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Table of Contents1.0 Abstract......................................................................................................................................3
2.0 Introduction................................................................................................................................4
3.0 Theory........................................................................................................................................5
4.0 Apparatus...................................................................................................................................9
5.0 Procedure...................................................................................................................................9
6.0 Results......................................................................................................................................10
7.0 Discussion................................................................................................................................14
8.0 Conclusions..............................................................................................................................16
9.0 References................................................................................................................................17
Appendix I.....................................................................................................................................18
Appendix II....................................................................................................................................22
1.0 Abstract
This beam bending experiment was performed using the Vishay ‘FLEXOR’ Cantilever Flexure Frame as well as the B.L.H. Digital Strain Indicator located in the Engineering Department at UNBSJ.
Aluminum, Oak and Pine specimens were tested to determine the modulus of elasticity using both a strain-based method, as well as calculating the modulus of elasticity using the deflection of the specimen.
Through calculations, it was determined that the Aluminum specimen, as expected, held a much higher modulus of elasticity than either Oak or Pine. Calculated results for each specimen also resulted in very similar values to published values for each material, with strain method results all being within 1 GPa of their respective published values.
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2.0 IntroductionBeam bending is an important design consideration in the design and construction of any
load bearing structure, including structures only charged with bearing their own weight. Whether the load is constant or variable, static or dynamic, pointed or distributed, we are able to calculate the actual load applied on various members of the structure and plan accordingly.
Deflection in a beam is caused by the presence of an internal moment, with deflection occurring along the direction of action of the moment. This is pivotal in building support structures, as excessive bending must be avoided.
This beam bending experiment was performed on the 25th of November, 2013 to observe elastic beam buckling in pine, oak and aluminum under various load increments, and to compare obtained results to theoretical values.
The main objectives of the Beam Bending Laboratory were to determine the linear correlation of load applied to a beam and the internal strain experienced, as well as exploring the differences between metallic and wooden beams. Similarities were observed in the two wooden beams, and comparatively high tensile strength was observed in the aluminum.
Please note that a full colour version of this report is available on the internet at http://bit.ly/18wOW5d.
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3.0 Theory
Structural elements such as beams deflect when a load is applied. Deflection is measured as a distance or an angle. The load determines how and how much the beam deflects. There are many different types of loading such as point, uniformly distributed, and uniformly varying loads.
A point load is a load is applied at a certain point on the beam. Common examples are at the center or end of a beam. A uniformly distributed load is where a load is applied evenly over the whole beam. A uniformly varying load is where a load is increasing or decreasing over the length of the beam. Beams may have more than just one of these load cases applied.
Point Load
Figure 1 – A beam with a point load applied at the center
1
Uniformly Distributed Load
Figure 2 – A beam with a uniformly distributed load applied
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1 "Civil Engineering." : TYPES OF BEAMS & TYPES OF LOADINGS. N.p., n.d. Web. 1 Dec. 2013. <http://civilengineerworks.blogspot.ca/2011/12/types-of-beams-types-of-loadings.html>.2 "Uniformly Distributed Load on a Beam." efunda.com. N.p., n.d. Web. 1 Dec. 2013. <http://www.efunda.com/glossary/formulas/beams/images/diagram_ss_uniform_1s.gif>.
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Uniformly Varying Load
Figure 3 – A beam with a uniformly varying load applied
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The amount of deflection can be calculated using three different methods. The first method is slopes by integration. The second method is known as superposition. Superposition is when “the deflections for a series of separate loadings acting on a beam may be superimposed. For example, if v1 is the deflection for one load and v2 is the deflection for another load, the total deflection for both loads acting together if the algebraic sum v1 + v2.”4 The third method to calculating the deflection is the moment area method. This method is a somewhat graphical way to find the deflection. Bending moment diagrams are used.
The rearranged formulae used to obtain calculated values for Modulus of Elasticity are as follows:
3 "Civil Engineering." : TYPES OF BEAMS & TYPES OF LOADINGS. N.p., n.d. Web. 1 Dec. 2013. <http://civilengineerworks.blogspot.ca/2011/12/types-of-beams-types-of-loadings.html>.4 Hibbeler, Russell. Mechanics of Materials + Masteringengineering With Pearson Etext Standalone Access Card. USA: Pearson College Div, 2013. Print.
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Calculating the Modulus of Elasticity using a cantilever beam is somewhat more straightforward, where the following formula applies:
Equation 1:
E=6 P L1
b t 2∗ϵ
Where: E = Modulus of Elasticity P = Point Load L1 = Length from Strain Gauge to Point Load b = Base, or Width of Beam t = Thickness of Beam
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- Illustration of deflection setup, depicting P and L1.
