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Faculty of Engineering and Industrial Sciences HES 2120 - STRUCTURAL MECHANICS

LAB. 2 - THIN WALL CYLINDER STUDENT NAME & No.:Nguyen Thanh Tung

747590x

Lab. Date & Time / Demonstrator: 04-05-2012 / Hoss

INTRODUCTION: The aim of this Laboratory is to compare experimental strain measurements and theoretical strain calculations for an internally pressurised aluminium thin-walled cylinder, GUNT item FL 130. Experimental strain readings will be made using a Data Logger, GUNT item FL 151. A generic view of the equipment is shown in Fig. 1 below. Two slightly different sets of equipment in use – they are functionally equivalent. A sectioned view of the thin-walled cylinder, GUNT item FL 130, may be found in Fig. 3 overleaf. The equipment can act as either a closed cylinder or an open cylinder. Strain gauges provide a convenient means of measuring very small strains. A typical gauge is shown in Fig. 2a – it measures linear strain ε only along its longitudinal axis. A number of strain gauges are installed at different angles to the cylinder centreline as shown in Fig. 2b. Readings from these strain gauges are monitored by FL 151 Data Logger directly as microstrain (10 -6 m/m or μm/m).

Fig. 1 – FL 130 Thin-wall Cylinder and FL 151 Data Logger (Generic)

Fig. 2a – A single Strain Gauge

Fig. 2 b – Strain Gauge Orientation S.G. – Vers. 2A, April 2012 1 of 9

OPEN / CLOSED CYLINDER : Sectioned details of FL 130 Thin-wall Cylinder Apparatus are shown in Fig. 3. The equipment can act as either a closed or an open cylinder as follows: Screwing out the Plunger (5) seals against Collar (7) which effectively closes end of cylinder (1). Conversely, if Plunger is screwed in, cylinder sees no axial force and effectively acts as an open cylinder – axial reactions are carried by Frame (6).

Fig. 3 – Sectioned FL 130 Thin-wall Cylinder Apparatus (Generic)

STRAINS, STRESSES and REFERENCE AXES: For this experiment we will use right handed reference axes as follows: centreline of cylinder is X axis (positive to the right) and Y axis is perpendicular to this (positive up). Strain gauge angle of inclination θ (see Fig. 4a) must be measured anti-clockwise to the positive X axis. The Axial and Tangential (or Hoop) Stresses induced in an internally pressurised closed thin-walled cylinder (see Fig. 4b) will therefore be in the X and Y directions respectively.

Fig. 4a – Strain Gauge Reference Axes

Fig. 4b – Axial and Tangential Stresses

NOTE: x – Axis is Axial dir’n ; 2 of 9

y – Axis is Transverse dir’n.

EXPERIMENTAL RESULTS: (a) OPEN CYLINDER : (i)

Release hydraulic pump Bleed Valve (if present) fully. Screw in Plunger until pressure gauge indicates a small pre-load (say 1 Bar); this will be taken as Nominal Zero. Screw in Bleed Valve (if present) until tight. Take strain gauge readings for all channels using FL 151 Data Logger. (Allow readings to “settle”.)

(ii)

Use hyd. pump Handle / Handwheel to increase pressure by 20 bar and hold. Take all strain gauge readings – subtract (i) from (ii). [CAUTION : Never exceed 35 bar.]

(iii)

Reduce indicated cylinder pressure to Zero.

Strain Gauge

Channel

1 2 3 4 5

A1 A2 A3 A4 A5

Angle

Strain

Strain

Strain

θ

p= 1 bar (μm/m)

p=21 bar (μm/m)

∆p= 20 bar (μm/m)

0o - 30 o - 45 o - 60 o - 90 o

-33 -17.7 0.4 18 41.7

-144.1 -34.5 93 216 368.8

-111.1 -16.8 92.4 197.6 327.1

.

. (b) CLOSED CYLINDER: Ensure indicated pressure zero prior to start. (iv)

Release hydraulic pump Bleed Valve (if present) fully, screw out Plunger. Screw in Bleed Valve (if present) until tight. Apply 10 bar pressure using hyd. pump Handle / Handwheel (to ensure Plunger seated), release Bleed Valve (if present) fully. Screw in Plunger roughly half a turn. Screw in Bleed Valve (if present) until tight. Apply small pre-load (say 1 Bar) using hyd. pump Handle / Handwheel; this will be taken as our Nominal Zero. Take strain gauge readings for all channels using FL 151 Data Logger. (Allow readings to “settle”.)

