Buckling Analysis of a Submarine with
Hull Imperfections
by
Harvey C. Lee
A Seminar Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF MECHANICAL ENGINEERING
Approved:
_________________________________________
Dr. Ernesto Gutierrez-Miravete, Seminar Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
April, 2007
© Copyright 2007
by
Harvey C. Lee
All Rights Reserved
ii
TABLE OF CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF EQUATIONS ................................................................................................... vii
LIST OF SYMBOLS ...................................................................................................... viii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. INTRODUCTION ....................................................................................................... 1
1.1
PROBLEM STATEMENT ................................................................................ 3
1.2
PURPOSE .......................................................................................................... 3
1.3
METHODOLOGY ............................................................................................. 4
1.4
EXPECTED RESULTS ..................................................................................... 5
2. SUBMARINE DESIGN .............................................................................................. 6
3. EIGENVALUE BUCKLING ANALYSIS OF THE MAIN CYLINDRICAL
SECTION .................................................................................................................... 8
4. NONLINEAR LARGE DISPLACEMENT STATIC BUCKLING ANALYSIS OF
THE MAIN CYLINDRICAL SECTION WITH HULL OUT-OF-ROUNDNESS .. 11
5. BUCKLING ANALYSIS OF THE SUBMARINE .................................................. 15
6. PLASTICITY EFFECTS ........................................................................................... 24
7. CONCLUSIONS ....................................................................................................... 31
7.1
RECOMMENDATIONS ................................................................................. 32
8. REFERENCES .......................................................................................................... 33
9. APPENDIX A – MATERIAL PROPERTIES .......................................................... 34
10. APPENDIX B – MAIN CYLINDRICAL SECTION ANSYS MACRO ................. 36
11. APPENDIX C – SUBMARINE ANSYS MACRO................................................... 40
12. APPENDIX D – MAIN CYLINDRICAL SECTION EIGENVALUE BUCKLING
RESULTS .................................................................................................................. 50
iii
13. APPENDIX E - MAIN CYLINDRICAL SECTION NONLINEAR BUCKLING
RESULTS .................................................................................................................. 53
14. APPENDIX F – SUBMARINE EIGENVALUE BUCKLING RESULTS .............. 55
15. APPENDIX G - SUBMARINE NONLINEAR BUCKLING RESULTS ................ 58
16. APPENDIX H - SUBMARINE NONLINEAR BUCKLING RESULTS WITH
PLASTICITY ............................................................................................................ 66
iv
LIST OF TABLES
Table 1 – Main Cylindrical Section Eigenvalue Buckling Results ................................... 9
Table 2 – Nonlinear Buckling results of the main cylindrical section ............................ 12
Table 3 – Eigenvalue Buckling results of the submarine ................................................ 16
Table 4 – Nonlinear Buckling results of the submarine .................................................. 19
Table 5 – Submarine depth capability vs. hull out-of-roundness .................................... 22
Table 6 – Submarine buckling results ............................................................................. 26
Table 7 – Submarine depth capability vs. hull out-of-roundness with plasticity ............ 29
v
LIST OF FIGURES
Figure 1 – Submarine Design Configuration and Dimensions .......................................... 7
Figure 2 – FEA model of the main cylindrical section with boundary conditions ............ 9
Figure 3 – Convergence of main cylindrical section Eigenvalue Buckling results ......... 10
Figure 4 – Buckled mode shape of 2 nodal diameters of the main cylindrical section ... 10
Figure 5 – Definition of out-of-roundness ....................................................................... 11
Figure 6 – Nonlinear Buckling of the main cylindrical section with 4” OOR ................ 12
Figure 7 – Southwell Plot of the main cylindrical section with 4” OOR ........................ 13
Figure 8 – Comparison of ANSYS and Southwell method in determining the critical
buckling pressure of the main cylindrical section as a function of OOR ................ 14
Figure 9 – FEA model of the submarine with boundary conditions................................ 15
Figure 10 – Convergence of submarine Eigenvalue Buckling results ............................. 16
Figure 11 – Submarine buckled mode shape of 2 nodal diameters ................................. 17
Figure 12 – Buckled mode shape of main cylindrical section with internal stiffeners .... 18
Figure 13 – Main cylindrical section OOR of 4” with eccentricities shown ................... 18
Figure 14 – Nonlinear Buckling of the submarine with 1” OOR .................................... 19
Figure 15 – Southwell Plot of the submarine with 1” OOR ............................................ 20
Figure 16 – Comparison of ANSYS and Southwell method in determining the critical
buckling pressure of the submarine as a function of OOR ...................................... 21
Figure 17 – Graph of Bernoulli’s equation plotting ocean pressure against depth ......... 22
Figure 18 – Submarine depth capability vs. hull out-of-roundness ................................. 23
Figure 19 – Hull stresses for 4” OOR .............................................................................. 24
Figure 20 – Internal stiffener stresses in the main cylindrical section for 4” OOR ......... 24
Figure 21 – Bilinear True Stress-Strain Curve for AISI 4340 Steel ................................ 25
Figure 22 – Multilinear Isotropic Hardening curve for AISI 4340 Steel ........................ 26
Figure 23 – Submarine buckling strength as a function of out-of-roundness.................. 27
Figure 24 – Buckled mode shape for 4” OOR with elastic-plastic material ................... 27
Figure 25 – Hull stresses for 4” OOR with elastic-plastic material ................................ 28
Figure 26 – Equivalent plastic strain of the internal stiffeners for 4” OOR .................... 29
Figure 27 – Submarine depth capability vs. hull out-of-roundness (Final Summary) .... 30
vi
LIST OF EQUATIONS
Equation 1 – Bernoulli's Equation ..................................................................................... 6
Equation 2 – Flugge's Theoretical Buckling Solution for a Simply Supported Cylinder
Under Uniform External Pressure ............................................................................. 8
Equation 3 – Relation between Critical Buckling Pressure and Ocean Depth ................ 21
Equation 4 – Relation between True Strain and Engineering Strain ............................... 25
Equation 5 – Relation between True Stress and Engineering Stress ............................... 25
vii
LIST OF SYMBOLS
FEA
–
Finite Element Analysis
Pcrit
–
Critical Buckling Pressure, Buckling Strength (psi)
DOF
–
Finite Element Degree of Freedom
Esize –
Finite Element Mesh Density Uniform Element Size
OOR –
Out-of-Roundness (in.)
e
–
Out-of-Roundness Eccentricity (in.)
BF
–
Eigenvalue Buckling Factor
R2
–
Linear Curve Fitting Accuracy
εTrue
–
True Strain (in./in.)
εEng
–
Engineering Strain (in./in.)
σTrue
–
True Stress (psi)
σEng
–
Engineering Stress (psi)
viii
ACKNOWLEDGMENT
To my loving wife Jennifer, whose very patience and
unwavering support, has encouraged me to bring this paper to its
final completion.
ix
ABSTRACT
The design of submarines for deep sea exploration has many challenges.
The greatest challenge is its buckling strength against the crushing pressures
of the ocean depth. The problem lies in the fact that there are no theoretical
solutions for such complex geometry. To further complicate the problem,
the out-of-roundness of the cylindrical hull due to manufacturing tolerances
as well as material nonlinearity must also be considered. To overcome
these issues, Finite Element Analysis will be used to determine the crushing
depth of a given submarine design once its buckling strength has been
found.
x
1. Introduction
Although there is much literature dealing with the stability of circular cylindrical
shells under uniform external pressure, only a few are devoted to numerical methods of
analysis with consideration to geometric imperfections and material nonlinearity. Of the
few are Forasassi and Frano’s [7] test and modeling report, in which they found the
length, thickness-to-diameter ratio, modulus and yield stress of the material, and initial
imperfections in the form of ovalization to be major factors that affected the collapse
pressure of pipes. These factors, as well as the end constraints, are key determinants in
the collapse pattern, or buckled mode shape, which are in the form of nodal diameters or
lobes.
The difficulty faced by the structural analyst is determining the critical buckling
pressure of the real cylindrical structure of concern. These structures are typically
complex whereby there are no derived theoretical solutions, such as the case with our
deep sea exploration submarine.
Even if the real cylindrical structure can be
approximated to a more simplistic geometry, the theoretical solution can still be quite
complex. This is evident in Flugge’s [9] derivation for a simply supported cylinder
under uniform external pressure as shown in Eqn 2.
To overcome these challenges require the use of numerical methods or Finite
Element analysis.
The least intensive, from a computational standpoint, is the
Eigenvalue Buckling analysis. This method, as defined by Brown [3], predicts the
theoretical buckling strength of an ideal linear elastic structure. As an example, the
Eigenvalue Buckling solution of an Euler column would match the classical Euler
solution. Although it is relatively simple to execute, its limitations restrict it from
modeling the true nature of real structures, which have geometric imperfections and
material nonlinearity amongst other non-ideal characteristics. It is anticonservative, but
does provide a good start for preliminary assessments.
