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.............................................................................3

2. Equipment and Procedure................................................................................4

3. Data, Analysis & Calculations.........................................................................7

4. Results..............................................................................................................8

5. Conclusions......................................................................................................9

6. References........................................................................................................9

7. Raw Notes......................................................................................................10

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1. Introduction & Background

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

of as the amount of deformation a material undergoes when subjected to a 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 begins 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 stress can be calculated. To

do so, the flexure formula can be used.

Equation 1

Where M is the bending moment at the point of interest (measured in inch-pounds

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

inches or meters), and I is the centroidal moment of inertia measured 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 is dependant on the beam geometry, and in this

case is equal to:

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Equation 2

where b is the width and t is the thickness.

c is replaced by half of the beam's thickness.

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

the applied force P multiplied by the effective length, Le.

Putting all three terms together, equation 1 becomes:

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. Equipment and Procedure

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 beam

should be fairly rectangular, thin, and long. Specific dimensions 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 performing the experiment, it is important to accurately 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:

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.

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.

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3. Data, Analysis & Calculations

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 centerline 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.

Table 1

The load was supplied using known 5 N weights. The conversion from Newtons

to pounds is:

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.

Equation 6

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

pounds.

4. Results

Stress vs Strain

y = 10.091e6 x

y = 10.059e6 x

0

500

1000

1500

2000

2500

3000

3500

0 50 100 150 200 250 300 350

Strain (με)

Str

ess

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% less than the known 10.1 x106 modulus of elasticity that is

generally accepted in the material science community.

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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. Conclusions

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. References

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

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

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7. Raw Notes

Figure 3

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Figure 4

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