Strain gage and cantilever beam experiment

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DescriptionStrain Gage on a Cantilever Beam.
Experiment # 2
SAFETY FIRST!
1. Must not wear open-toed shoes.
Objectives
1. Familiarity with the underlying principles of electrical resistance strain gages.
2. To set up a Wheatstone half-bridge circuit.
3. To measure strain with a strain gage in three cantilevered beams (acrylic,
aluminum, steel) loaded at the free end with a range of masses.
4. Calculate the maximum normal strain and Young’s modulus for the three beams,
and compare the calculated values with the theoretical values.
Background
Strain Gage:
The basic definition of axial strain is the change in length divided by the original length.
Strain may be measured with mechanical, optical, electrical, and acoustical strain gages.
This lab focuses on electrical resistance strain gages. When thin foil or wire is stretched,
its resistance increases. The axial strain, , experienced by the material is related to the
change in its resistance, R, by the following equation:
=
1 R
GF R
where R is the original resistance and GF is the strain gage factor. This resistance may be
measured directly with a digital multimeter. However, since the change in resistance is
small, the resolution in determining the strain is also small. To obtain a more robust
measurement of strain, a Wheatstone half-bridge, as shown in Figure 1, is used. Two
resistors and two strain gages are used to complete the circuit. The Wheatstone halfbridge strain gage produces a resistance change that is linearly proportional to axial
strain. Vs is the supply voltage (in our case, 5 volts) and Vo is the output voltage, which
will be measured after the material is loaded. For the half-bridge shown, the axial strain is
given as:
=
2 V0
GF Vs
where GF is the strain gage factor. Note that the strain is independent of the resistor
values in the circuit. With the variable resistor one should be able to come close to
balancing the circuit when it is not loaded, i.e., when Vo = 0 →  = 0.
1
R2 Strain Gage
R3
==
+
Vs
–
=
– Vo +
R4 (Variable Resistor)
R1 Strain Gage
Figure 1: Wheatstone half-bridge.
For more information, please study the following resources
•
•
National Instruments publication on strain gages (http://www.ni.com/whitepaper/3642/en/)
Lab handout with more information on the Wheatstone bridge from Penn State
(https://www.mne.psu.edu/cimbala/me345/Lectures/Strain_gages.pdf)
Cantilever Beam Bending:
For a cantilevered beam with a point load applied at the free end, the normal stress at the
outermost surface of the beam, , is
Mc
I
Hère M represent the internal moment that can be calculated from the applied load, c is
the distance from the neutral axis of the beam to the outer surface, and I is the moment of
inertia of the beam’s cross-section about the centroid.
The centroid is the geometric center of an area and corresponds to the neutral axis for
bending in the elastic region.
=
2
For composite geometries:
x A
x =
A
i
i
i
i
i
and
y A
y =
A
i
i
i
i
i
The moment of inertia is a geometric property that determines the ability of the beam to
resist rotation or bending.
y
b
y
bh3
I = I x =
12
h
x
centroid
o
x
Figure 2: Cross-section view of beam.
From Hooke’s Law, the strain, , at the surface is equal to
=

E
.
Test Procedure
Before the test
1. Set up the Wheatstone bridge, as shown in Figure 1, on the Bread board.
Make sure to turn off the power when constructing the Wheatstone half-bridge
to avoid short circuit. Two strain gages representing R1 and R2 will already be
glued to the top and bottom faces of the beams. Use an input voltage of 5 V.
2. Use a ruler and Vernier calipers to measure the dimensions of the beam crosssection, and the location of the strain gage with respect to the applied load.
These measurements will be used to calculate the moment of inertia and the
internal bending moment at the strain gage location.
3. Clamp the beam to the countertop 10 mm from the strain gages and then
balance the Wheatstone half-bridge using the potentiometer so that the output
voltage V0 is as close to zero as possible.
3
During the test
1. Apply loads from 100 g to 500 g in 100 g increments to the end of the beam.
*Be careful to not break the acrylic beam; use your judgment regarding
the appropriate maximum load.
