Post on 03-Apr-2018
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Strain Measurements
Engineering calculations are often based on stress. If we want to do
experiments to confirm our theory, we need to measure the result ofstress rather than stress directly. Stress results in the deformation of
material, which is called strain. For most engineering materials, there
is a rather simple relationship between stress and strain.
a Ea
a dL
LL
2L
1
L1
LL
1
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Lateral Strain, Poissons Ratio
If we stress a rod by pulling on it,and is stretches axially as a result, it
will also get thinner. This behavior
is quantified by Poissons ratio:
lateral strain
axial strain
L
a
This is a property of the material.
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General Stress States
y y
E
x
E
x x
E
y
E
x E x y
1 2
y
E y x
1 2
These equations relate the 2-D stress field to the
2-D strain field. I will assume that you alreadyknow this.
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We measure strain in one or more directions and infer the stress state
from that. In general, in order to know the 3-D stress state, wewould need 3 components of strain. In some cases (like pure axial
stress) we may be able to reduce the number of required
components. I will teach you more about the instrumentation side of
this topic, and it will be left to you to figure out how to get the stress
state from the measurements.
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12.3 Electrical Resistance Strain Gage
Ruge, 1940s
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Rosettes
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Installation
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The gauge length limits the spatial
resolution of the sensor.
Connection to the bridge is made
at the solder tabs.
The backing material needs to be
made of something that can:
Withstand the temperatures
encountered
Transmit strain but electrically
insulate
Accept the bonding adhesive
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12.4 Metallic Gauges
R LA
LCD2
If you have a conductor of resistivity , the resistance across that
conductor is
If you strain this conductor axially, its length will increase while its
cross sectional area will decrease. Taking the total differential ofR,
dR R
d
R
LdL
R
CD2 d CD2
1CD2
Ld dL 2L dDD
dR
R
dL
L 2
dD
D
d
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Metallic Gauges
dR /R
dL /L1 2
dD/D
dL /Ld/
dL /L
dR
RdL
L 2
dD
Dd
a dL
L
L dD
D
L
a
FdR /R
dL /LdR /R
a1 2v
d/
dL /L
For most strain gauges, = 0.3. If the resistivity is not a function of
strain, thenFonly depends on poissons ratio, andF~ 1.6.
Gage factor
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dRR dLL 1
d
1
1
E
d/
dL1/L
dR /R
dL /L1 2
1E
Piezoresistance
Coefficient
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Gage Factor
1
F
R
R
F dR /RdL /L
dR /Ra
1 2v d/dL /L
FandR are supplied by the
manufacturer, and we measureR.
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Example
A typical strain gauge uses constantan (55% copper, 45% nickel)
which has a resistivity of 49 X 10-8
W
m. The strain gauge must be120W nominally (why?). If the diameter is 0.025 mm, how long
does it need to be?
R eL
AcL = 12 cm
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12.5 Selection and Installation
Read on your own
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12.6 Circuitry for Metallic Strain Gage
Most commercial strain gages are 120 W, have a gage factor near 2, and
can measure 1 microstrain (1 part in a million).
1
F
RR
R 120 2 1E 6 0.00024W
Clearly, our work is cut out for us in terms of the measurement.
h i id i i
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12.8 The Strain Gage Bridge Circuit
eo
ei
R
1/R
4
2
R1/R
1
F
RR
eo eiF
4 2F
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eo eiF
4 2F
If we assume some typical values for the excitation voltage (8V) and the
gage factor (2), then we can see that the second term in the denominator
is not significant:
eo 16
4 4
eo eiF
4
so
id i h d i
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12.8.1 Bridges with 2 and 4 strain gages
The bending strain on the top gage is equal
and opposite of the one on the bottom.
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eo eiR
2
R1
R2
R
4
R3
R4
makeR2 =R4 =R
eo eiR
R1
R
R
R3
R
eo eiR
R1
1R
R3
1
eo
ei R
1
R1
1
R3
M l i l G B id
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Multiple Gauge BridgeMost strain gauge measurement systems allow us to make 1, 2, 3 or all 4
legs of the bridge strain gauges. There are many reasons to do this that
we will talk about now.
Going back to our fundamental bridge equations from chapter 6,
Eo EiR
1
R1
R2
R
3
R3
R4
Say that unstrained, all of these have the
same value. If they are then strained,
the resultant change isEo
is
dEo Eo
Rii1
4
dRi
Eo
M l i l G
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Multiple GaugesMake the following assumptions:
All gauges have the same nominal resistance (generally true)All gauges have matched gauge factors (must be purchased as set)
Then:
EoEi
F
4
1
2
4
3 Eo
A S i d C i
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Apparent Strain and Compensation
Things like temperature can change the resistance of a gauge and our
system may interpret this as strain. Sometimes our gauge may be
subject to strains other than the one we are interested in.
Compensation is removing these effects by using multiple gauges. As
an example, say you have a beam under axial stress and a bending
moment, and you are interested in the axial stress only:
x 12My /bh3 FN /bhThe two gauges see the
same axial strain but
opposite bending
strains eo
ei F
41 4
1
a1 b14
a4 b4e
oei
F
2a
T t C ti
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Temperature CompensationThe resistance of a strain gauge changes with temperature. In addition,
changing its temperature may cause strain in the gauge making it even
more sensitive to temperature.
C ti
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Compensation
12 8 2 B id C t t
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12.8.2 Bridge Constant
kA
B
k= the bridge constant
A = the actual bridge output
B = the output you would get with a single gage.
12 8 3 L d i E
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12.8.3 Lead-wire Error
Since we are looking at very small changes in resistance, the lead wires
can create significant errors. We handle this the same way we discussedfor RTDs.
We have wire especially made for strain gagemeasurements which has three conductors
12 10 T t C ti
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12.10 Temperature Compensation
R1
R2
R
3
R4
R1
RtR
2 Rt
R
3
R4
If the temperature of the
specimen changes, then
both gages will change their
resistance similarly
12 11 C lib ti
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12.11 Calibration
R RgRs
Rg Rs Rg
Rg2
Rg Rs
1
F
Rg
Rg
e 1
F
Rg
Rg Rs
12 16 1
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12.16.1
Multiple gages in series
12 17 1 C S iti it
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12.17.1 Cross Sensitivity
eL
Kt L /a po
1poKt100
Semicond ctor Strain Ga ges
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Semiconductor Strain Gauges
The gauges we have been talking about are made of metal. We can
also make them out of semiconductors, which is how the strain
gauges in our pressure sensors are made. These are dominated by the
piezoresistive component of the change in resistance and have
several advantages and disadvantages:
Pros:
Very high gauge factors (up to 200)
Higher resistance
Longer fatigue life
Lower Hysteresis
Smaller
High frequency response
Cons:
Temperature sensitivity
Nonlinear output
More limited on maximum strain
Mostly used for construction of
transducers
Hysteresis of Strain Gauges
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Hysteresis of Strain Gauges