MULTIDISCIPLINARY DESIGN AND OPTIMIZATION OF A COMPOSITE WING BOX A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES OF THE MIDDLE EAST TECHNICAL UNIVERSITY BY MUVAFFAK HASAN IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN THE DEPARTMENT OF AEROSPACE ENGINEERING SEPTEMBER 2003
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MULTIDISCIPLINARY DESIGN AND OPTIMIZATION OF A COMPOSITE WING BOX
A THESIS SUBMITTED TO THE GRADUATE SCHOOL OF NATURAL AND APPLIED SCIENCES
OF THE MIDDLE EAST TECHNICAL UNIVERSITY
BY
MUVAFFAK HASAN
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN
THE DEPARTMENT OF AEROSPACE ENGINEERING
SEPTEMBER 2003
Approval of the Graduate School of Natural and Applied Sciences
Prof. Dr. Canan Özgen
Director
I certify that thesis satisfies all the requirements as a thesis for the degree of
Doctor of Philosophy.
Prof. Dr. Nafiz Alemdaroğlu
Head of Department
This is to certify that we have read this thesis and that in our opinion it is
fully adequate, in scope and quality, as a thesis for the degree of Doctor of
Philosophy.
Prof. Dr. Yavuz Yaman
Supervisor
Examining Committee Members
Prof. Dr. Mehmet A. Akgün
Prof. Dr. Yavuz Yaman
Prof. Dr. Haluk Darendeliler
Assoc. Prof. Dr. Nizami Aktürk
Dr. Fatih Tezok
iii
ABSTRACT
MULTIDISCIPLINARY DESIGN AND OPTIMIZATION
OF A COMPOSITE WING BOX
Hasan, Muvaffak
Ph.D., Department of Aerospace Engineering
Supervisor: Prof. Dr. Yavuz Yaman
September 2003, 218 pages
In this study an automated multidisciplinary design optimization code is
developed for the minimum weight design of a composite wing box. The
multidisciplinary static strength, aeroelastic stability, and manufacturing
requirements are simultaneously addressed in a global optimization environment
through a genetic search algorithm.
The static strength requirements include obtaining positive margins of safety for
all the structural parts. The modified engineering bending theory together with the
coarse finite element model methodology is utilized to determine the stress
distribution. The nonlinear effects, stemming from load redistribution in the
structure after buckling occurs, are also taken into account. The buckling analysis
is based on the Rayleigh-Ritz method and the Gerard method is used for the
crippling analysis.
iv
The aeroelastic stability requirements include obtaining a flutter/divergence free
wing box with a prescribed damping level. The root locus method is used for
aeroelastic stability analysis. The unsteady aerodynamic loads in the Laplace
domain are obtained from their counterparts in the frequency domain by using
Rogers rational function approximations.
The outer geometry of the wing is assumed fixed and the design variables
included physical properties like thicknesses, cross sectional dimensions, the
number of plies and their corresponding orientation angles.
The developed code, which utilizes MSC/NASTRAN® as a finite element solver,
is used to design a single cell, wing box with internal metallic substructure and
Stacking sequence design of composite plates involves the determination of the
number of plies and their orientations. Because continuous optimizers have a low
computational cost and are widely available, stacking sequence problems have been
traditionally treated using continuous optimization techniques. Usually the laminate
is assumed to be made of stacks of plies and the thickness and/or orientation of
these stacks is treated as continuous design variables. After the optimization process
is completed, the thicknesses and/or orientations are rounded to integer values. This
is the standard approach to the composite laminate optimization in ASTROS and
MSC/NASTRAN. However, this assumed laminate stacking sequence might not
produce the optimal laminate design for a composite structure. Riche and Haftka
[27] identified the shortcomings and pitfalls implied in the gradient based approach.
The flexural and the in-plane response of laminates are nonlinear functions of the
number of plies, the ply thicknesses, and the fiber orientations. Therefore, for
problems involving this type of response, the design space contains local optima in
which continuous optimization strategies may get trapped. Second, composite
structures often exhibit many optimal designs. The reason is that composite
laminate performance is characterized by a number of parameters which is smaller
than the number of design variables. Different sets of design variables can produce
similar results, i.e., there are many optimal and near optimal designs. Traditional
design approaches not only have the drawback of sometimes converging to the
suboptimal designs, but also of yielding only one solution. Finally, rounding off
design variables may produce suboptimal or even infeasible designs.
Much effort has been devoted to the stacking sequence design of composite plates.
In response to the discrete nature of the problem, integer programming strategies
16
based on Branch and Bound algorithm have been used. Branch and Bound is
basically an enumeration method where one first obtains a minimum point for the
problem assuming all the variables to be continuous. Then each variable is assigned
a discrete value in sequence and the problem is solved again in the remaining
variables. This method was originally developed for linear programming problems.
However, in general, designing a composite laminate is a nonlinear integer
programming problem, Nagendra et al. [28].
Stochastic search methods on the other hand offer an advantage over mathematical
programming techniques, Hajela [29]. These methods are global search techniques
which work on function evluations only and do not require any gradient
information. Stochastic search methods can be easily applied to problems where the
design space consists of a mix of continuous, discrete, and integer variables. Their
lack of dependence on function gradients makes stochastic search methods less
susceptible to pitfalls of convergence to a local optimum and have better probability
in locating the global optimum. Among stochastic search methods, genetic
algorithms and simulating annealing are the most popular. Genetic algorithms are a
class of evolutionary strategies that derive their principle from Darwin’s theory of
the survival of the fittest. Simulated annealing algorithms are based on the
principles of statistical mechanics. Arora and Huang [30] presented a review on the
methods for optimization of non-linear problems with discrete-integer-continuous
variables.
As quoted by Goldberg [31], genetic algorithms were first introduced by Holland in
1975. They are based on Darwin’s theory of survival of the fittest. In a genetic
algorithm one starts with a set of designs. From this set, new and better designs are
reproduced using the fittest members of the set. Each design is represented by a
finite length string. Usually binary strings have been used for this purpose although
other representations are possible as well [32]. The entire process is similar to a
17
natural population of biological creatures; where successive generations are
conceived, born and raised until they are ready to reproduce.
Hajela [33] demonstarted the effectiveness of genetic search methods in the
optimization of problems with nonconvex and disjoint design spaces. The principal
drawback he identified in genetic search methods is the increase in function
evaluations necessary to obtain an optimum. He sugguested that a hybrid scheme
that switches from the genetic search approach to a conventional nonlinear
programming approach after a few generations might overcome this limitation.
Kogiso et al. [34] applied the genetic algorithm to the stacking sequence design of
laminated composite plates to maximize the buckling loads. To reduce the number
of analyses required by the genetic algorithm, a binary tree is used to store designs
and retrieve them and therefore avoid repeated analysis of design that appeared in
previous generations. Linear approximation based on lamination parameters was
used to reduce the cost of genetic optimization.
A two level optimization procedure for composite wing design subject to strength
and buckling constraints was presented by Liu et al. [35], [36]. At wing-level
design, continuous optimization of ply thicknesses with preassumed orientations of
0°, 90°, and ±45° is performed to minimize weight. At panel level, the number of
plies of each orientation (rounded to integers) and in-plane loads are specified, and
a permutation genetic algorithm was used to optimize the stacking sequence in
order to maximize the buckling load.
Upaadhyay and Kalayanaraman [37] developed a general optimization procedure
for the design of layered composite stiffened panels subject to longitudinal
compression and shear loading based on genetic algorithms. Stability and strength
considerations, expressed in the form of simplified equations served as constraints
in the optimal design method.
18
The original stacking sequence problem was solved by Leiva [38] using an
equivalent sizing optimization problem with continuous design variables.
Global optimization algorithms are much developed for unconstrained problems
than for constrained problems. Often these algorithms deal with constraint via
penalty functions, but this treatment may cause substantial degradation in
performance. Liu et al. [35], and Todoriki and Haftka [39] showed the advantage of
their repair mechanism and permutation genetic algorithm for handling constraints
on improving the performance of genetic algorithms.
The usefulness of stochastic search methods in MDO problems is severely limited
without the use of global function approximations. Given that these methods are
primarily based on the use of function information only, the use of response
surface-based approximation is a viable option. Response surfaces are obtained by
fitting a chosen-order polynomial model to a given experimental or numeric data.
The principal drawback of using the approach is that the user must specify the order
of the fit. Further, as problem dimensionality increases, response surface models are
imprecise and very difficult to generate [29]. References [35] and [36] demonstrated
that the response surface can be used effectively to find a near optimal wing design
and [40] used them in the approximation of a composite objective function that
included the weight of the structure, the manufacturing cost, the static response and
the aeroelastic response of a mettalic wing box. Unal et al. [41] discussed response
surface methods for approximation model building and multidisciplinary design
optimization problems.
Aeroelastic and stress analysis disicplines are treated by large-order finite element
models with thousands of degrees of freedom. The computational costs associated
with repeated construction of the full finite element model and the large-order
analysis degrade the usefulness of the optimization scheme, particularly in the
conceptual design stages when extensive trade-off studies for various design
19
concepts are needed. Several studies have focused on developing simplified
analysis models and tools to reduce the computation time required at the conceptual
design stage.
Giles [42] described an equivalent plate analysis formulation that is capable of
modeling aircraft wing structures with general planform geometry such as cranked
wing boxes. He used a Ritz solution technique to determine the static deflection,
stresses, and frequencies of an example wing configuration. The same author, [43],
generalized the method to provide capability to model aircraft wing structures with
unsymmetric cross sections. Livne [44] refined the method further by taking
transverse shear effects into account. He used the first order shear deformation
theory instead of the classical plate theory used by Giles [42,43]. Mukhopadhyay
[45] described an interactive wing flutter analysis program that is applicable for the
conceptual design stage. A comparison study for the results of two multidisciplinary
design optimization programs is given by Butler [46]. The first program uses a
simple beam model and is suitable for conceptual design phase. The second
program uses three dimensional finite element model and is suitable for preliminary
design stages. Some specialized tools like and ADOP (Aeroelastic Design
Optimization Program) and HpyerSizer were introduced to the aerospace
community in [47] and [48]. ADOP is an interdisciplinary optimization program for
the static, dynamic, and aeroelastic analysis of finite element models which was
developed at Douglas Aircraft Company, and HyperSizer is a structural
optimization system specifically designed for aerospace apllications.
Striz and Venkayya [49] investigated the influence of structural and aerodynamic
modeling of various fully built-up finite element wing models on flutter analysis.
They concluded that a reasonably coarse grid for both the structure and the
aerodynamics will result in natural frequencies and mode shapes that are close to
those obtained from more detailed models, whereas this evaluation will also result
in flutter speeds that are conservative.
20
The accuracy of the mathematical models for aeroelastic analysis, design, and
simulation is increased with the number of vibration modes chosen to represent the
structure. However, the associated increase in the model size adversely affects
calculation effeciency. Karpel [50] presented a dynamic residualization method
with which important structural and unsteady aerodynamic effects associated wtith
high-frequency vibration modes are retained without increasing the model size. The
formulation is based on state-space equations of motion where the unsteady
aerodynamic force coefficients are represented by a minimum-state rational
approxiamtion function. Later he applied the method for the multidiciplinary
optmimization of an active flexible wing (AFW), Karpel [51]. A gradient-based
constrianed optimization algorithm was used to minimize the weight subject to
constraints on flutter speed and control stability margins.
Various modal-based static and time-domain aeroservoelastic model size reduction
techniques were reviewed by Karpel [52]. These techniques are combined for an
integrated design optimization scheme where stress, closed loop flutter, control
margins, and time response are treated with a common baseline model. The
structure is represented by a relatively large number of low frequency modes of the
basic design (30-50 modes) and design changes are addressed without changing the
generalized coordinates. Less important modes are then eliminated using truncation,
static residualization, and dynamic residualization reduction methods. Karpel [53]
and Karpel and Brainin [54] expanded the modal approach to deal with stress
considerations in an optimization scheme. Fictitious masses were used to account
for local effects caused by high concentrated loads.
The k-method (American method) and the pk-method (British Method) have been
the standard analysis tools for aeroelastic stability analysis [55]. However, there are
many assumptions implicit in these methods that prohibit their use in an automated
design process.
21
In the k-method, the aerodynamic forces are presented in terms of complex inertia
terms and instability is described in terms of an artificial damping coefficient that is
valid only at the point of instability. This approach results in a linear eigenvalue
problem that is relatively easy to solve, nevertheless, damping information
produced by this method at subcritical speeds has no valid physical meaning
[11,56,57]. Thus, it is not possible to formulate a constraint which prescribes a
measure of viscous damping in the final design [11]. Furthermore, reliable damping
values at subcritical flight speeds are needed as a guideline for conducting wing
tunnel or flight flutter tests [57]. Another drawback found in this method is that it
occasionally produces multiple valued function of damping vs. velocity, making it
difficult to order the roots in an automated process to determine the flutter speed
[11,56]. The coefficient matrix of the eigenvalue problem of the k-method is
singular for zero frequency. Thus, the k-method can not predict the divergence
aeroleastic instability [56,57].
As quoted by Hassig [58], the pk-method or the British method was first proposed
by Irwin and Guyet in 1965. In the original method proposed by Irwin and Guyet
solutions were obtained using graphical methods to match the assumed reduced
frequency with the complex part of the computed complex eigenvalue. Hassig [58]
used a determinant iteration method to match the assumed reduced frequency with
the complex part of the computed complex eigenvalue and Rodden [55] modified
the equation further by introducing a damping terms that is dependent on the
frequency. The pk-method is an approximation of the p-method (transient method).
