Mechanics of Materials Laboratory

Torsion Test

David Clark

Group C:

David Clark

Jacob Parton

Zachary Tyler

Andrew Smith

10/27/2006

Abstract

The following experiment outlines the proper procedure for determining the shear

modulus for a material. During this exercise, aluminum and brass were both used as

samples to demonstrate how materials behave during testing conditions. By measuring

the applied torque with respect to the angle of twist, the shear modulus, shear stress at the

limit of proportionality, and failure conditions can be found. Ultimately, the shear

modulus for Aluminum and brass was determined to be 3078 and 3708 ksi respectively.

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Table of Contents

1. Introduction & Background.............................................................................3

1.1. General Background..............................Error! Bookmark not defined.

2. Equipment and Procedure................................................................................5

2.1. Equipment................................................................................................5

2.2. Experiment Setup.....................................................................................6

2.3. Procedure.................................................................................................6

3. Data, Analysis & Calculations.........................................................................7

4. Results............................................................................................................14

5. Conclusions....................................................................................................14

6. References......................................................................................................14

7. Raw Notes......................................................................................................15

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1. Introduction & Background

The following experiment is designed to determine the modulus of rigidity.

Utilizing test specimens with a known geometry, specimens can be twisted with the

values for torque simultaneously measured. To perform this task, a torsion testing

machine can be used. The torsion apparatus used in this exercise accepts standard 0.2525

inch diameter cylindrical specimens with hexagonal ends. With the sample secured and

clamped within the machine, the specimen can be twisted by applying a rotational torque

to one end, while the opposing end is kept straight by attaching it to the center of an arm

structure. This arm is held horizontal throughout the procedure by a torque gage

specifically calibrated to read torque for this configuration.

The ultimate goal, the determination of the modulus of rigidity, G, can be

expressed as

Equation 1

where T is the torque applied, L is the length of the specimen, φ is the angle of twist in

radians, J is the polar moment of inertia (see below).

Equation 2

R and d is the radius and diameter of the cross-sectional test area respectively. Equation 2

is only valid for solid circular regions.

T/φ is the slope of the linear portion of the curve when torque is plotted against

the angle of twist. A typical curve is drawn in Figure 1.

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Figure 1

Another method of determining G comes from Hooke's law, where

Equation 3

The value of shear stress is easily expressed by

Equation 4

Replacing J in Equation 4 with Equation 2, as well as converting the radial

dimension to the diameter, shear stress can be expressed as

Equation 5

where K1 is given by

Equation 6

Shear stress produced a shear strain expressed as

4

Equation 7

Converting the radial dimension to a diameter, the expression for shear strain

becomes

Equation 8

where K2 is expressed as

Equation 9

The apparatus used within this experiment measures the angle of twist in degrees

rather than radians, therefore Equations 8 and 9 can be determined to be

Equation 10

Equation 11

2. Equipment and Procedure

2.1. Equipment

1. Torsion Testing Machine

2. Specimens: In this experiment, aluminum and brass specimens were used.

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3. Calipers: Calipers should be used to measure the diameter and length of

the test specimen.

2.2. Experiment Setup

Measure and record the length and diameter of specimen, as estimated in Figure

1.

Figure 2

The specimen should be secured in the torsion testing machine. The control used

to generate twisting can be used to balance the arm and torque gage. When the apparatus

is set to read zero degrees twist and torque, the angle of twist can be induced.

2.3. Procedure

Twist is then induced in quarter degree increments. After each application, the

arm should be balanced and the resulting torque recorded.

When 8 degrees of twist has been applied, the twist should be increased in 2

degree increments. Again, the arm should be balanced after each step. Record the torque

and record if any hysteresis effects are experienced.

After 30 degrees of twist has been applied, the twist should be increased first to a

total twist of 90 degrees. Record the torque and note any hysteresis effects. Next,

increment the angle of twist in 90 degree increments until the specimen breaks. After

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every increment, record the torque. Usually this process needs to be done fairly quickly to

ensure minimal error due to relaxation in the specimen.

