young’s modulus experiment 1

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Young's Modulus Statics Experiment Full Report

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Experiment 1: Youngs modulusObjective(s)I.

II.

To investigate the relationship between load , span, width, height and the deflection of a beam that placed on two bearers and affected by a concentrated load at the centre. To ascertain the coefficient of elasticity for stainless steel, brass and mild steel.

Abstract(s)In the experiment, the material such as stainless steel, brass and Aluminium was used as a beam which is broken into two parts. The part one is where the beam is fixed at one end and one simple support end whereas on part two the setup was set to two simple support ends. Through these, the deflection value from the three kind of beam was measured when the load was applied on the center of the beam. The amount of loads was varied throughout both parts in order for us to investigate the relationship between the loads, span width, height and the deflection. Through this, we are able to measure the deflection of the beam(s) with a dial gauge where each experiment done on the beam was repeated three times. The reason different beam(s) was used is to compare the Coefficient of Elasticity between the different types of material used on the beam. During the experiment we also required to measure the width and the thickness of the beam used in order to get the cross sectional area so that the moment of inertia could be determined.

IntroductionThe Youngs Modulus Apparatus is a bench top model designed to understand and to determine the Youngs Modulus of given material sample(s). It consists of an epoxy coated steel reaction frame complete with a meter long linear scale. Two adjustable supports provide the variable span needed to perform the experiment. The weights and hanger are provided for loading of the beams. One set of dial gauges to 0.01mm resolutions complete with mounting brackets are employed for the measurement of the beam deflection. Youngs modulus The elastic modulus is one of the most vital properties involved in various aspects of material engineering for design purposes. Every material undergoes elastic deformation. Elastic deformation is mostly defined as temporary deformation of its physical shape and will able to return to its original state. For elastic deformation, the material undergoes an amount of stress without exceeding the elastic limit. Any deformation caused by further increases in load or stress beyond the yield point of a certain material will be plastic permanent.

The Youngs modulus (elastic modulus) is the measurement of the stiffness of a given material. It is defined as the limit for small strains of the rate of change of stress with strain. Besides using the stress and strain graphs, the Youngs Modulus of any material can also be determined by using the deflection of the material (beam) when subjected to load. Moment of Inertia, I Moment of Inertia, I, is the property of an object associated with its resistance to rotation. It depends on the objects mass and the distribution of mass with respect to the axis of rotation. For any beam, the inertia is calculated based on the cross sectional shape and the thickness. It does not depend on the length and material of the beam. For a rectangular section beam, I = bh/12, where b = width of beam and h = height of beam.

Material and Apparatusa) b) c) d) Set of hanger and weights Set of dial gauge (0.01 mm resolution) Four leveling feet with built in spirit level Stainless steel specimen e) Aluminium specimen f) Brass specimen

Procedure(s)Part One: (a)One fixed end and one simple support end 1. The clamping length (L) was set to 800mm. 2. The width and height of the test specimen was measured by using a vernier caliper and the values were recorded. 3. The test specimen was placed on the bearers. 4. One of the ends was set as fixed end and was tighten by using Allen key. 5. The load (F) hanger was mounted on the center of the test specimen. 6. The dial gauge was moved to the center of the test specimen. The height of the gauge was adjusted in order for the needle to touch the test specimen. The initial reading of gauge was recorded. 7. The load of 5N weight was loaded onto the weight hanger and the dial gauge reading was recorded 8. The experiment was then continued by varying the loads every once by increment of 5N until 25N. All the dial gauge readings were recorded. 9. All the loads were removed after the results were taken. 10. The experiment was repeated for another two times in order to obtain an average deflection value. 11. The graph of force versus deflection was plotted. 12. The experimental value of Young modulus for the respective was calculated for the respective beam by comparing the theoretical value.

13. The experiment was repeated by using different material beam (i.e. Aluminium,

brass, stainless steel) Part Two: (b)Two simple supports end. 1. The clamping length (L) was set to 600mm. 2. The width and height of the test specimen was measured by using a vernier caliper and the values were recorded. 3. The test specimen was placed on the bearers. 4. Both of the ends wont be tighten since both ends are simple support. 5. The load (F) hanger was mounted on the center of the test specimen. 6. The dial gauge was moved to the center of the test specimen. The height of the gauge was adjusted so that the needle touched the test specimen. The initial reading of gauge was recorded. 7. The load of 5N weight was loaded onto the weight hanger and the dial gauge reading was recorded. 8. The experiment was repeated for another two times in order to obtain an average deflection value. 9. All the loads were removed after the results were taken. 10. The graph of force versus deflection was plotted. 11. The experimental value of Young modulus for the respective was calculated for the respective beam by comparing the theoretical value. 12. The experiment was repeated by using different material beam (i.e. Aluminium, brass, stainless steel)

RESULT(s)Length, L in Part I = 800 mm Length, L in Part II = 600 mm Thickness, h (mm) Stainless Aluminium Brass Steel 4.35 3.80 4.33 4.40 4.28 4.34 4.00 3.90 4.32 4.40 Width, b (mm) Stainless Aluminium Brass Steel 23.50 23.90 23.70 23.50 23.50 23.50 23.90 23.90 23.90 23.70 23.70 23.70

