Problems of Chapter 3-4. Strain Gages – Introduction. Strain Gages Instrumentation
Problems to be solved
3-4-1
To determine the gage factor of a nominal resistance gage (120 Ω) a beam under pure bending is
utilized. The beam has a cross-section of b=1” and h= ¼”. The span of the region subjected to
pure bending is 24” and the material of the beam is steel. A dial gage measures the deflection of
the region of the beam subjected to pure bending and a high precision resistance meter is utilized
to record the changes of resistance of the gage. The strain gage is glued on the compression side.
The following data are collected.
Resistance (Ω) Beam deflection (mm)
122.5
0
121.9
0.140
121.4
0.280
120.6
0.420
From the given information provide the gage factor of the strain gage.
3-4 -2
A cantilever beam is utilized as a force transducer. The geometry of the beam is indicated in
Figure P3-4.1,
Figure P3-4.1Cantilever beam arrangement
Length
Width
Thickness
Young's
modulus
Gauge factor
Resistance
Gage position
L
b
h
E
50 cm
5 cm
2.5 cm
200 GPa
FG
2
R0 120 Ω
d 25 cm
The strain gage is in the arm of a full Wheastone bridge shown in Figure P3-4.2.
Figure P3-4.2 Wheatstone bridge, strain gage of the cantilever beam is R2. The voltage applied is
12 V.
1. Calculate the output of the bridge if a load of 100 N is applied at the tip of the cantilever
beam.
2. The gage is going to be used as the active element of a force gage and the output of the
instrument is a voltmeter that can measure plus minus 1 V. The scale of the voltmeter is
divided in plus minus 100 parts. Calculate the gain of the amplifier.
3. Calculate the required resolution of the voltmeter to read 1 N.
3-4 -3
A bridge with the configuration shown in Figure P3-4.3 is in the process of calibration. All the
arms have 120 Ω resistances. What strains are simulated if in one of the bridge arms the
following resistors are set in parallel, Rc1=1 Meg Ω, Rc2=0.5 Meg Ω , Rc3=1/4 Meg Ω. Compute
the output voltages if the applied voltage is 12 V.
Figure P3-4.3. Bridge calibration
3-4 -4
The cantilever beam of problem 3-4 -2 introduces four gauges of 120 Ω so that a maximum
output can be obtained and at the same time full temperature compensation can be achieved.
Figure P3-4.4
A Wheastone bridge has a configuration shown in Figure P3-4.4. The bridge is made of 120 Ω
gauges, FG=2. The bridge is designed to keep the gauge current Ig=20 mA. Compute the
sensitivity of the bridge if a very high impedance meter is utilized to read the output of the
bridge.
3-4 -5
In a Wheastone bridge three of the arms have gauges with the resistance 120 Ω, one of these
gauges is the active gauge. The fourth arm is utilized to balance the bridge. The bridge voltage is
6 V, and a high impedance instrument is utilized to measure the bridge output voltage.
1. What is the strain measure by the active gauge if the measured output is 800 mV?
2. What is the correction in the measured strain if Table 4.1 is applied?
3-4 -6
Compute the change of sensitivity of the bridge if in problem 3-4-4 the reading instrument is a
microammeter with an internal resistance R=120 Ω.
3-4 -7
Two strain gauges of 120 Ω and FG=2 are used to measure the axial force on a steel bar of d = 2”
1. How should these strain gauges be connected in the Wheastone Bridge to get optimum
results?
2. Compute the voltage to be applied to the bridge if the current in the gauges should be
limited to 15 mA.
3. A recording ammeter that has an internal resistance of 150 Ω registers a total
displacement of 10 units in a scale of 100 when a 0.5 megΩ is set in parallel with one of
the active gauges.
a. What is the sensitivity of the bridge in units of the scale per microstrain?
4. If a load P is applied what will be the largest load that can be measured?
3-4 -8
Figure P3-4.5 shows an instrumented bolt to measure the stresses caused by tightening of the
bolt. A strain indicator that is based on a Wheatstone Bridge is utilized to measure the output of
the bridge.
