efficient methods for structural …...ii efficient methods for structural analysis of built-up...

EFFICIENT METHODS FOR STRUCTURAL ANALYSIS OF BUILT-UP WINGS

by

Youhua Liu

Dissertation Submitted to the Faculty of Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

In

Aerospace Engineering

Approved:

Rakesh K. Kapania, Chairman

Romesh C. Batra Zafer Gürdal

Eric R. Johnson Efstratios Nikolaidis

April 2000Blacksburg, Virginia

Keywords: Built-Up Wing, Structural Analysis, Continuum Model, Equivalent Plate Model,Mindlin-Plate Theory, Ritz-Method, Neural Network, Sensitivity

Copyright ¶ 2000, Youhua Liu

ii

Efficient Methods for Structural Analysis of Built-Up Wings

by

Youhua Liu

Committee Chairman: Rakesh K. Kapania

Aerospace and Ocean Engineering

(ABSTRACT)

The aerospace industry is increasingly coming to the conclusion that physics-based high-

fidelity models need to be used as early as possible in the design of its products. At the preliminary

design stage of wing structures, though highly desirable for its high accuracy, a detailed finite

element analysis(FEA) is often not feasible due to the prohibitive preparation time for the FE

model data and high computation cost caused by large degrees of freedom. In view of this situation,

often equivalent beam models are used for the purpose of obtaining global solutions. However, for

wings with low aspect ratio, the use of equivalent beam models is questionable, and using an

equivalent plate model would be more promising.

An efficient method, Equivalent Plate Analysis or simply EPA, using an equivalent plate

model, is developed in the present work for studying the static and free-vibration problems of built-

up wing structures composed of skins, spars, and ribs. The model includes the transverse shear

effects by treating the built-up wing as a plate following the Reissner-Mindlin theory (FSDT). The

Ritz method is used with the Legendre polynomials being employed as the trial functions.

Formulations are such that there is no limitation on the wing thickness distribution. This method is

evaluated, by comparing the results with those obtained using MSC/NASTRAN, for a set of

examples including both static and dynamic problems.

iii

The Equivalent Plate Analysis (EPA) as explained above is also used as a basis for generating

other efficient methods for the early design stage of wing structures, such that they can be

incorporated with optimization tools into the process of searching for an optimal design. In the

search for an optimal design, it is essential to assess the structural responses quickly at any design

space point. For such purpose, the FEA or even the above EPA, which establishes the stiffness and

mass matrices by integrating contributions spar by spar, rib by rib, are not efficient enough.

One approach is to use the Artificial Neural Network (ANN), or simply called Neural Network

(NN) as a means of simulating the structural responses of wings. Upon an investigation of

applications of NN in structural engineering, methods of using NN for the present purpose are

explored in two directions, i.e. the direct application and the indirect application. The direct method

uses FEA or EPA generated results directly as the output. In the indirect method, the wing inner-

structure is combined with the skins to form an "equivalent" material. The constitutive matrix,

which relates the stress vector to the strain vector, and the density of the equivalent material are

obtained by enforcing mass and stiffness matrix equities with regard to the EPA in a least-square

sense. Neural networks for these material properties are trained in terms of the design variables of

the wing structure. It is shown that this EPA with indirect application of Neural Networks, or

simply called an Equivalent Skin Analysis (ESA) of the wing structure, is more efficient than the

EPA and still fairly good results can be obtained.

Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used

in structural design practices for searching the optimal solutions near a baseline design. In the

present work, the modal response of general trapezoidal wing structures is approximated using

shape sensitivities up to the second order, and the use of second order sensitivities proved to be

yielding much better results than the case where only first order sensitivities are used. Also

different approaches of computing the derivatives are investigated. In a design space with a lot of

design points, when sensitivities at each design point are obtained, it is shown that the global

variation in the design space can be readily given based on these sensitivities.

v

Acknowledgments

This work would not have been accomplished without the support and guidance of my advisor

and committee chairman, Dr. Rakesh K. Kapania. Dr. Kapania's professional attitude influenced me

a lot, and his prompt responses to my questions and submitted work, encouragement during all

phases of my work, and his understanding are greatly appreciated. I am grateful to Dr. Romesh C.

Batra, Dr. Zafer Gürdal, Dr. Eric R. Johnson, and Dr. Efstratios Nikolaidis for serving in my

committee. I would like to thank the financial support of NASA Langley Research Center on this

research through Grant NAG-1-1884 with Dr. Jerry Housner and Dr. John Wang as the Technical

Monitors. I am also thankful to other students for the helps I have received, especially Dr. Daniel

Hammerand, Dr. Luohui Long, and Mr. Erwin Sulaeman.

Finally, I would say this work could not have got started, let alone been finished, without the

unconditional support, trust and love of my wife, Ting, and my daughter, Lisa. I owe them a lot.

vi

Contents

List of Tables x

List of Figures xi

Nomenclature xvi

1. Introduction 1

1.1 The Trend of Early Analysis in Product Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 History and Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 Applications in Structural Engineering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Plate Theories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 Sensitivity Techniques. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8

1.6 Scope of the Present Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2. Neural Networks and Its Applications 11

2.1 Two Important Types of NN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

2.1.1 Feed-Forward Multi-Layer Neural Network. . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.2 Radial Basis Function Neural Network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Features of ANN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Algorithms in the MATLAB Neural Network Toolbox. . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Ways of Application of Neural Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

2.4.1 Direct Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vii

2.4.2 Indirect Application. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3. Continuum Model Approaches 21

3.1 Methods of Obtaining Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 An Example of NN Modeling of Continuum Models. . . . . . . . . . . . . . . . . . . . . . . . . .23

3.2.1 Neural Network with 2 Input Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24



4. An Approach for the Solution of Mindlin Plates 32

4.1 Assumptions and Formulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

4.2 Strain Energy and Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.3 Kinetic Energy and Mass Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .40

5. Equivalent Plate Analysis of Built-Up Wing Structures 42

5.1 Numerical Integration of Stiffness and Mass Matrices. . . . . . . . . . . . . . . . . . . . . . . . .42

5.1.1 Skins. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43

5.1.2 Spars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

5.1.3 Ribs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

5.2 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

5.3 Formulation for Vibration Problem of Wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50

5.4 Convergence Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.5 Static Problem Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.6 Results and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6.1 Free Vibration Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.6.1.1 A Trapezoidal Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

5.6.1.2 A Trapezoidal Shell with a Camber. . . . . . . . . . . . . . . . . . . . . . . . .57

5.6.1.3 A Solid Wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .59

5.6.1.4 A Built-up Wing Composed of Skins, Spars and Ribs. . . . . . . . . . 61

5.6.1.5 A Box Wing used as a test case in Livne. . . . . . . . . . . . . . . . . . . . .64

5.6.2 Displacement under Static Loads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

viii

5.6.2.1 Tip Point Force. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6.2.2 A Force Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.6.2.3 Tip Torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

5.6.2.4 The Box Wing in Livne. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

5.6.3 Skin Stress Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .69

5.6.4 On Efficiency of EPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73

6. Modal Response Using Sensitivity Technique

and Direct Application of Neural Networks 75

6.1 Shape Sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

6.2 An Issue in Equivalent Plate Analysis (EPA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .77

6.3 Approaches to Sensitivity Evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.4 Application of Sensitivity Technique (ST) in Multi-variable Optimization. . . . . . . . 80

6.5 Application of Neural Networks (NN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.6 Examples and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82

6.6.1 Results on sensitivity evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

6.6.2 Application of Sensitivity Technique (ST) and Neural Networks (NN) . . . .89

7. Equivalent Skin Analysis Using Neural Networks 95

7.1 Equivalent Skin Analysis (ESA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

7.1.1 The Constitutive matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7.1.2 Mass distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

7.2 Examples and Discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7.2.1 Results at a design point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .99

7.2.2 Three-variable case: design space I. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.2.3 Four-variable case: design space II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .115

7.2.4 Six-variable case: design space III. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .126

7.2.5 Design space IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.3 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .148

8. Conclusions and Future Work 149

ix

8.1 Conclusions of the Present Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.2 Recommendations for Future Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .151

References 153

Appendix A The Constitutive Matrix for Various Cases 160

A.1 Rotation along z-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .160

A.2 Rotation along y -axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

A.3 Skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .161

A.4 Spar and Rib Cap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.5 Spar and Rib Web. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163

Appendix B Formulation for Multi-Plane Problem Using EPA 164

B.1 Strain Energy and Stiffness Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .165

B.2 Kinetic Energy and Mass Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

Appendix C Airfoil Sections Generated with Karman-Trefftz Transformation 167

Vita 171

x

List of Tables

Table 3.1 Comparison of Continuum Model Properties for a Lattice Repeating Cell. . . . . . . . . . 31

Table 3.2 Comparison of Continuum Model Properties for a Lattice Repeating Cell. . . . . . . . . . 31

Table 5.1 Natural frequencies (Hz) of the cantilevered swept-back box wing. . . . . . . . . . . . . . . . 64

Table 5.2 Displacement (in) of the cantilevered swept-back box wing. . . . . . . . . . . . . . . . . . . . . 69

Table 5.3 Comparison of FEA and EPA in terms of DOF and Number of Elements. . . . . . . . . . .74

Table 7.1 Differences between the natural frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . 101

Table 7.2 Natural frequencies (rad/sec) of the wing in Fig. 7.20. . . . . . . . . . . . . . . . . . . . . . . . .148

Table 7.3 Natural frequencies (rad/sec) of the wing in Fig. 7.29. . . . . . . . . . . . . . . . . . . . . . . . .148

xi

List of Figures

Fig. 2.1 A feed-forward multi-layer neural network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Fig. 2.2 Details of a neuron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13

Fig. 2.3 Transfer functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14

Fig. 2.4 Radial basis function neural network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

Fig. 3.1 Geometry of repeating cells of a single-bay double laced lattice structure. . . . . . . . . . . . 22

Fig. 3.2 Evaluating continuum model properties for a repeating cell. . . . . . . . . . . . . . . . . . . . . . .24

Fig. 3.3 Training data for )( , cc LAfGA= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 3.4 Distributions of training and testing points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Fig. 3.5 Feed-forward NN simulation for )( , cc LAfGA= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 3.6 Feed-forward NN simulation errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Fig. 3.7 Radial-basis function NN simulation for )( , cc LAfGA= . . . . . . . . . . . . . . . . . . . . . . . . . .28

Fig. 3.8 Radial-basis function NN simulation errors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28

Fig. 3.9 Training history of a 3-10-1 feed-forward NN by trainbp. . . . . . . . . . . . . . . . . . . . . . . . .30

Fig. 3.10 Training history of a 3-10-1 feed-forward NN by trainbpa. . . . . . . . . . . . . . . . . . . . . . .30

Fig. 3.11 Training history of a 3-10-1 feed-forward NN by trainlm. . . . . . . . . . . . . . . . . . . . . . . .31

Fig. 4.1 The coordinate system and its transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Fig. 4.2 The Legendre polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36

Fig. 4.3 The Chebyshev polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Fig. 5.1 Wing skin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44

xii

Fig. 5.2 Sketches for wing spar and rib. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45

Fig. 5.3 The first 10 natural frequencies of wing I as functions of boundary-condition-

simulating spring value, when 6 terms of Legendre polynomials are used. . . . . . . . . . . . . .48


simulating spring value, when 8 terms of Legendre polynomials are used. . . . . . . . . . . . . .49

Fig. 5.5 Natural frequencies of wing I with regard to number of trial function terms. . . . . . . . . . 52

Fig. 5.6 Natural frequencies of wing II with regard to number of trial function terms. . . . . . . . . .53

Fig. 5.7 Mode Shapes and Natural Frequency f )/( srad for a Trapezoidal Plate. . . . . . . . . . . .56

Fig. 5.8 Mode Shapes and Natural Frequency f )/( srad for Wing-Shaped Shell

with a Camber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Fig. 5.9 Mode Shapes and Natural Frequency f )/( srad for the Solid Wing. . . . . . . . . . . . . . .60

Fig. 5.10 Wing cross-sections at rib positions and spar positions. . . . . . . . . . . . . . . . . . . . . . . . . .62

Fig. 5.11 Mode Shapes and Natural Frequency f )/( srad for a Built-up Wing

Composed of Skins, Spars and Ribs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Fig. 5.12 A box wing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Fig. 5.13 Comparison of Displacements for Load Case of Tip Point Force. . . . . . . . . . . . . . . . . .66

Fig. 5.14 Comparison of Displacements for Load Case of a Force Distribution. . . . . . . . . . . . . . 67

Fig. 5.15 Comparison of Displacements for Load Case of Tip Torque. . . . . . . . . . . . . . . . . . . . . .68

Fig. 5.16 Comparison of Von Mises Stress on the Upper and Lower Skins

of a Wing under a Point Force at the Wing Tip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .70

Fig. 5.17 Distribution of Von Mises Stress on the Upper Skin


Fig. 5.18 Distribution of Von Mises Stress on the Lower Skin


Fig. 6.1 Plan configuration of a trapezoidal wing: .,),( 221 baAsbasA ==+= τα . . . . . . . . . .76

Fig. 6.2 Natural frequencies using equivalent plate analysis with mode tracking. . . . . . . . . . . . .84

Fig. 6.3 Effect of the finite difference step size on the sensitivities

of various natural frequencies to taper ratio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xiii

Fig. 6.4 The 2nd natural frequency w.r.t. wing plan area

using 1st and 2nd order sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Fig. 6.5 The 3rd natural frequency w.r.t. wing sweep angle

using 1st and 2nd order sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Fig. 6.6 Comparison of the natural frequencies of the first 6 modes for wing structures

randomly chosen inside the box of design space, as obtained by the NN and ST

w.r.t. those obtained using a full-fledged EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92


along a path inside the box of design space (n1=0.945, n2 =8.200, n3=3.203) using

only the 1st order sensitivities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


along a path inside the box of design space (n1=0.945, n2 =8.200, n3=3.203) using

sensitivities up to the 2nd order. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Fig. 7.1 An example of mass density distribution generated using Eq. (7.8) . . . . . . . . . . . . . . . .101

Fig. 7.2 The stiffness matrix given by EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .102

Fig. 7.3 The stiffness matrix given by ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103

Fig. 7.4 Difference between stiffness matrices given by EPA and ESA. . . . . . . . . . . . . . . . . . . .104

Fig. 7.5 The mass matrix given by EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Fig. 7.6 The mass matrix given by ESA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Fig. 7.7 Difference between mass matrices given by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . 107

Fig. 7.8 49 randomly chosen wing plan forms in design space I. . . . . . . . . . . . . . . . . . . . . . . . . .110

Fig. 7.9 Comparison of the first 10 frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . . .111

Fig. 7.10 The relative errors in Fig. 7.9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112

Fig. 7.11 25 wing plan forms systematically varying through design space I. . . . . . . . . . . . . . . .113


Fig. 7.13 25 randomly chosen wing plan forms in design space II. . . . . . . . . . . . . . . . . . . . . . . .117

Fig. 7.14 Comparison of the first 10 frequencies by EPA and ESA. . . . . . . . . . . . . . . . . . . . . . .118

Fig. 7.15 The relative errors in Fig. 7.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119

xiv

Fig. 7.16 16 wing plan forms systematically varying through design space II. . . . . . . . . . . . . . .120


Fig. 7.18 An arbitrarily chosen wing plan form in design space II. . . . . . . . . . . . . . . . . . . . . . . .122

Fig. 7.19 Comparison of displacements by EPA and ESA for 1lb tip force . . . . . . . . . . . . . . . .123

Fig. 7.20 Comparison of the Von Mises stress at wing root by EPA and ESA

under 1lb tip force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .124

Fig. 7.21 Comparison of the Von Mises stress along central spar by EPA and ESA


Fig. 7.22 25 randomly chosen wing plan forms in design space III. . . . . . . . . . . . . . . . . . . . . . . 128



Fig. 7.25 16 wing plan forms systematically varying through design space III. . . . . . . . . . . . . . 131


Fig. 7.27 An arbitrarily chosen wing plan form in design space III. . . . . . . . . . . . . . . . . . . . . . . 133

Fig. 7.28 Comparison of displacements by EPA and ESA at 1lb tip force . . . . . . . . . . . . . . . . .134





Fig. 7.31 16 randomly chosen wing designs in design space IV. . . . . . . . . . . . . . . . . . . . . . . . . .139



Fig. 7.34 16 wing designs systematically varying through design space IV. . . . . . . . . . . . . . . . .142


Fig. 7.36 An arbitrarily chosen wing design in design space IV. . . . . . . . . . . . . . . . . . . . . . . . . .144

Fig. 7.37 Comparison of displacements by EPA and ESA at 1lb tip force . . . . . . . . . . . . . . . . .145



xv



Fig. B.1 Sketch for a wing composed of main-body and wing-let. . . . . . . . . . . . . . . . . . . . . . . . 164

Fig. C.1 The Karman-Trefftz transformation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

Fig. C.2 Airfoils shapes obtained using Karman-Trefftz transformation. . . . . . . . . . . . . . . . . . . 170

xvi

Nomenclature

a

A

gdc AAA ,,

ANN

b

{ B}

c

rcc ,0

1c

[C]

jib

b1,b2,b3

[D]

}{d

DOF

pqD

E

EA

EI

chord-length at wing tip

wing plan area

area of longitudinal bars, diagonal bars, and battens of a repeating cell

Artificial Neural Network

chord-length at wing root

Ritz base function vector defined in Eq. (4.12)

chord-length

chord-length at root

chord-length at tip

matrix defined in Eq. (4.20)

bias (threshold) of the i -th neural in the j -th layer

bias (threshold) vectors

constitutive matrix

displacement vector

number of Degrees Of Freedom

p -th row, q -th column term of constitutive matrix

Young’s Modulus

axial rigidity

bending rigidity

xvii

EPA

ESA

)(•f

FEA

FEM

FF

FSDT

GA

[G]

[H]

2,1h

i, j

I,J,K,L,M,N,P,Q,R,S

initff

[J]

22211211 ,,, JJJJ

J

k

[K]

]~

[K

L

gdc LLL ,,

'logsig'

2,1l

[M]

Equivalent Plate Analysis

Equivalent Skin Analysis

transfer function

Finite Element Analysis

Finite Element Method

feed-forward

First-order Shear Deformation Theory

shear rigidity



spar, rib cap height

integers

integers

MATLAB NN Toolbox feed-forward network initialization program

Jacobian matrix

terms of the inverse of Jacobian matrix

determinant of Jacobian matrix

integer

stiffness matrix based on {q}

stiffness matrix simulated by continuum model

Lagrangian, defined in Eq. (5.14)

length of longitudinal bars, diagonal bars, and battens of a repeating cell

Sigmoid transfer function

spar, rib cap width

mass matrix based on }{•q

xviii

]~

[M

MAC

m, n

N

NN

zN

n1, n2

),(4~1 ηξN

ribn

sparn

p

{ P}

)(xPi

p, q

'purelin'

zyx PPP ,,

}{q

jir

RBF

s

simuff

simurb

solverb

ST

t

mass matrix simulated by continuum model

modal assurance criterion

integers

dimension of [K] and [M], 25k=

Neural Network

number of integration zones in z-direction

number of neurons in the 1st and 2nd hidden layer

transformation functions

number of ribs

number of spars

input training data matrix

generalized load vector defined in Eq. (5.22)

Legendre polynomials

integers

linear transfer function

force components

generalized displacement vector defined in Eq. (4.11)

input of the i -th neural in the j -th layer

Radial Basis Function

length of semi-span of wing

MATLAB NN Toolbox FF network simulation program

MATLAB NN Toolbox RBF network simulation program

MATLAB NN Toolbox RBF network training program

sensitivity techniques

output training data matrix; time

xix

0t

2,1t

T

[T]

'tansig'

trainbp

trainbpa

trainlm

)(xTi

{ x}

x,y,z

jix

4~1x

4~1y

[ZZ]

U

u,v,w

000 ,, wvu

V

}{v

1−jkiw

Mij

Kij ww ,

w1,w2,w3

)( piw

skin thickness

spar, rib thickness

kinetic energy


hyperbolic tangent sigmoid transfer function

MATLAB NN Toolbox FF network training program with back-propagation

MATLAB NN Toolbox FF network training program with back-propagation andadaptive learning

MATLAB NN Toolbox FF network training program with Levenberg-MarquardtAlgorithm

Chebyshev polynomials

eigenvector

Cartesian coordinates

output of the i -th neural in the j -th layer

x-coodinates at quadrilateral wing plan corners

y-coodinates at quadrilateral wing plan corners


strain energy

displacements in x,y,z directions

displacements in x,y,z directions at plane 0=z

integration domain for a structure

velocity vector

weight between node k of the )1( −j -th layer and node i of the j -th layer

weight coefficients defined in Eq. (7.9)

weight matrices of the 1st, 2nd, and 3rd layer

weight coefficient defined in Eq. (6.15)

xx

w.r.t.

α

,,, zyx ααα

yx φφ αα ,

}{ε

}{ε

,,, zyx εεε

zxyzxy εεε ,,

φ

}{ iφ

}{ jφ

xφ

yφ

)(ξη r

λ

Λ

ν

θ

ρ

}{σ

τ

ω

ηξ ,

)(ηξ s

with regard to

wing aspect ratio

linear spring coefficients

strain vector

vector defined in Eq. (4.18)

strain tensors

bending angle in x direction

the i -th eigenvector for baseline design

the j -th eigenvector for perturbed design

rotation about the y direction

rotation about the x− direction

rib position function

eigenvalue

wing sweep angle at leading-edge

Poisson's ratio

shear angle in x direction

mass density; shape variable

stress vector

the taper ratio

frequency, rad/sec

transformed plane variables

spar position function

1

Chapter 1

Introduction

1.1 The Trend of Early Analysis in Product Design

To reduce product development cycle is essential to a nowadays manufacturing enterprise not

only on economic savings in the process itself, but also to a broad business advantage in getting

product innovations to customers faster, and thereby increasing the company's market share1 .