In order to model the behavior of real structures, a more advanced and
computationally intensive method is required, which is the Nonlinear Large
Displacement Static Buckling analysis. This approach seeks the load level at which the
1
structure becomes unstable by gradually increasing the load [3].
The equilibrium
equation can be rewritten in the form {U} = {F}/[K], where [K] is the global stiffness
matrix, {U} is the displacement vector and {F} is the load vector. The Nonlinear
method requires an iterative process to solve for {U} since information about [K] and
{F} are not known. Instability occurs within this iterative process when [K] approaches
zero. In ANSYS, the Finite Element Analysis software used in this study, the instability
manifests itself as an unconverged solution, indicating that the cylindrical structure can
no longer carry any more external pressure load because buckling has occurred. In
general practice, the solution previous to the last unconverged solution is the buckling
strength.
This paper, however, will use the Southwell method in determining this limit.
Ko’s [6] NASA report describes how the Southwell plot is generated as well as its
limitations. He states:
“The well-known graphical method of predicting buckling loads is the
Southwell method.
In the Southwell plot, the compliance (that is,
deflection/load) is plotted against deflection, and the buckling load is
determined from the inverse slope of the plot. The Southwell method has
been successful in predicting the classical buckling of simple structures
such as columns and plates. For complex structures exhibiting complex
buckling behavior (for example, local instabilities and plasticity effect),
the Southwell plot may not be a straight line, and therefore no discernable
slope may be obtained for accurately determining the buckling loads.”
Therefore, the Southwell method will be applied only to the Nonlinear Buckling analysis
with full elasticity and the study will show whether it is reliable or not in determining
the buckling strength of the submarine.
It is possible that a cylindrical structure under uniform external pressure
experience inelastic buckling. This occurs when the hoop stress exceeds the yield
strength before the critical buckling pressure is reached. Beyond the yield point, the
material’s stiffness, or modulus of elasticity, reduces significantly and thus the buckling
strength. Therefore, it is important that plasticity is considered in the analysis of our
2
submarine. This will be accomplished by executing a Nonlinear Large Displacement
Static Buckling analysis with elastic-plastic material properties.
1.1 Problem Statement
The problem with deep sea exploration is designing a submarine with a
sufficiently high buckling strength in order to withstand the crushing pressures of the
ocean depth. However, determining its buckling strength is far from trivial. As a result,
Finite Element Analysis will be required since there are no theoretical solutions to such
complex geometry and its inherent imperfections due to manufacturing limitations.
Many methods are utilized in industry and information to its validation is usually
proprietary. This paper provides general methods to this endeavor and will show the
advantages and disadvantages of each.
No amount of analysis or sophistication thereof should ever replace testing.
Unfortunately, it is not possible to perform non-destructive testing to determine the
submarine’s buckling strength since it is a catastrophic failure mode. Smaller scale
models would have to be devised that can be readily sacrificed without substantial
impact to cost.
1.2 Purpose
The purpose of this study is several folds, all related to determining the critical
buckling pressure, or buckling strength, of the submarine using Finite Element Analysis
(FEA). First, is to understand the effects of mesh density on the accuracy of the
solution. Second, is to understand the relationship, differences and advantages and
disadvantages between an Eigenvalue Buckling analysis and a Nonlinear Large
Displacement Static Buckling analysis. Lastly is to understand the effects of plasticity if
the stresses in the hull and internal stiffeners exceed the yield strength of the material.
3
1.3 Methodology
The commercial code ANSYS will be used to conduct all Finite Element analyses.
All Finite Element models will be generated with Shell 181 elements. This element is
based on the Reissner/Mindlin thick shell theory which includes bending, membrane and
transverse shear effects.
This theory is suitable in modeling the thick hull of the
submarine and its associated internal stiffeners.
The first stage is to calibrate the analysis by modeling just the main cylindrical
section of the submarine without internal stiffeners and simply supporting it at its ends.
An Eigenvalue Buckling analysis will then be conducted with several iterations of mesh
refinement until the solution converges to the theoretical critical buckling pressure to
within 5% error. As it was previously stated, this type of analysis predicts the theoretical
buckling strength of an ideal linear elastic structure.
The second stage is to take the model with the mesh density that converged to the
theoretical critical buckling pressure and conduct a Nonlinear Large Displacement Static
Buckling analysis with several iterations of various prescribed out-of-roundness or
“ovalization”. A perfect hull would be perfectly cylindrical. But in reality it will be
imperfect, having a certain amount of out-of-roundness governed by manufacturing
tolerances and capability. The Southwell method will be used to determine the critical
buckling pressure from the Nonlinear analysis.
The third stage is to apply the methods from Stages 1 and 2 to the submarine.
Once the critical buckling pressures have been found based on the various prescribed
out-of-roundness of the hull, the crushing depth capability of the submarine will then be
calculated as a function of hull out-of-roundness. It is predicted that as the prescribed
hull out-of-roundness increases, the buckling strength decreases.
The fourth and final stage is to analyze the hull and internal stiffener stresses at the
critical buckling pressure to determine if they have exceeded the yield strength of the
material. (To be technically accurate, the stresses should be compared against the
proportional limit of the material since the onset of plasticity occurs from this point.
However, for the purposes of this study and because most material data does not list the
proportional limit, the yield strength will be used instead.) If not, then the analysis is
4
complete. If so, then the method in Stage 2 will be re-executed but with elastic-plastic
material properties.
1.4 Expected Results
Its is expected that as the mesh density of the Finite Element model increases, the
buckling solution will converge to the exact solution or to a value within 5% error. It is
also expected that the Eigenvalue Buckling solution will produce the highest value since
it assumes an ideal geometry of the submarine with no imperfections. This will be a
solid baseline and reference point with which to compare the Nonlinear results to, which
takes into account the imperfections or out-of-roundness in our particular case. It is
anticipated that the buckling strength of the submarine behaves adversely as the out-ofroundness increases. Furthermore, with plasticity considered, it should be no surprise
that the stiffness of the material drops considerably beyond the yield point, thus leading
to an even further reduction in buckling strength. The solutions from the Eigenvalue, the
Nonlinear Elastic and the Nonlinear Elastic-Plastic will be compared and the effects on
the ocean depth capability of the submarine will be shown.
5
2. Submarine Design
Our deep sea exploration submarine was designed with the intent to have a
maximum crew capacity of 12 and a depth capability of 4 to 5 miles. The general layout
would be similar to a military submarine but on a much smaller scale. To support the
crew and all the necessary controls and instrumentation, the mean hull diameter was set
at 12 ft.
The main cylindrical section was divided into the fwd, mid and rear
compartments which are the control room, the research and analysis room and the engine
room, respectively.
Sonars and fwd ballast tanks are situated in the nose of the
submarine whereas the propulsion system and aft ballast tanks are mounted inside the
conical tail section. Two vertical and two horizontal fins that are welded onto the tail
provide stability and maneuverability. The fwd and aft bulkheads separate the nose and
tail section from the main compartments. Internals stiffeners welded onto the hull
provide additional strength for the submarine.
A very strong material is required if our submarine is to withstand the crushing
pressures of the ocean floor. As a result, AISI 4340 Steel, oil quenched at 845C and
tempered at 425C, was selected.
Although its tensile strength is higher at lower
temperatures, which is typical of the ocean floor environment, room temperature
properties were conservatively used for additional safety margin.
With the general layout defined and material selected, some preliminary analyses
were required in order to size the hull thickness as well as the internal stiffeners. A
finite element model was created and an Eigenvalue Buckling Analysis was conducted
to determine the critical buckling pressure (Pcrit). The critical buckling pressure was then
used to back calculate the depth capability using Bernoulli’s equation.
P = Po + gh
Equation 1 – Bernoulli’s Equation
where
P = Ocean Depth Pressure
Po = Atmospheric Pressure
6
 = Density of Seawater
g = Gravitational Acceleration
h = Ocean Depth
Several iterations were performed until reasonable sizes for the hull and internal
stiffeners were determined such that the 4 to 5 mile depth capability of the submarine
can be achieved. (A thickness of 1 ft. was prescribed for the bulkheads and remained
constant through each iteration) The final dimensions of our deep sea exploration
submarine structure as a finite element model is shown in Figure 1 below. Preliminary
analysis shows that its buckling strength is 11,219 psi, yielding a maximum ocean depth
capability of 4.9 miles.
