Numerical Solutions for Stress and
Displacement of Pressure Vessels
MANE 4240 – Introduction to Finite Elements
Instructor: Professor Ernesto Gutierrez-Miravete
Joseph Misulia
04/28/2014
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Table of Contents
Section
Page
1. Abstract………………………………………………………
1
2. Introduction………………………………………………….
2
3. Formulation…………………………………………………..
3
4. Solution………………………………………………………
5
5. Results………………………………………………………..
7
6. Discussion……………………………………………………
7. Conclusion……………………………………………………
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1.0 - Abstract
This paper presents analyses performed to determine the stresses and displacements
developed in pressure vessels of varying geometries when subject to a 1 Mpa (145.04psi)
internal pressure. The pressure vessel models analyzed were created using the material
properties of CRES 316 stainless steel and were cylindrical in shape having varying cap
geometries. This paper presents the results of numerical analyses performed using the Finite
Element Method (FEM) program COMSOL for multiple geometries as well as results of a
calibration case found analytically.
The COMSOL analyses were performed using the two dimensional symmetrical
mechanical structures suite native to COMSOL. As such, the models were created using two
dimensional geometries and assumed a cylindrical symmetry. A calibration case was examined
first to ensure the validity of the numerical solutions. A long cylinder having no caps was
analyzed using the COMSOL FEM approach and compared to an exact solution found
analytically. An extremely fine mesh applied in COMSOL was found to produce results that
compared favorably with the exact solution. Accordingly an extremely fine mesh was applied
when analyzing the pressure vessel models of interest.
The COMSOL calculation showed that cylindrical pressure vessels having spherical end
caps had lower von Mises stresses in the vessel when compared to vessels having flat type end
caps. A maximum von Mises stress of 24.39 MPa was found in a cylinder having a flat end cap
with a sharp corner transition versus a maximum von Mises stress of 4.85 MPa in a pressure
vessel of similar geometry with a spherical end cap. It was also observed that caps having
smooth geometries and transitions into the end cap had lower von Mises stresses. Similarly, the
maximum displacement and strain developed in the pressure vessels was reduced when a
spherical end cap was used versus a flat type end cap. A pressure vessel with a flat end cap
exhibited a maximum displacement of 13.8mm versus the maximum displacement of 2.43mm
found in the pressure vessel with a spherical end cap and similar geometry.
A difference in displacement and stress distribution was also found; pressure vessels with sharp
transitions and corners were found to have high, local stresses. Pressure vessels with gradual
transitions in geometry were found to have more even stress distributions. In each case the
tensile strength of the material, 480 MPa, was not exceeded showing that any of the geometries
analyzed would be sufficient given the 1 MPa internal pressure.
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2.0 – Introduction
Figure 1 below shows an example of two of the cylindrical pressure vessels that analyzed
herein. The model geometry shown is assumed to have cylindrical symmetry about the red
dotted line shown, making a cylinder. The material properties of CRES 316 stainless steel was
used for each model. In the cases analyzed, it was important to determine the maximum stress
developed in the material to ensure that the mechanical limits (tensile strength, 480MPa) of the
pressure vessels were not exceeded.
Figure 1: The 2D model created of the calibration case (left) and the 2D model created for a
cylinder having a flat cap (right)
When creating the models shown in Figure 1, a prescribed displacement of zero meters
in the r-direction was applied to the lower face to simulate the portion of the cylinder not shown.
Then a boundary load was applied to the inward facing surfaces in the form of a 1MPa pressure.
The model shown on the left was constructed to represent a calibration case, shown in Figure 2
below. For the calibration case shown on the left of Figure 1, the stresses at the inner and outer
surfaces take the form given below in Eq. 1 and Eq. 2 when the pressure outside the cylinder is
assumed to be zero.
