CONTENTS
1. INTRODUCTION ...................................................................................................... 3
1.1
Relevance of research topic .................................................................................. 3
1.2
Level of scientific development of research topic ................................................ 9
1.3
Research objectives ............................................................................................ 10
1.4
Scientific novelty of research ............................................................................. 10
1.5
Theoretical value of and the practical significance of the research ................... 11
1.6
Research methods ............................................................................................... 11
1.7
Theses ................................................................................................................. 11
1.8
Validity of data ................................................................................................... 12
1.9
Structure of dissertation ...................................................................................... 12
2. LITERATURE REVIEW ......................................................................................... 13
2.1
Optimum design.................................................................................................. 13
2.2
Research methodology........................................................................................ 16
2.3
Researches of laminated composite materials .................................................... 18
3. RESEARCH METHODOLOGY.............................................................................. 21
3.1
Composite panel with an asymmetrical thickness package structure ................ 28
3.2 Composite panel eccentrically supported by longitudinal-transverse set of
stiffeners ....................................................................................................................... 31
3.3
Symbolic integration method.............................................................................. 35
4. PROBLEM SOLUTION IN DOUBLE TRIGONOMETRIC SERIES ................... 39
5. CALCULATION EXAMPLE OF FLAT RECTANGULAR MULTYLAUER
PANEL ............................................................................................................................ 43
5.1
Initial Data .......................................................................................................... 43
5.2
Calculation Results ............................................................................................. 44
5.2.1
Deflection Calculations ................................................................................ 45
5.2.2
Stress Calculation ......................................................................................... 46
5.2.3
Equivalent Stress .......................................................................................... 48
5.2.4
Stress diagrams ............................................................................................. 51
1
6. CALCULATION EXAMPLE OF STIFFENED PANEL ........................................ 55
6.1
Initial Data .......................................................................................................... 55
6.2
Deflection Calculation ........................................................................................ 59
6.3
Stress Calculation ............................................................................................... 60
7. CONCLUSION ......................................................................................................... 61
REFERENCES ................................................................................................................ 62
2
1. INTRODUCTION
1.1 Relevance of research topic
Fiber-reinforced composites and in particular stiffened composite panels are
extensively used in aviation industry. The primary advantages of composites are their
excellent mechanical, thermal, and chemical properties, e.g. high stiffness-to-weight and
strength-to-weight ratios, corrosive resistance, low thermal expansion, vibration
damping.
Key benefits of using composites for aerospace applications include the following:
 weight reduction up to 20% – 50%;
 single-shell molded structures provide higher strength at lower weight;
 high impact resistance;
 high thermal stability;
 damage tolerant;
 resistant to fatigue/corrosion;
 structural components made from composite materials are easy to assemble.
The use of composite materials allows designing structures with predetermined
characteristics and makes it possible to significantly increase their weight efficiency and
reduce metal consumption.
The development of aircraft constructions is associated with the continuous struggle
to reduce the weight of the structure. Weight reduction can be achieved by a rational
choice of materials, structural scheme, and technological processes. The use of
composite materials makes it possible to reduce the weight of products and,
accordingly, reduce fuel consumption. Table 1 illustrates cost savings (in dollars) by
reducing the mass of a structure by 1 kg. In high-speed aviation, from 7% to 25% (by
weight) of polymer composites are used, thus the weight of products is reduced from 5
to 30%.
3
Table 1
cost saving, $
Space Shuttle
10,000 – 15,000
Satellite in Synchronous Orbit
10,000
Satellites in low-Earth Orbit
1,000
Supersonic Passenger Aircraft
200 – 500
Interceptor Aircraft
150 – 200
Boeing-747
150 – 200
Aircraft Engines
100 – 200
Passenger Aircraft
100
Transport Aircraft
50 – 75
Calculated data, experimental results and flight tests, show the use of composite
materials can reduce the weight of the aircraft airframe by 30-40% in comparison with
the use of traditional metallic materials. This provides a weight reserve, which can be
used to increase the flight range or payload.
Fiberglass was the first material used in aviation, by Boeing in 1950. Currently, the
use of composite materials in the aircraft design increases. For example, in aircraft
A320, A340 (Airbus) and B777 (The Boeing Company), 10-15% of composite
materials were used by weight. In these aircraft, composite materials were used mainly
for finishing work. In the modern aircraft of these two corporations, A350 and B787
Dreamliner, the proportion of composite materials by mass increased significantly. In
the design of the A350, composite materials make up 52% of the aircraft weight.
Boeing 787 consists of almost 50% of composites, with average weight savings of 20%.
Prior to the mid-1980s, aircraft manufacturers used composite materials in transport
aircrafts only in secondary structures (e.g., wing edges and control surfaces), but as
knowledge and development of the materials were improved, their use in primary
structure was increased. In 1988, Airbus introduced A320. Composite materials were
used for the entire tail structure, and also for fin/fuselage fairings, trailing-edge flaps
and flap-track fairings, spoilers, ailerons, and nacelles. In total, composites constituted
28% of the weight of the A320 airframe.
4
In 1995 the Boeing Company introduced Boeing 777. Boeing 777 consisted of
around 20% of composites by weight. Composite materials were used for wing‟s
leading and trailing edge panels, flaps and flaperons, spoilers, and outboard ailerons.
They were also used for floor beams, wing-to-body fairings, and landing-gear doors.
Using composite materials for the empennage saves approximately 1,500 lb in weight.
On the Figure 1, the commercial airplane models over the time by the percentage of
composites are shown.
Figure 1 Commercial airplane models over the time by percentage of composites
The following Table 2 lists some aircraft in which significant amounts of composite
materials are used in the airframe. The Figure 2 below shows the distribution of
materials in the F18E/F aircraft.
5
Table 2
Figure 2 Distribution of materials in the F18E/F aircraft
Composite materials are extensively used in the Eurofighter: wing skin, forward
fuselage skin, flaperons, and rudder are made from composites. Toughened epoxy skin
constitutes about 75 percent of the exterior area. In total, about 40 percent of the
structural weight of the Eurofighter is carbon-fibre reinforced composite material. Other
European fighters typically feature between 20 and 25 percent composites by weight.
Figure 3 shows percentages of composites used in the aircraft structures.
6
Figure 3 Percentages of composites used in the aircraft structures
Composite materials are also widely used in of Russian aircraft constructions. The
use of composites in MC-21 aircraft is 35%. Wings are entirely made from composite
materials. MС-21 composite design of wing and tail structures is shown on the Figure 4.
Sukhoi Superjet 100 includes composite parts such as wing mechanization, control
surfaces, landing gear flaps, and aircraft fairings.
A High strength carbon tape for primary structures
B1 Carbon fabric for secondary structures in combination with fiberglass to form a skin joint
with honeycomb
7
D Fiberglass for secondary structures
D1 Fiberglass for secondary structures in combination with fiberglass to form a skin joint with
honeycomb
Polymer composite floor panels
Metals
Distribution of Materials
Distribution of PCM
Composite materials
Glass
Enamel
Thermal Acoustic Insulation
Aluminum Alloys
Titanium Alloys
Steel
Other materials
Figure 4 Distribution of the material in MС-21 construction
Composites are also used in the production of passenger and transport aircraft such
as IL-86, IL-96-300, TU-204, TU-334, AN-124, AN-225, AN-70, AN-140, AN-148,
IL-114, fighters MIG-29, SU-27, YAK-36, YAK-130, SU-47, light sport aircraft SU-29,
SU-31m. The use of polymer composite materials in the Russian aircraft is shown on
Volume of PCM in constructions, %
the Figure 5 below.

