124657335 combined bending torsion

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ME2113 - 2

TORSION OF CIRCULAR SHAFTS

(EA-02-21)

National University of Singapore

SEMESTER 3

2012/2013

Students Profile

Name: Gilbert Lim Lee Hock

Matric No.: A0097755

Group: 2O2

Date & Time: 2pm5pm, 5th

October 2012


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NOMENCLATURE

Dh outer diameter of hollow shaft

Ds diameter of solid shaft

dh inner diameter of hollow shaft

G shear modulus of the shaft material

Ip polar second moment of area about the shaft axis

K torsional stiffness

L length of the shaft

T applied torque

total angle of twist of the shaft over L

induced shear stress at any point on the shaft and

radial distance of that point on the shaft at which the shear stress is measured


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Table of Contents

1. Objective 12. Introduction 1

3. Experimental Procedures 1

4. Sample Calculations 1

5. Results and Discussions 3

6. Conclusion 4

7. Appendix 5


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1

1. ObjectiveThe objective of this experiment is to:

a) To analyse surface stresses of a shaft when subjected to combined bending and torsionloads thru measurements via the strain gauge technique

b) To compare the experimental values with theoretical values using bending moment andtorsion equations.

2. IntroductionShafts subjected to both bending and twisting are frequently encountered in engineering,

applications. By applying St. Venant's principle and the principle of superposition, the stresses at

the surface of the shaft may be analysed.

The main purpose of this experiment is to analyse problems of this kind using, the strain gauge

technique and to compare the experimental results with theoretical results.

As the strain gauge technique enables only the determination of states of strain at about a point.

Hooke's law equations are used to calculate the stress components. In this experiment, the elastic

constants of the test material are first determined.

3. Experimental ProceduresA. Determination of elastic constants

(1) Measure the diameter of the tensile test piece and mount it on the tensometer.

(2) Use a quarter bridge configuration and for each tensile load applied to the testpiece, record

the longitudinal and transverse strains in order to evaluate the Young's modulus and Poisson's

ratio.

B. Combined bending and torsion test

1. Measure the dimensions of a and b.2. Connect the strain gauges causes to the strain-meter using, a quarter bridge configuration and

balance all the gauges.

3. For each loading, on the shaft record the strain readings.4. From the strain readings compute the stresses.5. Using, a full bridge configuration in a manner illustrated in Figures (3a) & (3b) record the

strain-meter reading for each applied load.


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a = (1+4)- (2+3) b = (1+2)- (3+4)

4. Sample Calculations4.1. Theoretical Values:

Bending Stress:

The average measurement for diameter of test specimen d, was found to be 16mm with

reference to table 1. Distance of b, was given as 100mm. Thus, for load P at 0.5kg,

1.22MPa

Shear Stress:

Similarly for shear stress, distance of a was given as 150mm. Thus for load P at 0.5kg,

4.2. Experimental Values

Bending Stress:

For load P at 0.5kg,


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Shear Stress:

For load P at 0.5kg,

= 1.99Mpa

5. Results

Diameter of Tensile Test Piece (mm) Cross sectional area (mm2)

D1 D2 Davg71.7

9.58 9.53 9.555Table 1 - Physical dimensions of test specimen

Load P (N) Direct Stress, x (MPa) Longitudinal Strain, x (10-6) Transverse Strain, y (10

-6)

200 2.79 55.00 -16

400 5.58 93.00 -26

600 8.37 131.00 -38

800 11.16 164.00 -50

1000 13.95 177.00 -59

1200 16.74 226.00 -68

Table 2 - Results of tensometer test

Graph 1To determine Youngs Modulus

y = 0.0843x - 2.1302

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

0 50 100 150 200 250

Longitudinal Stress (MPa) vs Longitudinal Strain (10-6)

Youngs Modulus, E = 84.3Gpa


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Graph 2To determine Poissons Ratio

Load P (kg)Strain (10

-6)

