Title: Modeling, analysis and optimization of cylindrical stiffened panels for reusable launch vehicle structures
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Title: Modeling, analysis and optimization of cylindrical stiffened panels for reusable launch vehicle structures
Alternate Title: Modeling, analysis and optimization of stiffened cylindrical panels for reusable launch vehicle structures
Physical Description: Book
Language: English
Creator: Venkataraman, Satchithanandam, 1969-
Publisher: State University System of Florida
Place of Publication: <Florida>
<Florida>
Publication Date: 1999
Copyright Date: 1999
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Subject: Aerospace Engineering, Mechanics and Engineering Science thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Aerospace Engineering, Mechanics and Engineering Science -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
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Summary: ABSTRACT: The design of reusable launch vehicles is driven by the need for minimum weight structures. Preliminary design of reusable launch vehicles requires many optimizations to select among competing structural concepts. Accurate models and analysis methods are required for such structural optimizations. Model, analysis, and optimization complexities have to be compromised to meet constraints on design cycle time and computational resources. Stiffened panels used in reusable launch vehicle tanks exhibit complex buckling failure modes. Using detailed finite element models for buckling analysis is too expensive for optimization. Many approximate models and analysis methods have been developed for design of stiffened panels. This dissertation investigates the use of approximate models and analysis methods implemented in PANDA2 software for preliminary design of stiffened panels. PANDA2 is also used for a trade study to compare weight efficiencies of stiffened panel concepts for a liquid hydrogen tank of a reusable launch vehicle. Optimum weights of stiffened panels are obtained for different materials, constructions and stiffener geometry. The study investigates the influence of modeling and analysis choices in PANDA2 on optimum designs. Complex structures usually require finite element analysis models to capture the details of their response. Design of complex structures must account for failure modes that are both global and local in nature. Often, different analysis models or computer programs are employed to calculate global and local structural response.
Summary: ABSTRACT (cont.): Integration of different analysis programs is cumbersome and computationally expensive. Response surface approximation provides a global polynomial approximation that filters numerical noise present in discretized analysis models. The computational costs are transferred from optimization to development of approximate models. Using this process, the analyst can create structural response models that can be used by designers in optimization. It allows easy integration of analysis models in optimization. The dissertation investigates use of response surface approximations for integrating structural response obtained from a global analysis in the local optimization of stiffened panels. In addition, response surfaces are used for correcting structural response predictions from a low fidelity model with a few expensive detailed finite element analyses.
Summary: KEYWORDS: structural optimization, reusable launch vehicles, stiffened panels, approximate analysis, buckling, response surface approximations, composite structures
Thesis: Thesis (Ph. D.)--University of Florida, 1999.
Bibliography: Includes bibliographical references (p. 180-192).
System Details: System requirements: World Wide Web browser and Adobe Acrobat Reader to view and print PDF files.
System Details: Mode of access: World Wide Web.
Statement of Responsibility: by Satchithanandam Venkataraman.
General Note: Title from first page of PDF file.
General Note: Document formatted into pages; contains xv, 209 p.; also contains graphics.
General Note: Vita.
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Bibliographic ID: UF00100712
Volume ID: VID00001
Source Institution: University of Florida
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MODELING, ANALYSIS AND OPTIMIZATION OF CYLINDRICAL STIFFENED
PANELS FOR REUSABLE LAUNCH VEHICLE STRUCTURES











By

SATCHITHANANDAM VENKATARAMAN


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE
UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1999
































This dissertation is dedicated to my parents, S. Venkataraman and Gowri Venkataraman.


















TABLE OF CONTENTS

A C K N O W L E D G E M EN T S ..................................................................... ..................... v

LIST OF TABLES ............................... ............ .............................vii

LIST OF FIGURES ...................................................... ii

A B S T R A C T ..................................................................................................................... x iv

CHAPTERS

1 IN TR O D U C TIO N ................................................................... ......... ..... ................

Sim ple M odels for Panel D esign ...................................... ...................... ............... 2
Panel M odeling Issues in Optim ization.................. ............................................... 3
M ulti-Fidelity A pproxim nations ....................................... ....................... ............... 4
O objectives of the D issertation ........................................ ......................... .............. 6
Outline of Dissertation ............. .................... ........ ............ ..............7
2 PANEL ANALYSIS AND DESIGN METHODS...................................................9...

Stiffened Panel A analysis M ethods............................................................ .............. 10
O ptim ization of Stiffened Panels............................................................. ............... 23
3 PANEL DESIGN STUDIES USING PANDA2 .............. ....................................34

The PA N D A 2 Program ..... ................................................................. .............. 35
PA N D A 2 C capabilities ............................................................. ........ .............. 36
Reusable Launch Vehicle Propellant Tank Design ................................................ 48
M modeling Issues ................. .... .. .. .. .. .. ................. ............. 58
Comparison of Weight Efficiency of Stiffened Panel Concepts................................... 85
4 HOMOGENIZATION APPROXIMATIONS IN COMPOSITE LAMINATES ...... 101

Errors from Use of Equivalent Properties for Sublaminates................................ 101
Facesheet Wrinkling of Sandwich Panels .......... ....................... 122
5 RESPONSE SURFACE APPROXIMATION...... .... .................................... 134

Response Surface Approximation Construction ............................................ 135
D esign of Experim ents ... .................................................................. .............. 136
E rror E stim ation ......................................................................................................... 13 8









Confidence Measures for Approximation Model...... ...................................... 140
S tep w ise R eg ressio n .................................................................................. .............. 14 1
6 ANALYSIS INTEGRATION USING APPROXIMATIONS ................................... 144

Integration of Overall and Panel level Analyses for a RLV Tank Optimization ....... 145
Integration of Low and High Fidelity Models with the Use of Correction Response
Surface A pproxim nations ................................. ...................... .............. 161
7 CON CLUD IN G REM ARK S ................................... ........................ .............. 175

Homogenization Approximation in Composites...... ....................................... 175
Stiffened Panel Optimization using PANDA2....................................... .............. 177
Response Surface Approximations for Model or Analysis Integration................... 179
REFERENCES ................................................... .......................... 180

APPENDICES

A M ATERIAL PROPERTIES ............................................................. .............. 193

M etallic A llo y s ........................................................................................................... 19 3
L am inated C om posite ..... .. ........................................ ........................ ............ .. 193
H on ey com b S an dw ich ................................................................................................ 194
B OPTIMUM DESIGNS OF METALLIC STIFFENED PANELS............................. 196

Stringer-ring stiffened Panel..................................... ........................ .............. 196
G rid-Stiffened Panels ................................................. .................................... 197
Titanium Honeycomb Core Sandwich Panels...... .... .................................... 200
Truss-C ore Sandw ich Panels.................................... ....................... ............... 203
C OPTIMUM DESIGNS OF COMPOSITE PANELS .............................................. 204

Stringer-R ing Stiffened P anels ................................................................................... 204
Honeycomb Core Sandwich Panels...... ........... ........ ..................... 206
Truss-C ore Sandw ich Panels .................................... ....................... ............... 207
BIO GRAPH ICA L SK ETCH ....................................... ......................... ............... 209















ACKNOWLEDGEMENTS


I want to thank Dr. Raphael T. Haftka for providing me with the opportunity to

complete my Ph.D. studies under his exceptional guidance. Without his untiring patience,

constant encouragement, guidance and knowledge this work would not have been

possible. Not only did Dr. Haftka direct my learning and research as a student, but he also

played the very important role of mentor. I want to thank him for the financial support he

has provided me and for all the opportunities he has created during the last five years for

me to grow as a professional and as a person. My academic and general with discussions

Professor Haftka have broadened my vision and given me new insight in to many areas of

engineering design, philosophy and history.

I would also like to thank my Ph.D. supervisory committee members, and Dr.

David Bushnell, Dr. Theodore F. Johnson, Dr. Bhavani Sankar, Dr. Andrew Rapoff, Dr.

Gary Consolazio. I am grateful for their willingness to serve on my committee, providing

me help whenever needed and for reviewing this dissertation. Special thanks to Dr. David

Bushnell for his help with the PANDA2 program and for educating me on the various

aspects of stiffened shell analysis and design. Special thanks to Dr. Bhavani Sankar for

the many discussions I have had with him and for the knowledge he has imparted to me

in the area of composite materials and fracture mechanics. I would like to mention my

special thanks to Dr. Theodore F. Johnson, for sponsoring my studies, co-authoring and

reviewing some of my publications, and for the weekly teleconferences, where his









practical experience and technical knowledge made this research more interesting and

relevant.

I would like to thank my colleagues, Dr. Peter Harrison, Dr. Willem Roux and

Luciano Lamberti with whom I have collaborated while working on my Ph.D.

dissertation, for the interaction and many fruitful discussions. Special thanks to Luciano

Lamberti, without his help this dissertation would not have been completed.

I would like to thank all my colleagues in the Structural and Multidisciplinary

Design Optimization Research Group for their help and support. I am grateful for the

friendship of Dr. Gerhard Venter, Roberto Vitali, Boyang Liu, Raluca Rosca, Melih

Papila, and Steven Cox, which made my doctoral studies a pleasurable experience.

The financial support provided by NASA grants NAG-1-669 and NAG-1-1808 is

gratefully acknowledged.

My parents deserve my deepest appreciation. I am especially grateful for the

countless sacrifices they made to ensure that I can pursue my dreams and for always

being there for me.

Lastly, I would like to thank my dear friend, confidante and true love Beth. Her

love, support and encouragement has had made my life rich and complete. I thank her for

helping me discover the meaning of companionship and sharing, and for teaching me to

enjoy the simple things in life.















LIST OF TABLES


Table Page

3.1: Panel concepts, stiffener locations, and materials considered for the RLV
liquid hydrogen tank design ................................................... ................ 49

3.2: Different laminate lay-up used for composite panel design...............................50

3.3: Design variables and bounds (in inches) used in the optimization of stringer-
ring stiffened panels (Values shown in parentheses correspond to
composite panels when different from metallic designs)......................... 53

3.4: Design variables and bounds (in inches) used in the optimization of
alum inum isogrid stiffened panels ......................................... ................ 53

3.5: Design variables and bounds (in inches) used in the optimization of
alum inum orthogrid stiffened panels...................................... ................ 54

3.6: Design variables and bounds (in inches) used in the optimization of
honeycom b core sandw ich panels.......................................... ............... 54

3.7: Design variables and bounds (in inches) used in the optimization of truss
core sandw ich pan els.............................................................. ................ 56

3.8: Buckling loads and analysis time for PANDA2, BOSOR4 and STAGS
models for an optimized stringer stiffened plate................ ................60

3.9: Comparison of buckling load factors obatined from PANDA2 and finite
element analysis for cylindrical panels optimized using PANDA2............ 62

3.10: Optimum weight of panel designed using IQUICK=0 and IQUICK= 1
an aly se s ....................................................................................................... 6 4

3.11: Critical margins of design obtained optimized using IQUICK=1 and
analyzed using IQ U IC K =0..................................................... ................ 64

3.12: Analysis of stiffened panels with use of PANDA2 local postbuckling
a n a ly sis ........................................................................................................ 6 6

3.13: Comparison of aluminum stringer-ring stiffened panel designs optimized
with and without local postbuckling effects........................... ................ 68









3.14: Comparison of aluminum orthogrid stiffened panel designs optimized with
and without local postbuckling effects................................... ............... 69

3.15 Comparison of composite stringer-ring stiffened panel designs optimized
with and without local postbuckling effects........................... ................ 69

3.16: Optimum weight of panels optimized with and without imperfections and/or
w ide colum n buckling constraint ........................................... ................ 73

3.17: Comparison of margins of perfect composite stringer-ring stiffened panels
analyzed w ith im perfections................................................... ............... 73

3.18: Effect of corrugation angle on optimum weight of truss core panel (Internal
pressure load case and axial compression load case are indicated by
the num bers 1 and 2 in parenthesis) ....................................... ............... 79

3.19: Comparison of CPU times required for one SUPEROPT execution for
global optimization of an aluminum stringer-ring stiffened panel using
IQUICK=0 and IQUICK=1 analysis options.........................................81

3.20: Number of optimization iterations and optimized weight of isogrid stiffened
panels with different initial designs (design vectors of the optimum
designs are presented in Table 3.21) ...................................... ................ 81

3.21: Isogrid stiffened panel designs obtained from different initial designs. ............ 82

3.22: Constraint margins (%) for optimized designs of isogrid panels......................82

3.23: Titanium honeycomb core sandwich panel optimization: optimum weights
and iterations for global optim um .......................................... ................ 84

3.24: Estimated number of panel analyses for stiffened panel trade study ................85

3.25: Comparisons of weight efficiencies of metallic stiffened panel concepts
(Sandwich panel weights shown in parenthesis do not include core
w e ig h t) ......................................................................................................... 8 6

3.26: Comparisons of weight efficiencies of composite stiffened panel concepts........90

3.27: Optimal weights (lbs/ft2), stiffener percentages in weight for the optimized
d e sig n s ......................................................................................................... 9 2

3.28: Active failure mechanisms for the optimized designs.....................................93

3.29: Sensitivity of optimum weight of metallic panels to geometric imperfections.... 95

3.30: Active constraints of metallic panels designed with and without
im perfections ......................................................................................... 96









3.31: Sensitivity of optimum weight of composite panels to geometric
im perfections ...................................................................................... . 97

3.32: Active failure margins of composite panels designed with and without
im perfections ...................................................................................... . 98

3.33: Imperfection sensitivity of composite stringer-ring stiffened panels
optimized with varying levels of design freedom .................................... 99

3.34: Optimized lay-ups obtained for composite stringer-ring stiffened panels
optimized with varying levels of design freedom ................................... 100

4.1: Initial design of panel (Subscript T and F denote pre-impregnated Tape and
woven Fabric lamina, respectively)...... .... .................. ................. 104

4.2: Material properties of tape and fabric plies used for panel design.................. 104

4.3: Optimized panel designs obtained from models using equivalent properties
and ply layup ................................................................. . ........... 105

4.4: Buckling load factors for stiffened panel designs (from Table 4.3) obtained
using exact and equivalent properties (values in paranthesis indicate
e rro rs) ........................................................................................................ 1 0 5

4.5: Maximal errors in bending stiffness of a laminate calculated using
equivalent properties ................................. ...................... .............. 109

4.6: Error in bending curvatures due to using equivalent property for graphite-
epoxy (E1=18.5 Mpsi, E2=1.89 Mpsi, G12=0.93 Mpsi and v 12=0.3) ........109

4.7: Maximal error in Dij terms for n full-sublaminate repetitions, as in [t /t2]sn
lam in ate ..................................................................................................... 1 16

4.8: Maximal error in Dj terms for n half-sublaminate repetitions, as in [(t /t2)ns
lam in ate ..................................................................................................... 1 17

4.9: Maximal buckling load errors (percent) of a single sublaminate (The coding
A-P refers to the stacking sequences in Table 4.10) .............................119

4.10: Stacking sequence of sublaminates optimized for maximal error in buckling
load for 1 sublaminate in total laminate....... .................. .................. 119

4.11: Effect of number of sublaminates in half sublaminate repetition on error in
buckling load calculated using equivalent properties............................. 121

4.12: Effect of number of sublaminates in full sublaminate repetition on error in
buckling load calculated using equivalent properties..............................121

4.13: Effect of wrinkling wavelength constraint on optimal facesheet thickness....... 124

ix









6.1: Optimization of tank wall laminate using local linear approximations for the
case with three design variables(ply thicknesses of a [45/0/90]2s
lam inate, ply thickness in m ils)........... ... ...................................... 150

6.2: Optimization of tank wall laminate using local linear approximations for the
case with six design variables (ply thicknesses of a
[+45/0/90/+45/0/90]S laminate, ply thickness in mils )........................ 153

6.3: Error analysis for the response surface approximations used in PANDA2
design optim izations .................................. .. .......... .......... .. ........ .... .. 155

6.4: Optimization of tank wall laminate using response surface approximations
for case with three design variables(ply thickness of a [+45 / 0 /9012S,
lay-up, ply thickness in mils )............... .................. ................. 157

6.5: Optimization of tank wall laminate using response surface approximations
for the case with five variables (ply thicknesses for a
[+45/0/90/+45/0/90]s lay-up, ply thickness in mils)............................ 160

6.6 : D esign v ariables ...... .. ........................................ ........................ .. . ....... .. 166

6.7: Comparison of local buckling load calculated from different programs......... 167

6.8: Regression statistics for response surface fitted to PANDA2 buckling load
fa cto r .......................................................................................................... 1 6 9

6.9: Correction response surface model ....................................... 171

6.10: Optimized designs for ring-stiffened cylinder (Units of thickness and length
are in inch) .................................................................... .. . ......... 173

A.1: Material properties for metallic panels (at 500 F) .................... ..................193

A.2: Material properties for IM7/977-2 composite panels (at 1900 F) ....................194

B. 1: Optimum designs of metallic stringer-ring stiffened panels ............................197

B.2: Optimum design of isogrid panels optimized with and without imperfections .198

B.3: Optimum design of orthogrid panels optimized with and without
im perfections .................................................................. . .......... 198

B.4: Effect of stiffener profile on optimum weight of grid-stiffened panels
optim ized w ith im perfections....... ... .......................................... 200

B.5: Grid stiffened panel: web thickness and failure margins for different
stiffener profiles .............. ............. ............................................. 200

B.6: Symmetric sandwich panel: optimized designs...... ................. ...................201

x









B.7: Asymmetric sandwich panel: optimized designs .................... ...................202

