5
ii
Department of Civil Engineering
Auburn University
Laboratory Report
on
Thin – Walled Pressure Vessel Assignment
Submitted to
Golpar Garmestani
Submitted by
Michael Harris
Group Members
Jake Brown, Cameron Bonner
April 8, 2015
-Table of Contents-
Theory…………….....................................................................................................1
Results and discussion……………………………………………………………….3
References……………………………………………………………………………
Appendix: A Notation ………………………………………………………………A – 1
Appendix: B Spreadsheets…………………………………………………………..B – 1
-Theory-
The experiment further increased the knowledge about thin-walled pressure vessels. These are containers that are defined as closed cylinders in which the internal pressure is different from the external pressure that have a wall thickness smaller than 10% of the internal radius. Due to the walls being reasonably thin, any bending stresses about the cylinder walls can be neglected, forming a two dimensional state of stress. The two types of stress that correspond to these two dimensions are a longitudinal normal stress (σl), hoop normal stress (σh), and a shear stress (τlh). This can be seen pictorially below in figure: 1.
Figure 1: Hoop stress and longitudinal stress shown on the face and cut sections of a thin-walled pressure vessel.
Using the following equations the theoretical values for the stresses above can be calculated with the force as p the gage pressure, r as the internal radius and t as the wall thickness.
σ1= pr2t σh= prt
In the experiment, the stresses were found both theoretically and measured with a "strain-rosette" (figure 2) constructed of 3 electrical-resistance strain (ε) gages attached to the outer wall of the thin-walled pressure vessel. The vessel is filled with air until it reaches a given pressure in the lab manual (Marshall), then the 3 strains are read off of a monitor given as εA, εB, and εC. Each represents a different angle (θ) on the stain-rosette with angles of θA is at 45o, θB is at 0o, and θC is at -45o or 315o. Using the measured strains one can find two perpendicular normal strains (εx and εy) along with a shear strain (τxy) with the equations in figure 3.
Figure 2: Example of a 45o strain-rosette. Figure 3: Equations for calculating strain.
For this experiment the equations are solved using a matrix that is shown below in figure 4.
Figure 4: Example of figure 3 in matrix form.
After the values of εx, εy and ϒxy are calculated then their corresponding stresses (σx, σy, τxy) can be found using the equations known as stress strain relationships for two-dimensional stress (figure 5). Modulus of elasticity (E), poisons ratio (v) and shear modulus (G) are also used in the equations.
Figure 5: Stress strain relationships where σx is σl and σy is σh
Once the values of the stresses and strains were calculated Mohr's Circle for stress and strain could be drawn. Using Mohr's Circle with equations in figure 6 the principal stresses (σ1, σ3) and maximum in-plane shear strain (τmax). To solve for the strain values one would simply replace the σ with ε.
Figure 6: Mohr's Circle
-Results and Discussion-
The experiment involved pumping air into a pressure vessel until it reached pressures of 250, 500 and 750 pound per square inch (psi). Once each level was reached individually the reading on the strain indicators was recorded. Table 1 is the recoded data along with the computed averages.
Table 1: In lab recorded data and the average of the 3 runs.
Run 1
Pressure (psi)
εA
εB
εC
250
0.000089
0.00003
0.000098
500
0.000194
0.000069
0.000207
750
0.000301
0.000107
0.000314
Run 2
Pressure (psi)
εA
εB
εC
250
0.000084
0.00003
0.00009
500
0.000191
0.00007
0.000102
750
0.000295
0.000105
0.000305
Run 3
Pressure (psi)
εA
εB
εC
250
0.000091
0.000034
0.000097
500
0.000195
0.000071
0.000202
750
0.000302
0.000108
0.00031
Average
Pressure (psi)
εA
εB
εC
250
0.000088
3.13E-05
0.000095
500
0.000193
0.00007
0.00017
750
0.000299
0.000107
0.00031
One the data was reduced from the table above the theoretical values for σl and σh were computed and shown in table 2 below.
Calculated Theoretical Stress
Pressure (psi)
σl
σh
250
2510.54
5021.084
500
5021.08
10042.17
750
7531.63
15063.25
Table 2: Theoretical stresses.
Then the measured σl, σh, εl and εh were computed using the experimentally measured strains (table 3).
Table 3: Calculated stress and strain using the experimental data.
Measured Stress & Strain
250 (psi)
500 (psi)
750 (psi)
εl ( in/in )
3.13E-05
7.00E-05
1.07E-04
εh ( in/in )
1.52E-04
2.94E-04
5.02E-04
σl (psi)
2183.08
4508.81
7317.77
σh (psi)
4836.10
9440.05
16041.13
-References-
Marshall, Justin. Introductory Assignment from Canvas. Auburn University, Auburn, AL
Ramey, G. Ed. (1993). Revised (Aug. 2012) by Martina Svyantek and Justin D. Marshall. Mechanics of solids and Structures Laboratory Manual. Auburn University, Auburn, AL
-Appendix A: Notation-
Symbol Description Page Number
Ix Moment of inertia about x axis 2
Iy Moment of inertia about y axis 2
Ixy Product of inertia 3
Imin Minimum amount of inertia 3
rx Radius of gyration about x axis 2
ry Radius of gyration about y axis 2
rmin Minimum radius of gyration 3
X Distance from y axis to centroid 2
Y Distance from x axis to centroid 2
AISC American Institute of Steel Construction 2
Θp Angle of principle axis 3
" Inches 3
A - 1
