Rigidity Modulus of a Mild Steel Bar PDF

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Rigidity and Elastic Modulus of a Mild Steel Bar...

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Structures Lab - Experiment 2 Name: A Chetan Gowda Roll Number: 19AE30001 Date of conduction of Experiment: 09/03/2021 Aim of the experiment: We need to find the modulus of rigidity and elastic modulus of a mild steel bar by subjecting it to the torsion test. Equipment Required: Torsion testing machine, Vernier Caliper, Mild steel specimen, Measuring tape, Marker Theory: In torsion testing the circular bar is placed in the machine such a way that its longitudinal axis coincides with the axis of the grips and so that it remains straight during the test. Then rotate one grip at a reasonable constant speed until the test piece breaks, here the shearing stresses will develop in any cross section of the bar whose value increases linearly from zero at the center to a maximum at the outer periphery. The assumptions made in this experiment include but are not limited to the following: 1) The torque is applied along the center of axis of the shaft. 2) The material is tested at steady state (absence of strain rate effects). 3) Plane sections remain plane after twisting. Consider now the solid circular shaft of radius R subjected to a torque T at one end, the other end being fixed under the action of this torque. Here, let the initial mark be at I. After applying a torsion T the final marker mark moves to F.

The arc length from I to F is θ*R From the diagram we get, γ =(θ*R)/L Relation between shear strain and shear stress is (G*θ)/L = τ/γ = T/J The modulus of Rigidity of the specimen is G = (T*L)/(J*θ)

Where, T = Torque applied τ = Shear stress θ = Twist angle L = Gauge length R = radius of the specimen J = Polar Moment of inertia G = Modulus of rigidity Modulus of rigidity is the measure of the elastic shear stiffness of the body, given by the ratio of shear stress to shear strain. Its SI unit is Pascal or N/m2 Modulus of elasticity is a quantity that measures an object or substance's resistance to being deformed elastically when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region. A stiffer material will have a higher elastic modulus. Its SI unit is Pascal or N/m2

Procedure: 1. Find the average diameter of the given specimen by measuring the diameter at different places and then averaging it. 2. Measure the gauge length of the given specimen. 3. Mark the specimen with markers wherever required. 4. Secure the specimen in the torsion meter. 5. Set the load pointer on the dial to zero. 6. Set the pointer to coincide with zero reading on the protractor of the torsion meter. 7. Rotate the wheels by a small amount which will gradually increase the torsion on the rod. 8. Keep noting the values of the torque as you keep increasing the angle of twist.

Observations: Diameter of the specimen (in mm) Reading-1 Reading-2 Reading-3 Reading-4 Reading-5 Reading-6

9.23 mm 9.13 mm 9.14 mm 9.14 mm 9.14 mm 9.12 mm

Angle of twist and corresponding Torque until specimen ruptures

Angle θ1

Angle θ2

Angle of twist(θ)

θ in Radian

5 7 7.5 8 9 10 10.5 11 11 11 11.5 12 12 12 12 12 12 12 12 12 12 12 12 12

6 9 11 14 17 21 24 30 60 90 140 180 270 360 540 720 900 1080 1260 1440 1620 1800 1980 2214

1 2 3.5 6 8 11 13.5 19 49 79 128.5 168 258 348 528 708 888 1068 1248 1428 1608 1788 1968 2202

0.017453 0.034907 0.061087 0.10472 0.139626 0.191986 0.235619 0.331613 0.855211 1.37881 2.242748 2.932153 4.502949 6.073745 9.215337 12.35693 15.49852 18.64011 21.78171 24.9233 28.06489 31.20648 34.34807 38.43214

Torque 7.6 17.2 24.8 33.6 43.8 53.8 58.8 61.6 65.8 69 73 74.6 76 76.8 77.6 77.8 78.2 78.4 78.7 78.8 79 79 79.1 79.2

Angle of twist and corresponding Torque while Loading and Unloading

Angle θ1

Angle θ2

Angle of twist(θ)

θ in Radian

Torque

0 4 5 6 8 9 10 11 12 11 10 8 7 6 5 4 0

0 4 5 7 9 11 13 16 17 16 14 10 8 7 5 4 0

0 0 0 1 1 2 3 5 5 5 4 2 1 1 0 0 0

0 0 0 0.0175 0.0175 0.0349 0.0523 0.0872 0.0872 0.0872 0.0698 0.0349 0.0175 0.0175 0 0 0

0 2 4.5 9.5 13.8 18.3 22.4 26.8 27.2 27 23.9 16.3 12.5 8.3 4.5 0.8 0.2

Calculation: Average diameter = 9.15 mm Gauge length (L) = 180 mm Polar Moment of inertia (J) = 688.15183 mm4 ((π*D4)/32) Modulus of rigidity (G) = T*L / J*θ Taking θ = 0.0872 radian, T = 27.2 N-m, L = 180 mm, J = 688.15183 mm4 Modulus of rigidity (G) = 81.051 GPa Relationship between Modulus of rigidity (G) and Modulus of elasticity (E). G(2*(1+μ)) = E µ is Poisson’s ratio Poisson’s ratio = Lateral strain / Longitudinal strain For mild steel, µ = 0.3 Modulus of elasticity (E) = 210.731 GPa

Result: Modulus of rigidity (G) = 81.051 GPa Modulus of elasticity (E) = 210.731 GPa

Graphs:

Torque vs Angle of Twist (Continuous loading) 90

80

70

Torque (N-m)

60

50

40

30

20

10

0 0

5

10

15

20

25

30

35

Angle of Twist (Radians)

Continuous loading until the specimen ruptures

40

45

Torque vs Angle of Twist (Loading-Unloading) 30

25

Torque (N-m)

20

15

10

5

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Angle of Twist (Radians)

Loading and Unloading the specimen

Conclusion: We got the values of modulus of rigidity and modulus of elasticity. I went and looked on the internet and found the modulus of rigidity to be 80 GPa and modulus of elasticity to be 210GPa (According to https://www.schoolphysics.co.uk). This validates our experiment and tells us that the values we got were correct and the experiment was successful.

Precautions: We have to be careful about the backlash error when measuring the loading-unloading. Cross sectional area of the rod must be uniform....