Optimization of a cylindrical shell of reinforced plastic with a filler under dynamic loading

Conclusions 1. The numerical investigation of profiles of functions of the physical constraints carried out in this work allows us to assume that problems of optimal design of shells of reinforced plastics, strengthened by an elastic filler, for purposes of stability (static and dynamic) under axial compression, in the given formulation, are problems of convex programming. This guarantees uniqueness of their solution and allows us to use gradient methods for a numerical realization.2. Optimal shells of a composite material have a smaller mass than equivalent shells of high-strength metal alloys. The gain in the expenditure of the material is ensured not only as a result of higher specific characteristics of the composite, but basically as a result of optimizing the reinforcement structure of the pack of orthotropic layers of the shell.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 879–885, September–October, 1977.     

Thermoelastoplastic deformation of complexly reinforced shells

The problem on the elastoplastic deformation of reinforced shells of variable thickness under thermal and force loadings is formulated. A qualitative analysis of the problem is carried out and its linearization is indicated. Calculations of isotropic and metal composite cylindrical shells have shown that the load-carrying capacity of shell structures under elastoplastic deformations is several times (sometimes by an order of magnitude) higher than under purely elastic ones; the heating of shells with certain patterns of reinforcement sharply reduces their resistance to elastic deformations, but only slightly affects their resistance to elastoplastic ones; not always does the reinforcement in the directions of principal stresses and strains provide the greatest load-carrying capacity of a shell; there are reinforcement schemes that ensure practically the same resistance of shells at different types of their fastening. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 42, No. 6, pp. 707–728, November–December, 2006.     

Dynamic stability of glass-reinforced plastic shells

The problem of the dynamic stability of circular-cylindrical glass-reinforced plastic shells subjected to external transverse pressure is examined in the nonlinear formulation. After the Lagrange equations have been constructed, the problem reduces to the integration of a system of ordinary differential equations with aperiodic coefficients. The integration has been carried out numerically on a computer for various loading rates and shell parameters. Analogous problems for isotropic metal shells were examined in [1–4]. A review of the subject may be found in [5].Mekhanika Polimerov, Vol. 4, No. 1, pp. 109–115, 1968     

Buckling and the initial postbuckling behavior of cylindrical shells made of composites with one plane of symmetry

A method for calculating the buckling stability of layered cylindrical shells made of composite materials with one plane of symmetry of mechanical characteristics is worked out. As a special case, shells made of fibrous materials by winding in directions not coinciding with coordinate axes are considered. An analysis of stability of shells under an axial compression, external pressure, and torsion is carried out. It is shown that, at a great number of layers and appropriate reinforcing angles, the shells can be considered orthotropic. The solution to the problem of the initial postbuckling behavior of shells made of composites with one plane of symmetry is also obtained. It is found that shells of this type can be less sensitive to geometrical imperfections. This fact is important from the practical point of view. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 2, pp. 213–236, March–April, 2007.     

Nonlinear free vibration analysis of SSMFG cylindrical shells resting on nonlinear viscoelastic foundation in thermal environment

In this paper, a semi-analytical method for the free vibration behavior of spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells under the thermal environment is investigated. The distribution of linear and uniform temperature along the direction of thickness is assumed. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The cylindrical shell has three layers consist of ceramic, FGM, and metal in two cases. In the first model i.e. Ceramic-FGM-Metal (CFM), the exterior layer of the cylindrical shell is rich ceramic while the interior layer is rich metal and the functionally graded material is located between these layers and the material distribution is in reverse order in the second model i.e. Metal-FGM-Ceramic (MFC). The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von Kármán equations and the smeared stiffeners technique, the problem of nonlinear vibration has been solved. In order to find the nonlinear vibration responses, the fourth order Runge–Kutta method is utilized. The results show that the different angles of stiffeners and nonlinear elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells. Also, the results illustrate that the vibration amplitude and the natural frequency for CFM and MFC shells with the first longitudinal and third transversal modes (m = 1, n = 3) with the stiffeners angle θ = 30°, β = 60° and θ = β = 30° is less than and more than others, respectively.     

Failure of water-filled cylindrical glass-reinforced epoxy shells under internal impulsive loading

Conclusions 1. The explosive failure of water-filled, geometrically similar glass-reinforced epoxy shells, whose dimensions increase by a factor of 1.5–4.4, is not associated with an energetic scale effect. This is attributable to the imperfect similarity of the shells, since although their dimensions changed, the diameter of the glass reinforcing fibers remained the same.2. The relative mass of explosive required to fracture the shells is approximately 0.4%. With respect to this index, glass-reinforced epoxy is comparable to structural steels 20 and 17Mn1Si at R₀160 mm.3. The circumferential deformation of the shells at failure is about 4% and does not depend on the thickness of the shell wall, the dimensions of geometrically similar shells, or the initial strain rate on the interval 0.21 · 10³–1.2 · 10³ sec^–1. Thin-walled shells begin to fail from the outside surface, thick-walled shells from the inside surface.4. The modulus of elasticity of the GRE is 2.4 · 10⁵ kgf/cm² and does not depend on the strain rate on the interval 10^–3–1.5 · 10³ sec^–1. The material deforms elastically up to failure.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 283–289, March–April, 1978.     