The p-method requires availability of the unsteady aerodynamic forces in the time
domain which are not usually available at low cost and require computational fluid
dynamics methods. Instead these forces are usually calculated in the frequency
domain for a discrete set of reduced frequencies on the assumption of undamped
harmonic motion utilizing simplified unsteady aerodynamic methods like the
Doublet Lattice Method (DLM) [59]. The rationale for the pk-method is that for
22
sinusoidal motions with slowly increasing or decreasing amplitude, the
aerodynamic forces based on constant amplitude (undamped harmonic motion) are
a good approximation. Thus, although the response of the system is assumed to be
damped in the pk-method, the aerodynamic forces are calculated based on the
assumption of undamped harmonic motion. Thus, the pk-method yields
approximate subcritical trends in terms of damping, although it does not lead to
double valued functions of damping vs. velocity [56].
For n structural modes, the pk-method and the k-method normally provide only n
roots of the flutter equation. However, the number of roots could exceed the number
of the structural modes if some aerodynamic lag roots appear. If the exact
Theodorsen function is used, the number of the aerodynamic lag roots that would
appear in the solution is expected to be infinite [60]. The inclusion of all the
activated aerodynamic lag roots could provide important physical insight into the
the flutter solution [56].
Rodden and Bellinger [61] compared the p-method with the pk-method for the
divergence analysis of a two degrees of freedom airfoil. They concluded that the pk-
method predicts the aerodynamic lag roots adequately and there is no need for
approximations to the aerodynamic transfer funcions and the use of the p-method.
This view is not necessarily shared by others. Zyl and Maserumule [62] analyzed
the same problem and concluded that what Rodden and Bellinger [61] have called
an aerodynamic lag root is nothing but a logical continuation of the structural roots
after their frequencies have gone to zero. Their argument was that whenever the
frequency of a structural mode goes to zero, one would expect the associated
complex root to be replaced by two real roots at higher speeds. A mode tracking
procedure like that used in the pk-method will track only one of the modes and
leave the other which implies that the solution would be incomplete.
23
A new aeroelastic stability analysis method, called the g-method, was recently
introduced by Chen [63]. The g-method is a generalization of the k-method and the
pk-method for reliable damping prediction that is valid in the entire reduced
frequency range. The g-method utilizes a damping perturbation method to include a
first order damping term in the flutter equation. This added damping term is
rigorously derived from Laplace-domain aerodynamics. The same reference proved
analytically that the added aerodynamic damping matrix by Rodden and Johnson
[55] is only valid for small values of the reduced frequency, k, or for cases where
the generalized aerodynamic forces are linear functions of the reduced frequency.
Thus, it is concluded that the pk-method is valid only under these limitations or at
the instability point where the damping is zero. The reduced frequency technique
used by the g-method potentially gives an unlimited number of roots which provide
a better insight into the mechanism resulting in the instability. Some interesting
results obtained by Chen [63] worth mentioning. Flutter is due to the aeroelastic
coupling of structural modes, but the coupling mechanism of the divergence speed
instability is not well understood. For restrained structures, it seems that the
divergence speed is a static aeroelastic instability since its associated frequency is
zero. However, results of the g-method sugguest that the divergence speed is caused
by the coupling of a structural mode and an aerodynamic lag root and should be
considered as a special case of flutter instability. The zero flutter frequency of the
divergence speed is caused by the zero-frequency aerodynamic lag root associated
with the restrained structure. For unrestrained structures the so called “dynamic
divergence” is again a special case of flutter instability caused by the coupling of
the aerodynamic lag root and structural modes but with non-zero frequency. Such
an interpretation could hardly be supported by the pk-method since it is not capable
of generating the non-zero-frequency aerodynamic lag roots.
Another new method which is based on tracking the orientation anlgles of the
eigenvectors was developed by Afolabi et al. [64]. The EVO (Eigen Vector
Orientation) method is based on the fact that the eigenvectors are initially real and
24
orthogonal to each other and lose their orthogonality at the flutter instability point.
Pidaparti et al. [65] reviewed this method and applied it to the flutter analysis of an
intermediate complexity wing (ICW) model. Results obtained using this method are
compared to the flutter prediction results obtained using the pk-method. A
reasonably good comparison between the EVO method and the pk-method was
obtained.
Zyl and Maserumule, [66,67], used three different forms of the pk-method to
determine the divergence speed of a single degree of freedom airfoil. The first was
that used by Hassig [58], the second was that used by Rodden and Johnson [55] and
the third was a form that is equivalent to the g-method of Chen [63]. Although the
three methods predicted the same divergence speed (which is expected since for
zero eigenvalue all the considered three forms are equivalent), the subcritical
damping and frequency behaviour predicted by the three methods were different.
Since aeroelastic divergence of a free flying aircraft does not occur at zero
eigenvalue, they concluded that the three different forms might predict different
divergence speeds.
Alternatively, a root locus solution of the flutter equation in Laplace domain
provides an insight to the aeroleastic stability problem and has been implemented
successfully in the flutter redesign problem by [11,26,68]. Brase and Eversman [68]
used this method to solve the structurally nonlinear flutter problem and the work
performed in [11] and [26] has been described early in the text. The basic advantage
of this method is that it provides valid damping behaviour for the velocity range of
interest. However, one difficulty associated with this method is that it requires the
availability of aerodynamic forces in the Laplace or time domain. An important
feature of these forces is the lag associated with the circulatory wake, where
disturbances shed to the flow by the wing motion continue to affect the loads at a
later time [69]. Theodorsen [60] employed a lift deficiency function in the reduced
frequency domain to represent this effect for the oscillatory flow over an airfoil. In
25
1940, Jones used a two term series of decaying exponentials in the time domain to
approximate the effect of circulation for the transient aeroelastic motion. This led to
the well known Jones rational function approximation of the Theodorsen function
[70].
Several methods had been developed to express the aerodynamics of general
planforms in the Laplace domain using rational function approximations. Among
them the Rogers method [71] and the minimum state method of Carpel [72] are the
most popular and widely used ones. Rogers method relies on approximating each
term of the generalized aerodynamic forces in the form of rational functions with
common denominator roots. The minimum state method is based on a more general
approximation function with coupled terms and constraints on the coefficient
matrices. Consequently this method requires computationally heavier, iterative,
nonlinear least square solutions. The computational time was nearly 1000 times
greater than that of Roger’s method. Karpel and Strul [73] modified this method to
improve its performance by introducing a new solution strategy and relieving some
of the constraints.
In a conventional Roger’s approximation the aerodynamic lag roots are usually
chosen based on experience from the reduced frequency range of interest. Eversman
and Tewari [69] introduced an improved method for the rational function
approximation of unsteady aerodynamics that is based on Roger’s method with the
aerodynamic lag roots chosen by an optimizer to minimize the total fit error. The
optimization method utilized was the simplex non-gradient method.
26
1.3 Scope and Contents of the Study
In this study an automated multidisciplinary design optimization code is developed
for the minimum weight design of a composite wing box. The multidisciplinary
static strength, aeroelastic stability, and manufacturing requirements are
simultaneously addressed in a global optimization environment through a genetic
search algorithm. The intention is to obtain a minimum weight final design that
complies with the existing compliance requirements (FAR/JAR) in less time than
what is currently needed while taking aeroelastic stability constraints into account at
the early stages of the design. This would eliminate the need for extensive design
modifications at later stages of the design which may result in weight penalties and
failure to deliver product on time.
The static strength requirements specify obtaining positive margins of safety for all
of the structural parts of the wing box taking into account all potential failure
modes. Besides to classical failure modes (material failure), specialized failure
modes (buckling and crippling) are taken into account in the optimization process.
The buckling analysis is based on the Rayleigh-Ritz method and the Gerard method
is used for crippling analysis. The static strength analysis procedure is based on a
refined process that is consistent with the aerospace industry approach to the
analysis of this type of structures. A coarse mesh finite element model is utilized to
determine the internal load distribution in the wing box. The modified engineering
bending theory is then utilized to calculate the stress distribution taking into account
nonlinear effects that result from redistribution of the load in the wing box after
buckling occurs in the structure. In this procedure MSC/NASTRAN® is used as
finite element solver.
The aeroelastic stability analysis requirements specify obtaining flutter and
divergence free wing box for a range of prescribed flight conditions and with
27
required damping level in the final design. The aeroelastic stability analysis
procedure is based on a root locus method. The unsteady aerodynamic forces in
Laplace domain are obtained from their counterparts in the frequency domain using
Rogers rational function approximations. In this procedure, MSC/NASTRAN® is
used to perform free vibration analysis to determine the generalized mass, stiffness,
damping, and aerodynamic forces in the frequency domain.
The optimization procedure utilized in this thesis relies on a genetic search
algorithm that is suitable for the design of the wing box problem under
consideration. It was courtesy of Prof. Prabhat HAJELA of the Rensselaer
Polytechnic Institute to provide the optimizer (EVOLVE) to be utilized in this
thesis.
In Chapter 2 the main components of a typical wing box, their functions and their
failure modes are described first. The stress analysis, based on the modified
engineering bending theory together with the classical laminated plate theory and
the coarse mesh finite element analysis methodology, is then explained. Then the
procedure for calculating the allowable stresses under the effect of combined
loading follows. Special attention is given to the buckling analysis based on the
Rayleigh-Ritz method. Buckling stress analysis results for two representative
metallic and composite panels, which are under the effect of the combined loading
conditions, are consequently given. The buckling allowable stresses obtained by the
Rayleigh-Ritz method are compared to those obtained with the specially orthotropic
plate assumption. Buckling analysis results of the Rayleigh-Ritz method are verified
using a fine mesh finite element analysis utilizing MSC/NASTRAN®. The strength
analysis procedure to determine the minimum margins of safety, the critical load
cases and the corresponding failure modes are then discussed. An illustrative test
case for a simple wing box with metallic internal structure and composite skins is
28
then analyzed. Justification to use the modified bending theory methodology is
illustrated with a fine mesh finite element analysis of the considered wing box.
In chapter 3 the aeroelastic phenomena in general with emphasis on flutter and
divergence phenomenon is first described. The mathematical formulation of the
aeroelastic stability problem is then discussed. Aeroelastic analysis methods based
on the k-method, pk-method, p-method, and the root locus method are also
explained. The differences between these methods are identified. The method of
obtaining the generalized aerodynamic forces in Laplace domain using Rogers
rational function approximations is explained. Two test cases are studied to see the
difference between the pk-method and the root locus method and to verify the
adopted methodology for approximating the generalized aerodynamic forces in
Laplace domain by using the rational function approximations. The first case study
is the BAH wing and the second is an intermediate complexity wing (ICW) model.
Chapter 4 explains the automated multidisciplinary design and optimization
procedure of the composite wing box. The optimization problem attempted is first
described. The problem is then mathematically formulated in terms of the objective
function, the static strength and aeroelastic stability constraints, and the
manufacturing constraints on the design variables. The solution procedure for the
optimization problem is explained. The static strength analysis, aeroelastic stability
analysis, and optimization methods utilized in the procedure are then discussed. The
developed code for the automated procedure with its features and limitations are
then described in detail.
In chapter 5 the developed multidisciplinary design and optimization code is
applied to the design of a rectangular wing box. The wing box is considered in three
separate case studies. The first case study aims at verifying the developed code and
studies the capability of the genetic algorithm in optimization for aeroelastic
constraints with manufacturing constraints imposed on the design variables. Thus,
29
an all metallic wing box which is fully described and has available optimization
results in literature is optimized to meet the aeroelastic stability constraints with
manufacturing constraints imposed on the thicknesses of the spars webs, ribs webs,
and spars caps areas. The second case study aims at studying the capability of the
developed code in the optimization of representative “real-life” composite wing
structures. Hence the wing box considered in the first case study is modified to have
composite skin panels and ribs chords and is then optimized to meet static strength
requirements subject to manufacturing constraints on the thicknesses, ply
orientations and cross sectional dimensions of the spars caps and the ribs chords.
The third case study aims at analyzing the advantages of considering the aeroelastic
stability constraints at the early stages of the design. Thus, the optimized wing
considered in the second case study is first analyzed to determine its
flutter/divergence speeds. Then a 20% increase in the flutter/divergence speed is
imposed on the design and the wing box is optimized to meet the aeroelastic
constraints, static strength constraint, and manufacturing constraints simultaneously.
Chapter 6 gives the conclusions and recommendations for future work and further
studies.
30
1.4 Limitations of the Study
The study is limited to fixed configuration design variables. Thus, the wing
planform, number of ribs, number of spars, and their corresponding locations are
assumed to be fixed.
The study is limited to the analysis of unstiffened skin panels. Although the skin
panels can be of laminated composite or metallic types, the spars and ribs are
limited to metal construction.
The developed code utilizes MSC/NASTRAN® (v75.7) as a finite element solver
and EVOLVE as an optimizer. Thus both of these packages are required for running
the code.
31
CHAPTER 2
STATIC STRENGTH ANALYSIS
2.1 Introduction
The wing of an aircraft provides the aerodynamic lift force necessary to carry the
payload and supports the fuselage together with any undercarriage loads. It consists
of the wing box, the leading edge, the trailing edge, and the control surfaces (flaps,
ailerons and spoilers). The wing box is the main load carrying structural component
of the wing.
In the design of a wing box, as with any other aircraft component, adequate strength
and stiffness has to be assured to demonstrate the compliance with the existing
requirements (FAR/JAR). There are mainly two requirements. The first requirement
is that under the effect of the applied or limit loads, no structural member shall be
stressed above the material yield point, or in other words there must be no
permanent deformation of any part of the structure. The second requirement is that
under the design or ultimate loads, which are equal to the applied loads times a
factor of safety, no failure of the structure should occur.
