Mechanics Of Materials - Cantilever Flexure Test

Mechanics of Materials Laboratory Cantilever Flexure Test

David Clark Group C: 9/15/2006

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............................................................4 1.1. General background...............................................................4 1.2. Calculating stress using Statics..............................................4 1.3. Load Estimate by Strain Relations and Hooke's Law............5 1.4. Load Estimation by Deflection................................................6 2. Equipment and Procedure............................................................7 2.1. Equipment and Setup.............................................................7 2.2. Test procedure for measuring the difference of two strain gages 8 2.3. Test procedure for measuring individual strain gages............9 3. Data, Analysis & Calculations.....................................................10 3.1. Known information................................................................10 3.2. Results..................................................................................10 3.3. Load calculations..................................................................11 3.4. Stress calculations................................................................11 4. Results........................................................................................13 4.1. Graphical Results.................................................................13 4.2. Comparison of Results.........................................................13 5. Conclusions.................................................................................14 6. References..................................................................................15 7. Raw Notes...................................................................................16

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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.

σ =−

M⋅y I

Equation 1 M is the moment at the point of loading. For a steady cantilever beam, it is expessed as: M = − P( L − x ) 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: y=

t 2

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Equation 3 where t is the thickness of the beam. Finally, I, is the centroidal moment of inertia for the beam. This is expressed as: I=

( )

1 b t3 12

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.

σx =

[ − P( L − x ) ] t

( )

1 b t3 12

2 = 6 ⋅ P ⋅ ( L − x) b ⋅t3

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:

σ x = E ⋅ε x Equation 6 Combining Equations 5 and 6, the strain can be expressed using Equation 7.

εx =

6 ⋅ P ⋅ ( L − x) E ⋅b ⋅t2

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 5

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. ∆ε x 6⋅ P =− ∆x E ⋅b ⋅t2 Equation 8 P=−

E ⋅ b ⋅ t 2 ∆ε x 6 ∆x

Equation 9 This difference of two strain gages can then be used to find the force, P. P1, 2 = −

E ⋅ b ⋅ t 2 ( ε1 − ε 2 ) E ⋅ b ⋅ t 2 (ε 2 − ε 3 ) and P2,3 = − 6 ( x1 − x2 ) 6 ( x 2 − x3 ) Equation 10

1.4.Load Estimation by Deflection The load can also be calculated in terms of deflection. This is derived from the expression, d 2 y M ( x) = E⋅I dx 2 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, y=

P (x3 − 3 ⋅ L ⋅ x 2 ) 6⋅ E ⋅ I Equation 12

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The deflection in the following experiment is easily known. Therefore, substituting δend for the difference in deflection and solving for P, P=

3 ⋅ δ end ⋅ E ⋅ I L3

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.

Beam Dimensions (inches) Base (b) Thickness (t) Length (L) Distance from the clame to the center of Gage 1 (x1) Distance from the clame to the center of Gage 2 (x2) Distance from the clame to the center of Gage 3 (x3)

1.000 0.250 8.969 1.000 4.000 7.000

Table 1

3.2.Results Differential Strain Measurements ε1-ε2 (με) ε2-ε3 (με) Initial Deflection 0 -649 Final Deflection 306 -349 Net Strain: 306 300

Table 2

Individual Strain Measurements ε1 (με) ε2 (με) ε3 (με) Initial Deflection 0 -359 -71 Final Deflection -903 -960 -364 Net Strain: 903 601 293

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. P1, 2 = −

E ⋅ b ⋅ t 2 ( ε 1 − ε 2 ) − 10.4 ×10 6 ⋅1 ⋅ 0.25 2 ( 306) = ⋅ = 10.906 lb 6 ( x1 − x2 ) 6 (1 − 4 ) 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. P=−

E ⋅ b ⋅ t 2 ∆ε x 10.4 ×10 6 ⋅1⋅ 0.25 2 ⋅ =− ⋅ ( − 101.67 ) = 11.014 lb 6 ∆x 6 Equation 15

The third load estimate was calculated using Equation 13. P=

3 ⋅ δ end ⋅ E ⋅ I = 16.387 lb L3 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.

σx =

6 ⋅ P ⋅ ( L − x ) 6 ⋅11.014 ⋅ ( 8.969 − 1) = = 8425 psi b ⋅t 2 1 ⋅ 0.25 2 Equation 17

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A second stress estimate can be determined using the calculated point load found in Equation 16.

σx =

6 ⋅ P ⋅ ( L − x ) 6 ⋅16.387 ⋅ ( 8.969 − 1) = = 12535 psi b ⋅t 2 1 ⋅ 0.25 2 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:

σ x = E ⋅ ε x = 10.4 × 10 6 ⋅ 903 × 10 −6 = 9391.2 psi Equation 19

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4. Results 4.1.Graphical Results Strain vs Position 1000 900 800 y = -101.67x + 1005.7

Strain (με)

700 600 500 400 300 200 100 0 0.000

1.000

2.000

3.000

4.000

5.000

6.000

7.000

8.000

Position

4.2.Comparison of Results Figure 4 The table below lists the three load estimates using all three methods. Differential Strain Method (με) 11.014

Calculation by Slope (με) 11.014

End Deflection Method (με) 16.387

Table 4 Table 5 contains the first and second stress estimate using both calculated load points. Station 1 2 3

(L - x) 7.96875 4.96875 1.96875

σ x (psi) (P = 11.014) 8426 5254 2082

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σ x (psi) (P = 16.387) 12536 7816 3097

Table 5 Table 6 catalogs the results from using Hooke's Law. Station 1 2 3

x (in) 1 4 7

σ (psi) 9391.2 6250.4 3047.2

Table 6 Table 7 summarizes and compares the different stress values generated throughout the experiment. Stress Summary and Comparison σ1 (psi) σ2 (psi) σ3 (psi) P = 11.014 8426 5254 2082 P = 16.387 12536 7816 3097 Hooke's Law 9391 6250 3047

Table 7 Table 8 shows the error between measured and calculated strain. Calculated Measured Error

ε1 1205 903 25.08%

ε2 752 601 20.04%

ε3 298 293 1.61%

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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