Figure 1 - Submarine Design Configuration and Dimensions
7
3. Eigenvalue Buckling Analysis of the Main Cylindrical
Section
The first stage was to calibrate the analysis by modeling just the main cylindrical
section of the submarine without internal stiffeners and simply supporting it at its ends.
Flugge [9] derives the theoretical solution for such a cylinder (Eqn 2).
Pcrit
=
Et
r (1 - υ2)
{
(1 - υ )λ4 + k [(λ2 + m ) - 2 (υλ + 3λ4 m + (4 - υ)λ2 m + m ) + 2 (2 - υ) λ2 m + m ]
m2 (λ2 + m2) 2- m2(3λ2+ m2)
2
2 4
6
2
4
6
2
4
}
Equation 2 – Flugge's Theoretical Buckling Solution for a Simply Supported
Cylinder Under Uniform External Pressure
where
E = Modulus of Elasticity
r = Mean hull radius
t = Hull thickness
 = Poisson’s ratio
m = Nodal diameters
t2
πr &
k
=
λ=
12 r 2
l
With the dimensions and material properties of our submarine section, the minimum
critical buckling pressure or buckling strength was calculated to be 4,097 psi with a 2
nodal diameter mode shape (m = 2).
The FEA model, shown in Figure 2, was set up in the global cylindrical
coordinate system and an external reference pressure of 12,000 psi was applied. An
Eigenvalue Buckling analysis was then conducted with several iterations of mesh
refinement until the solution converged to the theoretical solution with an error of
0.09%. The results are shown in Table 1 and Figure 3 plots the convergence to the exact
solution. Also, the buckled mode shape was found to be 2 nodal diameters (Figure 4),
8
confirming Flugge’s theoretical equation. As a result, the FEA model of the main
cylindrical section of our submarine has been calibrated.
Esize
6
5
4
3
2
1
DOF
336
480
720
1248
2280
8880
Pcrit (psi)
5,442
5,356
4,555
4,319
4,211
4,093
Flugge (psi)
4,097
4,097
4,097
4,097
4,097
4,097
Error
32.82%
30.73%
11.17%
5.41%
2.78%
0.09%
Table 1 – Main Cylindrical Section Eigenvalue Buckling Results
Isometric View
Side View
Front View
Figure 2 – FEA model of the main cylindrical section with boundary conditions
9
Cylindrical Hull Section Eigenbuckling Results
6,000
5,000
Pcrit (psi)
4,000
3,000
2,000
1,000
0
0
2000
4000
6000
8000
10000
DOF
Figure 3 – Convergence of main cylindrical section Eigenvalue Buckling
results
Figure 4 – Buckled mode shape of 2 nodal diameters of the main cylindrical
section
10
4. Nonlinear Large Displacement Static Buckling Analysis of
the Main Cylindrical Section with Hull Out-of-Roundness
The next step was to take the main cylindrical section, with the mesh density that
converged to the theoretical solution, and conduct a Nonlinear Large Displacement
Static Buckling analysis with several iterations of various prescribed out-of-roundness or
“ovalization” in our particular case. Out-of-roundness (OOR) is best defined by the
following figure.
e
e
e
Eg: If OOR = 4” then e = 2”
e
Figure 5 – Definition of out-of-roundness
An out-of-roundness of 1”, 2”, 3” and 4” were considered for all nonlinear
analyses conducted throughout this report. This geometric imperfection was created by
using the eigenvectors or nodal displacements, from the previously run Eigenvalue
Buckling analysis, with a scale factor to update the nodal coordinates of the Nonlinear
model. As an example, if we were to run an analysis with an OOR of 3”, the updated
nodal coordinates in our Nonlinear model would have the same contour plot as that
shown in Figure 4, except that the displacement scale range of –1 ft. to 1 ft. would run
from –0.125 ft. to 0.125 ft. instead. Here, the scale factor would be the eccentricity (e),
having the value of (3/12)/2 or 0.125. Another advantage in using this method is that
there is consistency in the OOR angle, which is desirable. The OOR angle is defined as
the maximum or minimum eccentricity circumferential location with respect to the
horizontal or vertical axis. For our case, the OOR angle is 45 degrees.
11
The main cylindrical section FEA model that converged to the theoretical
solution had a uniform mesh density based on an element size of 1 (See Table 1). The
boundary conditions of simply supported ends and a reference pressure of 12,000 psi
were maintained. A Nonlinear Large Displacement Static Buckling analysis was then
conducted for all four prescribed out-of-roundness using very small incremental load
steps. The results are shown in the table below and compared against the Eigenvalue
solution, which assumes perfect geometry with zero out-of-roundness.
OOR (in.)
0
1
2
3
4
ANSYS (psi)
4,093
3,591
3,324
3,117
2,898
Southwell (psi)
4,093
4,000
3,894
3,711
3,619
Eigenvalue
Table 2 – Nonlinear Buckling results of the main cylindrical section
The last converged solution in ANSYS represents the critical buckling pressure,
which signifies that the hoop stiffness of the cylinder approaches zero and can no longer
carry any more load. Figure 6 below shows the final buckled shape for the 4” out-ofroundness condition. To reiterate, these displacement scales are in feet.
Figure 6 – Nonlinear Buckling of the main cylindrical section with 4” OOR
12
Southwell plots were generated for each OOR case using the peak nodal
deflection (In Figure 6, the peak nodal deflection would be –0.657806 ft.). This is
possible because the load and deflection history in the Nonlinear analysis were recorded.
Figure 7 below shows the Southwell plot for the 4” out-of-roundness condition. A linear
trendline, shown in red, was fitted through the points and its equation and R2 value
given.
In the Southwell method, the inverse slope of this trendline is the critical
buckling pressure. For an OOR of 4”, Pcrit was calculated to be 3,619 psi.
Southwell Plot
OOR = 4"
Defl / Pressure (in. / psi)
3.00E-04
y = 0.0002763x + 0.0000447
2.50E-04
R2 = 0.9989078
2.00E-04
1.50E-04
1.00E-04
5.00E-05
0.00E+00
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Deflection (in.)
Figure 7 – Southwell Plot of the main cylindrical section with 4” OOR
It was interesting to observe that for each and every one of the cases analyzed,
the Southwell method consistently calculated the critical buckling pressure much greater
than that of ANSYS. Figure 8 shows this comparison. Also, the trend appears to show
that the differences widen as the out-of-roundness increases. Nevertheless, the overall
results are in agreement to what was expected, which is the fact that hull imperfections
reduce the buckling capability of the pressure vessel. In the case of the highest out-ofroundness analyzed, the buckling strength was knocked down by 11.7% (Southwell) and
by as much as 29.3% (ANSYS) with respect to the theoretical solution. It must be
reclarified that in Figure 8, which is a graphical plot of Table 2, the critical buckling
pressure for the out-of-roundness of 0” is based on the Eigenvalue Buckling analysis.
13
Critical Buckling Pressure vs. OOR
4500
Critical
Buckling Pressure (psi)
4000
ANSYS
Southwell
3500
3000
2500
0
1
2
3
4
5
OOR (in.)
Figure 8 – Comparison of ANSYS and Southwell method in determining the
critical buckling pressure of the main cylindrical section as a function of OOR
14
5. Buckling Analysis of the Submarine
With the main cylindrical section FEA model calibrated and the effects of out-ofroundness known, the buckling analysis of our deep exploration submarine can begin.
First, an Eigenvalue Buckling analysis was conducted with several iterations of
mesh refinement until the solution converged to within an error of 5% with respect to the
final iteration. Figure 9 shows the FEA model of the submarine with a reference
hydrostatic pressure of 12,000 psi applied and the center node of the aft bulkhead
grounded to prevent rigid body motion.
All DOF = 0
Figure 9 – FEA model of the submarine with boundary conditions
Each iteration generated the buckling factor (BF) and when multiplied by the
reference hydrostatic pressure, the critical buckling pressure (Pcrit) was determined.
The submarine FEA model converged to a critical buckling pressure of 11,219 psi. Its
uniform mesh density is based on an element size of 1, generating a DOF (degree of
15
freedom) of 28,200. The results are shown in Table 3. Figure 10 plots the convergence
of the solution and Figure 11 shows the final buckled mode shape of 2 nodal diameters.
Esize
6
5
4
3
2
1
DOF
1,032
2,760
3,000
3,384
7,512
28,200
Pcrit (psi)
23,855
12,924
12,905
12,508
11,678
11,219
Error
112.62%
15.19%
15.02%
11.48%
4.09%
0.00%
Table 3 – Eigenvalue Buckling results of the submarine
Submarine Eigenbuckling Results
30000
Pcrit (psi)
25000
20000
15000
10000
5000
0
0
10000
20000
30000
DOF
Figure 10 - Convergence of submarine Eigenvalue Buckling results
The next step was to perform the Nonlinear Large Displacement Static Buckling
analysis using the converged FEA model of the submarine. The method used to create
the geometric imperfection of the hull is similar to what was done for the main
cylindrical section as described in Chapter 4, but with internal stiffeners. Therefore, the
16
main cylindrical section of the submarine with internal stiffeners was isolated,
everything else being deleted, and an Eigenvalue Buckling analysis was conducted.
