me2114e combined bending & torsion _ lab

Bachelor of Technology Programme

ME2114E

Mechanics of Materials II

LAB REPORT:

Combined Bending & Torsion

EA-02-21

NAME: LOH ENG ZHI

MATRIC: A0107520A

Lab Group: 1B

Date: 30 JAN 2013

A0107520A ME2114E COMBINED BENDING & TORSION LOH ENG ZHI

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Table of Contents

Objectives Pg 3

Introduction Pg 3

Experimental Procedures Pg 4

Sample Calculations Pg 5

Results (Tables & Graphs) Pg 6

Discussion Pg 11

Conclusion Pg 13


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Objectives

To learn the correct sequence to use tensometer, strain-meter, to obtain the bending and

torsional stresses results, as well as trouble shooting.

To analyze and calculate the bending and shear stresses at the surface of the shaft

subjected to both bending and twisting using strain gauge technique.

To compare the experimental stresses results with theoretical stresses results, study what

the reasons behind will be.

Introduction

Frequently, shaft subjected to both bending and twisting are encountered in real life

engineering applications. By applying St.Venant’s principle and the principle of

superposition, the stresses at the surface of the shaft may be analyzed and calculated.

To compare and study the experimental results with the theoretical results by using the strain

gauge technique are the main purposes of this lab experiment.

As the strain gauge technique only determine the states of the strain at about a point. Hook’s

law equations are also introduced to calculate the stress components. So, the elastic constants

of the test material are first determined and recorded during this experiment.


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Experiment Procedures

A. Determination of elastic constants

1) Measure the diameter of the tensile test piece (measure the values for a few times, taking

the average value) and mount it onto the electronic tensometer.

2) Use a quarter bridge configurations and for each tensile load applied to the test piece,

record the longitudinal and transverse strains in order to evaluate the Young’s modulus

(E) and Poisson’s ratio (ν).

B. Combined bending and torsion test

1) Measure the dimensions of the free end of both a and b shaft (measure the values for a

few times, taking the average value).

2) Connect the strain gauges to the strain-meter using a bridge configuration and balance all

the gauges.

3) Record the strain readings for each loading on the shaft.

4) Compute the stresses from the strain readings.

5) Using a full bridge configuration given, record the strain-meter reading for each applied

load.


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Sample Calculations:

Part B – Combined Bending and Torsion Test:

As we get the values of Ɛ1, Ɛ2, Ɛ3 & Ɛ4 from Table 2, we can get the value of Ɛa and Ɛb with the

formula:

Ɛa = (Ɛ1 + Ɛ4) – (Ɛ2 + Ɛ3)

Ɛb = (Ɛ1 + Ɛ2) – (Ɛ3 + Ɛ4)

e.g When load P = 0.5kg,

Ɛa = (Ɛ1 + Ɛ4) – (Ɛ2 + Ɛ3) = (24 + 15) x 10

-6 – (-11 - 23) x 10

-6

= 73 x 10-6

Ɛb = (Ɛ1 + Ɛ2) – (Ɛ3 + Ɛ4) = (24 - 11) x 10

-6 – (-23 + 15) x 10

-6

= 21 x 10-6

Comparing Experimental Stresses & Theoretical Stresses:

1. The Experimental Bending Stress is calculated using the following formula:

σx = E(Ɛ1 – Ɛ4) / (1- = [(70.1 x 10

9)( 24–15)x10

-6] / (1 - 0.3)

= 0.901 MPa

*While E = 70.1GPa and v = 0.3 which can get from the Graph 1 and Graph 2

2. The Experimental Shear Stress is calculated using the following formula:

τxy = E(Ɛ1 – Ɛ2) / 2(1- = [(70.1 x 10

9)( 24+11)x10

-6] / [2 x (1 - 0.3)]

= 0.944 MPa

3. The Theoretical Bending Stress is calculated using the following formula:

σx = 32bP / πd3

= (32 x 0.1 x 0.5 x 9.81) / [π x (15.86 x 10-3

)3]

= 1.252 MPa

4. The Theoretical Shear Stress is calculated using the following formula:

τxy = 16aP / πd3

= (16 x 0.15 x 0.5 x 9.81) / [π x (15.86 x 10-3

)3]

