Background reading for Lab 5 (Calibration of a Cantilever Beam / Wheatstone Bridge
system)
1.0 Introduction
1.1 Motivation for resistance bridges – measuring small changes in resistance
A large number of sensors that are used in engineering applications convert the desired
measurement signal into a variable resistance. This was done for temperature in recent labs
using thermistors, but it can also be done for other physical parameters such as strain, pressure,
displacement, etc. However, most often we need the sensor signal to be in the form of a voltage
that can be easily measured and processed using an analog to digital signal converter and a data
acquisition system. We have used simple resistive voltage divider circuits in the past to
accomplish this, but these have some inherent drawbacks. A resistance bridge configuration can
get around many of these issues by allowing for automatic compensation for resistance
tolerances, variations with temperature (if we don’t want to measure such variations), and
resistance in the wires leading to the sensor. Also, resistance bridges can remove the offset (DC
component) of a signal and allow closer examination of small changes relative to this offset.
1.2 Intro to strain gages, sensors commonly connected to resistance bridges
Strain gages operate on one of four different principles: changes in semiconductor
characteristics, capacitance, wire vibration, or electrical resistance when the test piece to which
they are mounted undergoes a strain. Use of the resistance strain gage surpasses the others in
engineering practice and is therefore the focus in this course. The relationship between the
electrical resistance of a wire and the corresponding strain was first observed by Lord Kelvin in
1856. Although the effect would become a great contribution to the engineering field, Kelvin
regarded it as an experimental nuisance. During the early 1930's, the first resistance strain gage
was developed for industry, consisting of a carbon strip surrounded by an insulating pad. Its first
use was to measure vibrating strains on aircraft propellers. The rapidly growing aircraft industry
during WWII had the largest influence on strain gage development, and in 1952 the first modern
foil resistance strain gage was produced.
2.0 Strain Gages
2.1 Gage Construction
The components of a resistive strain gage are shown in Figure 1. The manufacture of the foil
element requires great precision. An alloy (usually constantan - 55% Cu, 45% Ni ) is rolled to a
constant thickness (typically 0.003 mm to 0.005 mm) and the pattern is stamped out. Foil
elements commonly have unstrained resistance values of 60, 120, 350, 500 and 1000 Ω. In our
experiment, the unstrained resistance is approximately 350 Ω. By adjusting the characteristics of
the strain gage (e.g., the alloy or geometry), it is possible to obtain a gage that is customized for
unique applications (e.g., an engineer measuring the thermal expansion of a combustor would be
concerned with the temperature-dependence of the gage's resistance).
Figure 1: Resistance strain gage
The backing of the strain gage is the portion to which the foil element is attached. The backing is
directly fastened to the test piece by an adhesive. Epoxy resins and polymides are the two most
commonly used materials for the backing because they meet the following specifications:
1. The backing must electrically insulate the foil element from the test piece.
2. The backing must transmit nearly 100 percent of the strain from the test piece to the foil
element. This includes both elongation and compression strains.
3. The backing must provide an effective heat sink to avoid self-heating resulting from the
current flowing through the foil element during resistance measurement.
2.2 Strain Gage Operation
The operation of a strain gage depends on the relationship between the resistance of a conductor
(R) and its electrical resistivity (ρ, which is a material property), cross-sectional area (Awire),
and length (Lwire),
L
(1)
R  wire .
Awire
The geometric deformation of the wire as it is stretched will change its resistance. When a wire is
in tension, its length increases and its cross-sectional area decreases, see Figure 2. The net effect,
according to Equation (1), will be an increase in the wire's resistance (the numerator will increase
and the denominator will decrease). When the wire is compressed, the opposite geometrical
changes occur and its resistance decreases. This principle allows mechanical strain to be
converted into changes in electrical resistance that can subsequently be converted to voltage
changes. Strain, ε, is the fractional change in length of an object (ΔL/L, where ΔL is the
change in length and L is its original length).
L

