363659998 Thin Cylinder Report PDF

Preview

Summary

report ...

Description

Table of Contents Summary ....................................................................................................................................................... 2 Introduction .................................................................................................................................................. 3 Background ................................................................................................................................................... 4 Experimental Procedure ............................................................................................................................... 5 Results ........................................................................................................................................................... 6 Open Ends Condition. ............................................................................................................................... 6 Strain Gauge 1 ....................................................................................................................................... 7 Strain Gauge 2 (Longitudinal Strain) ..................................................................................................... 7 Strain Gauge 3 ....................................................................................................................................... 8 Strain Gauge 4 ....................................................................................................................................... 8 Strain Gauge 5 ....................................................................................................................................... 9 Strain Gauge 6 ....................................................................................................................................... 9 Hoop Stress and Strain Relationship ................................................................................................... 10 Longitudinal and Hoop Strain Relationship ........................................................................................ 10 Theoretical Principal Strain ................................................................................................................. 11 Mohr’s Circle Readings........................................................................................................................ 12 Closed Ends Condition ............................................................................................................................ 13 Strain Gauge 1 ..................................................................................................................................... 13 Strain Gauge 2 ..................................................................................................................................... 14 Strain Gauge 3 ..................................................................................................................................... 14 Strain Gauge 4 ..................................................................................................................................... 15 Strain Gauge 5 ..................................................................................................................................... 15 Strain Gauge 6 ..................................................................................................................................... 16 Theoretical Principal Strain ................................................................................................................. 16 Mohr’s Circle Readings........................................................................................................................ 17 Discussion.................................................................................................................................................... 18 Conclusion ................................................................................................................................................... 19

Summary This experiment was conducted to analyze stress and strain in a thin walled cylinder using a thin cylinder device (SM1007). It shows the strain on a thin cylinder wall as it is being stressed by internal pressure. It then compares the laboratory results to the results calculated.

Introduction Cylinders are used in many engineering applications and they are all subjected to fluid pressure, when a cylinder is experiencing internal pressure there are three types of stresses which are acting on the cylinder wall, hoop stress, longitudinal stress, and radial pressure (this is negligible because it is insignificant when dealing with thin cylinders). Analyzation of the distribution of stress in a thin cylinder is important in pressure vessels because of its ability to detect if a structure will be able to withstand the forces both internally and externally expected of it, this can be used to detect structural failure. This experiment demonstrates the stress in a thin cylinder.

Background A cylinder is considered thin when its wall thickness is smaller than 10% of its internal radius. Being so thin its bending stresses can be ignored leaving only two types of stress, longitudinal and hoop stress. Finding the stresses in a cylinder experiencing internal pressure is arduous. Solving it requires considering equilibrium forces, displacement compatibility, stress and strain relationship, and the boundary considerations. For thin cylinders, however, a satisfactory solution can be found by some simplifying calculations.

Experimental Procedure The experiment was carried out using a SM1007 thin cylinder. Two experiments are carried out the first being the open ends experiments and the other being the closed ends experiment. Open Ends When the thin cylinder is switched on it is allowed to reach a stable temperature by letting it run for about 5 minutes, this gives more accurate readings. The pressure control is then opened and the hand wheel is screwed in to set the open ends condition. The pressure control is then shut and the and the strain gauge and pressure readings are reset. The readings are then taken starting at 0MN.m-2 in increments of 0. 5MN.m-2 until 3MN.m-2. Open the pressure control to reset the pressure back to 0MN.m-2. Closed Ends The cylinder is switched on and is allowed to reach a stable temperature by letting it run for about 5 minutes, this gives more accurate readings. The pressure control is opened and the hand wheel is then unscrewed to set up the closed ends condition. Check that the frame is not taking any load by turning the pressure control valve and pumping until the pressure gauge reaches 3MN.m-2 then push and pull the cylinder gently along its axis, if the cylinder moves then it is not taking any load. If it does not, then the hand wheel must be wound out and redone. The pressure control is then shut and the strain gauge and pressure readings are reset. The readings are then taken starting at 0MN.m-2 in increments of 0. 5MN.m-2 until 3MN.m-2. Open the pressure control to reset the pressure back to 0MN.m-2.

Results Open Ends Condition. Pressure 0 0.5 1 1.5 2 2.5 3

1 1 112 210 311 409 505 605

2 -1 -35 -67 -97 -130 -162 -194

Strain (Gauges) 3 4 1 2 2 39 4 73 4 104 3 137 5 170 3 203

Direct hoop stress (σH) = 𝑝𝑑/2𝑡 Where; p = Internal Pressure d = Internal diameter of the cylinder t = Wall thickness of the cylinder Using the formula above direct hoop stress is then calculated.

