ME2114-2 Combined Bending & Torsion

8/10/2019 ME2114-2 Combined Bending & Torsion

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ME2114 Mechanics of Materials II

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ME2114-2 Combined Bending & Torsion (D2)

Formal Report

Matriculation Number: A0101934

Name : Ong Wei Quan

Subgroup : 2K1


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Objectives

To analyse the stresses at the surface of shaft subjected to combined bending and twisting

using strain gauge technique.

To compare the experimental results with theoretical results.

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.

Experimental Procedures

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.

A. Combined bending and torsion test

1. Measure the dimensions of a and b.

2. Connect the strain gauges 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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Sample Calculations

Table 1 (Using load of 200N)

Direct Stress MPa88.210940.6

200

AreaSectionalCross

loadTensile5

Table 3 (Using load of 0.5kg for Quarter Bridge Configuration)

6

4321

6

3241

1027)1225()1024()()(

1071)2510()1224()()(

b

a

Table 4 (Using load of 0.5kg)

MPaEalExperiment

MPaD

aPlTheoretica

MPaE

alExperiment

MPaD

bPlTheoretica

xy

xy

x

x

654.0)3182.01(2

10)]10(24[(107.50

)1(2

)(

952.0)0159.0(

)81.95.0)(15.0(1616

892.03182.01

10)1224(107.50

1

)(

26.1)0158.0(

)81.95.0)(9945.0(3232

6921

33

69

41

33


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Results

A. Determination of elastic constants

Diameter of Tensile Test Piece (mm) Cross Sectional Area (mm2

)D1 D2 Daverage

69.49.37 9.42 9.40

mmN

DDDaverage 40.9

2

42.937.921

Cross Sectional Area 2522

10940.64

)0094.0(

4m

d

Table 1Load (N) Direct Stress, x(MPa) Longitudinal Strain, x(10

-

6)

Transverse Strain, y(10-6

)

200 2.88 61 -19

400 5.76 120 -37

600 8.65 177 -55

800 11.53 232 -73

1000 14.41 282 -91

1200 17.29 334 -107

Youngs modulus

x

xE

= Gradient of Graph 1

GPa7.50

y = 0.0507x

0

2

4

6

8

10

12

14

16

18

20

0 50 100 150 200 250 300 350 400

DirectStress(MPa)

Longitudinal Strain (x10-6)

Graph 1: Direct Stress vsLongitudinal

Strain


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Poissons ratiox

y

=Gradient of Graph 23182.0)3182.0(

B. Combined Bending and Torsion Test

Table 2

Load P (kg)Strain (10

-6) [Quarter Bridge Configuration]

1 2 3 4

0.00 0 0 0

0.524 -10 -25 12

1.0 45 -20 -48 26

1.568 -30 -73 38

2.088 -40 -98 50

2.5111 -50 -122 62

3.0132 -60 -147 75

Table 3: Comparing Quarter Bridge Configuration with Full Bridge Configuration

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 71 27 69 24

1.0 139 47 138 48

1.5 209 73 205 72

2.0 276 96 274 95

2.5 345 121 340 120

3.0 414 144 410 142

y = -0.3182x

-120

-100

-80

-60

-40

-20

0

0 50 100 150 200 250 300 350 400

TransverseStrain(x10

-6)

Longitudinal Strain (x10-6)

Graph 2: Transverse Strain vsLongitudinal

Strain


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a b D (diameter of the shaft)

0.150m 0.995m 0.0158m

y = 0.0072x

y = 0.0073x

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450

Load(kg)

a(x10-6)

Graph 3: Load vs a

Quarter Bridge

Full Bridge

y = 0.0207x

y = 0.021x

0

0.5

1

1.5

2

2.5

3

0 20 40 60 80 100 120 140 160

Load(kg)

b(x10-6)

Graph 4: Load vs b

Quarter Bridge

Full Bridge


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Table 4

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

Theoretical Experimental Theoretical Experimental

0.0 0.00 0.000 0.00 0.000

0.5 1.23 0.918 0.93 0.796

1.0 2.47 1.938 1.85 1.546

1.5 3.70 3.162 2.78 2.319

2.0 4.93 4.284 3.70 3.138

2.5 6.17 5.406 4.63 3.958

3.0 7.40 6.324 5.55 4.731

y = 2.5194x

y = 1.4366x

0

1

2

3

4

5

6

7

8

0 0.5 1 1.5 2 2.5 3

BendingStress(MPa)

Load (kg)

Graph 5: Bending Stress vs Load

Theoretical

Experimental

y = 1.904x

y = 1.2371x

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3

ShearStress(MPa)

Load (kg)

Graph 6: Shear Stress vs Load

Theoretical

Experimental


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Discussion

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

the deviation if any, in the results obtained.

