Combined Bending and Torsion Lab report

ME2114 - 2 Combined Bending & Torsion by

Lin Shaodun Student ID: A0066078X Sub Group: Lab 2B Date: 5th Feb 2010

TABLE OF CONTENTS

1

OBJECTIVES

1

INTRODUCTION

1

EXPERIMENTAL PROCEDURES

2

SAMPLE CALCULATIONS

6

RESULTS (TABLES & GRAPHS)

7

DISCUSSION

11

CONCLUSION

13

Bending

Torsion

Combined Stress

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 Barréde 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."

2

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) Grips

Specimen

Figure 1

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

Zeroing Knob

Figure 2

3

Grips

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)

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.

Figure 5

6) Repeat above test for both longitudinal and transverse strains (Figure 4) in order to evaluate the Young’s modulus and Poisson’s ratio. Longitudinal Strain Gauge

Transverse Strain Gauge

4

Figure 4

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

a

b

d

Figure 6

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. 1

2

P+

1

S- S+ 3

S- S+ 4

4

P-

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3 Connection for 𝜺𝒂 Strain Gauge No. 1+2 3+4 2+4 1+3

5

P+

3

Gauge Wire Color Red & Red Red & Red White & White Black & Black

2

P-

𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4 Figure 7

Terminal P+ PS+ S-

Connection for 𝜺𝒃 Strain Gauge No. 1+3 2+4 2+3 1+4

Gauge Wire Color Red & Red Red & Red White & White Black & Black

Terminal P+ PS+ S-

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 =

𝜀1 − 𝜀4

𝐸

1−𝜐

=

69.5 × 109 23 − 13 × 10−6 = 𝟏. 𝟎𝟔𝟕𝑴𝒑𝒂 1 − 0.3489

The experimental shear stress is calculated using the following formula: 𝜏𝑥𝑦 =

𝐸 2 1+𝜐

𝜀1 − 𝜀2 =

69.5 × 109 23 + 10 × 10−6 = 𝟎. 𝟖𝟓𝟎𝑴𝒑𝒂 2(1 − 0.3489)

The theoretical bending stress is calculated using the following formula: 𝜍𝑥 =

32𝑏𝑃 32 ∙ 0.1 ∙ 0.5 × 9.8 = 𝜋𝐷3 3.1416 ∙ 14.91 × 10−3

3

= 𝟏. 𝟓𝟎𝟔𝑴𝒑𝒂

The theoretical shear stress is calculated using the following formula: 𝜏𝑥𝑦 =

16𝑎𝑃 16 ∙ 0.15 ∙ 0.5 × 9.8 = 𝜋𝐷3 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

6

3

= 𝟔. 𝟕𝟕𝟔𝑴𝒑𝒂

RESULTS (TABLES & GR APHS) A. DETERMINATION OF ELASTIC CONSTANTS

Diameter of Tensile Test Piece (mm) D1

D2

Daverage

9.40

9.42

9.41

Cross Sectional Area (mm2) 69.5455

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

B. COMBINED BENDING AND TORSION TEST

Table : 2 Load P ( kg )

7

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

Table : 3 Load P ( kg )

Strain (10-6 ) [ Quarter Bridge Configuration ] 𝜺𝒂 𝜺𝒃

Strain (10-6 ) [ Full Bridge Configuration ] 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

𝜀𝑎 = 𝜀1 + 𝜀4 − 𝜀2 + 𝜀3

𝜀𝑏 = 𝜀1 + 𝜀2 − 𝜀3 + 𝜀4

Diameter of Shaft (mm) D1

D2

Daverage

14.92

14.90

14.91

a

b

0.15m

0.10m

Table : 4 Load P ( kg )

8

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

C. GRAPHS

Young’s Modulus: 69.5Gpa

Poisson’s Ratio: 0.3489

Theoretical Stress result is 43% higher than Experimental Stress result

9

Theoretical Stress result is 34% higher than Experimental Stress result

Quarter bridge result matches Full bridge result

Quarter bridge result matches Full bridge result

10

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)

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 9

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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/ndttheory/elastic-modulus-measurement/

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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 𝜺𝟑 − 𝜺𝟒 𝜸𝟏𝟐 =− 𝒔𝒊𝒏𝟐𝜽 + 𝒄𝒐𝒔𝟐𝜽, 𝒔𝒊𝒏𝒄𝒆 𝜽 = 𝟒𝟓°, 𝜸𝒙𝒚 = 𝜺𝟒 − 𝜺𝟑 𝟐 𝟐 𝟐 𝝉𝒙𝒚 = 𝑮𝜸𝒙𝒚 =

𝑬 𝟐 𝟏+𝝊

𝑺𝒊𝒎𝒊𝒍𝒂𝒓𝒚 , 𝒘𝒆 𝒉𝒂𝒗𝒆 𝝈𝒙 =

𝜺𝟒 − 𝜺𝟑 𝑬 𝜺𝟐 − 𝜺𝟑 𝟏−𝝊

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

𝝈𝒙 =

𝟑𝟐𝑲𝑫𝒊 𝒃𝑷 𝝅 𝑲𝑫𝒊

𝟒

− 𝑫𝒊

𝟒

,

𝝉𝒙𝒚 =

𝟏𝟔𝑲𝑫𝒊 𝒂𝑷 𝝅 𝑲𝑫𝒊

𝟒

− 𝑫𝒊 𝟒

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.

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