefficient of resistance. They should not
introduce significant resistance change, or generate or transmit electrical noise.
As copper has low specific resistance, it is commonly used as material for lead
wires. However, it should be noted that copper has a large temperature coefficient of
resistance and has poor corrosion and fatigue resistance. Tinned, plated or metal clad
solid copper wires have superior corrosion resistance.

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Stranded copper wires are flexible. Hence they are used when relative motion
between the lead wire and the component is expected. Nickel - chromium alloy lead
wires are prescribed in high-temperature applications.
These wires are suitable for temperatures up to about 370°C. As it has a high specific
resistance, only short-length lead wires should be used.
6. Environmental protection
Strain-gauge installations are affected by environments containing water or
moisture or chemical vapours. They are also susceptible to mechanical damage.
Therefore unless tests are to be conducted under laboratory conditions within a short
time after installation, protective coatings are essential. The important considerations
that influence the selection of a coating are test environment, test duration and the
degree of accuracy required. Though several commercial coatings are available, only
those proven by tests to be electrically and chemically compatible with a gauge
installation should be used in any application.
5. GAGE SENSITIVITIES AND GAGE FACTOR
The strain sensitivity of a single, uniform length of a conductor was preciously
defined as



(1)
Where ϵ is a uniform strain along the conductor and in the direction of the axis of the
conductor, this sensitivity FA is strain sensitivity to the axial strain, whenever the
conductor is wound into a strain gauge grid to yield the short gage length required for
measuring strain, the gage exhibits sensitivity to both axial and transverse strain.
The change is introduced by end loops, which are transverse to the straight portion of
the grid. Thus the gauge in addition to measures the strain reading. This is known as
the transverse or cross sensitivity of the gauge.
Transverse sensitivity of a strain gauge is a
measure of its response to strains
perpendicular to a primary sensing axis, a-a
(fig 9) however, their transverse sensitivity is
a small fraction of their axial sensitivity
Fig (9) Biaxial Strain Gauge

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The response of a bonded strain gage to a biaxial strain field can be expressed as

Where ϵa = normal strain along axial direction of gage
ϵt = normal strain along transverse direction of gage
at = shearing strain
Fa = sensitivity of gage to axial strain
Ft = sensitivity of gage to transverse strain
Fs = sensitivity of gage shearing strain
In general, the gage sensitivity to shearing strain is small and can be neglected. The
response of the gage can then be expressed as




Where is defined as the transverse sensitivity factor for the gage.
Now it is common practice to calibrate a strain gauge in a uniaxial stress field, i.e. in a
biaxial strain field with the ratio of the transverse-to-axial strain equal to the Poisson’s
ratio of the specimen material Eq (a). Thus Strain-gage manufacturers provide a
calibration constant known as the gage factor ‘F’ for each gage. The gage factor “F”
relates the resistance change to the axial strain Eq (4).
With this method of calibration, the strain field experienced by the gage is
biaxial, with
ϵt = -o ϵa (a)

where o = 0.285 is Poisson’s ratio of the material. If Eq.(a) is substituted into Eq.(3),
the resistance change in the calibration process is



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

Since the resistance changes given by Eqs. (4) and (5) are identical, the gage factor
is related to both and by the expression


 (6)
It may be noted that even when k t is as high as 10% and o = 0.3 is Poisson’s ratio
from Eq (6) Fa is only 1.03 times of ‘F’. It should be realized that Eq (4) is valid for
the uniaxial stress field used by the manufacturer to calibrate the strain gauge.
6. PERFORMANCE CAHRECTERISTICS OF FOIL STRAIN GAGES.
Foil strain gages are small precision resistors mounted on a flexible carrier that
can be bonded to a component part in a typical application. The gage resistance is
accurate to ±0.4 percent, and the gage factor, based on a lot calibration, is certified to
±1.5 percent. These specifications indicate that foil-type gages provide a means for
making precise measurements of strain. The results actually obtained however, are a
function of the installation procedures, the state of strain being measured, and
environment conditions during the test. All these factors affect the performance of a
strain- gage system, in general.
a. Strain- Gage Linearity, Hysteresis, and Zero Shift
b. Temperature Compensation
c. Elongation Limits
d. Stability
A. Strain- Gage Linearity, Hysteresis, and Zero Shift
One measure of the performance of a strain gage system (system here implies gage,
adhesive, and instrumentation) involves considerations of linearity, hysteresis, and
zero-shift. If gage output, in terms of measured strain, is plotted as a function of
applied strain as the load on the component is cycled, results similar to those shown in
fig.4. Will be obtained
A slight deviation from linearity is typically observed, and the unloading curve
normally falls below the loading curve to form a hysteresis loop. Also, when the
applied strain is reduced to zero, the gage output indicates a small negative strain,

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termed zero shift. The magnitude of the deviation from linearity, hysteresis, and zero
shift depends upon the strain level, the adequacy of the bond, the degree of cold work
of the foil material, and the carrier material.
Fig 10: A typical strain cycle showing nonlinearity, hysteresis, and zero
shift (scale exaggeration)
For properly installed gages, deviations from linearity should be
approximately 0.1 percent of the maximum strain for polyimide carriers and 0.05
percent for epoxy carriers. First cycle hysteresis and zero shifts are more frequently
observed in typical applications If possible, shift cycling to 125 percent of the
maximum test strain is recommended since the amount of hysteresis and zero shift
will decrease to less than 0.2 percent of the maximum strain after 4 or 5 cycles.
B. Temperature Compensation
A very important problem in strain measurement is the effect of temperature
on the performance. Strain gauge installation is subjected to temperature changes
during the test period, and careful consideration must be given for determining
whether the change in resistance is due to applied strain or temperature change. When
the ambient temperature changes four effects occur which may alter the performance
characteristics of the gauge
1. The strain sensitivity of the metal alloy used for the grid changes
2. The gage grid either elongates or contracts (∆L/L = α∆T)

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3. The base material upon which the gage is mounted either elongates or contracts
(∆L/L = β∆T).
4. The resistance of the gage changes because of the influence of the temperature
Co - efficient of resistivity of the gage material (∆R/R = υ∆T)
Where α = Thermal Co efficient of expansion of gage material
β = Thermal Co efficient of expansion of base material
ρ = Thermal Co efficient of resistivity of gage material
The combined effect of these three factors will produced a temperature induced
change in resistance of the gauge (∆R/R) ∆T which may be expressed as
(∆R/R) ∆T = (β – α) ∆T.F + υ∆T -------- (1)
Where F = gauge factor and ∆T is rise in temperature
In order to prevent significance errors due to this effect some form of temperature
compensation is usually employed when strain gauges are used in applications where
steady state strain must be measured.
There are two methods are available for effecting temperature compensation in a
gauge installation. In the first method, the gauge alloy coefficients α & β are adjested
to minimize the range when bonded to a test material with a matching coefficient for
linear expansion α. Such gauge is called a Self – temperature compensated gauge
The second method involves in general, the use of a compensating (dummy) gauge in
the electrical system to eliminate the error due to the apparent strain. In theis method
for zero error, the apparent strains be exactly equal i.e. in turns to temperature effect is
approximated.
C. Elongation Limits
The maximum strain that can be measured with a foil strain gage depends on the
gage length, the foil alloy, the carrier material, and the adhesive. The Advance and
Karma alloys with polyimide carriers, used for general-purpose strain gages, can be
employed to strain limits of ±5 and ±1.5 percent strain, respectively. This strain gage
is adequate for elastic analyses on metallic and ceramic components, where yield or
fracture strains rarely exceed 1 percent; however, these limits can easily be exceeded
in plastic analyses, where strains in the post yield range can become large. In these
instances, a special post-yield gage is normally employed, it is fabricated using a
double annealed Advance foil grid with a high-elongation polyimide carrier.
Urethane-modified epoxy adhesives are generally used to bond post-yield gages to the

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structure. If proper care is exercised in preparing the surface of the specimen,
roughening the back of the gage, formulating a high-elongation plasticized adhesive
system, and attaching the lead wires without significant stress raisers, it is possible to
approach strain levels of 20 percent before cracks begin to occur in the solder tabs or
at the ends of the grid loops.
Special purpose strain gage alloys are not applicable for the measurements of
large strains. The Isoelastic alloy will withstand ±2 percent strain; however, it
undergoes a change of sensitivity at strains larger than 0.75 percent (see fig. below).
Armour D and Ni chrome V are primarily used for high temperature measurements
and are limited to maximum strain levels of approximately ±1 percent.
Fig 11: A liquid metal electrical resistance strain gage.
For very large strains, where specimen elongations of 100 percent may be
encountered, liquid-metal strain gages can be used. The liquid-metal strain gage is
simply a rubber tube filled with mercury or a gallium-indium-tin alloy, as indicated in
fig. 12. When the specimen to which the gage is attached is strained, the volume of
the tube cavity remains constant since Poisson’s ratio of the rubber is approximately
0.5. Thus the length of the tube increases (l = ϵl) while the diameter of the tube
decreases (d = -ϵd). The resistance of such a gage increases with strain, and it can
be shown that the gage factor is given by
(1)
Performance characteristics response of a liquid metal gauge shows lightly nonlinear
with increasing strain due to the increase in gauge factor with strain and due to change
in resistance of gauge.
Rubber capillary tube
Gallium Indium tin Lead wire

