ME 220 University of Idaho Combined Stress & Pop Can Experiments Lab Report

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ME 220

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ME220 – Mechanics of Materials Laboratory TEST TITLE: Combined Stress & Pop Can Experiments NAME: (refer to lab manual pp. 74-90) 1. Summary (1/12) (The summary should be succinct (limited to one page), but contain the following four pieces of information, namely, the purpose of the experiment; experimental methods; results; and conclusion.) 2. Results (5/12) 1. Tabulate the test results as follows Torque only (a) ( in-lb) (experiment) Pressure only (b) (psi) (experiment) Sum of (a) and (b) Combined Torque and Pressure (experiment) gage 1 gage 2 gage 3 gage 4 gage 5 gage 6 4. Tabulate the principal strains measured from delta and rectangular strain gage rosettes as follows (for the combined loading only). ε1 ε2 Delta rosette Rectangular rosette 5. For each strain gage rosette, calculate the corresponding σ 1 , σ 2 , and τmax using the measured ε1 and ε 2 . The Young’s modulus (E) and Poisson’s ratio for steel are 30x106 psi and 0.25 respectively. 6. Calculate the theoretical σ 1 , σ 2 , and τmax using the thick and thin-walled pressure vessel equations as described below. 2 stress due to P Stress due to T Thick-walled tube σ a = pri2 / (ro2 − ri2 ) τ = Tro / J (outer surface) σ c = 2σ a Thin-walled tube σ a = Prm / (2t ) (outer surface) σ c = 2σ a ( τ = T / 2πrm2t Principal stresses: σ 1, 2 = (σ a + σ c ) / 2 ± {[(σ a − σ c ) / 2] + τ 2 2 } 1/ 2 Maximum shear stress: τ max = {[σ a − σ c ) / 2] + τ 2 } 2 1/ 2 [ ( Where: ro = outer radius ri = inner radius rm = mean radius [= (ro + ri ) / 2] t = wall thickness J = polar moment of inertia of the tube = (π / 2) ro4 − ri4 P = pressure T = torque σ a = axial stress σ c = circumferential stress τ = shear stress 3 )] ) 7. Tabulate the results from steps 5 and 6 as follows. σ1 σ2 τ max Experiment (delta) Experiment (rectan.) Thick-wall calc. Thin-wall calc. 3. Discussion (2/12) 1. Comment on the superposition principle applied in the current experiment. 2. Compare analytical and experimental results. 3. Discuss any sources of error in the analytical calculations. 4. Discuss any sources of error in the experimental method. 4. Conclusion (1/12) (Continue to next page) 4 THIN-WALLED PRESSURE VESSEL EXPERIMENT Lab Report 1. Summary (0.5/12) 2. Results (1/12) The results of the pop can experiment are as follows: Diameter before opening: 2.6005 inches Strain value: −840 ×10−6 in/in Wall Thickness: 0.0038 inches Calculate the pressure based on the measured strain (E and ν for aluminum are 10x106 psi and 0.33 respectively) 3. Discussion (1/12) 1. Do some research to see if you could find some information on the range of pressure inside the beverage can and comment on your results. 