A Study of the Effect of Imperfections on Buckling Capability in Thin
Cylindrical Shells Under Axial Loading
by
Lauren Kougias
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Gutierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2009
© Copyright 2009
by
Lauren Kougias
All Rights Reserved
2
CONTENTS
LIST OF TABLES ............................................................................................................. 4
LIST OF FIGURES ........................................................................................................... 5
LIST OF SYMBOLS ......................................................................................................... 6
ACKNOWLEDGEMENT ................................................................................................. 7
ABSTRACT ...................................................................................................................... 8
1. Introduction.................................................................................................................. 1
1.1
Problem Description........................................................................................... 2
1.2
Methodology ...................................................................................................... 3
1.3
Expected Outcomes ............................................................................................ 4
2. Buckling of a Thin Cylinder Under Axial Compression ............................................. 5
2.1
Modeling ............................................................................................................ 5
2.2
Theoretical Solution ........................................................................................... 6
2.3
Eigenvalue Buckling Solution and Mesh Density Study ................................... 8
2.4
Nonlinear Buckling Analysis ........................................................................... 12
3. Conclusions................................................................................................................ 17
4. References.................................................................................................................. 18
5. Appendix A – Nonlinear Buckling Results ............................................................... 19
3
LIST OF TABLES
Table 1. Material Properties for AMS 4928 (Ti 6-4)........................................................ 5
Table 2. Eigenvalue Buckling Mesh Density Study Results .......................................... 12
Table 3. Nonlinear Buckling Results .............................................................................. 14
4
LIST OF FIGURES
Figure 1. Definition of Ovalized, or "Out of Round" ....................................................... 3
Figure 2. Deformation of a fully integrated, linear element (a) and a fully integrated,
quadratic element (b) subjected to bending moment M .................................................... 9
Figure 3. Finite Element Model Boundary Conditions .................................................... 6
Figure 4. Proportional loading with unstable response. ................................................... 7
Figure 5. First Six Eigenvalue Mode Shapes .................................................................. 10
Figure 6. Eigenvalue Buckling Mode Four Deflections ................................................. 11
Figure 7. Convergence of Eigenvalue Buckling Results vs. Element Size .................... 12
Figure 8. Displacement and Von Mises Stress for Nonlinear Buckling Analysis in
Perfect Cylinder ............................................................................................................... 13
Figure 9. Effect of Ovalization on Buckling Capability ................................................. 14
Figure 10. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in
Cylinder, e = 50% shell thickness.................................................................................... 15
Figure 11. Load vs. Displacement Curve for Nonlinear Buckling Analysis in Cylinder, e
= 50% shell thickness ...................................................................................................... 16
Figure 12. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in
Cylinder, e = 1% shell thickness...................................................................................... 19
Figure 13. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in
Cylinder, e = 10% shell thickness.................................................................................... 19
Figure 14. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in
Cylinder, e = 100% shell thickness.................................................................................. 20
5
LIST OF SYMBOLS
E
ν
ρ
Modulus of Elasticity (Young’s Modulus)
Poisson’s Ratio
Density
e
t
R
L
Eccentricity or Imperfection Size (Half of out of roundness
value)
in
Cylinder wall thickness
in
Cylinder radius
in
Length of cylinder
in
kc
Buckling coefficient
N/A
N, Fcr
Critical buckling stress
psi
σy
Critical buckling stress
psi
6
psi
N/A
lb/in
ACKNOWLEDGEMENT
I would like to thank my husband, family and friends for all of their support.
Without their help and moral support, completing this project would have been nearly
impossible. Thank you to Professor Gutierrez-Miraverte for all of his guidance and
patience throughout these last few months. He has truly been an educational mentor and
an advisor. Thank you to my colleagues and technical mentors at Pratt & Whitney,
Richard Monahan and Bessem Jlidi, for their technical guidance and support during the
execution of this study. Special thanks also to my good friends John Battye and Susan
Smith for taking the time to read through this paper and offer constructive criticisms for
its improvement.
