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MULTIDISCIPLINARY DESIGN AND OPTIMIZATION OF A COMPOSITE WING BOX

A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES

OF THE MIDDLE EAST TECHNICAL UNIVERSITY

BY

MUVAFFAK HASAN

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

IN

THE DEPARTMENT OF AEROSPACE ENGINEERING

SEPTEMBER 2003

Approval of the Graduate School of Natural and Applied Sciences

Prof. Dr. Canan Özgen

Director

I certify that thesis satisfies all the requirements as a thesis for the degree of

Doctor of Philosophy.

Prof. Dr. Nafiz Alemdaroğlu

Head of Department

This is to certify that we have read this thesis and that in our opinion it is

fully adequate, in scope and quality, as a thesis for the degree of Doctor of

Philosophy.

Prof. Dr. Yavuz Yaman

Supervisor

Examining Committee Members

Prof. Dr. Mehmet A. Akgün

Prof. Dr. Yavuz Yaman

Prof. Dr. Haluk Darendeliler

Assoc. Prof. Dr. Nizami Aktürk

Dr. Fatih Tezok

iii

ABSTRACT

MULTIDISCIPLINARY DESIGN AND OPTIMIZATION

OF A COMPOSITE WING BOX

Hasan, Muvaffak

Ph.D., Department of Aerospace Engineering

Supervisor: Prof. Dr. Yavuz Yaman

September 2003, 218 pages

In this study an automated multidisciplinary design optimization code is

developed for the minimum weight design of a composite wing box. The

multidisciplinary static strength, aeroelastic stability, and manufacturing

requirements are simultaneously addressed in a global optimization environment

through a genetic search algorithm.

The static strength requirements include obtaining positive margins of safety for

all the structural parts. The modified engineering bending theory together with the

coarse finite element model methodology is utilized to determine the stress

distribution. The nonlinear effects, stemming from load redistribution in the

structure after buckling occurs, are also taken into account. The buckling analysis

is based on the Rayleigh-Ritz method and the Gerard method is used for the

crippling analysis.

iv

The aeroelastic stability requirements include obtaining a flutter/divergence free

wing box with a prescribed damping level. The root locus method is used for

aeroelastic stability analysis. The unsteady aerodynamic loads in the Laplace

domain are obtained from their counterparts in the frequency domain by using

Rogers rational function approximations.

The outer geometry of the wing is assumed fixed and the design variables

included physical properties like thicknesses, cross sectional dimensions, the

number of plies and their corresponding orientation angles.

The developed code, which utilizes MSC/NASTRAN® as a finite element solver,

is used to design a single cell, wing box with internal metallic substructure and

composite skins.

Keywords: multidisciplinary design, optimization, static Strength, flutter,

divergence, composite wing.

v

ÖZ

BİR KOMPOZİT KANAT KUTUSUNUN ÇOK YÖNLÜ

TASARIMI VE ENİYİLEŞTİRMESİ

Hasan, Muvaffak

Doktora, Havacılık ve Uzay Mühendisliği Bölümü

Tez Yöneticisi: Prof. Dr. Yavuz Yaman

Eylül 2003, 218 sayfa

Bu çalışmada, kompozit bir kanat kutusunun minimum ağırlıkla tasarımını elde

etmek için, otomatik çok disiplinli tasarım eniyileştirmesi yapan bir program

geliştirilmiştir. Çok disiplinli statik mukavemet, aeroelastik kararlılık ve üretim

gereksinimleri, bir genetik algoritma çerçevesinde oluşturulmuş olan eniyileştirme

ortamında eşzamanlı olarak ele alınmıştır.

Statik mukavemet gereksinimleri ile bütün yapısal parçalar için pozitif güvenlik

sınırlarının sağlanması amaçlanmaktadır. Gerilme dağılımını elde etmek için,

seyrek sonlu eleman modelleme tekniği ile beraber geliştirilmiş eğilme yöntemi

kullanılmıştır. Böylece, yapının burkulması neticesinde ortaya çıkan ve yükün

yeniden dağılımının sonucu olan doğrusal olmayan etkiler de hesaba katılmıştır.

Burkulma analizi Rayleigh-Ritz yöntemi kullanılarak, kesit kırışması analizi de

Gerard yönteminden yararlanarak gerçekleştirilmiştir.

vi

Aeroelastik gereksinimler, sönümleme seviyesi tanımlanmış bir kanat kutusunun

çırpınmadan ve ıraksamadan uzak olduğunun gösterilmesini amaçlamaktadır.

Aeroelastik kararlılık analizi için köklerin geometric yeri yöntemi kullanılmıştır.

Laplace ortamındaki kararsız aerodinamik yükler, yaklaşık rasyonel Rogers

fonksiyonları kullanılarak, frekans ortamındaki eşleniklerinden elde edilmiştir.

Kanadın dış geometrisinin değişmediği varsayılmış ve tasarım değişkenleri olarak

kalınlık, kesit boyutları, katman sayıları ve karşılık gelen katman açıları gibi

fiziksel özellikler göz önünde tutulmuştur.

Sonlu eleman çözücüsü olarak MSC/NASTRAN®’ı kullanmakta olan program tek

hücreli, iç yapısı metalik, kanat yüzeyleri kompozit olan bir kanat kutusunun

tasarımında kullanılmıştır.

Anahtar kelimeler: çokyönlü tasarım, eniyileştirme, statik mukavemet,

çırpınma, Iraksama, kompozit kanat.

vii

This work is dedicated to my mother and my father.

Bu çalışma annem ve babama ithaf edilmiştir

viii

ACKNOWLEDGMENTS

The author expresses his appreciation, indebtedness, and gratitude to his

supervisor Prof. Dr. Yavuz Yaman for his guidance, insight, encouragement, and

support which were vital for the success of this thesis.

I would like to take this opportunity to thank my thesis committee members, Prof.

Dr. Haluk Darendeliler and Prof Dr. Mehmet Akgün for their fruitful comments

and criticisms.

The technical discussions and invaluable support of Prof. Dr. Prabhat Hajela of

the Rensselaer Polytechnic Institute and the assistance of Dr. Raymond Kolonay

of General Electric are gratefully acknowledged.

Thanks go to my examining committee members Assoc. Prof. Dr. Nizami Aktürk

and Dr. Fatih Tezok, executive director of the design and engineering directorate

of the Turkish Aerospace Industries (TAI), for their beneficial comments and

suggestions.

I would like to express my deepest gratitude to my family in my homeland

Palestine for their endless love, trust and motivation. Special thanks go to my

brother Dr. Said Al Hasan of the University of Glamorgan, England for his moral

support and continuous encouragement.

ix

Special thanks also go to my wife Suzan for her endless love and motivation. She

always believed in me and whenever I was about to give up, she always persisted.

She shared all of the difficulties of this work with me and was always ready to

concess.

The technical and moral support of my TAI colleagues, Dr. Mustafa Usta, Serdar

Dilaver and Burak Soydan and the understanding and tolerance of my chief

engineer, Mrs. Aylin Barlas are gratefully acknowledged. The technical

discussions with TAI consultants Dr. Zoran Rudiç and Dr. Saied Ahmed were

helpful.

x

TABLE OF CONTENTS

ABSTRACT ...........................................................................................................iii

ÖZ............................................................................................................................ v

ACKNOWLEDGMENTS....................................................................................viii

TABLE OF CONTENTS ........................................................................................ x

LIST OF TABLES ...............................................................................................xiii

LIST OF FIGURES............................................................................................... xv

LIST OF SYMBOLS............................................................................................. xx

CHAPTER

1. INTRODUCTION............................................................................................... 1

1.1 Background to the Study ............................................................................... 1

1.2 Literature Survey........................................................................................... 6

1.3 Scope and Contents of the Study................................................................. 26

1.4 Limitations of the Study .............................................................................. 30

2. STATIC STRENGTH ANALYSIS .................................................................. 31

2.1 Introduction ................................................................................................. 31

2.2 Description of the Wing Box....................................................................... 33

2.3 Failure Modes of the Wing Box Components............................................. 35

2.4 Stress Analysis of the Wing Box................................................................. 37

2.4.1 Sectional Loads .................................................................................... 38

xi

2.4.2 Classical Laminated Plate Theory (CLPT) .......................................... 40

2.4.3 Equivalent Axial and Bending Stiffness Properties ............................. 45

2.4.4 Normal Stress Analysis ........................................................................ 48

2.4.5 Shear Stress Analysis ........................................................................... 51

2.5 Allowable Stresses ...................................................................................... 53

2.5.1 Crippling Allowable Stress .................................................................. 53

2.5.2 Allowable Buckling Stress ................................................................... 56

2.6 Static Strength Analysis .............................................................................. 64

2.7 Case Studies ................................................................................................ 65

2.7.1 Allowable Buckling Stress of a Typical Panel ..................................... 65

2.7.2 Stress Analysis of a Typical Wing Box................................................ 72

2.8 Conclusion................................................................................................... 79

3. AEROELASTIC STABILITY ANALYSIS ..................................................... 80

3.1 Introduction ................................................................................................. 80

3.2 Theory of Aeroelastic Stability ................................................................... 87

3.3 Frequency Domain Solution Methods......................................................... 93

3.3.1 The k-Method ....................................................................................... 93

3.3.2 The pk-Method ..................................................................................... 95

3.4 Laplace Domain Solution Methods............................................................. 97

3.4.1 The p-Method ....................................................................................... 97

3.4.2 The Root Locus Method....................................................................... 98

3.5 Case Studies .............................................................................................. 103

3.5.1 BAH Wing.......................................................................................... 103

3.5.2 ICW Wing .......................................................................................... 119

3.6 Conclusion................................................................................................. 134

4. MULTIDISCIPLINARY DESING AND OPTIMIZATION.......................... 135

4.1 Introduction ............................................................................................... 135

4.2 Statement of the Optimization Problem .................................................... 137

4.3 Formulation of the Optimization Problem ................................................ 138

xii

4.3.1 Objective Function ............................................................................. 138

4.3.2 Static Strength Constraints ................................................................. 138

4.3.3 Aeroelastic Stability Constraints ........................................................ 139

4.3.4 Design Variables ................................................................................ 140

4.4 Solution Procedure .................................................................................... 142

4.4.1 Static Strength Analysis Procedure .................................................... 142

4.4.2 Aeroelastic Stability Analysis Procedure ........................................... 143

4.4.3 Optimization Procedure...................................................................... 143

4.5 Code Description ....................................................................................... 145

4.5.1 Processing Module ............................................................................. 147

4.5.2 Analysis Module................................................................................. 149

4.5.3 General Features and Limitations of the Code................................... 155

4.6 Conclusion................................................................................................. 157

5. CASE STUDIES ............................................................................................. 158

5.1 Introduction ............................................................................................... 158

5.2 Wing Box Model Description ................................................................... 159

5.3 Case Study I............................................................................................... 163

5.4 Case Study II ............................................................................................. 181

5.5 Case Study III ............................................................................................ 190

5.6 Conclusion................................................................................................. 202

6. CONCLUSIONS ............................................................................................. 203

6.1 General Conclusions.................................................................................. 203

6.2 Recommendations for Future Work .......................................................... 207

REFERENCES.................................................................................................... 208

VITA ................................................................................................................... 218

xiii

LIST OF TABLES

2.1 Cut-Off Crippling Stresses .............................................................................. 55

2.2 Panel Material Properties ................................................................................ 66

2.3 Buckling Load Cases....................................................................................... 68

2.4 Critical Buckling Load Factors (λcr)................................................................ 69

2.5 Critical Buckling Load Factors (λcr)................................................................ 69

3.1 Natural Frequencies of the BAH Wing ......................................................... 107

3.2 ICW Material Properties ............................................................................... 119

5.1 Material Properties of the Rectangular Wing Box (Aluminum) ................... 160

5.2 Design Variables of the Rectangular Wing Box ........................................... 160

5.3 Design Variables Sets.................................................................................... 174

5.4 Design Variables Values for Flutter and Divergence Speeds Constraints .... 176

5.5 Material Properties and Allowable Stresses for the Modified Rectangular

Wing Box ..................................................................................................... 182

5.6 Final Design Variables Values for Optimum Design with Static Strength

Constraints.................................................................................................... 187

5.7 Summary of Skin Margins of Safety (Satic Strength Constraints) ............... 189

5.8 Summary of Spars Margins of Safety (Satic Strength Constraints) .............. 189

5.9 Summary of Spars Margins of Safety (Satic Strength Constraints) .............. 190

5.10 Natural Frequencies of the Composite Rectangular Wing

(Strenght Based Design).............................................................................. 191

5.11 Final Design Variables Values for Optimum Design with Static Strength

and Aeroelastic Constraints......................................................................... 196

xiv

5.12 Summary of Skin Margins of Safety

(Satic Strength and Aeroelastic Constraints)............................................... 198

5.13 Summary of Spars Margins of Safety


5.14 Summary of Spars Margins of Safety


xv

LIST OF FIGURES

1.1 Conventional Design Stage ............................................................................... 3

2.1 Structural Details of a Typical Wing Box ....................................................... 34

2.2 Calculation of the Sectional Loads.................................................................. 40

2.3 Positive Sign Convention of Stress Resultants and Ply Orientation Angle .... 43

2.4 Laminate Equivalent Stiffness......................................................................... 46

2.5 Typical Spar Cross Section ............................................................................. 50

2.6 Strain Distribution Over the Spar Section....................................................... 51

2.7 Tapered Section Shear Stress .......................................................................... 52

2.8 Crippling Failure ............................................................................................. 54

2.9 Plate Layout and Positive Sign Convention of Applied Loads ....................... 59

2.10 Convergence of the Buckling Load Factor.................................................... 67

2.11 Buckling Mode Shapes of the Composite Panel ........................................... 70

2.12 Buckling Mode Shapes of the Metallic Panel ............................................... 71

2.13 Structural Arrangement of the Rectangular Wing......................................... 73

2.14 Coarse Mesh Finite Element Models of the Rectangular Wing.................... 74

2.15 Fine Mesh Finite Element Models of the Rectangular Wing

(Total 864 Elements) ..................................................................................... 75

2.16 Fine Mesh Finite Element Models of the Rectangular Wing

(Total 2997 Elements) ................................................................................... 75

2.17 Front Spar Upper Cap Stress Distribution of the Rectangular Wing ............ 77

2.18 Front Spar Lower Cap Stress Distribution of the Rectangular Wing............ 78

3.1 Collar’s Aeroelastic Triangle .......................................................................... 81

xvi

3.2 Functional Diagram of an Aeroelastic System................................................ 89

3.3 Wing Planform of the BAH Jet Transport..................................................... 104

3.4 Aerodynamic Model of the BAH Jet Wing................................................... 105

3.5 Structural Model of the BAH Jet Wing......................................................... 106

3.6 Rogers Rational Function Approximations for the Real Part of the

Generalized Aerodynamic Forces of the BAH wing (Q11,Q12)................... 109



3.8 Rogers Rational Function Approximations for the Imaginary Part of the




3.10 Velocity vs. Damping Plot of the BAH Wing (10 Modes) ......................... 114

3.11 Velocity vs. Frequency Plot of the BAH Wing (10 Modes)........................ 115

3.12 Velocity vs. Damping Plot of the BAH Wing (2 Modes) ........................... 117

3.13 Velocity vs. Frequency Plot of the BAH Wing (2 Modes).......................... 118

3.14 Aerodynamic Configuration and Structure of the Intermediate

Complexity Wing (ICW)............................................................................. 120

3.15 Structural Model of the Intermediate Complexity Wing (ICW) ................. 121

3.16 Aerodynamic Model of the Intermediate Complexity Wing (ICW) ........... 122

3.17 ICW Structural & Aerodynamic Models Joined by Surface Spline

Elements ...................................................................................................... 122

3.18 First Mode Shape of the ICW (f=10.3 Hz).................................................. 124

3.19 Second Mode Shape of the ICW (f=29.5 Hz) ............................................. 124

3.20 Third Mode Shape of the ICW (f=41.8 Hz) ................................................ 125

3.21 Fourth Mode Shape of the ICW (f=62 Hz) ................................................. 125

3.22 Fifth Mode Shape of the ICW (f=91.4 Hz) ................................................. 126

3.23 Sixth Mode Shape of the ICW (f=99.6 Hz)................................................. 126


Generalized Aerodynamic Forces of the ICW wing (Q11,Q12) ................... 127

xvii







3.28 Velocity vs. Damping Plot of the Intermediate Complexity Wing ............. 132

3.29 Velocity vs. Frequency Plot of the Intermediate Complexity Wing ........... 133

4.1 The Basic Genetic Algorithm........................................................................ 144

4.2 General Flowchart of the Developed Code ................................................... 146

4.3 Flowchart of the Processing Module............................................................. 148

4.3 Flowchart of the Analysis Module ................................................................ 150

4.5 Flowchart of the Static Strength Analysis Module........................................ 152

4.6 Flowchart of the Aeroelastic Stability Analysis Module .............................. 154

4.5 Supported Spars Caps/Ribs Chords and the Corresponding Design

Variables...................................................................................................... 156

5.1 Layout and Aerodynamic Configuration of the Rectangular Wing .............. 161

5.2 Structural Model of the Rectangular Wing Box............................................ 162

5.3 Aerodynamic Model of the Rectangular Wing ............................................. 164

5.4 Rectangular Wing Structural & Aerodynamic Models Joined by Surface

Spline Element ............................................................................................ 164

5.5 First Mode Shape of the Rectangular Wing (f=6.4 Hz) ................................ 165

5.6 Second Mode Shape of the Rectangular Wing (f=24.7 Hz).......................... 166

5.7 Third Mode Shape of the Rectangular Wing (f=37.9 Hz)............................. 166

5.8 Fourth Mode Shape of the Rectangular Wing (f=71.1 Hz) ........................... 167

5.9 Fifth Mode Shape of the Rectangular Wing (f=110.7 Hz)............................ 167

5.10 Sixth Mode Shape of the Rectangular Wing (f=120.7 Hz) ......................... 168

5.11 Velocity vs. Damping Plot of the Rectangular Wing for Maximum

Values of the Design Variables ................................................................... 169

xviii

5.12 Velocity vs. Frequency Plot of the Rectangular Wing for Maximum


5.13 Velocity vs. Damping Plot of the Rectangular Wing for Minimum


5.14 Velocity vs. Frequency Plot of the Rectangular Wing for Minimum Values of

the Design Variables ................................................................................... 173

5.15 Convergence History for the Rectangular Wing Weight

(Flutter Speed Constraint Only) .................................................................. 175

5.16 Convergence History for the Rectangular Wing Weight

(Flutter and Divergence Speed Constraints) ............................................... 177

5.17 Velocity vs. Damping Plot of the Rectangular Wing Optimized

for Flutter and Divergence .......................................................................... 179

5.18 Velocity vs. Damping Plot of the Rectangular Wing Optimized

for Flutter and Divergence .......................................................................... 180

5.19 Static Strength Sizing Load Case ................................................................ 183

5.20 Convergence History for the Composite Rectangular Wing Weight

(Static Strength Constraints Only) .............................................................. 186

5.21 Spanwise Variation of the Spars Caps Width

(Static Strength Constraints) ....................................................................... 188

5.22 Velocity vs. Damping Plot of the Rectangular Wing

(Static Strength Based Design).................................................................... 192


(Static Strength Based Design).................................................................... 193

5.24 Convergence History for the Composite Rectangular Wing Weight

(Static Strength and Aeroelastic Constraints) ............................................. 195

5.25 Spanwise Variation of the Spars Caps Width




xix

5.27 Velocity vs. Frequency Plot of the Rectangular Wing


xx

LIST OF SYMBOLS

iF free body force vector of the ith finite element

iM free body moment vector of the ith finite element

t tangential unit vector

n normal unit vector

tuF , tangential sectional force at the upper grid

nuF , normal sectional force at the upper grid

tF ,� tangential sectional force at the lower grid

nF ,� normal sectional force at the lower grid

ouM , bending moment at the upper grid

oM ,� bending moment at the lower grid

N centroidal normal force

V centroidal shear force

Mo centroidal bending moment

H section height

Z neutral axis location

σ normal stress

τ shear stress

ε normal strain γ shear strain

Qij reduced stiffness terms

xxi

ijQ transformed stiffness terms

oiε midplane strain

iκ midplane curvature

Nx, Ny, Nxy force resultants

Mx, My, Mxy moment resultants

[A] extensional stiffness matrix

[B] extension-bending coupling stiffness matrix

[D] flexural bending stiffness matrix

[a] extensional compliance matrix

[d] bending compliance matrix

A cross sectional area

I moment of inertia

beff effective width

t thickness

E Young's modulus

ν poisson's ratio

Exx equivalent (average) axial modulus

Axx finite width axial stiffness

Dxx finite width bending stiffness

a panel length

b panel width

θ fiber orientation angle

θu taper angle of the upper cap

θu taper angle of the lower cap

Vweb web shear force

Pu normal force in the upper cap/chord and skin

Pℓ normal force in the lower cap/chord and skin ( )avewebτ average shear stress of the web

( ) maxwebτ maximum shear stress of the web

xxii

Fcs crippling stress

Fcy compression yield stress

Π total potential energy

w transverse displacement

U strain energy

V work done by external forces

Amn undetermined coefficients

λ load multiplier

λcr critical buckling load multiplier

R plate aspect ratio

[Ke] elastic stiffness matrix

[Kd] differential (geometric) stiffness matrix

M.Sb margin of safety for buckling

ρ density

[M] mass matrix

[K] stiffness matrix

[C] damping matrix

{ })(tx structural deformation vector

{ }),( txF applied aerodynamic load vector

M∞ free stream Mach number

V∞ free stream velocity

( ){ })(txFa aeroelastic force vector

{ })(tFe externally applied non-aeroelastic force vector

∞q free stream dynamic pressure

c mean aerodynamic chord

L reference length

( )[ ]pQ aerodynamic force matrix evaluated in Laplace

domain

s Laplace variable

xxiii

p non-dimensional Laplace variable

[ ]φ modal matrix

[ ]M~ generalized mass matrix

[ ]C~ generalized damping matrix

[ ]K~ generalized stiffness matrix

( )[ ]pQ~ generalized aerodynamic force matrix

ω angular frequency [rad/sec]

f oscillation frequency [Hz]

{ }q amplitude of the generalized coordinates

k reduced frequency

gs assumed structural damping

g structural damping γ coefficient of transient decay rate

an amplitude of the nth cycle of oscillation IR QQ ~,~ real and imaginary parts of the generalized force

matrix

( )[ ]apppQ~ Rogers approximation of the generalized

aerodynamic force matrix

{ })( pq structure state vector

{ })( pη aerodynamic lag state vector

M.Si margin of safety of the ith structural element

x design variable vector

W structural weight

allσ allowable stress

appσ applied stress

γjl damping for the lth mode calculated at the jth

velocity

γjREQ required damping level at the jth velocity

xxiv

GFACT damping scaling factor Ujx upper bound on the jth continuous design variable

Ljx lower bound on the jth continuous design variable

md number of discrete design variables

mi number of integer design variables

mc number of continuous design variables

xyz local coordinate system

XYZ global coordinate system

{} Column matrix

[] Square matrix . t∂∂ .. 22 t∂∂

Other parameters are clearly defined wherever applicable.

1

CHAPTER 1

INTRODUCTION

1.1 Background to the Study

Aircraft design is a complex and iterative task, which requires a trade off between

many conflicting requirements. These include, but are not limited to, high

flutter/divergence speeds, adequate static strength and minimum weight.

During the last decades aircraft manufacturers have begun to use composite

materials in primary structural parts of an aircraft. An important advantage of

composite materials is the freedom available to tailor material properties in a

beneficial way. Composite materials, with their unique stiffness and strength

properties, can provide the necessary strength and stiffness to satisfy conflicting

design requirements with substantially lower weights than would be required in

conventional metallic designs.

The conventional design cycle of a new aircraft component is composed of three

design stages. These stages are the conceptual, preliminary, and detailed design

stages. In the conceptual design stage, material selection is performed and various

design alternatives are analyzed to determine configuration design variables like the

necessary number of ribs and spars and their locations. At this stage, fairly simple

2

analysis models in the form of equivalent beam and/or plate models are usually

adequate for rough initial stress estimates. In the preliminary design stage, detailed

finite element models are constructed based on initial designs and the structure is

analyzed to determine the shape, thicknesses, and dimensions of structural members

like stringers, skins, ribs, and spars. The results of this analysis are then used to

update the initial design. The loads analysis group then calculates new loads based

on the updated design and provide them to the structural analysis group to perform

the detailed analysis stage which completes the design cycle.

Each of the design stage composing the design cycle is a looping process that

requires frequent update of the finite element model. This process is illustrated in

Figure 1.1. The design is first analyzed by using finite element models and the

strength requirements are checked. If the design is unsatisfactory, appropriate

changes are made and the process is repeated until a satisfactory design is reached.

Such a process is a lengthy process to do by hand and becomes even more complex,

if not impossible, when other constraints such as aeroelastic stability constraints

apply.

During the conceptual and preliminary structural sizing stages, design constraints

relating to the interaction of the structure and aerodynamics such as flutter and

divergence are frequently neglected. As a consequence, these effects often cause

problems later in the design cycle and result in extensive design modifications and

weight penalties.

Design of a composite aircraft wing, in order to achieve strength, buckling, and

aeroelastic stability requirements with minimum weight, is a multidisciplinary

design and optimization problem. It involves the interaction of the structural and

aeroelastic analysis disciplines with conflicting requirements on strength, stiffness,

and manufacturing limitations.

3

Figure 1.1 Conventional Design Stage

A variety of multidisciplinary optimization softwares that include aeroelastic

constraints in the optimization cycle have been developed for structural sizing.

Some of these involve fairly simple models that are suitable for conceptual

structural design while other, like FASTOP, ASTROS and MSC/NASTRAN, make

use of the finite element method and may be applied at the preliminary/detailed

design stages. However, the capability of these state of the art tools is limited in

many aspects. Their formulation relies on the fully stressed design concept which

does not necessarily ensure an optimum design. They can not account for

specialized potential failure modes like crippling that is based on emprical analysis

New Design

Analysis

FEM

Optimum Reached ? Final Design

New/ModifiedDesign

Yes

No

4

methods. The accuracy of an optimization depends on the accuracy of the analysis.

Since these softwares rely on the finite element method in calculating the stresses,

the accuracy of the analysis is limited by the mesh size of the finite element model

that is limited by hardware requirements. The optimization algorithms utilized by

these softwares rely on traditional mathematical programming methods. These

methods are gradient-based which are suitable for treating design problems with

continuous design variables. There are many problems which are inherent in the

gradient-based optimization techniques. A basic disadvantage of gradient-based

methods is their convergence to the optimum closest to the starting point in the

design space which might not be the global optimum. Since they use the gradient

information to advance in the design space, they require the design space to be

continuous and convex. They are inefficient when the number of design variables

involved is large.