One of the most valuable CAE (Computer Aided Engineering) tools is finite-element analysis

(FEA), which assists in analyzing structures to detect areas that might undergo excessive stress,

deformation, vibration, or other potential problems. Yet, instead of assisting in reducing time to

market, the traditional, full-blown FEA actually became a bottleneck and was often done only

toward the end of product design.

The experience of manufacturers in many industries has shown that 85~90% of the total time

and cost of product development is defined in the early stages of product development, when only

5~10% of project time and cost have been expended2,1 . This is because in the early concept stages,

fundamental decisions are made regarding basic geometry, materials, system configuration, and

manufacturing processes.

The process, however, can be re-oriented so that analysis is performed much earlier to shorten

the product development cycle. This moves CAE/analysis forward into conceptual design, where

changes are much easier and more economical to make in correcting poor designs earlier. The

CHAPTER 1 INTRODUCTION 2

major benefits of up-front analysis includes giving designers the ability to perform "what-if?"

simulations that enable them to evaluate alternative approaches and explore options early in the

design cycle to arrive at a superior design. This methodology employs CAE to help avoid "fires" in

the early design stage, rather than uses CAE to put out "fires" in the later design stage as the

traditional practice does1 .

Therefore, instead of being the last thing to do, CAE is now one of the first things for a

designer to do to make sure that the best design possible is to be obtained3 .To facilitate this

methodology of early analysis in product design, there have emerged the following two issues

concerning the development of CAE.

The first issue is the lack of integration between CAD and analysis programs. The need to

translate, clean up, and further process design data for use in analysis has limited the effectiveness

of both CAD and CAE software. Over the past few years, software vendors have been moving to

tightly couple CAD and CAE software programs by tying them into suites using a shared database

and a single user interface. Sharing database means that engineers no longer have to translate

design data to formats that the analysis program can recognize, and vice versa. It also allows

updates in one system to be reflected immediately in the other. CAD and CAE sharing the same

user interface makes it easier for a user to switch from one program to the other.

The second issue is the inappropriateness of FEA as the tool of CAE in many cases. Usually

FEA can only be integrated in the early design stage of structurally simple products or components

of a structurally complex product. For instance, at the preliminary design stage of built-up wing

structures, though highly desirable for its high accuracy, a detailed finite element analysis(FEA) is

often not feasible because: (i) the preparation time for the FEM model data may be prohibitive,

especially when there is little carry-over from design to design; (ii) for complex structures

composed of large number of components, a detailed FEA involves huge number of degrees of

freedom, and needs large amount of CPU time and computation capacity, which makes the cost too

high. For such cases, unconventional methods that are more efficient than FEA are needed.


People have employed continuum models, assuming the complex structures to behave

similarly, for analysis at the early stage of the design process of a complex product. This includes

using beam, plate or shell models to simulate complex structures. In the present work,

methodologies are developed in employing the first-order shear deformation theory (the Mindlin

plate) to simulate the structural responses of built-up wing structures, incorporating neural

networks and other tools to further enhance analysis efficiency. It is hoped that the methodologies

developed in the present work can be used in the early design stages of aerospace wings and other

plate-like complex structures, therefore a superior design can be obtained in a development process

of shorter cycle and less expenses.

1.2 Neural Networks

1.2.1 History and Concepts

The working mechanism in brains of biological creatures has long been an area of intense

study. It was found around the first decade of the 20-th century that neurons (nerve cells) are the

structural constituents of the brain. The neurons interact with each other through synapses, and are

connected by axons (transmitting lines) and dentrites (receiving branches). It is estimated that there

are on the order of 10 billion neurons in the human cortex, and about 60 trillion synapses4 .

Although neurons are 5~6 orders of magnitude slower than silicon logic gates, the organization of

them is such that the brain has the capability of performing certain tasks (for example, pattern

recognition, and motor control etc.) much faster than the fastest digital computer nowadays.

Besides, the energetic efficiency of the brain is about 10 orders of magnitude lower than the best

computer today. So it can be said, in the sense that a computer is an information-processing system,

the brain is a highly complex, nonlinear, and efficient parallel computer.

Artificial Neural Networks (ANN), or simply Neural Networks (NN) are computational

systems inspired by the biological brain in their structure, data processing and restoring method,

and learning ability. More specifically, a neural network is defined as a massively parallel


distributed processor that has a natural propensity for storing experiential knowledge and making it

available for future use by resembling the brain in two aspects: (a) Knowledge is acquired by the

network through a learning process; (b) Inter-neuron connection strengths known as synaptic

weights (or simply weights) are used to store the knowledge4 .

With a history traced to the early 1940s, and two periods of major increases in research

activities in the early 1960s and after the mid-1980s, ANNs have now evolved to be a mature

branch in the computational science and engineering with a large number of publications, a lot of

quite different methods and algorithms and many commercial software and some hardware. They

have found numerous applications in science and engineering, from biological and medical

sciences, to information technologies such as artificial intelligence, pattern recognition, signal

processing and control, and to engineering areas as civil and structural engineering.

1.2.2 Applications in Structural Engineering

In the field of structural engineering, there have been a lot of attempts and researches making

use of NN to improve efficiency or to capture relations in complex analysis or design problems.

The following are a few examples. Abdalla and Stavroulakis5 applied NN to represent

experimental data to model the behavior of semi-rigid steel structure connections, which are related

to some highly nonlinear effects such as local plastification etc. Several cases of neural network

application in structural engineering can be found in Vanluchene and Sun6 . All the problems

treated in Ref. 6 had been reproduced in Gunaratnam and Gero7 with a conclusion that

representational change of a problem based on dimensional analysis and domain knowledge can

improve the performance of the networks. There is a summary of applications of NN in structural

engineering in Ref. 8. In Liu, Kapania and VanLandingham9, methodologies of applying Neural

Networks and Genetic Algorithms to simulate and synthesize substructures were explored in the

solution of 1-D and 2-D beam problems.


1.3 Continuum Models

As has been indicated in 1.1, it is estimated that about 90% of the cost of an aerospace product

is committed during the first 10% of the design cycle2 . As a result, the aerospace industry is

increasingly coming to the conclusion that physics-based high fidelity models (Finite Element

Analysis for structures, Computational Fluid Dynamics for aerodynamic loads etc.) need to be used

earlier at the conceptual design stage, not only at a subsequent preliminary design stage. But an

obstacle to using the high fidelity models at the conceptual level is the high CPU time that are

typically needed for these models, despite the enormous progress that has been made in both the

computer hardware and software.

In view of this situation, often equivalent continuum models are used to simulate complex

structures for the purpose of obtaining global solutions in the early design stages. This idea is

reasonable as long as the complex structure behaves physically in a close manner to the continuum

model used and only global quantities of the response are of concern. During the late seventies and

early eighties, there was a significant interest in obtaining continuum models to represent discreet

built-up complex lattice, wing, and laminated wing structures. These models use very few

parameters to express the original structure geometry and layout. The initial model generation and

set-up is fast as compared to a full finite element model. Assembly of stiffness and mass matrices

and solution times for static deformation and stresses or natural modes are significantly less than

those needed in a finite element analysis. All these make continuum models very attractive for

preliminary design and optimization studies.

Despite its great potential, however, the continuum approach has gained a limited popularity in

the aerospace designers community. This might be due to the fact that, all the developments have

been made by keeping specific examples (e.g. periodic lattices or specific wings) in mind. Also,

with some exceptions, most of these approaches were rather complex. The key obstacle, though,

appears to be the fact that if the designer makes a change in the actual built-up structures, the

continuum model has to be determined from scratch.


The complex nature of the various methods and the large number of problems encountered in

determining the equivalent models are not surprising given the fact that determining these models

for a given complex structure (a large space structure or a wing) belongs to a class of problems

called inverse problems. These problems are inherently ill-posed and it is fruitless to attempt to

determine unique continuum models. The present work deals with investigating the possibility that

a more rational and efficient approach of determining the continuum models can be achieved by

using artificial neural networks.

The following are examples of work on using beam or plate models to simulate repetitive

lattice structures: Noor, Anderson, and Greene10; Nayfeh, and Hefzy11; Sun, Kim, and

Bogdanoff12; Noor13; Lee 16~14 . Specifics of these methods will be discussed in Chapter 3.

In the area of analyzing aerospace wing structures, a number of studies have been conducted on

using equivalent beam models to represent simple box-wings composed of laminated or anisotropic

materials, which include Kapania and Castel17, Song and Librescu18, and Lee19. They have given

some fine results for the specific problems. However, for wings with low aspect ratio, the use of

equivalent beam models is questionable, and using an equivalent plate model would be more

promising.

1.4 Plate Theories

There exists a considerable body of work on the static or dynamic behaviors of all kinds of

plates. A thorough description of literature on the study of plates was given by Lovejoy and

Kapania 21,20 , where more than 300 references has been listed about all plates. The plates studied

include thin, thick, laminated or composite, whose geometry can be rectangular, skew, or

trapezoidal, and the lamina can be of similar or dissimilar material and isotropic, orthotropic, or

anisotropic in nature

One way of classifying existing methods for the solution of plates is according to the

deformation theory used, namely: the Classical Plate Theory (CPT), the First-order Shear


Deformation Theory (FSDT), or the Higher-order Shear Deformation Theory (HSDT) etc. The

CPT is based on the Kirchhoff-Love hypothesis, that is, a straight line normal to the plate middle

surface remains straight and normal during the deformation process. This group of theories work

well for truly thin isotropic plates, but for thick isotropic plates and for thin laminated plates they

tend to overestimate the stiffness of the plate since the effects of through-the-thickness shear

deformation are ignored23,22 . The FSDT is based on the Reissner-Mindlin model25,24 , where the

constraint that a normal to the mid-surface remains normal to the mid-surface after deformation is

relaxed, and a uniform transverse shear strain is allowed. The FSDT is the most widely used theory

for thick and anisotropic laminated plates owing to its simplicity and its low requirement for

computation capacity. For more accurate results or more realistic local distributions of the

transverse strain and stress, one should use the HSDT26 , or the CFSDT (Consistent First-order

Shear Deformation Theory) proposed by Knight and Qi27 .

Methods of solving the CPT, FSDT or HSDT mainly include finite element, Galerkin, and

Rayleigh-Ritz methods 21,20 . In the context of using equivalent plate to represent the behaviors of

wing structures at the conceptual stage at least, it is obvious that while the computationally costly

finite element method is to be avoided, the Rayleigh-Ritz method is attractive.

There have been several studies using equivalent plate models to model wing structures.

Giles 29,28 developed a Ritz method based approach, which considers an aircraft wing as being

formed by a series of equivalent trapezoidal segments, and represents the true internal structure of

aircraft wings in the polynomial power form. In Giles28 the CPT was used, but this shortcoming

was removed subsequently29 . Tizzi30 presented a method whose many aspects are similar to that

of Giles. In Tizzi's work several trapezoidal segments in different planes can be considered, but the

internal parts of wing structures (spars, ribs, etc.) were not considered. Livne31 formulated the

FSDT to be used for modeling solid plates as well as typical wing box structures made of cover

skins and an array of spars and ribs based on simple-polynomial trial functions, which are known to

be prone to numerical ill-conditioning problems. Livne and Navarro then further developed the

method to deal with nonlinear problems of wing box structures32 .


1.5 Sensitivity Techniques

Sensitivity techniques are frequently used in structural design practices for searching the

optimal solutions near a baseline design35~33 . The design parameters for wing structure include

sizing-type variables (skin thickness, spar or rib sectional area etc.), shape variables (the plan

surface dimensions and ratios), and topological variables (total spar or rib number, wing topology

arrangements etc.). Sensitivities to the shape variables are extremely important because of the

nonlinear dependence of stiffness and mass terms on the shape design variables as compared to the

linear dependence on the sizing-type design variables.

Kapania and coworkers have addressed the first order shape sensitivities of the modal response,

divergence and flutter speed, and divergence dynamic pressure of laminated, box-wing or general

trapezoidal built-up wing composed of skins, spars and ribs using various approaches of

determining the response sensitivities42~36 .

1.6 Scope of the Present Work

The aim of the present work is trying to develop efficient methods for the structural analysis of

built-up wings at the early design stage, such that with a fraction of the computational cost of a

detailed FEA, sufficiently accurate results for the global properties of the wing can be obtained. In

the present study, continuum models, neural networks and some other efficient simulation tools are

going to be used to make the objective possible.

As a preparation for application in later chapters, basic concepts and formulations about two

most commonly used neural networks, the Feed-Forward NN and Radial Basis Function NN, are

described in Chapter 2. Details of how to use some basic functions in the MATLAB NN Toolbox

for training and testing networks are provided, together with two ways of application of neural

networks: the direct approach and the indirect approach.


Chapter 3 is composed of an introduction of the continuum models used by several authors,

and an example of treating a lattice structure with repeating cells by continuum modeling applying

neural networks, compared with results obtained by other authors.

The present study is an extension of the previous works of Kapania and Singhvi44,43 , Kapania

and Lovejoy 46,45,21,20 , and Cortial47 , who all used the Rayleigh-Ritz method with the Chebyshev

polynomials as the trial functions, and applied the Lagrange’s equations to obtain the stiffness and

mass matrices. In Kapania and Singhvi44,43 , the CPT was used to solve generally laminated

trapezoidal plates, while in Kapania and Lovejoy 46,45,21,20 , the FSDT was used. In all these studies,

only uniform plates were considered. In Cortial47 , efforts were made to use the method of Kapania

and Lovejoy 46,45,21,20 to calculate natural frequencies of box-wing structures, but an assumption of

constant wing thickness makes it difficult to apply the method to general wing structures.

In the present work, it is assumed that the wing plan form is quadrilateral, and the wing

structure is composed of skins, spars and ribs. The wing is represented as an equivalent plate

model, and the Reissner-Mindlin displacement field model is used. The Rayleigh-Ritz method is

applied to solve the plate problem, with the Legendre polynomials being used as the trial functions.

After the stiffness matrix and mass matrix are determined by applying the Lagrange’s equations,

static analysis can be readily performed and the natural frequencies and mode shapes of the wing

can be obtained by solving an eigenvalue problem. Formulations are such that there is no limitation

on the wing thickness distribution as was the case in Cortial47 . This basic part of work, a method to

solve the Mindlin plates, is contained in Chapter 4. Then the method, being called the Equivalent

Plate Analysis (EPA), is applied for solving built-up wing structures in Chapter 5. As examples of

verifying EPA, a wing-shaped plate, a wing-shaped plate with camber, a solid wing and a general

built-up wing are analyzed respectively, and the results are compared with those obtained from a

detailed FE analysis using MSC/NASTRAN.

The EPA as explained above can also be used as a basis for generating other efficient methods

in the design of wing structures, such that can be incorporated with optimization tools into the

process of searching for an optimal design. In the search for an optimal design, it is essential to

CHAPTER 1 INTRODUCTION10

assess the structural responses quickly at any design space point. For such purpose, the FEM or

even the above EPA, which establishes the stiffness and mass matrices by integrating contributions

spar by spar, rib by rib, are not efficient enough.

One approach is to use Neural Networks as a means of simulating the structural responses of

wings. This is the so called direct application of neural networks, as discussed in Chapter 2.

Another approach is to use the sensitivity techniques. Sensitivity techniques are frequently used in

structural design practices for searching the optimal solutions near a baseline design34,33 . In the

present work, the modal response of general trapezoidal wing structures is approximated using

shape sensitivities up to the 2nd order, and the use of second order sensitivities proved to be

yielding much better results than the case where only first order sensitivities are used. Also

different approaches of computing the derivatives are investigated. These two approaches of direct

simulation of modal wing responses are described in Chapter 6, along with an example showing

results giving by both approaches.

Finally, in Chapter 7, a method more efficient than the EPA with indirect application of neural

networks is developed. Instead of evaluating the matrices over all components of the wing

structure, evaluation is performed only over the skins, whose "equivalent" material constitutive

matrix and mass density distribution are changed accordingly to incorporate the effects of spars and

ribs. The new skin material properties are simulated using Neural Networks in terms of the wing

design variables. As it is shown, while the Neural-Network-aided EPA, which can be called

Equivalent Skin Analysis (ESA), gives almost equally good results, it uses only a fraction of the

CPU time spent in the ordinary EPA in generating the matrices.

Major parts of the present work are published. They include Chapter 4 and 5 in Ref. 48 and 49,

Chapter 6 in Ref. 50 and 51, and Chapter 7 in Ref. 52.

11

Chapter 2

Basics of Neural Networks

In this chapter a brief description is given to the most extensively used neural network in civil and

structural engineering, Multi-Layer Feed-Forward NN, and another kind of NN, Radial Basis

Function NN, which is very efficient in some cases. Some conceptual features of NN are listed.

Several functions of MATLAB NN Toolbox are introduced, which will be used as the major tools in

the present work. At the end of this chapter a brief discussion is made on approaches of application

of neural networks.

2.1 Two Important Types of NN

As simplified models of the biological brain, ANNs have lots of variations due to specific

requirements of their tasks by adopting different degree of network complexity, type of inter-

connection, choice of transfer function, and even differences in training method.

According to the types of network, there are Single Neuron network (1-input , 1-output, and no

hidden layer), Single-Layer NN or Percepton (no hidden layer), and Multi-Layer NN (1 or more

hidden layers). According to the types of inter-connection, there are Feed-Forward network (values

can only be sent from neurons of a layer to the next layer), Feed-Backward network (values can

only be sent in the different direction, i.e. from the present layer to the previous layer), and

Recurrent network (values can be sent in both directions).

CHAPTER 2 BASICS OF NEURAL NETWORKS 12

In the following a brief description is given to two kinds of extensively used neural networks

and some of the pertinent concepts.

2.1.1 Feed-Forward Multi-Layer Neural Network

An example of feed-forward multi-layer neural network is shown in Fig. 2.1, where the

numbers of input and output are 3 and 2 respectively, and there are two hidden layers with 5

neurons in the first hidden layer, and 3 neurons in the second hidden layer. The details of a neuron

is illustrated in Fig. 2.2. As shown in Fig. 2.2, in the j -th layer, the i -th neuron has inputs from

the )1( −j -th layer of value ),,1( 11

−− = j

jk nkx � , and has the following output

)( ji

ji rfx = (2.1)

where

ji

n

k

jk

jki

ji bxwr

j

−= ∑−

=

−−1

1

11 (2.2)

in which 1−jkiw is the weight between node k of the )1( −j -th layer and node i of the j -th layer,

and jib is the bias (also called threshold). The above relation can also be written as

∑−

=

−−=1

0

11jn

k

jk

jki

ji xwr (2.3)

where ji

j bx =−10 and 11 −=−j

oiw , or 110 −=−jx and j

ij

oi bw =−1 .


inputlayer

hiddenlayers

outputlayer

Fig. 2.1 A feed-forward multi-layer neural network

Σ f ( ).xi

jr ij

w k ij - 1

-1

.

.

.

.

.

.

bij

x1j -1

x2j-1

k= n j-1

w 1 ij - 1

w 2 ij - 1

xkj -1

T rans ferfunc tio n

S ummingjunc tion

O utput

Inputs ignals

Synapticweights

T hresho ld

Fig. 2.2 Details of a neuron


The transfer function (also called activation function or threshold function) is usually specified

as the following Sigmoid function

rerf −+

=1

1)( . (2.4)

Other choices of the transfer function can be the hyperbolic tangent function

r

r

e

erf −

−

+−=

1

1)( , (2.5)

the piece-wise linear function

≤≤≤−+

≥=

.5.0,0

;5.05.0,5.0

;5.0,1

)(

r

rr

r

rf (2.6)

and, sometimes, the 'pure' linear function

.)( rrf = (2.7)

These transfer functions are displayed in Fig. 2.3.

r

f(r)

-5 0 5-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

Linear

Hyperbolic tangent

Sigmoid

Piecewise-linear

Fig. 2.3 Transfer functions


n1 n2 n3number ofneurons:

inputlayer

hiddenlayer

outputlayer

Fig. 2.4 Radial basis function neural network

2.1.2 Radial Basis Function Neural Network

Radial Basis Function (RBF) NN usually have one input layer, one hidden layer and one output

layer, as shown in Fig. 2.4.

For the RBF network in Fig. 2.4, we have the relations between the input 1ix (here 1,,1 ni �= )

and the output 2kx (here 3,,1 nk �= ) as follows:

2

1

2,

22

k

n

jjjkk brwx += ∑

=

(2.8)

∑=

=1

1

1,

11 ),,(n

ijijij bwxGr (2.9)

where 2w , 2b are the weights and bias respectively, and the Gaussian function is used as the

transfer function:

)}{}{exp(),,( 21121,

1,

11jijijiji wxbbwxG −−= (2.10)

where 1w is the center vector of the input data, and 1b is the variance vector.


2.2 Features of ANN

Some important features of NN are briefed as follows.

• Many NN methods are universal approximators, in the sense that, given a dimension (number

of hidden layers and neurons of each layer) large enough, any continuous mapping can be

realized. Fortunately, the two NNs we are most interested in, the multi-layer feed-forward NN

and the radial basis function NN, are examples of such universal approximators54,53 .

• Steps of utilizing NN: specification of the structure (topology)→ training (learning)

→ simulation (recalling).

