Experimental Stress Analysis
10ME761
EXPERIMENTAL STRESS ANALYSIS
Subject Code : 10ME761
IA : Marks : 25
Hours/Week : 04
Exam Hours : 03
Total Hours : 52
Exam Marks : 100
PART – A
UNIT-1
Electrical Resistance Strain Gages: Strain sensitivity in metallic alloys, Gage construction,
Adhesives and mounting techniques, Gage sensitivity and gage factor, Performance
Characteristics, Environmental effects, Strain Gage circuits. Potentiometer, Wheatstone’s
bridges, Constant current circuits.
06 Hours
UNIT-2
Strain Analysis Methods: Two element, three element rectangular and delta rosettes, Correction
for transverse strain effects, Stress gage, Plane shear gage, Stress intensity factor gage. 06 Hours
UNIT-3
Photo-elasticity: Nature of light, Wave theory of light - optical interference , Stress optic law
effect of stressed model in plane and circular polariscopes, Isoclinics & Isochromatics, Fringe
order determination Fringe multiplication techniques, Calibration photoelastic model materials
08 Hours
UNIT-4
Two Dimensional Photo-elasticity: Separation methods: Shear difference method, Analytical
separation methods, Model to prototype scaling, Properties of 2D photo-elastic model materials,
Materials for 2D photoelasticity
06 Hours
PART –B
UNIT-5
Three Dimensional Photo elasticity: Stress freezing method, Scattered light photo-elasticity,
Scattered light as an interior analyzer and polarizer, Scattered light polariscope and stress data
Analyses.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
06 Hours
Page 1
Experimental Stress Analysis
10ME761
UNIT-6
Photoelastic (Birefringent) Coatings: Birefringence coating stresses, Effects of coating
thickness: Reinforcing effects, Poisson's, Stress separation techniques: Oblique incidence, Strip
coatings.
08 Hours
UNIT-7
Brittle Coatings: Coatings stresses, Crack patterns, Refrigeration techniques, Load relaxation
techniques, Crack detection methods, Types of brittle coatings, Calibration of coating.
Advantages and brittle coating applications.
06 Hours
UNIT-8
Moire Methods: Moire fringes produced by mechanical interference. Geometrical approach,
Displacement field approach to Moire fringe analysis out of plane displacement measurements,
Out of plane slope measurements .Applications and advantages
06 Hours
TEXT BOOKS:
1. "Experimental Stress Analysis", Dally and Riley, McGraw Hill.
2. "Experimental Stress Analysis". Sadhu Singh, Khanna publisher.
3. Experimental stress Analysis, Srinath L.S tata McGraw Hill.
REFERENCES BOOKS :
1. "Photoelasticity Vol I and Vol II, M.M.Frocht, John Wiley & sons.
2. "Strain Gauge Primer", Perry and Lissner,
3. "Photo Elastic Stress Analysis", Kuske, Albrecht & Robertson John Wiley & Sons.
4. "Motion Measurement and Stress Analysis", Dave and Adams,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 2
Experimental Stress Analysis
10ME761
TABLE OF CONTENT
SL.NO
PERTICULERS
PAGE NO
01
Electrical Resistance Strain Gages
4-39
02
Strain Analysis Methods
40-50
03
Photo-elasticity
51-79
04
Two Dimensional Photo-elasticity
80-86
05
Three Dimensional Photo elastic
87-96
06
Photoelastic (Birefringent) Coatings
97-103
07
Brittle Coatings
104-113
08
Moire Methods
114-127
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 3
Experimental Stress Analysis
10ME761
PART – A
UNIT-1
Electrical Resistance Strain Gages
In the electrical resistance strain gauges the displacement or strain is measured as a
function of the resistance change produced by the displacement in the gauging circuit. An ideal
strain gauge should have the following basic characteristics:
1. The gauge should be of extremely small size (gauge length and width) so as to adequately
estimate strain at a point.
2. The gauge should be of significant mass to permit the recording of dynamic strains.
3. The gauge should be easy to attach to the member being analysed and easy to handle.
4. The strain sensitivity and accuracy of the gauge should be sufficiently high.
5. The gauge should be unaffected by temperature, vibration, humidity or other ambient
conditions.
6. The calibration constant for the gauge should be stable over a wide range of temperature
and time.
7. The gauge should be capable of indicating both static and dynamic strains.
8. It should be possible to read the gauge either on location or remotely.
9. The gauge should exhibit linear response to strain.
10. The gauge and the associated equipment should be available at a reasonable cost.
11. The gauge should be suitable for use as a sensing element or other transducer systems.
Types of Resistance Strain Gauges
There are basically four types of electrical resistance strain gauges as classified below:
1. Unbonded gauges:
(a) Non-metallic
(b) Metallic
2. Bonded gauges
(a) Non-metallic
(b) Metallic
(i) Wire type
(ii) Foil type.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 4
Experimental Stress Analysis
10ME761
3. Weldable gauges.
4. Piezoresistive gauges.
1. (b) Unbonded-Metallic Gauges.
The unbonded nonmetallic gauge is a mechanically actuated gauge that contains a
resistance element so arranged that when one part of the gauge is displaced with respect to
another there is developed a change in pressure on the measuring element of the gauge. This
change in pressure changes the resistance of the element which may be recorded by electrical
means. A gauge of this type was developed in 1923 and 1924 by McCollum and Peters and is
shown in Fig. This gauge is composed of a series of carbon plates arranged in a stack. The stack
is so adjusted that a displacement of one part of the gauge relative to another changes the
pressure, on the stack of plates. When the strain is applied in the structure to which the gauge is
attached, the change in length is communicated to the carbon-plate stack. This change in length
requires a change in pressure in the stack, and the resistance of the stack changes.
Figure: Unbonded non-metallic strain gauge.
With an increase in pressure, the areas of contact between the plates are enlarged and new areas
come into contact, thus decreasing the resistance of the element. If the pressure is released, the
areas of contact are reduced, and some of the areas lose contact, thus increasing the resistance of
the element. If the pressure becomes excessive, so that the elastic limit of the carbon in the
gauges is exceeded or the carbon is even crushed, or if the plates are allowed to shift in the
lateral direction with respect to each other, the results become erratic. Besides these difficulties,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 5
Experimental Stress Analysis
10ME761
there is a further defect of mechanical friction and hystertsis in the mechanical parts of the
gauge.
Gauge of this kind have been used to determine displacements, loads and strains in
flexible cables, airplanes, bridges, vibrating members, dynamometers and pressure gauges.
However, with the advancement of metallic gauges the usefulness of these type of gauges has
reduced materially.
1. (b). Unbonded-Metallic Gauges.
The principle of the unbonded-metallic gauges is based on the change in electrical
resistance of a metallic wire due to the change in tension of the wire. The first device of this
kind was designed by Carlson and Eaton in 1930. This type of gauge is constructed by winding
wire in three coils, the first providing a coil unaffected by the gauge motion, and the other two
having tensions altered by the gauge motion, each in an opposite manner. The whole is mounted
in a sleeve that allows only longitudinal movement. The coils are placed under initial tension
into a four arm Wheatstone bridge. As the compressive strain is applied, the prestrain would
simply be relieved and the unbonded element would remain taut. For the gauge to register
compressive strains, the initial assembly must include a built-in tensile prestrain in the coils
greater than the maximum compressive strain to be measured. A gauge of this type is shown in
Fig. These type of gauges are rarely used for experimental stress analysis. However, these type
of gauges have been incorporated into accelerometers and pressure pickups.
Figure: Unbonded metallic strain gauge.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 6
Experimental Stress Analysis
10ME761
2.(a) Bonded Non-Metallic Gauges.
A strain gauge using direct bonding of a non-metallic resistor element to a material in
which the strain is so to be measured was reported by Bloach in 1935. In this gauge a carbon
coating is applied directly to the surface of the structure in which strain is to be measured. For
metallic structures the surface is first coated with a non-conducting material. If the underlying
surface of such a coating is stretched, the carbon particle would move apart, and the undercoating is compressed, the particles would move closer together, and the resistance will change.
This resistance change can be interpreted in terms of strain.
Generally these type of gauges are made by impregnating carbon particles in plastic
sheets. These sheets are then cut into strips about 6 mm wide and 25 mm long. On each end of
the strip a silver band is plated so that lead wires may be attached (fig). The gauge is bonded
directly to the surface to be strained with a common glue.
Figure: Bonded non-metallic strain gauge.
These sensitivity and resistance of the gauge are affected by temperature and humidity.
This gauge is of rugged construction and can withstand rough handling. However, the crosssensitivity of the gauge is quite high.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 7
Experimental Stress Analysis
10ME761
2. (b) Bonded Metallic Gauges.
The bonded metallic type of strain gauge consists of a length of a strain-sensitive
conductor mounted on a small piece of paper or plastic backing. In use this gauge is cemented to
the surface of the structural member to be tested. These gauges may be either of the wire or foil
type. In the case of wire strain gauges, the filament consists of a long length of wire in the form
of a grid fixed in place with a suitable cement. The wire grid may be either of the flat type (fig.
a) or wrap-around type (Fig. b). After attaching the lead wires to the two ends of the grid, a
second piece of paper is cemented over the wire as a cover. In the wrap around type of wire
gauges, the strain-sensitive wire is wound around a cylindrical core in the form of a close-wound
helix. This core is then flattened and cemented between layers of paper for purpose of protection
and insulation. Fig.(c) shows a flat wire grid free filament construction.
Figure (a): Bonded wire flat grid gauge.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 8
Experimental Stress Analysis
10ME761
Figure (b): Bonded wire wrap-around gauge.
Flat wire grid free filament construction.
Bonded flat foil grid gauge.
Figure: Types of bonded metallic gauges.
The foil type of stain gauge has a grid made from a very thin strain-sensitive foil (fig d).
The width of foil is very large as compared to the thickness so that the gauge provide a much
larger area for cementing the gauge. The gauge configuration is obtained by printing the desired
pattern on a sheet of foil with acid resistant ink and subsequently etching away the unprotected
metal. Another method of manufacture involves precision punching of the gauges from a foil
sheet. The foil type of gauges have the following advantages over the wire type gauges.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 9
Experimental Stress Analysis
10ME761
1. The width of the foil at the end of each loop can be greatly increased to reduce the
sensitivity of the sensitivity of the gauge to transverse strains.
2. The cross-section of the gauge conductor is rectangular, resulting in the high ratio of
surface area to cross-section area. This increases heat dissipation and avoids adhesion
between the grid and the backing material.
3. The gauge factor is higher by 4 to 10 per cent that other gauges.
4. These gauges are easier to manufacture.
5. These gauges can be used to measure strain on curved surfaces.
6. These gauges are suitable for static and dynamic strain measurements.
7. They have very good fatigue properties.
8. Stress relaxation and hysterisis is very less in these gauges.
9.
3. Weldable Strain gauges.
Some of the limitations of the bonded type of metallic gauges are their comparatively
costly, time consuming and complicated method of bonding.
This realization led to the
development of the weldable wire resistance strain gauge-a strain gauge capable of being
installed in minutes and in any environment.
This unique technique, utilizing capacitive
discharge spot welding equipment eliminates the need for all bonding materials.
The weldable strain gauge consists of a strain sensitive element, the Nickel Chrome or
Platinum Tungsten, housed within a small diameter stainless steel tube. The strain element is
insulated from the tube with highly compacted ceramic insulation or metallic oxide powder,
normally high purity magnesium oxide, which also serves as a strain transfer medium from the
housing to strain element. This weldable gauges are equipped with a thin flange spot welded to
the strain tube. This flange is subsequently spotwelded to the structure under test and provides
the bond required to transfer strain.
Integral leads are attached to the basic gauge by welding. When the gauge is welded to a
specimen and the test specimen put into tension or compression, the stress is transmitted through
the weld to the mounting flange, into the strain tube, and through the magnesium oxide powder.
The basic construction of a quarter-bridge or half-bridge, self-temperature compensated
gauge is shown in Fig. and includes integral metal sheathed or flexible lead wire configurations.
This gauge construction provides inherent mechanical and environmental protection for both the
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 10
Experimental Stress Analysis
10ME761
main filament and lead wires and is used over a broad temperature range from cryogenic to 65oC.
Weldable strain gauges can be used for a wide range of static and dynamic measurement
applications. Their rugged construction and positive attachment make it possible to measure
strain at higher or low temperatures and in server environments, including shock and vibration,
steam, salt water, chemicals, and other corrosive atmospheres.
(a) Quarter bridge gauge.
(b) Half bridge gauge.
Figure: Weldable strain gauges
4. Piezo-resistive strain gauges.
Crystals of silicon, germanium, quarts and Rochelle salt show a change in resistance
when deformed by applying pressure. This effect can be utilized to measure strain. Such like
gauges are called piezo-resistance strain gauges. We shall discuss these gauges in details.
Materials for Gauges
A good gauge material should have the following qualities:
1. High gauge factor
2. High resistance
3. Low temperature sensitivity
4. High electrical stability
5. High yield point stability
6. High endurance limit
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 11
Experimental Stress Analysis
10ME761
7. Good workability
8. Good solderability and workability
9. Low hysteresis
10. Good corrosion resistance
11. Low thermal e.m.f. when joined with other metals.
The important properties of the most commonly used materials for strain gauges are
given in Table.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 12
Experimental Stress Analysis
10ME761
Carrier Materials
A strain-gauge grid is normally supported on some form of carrier material.
This
provides the necessary electrical insulation between the grid and the material to be tested,
dimensional stability, and also provides some degree of mechanical protection for the delicate
sensing element. A good carrier material should have the following desirable characteristics:
1. Minimum thickness
2. High mechanical strength
3. High dielectric strength
4. Minimum temperature restrictions
5. Good adherence to cement used
6. Non-hygroscopic.
Temperature Compensation
The ideal strain gauge would change resistance in accordance with stress-producing
deformations in the structural surface to which it was bonded and for no other reason.
Unfortunately, gauge resistance is affected by many other factors, out of which temperature is
very important.
The total indicated strain occurring at a point in a structure is made up of mechanical
strain and apparent strain. The mechanical strain is that produced by external forces. The
apparent strain is the portion of the total indicated strain induced by thermal effects including
expansion of the base metal, expansion of the gauge metal and change in electrical resistance of
the gauge. Thus, when the ambient temperature increases (say), then
∆l
= α.∆T.
1. The gauge grid will elongate so that l
∆l
= β.∆T.
2. The base material on which the gauge is mounted will elongate so that l
3. The resistance of the gauge metal will increase because of the influence of the
∆R
= γ.∆T.
temperature coefficient of resistivity of the gauge material so that R
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 13
Experimental Stress Analysis
10ME761
The combined effect of these three factors will produce a temperature induced change in
 ∆R 


resistance of the gauge,  R  ∆T with may be expressed as:
 ∆R  = β − α ∆T.F + γ .∆T
(
)


