Strength of materials Advanced

8/10/2019 Strength of materials Advanced

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ABERDEEN COLLEGE

Stress and Strain Analysis of

a Thin CylinderLO2-b

Joseph Olsen

6/6/2013


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Unit: DV01 35 Strength of Materials Advanced

Aberdeen College - 2013 1 Mechanical Engineering

Contents

Introduction ................................................................................................................................................. 2

Objectives ................................................................................................................................................... 2

Equipment ................................................................................................................................................... 3

Method/Test Procedure ............................................................................................................................... 5

Closed End Condition.............................................................................................................................. 5

Results- Open End Condition................................................................................................................... 6

Closed End Condition.............................................................................................................................. 7

Calculations -Youngs modulus.................................................................................................................. 9

Poissons ratio ........................................................................................................................................... 10

Principle stresses (closed ends condition) ................................................................................................. 11

Mohrs circle of strain (closed ends condition)........................................................................................ 12

Factor of safety ......................................................................................................................................... 13

Discussion/Comparison of Values ............................................................................................................ 14

Conclusion ................................................................................................................................................ 16

References ................................................................................................................................................. 16

Appendix 1 ................................................................................................................................................ 17

Appendix 2 ................................................................................................................................................ 18

Appendix 3 ................................................................................................................................................ 19

Appendix 4 ................................................................................................................................................ 20


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Introduction

An experiment was carried out utilising the TQ SM1007 Thin cylinder testing machine. The cylinderwas pressurised and the stress and strain that the cylinder was exposed to was measured and recorded.

The relationship between the stress and strain was then analysed and using a number of different

scientific principles, compared to the theory.

Objectives

Using experimental apparatus to measure strains on the surface of a thin cylinder which is subjected to

an internal pressure, and incorporating theory this assignment had the following objectives:

a. Determine Youngs Modulus for the cylinder material and compare this to the stated value.

b. Determine Poissons Ratio for the cylinder material and compare this to the stated value.

c. Calculate experimental values for the principal stresses created in the cylinder walls (using the

measured principal strain values) and comparing these to the theoretical values (calculated

using thin cylinder theory).

d. Construct a Mohrs circle of strain using the principle strain gauge values, and then compare the

non-principal strain gauge readings to those found from using the Mohrs circle.

e. Apply appropriate theories of failure to determine a factor of safety for the cylinder at the

maximum level of internal pressure.f. Explain why the six strain gauges did what they did.


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Equipment

TQ SM1007 Thin cylinder testing machine (see appendices 1)

The Pressure Gaugeindicates the pressure within the cylinder.

The Hand Pump is pumped to increase the pressure within the cylinder.

The Hand Wheel winds in so that the axial stress can be transferred from the cylinder to the frame via

the movable pistons within the cylinder. The hand wheel can also be wound out so that the cylinder

takes all axial stress.

The Pressure Release Valve releases the pressurewithin the cylinder when turned anti clockwise.

The computer connection connects the experimental apparatus to the computer so that the apparatus

software can record the pressure and strain data.

Computer runs SM1007 software to record data.

The Frame takes the axial stress when in the closed ends configuration.

Han

Whe

Computer

Connection

Strain

Gauge

Pressure

Release

Valve

Hand

Pump

Pressure

GaugeFrame

ArrangemenStrain Gaug

and Gaug

Factor


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The Strain gauge measures the strain in the cylinder

across five different directions with six individual

gauges contained within it.

Closed endcondition

The pressurised oil creates a force in all directions.Under all conditions hoop stress and radial stress

will be created as the oil pushes out on the

cylinder. Although the radial stress is very small incomparison so in this case is insignificant. But inthe closed ends condition there is also axial stress.

This is because the pistons are free to move.

Therefore the oil pushes them into the end capswhich in turn transfer the strain to the cylinder.

Because there is stress in more than one direction

this can be classed as a complex stress situation.

Open endcondition

In the open ends condition the hand wheel is

tightened up so that there is a gap between the

back of the pistons and the end caps. Thisallows the axial stress to be transferred

through the pistons and to the frame.