B.8: Optimum design of metallic truss-core sandwich panels optimized with and
w without im perfections...... ............. ............ ...................... 203

C.1: Optimum designs of composite stringer-ring stiffened panels.........................205

C.2: Optimum designs of honeycomb core sandwich panels: fixed core thickness ..205

C.3: Optimum designs of composite symmetric honeycomb core sandwich
panels: optim ized core thickness....... ... ........................................ 206

C.4: Optimum design of composite asymmetric honeycomb core sandwich
p a n e ls ......................................................................................................... 2 0 7

C.5: Optimum designs of composite truss-core sandwich panels............................208
















LIST OF FIGURES


Figure Page

2.1: Panel show ing global im perfection .................................................. .............. 13

2.2: Schematic of wing showing skewed panels ..................................... ............... 15

2.3: Stiffened panel analysis m odel of VICON ....................................... .............. 15

2.4: B lade stiffened panel w ith voids ..................................................... ................ 25

3.1: View of single module of panel with T-shaped stringer, showing layer
numbering convention and discretization used. .....................................37

3.2: PANDA2 repeating module for a panel with T-shaped stiffeners ....................37

3.3: Schem atic of a Reusable Launch Vehicle....................................... ................ 48

3.4: Schematic of a T, J, blade, and Hat stiffener geometry...................................51

3.5: Cylinder with internal isogrid and external rings, circumferential Isogrid
p atte rn .......................................................................................................... 5 2

3.6: Schem atic of orthogrid stiffening concept ...................................... ................ 52

3.7: H oneycom b sandw ich lam inate....................................................... ............... 54

3.8: Schematic of the truss core sandwich module.................................................55

3.9: Comparison of the buckling mode shapes obtained with BOSOR4 and
S T A G S ........................................................................................................ 6 0

3.10: Critical buckling mode for the cylindrical stringer-ring stiffened panel ........... 61

3.11: Mode shape for limit buckling of composite sandwich cylinder ......................62

3.12: Effect of axial length on optimized weight for panels without rings................ 76

4.1. Stiffener geometry and associated terminology...... .................... ..................104

4.2: Sublam inate w ith layers t1 and t2 ................................................... .............. 106

xii









4.3: Ratio of bending stiffness from equivalent and exact properties .....................108

4.4: Schemes for stacking sublaminates: full-sublaminate and half-sublaminate
repetitions ............................................................ ......................... 111

4.5: Honeycomb sandwich laminate showing core (cell) dimensions ....................122

4.6: W rinkling of face sheet on sandwich core ..................................... ............... 123

4.7: Schematic of beam on an elastic (discrete spring) foundation....................... 125

4.8: Percentage error in wrinkling load factor due to using smeared properties for
beam on elastic foundation...... ........ ..... .................... 128

4.9: Facesheet wrinkling from discrete and continuum models at different
w wavelengths ......................................................................................... 129

4.10: Ratio of wrinkling loads from continuum and discrete models for ac < 2.0........ 130

4.11: Possible half wavelengths for wrinkling us facesheet at lower wavelengths.... 131

6.1: Finite element analysis model of the RLV ...... .... ..................................... 146

6.2: Schematic of local and general buckling of a ring stiffened cylinder under
compression loads and simple support (pinned) boundary conditions ..... 163

6.3: PANDA2 analysis model for local buckling load factor.............................. 164

6.4: B O SO R 4 1-D m odel ...................................... .......................... .............. 165

6.5: Finite element mesh used for STAGS model and the first critical mode from
a linear buckling analysis. ...... ........ ........ .................... 165

6.6: Schematic of a T-ring stiffener showing design variables and terminology...... 166

6.7: Response surface approximation prediction compared with PANDA2
p red ictio n ................................................................................................... 17 0

6.8: Comparison of PANDA2 and corrected PANDA2 local buckling load
prediction with STAGS local buckling prediction..............................172
















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy


MODELING, ANALYSIS AND OPTIMIZATION OF STIFFENED CYLINDRICAL
PANELS FOR REUSABLE LAUNCH VEHICLE STRUCTURES

By

Satchithanandam Venkataraman

December 1999


Chairman: Dr. Raphael T. Haftka
Major Department: Aerospace Engineering, Mechanics and Engineering Science


The design of reusable launch vehicles is driven by the need for minimum weight

structures. Preliminary design of reusable launch vehicles requires many optimizations to

select among competing structural concepts. Accurate models and analysis methods are

required for such structural optimizations. Model, analysis, and optimization complexities

have to be compromised to meet constraints on design cycle time and computational

resources.

Stiffened panels used in reusable launch vehicle tanks exhibit complex buckling

failure modes. Using detailed finite element models for buckling analysis is too

expensive for optimization. Many approximate models and analysis methods have been

developed for design of stiffened panels.









This dissertation investigates the use of approximate models and analysis methods

implemented in PANDA2 software for preliminary design of stiffened panels. PANDA2

is also used for a trade study to compare weight efficiencies of stiffened panel concepts

for a liquid hydrogen tank of a reusable launch vehicle. Optimum weights of stiffened

panels are obtained for different materials, constructions and stiffener geometry. The

study investigates the influence of modeling and analysis choices in PANDA2 on

optimum designs.

Complex structures usually require finite element analysis models to capture the

details of their response. Design of complex structures must account for failure modes

that are both global and local in nature. Often, different analysis models or computer

programs are employed to calculate global and local structural response. Integration of

different analysis programs is cumbersome and computationally expensive.

Response surface approximation provides a global polynomial approximation that

filters numerical noise present in discretized analysis models. The computational costs

are transferred from optimization to development of approximate models. Using this

process, the analyst can create structural response models that can be used by designers in

optimization. It allows easy integration of analysis models in optimization.

The dissertation investigates use of response surface approximations for

integrating structural response obtained from a global analysis in the local optimization of

stiffened panels. In addition, response surfaces are used for correcting structural response

predictions from a low fidelity model with a few expensive detailed finite element

analyses.

















CHAPTER 1
INTRODUCTION


Composite materials are desirable in lightweight structures due to their high

specific stiffness and strength. Laminated composite materials provide the designer with

freedom to tailor the properties and response of the structure for given loads to obtain the

maximum weight efficiency. However, high modulus and strength characteristics of

composites result in structures with very thin sections that are often prone to buckling.

Stiffeners are required to increase the bending stiffness of such thin walled members

(plates, shells). Consequently, stiffened panels are often used in aircraft and launch

vehicles to obtain lightweight structures with high bending stiffness. Stiffened shells are

also more tolerant to imperfections and resist catastrophic growth of cracks.

NASA is investigating the use of composite materials in the propellant tank

design for next generation reusable launch vehicles. In the design of such large structures,

many decisions have to be made regarding materials selection, fabrication techniques,

and stiffener types. These decisions affect structural weight and operational costs. A large

number of preliminary structure or substructure design optimizations need to be

performed in order to understand the impact that different design constraints have on the

optimum design. Often there are constraints on the available computational resources,

time and design cycle time. Designers are faced with the task of choosing between model,

analysis and optimization complexity. This introduction presents some issues in the areas

1






2


of stiffened panel modeling, analysis and optimization addressed in this dissertation, with

a brief outline of each chapter. Detailed literature review of stiffened panel design

methods is presented in Chapter 2.


Simple Models for Panel Design

Stiffened panels exhibit complex failure mechanisms and therefore require careful

selection of the model, theory and numerical procedure used for analysis. A variety of

analysis methods based on simple physical models, smeared models, finite strip models

and finite element models have been implemented in software. For example, BOSOR4

[24] and FASOR [44] analyze shells of revolutions; PANDA2 [25] uses closed form

solutions for individual failure modes and some 1-D discrete analysis models for design

of stiffened panels; PASCO [6], VICONOPT [126], and PANOPT [16] are finite-strip-

based design codes. A survey of these programs is presented in Chapter 2.

Finite element analysis provides the highest fidelity in modeling complex

structures as it can incorporate many local details such as stiffener termination, cutouts,

and local reinforcements. However, such analysis has several problems when directly

used in an optimization environment. Using finite element analysis for design

optimization of stiffened panels is often impractical due to high computational expense.

When the geometry of the shell and stiffeners is optimized, remeshing is frequently

required for acceptable accuracy. The numerical noise introduced by the discretization

makes it difficult to use gradient based optimization methods.

Kaufman et al. [67] investigated the numerical noise and human errors that are

present in results obtained using large and complex analysis codes. Numerical noise

introduced by the discretization (meshing) and round off, in finite element analyses, was









discussed by Burden and Faires [21, p. 10]. Venter et al. [120] described the noise

introduced by finite element discretization in a stepped plate optimization problem.

Giunta et al. [57] and Dudley et al. [52] investigated such noise in aerodynamic analyses.

Significant time and effort are often required to interface analysis and

optimization when using finite element analysis models. In particular, developing design

constraints for use with general-purpose analysis codes in optimization is a time-

consuming task. Furthermore, using general-purpose analysis software requires a good

understanding of the physical problem and the limitations of the model or theory being

used. Some panel design software implement a variety of design constraints that serve as

an expert system and thereby significantly reduce the modeling effort and chances of

obtaining unreasonable designs from the optimization.

Performing design optimization using finite element based non-linear analyses is

often too expensive to be practical. Aside from computational cost, the complexity of

analysis models presents a major challenge in design. Simpler analysis models such as

those used by PANDA2, PASCO, or VICONOPT are therefore very useful for

optimization purposes.


Panel Modeling Issues in Optimization

The choice of modeling details, analysis methods, and optimization techniques

determines the computational cost of the design. To compensate for an expensive

analysis (e.g., nonlinear response) or expensive optimization (e.g., global optimization),

the designer may have to introduce approximations to the modeling of the designed

structure.









Optimization of composite structures often involves a large number of design

variables. Laminate stacking sequence optimization using discrete thickness and ply

angles is a combinatorial design problem. Laminates optimized using a small number of

load cases often are highly tailored to the loading and can perform very poorly in off

design conditions. Furthermore, composite laminates exhibit a variety of failure modes

that are difficult to model or analyze. Designers in such cases have to rely on data

obtained from experiments.

In order to reduce model complexity, composite laminates are often approximated

using homogenized models or other approximations. Such approximations are valid

under certain assumptions and hence should be used with caution in optimization. In

particular, it is known that optimization programs are very unforgiving in exploiting the

weaknesses and limitations of the analysis models or theories used. They often find

designs in regions where the assumed model is no longer valid. Knowledge of the

validity and accuracy of such approximations is therefore required in order to develop

appropriate constraints to be imposed in the optimization to avoid serious design errors.


Multi-Fidelity Approximations

The limitations of simple models often require the use of more complex and

expensive analyses, which cannot be practically coupled with optimization programs.

This has led to the development of approximation methods based on function evaluation

at one or more points. These approximations may be accurate in a small region or in a

larger region of design space based on the data used in their construction.

Response surface methods typically fit low order polynomials sense to function

values over large regions of design space using least square fitting procedure. The fitted









approximation model is used to replaces the expensive analysis program in the

optimization. An approximation can be developed independently for each failure mode

from different analyses, so that the designer can readily integrate the various analysis

models and codes into the optimization.

Venter and Haftka [119] demonstrated the usefulness of response surface

approximations for expensive structural optimization in design for uncertainty. Ragon et

al. [88] used response surface methods for a global-local design study of an aircraft

wing.

Response surface approximations have their limitations. The cost of fitting the

approximation increases rapidly with the number of design variables (the curse of

dimensionality). Fitting a good global approximation that is valid over the entire design

space is not possible with low order polynomials. To address these problems, correction

response surfaces may be used to correct the less expensive analyses (lower fidelity

models), rather than fit approximations directly to the response functions obtained from

the more accurate analyses (higher fidelity models). The correction response surface

model is fitted using a small number of higher fidelity analyses.

Recently, different methods of using correction response have been proposed and

tested in the area of structural optimization. Mason et al. [78] used a 2-D finite element

model as a lower fidelity model to predict failure stresses and applied corrections

calculated using a full 3-D finite element analysis. Vitali et al. [123] used the approach

for optimizing a blade stiffened composite panel with cracks. The lower fidelity model

employed an infinite plate model to predict the stress intensity factor. The higher fidelity

model used a detailed finite element model with a crack to calculate the stress intensity









factor at the crack tip. Knill et al. [69] demonstrated the use of correction response

surfaces for the aerodynamic configuration design of a high-speed civil transport

airplane. Toropov and Markine [109] demonstrated the approach for the design of a four-

bar mechanism.


Objectives of the Dissertation

The first objective of the present research was to investigate the efficiency,

accuracy and advantages of using design codes based on simple methods for stiffened

panel design. The PANDA2 program was used as an example of such a design code.

PANDA2 uses a variety of simple physical models (smeared, finite strip, and 1-D

discretized) for capturing the failure of stiffened panels. The PANDA2 program was used

for design studies on stiffened panels. The effects of modeling choice and analysis

methods on optimum design of panels were investigated. The optimized designs were

verified using detailed finite element analysis.

The second objective was to illustrate the effect of laminate model choice on

design optimization. Stiffened composite panels often require a variety of global and

local analysis models and approximations. Simplified modeling and approximation can

affect the optimization results if used without proper constraints.

The third objective was to demonstrate the use of approximation techniques for

integrating analysis models or codes. Stiffened panels are substructures that form a

larger tank or vehicle structure. Interaction of the local (panel) design on the global

(vehicle structure) design should be taken into account in the design optimization of

panels. Use of approximations for integration of global and local analysis models is

presented. In addition, designers are often faced with the situation where the currently









available analysis code or model has certain limitations and a more detailed analysis is

required. Integrating the more complex analysis technique into the optimization can be

costly and cumbersome. Correction response surface techniques allow the use of a simple

model with a correction function fitted using a small number of "exact" (costly) analyses.


Outline of Dissertation

Panel analysis and design methods, which are key aspects of the present

dissertation, are reviewed in Chapter 2. The review focuses on design and analysis of

panels using specialized programs. A few examples of stiffened panel optimizations

using finite element analysis that employ special techniques to reduce optimization costs

such as sub-structuring or that use easily available analytic sensitivities are included.

Also, panel design methods that address global-local design issues are discussed.

Chapter 3 presents design optimization of various metallic and composite

stiffened panels using the PANDA2 program. The choice of models for analysis (closed

form, 1-D discretization), choice of shell theories (Donnell, Sanders), and inclusion of

details such as the effects of geometric imperfections and post-local buckling response

are investigated for the designed panels. The designs obtained from PANDA2 are

analyzed using more rigorous finite element analysis. The assistance of Mr. Luciano

Lamberti [72] in performing the numerous structural optimizations required for this study

is gratefully acknowledged.

Chapter 4 discusses the effects of using simplified laminate models in

optimization. Examples of modeling composite laminates as an orthotropic material and a

sandwich hexagonal core as a continuum foundation are presented. The tendency of

optimization programs to exploit modeling weaknesses is demonstrated.






8


In Chapter 5, approximation methods are reviewed. Response surface

approximation techniques are presented. These techniques are then used for a global-local

design problem and analysis model integration problem in Chapter 6. Finally, Chapter 7

presents some conclusions.

















CHAPTER 2
PANEL ANALYSIS AND DESIGN METHODS


The use of composite materials in aircraft and space vehicles is becoming

prevalent. The need to reduce weight makes composite materials a very attractive choice.

Using composite materials with very high specific modulus and strength often results in

structures with thin members. While the structures so designed may be sufficient to carry

the in-plane tensile loads and satisfy strength requirements, they are often prone to

buckling failures under compression or shear loads.

Stiffeners are used to increase the bending stiffness of thin-walled members

(plates and shells). The stiffeners add an extra dimension of complexity to the model

compared to unstiffened plates and shells. However, stiffened structures often employ a

repeating stiffener pattern. The repeated (periodic) nature of the geometry allows the use

of simplifying assumptions to obtain approximate analyses. The design of composite

stiffened panels present even more challenges due the additional failure modes and

anisotropy effects introduced by composite materials.

A variety of methods and programs are available for the analysis of stiffened

panels, ranging from simple closed form solutions to complicated 3-D discretized

solutions. The analysis costs typically increase with the level of detail modeled (e.g.,

branched shells vs. smeared representation of stiffeners) and the fidelity of the analysis

procedure used (e.g., linear vs. nonlinear analysis).

9









Stiffened Panel Analysis Methods

A variety of simple analysis methods have been developed for the analysis of

stiffened panels. These methods typically belong to one of the following classes: analysis

based on smeared properties, simple plate analysis under simple supports, and accurate

linked or segmented plate analysis or finite strip analysis. The more complicated or

detailed modeling usually employs discretized models such as finite element and

boundary element analysis. In the next section, some programs or software for design

optimization of composite stiffened panels that use the simplified analysis methods will

be reviewed. Examples of design optimization techniques and design for various

performance factors such as structural instability, crashworthiness, damage tolerance, and

postbuckling strength will be presented. The discussion will focus mainly on the design

optimization of stiffened plates and stiffened cylindrical shells or panels.


Stiffened Panel Analysis using Closed Form Solutions and Smeared Models

Agarwal and Davis [1] optimized composite hat stiffened panels for buckling

(local and Euler column buckling) and strength requirements. The buckling analysis was

based on simplified assumptions. The local buckling load factor was calculated by

assuming that panel segments were simply supported. The load distribution on the

different segments of the panel (skin, stiffener segments) was obtained using the effective

elastic modulus of the different segments. The Euler column buckling was obtained from

a smeared model. The results were verified using the BUCLASP-2 [111, 121] linked

plate analysis program that treats stiffeners as branched shells.