 R ∆T
Where
α = thermal coefficient of expansion of the gauge material
β = thermal coefficient of expansion of the base material
γ = temperature coefficient of resistivity of the gauge material
F = gauge factor
R = resistance of gauge
∆T = rise in temperature.
Equation holds only for small values of ∆T , where α,β and γ can be considered constant.
For large values of ∆T , average values of these factors might be used without introducing large
errors.
β − α ) ∆T , which
If α ≠ β , then the gauge will be subjected to a mechanical strain, (
does not occur in the specimen.
If α = β, then this component of apparent strain vanishes. However, the gauge will still
register a change of resistance with temperature if γ is not zero. In order to prevent significant
errors due to this effect, some form of “temperature compensation” is usually employed when
strain gauges are used in applications where the steady state or static component of strain must
be measured. Currently available methods of compensation for the apparent strain include the
use of a dummy or compensating strain gauge, self-temperature compensating (STC) gauge,
compensation by dissimilar or similar gauges in the Wheatstone bridges and compensation by
computation.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 14
Experimental Stress Analysis
10ME761
1. Compensating Dummy Gauge:
The earliest form of temperature compensation makes use of the electrical bridge circuit
in which the active gauge is connected to balance out unwanted temperature induced resistance
change. This is usually called the “compensating dummy” arrangement. The “dummy gauge”
identical to the active gauge in type and lot number, is mounted on an unstressed piece of the
specimen material and placed in the same thermal environment as the active gauge. The active
and compensating gauges are then connected as adjacent arms of the bridge circuit in the readout
instrument. Effects common to both gauges will preserve bridge – balance conditions, and no
output signal results. Since only the active gauge is exposed on mechanical or thermal strain
caused by specimen stress, bridge unbalance is proportional to the magnitude of specimen stress
producing strain. The method fails entirely if the temperature does not vary in an identical
fashion at both gauge locations.
2. Self-temperature Compensated Gauge:
The terms ”temperature compensated” is applied to strain gauges in which the resistance
change due to temperature is equal to zero. Self-temperature compensated gauges will perform
properly only when used on materials having the specific value of thermal expansion coefficient
for which they are designed. STC gauges can be obtained for use on materials having thermal
expansion coefficients from zero to 25 ppm/oC.
Two method are used for obtaining self-temperature compensation. In the first method,
self-temperature compensation is created by altering the temperature coefficient of resistance of
the grid material so that, when mounted on materials having a certain thermal expansion
coefficient, the apparent strain will be a suitably low value. This is done, in most cases, by
special selection or thermal processing of the grid alloy. The two principal classes of straingauge alloys susceptible to such treatment are Constantan and Karma. The second method
includes forming a grid with two lengths of gauge wires joined suitably in series so that the
resultant apparent strain is zero.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 15
Experimental Stress Analysis
Dual-element self temperature
10ME761
Compensation by dissimilar gauges.
Compensated gauge.
3. Compensation by Dissimilar Gauges:
Compensation of the temperature effect in a bridge network is accomplished by putting
dissimilar gauges into adjacent bridge arms as shown in Figure. The gauge in the first arm
should have a relatively small temperature effect in the same direction. With proper, fixed series
and shunt resistances for the gauge in the second arm, it is possible to obtain an overall
temperature effect for the second arm, that is equal to that of the first arm.
Hence, the
temperature effects of the two arms will cancel each other with a relatively small loss in the
strain sensitivity of the network.
This method would appear to have a better chance of success than the self-temperature
compensated gauge because the relative resistance of the filament is not critical. If will always
be possible after a gauge has been made, to select the fixed resistance for proper compensation.
Furthermore, compensation over a greater temperature increases. in this case, temperature would
not have to be known very accurately.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 16
Experimental Stress Analysis
10ME761
4. Compensation by Similar Gauges:
Best possible temperature compensation is obtained for unpredictable effects as for
predictable effects with two similar gauges in adjacent arms of a Wheatstone bridge. However,
this circuit arrangement eliminates the hydrostatic component of stress from the reading and only
the shear component of stress is reflected. Hence, the gauges should be arranged so as to pick up
the greatest signal from the shear component of stress. This means that one gauge should be
positioned in the direction of the maximum principal strain, the other in the direction of
minimum principal strain. This method is likely to give best results when the direction of the
principal strains is known.
5. Compensation by Computation:
By knowing the temperature characteristics of a strain gauge and the base metal, and if
the temperature can be observed separately, a correction can be calculated theoretically from
Equation and applied to the observed strain.
Figure: Compensation by similar gauges.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 17
Experimental Stress Analysis
10ME761
Adhesives
The bonded wire or foil should be bonded to the component with a suitable adhesive. The strain
gauge should be sufficuently elastic to faithfully transfer strain in the test compon ent to the
gauge sensing element .
Several important factors have to be considered while selecting the adhesive for a perticular
strain gauge and test component combinatio. It is very important to ensure that the adhesive is
compatible with both the gauge backing material and test material
Some of the common type of adhesives are as follows
1.Nitro-Cellulose Cement
Nitro-cellulose cement is commonly used to mount paper-backed gauges. Since these cements
contain a very large fraction of solvent (about 85%), the bonded gauge should be cured to
remove all the solvents by evaporation. The curing time varies with the percentage of solvent,
relative humidity, curing temperature and also the purpose for which the gauge is used. For
short-time tests using thin paper gauges, curing at room temperature for several hours may
suffice. The curing time for wrap-around gauges is 5 to 10 times longer than for flat-grid gauges.
In the case of long-term tests where stability is an important requirement, a ten-day roomtemperature Cure may be required. This time can be reduced to a day or two by circulating air at
about 55°C over the gauge installation. As nitro-cellulose cements .are hygroscopic, the cured
gauge should be immediately protected by resistance coating. This will ensure electrical and
dimensional of the gauge installation.
2.Epoxy Cements
Two types of epoxy cements-room-temperature epoxies and thermosetting epoxies-are
commonly used. Both types have two constituents-a monomer and a hardening agent. Mixing of
a monomer with the hardening agent inducts polymerization. Room-temperature epoxies use
amine-type hardening agent while thermosetting epoxies require anhydride type of hardening
agent. In the case of room temperature epoxy polymerization takes place at room temperature or
at a temperature slightly above room temperature. Thermosetting epoxies need a curing
temperature in excess of I20°C for several hours for complete polymerization.
The adhesive curing temperature and the residual stresses generated during curing are highly
sensitive to the ratio of the monomer to the hardener in the epoxy. For this reason, only the
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 18
Experimental Stress Analysis
10ME761
proportions of the monomer and the hardening agent recommended by the gauge manufacturer
should be used for preparing the adhesive. As the quantity of the adhesive required for installing
strain gauges is quite small, carefully weighed quantities of the monomer and the hardening
agent should be taken and mixed thoroughly. Organic fillers in moderate quantities are added to
improve the adhesive strength and to reduce the coefficient of expansion of the epoxy.Clamping
pressures and curing cycles differ for various adhesives and should be chosen to suit test
conditions. A clamping pressure of 0.35 to 1.4 kgf/cm2 (35 to 140 kPa) during the curing period
ensures a thin adhesive layer or bond line in the case of epoxy cements. Thin bond lines are
desirable as they tend to reduce creep, hysteresis and nonlinearity. Epoxy cements resist moisture
and chemicals and are useful for test temperatures in the range - 230° to +300°C.
3.Cyanoacrylate Cement
Cyanoacrylate cement cures rapidly at room temperature. It is an excellent general-purpose
adhesive for laboratory and short-term field applications. It is compatible with most of the test
materials and strain gauge –backing materials. A firm thumb pressure for about a minute is
sufficient to induce polymerization at room temperature. A strain gauge bonded with this
adhesive can be used approximately 10 min after bonding. Its performance deteriorates with
time, elevated temperature and moisture absorption. When protected with coatings like
microcrystalline wax or silicone rubber, the life of the strain gauge bonded with this adhesive can
be extended to 1 or 2 years. The typical operating temperature range for this adhesive is _75° to
+65°C.
4.Phenolic Adhesives
Phenolics are single component thermosetting adhesives requiring curing at an elevated
temperature of 120° to 175°C for a fairly long period. In most applications these adhesives have
been replaced by epoxy adhesives.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 19
Experimental Stress Analysis
10ME761
Gauge Installation steps
Fig: Gauge installation procedure
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 20
Experimental Stress Analysis
10ME761
The Transverse Sensitivity:
The strain sensitivity SA of a single, uniform length of a conductor is given by
∆R / R
SA =
ε
Where ε is the uniform strain along the conductor and in the direction of the conductor.
Whenever the conductor is wound into a strain-gauge grid, certain effects take place which alter
to a certain degree this value of sensitivity of the gauge. The change is introduced by the end
loops, which are transverse to the straight portion of the grid. Thus the gauge in addition to
measuring the strain along its axis also measures the strain transverse to it.
This affect is
reflected as an error in the strain gauge reading. This is known as the transverse or crosssensitivity of the gauge. Now
Axial (parallel) strain sensitivity
=
S11
∆R / R
=
when ε y 0
εx
Normal (perpendicular) strain sensitivity
=
S⊥
∆R / R
=
when ε x 0
εy
Transverse sensitivity factor K is defined as
K=
S⊥
S
∆R / R
=
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
εy
ε x =0
εx
ε y =0
∆R / R
Page 21
Experimental Stress Analysis
10ME761
Gauge factor F as specified by the manufacturer is
F=
∆R / R
εx
when ε y = −0.285 ε x
Assuming that the gauge has been calibrated on steel whose Poisson’s ratio is 0.285.
Gage sensitivity and Gage Factor
The strain sensitivity of a metral, FA is defined as the ratio of the resistance change in a
conductor per unit of its initial resistance to the applied axial strain.
An expression for FA can be derived as follows: the resistance of a wire is given by,
Where ,
l= length of wire
ρ= Specific resistance of wire
cD2= area of cross section of wire , A
Here D is a sectional dimention and c is a proportionality constant. For example. C= 1and π/4 for
square and circular cross sections respectively.
Taking logarithms,
When wire is strained axially, each of the variables in above equation may a chang.
Differentiation of equation (a) gives
This may be written as
Here
dl/l = €a = Axial strain in the wire
dD/D = €t = lateral strain in the wire
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 22
Experimental Stress Analysis
10ME761
Also,
v= Poission’s ratio = - (dD/D) / dl/l
(d)
Substituting in euqtion (c ) gives
In this equation the first term reprents the change in resistance due to geometrical changes in the
wire due to strain. It is usually between 1.4 and 1.7 for elastic strains.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 23
Experimental Stress Analysis
10ME761
The Wheatstone Bridge circuit
(a) Balanced bridge. Figure shows the basic circuit for the Wheatstone bridge. For a balanced
bridge the current IG through the galvanometer is zero.
Hence
I1 = I2, I4 = I3
The potential drops across the individual elements are:
=
E AB I=
I4R 4
1R 1 , E AD
=
E BC I=
I3R 3
2 R 2 , E DC
Hence
E BC
E AB
=
, I2
R1
R2
E DC
E AD
=
I4 =
, I3
R4
R3
=
I1
If EBD = 0, the potential at B must equal that at D, hence the drop from A to B must equal
that from A to D and the drop from B to C must be equal to D to C, i.e.
=
E AB E=
E DC
AD and E BC
∴
and
∴
or
or
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
E AB R 1
=
E BC R 2
E AD R 4
=
E DC R 3

R1 R 4
=

R2 R3


R1 R 2
=

R4 R3

R2 
R1 =
R4 
R3 
Page 24
Experimental Stress Analysis
10ME761
which is the condition for a balanced bridge.
Let ∆R1 be the change in R1 due to straining, then
∆R 1= R 1 .F.ε
To measure the unknown strain, R4 can be calibrated directly in terms of strain or instead
of balancing the bridge after straining the galvanometer deflection itself might be taken as a
measure of the strain.
(b) Unbalanced bridge. For the unbalanced bridge as shown in figure, at the point B.
I2 = I1 + IG
At the point D
I4 – IG = I2
Figure: Null balance Wheatstone bridge
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 25
Experimental Stress Analysis
10ME761
Null-Balanced Bridges
In static applications it is possible to employ a null balance bridge where the resistance of
one or more arms in the bridge is changed to match the effect of the change in resistance of the
active gauge. The null balance system is usually more accurate than the direction-readout bridge
and requires less expensive equipment for its operation. A relatively simple null-balance type of
Wheatstone bridge is shown in Figure. A slide wire resistance potentiometer is placed across the
bridge from B to D and the point C is connected to a point C’ on the balance resistor. Assume
that initially the bridge is balanced with active gauge in arm 1 so that R1R3 = R2R4 and R5 = R6.
The meter G is at null or zero voltage. Now consider a resistance change in R1, which upsets this
balance, causing a voltage indication on meter G. The slide wire on the potentiometer is
adjusted, making R5 ≠ R6, until the bridge is again balanced. The potentiometer adjustment,
which is calibrated, is proportional to the resistance change in the active gauge. Thus the
mechanical movement of the potentiometer serves as the readout means, and the voltage is
measured only to establish a zero or null point.
Now the equivalent resistances are
R 2e =
(a) Parallel balancing circuit.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
R 2R 5
R2 + R5
(b) Equivalent circuit.
Page 26
Experimental Stress Analysis
10ME761
R 3e =
R 3R 6
R3 + R6
An adjustment of the potentiometer will produce a change in the resistances of R5 and R6 equal
to ∆R5 and ∆R6, so that ∆R5 = -∆R6, since the total resistance of potentiometer remains constant.
The change in equivalent resistances become,
[R 2 (R 5 + ∆R 5 ) /(R 2 + R 5 + ∆R 5 )] − R 2 R 5 (R 2 + R 5 )
∆R 2e / R 2e =
R 2 R 5 /(R 2 + R 5 )
= (∆R 5 / R 5 ) /[1 + (R 5 / R 2 ) (1 + ∆R 5 / R 5 )]
∆R 3e / R 3e =
(∆R 6 / R 6 ) /[1 + (R 6 / R 3 ) (1 + ∆R 6 / R 6 )]
Hence by using an active gauge in arm 1 and a dummy gauge in arm 4, the change in
voltage output becomes,
=
∆E [VR 1R 2e /(R 1 + R 2e )2 ] [∆R 1 / R 1 − ∆R 2e / R 2e − ∆R 3e / R 3e ]
=0
∆R 1 / R 1 = Fε = ∆R 2 / R 2e − ∆R 3e / R 3e
= (∆R 5 / R 5 ) /[(1 + R 5 / R 2 )(1 + ∆R 5 / R 5 )]
- (∆R 6 / R 6 ) /[(1 + R 6 / R 3 ) (1+∆R 6 / R 6 )]
For an initially balanced bridge,
When
R5 = R6, R2 = R3
and
∆R5 = -∆R6, we get
∆R 1 / R 1 =
Fε
2 (1+R 5 / R 2 )( ∆R 5 / R 5 )
=
1 + 2(R 5 / R 2 ) + (R 5 / R 2 )2 [1 − ∆R 5 / R 5 )2 ]
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 27
Experimental Stress Analysis
10ME761
Equation indicates that the strain reading ε obtained by using a parallel-balance circuit is
nonlinear in terms of the adjustment of R5.

