## Abstract

The experiment involved the investigation of the young’s modulus of different beams made from different materials. The beams that were used for the test were steel, aluminum and timber. The experiment involved placing of the beams on the measurement machine shown below and addition of loads at the center of the beam. The loads were then varied across the beams while investigating the relationship that existed between the beams when different loads were added. The height, width, and the respective deflections were therefore recorded for analysis. The dial gauge was used to measure the deflection of the beam at the different loading points and the average of the measurements taken for use in analysis. All the three beams were put to test and the results recorded for calculation of the coefficient of elasticity. The moment of area of each beam used was calculated from the respective measurements of the height and widths measured.

## Introduction

The experiments main aim was to relate the Young’s modulus of three selected beams which were made of different materials. By definition the young’s modulus is the measure of stiffness of the material under test (Yamasaki and Sasaki, 2010). Steel beam, aluminum beam and timber beam were used as test pieces and put under test.

The experiment was set and the deflection of each beam was measured by a set of equipment’s were used in order to accomplish the various measurements. They included the test specimen beams that were made of steel aluminum and timber. The dial gauge was used in measuring the various deflections at different loads (Yamasaki and Sasaki, 2010). Different weights were also used to measure the various deflections.

## Theoretical Background

As stated in the introduction, the young’s modulus is the measure of materials stiffness when subjected to external tensile stress or compression. Materials stiffness can be quantified by calculating the ratio of the applied stress to the deformation undergone by a body in cases where the material experiences linear behavior (Yamasaki and Sasaki, 2010). The higher the calculated value of the Young’s modulus, the greater the strain requirements to acquire same degree of deformation, meaning that the material is more rigid.

The deflection of the beam (y) depends on; the applied load, that is calculated by getting the product of the mass and gravitational pull (W = m.g). Other factors that influence the value of the deflection are the lengths of the beam (L), the width of the beam (b) and the thickness of the beam (h). The Young’s modulus can thus be calculated from the formula used for calculating deflection. For a centrally loaded beam, the deflection can be calculated from the formula;

δ=WL348E!

It then follows that the young’s modulus of a beam or any other material can thus be calculated from the formula;

E=WL348δI

When the load is applied to the test beam, the beam deflects at the load point of the dial gauge and thus indicating the value of the deflection measured. The beams used as test pieces were made from steel, aluminum and timber.

The length L used was 600 mm, using different lengths in calculation of the various deflection results in different maximum loads allowable for use. To achieve a reasonable deflection, the load that results in the maximum deflection should be used (Yamasaki and Sasaki, 2010). Triangular beams that were used provided an easy and faster way of calculating the moment of area I for each beam. The width and the height of the beams used were used in calculating the moment of areas of the respective beams. the mean value of the calculated moment of area for each beam was calculated and thus used in further analysis.

## The moment of area for a triangular beam can thus be calculated from the formula below;

I=bd312

## Procedure

Measurements of the length, width and the height of the beam under test were taken in three consecutive times and the mean value taken for application to the subsequent calculations. A distance of 600 mm was measured and marked from the edge of the beam to show the Centre of the beam. Using the deflection measuring device, the heights of central point were measured at spans of 10 mm lengths. The loads were increased according to table 1 shown below and the results recorded for purposes of analysis. The applied load (N) was plotted against the resulting deflection (mm) and the gradient of the graph calculated and used in the subsequent analysis and inferences. The above outlined procedure was repeated for all the beams that were put under test and the results recorded for respective graph plotting’s and analysis. The theoretical values of the deflections of the different beams were calculated and the results compared to the experimental results.

## Results and calculations

Timber

## Observations

The values of the deflection noted were high and fast thus effecting the dial. Because of the high values of the deflection the reading will have error. In order to adjust the error resulting from the error, the base of the tool was hit. It should also be noted that the reading was dial dependent but not material dependent.

## Timber beam calculations

Rewriting the equation of the Young’s modulus that is given by;

E=WL348δI

## The equation can be written as the product of two terms separated as shown below;

E=Wδ×L348I ,

As defined earlier, W is the weight, δ the deflection of the beam, L is the length of the beam under test and in this experiment is given by 600 mm. Finally I, is the second moment of area and is calculated from the formula below;

I=bd312

The calculated mean value of the width (b) of the beam was 25.28 mm and the mean height also calculated and found to be 6.59 mm. the second moment of area of the timber beam was then calculated as;

I=25.28 ×6.59312=602.91 mm4

The gradient of the plot of the applied load versus the corrected value of the deflection was calculated and the result was equated to wδ. Where W is the applied load given in Newton’s while δ is the value of the adjusted deflection.

## The value of the Young’s modulus can thus be calculated from the corrected formula as shown below;

E = the gradient of the graph X L3/48I.