𝜎𝑟 =
𝑟𝑖 2 𝑝
𝑟𝑜 2
(1
−
)
𝑟𝑜 2 − 𝑟𝑖 2
𝑟2
Eq. 1
𝜎𝜃 =
𝑟𝑖 2 𝑝
𝑟𝑜 2
(1
+
)
𝑟𝑜 2 − 𝑟𝑖 2
𝑟2
Eq. 2
Where ro represents the inner radius, ri represents the inner radius, and p represents the pressure
inside the cylinder as shown in Figure 2 below. For a full derivation see Ref (a).
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ro
p
ri
Figure 2: Representation of the calibration case solved analytically to ensure numerical solution
is accurate
3.0 – Formulation
3.1 – Calibration/Analytical Solution
In FEA analysis it is important to validate all results obtained. By comparing results of a
known case solved analytically to results obtained using the FEM model it is possible to gain
some confidence in the accuracy of the FEM solution. As such, a calibrations study was also
performed in COMSOL using a well-known case of a long cylinder with open ends and an
internal pressure with zero external pressure. This case was selected because an analytical
solution is readily available, and is shown in Ref (a).
The solution found in Ref (a) begins with the stress distribution about a symmetrical axis,
and is given by Eq. 3 below. The stress components do not change with Ө and are found to be
functions of the radius r from the axis of symmetry perpendicular to the xy-plane.
𝜕𝜎𝑟 𝜎𝑟 − 𝜎𝜃
+
+𝑅 =0
𝜕𝑟
𝑟
Eq. 3
Where the body force R is assumed to be zero. The analytical solution reduces to a linear
differential equation with constant coefficients. A general solution is then found with four
constants of integration, which are obtained from the boundary conditions of the system. The
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solution for a cylinder of inner radius ri, outer radius ro, internal pressure pi, and external
pressure po and angle Ө is found to be Eq. 4 and Eq. 5 below:
2
𝑟𝑖 2 𝑟𝑜 (𝑝𝑜 − 𝑝𝑖 ) 1 𝑝𝑖 𝑟𝑖 2 − 𝑝𝑜 𝑟𝑜 2
𝜎𝑟 =
∙ 2+
𝑟𝑜 2 − 𝑟𝑖 2
𝑟
𝑟𝑜 2 − 𝑟𝑖 2
Eq. 4
2
𝑟𝑖 2 𝑟𝑜 (𝑝𝑜 − 𝑝𝑖 ) 1 𝑝𝑖 𝑟𝑖 2 − 𝑝𝑜 𝑟𝑜 2
𝜎𝜃 = −
∙ 2+
𝑟𝑜 2 − 𝑟𝑖 2
𝑟
𝑟𝑜 2 − 𝑟𝑖 2
Eq. 5
When po is assumed to be zero, Eq. 4 and Eq. 5 are shown to become Eq. 1 and Eq. 2, repeated
again below for clarity as Eq. 6 and Eq. 7.
𝜎𝑟 =
𝑟𝑖 2 𝑝
𝑟𝑜 2
(1
−
)
𝑟𝑜 2 − 𝑟𝑖 2
𝑟2
Eq. 6
𝜎𝜃 =
𝑟𝑖 2 𝑝
𝑟𝑜 2
(1
+
)
𝑟𝑜 2 − 𝑟𝑖 2
𝑟2
Eq. 7
3.2 – Numerical Approximation
Numerical approximations were made using the FEA program COMSOL Multiphysics.
The program allows the user to build a model, apply constraints and loads then finally apply a
mesh to be solved by the program. The analyses conducted of the pressure vessel systems
assumed a two dimensional axisymmetric model. Accordingly, a prescribed displacement of
zero meters in the direction of the cylinder radius was applied to the lower face of each cylinder
model as shown in Figure 3. To simulate an internal pressure, a boundary load of 1 MPa was
applied to the innermost diameter of the cylinder, the surface corresponding to ri in Figure 2
above. A mesh was selected using the COMSOL meshing program, using the “Extremely Fine”
setting. This was determined by a mesh extension study to have the lowest amount of error when
compared to the calibration case that was solved analytically.
