Effect of the stiffness of the elastic core on the buckling mode and the critical load for glass-reinforced plastic cylindrical shells in axial compression

The effect of the stiffness of the core, treated as a three-dimensional elastic body, on the buckling mode and the critical load is investigated for orthotropic cylindrical shells in axial compression. Parametric equations that permit the easy determination of the critical loads are obtained together with an expression for the stiffness parameter at which the buckling mode changes. The effect on the critical load of a central opening in the core is considered.Translated from Mekhanika Polimerov, No. 5, pp. 931–939, September–October, 1971.     

Local buckling of compressed elements of layered viscoelastic material

The problem of local surface buckling of compressed elements of layered material is solved using the equations of the theory of multilayer plates [1, 2, 5]. The local buckling deformations are described by means of the solutions of differential-difference equations damping into the interior of the medium. The effect of compressive forces, buckling wavelength, and the elastic and viscoelastic constants of the material on the behavior of the deformations and on the thickness of the layer to which significant buckling extends is investigated. Numerical estimates are presented for the buckling parameters of compressed laminated-plastic elements.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 816–821, September–October, 1968.     

Optimal winding of filament-wound reinforced plastic shells designed to resist buckling under external pressure or axial compression

The optimal method of winding the reinforcement of thin cylindrical shells subjected to external pressure or axial loading is investigated. Various types of winding — normal, simple oblique, crossed, and isotropic — are considered. An approximate formula is obtained for the critical external pressure. This formula is used to analyze the possibility of raising the critical load by adopting oblique winding. In the case of axial compression different buckling modes are examined and the optimal winding determined.Central Scientific-Research Institute of Machine Building, Moscow. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 864–875, September–October, 1968.     

Buckled shapes of polymeric cylindrical shells under long-time loading

The creep buckling of polyethylene cylindrical shells in axial compression has been investigated. The changes in the shape of the shell surface up to loss of stability were measured with a special radial deflection gauge. The experimentally determined shape of the buckled surface at discrete moments of time is approximated by a double Fourier series. The characteristic coefficients of the series of importance in creep buckling are established. From an analysis of the coefficients of the series it follows that the amplitudes of the axisymmetric coefficients diminish with time, while those of the coefficients giving a nonaxisymmetric buckled shape grow.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 269–274, March–April, 1971.     

Creep strength of anisotropic plates taking account of transverse shear strains

Cylindrical creep buckling of an orthotropic glass-reinforced plastic plate is investigated with consideration of shear creep in the plane xz. In this case the rheology of buckling is not described by the Kirchhoff-Love model, therefore the investigation is carried out using the modified theory of plates [3]. The long-time and instantaneous critical forces are obtained.Mekhanika Polimerov, Vol. 1, No. 5, pp. 114–117, 1965     

Stability of a transversely isotropic cylindrical shell in axial compression

The problem of the stability of a transversely isotropic cylindrical shell subjected to axial compression is considered with the aid of three-dimensional linearized equations for small subcritical strains. Transcendental equations are obtained for the critical loads corresponding to axisymmetric and nonaxisymmetric deformation. It is shown that in the case of the axisymmetric buckling mode the Kirchhoff-Love hypothesis is asymptotically exact irrespective of the properties of the material. The dependence of the critical load on the properties of the shell material is investigated numerically. Graphs of the variation of the critical load with relative shear modulus are presented.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1068, November–December, 1969.     

The effect of transverse shear upon the buckling stability of inelastic orthotropic plates

The problem of buckling of inelastic orthotropic plates is analyzed. The phenomena associated with the transverse strains of the plate are considered in the framework of the Ambartsumyan theory of plates, which excludes the hypothesis of undeformed normals. The problem is solved with the aid of the localized strains theory, and it is shown that in the case of orthotropic materials, i.e., those characterized by a large E/G ratio, the effect of the transverse shear must be taken into account in determining the critical stress for a plate under compressive loads parallel to its surface.Mekhanika Polimerov, Vol. 1, No. 2, pp. 133–136, 1965     

Shear buckling modes of cylindrical sandwich shells under external pressure, tension-compression of bearing layers with unequal forces, and inhomogeneous across-the-thickness temperature

Equations are set up for describing, in a correct statement and with an accuracy sufficient in actual practice, the shear buckling modes (BMs) of cylindrical sandwich shells with a transversely soft core of arbitrary thickness. Based on them, solutions are obtained to a number of problems on the buckling instability according to shear modes under some force and thermal loadings. It is found that the BMs occur in the shell along the circumferential and axial directions if, in the precritical state, a normal compressive stress arises in the transverse direction. It is shown that this condition is fulfilled in the following cases: in axial tension of the shell with unequal forces applied to the end faces of bearing layers (the parameter of critical load is maximum if the tensile forces are equal); under external (internal) pressure; on cooling the outer and heating the inner layers. The results obtained are presented in the form of simple analytical formulas for determining the corresponding critical parameters of the force and thermal actions.Translated from Mekhanika Kompozitnykh Materialov, Vol. 41, No. 1, pp. 37–48, January–February, 2005.     

