CHAPTER 6
Strain gage
1
Introduction
⚫ A strain-sensitive material is one whose electrical resistance is
proportional to the instantaneous spatial-average strain over its
surface.
⚫ two types: metallic (i.e., foil or wire) or semiconductor
⚫ When such a material is stretched, its length increases and its cross-
section decreases; consequently, there is an increase in its electrical
resistance.
⚫ This change in resistance is a measure of its mechanical motion. Thus,
a strain gage is a device which uses change in electrical resistance to
measure strain.
2
Introduction
⚫ The electrical resistance strain gage is basically a piece of very thin foil
or fine wire which exhibits a change in resistance proportional to the
mechanical strain imposed on it.
⚫ In order to handle such a delicate filament, it is either mounted on,
encapsulated in, or bonded to some type of carrier material and is
known as the bonded strain gage.
⚫ Bonded strain gages are available in a wide range of sizes and
resistances.
⚫ Un-bonded strain gages, where the wire is free, are rarely used
because of their limited frequency range and lack of sensitivity.
3
STRAIN-GAGE CONSTRUCTION
⚫ Most strain gages are of foil construction
⚫ Although fine-wire strain gages are used for special purposes,
such as at high temperatures
⚫ Foil strain gages are usually made by a printed-circuit
process.
4
5
STRAIN-GAGE CONSTRUCTION
⚫ Since the foil used in a strain gage must be very fine or thin to have a
sufficiently high electrical resistance (usually between 60 and 350
ohms)
⚫ it is difficult to handle.
⚫ In order to handle this foil, it must be provided with a carrier medium or
backing material, usually a piece of paper, plastic, or epoxy.
⚫ The backing material performs another very important function in
addition to providing ease of handling and simplicity of application. It
provides so much lateral resistance to the foil that it can be shortened
significantly without buckling; then compressive as well as tensile
strains can be measured. 6
Selection and installation of strain gage is influenced by following factors:
1. Grid materials and construction
2. Backing material
3. Bonding material
4. Gage protection
5. Gage configuration
Selection of grid material is based on compromise of following main desirable factors:
1. High gage factors
2. High resistivity
3. Low temperature sensitivity
4. High electrical sensitivity
5. High endurance limit
6. Low hysteresis
7
The strain gage grid is normally supported on some form of
backing material. It provides insulation between grid and
tested material and makes easy to handle the un-mounted
gage. The main desirable properties for backing materials are:
1. Minimum thickness consistent with other factors
2. High mechanical strength
3. Minimum temperature restrictions
4. Good adherence to cements used
5. Non hygroscopic characteristics
8
The adhesives used to cement or attach the gage to test item
fall into one of following categories: cellulose, phenolic epoxy,
cyanoacrylate or ceramic. The main desirable characteristics
of adhesive are:
1. High mechanical strength
2. High creep resistance
3. Minimum temperature restrictions
4. Good adherence
5. Minimum moisture attractions
6. Ease of application
7. The capacity to set up fast
9
Gage Factor
Ratio of per unit change in resistance to per unit change in length.
ΔR
Gage factor G= Δ𝑅L
𝐿
⚫ Let us consider a strain gage made of circular wire
Length=L
Cross sectional area=A
Diameter before being strained=D
Resistivity of strain gage material = ρ
ρ∗ L
Resistance of unstrained gage is =
𝐴
10
⚫ Let us consider tensile stress S is applied to the wire.
⚫ Stress produces positive strain causing length to increase and
area to decease.
⚫ When the wire is strained, there is change in dimensions and
they are
ΔL=Change in Length
ΔA=Change in area
ΔD=Change in diameter
ΔR=Change in resistance
11
⚫ To find how ΔR depends on other materials physical
quantities, differentiating w.r.t Stress S
𝑑𝑅 ρ 𝜕𝐿 ρ𝐿 𝜕𝐴 𝐿 𝜕ρ
= ∗ − 2 +
𝑑𝑆 𝐴 𝜕𝑆 𝐴 𝜕𝑆 𝐴 𝜕𝑆
ρ∗L
Dividing both side by R=
𝐴
1 𝑑𝑅 1 𝜕𝐿 1 𝜕𝐴 1 𝜕ρ
= ∗ − +
𝑅 𝑑𝑆 𝐿 𝜕𝑆 𝐴 𝜕𝑆 ρ 𝜕𝑆
It is evident that per unit resistance is due to per unit change in
length, per unit change in area and per unit change in resistivity.
