Buckling Analysis of a Submarine With Hull Imperfections

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

ii

© Copyright 2007

by

Harvey C. Lee

All Rights Reserved

iii

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

iv

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

v

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

vi

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

vii

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

viii

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)

ix

ACKNOWLEDGMENT

To my loving wife Jennifer, whose very patience and

unwavering support, has encouraged me to bring this paper to its

final completion.

x

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.

1

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

2

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

3

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.

4

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

5

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-of-

roundness 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.

6

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 + ρρρρgh

Equation 1 – Bernoulli’s Equation

where P = Ocean Depth Pressure

Po = Atmospheric Pressure

7

ρ = 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

8

3. Eigenvalue Buckling Analysis of the Main CylindricalSection

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).

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

ν = Poisson’s ratio

m = Nodal diameters

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),

π r

lλ =

t

12 r 2

2

k =&

Pcrit = Et (1 - υ )λ + k [(λ + m ) - 2 (υλ + 3λ m + (4 - υ)λ m + m ) + 2 (2 - υ) λ m + m ]

r (1 - υ ) m (λ + m ) - m (3λ + m ) { }2

4 62 2 2 2 2 2 2

2 2 2

4 4 4 46

2 2 22

9

confirming Flugge’s theoretical equation. As a result, the FEA model of the main

cylindrical section of our submarine has been calibrated.

Table 1 – Main Cylindrical Section Eigenvalue Buckling Results

Figure 2 – FEA model of the main cylindrical section with boundary conditions

Esize DOF Pcrit (psi) Flugge (psi) Error

6 336 5,442 4,097 32.82%

5 480 5,356 4,097 30.73%

4 720 4,555 4,097 11.17%

3 1248 4,319 4,097 5.41%

2 2280 4,211 4,097 2.78%

1 8880 4,093 4,097 0.09%

Isometric View

Side View

Front View

10

Figure 3 – Convergence of main cylindrical section Eigenvalue Buckling

results

Figure 4 – Buckled mode shape of 2 nodal diameters of the main cylindrical

section

Cylindrical Hull Section Eigenbuckling Results

0

1,000

2,000

3,000

4,000

5,000

6,000

0 2000 4000 6000 8000 10000

DOF

Pcrit (psi)

11

4. Nonlinear Large Displacement Static Buckling Analysis ofthe 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.

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.

e

ee

e

Eg: If OOR = 4” then e = 2”

12

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.

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-of-

roundness condition. To reiterate, these displacement scales are in feet.

Figure 6 – Nonlinear Buckling of the main cylindrical section with 4” OOR

OOR (in.) ANSYS (psi) Southwell (psi)

0 4,093 4,093 Eigenvalue

1 3,591 4,000

2 3,324 3,894

3 3,117 3,711

4 2,898 3,619

13

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.

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-of-

roundness 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.

Southwell PlotOOR = 4"

y = 0.0002763x + 0.0000447

R2 = 0.9989078

0.00E+00

5.00E-05

1.00E-04

1.50E-04

2.00E-04

2.50E-04

3.00E-04

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Deflection (in.)

Defl / Pressure (in. / psi)

14

Figure 8 – Comparison of ANSYS and Southwell method in determining the

critical buckling pressure of the main cylindrical section as a function of OOR

Critical Buckling Pressure vs. OOR

2500

3000

3500

4000

4500

0 1 2 3 4 5

OOR (in.)

Critical

Buckling Pressure (psi)

ANSYS

Southwell

15

5. Buckling Analysis of the Submarine

With the main cylindrical section FEA model calibrated and the effects of out-of-

roundness 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.

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

All DOF = 0

16

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.

Table 3 – Eigenvalue Buckling results of the submarine

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

Submarine Eigenbuckling Results

0

5000

10000

15000

20000

25000

30000

0 10000 20000 30000

DOF

Pcrit (psi)

Esize DOF Pcrit (psi) Error

6 1,032 23,855 112.62%

5 2,760 12,924 15.19%

4 3,000 12,905 15.02%

3 3,384 12,508 11.48%

2 7,512 11,678 4.09%

1 28,200 11,219 0.00%

17

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.