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D. Stability:
In some applications strains in the test components may have to be recorded over a
long period of time running into months and years with no freedom to unloading the
component for checking the zero reading in such applications maximum stability of
the strain gauge insulation is very important requirement.
Moisture and humidity effects on the backing, stress relaxation in the adhesive,
backing and grid material and instabilities in the resistance in the inactive arm of the
strain measuring bridge are the primary causes for drift in the zero reading of a strain
gauge installation.
7. ENVIRONMENTAL EFFECTS
The performance of resistance strain gages is markedly affected by the environment.
Moisture, temperature extremes, hydrostatic pressure, nuclear radiation, and cyclic
loading produce changes in gage behavior which must be accounted for in the
installation of the gage and in the analysis of the data to obtain meaningful results.
Each of these parameters is discussed in the following subsections.
(a) Moisture and humidity:
Absorption of moisture by the backing material and the adhesive causes significant
degradation of short term and long term performance of the strain gauge installation.
Moisture absorption causes are
 A decrease in insulation resistance of the gauge which manifests itself in the
form of an apparent strain due to change in the effective resistance of the
gauge.
 Variation in the gauge factor due to decrease in the strength and rigidity of the
bond
 Apparent strain due to strain in the grid induced by stress in the adhesive
 Apparent tensile strain resulting from the thing of the grid elements caused by
gauge cussed induced electrolysis in the moisture laden adhesive.
Effective protection of the gauge installation from moisture will prevent the
absorption of moisture and avoid the consequent degradation in the performance by
using protective coatings.

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Protective coatings:
Strain-gauge installations are affected by environments containing water or
moisture or chemical vapours. They are also susceptible to mechanical damage.
Therefore unless tests are to be conducted under laboratory conditions within a short
time after installation, protective coatings are essential. The important considerations
that influence the selection of a coating are test environment, test duration and the
degree of accuracy required. Though several commercial coatings are available, only
those proven by tests to be electrically and chemically compatible with a gauge
installation should be used in any application.
Fig (12) Waterproofing a strain gauge in severe environment
Wax coatings like microcrystalline wax, are melted and poured directly over
the strain gauge installation to form a coating of thickness in the range 0.5 to 1.5 mm.
These are good barriers to water or moisture but provide little mechanical protection.
It is well-suited for general-purpose laboratory work and field applications. It can
be used at temperatures up to 60°C. An air-drying polyurethane coating or acrylic
coating can also be used in the laboratory for protection against moisture over a
temperature range of -40° to +100°C.
Both single-component and two-component epoxy plastics are available for
use as protective coatings. For example, a solvent-thinned single – component epoxy
resin compound can be brushed on the gauge and cured to provide an excellent
chemical and mechanical protection. It is an excellent coating for transducer
applications up to 120°C.
Synthetic rubber coatings such as nitrile rubber, silicone rubber, poly-
sulphides, butyl polymers, etc. provide good protection against moisture, fresh and

22
salt water immersion or spray. The operating temperature range varies according to
the polymer selected over the range of -150° to +300°C.
A two-part 100% solids polysulphide modified epoxy compound is available
as a general-purpose coating. It gives good protection against oils, greases, gasolines,
most acids, alkalis and solvents. However, the operating temperature range for this
compound is only 0° to 100°C.
Metal-foil tapes when used with teflon or nylon padding and a sealing material
provide protection against mechanical damage and moisture. In case of gauge
installations subjected to long-term exposure to adverse environment, such as sea
water, several layers of different types of coatings are needed. A cross-sectional view
of a gauge instal1ation protected for long-term use is shown in Fig (12).
(b) Effects of Hydrostatic Pressure:
In the stress analysis of pressure vessels and piping systems, strain gages are
frequently employed on interior surfaces where they are exposed to a gas or fluid
pressure which acts directly on the sensing element of the gage. Under such
conditions, pressure-induced resistance changes occur which must be accounted for in
the analysis of the strain gage data.
Milligan and Brace independently studied this effect of pressure by mounting
a gage on a small specimen, placing the specimen in a special high-pressure vessel,
and monitoring the strain as the pressure was increased to 140,000Ib/in2
(965MPa). In
this type of experiment, the hydrostatic pressure p produces a strain in the specimen
which is given by
W.N.T Hook’s law

Hydrostatic pressure p produces a strain in the specimen which is given by


(2)
Where KT = - (1-2)/E is often referred to as the compressibility constant for a
material. The strain gages were monitored during the pressure cycle, and it was
observed that the indicated strains were less than the true strains predicted by Eq. (2).

23
The difference between the true strains and the indicated strains was attributed to the
pressure effect.
(c) Effects of High Temperature
Resistance-type strain-gages can be employed at elevated temperatures for both static
and dynamic stress analyses; however, the measurements require many special
precautions which depend primarily on the temperature and the time of observation.
At elevated temperatures, the resistance R of a strain gage must be considered to be a
function of temperature t and time t in addition to strain ϵ
Thus R=f (ϵ, T, t) (2)
The resistance change R/R is then given by

   (3)
Where
= =gage sensitivity to strain (gage factor)
= = gage sensitivity to temperature
= = gage sensitivity to time
The combined effect of this three factor will produced a temperature induced change
in strain with respect to time; this induced change in resistance of the gauge

can then be expressed in terms of the three sensitivity factor as

   (4)
The discussion of performance characteristics of foil strain gages. It was shown that
sensitivity of the gages t p temperature and time was minimized at normal operating
temperatures of 0 to 1500
F (-18 to 650
) by proper selection of the strain-gage alloy
and carrier materials. As the test temperature increases above this level, however, the
performance of the gage changes, and FT and Ft are not usually negligible.

24
8. STRAIN GAUGE CIRCUITS
The change in the resistance due to variation in applied strain is extremely small.
Two electrical circuits – the potentiometer circuit and Wheatstone bridge – are used to
measure such small changes in resistance. Mostly, some variation of the Wheatstone
bridge is used for this purpose.
I. The potentiometer and its applications to strain measurements:
The potentiometer circuit is well suited for dynamic measurements. An attractive
feature of the circuit is its extreme simplicity. The potentiometer circuit, which is
often employed in dynamic strain-gage applications to convert the gage output R/R
to a voltage signal E, is shown in Fig.7. Small increment in the open circuit voltage
E of the potentiometer circuit can be derived as follows.
Fig.13: potentiometer circuit.
Consider the resistance in the circuit R1 & R2 Apply krickoff’s-voltage law to circuit
Consider the resistance in the circuit R1 & R2 the open-circuit voltage E across AB is
Where V is the input (excitation) voltage and r = R2 / R1 is the resistance ratio for the
circuit. If incremental change R1 and R2 occur in the value of the resistors R1 and
R2, the change E of the output voltage E can be computed by using Eq. (1) as
follows;


 
Solving Eq. (a) for E gives

25


 


 


 

  
 

Numerator=   
  
  
  
   
 
 
 
 
 
Where resistance ratio
 
 

26
Substituting A & B in Eq (b) Which can be expressed in the following form by
introducing r = R2/R1

 
 

 
 

 
 
Examination of Eq.(2) shows that the voltage signal E from the potentiometer
Circuit is a nonlinear function of R1/R1 and R2/R2.To inspect the nonlinear aspects
of this circuit further, it is possible to rewrite Eq. (2) in the form

 
Where nonlinear term is expressed as
 
 
Equations (3) are the basic relationships which govern the behavior of the
potentiometer circuit, and as such they can be used to establish the applicability of
this circuit for strain-gage measurements. Error due to nonlinearity of the circuit can
be estimated with R1 as the resistance due to string gauge, R2 as a resistor of fixed
resistance and  R1 as the change in the resistance of the gauge due to a strain .
Where R2 as a resistor of fixed  R2 = 0 Equation 3b becomes

27

It shows that the nonlinear term is depends on magnitude of strain , gauge factor ‘F’
and ratio of m = R2 / R1. In most strain measurements the nonlinearity term can be
neglected & E [Eq (2)] can be determined from

 
However, if high accuracy in strain measurement is required or larger strains are to be
measured. Then plastic strain determinations in metallic materials the output signal
determined through Eq (5) can be corrected for error due to nonlinearity.
(a) Range and sensitivity of the circuit:
The output signal per unit strain is known as circuit sensitivity Sc of the
potentiometer circuit is given by

 
E output signal per unit strain =

  
 
With an active strain gauge R1 and fixed-blast resistor R2
i.e.   