o

Civil
Military
Fighter
year
Figure 5 Use of polymer composite materials in the aerospace structures
8
In AN-124 airplane, the mass of the structure made from PCM is 5500 kg, and their
area is more than 1,500
. In AN-70 construction, 2,900 kg of carbon fibers and
1,500 kg of organoplastic were used to produce parts for fuselage, wing, control
surfaces, and engine nacelle. In the construction of MIG-29, the use of composites
reduced the weight by 100 kg.
1.2 Level of scientific development of research topic
In fact, the application of composite materials in the primary structures is still
limited. Drawbacks of such a choice are that the design process becomes harder than for
classical metallic structures and the cost of composite structures is far bigger. Therefore,
appropriate design and optimization process are essential in order to minimize weight
and meet all structural design constraints. The design variables are: skin thickness and
stringer cross section for different sets of stringer spacing.
Stiffened composite panels have been widely used in high-performance aerospace
structures. In fact, although the use of stiffened composite panels is not a recent
achievement in structural mechanics, up to now there are no exact general methods for
their optimum design. In practical applications, engineers always use some simplifying
rules to take some relevant properties into account. Level of scientific development is
shown in Section II, Literature Review.
9
1.3 Research objectives
The purpose of the work concerns a development of a new approach for an
optimization of stiffened composite panels applied to a flap structure of a civil aircraft.
The approach is based on the design of a refined design model for the stress-strain state
calculation.
A flat rectangular multilayer panel made of polymer fibrous composite materials
eccentrically supported by the longitudinal-transverse set of stiffeners and flat
rectangular composite panel with an asymmetrical thickness package structure are
considered. The panels are subjected to the action of an arbitrarily distributed transverse
load and the thermal loading.
Optimal structure of a composite material with minimal equivalent stresses and
deformations is determined using computer parametric optimization.
1.4 Scientific novelty of research
Scientific novelty value is provided by:
1) the solution of eighth order equilibrium equation in trigonometric series for the
calculation of stress-strain state;
2) impact assessment of the stiffness of the longitudinal set in different directions
on the stress-strain state;
3) found patterns between geometry of the panels, a layers reinforcement structure,
and deflections;
4) computer optimization of flat and stiffened composite panels.
10
1.5 Theoretical value of and the practical significance of the research
1) Computer optimization program for stiffened anisotropic panels subjected to the
action of transverse and thermal loads was developed.
2) An analysis of the impact assessment of stiffness, geometric, and structural
parameters on the stress-strain of the panels was performed.
Calculation results make it possible to reduce and optimize weight characteristics of
a structure. The proposed algorithms and the developed program are intended to be used
in the aircraft design.
1.6 Research methods
A research method is based on the principles of anisotropy. A thin-walled flap
structure is reduced to the form of the anisotropic panel. The problems are solved by
displacement method using hypotheses of the theory of thin plates for the skin.
Symbolic integration method makes it possible to get equilibrium equations of the
eighth order. Then, the solution is in double trigonometric series.
1.7 Theses
1) An optimization approach for a flat rectangular multilayer stiffened panel and a
flat rectangular composite panel with an asymmetrical thickness package made
of polymer composite materials.
2) Impact assessment of the stiffness of the longitudinal set in different directions
on the stress-strain state of the stiffened panels.
3) Patterns between geometry of the panels, layers reinforcement structure and
deflections.
11
1.8 Validity of data
The results reliability of the thesis is ensured by the use of generally accepted
relations of the structural mechanics of thin-walled structures and the mechanics of
composites using numerical methods, and it is confirmed by comparison with published
solutions.
The literature review of the thesis haы been presented on XLV Gagarin Science
Conference, Moscow, 2019 [1].
1.9 Structure of dissertation
Section II is a literature review that includes three parts: optimum design, research
methodology, and researches in the fields of laminated composite materials. Section III
describes research methodology and mathematical model. Solution of the problems is
represented in Section IV. The sections V and VI present calculation examples. The
research conclusion is represented in section VII.
12
2. LITERATURE REVIEW
2.1 Optimum design
In the modern era, optimization is one of the most crucial issues associated with
engineering designs. The use of reinforced composite panels for the airfoil of the wing
structures is the way to increase the weight efficiency of the aircraft structures. The
optimization methods that are used to design composite stiffened panels usually derive
from laminated plate‟s optimization methods. Some problems arising in the design of
composite structures and the composite wing box were considered by Grishin et al. [2].
Some works on the optimum design of composite stiffened panels can be found in
literature. Nagendra et al. [3] have used a standard genetic algorithm (GA) to find a
solution for the problem to minimize the mass of a composite stiffened panel subject to
constraints on maximum allowable strains and on ply orientation angles. Chernyaev [4]
also applied genetic algorithms to consider an optimum design of composite plates and
composite stiffened panels. In the Reznik et al. paper [5] mass optimization of hybrid
composites with a filler of glass and carbon fibers was carried out. As a result, variants
of hybrid composite structures of a wing were obtained which had a good agreement
with the set of requirements for such a design. The conditions for constructive
optimization of laminated composite plates subjected to lateral loads that are
encountered in aircraft internal joints and aerospace structures were studied by Shafei
Erfan et al. [6]. In the work of Chedric [7], optimization algorithms based on the
method of optimal criteria and methods of mathematical programming for calculating
the stress-strain state of composite panels are presented. The results of optimization of
thicknesses distribution and angles of layers are obtained. In another work of Chedric
[8], the basic stress-strain analysis relations of composite panels were described. The
efficiency of the presented algorithms was carried on tests on composite panels and on
wings. A global-local approach to solve the problems of optimization, that takes into
account the relationship between computational models, was described by Chedric et al.
[9]. This approach was carried on the composite wing optimization and the calculation
13
of stress-strain states its lower panel. It was shown that this approach allows obtaining
lower mass design in comparison with traditional design methods. In the work of
Dudchenko et al. [10] the design of a contour supported composite panel loaded with a
transverse load was considered. The solution was represented in an analytical form
using the Vlasov‟s variation principle. The resulting solution is the basis for satisfying
the strength condition in the design problem.
Barkanov et al. [11] dealt with a problem of the optimum design of lateral wing
upper covers by considering different kinds of stiffeners and loading conditions. Liu et
al. [12] utilized the smeared stiffness-based method to find the best stacking sequences
of composite wings with blending and manufacturing constraints by considering a set of
pre-defined fibre angles, i.e. 0°, 90° and ± 45°. In [13] López et al. proposed a
deterministic and reliability-based design optimization of composite stiffened panels
considering a progressive failure analysis to minimize the weight of laminated
composite plates. A common limitation of the previous works is the utilization of
simplifying hypotheses and rules in the formulation of the stiffened panel design
problem. These restrictions mainly focus on the nature of the stacking sequence of the
laminates constituting the panel.
The optimal orientation angles of the composite layers for different anisotropy
variants of the wing of an UAV were obtained by Nguyen Hong Fong and Biryuk [14].
Christos Kassapoglou [15] presented an approach to determine the configuration that
simultaneously minimizes the cost and weight of composite-stiffened panels under
compression and shear, strength and manufacturing constraints. A variety of stiffener
cross-sectional shapes was examined. It was found that „J‟ stiffeners give the lowest
weight configurations while „T‟ stiffeners give the lowest cost configurations. The
optimum configuration for both cost and weight was obtained for a panel with „T‟
stiffeners.
