ESA Module 1 Part-B ME832. by Dr. Mohammed Imran

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GHOUSIA COLLEGE OF ENGINEERING
RAMANAGARAM-562159
EXPERIMENTAL STRESS ANALYSIS
[15ME832]
Dr. MOHAMMED IMRAN
ASST PROFESSOR
DEPARTMENT OF MECHANICAL ENGINEERING

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

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

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

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

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

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

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

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

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

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

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

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

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

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The response of a bonded strain gage to a biaxial strain field can be expressed as
D
= ϵ + ϵ + g (2)
Where ϵa = normal strain along axial direction of gage
ϵt = normal strain along transverse direction of gage
at = shearing strain
Fa = sensitivity of gage to axial strain
Ft = sensitivity of gage to transverse strain
Fs = sensitivity of gage shearing strain
In general, the gage sensitivity to shearing strain is small and can be neglected. The
response of the gage can then be expressed as
D
= ϵ + ϵ + 0
D
= ϵ + ϵ (2′)
D
= (ϵ + ϵ )
D
= ϵ 1 +
ϵ
ϵ
(3)
Where = is defined as the transverse sensitivity factor for the gage.
Now it is common practice to calibrate a strain gauge in a uniaxial stress field, i.e. in a
biaxial strain field with the ratio of the transverse-to-axial strain equal to the Poisson’s
ratio of the specimen material Eq (a). Thus Strain-gage manufacturers provide a
calibration constant known as the gage factor ‘F’ for each gage. The gage factor “F”
relates the resistance change to the axial strain Eq (4).
With this method of calibration, the strain field experienced by the gage is
biaxial, with
ϵt = -o ϵa (a)
D
= F . ϵ (4)
where o = 0.285 is Poisson’s ratio of the material. If Eq.(a) is substituted into Eq.(3),
the resistance change in the calibration process is
D
= ϵ (1 + (− ))

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D
= ϵ (1 −  ) (5)
Since the resistance changes given by Eqs. (4) and (5) are identical, the gage factor
is related to both and by the expression
D
= ϵ (1 −  ) = F . ϵa
= ( − n ) (6)
It may be noted that even when k t is as high as 10% and o = 0.3 is Poisson’s ratio
from Eq (6) Fa is only 1.03 times of ‘F’. It should be realized that Eq (4) is valid for
the uniaxial stress field used by the manufacturer to calibrate the strain gauge.
6. PERFORMANCE CAHRECTERISTICS OF FOIL STRAIN GAGES.
Foil strain gages are small precision resistors mounted on a flexible carrier that
can be bonded to a component part in a typical application. The gage resistance is
accurate to ±0.4 percent, and the gage factor, based on a lot calibration, is certified to
±1.5 percent. These specifications indicate that foil-type gages provide a means for
making precise measurements of strain. The results actually obtained however, are a
function of the installation procedures, the state of strain being measured, and
environment conditions during the test. All these factors affect the performance of a
strain- gage system, in general.
a. Strain- Gage Linearity, Hysteresis, and Zero Shift
b. Temperature Compensation
c. Elongation Limits
d. Stability
A. Strain- Gage Linearity, Hysteresis, and Zero Shift
One measure of the performance of a strain gage system (system here implies gage,
adhesive, and instrumentation) involves considerations of linearity, hysteresis, and
zero-shift. If gage output, in terms of measured strain, is plotted as a function of
applied strain as the load on the component is cycled, results similar to those shown in
fig.4. Will be obtained
A slight deviation from linearity is typically observed, and the unloading curve
normally falls below the loading curve to form a hysteresis loop. Also, when the
applied strain is reduced to zero, the gage output indicates a small negative strain,

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

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

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

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

21
Protective coatings:
Strain-gauge installations are affected by environments containing water or
moisture or chemical vapours. They are also susceptible to mechanical damage.
Therefore unless tests are to be conducted under laboratory conditions within a short
time after installation, protective coatings are essential. The important considerations
that influence the selection of a coating are test environment, test duration and the
degree of accuracy required. Though several commercial coatings are available, only
those proven by tests to be electrically and chemically compatible with a gauge
installation should be used in any application.
Fig (12) Waterproofing a strain gauge in severe environment
Wax coatings like microcrystalline wax, are melted and poured directly over
the strain gauge installation to form a coating of thickness in the range 0.5 to 1.5 mm.
These are good barriers to water or moisture but provide little mechanical protection.
It is well-suited for general-purpose laboratory work and field applications. It can
be used at temperatures up to 60°C. An air-drying polyurethane coating or acrylic
coating can also be used in the laboratory for protection against moisture over a
temperature range of -40° to +100°C.
Both single-component and two-component epoxy plastics are available for
use as protective coatings. For example, a solvent-thinned single – component epoxy
resin compound can be brushed on the gauge and cured to provide an excellent
chemical and mechanical protection. It is an excellent coating for transducer
applications up to 120°C.
Synthetic rubber coatings such as nitrile rubber, silicone rubber, poly-
sulphides, butyl polymers, etc. provide good protection against moisture, fresh and