7
ABSTRACT
The use of cylindrical shells in the aerospace industry is widespread as load carrying
structures. In military engines, thin cylindrical shells are used for bypass ducts. As a
military jet moves through the sky, the bypass ducts are subjected to several maneuver
loads, including the axial load from the engine thrust. This load puts the ducts into
compression. This paper addresses the buckling capability of a cylindrical shell under a
compressive axial load using finite element analysis and how variability in
manufacturing processes, such as ovalization of a cylindrical duct, can affect the
buckling capability of these parts.
Results show that, as expected, buckling capability of a thin cylinder is significantly
affected by out of roundness. Out of roundness of 1%, 10%, 50% and 100% of the shell
thickness resulted in a reduction in buckling capability of 8.3%, 37.8% 65.9% and 75%,
respectively. Finite element results were calibrated to and were in good agreement with
the theoretical solution for a cylinder under axial compression.
8
1. Introduction
The use of thin-walled cylinders is widespread in many engineering industries. In
military engines, thin cylindrical shells are used for bypass ducts. As a military jet
moves through the sky, the bypass ducts are subjected to several maneuver loads,
including the axial load from the engine thrust. In the aerospace industry, the fuselage of
a commercial jet is subject to inertial and pressure loads during takeoff and landing. The
bypass ducts of a military engine must withstand severe compressive loading due to the
many quick maneuvers that a fighter jet makes in a given mission. Submarines are
subject to large external pressure loads as they dive into the depths of the ocean.
Although the loading of these thin-walled cylinders varies, the similarity between all of
them is that the buckling capability of these structures is an extremely important factor
in their design.
As industry becomes more competitive and the need to design
lightweight, low-cost parts increases, creating an optimized design is becoming more
and more important. In order to design a part to its true limit, a more detailed analysis
must be conducted, and factors such as part variations due to manufacturing must be
considered.
In designing thin cylindrical shells, it is typically assumed that the cylinder is
perfectly round and the imperfections found in an actual manufactured part are not
captured. It is commonly found that results obtained from experiment are significantly
different from the theoretical solution.
The discrepancy is thought to be due to
imperfections in an actual manufactured part that are not accounted for in theory. A thin
cylindrical shell is extremely sensitive to initial geometric imperfections, which are
typically defined as geometric shape imperfections and load eccentricities. Slight part
imperfections are difficult to avoid and make the job of determining the actual buckling
capability of a part difficult for the structural designer and analyst. This study will
examine how imperfections in a thin cylinder, specifically slight ovalization due to
manufacturing, can affect its buckling load capacity.
The effect of this ovalization will be studied using finite element analysis. The least
expensive way to study this effect would be by conducting a linear eigenvalue buckling
analysis. This analysis, however, is known to be anticonservative [7]. The eigenvalue
method predicts the buckling strength of an ideal linear structure. Since it is a linear
solution, the stiffness matrix is not updated during the solution and the results predict a
1
load carrying capability larger than the structure could actually sustain. Therefore, it is
common to perform a nonlinear, large displacement static buckling analysis. Although
this method is more time consuming and computationally expensive, it is typically a
more accurate method for determining buckling capability of a part. The nonlinear large
displacement method gradually increases the load in steps. The equilibrium equation,
{F}=[K]{U}, is solved for displacement, {U}, in each step by an iterative process. As
the load increases in each step, the stiffness matrix, [K], is updated to reflect the new
stiffness under the current loading. The load is increased until instability occurs and the
stiffness approaches zero. Once this occurs the finite element package is unable to find a
solution and the job is aborted. The unconverged solution typically indicates that the
structure is unable to carry any more load and buckling has occurred. The load applied
to the structure in the last converged step previous to the unconverged solution is
typically used as the buckling capability of the structure.
For this study, a combination of linear eigenvalue and nonlinear buckling analysis
will be performed. Abaqus, a finite element code, has the capability of using the results
of an eigenvalue buckling analysis to impart imperfections into a part for a buckling
analysis. The eigenvalue mode shape that reflects an ovalized cylinder will be used to
impart slight deflections to the part. A nonlinear buckling analysis will then be
performed on the cylinder using a static Rik's solution, which models large deflections
and post-buckling behavior.