In aircraft structural design some other rules are imposed to the design of composite
stiffened panels, however some of them are not mechanically well justified, for instance
[3, 16] Among these rules, the most significant restriction is represented by the
14
utilization of a limited set of values for the layers orientation angles which are often
limited to the canonical values of 0°, 90° and ± 45°. To overcome the previous
restrictions, in the study [17] the multi-scale two-level (MS2L) optimization approach
for designing anisotropic complex structures [18–20] is utilized in the framework of the
multi-scale optimization of composite stiffened panels. The proposed MS2L design
approach aims of proposing a very general formulation of the design problem without
introducing simplifying hypotheses and by considering, as design variables, the full set
of geometric and mechanical parameters defining the behavior of the panel at each
characteristic scale.
Regarding wing design and modeling, Soloshenko and Popov [21] analyzed
composite wing structures of a transport aircraft. The choice of the allowable stresses
and the basic concepts of the wing layout were shown. Two analytical approaches to
model a composite wing like an anisotropic beam with and without consideration of the
limitation of transverse deformations were presented by Tuktarov and Chedric [22].
Using parametric modeling tools, a multi-spar wing design with a span of 12 meters was
developed by V. K. Belov and V. V. Belov [23]. A study of the stress-strain state of a
composite wing was conducted using finite element analysis. In the paper of Dubikov
and Penkov [24], the algorithm for designing a composite wing, taking into account the
aileron efficiency limitations and technological factors of composite structures, was
developed. The problem and the way of defining the stress-strain state of a composite
wing were formulated in the work of Kilasoniya and Beridze [25]. Relations for a
formation of a stiffness matrix for the plate finite element of an orthotropic laminated
composite material were derived. Moreover, they allow taking into account flexural and
membrane effects of thin-walled wing elements. In the work of Klyuchnik [26], the
stress-strain state of a modernized flap beam was determined using software systems.
The flap was calculated and the most dangerous sections were identified for the three
cases - “take-off”, “landing” and “removed”.
15
2.2 Research methodology
The widespread introduction of reinforced composite panels in aviation leads to the
need to develop methods for assessing the stress-strain state of a structure at the design
stage. In recent years, new methods have been developed for calculation of reinforced
composite panels.
Some common methods for designing and manufacturing structures made of glassreinforced plastic materials were stated in the work of Potter [27]. Mitrofanov and
Strelyaev [28] investigated the optimality of reinforced composite panels based on the
developed method of rational design. Sementsova [29] presented a method for
calculating the residual thermal stresses and strains that occur in composite wing box
structures after technological process. A new model for studying the stress-strain state
of stiffened anisotropic panels taking into account manufacturing technologies and
general boundary conditions was developed by Gavva and Lurie [30]. The solution was
constructed based on the Lagrange‟s variation principle. The mathematical model
developed by V. V. Firsanov [31] assesses the strength of a wing more accurately. The
asymptotic integration method of differential equations of the theory of elasticity was
used. It was shown that the transverse normal and tangential stresses make a significant
contribution to the stress-strain states which are neglected in the classical plate theory.
The theory of scale-dependent rods and plates, derived from the gradient theory of
elasticity, was considered by Lurie et al. [32]. The theory was based on the reduction of
the potential energy of the gradient theory, written with kinematic hypotheses of the
theory of rods (plates). Val. V. Firsanov in the paper [33] discussed the threedimensional theory of rectangular composite plates that does not include Kirchhoff
hypotheses. In the work of Menkov [34], the analytical method, to get an effective and
exact solution of orthotropic laminate plate equations, was considered. Soroka and
Shaldyrvan [35] proposed a method for designing three-dimensional unidirectionalreinforced fiber composites plates. The effectiveness of this approach was shown on
model problems. Studies of the stress-strain state were carried out using homogeneous
Lurier-Vorovich‟s solutions. In the work of Popov [36], a method of calculating
16
stiffened plates with holes was presented, taking into account physical, geometric nonlinearity and different modulus under static loading and thermal effects. The influence
of an imbalance structure of multilayer composite materials was studied by Pervushin
and Solovyov [37]. Simply supported and fixed rectangular plates under an action of a
distributed load over the surface with a different number of layers were investigated.
Kovalenko et al. [38] found exact solutions of the problem where loads transfer through
stiffeners to a plate. The problem was solved in the form of the explicit expansions in
Fadl-Pakovich‟s functions. Gorbachev [39] proposed an integral theory of the
deformation of inhomogeneous composite plates. The integral formula was used to
construct the theory. In the paper of Osyaev et al. [40], a mathematical model for an
analysis of the stress-strain state of multilayer structures was proposed. This model was
presented in a form of a system of equations of the loading parameters, stresses and
strains. A methodology for computation a wing box and an open section were proposed
by Shataev [41]. There was the description of the algorithm for a design of a composite
structure, taking into account the constrained deplanation. This algorithm allows taking
into account bend and torsion of an open section of a structure. Mathematical models of
thin-walled composite structures that allow determining some of their parameters taking
into account the action of force system and temperature field in static and dynamic
processes were created by Kasumov [42].
Some authors have used global approximation techniques to reduce function
evaluation computational time by using data previously obtained with analytical or
numerical methods. In this direction, Bisagni and Lanzi [43] developed an optimization
procedure with a global approximation strategy based on obtaining the structure
response by means of a system of artificial neural networks (ANNs) and GA. Lanzi et
al. [44] performed a comparative study between three different global approximation
techniques: ANN, cringing method and radial basis functions. All the techniques
showed a similar behavior that the dynamic finite element (FE) analysis and
computational time was satisfactorily reduced.
17
2.3 Researches of laminated composite materials
Laminated composites are the most frequent type of composite structures. Therefore
many researchers are interested in analyzing their characteristics. In terms of optimizing
laminated composite plates, many studies have been carried out.
Optimization of ply orientations of composite laminated plates was considered by
Gillet et al. [45]. They provided single and multi-objective optimization solutions to
minimize the mass and strain energy. The plate was assumed to be under traction and
compressive loads. GA and FEM were utilized in the procedure of solution method. To
minimize the mass of composite laminates, Satheesh et al. [46] conducted GA to
optimize the stacking sequence regarding to various failure criterions such as Tsai-Wu,
maximum stress and mechanism-based failure criteria as the constraints. In order to
reduce the weight of composite structures, An et al. [47] optimized the stacking
sequence through a modified version of GA. The structural strength was chosen as the
constraint in their study. They concluded that this method requires less computational
costs (nearly 90% computational costs savings) in comparison with traditional GA.
Considering flexural strength as the constraint, Kalantari et al. [48] minimized the
weight and cost of composite plates made up of carbon and glass fiber reinforcements
by implementing a combined optimization method consisting NSGA and fractional
factorial design method. The ply thicknesses and angles were assumed to be the design
variables in their study. Considering the material, thickness of plies and stacking
sequence of laminated composite plates as the design parameters, Ghasemi and Behshad
[49] attempted to minimize the cost and weight of the structures through GA. They
utilized element-free Galerkin method for optimization process in order to analyze
laminated composite and isotropic plates. Bloomfield et al. [50] applied two level
optimization approach to laminated composite plates to minimize the structural mass.
