On the theory of layered anisotropic cylindrical shells stiffened with ribs

The deformation, stability and vibration equations for anisotropic cylindrical shells stiffened with individual longitudinal and circumferential ribs are derived without introducing the hypothesis of nondeformable normals. The more general assumption adopted for layered materials (for example, glass-reinforced plastics) of a linear variation of the displacements over the thickness of the shell and the height of the ribs is used; in this case for the points of contact of the shell and the ribs after deformation the common normals form broken lines. The solution of the problem of the stability of a cylindrical shell stiffened with circumferential ribs is examined. For a shell with different, arbitrarily located ribs the problem is reduced to a homogeneous algebraic system of equations equal in number to three times the number of ribs.Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 647–654, July–August, 1974.     

The dynamic stability and nonlinear vibration analysis of stiffened functionally graded cylindrical shells

A theoretical model is developed to study the dynamic stability and nonlinear vibrations of the stiffened functionally graded (FG) cylindrical shell in thermal environment. Von Kármán nonlinear theory, first-order shear deformation theory, smearing stiffener approach and Bolotin method are used to model stiffened FG cylindrical shells. Galerkin method and modal analysis technique is utilized to obtain the discrete nonlinear ordinary differential equations. Based on the static condensation method, a reduction model is presented. The effects of thermal environment, stiffeners number, material characteristics on the dynamic stability, transient responses and primary resonance responses are examined.     

Dynamics and optimization of cylindrical composite shells

The stability of cylindrical composite shells under dynamic external pressure is discussed. A criterion for determining the load-carrying capacity based on Malmeister's equation with respect to bending parts of deformation is proposed. Optimization of the shell mass relative to various structural parameters has been carried out as a nonlinear programming problem. Numerical results are given.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 81–87, January–February, 1995.     

Numerical analysis method for studying local forms of stability loss of bearing layers of three-layered shells using mixed forms

A revised formulation of linearized stability problems of three-layered shells with a sofi filler has been presented. The form of stability loss of the rigid layers is mixed in the shells when the moment precritical stress-strain state (SSS) is reached and is localized in the principal moment SSS zones. If the filler thickness is much greater than the thickness of the rigid layers, the size of the bulges and thickness of the filler have the same order of magnitude. Thus, a very fine grid must be used for a numerical solution of the stability loss equations, which poses considerable computational difficulties. A numerical analysis method is proposed for the local forms of mixed mode stability loss of the rigid layers of a three-layered shell. Using this method, the solution of equations for the precritical SSS by the finite element scheme is found but an analytical solution of reduced stability loss equations is presented for estimating the critical load. This solution is an asymptotic approximation for local modes of stability loss implemented into design.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 1, pp. 88–100, January–February, 1995.     

Creep of a thermoplastic cylindrical shell under axisymmetric loading

The solution of the problem of creep of an axisymmetrically loaded thermoplastic cylindrical shell is considered. The strains and stresses for the zero-moment zone and with allowance for the edge effect are predicted on the basis of Kachanov's variational method using a computer. An algorithm is constructed for polyethylene and PVC shells at various values of the load. The analysis of the results obtained is illustrated by the calculation data for individual variants of the program.Leningrad Mozhaiskii Military Engineering Academy. Translated from Mekhanika Polimerov, No. 3, pp. 512–518, May–June, 1969.     

Difference spline scheme for modeling of convective flows using full Navier — Stokes equations

A difference scheme is constructed, in which enhanced stability is achieved by simultaneous solutions of the equations of motion, energy, and continuity. Spline approximations of spatial derivatives (with the original equations written in divergence form) substantially improve the accuracy of the scheme compared with the standard difference scheme using symmetric differences. The efficiency of the scheme is demonstrated for some problems of convective flow of compressible gas with lateral and bottom heating.Translated from Matematicheskoe Modelirovanie i Reshenie Obratnykh Zadach. Matematicheskoi Fiziki, pp. 38–45, 1993.     

Strain of a flexible orthotropic cylindrical shell as a function of orthotropy parameters

A procedure is proposed for calculating the stress-strain state of flexible orthotropic cylindrical shells of constant thickness with unsymtnetric load and nonhomogeneous boundary conditions. The system of nonlinear partial differential equations is solved by the method of lines. The system of nonlinear ordinary differential equations is reduced by linearization to a sequence of linear systems. The sequence of linear boundary-value problems is solved by the discrete orthogonalization method.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 59, pp. 57–61, 1986.     