Therefore there is no axial stress acting on thecylinder. As the only significant stress is axial

this situation can be defined as a simple stress

situation.

Axial

stress

Hoop str

Cylinder


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Method/Test Procedure

Open EndCondition

The pressure relief valve was opened and then the hand wheel was turned clockwise until it stoppedturning. At this point the pistons were pushed in so that the frame would take axial stress. This meant

that the cylinder was in open ends condition and would only be significantly subjected to hoop stress.

The apparatus was then connected to the computer and the pressure and strain gauges were zeroed by

pressing F4. The first strain and pressure readings were then taken at 0 MN/by pressing F2 whichrecorded them in a table.

The pressure relief valve was then tightened and the hand pump was slowly pumped until the pressure

reading was 0.5 MN/. It was necessary to wait a couple of seconds after the final pump as it took awhile for the readings to stabilise. If the pressure exceeded this number by too much then some pressure

was released using the pressure release valve. The pump could be pumped again if too much pressure

was lost. If the pressure reading was within 0.02 of the desired pressure then strain and pressure readingscould still be taken. After the readings were taken the pressure was taken up in 0.5 MN/incrementsand readings taken until the pressure read 3 MN/.

Closed EndCondition

The pressure relief valve was undone so that there was no pressure in the cylinder and the hand wheelwas unwound so that the pistons would push against the end caps and transfer the axial stress to the

cylinder. This meant the cylinder was subject to bi-axial loading as it would be subject to hoop stress

and axial stress.

The apparatus was then connected to the computer and the pressure and strain gauges were zeroed by

pressing F4. The first strain and pressure readings were then taken at 0 MN/by pressing F2 whichrecorded them in a table. The pressure relief valve was then tightened and the hand pump was slowly

pumped until the pressure reading was 3 MN/. It was necessary to wait a couple of seconds after thefinal pump as it took a while for the readings to stabilise. If the pressure exceeded this number by too

much then some pressure was released using the pressure release valve. The pump could be pumped

again if too much pressure was lost. If the pressure reading was within 0.02 of the desired pressure thenstrain and pressure readings could still be taken.


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Results- Open End Condition

Pressure

/ MNmStrain gauge readings / 1 2 3 4 5 6

0.00 0.0 0.0 0.0 0.0 0.0 0.0

0.50 95.8 -29.0 1.9 34.0 64.5 97.81.02 197.1 -60.9 2.1 69.6 133.3 199.8

1.50 291.2 -91.2 1.7 102.2 196.9 294.5

2.00 386.5 -122.3 1.2 135.1 260.9 390.7

2.50 483.7 -154.2 0.4 168.7 326.4 489.3

3.00 579.9 -187.8 -0.4 201.9 390.9 586.9

In a simple stress situation such as this, stress is only acting in one direction.Although the hoop stress acts in only one direction strain will act in two directions.

This is because as the stress makes an object lengthen (longitudinal strain) in the

direction of stress it will also cause the object to narrow in a direction 90(transverse

strain) to the direction of stress. This is because the object will maintain the same

volume/area unless the density changes. All longitudinal strain is therefore given apositive value as its size is increasing whilst all transverse strain is given a negative

value as its decreasing. Longitudinal strain will also always be higher than transverse

strain and they are proportional to one another as they are linked by Poissons ratio.

Strain gauges are designed to measure strain along their length. If this length

increases the resistance in the gauge increases, leading to higher strain readings. Ifthis length decreases the resistance decreases leading to negative strain readings.

Strain gauge one and six travel in the direction of the hoop stress. Therefore they had

the highest strain values as they would lengthen the most longitudinally whilst

narrowing the least transversely. They are very close to one another in value but are slightly different asthe cylinder wont uniformly deform if there are any variances in the thickness or structure.

Strain gauge two had negative readings as the hoop stress caused the strain gauge to shorten due to the

transverse strain whilst the longitudinal strain would have an insignificant effect on it.

Strain gauge three had readings around zero. This would indicate that at an angle of 60 to the hoop

stress, the longitudinal and transverse strain cancel each other out. As this angle is halfway between the

angle of gauges two and four it was expected that the gauge reading would be halfway between thesetwo figures which it was.

