Fall 2012
Lab 7: Report (34 Points)
Name:
MAE 323
User ID:
Cylindrical Pressure Vessel
Problem Statement:
Determine how different element types perform for modeling a cylindrical pressure vessel over a
wide range of r/t ratios, and how the hoop stress () in the cylindrical section compares to
theoretical equations for thin and thick cylinders under internal pressure, p.
The analysis will be performed using 4 different element types and 4 different r/t ratios based on
the base configuration given in figure below. The model dimensions should be parameterized in
such a way that the thickness can be changed without re-creating the geometry. Maintain a
constant element size of .2 in and use triangle/tetrahedral elements for all models.
The pressure vessel possesses the following characteristics for all models: Inner radius (r0)=3.5”,
p=1500 psi. Four different values of thickness should be investigated: t=.25”, .5”, 1”,1.5”.
Analytical Equations And Comparison
The following equations should be used to fill out table 1. For a thick-walled cylinder:
pr
 
r
2
i i
2
0
where:
©2012 Alex Grishin/Prashant Mohan
 p0 r02 
 ri 2 

ri 2 r02  p0  pi 
r 2  r02  ri 2 
   hoop stress
pi  pressure at inner radius
p0  pressure at outer radius
ri  inner radius
r0  outer radius
r  radius of interest*
*For r (the radius of interest), use the inner radius (ri) for maximum hoop stress.
For a thin-walled cylinder, the following equation should be used:
pr
 
t
Where t is the thickness of the cylinder. Use the inner radius for r in all cases.
Table 1 Hoop Stress for different thickness values (6 Points)
Thickness
r/t ratio
FE shell
FE solid
FE plane strain
FE axisymmetric
Theoretical (thin wall)
Theoretical (thick wall)
0.25
14.0
0.50
7.0
1.00
3.5
1.50
2.3
Computational Expense
Fill out table 2, reporting the total number of DoF’s in each model (remember, this can be
obtained by multiplying the number of nodes by the number of number of DoFs per node).
Table 2 # DoFs for each model (6 points)
FE shell
Thickness # DoF
0.25
0.5
1
1.5
FE Solid
# DoF
FE Plane Strain
# DoF
FE Axisymmetric
# DoF
Reference:
For thick cylinders: http://www.engineeringtoolbox.com/stress-thick-walled-tube-d_949.html
©2012 Alex Grishin/Prashant Mohan
For thin cylinders: http://en.wikipedia.org/wiki/Cylinder_stresses
Questions:
Comment on the overall limitations and advantages of each element type for this problem
(3 points)
Axisymmetric elements are planar elements, as are plane strain elements. Comment on the
differences or similarities between the axisymmetric and plane strain solutions (Hint: look
through the lecture notes. 3 points)
Figures
Figure 1: Boundary conditions for shell model (show me the model with the smallest thickness
(.25”) used in the study. 2 points). .
Figure 2: Boundary conditions for solid model (show me the model for the smallest thickness
(.25”) used in the study. 2 points). .
Figure 3: Boundary conditions for plane strain model (show me the model for the smallest
thickness (.25”) used in the study. 2 points).
Figure 4: Boundary conditions for axisymmetric model (show me the model for the smallest
thickness (.25”) used in the study. 2 points). .
Figure 5: Hoop stress for shell model (show me the model with the smallest thickness (.25”) used
in the study. 2 points).
Figure 6: Hoop stress for solid model (show me the model with the smallest thickness (.25”)
used in the study. 2 points).
Figure 7: Hoop stress for plane strain model (show me the model with the smallest thickness
(.25”) used in the study. 2 points).
Figure 8: Hoop stress for axisymmetric model (show me the model with the smallest thickness
(.25”) used in the study. 2 points).
©2012 Alex Grishin/Prashant Mohan