Аннотация (русский):
Приводится формулировка, алгоритм проектировочного расчета прочного корпуса из полимерных композиционных материалов для подводной техники на основе требований прочности и устойчивости, формализованных в виде основного критерия качества прочного корпуса - отношения массы корпуса к его водоизмещению W/D , и пример расчета рациональной структуры композитного корпуса новой конструкции - корпуса в форме кругового замкнутого тороида. В проектировочном расчете учтена переменность толщины корпуса, которая является результатом производства тороида методом намотки. Ограничения по прочности приняты в виде критерия Мизеса - Хилла, а по устойчивости - по верхнему критическому давлению. В качестве примера для расчета был взят тороидальный прочный корпус, выполненный продольно-поперечной намоткой из полимерных композиционных материалов.

Ключевые слова:
подводная техника, проектирование, тороидальный прочный корпус, намотка, внешнее гидростатическое давление, потеря устойчивости, прочность, углепластик, стеклопластик

Introduction Ocean engineering is an intensively developing area. Emergence of new technologies and materials contributes to realization of various architectural and constructive configurations of submersible structures and underwater vehicles by designers. For the last decades the most various shapes and sizes of autonomous underwater vehicles, including torpedo shapes, laminar flow bodies, streamlined rectangular styles and multihull vehicles, are tested [1]. The original projects of underwater drilling rigs, stations and submersibles with a pressure hull of toroidal shape [2], for example, underwater vehicle type "diving saucer" with increased maneuverability (Fig. 1) [3] can become one of the new solutions. Complex combination of design and constructive solutions in the underwater vehicle with toroidal pressure hull (Fig. 1) will provide the improved maneuvering motions, the stability of motion, the dynamic positioning control and will give possibility to change sharply a trajectory of motion both at the selected depth and in depth [3]. The selection of toroid as the shape of pressure hull by the designers is conditioned by advantageous characteristics, namely, a toroid possesses a good hydrodynamic shape, static and dynamic stability on the surface and in underwater positions; toroidal shell due to double curvature is more rigid and the wall thickness of the toroid as compared to cylindrical shape can be reduced in 1,5-2 times that depends upon geometry of toroid and the immersion depth [4]. To realize these original project solutions of architectural and constructive configurations of submersible structures and underwater vehicles [2], it is necessary to design the pressure hull of toroidal shape as one of their main structural parts. Fig. 1. Underwater vehicle type "diving saucer" with increased maneuverability [3]: 1 - main toroidal pressure hull; 2 - hydrodynamic hull fairing; 3 - pressure hull in the shape of ellipsoid; 4 - toroidal tank; 5 - vectorable nozzles The expanded methodology framework [5] of pressure hull design for submersible structure and underwater vehicles is shown in Fig. 2. The main stressed state of the pressure hull in operational environment is compression. Therefore, the task of the rational design of the toroidal pressure hull as pressure vessel subjected to external hydrostatic pressure can be formulated as follows: for the given hydrostatic pressure and volume of pressure hull to determine the hull thickness and to choose the fiber-reinforced material, so that along with achieving the minimum weight-to-displacement ratio W/D of the hull, as the main criterion of quality of pressure hull for underwater applications, such requirement as manufacturability must be also taken into account (Fig. 2). Fig. 2. Methodology framework of pressure hull design for submersible structure and underwater vehicles It is today one of the perspective ways of reducing the weight of the main pressure hulls along with providing high operational performance is their shaping by automated filament winding from FRP, GFRP and CFRP, that has already been successfully applied in such underwater vehicles with composite pressure hulls [2]. High specific strength and stiffness and also composite manufacturability provide significant advantages for FRP over conventionally used metals in pressure hull design, such as high strength steel, titanium and aluminum alloys. This was facilitated by the accumulation of sufficient knowledge about composite properties and wide variety of high strength and high modulus fibers (carbon and glass), the development of manufacturing procedures and increased experience in production of filament wound structure, especially shells for internal pressure applications, the