28

Where gauge factor

for strain gauge, thus the circuit sensitivity of the
potentiometer circuit is depends on the voltage V and ratio m = R2/R1, the Sc is limited
by the maximum power Pg that can be dissipated by the gauge without unfavorable
effect of performance.
As the power dissipated in the gauge is equal to I2
g Rg i.e. Pg = I2
g Rg input voltage.
Where m = R2/R1
Substituting Eq (a) in Eq (3) we get
Circuit sensitivity of the potentiometer circuit is depends on and the
term is completely depends on m = R2/R1 resistance ratio limited value is 9.
Therefore, , the term is depends on the characteristics of the
strain gauge range between 3 to 700 and Sc range 5 to 10µV per micro-strain.

29
II. WHEATSTONE BRIDGE:
 Wheatstone bridge is 2nd
circuit which can be employed to determine the
change in resistance to a strain.
 Wheatstone bridge can be used to determine both dynamic and static strain gauge
readings.
 The bridge as a direct readout device where the output voltage E is measured &
related to strain.
 The bridge may be used as a null balance system, where the output voltage E is
adjusted to zero value by adjusting the resistive balance of the bridge.
 There are two types of Wheatstone bridge circuits are used for the strain
measurement.
o Null balance type (balanced E = 0)
o Out of balance type (unbalance E ≠ 0)
A dc Wheatstone bridge consisting of four resistance arms with a battery and a meter
is shown in figure (14).
Fig (14) Wheatstone bridge
In this bridge the resistance shown in each of the four arms of the bridge can represent
a strain gauge. A voltage V is applied to the bridge. Some measuring instrument or
meter such as a galvanometer is used to measure the output of the bridge.

30
(a) Null Balance Type (balanced E = 0):
Condition for balancing of Wheatstone bridge. The requirement for balance, i.e.
zero potential difference E between points B and D for the bridge shown in fig (8) can
be determined as follows:
The voltage drop VAB across R1 is
VAB = i1R1= V / (R1 + R2) . R1 --------------------- (1)
Similarly the voltage drop VAD across R4 is
VAD = i4R4= V / (R3 + R4) R4 ---------- (2)
The potential difference between B and D, VBD, is
VBD = VAB - VAD = E ------------ (3)
Substituting of equations. (1) and (2) in (3), we get
–
The condition for balance is that the voltage E should be zero i.e. the numerator in
Eq. (4) should be zero:
i.e
Or
Or
Equation (5) or (6) gives the condition for the Wheatstone bridge to balance, that is
the ratio of resistances of any two adjacent arms of the bridge must be equal to the
ratio of the resistances of the remaining two arms taken in the same order.

31
Balancing Wheatstone Bridge:
Consider an initially balanced bridge i.e. consider equation for condition of balancing
R1 / R2 = R4 / R3 satisfied, and then change R1 and/or R2 by a small increment. Any
imbalance that may result from these changes can be removed and the bridge can be
again balanced by changing, say the ratio R4 / R3 in such a way that the above
condition for balancing is again satisfied.
For example, assume that the resistance R1 is a strain gauge mounted on a specimen.
The bridge can be balanced first under no load by altering the ratio of resistances R4 /
R3 suitably. After the specimen is loaded, the bridge can be balanced again by
adjusting the ratio of resistances R4 / R3. if the change in this ratio is known, then the
change in the strain gauge resistance ∆R1 due to the load can be determined. The
corresponding strain can be calculated from equation ε = ∆ R/R / F
(b) Out of Balance Type: (unbalance E ≠ 0)
It is this feature balancing which permits the Wheatstone bridge to he
employed for static strain measurements. The bridge is initially balanced before
strains are applied to the gages in the bridge; thus the voltage E is initially zero, and
the strain-induced voltage E can be measured directly for both static and dynamic
applications.
Consider an initially balanced bridge with R1R3 = R2R4 so that E = 0 and then
change each value of resistance R1 R2 R3and R4 by an incremental amount R1  R2
R3and R4.The voltage output E of the bridge can be obtained from Eq. (4) which
becomes

 
 
 
 

Where A is the determinant in the numerator and B is the determinant in the
denominator. By expanding each of these determinants, neglecting second-order
terms, and noting R1R3 = R2 R4 it is possible to show that

32
Numerator A
 
 
   
       
   
   
   
= (b)
 
 
   
= (c)
Substituting Eqs. (b) to (d) yields

 (7)
By letting R2/R1 = r it is possible to rewrite Eq.(8.19) as
 (8)
In reality, Eqs, (7) and (8) both carry a nonlinear term 1 - , However, the influence
of the nonlinear term is quite small and can be neglected, provided the strains being
measured are less than 5 percent Equation(8) thus represents the basic equation which
governs the behavior of the Wheatstone bridge in strain measurement.

33
B. Wheatstone-Bridge Sensitivity:
The sensitivity of the Wheatstone bridge must be considered from two points of view:
(1) With a fixed voltage applied to the bridge regardless of gage current (a condition
which exists in most commercially available instrumentation) and
(2) With a variable voltage whose upper limit is determined by the power
dissipated the particular arm of the bridge which contains the strain gage.
By recalling the definition for the circuit sensitivity given, and using the basic bridge
relationship given in Eq. (8), it is clear that the circuit sensitivity is


(9)
If all strain have equal value
If a multiple-gage circuit is considered with n gages (where n = I. 2. 3, or 4) whose
out puts sum when placed in the bridge circuit. it is possible to write
Which by Eq. using
Substituting Eq. (11) into Eq. (9) gives the circuit sensitivity as
(12)
This sensitivity equation is applicable in those cases where the bridge voltage V is
fixed and independent of gage current. The equation shows that the sensitivity of the
bridge depends upon the number n of active arms employed, the gage factor F , the

34
input voltage, and the ratio of the resistances R1 /R2 . A plot of r versus (I + r)2
(the
circuit efficiency) that maximum efficiency and hence maximum circuit sensitivity
occur when r = 1. With four active arms in this bridge a circuit sensitivity of F×V
can be achieved, whereas with one active gage a circuit sensitivity of only F × (V/4)
can be obtained.
9. CONSTANT-CURRENT CIRCUITS:
The potentiometer and Wheatstone bridge circuits driven with a voltage source which
ideally remains constant with changes in the resistance of the circuit. These voltage-
driven circuits exhibit nonlinear output whenever R/R is large. This nonlinear
behavior limits their applicability to semiconductor strain gages. It is possible to
replace the constant- voltage source with a constant-current source, and it can be
shown that improvements in both linearity and sensitivity result.
Constant-current power supplies with sufficient regulation for strain-gage applications
are relatively new and have been made possible by advances in solid-state electronics,
basically the constant-current power supply is a high impedance (1 to 10M) device
which changes output voltage with changing resistive load to maintain a constant
current.
(a) Constant-Current Potentiometer Circuit:
Consider the constant-current potentiometer circuit shown in Fig, 34a. When a very
high impedance meter is placed across resistance R1 the measured output voltage E is
E = I R (1)
When resistances R 1 =R and R2 =R b change by R1 =R and R2 = R b =0 the
output voltage becomes
E + E = I(R+ R) (a)
E = I(R+ R) – E (a’)
Thus from Eq’s, (1) and (a)
E = I(R+ R) - IR = I R = I R

(2)

35
Fig (15): constant current potentiometer circuits.
It should be noted that R, does not affect the signal output. Indeed. Even R2 is not
involved in the output voltage, and hence it can be eliminated to give the very simple
potentiometer circuit shown in Fig. 15.
If R= Rg is the resistance of a strain gauge with gauge factor F and Rg is the change
in resistance corresponding to strain
Substituting Eq.

into Eq. (2) yields
 (3)
By increasing the gauge current Ig to the maximum value dictated by power
dissipation considerations, the circuit sensitivity can be maximized thus,
The circuit sensitivity Sc = E/ reduces to
(4)
If the constant-current Source is adjustable, so that the current I can be increased to
the power-dissipation limit of the strain gage, then I = I g and Eq (4) can be rewritten
as
(5)
Thus, the circuit sensitivity is totally dependent on the strain-gage parameters Pg and
R g and S g and is totally independent of circuit parameters except for the capability to
adjust the current source. Comparison of Eqs and (5) shows
that the sensitivities differ by the r / (1+ r) multiplier for the constant-voltage
potentiometer: thus. Sc will always be higher for the constant-current potentiometer.

