## Abstract by simple bending. Young’s modulus is

Abstract  The purpose of theexperiment was to determine a value for Young’s modulus E for a steel beam bysimple bending. Young’s modulus is used to describe and measure deflection of amaterial. A simple supported steel beam of different thicknesses was provided.

The base and height of the steel beam was measured at three different points,then an average was determined using these. The main testwas conducted to determine the amount of deflection in the steel beam. Couples of2N were placed on both sides of the beam simultaneously and the deflection wasmeasured. This was repeated till the load reached 8N on both sides of the beam(16N in total).

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The deflection was also measured as the couples were unloaded. Theseresults were then put into an equation used to determine a value for Young’sModulus E. 1.) Introduction and Background   The theorystates that when couples (a pair of forces, in this case weights) are appliedto the end of a beam, the beam will deflect into a circular arc, this resultsin the beam being in a state of simple bending.

The aim of theexperiment (as mentioned in the abstract) was to determine Young’s modulus fora steel beam by simple bending. Conducting this would describe the elasticityof the material (steel), and how much force it could withhold before deflecting.This isimportant as in manufacturing, engineers may require specific properties (e.g. morebrittle, high ductility etc.

) and conducting these experiments will provide thespecific properties of the material.  DeflectionDeflection is the movementof a body (material) from its original position when a force, load or weight isapplied directly to it. If for example anengineer is constructing a bridge, the engineer would want the material used inthe structure to hold the bridge up, conducting these experiments on thematerial would help to realise the amount of load the material can handlebefore deflecting.  2.

) Experiential procedure  EquipmentThe experiment wasconducted to measure deflection on a steel beam by bending. The equipment usedto carry out the experiment includes: ·     A steel beam of different thickness – used todetermine Young’s modulus E  ·     Rule – used to measure the steel beam ·     A set of couples (16N in total) – used to applyload onto the steel beam·     Dial gauge – used to measure deflection of thesteel beam    MethodInitially the steel beamwas placed onto a support structure which held it in place. A dial gauge wasplaced with the steel beam to measure deflection. Firstly, the base andheight of the steel beam was measured at three different points, using these anaverage was determined. Two hooks were placed on both sides of the steel beam, 200mmfrom the end.

Making sure the dial gaugewas set to zero, couples (2N) were placed simultaneously on both sides of thehook. As the load was applied, the measurements displayed on the dial gauge werenoted. This was repeated 8 times till the load reached 16N, at this point theload was unloaded 2N at a time and the measurements were recorded till it wasfully unloaded.      Figure 1 3.) Results and CalculationsInitially the steel beamwas measured for its base and height at three different points. The resultsgathered are as follows: Table 1 All measurements are in mm Point 1 Point 2 Point 3 Average Base 24.

98 25.23 25.01 25.07 Height 4.00 4.07 4.

17 4.08 Working out for theaverage: Base- 24.98 + 25.

23 + 25.01= 75.22 ¸ 3 = 25.07Height- 4.

00 + 4.07 + 4.17= 12.24 ¸ 3 = 4.08 Once the steel bar wasmeasured, deflection was measured using a dial gauge, couples of 2N were placedat a time till the load reached 16N, at this point the load was unloaded 2N ata time and the measurements were recorded till it was fully unloaded.

Theresults gathered are as follows:Table 2 Load (N) Deflection (mm) D (actual deflection) (mm) 0 8.27 0 2 8.92   8.92 – 8.27 = 0.65 4 9.58   9.

58 – 8.27 = 1.31 6 10.23 10.23 – 8.27 = 1.96 8 10.

89 10.89 – 8.27 = 2.62 10 11.55 11.55 – 8.

27 = 3.28 12 12.20 12.20 – 8.27 = 3.93 14 12.89 12.89 – 8.

27 = 4.62 16 13.53 13.53 – 8.27 = 5.26   Table 3  Load (N) Deflection (mm) D (actual deflection) (mm) 14 12.98 12.98 – 8.

27 = 4.71 12 12.31 12.

31 – 8.27 = 4.04 10 11.65 11.65 – 8.27 = 3.38 8 10.

98 10.98 – 8.27 = 2.71 6 10.29 10.

29 – 8.27 = 2.02 4 9.

63 9.63 – 8.27 = 1.

36 2 8.96 8.96 – 8.27 = 0.69 0 8.

28 8.28 – 8.27 = 0.01 The theory of bending givesthe relationship: =  =   s is the longitudinal stressin the beam at a distance y from its axis; R is the radius of the curve, M isthe bending moment, E is the Young’s Modulus for the steel beam.

is the second moment of area of the crosssection of the beam about its neutral axis and: = Using this formula willprovide the second moment of area. As mentioned previously the average basemeasures at 25.07mm and the average height measures at 4.08mm.

Adding these tothe formula will give. = =  =  With this equation  , d (deflectionof the beam) and W (Weight), are two constants, these are instead representedas k, this is also the point the gradient is discovered on the graph. The newequation with k as the subject becomes,  This equation can be rearranged to make E (Young’s modulus) the subject, this will give a value for Young’s modulus E for the steel beam. The new equation with E as the subject becomes,   The gradient of the graph equals   = 0.

33Nmm, this can also be written as.   The value for L as shown in Figure 1 measures at 200mm or 0.2m. Similarly, the value for W measures at 600mm or 0.

6m.       Putting these values into the equation will give a value for Young’s Modulus.                       Figure 2   d         4.) Discussion The aim of the experimentwas to measure Young’s modulus E for asteel beam by simple bending. Conducting this experiment required bending thesteel beam using couples (up to 16N). Theresults of this experiment demonstrate that there is a positive correlationbetween the load of the couples and the deflection of the steel beam. As theload applied to the steel beam increases, the amount of deflection on the steelbeam also increases.

When the initial load of 2N was applied, the beamdeflected by 0.65mm, and when the load of 4N was applied, the beam deflected bya 0.66mm. When the load reached 6N however, the beam deflected by another0.

65mm. These results show that the deflection of the beam is constant when thesame amount of load is applied. One source of error which could havepotentially affected the results would be the dial gauge, to get an exactreading, the dial gauge has to be set to 0 exactly, any small objects near theset up (i.e. on the table) could potentially affect the results. Another sourceof error could be due to the way couples were applied to the beam, as the loadhas to be applied simultaneously, any small difference in time between the twoweights could potentially affect the results. One potential solution to getmore of an accurate reading would be to conduct the experiment multiple times. 5.

) Conclusion In conclusion, the experiment proves that thereis a positive correlation between the deflection of a steel beam and loadsapplied. This shows that, the higher the number of loads applied onto the beam,the deflection of the beam is greater. The graphs as shown in the results pageas Figure2 show that the trend lines are drawn in ascending order which shows apositive relationship between the loads applied and the deflection of the beam.The gradient of the graph, k, is determined to be . Therefore, using the formula,  elastic modulus of thebeam is determined as whichwas simplified to. An online database whichmainly holds measurements of Young’s modulus E for various materials shows thatthis steel beam has a higher value for Young’s modulus E than aluminium, iron, copperand titanium alloys. This means that this steel beam is brittle yet strong.