development of design and analysis of composite structure, especially in aerospace engineering. The research object Due to experience of design of toroidal pressure vessels by filament winding [6, 7], as a result the shell is formed with variable wall thickness (Fig. 3), it is possible to realize the minimum mass toroidal pressure hulls, that can be manufactured easily, and to estimate their potential for ocean engineering. Fig. 3. Geometrical parameters of filament-wound toroidal pressure hull with closed circular cross-section: R - distance from the center of the cross-section to the axis of rotation, r0 - maximum radius of rotation, а - inner radius of tube’s circular cross-section, hmax and hmin - maximum and minimum wall thickness, r - distance from the axis of rotation to a point on the surface of toroid The principal failure mode for a shell of pressure hull, which is under hydrostatic pressure, is either for a thin-walled shell by buckling, determined by the geometry of the hull and elastic modulus of the material, or for a thick-walled shell by compressive failure of the material when compressive stress reaches compressive strength of the material [8, 9]. Today, the works about the toroidal pressure hulls either describe only variations of the original architectural and constructive configurations of the submersible structures and the underwater vehicles with toroidal pressure hulls [2], or concentrate on the analysis of the stress-strain state and the strength of isotropic and orthotropic toroidal shells loaded with external pressure without taking into account their structural and manufacturing parameters (O. Makhning, P. F Jordan, L. H. Sobel, W. Flugge, Y. A. Fedosov, A. V. Netrebko, K. F Chernykh, V. A. Shamina, T. I. Kosheleva, V. V. Gulyaev, V. V. Gaydaychuk, N. A. Chaplin, Y. M. Grigorenko, M. S. Ganeeva, L. A. Kosolapova, A. S. Volmir, K. Z. Hayrnasov, G. S. Pogosyan, M. V. Goldmanis, J. Blachut, etc.). The lack of the reasonable recommendations about the choice of composite components, in particular reinforcing filaments, and the construction of the FRP laminate (a layered construction shaped as a shell) taking into account the manufacturing techniques of the closed circular toroid by filament winding, and the lack of calculations of load carrying capacity of the toroidal shell as pressure hull on minimum weight criteria make a subject of this work actual. The aim of this work is to develop the mathematical models and algorithms for rational design of a toroidal PH, limited to its manufacturing by filament winding and specificity of FRP application, based on the requirements of strength and static stability, formalized as weight-to-displacement ratio W/D of the hull. The mathematical model of the designing of laminate construction for the filament wound toroidal pressure hull with circular cross-section taking into account the resulting wall thickness variation hΣ(θ) in a meridian direction (Fig. 3) only from the position of elastic state stability (1) is presented in Ref. [10]. In Ref. [10] the designing variables (thickness of monolayers (laminas), their numbers and orientations) have discrete values; the capacities of winding equipment for manufacturing the closed toroids, namely the winding along geodesic fiber trajectories - longitudinal-transverse, cross helical and combined reinforcement scheme are taken into account as the most commonly used one. Elastic state stability condition: , (1) where qSW - working pressure; q - upper critical pressure; fs - safety factor, fs = 2. Because of the large number of design variables at the design of composite structures, it is difficult to specify a rational solution of the task. For FRP the structural strength and stiffness are mainly determined by strength and elastic modulus of reinforcing fibers. Comparison of efficiency of various reinforcing fibers for filament wound toroidal pressure hull based on the requirements of strength and static stability is exemplified by the hull with geometry k = a/R = 1/3, made by longitudinal-transverse filament winding. As it was established in [11], the hull, made by longitudinal-transverse winding with equal numbers of longitudinal and transverse monolayers , at restriction on upper critical pressure (Fig. 4) will have the minimum weight. CFRP and GFRP with epoxy matrix is considered as the hull materials, because epoxy composites have high compressive strength and shear. Properties of fibers Еf and matrix Еm and based on them unidirectional epoxy composite properties used in the design analysis are given in Tab. 1 and Tab. 2. Table 1 Properties of fibers and matrix used in the design analysis of FRP toroids Components of FRP Tensile modulus, GPa Tensile strength, GPa High modulus and high strength glass fiber