A similar study, “Buckling Analysis of a Submarine with Hull Imperfections”, was
completed by Lee in 2007. Lee explored the advantages and disadvantages of using
eigenvalue buckling analysis and nonlinear large displacement static buckling analysis to
evaluate a thin cylindrical shell subject to an external pressure load. He also studied the
effect of mesh density and material nonlinearity in his study. Lee determined that out of
roundness in a cylinder significantly decreases its capability to withstand an external
pressure load. This study does the same for a cylinder under axial loading.
1.1 Problem Description
The purpose of this project is to study how imperfections in a thin-walled cylinder
affect buckling capability. There are many imperfections that can affect the buckling
2
capability of a thin-walled cylinder, including ovalization, variation in thickness,
material imperfections, etc.
This paper will focus on the ovalization, or out-of-
roundness, that can result from manufacturing processes in the production of thin
cylinders. Figure 1 shows what is meant by ovalized. The solid blue line is a cylindrical
duct, while the dashed line is ovalized. The effect of ovalization and the degree of
ovalization required to significantly affect the critical buckling load for a cylinder in
pure axial compression will be studied.
e
e
e
e
Figure 1. Definition of Ovalized, or "Out of Round"
1.2 Methodology
The finite element method is used to conduct the analysis for this project. The
software used is Abaqus, a product of D'Assault Systemes. Abaqus is an industry leader
in field of finite element analysis. This project requires nonlinear large displacement
analysis, for which the Abaqus solver is known as best in class.
A perfectly round (no imperfections), simply supported cylinder is modeled in
Abaqus. The element size used is determined by conducting a mesh density study. An
eigenvalue buckling solution is conducted several times, iterating on element size until
the solution converges. The largest element that produces accurate results is used to
produce accurate results in a model that runs as quickly as possible.
Once an element size is determined, the nonlinear buckling analysis is performed.
The resulting buckling load obtained from this analysis is validated by comparing it to a
simple hand calculation using a simple equation derived using small deflection theory.
Once the solution is validated, a study is conducted by imparting various degrees of
imperfection into the cylinder.
The degree of ovalization, or out of roundness, is
3
measured by the total diametric deformation; as seen in Figure 1, the total out of
roundness is equal to e + e, or 2e. So if e is one inch, the total out of roundness is two
inches.
1.3 Expected Outcomes
It is expected that this study will reveal that the ovalization of a cylindrical duct has
a very significant effect on its buckling capability under axial compression. The stresses
in the cylinders will be examined to explain the premature collapse under axial loading.
It is unknown whether or not the trend between load carrying capability and out of
roundness will be linear or exponential.
4
2. Buckling of a Thin Cylinder Under Axial Compression
2.1 Modeling
A baseline model is created of a perfectly round cylinder. The cylinder is 80 inches
in diameter and 80 inches in length. The thickness is 0.15 inches. Symmetry along the
axial direction is used on one end of the cylinder. That is, it is held from translating in
the axial (z) direction and from rotating in the radial (x) and circumferential (y)
directions.
The opposite end is simply supported.
The model is run at room
temperature, or 70°F. Titanium, specifically AMS 4928, is used for the material in this
study, since information about its properties is widely available. The material properties
for AMS 4928 can be found in Table 1. Typically buckling occurs before the material
begins to yield, therefore linear elastic material properties are used for all analyses.
Table 1. Material Properties for AMS 4928 (Ti 6-4) [9]
E
1.69E+07 psi
ν
0.32
ρ
0.158 lb/in3
σy
86,000 psi
The mesh boundary conditions can be seen in Figure 2. The axial load is applied to
the simply supported edge of the cylinder in the +z direction. The value of the axial load
is 10,000 lb/node, which equates to a total load of 124,000 lb. The load is ramped up
throughout the solution in increments and may or may not exceed the assigned max
applied load value. The load proportionality factor, a ratio of load capability to applied
load, can exceed 1.0, which would mean that the load carrying capability of the cylinder
is greater than the load applied to the model. The boundary conditions can be seen in
Figure 2.
5
Axial Load Applied and
Simply Supported Along Edge
Symmetry Boundary Conditions Along Edge
Figure 2. Finite Element Model Boundary Conditions
2.2 Theoretical Solution
The behavior of a cylinder under axially compressive loading is complex. The
displacement and load have a linear relationship – as the load increases, the
displacement also increases and stiffness increases. However, as seen in Figure 3 at
Point A, once a buckling begins to occur, the displacement increases more quickly with
the same rate of loading until the critical buckling load is reached. At this point, the
structure essentially has no stiffness. Buckling is not typically caused by overstress of a
structure as it occurs while stresses are still below the yield strength of the material.
