# Mechanics of Materials - Modulus of Elasticity Flexure Test

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## Abstract

Since all materials experience some type of deformation when external forces act Since all materials experience some type of deformation when external forces act upon them, it is important to understand the behavior and limitations of these materials. upon them, it is important to understand the behavior and limitations of these materials. The stiffness can be characterized by a parameter known as the modulus of elasticity, or  The stiffness can be characterized by a parameter known as the modulus of elasticity, or  Y

Young's modulus. This number, in units of oung's modulus. This number, in units of pressure, can be pressure, can be used to predict such used to predict such behaviorsbehaviors as deflection, stretching, and buckling. The following experiment demonstrates how to as deflection, stretching, and buckling. The following experiment demonstrates how to ascertain the modulus of elasticity for a material by determining this characteristic for  ascertain the modulus of elasticity for a material by determining this characteristic for  2024-T6 aluminum.

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1. Introduction & Background

1. Introduction & Background ...44

2. Equipment and

2. Equipment and Procedure Procedure ...55

3. Data,

3. Data, Analysis & Calculations Analysis & Calculations ...88

4. 4. Results Results ...99 5. Conclusions 5. Conclusions ...1010 6. R 6. References eferences ...1010 7. Raw

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The modulus of elasticity refers to a material's stiffness. This can also be thought The modulus of elasticity refers to a material's stiffness. This can also be thought of

of as as ththe e amoamoununt t of of defdeforormamatition on a a mamateteririal al undunderergogoes es whwhen en susubjbjectected ed to to a a loloadad.. Experimentally, the modulus of elasticity, or Young's modulus, is found by determining Experimentally, the modulus of elasticity, or Young's modulus, is found by determining the slop of the stress versus strain curve.

the slop of the stress versus strain curve.

With excessive loading, the stress-strain curve initially begins linearly, followed With excessive loading, the stress-strain curve initially begins linearly, followed  by a dramatic change of slope. The phenomena occurring during this sudden change in  by a dramatic change of slope. The phenomena occurring during this sudden change in slope is known as plastic deformation and is beyond the scope of this lab. For the slope is known as plastic deformation and is beyond the scope of this lab. For the  purposes of testing the Young's modulus, the applied load should be kept below the yield  purposes of testing the Young's modulus, the applied load should be kept below the yield

strength, the pressure as which a material beg

strength, the pressure as which a material begins to experience plastic deformation.ins to experience plastic deformation.

A simple way of determining the Young's modulus is to create a uniaxial stress A simple way of determining the Young's modulus is to create a uniaxial stress state. This is achieved by supporting a beam in a cantilever setup while applying pressure state. This is achieved by supporting a beam in a cantilever setup while applying pressure to a point on the beam. A strain gage should be located perpendicular, as well as a known to a point on the beam. A strain gage should be located perpendicular, as well as a known distance, from the applied force.

distance, from the applied force. W

With a ith a known force, beam, known force, beam, and strain, and and strain, and resulresulting stress can be ting stress can be calculcalculated. Tated. Too do so, the flexure formula can be used.

do so, the flexure formula can be used.

I   I  cc M  M ⋅⋅ = = σ   σ   Equation 1 Equation 1 Where

Where M M is the bending moment at the point of interest (measured in inch-poundsis the bending moment at the point of interest (measured in inch-pounds or Newton-meters),

or Newton-meters), cc is the distance from the neutral axis to the surface (measured inis the distance from the neutral axis to the surface (measured in inc

inches hes or or metmetersers), ), andand  I  I  is is the the cencentrotroidaidal l mommoment ent of of ineinertirtia a meameasursured ed aroaround und thethe horizontal axis (inches

horizontal axis (inches44 or metersor meters44).).

Since all three terms are calculated, it is easier to replace each term with terms Since all three terms are calculated, it is easier to replace each term with terms representing terms physically measured.

representing terms physically measured. I  I is dependant on the beam geometry, and in thisis dependant on the beam geometry, and in this case is equal to:

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12 12 3 3 bt  bt   I   I == Equation 2 Equation 2 where

where bb is the width andis the width and t t is the thickness.is the thickness. cc is replaced by half of the beam's thickness.is replaced by half of the beam's thickness. M

M refers to the bending moment and in an elementary uniaxial setup is equal torefers to the bending moment and in an elementary uniaxial setup is equal to the applied force

the applied force P  P multiplied by the effective length,multiplied by the effective length, L Lee.. Putting all three terms together, equation 1 becomes: Putting all three terms together, equation 1 becomes:

2 2 3 3 6 6 12 12 2 2 bt  bt   L  L  P   P  bt  bt  t  t   L  L  P   P  ee ee ⋅⋅ = = ⋅⋅ = = σ   σ   Equation 3 Equation 3

Equation 3 is only valid for the surface of an end-loaded cantilever beam with a Equation 3 is only valid for the surface of an end-loaded cantilever beam with a rectangular cross-section.

rectangular cross-section.