(1) Choosing structural and initial parameters (number of layers, number of neurons of each

layer, and initial values of weights and thresholds, and the kind of transfer function) is usually

from experiences of the user and sometimes can be provided by the algorithms. (2) The training

process uses given input and output data sets to determine the optimal combination of weights

and thresholds. It is the major and the most time-consuming part of NN modeling, and there are

lots of methods regarding different types of NN. (3) Simulation means using the trained NN to

predict output according to new inputs (This corresponds to the 'recall' function of the brain).

• The input and output relationship of NN is highly nonlinear. This is mainly introduced by the

nonlinear transfer function. Some networks, e.g. the so-called "abductive" networks, use double

even triple powers besides linear terms in their layer to layer input-output relations55 .

• A NN is parallel in nature, so it can make computation fast. Neural networks are ideal for

implementation in parallel computers. Though NN is usually simulated in ordinary computers

in a sequential manner.

• A NN provides general mechanisms for building models from data, or give a general means to

set up input-output mapping. The input and output can be continuous (numerical), or not

continuous (binary, or of patterns).

• Training a NN is an optimization process based on minimizing some kind of difference

between the observed data and the predicted while varying the weights and thresholds. For


numerical modeling, which is of our major concern for the present study, there is a great

similarity between NN training and some kind of least-square fitting or interpolation.

• Simulation using NN gives better results in interpolation than in extrapolation, the same as any

other data fitting or mapping methods.

• Where and when to use NN depend on the situation, and NN is not a panacea. The following

comment on NN application on structural engineering seemingly can be generalized in other

areas:

"The real usefulness of neural networks in structural engineering is not in reproducing existing

algorithmic approaches for predicting structural responses, as a computationally efficient

alternative, but in providing concise relationships that capture previous design and analysis

experiences that are useful for making design decisions"7 .

Despite the above features and wide application in a lot of areas, there seems to be no evidence

for neural networks to claim superiority over some other mapping tools. For instance, in a recent

paper of Nikolaidis, Long, and Ling 73, it is claimed that the response surface polynomials with

stepwise regression and the neural network models appear to be almost equally accurate, but it took

considerably less time to develop the polynomials than the neural networks.

2.3 Algorithms in the MATLAB Neural Network Toolbox

When using MATLAB NN Toolbox, one should first choose the number of input and output

variables. This is accomplished by specifying the two matrices p and t , where p is a nm×

matrix; m is the number of input variables, and n the number of sets of training data; and t is a

nl × matrix; l is the number of output variables. The number of network layers, and numbers of

neurons of each layer are other factors that need to be specified.

MATLAB gives algorithms for specifying initial values of weights and thresholds in order that

training can be started. For feed-forward NN, function initff is given for this purpose. The

following is an example of using the algorithm


[w1,b1,w2,b2,w3,b3]=initff(p,n1,'logsig',n2,'logsig',t,'logsig');

where w1, w2, and w3 are initial values for the weight matrices of the 1st (hidden), 2nd (hidden)

and 3rd (output) layer respectively, b1, b2, and b3 are the bias (threshold) vectors, n1 and n2 the

number of neurons in the 1st and 2nd hidden layer respectively, and 'logsig' means that the Sigmoid

transfer function is used.

The present version of MATLAB NN Toolbox can support only 2 hidden layers, but the number

of neurons is only limited by the available memory of the computer system being used. For the

transfer function, one can also use other choices, such as 'tansig' (hyperbolic tangent sigmoid),

'radbas' (radial basis) and 'purelin' (linear) etc.

Experiences of using initff indicated that it seems to be a random process since it is found that

the result of the execution of this algorithm each time is different. And other conditions kept the

same, two executions of this function usually give quite different converging histories of training

by the training algorithm8 .

Shown in the following is the MATLAB algorithm for training feed-forward network with back-

propagation:

[w1,b1,w2,b2,w3,b3,ep,tr]=trainbp(w1,b1,'logsig',w2,b2,'logsig',w3,b3,'logsig',p,t,tp);

where most of the parameters which the user should take care of have been mentioned in the above

paragraphs. The only parameter that the user sometimes need to specify is the 41× vector tp,

where the first element indicates the number of iterations between updating displays, the second the

maximum number of iterations of training after which the algorithm would automatically terminate

the training process, the third the converging criterion (sum-squared error goal), and the last the

learning rate. The default value of tp is [25, 100, 0.02, 0.01].

Other algorithms for training include trainbpa (training feed-forward NN with back-

propagation and adaptive learning), solverb (designing and training radial basis network), and

trainlm (training feed-forward NN with Levenberg-Marquardt algorithm) etc.

trainbpa and trainlm have very similar formats for using as that of trainbp. The radial basis

network designing and training algorithm has the following format


[w1,b1,w2,b2,nr,err]=solverb(p,t,tp);

where the algorithm chooses centers for the Gaussians and increases the neuron number of the

hidden layer automatically if the training cannot converge to the given error goal. So it is also a

designing algorithm.

After the NN is trained, one can predict output from input by using simulation algorithms in

terms of the obtained parameters w1, b1, w2, b2, etc. For feed-forward network one use

y=simuff(x,w1,b1,'logsig',w2,b2,'logsig',w3,b3,'logsig');

where x is the input matrix, and y the predicted output matrix. Similarly, after a radial basis

network has been trained one uses

y=simurb(x,w1,b1,w2,b2);

to predict the output.

Once a NN is trained, we can use the formulations in 2.1 or 2.2 together with the obtained

parameters (weights etc.) to setup the network to do prediction anywhere and not necessarily within

the MATLAB environment.

2.4 Ways of Application of Neural Networks

For the efficient simulation of the structural performances of complex wings, there can be two

directions to apply NN as specified in the following:

2.4.1 Direct Application

In this case, the input layer includes all the design variables of interest (for instance, the four

shape parameters of the wing plan form: the sweep angle, the aspect ratio, the taper ratio, and the

plan area). The output layer gives the desired structural responses, such as natural frequencies etc.

The EPA is being used as the training data generator, though if necessary, results obtained using

the FEA can also be used as the training data. Preparation of training data is very important, and the

training algorithm used also greatly impacts the process of training8 . Caution must be taken in

specifying the network parameters and training criterion, such that the results of the trained


network would not oscillate around the training data. Once the networks are trained, structural

responses at any design point can be recalled in a fraction of a second and this is really favorable in

a design situation51.

2.4.2 Indirect Application

Here it is desired to find a way of incorporating NN into the application of the equivalent plate

analysis (EPA) of complex wing structures, other than just making use of results generated by EPA

as the training data base. Note that in the EPA of a complex wing, the computational effort is

mainly spent on integrals for generating the contribution from the inner-structural components of

the wing, i.e. the spars and the ribs, in the stiffness and mass matrices. If an anisotropic material

can be found to replace the inner components, in terms of an equivalent skin, such that the new

composite wing has very close global properties as the original one, then the EPA can be

performed more efficiently. Solution of the adequate material properties of the anisotropic material

is the major obstacle here. The role of NN will be relating the material properties to all kinds of

wing design parameters, and it can be trained when there exists enough data base for training. This

way of applying NN has been claimed to be the best use of the Neural Networks in structural

engineering7 . This is the path that is to be followed in Chapter 7.

21

Chapter 3

Continuum Model Approaches

3.1 Methods of Obtaining Continuum Models

A lot of methods have been used to develop continuum models to represent complex structures.

Many of these methods involve the determination of the appropriate relationships between the

geometric and material properties of the original structure and its continuum models. An important

observation is that the continuum model is not unique, and determining the continuum model for a

given complex structure is inherently ill-posed therefore diverse approaches can be used. This can

be clearly shown in the following example of determining continuum models for a lattice structure.

The single-bay double-laced lattice structure shown in Fig. 3.1 has been studied in Ref. 10, 12,

and 14 with different approaches to the continuum modeling. This lattice structure with repeating

cells can be modeled by a continuum beam if the beam's properties is properly provided.

Noor et al's method include the following steps10: (1)introducing assumptions regarding the

variation of the displacements and temperature in the plane of the cross section for the beamlike

lattice, (2)expressing the strains in the individual elements in terms of the strain components in the

assumed coordinate directions, (3)expanding each of the strain components in a Taylor series, and

(4)summing up the thermoelastic strain energy of the repeating elements which then gives the

thermoelastic and dynamic coefficients for the beam model in terms of material properties and

geometry of the original lattice structure.

CHAPTER 3 CONTINUUM MODEL APPROACHES 22

Lg

Lc

longitudinal bar

diagonal bar

batten

In Sun et al12, the properties of the continuum model is obtained respectively by relating the

deformation of the repeating cell to different load settings under specified boundary conditions. For

example, the shear rigidity GA is obtained by performing a numerical shear test in which a unit

shear force is applied at one end of the repeating cell and the corresponding shear deformation is

calculated by using a finite element program. The mass and rotatory inertia are calculated with a

averaging procedure.

Lee put forward a method that he thought to be more straightforward14. He used an extended

Timoshenko beam to model the equivalent continuum beam. By expressing the total strain and

kinetic energy of the repeating cell in terms of the displacement vector at both ends of the

continuum model, and equating them to those obtained through the extended Timoshenko beam

Length of longitudinal bars: cL Length of battens: gL

Length of diagonal bars: 21

)( 22gcd LLL += Areas: ,cA ,gA dA

Fig. 3.1 Geometry of repeating cells of a single-bay double laced lattice structure


theory, he got a group of relations. The number of these relations, 2N(1+2N), where N is the degree

of freedom of the continuum model, is usually larger than that of the equivalent continuum beam

properties to be determined. Lee then introduced a procedure in which the stiffness and mass

matrices for both the lattice cell and the continuum model are reduced and so is the number of

relations. Yet how to reduce the number of relations to be equal to the number of unknowns seems

to depend on luck.

All the above three methods give close results for the continuum model properties, and the

continuum models also generate promising global results for the lattice structure.

3.2 An Example of NN Modeling of Continuum Models

Emphasizing the application of NN, we choose an approach similar to that in Ref. 12, that is, to

derive the properties of the beam by investigating the force-deformation relationships of the

repeating cell in certain boundary conditions. The approach is illustrated in Fig. 3.2, where the

beam's axial rigidity EA, bending rigidity EI, and shearing rigidity GA are calculated respectively

by using the results of finite element analysis of the repeating cell in different load conditions.

Concerning the finite element analysis of 3-D lattice structures one can consult Ref. 56.

There are five parameters of the repeating cell for the lattice structure in Fig. 3.1 that can be

varied, the longitudinal bar length cL , the batten length gL , and the longitudinal, batten and

diagonal bar area, cA , gA and dA . Generally, a function with more variables will be more complex

and it will be more difficult for a neural network to simulate its performance. A NN with more

input variables needs much more training data since in the training data each variable should vary

separately. As can be shown in the following, this kind of "coarse" training data pose an obstacle to

most of the training algorithms.

Three scenarios were investigated, with the number of input variables set to be 2, 3 and 4

respectively.


1

2

3

4

5

6

x, u

y, v

1/31/3

1/3

z, w

1

2

3

4

5

6

x, u

y, v

1/31/3

1/3

z, w

1

2

3

4

5

6

x, u

y, v

1/21/2

1

z, w

3.2.1 Neural Network with 2 Input Variables

The input variables are cL and cA . The number of training data sets is 400=20× 20. The

number of testing data, most of which are located at centers among the training data mesh, is also

400=20× 20. Part of the results, as the training data, about GA, is shown in Fig. 3.3.Positions of the

training as well as the testing data points are shown in Fig. 3.4. Simulations on the testing data and

the relative errors of a 2-10-1 FF NN (feed-forward neural network with 2 inputs, one hidden layer

of 10 neurons, and 1 output) trained with Levenberg-Marquardt algorithm (trainlm) are shown in

Axial rigidity EA:

641

521

441 vvv

L

v

LEA cc

++≅

∆=

Bending rigidity EI:

45

243

23

231

vv

LLLLEI

LEILM

cgcg

c

g

−≅=⇒

⋅=×=

φ

φShear rigidity GA:

641

521

441

641

521

441

11

www

L

GA

c

++=

++≅=

θθθθ

Fig. 3.2 Evaluating continuum model properties for a repeating cell


Figs. 3.5 and 3.6. Results of a radial-basis-function (RBF) NN doing the same job are shown in

Figs. 3.7 and 3.8. In both cases the training error criteria were set to be 3104.0 −× .

From Figs 3.5 and 3.7 we can see that both the FF NN and RBF NN give a very good

simulation of the relation ),( cc LAfGA= , except at points outside the training data range of

variable cL ( cf. Fig. 3.4). At the "inside" points, or positions where interpolations are made, the

abstract values of the relative errors are well below the 1%. On the other hand, at points outside the

training data range of variable cL , the relative errors can be as high as 3~5%. This provides another

proof to the fact mentioned before, that interpolation using NN will give results more accurate than

extrapolation.


1.5

2

2.5

3

GA

0

1

2Ac(m

2 )

0

2

4

6

8

10

12

Lc(m)

X Y

Z

GA Training Data

x10-4

x106

Lc(m)

Ac(

m2 )

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4Testing pointsTraining points

Training & Testing Points

x10-4

Fig. 3.3 Training data for )( , cc LAfGA=

Fig. 3.4 Distributions of training and testing points


2

3

GA

(FF

)

0

0.5

1

1.5

2Ac(m

2 )

0

2

4

6

8

10

12

Lc(m)

X Y

Z

FF Simulation on Testing Data

x10-4

x106

-4

-2

0

2

{GA

-GA

(FF

)}/G

A

0.5

1

1.5

2

Ac(m2 )

2

4

6

8

10

Lc(m)

0.49

-0.070.49

-2.85

-1.18-0.62 -0.07-0.07

-0.07

-0.07-0.07

X Y

Z

FF Simulation Errors

x10-4

x10-2

Fig. 3.5 Feed-forward NN simulation for )( , cc LAfGA=

Fig. 3.6 Feed-forward NN simulation errors


2

3

GA

(RB

F)

0

1

2Ac(m

2 )

0

2

4

6

8

10

12

Lc(m)

X Y

Z

RBF Simulation on Testing Data

x10-4

x106

0

2

4

{GA

-GA

(RB

F)}

/GA

0

1

2Ac(m

2 )

0

2

4

6

8

10

12

Lc(m)

0.90

0.90

-0.15

1.96

0.380.901.96

-0.15

1.43

1.431.962.48

0.901.43

-0.15

-0.15

-0.15

-0.1

5

X Y

Z

RBF Simulation Errors

x10-2

x10-4

Fig. 3.7 Radial-basis function NN simulation for )( , cc LAfGA=

Fig. 3.8 Radial-basis function NN simulation errors



The input variables were chosen as cL , cA and dA . The number of training data sets is

343=7× 7× 7. For this case, the effectiveness of different training algorithms can be seen clearly in

Figs. 3.9~3.11. When ordinary back-propagation training algorithm, i.e. trainbp is used, it is very

hard to train the NN to the error level of 110− , as shown in Fig. 3.9. When the adaptive learning

technique is included, an improvement can be made, but it is still hard to reach the 210− error level,

as can be seen in Fig. 3.10. Now if the algorithm with Levenberg-Marquardt (trainlm) is used, it is

quite easy to push the training error level to the order of 510− , as shown in Fig. 3.11.

The improvement by trainlm is really amazing. All the training algorithms carry out an

optimization process. While trainbp uses steepest-descent method with constant step size, trainbpa

accelerates the process by adjusting the step size. On the other hand, trainlm adopts a kind of

modified Newton's Methods, which adjusts both the searching direction and the step size.

Concerning the optimization methods one can consult Ref. 57.

Samples of the NN simulation results are given in Table 3.1, where the desired values and

values obtained by Noor et al10 and Lee14 are also presented.


The input variables were chosen as cL , cA , gA and dA .The number of training data sets is

625=5× 5× 5× 5. For this case only trainlm could train a FF NN that could give reasonable results.

Samples of the NN simulation results are given in Table 3.2.


Epoch

Sum

-Squ

ared

Err

or

0 5000 10000 15000 20000

10-1

100

101

Epoch

Sum

-Squ

ared

Erro

r

0 5000 10000 15000 20000

10-1

100

101

102

Fig. 3.9 Training history of a 3-10-1 feed-forward NN by trainbp

Fig. 3.10 Training history of a 3-10-1 feed-forward NN by trainbpa


Epoch

Sum

-Squ

ared

Err

or

0 50 100 150 200 250

10-5

10-4

10-3

10-2

10-1

100

101

Table 3.1 Comparison of Continuum Model Properties for a Lattice Repeating CellFeatures of the cell: 25210 106,0.5,/1017.7 mAmLmNE gg

−×==×=

Variables specified: 2525 104,108,5.7 mAmAmL dcc−− ×=×==

Present(FEA)

present (FFsimulated, trained

by trainbpa, 3variables)

present(RBF

simulated,3vaiables)

present (FFsimulated, trained

by trainlm, 3variables)

U. Lee Noor etal

EA( )107 N 2.659 2.60 2.66 2.656 2.71 2.53

GA( )106 N 2.183 2.24 2.17 2.186 2.2 2.2

EI( )10 27 mN ⋅ 8.147 8.08 8.17 8.148 8.20 8.01

Table 3.2 Comparison of Continuum Model Properties for a Lattice Repeating CellFeatures of the cell: mLmNE g 0.5,/1017.7 210 =×=

Variables specified: 252525 106,104,108,5.7 mAmAmAmL gdcc−−− ×=×=×==

Present(FEA)

present (FF simulated, trainedby trainlm, 4 variables)

U. Lee Noor et al

EA( )107 N 2.659 2.682 2.71 2.53

GA( )106 N 2.183 2.134 2.2 2.2

EI( )10 27 mN ⋅ 8.147 8.181 8.20 8.01

Fig. 3.11 Training history of a 3-10-1 feed-forward NN by trainlm

32

Chapter 4

An Approach for the Solution of Mindlin

Plates

In this chapter a FSDT (the Reissner-Mindlin theory) method, using the Ritz method with the

Legendre or the Chebyshev polynomials being employed as the trial functions, is derived to solve

plate problems. Formulations are such that there is no limitation on the plate thickness variation

and therefore can be used to deal with real-life wings in the next chapter.

4.1 Assumptions and Formulations

For the solution of a plate under static or dynamic deformation, the Reissner-Mindlin method, a

First-order Shear Deformation Theory (FSDT), is based on two assumptions for the displacement

field: (1) A normal line to the non-deformed middle surface remains to be a straight line; (2) The

transverse normal stress can be neglected in the constitutive relations.

According to these assumptions, and assuming linearity, the displacement field of the plate is

given as:

=+=+=

),,(),,,(

),,(),,(),,,(

),,(),,(),,,(

0

0

0

tyxwtzyxw

tyxztyxvtzyxv

tyxztyxutzyxu

y

x

φφ

(4.1)

CHAPTER 4 AN APPROACH FOR MINDLIN PLATES 33

where as shown in Fig. 4.1, wvu ,, are displacements in the zyx ,, direction respectively, subscript

0 refers to quantities associated with the plane 0=z , xφ and yφ are the rotations about the y and

x− direction respectively. It is assumed here that the middle surface of the plate is without or with

a very small curvature, therefore 0=z can be considered to be the middle surface.