2. A circular shaft with radius r is subjected to a torque T. The modulus of elasticity and Poisson's ratio of the shaft are E and ν respectively. Show how a single strain gage mounted in the 450 direction with respect to the longitudinal axis can be employed to determine the applied torque. In other words, write T in terms of E, ν , r, and ε, where ε is the reading of the strain gage. 4. Conclusion (0.5/12) 5 Combined Strain experiment Date 26-Jul-20 group gage factor unload voltage (V) unload strain 2.070 Delta Rosette 0 deg 60 deg 1 2 -4.34160E-03 -0.0014514 0 0 120 deg 3 -0.0035962 0 2.095 2.100 2.095 Rectangular Rosette 0 deg 45 deg 90 deg 1 2 3 -0.0041220 -0.0028902 -0.0026624 0 0 0 torsion load (lbs) = voltages (V) Vr Strain 20 -4.34780E-03 -0.0000012 0.0000023 -0.0041192 -0.0001020 0.0001971 -0.0041786 -0.0000110 0.0000211 principle strain + principle strain acute angle (rad) acute angle (deg) 0.000229157 -0.000229706 -0.779709428 -44.67405945 pressure (psi) = voltages (V) Vr strain 2000 -4.57300E-03 -0.0000451 0.0000872 principle strain + principle strain acute angle (rad) acute angle (deg) 0.00031718 8.71886E-05 0.006056906 0.347035173 combined load voltages (V) Vr strain -4.57580E-03 -0.0000457 0.0000883 principle strain + principle strain acute angle (rad) acute angle (deg) 2.070 0.0004589 -0.0000563 0.5581708 31.9808310 -0.0009198 0.0001037 -0.0002003 2.070 -0.0022804 0.0001189 -0.0002264 -0.0025956 0.0000130 -0.0000249 -0.0033760 -0.0000947 0.0001805 -0.0034854 -0.0001605 0.0003065 -0.0027634 0.0000247 -0.0000471 -0.0034158 -0.0001469 0.0002806 0.000223806 -0.0002276 -0.73442375 -42.0793811 -0.0021372 -0.0001337 0.0002585 -0.0042884 -0.0001350 0.0002609 -0.0043142 -0.0000375 0.0000716 0.000306808 7.12484E-05 0.036389394 2.084958689 -0.0016038 -0.0000297 0.0000574 -0.0048120 -0.0002371 0.0004583 -0.0043682 -0.0000480 0.0000917 0.0004377 -0.0000655 0.5929928 33.9759842 A C B A B C shear Excitation Voltage (Volts) 5.1285 5.1285 -0.0001 -0.0002 -0.0003 5.1285 5.1285 Mohr's strain circle (combined) delta rosette Mohr's strain circle (torsion) delta rose 0.0002 0.0002 0.0001 0.0001 shear 0.0003 shear 0.0003 0 0 -0.0001 -0.0001 -0.0002 -0.0002 -0.0003 -0.0001 0.0001 0.0003 0.0005 -0.0003 -0.0003 -0.0001 normal Mohr's strain circle (combined) rectangular rosette 0.0003 0.0003 0.0002 0.0002 0.0001 0.0001 shear shear Mohr's strain circle (torsion) rectangular ros 0 0 -0.0001 -0.0001 -0.0002 -0.0002 -0.0003 -0.0001 0 0.0001 0.0002 normal 0.0003 0.0004 0.0005 -0.0003 -0.0003 -0.0001 Mohr's strain circle (pressure) delta rosette ohr's strain circle (torsion) delta rosette 0.0002 0.00015 0.0001 shear 0.00005 0 -0.00005 -0.0001 -0.00015 -0.0002 -0.0001 0.0001 0 0.0003 0.0001 normal 0.0002 0.0003 normal Mohr's strain circle (pressure) rectangular rosette strain circle (torsion) rectangular rosette 0.0002 0.00015 0.0001 shear 0.00005 