Dynamic stability of nearly cylindrical orthotropic shells of revolution subjected to meridional forces

We study the natural vibrations and the dynamic stability of nearly cylindrical orthotropic shells of revolution subjected to meridional forces uniformly distributed over the shell ends. We consider shells of medium length for which the shape of the midsurface generatrix is described by a parabolic function. Using the theory of shallow shells, we obtain the resolving equation for the vibrations of the corresponding prestressed shell. In the isotropic case, this equation differs from the well-known equation [1] by an additional term, which can be of the same order as the other terms taken into account. We consider shells of both positive and negative Gaussian curvature. We assumed that the shell ends are freely supported. The formulas and universal curves describing the dependence of the minimum frequency, the wave generation shape, and the dynamic instability domain boundaries on the orthotropy parameters, the preliminary stress, the Gaussian curvature, and the amplitude of the shell deviation from the cylinder are given in dimensionless form. We find that in the case of prestresses the orthotropy parameters and the shell deviation from the cylinder (of the order of thickness) can significantly change the least frequencies, the wave generation shape, and the dynamic instability domain boundaries of the corresponding prestressed orthotropic cylindrical shell.In this case, we note that for convex shells under preliminary compression the influence of the elastic parameter in the axial direction is stronger than the influence of the elastic parameter in the circular direction, while the situation is opposite in the case of concave shells. In the case of preliminary extension, the leading role of any orthotropy parameter can vary depending on the value of the preliminary stress and the Gaussian curvature.     

On stability of orthotropic cylindrical shells

In contrast to [1–3], the present paper obtains a system of stability equations and the corresponding resolving equation for orthotropic cylindrical shells of any but very short length in the case where the precritical stress state cannot be treated as the zero-moment state. These equations are a generalization of the results obtained in [4]. On the basis of these equations, one can obtain both the well-known formulas [1–3] and, for medium-length shells, some new expressions of the critical load in longitudinal compression and that under the joint action of torsionalmoments, normal pressure, and longitudinal compression. Some estimates are performed and the determination of the domain of application of some formulas given in [2] and in the present paper is attempted. For an orthotropic shell, a relationship between the elastic parameters and the shear modulus is established for axisymmetric and nonaxisymmetric buckling mode shapes in longitudinal compression.     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

On the theory of loaded general cylindrical Cosserat elastic shells

We study the equilibrium of cylindrical Cosserat elastic shells under the action of body loads and tractions and couples distributed along its edges. The shells have arbitrary open or closed cross-sections and are made from an isotropic and homogeneous material. On the end edges, the appropriate resultant forces and resultant moments are prescribed. We consider the problem of Almansi for cylindrical Cosserat shells and obtain a solution expressed in the form of the displacement field.     

Torsional,flexural, and torsional-flexural buckling modes of a cylindrical shell under combined loading

Stability problems for cylindrical shells under various loading modes were considered in numerous papers. A detailed analysis of such problems can be found, e.g., in the monograph [1]. We refer to the solutions presented in this monograph as classical.For long cylindrical shells in axial compression, one of the buckling modes is the purely beam flexural mode similar to the classical buckling mode of a straight rod. It is well known that it can be studied by using the nonlinear or linearized equations of the membrane theory of shells. In [2], it was shown that, on the basis of such equations constructed starting from the noncontradictory version of geometrically nonlinear elasticity relations in the quadratic approximation [3], under the separate action of the axial compression, external pressure, and torsion, there are also previously unknown nonclassical buckling modes, most of which are shear ones.In the present paper, we show that the use of the above equations for cylindrical shells under compression and external pressure with simultaneous pure torsion or bending permits revealing the earlier unknown torsional, beam flexural, and beam torsional-flexural buckling modes, which are nonclassical, just as those found in [2]. The second of these buckling modes is realized when axially compressing forces are formed in the shell with simultaneous torsion, and the third of them is realized under compression combined with pure bending.It was found that, earlier than the classical buckling modes, the torsional buckling modes can be realized for relatively short shells with small shear rigidity in the tangent plane, while the second and third buckling modes can be realized for relatively long shells.     