 ∆R t 
1 ∆R g
= (1 − n)∆T  α p −



∆T R g
 R t extra

1 ∆R g
.
∆T R g



represents the combined effect of temperature on actual gauge installation.
Calibration Methods
Necessity of calibration
1. By knowing the output of a constant voltage or constant gauge current bridge circuit
and sensitivity (units of deflection/mV or/µA) of the indicator, the indicator deflection can be
related to strain. This is the method generally used for designing strain gauge circuits and
selecting recording or indicating equipment. However, it is not suitable for final calibration of
system, since with this method the final accuracy of the system would depend upon the accuracy
with which all of the parameters involved could be measure and held constant during the period
of test. Hence a handy, reliable, and direct calibrations is extremely important.
2. Since the final output of the strain amplifiers is an electrical signal whose magnitude
depends on the strain to which the gauge is subjected, the strain appears as nothing more than a
wave or series of waves on an oscilloscope screen, and some means of judging its absolute
magnitude must be provided. The following methods may be used for calibrating a strain gauge.
1. Electrical calibration. (a) Static Strain Calibration
First Method : Shunt resistor. The strain-gauge resistance in one leg of the Wheatstone bridge
circuit is shunted by an open-circuited resistor of considerably high value as shown in Figure.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 28
Experimental Stress Analysis
10ME761
Constant-Current circuits:
Upto now we have discussed the wheatstone and Pot circuits driven by a constant voltage
source which ideally remains constant with change in resistance of the circuit. The output of
these circuits is non-linear and the non-linearity factor η increases as ∆ R/R increases. To
improve upon the circuit performance a constant current source instead of the constant voltage
source may be used. A constant-current power supply is a high impedance device (of the order
of 1 to 10 MΩ) which changes output voltage with changing resistive load to maintain a constant
current.
Wheatstone bridge circuit:
Consider the constant-current source Wheatstone bridge circuit as shown in Figure At
point A
I= I1 + I 2
Also
VAB = I1R 1
VAB = I 2 R 4
The output voltage E from the bridge can be expressed as
=
E V=
VAB − VAD
BD
= I 1R 1 − I 2 R 4
For the balanced bridge under no-load conditions,
E=0
Figure: Constant current Wheatstone bridge circuit.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 29
Experimental Stress Analysis
10ME761
Hence
I 1R 1 = I 2 R 4
=
VAC I1 ( R 1 + R 2 )
= I2 ( R 3 + R 4 )
Now
I1  R 3 + R 4 
=

I2  R1 + R 2 
Hence
R + R4
I1
=1 + 3
I2
R1 + R 2
I1 + I 2 R 1 + R 2 + R 3 + R 4
=
I2
R1 + R 2
1+
Or


R1 + R 2
I2 =

I
 R1 + R 2 + R 3 + R 4 


R3 + R4
I1 = 
I
 R1 + R 2 + R 3 + R 4 
∴
and
Thus the output voltage becomes
E=
I ( R 1R 3 − R 2 R 4 )
( R1 + R 2 + R 3 + R 4 )
When the bridge is balanced, E = 0 we get
R 1R 3 = R 2 R 4
If the resistances. R 1 , R 2 , R 3 and R 4 change by the amounts ∆R 1 , ∆R 2 , ∆R 3 and ∆R 4 ,
the output voltage E + ∆E measured with a very high impedance meter is
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 30
Experimental Stress Analysis
E + ∆E
=
1
4
∑ (R
i =1
i
+ ∆R i )
10ME761
( R 1 + ∆R 1 ) ( R 3 + ∆R 3 ) − ( R 2 + ∆R 2 )( R 4 + ∆R 4 ) 
For an initially balanced bridge, we get
=
∆E
 ∆R 1 ∆R 2 ∆R 3 ∆R 4 ∆R 1 ∆R 3 ∆R 2 ∆R 4 
−
+
−
+
−
−


R1
R2
R3
R4
R1 R 3
R2
R4 

+
∆
R
R
(
)
∑ i
i
4
IR 1R 3
i =1
4
The output given by Eq. is nonlinear with respect to ∆R because of the term
∑ ∆R
i =1
i
in the
denominator and the last two terms in the bracketed quantity. However, this non-linearity is
much smaller than the constant voltage source circuit.
Let
R=
rR g ,
R=
R=
R g , R=
2
3
1
4
∆R 2 =
∆ R 3 = 0, i.e. R 1 is an active gauge and R 4 is temperature-compensating dummy gauge,
then
∆R 1 =
∆R g , ∆R 4 =
0,
Eq. then reduces to
∆R g
Ir R g
∆E =
.
∆R g R g
2 ( 1+r ) +
Rg
=
Ir R g ∆R g
.
. (1 −η )
2 ( 1+r ) R g
Where the non-linear term η , is
=
η
∆R g /R g
Fε
=
2 ( 1+r ) + ∆R g /R g 2 ( 1+r ) + Fε
The nonlinear term can be minimised by increasing r as is obvious from Equation .
Generally r is taken to be equal to 9.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 31
Experimental Stress Analysis
10ME761
Potentiometer circuit:
Consider the constant-current potentiometer circuit shown in Figure. When a very high
impedance meter is placed across resistance R 1 , the output is
E = IR 1
When resistance R 1 and R 2 change by ∆R 1 and ∆R 2 , then output voltage becomes
E +=
∆E I ( R 1 + ∆r1 )
=
∆E I ( R 1 + ∆R 1 ) − IR 1
= IR 1 .
Thus
When
R1 = Rg
∆R 1
R1
then Equation becomes
E = IR g Fe
Therefore, the output voltage is linear with respect to resistance change ∆R and strain ε.
Circuit sensitivity
=
Sc
∆E
= IR g F
ε
= Pg R g F
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
when I=Ig
Page 32
Experimental Stress Analysis
10ME761
Figure: Constant current potentiometer circuit.
It can be easily seen that the circuit sensitivity of the constant current is higher than that
of the constant-voltage circuit.
Associated Instrumentation:
Static strains:
For most analysis work, static strains may be measured by a wheatstone bridge. The
bridge may be either operated on D.C. or A.C. When A.C is employed then a carrier system has
to be used. Generally null balance system is preferred over the out-of-balance method because
the null balance system is more accurate than the direct readout and is less expensive. The
earlier commercial strain indicators use a reference bridge circuit to provide the null-balance
system as shown in Figure. With commercial strain indicators, the adjustment resistance gives a
direct readout in strain. A commercial null-balance system is shown in Figure(a) and (b). These
models can operated either on A.C. or battery. Figure shows a direct readout amplifier and
junction box. Switching and balancing units are available for multiple gauge installations.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 33
Experimental Stress Analysis
10ME761
Figure: Direct readout amplifier and junction box.
In the manual null-balance strain indicators the output from each gauge is recorded
manually on a data sheet.
The data are usually processed by hand or with a simple
programmable calculator. Figure shows a typical strain indicator system. Figure shows the D.C.
operation of the Wheatstone bridge and Figure shows the A.C. operation. Figure shows an
automatically balanced bridge, it is necessary to build a phase-sensitive detector into the
rectifying circuit.
The manual direct reading strain indicator is shown in Figure. With this instrument, the
Wheatstone bridge is initially balanced and then the voltage output due to stain is amplified and
read out on a digital voltmeter.
Figure: Switching and balancing unit.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 34
Experimental Stress Analysis
10ME761
Figure: Direct digital readout strain indicator.
Figure: A typical strain indicating system.
When relatively large strain-gauge installations are used to analyse a problem then automatic
data-acquisition systems should be employed. The automatic data-acquisition system consists of
four basic sub-system which include the controller, the signal conditioner-scanner, the analog-todigital converter and the readout devices. The signal conditioner-scanner consists of the power
supply, the Wheatstone bridges and the switches used to connect a large number of gauges in
turn to the single voltage recording instrument. Each bridge has a small control panel with a
balance adjustment, a span adjustment and a calibration switch. The output from the wheatstone
bridge is switched into a high quality digital voltmeter, which measures the average of the input
voltage over a fixed measuring period.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 35
Experimental Stress Analysis
10ME761
Figure: D.C. operation of Wheatstone bridge.
Dynamic Strains:
While recoding dynamic strains, the frequency of the strain signal is an important
consideration in selecting the recording system. The help of the following table may be taken in
selecting the recording system depending upon the strain frequency.
Frequency Range
Recording System
Very low (0-3 Hz)
Integrating
digital
voltmeter,
potentiometer
recorder and xy-recorder.
Intermediate (0-10 kHz)
Oscillograph with a pen (0-100 Hz) and a light
writing (0-10 kHz) galvanometer.
High (0-20 kHz)
FM instrument tape recorder.
Very high (above 20 kHz)
Cathode ray oscillosope.
For very low frequencies, instruments such as potentiometer (or strip chart) recorders and xyrecorders, which employ servo-motors together with feedback control and null-balance
positioning can be used to measure the output voltage from the strain gauge bridge. The
operating principle of such an instrument is shown in Figure.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 36
Experimental Stress Analysis
10ME761
Figure: Operating principle of a servo-driven null-balance circuit (potentiometer) for
voltage measurement.
Potentiometer recorders can be used to measure voltages from 1 µV to 100 v. The chart
speeds can be varied over a wide range of 25 mm/h to 50 mm/s. Because of their low frequency
response, the potentiometer cannot be used in strain gauge applications where the strain signal
has frequency components greater than 1 Hz. Figure shows a double beam oscilloscope.
For recording strains at high frequencies magnetic-tape analog data recording systems are used.
Data recorded and strode on magnetic tape are usually played back and displayed on an
oscillograph. By varying the tape speed during playback, the time base can be extended or
compressed. Information stored on magnetic tape can be reliably retrieved any number of times
and different analysis made. Figure shows the schematic sketch for a magnetic tape recorder. In
a magnetic tape recorder, the tape (1.27 to 25.4 mm wide) is driven at a constant speed by a
served d.c. capstan motor over either the record or reproduce heads. The speed of the capstan
motor is monitored with a photocell and compared with the frequency from a crystal oscillator to
provide the feedback signal in a closed loop servo system designed to maintain constant tape
speeds. The signal written on the tape by the record magnetic head assemblies is in the form of
variations in the level of magnetism imposed on the magnetic coating of the tape. The reproduce
head converts these variations in magnetism back into electrical signals.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 37
Experimental Stress Analysis
10ME761
Figure: X-Y Recorder.
(Courtesy: Hewlett Packard, Palo Alto, California, USA)
Figure: Magnetic tape recorder.
Most instrument tape systems can be used in either direct recording or frequency
modulation (FM) modes. With direct recording, the intensity of magnetization on the tape is
proportional to the instantaneous amplitude of the input signal. Direct recording is usually
limited to audio recording where the human ear, on playback, can average the amplitude errors
or to recordings where the signal frequency and not the signal amplitude is of primary
impotence.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 38
Experimental Stress Analysis
10ME761
With FM recording, a carrier oscillator is frequency modulated by the input signal. The
oscillator has a centre frequency which corresponds to a zero input signal. Deviations from the
centre frequency are proportional to the input signal. The polarity of the input signal determines
the direction of deviation FM recording preserves the d.c. component in the signal and is much
more accurate than direct recording. Figure shows a basic FM system.
Figure: Basic F.M. system
Very high frequency recording can be accomplished with cathode ray oscilloscopes
which have bandwidths upto 500 MHz. The CRO is, in effect, a voltmeter which can be
employed to measure transient voltage signals. The heart of the CRO is the cathode ray tube
(CRT). The stream of electrons permits the CRT to be employed as a dynamic voltmeter with an
inertialess indicating system. Now-a-days, storage oscilloscopes are available which retain the
display the image of an electrical wave-form on the tube face after the waveform ceases to exist.
The stored display can be instantaneously erased to ready the tube for display of the next
waveform.
For dynamic strain-gauge applications, the CRO is an ideal voltage-measuring
equipment. The input impedance of the instrument is quite high (about 1 MΩ); thus, there is no
appreciable interaction between THE Wheatstone bridge and the measuring instrument. The
frequency response of an oscilloscope is usually quite high, and even a relatively low-frequency
model (800 kHz bandpass) greatly exceeds the requirements for mechanical strain
measurements, which are rarely more than 50 kHz. The sensitivity of the CRT is normally quite
low and it often requires 100 V to produce 25 mm of deflection on the face of the tube. This
difficulty is generally overcome by providing a built in amplifier in the CRO.The strain gauge
record showing strain as a function of time is obtained by photographing the face of the CRT as
the Spot of light traverses the fluorescent screen still camera or a movie camera may be used to
obtain a permanent strain signal record.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 39
Experimental Stress Analysis
10ME761
UNIT-2
Strain Analysis Methods
Introduction:
When the state of strain at a point and the direction of principal strains is known, then the
strain gauges can be oriented along these directions, and strain measurements may be made.
However, when the state of strain is not known, then three or more gauges may be used at the
point to determine the state of strain at the point. The resulting configuration is termed a strain
rosette. Strain-rosette analysis is the art of arranging strain gauges as rosettes at a number of
points on the object to be investigated, taking the measurements, and computing the state of
stress at these points.
Strain rosette analysis is based on the assumptions of isotropic, homogeneous and linear
material and of strain gradients so small that the strains can be considered as substantially
uniform over the area covered by the rosette gauges. In this chapter, we shall study strain
rosettes of various configurations currently in use.
Delta rosette
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Three gauge rectangular rosette
Page 40
Experimental Stress Analysis
10ME761
In one form of the rectangular rosette the gauges are in one plane and in the other on top
of one another. The rosette in the first case will cover smaller area than that of the latter and
hence will give more accurate results in a region in which the strains are varying. However, if
this rosette is mounted on a thin member subjected to severe bending, a considerable error will
be introduced since each gauge is at a different distance from the neutral axis. In this case the
use of the rosette having all the gauges in one plane and of the foil type would give better results.
In delta and T-delta rosettes, all the gauges are generally arranged in one plane and not on top of
another.
For completely defining the strain or stress at a component or structures generally it point on the
surface of a necessary to me three different directions at that measure strains along rosettes with
strain gauge element strain gauges or purpose are used for this.
Several multiple-element rosettes with gauges oriented along specified directions are
commercially available. These rosettes are denoted by the angles along which the gauges are
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 41
Experimental Stress Analysis
10ME761
oriented in them as the three-element rectangular rosette, delta rosette, four-element rectangular
rosette and three delta rosette. Some of the commercially available types of rosette gauges are
shown in Fig.
The principal stress directions may be in some cases determined by the brittle-lacquer coating
method. In such cases, i.e. when principal stress directions are known, two-element rectangular
rosettes are used for determining the principal strains/stresses. Some two-element rectangular
rosettes are shown in Fig.
THREE-ELEMENT RECTANGULAR ROSETTE
In this rosette the three gauges are laid out so that the axes of gauges Band C are at 45° and 90°
respectively to the axis of gauge A
Taking the OA axis (Fig) to be coincident with the Ox-axis, the angles corresponding to the
gauges A, Band C in the three-element rectangular rosette are
Hence from Eq
Substituting values of above equation we get
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 42
Experimental Stress Analysis
10ME761
Substituting the values of strains we get
Substituting the values of above equations the values of principal stresses in terms of the
measured strains can be obtained.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 43
Experimental Stress Analysis
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
10ME761
Page 44
Experimental Stress Analysis
10ME761
THREE-ELEMENT DELTA ROSETTE
In the three element delta rosette three gauges are placed at angular dispositions of 0, 120 and
240 degrees. For a delta rosette, one obtains from
Substituting these values in Eq.principal strains
Also, from Eq. the principal angle is obtained as
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 45
Experimental Stress Analysis
10ME761
From Eq.the corresponding principal stresses are
With the aid of Mohr's diagram, it is possible to show that the principal angle associated with the
algebraically large stress is
Four element rectangular rosette and Tee delta rosette
The four element rectangular rosette has four gagues with their axes 45 deg apart shown in fig.
Though theoretically strain readings from three gauges are suffient for analysis the strain reading
from the fourth gauges provides a convenient means of checking the readings since the sum of
the strains in any two directions at right angles should be an invariant.
In a tee-delta rosette, three gauges A, B and C (Fig. 20.7) are arranged as in the delta rosette such
that the included angle between any two axes is 60°. The fourth gauge D is placed with its axis at
right angles to the axis of anyone of the other three. The strain readings from all the four gauges
are utilized in the computation of principal strains and stresses and principal directions.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 46
Experimental Stress Analysis
10ME761
Equations for principal strains, principal stresses and principal directions have been derived for
both the tee-delta rosette and the four element rectangular rosette and are discussed in detail in
Ref. 1. It may be pointed out that the commonly used rosettes are the three-element rectangular
rosette and delta rosette.
Correction for transverse strain effects( Transverse sensitivity error correction )
1) Two element rectangular rosette
The principal strain can be obtained by
2) Three element rectangular rosette
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 47
Experimental Stress Analysis
10ME761
3) Delta rosette
Shear strain gagues
Strain gagues do not respond to shear strains. However the relationship between shear and
normal strains can be utilized to obtained from a strain rosette an output directly proportional to
the shear strain in the surface.
Let us consider two strain gagues as A and B oriented in x and y axis as shown in figure.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 48
Experimental Stress Analysis
10ME761
Strain along two direction
Shear strain is obtained by
Stress gagues
Applications it may be desirable to have an output from a single strain gagues directly
proportional to axial stress in a particular direction. Such gagues are known as stress gagues.
The principle of operation of a stress gagues as follows
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 49
Experimental Stress Analysis
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
10ME761
Page 50
Experimental Stress Analysis
10ME761
UNIT-3
Photo-elasticity
Stress-Optic Law
Maxwell reported in 1853 that the changes in the indices of refraction were linearly
proportional to the loads (thus to the stresses or strains for a linearly elastic material) and
followed the relationship:
n1 − n 0= C1σ1 + C 2 (σ2 + σ3 ) 