18

Figure 12 – Buckled mode shape of main cylindrical section with internal

stiffeners

Figure 13 – Main cylindrical section OOR of 4” with eccentricities shown

+ 0.166667- 0.166667

- 0.166667+ 0.166667

19

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.

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

OOR (in.) ANSYS (psi) Southwell (psi)

0 11,219 11,219 Eigenvalue

1 9,796 10,132

2 8,450 10,111

3 7,950 10,417

4 6,950 10,537

20

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

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,


y = 0.0000987x + 0.0000012

R2 = 0.9972790

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

0 0.05 0.1 0.15 0.2 0.25 0.3

Deflection (in.)


21

which is a graphical plot of Table 4, the critical buckling pressure for the out-of-

roundness of 0” is based on the Eigenvalue Buckling analysis.

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.

Pritical vs. OOR

0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5

OOR (in.)

Pcritical (psi)

ANSYS

Southwell

22

Figure 17 – Graph of Bernoulli’s equation plotting ocean pressure against depth

Table 5 - Submarine depth capability vs. hull out-of-roundness

Pressure of Ocean Water at Depth

14.7

2304

4593

6882

9171

11460

13749

16038

18327

20616

22905

0

5000

10000

15000

20000

25000

0 1 2 3 4 5 6 7 8 9 10 11

Depth (mi)

Pressure (psi)

ANSYS

OOR (in.) Pcrit (psi) Depth Capability

0 11,219 4.9 miles

1 9,796 4.3 miles

2 8,450 3.7 miles

3 7,950 3.5 miles

4 6,950 3.0 miles

23

Figure 18 - Submarine depth capability vs. hull out-of-roundness

Submarine Depth Capabilty vs Hull OOR

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5

Hull OOR (in.)

Ocean Depth (mi)

24

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

25

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:

εεεεTrue = ln (1 + εεεεEng)

Equation 4 – Relation between True Strain and Engineering Strain

σσσσTrue = σσσσEng (1 + εεεεEng)

Equation 5 – Relation between True Stress and Engineering Stress

Figure 21 – Bilinear True Stress-Strain Curve for AISI 4340 Steel

Bilinear True Stress-Strain Curve

0

50000

100000

150000

200000

250000

300000

0 0.05 0.1 0.15

Strain (in./in.)

Stress (psi)

26

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.

Table 6 – Submarine buckling results

OOR (in.) Elastic (psi) Elastic-Plastic (psi)

0 11,219 11,219 Eigenvalue

1 9,796 8,262

2 8,450 7,166

3 7,950 6,3304 6,950 5,724

ANSYS Nonlinear Large Displacement Static

27

Figure 23 – Submarine buckling strength as a function of out-of-roundness

Figure 24 – Buckled mode shape for 4” OOR with elastic-plastic material

Pcritical vs. OOR

0

2000

4000

6000

8000

10000

12000

0 1 2 3 4 5

OOR (in.)

Pcritical (psi)

ANSYS - Elastic

ANSYS - Elastic-Plastic

28

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.

29

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.

Table 7 - Submarine depth capability vs. hull out-of-roundness with plasticity


OOR (in.) Pcrit (psi) Depth Capabilty

0 11,219 4.9 miles

1 8,262 3.6 miles

2 7,166 3.1 miles

3 6,330 2.8 miles

4 5,724 2.5 miles

30

Figure 27 - Submarine depth capability vs. hull out-of-roundness (Final Summary)

Submarine Ocean Depth Capability vs. Hull OOR

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0 1 2 3 4 5

Hull OOR (in.)

Ocean Depth (mi)

ANSYS - Elastic


31

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-of-

roundness, 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.

32

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.

33

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.