Strain gauge four is at an angle of 45 to the hoop stress. Therefore itwould be expected that the gauge readings would be halfwaybetween the readings for gauges one/six and gauge two. These

results confirmed what was expected.

Strain gauge five is at an angle of 30 to the hoop stress. Therefore itwould be expected that the gauge readings would be halfway

between gauge four and gauge six. This was again confirmed by the

results.


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Closed EndCondition

Pressure

/ MNm

Strain gauge readings / 1 2 3 4 5 6

3.00 476.1 103.0 191.1 297.8 383.2 493.5

In the closed end condition the thin walledcylinder is subject to both axial and hoop stress

so this is a complex stress situation. Hoop stress

is double the size of the axial stress as and .

For this situation I have labelled my stresseshoop stress and axial stress so that it is clearwhat is causing the stress, but for calculations

sake it is easier to label them and respectively.

The hoop stress will primarily cause a positive

strain in the y direction but will also cause anegative strain in the x direction but at a third of

the magnitude as calculated previously with

Poissons ratio. Conversely the axial stress will primarily cause a positive strain in the x direction and

secondly a negative strain at a third of the magnitude in the y direction.

To calculate the strain in the direction of hoop stress both the strains in the y direction are added

together. Thus to calculate the strain in the direction of the axial stress both the strains in the x directionare added to together.

It was expected that gauge one and six would have the highest readings but that they would not be as

large as with the open ends cylinder. This was because the axial stress would also cause a negative strainin the y direction. As the axial stress is half the hoop stress and Poissons ratio is about a third it would

be expected that the negative strain caused by the axial stress would be about a sixth of the positive

strain in the y direction caused by the hoop stress. Therefore it would be expected that the strain gauge

readings for one and six would be a sixth smaller than the readings at the same pressure for the simplestress situation. The readings on the gauge confirmed what was expected, although there was a

significant difference between the two readings signifying that either the gauges werent manufactured

properly, or that the cylinder reacts to stress differently in different areas.


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Taking in the hoop to axial stress ratio and poisons ratio it could have been expected that gauge two

would be about a sixth of the strain value of gauge one at the same pressure in the open ends condition.

Although the actual gauge reading is less than this it was pretty close at 103.0 compared to 96.5.

It was expected that gauge four would be halfway between the readings for gauges one/six and gauge

two. This would be around the 300 mark and at 297.8 it was very close.

Gauge three should be halfway between the gauge two and the gauge four readings at about 200. At

191.1 this is reasonably close.

Gauge five should be halfway between the values for gauge four and gauges one/six. This would be

around 395, but the actual reading was 383.2 which wasnt too bad.


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Calculations -Youngs modulus

For simple stress

To calculate youngs modulus of elasticity using this formula itwas imperative that the cylinder was only exposed to simple

stress. The stress and the strain must also be running in thesame direction. To ensure this was the case and that there was

only one significant direction of stress, the apparatus was used

in the open ends condition. This meant that axial stress (a)could be eradicated as all axial forces where taken by the

pistons in the ends of the cylinder and transferred to the frame

ensuring that the only significant stress acting upon the cylinder

was the hoop stress (h).

To measure the strain running in the direction of the hoop stress, strain gauges running in the samedirection must be used. Therefore strain gauges one and six were used to record the necessary strainreadings.

As the hoop stress was created by the high pressure of the oil within the cylinder it was hard to directly

measure the hoop stress. Therefore it had to be calculated using with p being the pressurewithin the cylinder which was read off of the gauge, d being the inside diameter of the cylinder and t

being the thickness of the cylinder wall.

when thickness (t) = 0.003m and internal diameter (d)= 0.080m. Therefore Pressure (p)(MN/m)

Hoop stress (h)(MN/m)

Gauge one strain(1x10) Gauge six strain(1x10) Average strain gaugereading (1x10)

((G1+G2)/2)

0 0 0 0 0

0.5 6.666666667 95.8 97.8 96.8

1.02 13.6 197.1 199.8 198.45

1.5 20 291.2 294.5 292.85

2 26.66666667 386.5 390.7 388.6

2.5 33.33333333 483.7 489.3 486.5

3 40 579.9 586.9 583.4

From graph (appendix 2)

y1=0 y2= 32.5 x1=0, x2=475

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Poissons ratio

For simple stress

To calculate poisons ratiousing this formula it is imperative thatthe cylinder be experiencing simple stress and not complex stress.