0 -0.00005 -0.0001 -0.00015 -0.0002 -0.0001 normal 0.0001 0.0003 0 0.0001 0.0002 normal 0.0003 delta rosette 0.0004 ctangular rosette 0.0004 Data tables used to gen max strain min strain angle radius center angle(radians) 0.00 0.39 0.79 1.18 1.57 1.96 2.36 2.75 3.14 3.53 3.93 4.32 4.71 5.11 5.50 5.89 6.28 line a data line b data line c data line A data line B data line C data Data tables used to generate mohr's circles delta rossette torsion max strain 2.29157E-04 min strain -2.29706E-04 angle -7.79709E-01 radius 0.000229432 center -2.74191E-07 x angle(radians) X for plotting 0.00 2.29157E-04 0.39 2.11693E-04 0.79 1.61958E-04 1.18 8.75255E-05 1.57 -2.74191E-07 1.96 -8.80738E-05 2.36 -1.62507E-04 2.75 -2.12241E-04 3.14 -2.29706E-04 3.53 -2.12241E-04 3.93 -1.62507E-04 4.32 -8.80738E-05 4.71 -2.74191E-07 5.11 8.75255E-05 5.50 1.61958E-04 5.89 2.11693E-04 6.28 2.29157E-04 y Y for plotting 0.00000E+00 8.77996E-05 1.62233E-04 2.11967E-04 2.29432E-04 2.11967E-04 1.62233E-04 8.77996E-05 2.81088E-20 -8.77996E-05 -1.62233E-04 -2.11967E-04 -2.29432E-04 -2.11967E-04 -1.62233E-04 -8.77996E-05 -5.62175E-20 rectangular rosette torsion 2.23806E-04 -2.27602E-04 -7.34424E-01 0.000225704 -1.89814E-06 x y X for plotting Y for plotting 2.23806E-04 0.00000E+00 2.06625E-04 8.63732E-05 1.57699E-04 1.59597E-04 8.44751E-05 2.08523E-04 -1.89814E-06 2.25704E-04 -8.82713E-05 2.08523E-04 -1.61495E-04 1.59597E-04 -2.10421E-04 8.63732E-05 -2.27602E-04 2.76521E-20 -2.10421E-04 -8.63732E-05 -1.61495E-04 -1.59597E-04 -8.82713E-05 -2.08523E-04 -1.89814E-06 -2.25704E-04 8.44751E-05 -2.08523E-04 1.57699E-04 -1.59597E-04 2.06625E-04 -8.63732E-05 2.23806E-04 -5.53042E-20 line a data 2.33610E-06 2.75318E-04 2.33610E-06 -2.75318E-04 2.10723E-05 2.10723E-05 2.70845E-04 -2.70845E-04 line b data -2.00260E-04 2.75318E-04 -2.00260E-04 -2.75318E-04 -2.26430E-04 -2.26430E-04 2.70845E-04 -2.70845E-04 line c data 1.97101E-04 2.75318E-04 1.97101E-04 -2.75318E-04 -2.48686E-05 -2.48686E-05 2.70845E-04 -2.70845E-04 line A data -2.74191E-07 0.00000E+00 -2.88449E-06 -0.00022942 -1.89814E-06 2.10723E-05 0.00000E+00 2.24532E-04 line B data -2.74191E-07 0.00000E+00 1.99712E-04 0.000112448 -1.89814E-06 -2.26430E-04 0.00000E+00 2.29704E-05 line C data -2.74191E-07 0.00000E+00 -1.97650E-04 0.000116969 -1.89814E-06 -2.48686E-05 0.00000E+00 -2.24532E-04 delta rossette y Y for plotting 0.00000E+00 4.40070E-05 8.13143E-05 1.06242E-04 1.14996E-04 1.06242E-04 8.13143E-05 4.40070E-05 1.40887E-20 -4.40070E-05 -8.13143E-05 -1.06242E-04 -1.14996E-04 -1.06242E-04 -8.13143E-05 -4.40070E-05 -2.81774E-20 rectangular rosette pressure 3.06808E-04 7.12484E-05 3.63894E-02 0.00011778 1.89028E-04 x y X for plotting Y for plotting 3.06808E-04 0.00000E+00 2.97842E-04 4.50723E-05 2.72311E-04 8.32828E-05 2.34100E-04 1.08814E-04 1.89028E-04 1.17780E-04 1.43956E-04 1.08814E-04 1.05745E-04 8.32828E-05 