S-2 GLASS [12] 93 4.5-4.75 High modulus and high strength carbon fiber M35J (TORAYCA®) [13] 343 4.7 Intermediate modulus and high strength carbon fiber IM10 (HexTow®) [14] 303 6.964 Ultra-high modulus carbon fiber CN-90 (Nippon Graphite Fiber Corporation) [15] 860 3.43 Epoxy resin ЕDТ-10 [6] 3 0.07 Table 2 Unidirectional epoxy composite properties used in the design analysis of FRP toroids Properties GFRP СFRP S-2 GLASS/ Epoxy [12] IM10/ Epoxy [13] M35J / Epoxy [14] CN-90/ Epoxy [15] Fiber volume fraction Vf 0.6 0.6 0.6 0.6 Longitudinal modulus Е1, GPa 59 190 207 550 Transverse modulus Е2, GPa 20 7.39 7.4 5,4 Axial Poisson’s ratio μ12 0.28 0,348 0.348 0.344 Longitudinal tensile strength F1t, MPa 2000 3310 2470 1800 Longitudinal compressive strength F1c, MPa 1240 1793 1270 370 Transverse compressive strength F2c, MPa 200 180 180 180 To carry out the checking calculation of hull strength, as it is presented in [11], with thickness and laminate construction satisfying only the stability condition (1), it is necessary to know the values of the acting stresses in the laminate, the ultimate strength of the laminate and to select safety factor. According to [16, 17] at k ≤ 0.5 for determination of stresses σθ, σψ (Fig. 4b) in the toroidal shell subjected to axisymmetric loading solution of membrane theory of shells - Feplya solution - can be used. According to this solution the stresses from membrane efforts Nθ, Nψ are: ; , where hΣ (θ, k) - variable thickness of composite materials [6]. The selection of composite materials, providing minimum weight of the structure, is not as obvious as in uniaxial tension or compression along the fiber direction, when you can use the traditional estimation of the material on the value of its specific strength [18]. To determine the strength properties of the laminate it is necessary to know the construction of the laminate and strength and elastic properties of the constituent unidirectional composite (lamina). The construction of the laminate includes the number and type of different laminas, their position and orientation within the laminate [19]. Then it is necessary to choose the criterion for the strength of the laminate. For estimating the strength of orthogonal reinforced composite materials, the strength conditions for the chosen Mises - Hill criterion are generally the following [18]: ; ; (2) . For keeping the strength balance condition by stability and stress according to the recommendations [20] and designing experience of composite wound cylindrical pressure hulls [21] the safety factor for a hull design in (1) and (2) was taken as fs = 2. Fig. 4. Dependences of the relative averaged thickness ∆m in a meridian direction (a) and the weight-to-displacement ratio W/D (b) of the toroidal hull (k = 1/3), made by longitudinal-transverse filament winding on operating depth Hsw and failure modes of the hulls for different reinforcing fibers: 1 - СFRP CN-90/Epoxy; 2 - СFRP M35J/ Epoxy; 3 - СFRP IM10/ Epoxy; 4 - GFRP S-2 GLASS/Epoxy ── - strength (2); - - - - - elastic state buckling [11] Ultimate strengths of the laminas F1, F2 in (2) are taken from the supplier’s technical data bases (Tab. 2). Ultimate strengths Fθ(θ), Fψ(θ), elastic modules Еθ(θ), Еψ(θ) and μθψ(θ) of the laminate in a-meridional and R-circumferential directions correspondingly are determined by the prediction methods for elastic and strength properties of the laminated composites based on Mises - Hill criterion presented in [18] (Tab. 3). Table 3 The laminate properties of filament-wound toroidal pressure hull (k = 1/3) calculated by prediction methods for elastic and strength properties of laminated composites presented in [18] and used in design analysis Properties Angle Hull material GFRP S-2 GLASS/ Epoxy СFRP IM10/ Epoxy СFRP M35J / Epoxy СFRP CN-90/ Epoxy Modulus in a-meridional direction Еθ, GPa θ = 0° 39.76 99.09 107.6 262.8 θ = 180° 46.12 129.6 141 347.8 Modulus in R-circumferential direction Еψ, GPa θ = 0° 39.76 99.09 107.6 262.8 θ = 180° 33.2 68.5 74.2 177.7 Major Poisson's ratio μθψ θ = 0° 0.138 0.026 0.024 0.01 θ = 180° 0.119 0.02 0.018 0.0074 Ultimate compressive strength in a-meridional direction Fθc, MPa θ = 0° 400 986.4 686.8 188.8 θ = 180° 462.6 1292 900.4 249.7 Ultimate compressive strength in R-circumferential direction Fψc, MPa θ = 0° 400 986.4 686.8 188.6 θ = 180° 331.6 683.8 474.4 127.5 Substituting the data from Tab. 3 to the strength conditions (2), it is seen that the strength conditions by Mises - Hill criterion (2) is satisfied not for all the hulls, designed in [11] only by the stability condition (1) for the operating depth up to 3000 m (Fig. 5). 