(v)

Increase pressure by 20 bars and hold using hyd. pump Handle / Handwheel. Take all strain gauge readings – subtract (IV) from (v). [CAUTION: Never exceed 35 bars.]

(vi)

Reduce indicated cylinder pressure to Zero.

Strain Gauge

Channel

1 2 3 4 5

A1 A2 A3 A4 A5

Angle

Strain

Strain

Strain

θ

p= 1 bar (μm/m)

p= 21 bar (μm/m)

∆p= 20 bar (μm/m)

0o - 30 o - 45 o - 60 o - 90 o

120.6 22.9 -90.6 -199.2 -333

174.9 124 69.6 12 -66.6

54.3 101.1 160.2 211.2 267

3 of 9

ANALYSIS INSTRUCTIONS: Analyse both open and closed cylinders as follows, filling out Table overleaf: For background Theory, meaning of symbols etc. - refer to Beer, Johnston ET. Al. – Mechanics of Materials – 5th Edition. A.

Determine Theoretical STRESS : As the cylinder is thin-walled, the radial stress can be neglected ( σ R = σ Z = 0) and we have a case of Plane Stress. Hoop or Tangential stress for a cylinder of internal radius r is determined by:

σ HOOP

σY

p r t

(7.30)

Axial stress is determined: [This will of course be Zero for the open cylinder case.]

σ AXIAL

σX

p r 2 t

(7.31)

Draw a Mohr’s Circle to scale of in-plane Stress for both open and closed cylinders (two graphs required) – refer to Section 7.9 in Text for guidance. B.

Determine Theoretical STRAIN : Convert stress to strain using three dimensional Hooke’s Law [Recall

εX εY C.

σX E σY E

σY E σ ν. X E ν.

σZ E σ ν. Z E ν.

σ Z = 0]

(2.38)

TRANSFORM Theoretical STRAIN : For a strain gauge inclined at an angle θ to the X axis, determine linear strain along its own axis,

εθ

εθ

,

using following equation :

ε X .cos 2θ ε Y .sin 2θ

γ XY .cosθ sinθ

(7.60)

[ Note : For this particular loading case shear stress, τXY, and hence shear strain, γXY, will both be Zero.] D.

COMPARE with Experimental STRAIN : At selected locations (strain gauges 1, 3 and 5), recall Experimental strains from Results section and compare with above Theoretical strains. Note re. Terminology used in Table overleaf: ε θ3 , θ in strain gauge No. 3, which is inclined at θ

4 of 9

- 450 .

- 45 0 is the linear strain

ANALYSIS SUMMARY TABLE: REF. A A B

OPEN CYLINDER - THEORY QUANTITY VALUE UNITS

σ AXIAL σ HOOP

σX σY

0 23.3 -107

εX

(MPa) (MPa) (μm/m)

324 -107

(μm/m)

C

εY ε θ1 , θ 00

C

ε θ3 , θ - 45 0

108.5

(μm/m)

C

ε θ5 , θ - 90 0

324

(μm/m)

B

(μm/m)

OPEN CYLINDER - EXPERIMENT REF. QUANTITY VALUE UNITS D (μm/m) -111.1 ε , θ 00 θ1

D

ε θ3 , θ - 45 0

92.4

(μm/m)

D

ε θ5 , θ - 90 0

327.1

(μm/m)

CLOSED CYLINDER - THEORY REF. QUANTITY VALUE UNITS A 11.7 (MPa) σ AXIAL σ X A B

σ HOOP

σY

23.3 55.7

εX

(MPa) (μm/m)

270 55.7

(μm/m)

C

εY ε θ1 , θ 00

C

ε θ3 , θ - 45 0

162.85

(μm/m)

C

ε θ5 , θ - 90 0

270

(μm/m)

B

(μm/m)

CLOSED CYLINDER - EXPERIMENT REF. QUANTITY VALUE UNITS D (μm/m) 54.3 ε , θ 00 θ1

D

ε θ3 , θ - 45 0

160.2

(μm/m)

D

ε θ5 , θ - 90 0

267

(μm/m)

5 of 9

.

.

DATA: 1 Microstrain = 10 -6 m/m or μm/m Cylinder dimensions:

Length = 400 mm;

1 bar = 105 Pa Outer diameter = 76 mm;

Thickness = 3.0 mm.

Aluminium properties: Young’s Modulus, E = 72 GPa; Poisson’s ratio, ν = 0.33. .

.