Stability of shells of revolution with curvilinear generators in axial compression

Summary Curvilinearity of the generators forces the structure to behave in a qualitatively different way under the action of axial forces. Firstly, up to loss of stability the individual generators of the system are in a state of longitudinal-transverse bending. Secondly for a shell with negative Gaussian curvature there may be a sharp drop in the critical axial compressive loads even in structures that deviate only slightly from the cylindrical.All this means that shells of revolution with curvilinear generators in axial compression cannot be designed for stability using the formulas derived for cylindrical systems.Prikladnaya Mekhanika, Vol. 2, No. 1, pp. 59–68, 1966     

Static and dynamic buckling modes of a cylindrical shell under external pressure

We consider the problem of static and dynamic buckling modes of thin shells under external hydrostatic pressure. If the statement of the problem uses the linearized equations of motion obtained in the moderately large bending theory of shells according to the classical or refined model, then part of terms related to the external load in these equations are assumed to be conservative, and the other terms are assumed to be nonconservative. In this connection, we study four statements of the elastic stability problem for a cylindrical shell with hinged faces. The first of them is the statement of the static boundary value problem in the sense of Euler, where the action of external pressure is assumed to be conservative. The second statement is used to study small vibrations near the static equilibrium by a dynamic method for the same conservative load. The third and fourth statements of the problem correspond to the action of a nonconservative load and are similar to the first and second statements, respectively. They use the linearized equations of equilibrium and motion constructed earlier in a consistent version on the basis of a Timoshenko type model and allowing one to reveal all classical and nonclassical shell buckling modes.     

On the influence of a transverse magnetic field on the vibration spectra of shallow shells

In the design of electric machines, devices, and plasma generator bearing constructions, it is sometimes necessary to study the influence of magnetic fields on the vibration frequency spectra of thin-walled elements. The main equations of magnetoelastic vibrations of plates and shells are given in [1], where the influence of the magnetic field on the fundamental frequencies and vibration shapes is also studied. When studying the higher frequencies and vibration modes of plates and shells, it is very efficient to use Bolotin’s asymptotic method [2–4]. A survey of studies of its applications to problems of elastic system vibrations and stability can be found in [5, 6]. Bolotin’s asymptotic method was used to obtain estimates for the density of natural frequencies of shallow shell vibrations [3] and to study the influence of the membrane stressed state on the distribution of frequencies of cylindrical and spherical shells vibrations [7, 8]. In a similar way, the influence of the longitudinal magnetic field on the distribution of plate and shell vibration frequencies was studied [9, 10]. It was shown that there is a decrease in the vibration frequencies of cylindrical shells under the action of a longitudinal magnetic field, and the accumulation point of the natural frequencies moves towards the region of lower frequencies [10]. In the present paper, we study the influence of a transverse magnetic field on the distribution of natural frequencies of shallow cylindrical and spherical shells, obtain asymptotic estimates for the density of natural frequencies of shell vibrations, and compare the obtained results with the empirical numerical results.     

Flow around a flexible cylindrical shell by a plane stream of an ideal liquid

The flow by a plane stream of an ideal liquid around a cylindrical shell of zero flexural stiffness (a soft cylindrical shell), or a gas bubble on the boundary of which forces of tension act, was studied in [1–6]. The flow around an elastic plate in a linear formulation was considered in [7, 8]. We consider the flow, around a flexible cylindrical shell which possesses a flexural stiffness and at the same time admits large displacements, by a plane system of an ideal incompressible liquid. An application of methods of the theory of functions of a complex variable leads to an effective solution of the problem. The shape of the shell, the forces in it, the forces acting on the shell, and the field of velocities of the flow of the liquid are determined.     

On stability of cylindrical shells of variable thickness with initial imperfections

The paper outlines a numerical method for stability analysis of cylindrical shells with initial imperfections. We solve a nonlinear buckling problem for a cylindrical shell with variable wall thickness under surface pressure. The imperfections of the shell are modeled as the first buckling mode. A probabilistic approach is used to determine the reliability against buckling of the cylindrical shell with the probability density of initial imperfections represented by uniform distribution, triangular distribution, or Gaussian distribution     

New matrix method for response analysis of circumferentially stiffened non-circular cylindrical shells under harmonic pressure

Based on the governing equation of vibration of a kind of cylindrical shells written in a matrix differential equation of the first order,a new matrix method is pre- sented for steady-state vibration analysis of a noncircular cylindrical shell simply sup- ported at two ends and circumferentially stiffened by rings under harmonic pressure.Its difference from the existing works by Yamada and Irie is that the matrix differential equation is solved by using the extended homogeneous capacity precision integration ap- proach other than the Runge-Kutta-Gill integration method.The transfer matrix can easily be determined by a high precision integration scheme.In addition,besides the normal interacting forces,which were commonly adopted by researchers earlier,the tan- gential interacting forces between the cylindrical shell and the rings are considered at the same time by means of the Dirac-δfunction.The effects of the exciting frequencies on displacements and stresses responses have been investigated.Numerical results show that the proposed method is more efficient than the aforementioned method.     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressures

This paper treats the classical problem of radial motions of cylindrical and spherical shells under pulsating pressures. The novelty in this work is that the shells are taken to be non-linearly viscoelastic (of strain-rate type). It is remarkable that this classical problem, which does not treat the loss of stability to non-radial motions (but which is essential for such treatments), has such a rich dynamics due to the often neglected effects of non-linear material response, to the role of prestress under the action of the mean pressure, and to the different effects of pressure on cylindrical and spherical shells. The study of radial motions near primary resonance (when the frequency of the pulsating pressure is near the natural frequency about an equilibrium state under a constant pressure) gives formulas ensuring that the motions are of hardening or softening type depending on the constitutive functions and whether the constant mean pressure is compressive or inflational. The method of multiple scales gives asymptotic formulas for the principal parametric instability regions (Mathieu tongues) and for the stable and unstable motions at twice the forcing frequency, which closely agree with those obtained by numerical continuation methods. The dependence of frequency on amplitude and the form of instability regions are critically influenced by deviations (even very slight deviations) of material response from that of linearly viscoelastic shells, by the constant mean pressure, and by the type of shell. This paper exhibits the rich diversity of postcritical periodic motions.     

Thermal stresses in cylindrical Cosserat elastic shells

We consider the problem of thermal stresses in cylindrical elastic shells, modelled as Cosserat surfaces. In the theory of Cosserat shells, the thermal effects are described generally by means of two temperature fields. The problem consists in finding the equilibrium of the shell under the action of a given temperature distribution. The temperature fields are assumed to be general polynomial functions in the axial coordinate, whose coefficients depend on the circumferential coordinate.     

圆柱薄壳稳定性的一个修正理论

著名的唐乃尔(Donnell)——穆什塔利的简化壳体理论只能较精确地适用于较短圆柱壳稳定性计算.其近似性误差随长度与半径之比的增加而增大.本文考虑了横向切力的影响,对非完善型圆柱壳体推导了几何非线性理论的基本方程,建立了对各种长度半径比的圆柱壳体稳定性均适用的修正理论.     

Some problems of optimizing shell shape and thickness distribution on the basis of a genetic algorithm

We consider problems related to designing axisymmetric shells of minimal weight (mass) and the development of efficient nonlocal optimization methods. The optimization problems under study consist in simultaneous search for the optimal geometry and the shell thickness optimal distribution from the minimal weight condition under strength constraints and additional geometric constraints imposed on the thickness function, the transverse cross-section radii distribution, and the volume enclosed by the shell. Using the method of penalty functions, we reduce the above optimal design problem to a nonconvex minimization problem for the extended Lagrange functional. To find the global optimum, we apply an efficient genetic algorithm. We present the results of numerical solution of the optimal design problem for dome-like shells of revolution under the action of gravity forces. We present some data characterizing the convergence of the method developed here.     

Buckling of thick cylindrical shells under external pressure: A new analytical expression for the critical load and comparison with elasticity solutions

In this paper a set of stability equations for thick cylindrical shells is derived and solved analytically. The set is obtained by integration of the differential stability equations across the thickness of the shell. The effects of transverse shear and the non-linear variation of the stresses and displacements are accounted for with the aid of the higher order shell theory proposed by [Voyiadjis, G.Z. and Shi, G., 1991, A refined two-dimensional theory for thick cylindrical shells, International Journal of Solids and Structures, 27(3), 261–282.]. For a thick shell under external hydrostatic pressure, the stability equations are solved analytically and yield an improved expression for the buckling load. Reference solutions are also obtained by solving numerically the differential stability equations. Both the full set that contains strains and rotations as well as the simplified set that contains rotations only were solved numerically. The relative magnitude of shear strain and rotation was examined and the effect of thickness was quantified. Differences between the benchmark solutions and the analytic expressions based on the refined theory and the classical shell theory are analysed and discussed. It is shown that the new analytic expression provides significantly improved predictions compared to the formula based on thin shell theory.     

局部轴压下圆柱壳的失稳分析

本文用传递矩阵法在初始缺陷法的概念下对圆柱壳的一端有四个类似集中力的局部轴压作用下的失稳问题进行了研究。     

可渗透圆柱壳流固耦合问题的流场与内力分析

在弹性薄壳的非线性理论和流体力学基本方程的基础上,研究了可渗透圆柱壳的流固耦合问题.假定壳体具有均布孔隙且孔的面积很小,不考虑其阻力,忽略对弯曲刚度和壳体腔内流体微小运动影响,应用相容欧拉--拉格朗日法建立了带孔的圆柱壳在流体中相互作用的基本方程.通过具体算例求解,给出了流场速度与压力的变化、圆柱壳的变形及内力分布,并对相关参数进行了讨论.     

Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related