3 2 1 а б Fig. 5. Stress distribution diagrams from membrane efforts for filament wound toroidal hulls (k = 1/3) with orthogonal reinforcement , and the wall thickness satisfying the stability condition by upper critical pressure (1) at the operating depth Hsw = 1000 m (а) and 3000 m (b): 1 - GFRP S-2 GLASS/Epoxy; 2 - СFRP M35J/Epoxy; 3 - СFRP CN-90/Epoxy For the considered hull with variable wall thickness stresses σθ and σψ (Fig. 5) reach maximum values at the external equatorial perimeter (θ = 0°, Fig. 2), whereas for toroidal shell with constant wall thickness the points at the internal equatorial perimeter (θ = 180°, Fig. 2) will be the most loaded [16, 22]. The thickness of the considered toroidal hull can be determined from the strength conditions (2). The results of the design analysis of the GFRP and CFRP toroidal pressure hull (k = 1/3) made by longitudinal-transverse winding under restrictions on stability (1) and strength (2) are provided in Fig. 4. The algorithm of the design analysis of the rational laminate construction for filament wound toroidal pressure hull is presented in Fig. 6. As shown in Fig. 4b, the toroidal hulls from carbon epoxy composite CN-90/Epoxy should be calculated from the strength conditions. Although FRP based on ultra-high modulus carbon fibers CN-90 provides the most minimum weight of the hull as compared to FRP with other fibers when the design analysis was only under the restrictions on stability (1) (Fig. 4b) [11]. It can be explained by the fact that the compressive strength of high modulus carbon fiber (Еf = 345-600 GPa) is 1.7 GPa, but the compressive strength of ultra-high modulus carbon fiber (Еf = 600-965 GPa) is 0.5 GPa, although the tensile strengths of high and ultra-high modulus carbon fibers are equal or lower than the strengths of standard (Еf = 200-275 GPa) and intermediate modulus fibers (Еf = 275-345 GPa) [23]. CFRP (M35J/Epoxy, IM10/Epoxy - Fig. 4b) based on intermediate modulus carbon fibers (Еf = 275-345 GPa, tensile strength - 4500-7000 MPa) will be able to realize W/D ≤ 0.5 for toroidal pressure hull with operating depth below 3000 m, providing satisfaction of stability and strength conditions (1), (2). As it is shown in Fig. 4b, the toroids from CFRP M35J/Epoxy should be calculated from stability conditions (1) to Hsw = 1550 m, from CFRP IM10/Epoxy - to Hsw = 4100 m, from GFRP S-2 GLASS/Epoxy - to Hsw = 1450 m. For the most large depths, the toroids should be calculated from the strength conditions for stress (2). Fig. 6. Flow chart for minimum weight design analysis of composite toroidal pressure hull for ocean engineering In those cases, where the strength condition is more rigid than the restriction on stability (curves 1, 2, 4 in Fig. 4b), it is necessary to investigate other structure of composite materials taking into account the structural and manufacturing constraints [21, 24] in order to achieve hull thickness reduction, respectively and W/D ratio of the hull, for example, to consider the laminate construction with more numbers of plies in a-meridional direction ( > 1) - , and to accept the optimum structure of composite materials, which provides minimum weight of hull at satisfaction of stability and strength conditions. So, for example, the replacement (01°/901°)l1 (δ = 1) for the toroid with k = 1/3, designed from CFRP M35J/Epoxy for the operating depth to 3000 m, into the structure (δ = 2) will bring to reduction of W/D by 10 % at satisfaction of strength and stability conditions by upper critical pressure. Conclusions 1. Application of various types of glass and carbon fibers for toroidal pressure hull design on the example of the hull made by longitudinal-transverse filament winding are investigated. It is shown that the choice of reinforcing fiber, providing minimum weight and maximum of load carrying capacity of the hull, is rather ambiguous task. While designing a toroidal pressure hull for ocean engineering an attention should be paid to the coordinated increase in both strength and stiffness of constituent unidirectional composite. Carbon epoxy composite based on intermediate modulus carbon fibers (Еf = 275-345 GPa, tensile strength - 4500-7000 MPa) is the most suitable for toroidal pressure hulls of deepwater submersibles. 2. To evaluate the effectiveness of the pressure hull, developed by using the other schemes of winding, according to the proposed algorithm, it is necessary to make a complete analysis of the influence of all designing parameters such as geometry of the toroid, type of the structure and type of the components of FRP laminate, taking into account the structural and manufacturing constraints at production of the toroidal pressure hull by filament winding. It will help design a rational laminate construction to achieve minimum weight and maximum of load carrying capacity of the toroidal pressure hull.

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