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.     

Deformation of cylindrical composite shells under combined dynamic loading

Conclusions A procedure has been shown for calculating the stress-strain state of cylindrical multilayer shells made from composite materials under the combined action of dynamic axial compression and dynamic external pressure, as well as with different variants of combined loading with static and dynamic forces. An investigation has been made of the effect on the mode of the buckled shell surface of the ratio of the application rate of dynamic loads; ranges of loading rates have been established in which stresses predominate caused either by axial compression or external pressure. It has been shown that, as a result of preliminary static loading, a marked change occurs in the initial imperfections of the shell mode which affects subsequent dynamic buckling. To calculate the time when the first defect occurs and its location in the shell body, a procedure has been devised for layer-by-layer strength analysis employing a tensor-polynomial criterion. It was demonstrated that the level of preliminary static loading noticeably affects the time until the first failure of the layer, not only a reduction of this time being possible with an increase in the static loads, but also an increase in it.We should also point out the work in [10] where it is shown that it is possible to weaken the susceptibility of the shell to initial imperfections when internal pressure is applied.Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 461–473, May–June, 1981.     

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.     

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.     

Strength of cylindrical shells of laminar composites under dynamic loading

Conclusion The methods described in the present review paper for the computation of cylindrical shells of laminar composites under various types of dynamic loads make it possible to calculate the stress-strain state at any point of the body of the shell at an arbitrary time and to determine the time of the first failure of a layer (including fatigue failure) using strength criterion of an anisotropic body for the case of a plane stressed state, and also the location of the first flaw in the shell. This makes it possible to determine the magnitude of the applied pulse (the amplitude of a pulsed loading), or the amplitude of a vibratory loading for which the first centers of failure appear in the design.Presented at the Second Soviet-American Symposium on the Problem Failure of Composite Materials (Bethlehem, Pa., USA, March, 1981).Translated from Mekhanika Kompozitnykh Materialov, No. 3, pp. 460–467, May–June, 1982.     

Long-time stability of orthotropic cylindrical shells under external pressure

The problem is solved using a refined theory of shells that takes shear strains into account. The shell deformations are described by means of the relations for an orthotropic material, it being assumed that creep strains develop only as a result of shear forces. The geometrically linear problem is considered. For the sake of comparison, the long-time critical load is calculated on a Minsk-22 computer using the Kirchhoff-Love and refined models. It is shown that when shears are taken into account, in certain cases the critical load may be reduced by 30%.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 5, No. 2, pp. 315–320, March-April, 1969.     

The effect of a reinforcing ring on the stress-strain state of flexible cylindrical shells

We investigate the effect of a reinforcing ring on the stress-strain state of a cylindrical shell in the geometrically nonlinear problem with a nonaxisymmetric load on the edge. The nonlinear boundary-value problem is reduced to a sequence of linear problems by the quasilinearization method. The linear problems are solved by the discrete orthogonalization method. The results obtained using linear and nonlinear theory are compared.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 55, pp. 92–96, 1985.     

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.     

Minimizing the mass of cylindrical shells formed from a composite material with an elastic filler and designed for strength and stability under a combined loading

The mass of a multilayer cylindrical shell, formed from a composite material with an elastic filler and designed for strength and stability under the combined action of axial compression and external pressure, is minimized. The problem is formulated as one of nonlinear programming and is solved by Rossen's method of projection gradients. The strength of the material is established from analysis of the strength of the layers making up the entire bundle. Failure of an individual layer is determined from Malmeister's criterion. The structure of a shell with different external loads and the dependence of minimal mass on the stiffness of the filler and on the volume coefficient of reinforcement are investigated in numerous examples.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. K. Preikshas Shyaulyaisk Pedagogical Institute. Translated from Mekhanika Polimerov, No. 2, pp. 289–297, March–April, 1976.     

Stability loss of reinforced cylindrical shells under isotropic external pressure

Conclusions 1. Upon the loading of a composite shell having a metallic matrix the shear effects are insignificant even in the case of comparatively large volume reinforcement contents and wall thicknesses.2. The principal modulus of a material which determines the stability of a reinforced shell upon isotropic external pressure is the secant modulus in the circumferential direction.3. In the case of complex reinforcement schemes some decrease in the stability of the shell is possible, probably due to an imperfection in reinforcement technology.Institute of Solid State Physics, Academy of Sciences of the USSR, Moscow Region. Translated from Mekhanika Polimerov, No. 1, pp. 90–95, January–February, 1977.     

Static and dynamic beam forms of the loss of stability of a long orthotropic cylindrical shell under external pressure

A cylindrical shell with end sections which are closed and supported by hinges, in accordance with the concepts of the rod theory, is considered to be under the action of an omnidirectional external pressure which remains normal to the lateral surface during the deformation process. It is shown that, for such shells, the previously constructed consistent equations of the momentless theory, reduced using the Timoshenko shear model to the one-dimensional equations of the rod theory, describe three forms of loss of stability: (1) static loss of stability, which occures through a bending mode from the action of the total end axial compression force since, under the clamping conditions considered, its non-conservative part cannot perform work on deflections of the axial line; (2) also a static loss of stability but one which occurs through a purely shear mode with the conversion of a cylinder with normal sections into a cylinder with parallel sloping sections and a corresponding critical load which is independent of the length of the shell; (3) dynamic loss of stability which occurs through a bending-shear form and can only be revealed by a dynamic method using an improved shear model.     

On stability of axisymmetric forced vibration of imperfect cylindrical shells

This paper presents the analysis of forced vibrations of a cylindrical shell with axisymmetric initial imperfection subjected to hydrodynamic pulsating pressure. The stability of steady state harmonic response is studied with respect to both axisymmetric and asymmetric perturbations. The analysis of stability is based on the nonlinear von Kármán-Donnell equations linearized with respect to perturbations. An interesting conclusion of this paper is that although the axisymmetric steady state motion is linear it can be unstable due to asymmetric perturbations.

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Zusammenfassung Die vorliegende Arbeit behandelt erzwungene Schwingungen zylindrischer Schallen mit axialsymmetrischen Imperfektionen, welche einem hydrostatischen pulsierenden Druck ausgesetzt sind. Die Stabilität in Bezug auf axialsymmetrische und nicht-axialsymmetrische Störungen wird untersucht. Diese Untersuchung basiert auf die nicht-linearen von Kármán-Donnell Gleichungen, welche für kleine Störungen linearisiert worden sind. Eine interessante Schlufolgerung dieser Arbeit ist, daß, obwohl die axialsymmetrische stationäry Bewegung linear ist, sie wegen asymmetrischen Storungen instabil werden kann.

     

Nonaxisymmetric bulging of cylindrical shells made from polymeric material on prolonged loading

Forms of bulging of cylindrical shells prepared from polymeric materials caused by the action of a constant compressing load in the direction of the generatrix are discussed. The functions of the initial irregularities and the additional radial deflection and stresses are characterized by the Fourier series in a complex representation. When the problem in a geometrically and physically nonlinear formulation was solved, it was found that the form of the deformed surface is transformed, and the bulging of the shell at the critical instant takes place asymmetrically. This agrees with the experimental results.