The wing of an aircraft is generally subject to two types of loading. These are
basically ground and flight loads. Ground loads include loads due to landing,
32
taxiing and towing. Flight loads include aerodynamic loads in a cruise flight,
inertial loads during maneuver and gust loads. In a typical design, the wing is
analyzed for hundreds of load cases and the integrity of the wing structure has to be
assured under the effect of all these load cases. Thus, for each load case, the stress
acting on each element has to be determined. Depending on whether the stress is
compression or tension, the relevant allowable stress is determined and the margin
of safety is calculated. After analyzing the entire load cases, the critical load case
(the load case with the minimum margin of safety) and the corresponding failure
mode is identified. Structural tests are finally performed for the most critical load
case(s) to demonstrate the compliance with the certification requirements.
Typical wing boxes exhibit thin metal/composite panels joined together to form the
structure. Since these panels are very thin, they usually buckle at very low stress
amplitudes causing redistribution of the load in the structure. Besides, the stress
levels usually exceed the linear material range and plasticity effects start gaining
importance. A detailed linear finite element analysis of the structure would not
account for such nonlinear effects. Instead, simple finite element models are usually
used to get the internal load distribution in the unbuckled structure. This is then
followed by post processing of the internal loads to simulate the accurate buckling
behavior of the structure. This approach significantly reduces the manpower and
resources required to analyze the structure and yields quite adequate results.
In this chapter the main components of a typical wing box, their functions and their
failure modes are first described. The stress analysis based on the modified
engineering bending theory together with classical laminated plate theory and the
coarse mesh finite element analysis methodology is then explained. The procedure
for calculating the allowable stresses under the effect of combined loading is then
explained. Special attention is given to the buckling analysis based on the Rayleigh-
Ritz method. Buckling stress analysis results for two representative metallic and
composite panels, which are under the effect of combined loading conditions, are
33
then given. The buckling allowable stresses obtained by the Rayleigh-Ritz method
are compared to those obtained with the specially orthotropic plate assumption. The
strength analysis procedure to determine the minimum margins of safety, the critical
load cases and the corresponding failure modes are then discussed. An illustrative
test case for a simple wing box with metallic internal structure and composite skins
is then analyzed. Justification to use the modified bending theory methodology is
illustrated with a fine mesh finite element analysis of the considered wing box. The
chapter ends with some concluding remarks and discussions.
The discussions applied to this chapter are based on the aerospace industry
approach for the analysis of wing boxes and stem from the author’s work
experience in this field. They are partially discussed in [74-77].
2.2 Description of the Wing Box
A wing box is typically made of skin panels, spars, and ribs. The skins provide the
contour necessary to generate the aerodynamic force and transfer these loads to the
spars, ribs, and stringers. Thus the skins are usually subject to a combination of
shear and axial stresses. The skins can be either stiffened or unstiffened. Stiffened
skin panels have stringers attached to them. The stringers resist the compressive
stresses due to wing bending. They also divide the skins and thus increase the
allowable buckling stress of the panels. The spars are made of the spar caps and
web. They carry the shear and bending stresses. The caps resist the bending stresses
and the web carries the shear. The ribs are usually shear tied to the skins. They
support the skins in resisting the aerodynamic loads, help the wing to maintain its
aerodynamic shape, support the stiffeners to prevent global buckling and transform
any concentrated loads coming from any attached fittings (for example the engine
fittings) to the wing box. The construction of the ribs is similar to the spars. They
are basically made of chords and webs. The chords resist the induced bending
stresses and the web resists the crushing aerodynamic loads and shear stresses. The
34
structural details of a typical wing box considered in this work are illustrated in
Figure 2.1. The structure has metallic ribs and spars. The skins can be composite or
metallic.
Figure 2.1 Structural Details of a Typical Wing Box
Rib Chords Rib Web
Rib Details
Spar Caps
Spar Details
Spar Web
FRONT SPAR (Metallic)
REAR SPAR (Metallic)
RIBS (Metallic)
UPPER SKIN (Metallic/Composite) INTERMEDIATE SPAR
(Metallic)
LOWER SKIN (Metallic/Composite)
35
2.3 Failure Modes of the Wing Box Components
This section describes the potential failure modes of the skins, ribs webs, ribs
chords, spar webs, and spars caps.
The failure mode of a wing structural part depends on its stress state and
construction material. Depending on the load case under consideration, some parts
of the structure may be in compression while others are in tension. A structural
element may be in compression for a certain load case and in tension for another.
For example, while the upper cap of a spar is subjected to a compressive stress in
cruise flight condition, it is subject to tensile stress in a hard landing load case.
The skins of the wing box resist both shear and in-plane axial loads. The skins can
be either of composite or metallic type. Thus, their failure modes change depending
on their material type and loading condition.
If the skins are made of composite material and their stress state is due to a
combination of shear and compression loads, then their probable failure mode is
buckling. Since buckling would cause delamination of the plies, buckling of
composite skins is not usually allowed up to ultimate load. If the skins are subject to
a combination of shear and tension loads, then their probable failure mode is fiber
breakage, if the principal normal stress/strain in the ply exceeds the maximum
allowable ultimate tensile stress/strain. Although buckling is not likely to happen,
since the tensile stress helps to prevent buckling, the skin panel may buckle due to
the shear effect and buckling analysis has to be performed to check for buckling.
Independent of whether the load case is a combination of shear and compression or
shear and tension, the maximum stress/strain in each ply has to be checked against
the relevant material allowable stress/strain. These allowable stresses/strains are
usually reduced to account for fatigue effects and manufacturing defects.
36
If the skins are metallic and subject to a combination of shear and compression
loads, then their potential failure mode is buckling. However, since buckling would
not cause failure of the structure (it causes only redistribution of the load in the
structure), it is usually allowed above certain percentage of the ultimate load
(typically 60%). The main reason for not allowing buckling below this percentage
of the ultimate load is due to fatigue issues. Under the effect of combined axial and
shear stresses, metallic skin panels usually fail in rupture if they are thin and the
maximum tensile stress has to be checked against the maximum allowable tensile
stress coming from the material allowable data or fatigue considerations. The
maximum shear stress in the panel has also to be checked against the allowable
shear stress of the material.
The spars are designed to be shear resistant. In a shear resistant design the web is
not allowed to buckle and must support the induced stress under the effect of the
combined action of the bending, axial and shear loads without failure. Thus the
possible failure modes for the web are buckling under the effect of the combined
stress state, web rupture due to the effect of combined axial, bending and shear
stresses (Von-mises), the tensile stress in the extreme fiber of the web exceeding the
allowable ultimate stress of the material and the maximum shear stress on the web
exceeding the ultimate shear stress of the material. The caps of the spars are
supported by both the skins and spar web. Thus, the spar caps can not buckle as a
column and the only possible failure modes for the spar caps are crippling for the
part under compression and the maximum allowable ultimate tensile strength for the
part under tension stress.
The ribs are similar to the spars in construction. However, since they support the
skin in resisting the aerodynamic loads, they are subject to bending, shear and
biaxial in-plane loads. Special purpose ribs have fittings attached to them to support
the control surfaces and engines. So they act as load transfer member that transfer
37
loads coming from the engines and control surfaces and dump them to the main
box. The rib failure modes are similar to the spars.
2.4 Stress Analysis of the Wing Box
This section gives the details of the stress analysis method for various components
of the wing box.
The wing box of an aircraft wing is a thin walled structure. Nonlinear effects such
as buckling, post buckling and material non-linearity has to be taken into
consideration when analyzing thin walled structures. Such effects can be taken into
consideration using two different approaches. In the first approach, the structure can
be analyzed using nonlinear finite element methods with detailed finite element
models. The results of a stress analysis that utilizes the finite element method to
determine the stress distribution in a structure depend to a great extent on the type
of elements utilized in the model and the mesh density. An alternative to this
approach is to use a coarse mesh finite element model is utilized to determine the
internal load distribution in the structure and then post processing of these results to
simulate the correct behavior of the structure. This second option is not very much
sensitive to the type of element chosen and would always yield results with good
accuracy. Since the structure has to be analyzed for hundreds of load cases, and
considering that a typical coarse mesh global finite element model (a model which
includes the wing, fuselage and the tail) of an airframe structure has millions of
degrees of freedom, these make the first approach non practical and would result in
finite element models that are practically impossible to handle and analyze with
good accuracy due to the large sizes of the resulting matrices. In this work, the
analysis methodology for determining the stress distribution in the structural parts
of the wing box is based on the second approach.
38
For the stress analysis of the wing box a coarse mesh finite element model is first
used to determine the internal load distribution in the box. In a coarse mesh finite
element model, the skin, for example, is modeled by only one element between its
adjacent ribs and spars. Thus, grids are only created at the intersection points of the
structural components. After the internal loads are determined, they are summed to
determine the sectional forces (i.e., the shear force, the normal force and the
bending moment) acting on the section. The section axial and bending stiffnesses
are determined and the modified engineering bending theory is then used to
determine the stress distribution over the section. The classical laminated plate
theory is used in both determining the equivalent stiffnesses and the analysis of the
composite skins over the section.
2.4.1 Sectional Loads
The sectional loads are obtained from the finite element model using the grid point
force balance output of MSC/NASTRAN. The free body forces and moments of the
elements adjacent to the section on either side ( )ii MF , are summed and reduced to
force-couple systems at the upper and lower grids of the section in a coordinate
frame that is normal and tangential to the section.
, tFFui
itu ⋅
= ∑ (2.1)
, nFFui
inu ⋅
= ∑ (2.2)
( ) tnMMui
io,u ×⋅
= ∑ (2.3)
, tFFi
it ⋅
= ∑}
} (2.4)
39
nFFi
in ⋅
= ∑}
}, (2.5)
( ) tnMMi
io, ×⋅
= ∑}
} (2.6)
The sectional normal force (N), the shear force (V) and the bending moment (M)
acting on the section are then obtained by reducing the force-couple systems at the
upper and lower grids to a force couple system at the section centroid.
nnu FFN ,, �+= (2.7)
ttu FFV ,, �+= (2.8)
( ) nnuoouo FZFZHMMM ,,,, ��−−++= (2.9)
The procedure for calculating the sectional loads is illustrated in Figure 2.2. Note
that the bending moment acting on the section is a function of the neutral axis
location Z .
40
Figure 2.2 Calculation of the Sectional Loads
2.4.2 Classical Laminated Plate Theory (CLPT)
For the analysis of laminated skin panels, the classical laminated plate theory
(CLPT) is used. The CLPT is based on the thin plate theory with the Kirchoff
assumptions (i.e., plane sections remain plane after deformation) and plane stress.
Besides each lamina (layer) of the laminate is assumed orthotropic, linear elastic
and has constant thickness.
n ≡≡≡≡
Fℓ,t
t
Fu,n
Fℓ,n
Fu,t
θu
θℓ
N
V
Mo H
Mu,o
Mℓ,o
Z
41
The stress-strain relations for an orthotropic lamina are given by
=
12
2
1
66
2212
1211
12
2
1
00
00
γεε
τσσ
QQQQQ
(2.10)
where Qij are the reduced stiffness terms which are obtained from the lamina
material properties.
2
21
1
121266
2112
1122112
2112
222
2112
111
;
1
1 ;
1
EEGQ
EQQ
EQEQ
νννν
ν
νννν
==
−==
−=
−=
In the laminate coordinate system (xyz), this equation transforms to the following
form,
=
xy
y
x
xy
y
x
QQQQQQQQQ
γεε
τσσ
662616
262212
161211
(2.11)
Here ijQ are known as the transformed stiffness terms and are given by the
following set of equations,
42
)()22(
)2()2(
)2()2(
)()4(
)2(2
)2(2
4466
226612221166
3662212
366121126
3662212
366121116
4412
2266221112
422
226612
41122
422
226612
41111
mnQmnQQQQQ
mnQQQmnQQQQ
mnQQQmnQQQQ
nmQnmQQQQ
mQnmQQnQQ
nQnmQQmQQ
++−−+=
+−+−−=
+−+−−=
++−+=
+++=
+++=
(2.12)
where θ= sinn and θ= cosm . The positive sign convention for θ and the stress
resultants is illustrated in Figure 2.3. The assumption of linear strain distribution
through the laminate results in the following equation
+
=
xy
y
xx
xy
y
x
z
xy
y
κκκ
γεε
γεε
�
�
�
(2.13)
where oiε and iκ are the mid-plane strains and curvatures respectively. Substituting
equation (2.13) into equation (2.11) gives,
+
=
xy
y
xx
xy
y
x
zQQQQQQQQQ
xy
y
κκκ
γεε
τσσ
�
�
�
662616
262212
161211
(2.14)
43
Figure 2.3 Positive Sign Convention of Stress Resultants and Ply Orientation
Angle
+θ
σx
σxy
σy
σ1 σ12 σ2
1 (Lamina axis)
x (Laminate axis)
y
2
Nx
Ny
Nxy
x
y Mxy
x
y
Mx
My
Moment Resultants Force Resultants
44
The stress resultants are obtained by integrating the stress through the thickness of
the laminate.
∫ ∑ ∫− =
=
=
2/
2/ 1
z
zi
i
1-i
t
t
N
ixy
y
x
xy
y
x
xy
y
x
dzdzNNN
τσσ
τσσ
(2.15)
∫ ∑ ∫− =
τσσ
=
τσσ
=
2
2 1
z
zi
i
1-i
/t
/t
N
ixy
y
x
xy
y
x
xy
y
x
dzz z dzMMM
(2.16)
Substituting equation (2.14) into equations (2.15) and (2.16) results in the following
load-strain relation matrix equation,
=
xy
y
x
oxy
oy
ox
xy
y
x
xy
y
x
κκκ
εε
DDDBBBDDDBBBDDDBBBBBBAAABBBAAABBBAAA
MMMNNN
γ
662616662616
262212262212
161211161211
662616662616
262212262212
161211161211
(2.17)
where,
( ) ( )∑=
−−=N
kkkkijij zzQA
11 (2.18)
( ) ( )∑=
−−=N
kkkkijij zzQB
1
21
2
21 (2.19)
( ) ( )∑=
−−=N
kkkkijij zzQD
1
31
3
31 (2.20)
45
Equation (2.17) can be written in a compact form as,
=
κε o
DBBA
MN
(2.21)
Here [A] is the extensional stiffness matrix, [B] is the extension-bending coupling
stiffness matrix and [D] is the flexural bending stiffness matrix. For any symmetric
laminate the coupling stiffness matrix [B] is always zero.