1 2 3 4

0.0 0 0 0 0

0.5 23 -9 -20 9

1.0 46 -19 -41 20

1.5 68 -29 -63 29

2.0 92 -39 -85 39

2.5 116 -49 -105 49

3.0 138 -59 -126 59Table 3Direct strain data from the 4 strain gauges

Load P (kg)Quarter Bridge Configuration Full Bridge Configuration

a (10-6) b (10

-6) a (10-6) b (10

-6)

0.0 0 0 0 0

0.5 61 25 65 23

1.0 126 48 128 45

1.5 189 73 193 68

2.0 255 99 256 91

2.5 319 123 320 115

3.0 382 146 387 138Table 4 - Comparison of Quarter bridge and Full bridge configuration

y = -0.321x + 2.4213

-80

-70

-60

-50

-40

-30

-20

-10

0

0.00 50.00 100.00 150.00 200.00 250.00

Transverse Strain (10-6) vs Longitudinal Strain (10-6)

Poisson's Ratio, = -0.321


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Graph 3Plot of Load vs afor Quarter Bridge and Full Bridge

Graph 4 - Plot of Load vs bfor Quarter Bridge and Full Bridge

Load P (kg)Bending Stress, x (Pa) Shear Stress, xy (Pa)

Theoretical Experimental Theoretical Experimental

0.0 0.00E+00 0.00E+00 0.00E+00 0.00E+00

0.5 1.22E+06 8.93E+05 9.15E+05 1.99E+06

1.0 2.44E+06 1.66E+06 1.83E+06 4.03E+06

1.5 3.66E+06 2.49E+06 2.74E+06 6.02E+06

2.0 4.88E+06 3.38E+06 3.66E+06 8.13E+06

2.5 6.10E+06 4.28E+06 4.57E+06 1.02E+07

3.0 7.32E+06 5.04E+06 5.49E+06 1.22E+07Table 5 - Theoretical and Experimental Stresses

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 50 100 150 200 250 300 350 400

Load

P(

kg)

a (x10-6)

Load vs a

Quarter Bridge

Full Bridge

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0 20 40 60 80 100 120 140

Load

P(

kg)

b (x10-6)

Load vs b

Quarter Bridge

Full Bridge


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Graph 5 - Bending Stress vs Load

Graph 6 - Shear Stress vs Load

6. Discussion6.1. Compare the theoretical stresses with the experimental values. Discuss possible reasons for

the deviations if any, in the results obtained.

6.2. From the results of step (B5), deduce the type of strain the strain-meter readings represent.

6.3. Apart from the uniaxial tension method used in this experiment, how can the elasticconstants be determined.

0.00E+00

1.00E+06

2.00E+06

3.00E+06

4.00E+06

5.00E+066.00E+06

7.00E+06

8.00E+06

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Bending

Stress

(P

a)

Load P (kg)

Bending Stress vs Load

Theoretical

Experimental

-2.00E+06

0.00E+00

2.00E+06

4.00E+06

6.00E+06

8.00E+06

1.00E+07

1.20E+07

1.40E+07

0.0 0.5 1.0 1.5 2.0 2.5 3.0

ShearStress

(Pa)

Load P (kg)

Shear Stress vs Load

Theoretical

Experimental


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6.4. Instead of using Equations (3) and (8) for strains, develop alternative equations to enable thedetermination of strains from the four gauges readings.

6.5. Develop stress equations for combined bending, and twisting, of hollow shafts with K as theratio of inside to outside diameter.

6.6. In certain installations shafts may be subjected to an axial load F in addition to torsional andbending loads. Would the strain gauge arrangement for this experiment be acceptable to the

determination of stresses?

Give reasons for your answer. For simplicity, a solid shaft may be considered.


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Appendix

8

Graph 7 Torsional Stiffness for Solid Shafts

Graph 8 Torsional Stiffness for Hollow Shafts


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Appendix

9

a

Graph 9

Strength and Stiffness of Hollow and Solid Shafts with the Same Volume

Graph 10 Strength and Stiffness of Hollow and Solid Shafts with the Same Outer Diameter

124657335 combined bending torsion

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