Mechanics of MaterialsECVL 268

Lab Report #3

Extracting the Modulus of Rigidity of Materials through Torsion Test

ByMalek Al Bayyari,

Ahmed KhalafAtanaz Hatamifardi

4th November, 2013

Table of Contents

Table of Contents …………………………………………………………………………………. iList of Tables …………………………………………………………………………………….. iiList of Figures …………………………………………………………………………………… iiiObjectives ………………………………………………………………………………………... 1Background and Theory ………………………………………………………………………….. 1Materials and Apparatus………………………………………………………….………….…… 3Test Procedure………………………………………………………….……………………....… 3Data ……………………………………………………………………………………………… 4Data Analysis ………………………………………………...………………….……………….. 5Error Analysis ………………………………………...…….………………….……………..….. 8Conclusion …...……………………………………………………………….………....……….. 8References …..……………………………………………………………….……….….…........ 8

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List of Tables

Table 1: Force and Angle of Twist for Aluminum rod …...…………………………...………… 2Table 2: Force and Angle of Twist for Plastic rod ……………………...……………………..… 3Table 3: Angle of Twist and Torque of the Aluminum Rod ……………...……………………… 5Table 4: Angle of Twist and Torque of the Plastic Rod ………………………….…....………… 6Table 5: Modulus of Rigidity of Aluminum Rod ……...……………..……….……...………….. 7Table 6: Modulus of Elasticity of Plastic Rod ………………………………...…………….…… 7

ii

List of Figures

Figure 1: Universal Testing Machine for Torsion………….…………...………………………... 4Figure 2: HI-Tech Interface Unit (MEC-16) for showing load and angle of twist ...…..………... 4Figure 3: Angle of Twist Vs. Torque for Loading Aluminum Rod ………………...…….……... 5Figure 4: Angle of Twist Vs. Torque for Loading Plastic Rod ………………………………….. 6

iii

Extracting the Modulus of Rigidity of Materials through Torsion Test

Objectives

The main objective of this experiment is to be able to calculate the shear modulus of elasticity for stainless steel and plastic by calculating the angle of twist that was resulted from the torsion force

Background and Theory

The modulus of rigidity also known as shear modulus of elasticity is an important mechanical property that indicates the elasticity of a material for a torsion force. It indicates how elastic or bendable a material is, as well as Young’s modulus of elasticity all materials have a shear modulus of elasticity. Materials having this property should have a linear elastic behavior, it’s the ratio of shear stress to within the elastic behavior of the material (Hibbeler). The modulus of elasticity is calculated using the following equation (Also known as Hooke’s Law):

G = τγ (1)

Where G = Shear Modulus of Elasticity (GPa)τ = Shear Stress GPaγ = Shear Stain (mm/mm)

It is the rate of change of strain as a function of stress, to obtain the stiffness of the elastic material. The modulus of rigidity can be determined via a torsion test of rods, by measuring the angle of twist of the rod, by applying a load F at any point (Hibbeler). By applying load and torque to the fixed rod, by plotting the angle of twist and torque diagram and using its slope the modulus of rigidity can be determined using the following equation:

G = (

Tϕ

)( L)

(J ) (2)

Where G = Shear Modulus of Elasticity (GPa)J = Polar Moment of Inertia of the Cross-Section (mm4)

(Tϕ

) = Slope of the torque vs angle of twist graph ((N*mm)/rad)

L= The length of rod (mm)

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1

The moment of inertia (solid rectangular cross-section) of the rod is calculated using the following equation:

J = π(d)4

32 (3)

Where J = Polar Moment of Inertia of the Cross-Section (mm4)d= Diameter of rod (mm)

The torque and angle of twist have a linear relationship due to the fact that the material of the rod are elastic, and thus comply by Hooke’s Law. The torsional load produces shear stress, by twisting the rod along its longitudinal axis. It creates an angle of twist occurs as the material is acting in a elastic and linear manner. Since the material is homogenous the modulus of rigidity. As it can be seen there is a direct proportion between torque and the angle of twist of the rod. There is a maximum load of about 10N applied to the rod. Though the modulus of elasticity of the beam should remain the same, the parameters that effect the deflection are; the load applied, the type of materials (it’s elasticity), and its cross sectional-area.

Figure 1: Universal Testing Machine for Torsion

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Figure 2: HI-Tech Interface Unit (MEC-16) for showing load and angle of twist

Materials and Apparatus

1) Stainless steel rod.2) Plastic rod (made from Polycarbonate) 3) Universal testing frame4) Caliper5) Measuring tape6) Knob7) HI-Tech Interface Unit (MEC-16)

Test Procedure

1) The length and diameter of the stainless steel rod were measured from three different places and the average was taken.

2) The stainless steel rod was placed into the testing frame.

3) The force was applied to the stainless steel rod by twisting the knob.

4) The torsion force and the angle were shown on the screen.

5) Twisting was stopped when the force reached 10 N.

