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Page 1 of 21 Ian Ip 10011223 Lab #1: Cantilevered Beams March 1, 2012 Ian Ip 10011223 Section 5 Email: 0imhi@queensu Lab partner: Blair Hanbury APSC100 Module 2 ABSTRACT An experiment was designed and conducted to investigate the properties of a cantilevered aluminum beam. The beam went under testing in an established experimental set up with one end anchored to a rigid support, and the other side unsupported and point mass M attached onto it. The experiment aimed to determine the specific deflection equation to the aluminium beam by solving for the power m and n in the equation. The values m and n were established to be 1 and 3 respectively. Once the integer values to the power were obtained, it was used to find the Young’s modulus of the aluminium beam using each of the two methods available. Method one involves using the slope of the d vs. l 3 plot and method two made use of the intercept of the Ln (d) vs. Ln (l) plot. The E values from method 1 and 2 are 610 10 ± 210 10 Pa and510 10 ± 110 10 Pa. Both values agree and confirmed the accuracy of the experiment; the two values are also inside the range of

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Page 1: APSC 100 Module 2 Lab 1 - Cantilevered Beams.docx

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Lab #1: Cantilevered Beams

March 1, 2012

Ian Ip10011223Section 5

Email: 0imhi@queensu

Lab partner: Blair Hanbury

APSC100 Module 2

ABSTRACT

An experiment was designed and conducted to investigate the properties of a cantilevered aluminum beam. The beam went under testing in an established experimental set up with one end anchored to a rigid support, and the other side unsupported and point mass M attached onto it. The experiment aimed to determine the specific deflection equation to the aluminium beam by solving for the power m and n in the equation. The values m and n were established to be 1 and 3 respectively. Once the integer values to the power were obtained, it was used to find the Young’s modulus of the aluminium beam using each of the two methods available. Method one involves using the slope of the d vs. l3 plot and method two made use of the intercept of the Ln (d) vs. Ln (l) plot. The E values from method 1 and 2 are 6∗1010±2∗1010Pa and5∗1010±1∗1010Pa. Both values agree and confirmed the accuracy of the experiment; the two values are also inside the range of accepted Young’s modulus values found a published academic source.

I verify that this formal report is my own individual work and has not been copied in whole or in part from another source (with the possible exception of equations, tables and/or diagrams from the experimental descriptions on the APSC100-2 website). Furthermore, I have not and will not lend this report (electronic or hardcopy) to any other student, either now or in the future. Signed:__________________________

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1.0 Introduction

A cantilevered beam describes a rigid beam supported at one end and is free to move at the opposite end. A force known as a point load can then act at a point on the beam rather than distributed across the entire surface of the object. In response to the point load, the beam responds by a deflection on the unsupported end as shown in figure 1.

The stiffness or young’s modulus of the cantilevered beam can be determined by measuring the deflection, unsupported length, width, thickness of the cantilevered beam. All of these quantities can be related by the following equation defining the deflection of a rectangular beam. 1

d= 4 Pm ln

Ew t 3

Where d is the deflection of the blank in metre, P is the magnitude of the point load in Kg; l is the unsupported length, w is the width and t is the thickness of the beam all in metres. E is the young’s modulus. The power m and n can be determined by respectively plotting d vs. p and d vs. l.

There are two ways that the power m and n can be determined. The first method involves plotting d versus p and d versus l until a straight line is obtained. The second method involves plotting Ln(d) versus Ln(p) (or Ln(l)). In this method, a linear equation is obtained in the form of Ln(y) =m*Ln(x) +Ln(A) where m and n is the slope and the power (integer value) from equation 1 and Ln(A) is the y-intercept.

Figure 1: A simple explanation of point load and cantilevered beam in a diagram. (courtesy of APSC 100 Module 2)

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The power m and n along with measurements made on the beam can then be used to calculate E by rearranging equation 1.

2

E=4 Pm ln

dw t3

Measuring the Young’s modulus is extremely common in the discipline of civil engineering where civil engineers has to use the Cantilevered beam principles to determine the safety limit of load on a bridge.

2.0 Apparatus and Procedure

This experiment consist of two main parts to find the power m and n respectively and then using the different elements from equation 2 to find E. The basic set up of the experiment involves one end of the aluminium beam anchored in the support jig on the table and weights to be hung on to the unsupported end. A measurement pole is then placed next to the unsupported end to measure the deflection as shown in figure 2. Prior to starting part 1 of the experiment, measurements of the constant variables

were taken. The thickness and width of beam were measured using a Vernier caliper.