In practical applications, like the design of a composite wing box, the design

variables are not all continuous and some of them must be selected from a set of

integer or discrete values. The structural members may have to be chosen from

standard sizes and member thicknesses may have to be selected from commercially

available ones. Stacking sequence design of composite plates involves the

determination of the number of plies and their orientations. The stacking sequence

design problem is discrete in nature. Due to manufacturing limitations, the plies are

fabricated at certain thicknesses and the orientations are limited to a small set of

discrete angles. For extracting the best performance from a composite laminate, the

stacking sequence and ply orientations have to be included as design variables in

the optimization process. Buckling and aeroelastic response depends on the stacking

sequence of the laminate. Standard industry tools like ASTORS and

MSC/NASTRAN work on pre-assumed stacking sequence and treat the ply

orientations and/or thicknesses as continuous design variables. The results are then

rounded to closest integer values. However, rounding off design variables may

5

produce suboptimal designs and the assumed laminate stacking sequence might not

produce the optimal laminate design for a composite structure.

In recent years, a lot of research has been devoted to developing optimization

methods that are capable of treating optimization problems with mixed continuous,

discrete, and integer design variables. Among these methods, stochastic search

methods such as simulated annealing and genetic algorithms are the most popular.

Such optimization algorithms offer an advantage over mathematical programming

techniques since they work on function evluations only and do not require any

gradient information. Their lack of dependence on function gradients makes

stochastic search methods less susceptible to pitfalls of convergence to a local

optimum and have better probability in locating the global optimum. Genetic

algorithms have been successfully applied to the stacking sequence design of

composite laminates.

Motivated by providing improved designs in less time than what is currently needed

and make use of composite materials to satisfy or balance conflicting requirements

on strength and stiffness, Multidisciplinary Design Optimization (MDO) has been

the subject of numerous investigations in recent years. The objective of MDO is to

obtain an “optimum” design satisfying performance requirement from various

disciplines such as materials, structures, aerodynamics, control and propulsion. The

objective function may be the weight of the structure, manufacturing cost, or some

performance index.

Integrating different disciplines in the design approach has the advantage that the

final design would be obtained by proper trade-offs between design requirements

from various disciplines. Nevertheless, considering too many design requirements

simultaneously usually makes the design problem too complicated and sometimes

not feasible for solution.

6

1.2 Literature Survey

This section gives a detailed literature survey on the use of multidisciplinary

optimization in aircraft design. The survey focus on optimization with static

strength and aeroleastic constraints, recent developments in the field of aeroelastic

stability analysis and modeling techniques.

The subject of multidisciplinary design optimization has been the objective of

numerous investigations in recent years. Design problems which have been

traditionally solved one discipline at a time, are being analyzed from a

multidisciplinary point of view. Haftka et al. [1] discussed the multidisciplinary

optimization of engineering systems from the standpoint of the computational

alternatives available to the designer. They emphasized that the solution procedure

is necessarily iterative in nature. Sobieski and Haftka [2] presented a survey on the

methods that have been used for the modeling of multidisciplinary design

optimization problems.

Early attempts to optimize structure with aeroelastic constraints were based on

optimality criterion methods. Optimality criterion techniques were generally

efficient and had the ability to handle large number of design variables. But, they

had the drawback that they could not simultanously handle multidiscipinary

constraint types such as strength and flutter requirements. Such constraints could

only be handled in a sequential optimization process. One of the earliest attempts in

this field was performed by Turner [3] in 1969. Turner used a Lagrangian function

consisting of the total mass of the system and an expression that introduced the

flutter constraint in the form of Lagrangian multipliers. For a given frequency, the

stationary points of this function were determined by solving a set of nonlinear

equations using the Newton-Raphson method. This yielded the minimum mass for

the selected frequency. The technique was successfully applied to a rectangular

7

wing with the structural model represented as an elastic beam and the unsteady

aerodynamics calculated using the strip theory.

Because of inefficiencies in existing flutter-prevention procedures at that time,

where a flutter analyst relied largely on judgment in pursuing a flutter “fix”, need

arose to develop the Flutter and Strength Optimization Program (FASTOP) in 1975

[4]. FASTOP was mainly composed of two sub-programs, SOP and FOP, which

were coupled sequentially. The Strength Optimization Program (SOP) was based on

an early developed Automated Structural Optimization Program (ASOP) [5]. SOP

performed automated resizing to achieve a fully stressed (near-minimum-weight)

design. That is, a design in which each element is either subjected to its maximum

allowable stress under at least one loading condition, or is at a prespecified

minimum permitted gauge. The Flutter Optimization Program (FOP) addressed

dynamic analysis requirements to calculate the flutter speed, and if required perform

resizing to increase the flutter speed. The procedure employed to resize the structure

to meet a minimum flutter-speed requirement was based on the criterion that, for

minimum weight, the derivatives of the flutter speed with respect to element weight

must be equal for all elements that had been resized to meet the flutter speed

requirement. FASTOP was used successfully to achieve a 30% increase in the

flutter speed of a strength based design.

In 1971 Rudisill and Bhatia [6] made a major contribution to the field of structural

optimization with flutter constraints by deriving analytical expressions for the first

partial derivative of the flutter velocity and reduced frequency with respect to

structural parameters. They also presented a search procedure that utilized two

gradient search methods and a gradient projection method. A velocity gradient

search method was employed when it was desired to increase the flutter velocity, a

mass gradient search was employed whenever it was necessary to reduce the flutter

velocity, and a gradient projection search method was employed to find a relative

maximum of the flutter velocity while the total mass of the structure is held

8

constant. The step size in the projected gradient search was selected by trial and

convergence proved slow. The method was applied for the structural optimization

of a rectangular box beam subject to a flutter constraint. The same authors [7],

derived equations for finding the second derivative of the flutter velocity with

respect to design parameters. The desired step size in the projected gradient search

was then determined utilizing first and second derivatives of the flutter speed. This

resulted in a significant decrease in the required redesign cycles in the projected

gradient search.

With the contribution of Rudisill and Bhatia in deriving analytical expressions for

the derivatives of the flutter velocity and reduced frequency with respect to

structural parameters [6], [7], gradient based or mathematical programming

techniques started gaining importance in this field. Gwin and Taylor [8] used a

gradient based feasible direction method for the optimization of wing structures

subject to a lower bound on the flutter speed. They also used the natural modes of

the initial design as primitive modes for subsequent designs. This eliminated the

requirement of calculating the sensitivities of the eigenvectors with respect to the

design variables.

Although gradient based techniques had the advantage of handling optimization

problems with multiple constraint types such as stress and flutter simultaneously,

they were restricted in the number of design variables that could be considered and

had the disadvantage of being computationally expensive, requiring many

engineering analyses and gradient calculations to perform the redesign.

Haftka et al. [9] compared mathematical programing and optimality criteria

procedures for the weight optimization of typical aircraft wing structures to satisfy

prescribed flutter requirements. The mathematical programing method selected was

based on an interior penalty function approach. A Lagrangian optimality criterion

and an intuitive optimality criterion based on uniform strain energy density were

9

considered. They concluded that both the mathematical programing and Lagrangian

optimality criterion techniques were reliable and compred well. However, the

optimality criterion based on uniform strain energy density was found to yield less

reliable results. A similar work was performed by McIntosh and Ashley [10] who

applied three different search schemes for the optimization of a simple rectangular

wing model with flutter constraint. The first scheme was based on the method of

feasible directions that is representative of mathematical programming methods.

The other two schemes were derived from necessary conditions for a local optimum

and can be classified as optimality criteria schemes. Results sugguested that

optimality criteria based schemes may be better than the mathematical

programming scheme when designing for multiple constraints with a large number

of design variables.

A major contribution in the field of structural optimization with aeroelastic

constraints is attributed to Hajela [11]. One difficulty with all of the methods used

for flutter optimization was associated with the mode tracking process. Optimizers

used to track one mode to determine the onset of flutter and when the modes

switched they could not adjust to track the new mode. Another difficulty was

associated with “hump” modes that were difficult to track. Hajela solved these

problems by placing the constraints on damping calculated over a series of

velocities rather than placing the constraint on the flutter velocity. Treating the

aeroelastic stability constraints in this manner has the advantage of insuring a

prescribed value of damping in the final design and effectively handles “hump”

modes type of instability. This approach has become a standard process in both

ASTROS and MSC/NASTRAN.

The search for optimal structures is intimately connected with a sensitivity analysis

of the structure with repsect to the design variables determining the aeroelastic

behaviour of the structure. Sensitivity analysis itself provides the designer with

10

some important information and indicates ways of improving the structure in a

rational manner.

Seyranian [12] used variational principles to determine the characteristics of

aeroleastic stability with respect to changes in design parameters. A long and thin

straight wing flying in incompressible air was considered. The wing was treated as a

continuous slender elastic beam and the sensitivity of flutter and divergence speeds

with respect to changes in stiffness and mass distributions was determined. For

appropriate calculation of the sensitivity gradients, he showed that both the main

flutter and the so called adjoint flutter problems have to be solved simultanously. It

was demonstrated that there exists some stiffness and mass distributions for which

removal of some structural mass may increase the critical flutter speed. It was also

shown that the optimization problem of maximizing the flutter/divergence speed for

a given total mass possesses at least two extrema, but that one of them is a local

maximum. Isaac and Kapania [13] studied the sensitivity of flutter speed with

respect to shape and modal parameters using a combination of central difference

scheme and the automatic differentiation software ADIFOR. Kapania and Bergen

[14] calculated the sensitivities of the flutter speed and frequency with respect to

wing shape parameters (aspect ratio, surface area, taper ratio, and sweep angle)

using the finite difference method.

Mathematical nonlinear programming algorithms provide a significant capability

for the automated optimal structural design problem. These algorithms are generally

gradient based and require at least the first derivative of the objective function and

constraints with respect to the design variables. Such algorithms are efficient in

locating a relative optimum closest to the starting point in the design space. With

the advances in mathematical programming techniques, automated analysis

procedures started emerging. The Automated Structural Optimization System

ASTROS [15] was developed to be used in the preliminary design phases of

aerospace structures. This state of the art tool integrated existing methodologies into

11

a unified multidisciplinary package. In this software, concepts like design variable

linking, are employed to reduce the number of design variables and keep the design

from specifying structural sizes that are unrealistic from manufacturing standpoint

of view. Composite materials are treated as having independent physical variables

for each of the user specified ply directions (the same as FASTOP program),

however, the selection of ply orientations as a design variables could not be

specified. In ASTROS constraints on stress/strain (based on Von Mises and/or Tsai-

Wu criteria), displacements, modal frequency, and flutter can be imposed. The

flutter stability analysis is based on the pk-method with the constraint imposed on

the damping. The optimization technique employed is the modified feasible

directions method with a polynomial interpolation used in the one-dimensional

search.

With the advent of composite materials and their introduction into the aircraft

industry, research was directed on making use of these materials in aircraft

structural design. An important advantage of composite materials is the freedom

available to tailor material properties in a beneficial way. Ply thicknesses and

orientation angles may be changed to acheive required stiffness and strength

properties. Concepts like aeroelastic tailoring started emerging. Shirk et al. [16]

gave a definition for aeroelastic tailoring and presented a historical background on

the theory underlying it. They defined aeroelastic tailoring as the embodiment of

directional stiffness into an aircraft structural design to control the aeroelastic

deformation, static or dynamic, in such a way to affect the aerodynamic and

structural performance of that aircraft in a beneficial way.

Lerner and Markowitz [17] developed finite element resizing procedures for

determining the lightest way to stiffen a strength-designed structure to meet static

aeroelastic design objectives like control surface effectiveness, flexible-surface lift-

curve slope, and static divergence velocity. The procedure is demonstrated on a

preliminary design of a wing having composite cover skins.

12

The effect of the fiber orientation angle on the aeroelastic characteristics (flutter and

divergence) was demonstrated by Weisshaar and Foist [18] who showed in their

parametric study of a swept back wing that the flutter and divergence speeds of the

wing are very sensitive to change in the fiber orientation angle with the magnitude

of this change being very large for a certain range of the fiber orientation angle.

Lottati [19] investigated the critical flutter and divergence velocities of a swept

wing as influenced by the bending-torsion stiffness coupling of a composite

cantilevered wing. A high aspect ratio forward swept wing, idealized by a box beam

was considered. His results indicated that the flutter and divergence of a fixed-root

wing involve a compromise. The bending-torsion stiffness that maximizes the

flutter velocity tends to minimize the divergence speed and vice versa.

Ringertz [20] considered the optimal design of a cantilevered wing in

incompressible flow. The wing was modeled as a full depth sandwich wing using

finite element analysis. The design variables were the thicknesses of the composite

face sheets and the objective was to minimize the weight. He concluded that despite

the apparently simpler analysis involved, the divergence speed constraint may be

more difficult for the optimization than the flutter constraint. He also illustrated that

it is important to formulate the optimization problem such that the final design is

not sensitive to imperfections in the design.

A feasibility study that addressed the effect of composite tailoring on the aeorelastic

stability margins of the V-22 composite tiltrotor wing was performed by Popelka et

al. [21]. They concluded that the gain in stability margins is affected by the

conflicting requirements of the torsional and bending modes of the wing.

Eastep et al. [22] explored the benefits of defining the ply orientation as a variable

in the design of composite structures. The implication being that at any given point

13

in the structure all of the fibers are oriented in the same direction and the angle of

the orientation and the total thickness of the fibers are the two variables. In contrast,

the general practice in the design of composite structures is to assign a fixed lay-up

consisting of a number of fiber directions (four or more) and the optimization

selects the percentage of fibers in each direction. An optimization study was

conducted for a composite wing. The effect of ply orientations of the composite

skins on the optimized wing weight subjected to constraints on strength, and

aeroelastic constraints are presented. The study indicated that optimal design of

composite wings is relatively insensitive to the orientation of the laminate lay-up

when the wing is subjected to multiple structural constraints.

Khot and Kolonay [23] used a two level approach to design a composite wing

structure with enhanced roll maneuver capability using the control system to twist

and recamber the wing. In the first step, optimum designs satisfying requirements

on strength, aileron efficiency and flutter for a specified fiber direction were

obtained using ASTROS. The control system was then designed to retwist and

recamber the optimum wing to achieve the target flexible roll rate.

Venkataraman and Haftka [24] presented an overview of the combination of model

complexity, analysis complexity, and optimization complexity in the design of

composite panels and identified areas where more research and development is

needed. They concluded that all applications of expensive optimization (e.g., global

optimization) with the most expensive analysis (analysis under uncertainty) have

been used with the simplest analysis models of single laminate.

Zero-order methods of mathematical programming have been successfully applied

to the optimization of aircraft structures subject to aeroelastic constraints. Such

methods are local search methods that work directly on function evaluations and do

not require the gradients of the objective function and/or constraints. Nevertheless,

they are applicable only to optimization problems with continuous design variables.

14

Since they are local search methods, they locate the relative optimum from the

starting point. An example of the application of such methods is the work

performed by Isogai [25] who developed a preliminary design code that utilized the

non-gradient complex method. He presented a preliminary design study of high-

aspect ratio forward/aft swept composite wings. The thickness distribution and the

fiber orientation angle of the upper/lower skin panels of the simplified wing box

were taken as the active design variables. It was shown that about 47% of weight

reduction of box structure is obtained (both for the aft/forward swept wings

considered) by the optimization compared with those of the non optimized designs.

This supported the result obtained by Weisshaar and Foist [18]. He concluded that

the complex method is very effective and robust when the fiber orientation angle is

taken as one of the design variables and when the flutter velocity is one of the

constraints. However, the rate of convergence of the complex method rapidly

degraded with increasing the number of design variables.

Jha and Chattopadhyay [26] developed an integrated multidisciplinary procedure

for structural and aeroelastic optimization of composite wings based on refined

analysis techniques. The structural analysis is based on a composite box beam

model with each wall of the beam analyzed as a composite plate using a refined

high order displacement field. Unsteady aerodynamic computations are performed

using a panel code based on the constant pressure lifting surface method.

Flutter/divergence dynamic pressure is obtained by the Laplace domain method

through rational function approximation of unsteady aerodynamic loads. The

objective of the optimization procedure was to minimize wing structural weight

with constraints on flutter/divergence speed and stresses at the root due to a static

load condition. Composite ply orientations and laminate thickness were selected as

design variables (assumed constant along the span and chord and the same on top

and bottom walls and side walls giving a total of 12 design variables). The

Kreisselmeier-Steinhauser (K-S) function approach was employed to integrate the

objective function and constraints into a single envelope function and the resulting

15

unconstrained optimization problem was solved using the non-gradient Davidson-

Fletcher-Powell algorithm. They concluded that the optimized design has

significantly lower wing weight (8%) and higher flutter dynamic pressure (23%)

relative to the selected base design.

Stacking sequence design of composite plates involves the determination of the

number of plies and their orientations. Because continuous optimizers have a low

computational cost and are widely available, stacking sequence problems have been

traditionally treated using continuous optimization techniques. Usually the laminate

is assumed to be made of stacks of plies and the thickness and/or orientation of

these stacks is treated as continuous design variables. After the optimization process

is completed, the thicknesses and/or orientations are rounded to integer values. This

is the standard approach to the composite laminate optimization in ASTROS and

MSC/NASTRAN. However, this assumed laminate stacking sequence might not

produce the optimal laminate design for a composite structure. Riche and Haftka

[27] identified the shortcomings and pitfalls implied in the gradient based approach.

The flexural and the in-plane response of laminates are nonlinear functions of the

number of plies, the ply thicknesses, and the fiber orientations. Therefore, for

problems involving this type of response, the design space contains local optima in

which continuous optimization strategies may get trapped. Second, composite

structures often exhibit many optimal designs. The reason is that composite

laminate performance is characterized by a number of parameters which is smaller

than the number of design variables. Different sets of design variables can produce

similar results, i.e., there are many optimal and near optimal designs. Traditional

design approaches not only have the drawback of sometimes converging to the

suboptimal designs, but also of yielding only one solution. Finally, rounding off

design variables may produce suboptimal or even infeasible designs.

Much effort has been devoted to the stacking sequence design of composite plates.

In response to the discrete nature of the problem, integer programming strategies

16

based on Branch and Bound algorithm have been used. Branch and Bound is

basically an enumeration method where one first obtains a minimum point for the

problem assuming all the variables to be continuous. Then each variable is assigned

a discrete value in sequence and the problem is solved again in the remaining

variables. This method was originally developed for linear programming problems.

However, in general, designing a composite laminate is a nonlinear integer

programming problem, Nagendra et al. [28].

Stochastic search methods on the other hand offer an advantage over mathematical

programming techniques, Hajela [29]. These methods are global search techniques

which work on function evluations only and do not require any gradient

information. Stochastic search methods can be easily applied to problems where the

design space consists of a mix of continuous, discrete, and integer variables. Their

lack of dependence on function gradients makes stochastic search methods less

susceptible to pitfalls of convergence to a local optimum and have better probability

in locating the global optimum. Among stochastic search methods, genetic

algorithms and simulating annealing are the most popular. Genetic algorithms are a

class of evolutionary strategies that derive their principle from Darwin’s theory of

the survival of the fittest. Simulated annealing algorithms are based on the

principles of statistical mechanics. Arora and Huang [30] presented a review on the

methods for optimization of non-linear problems with discrete-integer-continuous

variables.

As quoted by Goldberg [31], genetic algorithms were first introduced by Holland in

1975. They are based on Darwin’s theory of survival of the fittest. In a genetic

algorithm one starts with a set of designs. From this set, new and better designs are

reproduced using the fittest members of the set. Each design is represented by a

finite length string. Usually binary strings have been used for this purpose although

other representations are possible as well [32]. The entire process is similar to a

17

natural population of biological creatures; where successive generations are

conceived, born and raised until they are ready to reproduce.

Hajela [33] demonstarted the effectiveness of genetic search methods in the

optimization of problems with nonconvex and disjoint design spaces. The principal

drawback he identified in genetic search methods is the increase in function

evaluations necessary to obtain an optimum. He sugguested that a hybrid scheme

that switches from the genetic search approach to a conventional nonlinear

programming approach after a few generations might overcome this limitation.

Kogiso et al. [34] applied the genetic algorithm to the stacking sequence design of

laminated composite plates to maximize the buckling loads. To reduce the number

of analyses required by the genetic algorithm, a binary tree is used to store designs

and retrieve them and therefore avoid repeated analysis of design that appeared in

previous generations. Linear approximation based on lamination parameters was

used to reduce the cost of genetic optimization.

A two level optimization procedure for composite wing design subject to strength

and buckling constraints was presented by Liu et al. [35], [36]. At wing-level

design, continuous optimization of ply thicknesses with preassumed orientations of

0°, 90°, and ±45° is performed to minimize weight. At panel level, the number of

plies of each orientation (rounded to integers) and in-plane loads are specified, and

a permutation genetic algorithm was used to optimize the stacking sequence in

order to maximize the buckling load.

Upaadhyay and Kalayanaraman [37] developed a general optimization procedure

for the design of layered composite stiffened panels subject to longitudinal

compression and shear loading based on genetic algorithms. Stability and strength

considerations, expressed in the form of simplified equations served as constraints

in the optimal design method.

18

The original stacking sequence problem was solved by Leiva [38] using an

equivalent sizing optimization problem with continuous design variables.

Global optimization algorithms are much developed for unconstrained problems

than for constrained problems. Often these algorithms deal with constraint via

penalty functions, but this treatment may cause substantial degradation in

performance. Liu et al. [35], and Todoriki and Haftka [39] showed the advantage of

their repair mechanism and permutation genetic algorithm for handling constraints

on improving the performance of genetic algorithms.

The usefulness of stochastic search methods in MDO problems is severely limited

without the use of global function approximations. Given that these methods are

primarily based on the use of function information only, the use of response

surface-based approximation is a viable option. Response surfaces are obtained by

fitting a chosen-order polynomial model to a given experimental or numeric data.

The principal drawback of using the approach is that the user must specify the order

of the fit. Further, as problem dimensionality increases, response surface models are

imprecise and very difficult to generate [29]. References [35] and [36] demonstrated

that the response surface can be used effectively to find a near optimal wing design

and [40] used them in the approximation of a composite objective function that

included the weight of the structure, the manufacturing cost, the static response and

the aeroelastic response of a mettalic wing box. Unal et al. [41] discussed response

surface methods for approximation model building and multidisciplinary design

optimization problems.

Aeroelastic and stress analysis disicplines are treated by large-order finite element

models with thousands of degrees of freedom. The computational costs associated

with repeated construction of the full finite element model and the large-order

analysis degrade the usefulness of the optimization scheme, particularly in the

conceptual design stages when extensive trade-off studies for various design

19

concepts are needed. Several studies have focused on developing simplified

analysis models and tools to reduce the computation time required at the conceptual

design stage.

Giles [42] described an equivalent plate analysis formulation that is capable of

modeling aircraft wing structures with general planform geometry such as cranked

wing boxes. He used a Ritz solution technique to determine the static deflection,

stresses, and frequencies of an example wing configuration. The same author, [43],

generalized the method to provide capability to model aircraft wing structures with

unsymmetric cross sections. Livne [44] refined the method further by taking

transverse shear effects into account. He used the first order shear deformation

theory instead of the classical plate theory used by Giles [42,43]. Mukhopadhyay

[45] described an interactive wing flutter analysis program that is applicable for the

conceptual design stage. A comparison study for the results of two multidisciplinary

design optimization programs is given by Butler [46]. The first program uses a

simple beam model and is suitable for conceptual design phase. The second

program uses three dimensional finite element model and is suitable for preliminary

design stages. Some specialized tools like and ADOP (Aeroelastic Design

Optimization Program) and HpyerSizer were introduced to the aerospace

community in [47] and [48]. ADOP is an interdisciplinary optimization program for

the static, dynamic, and aeroelastic analysis of finite element models which was

developed at Douglas Aircraft Company, and HyperSizer is a structural

optimization system specifically designed for aerospace apllications.

Striz and Venkayya [49] investigated the influence of structural and aerodynamic

modeling of various fully built-up finite element wing models on flutter analysis.

They concluded that a reasonably coarse grid for both the structure and the

aerodynamics will result in natural frequencies and mode shapes that are close to

those obtained from more detailed models, whereas this evaluation will also result

in flutter speeds that are conservative.

20

The accuracy of the mathematical models for aeroelastic analysis, design, and

simulation is increased with the number of vibration modes chosen to represent the

structure. However, the associated increase in the model size adversely affects

calculation effeciency. Karpel [50] presented a dynamic residualization method

with which important structural and unsteady aerodynamic effects associated wtith

high-frequency vibration modes are retained without increasing the model size. The

formulation is based on state-space equations of motion where the unsteady

aerodynamic force coefficients are represented by a minimum-state rational

approxiamtion function. Later he applied the method for the multidiciplinary

optmimization of an active flexible wing (AFW), Karpel [51]. A gradient-based

constrianed optimization algorithm was used to minimize the weight subject to

constraints on flutter speed and control stability margins.

Various modal-based static and time-domain aeroservoelastic model size reduction

techniques were reviewed by Karpel [52]. These techniques are combined for an

integrated design optimization scheme where stress, closed loop flutter, control

margins, and time response are treated with a common baseline model. The

structure is represented by a relatively large number of low frequency modes of the

basic design (30-50 modes) and design changes are addressed without changing the

generalized coordinates. Less important modes are then eliminated using truncation,

static residualization, and dynamic residualization reduction methods. Karpel [53]

and Karpel and Brainin [54] expanded the modal approach to deal with stress

considerations in an optimization scheme. Fictitious masses were used to account

for local effects caused by high concentrated loads.

The k-method (American method) and the pk-method (British Method) have been

the standard analysis tools for aeroelastic stability analysis [55]. However, there are

many assumptions implicit in these methods that prohibit their use in an automated

design process.

21

In the k-method, the aerodynamic forces are presented in terms of complex inertia

terms and instability is described in terms of an artificial damping coefficient that is

valid only at the point of instability. This approach results in a linear eigenvalue

problem that is relatively easy to solve, nevertheless, damping information

produced by this method at subcritical speeds has no valid physical meaning

[11,56,57]. Thus, it is not possible to formulate a constraint which prescribes a

measure of viscous damping in the final design [11]. Furthermore, reliable damping

values at subcritical flight speeds are needed as a guideline for conducting wing

tunnel or flight flutter tests [57]. Another drawback found in this method is that it

occasionally produces multiple valued function of damping vs. velocity, making it

difficult to order the roots in an automated process to determine the flutter speed

[11,56]. The coefficient matrix of the eigenvalue problem of the k-method is

singular for zero frequency. Thus, the k-method can not predict the divergence

aeroleastic instability [56,57].

As quoted by Hassig [58], the pk-method or the British method was first proposed

by Irwin and Guyet in 1965. In the original method proposed by Irwin and Guyet

solutions were obtained using graphical methods to match the assumed reduced

frequency with the complex part of the computed complex eigenvalue. Hassig [58]

used a determinant iteration method to match the assumed reduced frequency with

the complex part of the computed complex eigenvalue and Rodden [55] modified

the equation further by introducing a damping terms that is dependent on the

frequency. The pk-method is an approximation of the p-method (transient method).

The p-method requires availability of the unsteady aerodynamic forces in the time

domain which are not usually available at low cost and require computational fluid

dynamics methods. Instead these forces are usually calculated in the frequency

domain for a discrete set of reduced frequencies on the assumption of undamped

harmonic motion utilizing simplified unsteady aerodynamic methods like the

Doublet Lattice Method (DLM) [59]. The rationale for the pk-method is that for

22

sinusoidal motions with slowly increasing or decreasing amplitude, the

aerodynamic forces based on constant amplitude (undamped harmonic motion) are

a good approximation. Thus, although the response of the system is assumed to be

damped in the pk-method, the aerodynamic forces are calculated based on the

assumption of undamped harmonic motion. Thus, the pk-method yields

approximate subcritical trends in terms of damping, although it does not lead to

double valued functions of damping vs. velocity [56].

