1. EXPERIMENT A2
Strain Gage Measurements
Summary:
Welcome to the ME-242 laboratory! In this laboratory you will perform hands-on experiments with
strain gages. Strain gages are a key tool in the testing of materials. This includes everything from simple
test coupons to advanced and very complex structures. This experiment will provide you with an
understanding of:
• the principles involved in strain gage measurements
• the factors to be considered in selecting a strain gage for an actual application
This experiment will also use supplemental information contained in the Student Manual For Strain
Gage Technology booklet that is supplied by the Department of Mechanical Engineering. This experiment
should give you the tools necessary to instrument and conduct your own strain gage experiments if called
upon in future projects.
In this experiment you will be examining the stress state in cantilever beams. You will also use a
strain gage rosette that has already been mounted for you to calculate principal strains in the cantilevered
beam through the use of Mohr’s circle.
Instructions:
Your key to success in this lab is to come prepared!
• Before arriving at the lab, read through this lab module so that you will understand what the lab
procedure is and how the lab equipment is used.
• Before coming to lab read through this handout to familiarize yourself with what will be expected
of you in the lab.
• Read also the ‘Student Manual for Strain Gage Technology’. It contains excellent descriptions of
how to select, mount and attach (solder) strain gages. The mounting section has simple
explanations with very good illustrations of these procedures.
• Background material on strain gages and Wheatstone bridges can be found in most introductory
textbooks dealing with mechanical measurements including your course textbook. A good
reference is Introduction to Engineering Experimentation by Wheeler and Ganji.
• Background material on the behavior of cantilever beams may be found in most introductory
mechanics of materials texts such as Mechanics of Materials, by Hibbler, and your ME 242
textbook.
• In your report discussion be sure to answer the questions included in the lab description below,
as well as the points described at the end (i.e. calculation of Poisson's ratio, etc.).
• At key points in the discussion questions will be posed that you should answer. These questions
will be numbered and the questions are in bold print surrounded by a frame.
• In your report discussion be sure to answer the questions included in the lab description below,
as well as the points described at the end (i.e. calculation of Poisson's ratio, etc.).
• Suggestions and reminders on report content are offered in the Report Notes sections.
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2. What’s a Strain Gage Used For? Structural Element of
The Birdman Contest is an annual event held on Lake Biwa near Benicia-Martinez Bridge in
Kyoto, Japan. In this contest cleverly designed human-powered Southern California
airplanes and gliders fly several hundred meters across the lake.
Aside from the great spectacle of this event, it is a wonderful view of
Undergoing a Load Test
engineering experimentation and competition. Despite the careful
designs and well-balanced airframes occasionally the wings of these
vehicles fail and crash into the lake. There have been some
spectacular crashes but few, if any, injuries to the contestants.
Increasingly, each time a new airplane, automobile, or other vehicle is
introduced, the structure of such vehicles is designed to be lighter to
attain faster running speeds and less fuel consumption. It is possible
to design a lighter and more efficient product by selecting light-weight
materials. However, as with all technology, there are plusses and
minuses to be balanced. If a structural material is made lighter or
thinner the safety of the vehicle is compromised unless the required
strength is maintained. By the same token, if only the strength is
taken into consideration, the vehicle’s weight will increase and its
economic feasibility is compromised. Picture of A Strain Gage
Attached to Bridge Element
In engineering design the balance between safety and economics is
one variable in the equation of creating a successful product. While
attempting to design a component or vehicle that provides the
appropriate strength it is important to understand the stress borne by
the various parts under different conditions. However, there is no
technology or test tool that allows direct measurement of stress.
Thus, strain on the surface is frequently measured in order to
determine internal stress. Strain gages are the most common
instrument to measure surface strain.
Background:
Introduction:
Experimental stress analysis is an important tool in the design and testing of many products.
Several practical techniques are available including photoelastic coatings and models, brittle coatings,
moiré, and electrical resistance strain gages.
In this experiment the strain gage will be utilized. There are three steps in obtaining experimental
strain measurements using a strain gage:
1. Selecting a strain gage
2. Mounting the gage on the test structure and
3. Measuring strains corresponding to specific loads.
The operation and selection criteria for strain gages will be discussed in this introduction. In Part I
of this experiment, you will mount a strain gage on a beam and test its accuracy. Measurements will be
made with a strain gage rosette in Part II of this experiment to obtain the principal stresses and strains on
a cantilevered beam.
Strain Gages:
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3. There are many types of strain gages. The fundamental structure of a strain gage consists of a
grid-shaped sensing element of thin metallic resistive foil (3 to 6 microns thick) that is sandwiched
between a base of thin plastic film (12-16 micron thick) and a covering or lamination of thin film.