A variational principle in a geometrically nonlinear theory of heterogeneous viscoelastic shells

The use of the hereditary theory for shells heterogeneous across their thickness is considered. A variational method is formulated for calculating thin anisotropic shells made of a material whose deformation behavior can be described by relations of the linear theory of viscoelasticity. In order to transform the corresponding functional into a form suitable for shells, some assumptions related to concepts of the theory of thin shells are introduced. In the capacity of Euler equations, physical relations, nonlinear equilibrium equations, and nonlinear boundary conditions are derived. The state equations are deduced for a multilayered shell. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 2, pp. 231–240, March–April, 2009.     

Design of anisotropic structures: Calculation,optimization, and tests

Conclusions A computational procedure based on a limiting state has been worked out which proposes the performance of tests of samples, small-scale models, and full-scale structures and a statistical analysis of the experimental data. Advantages of the method are shown in comparison with the computational method based on fracturing loads which pertain to estimating the accuracy of the computational scheme of structures and determining their safety coefficients. An optimality criterion is formulated within the framework of the method which is related to ensuring a maximum of the limiting (fracturing or critical) load with a constant mass of the structure.Applications of the method are given to estimation of the strength and stability of structures made out of polymeric and metal CM with the use of the Gol'denblat-Kopnov criterion and solutions based on linear shell theory. The conditions for realization of the optimality criterion of structures are determined. Recommendations are made for the rational reinforcement of structures made out of filamentary CM, and a procedure is proposed for the rational design of reinforced structures.As an example of the tests for stability of cylindrical shells made out of aluminum-magnesium alloy under external pressure, the legitimacy of the cause of the disagreement between the calculated and experimental critical loads, which consists of a discrepancy in the nature of wave formation at the instant of stability loss and initial imperfections, and the effectiveness of the computational method based on a limiting state are confirmed. Refined computational formulas for the critical loads of cylindrical shells made out of fiberglass, carbon fiber, and boraluminum under typical kinds of loading are proposed within the framework of the method and on the basis of an analysis of the results of tests, and a comparison is made of the effectiveness of the reinforcement of shells made out of aluminum-magnesium alloy and carbon fiber. The values of the safety coefficients of structures which guarantee their reliability are determined.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 262–271, March–April, 1980.     

Numerical Investigation of the Dynamics of Thick-Walled Cylindrical Shells with a Helical Reinforcement

The Wilkins method is modified for numerically investigating the dynamics of thick-walled cylindrical shells with a helical orthotropy. The results of calculations by this algorithm agree well with known analytical solutions. Some new physical effects, which are absent in isotropic and cylindrically orthotropic shells, are investigated.__________Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 225–236, March–April, 2005.     

Free vibration of four-parameter functionally graded moderately thick doubly-curved panels and shells of revolution with general boundary conditions

This paper aims to present a unified vibration analysis approach for the four-parameter functionally graded moderately thick doubly-curved shells and panels of revolution with general boundary conditions. The first-order shear deformation theory is used in this formulation. The functionally graded panels structures consists of ceramic and metal which are set to vary continuously in the thickness direction according to the general four-parameter power-law distribution, and six types of power-law distributions are considered for the ceramic volume fraction. The admissible function of the FG panels and shells of revolution is obtained by the improved Fourier series with the help of the governing equations and the boundary conditions. The solution is obtained by using the variational operation in terms of the unknown expanded coefficients. By a great many numerical examples, the rapid convergence and good reliability and accuracy of the proposed approach are validated. A variety of new results for vibration problems of the FG doubly-curved shells and panels with different elastic restraints, geometric and material parameters are presented. The effects of the elastic restraint parameters, power-law exponent, circumference angle and power-law distributions on the free vibration characteristic of the panels are also presented, which can be served as benchmark data in the research and the actual production process.     

Application of the conditional moments method to multi-layered anisotropic shells

In this paper the so-called Conditional Moments Method (CMM) is explained, which allows investigating the stress-strain state of stochastic composites under externally applied loads. This technique is used to determine effective (homogenized) material properties and, in particular, preparatory measures are taken, which eventually will allow to assess the deformation behavior of tires. The tire shells are modeled as a rubber matrix strengthened by metal or cord fibers. For this reason general CMM results will be specialized to the case of ellipsoidal inclusions of arbitrary aspect ratios in a matrix of various stiffness. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

圆环壳在离心力作用下的应力和位移

本文应用文[1]、[2]一般轴对称问题的结果计算了圆环壳在离心力作用下的应力和位移,解答以Fourier级数形式给出,文中分别给出了部分环壳和闭合环壳的算例。     

Dynamic stability of axisymmetrically heated glass-reinforced plastic cylindrical shells bonded to elastic cylinders

The dynamic stability of glass-reinforced plastic shells subjected to the action of a longitudinal pulsating force and an external, uniformly distributed load is examined. The shells are orthotropic, and their elastic properties are temperature-dependent and vary over the thickness of the wall. It is assumed that the elastic cylinder forms a Winkler foundation. Three boundary conditions are investigated.Moscow Institute of Chemical-Machine Building. Translated from Mekhanika Polimerov, No. 2, pp. 331–337, March–April, 1972.     