34

9. Appendix A – Material Properties

KeyWords:

SubCat: Low Alloy Steel, AISI 4000 Series Steel,

Medium Carbon Steel, Metal, Ferrous Metal

Component Value Min Max

Carbon, C 0.37 0.43

Chromium, Cr 0.7 0.9

Iron, Fe 96

Manganese, Mn 0.7

Molybdenum, Mo 0.2 0.3

Nickel, Ni 1.83

Phosphorous, P 0.035

Sulfur, S 0.04

Silicon, Si 0.23

Properties Metric English

Physical Value Value Min Max Comment

Density, g/cc 7.85 0.284 -- -- density is in lb/in^3 for english units

Mechanical

Tensile Strength, Ultimate, MPa 1595 231 -- -- all stresses are in ksi for english units

Tensile Strength, Yield, MPa 1475 214 -- --

Elongation at Break, % 12 12 -- --

Reduction of Area, % 46 46 -- --

Modulus of Elasticity, GPa 212 30700 -- --

Bulk Modulus, GPa 140 20300 -- -- Typical for steel.

Poissons Ratio 0.3 0.3 -- -- Calculated

Machinability, % 50 50 -- --

Shear Modulus, GPa 81.5 11800 -- -- Estimated from elastic modulus

Electrical

Electrical Resistivity, ohm-cm 2.48E-05 -- -- --

Electrical Resistivity at Elevated Temperature, ohm-cm 5.52E-05 -- -- --



Thermal

CTE, linear 20°C, µm/m-°C 12.7 -- -- -- specimen oil hardened, 600°C (1110°F) temper



CTE, linear 250°C, µm/m-°C 12.6 -- -- -- 1.88% Ni, normalized, tempered

CTE, linear 500°C, µm/m-°C 13.7 -- -- -- 1.88% Ni, normalized and tempered

CTE, linear 500°C, µm/m-°C 13.9 -- -- -- 1.90% Ni, quenched, tempered


Specific Heat Capacity, J/g-°C 0.475 -- -- -- Typical 4000 series steel

Thermal Conductivity, W/m-K 44.5 -- -- -- Typical steel

AISI 4340 Steel, oil quenched 845°C, 425°C

(800°F) temper, tested at 25°C (77°F)

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

annealed and cold drawn. Based on 100%

machinability for AISI 1212 steel.

Date: 2/10/2007 2:21:07 PM

35

KeyWords:

SubCat: Low Alloy Steel, AISI 4000 Series Steel,

Medium Carbon Steel, Metal, Ferrous Metal

Component Value Min Max

Carbon, C 0.37 0.43

Chromium, Cr 0.7 0.9

Iron, Fe 96

Manganese, Mn 0.7

Molybdenum, Mo 0.2 0.3

Nickel, Ni 1.83

Phosphorous, P 0.035

Sulfur, S 0.04

Silicon, Si 0.23

Properties Metric English

Physical Value Value Min Max Comment

Density, g/cc 7.85 0.284 -- -- density is in lb/in^3 for english units

Mechanical

Tensile Strength, Ultimate, MPa 1985 288 -- -- all stresses are in ksi for english units

Tensile Strength, Yield, MPa 1840 267 -- --

Elongation at Break, % 4 4 -- --

Reduction of Area, % 11 11 -- --

Modulus of Elasticity, GPa 213 30900 -- --

Bulk Modulus, GPa 140 20300 -- -- Typical for steel.

Poissons Ratio 0.3 0.3 -- -- Calculated

Machinability, % 50 50 -- --

Shear Modulus, GPa 82 11900 -- -- Estimated from elastic modulus

Electrical

Electrical Resistivity, ohm-cm 2.48E-05 2.48E-05 -- --

Electrical Resistivity at Elevated Temperature, ohm-cm 2.98E-05 2.98E-05 -- --



Thermal

CTE, linear 20°C, µm/m-°C 10.4 -- -- specimen oil hardened, 630°C (1110°F) temper

CTE, linear 250°C, µm/m-°C 12.6 -- -- 1.88% Ni, normalized, tempered

CTE, linear 500°C, µm/m-°C 13.7 -- -- 1.88% Ni, normalized and tempered

CTE, linear 500°C, µm/m-°C 13.9 -- -- 1.90% Ni, quenched, tempered

Specific Heat Capacity, J/g-°C 0.475 -- -- Typical 4000 series steel

Thermal Conductivity, W/m-K 44.5 -- -- Typical steel

annealed and cold drawn. Based on 100%

machinability for AISI 1212 steel.

