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 ultiFidelity 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
Stringerring stiffened Panel..................................... ........................ .............. 196
G ridStiffened Panels ................................................. .................................... 197
Titanium Honeycomb Core Sandwich Panels...... .... .................................... 200
TrussC ore Sandw ich Panels.................................... ....................... ............... 203
C OPTIMUM DESIGNS OF COMPOSITE PANELS .............................................. 204
StringerR ing Stiffened P anels ................................................................................... 204
Honeycomb Core Sandwich Panels...... ........... ........ ..................... 206
TrussC 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, coauthoring 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 NAG1669 and NAG11808 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 layup 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 stringerring 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 stringerring 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 stringerring 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 stringerring 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 stringerring stiffened panels
optimized with varying levels of design freedom .................................... 99
3.34: Optimized layups obtained for composite stringerring stiffened panels
optimized with varying levels of design freedom ................................... 100
4.1: Initial design of panel (Subscript T and F denote preimpregnated 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 fullsublaminate repetitions, as in [t /t2]sn
lam in ate ..................................................................................................... 1 16
4.8: Maximal error in Dj terms for n halfsublaminate repetitions, as in [(t /t2)ns
lam in ate ..................................................................................................... 1 17
4.9: Maximal buckling load errors (percent) of a single sublaminate (The coding
AP 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,
layup, 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 layup, 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 ringstiffened 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 IM7/9772 composite panels (at 1900 F) ....................194
B. 1: Optimum designs of metallic stringerring 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 gridstiffened 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 trusscore sandwich panels optimized with and
w without im perfections...... ............. ............ ...................... 203
C.1: Optimum designs of composite stringerring 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 trusscore 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 Tshaped stringer, showing layer
numbering convention and discretization used. .....................................37
3.2: PANDA2 repeating module for a panel with Tshaped 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 stringerring 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: fullsublaminate and halfsublaminate
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 1D 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 Tring 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 1D discrete analysis models for design
of stiffened panels; PASCO [6], VICONOPT [126], and PANOPT [16] are finitestrip
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 generalpurpose analysis codes in optimization is a time
consuming task. Furthermore, using generalpurpose 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 nonlinear 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.
MultiFidelity 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 globallocal 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 2D finite element
model as a lower fidelity model to predict failure stresses and applied corrections
calculated using a full 3D 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 highspeed 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 1D
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 substructuring or that use easily available analytic sensitivities are included.
Also, panel design methods that address globallocal 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, 1D discretization), choice of shell theories (Donnell, Sanders), and inclusion of
details such as the effects of geometric imperfections and postlocal 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 globallocal
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 inplane 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 thinwalled 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 3D 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 BUCLASP2 [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 BUCLASP2 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 stiffenerintroduced 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 segmentedplate 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
[4751] 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)
1y 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 bowtype 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 multifidelity 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
suboptimal 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 Jstiffened) 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 57%
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 inplane
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 HitandRun" [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 semianalytical approach for nonlinear 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 nonlinear plate theory applied to each
component platestrip accounts for the large inplane 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 nonlinear
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 [2533] 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 generalpurpose FE codes). PANDA2 uses a combination of approximate
physical models, exact closed form (finite strip analysis) models and a 1D 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
Level1 analysis uses the GANBIF [12] analysis routine. The analyses in level1 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 level2 analysis accounts for the primary nonlinear effects such as edge
restraints and can satisfy the prescribed boundary conditions exactly. The level2
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 level2 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 level3 analyses are the most accurate and use twodimensional 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
reliabilitybased design, localglobal design, and multicriteria (costweight
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 builtup 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
Tstringer 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 crosssections
for panels used in the wing structure. In order to provide compatibility between panels,
the stiffener spacing and crosssection 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 gradientbased 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 bladestiffened panels with centrally located voids. Figure 2.4
shows a stiffener crosssection 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 crosssection. 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 Jstiffener 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 Paretooptimal 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 prechosen stacking sequence are used as
optimization variables. Composite laminates that are manufactured from prepregs (tape
layups) 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 nonconvex 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 suboptimal 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: substring crossover, stack deletion,
stack addition, orientation mutation, interlaminate swap and intralaminate 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 semiempirical 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, Isection, hat
section, Jsection and anglesection 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 nonintegral 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 crosssection geometry, and laminate stacking sequence. A global
smeared model was used to calculate general instability loads. Local buckling was
calculated using a RayleighRitz 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 worstcase 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 nonlinear 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 2D 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, outofplane 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 [2533] 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 Zshaped crosssections.