Agarwal and Sobel [2] presented weight comparisons for stiffened, unstiffened

and sandwich cylinders under axial compression loading. The overall buckling load was









captured using a smeared representation. The stiffeners in this case were treated as one-

dimensional beams with twisting taken into account. The resulting designs were also

analyzed with the BUCLASP-2 program. It was shown that smeared models

underestimate buckling loads by 30%.

Stroud and Agranoff [99] extended the approach of using simplified buckling

equations to the design of more general hat and corrugated panels under axial

compression and shear loads. The analyses treated individual segments as simply

supported flat plates, calculating buckling loads of each individual element. The authors

investigated the weight efficiencies of the different stiffeners and also calculated weight

penalties that were due to the imposition of manufacturing constraints (discrete ply

thickness, stiffener dimensions). The global buckling was analyzed using a smeared

model similar to that used by Agarwal and Davis [1]. The failure margins obtained using

simple analysis were found to be conservative for most cases due to the assumption of

simple supports for the different members. The simplification also could not account for

stiffener rolling modes that were critical for some of the designed panels.

Simple smeared representations of stiffened panels are used often for optimizing

them with general instability constraints. Tomashevskii et al. [107, 108] used such a

smeared model and simple expressions for stiffener buckling to optimize stiffened

composite cylinders. Sun and Mao [104] optimized stiffened composite cylinders to

investigate the effects of shell geometry, stiffener eccentricity and laminate stacking

sequence under axial compression and hydrostatic pressure loads. A smeared

representation taking into account the stiffener-introduced eccentricity was used. The

initial postbuckling was investigated using Koiter's general theory of general instability









[62, 70]. The analysis method provided an efficient tool to investigate the effects of

different design variables on general instability.


Analysis using Finite Strip Methods

Finite strip methods (FSM) represent a class of analysis that has an accuracy (and

computational expense) that lies between closed form solution of the type seen in linked

or segmented plate analysis models and the finite element method. Linked plate analysis

solves the exact plate equation in the segmented-plate model and hence is considered as

the "exact" finite strip method. Inexact finite strip analysis divides the panel segments

into strips and approximates the displacement field in the strips. Within each strip, the

displacement field is expressed using polynomials to describe the widthwise variation

and trigonometric functions to describe the displacement field. Dawe and his coworkers

[47-51] published several papers on such approximate finite strip methods for buckling

and postbuckling analysis of prismatic composite structures.

Williams and his coworkers [130] developed a rigorous analysis procedure

(VIPASA, Vibration and Instability Analysis of Plate Assemblies including Shear and

Anisotropy), where the buckling loads of stiffened panels were calculated by treating

them as plate assemblies. The procedure accounts for the physical connections between

the adjacent members and permits a buckling pattern that is continuous along the

connection. The buckling solutions are based on exact plate analysis equations. This

procedure was implemented into the VIPASA analysis code. The analysis was restricted

to uniform transverse or edge loading with simple supports at the panel ends.

Stroud and Anderson coupled the VIPASA analysis code with the CONMIN

[115] optimization program for design optimization of hat and corrugated stiffened









panels [101]. CONMIN used the method of feasible directions for the optimization, and

Taylor series approximation for its constraint approximation. The designs obtained were

compared with those obtained from the simplified analysis described in Stroud and

Agranoff [99]. It was shown that for hat stiffened panels loaded in axial compression, the

simplified analysis was conservative (2% to 9% heavier). The simplified analysis was

found to be inadequate when local buckling wavelengths coincided with the short

buckling wavelengths of overall buckling due to anisotropy effects. It was found that

under combined shear and axial loading, inclusion of anisotropy terms (D16 and D26) was

very important (i.e., replacing laminates with orthotropic representations can lead to

incorrect buckling loads). The authors demonstrated that while the simplified models

provided reasonably accurate predictions of response, using such models for optimization

or sizing without proper design constraints resulted in very unconservative designs.

X






--Ny




Nx
Z


Figure 2.1: Panel showing global imperfection

Anderson and Stroud [6, 101] implemented the VIPASA analysis into an

automated panel sizing code (PASCO). The VISPASA analysis was enhanced to include









global (overall bowing type) imperfection as shown in Figure 2.1. The maximum bending

moment on the panel was calculated by using the classical beam formula


M = PL2 sec(- 1 (2.1)
1-y a y 2


where, y=N (2.2)
NxE

in which Nx is the applied axial load per unit width, NxE is the Euler buckling load for the

panel, and P is the pressure acting on the panel. The maximum bending moment was then

used to calculate stress resultants on the skin and stiffener segments to determine the

buckling load factors. Weight efficiencies of hat and blade stiffened panels were

compared at different values of global imperfection amplitude. Small geometric

imperfections had a big influence on the optimal weight. Comparison with experimental

results showed the importance of including imperfections. The PASCO code provided a

rational method for including the effect of bow-type imperfections, unlike other

simplified analysis methods described so far that required some empirical knockdown

factors to account for imperfections.

A shortcoming of PASCO or VIPASA analysis [102] was the inexact matching of

boundary conditions under shear load. The displacement field used under shear loads

results in a skewed plate. The analysis model used in VIPASA and PASCO was found to

be inaccurate for shear buckling with long wavelength. The VICON (VIPASA with

Constraints) program developed by Williams and coworkers [126, 127] overcame the

problem by using Lagrange multipliers to provide arbitrary supports or boundary

conditions. Swanson et al. [105] incorporated this model into design optimization via an

ad hoc correction. An iterative approach was used where the PASCO smeared









(approximate) model was used to size panels. A correction factor was obtained using a

VICON analysis for the next PASCO optimization cycle. The approach illustrates the

use of multi-fidelity models using corrections based on a small number of accurate and

expensive function evaluations during the optimization. The procedure was used to

perform weight comparisons of hat, blade stiffened, and trapezoidal corrugated panels.

The correction approach developed by Swanson et al. [105] was found to be quite

inefficient. To overcome this deficiency and to implement the VICON directly into the

sizing code and refine some analysis features [128] the VICONOPT program [42, 43]

was developed.
















Figure 2.2: Schematic of wing showing skewed panels


Figure 2.3: Stiffened panel analysis model of VICON









A similar problem in using approximate end conditions for stiffened panels is

encountered in aircraft wing design. Stiffened panels used in aircraft wing structures have

skewed (Figure 2.2) edges. However, they are often approximated as rectangular panels

in the design process. York and Williams [131] showed that the approximation led to

sub-optimal panels, because response of skew panels under shear load is very different

from that of rectangular panels. An accurate method of analyzing skew panels was

implemented in the VICON program by use of Lagrange multipliers [127]. The dotted

lines in Figure 2.3 indicate support lines along which simple support boundary condition

are enforced in VICON using the Lagrange multipliers. This provided a more accurate

representation of the boundary conditions for skew panels and rectangular panels under

shear loads.

Analysis costs typically increase with increase in the fidelity and accuracy of the

analysis model. York and Williams [132] demonstrated the use of infinite width model

for analysis of the skewed panels and compared the results to those obtained using exact

models. It was shown that the infinite width model provides conservative estimates for

buckling loads of skewed plates. The optimizations were performed for two metallic

panels (blade and J-stiffened) and for two composite panels (blade and hat stiffened). The

analyses were performed for different skew angles and aspect ratios and the buckling

loads were compared. The errors in buckling loads were typically in the range of 5-7%

with the worst case being 15%. However, the CPU run times for the infinite width

analysis were typically 7% to 30% of those required by the exact analysis. The efficiency

of such approximations is of great value in performing structural optimizations.









The PANOPT program developed at the National Aerospace labs, Netherlands, by

Arendsen and coworkers [16] is capable of design optimization of stiffened composite

panels for buckling and postbuckling. The PANOPT program is an extension of the finite

strip analysis method proposed by Riks [94] enhanced by Arendsen [15] to include higher

order energy terms.

Several other programs that use similar analysis approaches have been developed

for design optimization of stiffened composite panels. The analysis and sizing program

STIPSY developed by Baburaj and Kalyanaraman [17] uses approximate finite strip

models similar to those presented by Agarwal and Davis [1] and by Stroud and Agranoff

[99]. The authors included a more rigorous model for the torsional buckling of the

stringers. Postbuckling strength is considered by using the effective width of elements

undergoing local buckling. The effective width calculations of composite panels

implemented in STIPSY were originally developed by Rhodes and Marshall [92]. The

analysis code was coupled with an optimizer based on Rosen's gradient projection

method and an indirect method based on Sequential Unconstrained Minimization

Technique (SUMT). Optimal designs were obtained for various combinations of in-plane

compression and shear loads. The effect of allowing local postbuckling on the optimal

design weight was also investigated.

Zabinsky, Graesser, Kim and Tuttle [59, 68, 112, 133, 134] at the University of

Washington collaborated with researchers at Boeing [76] in the development of the

program COSTADE (Composite Optimization Software for Transport Aircraft Design

Evaluation) program. The structural analysis and optimization module of COSTADE was

based on the UWCODA code (University of Washington Composite Optimization and









Design Algorithm) [59]. The UWCODA program was developed for global optimization

[133] of composite stiffened panel structures with continuous and discrete variables.

COSTADE used a smeared stiffness approach for calculation of critical buckling and

stress margins. A stochastic algorithm denoted "Improving Hit-and-Run" [134] was

employed for the design optimization of the composite laminates. The program can

perform preliminary design optimization of stiffened composite panels with strength

constraints. COSTADE is capable of point design where a single panel type spans the

whole structure or can perform optimization of a blended design. In the latter,

compatibility constraints need to be imposed for continuity of plies and structural

members from one panel to the other. The buckling analysis of COSTADE as based on a

finite strip analysis method developed by Kim and Tuttle [68]. Three different strength

criteria are used. A pristine strength criterion calculated first ply failure load using

maximum strain failure. The ultimate damage tolerance criterion obtained strain

constraints using the ultimate loads. The failsafe damage tolerance criteria were based on

proof load conditions. COSTADE was further augmented using more detailed finite

element models for other damage scenarios (transverse crack) [58].

More recently researchers have started investigating methods for designing

stiffened shell structures with nonlinear analysis. Stoll and Guirdal [96] developed an

approximate semi-analytical approach for non-linear analysis of stiffened plates. The

NonLinear Panel ANalysis code (NLPAN) can predict postbuckling stresses and

deformations, elastic limit points, and imperfection sensitivities of linked plate

assemblies subject to inplane axial and pressure loads with temperature effects. The

program was developed as an extension of VIPASA so that it could be used in an









automated fashion with the VIPASA. The non-linear plate theory applied to each

component plate-strip accounts for the large in-plane rotations that occur in the

postbuckling response. Buckling eigensolutions with the second order contributions from

VIPASA were used to describe the displacements. NLPAN uses a stationary potential

energy condition to obtain a set of nonlinear algebraic equations governing equilibrium.

These equations are load independent with a relatively small number of variable modal

amplitudes, allowing a rapid exploration of the nonlinear regime. Stoll [97] extended the

capabilities of NLPAN to include various end support boundary conditions. The load

deflection and load vs. end shortening are compared with experimental test results.

Comparisons of the analysis results obtained using NLPAN with the STAGS [4, 90]

analysis code was presented by Stoll et al. [98]. The NLPAN program was shown to

provide nonlinear analysis capability at a small fraction of the cost of performing a

nonlinear finite element analysis.


Analysis of Cylindrical Stiffened Panels

Cylindrical stiffened panels are as important as stiffened plates. Stiffened

cylindrical panels or cylinders are used in aircraft fuselage, rocket and missile structural

components, and launch vehicle tank structures. The curvature of circular cylindrical

shells increases their load carrying capacity. However, the nonlinearities introduced by

the curvature result in unstable postbuckling behavior and make the shell structures more

sensitive to geometric imperfections. Therefore, much research has been done in the area

of stability of unstiffened and stiffened cylinders made of isotropic and anisotropic

materials. Examples discussed here only include analysis methods and programs that

have been used for obtaining optimum designs of curved stiffened panels.









The VICONOPT program was discussed in detail for analysis of stiffened plates.

Until recently, stiffened curved panels were modeled only approximately in VICONOPT

by discretizing it into a series of flat plates. While this approach gave quite accurate

results, it required a large number of plate elements to approximate the curved surface.

Since this was not very efficient, a curved plate element was developed [79] for inclusion

in VICONOPT. The curved plate element was developed using the non-linear

equilibrium equations that include transverse shear deformation effects. The plate

element can be used for obtaining linear bifurcation buckling and postbuckling responses.


Analysis of Shell of Revolution

Complex analysis methods and programs that can analyze shells of revolution

have been used extensively for shell analysis and design. The BOSOR4 program [24] is

an energy based discrete analysis method where the model is discretized along the

meridian of a shell of revolution. BOSOR4 can perform linear and nonlinear static and

buckling analysis of shells. For axisymmetric geometry and loads BOSOR4 can obtain

very accurate estimates of displacements, stress and buckling loads. BOSOR4 can also be

used for analysis of prismatic structures such as stiffened panels. The FASOR program

[44] provides similar capabilities using direct integration of the governing ordinary

differential equations for a shell of revolution.


Analysis using Multiple Models

Cylindrical stiffened shells exhibit complex failure modes and hence often require

multiple models for efficient analysis. An important and significant contribution to the

area of preliminary design of stiffened cylinders by Bushnell [25-33] at Lockheed Palo









Alto Research Laboratory has led to the development of PANDA2. The PANDA2

program finds the minimum weight designs for laminated composite plates, cylindrical

panels or cylinders with and without stiffeners that can run along one or two orthogonal

directions. The philosophy of PANDA2 is to provide optimum preliminary design of

stiffened panels that experience complex and nonlinear behavior without resorting to the

use of general purpose (finite element) analysis codes that require elaborate database

management systems. Instead, PANDA2 uses several separate relatively simple models,

each designed to capture specific failure modes or mechanisms (often with the same

accuracy as general-purpose FE codes). PANDA2 uses a combination of approximate

physical models, exact closed form (finite strip analysis) models and a 1-D discretized

branched shell analysis models to calculate prebuckling, buckling and postbuckling

responses.

The most challenging task in stiffened shell optimization is the selection of

appropriate modeling, analysis theories, and design constraints. PANDA2 has automated

this task and therefore can be regarded as an expert system for stiffened shell design. The

capabilities of PANDA2 and its efficiency allow it to perform global optimization of

stiffened panels. A more detailed overview of PANDA2 theory, analysis models,

procedure and optimization capabilities is presented in Chapter 3.

Arbocz and Hol [11, 12] at the Technical University of Delft, have developed an

expert system (DISDECO, Delft Interactive Shell Design Code) for design of anisotropic

stiffened panels and shells. The program uses a hierarchial analysis approach for accurate

prediction of buckling load and a reliable estimation of imperfection sensitivity. With this

tool, the designer can access a series of analysis programs of increasing complexity. The









analysis modules of DISDECO can calculate the critical buckling loads of stiffened

anisotropic shells subject to combined loads, investigate the effects of various types of

boundary conditions on critical buckling load, and obtain perspective of degrading effects

of possible initial imperfections on buckling loads.

Three hierarchical analysis models were implemented in the DISDECO code. The

Level-1 analysis uses the GANBIF [12] analysis routine. The analyses in level-1 are the

most approximate and are to be used for investigating overall or general buckling

characteristics of the shell. The solutions are based on membrane prebuckling solutions

and linearized stability equations reduced to algebraic eigenvalue problems by the use of

a truncated double Fourier series. The program can also be used to determine the

coefficients in Koiter's asymptotic expansion for initial postbuckling response.

The level-2 analysis accounts for the primary nonlinear effects such as edge

restraints and can satisfy the prescribed boundary conditions exactly. The level-2

analysis, based on the ANLISA code [10, 11], uses a rigorous prebuckling solution and

reduces the stability problem to the solution of a set of ordinary differential equations.

ANILISA solves the Donnell type anisotropic shell equations. The buckling analysis can

include geometric imperfections that are of the form of a double periodic trigonometric

(sin(mx)sin(ny)) function or axisymmetric imperfections that are similar to the critical

buckling mode of the perfect shell. The solutions from this level are very accurate for

axisymmetric geometry and loading. Also present in level-2 are other computational

modules such as COLLAPSE [13] that can handle axisymmetric initial imperfections,

and ANSOVI [91] that solves Novozhilov type anisotropic shell equations.









The level-3 analyses are the most accurate and use two-dimensional finite element

analysis codes with advanced analysis capabilities that include geometric and material

nonlinearities. At present different members of the STAGS family [3, 5] are used in this

level.

When fully implemented DISDECO will provide a comprehensive interactive

analysis and design tool that will allow the designer to explore the effects of choice of

shell theory, analysis models, and edge conditions. Also, the database of imperfections

and other stochastic methods for determining effects of imperfections will enable the

designer to obtain reliable designs for stiffened composite shells.


Optimization of Stiffened Panels


Stiffened Panel Optimization Using Approximate Analyses

In this section, design optimization techniques and designs obtained using the

analysis methods or programs described in the previous section are reviewed. Even with

the availability of fast computers, simple analysis methods continue to be used. The

simplicity of the analysis makes it possible to do more complicated optimization, such as

reliability-based design, local-global design, and multi-criteria (cost-weight

minimization) optimization.