n 2 − n 0= C1σ 2 + C 2 (σ 3 + σ1 ) 
n 3 − n 0= C1σ3 + C 2 (σ1 + σ 2 ) 

where σ1, σ2, σ3 = principal stresses at the point
n0 = index of refraction of material in the unstressed state.
n1, n2, n3= principal refractive indices of the material in the stressed state associated with the
principal stresses, σ1, σ2 and σ3 respectively.
C1, C2 = stress-optic coefficients, which depend on the material.
Equations are the fundamental relationships between stress and optical effect and are known as
the stress-optic law.
Eliminating n0 from equations, we get
n 2 − n1= (C 2 − C1 )(σ1 − σ2 =
) C(σ1 − σ2 ) 

n 3 − n 2= (C 2 − C1 )(σ2 − σ3 =
) C(σ2 − σ3 ) 
n1 − n 3= (C 2 − C1 )(σ3 − σ1 =
) C(σ3 − σ1 ) 
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 51
Experimental Stress Analysis
10ME761
where C = C2 – C1 is the relative or differential stress-optic coefficient expressed in terms of
Brewster’s (1 Brewster = 10-12 cm2/dyn = 10-12 m2/N).
Now the wave equation is,
2π
E=
α cos (z − ct)
λ
= α cos φ
Angular phase shift between two waves,
∆ = φ2 - φ1
Since the stressed photoelastic models behaves like temporary wave plate, hence
2 πh
(n1 − n 0 )
λ
2 πh
=
φ2
(n 2 − n 0 )
λ
2 πh
∆ = φ=
(n 2 − n1 )
2 − φ1
λ
=
φ1
∴
Therefore, if a beam of plane-polarized light is passed through a slice of thickness h at normal
incidence, the relative retardation ∆ accumulated along each of the principal stress directions
becomes
2 πhC
( σ1 − σ 2 )
λ
2 πhC
=
∆ 23
(σ2 − σ3 )
λ
2πhC
=
∆ 31
( σ 3 − σ1 )
λ
=
∆12
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 52
Experimental Stress Analysis
10ME761
where ∆12, ∆23, ∆31 is the magnitude of the relative retardation developed between components of
light beam propagating in the σ3, σ1, σ2 directions respectively.
For two-dimensional or plane-stress problems (σ3 = 0) and we get
2 πhC
( σ1 − σ 2 )
∆= λ
Nfσ
σ1 - σ2 = h
or
∆
N = 2π
where
is the relative retardation in terms of a complete cycle of retardation and is termed the fringe
order.
fσ =
λ
C
is a property of the model material for a given wavelength of light and is called the material
fringe value in terms of normal stress and h is the model thickness.
Equation may be written as
=
τ
σ1 − σ 2 Nfτ
=
2
h
where fτ is the material fringe value in terms of shear and is equal to one-half of fσ.
For a perfectly linear photoelastic material,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 53
Experimental Stress Analysis
10ME761
1
(σ1 − vσ 2 )
E
1
ε2 =
(σ2 − vσ1 )
E
ε1=
ε1 =
− ε2
Thus
1+ v
( σ1 − σ 2 )
E
Equation becomes
 E  (ε − ε ) = Nfσ

 1 2
h
 1+ v 
or
ε1 − ε 2 =
Nfε
h
1+ v 
fε = 
 fσ
 E 
where
is called the material fringe value in terms of strain equation can also be written as
ε1 - ε2 = NFε
where
Fε =
fε
h is the model fringe value in terms of strain.
Equation states that in a transparent and isotropic model in which the stresses are twodimensional, the angular phase difference between the two rectangular wave components
travelling through the model is directly proportional to the difference of the principal stresses.
At those points in a stressed model where σ1 = σ2, the fringe order is zero and permanent
black dots appear at these points. Such points are called isotropic points. If σ1 = σ2 = 0 then also
the fringe order is zero at these points and permanent black dots appear. Such points are called
singular points.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 54
Experimental Stress Analysis
10ME761
Basic Elements of a Polariscope
The polariscope is an optical instrument containing polaroids that utilizes the properties
of polarized light in its operation. For photoelastic investigations two types of polariscopes are
used:
1. Plane polariscope
2. Circular polariscope.
In the plane polariscope, plane-polarized light is used and in the circular polariscope, circularly
polarized light is used. When the light is transmitted through the model then the polariscope is
called of the transmission type. The polariscope may also be either of the lens type or diffused
light type.
Plane polariscope
The basic arrangement of a lens type plane polariscope is shown in figure shows the set
up for a diffused light polariscope.
Figure: Lens type plane polariscope.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 55
Experimental Stress Analysis
10ME761
Figure: Diffused light plane polariscope.
The light source may be a mercury or a sodium vapour lamp, an incandescent filament
lamp or a bank of bulbs. Mercury or sodium vapour lamps are used as monochromatic light
sources and incandescent filament lamp is used as a white light source, for the lens type in an
opaque box with a ground glass on one side to give diffused light. The filter F is generally a


Wratten filter No. 77 to give a particular wavelength of 5461 A (green) or 5893 A (yellow).
The first field lens (FL1) gives a parallel beam of light in the field of view.
The function of the polarizer (P) is to produce plane-polarized light. Polarizers are nowa-days made from thin sheets of Polaroid.
The model M made out of a photoelastic material is loaded in a loading frame by which
various types of loads can be applied. The load is applied either by means of dead weight
through a level or by means of a screw and measured by a spring balance.
The analyzer (A) is similar to the polarizer and is used to combine the two beams coming
from the model. The polarizer and analyzer are generally coupled together by a flexible coupling
to achieve their simultaneous rotation.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 56
Experimental Stress Analysis
10ME761
The second field lens (FL2) is used to make the parallel beam of light converse on the
projection lens (PL), which finally projects the interference fringes onto the screen or camera
(C). The aperture of the projection lens may be controlled to obtain a part or full view of the
model.
Two types of set up are possible with the plane polariscope, i.e. bright, when polarizer
and analyzer are parallel and dark when polarizer and analyzer are crossed.
Circular polariscope
In addition to all the elements of a plane polariscope, the circular polariscope has two
more quarter wave plates (QWP), the first between the polarizer and model and the second
between the model and the analyzer as shown in figure. The fast and slow axes of the QWP’s
are inclined at 45° with the polarizer or the analyzer. The QWP’s are made of Polaroid film and
produces a path difference of λ/4 or a phase difference of 90° in the two light vectors passing
through them. Four different set ups are possible with the circular polariscope as depicted
below:
Set up
Polarizer-Analyzer
Quarter-wave plates
Field
1
Crossed
Parallel
Bright
2
Crossed
Crossed
Dark
3
Parallel
Crossed
Bright
4
Parallel
Parallel
Dark
The crossed-crossed set up is called the standard set up of the circular polariscope. The
first QWP converts plane polarized light into circularly polarized light and the second QWP
converts circularly polarized light into plane polarized light.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 57
Experimental Stress Analysis
10ME761
Fig: A
Figure:(B) Circular Polariscope
The diffused light plane polariscope can be easily converted into a circular polariscope by
interposing two QWP’s as done for the lens polariscope. Diffused light polariscope is generally
used for preliminary or rough work and lens type polariscope is used for more accurate work.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 58
Experimental Stress Analysis
10ME761
Effect of a Stressed Model in a Plane Polariscope
Dark-Field set up
Consider the dark-field set up of the plane polariscope (Fig) when the polarizer and
analyzer are crossed. The plane polarized light beam emerging from the polarizer can be
represented by
E = α cos wt
The light vector on entering the two dimensional stressed model will be decomposed into
two vectors along the two principal directions, one along the fast (or σ1) axis and the other along
the slow (or σ2) axis. Light vector (electric) along the fast axis on entering the model
E1e = α cos wt cos θ,
and along the slow axis
E2e = α cos wt sin θ
where θ is the angle between the axis of polarizer and maximum principal stress σ1 and the
subscript ‘e’ stands for entering.
Figure: Effects of a stressed model in a plane polariscope
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 59
Experimental Stress Analysis
10ME761
Since the light vector E1e travels faster than E2e, therefore, on emerging out from the model they
develop a phase difference. Hence the light vector leaving along the fast axis of the model E1l
and falling on the analyzer becomes,
E1l = α cos(wt + ∆ ) cos θ
Whereas the light vector leaving along the slow axis of the model and falling on the analyzer will
be given by
E 2l =
E 2e =
α cos wt sin θ
where the subscript l stands for leaving.
Since the axis of the analyzer is oriented at right angles to that of the polarizer, the light
vector transmitted through the analyzer is
=
E t E1l sin θ − E 2l cos θ
= α cos(wt + ∆) cos θ sin θ - α cos ωt cos θ sin θ
= α sin θ cos θ[cos(ωt + ∆) − cos ωt]
∆
∆
= -α sin 2θ sin  ωt +  sin
2
2

Intensity of light I is proportional to the square of the amplitude Et. Hence
∆
∆
I ∝ a 2 sin 2 2θ sin 2  ωt +  sin 2
2
2

∆
∆
= I 0 sin 2 2θ.sin 2  ωt +  sin 2
2
2

where I0 = maximum transmitted light intensity.
Light intensity I will be zero or extinction can be obtained in the following three ways:
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 60
Experimental Stress Analysis
10ME761
(a) Effect of frequency
 ωt + ∆  = nπ, n = 0,1, 2,.....


2
When 
 ωt + ∆ 


2  = 0 and I = 0.
Then sin2 
However, the circular frequency ω for light in the visible spectrum is approximately 1015
rad/s and neither the eye nor any type of existing high speed photographic film can detect the
periodic extinction associated with the ωt term and thus this factor can be ignored.
Hence we are left with
=
I I 0 sin 2 2θ sin 2
∆
2
(b) Effect of principal stress directions.
When 2θ = nπ, n = 0, 1, 2, …… sin2 2θ = 0 and I = 0
Therefore, when one of the principal stress directions coincides with the axis of the
polarizer, extinction occurs. When the entire model is viewed in the polariscope, a fringe pattern
is observed; the fringes are loci of points were the principal stress directions coincide with the
axis of the polarizer. The fringe pattern formed by the sin2 2θ term is known as the isoclinic
fringe pattern. These are the loci of points having constant stress directions. The isoclinic fringe
patterns are employed to determine the principal stress directions in the in photoelastic model.
(c) Effect of principal stress difference.
∆
∆
= nπ, n = 0, 1, 2, ...... sin 2
2 =0
When 2
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 61
Experimental Stress Analysis
10ME761
and extinction occurs. Therefore, when the principal stress difference is either zero (n = 0) or
sufficient to produce an integral number of wavelengths of retardation (n = 1, 2, 3….), extinction
occurs. When a complete model is viewed in the polariscope a fringe pattern is observed which
are the loci of points exhibiting the same order of extinction (n = 0, 1, 2, 3,…). The fringe
∆
pattern produced by the sin 2 term is known as the isochromatic (same colour) fringe pattern.
2
=
∆
Now
2 πhC
( σ1 − σ 2 )
λ
∆ hC
=
( σ1 − σ 2 )
Hence 2 π λ
or
h
(σ1 − σ 2 ), n = 0,1, 2,....
f
σ
n=N=
when a model is viewed with white light the isochromatic fringe pattern appears as a series of
coloured bands. Thus we find that in a plane polariscope the isoclinic and isochromatic fringe
pattern are obtained, superimposed on each other.
Bright-Field set up
In the bright-field set up the axis of the analyzer is parallel to that of the polarizer. Hence
=
E t E1l cos θ + E 2l sin θ
= α cos (ωt + ∆) + cos 2 θ + α cos ωt sin 2 θ
1 + cos 2θ 
 1 − cos 2θ 
= α cos(ωt + ∆) 
 + α cos ωt 

2
2




α
α
[cos(ωt + ∆ ) + cos ωt] + cos 2θ[cos(ωt + ∆ ) − cos ωt]
=
2
2
∆
∆
∆
∆
=α cos  ωt +  cos − α cos 2θ sin  ωt +  sin
2
2
2
2


DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 62
Experimental Stress Analysis
10ME761
∆
∆
+ sin 2 cos 2 2θ
2
2
∆
= α 1-sin 2 sin 2 2θ
2
α cos 2
Et =
Hence
∆
I ∝ α 2  1 − sin 2 sin 2 2θ 
2


∆
= I 0  1 − sin 2 sin 2 2θ 
2


I = 0 when sin2 ∆/2 sin2 2θ = 1 or the intensity I is maximum when sin2 ∆/2 sin2 2θ = 0.
Therefore, the conditions for maximum intensity of the transmitted light are now the same as
those for extinction for the dark field set up.
Effect of a Stressed Model in a Circular Polariscope
Dark-field set up
Consider the standard set up (crossed-crossed) of the circular polariscope as shown in
figure. The light vector leaving the polarizer can be written as
E = α cos ωt
Components of light vector on entering the first QWP become
π α
= cos ωt
2
2
π α
E 2e =
α cos ωt.sin = cos ωt
2
2
E1e =
α cos ωt cos
The first QWP produces a phase deference of π/2 and converts plane polarized light into
circularly polarized light. Components of light vector on leaving the first QWP become
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 63
Experimental Stress Analysis
10ME761
α
π  −α
cos  ωt +
sin ωt
=

2
2
2

α
E=
E=
cos ωt
2l
2e
2
E1l
=
If the principal axes of the model are inclined at angle θ with the axis of the first QWP, the
components of light vector along the principal axis of the model on entering are
=
E αe E1l cos θ − E 2l sin θ
α
α
=sin ωt cos θ −
cos ωt sin θ
2
2
=
E be E1l sin θ + E 2l cos θ
α
α
=sin ωt sinθ+
cos ωt cos θ
2
2
The model introduces a phase difference of∆. Therefore, the components of light vector on
leaving the model and entering the second QWP become,
α
[sin(ωt + ∆) cos θ + cos(ωt + ∆) sin θ]
2
α
=−
[sin(ωt + ∆ + θ) cos θ − cos(ωt + θ) sin θ]
2
=
E 4e E bl cos θ − E αl sin θ
α
=
[cos(ωt + θ) cos θ + sin(ωt + ∆ + θ) sin θ]
2
E αl = −
The second QWP also produces a phase difference of π/2. Therefore, the components of light
vector on leaving the second QWP and entering the analyzer become
E 3l = E 3e
=
E 4l
α 
π
π

cos  ωt + θ +  cos θ + sin  ωt + ∆ + θ +  sin θ 

2
2
2



α
=
[− sin(ωt + θ) cos θ + cos(ωt + ∆ + θ) sin θ]
2
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 64
Experimental Stress Analysis
10ME761
Figure: Effect of a stressed model in a standard circular polariscope
The resultant light vector transmitted through the crossed analyzer become,
π
π
=
E t E 3l cos − E 4l cos
4
4
1
=
(E 3l − E 4l )
2
α
= [cos(ωt + θ) sin θ − sin(ωt + ∆ + θ) cos θ
2
+ sin(ωt+θ) cos θ - cos(ωt+∆ +θ)sinθ]
α
= [sin(ωt + 2θ) − sin(ωt + ∆ + 2θ)]
2
∆
∆
= α cos  ωt + 2θ+  sin
2
2