Strength and buckling resistance were taken as the constraints of the optimization
procedure. The design variables in the first level were laminate thickness and lamination
parameters. Fakhrabadi et al. [51] employed the discrete shuffled frog leaping algorithm
18
to optimize fiber orientations, ply thicknesses and the number of layers of a laminated
composite plate. Minimizing the mass and cost of the structure simultaneously form the
objective functions of their study. Sørensen and Lund [52] developed a novel Gradient
Based Method (GBM) to optimize the thickness of layers of laminated composite plates
leading to minimum structural weight. They focused on displacements and
manufacturing constraints in their study. Using ANSYS software, Marannano and
Mariotti [53] optimized the fiber orientations of composite panels in order to maximize
the stiffness and minimize the weight. The numerical results were validated by
conducting experiments including bending, shear and traction tests.
Eremin and Chernyshov [54] considered theoretical foundations of the manufacture
of an orthotropic composition package consisting of different orientation orthotropic
monolayers. Implementation of the effect of polymer “self-reinforcing” in the
traditional technological scheme for creating layered composite material was
theoretically substantiated by Kleimenov [55]. The contact-conjugate bending problem
of Kirchhoff plates with equally stressed reinforcement was formulated by Nemikovsky
and Yankovsky. [56]. An iterative method for solving the problem of rational design of
rectangular and polygonal plates which are simply connected and doubly connected
with straight and curvilinear sides was proposed. Kundrat [57] proposed a solution of
the stress-strain problem of an orthotropic semi-infinite plate supported by a periodic
system of rigid linings. The localized pre-fracture zones were modeled by tangential
displacement discontinuity lines. Numerical results were shown for the sizes of prefracture zones and contact stresses. In the work of Nemikovsky and Yankovsky [58],
the problem of elastoplastic bending of multilayer plates of variable thickness
reinforced with fibers of constant cross section was formulated. For rectangular plates,
the problem was solved by the Bubnov-Galerkin method. Calculations showed that the
load capacity of reinforced plates for elastoplastic bending is much higher than for
elastic bending. Moreover, it was shown that the load capacity of the structure can be
increased due to the separation of the supporting layers of the plate with the same
reinforcement consumption. Detailed algorithms for constructing the stiffness matrix of
19
a rectangular plate finite element of layered composites, taking into account the bending
and membrane effects, were developed by Kilasoniya and Beridze [59]. In the work of
Kutyinov and Chedric [60], the behavior of the composite wing box structure under the
action of loads that cause its bending and torsion was described. The calculated and
experimental results were compared and the interaction of bending and torsional
deformations was analyzed. The design formulas for designing reinforced composite
walls during shear and compression as well as the problem of rational reinforcement
and weight analysis of walls loaded with tangential flows are given by Mitrofanov [61].
Some researchers have studied optimization problems of laminated composite plates
with thermal effects to maximize the critical thermal capacity with uniform [62, 63] or
nonuniform thermal distribution [64]. In addition, Ijsselmuiden et al. [65] carried out a
thermo mechanical design optimization of composite panels and Cho [66] studied the
hygrothermal effects in optimization problems of dynamic behavior, where temperature
and moisture are assumed to be uniform once they have reached equilibrium.
20
3. RESEARCH METHODOLOGY
The use of stiffened composite panels for wing surface is the way to increase the
weight efficiency of aircraft structures.
The stress-strain state of a flat rectangular multilayer panel made of polymer fibrous
composite materials eccentrically supported by the longitudinal-transverse set of
stiffeners and a flat rectangular composite panel with an asymmetrical thickness
package structure is considered. The panels are subjected to the action of an arbitrarily
distributed transverse load q (x, y) in a stationary temperature field of intensity ΔT.
The problems are solved by displacement method using hypotheses of the thin plate
theory. The problems are reduced to finding the displacements of a single base surface.
As a design model, it is proposed to map stiffened panels as anisotropic with
“smearing” of the stiffness of thin-walled reinforcing elements.
The stress-strain state determination of the panels is reduced to solving a boundary
value problem for an equation of the eighth order in partial derivatives. The closed-form
solution is represented in double trigonometric series.
Basic equations of two-dimensional problem by displacement method are presented
below. Stiffened composite panel is shown on the Figure 6. Axes direction of the panel
is shown on the Figure 7.
Figure 6 Stiffened composite panel
21
Figure 7 Axes direction of the panel
According to Kirchhoff–Love plate theory, the deformations have the form:
Kinematic (geometric) relations:
where w – displacement in z direction (deflection); u – displacement in x direction; v –
displacement in y direction.
According to geometric relations (2), the deformations (1) have the following form:
22
After integrating (3), according to the Kirchhoff theory for the components of the
displacement vector; displacements of the panel are in the form:
where
,
are the displacements
and
when z = 0.
Since the structure of the panels is asymmetric, due to realization of the normal
element hypothesis, a plane in which the coordinate axes are located and the origin of
the z coordinate can be chosen arbitrarily. Then, unknowns for a two-dimensional
problem will be in the form:
Kinematic (geometric) relations for a two-dimensional problem in x0y coordinate
system are in the form:
23
Taking into account equations (4), geometric relations (6) have the form:
(
)
Hooke's law for a flat isotropic panel has a form:
{
where
[
}
is the stress in x direction,
]{
}
is the stress in y direction,
is the shear stress,
is the Poisson's ratio. E is elastic modulus.
After substitution deformations (7) into Hooke's law (8), stresses will have the form:
(
)
.
/
(
)
.
/
(
)
In the matrix form:
{
}
[
]
{
where
}
is the stiffness matrix for the isotropic panel, which has a form:
24
[
]
For the k-th layer of a composite panel, stress is in the form:
{
}
̅
[̅
̅
̅
̅
̅
̅
̅ ]
̅
(
{
)
}
where:
{
}
[̅ ]
Deformations in the plane of the panel:
Curvatures of the panel:
25
Deformations of the k-th layer:
Coordinate systems of the layer (102) and of the panel (x0y) are shown on the Figure
8.
Figure 8 Coordinate systems
The conversion of the stiffness of the k-th layer from the coordinate system of the
layer (102) to the coordinate system of the panel (x0y) is shown below.
̅
̅
̅
̅
̅
̅
{̅ }
{
[
}
]
where
.
26
Stiffness of the k-th layer in the coordinate system of the layer (102) is in the form:
where
– elastic modulus in longitudinal direction,
– shear modulus,
transverse direction,
– elastic modulus in
– Poisson's ratio,
.
Using the geometric relations, Hooke's law, and taking into account a temperature
influence, the components of the stress tensor of the k-th layer are determined by (17).
{
}
̅
[̅
̅
̅
̅
̅
̅
̅ ]
̅
(
{
)
}
where
̅
, i,j = 1, 2, 6
, i,j = 1, 2, 6
α T – thermal deformation,
Coefficients of thermal expansion for the k-th layer are determined by the formula:
27
̅̅̅
{̅̅̅}
̅̅̅
[
]
,
-
3.1 Composite panel with an asymmetrical thickness package structure
For the flat rectangular composite panel with an asymmetrical thickness package
structure internal forces are in the form:
∫
∫
∫
∫
∫
∫
∫
∫
∫
where
– longitudinal forces,
– tangential forces,
– bending moments,
– torsion moments.
28
– thickness of a panel
Internal forces are connected with the strain vector the by the equation (18).
{
}
[
]
{
{
}
}
where the thermal forces are shown below.
∑ ̅
∑ ̅
∑ ̅
∑ ̅
̅
29
∑ ̅
̅
Layer characteristics are represented below:
(
∑
(
)
∑
)
Matrix of generalized stiffness has the form:
[
]
In the orthotropic panel, stiffness
expressions of
,
,
in the
due to their smallness in comparison with the other stiffness
characteristics.
