Experiment #5

Cantilever Beam

Stephen Mirdo

Performed on November 1, 2010

Report due November 8, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………..…pp. 1 - 2 Procedure ………………………….…………………………………...……..……..... p. 3 Results …................................................................................................................ pp. 4 - 5 Discussion and Conclusion …………………….......…………………….......…... pp. 6 - 7 Appendix ……………………………………..…………………..….……..…... pp. 8 - 10

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Object

The object of this experiment was to interpret the indicated strain experienced by a cantilever beam under flexural stress via Hooke’s Law and verify the flexural formula for the cantilever beam.

Theory

For this experiment, the flexural strain induced by a loading force on a cantilever

beam will be used to determine the flexural stress of the beam. By Hooke’s Law, the stress acting in the beam is equal to the beam material’s modulus of elasticity, E, multiplied by the induced strain, ε, as seen in Equation 1 below. Because the stress is a flexural stress, meaning the material is bending due to the stress, the flexural stress equation must be used (Equation 2). The components of the flexural stress equation are as follows: Mmax is the moment induced by the loading force, cmax is the distance from the neutral axis of the beam to the outer edge of its cross-section, and Ixx,c is the moment of inertia of the cross-section of the beam.

σ = Eε (Equation 1)

σflexural, max = Mmax cmax / Ixx,c (Equation 2)

Figure 1: Diagram of dimension of the cantilever beam experiment exposing the induced moment and the loading force due to the screw.

Figure 1 illustrates how the load is applied to the cantilever beam in this experiment. For every complete tightening rotation of the screw, a proportional increase in the loading force will occur. This loading causes a deflection of the beam, δ, and a moment where the beam is attached to the testing apparatus.

δ = WL3 / 3EIxx,c (Equation 3) where W is the loading force, L is the length of the cantilever beam, E is the elastic modulus of the beam’s material and Ixx,c is the moment of inertia of the beam’s cross-section. To calculate the moment of inertia for a beam of rectangular cross-section, employ the following equation:

Ixx,c = bh3/12 (Equation 4)

2

To determine the moment the loading force, W, of the screw exerts on the beam, the magnitude of the loading force must be calculated. The loading force can be determined by using the following equation:

W = 3δE(1/12)bh3 / L3 (Equation 5) where δ is the deflection of the beam, E is the elastic modulus of the beam’s material, b is the width of the cross-section of the beam, h is the height of the cross-section of the beam, and L is the length of the beam. Figure 2 diagrams a rectangular beam’s cross section.

Figure 2: Diagram of a rectangular beam’s cross-section exposing the neutral axis x-x. Once the loading force W of the screw has been determined, the moment, Mmax, that the force exerts can then be calculated. To determine the moment, use the following equation:

Mmax = WL (Equation 6)

where W is the loading force and L is the length of the cantilever beam. The remaining term of the flexural stress formula (Equation 2) left to be determined is the distance, cmax, from the neutral axis x-x of the cross section of the cantilever beam to the fibers of the material at the outermost reaches of its geometry. For a rectangular specimen, cmax can be calculated using the following equation:

cmax = h/2 (Equation 7)

where h is the height of the rectangular cross-section of the cantilever beam. To calculate the flexural stress at any height above the neutral axis, any value that is greater than zero and less than cmax may be used.

Because the cantilever beam in this experiment has a doubly symmetric cross-section, the flexural stress formula derived above will yield the stress at the top and bottom of the beam. The sense of the stress, whether it is in compression or tension, will be determined by the sense of the bending moment. For the testing apparatus featured in Figure 3 below, the flexural stress on the top of the beam will be positive as it is in tension and the stress on the bottom of the beam will be negative, as it is in compression.

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Procedure

Equipment Cantilever beam experiment apparatus Strain Gage Strain Indicator Experiment

1) Measure the width of the beam, thickness of the beam, and the length of the beam.

2) Lower the crew until it touches the beam, as seen in Figure 1. Use a scrap of paper between the beam and the screw to just tighten the screw until it touches the beam.

3) Power up the strain indicator. 4) Set the gage factor of the strain indicator to 2.5 5) Set the quarter bridge of the strain indicator for channel 1. 6) Turn calibration on and balance channel 1 to zero the display of the strain

indicator. 7) Mark the position of the screw’s eye. 8) Turn one complete turn of the screw and record the indicated microstrain.

Note: One complete turn of the screw produces a deflection of 0.05 in. 9) Record the indicated microstrain for deflections of 0.05 in up to 1.00 inch.

This is equivalent of 20 turns of the screw. 10) Repeat steps 8 and 9 for a total of three trials.

Figure 3: Diagram of the cantilever beam experiment apparatus. (Adapted from Materials Laboratory Manual, Fall 2010, University of Memphis, Department of M.E.)

