Experiment #3

Stresses in a Thin-walled Cylindrical Pressure Vessel

Stephen Mirdo

Performed on October 11, 2010

Report due October 25, 2010

Table of Contents Object ………………………………………..………………………….………….…. p. 1 Theory …………………………………………………………………………..…pp. 1 - 3 Procedure ………………………….…………………………………...……..……..... p. 4 Results ….................................................................................................................. p. 5 - 6 Discussion and Conclusion …………………….......…………………….......…... pp. 7 - 8 Appendix ……………………………………..…………………..….……..…... pp. 9 - 11

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Object The object of this experiment was to determine the stresses developed in a thin-walled pressure vessel when subjected to a uniform internal pressure.

Theory

Stress is defined as the intensity of a force per unit area. A normal stress is a stress that acts perpendicular to the cross-sectional area of a member. In a thin-walled, cylindrical pressure vessel, the normal stress has two components. One component is the normal stress acting on a transverse section and is known as a longitudinal stress, σL. The other component is the normal stress acting on a longitudinal section and is known as the circumferential, or hoop, stress, σh.

Theoretically, the normal stresses σL and σh can be determined by employing Newton’s First Law and a force balance equation. Figure 1 below is a free body diagram of a cylindrical pressure vessel with hemispherical ends. By using method of section, an equilibrium equation for this scenario can be written as follows:

ΣFx = σL(πDt) – P(π/4)D2 = 0 (Equation 1)

where σL is the longitudinal stress, D is the diameter of the pressure vessel, P is the pressure present in the vessel and t is the thickness of the vessel’s walls. Rearranging Equation 2 and solving for σL yields the following equation that solves for the theoretical value of the longitudinal stress.

σL = PD/4t (Equation 2)

Figure 1: Free body diagram of a pressure vessel exposing σL (Adapted from Mechanics of

Materials, T.A. Philpot, 2011)

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Similarly, the theoretical hoop stress, σh, can also be determined by employing Newton’s First Law and a force balance equation. Figure 2 below shows a free body diagram exposing the hoop stress. Again, using method of section, an equilibrium equation yields the following:

ΣFz = σh(2tΔx) – P(DΔx) = 0 (Equation 3)

where σh is the hoop stress, t is the thickness of the vessel walls, Δx is a finite segment of the vessel wall, P is the pressure acting inside the vessel and D is the diameter of the pressure vessel. Rearranging Equation 4 to solve for the theoretical hoop stress yields the following:

σh = PD/2t (Equation 4)

Figure 2: Free body diagram of a pressure vessel exposing σh (Adapted from

Mechanics of Materials, T.A. Philpot, 2011)

Another method used to express normal stress is to employ Hooke’s Law. Hooke’s Law is defined as the proportionality of a load to deflection incurred by the load. For a uniaxial loading scenario, Hooke’s Law is written as:

σn = Eε (Equation 5)

where σn is the normal stress, E is the modulus of elasticity of a material and ε is the uniaxial strain.2

For a cylindrical pressure vessel with hemispherical ends, the loading scenario is biaxial. It is termed biaxial because the longitudinal stress, σL, and the hoop stress, σh, are acting on the pressure vessel simultaneously. Thus, the deformation of the vessel must be determined by summing both stresses by employing the principle of superposition. The principle of superposition states that the effects of separate loadings can be added algebraically if each effect is linearly related to the load that produced it and

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the effect of the first load considered does not significantly change the effect of the second load.1 Employing Hooke’s Law and the principle of superposition, the value of the longitudinal strain, εL, of the pressure vessel can be computed using the following equation:

εL = (1/E)(σL – υσh) (Equation 6)

where εL is the longitudinal strain, E is the elastic modulus of the pressure vessel, σL is the longitudinal stress, υ is Poisson’s ratio of the vessel’s material and σh is the hoop stress of the vessel. Again, employing Hooke’s Law and the principle of superposition for the hoop stress, Equation 6 becomes the following: εL = (1/E)[σL – υ(Eεh + υσL)] = σL/E – (1/E)(Eυεh + υ2σL) = σL/E – υεh – (υ2σL)/E

(Equation 7) where εh is the hoop strain. Further simplifying Equation 7 yields:

εL + υεh = [σL(1-υ2)] / E (Equation 8)

Rearranging Equation 8 and solving for the longitudinal stress, σL, yields:

σL = E(εL + υεh) / (1-υ2) (Equation 9)

Similarly, the hoop stress, σh, can be derived using the method described through Equations 6 through 9 and yields:

σh = E(εh + υεL) / (1-υ2) (Equation 10)

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Procedure Equipment:

Thin-walled cylindrical pressure experiment apparatus Strain indicator

Experiment:

1) Power on the strain indicator. Set the gage factor to 2.5 and set channels 1 and 2 to quarter bridge.

2) Calibrate the strain indicator for channels 1 and 2 to zero the display. Channel 1 will read the longitudinal strain and channel 2 will read the hoop strain.

3) Ensure that the dump valve is secured. 4) Use the pump to increase the pressure of the cylinder to 100 psi. 5) Once the readings of the strain indicator have become steady, record the

values indicated for longitudinal and hoop strain. These values are indicated in microstrain, or x10-6.

6) Use the pump to increase the pressure of the cylinder to 200 psi. 7) Again, allow the indicated strain readings to become steady and record these

values. 8) Repeat steps 6 and 7 until a final pressure of 500 psi has been reached. 9) Open the dump valve of the apparatus to release the pressure of the cylinder. 10) Repeat steps 3 through 9 for three more trials.

Figure 1: Thin-walled cylindrical pressure experiment apparatus (Adapted from

Materials Laboratory Manual, Fall 2010, University of Memphis, Department of M.E.)3

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Results

Table 1: Properties of the cylindrical pressure vessel Diameter (in) 4.08 Thickness (in) 0.39 Poisson's Ratio 0.285

Elastic Modulus (psi) 2.30E+07 Table 2: Recorded longitudinal and hoop microstrain gathered over four trials at incremental pressures. Trial 1 Trial 2

Pressure (psi)

Longitudinal Strain (με)

Hoop Strain

(με) Pressure

(psi) Longitudinal Strain (με)

Hoop Strain

(με) 100 6 17 100 7 17 200 11 36 200 10 37 300 11 54 300 15 56 400 21 74 400 20 77 500 26 94 500 24 95

Trial 3 Trial 4

Pressure (psi)

Longitudinal Strain (με)

Hoop Strain

(με)

Pressure (psi)

Longitudinal Strain (με)

Hoop Strain

(με) 100 4 18 100 5 19 200 8 36 200 9 38 300 12 54 300 13 57 400 17 73 400 18 76 500 21 93 500 22 96

Table 3: Calculated experimental and theoretical longitudinal and hoop stresses using Equations 2, 4, 9 and 10. Note: Microstrain values are the averages of four trials.

Pressure (psi)

Longitudinal Strain (με)avg

Hoop Strain (με)avg

Experimental σL (psi)

Experimental σh (psi)

Theoretical σL (psi)

Theoretical σh (psi)

100 5.50 17.75 264.32 483.58 261.54 523.08 200 9.50 36.75 500.01 987.75 523.08 1046.15 300 12.75 55.25 713.36 1474.06 784.62 1569.23 400 19.00 75.00 1010.72 2013.06 1046.15 2092.31 500 23.25 94.50 1256.24 2531.53 1307.69 2615.38

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Longitudinal Stress vs. Pressure in a Thin-Walled Pressure Vessel

0.00

200.00

400.00

600.00

800.00

1000.00

1200.00

1400.00

0 100 200 300 400 500 600

Pressure in Vessel (psi)

Long

itudi

nal S

tres

s of

Ves

sel (

psi)

Theoretical Experimental

Figure 3: Graph of experimental and theoretical longitudinal stress over pressure increments of 100 psi in a cylindrical pressure vessel.

Hoop Stress vs. Pressure in a Thin-Walled Pressure Vessel

0.00

500.00

1000.00

1500.00

2000.00

2500.00

3000.00

0 100 200 300 400 500 600

Pressure in Vessel (psi)

Hoo

p St

ress

of V

esse

l (ps

i))

Theoretical Experimental

Figure 4: Graph of experimental and theoretical hoop stress over pressure increments of 100 psi in a cylindrical pressure vessel.