Thin Film Laminate
Figure 1: Strain Gage
Construction
Resistive Foil
Plastic Base
Strain Gage Operation:
When needed for testing the strain gage is tightly bonded to the structural element under test.
This ensures that the sensing element (the metallic resistive foil at the center of the “sandwich”) may
elongate or contract in the same manner as the strain experienced by the test article. Typically the
sensing element is made of a copper-nickel alloy foil. When experiencing a contraction or elongation,
most metals undergo a change in electrical resistance. The alloy foil has a rate of resistance change that,
with a certain constant, is proportional to the strain. The strain gage is therefore a measuring device that
applies the principle of resistance change as a means to effectively sense strain.
Strain Measurement
It should be noted that there are various types of strain measuring
methods available. These may be roughly classified into
mechanical, electrical, and even optical techniques.
From a geometric perspective, strain recorded during any test
may be regarded as a distance change between two points on a
test article. Thus all techniques are simply a way of measuring
this change in distance.
If the elastic modulus of the test article’s constituent material is
known, strain measurement will allow calculation of stress. As
you have learned from your studies and prior labs strain
measurement is often performed to determine the stress created This LVDT, attached to a tensile
in a test article by some external force, rather than to simply gain specimen, is also a common tool
knowledge of the strain value itself. for measuring strain.
A resistance strain gage consists of a thin strain-sensitive wire mounted on a backing that insulates
the wire from the test structure. Strain gages are calibrated with a gage factor F, which relates strain to
the resistance change in the wire by
F = ∆R/R F = ∆R/R
∆L/L ∆L/L
where R is the resistance and L is the length of the wire. The change in resistance corresponding
to typical values of strain is usually only a fraction of an ohm.
(1) What parameters of the strain gage can be altered to increase its sensitivity?
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4. The gage factor, F, differs depending on the material used for the resistive foil. Typically, the gage
factors of the strain gages you will use in this experiment are between 2 and 3. It should be noted that the
resistance change caused by strain is extremely small. For example, assume a structural element in a
bridge is instrumented with a strain gage. The structural element experiences a strain of 1000 x 10 -6. If the
strain gage used to measure this has a resistance of 120Ω and a gage factor of 2 the following
relationship is established:
(∆R)/ 120Ω = 2 (1000 x 10-6)
and
(∆R)=120(2)(1000 x 10-6) = 0.24Ω
and the rate of resistance change may be shown as:
(∆R)/R = 0.24/120 = 0.002 = 0.2%
It is very difficult to accurately measure such a small resistance change. Thus, these resistance
changes are typically measured with a dedicated strain amplifier that uses an electric circuit called a
Wheatstone Bridge.
The Wheatstone Bridge
A Wheatstone bridge is a measuring instrument that, despite
popular myth, was not invented by Sir Charles Wheatstone,
but by Samuel H. Christie in 1833. The device was later
improved upon and popularized by Wheatstone. The bridge is
used to measure an unknown electrical resistance by
balancing two legs of a circuit, one leg of which includes the
unknown component that is to be measured. The Wheatstone
bridge illustrates the concept of a difference measurement,
which can be extremely accurate. Variations on the
Wheatstone bridge can be used to measure capacitance,
inductance, and impedence.
In a typical Wheatstone configuration, Rx is the unknown
resistance to be measured; R1, R2 and R3 are resistors of
known resistance and the resistance of R2 is adjustable. If
the ratio of the two resistances in the known leg (R2/R1) is
equal to the ratio of the two in the unknown leg (Rx/R3), then
the voltage between the two midpoints will be zero and no Typical Wheatstone Bridge
current will flow between the midpoints. R2 is varied until this diagram with strain gage at Rx
condition is reached. The current direction indicates if R2 is
too high or too low.
Detecting zero current can be done to extremely high
accuracy. Therefore, if R1, R2 and R3 are known to high
precision, then Rx can be measured to high precision. Very
small changes in Rx disrupt the balance and are readily
detected.
Alternatively, if R1, R2, and R3 are known, but R2 is not
adjustable, the voltage or current flow through the meter can
be used to calculate the value of Rx. This setup is what you
will use in strain gage measurements, as it is usually faster to
read a voltage level off a meter than to adjust a resistance to
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5. zero the voltage.