2. Record the applied load and the output voltage for each load. Repeat this
procedure for the other beams.
Report
A full lab report is to be submitted. The report should include the following in the Results
section:
a) A table for each beam that includes load, output voltage, and experimental strain.
b) A table for each beam that includes the internal bending moments, theoretical
stress, and theoretical strain at each load. Use the theoretical Young’s modulus for
the theoretical strain calculation.
c) A plot of the theoretical stress versus experimental strain for each beam. Use
regression analysis in Excel or MATLAB to draw a best-fit straight line through
the data points. From the regression analysis, determine the experimental Young’s
modulus for the three beams used in this lab.
d) A plot of experimental versus theoretical values of strain for each beam.
e) Show a sample calculation
The report should include a discussion about strain measurement uncertainty as well as a
comparison of the theoretical and experimental results found in this lab (strain and
Young’s modulus).
Reference
Holman, J.P., Experimental Methods for Engineers, 7th ed,. McGraw-Hill, New York,
2001.
4
MECHANICS OF MATERIALS LAB
LAB REPORT GUIDELINES
1
LABORATORY REPORT PREPARATION
The ability to communicate clearly both orally and in writing is of great importance to
professional engineers. After graduation you will spend a good part of your time explaining
your ideas and points of view both to your superiors and to the technicians under
your supervision. You are, therefore, advised to work on improving your written communication
skills, and the preparation of laboratory reports provides you with an excellent opportunity to
do so. Unlike a newspaper article, a technical report has a certain standard format that must be
adhered to. In addition, the style of technical and scientific writing is different from that used in,
say, books of literature. Colloquial language should not be used; instead, explain your ideas in
simple and grammatically correct English using clear and short sentences. Try to get to the point
directly and avoid unnecessary elaborations and word text. Also, it is a well-established
tradition in scientific writing to report in the third person (i.e., avoid using “I” and “we”).
Note on verb tense: The experiment is already finished. Use the past tense when talking about
the experiment. The report still exists; use the present tense when talking about the report.
Note on graphics: Figures and Tables are used in technical reports (not Graphs and Charts).
Figure captions should be numbered consecutively and placed under each figure. Table captions
should be placed above each table.
Full Reports
The audience for this type of report is a practicing mechanical engineer who knows something
about the topic, but has probably forgotten much of it so he or she needs to have the relevant
material reviewed. It should be neat, legible, well organized, and include the following:
1. Cover Sheet: Report title, names of all group members, course number, and date.
2. Abstract: The abstract is an executive summary that briefly describes the experiment and
states the main findings. It summarizes the entire report in one main paragraph. Your
abstract should emphasize the objective, procedure, results, and significance. Be precise
and specific. A technical report is not a mystery novel – state your conclusion as soon as
possible!
3. Introduction: A brief introduction that explains the purpose of the experiment. The
introduction should include any other introductory/background information or theory that
the reader needs to know. This is where you tell the reader what you did and why you did
it.
4. Methods and Materials: In paragraph form, describe the steps taken to perform the
experiment, measurement techniques, and the apparatus. Include photos or diagrams of
the apparatus as appropriate. Use your own words. Do not copy the procedure from the
lab handout. This is where you tell the reader how you did the experiment and you
describe the equipment and materials used to conduct the experiment. You should
provide enough information so that another researcher in your field could use your
description to replicate the experiment.
2
5. Results: Present your results to the reader. Although results are usually presented
quantitatively, you should always introduce each block of information with simple clear
language. Include measured results, an estimate of the experimental uncertainty, and any
calculations used. In most cases it is sufficient to provide a sample calculation with clear
explanation of the equations. Use tables and figures as necessary. All tables and figures
should be labeled with a Figure/Table number and a descriptive caption. Presentation of
results is extremely important so take time to determine the best way to present them.
Compare your data with theoretical or empirical results.
