Mechanics of Materials - Modulus of Elasticity Flexure Test

Mechanics of Materials Laboratory Lab #4 Modulus of Elasticity Flexure Test

David Clark  Group C 9/8/2006

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

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Table of Contents 1. Introduction & Background ............ ........................ ........................ ........................ ................... ............ ..... 4 2. Equipment and Procedure........... ........................ .............................. ...................... ........... ............ ........ .. 5 3. Data, Analysis & Calculations........... ....................... ........................... ..................... ........... ........... ...... 8 4. Results............. ......................... ........................ ........................ ................................. ........................... ........... ........... ......... ... 9 5. Conclusions............ ........................ ........................ ........................ ........................ ........................ .............. ....... ....... .. 10 6. References References............. ......................... ........................ ........................ ........................ ................... ............. ........... ........ ... 10 7. Raw Notes........... ....................... ........................ ........................ ........................ .................... ............. ........... ........... ..... 11

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1. Introduct Introduction ion & Backg Backgrou round nd The modulus of elasticity refers to a material's stiffness. This can also be thought of as the the amoun amountt of defor deforma mati tion on a mate materi rial al under undergo goes es when when subj subject ected ed to a load load.. Experimentally, the modulus of elasticity, or Young's modulus, is found by determining the slop of the stress versus strain curve. 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 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 strength, the pressure as which a material beg ins to experience plastic deformation. 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 to a point on the beam. A strain gage should be located perpendicular, as well as a known distance, from the applied force. With a known force, beam, and strain, and resulting resulting stress can be calculated. calculated. To To do so, the flexure formula can be used. σ  

=

M  ⋅ c  I 

Equation 1 M is the bending moment at the point of interest (measured in inch-pounds Where M is or Newton-meters), c is the distance from the neutral axis to the surface (measured in inches inches or meters meters), ), and  I  is the centro centroida idall moment moment of inerti inertiaa measur measured ed around around the horizontal axis (inches4 or meters4). Since all three terms are calculated, it is easier to replace each term with terms representing terms physically measured. I  measured. I is is dependant on the beam geometry, and in this case is equal to:

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 I  =

bt 3 12

Equation 2 where b is the width and t is t is the thickness. c is replaced by half of the beam's thickness. M refers M  refers to the bending moment and in an elementary uniaxial setup is equal to the applied force P  force P multiplied multiplied by the effective length, L length, Le. Putting all three terms together, equation 1 becomes:  P ⋅ Le σ  

=

t  2

bt 3

=

6 P ⋅ Le bt 2

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

2. Equipmen Equipmentt and and Procedu Procedure re This experiment was conducted using the following equipment: 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 b eam should should be fairly fairly rectangu rectangular lar,, thin, thin, and long. long. Specif Specific ic dimens dimension ionss are dependant to the size of the cantilever flexure frame and available weights.

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3. P-3500 strain indicator: Any equivalent device that accurately translates to the output of strain gages into units of strain. 4. Strain gages: 5. Micrometers and calipers: 6. Hanger and known weights: Before Before perfor performin ming g the experi experimen ment, t, it is import important ant to accura accurately tely measure measure the 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 defined as the distance between the strain gage and the location where the load will be applied.

Figure 1 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 length. The strain gages used in this experiment have three leads to effectively eliminate any inaccuracies that would occur do to the length of the lead wires. Two lead wires connect to the first side of the gage where the third lead, known as the independent lead, connects to the opposing side. It is important to note the independent lead cannot be interchanged with either of the other two leads in connecting into the strain indicator.

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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 configuring the strain indicator (Omega). The strain gage should be connected to the indicator as specified: •

The independent lead to the P+

•

One dependent lead to the S-

•

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 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 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: 2

 P max

σ  bt  =

6 Le

Equation 4 where P is the applied load, Le is the effective length, b is the base width of the specimen, and t is the thickness. 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 added, each weight should be removed one-by-one. The strain should be recorded for  each decrement. 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 change of strain, represents the modulus of elasticity. elasticity. 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  the trend line equation represents the modulus of elasticity. elasticity.

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3. Data, Data, Analysis Analysis & Calculati Calculations ons The dimensions of the beam were as follows: •

 b = 1.000 inches (width)

•

t = 0.250 inches (thickness)

•

Le = 6.125 inches (effective length; from gage c enterline to applied load)

The gage factor for the strain gage used is 2.08.

The following table catalogs the applied loads, resulting strain, and calculated stress.

Load, Strain, and Stress Data Da ta Load (lb)

Strain (με)

Stre ss ss (psi)

0.000

0

0.000

1.124

66

660.912

2.248

132

1321.824

3.372

198

1982.736

4.496

262

2643.648

5.620

326

3304.560

4.496

262

2643.648

3.372

198

1982.736

2.248

132

1321.824

1.124

65

660.912

0.000

0

0.000

Table 1 The load was supplied using known 5 N weights. The conversion from Newtons to pounds is: 1lbf  = 4.448 N 

Equation 5 The strain was taken from the readout on the P-3500 strain indicator.

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

=

6 ⋅ P ⋅ Le b ⋅ t 2

=

6 ⋅ (1.124lb ) ⋅ ( 6.125in )

(1.000in ) ( 0.250in ) 2

=

660.912 psi

⋅

Equation 6 To produce units of PSI, all lengths were in inches and the applied load was in  pounds.

4. Result sultss Stress vs Strain 3500

y = 10.091e6 x 3000

2500

y = 10.059e6 x      s 2000

     s      e      r       t        S 1500

1000

500

0 0

50

100

150

200

250

300

350

Strain (με)

Figure 2 The modulus of elasticity for the points generated from loading and unloading the   beam was 10.091x106 and 10.059x106 respectively. The average of these two figures, 10.075x106 is 0.248 0.248% % less less than the the known known 10.1 10.1 x10 x106 modulu moduluss of elasti elasticit city y that that is generally accepted in the material science community co mmunity..

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Sources of error within this experiment occur with all linear measurements of the specimen as well as uncertainty in the weights creating the applied force.

5. Conc Conclu lusi sion onss Utilizing a cantilever beam setup and strain gauges, the modulus of elasticity for  2024-T6 aluminum was found to be 10.075 x106. This result is acceptable and is deviates only 0.248% of the scientifically acknowledged value.

6. Refe Referren ence cess "The Strain Gage." Omega Engineering. 5 Sept. 2006.

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7. Raw Raw Note Notess

Figure 3

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Figure 4

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