8.02139E-05 4.50723E-05 7.12484E-05 1.44298E-20 8.02139E-05 -4.50723E-05 1.05745E-04 -8.32828E-05 1.43956E-04 -1.08814E-04 1.89028E-04 -1.17780E-04 2.34100E-04 -1.08814E-04 2.72311E-04 -8.32828E-05 2.97842E-04 -4.50723E-05 3.06808E-04 -2.88595E-20 combined 4.58934E-04 -5.62694E-05 5.58171E-01 0.000257602 2.01332E-04 x X for plotting 4.58934E-04 4.39325E-04 3.83484E-04 2.99912E-04 2.01332E-04 1.02752E-04 1.91804E-05 -3.66606E-05 -5.62694E-05 -3.66606E-05 1.91804E-05 1.02752E-04 2.01332E-04 2.99912E-04 3.83484E-04 4.39325E-04 4.58934E-04 8.71971E-05 1.37995E-04 8.71971E-05 -1.37995E-04 7.15602E-05 1.41336E-04 7.15602E-05 -1.41336E-04 8.82523E-05 8.82523E-05 2.58472E-04 1.37995E-04 2.58472E-04 -1.37995E-04 1.80464E-04 1.41336E-04 1.80464E-04 -1.41336E-04 5.74262E-05 5.74262E-05 2.60884E-04 1.37995E-04 2.60884E-04 -1.37995E-04 3.06496E-04 1.41336E-04 3.06496E-04 -1.41336E-04 4.58319E-04 4.58319E-04 2.02184E-04 0.00000E+00 8.71971E-05 1.39300E-06 1.89028E-04 0.00000E+00 7.15602E-05 8.56430E-06 2.01332E-04 8.82523E-05 2.02184E-04 0.00000E+00 2.58472E-04 -1.00278E-04 1.89028E-04 0.00000E+00 1.80464E-04 -1.17468E-04 2.01332E-04 5.74262E-05 2.02184E-04 0.00000E+00 2.60884E-04 1.00278E-04 1.89028E-04 0.00000E+00 3.06496E-04 -8.56430E-06 2.01332E-04 4.58319E-04 pressure 3.17180E-04 8.71886E-05 6.05691E-03 0.000114996 2.02184E-04 x X for plotting 3.17180E-04 3.08427E-04 2.83499E-04 2.46191E-04 2.02184E-04 1.58177E-04 1.20870E-04 9.59422E-05 8.71886E-05 9.59422E-05 1.20870E-04 1.58177E-04 2.02184E-04 2.46191E-04 2.83499E-04 3.08427E-04 3.17180E-04 delta rossette y Y for plotting 0.00000E+00 9.85799E-05 1.82152E-04 2.37993E-04 2.57602E-04 2.37993E-04 1.82152E-04 9.85799E-05 3.15600E-20 -9.85799E-05 -1.82152E-04 -2.37993E-04 -2.57602E-04 -2.37993E-04 -1.82152E-04 -9.85799E-05 -6.31201E-20 rectangular rosette combined 4.37728E-04 -6.54925E-05 5.92993E-01 0.00025161 1.86118E-04 x y X for plotting Y for plotting 4.37728E-04 0.00000E+00 4.18576E-04 9.62872E-05 3.64033E-04 1.77915E-04 2.82405E-04 2.32458E-04 1.86118E-04 2.51610E-04 8.98308E-05 2.32458E-04 8.20250E-06 1.77915E-04 -4.63398E-05 9.62872E-05 -6.54925E-05 3.08260E-20 -4.63398E-05 -9.62872E-05 8.20250E-06 -1.77915E-04 8.98308E-05 -2.32458E-04 1.86118E-04 -2.51610E-04 2.82405E-04 -2.32458E-04 3.64033E-04 -1.77915E-04 4.18576E-04 -9.62872E-05 4.37728E-04 -6.16520E-20 3.09122E-04 -3.09122E-04 9.16675E-05 3.01933E-04 9.16675E-05 -3.01933E-04 3.09122E-04 -3.09122E-04 -4.70921E-05 3.01933E-04 -4.70921E-05 -3.01933E-04 3.09122E-04 -3.09122E-04 2.80568E-04 3.01933E-04 2.80568E-04 -3.01933E-04 0.00000E+00 2.31455E-04 1.86118E-04 0.00000E+00 9.16675E-05 2.33210E-04 0.00000E+00 -2.13658E-04 1.86118E-04 0.00000E+00 -4.70921E-05 -9.44505E-05 0.00000E+00 -1.77974E-05 1.86118E-04 0.00000E+00 2.80568E-04 -2.33210E-04
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Explanation & Answer