Stability of axially compressed cylindrical composite shells in the presence of nonstationary heating

The problem of smooth cylindrical composite shells uniformly compressed in the axial direction and subjected to nonstationary heating is solved in the linear quasi-static formulation. Expressions are obtained for the critical loads and their regions of application are determined. The calculations are compared with experimental data obtained by linearly heating the outer surface of axially compressed smooth cylindrical shells of glass-reinforced plastic based on phenol-formaldehyde resin.Zhukovskii Central Aerohydrodynamic Institute, Moscow Region. Translated from Mekhanika Polimerov, No. 2, pp. 289–297, March–April, 1973.     

Experimental-theoretical studies of the stability of orthotropic shells with a filler in axial compression

Conclusions 1. An analysis has been made of the solution to the problem of the stability of multilayer cylindrical shells having a filler and simple calculation formulas have been obtained for determining the critical forces.2. The stability of fiberglass-plastic shells with rubber-like fillers has been studied experimentally.3. Comparative experimental-theoretical studies of critical forces have been made, and the stability coefficients have been ascertained for the shell class under consideration.Translated from Mekhanika Polimerov, No. 3, pp. 485–489, May–June, 1978.     

Heat conduction of shells reinforced by fibers with constant and variable cross sections

We propose a model for heat conduction of a spatially reinforced medium and present its generalization to the case of a polyreinforced layer. We consider the heat-conduction equations for fibrous shells and construct a procedure for reduction of a three-dimensional problem of heat conduction to a two-dimensional one. Analytic solutions of a stationary problem of heat conduction are found for thin conic shells of revolution for various structures of reinforcement, and a graphical comparison of the corresponding results is performed. We study one of the approaches to rational reinforcement of thin shells, according to which the thermal “transparency” of a shell in the transverse direction is taken as a criterion of rational design. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 132–150, April–June, 1998.     

Reaction of sloping spherical shells of glass-fiber-reinforced plastic during nonstationary loading

The behavior of spherical shells of glass-fiber-reinforced plastic under the action of exponentially changing dynamic loading was studied in a nonlinear formulation. The method of finite differences, used in the form of an explicit difference scheme, was used to solve the differential equations of the dynamics of sloping shells based on the Kirchhoff-Love hypotheses. The characteristic features of the deformation process and the influence of the degree of anistropy of the shell material operating under conditions of dynamic loading are noted.Moscow. Translated from Mekhanika Polimerov, No. 2, pp. 311–314, March–April, 1975.     

Numerical solution of multipoint boundary-value problems in improved shell theory

For some classes of stratified shells subject to transverse shear (shells of revolution with parameters varying along the generatrix, noncircular cylindrical shells with characteristics varying along the directrix, rectangular shallow shells), we propose an approach that reduces the solution of multipoint boundary-value problems to a number of two-point problems. As an example, we consider the stressed state of an open noncircular cylindrical shell supported in some section along the generatrix.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 35–38, 1987.     

The effect of the nonuniformity properties of a material in problems of elastic equilibrium and stability of shells

On the basis of two improved models of shells that are nonuniform by thickness (based on assumptions of Timoshenko type stated separately for each layer and assuming a uniform stress-deformed state of a thin-walled element of the layered structure) we study the stressed state and stability of three-layer cylindrical shells with a nonsymmetric packet. We show the equivalence of the two models in describing the reaction of layered shells to uniform external actions. We give an analysis of the dependence of the stress-deformed state and the reserve stability of shells on the structure of the packet.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 27, 1988, pp. 89–93.     

A direct method for determining the mass of gases in ICF shell

In the present paper, the defects of dew point method for measuring the mass of gas filled in ICF shells are analyzed. An accurate state equation for gas D2 is deduced from Benedict-Webb-Rubin (BWR) equation and experimental data in planar phase. A direct method to determine gas mass in ICF shells via measuring the temperature and pressure outside the shells and solving the equation of state by numerical method is proposed. It overcomes the theoretical defects of dew point method and the complexities of equipment. In the present method, the state equation can be improved by more accurately measuring P-V-T values of gas D2, so the measuring precision of the mass of gas in the shells can also be improved. The present method is effective for treating mix gases filled in the shells as well. The errors between the computational results and experimental data are very small. Some cases in the filling process are predicted, and the proper temperature and pressure for filling gases effectively are also suggested.