For n structural modes, the pk-method and the k-method normally provide only n

roots of the flutter equation. However, the number of roots could exceed the number

of the structural modes if some aerodynamic lag roots appear. If the exact

Theodorsen function is used, the number of the aerodynamic lag roots that would

appear in the solution is expected to be infinite [60]. The inclusion of all the

activated aerodynamic lag roots could provide important physical insight into the

the flutter solution [56].

Rodden and Bellinger [61] compared the p-method with the pk-method for the

divergence analysis of a two degrees of freedom airfoil. They concluded that the pk-

method predicts the aerodynamic lag roots adequately and there is no need for

approximations to the aerodynamic transfer funcions and the use of the p-method.

This view is not necessarily shared by others. Zyl and Maserumule [62] analyzed

the same problem and concluded that what Rodden and Bellinger [61] have called

an aerodynamic lag root is nothing but a logical continuation of the structural roots

after their frequencies have gone to zero. Their argument was that whenever the

frequency of a structural mode goes to zero, one would expect the associated

complex root to be replaced by two real roots at higher speeds. A mode tracking

procedure like that used in the pk-method will track only one of the modes and

leave the other which implies that the solution would be incomplete.

23

A new aeroelastic stability analysis method, called the g-method, was recently

introduced by Chen [63]. The g-method is a generalization of the k-method and the

pk-method for reliable damping prediction that is valid in the entire reduced

frequency range. The g-method utilizes a damping perturbation method to include a

first order damping term in the flutter equation. This added damping term is

rigorously derived from Laplace-domain aerodynamics. The same reference proved

analytically that the added aerodynamic damping matrix by Rodden and Johnson

[55] is only valid for small values of the reduced frequency, k, or for cases where

the generalized aerodynamic forces are linear functions of the reduced frequency.

Thus, it is concluded that the pk-method is valid only under these limitations or at

the instability point where the damping is zero. The reduced frequency technique

used by the g-method potentially gives an unlimited number of roots which provide

a better insight into the mechanism resulting in the instability. Some interesting

results obtained by Chen [63] worth mentioning. Flutter is due to the aeroelastic

coupling of structural modes, but the coupling mechanism of the divergence speed

instability is not well understood. For restrained structures, it seems that the

divergence speed is a static aeroelastic instability since its associated frequency is

zero. However, results of the g-method sugguest that the divergence speed is caused

by the coupling of a structural mode and an aerodynamic lag root and should be

considered as a special case of flutter instability. The zero flutter frequency of the

divergence speed is caused by the zero-frequency aerodynamic lag root associated

with the restrained structure. For unrestrained structures the so called “dynamic

divergence” is again a special case of flutter instability caused by the coupling of

the aerodynamic lag root and structural modes but with non-zero frequency. Such

an interpretation could hardly be supported by the pk-method since it is not capable

of generating the non-zero-frequency aerodynamic lag roots.

Another new method which is based on tracking the orientation anlgles of the

eigenvectors was developed by Afolabi et al. [64]. The EVO (Eigen Vector

Orientation) method is based on the fact that the eigenvectors are initially real and

24

orthogonal to each other and lose their orthogonality at the flutter instability point.

Pidaparti et al. [65] reviewed this method and applied it to the flutter analysis of an

intermediate complexity wing (ICW) model. Results obtained using this method are

compared to the flutter prediction results obtained using the pk-method. A

reasonably good comparison between the EVO method and the pk-method was

obtained.

Zyl and Maserumule, [66,67], used three different forms of the pk-method to

determine the divergence speed of a single degree of freedom airfoil. The first was

that used by Hassig [58], the second was that used by Rodden and Johnson [55] and

the third was a form that is equivalent to the g-method of Chen [63]. Although the

three methods predicted the same divergence speed (which is expected since for

zero eigenvalue all the considered three forms are equivalent), the subcritical

damping and frequency behaviour predicted by the three methods were different.

Since aeroelastic divergence of a free flying aircraft does not occur at zero

eigenvalue, they concluded that the three different forms might predict different

divergence speeds.

Alternatively, a root locus solution of the flutter equation in Laplace domain

provides an insight to the aeroleastic stability problem and has been implemented

successfully in the flutter redesign problem by [11,26,68]. Brase and Eversman [68]

used this method to solve the structurally nonlinear flutter problem and the work

performed in [11] and [26] has been described early in the text. The basic advantage

of this method is that it provides valid damping behaviour for the velocity range of

interest. However, one difficulty associated with this method is that it requires the

availability of aerodynamic forces in the Laplace or time domain. An important

feature of these forces is the lag associated with the circulatory wake, where

disturbances shed to the flow by the wing motion continue to affect the loads at a

later time [69]. Theodorsen [60] employed a lift deficiency function in the reduced

frequency domain to represent this effect for the oscillatory flow over an airfoil. In

25

1940, Jones used a two term series of decaying exponentials in the time domain to

approximate the effect of circulation for the transient aeroelastic motion. This led to

the well known Jones rational function approximation of the Theodorsen function

[70].

Several methods had been developed to express the aerodynamics of general

planforms in the Laplace domain using rational function approximations. Among

them the Rogers method [71] and the minimum state method of Carpel [72] are the

most popular and widely used ones. Rogers method relies on approximating each

term of the generalized aerodynamic forces in the form of rational functions with

common denominator roots. The minimum state method is based on a more general

approximation function with coupled terms and constraints on the coefficient

matrices. Consequently this method requires computationally heavier, iterative,

nonlinear least square solutions. The computational time was nearly 1000 times

greater than that of Roger’s method. Karpel and Strul [73] modified this method to

improve its performance by introducing a new solution strategy and relieving some

of the constraints.

In a conventional Roger’s approximation the aerodynamic lag roots are usually

chosen based on experience from the reduced frequency range of interest. Eversman

and Tewari [69] introduced an improved method for the rational function

approximation of unsteady aerodynamics that is based on Roger’s method with the

aerodynamic lag roots chosen by an optimizer to minimize the total fit error. The

optimization method utilized was the simplex non-gradient method.

26

1.3 Scope and Contents of the Study

In this study an automated multidisciplinary design optimization code is developed

for the minimum weight design of a composite wing box. The multidisciplinary

static strength, aeroelastic stability, and manufacturing requirements are

simultaneously addressed in a global optimization environment through a genetic

search algorithm. The intention is to obtain a minimum weight final design that

complies with the existing compliance requirements (FAR/JAR) in less time than

what is currently needed while taking aeroelastic stability constraints into account at

the early stages of the design. This would eliminate the need for extensive design

modifications at later stages of the design which may result in weight penalties and

failure to deliver product on time.

The static strength requirements specify obtaining positive margins of safety for all

of the structural parts of the wing box taking into account all potential failure

modes. Besides to classical failure modes (material failure), specialized failure

modes (buckling and crippling) are taken into account in the optimization process.

The buckling analysis is based on the Rayleigh-Ritz method and the Gerard method

is used for crippling analysis. The static strength analysis procedure is based on a

refined process that is consistent with the aerospace industry approach to the

analysis of this type of structures. A coarse mesh finite element model is utilized to

determine the internal load distribution in the wing box. The modified engineering

bending theory is then utilized to calculate the stress distribution taking into account

nonlinear effects that result from redistribution of the load in the wing box after

buckling occurs in the structure. In this procedure MSC/NASTRAN® is used as

finite element solver.

The aeroelastic stability analysis requirements specify obtaining flutter and

divergence free wing box for a range of prescribed flight conditions and with

27

required damping level in the final design. The aeroelastic stability analysis

procedure is based on a root locus method. The unsteady aerodynamic forces in

Laplace domain are obtained from their counterparts in the frequency domain using

Rogers rational function approximations. In this procedure, MSC/NASTRAN® is

used to perform free vibration analysis to determine the generalized mass, stiffness,

damping, and aerodynamic forces in the frequency domain.

The optimization procedure utilized in this thesis relies on a genetic search

algorithm that is suitable for the design of the wing box problem under

consideration. It was courtesy of Prof. Prabhat HAJELA of the Rensselaer

Polytechnic Institute to provide the optimizer (EVOLVE) to be utilized in this

thesis.

In Chapter 2 the main components of a typical wing box, their functions and their

failure modes are described first. The stress analysis, based on the modified

engineering bending theory together with the classical laminated plate theory and

the coarse mesh finite element analysis methodology, is then explained. Then the

procedure for calculating the allowable stresses under the effect of combined

loading follows. Special attention is given to the buckling analysis based on the

Rayleigh-Ritz method. Buckling stress analysis results for two representative

metallic and composite panels, which are under the effect of the combined loading

conditions, are consequently given. The buckling allowable stresses obtained by the

Rayleigh-Ritz method are compared to those obtained with the specially orthotropic

plate assumption. Buckling analysis results of the Rayleigh-Ritz method are verified

using a fine mesh finite element analysis utilizing MSC/NASTRAN®. The strength

analysis procedure to determine the minimum margins of safety, the critical load

cases and the corresponding failure modes are then discussed. An illustrative test

case for a simple wing box with metallic internal structure and composite skins is

28

then analyzed. Justification to use the modified bending theory methodology is

illustrated with a fine mesh finite element analysis of the considered wing box.

In chapter 3 the aeroelastic phenomena in general with emphasis on flutter and

divergence phenomenon is first described. The mathematical formulation of the

aeroelastic stability problem is then discussed. Aeroelastic analysis methods based

on the k-method, pk-method, p-method, and the root locus method are also

explained. The differences between these methods are identified. The method of

obtaining the generalized aerodynamic forces in Laplace domain using Rogers

rational function approximations is explained. Two test cases are studied to see the

difference between the pk-method and the root locus method and to verify the

adopted methodology for approximating the generalized aerodynamic forces in

Laplace domain by using the rational function approximations. The first case study

is the BAH wing and the second is an intermediate complexity wing (ICW) model.

Chapter 4 explains the automated multidisciplinary design and optimization

procedure of the composite wing box. The optimization problem attempted is first

described. The problem is then mathematically formulated in terms of the objective

function, the static strength and aeroelastic stability constraints, and the

manufacturing constraints on the design variables. The solution procedure for the

optimization problem is explained. The static strength analysis, aeroelastic stability

analysis, and optimization methods utilized in the procedure are then discussed. The

developed code for the automated procedure with its features and limitations are

then described in detail.

In chapter 5 the developed multidisciplinary design and optimization code is

applied to the design of a rectangular wing box. The wing box is considered in three

separate case studies. The first case study aims at verifying the developed code and

studies the capability of the genetic algorithm in optimization for aeroelastic

constraints with manufacturing constraints imposed on the design variables. Thus,

29

an all metallic wing box which is fully described and has available optimization

results in literature is optimized to meet the aeroelastic stability constraints with

manufacturing constraints imposed on the thicknesses of the spars webs, ribs webs,

and spars caps areas. The second case study aims at studying the capability of the

developed code in the optimization of representative “real-life” composite wing

structures. Hence the wing box considered in the first case study is modified to have

composite skin panels and ribs chords and is then optimized to meet static strength

requirements subject to manufacturing constraints on the thicknesses, ply

orientations and cross sectional dimensions of the spars caps and the ribs chords.

The third case study aims at analyzing the advantages of considering the aeroelastic

stability constraints at the early stages of the design. Thus, the optimized wing

considered in the second case study is first analyzed to determine its

flutter/divergence speeds. Then a 20% increase in the flutter/divergence speed is

imposed on the design and the wing box is optimized to meet the aeroelastic

constraints, static strength constraint, and manufacturing constraints simultaneously.

Chapter 6 gives the conclusions and recommendations for future work and further

studies.

30

1.4 Limitations of the Study

The study is limited to fixed configuration design variables. Thus, the wing

planform, number of ribs, number of spars, and their corresponding locations are

assumed to be fixed.

The study is limited to the analysis of unstiffened skin panels. Although the skin

panels can be of laminated composite or metallic types, the spars and ribs are

limited to metal construction.

The developed code utilizes MSC/NASTRAN® (v75.7) as a finite element solver

and EVOLVE as an optimizer. Thus both of these packages are required for running

the code.

31

CHAPTER 2

STATIC STRENGTH ANALYSIS

2.1 Introduction

The wing of an aircraft provides the aerodynamic lift force necessary to carry the

payload and supports the fuselage together with any undercarriage loads. It consists

of the wing box, the leading edge, the trailing edge, and the control surfaces (flaps,

ailerons and spoilers). The wing box is the main load carrying structural component

of the wing.

In the design of a wing box, as with any other aircraft component, adequate strength

and stiffness has to be assured to demonstrate the compliance with the existing

requirements (FAR/JAR). There are mainly two requirements. The first requirement

is that under the effect of the applied or limit loads, no structural member shall be

stressed above the material yield point, or in other words there must be no

permanent deformation of any part of the structure. The second requirement is that

under the design or ultimate loads, which are equal to the applied loads times a

factor of safety, no failure of the structure should occur.

The wing of an aircraft is generally subject to two types of loading. These are

basically ground and flight loads. Ground loads include loads due to landing,

32

taxiing and towing. Flight loads include aerodynamic loads in a cruise flight,

inertial loads during maneuver and gust loads. In a typical design, the wing is

analyzed for hundreds of load cases and the integrity of the wing structure has to be

assured under the effect of all these load cases. Thus, for each load case, the stress

acting on each element has to be determined. Depending on whether the stress is

compression or tension, the relevant allowable stress is determined and the margin

of safety is calculated. After analyzing the entire load cases, the critical load case

(the load case with the minimum margin of safety) and the corresponding failure

mode is identified. Structural tests are finally performed for the most critical load

case(s) to demonstrate the compliance with the certification requirements.

Typical wing boxes exhibit thin metal/composite panels joined together to form the

structure. Since these panels are very thin, they usually buckle at very low stress

amplitudes causing redistribution of the load in the structure. Besides, the stress

levels usually exceed the linear material range and plasticity effects start gaining

importance. A detailed linear finite element analysis of the structure would not

account for such nonlinear effects. Instead, simple finite element models are usually

used to get the internal load distribution in the unbuckled structure. This is then

followed by post processing of the internal loads to simulate the accurate buckling

behavior of the structure. This approach significantly reduces the manpower and

resources required to analyze the structure and yields quite adequate results.

In this chapter the main components of a typical wing box, their functions and their

failure modes are first described. The stress analysis based on the modified

engineering bending theory together with classical laminated plate theory and the

coarse mesh finite element analysis methodology is then explained. The procedure

for calculating the allowable stresses under the effect of combined loading is then

explained. Special attention is given to the buckling analysis based on the Rayleigh-

Ritz method. Buckling stress analysis results for two representative metallic and

composite panels, which are under the effect of combined loading conditions, are

33

then given. The buckling allowable stresses obtained by the Rayleigh-Ritz method

are compared to those obtained with the specially orthotropic plate assumption. The

strength analysis procedure to determine the minimum margins of safety, the critical

load cases and the corresponding failure modes are then discussed. An illustrative

test case for a simple wing box with metallic internal structure and composite skins

is then analyzed. Justification to use the modified bending theory methodology is

illustrated with a fine mesh finite element analysis of the considered wing box. The

chapter ends with some concluding remarks and discussions.

The discussions applied to this chapter are based on the aerospace industry

approach for the analysis of wing boxes and stem from the author’s work

experience in this field. They are partially discussed in [74-77].

2.2 Description of the Wing Box

A wing box is typically made of skin panels, spars, and ribs. The skins provide the

contour necessary to generate the aerodynamic force and transfer these loads to the

spars, ribs, and stringers. Thus the skins are usually subject to a combination of

shear and axial stresses. The skins can be either stiffened or unstiffened. Stiffened

skin panels have stringers attached to them. The stringers resist the compressive

stresses due to wing bending. They also divide the skins and thus increase the

allowable buckling stress of the panels. The spars are made of the spar caps and

web. They carry the shear and bending stresses. The caps resist the bending stresses

and the web carries the shear. The ribs are usually shear tied to the skins. They

support the skins in resisting the aerodynamic loads, help the wing to maintain its

aerodynamic shape, support the stiffeners to prevent global buckling and transform

any concentrated loads coming from any attached fittings (for example the engine

fittings) to the wing box. The construction of the ribs is similar to the spars. They

are basically made of chords and webs. The chords resist the induced bending

stresses and the web resists the crushing aerodynamic loads and shear stresses. The

34

structural details of a typical wing box considered in this work are illustrated in

Figure 2.1. The structure has metallic ribs and spars. The skins can be composite or

metallic.

Figure 2.1 Structural Details of a Typical Wing Box

Rib Chords Rib Web

Rib Details

Spar Caps

Spar Details

Spar Web

FRONT SPAR (Metallic)

REAR SPAR (Metallic)

RIBS (Metallic)

UPPER SKIN (Metallic/Composite) INTERMEDIATE SPAR

(Metallic)

LOWER SKIN (Metallic/Composite)

35

2.3 Failure Modes of the Wing Box Components

This section describes the potential failure modes of the skins, ribs webs, ribs

chords, spar webs, and spars caps.

The failure mode of a wing structural part depends on its stress state and

construction material. Depending on the load case under consideration, some parts

of the structure may be in compression while others are in tension. A structural

element may be in compression for a certain load case and in tension for another.

For example, while the upper cap of a spar is subjected to a compressive stress in

cruise flight condition, it is subject to tensile stress in a hard landing load case.

The skins of the wing box resist both shear and in-plane axial loads. The skins can

be either of composite or metallic type. Thus, their failure modes change depending

on their material type and loading condition.

If the skins are made of composite material and their stress state is due to a

combination of shear and compression loads, then their probable failure mode is

buckling. Since buckling would cause delamination of the plies, buckling of

composite skins is not usually allowed up to ultimate load. If the skins are subject to

a combination of shear and tension loads, then their probable failure mode is fiber

breakage, if the principal normal stress/strain in the ply exceeds the maximum

allowable ultimate tensile stress/strain. Although buckling is not likely to happen,

since the tensile stress helps to prevent buckling, the skin panel may buckle due to

the shear effect and buckling analysis has to be performed to check for buckling.

Independent of whether the load case is a combination of shear and compression or

shear and tension, the maximum stress/strain in each ply has to be checked against

the relevant material allowable stress/strain. These allowable stresses/strains are

usually reduced to account for fatigue effects and manufacturing defects.

36

If the skins are metallic and subject to a combination of shear and compression

loads, then their potential failure mode is buckling. However, since buckling would

not cause failure of the structure (it causes only redistribution of the load in the

structure), it is usually allowed above certain percentage of the ultimate load

(typically 60%). The main reason for not allowing buckling below this percentage

of the ultimate load is due to fatigue issues. Under the effect of combined axial and

shear stresses, metallic skin panels usually fail in rupture if they are thin and the

maximum tensile stress has to be checked against the maximum allowable tensile

stress coming from the material allowable data or fatigue considerations. The

maximum shear stress in the panel has also to be checked against the allowable

shear stress of the material.

The spars are designed to be shear resistant. In a shear resistant design the web is

not allowed to buckle and must support the induced stress under the effect of the

combined action of the bending, axial and shear loads without failure. Thus the

possible failure modes for the web are buckling under the effect of the combined

stress state, web rupture due to the effect of combined axial, bending and shear

stresses (Von-mises), the tensile stress in the extreme fiber of the web exceeding the

allowable ultimate stress of the material and the maximum shear stress on the web

exceeding the ultimate shear stress of the material. The caps of the spars are

supported by both the skins and spar web. Thus, the spar caps can not buckle as a

column and the only possible failure modes for the spar caps are crippling for the

part under compression and the maximum allowable ultimate tensile strength for the

part under tension stress.

The ribs are similar to the spars in construction. However, since they support the

skin in resisting the aerodynamic loads, they are subject to bending, shear and

biaxial in-plane loads. Special purpose ribs have fittings attached to them to support

the control surfaces and engines. So they act as load transfer member that transfer

37

loads coming from the engines and control surfaces and dump them to the main

box. The rib failure modes are similar to the spars.

2.4 Stress Analysis of the Wing Box

This section gives the details of the stress analysis method for various components

of the wing box.

The wing box of an aircraft wing is a thin walled structure. Nonlinear effects such

as buckling, post buckling and material non-linearity has to be taken into

consideration when analyzing thin walled structures. Such effects can be taken into

consideration using two different approaches. In the first approach, the structure can

be analyzed using nonlinear finite element methods with detailed finite element

models. The results of a stress analysis that utilizes the finite element method to

determine the stress distribution in a structure depend to a great extent on the type

of elements utilized in the model and the mesh density. An alternative to this

approach is to use a coarse mesh finite element model is utilized to determine the

internal load distribution in the structure and then post processing of these results to

simulate the correct behavior of the structure. This second option is not very much

sensitive to the type of element chosen and would always yield results with good

accuracy. Since the structure has to be analyzed for hundreds of load cases, and

considering that a typical coarse mesh global finite element model (a model which

includes the wing, fuselage and the tail) of an airframe structure has millions of

degrees of freedom, these make the first approach non practical and would result in

finite element models that are practically impossible to handle and analyze with

good accuracy due to the large sizes of the resulting matrices. In this work, the

analysis methodology for determining the stress distribution in the structural parts

of the wing box is based on the second approach.

38

For the stress analysis of the wing box a coarse mesh finite element model is first

used to determine the internal load distribution in the box. In a coarse mesh finite

element model, the skin, for example, is modeled by only one element between its

adjacent ribs and spars. Thus, grids are only created at the intersection points of the

structural components. After the internal loads are determined, they are summed to

determine the sectional forces (i.e., the shear force, the normal force and the

bending moment) acting on the section. The section axial and bending stiffnesses

are determined and the modified engineering bending theory is then used to

determine the stress distribution over the section. The classical laminated plate

theory is used in both determining the equivalent stiffnesses and the analysis of the

composite skins over the section.

2.4.1 Sectional Loads

The sectional loads are obtained from the finite element model using the grid point

force balance output of MSC/NASTRAN. The free body forces and moments of the

elements adjacent to the section on either side ( )ii MF , are summed and reduced to

force-couple systems at the upper and lower grids of the section in a coordinate

frame that is normal and tangential to the section.

, tFFui

itu ⋅

= ∑ (2.1)

, nFFui

inu ⋅

= ∑ (2.2)

( ) tnMMui

io,u ×⋅

= ∑ (2.3)

, tFFi

it ⋅

= ∑}

} (2.4)

39

nFFi

in ⋅

= ∑}

}, (2.5)

( ) tnMMi

io, ×⋅

= ∑}

} (2.6)

The sectional normal force (N), the shear force (V) and the bending moment (M)

acting on the section are then obtained by reducing the force-couple systems at the

upper and lower grids to a force couple system at the section centroid.

nnu FFN ,, �+= (2.7)

ttu FFV ,, �+= (2.8)

( ) nnuoouo FZFZHMMM ,,,, ��−−++= (2.9)

The procedure for calculating the sectional loads is illustrated in Figure 2.2. Note

that the bending moment acting on the section is a function of the neutral axis

location Z .

40

Figure 2.2 Calculation of the Sectional Loads

2.4.2 Classical Laminated Plate Theory (CLPT)

For the analysis of laminated skin panels, the classical laminated plate theory

(CLPT) is used. The CLPT is based on the thin plate theory with the Kirchoff

assumptions (i.e., plane sections remain plane after deformation) and plane stress.

Besides each lamina (layer) of the laminate is assumed orthotropic, linear elastic

and has constant thickness.

n ≡≡≡≡

Fℓ,t

t

Fu,n

Fℓ,n

Fu,t

θu

θℓ

N

V

Mo H

Mu,o

Mℓ,o

Z

41

The stress-strain relations for an orthotropic lamina are given by

=

12

2

1

66

2212

1211

12

2

1

00

00

γεε

τσσ

QQQQQ

(2.10)

where Qij are the reduced stiffness terms which are obtained from the lamina

material properties.

2

21

1

121266

2112

1122112

2112

222

2112

111

;

1

1 ;

1

EEGQ

EQQ

EQEQ

νννν

ν

νννν

==

−==

−=

−=

In the laminate coordinate system (xyz), this equation transforms to the following

form,

=

xy

y

x

xy

y

x

QQQQQQQQQ

γεε

τσσ

662616

262212

161211

(2.11)

Here ijQ are known as the transformed stiffness terms and are given by the

following set of equations,

42

)()22(

)2()2(

)2()2(

)()4(

)2(2

)2(2

4466

226612221166

3662212

366121126

3662212

366121116

4412

2266221112

422

226612

41122

422

226612

41111

mnQmnQQQQQ

mnQQQmnQQQQ

mnQQQmnQQQQ

nmQnmQQQQ

mQnmQQnQQ

nQnmQQmQQ

++−−+=

+−+−−=

+−+−−=

++−+=

+++=

+++=

(2.12)

where θ= sinn and θ= cosm . The positive sign convention for θ and the stress

resultants is illustrated in Figure 2.3. The assumption of linear strain distribution

through the laminate results in the following equation

+

=

xy

y

xx

xy

y

x

z

xy

y

κκκ

γεε

γεε

�

�

�

(2.13)

where oiε and iκ are the mid-plane strains and curvatures respectively. Substituting

equation (2.13) into equation (2.11) gives,

+

=

xy

y

xx

xy

y

x

zQQQQQQQQQ

xy

y

κκκ

γεε

τσσ

�

�

�

662616

262212

161211

(2.14)

43

Figure 2.3 Positive Sign Convention of Stress Resultants and Ply Orientation

Angle

+θ

σx

σxy

σy

σ1 σ12 σ2

1 (Lamina axis)

x (Laminate axis)

y

2

Nx

Ny

Nxy

x

y Mxy

x

y

Mx

My

Moment Resultants Force Resultants

44

The stress resultants are obtained by integrating the stress through the thickness of

the laminate.

∫ ∑ ∫− =

=

=

2/

2/ 1

z

zi

i

1-i

t

t

N

ixy

y

x

xy

y

x

xy

y

x

dzdzNNN

τσσ

τσσ

(2.15)

∫ ∑ ∫− =

τσσ

=

τσσ

=

2

2 1

z

zi

i

1-i

/t

/t

N

ixy

y

x

xy

y

x

xy

y

x

dzz z dzMMM

(2.16)

Substituting equation (2.14) into equations (2.15) and (2.16) results in the following

load-strain relation matrix equation,

=

xy

y

x

oxy

oy

ox

xy

y

x

xy

y

x

κκκ

εε

DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA

MMMNNN

γ

662616662616

262212262212

161211161211

662616662616

262212262212

161211161211

(2.17)

where,

( ) ( )∑=

−−=N

kkkkijij zzQA

11 (2.18)

( ) ( )∑=

−−=N

kkkkijij zzQB

1

21

2

21 (2.19)

( ) ( )∑=

−−=N

kkkkijij zzQD

1

31

3

31 (2.20)

45

Equation (2.17) can be written in a compact form as,

=

κε o

DBBA

MN

(2.21)

Here [A] is the extensional stiffness matrix, [B] is the extension-bending coupling

stiffness matrix and [D] is the flexural bending stiffness matrix. For any symmetric

laminate the coupling stiffness matrix [B] is always zero.