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

Date: 2/10/2007 2:27:56 PM

AISI 4340 Steel, oil quenched 845°C, 425°C

(800°F) temper, tested at -195C

36

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,0KEYW,PR_SET,1KEYW,PR_STRUC,1KEYW,PR_THERM,0KEYW,PR_FLUID,0KEYW,PR_MULTI,0/GO!*/COM,/COM,Preferences for GUI filtering have been set to display:/COM, Structural!*/PREP7!*ET,1,SHELL181!*KEYOPT,1,1,0KEYOPT,1,3,2KEYOPT,1,8,0KEYOPT,1,9,0KEYOPT,1,10,0!*R,1,4/12, , , , , ,RMORE, , , , , , ,!*MPREAD,'matprop','mp',' 'csys,0K,1,0,0,0,K,2,0,0,12,K,3,0,0,24,K,4,0,0,36,K,5,0,6,0,kplotLSTR, 1, 2LSTR, 2, 3LSTR, 3, 4!

37

FLST,2,1,3,ORDE,1FITEM,2,5FLST,8,2,3FITEM,8,1FITEM,8,2LROTAT,P51X, , , , , ,P51X, ,360,4,!FLST,2,4,4,ORDE,2FITEM,2,4FITEM,2,-7ADRAG,P51X, , , , , , 1!FLST,2,4,4,ORDE,4FITEM,2,8FITEM,2,11FITEM,2,13FITEM,2,15ADRAG,P51X, , , , , , 2!FLST,2,4,4,ORDE,4FITEM,2,16FITEM,2,19FITEM,2,21FITEM,2,23ADRAG,P51X, , , , , , 3!/REPLOT!/SOLUFLST,2,8,4,ORDE,6FITEM,2,4FITEM,2,-7FITEM,2,24FITEM,2,27FITEM,2,29FITEM,2,31!*/GODL,P51X, ,UX,0FLST,2,8,4,ORDE,6FITEM,2,4FITEM,2,-7FITEM,2,24FITEM,2,27FITEM,2,29FITEM,2,31!*/GODL,P51X, ,UY,0FLST,2,2,3,ORDE,2FITEM,2,6FITEM,2,8!*/GODK,P51X, ,0, ,1,UZ, , , , , ,!

38

FLST,2,2,3,ORDE,2FITEM,2,18FITEM,2,20!*/GODK,P51X, ,0, ,1,UZ, , , , , ,!/VIEW,1,,,-1/ANG,1/REP,FAST/prep7/TITLE,Cylindrical Hull Section (Esize = 1)!*TYPE, 1MAT, 1REAL, 1ESYS, 0!esize,1!*amesh,allcsys,1nrotat,allsfe,all,2,pres,,12000,,,/SOLUSBCTRAN!/DIST, 1, 27.1280083138/FOC, 1, -4.93790132953 , 4.04348334897 , 16.2225589785/VIEW, 1, -0.446499709800 , 0.488816565998 , -0.749464057814/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!*

39

/AUTO,1/REP,FAST!eplot/replotFINISH! Run the Eigenvalue Buckling Analysis for the first 7 modes/SOL!*allselANTYPE,0pstres,onsolve!*FINISH/SOLUTIONANTYPE,1BUCOPT,LANB,7,0,0MXPAND,7,0,100000,1,0.001,solveFINISH/POST1allseleplotSET,FIRSTrsys,1/contour,0,12plnsol,u,x,0,1/ANG,1/REP,FAST/DIST,1,1.37174211248,1/STAT,GLOBALFINISH

40

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,0KEYW,PR_SET,1KEYW,PR_STRUC,1KEYW,PR_THERM,0KEYW,PR_FLUID,0KEYW,PR_MULTI,0/GO!*/COM,/COM,Preferences for GUI filtering have been set to display:/COM, Structural!*/PREP7!*ET,1,SHELL181!*KEYOPT,1,1,0KEYOPT,1,3,2KEYOPT,1,8,0KEYOPT,1,9,0KEYOPT,1,10,0!*R,1,4/12, , , , , ,R,2,6/12, , , , , ,R,3,1, , , , , ,RMORE, , , , , , ,!*MPREAD,'matprop','mp',' 'csys,0K,1,0,0,0,K,2,0,0,12,K,3,0,0,24,K,4,0,0,36,K,5,0,6,0,kplotLSTR, 1, 2LSTR, 2, 3

