MMEE22111144 -- 22 CCoommbbiinneedd

BBeennddiinngg && TToorrssiioonn

by

Lin Shaodun

Student ID: A0066078X

Sub Group: Lab 2B

Date: 5th Feb 2010

1

TABLE OF CONTENTS

OBJECTIVES 1

INTRODUCTION 1

EXPERIMENTAL PROCEDURES 2

SAMPLE CALCULATIONS 6

RESULTS (TABLES & GRAPHS) 7

DISCUSSION 11

CONCLUSION 13

2

OBJECTIVES

A) To familiarize the operation of lab equipments including Manual Hounsfield

Tensometer and SB10 Strain Gauge Switch and Balance unit.

B) To analyze stresses at surface of shaft subjected to combined bending and

twisting using strain gauge technique.

C) To compare experimental results with theoretical results.

INTRODUCTION

Shaft subjected to both bending and twisting are frequently encountered in

engineering applications. By apply Saint Venant’s principle and the principle of

superposition, the stress at the surface of the shaft may be analyzed.

The main purpose of the experiment is to analyze this kind of problems using the

strain gauge technique and to compare the experimental results with theoretical

result.

As the strain gauge technique enables only the determination of state of strain at

certain point, Hooke’s law equations are used to calculate the stress components. In

this experiment, the elastic constants of the test material are firstly determined.

Saint-Venant's principle, named after the French elasticity theorist

Jean Claude Barré de Saint-Venant can be stated as follow:

"The stresses due to two statically equivalent loadings applied over a

small area are significantly different only in the vicinity of the area on

which the loadings are applied; and at distances which are large in

comparison with the linear dimensions of the area on which the

loadings are applied, the effects due to these two loading area are the

same."

Bending TorsionCombined

Stress

3

Superposition principle, states that, 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. So that if input A produces response X and input B

produces response Y then input (A + B) produces response (X + Y).

In mechanical engineering, superposition is used to solve for beam and

structure deflections of combined loads when the effects are linear (i.e.,

each load does not affect the results of the other loads, and the effect of

each load does not significantly alter the geometry of the structural

system)

EXPERIMENTAL PROCEDURES

A. DETERMINATION OF ELASTIC CONSTANTS.

1) Measure the diameter of the tensile test specimen with a venire caliper.

2) Turn the Tensometer hand wheel in clock-wise direction until the specimen

is firmly supported by two grips (no free play). Do not apply extra tensile

load on the specimen, this is to ensure the whole measurement process is

performed within material elastic range. (Figure 1)

3) Adjust the knob and set the mercury tube to zero position. (Figure 2)

Grips

Figure 1

Zeroing Knob

Grips Specimen

Figure 2

4

4) Connect strain gauge terminal wires to SB10 Switch and Balance unit

Channel 10 using a quarter bridge configuration, Red wire to P+ terminal ,

White wire to S- terminal, Black wire to D terminal ( Yellow) (Figure 3)

5) Adjust the Channel 10 VR until the Strain Indicator display value is zero,

(Figure 5) apply load to specimen by gradually turning the hand wheel. The

load applied can be read from the mercury tube. Record down the strain

value for every 0.2KN tensile load applied until the final load reaches

1.2KN.

6) Repeat above test for both longitudinal and transverse strains (Figure 4) in

order to evaluate the Young’s modulus and Poisson’s ratio.

Figure 3

Longitudinal

Strain Gauge

Transverse

Strain Gauge

Figure 5

Figure 4

5

B. COMBINED BENDING AND TORSION TEST

1) Measure the shaft diameter d and dimension a and b with a venire caliper.()

2) Connect strain gauge terminal wires to SB10 Switch and Balance unit

Channel 10 using a quarter bridge configuration, Red wire to P+ terminal ,

White wire to S- terminal, Black wire to D terminal ( Yellow) ()

3) Adjust the Channel 10 VR until the Strain Indicator display value is zero (),

Apply weight at the end of shaft b, record down the strain value for every

0.5kg load applied until the final load reaches 3.0kg.

4) Repeat above test for all four channels, and record strain value 𝜀1~ 𝜀4 under

varies loads.

5) Using a full bridge configuration in a manner illustrated in (Figure 7), record

the strain-meter reading for each load applied.

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3 𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4

Connection for 𝜺𝒂 Strain

Gauge No. Gauge Wire

Color Terminal

1 + 2 Red & Red P +

3 + 4 Red & Red P -

2 + 4 White & White S +

1 + 3 Black & Black S -

Connection for 𝜺𝒃 Strain

Gauge No. Gauge Wire

Color Terminal

1 + 3 Red & Red P +

2 + 4 Red & Red P -

2 + 3 White & White S +

1 + 4 Black & Black S -

a

b d

P+

P-

S- S+

1 3

2 4

P+

P-

S- S+

1 2

4 3

Figure 6

Figure 7

6

SAMPLE CALCULATIONS

A. COMPARING EXPERIMENTAL STRESSES WITH THEORETICAL

STRESS (FROM QUARTER BRIDGE READING, P=0.5KG)