A number of papers have been published on design optimization of panels using

PASCO and VICONOPT, but only a sample of those are reviewed here. Butler and his

coworkers [40, 41, 42, 43] have published several papers on the use of VICONOPT [42,

43] for stiffened panel optimizations in aircraft wing design. Butler performed design

optimizations for stiffened panels [40] using different loads as encountered in practical

wing design. The results obtained were then verified using a finite element analysis









program and the PANDA2 program. Integrally stiffened and built up panels were

optimized using VICONOPT. The integrally stiffened (machined) stiffeners provided a

significant weight saving (20%) compared with the conventional built-up stiffeners (with

a bottom flange) when designed for buckling. The weight difference was smaller (3%) for

designs optimized for postbuckling strength. In addition, comparison with finite element

analysis showed very good agreement except for cases with substantial shear loading.

The optimal weights obtained were compared with designs obtained using PANDA2 for a

T-stringer stiffened panel and were almost identical.

Butler [41] demonstrated the use of VICONOPT [42, 43] for an aircraft wing

design. The wing structure was divided into six different panels. The skew panels were

approximated by treating them as infinitely long. Initial minimum weight optimizations

of panels were performed, for a range of loads and panel lengths, to generate design

charts. The design charts were used for selection of stiffener spacing and cross-sections

for panels used in the wing structure. In order to provide compatibility between panels,

the stiffener spacing and cross-section were fixed and the thicknesses optimized in the

wing structure design. Four different panel concepts were considered: metallic blade

stiffener, composite blade stiffener, composite hat stiffener, and composite foam

sandwich with blade stiffener. VICONOPT uses a gradient-based optimizer and can get

trapped in local optima. Butler further showed that design charts obtained for minimum

weight designs by varying loads or panel size helped to identify panels in the wing design

that were trapped in local optima.
















4


Figure 2.4: Blade stiffened panel with voids

Williams and his coworkers presented further demonstrations of the VICONOPT

capabilities for stiffened panel optimization. Williams and Jianqiao [129] used

VICONOPT to optimize blade-stiffened panels with centrally located voids. Figure 2.4

shows a stiffener cross-section with a series of equally spaced voids in its central core

layer, which run for the full length of the plate and have a constant cross-section. The

longitudinal voids were used, in a manner analogous to a core material in sandwich

composite laminates, to increase in the bending stiffness of the laminates used for the

panel skin and blade stiffeners. In addition to optimizing the blade spacing, stiffener

segment width and thickness the optimization can also size the length of voids in the skin

and stiffener segments. This work demonstrated the design features of VICONOPT and

the use of voids for weight savings in composite panels.

Structural optimization of panels can often result in designs that are either

impractical or too expensive to manufacture. There is an increased interest in combining

cost constraints with structural constraints during the optimization. Edwards et al. [53]

replaced the minimum mass objective with minimum cost or a combination of cost and

mass and obtained optimal designs. The VICONOPT program was used for optimization

with new cost functions (material, manufacturing, and operational costs) incorporated.

Panels can be optimized for different combinations of cost and weight. This example









demonstrates the advantage of having inexpensive analysis models for use in expensive

(multiobjective) optimizations.

Kassapoglou [66] performed simultaneous cost and weight optimization of

composite panels under compression and shear loading using a simple analysis. The

panels were analyzed for global and local buckling. The simple analysis used smeared

stiffness for global buckling and simply supported plate models for local buckling.

Analysis results were compared with more detailed finite strip based solution procedures

developed by Peshkam and Dawe [87] and were shown to be reasonably accurate. The

simple analysis procedure enabled the author to perform several design optimizations for

different stiffener geometries at various fixed spacings. The panels were optimized

separately for cost and weight. Cost functions were developed that accounted for

material, manufacturing and operational costs. It was found that the blade stiffener design

was the best minimum cost concept, whereas the J-stiffener design was the best minimum

weight concept. A multiobjective optimization was performed to minimize weight and

cost objectives that were combined using a penalty function. A Pareto-optimal curve was

obtained for the panels that represented a compromise solution between minimum weight

and cost. Also information about dominating failure mode along the Pareto curve was

obtained. Such qualitative information is often useful to the designer in concept selection.


Panel Optimization for Global Optimum and/or Discrete Variables

A distinctive feature of using composite materials is the ability to tailor the

material properties. Typically, this is achieved by optimizing the ply stacking sequence in

the laminate. The programs discussed thus far have all used gradient based optimization

routines. In all cases, the ply thicknesses for a pre-chosen stacking sequence are used as









optimization variables. Composite laminates that are manufactured from pre-pregs (tape

lay-ups) have plies of discrete thickness. Hence, the ply optimization must use discrete

variables. Since, continuous optimization problems are easier to solve often continuous

values are used for ply thicknesses and orientations. This can lead to several problems.

Using ply angles and thicknesses as optimization variables result in non-convex design

space and also lead to multiple local optima. It has also been found that rounding the ply

thickness to thickness of discrete number of plies can lead to sub-optimal designs.

Genetic algorithms (GAs) have been developed for the stacking sequence

optimization of laminates with discrete ply angles and thicknesses [73, 75, 83]. In

addition to having capability to use discrete variables, GAs also provide global

optimization capabilities. The optimization results in a family of optimum designs, with

comparable performance, rather than a single optimum as in the case of gradient based

methods.

Nagendra et al. [81, 82] applied a simple genetic algorithm with crossover,

mutation and permutation operators for blade stiffened composite panel design. The ply

thicknesses, orientations, and blade height were also coded using an integer

representation. The stiffened panel was optimized for minimum weight under inplane

axial compression and shear loads. The panel was required to satisfy buckling, strain, and

contiguity (no more than four plies of the same orientation are stacked together, to

prevent matrix cracking) constraints. The PASCO program was used for analysis of the

stiffened panel. An eccentricity (axial bowing) of 3% was used for the initial geometric

imperfection. To avoid mode interaction effects, penalty parameters were introduced in

the optimization to obtain buckling mode separation. Besides optimizing for discrete









variables, the genetic optimization searches for the global optimum in the design space.

Furthermore, it can provide a family of near optimal solutions (unlike a single optimum

obtained from gradient based optimization). The design obtained using the simple GA

was lighter (25.19 lb) compared to the rounded design (26.08 lb) obtained using a

continuous optimizer. The advantages of global optima and family of near optimal

designs come at a very expensive computational cost because the simple GA requires

tens of thousands of analyses.

Nagendra et al. later presented an improved genetic algorithm [83] that introduced

new genetic operators. In addition to the basic crossover, mutation, and permutation

operators, the following new operators were added: sub-string crossover, stack deletion,

stack addition, orientation mutation, inter-laminate swap and intra-laminate swap

operators. The new features of the improved GA were found to improve the convergence

and reliability of the algorithm. The optimal design weight (24.2 lb) obtained using the

improved GA was lower than the optimal design weight (25.2 lb) obtained using the

simple GA. However GA's are still expensive to use for practical design applications.

Even with the improved efficiency of the genetic algorithm and the efficient analysis

methods (such as in PASCO or PANDA2), performing discrete optimization of stiffened

panels remains a challenging task.

Harrison et al. [61] investigated the use of response surface (RS) approximations

to replace the analyses program in the genetic optimization. A hat stiffened cylindrical

panel under axial compression load was optimized. The stacking sequence of the wall and

stringer segments and the stiffener geometry parameters were used as optimization

variables. In order to fit a response surface, the buckling load and stress failure margins









of panels were obtained using the PANDA2 program. A weighted least square procedure

was used to fit a polynomial approximation. Each laminate stacking sequence was

represented using five lamination parameters. The lamination parameters of the three

laminates and four hat stiffener geometry variables resulted in a total of 19 variables that

were required to describe the response of the panel. A set of 450 points obtained from the

first 15 generations of the genetic optimization were used as fitting points. Once the

response surface was obtained, it replaced the PANDA2 analysis for a number of

generations and then the process was repeated. The polynomial (response surface)

captured the failure envelope of the design space for a large number of designs.

Convergence histories averaged over six trials showed that the RS approximation reduced

the number of generations required for convergence to about 30, from about 95 required

for optimization with exact analysis.

Crossley and Laananen [45] used genetic optimization for design of stiffened

panels for maximum energy absorption. Constraints were imposed on local buckling and

stiffener buckling. Simple closed form solutions were used to obtain buckling loads.

Energy absorption was estimated by a semi-empirical method based on an analogy to the

crippling of metal cross sections [19] with empirical corrections obtained from

experiments. The novel feature of this genetic optimization was that the stiffener type

was also encoded as a design variable. In past work, the geometry was chosen a priori

and then optimized for size. Here, stiffener types such as blade, channel, I-section, hat

section, J-section and angle-section were represented by a binary string along with the

stacking sequence for the different segments. The emphasis was on crashworthiness

design rather than design for instability under inplane flight loads.









Programs such as PANDA2 treat stiffener spacing as continuous design variables.

This can often lead to non-integral number of stiffeners in a cylinder. Jaunky et al. [63,

64] used genetic algorithms for grid stiffened cylinders to optimize the number of

stiffeners, stiffener cross-section geometry, and laminate stacking sequence. A global

smeared model was used to calculate general instability loads. Local buckling was

calculated using a Rayleigh-Ritz analysis of individual skin and stiffener segments.


Panel Optimization for Imperfections, Postbuckling and Damage Tolerance

Buckling loads of thin walled composite stiffened panels are highly sensitive to

geometric imperfections. It has been shown that optimization can heighten imperfection

sensitivity by driving multiple modes to have similar buckling modes, thus increasing

mode interactions. Researchers have investigated extensively the effect of geometric

imperfections on buckling loads and have developed a variety of analysis methods and

tools to incorporate imperfections into the analysis. However, there is still no good way

to incorporate imperfections into design because the imperfection amplitudes and shapes

of panels being designed are not known. Designers often assume worst case scenarios for

the design optimization.

Elseifi, Gurdal, and Nikolaidis [54] developed an elliptical convex model to

represent worst-case geometric imperfections. The convex model provides the

imperfection corresponding to the weakest panel profile and the minimum elastic limit

load. The nonlinear analysis for the panels with imperfections was performed using

NLPAN. Response surfaces were used to approximate the elastic loads. This was done so

as to smooth out the noise from the analysis. Two response surfaces were required (one

each for positive and negative imperfection amplitudes). The results from the convex









model were compared to those obtained using a Monte Carlo simulation. It was shown

that the convex models provide significant computational cost savings over traditional

probabilistic analysis methods.

Elseifi, Gurdal, and Nikolaidis [55] further developed a manufacturing model for

predicting imperfection profiles of composite laminate panels. The manufacturing model

was successfully implemented into the optimization for postbuckling design. It was

demonstrated that the laminate design influences the manufacturing imperfections in the

panel. Therefore, if an arbitrary imperfection is used for designing a panel, the designed

panel will have a very different imperfection when manufactured. The panel so designed

will fail because the imperfection that results from manufacturing was not taken into

account during the design. A closed loop design procedure, where the manufacturing

model is used to obtain the imperfection mode of the designed panel, was implemented.

The resulting designs were compared to those obtained using an empirical imperfection

with non-linear analysis using NLPAN. The designs obtained using arbitrary or empirical

imperfection could still fail at loads smaller than the design loads.

Perry and Gurdal [85, 86] coupled the nonlinear stiffened panel analysis program

NLPAN developed by Stoll and Gurdal [96] with the ADS [116] optimization routine and

used it to design panels for postbuckling response. Weight minimization of stiffened

panels was performed with limit point stress and strain constraints for panels with and

without initial geometric imperfections. The maximum strain failure criterion was used

for limiting strains in the panel segments. Results were compared with designs obtained

using PASCO for buckling constraints. The weight savings were 28% for panels

optimized without initial imperfection and 34% for panels optimized assuming a bowing









imperfection of 1% of panel length. Panels designed for buckling were more sensitive to

imperfections and did not carry the design load even for small values of imperfection.

Panels designed for buckling loads often have poor load carrying capabilities in the

postbuckling regime due to mode interactions. Consideration of the postbuckling

response in the optimization of stiffened shell structures can be greatly beneficial, and the

availability of an efficient code such as NLPAN is very valuable to designers.

Damage tolerance of optimized stiffened panel designs is of practical importance.

Stiffened panels have been optimized for different damage scenarios such as through the

thickness crack, delamination, stiffener failure and other problems. The vast majority of

this work on damage tolerance has been accomplished using finite element analysis

programs. Wiggenraad and Arendsen [125] have demonstrated the modeling of damage

scenarios using finite strip analysis methods. The PANOPT program was used to analyze

panels with damage models. Stress, buckling and postbuckling responses were obtained

to enforce the damage constraints. Optimizations were performed with a single model

and with multiple models that included the damage. The accuracy and effectiveness of

the approach was demonstrated by validating the results from optimization with

experiments.


Stiffened Panel Optimization with Finite Element Analysis Models

Finite element analysis programs that use 2-D discretization (plate and shell

elements) are commonly used for analysis purposes. However, fewer examples are

available for finite element analysis based optimization of stiffened panels. Using finite

element analysis for design optimization of stiffened panels expensive. Stiffened panel

design often involves optimizing for the geometry of the shell and stiffeners, which









frequently requires remeshing for acceptable accuracy. The numerical noise introduced

from the discretization makes it difficult to use gradient based optimization methods.

Often significant time and effort are required to create analysis models and even more

effort is required to implement design constraints for optimization when using FE

models. However, with the availability of modern computers and advances in modeling

and approximation techniques, more papers are beginning to appear that discuss

optimization of stiffened panels coupled with general purpose finite element analysis

software. A few examples are mentioned here. Tripathy and Rao [110] used finite

element model for optimization of stiffened panels for maximum buckling strength

obtained using a linear bifurcation buckling analysis. Eschenauer et al. [56] have

developed new modeling techniques that minimize the effort required for adaptive mesh

generation required in shape optimization of stiffened panels. More recently Vitali et al.

[122, 123] have used finite element analysis for optimization, with correction response

surface techniques. The methodology employs detailed finite element analysis models in

tandem with the approximate low cost analysis models. A correction function is fitted to

the ratio of the accurate result from the expensive analysis and less accurate analysis at a

small number of points using a least square regression. The obtained correction function

is used to correct the less approximate analysis. The corrected model was then used in the

design optimization of stiffened panels for buckling strength [122] and crack resistance

[123].

















CHAPTER 3
PANEL DESIGN STUDIES USING PANDA2


Preliminary design optimizations of stiffened panels for propellant tanks or launch

vehicles are often performed to obtain accurate weight estimates in the concept selection

phase. A large variety of analysis and design codes are available for stiffened panel

optimization. Chapter 2 presents an overview of available programs and methods for

stiffened shell design and analysis. The present section is focuses on PANDA2, one of

the premier programs for analysis and optimization of cylindrical composite stiffened

panels under buckling, out-of-plane displacement and stress constraints. The program

uses a combination of closed form solutions and discrete models for buckling load

prediction, and it accounts for a large number of failure modes. A significant feature of

PANDA2 is the large number of design constraints it has implemented that are based on

the developer's expertise in the design of shell structures.

Many papers describing PANDA2 have been published by its developer, David

Bushnell. However, much less literature is available on its use by others. Here, the

present capabilities, modeling, and theories of PANDA2 are summarized with references

to original publications for additional details. The objective is to report on the usefulness

of the PANDA2 program for the large number of stiffened panel optimizations required

at the preliminary design stage.









Computational resources, cost, and design cycle time are often limited.

Compromises are required in the model, analysis and optimization complexities to meet

such restrictions. The present work presents issues of the modeling and analysis

complexity involved in stiffened panel design. Modeling choices and features in

PANDA2 are explored. In particular, the difference in the use of closed form and

discretized stiffener models in PANDA2 and the different options to account for

geometric imperfections are investigated. This chapter also presents the results from

optimization of metallic and composite stiffened panel concepts considered for a liquid

hydrogen RLV tank. Weight efficiencies and sensitivity to geometric imperfections are

compared for the different concepts.


The PANDA2 Program

PANDA2, developed by Bushnell [25-33] at Lockheed Palo Alto Research

Laboratory, is a program for preliminary design of cylindrical composite stiffened or

unstiffened panels for minimum weight or distortion. Panels can have stiffeners along

one or two orthogonal directions with blade-, hat-, T-, J- or Z-shaped cross-sections.

PANDA2 is also capable of analyzing panels with Z-shaped stiffeners that are attached

by riveting. The riveted attachment significantly affects the buckling response of the

panels. PANDA2 can also be used to analyze and optimize sandwich panels with either

hexagonal core or foam core, truss core sandwich panels, and isogrid panels.

The philosophy of PANDA2 is to provide at low cost optimum preliminary design

of stiffened panels that experience complex and nonlinear behavior. PANDA2 uses

several relatively simple models each designed to capture a specific failure mode or

mechanism. PANDA2 uses a combination of approximate physical models, exact closed









form (finite strip type analysis) models and 1-D discrete branched shell analysis models

(based on one dimensional discretization) to calculate prebuckling, buckling and

postbuckling response. For details of the models see Bushnell [26], [27], and Bushnell

and Bushnell [35] and [37].


PANDA2 Capabilities


Analysis Models

There are five model types. The first model is based on closed form models

(PANDA type, [26]) for general, local and panel buckling, bifurcation buckling of

stiffener parts and rolling of stiffeners with and without participation of the skin. This

model represents the buckling mode displacement components u, v, w as one-term

Rayleigh-Ritz expansions. Buckling load factors are computed for each combination of

modal wave number and slope of buckling nodal lines, (m,n,s), from Eq. (57) in Bushnell

[26].