After the mid-plane strains and curvatures due to an applied stress resultant loads
set are determined from equation (2.21) the ply strains and stresses in the laminate
coordinate system can be determined utilizing equations (2.13) and (2.14). These
strains/stresses are then transformed into the lamina principal axis to calculate the
principal ply strains and stresses.
2.4.3 Equivalent Axial and Bending Stiffness Properties
The segments forming the cross section of a spar or rib together with the effective
skins forms a section that is made of a combination of composite and isotropic
materials that exhibit different elastic properties. In analyzing this type of sections,
the axial and bending stiffnesses (EA,EI) should be used rather than the area and
moment of inertia (A,I) of the section. The calculation of the stiffness for a segment
that is made of an isotropic material is straightforward. However, for a segment
which is made of a laminated composite material the approach is different.
Consider a symmetrically laminated plate of width (beff) and thickness (t) shown in
Figure 2.4. This plate represents the skin segment on the section. The equivalent
axial and bending stiffness terms which relate the applied loads to the mid-plane
strain and curvature can be determined in the following way.
46
Figure 2.4 Laminate Equivalent Stiffness
The axial and bending stiffness matrices of this laminate with respect to its local
centroidal axes system (xyz) are first obtained using the classical lamination theory
as outlined in the previous section. Since the laminate is symmetric, the axial-
bending coupling stiffness matrix [B] is zero. The compliance matrices for the
laminate are then obtained by inverting the stiffness matrix.
[ ] [ ]1−
= Aa (2.22)
[ ] [ ]1−
= Dd (2.23)
Ply4
Ply3
Ply2 Ply1
yy
z Z
Zo
Y
beff
t
X
x
47
The equivalent or average axial modulus for the laminate is then calculated as
taExx
11
1= (2.24)
The finite width axial stiffness (Axx) and bending stiffness (Dyy) for the laminate are
then calculated from the following equations.
11ab
A effxx = (2.25)
11db
D effyy = (2.26)
The stiffness is then transformed to the global coordinate system (XYZ) using the
following equations.
xxXX AA = (2.27)
xxoyyYY AZDD 2+= (2.28)
48
2.4.4 Normal Stress Analysis
The normal stress acting on the section is due to the bending stress induced by the
bending moment (M) acting on the section and the axial stress that is due to the
axial load (N). The principle of superposition is used to determine the normal stress
distribution over the section. Since the section is, generally, composed of different
materials the linear stress assumption is not valid. Nevertheless, the strain
distribution over the section is still valid if the different components forming the
section are perfectly fastened to each other. Hence the strain distribution rather than
the stress distribution is determined and the stresses are calculated from the strains.
XXYY AN
DzM +=ε (2.29)
where XXA is the axial and YYD is the centroidal bending stiffness of the section.
The total stiffness of the section is the summation of the stiffnesses of the individual
segments forming the section.
∑=i
iXXXX AA , (2.30)
XXi
iYYYY AZDD2
, −=∑ (2.31)
∑
∑=
iiXX
iiXXio
A
AZZ
,
,,
(2.32)
where Z denotes the position of the neutral axis with respect to the reference
coordinate frame (XYZ).
49
Bruhn [74] modified the engineering bending theory to account for load
redistribution resulting from buckling of the skins. The engineering bending theory
is modified in the sense that skins working in compression are only partially
effective in carrying stress. Thus, the width of the skin that is considered effective
in carrying compression load is given by the following equation
( )2,9.1min btb skskeff ε= (2.33)
where tsk is the skin thickness, εsk is the strain level in the skin and b is the total
width of the skin panel.
Since the effective skin width is a function of its strain level, the process of
calculating the section stiffnesses requires an iterative process. First, the skin is
assumed to be fully effective in carrying compression loads and the section stiffness
is calculated based on this assumption. Then, the part of the skin which is under
compression effect is identified. This can be the upper or lower skin depending on
the load case under consideration. The effective skin width is then calculated from
equation (2.33) and the position of the centroid together with the section stiffness is
recalculated. The sectional loads are then reduced to a force-couple system at the
new calculated centroid location and the strains in the skins are calculated again.
This process is repeated until convergence of the centroid location is achieved.
50
N.A Web
Lower Right Skin Lower Left Skin
Upper Left Skin Upper Right Skin
Upper Cap
Lower Cap
Lower Grid Point
Upper Grid Point
Y
Z
Y
A
A
Z
Section A-A
Figure 2.5 Typical Spar Cross Section
51
Figure 2.6 Strain Distribution Over the Spar Section
2.4.5 Shear Stress Analysis
The shear force acting on the section is assumed to be carried by the web only.
Nevertheless, the caps and skins of tapered box beams carry a part of the shear force
and help in relieving the shear stress acting on the web, Niu [75]. This effect is
illustrated in Figure 2.7. For the most general case of a tapered cross section, the
force acting on the web is given by
��θθ tantan PPVV uuweb −+= (2.34)
θu
θℓ
N.A H
N
V
M
Centroid Force Resultant Strain Distribution
52
where P is the total normal force in the skin and cap/chord and θ is the taper angle.
The value of P is obtained by integrating the normal stress distribution over the skin
and cap/chord area and will be negative if the integration result is negative.
The average and maximum shear stress on the web are then calculated from the
following equation.
( )webweb
webaveweb ht
V =τ (2.35)
( ) ( )avewebweb ττ23 max = (2.36)
While the average stress value is used in buckling analysis, the maximum shear
stress is utilized in the strength checking.
Figure 2.7 Tapered Section Shear Stress
Pu tan θu
Zθu
θℓ
Vw
Pu
Pℓ
Pℓ tan θℓ
V
53
2.5 Allowable Stresses
The allowable stresses for tension load cases are usually specified in terms of
material ultimate stresses and/or reduced allowable stresses to account for fatigue
considerations and manufacturing defects. Thus these values are usually specified
based on prescribed inspection programs, crack growth and damage tolerance
analysis results. The procedure of calculating these allowable stresses is beyond the
scope of the current work. For the preliminary sizing of the structure, usually the
allowable fatigue stress values are specified based on past experience and similar
designs.
The main allowable stresses necessary for the strength analysis of structural
members working under compressive loads are the crippling and buckling allowable
stresses
2.5.1 Crippling Allowable Stress
Crippling is defined as an inelastic distortion of the cross-section of a structural
element in its own plane resulting in permanent deformation of the section. This
behavior is one of the most common failure mechanisms encountered by aerospace
structures under compressive loads. The crippling phenomenon is quite complex.
There are no analytical equations to describe crippling. The crippling failure is
illustrated in Figure 2.8. Crippling is always preceded by local buckling of the
segments forming the section.
Empirical techniques have been developed by using coefficients derived from
various tests since there is no analytical basis for the prediction of the crippling
strength. The crippling stress for a particular cross-section area is calculated as if
the stress was uniform over the entire section. Furthermore, the maximum crippling
strength of a member is calculated as a function of its cross-section rather than its
54
length. In reality, parts of the section may buckle well below the crippling stress.
This results in the more stable areas, such as corners and intersections, reaching a
higher stress than buckled elements. At failure, the stress in the corners and
intersections is always above the material yield stress although the “crippling”
stress, which is an average value, may be considerably less than the yield stress.
Figure 2.8 Crippling Failure
Bruhn [74] presented several methods for calculating the allowable crippling stress
of a section. These methods include the Gerard method, the Needham method and
the Modified Needham method. The Gerard method is generalization or broader
application of the Needham Method. In this work the Gerard method is adopted for
the calculation of the allowable crippling stresses of the spar caps and rib chords.
Initial Structure Local Buckling Crippling
P1 P2 P3
P1 < P2 < P3
55
For angle sections, the following crippling stress equation applies within ± 10
percent limits,
85.0 2/1
2256.0
=
cycy
cs
FE
At
FF (2.37)
For tee sections the crippling stress is obtained from the following equation with an
error limit of ± 5.
40.0 2/1
2367.0
=
cycy
cs
FE
At
FF (2.38)
where Fcs is crippling stress of the section, Fc is the compression yield stress, t is
the average thickness, A is the cross sectional area and E is the Young’s modulus of
elasticity.
Some cutoff values are used as the crippling strength cutoff since there is not
sufficient data to permit an exact solution at higher stress values for most materials.
Table 2.1 gives the cut-off crippling values for commonly used cross-sections.
Table 2.1 Cut-Off Crippling Stresses
Type of Section Max. Fcs
Angles 0.7*Fcy
T-Sections 0.8*Fcy
Zee, J, Channels 0.9*Fcy
56
2.5.2 Allowable Buckling Stress
The calculation of the buckling allowable stress is necessary for the strength
analysis of the skins, spar webs, and rib webs since these parts of the wing box are
not usually allowed to buckle up to the ultimate load or a certain percentage of the
limit load. Different parts of the structure are usually subject to different kinds of
combined load systems. The skins are subject to a combination of in-plane axial and
shear stress. The spar webs carry bending, longitudinal axial and shear stresses. The
ribs support in-plane transverse axial and shear stresses.
Practically for metallic structures the allowable buckling stresses under the effect of
compression and shear stresses are calculated separately using simple equations that
are based on buckling coefficients obtained from tabulated or graphical data. The
allowable buckling stress under the combined effect of axial, biaxial, bending and
shear stresses (depending on the structure under consideration) is then obtained
from what is known as interaction equations. However, the use of this method is
complicated since the analyst is required to read buckling coefficients from curves
or tables and may result in conservative results in some cases where tension stress
effects are usually neglected due to the lack of appropriate interaction equations and
for the sake of simplifying the analysis. For example, in calculating the allowable
buckling stress of fuselage skin panels, the effect of the transverse tensile hoop
stress that is due to the internal pressure load effect is usually neglected and only
the effect of longitudinal compression acting with shear stress is considered. This is
basically due to the lack of an interaction equation that would consider the effect of
combined tension, compression and shear buckling stresses simultaneously.
Obviously, considering the effect of the tensile hoop stress would have resulted in a
higher allowable buckling stress which in turn would result in weight saving.
For composite panels buckling analysis the problem of calculating the allowable
buckling stress is more difficult. Practically used composite panels are generally
57
symmetrically laminated anisotropic panels that exhibit bending-twisting coupling
effects. Several research works has been done on the optimization of composite
panels with buckling constraints. There, the use of the interaction equations with the
assumption specially orthotropic material is a common practice. A specially
orthotropic plate has either a single layer of specially orthotropic material or
multiple specially orthotropic layers that are symmetrically arranged about the
laminate middle surface to form a symmetric cross-ply laminate [78]. Such plates
do not exhibit any bending-extension or bending-twisting stiffness coupling terms.
However, and as will be shown later in this section, this approach will result in a
non conservative result for the allowable buckling stress and most practical
laminated plates do not satisfy the requirements of specially orthotropic plates [79].
Besides, for a generally anisotropic material the allowable buckling stress becomes
a function of the direction of the applied shear stress which makes the use of the
interaction equation invalid. In industry the use of the energy methods, such as the
Galerkin and Rayleigh-Ritz methods, to determine the allowable buckling stress of
a composite panel is the adopted approach.
In this study, the Rayleigh-Ritz method is used to determine the allowable buckling
stress of a symmetrically laminated anisotropic plate with four sides simply
supported. The plate is assumed subject to the most general in-plane stress state
(a combination of bending, biaxial and shear stresses).
Consider a symmetrically laminated composite rectangular plate of length (a) and
width (b) that is simply supported along all edges. The plate is subjected to a
combination of linearly varying in-plane axial stress and constant shear stress
resultants acting on its boundaries. This plate with the positive sign convention for
the stress resultants is illustrated in Figure 2.9. The in-plane axial stress resultants
can be written as,
58
( ) yb
NNNyNoxx
bxxo
xxxx−+= (2.39)
( ) xa
NNNxN
oyy
ayyo
yyyy
−+= (2.40)
The governing differential equation for the transverse buckling analysis of this plate
is [78],
4
4
113
4
2622
4
662
2
123
4
164
4
11 4224xwD
yxwD
yxwD
xwD
yxwD
xwD
∂∂+
∂∂∂+
∂∂∂
+
∂∂+
∂∂∂+
∂∂
2 2
22
2
2
ywN
yxwN
xwN yyxyxx ∂
∂+∂∂
∂+∂∂= (2.41)
Subject to the boundary conditions,
02,0:,02
162
2
122
2
11 =∂∂
∂−∂∂−
∂∂−===
xywD
ywD
xwDMwax x (2.42)
02,0:,02
262
2
222
2
12 =∂∂
∂−∂∂−
∂∂−===
xywD
ywD
xwDMwby y (2.43)
Here, Dij are the bending stiffness terms as obtained from the classical laminated
plate theory. An exact solution to this equation is not possible due to the presence of
the bending-twisting stiffness coupling terms D16, D26, the problem of presence of
odd derivatives in the shear stress terms, and the applied axial loads Nxx, Nyy
generally being functions of x and y.
An approximate solution to the problem can be obtained using the Rayleigh Ritz
method [80]. Such a solution will approach the exact solution of the problem and
will handle the type of loading under consideration.
