ME2113 - 2

TORSION OF CIRCULAR SHAFTS

(EA-02-21)

National University of Singapore

SEMESTER 3

2012/2013

Student’s Profile

Name: Gilbert Lim Lee Hock

Matric No.: A0097755

Group: 2O2

Date & Time: 2pm – 5pm, 5th October 2012

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

Table of Contents

1. Objective 1

2. Introduction 1

3. Experimental Procedures 1

4. Sample Calculations 1

5. Results and Discussions 3

6. Conclusion 4

7. Appendix 5

1. Objective

The objective of this experiment is to:

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

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

2. Introduction

Shafts 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 Procedures

A. 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.

1

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

4. Sample Calculations

4.1. Theoretical Values:

Bending Stress: σ x=32bP

π d3

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,

σ x=32∗0.1∗0.5∗9.81

π ¿0.0163=¿1.22MPa

Shear Stress: τ xy=16aP

π d3

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

τ xy=16∗0.15∗0.5∗9.81

π 0.0163=0.915MPa

4.2. Experimental Values

Bending Stress: σ x=E

1−ν(ε1−ε 4)

For load P at 0.5kg,

2

σ x=84.3×109

1−(−0.321 )(23−9 )×10−6=0.89MPa

Shear Stress: τ xy=E

2(1+ν)(ε 1−ε2)

For load P at 0.5kg,

τ xy=84.3×109

2 (1−0.321 )(23−(−9))×10−6 = 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, σx (MPa)Longitudinal Strain, εx (10-

6)Transverse Strain, εy (10-6)

200 2.79 55.00 -16

400 5.58 93.00 -26600 8.37 131.00 -38

800 11.16 164.00 -50

1000 13.95 177.00 -591200 16.74 226.00 -68

Table 2 - Results of tensometer test

40 60 80 100 120 140 160 180 200 220 2400.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

f(x) = 0.0843479129440751 x − 2.1301547488245

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

Graph 1 – To determine Young’s Modulus

3

Young’s Modulus, E = 84.3Gpa

40.00 60.00 80.00 100.00 120.00 140.00 160.00 180.00 200.00 220.00 240.00

-80

-70

-60

-50

-40

-30

-20

-10

0

f(x) = − 0.320954907161804 x + 2.42130857648098

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

Poisson's Ratio, υ = -0.321

Graph 2 – To determine Poisson’s Ratio

Load P (kg)Strain (10-6)

ε1 ε2 ε3 ε4

0.0 0 0 0 00.5 23 -9 -20 9

1.0 46 -19 -41 20

1.5 68 -29 -63 292.0 92 -39 -85 39

2.5 116 -49 -105 49

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

Load P (kg)Quarter Bridge Configuration Full Bridge Configuration

εa (10-6) εb (10-6) εa (10-6) εb (10-6)

0.0 0 0 0 00.5 61 25 65 23

1.0 126 48 128 45

1.5 189 73 193 682.0 255 99 256 91

2.5 319 123 320 115

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

4

0 50 100 150 200 250 300 350 4000.0

0.5

1.0

1.5

2.0

2.5

3.0

Load vs εa

Quarter BridgeLinear (Quarter Bridge)Full BridgeLinear (Full Bridge)

εa (x10-6)

Load

P (k

g)

Graph 3 – Plot of Load vs εa for Quarter Bridge and Full Bridge

0 20 40 60 80 100 120 1400.0

0.5

1.0

1.5

2.0

2.5

3.0

Load vs εb

Quarter BridgeLinear (Quarter Bridge)Full BridgeLinear (Full Bridge)

εb (x10-6)

Load

P (k

g)

Graph 4 - Plot of Load vs εb for Quarter Bridge and Full Bridge

Load P (kg)Bending Stress, σx (Pa) Shear Stress, τxy (Pa)

Theoretical Experimental Theoretical Experimental

0.0 0.00E+00 0.00E+00 0.00E+00 0.00E+000.5 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+062.0 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

5

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00E+00

1.00E+06

2.00E+06

3.00E+06

4.00E+06

5.00E+06

6.00E+06

7.00E+06

8.00E+06

Bending Stress vs Load

TheoreticalLinear (Theoretical)ExperimentalLinear (Experimental)

Load P (kg)

Bend

ing

Stre

ss (

Pa)

Graph 5 - Bending Stress vs Load

0.0 0.5 1.0 1.5 2.0 2.5 3.00.00E+00

2.00E+06

4.00E+06

6.00E+06

8.00E+06

1.00E+07

1.20E+07

1.40E+07

Shear Stress vs Load

TheoreticalLinear (Theoretical)ExperimentalLinear (Experimental)

Load P (kg)

Shea

r Str

ess

(Pa)

Graph 6 - Shear Stress vs Load

6. Discussion

6.1. 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 elastic constants be determined.

6

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

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

6.6. In certain installations shafts may be subjected to an axial load F in addition to torsional and bending 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.

7

Appendix

8

Graph 7 – Torsional Stiffness for Solid Shafts

Graph 8 – Torsional Stiffness for Hollow Shafts

Appendix

a

9

Graph 10 – Strength and Stiffness of Hollow and Solid Shafts with the Same Volume

Graph 9 – Strength and Stiffness of Hollow and Solid Shafts with the Same Outer Diameter