The stressed state of elastoplastic shells with nonthrough cracks

We study the problem of the stressed state and limiting equilibrium of shells with nonthrough surface cracks. Several assumptions make it possible to reduce the three-dimensional problem to the two-dimensional problem and the latter to a system of singular integral equations whose solution is constructed using numerical methods. For cylindrical and spherical shells weakened by nonthrough cracks situated along the coordinate lines we carry out a numerical analysis of the dependence of the crack opening on the load and the geometric and physico-mechanical parameters of the shells.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 147–151.     

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     

Propagation of Transient Hydroelastic Waves in a Semiinfinite, Piecewise-Constant Cylindrical Shell Containing a Fluid

A solution is formulated for a new problem of wave propagation in a semiinfinite cylindrical shell with a junction connecting two shells of different radii. The material of the shell is assumed to be viscoelastic, and the fluid is assumed to be viscous. The motion of the shell is described by Kirchhoff–Love theory, and the motions of the fluid are described by equations averaged over the cross section. The problem is solved by means of the time Laplace transform and subsequent numerical inversion. The numerical results for the pressure and radial displacement of the shell are analyzed for various values of the parameters.     

Compact difference methods applied to initial-boundary value problems for mixed systems

Summary. An initial--boundary value problem to a system of nonlinear partial differential equations, which consists of a hyperbolic and a parabolic part, is taken into consideration. The problem is discretised by a compact finite difference method. An approximation of the numerical solution is constructed, at which the difference scheme is linearised. Nonlinear convergence is proved using the stability of the linearised scheme. Finally, a computational experiment for a noncompact scheme is presented. Received May 20, 1995     

Analysis of free damped vibrations of laminated composite cylindrical shells

Damped free vibrations of multilayered composite cylindrical shells are investigated. Vibration and damping analysis of cylindrical shells is performed by using the first-order shear deformation theory (FOSDT). Based on other researchers' works, two damping models are developed, i.e., the energy method (EM), and the method of complex eigenvalues (MCE). Several numerical examples of the damped free vibration problem of laminated composite cylindrical shells have been solved and comparison has been made with the results of other authors.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 646–659, September–October, 1995.     

扁柱面网壳的非线性动力学行为

用连续化法建立了正三角形网格的三向单层扁柱面网壳的非线性动力学方程和协调方程．在两对边简支条件下用分离变量函数法给出扁柱面网壳的横向位移．由协调方程求出张力,通过Galerkin作用得到了一个含二次、三次的非线性动力学微分方程．通过求Floquet指数讨论平衡点邻域的稳定性,用复变函数留数理论求出Melnikov函数,可得到该动力学系统发生混沌运动的临界条件．通过数值计算模拟和Poincaré映射也证明了混沌运动存在．     

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.     

A novel numerical solution strategy for solving nonlinear free and forced vibration problems of cylindrical shells

Nonlinear vibration analysis of circular cylindrical shells has received considerable attention from researchers for many decades. Analytical approaches developed to solve such problem, even not involved simplifying assumptions, are still far from sufficiency, and an efficient numerical scheme capable of solving the problem is worthy of development. The present article aims at devising a novel numerical solution strategy to describe the nonlinear free and forced vibrations of cylindrical shells. For this purpose, the energy functional of the structure is derived based on the first-order shear deformation theory and the von–Kármán geometric nonlinearity. The governing equations are discretized employing the generalized differential quadrature (GDQ) method and periodic differential operators along axial and circumferential directions, respectively. Then, based on Hamilton's principle and by the use of variational differential quadrature (VDQ) method, the discretized nonlinear governing equations are obtained. Finally, a time periodic discretization is performed and the frequency response of the cylindrical shell with different boundary conditions is determined by applying the pseudo-arc length continuation method. After revealing the efficiency and accuracy of the proposed numerical approach, comprehensive results are presented to study the influences of the model parameters such as thickness-to-radius, length-to-radius ratios and boundary conditions on the nonlinear vibration behavior of the cylindrical shells. The results indicate that variation of fundamental vibrational mode shape significantly affects frequency response curves of cylindrical shells.     