= 0.939 MPa


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Results (Tables & Graphs):

Part A - Determination of Elastic Constants

Diameter of Tensile Test Piece (mm) Cross Section Area

(mm2)

D1 D2 Daverage

71.03 9.52 9.49 9.51

Cross Sectional Area, A = πd2/4

= π(9.512)/4

= 71.03 mm2

Table 1

Load (N) Direct Stress σx

(MPa)

Longitudinal / Axial

Strain, εx (10-6

)

Transverse Strain,

εy (10-6

)

200 2.816 45 -14

400 5.631 83 -25

600 8.447 122 -37

800 11.263 162 -49

1000 14.079 202 -61

1200 16.894 244 -73

Direct Stress, σx = Tensile Load / Cross Sectional Area

= 200 N / 71.03 mm2

= 2.816 MPa

Part B - Combined Bending and Torsion Test

a b d (diameter of the shaft)

0.15 m 0.10 m 15.86 x 10-3

m

Table 2

Load P (kg) Strain (10

-6) [Quarter Bridge Configuration]

ε1 ε2 ε3 ε4

0.0 0 0 0 0

0.5 24 -11 -23 15

1.0 46 -23 -46 25

1.5 68 -33 -69 37

2.0 90 -43 -91 49

2.5 115 -54 -112 61

3.0 136 -66 -136 73


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Table 3: Comparing Quarter Bridge Configuration with Full Bridge Configuration

Load P (kg) Quarter Bridge Configuration Full Bridge Configuration

εa εb εa εb

0.0 0 0 0 0

0.5 73 21 67 25

1.0 140 44 135 47

1.5 207 67 200 68

2.0 273 89 267 91

2.5 342 112 336 115

3.0 411 133 406 135

Table 4

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

Theoretical Experimental Theoretical Experimental

0.0 0.000 0.000 0.000 0.000

0.5 1.252 0.901 0.939 0.944

1.0 2.505 2.103 1.879 1.860

1.5 3.757 3.104 2.818 2.723

2.0 5.009 4.106 3.757 3.586

2.5 6.262 5.408 4.696 4.557

3.0 7.514 6.309 5.636 5.446


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Discussion

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

reasons for the deviations if any, in the results obtained.

Answer:

From Table 4 and Graph 3, the theoretical stresses results are around 22% higher than the

experimental stress values, for both the bending stress and shearing stress.

Possible reasons for deviations:

a) Strain hardening of the shaft

b) Experimental error when doing the measurement

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

represent.

Answer:

Ɛa is the axial strain from combined bending and torsion, while Ɛb is the lateral strain.

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

constants be determined?

Answer:

We can determine the elastic constants by using compressive load method (To apply a

compressive loading on the material).

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

enable the determination of strains from the four gauges readings.

Answer:

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

γxy /2 = -[(ε1 – ε2) x sin2 ]/2 + (γ12 x cos2 )/2

when = 450,

γxy = ε1 – ε2

2ε1 = +

+ γxy

And,

2ε2 = +

- γxy

2ε3 = +

- γxy

2ε4 = +

+ γxy

So, the alternative equations to enable the determination of strains: εx = [(ε1 – ε4) + (ε2 – ε3)] / [2(1-v)]


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

K as the ratio of inside to outside diameter. Answer:

So:

Where K = (Di / D0) *Which Di = Inside Diameter, D0 = Outside Diameter

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. Answer:

Yes, the strain gauge arrangement for this experiment is acceptable to the determination

of stresses.

Reasons: a) According to the principle of superposition, for all linear systems, the net response at a

given place and time caused by two or more stimuli is the sum of the responses which

would have been caused by each stimulus individually.

b) For this experiment, the strain gauges make an angle of 45O with the axis of the shaft,

when calculating the strain value, need to times sin45O to find out the resultant strain in axial

direction.


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Conclusion

From this experiment, it lets me to have a better understanding of the stresses at the surface of

the shaft can be analyse by applying St. Venant’s principle and the principle of superposition. It

also uses the strain gauge technique to determine the states of strain at about a point. And from

the results achieve between the experimental and theoretical, it is found to be different.

me2114e combined bending & torsion _ lab

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