Because conventional ohmmeters are not capable of measuring these small changes in resistance
accurately, a Wheatstone bridge is usually employed. It can be operated in either a balanced or
unbalanced configuration. The configuration for an unbalanced bridge is shown in Figure 2. For an
unbalanced bridge, a change in resistance is measured as a non-zero voltage Vo which, can be
calibrated in standard strain units (∆L/L x 10-6) or micro strain. A balanced bridge is rebalanced after
each load increment so that the output voltage Vo is zero. The appropriate changes in resistance are
then noted and strain calculated using the gage factor.
Digital Strain Indicator
2.065 BLH
P2 S1P2 D1-120
Figure 2 - An unbalanced Wheatstone bridge with strain gage
(2) Does the BLH meter use a balanced or out- of-balance bridge?
(3) In Figure 1, what is the relationship between R1, R2, R3, and R4 for Vo=0?
Strain gage Wheatstone bridges are usually described by the number of active strain gage
elements vs. the number of fixed resistors. The various common configurations and their relationship to
Figure 1 are described in Table 1, below:
Table 1: Wheatstone Bridge Summary
Type of Bridge Active Resistance Precision Fixed Usual Application
Typical Designation Elements Resistances
Quarter Bridge R1 R2, R3, R4 Easiest to use, requires
material matching
Half Bridge R1, R3 R2, R4 Cancels unwanted
thermal effects or
bending effects
Full Bridge R1, R2, R3, R4 - Increased sensitivity
There are many advantages and disadvantages with each of these systems. These
include self compensation for resistance changes in the ‘active’ elements due to temperature and self-
heating effects, cancellation of unwanted forces or bending effects in measurements, and the ability to
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6. multiply and increase our strain sensitivity in applications where only very small strains are present.
These are discussed in more detail in the strain gage manuals, in your textbook, and in the literature.
In our experiment, quarter bridges will be used since we can use gages that are
compensating for temperature with respect to the tested material. In our case, an Al gage on an Al beam.
For gages and materials that are not self-compensating (i.e. a poor match of thermal expansion
coefficients), a half bridge or full bridge could be used to compensate for temperature changes.
(4) Briefly describe how to use the half bridge to compensate for temperature
changes. To achieve temperature compensation with two active gages, where
should the second strain- sensitive resistance be mounted?
Strain Gage Selection:
The process of selecting the proper strain gage for an application is not easy. Many interacting
factors must be considered together if accurate measurements are to be obtained. The following define
the general parameters that must first be considered.
1. The temperature range - This is the gage operating temperature range which can be a
function of the ambient temperature of the experiment and the power dissipation and thermal
conduction ratios of the substrate and base materials.
2. Magnitude of strain - The amount and type (elastic and/or inelastic) must be considered.
3. Principal axis of strain to be measured - An analysis of the stress vectors must be done to
determine proper gage mounting axes. If the principal axes are not known, a 3-element
rosette gage must be used.
4. Strain gradients - This effects the size of the selected gage in many cases.
5. Duration of measurement - Gages and substrate mounting materials have different degrees
of durability.
6. Approximate number of cycles - All gages experience fatigue and are rated for various fatigue
lives.
7. Accuracy requirements - All materials differ to some extent in linearity.
8. Any special ambient conditions which would adversely affect installation (magnetic fields,
radiation, etc.)
(5) What is the expression for the axial strain gradient along a cantilevered
beam’s upper surface? From observing this expression, explain why the strain
gage length is not an important factor here.
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7. Experimental Apparatus:
Strain gage and terminals
Aluminum beam (with no gage attached yet)
M-Line Neutralizer (or isopropyl alcohol)
Cotton swabs and gauze sponges
clean strain gage box (or glass slide)
tweezers
tape
M-Bond 200 adhesive and catalyst
degreaser
sandpaper
conditioner
medium hard pencil and ruler
wire
micrometer
BLH 1200B Portable Digital Strain Indicator
Flexor Cantilever Flexure Frame
BLH 1225 Switching and Balancing Unit
Beam with pre-mounted strain gage rosette (Beam E-103)
Ohmmeter
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8. Experimental Procedure:
This lab has two distinct portions.
In the first you will mount a strain gage on a beam and make measurements with it. In the
second you will be using a pre-mounted strain gage rosette on another beam to make measurements.
PART I - MOUNTING AND USE OF A STRAIN GAGE
I.a) Mounting
Surface preparation has five component steps:
1) solvent degreasing
2) surface abrasion
3) application of layout lines for gage
4) surface conditioning
5) neutralizing
Procedures for preparing the surface and mounting the strain gage are described in the
Student Instruction Manual, Strain Gage Installations with M-Bond 200 and AE-10 Adhesive Systems.
Prepare the surface as described in sections 2 and 3. Then immediately begin the bonding
procedure described in section 4. Soldering of leads onto the gage is described in section 5.
