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 Lab: Deflection of Aluminium and Pine Wood May 12 2000

 Properties

Properties of Aluminum beam:

 Gauge Length 25.9 cm Width 3.8 cm Thickness 0.32 cm Weight 99.67 g

Table 1:  Properties of the aluminum beam

 Gauge Length 25.9 cm Width 3.8 cm Thickness 0.624 cm Weight 28.66 g

Table 1:  Properties of the pinewood beam

 Load (kg) Deflection of Aluminum beam (inches) Deflection of wood beam (inches) 0.1 0.035 0.034 0.622 0.216 0.234 1.122 0.382 0.42 1.622 0.548 0.596 2.122 0.712 0.794 2.622 0.877 0.984

Table 3:  Observed results

 Calculations

Elastic modulus of aluminum Density of Aluminium:

s = s = s = 3.164Mg.m-3

Density of pinewood:

s = = s = 0.467Mg.m-3

Specific modulus for Aluminum:

Specific modulus = = = 21.807 x 106 m2s-2

Specific modulus for Pinewood:

Specific modulus = = = 17.709 x 106 m2s-2

Thickness of pine beam if beam weight 99.67g:

s = 0.467 = t  = 0.022m

Deflection for an aluminum beam with a 20N load: Deflection for a pinewood beam of thickness 2.2 cm, with a 20N load: Load (kg) Actual Deflection (mm) Calculated Modulus of Elasticity Aluminium Pine Wood Aluminium Pine Wood 0.1 0.889 0.8636 6.15E+10 8.54E+09 0.622 5.486 5.9436 6.20E+10 7.72E+09 1.122 9.703 10.668 6.32E+10 7.76E+09 1.622 13.919 15.1384 6.37E+10 7.90E+09 2.122 18.085 20.1676 6.42E+10 7.76E+09 2.622 22.276 24.9936 6.44E+10 7.74E+09 Average Modulus 6.32E10 Pa 7.9E09 Pa

 Discussion where:

Y = deflection (m)

l  = guage length of beam [m]

E = modulus of elasticity [Pa]

w = beam width [m]

t = beam thickness [m]

The purpose of this lab was to find the dimensions for a beam of wood that had the same deflection properties of a dimensioned aluminium beam.  It was decided that the dimensions of the wood were to be the same as the dimensions of the aluminium beam, except for the thickness.  In order to determine the thickness of the wood, the observed deflections of the aluminium beam under various forces was used in equation 1.

The measurements of the wooden beam were calculated using the experimental values of the deflection of the aluminum beam.  More specifically, it was a rational decision to cut the wood to the same length and width as that of the aluminum beam.   The thickness of the wood was calculated using the various experimental deflection values for the aluminum beam.  Since the purpose of this lab has been to compare and study the elastic behaviour of the two materials, it was yet another rational decision to produce the data such that the deflection values for both materials would be the same for each mass.  This was accomplished by equating the deflection equations (equation 1) for both wood and steel and the resulting equation was equation 2.

Equation 1:    Y = FL3

3EI

‘E’ is the elastic modulus of the material and ‘t’ is its thickness.  Detailed calculations are shown in Appendix A.  Equation 1 accurately predicts the behavior of the aluminum, but it does not do so for the wooden beam.  This is so because the wooden beam

The figure below shows the load-deflection curve for the aluminium and the pinewood beams.  It shows both the theoretical and the actual plots.  All four lines are linear and converge from a single point and projects outwards, in a common direction, with different slopes.   The slope of the curves indicated the strength of the material.  The steeper the slope the stronger the material.  For both materials the theoretical slope is steeper than the actual slope.     The comparison between actual and the theoretical load-deflection curves for the aluminium beam shows that equation 1 is not accurate to a high degree.  The theoretical curve has a steeper slope than the actual curve.  This relates that as the load increases the difference between the slopes increases.  The difference in slope also indicates that the accuracy of equation 2 decreases as the load increases.  There are two reasons that can be accountable for this deviation, experimental error and “data” error.  Inaccurate instruments, reading errors, and other human errors could have caused the experimental errors.  The data error, relates to the value of the modulus of elasticity that was provide.  The modulus of elasticity varies with the type of aluminium alloy being used.  For the experiment, the type of alloy used was not known.  Thus a general value for the modulus of elasticity of aluminium was used. The comparison between the actual and the theorectial cuvres for the pine wood once again shows that equation 2 is not accurate to a high degeree.  Once again, the theoretical curve has a steeper slope than the actual curve.  This relates that as the load increases the difference between the slopes increases.  The difference in slope also indicates that the accuracy of equation 1 decreases as the load increases.  The two reasons accountable for this deviation are experimental error and “data” error.   The same experimental errors applies to the wood as it did to the aluminium. The data error in this case, relates to the value of the modulus of elasticity that was provided.  The modulus of elasticity varies with the species of pine, and its moisture content.  For the experiment, a general value for the modulus of elasticity was used because the type and the moisture content of the wood was not known.  Another trend observed on the actual load-deflection curve for the wooden beam is a gradual change of slope after a load of 1.6 kg was added.  The new slope bends toward the right.  This curvature of the line indicates permanent bending .

Ideally the theoretical load-deflection curve for aluminium, and the theoretical load-deflection curve for the wooden beam should be the same curve.  is a linear curve and thus it agrees, in close approximation, with the theoretical curve.  The actual curve for the wooden beam also agrees with its theoretical values.  However, in this case the curve was not entirely linear.  With the load of 1.5 kg, the wood was permanently deformed by a small amount.  This will be discussed further on in the section of error analysis.  The two theoretical curves are not accurate, in that according to Equation 1 (stated below), the two curves should superimpose upon one another.  However,

Further calculations show that while a beam of aluminum, weighing 9.967 X 10-2 kg, deflects an amount x, a beam of wood weighing slightly more than one third of that mass, 2.866 X 10-2 kg, also deflects the same amount.  The density of the materials was calculated to be 3164.73 kg/m3 for aluminum and 455.00 kg/m3 for wood.  Using equation 3, the specific modulus was calculated to be 2.18 X 106 Pa m3/kg for aluminum and 1.818 X 108 Pa m3/kg for wood.  This is a significant difference and it is due to the different properties of these two materials.  While aluminum comprises of compact molecules, wood comprises of plant cells.  Aluminum has a compact atomic structure, while wood contains air and water within its cells.  Due to this, aluminum has a higher modulus of elasticity than that of wood.

Equation 3:     Specific modulus = E = Elastic Modulus (Pa)

P        Density (kg/m3)

 Result and Observations

The results obtained from this report are accurate to a certain degree.  In conducting this experiment there are several sources of errors that should be accounted for.

Instrumental and Reading errors

The values obtained from the experiment are subject to errors.  Errors in both the accuracy of the instruments and the reading of the instruments are possible.

Other Errors:

It is uncertain what type of aluminum alloy was used in the experiment.  Different alloys of aluminum have different values for their modulus of elasticity.  For the purpose of this experiment, a general value for the modulus of elasticity, for aluminum, were used.

The modulus of elasticity of pine wood depends on several factors such as the spicies of pine, the moisture content, and the orientation of the grains .  For the ourpose of this experiment a general value for the modulus of elasticity was used