1st reading 2nd reading 3rd reading Average reading

3.90 4.35 Table 1

Part I One fixed end and one simple support end. Deflection #1 (mm) Deflection #2 (mm) Deflection #3 (mm) Average Deflection (mm) Load Alumi- Stainless Alumi- Stainless Alumi- Stainless Alumi- Stainless (N) Brass Brass Brass Brass nium Steel nium Steel nium Steel nium Steel 5 0.52 0.10 0.22 0.53 0.14 0.20 0.52 0.13 0.20 0.52 0.12 0.21 10 1.18 0.28 0.58 1.18 0.34 0.60 1.18 0.34 0.58 1.18 0.32 0.59 15 1.82 0.53 1.03 1.83 0.59 1.03 1.83 0.60 1.03 1.83 0.57 1.03 20 2.47 0.95 1.49 2.48 0.90 1.48 2.48 0.89 1.48 2.48 0.91 1.48 25 3.11 1.27 1.92 3.14 1.21 1.93 3.09 1.19 1.92 3.11 1.22 1.92 Table 2 Part II Two simple support end.

Deflection #1 (mm) Deflection #2 (mm) Deflection #3 (mm) Average Deflection (mm) Load Alumi- Stainless Alumi- Stainless Alumi- Stainless Alumi- Stainless (N) Brass Brass Brass Brass nium Steel niun Steel nium Steel nium Steel 5 0.66 0.29 0.34 0.66 0.28 0.33 0.65 0.28 0.33 0.66 0.28 0.33 10 1.31 0.59 0.74 1.31 0.58 0.74 1.31 0.58 0.74 1.31 0.58 0.74 15 1.96 0.88 1.15 1.95 0.86 1.15 1.96 0.87 1.15 1.96 0.87 1.15 20 2.61 1.15 1.56 2.61 1.15 1.57 2.61 1.14 1.56 2.61 1.15 1.56 25 3.27 1.43 1.97 3.27 1.43 1.98 3.26 1.43 1.99 3.27 1.43 1.98 30 3.90 1.70 2.39 3.91 1.71 2.40 3.92 1.71 2.40 3.91 1.71 2.40 35 4.56 2.00 2.81 4.55 2.00 2.81 4.59 2.00 2.81 4.57 2.00 2.81 40 5.20 2.28 3.25 5.20 2.27 3.25 5.20 2.27 3.24 5.20 2.27 3.25Table 3

Calculation(s)

Part 1 : One fixed end and one simple support end

The deflection at length a from the fixed support: = F a (L a) (4L a) / 12EIL For a load in the centre of the beam, substituting a = L/2 in the above equation, the deflection is: = 3.5FL / 384 EI E = 3.5FL / 384I E = (F / ) (3.5L / 384I) The moment of inertia for a rectangular section beam, I = bh / 12

The moment of Inertia, I , of rectangular section of brass beam, I = [ 23.70 x 10 ^(-3)m] [4.35x 10 ^(-3)m] / 12 = 1.63 x 10^(-10) m^(4) F / = 20.00N/0.00171m =11696.00 N/m E = 11696.00 N/m [3.50(0.80m) / 384(1.63 x 10^(-10) m^(4)] = 3.35 x 10^(11) Pa = 335.00 GPa Percentage Error = | Theoretical Value Experimental Value | Theoretical Value = | 100.00 GPa 335 GPa | 100.00 GPa = 2.35 % The moment of Inertia, I , of rectangular section of aluminium beam, x 100% x 100%

I = [ 23.50 x 10 ^(-3)m] [4.34 x 10 ^(-3)m] / 12 = 1.60 x 10^(-10) m^(4) F / = 20.00N/0.00259m =7722.00 N/m E = 7722.00 N/m [3.50(0.80m) / 384(1.60x 10^(-10) m^(4)] = 2.25 x 10^(11) Pa = 225.00 GPa Percentage Error = | Theoretical Value Experimental Value | Theoretical Value = | 210.00 GPa 225.00 GPa | 210.00 GPa = 0.07 % x 100%

x 100%

The moment of Inertia, I, of rectangular section of stainless steel beam, I = [23.90 x 10 ^ (-3) m] [3.90 x 10 ^ (-3) m] / 12 = 1.18x 10^ (-10) m^ (4) F / = 20.00N/0.0011m =18182.00N/m E = 18182.00 N/m [3.50(0.80m) / 384(1.18x 10^(-10) m^(4)] = 7.19 x 10^(11) Pa = 719.00 GPa Percentage Error = | Theoretical Value Experimental Value | Theoretical Value = | 195.00 GPa 719GPa | 195.00 GPa =2.69 % x 100% x 100%

Part 2 : Two simple support end

The deflection at distance a from the left-hand hand support is: = F a (L a) / 3EIL For a load in the centre of the beam, substituting a = L/2 in the above equation, the deflection is: = FL / 48EI E = FL / 48I E = (F / ) (L / 48I) The moment of inertia for a rectangular section beam, I = bh / 12

The moment of Inertia, I , of rectangular section of brass beam, I = [ 23.70 x 10 ^(-3)m] [4.35 x 10 ^(-3)m] / 12 = 1.63 x 10^(-10) m^(4) F / = 35.00N/0.00292m =11986.00 N/m E = 11986.00 N/m [(0.6m) / 48(7.12 x 10^(-11) m^(4)

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