Again, the ends were simply supported and a reference pressure of 12,000 psi was
applied. Figure 12 shows the buckled mode shape.
Figure 11 – Submarine buckled mode shape of 2 nodal diameters
The nonlinear model’s nodal coordinates were updated using the nodal
displacements from the buckling analysis with a scale factor applied. This simulated the
desired preconditioned out-of-roundness effect. Different scale factors were used for the
1”, 2”, 3” and 4” out-of-roundness conditions analyzed. Figure 13 shows a scale factor
of (4/12)/2 or 0.166667 used to preset the main cylindrical section with an OOR of 4”.
Eccentricities (e) are also shown. The OOR angle of 45 degrees was consistent with the
buckled mode shape of the full submarine (See Figure 11), which is desirable.
17
Figure 12 – Buckled mode shape of main cylindrical section with internal
stiffeners
- 0.166667
+ 0.166667
+ 0.166667
- 0.166667
Figure 13 – Main cylindrical section OOR of 4” with eccentricities shown
18
A Nonlinear Large Displacement Static Buckling analysis was then conducted
for all four out-of-roundness conditions using very small incremental load steps. The
results are shown in Table 4 below and compared against the Eigenvalue solution, which
assumes perfect geometry with zero out-of-roundness.
OOR (in.)
0
1
2
3
4
ANSYS (psi)
11,219
9,796
8,450
7,950
6,950
Southwell (psi)
11,219
10,132
10,111
10,417
10,537
Eigenvalue
Table 4 - Nonlinear Buckling results of the submarine
The last converged solution in ANSYS represents the critical buckling pressure,
which signifies that the hoop stiffness of the submarine approaches zero and can no
longer carry any more load. Figure 14 below shows the final buckled mode shape for
the 1” out-of-roundness condition.
Figure 14 - Nonlinear Buckling of the submarine with 1” OOR
19
Southwell plots were generated for each OOR case using the peak nodal
deflection. This is possible because the load and deflection history in the Nonlinear
analysis were recorded. Figure 15 shows the Southwell plot for the 1” out-of-roundness
Southwell Plot
OOR = 1"
Defl / Pressure (in. / psi)
3.00E-05
y = 0.0000987x + 0.0000012
R2 = 0.9972790
2.50E-05
2.00E-05
1.50E-05
1.00E-05
5.00E-06
0.00E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
Deflection (in.)
Figure 15 - Southwell Plot of the submarine with 1” OOR
condition.
A linear trendline, shown in red, was fitted through the points and its
equation and R2 value given. In the Southwell method, the inverse slope of this trendline
is the critical buckling pressure. For an OOR of 1”, the critical buckling pressure was
calculated to be 10,132 psi.
The buckling strength of the submarine calculated from the Southwell plots for
each case (See Table 4) was found to be inconsistent and erroneous. The trend shows
that as the out-of-roundness increases from 2” to 4” the buckling strength becomes
relatively level with a slight increase, which of course is not possible. Figure 16 shows
the trend against that of ANSYS. Because the Southwell method was found to be
inaccurate and thus unreliable in this particular study, the buckling strength determined
by ANSYS was used from this point forward. It must be reclarified that in Figure 16,
20
which is a graphical plot of Table 4, the critical buckling pressure for the out-ofroundness of 0” is based on the Eigenvalue Buckling analysis.
Pritical vs. OOR
12000
Pcritical (psi)
10000
8000
ANSYS
6000
Southwell
4000
2000
0
0
1
2
3
4
5
OOR (in.)
Figure 16 - Comparison of ANSYS and Southwell method in determining the
critical buckling pressure of the submarine as a function of OOR
With Pcrit found, the ocean depth capability of the submarine can be calculated
using Bernoulli’s equation (Eqn 1). Figure 17 is a graph of this equation where the
ocean pressure is plotted against depth. From this graph, the relationship between
critical buckling pressure and ocean depth capability was created and is shown in Eqn 3.
Pcrit = 2289(depth) + 14.696
Equation 3 – Relation between Critical Buckling Pressure and Ocean Depth
From this equation the ocean depth capability of our deep sea exploration submarine was
then calculated as a function of out-of-roundness. The results are shown in Table 5 and
Figure 18.
21
Pressure of Ocean Water at Depth
25000
22905
20616
20000
Pressure (psi)
18327
16038
15000
13749
11460
10000
9171
6882
5000
4593
2304
0
14.7
0
1
2
3
4
5
6
7
8
9
10
11
Depth (mi)
Figure 17 – Graph of Bernoulli’s equation plotting ocean pressure against depth
OOR (in.)
0
1
2
3
4
ANSYS
Pcrit (psi)
11,219
9,796
8,450
7,950
6,950
Depth Capability
4.9
4.3
3.7
3.5
3.0
Table 5 - Submarine depth capability vs. hull out-of-roundness
22
miles
miles
miles
miles
miles
Submarine Depth Capabilty vs Hull OOR
Ocean Depth (mi)
6.0
5.0
4.0
3.0
2.0
1.0
0.0
0
1
2
3
Hull OOR (in.)
Figure 18 - Submarine depth capability vs. hull out-of-roundness
23
4
5
6. Plasticity Effects
The Nonlinear Large Displacement Static Buckling analysis that was performed in
the previous chapter assumed perfectly elastic material behavior. Unfortunately, what
was found was that the stresses in the hull and internal stiffeners exceeded the material’s
yield strength of 214 ksi (See Figures 19 & 20), rendering the submarine’s buckling
Figure 19 – Hull stresses for 4” OOR
Figure 20 – Internal stiffener stresses in the main cylindrical section
for 4” OOR
24
strength inaccurate. As a result, the Nonlinear Large Displacement Static Buckling
analysis was re-executed using elastic-plastic material properties. These properties were
simulated by generating a bilinear true stress-strain curve (Figure 21) based on the
material’s yield strength, ultimate tensile strength, elastic modulus and percent
elongation at break, which was assumed as the strain at ultimate. Furthermore, because
these properties are from the engineering stress-strain curve, corrections were made to
create the true stress-strain curve. The relation between engineering and true stress and
strain is given by the following:
True = ln (1 + Eng)
Equation 4 – Relation between True Strain and Engineering Strain
True = Eng (1 + Eng)
Equation 5 – Relation between True Stress and Engineering Stress
Bilinear True Stress-Strain Curve
300000
Stress (psi)
250000
200000
150000
100000
50000
0
0
0.05
0.1
0.15
Strain (in./in.)
Figure 21 – Bilinear True Stress-Strain Curve for AISI 4340 Steel
25
To analyze for plasticity in ANSYS, the multilinear isotropic hardening (MISO) rule was
used (Figure 22). Brown [3] recommends this option for proportional loading and large
strain applications of metal plasticity.
Figure 22 – Multilinear Isotropic Hardening curve for AISI 4340 Steel
The results from the Nonlinear Large Displacement Static Buckling analysis with
elastic-plastic material properties for the four out-of-roundness conditions are shown in
Table 6 below and its graph in Figure 23.
They are compared against the Eigenvalue
Buckling solution as well as the previous Nonlinear elastic solutions.
ANSYS Nonlinear Large Displacement Static
OOR (in.)
Elastic (psi) Elastic-Plastic (psi)
0
11,219
11,219
1
9,796
8,262
2
8,450
7,166
3
7,950
6,330
4
6,950
5,724
Table 6 – Submarine buckling results
26
Eigenvalue
Pcritical vs. OOR
12000
Pcritical (psi)
10000
8000
ANSYS - Elastic
ANSYS - Elastic-Plastic
6000
4000
2000
0
0
1
2
3
4
5
OOR (in.)
Figure 23 – Submarine buckling strength as a function of out-of-roundness
Figure 24 – Buckled mode shape for 4” OOR with elastic-plastic material
27
From Table 6 and Figure 23, it can be clearly seen how plasticity effects reduce
the submarine’s buckling strength even further, due primarily to the tangent modulus
once the yield strain has been exceeded. Furthermore, when plasticity is considered, the
stresses yield off and redistribute over a larger area of the submarine. Figure 24 shows
the buckled mode shape and Figure 25 shows the dramatic difference in stress compared
to that in Figure 19. Both figures are for an out-of-roundness of 4”.