In part one, the objective is to determine the power m by making the length of the unsupported beam constant. By adding mass M to the unsupported end of the beam, we realize equilibrium is established and can be given by this following equation.

3

P=Mg

Where point load P is equal to the product of mass M and the gravitational accleration constant of 9.8m/s2. The defection d can be determined by taking the height of the beam with the point load exerted on the beam in reference to a reference point, subtracting that by the initial height of the beam with no point load in reference to the common reference point. The aim of part 1 is to vary load mass M from a range 100g to 900g on a cantilevered beam with an unsuported length of about 0.9m. As mass M varies, the deflection will change. Record the mass M and plot pointload P vs. deflection d using one of the two methods stated above and determine the interger value of the power m.

Part two of the experiment involves making measurements in similar style to that of part one to determine the power n. Instead of a constant length and variable mass in part one, part two make use of a constant mass (and effectively pointload) and vary the length of the unsupported beam. Using a load mass of 700g, the length of the unsupport beam range from 0.2m to 0.7m at intervals of 0.1. At

Figure 2: The basic experiment set up

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each length interval, the heightof the unsupported end without load mass is measured and subtracted from the measured height of the unsupported end with load mass attached to obtain deflection d. The recorded data of d and l will be plotted using one of the two methds stated aboce and the integer value of power n can be determined.

After obtaining power m and n, Young`s modulus E can now be calculated using equation 2. Using a graph with a straight line from part 2, we can determine the value of E by determining the slope of a d vs. power or l (n) graph. Otherwise, E can be obtained from the antilog of the intercept of a ln – ln plot.

3.0 Results and Analysis

From equation 1, it is important to note that some of the variables in the equation are constant throughout the lab.

Width w: 2.59*10-2 ± 5*10-4m Thickness t: 5.00*10-3 ± 5*10-4m Length l: 89.6*10-2 ± 5*10-4m

3.1 Part 1: Determining the power m in equation 1

The first part involves plotting the point load and deflection, as well as an Ln-Ln plot. The graphs are shown in figure 3 and 4 respectively. The initial height without any load mass was measured to be 85.2*10-2 ± 5*10^-4 m.

Figure 3: The d vs. p graph

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Figure 4: The Ln-Ln plot of d vs. p

In figure three, the tread line of the relationship is indeed linear and does not require figure 4 to help determine the integer value of m, although figure 4 was used to confirm that the figure 3 is indeed linear. The graph showed a linear relation without any alteration and therefore the power m is determined to be 1. Further regression analysis (refer to the appendix) verified the linear pattern.

3.2 Part 2: Determining power of n in equation 1

The second part involves plotting the unsupported length and deflection, as well as an Ln-Ln plot. The graphs are shown in figure 5 and 6 respectively. The constant load mass in the experiment was weighted to be 0.7kg.

First, a plot of l vs. d was plotted as shown in figure 5 and the linear relationship in part 1 was not seen in part 2.

Because d vs. l was not linear, a second graph was plotted. This time, the Ln of d and l were plotted and it yielded a linear relationship after altercation as shown in figure 7. From the slope value obtained from its tread line, it was clear that 3 was the value of n. To verify, d vs. l3 was plotted in figure 6and if n is indeed 3, the graph should be a straight line.

Figure 5: d vs. l plot

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Figure 6: The d vs. L^3 plot

Figure 7: The Ln-Ln plot of d vs. l

3.3 Part 3: Determine the Young’s modulus E of the beam

There are two E values that can be obtained using two different methods. The E values should be in close proximity to each other and can be calculated by each of the following equations derived from d= (slope)*ln and the plot of d vs. ln: (Note the length L originated from part 1)

4

E= 4 Ln

( slope )w t 3

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5

E= 4 ln

w t3 e∫¿¿ From the regression analysis of the d vs. l^3 plot in figure 6, the slope of the d vs. l^3 plot was determined to be 0.147 ± 0.004 and the intercept of the Ln-Ln plot was determined to be -1.81 ± 0.140.

3.31 Calculate E using equation 4

E= 4 l3

( slope )w t 3

E=4 (0.896 )3

0.147∗.0259∗( .005 )3

E=6∗1010±2∗1010Pa

3.32 Calculate E using equation 5

E= 4 l3

w t3 e∫¿¿

E=4 (0.896 )3

.0259∗( .005 )3∗e−1.81

E=5∗1010±1∗1010Pa

The final results with error calculations (located in the appendix) are listed in the conclusion.