The experimental results are much smaller than the theoretical results. This may be due to

the following:

The metal used in the tensile test piece may have different properties (youngs

modulus) to the one used in the hanger causing the theoretical and experimental

graphs to deviate for example in graphs 5 and 6.

The orientation of each rosette is such that the axis of any one gauge makes an angle

of 45 with the axis of the shaft. The angle may not be exactly 45 degrees as the strain

gauges are taped using scotch tape and the orientation of the gauges may change

over time.

When the hanger is loaded, there is some oscillation which contributes to the

inaccuracy of the readings.

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

The shear strain, xy, by transformation of axes, is given by

2cos2

2sin22

3443

xy

Where the subscripts 3 and 4 refer to the direction of the gauge numbers 3 and 4 respectively.

Since = -45o,

2134 xy

Hence, xya 2)()()()( 34213241

by transformation of axes,at point B,

224

xy

B

y

B

x

or

xB xy

2

14

Where the superscript B refers to the point B.

Thus, the bending strain at point A is then given by

12

)( 41B

x

A

x

x

So,

)1(41 x and similarly, )1(32 x

Hence, )1(2)()()()( 32414321 xb

Therefore, configuration bmeasures the product of strain in x-direction with the term 2(1-).


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3. Apart from the uniaxial tension method used in this experiment, how can the elastic

constants be determined.

Poisson's Ratio, can be determined by using the expressionG E

v

2 1( ) To determine Young's modulus, E, a cantilever beam with weights hung from the beam at a

few locations along the beam. The resulting deflections v wL

EI

3

3caused by the weights

can be

measured using a dial-gauge, where I is the second moment of area. By varying v with L, a

graph of v against L3could be plotted and the gradient obtained. The value of the youngs

modulus can be obtained from the gradient.

To determine Shear modulus, G, the torsion test could be used. The angle of twist is given by

TL

GI. By varying the torque T with the angle , a graph of against T could be plotted and

the gradient obtained. Since L and I are known by measuring the dimensions of the bar, G can

be obtained.

4. Instead of using Equations (3) and (8) for strains, develop alternative equations to enable

the determination of strains from the four gauges readings.

Since the full bridge configuration is quite similar to the Wheatstone bridge, 2 expressions can

be derived

xya 2)()()()( 34213241

)1(2)()()()( 32414321 xb

where 1, 2, 34, are the four gauge readings and a, bare the resulting strains.

5. Develop stress equations for combined bending, and twisting, of hollow shafts with K as the

ratio of inside to outside diameter.


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appliediswherepoint toonefromdistancewhere

1

32=

164

2

164

16464

=areaofmomentSecond=

4344

44

4

4444

PbKD

bP

KD

DbP

KDD

dDdDI

I

yM

x

z

z

xy

x

armtorsiontheoflengthwhere

1

16=

132

2

132

13232

=areaofmomentsecondPolar=

4344

444

4

444

aKD

aP

KD

DaP

KDDdDdDJ

J

rT

xy

x

x

xxxy

D = outer diameter of shaft; d = inner diameter of shaft;D

dK

6.

In certain installations shafts may be subjected to an axial load F in addition to torsionaland 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.

The shaft will experience a higher axial strain and axial stress when an additional axial load F

is applied. This additional strain caused by the load can still be measured by the strain gauge

and the readings show the resultant axial and shear strain due to the combined bending and

torsion and axial load (principle of superposition). Therefore, the strain gauge arrangement

for this experiment is acceptable for the determination of both the axial and shear strain.

Conclusion

The stresses and strains due bending and twisting were determined using the strain gauge

technique.


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The deviation of experimental results to theory may be due to the different elastic properties

of the test piece and the experimental setup.

The Youngs modulus, E and the Poissons ratio, were determined to be 50.7GPa and 0.32

respectively.

ME2114-2 Combined Bending & Torsion

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