Title: Bending of Beam and Coefficient of Elasticity.Objective:1. To investigate the relationship between loads, span width, height and deflection of a simply supported beam.2. To ascertain the Coefficient of Elasticity (Youngs Modulus) for brass and aluminium.Introduction:Young ModulusYoungs modulus is also the modulus of elasticity, E. It is used to describe a solid materials property, or more specifically, its stiffness. Every material experiences a deformation of some sort. Mechanical deformation gives energy to a material which is either stored elastically or dissipates plastically. Elastic deformation is a temporary deformation which causes a materials physical shape to alter for a period of time and then it will return to its original state. It can return to its original state because the load that causes the deformation or stress has not exceeded its elastic limit. If the limit is exceeded, it will cause a plastic deformation which is permanent. Based on Hookes law, the modulus of elasticity is the ratio of the stress to the strain.E = /Where stress can be calculated from different formulas for different types of loading and strain is the change of length divided by the initial length. Youngs modulus is essentially a measure of the resistance of a material to elastic deformation under a certain load. A stiff material has a high youngs modulus, while a flexible material has a low youngs modulus. Youngs modulus can also be calculated based on the deflection of a material under a certain load. This is turn is affected by the length, cross-sectional shape, and material of the beam.

Moment of InertiaMoment of inertia is rotational inertia. It is the rotational analogue of mass for linear motion. In a scalar situation, moment of inertia can be described as the product of mass and the square of perpendicular distance to the rotation axis.I = mr2This can be used as a basis for all other moments of inertia regardless of shape since all objects are composites of point masses. However, moments of inertia vary according to its geometry. This general form can only be used for principle axes which include all axes of symmetry objects only.Apparatus and setup: Set of stainless steel hanger and weights (approximately 50 kg)Set of dial gauges (0.01 mm resolution)4 levelling feet with built in spirit level (1050 x 400 x 300 mm)

Procedure:

Part Ia) One fixed end and one simple support end.1) The clamping length (L) was set to 800 mm.2) The width and height of the test specimen was measures using a calliper and the values were recorded.3) The specimen was placed on the bearer.4) One end was set as a fixed end and the screw was tightened.5) The load (F) hanger was mounted on the centre of the test specimen.6) The dial gauge was moved to the centre of the test specimen. The height of the gauge was adjusted so that the needle touched the specimen. The initial reading of the gauge was recorded.7) The 5N weight was loaded onto the weight hanger and the dial gauge reading was recorded.8) Procedure (7) was repeated for loads of 10, 15 and 20N. 9) The loads were removed after all the results were taken.10) The experiment was repeated once more to obtain the average deflection value.11) A graph of force versus deflection was plotted.12) The experimental young modulus was calculated and compared with the theoretical value.13) The experiment was repeated by using a different material beam. (i.e. brass, aluminium)Part IIb) Two simple support end1) The clamping length was set to 600 mm.2) The width and height of the test specimen was measures using a calliper and the values were recorded.3) The specimen was placed on the bearer.4) The screws were not tightening to ensure both ends are simple support.5) The load (F) hanger was mounted on the centre of the test specimen.6) The dial gauge was moved to the centre of the test specimen. The height of the gauge was adjusted so that the needle touched the specimen. The initial reading of the gauge was recorded.7) The 5N weight was loaded onto the weight hanger and the dial gauge reading was recorded.8) Procedure (7) was repeated for loads of 10, 15 and 20N. 9) The loads were removed after all the results were taken.10) The experiment was repeated once more to obtain the average deflection value.11) A graph of force versus deflection was plotted.12) The experimental young modulus was calculated and compared with the theoretical value.13) The experiment was repeated by using a different material beam. (i.e. brass, aluminium)Results:BrassLength, L (mm)Thickness, h (mm)Width, b (mm)

1st reading8004.1625.12

2nd reading8004.1725.12

Average8004.16525.12

Mass (Brass) (pin fix)(N)1st (mm)2ndAverage (/mm)Young modulus (E/Pa)F/(N/mm)

50.920.930.925166778.355.41

101.881.861.87164994.625.35

152.772.772.77167079.395.41

203.753.723.735165215.505.35

Mass (Brass) (pin pin) (N)1st (mm)2ndAverage (/mm)Young modulus (E/Pa)F/(N/mm)

51.901.881.89186569.892.65

103.923.923.92179906.672.55

155.995.965.98176898.202.51

207.057.017.03200635.612.84

AluminiumLength, L (mm)Thickness, h (mm)Width, b (mm)

1st reading6003.022.34

2nd reading6003.002.44

Average6003.012.35

Mass (Aluminium) (pin pin) (N)1st (mm)2nd (mm)Average (/mm)Young modulus (E/Pa)F/(N/mm)