Note: the gage and terminal is first transferred to a clear glass plate using the tweezers.
Figure 4 shows the strain gage and tabs under the tape that is then used to transfer them to (and
align them on) the beam. After aligning the gage, carefully lift one side of the tape so the gage is off
the surface of the beam. Apply M-Bond 200 catalyst onto the gage and along the tape-beam junction
and press the gage onto the beam as described in the manual. Remove the tape.
Strain Gage
Connector Tabs
Figure 4: Strain Gage and Tabs under tape on beam
(6) List the properties of the mounting surface that are important for accurate strain
gage measurement.
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9. I.b) Testing of Strain Gage
Before proceeding perform the following tests to verify the gage and mounting procedure.
(7) Check gage resistance with the manufacturer’s value. Include both values and
the percent difference in your report.
(8) Check the resistance between the strain gage and the beam. A thoroughly dried
gage should show a resistance of 1,000 megaohms or more. Lower resistances can
be tolerated, but a minimum of 50 megaohms is necessary for accurate stable
functioning of the gage. Include the measured value in your report and
explain/discuss why a large resistance is necessary.
I.c) Making Measurements with the Strain Gage
The beam with the strain gage you have just attached will be placed in the Flexor Cantilever
Flexure Frame to take strain measurements. The arrangement is schematically shown in Figure 2.
(Note: Portions of this procedure are taken directly from the Experiments in Mechanics strain Gage series
entitled “E-101 Modulus of Elasticity - Flexure".)
P, δ
Fixed end
t
L
b
L = length between gage an applied displacement
= length between clamp and applied displacement
δ = applied displacement
P = load corresponding to the applied displacement
b = beam width
t = beam thickness
Figure 3: Beam with Strain Gage in Flexure Fixture
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10. To prepare the experiment, back the micrometer out of the way and insert the beam into the Flexor, with
the gage on the top surface and the gage end in the clamp. Center the free end of the beam between the
sides of the Flexor and firmly clamp the beam in place with the knurled clamping screw.
(9) Write down the equation which relates the axial stress on the upper surface (as a
function of position along the beam) to the applied load for a cantilever beam.
(10) Write down the equation which relates end displacement and applied load for a
cantilever beam. The location of the displacement should be at the point of
application of the load.
Connect the lead wires from the strain gage to the binding posts on the Flexor and the
appropriate wires from the Flexor to the BLH Digital Strain Indicator so that the strain gage is
connected to the BLH Indicator as shown in Figure 5.
(11) Write down the equation which relates axial stress to axial strain.
long jumper,
short jumper connecting at gage
OK BETTER
Figure 5: Schematic of BLH Digital Strain Indicator
Measure the distance from the centerline of the strain gage grid to the point of load
application at the free end of the beam, using an accurate scale. Also, measure the distance from the
clamp to the point of load application. Measure the width and thickness of the beam. Record, not only
the measured value, but also the resolution (estimated error) for each value.
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11. With the beam unloaded (except for its own weight), set the gage factor on the strain indicator
to the value given on the strain gage package data form. Balance the bridge using the coarse and
fine bridge balance controls.
To prevent plastic deformation, the maximum deflection for the thick beam should be 0.1
inches and for the thin beam should be 0.3 inches. To prevent damage (permanent bending) of the
beam do not exceed these limits.
Apply the maximum displacement in ten equal increments. At each displacement increment
record the micrometer reading and strain. (The display will be in micro inches/inch if the X1 switch is
depressed.) Unload the beam in 10 equal decrements and again record the micrometer reading and
strain at each decrement.
Report Note:
• Be sure to include a graph of strain versus displacement in your report.
• Compare your measurements (quantitatively) to theoretical predictions.
• Discuss the sources of error in the experimental data.
• Were any problems encountered in your experiment?
• Did they cause errors in the measurement?
• Did you observed any hysteresis - and if so, what % and what was the source?
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12. PART II - PRINCIPAL STRESSES AND STRAINS ON A CANTILEVER BEAM
In this part of the experiment you will use a beam which has already had a rectangular
rosette, consisting of 3 strain gages mounted at 45 degree angles, to measure the strain on the upper
surface of a cantilever beam. The principal strains and stresses will then be calculated and compared
with simple beam theory. (Note: Portions of this procedure are taken directly from the Experiments in
Mechanics strain Gage series entitled “E-101 Modulus of Elasticity - Flexure".)
Measure the distance from the centerline of the rosette to the loading point on the free end of
the beam, using an accurate scale. Measure the width and thickness of the beam with a micrometer.