41

LSTR, 3, 4!FLST,2,1,3,ORDE,1FITEM,2,5FLST,8,2,3FITEM,8,1FITEM,8,2LROTAT,P51X, , , , , ,P51X, ,360,4,!FLST,2,4,4,ORDE,2FITEM,2,4FITEM,2,-7ADRAG,P51X, , , , , , 1!FLST,2,4,4,ORDE,4FITEM,2,8FITEM,2,11FITEM,2,13FITEM,2,15ADRAG,P51X, , , , , , 2!FLST,2,4,4,ORDE,4FITEM,2,16FITEM,2,19FITEM,2,21FITEM,2,23ADRAG,P51X, , , , , , 3!/VIEW,1,,,-1/ANG,1/REP,FAST/replot!!!/PREP7csys,1LSTR, 5, 1LSTR, 1, 7LSTR, 1, 8LSTR, 1, 6LSTR, 17, 4LSTR, 4, 19LSTR, 4, 20LSTR, 4, 18!FLST,3,2,3,ORDE,2FITEM,3,9FITEM,3,13KGEN,2,P51X, , ,-1, , , ,0LSTR, 9, 21LSTR, 13, 22!ADRAG, 40, , , , , , 8ADRAG, 42, , , , , , 11ADRAG, 45, , , , , , 13

42

ADRAG, 48, , , , , , 15ADRAG, 41, , , , , , 16ADRAG, 54, , , , , , 19ADRAG, 57, , , , , , 21ADRAG, 60, , , , , , 23!FLST,2,3,4FITEM,2,32FITEM,2,7FITEM,2,34AL,P51XFLST,2,3,4FITEM,2,34FITEM,2,6FITEM,2,33AL,P51XFLST,2,3,4FITEM,2,33FITEM,2,5FITEM,2,35AL,P51XFLST,2,3,4FITEM,2,35FITEM,2,4FITEM,2,32AL,P51XFLST,2,3,4FITEM,2,36FITEM,2,31FITEM,2,38AL,P51XFLST,2,3,4FITEM,2,38FITEM,2,29FITEM,2,37AL,P51XFLST,2,3,4FITEM,2,37FITEM,2,27FITEM,2,39AL,P51XFLST,2,3,4FITEM,2,39FITEM,2,24FITEM,2,36AL,P51Xaplot!FLST,3,1,3,ORDE,1FITEM,3,4KGEN,2,P51X, , , , ,8, ,1kplott,,,,,,,,,1FLST,3,1,3,ORDE,1FITEM,3,39KGEN,2,P51X, , , , ,8, ,1kplott,,,,,,,,,1

43

LSTR, 4, 39LSTR, 39, 40/replotlplot!FLST,3,1,3,ORDE,1FITEM,3,40KGEN,2,P51X, , ,2, , , ,1FLST,3,1,3,ORDE,1FITEM,3,40!LSTR, 40, 41FLST,2,1,4,ORDE,1FITEM,2,68FLST,8,2,3FITEM,8,39FITEM,8,40AROTAT,P51X, , , , , ,P51X, ,360,4,!FLST,3,1,3,ORDE,1FITEM,3,39KGEN,2,P51X, , ,4, , , ,1FLST,2,1,3,ORDE,1FITEM,2,45FLST,8,2,3FITEM,8,4FITEM,8,39LROTAT,P51X, , , , , ,P51X, ,360,4,!LSTR, 17, 46LSTR, 46, 42LSTR, 20, 45LSTR, 45, 41LSTR, 19, 48LSTR, 48, 44LSTR, 18, 47LSTR, 47, 43/replotFLST,2,4,4FITEM,2,31FITEM,2,82FITEM,2,76FITEM,2,80AL,P51XFLST,2,4,4FITEM,2,81FITEM,2,76FITEM,2,83FITEM,2,72AL,P51XFLST,2,4,4FITEM,2,24FITEM,2,80FITEM,2,77FITEM,2,86AL,P51X