(2)
L
tension along longest dimension
area  length  = resistance 
compression along longest dimension
area  length  = resistance 
Figure 2: The effect of geometrical changes on resistance
Changes in the resistance of a strain gage (ΔR) can be related to the corresponding strain by a
constant called the gage factor, GF (this often appears as just F or G in many references).
R
L
 GF
 GF
(3)
R
L
The actual strain experienced by the gage is a function of the strain which the test piece
experiences and the gage's orientation on the test piece with respect to the direction of the load.
Figure 3 illustrates these relationships, where the subscript sg indicates strain gage and bar
indicates test piece.
Figure 3: Strain of a strain gage as a function of the test piece’s strain for two orientations.
Poisson's ratio, , is the ratio of lateral strain to axial strain. (Note that Poisson’s ratio appears as
 in many references.) Axial refers to the direction along which the stress is applied (e.g., the
arrows labeled “load” in Figure 3). The negative sign indicates that if the axial strain is positive
(elongation), the lateral (or perpendicular) strain will be negative (compression).
 bar  
 lateral
 axial
(4)
If axially- and laterally-oriented strain gages have the same gage factor and the same unstrained
resistance (R), then Equations (3) and (4) can be combined to give the following relation.
Rlateral   bar Raxial
(5)
The stress, σ, on an object is the load per unit area. Strain is related to stress by a factor called
the Modulus of Elasticity, E.


The nominal properties of some common metals are listed in Table 1.
E
(6)
Table 1. Nominal properties for some common metals
metal
aluminum
copper
steel
Poisson’s ratio
0.33
0.36
0.30
Modulus of Elasticity
70,000 MPa
110,000 MPa
205,000 MPa
3.0 Wheatstone Bridge
3.1 Key concepts
Now that we understand why the resistance of a strain gage will change in response to strain, the
question becomes how to detect this change in resistance accurately and precisely. The
measurement of ΔR is generally not made directly using an ohmmeter, because the changes in
resistance are often extremely small, thereby requiring more specific instrumentation.
Example
Calculate the value of R for a typical strain gage with a nominal (unstrained) resistance of R =
350 , a gage factor (GF) of 2.0 and an applied strain of  = 2  10-6 m/m.
R  R GF

R  (350 )( 2.0) 2  10 6 m
m

R  0.0014 
The 1.4 milliohms calculated above is generally too small to measure accurately using a standard
multimeter. As a result, a more sophisticated method is needed. A Wheatstone bridge is
commonly employed. In its original form, the Wheatstone Bridge is a "null" instrument, which
means that the measured voltage, Vout, will ideally equal zero when no strain exists. This
technique allows more accurate observation of the effects that small changes in resistances have
on the measured voltage (i.e., it is easier to compare 0.002 Ω to 0.000 Ω than to compare
100.002 Ω to 100.000 Ω).
3.2 Basic Analysis
A conventional Wheatstone bridge, shown in Figure 4, provides a constant voltage, Vin, across
two parallel branches of the bridge, (RA-RD and RB-RC); the output of the bridge, Vout, is measured
at the other two terminal points of the bridge. For strain measurements, RA, RB, RC and RD are
frequently all strain gages. Depending on the application, one or all four resistors can be active
strain gages mounted on the test piece. Strain gages not mounted on the test piece, but connected
to the bridge are referred to as dummy loads (we'll see why this is a good idea later on). Using
basic circuit analysis, you should be able to prove to yourself that
Vout
RC
RD


Vin
R A  RD RB  RC
RA
RB
+ Vout -
RD
(7)
+
Vin
RC
-
Figure 4: Conventional Wheatstone bridge
If the 4 strain gages have exactly the same resistance in their unstrained state (R) then the output
voltage in the unstrained state must be zero. In reality, slight differences in the strain gages will
make the output voltage in the unstrained state different from zero, so sometimes a potentiometer
is added somewhere in the circuit that allows us to "balance the bridge", that is, adjust one of the
legs of the circuit until the output voltage is very nearly zero.
If gage D is strained so that its resistance increases and the other gages are not strained then the
voltage will increase to the value given by (to derive this equation, just realize that both sides of
the Wheatstone bridge are voltage dividers):
Vout
RC
RD  RD


Vin
R A  RD  RD RB  RC
(8)
Likewise, if strain gage B is strained so that its resistance increases and the other gages are not
strained, then the output voltage will decrease to the value given by:
Vout
RC
RD