59
The Rayleigh-Ritz method is based on the principle of stationary value of the total
potential energy of an elastic body. The total potential energy of an elastic body is
the summation of the strain energy stored in the body, U, and the work done by the
external forces, V.
valuestationary =+=Π VU (2.44)
Figure 2.9 Plate Layout and Positive Sign Convention of Applied Loads
In the Rayleigh-Ritz method a solution is sought in the form
( )∑∑= =
=M
m
N
nmnmn yxWAw
1 1, (2.45)
X
Y
a
b
oxxN
bxxN
ayyNo
yyN
xyN
xyN
ss
ssss
ss
60
where Amn are undetermined coefficients and the functions Wmn(x,y) are chosen in a
variable separable form and must at least satisfy the geometric boundary conditions
of the problem under consideration. Thus, the energy criterion reduces to satisfying
the condition,
( ) [ ] 0)()( =∂+∂=
∂Π∂
mnmn AwVwU
Aw (2.46)
The strain energy for the transverse bending of a symmetrically laminated
composite plate is [80],
∫ ∫
∂∂+
∂∂
∂∂+
∂∂=
b a
ywD
yw
xwD
xwDU
0 0
2
2
2
222
2
2
2
12
2
2
2
11 221
dydxyx
wDyx
wywD
xwD
∂∂∂+
∂∂∂
∂∂+
∂∂+
22
66
2
2
2
262
2
16 44 (2.47)
The work done by the external in-plane loads, V, is,
dydxyw
xwN
ywN
xwNV
b a
xyyyxx∫ ∫
∂∂
∂∂+
∂∂+
∂∂=
0 0
22
22λ (2.48)
where λ is a load multiplier.
For the simply supported plate under consideration, a solution in the following form
is assumed,
=∑∑= = b
yna
xmAwM
m
N
nmn
ππ sinsin1 1
(2.49)
61
Note that this solution satisfies the geometrical boundary conditions and the natural
boundary conditions are approximated by the Rayleigh-Ritz process (minimization
of the total potential energy of the system). Substituting the assumed solution into
the strain energy and work expressions and making use of the energy criterion
results in the following set of linear equations.
( )
(2.50) ,,2,1,,2,1
0
cossincossin
cossincossin2
coscossinsin
sinsincoscos
cossin cossin
cossincossin2
cossin cossin
cossincossin2
coscoscoscos
sinsinsinsin
00
00
2
002
2
002
2
00
2
00
2263
4
00
2
00
2163
4
006622
4
00
1 1224
422
122222
22
4
114
422
==
=
+
+
−+
+
−+
+
+
−
+
+
+
+++
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∫∫
∑∑= =
NjMi
dyyydxxxjm
dyyydxxxinNab
dyyydxxxxa
NNN
bjn
dyyyyb
NNNdxxxa
mi
dyyydxxxjnm
dyyydxxxjinDab
dyyydxxxjim
dyyydxxxmniDba
dyyydxxxDba
mnij
dyyydxxx
DbjnDjmin
baD
aim
j
b
nm
a
i
n
b
ji
a
mxy
n
b
jm
a
i
oyy
ayyo
yy
n
b
j
oxx
bxxo
xxm
a
i
n
b
jm
a
i
j
b
ni
a
m
j
b
ni
a
m
n
b
jm
a
i
n
b
jm
a
i
n
b
jm
a
i
M
i
N
j
l
l
ββαα
ββααλπ
ββααλπ
ββααλπ
ββαα
ββααπ
ββαα
ββααπ
ββααπ
ββαα
πππ
62
where,
nmjikb
ka
kkk ,,, : ; === πβπα
Performing the integration and simplifying terms results in the following set of
linear homogenous equations,
( )[ ]( ) ( )[ ]
( ) ( )[ ]
==
=
−+−++
−+
−
++++
−+++
∑∑
∑∑
= =
= =
NnMm
ANNLRNNKNMRa
ANN
aRnNNam
ADjnmnRDmiMR
ADRnDDRnmDm
M
i
N
jij
oyy
ayymnij
oxx
bxxmnijxymnij
mn
oyy
ayy
oxx
bxx
ij
M
i
N
jmnij
mn
,,2,1,,2,1
for 0
48
22
32
22
1 1
22
2222222
1 126
22216
222
2244
6612222
1144
l
l
ππλ
π
π
(2.51)
where,
≠=−=
≠=+−=
±±≠≠−−=
whereelse ; 0
m)(iand ,oddm)(i ,n)(j ; )(
whereelse ; 0
n)(jand ,oddn)(j ,m)(i ; )()(
whereelse; 0
oddj)(n and odd,i)(m j),(n i),(m ;))((
222
22
2222
�
�
mimnij
L
njnjmnij
K
jnimmnij
M
mnij
mnij
mnij
63
and R is the plate aspect ratio (a/b). Equation (2.51) can be cast into the form of a
classical eigenvalue problem.
[ ] [ ][ ] { } 0=− AKK de λ (2.52)
where [Ke] represents the elastic stiffness matrix, [Kd] is the differential (also known
as the geometric) stiffness matrix and {A} is a column matrix of the undetermined
coefficients Amn. Solving the eigenvalue problem yields the critical load multiplier,
λcr, which will cause buckling of the plate. Then the margin of safety for the
buckling strength of the plate, M.Sb, can be obtained as,
1. −= crbSM λ (2.53)
For a metallic panel, the critical buckling stress can be obtained by replacing the
relevant bending stiffness terms by their equivalents for an isotropic material. For a
metallic plate made of an isotropic material, the bending stiffness terms are given
by,
)1(24
0)1(12
)1(12
3
66
2616
2
3
12
2
3
2211
ν
νν
ν
+=
==−
=
−==
tED
DD
tED
tEDD
(2.54)
The solution obtained by the Rayleigh-Ritz method is always in the direction of
stiffer plate. Thus the buckling loads obtained by the Ritz method are always higher
than the true solution. This is due to the fact that the obtained solutions involve
additional constraints on the energy criterion which are beyond the physical
constraints of the problem [79]. The solution is also approximate since it is
64
restricted to the pre-selected set of functions, Wmn(x,y). However, the energy
criterion is sufficient to select the most accurate set of these functions that represent
the most accurate solution. As the number of functions selected is increased (i.e, the
number of terms in the assumed series) the accuracy of the solution obtained should
increase or remain the same. Furthermore, if the selected functions form a complete
mathematical set, then the obtained solution must approach the correct one as the
number of terms in the series is increased [79].
2.6 Static Strength Analysis
The static strength analysis of the wing box is performed to demonstrate compliance
with the certification requirements, i.e., the structure has adequate strength to carry
the loads without failure or loss of strength.
For each load case, the stress/strain distribution in every structural element of the
wing box is first determined using the methods explained in section 2.4. Then the
relevant allowable stresses are determined based on the methods outlined in section
2.5. The margin of safety and the corresponding failure mode for the load case
under consideration are then evaluated. The process is repeated for all of the load
cases to identify the minimum margin of safety, the critical load cases and the
corresponding failure modes.
The output of the static strength analysis is a summary of the minimum margins of
safety, the critical load cases and the corresponding failure modes.
65
2.7 Case Studies
In this section two test cases are considered. The first one is the buckling analysis of
a typical skin panel and it aims at verifying the Rayleight-Ritz for buckling analysis
using MSC/NASTRAN and to show that the specially orthotropic assumption is not
generally a justified assumption. The second test case considers the stress analysis
of a simple wing box and it aims at justifying the methodology of using the
modified engineering theory combined with the coarse mesh finite element analysis
and compares the results thus obtained with a detailed mesh finite element model
analysis.
2.7.1 Allowable Buckling Stress of a Typical Panel
For verification and illustration purposes, the buckling stresses for a typical panel
under the effect of various combined loading conditions are determined using both
the Rayleight-Ritz and the finite element methods. Two versions of the panel are
considered. The first version is made of a unidirectional tape graphite-epoxy
composite material with a ply stacking sequence of [ ] s45/0/45 22 ±°° . This stacking
sequence is typical for a composite skin panel and results in an anisotropic laminate.
The second version is a metallic one that is made of an isotropic material. The panel
considered has a length of 600 mm, a width of 300 mm and a thickness of 1.6 mm.
The panel material properties are given in Table 2.2.
A convergence study is first done for the composite version of the panel loaded in
axial compression with an axial load intensity of == bxx
oxx NN .10 [N/mm]. The
panel is first analyzed as anisotropic and then as specially orthotropic by setting the
stiffness coupling terms D16 =D26 =0 in equation (2.49). The panel critical buckling
load is also determined using a fine mesh MSC/NASTRAN® finite element model
with a mesh density of 2557 total quadrilateral elements. The convergence history
66
of the buckling load factor λcr is illustrated in Figure 2.10. The anisotropic solution
approaches the finite element method solution with ten terms in the series and an
error of 1.6% relative to the finite element method solution. The specially
orthotropic solution convergence is very fast (2 terms in the series), however, with a
non-conservative result and an error of 45.3% relative to the finite element method
solution.
Table 2.2 Panel Material Properties
Isotropic Material (Aluminum)
Composite Material (Graphite/epoxy)
E = 10.5x106 [psi] E1 = 18.5x106 [psi]
E2 = 1.6x106 [psi]
ρ = 0.1 [lb/in3] ρ = 0.055 [lb/in3]
ν = 0.3 ν12 = 0.25
G12 = 0.65x106 [psi]
The buckling stress for both versions of the panel was then determined using both
MSC/NASTRAN® and the developed Rayleigh-Ritz approach for six representative
load cases that a typical skin, rib web and spar web would be subjected to in real
life. The different load conditions considered and their magnitudes are given in
Table 2.3.
67
Figure 2.10 Convergence of the Buckling Load Factor
0.50
0.75
1.00
1.25
1.50
1 6 11 16 21M=N
Orthotropic Solution (Rayleigh-Ritz; D16=D26=0) Anisotropic Solution (Rayleigh-Ritz) FEM (MSC/NASRAN)
64.0=crλ
65.0=crλ
93.0=crλ
crλ
10 [N/mm]
Graphite Epoxy
[(45°)2,(0°)2,±45]s
68
Results obtained from MSC/NASTRAN and Rayleigh-Ritz methods are tabulated
in Table 2.4. Note that both approaches give more or less the same result. This is no
surprise, since both methods are based on the energy principles. Comparison
between the buckling load multipliers for the anisotropic and specially plate
solutions is depicted in Table 2.5. It is worth to note the huge difference between
the two approaches and the error that would be involved if this assumption is made.
Another interesting point is the difference between results obtained for the buckling
load factor for the positive and negative shear load cases. For an anisotropic plate
the direction of the shear has to be taken into account in the analysis. An important
aspect that has no effect in the specially orthotropic case. The buckling mode shapes
are illustrated in Figures 2.11 and 2.12 for the composite and metallic panels
respectively.
Table 2.3 Buckling Load Cases
L.C # oxxN
[N/mm]
bxxN
[N/mm]
oyyN
[N/mm]
ayyN
[N/mm]
xyN [N/mm]
1 0.0 0.0 0.0 0.0 5.0
2 0.0 0.0 0.0 0.0 -5.0
3 5.0 -5.0 0.0 0.0 5.0
4 -10.0 -10.0 0.0 0.0 5.0
5 -10.0 -10.0 5.0 5.0 5.0
6 10.0 -20.0 -5.0 -5.0 5.0
69
Table 2.4 Critical Buckling Load Factors (λcr)
Metallic Plate Composite Plate
L.C # Rayleigh-Ritz NASTRAN Rayleigh-Ritz NASTRAN
1 3.90 3.90 0.98 0.97
2 3.90 3.90 3.69 3.65
3 3.72 3.72 0.96 0.95
4 1.10 1.10 0.43 0.42
5 1.63 1.60 0.71 0.70
6 0.72 0.72 0.28 0.27
Table 2.5 Critical Buckling Load Factors (λcr)
L.C # Orthotropic Anisotropic Ratio
1 2.46 0.98 2.51
2 2.46 3.69 0.67
3 2.39 0.96 2.49
4 0.84 0.43 1.95
5 1.33 0.71 1.87
6 0.43 0.28 1.54
70
Figure 2.11 Buckling Mode Shapes of the Composite Panel
LOAD CASE 2 LOAD CASE 1
LOAD CASE 3 LOAD CASE 4
LOAD CASE 5 LOAD CASE 6
71
Figure 2.12 Buckling Mode Shapes of the Metallic Panel
LOAD CASE 2 LOAD CASE 1
LOAD CASE 3 LOAD CASE 4
LOAD CASE 5 LOAD CASE 6
72
2.7.2 Stress Analysis of a Typical Wing Box
In this case study, the stress analysis for a typical wing is performed using four
different approaches. In the first approach, the wing box is modeled with a coarse
mesh finite element model and the stresses were directly obtained from the finite
element model output. In the second approach, the coarse mesh finite element
model was used to determine the internal load distribution only. These loads were
then used to calculate the sectional loads and the stress analysis was performed
using the engineering bending theory. The third approach is the same with the
second, however, with the modified version of the engineering bending theory used
to account for the load redistribution in the structure after skin buckling. In the
fourth approach, a detailed mesh finite element model is constructed to determine
the stress distribution in the structure.
The typical rectangular wing box analyzed is shown in Figure 2.15 with all the
relevant dimensions. It is a single cell wing box that has three metallic ribs, two
spars and coverage skins. The skins are laminated composite with a layup of
[±45°]3S. The material properties are given in table 2.2. The wing box is assumed
subjected to a concentrated tip load of 2000 [N] acting up.