4.0 Discussion

Overall, the lab was conducted in a timely and somewhat accurate manner due to the precision of the Vernier caliper. After obtaining the m and n integer values, the final version of equation one for the aluminum cantilevered beam used in the lab is:

6

d= 4 P1l3

Ew t3

The power m has an integer value of 1 and both the d vs. p showed a straight line and Ln (d) vs. Ln (p) graph returned with a slope of 1. The power n has an integer value of three; the Ln (d) vs. Ln (l) plot has

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a tread line slope value of 2.95, rounded upward to 3. To confirm, d vs. l3 was plotted and the plot it returned a linear relationship.

There are many advantages and disadvantages for using method 1 or method 2 to determine power. In method 1, it is easier if the integer value is 1. If it is greater than 1, it will take some time to increase the power of the x variable and finally obtain a linear graph. Method 1 however, does not require a regression analysis to determine the value. In method 2, the integer value is obtained by plotting a Ln – Ln plot and using regression analysis to determine the slope, which is in turn the integer value of m/n. While method 2 require more work and take a little more time, it is great in determining the integer value if it is greater than 1, and also to confirm the integer value obtained from a d vs. x^power plot. Another strategy for dealing with integer values that are greater than 1 is to plot the Ln –Ln plot. Once the slope is obtained, to plot a d vs. x to the slope plot to get a straight line, and therefore confirming the integer values are correct.

The two E values are within range of one another and within range to the published range of E values. Depending on composition, the Young’s modulus of aluminium ranges from 4.7*1010 to 7.4*1010. (MEMSNET, nd) Both E values obtained from the experiment fit into the mid of the published range.

5.0 Conclusion

In conclusion, the experiment used an aluminium cantilevered beam to determine the specific deflection equation. As well as calculating the Young’s modulus using two separate methods, both returned values that are in close proximity to each other and within range of the published values. The final equation was determined to be:

d= 4 P1l3

Ew t3

And the two E values are 6∗1010±2∗1010Pa from method 1 and 5∗1010±1∗1010Pa from method 2. Because of this, the E values are considered to be within range of each other and agree with each other because of the uncertainities.

6.0 Reference

Material: Aluminum (Al), film. (n.d.). MEMSnet. Retrieved March 4, 2012, from https://www.memsnet.org/material/aluminumalfilm/

Clapham, Lynann. APSC 100 Practical Engineering Module 2. Kingston: Queen's University, 2011. Print.

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7.0 Appendix

7.1 Part 1: Determine m

7.11 Table of values

Ln(d)

Ln(P) SLn(d)

-0.0202 -0.0202 0.0081970.672944 0.672944 0.006411.07841 1.07841 0.0053191.366092 1.366092 0.0044641.589235 1.589235 0.0039371.771557 1.771557 0.0034971.925707 1.925707 0.0031852.059239 2.059239 0.0028572.177022 2.177022 0.002674

Load Mass (Kg)

Distance from end of beam to ground (m)

Deflection (m)

Point Load (N)

0.1 0.835 0.061 0.980.2 0.818 0.078 1.960.3 0.802 0.094 2.940.4 0.784 0.112 3.920.5 0.769 0.127 4.90.6 0.753 0.143 5.880.7 0.739 0.157 6.860.8 0.721 0.175 7.840.9 0.709 0.187 8.82Initial Height(m)

85.2

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7.12 Regression Analysis from d vs. p plot

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.99944R Square 0.99888Adjusted R Square 0.99872Standard Error 0.001555Observations 9

ANOVA

df SS MS FSignificance

Regression 1 0.015105 0.015105 6244.22Residual 7 1.69E-05 2.42E-06Total 8 0.015122

CoefficientsStandard

Error t Stat P-value Lower 95%Intercept 0.046667 0.00113 41.30085 1.27E-09X Variable 1 0.01619 0.000205 79.02038 1.37E-11

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 0.062533 -0.001532 0.0784 -0.00043 0.094267 -0.000274 0.110133 0.0018675 0.126 0.0016 0.141867 0.0011337 0.157733 -0.000738 0.1736 0.00149 0.189467 -0.00247

0 1 2 3 4 5 6 7 8 9 10

-0.003-0.002-0.001

00.0010.0020.003

X Variable 1 Residual Plot

X Variable 1

Resid

uals

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7.13 Regression Analysis from Ln (d) - Ln (p) plot

SUMMARY OUTPUT

Regression StatisticsMultiple R 1R Square 1Adjusted R Square 1Standard Error 0Observations 9