After the mid-plane strains and curvatures due to an applied stress resultant loads

set are determined from equation (2.21) the ply strains and stresses in the laminate

coordinate system can be determined utilizing equations (2.13) and (2.14). These

strains/stresses are then transformed into the lamina principal axis to calculate the

principal ply strains and stresses.

2.4.3 Equivalent Axial and Bending Stiffness Properties

The segments forming the cross section of a spar or rib together with the effective

skins forms a section that is made of a combination of composite and isotropic

materials that exhibit different elastic properties. In analyzing this type of sections,

the axial and bending stiffnesses (EA,EI) should be used rather than the area and

moment of inertia (A,I) of the section. The calculation of the stiffness for a segment

that is made of an isotropic material is straightforward. However, for a segment

which is made of a laminated composite material the approach is different.

Consider a symmetrically laminated plate of width (beff) and thickness (t) shown in

Figure 2.4. This plate represents the skin segment on the section. The equivalent

axial and bending stiffness terms which relate the applied loads to the mid-plane

strain and curvature can be determined in the following way.

46

Figure 2.4 Laminate Equivalent Stiffness

The axial and bending stiffness matrices of this laminate with respect to its local

centroidal axes system (xyz) are first obtained using the classical lamination theory

as outlined in the previous section. Since the laminate is symmetric, the axial-

bending coupling stiffness matrix [B] is zero. The compliance matrices for the

laminate are then obtained by inverting the stiffness matrix.

[ ] [ ]1−

= Aa (2.22)

[ ] [ ]1−

= Dd (2.23)

Ply4

Ply3

Ply2 Ply1

yy

z Z

Zo

Y

beff

t

X

x

47

The equivalent or average axial modulus for the laminate is then calculated as

taExx

11

1= (2.24)

The finite width axial stiffness (Axx) and bending stiffness (Dyy) for the laminate are

then calculated from the following equations.

11ab

A effxx = (2.25)

11db

D effyy = (2.26)

The stiffness is then transformed to the global coordinate system (XYZ) using the

following equations.

xxXX AA = (2.27)

xxoyyYY AZDD 2+= (2.28)

48

2.4.4 Normal Stress Analysis

The normal stress acting on the section is due to the bending stress induced by the

bending moment (M) acting on the section and the axial stress that is due to the

axial load (N). The principle of superposition is used to determine the normal stress

distribution over the section. Since the section is, generally, composed of different

materials the linear stress assumption is not valid. Nevertheless, the strain

distribution over the section is still valid if the different components forming the

section are perfectly fastened to each other. Hence the strain distribution rather than

the stress distribution is determined and the stresses are calculated from the strains.

XXYY AN

DzM +=ε (2.29)

where XXA is the axial and YYD is the centroidal bending stiffness of the section.

The total stiffness of the section is the summation of the stiffnesses of the individual

segments forming the section.

∑=i

iXXXX AA , (2.30)

XXi

iYYYY AZDD2

, −=∑ (2.31)

∑

∑=

iiXX

iiXXio

A

AZZ

,

,,

(2.32)

where Z denotes the position of the neutral axis with respect to the reference

coordinate frame (XYZ).

49

Bruhn [74] modified the engineering bending theory to account for load

redistribution resulting from buckling of the skins. The engineering bending theory

is modified in the sense that skins working in compression are only partially

effective in carrying stress. Thus, the width of the skin that is considered effective

in carrying compression load is given by the following equation

( )2,9.1min btb skskeff ε= (2.33)

where tsk is the skin thickness, εsk is the strain level in the skin and b is the total

width of the skin panel.

Since the effective skin width is a function of its strain level, the process of

calculating the section stiffnesses requires an iterative process. First, the skin is

assumed to be fully effective in carrying compression loads and the section stiffness

is calculated based on this assumption. Then, the part of the skin which is under

compression effect is identified. This can be the upper or lower skin depending on

the load case under consideration. The effective skin width is then calculated from

equation (2.33) and the position of the centroid together with the section stiffness is

recalculated. The sectional loads are then reduced to a force-couple system at the

new calculated centroid location and the strains in the skins are calculated again.

This process is repeated until convergence of the centroid location is achieved.

50

N.A Web

Lower Right Skin Lower Left Skin

Upper Left Skin Upper Right Skin

Upper Cap

Lower Cap

Lower Grid Point

Upper Grid Point

Y

Z

Y

A

A

Z

Section A-A

Figure 2.5 Typical Spar Cross Section

51

Figure 2.6 Strain Distribution Over the Spar Section

2.4.5 Shear Stress Analysis

The shear force acting on the section is assumed to be carried by the web only.

Nevertheless, the caps and skins of tapered box beams carry a part of the shear force

and help in relieving the shear stress acting on the web, Niu [75]. This effect is

illustrated in Figure 2.7. For the most general case of a tapered cross section, the

force acting on the web is given by

��θθ tantan PPVV uuweb −+= (2.34)

θu

θℓ

N.A H

N

V

M

Centroid Force Resultant Strain Distribution

52

where P is the total normal force in the skin and cap/chord and θ is the taper angle.

The value of P is obtained by integrating the normal stress distribution over the skin

and cap/chord area and will be negative if the integration result is negative.

The average and maximum shear stress on the web are then calculated from the

following equation.

( )webweb

webaveweb ht

V =τ (2.35)

( ) ( )avewebweb ττ23 max = (2.36)

While the average stress value is used in buckling analysis, the maximum shear

stress is utilized in the strength checking.

Figure 2.7 Tapered Section Shear Stress

Pu tan θu

Zθu

θℓ

Vw

Pu

Pℓ

Pℓ tan θℓ

V

53

2.5 Allowable Stresses

The allowable stresses for tension load cases are usually specified in terms of

material ultimate stresses and/or reduced allowable stresses to account for fatigue

considerations and manufacturing defects. Thus these values are usually specified

based on prescribed inspection programs, crack growth and damage tolerance

analysis results. The procedure of calculating these allowable stresses is beyond the

scope of the current work. For the preliminary sizing of the structure, usually the

allowable fatigue stress values are specified based on past experience and similar

designs.

The main allowable stresses necessary for the strength analysis of structural

members working under compressive loads are the crippling and buckling allowable

stresses

2.5.1 Crippling Allowable Stress

Crippling is defined as an inelastic distortion of the cross-section of a structural

element in its own plane resulting in permanent deformation of the section. This

behavior is one of the most common failure mechanisms encountered by aerospace

structures under compressive loads. The crippling phenomenon is quite complex.

There are no analytical equations to describe crippling. The crippling failure is

illustrated in Figure 2.8. Crippling is always preceded by local buckling of the

segments forming the section.

Empirical techniques have been developed by using coefficients derived from

various tests since there is no analytical basis for the prediction of the crippling

strength. The crippling stress for a particular cross-section area is calculated as if

the stress was uniform over the entire section. Furthermore, the maximum crippling

strength of a member is calculated as a function of its cross-section rather than its

54

length. In reality, parts of the section may buckle well below the crippling stress.

This results in the more stable areas, such as corners and intersections, reaching a

higher stress than buckled elements. At failure, the stress in the corners and

intersections is always above the material yield stress although the “crippling”

stress, which is an average value, may be considerably less than the yield stress.

Figure 2.8 Crippling Failure

Bruhn [74] presented several methods for calculating the allowable crippling stress

of a section. These methods include the Gerard method, the Needham method and

the Modified Needham method. The Gerard method is generalization or broader

application of the Needham Method. In this work the Gerard method is adopted for

the calculation of the allowable crippling stresses of the spar caps and rib chords.

Initial Structure Local Buckling Crippling

P1 P2 P3

P1 < P2 < P3

55

For angle sections, the following crippling stress equation applies within ± 10

percent limits,

85.0 2/1

2256.0

=

cycy

cs

FE

At

FF (2.37)

For tee sections the crippling stress is obtained from the following equation with an

error limit of ± 5.

40.0 2/1

2367.0

=

cycy

cs

FE

At

FF (2.38)

where Fcs is crippling stress of the section, Fc is the compression yield stress, t is

the average thickness, A is the cross sectional area and E is the Young’s modulus of

elasticity.

Some cutoff values are used as the crippling strength cutoff since there is not

sufficient data to permit an exact solution at higher stress values for most materials.

Table 2.1 gives the cut-off crippling values for commonly used cross-sections.

Table 2.1 Cut-Off Crippling Stresses

Type of Section Max. Fcs

Angles 0.7*Fcy

T-Sections 0.8*Fcy

Zee, J, Channels 0.9*Fcy

56

2.5.2 Allowable Buckling Stress

The calculation of the buckling allowable stress is necessary for the strength

analysis of the skins, spar webs, and rib webs since these parts of the wing box are

not usually allowed to buckle up to the ultimate load or a certain percentage of the

limit load. Different parts of the structure are usually subject to different kinds of

combined load systems. The skins are subject to a combination of in-plane axial and

shear stress. The spar webs carry bending, longitudinal axial and shear stresses. The

ribs support in-plane transverse axial and shear stresses.

Practically for metallic structures the allowable buckling stresses under the effect of

compression and shear stresses are calculated separately using simple equations that

are based on buckling coefficients obtained from tabulated or graphical data. The

allowable buckling stress under the combined effect of axial, biaxial, bending and

shear stresses (depending on the structure under consideration) is then obtained

from what is known as interaction equations. However, the use of this method is

complicated since the analyst is required to read buckling coefficients from curves

or tables and may result in conservative results in some cases where tension stress

effects are usually neglected due to the lack of appropriate interaction equations and

for the sake of simplifying the analysis. For example, in calculating the allowable

buckling stress of fuselage skin panels, the effect of the transverse tensile hoop

stress that is due to the internal pressure load effect is usually neglected and only

the effect of longitudinal compression acting with shear stress is considered. This is

basically due to the lack of an interaction equation that would consider the effect of

combined tension, compression and shear buckling stresses simultaneously.

Obviously, considering the effect of the tensile hoop stress would have resulted in a

higher allowable buckling stress which in turn would result in weight saving.

For composite panels buckling analysis the problem of calculating the allowable

buckling stress is more difficult. Practically used composite panels are generally

57

symmetrically laminated anisotropic panels that exhibit bending-twisting coupling

effects. Several research works has been done on the optimization of composite

panels with buckling constraints. There, the use of the interaction equations with the

assumption specially orthotropic material is a common practice. A specially

orthotropic plate has either a single layer of specially orthotropic material or

multiple specially orthotropic layers that are symmetrically arranged about the

laminate middle surface to form a symmetric cross-ply laminate [78]. Such plates

do not exhibit any bending-extension or bending-twisting stiffness coupling terms.

However, and as will be shown later in this section, this approach will result in a

non conservative result for the allowable buckling stress and most practical

laminated plates do not satisfy the requirements of specially orthotropic plates [79].

Besides, for a generally anisotropic material the allowable buckling stress becomes

a function of the direction of the applied shear stress which makes the use of the

interaction equation invalid. In industry the use of the energy methods, such as the

Galerkin and Rayleigh-Ritz methods, to determine the allowable buckling stress of

a composite panel is the adopted approach.

In this study, the Rayleigh-Ritz method is used to determine the allowable buckling

stress of a symmetrically laminated anisotropic plate with four sides simply

supported. The plate is assumed subject to the most general in-plane stress state

(a combination of bending, biaxial and shear stresses).

Consider a symmetrically laminated composite rectangular plate of length (a) and

width (b) that is simply supported along all edges. The plate is subjected to a

combination of linearly varying in-plane axial stress and constant shear stress

resultants acting on its boundaries. This plate with the positive sign convention for

the stress resultants is illustrated in Figure 2.9. The in-plane axial stress resultants

can be written as,

58

( ) yb

NNNyNoxx

bxxo

xxxx−+= (2.39)

( ) xa

NNNxN

oyy

ayyo

yyyy

−+= (2.40)

The governing differential equation for the transverse buckling analysis of this plate

is [78],

4

4

113

4

2622

4

662

2

123

4

164

4

11 4224xwD

yxwD

yxwD

xwD

yxwD

xwD

∂∂+

∂∂∂+

∂∂∂

+

∂∂+

∂∂∂+

∂∂

2 2

22

2

2

ywN

yxwN

xwN yyxyxx ∂

∂+∂∂

∂+∂∂= (2.41)

Subject to the boundary conditions,

02,0:,02

162

2

122

2

11 =∂∂

∂−∂∂−

∂∂−===

xywD

ywD

xwDMwax x (2.42)

02,0:,02

262

2

222

2

12 =∂∂

∂−∂∂−

∂∂−===

xywD

ywD

xwDMwby y (2.43)

Here, Dij are the bending stiffness terms as obtained from the classical laminated

plate theory. An exact solution to this equation is not possible due to the presence of

the bending-twisting stiffness coupling terms D16, D26, the problem of presence of

odd derivatives in the shear stress terms, and the applied axial loads Nxx, Nyy

generally being functions of x and y.

An approximate solution to the problem can be obtained using the Rayleigh Ritz

method [80]. Such a solution will approach the exact solution of the problem and

will handle the type of loading under consideration.

59

The Rayleigh-Ritz method is based on the principle of stationary value of the total

potential energy of an elastic body. The total potential energy of an elastic body is

the summation of the strain energy stored in the body, U, and the work done by the

external forces, V.

valuestationary =+=Π VU (2.44)

Figure 2.9 Plate Layout and Positive Sign Convention of Applied Loads

In the Rayleigh-Ritz method a solution is sought in the form

( )∑∑= =

=M

m

N

nmnmn yxWAw

1 1, (2.45)

X

Y

a

b

oxxN

bxxN

ayyNo

yyN

xyN

xyN

ss

ssss

ss

60

where Amn are undetermined coefficients and the functions Wmn(x,y) are chosen in a

variable separable form and must at least satisfy the geometric boundary conditions

of the problem under consideration. Thus, the energy criterion reduces to satisfying

the condition,

( ) [ ] 0)()( =∂+∂=

∂Π∂

mnmn AwVwU

Aw (2.46)

The strain energy for the transverse bending of a symmetrically laminated

composite plate is [80],

∫ ∫

∂∂+

∂∂

∂∂+

∂∂=

b a

ywD

yw

xwD

xwDU

0 0

2

2

2

222

2

2

2

12

2

2

2

11 221

dydxyx

wDyx

wywD

xwD

∂∂∂+

∂∂∂

∂∂+

∂∂+

22

66

2

2

2

262

2

16 44 (2.47)

The work done by the external in-plane loads, V, is,

dydxyw

xwN

ywN

xwNV

b a

xyyyxx∫ ∫

∂∂

∂∂+

∂∂+

∂∂=

0 0

22

22λ (2.48)

where λ is a load multiplier.

For the simply supported plate under consideration, a solution in the following form

is assumed,

=∑∑= = b

yna

xmAwM

m

N

nmn

ππ sinsin1 1

(2.49)

61

Note that this solution satisfies the geometrical boundary conditions and the natural

boundary conditions are approximated by the Rayleigh-Ritz process (minimization

of the total potential energy of the system). Substituting the assumed solution into

the strain energy and work expressions and making use of the energy criterion

results in the following set of linear equations.

( )

(2.50) ,,2,1,,2,1

0

cossincossin

cossincossin2

coscossinsin

sinsincoscos

cossin cossin

cossincossin2

cossin cossin

cossincossin2

coscoscoscos

sinsinsinsin

00

00

2

002

2

002

2

00

2

00

2263

4

00

2

00

2163

4

006622

4

00

1 1224

422

122222

22

4

114

422

==

=

+

+

−+

+

−+

+

+

−

+

+

+

+++

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∫∫

∑∑= =

NjMi

dyyydxxxjm

dyyydxxxinNab

dyyydxxxxa

NNN

bjn

dyyyyb

NNNdxxxa

mi

dyyydxxxjnm

dyyydxxxjinDab

dyyydxxxjim

dyyydxxxmniDba

dyyydxxxDba

mnij

dyyydxxx

DbjnDjmin

baD

aim

j

b

nm

a

i

n

b

ji

a

mxy

n

b

jm

a

i

oyy

ayyo

yy

n

b

j

oxx

bxxo

xxm

a

i

n

b

jm

a

i

j

b

ni

a

m

j

b

ni

a

m

n

b

jm

a

i

n

b

jm

a

i

n

b

jm

a

i

M

i

N

j

l

l

ββαα

ββααλπ

ββααλπ

ββααλπ

ββαα

ββααπ

ββαα

ββααπ

ββααπ

ββαα

πππ

62

where,

nmjikb

ka

kkk ,,, : ; === πβπα

Performing the integration and simplifying terms results in the following set of

linear homogenous equations,

( )[ ]( ) ( )[ ]

( ) ( )[ ]

==

=

−+−++

−+

−

++++

−+++

∑∑

∑∑

= =

= =

NnMm

ANNLRNNKNMRa

ANN

aRnNNam

ADjnmnRDmiMR

ADRnDDRnmDm

M

i

N

jij

oyy

ayymnij

oxx

bxxmnijxymnij

mn

oyy

ayy

oxx

bxx

ij

M

i

N

jmnij

mn

,,2,1,,2,1

for 0

48

22

32

22

1 1

22

2222222

1 126

22216

222

2244

6612222

1144

l

l

ππλ

π

π

(2.51)

where,

≠=−=

≠=+−=

±±≠≠−−=

whereelse ; 0

m)(iand ,oddm)(i ,n)(j ; )(

whereelse ; 0

n)(jand ,oddn)(j ,m)(i ; )()(

whereelse; 0

oddj)(n and odd,i)(m j),(n i),(m ;))((

222

22

2222

�

�

mimnij

L

njnjmnij

K

jnimmnij

M

mnij

mnij

mnij

63

and R is the plate aspect ratio (a/b). Equation (2.51) can be cast into the form of a

classical eigenvalue problem.

[ ] [ ][ ] { } 0=− AKK de λ (2.52)

where [Ke] represents the elastic stiffness matrix, [Kd] is the differential (also known

as the geometric) stiffness matrix and {A} is a column matrix of the undetermined

coefficients Amn. Solving the eigenvalue problem yields the critical load multiplier,

λcr, which will cause buckling of the plate. Then the margin of safety for the

buckling strength of the plate, M.Sb, can be obtained as,

1. −= crbSM λ (2.53)

For a metallic panel, the critical buckling stress can be obtained by replacing the

relevant bending stiffness terms by their equivalents for an isotropic material. For a

metallic plate made of an isotropic material, the bending stiffness terms are given

by,

)1(24

0)1(12

)1(12

3

66

2616

2

3

12

2

3

2211

ν

νν

ν

+=

==−

=

−==

tED

DD

tED

tEDD

(2.54)

The solution obtained by the Rayleigh-Ritz method is always in the direction of

stiffer plate. Thus the buckling loads obtained by the Ritz method are always higher

than the true solution. This is due to the fact that the obtained solutions involve

additional constraints on the energy criterion which are beyond the physical

constraints of the problem [79]. The solution is also approximate since it is

64

restricted to the pre-selected set of functions, Wmn(x,y). However, the energy

criterion is sufficient to select the most accurate set of these functions that represent

the most accurate solution. As the number of functions selected is increased (i.e, the

number of terms in the assumed series) the accuracy of the solution obtained should

increase or remain the same. Furthermore, if the selected functions form a complete

mathematical set, then the obtained solution must approach the correct one as the

number of terms in the series is increased [79].

2.6 Static Strength Analysis

The static strength analysis of the wing box is performed to demonstrate compliance

with the certification requirements, i.e., the structure has adequate strength to carry

the loads without failure or loss of strength.

For each load case, the stress/strain distribution in every structural element of the

wing box is first determined using the methods explained in section 2.4. Then the

relevant allowable stresses are determined based on the methods outlined in section

2.5. The margin of safety and the corresponding failure mode for the load case

under consideration are then evaluated. The process is repeated for all of the load

cases to identify the minimum margin of safety, the critical load cases and the

corresponding failure modes.

The output of the static strength analysis is a summary of the minimum margins of

safety, the critical load cases and the corresponding failure modes.

65

2.7 Case Studies

In this section two test cases are considered. The first one is the buckling analysis of

a typical skin panel and it aims at verifying the Rayleight-Ritz for buckling analysis

using MSC/NASTRAN and to show that the specially orthotropic assumption is not

generally a justified assumption. The second test case considers the stress analysis

of a simple wing box and it aims at justifying the methodology of using the

modified engineering theory combined with the coarse mesh finite element analysis

and compares the results thus obtained with a detailed mesh finite element model

analysis.

2.7.1 Allowable Buckling Stress of a Typical Panel

For verification and illustration purposes, the buckling stresses for a typical panel

under the effect of various combined loading conditions are determined using both

the Rayleight-Ritz and the finite element methods. Two versions of the panel are

considered. The first version is made of a unidirectional tape graphite-epoxy

composite material with a ply stacking sequence of [ ] s45/0/45 22 ±°° . This stacking

sequence is typical for a composite skin panel and results in an anisotropic laminate.

The second version is a metallic one that is made of an isotropic material. The panel

considered has a length of 600 mm, a width of 300 mm and a thickness of 1.6 mm.

The panel material properties are given in Table 2.2.

A convergence study is first done for the composite version of the panel loaded in

axial compression with an axial load intensity of == bxx

oxx NN .10 [N/mm]. The

panel is first analyzed as anisotropic and then as specially orthotropic by setting the

stiffness coupling terms D16 =D26 =0 in equation (2.49). The panel critical buckling

load is also determined using a fine mesh MSC/NASTRAN® finite element model

with a mesh density of 2557 total quadrilateral elements. The convergence history

66

of the buckling load factor λcr is illustrated in Figure 2.10. The anisotropic solution

approaches the finite element method solution with ten terms in the series and an

error of 1.6% relative to the finite element method solution. The specially

orthotropic solution convergence is very fast (2 terms in the series), however, with a

non-conservative result and an error of 45.3% relative to the finite element method

solution.

Table 2.2 Panel Material Properties

Isotropic Material (Aluminum)

Composite Material (Graphite/epoxy)

E = 10.5x106 [psi] E1 = 18.5x106 [psi]

E2 = 1.6x106 [psi]

ρ = 0.1 [lb/in3] ρ = 0.055 [lb/in3]

ν = 0.3 ν12 = 0.25

G12 = 0.65x106 [psi]

The buckling stress for both versions of the panel was then determined using both

MSC/NASTRAN® and the developed Rayleigh-Ritz approach for six representative

load cases that a typical skin, rib web and spar web would be subjected to in real

life. The different load conditions considered and their magnitudes are given in

Table 2.3.

67

Figure 2.10 Convergence of the Buckling Load Factor

0.50

0.75

1.00

1.25

1.50

1 6 11 16 21M=N

Orthotropic Solution (Rayleigh-Ritz; D16=D26=0) Anisotropic Solution (Rayleigh-Ritz) FEM (MSC/NASRAN)

64.0=crλ

65.0=crλ

93.0=crλ

crλ

10 [N/mm]

Graphite Epoxy

[(45°)2,(0°)2,±45]s

68

Results obtained from MSC/NASTRAN and Rayleigh-Ritz methods are tabulated

in Table 2.4. Note that both approaches give more or less the same result. This is no

surprise, since both methods are based on the energy principles. Comparison

between the buckling load multipliers for the anisotropic and specially plate

solutions is depicted in Table 2.5. It is worth to note the huge difference between

the two approaches and the error that would be involved if this assumption is made.

Another interesting point is the difference between results obtained for the buckling

load factor for the positive and negative shear load cases. For an anisotropic plate

the direction of the shear has to be taken into account in the analysis. An important

aspect that has no effect in the specially orthotropic case. The buckling mode shapes

are illustrated in Figures 2.11 and 2.12 for the composite and metallic panels

respectively.

Table 2.3 Buckling Load Cases

L.C # oxxN

[N/mm]

bxxN

[N/mm]

oyyN

[N/mm]

ayyN

[N/mm]

xyN [N/mm]

1 0.0 0.0 0.0 0.0 5.0

2 0.0 0.0 0.0 0.0 -5.0

3 5.0 -5.0 0.0 0.0 5.0

4 -10.0 -10.0 0.0 0.0 5.0

5 -10.0 -10.0 5.0 5.0 5.0

6 10.0 -20.0 -5.0 -5.0 5.0

69

Table 2.4 Critical Buckling Load Factors (λcr)

Metallic Plate Composite Plate

L.C # Rayleigh-Ritz NASTRAN Rayleigh-Ritz NASTRAN

1 3.90 3.90 0.98 0.97

2 3.90 3.90 3.69 3.65

3 3.72 3.72 0.96 0.95

4 1.10 1.10 0.43 0.42

5 1.63 1.60 0.71 0.70

6 0.72 0.72 0.28 0.27

Table 2.5 Critical Buckling Load Factors (λcr)

L.C # Orthotropic Anisotropic Ratio

1 2.46 0.98 2.51

2 2.46 3.69 0.67

3 2.39 0.96 2.49

4 0.84 0.43 1.95

5 1.33 0.71 1.87

6 0.43 0.28 1.54

70

Figure 2.11 Buckling Mode Shapes of the Composite Panel

LOAD CASE 2 LOAD CASE 1



71

Figure 2.12 Buckling Mode Shapes of the Metallic Panel




72

2.7.2 Stress Analysis of a Typical Wing Box

In this case study, the stress analysis for a typical wing is performed using four

different approaches. In the first approach, the wing box is modeled with a coarse

mesh finite element model and the stresses were directly obtained from the finite

element model output. In the second approach, the coarse mesh finite element

model was used to determine the internal load distribution only. These loads were

then used to calculate the sectional loads and the stress analysis was performed

using the engineering bending theory. The third approach is the same with the

second, however, with the modified version of the engineering bending theory used

to account for the load redistribution in the structure after skin buckling. In the

fourth approach, a detailed mesh finite element model is constructed to determine

the stress distribution in the structure.

The typical rectangular wing box analyzed is shown in Figure 2.15 with all the

relevant dimensions. It is a single cell wing box that has three metallic ribs, two

spars and coverage skins. The skins are laminated composite with a layup of

[±45°]3S. The material properties are given in table 2.2. The wing box is assumed

subjected to a concentrated tip load of 2000 [N] acting up.

73

Figure 2.13 Structural Arrangement of the Rectangular Wing

90°

Spar cross section

101.6

1524 15241524

635

All dimensions in millimeters

Y

X

Z

Y

Rib 3 Rib 2Rib 1

0°

20

20

2.0

20

20

1.0

Rib cross section

0.8 1.2

74

The coarse mesh finite element model is illustrated in Figure 2.16. This model has a

total of 6 membrane-bending CQUAD4 elements to model the skins, 6 CSHEAR

elements to represent the spars webs, 3 membrane CQUAD4 elements to model the

three ribs and 18 axial CROD elements to model the spars caps and ribs chords. The

total number of elements in the model is thus 33 elements. Two fine mesh finite

element models that use the same same kind of elements as the coarse mesh finite

element model are constructed. The first model is illustrated in Figure 2.15 and has

a total of 864 elements. The second model is illustrated in Figure 2.16 and has a

total of 2997 elements. All models are constrained at the root by fixing the

translational degrees of freedom (Tx=Ty=Tz=0).