5 (Prof. Byron Walton, 2013)
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By Deflection:
Equation 2:
E=(PY )( X3−3L2∗X2 )
[6 ( b t3 )12 ]
Where:
E = Modulus of Elasticity P = Point Load Y = Deflection X = Deflection (measured at Micrometer) L2 = Distance from Fixed end to Location of Point Load b = Base, or Width of Beam t = Thickness of Beam
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- Illustration of deflection setup, depicting P, Y, X, and L2.
6 (Prof. Byron Walton, 2013)
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4.0 Apparatus
Vishay FLEXOR Cantilever Flexure Frame B.L.H. Digital Strain Indicator Vernier Calipers Set of Weights with Hanger Meter Stick 1/1000 Inch Precision Micrometer Prepared Specimen Beams of Aluminum, Pine, and Oak
5.0 Procedure
Cross-Sectional dimensions of each specimen were found, as well as the following lengths:o L1: Strain Gauge to Point Loado L2: Support Point to Point Loado X: Support Point to Screw Micrometer
Beam was placed into the Cantilever Flexure Frame for the group by lab coordinator. Hanger was hung from end of beam. A zero reading was taken on the screw micrometer, and the strain gauge was set at zero. 1N weight was placed on the hanger. Strain measurement was recorded, as well as a deflection measurement on the micrometer. Weight placement was repeated with measurements taken to a total of 6N for wooden
beams, 9N for aluminum beam.
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6.0 Results
The experimental values for the modulus of elasticity for all three samples using both the deflection and strain methods can be found in figure 6.0.1. Figures 6.0.2 to 6.0.4 are plots of the strain method used in this experiment derived from equation 2, where 6PL1 is the y-axis and bt2ϵ is the x-axis. The slope of the line of best fit is the modulus of elasticity. Figures 6.0.5 to 6.0.7 are plots of the deflection method, where the point load, P is plotted against deflection at the deflection gauge, yGAUGE. The slopes of these plots are multiplied by a function of the area of the moment divided by six times the moment of inertia, as seen in equation 1. Sample calculations for determining the modulus of elasticity using the deflection method can be found in section 6.1. Raw measurement data can be found in Appendix I.
Experimental Moduli of Elasticity (GPa)
Specimen Deflection Method Strain MethodAluminum 66.838 70.341Oak 9.317 10.054Pine 8.143 9.396
Figure 6.0.1
0.000E+00 5.000E-05 1.000E-04 1.500E-04 2.000E-040
2000400060008000
10000120001400016000
f(x) = 70340741.3017554 x + 514.61502014375R² = 0.99768033553854
Strain Deflection in Aluminum
bt^2*ϵ
6PL
Figure 6.0.2 – Plot of the strain measurement method for Aluminum
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0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-030
100020003000400050006000700080009000
10000
f(x) = 10054096.0462177 x − 1.25956614007009R² = 0.999986817310936
Oak By Strain Measurement
bt^2*ϵ
6PL
Figure 6.0.3 – Plot of the strain measurement method for oak wood.
0.000E+00 2.000E-04 4.000E-04 6.000E-04 8.000E-04 1.000E-03 1.200E-030
100020003000400050006000700080009000
10000
f(x) = 9396853.67883706 x + 13.8500367529123R² = 0.999979258116825
Pine By Strain Measurement
bt^2*ϵ
6PL
Figure 6.0.4 – Plot of the strain measurement method for pine wood.
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0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.8000123456789
10
f(x) = 5.29612186165098 x + 0.260666294622856R² = 0.998373537025239
Aluminum By Deflection Measurement
Deflection at Gauge (mm)
Forc
e (N
)
Figure 6.0.5 – Plot of the deflection measurement method for aluminum.
0.000 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.0000
1
2
3
4
5
6
7
f(x) = 0.366123649321326 x + 0.00829905463849912R² = 0.999983062715642
Oak By Deflection Measurement
Deflection at Gauge (mm)
Forc
e (N
)
Figure 6.0.6 – Plot of the deflection measurement method for oak wood.
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0.000 5.000 10.000 15.000 20.000 25.0000
1
2
3
4
5
6
7
f(x) = 0.313677050880953 x + 0.0337891470566265R² = 0.999908335093213
Pine By Deflection Method
Deflection at Gauge (mm)
Forc
e (N
)
Figure 6.0.7 – Plot of the deflection measurement method for pine wood.
6.1 Sample CalculationsThe following is an example calculation to determine the modulus of elasticity using
equation 2. Experimental values from the test on the aluminum sample will be used.
