Experiment 1: Young’s modulus Objective(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.
Introduction The Young’s Modulus Apparatus is a bench top model designed to understand and to determine the Young’s 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. Young’s 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 Young’s 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 Young’s 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 = bh³/12, where b = width of beam and h = height of beam.
Material and Apparatus a) 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 won’t 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
1st reading
Thickness, h (mm) Stainless Aluminium Brass Steel 4.35 3.80 4.33
Width, b (mm) Stainless Aluminium Brass Steel 23.50 23.90 23.70
2nd reading
4.40
4.00
4.32
23.50
23.90
23.70
3rd reading
4.28
3.90
4.40
23.50
23.90
23.70
Average reading
4.34
3.90 4.35 Table 1
23.50
23.90
23.70
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 3.25 Table 3
Calculation(s)
Part 1 : One fixed end and one simple support end
The deflection at length a from the fixed support: δ = F a³ (L – a) ² (4L – a) / 12EIL³ For a load in the centre of the beam, substituting a = L/2 in the above equation, the deflection is: δ = 3.5FL³ / 384 EI E = 3.5FL³ / 384I δ E = (F / δ) (3.5L³ / 384I) The moment of inertia for a rectangular section beam, I = bh³ / 12
The moment of Inertia, I , of rectangular section of brass beam, I = [ 23.70 x 10 ^(-3)m] [4.35x 10 ^(-3)m]³ / 12 = 1.63 x 10^(-10) m^(4) F / δ = 20.00N/0.00171m =11696.00 N/m E = 11696.00 N/m [3.50(0.80m)³ / 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 |
x 100%
x 100%
100.00 GPa = 2.35 % The moment of Inertia, I , of rectangular section of aluminium beam,
I = [ 23.50 x 10 ^(-3)m] [4.34 x 10 ^(-3)m]³ / 12 = 1.60 x 10^(-10) m^(4) F / δ = 20.00N/0.00259m =7722.00 N/m E = 7722.00 N/m [3.50(0.80m)³ / 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 |
x 100%
x 100%
210.00 GPa = 0.07 %
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] ³ / 12 = 1.18x 10^ (-10) m^ (4) F / δ = 20.00N/0.0011m =18182.00N/m E = 18182.00 N/m [3.50(0.80m)³ / 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: δ = F a² (L – a)² / 3EIL For a load in the centre of the beam, substituting a = L/2 in the above equation, the deflection is: δ = FL³ / 48EI E = FL³ / 48Iδ E = (F / δ) (L³ / 48I) The moment of inertia for a rectangular section beam, I = bh³ / 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]³ / 12 = 1.63 x 10^(-10) m^(4) F / δ = 35.00N/0.00292m =11986.00 N/m E = 11986.00 N/m [(0.6m)³ / 48(7.12 x 10^(-11) m^(4)] = 7.58 x 10^(11) Pa = 758.00 GPa Percentage Error = | Theoretical Value – Experimental Value | Theoretical Value = | 100.00 GPa – 758.00 GPa |
x 100%
x 100%
100.00 GPa = 6.58 % 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]³ / 12
= 1.18x 10^(-10) m^(4) F / δ = 35.00N/0.00199m =17588.00 N/m
E = 17588.00 N/m [(0.6m)³ / 48(4.26 x 10^(-11) m^(4)] = 1.858 x 10^(12) Pa = 1858.00 GPa Percentage Error = | Theoretical Value – Experimental Value | Theoretical Value = | 195.00 GPa – 1858.00 GPa |
x 100%
x 100%
195.00 GPa = 8.53 % The moment of Inertia, I , of rectangular section of aluminium beam, I = [23.5 x 10 ^(-3)m] [4.34 x 10 ^(-3)m]³ / 12 = 1.60x 10^(-10) m^(4) F / δ = 35.00N/0.00454m =7709.00 N/m E = 7709.00 N/m [(0.6m)³ / 48(4.818 x 10^(-11) m^(4)] = 7.20x 10^(11) Pa = 720.00 GPa Percentage Error = | Theoretical Value – Experimental Value | Theoretical Value = | 210.00 GPa – 720.00 GPa | 210.00 GPa = 2.43 %
Notation:
x 100%
x 100%
E = Young modulus (Pa) F = Force/load applied (N) δ = Deflection (m) L = Beam length (m) I = Moment of Inertia (m4) F/δ = Slope of graph line force versus deflection
Discussion Based on the experiment, Young’s Modulus, It is also known as the Young modulus, modulus of elasticity, elastic modulus or tensile modulus. The SI unit of modulus of elasticity (E) is the Newton over meter square (N/m²); the practical unit is gigapascals (GPa). Hence, it is a measure of the stiffness of an isotropic elastic material. So, we used three types of materials which consists of the Aluminium (Al), the Brass, and the Stainless Steel. However, these materials are used to measure the Young’s Modulus by using a bench top model of Young’s Modulus Apparatus. There are two conditions apply to the specimens which are; (1) One of the end of the beam was fixed and another end with the simple support only. (2) Both the ends of the beam were simple supports only. Both of the conditions are with the loads mounted on the middle of the test specimens.