PANDA2 is also capable of analyzing panels with Zshaped 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 1D 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 oneterm
RayleighRitz 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 skinstringer "panel module" or "repeating module" model [25]. A
module (Figure 3.1) includes a length of panel between rings and a crosssection of a
stringer plus a portion of the panel skin of width equal to the spacing between stringers.
The segments of the skinstringer module crosssection 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 Tshaped 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 Tshaped stiffeners.
A repeating module of a Tshaped 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 skinstringer module of the
second model. In the fourth model, the ring crosssection 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 interring
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 inplane 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 1D
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 skinstringer module
(Model 2) is included in the buckling analysis when IQUICK=0. The 1D skinring
discrete module (Model 4) in the length direction is used to calculate interring buckling
analysis with both IQUICK= 0 and IQUICK=1. The analysis performed with the 1D
skinstringer 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 ringstiffened shell,
the models used for prebuckling analysis calculate stresses at the midbay 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, interring and local buckling loads.
An outofroundness (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 stiffenerskin 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 skinstringer 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 outofroundness 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), interring 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, interring, local and out
ofroundness) 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
nonconservative or overconservative 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, interring, and local buckling models. The shapes of the imperfections
change during optimization as the critical buckling mode changes. The outofroundness
does not have wavelength and therefore stays the same. The buckling mode and the
corresponding number of halfwaves is determined from closed form solution [26]. For
discrete models, the local buckling mode is predicted from the single skinstringer
module; the interring 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 halfwaves 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 stressstrain relation. The buckling
analysis is carried out for the imperfect structure with closed form solutions [26] or 1D
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 inplane 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, crosssectional 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 prepreg ply thicknesses. Rounding ply thicknesses often results in suboptimal
designs. Hence, ply thickness of the preliminary optimum designs were rounded to the
nearest integer ply thickness value and reoptimized 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 userspecified 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
stringerskin 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 interring 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 TStringer and Jrings Internal External Al2219 T87
Aluminum isogrid stiffened panel Internal blade External Al2219 T87
isogrid Jrings
Aluminum orthogrid stiffened panel Internal blade Al2219 T87
stiffeners
Titanium symmetric sandwich T rings N/A Internal Ti6A14V
Titanium asymmetric sandwich T rings N/A Internal Ti6A14V
Titanium truss core sandwich Hatshaped N/A Ti6A14V
corrugation
Composite T stringer and T rings External External IM7/9772
Composite symmetric sandwich T rings N/A Internal IM7/9772
Composite asymmetric sandwich T rings N/A Internal IM7/9772
Composite truss core sandwich External corrugated N/A IM7/9772
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 IM7/9772 graphite epoxy system.
Composite panels were optimized with fixed layups, 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 layup used for composite panel design
pt Shell segment Layup design
Panel with
Tstringers and Trings
Sandwich Panel
with Trings
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/On45/+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/On45/+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 layups 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 prepreg 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 stringerstiffened panels have external Jrings 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 stringerring 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 Jring
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 crosssection 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
stringerring stiffened panels (Values shown in parentheses correspond to
composite panels when different from metallic designs)
Tstiffener stringer geometry variables
Jstiffener 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 hexcell 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 (Ti6A14V) 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 103 for the ratio of initial facesheet
waviness to facesheet wrinkling halfwavelength, as recommended for panels of good
quality [31 ].