LABORATORY REPORT: Hand in your individual Lab. Report by due Date – 1 week after conducting Lab. Please scan your report and save whole report as one PDF file (not per page), email to your demonstrator. Please see Study Guide or Blackboard for your demonstrator’s email address. Also please refer to Subject Outline for penalties etc. The Report must include (in order): (i) This 6 page handout as cover sheet /results for your report. (2 marks) Experimental results should be recorded on page 4. Conduct analysis per enclosed instructions for both open and closed cylinders and fill in summary of analysis in table on page 6. Attach sample calculations as noted below. (ii) Two Mohr’s Circles of in-plane Stress to scale (one each for open and closed cylinders) (1 mark). (iii) Discussion and Conclusion. Compare theoretical results with those measured and comment on potential sources of errors (1 mark). (iv) Sample Calculations should be included in an Appendix, attached to the end (1 mark). The report should be brief and to the point, it is not necessary to include diagrams or a procedure unless directly relevant to your discussion.

6 of 9

Mohr’s Circle in plane Stress: 1. Mohr’s Circle of Open Cylinder in plan Stress

2. Mohr’s Circle of Close Cylinder in plan Stress

7 of 9

Discussion During the experience the data between theory and the measurement result appear a lot of different, the data from measurement is higher or lower than theory data but that is not much. For example, in open cylinder at angle 0 the strain is -111.1µm/m in measurements and lowers than theory result that is -107µm/m but at angle 90 the measurement is higher than theory (327.1µm/m and 324µm/m). In closed cylinder at angle 0 the strain is 54.3µm/m lower than 55.7µm/m of theory. Moreover, there are some impact that lead to the error of the measurement result, the most normally result is because of the machine is not running well every time if look around and compare to other machines people can see the different of the result. Secondly, when we applied the pressure at 1bar and 21bar is not exactly that also given a little bit of impact to system. Addition, when we do experiment, the hand wheel for plunger is not maximum open or tightly closed for open and closed cylinder. The second reason is that the FL 130 Thin-wall Cylinder Apparatus is used in several years. As the result, the values which we collected from the machine will be not accurate.

Conclusion In conclusion, to improve the accuracy, the machine should use auto electrical pump, not manual pump (hydraulic pump) which makes more accuracy. Moreover, we also need to maintain the machine to make sure the values taken from the machine are similar to the first times.

8 of 9

Appendix For closed cylinder

9 of 9

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LAB. 2 - THIN WALL CYLINDER STUDENT NAME & No.:Nguyen Thanh Tung

747590x

Lab. Date & Time / Demonstrator: 04-05-2012 / Hoss

INTRODUCTION: The aim of this Laboratory is to compare experimental strain measurements and theoretical strain calculations for an internally pressurised aluminium thin-walled cylinder, GUNT item FL 130. Experimental strain readings will be made using a Data Logger, GUNT item FL 151. A generic view of the equipment is shown in Fig. 1 below. Two slightly different sets of equipment in use – they are functionally equivalent. A sectioned view of the thin-walled cylinder, GUNT item FL 130, may be found in Fig. 3 overleaf. The equipment can act as either a closed cylinder or an open cylinder. Strain gauges provide a convenient means of measuring very small strains. A typical gauge is shown in Fig. 2a – it measures linear strain ε only along its longitudinal axis. A number of strain gauges are installed at different angles to the cylinder centreline as shown in Fig. 2b. Readings from these strain gauges are monitored by FL 151 Data Logger directly as microstrain (10 -6 m/m or μm/m).

Fig. 1 – FL 130 Thin-wall Cylinder and FL 151 Data Logger (Generic)

Fig. 2a – A single Strain Gauge

Fig. 2 b – Strain Gauge Orientation S.G. – Vers. 2A, April 2012 1 of 9

OPEN / CLOSED CYLINDER : Sectioned details of FL 130 Thin-wall Cylinder Apparatus are shown in Fig. 3. The equipment can act as either a closed or an open cylinder as follows: Screwing out the Plunger (5) seals against Collar (7) which effectively closes end of cylinder (1). Conversely, if Plunger is screwed in, cylinder sees no axial force and effectively acts as an open cylinder – axial reactions are carried by Frame (6).

Fig. 3 – Sectioned FL 130 Thin-wall Cylinder Apparatus (Generic)

STRAINS, STRESSES and REFERENCE AXES: For this experiment we will use right handed reference axes as follows: centreline of cylinder is X axis (positive to the right) and Y axis is perpendicular to this (positive up). Strain gauge angle of inclination θ (see Fig. 4a) must be measured anti-clockwise to the positive X axis. The Axial and Tangential (or Hoop) Stresses induced in an internally pressurised closed thin-walled cylinder (see Fig. 4b) will therefore be in the X and Y directions respectively.