L = 600 mm, I = 602.91 mm4

It therefore means that the value of the Young’s modulus can be found by replacing the respective values thus resulting with;

E = 0.995 ×600348×602.91=7.426 N/mm2

## Aluminum

Aluminum beam calculations

Rewriting the equation of the Young’s modulus that is given by;

E=WL348δI

## The equation can be written as the product of two terms separated as shown below;

E=Wδ×L348I ,

As defined earlier, W is the weight, δ the deflection of the beam, L is the length of the beam under test and in this experiment is given by 600 mm. Finally I, is the second moment of area and is calculated from the formula below;

I=bd312

The calculated mean value of the width (b) of the beam was 25.63 mm and the mean height also calculated and found to be 6.34 mm. the second moment of area of the timber beam was then calculated as;

I=25.28 ×6.59312=544.30 mm4

The gradient of the plot of the applied load versus the corrected value of the deflection was calculated and the result was equated to wδ. Where W is the applied load given in Newton’s while δ is the value of the adjusted deflection.

## The value of the Young’s modulus can thus be calculated from the corrected formula as shown below;

E = the gradient of the graph X L3/48I.

L = 600 mm, I = 544.30 mm4

It therefore means that the value of the Young’s modulus can be found by replacing the respective values thus resulting with;

E = 8.3655 ×600348×544.30=69.162 N/mm2

## Steel

Steel beam calculations

Rewriting the equation of the Young’s modulus that is given by;

E=WL348δI

## The equation can be written as the product of two terms separated as shown below;

E=Wδ×L348I ,

As defined earlier, W is the weight, δ the deflection of the beam, L is the length of the beam under test and in this experiment is given by 600 mm. Finally I, is the second moment of area and is calculated from the formula below;

I=bd312

The calculated mean value of the width (b) of the beam was 25.55 mm and the mean height also calculated and found to be 6.47 mm. The second moment of area of the timber beam was then calculated as;

I=25.28 ×6.59312=576.66 mm4

The gradient of the plot of the applied load versus the corrected value of the deflection was calculated and the result was equated to wδ. Where W is the applied load given in Newton’s while δ is the value of the adjusted deflection.

## The value of the Young’s modulus can thus be calculated from the corrected formula as shown below;

E = the gradient of the graph X L3/48I.

L = 600 mm, I = 576.66 mm4

It therefore means that the value of the Young’s modulus can be found by replacing the respective values thus resulting with;

E = 8.3655 ×600348×544.30=252.32 N/mm2

## Discussion

The weight of the beam can be calculated by finding the product of the mass and the gravitational pull on the beam. The weight of the beam acts towards the Centre of the gravity and thus the beam should be able to withstand its weight. Deflection of a beam is as result of the action of this force (Liu and Yu, 2011). The readings of the deflection are noted and read from the dial gauge. The property that makes an object resistive to the force of gravity is the stiffness of a material and is measured by the young’s modulus of a material.

The calculated values of the young’s modulus for the three different types of beams used were found as follows. Timber had the least value of the young’s modulus of 7.426 N/mm2, followed by aluminum that had a calculated young’s modulus of 69.16 N/mm2. Steel had the highest value of the calculated young’s modulus that was found to be 252.32 N/mm2.

The theoretical values of young’s modulus differed with the experimental values, for example the Young’s modulus of steel ranged from 180 to 200 N/mm2 while the experimental young’s modulus value was calculated to be 252.32 N/mm2. The calculated young’s modulus of aluminum had also a slight difference from the theoretical value that is to say the experimental value was found to be 69.16 N/mm2 while the theoretical value stands at 68 N/mm2. Finally timber also showed a slight difference in the theoretical and experimental results of young modulus. The experimental value of the young’s modulus was calculated to be 7.426 N/mm2 while the theoretical value stands at 8.537 N/mm2 (Liu and Yu, 2011).

The material young’s modulus being a measure of the materials stiffness is therefore a key indicator of materials strength. From the calculated results it is clear that steel is the stiffest of the three beams that were employed because of its high value of the young’s modulus (Liu and Yu, 2011). Timber on the other hand is seen as the beam that is less stiff and therefore can be said to be less strong compared to the other two beams. there was a slight deviation between the theoretical and the experimental results, this may mainly be due to errors due to equipment’s, errors due to observation and errors due to the difference in the conditions the experiment was performed.

The theoretical values and the calculated values differed with minimal ranges and the difference could be attributed to the difference in the conditions and errors due to measurement observation.

## Conclusion

In conclusion, it is evident that a linear relationship exists between the applied load and the adjusted deflection for timber, aluminum and steel. The deflection of the materials was found not to depend upon the position the beams are fixed at the ends. The main objective of the experiment was to calculate the young’s modulus of the three beams that were presented. This was achieved; further the beam deflections and their resulting young’s modulus were compared to the theoretical values (Liu and Yu, 2011). The steel beam had the highest value of the young’s modulus meaning that it was the strongest beam out of all the three beams presented for test. Timber on the other hand had the least young’s modulus meaning it was the least stiff this is because of its low value of young’s modulus. The widths and heights of the beams were very instrumental in the calculation of the beams young’s modulus as they had a direct influence on the value of the second moment of area.

## References

Liu, B. and Yu, J., 2011. The Effect of Temperature and Pulling Rate on Young's Modulus of Steel FAS390Q. AMR, 299-300, pp.337-340.

Yamasaki, M. and Sasaki, Y., 2010. Determining Young’s modulus of timber on the basis of a strength database and stress wave propagation velocity I: an estimation method for Young’s modulus employing Monte Carlo simulation. Journal of Wood Science, 56(4), pp.269-275.