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

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

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

25
D =
+ D
+ D + R + D
V −
D =
+ D
+ D + R + D
. V −
+ R
. V
D =
+ D
+ D + R + D
−
+ R
. V
D =
( + D )( + R ) − ( + D + R + D )
( + D + R + D )( + R )
. V (b)
D = . V (b)
Numerator= = ( + D )( + R ) − ( + D + R + D )
= ( + D )( + R ) − ( + R ) − (D + D )
= ( + R )( + D − ) − (D + D )
= ( + R )D − (D + D )
= D + R D − D − D
= R D − D
=
D
−
D

= = ( + D + R + D )( + R )
= ( + R ) 1 +
D
( )
+
D
( )
( + R )
= ( + R ) 1 +
D
1 +
R
+
D
R
R
+ 1
Where = resistance ratio
= (1 + m) 1 +
D
(1 + m)
+
D
R
1
m
+ 1
= ( + ) +
( + )
D
+
D

26
Substituting A & B in Eq (b) Which can be expressed in the following form by
introducing r = R2/R1
D =
⎝
⎜
⎛
D
−
D
( + ) +
( + )
D
+
D
⎠
⎟
⎞
V
D =
⎝
⎜
⎛
( + )
D
−
D 1
+
( + )
D
+
D
⎠
⎟
⎞
V
D =
⎝
⎜
⎛
( + )
D
−
D 1
+
( + )
D
+
D
⎠
⎟
⎞
V (2)
Examination of Eq.(2) shows that the voltage signal E from the potentiometer
Circuit is a nonlinear function of R1/R1 and R2/R2.To inspect the nonlinear aspects
of this circuit further, it is possible to rewrite Eq. (2) in the form
D =
(1 + m)
D
−
D
(1 − h) (3a)
Where nonlinear term h is expressed as
(1 − h) =
1
1 +
1
(1 + m)
D
+ m
D
R
h
= 1 −
1
1 +
1
(1 + m)
D 1
1
−
D 2
2
(3b)
Equations (3) are the basic relationships which govern the behavior of the
potentiometer circuit, and as such they can be used to establish the applicability of
this circuit for strain-gage measurements. Error due to nonlinearity of the circuit can
be estimated with R1 as the resistance due to string gauge, R2 as a resistor of fixed
resistance and  R1 as the change in the resistance of the gauge due to a strain .
Where R2 as a resistor of fixed  R2 = 0 Equation 3b becomes

27
h = 1 −
1
1 +
1
(1 + m)
D

h = 1 −
1
1 +
1
(1 + m)
. ∈
(4)
It shows that the nonlinear term is depends on magnitude of strain∈, gauge factor ‘F’
and ratio of m = R2 / R1. In most strain measurements the nonlinearity term h can be
neglected & E [Eq (2)] can be determined from
D =
( + )

D
−
D
− −( )
However, if high accuracy in strain measurement is required or larger strains are to be
measured. Then plastic strain determinations in metallic materials the output signal
determined through Eq (5) can be corrected for error due to nonlinearity.
(a) Range and sensitivity of the circuit:
The output signal per unit strain is known as circuit sensitivity Sc of the
potentiometer circuit is given by
D =
m
(1 + m)2

D
−
D
V (1)
E output signal per unit strain = =

∈
D
∈
=
m
(1 + m)2

D
−
D

V
∈

=
m
(1 + m)2

D 1
1
−
D 2
2

V
∈
(2)
With an active strain gauge R1 and fixed-blast resistor R2
i.e. = ; D = D ; = ; D =
=
m
(1 + m)2