Rather, buckling is caused by imperfections/instabilities in the structure. Figure 3 shows
an example of a complex, unstable response of a structure. The nonlinear buckling
analysis should exhibit similar behavior, at least up to point B on the curve.
6
Figure 3. Proportional loading with unstable response [5].
A simple hand calculation can be used to find the critical buckling load, or buckling
capability, of a thin cylindrical shell. The classic solution for the maximum stress in a
cylinder under axial compression, using small deflection theory is shown in Equation 1
[8].
Equation 1.
Fcr 
where
E
t2
31   R
E = Modulus of Elasticity
ν = Poisson's ratio
t = Wall thickness
R = radius of shell
The resulting stress from this equation is multiplied by the cylinder's perimeter to
obtain the total load carrying capability of the structure. For the cylinder used in this
study, the calculated theoretical critical load is 1,455,952 lb.
As described above, small deflection theory has proven anticonservative for
determining the buckling strength of thin-walled cylinders or thin-curved panels. It is
generally accepted that this is due to geometrical imperfections and associated stress
concentrations.
Large deflection theory shows better correlation with experimental
results; however, it is difficult to find solutions to these problems without knowledge of
7
imperfections due to material grain structure and manufacturing techniques. Therefore,
design is typically based on best fit curves for experimental or test results.
The
calculations below use design curves provided in E.F. Bruhn’s book [3]. Bruhn derives
the theoretical solution for a monocoque cylinder under axial compression. Equation 2
from Bruhn [3] is used to find the critical stress for cylinder under axial compression.
Equation 2.
Fcr 
where
k c 2 E  t 
 
12 1   2  L 

2

kc = buckling coefficient
E = Modulus of Elasticity
t = Wall thickness
L = Length of cylinder
ν = Poisson’s ratio
Curves relating cylinder dimensions and Young’s modulus are used to find the buckling
coefficient kc.
The theoretical curve results in a kc of 750 and a load of 1,539,279 lb,
which is only 2% higher than the theoretical solution from Reference 8.
The
experimental curves are based on 90% probability. The solution using the experimental
curves is 420,736 lb, or a 71% reduction in buckling capability as compared to the
theoretical solution. This is a significant reduction in capability, which is based on
experimental data, and proves that taking imperfections into account when designing a
thin cylindrical shell is extremely important.
2.3 Eigenvalue Buckling Solution and Mesh Density Study
At the beginning of this study linear shells (called S4 elements) were used for the
mesh due to their fast computational time. However, problems were encountered using
this type of element because linear shells are prone to shear locking. Shear locking can
affect the performance of fully integrated, linear elements subjected to bending loads.
Since the edges of quadratic elements are able to curve, shear locking is not typically an
issue. They may exhibit some locking if they are distorted or if the bending stress has a
gradient, but typically don’t have as many issues as linear shells. Figure 4, from the
8
Abaqus 6.9 Documentation [5], shows a comparison of how linear and quadratic shells
behave under a bending load.
Figure 4. Deformation of a fully integrated, linear element (a) and a fully integrated, quadratic
element (b) subjected to bending moment M
Quadratic shells (S8 elements) were then used to mesh the part. However, the
solution time increased tremendously, and for an unknown reason, the structure was not
undergoing collapse. With further reading through the Abaqus documentation, it was
found that S4R5 elements, which are reduced integration linear shells with 5 degrees of
freedom per node, are recommended for modeling thin shell structures. These elements
are used for the Abaqus benchmarking studies and example problems for use in
modeling thin cylinders. Therefore, these elements were utilized for this study and
yielded accurate results.
While performing a finite element analysis, it is important to ensure that the results
obtained are reliable and accurate by iterating on the element size, or mesh density. In
order to ensure that the results obtained during this analysis are correct, this study is
conducted using the eigenvalue buckling analysis.