30
3.2 Composite panel eccentrically supported by longitudinal-transverse set of
stiffeners
For the calculation of the stress-strain state of the stiffened panel, it is proposed to
map the panel as anisotropic with “smearing” of the stiffness of thin-walled reinforcing
elements. Therefore, the internal forces have the form:
∫
∫
∫
∫
∫
∫
∫
∫
– longitudinal forces,
– tangential forces,
– bending moments,
– torsion moments,
– distance between longitudinal elements,
– distance between transverse elements,
– area of skin cross section between longitudinal (transverse) elements,
– area of longitudinal element cross section
– area of transverse element cross section
31
Internal forces are connected with the strain vector in the form:
{
}
[
]
{
{
}
}
Generalized stiffness is determined by material characteristics and panel geometry:
The thermal forces have the form:
(
)
(
)
(
)
32
(
)
.
Stiffness of longitudinal elements:
∫ ̅
∫ ̅
∫ ̅
̅
∫ ̅
̅
∫ ̅
Such as a stringer consists of m elements (web, flanges) and each element has its
own number of layers, the element stiffness in this case have double sum (over layers
and over elements).
Layer characteristics for the cross section shown on the Figure 9 are presented
below.
C1
H(
H(
𝑘)
𝑘)
C3
H(
𝑘)
C4
Figure 9 Geometric cross section characteristics
33
a) Web
(
)
(
)
b) Flange
(
∑
(
In the orthotropic panel, stiffness
expressions of
,
,
∑
)
)
in the
due to their smallness in comparison with the other stiffness
characteristics.
34
3.3 Symbolic integration method
Equilibrium equations for the panel have a form:
.
/
The equilibrium equations of the panel under the action of external transverse load
are the system of three differential equations for the three desired functions –
(x,y),
(x,y), w(x,y), which have the following operator form:
(24)
where
- linear differential operators for orthotropic panel,
,
– displacements,
(
)
35
(
)
(
(
(
)
)
)
The system of differential equilibrium equations (24) can be reduced to one
differential equation for the potential function
through which all the calculated
values of the problem can be expressed.
In the symbolic integration method, displacements are determined by the minors of
the determinant det [
], i, j = 1, 2, 3, made up of its third line corresponding to the
third heterogeneous equation of system (24); the first two homogeneous equations are
satisfied identically.
(25)
After substitution
into (23), obtain displacements in the form:
.
/
36
where
.
/
where
.
/
where
37
The third equation of the system (24) based on the formulas for
,
,
, is reduced
to the heterogeneous linear eighth-order partial differential equation for the desired
potential function Ф(x,y):
.
/
where
The coefficients
are the constant values that depend on elastic properties
of a material and geometrical parameters of a structure.
38
4. PROBLEM SOLUTION IN DOUBLE TRIGONOMETRIC SERIES
Dimensions of the panel are shown on the Figure 10.
Figure 10 Geometry of the panel
To find
, use Fourier series, therefore
has the form:
∑∑
or in a different coordinate system:
∑∑
where
(
)
(
)
– dimensionless coordinates.
Boundary conditions correspond to the simply supported edges in the bending
problem, and to the fixed edges in the flat problem in the tangential direction, when the
panel contour is loaded by flows of tangential forces. The panel edge perpendicular to y
axis is loaded by tangential force
, which are balanced by normal forces
on the
boundary and areas perpendicular to the x axis. Thus, boundary conditionals have a
type:
39
:
:
Transverse load q (x, y) in double trigonometric series has a form:
∑∑
∑∑
[
(
(
) (
)
(
)
(
) (
)
(
) (
)
) ]
∑∑
After the orthogonalization procedure of the external load (32) in double
trigonometric series, obtain the external load in the form:
Conduct the orthogonalization procedure for the equation (33). Therefore,
has
the form:
⌈
(
)
(
) (
)
(
) (
)
(
) (
)
(
) ⌉
40
Displacements in double trigonometric series according to equations (26), (27), (28)
have the form:
∑∑
∑∑
∑∑
where
[
(
)
[
(
) (
[
(
(
) (
)
(
) (
)
)
(
) (
(
)(
)
)
(
(
) ]
) ]
) ]
Deformations have the form:
∑ ∑(
)
∑ ∑(
)
∑ ∑ *(
)
(
(
)
(
)
)
+
41
*(
)
(
)
+
Curvatures are in the form:
∑ ∑(
)
∑ ∑(
)
∑ ∑(
)(
(
)
(
)
(
)(
)
)
Kirchhoff theory is applicable for a flat rectangular composite panel. Internal forces
of the flat composite panel with an asymmetrical package structure may be obtained by
integrating the corresponding components of the stress tensor (17) by the coordinate z.
In this case, system (24) follows from the equilibrium equations in terms of forces and
moments.
According to consideration for the orthotropic panel, linear differential operators
(26), (27), (28) for symmetric components of the stress-strain state and linear
differential operator of the equation (29) contain only even degree derivatives for each
coordinate. The asymmetric components of the stress-strain state are determined by odd
derivatives.
42
5. CALCULATION EXAMPLE OF FLAT RECTANGULAR
MULTYLAUER PANEL
As an example problem, determine the stress-strain state of the multilayer panel,
package of the panel is shown on the Figure 11. Consider panels with dimensions 600 x
300, 450 x 300, and 300 x 300 mm.
In the considered panel, stiffness
is too small in
comparison with the other stiffness characteristics. Therefore, let:
.
Figure 11 Panel Package
5.1 Initial Data
The panel consists of 7th layers; the package is [0/+φ/-φ/90/0/90/0]. Each layer has
the same thickness and made from the same material. Consider the problem without
thermal loading.
*
+
*
+
*
+
– thickness of each layer
43
[
]
5.2 Calculation Results
Deflection, stresses, and equivalent stresses were calculated. Table 3 shows
geometric characteristics of each layer, Table 4 – stiffness of the panel.
Table 3
# of
layer, k
Layer
boundary
points,
1
2
2
3
3
4
4
5
5
6
6
7
7
8
1
2
3
4
5
6
7
Coordinate
of a layer,
z
0
0.52
0.52
1.04
1.04
1.56
1.56
2.08
2.08
2.60
2.60
3.12
3.12
3.64
,
,
,
0.52
0.135
0.047
0.52
0.406
0.328
0.52
0.676
0.891
0.52
0.946
1.734
0.52
1.217
2.859
0.52
1.487
4.265
0.52
1.758
5.952
Table 4
∑
̅
,
⁄
∑
̅
,
∑
̅
,
44
5.2.1 Deflection Calculations
Deflection of the panels is presented in Table 5 and on Figure 12.
Table 5
,*
+
φ, deg 600 x 300 450 x 300 300 x 300
0
9620
6150
2146
15
9069
5778
2057
30
7954
5114
1916
45
6762
4579
1856
60
5917
4309
1899
75
5591
4310
2008
90
5552
4373
2075
𝐰/𝒒
φ, deg
0
15
30
45
60
75
90
(𝐰(𝐱,𝐲))/𝒒, [〖𝒎𝒎〗^𝟑/𝒌𝒈]
1150
2150
3150
600 x 300
4150
450 x 300
5150
300 x 300
6150
7150
8150
9150
10150
Figure 12 Deflections of the panels
45
5.2.2 Stress Calculation
The maximum tensile stress in x direction (
is in the 7th layer, point 8. The stress
is shown in the Table 6 and on the Figure 13.