44

FLST,2,4,4FITEM,2,77FITEM,2,81FITEM,2,73FITEM,2,87AL,P51XFLST,2,4,4FITEM,2,82FITEM,2,29FITEM,2,84FITEM,2,79AL,P51XFLST,2,4,4FITEM,2,79FITEM,2,85FITEM,2,75FITEM,2,83AL,P51XFLST,2,4,4FITEM,2,86FITEM,2,27FITEM,2,84FITEM,2,78AL,P51XFLST,2,4,4FITEM,2,87FITEM,2,78FITEM,2,85FITEM,2,74AL,P51X!FLST,3,1,3,ORDE,1FITEM,3,46KGEN,2,P51X, , ,-1, , , ,1LSTR, 46, 49ADRAG, 88, , , , , , 77ADRAG, 89, , , , , , 78ADRAG, 92, , , , , , 79ADRAG, 95, , , , , , 76!FLST,3,4,3,ORDE,2FITEM,3,41FITEM,3,-44KGEN,2,P51X, , ,6, , , ,1kplott,,,,,,,,,1!FLST,3,4,3,ORDE,2FITEM,3,58FITEM,3,-61KGEN,2,P51X, , , , ,-3, ,1LSTR, 59, 63LSTR, 58, 62LSTR, 61, 65LSTR, 60, 64lplotLSTR, 42, 59

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LSTR, 41, 58LSTR, 44, 61LSTR, 43, 60LSTR, 63, 46LSTR, 62, 45LSTR, 65, 48LSTR, 64, 47NUMMRG,KP,.001,.001, ,LOW/replotFLST,2,4,4FITEM,2,105FITEM,2,101FITEM,2,109FITEM,2,81AL,P51XFLST,2,4,4FITEM,2,106FITEM,2,102FITEM,2,110FITEM,2,83AL,P51XFLST,2,4,4FITEM,2,107FITEM,2,103FITEM,2,111FITEM,2,85AL,P51XFLST,2,4,4FITEM,2,108FITEM,2,104FITEM,2,112FITEM,2,87AL,P51Xaplot!FLST,3,1,3,ORDE,1FITEM,3,1KGEN,2,P51X, , , , ,-9, ,1kplott,,,,,,,,,1LSTR, 1, 23!csys,0! Create NoseK,next,0,5.963,-1K,next,0,5.850,-2K,next,0,5.657,-3K,next,0,5.375,-4K,next,0,4.989,-5K,next,0,4.472,-6K,next,0,3.771,-7K,next,0,3.317,-7.5K,next,0,2.749,-8K,next,0,2.398,-8.25K,next,0,1.972,-8.5K,next,0,1.404,-8.75K,next,0,1.258,-8.8

46

K,next,0,1.091,-8.85K,next,0,0.892,-8.9K,next,0,0.632,-8.95K,next,0,0.000,-9!FLST,3,18,3FITEM,3,5FITEM,3,25FITEM,3,27FITEM,3,29FITEM,3,30FITEM,3,31FITEM,3,33FITEM,3,35FITEM,3,37FITEM,3,38FITEM,3,50FITEM,3,52FITEM,3,54FITEM,3,56FITEM,3,57FITEM,3,66FITEM,3,67FITEM,3,68BSPLIN, ,P51X/replot!FLST,2,1,4,ORDE,1FITEM,2,46FLST,8,2,3FITEM,8,1FITEM,8,23AROTAT,P51X, , , , , ,P51X, ,360,4,!NUMMRG,KP,0.001,0.001, ,LOWlplott!FLST,5,28,5,ORDE,6FITEM,5,1FITEM,5,-12FITEM,5,33FITEM,5,-40FITEM,5,45FITEM,5,-52ASEL,R, , ,P51Xlslaksll!cm,externalshell.a,area! Define area attributesFLST,5,8,5,ORDE,2FITEM,5,21FITEM,5,-28CM,_Y,AREAASEL, , , ,P51XCM,_Y1,AREA