Once the element size is small
enough that the solution begins to converge, that element size is used. This analysis is
performed in Abaqus using a perfectly round cylinder, with only a small imperfection (e
= 0.1% shell thickness, or 1.5E-4”) to aid in buckling. The first six eigenvalue buckling
mode shapes can be seen in Figure 5.
9
Figure 5. First Six Eigenvalue Mode Shapes
The mode of interest is mode four. This mode represents the ovalization of the duct,
so it is used to impart imperfections into the structure. The mode shape can be seen in
Figure 6 below. The eigenvalue for this mode is 11,776, which equates to a buckling
load of 1,460,224 lb. This is within 1% of the theoretical solution of 1,455,592 lb.
10
Figure 6. Eigenvalue Buckling Mode Four Deflections
Table 2 shows the eigenvalue buckling analysis results for various element sizes.
Figure 7 shows how the solution begins to converge to approximately 0.5% once the
element size reaches two inches. Therefore, an element size of two inches is used for all
the modeling completed hereafter. A more refined mesh would be acceptable but the
increased computing time would be of little benefit.
11
Table 2. Eigenvalue Buckling Mesh Density Study Results
Element Size
(inches)
10
5
4
3
2
1.5
1.25
1
0.5
1
2
12045
11679
11394
11187
11109
11081
11057
11057
12045
11679
11394
11187
11109
11081
11057
11057
3
15246
12089
11926
11822
11769
11753
11746
11743
11743
4
15504
12132
11954
11838
11776
11757
11749
11744
11744
5
15504
12132
11954
11838
11776
11757
11749
11744
11744
% Change
27.79%
1.49%
0.98%
0.53%
0.16%
0.07%
0.04%
0.00%
% Error vs. Element Size
100.50%
Percent Change in Mode 4 Eigenvalue
100.00%
99.50%
99.00%
98.50%
98.00%
97.50%
97.00%
96.50%
0
1
2
3
4
5
6
Element Size (inches)
Figure 7. Convergence of Eigenvalue Buckling Results vs. Element Size
2.4 Nonlinear Buckling Analysis
A nonlinear buckling analysis is performed using the “modified Riks method”. The
modified Riks method is a solution method in Abaqus used for load cases where the
loading is proportional to, or governed by a single scalar parameter. In this study, this
scalar parameter is the axial compressive load. The result of the method is loads and
12
displacements. Since both loads and displacements are unknown, another quantity,
called arc length, along the static equilibrium path in load-displacement space, is used to
measure the progress of the solution. This allows the modeling of both stable and
unstable structures. For more information on the modified Riks method, see Reference
5.
A baseline analysis of a perfect cylinder is conducted. In order to aid in the
buckling of the structure, a very small imperfection (0.00015”, or 0.0003” out of
roundness) is introduced into the model. Figure 8 shows displacement and von Mises
stress plots of this analysis. The stress plot shows that stresses are high in the area
around the imperfections, or where the cylinder is out of round.
Figure 8. Displacement and Von Mises Stress for Nonlinear Buckling Analysis in Perfect Cylinder
Next, several degrees of out of roundness are analyzed up to 100% of the shell
thickness. The results can be seen in Table 2. Figure 9 illustrates how even a slight out
of roundness can affect buckling capability. A cylinder that is out of round by only 1%
13
of the shell thickness reduces the buckling capability by 8.3%. This is a significant
reduction in capability when trying to design a structure. The trend appears to trend
exponentially, rather than linearly. The column labeled LPF in Table 3 is a ratio of the
load carrying capability to the applied load. This parameter is used to find the load
carrying capability of the structure.
Table 3. Nonlinear Buckling Results
Imperfection:
% Shell
Thickness
0.1%
1%
10%
50%
100%
Distance OOR
(inches)
0.0003
0.003
0.03
0.15
0.3
Load
Capability
(lb)
1,431,729
1,312,689
891,003
488,703
357,589
LPF
1.155
1.059
0.719
0.394
0.288
% Error
1.66%
9.84%
38.80%
66.43%
75.44%
% Reduction in
Capability
0.0%
8.3%
37.8%
65.9%
75.0%
Effect of Ovalization on Buckling Capability for a Thin
Cylindrical Shell
1,600,000
Buckling Load Capability (lb)
1,400,000
1,200,000
1,000,000
800,000
600,000
400,000
200,000
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Distance Out of Round (in)
Figure 9. Effect of Ovalization on Buckling Capability
As the imperfection, or out of roundness, increases in the structure, the shape of the
buckled structure becomes more representative of an ovalized structure. Figure 10
shows the displacement and stress plots for a cylinder with an e equal to 50% of the shell
thickness, or out of round by 0.15”. The von Mises stress is higher near the “corners” of
14
the ovalized shape, which is what would be expected. Appendix A shows displacement
and stress results for e = 1%, 10% and 100%.