6. Discussion: Interpret the results of the experiment. This is arguably the most important
part of your report. Here you have the opportunity to show that you understand the
experiment. You must explain, analyze, and interpret your results. Discuss experimental
and theoretical results and why they do or do not agree. Explain any errors. Focus your
discussion on the following questions:
o
What results were expected? What results were obtained? If there were any
discrepancies, how can you account for them?
o
Do any of your results have particular technical or theoretical interest?
o
How do your results relate to your experimental objective(s)?
o
How do your results compare to those obtained in similar investigations?
o
What are the strengths and limitations of your experimental design?
o
If you encountered difficulties in the experiment, what were their sources? How
might they be avoided in future experiments?
7.
Conclusion: Present the conclusions you draw from the results. All conclusions should
be clearly stated and supported with evidence. Cite specific results and observations from
the experiment and tie them to your conclusions. Summarize reasons for any
disagreement between your results and the expected results. Recommend ways to correct
problems that may have led to discrepancies or bad data points. Recommend any
practical way of improving the experiment.
8.
References: Provide bibliographical information for any material that is not original that
you cited in your report. For example: technical specifications, equations, tables, or
figures used from another source.
9.
Appendices: Appendices should include raw data, calculations, graphs, and other
quantitative materials that were part of the experiment, but not detailed in any of the
above sections. Refer to each appendix at the appropriate point (or points) in your report.
For example, at the end of your results section, you might have the note, “See Appendix
A: Raw Data”. Appendices are optional, and are useful for keeping the body of the main
report from becoming hard to read.
3
LABORATORY REPORT WRITING SUGGESTIONS
1. Write for a specific audience. This will dictate how much background information and
detail need to be provided.
2. Be brief, yet complete.
3. Use quantitative information wherever possible in results, conclusions, and
discussion. Avoid imprecise words such as “good, bad, less, more” without numerical
support.
4. When referring to equipment and circuits, include diagrams so the reader can
understand what is going on.
5. Do not be intentionally wordy. Use short sentences most of the time with an elegant
long one at times for variety.
6. Be definite and forceful in your reporting.
7. Put topics in the correct section, for example, do not put discussion topics in the Results
section.
8. The experimental and reporting activities are part of your preparation for professional
engineering practice; avoid references to instructors or lab handouts in your report. Write as
if this is your original work.
9. Engineering reports are almost always written in the third person singular because it is
the results of the experiment that are important to communicate, not who performed
the experiment. Therefore, avoid using “I” or “we”, except, perhaps, for an opinion
expressed in the discussion.
10. Write in the past tense.
11. Avoid awkward possessive (“the machine’s controller).
12. Avoid contractions (“it would’ve”).
13. Avoid slang expressions (“instrument was hooked up to”, “thermometer was stuck into”).
14. As much as possible use active rather than passive language.
15. Number every page of the report sequentially.
4
Gauge Factor (GF)
Vs (Supplied Voltage)
2
5
Dimensions
Base (mm)
Height (mm)
Length (mm)
c
Moment of Inetia (mm^4)
Acrylic Beam
37.6
5.3
190
2.65
466.4812667
27
3.2
200
1.6
73.728
28.5
1.6
200
0.8
9.728
Polycarbonate Beam
Aluminum Beam
Acrylic Beam
Mass (g)
Polycarbonate Beam
Vo (Voltage Output)Mass (g)
Aluminum Beam
Vo (Voltage Output) Mass (g)
100 1.5 mV
50 1.5 mV
100
200 3 mV
100 3.3 mV
200
300 5 mV
150 4.5 mV
300
400 7 mV
200 5.8 mV
400
500 9 mV
250 6.8 mV
500
t of Inetia (mm^4)
Acrylic moment (N.mm)
Polycarbonated moment (N.mm)
Vo (Voltage Output)
0.8 mV
186200
98000
1.9 mV
372400
196000
3.3 mV
558600
294000
3.8 mV
744800
392000
4.8 mV
931000
490000
Aluminum moment (N.mm)
Acrylic Normal Stress (Mpa)
Polycarbonated Normal Stress (Mpa)
196000
1057.770237
2126.736111
392000
2115.540474
4253.472222
588000
3173.310711
6380.208333
784000
4231.080948
8506.944444
490000
5288.851185
10633.68056
Aluminum Normal Stress (Mpa)
16118.42105
32236.84211
48355.26316
64473.68421
40296.05263