Vin
R A  RD RB  RB  RC
(9)
Example
Given a cantilever beam in bending like that shown below,
 Where should the strain gages be placed in the Wheatstone bridge?
 What is the relationship between the measured voltage and the applied strain?
 What is the relationship between the measured voltage and the applied voltage?
Load P
L
S2
nd
de
Fixe
t
S1
S3
S4
b
When load P is applied to the end of the cantilever beam, strain gages S1 and S2 will be in
tension while strain gages S3 and S4 will be in compression. This means that the resistance of
S1 and S2 will increase and the resistance of S3 and S4 will decrease. If the gages were placed
carefully such that they are exactly opposite each other on the beam and the beam is of uniform
properties, the tension strain () on the top of the beam will be equal to the compressive strain () on the bottom at the strain gage locations. Thus, dR1 = dR2 = -dR3 = -dR4. Looking at
Equation (7), if
S1  R A
S 2  RB
S 3  RC
S 4  RD
Vout will be 0 for all bending loads (you should prove this to yourself; also how would this
configuration work for pure axial tension or compression loads?). Now, let’s let
S1  R B
S 2  RD
S 3  RC
.
S 4  RA
Using Equation (7), we find that this configuration gives an output voltage of
V
Vout  in dR  (dR)  dR  (dR)
4R
dR
Vout  Vin
 Vin GF
R
A brief perusal of your strength of materials book should inform you that the strain at the
location of the strain gages on a rectangular cantilever beam under bending is
6 PL

Ebt 2
where P, L, b and t are as defined in the figure. E is the Young’s modulus for the material.
Finally, Vout can be related directly to force P by
Vout  Vin GF
6 PL
Ebt 2
Some typical numbers might be GF = 2.10, L = 0.15 m, b = 0.025 m, and t = 0.003 m. For this
aluminum beam,
Vout
6(0.15)
 2.10
P  0.0001P .
10
Vin
(7 10 )(0.025)(0.003) 2
If Vin = 8 V, then this setup would give an output of 0.0008 V per Newton of applied force.
4.0 Effects of Temperature
Many strain gages can withstand extreme temperatures; however, when there is a temperature
change while a measurement is being made, the effects can cause large errors in data unless
proper precautions are taken. The reason is that measurements with a strain gage depend on
changes in resistance leading to a change in voltage. But changes in resistance can also be caused
by changes in temperature due to temperature versus resistance relationships. There are several
temperature effects which the engineer must take into consideration:
1. The thermocouple effect.
2. Differences in thermal expansion of the gage and test piece.
3. Changes in resistance of the wires leading to the strain gage.
4. Changes in gage resistance.
4.1 Thermocouple Effect
The connection between the copper leads and the constantan foil element creates a thermocouple.
The induced voltages produced are about 4 mv/100 °C. Because there are two junctions per
strain gage, ordinarily the voltages should cancel each other and the effect will not be noticed.
However, if the test piece has severe temperature gradients, the temperature at one junction could
be different than at the other and the two induced voltages will not be equal; this will affect the
measured voltage. To correct for this effect, a minimum of two strain gages should be used in the
Bridge circuit if at all possible.
4.2 Differences in Thermal Expansion
When the coefficients of expansion for the test piece and the strain gage differ, there can be large
strain error when the temperature changes during measurement because the strain gage and the
test piece will expand at different rates. Assuming that the two are rigidly connected, an induced
strain will appear in the gage and a false output voltage will be produced. One way to avoid this
problem is to purchase strain gages that have been heat treated to match their coefficient of
expansion to that of the test piece. These gages are referred to as Self Temperature
Compensating (STC) gauges.
4.3 Changes in Lead Wire Resistance
If the Wheatstone bridge consists of only one strain gage and three fixed resistances, then the
change in the resistance of the lead wires to the gage must be taken into account. For example, a
two conductor copper cable of 1 mm cross section and 10 m length will experience a change in
resistance of approximately 0.0128 Ω for a temperature change of 10 °C. This is far larger than
typical changes in resistance caused by strains, as seen in a previous example. A quarter bridge,
three lead system corrects this problem (see Figure 5). By using this system, the two induced
lead resistances, RLA and RLD will leave the voltage Vout, in Figure 5, unchanged. The third
induced resistance, RLE, will not effect Vout since no current flows through this branch (when the
bridge is balanced and the current would be very small for a non-balanced bridge).
Figure 5: Quarter bridge, three lead system
4.4 Change in Gage Resistance
The resistance of the foil element, like any conductor, is a function of temperature. Equation (10)
describes the relationship between the resistance R, the standard resistance at 0 °C, Ro, the
temperature coefficient of resistance, α, and the change in temperature, ΔT. (Note that this
relationship also governs RTDs, which are resistors meant to measure temperature.)
R  Ro (1  T )
(10)
Example
Calculate a typical value of change in gage resistance due to temperature change.
Use Ro = 350 Ω, α = 0.00393 / °C (for constantan) and ΔT = 1 °C.