36
It was noted in deriving Eq. (2) that R2 and R2 did not affect the signal output of the
constant-current potentiometer. This indicates that temperature compensation by
signal cancellation in the strain-gage circuit or signal addition cannot be
performed. It is possible to maintain the advantages of high sensitivity and perfect
linearity of this circuit and to obtain the capability of signal addition or subtraction
by using a double constant-current potentiometer circuit.
(b) Constant-Current Wheatstone bridge Circuits:
To consider a bridge driven by a constant current supply as shown in fig below. The
current I delivered by the supply divides at point A or the bridge into currents I1 and I2
where I=I1+I2 the voltage drop between points A and B of the bridge is
(i)
And the voltage drop between points A and D is
(ii)
Thus the output voltage E from the bridge can be expressed as
(1)
Fig: Constant Wheatstone bridge with constant current supply.
Far the bridge to be in balance (E=0) under no-load conditions,
(2)

37
Consider next the voltage and note the
(iii)
From which
(v)
Eq. (IV) can be substituted in to Eq. (v)
Substituting eqs. (vi) in to (1)
(3)
From Eq. (3) It is evident that the balance condition (E = 0) for the constant-current
Wheatstone bridge is the same as that for the constant-voltage Wheatstone bridge.
namely.
(4)
If resistance R1, R2, R3, and R4 change by the amounts R1, R2, R3, and R4, the
voltage E+E measured with a very high impedance meter is
 
    (vii)
    
Expanding Eq. (vii) and simplifying after assuming the initial balance condition gives
Fig: constant Wheatstone bridge designed to minimize nonlinear effect

38


Inspection of Eq. (5) shows that the output signal E, is nonlinear with respect to R
because of the term ƩR in the denominator and because of the second-order terms
and within the bracketed quantity The nonlinear, of the
constant-current Wheatstone bridge, however is less than that with the constant-
voltage bridge Indeed, if the constant-current Wheatstone bridge is properly designed,
the nonlinear terms can he made insignificant even for the large encountered
with semiconductor strain gages.
The nonlinear effects in a typical situation call he evaluated by considering the
constant-current Wheatstone bridge shown in Fig below. A single active gage is
employed in arm R1, and it temperature-compensating dummy gage is employed in
arm R4 Fixed resistors arc employed in arms R2 and R3. Thus
 
Under stable thermal environments,   Equation (5) then
reduces to

Again it is evident that. Eq. (6) is nonlinear due to the presence of the term in
the denominator to determine the degree of the nonlinearity let
Inspection of Eq. (7) shows that the nonlinear term can be minimized by increasing
m in this case; nonlinear term will depend on the gage factor F and on the
magnitude of the strain .

39
Unit-2 STRAIN ANALYSIS METHOD
TWO-ELEMENT RECTANGULAR ROSETTE
Electrical-resistance strain gages are normally employed on the free surface of a
specimen to establish the stress at a particular point on this surface. In general it is
necessary to measure three strains at a point to completely define either the stress or
the strain field, In terms of principal strains it is necessary to measure and the
direction of relative to the x-axis as given by the principal angle . Conversion of
the strains in to stresses requires, in addtion, knowledge of the elastic constants E and
 of the specimen material.
Where considering only one direction of stress (x-axis) and will
be
We know Hook’s law
Let as assume isotropic state of stress where then the
magnitude of stress can be established from
(1)
Consider a two-element rectangular rosette Similar to those illustrated in fig:1 is
mounted on the specimen with its axes coincident with the principal directions. The
two principal strains, and obtained from the gages can be employed to give the
principal stresses and .
Fig 1:Two-element rectangular rosette.
These relations given the complete state of stresses at a point only.

40
THREE-ELEMENT ROSETTE
Fig 2:Three gage elements placed at arbitrary angles relative to the x and y axes.
In the most general case, no knowledge of the stress field or its directions is available
before the experimental analysis is conducted. Three-element rosettes are required in
these instances to completely establish the stress field. To show that three strain
measurements are sufficient, consider three strain gages aligned, along axes A, B, and
C, as shown in Fig. (2).
From Equations of stress transformation where element is rotated on z-axis
therefore stresses along z direction is zero. Using fig 2: equation is evident that
(3)
The Cartesian components of strain can be determined from a
simultaneous solution of Eq. (3). The principal strains and the principal directions can
then be established by employing Eq. Principal stresses. The results are
Using
(4)
Where is the angle between the principal axis ( ) and the x axis. The principal
stresses can then be computed from the principal strains by utilizing Eqs. (2).

41
THREE-ELEMENT RECTANGULAR ROSETTE
In actual practice, three-element rosettes with fixed angles (that are, and
fixed at specified values) are employed to provide sufficient data to completely
define the stress field. These rosettes are defined by the fixed angles as the rectangular
rosette, the delta rosette, and the tee-delta rosette.
The three-element rectangular rosette employs gages placed at the 00
,450
, and 90°
positions, as indicated in Fig. 3.
Fig 3: Gage position in a three element rectangular rosette.
For this particular rosette it is clear from Eqs. (3) that
(a)
Thus by measuring the strains the Cartesian components of
strain can be quickly and simply established through the use of
equation (a).Next, by utilizing Eqs. (4), the principal strains and can
beestablished as
(b)
And the principal angle is given by

42
The solution of equation (C) gives two values for the angle , namely, which refers
to the angle between thex-axis and the axis or the maximum principal strain and
which is the angle between the x axis and the axis of the minimum principal strain
. It is possible to show that the principal axes can be identified by applying the
following rules
(d)
Finally, the principal stresses occurring in the component canbe established
byemploying (b) (c) together with (2) to obtain
(e)
The use of Eqs. (a) To (e) permits a determination of the Cartesian components of
strain, the principal strains and their directions, and the principal stresses by a totally
analytical approach.
THREE ELEMENT DELTA ROSETTES:
The delta rosette employs three gages placed at the 0. 120 , and 240 positions, as
indicated in Fig.4.
Figure4. Gage positions in a three-element delta rosette.

43
For the angular layout of the delta rosette it is clear from Eqs. (3) That
(a)
Solving Eq.(a) for in terms of gives
Also from the Eq. (4) the principal strains can be written in terms
of gives
(C)
The principal angle can be determined from Eq (4) as
The solution of Eq. (d) gives two values for the principal angle , as was the case for
the rectangular rosette.
Principal angles can be identifying by applying the following rules:
(e)
Finally, the principal stresses can determine from the principal strain by employing
Eq. (2) to obtain
(f)

44
By employing Eq. (e) to (f), it is possible to determine the Cartesian components of
strain, the principal strain and their directions, and the principal stress from the three
observation strain made with a delta rosette.
CORRECTIONS FOR TRANSVERSE STRAIN EFFECTS
It was noted that foil-type resistance strain gages exhibit a sensitivity S, to transverse
strains. Fig: 5. shows that in certain instances this transverse sensitivity can lead to
large errors, and it is important to correct the data to eliminate this effect. Two
different procedures for correcting data have been developed.

ϵ
ϵ
ϵ 
ϵ

The first procedure requires a priori knowledge of the ratio
ϵ
ϵ
of the strain field. The
correction factor is evident in Eq. (b)below. Where
ϵ ϵ

ϵ
ϵ
(a)
Fig5: Error as a function of transverse sensitivity factor with the biaxial strain
ratio as a parameter.
The termϵ is the apparent strain, and the correction factor CF is given by
ϵ

ϵ
ϵ
(b)
It is possible to correct the strain gage for this transverse sensitivity by adjusting its
gage factor. The corrected gage factor which should be dialed into the measuring
instrument is

45
ϵ
ϵ

(c)
Correction for the cross-sensitivity effect when the strain field is unknown is more
involved and requires the experimental determination of strain in both the x and y
directions. If ϵ and ϵ are the apparent strains recorded in the x and y directions,
respectively. Then from Eq. (a) it is evident that
ϵ

ϵ ϵ ϵ

ϵ ϵ (d)
Where the unprimed quantities ϵ andϵ are the true strains, Solving Eqs. (d) Forϵ
andϵ gives
ϵ

ϵ ϵ ϵ

ϵ ϵ (e)
Equation (e) gives the true strainsϵ andϵ in terms of the apparent strains ϵ and
ϵ Correction equations for transverse strains in two- and three-element rosette.
Correction equations for transverse sensitivity effect in three-element rosette.
Transverse sensitivity effect for three element of rectangular strain gage as follows.



Where are indicated strains and ЄA ,ЄB , ЄC are corrected strains
Transverse sensitivity effect for three element of delta strain gage as follows.