The second model type obtains buckling load factors and local postbuckling

response from a skin-stringer "panel module" or "repeating module" model [25]. A

module (Figure 3.1) includes a length of panel between rings and a cross-section of a

stringer plus a portion of the panel skin of width equal to the spacing between stringers.

The segments of the skin-stringer module cross-section are discretized. Buckling load

factors and mode shapes are obtained via a finite difference energy formulation

analogous to that used in BOSOR4 [23, 24].















S



SEGM






Figu


LAYER 1

w (SEGMENT, %t(4,11)
NODE) = (4,1) (311) LAYERj
SEGMENT NO 4
SEGMENT NO 4
SEGMENT NO 3 L 1A
LAYER 1 4 LAYER k
SEGMENT NO 2
ENT NO 1 SEGMENT NO 1
AGAIN LAYER 1 (3,1) LAYER 1
b2
1-* -- b2 --(1,1) (1,11) (2,)- -(2,I ) (5'1) (5,11)
| lMODI FIn WIDTH STTFFRNFR LAYER m
SPACING, B LAYER m LAYER n


re 3.1: View of single module of panel with T-shaped stringer, showing layer
numbering convention and discretization used.


* . *I
*
*


4- MODULE NO. 1 -4- MODULE NO. 2 - MODULE NO. 3 -


Figure 3.2: PANDA2 repeating module for a panel with T-shaped stiffeners.


A repeating module of a T-shaped stiffener is shown in Figure 3.1. The panel


module treats the stiffener as a branched shell assembly. Each segment of the panel


module can have a different laminate. The exploded view of the panel module in Figure


3.1 shows the node numbering used in the discretized analysis as well as the convention


of layer numbering (stacking sequence specification). The numbers in parentheses in the


exploded view indicate the segment and node numbers. Symmetry boundary conditions


are used at Segment 1, Node 1 and Segment 5, Node 11. The variation of deflections in


the direction transverse to the discretization plane (normal to the plane of the paper) is


assumed to be harmonic. This discretized model is similar to that used in BOSOR4 for


analysis of shells of revolution [24]. The PANDA2 model for stiffened panels is valid









only when there are more than two repeating elements as shown in Figure 3.2. Panels

with a single stiffener cannot be handled using PANDA2. The single module gives good

approximations for local skin buckling, wide column buckling, and buckling of stiffener

parts.

In the third model, the entire width of the panel is discretized. The stiffeners are

smeared out. This model is used to capture the effect of loads that vary along the width of

the panel or to capture the prebuckling behavior of flat panels if normal pressure is

present. Stresses and displacements calculated from this model are used for buckling

analyses.

The fourth model [33] is analogous to the discretized skin-stringer module of the

second model. In the fourth model, the ring cross-section is discretized and a segment of

panel skin with smeared stringers equal to the spacing between rings is also discretized.

Buckling load factors and mode shapes are obtained via the finite difference energy

formulation that is used in BOSOR4 [24]. The model is useful in obtaining inter-ring

buckling loads, ring segment buckling loads, and prebuckling displacement along the

meridional length.

The fifth model [33] is analogous to the first model. This model is based on a

double trigonometric series expansion of the buckling modal displacement components,

u, v, w. This model is especially useful for panels in which in-plane shear loading plays a

significant role or the panel has significant anisotropic terms, that is, terms that govern

coupling between normal and shear stress and moment resultants.

PANDA2 performs the buckling analysis using closed form solutions [26] or 1-D

discrete models (e.g., Figure 3.1) [25, 33]. The type of analysis used in optimizations is









chosen by setting the value of the IQUICK flag in the "mainsetup" processor execution.

When IQUICK=1, closed form solutions (Models 1 and 5) are used for the buckling

analysis. The discretized BOSOR4 type model of the repeating skin-stringer module

(Model 2) is included in the buckling analysis when IQUICK=0. The 1-D skin-ring

discrete module (Model 4) in the length direction is used to calculate inter-ring buckling

analysis with both IQUICK= 0 and IQUICK=1. The analysis performed with the 1-D

skin-stringer discrete module is computationally more expensive and is not available for

all types of stiffeners.


Analysis Procedure

Bushnell describes the analysis features of PANDA2 in detail in references [26-

33]. However, for the sake of convenience some of the essential features of the analysis

are summarized here. The analysis summary includes calculation of constitutive relations,

equilibrium behavior, and buckling analysis.

The constitutive relations are developed in stages and account for general

laminated components of each segment of a module. The constitutive matrix [C] is a 6x6

matrix that relates the reference surface strains and changes in curvature and twist to the

force and moment stress resultants. PANDA2 computes the integrated constitutive law

for each segment of a panel module. In addition, thermal resultants and strains from

curing and from applied thermal loading are calculated for each segment. For stiffened

panels, a smeared stiffener theory is used and an integrated equivalent constitutive matrix

Cs is computed for the smeared approximation. The tangent stiffness of the locally

buckled skin Ctan is also computed in PANDA2. In case of stiffened panels with locally









buckled skin, the reduced stiffness for the smeared model due to local buckling Cstan is

computed using Ctan for the panel skin between the stiffeners.

The prebuckling analysis of PANDA2 includes the membrane and, if the panel is

cylindrical, axisymmetric bending effects, using nonlinear or linear analysis for response

to pressure loads. The prebuckling analysis uses a smeared model of the global model

and a discrete model for the response of a single panel module. In a ring-stiffened shell,

the models used for prebuckling analysis calculate stresses at the mid-bay and at ring-

web intersection in order to obtain buckling loads. The overall static response of the

global model (smeared) and the local static response of the panel module are combined

together to get the total state due to applied pressure. Strain and stress resultant

distributions in all panel modules are determined using (i) all loads except normal

pressure and (ii) only normal pressure load. The effect of bowing of the panel due to

curing, applied thermal loads, normal pressure and edge moments is included along with

initial geometric imperfections in the form of general, inter-ring and local buckling loads.

An out-of-roundness (ovalization) type imperfection is also included.

Stresses are calculated for all layers of each segment in the material direction. The

stress states of the panel module are calculated either for an unbuckled or postbuckled

state, whichever is applicable. Tensile forces in parts of the stiffener web that tend to pull

the web from the panel are calculated to provide safety against stiffener-skin separation.

Von Mises stress criterion or maximum stress criterion can be chosen for isotropic

materials. For composite materials, maximum stress criterion is used for ply failure.

PANDA2 also models transverse ply cracking. The transverse stiffness of plies that have

exceeded the allowable stress is reduced to zero to account for transverse ply cracking.









PANDA2 computes buckling load factors using closed form solutions (called

"PANDA type" [26]) for general instability, local buckling of the panel skin, local

buckling of stiffener segments, rolling of stiffeners with and without stiffener

participation. General instability is predicted using a model in which the stiffeners are

smeared out as prescribed by Baruch and Singer [18]. The local instability load is also

obtained using a discrete model (Figure 3.1) of a single module along either the

circumferential or axial direction. The discretized model is also used for calculating wide

column bucking. The discretized model accounts for local bending of the skin and

deformation of the stiffener parts in the wide column buckling mode. These details

cannot be captured using the smeared model. PANDA2 also allows design optimization

of panels with local postbuckling. The theory used in PANDA2 for postbuckling is

similar to that formulated by Koiter for panels loaded into the far postbuckling regime

[71]. Local skin postbuckling is captured in PANDA2 [29] by the second model, the

discretized skin-stringer repeating module.

PANDA2 subroutines, PANEL and STAGSMODEL, can generate input files for

more detailed BOSOR4 [26, 33] and STAGS [34, 90] analysis. This provides the user

with an easy way to check the final optimum using more rigorous analysis models.


Analysis and Design with Effects of Initial Imperfections

Shape imperfections greatly affect the load carrying capacity of cylindrical shells

under axial compression. However, it is often difficult or impossible to choose a priori an

imperfection shape for design optimization of the shell structure. A considerable amount

of research has been done in the area of imperfections. Stochastic approaches to account

for the effect of imperfections on panel instability have been developed [46, 54, 74].









These methods are expensive to use and require a database of imperfections based on the

manufacturing process. More recently, manufacturing models have been proposed to

predict manufacturing induced geometric imperfections [55].

PANDA2 considers a variety of modal imperfections, such as global, local, inter-

ring, and out-of-roundness imperfection [35, 37]. A user supplied imperfection amplitude

is applied to the corresponding critical buckling mode shape of the panel. For cylindrical

panels, the imperfections affect the buckling load factors calculated by closed form

solutions directly by changing the effective radius of the shell. Additionally, global or

bowing imperfections affect the distributions of the prebuckling stress resultants over the

various segments of the panel, which in turn affects the buckling loads.

PANDA2 calculates knockdown factors to reduce buckling loads due to

imperfections. The buckling loads are calculated for general instability (skin, stringers,

rings all buckle together, smeared), inter-ring buckling (skin and stringers buckle

between rings), and local buckling (skin buckles between adjacent stringers and rings).

The knockdown factors are calculated using closed form solutions of both perfect and

imperfect cylindrical panels, in which the imperfections (global, inter-ring, local and out-

of-roundness) are used to reduce the radius of curvature of the portion of the panel used,

thereby reducing the buckling loads. Imperfections also cause stress redistribution due to

prebuckling bending caused by the load eccentricity in initially imperfect panels. More

details of the implementation can be obtained from Bushnell and Bushnell [35, 37]. The

knockdown factors are reduced further by the ratio Arbocz/PANDA for the perfect panel

if the ratio is less than unity. Arbocz theory [9] accounts for the induced prebuckling

membrane hoop compression generated in a cylindrical shell with axisymmetric









imperfections subjected to uniform axial compression. More details of the Arbocz theory

implementation in PANDA2 can be found in Bushnell [30].

In practice, since the shapes of the initial imperfections are unknown (buckling

modes), it is difficult for the user to judge whether or not the chosen imperfection

amplitude is reasonable. Use of a fixed value for the imperfection amplitude can lead to

non-conservative or over-conservative designs. To overcome this, PANDA2 provides an

option to automatically adjust the value of the buckling modal imperfection amplitude

supplied to the program. PANDA2 reduces the value of the imperfection amplitude that it

judges to be larger than that which would be easily detectable by the most casual

inspection. For a given value of imperfection amplitude, imperfection shapes with shorter

(axial and circumferential) wavelength are more easily detectable than imperfection

shapes with long wavelengths because the high curvatures are easily detected. PANDA2

reduces the imperfection amplitude to a value that will give no more than 0.1 radians wall

rotation if the user selects the option to adjust imperfection amplitudes. This option is

used in the present design optimizations of panels with imperfections.

PANDA2 applies the given amplitudes to the shape of the critical buckling modes

for general, inter-ring, and local buckling models. The shapes of the imperfections

change during optimization as the critical buckling mode changes. The out-of-roundness

does not have wavelength and therefore stays the same. The buckling mode and the

corresponding number of half-waves is determined from closed form solution [26]. For

discrete models, the local buckling mode is predicted from the single skin-stringer

module; the inter-ring buckling mode is predicted from the single "skin"-ring module,

where "skin" denotes skin with smeared stringers.









The buckling mode is introduced in the imperfection sensitivity analysis as a

harmonic function of the x and y coordinates and the unknown number of half-waves m

(axial) and n (circumferential) [26]. The amplitude of the harmonic function is set to the

modal imperfection amplitude (provided by the user and possibly adjusted by PANDA2).

The strains are then calculated as the sum of two distinct contributions: the strain induced

by the applied loads acting on the perfect shell and the strains induced by amplification of

the modal imperfection shape during loading. Then the stresses and the resultant re-

distributed stress resultants are calculated using the stress-strain relation. The buckling

analysis is carried out for the imperfect structure with closed form solutions [26] or 1-D

discrete models [25, 33].

Designing shell structures without considering the effects of initial imperfections

can lead to critical designs. Initial imperfections have two effects. First, they reduce the

effective radius of curvature of cylindrical panels leading to lower buckling loads.

Second, they give rise to prebuckling bending which leads to local increases in

destabilizing stress resultants in parts of the segmented structure. The wide column

buckling model ignores curvature and calculates the column buckling failure load. The

wide column buckling model compensates for the first effect and can be used as an

alternative to designing without imperfections. However, there are significant modal

interaction effects in flat panels that are not captured by the wide column buckling model

(see Van der Neut [113]). A comparison of designs obtained without imperfections to

those obtained with wide column buckling and with imperfections is presented in the

Modeling Issues section in this chapter.









Optimization Capabilities

PANDA2 can analyze and optimize panels with up to five different load sets that

are combinations of in-plane loads, edge moments, normal pressure and temperature

gradients. Design constraints include general, local, and stiffener buckling; lateral

displacement under pressure; and stress failure (in plies). PANDA2 can optimize stiffener

spacing, cross-sectional dimensions, ply thicknesses and orientation angles of the

laminates. The PANDA2 optimization is performed using the gradient based ADS

optimization subroutine developed by Vanderplaats [116].

PANDA2 performs gradient based optimization and therefore can use continuous

design variables. Optimum designs obtained with continuous values for ply thicknesses

are not practical as they have to be integer multiples of the commercially available tape

or pre-preg ply thicknesses. Rounding ply thicknesses often results in sub-optimal

designs. Hence, ply thickness of the preliminary optimum designs were rounded to the

nearest integer ply thickness value and re-optimized again for the other design variables

(excluding ply thickness variables).

PANDA2 uses the method of feasible directions [116] for constrained

optimization. The line search (gradient based) is controlled by the PANDAOPT processor

of PANDA2. PANDAOPT performs successive line searches with a new search direction

vector calculated at the new design points. The gradient vector required to obtain the

search direction is calculated from a forward finite difference calculation. In order to

ensure convergence, the move limit is reduced by 50% after each successive iteration.

The iterations are stopped when the line search converges to an optimum or when the

number of iterations has reached the user-specified maximum. Bushnell [25]

recommends that a small number of iterations be used in each PANDAOPT execution









and that many PANDAOPT executions be performed to obtain an optimum. However,

for cases where the constraints are highly oscillatory, the user should specify a large

number of iterations in PANDAOPT to ensure convergence.

PANDA2 also has the capability to perform global optimization [30] with the use

of the SUPEROPT processor. The global optimization strategy of PANDA2 is based on

automated random multiple restarts of gradient based (line) searches to locate the global

optimum. SUPEROPT performs a user specified number of PANDAOPT executions with

an intermittent restart (AUTOCHANGE process). A total of 275 iterations (line searches)

are performed for each SUPEROPT execution.

The AUTOCHANGE process of PANDA2 randomly changes the vector of design

variables (Item 51 of "panda2.news" [25]). The AUTOCHANGE processor changes the

design variables as follows:

y, = (1+ &, ), i= 1,2,3,..., number of design variables (3.1)

in which x, is the old value of the ith design variable, y, is the new value and x&, is the

relative change in the value of the variable, and takes a random value between -0.5 and

1.5 (except for stiffener spacing, where a range between -1.0 and 1.0 is used). The

perturbed design then provides a new initial point for the optimization.

Bushnell recommends that initial panel optimization in PANDA2 be performed

with closed form solutions (IQUICK=1) to locate a near optimal design with small

computational effort. The obtained results must be analyzed using discrete models (of

stringer-skin module in IQUICK=0 option), and if they are not satisfactory, optimized

again using the IQUICK=0 option. The recommended practice is useful because the

discretized solutions can result in discontinuous behavior at the critical loads because the









failure mechanisms change significantly when small perturbations are made to the design.

Such changes in failure mechanisms cause problems for gradient based optimization.

Another advantage of using the IQUICK=1 option is that when geometric

imperfections (in the shape of critical modes) are included in the design, the analysis

estimates stresses and buckling loads for different combinations of positive and negative

imperfection amplitudes. The IQUICK=0 option requires the designer to add new load

cases that have different combinations of the positive and negative imperfection

amplitudes. Modeling general and inter-ring imperfection with both positive and negative

signs therefore results in four different load cases. Optimum designs obtained using

IQUICK=0 and IQUICK=1 are compared in the Modeling Issues section. The required

computation (CPU) time is also presented for the optimizations. Computers have become

faster and cheaper than they were at the time when PANDA2 was initially developed. It

appears that with the availability of faster computers, the user should use the IQUICK=0

option whenever possible

Papers comparing PANDA2 analysis and design optimization results using either

the STAGS program [34] or experiments have been published. Bushnell and Bushnell

[36] optimized composite stiffened panels under combined loads and verified the

optimized designs using STAGS. Bushnell et al. [38] also validated PANDA2 designs

with experiments. The Modeling Issues section presents finite element analysis

verification of panels optimized using PANDA2.

The most challenging task in stiffened shell optimization is the selection of

appropriate modeling, theories and design constraints. Programs such as PANDA2









partially automate this task and thereby serve as an expert system for stiffened shell

design.


Reusable Launch Vehicle Propellant Tank Design

Unlike expendable launch vehicles, reusable launch vehicles (RLV) have tanks

that are an integral part of the vehicle structure. RLVs require extra fuel to carry the tank

structures through the entire mission. It is hence of paramount importance to reduce the

structural weight of the tanks. Selection of structural concepts for RLV tanks is therefore

driven primarily by minimum weight design. In the present work, different stiffened

cylindrical panels were optimized for use in the liquid hydrogen tank of RLVs. The

minimum weight structure is obtained for given loads and is designed with buckling,

strength (stress failure), and strain constraints.