1. Choose the arrangement of the gauges in such a way that they only measure the axial
strain.
2. Choose the arrangement of the gauges in such a way that they only measure the bending
strain.
Figure P3-4.5 Bolt with strain gages bolted in the shank
3-4 -9
A shaft subjected to torsion is instrumented with strain gages to measure the applied torque.
Utilize a quarter bridge configuration, a half bridge configuration and a full bridge configuration.
Indicate the output in each case.
3-4 -10
A cantilever beam is loaded with a force F as indicated in Figure P3-4.6. The cantilever beam is
instrumented with four gauges distributed a shown. Connect the gauges in such a way that only
the axial component of the force is measured and the bending deformations removed.
Figure P3-4.6. Instrumented cantilever beam.
3-4 -11
A support pedestal of a large structure is instrumented to measure the total load sensed utilizing
strain gauges. Choose a Wheatstone bridge configuration such that thermal strains are
compensated and the bending stresses are removed.
Figure P3-4.7 Instrumented support of a large structure
3-4 -12
A second set of gauges is introduced in the support to measure the vertical component. Configure
the Wheastone Bridge in such an arrangement that only the bending stresses are measured.
3-4 -13
Figure P3-4.8 is the screen of an oscilloscope showing two traces. The lower trace represents the
voltage displayed when the Wheatstone Bridge that contains 4 strain gauges glued to a circular
bar is calibrated, with a resistor of 0.5 MΩ. The upper trace indicates the passage of a wave
along two sets of strain gauges as it propagates along the bar after the bar is impacted by a
projectile.
Figure P3-4.8. Digital oscilloscope trace. The dimensions of the squares is 1cmx1cm. The sweep
scale of the oscilloscope has been set to 10 μs/cm. The vertical scale is 1mV/cm.
It is a steel bar of the dimensions given in Figure P3-4.9.
Figure P3-4.9.Bar subjected to impact.
The bar has an area of 6.45 cm2. The gages are 120 Ω and FG=2.1.
1. Compute the speed of propagation of the axial wave.
2. Compute the amount of stress produced by the impact.
3-4 -14
Figure P3-4.10 represents a typical strain gage circuit that utilizes a PC as a reading device for a
strain gauge circuit. The gauge is connected to a Wheastone bridge; the output goes to an
operational amplifier. The operational amplifier feeds an adapting board that acts as interface
with the PC circuit. Through software the voltage is read by the PC and display in the PC
monitor.
Figure P3-4.10.Circuit to read a strain gage output utilizing a PC.
The amplifier shown in Figure 4.27 is utilized to amplify the signal coming from the Wheastone
Bridge. Vout of the Figure 4.27 is Vamp of Figure P3-4.10. Assume that the voltage has to be
amplified 10 times. Apply (4.80) and select the resistances that will provide the necessary
amplification. To satisfy circuit requirements of the opamp select k resistors.
3-4.15
Figure P3-4.11 shows a vibrating beam and a potentiometer circuit where Rb=Rg. That is the
ballast resistor has been replaced by another gauge. The output E because the circuit has been
grounded is E+ΔE. The output is a sinusoidal wave of amplitude E.  Fg  R g Pg that adds and
subtract to the voltage E.
Figure P3-4.11. Vibrating beam with two strain gauges bonded to measure the vibration of the
beam forming a potentiometer circuit.
Following the derivations in 4.11 the value of ΔE is,
E  Fg  R g Pg
An RC circuit is added to remove the constant voltage and allow passing the frequency of the
beam oscillations 10 Hz. Select the values of R and C such that the beams’ vibration frequency is
allow to pass without distortion.
3-4.16
Assuming the set up shown in Figure P3-4.10, add the filtering circuit. Assume that the strain
gauges have Rg=120 Ω, gauge factor FG= 2. The bridge voltage is E=5 V. Analyze the different
steps and compute the output of the complete circuit.