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Results

Table 1: Screw and Beam properties required to calculate flexural stress. Δ Deflection of Each Rotation of Screw (in) 0.05 L Length of Beam (in) 16.375 b Width of Beam (in) 1.494 h Thickness of Beam (in) 0.369 E Elastic Modulus of Beam (psi) 1.06E+07

Table 2: Experimental data points with the flexural stress calculated using Equation 2 and experimental flexural stress determined from Equation 1. Note: The average microstrain in this table is the average over three trials. Number of Complete

Twists

Deflection of Beam (in)

Average με

Experimental σmax (psi)

Weight (lbf)

Calculated σmax (psi)

1 0.050 64.0 678.4 2.265 1094.0 2 0.100 128.7 1363.9 4.530 2188.1 3 0.150 195.0 2067.0 6.796 3282.1 4 0.200 260.7 2763.1 9.061 4376.1 5 0.250 326.0 3455.6 11.326 5470.2 6 0.300 392.0 4155.2 13.591 6564.2 7 0.350 458.3 4858.3 15.856 7658.2 8 0.400 524.7 5561.5 18.121 8752.3 9 0.450 590.7 6261.1 20.387 9846.3

10 0.500 657.3 6967.7 22.652 10940.3 11 0.550 725.7 7692.1 24.917 12034.4 12 0.600 793.7 8412.9 27.182 13128.4 13 0.650 862.7 9144.3 29.447 14222.4 14 0.700 931.3 9872.1 31.712 15316.5 15 0.750 1000.0 10600.0 33.978 16410.5 16 0.800 1069.0 11331.4 36.243 17504.5 17 0.850 1138.0 12062.8 38.508 18598.6 18 0.900 1206.7 12790.7 40.773 19692.6 19 0.950 1276.3 13529.1 43.038 20786.6 20 1.000 1345.0 14257.0 45.303 21880.7

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Stress Induced by Loading

0.0

5000.0

10000.0

15000.0

20000.0

25000.0

0 5 10 15 20 25

Number of Twists of Screw

Stre

ss (p

si)

Experimental Calculated

Figure 4: Graph of the induced stress plotted against the number of twists of the screw displaying the calculated theoretical values and the experimental values.

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Discussion & Conclusion

It was noted that there was a discrepancy between the calculated and experimental values for the stress experienced by the cantilever beam. To determine the amount of error between the values, a percent error analysis was performed. The table below indicates ~35% error throughout the entire experiment between the calculated and experimental values. Table 3: Percent error analysis of the calculated and experimental flexural stress of the cantilever beam.

Calculated σmax (psi)

Experimental σmax (psi) % Error

1094.0 678.4 38.0% 2188.1 1363.9 37.7% 3282.1 2067.0 37.0% 4376.1 2763.1 36.9% 5470.2 3455.6 36.8% 6564.2 4155.2 36.7% 7658.2 4858.3 36.6% 8752.3 5561.5 36.5% 9846.3 6261.1 36.4% 10940.3 6967.7 36.3% 12034.4 7692.1 36.1% 13128.4 8412.9 35.9% 14222.4 9144.3 35.7% 15316.5 9872.1 35.5% 16410.5 10600.0 35.4% 17504.5 11331.4 35.3% 18598.6 12062.8 35.1% 19692.6 12790.7 35.0% 20786.6 13529.1 34.9% 21880.7 14257.0 34.8%

The amount of error between the calculated and experimental values indicates that Hooke’s Law (Equation 1) is not the best choice for a scenario where there is flexure of a beam. Equation 1 is better suited for applications where the deformation of a specimen is uniform along an axis. In the case of a bending beam, as in this experiment, the deformation is not the same along the x-x axis.

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Figure 5: Diagram of the bending moment of a beam illustrating flexural stress in tension and compression, varying by the distance from the neutral axis.

As seen in Figure 5 above, the deformation of the uppermost limit of the beam is an elongation of the member. Therefore, the strain will be a positive value. The deformation of the lowermost limit of the beam indicates a contraction of the member due to the bending moment of the cantilever beam. The strain for the lowermost limit will be negative in value. For any distance c from the neutral axis, there will be a corresponding strain and therefore flexural stress. Due to the bending moment of the cantilever beam and the presence of stress due to this bending, the flexural stress equation (Equation 2) is the correct tool to solve this scenario.

There were sources of error in this experiment. One source of error was the

assumed value of deflection, 0.05 inch, for every twist of the screw of the apparatus. Due to wear of the metal components of the apparatus, such as the threads of the screw, it is indeterminable how much deflection was incurred. The value of 0.05 inch is the best estimate available. Another source of error would be in the modulus of elasticity of the beams material. The accepted value of the elastic value for the beam’s material was used to calculate the induced stress. The actual modulus of the testing apparatus may differ from the accepted value. Another source of error is human error incurred by an inability to ensure that each rotation of the screw of the testing apparatus was precisely 360 degrees.

A few improvements to this experiment are in order. One such improvement

would be to use a type of screw that has a locking position such as those found on the volume knob of some stereos. The “click” of the knob enables users to know when the center position has been reached. A similar apparatus would be useful so as to let the user know when exactly 360 degrees of rotation of the apparatus’ loading screw has been achieved. Another improvement would require the student to calculate the flexural strain incurred by the member by inducing stress in precise increments.

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Appendix Data Usage Sample calculation of experimental flexural stress at 1 complete rotation of screw:

10.6 x 106 psi * 64 με = 678.4 psi Sample calculation of calculated flexural stress at 1 complete rotation of screw:

[2.265 lbf * 16.375 in * (.369/2) in] / [(1.494 in * (0.369 in)3)/12] = 1094.0 psi Sample calculation of percent error between calculated and experimental stress at 1 complete rotation of screw:

| 1094 psi – 678.4 psi | / 1094 psi * 100 = 38.0%

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Bibliography

Mechanics of Materials, 2nd Edition Timothy A. Philpot (2011)

Fundamentals of Material Science and Engineering: An Integrated Approach

W.D. Callister, Jr and D.G. Rethwish (2008)

Materials Laboratory Manual, Fall 2010 University of Memphis, Department of Mechanical Engineering