Fig. 4a – Strain Gauge Reference Axes

Fig. 4b – Axial and Tangential Stresses

NOTE: x – Axis is Axial dir’n ; 2 of 9

y – Axis is Transverse dir’n.

EXPERIMENTAL RESULTS: (a) OPEN CYLINDER : (i)

Release hydraulic pump Bleed Valve (if present) fully. Screw in Plunger until pressure gauge indicates a small pre-load (say 1 Bar); this will be taken as Nominal Zero. Screw in Bleed Valve (if present) until tight. Take strain gauge readings for all channels using FL 151 Data Logger. (Allow readings to “settle”.)

(ii)

Use hyd. pump Handle / Handwheel to increase pressure by 20 bar and hold. Take all strain gauge readings – subtract (i) from (ii). [CAUTION : Never exceed 35 bar.]

(iii)

Reduce indicated cylinder pressure to Zero.

Strain Gauge

Channel

1 2 3 4 5

A1 A2 A3 A4 A5

Angle

Strain

Strain

Strain

θ

p= 1 bar (μm/m)

p=21 bar (μm/m)

∆p= 20 bar (μm/m)

0o - 30 o - 45 o - 60 o - 90 o

-33 -17.7 0.4 18 41.7

-144.1 -34.5 93 216 368.8

-111.1 -16.8 92.4 197.6 327.1

.

. (b) CLOSED CYLINDER: Ensure indicated pressure zero prior to start. (iv)

Release hydraulic pump Bleed Valve (if present) fully, screw out Plunger. Screw in Bleed Valve (if present) until tight. Apply 10 bar pressure using hyd. pump Handle / Handwheel (to ensure Plunger seated), release Bleed Valve (if present) fully. Screw in Plunger roughly half a turn. Screw in Bleed Valve (if present) until tight. Apply small pre-load (say 1 Bar) using hyd. pump Handle / Handwheel; this will be taken as our Nominal Zero. Take strain gauge readings for all channels using FL 151 Data Logger. (Allow readings to “settle”.)

(v)

Increase pressure by 20 bars and hold using hyd. pump Handle / Handwheel. Take all strain gauge readings – subtract (IV) from (v). [CAUTION: Never exceed 35 bars.]

(vi)

Reduce indicated cylinder pressure to Zero.

Strain Gauge

Channel

1 2 3 4 5

A1 A2 A3 A4 A5

Angle

Strain

Strain

Strain

θ

p= 1 bar (μm/m)

p= 21 bar (μm/m)

∆p= 20 bar (μm/m)

0o - 30 o - 45 o - 60 o - 90 o

120.6 22.9 -90.6 -199.2 -333

174.9 124 69.6 12 -66.6

54.3 101.1 160.2 211.2 267

3 of 9

ANALYSIS INSTRUCTIONS: Analyse both open and closed cylinders as follows, filling out Table overleaf: For background Theory, meaning of symbols etc. - refer to Beer, Johnston ET. Al. – Mechanics of Materials – 5th Edition. A.

Determine Theoretical STRESS : As the cylinder is thin-walled, the radial stress can be neglected ( σ R = σ Z = 0) and we have a case of Plane Stress. Hoop or Tangential stress for a cylinder of internal radius r is determined by:

σ HOOP

σY

p r t

(7.30)

Axial stress is determined: [This will of course be Zero for the open cylinder case.]

σ AXIAL

σX

p r 2 t

(7.31)

Draw a Mohr’s Circle to scale of in-plane Stress for both open and closed cylinders (two graphs required) – refer to Section 7.9 in Text for guidance. B.

Determine Theoretical STRAIN : Convert stress to strain using three dimensional Hooke’s Law [Recall

εX εY C.

σX E σY E

σY E σ ν. X E ν.

σZ E σ ν. Z E ν.

σ Z = 0]

(2.38)

TRANSFORM Theoretical STRAIN : For a strain gauge inclined at an angle θ to the X axis, determine linear strain along its own axis,

εθ

εθ

,

using following equation :

ε X .cos 2θ ε Y .sin 2θ

γ XY .cosθ sinθ

(7.60)

[ Note : For this particular loading case shear stress, τXY, and hence shear strain, γXY, will both be Zero.] D.