From Eq. (4.1) we can get the strains:

∂∂+=

∂∂+

∂∂==

∂∂+=

∂∂+

∂∂==

∂∂

+∂∂+

∂∂+

∂∂=

∂∂+

∂∂==

=∂∂=

∂∂

+∂∂=

∂∂=

∂∂+

∂∂=

∂∂=

x

w

z

u

x

wy

w

y

w

z

vxy

zx

v

y

u

x

v

y

uz

wy

zy

v

y

vx

zx

u

x

u

xzxzx

yyzyz

yxxyxy

z

yy

xx

0

0

00

0

0

2

2

)(2

0

φεγ

φεγ

φφεγ

ε

φε

φε

(4.2)

For the convenience of calculation, a transformation from ),( yx to ),( ηξ is performed, with

the plate configuration in the ),( yx plane being transformed to a square in the ),( ηξ plane, as

shown in Fig. 4.1. The plate configuration in the ),( yx plane can be any quadrilaterals, of which a

special case, the skewed trapezoidal, is frequently used in aircraft wing configurations.

x, u

z, w y, v

1

2

4

3

z

ξ

η

(1,-1) (1,1)

(-1,-1) (-1,1)

Fig. 4.1 The coordinate system and its transformation


The transformation can be formulated as

=

=

∑

∑

=

=4

1

4

1

),(

),(

iii

iii

yNy

xNx

ηξ

ηξ (4.3)

where

+−=++=−+=−−=

)1)(1(),(

)1)(1(),(

)1)(1(),(

)1)(1(),(

41

4

41

3

41

2

41

1

ηξηξηξηξηξηξηξηξ

N

N

N

N

(4.4)

The Jacobian matrix for this transformation is

∂∂

∂∂

∂∂

∂∂

=

ηξ

ηξyy

xx

J ][ (4.5)

where from Eqs. (4.3) and (4.4) it is ready to obtain

−−+−+=∂∂

−−+−+=∂∂

−−+−+=∂∂

−−+−+=∂∂

)])(1())(1[(

)])(1())(1[(

)])(1())(1[(

)])(1())(1[(

142341

124341

142341

124341

yyyyy

yyyyy

xxxxx

xxxxx

ξξη

ηηξ

ξξη

ηηξ

. (4.6)

The inverse of the Jacobian matrix:

=

−

−=

∂∂

∂∂−

∂∂−

∂∂

=−

2221

1211

1121

12221 1][

JJ

JJ

JJ

JJ

JJ

xy

xy

Jξξ

ηη

(4.7)

where J is the determinant of the Jacobian:


∂∂

∂∂−

∂∂

∂∂=

ξηηξyxyx

J . (4.8)

We express the terms on the plane 0=z in Eq. (4.1), i.e. 0u , 0v , 0w , xφ and yφ , in the

following forms

==

==

==

==

==

∑∑

∑∑

∑∑

∑∑

∑∑

= =

= =

= =

= =

= =

R

r

S

ssrrsY

TRSy

P

p

Q

qqppqX

TPQx

M

m

N

nnmmnW

TMN

K

k

L

llkklV

TKL

I

i

J

jjiijU

TIJ

BBtYqB

BBtXqB

BBtWqBw

BBtVqBv

BBtUqBu

1 1

1 1

1 10

1 10

1 10

)()()(}{}{

)()()(}{}{

)()()(}{}{

)()()(}{}{

)()()(}{}{

ηξφ

ηξφ

ηξ

ηξ

ηξ

(4.9)

or

}{}{

}{

}{

}{

}{

}{

}}{,}{,}{,}{,}{{0

0

0

qB

q

q

q

q

q

BBBBBw

v

u

T

Y

X

W

V

U

TRS

TPQ

TMN

TKL

TIJ

y

x

=

=

φφ

(4.10)

where RQPNMLKJI ,,,,,,,, and S are integers,

====

==

TRSY

TPQX

TMNW

TKLV

TIJIJJU

TTY

TX

TW

TV

TU

YYqXXq

WWqVVq

UUUUUUUq

qqqqqq

},,{}{,},,{}{

},,{}{,},,{}{

},,,,,,,,,,{}{

}}{,}{,}{,}{,}{{}{

1111

1111

122111211

��

��

�� (4.11)

is called the generalized displacement vector, and

=

=

=

RSPQMNKLIJ

BBBBB

BBBBBBT

TTRS

TPQ

TMN

TKL

TIJ

,,,,

)}()(,),()({}{

}}{,}{,}{,}{,}{{}{

11

µνηξηξ νµµν � (4.12)


is the Ritz base function vector, in which )(xBi can either be chosen to be the Legendre

polynomials or the Chebyshev polynomials:

)()( 1 xPxB ii −= or )()( 1 xTxB ii −=

where

=+

−++=

==

−+ .,1),(1

)(1

12)(

)(

1)(

11

1

0

�nxPn

nxxP

n

nxP

xxP

xP

nnn

(4.13)

and

=−===

−+ .,1),()(2)(

)(

1)(

11

1

0

�nxTxxTxT

xxT

xT

nnn

(4.14)

The foremost 10 non-constant items of the Legendre polynomials and the Chebyshev

polynomials are shown in Figs. 4.2 and 4.3 respectively.

x

Pn(

x)(n

=1

,...,1

0)

-1 -0.5 0 0.5 1-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

First 10 Lengendre polynomials

Fig. 4.2 The Legendre polynomials


x

Tn(

x)(n

=1

,...,1

0)

-1 -0.5 0 0.5 1-1

-0.75

-0.5

-0.25

0

0.25

0.5

0.75

1

First 10 Cheb yshev pol ynomials

4.2 Strain Energy and Stiffness Matrix

The strain energy of a plate or a plate-like structure (for instance, a wing) is

∫∫∫=V

T dVU }{}{21 εσ (4.15)

Introducing the stress-strain relations by the generalized Hooke’s law, we have for Eq. (4.15)

∫∫∫=V

T dVDU }]{[}{21 εε (4.16)

Note that }]{[}{ εσ D= and ][][ DD T = is assumed, and the integration domain V in Eqs. (4.15)

and (4.16) includes all and only the spaces the components of the wing occupy.

Using Eqs. (4.5) and (4.7), we can write

Fig. 4.3 The Chebyshev polynomials


∂∂∂∂

=

∂∂∂∂

∂∂

∂∂

∂∂

∂∂

=

∂∂∂∂

y

fx

f

J

y

fx

f

yx

yx

f

f

T][

ηη

ξξ

η

ξ

∂∂∂∂

=

∂∂∂∂

=

∂∂∂∂

=

∂∂∂∂

∴ −−

η

ξ

η

ξ

η

ξf

f

JJ

JJf

f

Jf

f

J

y

fx

f

TT

2212

211111 )]([)]([ (4.17)

From Eqs. (4.2) and (4.17), we have

=

=

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

y

x

w

w

v

v

u

u

zx

yz

xy

y

x

y

y

x

x

JJ

JJ

JzJzJzJzJJJJ

JzJzJJ

JzJzJJ

φφ

εεεεε

ε

ηφ

ξφ

ηφ

ξφ

η

ξ

η

ξ

η

ξ

0

0

0

0

0

0

0100000000

1000000000

0000

00000000

00000000

}{

2111

2212

2111221221112212

22122212

21112111

= }]{[ εT (4.18)

From Eq. (4.9), we can write


}]{[

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

)()()(

}{

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

0

0

0

0

0

0

qC

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

BBtU

R

r

S

ssrrs

P

p

Q

qqppq

R

r

S

ssrrs

R

r

S

ssrrs

P

p

Q

qqppq

P

p

Q

qqppq

M

m

N

nnmmn

M

m

N

nnmmn

K

k

L

llkkl

K

k

L

llkkl

I

i

J

jjiij

I

i

J

jjiij

y

x

w

w

v

v

u

u

y

y

x

x=

′

′

′

′

′

′

′

′

′

′

=

=

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

∑∑

= =

= =

= =

= =

= =

= =

= =

= =

= =

= =

= =

= =

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

ηξ

φφ

ε

ηφ

ξφ

ηφ

ξφ

η

ξ

η

ξ

η

ξ

(4.19)

where

=

TRS

TPQ

RS

PQ

MN

KL

IJ

B

B

B

B

B

B

B

C

}{0000

0}{000

][0000

0][000

00][00

000][0

0000][

][

,

,

,

,

,

ξη

ξη

ξη

ξη

ξη

(4.20)

′′′

′′′=

)()()()()()(

)()()()()()(][

2111

2111

, ηξηξηξηξηξηξ

νµ

νµξηµν BBBBBB

BBBBBBB

�

�

TBBBBBBB })()()()()()({}{ 2111 ηξηξηξ νµµν �=

.,,,, RSPQMNKLIJ=µν

and }{q is the general displacement vector shown in Eq. (4.10).


Substitute Eqs. (4.18), (4.19) into Eq. (4.16), and we have

∫∫∫=V

TTT dVqCTDTCqU }]{][][[][][}{21 (4.21)

If we write

}]{[}{21 qKqU T= (4.22)

Then comparison of Eqs. (21) and (22) gives

∫∫∫=V

TT dVCTDTCK ]][][[][][][ (4.23)

This is the stiffness matrix of the plate in terms of }{q . Details for the constitutive matrix ][D

for different parts of the wing structure can be found in Appendix A.

4.3 Kinetic Energy and Mass Matrix

The kinetic energy of a plate or a plate-like structure (wing) is

∫∫∫ ∫∫∫==V V

T dVvvdVvT }{}{21

2

21 ρρ (4.24)

where the velocity vector

}]{][[

00100

0010

0001

}{ 0

0

0

0

0

0

•

∂∂

∂∂

∂∂

∂∂

∂∂

=

=

∂∂

∂∂

+∂∂

∂∂+

∂∂

=

∂∂= qHZz

z

t

wt

zt

vt

zt

u

t

dv

t

t

tw

tv

tu

y

x

y

x

φ

φ

φ

φ

(4.25)

where

=

00100

0010

0001

][ z

z

Z

=w

v

u

d}{ is the displacement vector, and


=

TRS

TPQ

TMN

TKL

TIJ

B

B

B

B

B

H

}{0000

0}{000

00}{00

000}{0

0000}{

][ (4.26)

and }{•q is the time-derivative of }{q .

Then we have

∫∫∫••

=V

TT dVqHZZHqT }]{][[][}{21 ρ (4.27)

where

==

2

2

000

000

00100

0010

0001

][][][

zz

zz

z

z

ZZZZ T (4.28)

Compare

}]{[}{21

••= qMqT T (4.29)

with Eq. (4.27) we have

∫∫∫=V

T dVHZZHM ]][[][][ ρ (4.30)

which is the mass matrix of the plate in terms of the general velocity vector, }{•q

It should be pointed out that the present formulation can deal with quadrilaterals of any shape

(see Fig.4.1), which include a skewed trapezoid, the usual shape for a wing plan form. Also, the

formulation can be extended to more complicated cases. When the wing plan form is composed of

several quadrilaterals or trapezoids, one can obtain the whole stiffness and mass matrices by

assembling the corresponding matrices of the component elements, as has been done in Tizzi30 . A

brief description of how to do in this kind of scenario can be found in Appendix B.

42

Chapter 5

Equivalent Plate Analysis of Built-Up Wing

Structures

Now we want to solve a wing problem by assuming that the wing behaves like a plate. This

assumption is very reasonable as long as the wing has a small thickness-chord ratio. The

formulations for a general quadrilateral plate derived in Chapter 4 will be used to deal trapezoidal

built-up wing structures in this chapter. This is accomplished by evaluating Eqs. (4.20) and (4.27)

for all the wing components. How to deal with the boundary conditions and the convergence

problem are discussed briefly. Once the total stiffness and mass matrices are determined, various

static and dynamic problems can be solved based on their utilization.

5.1 Numerical Integration of Stiffness and Mass Matrices

For a specific wing, now we want to evaluate the integrals in Eqs. (4.20) and (4.27).

Assume kSRQPNMLKJI ========== , then we know ][K and ][M will be

matrices of dimension NN × , where 25kN = .

We know coordinates in ),( yx plane are transformed to ),( ηξ , but coordinate z remains the

same. Therefore, for an integral in space ),,( zyx , we have

CHAPTER 5 EQUIVALENT PLATE ANALYSIS OF WING 43

∫ ∫∫∫∫ − −==

1

1

1

1),(),,( ηξηξ ddGdVzyxFI

V

(5.1)

where

∑∫=

⋅=z

i

i

N

i

z

zdzJzyxFG

1

2

1

]),,(),,([),( ηξηξηξ (5.2)

here zN is the number of integration zones in z-direction, and 1iz and 2iz are integration limits of

the i -th zone.

Using the Gaussian quadrature, we can get the numerical value of integral in Eq. (5.1) as

∑∑= =

≅g g

gggg

M

i

N

j

Nj

Mi

Nj

Mi GggI

1 1

)()()()( ],[ ηξ (5.3)

where )()( , gg N

j

M

i gg are the weights , )()( , gg N

j

M

i ηξ are the sampling points, and gM and gN

represent the number of sampling points used in the ξ and η directions respectively.

For a wing composed of skins, spars and ribs, the integrals in Eq. (5.1) can be detailed as

follows:

5.1.1 Skins

The sketch of skins at a wing section is shown in Fig. 5.1. Particulars of integration for skins:

ηξdddzJFdzJFdVzyxFUU

UU

LL

LL

tz

tz

tz

tzV

∫ ∫ ∫∫∫∫∫ − −

+

−

+

−

⋅+⋅=

1

1

1

1

21

21

21

21

),,( (5.4)

where ULUL tt ,2

, tan1 α+= , subscript UL, indicate the lower and upper skin respectively. It

should be noted that for skins composed of laminated layers the skin contributions can be obtained

by simply adding up the efforts of all the layers, with the material constitutive matrix of each layer

being treated separately (see Appendix A).


α

ZU

LZ

c

t0

Fig. 5.1 Wing skin

Calculation of Eqs. (4.23) and (4.30) using Eq. (5.4) give the stiffness and mass matrices of the

skins: ][ skinK and ][ skinM .

5.1.2 Spars

Their contribution to the stiffness and mass matrices can be calculated by performing the

integrals for each spar. Representative dimensions of a spar are shown in Fig. 5.2. Thus, for a spar

cap, we have:

( ) ( )[ ] ( )[ ]{ } dzJzclyclxFdcld

dzJzyxFdddxdydzzyxF

ss

tz

htz

htz

tz

tz

htz

c

l

c

l

htz

tzV

LU

UU

LL

LL

LU

UU

s

s

LL

LL

⋅++

+=

⋅

+=

∫∫ ∫∫

∫∫ ∫∫∫∫∫−

−−−

++

+−

−

−−

+

−

++

+−

,),(,),(

]),,(),,([),,(

11

1

1 1

1

1

)(

)(

1

1

21

121

21

21

21

121

1

1

21

21

ηηξξηηξξξη

ηξηξξηηξ

ηξ (5.5)

where c is the chord-length at η : ),1()1( 121

021 ηη ++−= ccc :0c the chord-length at wing root,

:1c the chord-length at wing tip, and )(ηξ s is the spar position function. Eq. (5.5) can be easily

computed numerically using Eq. (5.3).

For the spar web:

( ) ( )[ ] ( )[ ]{ }∫∫∫

∫ ∫ ∫∫∫∫−−

++−−

+

− −

−−

++

⋅++=

⋅=

121

21

1

1

121

121

,),(,),(

]),,(),,([),,(

11

1

1 1

1

1

1

1

htz

htz ss

c

t

c

t

htz

htzV

UU

LL

s

s

UU

LL

dzJzctyctxFdctd

dzddJzyxFdxdydzzyxF

ηηξξηηξξξη

ηξηξηξξ

ξ (5.6)


Calculation of Eqs. (4.23) and (4.30) using Eqs. (5.5) and (5.6) give the stiffness and mass

matrices of the skins: ][ sparK and ][ sparM .

l l1 2,

h ,h

t ,t

1 2

1 2

Skin

Spar or rib cap

Spar or rib web

Spar or rib center surface

Rib,

Spar,x,

y,

η (ξ)

ξ (η)

r

s

η

ξ

Fig. 3 Wing Spar or Rib

Λ

5.1.3 Ribs

The contributions of the ribs to the stiffness and mass matrix can be calculated in a manner

similar to the one used for spars. The dimensions of a rib are also given in Fig. 5.2.

For a rib cap:

( ) ( )[ ] ( )[ ]{ } dzJzslyslxFdsld

dzddJzyxFdxdydzzyxF

rr

tz

htz

htz

tz

s

l

s

l

htz

tz

tz

htzV

LU

UU

LL

LL

r

r

LU

LL

UU

LU

⋅++

+=

⋅

+=

∫∫ ∫∫

∫ ∫ ∫ ∫∫∫∫−

−−−

++

+−

−

+

−

++

+

−

−−

,)(,,)(,

]),,(),,([),,(

22

1

1 2

1

1

1

1

21

221

221

21

2

2

221

21

21

221

ξηηξξηηξηξ

ηξηξηξη

η (5.7)

Fig. 5.2 Sketches for wing spar and rib


where s is the wing span, and )(ξη r is the rib position function.

For the rib web:

( ) ( )[ ] ( )[ ]{ }∫∫∫

∫ ∫ ∫∫∫∫−−

++−−

−

+

−

−−

++

⋅++=

⋅=

221

221

2

2

221

221

,)(,,)(,

]),,(),,([),,(

22

1

1 2

1

1

1

1

htz

htz rr

s

t

s

t

htz

htzV

UU

LL

r

r

UU

LL

dzJzstystxFdstd

dzddJzyxFdxdydzzyxF

ξηηξξηηξηξ

ηξηξηξη

η (5.8)

The same as for the spars, integration on ribs can be obtained by summing up contributions from all

the ribs.

Calculation of Eqs. (4.23) and (4.30) using Eqs. (5.7) and (5.8) give the stiffness and mass

matrices of the skins: ][ ribK and ][ ribM .

5.2 Boundary Conditions

The boundary conditions can be approximated using artificial springs on the boundary.

Applying linear springs with very large magnitudes of stiffness on the boundaries can approximate

the boundary conditions of simply supported edge. Applying linear and rotational springs with very

large magnitudes of stiffness on the boundaries can approximate the boundary conditions of

clamped edge. While details of these practices for a general case can be found in Lovejoy and

Kapania 21,20 , in the following the special case with the wing being clamped at its root will be

treated.

Assume that in Fig. 4.1, there are artificial springs distributed along the side of 12

)1,11( −=≤≤− ηξ . These are linear springs with constant stiff coefficients of xα , yα , zα , xφα ,

and yφα respectively, which are responsive only to displacements of 0u , 0v , 0w , xφ and yφ

respectively. The strain energy that these springs possess can be written as

∫ ++++=12

2220

20

202

1 )( dlwvuU yyxxzyxspring φαφαααα φφ (5.9)


Since at side 12 0,2/)}1()1({ 21 =++−= yxxx ξξ , and rcxx == 21 ,0 , here rc is the

chord-length at the root, we have ξξ dcdxxdxdl r21

12 2/)( =−== , therefore Eq. (5.9) becomes

∫−++++=

1

1

2220

20

204

1 )( ξφαφαααα φφ dwvucU yyxxzyxrspring (5.10)

Considering Eqs. (4.9) and (4.11), and comparing Eq. (5.10) with }]{[}{21 qKqU spring

Tspring = ,

we obtain the stiffness matrix for the springs as

])[],[],[],[],([][ yxzyxspring KKKKKdiagK φφααα= (5.11)

where

−−=

−−=

−−=

−−=

−−=

∫∫∫∫∫

−

−

−

−

−

1

121

1

121

1

121

1

121

1

121

)}1,()}{1,({][

)}1,()}{1,({][

)}1,()}{1,({][

)}1,()}{1,({][

)}1,()}{1,({][

ξξξα

ξξξα

ξξξα

ξξξα

ξξξα

φφ

φφ

α

α

α

dBBcK

dBBcK

dBBcK

dBBcK

dBBcK

TRSRSyry

TPQPQxrx

TMNMNzrz

TKLKLyry

TIJIJxrx

(5.12)

The magnitudes of xα , yα , zα , xφα , and yφα must be large enough such that the boundary

conditions are properly simulated. But they cannot be too large, or else all the stiffness values other

than those of these springs will be pushed beyond the significant figures of the computation system.

This is to say, the spring magnitudes need to be within a range in order that the boundary condition

is properly simulated. This range depends on the specifics of the computation environment and the

computer system. In Figs. 5.3 and 5.4 the first 10 natural frequencies of a wing that we shall

specify in 5.4 , obtained using a method to be explained in 5.3 , are shown with regard to the spring

magnitudes. In these cases, it is assumed valueSpringyxzyx ===== φφ ααααα , and 6 and 8

terms of the Lengendre polynomials are used respectively. Since the computation is performed

using MATLAB 5.2, double precision is used. It can seen from both Figs. 5.3 and 5.4 that within the

spring value of 216~8 /(10 inlb or )/ inlb , all of the natural frequencies are stable. In all the

following cases, the spring value will be specified as 1210 .


Log10 (Spring value)

W1

,...,W

10

0 5 10 15 200

500

1000

1500

2000

2500

K=6


simulating spring value, when 6 terms of Legendre polynomials are used


Log10 (Spring value)

W1,

...,W

10

0 5 10 15 200

500

1000

1500

2000

2500

K=8


simulating spring value, when 8 terms of Legendre polynomials are used


5.3 Formulation for Vibration Problem of Wing

Under the assumption that the wing is a conservative system, the Lagrange's equations58 for

free vibration are

0=∂∂−

∂∂

jj q

L

q

L

dt

d�

µν,,1 �=j , .,,,, RSPQMNKLIJ=µν (5.13)

Since the Lagrangian for the wing is

UTVTL −=−= (5.14)

where V is the potential energy, and by using Eqs. (4.22), (4.29) and (5.13), we can find the

natural frequencies and mode shapes for the free vibrating wing by solving the following

eigenvalue problem

[ ] 0}{ =− xMK totaltotal λ (5.15)

where

][][][][],[][][ ribsparskinstrainspringstraintotal KKKKKKK ++=+= , (5.16)

][][][][ ribsparskintotal MMMM ++= , (5.17)

2ωλ = is an eigenvalue of the system of equations, ω is the corresponding frequency in

radians/second, and }{ x is the corresponding eigenvector.

5.4 Convergence Test

Since in Eq. (4.9) only a combination of finite terms of trial functions are used to represent the

deformation of the wing, it is obvious that any results coming from this representation would be an

approximation. This is especially true for the higher modes of the free vibration problem if the

Legendre or Chebyshev polynomials are being employed as the trial functions. The validity of the

method can be established only if the results are converging when more trial function terms are

used. For this purpose, two wings with the following configuration particulars (more details will be

given in 5.6)are used:


Wing I: sweep angle $30=Λ , span in192= , chord length at root in72= , chord length at tip

in36= ;

Wing II: sweep angle $891.24=Λ , span in857.72= , chord length at root in627.49= , chord

length at tip in332.24= .

Figs. 5.5 and 5.6 show the first 10 natural frequencies of Wing I and II when 4 to 10 terms of

the Legendre polynomials are used. More cases of convergence tests for plates are reported in

Lovejoy and Kapania 21,20 . We can have the following rule of thumb: when K terms of trial

functions are employed, the first K natural frequencies will have converged or nearly converged

values. Based on this rule, either 6=K or 8=K is used in all calculations in following sections

and chapters.