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Discussion & Conclusion

It was noted that the graphs produced for stress due to pressure, as seen in Figures 3 and 4, generated a linear function. It can be concluded that the pressure in the cylindrical vessel did not exceed the proportional limit, or yield strength, of the vessel’s material. The deformation incurred by the pressure was elastic and completely recoverable. Therefore, the use of Hooke’s law was applicable. If the plotted data had yielded an exponential line portion, Hooke’s Law would only be applicable to the data points that occur before the curvature.

The experimental values calculated for longitudinal stress and hoop stress are

within an acceptable range of their respective theoretical values. However, due to the fact that the stresses were measured indirectly by the test apparatus, a percent difference analysis was conducted to conclude if the results produced by this experiment were accurate. The values for the percent difference of experimental and theoretical calculations of longitudinal and hoop stress can be seen in Table 4 and Table 5.

Table 4: Percent Difference between calculated theoretical and experimental longitudinal stress of cylindrical pressure vessel.

Pressure (psi)

Experimental σL (psi)

Theoretical σL (psi)

% Difference of σL

100 264.32 261.54 1.06% 200 500.01 523.08 4.51% 300 713.36 784.62 9.51% 400 1010.72 1046.15 3.45% 500 1256.24 1307.69 4.01%

Table 5: Percent Difference between calculated theoretical and experimental hoop

stress of cylindrical pressure vessel. Pressure

(psi) Experimental

σh (psi) Theoretical

σh (psi) % Difference

of σh

100 483.58 523.08 7.85% 200 987.75 1046.15 5.74% 300 1474.06 1569.23 6.25% 400 2013.06 2092.31 3.86% 500 2531.53 2615.38 3.26%

The percent difference analysis of the theoretical and experimental results of the stresses calculated from the experimental data allowed for the conclusion that they are relatively numerically equivalent and therefore accurate.

There were a few sources of error in this experiment. One source of error was due to the readings of the indicated strain. The strain indicator values fluctuated very

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slowly around a central value. This was most likely due to the need of new wiring between the strain gage and strain indicator. Another source of error was incurred by the readings of the pressure gauge of the thin-walled cylindrical pressure vessel apparatus. Without automation, it was difficult to bring the pressure of the vessel to an exact indicated pressure. The error of the pressure recordings will cause a slightly skewed calculated value for the theoretical longitudinal and hoop stresses.

A few improvements for this experiment can be made. One such improvement

would be to employ a puzzle-like, educational approach. Instead of simply calculating the stresses present in the vessel, it would be interesting to use the same approach as in this experiment to calculate experimental stresses, but the objective would be to destroy the pressure vessel. By adding stresses that are beyond the proportional limit of the material and recording the indicated strains, one performing this experiment could identify the material used in the construction of the pressure vessel. Another improvement would be to ensure that the wiring used on the testing apparatus is properly grounded and there is no damage to the wiring. Damaged wiring generates line noise and skews the actual indicated strains produced by the strain indicator.

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Appendix Data Usage Sample calculation of average longitudinal strain at 100 psi:

(6 + 7 + 4 +5) με / 4 = 5.5 με

Sample calculation of theoretical longitudinal stress, σL, at 300 psi using Equation 2:

σL = (300 lbf/in2 * 4.08 in) / (4 * 0.39 in) = 784.62 psi Sample calculation of experimental hoop stress, σh, at 300 psi using Equation 10:

σh = [23 x 106 lbf/in2 * (55.25με + 0.285 * 12.75με)] / (1 – 0.2852) = 1474.06 psi Sample calculation of % Difference of theoretical and experimental values of longitudinal stress, σL, at 500 psi:

| (1307.69 psi – 1256.24 psi) / [(1256.24 psi + 1307.69 psi) / 2] | * 100 = 4.01%

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Bibliography 1. Mechanics of Materials, 2nd Edition

Timothy A. Philpot (2011)

2. Fundamentals of Material Science and Engineering: An Integrated Approach W.D. Callister, Jr and D.G. Rethwish (2008)

3. Materials Laboratory Manual, Fall 2010

University of Memphis, Department of Mechanical Engineering