ANOVA

df SS MS FSignificance

FRegression 1 4.138703 4.138703 #NUM! #NUM!Residual 7 0 0Total 8 4.138703

CoefficientsStandard

Error t Stat P-value Lower 95%Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 0 0 65535 #NUM! 0 0 0 0X Variable 1 1 0 65535 #NUM! 1 1 1 1

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 -0.0202 02 0.672944 03 1.07841 04 1.366092 05 1.589235 06 1.771557 07 1.925707 08 2.059239 09 2.177022 0

-0.5 0 0.5 1 1.5 2 2.50

0.20.40.60.8

1

X Variable 1 Residual Plot

X Variable 1Re

sidua

ls

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7.14 Error Propagation

Vertical:S¿

7.2 Part 2: Determine n

7.21 Table of values

Unsupported Length (m)

Height without Mass (m)

Height with Mass (m)

Deflection (m)

Length^3

0.2 0.872 0.871 0.001 0.0080.3 0.871 0.865 0.006 0.0270.4 0.868 0.854 0.014 0.0640.5 0.867 0.842 0.025 0.1250.6 0.864 0.829 0.035 0.2160.7 0.861 0.811 0.05 0.3430.8 0.859 0.781 0.078 0.5120.9 0.85 0.74 0.11 0.729

Ln(d) Ln(l) SLn(d) SLn(l)

-6.90776 -1.60944 0.0000005 0.0001

-5.116 -1.20397 0.000003 0.00015

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-4.2687 -0.91629 7E-06 0.0002

-3.68888 -0.69315 0.0000125 0.00025

-3.35241 -0.51083 0.0000175 0.0003

-2.99573 -0.35667 0.000025 0.00035

-2.55105 -0.22314 0.000039 0.0004

-2.20727 -0.10536 0.000055 0.00045

7.22 Regression Analysis from d vs. l3 plot

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.998125R Square 0.996253Adjusted R Square 0.995628Standard Error 0.002505Observations 8

ANOVA

df SS MS FSignificanc

e F

Regression 1 0.0100090.01000

9 1595.14 1.65E-08Residual 6 3.76E-05 6.27E-06Total 7 0.010047

Coefficients

Standard Error t Stat P-value Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 0.002809 0.0012832.18986

90.07108

8 -0.000330.00594

8 -0.00033 0.005948

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

-0.004-0.002

00.0020.0040.006

X Variable 1 Residual Plot

X Variable 1

Resid

uals

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X Variable 1 0.146505 0.003668 39.9392 1.65E-08 0.1375290.15548

1 0.137529 0.155481

RESIDUAL OUTPUT

Observation Predicted Y

Residuals

1 0.003981 -0.002982 0.006765 -0.000763 0.012186 0.0018144 0.021122 0.0038785 0.034454 0.0005466 0.05306 -0.003067 0.07782 0.000188 0.109611 0.000389

7.23 Regression Analysis from Ln (d) - Ln (l) plot

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.990769R Square 0.981623Adjusted R Square 0.97856Standard Error 0.22485Observations 8

ANOVA

df SS MS FSignificance

FRegression 1 16.20338 16.20338 320.4948 1.95E-06Residual 6 0.303344 0.050557Total 7 16.50673

CoefficientsStandard

Error t Stat P-value Lower 95%Upper 95%

Lower 95.0%

Upper 95.0%

Intercept -1.81434 0.140394 -12.9232 1.32E-05 -2.15787 -1.47081 -2.15787 -1.47081

-1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0

-0.4-0.3-0.2-0.1

00.10.20.3

X Variable 1 Residual Plot

X Variable 1

Resid

uals

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X Variable 1 2.949545 0.164757 17.90237 1.95E-06 2.546398 3.352691 2.546398 3.352691

RESIDUAL OUTPUT

Observation Predicted Y Residuals1 -6.56145 -0.34632 -5.36551 0.2495173 -4.51698 0.2482844 -3.85881 0.169935 -3.32104 -0.031366 -2.86637 -0.129367 -2.47251 -0.078538 -2.12511 -0.08217

7.24Error Propagation

Vertical:S¿

Horizontal: S¿

7.3 Part 3: Determine E

7.31 Method 1 Equation and Error Propagation

E= 4 Ln

( slope )w t 3

SE=E∗(S ( slope )slope

+ Sww

+ 3∗Stt

)

7.32 Method 2 Equation and Error Propagation

E= 4 ln

w t3 e∫¿¿

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SE=E∗¿

-End