K (Number of Lengendre plynomials)

Nat

ural

frequ

enci

es(r

ad/s

ec)

3 4 5 6 7 8 9 100

1000

2000

3000

4000

N=1N=2N=3N=4N=5N=6N=7N=8N=9N=10

Fig. 5.5 Natural frequencies of wing I with regard to number of trial function terms


K (Number of Lengendre plynomials)

Nat

ural

frequ

enci

es(r

ad/s

ec)

3 4 5 6 7 8 9 100

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

N=1N=2N=3N=4N=5N=6N=7N=8N=9N=10

Fig. 5.6 Natural frequencies of wing II with regard to number of trial function terms


5.5 Static Problem Solutions

Assume that an external, distributed force with components ),,( tyxPx , ),,( tyxPy and

),,( tyxPz is applied on the wing structure, then the virtual work done by this load on the

infinitesimal area dydx ⋅ is

zyx QQQQ δδδδ ++= (5.18)

where

dydxtzyxwtyxPQ

dydxtzyxvtyxPQ

dydxtzyxutyxPQ

zz

yy

xx

⋅⋅⋅=⋅⋅⋅=⋅⋅⋅=

),,,(),,(

),,,(),,(

),,,(),,(

δδδδδδ

(5.19)

and by using Eq. (4.1) we obtain

dydxwPQ

dydxzvPQ

dydxzuPQ

zz

yyy

xxx

⋅⋅⋅=⋅⋅⋅+⋅=⋅⋅⋅+⋅=

0

0

0

)(

)(

δδδφδδδφδδ

(5.20)

Using Eq. (4.6), that is, approximating the displacements yxwvu φφ ,,,, 000 in terms of the Ritz

functions, the total work done by the external force on the whole wing surfaces is given by

}{}{

}{}{}{}{

}{}{}{}{}{}{

qP

dxdyBzPBzP

qBPqBPqBPQ

T

yT

RSyxT

PQx

WT

MNzVT

KLyUT

IJx

δ

δφδφ

δδδδ

=

++

++= ∫∫ (5.21)

where }{P is the generalized load vector

TPPPPPP }}{}{}{}{}{{}{ 54321= (5.22)

in which

∫∫∫∫

=

=

dxdyBtyxPzP

dxdyBtyxPP

TRSPQyx

TMNKLIJzyx

}){,,(}{

}){,,(}{

,,5,4

,,,,3,2,1 (5.23)

If the external force is a concentrated force, the above derivations can be simplified. For instance

for the first component of the generalized load vector we have


TaaIJaaaax BtyxPP )},(){),,(),,((1 ηξηξηξ= (5.24)

where ),( aa ηξ is the transformed coordinates of the point where the load is applied.

Using the principle of virtual work, we have, for the static case, the following relation for the

generalized displacement vector }{q and generalized load vector }{ P

}{}]{[ PqK = (5.25)

5.6 Results and Discussion

In order to assess the accuracy of the present method and test its performance in various

situations, a series of calculations were carried out for several wing-shaped structures clamped at

the root. Results using MSC/NASTRAN, a commercial finite element code, are provided for

comparison. Finally, for the comparison between the present method and an existing FSDT

method, a swept-back box wing used as a test case in Livne31 were calculated for its free vibration

and static response analyses.

5.6.1 Free Vibration Analysis

5.6.1.1 A Trapezoidal Plate

The geometric and material parameters for the plate are given as: Span=192 inches, Root

width=72 inches, Tip width=36 inches, Sweep angle (leading edge)=$30 , Thickness=1.8 inch

(thickness ration at tip=1/20); Mass Density 424 /sec10526.2 inlb ⋅×= −ρ , Young’s Modulus

27 /10025.1 inlbE ×= , Poisson's Ratio 3.0=ν . The plate is clamped at the root.

Comparisons are made in Fig. 5.7 between the mode shapes as obtained by the present method

and those by the FEA calculations using MSC/NASTRAN employing 200 shell elements

(CQUAD4). The comparison of the natural frequencies is also shown in Fig. 5.7. It can be seen that

both the mode shapes and natural frequencies as obtained using the present method, are in good


agreement with those obtained using the FEA. The relative differences of the natural frequencies

for the first 8 modes are within -0.62~2.12%.

by FEA Equivalent Plate Model

Fig. 5.7 Mode Shapes and Natural Frequency f )/( srad for a Trapezoidal Plate


5.6.1.2 A Trapezoidal Shell with a Camber

All parameters are the same as with the previous case except that there is a camber with the

camber-chord ratio varying from 2.345% at the root to 0.938% at the tip.

Comparisons are made between the results as obtained by the present method with those

obtained using the FEA in MSC/NASTRAN employing 200 shell elements(CQUAD4), as shown

in Fig. 5.8.

It can be seen that, although this case is very similar to the previous one except for a small

camber, there are significant differences in the natural frequencies of a number of modes. Most of

the variations were predicted quite accurately by the present method, as shown clearly in the

comparison with the FEA results in Fig. 5.8. But the relative differences were slightly higher than

the ones in the previous case, varying in a range of -1.31% to 5.26% for the first 8 modes. Larger

differences for the present case can be attributed to the fact that the present method ignores the

coupling between the in-plane and transverse displacements caused by the mid-surface curvature.



Fig. 5.8 Mode Shapes and Natural Frequency f )/( srad

for Wing-Shaped Shell with a Camber


5.6.1.3 A Solid Wing

The middle surface of this wing is the same as that of the previous case. Its thickness-chord

ratio is varied from 0.15 at the root to 0.06 at the tip. The sections were generated by the Karman-

Trefftz transformation (see Ref. 59 and Appendix C).

Comparisons are made in Fig. 5.9 between the results as obtained by the present method with

those obtained by the FEA calculations using MSC/NASTRAN employing 250 solid elements

(CHEXA and CPENTA) and 572 nodes. It can be seen that although there are thickness variations

as well as a camber, the present method yields results that compare quite well with those obtained

using the FEA. The relative differences for the first 8 modes were within -5.82~1.42%, comparable

to those in the previous case.



Fig. 5.9 Mode Shapes and Natural Frequency f )/( srad for the Solid Wing


5.6.1.4 A Built-up Wing Composed of Skins, Spars and Ribs

The outside geometrical shape is the same as in the previous case, the solid wing. There are 4

spars and 10 ribs distributed uniformly in the wing. Sketch of the wing cross-section shapes and rib

and spar positions is shown in Fig. 5.10. Particulars of the wing are: Skin Thickness int 118.0 = ;

Spar Cap Height inh 197.1 = , Spar Cap Width inl 373.1 = , Spar Web Thickness int 058.1 = ; the

ribs have the same cap dimensions and web thickness as the spars.

The FEA calculations are made by using MSC/NASTRAN employing 370 elements and 110

nodes. The wing skins were modeled using shell elements (CQUAD4), the spar and rib caps were

modeled using bar elements (CBAR), and the spar and rib webs were modeled using shear panel

elements (CSHEAR). Comparison between the mode shapes as well as the corresponding natural

frequencies as obtained by the two methods are shown in Fig. 5.11. It can be seen that the mode

shapes were simulated equally well by the present method as compared to the FEA, and it is found

that the relative differences for the first 8 modes were within -4.79~2.15 %, comparable to those in

the previous cases.


X Y

Z

Fig. 5.10 Wing cross-sections at rib positions and spar positions



Fig. 5.11 Mode Shapes and Natural Frequency f )/( srad

for a Built-up Wing Composed of Skins, Spars and Ribs


5.6.1.5 A Box Wing used as a test case in Livne21 .

This is a cantilevered all aluminum wing swept back by $30 . It has a constant thickness and

constant chord length. Its 5 spars and 3 ribs with identical cross sections are bonded to the top and

bottom cover skins (see Fig. 5.12). Details of this box wing can be found in Refs. 31 and 60. The

same kinds of elements were employed as in the previous case. Results for the natural frequencies

by the FEA using MSC/NASTRAN and the present FSDT are shown in Table 5.1, in comparison

with those given in Livne31 by the FEA using ELFINI and a FSDT based on simple-polynomial

trial functions. While there are some differences between the two FEA calculations, which may

have been caused by different discretization and element choices, the accuracy of the present FSDT

results are promising.

Fig. 5.12 A box wing

Table 5.1 Natural frequencies (Hz) of the cantilevered swept-back box wing

Mode No.

Description ofMode Shape

FEA(Livne31)

FSDT(Livne31)

FEA(present)

FSDT(present)

1 1st bending 115.6 114.7 116.6 118.02 In plane 317.6 312.4 327.9 349.73 1st torsion 418.4 428.9 409.4 419.14 2nd bending 576.4 575.3 572.1 571.15 2nd torsion 1086 1125 1064 1090


5.6.2 Displacement under Static Loads

The built-up wing in the 4-th case (5.4.1.4) of the free vibration analysis is used here. Three

cases of static load were considered as shown as follows.

5.6.2.1 Tip Point Force

A downward (-z-direction) force of magnitude of 1lb is applied at the middle point of the

wing tip. The displacements along the leading and trailing edge of the wing are shown in Fig. 5.13.

It can be seen that the present method calculated the vertical displacement w accurately compared

with the FEA, and also predicted quite well the trends of variation for the other two displacement

components, u and v .

5.6.2.2 A Force Distribution

A downward (-z-direction) force of magnitude of 1lb is applied at every upper-surface nodes

of the FEA model. This is a case similar to the wing being under uniform pressure difference

between its upper and lower surfaces. The displacements along the leading and trailing edge of the

wing are shown in Fig. 5.14. Quite similar results to those in load case 5.4.1.2 were obtained.

5.6.2.3 Tip Torque

A downward (-z-direction) force of magnitude of 1lb is applied at the tip of the fore-most

spar, while an upward (z-direction) force of magnitude of 1lb is applied at the tip of the aft-most

spar. This is a case in which the wing tip is subjected to a torque. The displacements along the

leading and trailing edges of the wing are shown in Fig. 5.15. The relative difference for w at the

tip/leading-edge corner and tip/trailing-edge corner are 26.4% and 2.64% respectively. Note that

the absolute differences are 410397.0 −× inch and 410181.0 −× inch respectively; therefore the large

relative difference at the tip/leading-edge corner is because of the small magnitude of w along the

leading edge, and the difference between the twist angles predicted by the two methods would be

small.


Y (inch)

100

u,10

0v,w

(inch

)

0 50 100 150-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

100u(FEM)100v(FEM)w(FEM)100u(present cal.)100v(present cal.)w(present cal.)

x 10-3

(a) leading edge

Y (inch)

10

0u,

10

0v,

w(in

ch)

0 50 100 150-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5


x 10-3

(b) trailing edge

Fig. 5.13 Comparison of Displacements for Load Case of Tip Point Force


Y (inch)

100u

,100

v,w

(inch

)

0 50 100 150-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01


(a) leading edge

Y (inch)

100

u,10

0v,

w(in

ch)

0 50 100 150-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01


(b) trailing edge

Fig. 5.14 Comparison of Displacements for Load Case of a Force Distribution


Y (inch)

100

u,10

0v,w

(inch

)

0 50 100 150-5

0

5

10

15

20


x 10-5

(a) leading edge

Y (inch)

100u

,100

v,w

(inch

)

0 50 100 150-1

0

1

2

3

4

5

6

7

8


x 10-4

(b) trailing edge

Fig. 5.15 Comparison of Displacements for Load Case of Tip Torque


5.6.2.4 The Box Wing in Livne21

The swept-back box wing is under a downward point force of 1lb at the tip of the rear spar.

Displacements at the wing tip from measurement, FEA calculation and present method are shown

in Table 5.2. It can be seen that the present method yielded very good results for this test case.

Table 5.2 Displacement (in) of the cantilevered swept-back box wing

Position Measured 60,31 FEA(present) FSDT(present)Front spar tip 41081.1 −× 41079.1 −× 41074.1 −×Rear spar tip 41021.2 −× 41019.2 −× 41020.2 −×

5.6.3 Skin Stress Distributions

The upper and lower skin stress of the wing in the above static case (5.4.2.1) were calculated

using the present method. The Von Mises stress distribution along a line with a distance of 5% span

to the root chord is shown in Fig. 5.16(a) in comparison with points obtained using

MSC/NASTRAN. Also the Von Mises stress distribution along a span-wise line with a distance of

37.5% chord length to the leading edge is shown in Fig. 5.16(b) in comparison with points obtained

using MSC/NASTRAN. It can be seen that, although there are substantial differences (the largest

one is about 15%) between the present calculations and the stresses determined using the FEA, the

variation trends of the stresses from both methods are quite similar. This means that the position of

the largest stress determined by the present method will be reliable. The Von Mises stress

distributions on the upper and lower skins calculated using the present method are shown in Figs.

5.17 and 5.18 respectively.


Relative Distance to Leading-Edge

Von

Mis

esS

tress

(psi

.)

0 0.25 0.5 0.75 10

1

2

3

4

5

Lower Skin (FEM)Upper Skin (FEM)Lower Skin (present cal.)Upper Skin (present cal.)

(a) Near the Root Chord

Relative Distance to Wing Root

Von

Mis

esS

tress

(psi

.)

0 0.25 0.5 0.75 10

1

2

3

4

5

6

Lower Skin (FEM)Upper Skin (FEM)Lower Skin (present cal.)Upper Skin (present cal.)

(b) Near the Central Spar

Fig. 5.16 Comparison of Von Mises Stress on the Upper and Lower Skins

of a Wing under a Point Force at the Wing Tip


0

2

4

6

Von

Mis

esst

ress

(psi

)

0

0.2

0.4

0.6

0.8

1 Relative dist ance to the root

0

0.2

0.4

0.6

0.8

1

Relative dist ance to leading-edge

0.85

5.04

0.50

3.65

3.65

2.25

2.95

4.70

2.252.95

3.65

2.60

3.30

0.85

4.704.70

3.65

4.35

2.60 2.60

2.25

1.55

0.50

X Y

Z

Von Mises Stress on Upper skin

Fig. 5.17 Distribution of Von Mises Stress on the Upper Skin



0

2

4

6

Von

Mis

esst

ress

(psi

)

0

0.2

0.4

0.6

0.8

1 Relative dist ance to the root

0

0.2

0.4

0.6

0.8

1

Relative dist ance to leading-edge

4.00

2.95

4.35

1.90

2.953.30

1.90

1.551.20

1.90

X Y

Z

Von Mises Stress on Lower Skin

Fig. 5.18 Distribution of Von Mises Stress on the Lower Skin



5.6.3 On Efficiency of EPA

To show the efficiency of the present equivalent plate analysis of built-up wings, the ideal

situation would be comparison of the CPU times used in EPA and other analysis methods, such as

FEA. But it seems to the author that only under the following situations the comparison can be

made meaningful: (1) The two calculations are to be carried out in the same computing

environment to exclude the difference that different environments may cause, that is to say, both

should be coded using the same language, such as in FORTRAN or the MATLAB m file; (2) Similar

linear algebraic solvers and other type of supplementary algorithms, be used. Unfortunately, the

author could not establish a comparison based upon such requirements. While the EPA was coded

in the MATLAB m file, the FEA was carried out using a commercial package, MSC/NASTRAN.

But we still can perform a comparison of efficiency in an indirect way, that is, by comparing

the number of degrees of freedom (DOF), and elements used. DOF determines the dimensions of

the stiffness and mass matrices, therefore directly influences the CPU time required to solve a static

or dynamic problem through operations upon these matrices. The number of elements, on the other

hand, represents the effort needed to set up element matrices and assemble them together to create

the global matrices. For the case of EPA, the number of elements can be deemed as k2.

Details of DOF and number of elements for several of the previous example cases are displayed

in Table 5.3. As shown in Table 5.3, for these simple cases, the FEA uses thousands or hundreds of

DOF, while for the EPA it is always 180 or 360, depending on whether 6 or 8 terms of polynomials

are used. That means a very big difference on computational effort for either static or dynamic

problems, if the same linear algebraic solvers are used. However, in reality the more efficient

sparse matrix solvers can be used for FEA, therefore the difference of efficiency would not be a

direct comparison of the DOFs.

Similar situation is shown for the comparison of numbers of elements in Table 5.3. Still we

cannot say that the ratio of the computational effort for the element matrix terms for FEA and EPA

is the ratio of the numbers of elements, because for EPA the elements are global, and more effort is


needed for calculating the element matrices. It is based on this reasoning that a more efficient

approach is pursued in Chapter 7.

It is worth noting that for the FEA of the built-up wing the ratio of number of elements to DOF

is the largest, more than 3 times higher than the other cases. This indicates a relative larger

computational effort on calculating and assembling the element matrices in the FEA, also more

efforts for calculating the contributions from the structure components in the EPA. Therefore it

seems that the difference in efficiency of FEA and EPA is more reliably reflected in the difference

between the DOFs.

Table 5.3 Comparison of FEA and EPA in terms of DOF and Number of Elements

FEA EPAExample

DOF

No. ofElements

DOF

k=6DOF

k=8No. of

Elementsk=6

No. ofElements

k=8

Trapezoidal plate & shell 1350 200 180 320 36 64

Solid wing 3300 250 180 320 36 64

Built-up wing 600 370 180 320 36 64

75

Chapter 6

Modal Response Using Sensitivity Techniques

and Direct Application of Neural Networks

The modal response of wing structures is very important for assessing their dynamic response

including dynamic aeroelastic instabilities. Moreover, in a recent study61 an efficient structural

optimization approach was developed using structural modes to represent the static aeroelastic

wing response (both displacements and stresses).

In this chapter, the natural frequencies of general trapezoidal wing structures are to be

approximated using shape sensitivities up to the 2nd order, and different approaches of computing

the derivatives are investigated. The baseline design and shape sensitivities are calculated based on

the equivalent plate-model analysis (EPA) method developed by Chapter 4 and 5. For

comparison, an efficient method that employs the artificial neural networks to relate the natural

frequencies of a wing to its shape variables is also established. An example of a 34 full factorial

experimental design, i.e., 4 levels in 3 variables, is treated by these methods to display their

respective merits.

CHAPTER 6 MODAL RESPONSE USING ST AND NN 76

6.1 Shape Sensitivities

For a trapezoidal wing, there are four major independent shape variables: 1) the sweep angleΛ ,

2) the aspect ratioα , 3) the taper ratio τ , and 4) the plan area A . All the other dimensions of the

wing plate configuration can be calculated using these parameters as follows:

)1(2,)1(2, τατατα +=+== sbsaAs (6.1)

where s is the length of semi-span, a and b are the chord-length at wing tip and root

respectively, as shown in Fig. 6.1.

x

y

sa

b

Λ

Fig. 6.1 Plan configuration of a trapezoidal wing: .,),( 221 baAsbasA ==+= τα

The sensitivities for the design parameters at a baseline design point indicate trends in the

response of the baseline design if the parameters are perturbed. Usually, only the first order

derivatives are used:

)(),,,(),,,( 01

00

20

10

21 iin

ii

nn xxx

fxxxfxxxf −

∂∂+≅ ∑

=

�� (6.2)


where ),,,(

0

020

10

nxxxii x

f

x

f

�

∂∂=

∂∂

is the sensitivity at the baseline point with respect to the i -th design

parameter. For a more accurate approximation, we can use higher-order derivatives in the Taylor’s

expression:

),,,()(

),,,()(),,,(),,,(

020

10

2

102

1

10

20

1000

20

10

21

nn

ii

ii

n

i

ni

iinn

xxxfx

xx

xxxfx

xxxxxfxxxf

�

��

∂∂−+

∂∂−+≅

∑

∑

=

= (6.3)

where besides the first order derivatives, second order derivatives ),1,(02

njixx

fji

�=∂∂

∂ are also

used.

6.2 An Issue in Equivalent Plate Analysis (EPA)

Due to its efficiency in determining the natural frequencies and mode shapes of wings, the

Equivalent Plate Analysis (EPA) described in Chapters 4 and 5 can be used to investigate the

variation of modal response, that is, to evaluate the sensitivities of the natural frequencies with

respect to trapezoidal wing structures shape changes. For determining the response of the baseline

design, the EPA can be used, or the FEA employing a commercial package such as

MSC/NASTRAN can be used for better accuracy.

A key problem that needs to be addressed before this evaluation can be made is mode tracking.

The natural frequencies given by an ordinary eigenvalue solver are usually ranked by magnitude

but not by the modal content. As design variables are perturbed, frequencies drift and mode

crossing may occur. An algorithm for mode tracking is needed to maintain the correspondence

between eigenpairs of the baseline and the perturbed design. Several methods for such purpose

have been given by Eldred et al for self-adjoint62 and nonself-adjoint63 eigenvalue problems.


In the present study, a simple yet effective method is used. In this method, any ordinary

eigenvalue solver can be used, and the modes of the baseline structure are chosen as the

benchmarks. By using the modal assurance criterion (MAC) defined as

}){}})({{}({

}){}({ 2

iT

ijT

j

iT

jjiMAC

φφφφφφ

= (6.4)

where }{ jφ and }{ iφ are the eigenvector for the perturbed and the baseline design respectively, if

)(max lil

ji MACMAC = , we say that the j -th mode of the perturbed design corresponds to the i -th

mode of the baseline structure.

6.3 Approaches to Sensitivity Evaluation

There can be three kinds of approach for obtaining sensitivity derivatives: the finite difference

approach, the analytical approach, and the semi-analytical approaches. The finite difference

approach is very simple to formulate and implement, but is numerically inefficient and is sensitive

to the step-size used. A too-large step size usually causes significant truncation errors and a too-

small step size may lead to large round-off errors. As a result, the more elegant and accurate

analytical approach is used if it does not involve complex mathematical derivation. But for most

practical problems, the derivation of analytical derivatives is too formidable to handle manually.

The basic idea behind the Automatic Differentiation (AD) is to let a computer to perform such

extensive tasks. The advantage of AD is to avoid truncation errors. The method has found

applications in sensitivity evaluation42,41 . For the basic theory of AD one can consult Ref. 64, and

for the state-of-the-art of AD one can refer to Ref. 65. If an approach uses both analytical and

finite-difference solutions to obtain the derivative, then it can be called a semi-analytical one.