Intensity of light I ∝ E2. Hence
∆
∆
I ∝ α 2 cos 2  ωt + 2θ +  sin 2
2
2

∆
∆
I I 0 cos 2  ωt + 2θ +  sin 2
=
2
2

DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 65
Experimental Stress Analysis
10ME761
I = 0, i.e., extinction can be obtained in two ways.
(a) Effect of frequency
 ωt + 2θ + ∆=
 (2n + 1) π


2
2

When
then
Hence
∆
cos 2  ωt + 2θ +  = 0, n = 0, 1, 2,....
2

I = 0.
But the frequency ω is very high and any extinction produced by it cannot be detected by
eye or any photographic equipment. Hence the isoclinics are automatically eliminated. Thus
I = I 0 sin 2
∆
2
(b) Effect of stress difference.
∆
When 2 = nπ, n = 0, , 2,….
Then
and
∆
sin 2 = 0
2
I=0
This type of extinction is identical with that for the plane polariscope and referred to as
isochromatic fringe pattern.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 66
Experimental Stress Analysis
10ME761
Thus
or
∆ h
=
( σ1 − σ 2 )
2π fσ
h
n = N = ( σ1 − σ 2 )
fσ
n=
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 67
Experimental Stress Analysis
10ME761
Fractional Fringe Order Determination
1
We can determine the isochromatic fringe order to the nearest 2 order by using both the
dark and bright-field arrangements of a polariscope. Further improvements on the accuracy of
the fringe order determination can be achieved either by using the mixed-field patterns or by
using Post’s fringe multiplication method. In order to achieve higher accuracy, as is desirable in
many applications, the following methods may be used:
1. Compensation techniques
2. Colour matching techniques
3. Equidensometry method
Compensation Techniques
Compensation is a technique in which partial modification of relative retardation either
by addition or subtraction is brought about so that the fractional fringe order at a point become
integral. Then by knowing the amount of relative retardation added or substracted the actual
fringe order at that point can be ascertained.
The following methods for compensation
techniques are most commonly used:
1. The Babinet compensation method.
2. The Babinet Soleil compensation method.
3. Tension or compression strip method.
4. Tardy method of compensation.
5. Senarmont method of compensation
6. Photometric method.
1. The Babinet Compensation Method. The Babinet compensator uses two wedges of quarts,
which is a naturally double refracting material. As shown in Figure, one of the wedges is fixed
in the instrument, while the other can be displaced relative to the first so as to alter combined
thickness by means of a fine micrometer screw with graduated drum head. With micrometer
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 68
Experimental Stress Analysis
10ME761
screw at zero, the compensator is said to be in the neutral position. The compensator is placed in
the polariscope in between the model and second quarter wave plate. The optic axis of the two
wedges are orthogonal to each other. The polarized light beam in one and retarded in the other
wedge.
The relative retardation R produced when the two wedges have been displaced from their
neutral position is given by,
K
(d − d0 )
λ
K
= (t + d0 − d0 )
λ
Kt K
=
=
x tan α
λ λ
=
R
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 69
Experimental Stress Analysis
10ME761
Figure: The Babinet Compensator
where K = n1 – n2
α = angle of wedge
≈ 2.5°
x = horizontal displacement, which is equal to the micrometer reading
or
 K tan α  .x


R=  λ 
= C. x
 K tan α  is a constant.


where C =  λ 
thus
Micrometer reading
m
=
m 0  Number of turns necessary to 


 produce a retardation of one 


wavelength
R=
The Babinet compensator allows fringe orders, to be determined to within 0.01 fringe.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 70
Experimental Stress Analysis
10ME761
2. The Babinet Soleil Compensation Method. The Babinet-Soleil compensator shown in
figure is an improvement upon the Babinet compensator. This instrument consists of a quartz
plate of uniform thickness and two quartz wedges. The optical axes of the quartz crystals
employed in the plate and the wedges are mutually orthogonal. The birefringence exhibited by
the compensator can be controlled by adjusting the thickness of the two wedges by turning a
calibrated micrometer screw. When t1 = t2, no relative retardation takes place, however for
t 2 < t1
>
, both positive and negative retardation can be produced over the whole area of the
compensator plate. This compensator is very useful for measuring boundary stresses.
Figure: The Babinet-Soleil Compensator.
In practice, a point is selected on the model where the fringe order is to be established
precisely. Then isoclinic parameters are established for this point to give the direction to either
σ1 or σ2. The compensator is then aligned with the principal stress direction and adjusted to
cancel out the model retardation. The reading of the screw micrometer is proportional to the
fringe order at that point. Like this fringe order at a point can be ascertained to within 0.001
fringe.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 71
Experimental Stress Analysis
10ME761
3. Tension or Compression Strip Method. In a standard circular polariscope, at an
isotropic point the fringe order is always zero.
Based upon this fact a method for the
determination of (σ1 - σ2) has been suggested by Wetheim and developed by Coker. In this
method a pure tensile or compressive stress is superimposed over an arbitrary system of σ1 and
σ2 in such a way as to convert the given stress system into one which is optically equivalent to an
isotropic point. White light is exclusively used in this method.
Figure shows how the plane stress system at a point can be converted to an isotropic
system plus a tension of (σ1 - σ2). A tension compensator may be placed parallel to the
minimum stress σ2 and the compression compensator must be placed parallel to the maximum
stress σ1. The value of (σ1 - σ3) equals numerically the stress in the compensator at extinction.
If the fringe order at a point by placing a tension compensator increases, then that point is having
tensile stress.
Figure: Superposition of retardation exhibited by model and compensator
4. The Tardy Method of Compensation.
The Tardy method of compensator is
generally preferred over the Babinet-Soleil method since no auxiliary equipment is required and
the analyzer of the polariscope serves as the compensator. In this method the polarizer of the
polariscope is aligned with the direction of the principal stress σ1 at the point of interest and all
other elements of the polariscope are rotated relative to the polarizer so that a standard dark-field
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 72
Experimental Stress Analysis
10ME761
polariscope exists. Then the analyzer alone is rotated to obtain extinction. The rotation of the
analyzer gives the fractional fringe order.
As shown in Figure, here θ = -π/4 and the light vector emerging out from the second
QWP becomes (see Art).
a
2
π
π
 
 π

 π 
sin  ωt + ∆ − 4  cos  − 4  − cos  ωt − 4  sin  − 4  


a
π
π 
= - sin  ωt + ∆ −  + cos  ωt −  
2 
4
4 

E 3l =−
Figure: The Tardy compensation method.
=
E 4l
a
2
π
π
 
 π

 π 
 sin  ωt − 4  cos  − 4  − cos  ωt + ∆ − 4  sin  − 4  


π
π 
a
=  − sin  ωt −  − cos  ωt + ∆ −  
2
4
4 


Let γ be the angle through which analyzer should be rotated to obtain extinction, i.e. Et = 0, then
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 73
Experimental Stress Analysis
10ME761
π
π
=
E t E 4l cos  − γ  − E 3l cos  + γ 
4

4

a
π
π 
π
= - sin  ωt −  + cos  ωt + ∆ −   cos  − γ 
2 
4
4 

4

a
π
π 
π
+ sin  ωt + ∆ −  + cos  ωt −   cos  + γ 
2 
4
4 

4

Simplifying, we get
∆ 
∆

=
El a sin  ωt +  sin  −=
γ   0
2   2


∆
sin  − γ  = 0
2

Hence
∆
− γ = nπ, n = 0, 1, 2, .......
2
or
∆
= nπ + γ
2
or
N=
or
∆
γ
= n+
2π
π
If the analyzer is rotated in the opposite direction then
γ
N = (n + 1) - π
Thus the Tardy method of compensation can be accomplished in the following way:
1. Using a plane-polariscope set up, determine the principal stress directions at the point of
interest by rotating the crossed polarizer and analyzer until an isoclinic passes through
that point.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 74
Experimental Stress Analysis
10ME761
2. Now rotate only the quarter wave plates of a circular polariscope to obtain a standard
dark-field arrangement.
3. Rotate only the analyzer then until an isochromatic fringe coincides with the point.
Determine the angle γ that the analyzer has rotated.
4. If the Nth order fringe moves to the point as the analyzer rotates through the angle γ, the
fringe order N0 at the point is
N=
N+
0
γ
π
If the (N + 1)st order fringe moves to the point as the analyzer rotates through the angle γ,
the fringe order N0 at the point is
N0 = (N + 1) −
γ
π
To account for the finite fringe width, the following procedure, as illustrated in fig, may be
followed:
1. The angle of analyzed ra = 0 [Fig (a)].
2. Rotate the analyzer until the fringe N just touches the boundary at the point of interest.
This is angle rb. [Fig (b)].
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 75
Experimental Stress Analysis
10ME761
Figure: Illustration of Tardy method of fringe order determination.
3. Continue to rotate the analyzer until N vanishes from the field of view. This is angle rc
[Fig.]
Then the fringe order at the point of interest of the free boundary is
N=
N+
0
rb + rc
360
This method is commonly referred to as the Tardy in-out method.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 76
Experimental Stress Analysis
10ME761
5. The Senarmont Method of Compensation.
(Friedel’s Method)
The following steps are involved for this method:
1. Remove first quarter wave plate.
2. Rotate system of polarizer and analyzer so that their axes make angles of 45° with the
principal directions in the modal at the point of interest.
3. Rotate second quarter-wave plate until one axis is parallel to the axis of the polarizer.
4. Rotate the analyzer until extinction is obtained at the point of interest.
The arrangements of the elements of the polariscope are shown in Fig.
Let the light vector from the polarizer be given by
E = a cos ωt
Since the polarizer is set as 45° to the principal directions in the model hence on entering the
model the light vector is resolved into two components, given by
1
E1e =
α cos ωt.cos 45 = α cos ωt
2
1
E 2e =
α cos ωt.sin 45 = α cos ωt
2
The model introduces a phase different of ∆. Therefore, on leaving the model, the components
of light vector become,
α
cos(ωt + ∆)
2
α
E=
E=
cos ωt
2e
2e
2
=
E1l
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 77
Experimental Stress Analysis
10ME761
The fast axis of the QWP is set at 90° to the polarizer axis. Hence on entering the QWP, the
light components become
=
Eac E1l cos 45 − E 2l cos 45
α
= [cos(ωt + ∆) − cos ωt]
2
=
E bc E1l cos 45 + E 2l cos 45
α
= [cos(ωt + ∆) − cos ωt]
2
Figure: Senarmont compensation method
The QWP introduces a phase difference of π/2. Hence on leaving the QWP , the light
vectors become,
α
π
π 
cos  ωt + ∆ +  − cos  ωt +  

2
2
2 


α
= [− sin(ωt + ∆) + sin ωt]
2
α
E=
E=
[cos(ωt + ∆) + cos ωt]
bl
bc
2
=
Eal
Now the analyzer is rotated through an angle γ to obtain extinction at the point of interest. Light
transmitted through the analyzer is
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 78
Experimental Stress Analysis
10ME761
=
E t Eal cos γ − E bt sin γ
α
= − [{sin(ωt + ∆) − sin ωt} cos γ + {cos(ωt + ∆) + cos ωt} sin γ ]
2
α
∆
∆
∆
∆