COMPARE with Experimental STRAIN : At selected locations (strain gauges 1, 3 and 5), recall Experimental strains from Results section and compare with above Theoretical strains. Note re. Terminology used in Table overleaf: ε θ3 , θ in strain gauge No. 3, which is inclined at θ

4 of 9

- 450 .

- 45 0 is the linear strain

ANALYSIS SUMMARY TABLE: REF. A A B

OPEN CYLINDER - THEORY QUANTITY VALUE UNITS

σ AXIAL σ HOOP

σX σY

0 23.3 -107

εX

(MPa) (MPa) (μm/m)

324 -107

(μm/m)

C

εY ε θ1 , θ 00

C

ε θ3 , θ - 45 0

108.5

(μm/m)

C

ε θ5 , θ - 90 0

324

(μm/m)

B

(μm/m)

OPEN CYLINDER - EXPERIMENT REF. QUANTITY VALUE UNITS D (μm/m) -111.1 ε , θ 00 θ1

D

ε θ3 , θ - 45 0

92.4

(μm/m)

D

ε θ5 , θ - 90 0

327.1

(μm/m)

CLOSED CYLINDER - THEORY REF. QUANTITY VALUE UNITS A 11.7 (MPa) σ AXIAL σ X A B

σ HOOP

σY

23.3 55.7

εX

(MPa) (μm/m)

270 55.7

(μm/m)

C

εY ε θ1 , θ 00

C

ε θ3 , θ - 45 0

162.85

(μm/m)

C

ε θ5 , θ - 90 0

270

(μm/m)

B

(μm/m)

CLOSED CYLINDER - EXPERIMENT REF. QUANTITY VALUE UNITS D (μm/m) 54.3 ε , θ 00 θ1

D

ε θ3 , θ - 45 0

160.2

(μm/m)

D

ε θ5 , θ - 90 0

267

(μm/m)

5 of 9

.

.

DATA: 1 Microstrain = 10 -6 m/m or μm/m Cylinder dimensions:

Length = 400 mm;

1 bar = 105 Pa Outer diameter = 76 mm;

Thickness = 3.0 mm.

Aluminium properties: Young’s Modulus, E = 72 GPa; Poisson’s ratio, ν = 0.33. .

.

LABORATORY REPORT: Hand in your individual Lab. Report by due Date – 1 week after conducting Lab. Please scan your report and save whole report as one PDF file (not per page), email to your demonstrator. Please see Study Guide or Blackboard for your demonstrator’s email address. Also please refer to Subject Outline for penalties etc. The Report must include (in order): (i) This 6 page handout as cover sheet /results for your report. (2 marks) Experimental results should be recorded on page 4. Conduct analysis per enclosed instructions for both open and closed cylinders and fill in summary of analysis in table on page 6. Attach sample calculations as noted below. (ii) Two Mohr’s Circles of in-plane Stress to scale (one each for open and closed cylinders) (1 mark). (iii) Discussion and Conclusion. Compare theoretical results with those measured and comment on potential sources of errors (1 mark). (iv) Sample Calculations should be included in an Appendix, attached to the end (1 mark). The report should be brief and to the point, it is not necessary to include diagrams or a procedure unless directly relevant to your discussion.

6 of 9

Mohr’s Circle in plane Stress: 1. Mohr’s Circle of Open Cylinder in plan Stress

2. Mohr’s Circle of Close Cylinder in plan Stress

7 of 9

Discussion During the experience the data between theory and the measurement result appear a lot of different, the data from measurement is higher or lower than theory data but that is not much. For example, in open cylinder at angle 0 the strain is -111.1µm/m in measurements and lowers than theory result that is -107µm/m but at angle 90 the measurement is higher than theory (327.1µm/m and 324µm/m). In closed cylinder at angle 0 the strain is 54.3µm/m lower than 55.7µm/m of theory. Moreover, there are some impact that lead to the error of the measurement result, the most normally result is because of the machine is not running well every time if look around and compare to other machines people can see the different of the result. Secondly, when we applied the pressure at 1bar and 21bar is not exactly that also given a little bit of impact to system. Addition, when we do experiment, the hand wheel for plunger is not maximum open or tightly closed for open and closed cylinder. The second reason is that the FL 130 Thin-wall Cylinder Apparatus is used in several years. As the result, the values which we collected from the machine will be not accurate.

Conclusion In conclusion, to improve the accuracy, the machine should use auto electrical pump, not manual pump (hydraulic pump) which makes more accuracy. Moreover, we also need to maintain the machine to make sure the values taken from the machine are similar to the first times.

8 of 9

Appendix For closed cylinder

9 of 9

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