There is a force always acting towards the Earth which is the gravity force. Therefore, there is a force acting downwards which is applied on the surface of the beam, F=mg, where m is the mass of the loads and g is the gravitational acceleration. This force causes the beam to deflect and gives readings to the dial gauge. However, there is always a resisting force acting against the gravitational force which is caused by the stiffness of the isotropic elastic materials that will react by applying a reaction force back to the downward force as action of resistance to the deflection which is called the Young Modulus.
Hence, we have conducted the experiments using the formulas of
E = (F / δ) (3.5L³ / 384I) which is used for part 1 whereby one end of the beam is fixed and one simple support end. In Part 2, both ends are just simple support of the beam, we used the formula of
E = (F / δ) (L³ / 48I) With the width and the thickness measured by using the meter ruler, we can calculate the moment of inertia for all the rectangular beams by the formula of
Moment of Inertia, I = bh x h² / 12 where b is the width of the beams and h is the thickness of the beams. From the equation above, we can know that the moment of inertia, I is directly proportional to the width and the thickness of the rectangular beams. Moreover, there is a general formula of Young Modulus which is;
where, E
= the Young's modulus (modulus of elasticity)
F
= the force applied to the object;
A0 = the original cross-sectional area through which the force is applied; ΔL = the amount by which the length of the object changes; L0 = the original length of the object.
The length of the deflection is proportional to the amount of the loads that are loaded on the surface of the beams whenever the end of the beams is fixed or both are just simple support according to the experiments conducted. For Part 1 results, brass beam =335.00Gpa, stainless steel = 719.00Gpa, and aluminium = 225.00Gpa. Whereas in Part 2, brass beam =758.00Gpa, stainless steel = 720.00Gpa, and aluminium = 1858.00Gpa. Thus, we can clearly see that both simple supports of the beams’ ends are larger than the one fixed end. Hence, we can conclude that the more the loads are added to the beams the deflection of the beams will also increase. However, when the loads added to the beams have reached to the maximum or limit of the deflection of the beams, there will be no further deflection when loads are added on it. Nevertheless, we can find another suitable and the limitation of stress on the materials with the measurements of the Young Modulus which is the carbon fibre. It consists of thin fibres with 0.005-0.010mm in diameter because mostly composed of carbon atoms. It has high tensile strength, low weight, and
low thermal expansion which can withstands an impact of many tons and deform minimally. There are several precaution steps that should be avoided during the experiments. Firstly, the vernier calliper and the dial gauge used should always directly to the zero readings in order to prevent zero errors. Secondly, we have to prevent parallax errors while taking the readings in the dial gauge and the vernier calliper. Thirdly, the loads hanger should be directly in the middle of the beams so that there is less errors in the readings.
Conclusion As a conclusion, the length of deflection is directly proportional to the magnitude of the force that applied on the surface of the beams, whether the ends of the beam are fixed or both of the ends are simply supported. The condition of Young’s modulus for both ends using simple supported bearers is larger than the first one fixed end. The length of the beams is inversely proportional to the Young’s modulus. Furthermore, stainless steel has higher value of Young’s modulus compared to the rest and this makes it much stiffer than other specimen beam(s). Finally, the width and height of a rectangular section beam is directly proportional to its moment of inertia.
Reference(s): 1. Wikipedia, The Online Cyclopedia, Young’s Modulus ,20 June 2009
(http://en.wikipedia.org/wiki/Young's_modulus) 2. Wikipedia, The Online Cyclopedia, Mild Steel ,20 June 2009 (http://en.wikipedia.org/wiki/Mild_steel) 3. Online calculator counting Young Modulus, 20 June 2009 (http://www.allmeasures.com/Formulae/static/materials/5/youngs_modulus.htm) 4. The CODECOGS Engineering ,Shear force and bending moment, 21 June 2009 (http://www.codecogs.com/reference/engineering/materials/shear_force_and_bending_ moment.php)
5. Wikipedia, The Online Cyclopedia, Carbon fiber ,21 June 2009 (http://en.wikipedia.org/wiki/Carbon_fiber)
6. Wikipedia, The Online Cyclopedia, Dial indicator ,21 June 2009 (http://en.wikipedia.org/wiki/Dial_indicator) 7. Wikipedia, The Online Cyclopedia, Moment of inertia ,21 June 2009
(http://en.wikipedia.org/wiki/Moment_of_inertia)
Appendix