47

CMSEL,S,_Y!*CMSEL,S,_Y1AATT, 1, 3, 1, 0,CMSEL,S,_YCMDELE,_YCMDELE,_Y1!* Define area attributesFLST,5,16,5,ORDE,6FITEM,5,13FITEM,5,-20FITEM,5,29FITEM,5,-32FITEM,5,41FITEM,5,-44CM,_Y,AREAASEL, , , ,P51XCM,_Y1,AREACMSEL,S,_Y!*CMSEL,S,_Y1AATT, 1, 2, 1, 0,CMSEL,S,_YCMDELE,_YCMDELE,_Y1! Define area attributescmsel,s,externalshell.alslaksllaplotFLST,5,28,5,ORDE,6FITEM,5,1FITEM,5,-12FITEM,5,33FITEM,5,-40FITEM,5,45FITEM,5,-52CM,_Y,AREAASEL, , , ,P51XCM,_Y1,AREACMSEL,S,_Y!*CMSEL,S,_Y1AATT, 1, 1, 1, 0,CMSEL,S,_YCMDELE,_YCMDELE,_Y1! Create meshallsel,allESIZE,1MSHKEY,1amesh,all!* Reverse area normalsasel,s,,,21asel,a,,,25asel,a,,,29

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asel,a,,,30asel,a,,,31asel,a,,,32asel,a,,,39asel,a,,,40asel,a,,,47asel,a,,,48lslakslleslansleAREVERSE,all!FINISH/SOLFLST,2,1,3,ORDE,1FITEM,2,40!*/GODK,P51X, ,0, ,1,ALL, , , , , ,FINISH/PREP7allselcsys,1nrotat,all!FLST,5,28,5,ORDE,6FITEM,5,1FITEM,5,-12FITEM,5,29FITEM,5,-40FITEM,5,49FITEM,5,-52ASEL,R, , ,P51Xeslansleeplot!cm,externalshell.e,elements!sfe,all,2,pres,,12000,,,!FINISH! Run the Eigenvalue Buckling Analysis for the first 7 modes/SOL!*allselANTYPE,0pstres,onsolve!*FINISH/SOLUTIONANTYPE,1BUCOPT,LANB,7,0,0MXPAND,7,0,100000,1,0.001,

49

solveFINISH/POST1allseleplotSET,FIRSTSET,NEXTrsys,1/contour,0,12plnsol,u,x,0,1/ANG,1/REP,FAST/DIST,1,1.37174211248,1/DIST, 1, 27.1280083138/FOC, 1, -4.93790132953 , 4.04348334897 , 16.2225589785/VIEW, 1, -0.446499709800 , 0.488816565998 , -0.749464057814/ANG, 1, 0.415875984041/DIST,1,0.924021086472,1/REP,FAST/STAT,GLOBALFINISH

50

12. Appendix D – Main Cylindrical Section EigenvalueBuckling Results

Buckled mode shape for Element size = 6 (DOF = 336)


51


Buckled mode shape for Element size = 3 (DOF = 1,248)

52


53

13. Appendix E - Main Cylindrical Section NonlinearBuckling Results

Buckled mode shape for OOR = 1”


54



55

14. Appendix F – Submarine Eigenvalue Buckling Results



56



57


58

15. Appendix G - Submarine Nonlinear Buckling Results


Hull stresses for OOR = 1”

59

Internal stiffener stresses for OOR = 1”


60



61



62



63



64

Southwell Plot for OOR = 1” (Pcrit = 10,132 psi)



y = 0.0000989x + 0.0000046

R2 = 0.9978519

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

0 0.05 0.1 0.15 0.2 0.25

Deflection (in.)



y = 0.0000987x + 0.0000012

R2 = 0.9972790

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

0 0.05 0.1 0.15 0.2 0.25 0.3

Deflection (in.)


65




y = 0.0000949x + 0.0000116

R2 = 0.9996126

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

4.00E-05

0 0.05 0.1 0.15 0.2 0.25

Deflection (in.)



y = 0.0000960x + 0.0000081

R2 = 0.9992345

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

0 0.05 0.1 0.15 0.2 0.25 0.3

Deflection (in.)


66

16. Appendix H - Submarine Nonlinear Buckling Results withPlasticity



67

Internal stiffener strains for OOR = 1”


68


Internal stiffener strains for OOR 2”

69



70



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Buckling Analysis of a Submarine With Hull Imperfections

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