The finite difference approaches can be constructed using the following formulas:

)(2

)()()( 2xO

x

xxfxxfxf ∆+

∆∆−−∆+=′ (6.5)


)()()(2)(

)( 2

2xO

x

xxfxfxxfxf ∆+

∆∆−+−∆+=′′ (6.6)

where

xx ⋅=∆ ε (6.7)

in which ε is the relative step size, but herein it is simply called the step size. Eq. (6.5) can be

applied twice for evaluating the mixed second order derivatives such as )(02

jixx

fji

≠∂∂

∂.

The analytical approaches for shape sensitivities of modal response can be based on the

following equations

}{][][

}{ iiT

ii MK φ

θλ

θφ

θλ

∂∂−

∂∂=

∂∂

(6.8)

∑=

=∂

∂ n

jjij

i

1

}{}{

φαθφ

(6.9)

where

∂∂−=

≠

∂∂−

∂∂

−=

}{][

}{2

1

},{][][

}{)(

1

iT

iii

iiT

iji

ij

M

ijMK

φθ

φα

φθ

λθ

φλλ

α (6.10)

here θ is the shape variable, iλ and }{ iφ are the i -th eigenvalue and eigenvetor, and }{ iφ is mass-

normalized such that 1}]{[}{ =iT

i M φφ . Eqs. (6.8) and (6.9) were first derived by Wittrick66 and

Fox and Kapoor67 respectively. One can find more on this topic in Ref. 68.

The major difficulty of applying Eqs. (6.8) and (6.9) lies in the calculation of θ∂

∂ ][ K and

θ∂∂ ][ M

. For instance, consider θ∂

∂ ][ K. According to Chapter 4, the stiffness matrix ][K is

formulated as an integral


ηξ

ηξ

ddCGC

ddCdzJTDTCdVCTDTCK

T

z

z

TT

V

TT

][][][

][][][][][]][][[][][][

1

1

1

1

1

1

1

1

2

1

∫ ∫

∫ ∫ ∫∫∫∫

− −

− −

=

==

(6.11)

where only the inner part dzJTDTGz

z

T ][][][][2

1∫= is a function of the shape variables, and the

Gaussian quadrature is used to obtain the integration on ξ and η . Therefore,

ηξθθ

ddCG

CK T ][

][][

][ 1

1

1

1∫ ∫− −

∂∂=

∂∂

(6.12)

in which θ∂

∂ ][Gcan either be determined analytically or numerically.

People often make use of the advantages of both the finite difference and analytical approaches

in different stages of obtaining some complicated sensitivities. While trying to use the analytical

approach as much as possible, in other parts of the process the finite difference is used, as in the

cases of Refs. 40 and 42. This kind of approach is usually called semi-analytical.

In summary, there are three approaches to calculate the first order modal sensitivities:

(i) analytical approach: Eqs.(6.8)~(6.10) are used, and θ∂

∂ ][ K and

θ∂∂ ][ M

are determined

analytically.

(ii) semi-analytical approach: Sensitivities θ∂

∂ ][K and

θ∂∂ ][M

in Eqs.(6.8)~(6.10) are

determined numerically, that is, for the case of θ∂

∂ ][K, Eq. (6.12) is used where

θ∂∂ ][G

is calculated

using a finite difference scheme.

(iii) finite difference approach: θλ

∂∂ i and

θφ

∂∂ }{ i are determined using Eq. (6.5) directly.

For the second order sensitivities, there can still be three approaches as specified above. While

the formulation for the analytical approaches is becoming more complicated, a scheme as simple as

Eq. (6.6) can be used for the finite difference approach.


6.4 Application of Sensitivity Technique (ST) in Multi-variable

Optimization

In a multi-variable case, the following formulation is used instead of Eq. (6.3):

RR

RR TTi

2

)(2

1)()()(

∂∂−+

∂∂−+≅

ii

iii p

ppp

pppp (6.13)

where Tnvvv ),,,( 21�=p is an arbitrary point in the design space, Tn

iii vvv ),,,( 21�=ip is the i -

th node point in the design space, )( piR is the response at p estimated by using the response and

its sensitivities at ip , )( ipR is the response at the i -th node point ip , and

ippip

=

∂∂

∂∂

∂∂=

∂∂

T

nvvv,,, 21 � .

Once there are enough estimates for the response at p using Eq. (6.13), a more accurate

evaluation of response at p can be determined using the following weighting procedure involving

the so-called exponentially decaying influence function35 :

∑=i

ii RwR )()()( ppp (6.14)

where i ranges through the wN design points which are closest to p , and the weight coefficients

)( piw are determined such that its sum is unity:

∑ −−−−

=

ip

pi C

Cw

)exp(

)exp()(

i

i

pp

ppp (6.15)

in which pC is an empirical constant, and the distance between p and ip is defined as

∑=

−=−n

j

ji

j vv1

2)(ipp . It can be seen that 1)( =∑i

iw p .


6.5 Application of Neural Networks (NN)

In this case, the input layer includes all the design variables of interest (for instance, the four

shape parameters of the wing plan form). The output layer gives the desired structural responses,

such as natural frequencies etc. The EPA is being used as the training data generator, though if

necessary, results obtained using FEM can also be used as the training data. Preparation of training

data is very important, and the training algorithm used also greatly impacts the training process23 .

Caution must be exerted in specifying the network parameters and training criterion so that the

results of the trained network would not oscillate around the training data. The direct application is

what we do in this paper.

6.6 Examples and Discussion

6.6.1 Results on sensitivity evaluation

Particulars of the baseline wing structure are as follows: the sweep angle $30=Λ , the aspect

ratio 5.3=α , the taper ratio 5.0=τ , the plan area 25832inA = . The wing sections are generated

using the Karman-Trefftz transformation (Ref. 59, and details in Appendix C) and has a thickness-

chord ratio of 0.15 at the wing root and 0.06 at the tip. The skin thickness int 118.0 = . There are 4

spars and 10 ribs distributed uniformly under the skins. Particulars of the spars and ribs are the

same: the cap height inh 197.1 = , the cap width inl 373.1 = , and the web thickness int 059.1 = .

There is only one material used with mass density 424 /sec10526.2 inlb ⋅×= −ρ , Young’s modulus

27 /10025.1 inlbE ×= , and Poisson's ratio 3.0=ν . The wing is clamped at the root.

An example of using EPA to calculate the natural frequencies with regard to shape variables

while tracking modes by evaluating MACs is provided in Fig. 6.2, where the variation of the

natural frequencies of the first 10 modes w.r.t. the aspect ratio are shown. It can be seen that for

most cases the intersection of natural frequencies has been treated well, and only in a few cases the

frequency variations near the intersection point seem to have a minor problem, probably due to


some kind of interaction between the two modes. If an eigenvalue solver that can work more

accurately with repeated eigenvalues is made use of, the situation can be improved.

The effect of step size on the finite difference approach for sensitivities was investigated for all

the four shape variables. The case with the taper ratio is shown in Fig. 6.3. From all the cases, it is

seen that for the best results for both the 1st and 2nd order sensitivities, the step size ε defined in

Eq. (6.7), should be between 0.005~0.015, and for fairly accurate results ε can be between

0.0017~0.045.

To evaluate θ∂

∂ ][ G analytically proved to be formidable except only in some simplified cases.

In order to compare the sensitivities using the analytical, semi-analytical and finite difference

approach, a special case of the above baseline wing with a constant thickness was considered so

that the analytical derivation of θ∂

∂ ][G in Eq. (6.12) is not formidable. When ε is specified as

0.005, it is found that for the 1st order sensitivities to the four shape variables ( ,,, ταΛ and A ) the

relative difference (averaged for the first 10 modes) between the finite difference and analytical

approach is 0.003%, 0.003%, 0.002% and 0.003% respectively. The relative difference between the

semi-analytical and analytical approach is 0.14%, 0.04%, 0.02% and 0.01% respectively. Therefore

in this case the finite difference approach is more accurate than the semi-analytical one, however

both the approaches yield quite accurate results.


Aspect Ratio

Nat

ural

Fre

quen

cies

(rad

/sec

)

2 3 4 50

500

1000

1500

2000

2500

3000

3500

Fig. 6.2 Natural frequencies using equivalent plate analysis with mode tracking


Log (step size)

Sen

sitiv

ities

-4 -3 -2 -1-3000

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

1st Order(2nd Order)/200

10

step size =0.005-0.015

step size =0.0017-0.045

Fig. 6.3 Effect of the finite difference step size on the sensitivities

of various natural frequencies to taper ratio


For the original baseline wing, since the derivation of the analytical derivatives for θ∂

∂ ][G is

too formidable, only the comparison of the 1st order sensitivities using the finite difference and the

semi-analytical approach was made. It is found in this case the sensitivities to the aspect ratioα ,

taper ratio τ and plan area A using both approaches are quite close, the average difference for the

first 10 modes being in the range of 0.5~1.4%. As an example, the 2nd natural frequency w.r.t. A is

shown in Fig. 6.4, where it can be seen that the 1st order sensitivities using the finite difference and

the semi-analytical approach almost coincide with each other. On the other hand, sensitivities to the

sweep angle Λ using the two approaches have had some quite large relative differences especially

for modes whose sensitivity to Λ is small. One such example, the 3rd natural frequency w.r.t. Λ , is

shown in Fig. 6.5, where attention should be paid to the scale for the vertical coordinate to see how

small the sensitivity to Λ is.

It is observed in Fig. 6.5 that, as in the case of the constant-thickness wing, the finite difference

approach has a better performance than the semi-analytical one. In fact, in some extreme cases, the

linear approximation using the first order sensitivity obtained using the semi-analytical approach is

not at all tangent to the actual variation at the baseline point. This is not the case for that using the

finite difference approach, if the step size chosen is not too large. However, the computational

effort for the semi-analytical approach might be less than that for the finite difference approach,

since in the former case the eigenvalue problem needs to be solved only once while in the latter it

needs to be solved twice.


Plane Area (in )

Nat

ural

Fre

quen

cy(r

ad/s

ec)

4500 5000 5500 6000 6500 7000600

620

640

660

680

700

720

740

760

780

Actutal VariationLinear Approx. (fd)2nd-Order Approx.Linear Approx. (semi-an)

2

Bas

elin

e

Fig. 6.4 The 2nd natural frequency w.r.t. wing plan area

using 1st and 2nd order sensitivities


Sweep Angle (deg)

Nat

ural

Fre

quen

cy(r

ad/s

ec)

25 30 35766

767

768

769

770

771

772

773

774

775

776

Actual VariationLinear Approx. (fd)2nd-Order Approx.Linear Approx. (semi-an)

Bas

elin

e

Fig. 6.5 The 3rd natural frequency w.r.t. wing sweep angle

using 1st and 2nd order sensitivities


It is obvious from observing Figs. 6.4 and 6.5 that the approximation using sensitivities up to

the second order has much improved the results compared with the case where only the first order

sensitivity is used. Similarly it has been shown in Haftka and Gurdal69 that, for the stress-ratio in a

three-bar truss, the quadratic approximation is much more accurate than the linear one. Also it can

be seen that the second order sensitivities using the finite difference scheme of Eq. (6.6) are fairly

accurate, at least for the purpose of engineering application. Another advantage of this scheme is

that it shares the perturbation data with the first order sensitivity scheme Eq. (6.5), therefore its

evaluation has no increase in the computational effort at all.

Using the finite difference approach based on Eq. (6.5) the mixed second order sensitivities

)(02

jixx

fji

≠∂∂

∂can be readily determined. As an example, the mixed second order sensitivity on τ

and A for the first five natural frequencies were calculated, and the results are listed as follows:

0.0099, 0.0153, 0.0353, 0.0494 and 0.0156.

6.6.2 Application of Sensitivity Technique (ST) and Neural Networks (NN)

For a trapezoidal wing, there are four major independent shape variables, i.e. the sweep angle

Λ , the aspect ratio α , the taper ratio τ , and the plan area A . As an example , a 34 full factorial

experimental design with 4 levels in α,Λ , and τ respectively, was used. Particulars of the levels

of every variable are as follows: ]30,20,10,0[ $$$$=Λ , ]5.2,0.2,5.1,0.1[=α , and

]6.0,5.0,4.0,3.0[=τ . The plan area is chosen to be a constant: 23500inA = . The other particulars

are the same as in 6.7.1.

The natural frequencies of the wing structure at the 64 node points in the design space were

calculated using EPA, and the 1st and 2nd order sensitivities at these points were also determined by

finite difference using EPA3. For each mode, a feed-forward neural network with a structure of

110153 ××× , i.e. 3 inputs, 15 neurons in the first hidden layer, 10 neurons in the second hidden

layer, and 1 output, is trained using the MATLAB NN Toolbox function trainlm that trains feed-

forward network with the Levenberg-Marquardt algorithm8 . There are 64 sets of training data,


which are non-dimensionalized before the training process. Once the networks are trained, the

input-output relationships can be readily retrieved by using the function simuff.

For the application of sensitivity technique, the major task is to evaluate the sensitivities, and

to generate responses at an arbitrary design point using Eqs. (6.13) and (6.14) does not need large

amount of CPU time. The constantpC in Eq. (6.14) was specified to be 10, and 10=wN was used.

Shown in Fig. 6.6 are the first 6 natural frequencies of 20 randomly chosen wing structures

inside the box defined in terms of lower and upper bounds on the design variables specified above.

From the figure it can be seen that both of the results given by NN and ST are in very good

agreement with the desired values (those given by the EPA) except for a few cases where there are

some differences. These cases might be caused by the unstable performance of the algorithm used

for extracting eigenvalues in the EPA near the mode-crossing points, as shall be shown in Figs 6.7

and 6.8. In order to see the effects of sensitivity order, a randomly chosen path inside the design

space box is defined as

−==

==Λ=

=+−=

).1/(,

,,,

3,2,1,)1(321

10

jjj

nj

jjjjj

rrnsa

vvv

javavv

j

τα (6.16)

where jv0 and jv1 are lower and upper bounds of variable jv , for instance, $010 =v , $301

1 =v etc.,

]1,0[∈s is the range of a shape variable, and )3,2,1( =jr j are randomly determined values between

0 and 1. Results of natural frequencies of the first 4 modes for wing structures defined by points

along a path with n1=0.945, n2 =8.200, and n3=3.203 are shown in Fig. 6.7, where only the 1st

order sensitivities were used, and in Fig. 6.8, where sensitivities up to the 2nd order were used. It

can be seen that when sensitivities up to the 2nd order are used, results are effectively improved.

Generally speaking, neural networks and sensitivity technique can give equally good results,

and the former uses less time than the latter. But both methods, once the NNs are trained or the

sensitivities are obtained, are much more efficient than the EPA . For instance, the CPU times


consumed by the EPA, the sensitivity based method and the NN based method are in the ratio of

06.0:1:55 .

The example used above has only three variables. For design problems with more variables, the

method of NN and ST can still be applied in general, only at the expense of more computing time.

We can expect that similar conclusions to those obtained above still apply to these cases. For a

design problem with very large number of variables, in combination with the NN or ST method,

methodologies to shrink the design space, such as the reasonable design space approach described

in Balabanov et al70 , can be used. This can make the search of optimal design easier and at the

same time the application of NN or ST more accurate, just as the case in Ref. 26 where the

response surface approximation was used to simulate high-fidelity models. Also for this kind of

high dimensionality design problems, a full multi-level factorial experimental design is almost

impossible to use hence the methods of either NN, or ST, or even response surface are hard to

apply because the cost would be too high. In such a case, an incomplete block statistical

experimental design using the D-optimal criterion72,71 can be used, which, with much reduced

number of design node points, makes the application of NN or ST possible.


Frequencies by EPA (rad/sec)

Sim

ulat

edF

requ

enci

es(r

ad/s

ec)

0 1000 2000 3000 40000

500

1000

1500

2000

2500

3000

3500

4000

by Neural Networksby Sensitivity Technique


randomly chosen inside the box of design space, as obtained by the NN and ST

w.r.t. those obtained using a full-fledged EPA


+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

Shape Variable Range

Sim

ulat

edF

requ

enci

es(r

ad/s

ec)

0 0.25 0.5 0.75 10

500

1000

1500

2000

2500

3000

3500

4000 by EPAby Neural Networksby ST, 1st modeby ST, 2nd modeby ST, 3rd modeby ST, 4th mode

+

0 0.25 0.5 0.75 10

500

1000

1500

2000

2500

3000

3500

4000


along a path inside the box of design space (n1=0.945, n2 =8.200, n3=3.203)

using only the 1st order sensitivities


+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +

0 0.25 0.5 0.75 10

500

1000

1500

2000

2500

3000

3500

4000 by EPAby Neural Networksby ST, 1st modeby ST, 2nd modeby ST, 3rd modeby ST, 4th mode

+

Shape Variable Range

Sim

ulat

edF

requ

enci

es(r

ad/s

ec)

0 0.25 0.5 0.75 10

500

1000

1500

2000

2500

3000

3500

4000


along a path inside the box of design space (n1=0.945, n2 =8.200, n3=3.203)

using sensitivities up to the 2nd order

95

Chapter 7

Equivalent Skin Analysis Using Neural

Networks

Since the calculation of various integrals in Eqs. (4.23) and (4.30) are time-consuming, it is desired

to replace the actual wing structure by an equivalent continuum model, that is, one that is

composed of a skin-like material, whose constitutive matrix ][D and distribution of mass ρ in

Eqs. (4.23) and (4.30) respectively are to be decided.

7.1 Equivalent Skin Analysis (ESA)

The method is actually a Neural-Network-Aided Equivalent Plate-Model Analysis. For

simplicity, we can call it an Equivalent Skin Analysis (ESA) of wing structures.

It is assumed that the mass density ρ is a function of position in the plan form while each term

of ][D is a constant throughout the wing area. There can be other choices, as will be discussed

later.

We are going to solve the above problem by requiring that the stiffness and mass matrices of

the equivalent model are most approximate to those of the actual wing in a least squares sense. This

gives the following procedures.

CHAPTER 7 EQUIVALENT SKIN ANALYSIS 96

7.1.1 The Constitutive matrix

Let's write ][][][ ijstrain KKK == as the target matrix, and the stiffness matrix of the equivalent

continuum model is

][])([]~

[

][]][[][][]~

[

,,

ijpq

qppq

p q m n

ijmnpqpqij

mnmnTmn

Tmn

m nnm

GDGDK

CTDTCggK

∑∑∑ ∑∑∑∑

===

= (7.1)

where mg and ng are the Gauss quadrature weights; the constitutive matrix ][D relates the stress

and strain vectors by }]{[}{ εσ D= , and pqD is the p -th row, q -th column term of the constitutive

matrix;

m ( Km ,,1�= ) corresponds to the m -th Gauss integration position in the x-direction,

K is an integer with a usual value of 6 or 8;

n ( Kn ,,1�= ) corresponds to the n -th Gauss integration position in the y-direction;

5,,1�=p is the row number of ][D ;

5,,1�=q is the column number of ][D ;

51

][][

5

1

0

010

0

][][,

��

�

�

�

��

�

q

CTpTCggG mnmnTmn

Tmnnm

ijmnpq

= , ∑=nm

ijmnpq

ijpq GG

,, . (7.2)

By constructing an error function

[ ]2,

)(~∑ −=

jiijpqij

KijK KDKwE (7.3)

where Kijw are weight coefficients, and by requiring

[ ] 0~

)(~

2,

∑ =∂∂

−=∂∂

ji pq

ijijpqij

Kij

pq

K

D

KKDKw

D

E,


and noting

ijpq

pq

ij GD

K=

∂∂ ~

( ijpqG means the ji, -th term of matrix ][ pqG ),

we can obtain the constitutive matrix term ][ pqD by solving the following linear equation set:

∑∑ ∑ =ji

ijpqij

Kij

qpqp

ji

ijpq

ijqp

Kij GKwDGGw

,','''

,'' )( (7.4)

Kji ,,1, �= and 5,,1',',, �=qpqp .

This is an equation set with 25unknowns. Since Nji ,,1, �= and N is usually very large (if

use the Legendre polynomials of 6 terms as the basis functions, 180=N , if 8 terms are used, then

320=N ), the job of generating the matrix in Eq. (7.4) is quite extensive.

If ][D is assumed to be symmetrical, then Eq. (7.4) will become

∑∑ ∑ =

−+

ji

ijpqij

Kij

qpqp

ijpq

ijpqqp

ijqp

ji

Kij GKwDGGGw

,)','(''''''''

,

})1({ δ (7.5)

where )','( qp and ),( qp have the followings 15 combinations instead of 25: (1,1), (1,2), (1,3),

(1,4), (1,5), (2,2), (2,3), (2,4), (2,5), (3,3), (3,4), (3,5), (4,4), (4,5), and (5,5).

7.1.2 Mass distribution

Let's write ][][][ ijtotal MMM == as the target matrix, and

∑∑∑∑ ==m n

mnmnmnmnTmn

Tmn

m nnmmn FHZZHggM ][][][][][]

~[ ρρ (7.6)

as the mass matrix of the continuum model, where m ( Km ,,1�= ) corresponds to the m -th Gauss

integration position in the x-direction, n ( Kn ,,1�= ) corresponds to the n-th Gauss integration

position in the y-direction, mnρ is the mass density of the equivalent model at position (m ,n ), mρ

and nρ are integration weights, and mnmnTmn

Tmnmn HZZHF ][][][][][ = is a NN × matrix varying

with position ( nm, ) ( 25KN = ).