D g
g

V
∈

28
=
m
(1 + m)2
V
⎣
⎢
⎢
⎡
D g
g
∈
⎦
⎥
⎥
⎤

=
m
(1 + m)2
. V . F (3)
Where gauge factor =
D
∈
for strain gauge, thus the circuit sensitivity of the
potentiometer circuit is depends on the voltage V and ratio m = R2/R1, the Sc is limited
by the maximum power Pg that can be dissipated by the gauge without unfavorable
effect of performance.
As the power dissipated in the gauge is equal to I2
g Rg i.e. Pg = I2
g Rg input voltage.
= ( + )
= (1 + )
Where m = R2/R1
= (1 + )
= (1 + )
= (1 + ) ( )
Substituting Eq (a) in Eq (3) we get
=
m
(1 + m)2
. (1 + ) . F
=
( + )
. .
Circuit sensitivity of the potentiometer circuit is depends on
m
( )
and . F the
term
m
( )
is completely depends on m = R2/R1 resistance ratio limited value is 9.
Therefore,
m
( )
≈ 0.9, the term . F is depends on the characteristics of the
strain gauge range between 3 to 700 and Sc range 5 to 10µV per micro-strain.

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

30
(a) Null Balance Type (balanced E = 0):
Condition for balancing of Wheatstone bridge. The requirement for balance, i.e.
zero potential difference E between points B and D for the bridge shown in fig (8) can
be determined as follows:
The voltage drop VAB across R1 is
VAB = i1R1= V / (R1 + R2) . R1 --------------------- (1)
Similarly the voltage drop VAD across R4 is
VAD = i4R4= V / (R3 + R4) R4 ---------- (2)
The potential difference between B and D, VBD, is
VBD = VAB - VAD = E ------------ (3)
Substituting of equations. (1) and (2) in (3), we get
=
( + )
–
( + )
=
( + ) − ( + )
( + )( + )

=
−
( + )( + )
( )
The condition for balance is that the voltage E should be zero i.e. the numerator in
Eq. (4) should be zero:
i.e − = 0
=
Or
= (5)
Or
= (6)
Equation (5) or (6) gives the condition for the Wheatstone bridge to balance, that is
the ratio of resistances of any two adjacent arms of the bridge must be equal to the
ratio of the resistances of the remaining two arms taken in the same order.

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

D =
D D
D D
D D
D D

D = . (a)
Where A is the determinant in the numerator and B is the determinant in the
denominator. By expanding each of these determinants, neglecting second-order
terms, and noting R1R3 = R2 R4 it is possible to show that

32
Numerator A
=
+ D + D
+ D + D

= ( + D )( + D ) − ( + D )( + D )
= . + .D + . D + D D − − D − D − D D
= .D + . D − D − D
= .D + . D − D − D
= .
D
+
D
− .
D
+
D

=
∆
−
∆
+
∆
−
∆
(b)
=
+ D + + D 0
0 + D + + D

= ( + D + + D )( + D + + D )
= + + +
= ( + ) + ( + )
= ( + )( + )
= ( + ) 1 +
= ( + ) 1 +
= ( + )
+

=
( + )

=
( )
(c)
Substituting Eqs. (b) to (d) yields
D = .
∆
−
∆
+
∆
−
∆
1 3( 1 + 2)2
1 2

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

33
B. Wheatstone-Bridge Sensitivity:
The sensitivity of the Wheatstone bridge must be considered from two points of view:
(1) With a fixed voltage applied to the bridge regardless of gage current (a condition
which exists in most commercially available instrumentation) and
(2) With a variable voltage whose upper limit is determined by the power
dissipated the particular arm of the bridge which contains the strain gage.
By recalling the definition for the circuit sensitivity given, and using the basic bridge
relationship given in Eq. (8), it is clear that the circuit sensitivity is =
D
=
D
= ( )
∆
−
∆
+
∆
−
∆
(9)
=
ϵ (1 + )
(∈ −∈ +∈ −∈ )
If all strain have equal value ∈ = −∈ =∈ = −∈ =∈
=
ϵ (1 + )
(4 ∈)
= 4.
(1 + )
If a multiple-gage circuit is considered with n gages (where n = I. 2. 3, or 4) whose
out puts sum when placed in the bridge circuit. it is possible to write

∆
=
∆
(10)
Which by Eq. using

∆
= . . ϵ (11)
Substituting Eq. (11) into Eq. (9) gives the circuit sensitivity as
=
( )
(12)
This sensitivity equation is applicable in those cases where the bridge voltage V is
fixed and independent of gage current. The equation shows that the sensitivity of the
bridge depends upon the number n of active arms employed, the gage factor F , the