Compressive stability of multilayered delaminations in composites

The behavior of defects of the near-surface crack type was investigated in laminated composites. Multilayered delaminations in compressed elements were examined on the assumption that cylindrical bending of the delamination is accompanied by shear within the framework of a model similar to the Timoshenko model. The effect of the mechanical properties and thickness and number of layers in the delaminated section on the critical buckling strain and strain corresponding to the beginning of growth of the delamination was investigated. It was shown that when the delaminated section contains several bearing layers and the rigidity of the composite with respect to interlaminar shear is small, consideration of the shear in the delaminated section can lead to a significant change in the critical compressive strains and/or critical size of the delamination.Translated from Mekhanika Kompozitnykh Materialov. Vol. 33, No. 3, pp. 312–320, May–June, 1997.     

Polarized Hessian covariant: Contribution to pattern formation in the Föppl–von Kármán shell equations

We analyze the structure of the Föppl–von Kármán shell equations of linear elastic shell theory using surface geometry and classical invariant theory. This equation describes the buckling of a thin shell subjected to a compressive load. In particular, we analyze the role of polarized Hessian covariant, also known as second transvectant, in linear elastic shell theory and its connection to minimal surfaces. We show how the terms of the Föppl–von Kármán equations related to in-plane stretching can be linearized using the hodograph transform and relate this result to the integrability of the classical membrane equations. Finally, we study the effect of the nonlinear second transvectant term in the Föppl–von Kármán equations on the buckling configurations of cylinders.     

Stability of cylindrical glass-reinforced plastic shells

The stability of cylindrical shells in axial compression has been experimentally investigated. The glass-reinforced plastic shells were fabricated in various ways: by winding glass strand, from glass cloth, and by a combination of the first two methods. The buckling mode and critical load were determined as functions of the ratio of shell radius to shell thickness. A statistical analysis of the test data leads to a formula with an empirical coefficient for use in practical stability calculations.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 5, pp. 886–891, September–October, 1969.     

Methods of finite element analysis of arbitrary buckling forms of sandwich plates and shells

Two statements of the problem of arbitrary buckling forms (BFs) (including synphasic, antiphasic, mixed flexural, flexural-shear, and shear forms in the tangential directions) of general-form sandwich shells and two schemes of its solution by the FEM are given. The first of the schemes is based on the use of refined linear equations for determination of the precritical stress-strain state and linearized equations of neutral equilibrium with all parametric addends necessary to determine the critical loads and reveal the possible BFs. The second one uses the general geometrically nonlinear relations of elasticity theory for investigation of the whole deformation process up to buckling in terms of a modified incremental (stepwise) statement of the problem. Examples of solution of particular problems are given.Center for Study of Dynamics and Stability, Tupolev Kazan' State Technical University, Kazan', Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 473–486, July–August, 2000.     

Stability of a laminated spherical shell

The stability "in the small" of a spherical shell consisting of alternating stiff and soft layers is investigated. The spectra of the bifurcation values of the load are found for a closed shell subject to hydrostatic pressure and their dependence on the buckling mode is studied. The effect of the relative stiffness of the layers on the buckling of the shell is investigated.Moscow Power Engineering Institute. Translated from Mekhanika Polimerov, No. 3, pp. 459–464, May–June, 1976.     

Application of the variation principle for the problem of buckling of a nonlinearly elastic ring nonuniform along its thickness

There has been considerable recent attention given to the stressed and buckled states of items with complicated configuration made of different nonlinearly elastic materials joined by complete adhesion. However, effective analytical solutions for such problems have been hindered by mathematical difficulties. Approximate methods have thus been developed for such problems. A variational combined principle has been formulated in this communication. A nonlinear geometrical approach has been used for formulating a mixed-type functional with physical relationships given by Euler equations, nonlinear equilibrium equations, and nonlinear boundary conditions for a piecewise-nonuniform nonlinearly elastic body composed of finite elements (particles). As an example, buckling along the nonuniform thickness of nonlinearly elastic rings was analyzed hypothetically assuming plane cross-sections. Options for two-, three-, four-, five-, and six-layered rings in a periodical structure have been reviewed. The critical buckling forces for an even number of layers have been found to be equal to each other. The ratios of the critical forces, elasticity moduli, and proportionality levels were determined for all five variants by the Runge-Kutta method.Presented at the Ninth International Conference on the Mechanics of Composite Materials (Riga, October, 1995).Translated from Mekhanika Kompozitnykh Materialov, Vol. 31, No. 2, pp. 262–268, March–April, 1995.     