E=( PyGAUGE ) x3−3 L2 x2
6 I=( P
yGAUGE ) (254 mm)3−3 (287 mm)(254 mm)2
6(517.171mm4)
E=12630.33( PyGAUGE )=12620.33 (5.2961 )
E=66.839GPa
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7.0 Discussion
This experiment to determine the moduli of elasticity via the strain and deflection methods for beams went very well. The results were very close to the theoretical values and most of the sources of error were strictly measurement errors. The theoretical modulus of elasticity for each material was found at “http://www.engineeringtoolbox.com/young-modulus-d_417.html”, however the type of aluminum and wood samples was unknown. A general value for the modulus of elasticity of any aluminum is approximately 69 GPa, for oak it is 11 GPa and for pine it is 9 GPa.
In each case, the strain method was closer to the actual values than the deflection method, and the deflection method consistently gave a lower modulus of elasticity than both the strain method and the actual modulus of elasticity. The deflection method for aluminum was 3% lower than the theoretical value of 69 GPa and the strain method was 1.9% higher. In the case of the oak specimen, the deflection method yielded 9.3 GPa, a 15% lower result than the actual value of 11 GPa, whereas the strain method resulted in 10 GPa, 8.6% higher than theoretical. The pine sample’s experimental modulus of elasticity from the deflection method was 9.5% lower than the actual value of 9 GPa while the strain method yielded a 4.4% higher result at 9.4 GPa.
The variation in the results is almost exclusively due to measurement errors. The electronic strain meter’s numbers could never be completely stabilized so an average was taken for each result. This could be partially attributed to the weights applying varying force due to the slight motion of the weights. Another possibility is that the measurements of the cross-sectional area and the distances for the deflection method were not accurate. The ruler went only to millimeters which could have caused variation in the results. Another variation from the published values is that the exact type of material is unknown. The published moduli of elasticity presented are a general representation of the material, however for aluminum it ranges from 68.9 GPa for 2014-T6 to 73.1 GPa for 6061-T6. This is also the case for the wood samples. The exact species and dryness of the samples are unknown so the generally recognized values of 11 GPa for oak and 9 GPa for pine has been used, which could lead to skewed results.
It is interesting that the strain method consistently yielded higher results than the deflection method. The results from the deflection method indicate that the beam is more flexible than it actually is. This could be due to not correctly measuring the deflection. A higher greater than actual deflection would result in a lower modulus of elasticity. It could also be that the strain gauge is located closer to the support and impurities in the samples would not affect the reading as much as the deflection method reading.
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Experimental Moduli of Elasticity (GPa)Specimen Deflection Method Strain Method Published Value
Aluminum 66.838 70.341
69
Oak 9.317 10.054 11Pine 8.143 9.396 9
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8.0 ConclusionsThe beam bending laboratory session was generally successful, as observed values for
modulus of elasticity were very close to published values, falling within between 1.9% and 9.0%. The objectives of this experiment were fulfilled. It was determined that the experimental values for the Modulus of Elasticity by strain method compared to their respective published values are as follows:
- For Aluminum: 70.341GPa Experimentally, compared to 69 GPa Published- For Oak: 10.054 GPa Experimentally, compared to 11 GPa Published- For Pine: 9.396 GPa Experimentally, compared to 9 GPa Published
While we are generally pleased with the observed results in Strain Method, the Deflection Method results are subject to more uncertainties and possibilities for error, contributing to the expected and observed loss of accuracy.
Group 2 is confident that the laboratory session was completed successfully, as evidenced by observed values closely shadowing published values.
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9.0 References
"Civil Engineering." : TYPES OF BEAMS & TYPES OF LOADINGS. (2011, 12 01). Retrieved 12 01, 2013, from blogspot.ca: <http://civilengineerworks.blogspot.ca/2011/12/types-of-beams-types-of-loadings.html
blogspot.ca. (n.d.). Civil Engineering." : TYPES OF BEAMS & TYPES OF LOADINGS. Retrieved 12 01, 2013, from blogspot.ca: <http://civilengineerworks.blogspot.ca/2011/12/types-of-beams-types-of-loadings.html
efunda.com. (n.d.). "Uniformly Distributed Load on a Beam.". Retrieved 12 01, 2013, from efunda.com: <http://www.efunda.com/glossary/formulas/beams/images/diagram_ss_uniform_1s.gif>.
Hibbeler, R. (2014). Mechanics of Materials (9 ed.). Montreal, Quebec: Pearson Prentice Hall.