Purchase answer to see full
attachment

DescriptionStrain Gage on a Cantilever Beam. Experiment # 2 SAFETY FIRST! 1. Must not wear open-toed shoes. Objectives 1. Familiarity with the underlying principles of electrical resistance strain gages. 2. To set up a Wheatstone half-bridge circuit. 3. To measure strain with a strain gage in three cantilevered beams (acrylic, aluminum, steel) loaded at the free end with a range of masses. 4. Calculate the maximum normal strain and Young’s modulus for the three beams, and compare the calculated values with the theoretical values. Background Strain Gage: The basic definition of axial strain is the change in length divided by the original length. Strain may be measured with mechanical, optical, electrical, and acoustical strain gages. This lab focuses on electrical resistance strain gages. When thin foil or wire is stretched, its resistance increases. The axial strain, , experienced by the material is related to the change in its resistance, R, by the following equation: = 1 R GF R where R is the original resistance and GF is the strain gage factor. This resistance may be measured directly with a digital multimeter. However, since the change in resistance is small, the resolution in determining the strain is also small. To obtain a more robust measurement of strain, a Wheatstone half-bridge, as shown in Figure 1, is used. Two resistors and two strain gages are used to complete the circuit. The Wheatstone halfbridge strain gage produces a resistance change that is linearly proportional to axial strain. Vs is the supply voltage (in our case, 5 volts) and Vo is the output voltage, which will be measured after the material is loaded. For the half-bridge shown, the axial strain is given as: = 2 V0 GF Vs where GF is the strain gage factor. Note that the strain is independent of the resistor values in the circuit. With the variable resistor one should be able to come close to balancing the circuit when it is not loaded, i.e., when Vo = 0 →  = 0. 1 R2 Strain Gage R3 == + Vs – = – Vo + R4 (Variable Resistor) R1 Strain Gage Figure 1: Wheatstone half-bridge. For more information, please study the following resources • • National Instruments publication on strain gages (http://www.ni.com/whitepaper/3642/en/) Lab handout with more information on the Wheatstone bridge from Penn State (https://www.mne.psu.edu/cimbala/me345/Lectures/Strain_gages.pdf) Cantilever Beam Bending: For a cantilevered beam with a point load applied at the free end, the normal stress at the outermost surface of the beam, , is Mc I Hère M represent the internal moment that can be calculated from the applied load, c is the distance from the neutral axis of the beam to the outer surface, and I is the moment of inertia of the beam’s cross-section about the centroid. The centroid is the geometric center of an area and corresponds to the neutral axis for bending in the elastic region. = 2 For composite geometries: x A x = A i i i i i and y A y = A i i i i i The moment of inertia is a geometric property that determines the ability of the beam to resist rotation or bending. y b y bh3 I = I x = 12 h x centroid o x Figure 2: Cross-section view of beam. From Hooke’s Law, the strain, , at the surface is equal to =  E . Test Procedure Before the test 1. Set up the Wheatstone bridge, as shown in Figure 1, on the Bread board. Make sure to turn off the power when constructing the Wheatstone half-bridge to avoid short circuit. Two strain gages representing R1 and R2 will already be glued to the top and bottom faces of the beams. Use an input voltage of 5 V. 2. Use a ruler and Vernier calipers to measure the dimensions of the beam crosssection, and the location of the strain gage with respect to the applied load. These measurements will be used to calculate the moment of inertia and the internal bending moment at the strain gage location. 3. Clamp the beam to the countertop 10 mm from the strain gages and then balance the Wheatstone half-bridge using the potentiometer so that the output voltage V0 is as close to zero as possible. 