By constructing an error function


[ ]2

,

)(~∑ −=

jiijmnij

MijM MMwE ρ (7.7)

where Mijw are weight coefficients, and by requiring

[ ] 0~

)(~

2,

∑ =∂∂

−=∂∂

ji mn

ijijmnij

Mij

mn

MM

MMwE

ρρ

ρ,

and noting

ijmn

mn

ij FM

=∂∂ρ

~ ( ij

mnF is the ji, -th term of matrix mnF ),

we can obtain the mass distribution mnρ by solving the following linear equation set with 2K

unknowns:

KnmFMwFFwji

ijmnij

Mij

nmnm

ji

ijmn

ijnm

Mij ,,1,,)(

,','''

,'' �== ∑∑ ∑ ρ , Knm ,,1',' �= . (7.8)

In the present study the following weight coefficients are used:

+=

+=

1)(max10

1)(max10

,

,

Skinij

ji

Skinij

Mij

Skinij

ji

Skinij

Kij

MMw

KKw (7.9)

The basic idea behind this choice is that we want to form the equivalent matrices more in the way

of the skin's, which is more like a plate than the other components of the wing, i.e. spars and ribs.

Several choices about the variation of ρ and ][D have been tried, but it is found that the

present assumptions give the best results in terms of feasibility and accuracy. For instance, to be

consistent with the assumption that each term of ][D is a constant throughout the wing area, ρ can

also be assumed a constant. This certainly decreases the accuracy of the method due to the loss of

flexibility in varying ρ to simulate the target mass matrix][M , but the resultant reduction in

computational effort is small since in the first place, forming Eq. (7.8) and training the ρ -related

neural networks do not need much CPU time. In other cases, ][D was assumed to be variable in the

span-wise direction or throughout the wing area, but it is found that although the equivalent

material is more flexible to simulate the target stiffness matrix][K , the resultant ]~

[K usually has a


larger abstract error and the solution of the free vibration problem usually gives worse natural

frequencies. Moreover, the CPU time needed for generating Eq. (7.5), which requires the major

computational effort in our method, increases in a factor of about K (number of Gauss integration

points, usually with a value of 6) in the case of ][D being variable in the span-wise direction. In the

case of ][D being variable throughout the wing area, the increase can be as large as 2K times. As

we shall see in the following examples, these kinds of increase in CPU time are formidable.

7.2 Examples and Discussion

The Neural-Network-aided equivalent plate analysis (which can also be called the Equivalent

Skin Analysis, or ESA) method is compared with the ordinary equivalent plate analysis (EPA)

described in Chapter 4 and 5 for four cases where 3 to 6 design variables are involved

respectively. In some of the results, FEA results employing MSC/NASTRAN are also provided as

benchmarks.

Some common parameters of the built-up wing structures will be specified if not specified

otherwise. The sections were generated by the Karman-Trefftz transformation59 . The thickness-

chord ratio is varied from 0.15 at the root to 0.06 at the tip. Skin Thickness int 118.0 = ; Spar Cap

Height inh 197.1 = , Spar Cap Width inl 373.1 = , Spar Web Thickness int 058.1 = (for definition of

1h etc. one can see Chapter 5); the ribs have the same cap dimensions and web thickness as the

spars. Mass Density 424 /sec10526.2 inlb ⋅×= −ρ , Young’s Modulus 27 /10025.1 inlbE ×= ,

Poisson's Ratio 3.0=ν . The wing is clamped at the root.

7.2.1 Results at a design point

Before we explore the design spaces, let's show in detail how the present method works at one

specific design point. A point, included as a node point in all the design spaces that will be

considered in the following sections, is specified as:


==

====Λ

10

4

3500

45.0

75.1

15

2

rib

spar

n

n

inA

τα

$

The "equivalent skin" constitutive matrix and mass density distribution solved using Eqs. (7.5)

and (7.8) are shown in the following and in Fig. 7.1.

.

0.1481 0.0024 0.0123 0.0008- 0.0157

0.0024 0.0294 0.0041 0.0203- 0.0035

0.0123 0.0041 0.3836 0.0091 0.0040

0.0008- 0.0203- 0.0091 1.1956 0.2849

0.0157 0.0035 0.0040 0.2849 1.2515

=E

D

.

1.2034 1.4531 1.5221 1.3742 1.4151 1.3450

1.1338 1.4525 1.2346 1.1415 1.1490 1.2637

1.2507 1.4135 1.1783 1.2193 1.1939 1.1551

1.2941 1.2606 1.1869 1.2303 1.2084 1.0649

1.3352 1.2368 1.2436 1.2890 1.2629 1.0852

1.1687 1.1124 1.1230 1.1451 1.1337 0.9969

0

=ρρ

Comparison of the target stiffness and mass matrices, ][ tK and ][ tM , with their simulated

counterparts, ][ sK and ][ sM , is shown in the following and Figs. 7.2~7.7. The relative differences

for the first 10 natural frequencies by EPA and ESA are shown in Table 7.1.

%.95.4]max[

]max[%,49.2

]max[

]max[=

+−=

−

t

st

t

st

K

KK

K

KK

%.75.0]max[

]max[%,20.1

]max[

]max[=

+−=

−

t

st

t

st

M

MM

M

MM


Table 7.1 Differences between the natural frequencies by EPA and ESA

Mode number 1 2 3 4 5 6 7 8 9 10

100×−

EPA

EPAESA

f

ff0.07 2.56 -0.08 1.26 5.41 -1.37 4.24 10.05 2.50 1.40

1.25

1.5

Nor

mal

ize d

Den

sit y

-1

-0.5

0

0.5

1

ξ

-1

-0.5

0

0.5

1

η

1.23

1.261.26

1.131.101.16 1.19

1.261.33

1.16

1.19

1.361.191.331.42

1.36

1.23

1.33

1.261.29

1.161.19

1.331.39

1.16

1.29

1.491.461.421.391.361.331.291.261.231.191.161.131.101.061.03

Mass Density DistributionMass Density Distribution

Fig. 7.1 An example of mass density distribution generated using Eq. (7.8)


-6E+08

-4E+08

-2E+08

0

2E+08

4E+08

6E+08

8E+08

1E+09

[K] t

050

100150

I

0

50

100

150

J

XY

Z

Target Stiffness Matrix

Fig. 7.2 The stiffness matrix given by EPA


-6E+08

-4E+08

-2E+08

0

2E+08

4E+08

6E+08

8E+08

1E+09

[K] s

050

100150

I

0

50

100

150

J

XY

Z

Simulated Stiffness Matrix

Fig. 7.3 The stiffness matrix given by ESA


-0.04

-0.02

0

0.02

([K] t-[

K] s)

/max

([K] t)

050

100150

I

0

50

100

150

J

XY

Z

Difference of Stiffness Matrix

Fig. 7.4 Difference between stiffness matrices given by EPA and ESA


-1

-0.5

0

0.5

1

[M] t

050

100150

I

0

50

100

150

J

XY

Z

Target Mass Matrix

Fig. 7.5 The mass matrix given by EPA


-1

-0.5

0

0.5

1

[M] s

050

100150

I

0

50

100

150

J

XY

Z

Simulated Mass Matrix

Fig. 7.6 The mass matrix given by ESA


-0.02

-0.01

0

0.01

0.02

([M] t-[

M] s)

/max

([M] t)

050

100150

I

0

50

100

150

J

XY

Z

Difference of Mass Matrix

Fig. 7.7 Difference between mass matrices given by EPA and ESA


7.2.2 Three-variable case: design space I

In this case 4 spars and 10 ribs are evenly distributed inside the wing plan form under the skins.

For a trapezoidal wing, there are four major independent shape variables: sweep angleΛ , aspect

ratioα , taper ratio τ , and plan area A (see Fig. 6.1). A 33 full factorial experimental design with

3 levels in Λ , α , and τ respectively, was used. Particulars of the levels of every variable are as

follows:

===Λ °°°

}.6.0,45.0,3.0{

},5.2,75.1,0.1{

},30,15,0{

τα

For each point in this design space, the EPA is carried out, then Eqs. (7.5) and (7.8) are used to

generate the 15 constitutive matrix terms and mass densities at 36 (66× ) Gauss sampling points.

Upon the obtained "equivalent skin" constitutive matrix [D] and mass density distribution ][ρ , the

ESA is performed based on the simulated stiffness and mass matrices. For each of these

parameters, a feed-forward neural network with a structure of 110154 ××× , i.e. 4 inputs, 15

neurons in the first hidden layer, 10 neurons in the second hidden layer, and 1 output, is trained

using the MATLABµ NN Toolbox function trainlm that trains feed-forward network with the

Levenberg-Marquardt algorithm8 . Therefore, there are totally 15+36=51 networks to be trained.

There are 81 (43 ) sets of training data, which are non-dimensionalized before the training process.

Once the networks are trained, the input-output relationships can be readily retrieved by using the

function simuff.

The major computational effort was spent in generating the 81 sets of training data, with about

15 hours of CPU time being spent on a PII/350 personal computer, while less than 1 hour of CPU

time being used in training the neural networks. A set of results are given in Figs. 7.8 to 7.10 where

49 points, which mean 49 new designs, were randomly chosen within the design space box. Upon

each new design both the EPA and the ESA are performed. The plan forms of the new design are

shown in Fig. 7.8. The first 10 natural frequencies by the EPA and the ESA are compared in Fig.


7.9 and their relative differences (based on the EPA results) are shown in Fig. 7.10. It can be seen

that except for a very few cases (2 out of 490), the relative difference is within -10%~10%.

Fig. 7.11 shows 25 new designs through a randomly chosen path inside the design space box

which is defined as

−==

==Λ=

=+−=

).1/(,

,,,

3,,1,)1(321

10

jjj

nj

jjjjj

rrnsa

vvv

javavv

j

τα�

(7.10)

where jv0 and jv1 are the lower and upper bounds of variable jv , for instance, $010 =v , $301

1 =v

etc., ]1,0[∈s is the range of a shape variable, and )3,2,1( =jr j are randomly determined values

between 0 and 1. Results of natural frequencies of the first 6 modes for wing structures defined by

points along a path with n1=0.945, n2 =8.200, n3=3.203 and n4 =1.778 are shown in Fig. 7.12,

where it can be seen that results by the EPA and the ESA agree with each other quite well.


Fig. 7.8 49 randomly chosen wing plan forms in design space I


Frequency by EPA

Fre

quen

cyby

NN

-aid

edE

PA

0 1000 2000 3000 4000 5000 6000 70000

1000

2000

3000

4000

5000

6000

7000

Fig. 7.9 Comparison of the first 10 frequencies by EPA and ESA


Frequency by EPA

Rel

ativ

eer

ror

0 1000 2000 3000 4000 5000 6000 7000

-0.1

-0.05

0

0.05

0.1

0.15

Fig. 7.10 The relative errors in Fig. 7.9


Fig. 7.11 25 wing plan forms systematically varying through design space I


+ + + + + + + + + + + + + + + + + + + + + + + ++

x x x x x x x x x x x x x x x x x x x xx

xx

xx

Shape variable range

Nat

ural

frequ

ency

(rad

/sec

)

0 0.25 0.5 0.75 10

1000

2000

3000

4000

5000

6000

by EPA1st mode by NN-aided EPA2nd mode by NN-aided EPA3rd mode by NN-aided EPA4th mode by NN-aided EPA5th mode by NN-aided EPA6th mode by NN-aided EPA

+x



7.2.3 Four-variable case: design space II

In this case 4 spars and 10 ribs are evenly distributed inside the wing plan form under the skins.

A 43 full factorial experimental design with 3 levels in Λ , α , τ and A respectively, was used.

Particulars of the levels of every variable are as follows:

====Λ °°°

.}5000,3500,2000{},6.0,45.0,3.0{

},5.2,75.1,0.1{},30,15,0{2inAτ

α

For each point in this design space, the EPA is carried out, then Eqs. (7.5) and (7.8) are used to

generate the 15 constitutive matrix terms and mass densities at 36 (66× ) Gauss sampling points,

and the ESA is performed. For each of these parameters, a feed-forward neural network with a

structure of 110154 ××× , i.e. 4 inputs, 15 neurons in the first hidden layer, 10 neurons in the

second hidden layer, and 1 output, is trained using the MATLAB NN Toolbox function trainlm that

trains feed-forward network with the Levenberg-Marquardt algorithm. Therefore, there are totally

15+36=51 networks to be trained. There are 81 (43 ) sets of training data, which are non-

dimensionalized before the training process. Once the networks are trained, the input-output

relationships can be readily retrieved by using the function simuff.

The major computational effort was spent in generating the 81 sets of training data, with about

45 hours of CPU time being spent on a PII/350 personal computer, while less than 1 hour of CPU

time being used in training the neural networks. A set of results are given in Figs. 7.13 to 7.15

where 25 points, which mean 25 new designs, were randomly chosen within the design space box.

Upon each new design both the EPA and the ESA are performed. The plan forms of the new design

are shown in Fig. 7.13. The first 10 natural frequencies by the EPA and the ESA are compared in

Fig. 7.14 and their relative differences (based on the EPA results) are shown in Fig. 7.15. It

can be seen that except for a very few cases (3 out of 250), the relative difference is within -

10%~10%.


which is defined as


−==

===Λ=

=+−=

).1/(,

,,,

4,,1,)1(4321

10

jjjnj

jjjjj

rrnsa

Avvv

javavv

j

ντα�

(7.11)

where jv0 and jv1 are the lower and upper bounds of variable jv , for instance, $010 =v , $301

1 =v

etc., ]1,0[∈s is the range of a shape variable, and )4,1( �=jr j are randomly determined values

between 0 and 1. Results of natural frequencies of the first 6 modes for wing structures defined by

points along a path with n1=0.945, n2 =8.200, n3=3.203 and n4 =1.778 are shown in Fig. 7.17,

where it can be seen that results by the EPA and the ESA agree with each other quite well.

While the former results are about free vibration frequencies, Figs. 7.18 to 7.21 show some

static results. For an arbitrary new design whose plan form is shown in Fig. 7.18, a down-ward (-z

direction) point force of 1lb is applied at the mid-point of the wing tip (actually the force is

divided into components acting on the two spar tips close to the mid-point). Fig. 7.19 shows

displacements along the leading-edge by the EPA and the ESA, where wvu ,, are displacement

components in the chord-wise, span-wise, and vertical directions respectively. Figs. 7.20 and 7.21

show the Von Mises stress distributions at the wing root and the central spar respectively. It also

can be seen that the EPA and the ESA give very compatible static results.


Fig. 7.13 25 randomly chosen wing plan forms in design space II


Frequency by EPA

Fre

quen

cyby

ES

A

0 1000 2000 3000 4000 5000 6000 70000

1000

2000

3000

4000

5000

6000

7000



Frequency by EPA

Rel

ativ

eer

ror

0 1000 2000 3000 4000 5000 6000 7000

-0.1

-0.05

0

0.05

0.1

0.15



Fig. 7.16 16 wing plan forms systematically varying through design space II


+ + + + + + + + + + + ++

+ ++

x x x x x x x x x xx

xx

x

xx


Nat

ural

frequ

ency

(rad

/sec

)

0 0.25 0.5 0.75 10

1000

2000

3000

4000

5000

6000 by EPA1st Bending mode by ESA1st Torsion mode by ESA2nd Bending mode by ESAIn plane mode by ESA2nd Torsion mode by ESA3rd Bending mode by ESA

+x



Fig. 7.18 An arbitrarily chosen wing plan form in design space II


Distance from root (inch)

10u,

10v,

w(in

ch)

0 10 20 30 40 50 60 70 80

-3.5E-04

-3.0E-04

-2.5E-04

-2.0E-04

-1.5E-04

-1.0E-04

-5.0E-05

0.0E+00

5.0E-05

10u (EPA)10v (EPA)w (EPA)10u (ESA)10v (ESA)w (ESA)

Fig. 7.19 Comparison of displacements by EPA and ESA for 1lb tip force


Relative distance to leading-edge

Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3

3.5

Upper skin (EPA)Lower skin (EPA)Upper skin (ESA)Lower skin (ESA)


under 1lb tip force


Relative distance to root

Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 11.5

2

2.5

3

3.5

4

Upper skin (EPA)Lower skin (EPA)Upper skin (ESA)Lower skin (ESA)


under 1lb tip force


7.2.4 Six-variable case: design space III

In this case spars and ribs are evenly distributed inside the wing plan form but their numbers

are design variables. A 63 full factorial experimental design with 3 levels in Λ , α , τ , A and

numbers of spars and ribs, sparn and ribn respectively, was used. Particulars of the levels of every

variable are as follows:

====

==Λ °°°

}.13,10,7{},6,4,2{

,}5000,3500,2000{},6.0,45.0,3.0{

},5.2,75.1,0.1{},30,15,0{2

ribspar nn

inAτα

Similar to case I, for each point in design space II, the EPA is carried out and Eqs. (7.5) and

(7.8) are used to generate the 15 constitutive matrix terms and the 36 mass densities which are then

used to perform the ESA. 51 feed-forward neural networks with the structure of 110156 ××× are

trained using the MATLAB NN Toolbox function trainlm. There are 729 (63 ) sets of data that could

be used for training, but it was found that at some design points the differences between the natural

frequencies by the EPA and the ESA become too large. Therefore a screening process was

introduced, in which any point where the maximum relative difference between the first 10 natural

frequencies by the EPA and the ESA surpasses 20% will be discarded. 28 points were removed

through the process, therefore 701 sets of data were used for training.

Generating the 729 sets of pre-training data used about 152 hours of CPU time on the Crunch

(SGI Origin 2000 with eight R10000 processors) of the College of Engineering, Virginia Tech, and

training the neural networks spent about 2 hours on a PII/350 PC. A set of results are given in Figs.

7.22 to 7.24 where 25 points were randomly chosen within the design space box. The plan forms of

the new designs are shown in Fig. 7.22, where dashed lines indicate the spar or rib positions. The

first 10 natural frequencies by the EPA and the ESA are compared in Fig. 7.23 and their relative

differences (based on the EPA results) are shown in Fig. 7.24. It can be seen that the relative

difference is within -5%~15%.


which is defined as


−==

=====Λ=

=+−=

).1/(,

,,,,,

6,,1,)1(654321

10

jjj

nj

ribspar

jjjjj

rrnsa

nnAvvv

javavv

j

ννντα�

(7.12)

where )6,,1( �=jr j are randomly determined values between 0 and 1, and see Eq. (7.10) for the

definition of other symbols. Results of natural frequencies of the first 6 modes for wing structures

defined by points along a path with n1=0.2243, n2 =0.8591, n3=0.2064, n4 =3.0700, n5 =2.2196

and n6=0.9440 are shown in Fig. 7.26, where it can be seen that results by the EPA and the ESA

agree with each other quite well.

Now some static results. For an arbitrary new design whose plan form is shown in Fig. 7.27, a

down-ward (-z direction) point force of 1lb is applied at the mid-point of the wing tip. Fig. 7.28

shows displacement components along the leading-edge by the EPA and the ESA, compared FEA

using MSC/NASTRAN. Figs. 7.29 and 7.30 show the Von Mises stress distributions at the wing

root and the central spar respectively together with the FEA results. Comparison of the natural

frequencies of this wing as given by the EPA, the ESA and the FEA is shown in Table 7.1. It can

be seen that the EPA and the ESA results are close, and they all agree quite well with the FEA

results.


Fig. 7.22 25 randomly chosen wing plan forms in design space III


Frequency by EPA

Fre

quen

cyby

ES

A

0 1000 2000 3000 4000 5000 6000 70000

1000

2000

3000

4000

5000

6000

7000



Frequency by EPA

Rel

ativ

eer

ror

0 1000 2000 3000 4000 5000 6000 7000-0.05

0

0.05

0.1

0.15

0.2



Fig. 7.25 16 wing plan forms systematically varying through design space III


++

+ + + + + + + + + + + + + +

x

xx

xx

x x xx x x x x x x x


Nat

ural

frequ

ency

(rad

/sec

)

0 0.25 0.5 0.75 10

1000

2000

3000

4000

5000

by EPA1st Bending mode by ESA1st Torsion mode by ESA2nd Bending mode by ESAIn plane mode by ESA2nd Torsion mode by ESA3rd Bending mode by ESA

+x



Fig. 7.27 An arbitrarily chosen wing plan form in design space III



10u,

10v,

w(in

ch)

0 10 20 30 40 50 60 70 80

-3.5E-04

-3.0E-04

-2.5E-04

-2.0E-04

-1.5E-04

-1.0E-04

-5.0E-05

0.0E+00

5.0E-05

10u (EPA)10v (EPA)w (EPA)10u (ESA)10v (ESA)w (ESA)10u (FEM)10v (FEM)w (FEM)

Fig. 7.28 Comparison of displacements by EPA and ESA at 1lb tip force



Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 10

0.5

1

1.5

2

2.5

3

3.5

Upper skin (EPA)Lower skin (EPA)Upper skin (ESA)Lower skin (ESA)Upper skin (FEM)Lower skin (FEM)


under 1lb tip force



Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 11.5

2

2.5

3

3.5

4



under 1lb tip force


7.2.5 Design space IV

In this case a wing plan with $30=Λ , 192=s in, 72=b in, and 36=a in (see Fig. 6.1 for

definitions of s,b , and a ) is used. A 24 32 × full factorial experimental design with 2 levels in tt0

(skin thickness at wing tip), rta (skin thickness increment ratio at root over the tip), 1h (spar cap

height) and 2h (rib cap height), and 3 levels in sparn and ribn , is carried out. The skins are assumed

to vary linearly from the root to the tip. Particulars of design space III are as follows:

==×=×=

=×=

}.14,10,6{},6,4,2{

,197.0}3,1{,197.0}3,1{

},2,0{,118.0}3,1{

21

0

ribspar

rtt

nn

inhinh

aint

There are 144 sets of data for training. Generating these data sets used much less CPU time

than in the case of design space III. A set of results are given in Figs. 7.31 to 7.33 where 16 points

were randomly chosen within the design space box. The plan forms of the new designs are shown

in Fig. 7.31, where dashed lines indicate the spar or rib positions, and the skin thickness at the wing

root and tip, and cap heights of spars and ribs are represented as shown in Fig. 7.36. The first 10

natural frequencies by the EPA and the ESA are compared in Fig. 7.32 and their relative

differences (based on the EPA results) are shown in Fig. 7.33. It can be seen that the relative

difference is within -5%~15%.


which is defined as

−==

======

=+−=

).1/(,

,,,,,

6,,1,)1(65

24

132

01

10

jjj

nj

ribsparrtt

jjjjj

rrnsa

nnhhvavtv

javavv

j

ννν�

(7.13)

where )6,,1( �=jr j are randomly determined values between 0 and 1, and see Eq. (7.10) for the

definition of other symbols. Results of natural frequencies of the first 6 modes for wing structures

defined by points along the path with n1=0.0031, n2 =0.9999, n3=0.2089, n4 =64.7024, n5 =0.9067


and n6=0.5325 are shown in Fig. 7.35, where it can be seen that results by the EPA and the ESA

agree with each other quite well.

