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engineering principles lab report on beams and young modulus...

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Mechanics Case Study By Aleksandra Luczyk Word count: 1056

Abstract In these experiments, we have looked at different materials and how their properties affect stress and strain of beams. The beams were made out of three different materials and were all tested to show how deflection and extension changes when different loads are placed on the beam. From these result’s we were able to calculate the stress and strain of each material and hence young’s modulus. In both experiments we found out that the best material used for safe structures would be steel as it had the highest young’s modulus in comparison to the other materials.

Contents 1. 2. 3. 4. 5. 6.

Introduction Beam Material Tensile Tests Discussion Conclusion References

Introduction Many buildings and bridges consist of components such as beams. Beams are types of structures which support loads. These structures need to be strong to ensure they support different loads which may be placed on them as well as their own weight. As beams are used within many real-life examples, they need to be safe for users. Deflection is a factor which needs to be considered when placing loads on beams. The deflection can be calculated which allows engineers to know whether a material is suitable to use on for example; a bridge. If a beam deflects excessively under a load, it can cause damage to the structure or break eventually. Different materials have been used within the experiment to determine which one deals best under different loads. The formula for deflection is as follows: 𝑊𝐿3 𝛿= 𝐾𝐸𝐼 The beam stiffness measure how the beam will resist deflection under different loads. Young’s modulus determines the stiffness of a material and therefore is important when considering materials for structures. Young’s modulus is the ratio of stress to strain. The formulas for stress, strain and Young’s modulus are as follows: 𝜎=

𝐹 𝐴

𝜀=

∆𝐿 𝐿

𝐸=

𝜎 𝜀

Beam Material

Figure 1: TecQuipment experiment setup for Beam Material.

Aim The aim of this experiment is to determine whether beams of different materials deflect different amounts when the same amount of force is applied on them and also show the relationship between beam material and deflection. Method 1. Select 3 beams of similar dimensions but of different materials (steel, aluminium and brass). 2. Measure the beam dimensions (d) and (b) and calculate (I) value. 3. Rest aluminium beam across supports then adjust dial indicator and wire stirrup to the centre of the beam (180mm). 4. Set dial indicator to zero. 5. Starting from zero, add weights onto the wire stirrup in increments of 100g, up to a maximum of 500g. 6. Tap the work panel to reduce friction. 7. Record deflection at each weight. 8. Calculate load (W) and results for 48𝛿𝐼/𝐿3 9. Repeat for other two materials.

10. Plot a graph of load (W) against 48𝛿𝐼/𝐿3 for each material. The gradient of the graph will give the value of Young’s modulus (E). Results

Load (m) / g

Load (W) / N

0 100 200 300 400 500

0 0.98 1.96 2.94 3.92 4.90

Beam Modulus Deflection / mm

0 0.58 1.10 1.21 2.14 2.87 Material: Aluminium Beam Size: 3.2 x 9.5 x 372 (mm)

Deflection (δ) / m

48𝛿𝐼/𝐿3

0 5.80x10−4 1.10x10−3 1.21x10−3 2.14x10−3 2.87x10−3

0 1.37x10−11 2.61x10−11 2.87x10−11 5.07x10−11 6.80x10−11

I: 2.54 × 10−11

Graph showing Load against 48𝛿𝐼/𝐿^3 for aluminium 6 5

Load (N)

4 3 2

1 0 0

10

20

30

40

48𝛿𝐼/𝐿^3 (x10^-12)

50

60

70

80

Load (m) / g

Load (W) / N

0 100 200 300 400 500

0 0.98 1.96 2.94 3.92 4.90

Beam Modulus Deflection / mm

0 0.18 0.38 0.56 0.73 0.92 Material: Steel Beam Size: 3.2 x 9.5 x 372 (mm)

Deflection (δ) / m

48𝛿𝐼/𝐿3

0 1.80x10−4 3.80x10−4 5.60x10−4 7.30x10−4 9.20x10−4

0 5.90x10−13 1.25x10−12 1.84x10−12 2.39x10−12 3.02x10−12

I: 2.54 × 10−11

Graph showing Load against 48𝛿𝐼/𝐿^3 for steel 6 5

Load (N)

4 3 2 1 0 0

0.5

1

1.5

2

-1

48𝛿𝐼/𝐿^3 (x10^-12)