Table 6
φ, deg 600 x 300
0
6143
15
6222
30
6066
45
5146
60
3904
75
2916
90
2499
450 x 300
7176
6997
6560
5833
4970
4291
4015
300 x 300
5620
5422
5054
4750
4565
4496
4485
Sigma x
8000
7000
𝝈_𝒙/𝒒
6000
5000
600 x 300
4000
450 x 300
3000
300 x 300
2000
0
15
30
45
60
75
90
φ, deg
Figure 13 Stress in x direction
46
The maximum tensile stress in y direction is in the 6th layer, point 7. Stress in y
direction is shown in the Table 7 and on the Figure 14.
Table 7
φ, deg 600 x 300 450 x 300 300 x 300
0
11140
6999
2198
15
10700
6836
2339
30
9969
6638
2638
45
9609
6769
2978
60
9664
7196
3288
75
9932
7677
3512
90
10110
7912
3601
Sigma y
12150
10150
𝝈_𝒚/𝒒
8150
6150
600 x 300
4150
450 x 300
300 x 300
2150
150
0
15
30
45
60
75
90
φ, deg
Figure 14 Stress in y direction
47
5.2.3 Equivalent Stress
To assess the strength of a multilayer panel made from composite material, relative
equivalent stresses are determined. For the calculation, the tensor strength criterion in
the form of Goldenblat-Kopnov is used. According to this strength criterion for a
unidirectional layer of composite material, the cracking or destruction of the structure
are not occur until the following inequality is right in each layer:
[ (
̅
̅
(
)
̅
̅
(
)
(
̅
̅
̅
̅
)
(
̅
) ]
)
where
̅
̅
̅
̅
̅
0
,
1
̅
[
]
̅
[
]
̅
[
]
̅
[
]
̅
[
]
48
The equivalent stress for 7th layer, point 8 is shown in the Table 8 and on the Figure
15.
Table 8
,*
+
φ, deg 600 x 300 450 x 300
300 x300
0
384.5
261.7
119.4
15
368
252.6
118
30
337.5
237.4
116.8
45
310.3
228.3
118.4
60
295.8
227.8
122.3
75
295
234
126.8
90
297.8
238.6
129
Stress, 7 layer, point 8
450
𝝈_𝒆𝒒, [〖𝒎𝒎〗^𝟐/𝒌𝒈]
400
350
300
250
600 x 300
200
450 x 300
150
300 x300
100
50
0
15
30
45
60
75
90
φ, deg
Figure 15 Equivalent Stress for 7th layer
Determine the allowable load for a flap construction.
0
1
Then
[
]
49
The equivalent stress for 6th layer, point 7 is shown in the Table 9 and on the Figure
16.
Table 9
,*
+
φ, deg 600 x 300 450 x 300
300 x 300
0
209.3
163.96
95.5
15
205.1
161.1
93.9
30
195.7
155.5
91.3
45
182.5
149.2
89.9
60
171.7
144.6
89.5
75
167.1
143.3
89.6
90
166.4
143.5
89.7
valent Stress, 6 layer, point 2
𝝈_𝒆𝒒, [〖𝒎𝒎〗^𝟐/𝒌𝒈]
250
200
150
600 x 300
450 x 300
300 x 300
100
50
0
15
30
45
60
75
90
φ, deg
Figure 16 Equivalent Stress in 6th layer
50
5.2.4 Stress diagrams
Stress diagrams for the 600 x 300 mm panel are shown on the figures below for the
different layer package.
, 600 x 300
[0/+φ/-φ/90/0/90/0]
𝝈𝒙 𝒒, 𝝋
𝟎 𝒅𝒆𝒈
𝝈𝒙 𝒒, 𝝋
𝟑𝟎 𝒅𝒆𝒈
Layer #1
Layer #1
Layer #2
Layer #2
Layer #3
Layer #3
Layer #4
Layer #4
Layer #5
Layer #5
Layer #6
Layer #6
Layer #7
Layer #7
Figure 17 Stress in x direction
51
𝝈𝒙 𝒒, 𝝋
𝟒𝟓 𝒅𝒆𝒈
𝝈𝒙 𝒒, 𝛗
𝛗
𝟗𝟎 𝐝𝐞𝐠
Layer #1
Layer #1
Layer #2
Layer #2
Layer #3
Layer #3
Layer #4
Layer #4
Layer #5
Layer #5
Layer #6
Layer #6
Layer #7
Layer #7
Figure 18 Stress in x direction
52
, 600 x 300
[0/+φ/-φ/90/0/90/0]
𝝈𝒚 𝒒, 𝝋
𝟎 𝒅𝒆𝒈
𝝈𝒚 𝒒, 𝝋
𝟑𝟎 𝒅𝒆𝒈
Layer #1
Layer #1
Layer #2
Layer #2
Layer #3
Layer #3
Layer #4
Layer #4
Layer #5
Layer #5
Layer #6
Layer #6
Layer #7
Layer #7
Figure 19 Stress in y direction
53
𝝈𝒚 𝒒, 𝝋
𝟒𝟓 𝒅𝒆𝒈
𝝈𝒚 𝒒, 𝛗
𝛗
𝟗𝟎 𝐝𝐞𝐠
Layer #1
Layer #1
Layer #2
Layer #2
Layer #3
Layer #3
Layer #4
Layer #4
Layer #5
Layer #5
Layer #6
Layer #6
Layer #7
Layer #7
Figure 20 Stress in y direction
54
6. CALCULATION EXAMPLE OF STIFFENED PANEL
Consider the panel with longitudinal set of stiffened elements; the panel dimension is
600 x 300 mm. The configuration of the stiffened panel is shown on the Figure 21.
Figure 21 Skin – stringer panel configuration
6.1Initial Data
The panel consists of 7th layers; the package is [0/+φ/-φ/90/0/90/0]. Each layer has
the same thickness and made from the same material. Consider the problem without
thermal loading. Geometrical dimensions are shown on the Figure 22.
*
+,
*
+,
*
+,
,
– thickness of each layer,
,
55
,
[
]
Figure 22 Geometrical characteristics
,
,
,
In the problem, different cases of stringer stiffness considerations were taken into
account.
56
Case 1 (x11): take into account only longitudinal stiffness of the stiffened element.
Therefore, the stiffness of the whole panel has the following form:
Case 2 (x11, x12): consider the longitudinal stiffness and stiffness in 12 direction of
the stiffened element. Therefore, the stiffness of the whole panel has the following
form:
57
Case 3 (x11, x12, x66): take into account longitudinal stiffness, stiffness in 12
direction, and torsional stiffness of the longitudinal element. Therefore, the stiffness of
the whole panel has the following form:
58
6.2 Deflection Calculation
Deflections according to stiffness consideration are presented in the Table 10 and on
the Figure 23.
Table 10
,*
φ, deg
x11
0 212.45
15 216.49
30
232.2
45
261
60
294
75
322.7
90 335.957
x11, x12
+
x11, x12, x66
195
186.5
172.8
173.3
198.6
249.192
282.924
108.426
93.67
74.563
69.256
79.585
107.257
129.674
Назва
φ, deg
0
15
30
45
60
75
90
(𝐰(𝐱,𝐲))/𝒒, [〖𝒎𝒎〗^𝟑/𝒌𝒈]
50
100
x11
150
x11, x12
200
x11, x12, x66
250
300
350
Figure 23 Deflections of the stiffened panel
59
6.3 Stress Calculation
The maximum tensile stress is in x direction is in the 1st layer, point 1. It is shown in
the Table 11 and on the Figure 24.