=−  2 cos  ωt +  sin cos γ + 2 cos  ωt +  cos sin γ  =0
2
2
2
2
2



∆
∆
Hence sin 2 cos γ + cos 2 sin γ = 0
∆
sin  + γ  =0
2

or
∆
+ γ = nπ , n = 0,1, 2,....
2
f
∆
= nπ − γ
2
∆
γ
= n−
N=
π
2π
Tardy and Senarmont compensation methods are called the ‘goniometric’ or ‘null location’
methods.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 79
Experimental Stress Analysis
10ME761
UNIT-4
Two Dimensional Photo-elasticity
Stress components
From the analysis of Mohr’s circle for a plane state of stress the following results are obtained
Hence' from photo elastic observations one can determine (σx-σy) and τxy. The values of (σx-σy)
and τxy or equivalently, (σ1-σ2) and θ enable one to construct a Mohr's circle, but the origin
cannot be located. Consequently, the individual values of σx and σy cannot be determined, unless
recourse is made to other techniques. These are generally called as separation techniques.
Separation techniques
Many techniques have been proposed to determine the individual values of σ1 and σ2. Such
techniques which are commonly used are described .here.
A) Use of lateral extensometer
B) At the free boundary
C) Use of Laplace Equation
D) Shear difference method
E) Oblique incidence method
A . Use of lateral extensometer
From Hooke's law, for the plane state of stresses,
We can write as
If h is the original thickness of the model before being stressed and h + ∆h is the thickness after
loading, then
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 80
Experimental Stress Analysis
10ME761
if the change in thickness can be determined, then a knowledge of v and E of the model material
will enable-one to determine (σx+σy) from above set of equations
The instrument that is used to determine the change in thickness, i.e. the change in the lateral
dimension of the model, is called the lateral extensometer.
Since (σx+σy) and (σx-σy) are known, the individual values of σx and σy can be calculated.
Alternatively, as (σ1+σ2) = (σx+σy) and (σ1-σ2) are known, σ1 and σ2 can be determined.
B. At the free boundary
When a boundary of the model is not loaded directly, it is called a free boundary. The normal
and shear stress on a plane tangential to a free boundary are therefore zero. The principal stress
axes are normal and tangential to the boundary as shown in figure.
D. Shear difference Method
This is a step-by step integration process along a selected line starting from a point where one
of the normal stress is known. Generally, the initial point1ies'on the boundary where the
individual values of σ1 and σ2 are known. The method makes use of one of the differential
equations of equilibrium.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 81
Experimental Stress Analysis
in the absence of body forces. The line of integration called the x and
10ME761
axis, joins the point of
interest where the individual values of σx and σy are known. Assuming that this line is called the
x-axis, the first equation above can be integrated from the initial point i up to the desired point j.
Thus
Using the finite difference form, the above equation can be written as
The values of ∆τxy/∆y are calculated from the values of τxy determined along two line CD and
EF, which are ∆y/2 parts from the line of integration i e x-axis.
In general, the boundary at the initial point i will not be normal to the x-axis.
Let θ be the angle between the tangent to the boundary and the x-axis (shown below fig).
At the boundary, one of the .principal stresses (say σ2) is zero and the other stress (i.e. σ1)
tangential to the boundary can be evaluated from the isochromatics.
Then from the equation is given as,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 82
Experimental Stress Analysis
10ME761
E. Oblique incidence method
Let the incident ray be in the xz-plane and let Φ be the angle between the z-axis and the incident
ray. As shown in below figure. For normal incidence along the z-axis, the stress σ1 and σ2
corresponding to these stresses, However, experiments reveals that for oblique incidence , the
stress components which cause photo elastic effect are not the primary stresses rbut the
secondary stresses.
The secondary stresses corresponding to a given direction are those rectangular stress
components whose vectors lie completely in a plane perpendicular to the given direction. for
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 83
Experimental Stress Analysis
10ME761
example, for the normal incidence ZO, the stress components w se vectors lie completely in the
plane perpendicular to ZO, i.e. in the xy-plane, are σx , σy and τxy. For the direction Z'O, the
stress vectors which lie completely in a plane, perpendicular to Z' 0 are σ’x , σ’y = σy and τx'y' . For
incidence along the y-axis, t'he appropriate stress component is σx only since σz and τxy are zero
for a plane stress cases. For incidence along the x-axis, the appropriate stress component is σy.
These are called secondary
rectangular stress components corresponding to direction of
incidence The principal stresses corresponding to the secondary rectangular stress components
are called secondary principal stress. There are denoted usually by principal stress one and
second.
Stress-Optic Law for Oblique incidence.
The stress-optic law for oblique incidence is similar in form to the stress optic law for normal
incidence. It states that relative retardation is proportion to (σ’1-σ’2) and the length of the light
path for the given direction of incidence.
Where h' is the length of the light path and σ’1and σ’2 are the secondary principal stresses for the
given direction of incidence. The number of wavelengths of relative path difference is,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 84
Experimental Stress Analysis
10ME761
where f' is the model fringe constant in tension for the given oblique incidence. If one uses the
material fringe constant F, then
If the oblique incidence is at an angle Φ to the z-axis in the xz-plane, then
And
In the case of plane stress under normal incidence, the state of stress does not change from point
to point along the light path such as AB.
The stress optic law as expressed by previous set of equation can be considered as a generalized
law if one ignores the variation in the directions of the secondary principal stresses along the
light path and if their directions do not change then,
If (σ’1-σ’2) varies linearly or if it is constant, then the above equation can written as
Where (σ’1-σ’2) represents the main value or the value at the midpoint of the light path.
Or
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 85
Experimental Stress Analysis
10ME761
Or
From normal incidence along path AB,
Where θ is the isoclinic parameter under normal incidence. One can solve the above three
equations to σx,σy and τxy. One can determine the isoclinic parameter θ’ for the oblique
incidence and use Mohr’s circle to write
From the set of equation by solving for σx and σy.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 86
Experimental Stress Analysis
10ME761
PART –B
UNIT-5
Three Dimensional Photo elasticity
Introduction
Many stress analysis problems are three-dimensional in nature for which the twodimensional photoelastic method cannot be employed. These problems, however, can be solved
either by locking in the stresses in the model or a multilayer reflection technique may be used to
determine the stresses at the inner layers of the body. Three-dimensional stress distribution in
the body can also be determined by the scattered light method. In this chapter we shall discuss
the locking-in method the stresses in three-dimensional models in detail and the multilayer
reflection technique in brief.
The stresses in a three-dimensional model can be locked-in either by stress-freezing, by
curing method, by creep method or by gamma-ray irradiation method. Out of these methods, the
stress-freezing method is most widely used. This method consists in loading the model at room
temperature (at which the primary secondary bonds break down), keeping at that temperature for
few hours and then cooling to room temperature at a slow rate. The stresses thus frozen in the
model can be analyzed by slicing and viewing in a polariscope.
Maxwell in 1853, and Filon and Harris in 1923 had both obtained the stress-frozen effect
accidentally, while Tuzi in 1927 and Solakian in 1935 both attempted to calculate the residual
stresses and to relate them to the applied loads. None of these, however, fully appreciated the
significance of the phenomenon, and it was Solakian who in 1935 and Oppel in 1936 first
produced a quantitative solution of a three dimensional problem, followed by He’tenyi, who in
1938 first established the strict proportionality of stresses. This discovery opened the way to the
complete solution of the three-dimensional problems and resulted in a very sharp increases in
activity in photoelastic research.
Hetenyi, O’ Rourke, Drucker and Mindlin, Drucker and
Woodward, Frocht and Mindlin developed methods of measurement of the frozen stresses and
Jessop in 1949 devised integration methods for determining the separate stresses. Jessop and
Stableford in 1953 published an extension to three dimensions of the Lame-Maxwell equations,
which made it possible to determine the principal stresses along a stress trajectory in a plane of
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 87
Experimental Stress Analysis
10ME761
symmetry. The stress-freezing method of three-dimensional photoelasticity today is a practical
instrument of stress analysis, particularly in complex structures.
Stress Freezing Method
In the stress-freezing method, the model deformations caused by the applied loads are locked in
the model. This is made possible by the diphase behavior of many polymeric materials when
they are heated. Polymeric materials are composed- of hydrocarbon molecular chains. These
molecular chains exist in the material in two essential forms. One form is a well-bonded, form
three dimensional network, called primary bonds. The second one called the secondary bonds,
occur in a form which is less solidly bonded and are shorter compared to primary bonds.
At room temperature, both bonds are firm and resist deformation when loads are-applied.
However, as the temperature is increased, the Secondary bonds loose gradually their ability to
resist" deformation" At a particular temperature called the critical temperature, the secondary
bonds break down completely and the applied load is carried entirely by the primary bonds.
Consider a model made of such a diphase polymeric material and subjected to a given system of
loading. Initially, at room temperature this load is carried by the primary bonds and the
secondary bonds together. Let the temperature be raised gradually until the critical temperature
for - the particular material is reached. At this temperature, the secondary bonds break down,
becoming a soft –jelly like material. The load is now taken up entirely by the primary bond. With
the load still on, the temperature is gradually reduced to the room temperature. During this
process the secondary bonds gradually solidify and lock the primary bonds in their deformed
configuration. If the load is now removed, primary bonds tend to regain their original
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 88
Experimental Stress Analysis
10ME761
undeformed configuration, but this is however prevented by the secondary bongs is reached
which does not differ appreciably from the deformed configuration. Hence, the deformations are
locked inside the model. These steps are shown in fig below
(a) Unloaded model at room temperature: primary & secondary bonds
(b) Loaded model at' room temperature: primary &secondary bonds
(c) Loaded model at critical temperature primary bonds
(d) Loaded model cooled to room temperature primary bonds
(e) Unloaded rnodel at room' temperature; primary & secondary bonds
The importance of this locking in the deformation process lies in a very useful aspect which can
be seen from the spring-ice analogy. If the assembly as shown in Fig. (e) is now cut into thin
slices, each slice will retain the corresponding parts of the deformed springs. Experiments With
diphase materials reveal that even very thin slices can be carefully cut. and polished without
destroying their locked-in birefringent characteristics. It is thus important aspect that is
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 89
Experimental Stress Analysis
10ME761
extensively used in three-dimensional photoelectic - analysis. Below figure shows the stress
pattern in a slice taken from a stress frozen model.
Scattered light photo-elasticity
The techniques of scattered light photo elasticity can be used with great advantage for the
analysis of stresses in general three-dimensional photoelastic models. The technique is
nondestructive in nature and the model can be subjected to static loading, transient loading,
prestressing and thermal loading. Further, the tests can be conducted at room temperature
without resorting to freezing techniques.
Scattered light photo elasticity differs from conventional photo elasticity of the-transmissiontype in several ways.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 90
Experimental Stress Analysis
10ME761
Scattering
When a wave of light which is traversing space meets an obstacle, such as a small water particle
or a dust particle or even a gaseous molecule like that of oxygen, the obstacle acts as a secondary
source and scatters some of the light. If the wave passes through a cloud of such obstacles its
intensity is weakened owing to the loss of light by scattering. The percentage of light scattered is
proportional to A-4, where A is the wavelength. Considering red light (A = 7200 A or 720 nm)
and violet light (A =d 4000 A 400 nm), the law predicts that violet light is scattered (720/400)4
or ten times ~ore than red light from particles which are smaller than the wavelength of either
colour. When a clear sky is observed at an angle to the direction of the sun's rays, the molecules
of air scatter light, and the sky appears blue. This is because the violet end of the spectrum is
scattered more than the red end. At the time of sunrise or sunset the surrounding region appears
reddish or orange because while travelling the great thickness of the atmosphere the scattering
removes the blue rays more effectively than the red. Hence the transmitted light observed
appeased.
Polarization associated with Scattering
Let us suppose that a ray passing through a scattering medium is viewed transversely so that the
line of vision is at right angles to it. The electromagnetic vibrations corresponding to the
scattered light are several incident rays. If the incident ray is unpolarized, the scattered Light
vibration in the transverse plane will be as shown in Fig. The scattered light that the eye
perceives arises entirely from the component vibrating at right angles to the line of vision,
whereas the component of vibration along the line of vision will have no effect. Consequently,
for observation along OE, the apparent vibration is along MINI. and for observation along OF,.
The apparent vibration is along M2N2. Therefore, for directions of observations at right angles to
the incident ray, the scattered light is completely polarized. This phenomenon of polarization
associated with scattering can be made use of as either a polarizer or an analyzer
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 91
Experimental Stress Analysis
10ME761
Scattering as Polarizer
Consider a beam of unpolarized light is travelling. Through-a-material fig below. The incident
light has all possible directions of vibrations in the wave front but light scattered in a particular
direction at right angles to the direction of incidence always vibrates at right angles to the plane
containing the direction of incidence and direction of observation.
Consider the li.ght scattered from point C. If the model considered is stressed, the polarized light
scattered from this point gets divided into two components along the directions of the secondary
principal stresses in the plane transverse to CE. In traversing the stressed medium CD. These
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 92
Experimental Stress Analysis
10ME761
two vibratory components acquire a certain relative phase difference. If an analyzer is now
interposed between the eye E and D, the resulting picture is either bright or dark depending upon
the phase difference acquired over the distance CD. If the section that is illuminated is moved
from BC to .B'C', the resulting picture as observed from the same E will be different since the
polarized light now has traversed' the stressed medium C'D. Let us assume that the directions of
the secondary principal stresses one and two do not change over the length CD or C'D. If N is the
birefringence caused over the distance CD, and N + dN the birefringence caused over the
distance C’D, then the difference AN is due to the slice C'C = Ay. Hence, AN/Ay is the space
rate of formation of fringes (or birefringence) in successive planes normal to the direction of
observation. It should be noted that the photoelastic effect one observes is entirely due to the
retardations acquired over CD or CD and the distance BC or' B'C' has no effect (other than
diminishing the intensity.
In this respect, the stress-optic law does not differ from the one used in the conventional
transmission type. However, the major advantage of scattered light is in being able to obtain the
values of pp one dash and pp second dash at every, point along the path C'D. This can be seen
from Fig. 10.3. By moving the incident light successively along B1 B2' B3, .. the birefringence
caused over the path lengths CID, C2D, C3D, ... , are measured at E. If NI is the birefringence
due to CID, N2 the birefringence due to C2D, N3 due to C3D, etc.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 93
Experimental Stress Analysis
10ME761
Scattering as Analyzer
Let the incident light be now linearly polarized. This light gets divided into two components
along the secondary principal axes at the point of entrance. In travelling from B to e, these
components pass through a stressed medium Be and at point e, the vibratory components along
the secondary axes σ1 and σ2 will have a certain relative phase difference.
The particle at e acts as an analyzer (because the light scattered from point e effectively vibrates
at right angles to the plane containing the direction of incidence and the line of vision); and for
an observer along DE, the resulting point image will have a definite light intensity depending on
the relative retardation acquired by the light components over the distance Be. If the line of
observation is moved to D'E', the resulting picture varies depending upon the additional
retardation acquired over the distance. Now, if the additional birefringence added over that
distance is .1N, then is the space rate of formation of birefringence in successive planes normal
to the incident light beam. The resulting light intensity as seen from E or E' is not affected by the
birefringence.picked up in the distance Cl) or C'D', since there is no analyzer between DE It isagain assumed as that the secondary axes do not rotate along the path BC or BC'.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 94
Experimental Stress Analysis
10ME761
Scattered light polariscope
Figure shows the sketch=of-a-scattered light polariscope and the essential elements involved.
The source of light can be either a high-intensity mercury arc lamp or a laser. A 15 mW, He-Ne
laser is quite adequate are most of the techniques described in this book the source is a mercury
lamp it is necessary to collimate the incident light beam using suitable lens systems. The laser
beam may have to be expanded and collimated again (after suitable filtering) photography
purpose for investigation, once either a sheet of light or a thin pencil 0 beam. The model is
attached to an x-y translating stage. The translating stage is fixed to a rotating stage which can be
raised or lowered. Thus the model can be rotated about the axis of the incident ray, and any point
of the model can be brought under observation through the X y- and z-motions. The line of
observation is perpendicular to the incident ray.
The compensator can be fixed to the rotating stage and can be rotated independently or with the
rotating stage and the model. The polarizer and plate units can be fixed separately or to the
rotating stage Fig. shows the photograph of a scattered light polariscope and the photometer to
monitor the intensity of scattered light.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 95
Experimental Stress Analysis
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
10ME761
Page 96
Experimental Stress Analysis
10ME761
UNIT-6
Photoelastic (Birefringent) Coatings
Separation of Principal Stresses
From the isoclinics and isochromatic data, it is possible to determine the directions and
difference of principal stresses in the interior of the coating cemented to the machine part, but
not the individual principal stresses. Therefore additional information is required to find the
individual principal stresses: The two important methods among several methods developed for
separation of principal stresses are: (a) the oblique incidence method, and (b) the strip coat
method, which give the additional data required.
a) Oblique-Incidence Method
Separation of principal stresses at any point on the surface of the machine part is accomplished
by using the isochromatic patterns from normal incidence and oblique incidence. It is assumed
that the incident ray and reflected ray in the reflection polariscope do not undergo any refraction
effects. Figure shows the schematic diagram of measurement of fringe order III from oblique
incidence. The rectangular stress components which cause photoelastic effect are σ1 and