2.5

3

3.5

Load (m) / g

Load (W) / N

0 100 200 300 400 500

0 0.98 1.96 2.94 3.92 4.90

Beam Modulus Deflection / mm

Deflection (δ) / m

48𝛿𝐼/𝐿3

0 3.60x10−4 6.80x10−4 1.09x10−3 1.43x10−3 1.71x10−3

0 1.18x10−12 2.23x10−12 3.57x10−12 4.69x10−12 5.60x10−12

0 0.36 0.68 1.09 1.43 1.71 Material: Brass Beam Size: 3.2 x 9.5 x 372 (mm)

I: 2.54 × 10−11

Graph showing Load against 48𝛿𝐼/𝐿^3 for brass 6

5

Load (N)

4 3 2 1 0 0

1

2

3

-1

48𝛿𝐼/𝐿^3 (x10^-12)

4

5

6

Tensile Tests

Figure 1: TecQuipment experiment setup for Tensile Tests.

Aim The aim of this experiment is to test the tensile properties of different materials and hence calculate their stress and strain. Method 1. 2. 3. 4. 5. 6. 7. 8. 9.

Select three specimens of different materials (PVC, aluminium and steel). Measure dimensions of test specimens and find cross-sectional area. Place PVC specimen in the tensile tester kit and set dial indicator to zero. Turn the load nut clockwise slowly in increments of 0.2mm until the specimen breaks. Measure the elongation by pushing the broken specimen together and measuring its final length. Convert dial indicator values into force values and calculate the extension. Calculate stress (σ) and strain (ε) values. Repeat for other two materials. Plot a graph of stress (σ) against strain (ε) for each material. The gradient of the graph will give the value of Young’s modulus (E).

Results Specimen: PVC Original cross-sectional area ( 𝑚𝑚2 ): 2 Original length (mm): 31 Load nut Dial Indicator Force (N) movement (mm) (mm) 0 0 0 0.2 0.03 3 0.4 0.03 3 0.6 0.03 3 0.8 0.04 4 1.0 0.1 10 1.2 0.15 15 1.4 0.22 22 1.6 0.28 28 1.8 0.34 34 2.0 0.39 39 2.2 0.42 42 2.4 0.44 44 2.6 0.32 32 2.8 0.34 34 3.0 0.35 35 3.2 0.34 34 3.4 0.33 33 3.6 0.33 33 3.8 0.33 33 4.0 0.33 33 4.2 0.33 33 4.4 0.34 34 4.6 0.33 33 4.8 0.33 33 5.0 0.33 33 6.0 0.33 33 7.0 0.36 36 8.0 0.37 37 9.0 0.37 37 10.0 0.37 37 20.0 BREAK Tensile Strength ( 𝑁𝑚−2 ): 1.85x107 Yield Stress (𝑁𝑚−2 ): 1.95x107 % Elongation: 31.06

Extension (mm)

Stress (σ) (𝑁𝑚−2 )

Nominal Strain (ε)

0 0.17 0.37 0.57 0.76 0.90 1.05 1.18 1.32 1.46 1.61 1.78 1.96 2.28 2.46 2.65 2.86 3.07 3.27 3.47 3.67 3.87 4.06 4.27 4.47 4.67 5.67 6.64 7.63 8.63 9.63

0 1.5x106 1.5x106 1.5x106 2.0x106 5.0x106 7.5x106 1.1x107 1.4x107 1.7x107 1.95x107 2.1x107 2.2x107 1.6x107 1.7x107 1.75x107 1.7x107 1.65x107 1.65x107 1.65x107 1.65x107 1.65x107 1.7x107 1.65x107 1.65x107 1.65x107 1.65x107 1.8x107 1.85x107 1.85x107 1.85x107

0 5.48x10−3 1.19x10−2 1.84x10−2 2.45x10−2 2.9x10−2 3.39x10−2 3.81x10−2 4.26x10−2 4.71x10−2 5.19x10−2 5.74x10−2 6.32x10−2 7.35x10−2 7.94x10−2 8.54x10−2 9.22x10−2 9.90x10−2 1.05x10−1 1.11x10−1 1.18x10−1 1.25x10−1 1.31x10−1 1.38x10−1 1.44x10−1 1.51x10−1 1.83x10−1 2.14x10−1 2.46x10−1 2.78x10−1 3.11x10−1

Graph showing stress against strain graph for PVC 25

Stress (MPa)