Therefore the cylinder is used in the open ends condition. By

using the cylinder in the open ends condition the axial stress iseradicated leaving only hoop stress and an insignificant amount of

radial stress.

To measure the transverse strain, (the strain at 90 to the

direction of stress which in this case is hoop stress) the gauge that runs in the same direction must beused. Therefore strain gauge two was used.

To measure the longitudinal strain (the strain running in the same direction as the stress) the gauges that

run in the same direction must be used. Therefore gauges one and six were used.

Strain gauge one (1x10)

Strain gauge six (1x10)

Average strain gauge reading (1x10)

Strain gauge two (1x10)

0 0 0 0

95.8 97.8 96.8 -29

197.1 199.8 198.45 -60.9

291.2 294.5 292.85 -91.2

386.5 390.7 388.6 -122.3

483.7 489.3 486.5 -154.2

579.9 586.9 583.4 -187.8

From graph (appendix 3)

y1=32 y2= 164 x1=, x2=515

T

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Principle stresses (closed ends condition)

Experimental principle stress values

The principle stresses in a thin cylinder in the closed ends condition are hoop stress and axial stress. Themain difference between this and the open ends condition being that the pistons push against the ends of

the cylinder as the hand wheal isnt screwed in to take the axial stress. Therefore the cylinder is exposed

to hoop and axial stress.

E is the youngs modulus of elasticity determined during the open ends condition and is Poissons ratio

also determined at that time. is the hoop strain that was an average of the hoop strain measured bygauges one and six. is the axial strain that was measured by gauge two. Using these values and theformula above it was possible to calculate.

=

=484.8

() =

[]

Calculation of theoretical principle stress value using thi n cylinder theory

40 = 40 MN/


13/21



Mohrs circle of strain (closed ends condition)

To construct a Mohrs circle of strain,

a reference line with a point in the

middle was marked. This line wouldis normally used to measure shear

strain but the strain gauges used onlymeasured direct strain.

From this centre point a horizontal

line was drawn (direct strain line).Utilising an appropriate scale the

minor principle strain (average axial

strain reading from gauges one and

six) was marked on the horizontalline. The major principle strain (hoop

strain measured by gauge two) wasthen marked upon the line. These two marks were used

as two points on the diameter of a circle so that the

circle could be drawn in with a Centre point. The line

from the centre point to the marked on minor principalstrain point was coloured in blue. This corresponded to

the axial strain line on the above diagram of gauges.

From the centre point to the major principle strainpoint was coloured in red as this corresponded to the

hoop strain lines on the gauge diagram. To find the

theoretical strain gauge value for a non-

principal strain value a line had to be drawnfrom the centre of the circle to its

circumference. The angle of this line was double the

angle of the strain gauge either in relation to the hoopstrain line or axial strain line. So if the gauge angle was

measured from the horizontal blue axial strain line on

the gauge diagram then it would have to be measuredfrom the blue line in the Mohrs circle. The angle

measured on the circle was double the angle measured on the strain gauge diagram. If the angle of the

strain gauge had been measured from a vertical red hoop strain line then this angle would be doubled

and measured from the red line in the Mohrs circle.

Utilising the scale used to mark on the major and minor strain points, the direct strain for the non-principal gauge could be measured. The horizontal distance from the circumference point to the

reference line (shear strain line) was a graphical representation of the direct strain. Therefore it could be

measured with a ruler.