环肋加劲圆柱壳在静水压力作用下的初始后屈曲分析

本文用Koiter理论分析环肋加劲圆柱壳在静水压力作用下的后屈曲性能.前屈曲状态采用与边界条件一致的非线性有矩方程,本征值问题的解用伽辽金方法求出,得到的临界载荷与经典线性解作了比较.具体计算了三种不同环肋参数的外肋加劲圆柱壳.结果表明,肋的强弱不仅显著影响临界载荷值,同时也改变了柱壳的缺陷敏感度.     

Numerical modeling of nonlinear deformation and buckling of composite plate-shell structures under pulsed loading

Nonlinear three-dimensional problems of dynamic deformation, buckling, and posteritical behavior of composite shell structures under pulsed loads are analyzed. The structure is assumed to be made of rigidly joined plates and shells of revolution along the lines coinciding with the coordinate directions of the joined elements. Individual structural elements can be made of both composite and conventional isotropic materials. The kinematic model of deformation of the structural elements is based on Timoshenko-type hypotheses. This approach is oriented to the calculation of nonstationary deformation processes in composite structures under small deformations but large displacements and rotation angles, and is implemented in the context of a simplified version of the geometrically nonlinear theory of shells. The physical relations in the composite structural elements are based on the theory of effective moduli for individual layers or for the package as a whole, whereas in the metallic elements this is done in the framework of the theory of plastic flow. The equations of motion of a composite shell structure are derived based on the principle of virtual displacements with some additional conditions allowing for the joint operation of structural elements. To solve the initial boundary-value problem formulated, an efficient numerical method is developed based on the finite-difference discretization of variational equations of motion in space variables and an explicit second-order time-integration scheme. The permissible time-integration step is determined using Neumann's spectral criterion. The above method is especially efficient in calculating thin-walled shells, as well as in the case of local loads acting on the structural element, when the discretization grid has to be condensed in the zones of rapidly changing solutions in space variables. The results of analyzing the nonstationary deformation processes and critical loads are presented for composite and isotropic cylindrical shells reinforced with a set of discrete ribs in the case of pulsed axial compression and external pressure.Scientific Research Institute of Mechanics, Lobachevskii Nizhegorodsk State University, N. Novgorod, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 757–776, November–December, 1999.     

Mechanical stability of functionally graded stiffened cylindrical shells

This paper is concerned with the elastic buckling of stiffened cylindrical shells by rings and stringers made of functionally graded materials subjected to axial compression loading. The shell properties are assumed to vary continuously through the thickness direction. Fundamental relations, the equilibrium and stability equations are derived using the Sander’s assumption. Resulting equations are employed to obtain the closed-form solution for the critical buckling loads. The results show that the inhomogeneity parameter and geometry of shell significantly affect the critical buckling loads. The analytical results are compared and validated using the finite element method.     

Method of solving nonlinear dynamic problems in viscoelasticity

The authors propose a method of solving Volterra's system of nonlinear integrodifferential equations. This method is based on the use of a power series. As an illustration, the authors consider the vibration of flexible viscoelastic cylindrical shells under impulsive and periodic loads.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 554–558, May–June, 1973.     

Study of wave processes in shells and plates with regard to transverse normal and shear deformation

A variant of the theory of orthotropic plates and cylindrical shells taking account of transverse normal and shear deformation was examined. Independent approximations were adopted for distribution of displacements and stresses over the thickness of the shell. One of the requirements for constructing the theory is physical correctness, which is achieved by utilizing variational methods for formulating the mathematical model. The Reissner principle for dynamic processes was used for derivation of the equations. The elliptical part of the starting differential operator was shown to be symmetrical and positive in the space of the integrate of square functions. We examined the problem of the propagation of axially symmetric harmonic waves in the cylinder using the starting differential equations. These results were compared with those obtained equations derived in elasticity theory. Analysis of induced vibration was carried out for the case of a square plate upon the action of a suddenly applied load.Presented at the Ninth International Conference on the Mechanics of Composite Materials, Riga, Latvia, October, 1995.Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 6. pp. 816–823, November–December, 1995.