Figure 3.3: Schematic of a Reusable Launch Vehicle

The RLV considered here is a lifting body (see Figure 3.3) which has a circular

tank structure with wings attached. The radius of the tank is 160 inches. The cylindrical

portion of the tank is manufactured in short barrel sections and assembled to form the









entire tank. The attachment line between the barrel sections has a substantial ring frame.

For the preliminary design trade studies, a barrel length of 300 inches was used. The

cylinder geometry is modeled with different longitudinal (stringers) and circumferential

(ring) stiffeners.


'able 3.1: Panel concepts, stiffener locations, and materials considered for the RLV
liquid hydrogen tank design

Panel Type Stringers Rings Material
Aluminum T-Stringer and J-rings Internal External Al-2219 T87
Aluminum isogrid stiffened panel Internal blade External Al-2219 T87
isogrid J-rings
Aluminum orthogrid stiffened panel Internal blade Al-2219 T87
stiffeners
Titanium symmetric sandwich T rings N/A Internal Ti-6A1-4V
Titanium asymmetric sandwich T rings N/A Internal Ti-6A1-4V
Titanium truss core sandwich Hat-shaped N/A Ti-6A1-4V
corrugation
Composite T stringer and T rings External External IM7/977-2
Composite symmetric sandwich T rings N/A Internal IM7/977-2
Composite asymmetric sandwich T rings N/A Internal IM7/977-2
Composite truss core sandwich External corrugated N/A IM7/977-2


skin


The design concepts, stiffener location, and materials used in the design of RLV

liquid hydrogen (cylindrical) tanks are shown in Table 3.1. The selection of stiffener

types and their positioning (external vs. internal) is based on manufacturing

consideration, type of thermal protection system (TPS), and the TPS attachment method

used for the vehicle. Both metallic and composite panels were designed. Material

properties are given in Appendix A. The aluminum and titanium alloys used for metallic


I









concepts had the same specific stiffness, but the titanium alloy had 45% higher specific

strength. The composite material used was the IM7/977-2 graphite epoxy system.

Composite panels were optimized with fixed lay-ups, or with some ply

thicknesses included as design variables. Table 3.2 shows the laminates used in the

optimization and the ply thicknesses that were used as variables. The laminate thickness

bounds are also shown. Skin laminate thickness of stiffened panels (thickness of inner

facesheets in the case of sandwich) was required to be at least 12 plies (0.06 inch thick) in

order to avoid the liquid hydrogen permeation.


Different laminate lay-up used for composite panel design

pt Shell segment Lay-up design


Panel with
T-stringers and T-rings


Sandwich Panel
with T-rings





Truss core Panel


Wall
Wall
Stiffeners
Stiffeners
Stiffeners
Inner facesheet
Outer facesheet
Wall
Stiffeners
Stiffeners
Skin


[(+65/-65)3 S
[+45/903/-45/03/-45/903/45] T
[+45/-45/On-45/+45/0js
[+45/-45/0 n/-45/+45/0m]A
[+45/90 n /-45/0m]s
[+45,n/902/-45n3/0ms
[+45,n/902/-45n3/0ms
[(+65/-65)3] S
[+45/-45/On-45/+45/0m]s
[+45/90n /-45/0m]s
[(+65/-65)3]


Laminate
thickness
(inch)
0.06
0.065
0.055 to 0.100
0.055 to 0.100
0.035 to 0.080
0.06 to 0.14
0.035 to 0.14
0.06
0.055 to 0.100
0.035 to 0.080
0.06


In the design, 0.005 in thick plies were used and lay-ups were chosen so that no

more than four contiguous plies have the same orientation (n, n], n2 and n3 in Table 3.2

can be up to 4 while m is limited to 2 due to symmetry). Continuous optimization of ply

thicknesses was performed first. A new optimization was performed after rounding the


Table 3.2:


Conce









ply thicknesses to an integer multiple of 0.005 inch (thickness of pre-preg plies used).

This strategy was applied to all the composite panel concepts studied in this work.





Stiffening Concepts, Geometry and Design Variables

PANDA2 analysis uses a single repeating element module along the longitudinal

and circumferential directions. Figure 3.4 shows the stiffeners that were used in the

optimizations. The isogrid and the stringer-stiffened panels have external J-rings in order

to provide attachments for the TPS. Internal rings were used for the sandwich case

because the foam type insulation used requires a smooth outer surface.

Stiffener top flange

IStiffener web
Skin Stiffener bottom flange



Figure 3.4: Schematic of a T, J, blade, and Hat stiffener geometry.

Table 3.3 lists the design variables for stringer-ring stiffened panels. The upper

and lower bounds are also provided. Design variable bounds for composite panels are

shown in parentheses only when they differ from the corresponding bounds used for

metallic panels. This notation will be used in the rest of the paper.

Aluminum isogrid panel (see Figure 3.5) has blade stiffeners that delimit

equilateral triangles. In this study, the isogrid has stiffeners running along the

circumferential direction because the load cases considered produce substantial hoop

stresses. In addition to the internal isogrid stiffeners, the panel also has external J-ring








stiffeners for attaching the TPS. The design variables and their bounds are described in

Table 3.4.

Aluminum orthogrid panels have blade stiffeners that run along the longitudinal

and circumferential directions, as shown in Figure 3.6. Design variable linking is used to

maintain identical stiffener cross-section along the longitudinal and circumferential

directions. However, stiffener spacings are not linked. The design variables and their

bounds are listed in Table 3.5.


Circumferential

Cylinder with internal isogrid and external rings, circumferential Isogrid
pattern


\x


y __TII


Figure 3.6: Schematic of orthogrid stiffening concept


Figure 3.5:










Design variables and bounds (in inches) used in the optimization of
stringer-ring stiffened panels (Values shown in parentheses correspond to
composite panels when different from metallic designs)


T-stiffener stringer geometry variables


J-stiffener ring geometry variables


Description


stringer spacing

width of stringer bottom
flange
height of stringer
width of stringer top flange

skin thickness

stringer bottom flange
thickness
stringer web thickness
stringer top flange thickness


Lower Upper
bound bound
5.0 30.0


0.4
(1.0)
1.0
0.4
(1.0)
0.08
(0.06)


Description

ring spacing


3.0 width of ring bottom
flange
2.0 height of ring
3.0 width of ring top flange


0.25
(0.06)


skin thickness


ring bottom flange
0.05 0.25 thickness
(0.055) (0.10) ring web thickness
ring top flange thickness


Lower Upper
bound bound


10
(12)
0.4
(1.5)
2
0.4
1.5
0.08
(0.06)


30
(36)
3.0
(2.5)
4
(3.0)
2.5
0.25
(0.06)


0.05 0.25
(0.055) (0.10)


Design variables and bounds (in inches) used in the optimization of
aluminum isogrid stiffened panels


Description

Skin thickness
Isogrid module size (spacing)
Isogrid web height
Isogrid web thickness
ring spacing
Width of ring bottom flange
Height of ring
Width of ring top flange
Thickness of ring bottom
flange
Thickness of ring web
Thickness of ring too flange


Lower
bound
0.08
10
1
0.03
30
0.4
2
0.3
0.03

0.03
0.03


Upper
bound
0.25
25
5
0.25
120
4
5
3
0.25

0.25
0.25


Table 3.3:


Table 3.4:










Design variables and bounds (in inches) used in the optimization of
aluminum orthogrid stiffened panels


Description
Stringer spacing
Ring spacing
Height of stringer
Skin thickness
Thickness of stiffener bottom flange
Thickness of stiffener web


Lower bound
5.0
5.0
1.0
0.08
0.05
0.05


Upper bound
20.0
60.0
2.0
0.25
0.25
0.25


7C


%t =0.002 inch

0'S


L direction


Figure 3.7: Honeycomb sandwich laminate


Design variables and bounds (in inches) used in the optimization of
honeycomb core sandwich panels


Description
Facesheet thickness

Core thickness
Ring spacing
Ring bottom flange width
Ring height
Ring top flange width
Ring bottom flange thickness
Ring web thickness
Ring top flange thickness
Honeycomb core cell diameter
Thickness of hex-cell wall


Lower bound


0.001
(0.06 internal; 0.035 external)
0.25
30(10)
1.5
2
1.5
0.01 (0.035)
0.01 (0.035)
0.01 (0.035)
0.125
0.002


Upper bound
0.1
(0.14)
2
120
2.5
4
2.5
0.25 (0.1)
0.25 (0.1)
0.25 (0.1)
0.375
0.002


Table 3.5:


Table 3.6:









Ring stiffened sandwich panels with titanium or composite facesheets having an

(expanded ribbon) honeycomb hexagonal core made of Titanium (Ti-6A1-4V) and brazed

to the facesheets were also studied. Manufacturing imposes a constraint that the core cell

wall (to) has to be at least 0.002 inch. PANDA2 can size the core diameter and cell wall

thickness. A typical sandwich laminate and a transverse section of its hexagonal

honeycomb core are shown in Figure 3.7. The design variables and their bounds are listed

in Table 3.6. Panels were optimized using a value of 10-3 for the ratio of initial facesheet

waviness to facesheet wrinkling half-wavelength, as recommended for panels of good

quality [31 ].



Web-skin contact segment Upper skin


Height
-I -- --- -- --

Pitch Lower skin

Figure 3.8: Schematic of the truss core sandwich module

Sandwich panels were designed with both fixed and varying core thickness. The

sandwich core for fixed thickness may add to the weight of the panel. However, the

increased thickness gives higher insulation capacity for the wall construction and can

result in total weight savings from the reduced external insulation required for the

hydrogen tank. Therefore, for RLV tank designs, core weight is not considered as

structural weight but as insulation weight. In this study, for structural design

comparisons, this weight is part of the panel weight.

Asymmetric sandwich wall constructions were also investigated. To minimize

hydrogen permeation internal facesheets were required to have a minimum thickness of









12 plies. Since the outer facesheet was not required to have the 12-ply thickness,

optimizations were performed for asymmetric sandwich constructions. In order to

provide comparisons to the composite designs, similar optimizations were also performed

for the titanium honeycomb core panels.

The metallic truss core panel is a corrugated panel with smooth face sheets and is

designed using the titanium alloy to keep it consistent with metallic sandwich concepts.

The schematic of the truss core concept as treated in PANDA2 is shown in Figure 3.8.

The design variables and their bounds are listed in Table 3.7.


Table 3.7:


Design variables and bounds (in inches) used in the optimization of truss
core sandwich panels


Description
Pitch
Length of the contact segment skin-web
Height
Thickness of lower skin
Thickness of web
Thickness of upper skin
Web or corrugation angle cx


Lower bound
1.0
0.2
0.5
0.01 (0.06)
0.01 (0.035)
0.01
450


Upper bound
5.0
1
1.5
0.1 (0.06)
0.1 (0.1)
0.1
450


Design Loads and Safety Margins

Only a small number of load cases are used for preliminary design optimizations.

In the present study two loading conditions were selected: (i) Internal proof pressure of

35 psi; (ii) Axial compressive load Nx = 1000 lb/in, with an internal (stabilizing) pressure

of 5 psi. The selected load cases were the most critical for stress failure and bucking

failure, respectively.









The following safety margins were used in the optimization: general buckling

safety factor of 1.4; all other buckling and stress safety factors equal to 1.2. Stress

margins are calculated using the Von Mises stress criterion for metallic alloys and

maximum stress failure for composite materials. In case of metals, stress margins are

calculated taking the yield stress as stress limit. See Appendix A for stress limits.

For the present design study, the general buckling load factor is higher than the

local buckling load factor. This implies that the panel can undergo local buckling before

it fails in general buckling mode. Local buckling results in softening of the skin, increase

in stresses, and amplification of initial imperfections, thereby reducing the general

buckling load.

PANDA2 is capable of capturing the effects of local buckling on general

instability for panels with stringers or both stringers and rings. Local postbuckling

analysis is performed with the use of a discretized stringer-skin module (IQUICK=0) in

PANDA2. This model ignores the curvature of the stiffened shell. Therefore, PANDA2

local postbuckling analysis cannot be used for deeply curved shells. In addition, the

discretized analysis (IQUICK=0) is not available for shells without stringers. In the

present study, stringer-ring stiffened panels optimized with effects of local buckling on

general instability were compared with panels designed with safety factors applied to

buckling loads from linear analyses. These results are presented in the next section.

However, local postbuckling was not permitted in panels that were optimized for

comparing the different stiffened panel concepts.









Modeling Issues

With PANDA2 employing a variety of approximate models, one may ask what

are the effects of the approximations. In addition, PANDA2 offers the user the choice

between more approximate and faster analyses (IQUICK=1) and accurate and slower

analyses (IQUICK=0). The choice could depend on the required accuracy and

computational effort. The PANDA2 accuracy issues have been addressed by Bushnell

[34] and his coworkers [38]. Here, the effect of the different approximations and

modeling choices available in PANDA2 on optimum stiffened panel designs are

investigated. PANDA2 analysis results are also compared with results from detailed finite

element analysis models.


Comparison of PANDA2, BOSOR4 and STAGS Analysis Models

PANDA2, BOSOR4, and STAGS represent programs in increasing order of

modeling fidelity, model time preparation, and computational expense. The PANDA2

1-D discretized models [20, 34] are similar to those developed for the BOSOR4 program.

However, unlike the BOSOR4 program in which the entire shell along one direction can

be modeled, PANDA2 assumes a repeating pattern for the stiffeners and therefore uses

only a single repeating (skin and stiffener) module with appropriate boundary conditions.

BOSOR4 is an energy-based discrete analysis method where the model is

discretized along the meridian of a shell of revolution. BOSOR4 can also be used for

prismatic structures such as stiffened panels [25]. For axisymmetric geometry and loads,

BOSOR4 can obtain very accurate solutions. The 1-D discrete models in PANDA2 are

based on BOSOR4 analysis models.









Using non-linear 2-D finite element models provides a more detailed analysis of

stiffened shells. Finite element analysis allows the designer to model the shell structure

and support conditions more accurately. However, the cost of analysis and modeling

increases sharply with addition of such details. A variety of commercial finite element

programs are available to users. In the present study the STAGS program [4, 90] was

used. STAGS is a finite element code for general-purpose nonlinear static and dynamic

analysis of shell structures of arbitrary shape and complexity. Its capabilities include

stress, stability, vibration and transient analysis using both material and geometric

nonlinearities. The element independent fully co-rotational procedure [89] implemented

in STAGS allows large rotations required for nonlinear analysis. In addition,

incorporation of the Riks arc-following algorithm [93] permits STAGS to analyze

stiffened panels in the nonlinear regime.

Three examples are used here to illustrate the accuracy and computational effort

of PANDA2 analysis. The examples chosen to investigate PANDA2 analysis are a

stringer stiffened plate, a cylindrical stringer-ring stiffened panel, and a cylindrical

honeycomb core sandwich panel with ring stiffeners. The second and third examples are

panels designed for the reusable launch vehicle liquid hydrogen tank. Optimized stiffened

panel designs are chosen, as it is known that optimizers often exploit model/analysis

weaknesses and obtain poor designs.

The first example is the analysis of a T-stiffened flat panel with three stringers

optimized for minimum weight and designed to resist buckling under a compressive axial

load of 1000 lb/in. The predicted buckling loads and corresponding computational times

required by PANDA2, BOSOR4, and STAGS are compared in Table 3.8. The results









demonstrate the accuracy of PANDA2 for regularly stiffened panel structures under

uniform loads and gives evidence of its high computational efficiency. Figure 3.9 shows

the buckling mode shapes from BOSOR4 and STAGS. A more significant difference that

cannot be quantified is the effort required to generate the analysis models. PANDA2 has

implemented in its library a variety of routines for the different stiffened panel models

with the design constraints necessary to prevent generation of poor designs. The

implementations of such constraints for use with general-purpose finite element programs

would require substantial investment of time by the designer.


Table 3.8: Buckling loads and analysis time for PANDA2, BOSOR4 and STAG
models for an optimized stringer stiffened plate.

PANDA2 BOSOR4 STAGS
Buckling load factor 0.7781 (closed form) 0.7821 0.7848
0.7751 (1-D discrete module)
CPU time per analysis (s) 2.49 7.30 970.1










Figure 3.9: Comparison of the buckling mode shapes obtained with BOSOR4 and
STAGS

The second example is an aluminum stringer-ring stiffened panel optimized using

PANDA2 with imperfections (Column 3, Table B.1 in Appendix-B). The stringer-ring

stiffened panel was modeled in detail, with stringers and rings modeled using branched

shell assembly. A one-quarter finite element model (MSC/NASTRAN [8]) of the cylinder

with symmetry boundary conditions applied to the structure was used for the linear






61


bifurcation buckling analysis. The NASTRAN finite element analysis program was

chosen for the stringer-ring-stiffened panel, as it had a more user-friendly preprocessor

than STAGS. Linear bifurcation buckling loads were obtained for a uniform axial

compression load of 600 lb/inch.

The third example chosen for verification with STAGS, is the symmetric titanium

sandwich panel (symmetric thick core) optimized using PANDA2 with imperfections

(Column 5, Table B.6 in Appendix-B). The sandwich walls were modeled using the first

order shear deformable shell elements (480 elements) in STAGS. To reduce the

computations, symmetry was used, and only one half of the cylinder was modeled. The

ring stiffeners were modeled as branched shell elements (as opposed to approximating

them with beam elements). An axial compression load case with Nx = -1000 lb/in and 5

psi internal pressure was used for the analysis. A linear bifurcation buckling analysis was

performed. The critical mode shape obtained from linear analysis was used as an initial

imperfection with amplitude of 0.8 inch (0.5% of cylinder radius) to obtain the limit

buckling load from a non-linear analysis.
