12
π𝐷^2
Area=
4
𝜕𝐴 π 𝜕𝐷
=2∗ ∗𝐷∗
𝜕𝑆 4 𝜕𝑆
So,
2𝜋
1 𝜕𝐴 4
𝐷 𝜕𝐷 2 𝜕𝐷
= 𝜋 =
𝐴 𝜕𝑆 𝐷^2 𝜕𝑆 𝐷 𝜕𝑆
4
Now previous equation becomes
1 𝑑𝑅 1 𝜕𝐿 2 𝜕𝐷 1 𝜕ρ
= ∗ − +
𝑅 𝑑𝑆 𝐿 𝜕𝑆 𝐷 𝜕𝑆 ρ 𝜕𝑆
13
⚫ Poisson’s ratio v=lateral strain/longitudinal strain
𝜕𝐷/𝐷
𝑣 =-
𝜕𝐿/𝐿
Or, 𝜕𝐷/𝐷=- 𝑣 * 𝜕𝐿/𝐿
Therefore,
1 𝑑𝑅 1 𝜕𝐿 2 𝜕𝐿 1 𝜕ρ
= ∗ +𝑣∗ +
𝑅 𝑑𝑆 𝐿 𝜕𝑆 𝐿 𝜕𝑆 ρ 𝜕𝑆
For small variations, above relation can be written as:
ΔR ΔL ΔL Δρ
= + 2𝑣 +
𝑅 𝐿 𝐿 ρ
14
Gage factor
Ratio of per unit change in resistance to per unit change in
length.
ΔR
Gage factor G= Δ𝑅L
𝐿
The gage factor can be written as
Δρ
ρ
G= 1+2v+ 𝜀
= resistance change due to length change + Due to
area change + Due to piezo-resistive effect (Change in resistivity)
ΔL
Where, = strain=
𝐿 15
⚫When change in resistivity is neglected,
G= 1+ 2v
Poisson’s ratio of all metal is small. This gives small gauge factor.
16
Required characteristics for strain
measuring device
⚫ Calibration constant for the gage should be stable with respect to both
temperature and time
⚫ The gage should be capable of measuring ±1μm/m over a range of ±5%
strain (50mm/m)
⚫ The gage length and width should be small so that the measurement
approximates strain at a point.
⚫ The inertia of gage should be minimal to permit the recording of high
frequency dynamic strain.
⚫ The input output relationship should be linear over the entire range of gage.
⚫ The gage and associated electronics should be economical.
⚫ Installation and read-out of the gage should require minimal skills and
understanding.
17
Strain gage Ballast and Bridge circuit
Current sensitive circuit
Uses the flow of current in passive resistance transducer as an indication of
value of resistance . Transducer may be any type of variable form of variable
resistance element. When resistance is changed due to physical quantity,
resistance changes and hence current flowing through.
18
Ballast circuit
It is only a simple variation of current sensitive circuit. In this
case voltage sensitive device is connected across the
transducer in place of series connected current sensitive
device.
19
Strain gage ballast circuit
If Rg is the resistance of strain gage , the output voltage
when gage is not strained is,
Rg
e0 = *e
R𝑏+Rg i
Change in output voltage when gage is strained is
calculated by differentiating e0
ei Rg∗ei R𝑏 * e
deo = ( - ) ∗ 𝑑R = dR
(R𝑏+Rg) (R𝑏+Rg) 2 g (R +R )
𝑏 g
2 g i
Rg∗R𝑏 𝑑Rg *
= ei
(R𝑏+Rg)2 Rg
20
Rg∗R𝑏 ΔR ΔL
=
(R𝑏+Rg) 2 G* ε * ei , since Gage factor G= ΔL and
𝑅
𝐿
=ε
𝐿
Hence change in resistance is directly proportional to the strain.