Figure 2.14 Coarse Mesh Finite Element Models of the Rectangular Wing

Upper skin is removed for better visibility

1000 [N] 1000 [N]

75

Figure 2.15 Fine Mesh Finite Element Models of the Rectangular Wing

(Total 864 Elements)

Figure 2.16 Fine Mesh Finite Element Models of the Rectangular Wing

(Total 2997 Elements)

1000 [N] 1000 [N]



1000 [N]

1000 [N]

76

The stress analysis results for the upper and lower spar caps using the four different

approaches are depicted in Figures 2.17 and 2.18 respectively. Obviously the worst

approach is to rely on direct stress output of the coarse mesh finite element model.

The stress output obtained by the second approach, i.e., using a coarse mesh to

determine the internal load distribution and then utilizing the engineering bending

theory to determine the stresses, compares well with the detailed mesh finite

element model stress output. This shows that the use of engineering bending theory

with the loads obtained from a coarse mesh finite element model yields results that

are accurate enough as compared to the fine mesh finite element analysis. Note that

the use of a fine mesh in large scale finite element models might be computationally

expensive. Nevertheless, the use of fine mesh linear finite element models, would

not account for nonlinear effects stemming from load redistribution in the structure

after buckling occurs. Note the large jump on the compressive stress on the upper

cap after the skin buckles as predicted by the modified engineering bending theory.

Thus, the use of coarse mesh finite element models to determine the load

distribution in the structure and using the modified engineering bending theory to

determine the stresses is a good approach since it yields fairly good results at low

cost and can be used to accurately simulate the behavior of the structure.

77

Figure 2.17 Front Spar Upper Cap Stress Distribution of the Rectangular

Wing

-500

-450

-400

-350

-300

-250

-200

-150

-100

-50

00 1000 2000 3000 4000 5000

Span, Y [mm]

Stre

ss [

MPa

]

Modified Engineering Bending TheoryEngineering Bending TheoryCoarse Mesh Finite Element ModelFine Mesh Finite Element Model (864 Elements)Fine Mesh Finite Element Model (2997 Elements)

78

Figure 2.18 Front Spar Lower Cap Stress Distribution of the Rectangular

Wing

0

25

50

75

100

125

150

175

200

225

250

0 1000 2000 3000 4000 5000

Span, Y [mm]

Stre

ss [

MPa

]Modified Engineering Bending TheoryEngineering Bending TheoryCoarse Mesh Finite Element ModelFine Mesh Finite Element Model (864 Elements)Fine Mesh Finite Element Model (2997 Elements)

79

2.8 Conclusion

The strength analysis of a wing box for demonstrating the compliance with the

certification requirements is not an easy task to achieve. It is not just constructing

finite element models and reading stress output. Several failure modes have to be

considered and nonlinear effects such as buckling and crippling have to be taken

into consideration.

The use of coarse mesh finite element models for the determination of the internal

load distribution in the structure and then post processing these loads to determine

the stress distribution is a justified approach. However, such models shall never be

used to read stress outputs directly from the finite element model. Simple theories

such as the modified engineering bending theory can then be used to simulate the

correct behavior of the structure.

The buckling analysis based on the specially orthotropic plate assumptions is not a

generally valid one and can only be justified if the laminate is cross-ply symmetric.

In practical applications most laminates would not qualify to this assumption. On

the other hand using the energy methods such as the Rayleigh-Ritz method would

result in accurate and acceptable results at low cost compared to the finite element

method and without any need to prepare separate finite element models for buckling

analysis purposes. For anisotropic laminates, care should be taken in the correct

sign for the applied shear stress, since positive and negative shear loads would

result in completely two different results.

80

CHAPTER 3

AEROELASTIC STABILITY ANALYSIS

3.1 Introduction

Aeroelasticity is defined as the science, which studies the behaviour of an elastic

body under the effect of aerodynamic, elastic, and inertia forces. An aircraft

structure immersed in an air flow is subjected to surface pressures induced by that

flow. If the incident flow is unsteady or the boundary conditions are

time-dependent, these pressures become time dependent. Moreover, if the structure

undergoes dynamic motions, it changes the boundary conditions of the flow and the

resulting fluid pressures which in turn changes deflections of the structure.

The importance of aeroelasticity has been widely recognized in the aerospace

industry. The missions of aircraft structures are becoming increasingly more

complex. New developed aircrafts are larger in size and more flexible. Aeroelastic

effects due to aircraft flexibility may significantly alter the performance and safety

of a new developed aircraft. Hence considering aeroelastic effects at the early stages

of design to produce competitive and safe aircraft is a vital task.

The whole spectrum of aeroelastic phenomena to be considered during the design

process can be classified by means of the expanded Collar’s aeroelastic triangle

81

illustrated in Figure 3.1. Three types of forces are mainly involved in the aeroelastic

analysis process. These are aerodynamic, elastic and inertial forces. Accordingly,

the aeroelastic phenomena can be divided into two main groups, static and dynamic

types.

Figure 3.1 Collar’s Aeroelastic Triangle

Static aeroelastic phenomena lie outside of the triangle and includes phenomenon

like wing divergence, control surface effectiveness, static stability, and load

distribution.

Aerodynamic Forces

Elastic Forces

Inertial Forces

Vibrations

Static Aeroelasticity

Flight Mechanics

Dynamic Aeroelasticity

82

Divergence is a nonoscillatory instability phenomenon that occurs when the

restoring elastic moments within a wing are exceeded by the aerodynamic moments.

When a wing twists, an extra lift force is developed by the wing due to the increase

of incidence. This force acts at the aerodynamic center which is at or near the

quarter-chord for a subsonic flow. If the flexural axis lies aft of the aerodynamic

center (as in most actual wings), the increase in lift will tend to increase the twist

which in tern increases the incidence again resulting in more lift and hence more

incidence. At speeds below a critical speed known as the divergence speed the

increments in lift converge to a condition of stable equilibrium in which the

torsional moment of the aerodynamic forces about the flexural center is balanced by

the torsional rigidity of the wing. Divergence can be completely prevented by

placing the flexural axis at or forward of the aerodynamic center. Divergence is not

usually a consideration for swept back wings but it can be a critical design problem

on some slender straight or swept forward wing configurations.

Control surface effectiveness is a phenomenon that is related to ailerons. Ailerons

of an elastic wing are less effective than those of a rigid wing. The effectiveness

drops as the speed increases. At a critical speed that is known as the aileron reversal

speed, the aileron twists the wing to such an extent that the gain in the rolling

moment due to aileron rotation is less than the loss due to wing twist causing the

aircraft to roll in the opposite direction.

Since aircraft structures are flexible in nature, aerodynamic loads applied to them

produce deformation that might be significant enough to change the static stability

and load distribution due to changes in the shape of the aircraft.

Dynamic aeroelastic phenomena lie within the triangle since it involves the

interaction of all of the three types of forces and includes phenomenon like flutter,

buffeting, and dynamic response. Among all of these phenomena flutter is the most

83

important. The occurrence of flutter within the flight envelope leads to a

catastrophic structural failure and loss of the aircraft.

Flutter is defined as the sustained oscillation of the lifting surface under the effect of

high-speed air passage. The occurrence of flutter does not require any external

forcing agency. Any initial disturbance like the engine sound or control surface

movement is enough to trigger flutter. When a lifting surface starts to vibrate during

flight the oscillations usually die out due to the presence of structural and

aerodynamic damping. The structural damping of the lifting surface is constant.

Nevertheless, the aerodynamic damping is not constant and depends on the flight

speed. As the flight speed is increased the aerodynamic damping first increases then

starts dropping reaching negative values. At a certain speed the summation of the

structural and aerodynamic damping becomes zero. At this speed any disturbance

introduced to the structure will cause self sustained oscillations. This speed and the

corresponding oscillation frequency are known as the critical flutter speed and

flutter frequency respectively. At any speed that is equal to or above this speed,

disturbing the structure will cause it to start extracting energy from the air stream

and oscillations grow up indefinitely resulting in failure of the structure. In order for

the structure to start extracting energy from the air stream there must be a

significant phase difference between the coupled modes involved. This phase

difference is provided by the aerodynamic damping inherent in the aeroelastic

system. Flutter can be classified mainly into two types, classical and stall flutter.

Classical flutter involves the coupling of at least two structural modes. Wing

bending-torsion flutter and control surface flutter are typical types of classical

flutter. Control surface flutter involves the coupling between the control surface

mode and the wing torsion and/or bending modes. A wing equipped with an aileron

can flutter at a speed that is much lower than the wing bending-torsion flutter speed.

If the control surface flutter involves coupling of two modes then the flutter is

termed as binary flutter. Control surface flutter which involves the interaction of all

84

of the three modes is known as ternary flutter. Classical flutter can be eliminated if

the aerodynamic center, the flexural center, and mass center coincide.

Stall flutter on the other hand is associated with the flow separation and

reattachment at high angles of attack in the transonic and supersonic flow regimes.

It does not involve any structural modes coupling and happens when the torsional

structural mode becomes unstable. This aeroelastic instability phenomenon is

critical for rotating machineries such as helicopter rotors and turbine blades. Since

aircrafts rarely come close to stall when flying at the maximum velocities and

dynamic pressures for which they are designed, this phenomenon is not a serious

one on wings and tails. Nevertheless, it is an important aspect for turbojet engines

operating off their design speeds.

Buffeting is the transient vibration of aircraft structural components due to

aerodynamic impulses produced by the wake behind wings and engine nacelles. A

serious buffeting problem is encountered by aircraft during pull-up maneuvers to

maximum lift coefficients at high speed. This often results in rugged transient

vibrations in the tail due to aerodynamic impulses from the wing wake. The

problem of determining the dynamic stresses to provide adequate strength is very

difficult. The principal obstacle has been the lack of knowledge of the properties of

the wake behind the partially stalled wing. Buffeting problems are usually alleviated

by proper positioning of the tail and aerodynamic design to prevent flow separation.

This is the main reason for having fairings attached at the wing fuselage junctions.

All of these static and dynamic aeroelastic phenomena have profound effects on the

aircraft design and can only be solved in concurrent consideration by all disciplines

involved. Nevertheless, among them flutter and divergence acquire special

importance since the occurrence of any of them will lead to catastrophic structural

failure and loss of the aircraft.

85

The wing of an aircraft structure plays an important role in the aeroelastic design

and analysis process. Aeroelastic stability analysis of an aircraft is concerned with

determining the stability boundaries of the aircraft structure. It identifies the flight

conditions in terms of flight speed and altitude or density at which the structure

becomes unstable. Several aeroelastic instability problems like flutter and

divergence are highly influenced by the stiffness and mass distribution on the wing.

In the development of a new aircraft, enough stiffness has to be provided to insure

that the aircraft is free from any sort of aeroelastic instability in its design flight

envelope. JAR25.629 specifies the certification requirements for a new developed

aircraft. The compliance requirements specify that the aircraft must be free from

flutter and divergence with adequate damping margins for speeds up to 1.2VD,

where VD is the design diving speed. Furthermore, it must be free from flutter at

speeds up to VD after certain specified structural failures. This must be

demonstrated by analysis substantiated by full scale flutter test up to VD.

Flutter clearance for a particular design does not rely completely upon analysis

alone. Ground vibration and wind tunnel tests are first performed both to confirm

the analysis and to provide extensive information on the effects of varying a number

of important parameters such as fuel quantity, engine location, and stiffness

distributions. The effect of structural failures is also simulated in the wind tunnel.

Then the actual airplane is tested. Accelerometers are located on all the principal

components of the airframe and position indicators are provided on the control

surfaces. The airplane is flown at specific altitudes at incrementally increasing

speeds. The structure is excited by means of pilot-input, control surface pulses or

through the use of wing or empennage tip “shakers”. For each speed/altitude

combination up to VD, the airplane’s motions and the decay of these motions are

measured. If adequate damping exists at all speeds up to VD, the results of the flight

flutter test, together with the analysis and its supporting wind tunnel data provide

the basis for the flutter clearance of the airplane. Thus, it is necessary to obtain a

86

valid damping history to prevent loss of the aircraft or wind tunnel test model

during the flight and wind tunnel tests.

Traditionally frequency domain methods like the k-method and pk-method are used

for aeroelastic stability analysis. However, these methods produce damping

information that is valid only at the instability point or near to the instability point.

To gain an insight into the physical phenomena leading to aeroelastic instability it is

necessary to obtain valid damping and frequency history. Laplace domain methods

like the p-method and root locus method provide such an insight. However, the

main difficulty in implementing this method lies in obtaining the aerodynamic loads

for an arbitrary motion in Laplace domain. Unsteady aerodynamic forces are only

well developed for simple harmonic motions in the frequency domain. This problem

is circumvented using rational function approximations for the aerodynamics in

Laplace-domain.

In this chapter, the mathematical formulation of the aeroelastic stability problem is

first discussed. Aeroelastic analysis methods based on the k-method, pk-method, p-

method, and the root locus method are then explained. The differences between

these methods are identified. The method of obtaining the generalized aerodynamic

forces in Laplace domain using Rogers rational function approximations is

explained. The aerodynamic lag roots necessary for this approximation are obtained

using a direct search optimizer that is based on the complex method. Two test cases

are studied to study the difference between the PK-method and P-method and verify

the adopted methodology for approximating the generalized aerodynamic forces in

Laplace domain using rational function approximations. Finally, the chapter ends

with some concluding remarks and discussions.

87

3.2 Theory of Aeroelastic Stability

In this section the classical aeroelastic stability equation is derived and various

solution methods with their advantages and disadvantages are discussed.

The equation of motion of a multi degree of freedom, discrete and damped

aeroelastic system can be derived based on the dynamic equilibrium of forces. The

time-domain equation of motion in matrix form is given as

[ ]{ } [ ]{ } [ ]{ } { }),()()()( txFtxKtxCtxM =++ �� (3.1)

where [M], [C] and [K] denote the mass, damping and stiffness matrices

respectively and { })(tx is the structural deformation vector. The applied

aerodynamic loads vector { }),( txF is in general a time function of the structural

deformation and the free stream Mach number ∞M defined as

aVM ∞

∞ = (3.2)

where a is the speed of sound, which is a function of the flow temperature and

density, and ∞V is the free stream velocity. The applied aerodynamic loads vector

{ }),( txF can be split into mainly two parts in the following form

{ } { } ( ){ })()(),( txFtFtxF ae += (3.3)

where the vector ( ){ })(txFa represents aeroelastic forces which are the induced

aerodynamic forces due to the deformation of the structure and the vector { })(tFe

represents the externally applied non-aeroelastic forces to the structure such as gust

88

and control surface loads. These aeroelastic forces are function of the flight speed

and altitude and the calculation of them relies on theoretical predictions that require

unsteady aerodynamic computations. Since the aeroelastic forces are function of the

structural deformations, equation (3.1) can be interpreted as an aerodynamic

feedback relationship. Figure 3.2 illustrates the functional diagram of this

relationship. Without an aerodynamic feedback, this relationship reduces to that of

an open-loop forced vibration system with finite amplitude response. With the

inclusion of the aerodynamic forces, the relationship represents a closed-loop

dynamic response system that can be described by the following equation

[ ]{ } [ ]{ } [ ]{ } ( ){ } ( ){ })()()()()( txFtxFtxKtxCtxM ea =−++ DDD (3.4)

For stability analysis the non-aeroelastic forces are ignored resulting in the

following equation

[ ]{ } [ ]{ } [ ]{ } ( ){ } 0)()()()( =−++ txFtxKtxCtxM a�� (3.5)

Equation (3.5) is the generally non-linear time domain aeroelastic stability equation.

It defines a closed-loop aeroelastic structure that can be self excited in nature and

gives rise to aeroelastic stability problems like flutter and divergence. Aeroelastic

stability analysis involves the search for the flight speeds and corresponding

altitudes at which the structure becomes unstable.

89

Figure 3.2 Functional Diagram of an Aeroelastic System

If the induced aerodynamic forces form a non-linear function of the structural

deformation, then a time-marching solution technique must be used to determine the

aeroelastic stability of the system. In this case equation (3.5) must be solved with

the initial conditions specified to determine the structural response of the system

with time. The stability boundary of the aeroelastic system is then determined by

examining the decay or growth rate of the structural response { })(tx as a function

of the flight speed. While a decaying response indicates a stable system, a growing

one implies an instable one. This time-marching computational procedure is rather

costly since it generally requires the employment of Computational Fluid Dynamics

(CFD) methods to determine the nonlinear time-domain unsteady aerodynamic

forces.

The conventional industrial practice for stability analysis is to recast equation (3.5)

into a linear system and to determine the stability boundary by solving the complex

eigenvalues of this system. Such a procedure involves the assumption of amplitude

linearization. The amplitude linearization states that the aerodynamic response

varies linearly with respect to the amplitude of the structural deformation if this

amplitude is sufficiently small at all times. For flutter and divergence analysis the

amplitude of the structural deformation can always be assumed to be

( ){ })(txFa

{ })(tFe {x(t)} [[[[ ]]]] {{{{ }}}} [[[[ ]]]] {{{{ }}}} [[[[ ]]]] {{{{ }}}})()()( txKtxCtxM ++++++++ ��

90

mathematically infinitesimal up to failure. Thus, the aeroelastic loads can be

expressed as a linear combination of the structural deformations by means of the

following convolution integral

( ){ } { }∫

−= ∞∞

t

a dttxtL

VQqtxF0

)()()( τ (3.6)

where,

[ ]Q the aerodynamic transfer function defined in time domain

∞V the free stream velocity

∞q the dynamic pressure ( 2

21

∞∞ = Vq ρ where ρ is the density of air)

L a reference length (L=c/2 where c is the mean aerodynamic chord)

Taking the Laplace transform of equation (3.6) result in the following equation

which defines the aeroelastic forces in the Laplace domain

( ){ } ( )[ ]{ })()( sxpQqsxFa ∞= (3.7)

where ( )[ ]pQ is the aerodynamic matrix evaluated in the Laplace domain, s is the

Laplace variable and p is a non-dimensional Laplace variable defined as,

∞

=VsLp (3.8)

Transforming equation (3.5) into Laplace domain and making use of equation (3.7)

yields,

91

[ ] [ ] [ ] ( )[ ][ ]{ } 0)(2 =−++ ∞ sxpQqKCsMs (3.9)

A typical finite element model of an aircraft structure exhibits a large number of

degrees of freedom. Thus the size of the stiffness, damping and mass matrices

involved will be large and direct solution of this eigenvalue problem to determine

the system stability is computationally expensive. To circumvent this problem, the

modal approach based on the principle of superposition is usually used. In the

modal approach the response of the system is described in terms of a linear

combination of the lower order natural modes of the system. The generalized

(modal) coordinates { })(q s describe the contribution of each natural mode to the

total response of the system and are defined as

{ } [ ]{ })(q)( ssx φ= (3.10)

where [ ]φ is the modal matrix whose columns contain the lower order natural

modes obtained from a free vibration analysis of the system neglecting damping and

aerodynamic terms. The number of modes included in the modal matrix depends on

the frequency range of interest, but usually, no more than the first lowest ten natural

modes are required for the flutter analysis of a wing structure [70]. The rationale of

the modal reduction approach is based on the premises that the critical aeroelastic

modes are usually due to the coupling of the lower order structural modes. Thus, the

structural deformation of the aeroelastic mode can be sufficiently represented by the

superposition of lower order structural modes. Substituting equation (3.10) into

(3.9) and pre-multiplying by the transpose of the modal matrix yields

[ ] [ ] [ ] ( )[ ][ ]{ } 0)(q~~~~2 =−++ ∞ spQqKCsMs (3.11)

92

where [ ]M~ , [ ]C~ , [ ]K~ and ( )[ ]pQ~ denote the generalized (modal) mass, damping,

stiffness and aerodynamic force matrices respectively. These matrices are defined

as

[ ] [ ] [ ] [ ]φφ MM T=~ (3.12)

[ ] [ ] [ ] [ ]φφ CC T=~ (3.13)

[ ] [ ] [ ] [ ]φφ KK T=~ (3.14)

( )[ ] [ ] ( )[ ] [ ]φφ pQpQ T=~ (3.15)

Equation (3.11) is known as the classical aeroelastic stability equation that defines a

classical eigenvalue problem which can be solved to determine the stability region

of the aeroelastic system. However, the solution of this eigenvalue problem depends

on the availability of the generalized aerodynamic forces matrix in Laplace domain.

Unfortunately, this requires the determination of the unsteady aerodynamics in

time-domain which is a complicated process. Instead equation (3.11) is usually cast

into the frequency domain and the generalized aerodynamic forces are calculated in

the frequency domain using simplified aerodynamic theories such as the DLM

(Doublet Lattice Method) for subsonic flows and the MBM (Mach Box Method) for

supersonic flows.

93

3.3 Frequency Domain Solution Methods

There are mainly two methods for solving the aeroelastic stability equation in the

frequency domain. These are the k-method (also known as the American method)

and the pk-method (which is also known as the British method). These methods are

described in this section.

3.3.1 The k-Method

The k-method or American method for instability analysis is used to determine

flutter type of aeroelastic instability only. It was first developed by Theodorsen in

the year 1935 [55]. Theodorsen first introduced an artificial structural damping

coefficient (gs) to sustain the assumed harmonic motion in the governing equation

and then expressed the aerodynamic force term as a complex inertia term.

Assuming undamped harmonic motion such that

{ } { } tieqq ω= (3.16)

where ω is the oscillation frequency and { }q is the amplitude of the assumed

undamped harmonic motion. The governing equation of motion in the frequency

domain can be obtained from its counterpart in the Laplace-domain, i.e., equation

(3.11), by replacing the Laplace variable (s) by (iw) and noting that p=ik. Thus,

[ ] [ ] [ ] ( )[ ][ ]{ } 0q~~~~2 =−++− ∞ ikQqKCiM ωω (3.17)

where k is the reduced frequency defined as

94

∞

=V

Lk ω (3.18)

The dynamic pressure can then be expressed in terms of the reduced frequency in

the following form

22

21

21

== ∞∞ kLVq ωρρ (3.19)

Substituting equation (3.19) into equation (3.17) and introducing the artificial

damping coefficient in the resulting equation yields the fundamental equation for

flutter analysis by the k-method.

[ ] ( )[ ] [ ] [ ] { } 0q~~1

,~2

~1

22

=

+

++

++

− ∞ KCig

iMikQkLM

ig ss

ωρω (3.20)

where gs is the assumed structural damping. In equation (3.20) the coefficient of the

modal damping matrix [ ]C~ has been multiplied by sig+1 for mathematical

convenience, and is valid only when gs=0. Equation (3.20) is solved as an

eigenvalue problem for a series of values of the parameters M∞, k, and ρ. The

complex eigenvalue, ( )sig+12ω , is interpreted as real values of ω and gs. The

velocity, ∞V , is recovered from kLV /ω=∞ . The k-method of flutter analysis is a

looping procedure. The values of ∞V , gs, and ω are solved for various values of M∞,

k, and ρ. Plots of ∞V versus gs can then be used to determine the flutter speed.

Flutter occurs for values of M∞, k, and ρ for which gs =0 where gs goes from

negative to positive values indicating instability. Since the k-method’s numerical

procedure requires only a straightforward complex eigenvalue analysis of each

reduced frequency, its solution technique is efficient and robust. However, several

drawbacks of the k-method make it a less attractive method for flutter analysis.

95

The solution is valid only at gs =0. Other non-zero damping values are artificial and

do not have any significant physical meaning. Sometimes the frequency and

damping values “loop” around themselves and yield multi-value frequency and

damping as a function of velocity. This gives difficulty in tracking the eigenvalue in

the reduced frequency list [11], [57]. The term 1/k involved in equation (3.18)

indicates that the k-method can not generate flutter solution at k = 0. This is the

reason why this method excludes the rigid body modes from its flutter equation.

The failure at k = 0 also implies that the k-method can not predict the divergence

speed instability which is an important aeroelastic instability phenomena [57].

3.3.2 The pk-Method

The pk-method or the British method was first proposed by Irwin and Guyet in 1965

[55]. In the pk-method the generalized coordinates are assumed to be damped

harmonic functions with the aerodynamic forces still obtained from undamped

harmonic solutions. In the original method proposed by Irwin and Guyet solutions

were obtained using graphical methods to match the assumed reduced frequency, k

with the imaginary part of the computed complex eigenvalue p. A variation of this

method was then introduced by Hassig [58] who used a determinant iteration

method to match the assumed reduced frequency with the calculated one. With the

assumed damped harmonic motion for the generalized coordinates, equation (3.11)

can be expressed as

[ ] [ ] [ ] ( )[ ] { } 0q~2

~~~ 22

2

=

−+

+

∞∞∞ ikQVKpCL

VpML

V ρ (3.21)

where { }q is the amplitude of the assumed generalized coordinates and p is a non-

dimensional parameter defined as

96

)( ikVsLp +==∞

γ (3.22)

Equation (3.21) is the equation of the pk-method. In this equation the damping is

expressed in terms of γ which is the coefficient of transient decay rate. It is related

to the structural damping coefficient (g) by the following relation

== +

n

n

aag 1ln12

πγ (3.23)

where an and an+1 represent the amplitudes of successive cycles of oscillation. Note

that in the original pk-method equation the aerodynamic forces are only dependent

on the imaginary part of the eigenvalue. Rodden [55] modified the equation of the

pk-method by dividing the aerodynamic force matrix into an aerodynamic stiffness

matrix and an aerodynamic damping matrix and introduced dependence on the real

part of the eigenvalue. The modified equation of the pk-method given by Rodden is

given as

[ ] [ ] [ ] { } 0q~~2

~~~ 22

2

=

+−+

+

∞∞∞ RI QQkpVKpC

LVpM

LV ρ (3.24)

where IR QQ ~,~ are the real and imaginary parts of ( )∞MikQ ,~ .

Equation (3.24) is solved at several given values of V∞ and ρ, for the complex roots

p associated with the modes of interest. This is accomplished by an iterative

procedure that matches the reduced frequency k to the imaginary part of p for every

structural mode. This iterative procedure is called the reduced frequency “lining-

up” process and requires the repeated interpolation of ( )∞MikQ ,~ from its discrete

values calculated at a prescribed discrete reduced frequency list.

97

The principal advantage of the pk-method is that it produces results directly for

given values of velocity, whereas the k-method requires iteration to determine the

reduced frequency of flutter. In addition, the damping found from the pk-method

equation is believed to give a more realistic estimate of the physical damping than

the damping parameter gs utilized by the k-method which is a mathematical artifice

[55]. Since this method is valid for all range of reduced frequencies including k = 0

it can predict divergence. However, Chen [63], proved analytically that this method

is only valid for small values of the reduced frequency, k, or for cases where the

generalized aerodynamic forces are linear functions of the reduced frequency.

3.4 Laplace Domain Solution Methods

There are mainly two methods for solving the aeroelastic stability equation in the

Laplace domain. These are the p-method and the root locus method. Both methods

are described in this section.

3.4.1 The p-Method

The p-method equation of motion is deduced from general aeroelastic stability

equation by introducing a non-dimensional Laplace parameter defined by equation

(3.22) into equation (3.11). The resulting equation is

[ ] [ ] [ ] ( )[ ] { } 0)(q,~~~~ 22

=

−+

+

∞∞

∞∞ pMpQqKpCL

VpML

V (3.25)

The solution procedure for the p-method is an iterative process due to the implicit

dependence of generalized aerodynamic forces matrix on the Laplace parameter p.

The solution process is similar to that of the pk-method.

98

Solution of equation (3.25) provides the “true” damping of the aeroelastic system.