Figure 10. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in Cylinder, e
= 50% shell thickness
The load vs. displacement curve for this analysis, seen in Figure 11, also closely
represents the theoretical prediction shown in Figure 3. The load slowly increases with
slight slope changes as the structure becomes unstable. Once the peak load capability is
reached, the stiffness of the structure is zero and collapse occurs.
15
Figure 11. Load vs. Displacement Curve for Nonlinear Buckling Analysis in Cylinder, e = 50% shell
thickness
16
3. Conclusions
The results of this project meet the expected outcome. It was clearly evident that by
decreasing the element size for the eigenvalue buckling analysis, the accuracy of the
solution was increased. This helped to calibrate the model and ensure that the element
size used for all analyses was appropriate.
The baseline nonlinear buckling analysis resulted in a solution within 2% of the
theoretical solution. It was shown that adding imperfections in the form of out of
roundness, or ovalization, significantly reduced the load carrying capability of the
structure. An e of 1%, 10%, 50% and 100% of the shell thickness resulted in a reduction
in buckling capability of 8.3%, 37.8% 65.9% and 75%, respectively. The stresses in the
nonlinear buckling analysis did not exceed the yield strength of the material, 86 ksi.
Therefore, using elastic material properties for Ti 6-4 was an appropriate assumption.
In addition, the hand calculation based on experimental data showed that, based on
90% probability, the actual load carrying capability of a thin cylindrical shell under axial
loading is 70% less than the theoretical value. This result is close to the 65.9% reduction
from an e of 50% of the shell thickness. The FE solution, however, does not take into
account any other material or geometric imperfections that will be present in a structure.
Therefore, it would be expected that for an out of round structure the actual load
carrying capability would be less than the values reported here due to other
imperfections in the structure.
In conclusion, the intent of this study was to show how ovalization, or out of
roundness, of a cylinder affected the buckling capability. It was shown that this type of
imperfection can significantly reduce the capability of a thin shell. However, further
studies that take other imperfections into account must be addressed. It should also be
noted that this study only addresses isotropic materials and the results should not be
assumed to be the same for a composite structure.
17
4. References
[1] Buckling of Thin Shells: Recent Advances and Trends
Jin Guang Teng, Appl. Mech. Rev. 49, 263 (1996), DOI:10.1115/1.3101927
[2] Young, W.C.,1989, Roarks Formulas for Stress and Strain, McGraw Hill Inc., NY,
pp 714-717, Chap. 15.
[3] Bruhn, E.F., 1973, Analysis and Design of Flight Vehicle Structures, Jacobs
Publishing, Inc., ppC8.1-C8.26.
[4] Buckling analysis of a submarine with hull imperfections
Lee, Harvey C., 2007
[5] Abaqus 6.9 Users Manual
[6] Harris, Seurer, Skeen and Benjamin. The Stability of Thin-Walled Unstiffened
Circular Cylinders Under Axial Compression. Jour. Aero. Sciences. Vol. 24, August,
1957.
[7] ANSYS 11.0 Users Manual
[8] Broggi, Matteo, 2008, Buckling of Cylindrical Shells with Random Imperfections –
Revisited. Institute of Engineering Mechanics, University of Innsbruck. Innsbruck,
Austria, EU.
[9] Efunda. “Ti 6Al-4V” http://www.efunda.com/materials/alloys/titanium/
18
5. Appendix A – Nonlinear Buckling Results
Figure 12. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in Cylinder, e
= 1% shell thickness
Figure 13. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in Cylinder, e
= 10% shell thickness
19
Figure 14. Axial displacement and Von Mises Stress for Nonlinear Buckling Analysis in Cylinder, e
= 100% shell thickness
20