73
Figure 2.13 Structural Arrangement of the Rectangular Wing
90°
Spar cross section
101.6
1524 15241524
635
All dimensions in millimeters
Y
X
Z
Y
Rib 3 Rib 2Rib 1
0°
20
20
2.0
20
20
1.0
Rib cross section
0.8 1.2
74
The coarse mesh finite element model is illustrated in Figure 2.16. This model has a
total of 6 membrane-bending CQUAD4 elements to model the skins, 6 CSHEAR
elements to represent the spars webs, 3 membrane CQUAD4 elements to model the
three ribs and 18 axial CROD elements to model the spars caps and ribs chords. The
total number of elements in the model is thus 33 elements. Two fine mesh finite
element models that use the same same kind of elements as the coarse mesh finite
element model are constructed. The first model is illustrated in Figure 2.15 and has
a total of 864 elements. The second model is illustrated in Figure 2.16 and has a
total of 2997 elements. All models are constrained at the root by fixing the
translational degrees of freedom (Tx=Ty=Tz=0).
Figure 2.14 Coarse Mesh Finite Element Models of the Rectangular Wing
Upper skin is removed for better visibility
1000 [N] 1000 [N]
75
Figure 2.15 Fine Mesh Finite Element Models of the Rectangular Wing
(Total 864 Elements)
Figure 2.16 Fine Mesh Finite Element Models of the Rectangular Wing
(Total 2997 Elements)
1000 [N] 1000 [N]
Upper skin is removed for better visibility
Upper skin is removed for better visibility
1000 [N]
1000 [N]
76
The stress analysis results for the upper and lower spar caps using the four different
approaches are depicted in Figures 2.17 and 2.18 respectively. Obviously the worst
approach is to rely on direct stress output of the coarse mesh finite element model.
The stress output obtained by the second approach, i.e., using a coarse mesh to
determine the internal load distribution and then utilizing the engineering bending
theory to determine the stresses, compares well with the detailed mesh finite
element model stress output. This shows that the use of engineering bending theory
with the loads obtained from a coarse mesh finite element model yields results that
are accurate enough as compared to the fine mesh finite element analysis. Note that
the use of a fine mesh in large scale finite element models might be computationally
expensive. Nevertheless, the use of fine mesh linear finite element models, would
not account for nonlinear effects stemming from load redistribution in the structure
after buckling occurs. Note the large jump on the compressive stress on the upper
cap after the skin buckles as predicted by the modified engineering bending theory.
Thus, the use of coarse mesh finite element models to determine the load
distribution in the structure and using the modified engineering bending theory to
determine the stresses is a good approach since it yields fairly good results at low
cost and can be used to accurately simulate the behavior of the structure.
77
Figure 2.17 Front Spar Upper Cap Stress Distribution of the Rectangular
Wing
-500
-450
-400
-350
-300
-250
-200
-150
-100
-50
00 1000 2000 3000 4000 5000
Span, Y [mm]
Stre
ss [
MPa
]
Modified Engineering Bending TheoryEngineering Bending TheoryCoarse Mesh Finite Element ModelFine Mesh Finite Element Model (864 Elements)Fine Mesh Finite Element Model (2997 Elements)
78
Figure 2.18 Front Spar Lower Cap Stress Distribution of the Rectangular
Wing
0
25
50
75
100
125
150
175
200
225
250
0 1000 2000 3000 4000 5000
Span, Y [mm]
Stre
ss [
MPa
]Modified Engineering Bending TheoryEngineering Bending TheoryCoarse Mesh Finite Element ModelFine Mesh Finite Element Model (864 Elements)Fine Mesh Finite Element Model (2997 Elements)
79
2.8 Conclusion
The strength analysis of a wing box for demonstrating the compliance with the
certification requirements is not an easy task to achieve. It is not just constructing
finite element models and reading stress output. Several failure modes have to be
considered and nonlinear effects such as buckling and crippling have to be taken
into consideration.
The use of coarse mesh finite element models for the determination of the internal
load distribution in the structure and then post processing these loads to determine
the stress distribution is a justified approach. However, such models shall never be
used to read stress outputs directly from the finite element model. Simple theories
such as the modified engineering bending theory can then be used to simulate the
correct behavior of the structure.
The buckling analysis based on the specially orthotropic plate assumptions is not a
generally valid one and can only be justified if the laminate is cross-ply symmetric.
In practical applications most laminates would not qualify to this assumption. On
the other hand using the energy methods such as the Rayleigh-Ritz method would
result in accurate and acceptable results at low cost compared to the finite element
method and without any need to prepare separate finite element models for buckling
analysis purposes. For anisotropic laminates, care should be taken in the correct
sign for the applied shear stress, since positive and negative shear loads would
result in completely two different results.
80
CHAPTER 3
AEROELASTIC STABILITY ANALYSIS
3.1 Introduction
Aeroelasticity is defined as the science, which studies the behaviour of an elastic
body under the effect of aerodynamic, elastic, and inertia forces. An aircraft
structure immersed in an air flow is subjected to surface pressures induced by that
flow. If the incident flow is unsteady or the boundary conditions are
time-dependent, these pressures become time dependent. Moreover, if the structure
undergoes dynamic motions, it changes the boundary conditions of the flow and the
resulting fluid pressures which in turn changes deflections of the structure.
The importance of aeroelasticity has been widely recognized in the aerospace
industry. The missions of aircraft structures are becoming increasingly more
complex. New developed aircrafts are larger in size and more flexible. Aeroelastic
effects due to aircraft flexibility may significantly alter the performance and safety
of a new developed aircraft. Hence considering aeroelastic effects at the early stages
of design to produce competitive and safe aircraft is a vital task.
The whole spectrum of aeroelastic phenomena to be considered during the design
process can be classified by means of the expanded Collar’s aeroelastic triangle
81
illustrated in Figure 3.1. Three types of forces are mainly involved in the aeroelastic
analysis process. These are aerodynamic, elastic and inertial forces. Accordingly,
the aeroelastic phenomena can be divided into two main groups, static and dynamic
types.
Figure 3.1 Collar’s Aeroelastic Triangle
Static aeroelastic phenomena lie outside of the triangle and includes phenomenon
like wing divergence, control surface effectiveness, static stability, and load
distribution.
Aerodynamic Forces
Elastic Forces
Inertial Forces
Vibrations
Static Aeroelasticity
Flight Mechanics
Dynamic Aeroelasticity
82
Divergence is a nonoscillatory instability phenomenon that occurs when the
restoring elastic moments within a wing are exceeded by the aerodynamic moments.
When a wing twists, an extra lift force is developed by the wing due to the increase
of incidence. This force acts at the aerodynamic center which is at or near the
quarter-chord for a subsonic flow. If the flexural axis lies aft of the aerodynamic
center (as in most actual wings), the increase in lift will tend to increase the twist
which in tern increases the incidence again resulting in more lift and hence more
incidence. At speeds below a critical speed known as the divergence speed the
increments in lift converge to a condition of stable equilibrium in which the
torsional moment of the aerodynamic forces about the flexural center is balanced by
the torsional rigidity of the wing. Divergence can be completely prevented by
placing the flexural axis at or forward of the aerodynamic center. Divergence is not
usually a consideration for swept back wings but it can be a critical design problem
on some slender straight or swept forward wing configurations.
Control surface effectiveness is a phenomenon that is related to ailerons. Ailerons
of an elastic wing are less effective than those of a rigid wing. The effectiveness
drops as the speed increases. At a critical speed that is known as the aileron reversal
speed, the aileron twists the wing to such an extent that the gain in the rolling
moment due to aileron rotation is less than the loss due to wing twist causing the
aircraft to roll in the opposite direction.
Since aircraft structures are flexible in nature, aerodynamic loads applied to them
produce deformation that might be significant enough to change the static stability
and load distribution due to changes in the shape of the aircraft.
Dynamic aeroelastic phenomena lie within the triangle since it involves the
interaction of all of the three types of forces and includes phenomenon like flutter,
buffeting, and dynamic response. Among all of these phenomena flutter is the most
83
important. The occurrence of flutter within the flight envelope leads to a
catastrophic structural failure and loss of the aircraft.
Flutter is defined as the sustained oscillation of the lifting surface under the effect of
high-speed air passage. The occurrence of flutter does not require any external
forcing agency. Any initial disturbance like the engine sound or control surface
movement is enough to trigger flutter. When a lifting surface starts to vibrate during
flight the oscillations usually die out due to the presence of structural and
aerodynamic damping. The structural damping of the lifting surface is constant.
Nevertheless, the aerodynamic damping is not constant and depends on the flight
speed. As the flight speed is increased the aerodynamic damping first increases then
starts dropping reaching negative values. At a certain speed the summation of the
structural and aerodynamic damping becomes zero. At this speed any disturbance
introduced to the structure will cause self sustained oscillations. This speed and the
corresponding oscillation frequency are known as the critical flutter speed and
flutter frequency respectively. At any speed that is equal to or above this speed,
disturbing the structure will cause it to start extracting energy from the air stream
and oscillations grow up indefinitely resulting in failure of the structure. In order for
the structure to start extracting energy from the air stream there must be a
significant phase difference between the coupled modes involved. This phase
difference is provided by the aerodynamic damping inherent in the aeroelastic
system. Flutter can be classified mainly into two types, classical and stall flutter.
Classical flutter involves the coupling of at least two structural modes. Wing
bending-torsion flutter and control surface flutter are typical types of classical
flutter. Control surface flutter involves the coupling between the control surface
mode and the wing torsion and/or bending modes. A wing equipped with an aileron
can flutter at a speed that is much lower than the wing bending-torsion flutter speed.
If the control surface flutter involves coupling of two modes then the flutter is
termed as binary flutter. Control surface flutter which involves the interaction of all
84
of the three modes is known as ternary flutter. Classical flutter can be eliminated if
the aerodynamic center, the flexural center, and mass center coincide.
Stall flutter on the other hand is associated with the flow separation and
reattachment at high angles of attack in the transonic and supersonic flow regimes.
It does not involve any structural modes coupling and happens when the torsional
structural mode becomes unstable. This aeroelastic instability phenomenon is
critical for rotating machineries such as helicopter rotors and turbine blades. Since
aircrafts rarely come close to stall when flying at the maximum velocities and
dynamic pressures for which they are designed, this phenomenon is not a serious
one on wings and tails. Nevertheless, it is an important aspect for turbojet engines
operating off their design speeds.
Buffeting is the transient vibration of aircraft structural components due to
aerodynamic impulses produced by the wake behind wings and engine nacelles. A
serious buffeting problem is encountered by aircraft during pull-up maneuvers to
maximum lift coefficients at high speed. This often results in rugged transient
vibrations in the tail due to aerodynamic impulses from the wing wake. The
problem of determining the dynamic stresses to provide adequate strength is very
difficult. The principal obstacle has been the lack of knowledge of the properties of
the wake behind the partially stalled wing. Buffeting problems are usually alleviated
by proper positioning of the tail and aerodynamic design to prevent flow separation.
This is the main reason for having fairings attached at the wing fuselage junctions.
All of these static and dynamic aeroelastic phenomena have profound effects on the
aircraft design and can only be solved in concurrent consideration by all disciplines
involved. Nevertheless, among them flutter and divergence acquire special
importance since the occurrence of any of them will lead to catastrophic structural
failure and loss of the aircraft.
85
The wing of an aircraft structure plays an important role in the aeroelastic design
and analysis process. Aeroelastic stability analysis of an aircraft is concerned with
determining the stability boundaries of the aircraft structure. It identifies the flight
conditions in terms of flight speed and altitude or density at which the structure
becomes unstable. Several aeroelastic instability problems like flutter and
divergence are highly influenced by the stiffness and mass distribution on the wing.
In the development of a new aircraft, enough stiffness has to be provided to insure
that the aircraft is free from any sort of aeroelastic instability in its design flight
envelope. JAR25.629 specifies the certification requirements for a new developed
aircraft. The compliance requirements specify that the aircraft must be free from
flutter and divergence with adequate damping margins for speeds up to 1.2VD,
where VD is the design diving speed. Furthermore, it must be free from flutter at
speeds up to VD after certain specified structural failures. This must be
demonstrated by analysis substantiated by full scale flutter test up to VD.
Flutter clearance for a particular design does not rely completely upon analysis
alone. Ground vibration and wind tunnel tests are first performed both to confirm
the analysis and to provide extensive information on the effects of varying a number
of important parameters such as fuel quantity, engine location, and stiffness
distributions. The effect of structural failures is also simulated in the wind tunnel.
Then the actual airplane is tested. Accelerometers are located on all the principal
components of the airframe and position indicators are provided on the control
surfaces. The airplane is flown at specific altitudes at incrementally increasing
speeds. The structure is excited by means of pilot-input, control surface pulses or
through the use of wing or empennage tip “shakers”. For each speed/altitude
combination up to VD, the airplane’s motions and the decay of these motions are
measured. If adequate damping exists at all speeds up to VD, the results of the flight
flutter test, together with the analysis and its supporting wind tunnel data provide
the basis for the flutter clearance of the airplane. Thus, it is necessary to obtain a
86
valid damping history to prevent loss of the aircraft or wind tunnel test model
during the flight and wind tunnel tests.
Traditionally frequency domain methods like the k-method and pk-method are used
for aeroelastic stability analysis. However, these methods produce damping
information that is valid only at the instability point or near to the instability point.
To gain an insight into the physical phenomena leading to aeroelastic instability it is
necessary to obtain valid damping and frequency history. Laplace domain methods
like the p-method and root locus method provide such an insight. However, the
main difficulty in implementing this method lies in obtaining the aerodynamic loads
for an arbitrary motion in Laplace domain. Unsteady aerodynamic forces are only
well developed for simple harmonic motions in the frequency domain. This problem
is circumvented using rational function approximations for the aerodynamics in
Laplace-domain.
In this chapter, the mathematical formulation of the aeroelastic stability problem is
first discussed. Aeroelastic analysis methods based on the k-method, pk-method, p-
method, and the root locus method are then explained. The differences between
these methods are identified. The method of obtaining the generalized aerodynamic
forces in Laplace domain using Rogers rational function approximations is
explained. The aerodynamic lag roots necessary for this approximation are obtained
using a direct search optimizer that is based on the complex method. Two test cases
are studied to study the difference between the PK-method and P-method and verify
the adopted methodology for approximating the generalized aerodynamic forces in
Laplace domain using rational function approximations. Finally, the chapter ends
with some concluding remarks and discussions.