However, the generalized aerodynamic forces are not usually available in the

Laplace-domain at a low computational cost. They are first determined in the time

domain using Computational Fluid Dynamics (CFD) methods and then transformed

to the Laplace domain using methods like the indicial response method [81].

3.4.2 The Root Locus Method

The root locus method relies on performing a root-loci analysis to determine the

variation of the aeroelastic system frequencies with the dynamic pressure. For this

purpose, the state space form of equation (3.25) is used with the generalized

aerodynamic forces represented by aerodynamic transfer functions in the Laplace

domain.

The generalized aerodynamic forces are usually calculated for a discrete set of

reduced frequencies assuming harmonic motion in the frequency-domain using

simplified unsteady aerodynamic methods like the doublet lattice method.

Nevertheless, since the generalized aerodynamic forces in the Laplace-domain are

analytic for a causal, stable, and linear system, they can be directly deduced from

their frequency-domain counterparts [69], [26], [63]. This is usually accomplished

by approximating the frequency-domain aeroelastic forces in terms of rational

functions of the Laplace variable.

Several methods have been developed to express the aerodynamics of general

planforms in the Laplace domain based on rational function approximations.

Among them the Rogers method [71] and the minimum state method of Carpel [72]

are the most popular and widely used ones.

In this thesis the Roger’s method due to its simplicity and foolproof is chosen. The

aerodynamic lag roots are obtained from a direct search complex method

99

optimization algorithm to minimize the fit error. This is similar to the approach

suggested by Eversman and Tewari [69].

The Roger’s approximation to the unsteady aerodynamics is given by

( )[ ] [ ] [ ] [ ] [ ]jN

j japp A

pppApAApQ ∑

= −++++=

3 2

2210

~γ

(3.26)

where p is the nondimensionalized Laplace variable p=ik=sL/V, γj-2 are the

aerodynamic lag parameters which are usually preselected in the range of reduced

frequencies of interest, and [Ai] are real coefficient matrices to be determined such

that the assumed matrix form approximates the tabulated matrices. Equation (3.26)

includes the noncirculatory static aerodynamic force [A0], the aerodynamic

damping [ ]pA1 , the apparent aerodynamic mass [ ] 22 pA , and the circulatory

aerodynamic lag terms represented by the the summation term. Defining the

calculated generalized aerodynamic matrix in the frequency domain as

( )[ ] ( )[ ] ( )[ ]pGipFpQ +=~ (3.27)

where [F(p)] is the real part and [G(p)] is the imaginary part of the matrix. Setting

equation (3.26) equal to equation (3.27) results in the following set of equations

( )[ ] [ ] [ ] [ ]j

N

j j

Ap

ppAApF ∑= −+−

−++=3

22

2

22

20 γ (3.28)

( )[ ] [ ] [ ]j

N

j j

j Ap

ppApG ∑

= −

−

+−+=

32

22

21 γ

γ (3.29)

The elements of the coefficient matrices are then obtained using a term by term

fitting of generalized aerodynamic forces matrix. Writing out equations (3.28) and

100

(3.29) for each element of the genralized aerodynamic force matrix and noting that

p=ik results in the following set of equations,

++

++

++

++−

++−

++−

=

−

−

−

−

−

−

−

−

−

mnN

mn

mn

mn

mn

Nn

nN

n

nn

N

N

N

N

Nn

n

n

nn

N

N

nmn

mn

mn

nmn

mn

mn

a

aaaa

kk

kkk

kk

kkk

kk

kkk

kk

kkk

kk

kkk

kk

kkk

kg

kgkgkf

kfkf

,

,3

,2

,1

,0

22

22

21

21

22

22

2221

22

212

22

21

1221

21

111

22

2

2

21

2

22

22

22

22

21

22

222

2

22

21

21

21

21

212

1

2

1

2

1

00

00

00

01

01

01

)(

)()()(

)()(

�

�

��

�

�

�

��

�

�

�

�

γγ

γγ

γγ

γγ

γγ

γγ

γγ

γγ

γγ

(3.30)

where hmn indicates the mnth element of the matrix [H] and ki indicates the ith

reduced frequency at which the genralized aerodynamic matrix is available.

Equation (3.30) represents an over determined system that can be solved by

standard least square technique. In the solution process the aerodynamic lag

parameters γj-2 are calculated using an optimization process to minimize the error

between the tabulated and fitted aerodynamic forces. The objective function to

minimize is defined as the sum of the squares of the errors between the tabulated

data and the coresspoding data obtained from the fitted functions.

Substituting the Roger’s function approximations into equation (3.25) results in the

following equation,

101

[ ] [ ] [ ] [ ] [ ] [ ] { }=

−+

−

+

−

∞∞

∞∞

∞ )(q~~~01

22

2

pAqKpAqCL

VpAqML

V

[ ]{ })(q3 2

pAp

pq j

N

j j∑= −

∞ + γ (3.31)

Defining each aerodynamic lag mode as a new state such that

( ){ } ( ){ }pp

ppj

q2−+

=γ

η (3.32)

Making use of equation (3.32) in equation (3.31) and writing the resulting equation

in the time domain results in the following time domain constant coefficient

equation

[ ]{ } [ ]{ } [ ]{ } [ ] { }η

=++ ∑

=∞

N

jjAqKCM

3

qqq �� (3.33)

where

[ ] [ ] [ ]2

2~ A

VLqMM

−=

∞∞ (3.34)

[ ] [ ] [ ]1~ A

VLqCC

−=

∞∞ (3.35)

[ ] [ ] [ ]0~ AqKK ∞−= (3.36)

102

The state space form of equation (3.33) can now be written as,

{ }{ }{ }

{ }

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

[ ] [ ] [ ] [ ]

{ }{ }{ }

{ }

−−=

−

−∞

−∞

−−

− 2

11

11

111

2

1

qq

00

00

000qq

NN

N

N BI

BIAMqAMqCMKM

I

η

η

η

η�

�

�

��

�

�

�

�

�

�

��

�

(3.37)

where [I] is the identity matrix, and the matrix [Bj] is a diagonal matrix defined as

[ ]

−= ∞

i

i

i LVB

γ

γ

�

��

�

0

0 (3.38)

A root-loci analysis of equation (3.37) yields the variation of the aeroelastic system

frequencies with the dynamic pressure for a given Mach number. It also provides

the ‘true’ viscous damping values for each of the structural modes. The variations of

the frequency and modal damping with the dynamic pressure provide an insight into

flutter and divergence onset. Flutter appears when the frequencies of any two modes

coalesce and the damping of either of the modes goes to zero. Divergence is

indicated when both the frequency and damping drop to zero. This state-space form

of the equation of motion is also suitable for aeroservoelastic analysis and control

system integration design.

103

3.5 Case Studies

Two test cases are studied to see the difference between the pk-method and the root

locus method and verify the adopted methodology for approximating the

generalized aerodynamic forces in Laplace domain using rational function

approximations. The first case study is the BAH wing and the second is an

intermediate complexity wing (ICW) model.

3.5.1 BAH Wing

The first test case is a hypothetical jet transport wing studied throughout

Bisplinghoff, Ashley, and Halfman [70] and hence named after them as the BAH

wing. This wing is also an MSC/NASTRAN® [55] flutter analysis test case

(HA145B) that has been studied extensively in literature (e.g., [61], [63]). The

planform of the BAH jet transport wing is illustrated in Figure 3.3. The elastic axis

of this wing is straight and perpendicular to the root at 35% of the chord.

This wing is assumed to be flying in incompressible air (Mach number, M∞=0) and

at a dynamic pressure q∞=4.0075 [psi]. Figure 3.4 illustrates the lifting surface

aerodynamic model of the wing. The aerodynamic model is divided into six strips

across the span and four equally spaced boxes chordwise. This idealization is not

representative of industrial practice were a finer mesh is preferred. Nevertheless, it

is the model used by MSC/NASTRAN to be consistent with the structural model in

which the wing is idealized by a small number of grids. The unsteady aerodynamic

forces are computed by the Doublet Lattice Method (DLM). The structural model of

the wing is given in Figure 3.5. The structural model is constrained at the root by

fixing the translational and rotational degrees of freedom in the global coordinate

system at the root. The stiffness is defined by direct matrix input of the flexibility

influence coefficients. The structural grid points are defined at one-quarter and

104

three-quarter of the chord. They are all restrained to move in all directions except

the transverse direction where they are allowed to translate freely. The wing inertial

data are derived from a system of concentrated and distributed mass elements

model. Each wing segment is represented by a ‘dumbbell’ mass unit consisting of

two concentrated and one distributed mass elements. These three mass elements

represent the inertia and total mass of the wing segment enclosing them. No

damping is assumed to exist in the model. The structural model and the

aerodynamic model are connected by using a linear spline element (SPLINE2

element of MSC/NASTRAN). The spline element is used to transform the loads

calculated on the aerodynamic mesh onto the structure and interpolate

displacements on the aerodynamic mesh using the structural displacements.

Figure 3.3 Wing Planform of the BAH Jet Transport

225 100

500

Elastic axis (35 %chord)

78.8

All dimensions in inches

Jet Engine

186

Flow direction (V∞)

105

Figure 3.4 Aerodynamic Model of the BAH Jet Wing

225


25% Chord

50% Chord

75% Chord

138

227

45

318

413

500

100

Y

X

106

Figure 3.5 Structural Model of the BAH Jet Wing


186

268

90

368

458

100

Fixed Elastic axis

Grid point Concentrated mass element Distributed mass element ‘Dumbbell’ mass unit

Y

X

107

A free vibration analysis is performed to determine the natural frequencies and

mode shapes of the wing. The first ten natural frequencies and mode shapes were

determined by using the modified Givens method of MSC/NASTRAN® and are

given in Table 3.1. The first two natural frequencies are 12.789 [rad/sec] = 2.04

[Hz] for the first bending and 22.322 [rad/sec] = 3.55 [Hz] for the first torsion.

These are in close agreement with the uncoupled bending and torsion frequencies of

12.799 [rad/sec] and 22.357 [rad/sec] obtained by Bisplinghoff et al [70].

Table 3.1 Natural Frequencies of the BAH Wing

Mode # Frequency [Hz]

1 2.04

2 3.55

3 7.28

4 11.70

5 14.88

6 21.15

7 24.65

8 32.66

9 39.05

10 48.23

The generalized mass, stiffness, and aerodynamic forces are then obtained from

MSC/NASTRAN® using Direct Matrix Abstraction Program (DMAP) statements.

The generalized aerodynamic force matrix was calculated using the DLM for five

values of reduced frequencies (k=0.001, 0.05, 0.2, 0.5, 1.0). This matrix is then used

to perform a Rogers rational function approximation for the generalized

aerodynamic forces using two aerodynamic lag parameters. The aerodynamic lag

108

parameters which give the best fit function are then determined using a non-gradient

direct search optimizer utilizing the complex method. The best found lag parameters

are determined by the optimizer to be γ1=0.2 and γ2=1.0. The first four real and

imaginary parts of the generalized aerodynmic forces ( )ikQ~ computed by the DLM

and the approximated generalized aerodynmic forces ( ) appikQ~ fitted by Rogers

rational functions approximation are shown in Figures 3.6-3.9. Both the calculated

and fitted values are in very good agreement with each other.

109

Figure 3.6 Rogers Rational Function Approximations for the Real Part of the

Generalized Aerodynamic Forces of the BAH wing (Q11,Q12)

-200

50

300

550

800

0.0 0.3 0.6 0.9 1.2Reduced Frequency (k)

Re[

Q11

]

-1800

-1350

-900

-450

00.0 0.3 0.6 0.9 1.2

Reduced Frequency (k)

Re[

Q12

]Tabulated (DLM) Approximated (Rogers)

110


Generalized Aerodynamic Forces of the BAH wing (Q21,Q22)

Tabulated (DLM) Approximated (Rogers)

-200

-50

100

250

400


Re[

Q21

]

600

800

1000

1200

1400


Re[

Q22

]

111

Figure 3.8 Rogers Rational Function Approximations for the Imaginary Part

of the Generalized Aerodynamic Forces of the BAH wing (Q11,Q12)


-1000

-750

-500

-250

00.0 0.3 0.6 0.9 1.2


Im[Q

11]

-2400

-1800

-1200

-600

00.0 0.3 0.6 0.9 1.2


Im[Q

12]

112


of the Generalized Aerodynamic Forces of the BAH wing (Q21,Q22)


-50

25

100

175

250

0.0 0.3 0.6 0.9 1.2


Im[Q

21]

-2000

-1500

-1000

-500

00.0 0.3 0.6 0.9 1.2


Im[Q

22]

113

An aeroelastic stability analysis using the pk-method of MSC/NASTRAN® is then

performed. In order to be consistent with the previous works [55,63], all of the ten

modes are first selected as generalized coordinates. The reduced frequencies chosen

are the same as those used in calculating the generalized aerodynamic forces, and

the first two flutter roots are requested for a series of speeds ranging from 4800 to

25000 at increments of 1200 [in/sec].

An aeroelastic stability analysis for the wing is then performed using the root locus

method. Comparison between the damping and frequency curves obtained by both

methods is depicted in Figures 3.10 and 3.11 respectively. Flutter and divergence

types of instabilities are detected by both methods. The flutter speed found by the

pk-method is determined as 1056 [ft/sec] while the one determined by the root locus

method is found as 1088 [ft/sec]. The corresponding flutter frequency found by the

pk-method is 3.09 [Hz] and that found by the root locus method is 3.06 [Hz]. The

divergence speed is determined as 1651 [ft/sec] by the pk-Method and 1663 [ft/sec]

by the root locus method. The flutter speed determined by Bisplinghoff, Ashley and

Halfman [70] was 865 [mph] = 1268.7 [ft/sec] and a corresponding flutter

frequency of 18.6 [rad/sec] = 2.96 [Hz]. Nevertheless, Bisplinghoff, Ashley and

Halfman used the strip theory for the aerodynamics which is significantly different

than the doublet lattice method. They also considered the first two modes only in

their solution. The divergence speed for this wing was determined by them as 1948

[ft/sec] using strip theory aerodynamics and 1910 [ft/sec] using lifting-line theory

for the aerodynamics [70].

114

Figure 3.10 Velocity vs. Damping Plot of the BAH Wing (10 Modes)

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0 500 1000 1500 2000 2500

Velocity [ft/Sec]

Dam

ping

, g .

Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)

115

Figure 3.11 Velocity vs. Frequency Plot of the BAH Wing (10 Modes)

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 500 1000 1500 2000 2500

Velocity [ft/Sec]

Freq

uenc

y [H

z]Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)

116

The same wing is then analyzed again by selecting the first two modes only as the

generalized coordinates. Comparison between the damping and frequency curves

obtained by the root locus and pk methods is shown in Figures 3.12 and 3.13

respectively. Although the damping curves for the first mode are given as discrete,

they are indeed continuous and are cut for better visibility in the velocity range of

1000-1700 [ft/sec] where they achieve relatively high negative damping values.

Flutter and divergence types of instabilities are detected by both methods. The

flutter speed found by the pk-method is determined as 1138 [ft/sec] while the one

determined by the root locus method is 1181 [ft/sec]. The corresponding flutter

frequency found by the pk-method is 3.05 [Hz] and that found by the root locus

method is 3.01 [Hz]. The divergence speed is determined as 1951 [ft/sec] by the pk-

Method and 1970 [ft/sec] by the root locus method. These are in good agreement

with the values found by Bisplinghoff, Ashley and Halfman [70].

117

Figure 3.12 Velocity vs. Damping Plot of the BAH Wing (2 Modes)

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0 500 1000 1500 2000 2500

Velocity [ft/Sec]

Dam

ping

, g .

Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)

118

Figure 3.13 Velocity vs. Frequency Plot of the BAH Wing (2 Modes)

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0 500 1000 1500 2000 2500

Velocity [ft/Sec]

Freq

uenc

y [H

z]Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)

119

3.5.2 ICW Wing

The intermediate complexity wing (ICW) is a test case of ASTROS [82] that has

been used in many applications of multidisciplinary design and optimization

[4,5,17,22]. It is a symmetric two-cell box beam having aluminum substructure and

graphite/epoxy cover skins with the zero degree fibers aligned along the midspar of

the wing. The material properties are given in Table 3.2. No information about the

physical properties of this model is available in literature. Thus, the following

properties have been assumed. The composite skins are assumed to have a layup of

[02/902/45/-45]. The spar webs are assumed to have a thickness of 0.04 [in], the ribs

are assumed to have a thickness of 0.02 [in], and the posts are assumed to have an

area of 0.04 [in2]. The wing geometry and aerodynamic planform of this wing are

shown in Figure 3.14. The finite element model of the wing has a total of 88 nodes

and consists of 62 quadrilateral and 2 triangular membrane elements (skins), 55

shear panels (ribs and spars) and 39 rod elements (posts). The wing is cantilevered

at the root. The finite element model is shown in Figure 3.15. For aerodynamics the

wing is modeled as a flat plate lifting surface with 72 boxes (9 spanwise and 8

chordwise with unequal spacing). The aerodynamic model is depicted in Figure

3.16. The aerostructural interconnection is defined by two surface spline elements

and is shown in Figure 3.17.

Table 3.2 ICW Material Properties

Isotropic Material (Aluminum)

Composite Material (Graphite/epoxy)

E = 10.5x106 [psi] E1 = 18.5x106 [psi]

E2 = 1.6x106 [psi]

ρ = 0.1 [lb/in3] ρ = 0.055 [lb/in3]

ν = 0.3 ν12 = 0.25

G12 = 0.65x106 [psi]

tply = 0.00525 [in]

120

Figure 3.14 Aerodynamic Configuration and Structure of the Intermediate

Complexity Wing (ICW)


Section A-A of the Wing Box

47

4.5 6

Section B-B of the Wing Box

29.33

2.25 3

108

48

300

90

AA

B B

X

Y

121

Figure 3.15 Structural Model of the Intermediate Complexity Wing (ICW)

LOWER SKIN (Composite)

Front Spar (Metallic)

Intermediate Spar (Metallic)

Ribs (Metallic)

Rear Spar (Mettalic)

Posts (Metallic)

Upper skin (Composite) is removed for better visibility

122

Figure 3.16 Aerodynamic Model of the Intermediate Complexity Wing (ICW)

Figure 3.17 ICW Structural & Aerodynamic Models Joined by Surface Spline

Elements

123


mode shapes of the wing. The first six natural frequencies and mode shapes are

determined by using the Lanczos method of MSC/NASTRAN®. The resulting mode

shapes and corresponding frequencies are shown in Figures 3.18-3.23. Guyan

reduction to only out-of-plane displacements was performed to eliminate the

inplane modes. This improves convergence in the flutter solution [49]. Again, the

generalized mass, stiffness, and aerodynamic forces are then obtained from

MSC/NASTRAN® using DMAP statements. The generalized aerodynamic force

matrix was calculated using the DLM for seven values of reduced frequencies

(k=0.001, 0.133, 0.182, 0.3, 0.4, 1.0, 2.0). This matrix is then used to perform a

Rogers rational function approximation for the generalized aerodynamic forces

using four aerodynamic lag parameters. The aerodynamic lag parameters which

give the best fit function are then determined using a non-gradient direct search

optimizer utilizing the complex method. The best found lag parameters are

determined by the optimizer to be γ1=0.25, γ2=0.75, γ3=1.25, and γ4=1.75. The real

and imaginary parts of the first four generalized aerodynmic forces ( )ikQ~

computed by the DLM and the approximated generalized aerodynmic forces

( ) appikQ~ fitted by Rogers rational functions approximation are shown in Figures

3.24-3.27. Again both the calculated and fitted values are found to be in very good

agreement with each other.

124

Figure 3.18 First Mode Shape of the ICW (f=10.3 Hz)

Figure 3.19 Second Mode Shape of the ICW (f=29.5 Hz)

125

Figure 3.20 Third Mode Shape of the ICW (f=41.8 Hz)

Figure 3.21 Fourth Mode Shape of the ICW (f=62 Hz)

126

Figure 3.22 Fifth Mode Shape of the ICW (f=91.4 Hz)

Figure 3.23 Sixth Mode Shape of the ICW (f=99.6 Hz)

127


Generalized Aerodynamic Forces of the ICW wing (Q11,Q12)


-1000

500

2000

3500

5000

0.0 0.5 1.0 1.5 2.0 2.5


Re[

Q11

]

2000

3500

5000

6500

8000

0.0 0.5 1.0 1.5 2.0 2.5


Re[

Q12

]

128


Generalized Aerodynamic Forces of the ICW wing (Q21,Q22)


-4000

-3000

-2000

-1000

00.0 0.5 1.0 1.5 2.0 2.5


Re[

Q21

]

0

1000

2000

3000

4000

0.0 0.5 1.0 1.5 2.0 2.5


Re[

Q22

]

129


of the Generalized Aerodynamic Forces of the ICW wing (Q11,Q12)


-12000

-9000

-6000

-3000

00.0 0.5 1.0 1.5 2.0 2.5


Im[Q

11]

0

1500

3000

4500

6000

0.0 0.5 1.0 1.5 2.0 2.5


Im[Q

12]

130


of the Generalized Aerodynamic Forces of the ICW wing (Q21,Q22)


-12000

-9000

-6000

-3000

00.0 0.5 1.0 1.5 2.0 2.5


Im[Q

22]

-1000

0

1000

2000

3000

0.0 0.5 1.0 1.5 2.0 2.5


Im[Q

21]

131

An aeroelastic stability analysis using the pk-method of MSC/NASTRAN® is then

performed. All of the six modes are selected as generalized coordinates. The

reduced frequencies chosen are the same as those used in calculating the

generalized aerodynamic forces, and all of the six flutter roots are requested for a

series of speeds ranging from 200 to 1000 [knot].

An aeroelastic stability analysis for the wing is then performed using the root locus

method. Comparison between the damping and frequency curves obtained by both

methods is depicted in figures 3.28 and 3.29 respectively. A flutter type of

instability is detected by both methods. The flutter speed found by the pk-method is

determined as 528.5 [knot] while the one determined by the root locus method is

found as 527.6 [knot]. The corresponding flutter frequency found by the pk-method

is 17.8 [Hz] and that found by the root locus method is 18.0 [Hz]. Although the

flutter speed determine by both methods is in good agreement, the damping

behavior shows some deviation between both methods specially for higher order

modes and high velocities.

132

Figure 3.28 Velocity vs. Damping Plot of the Intermediate Complexity Wing

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

200 300 400 500 600 700 800 900 1000

Velocity [knot]

Dam

ping

,g .

Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 3(PK-Method) Mode 4 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)Mode 3 (RL-Method Mode 4 (RL-Method)

133

Figure 3.29 Velocity vs. Frequency Plot of the Intermediate Complexity Wing

-10.0

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0 200 400 600 800 1000 1200

Velocity [knot]

Freq

uenc

y [H

z]Mode 1 (PK-Method) Mode 2 (PK-Method)Mode 3(PK-Method) Mode 4 (PK-Method)Mode 1 (RL-Method) Mode 2 (RL-Method)Mode 3 (RL-Method Mode 4 (RL-Method)

134

3.6 Conclusion

Flutter and divergence are the most important aeroelastic instability phenomena

since the occurrence of any of them would lead to catastrophic structural failure and

loss of the aircraft.

Obtaining valid damping history is required for the certification of a new developed

aircraft and is needed to prevent loss of the aircraft or wind tunnel test model in a

flutter clearance test.

Aeroelastic stability analysis to determine the onset of flutter and divergence can be

performed relatively easily in the frequency domain using either k-method or the

pk-method. However, these methods produce damping information that is either

invalid (k-method) or approximate (pk-method).

On the contrary, Laplace domain methods, i.e., the p-method and root locus method,

results in damping information that is valid for all of the speed range of interest and

provide better insight into the physical phenomena leading to aeroelastic instability.

However, the main difficulty in implementing these methods lies in obtaining the

aerodynamic loads for an arbitrary motion in Laplace domain. This problem can be

circumvented effectively through the use of rational function approximations.

135

CHAPTER 4

MULTIDISCIPLINARY DESING AND OPTIMIZATION

4.1 Introduction

Design of a composite aircraft wing, in order to achieve strength, buckling, and

aeroelastic stability requirements with minimum weight, is a multidisciplinary

design and optimization problem. It involves the interaction of the structural and

aeroelastic analysis disciplines with conflicting requirements on strength, stiffness,

and manufacturing limitations.

A variety of multidisciplinary optimization softwares that include aeroelastic and

static strength constraints in the optimization cycle have been developed for

structural sizing. Among them MSC/NASTRAN and ASTROS are the most

popular. However, the capability of these state of the art tools is limited in many

aspects. Their formulation relies on the fully stressed design concept which does not

necessarily ensure an optimum design. They cannot account for specialized

potential failure modes like crippling that is based on emprical analysis methods.

The accuracy of an optimization depends on the accuracy of the analysis. Since

these softwares rely on the finite element method in calculating the stresses, the

accuracy of the analysis is limited by the mesh size of the finite element model and

the requirements on the analysis (linear/nonlinear). Typical wing boxes exhibit thin

136

metal/composite panels joined together to form the structure. Since these panels are

very thin, they usually buckle at very low stress amplitudes causing redistribution of

the load in the structure. Tools like ASTROS and MSC/NASTRAN cannot account

for such effects.

In practical applications, like the design of a composite wing box, the design

variables are not all continuous and some of them must be selected from a set of

integer or discrete values. The structural members may have to be chosen from

standard sizes and member thicknesses may have to be selected from commercially

available ones. Stacking sequence design of composite plates involves the

determination of the number of plies and their orientations. The stacking sequence

design problem is discrete in nature. Due to manufacturing limitations, the plies are

fabricated at certain thicknesses and the orientations are limited to a small set of

discrete angles. Thus, the optimization problem of designing a composite wing box

subject to static strength and aeroelastic stability constraints together with

manufacturing constraints is a nonlinear optimization problem that envolves a

combination of continuous and discrete design variables.

The optimization algorithms utilized by softwares like ASTROS and

MSC/NASTRAN utilize gradient-based algorithms which are suitable for treating

design problems with continuous design variables. There are many problems which

are inherent in the gradient-based optimization techniques. A basic disadvantage of

gradient-based methods is their convergence to the optimum closest to the starting

point in the design space which might not be the global optimum. Since they use the

gradient information to advance in the design space, they require the design space to

be continuous and convex. They are inefficient when the number of the design

variables involved is large.

Genetic algorithms offer an alternative for solving wing box optimization problem.

Genetic algorithms work on function evluations only and do not require any

137

gradient information. Their lack of dependence on function gradients makes


optimum and have better probability in locating the global optimum. Genetic

algorithms have been successfully applied to the stacking sequence design of

composite laminates.

The automated MDO procedure developed for the multidisciplinary design and

optimization of the composite wing box is explained in this chapter. The

optimization problem attempted is first described. The problem is then

mathematically formulated in terms of the objective function, the static strength and

aeroelastic stability constraints, and the manufacturing constraints on the design

variables. The solution procedure for the optimization problem is explained. The

static strength analysis, aeroelastic stability analysis, and optimization methods

utilized in the procedure are then discussed. The developed code for the automated

procedure with its features and limitations are then described in detail.

4.2 Statement of the Optimization Problem

The problem attempted is to minimize the weight of a composite wing box subject

to static strength constraints (positive margins of safety), aeroelastic stability

constraints (flutter/divergence free structure), and side constraints (manufacturing

constraints) on the design variables.