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3 = 72 × 10−6 𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4 = 26 × 10−6

The experimental bending stress is calculated using the following formula:

𝜍𝑥 =𝐸 𝜀1 − 𝜀4

1 − 𝜐=

69.5 × 109 23 − 13 × 10−6

1 − 0.3489= 𝟏. 𝟎𝟔𝟕𝑴𝒑𝒂

The experimental shear stress is calculated using the following formula:

𝜏𝑥𝑦 =𝐸

2 1 + 𝜐 𝜀1 − 𝜀2 =

69.5 × 109

2(1 − 0.3489) 23 + 10 × 10−6 = 𝟎. 𝟖𝟓𝟎𝑴𝒑𝒂

The theoretical bending stress is calculated using the following formula:

𝜍𝑥 =32𝑏𝑃

𝜋𝐷3=

32 ∙ 0.1 ∙ 0.5 × 9.8

3.1416 ∙ 14.91 × 10−3 3= 𝟏. 𝟓𝟎𝟔𝑴𝒑𝒂

The theoretical shear stress is calculated using the following formula:

𝜏𝑥𝑦 =16𝑎𝑃

𝜋𝐷3=

16 ∙ 0.15 ∙ 0.5 × 9.8

3.1416 ∙ 14.91 × 10−3 3= 𝟏. 𝟏𝟐𝟗𝑴𝒑𝒂

B. COMPARING EXPERIMENTAL STRESSES WITH THEORETICAL

STRESS (FROM QUARTER BRIDGE READING, P=3.0KG)

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3 = 419 × 10−6 𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4 = 143 × 10−6

The experimental bending stress is calculated using the following formula:

𝜍𝑥 = 𝐸 𝜀1 − 𝜀4 1 − 𝜐 = 69.5 × 109 134 − 75 × 10−6 1 − 0.3489 = 𝟔. 𝟐𝟗𝟖𝑴𝒑𝒂

The experimental shear stress is calculated using the following formula:

𝜏𝑥𝑦 = 𝐸 𝜀1 − 𝜀2 2 1 + 𝜐 = 69.5 × 109 × 134 + 63 × 10−6 2(1 − 0.3489) = 𝟓. 𝟎𝟕𝟓𝑴𝒑𝒂

The theoretical bending stress is calculated using the following formula:

𝜍𝑥 = 32𝑏𝑃 𝜋𝐷3 = 32 ∙ 0.1 ∙ 3 × 9.8 3.1416 ∙ 14.91 × 10−3 3 = 𝟗.𝟎𝟑𝟓𝑴𝒑𝒂

The theoretical shear stress is calculated using the following formula:

𝜏𝑥𝑦 = 16𝑎𝑃 𝜋𝐷3 = 16 ∙ 0.15 ∙ 3 × 9.8 3.1416 ∙ 14.91 × 10−3 3 = 𝟔. 𝟕𝟕𝟔𝑴𝒑𝒂

7

RESULTS (TABLES & GRAPHS)

A. DETERMINATION OF ELASTIC CONSTANTS

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

D1 D2 Daverage 69.5455

9.40 9.42 9.41

B. COMBINED BENDING AND TORSION TEST

Table : 1

Load P ( kN ) Direct Stress 𝝈𝒙 ( MPa ) Longitudinal Strain

𝜺𝒙 𝟏𝟎−𝟔

Transverse Strain

𝜺𝒚 𝟏𝟎−𝟔

0.2 2.88 42 -13

0.4 5.75 85 -28

0.6 8.63 125 -43

0.8 11.50 168 -58

1.0 14.38 206 -71

1.2 17.25 249 -86

Table : 2

Load P ( kg ) Strain (10-6 ) [ Quarter Bridge Configuration ]

𝜺𝟏 𝜺𝟐 𝜺𝟑 𝜺𝟒

0.0 0 0 0 0

0.5 23 -10 -26 13

1.0 45 -22 -51 25

1.5 67 -31 -73 38

2.0 88 -43 -99 49

2.5 112 -52 -122 62

3.0 134 -63 -147 75

8

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3 𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4

Diameter of Shaft (mm) a b

D1 D2 Daverage 0.15m 0.10m

14.92 14.90 14.91

Table : 3

Load P ( kg )

Strain (10-6 ) [ Quarter Bridge Configuration ]

Strain (10-6 ) [ Full Bridge Configuration ]

𝜺𝒂 𝜺𝒃 𝜺𝒂 𝜺𝒃

0.0 0 0 0 0

0.5 72 26 70 24

1.0 143 49 139 47

1.5 209 71 206 70

2.0 279 95 275 93

2.5 348 120 342 117

3.0 419 143 408 142

Table : 4

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

Theoretical Experimental Theoretical Experimental

0.0 0 0 0 0

0.5 1.506 1.067 1.129 0.850

1.0 3.012 2.135 2.259 1.726

1.5 4.517 3.096 3.388 2.525

2.0 6.023 4.163 4.517 3.375

2.5 7.529 5.337 5.647 4.225

3.0 6.367 6.298 6.776 5.075

9

Young’s Modulus:

69.5Gpa

Poisson’s Ratio:

0.3489

Theoretical Stress

result is 43% higher

than Experimental

Stress result

C. GRAPHS

10

Quarter bridge

result matches Full

bridge result

Quarter bridge

result matches Full

bridge result

Theoretical Stress

result is 34% higher

than Experimental

Stress result

11

DISCUSSION

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

possible reasons for the deviations (if any) in the results obtained.