(See appendices 4 for Mohrs circle of strain diagramsand theoretical values)

Gauge diagram

Mohrs Circle Diagram

Nonprinci

gaug

strai


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Factor of safety

Tresca

=

Von Misses

=


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Discussion/Comparison of Values

Youngs modulus

(All sizes in GN/ The difference between the experimental value and the theoretical value is very small although the

experimental value is slightly smaller. This could be for a number of reasons. Either the theoretical

value for youngs modulus is wrong or mistakes such as the line of best fit being slightly wrong causedan error in the experimental value. Temperature can also affect this but this wouldnt have played amajor part as it was conducted at room temperature.

Poissons ratio

The experimental value was 2.424% smaller than the theoretical value. This could have been caused by

the line of best fit not being totally perfect. If there were more readings taken then perhaps theexperimental value would be closer to the theoretical value. The gauges might not be precise enough or

the thin cylinder might have a slightly different chemical and crystalline structure than the theoretical

one.

Principal stresses

(All sizes in

The percentage error in both the principle stresses is about 1%, this would therefore indicate that one or

more of the variables used to calculate the experimental value was wrong. Although at only 1% this

percentage error is relatively small.


16/21



Mohrs circle of strain

All sizes are

The percentage errors are both positive and negative which would give an indication that using Mohrscircle of strain gives a very close approximation. If more tests were carried out there is every chance

that the percentage error would average out to zero.

Factors of safety

Using Trescas method of finding the factor of safety it was calculated to be 6.025 whilst using Von

Misses method it was found that the FOS was 7.038. Both of these findings indicate that the cylinder

was nowhere near failure at 3MN/m as with Trescas method predicting it wouldnt fail till18.075MN/m and Von Misses method predicting failure wouldnt occur till pressure reached

21.114MN/m.

Trescas formula is more simplistic and conservative whereas Von Misses formula is more complicatedand gives a closer approximation. This would make Trescas method more favourable as a products

could be manufactured slightly weaker.

Overall accuracy

Overall the experiment got some good results and the percentage errors where relatively small. If the

experiment was repeated and there was more measurements taken with smaller increments there is no

doubt that the percentage errors could be reduced further. Although there is also a high likelihood that

the figures given by the manufacturer for the apparatus may be slightly wrong or that the strain orpressure gauges were slightly out.

Strain gauge Angle from the Theoretical value fromMohrs circle ( Actual value( 3 30 198.45 191.1

4 45 293.90 297.8

5 60 389.95 383.2


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Conclusion

The experimental values for Youngs modulus, Poissons ratio and the principal stresses were found andcompared successfully against the theoretical values. Mohrscircles of strain for the non-principal

gauges in the complex stress situation were constructed and proved reasonably accurate. The theories of

failure determined two factors of safety which both proved the cylinder was kept well within safe limits.Using the findings and the principles that were learnt it was possible to fully explain why the six strain

gauges read what they read. Therefore the experiment was a resounding success.

References

Matweb. (2011). aluminium 6063-T6.Available:http://asm.matweb.com/search/SpecificMaterial.asp?bassnum=MA6063T6. Last accessed 03 Jun 2013


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Appendix 1

Overall dimension 715mm x 310mm x 380mm high

Total mass of unit 26kg

Recommended oil Shell Tellus 37 (or equivalent)

Cylinder and reservoir

capacity

Approximately 2 litres

Pressure gauge 05.0 MN/m (Operating range 03.5 MN/m)

Nominal dimensions 80 mm internal diameter ( D )

3mm wall thickness ( t )

358.8mm length

Cylinder material Aged aluminium alloy 6063-T6Youngs modulus (E) 69GN/m

Poissons ratio 0.33

Strain gauges Electrical resistance and self-temperature compensation type.

Gauge factor 1.93


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Appendix 2


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Appendix 3


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Appendix 4

Strain gauge Angle from the Theoretical value fromMohrs circle (*

Actual value

(*3 30 198.45 191.1

4 45 293.90 297.8

5 60 389.95 383.2

direct strain

Shear strain

103

484.8

389.35

120

0direct strain

Shear strain

103

484.8

293.9

198.45

direct strain

Shear strain

103

484.8

60

90

0

Centre

0

Strain gauge three

Strain gauge four

Strain gauge five

Strength of materials Advanced

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