3 During the test 1. Apply loads from 100 g to 500 g in 100 g increments to the end of the beam. *Be careful to not break the acrylic beam; use your judgment regarding the appropriate maximum load. 2. Record the applied load and the output voltage for each load. Repeat this procedure for the other beams. Report A full lab report is to be submitted. The report should include the following in the Results section: a) A table for each beam that includes load, output voltage, and experimental strain. b) A table for each beam that includes the internal bending moments, theoretical stress, and theoretical strain at each load. Use the theoretical Young’s modulus for the theoretical strain calculation. c) A plot of the theoretical stress versus experimental strain for each beam. Use regression analysis in Excel or MATLAB to draw a best-fit straight line through the data points. From the regression analysis, determine the experimental Young’s modulus for the three beams used in this lab. d) A plot of experimental versus theoretical values of strain for each beam. e) Show a sample calculation The report should include a discussion about strain measurement uncertainty as well as a comparison of the theoretical and experimental results found in this lab (strain and Young’s modulus). Reference Holman, J.P., Experimental Methods for Engineers, 7th ed,. McGraw-Hill, New York, 2001. 4 MECHANICS OF MATERIALS LAB LAB REPORT GUIDELINES 1 LABORATORY REPORT PREPARATION The ability to communicate clearly both orally and in writing is of great importance to professional engineers. After graduation you will spend a good part of your time explaining your ideas and points of view both to your superiors and to the technicians under your supervision. You are, therefore, advised to work on improving your written communication skills, and the preparation of laboratory reports provides you with an excellent opportunity to do so. Unlike a newspaper article, a technical report has a certain standard format that must be adhered to. In addition, the style of technical and scientific writing is different from that used in, say, books of literature. Colloquial language should not be used; instead, explain your ideas in simple and grammatically correct English using clear and short sentences. Try to get to the point directly and avoid unnecessary elaborations and word text. Also, it is a well-established tradition in scientific writing to report in the third person (i.e., avoid using “I” and “we”). Note on verb tense: The experiment is already finished. Use the past tense when talking about the experiment. The report still exists; use the present tense when talking about the report. Note on graphics: Figures and Tables are used in technical reports (not Graphs and Charts). Figure captions should be numbered consecutively and placed under each figure. Table captions should be placed above each table. Full Reports The audience for this type of report is a practicing mechanical engineer who knows something about the topic, but has probably forgotten much of it so he or she needs to have the relevant material reviewed. It should be neat, legible, well organized, and include the following: 1. Cover Sheet: Report title, names of all group members, course number, and date. 2. Abstract: The abstract is an executive summary that briefly describes the experiment and states the main findings. It summarizes the entire report in one main paragraph. Your abstract should emphasize the objective, procedure, results, and significance. Be precise and specific. A technical report is not a mystery novel – state your conclusion as soon as possible! 3. Introduction: A brief introduction that explains the purpose of the experiment. The introduction should include any other introductory/background information or theory that the reader needs to know. This is where you tell the reader what you did and why you did it. 4. Methods and Materials: In paragraph form, describe the steps taken to perform the experiment, measurement techniques, and the apparatus. Include photos or diagrams of the apparatus as appropriate. Use your own words. Do not copy the procedure from the lab handout. This is where you tell the reader how you did the experiment and you describe the equipment and materials used to conduct the experiment. You should provide enough information so that another researcher in your field could use your description to replicate the experiment. 2 5. Results: Present your results to the reader. Although results are usually presented quantitatively, you should always introduce each block of information with simple clear language. Include measured results, an estimate of the experimental uncertainty, and any calculations used. In most cases it is sufficient to provide a sample calculation with clear explanation of the equations. Use tables and figures as necessary. All tables and figures should be labeled with a Figure/Table number and a descriptive caption. Presentation of results is extremely important so take time to determine the best way to present them. Compare your data with theoretical or empirical results. 