Figure 3.10: Critical buckling mode for the cylindrical stringer-ring stiffened panel

























Figure 3.11: Mode shape for limit buckling of composite sandwich cylinder


ble 3.9: Comparison of buckling load factors obatined from PANDA2 and Finite
element analysis for cylindrical panels optimized using PANDA2

Concept PANDA2 Finite Element Analysis
Cylindrical stringer-ring stiffened panel 1.2 (+10%) 1.07 (NASTRAN)
(analyzed without imperfections)
Cylindrical sandwich panel with T-rings 4.224 (-8%) 4.576 (STAGS)
(analyzed without imperfections)
Cylindrical sandwich panel with T-rings 1.251 (-20%) 1.562 (STAGS)
(analyzed with global imperfections)


Table 3.9 shows the results of the finite element analysis (NASTRAN or STAGS)

and compares it to the PANDA2 analysis results. For stringer-ring stiffened panels

PANDA2 results were about 10% higher from the finite element analysis results. The

lower value of finite element analysis results was due to buckling at the supports. For

stringer-ring stiffened panels PANDA2 optimizes using conditions at the ring and midbay

between rings. Panels exhibit localized bending at end supports where they will need


Ta









extra stiffening. Figure 3.10 shows the critical buckling mode shape of the stiffened

panel. The buckling failure mode is local.

For sandwich panels the buckling load predicted using PANDA2 analysis is lower

than that obtained from STAGS finite element analysis. Results are compared for linear

buckling analysis of a perfect structure and nonlinear analysis of the shell with a global

imperfection amplitude of 0.8 inch. The imperfection shape used in STAGS analysis was

the first critical buckling mode shape obtained from the linear analysis with amplitude of

0.8. The difference is 8% for linear analysis and 20% for nonlinear analysis with

imperfections. The reasons for the larger difference for the second case is be partly due to

the difference in the imperfections used for the buckling analysis and the conservative

approach PANDA2 takes in dealing with imperfections. The buckling mode shape

predicted by PANDA2 (general buckling with 11 axial halfwaves) was in good

agreement with the mode shapes obtained from the STAGS linear and nonlinear (Figure

3.11) analyses.

To summarize, three examples were presented. The first example, a flat stringer

stiffened panel that can be modeled almost identically in PANDA2 and STAGS, showed

excellent agreement in buckling loads calculated using the two programs. The PANDA2

model cannot capture the details at the end supports for the second example, a composite

stringer-ring stiffened panel, and therefore resulted in small differences in the predicted

buckling loads. The difference in buckling loads of the sandwich ring-stiffened panel

(example three) is attributed to three factors: the conservative knockdown factors applied

for transverse shear correction, the conservative approach PANDA2 takes to account for










effect of imperfections, and the differences in the initial imperfections used in the

analysis models in the two programs.



Comparison of Designs Obtained Using PANDA2 Analysis for IQUICK=0 and 1

Table 3.10 presents the optimized weights of aluminum and composite stringer-

ring stiffened cylindrical panels designed for the RLV tank.


Table 3.10:


Optimum weight of panel designed using IQUICK=0 and IQUICK=1
analyses


Panel concept


Aluminum stringer
and ring stiffened


Composite stringer
and ring stiffened


Table 3.11:


Without imperfections
IQUICK Panel Stiffener weight
FLAG weight fraction %
(lb/ft2) Stringers Rings
1 1.588 6.84 5.91
1-0 1.602 5.43 7.78
0 1.600 7.59 5.70
1 0.826 23.30 17.10
1-0 0.822 23.64 16.46
0 0.821 23.43 16.61


Critical margins of design obtained optimized
analyzed using IQUICK=0


With imperfections
Panel Stiffener weight
Weight fraction %
(lb/ft2) Stringers Rings
1.852 19.73 4.81
1.842 19.31 4.95
1.840 19.32 4.76
0.910 28.40 17.50
0.909 28.24 17.57
0.911 32.29 13.63




using IQUICK=1 and


Concept


Aluminum stringer
and ring stiffened

Composite
stringer-ring
stiffened


Without imperfections
Critical Failure mode
margin
%

-3.3 Local buckling
-11.1 Wide column buckling


Stress (pressure)


With imperfections
Critical Failure mode
margin
%

-0.1 Stringer web and top
flange buckling


Local buckling
General buckling









Optimum designs are obtained with and without initial imperfections. Stringer-

ring stiffened panels resulted in almost equal weights for the optimum designs obtained

using IQUICK=1 and IQUICK=0 analyses. The aluminum stringer-ring stiffened panel

without imperfections that resulted in lower weight also violated the buckling constraints

in IQUICK=0 analysis. For composite panels, there were no constraint violations (Table

3.11). The weights were almost identical.

Optimum designs obtained using IQUICK=1 analysis exhibited small violations

of buckling and stress constraints. The designs obtained from global optimization

(SUPEROPT) with IQUICK=1 analysis were re-optimized locally (PANDAOPT) with

IQUICK=0 analysis. The final designs obtained are shown in Table 3.10 on rows

indicated with IQUICK flag value of 1-0. The re-optimized designs satisfied all

constraints and had weights very similar to those obtained using IQUICK-0 analysis. The

small differences in the case of composite designs arise from the re-optimizations

performed after rounding ply thicknesses to discrete values.

It appears that the IQUICK=1 analysis is sufficiently accurate to use for

preliminary optimizations. Designs should, however, be analyzed and reoptimized using

IQUICK=0 models whenever possible to ensure that there are no small violations or

optimization failures.


Optimization of Panels with Local Postbuckling Effects

Safety factors were used in the design of panels for the RLV tank design for the

stability and stress constraints. Factors of safety equal to 1.2 and 1.4 were applied to the

local and general buckling load factors, respectively. In design of stiffened shell

structures, it is typical to have a lower safety factor for local buckling compared to









general buckling failure. This is because local buckling of the skin or wall often does not

significantly affect the structural integrity of the aircraft wing or launch vehicle structure.

However, using safety factors for buckling constraints can lead to critical designs.

The design constraints are calculated from the buckling load factors obtained using

service loads and the specified safety factors. PANDA2 has separate models to capture

local and general buckling failure modes. The buckling loads are obtained independently

and used as design constraints with different safety factors.



Table 3.12: Analysis of stiffened panels with use of PANDA2 local postbuckling
analysis


Concept


Aluminum stringer-ring
stiffened panel




Aluminum orthogrid
stiffened panel




Composite stinger and
ring stiffened panel


Optimum
weight
(lb/ft2)


1.840





2.147





0.911


Buckling loads (lb) and maximum stress (Kpsi)
Mechanism Without local With local
postbuckling postbuckling
Bending stress (Kpsi) 48.34 80.67
Local Buckling 1197 944
Stringer buckling 1682 1158
General Buckling 1666 1599
Bending stress (Kpsi) 48.07 201.4
Local Buckling 1225 602
Stringer buckling 1199 735
General Buckling 1410 1408
Bending stress (Kpsi) 73.86 249.7
Local Buckling 1500 875
Stringer buckling 1671 1029
General Buckling 1509 1565


The rationale of linear behavior of the structure, on which safety factors are

based, does not hold true in the case of buckling of stiffened shell structures in which

local buckling is allowed. Local buckling of the shell wall reduces the effective stiffness









of the structure. The reduced stiffness of the structure will result in general buckling

failure at a load somewhat lower than the design load factor applied to it. Furthermore,

the local buckling also generates additional stresses near local buckling creases that can

lead to stress failure and other local instabilities such as buckling of stringer segments at

load factors possibly well below those used in the design. It is therefore recommended

that the shell be designed for the ultimate load it will need to withstand and correctly

account for postbuckling response if the local buckling load is lower than the general

buckling failure load.

PANDA2 is capable of capturing the effects of local buckling on general

instability for panels with stringers and panels with both stringers and rings. The

IQUICK=0 (discrete) analysis model is used. However, the loads applied in the PANDA2

analysis must be set equal to the desired value of the general buckling load. The local

buckling load factor will then have a value of less than one, (e.g., 1.2/1.4 in the case of

RLV tank design).

Three panels (namely metallic stringer-ring stiffened panel [Table B.1]), orthogrid

stiffened panel [Table B.3] and composite stringer-ring stiffened panel [Table C.1] that

were optimized without permitting local postbuckling) are chosen for analysis with

postbuckling. The panels were designed with initial imperfections included in the

analysis model. Table 3.12 shows the maximum stress and buckling load of the designs

analyzed with an ultimate load of Nx=-1400 lb/inch and internal pressure of 5 psi, with

local postbuckling permitted. In the present study a stress safety factor of 1.0 is used for

analysis with ultimate loads.










Table 3.13: Comparison of aluminum stringer-ring stiffened panel
with and without local postbuckling effects


Variable


Stringer spacing
Stringer height
Stringer top flange width
(and thickness)
Stringer bottom flange
width (and thickness)
Ring spacing
Ring height
Ring bottom flange width
(and thickness)
Ring top flange width
(and thickness)
Wall thickness
Stringer web thickness
Ring web thickness
Panel weight
Stringer weight fraction %
Ring weight fraction %


Design obtained
without local
postbuckling
7.764
1.927
1.065 (0.071)

0.400 (0.050)

30.00
2.000
0.584 (0.104)

0.436 (0.050)

0.097
0.050
0.050
1.840
19.32
5.30


Design obtained
with local
postbuckling
7.287
2.000
1.107 (0.078)

0.400 (0.050)

30.00
2.141
0.686 (0.122)

0.474 (0.054)

0.096
0.053
0.050
1.915 (4.1%)
22.02
5.42


designs optimized



Design obtained
with local
factor= 1.4
6.545
1.936
1.037 (0.071)

0.400 (0.050)

30.00
2.3718
0.814 (0.137)

0.400 (0.05)

0.096
0.05
0.056
1.924 (4.6%)
21.92
6.57


Analysis of panels with postbuckling results in large increases in stress, and

significant decreases in local buckling and stringer buckling loads. The reduction in

general buckling loads is insignificant. The designs from Table 3.12 were re-optimized

with local postbuckling allowed. The axial compression load case was modified such that

the applied load was 1272.73 lb/inch, with safety factors of 0.9429, 1.1, and 1.1,

respectively, for local buckling, general buckling, and stress failure. The value of the

applied axial compression load (1272.73 lb/inch) is 10% lower than the required 1400

lb/inch load. This is because a load factor of 1.0 is changed by PANDA2 to 1.1 to avoid

numerical difficulty.










Table 3.14:


Comparison of aluminum orthogrid stiffened panel designs optimized with
and without local postbuckling effects


Variable



Stringer spacing
Stringer height
Ring spacing
Wall thickness
Stringer web thickness
Panel weight
Stringer Weight fraction %
Ring weight fraction %


Design obtained
without local
postbuckling
6.544
1.791
19.70
0.092
0.156
2.147
28.69
9.53


Design obtained
with local
postbuckling
6.851
2.000
29.71
0.093
0.205
2.400 (11.8%)
35.92
8.28


Design obtained with
local buckling factor
=1.4
6.266
2.000
24.33
0.093
0.166
2.295 (6.9%)
33.15
8.54


Comparison of composite stringer-ring stiffened panel designs optimized
with and without local postbuckling effects


Variable


Stringer spacing
Stringer height
Stringer bottom flange
width
Stringer top flange width
Ring spacing
Ring height
Ring bottom flange width
Ring top flange width
Panel weight
Stringer Weight fraction
Ring weight fraction


Design without local
postbuckling


8.545
1.656
1.000

1.170
34.49
3.221
1.500
1.799
0.911
32.3
13.6


Design obtained
with local
postbuckling
7.598
2.000
1.000

1.488
36.00
2.121
1.500
1.500
1.035 (13.6%)
42.2
10.2


Design obtained
with local buckling
factor 1.4
10.51
2.000
1.073

1.046
36.00
3.141
1.500
1.500
0.954 (4.7%)
33.7
14.7


Table 3.15









Table 3.13 compares an aluminum stringer-ring stiffened panel design obtained

using linear bifurcation buckling analysis with the design obtained where local

postbuckling was permitted. Tables 3.14 and 3.15 present similar results for an aluminum

orthogrid stiffened panel and composite stringer-ring stiffened panel. The composite

stringer-ring stiffened panels were designed with a [(+65/-65)3]s laminate for the skin and

a [45/-45/03/45/-45/0]s laminate for the stiffeners.

Optimized designs are also obtained for the case where the local buckling load is

increased to the value of the general buckling load factor (1.4). In the present case, the

applied load for axial compression is set at 1272.73 (1.4x1000/1.1) with general buckling

and stress safety factor set equal to 1.1.

The aluminum stringer-ring stiffened panels showed a smaller increase in weight

for metallic designs (4.1%) compared to their composite counterparts (13.6%) The

weight increase is primarily due to the increase in weight of the stringers and rings. The

reduced wall stiffness due to local buckling is compensated by the increased stiffness of

the stringers and rings.. The increase in weight is small for the present case, as the driving

factor for the designs was the hoop stress from the (35 psi) internal proof pressure load

case. The weight increase for the orthogrid panel was higher (11.8%) than for metallic

stringer-ring stiffened panel (4.1%).

Designs obtained using local buckling load factor set equal to the general

buckling load factor (1.4) were found to be lighter that those obtained with local

postbuckling permitted (except for aluminum stringer-ring stiffened panel for which the

weight increase was approximately equal). The higher weight of panels designed for local

postbuckling is due to the conservative approach used in PANDA2 analysis for the non-









linear response of panels after local buckling has occurred (when local buckling load

factor is less than 0.95). The analysis models reduce the effective stiffness of the skins,

which in turn result in higher amplification of the initial imperfections leading to lower

buckling loads.

The weight increase in panels optimized with a higher local buckling load factor

of 1.4 (also used for general buckling) was, 4.6%, 4.7%, and 6.9%, (see the third column

of Tables 3.13, 3.14 and 3.15) respectively, for aluminum stringer-ring stiffened panel,

aluminum orthogrid stiffened panel and composite stringer-ring stiffened panel. The

weight increase of panels designed with a higher load factor for local buckling) is smaller

than the weight increase for corresponding panels optimized with local postbuckling

effects.

It appears that higher weight increase in panels optimized with local postbuckling

permitted is due to the conservative models used for estimating the nonlinear effects in

the PANDA2 analysis. The optimizer does not have the freedom to choose a lighter

design that can be obtained by raising the local buckling load factor. The designer using

PANDA2 for stiffened panel design should therefore perform two optimizations, one

with local postbuckling included and another with a local buckling load factor raised to

the value of general buckling load, to select a design with lower weight.

Nevertheless, the model used in PANDA2 to design panels with both stringers

and rings for local postbuckling is a useful tool because the nonlinear analysis of

PANDA2 requires several orders of magnitude lower computational effort compared to

performing a full non-linear finite element analysis. The RLV panels designed showed

weight increases (5 to 14%) due to the more critical proof pressure load cases. For panels









designed without such a load case, local buckling could result in larger reductions in

general buckling load margins.


Modeling Geometric Imperfections vs. Using Wide Column Buckling

This section presents optimum designs of perfect and imperfect stringer-ring

stiffened panels. The perfect panels were optimized with and without the wide column

buckling constraint. The optimum weights and failure margins of the optimum designs

are compared.

PANDA2 permits use of wide column buckling constraint for curved panels. The

wide column model treats the cylindrical panel between rings as a flat panel and obtains

the inter-ring buckling load. Two models are available in PANDA2 [25] to capture the

wide column buckling failure. A 1-D discrete model (IQUICK=0) of the skin-stringer

module and a closed form solution (IQUICK=1) for inter-ring buckling where a large

shell radius replaces the actual radius. The inter-ring buckling load factor is also

computed with a 1-D skin-ring discrete model that accounts for the shell curvature [34].

PANDA2 optimizations use the lower bounds from the different analyses to ensure

conservative designs.

Optimizations were performed for panels with and without imperfections. Panels

designed without imperfections were optimized with and without wide column buckling

constraint. Imperfect panels were optimized with global and out-of-roundness

imperfection amplitudes of 0.8 inch (0.5% of the cylinder radius), inter-ring and local

imperfection amplitudes of 0.1 inch and 0.01 inch, respectively. PANDA2 was allowed to

adjust these values if the rotation of the wall exceeded 0.1 radian for the critical mode

shape used for imperfections.










Table 3.16:


Optimum weight of panels optimized with and without imperfections
and/or wide column buckling constraint


Optimum panel weight (lb/ft2)


Perfect panel
without wide
column buckling
constraint

1.5908
1.5840
1.5736
1.5688


Perfect panel
with wide
column
buckling
constraint
1.6157
1.6825
1.8640
2.1602


Panels designed
with initial
imperfections



1.8411
1.8640
1.9032
2.0890


Table 3.17:


Comparison of margins of perfect composite stringer-ring stiffened panels
analyzed with imperfections


Constraint margins (in percentage) of perfect panels


spacing obtained from an optimization without
wide column buckling constraint


(inch) Bending Local General
stress Buckling Buckling


Stringer
(Ring)
buckling


obtained from an optimization with wide
column buckling constraint


Bending Local General
stress Buckling Buckling


Stringer
(Ring)
buckling


30 -98.8 -99.0

60 -98.6 -98.6

90 -98.9 -98.5


-98.5


-98.5


-48.9 -98.0
(-97.3)
-49.2 -97.6
(-98.3)
-54.6 -97.2
(-98.3)
-49.2 -97.4
(-98.5)


-95.2 -96.0 -40.5 -91.4
(-85.1)
-67.3 -76.5 -23.6 -73.3
(-25.2)


-17.8

6.96


-38.0

-36.3


-0.75

17.1


12.6
(-36.9)
58.5
(-39.8)


Table 3.16 presents the optimized weight of panels at different ring spacings

obtained from optimization with the discretized analysis model (IQUICK=0). Table 3.17

compares the critical constraints for perfect panels designed without and with wide

column buckling constraint enforced, and analyzed with imperfections. Table 3.16 shows


Ring
spacing
(inch)



30
60
90
120


Ring









that perfect panels are insensitive to ring spacing if wide column buckling constraint is

not included. The resulting designs are critical in hoop stress due to proof pressure and

local buckling.