Figure 25 – Hull stresses for 4” OOR with elastic-plastic material
The majority of the backing strength against buckling is attributed to the internal
stiffeners in the main cylindrical section. Once they yield, their hoop stiffness that
provides ring stability begins to decline. Figure 26 shows how the high plastic strains
due to bending are concentrated at four local regions in the internal stiffeners. This is
caused by the 2 nodal diameter buckled mode shape.
28
Figure 26 - Equivalent plastic strain of the internal stiffeners for 4” OOR
The ocean depth capability of the submarine, with plasticity considered, was recalculated
using Equation 3. The final results are shown in Table 7 and Figure 27 comparing the
Eigenvalue, Nonlinear Elastic and Nonlinear Elastic-Plastic solutions. It must be
reclarified that in Figure 27, which is a graphical plot of Table 6, the critical buckling
pressure for the out-of-roundness of 0” is based on the Eigenvalue Buckling analysis.
ANSYS - Elastic-Plastic
OOR (in.)
Pcrit (psi)
0
11,219
1
8,262
2
7,166
3
6,330
4
5,724
Depth Capabilty
4.9
3.6
3.1
2.8
2.5
miles
miles
miles
miles
miles
Table 7 - Submarine depth capability vs. hull out-of-roundness with plasticity
29
Submarine Ocean Depth Capability vs. Hull OOR
6.0
Ocean Depth (mi)
5.0
4.0
ANSYS - Elastic
ANSYS - Elastic-Plastic
3.0
2.0
1.0
0.0
0
1
2
3
4
5
Hull OOR (in.)
Figure 27 - Submarine depth capability vs. hull out-of-roundness (Final Summary)
30
7. Conclusions
The buckling analysis results of our deep sea exploration submarine were overall
what was expected. First, it was clearly seen that by increasing the Finite Element mesh
density the buckling solution from the Eigenvalue Buckling analysis monotonically
converged to the exact solution, as in the case of the main cylindrical section study. This
approach defined the calibration of the model and was then applied to the more complex
submarine model, where a theoretical or exact solution does not exist. The buckling
solution of the submarine through mesh refinement showed the same behavior,
converging to a value within 5% error, which is acceptable by industry standards.
From the Eigenvalue Buckling analysis it was shown that an ideal geometry of
the submarine with no imperfections resulted in the highest buckling strength of 11,219
psi. Using Bernoulli’s equation, this translated to a crushing depth capability of 4.9
miles into the ocean. However, once imperfections were introduced via hull out-ofroundness, in our particular case “ovalization”, the depth capabilities were dramatically
different. In order to model this imperfection, a Nonlinear Large Displacement Static
Buckling analysis was required.
An out-of-roundness of 1”, 2”, 3” and 4” were
considered. As anticipated, these imperfections had an inverse effect on the submarine’s
ideal buckling strength, reducing it by approximately 13%, 25%, 29% and 38%,
respectively. This translated to a depth capability of 4.3, 3.7, 3.5 and 3.0 miles.
Although the original intent was to use the Southwell method in determining the
buckling strength of the submarine from the Nonlinear analysis, the results proved to be
inconsistent and erroneous. It was found that as the out-of-roundness increased from 2”
to 4”, the results became relatively level with a slight increase, which of course is not
possible. However, in the case of the main cylindrical section Nonlinear analysis, the
Southwell method was consistent and the trend was in alignment to what was expected,
even though the results were higher than that of ANSYS’s last converged buckling
solutions. Therefore, it was concluded that for complex geometries, as in the case of our
submarine, the Southwell method was not valid. As a result, the last converged buckling
solution in ANSYS was used instead to determine the buckling strength.
31
Finally, it was found that the stresses in the hull and internal stiffeners exceeded
the yield strength of the material for each out-of-round condition analyzed. Therefore,
the Nonlinear Large Displacement Static Buckling analyses had to be rerun, but with
elastic-plastic material properties in order to capture a better representation of its true
behavior. Indeed, what was found was that plasticity effects reduced the submarine’s
buckling strength even further, due primarily to the tangent modulus once the yield
strain had been exceeded. With respect to the ideal buckling strength of the submarine,
with plasticity considered, the actual reductions were approximately 26%, 36%, 44%
and 49% for the out-of-roundness of 1”, 2”, 3” and 4”, respectively, as compared to the
previous Nonlinear fully elastic results. These reductions translate to a more accurate
depth capability of 3.6, 3.1, 2.8 and 2.5 miles for our deep sea exploration submarine.
In conclusion, although the design intent of our deep sea exploration submarine
was to have a depth capability in the order of 4 to 5 miles, manufacturing limitations
leading to hull imperfections, in conjunction with real material behavior, proves more
challenging in achieving this endeavor.
7.1 Recommendations
Although this study provides a relatively reasonable method in analyzing the
buckling strength of a deep sea exploration submarine given the timeframe allowed,
further improvements can be made. For example, it was assumed that if the Finite
Element model from the Eigenvalue Buckling analysis converged with a particular mesh
density, it was also valid for the Nonlinear analysis. This may or may not be the case
and it is recommended that a convergence study be executed for the Nonlinear analysis
as well. Mesh refinement can be confined to the areas of concern (ie: main cylindrical
section and internal stiffeners) so that computational time can be reduced. Also, within
this convergence study, it is recommended that the mesh density be examined to
determine whether it is sufficient in capturing the actual stresses and strains since they
have a direct effect on the results of the Nonlinear analysis with plasticity. The Finite
Element model in this study was relatively coarse since displacements were of primary
concern and stresses and strains were of secondary interest.
32
8. References
[1] Warren C. Young and Richard Budynas, “Roark's Formulas for Stress and Strain,”
7th Edition, McGraw-Hill Companies, Inc., 2002.
[2] R. Cook, D. Malkus, M. Plesha and R. Witt, “Concepts and Applications of Finite
Element Analysis,” 4th Edition, John Wiley & Sons, Inc., 2002.
[3] K. Brown, “Advanced ANSYS Topics, V5.5,” CAEA, Inc., 1998.
[4] H. Schmidt, “Stability of Steel Shell Structures General Report,” Journal of
Constructional Steel Research 55 (2000) 159 – 181.
[5] F.B. Sealy, J.O. Smith, “Advanced Mechanics of Materials,” 2nd Edition, Wiley &
Sons, 1952.
[6] W. L. Ko, “Accuracies of Southwell and Force/Stiffness Methods in the Prediction
of Buckling Strength of Hypersonic Aircraft Wing Tubular Panels,” NASA Technical
Memorandum 88295, Nov 1987.
[7]
G. Forasassi, R. Lo Frano, “Buckling of Imperfect Thin Cylindrical Shell Under
Lateral Pressure,” Journal of Achievements in Materials and Manufacturing
Engineering, Vol. 18, Issue 1-2, Sept – Oct 2006.
[8]
E. Ventsel, T. Krauthammer, “Thin Plates and Shells – Theory, Analysis, and
Applications,” Mercel Dekker, Inc., 2001.
[9] W. Flugge, “Stresses in Shells,” Springer-Verlag, Berlin, 1960.
33
9. Appendix A – Material Properties
AISI 4340 Steel, oil quenched 845°C, 425°C
(800°F) temper, tested at 25°C (77°F)
Date: 2/10/2007 2:21:07 PM
KeyWords:
alloy steels, UNS G43400, AMS 5331, AMS 6359, AMS 6414, AMS 6415, ASTM A322, ASTM A331, ASTM A505, ASTM A519, ASTM A547,
ASTM A646, MIL SPEC MIL-S-16974, B.S. 817 M 40 (UK), SAE J404, SAE J412, SAE J770, DIN 1.6565, JIS SNCM 8, IS 1570
40Ni2Cr1Mo28, IS 1570 40NiCr1Mo15
SubCat: Low Alloy Steel, AISI 4000 Series Steel,
Medium Carbon Steel, Metal, Ferrous Metal
Component
Carbon, C
Chromium, Cr
Iron, Fe
Manganese, Mn
Molybdenum, Mo
Nickel, Ni
Phosphorous, P
Sulfur, S
Silicon, Si
Properties
Physical
Value
Min
0.37
0.7
Max
0.43
0.9
0.2
0.3
96
0.7
1.83
0.035
0.04
0.23
Metric
Value
English
Value
Min
Max
Density, g/cc
7.85
0.284
--
--
density is in lb/in^3 for english units
Mechanical
Tensile Strength, Ultimate, MPa
Tensile Strength, Yield, MPa
Elongation at Break, %
Reduction of Area, %
Modulus of Elasticity, GPa
Bulk Modulus, GPa
Poissons Ratio
Machinability, %
1595
1475
12
46
212
140
0.3
50
231
214
12
46
30700
20300
0.3
50
---------
---------
all stresses are in ksi for english units
Shear Modulus, GPa
81.5
11800
--
--
2.48E-05
5.52E-05
7.97E-05
2.98E-05
-----
-----
-----
12.7
12.3
13.7
12.6
13.7
13.9
14.5
0.475
44.5
----------
----------
----------
Electrical
Electrical Resistivity, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Thermal
CTE, linear 20°C, µm/m-°C
CTE, linear 20°C, µm/m-°C
CTE, linear 250°C, µm/m-°C
CTE, linear 250°C, µm/m-°C
CTE, linear 500°C, µm/m-°C
CTE, linear 500°C, µm/m-°C
CTE, linear 500°C, µm/m-°C
Specific Heat Capacity, J/g-°C
Thermal Conductivity, W/m-K
34
Comment
Typical for steel.