6) Same procedure was repeated for the plastic rod.

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Data

Table 1: Force and Angle of Twist for Aluminum rodDiameter: [d1=4.71mm, d2=4.70mm, d3=4.73mm]Length: 396mmArm: 68mm

F (N) () (rad.)0 282.4 4.92

0.9 284.6 4.961.5 284.9 4.972.3 285.5 4.983.5 286.6 5.004.8 287.9 5.026.4 289.4 5.057.9 290.8 5.088.8 291.9 5.09

Table 2: Force and Angle of Twist for Plastic rodDiameter: [d1=10.0mm, d2=9.98mm, d3=9.98mm]Length: 397cmArm: 68mm

F (N) () (rad.)0 273.8 4.78

1.2 275.4 4.812.2 276.5 4.833.3 277.7 4.854.3 279.0 4.875.3 280.4 4.896.1 281.2 4.917.8 283.6 4.958.9 284.9 4.9710 286.5 5.00

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Data Analysis

The data collected was analyzed, and the Angle of Twist vs Torque Graph was drawn using Excel. The Two diagrams show the Torque (rotation. Twist) of the circular rod due to the applied load. The modulus of elasticity for both materials was calculated using the slope of the Graphs, and Equation 5, respectively.

Table 3: Angle of Twist and Torque of the Aluminum Rod

(rad.) T (N*mm)

0 00.04 61.20.05 1020.06 156.40.08 2380.1 326.40.13 435.20.16 537.20.17 598.40.19 666.4

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

100

200

300

400

500

600

700

f(x) = 3757.21034870641 x − 56.0866141732283

Aluminium Rod Torque vs. Angle of Twist

ɸ(rad.)

T (

N*m

m)

Figure 3: Angle of Twist Vs. Torque for Loading Aluminum Rod

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Table 4: Angle of Twist and Torque of the Plastic Rod

(rad.) T (N*mm)

0 00.02 81.60.04 149.60.07 224.40.09 292.40.11 360.40.13 414.80.17 530.40.19 605.20.22 680

0 0.05 0.1 0.15 0.2 0.250

100

200

300

400

500

600

700

800

f(x) = 3061.10479285134 x + 15.5251015434606

Plastic Rod Torque vs. Angle of Twist

ɸ(rad.)

T (

N*m

m)

Figure 4: Angle of Twist Vs. Torque for Loading Plastic Rod

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Table 5: Modulus of Rigidity of Aluminum RodDiameter (mm) J (mm4) G (GPa) Average G (GPa)

4.71 48.32 26.9 26.83±0.354.70 47.91 27.134.73 39.14 26.45

Table 6: Modulus of Elasticity of Plastic Rod

Diameter (mm) J (mm4) G (GPa) Average G (GPa)

10.00 981.75 1.24 1.25±0.0058

9.98 973.92 1.25

9.98 973.92 1.25

Calculations Sample (For Aluminum Rod): d1=4.71mm

J = π(d)4

32

J = π(4.71)4

32

J=48.32 mm4

Slope (Tϕ

) = 3757.2 ((N*mm)/rad)

L= 200 mm

G = (3757.2 )( 200 )

( 48.32 )

G = 26903.79 MPa

G = 26.90 GPa

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Error Analysis

Using the equation: percent difference=|theoretical value-measured valuetheoretical value | ×100 the percentage

difference between the modulus of rigidity for the Aluminum and Plastic (Polycarbonate) rod (measured value) for the conditions of can be determined by comparing it to the theoretical value of the modulus of rigidity for each material, which is mentioned in the background and theory.

The theoretical value for modulus of rigidity for Aluminum is 27 GPa, and the value obtained from the results and calculations during loading and unloading is 26.83±0.35GPa, having a percentage difference of 0.63%.

The theoretical value for modulus of elasticity for Plastic (Polycarbonate) is 2.3 GPa, and the value obtained from the results and calculations during loading is 1.25±0.0058GPa, having a percentage difference of 45.65%.

Conclusion

The experiment went as expected as the shear modulus of rigidity was calculated for both the stainless steel and the plastic rod by applying the torsion force and calculate the angle of twist. Modulus of rigidity of aluminum is (26.83±0.35GPa), plastic (1.25±0.0058GPa).Errors may occur due to wrong diameter, length and angle of twist measurements.

References

Hibbeler, R. C. (2011). Mechanics of Materials Eight Edition in SI Units, Prentice Hall, New York.

Modulus of Rigidity; http://www.engineeringtoolbox.com/modulus-rigidity-d_946.html, November 3, 2013.

Torsional Loading Theory; http://classes.mst.edu/civeng120/lessons/torsion/theory_full/index.html, November 3, 2013.

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Work DistributionAuthor DutiesMalek Al Bayyari, Objective, Test Procedure, ConclusionAhmed Khalaf Materials, Data Atanaz Hatamifardi Background and Theory, Data Analysis, Figures, Error Analysis, References