Maximum sensitivity is attained when Rb and Rg are equal. For
this arrangement
1
e0 = G* ε * ei
4
21
The strain gage bridge circuits
⚫ The resistance bridge circuits are convenient for use with
strain gages for two reasons:
1. Null adjusting is easy
2. Temperature effect can be reduced or eliminated easily
⚫ The strain gage bridge circuits can be of three types
1. Quarter Bridge
2. Half Bridge
3. Full Bridge
22
Quarter Bridge
⚫ The quarter bridge contains only one strain
gage and the remaining elements are fixed
resistors as shown in figure below. Here, R4=
strain gage and R2, R3, R1 = fixed resistors. If
R1=R2=R3=R4 = R and the bridge is voltage
sensitive, if current flowing through
R1,R2,R3,and R4 be i1,i2,i3 and i4 and bridge
is balanced,
𝑒𝑖
⚫ i1=
𝑅1+𝑅3
𝑒𝑖
⚫ i2=
𝑅2+𝑅4
23
⚫ Now
𝑅1 𝑅2
eo = − ∗ 𝑒𝑖
𝑅1+𝑅3 𝑅2+𝑅4
𝑅1𝑅4−𝑅2𝑅3
eo = *ei
(𝑅1+𝑅3)(𝑅2+𝑅4)
When change of ΔR4 occurs at R4 this causes change in output as
Δeo
(𝑅4+𝛥𝑅4 )𝑅1−𝑅2𝑅3
e o + Δ e o= *ei
(𝑅1 +𝑅3)(𝑅2+𝑅4+𝛥𝑅4)
To simplify putting If R1=R2=R3=R4 = R, at the moment eo=0;
𝛥𝑅
(𝑅)
Δ eo = 𝛥𝑅 ei
4+2( 𝑅 )
24
⚫We know
ΔR4 = G*R *ε
𝐺∗𝜀
Δ eo = ei
4+2𝐺∗𝜀
Usually the second term in the denominator is negligible.
Then we can write,
𝐺∗𝜀
Δ eo = ei
4
25
Half bridge
⚫ The half bridge contains two strain gages in
the bridge, the other remaining elements
being fixed resistors. We know that for a
symmetrical beam section the tensile and
compressive strains are equal except for
the sign. Therefore one of the two strain
gages of a half bridge is bonded to tension
side and the other one to compression
side.
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⚫ The resistance change due to strain will be equal but
of opposite sign and the bridge output will be double
compared to Quarter Bridge which is given by:
𝐺∗𝜀
Δ eo = ei
2
27
Full bridge
⚫ All the elements in full bridge are
strain gages. In this case, the gages 1
and 4 are mounted to record tension
and 2 and 3 to record compression or
vice-versa as shown in figure. This
type of bridge gives the output which
is four times the output of Quarter
Bridge and is given by:
Δ eo = 𝐺 ∗ 𝜀 ∗ ei
28
Semiconductor strain gauge:
⚫ A typical semiconductor strain gauge is formed by the
semiconductor technology i.e., the semiconducting wafers
or filaments of length varying from 2 mm to 10 mm and
thickness of 0.05 mm are bonded on suitable insulating
substrates (for example Teflon). The gold leads are usually
employed for making electrical contacts. The electrodes are
formed by vapor deposition. The assembly is placed in a
protective box as shown in the figure.
⚫ The sensing element is rectangular filament made as a
wafer from silicon or geranium crystals.
⚫ To these crystals, boron is added to get some desired
properties and this process is called doping and the crystals
are called doped crystals.
29
Semiconductor strain gauge:
⚫ The strain sensitive, elements used by the semiconductor strain gauge are
the semiconductor materials such as, silicon and germanium. When the
strain is applied to the semiconductor element a large of change in resistance
occur which can be measured with the help of a Wheatstone bridge. The
strain can be measured with high degree of accuracy due to relatively high
change in resistance.