Table 11
φ, deg
0
15
30
45
60
75
90
x11
x11, x12
-677.3
-666.2
-656.3
-687.2
-782
-932.3
-1021
-653.608
-635.834
-621.225
-647.522
-727.139
-870.437
-966.449
x11, x12, x66
-441.06
-406.781
-368.144
-371.508
-419.493
-515.541
-585.41
x
-200
-300
0
15
30
45
60
75
90
-400
Sigma x
-500
-600
x11
-700
x11 x12
-800
x11 x12 x66
-900
-1000
-1100
φ, deg
Figure 24 Stress in x direction
60
7. CONCLUSION
The new approach for an optimization of stiffened composite panels was developed.
The main research results are listed below.
1. Method of the stress-strain state calculation of composite stiffened panels
subjected to the action of an arbitrarily distributed transverse load in a stationary
temperature field was proposed.
2. The design model was based on the principles of constructive anisotropy.
3. The solution of eighth order equilibrium equation in double trigonometric series
was obtained.
4. Computer optimization program for stiffened anisotropic panels subjected to the
action of transverse and thermal loads was developed.
5. Impact assessment of the stiffness of the longitudinal set in different directions on
the stress-strain state of the stiffened panels was considered.
6. An influence of the torsional stiffness of longitudinal stiffened elements on the
strength characteristics of the panels was investigated.
7. The results of the stress-strain state determination coincide with the accuracy of
18% with the results based on the solution of contact skin-stringer problem [30].
This statement is accurate when the torsional stiffness of the reinforcing element
is not taken into account.
8. Patterns between geometry of the panels, a layers reinforcement structure, and
deflections were obtained.
The results of the calculations make it possible to reduce and optimize weight
characteristics of a structure. The proposed algorithm and the developed program are
intended to be used in aircraft design.
61
REFERENCES
[1] A. Kovkuta, L. Gavva, “Review of optimum design methods and research
methodologies for calculation of stiffened composite panels,” XLI Gagarin Science
Conference, Moscow, 2019, pp. 565.
[2] V. Grishin, O. Mitrofanov, D. Bondarenko, “Problems of ensuring strength in the
design of composite bearing panels and composite wing boxes,” TsAGI, №2640, Vol.1,
pp. 793-797, 1999.
[3] Nagendra S, Jestin D, Gürdal Z, Haftka R, Watson L, “Improved genetic algorithm
for the design of stiffened composite panels,” Comput Struct, 58(3), pp. 543–55, 1996.
[4] A. Chernyaev, “Discrete optimization of elements of aircraft structures made of
composite materials,” 2009.
[5] S. Reznik, P. Prosuntsov, T. Ageeva, “Optimal design of the wing of a suborbital
reusable spacecraft made of a hybrid polymer composite material,” NPO S. Lavochkina,
Vol. 1, 2013.
[6] Shafei Erfan, Kabir Mohammad Zaman, “Dynamic Stability Optimization of
Laminated Composite Plates under Combined Boundary Loading,” Applied Composite
Materials, Vol.18, pp. 539–557, 2011.
[7] V. Chedric, “Optimization of structures made of composite materials,” TsAGI,
№2664, pp. 188-198, 2004.
[8] V. Chedric, “Practical methods for the optimal design of structures made of
laminated composite materials,” Mechanics of Composite Materials and Structures, №
2, Vol. 11, pp. 184-198, 2005.
[9] V. Chedric, S. Tuktarov, “The method of global-local calculation and optimization
of aircraft structures made of composite materials,” 2015.
[10] A. Dudchenko, L. Kyong, S. Lurie, “Calculation and design of contour supported
composite panel, loaded by lateral force,” Trudy MAI, №50, 2012.
[11] E. Barkanov, O. Ozolins, E. Eglitis, F. Almeida, M. Bowering, G. Watson,
“Optimal design of composite lateral wing upper covers,” Part I: linear buckling
analysis. Aerospace Sci Technol, 38:1–8, 2014.
62
[12] D. Liu, V. Toropov, M. Zhou, D. Barton, O. Querin, Collection of Technical
Papers – AIAA/ASME/ASCE/AHS/ASC Structures. Structural Dynamics and Materials
Conference; 2010. Ch., “Optimization of Blended Composite Wing Panels Using
Smeared Stiffness Technique and Lamination Parameters”.
[13] R. H. Lopez, D. Lemosse, de Cursi JES, J. Rojas, A. El-Hami, “An approach for
the reliability based design optimization of laminated composites,” Eng Optim
2011;43(10):1079–94.
[14] Nguyen Hong Fong, V. Biryuk, “Studies of the optimization of the structuralpower scheme of an aircraft with a straight wing made of composite materials,” Trudy
MFTI, Vol. 6, №2, pp. 133-141, 2014.
[15] Christos Kassapoglou, “Simultaneous cost and weight minimization of compositestiffened panels under compression and shear,” Composites Part A 28A, pp. 419-435,
1997.
[16] B. Liu, RT Hatfka, P. Trompette, “Maximization of buckling loads of composite
panels using flexural lamination parameters,” Struct Multidiscip Optim, 26: 28–36,
2004.
[17] Marco Montemurro, Alfonso Pagani, Giacinto Alberto Fiordilino, Jérôme Pailhès,
Erasmo Carrera, “A general multi-scale two-level optimisation strategy for designing
composite stiffened panels,” Composite Structures 201, 968 – 979, 2018.
[18] A. Catapano, M. Montemurro, “A multi-scale approach for the optimum design of
sandwich plates with honeycomb core,” Part I: homogenisation of core properties,
Compos Struct, 118:664–76, 2014.
[19] A. Catapano, M. Montemurro, “A multi-scale approach for the optimum design of
sandwich plates with honeycomb core,” Part II: the optimisation strategy, Compos
Struct, 118: 677–90, 2014.
[20] M. Montemurro, A. Catapano, D. Doroszewski, “A multi-scale approach for the
simultaneous shape and material optimisation of sandwich panels with cellular core,”
Compos Part B, 91:458–72, 2016.
[21] V. Soloshenko, Y. Popov, “Conceptual design of the wing box of medium-haul
aircraft made of composite materials,” Bulletin of Moscow Aviation Institute, №1,
2013.
63
[22] S. Tuktarov, V. Chedric, “Some aspects of modeling the composite wing box of
high elongation by an anisotropic beam,” TsAGI, №3, 2015.
[23] V. K. Belov, V. V. Belov, “Investigation of the stress-strain state and stiffness
characteristics of the wing,” SCIENCE BULLETIN OF THE NOVOSIBIRSK STATE
TECHNICAL UNIVERSITY, № 1, pp. 17-29, 2003.
[24] E. A. Dubovikov, E. A. Penkov, “Designing of a composite wing box with given
elastic characteristics,” 2013.
[25] D. N. Kilasoniya, N. M. Beridze, “The general scheme of determining of the stressstrain state of the structure of a wing made of composite materials,” J. Appl. Math.
Mech., №4, pp. 90-93, 2004.
[26] O.V. Klyuchnik, “The study of the stress-strain state of the modernized flap beam,”
2013.
[27] K. D. Potter, “The early history of the resin transfer modeling process for
aerospace applications,” Compos. A 5, Vol. 30, pp. 619-621, 1999.