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Combined Strain experiment
Date
28-Jul-20
group
gage factor

unload voltage (V)
unload strain

2,070
Delta Rosette
0 deg
60 deg
1
2
-4,34160E-03
-0,0014514
0
0

120 deg
3
-0,0035962
0

2,095
2,100
2,095
Rectangular Rosette
0 deg
45 deg
90 deg
1
2
3
-0,0041220
-0,0028902
-0,0026624
0
0
0

torsion load (lbs) =
voltages (V)
Vr
Strain

20
-4,34780E-03
-0,0000012
0,0000023

-0,0041192
-0,0001020
0,0001971

-0,0041786
-0,0000110
0,0000211

principle strain +
principle strain acute angle (rad)
acute angle (deg)

0,000229157
-0,000229706
-0,779709428
-44,67405945

pressure (psi) =
voltages (V)
Vr
strain

2000
-4,57300E-03
-0,0000451
0,0000872

principle strain +
principle strain acute angle (rad)
acute angle (deg)

0,00031718
8,71886E-05
0,006056906
0,347035173

combined load
voltages (V)
Vr
strain

-4,57580E-03
-0,0000457
0,0000883

principle strain +
principle strain acute angle (rad)
acute angle (deg)

2,070

0,0004589
-0,0000563
0,5581708
31,9808310

-0,0009198
0,0001037
-0,0002003

2,070

-0,0022804
0,0001189
-0,0002264

-0,0025956
0,0000130
-0,0000249

-0,0033760
-0,0000947
0,0001805

-0,0034854
-0,0001605
0,0003065

-0,0027634
0,0000247
-0,0000471

-0,0034158
-0,0001469
0,0002806

0,000223806
-0,0002276
-0,73442375
-42,0793811

-0,0021372
-0,0001337
0,0002585

-0,0042884
-0,0001350
0,0002609

-0,0043142
-0,0000375
0,0000716
0,000306808
7,12484E-05
0,036389394
2,084958689

-0,0016038
-0,0000297
0,0000574

-0,0048120
-0,0002371
0,0004583

-0,0043682
-0,0000480
0,0000917
0,0004377
-0,0000655
0,5929928
33,9759842

A

C

B

A

B

C

shear

Excitation
Voltage
(Volts)
5,1285

5,1285

-0,0001

-0,0002

-0,0003

5,1285

5,1285

Mohr's strain circle (combined) delta rosette

Mohr's strain circle (torsion) delta rose

0,0002

0,0002

0,0001

0,0001

shear

0,0003

shear

0,0003

0

0

-0,0001

-0,0001

-0,0002

-0,0002

-0,0003
-0,0001

0,0001

0,0003

0,0005

-0,0003
-0,0003

-0,0001

normal
Mohr's strain circle (combined) rectangular rosette
0,0003

0,0003

0,0002

0,0002

0,0001

0,0001

shear

shear

Mohr's strain circle (torsion) rectangular ros

0

0

-0,0001

-0,0001

-0,0002

-0,0002

-0,0003
-0,0001

0

0,0001

0,0002

normal

0,0003

0,0004

0,0005

-0,0003
-0,0003

-0,0001

Mohr's strain circle (pressure) delta rosette

ohr's strain circle (torsion) delta rosette

0,0002
0,00015
0,0001

shear

0,00005
0

-0,00005
-0,0001
-0,00015
-0,0002
-0,0001

0,0001

0

0,0003

0,0001

normal

0,0002

0,0003

normal
Mohr's strain circle (pressure) rectangular rosette

strain circle (torsion) rectangular rosette
0,0002
0,00015
0,0001

shear

0,00005
0

-0,00005
-0,0001
-0,00015
-0,0002
-0,0001

normal

0,0001

0,0003

0

0,0001

0,0002

normal

0,0003

delta rosette

0,0004

ctangular rosette

0,0004

Data tables used to gen

max strain
min strain
angle
radius
center
angle(radians)
0,00
0,39
0,79
1,18
1,57
1,96
2,36
2,75
3,14
3,53
3,93
4,32
4,71
5,11
5,50
5,89
6,28
line a data

line b data

line c data

line A data

line B data

line C data

Data tables used to generate mohr's circles
delta rossette
torsion
max strain
2,29157E-04
min strain
-2,29706E-04
angle
-7,79709E-01
radius
0,000229432
center
-2,74191E-07
x
angle(radians)
X for plotting
0,00
2,29157E-04
0,39
2,11693E-04
0,79
1,61958E-04
1,18
8,75255E-05
1,57
-2,74191E-07
1,96
-8,80738E-05
2,36
-1,62507E-04
2,75
-2,12241E-04
3,14
-2,29706E-04
3,53
-2,12241E-04
3,93
-1,62507E-04
4,32
-8,80738E-05
4,71
-2,74191E-07
5,11
8,75255E-05
5,50
1,61958E-04
5,89
2,11693E-04
6,28
2,29157E-04

y
Y for plotting
0,00000E+00
8,77996E-05
1,62233E-04
2,11967E-04
2,29432E-04
2,11967E-04
...


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I was having a hard time with this subject, and this was a great help.

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