Prof. Byron Walton, U. S. (2013). CE2023 Lab: Plastic Bending of a Beam. Saint John, NB: UNBSJ.
wikipedia.org. (n.d.). wikipedia.org: Deflection (Engineering). Retrieved 12 01, 2013, from wikipedia.org: http://en.wikipedia.org/wiki/Deflection_(engineering)
16
Appendix I
Measured Dimensions of Test SpecimensSpecimen
base(b, mm)
Thickness (t, mm)
Mom. of Inert. (I, mm^4)
Length 1 (L1, mm)
Length 2 (L2, mm) x (mm)
Aluminum 25.42 6.250 517.171 248.000 287.000 254.000Oak 24.5 5.000 255.208 253.000 286.000 254.000Pine 24.6 4.950 248.639 253.000 286.000 253.000
AluminumBy Deflection Measurement: By Strain Measurement
Loading (N)
y(in) (mm) y(final) (mm) y(gauge) (mm)
Strain (μmm/mm) 6PL1 bt^2*ϵ
0 0.61 0.610 0.000 0.000E+00 00.000E+
00
1 0.61 0.737 0.127 1.100E-08 14881.092E-
05
2 0.61 0.914 0.304 3.200E-08 29763.178E-
05
3 0.61 1.092 0.482 5.600E-08 44645.561E-
05
4 0.61 1.328 0.718 7.500E-08 59527.447E-
05
5 0.61 1.499 0.889 9.800E-08 74409.731E-
05
6 0.61 1.702 1.092 1.220E-07 89281.211E-
04
7 0.61 1.880 1.270 1.420E-07 104161.410E-
04
8 0.61 2.083 1.473 1.640E-07 119041.628E-
04
9 0.61 2.261 1.651 1.850E-07 133921.837E-
04
17
OakBy Deflection Measurement: By Strain Measurement
Loading (N)
y(in) (mm) y(final) (mm) y(gauge) (mm)
Strain (μmm/mm) 6PL1 bt^2*ϵ
0 6.8072 6.807 0.000 0.000E+00 00.000E+
00
1 6.8072 9.474 2.667 2.480E-07 15181.519E-
04
2 6.8072 12.268 5.461 4.910E-07 30363.007E-
04
3 6.8072 14.986 8.179 7.400E-07 45544.533E-
04
4 6.8072 17.704 10.897 9.890E-07 60726.058E-
04
5 6.8072 20.420 13.613 1.230E-06 75907.534E-
04
6 6.8072 23.190 16.383 1.480E-06 91089.065E-
04
PineBy Deflection Measurement: By Strain Measurement
Loading (N)
y(in) (mm) y(final) (mm) y(gauge) (mm)
Strain (μmm/mm) 6PL1 bt^2*ϵ
0 2.616 2.616 0.000 0.000E+00 00.000E+
00
1 2.616 5.639 3.023 2.640E-07 15181.591E-
04
2 2.616 8.839 6.223 5.310E-07 30363.201E-
04
3 2.616 11.989 9.373 8.000E-07 45544.822E-
04
4 2.616 15.291 12.675 1.072E-06 60726.462E-
04
5 2.616 18.466 15.850 1.341E-06 75908.083E-
04
6 2.616 21.666 19.050 1.603E-06 91089.662E-
04
18
12630.33
66.838 Gpa
25448.39.317 Gpa
25958.28
8.143 GPa
19
0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.8000123456789
10
f(x) = 5.29612186165098 x + 0.260666294622856R² = 0.998373537025239
Aluminum By Deflection Measurement
Deflection at Gauge (mm)
Forc
e (N
)
0.000 2.000 4.000 6.000 8.000 10.000 12.000 14.000 16.000 18.0000
1
2
3
4
5
6
7
f(x) = 0.366123649321326 x + 0.00829905463849912R² = 0.999983062715642
Oak By Deflection Measurement
Deflection at Gauge (mm)
Forc
e (N
)
0.000 5.000 10.000 15.000 20.000 25.00001234567
f(x) = 0.313677050880953 x + 0.0337891470566265R² = 0.999908335093213
Pine By Deflection Method
Deflection at Gauge (mm)
Forc
e (N
)
20
0.000E+00 5.000E-05 1.000E-04 1.500E-04 2.000E-040
2000400060008000
10000120001400016000
f(x) = 70340741.3017554 x + 514.61502014375R² = 0.99768033553854
Strain Deflection in Aluminum
bt^2*ϵ
6PL
0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-030
100020003000400050006000700080009000
10000
f(x) = 10054096.0462177 x − 1.25956614007009R² = 0.999986817310936
Oak By Strain Measurement
bt^2*ϵ
6PL
0.000E+00 2.000E-04 4.000E-04 6.000E-04 8.000E-04 1.000E-03 1.200E-030
100020003000400050006000700080009000
10000
f(x) = 9396853.67883706 x + 13.8500367529123R² = 0.999979258116825
Pine By Strain Measurement
bt^2*ϵ
6PL
Appendix II
- Cantilever Beam Bending Apparatus
21
- Digital Strain Indicator
- Aluminum Specimen, with Strain Gauge, Loaded in Frame
22
- Micrometer Measuring Location of Beam’s Surface
23
- Unloaded Hanger at Point-Load Position on Aluminum Beam
24