[28] O.V. Mitrofanov, D.V. Strelyaev, “Optimal reinforcement of composite stiffened
panels of the wing of the aerospace plane "Shuttle",” Scientific Bulletin of MSTU GA
13, pp. 31-34, 1999.
[29] A.N. Sementsova, “Analysis of temperature stresses and deformations in caisson
structures made of composite materials,” Trudy MAI, №65, 2013.
[30] L.M. Gavva, S.A. Lurie, “The method of calculating the stress-strain state of
structurally anisotropic panels made of composite materials, taking into account
manufacturing technologies and general boundary conditions,” All-Russian ScientificTechnical Journal "Polyot" ("Flight"), №7, 2018.
[31] V.V. Firsanov, “Mathematical model of the stress-strain state of a rectangular plate
of variable thickness, taking into account the boundary layer,” Mechanics of composite
materials and structures, №1, 2016.
[32] S.A. Lurie, E.D. Lykosova, E.I. Popova, “The theory of scale-dependent rods and
plates,” Mechanics of composite materials and structures, №4, 2015.
[33] Val. V. Firsanov, “Clarification of the classical theory of rectangular plates made
of composite materials,” Mechanics of composite materials and structures, №1, 2002.
64
[34] G. B. Menkov, “Analytical methods for investigation of the strength of thin-wall
structures made of composite materials,” Mechanics of composite materials and
structures, №1, 1996.
[35] V.A. Soroka, V.A. Shaldyrvan, “Calculation and design of unidirectionalreinforced plates,” Mechanics of composite materials and structures, №2, 2001.
[36] O. N. Popov, “Study of the stress-strain state of stiffened plates with ribs of
different stiffness, different aspect ratios, different flexibility and with eccentricity,”
Mechanics of composite materials and structures, №1, 2006.
[37] Yu. S. Pervushin, P.V. Solovyov, “Features of the deformation behavior and stress
state of unbalanced plates made of composite materials,” VESTNIK of Samara
University. Aerospace and Mechanical Engineering, №1, 2013.
[38] M. D. Kovalenko, S. N. Popov, A. A. Strelnikov, V. N. Tatarinov, N. N. Tsybin,
“Stiffened plates. Exact solutions,” Mechanics of composite materials and structures,
№2, 2006.
[39] V.I. Gorbachev, “Engineering theory of deforming inhomogeneous plates made of
composite materials,” Mechanics of composite materials and structures, №4, 2017.
[40] O. G. Osyaev, Yu. A. Taturin, A. M. Kostin, A.V. Zhukov, “Multilevel model of
strength analysis of structures made of polymer composites under multifactor loading,”
DSTU Bulletin, pp. 613-620, 2012.
[41] P. A. Shataev, “Calculation of composite structures in the design formulation,”
2015.
[42] E. V. Kasumov, “Construction of computational models and algorithms for
determining of the rational parameters of thin-walled structures,” 2001.
[43] C. Bisagni, L. Lanzi, “Post-buckling optimisation of composite stiffened panels
using neural networks,” Compos Struct, 58(2):237–47, 2002.
[44] L. Lanzi, C. Bisagni, S. Ricci, “Neural network systems to reproduce crash
behavior of structural components,” Comput Struct, 82(1):93–108, 2004.
[45] A. Gillet, P. Francescato, P. Saffre, “Single-and multi-objective optimization of
composite structures: the influence of design variables,” J Compos Mater 2010;
44(4):457–80.
65
[46] R. Satheesh, G. Narayana Naik, R. Ganguli, “Conservative design optimization of
laminated composite structures using genetic algorithms and multiple failure criteria,” J
Compos Mater 2010; 44(3):369–87.
[47] H. An, S. Chen, H. Huang, ”Laminate stacking sequence optimization with
strength constraints using two-level approximations and adaptive genetic algorithm,”
Struct Multidiscip Optim 2015; 51(4):903–18.
[48] M. Kalantari, C. Dong, IJ. Davies, “Multi-objective robust optimization of
unidirectional carbon/glass fibre reinforced hybrid composites under flexural loading,”
Compos Struct 2016; 138:264–75.
[49] M. R. Ghasemi, A. Behshad, “An element-free Galerkin-based multi objective
optimization of laminated composite plates,” J Optim Theory Appl 2013; 156(2):330–
44.
[50] M. W. Bloomfield, J. E. Herencia, P. M. Weaver, “Enhanced two-level
optimization of anisotropic laminated composite plates with strength and buckling
constraints,” Thin-Walled Struct 2009; 47(11):1161–7.
[51] MMS. Fakhrabadi, A. Rastgoo, M. Samadzadeh, “Multi-objective design
optimization of composite laminates using discrete shuffled frog leaping algorithm,” J
Mech Sci Technol 2013; 27(6):1791–800.
[52] R. Sørensen, E. Lund, “Thickness filters for gradient based multi-material and
thickness optimization of laminated composite structures,” Struct Multidiscip Optim
2015; 52(2):227–50.
[53] G. Marannano, G. V. Mariotti, “Structural optimization and experimental analysis
of composite material panels for naval use,” Meccanica 2008; 43(2):251–62.
[54] V. Yu. Eremin, S. M. Chernyshov, “Strength of orthotropic composite package,”
TsAGI, №3, pp. 15-23, 2009.
[55] V. D. Kleimenov, “Criteria of “self-reinforcing” of layered polymer composite
materials,” Composite materials constructions, №1, pp. 32-38, 2006.
[56] Yu. V. Nemikovsky, A. P. Yankovsky, “Design of rectangular and polygonal
transversely bent plates with equally stressed reinforcements,” Mechanics of composite
materials and structures, №4, 1998.
66
[57] N. M. Kundrat, “Localized pre-fracture zones in an orthotropic plate with a
periodic system of collinear reinforcements,” Mechanics of composite materials and
structures, №1, 2003.
[58] Yu. V. Nemikovsky, A. P. Yankovsky, “Elastoplastic transverse bending of
laminated reinforced plates,” Mechanics of composite materials and structures, №1,
2005.
[59] D. N. Kilasoniya, N. M. Beridze, “Algorithms for constructing stiffened matrices
of finite element for thin-walled aircraft structures made from composites,” Probl.
Mech., №1, pp. 74-82, 2005.
[60] V. F. Kutyinov, V. V. Chedric, “Computational and experimental study of the
stress-strain state of the composite wing box,” TsAGI, №2658, pp. 9-14, 2002.
[61] O.V. Mitrofanov, “Design of rib walls and spars made from composite materials,”
TsAGI, №3-4, Vol. 74, pp. 27-32, 2000.
[62] de Faria A, de Almeida S, “Buckling optimization of plates with variable thickness
subjected to nonuniform uncertain loads,” Int J Solids Struct 2003; 40(15):3955–66.
[63] U. Topal, U. Uzman, “Thermal buckling load optimization of laminated composite
plates,” Thin Wall Struct 2008; 46(6):667–75.
[64] L. Young-Shin, L. Yeol-Wha, Y. Myung-Seog, P. Bock-Sun, “Optimal design of
thick laminated for maximum thermal buckling load,” J Therm Stresses 1999;
22(3):259–73.
[65] S. T. Ijsselmuiden, M. Abdalla, Z. Gnrdal, “Thermomechanical design
optimization of variable stiffness composite panels for buckling,” J Therm Stresses
2010; 33(10):977–92.
[66] HK. Cho, “Optimization of dynamic behaviors of an orthotropic composite shell
subjected to hygrothermal environment,” Finite Elem Anal Des 2009; 45(11):852–60.
67