51.701.721.712464025.232.92

103.013.013.012799656.583.32

154.174.164.183024030.973.59

205.425.385.403121098.633.70

Calculation:I = bh3/12 = 3.5 FL3/384EI ----- E = 3.5FL3/384I (pin fix) = FL3/48EI ----------- E = FL3/48I (pin pin)BrassI = ((25.12) x (4.165)3) / 12I = 151.25 mm4Pin fixF = 5 NE = (3.5 x 5 x (800)3) / (384 x 0.925 x 151.25)E = 166778.35 PaAverage slope theoretical (F/) = (5.41 + 5.35 + 5.41 + 5.35) / 4 = 5.38 N/mmSlope from graph = 5.37 N/mmPercentage error= = =0.186%Pin pinF = 5 NE = (5 x (800)3) / (48 x 1.89 x 151.25)E = 186569.89 PaAverage slope theoretical (F/) = (2.65 + 2.55 + 2.51 + 2.84) / 4 = 2.64 N/mmSlope from graph = 2.68 N/mmPercentage error= = =0.149%AluminiumI = ((2.35) x (3.01)3) / 12I = 5.34 mm4Pin pinF= 5 NE = (5 x (600)3) / (48 x 1.71 x 5.34)E = 22464025.23 PaAverage slope theoretical (F/) = (2.92 + 3.32 + 3.59 + 3.70) / 4 = 3.38 N/mmSlope from graph = 3.57 N/mmPercentage error= = =5.32%

Discussion:Youngs modulus is linear elastic of the solid materials in mechanical property. It is used to measure the force that used to stretch or compress a solid material. When a load is applied on the solid, it will start to deform. After the load is removed, the body of the solid will back to its original shape, if the solid material is elastic. In the experiment, there is downward force applied on the beam and the reading that showed on the dial gauge is recorded. In this case, there is always a force of resistance against the gravitational force which causes the reaction force to react back to the downward force. This has shown a resistance action toward the deflection. This is also called as Youngs modulus. There are two part of experiment applied to the specimens. First is one fixed end and one simple support end. The second one is two simple supports end.Thus, the following formula is used for experiment in part 1:E = (F / ) (3.5L / 384I)

Meanwhile, the following formula is used for experiment in part 2:E = (F / ) (L / 48I)

Where E = Youngs modulusF = load applied

I = Moment of inertia

For experiment in part 1, the result of Youngs modulus for brass is 166778.35 Pa (pin fix). The average slope of the brass is 5.38N/mm. Besides, the percentage error of the slope of the brass in part 1 is 0.186%. For experiment in part 2, the result of Youngs modulus for brass is 186569.89 Pa (pin pin). The average slope for the slope of brass is 2.64 N/mm. The percentage error of the slope of the brass in part 2 is 0.149%. Meanwhile, the result of Youngs modulus for aluminium is 22464025.23 Pa (pin pin) and the average slope of the aluminium is 3.38 N/mm. The percentage error of the slope of the aluminium in part 2 is 5.32%. The percentage error of the slope of the brass is much lower than the percentage error of the slope of the aluminium.

From the results, we can see that the Youngs modulus in part 2 is larger than that in part 1. This is because the deflection of the beams will increase when more loads are applied on it. However, the beam will no longer deflect when the load added on the beams have over the limitation of the deflection of the beams. Thus, in order to solve this problem, carbon fibre can be used as the measurement of the Youngs modulus. This is because tensile strength of the carbon fibre is high. Besides, it has a lower weight. Next, it also has a lower thermal expansion which helps to withstand the material on impact and minimize the deformation.

After that, there are several precaution steps that should be considered throughout the Youngs modulus experiment. First of all, the dial gauge should always set to zero readings before start the experiment so that zero errors can be avoided. Next, the load hanger should be hanging at the middle of the beam, in order to get less errors readings. Besides, when we record the reading of the reading on the dial gauge, we should always remember to avoid the parallax error.

Conclusion:

In conclusion, according to the formula given in part 1 and part 2 of the experiment, the force applied on the surface of the beams is directly proportional to the length of the deflection. The value of the Youngs modulus in part 2 of the experiment is much larger the value of the Youngs modulus in part 1 of the experiment. This is because from the formula given, the deflection length is inversely proportional to the Youngs modulus. Besides, the Youngs modulus for the aluminium is larger than that in brass due to the stiffness of the materials. Lastly, according to the formula, the width and height of the test specimen is always directly proportional to the moment of inertia. Since the percentage error of the slope of the experiment in part 1 and part 2 is lower than 20%, the experiment is accepted.

Reference:1. Modulus of Elasticity. (2015). [online] Available at: http://tpm.fsv.cvut.cz/student/documents/files/BUM1/Chapter15.pdf [Accessed 3 Aug. 2015].2. Hyperphysics.phy-astr.gsu.edu, (2015). Moment of Inertia. [online] Available at: http://hyperphysics.phy-astr.gsu.edu/hbase/mi.html [Accessed 3 Aug. 2015].