Material properties of CRES 316 Stainless Steel were applied to the model and integrated
into the mesh by COMSOL. CRES 316 is a common steel used in many every day applications
and is used commonly in pressure vessels installed in environments where anti corrosive
properties are desirable.
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4.0 – Solution
4.1– Analytical/Calibration Solution
An analytical solution was generated in Maple; it was provided by Professor GutierrezMiravete and can be seen in full in Appendix A. The program calculates the stress distribution,
and displacement of the cylinder structure in the calibration case. The analytical solution
generated in Maple makes use of the analysis performed in Ref (a). The solution to determine
the stress distribution is identical to that found in Ref (a), and shown in Eq. 6 and Eq. 7. The
solutions for stress, 𝝈𝜽 and 𝝈𝒓 , are used along with Hook’s law, Eq. 8, to derive the equations for
strain as shown below in Eq. 9 and Eq. 10. The solutions for strain are used to find the
maximum displacement of the cylinder and compare that number to the number calculated by
COMSOL to validate the numerical solution.
𝜎 = 𝐸𝜖
Eq. 8
1
Eq. 9
[𝜎 − 𝑣𝜎𝜃 ]
𝐸 𝑟
1
Eq. 10
𝜖𝜃 = [𝜎𝜃 − 𝑣𝜎𝑟 ]
𝐸
Where E represents the modulus of elasticity and v represents the poison’s ratio of the material
𝜖𝑟 =
used to make up the pressure vessel, in this case CRES 316 stainless steel, and 𝝐 represents the
strain developed in the pressure vessel walls.
4.2 – Numerical Solution
The numerical analysis performed in COMSOL was similar for every cylinder analyzed.
The initial step was to create a model in the 2D axisymmetric structural suite of COMSOL, apply
the necessary boundary conditions and boundary loads (pressure), shown below in Figure 3,
then mesh the model using COMSOL’s physics controlled meshing program. Figure 3 and
Figure 4 show the models, boundary conditions, loads and associated mesh of a cylinder with a
flat end cap and a cylindrical cap respectively, additional models and associated meshes can be
found in Appendix B.
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Boundary Load
1 Mpa (145psi)
Prescribed Displacement
of 0 m along z-axis
Figure 3: Cylinder models modeled assuming 2D symmetry, boundary conditions and loading
Figure 4: Cylinder models after meshing is completed by COMSOL
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5.0 – Results
5.1 – Analytical/Calibration Solution Results
Results from the calibration analysis can be seen in Figure 5 below, additional plots can
be seen in Appendix B. The results of the calibration case showed that the FEM analyses were
accurate to within 7.1% for stress and 5.2% for displacement, as shown in Table 1. Having
verified the results of the FEM analysis for the calibration case where the exact solution is
known, the FEM analysis could be performed on the cylinders of interest with reasonable
confidence in the solution results.
Figure 5: This figure shows the total displacement and von Mises stress distribution of the
calibration case
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Table 1: Comparisons between the exact solution and numerical results for the calibration case
5.2 – Numerical Solution Results
Results from the numerical analysis can be seen below in Figure 6 and Figure 7. The
results of a cylinder having a flat cap and a cylinder having a spherical cap are shown in
Figure 6 and Figure 7, additional geometries were analyzed associated results are included in
Appendix B.
Figure 6: Von Mises stress distribution and displacement of the flat cap cylinder as a result of a
1 MPa internal pressure
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Figure 7: Von Mises stress distribution and displacements of a cylinder having a spherical cap
due to a 1 MPa internal pressure
6.0 – Discussion
Results of the analyses establish that a FEM analysis carried out in COMSOL is accurate
and confidence can be placed on the results based on the calibration case. A comparison between
the analytical solution to the calibration case and a FEM solution to the calibration case showed
that the FEM results were accurate to within 7.1% for stress concentrations found and 5.5% for
maximum displacements.