87
3.2 Theory of Aeroelastic Stability
In this section the classical aeroelastic stability equation is derived and various
solution methods with their advantages and disadvantages are discussed.
The equation of motion of a multi degree of freedom, discrete and damped
aeroelastic system can be derived based on the dynamic equilibrium of forces. The
time-domain equation of motion in matrix form is given as
n]Front Spar-Upper Cap Front Spar-Lower CapRear Spar-Upper Cap Rear Spar-Lower Cap
198
Table 5.12 Summary of Skin Margins of Safety (Satic Strength and Aeroelastic Constraints)
Skin Panel
Location
M.S Failure Mode
Upper Skin Root-Rib1 0.01 SKIN BUCKLING
Rib1-Rib2 3.73 SKIN BUCKLING
Rib2-Rib3 22.0 SKIN BUCKLING
Lower Skin Root-Rib1 1.90 MAX. STRESS-2 (T)
Rib1-Rib2 0.24 SKIN BUCKLING
Rib2-Rib3 55.8 MAX. STRESS-2 (T)
Table 5.13 Summary of Spars Margins of Safety (Satic Strength and Aeroelastic Constraints)
Spar Section
Location
M.S Failure Mode
Front Spar Root-Rib1 0.45 CRIPPLING
Rib1-Rib2 3.45 WEB RUPTURE (VM STRESS)
Rib2-Rib3 15.9 CRIPPLING
Rear Spar Root-Rib1 0.32 CRIPPLING
Rib1-Rib2 3.42 WEB RUPTURE (VM STRESS)
Rib2-Rib3 16.7 CRIPPLING
199
Table 5.14 Summary of Spars Margins of Safety (Satic Strength and Aeroelastic Constraints)
Rib Number M.S Failure Mode
Rib1 1.96 BUCKLING
Rib2 30.0 BUCKLING
Rib3 9.54 BUCKLING
200
Figure 5.26 Velocity vs. Damping Plot of the Rectangular Wing
(Static Strength and Aeroelastic Constraints)
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
500 800 1100 1400 1700 2000
Velocity [ft/sec]
Dam
ping
, g .
Mode 1 (RL Method) Mode 2 (RL Method)
201
Figure 5.27 Velocity vs. Frequency Plot of the Rectangular Wing
(Static Strength and Aeroelastic Constraints)
-5
0
5
10
15
20
25
30
35
40
500 800 1100 1400 1700 2000
Velocity [ft/sec]
Freq
uenc
y [H
z]Mode 1 (RL Method) Mode 2 (RL Method)
202
5.6 Conclusion
In this chapter the developed multidisciplinary design and optimization code has
been applied to the design of a rectangular wing box. The wing has been considered
in three different test cases.
In the first case study the developed code is verified and the capability of the
genetic algorithm in optimization for aeroelastic constraints with manufacturing
constraints imposed on the design variables is demonstrated.
In the second case study the capability of the developed code in the optimization of
“real-life” composite wing box is demonstrated using a generic composite wing box
model.
In the third case study the advantage of considering aeroelastic stability constraints
at early stages of the design is investigated. In the case study considered it is shown
that a 38% increase in the aeroelastic instability speed is acheivable at the cost of
3% icrease in the total strutural weight of the wing box considered. The capability
of the genetic algorithm in the optimization of composite wing box structures with
static strength, aeroelastic stability, and manufacturing constraints is also
demonstrated.
203
CHAPTER 6
CONCLUSIONS
6.1 General Conclusions
In this thesis an automated multidisciplinary design optimization code was
developed for the minimum weight design of a composite wing box. The
multidisciplinary static strength, aeroelastic stability, and manufacturing
requirements were simultaneously addressed in a global optimization environment
through a genetic search algorithm.
The aim was to obtain a minimum weight final design that complies with the
existing certification requirements (FAR/JAR) in a time, which is less than what is
currently needed, while taking aeroelastic stability constraints into account at the
early stages of the design. Consequently, the need for extensive design
modifications at later stages of the design, that may result in weight penalties, was
eliminated.
The static strength requirements specify obtaining positive margins of safety for all
of the structural parts of the wing box taking into account all potential failure
modes. Besides to classical failure modes (material failure), specialized failure
modes (buckling and crippling) were taken into account in the optimization process.
The aeroelastic stability analysis requirements specify obtaining flutter and
204
divergence free wing box for a range of prescribed flight conditions and with
required damping level in the final design.
The global optimization problem of a wing box in which the design variables are of
mixed-discrete type and the static strength and aeroelastic stability constraints are
considered simultaneously has never been attempted in previous studies. Very
simple models that rely on using direct stress output of coarse mesh finite element
models had been used and they did not account for specialized failure modes that
should be considered in the design cycle. In chapter 2 of this thesis, it was shown
that this approach would result in erroneous stress estimates.
Typical wing boxes exhibit thin metal/composite panels joined together to form the
structure. Since these panels are very thin, they usually buckle at very low stress
amplitudes causing redistribution of the load in the structure. Nonlinear effects that
result from load redistribution in the structure should be taken into account to insure
failure free structure. A detailed linear finite element analysis of the structure would
not account for such nonlinear effects. This effect can only be simulated using
nonlinear finite element analysis with fine mesh models. In this thesis, the
aerospace industry approach to this problem was used to circumvent this problem.
The approach relied on constructing coarse mesh finite element models to
determine the internal load distribution in the structure and then using simplified
theories, like the modified engineering bending theory, to determine the stresses and
simulate the correct behavior of the structure after buckling occurs.
Buckling analysis of composite plates is usually based on the specially orthotropic
plate assumptions and the use of interaction equations. It was shown in chapter 2
that the buckling analysis based on the specially orthotropic plate assumptions is not
a generally valid approach and can only be justified if the laminate is cross-ply
symmetric one. On the other hand using energy methods such as the Rayleigh-Ritz
method would result in accurate and acceptable results at low cost when compared
205
to the finite element method. It also eliminates the need to prepare specialized finite
element models for buckling analysis purposes. Another important result is related
to the buckling analysis of anisotropic laminates. For anisotropic laminates, it was
shown that care should be taken in the correct sign for the applied shear stress,
otherwise positive and negative shear loads would result in completely two different
results for the allowable buckling load.
Flutter and divergence are the most important aeroelastic instability phenomena
since the occurrence of any of them would lead to catastrophic structural failure and
loss of the aircraft. Obtaining valid damping history is generally required for the
certification of a new developed aircraft and it is also needed to prevent loss of the
aircraft or wind tunnel test-model in a flutter clearance test.
Aeroelastic stability analysis to determine the onset of flutter and divergence can be
performed relatively easily in the frequency domain using either k-method or the
pk-method. However, these methods produce damping information that is either
invalid (k-method) or approximate (pk-method).
On the contrary, the root locus method results in damping information that is valid
for all of the speed range of interest and provides better insight into the physical
phenomena leading to aeroelastic instability. The computational cost problem
associated with the calculation of the unsteady aerodynamic forces in the Laplace
domain is circumvented efficiently through the use of the Rogers rational function
approximations.
In the practical design of a composite wing box, the design variables are not all
continuous and some of them must be selected from a set of integer or discrete
values. The structural members may have to be chosen from standard sizes and
member thicknesses may have to be selected from commercially available ones.
Stacking sequence design of composite plates involves the determination of the
206
number of plies and their orientations. The stacking sequence design problem is
discrete in nature. Due to manufacturing limitations, the plies are fabricated at
certain thicknesses and the orientations are limited to certain sets of discrete angles.
Thus, the optimization problem is a nonlinear optimization problem that involves a
combination of continuous and discrete design variables. Commercial softwares
utilize gradient-based methods that can not treat this type of optimization problems
efficiently and may produce suboptimal or infeasible designs. In this thesis, the
problem was solved efficiently by using a genetic algorithm based optimizer.
The developed code was applied to the design of composite rectangular wing box
with metallic internal substructure. Hence the two spars and number of ribs are in
the form of conventional aluminum construction. The skin of the wing was taken as
composite. The wing box was considered in three different test cases. In the first
case study the developed code was verified and the capability of the genetic
algorithm in optimization for aeroelastic constraints with manufacturing constraints
imposed on the design variables was demonstrated. In the second case study the
capability of the developed code in the optimization of “real-life” composite wing
box was demonstrated. In the third case study the advantage of considering
aeroelastic stability constraints at early stages of the design was investigated. It was
shown that a 38% increase in the aeroelastic instability speed is acheivable at the
cost of 3% increase in the total structural weight of the wing box considered. The
capability of the genetic algorithm in the optimization of composite wing box
structures with static strength, aeroelastic stability, and manufacturing constraints
was demonstrated.
207
6.2 Recommendations for Future Work
In this thesis the global optimization problem was solved by using the genetic
algorithm. Although no attempt was done to tune the genetic algorithm to suite the
optimization problem under consideration, the computational cost involved was
found to be high. Nevertheless, a hybrid optimization scheme that uses the genetic
algorithm to locate the global minimum in the design space using the first few
generations and then switches to a conventional nonlinear programming approach
may be investigated.
The constraints were handled using the conventional penalty function approach.
The use of gene repair strategies and the K-S function approach in handling the
constraints may be investigated.
The developed code may be modified to analyze an all composite wing box.
208
REFERENCES
1. Haftka, R. T., Sobieski, S., and Padula, S. L., “On Options for Interdisciplinary Analysis and Design Optimization,” Structural Optimization, Vol. 4, pp. 65-74, 1992
2. Sobieszczanski-Sobieski, J., and Haftka, R. T., “Multidisciplinary Aerospace Design Optimization: Survey of Recent Developments,” Structural Optimization, Vol. 14, pp. 1-23, 1997
3. Turner, M. J., “Optimization of Structures to Satisfy Flutter Requirements,” AIAA Journal, Vol. 7, No. 5, pp. 945-951, May 1969
4. Wilkinson, K., Markowitz, J., Lerner, E., George, D., and Batill, S. M., “FASTOP: A Flutter and Strength Optimization Program for Lifting Surface Structures,” Journal of Aircraft, Vol. 14, pp. 581-587, 1977
5. Isakson, J., Pardo, H., Lerner, E., and Venkayya, V. B., “ASOP-3: A Program for Optimum Structural Design to Satisfy Strength and Deflection Constraints,” Journal of Aircraft, Vol. 15, pp. 422-428, 1978
6. Rudisill, C. S., and Bhatia, K. G., “Optimization of Complex Structures to Satisfy Flutter Requirements,” AIAA Journal, Vol. 9, No. 8, pp. 1487-1491, August 1971
7. Rudisill, C. S., and Bhatia, K. G., “Second Derivative of the Flutter Velocity and the Optimization of Aircraft Structures,” AIAA Journal, Vol. 10, No. 12, pp. 1569-1572, December 1972
209
8. Gwin, L. B., and Taylor, R. F., “A General Method for Flutter Optimization,” AIAA Journal, Vol. 11, No. 12, pp. 1613-1617, December 1973
9. Haftka, R. T., Starnes Jr., J. H., Barton, F. W., and Dixon, S. C., “Comparison of Two Types of Structural Optimization Procedures for Flutter Requirements,” Vol. 13, No. 10, pp. 1333-1339, October 1975
10. McIntosh, S. C., and Ashley, H., “On the Optimization of Discrete Structures with Aeroelastic Constraints,” Computers and Structures, Vol. 8, pp. 411-419, 1978
11. Hajela, P., “A Root Locus-Based Flutter Synthesis Procedure,” Journal of Aircraft, Vol. 20, No. 12, pp. 1021-1027, December 1983
12. Seyranian, A. P., “Sensitivity Analysis and Optimization of Aeroelastic Stability,” Inernational Journal of Solids, Vol. 18, No. 9, pp. 791-807, 1982