The perfect panels optimized with wide-column buckling constraint imposed were

critical in wide column buckling failure for all values of the ring spacing. However, for

larger ring spacing, the effect was more pronounced and resulted in optimum designs

with taller stringers to provide increased bending stiffness. The optimized weight is more

sensitive to ring spacing when the wide column model is used. This is expected because

the wide-column buckling uses the inter-ring portion of the stiffened shell and column

buckling is a function of the length.

Panels designed without imperfections are less sensitive to imperfections if

designed with the wide column buckling constraint. Introduction of imperfections in the

analysis of these panels (perfect panels optimized with wide column buckling constraint)

results in critical buckling and stress constraints. This is because in addition to reducing

the curvature, imperfections induce bending in the prebuckling phase that result in

redistribution of the stress resultants and increase stresses in the stiffener segments. Using

wide column constraint cannot simulate these conditions and hence cannot entirely

protect against the detrimental effects of imperfections.

Panels optimized without imperfections and without wide column constraint can

be extremely unconservative and can fail at loads much smaller than design loads in

actual use. Wide column constraint alleviates this problem but does not eliminate it.

However, at smaller ring spacing it appears that the wide column buckling constraint









alone is not enough to design the panels. Even if unknown, small values of imperfections

that are reasonable should be used in the optimization of panels with PANDA2.


Modeling Issues in Truss Core Panel Design

In preliminary comparisons of optimum weight of stiffened panel designs, it was

discovered that truss core panel weights were very different from weights of the

remaining concepts. The present section presents the results of the investigation

performed to explain the reason for heavier weights for truss core sandwich panels. In

particular, the effect of not including rings as clamped supports and the effect of having

different corrugation angles are investigated.


Optimization of panels without rings

The PANDA2 program has a limitation in the modeling of truss core panels: it

does not allow inclusion of ring frames in the analysis model. Figure 3.12 shows

optimum weight of panels designed with rings replaced by clamped supports for different

lengths. The optimizations were performed with the IQUICK=1 analysis.

Figure 3.12 shows that the optimum designs are lighter for panels of short lengths

(10 inches). For short panels, the bending boundary layer length (BLL) at supports is

comparable to the panel length (see Items 175, 242 and 378 of

.../panda2/doc/panda2.news file [25]). The bending effects help counteract hoop tensile

stresses that arise due to the clamped edge supports, resulting in a low weight for the

optimized panels. Local buckling failure and stiffener buckling drive the design at short

lengths.










8
+ T-stringer
7 -A-Isogrid
S6r --x- Sandwich
6 Truss-core




1 3





0 30 60 90 120 150 180 210 240 270 300
Axial length of panel (inch)


Figure 3.12: Effect of axial length on optimized weight for panels without rings

The sharp increase in weight observed when the panel length changes from 30 to

60 inches (or from 10 to 30 inches for sandwich and truss core panels) is due to the stress

field produced by local bending near the clamped edges. The clamped edges counteract

the radial expansion induced by the internal pressure and produce large compressive and

bending stresses at the supports. For panel lengths from 30 to 150 inches, the optimum

weight of panels remains approximately constant. The active constraints are those

corresponding to the internal proof pressure load case. Stress is the only critical margin

for the T stringer and isogrid concepts; upper skin buckling and core shear failure modes

are active for truss core and honeycomb-core sandwich panels, respectively.

Beyond this length (150 inches), the weight increases for stringer stiffened panels.

The weight increase for panels longer than 180 inches is attributed to column buckling

modes becoming active at large lengths. The weight of sandwich panels decreases

because PANDA2 neglects conditions at the support for long panels. When there are no

rings in the model, PANDA2 designs for conditions at the midsection of the panel and the









conditions at the supports. However, if the BLL is very small compared to the panel axial

length PANDA2 ignores the local bending at supports and the resulting stress field. The

panel designed thus will need to be stiffened locally at supports to carry the high local

stresses.

PANDA2 cannot design panels with varying cross-sections along the length.

PANDA2 works well for panels with many repeating stiffeners in the cylinder, where the

inter-ring buckling behavior of the shell can be characterized by extracting a segment

between two adjacent rings. The design is performed using the stress state at rings and at

mid section between rings. In the absence of ring stiffeners, PANDA2 uses the conditions

at mid panel and at end supports for the design. For long panels, large bending stresses

are present at the end supports. Using the local stresses to design the entire panel

produces overly heavy designs.

However, the designer can obtain designs for mid-bay conditions and end support

conditions by performing two sequential optimizations in PANDA2. The first

optimization uses a "complete" analysis, in which the margins are computed including

conditions at both mid-length and ring locations. The results of the first optimization are

used as input for a new optimization that neglects the conditions at the shell edges. The

optimum values of stiffener spacing, thicknesses, and cross-section dimensions obtained

from the initial optimization with complete analysis (using stress state at mid-bay and at

rings) are chosen as new lower bounds for the stiffener sizing in the optimization that

uses only the stress state at mid-bay (between rings). Stiffener spacing and cross-section

can no longer be optimization variables. A new optimization is performed and a lighter

design is obtained. The panel skin and stringer dimensions from the first optimum design









would be used for a certain axial length of panel near the rings; the panel skin and

stringer dimensions and ring dimensions from the second optimum design would be used

for the panel mid-length region and for the rings. The length of the "edge" design, i.e. the

length at which the cross-section changes, is chosen based on engineering judgment or

based on value of the boundary layer length. The "hybrid" design then must be verified

by finite element analysis of the detailed model.

For truss core panels this approach was not used because too few design variables

were used in the optimization, and there was not enough design freedom. It was felt that

the weight obtained for the 300 inch long panel was a reasonably accurate representation

of truss core panel, even though weight of local stiffening is ignored.


Effect of corrugation angles on optimum weight

Truss core stiffened panels behave differently from honeycomb or foam core

sandwich panels. In the case of truss core sandwich panels, the thickness and orientation

of the corrugation web affect the out-of-plane shear stiffnesses. The G13 shear

deformation of the sandwich is carried by inplane shear deformation of the corrugation

webs, whereas the bending and inplane (compression/tension) loading of the web provide

the G23 shear stiffness. In the case of truss core sandwich, internal pressure in the cylinder

causes transverse compression of the sandwich laminate. Since the webs of the sandwich

core are inclined, the compressive forces in the web induce compressive stresses in the

facesheets at the crown portion of the truss core corrugation. Optimized panels must

choose a corrugation angle that will minimize the compressive stresses induced in the

facesheets while maximizing the transverse shear stiffness.





Table 3.18 presents optimum weight and critical failure constraints obtained for

truss core panels with different corrugation angles for a 300 inch long panel clamped at

the supports. Heavier panels are designed for web angles that are smaller than 600. The

fixed value of corrugation angle (450) prescribed from manufacturing constraints is not

optimal for structural efficiency. The weight obtained for the very small web orientation

angle (150) is very high. The analysis of constraint margins indicates that small

corrugation angles result in critical margins for corrugation web buckling and upper skin

buckling, leading to thicker corrugation webs. For larger corrugation angles, general

buckling becomes more critical. This is due to the reduction in transverse shear stiffness

of the core when the corrugation angle becomes large. The optimum corrugation angle is

530 for panels optimized without imperfections and 620 for panel optimized with

imperfections.


Table 3.18: Effect of corrugation angle on optimum weight of truss core panel
(Internal pressure load case and axial compression load case are indicated
by the numbers 1 and 2 in parenthesis)

Web angle Optimum Critical constraints
a panel weight
(lb/ft2)
150 5.098 Corrugation (1) and Upper skin bucking (2)
300 2.926 Corrugation (1) and Upper skin bucking (2)
450 2.126 Corrugation (2) and Upper skin bucking (2)
600 1.735 Stress (1) and Corrugation buckling (2)
Optimized 1.705 Stress (1); Corrugation (2), Upper skin (2), and General
(62.3120) buckling (2)
750 1.744 Stress (1); Corrugation (2) and General buckling (2)
850 1.781 Stress (1). Corrugation (2) and General buckling (2)


,I


v









Optimization and Cost Issues

This section presents some optimization issues and the computational effort

required for such optimizations. The discussion will focus on the global optimization

capability of PANDA2 and the computational effort (CPU time) required for the

optimization.

Structural optimization of stiffened panels often involves large numbers of design

variables that are somewhat redundant. The redundancy creates different optimum

designs that have almost identical weights, that is, multiple local optima. Expensive

global optimization techniques are often required to obtain the best design. Introducing

optimization complexity such as global optimization requires an increased amount of

computational effort. However, global optimization algorithms often provide multiple

solutions with similar performance. The designer can use such information, if the weight

difference is small, to choose a design that satisfies other considerations (such as

manufacturing cost and damage tolerance) that were not included in the analysis.

The efficiency of PANDA2 analysis methods and models allows it to perform

global optimization of stiffened panels. Global optimization typically requires a

computational effort of magnitude one or more orders higher than that required for

locating local optima. Table 3.19 shows the computation (CPU) time required for one

SUPEROPT execution for global optimization of a stringer-ring stiffened panel. The

computations were performed on a 233 VIMHz Digital Alphastation-200 4/233. The weight

of the optimum design obtained using IQUICK=1 analysis is same as that obtained using

IQUICK=0 analysis at a fraction of the computation cost. The CPU time for a global

optimization is still much lower than that required for a single nonlinear finite element

analysis (Table 3.8).









Table 3.19: Comparison of CPU times required for one SUPEROPT execution for
global optimization of an aluminum stringer-ring stiffened panel using
IQUICK=0 and IQUICK=1 analysis options

IQUICK=1 IQUICK=0
CPU time (seconds) 134.2 394.2
Optimum weight of panel (lb/ft2) 1.852 1.840
Constraint violation of optimum 0.10% (stringer web and 0.24% (local buckling)
(IQUICK=0 analysis). top flange buckling)
Constraint violation of optimum 0.04% (stringer web and 2% (local buckling)
(IQUICK=1 analysis). flange buckling) 10% (stringer rolling)


Table 3.20: Number of optimization iterations and optimized weight of isogrid
stiffened panels with different initial designs (design vectors of the
optimum designs are presented in Table 3.21)


Number of
SUPEROPT
executions


Optimum design of perfect panel
All variables at lower bound
All variables at median value
All variables at upper bound


Number of
optimization
iterations to obtain:
Local Best
optimum optimum
47 284
267 267
33 593
259 292


Panel weight (lb/ft)

Local Best
optimum optimum
2.627 2.409
2.409 2.409
2.761 2.472
3.128 2.474


In the present study of stiffened panel designs, it was discovered that even with

the SUPEROPT procedure of PANDA2 it was not possible to obtain a global optimum.

Several examples of such failures in locating global optimum are presented and methods

were used to identify the failure is discussed.

Among stiffened panel concepts, optimization failure was encountered for isogrid

stiffened panels (with blade stiffeners) with imperfections. The inspection of the iteration

history file (*.OPP file) revealed that constraints were oscillatory near the optimum and


Initial design









the design shifted from feasible to infeasible domains. This was because some constraints

had very large gradients at the optimum. One way to deal with such oscillations is to

allow smaller move limits. This is achieved in PANDA2 by increasing the number of

iterations in each PANDAOPT execution. This has a disadvantage because with smaller

move limits, gradient based optimizations are likely to get trapped at a local optimum.



Table 3.21: Isogrid stiffened panel designs obtained from different initial designs.

Design variable Design 1 Design 2 Design 3 Design 4
Skin thickness 0.107 0.107 0.113 0.098
Isogrid module size 10.0 10.0 11.4 10.71
(spacing)
Isogrid web height 1.509 1.510 1.662 2.049
Isogrid web thickness 0.125 0.125 0.129 0.126
Ring spacing 30.00 30.00 42.01 113.9
Height of ring 2.160 2.160 2.084 2.000
Thickness of ring web 0.030 0.030 0.030 0.076
Panel weight (lb/ft2) 2.409 2.409 2.472 2.474
Isogrid weight (%) 33.9 33.9 32.8 42.1
Ring weight (%) 2.2 2.3 1.3 1.1




Table 3.22: Constraint margins (%) for optimized designs of isogrid panels

Design Optimized Stress in Triangular Isogrid web
weight (lb/ft2) the skin skin buckling buckling
1 2.409 27.0 -1.26 -0.483
2 2.409 27.0 -1.09 -0.596
3 2.472 31.9 -0.815 -0.432
4 2.474 21.1 0.578 8.84



Table 3.20 summarizes the number of iterations required to converge to a

practical optimum for optimizations with four different initial starting points. Tables 3.21









and 3.22, respectively, present the optimized designs and active failure mechanisms of

designs obtained from optimizations using the different starting points.

The optimum designs obtained for different starting points (Table 3.20) are

marginally different. There is also a wide scatter in the number of iterations required to

converge to a local optimum and locate a practical global optimum. The designs obtained

from the four optimizations are compared in Table 3.21. It can be seen that even though

the final weights of the optimum designs are almost equal, the design variables are quite

different. The final designs obtained for optimization from initial designs 1 and 2 (Table

3.20) were identical. Optimizations with initial Designs 3 and 4 were trapped in local

optima due to large gradients of some critical constraints. However, the optimum weight

obtained for initial designs 3 and 4 are only marginally higher (2.7%) than those obtained

for designs 1 and 2. For practical purposes all four designs are equally acceptable. The

results indicate that it often requires multiple SUPEROPT executions to find the global

optimum.

Similar results were also found in the design of titanium honeycomb core

sandwich panels (Table 3.23). Titanium honeycomb core sandwich panels were

optimized with different options of fixed or optimized core and with symmetric and

asymmetric facesheets. The optimum designs obtained from a single SUPEROPT

execution are shown in the second column of Table 3.23. Optimization failure was

detected when the sandwich panel optimized with increased design freedom (asymmetric

fixed core) resulted in a higher weight. Re-optimizing the designs with more executions

of the SUPEROPT process resulted in improved designs as shown in column 5 of Table

3.23.









Table 3.23: Titanium honeycomb core sandwich panel optimization: optimum weights
and iterations for global optimum

Panel type Best design from one Best design from two
SUPEROPT execution SUPEROPT executions
Optimum Optimization Optimum Optimization
weight (lb/ft2) iterations weight (lb/ft2) Iterations
Symmetric, fixed core 1.337 275 1.358 399
Asymmetric fixed core 1.383 275 1.354 361
Symmetric optimized core 1.323 275 1.323 275
Asymmetric optimized core 1.327 275 1.320 365


In the use of PANDA2 it is common to encounter small differences as shown in

Table 3.23. Such small differences could be due to small difference in the analysis

models, modeling philosophy, or occasionally failure of the optimization process.

Designers should pay attention to small differences to ensure that they are indeed

acceptable differences that reflect the analysis and optimization accuracy. The low cost

analysis of PANDA2 permits the designer to re-optimize or re-analyze the panel designs

at very little additional cost.

The designer does not have information on the actual global optimum and has to

rely on other methods to determine if the design obtained is a practical global optimum.

Often, multiple optimizations are performed either with different degrees of design

freedom or in different areas of the design space (using fixed ranges for a selected design

variable). In addition to helping identify convergence to local optima, such exercises also

provide the designer with multiple designs from which a selection can be made based on

considerations that were not included in the analysis. More accurate finite element

analysis models should be used to verify the designs. Replacing the PANDA2 analysis









with such detailed finite element analysis models for optimization, however, is neither a

feasible nor a justifiable option.


Table 3.24: Estimated number of panel analyses for stiffened panel trade study

Number of stiffened panel concepts (ne) 14
Variations of each concept (ny) 2
Average number of design variables (nd) 10
Number of line searches performed in PANDA2 global 275
optimization (n)
Number of analyses per search (n) 11
Total number of analyses performed (n x nv x n x n) 84700



In the next section, different stiffened panel concepts are optimized to compare

their weight efficiency and sensitivity to imperfections. Table 3.24 gives an estimate for

the minimum number of analyses required for the trade study. The numbers in

parentheses indicate the results when local searches make use of approximations. Using

detailed finite element models for such large number of analyses would be extremely

expensive and time consuming.


Comparison of Weight Efficiency of Stiffened Panel concepts

This section presents the results of the optimization performed for the various

stiffened panel concepts introduced earlier for the RLV tank design. The library of

analysis tools implemented in PANDA2 depends on the type of stiffeners. Optimizations

were performed with the IQUICK=0 option for stringer-ring stiffened panels and

orthogrid panels. Isogrid stiffened panels, truss core and honeycomb-core sandwich

panels were optimized with IQUICK=1. The choice reflects the best analysis choice

available in PANDA2 for each concept.




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