For an arbitrary new design whose plan form is shown in Fig. 7.36, a down-ward (-z direction)

point force of 1lb is applied at the mid-point of the wing tip. Fig. 7.37 shows displacement

components along the leading-edge by the EPA and the ESA, compared with FEA using

MSC/NASTRAN. Figs. 7.38 and 7.39 show the Von Mises stress distributions at the wing root and

the central spar respectively together with the FEA results. Comparison of the natural frequencies

of this wing as given by the EPA, the ESA and the FEA is shown in Table 7.2. Again, it can be

seen that the EPA and the ESA results are close, and they all agree quite well with the FEA results.

It is noted that a coarser design space III does not worsen the accuracy of the ESA.


Fig. 7.31 16 randomly chosen wing designs in design space IV


Frequency by EPA

Fre

quen

cyby

ES

A

0 500 1000 1500 20000

250

500

750

1000

1250

1500

1750

2000



Frequency by EPA

Rel

ativ

eer

ror

0 500 1000 1500 2000

-0.05

0

0.05

0.1

0.15



Fig. 7.34 16 wing designs systematically varying through design space IV


+ + + + + + + + + + + + + + + +x x x x x x x x x x x x x x x x


Nat

ural

frequ

ency

(rad

/sec

)

0 0.25 0.5 0.75 10

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

by EPA1st Bending mode by ESA2nd Bending mode by ESAIn plane mode by ESA1st Torsion mode by ESA3rd Bending mode by ESA2nd Torsion mode by ESA

+x



Spar cap height (x10)

Rib cap height (x10)

Skin thicknessat root (x10)

Skin thicknessat tip (x10)

Fig. 7.36 An arbitrarily chosen wing design in design space IV



10u,

10v,

w(in

ch)

0 50 100 150 200

-7.0E-04

-6.0E-04

-5.0E-04

-4.0E-04

-3.0E-04

-2.0E-04

-1.0E-04

0.0E+00

10u (EPA)10v (EPA)w (EPA)10u (ESA)10v (ESA)w (ESA)10u (FEM)10v (FEM)w (FEM)

Fig. 7.37 Comparison of displacements by EPA and ESA at 1lb tip force



Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1



under 1lb tip force



Von

Mis

esst

ress

(psi

)

0 0.25 0.5 0.75 10

0.5

1

1.5

2



under 1lb tip force


7.3 Conclusion

CPU time savings using the ESA are obvious. For instance, when 6 terms of the Legendre

polynomials ( 6=K ) are used, about 85% less CPU time is spent in evaluating the stiffness and

mass matrices compared with the EPA, where matrix evaluating takes about 68% of the total CPU

time when solving the free vibration problem. When 8=K , about 83% less CPU time is spent in

evaluating the matrices compared with the EPA, where matrix evaluating takes about 65% of the

total CPU time. Generally speaking, the results given by the ESA in design space II and III are as

good as those in design space I although the number of variables increases from 4 to 6.

Table 7.1 Natural frequencies (rad/sec) of the wing in Fig. 7.20

Mode No. 1 2 3 4 5Mode Shape 1st bending 2nd bending 1st torsion In plane 2nd torsion

EPA 279.3 982.8 1057.5 1447.4 1945.5ESA 274.5 984.1 1045.1 1440.3 1936.3FEM 279.9 965.6 973.5 1454.4 1830.8

Table 7.2 Natural frequencies (rad/sec) of the wing in Fig. 7.29

Mode No. 1 2 3 4 5Mode Shape 1st bending 2nd bending In plane 1st torsion 3rd bending

EPA 71.9 233.9 358.1 452.2 479.9ESA 70.8 239.4 358.4 469.4 504.8FEM 66.0 222.6 377.0 413.1 468.0

149

Chapter 8

Conclusions and Future Work

8.1 Conclusions of the Present Work

(1) An efficient method capable of static and vibration analyses of the built-up wing structures,

Equivalent Plate-model Analysis (EPA) method, has been developed and comparisons for a

series of examples with commercial FEA calculations have shown the accuracy of the

method for design purposes. On the assumption that the wing structure behaves like a plate

whose deformation can be modeled by the FSDT of Reissner-Mindlin, the Rayleigh-Ritz

method is applied to solve the plate problem, with the Legendre polynomials being used as

the trial functions. The stiffness matrix and mass matrix are determined by applying the

Lagrange’s equations, and can be calculated numerically by using the Gaussian integration

quadrature. Then static analysis can be readily performed and the natural frequencies and

the mode shapes of the wing can be obtained by solving an eigenvalue problem.

Comparison of results by the present method with those by the commercial finite

element analysis code MSC/NASTRAN for a series of 5 vibration problems, 4 static

loading problems, and 1 stress distribution problem showed an overall good agreement

between the two approaches with different methodologies. Mode shapes and natural

frequencies for cases from a thick wing-shaped plate, the same plate with a camber, a solid

wing, to built-up wing structures composed of skins, spars and ribs, have all shown that the

CHAPTER 8 CONCLUSIONS AND FUTURE WORK 150

present method has a fairly good correlation to the FEA, although results for simpler cases

seem to be more accurate. It is also shown that static displacements and stress variation

trends of wing structure can be predicted by the present method quite accurately.

The EPA is formulated mostly in matrix form and calculation can be readily carried out

in the MATLAB environment. It is suitable to be used for the early stages of wing design.

Due to the efficiency of the method, it can also be used as a means to analyze the shape

sensitivity of wing structures.

(2) Modal response of general trapezoidal wing structures was investigated based on an

equivalent model analysis and sensitivity techniques. The variations of the natural

frequencies w.r.t. shape design variables need to be coordinated with the baseline mode

shapes by mode tracking. The use of second order sensitivities proved to be yielding much

better results than the case where only first order sensitivities are used. Shape sensitivities

can be evaluated using analytical, finite difference and semi-analytical approaches. The

present research shows that when the analytical solution is not available, the finite

difference approach would be more accurate than the semi-analytical one provided the step

size is properly specified. But the semi-analytical approach might need less CPU time since

the eigenvalue problem is solved only once.

Neural networks can be trained to relate the natural frequencies of a wing structure to its

shape variables. In this approach the major efforts are in training the networks. Once the

networks are trained, there needs an almost negligible computational effort to obtain

equally good results for the natural frequencies for any given set of the wing shape

variables.

(3) The Equivalent Plate Analysis (EPA) of built-up wing structures is coupled, in an indirect

way, with Neural Networks (NN) to make an even more efficient method, the Equivalent

Skin Analysis (ESA). In the EPA, major part of computational effort is spent on calculating

contributions to the stiffness and mass matrices from each spar and rib. This can be avoided

by replacing the wing inner-structure with an "equivalent" material that combines to the


skin and whose properties are simulated by neural networks. The constitutive matrix, which

relates the stress vector to the strain vector, and the density of the equivalent material are

obtained by enforcing mass and stiffness matrix equities with regard to the EPA in a least-

square sense. Neural networks for the material properties are trained in terms of the design

variables of the wing structure. Examples show that ESA takes off more than 80% of the

CPU time that is spent in the EPA on computing the total stiffness and mass matrices, and

still fairly good results can be obtained. Therefore, the ESA is very promising to be used at

the early stages of wing structure design.

8.2 Recommendations for Future Work

Generally, the efficient methods developed in the present work (EPA and ESA) can be

extended to deal with all the wing structure problems the FEA can solve, except in cases where

localized solutions are important. Accordingly, these problems can be considered in the early stage

of wing design to shorten design cycle and make better choices. Specifically, developments can be

made at the following area:

(1) To solve complex wings whose planar configuration is composed of 2 or more

quadrilaterals/trapezoids. The extended method should also deal with 3-D cases, that is, the

trapezoidal components can be not in the same plane, as the tail structure considered in

Tizzi 30 . But unlike in Tizzi30 , now all the inner-structure components will be considered.

(2) To extend the present series of efficient methods to deal with steady-state and transient

response problems. Since the stiffness and mass matrices have been given, solution of these

kinds of problems without dissipation should be straightforward. For problems where

structural dissipation is to be considered, work should be done to set up the dissipation

matrix.


(3) To extend the present series of efficient methods to include geometrical nonlinearity, and

material nonlinearity (plasticity, strain hardening etc.), for problems of large deformation

and in extreme material conditions.

(4) To extend the present series of efficient methods to deal with all spectrum of problems that

are structure-concerned and need to be addressed in wing structure design, such as

aeroelasticity (divergence and flutter speeds), global buckling, and composites/structures

with imperfections or damages.

153

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160

Appendix A

The Constitutive Matrix for Various Cases

A.1 Rotation along z-axis

Assume that the coordinate system ),,( zyx is rotated along the z-axis an angle θ and becomes

)',','( zyx . If we write

Tzxyzxyyx

Tzxyzxyyx

}2,2,2,,{}{

},,,,{}{

εεεεεε

σσσσσσ

=

=

}']{'[}'{ εσ D= , }]{[}{ εσ D=

it can be derived that

1]][][[]'[ −= eTDTD (A.1)

where

)]([)]([][ 11 θθ −== −−eee TTT

−−

−=

θθθθ

θθθθθθ

θθθ

cossin000

sincos000

002cos2sin2sin

002sincossin

002sinsincos

][21

21

22

22

T (A.2)

APPENDIX A CONSTITUTIVE MATRIX 161

−−

−=

θθθθ

θθθθθθ

θθθ

cossin000

sincos000

002cos2sin2sin

002sincossin

002sinsincos

][2122

2122

eT (A.3)

A.2 Rotation along y -axis

Assume that the coordinate system ),,( zyx is rotated along the y -axis an angle θ and becomes

)',','( zyx , then the relationship of the constitutive matrices for the two systems can still be

described as in (A.1), but with (A.2) and (A.3) becoming

−

−

=

θθθθθθ

θθθ

2cos0002sin

0cossin00

0sincos00

2sin0010

2sin000cos

][

21

2

T (A.4)

−

−

=

θθθθθθ

θθθ

2cos0002sin

0cossin00

0sincos00

2sin0010

2sin000cos

][21

212

eT (A.5)

A.3 Skin

The local constitutive matrix is


−−

−−

=

55

44

33

22

22

0000

0000

0000

00011

00011

][

D

D

D

EE

EE

D ννν

νν

ν

(A.6)

where GD =33 , )1(2 ν+

= EG , E is the Young's modulus, ν is the Poisson's ratio, and

kGD =44 , kGD =55 ,6

5=k or 12

2π for thick plates and 044 =D , 055 =D for thin skins.

If the wing has a high aspect ratio and a small swept angle, then we can see ),,( 000 zyx (the

local coordinates) to rotate an angle αθ = (see Fig. 2 for the definition of α )along the y -axis

to coincide with ),,( zyx (the global coordinates), therefore the global constitutive matrix is

1)](][)][([][ −= αα eTDTD

where ][T and ][ eT are found in Eqs. (A.4) and (A.5). If the skin is composed of laminated

layers, for the i -th layer the global constitutive matrix is

1)](][)][([][ −= ieii TDTD θθ

where ][T and ][ eT are found in Eqs. (A.2) and (A.3), and iθ is the orientation angle of the i -

th layer.

A.4 Spar and Rib Cap

Since the cap is slim (i.e. 1/, 11 <<sth ), the spar cap behaves like a 1-D bar. That is, the local

constitutive matrix is

]0000[][ EdiagD = . (A.7)

After a rotation of an angle θ along the z-axis, the global constitutive matrix becomes

1)](][)][([][ −= θθ eTDTD


where ][T and ][ eT are found in Eqs. (A.2) and (A.3). The angle θ is the angle between the local

orientation of the spar or rib and the x -axis.

A.5 Spar and Rib Web

The local constitutive matrix is

]000[][ GEdiagD = . (A.8)

The global constitutive matrix can be obtained using

1)](][)][([][ −= θθ eTDTD

where ][T and ][ eT are found in Eqs. (A.2) and (A.3), and θ is the angle between the local

orientation of the spar or rib and the x -axis.

164

Appendix B

Formulation for Multi-Plane Problem Using

EPA

As shown in Fig. B.1, a wing is composed of a main-body and a wing-let. What is given in this

appendix can be used as a basis to solve wing structures with more than one wing-let or composed

of more than two planes.

x,

z

y,

1

2

1

2

(ξ ,η )

(ξ ,η )

ξη

Fig. B.1 Sketch for a wing composed of main-body and wing-let

APPENDIX B MULTI-PLANE PROBLEMS USING EPA 165

B.1 Strain Energy and Stiffness Matrix

The total strain energy of the structure:

}]{[}{}]{[}{}]{[}{ 1121

22221

11121 qKqUqKqqKqU BC

TJT

TT +++= (B.1)

where ][ 1K and ][ 2K are the stiffness matrices for the main-body and the wing-let respectively,

whose formulation can be found in Ref. 1, ][ BCK is the stiffness matrix for the large springs

simulating the boundary conditions at the root2 , and JTU is the strain energy relating to the joint

between the main-body and the wing-let, which will be treated as follows.

In the ),( 11 ηξ and ),( 22 ηξ planes, the relationships between the displacement vectors and the

general displacement vectors1 can be written as

==

}]{[}{

}]{[}{

222

111

qHu

qHu (B.2)

where ][ 1H and ][ 2H are functions of ),( 11 ηξ and ),( 22 ηξ respectively. For the joint joining the

main-body and the wing-let, we can have

1,11;1,11 2211 −=≤≤−=≤≤− ηξηξ

and by expressing the displacement vector in plane ),( 22 ηξ in terms of plane ),( 11 ηξ , we get

}]{[}'{ 22 uRu = (B.3)

where }'{ 2u is the displacements of the wing-let expressed in plane ),( 11 ηξ .

Now we can write the strain energy of the joint as

}']{[}'{ 212121 uuKuuU JT

TJT −−= (B.4)

where ][ JTK is the stiffness matrix for the joint, and springs with very large magnitude can be used

if the joint is rigid. Using Eqs. (B.2) and (B.3), Eq. (B.4) can be extended as

( ) ( )( )

( )}]{][][[][][}{

}]{[][][]][[][}{}]{][[][}{

}]{][[][}{}{][][]][[}{}]{[}{

}]{[}{][}]{[}{

222221

221121

111121

2221

2121

1121

212121

qHRKRHq

qHRKRKHqqHKHq

uRKRuuRKRKuuKu

uRuKuRuU

JTTTT

TJT

TJT

TTJT

TT

JTTTT

JTT

JTT

JTT

JTT

JT

+

−−+=

+−−+=

−−=

(B.5)

APPENDIX B MULTI-PLANE PROBLEMS USING EPA 166

A general displacement vector for the whole system can be constructed as

=}{

}{}{

2

1

q

qq

and the total strain energy of the structure can also be written as

( ) }]{[}{}{][][}{}]{[}{

}{

}{

][][

][][

}{

}{}]{[}{

222221

22112121

111121

2

1

2221

1211

2

121

21

qKqqKKqqKq

q

q

KK

KK

q

qqKqU

TTTT

T

T

+++=

== (B.6)

Comparing Eq. (B.6) with Eqs. (B.1) and (B.5), we can obtain the stiffness matrix of the whole

structure

=

][][

][][][

2221

1211

KK

KKK in terms of }{q :

( )

+−==

+=

++=

][][][]][[][][][

]][][[][][][][

][]][[][][][

2121

2112

22222

11111

HRKRKHKK

HRKRHKK

KHKHKK

TJT

TJT

TT

JTTT

BCJTT

(B.7)

B.2 Kinetic Energy and Mass Matrix

The total kinetic energy of the structure:

=+=}{

}{

][0

0][

}{

}{}]{[}{}]{[}{

2

1

2

1

2

121

22221

11121

q

q

M

M

q

qqMqqMqT

T

TT

�

�

�

�� (B.8)

where ][ 1M and ][ 2M are the mass matrices for the main-body and the wing-let respectively1.

Therefore the mass matrix of the whole structure is

=][0

0][][

2

1

M

MM (B.9)

167

Appendix C

Airfoil Sections Generated with the Karman-

Trefftz Transformation

The Karman-Trefftz transformation59 is defined as

n

n

c

c

ncz

ncz

)(

)(

+−=

+−

ςς

(C.1)

where iyxz += is a vector in the −z plane, 1−=i , and ηξς i+= is a vector in the −ς plane.

If 2=n , Eq. (C.1) becomes the Joukowski transformation:

2

2

)(

)(

2

2

c

c

cz

cz

+−=

+−

ςς

or ς

ς2c

z += (C.2)

As shown in Fig. C.1, the Karman-Trefftz transformation in Eq. (C.1) maps a circle in the ς -

plane:

µς θ += iae (C.3)

where )sin(

sin

sin

sin

δβδ

βδ

+⋅=⋅= cma is the radius,

ββδδµ δ sincossincos iaacimmmei +−=+== ,

APPENDIX C KARMAN-TREFFTZ TRANSFORMATION 168

τO

T T'

ξ

η

x

y

(-nc, 0) (nc, 0)

β

δC

C'

(ξ, η) −>(x, y)

The Karman-Trefftz Transformation

)arg()Im(

)Re(tan 1 µς

µηµξθ −=

−−= − ,

to a Karman-Trefftz airfoil section in the z-plane.

More specifically, the transformation in Eq. (C.1) can written as:

ς -plane z-plane

CTaOTcOCmOCCOTOTC ====∠=∠= ,,,,, µδβ

Fig. C.1 The Karman-Trefftz transformation


=−=

+==+−

⋅==

+−−⋅==

−

cTAN

crrrr

rnr

nrncyy

rnr

rncxx

nn

n

nn

n

#

#

ξηλλλλ

ηξλληξ

ληξ

12,121

222,121

2

2

2

,

}){(,/

cos21

cos2),(

cos21

1),(

21

(C.4)

in which, a general arc tangent function 1−TAN is defined according to the common arc tangent

function ]2

,2

[tan 1 ππ−∈− :

≥<

+

<<≤≥

+

>≥

=

−

−

−

−

.00,tan2

);00()00(,tan

;00,tan

1

1

1

1

xandyifx

y

xandyorxandyifx

y

xandyifx

y

x

yTAN

π

π (C.5)

If the common arc tangent function 1tan− instead of 1−TAN is used in Eq. (C.4), there would be

some abnormal kinks in the curve in the −z plane transformed from a circle in the −ς plane.

It can be proved that the trailing edge angle of the airfoil is )2( n−= πτ . When ,0,2 == τn so

the Joukowski airfoils have cusped trailing edges.

The camber of the airfoil is determined by βδµ sinsin)Im( am == , and the thickness by

βδµ coscos)Re( acm −== . The chord-length is

−++++=

nn

nn

nclεεεε

)2(

)2(1 (C.6)

where

−= 1cos2 βε

c

a is a quantity of small value having a close relationship with the

thickness. If ,0→ε from Eq. (C.6) we can see that ncl 2→ .

Fig. C.2 shows several airfoils obtained using the Karman-Trefftz transformation with different

combinations of ,, δβ and n.


I

II

III

When there is an incoming flow, it is ready to obtain the velocities and pressures at any point,

streamline patterns, and the lift coefficient etc., by using Eq. (C.1) or (C.4) and the velocity

potential of the flow past a circle with circulation.

Fig. C.2 Airfoil shapes obtained using Karman-Trefftz transformation

95.1,85.0,05.0 === nπδβ

9.1,996.0,001.0 === nπδβ

99.1,88.0,05.0 ==−= nπδβ

171

Vita

Youhua Liu was born on August 22, 1963, in Jing County, Hunan Province, China. He attended

the No. 1 Middle School of Jing County for high school education, from September, 1978 to July,

1980, among the first group of students enrolled county-wide since the Cultural Revolution. In

September, 1980, he enrolled at Huazhong (Central China) University of Science and Technology

(HUST), in Wuhan, capital of Hubei Province. Majoring in Naval Architecture and Ocean

Engineering, he earned a Bachelor's degree in engineering in July, 1984. He continued to study and

got a Master's degree in engineering in July, 1987. From 1991 to 1996 he worked at HUST as a

faculty member and research engineer. In 1994, he worked for two months as a visiting scientist at

Yokohama National University, Japan.

In August 1996, Youhua Liu began his study toward a PhD with the Department of Aerospace

and Ocean Engineering, Virginia Tech. Blacksburg is a wonderful place to live in and being a

Hokie is a fantastic experience to him. He is expected to obtain his PhD in aerospace structures in

early 2000.

efficient methods for structural …...ii efficient methods for structural analysis of built-up...

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