σ2cos2Φ. The length of the, oblique light path inside the coating is h/cosΦ .Hence, the secondary
principal stress difference.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 97
Experimental Stress Analysis
10ME761
Where n1 is the fringe order observed. Substituting in above equation and simplifying, one gets
The expressions for the principal stresses in the machine part are:
In the above expressions, n is the fringe order observed under normal incidence and nl the fringe
order under oblique incidence. The equation relating the strains in the machine part, the material
constants and the fringe order from oblique incidence is
Solving for €1p and €2p from above equations then the individual principal strains are
Thus, it is possible to determine the individual values of all principal stress and strains.
b) Birefringent Strip-Coating Method
This method is based on the anisotropic reaction of birefringent coating to strain when the
coating consists of a number of narrow parallel strips cemented to the surface of the metal part.
These closely spaced narrow strips whose thickness is several times the strip width respond
mainly to the strains (in the machine part) that are parallel to their long axes. Reflective surfaces
are provided at the junction of the metal part and the strips. The strain in the strips is
approximately uniaxial. The isochromatics when measured give the strain in the direction of the
strip. However, if one assumes that the strip responds to the strains in the longitudinal direction
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 98
Experimental Stress Analysis
10ME761
lateral-direction' and also to shear strain the strain optic .Jaw for a single narrow rectangular strip
cemented to the part can be expressed as
The axial, transverse and shear strain sensitivities SI and S. depend on the height (h,) and width
(IV) of the strips. If the ratio is very large and strips are closely spaced, these strips behave like a
continuous coating. When the height of strip is large compared to the width, i.e. when is very
small, the optical response is primarily due to strain along the length of the strip and the
transverse and shearing strain effects are very small and may be neglected. The strain optic law
then becomes
From the above equation it is clear that the fringe orders in the strips are proportional to the axial'
strain along the length of the strips. A fringe pattern observed from closely spaced strips on the
machine part appears as though the fringes are continuous.
A two-strip-coating method was proposed by E. Monch In this method a series of narrow strips
of rectangular cross-section are cemented over a continuous birefringent coating which in turn is
cemented to the part under strain Fig. Reflecting surfaces are provided at the junction between
the continuous coating and the part and also between the strips and continuous coating so that
light passage is as shown in Fig.above. The coloured fringe pattern of the isochromatic can be
photographed. From the continuous coating, the isochromatic fringe order gives the difference of
principal strains (strain difference) and the angle θ1 between the direction of principal strain 1
and the direction of the strips can be obtained from the isoclinics. From fringe order n, in the
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 99
Experimental Stress Analysis
10ME761
strip, the strain 1 along the length of the strip can be determined. From these three measured
values of strain and θ it is possible to determine the state of strain at the point.
The equations for principal strains appear as
The expression for principal stresses are as follows,
Two strips and the continuous-coating method can also be used to determine the principal strains
and stresses. In this method two birefringent strip coatings are cemented to the surface of a
continuous coating which in turn is cemented to the part. A reflecting surface is provided at the
junction of the strips, the continuous coating and the part. The two strips are orthogonal to each
other in the x- and y-directions and these measure linear strains. The continuous coating
measures
In order to measured from the--strip-coatings in the x- and y-directions respectively. Assuming
that the thicknesses of the coating and that of the strip are equal
The principal stresses from above Equation and from the biaxial relationship are stress-strain
The strip method can be used to determine the principal strains in a part as in the case of the
strain-gauge technique by employing strips in three different directions. The three different strip
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 100
Experimental Stress Analysis
10ME761
coatings oriented at 0°, 45° and 90° directions are cemented to the part to determine strains "45",
90", as in the case of rectangular rosettes. Let the strains along these directions be respectively
The conventional rectangular rosette equations can be used to compute the principal strains
The principal stresses are
From the three methods described above, it is seen that the individual principal stresses can be
determined by employing the strip-coat method. However, the strip-coat method is not popular
since birefringent strips are not commercially available in the market.
Effects of coating thickness: Reinforcing effects, Poisson's ratio
Residual stresses get developed in the coating due to: (i) time-edge effects, (ii) difference in
thermal expansion of the metal part and coating material, (iii) casting defects, and (iv) machining
and cementing. These are responsible for the presence of initial fringes in the coating. These
initial fringes due to residual stress can be eliminated by proper curing and also by taking care
during casting, cementing and machining the parts. Apart from the above errors, there are other
sources which cause inaccuracies in the measurement of fringes in thick coatings. To obtain a
reasonable number of fringes in analysing a problem, a thick coating is required. However,
thicker the coating, larger is the error introduced. The ideal coating has zero thickness which
does not produce any fringe at all. A compromise must therefore be made regarding thickness.
The inaccuracies due to thickness effects are:
i) Variation of stress difference through the thickness of the coating, which means variation
of strain difference,
(ii) Change in curvature of the part during loading,
(iii) coating-reinforcement effect,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 101
Experimental Stress Analysis
10ME761
(iv) Errors at the ends of the coating boundary, and
(v) Errors at the boundary due to mismatch of Poisson's ratio between the material of the part and
the coating. While the coating-reinforcement effect decreases the stress difference in the coating,
others produce stress variation in the coating. This in turn makes the strains vary through the
thickness of the coating.
Effects of the Thickness of Birefringent Coatings
It was assumed in the previous discussions that the strain was transmitted from the machine part
across the interface to the coating in an ideal manner. Unfortunately, this is not true in practice.
The observed birefringence is directly proportional to the surface strain at a point as long as the
strain is uniform over the surface. In such cases it is easy to determine the strain. However, in the
presence of strain gradients at the metal surface the deformation of the coating will vary across
its thickness and the observed birefringence will depend on the magnitude of strain and also on
the intensity of its gradient. In addition, the photoelastic pattern will be affected by any curvature
of the metal surface occurring under load. This distortion of the strain distribution in the coating
thickness due to strain gradients and curvature on the metal surface is known as the thickness
effect. The magnitude of error introduced due to thickness effect depends on the nature of the
problem. In some cases the error may be small and in some others it may be very large. This
problem has been analyzed fairly exhaustively and the reader is referred to appropriate
references.
Reinforcing Effects
Birefrigent coatings are elastic materials which are cemented to surfaces of machine parts or
structures. The structures or machine parts are subjected to loading. The load gets transmitted
from these machine parts to the coating by shear and normal traction developed at the interface.
When a prototype or actual structural member coated with birefringent coating is subjected to
complicated loads, the coating carries a portion of the load. Therefore, the strain in the machine
part is reduced to some extent. It is possible to calculate the reinforcing effect due to coating and
also correction factors which take care of the reinforcing effect. Zandman, Redner and Riegnerhave studied the reinforcing effects of birefringent coatings and established correction factors
for: (i) plane-stress problems, (ii) bending of plates, and (iii) torsion of circular shafts.
The curves in Fig.represent qualitatively the. plot of The Young's modulus for the material of
the machine part increases from a to d. An analysis of these curves shows that large errors
can be introduced if the results from the coatings are not corrected. These curves show decreases
and then increases with increase in he/hp. One of the two reasons for this error is the reinforcing
effect which reduces the curvature of the plate, when compared to the uncoated part. Hence, this
is responsible for reduction in strain at the interface. The other reason is that the average strain
determined over the thickness of the coating is based on
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 102
Experimental Stress Analysis
10ME761
the average fringe order measured by a reflection polariscope because the strain varies through
the thickness of the coating. This average strain is always higher than the strain at the interface.
Thus it is clear that the strain variations through the thickness counteract the reinforcement
effect. Therefore, in order to obtain the true difference of strains in the uncoated part, the
difference of strain measured in the coating must be corrected by using the correction factor.
Poisson's Ratio Variation
It is clear from the previous discussions that the inaccuracies which have been introduced in the
measurement of strains in birefringent coating on account of reinforcement, strain gradient and
curvature effects, are very small in the case of plane-stress problems. However, on the other
hand, the error introduced on account of mismatch in Poisson's ratio of birefringent coating and
the metal part is very high. It has been assumed that the deformation of the part and the coating is
identical in every respect. The deformation is identical both in the machine part and coating
provided the Poisson's ratio of the part and coating are the same. Usually the Poisson's ratio of
the coating (ve) is always greater than that of the metal part (vp). Since the coating is bonded to
the part, the strains in the coating and part will be same at the interface. The deformation of the
upper layer of the coating will be different from the deformation at the interface due to the
mismatch of Poisson's ratio. The conclusion drawn by several investigators is that the Poisson's
contraction of the coating is greater than that of the part when Poisson's ratio of the coating is
greater than that of the part.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 103
Experimental Stress Analysis
10ME761
UNIT-7
Brittle Coatings
Introduction
In the process of designing a machine or structural component, it is often desirable to resort to a
quick, easy and an inexpensive technique of evaluating qualitatively and quantitatively, the state
of surface strain in the component. Surface strains, obtained from whole-field techniques, such as
the brittle-coating method, serve the following broad objectives
(i) Diagnosis or failure-analysis of in-service failure of components, and
(ii) Determination of the location and orientation of strain-sensors
Before going into the theory' and application of the technique, it is necessary to understand what
brittle-coating is and what data are obtained from a brittle-coating test. Brittle coating is any thin
surface coating applied on the surface of a model or a component under test and which fractures
or cracks in response to the strain applied to the model on which it is coated, indicating
quantitatively the direction and magnitude of surface strains in the model. If these surface strains
are within the elastic limit of the material of the models, the resulting crack pattern provides an
overall graphical picture of the distribution, sequence and direction of surface strains. If the
coatings are carefully calibrated, the crack pattern also provides quantitative values for the
magnitudes of principal surface strains. The State of strain in the coating thus indicates, both
qualitatively and quantitatively, the state of strain in the model, and hence the state of stress.
The technique of brittle coating is applied to a number of stress-analysis problems, such as the
determination of stress concentrations in components under the influence of static, dynamic and
impact loads. It is also used for the measurement of thermal and residual stresses. The models or
prototypes on which brittle coatings .are applied can be made of any material plastics, wood,
paper, rubber, glass, bone and metals.
The major disadvantage of the brittle-coating method is the loss in the accuracy of results if
elaborate precautions are not taken in evaluating the sensitivity of the coating and the hazardous
nature of the chemicals involved in the application of this technique.
In order to understand the steps involved in a typical brittle-coating application, consider a flat
tension model as shown in Fig.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 104
Experimental Stress Analysis
10ME761
The width of the specimen varies along the length of the model, the minimum width being at the
middle of the model The model is first coated with a thin layer of brittle coating, 0.125 to 0.25
mm thick, which is later dried at room temperature and cured at an elevated temperature. The
details of the technique of applying brittle coating and curing are discussed in later sections. The
coated model is loaded in incremental steps. When the strain or stress in the coating at point A
exceeds a critical value, a crack develops in the coating. On further loading, this crack gets
extended and, in addition, new cracks are also formed. This crack pattern indicates the state of
strain in the coating from which one can obtain the state of strain and hence the state of stress in
the model.
Coating Stress in the Model
Stresses acting at a small region surrounding point A on the surface of the model are indicated in
Fig . The state of stress in this region of the model is biaxial.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 105
Experimental Stress Analysis
10ME761
In the derivation of the relations between the stresses in the coating and the model, the following
assumptions' are made:
(i) There is perfect adhesion between the coating and the model surface.
(ii) The thickness of the coating being small, there is no variation of stresses or strains through
the thickness.
(iii) The strains on the surface of the model due to' the applied loads are faithfully transmitted to
the coating.
(iv) There are no residual stresses in the coating.
Let €1m and €2m refer to the strains .in the model or the machine part, and €1c and €2c to the
corresponding strains in the coating. the stress normal to the model, i.e. σ3m, and the
corresponding stress σ3c 10 the coating are zero. Thus, the state corresponds to a plane state, and
one has
The stress-strain relations for the model and coating are
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 106
Experimental Stress Analysis
10ME761
Em and Vm in the above equations refer to the modulus of elasticity and the Poisson's ratio for
the material of the model. Ec and Vc refer to the corresponding parameters for the coating.
Since one is interested in finding the relation between the model and coating stresses, Equation
are combined, resulting in
In practice, the model stresses σ1m and σ2m are determined from the values of σ1c and σ2c To
do this, the model is loaded until cracks develop in the coating. Using an appropriate law of
failure for the coating material and a calibration process, σ1c and σ2c are established.
Relative Merits of Stress-Coat and All-Temp Coatings
All-Temp coatings have several advantages over the Stress-Coat coatings They are relatively
insensitive to minor changes to temperature, whereas the Stress-Coat material is very sensitive to
temperature variations. All-Temp coatings can be used for applications up to 400°C or more' but
Stress-Coat materials can be used only up to 40°C. The presence of oil and water does not
significantly influence the All-Temp coating but poses serious problems the case of Stress-Coat
materials.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 107
Experimental Stress Analysis
10ME761
Stress-coat materials, on the other hand, are very simple to coat, dry and calibrate. All-Temp
coatings require sophisticated temperature-controlled drying -ovens, having a capability up to
l000 C and often big enough to accommodate huge prototypes. The high temperature of curing
ceramic based coatings often results in damages, distortions and metallurgical changes in the
model or. prototype being investigated visual. Observation of cracks is not possible with AllTemp coatings and the StatifIux method of crack detection, described later, has to be adopted
invariably. This is cumbersome, costly and more time-consuming as compared to the simple
crack detection methods suitable for Stress-Coat materials.
Properties of Stress-Coat materials
Influence of Quantity of Plasticizer on Threshold Strain
The amount of plasticizer in the brittle coating (Stress-Coat) determines the threshold strain of
the materials. A higher percentage of plasticizer increases the strain required to crack the coating,
thus increasing the threshold strain or decreasing the sensitivity of the coating. Reduction in the
quantity of plasticizer in a coating is indicated by the increasing coating number given by the
manufacturer, as shown in Fig. It can be seen that with decreasing quantity of plasticizer, the
threshold strain of a particular Stress-Coat decreases drastically. Based on the threshold strain
an4 sensitivity of the coatings at different temperatures and humidity, the manufacturers of these
coatings specify the coating number for the required temperature and humidity conditions.
Figure is a representation of one such manufacturer's specifications for a brittle coating of known
threshold sensitivity. For any combination of dry- and wet-bulb temperature expected during a
test and humidity, one can select the correct brittle coating that gives the desired threshold
sensitivity.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 108
Experimental Stress Analysis
10ME761
Influence of Curing Time and Coating Thickness on Threshold Strain
The coatings are dried and cured at room temperature and humidity conditions which normally
takes around 20 h by which time most of the solvent (carbon disulphide) will be driven out. If a
small amount of this solvent is still. left in the coating, it softens the coating, increasing the
threshold strain. Therefore the coatings are often dried at elevated temperatures and lower
humidities to drive off the solvent completely and quickly, in about 2 to 3 h. however, when such
coatings are exposed to atmosphere, they absorb moisture from the atmosphere, the quantity
absorbed being a function of time elapsed before testing. Since moisture tends· to soften thinner
Coatings more readily than thicker coatings, a higher coating thickness results in a drop in the
value of the threshold strain. Thus, room-temperature curing produces a coating which has
decreasing sensitivity with increasing thickness due to the larger quantity of residual carbon
disulphide. High-temperature curing, on the other hand, produces a coating that has increasing
sensitivity with decreasing thickness. An optimum temperature suitable for a particular thickness
of coating should be used to achieve approximately uniform threshold strain over the range of
thicknesses.
Effect of Curing and Testing Temperature on Threshold Strain
The influence of curing and testing temperature on the value of the threshold strain is by far the
most important factor in the list of variables that affect the coating.
For any particular thickness of the coating, a higher test temperature reduces the threshold strain
and hence it is important to have uniform temperature and humidity all around the coated
surface. The results of the brittle-coating test can be misleading due to non-uniform temperature
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 109
Experimental Stress Analysis
10ME761
and humidity, resulting in varying threshold strains at different zones. The threshold strain also
increases very significantly due to the change in residual stress, when the coating expands or
contracts relative to the base material.
Influence of Rate of Loading on Threshold Strain
Normally, the time in which the full calibration load is applied on the calibration strip is around
one second. The threshold value €1 thus obtained is dependent on the hold time during which the
load is held constant. If the coating is loaded very slowly, it exhibits viscoelastic tendencies. On
account of this, a higher model strain is required to· crack the coating. Suitable corrections have
to be made to the threshold values of strain to account for slower rates of loading by using
correction charts and formulae supplied by the manufacturers. It is also observed that the
