Abstract
The aim of this experiment was to determine the relationship between the Young’s Modulus and the deflection of a beam. Since different materials have different values of Young’s Modulus, the students selected three different materials for the experiment. In this case, the materials used include timber, aluminium, and steel. All the materials were identical in terms of the dimensions such as width and length. The each beam of the three materials was subjected to some magnitude of force at the centre and deflection measured. Data concerning the magnitude of load and the corresponding defection were then gathered and displayed on a table. Based on the data, graphs of load versus deflection were produced. The gradients of these graphs were then used to calculate Young’s Modulus for each material. Based on Young’s Modulus values, the relationship between Young’s Modulus and deflection was then established. It was found that Young’s Modulus is directly proportional to the magnitude of defection.
Introduction
In a beam, deflection refers to the displacement due to loading that takes place in the direction that is not along the beam’s axis. It is an undesirable phenomenon in the construction of a beam. The tendency of material to resist deflection is called stiffness. Beams made from various materials have different degrees of stiffness. Indeed, even elastic materials have a particular degree of stiffness. Stiffer materials are more likely to resist deflection than less stiff ones.
The deflection of a beam is influenced by various factors that include applied load, the width of the beam, the thickness of the beam, and the length of the beam. Theoretically, it is known that increasing the magnitude of the load applied to the beam leads to an increase in the magnitude of deflection of the beam. Similarly, an increase in the length of a beam results in an increase in its deflection if the load applied to the beam remains constant. On the other hand, reducing the length of the beam has an effect of lowering its level of deflection. Beam deflection also varies depending on the material of which the beam is made. Young’s Modulus is a measure of elasticity or stiffness of an object when subjected to strain or compression (Katz, 2015, p. 405). Technically, Young’s Modulus is defined as the ratio of stress to strain (Lokensgard, 2010, p. 81; Sang et al., 2014, p. 106).
Theoretically, the deflection of a beam is inversely proportional to the magnitude of Young’s Modulus (Taylor, 2004, p. 403). The formula for the deflection of the beam is as shown below:
δ=WL348EI
In the above formula, W represents the load, L represents the length of the beam, I represent the second moment of Area for the rectangular beam, and E represents the value of Young’s Modulus. This formula shows that deflection is inversely proportional to the magnitude of Young’s Modulus. The second moment of Area is calculated using the following formula:
I=bd312
The above formula is based on the dimensions of the beam shown figure 1 below:
Figure 1: Diagram showing the dimensions used in calculating the second moment of area (I)
In this experiment, the students sought to determine the relationship between Young’s modulus and the deflection of a beam of the same material.
Procedure
The materials used in the experiment are as follows: weight, dial gauge, aluminium, steel and timber beams, and a device for measuring deflection. The experiment was performed as follows. First, the length, width, and height of each of the three beams (steel, aluminium, and timber) were measured three times and the average of each measurement calculated. The centre of the beam was then marked. Next, a distance of 600mm was marked on each side of the mark at the centre of the beam. The height of the central point of the deflection was then measured using a span of mm measuring device. Next, a central load of increments shown in table 1 was applied.
The results were then entered in tables 2-7 and the graph of Load (N) against adjusted defection (mm) plotted. The gradient of the graph was then calculated. Lastly, the observed and calculated values of deflections were compared.
Results
The measurements of steel beam dimensions and load deflections are shown in Tables 2 and 3 shown below:
The measurements of aluminium beam dimensions and load deflections are shown in Tables 2 and 3 shown below.
The measurements of timber beam dimensions and load deflections are shown in Tables 2 and 3 shown below:
It was observed that that the DTOA had to be hit to make the gauge stable.
Sample Calculations
Average value of b for steel beam
Average=25.62+25.62+25.503=25.58mm
Average value of d for aluminium
Average=6.36+6.31+6.353=6.34mm
Graph of Load (N) Against adjusted deflection (mm) for steel
Figure 2: Graph of Load (N) against adjusted deflection (mm) for steel
Graph of Load (N) Against adjusted deflection (mm) for aluminium
Figure 3: Graph of Load (N) against adjusted deflection (mm) for aluminium
Graph of Load (N) Against adjusted deflection (mm) for timber
Figure 4: Graph of Load (N) against adjusted deflection (mm) for timber
Young’s Modulus of timber
The gradient of the graph of load (N) against adjusted deflection (mm) gives the value of Wδ
Therefore, the equation for determining the value of Young’s Modulus is as follows:
E=gradient of loadvdeflection graph*L348I
Second moment of area I
I=bd312
Since the average values of b and d are 25.28mm and 6.59mm respectively, the second moment of area is calculated as follows:
I=25.28*6.59312=602.91mm4
Substituting L and I, and the gradient of the graph of load against deflection with their respective values, we obtain the expression below:
E=1.0115*600348*602.91=1.0115*2.16*10828939.68=1.0115*7463.8
E=7549.63kN/mm2
Therefore, the value of Young’s Modulus of timber is 7549.63kN/mm2.