34
input voltage, and the ratio of the resistances R1 /R2 . A plot of r versus (I + r)2
(the
circuit efficiency) that maximum efficiency and hence maximum circuit sensitivity
occur when r = 1. With four active arms in this bridge a circuit sensitivity of F×V
can be achieved, whereas with one active gage a circuit sensitivity of only F × (V/4)
can be obtained.
9. CONSTANT-CURRENT CIRCUITS:
The potentiometer and Wheatstone bridge circuits driven with a voltage source which
ideally remains constant with changes in the resistance of the circuit. These voltage-
driven circuits exhibit nonlinear output whenever DR/R is large. This nonlinear
behavior limits their applicability to semiconductor strain gages. It is possible to
replace the constant- voltage source with a constant-current source, and it can be
shown that improvements in both linearity and sensitivity result.
Constant-current power supplies with sufficient regulation for strain-gage applications
are relatively new and have been made possible by advances in solid-state electronics,
basically the constant-current power supply is a high impedance (1 to 10M) device
which changes output voltage with changing resistive load to maintain a constant
current.
(a) Constant-Current Potentiometer Circuit:
Consider the constant-current potentiometer circuit shown in Fig, 34a. When a very
high impedance meter is placed across resistance R1 the measured output voltage E is
E = I R (1)
When resistances R 1 =R and R2 =R b change by DR1 =DR and DR2 = DR b =0 the
output voltage becomes
E + DE = I(R+ DR) (a)
DE = I(R+ DR) – E (a’)
Thus from Eq’s, (1) and (a)
DE = I(R+ DR) - IR = I DR = I R
D
(2)

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

36
It was noted in deriving Eq. (2) that R2 and DR2 did not affect the signal output of the
constant-current potentiometer. This indicates that temperature compensation by
signal cancellation in the strain-gage circuit or signal addition cannot be
performed. It is possible to maintain the advantages of high sensitivity and perfect
linearity of this circuit and to obtain the capability of signal addition or subtraction
by using a double constant-current potentiometer circuit.
(b) Constant-Current Wheatstone bridge Circuits:
To consider a bridge driven by a constant current supply as shown in fig below. The
current I delivered by the supply divides at point A or the bridge into currents I1 and I2
where I=I1+I2 the voltage drop between points A and B of the bridge is
= +
= (i)
And the voltage drop between points A and D is
= (ii)
Thus the output voltage E from the bridge can be expressed as
= V = V − V = I R − I R (1)
Fig: Constant Wheatstone bridge with constant current supply.
Far the bridge to be in balance (E=0) under no-load conditions,
= (2)

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

38
+ D =
I
(ƩR + ƩR)
∆
−
∆
+
∆
−
∆
+
∆ ∆
−
∆ ∆
(5)
Inspection of Eq. (5) shows that the output signal E, is nonlinear with respect to DR
because of the term ƩDR in the denominator and because of the second-order terms
∆ ∆
and
∆ ∆
within the bracketed quantity The nonlinear, of the
constant-current Wheatstone bridge, however is less than that with the constant-
voltage bridge Indeed, if the constant-current Wheatstone bridge is properly designed,
the nonlinear terms can he made insignificant even for the large
∆
′ encountered
with semiconductor strain gages.
The nonlinear effects in a typical situation call he evaluated by considering the
constant-current Wheatstone bridge shown in Fig below. A single active gage is
employed in arm R1, and it temperature-compensating dummy gage is employed in
arm R4 Fixed resistors arc employed in arms R2 and R3. Thus
= = = = D = D = 0
Under stable thermal environments,D = D D = 0 Equation (5) then
reduces to
D =
I
2(1 + m) + ∆ ⁄
∆
(6)
Again it is evident that. Eq. (6) is nonlinear due to the presence of the term ∆ ⁄ in
the denominator to determine the degree of the nonlinearity let
I
2(1 + m) + ∆ ⁄
∆
=
I
2(1 + m)
∆
(1 − h)
h =
∆ ⁄
2(1 + m) + ∆ ⁄
=
∈
2(1 + m) + ∈
(7)
Inspection of Eq. (7) shows that the nonlinear term h can be minimized by increasing
m in this case; nonlinear term h will depend on the gage factor F and on the
magnitude of the strain ∈.

ESA Module 1 Part-B ME832. by Dr. Mohammed Imran

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