Mechanics of Materials - Cantilever Flexure Test

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Mechanics of Materials Laboratory Cantilever Flexure Test David Clark Group C: 9/15/2006

Transcript of Mechanics of Materials - Cantilever Flexure Test

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Mechanics of Materials Laboratory

Cantilever Flexure Test

David Clark

Group C:

9/15/2006

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Abstract

A cantilever beam, a beam supported at one point, has been used many times in

countless designs and structures. It is important to understand the behavior of this setup

to avoid any type of failure that might occur if this design is improperly used or executed.

The following experiment utilizes a cantilever test fixture, strain gages, and basic

principles of Statics to determine both theoretical and actual stress along a cantilever

beam. For the 2024-T6 aluminum beam tested, the measured strain was found to be 903,

601, and 293 microstrain along a 1, 4, and 7 inch spacing respectively. The calculated

strain was 1205, 751, and 297 microstrain along the same interval. This deviance in

measured and calculated values demonstrates the need to test all conditions and better

understand the limitations of calculations.

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Table of Contents

1. Introduction & Background.............................................................................3

1.1. General background.................................................................................3

1.2. Calculating stress using Statics................................................................3

1.3. Load Estimate by Strain Relations and Hooke's Law..............................4

1.4. Load Estimation by Deflection................................................................5

2. Equipment and Procedure................................................................................6

2.1. Equipment and Setup...............................................................................6

2.2. Test procedure for measuring the difference of two strain gages............7

2.3. Test procedure for measuring individual strain gages.............................8

3. Data, Analysis & Calculations.........................................................................9

3.1. Known information..................................................................................9

3.2. Results......................................................................................................9

3.3. Load calculations...................................................................................10

3.4. Stress calculations..................................................................................10

4. Results............................................................................................................12

4.1. Graphical Results...................................................................................12

4.2. Comparison of Results...........................................................................12

5. Conclusions....................................................................................................13

6. References......................................................................................................14

7. Raw Notes......................................................................................................15

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1. Introduction & Background

1.1. General background

A cantilever beam refers to any beam that is supported at only one point. This

type of design has been used many times in countless designs and structures. It is

important to understand the behavior of this setup to avoid any type of failure that might

occur if this design is improperly used or executed.

1.2. Calculating stress using Statics

Analysis utilizing basic principles of Statics establishes a cantilever beam

experiences a vertical, horizontal, and moment reaction at the point being supported and a

point force at a length, L.

The stress is found using the elastic flexure formula where terms M, y, and I are

explained below.

Equation 1

M is the moment at the point of loading. For a steady cantilever beam, it is

expessed as:

Equation 2

where P is the applied load, L is the length between the supporting and loading point, and

x is the distance between the clamp and the strain gage.

y is the distance measured from the neutral axis to the point under consideration.

For a simple cantilever setup, this is expressed as:

Equation 3

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where t is the thickness of the beam.

Finally, I, is the centroidal moment of inertia for the beam. This is expressed as:

Equation 4

where b is the length of the base and t is the thickness.

Combining Equations 1 through 4, the first relation utilizing measurable parameters

can be expressed.

Equation 5

Equation 5 returns units of pounds per square inch (psi) when P is in pounds and

L, x, b, and t are in units of inches.

1.3. Load Estimate by Strain Relations and Hooke's Law

Stress is directly proportional to strain by a constant, E, known as the elastic

modulus. This is expressed mathematically as:

Equation 6

Combining Equations 5 and 6, the strain can be expressed using Equation 7.

Equation 7

An important behavior studied in the following experiment is the correlation

between strain, ε, and the distance along with length, L. The first derivative of Equation 7

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with respect to distance, x, can be expressed as in Equation 8. Simple algebraic

manipulation can be used to solve for the applied force, P, as in Equation 9.

Equation 8

Equation 9

This difference of two strain gages can then be used to find the force, P.

Equation 10

1.4. Load Estimation by Deflection

The load can also be calculated in terms of deflection. This is derived from the

expression,

Equation 11

Equation 2 and 4 are utilized to express M and I respectively. E, the modulus of

elasticity, is known for 2024-T6 aluminum to be 10.4 × 106.

Integrating Equation 11 twice with the known conditions of dy/dx = 0 at x = 0, and

y=0 at x=0, y can be expressed as,

Equation 12

The deflection in the following experiment is easily known. Therefore,

substituting δend for the difference in deflection and solving for P,

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Equation 13

2. Equipment and Procedure

2.1. Equipment and Setup

1. Cantilever flexure frame: A simple apparatus to hold a rectangular beam

at one end while allowing flexing of the specimen upon the addition of a

downward force.

2. Metal beam: In this experiment, 2024-T6 aluminum was tested. The beam

should be fairly rectangular, thin, and long. Specific dimensions are

dependant to the size of the cantilever flexure frame and available weights.

3. P-3500 strain indicator: Any equivalent device that accurately translates

to the output of strain gages into units of strain.

4. Three strain gages:

5. Micrometers and calipers:

The specimen should be secured in the flexure frame such that an applied force

can be placed opposite of the securing end of the fixture. Three strain gages should be

mounted such that the long metal traces run parallel to the length of the beam. The center

of the three gages should be mounted one inch, four inches, and seven inches from the

end of the clamp in the fixture.