13. Isaac, J. C., and Kapania, K. K., “Aeroelastic Sensitivity Analysis of Wings Using Automatic Differentiation,” AIAA Journal, Vol. 35,No. 3, pp. 519-525, March 1997
14. Kapania, K. K., and Bergen, F. D., “Shape Sensitivity Analysis of Flutter Response of a Laminated Wing”, AIAA Journal, Vol. 29,No. 4, pp. 611-612, April 1991
15. Neill, D. J., Johnson, E. H., and Canfield, R., “ASTROS- A Multidisciplinary Automated Structural Design Tool,” Journal of Aircraft, Vol. 12, pp. 1021-1027, 1990
16. Shirk, M. H., Hertz, T. J., and Weisshaar, T. A., “Aeroelastic Tailoring- Theory, Practice, and Promise,” Journal of Aircraft, Vol. 23, pp. 6-18, 1986
17. Lerner, E., and Markowitz , J., “An Efficient Structural Resizing Procedure for Meeting Static Aeroelastic Design Objectives,” Journal of Aircraft, Vol. 16, pp. 65-71, 1979
210
18. Weisshaar, T. A., and Foist, B. F., “Vibration and Flutter of Advanced Composite Lifting Surfaces,” AIAA Paper 83-0961, 1983
19. Lottati, I., “Flutter and Divergence Aeroelastic Characteristics for Composite Forward Swept Cantilevered Wing,” Journal of Aircraft, Vol. 22, pp. 1001-1007, 1985
20. Ringertz, U. T., “On Structural Optimization with Aeroelasticity Constraints,” Structural Optimization, Vol. 8, pp. 16-23, 1994
21. Popelka, D., Lindsay, D., Parham Jr, T., Berry, V., and Baker, D. J., “Results of an Aeroelastic Tailoring Study for a Composite Tiltrotor Wing,” 51st American Helicopter Society Annual Forum, Fort Worth, Texas, May 9-11, 1995
22. Eastep, F. E., Tischler,V. A., Venkayya, V. B., and Khot, N. S., “Aeroelastic Tailoring of Composite Structures,” Journal of Aircraft, Vol. 36, pp. 1041-1047, 1999
23. Khot, N. S., and Kolonay, R. M., “Composite Wing Optimization for enhancement of the Rolling Maneuver in Subsonic Flow,” Structural Optimization, Vol. 17, 95-103, 1999
24. Venkataraman, S., and Haftka, R. T., “Optimization of Composite Panels-A Review”, Proceedings of the 14th Annual Technical Conference of the American Society of Composites, OH., September 1999
25. Isogai, K., “Direct Search method to Aeroelastic Tailoring of a Composite Wing under Multiple Constraints,” Journal of Aircraft, Vol. 26, pp. 1076-1080, 1989
26. Jha, R., and Chattopadhyay, A., “Multidisciplinary Optimization of Composite Wings Using Refined Structural and Aeroelastic Analysis Methodologies,” Engineering Optimization, Vol. 32, pp. 59-78, 1999
211
27. Riche, R. L., and Haftka, R. L., “Evolutionary Optimization of Composite Structures,” Evolutionary Algorithms in Engineering Optimization, March 1997
28. Nagendra, S., Haftka, R. T., and Gürdal, Z., “Stacking Sequence Optimization of Simply Supported Laminates with Stability and Strain Constraints,” AIAA Journal, Vol. 30, No. 8, pp. 2132-2137, August 1992
29. Hajela, H., “Nongradient Methods in Multidisciplinary Design Optimization-Status and Potential,” Journal of Aircraft, Vol. 36, pp. 255-265, 1999
30. Arora, J. S., and Huang, M. W., “Methods for Optimization of nonlinear problems with discrete variables: a review,” Structural Optimization, Vol. 8, pp. 69-85, 1994
31. Goldberg, D. E., “Genetic Algorithms in Search, Optimization, and Machine Learning,” Addison-Wesley Publishing Compoany, 1989
32. Kim, H., Adeli, H., “Discrete Cost Optimization of Composite Floors Using a Floating-Point Genetic Algorithm,” Engineering Optimization, Vol. 33, pp. 485-501, 2001
33. Hajela, P., “Genetic Search-An Approach to the Nonconvex Optimization Problem,” AIAA Journal, Vol. 28, No. 7, pp. 1205-1210, July 1990
34. Kogiso, N., Watson, L. T., Gürdal, Z., and Haftka, R. T., “Genetic Algorithms with Local Improvement for Composite Laminate Design,” Structural Optimization, Vol.7, pp 207-218, 1994
35. Liu, B., Haftka, R. T., Akgün, M. A., and Todoroki, A., “Permutation Genetic Algorithm For Stacking Sequence Design of Composite Laminates,” Proceedings of the 39th AIAA/ASME/ASCE/AHS Structures, Structure Dynamics and Material Conference, Long Beach, CA, April 20-23, 1998
212
36. Liu, B., Haftka, R. T., and Akgün, M. A., “Two-Level Composite Wing Structural Optimization Using Response Surfaces,” Structural and Multidisciplinary Optimization, Vol. 20, pp. 87-96, 2000
37. Upaadhyay, A., and Kalayanaraman, V., “Optimum Design of Fibre Composite Stiffened Panels Using Genetic Algorithms,” Engineering Optimization, Vol. 33, pp. 201-220, 2000
38. Leiva, J. P., Ghosh, D. K., and Rastogi, N., “A New Approach in Stacking Sequence Optimization of Composite Laminates Using Genesis Structural Analysis and Optimization Software,” 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optmization, 4-6 September 2002, Geogia, AIAA 2002-5451, 2002
39. Todoroki, A., and Haftka, R. T., “Stacking Sequence Optimization by a Genetic Algorithm with a New Recessive Gene Like Repair Strategy,” Composites Part B,Vol. 29, No. 3, pp. 277-285, 1998
40. Li, G., Wang, H., Aryasomayajula, S. R., and Grandhi, R. V. “Two-Level Optimization of Airframe Structures Using Response Surface Approximation,” Structural and Multidisciplinary Optimization, Vol. 20, pp. 116-124, 2000.
41. Unal, R., Lepsch, R. A., and McMillin, M. L., “Response Surface Model Building and Multidisciplinary Optimization Using D-Optimal Designs,” AIAA-98-4759, 1998
42. Giles, G.L., “Equivalent Plate Analysis of Aircraft Wing Box Structures with General Planform Geometry,” Journal of Aircraft, Vol. 23, No. 11, pp. 859-864 November 1986
43. Giles, G.L, “Further Generalization of an Equivalent Plate Representation for Aircraft Structural Analysis,” Journal of Aircraft, Vol. 26, No. 1, pp. 67-74, January 1989
213
44. Livne, E., “Equivalent Plate Structural Modeling for Wing Shape Optimization Including Transverse Shear,” AIAA Journal, Vol. 32, No. 6, pp. 1278-1288, June 1994
45. Mukhopadhyay, V., “Interactive Flutter Analysis and Parametric Study for Conceptual Wing Design,” AIAA 95-3943, 1995
46. Butler, R., Hansson, E., Lillico, M., and Dalen, F., “Comparison of Multidisciplinary Design Optimization Codes for Conceptual and Preliminary Wing Design,” Journal of Aircraft, Vol. 36, No. 6, pp. 934-940, November 1999
47. Dodd, A. J., Kadrinka, K. E., Loikkanen, M. J., Rommel, B. A., Sikes, G. D., Strong, R. C., and Tzong, T. J., “Aeroelastic Optimization Program”, Journal of Aircraft, Vol.27, No.12, pp 1028-1036, December 1990
48. Collier, C. S., Peckenheim, M., and Yarrington, P. W., “Next Generation Structural Optimization Today”, MSC Aerospace Users’ Conference Proceedings, September 1997
49. Striz, A. G., Venkayya, V. B., “Influence of Structural and Aerodynamic Modeling on Flutter Analysis,” Journal of Aircraft, Vol. 31, No. 5, Septemer 1994.
50. Karpel, M., “Reduced-Order Aeroelastic Models via Dynamic Residualization,” Journal of Aircraft, Vol. 27, No. 5, pp. 449-455, May 1990
51. Karpel, M., “Multidisciplinary Optimization of Aeroservoelastic Systems Using Reduced-Size Models,” Journal of Aircraft, Vol. 29, No. 5, pp. 939-946, October 1992
52. Karpel, M., “Reduced-Order Models for Integrated Aeroservoelastic Optimization,” Journal of Aircraft, Vol. 36, No. 1, pp. 146-155, January 1999
214
53. Karpel, M., “Modal-Based Enhancements of Integrated Design Optimization Schemes,” Journal of Aircraft, Vol. 35, No. 3, pp. 437-444, May 1998
54. Karpel, M., and Brainin, L., “Stress Considerations in Reduced-Size Aeroelastic Optimization,” AIAA Journal, Vol. 33, No. 4, pp. 716-722, April 1995
55. Rodden, W. P., and Johnson, E. H., “User’s Guide of MSC/NASTRAN Aeroelastic Analysis,” MSC/NASTRAN v68, 1994
56. Markowitz, J., and Gabriel, I., “An Automated Procedure for Flutter and Strength Analysis and Optimization of Aerospace Vehicles, Volume I. Theory and Application,” AFFDL-TR-75-137, December 1975
57. Theoretical Manual of ZAERO v4.0, ZONA 99-24.4, Zona Technology Incorporation, November 1999
58. Hassig, H. J., “An Approximate True Damping Solution of the Flutter Equation by Determinant Iteration,” Journal of Aircraft, Vol. 8, No 10, pp. 885-889, November 1971
59. Albano, E. And Rodden W. P, “A Doublet-Lattice Method for Calculating Lift Distributions on Oscillating Surfaces in Subsonic Flows,” AIAA Journal, Vol. 7, No. 2, pp. 279-285, February 1969
60. Theodorsen, T., “General Theory of Aerodynamic Instability and the Mechanism of Flutter,” NACA Report 496, 1935
61. Rodden, W. P., and Bellinger, E. D., “Aerodynamic Lag Functions, Divergence, and the British Flutter Method,” Journal of Aircraft, Vol. 19, No. 7, pp. 596-598, July 1982
62. Zyl, L. H., and Maserumule, M. S., “Aeroelastic Divergence and Aerodynamic Lag Roots,” Journal of Aircraft, Vol. 38, No. 3, pp. 586-588, May 2001
215
63. Chen, P. C., “Damping Perturbation Method for Flutter Solution,” AIAA Journal, Vol. 38, No. 9, pp. 1519-1524, September 2000
64. Alfolabi, D., Pidaparti, R. M. V., and Yang, H. T. Y., “Flutter Prediction Using Eigenvector Orientation Approach,” AIAA Journal, Vol. 36, No. 1, pp. 69-74, 1998
65. Pidaparti, R. M. V., Tischler, V. A., and Venkayya, V. B., “Flutter Prediction Methods for Aeroelastic Design Optimization,” Journal of Aircraft, Vol. 38, No. 3, pp. 557-559, May 2001
66. Zyl, L. H., and Maserumule, M. S., “Divergence and the p-k Flutter Equation,” Journal of Aircraft, Vol. 38, No. 3, pp. 584-586, May 2001
67. Zyl, L. H., and Maserumule, M. S., “Unrestrained Aeroelastic Divergence and the p-k Flutter Equation”, Journal of Aircraft, Vol.38, No.3, pp 588-590, May 2001
68. Brase, L. O., and Eversman, W., “Application of Transient Aerodynamics to the Structural Nonlinear Flutter Problem,” Journal of Aircraft, Vol. 25, No. 11, pp. 1060-1068, November 1988
69. Eversman, W. and Tewari, A., “Modified Exponential Series Approximation for the Theodorsen Function,” Journal of Aircraft, Vol. 28, No. 9, pp. 553-557, September 1991
70. Bisplinghoff, R. L., Ashley, H., and Halfman, R. L., “Aeroelasticity,” Dover Publications, Inc., 1955
71. Roger, K. L., “Airplane Math Modeling Methods for Active Control Design,” AGARD-CP-228, August 1977
216
72. Karpel, M., “Design for Active Flutter Suppression and Gust Alleviation Using State-Space Aeroelastic Modeling,” Journal of Aircraft, Vol. 19, No. 3, pp. 221-227, March 1982
73. Karpel, M. and Strul, E., “Minimum-State Unsteady Aerodynamic Approximations with Flexible Constraints,” Journal of Aircraft, Vol. 33, No. 6, pp. 1190-1196, November 1996
74. Bruhn, E. F., “Analysis and Design of Flight Vehicle Structures,” Jacobs Publishing, Inc., 1973
75. Niu, M. C. Y., “Airframe Stress Analysis and Sizing,” Conmilit Pres Ltd, 1999
76. Niu, M. C. Y., “Airframe Structural Design,” Conmilit Pres Ltd, 1999
77. Dinçer, S. Ö., “Development of A Wing Preliminary Structural Analysis Code,” MS Thesis, Middle East Technical University, 2000
78. Jones, R. M., “Mechanics of Composite Materials,” Taylor and Francis, Inc., 1999
79. Ashton, J. E., Whitney, J. M., “Theory of Laminated Plates,” Technomic Publishing Co., Inc., 1970
80. Whitney, J. M., “Structural Analysis of Laminated Anistropic Plates,” Technomic Publishing Co., Inc., 1987
81. Kolonay, R. M., “Unsteady Aeroelastic Optimization in the Transonic Regime,” Ph.D Thesis, Purdue University, December 1996
82. Neill, D. J., Herendeen, D. L., Venkayya, V. B.“ASTROS Enhancements, Volume III – Theoretical Manual,” AFWAL-TR-96-3006, Vol. 3, April 1996
217
83. Broadbent, E. G., “The Elemntary Theory of Aeroelasticity Part I. Divergence and Reversal of Control,” Aircraft Engineering, Vol. 26, pp 70-78, March 1954
84. Broadbent, E. G., “The Elemntary Theory of Aeroelasticity Part II. Wing Flutter,” Aircraft Engineering, Vol.26, pp 113-121, April 1954
85. Broadbent, E. G., “The Elemntary Theory of Aeroelasticity Part III. Flutter of Control Surfaces and Tabs,” Aircraft Engineering, Vol.26, pp 145-153, May 1954
86. Rodden, W. P., “ A Matrix Approach to Flutter Analysis,” Sherman M. Fairchild Fund Paper No. FF-23, Institute of the Aeronautical Sciences, May 1959
87. Duncan, W. J., “The Fundementals of Flutter”, Technical Report of the Aeronautical Research Council, Vol. II, No. 2417, pp 757-792, November 1948
88. Moore, G. J., “User’s Guide to MSC/NASTRAN Design Sensitivity and Optimization”, Macneal-Schwendler Corporation, Version 68, May 1994
89. Lin, C. Y., and Hajela, P., “Evolve, A Genetic Search Optimization Code, User’s Manual,” Department of Mechanical Engineering, Aerospace Engineering and Mechanics, Resselaer Ploytechnic Institute, January, 1993
90. Anderson Jr., J. D., “Fundementals of Aerodynamics”, McGraw-Hill, Inc., 1985
218
VITA
The author was born in Tubas (Palestine) on February 1970. He received his B.Sc.
degree in Aeronautical Engineering from the Middle East Technical University
(METU) in 1993 and his M.Sc. in Mechanical Engineering from METU in 1996.
He started working as a design engineer at the Turkish Aerospace Industries (TAI)
company in 1998 and is currently a structural analysis specialist at the same
company. His main areas of interest include stress analysis, structural dynamics,