Webskin 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 12ply 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 skinweb
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 stringerskin 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, stringerring 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
1D 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 energybased 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 1D discrete models in PANDA2 are
based on BOSOR4 analysis models.
Using nonlinear 2D 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 generalpurpose 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 corotational procedure [89] implemented
in STAGS allows large rotations required for nonlinear analysis. In addition,
incorporation of the Riks arcfollowing 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 stringerring 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 Tstiffened 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 generalpurpose 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 (1D 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 stringerring stiffened panel optimized using
PANDA2 with imperfections (Column 3, Table B.1 in AppendixB). The stringerring
stiffened panel was modeled in detail, with stringers and rings modeled using branched
shell assembly. A onequarter 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 stringerringstiffened panel, as it had a more userfriendly 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 AppendixB). 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 nonlinear analysis.
Figure 3.10: Critical buckling mode for the cylindrical stringerring 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 stringerring stiffened panel 1.2 (+10%) 1.07 (NASTRAN)
(analyzed without imperfections)
Cylindrical sandwich panel with Trings 4.224 (8%) 4.576 (STAGS)
(analyzed without imperfections)
Cylindrical sandwich panel with Trings 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 stringerring 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
stringerring 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
stringerring stiffened panel, and therefore resulted in small differences in the predicted
buckling loads. The difference in buckling loads of the sandwich ringstiffened 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
10 1.602 5.43 7.78
0 1.600 7.59 5.70
1 0.826 23.30 17.10
10 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
stringerring
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 stringerring 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 reoptimized locally (PANDAOPT) with
IQUICK=0 analysis. The final designs obtained are shown in Table 3.10 on rows
indicated with IQUICK flag value of 10. The reoptimized designs satisfied all
constraints and had weights very similar to those obtained using IQUICK0 analysis. The
small differences in the case of composite designs arise from the reoptimizations
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 stringerring
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 stringerring stiffened panel [Table B.1]), orthogrid
stiffened panel [Table B.3] and composite stringerring 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 stringerring 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 reoptimized
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 stringerring 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 stringerring 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 stringerring stiffened panel. The composite
stringerring 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 stringerring 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
stringerring 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 stringerring 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 stringerring stiffened panel,
aluminum orthogrid stiffened panel and composite stringerring 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 nonlinear 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 stringerring
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 interring buckling load. Two models are available in PANDA2 [25] to capture the
wide column buckling failure. A 1D discrete model (IQUICK=0) of the skinstringer
module and a closed form solution (IQUICK=1) for interring buckling where a large
shell radius replaces the actual radius. The interring buckling load factor is also
computed with a 1D skinring 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 outofroundness
imperfection amplitudes of 0.8 inch (0.5% of the cylinder radius), interring 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 stringerring 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 widecolumn 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 widecolumn buckling uses the interring 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
+ Tstringer
7 AIsogrid
S6r x Sandwich
6 Trusscore
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 honeycombcore 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 crosssections along the length.
PANDA2 works well for panels with many repeating stiffeners in the cylinder, where the
interring 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 midbay 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 midlength 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 crosssection dimensions obtained
from the initial optimization with complete analysis (using stress state at midbay and at
rings) are chosen as new lower bounds for the stiffener sizing in the optimization that
uses only the stress state at midbay (between rings). Stiffener spacing and crosssection
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 midlength region and for the rings. The length of the "edge" design, i.e. the
length at which the crosssection 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 outofplane 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 stringerring stiffened panel. The
computations were performed on a 233 VIMHz Digital Alphastation200 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 stringerring 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. Reoptimizing 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 reoptimize or reanalyze 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 stringerring stiffened panels and
orthogrid panels. Isogrid stiffened panels, truss core and honeycombcore sandwich
panels were optimized with IQUICK=1. The choice reflects the best analysis choice
available in PANDA2 for each concept.