3. Data, Analysis & Calculations

Table 1 catalogs angle and torque applied at each increment.

Table 1

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Table 2 lists measured and calculated dimensional data. K1 and K'2 are determined

using Equation 6 and 11.

Table 2

Table 3 contains the angle and torque for Aluminum as previously displayed in

Table 1. The shear stress and strain are calculated using Equation 5 and 10. Table 4

contains the same information for Brass.

As a sample, K1, K'2, the shear stress, and the shear strain are calculated in

Equation 12 through 15.

Equation 12

Equation 13

Equation 14

Equation 15

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Table 3

9

Table 4

10

Stress vs Strain

0

10000

20000

30000

40000

50000

60000

70000

0 50000 100000 150000 200000 250000 300000 350000 400000

Strain (ue)

Str

es

s (

ps

i)

Aluminum

Brass

Figure 3

As shown in Figure 4, the slope of the linear portion of the stress versus strain

curve is approximately 3072 ksi.

Linear Stress vs Strain (Aluminum)

y = 3.0715x - 947.2

-2000

0

2000

4000

6000

8000

10000

12000

14000

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Strain (ue)

Str

es

s (

ps

i)

Figure 4

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Linear Stress vs Strain - Brass

y = 3.7077x + 794.52

0

5000

10000

15000

20000

25000

0 1000 2000 3000 4000 5000 6000 7000

Strain (ue)

Str

es

s (

ps

i)

Figure 5

As shown in Figure 5, the slope of the stress versus strain curve for brass is 3708

ksi.

The limit of proportionality is determined by offsetting a line 0.2% right of the

linear stress versus strain plot, and finding where it crosses the original stress-strain

curve. For Aluminum, the limit of proportionality is graphically determined to be

approximately 22000 psi. Brass exhibited a higher limit of proportionality with a value of

approximately 26500 psi.

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Limit of Proportionality - Aluminum

0

5000

10000

15000

20000

25000

30000

4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000

Strain (ue)

Str

es

s (

ps

i)

Figure 6

Limit of Proportionality - Brass

10000

15000

20000

25000

30000

35000

40000

5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000

Strain (ue)

Str

es

s (

ps

i)

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4. Results

The results found within this experiment were somewhat innaccurate. The shear

modulus for Aluminum was determined to be 3072 ksi, which contains nearly 22% error

when compared against the standard 3920 ksi. The experiment using Brass contained

37% error, with 3708 ksi and 5080 ksi as the experimental and accepted values

respectively.

Sources for error occur within the testing apparatus. As noted in the experiment,

even with both degree indicators tightened, the 6 degree wheel and the 360 degree wheel

did not agree on angular measurement. For example, when 30 degrees had been achieved

on the 6 degree wheel, the 360 degree wheel did not read 30 degrees as well.

5. Conclusions

The results in this particular experiment may or may not be acceptable for

practical use. If an application requires a high degree of accuracy of the shear modulus or

shear behavior, more thorough testing must be done.

As an improvement to the experiment, the degree wheels should be further

analyzed to ensure that both agree accurately to the angular behavior of the specimen.

6. References

Gilbert, J. A and C. L. Carmen. "Chapter 4 – Column Buckling Test." MAE/CE 370 –

Mechanics of Materials Laboratory Manual. June 2000.

"Copper, UNS C27000 (Yellow brass), OSO70 Temper flat products" MatWeb. Nov 1,

2006. <http://www.matweb.com/search/SpecificMaterial.asp?

bassnum=MCUABQ01>

"Aluminum 2024-T6" MatWeb. Nov 1, 2006.

<http://www.matweb.com/search/SpecificMaterial.asp?bassnum=MA2024T6>

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7. Raw Notes

Figure 7

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Figure 8

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Figure 9

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