 1 
R1 C  350 1  0.00393  1 C   351.38 
 C


R  1.38 
Thus, even for a small change in temperature such as 1 °C, the resistance increases 1.38 Ω. This
change is enormous when compared to the typical ΔR of 0.0014 Ω in Example 1. The method
used to correct for this error is referred to as temperature compensation.
4.5 Temperature Compensation
From the above example, the effects of temperature change can be disastrous when only one
strain gage is used. The change in temperature will change the gage resistance causing a false
output voltage. To cancel this effect, it is recommended that at least one additional gage be
installed in an adjacent arm of the bridge. This additional gage will experience the same change
in resistance if it is located in the same environmental conditions (i.e., the same temperature) as
the gage that is mounted on the test piece. The changes in resistance, ΔRA and ΔRD, will lead to a
net change in output voltage of zero (Fig. 6). With this setup the bridge is said to be temperature
compensated.
Figure 6: Temperature compensation with two strain gages
If an active gage is in arm A then the additional gage must be placed in either arm B or D, but
not C. It is important to note that if the extra gage is placed on arm C, the voltage effect will not
be cancelled but will be doubled.
5.0 Cross-talk
Oftentimes, more than one type of loading (e.g., tension and bending) is applied to a test piece at
the same time. If a strain measurement is being made for only one of these types of loading, the
results can become contaminated. The reasoning is: if the additional load is large enough, it
might aid or hinder the loading of interest, and hence will change the amount of strain. This
situation is referred to as cross-talk, see Figure 7. To determine the amount of cross-talk:
1.
2.
3.
4.
Apply only the loading of interest to the test piece (e.g., bending).
Graph the strain versus the applied mass for this loading.
Apply an additional type of load (e.g., tension, if bending is the load of interest).
On the same graph, superimpose another plot of strain versus mass for the initial type of
loading of interest (e.g., bending, in the examples above).
Note: In Figure 7, the Bending only line and the Bending and Tension line intersect. This is
cross-talk. The Bending and Torque line which is parallel to the Bending only line is not crosstalk, but is a zero-offset problem (i.e., the applied torque does not affect the bending).
Figure 7: Evidence of cross-talk in strain-gage measurements
6.0 Measuring axial loads
The bridge configuration derived above has good temperature compensation and measures
bending while negating any effects of axial loads, either tension or compression. Can the strain
gage arrangement used for bending also be used for axial loads?
S2
d
Fixe
end
t
S1
S3
S4
Load P
b
Figure 8: Example of axial loading with parallel strain gages.
Recall that for a full bridge, if all the sensors have the same, nominal, unstrained resistance,
V
V
V
V
dVout  0  dVin  in dR A  in dR B  in dRC  in dR D
4R
4R
4R
4R
(11)
Vin
dRD  dRC  dRB  dR A 
dVout 
4R
In bending,
dRS 1 
dRS 2 
dRS 3 
dRS 4 
for the configuration discussed previously. For the configuration shown above, we expect
dRS 1 
dRS 2 
dRS 3 
dRS 4 
Looking at Equation (11), there is no configuration of the sensors in the Wheatstone bridge that
will give a signal for axial loads. Note, however, that if only two strain gages are used (a half
bridge), there are multiple configurations that will give an output signal for axial loading. One
problem with this is that the strain gages will need to be placed in “opposite” arms of the bridge
(i.e., RA and RC or RB and RD) and temperature compensation will be lost. For a pair of sensors to
compensate for temperature changes, they need to be positioned in the same branch of the bridge
so that both resistances are part of the same voltage divider.
One strain gage arrangement
to measure axial strain with
two sensors
RA
+
RB
+ Vout -
Vin
RC
RD
-
Figure 9: Strain gage placement to measure axial loading of a beam using only two strain gages
In order to be able to measure axial loads and have temperature compensation, the strain gages
can be placed as shown below
d
Fixe
S2
end
S1
S4
S3
t
Load P
b
Figure 10: Strain gage placement to measure axial loading of a beam using a Poisson
configuration
In this arrangement the resistance changes will be
dR S 1   dR
dR S 2  dR
dR S 3   dR
dR S 4  dR
Note that strain gages S2 and S4 will increase their resistance under compression because the bar
must expand slightly in the transverse direction as the beam is compressed axially. The amount
of this transverse, or lateral, strain, is determined by Poisson’s ratio:
 
 lateral
 axial
This strain gage configuration using a Wheatstone bridge is often called a Poisson bridge, or
Poisson’s bridge. For an initially balanced bridge,
Vout
(1   )GF

Vin
2  (1  )GF
(12)