20

15

10

5

0 0

0.05

0.1

0.15

Strain

0.2

0.25

0.3

0.35

Specimen: Steel Original cross-sectional area ( 𝑚𝑚2 ): 2 Original length (mm): 31 Load nut Dial Indicator Force (N) movement (mm) (mm) 0 0 0 0.2 0.008 0.8 0.4 0.009 0.9 0.6 0.01 1.0 0.8 0.023 2.3 1.0 0.034 3.4 1.2 0.051 5.1 1.4 0.068 6.8 1.6 0.086 8.6 1.8 0.104 10.4 2.0 0.118 11.8 2.2 0.134 13.4 2.4 0.158 15.8 2.6 0.172 17.2 2.8 0.19 19.0 3.0 0.21 21.0 3.2 0.227 22.7 3.4 0.245 24.5 3.6 0.261 26.1 3.8 0.281 28.1 4.0 0.3 30.0 4.2 0.318 31.8 4.4 0.327 32.7 4.6 0.354 35.4 4.8 0.327 32.7 5.0 0.354 35.4 6.0 0.445 44.5 7.0 0.511 51.1 8.0 0.555 55.5 9.0 0.58 58.0 10.0 0.588 58.8 20.0 BREAK Tensile Strength ( 𝑁𝑚−2 ): 2.94x107 Yield Stress (𝑁𝑚−2 ): 1.59x107 % Elongation: 30.36

Extension (mm)

Stress (σ) (𝑁𝑚−2 )

Nominal Strain (ε)

0 0.192 0.391 0.59 0.777 0.966 1.149 1.332 1.514 1.696 1.882 2.066 2.242 2.428 2.61 2.79 2.973 3.155 3.339 3.519 3.7 3.882 4.073 4.246 4.473 4.646 5.555 6.489 7.445 8.42 9.412

0 4x105 4.5x105 5x105 1.15x106 1.7x106 2.55x106 3.4x106 4.3x106 5.2x106 5.9x106 6.7x106 7.9x106 8.6x106 9.5x106 1.05x107 1.1135x107 1.225x107 1.305x107 1.405x107 1.5x107 1.59x107 1.635x107 1.77x107 1.635x107 1.77x107 2.225x107 2.225x107 2.775x107 2.9x107 2.94x107

0 6.19x10−3 1.26x10−2 1.90x10−2 2.51x10−2 3.12x10−2 3.71x10−2 4.30x10−2 4.88x10−2 5.47x10−2 6.07x10−2 6.66x10−2 7.23x10−2 7.83x10−2 8.41x10−2 9.0x10−2 9.59x10−2 1.02x10−1 1.08x10−1 1.14x10−1 1.19x10−1 1.25x10−1 1.31x10−1 1.37x10−1 1.44x10−1 1.50x10−1 1.79x10−1 2.09x10−1 2.40x10−1 2.72x10−1 3.04x10−1

Graph showing stress against strain for steel 35 30

Stress (MPa)

25 20 15

10 5 0 0

0.05

0.1

0.15

0.2

Strain

0.25

0.3

0.35

Specimen: Aluminium Original cross-sectional area ( 𝑚𝑚2 ): 2 Original length (mm): 31 Load nut Dial Indicator Force (N) movement (mm) (mm) 0 0 0 0.2 0.013 1.3 0.4 0.028 2.8 0.6 0.044 4.4 0.8 0.054 5.4 1.0 0.072 7.2 1.2 0.0875 8.75 1.4 0.104 10.4 1.6 0.12 12 1.8 0.136 13.6 2.0 0.153 15.3 2.2 0.168 16.8 2.4 0.183 18.3 2.6 0.19 19 2.8 0.203 20.3 3.0 0.215 21.5 3.2 0.228 22.8 3.4 0.234 23.4 3.6 0.242 24.2 3.8 0.2515 25.15 4.0 0.2545 25.45 4.2 0.262 26.2 4.4 0.265 26.5 4.6 0.271 27.1 4.8 0.274 27.4 5.0 0.275 27.5 6.0 BREAK Tensile Strength (𝑁𝑚−2 ): 1.375x107 Yield Stress (𝑁𝑚−2 ): 9.15x106 % Elongation: 15.24

Extension (mm)

Stress (σ) (𝑁𝑚−2 )

Nominal Strain (ε)