Pressure Direct Hoop Stress 0 0 0.5 6.6 1 13.3 1.5 20 2 26.6 2.5 33.3 3 40

5 2 73 139 202 266 331 395

6 4 113 211 306 405 501 596

Strain Gauge 1 700 600

Strain

500 400 300

200 100 0 0

0.5

1

1.5

2

2.5

3

3.5

2.5

3

3.5

Pressure Figure 1: Graph of Strain (1) against Pressure

Strain Gauge 2 (Longitudinal Strain) 0 0

0.5

1

1.5

-50

Strain

-100

-150

-200

-250

Figure 2: Graph of Strain (2) against Pressure

Pressure

2

Strain Gauge 3 6 5

Strain

4 3 2 1 0 0

0.5

1

1.5

2

2.5

3

3.5

2.5

3

3.5

Pressure Figure 3:Graph of Strain (3) against Pressure

Strain Gauge 4 250

200

Strain

150

100

50

0 0

0.5

1

1.5

2

Pressure

Figure 4:Graph of Strain (4) against Pressure

Strain Gauge 5 450 400 350

Strain

300 250 200 150 100 50 0 0

0.5

1

1.5

2

2.5

3

3.5

Pressure Figure 5:Graph of Strain (5) against Pressure

Strain Gauge 6 700 600

Strain

500 400 300

200 100 0 0

0.5

1

1.5

Pressure

Figure 6:Graph of Strain (6) against Pressure

2

2.5

3

Hoop Stress and Strain Relationship 40 y = 0.0673x - 0.6587

35 30

Stress

25 20

15 10 5 0 -5

0

100

200

300

400

500

600

700

Strain

Steel is three times harder than aluminum and its Young’s Modulus is 210 GN.m-2. If the cylinder had been made of steel the strain values would be lower for the same stress.

Longitudinal and Hoop Strain Relationship 0 0

100

200

300

400

500

Longitudinal Strain

-50

-100

-150

-200

-250

y = -0.3262x + 1.5468

Hoop Strain

600

700

Gradient of the graph = 0.3262 Given Poisson’s Ratio = 0.33 Percent error = 1.165%

Theoretical Principal Strain Poisson’s Ratio = 0.33 Young’s Modulus = 69GN.m-2 Hoop Stress at 3MN.m-2 = 40N.m-2 𝜀𝐻𝑜 = 𝜎𝐻𝑜/𝐸 𝜀𝐿𝑜 = −𝑣𝜎𝐻𝑜/𝐸 Principal Strain Hoop = 40/69*109 = 5.7971*10-10 Principal Strain Longitudinal = -0.33 * 40/69*109 = -1.913*10-10

Mohr’s Circle Readings Angle 30o 45o 60o

Direct Strain 2 200 400

Shear Strain 343 398 343

Closed Ends Condition Pressure 1 1 88 171 252 335 414 500

0 0.5 1 1.5 2 2.5 3

Strain (Gauges) 3 4 2 1 29 54 62 104 95 151 128 200 159 247 194 299

2 0 14 33 49 65 82 100

5 1 67 132 198 262 326 390

6 2 90 171 253 333 412 495

Strain Gauge 1 600 500

Strain

400 300 200 100 0 0

0.5

1

1.5

Pressure

2

2.5

3

Strain Gauge 2 120 100

Strain

80 60 40 20 0 0

0.5

1

1.5

2

2.5

3

2

2.5

3

Pressure

Strain Gauge 3 250

200

Strain

150

100

50

0 0

0.5

1

1.5

Pressure

Strain Gauge 4 350 300

Strain

250 200 150

100 50 0 0

0.5

1

1.5

2

2.5

3

2

2.5

3

Pressure

Strain Gauge 5 450 400 350

Strain

300 250 200 150 100 50 0 0

0.5

1

1.5

Pressure

Strain Gauge 6 600 500

Strain

400 300 200 100 0 0

0.5

1

1.5

Pressure

Theoretical Principal Strain Poisson’s Ratio = 0.33 Young’s Modulus = 69GN.m-2 Hoop Stress at 3MN.m-2 = 40N.m-2 𝜀𝐻𝑐 = (𝜎𝐻 − 𝑣𝜎𝐿)/𝐸 𝜀𝐿𝑐 = (𝜎𝐿 − 𝑣𝜎𝐻)/𝐸 Principal Strain Hoop = 40 – (0.33 * 20)/69*109 = 4.841*10-10 Principal Strain Longitudinal = 20 - (0.33 * 40)/69*109 = 0.986*10-11

2

2.5

3

Mohr’s Circle Readings Angle 30o 45o 60o

Direct Strain 200 300 400

Shear Strain 170 200 175

Discussion In the open ends experiment there is no direct longitudinal strain but gauge 2 measures longitudinal strain because the hoop stress causes an indirect longitudinal strain. The longitudinal strain is negative meaning that is was caused by a compressive stress. In the graph showing the relationship between longitudinal and hoop strain, the gradient for the line is almost exactly the Poisson’s ratio thereby showing that the longitudinal strain was caused by a compressive stress. Also from the analysis of the Mohr’s circle in the open ends experiment we can see the circle almost predicts the direct strain exactly. This is the same in the closed ends experiment as it is shown that the Mohr’s circle predicts the direct strain almost exactly.

Conclusion From the report above we observe the strain in a thin cylinder both experimentally and theoretically, using the apparatus and through calculations using the Mohr’s circle. The report show and explains internal pressure and complex stresses in a thin cylinder using the experiment and the calculations....