138

4.3 Formulation of the Optimization Problem

The optimization problem is mathematically formulated in this section. The

objective function, static strength constraints, aeroelastic stability constraints, and

the types of design variables involved in the optimization problem are identified.

4.3.1 Objective Function

The objective function is the structural weight of the wing box excluding any other

nonstructural weights, like fuel weight, and can be represented by

( ) )(1

xVxWelementN

lll∑

=

= ρ (4.1)

where Nelement is the number of elements in the finite element model, Vl is the

volume of the lth element, ρl is the corresponding material density, and x is design

variables vector.

4.3.2 Static Strength Constraints

The static strength constraints involve obtaining a positive margin of safety for all

of the structural elements of the wing box (spars, ribs, and skins) taking into

account all of their relevant potential failure modes as described in detail in section

2.3. Thus, the skins are analyzed for buckling under the effect of combined biaxial

and shear loads and for material failure under the effect of axial, shear, and

combined stresses. For composite skin panels, the principal stresses/strains in each

ply and the interlaminar shear stresses are checked against the material allowable to

insure integrity of the plies and failure free laminates. The spars are designed as

shear resistant beams. Thus, the spars webs are checked against material failure and

buckling under the effect of combined shear, bending, and biaxial stresses. The spar

139

caps are analyzed for crippling if they are in compression and for maximum stress

checked against the allowable material stress if they are in tension. The ribs are

checked for buckling under the effect of combined crushing, bending, and shear

stresses. Similar to the spars caps, the ribs chords are checked for crippling if they

are in compression and maximum stress if they are in tension.

Thus, the constraints on static strength requirements can be written as

( ) elements structural ofnumber ,,2,1,0. �=≥= iSMxg ii (4.2)

where M.Si is the minimum margin of safety of the ith structural element, analyzed

for all potential failure modes and under the effect of all load cases considered. The

margin of safety is defined as,

1. −=app

allSMσσ (4.3)

where allσ is the allowable stress, and appσ is the applied stress.

4.3.3 Aeroelastic Stability Constraints

The aeroelastic stability constraints are treated by constraining the damping rather

than the flutter/divergence speeds. Defining the constraint in this manner eliminates

the requirement for calculating the exact flutter/divergence speeds which can be a

computationally expensive task, and has the advantage of insuring a prescribed

damping level in the final design. This approach can also handle “hump” modes

effectively in an automated design process by evaluating the constraints (damping)

at a series of velocities and eliminates the necessity for the mode tracking process.

Treating the aeroelastic constraints in this manner was first proposed by Hajela [11]

140

and has become a standard process in both ASTROS [82] and MSC/NASTRAN

[88]. It is also the approach adopted by ADOP [47].

Thus, the aeroelastic constraints can be expressed as

( ) modes ofnumber ,21

s velocitieofnumber ,2,10

l

l

,,lj

GFACTxg jREQjl

jl ==

≤−

=γγ

(4.4)

where γjl is the damping for the lth mode calculated at the jth velocity and γjREQ is

the user-defined required damping level at the jth velocity. GFACT is a factor that is

used to scale the constraints and is typically equal to 0.1 [82,88]. Flutter analysis at

a series of flight speeds that are 0.5, 0.75, 0.9, 1.0, 1.1 times the required flutter

speed should be adequate to preclude “hump” mode type of behavior [15,82]. The

typical GFACT value of 0.1 and the flight speeds recommended by [82] are used in

this thesis.

4.3.4 Design Variables

The design variables include the cross sectional dimensions of the spars caps and

ribs chords, the thicknesses of the ribs webs and spars webs, and the number of

plies and their corresponding orientations of the composite skin elements. If the

skin elements are chosen as metallic, then the corresponding design variables are

the thicknesses of the skin elements.

Due to manufacturing limitations, the plies are fabricated at certain thicknesses and

the orientations are limited to a small set of discrete angles. Thus, the design

variables of a composite skin element are a combination of integer (number of

plies) and discrete (corresponding ply orientation angles) types. If the spar webs and

ribs webs are made of sheet metal, then their thicknesses has to be selected from

commercially available ones and thus the corresponding design variables are

141

discrete. Alternatively, if they are machined parts, then their thicknesses are

continuous and should be described by continuous design variables in the

optimization process. The same discussion applies for the spars caps and ribs

chords. Thus, the design variables for the spars and ribs can be of discrete,

continuous, or a mixture of discrete and continuous design variable types.

From the above discussions, it is evident that the optimization problem of the wing

box involves a combination of integer, discrete and continuous design variables.

The integer design variables are implicit design variables that define the number of

plies in the composite skin panels. Mathematically, the side constraints

(manufacturing constraints) on the design variables can be described as,

{ }{ }

+++=≤≤

++==∈

==∈

cididUjj

Lj

iddjjjj

dkjjjjj

mm,m,mmj,xxx

m,m,mj,n,nIx

m,,j,d,,d,dDx

h

h

hh

1

21maxmin

21

(4.5)

where Dj is the jth discrete set that contains k number of elements and from which

the jth discrete variable is chosen. The jth integer design variables is chosen from the

integer set Ij with an integer lower and upper bounds given by minjn and max

jn

respectively. The jth continuous design variable is described by lower and upper

bounds given by Ljx and U

jx respectively. The number of discrete, integer, and

continuous design variables is defined by md, mi, mc respectively.

The optimization problem defined by equations (4.1) through (4.4) forms a mixed-

discrete nonlinear programming problem. The term “mixed-discrete” implies that

the problem involves a combination of continuous and discrete design variables and

since integer design variables can be considered as discrete design variables, no

distinction is usually made between discrete and integer design variables. Solution

of this problem is described in the following section.

142

4.4 Solution Procedure

This section describes the methods used for evaluating the static strength

constraints, aeroelastic stability constraints, and the optimization procedure used.

4.4.1 Static Strength Analysis Procedure

The static strength analysis procedure is based on a refined analysis process that

takes into account all potential failure modes and the load redistribution in the

structure after buckling occurs. This process has been described in detail in chapter

2 and is summarized here for convenience.

The static strength analysis procedure relies on obtaining the internal loads

distribution from a coarse mesh finite element model. After the internal loads are

determined, they are summed to determine the sectional forces (i.e., the shear force,

the normal force and the bending moment) acting on each spar/rib section. The

section axial and bending stiffnesses are determined and the modified engineering

bending theory is then used to determine the stress distribution over the section. The

classical laminated plate theory is used in both determining the equivalent

stiffnesses and the analysis of the composite skins over the section. The allowable

crippling stresses are calculated based on the Gerard method and the allowable

buckling stresses for a general inplane stress state are obtained using the Rayleigh-

Ritz method. After the stress/strain distribution for all of the load cases under

consideration and the relevant allowable stresses are determined, the minimum

margins of safety that form the constraints on static strength requirements are

determined.

143

4.4.2 Aeroelastic Stability Analysis Procedure

The aeroelastic stability analysis procedure relies on a root locus method to

determine the damping level of each mode at a prescribed set of velocities. These

damping levels form the aeroelastic stability constraints. The Laplace domain

unsteady aerodynamic forces are deduced form their frequency domain counterparts

using Rogers function approximation for the generalized aerodynamic forces.

This aeroelastic stability analysis method has been described in detail in chapter 3

of this thesis.

4.4.3 Optimization Procedure

The optimization procedure utilized in this thesis is based on a genetic search

optimization method. This method has been chosen since it can adequately treat

mixed- discrete nonlinear optimization problems, it is less susceptible to pitfalls of

convergence to a local optimum, and have better probability in locating the global

optimum. Another adavantage of genetic algorithms is that since they work on a

population of designs they produce a family of designs in their final population with

similar performance results offering the designer several alternatives to chose the

design suitable to him.

Genetic Algorithms (GA) are based on Darwin’s theory of the survival of the fittest.

In a genetic algorithm one starts with a set of designs. From this set, new and better

design mare reproduced using the fittest members of the set. Each design is

represented by a finite length string. Usually binary strings have been used for this

purpose. The entire process is similar to a natural population of biological creatures;

where successive generations are conceived, born and raised until they are ready

reproduce. Three operators are needed to implement the algorithm. These are the

selection, crossover, and mutation operators. Selection process is one that biases the

144

search toward producing more fit members in the population and eliminating the

less fit ones. Crossover allows selected members of the population to exchange

characteristics of the design among themselves. Mutation is the third step in this

genetic refinement process, and is one that safeguards the process from a complete

premature loss of valuable genetic material during reproduction and crossover. The

basic genetic algorithm is illustrated in Figure 4.1.

Figure 4.1 The Basic Genetic Algorithm

Initialize Population

Evaluate (Selection)

Crossover

Mutation

Converged?

Start

Yes

No Stop

145

4.5 Code Description

This section describes the details, features, and limitations of the developed

multidisciplinary design and optimization code.

The code consists mainly of an analysis module that is connected to an optimizer

through a processing module as shown in Figure 4.2. The analysis module consists

of two parts, a static strength analysis module and an aeroelastic stability analysis

module. The optimizer (EVOLVE) uses genetic search techniques. EVOLVE has

been developed by Lin and Hajela (1993). It was courtesy of Prof. Prabhat

HAJELA of the Rensselaer Polytechnic Institute to provide EVOLVE to be utilized

in this thesis.

146

Figure 4.2 General Flowchart of the Developed Code

OPTIMIZER (EVOLVE)

ANALYSIS MODULE

STATIC STRENGTH ANALYSIS MODULE

AEROELASTIC STABILITY ANALYSIS MODULE

PROCESSING MODULE

DESIGN VECTOR

OBJECTIVE FUNCTION

& CONSTRAINTS

OUTPUT

Other party/Commercial Software

USER INPUT

NASTRAN BULK DATA FILE (*.bdf)

MSC/NASTRAN® SOL (145)


147

4.5.1 Processing Module

The processing module processes the user input files and MSC/NASTRAN® bulk

data input files. The flowchart of this module is shown in Figure 4.3. Based on user

input, it first identifies the type of design variables selected by the user in the

model. The type of design variables can be discrete, integer, continuous, and spatial

functions. Discrete design variables are chosen from a user supplied discrete set of

data. A typical example is the ply orientation angles set or a sheet metal part

thickness that is selectable from a standard list. Integer design variable describe

variable that are chosen from an integer set like the number of plies of a composite

skin panel. Continuous design variables describe variables that are continuous. A

typical example of this is the thickness or cross section dimensions of a machined

part. Spatial function type design variables describe design elements that are

function of span location. These can be of linear, quadratic, or a combination of

linear and quadratic functions of the span location. A typical type of this is the spars

caps width and height that usually vary linearly with the span location having their

maximum at the root of the wing where the bending moment attains its maximum

value. This type of design variables is intended to reduce the number of independent

design variable in the model by describing the properties of large number of

elements in a large scale finite element model by a small number of design

variables.

After identifying the number of design variables and types, the processing module

prepares an input file for EVOLVE and produces mapping information between the

design variables and the physical properties of those finite elements affected by the

change in design variables. Separate bulk data cards for each of these finite

elements are produced by this module and then supplied to the analysis module.

148

Figure 4.3 Flowchart of the Processing Module

IDENTIFY TYPE OF DESIGN ELEMENTS• DISCRETE • CONTINUOUS • INTEGER • FUNCTION

PREPARE INPUT FOR EVOLVE

MAPPING INFO BETWEEN DESIGN VARIABLES & ELEMENT PROPERTIES

PREPARE SEPARATE NASTRAN BULK DATA CARDS FOR EACH DESIGN

ELEMENT

ELEMENTS GEOMETRY, TYPE AND CONNECTIVITY INFORMATION

ELEMENTS MATERIAL AND AXIS ORIENTATION ANGLES

IDENTIFY SKINS, SPARS, RIBS ELEMENTS

INFORMATION NECESSARY FOR SECTIONAL LOADS CALCULATION

EVOLVE

ANALYSIS MODULE


149

Another function of the processing module is to produce the geometric and

connectivity information to be utilized by the analysis module. This information

includes elements local coordinate systems, material orientation angles, and

elements normals. It also identifies the type of elements used in modeling the

structure (e.g., CSHEAR, CQUAD, CROD), the grid numbers, and section normal

and tangential vectors necessary for calculating the sectional loads. The web, upper

skins, lower skins, and caps element identification numbers on each section are

identified by this module.

4.5.2 Analysis Module

The analysis module is illustrated in Figure 4.4. The main function of this module is

to calculate the objective function and constraint values for a given set of design

variables supplied by the optimizer (EVOLVE). The objective function is the

weight of the wing box. The constraint values are the static strength margins of

safety and the structural damping. It uses the mapping information between the

design variables and the physical properties produced by the processing module to

map the design variables into physical properties (i.e., areas, thicknesses, etc.) and

updates the finite element model bulk data file. The static strength analysis module

and the aeroelastic stability analysis module are then called by this module to

calculate the objective function and constraint values. These values are then

submitted to the optimizer for its decision making on a new set of design variables.

150

Figure 4.3 Flowchart of the Analysis Module

MAP DESIGN VARIABLES

INTO ELEMENT PROPERTY

INFORMATION

UPDATE STATIC ANALYSIS BULK

DATA FILE

UPDATE AEROELASTICANALYSIS BULK

DATA FILE

DAMPING AND INSTABILITY SPEED

INFORMATION

STRUCTURAL WEIGHT AND MARGINS OF

SAFETY INFORMATION

STATIC STRENGTH ANALYSIS MODULE

AEROELASTIC STABILITY

ANALYSIS MODULE

EVOLVE

PROCESSING MODULE


OBJECTIVE FUNCTION AND CONSTRAINTS VALUES

DESIGN VARIABLES

151

4.5.2.1 Static Strength Analysis Module

The static strength analysis module performs static strength analysis of the wing

box according to the analysis methods explained in chapter 2. It determines the

minimum margins of safety, the critical load cases, and the corresponding failure

modes. The flowchart of this module is depicted in Figure 4.5.

The static strength analysis module calls MSC/NASTRAN® (SOL 101) in batch

mode to perform a linear-static finite element analysis. After the finite element

analysis is completed, the output file generated by MSC/NASTRAN® is

manipulated by this module to determine the internal load distribution in the

structure. Elements centroidal stresses and strains, shear flows, and ply interlaminar

stresses are also extracted from MSC/NASTRAN® output file.

The grid point force balance output is used to determine the sectional shear force,

normal force, and bending moment loads at the end sections of spars and ribs for

each bay. These sectional loads are also interpolated to determine the sectional

loads at middle sections for further analysis.

The sections equivalent bending and axial stiffnesses are determined by this module

using the classical laminated plate theory. Based on the modified engineering

bending theory, the stress distribution over the section is determined in an iterative

manner taking into account the skin buckling effect and load redistribution in the

structure.

This module calculates the allowable stresses for each structural element in the

wing box. These include the crippling stress based on the Gerard method and

buckling loads based on the Rayleigh-Ritz method. The structure is then analyzed

for all load cases under consideration to determine the margins of safety, critical

load cases and the corresponding failure modes.

152

Figure 4.5 Flowchart of the Static Strength Analysis Module

SPAR STRENGTH ANALYSIS

STATIC ANALYSIS FEM MODEL


GRID POINT FORCE BALANCE

PROCESS NASTRAN OUTPUT (*.f06) FILE

ELEMENT STRAINS INTERLAMINAR STRESSES

ELEMENT FORCES AND STRESSES

GEOMETRY INFORMATIONMATERIAL ALLOWABLE

SKIN STRENGTH ANALYSIS

RIB STRENGTH ANALYSIS

BUCKLING ANALYSIS (RAYLEIGH-RITZ)

WEIGHT, MARGINS OF SAFETY & FAILURE

MODES INFORMATION

SECTIONAL LOADS (V, N, M)

CRIPLLING ANALYSIS (GERARD)


153

4.5.2.2 Aeroelastic Stability Analysis Module

The aeroelastic stability analysis module performs an aeroelastic stability analysis

to determine the damping values of each mode for a prescribed set of velocities

based on the root locus method as explained in chapter 3. The flowchart of this

module is shown in Figure 4.5. It calls MSC/NASTRAN® (SOL 145) in batch mode

to determine the generalized stiffness, mass, damping, and aerodynamic forces

matrices. MSC/NASTRAN® DMAP (Direct Matrix Abstraction Program)

statements are used to extract these matrices. The method used for the free vibration

analysis is specified by the user. Several methods are available in

MSC/NASTRAN® to perform free vibration analysis. Among them Lancsoz

method is the most popular and widely used one. The unsteady aerodynamic forces

are calculated using the theory specified by the user. This can be the DLM (Doublet

Lattice Method) for subsonic flow regimes or the MBM (Mach Box Method) for

supersonic flow regimes.

After extracting the generalized aerodynamic forces, this module performs Rogers

least square rational function approximations for the unsteady aerodynamic forces.

The aerodynamic lag roots are selected by non-gradient direct search optimizer that

is based on the complex method. This module then solves the quadratic eigenvalue

problem to determine the generally complex roots. The damping and frequency are

then extracted from these roots.

The output of this module is the damping of each mode calculated at the user

specified set of velocities. Although not required by the optimizer, the instability

speed(s) are also calculated and output by this module if requested by the user.

154

Figure 4.6 Flowchart of the Aeroelastic Stability Analysis Module

AEROELASTIC FEM MODEL


GENERALIZED MASS MATRIX

PROCESS NASTRAN OUTPUT FILES

GENERALIZED UNSTEADY AERODYNAMIC FORCES

GENERALIZED STIFFNESS MATRIX

DAMPING AND INSTABILITY SPEED(S)

INFORMATION

GENERALIZED DAMPING MATRIX

LEAST SQUARE ROGERS RATIONAL FUNCTION

APPROXIMATION

COMPLEX METHOD OPTIMIZATION FOR THE

LAG ROOTS (γ)

(Root Locus Method) STATE SPACE FORM OF THE EQUATION

OF MOTION IN LAPLACE DOMAIN


155

4.5.3 General Features and Limitations of the Code

The code developed can be used for the design wing boxes as well as any wing like

box structures. These include control surfaces such ailerons and horizontal and

vertical stabilizers. There is no restriction on the number of ribs and spars. The

number of ribs and spars is defined by parameters that can be changed to suit the

problem analyzed. Thus, multi cell wing boxes can be analyzed by the code.

Aeroelastic stability analysis can be performed in both the subsonic and supersonic

regimes by selecting the appropriate aerodynamic theory in the user supplied

MSC/NASTRAN® input file. The design variables for the wing part structures can

be specified as integer, discrete, continuous and as function of the span location.

Design variable linking is also supported. Thus, different parts of the structure can

be described by common variables.

The code includes the following limitations. Only the skins can be analyzed as

composite in structural analysis module of the current version of the code, although

modification to include the spars and ribs as composite is quite straight forward due

to modularity of the code and the similarity of the analysis procedure involved. The

analysis of stringers is not supported, so only unstiffened skin panels can be

analyzed. The shape of the supported spars caps, ribs chords, and their

corresponding design variables are illustrated in Figure 4.5. Running the code

requires MSC/NASTRAN® (v75.7) and EVOLVE.

156

Figure 4.5 Supported Spars Caps/Ribs Chords and the Corresponding Design

Variables

W

t H

W

t

W

t

tH

W

t

t H

Extruded Tee Formed Angle

Formed Double Angle

157

4.6 Conclusion

The optimization problem of designing a composite wing box subject to static

strength and aeroelastic stability constraints together with manufacturing constraints

is a nonlinear optimization problem that envolves a combination of continuous and

discrete design variables. Gradient-based optimization algorithms are suitable for

solving optimization problems with continuous design variables and has many

problems inherent in them. Genetic algorithms offer an alternative for solving this

optimization problem. Their lack of dependence on function gradients makes


optimum and have better probability in locating the global optimum.

The automated MDO procedure developed for the multidisciplinary design and

optimization of a composite wing box was explained in this chapter. The problem

was first mathematically formulated in terms of the objective function, the static

strength and aeroelastic stability constraints, and the manufacturing constraints on

the design variables. The solution procedure for the optimization problem was

explained. The static strength analysis, aeroelastic stability analysis, and

optimization methods utilized in the procedure were described. The developed code

for the automated procedure with its features and limitations was then described in

detail.

158

CHAPTER 5

CASE STUDIES

5.1 Introduction

In this chapter the developed multidisciplinary design and optimization code is

applied to the design of a rectangular wing box. The wing box is considered in three

different case studies. The first case study aims at verifying the developed code and

studying the capability of the genetic algorithm in optimization for aeroelastic

constraints with manufacturing constraints imposed on the design variables. Thus,

an all metallic wing box which is fully described and has available optimization

results in literature is optimized to meet aeroelastic stability constraints with

manufacturing constraints imposed on the thicknesses of the spars webs, ribs webs,

and spars caps areas. Although the design variables for this problem were

traditionally treated as continuous design variables in previous investigations, they

are chosen to be of discrete type to represent manufacturing constraints in this

study. In the second case study the wing box considered in the first case study is

modified to have composite skin panels and ribs chords and is then optimized to

meet static strength requirements subject to manufacturing constraints on the

thicknesses, ply orientations, and cross sectional dimensions of the spars caps and

the ribs chords. Thus the purpose of this case study is to study the capability of the

developed code in the optimization of representative “real-life” composite wing

159

structures and to form a basis for the third case study. The third case study aims at

studying the advantage of considering the aeroelastic stability constraints at early

stages of the design. Thus, the optimized wing considered in the second case study

is first analyzed to determine its flutter/divergence speeds. Then a 20% increase in

the flutter/divergence speed is imposed on the design and the wing box is optimized

to meet aeroelastic constraints, static strength constraint, and manufacturing

constraints simultaneously.

5.2 Wing Box Model Description

In this section, the wing box considered in the case studies is explained and

background on the model is given.

The wing considered in the case studies is illustrated in Figure 5.1. It is an unswept

cantilever wing with constant cross section. This wing box was first studied by

Rudisill and Bhatia [6], [7], and later by McIntosh and Ashley [10] among others,

[11], [49]. As indicated by Striz and Venkayya [49], this wing box model represents

one of the very few cases in the flutter optimization literature where all structural,

material, and environmental data were given to allow for direct comparison of

results. Hence, it is chosen as a base model for verification purposes of the

developed code in this study. The material properties as used by Rudisill and Bhatia

are given in Table 5.1.

The structural model of this wing box is shown in Figure 5.2. It has three bays and

in each bay there are two skin elements, two spar webs, and one rib, all modeled by

quadrilateral membrane elements. In each bay there are four spar caps that are

modeled by axial rod elements.

The flutter optimization problem of this wing first considered by Rudisill and

Bhatia [6], [7], and later by Hajela [11] involves a total of twelve design variables.

160

These design variables are numbered as given in Table 5.2. The initial design values

were selected as: x1-x3 = 0.04 [in], x4-x6 = 0.08 [in], x7-x9 = 0.04 [in], and

x10-x12 = 2.0 [in2]. The initial weight of the wing box is 196 [lb]. Minimum-gage

constraints were imposed at one-sixth of the initial design values as side constraints

(manufacturing constraints). The optimization problem considered was to minimize

the weight of the wing subject to a flutter speed constraint of 600 [ft/sec] at an

altitude of 10,000 [ft]. As given in reference [7], the true optimum for this problem

corresponds to a wing structure with all of the design variables at their lower

bounds and a corresponding weight of 32.7 [lb].

Table 5.1 Material Properties of the Rectangular Wing Box (Aluminum)

Property Value

Modulus of Elasticity (E) 10.5x106 [psi]

Modulus of Rigidity (G) 4.0x106 [psi]

Density (ρ) 5.46 [slugs/ft3]

Table 5.2 Design Variables of the Rectangular Wing Box

Bay Number Skin Thickness

Web Thickness

Rib Thickness

Spar-Cap Areas

1 x1 x4 x7 x10

2 x2 x5 x8 x11

3 x3 x6 x9 x12

161

Figure 5.1 Layout and Aerodynamic Configuration of the Rectangular Wing

Front Spar Rear Spar

Leading Edge Trailing Edge

4

Rib 1 Rib 2 Rib 3

60 60 60

10

25

15


V∞

Bay 1 Bay 2 Bay 3

162

Figure 5.2 Structural Model of the Rectangular Wing Box

SKIN (Membrane)

Rib (Membrane)

Spar Web (Membrane)

Spar Cap (Rod)


163

5.3 Case Study I

The purpose of this case study is to verify the developed code and illustrate the

capability of the genetic algorithm in optimization with aeroelastic stability and

manufacturing constraints.

In this case study, the developed code is applied to the design of the wing box

discussed in section 5.2. Different than the work done by Rudisill and Bhatia [6],

[7], and later by Hajela [11] who used gradient-based optimization methods to

optimize this wing box, the genetic algorithm based code developed in this study is

used to optimize the wing box. Manufacturing constraints are imposed on the

thicknesses of the spars webs, ribs webs, and spars caps areas. Thus, instead of

using continuous type design variables, all design variables are chosen to be of

discrete type.

Aeroelastic stability analysis is first performed to study the behavior of this model.

The wing is modeled as a flat plate lifting surface with a total of 80 aerodynamic

boxes (8 chordwise and 10 spanwise with equal spacing). The aerodynamic model

is shown in Figure 5.3. The doublet lattice method of MSC/NASTRAN® is used to

calculate the unsteady aerodynamic forces for an input Mach number of 0.557. This

Mach number corresponds to a speed of 600 [ft/sec] at an altitude of 10,000 [ft].

The structural model and the aerodynamic model are connected by using a surface

spline element as shown in Figure 5.4.

164

Figure 5.3 Aerodynamic Model of the Rectangular Wing

Figure 5.4 Rectangular Wing Structural & Aerodynamic Models Joined by

Surface Spline Element

165

A free vibration analysis is then performed to determine the natural frequencies and

mode shapes of the wing. The first six natural frequencies and mode shapes are

determined by using the Lanczos method of MSC/NASTRAN®. The resulting mode

shapes and corresponding frequencies are shown in Figures 5.5-5.10. Guyan

reduction to only out-of-plane displacements is performed to eliminate the inplane

modes and improve convergence in the flutter solution. The resulting natural

frequencies are in very good agreement with the results of Striz and Venkayya [49]

who used ASTROS in their work. Note that the natural frequencies and mode

shapes are identical for both the initial and final (optimum) designs for this

particular case. Since the stiffness and mass matrices are linear functions of the

chosen design variables, they are scaled by the same factor (1/6) relative to the

initial design.

Figure 5.5 First Mode Shape of the Rectangular Wing (f=6.4 Hz)

166

Figure 5.6 Second Mode Shape of the Rectangular Wing (f=24.7 Hz)

Figure 5.7 Third Mode Shape of the Rectangular Wing (f=37.9 Hz)

167

Figure 5.8 Fourth Mode Shape of the Rectangular Wing (f=71.1 Hz)

Figure 5.9 Fifth Mode Shape of the Rectangular Wing (f=110.7 Hz)

168

Figure 5.10 Sixth Mode Shape of the Rectangular Wing (f=120.7 Hz)

Aeroelastic stability analysis is then performed using both the root locus method of

the developed code and the pk-method of MSC/NASTRAN®. All of the six modes

are selected as generalized coordinates.