Calculated
annealed and cold drawn. Based on 100%
machinability for AISI 1212 steel.
Estimated from elastic modulus
specimen oil hardened, 600°C (1110°F) temper
specimen oil hardened, 600°C (1110°F) temper
specimen oil hardened, 600°C (1110°F) temper
1.88% Ni, normalized, tempered
1.88% Ni, normalized and tempered
1.90% Ni, quenched, tempered
specimen oil hardened, 600°C (1110°F) temper
Typical 4000 series steel
Typical steel
AISI 4340 Steel, oil quenched 845°C, 425°C
(800°F) temper, tested at -195C
Date: 2/10/2007 2:27:56 PM
KeyWords:
alloy steels, UNS G43400, AMS 5331, AMS 6359, AMS 6414, AMS 6415, ASTM A322, ASTM A331, ASTM A505, ASTM A519, ASTM A547,
ASTM A646, MIL SPEC MIL-S-16974, B.S. 817 M 40 (UK), SAE J404, SAE J412, SAE J770, DIN 1.6565, JIS SNCM 8, IS 1570
40Ni2Cr1Mo28, IS 1570 40NiCr1Mo15
SubCat: Low Alloy Steel, AISI 4000 Series Steel,
Medium Carbon Steel, Metal, Ferrous Metal
Component
Carbon, C
Chromium, Cr
Iron, Fe
Manganese, Mn
Molybdenum, Mo
Nickel, Ni
Phosphorous, P
Sulfur, S
Silicon, Si
Value
Properties
Physical
Density, g/cc
Metric
Value
7.85
English
Value
0.284
Min
--
1985
1840
4
11
213
140
0.3
50
288
267
4
11
30900
20300
0.3
50
---------
---------
82
11900
--
--
2.48E-05
2.98E-05
5.52E-05
7.97E-05
2.48E-05
2.98E-05
5.52E-05
7.97E-05
-----
-----
-------
-------
Mechanical
Tensile Strength, Ultimate, MPa
Tensile Strength, Yield, MPa
Elongation at Break, %
Reduction of Area, %
Modulus of Elasticity, GPa
Bulk Modulus, GPa
Poissons Ratio
Machinability, %
Shear Modulus, GPa
Electrical
Electrical Resistivity, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Electrical Resistivity at Elevated Temperature, ohm-cm
Thermal
CTE, linear 20°C, µm/m-°C
CTE, linear 250°C, µm/m-°C
CTE, linear 500°C, µm/m-°C
CTE, linear 500°C, µm/m-°C
Specific Heat Capacity, J/g-°C
Thermal Conductivity, W/m-K
Min
0.37
0.7
Max
0.43
0.9
0.2
0.3
96
0.7
1.83
0.035
0.04
0.23
10.4
12.6
13.7
13.9
0.475
44.5
35
Max
Comment
-- density is in lb/in^3 for english units
all stresses are in ksi for english units
Typical for steel.
Calculated
annealed and cold drawn. Based on 100%
machinability for AISI 1212 steel.
Estimated from elastic modulus
specimen oil hardened, 630°C (1110°F) temper
1.88% Ni, normalized, tempered
1.88% Ni, normalized and tempered
1.90% Ni, quenched, tempered
Typical 4000 series steel
Typical steel
10. Appendix B – Main Cylindrical Section ANSYS Macro
!This macro recreates the main cylindrical section without stiffeners
!and runs an Eigenvalue Buckling Analysis with an element size of 1 for
!the first 7 modes
!
!Author: Harvey C. Lee
!Date created: March 17, 2007
!
!Directions: Create this macro and call it
!create_cylinder&run_eigenbuckling.mac. Then launch ANSYS and in the
!command prompt, type create_cylinder&run_eigenbuckling
!
/COM,ANSYS RELEASE 10.0A1 UP20060105
12:46:41
03/14/2007
!*
!*
/NOPR
/PMETH,OFF,0
KEYW,PR_SET,1
KEYW,PR_STRUC,1
KEYW,PR_THERM,0
KEYW,PR_FLUID,0
KEYW,PR_MULTI,0
/GO
!*
/COM,
/COM,Preferences for GUI filtering have been set to display:
/COM, Structural
!*
/PREP7
!*
ET,1,SHELL181
!*
KEYOPT,1,1,0
KEYOPT,1,3,2
KEYOPT,1,8,0
KEYOPT,1,9,0
KEYOPT,1,10,0
!*
R,1,4/12, , , , , ,
RMORE, , , , , , ,
!*
MPREAD,'matprop','mp',' '
csys,0
K,1,0,0,0,
K,2,0,0,12,
K,3,0,0,24,
K,4,0,0,36,
K,5,0,6,0,
kplot
LSTR,
1,
2
LSTR,
2,
3
LSTR,
3,
4
!
36
FLST,2,1,3,ORDE,1
FITEM,2,5
FLST,8,2,3
FITEM,8,1
FITEM,8,2
LROTAT,P51X, , , , , ,P51X,
!
FLST,2,4,4,ORDE,2
FITEM,2,4
FITEM,2,-7
ADRAG,P51X, , , , , ,
!
FLST,2,4,4,ORDE,4
FITEM,2,8
FITEM,2,11
FITEM,2,13
FITEM,2,15
ADRAG,P51X, , , , , ,
!
FLST,2,4,4,ORDE,4
FITEM,2,16
FITEM,2,19
FITEM,2,21
FITEM,2,23
ADRAG,P51X, , , , , ,
!
/REPLOT
!
/SOLU
FLST,2,8,4,ORDE,6
FITEM,2,4
FITEM,2,-7
FITEM,2,24
FITEM,2,27
FITEM,2,29
FITEM,2,31
!*
/GO
DL,P51X, ,UX,0
FLST,2,8,4,ORDE,6
FITEM,2,4
FITEM,2,-7
FITEM,2,24
FITEM,2,27
FITEM,2,29
FITEM,2,31
!*
/GO
DL,P51X, ,UY,0
FLST,2,2,3,ORDE,2
FITEM,2,6
FITEM,2,8
!*
/GO
DK,P51X, ,0, ,1,UZ, , , , ,
!
,360,4,
1
2
3
,
37
FLST,2,2,3,ORDE,2
FITEM,2,18
FITEM,2,20
!*
/GO
DK,P51X, ,0, ,1,UZ, , , , , ,
!
/VIEW,1,,,-1
/ANG,1
/REP,FAST
/prep7
/TITLE,Cylindrical Hull Section (Esize = 1)
!*
TYPE,
1
MAT,
1
REAL,
1
ESYS,
0
!
esize,1
!*
amesh,all
csys,1
nrotat,all
sfe,all,2,pres,,12000,,,
/SOLU
SBCTRAN
!
/DIST, 1,
27.1280083138
/FOC,
1, -4.93790132953
,
4.04348334897
/VIEW, 1, -0.446499709800
, 0.488816565998
/ANG,
1, 0.415875984041
/DIST,1,0.924021086472,1
!
/PSF,PRES,NORM,2,0,1
/PBF,TEMP, ,1
/PIC,DEFA, ,1
/PSYMB,CS,0
/PSYMB,NDIR,0
/PSYMB,ESYS,0
/PSYMB,LDIV,0
/PSYMB,LDIR,0
/PSYMB,ADIR,0
/PSYMB,ECON,0
/PSYMB,XNODE,0
/PSYMB,DOT,1
/PSYMB,PCONV,
/PSYMB,LAYR,0
/PSYMB,FBCS,0
!*
/PBC,ALL,,1
/PBC,NFOR,,0
/PBC,NMOM,,0
/PBC,RFOR,,0
/PBC,RMOM,,0
/PBC,PATH,,0
!*
38
,
16.2225589785
, -0.749464057814
/AUTO,1
/REP,FAST
!
eplot
/replot
FINISH
! Run the Eigenvalue Buckling Analysis for the first 7 modes
/SOL
!*
allsel
ANTYPE,0
pstres,on
solve
!*
FINISH
/SOLUTION
ANTYPE,1
BUCOPT,LANB,7,0,0
MXPAND,7,0,100000,1,0.001,
solve
FINISH
/POST1
allsel
eplot
SET,FIRST
rsys,1
/contour,0,12
plnsol,u,x,0,1
/ANG,1
/REP,FAST
/DIST,1,1.37174211248,1
/STAT,GLOBAL
FINISH
39
11. Appendix C – Submarine ANSYS Macro
!This macro recreates the submarine and runs an Eigenvalue Buckling
!Analysis with an element size of 1 for the first 7 modes of which the
!2nd mode (2ND) is of interest
!