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Figure 1

Whenever taking a reading from a strain gage, consult the strain gage

measurement device for optimal setup. The instructions below explain the setup used in

using the P-3500 strain indicator.

2.2. Test procedure for measuring the difference of two strain

gages

The first setup creates a half-bridge setup to find the difference between two

strain gages.

Figure 2

As shown in the diagram, the direction of stress is opposite of the other to read

positive strain. This returns the difference of the two strains.

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With a known initial deflection, the strain indicator was balanced to read zero

strain. Without adjusting the balance, the difference in strain for gage 2 to 3 was

measured and recorded.

To generate a change in strain, a 300 με increase was added by applying a point

load on the bar. The deflection was recorded and the difference in strain for 1 to 2, as

well as 2 to 3, was measured and recorded.

2.3. Test procedure for measuring individual strain gages

To demonstrate how this differential method of calculating strain is equivalent to

the individual strain measurements, the net strain reading for each single strain gage was

measured and recorded using a quarter-bridge configuration.

Figure 3

D, the "dummy" resistance, is needed to balance the bridge. Any uncertainty

within the accuracy of this resistance greatly influences the accuracy of the strain

indicator and should have the same impedance as R3.

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3. Data, Analysis & Calculations

3.1. Known information

The gage factor for the strain gage used was 2.085 and the transverse sensitivity

was 1.0. Both these factors are dependant upon the strain gage used and are generally

given by the manufacturer.

Table 1

3.2. Results

Table 2

Table 3

The difference in deflection between the initial and final position was 0.291

inches.

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3.3. Load calculations

The first load estimate is calculated using Equation 10 and the differences in

strain gages, as recorded in table 1. An example calculation is shown below.

Equation 14

The strain gradient is determined by finding the slope of the strain versus position

graph. For these results, see section 4.

A second load estimate was calculated using Equation 9 and the slope of the strain

versus position graph.

Equation 15

The third load estimate was calculated using Equation 13.

Equation 16

The comparison of the three load estimates is listed in the results section.

3.4. Stress calculations

Three estimates for stress can be determined. The first stress estimate is found

using the calculated force from the differential strain gage measurement setup. This is

expressed using Equation 5. Equation 17 is a sample calculation for the determination of

the stress at gage 1.

Equation 17

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A second stress estimate can be determined using the calculated point load found

in Equation 16.

Equation 18

The third stress estimate uses Hooke's Law to correlate strain to stress. Using

Equation 6, the stress can be found as follows:

Equation 19

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4. Results

4.1. Graphical Results

Strain vs Position

y = -101.67x + 1005.7

0

100

200

300

400

500

600

700

800

900

1000

0.000 1.000 2.000 3.000 4.000 5.000 6.000 7.000 8.000

Position

Str

ain

ε)

4.2. Comparison of Results

Figure 4

The table below lists the three load estimates using all three methods.

Table 4

Table 5 contains the first and second stress estimate using both calculated load points.

1 7.96875 8426 125362 4.96875 5254 78163 1.96875 2082 3097

σ x (psi) (P = 11.014)

σ x (psi) (P = 16.387)(L - x)Station

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Table 5

Table 6 catalogs the results from using Hooke's Law.

Station x (in) σ (psi)1 1 9391.22 4 6250.43 7 3047.2

Table 6

Table 7 summarizes and compares the different stress values generated

throughout the experiment.

σ1 (psi) σ2 (psi) σ3 (psi)P = 11.014 8426 5254 2082P = 16.387 12536 7816 3097

Hooke's Law 9391 6250 3047

Stress Summary and Comparison

Table 7

Table 8 shows the error between measured and calculated strain.

Table 8

5. Conclusions

Due to the large margin of error from the measured and calculated results, the

experimental results are not acceptable for practical application. Any design utilizing a

cantilever setup that experiences stresses close to the yield point of the material need to

be more rigorously tested. At maximum deflection, strain gage 1 exhibited a 25% error

from the calculated value. One cause for this error occurs because the equations used are

accurate in small deflections and loads easily handled by the material tested. Also,

Hooke's law is only valid for a portion of the elastic range for some materials, including

aluminum (Wikipedia). Although the net deflection in this experiment was small, the

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stress put upon the material in testing was theoretically 12,536 psi, nearly 84% of the

yield stress. Strain gage three experienced approximately 2% error, whereas the stress

that that point was theoretically 3,097 psi, or 21% of the yield stress. Therefore,

whenever a cantilever setup is used in high stress or deflection applications, thorough

testing and a suitable safety factor must be considered.

6. References

Gilbert, J. A and C. L. Carmen. "Chapter 8 – Cantilever Flexure Test." MAE/CE 370 –

Mechanics of Materials Laboratory Manual. June 2000.

Kuphaldt, Tony R. (2003). "Chapter 9 – Electrical Instrumentation Signals."

AllAboutCircuits.com. Retrieved September 19, 2006, from Internet:

http://www.allaboutcircuits.com/vol_1/chpt_9/7.html

"Hooke's Law." Wikipedia. Retrieved September 22, 2006, from Internet:

http://en.wikipedia.org/wiki/Hooke%27s_law

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7. Raw Notes

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