6. Discussion: Interpret the results of the experiment. This is arguably the most important part of your report. Here you have the opportunity to show that you understand the experiment. You must explain, analyze, and interpret your results. Discuss experimental and theoretical results and why they do or do not agree. Explain any errors. Focus your discussion on the following questions: o What results were expected? What results were obtained? If there were any discrepancies, how can you account for them? o Do any of your results have particular technical or theoretical interest? o How do your results relate to your experimental objective(s)? o How do your results compare to those obtained in similar investigations? o What are the strengths and limitations of your experimental design? o If you encountered difficulties in the experiment, what were their sources? How might they be avoided in future experiments? 7. Conclusion: Present the conclusions you draw from the results. All conclusions should be clearly stated and supported with evidence. Cite specific results and observations from the experiment and tie them to your conclusions. Summarize reasons for any disagreement between your results and the expected results. Recommend ways to correct problems that may have led to discrepancies or bad data points. Recommend any practical way of improving the experiment. 8. References: Provide bibliographical information for any material that is not original that you cited in your report. For example: technical specifications, equations, tables, or figures used from another source. 9. Appendices: Appendices should include raw data, calculations, graphs, and other quantitative materials that were part of the experiment, but not detailed in any of the above sections. Refer to each appendix at the appropriate point (or points) in your report. For example, at the end of your results section, you might have the note, “See Appendix A: Raw Data”. Appendices are optional, and are useful for keeping the body of the main report from becoming hard to read. 3 LABORATORY REPORT WRITING SUGGESTIONS 1. Write for a specific audience. This will dictate how much background information and detail need to be provided. 2. Be brief, yet complete. 3. Use quantitative information wherever possible in results, conclusions, and discussion. Avoid imprecise words such as “good, bad, less, more” without numerical support. 4. When referring to equipment and circuits, include diagrams so the reader can understand what is going on. 5. Do not be intentionally wordy. Use short sentences most of the time with an elegant long one at times for variety. 6. Be definite and forceful in your reporting. 7. Put topics in the correct section, for example, do not put discussion topics in the Results section. 8. The experimental and reporting activities are part of your preparation for professional engineering practice; avoid references to instructors or lab handouts in your report. Write as if this is your original work. 9. Engineering reports are almost always written in the third person singular because it is the results of the experiment that are important to communicate, not who performed the experiment. Therefore, avoid using “I” or “we”, except, perhaps, for an opinion expressed in the discussion. 10. Write in the past tense. 11. Avoid awkward possessive (“the machine’s controller). 12. Avoid contractions (“it would’ve”). 13. Avoid slang expressions (“instrument was hooked up to”, “thermometer was stuck into”). 14. As much as possible use active rather than passive language. 15. Number every page of the report sequentially. 4 Gauge Factor (GF) Vs (Supplied Voltage) 2 5 Dimensions Base (mm) Height (mm) Length (mm) c Moment of Inetia (mm^4) Acrylic Beam 37.6 5.3 190 2.65 466.4812667 27 3.2 200 1.6 73.728 28.5 1.6 200 0.8 9.728 Polycarbonate Beam Aluminum Beam Acrylic Beam Mass (g) Polycarbonate Beam Vo (Voltage Output)Mass (g) Aluminum Beam Vo (Voltage Output) Mass (g) 100 1.5 mV 50 1.5 mV 100 200 3 mV 100 3.3 mV 200 300 5 mV 150 4.5 mV 300 400 7 mV 200 5.8 mV 400 500 9 mV 250 6.8 mV 500 t of Inetia (mm^4) Acrylic moment (N.mm) Polycarbonated moment (N.mm) Vo (Voltage Output) 0.8 mV 186200 98000 1.9 mV 372400 196000 3.3 mV 558600 294000 3.8 mV 744800 392000 4.8 mV 931000 490000 Aluminum moment (N.mm) Acrylic Normal Stress (Mpa) Polycarbonated Normal Stress (Mpa) 196000 1057.770237 2126.736111 392000 2115.540474 4253.472222 588000 3173.310711 6380.208333 784000 4231.080948 8506.944444 490000 5288.851185 10633.68056 Aluminum Normal Stress (Mpa) 16118.42105 32236.84211 48355.26316 64473.68421 40296.05263 Purchase answer to see full attachment

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