MMEE22111144 -- 22 CCoommbbiinneeddBBeennddiinngg && TToorrssiioonn

by

Lin ShaodunStudent ID: A0066078X Sub Group: Lab 2B

Date: 5th Feb 2010

T AB L E O F CO N TE N TS

O B J ECTI VES 1

I NTRO D UC TI O N 1

EXP ERI M E NTA L P R O CEDU RES 2

S AM P LE C ALC UL AT I O NS 6

RES ULT S ( TAB L ES & GR AP H S ) 7

DI S C US S I O N 11

CO NCL US I O N 1 3

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Bending Torsion CombinedStress

O BJEC TI V ES

A) To familiarize the operation of lab equipments including Manual HounsfieldTensometer 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.

I NTRO DU C TI O N

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 theoristJean 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 thesame."

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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 structuralsystem)

EXP ERI MEN T AL P R O CEDU RE S

A. DET ERM INAT ION OF ELAS T IC C ONS T A NT S .

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 isperformed within material elastic range. (Figure 1)

Grips Specimen Grips

Figure 1

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

Zeroing Knob

Figure 2

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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 reaches1.2KN.

Figure 5

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

LongitudinalStrain Gauge

TransverseStrain Gauge

Figure 4

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Gauge No. Color Terminal Gauge No. Color Termi

1 + 2 Red & Red P + 1 + 3 Red & Red P +3 + 4 Red & Red P - 2 + 4 Red & Red P -2 + 4 White & White S + 2 + 3 White & White S +1 + 3 Black & Black S - 1 + 4 Black & Black S -

B. COM B INE D BEN D IN G A ND T ORS IO N T ES T

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

d b

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 every0.5kg load applied until the final load reaches 3.0kg.

4) Repeat above test for all four channels, and record strain value ��1 ~ ��4 undervaries 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+

S- S+1 3

P+

S- S+

3 4 P- 4 2

P-��� = ��1 + ��4 − ��2 + ��3 ��� = ��1 + ��2 − ��3 + ��4

Figure 7Connection for ���

Connection for ���Strain Gauge Wire Strain Gauge Wire nal

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−

S AM PL E C AL CU L AT I O NS

A. COM P A R ING EX P ER I M ENT AL S T RES S E S W IT H T HE ORET ICA L S T RES S ( F R OM QU A RT ER BR ID GE REA D I NG, P = 0 . 5 K G)

��� = ��1 + ��4 − ��2 + ��3 = 72 × 10−6 ��� = ��1 + ��2 − ��3 + ��4 = 26 × 10−6The experimental bending stress is calculated using the following formula:

� � = � 𝜀 1 − 𝜀 4 =1 − �

69 .5 × 109 23 − 13 × 10−6 = �. �������1 − 0.3489The experimental shear stress is calculated using the following formula:

� 69.5 × 109�� � = 2 1 + � ��1 − ��2 = 2(1 − 0.3489) 23 + 10 × 10 = �. �������

The theoretical bending stress is calculated using the following formula:32�𝑃 32 ∙ 0.1 ∙ 0.5 × 9.8� � = ��3 = 3.1416 ∙ 14.91 × 10−3 3 = �. �������

The theoretical shear stress is calculated using the following formula:16�𝑃 16 ∙ 0.15 ∙ 0.5 × 9.8�� � = ��3 = 3.1416 ∙ 14.91 × 10−3 3 = �. �������

B. COM P A R ING EX P ER I M ENT AL S T RES S E S W IT H T HE ORET ICA L S T RES S ( F R OM QU A RT ER BR ID GE REA D I NG, P = 3 . 0 K G)

��� = ��1 + ��4 − ��2 + ��3 = 419 × 10−6 ��� = ��1 + ��2 − ��3 + ��4 = 143 × 10−6The experimental bending stress is calculated using the following formula:

� � = � ��1 − ��4 1 − � = 69.5 × 109 134 − 75 × 10−6 1 − 0.3489 = �. �������

The experimental shear stress is calculated using the following formula:

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�� � = � ��1 − ��2 2 1 + � = 69.5 × 109 × 134 + 63 × 10−6 2(1 − 0.3489) = �. �������

The theoretical bending stress is calculated using the following formula:

� � = 32�𝑃 ��3 = 32 ∙ 0.1 ∙ 3 × 9.8 3.1416 ∙ 14.91 × 10−3

3 = �. �������

The theoretical shear stress is calculated using the following formula:

�� � = 16�𝑃 ��3 = 16 ∙ 0.15 ∙ 3 × 9.8 3.1416 ∙ 14.91 × 10−3

3 = �. �������

��� ��− � ��� ��−

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

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RES UL TS (T AB L ES & G R APH S)

A. DET ERM INAT ION OF ELA S T IC C ONS T A NT S

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

D1 D2 Daverage69.5455

9.40 9.42 9.41

Table : 1

Load P ( kN ) Direct Stress 𝝈� ( MPa ) Longitudinal Strain Transverse Strain�

B. COM B INE D BEN D IN G A ND T ORS IO N T ES T

Table : 2Load P ( kg )

Strain (10-6 ) [ Quarter Bridge Configuration ]

��� ��� ��� ���

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

0.15m 0.10m14.92 14.90 14.91

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

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Table : 3

Load P ( kg )Strain (10-6 )

[ Quarter Bridge Configuration ]Strain (10-6 )

[ Full Bridge Configuration ]��� ��� ��� ���

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

��� = ��1 + ��4 − ��2 + ��3 ��� = ��1 + ��2 − ��3 + ��4Diameter of Shaft (mm) a b

D1 D2 Daverage

Table : 4

Load P ( kg )

Bending Stress 𝝈� ( MPa ) Shear Stress �� � ( MPa )

C. GRA P HS

Young’s Modulus:

69.5Gpa

Poisson’s Ratio:

0.3489

Theoretical Stress result is 43% higher than Experimental

Stress result

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Theoretical Stress result is 34% higher than Experimental

Stress result

Quarter bridge result matches Full

bridge result

Quarter bridge result matches Full

bridge result

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DI SCU SSI O N

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 goodcorrelation. (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 bridgeresult 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 aredefined 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∆�/�𝑔compensated by a new Gauge Factor (�� = ).𝜀

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

��� is the axial strain from combined bending and torsion, ��� is the lateral strain from combined bending and torsion . Hence Poisson’s ratio can beobtained by this equation: � = ��� /���

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:

� = 1 − 2 �𝑠 2�𝐿

2 − 2

�𝑠 2�𝐿

� = 2��2

1 + �

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: ht t p : / / www . oly m pu s - ims. c om/ e n/ap p l i c at i on s - and - solu t ions / n d t - theor y /elastic - modulu s - m e asure me nt/

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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.

����� = − ��� −

����

����𝜽 +

�

����

������, ����� 𝜽 = ��°, ���� = ��� − ���

��� � = ������ = � � + � ��� − ���

���������� , �� ���� 𝝈� = ��� − ���� − �

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

𝝈� = �

�����

��𝝅 ���� � − �� �

, ��� =

�

�����

��𝝅 ���� � − ���

������� �������� � � = 𝑲 × ������� �������� ��.

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.

CO NCL USI O N

Although there is big deviation between experimental and theoretical result, the experiment of combined bending and torsion help me better understand the straingauge technique as well the transformation equation of strains.

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