Where are indicated strains and ЄA ,ЄB , ЄC are corrected strains

46
THE STRESS GAGE:
The transverse sensitivity which was shown in the previous section to result in errors
in strain measurements can be employed to produce a special-purpose transducer
known as a stress gage.
The stress gage looks very much like a strain gages except that its grid is designed to
give a select value of Kt, so it the output R/R is proportional to the stress along the
axis of the gage. The stress gage serves a very useful purpose when a stress
determination in a particular direction is the ultimate objective of the analysis, for it
can be obtained with a single gage rather than a three-element rosette.
The principle upon which a stress gage is based is exhibited in the following
derivation. The output of a gage R/R as expressed by Equation given below is

ϵ ϵ (a)
The relationship between stress and strain for a plane state of stress is given by
Equations
Substituting Equation (a) and(b) yields


(c)
Examination of Eq. (c) indicates that the output of the gage R/R will be independent
of , if = . It can also be shown that the axial sensitivity Fa of a gage is related to
the alloy sensitivity Fa, by the expression
(d)
Substituting Eq. (d) into Eq. (c) and letting Kt=  leads to


(e)
Since the factor

is a constant for a given gage alloy and specimen material, the
gage: output in terms of R/R is linearly proportional to stress.

47
In practice the stress gage is made with a V-type grid configuration. As shown
in Fig. (6a) below Further analysis of the stress gage is necessary to understand its
operation in a strain field which is unknown and in which the strain gage is placed in
an arbitrary direction. Consider the placement of the gage, as shown in Fig. (6b) along
an arbitrary .x axis which is at some unknown angle with the principal axis
corresponding to .The grid elements an: at a known angle relative to the x axis.
Fig:(6a)
Fig(6b) the stress gage relative to the x
axis and the principal axis corresponding
to
The strain along the top grid clement is given by a modified form of Eq. (3) as
ϵ (f)
The strain along the lower grid element is
ϵ (g)
SummingEqs. (f) and(g) and expanding the cosine terms yield
ϵ ϵ (h)
Note from the Mohr’s strain circles given by
(i)
(j)
Substituting Eq. (i) and (j) into (h) gives
ϵ ϵ
ϵ ϵ
ϵ ϵ (k)

48
If the gage manufactured so that is equal to arctan
And Eq. (k) becomes
ϵ ϵ (l)
Substituting (l) into Eq. (e) gives

ϵ ϵ (m)
Where ϵ ϵ is the average strain indicated by the two elements of the gage
is equal to (R /R)/ .
The gage reading will give ϵ ϵ , and it is only necessary to multiply.this
by

obtain . The stress gage will thus give directly; with single gage.
However, it does not give any data regarding or theprincipal angle
Moreover, may not be the most important stress since itmay differ
appreciablyfrom . If the directions of the principal stresses areknownthe stress gage
may be used more effectively by choosing the x axis tocoin side with the principal
axis, corresponding to so that = .In fact, whenprincipal directions are known, a
conventional single-element strain gage canbe employed as a stress gage.
Fig: 7 A single element strain gage employed as a stress gage when the principal
directions are known.
This adaption is possible if the gage is located along a line which makes an angle
with respect to the principal axis, as shown fig 7:In this case thestrains will be
symmetrical about the principal axis; hence it is clear that
ϵ ϵ ϵ
and Eq. (m) reduces to

49

ϵ (n)
The value is recorded on the strain gage and converted to or directly by
multiplying by E / (1–v), this procedure reduces the number of gages necessary if
only the value of is to be determined. The saving of a gage is of particular
importance in dynamic work when the instrumentation required becomes Complex
and the number of available channels of recording equipment is limited.
PLANE-SHEAR GAGES OR TORQUE GAGE
Consider two strain gages A and B oriented at angles with respect to the x
axis, as shown in Fig. 8. The strains along the gage axes are given by a modified form
of Eqs. (a), below as
Figure 8 Positions of gagesA and B for measuring
Using equations
(a)
(b)
From Eq.(b) the shear strain is

50
(c)
If gages A and B are oriented such that
(d)
Then EQ. (c) reduce to
(e)
Since the cosine is an even function, satisfies Eq. (d).Thus, the shearing
strain is proportional to the difference between normal strains experienced by
gage A and B when they are oriented with respect to the x axis as shown in fig. 10
The angle can he arbitrary: however, for the angle eq. (e) reduces
simply to
(f)
Equation (f) indicates that the shearing strain , can be measured with a two element
rectangular rosette by orienting the gages at 45 and - 45 with respect to the x axis is
and connecting one gage in arm R1and the other in to arm R4Wheatstone bridge. The
subtraction will be performed automatically the bridge, and the output will
give , directly.
The stress intensity factor gage (KI):
Consider a two-dimensional body with a single-ended through crack as
shown in the fig: 9 below. The stability of this crack is determined by the opening-
mode stress intensity factor KI. If the specimen is fabricated from a brittle material,
the crack will be initiated when
KI > KIC (1)
Fig: 9 single edge crack

51
It is possible to determine KI as a function of loading on a structure by placing one or
more strain gages near the crack tip. To show an effective approach to this
measurement, consider a series of representation, using three terms, the three-term
representation of the strain field is
(2)
Where A0, B0, andA1 are known coefficients which depend on the geometry of the
specimen and loading. We known that A0 and KI are related by
(3)
Eq. (2) could be determine the known coefficient A0, B0, and A1 if three or more strain
gages were placed at approximate positions in the near field region.
However, the numbers of gages are required for the determination of A0 or KI
can be reduced to one considering a gage oriented at an angle α and positioned along
the axis as shown in the below figure 10, for the rotated co-ordinates, the strain
is obtained from equation (1)and equation (n)

52
Fig: 10 Rotated co-ordinate system positioned at point P.
(4)
The coefficient of the B0 term is eliminated by selecting the angle α as
Next, the coefficient of A1 vanishes if the angle is selected as
By the proper placement of a single strain gage with angles α and determined to
satisfy Eq. (6) and (7), the strain , is related directly to the stress intensity factor
KI by

53
The choice the angle α and depending only on Poisson’s ratio, as indicated in the
table below.
Table: Angle α and as a function of Poisson’s ratio
0.250 73.74 63.43
0.300 65.16 61.29
0.333 60.00 60.00
0.400 50.76 57.69
0.500 38.97 54.74
SOLVED PROBLEMS ON UNIT-2
PROBLEMS ON THREE ELEMENT STRAIN ROSETTE
1. A rectangular strain rosette is bonded at a critical point onto the surface of a
structural member. When the structural member is loaded, the strain gauge shows
the following readings.
Є0 = +850μm/m, Є45 = -50μm/m, Є90 = -850μm/m
The gauge factor and the cross sensitivity of the gauges are 2.80 and 0.06
respectively.
(i) Find the actual strains
(ii) Find the magnitudes and directions of corrected principal strains.
Poisson’s ratio of the material of the strain gauge is 0.285.
Given: μ
μ
μ
F = 2.80
Rectangular rosette
To Find:
1) Corrected strain:
2) Principal strain & direction:

54
Solution:
Formula:
ϵ

ϵ ϵ
ϵ

ϵ ϵ ϵ ϵ
ϵ

ϵ ϵ
Substituting given data in these equations we get:
ϵ ϵ ϵ
2) Principal strain & their direction:
Formula:
ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ
ϵ ϵ ϵ ϵ ϵ ϵ ϵ ϵ
ϵ ϵ ϵ
ϵ ϵ
Substituting corrected values of strain in these equations we get:

55
Direction:
2. A three element rectangular rosette is mounted on a steel component with E =
204 GPa and υ = 0.3. The manufacturers gauge factor F and the cross-sensitivity
Kt of this type of gauge is known to be 2.8 and 0.06 respectively. The readings
corresponding to the three gauges as indicated on a strain meter with the gauge
factor control set at 2.8 are
A = +700μ strain, b = -50μ strain, C = -700μ strain
(a) Find the actual strains ЄA,ЄB , ЄC.
(b) Find the magnitude of corrected principal stresses and their direction.
(c) What is the error if indicated strains AB, Care used to calculate the
principal stresses
Rectangular rosette
Given: μ
μ
μ
F = 2.80
To Find:

56
2) Principal stress & direction by using corrected strain:
3) Principal stress by using given strain & also % error between given and actual
stresses:
Solution:
Formula:
ϵ

ϵ ϵ
ϵ

ϵ ϵ ϵ ϵ
ϵ

ϵ ϵ
ϵ ϵ ϵ
2) Principal stress & their direction by using corrected strain:
Formula:
ϵ ϵ ϵ
ϵ ϵ

57
Direction:
3) Principal stress & their direction by using given strain:
To find % error between the corrected and given stress
Note: 4.53 % error between the corrected and given stress

58
3. A delta strain rosette bonded onto the surface of a structural member, made of
aluminum, yields the following strain when the structure is loaded,
Є0 = +500μm/m, Є120 = -250μm/m and Є240 = 250μm/m.
Given that Kt = - 0.07 & υ0 = 0.285. Determine the magnitudes and directions of
principal strain at the point where the strain rosette is bonded, also determine the
principal stresses if young’s modulus for aluminum is 80 GPa and Poisson’s ratio
is 0.3.
Given:  μ
μ
μ
F = 2.80
Delta rosette
To Find:
3) Principal stress:
Solution:
Formula:




59
Formula:
The principal angle can be given by
Substituting corrected values of strain in these equations we get

60
Direction:
3) Principal stress:
Formula:
PROBLEMS ON TWO ELEMENT STRAIN ROSETTE
4. A two element rectangular rosette was used to determine the two principal stresses at a
point . if Find and and take young’s
modulus E = 207GPa and Poisson’s ratio
Given:
E = 207GPa =
To find: Principal stresses
and
Formula:
Substituting values of strain in these equations we get

61
5. The following apparent strain data were obtained with two element rectangular
rosettes.
Rosettes number
1.
2.
600
-200
300
700
Determine the true strain and if . In each case, determine the
error which would have occurred if the cross sensitivity of the gage had been
neglected.
1. Rosettes number one
Given data:
Assume
Formula:
ϵ

ϵ ϵ ϵ

ϵ ϵ
Substituting given values of strain in these equations, we gettrue strain
ϵ
ϵ
To find % error between the corrected or true and given strain
Note: % error between the corrected and given stress

62
2. Rosettes number two
Given data:
Assume
Formula:
ϵ

ϵ ϵ ϵ

ϵ ϵ
Substituting given values of strain in these equations, we gettrue strain
ϵ
ϵ
To find % error between the corrected or true and given strain
error 0 005 3 100
error 0 5 3
Note: 0 5 3% error between the corrected and given stress

63
ASSIGNMENT PROBLEMS ON UNIT-2
1. Define a Strain rosette and mention the different types of strain rosette
configurations
2. Explain the construction of the three elements Delta rosette and derive the
expressions for the principal stresses and their orientations in terms of strain
measurement readings.
3. Explain the construction of the three elements rectangular rosette and derive
the expressions for the principal stresses and their orientations in terms of
strain measurement readings.
4. A rectangular strain gauge rosette is bonded at a critical point onto the surface of a
structural member. When the structural member is loaded, the strain gauges show
the following reading:
ε0 = 850 µm/m, ε45 = -50 µm/m, ε90 = -850 µm/m
The gauge factor and cross sensitivity of the gauges are 2.80 and 0.06
respectively. Find:
 Actual strains
 Magnitude and directions of principal strains.
 The error if indicated strains ε0,ε45, ε90 are used to calculate the principal
stresses.
Given E = 200GPa and Poisson’s ratio of the material of the strain gauge is 0.285.
5. The observations made with a delta rosette mounted on a steel specimen are
єA=400µm/m; єB=-200µm/m; єC=200µm/m. Determine the principal strains &
principal stresses & the principal angles ф1& ф2
6. The following observations were made with a delta rosette mounted on a steel
specimen εA = 4 0 µm/m ; εB = -200 µm/m ; εC = 200 µm/m Determine the
principal strain, the principal stresses and their orientations. Take µ = 0.3, E =
200×10
3
N/m
2
7. The following readings of strain were obtained on a three-element rectangular
strain rosette mounted on a Aluminum for which E=70GPa , ν =0.3 , εa= +285 µ
strains εB= + 5 µ strains εC= 102 µ strains

64
Determine:
 The Principal stresses and its direction
 The Principal strains and its direction
 The maximum shear stress
8. Three strain gauges are applied to an area; at a point in such a manner that gauge
“B” makes a +ve 30
0
with the gauge “A” and gauge “C” makes an angle of 45
0
with gauge “B”. The strains obtained are as follows.
εA= - 00 µm/m, εB= -400 µm/m, εC= 400 µm/m
Take E= 2X 10
5
N/mm
2
& Poisson’s ratio µ= 0.3. Calculate principal stresses,
strains and their directions.
9. A rectangular rosette mounted on the surface of a structural member indicates the
following reading, when the member is stressed ε0 = +500 strains ε45 =
+50 strains ε90 = -500 strains. Modulus of elasticity (E)=200×10
9
N/m
2
,
Poisson’s ratio (µ)= 0.30. Gauge factor and cross sensitivity of the strain gauge
are 2.80 and 0.06 respectively.
Determine:
 Actual strains along 0°, 45°, 90° directions.
 Principal strains and maximum shear strain.
 Principal stresses and maximum shear stress.
 Directions of principal stress.
10.A rectangular strain rosette is bonded at a critical point onto the surface of a
structural member. When the structural member is loaded, the strain gauge shows
the following readings.
Є0 = +850μm/m, Є45 = -50μm/m, Є90 = -850μm/m
The gauge factor and the cross sensitivity of the gauges are 2.80 and 0.06
respectively.
(iii) Find the actual strains
(iv) Find the magnitudes and directions of corrected principal strains.
Poisson’s ratio of the material of the strain gauge is 0.285.
11.A three element rectangular strain rosette bonded onto a machine component as
shown in figure. Yield strains indicated as shown below. Determine the magnitude
and direction of principal strains.
ЄA = +800μm/m, ЄB = -80μm/m, ЄC = -1000μm/m.

65
K = 0.0 and υ0 = 0.28.
12.A delta strain rosette bonded onto the surface of a structural member, made of
aluminum, yields the following strain when the structure is loaded,
Є0 = +500μm/m, Є120 = -250μm/m and Є240 = 250μm/m.
Given that Kt = - 0.07 & υ0 = 0.285. Determine the magnitudes and directions
of principal strain at the point where the strain rosette is bonded, also
determine the principal stresses if young’s modulus for aluminum is 80 GPa
and Poisson’s ratio is 0.3.
13.A three element delta rosette is bonded at a point onto the surface of a machine
element to determine the magnitudes and directions of strains at the point. The
strains indicated by a strain indicator are as follows.
Є0 = + 00μm/m, Є120 = +300μm/m and Є240 = - 00μm/m
Given that the gauge factor of the strain gauges is 2, and the Poisson’s ratio of
the material of the strain gauge is 0.28 and the cross sensitivity of the strain
gauge is 0.05. Determine the magnitudes and directions of the principal strains
at the point on the surface of the machine element.
14.A three element rectangular strain gauge rosette is bonded on the surface of
machine component as shown in figure. Yield strain as indicated below when the
machine component is under load
ЄA = 500μm/m, ЄB = -250μm/m
ЄC = 250μm/m E = 2.1 x 105
N/mm2
υ = 0.28. Determine the magnitude and directions of the principal stresses at
the point ‘O’ on the machine component. Assume the manufacturers gauge
factor and cross sensitivity as 2.8 and 0.06 respectively.

66
15.A three element delta rosette is bonded onto the surface of a machine element
made of aluminum for strain measurement. Strain gauge A is along X-axis and
strain gauge B and C are oriented along directions at angles of 1200
and 2400
from
X axis measured in anticlockwise direction. Strains measured are as follows
ЄA = 750μm/m, ЄB = -250μm/m, ЄC = +300μm/m
Other data supplied is Kt = - 0.07
υ0 = 0.30
υal = 0.33
Eal = 72 GPa
Find the magnitudes of the principal strains and the principal stresses and
orientations of the principal planes.
16.A three element rectangular rosette is mounted on a steel component such that the
gauges are separated by 450
with E = 200 GPa and υ = 0.3. The manufacturers
gauge factor F of this type of gauge is known to be 2.8. The reading
corresponding to the three gauges as indicated on a strain meter with gauge factor
control set at 2.8 are
ЄA = 1000μ strain, ЄB = -100μ strain, ЄC = -1000μ strain
Find the magnitude of principal stresses and their directions.
17. A three element delta rosette bonded onto a machine element yields strains as
shown below.
ЄA = 00μm/m, Єb = -300μm/m, ЄC = +300μm/m
Kt = -0.07 , υ 0 = 0.30, υ al = 0.33, Eal = 71.3 GPa
Find magnitudes and direction of principal strains and stresses.