Creep buckling of glass-reinforced plastic cyclindrical shells

The buckling modes of cylindrical shells made of a polymeric composite with creep properties are considered. The initial imperfections of the shell are characterized by a Fourier series. The time dependence of a large number of harmonics of the Fourier series is investigated by means of nonlinear equations of the Timoshenko type. It is established that some are damped, while others grow at an ever-increasing rate. It is shown that in the course of time the effective shape of the deflections is transformed and at the critical moment the shell buckles in a mode that differs from the shape of the initial deflections.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 697–703, July–August, 1971.     

Thermal Buckling Behavior of Hybrid-Composite Angle-Ply Laminated Plates with an Inclined Crack

A thermal buckling analysis of symmetric and antisymmetric angle-ply laminated hybrid composite plates with an inclined crack subjected to a uniform temperature rise is presented in this paper. The first-order shear deformation theory in conjunction with the variational energy method is employed in mathematical formulations. The eight-node Lagrange finite-element technique is used for determining the thermal buckling temperatures of hybrid laminates. The effect of crack size and stacking sequences on the temperatures is investigated.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 2, pp. 193–202, March–April, 2002.     

Effect of combined loading on the components of the compliance tensor in shell stability problems

The local strains theory is used to establish relations for determining the components of the compliance tensor in shell buckling with allowance for combined loading of the material at the buckling point and as a function of the precritical loading path in stress space.Mekhanika Polimerov, Vol. 2, No. 3, pp. 387–391, 1966     

Stability of transversely isotropic annular plates with low shear resistance

The stability differential equations of a cylindrically orthotropic circular plate are obtained on the assumption of an axisymmetric buckling mode with allowance for transverse shears. These equations are solved for the case of a transversely isotropic material when the inner and outer edges of the plate are identically loaded by uniformly distributed radial forces. The transcendental equations for the critical load parameter are constructed for various edge conditions. The dependence of this parameter on the boundary conditions and the relative thickness of the plate, Poisson's ratio, and the ratio of the Young's and shear moduli of the material are investigated. Certain conclusions are reached concerning the design of reinforced-plastic plates.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 872–880, September–October, 1969.     

General stability of composite panels reinforced with flexible rods taking account of the side boundary conditions

Reinforced panels are the basic load-bearing elements of various structures. Optimization of massive structures requires consideration of deformation of the panel cross-sections. This is particularly important in determining the bearing strength at buckling. The load scheme, conditions for fixation of the panel cross-section, and bend-torsional stiffness taking account of the deformation of the rod cross-section affect the buckling load in real structures. The stress distribution prior to buckling must be known to solve the buckling problem properly. The stress in the panel is proportional to the active load. The stress distribution is assumed to be known according to our previous method [1]. The load scheme and panel dimensions are shown in Fig. 1. The stress distribution in the panel prior to buckling can be found using Eqs. (1)-(3). A view of the cross-section is given in Fig. 1. The displacements in the panel at buckling for the boundary area are found using Eqs. (4)-(6), while the stresses in the skin and stiffness are found using Eq. (7). Roots k₁ and k₂ are those of the characteristic equation and is a dimensionless coordinate. The problem was solved using variational theory. The potential energy is given by Eqs. (8) and (9) by orihogonalization of Eqs. (5). The basic equations are converted to Eqs. (10) by evaluation of the components in Eqs. (8) and (9). Its calculation (11) gives the compression load. Optimization of parameter gives the critical strength P₁ = 6.93 kN (without taking account of the boundary area) and P₂ = 5.31 kN (taking account of the boundary area).Translated from Mekhanika Kompozitnikh Materialov, Vol. 30, No. 4, pp. 540–546, July–August, 1994.     

Application of the theory of creep stability to polymer materials

The buckling and loss of stability of a rod made from a nonlinear viscous-elastic polymer material are studied with due allowance for instantaneous nonlinear strains. The case of a hinge-fixed rod is considered for various initial deflections.Scientific-Research Institute of Mechanics of the M. V. Lomonosov Moscow State University. Translated from Mekhanika Polimerov, No. 1, pp. 98–104, January–February, 1972.