!Author: Harvey C. Lee
!Date created: March 17, 2007
!
!Directions: Create this macro and call it
!create_sub&run_eigenbuckling.mac. Then launch ANSYS and in the command
!prompt, type create_sub&run_eigenbuckling
!
/COM,ANSYS RELEASE 10.0A1 UP20060105
12:46:41
03/14/2007
!*
!*
/NOPR
/PMETH,OFF,0
KEYW,PR_SET,1
KEYW,PR_STRUC,1
KEYW,PR_THERM,0
KEYW,PR_FLUID,0
KEYW,PR_MULTI,0
/GO
!*
/COM,
/COM,Preferences for GUI filtering have been set to display:
/COM, Structural
!*
/PREP7
!*
ET,1,SHELL181
!*
KEYOPT,1,1,0
KEYOPT,1,3,2
KEYOPT,1,8,0
KEYOPT,1,9,0
KEYOPT,1,10,0
!*
R,1,4/12, , , , , ,
R,2,6/12, , , , , ,
R,3,1, , , , , ,
RMORE, , , , , , ,
!*
MPREAD,'matprop','mp',' '
csys,0
K,1,0,0,0,
K,2,0,0,12,
K,3,0,0,24,
K,4,0,0,36,
K,5,0,6,0,
kplot
LSTR,
1,
2
LSTR,
2,
3
40
LSTR,
3,
4
!
FLST,2,1,3,ORDE,1
FITEM,2,5
FLST,8,2,3
FITEM,8,1
FITEM,8,2
LROTAT,P51X, , , , , ,P51X,
!
FLST,2,4,4,ORDE,2
FITEM,2,4
FITEM,2,-7
ADRAG,P51X, , , , , ,
!
FLST,2,4,4,ORDE,4
FITEM,2,8
FITEM,2,11
FITEM,2,13
FITEM,2,15
ADRAG,P51X, , , , , ,
!
FLST,2,4,4,ORDE,4
FITEM,2,16
FITEM,2,19
FITEM,2,21
FITEM,2,23
ADRAG,P51X, , , , , ,
!
/VIEW,1,,,-1
/ANG,1
/REP,FAST
/replot
!
!
!
/PREP7
csys,1
LSTR,
5,
1
LSTR,
1,
7
LSTR,
1,
8
LSTR,
1,
6
LSTR,
17,
4
LSTR,
4,
19
LSTR,
4,
20
LSTR,
4,
18
!
FLST,3,2,3,ORDE,2
FITEM,3,9
FITEM,3,13
KGEN,2,P51X, , ,-1, , , ,0
LSTR,
9,
21
LSTR,
13,
22
!
ADRAG,
40, , , , , ,
ADRAG,
42, , , , , ,
ADRAG,
45, , , , , ,
,360,4,
1
2
3
8
11
13
41
ADRAG,
48, , , , , ,
ADRAG,
41, , , , , ,
ADRAG,
54, , , , , ,
ADRAG,
57, , , , , ,
ADRAG,
60, , , , , ,
!
FLST,2,3,4
FITEM,2,32
FITEM,2,7
FITEM,2,34
AL,P51X
FLST,2,3,4
FITEM,2,34
FITEM,2,6
FITEM,2,33
AL,P51X
FLST,2,3,4
FITEM,2,33
FITEM,2,5
FITEM,2,35
AL,P51X
FLST,2,3,4
FITEM,2,35
FITEM,2,4
FITEM,2,32
AL,P51X
FLST,2,3,4
FITEM,2,36
FITEM,2,31
FITEM,2,38
AL,P51X
FLST,2,3,4
FITEM,2,38
FITEM,2,29
FITEM,2,37
AL,P51X
FLST,2,3,4
FITEM,2,37
FITEM,2,27
FITEM,2,39
AL,P51X
FLST,2,3,4
FITEM,2,39
FITEM,2,24
FITEM,2,36
AL,P51X
aplot
!
FLST,3,1,3,ORDE,1
FITEM,3,4
KGEN,2,P51X, , , , ,8, ,1
kplott,,,,,,,,,1
FLST,3,1,3,ORDE,1
FITEM,3,39
KGEN,2,P51X, , , , ,8, ,1
kplott,,,,,,,,,1
15
16
19
21
23
42
LSTR,
4,
LSTR,
39,
/replot
lplot
!
FLST,3,1,3,ORDE,1
FITEM,3,40
KGEN,2,P51X, , ,2,
FLST,3,1,3,ORDE,1
FITEM,3,40
!
LSTR,
40,
FLST,2,1,4,ORDE,1
FITEM,2,68
FLST,8,2,3
FITEM,8,39
FITEM,8,40
AROTAT,P51X, , , ,
!
FLST,3,1,3,ORDE,1
FITEM,3,39
KGEN,2,P51X, , ,4,
FLST,2,1,3,ORDE,1
FITEM,2,45
FLST,8,2,3
FITEM,8,4
FITEM,8,39
LROTAT,P51X, , , ,
!
LSTR,
17,
LSTR,
46,
LSTR,
20,
LSTR,
45,
LSTR,
19,
LSTR,
48,
LSTR,
18,
LSTR,
47,
/replot
FLST,2,4,4
FITEM,2,31
FITEM,2,82
FITEM,2,76
FITEM,2,80
AL,P51X
FLST,2,4,4
FITEM,2,81
FITEM,2,76
FITEM,2,83
FITEM,2,72
AL,P51X
FLST,2,4,4
FITEM,2,24
FITEM,2,80
FITEM,2,77
FITEM,2,86
AL,P51X
39
40
, , ,1
41
, ,P51X, ,360,4,
, , ,1
, ,P51X, ,360,4,
46
42
45
41
48
44
47
43
43
FLST,2,4,4
FITEM,2,77
FITEM,2,81
FITEM,2,73
FITEM,2,87
AL,P51X
FLST,2,4,4
FITEM,2,82
FITEM,2,29
FITEM,2,84
FITEM,2,79
AL,P51X
FLST,2,4,4
FITEM,2,79
FITEM,2,85
FITEM,2,75
FITEM,2,83
AL,P51X
FLST,2,4,4
FITEM,2,86
FITEM,2,27
FITEM,2,84
FITEM,2,78
AL,P51X
FLST,2,4,4
FITEM,2,87
FITEM,2,78
FITEM,2,85
FITEM,2,74
AL,P51X
!
FLST,3,1,3,ORDE,1
FITEM,3,46
KGEN,2,P51X, , ,-1, , , ,1
LSTR,
46,
49
ADRAG,
88, , , , , ,
ADRAG,
89, , , , , ,
ADRAG,
92, , , , , ,
ADRAG,
95, , , , , ,
!
FLST,3,4,3,ORDE,2
FITEM,3,41
FITEM,3,-44
KGEN,2,P51X, , ,6, , , ,1
kplott,,,,,,,,,1
!
FLST,3,4,3,ORDE,2
FITEM,3,58
FITEM,3,-61
KGEN,2,P51X, , , , ,-3, ,1
LSTR,
59,
63
LSTR,
58,
62
LSTR,
61,
65
LSTR,
60,
64
lplot
LSTR,
42,
59
77
78
79
76
44
LSTR,
41,
58
LSTR,
44,
61
LSTR,
43,
60
LSTR,
63,
46
LSTR,
62,
45
LSTR,
65,
48
LSTR,
64,
47
NUMMRG,KP,.001,.001, ,LOW
/replot
FLST,2,4,4
FITEM,2,105
FITEM,2,101
FITEM,2,109
FITEM,2,81
AL,P51X
FLST,2,4,4
FITEM,2,106
FITEM,2,102
FITEM,2,110
FITEM,2,83
AL,P51X
FLST,2,4,4
FITEM,2,107
FITEM,2,103
FITEM,2,111
FITEM,2,85
AL,P51X
FLST,2,4,4
FITEM,2,108
FITEM,2,104
FITEM,2,112
FITEM,2,87
AL,P51X
aplot
!
