Numerical solution of problems in the dynamics of ribbed shells of revolution

We consider dynamic processes in shells of revolution reinforced with discrete ribs. The stress—strain state of the shell is determined in the framework of the linear theory of Timoshenko elastic thin shells. The ribs are described using the theory of curvilinear rods. The system of differential equations is solved by applying the variational principle to nonstationary problems.S. I. Subbotin Institute of Geophysics of the Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 45–48, 1992;     

Optimally reinforced shells of revolution

The design of filament-wound multilayer shells of revolution of arbitrary shape subjected to axisymmetric surface and body forces is examined. Three cases of existence of a solution of the optimal problem are established and certain particular types of solutions are presented.     

Stress analysis near holes in shells of revolution: Geometrically nonlinear problem

An approach is proposed to numerical analysis of the stress state of shells of revolution in the geometrically nonlinear setting. It is based on a combination of linearization and discrete orthogonalization methods. Stress concentration is analyzed for a flexible spherical shell with a central hole, depending on load intensity, hole size, and the boundary conditions on the contours in the precritical strain region.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 97–101, 1985.     

Numerical analysis of the spectrum of the Orr-Sommerfeld problem

A high-accuracy method for computing the eigenvalues λ _n and the eigenfunctions of the Orr-Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number R increases is studied in detail. For the Couette flow, it is shown that the eigenvalues λ _n regarded as functions of R have a countable set of branch points R _k > 0 at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.     

Nonlinear nonaxisymmetric deformation of composite shells of revolution

We discuss the results of the determination of the stress and displacement fields in nonaxisymmetrically loaded nonlinear-elastic shells of revolution. The original nonlinear system of equations is linearized in accordance with the method of variation of elastic parameters. The two-dimensional linear boundary-value problem is reduced to a sequence of one-dimensional problems, which are solved using a numerical method. We carry out an analysis of the stress-strain state of a conical shell made of a composite material of granular structure. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 80–83.     

Determination of the thermal stresses in anisotropic shells of revolution

Using the relations of the improved model of layered anisotropic shells based on the straight line assumption taking account of the thermal compression over the thickness we obtain a resolvent system of equations for shells whose properties depend on temperature. We carry out a study of the stresses in a two-layer cylindrical shell formed by winding as a function of the winding angle. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 82–85.     

Numerical analysis of laminated shells under in-plane axial compression

The authors of the paper present numerical stability analysis of axially compressed laminated shells. The analysis is carried out by using the commercial code NX-Nastran and authors' own program. In both cases the Equivalent Single Layer approach is adopted, in which the laminated shell is treated as an orthotropic single layer panel with the resultant stiffness calculated for the multilayered cross-section. Since the distribution of displacement in the thickness direction is assumed to be linear (First Order Shear Deformation), the shell kinematics is consistent with the Reissner-Mindlin type theory. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Yield surfaces for non-homogeneous anisotropic shells of revolution

The state of stress in a rotationally symmetric shell is characterized by the direct stresses and moments in the circumferential and longitudinal directions. It is assumed that the material of the shell is rigid perfectly plastic and that the yield stress of the material varies over the thickness of the shell. The material has different yield stresses in tension and compression and the yield stresses in the principal directions have different values. The yield condition for the shell is obtained in terms of the stress resultants assuming that the material of the shell obeys the maximum shear stress criterion.     

Haar wavelet discretization approach for frequency analysis of the functionally graded generally doubly-curved shells of revolution

This paper aims to investigate the free vibrational analysis of the generally doubly-curved shells of revolution made of functionally graded (FG) materials and constrained with different boundary conditions by means of an efficient, convenient and explicit method based on the Haar wavelet discretization approach. The FG materials of the shell consist of a combination of ceramic and metal, which four parameter power-law distribution functions have chosen for modeling of the smoothly and gradually variation of the material properties in the thickness direction. The theoretical model of the shell is formulated by employing of the first-order shear deformation theory. The rotation and displacement components of each point of the shell are expanded in the form of product of the Haar wavelet series in meridional direction as well as trigonometric series in the circumferential direction. By adding the boundary condition equations to the main system of equations, the constants appeared from the integrating of the Haar wavelet series are satisfied. In addition, with solving the characteristic equation, the vibrational results including the natural frequencies and the corresponding mode shapes are achieved. Then, the present results have been compared with those available in the literature. The results indicate that this method has high accuracy, high reliability and also a higher convergence rate in attaining the frequencies of the FG doubly-curved shells of revolution. Also, the effects of the main parameters such as power-law exponent, geometrical parameters, material distribution profiles and different types of boundary conditions, on the vibrational behavior of the FG doubly-curved shells of revolution, are investigated. Finally, taking into account the effects of geometrical parameters and material distribution profiles, for FG doubly-curved shells of revolution with different boundary conditions such as classic, elastic restraints and their combination, a variety of new frequency studies are provided which can be considered as proof results for further researches in this field.