## Abstract

In this experiment, the young modulus of three beams was investigated these were steel, aluminum and timber. The beams were placed and the deflections values upon addition of various loads at the Centre of the beam taken. There was a variation of the loads across the beam to investigate thee relationship that existed between the loads, the span width, height, and deflections. Measurement of the deflection was therefore done through the dial gauge with three sets of measurements being taken and the average calculated. Different beams were used to enable elaborate comparison of the coefficient of elasticity. Measurement of the width and height of the beams was also done to aid in the calculation of moment of inertia.

## Introduction

The main aim of this experiment was to relate the young’s modulus that is the measure of the stiffness of a selected material and the beams deflection of specific materials. The materials that were used in the experiment were as follows; steel beam, aluminum beam, timber beam, a device used for measuring deflection, weight, and a dial gauge.

## Background information

The young’s modulus of material is the measure of the stiffness of a solid material under subjection to external tensile stress or compression. The young’s modulus is the measure of the ratio between the applied stresses to the deformation the body undergoes in case the behavior is linear. When the calculated value of the young’s modulus is high, it means that there is greater strain requirement for the same degree of deformation, and thus, the material is more rigid.

The beam deflection y is dependent on several factors that include; the load applied normally given by the product of mass and the gravitational pull force (W=m.g), the length of the beam (L), the beam width (b) and lastly the beam thickness normally given by h.

δ=WL348E!

## And the young’s modulus can thus be stated as;

E=WL348δI

Application of load to the beam results into the deflection at the load point with a dial gauge. The materials of the beam that were used in the experiment were made from steel, aluminum and timber.

The length L was set at 600 mm, the use of different length may result into changes in the maximum load used. Maximum deflection is aimed at in trying to achieve reasonable deflection. The moment of Area I for a triangular beam having a leght of b and a width of d was calculated in three points along the beam. The average value of the moment of area was thus determined. Below was the formula used in calculating the moment of area;

I=bd312

Where L is the length of the beam in meters, δ is the deflection of the beam in meters, W is the force (N), E= Young’s Modulus (N/m2) and I the second moment of inertia of area in m4. The deflection of the beam may also be influenced by other factors such as the method of loading, the material of the beam and the position of loading.

## Procedure

The length, width and height were measured in three consecutive times for each of the beam and the average results calculated and used for the subsequent calculations. The Centre of each beam was then marked a distance of 600 mm from the from the edge of the beam. At spans of 10 mm, the height of the central point was measured using the deflection measuring device. Central load increments were applied according to the table 1, below. The results were then tabulated to complete the table. A plot of load (N) against adjusted deflection (mm) was made and the gradient of the plot calculated. The procedure was repeated for other beams that were to be tested which were aluminum and timber beams. A comparison of the observed and calculated values was done with the experimental values of the deflections. The set up of the experiment was done as shown in the figure below;

## Results and calculations

Steel

## Steel beam calculations

E=WL348δI This can also be written as;

E=Wδ×L348I , where I is the second moment of area in mm4 and can be calculated from the formula;

I=bd312

The average value of width b was calculated from the table above to be 25.55 mm and the average height d also calculated to be 6.47 mm. Thus the value of I can be calculated as;

I=25.55 ×6.47312=576.66 mm4

wδ=the gradient of the graph of applied load versus adjusted data values.

## W is the applied load in N and δ is the adjusted deflection values.

Thus;

E = gradient of the graph plotted ×L348I , where L = 600 mm, I = 576.66 mm4.

E = 32.334 ×600348×576.66=252.32 kN/mm4

## Aluminum

Aluminum beam calculations

E=WL348δI This can also be written as;

E=Wδ×L348I , where I is the second moment of area in mm4 and can be calculated from the formula;

I=bd312

The average value of width b was calculated from the table above to be 25.63 mm and the average height d also calculated to be 6.34 mm. Thus the value of I can be calculated as;

I=25.63 ×6.34312=544.30 mm4

wδ=the gradient of the graph of applied load versus adjusted data values.

## W is the applied load in N and δ is the adjusted deflection values.

Thus;

E = gradient of the graph plotted ×L348I , where L = 600 mm, I = 544.30 mm4.

E = 8.3655 ×600348×544.30=69.162 kN/mm4

## Timber

Timber beam calculations

E=WL348δI This can also be written as;

E=Wδ×L348I , where I is the second moment of area in mm4 and can be calculated from the formula;

I=bd312

The average value of width b was calculated from the table above to be 25.28 mm and the average height d also calculated to be 6.59 mm. Thus the value of I can be calculated as;

I=25.28 ×6.59312=602.91 mm4

wδ=the gradient of the graph of applied load versus adjusted data values.

## W is the applied load in N and δ is the adjusted deflection values.

Thus;

E = gradient of the graph plotted ×L348I , where L = 600 mm, I = 602.91 mm4.

E = 0.995 ×600348×602.91=7.426 kN/mm4

## Discussion

The action of gravity always pulls the beam towards the earth’s surface, gravitational pull is normally given by F = mg (that is force is the product of mass and gravitational acceleration. Action of the force of gravity results into the deflection of the beam resulting into readings at the dial gauge. The force resisting the action of the force of gravity is as a result of the beam’s stiffness and is normally a measure of the young’s modulus.

The theoretical value of steel ranges from 180 to 200 N/mm2, compared to the calculated value that was found to be 252.32 N/mm2. The theoretical young modulus E of aluminum E is 68 N/mm2 while the calculated value was found to be 69.16 N/mm2. The theoretical value of modulus of elasticity of timber is 8.537 N/mm2, while the calculated value of the young’s modulus was 7.426 N/mm2.

Young’s modulus of a material being the measure of a material stiffness, it can therefore be concluded that steel beam is stiffer than the other two beams that were also put to test. Aluminum was also found to be stiffer than timber. 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

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