0 0.187 0.372 0.556 0.746 0.928 1.1125 1.296 1.48 1.664 1.847 2.032 2.217 2.41 2.597 2.785 2.972 3.166 3.358 3.5485 3.7455 3.938 4.135 4.329 4.526 4.725

0 6.5x105 1.4x106 2.2x106 2.7x106 3.6x106 4.375x106 5.2x106 6.0x106 6.8x106 7.65x106 8.4x106 9.15x106 9.5x106 1.015x107 1.075x107 1.14x107 1.17x107 1.21x107 1.2575x107 1.2725x107 1.31x107 1.325x107 1.355x107 1.37x107 1.375x107

0 6.03x10−3 1.2x10−2 1.79x10−2 2.41x10−2 2.99x10−2 3.59x10−2 4.18x10−2 4.77x10−2 5.37x10−2 5.96x10−2 6.55x10−2 7.15x10−2 7.77x10−2 8.38x10−2 8.98x10−2 9.59x10−2 1.02x10−1 1.08x10−1 1.14x10−1 1.21x10−1 1.27x10−1 1.33x10−1 1.40x10−1 1.46x10−1 1.52x10−1

Graph showing stress against strain for aluminium 16 14

Stress (MPa)

12 10 8 6 4 2 0 0

0.02

0.04

0.06

0.08

Strain

0.1

0.12

0.14

0.16

Discussion Within the tensile tests, we view how different loads affect a material in terms of stress and strain. The stress distributed on a beam is a necessary factor for determining how well a material performs. When a force is applied to a material, it will stretch and return to its original shape if the material is perfectly elastic. After a certain stress, the material will no longer return to its original length. This is when the material reaches plastic deformation and will continue to stretch until it breaks. Looking at our graphs, we can see that both PVC and steel have a kink in their graph whereas aluminium does not. This shows that aluminium has a moderate capacity to withstand fracture within its plastic deformation zone. Aluminium however is not the best material to use within bridges or buildings as it did fracture the quickest. Steel performed the best as it is one of the toughest materials and it is able to absorb high energy before fracture. In conclusion, steel would be the best material as in comparison to aluminium, it can withstand double the amount of stress before fracture.

Graph of stress against strain for aluminium and steel 35 30

Steel

Stress (MPa)

25 20 15

Aluminium

10 5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Strain

Within the beam material experiment, the aim was to find out which material would be best when dealing with deflection under load. By comparing the graphs, it is clear that steel deals with deflection much better than brass and aluminium. At 4.9N of force applied to the beams, steel deflected 0.92mm whereas aluminium deflected by 2.87mm. Although the difference is only of 1.95mm, when dealing with heavy structures like bridges, deflection needs to be kept to an absolute minimum. Steel performed best and hence would be the most suited material. The Young’s modulus determines the stiffness of a material. It can be determined from a stress-strain graph however can also be calculated using deflection of a beam when subject to load, hence both experiments show how materials perform under stress. After looking at the graphs from both experiments it is clear that aluminium is a weak material that wouldn’t necessarily be used in structures needed to support large loads. Steel however is a material which could be used.

There are different types of beam supports which we come across every day. We have cantilevers which are fixed at one end and free to move at the other. These can be seen within balconies, traffic lights, shelves and many others. They are easy to construct and only require a fixed support at one end. The issue is that they can deflect however by increasing depth, cantilever beams can be made for rigid. In some examples, deflection is good like when using a pool diving board. Simply supported beams are used within bridges. This type of beam spans horizontally and has two supports at either end. Although there are two supports, deflection and fracture can still occur. When talking about large structures like bridges, the support on a beam must be strong enough to withstand a large amount of load as well as hold the structure up itself. Cantilever beam

Figure 17: TecQuipment cantilever.

Simply supported beam

Figure 15: TecQuipment simply supported beam.

Conclusion In conclusion, steel was the strongest material when tested against aluminium, brass and PVC in both deflection and tensile tests. It is able to withstand the most load with little deflection and has the highest resistance to fracture. This therefore makes it suitable in real life structures. Although human errors may have altered some results, in general steel performed the best.

References Anon., 2018. Cite a Website - Cite This For Me [online]. Raeng.org.uk. Available from: https://www.raeng.org.uk/publications/other/15-beam-deflection [Accessed 25 Jan 2018]. Anon., 2018. Where are overhang beams used? - Quora [online]. Quora.com. Available from: https://www.quora.com/Where-are-overhang-beams-used [Accessed 25 Jan 2018]....