calibration tests give consistent values of threshold strain if the hold time is around 15 s. The
normally used-loading-hold-unloading cycle is shown in Fig. If the time duration under load is
smaller, the required strain for cracking the coating will be higher.
Crack-Detection Techniques
Cracks that are developed in the coating remain open if the load is properly applied. For the
accurate use of the coatings for either directional information or stress-level data, it is necessary
to detect the cracks, which are normally V-shaped, having a depth equal to the coating thickness
and a width of approximately 0.05 to 0.075 mm. The following three methods are used for
detecting the cracks.
(i) If the strain sensitivity is less than 10-3 mm/mm, the cracks normally remain open and can be
easily seen and photographed. The use of a beam of light with a large angle of incidence and
directed perpendicular to the cracks helps appreciably in this task. If the strain sensitivity is
more than 10-3 mm / rnm, the cracks tend to close and other methods of crack detection must
be used
ii) The second method of crack detection is to use a red-dye etchant, which is a mixture of
turpentine, machine oil and red dye. The etcbant is applied to the surface of a cracked brittle
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 110
Experimental Stress Analysis
10ME761
coating for approximately one minute. During this period, the etchant attached rhecoating in the
neighbourhood of the cracks and widens them. The etchant is then wiped out from the surface of
the coating which is further cleaned with an emulsifier, a solution of soap in water. The dye that
penetrates into the cracks is not removed by this process and hence the cracks appear as bright
red lines. For the dye-penetrant method, normally an undercoat, usually a dull aluminum lacquer
is applied below the coating and the resulting yellow background provides a contrast with the
red-colored cracks.
The principal disadvantage of the dye etch ant is that it attacks and sensitizes the coating and
makes it unusable for further use at higher test loads. Thus it is the normal practice to make the
cracks visible for photography by dye etching with a red-dye etch ant after all the load
increments have been applied and crack ends have been marked at each successive application
iii) The most effective method of crack detection is by the technique called the StatifIux
technique. A mild electrolyte, usually water mixed with wetting agents, is applied to the coating
to be examined before the load is applied. Due to the application of the load, cracks develop and
the electrolyte flows into cracks, fills them, and makes electrical contact with the metal below
the coating. On .wiping the coating surface, the electrolyte is removed from the entire surface,
excepting the cracks. StatifIux powder, normally a special type of talcum powder is blown onto
the specimen with a special sprayer. These electrified particles are attracted by the electrolyte
remaining in the cracks and form white ridge-like lines over the cracks and provide an excellent
means of locating the crack pattern. A flat black undercoat lacquer is usually used to provide a
contrasting background for the white StatifIux powder.
StatifIux powder and the water-based electrolyte do not attack the coating and thus overcome the
disadvantage of the dye etchants, but the disadvantage of the Statiflux method is that the
electrolyte must be applied and removed from the coating for each load level. A nonuniform
application of this electrolyte results in temperature changes due to different evaporation rates
and this can result in errors if quantitative results are expected from the tests.
Calibration of brittle coating materials
The brittle-coating material is calibrated and its strain sensitivity is evaluated, as already
explained by coating calibration strips simultaneously along with the model or test structure
under identical conditions. The same curing and firing cycle is adopted for the calibration
models, and the load-duration, load-time relationships are to be the same as those relating to the
model or test structure.
The standard calibrating beams for static calibration are rectangular aluminum strips of
approximately 25 mm X 6 mm in cross-section and 300 mm in length. A maximum deflection of
around IS mm at an effective span of 250 mm can be imposed on the calibration beam by fixing
them as cantilever beams and operating a cam as shown in Fig.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 111
Experimental Stress Analysis
10ME761
These calibrating strips are assembled in a calibrating jig. The strain applied to produce cracks
on the calibrating beam can be varied by using different cams.
In order to calibrate the coating under dynamic conditions, a hollow tapered-wall shown in Fig.
is normally used. This calibration cylinder is supported rigidly at the upper end and a long guide
is attached to its lower end. Dynamic load is applied by dropping a top on a spring anvil, fixed to
the bottom end of the guide rod. The dynamic strain applied on the calibration cylinder can be
varied by varying the weight of the top. The resulting crack patterns are recorded using a highspeed camera, which is triggered on by an electronic circuit. The resulting crack pattern is related
to the dynamic strain applied, by using well-established empirical equations. These equations
take into account the curing temperature, coating thickness and coating number.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 112
Experimental Stress Analysis
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
10ME761
Page 113
Experimental Stress Analysis
10ME761
UNIT-8
Moire Methods
Moire Phenomenon
Two arrays of alternately placed transparent and opaque lines or dots, when moved relative to
each other, result in fringe patterns consisting of alternately placed bright and dark bands which
are termed moire fringes. An ensemble of equispaced opaque lines separated by transparent slits
or lines, which are used to obtain moire fringes is caned a grating. In parallel line gratings shown
in Fig which are most commonly used for the moire method of strain' analysis, the opaque and
transparent lines are perfectly parallel and equispaced. In a radial-line grating opaque and
transparent lines are alternate radial lines and in circular-line grating these form circles of
varying radii. The opaque and transparent lines in these line gratings can either .be equal or
unequal.
Figure below shows a parallel line grating. The distance between corresponding points in a
grating is called the pitch and is denoted by p. The intensity of the grating, which represents the
number of lines per unit length, is denoted by d. The direction perpendicular to the lines in the
plane of grating is called the primary direction, while the direction parallel to grating is called the
secondary direction.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 114
Experimental Stress Analysis
10ME761
Crossed gratings or dots are usually made of two systems of lines perpendicular to each other.
When two.gratings ofthe same category having a mismatch of pitches (i.e. with different pitches
and therefore densities) are placed one above the other, moire fringes are formed, even without
relative movement between the two of the two gratings used for fringe formation, one grating is
either bonded to, or etched or printed on the specimen being analyzed and is termed "model
grating" or "specimen grating”. The other grating is known as "master grating" or "reference
grating". The specimen' grating undergoes deformation depending on the state of strain on the
surface and is accompanied by a change of spacing (i.e. pitch) between the lines of grating. The
master grating, which is not strained, does not undergo any change in the spacing between lines
of grating. The strained specimen grating and the unstrained master grating interfere optically to
form moire fringes.
Figure below shows the cross-section of the specimen and master gratings
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 115
Experimental Stress Analysis
10ME761
Which are identical, superposed and aligned such that the opaque lines of the two gratings are
exactly one above the other. The emerging light should have a band width equal to the pitch of
the grating. However, owing to diffraction and the limiting resolution capacity of the human eye,
the entire field appears to be uniformly gray and the intensity of the transmitted light is only
about half that of the incident light. Further, no fringes are seen
If the specimen grating is now subjected to a linear deformation in ~he secondary direction, no
fringes are formed. However, when this deformation is along the primary direction, the pitch
changes to a new value p' as shown in Fig below The emerging light will have varying band
widths depending
upon the overlap of opaque bars over the transparent interspaced. Further, the intensity of light
that emerges will not constant over a .band .width. It will be uniformly varying from a low
intensity to a high intensity and back again to a low intensity. Maximum transmission of light
appears to occur at positions where the transparent interspaces of the two gratings are aligned
and a comparatively bright band is formed. At such ~laces, where an opaque line of one grating
is aligned with the transparent Interspaced of the other, the intensity of transmitted light IS a
minimum and a .dark band is formed. Alternating varying bright and dark bands constitute moire
fringes with fringe spacing.
When the specimen grates and master grating have a mismatch the mechanism of formation of
moire fringe will be similar to the mechanism shown in Fig. When there is no linear deformation
the specimen grating, the pitch of fringes would be. When the grating is subjected to a uniform
linear deformation along the primary direction, moire fringes of a different fringe spacing 52 are
observed. It is interesting to note that though the dark bands are usually counted for fringe order,
it is difficult for the human eye to locate the points of minimum intensity to identify the exact
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 116
Experimental Stress Analysis
10ME761
positions of fringe orders. This difficulty can be obviated by' resorting to counting brighter bands
since it is easy for the human eye to locate the points of maximum intensity.
Moire Fringe Analysis
(a) Geometric Approach
In general, at any point on the surface of the strained specimen on which a specimen grating has
been bonded, stretching and rotating occur simultaneously. Information readily available from
the moire fringes observed is the angle of inclination B of the moire fringe with respect to the
lines of master grating and the fringe spacing 8. The quantities to be determined at the point of
interest for the analysis of the state of strain are the angle of rotation .p, of the specimen grating
with respect to the lines of master grating, and the pitch p' of the specimen grating in the
deformed state. The cases considered for analyses are: (i) pure extension case with no rotation,
(ii) pure rotational case with no extension, (iii) shear strain, and (iv) combined normal and shear
strain.
(i)
Pure Extension with rotation
Let the specimen to which a grating is bonded undergo a uniform .uniaxial tensile strain E. The
pitch of the specimen grating increases from p to p' in the process. The specimen grating in this
state interferes with the master grating of pitch p and forms a fringe pattern. The distance
between two consecutive fringes so formed is delta. Since the strain field is uniform,
the change of displacement between two points on two consecutive fringes is p, and hence the
average strain between two consecutive fringes would be
and when the strain is compressive,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 117
Experimental Stress Analysis
10ME761
the final gauge length after undergoing a deformation equal to p. The undeformed gauge length
is therefore (delta - p), and the corresponding engineering strain is
For compression, the corresponding expression is
Above equation gives the Eulerian strains and the above two expressions give the Lagrangian
strains.
If there are m number of grating lines in the master grating between two consecutive fringes,
there would be (m - 1) number of grating lines in the specimen grating between the
corresponding fringes if the specimen grating is under tension. Hence
ii) Pure Rotational Case with no Extension
Let the specimen with a specimen grating of pitch p' bonded to it be rotated through an angle cp
with reference to the master grating line having a pitch p at point P . The moire fringes so formed
make an angle θ with reference to the master grating measured in the same direction as Φ From
the geometry of moire fringes shown in the figure
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 118
Experimental Stress Analysis
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
10ME761
Page 119
Experimental Stress Analysis
10ME761
If the angle θ between the fringe and the master grating line, the distance between the fringes,
and the pitch p of the master grating are known, then the angle of rotation Φ of the specimen
grating can be calculated as follows:
If the pitches of the two gratings are the same and equal to p, the geometry of fringe formation is
as shown in Fig. If delta xx is the distance measured between the fringes along the secondary
direction of the master grating, then the angle of rotation cp between the two gratings can be
calculated by the relation
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 120
Experimental Stress Analysis
10ME761
Displacement field approach to Moire fringe analysis out of plane displacement
measurements
Moire Techniques for out-of-Plane Problems
The problems solved in the previous section using various techniques referred to 'the plane-stress
case and therefore Ux and u displacement fields could be obtained. some techniques to determine
the u, displacements due to the Poisson’s effect were also discussed. The methods described in
the previous sections yield the isotropic representing Uz displacements.
In the present section some techniques to directly determine the lateral displacement u, in planestress problems due to Poisson's effect will be discussed. In the case of plate bending, the lateral
displacement becomes much larger compared to Ux and uy displacements. Techniques to
determine the deflection, slope and curvature in such problems for purposes of moment and
strain analysis will also be discussed.
a) Moire Techniques for Lateral Displacements due to Poisson's Ratio
Effect-shadow Moire Techniques
Moire patterns caused by the interference of a reference grating with its shadow on the reflecting
surface of the deformed specimen yield. The technique is referred to as the shadow-moire
technique. This technique is further categorized into, the following' types, depending upon the
complexity of the arrangement' of elements and the application.
i)
Collimated Illumination and Recording Moire
The mechanism of fringe formation by this technique is Illustrated A collimated beam of light
strikes a master grating of pitch p, points on the curved surface (caused by deformation due to
loading) of the specimen, at an angle is from the normal to the grating. The shadow of the master
grating on the specimen (which can be taken as the deformed specimen grating) and the master
grating are assumed as being viewed at infinity by a camera, the optical axis of which makes an
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 121
Experimental Stress Analysis
10ME761
angle of beeta from the normal to the master grating. The Image of line gratings in the AD part of
the master grating falls on the AB art of the carved surface and therefore the pitch of lines on the
curved surface Will be different from p. The camera collects the reflected be from the ~B part of
the surface which interferes-with-the CA art of the master grating and forms a moire pattern.
Assuming AC has m lines and ml lines,
where N is the fringe order of the moire pattern at C further,
where u, is the height of the master grating above the curved surface at O.
Hence
When the optical axis of the camera is placed normal, to, the master grating
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 122
Experimental Stress Analysis
10ME761
This is the most commonly used equation for the shadow moire method. Shadow moire method
is extended usefully to the solution of torsion problems through the use of membrane analogy. A
membrane is stretched over a boundary corresponding to the cross-section of the shaft. It is
subjected to a uniform pressure over its lower surface. A shadow-moire pattern is obtained from
the top reflective surface of the membrane. From the moire patterns, a curve of variation of
fringe order along the radius is obtained. From this, a curve of variation of u, displacement along
the radius is obtained. This curve will be common over any diametric cross-section through the
centre. The volume under the membrane surface is proportional to the tensional rigidity of the
shaft and the maximum slope of the surface is proportional to the maximum resultant shear.
Figure shows the stepwise procedure followed for a circular shaft.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 123
Experimental Stress Analysis
10ME761
b) Ligtenberg's Photoreflective Moire
In this technique, moire is produced-by the interference of the image of a master grating on the
reflective surface of the specimen in its unloaded state, with the image of the master grating on
the reflective surface of the specimen in its loaded state. The superposition of these images is
effected by exposing the same film twice to the' two images. The set-up used for this technique
shown in Fig consists of a curved screen made of equispaced dark strips of pitch p, on a white
background, brightly illuminated by an overhead light source.
The front surface of a specimen is made reflective and the specimen in a loading jig is placed at a
distance d from the curved screen. A camera is placed at the centre of the screen for viewing the
image of the specimen. When the specimen is in an unloaded state, a ray of light from P on the
master grating is reflected at any angle ex, at a point R, under consideration on the specimen, to a
point Q on the ground glass plate of the camera. When the specimen is loaded, the normal at R
gets tilted. The point Q on the ground glass plate does not correspond any more to the point P on
the master grating, but to a different point T. The ray RP has tilted through an angle 28 resulting
in a displacement or shift equal to PT. This displacement. PT is therefore a measure of the slope
at R on the specimen due to the load. For large d .
A shift of one pitch in the distance PT yields one fringe at point Q. If the shift PT = S = Np, then
Nth order fringe is formed at Q. Hence,
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 124
Experimental Stress Analysis
10ME761
The angle B is the partial slope of the deformed plate, depending upon the direction' of the
grating: Orientation of the lines along one direction gives the partial slope in the primary
direction of the gratings
Applications and advantages
MOIRE TECHNIQUES FOR INPLANE PROBLEMS
Moire fringes, i.e. the isothetics, in general, are obtained by optical interference between a
specimen grating which is either printed or bonded, and a closely placed master grating. Figure
indicates a general set-up
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 125
Experimental Stress Analysis
10ME761
used for this purpose. The specimen grating is either rigidly bonded to the specimen or
chemically etched or printed by a photographic process directly onto the transparent specimen.
The master grating is placed very close to the specimen grating. The primary directions of both
the specimen and master gratings are aligned to coincide with the imaginary x-axis in the
specimen. Collimated light passes the gratings. When the specimen is strained, the specimen
grating undergoes deformation which interferes with the unstrained master grating resulting in
moire fringes. This is photographed by a camera. The fringe pattern gives the uy-displacement
field. The specimen grating is removed from the specimen and another specimen grating is
bonded onto it such that its primary direction coincides with the y-axis of the coordinate system.
Alternatively, a new specimen similar to the earlier one with the primary direction of the
specimen grating bonded to it coinciding with the j-axis can be used. The master grating is now
aligned with the specimen grating. The fringe pattern now obtained gives the ux displacement
field. From the 11.< and u, displacement fields, derivatives of displacements are evaluated and
the strain components at the points of interest are determined using Equations Figures (a) and (b)
show the moire patterns for a diametrally compressed disk for
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 126
Experimental Stress Analysis
10ME761
Apart from this general technique of obtaining moire fringes for the strain analysis of in plane
problems, several other techniques have been developed to solve certain unique problems. The
various techniques used can be broadly classified as follows
(i) Contact-grating techniques, and
(ii) remote-grating techniques.
In the contact-grating technique, the specimen grating is placed very close to the master grating
so as to be thought of as being in contact. Alternatively, the image of the specimen grating is
superposed with the master grating to form a moire pattern. Two patterns can be obtained with
the primary direction of the two gratings oriented once each along the two reference axes. The
patterns give an. overall picture of the Ux and Uy, displacement fields.
DEPARTMENT OF MECHANICAL ENGG. SJBIT.
Page 127