Record, not only the measured value, but also the resolution (estimated error) for each value.
Back the micrometer out of the way and insert the beam into the Flexor with the gage on the
top surface and the gage end in the clamp. Connect the lead wires from the rosette to the binding
posts of the Flexor. Connect the strain gages between the G and W terminals as shown in Figure 6.
Note that the G (or GN) terminals are connected in series so therefore the common ground wire from
the gages must be connected to the G terminal. Use jumper wires between the W and W’ terminals
and between P1 and D1 as shown in Figure 6. Set the gage resistance to the HB position and the
digital strain indicator to 1/4 bridge operation.
NOTES:
All “GN” posts are interconnected inside unit.
Strain gages 1, 2, 3, etc.
Jumper all W to W´ posts externally.
Use channel selector knob.
Figure 6: Wiring schematic for the BLH Switching and Balancing Unit
Hook up the Switching and Balancing Unit to the Strain Indicator as shown in Figure 7. Additional
information can be obtained from the manuals provided.
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13. Figure 7 - Wire diagram for the connection between the
BLH Switching and Balancing Unit and the Digital Strain Indicator
With the beam unloaded, set the gage factor and zero the strain reading, recording the
readings for all three gages. Apply the maximum displacement of 0.3 in. in ten equal increments.
At each displacement increment record the micrometer reading and strain in the three gages. Unload
the beam in 10 equal decrements and again record the micrometer reading and strain in the three
gages at each decrement
Report Note: Be sure to address the following in your report….
• Be sure to include a graph of strain versus displacement in your report for each of the gages (in
one figure).
• Calculate the principal strains using the equation requested in question 6 of your prelab.
Compare (quantitatively) the theoretical predicted axial strain (question 3 of prelab) to the
appropriate principal strain.
• Use Hooke’s law for plane stress to calculate the principal stresses from the experimentally
determined principal strains.
• Also calculate Poisson’s ratio. Assume E = 71,700 MPa.
• Calculate the angle the rosette makes with the beam axis (be sure to define the angle using a
diagram) and quantitatively compare the measured angle to the calculated angle.
• Be sure to organize the analysis section so that it is easy to read and not excessively long.
Discuss any differences between theory and experiment. Discuss the sources of error in the
experimental data. Were any problems encountered in your experiment? Did they cause errors
in the measurement?
• Did you observed any hysteresis - and if so, what % and what was the source?
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14. EXPERIMENT A2 STRAIN GAGES
PRELIMINARY QUESTIONS
Group number (names):_____________________________ Date:______________
1. What are potential safety concerns for this experiment?
2. Sketch a simple (metal wire) strain gage. On this drawing indicate where the leads should be
attached. Also indicate the direction of "transverse sensitivity". Should the transverse sensitivity be high or
low? What is the difference between a strain gage and a rosette?
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15. 3. Using the variables defined for the cantilevered beam in Figure 2, write down an equation for
a) displacement δ at the loading point for an applied end load P. (Use tables in ME226 text.)
b) moment as a function of position x, M(x) for an applied end load P. (Apply static equilibrium to a
free-body diagram of a portion of the beam.)
c) axial stress σx in terms of moment (Consult your ME226 text.)
d) axial strain ex as a function axial stress (Reduce Hooke’s law.)
Combine these equations to obtain axial strain ex(x) on the upper surface of the beam as a function of x,
in terms of the applied displacement δ (i.e., eliminate P from the equation.). Use this equation during
the lab, to check whether your measured strains are accurate.
Use these equations to answer the following questions:
i) Where along the length of the beam will the maximum deflection occur?
ii)Where along the length of the beam will the maximum linear strain occur?
iii) Does this strain occur perpendicular or parallel to the axis of the beam?
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16. 4. Using the results of question 3, derive an expression for the axial strain gradient along the
beam’s upper surface. (Note that the strain gradient is de x/dx.)
5. Using the results of question 3, calculate the displacement to be applied at the free end of the
cantilever beam in order to produce an axial stress of approximately 5,000 psi at the fixed end of the
beam. Assume E = 71,700 MPa, L= 10 in, and t = 0.125 in. Compare to the value for t=0.25 in. Note that,
if the yield strength of the aluminum bar S=5000 psi, then this calculated displacement is the maximum
displacement before plastic deformation of the beam will occur. Avoid plastic deformation of the
beams.
6. Write down the equations for the principal strains in terms of the strains e 0, e45, and e90 measured
using a strain gage rosette. Also give the equation for the angle between the rosette and the principal
axes. Make sure you include a sketch defined the angle.
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