FLST,3,1,3,ORDE,1
FITEM,3,1
KGEN,2,P51X, , , , ,-9, ,1
kplott,,,,,,,,,1
LSTR,
1,
23
!
csys,0
! Create Nose
K,next,0,5.963,-1
K,next,0,5.850,-2
K,next,0,5.657,-3
K,next,0,5.375,-4
K,next,0,4.989,-5
K,next,0,4.472,-6
K,next,0,3.771,-7
K,next,0,3.317,-7.5
K,next,0,2.749,-8
K,next,0,2.398,-8.25
K,next,0,1.972,-8.5
K,next,0,1.404,-8.75
K,next,0,1.258,-8.8
45
K,next,0,1.091,-8.85
K,next,0,0.892,-8.9
K,next,0,0.632,-8.95
K,next,0,0.000,-9
!
FLST,3,18,3
FITEM,3,5
FITEM,3,25
FITEM,3,27
FITEM,3,29
FITEM,3,30
FITEM,3,31
FITEM,3,33
FITEM,3,35
FITEM,3,37
FITEM,3,38
FITEM,3,50
FITEM,3,52
FITEM,3,54
FITEM,3,56
FITEM,3,57
FITEM,3,66
FITEM,3,67
FITEM,3,68
BSPLIN, ,P51X
/replot
!
FLST,2,1,4,ORDE,1
FITEM,2,46
FLST,8,2,3
FITEM,8,1
FITEM,8,23
AROTAT,P51X, , , , , ,P51X, ,360,4,
!
NUMMRG,KP,0.001,0.001, ,LOW
lplott
!
FLST,5,28,5,ORDE,6
FITEM,5,1
FITEM,5,-12
FITEM,5,33
FITEM,5,-40
FITEM,5,45
FITEM,5,-52
ASEL,R, , ,P51X
lsla
ksll
!
cm,externalshell.a,area
! Define area attributes
FLST,5,8,5,ORDE,2
FITEM,5,21
FITEM,5,-28
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
46
CMSEL,S,_Y
!*
CMSEL,S,_Y1
AATT,
1,
3,
1,
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
!* Define area attributes
FLST,5,16,5,ORDE,6
FITEM,5,13
FITEM,5,-20
FITEM,5,29
FITEM,5,-32
FITEM,5,41
FITEM,5,-44
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
CMSEL,S,_Y
!*
CMSEL,S,_Y1
AATT,
1,
2,
1,
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
! Define area attributes
cmsel,s,externalshell.a
lsla
ksll
aplot
FLST,5,28,5,ORDE,6
FITEM,5,1
FITEM,5,-12
FITEM,5,33
FITEM,5,-40
FITEM,5,45
FITEM,5,-52
CM,_Y,AREA
ASEL, , , ,P51X
CM,_Y1,AREA
CMSEL,S,_Y
!*
CMSEL,S,_Y1
AATT,
1,
1,
1,
CMSEL,S,_Y
CMDELE,_Y
CMDELE,_Y1
! Create mesh
allsel,all
ESIZE,1
MSHKEY,1
amesh,all
!* Reverse area normals
asel,s,,,21
asel,a,,,25
asel,a,,,29
0,
0,
0,
47
asel,a,,,30
asel,a,,,31
asel,a,,,32
asel,a,,,39
asel,a,,,40
asel,a,,,47
asel,a,,,48
lsla
ksll
esla
nsle
AREVERSE,all
!
FINISH
/SOL
FLST,2,1,3,ORDE,1
FITEM,2,40
!*
/GO
DK,P51X, ,0, ,1,ALL, , , , , ,
FINISH
/PREP7
allsel
csys,1
nrotat,all
!
FLST,5,28,5,ORDE,6
FITEM,5,1
FITEM,5,-12
FITEM,5,29
FITEM,5,-40
FITEM,5,49
FITEM,5,-52
ASEL,R, , ,P51X
esla
nsle
eplot
!
cm,externalshell.e,elements
!
sfe,all,2,pres,,12000,,,
!
FINISH
! Run the Eigenvalue Buckling Analysis for the first 7 modes
/SOL
!*
allsel
ANTYPE,0
pstres,on
solve
!*
FINISH
/SOLUTION
ANTYPE,1
BUCOPT,LANB,7,0,0
MXPAND,7,0,100000,1,0.001,
48
solve
FINISH
/POST1
allsel
eplot
SET,FIRST
SET,NEXT
rsys,1
/contour,0,12
plnsol,u,x,0,1
/ANG,1
/REP,FAST
/DIST,1,1.37174211248,1
/DIST, 1,
27.1280083138
/FOC,
1, -4.93790132953
/VIEW, 1, -0.446499709800
/ANG,
1, 0.415875984041
/DIST,1,0.924021086472,1
/REP,FAST
/STAT,GLOBAL
FINISH
,
,
4.04348334897
0.488816565998
49
,
16.2225589785
, -0.749464057814
12. Appendix D – Main Cylindrical Section Eigenvalue
Buckling Results
Buckled mode shape for Element size = 6 (DOF = 336)
Buckled mode shape for Element size = 5 (DOF = 480)
50
Buckled mode shape for Element size = 4 (DOF = 720)
Buckled mode shape for Element size = 3 (DOF = 1,248)
51
Buckled mode shape for Element size = 2 (DOF = 2,280)
52
13. Appendix E - Main Cylindrical Section Nonlinear
Buckling Results
Buckled mode shape for OOR = 1”
Buckled mode shape for OOR = 2”
53
Buckled mode shape for OOR = 3”
Buckled mode shape for OOR = 4”
54
14. Appendix F – Submarine Eigenvalue Buckling Results
Buckled mode shape for Element size = 2 (DOF = 7,512)
Buckled mode shape for Element size = 3 (DOF = 3,384)
55
Buckled mode shape for Element size = 4 (DOF = 3,000)
Buckled mode shape for Element size = 5 (DOF = 2,760)
56
Buckled mode shape for Element size = 6 (DOF = 1,032)
57
15. Appendix G - Submarine Nonlinear Buckling Results
Buckled mode shape for OOR = 1”
Hull stresses for OOR = 1”
58
Internal stiffener stresses for OOR = 1”
Buckled mode shape for OOR = 2”
59
Hull stresses for OOR = 2”
Internal stiffener stresses for OOR = 2”
60
Buckled mode shape for OOR = 3”
Hull stresses for OOR = 3”
61
Internal stiffener stresses for OOR = 3”
Buckled mode shape for OOR = 4”
62
Hull stresses for OOR = 4”
Internal stiffener stresses for OOR = 4”
63
Southwell Plot
OOR = 1"
Defl / Pressure (in. / psi)
3.00E-05
2.50E-05
y = 0.0000987x + 0.0000012
R2 = 0.9972790
2.00E-05
1.50E-05
1.00E-05
5.00E-06
0.00E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
Deflection (in.)
Southwell Plot for OOR = 1” (Pcrit = 10,132 psi)
Southwell Plot
OOR = 2"
Defl / Pressure (in. / psi)
3.00E-05
2.50E-05
y = 0.0000989x + 0.0000046
R2 = 0.9978519
2.00E-05
1.50E-05
1.00E-05
5.00E-06
0.00E+00
0
0.05
0.1
0.15
Deflection (in.)
Southwell Plot for OOR = 2” (Pcrit = 10,111 psi)
64
0.2
0.25
Southwell Plot
OOR = 3"
Defl / Pressure (in. / psi)
3.50E-05
y = 0.0000960x + 0.0000081
R2 = 0.9992345
3.00E-05
2.50E-05
2.00E-05
1.50E-05
1.00E-05
5.00E-06
0.00E+00
0
0.05
0.1
0.15
0.2
0.25
0.3
Deflection (in.)
Southwell Plot for OOR = 3” (Pcrit = 10,417 psi)
Southwell Plot
OOR = 4"
Defl / Pressure (in. / psi)
4.00E-05
3.50E-05
y = 0.0000949x + 0.0000116
R2 = 0.9996126
3.00E-05
2.50E-05
2.00E-05
1.50E-05
1.00E-05
5.00E-06
0.00E+00
0
0.05
0.1
0.15
Deflection (in.)
Southwell Plot for OOR = 4” (Pcrit = 10,537 psi)
65
0.2
0.25
16. Appendix H - Submarine Nonlinear Buckling Results with
Plasticity
Buckled mode shape for OOR = 1”
Hull stresses for OOR = 1”
66
Internal stiffener strains for OOR = 1”
Buckled mode shape for OOR = 2”
67
Hull stresses for OOR = 2”
Internal stiffener strains for OOR 2”
68
Buckled mode shape for OOR = 3”
Hull stresses for OOR = 3”
69
Internal stiffener strains for OOR = 3”
Buckled mode shape for OOR = 4”
70
Hull stresses for OOR = 4”
Internal stiffener strains for OOR = 4”
71