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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. 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