To obtain the slope of all points, linear regression should be used to generate a To obtain the slope of all points, linear regression should be used to generate a linear function for a stress-strain curve. The first derivative of this equation will yield the linear function for a stress-strain curve. The first derivative of this equation will yield the modulus

modulus of elasticity.of elasticity.

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This experiment was conducted using the following equipment: This experiment was conducted using the following equipment:

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1. Cantilever flexure frameCantilever flexure frame: A simple apparatus to hold a rectangular beam: A simple apparatus to hold a rectangular beam at one end while allowing flexing of the specimen upon the addition of a at one end while allowing flexing of the specimen upon the addition of a downward force.

downward force. 2.

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

should uld be be faifairly rly rectrectanguangularlar, , thithin, n, and and lonlong. g. SpeSpecifcific ic dimdimensensionions s areare dependant to the size of the cantilever flexure frame and available weights. dependant to the size of the cantilever flexure frame and available weights.

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3.

3. P-3500 strain indicatorP-3500 strain indicator: Any equivalent device that accurately translates: Any equivalent device that accurately translates to the output of strain gages into

to the output of strain gages into units of strain.units of strain. 4.

4. Strain gagesStrain gages:: 5.

5. Micrometers and calipersMicrometers and calipers:: 6.

6. Hanger and known weightsHanger and known weights:: Bef

Before ore perperforforminming g the the expexperierimenment, t, it it is is impimportortant ant to to accaccuraurately measutely measure re thethe dimensions of the specimen to be tested. Using micrometers and/or calipers, the width, dimensions of the specimen to be tested. Using micrometers and/or calipers, the width, thickness, and effective length should be measured and recorded. The effective length is thickness, and effective length should be measured and recorded. The effective length is defined as the distance between the strain gage and the location where the load will be defined as the distance between the strain gage and the location where the load will be applied.

applied.

Figure 1 Figure 1

The specimen should then be secured in the flexure fixture. The strain gage The specimen should then be secured in the flexure fixture. The strain gage should be attached to the beam such that the long wires run parallel to the effective should be attached to the beam such that the long wires run parallel to the effective length.

length.

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The gage factor refers to the change in resistance of the gage with respect to the The gage factor refers to the change in resistance of the gage with respect to the change in length. The gage factor is usually supplied with strain gages and is important in change in length. The gage factor is usually supplied with strain gages and is important in configuring the strain indicator (Omega).

configuring the strain indicator (Omega).

The strain gage should be connected to the indicator as specified: The strain gage should be connected to the indicator as specified:

• The independent lead to the P+The independent lead to the P+ •

• One dependent lead to the S-One dependent lead to the S-•

• One dependent lead to a dummy connection (in this experiment, the D120)One dependent lead to a dummy connection (in this experiment, the D120)

With only the hook on the loading point, the strain indicator should read zero. If it With only the hook on the loading point, the strain indicator should read zero. If it does not, the balance should be adjusted such that a zero readout is achieved.

does not, the balance should be adjusted such that a zero readout is achieved.

Before loading weights, the maximum load to be tested should first be examined Before loading weights, the maximum load to be tested should first be examined to ensure that the yield stress is not surpassed. For 2024-T6 aluminum, the yield stress is to ensure that the yield stress is not surpassed. For 2024-T6 aluminum, the yield stress is 15,000 PSI. This applied stress is calculated using the following equation:

15,000 PSI. This applied stress is calculated using the following equation:

e e  L  L bt  bt   P   P  6 6 2 2 max max σ   σ   = = Equation 4 Equation 4 where P is the applied load, L

where P is the applied load, Leeis the effective length, b is the base width of is the effective length, b is the base width of the specimen,the specimen, and t is the thickness.

and t is the thickness.

Added weights at regular intervals should be placed on the hook one at a time, Added weights at regular intervals should be placed on the hook one at a time, recording the strain readout after each addition. After the maximum weight to be tested is recording the strain readout after each addition. After the maximum weight to be tested is added, each weight should be removed one-by-one. The strain should be recorded for  added, each weight should be removed one-by-one. The strain should be recorded for  each decrement.

each decrement.

If the applied loads are below the yield strength of the material tested, the plot of  If the applied loads are below the yield strength of the material tested, the plot of  stress versus strain should be linear. The slope, change of stress with respect to the stress versus strain should be linear. The slope, change of stress with respect to the change of

change of strain, represents the modulus of strain, represents the modulus of elasticityelasticity..