67
18. A three element rectangular rosette is mounted on a steel component with E =
204 GPa and υ = 0.3. The manufacturers gauge factor F and the cross-sensitivity
Kt of this type of gauge is known to be 2.8 and 0.06 respectively. The readings
corresponding to the three gauges as indicated on a strain meter with the gauge
factor control set at 2.8 are
A = +700μ strain, b = -50μ strain, C = -700μ strain
(a) Find the actual strains ЄA ,ЄB , ЄC.
(b) Find the magnitude of corrected principal stresses and their direction.
(c) What is the error if indicated strains AB, Care used to calculate the
principal stresses
19. A three element rectangular rosette is fixed to the fuselage of a jet airliner near a
window-opening. At altitudes above 3000m the airliner is pressurized to simulate
atmospheric conditions prevailing at 3000m altitude. The altitude at which the
airliner cruises is 12,000m. The strain indicator readings with the airliner on the
ground and cruising at 12,000m altitude are as follows
Gauge 00
Gauge 450
Gauge 900
Airliner on the ground 13 μm/m 0 μm/m 35
μm/m
At 12,000m altitude; 500 μm/m 3 0 μm/m -120μm/m

68
Determine the magnitude and direction of the principal stresses if the modulus of the
elasticity of the material, E = 70 GPa and the Poisson’s ratio υ = 0.33.
20. A three element delta rosette serves to find the state of stress at a point O on the
surface of a stressed – aluminum component. The measured strains referred to a
given direction Ox are:
Direction Strain
00
500 μm/m
1200
-250μm/m
2400
+250μm/m
kt= - 0.07, υ0 = 0.285, υal= 0.33, Eal = 70 GPa
Find the actual principal strains and stresses and their directions with respect to Ox.
21. A three element rectangular strain rosette is fixed on the inside surface of a steel
pressure vessel near an inspection hole. The strain indicator readings before and
after pressurizing the vessel to a pressure of 6 MPa were as follows:
Gauge 00
Gauge 450
Gauge 900
Before pressurizing 15 μm/m 50 μm/m 35
μm/m
After pressurizing -285 μm/m +200 μm/m +430
μm/m
Gauge factor F = 2, kt= - 0.03, υ0 = 0.205, υsteel= 0.3; Esteel= 200 Gpa
Determine the magnitude and directions of the principal stresses.
22.A three-element rectangular rosette was used to measure the strains at a point in a steel
component. The observed strains are
ЄA = 850μm/m, Єb = -1200μm/m, ЄC = 1000μm/m
Neglecting the transverse sensitivity effects, determine the principal stresses and their
directions.

69
23. The following observations are made with a three-element rectangular rosette mounted
on an aluminum component.
A = 900μm/m, B = 310μm/m, C = -200μm/m
Determine the principal strains, principal stresses and principal stress directions.
K= - 0.04, υ0 = 0.285, υal= 0.33
24.A two element rectangular rosette was used to determine the two principal stresses at a
point shown in fig. below. if Find and
and take young’s modulus E = 207GPa and Poisson’s ratio
25.The following apparent strain data were obtained with two element rectangular
rosettes.
Rosettes number
3.
4.
5.
6.
600
-200
1,200
600
300
700
400
-300
Determine the true strain and if . In each case, determine the
error which would have occurred if the cross sensitivity of the gage had been
neglected.

70
UNIT 3: PHOTOELASTICITY
1. NATURE OF LIGHT
It has been observed that the disturbance that is being propagated can be either
perpendicular to the direction of propagation or in the direction of propagation. If the
disturbance is normal to the propagation direction it is called a transverse wave and
when it is in the direction of propagation it is called a longitudinal wave.
Light-waves belong to the class of transverse waves and the disturbance can be
represented by means of a vector called the light-vector. This light vector is
perpendicular to the direction of propagation. Light is known to be an electromagnetic
disturbance propagated through space and two vectors, namely the electric force
vector E and the magnetic force vector H.
H
E
Z
Fig (3) Electric and magnetic vectors
are associated with it. These two vectors are mutually perpendicular as shown in Fig
(3) and either of these can be taken as the fundamental light-vector.
2. PROPERTIES OF LIGHT
The colour of the visible light is determined by the frequency of the components of
the light vector. The colours in the visible spectrum range from deep red to deep
violet with frequencies of 390 × 1012
Hz to 770 × 1012
Hz, respectively. Most
photoelastic studies are made by using light in the visible range.
When the light vector is composed of vibrations, all of them having the same
frequency, it is called monochromatic light, i.e. light of single colour. When the
components of the light vector are of different frequencies, the colours of all the
components are mixed and eye records this mixture as white light.

71
Ordinary light consists of electromagnetic waves vibrating in directions perpendicular
to the direction of propagation. When the vibration pattern of these waves exhibits a
preference as to the transverse direction of vibration, then the light is said to be
polarized. Two types of light, i.e. (i) plane polarized and (ii) circularly polarized light,
are used in photoelasticity.
Plane polarized light is obtained by restricting the light vector to vibrate in a single
plane known as the plane of polarization. Figure (a) shows that the tip of the light
vector sweeps out a sine curve as it propagates. The light is vibrating in the plane of
polarization.
Plane polarizers are optical elements which absorb the components of the light vector
not vibrating in the direction of the axis of the polarizer. When a light vector passes
through a plane polarizer, this optical element absorbs that component of the light
vector which is perpendicular to the axis of polarization and transmits the component
parallel to the axis of polarization as shown in Figure (b). Say the light
vector A = a sin ωt where a = amplitude and ω = frequency of light wave, and α =
angle which the light vector A makes with the axis of polarization. Then
Figure (a) Plane of polarization
A0 = Absorbed component= a sin ωt sinα
At = Transmitted component= a sin ωt sin α cosα (1).
In a plane or linear polarizer, H type polaroid film is used which is a thin sheet of
polyvinyl alcohol heated, stretched, and bonded to a supporting sheet of cellulose
acetate butyrate. The polyvinyl face of the assembly is then stained with a liquid rich
in iodine. The amount of iodine diffused into the sheet determines its quality which is
judged by its transmission ratio.

72
Figure (b) Plane polarizer
Circularly polarized light is obtained when the tip of the light vector describes a
circular helix as the light propagates along the z-axis as shown in Figure (C).
Circularly polarized light is obtained with the help of a quarter wave plate (QWP),
made of a double refracting material. It resolves the light vector into two orthogonal
components and transmits each of them at different velocities. The phase difference
between these two components is π/2, i.e. quarter of a cycle.
The light vector component transmitted by plane polarizer is
At = a sin ωt sin α cosα
There are two axes 1 and 2 of the QWP shown in Figure (D). At makes an
angle β with the axis 1 of the QWP. At is resolved into two components along two
axes 1 and 2, i.e. fast and slow axes of the QWP. Component At travels at a
velocity V1 which is more than the velocity V2 with which the component At2travels.
Figure (c) Circularly polarized light

73
Figure (d) Quarter wave plate
Now
Since V1 > V2, the two components emerge from the plate with a phase difference.
Let λ = wave length of light.
Change in refractive index in direction (1)
= n1 – n0
Change in refractive index in direction (2)
= n2 – n0
Then,
Wave plates employed in a photoelastic study may consist of a single plate of quartz
or calcite cut parallel to the optic axis; a single plate of mica, a sheet of oriented
cellophone, or a sheet of oriented polyvinyl alcohol. QWPs are designed for a
monochromatic light.
When angle β = 45° and δ = , a circularly polarized light is obtained.

74
The Wave Equation
Since the disturbance producing light can be represented by a transverse wave motion,
it is possible to express the magnitude of the light (electric) vector in terms of the
solution of the one-dimensional wave equation:
E =f(z -ct) + g(z + ct) (1)
E = magnitude of light vector
z = position along axis of propagation
t = time
f(z - ct)1 = wave motion in positive z direction
g(z + ct) = wave motion in negative z direction
Most optical effects of interest in experimental stress analysis can be described with a
simple sinusoidal or harmonic waveform. Thus, light propagating in the positive z
direction away from the source can be represented by Eq. (1) as
E =f(z - ct) =(K / z) z cos/(z - ct) (2)
where K is related to the strength of the source and K/z is an attenuation coefficient
associated with the expanding spherical wave front. At distances far from the source,
the attenuation is small over short observation distances, and therefore it is frequently
neglected. For plane waves, the attenuation does not occur since the beam of light
maintains a constant cross section. Equation (2) can then be written as
E = a cos(2 / )(z - ct) (3)
Where a is a constant known as the amplitude of the wave. A graphical representation
of the magnitude of the light vector as a function of position along the positive z axis,
at two different times, is shown for a plane light wave in Fig. a. The length from peak
to peak on the magnitude curve for the light vector is defined as the wavelength A.
The time required for passage of two successive peaks at some fixed value of z is
defined as the period T of the wave and is given by
T = / c (4)

75
Fig: a Magnitude or the light vector as a function
or position along the axis or propagation at two
different times.
Fig: b Magnitude of the light vector as a function
of position along the axis of propagation for two
waves with different initial phases.
The frequency of the light vector is defined as the number of oscillations per second.
Thus, the frequency is the reciprocal of the period, or
The terms angular frequency and wave number are frequently used to simplify the
argument in a sinusoidal representation of a light wave. The angular frequency cu and
the wave number are given by
(5)
(6)
Substituting Eqs. (5) and (6) into Eq. (3) yields
E = a cos( z - t) (7)
Two waves having the same wavelength and amplitude but a different phase are
shown in Fig b. The two waves can be expressed by
E1= a cos (z + 1 - ct) 2n E2= a cos (z + 2- ct)
Where 1= initial phase of wave E1
2= initial phase of wave E2
= 2- 1= the linear phase difference between waves
The linear phase difference 15is often referred to as retardation since wave 2 trails
wave 1.
The magnitude of the light vector can also be plotted as a function of time at a fixed
position along the beam. This representation is useful for many applications since the

76
eye, photographic films, and other light-detecting devices are normally located at
fixed positions for observations.
2. STRESS-OPTIC LAW-TWO DIMENSIONAL CASE:
Consider a model of uniform thickness made of glass, epoxy or some
transparent high polymer material. Let the model be loaded such that it is in a plane
state of stress. Then the state of stress at any point can be characterized by the three
rectangular stress components σx, σxandτxy or by the principal stresses σ1, σ2and their
orientations with reference to a set of axes. The situation is as shown in figure (1).
Let n0 be the refractive index of the material when it is in free (i.e. unstressed)
state. When the model is put in a state of stress, experiments show that:
Fig (1) Plane stress state and principal stresses.
(i) The model becomes doubly refractive;
(ii) the directions of the polarizing axes in the plane of the model at any
point P coincide with the directions of the principal stress axes at that
point; and
(iii) if n1 and n2 are the refractive indices for vibrations corresponding to
these two directions, then
– –
– –
c1is called the direct stress –optic coefficient and c2 the transverse stress-optic
coefficient. Since the stress vary uniformly, i.eσ1, σ2and θ are continuously distributed
functions over the model in the xy-plane, the directions of the polarizing axes as well
as the values of n1 and n2 vary uniformly over the xy-phase of the model.