The model is first analyzed for the case where all of the design variables are at their

upper bounds. The damping and frequency curves obtained by both methods are

depicted in Figures 5.11 and 5.12 respectively. A very good agreement between

both methods is noticed. Flutter and divergence types of aeroelastic instability are

detected by both methods. The flutter speed is predicted by both methods as 877

[ft/sec] with a corresponding flutter frequency of 13.2 [Hz]. The divergence speed

predicted by the root locus method was slightly less than obtained by the pk-

method. While the divergence speed predicted by the root locus method is

determined as 941.3 [ft/sec], the pk-method predicted a divergence speed of 944

[ft/sec]. These values are in very good agreement with those determined by Striz

and Venkayya [49] who determined a flutter speed of 875 [ft/sec] and a divergence

speed of 958 [ft/sec] for this model.

169

Figure 5.11 Velocity vs. Damping Plot of the Rectangular Wing for Maximum

Values of the Design Variables

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 200 400 600 800 1000 1200 1400

Velocity [ft/sec]

Dam

ping

, g .

Mode 1 (RL Method) Mode 2 (RL Method)Mode 1 (pk-Method) Mode 2 (pk-Method)

170

Figure 5.12 Velocity vs. Frequency Plot of the Rectangular Wing for Maximum


-5

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400

Velocity [ft/sec]

Freq

uenc

y [H

z]Mode 1 (RL Method) Mode 2 (RL Method)Mode 1 (pk-Method) Mode 2 (pk-Method)

171

The model is then analyzed for the case when all of the design variables are at their

lower bounds. For this case the frequency and damping curves obtained by both the

root locus method and the pk-method of MSC/NASTRAN® are presented in Figures

5.13 and 5.14 respectively. In this case, the model is again detected to have flutter

and divergence with the speeds predicted by both methods beeing identical. The

flutter and divergence speeds in this case are determined to be 601 [ft/sec] and

392.5 [ft/sec] respectively. The corresponding flutter frequency is determined as

11.7 [Hz]. Note that the divergence speed in this case is much lower than the flutter

speed. Thus, limiting the aeroelastic stability constraints into flutter only would

result in an unsafe final design that has already exceeded its divergence speed.

172

Figure 5.13 Velocity vs. Damping Plot of the Rectangular Wing for Minimum


-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 200 400 600 800 1000 1200 1400

Velocity [ft/sec]

Dam

ping

, g

.Mode 1 (RL Method) Mode 2 (RL Method)Mode 1 (pk-Method) Mode 2 (pk-Method)

173

Figure 5.14 Velocity vs. Frequency Plot of the Rectangular Wing for Minimum


-5

0

5

10

15

20

25

30

0 200 400 600 800 1000 1200 1400

Velocity [ft/sec]

Freq

uenc

y [H

z]Mode 1 (RL Method) Mode 2 (RL Method)Mode 1 (pk-Method) Mode 2 (pk-Method)

174

In order to check the capability of the developed code in determining the global

optimum design for this problem and in order to be consistent with the work done

performed in [6], [7], and [11], the wing box is first optimized with a flutter speed

constraint only. Thus, divergence is not taken into account in the optimization

process. The design variables are all selected to be of discrete type to represent

manufacturing constraints on the design variables. Each of the design variables is

selectable from a discrete set that has six members and includes the lower and upper

bounds on the corresponding continuous design variables as members of the set.

The design variables sets are given in Table 5.3.

Table 5.3 Design Variables Sets

Design Variable Design Variable Set

x1-x3 {0.00667,0.0133,0.02,0.0267,0.0334,0.04} [in]

x4-x6 {0.0133,0.0267,0.04,0.0533,0.0667,0.08} [in]

x7-x9 {0.00667,0.0133,0.02,0.0267,0.0334,0.04} [in]

x10-x12 {0.33333,0.667,1.0,1.33,1.67,2.0} [in2]

A population size of 20 is chosen and the constraints are augmented into the

objective function using the penalty function approach as available in EVOLVE

[89]. The penalty coefficient is chosen initially as twenty and a penalty adjuster is

added to it after every ten generations. The penalty adjuster prevents bias of the

reproduction plan in EVOLVE [89]. The probability of crossover is chosen as 0.8

and that of mutation is selected as 0.01. The initial population is selected randomly.

The convergence history for the best weight in a given generation is illustrated in

Figure 5.15. Note that the jump in the objective function is due to the penalty

adjuster. A converged result is reached at generation 51. The design variables

values are all at their lower bounds and the resulting weight is 32.4 [lb].

175

Figure 5.15 Convergence History for the Rectangular Wing Weight

(Flutter Speed Constraint Only)

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 20 40 60 80 100

Generation Number

Obj

ectiv

e Fu

nctio

n

176

Next, the wing is optimized for both flutter and divergence at the same flight

conditions. A minimum flutter/divergence speed of 600 [ft/sec] is imposed. The

convergence history for this test case is illustrated in Figure 5.16. Convergence is

achieved in generation 82 with a minimum weight of 46.3 [lb]. The design variable

values corresponding to the optimum design are given in Table 5.4.

Table 5.4 Design Variables Values for Flutter and Divergence Speeds Constraints

Design Variable Design Variable Value

x1 0.02 [in]

x2 0.0333 [in]

x3 0.00667 [in]

x4 0.04 [in]

x5 0.0133 [in]

x6 0.0133 [in]

x7 0.0133 [in]

x8 0.0333 [in]

x9 0.02 [in]

x10 0.333 [in2]

x11 0.333 [in2]

x12 0.333 [in2]

177

Figure 5.16 Convergence History for the Rectangular Wing Weight

(Flutter and Divergence Speed Constraints)

30.0

40.0

50.0

60.0

70.0

80.0

90.0

0 20 40 60 80 100Generation Number

Obj

ectiv

e Fu

nctio

n

178

In order to verify the final design, an aeroelastic stability analysis is performed for

these design values. The frequency and damping curves for this design are

illustrated in Figures 5.17 and 5.18 respectively. The flutter and divergence speeds

for this optimum design wing are determined as 830.1 [ft/sec] and 661.2 [ft/sec]

respectively. The corresponding flutter frequency is calculated as 19.3 [Hz]. Note

that in the developed code the constraints on aeroelastic stability are placed on

damping for a series of speeds that are 0.5, 0.75, 0.9, 1.0, 1.1 times the required

minimum instability speed. Thus, for the requested minimum instability speed of

600 [ft/sec], the resulting instability speed of 661.2 [ft/sec] is perfect agreement

with what has been requested from the genetic algorithm.

179

Figure 5.17 Velocity vs. Damping Plot of the Rectangular Wing Optimized for

Flutter and Divergence

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0 200 400 600 800 1000

Velocity [ft/sec]

Dam

ping

, g .

Mode 1 (RL Method) Mode 2 (RL Method)

180

Figure 5.18 Velocity vs. Damping Plot of the Rectangular Wing Optimized for

Flutter and Divergence

-5

0

5

10

15

20

25

30

35

40

45

0 200 400 600 800 1000

Velocity [ft/sec]

Freq

uenc

y [H

z]Mode 1 (RL Method) Mode 2 (RL Method)

181

5.4 Case Study II

The purpose of this case study is to study the capability of the developed code in the

optimization of representative real-life composite wing structures.

In this case study, the wing box in its original form is altered to have composite skin

panels. Ribs chords and spars caps are also added to the model to represent a real

wing box structure. The skin panels are assumed to be made of Hercules AS4/3502

graphite epoxy and the spars and ribs are assumed to be made of Al-2024

aluminum. The material properties and allowable stresses for these materials are

given in Table 5.5.

As a static strength sizing load case, a total lift force of 4500 [lbf] is obtained using

simple aerodynamic formulations [90]. This load is assumed to be uniformly

distributed along the span. While the front spar is assumed to carry 2/3 of this load,

the remaining load is assumed to be reacted by the rear spar. The resulting nodal

forces as applied to the finite element model are shown in Figure 5.19.

182

Table 5.5 Material Properties and Allowable Stresses for the Modified Rectangular Wing Box

Isotropic Material

(Al-2024 Aluminum) Composite Material

(Hercules AS4/3502 Graphite/epoxy) E = 10.5x106 [psi] E1 = 19x106 [psi]

ν = 0.3 E2 = 1.9x106 [psi]

ρ = 0.1 [lb/in3] G12 = 0.65x106 [psi]

allσ (tension) = 67 [ksi] ν12 = 0.3

allσ (compression) = 57 [ksi] ρ = 0.055 [lb/in3]

allτ =39 [ksi] tply = 0.00525 [in]

cyσ = 39 [ksi] all11σ (tension) = 203.5 [ksi]

all11σ (compression) = 165.1 [ksi]

all22σ (tension) = 11.7 [ksi]

all22σ (compression) = 27.4 [ksi]

all12σ = 10.0 [ksi]

arinterlaminτ = 10.0 [ksi]

183

Figure 5.19 Static Strength Sizing Load Case

For the composite skin panels, the maximum allowable number of plies is limited to

120 and the laminate is assumed to be made of stacks of two contiguous plies that

are laminated symmetrically. Thus the maximum number of stacks of plies in the

laminate is limited to a maximum of 60 and the optimization problem is to

determine the non-empty stacks and their corresponding orientations. Four each of

the stacks in the laminate two design variables are defined. The first design variable

is of discrete type that defines the orientation angle of the stack of plies. This design

variable is selectable from the discrete set {0°, 90°, +45°, -45°}. The second design

variable is an implicit design variable that defines whether the corresponding stack

of plies exists or not. Due to the imposed symmetry condition on the laminate,

optimizing half of the laminate is enough to describe the full laminate. Thus, for

each laminate, a total of sixty design variables thirty of which define the orientation

Loads in[lbf]

184

and the remaining define the existence or absence of the corresponding stacks of

plies are defined to describe the laminate.

Due to computational resources limitations, the upper and lower skin panels are

assumed to be continuous along the span. Thus a total of 120 design variables

describe the upper and lower skins.

The spars webs are assumed to be made of a single piece of sheet metal. Thus, a

total of two discrete design variables describe the webs thicknesses of the front and

rear spars. These thicknesses are selectable from the set {0.1, 0.2, 0.3, 0.4}.

The spars caps are selected to have a “double L” shape. The height of the caps is

fixed at 1.0 [in] and the width is selected as a continuous linear function of the span

location. The thickness of the spars caps is discrete and selectable from the set

{0.04, 0.08, 0.16, 0.24}. Thus, two design variables are necessary to describe each

of the spars caps, resulting in a total of eight design variables to describe the caps of

the front and rear spars.

The ribs are assumed to be formed sheet with integrated chords. The width of the

chord is fixed at 0.5 [in] and the only variable is the rib thickness. The thicknesses

of the three ribs are defined by three separate design variables that are selectable

from the set {0.05, 0.1, 0.15, 0.2}.

The total number of design variables for this case study is thus 133. A problem with

this number of design variables is a challenging large scale optimization problem

that is hard to solve using gradient-based methods and simple enumeration schemes.

The generation size and other relevant optimization parameters are the same as

those selected in the first case study.

185

The convergence history for the best weight in a given generation for this test case

is illustrated in Figure 5.20. Note that the jumps in the objective function values are

again due to the effect of the penalty coefficient adjuster. Convergence is achieved

in generation 154 with a minimum weight of 146.1 [lb]. The sizing summary for the

optimum design is given in Table 5.6. The variation of the spars caps width with

span location is illustrated in Figure 5.21. The static strength analysis results for the

skins, spars, and ribs are given in Tables 5.7-5.9. All of the structural parts are

found to have positive margins of safety in the final design with the margin of

safety being either nearly zero or the corresponding design variable is at lowest

permissible value.

186

Figure 5.20 Convergence History for the Composite Rectangular Wing Weight

(Static Strength Constraints Only)

120.0

130.0

140.0

150.0

160.0

170.0

180.0

190.0

0 20 40 60 80 100 120 140 160

Generation Number

Obj

ectiv

e Fu

nctio

n

187

Table 5.6 Final Design Variables Values for Optimum Design with Static Strength Constraints

Physical Design Variable Design Variable Value

Upper Skin Number of Plies 80

Upper Skin Stacking Sequence [(45°2/-45°2)s/-45°2/90°2/-5°2/

45°2/(0°2/45°2)2/0°4/90°2/0°2/

45°2/0°2/-45°2/45°2]s

Lower Skin Number of Plies 16

Lower Skin Stacking Sequence [90°2/45°2/0°2/90°2]s

Front Spar-Upper Cap Thickness 0.04 [in]

Front Spar-Lower Cap Thickness 0.04 [in]

Front Spar Web Thickness 0.10 [in]

Rear Spar-Upper Cap Thickness 0.08 [in]

Rear Spar-Lower Cap Thickness 0.04 [in]

Rear Spar Web Thickness 0.10 [in]

First Rib Thickness 0.15 [in]

Second Rib Thickness 0.10 [in]

Third Rib Thickness 0.05 [in]

188

Figure 5.21 Spanwise Variation of the Spars Caps Width

(Static Strength Constraints)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0

Span [in]

Spar

Cap

Wid

th [i

n] .

Front Spar-Upper Cap Front Spar-Lower CapRear Spar-Upper Cap Rear Spar-Lower Cap

189

Table 5.7 Summary of Skin Margins of Safety (Satic Strength Constraints)

Skin Panel

Location

M.S Failure Mode

Upper Skin Root-Rib1 0.01 SKIN BUCKLING

Rib1-Rib2 3.98 SKIN BUCKLING


Lower Skin Root-Rib1 0.79 MAX. STRESS-2 (T)


Rib2-Rib3 45.4 MAX. STRESS-2 (T)

Table 5.8 Summary of Spars Margins of Safety (Satic Strength Constraints)

Spar Section

Location

M.S Failure Mode

Front Spar Root-Rib1 0.04 WEB RUPTURE (VM STRESS)

Rib1-Rib2 2.51 WEB RUPTURE (VM STRESS)

Rib2-Rib3 8.98 CRIPPLING

Rear Spar Root-Rib1 0.25 WEB RUPTURE (VM STRESS)



190

Table 5.9 Summary of Spars Margins of Safety (Satic Strength Constraints)

Rib M.S Failure Mode

Rib1 3.50 BUCKLING

Rib2 14.75 ULTIMATE STRESS (SHEAR)

Rib3 7.75 BUCKLING

5.5 Case Study III

The purpose of this case study is to study the advantage of considering the

aeroelastic stability constraints at early stages of the design. Thus, the optimized

wing considered in the second case study is first analyzed to determine it’s

flutter/divergence speeds. Then a 20% increase in the flutter/divergence speed is

imposed on the design and the wing box is optimized to meet aeroelastic

constraints, static strength constraint, and manufacturing constraints simultaneously.


mode shapes of the strength based “optimum” wing. The first six natural

frequencies and mode shapes are determined by using the Lanczos method of

MSC/NASTRAN®. Guyan reduction to only out-of-plane displacements is

performed to eliminate the inplane modes and improve convergence in the flutter

solution. The resulting natural frequencies are given in Table 5.10. The

corresponding mode shapes are similar to those of the initial wing considered in the

first case study and are shown in Figures 5.5-5.10.

191

Table 5.10 Natural Frequencies of the Composite Rectangular Wing (Strenght Based Design)

Mode Number Natural Frequency [Hz]

1 6.4

2 36.1

3 54.3

4 114.8

5 124.4

6 161.1

Aeroelastic stability analysis is then performed for the strength based design using

the root locus method at sea level flight condition. Both flutter and divergence types

of aeroelastic instability are detected in the supersonic regime (M=1.2). The

ZONA51 method of MSC/NASTRAN® is used to calculate the unsteady

aerodynamic forces. The damping and frequency curves are shown in Figures 5.22

and 5.23 respectively. The flutter speed is determined as 1558.9 [ft/sec] with a

corresponding flutter frequency of 37.4 [Hz]. The divergence speed is calculated as

1275.7 [ft/sec].

192

Figure 5.22 Velocity vs. Damping Plot of the Rectangular Wing

(Static Strength Based Design)

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

500 800 1100 1400 1700 2000

Velocity [ft/sec]

Dam

ping

, g .


193


(Static Strength Based Design)

-5

0

5

10

15

20

25

30

35

40

500 800 1100 1400 1700 2000

Velocity [ft/sec]

Freq

uenc

y [H


194

The wing is thus determined to be critical in divergence and a 20% increase in this

speed is imposed on the design. Therefore, the minimum required aeroelastic

instability speed is selected as 1.2×1275.7 = 1531 [ft/sec] and the structure is

optimized to meet both static strength and aeroelastic constraints simultaneously

with the manufacturing constraints imposed on the design variables. The population

size and other relevant optimization parameters are the same as those chosen in the

first case study.

The convergence history for the best weight in a given generation for this test case

is illustrated in Figure 5.24. Convergence is achieved in generation 122 with a

minimum weight of 150.2 [lb]. The sizing summary for the optimum design is

given in Table 5.11. The variation of the spars caps width with span location is

illustrated in Figure 5.25. The static strength analysis results for the skins, spars,

and ribs are given in Tables 5.12-5.14. All of structural parts are found to have

positive margins of safety. Thus non of the static strength constraints is violated. To

verify the “optimum” for the aeroelastic constraints, an aeroelastic stability analysis

is performed for the final design. The damping and frequency curves are shown in

Figures 5.26 and 5.27 respectively. The flutter speed is determined as 1762 [ft/sec]

with a corresponding flutter frequency of 37.7 [Hz]. The divergence speed is

calculated as 1767.5 [ft/sec]. Note that a 38% increase in the aeroelastic instability

speed has been acheived at the cost of 3% icrease in the total strutural weight.

195

Figure 5.24 Convergence History for the Composite Rectangular Wing Weight

(Static Strength and Aeroelastic Constraints)

140.0

145.0

150.0

155.0

160.0

165.0

170.0

175.0

180.0

185.0

190.0

0 20 40 60 80 100 120 140 160

Generation Number

Obj

ectiv

e Fu

nctio

n

196

Table 5.11 Final Design Variables Values for Optimum Design with Static Strength and Aeroelastic Constraints

Physical Design Variable Design Variable Value

Upper Skin Number of Plies 80

Upper Skin Stacking Sequence [(45°2/-45°2)2/-45°2/45°6/-45°4/

90°2/0°4/45°2/90°4/0°2/90°4/0°2]s

Lower Skin Number of Plies 20

Lower Skin Stacking Sequence [45°2/90°2/0°4/-45°2]s

Front Spar-Upper Cap Thickness 0.04 [in]

Front Spar-Lower Cap

Thickness

0.04 [in]

Front Spar Web Thickness 0.10 [in]

Rear Spar-Upper Cap Thickness 0.04 [in]

Rear Spar-Lower Cap Thickness 0.08 [in]

Rear Spar Web Thickness 0.10 [in]

First Rib Thickness 0.10 [in]

Second Rib Thickness 0.10 [in]

Third Rib Thickness 0.05 [in]

197

Figure 5.25 Spanwise Variation of the Spars Caps Width


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

0.0 20.0 40.0 60.0 80.0 100.0 120.0 140.0 160.0 180.0

Span [in]

Spar

Cap

Wid

th [i

n]Front Spar-Upper Cap Front Spar-Lower CapRear Spar-Upper Cap Rear Spar-Lower Cap

198

Table 5.12 Summary of Skin Margins of Safety (Satic Strength and Aeroelastic Constraints)

Skin Panel

Location

M.S Failure Mode

Upper Skin Root-Rib1 0.01 SKIN BUCKLING



Lower Skin Root-Rib1 1.90 MAX. STRESS-2 (T)


Rib2-Rib3 55.8 MAX. STRESS-2 (T)

Table 5.13 Summary of Spars Margins of Safety (Satic Strength and Aeroelastic Constraints)

Spar Section

Location

M.S Failure Mode

Front Spar Root-Rib1 0.45 CRIPPLING



Rear Spar Root-Rib1 0.32 CRIPPLING



199

Table 5.14 Summary of Spars Margins of Safety (Satic Strength and Aeroelastic Constraints)

Rib Number M.S Failure Mode

Rib1 1.96 BUCKLING

Rib2 30.0 BUCKLING

Rib3 9.54 BUCKLING

200



-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

500 800 1100 1400 1700 2000

Velocity [ft/sec]

Dam

ping

, g .


201

Figure 5.27 Velocity vs. Frequency Plot of the Rectangular Wing


-5

0

5

10

15

20

25

30

35

40

500 800 1100 1400 1700 2000

Velocity [ft/sec]

Freq

uenc

y [H


202

5.6 Conclusion

In this chapter the developed multidisciplinary design and optimization code has

been applied to the design of a rectangular wing box. The wing has been considered

in three different test cases.

In the first case study the developed code is verified and the capability of the

genetic algorithm in optimization for aeroelastic constraints with manufacturing

constraints imposed on the design variables is demonstrated.

In the second case study the capability of the developed code in the optimization of

“real-life” composite wing box is demonstrated using a generic composite wing box

model.

In the third case study the advantage of considering aeroelastic stability constraints

at early stages of the design is investigated. In the case study considered it is shown

that a 38% increase in the aeroelastic instability speed is acheivable at the cost of

3% icrease in the total strutural weight of the wing box considered. The capability

of the genetic algorithm in the optimization of composite wing box structures with

static strength, aeroelastic stability, and manufacturing constraints is also

demonstrated.

203

CHAPTER 6

CONCLUSIONS

6.1 General Conclusions

In this thesis an automated multidisciplinary design optimization code was

developed for the minimum weight design of a composite wing box. The

multidisciplinary static strength, aeroelastic stability, and manufacturing

requirements were simultaneously addressed in a global optimization environment

through a genetic search algorithm.

The aim was to obtain a minimum weight final design that complies with the

existing certification requirements (FAR/JAR) in a time, which is less than what is

currently needed, while taking aeroelastic stability constraints into account at the

early stages of the design. Consequently, the need for extensive design

modifications at later stages of the design, that may result in weight penalties, was

eliminated.

The static strength requirements specify obtaining positive margins of safety for all

of the structural parts of the wing box taking into account all potential failure

modes. Besides to classical failure modes (material failure), specialized failure

modes (buckling and crippling) were taken into account in the optimization process.

The aeroelastic stability analysis requirements specify obtaining flutter and

204

divergence free wing box for a range of prescribed flight conditions and with

required damping level in the final design.

The global optimization problem of a wing box in which the design variables are of

mixed-discrete type and the static strength and aeroelastic stability constraints are

considered simultaneously has never been attempted in previous studies. Very

simple models that rely on using direct stress output of coarse mesh finite element

models had been used and they did not account for specialized failure modes that

should be considered in the design cycle. In chapter 2 of this thesis, it was shown

that this approach would result in erroneous stress estimates.

Typical wing boxes exhibit thin metal/composite panels joined together to form the

structure. Since these panels are very thin, they usually buckle at very low stress

amplitudes causing redistribution of the load in the structure. Nonlinear effects that

result from load redistribution in the structure should be taken into account to insure

failure free structure. A detailed linear finite element analysis of the structure would

not account for such nonlinear effects. This effect can only be simulated using

nonlinear finite element analysis with fine mesh models. In this thesis, the

aerospace industry approach to this problem was used to circumvent this problem.

The approach relied on constructing coarse mesh finite element models to

determine the internal load distribution in the structure and then using simplified

theories, like the modified engineering bending theory, to determine the stresses and

simulate the correct behavior of the structure after buckling occurs.

Buckling analysis of composite plates is usually based on the specially orthotropic

plate assumptions and the use of interaction equations. It was shown in chapter 2

that the buckling analysis based on the specially orthotropic plate assumptions is not

a generally valid approach and can only be justified if the laminate is cross-ply

symmetric one. On the other hand using energy methods such as the Rayleigh-Ritz

method would result in accurate and acceptable results at low cost when compared

205

to the finite element method. It also eliminates the need to prepare specialized finite

element models for buckling analysis purposes. Another important result is related

to the buckling analysis of anisotropic laminates. For anisotropic laminates, it was

shown that care should be taken in the correct sign for the applied shear stress,

otherwise positive and negative shear loads would result in completely two different

results for the allowable buckling load.

Flutter and divergence are the most important aeroelastic instability phenomena

since the occurrence of any of them would lead to catastrophic structural failure and

loss of the aircraft. Obtaining valid damping history is generally required for the

certification of a new developed aircraft and it is also needed to prevent loss of the

aircraft or wind tunnel test-model in a flutter clearance test.

Aeroelastic stability analysis to determine the onset of flutter and divergence can be

performed relatively easily in the frequency domain using either k-method or the

pk-method. However, these methods produce damping information that is either

invalid (k-method) or approximate (pk-method).

On the contrary, the root locus method results in damping information that is valid

for all of the speed range of interest and provides better insight into the physical

phenomena leading to aeroelastic instability. The computational cost problem

associated with the calculation of the unsteady aerodynamic forces in the Laplace

domain is circumvented efficiently through the use of the Rogers rational function

approximations.

In the practical design of a composite wing box, the design variables are not all

continuous and some of them must be selected from a set of integer or discrete

values. The structural members may have to be chosen from standard sizes and

member thicknesses may have to be selected from commercially available ones.

Stacking sequence design of composite plates involves the determination of the

206

number of plies and their orientations. The stacking sequence design problem is

discrete in nature. Due to manufacturing limitations, the plies are fabricated at

certain thicknesses and the orientations are limited to certain sets of discrete angles.

Thus, the optimization problem is a nonlinear optimization problem that involves a

combination of continuous and discrete design variables. Commercial softwares

utilize gradient-based methods that can not treat this type of optimization problems

efficiently and may produce suboptimal or infeasible designs. In this thesis, the

problem was solved efficiently by using a genetic algorithm based optimizer.

The developed code was applied to the design of composite rectangular wing box

with metallic internal substructure. Hence the two spars and number of ribs are in

the form of conventional aluminum construction. The skin of the wing was taken as

composite. The wing box was considered in three different test cases. In the first

case study the developed code was verified and the capability of the genetic

algorithm in optimization for aeroelastic constraints with manufacturing constraints

imposed on the design variables was demonstrated. In the second case study the

capability of the developed code in the optimization of “real-life” composite wing

box was demonstrated. In the third case study the advantage of considering

aeroelastic stability constraints at early stages of the design was investigated. It was

shown that a 38% increase in the aeroelastic instability speed is acheivable at the

cost of 3% increase in the total structural weight of the wing box considered. The

capability of the genetic algorithm in the optimization of composite wing box

structures with static strength, aeroelastic stability, and manufacturing constraints

was demonstrated.

207

6.2 Recommendations for Future Work

In this thesis the global optimization problem was solved by using the genetic

algorithm. Although no attempt was done to tune the genetic algorithm to suite the

optimization problem under consideration, the computational cost involved was

found to be high. Nevertheless, a hybrid optimization scheme that uses the genetic

algorithm to locate the global minimum in the design space using the first few

generations and then switches to a conventional nonlinear programming approach

may be investigated.

The constraints were handled using the conventional penalty function approach.

The use of gene repair strategies and the K-S function approach in handling the

constraints may be investigated.

The developed code may be modified to analyze an all composite wing box.

208

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218

VITA

The author was born in Tubas (Palestine) on February 1970. He received his B.Sc.

degree in Aeronautical Engineering from the Middle East Technical University

(METU) in 1993 and his M.Sc. in Mechanical Engineering from METU in 1996.

He started working as a design engineer at the Turkish Aerospace Industries (TAI)

company in 1998 and is currently a structural analysis specialist at the same

company. His main areas of interest include stress analysis, structural dynamics,

and aeroelasticity.

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