For the data analysis performed here, the data points were logged into excel, For the data analysis performed here, the data points were logged into excel, graphed, and a trend line with a linear equation were constructed. The first derivative of  graphed, and a trend line with a linear equation were constructed. The first derivative of  the trend line

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The dimensions of the beam were as follows: The dimensions of the beam were as follows:

•  b = 1.000 inches (width) b = 1.000 inches (width) •

• t = 0.250 inches (thickness)t = 0.250 inches (thickness) •

• LLee= 6.125 inches (effective length; from gage c= 6.125 inches (effective length; from gage centerline to applied load)enterline to applied load) The gage factor for the strain gage used is 2.08.

The gage factor for the strain gage used is 2.08.

The following table catalogs the applied loads, resulting strain, and calculated The following table catalogs the applied loads, resulting strain, and calculated stress. stress. L Looaad d ((llbb)) SSttrraaiin n ((μμεε)) SSttrre se ss s ((ppssii)) 0 0..00000 0 0 0 00..000000 1 1..11224 4 666 6 666600..991122 2 2..22448 8 11332 2 11332211..882244 3 3..33772 2 11998 8 11998822..773366 4 4..44996 6 22662 2 22664433..664488 5 5..66220 0 33226 6 33330044..556600 4 4..44996 6 22662 2 22664433..664488 3 3..33772 2 11998 8 11998822..773366 2 2..22448 8 11332 2 11332211..882244 1 1..11224 4 665 5 666600..991122 0 0..00000 0 0 0 00..000000

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Table 1 Table 1

The load was supplied using known 5 N weights. The conversion from Newtons The load was supplied using known 5 N weights. The conversion from Newtons to pounds is: to pounds is:  N   N  lbf  lbf  44..448448 1 1 == Equation 5 Equation 5 The strain was taken from the readout

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The stress was calculated using equation 3. Equation 6 demonstrates a sample The stress was calculated using equation 3. Equation 6 demonstrates a sample calculation to find stress for the 1.124 pound load.

calculation to find stress for the 1.124 pound load.

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psipsi in in lb lb t  t  b b  L  L  P   P  ee 912 912 .. 660 660 250 250 .. 0 0 000 000 .. 1 1 125 125 .. 6 6 124 124 .. 1 1 6 6 6 6 2 2 2 2 == ⋅⋅ ⋅⋅ ⋅⋅ = = ⋅⋅ ⋅⋅ ⋅⋅ = = σ   σ   Equation 6 Equation 6

To produce units of PSI, all lengths were in inches and the applied load was in To produce units of PSI, all lengths were in inches and the applied load was in  pounds.  pounds.

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Stress vs Strain Stress vs Strain y = 10.091e6 x y = 10.091e6 x y = 10.059e6 x y = 10.059e6 x 0 0 500 500 1000 1000 1500 1500 2000 2000 2500 2500 3000 3000 3500 3500 0 0 550 0 11000 0 11550 0 22000 0 22550 0 33000 0 335500 Strain (με) Strain (με)        S        S      t      t      r      r      e      e      s      s      s      s Figure 2 Figure 2

beam was 10.091x1066 and 10.059x10and 10.059x1066 respectively. The average of these two figures,respectively. The average of these two figures, 10.075x10

10.075x1066 is is 0.20.24848% % leless than ss than ththe e knoknown wn 10.10.1 1 x1x10066 modmoduluulus s of of elaelastisticitcity y thathat t isis generally accepted in

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Sources of error within this experiment occur with all linear measurements of the Sources of error within this experiment occur with all linear measurements of the specimen as well as uncertainty in the

specimen as well as uncertainty in the weights creating the applied force.weights creating the applied force.

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Utilizing a cantilever beam setup and strain gauges, the modulus of elasticity for  Utilizing a cantilever beam setup and strain gauges, the modulus of elasticity for  2024-T6 aluminum was found to be 10.075 x10

2024-T6 aluminum was found to be 10.075 x1066. This result is acceptable and is deviates. This result is acceptable and is deviates only 0.248% of the scientifically acknowledged value.

only 0.248% of the scientifically acknowledged value.

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"The Strain Gage." Omega Engineering. 5 Sept. 2006. "The Strain Gage." Omega Engineering. 5 Sept. 2006.

<http://www.omega.com/literature/transactions/volume3/strain.html> <http://www.omega.com/literature/transactions/volume3/strain.html>

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Figure 3 Figure 3

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Figure 4 Figure 4

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