77
If a plane polarized light is incident normally at any point P of the model, then the
incident light vector gets resolved along the directions of σ1, σ2 and these two
vibrating components travel through the thickness of the model with different
velocities. The velocities of propagation of these two components are governed by
Equation (1). When they emerge, there will be a certain amount of relative phase
difference between these two components. The relative phase difference ε is given by
–
Using Equation (1).
– – – –
–
If c1 + c2 is set equal to c, the stress-optic coefficient, the relative retardation ε is then
given by
– (3)
The number of wavelengths of relative path difference is given by
– (4)
Equations (3) & (4) are known as stress-optic relations. They relate the stresses to the
optical behavior of the model. According to these equations, the relative phase
difference is directly proportional to ( 1- 2) and model thickness d,and inversely
proportional to the wavelength of light used. In photoelastic analysis we try to
evaluate the value of ( 1- 2) at a point from the measured value of ε or N.
from Eq. (4), therefore,
– (5)
Denoting λ/cd by f, the principal difference is given by
– (6)
‘f ’is called the model fringe constant. Putting N =1, we can see that f expresses the
value of ( 1- 2) necessary to cause a relative path difference of one λ in a model of
given thickness d. This is also equal to the value ( 1- 2) necessary to cause a relative
phase difference of 2π radians in given model. Equation (5) can also be written in the
form
– (7)

78
F represents the material fringe constant. If d = 1, F becomes equal to f. Hence, F
represents the model fringe constant per unit thickness. From Equations (6) and (7)
3. POLARIZATION:
Light-waves belong to the class of transverse waves and the disturbance can
be represented by means of a vector called the light vector. In ordinary light, the tip of
the light-vector describes a random vibratory motion in a plane transverse to the
direction of propagation as shown in figure(4). If the tip of the light-vector is forced to
follow a definite law (or a pattern), the light is said to be polarized.
Light vector
(a)
(b)
Fig (4) Ordinary light - random vibratory motion
Types of polarization
i)Plane polarization or linear polarization
If thelight-vector is parallel to a given direction in the wave-front, it is said tobe
linearly or plane polarized as shown in Fig (5 c).
(a) (b) (c)
Fig (5) Circular, elliptical and linear polarizations

79
ii) Circular polarization
If the tip of the light is constrained to lie on circumference of a circle, it is said to be
circularly polarized. If the tip of the Light-vector in Fig. 5(a) describes the circle in a
counter-clockwise directionas shown, then it is said to be right-handedly circularly
polarized. Onthe other hand, if the path is traversed in a clockwise direction, then it
isleft-handedly circularly polarized. This notation is adopted so as to beconsistent
with the right-handed coordinate system. The positive z-axis is away from the source
and the vibrations are in planes parallel to the xyplane.
iii) Elliptical polarization
If the light beam is constrained to movein a pattern such that the tip of the
light vectordescribes an ellipse then the light is said to be elliptically polarized.
One can speak of right-handedly or left-handedly ellipticallypolarized light. It is
easily observed that an elliptically polarized lightis the most general form of polarized
light since a circle can be consideredas an ellipse with the major and minor axes being
equal. Similarly, astraight line is a degenerated form of an ellipse with the minor axis
being equal to zero.
4. PLANE POLARISCOPE, ISOCHROMATICS AND ISOCLINICS
Fig (2) Plane polariscope, Isochromatics and Isoclinics
Consider the arrangement shown in figure (2). S is a source of monochromatic
light, P is a polarizer, M is the model under a plane state of stress, A, called the

80
analyzer, is a second polarizing element kept at 900
to the polarizer and B is the
screen. We shall assume that through a suitable optical arrangement, the image of the
model is projected on the screen. We shall trace the passage of a typical ray of light
through the various optical elements in the assembly.
The arrangement shown in figure (2) is known as a plane polariscope. The
polarizer and the analyzer are always kept crossed, but their combined orientation can
be arbitrary. Now we can make a few important observations as follows.
When the model is stressed, it behaves as a crystal and at the point where the
ray passes, the polarizing axes coincide with the principal stress axes 1, 2 at that
point. In general, the polarizer (i.e the axis of the polarizing element) makes an angle
Φ with the 1axis. If Φhappens to be zero (or π/2), i.e. if the polarizer coincides with
either 1 (or 2), then a plane polarized light incident on the model at that point will
emerge as a plane polarized light. Since the analyzer is kept crossed with respect to
the polarizer, the light coming out of the analyzer is zero. Consequently, at all those
points of the model, where the directions of the principal stresses happen to coincide
with the particular orientation of the polarizer-analyzer combination, the light coming
out of the analyzer will be zero. If the polarizer-analyzer combination happens to
coincide with the directions of 1, 2stresses at one point of the model, then in general,
there will be a locus of points in the model along which this condition is satisfied.
This is so because, in general, the stresses are disturbed in a continuous manner in the
model.
The locus of points where the directions of the principal stresses coincide with
a particular orientation of the polarizer-analyzer combination is known as an isoclinic
(meaning same inclination). For example, if the polarizing element is kept vertical and
the analyzer is kept horizontal, then on the screen, a dark band will be seen which is
the locus of the points where the 1, 2directions happen to be vertical and horizontal.
If one measures angles from the vertical reference axis, this isoclinic will be called the
00
-isoclinic. If now, the polarizer is turned through say, 300
and the analyzer is also
rotated through an equal amount (so that the analyzer is always kept crossed with
respect to the polarizer) then the previously observed 00
-isoclinic vanishes and a new
dark band is observed on the screen. This is the 300
-isoclinic and it represents the
locus of points in the model where the principal stress axes are oriented at 300
and
300
+ (π/2) with respect to the vertical. In figure (2) one such isoclinic is marked on
the image.

81
Let us now consider another situation. Suppose at a particular point of the
model, the values of 1and 2 are such as to cause a relative phase difference of 2πm
where m is integer. The relative phase difference is related to 1 - 2. When the
relative phase difference is 2mπ, the model behaves as a full-wave plate at that
particular point. An incident linearly polarized light on a full-wave plate emerges as a
linearly polarized light and is cutoff by the analyzer, because of its crossed position.
Therefore, at all those points of the model where the values of 1- 2 are such to cause
a relative phase difference of 2mπ (m = 0, 1, 2…), the intensity of light on the screen
will be zero. On the screen, a series of dark bands corresponding to the loci of these
points are observed. These dark bands or fringes are known as isochromatics. An
isochromatic is a locus of points where the values of 1- 2 are such to cause a relative
phase difference of 2mπ (m = 0, 1, 2…), when the background is dark. The locus of
points where the values of 1- 2 are such to cause zero radians of phase difference (or
equivalently, zero number of wavelengths of relative path difference) is called the
zero-order fringe. The locus of points where the values of 1- 2 are such as to cause
2π radians phase difference (equivalently to a relative path difference of λ) is known
as the first-order fringe. Similarly, on the screen one can observe the second-order
fringe, third-order fringe, and so on. These are shown qualitatively in fig (2) on the
screen.
It should be observed that the background on the screen (i.e. the region outside
the image of the model) is dark, since the light coming out of the polarizing element is
cut off by the analyzer. Hence, the dark background corresponds to the zero-order
fringe.
ANALYSIS THROUGH TRIGONOMETRIC RESOLUTIONS
(Effects of a stressed model in a plane polariscope)
The concept of Isoclinic’s and Isochromatics can also be explained through
trigonometric resolution process as discussed below.
Consider fig (3a), which is equivalent to fig (2). Let the linearly polarized light
coming out of the polarizer will be
A1 = a cosωt

Experimental stress analysis BE notes by mohammed imran

Experimental stress analysis BE notes by mohammed imran

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