The theoretical stress result is 34% ~ 43% higher than experimental stress

result according to Graph 3 and 4, but these two data sets has very good

correlation. (Figure 8)

It seems the measure equipment has good linearity but there is an offset from

strain gauges. Same equipment set (SB-10 Switch and Balance Unit, Strain

Indicator) was used for measurement of Young’s modulus and Poisson’s ratio

during Determination of Elastic Constants test and the result matches with

actual data very well. Graph 5 and 6 also indicates that the Quarter bridge

result does not deviate from Full bridge result significantly.

In order to double confirm the theoretical calculation result, a FEA model

has been constructed using SolidWorks Simulation, materials properties are

defined using the result of Determination of Elastic Constants test. (Figure 9)

Figure 8

Figure 9

12

Above graphs shows, FEA result match with theoretical calculation very well,

deviation is less than 5% for both bending and shear stress. So the possible

reason of deviation between experimental and theoretical result could be the

strain gauge output drift, hence the strain measurement result needs to be

compensated by a new Gauge Factor (𝐺𝐹 =∆𝑅/𝑅𝑔

𝜀 ).

2) From the results of step (b5), deduce the type of stain the stain-meter reading

represent.

𝜀𝑎 is the axial strain from combined bending and torsion, 𝜀𝑏 is the lateral

strain from combined bending and torsion . Hence Poisson’s ratio can be

obtained by this equation: 𝜐 = 𝜀𝑏/𝜀𝑎

3) Apart from the uniaxial tension method used in the experiment, how can the

elastic constant be determined?

Ultrasonic method can be used to determine the elastic constant:

𝜐 = 1 − 2 𝐶𝑠

𝐶𝐿

2

2 − 2 𝐶𝑠

𝐶𝐿

2

𝐸 = 2𝜌𝐶𝑆2 1 + 𝜐

CS --- Speed of sound wave in longitudinal direction

CL --- speed of sound wave in shear direction

A commercial equipment of using this technique can be found in this

webpage: http://www.olympus-ims.com/en/applications-and-solutions/ndt-

theory/elastic-modulus-measurement/

13

4) Instead of stress Equations (3) and (8) for strain, develop alternative

equations to enable the determination of strains from the four gauges

readings.

𝜸𝒙𝒚

𝟐= −

𝜺𝟑 − 𝜺𝟒

𝟐𝒔𝒊𝒏𝟐𝜽 +

𝜸𝟏𝟐

𝟐𝒄𝒐𝒔𝟐𝜽, 𝒔𝒊𝒏𝒄𝒆 𝜽 = 𝟒𝟓°,𝜸𝒙𝒚 = 𝜺𝟒 − 𝜺𝟑

𝝉𝒙𝒚 = 𝑮𝜸𝒙𝒚 =𝑬

𝟐 𝟏 + 𝝊 𝜺𝟒 − 𝜺𝟑

𝑺𝒊𝒎𝒊𝒍𝒂𝒓𝒚 ,𝒘𝒆 𝒉𝒂𝒗𝒆 𝝈𝒙 =𝑬

𝟏 − 𝝊 𝜺𝟐 − 𝜺𝟑

5) Develop stress equations for combined bending and twisting of hollow shafts

with K as the ratio of inside to outside diameter.

𝝈𝒙 =𝟑𝟐𝑲𝑫𝒊𝒃𝑷

𝝅 𝑲𝑫𝒊 𝟒 − 𝑫𝒊𝟒

, 𝝉𝒙𝒚 =𝟏𝟔𝑲𝑫𝒊𝒂𝑷

𝝅 𝑲𝑫𝒊 𝟒 − 𝑫𝒊𝟒

𝑶𝒖𝒕𝒔𝒊𝒅𝒆 𝒅𝒊𝒂𝒎𝒆𝒕𝒆𝒓 𝑫𝒐 = 𝑲 × 𝑰𝒏𝒔𝒊𝒅𝒆 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 𝑫𝒊.

6) In certain installation shafts may be subjected to an axial load F in addition to

tensional and bending load , Would the strain gauge arrangement for this

experiment be acceptable the determination of stress? Give reasons for you

answer, for simplicity, as solid shaft may be considered.

According to Superposition principle, 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, that means the axial stress can be measured by strain gauge .

Since in this experiment, the strain gauges were installed in 45° direction,

the strain value need to times sin45° as the resultant strain in axial

direction.

CONCLUSION

Although there is big deviation between experimental and theoretical result, the

experiment of combined bending and torsion help me better understand the strain

gauge technique as well the transformation equation of strains.