Young’s Modulus of aluminium
The gradient of the graph of load (N) against adjusted deflection (mm) gives the value of Wδ
Therefore, the equation for determining the value of Young’s Modulus is as follows:
E=gradient of loadvdeflection graph*L348I
Second moment of area I
I=bd312
Since the average values of b and d are 25.63mm and 6.34mm respectively, the second moment of area is calculated as follows:
I=25.63*6.34312=544.30mm4
Substituting L and I, and the gradient of the graph of load (N) against deflection with their respective values, we obtain the expression below: E=0.1196*600348*544.30=0.1196*2.16*10826126.4=0.1196*8267.50
E=988.79kN/mm2
Therefore, the value of Young’s Modulus of Aluminium is 988.79kN /mm2.
Young’s Modulus of steel
The gradient of the graph of load (N) against adjusted deflection (mm) gives the value of Wδ
Therefore, the equation for determining the value of Young’s Modulus is as follows:
E=gradient of loadvdeflection graph*L348I
Second moment of area I
I=bd312
Since the average values of b and d are 25.58mm and 6.47mm respectively, the second moment of area is calculated as follows:
I=25.58*6.47312=577.34mm4
Substituting L and I, and the gradient of the graph of load (N) against deflection with their respective values, we obtain the expression below: E=0.031*600348*577.34=0.031*2.16*10827712.32=0.031*7794.37
E=241.63kN/mm2
Therefore, the value of Young’s Modulus of steel is 241.63 kN /mm2.
When a load of 10N is applied to the three materials used in this experiment (steel, aluminium, and timber), the deflection and Young’s Modulus are shown in the table below.
Based on the data shown in Table 8, a graph of deflection against Young’s Modulus can be produced. The graph is as shown in figure 5 below:
Figure 5: Graph showing the relationship between Young’s Modulus and deflection
Discussion
In the experiment, beams of different materials were used to investigate the relationship between Young’s Modulus and deflection of the material. The different materials were used because different materials have different values of Young’s Modulus. For instance, the literature value for Young’s Modulus for aluminium ranges from 66 to 68KN/mm2 (Jahnel, F., 2000, p. 200). On the other hand, the literature value for Young’s Modulus for S 275 steel (mild steel) is approximately 205,000 N/mm2 (Joannides & Weller, 2002, p. 8). While measuring the dimensions of the beams (width and height), three trials were performed and an average value of the dimensions calculated based on the three trials. The essence of the three trials was to minimise error that could emerge from the measurements. In this case, measuring the dimensions three times is important since it makes it possible to determine the value that is as accurate as possible.
In the experiment involving the measurement of the deflection produced by the respective load, it was observed that increasing the load lead to an increase in the deflection of the beam. This implies that the magnitude of deflection is directly proportional to the magnitude of the load causing that deflection. However, the extent to which a change in the magnitude of load leads to a corresponding change in the magnitude of the deflection varies form one material to the other. In the experiment, this is shown by the gradient of the graph of load against deflection. In this case, the gradient of the graph of load against deflection for timber was found to be the highest of the gradients of the three graphs produced in the experiment. In this case, the gradient was found to be 1.0115. On the other hand, the gradient of the graph of load versus deflection for steel was found to be the lowest (0.031). These observations imply that steel was the stiffest of the three materials while timber was the least stiff material. The stiffness of a material is an important property in determining the material’s use in engineering. Materials used in a structure tend to be subjected to different magnitude of forces. Consequently, they must be able to withstand the different forces. For example, materials used in making a bridge are often subjected to different magnitudes of force. Consequently, the choice of a material to be used in making a given part of the bridge is very important. If the material used is not suitable, then the bridge might collapse if a given load exceeds the capacity of the materials used in making the bridge to withstand.
Stiff materials are suitable in making various parts of a bridge. For example, the material for making the beam of a bridge should be stiff because the bean of a bridge is often subjected to forces that cause it to twist and bend. This bending and twisting tendency is undesirable. Therefore, a material used in making the beam should be able to resist forces that tend to cause bending and twisting. Indeed, if the material used in making the beam of a bridge is not stiff enough, the bridge might collapse. Steel is one of the stiffest materials known. It is widely used in making beams of various structures including those of bridges. The material is able to withstand high magnitude of forces causing bending.
The experiment found that Young’s Modulus is directly proportional to deflection. However, this observation is contrary to the expected relationship. Theoretically, Young’s Modulus is inversely proportional to the deflection (Borisenko, Gaponenko, and Gurin, 1999, p. 88; Nguyen, & Wereley, 2002, p. 383). This could be due to some errors in the experiment. Some of the sources of error in this experiment could include the changes in the properties of the beams due to changes in temperature in the course of the experiment. Such changes could have resulted in changes in the properties of the beams, hence resulting in errors. Systematic error could have played a major role in influencing these results.
Conclusion
The experiment was successfully conducted since all the aims of the experiment were achieved. The students found out that young’s Modulus is directly proportional to the magnitude of deflection if other factors remain constant. However, this finding is contrary to the theoretical fact about the relationship between the two variables (Young’s Modulus and deflection).The students gained many insights into the various concepts of the topic. Indeed, the experiment was a good experience since the students had the opportunity to practice the theoretical skills learnt in class.
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