⚫ A temperature compensated semiconductor strain gauge can be used to
measure small strains of the order of 10-6 i.e., micro-strain. This type of
gauge will have a gauge factor of 130 ± 10% for a semiconductor material of
dimension 1 x 0.5 x 0.005 inch having the resistance of 350 Ω.
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Advantages of Semiconductor Strain
Gauge:
1. The gauge factor of semiconductor strain gauge is very high, about ±130.
2. Semiconductor strain gauge exhibits very low hysteresis i.e., less than
0.05%.
3. They are useful in measurement of very small strains of the order of 0.01
micro-strains due to their high gauge factor.
4. The semiconductor strain gauge has much higher output, but it is as stable
as a metallic strain gauge.
5. It has a large fatigue life i.e., 10 x 106 operations can be performed.
6. It possesses a high frequency response of 1012 Hz.
7. They can be manufactured in very small sizes, their lengths ranging from 0.7
to 7.0 mm.
32
Orientation of gages in Wheatstone
bridge
⚫ Configuration Type I
⚫ Measures axial or bending strain
⚫ Requires a passive quarter-bridge
completion resistor known as a dummy
resistor
⚫ Requires half-bridge completion
resistors to complete the Wheatstone
bridge
⚫ R4 is an active strain gage measuring
the tensile strain (+ε)
37
Configuration II
⚫ Ideally, the resistance of the strain gage should change only in response to applied strain.
However, strain gage material, as well as the specimen material to which the gage is
applied, also responds to changes in temperature.
⚫ The quarter-bridge strain gage configuration type II helps further minimize the effect of
temperature by using two strain gages in the bridge.
⚫ As shown in Figure, typically one strain gage (R4) is active and a second strain gage(R3) is
mounted in close thermal contact, but not bonded to the specimen and placed transverse to
the principal axis of strain.
⚫ Therefore the strain has little effect on this dummy gage, but any temperature changes affect
both gages in the same way. Because the temperature changes are identical in the two
strain gages, the ratio of their resistance does not change, the output voltage (Vo) does not
change, and the effects of temperature are minimized.
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39
Half Bridge Configuration
Configuration Type I
⚫ Measures axial or bending strain
⚫ Requires half-bridge completion resistors to
complete the Wheatstone bridge
⚫ R4 is an active strain gage measuring the tensile
strain (+ε)
⚫ R3 is an active strain gage compensating for
Poisson’s effect (-νε)
⚫ This configuration is commonly confused with the
quarter-bridge type II configuration, but type I
has an active R3 element that is bonded to the
strain specimen.
40
Configuration Type II
⚫ Measures bending strain only
⚫ Requires half-bridge completion
resistors to complete the
Wheatstone bridge
⚫ R4 is an active strain gage
measuring the tensile strain (+ε)
⚫ R3 is an active strain gage
measuring the compressive strain (-
ε)
41
Full-Bridge Strain Gage
⚫ A full-bridge strain gage configuration has four active strain
gages and is available in three different types. Types 1 and 2
measure bending strain and type 3 measures axial strain.
Only types 2 and 3 compensate for the Poisson effect, but all
three types minimize the effects of temperature.
42
⚫ Configuration Type I
⚫ Highly sensitive to bending
strain only
⚫ R1 and R3 are active strain
gages measuring
compressive strain (–e)
⚫ R2 and R4 are active strain
gages measuring tensile
strain (+e)
43
Configuration Type II
⚫ Sensitive to bending strain only
⚫ R1 is an active strain gage measuring
the compressive Poisson effect (–νe)
⚫ R2 is an active strain gage measuring
the tensile Poisson effect (+νe)
⚫ R3 is an active strain gage measuring
the compressive strain (–e)
⚫ R4 is an active strain gage measuring
the tensile strain (+e)
44
⚫ Configuration Type III
⚫ Measures axial strain
⚫ R1 and R3 are active strain
gages measuring the
compressive Poisson effect
(–νe)
⚫ R2 and R4 are active strain
gages measuring the tensile
strain (+e)
45
Measurement of torsional stress
46
47