The FEM results of the cylinders of interest, flat cap cylinder and spherical cap cylinder
shown above, show that the spherical cap cylinder has a much lower max stress and
displacement than the flat cap cylinder. The flat cap cylinder has a localized stress distribution in
way of the sharp cylinder to cap transition, whereas the spherical cap cylinder has a more evenly
distributed stress concentration located in way of the same cylinder to cap transition. However,
stresses found did not exceed the yield strength of the cylinder material for either case.
Similarly it was found that spherical cap cylinder exhibited a lower maximum
displacement than the flat cap cylinder. The flat cap cylinder showed a very high displacement
in way of the center of the flat cap and a relatively low displacement elsewhere in the cylinder.
The spherical capped cylinder showed very little overall displacement in way of the cap, and the
maximum displacement was found in the body of the cylinder itself.
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7.0 – Conclusions
The FEM analyses performed herein show that a cylinder having a spherical cap is a
superior design to a cylinder having a flat cap. The stresses developed in the spherical capped
cylinder were significantly lower than those in the flat capped cylinder, 4.85 MPa and 24.39 MPa
respectively. Maximum stresses developed in the spherical capped cylinder were shown to be
evenly distributed around the cylinder to cap transition region whereas the flat capped cylinder
exhibited a high local stress in the same region.
Displacements calculated for the spherical capped cylinder were also shown to be
significantly lower than those found in the flat capped cylinder, 2.4mm and 13.8mm respectively.
The maximum displacements in the spherical capped cylinder were found to be in near the
cylinder to cap transition region; the maximum displacement in the flat capped cylinder was
found to be localized in the center of the flat capped cylinder.
Additional studies were performed on different cap geometries and can be seen in
Appendix B. These studies continue to enforce what was found when analyzing the flat capped
cylinder and spherical capped cylinder. When designing pressure vessels it is important to
reduce localized stress by incorporating smooth transitions in geometry, and incorporate a
spherical type end cap where practicable.
References
a) Timoshenko, Stephen, and J. N. Goodier. Theory of Elasticity. New York: McGraw-Hill, 1970.
Print.
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A- 1
Appendix A
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A- 2
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A- 3
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A- 4
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Appendix B
This appendix is included to show results of analyses performed that were not specifically
referenced in the body of this paper. All cases shown in Appendix B assumed the same material
properties, boundary constraints and loading conditions as the cases presented in the body of this paper.
Alternate Case A - A flat cap design having half the wall thickness of the cylinder body - Extremely
Fine Mesh (2884 Elements)
Figure B-1: Mesh applied to Alternate Case A
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B-2
Figure B-2: Solution for von Mises stress distribution in Alternate Case A
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B-3
Figure B-3: Solution for maximum displacement of stress the distribution in Alternate Case A
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B-4
Alternate Case B: Pressure vessel having smooth transitions in to a hemispherical type cap Extremely Fine Mesh (3687 Elements)
Figure B-4: Mesh applied to Alternate Case B
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B-5
Figure B-5: Solution for Von Mises stress distribution of Alternate Case B
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B-6
Figure B-6: Solution for maximum displacement of Alternate Case B
Alternate Case C: A cylinder having a reverse spherical cap - Extremely Fine Mesh (4122 Elements)
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B-7
Figure B-7: Geometry of Alternate Case C
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B-8
Figure B-8: Solution for von Mises stress found in Alternate Case C
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B-9
Figure B-9: Solution for maximum displacement found in Alternate Case C
Alternate Case D: A reverse hemispherical cap, with a uniform exterior - Extremely Fine Mesh (6102
Elements)
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B - 10
Figure B-10: Geometry for Alternate Case D
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B - 11
Figure B-11: Solution for von Mises stress distribution in Alternate Case D
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B - 12
Figure B-12: Solution for maximum displacement of Alternate Case D
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