正弦波纹扁球壳的非线性强迫振动

波纹壳是传感器弹性元件的一类重要形式,也是精密仪器仪表弹性元件中的一类重要形式。由于波纹壳形状复杂、参数众多、厚度薄,对其进行非线性分析非常重要同时也是十分困难的。本文考虑一种在传感器弹性元件中有重要应用价值的正弦波纹浅球壳体,将这种壳体视为结构上的圆柱正交异性扁球壳,根据Andryewa的思想,分别得到了正弦波纹壳径向、环向在拉伸、弯曲下的等价的四个各向异性参数;建立了正弦波纹扁球壳的非线性强迫振动微分方程;得到了正弦波纹扁球壳非线性强迫振动的共振周期解及幅频特性曲线。     

Nonlinear vibrations of cylindrical shells with initial imperfections in a supersonic flow

The paper studies the dynamics of nonlinear elastic cylindrical shells using the theory of shallow shells. The aerodynamic pressure on the shell in a supersonic flow is found using piston theory. The effect of the flow and initial deflections on the vibrations of the shell is analyzed in the flutter range. The normal modes of both perfect shells in a flow and shells with initial imperfections are studied. In the latter case, the trajectories of normal modes in the configuration space are nearly rectilinear, only one mode determined by the initial imperfections being stable __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 9, pp. 63–73, September 2007.     

Thermoelastic stability of bimetallic shallow shells of revolution

This article considers the thermoelastic stability of bimetallic shallow shells of revolution. Basic equations are derived from Reissner’s non-linear theory of shells by assuming that deformations and rotations are small and that materials are linear elastic. The equations are further specialized for the case of a closed spherical cup. For this case the perturbated initial state is considered and it is shown that only in the cases when the cup edge is free or simply supported buckling under heating is possible. Further the perturbated flat state is considered and the critical temperature for buckling is calculated for the case of free and simply supported edges. The temperature–deflection diagrams are calculated by the use of the collocation method for shallow spherical, conical and cubic shells.     

Nonlinear strain gradient elastic thin shallow shells

The governing equilibrium equations for strain gradient elastic thin shallow shells are derived, considering nonlinear strains and linear constitutive strain gradient elastic relations. Adopting Kirchhoff’s theory of thin shallow structures, the equilibrium equations, along with the boundary conditions, are formulated through a variational procedure. It turns out that new terms are introduced, indicating the importance of the cross-section area in bending of thin plates. Those terms are missing from the existing strain gradient shallow thin shell theories. Those terms highly increase the stiffness of the structures. When the curvature of the shallow shell becomes zero, the governing equilibrium for the plates is derived.     

THE CREEP EFFECTS ON THE BUCKLING OF SHALLOW SHELLS

An analysis of the creep effects on the buckling of concreteshallow shells is presented in this paper.Based on the non-linear theory of thin elastic shells.it is found that forthose of the elliptical paraboloidal type.the upper cri-tical load of the load-deflection curve will decrease whilethe lower limit Will increase with time.As to the problemof local buckling of the shell.the critical load is de-pendent only upon the modulus of elasticity at the instantit occurs.     

Nonlinear stability of functionally graded material (FGM) sandwich cylindrical shells reinforced by FGM stiffeners in thermal environment

In this paper, Donnell's shell theory and smeared stiffeners technique are improved to analyze the postbuckling and buckling behaviors of circular cylindrical shells of stiffened thin functionally graded material(FGM) sandwich under an axial loading on elastic foundations, and the shells are considered in a thermal environment. The shells are stiffened by FGM rings and stringers. A general sigmoid law and a general power law are proposed. Thermal elements of the shells and reinforcement stiffeners are considered. Explicit expressions to find critical loads and postbuckling load-deflection curves are obtained by applying the Galerkin method and choosing the three-term approximate solution of deflection. Numerical results show various effects of temperature, elastic foundation, stiffeners, material and geometrical properties, and the ratio between face sheet thickness and total thickness on the nonlinear behavior of shells.     

Thermoelastic stability of layered shallow shells

The phenomena associated with thermal snap-through and snap-buckling of symmetrically layered shallow shells of polygonal planform are studied by means of a two-degree-of-freedom model derived from a Ritz–Galerkin approximation. The composite structure is homogenized considering perfect bond and the kinematic assumptions of the first order shear deformation theory. The simply supported shell edges are assumed to be prevented from in-plane motions. The geometrically non-linear, quasi-static equilibrium conditions are derived according to the von Kármán–Tsien theory and simplified by the Berger-approximation. A unifying non-dimensional formulation of the elastic stability analysis is presented that turns out to be independent of the special polygonal planform of the simply supported shallow shell.     

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.     

Dynamic modeling thin skeletonal shallow shells as 2D structures with nonuniform microstructures

The subject of this consideration is a thin skeletonal elastic shallow shell with an orthogonal beam-grid microstructures. The important feature of the considered shells is that a dimension of the microstructure is of an order of the shell thickness. The formulation of 2D-macroscopic mathematical model of these shells, based on a tolerance averaging approximation (Wo?niak et?al., 2008), is the aim of the paper. During the modeling procedure, the shell under consideration is treated as a structure with a nonuniform microstructure. The general results of the contribution will be illustrated by the analysis of natural vibrations of a cylindrical thin skeletonal shallow shell.     

波纹壳的摄动解法

应用扁锥壳的非线性大挠度理论,研究了在均布载荷和中心集中载荷下,具有光滑中心的锯齿形和梯形波纹壳,采用摄动法和幂级数方法,得到了波纹壳的弹性特征。本文的解答符合实验结果。     

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.     

Vibrations of shallow shells that are circular in plan and have local supports

The paper deals with vibrations of doubly curved shallow shells that are circular in plan and are reinforced by local rod-type supporting elements or cylindrical shells. The coefficients of the frequency equation are found by using numerical-analytical methods to solve boundary-value problems for fixed values of frequency. The natural frequencies and modes of vibration of a system composed of a shell and elastic supports are determined in the course of solving the problem. It is shown that it is possible to also account for reactive moments and shearing forces, in addition to the normal reactions of an elastic support. The potential of the approach which is developed is illustrated by the solution of specific problems. Special Design Office of the S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Prikladnaya Mekhanika, Vol. 35, No. 3, pp. 69–75, March, 1999.     

双曲率壳的力学研究进展

双曲率壳是飞机、汽车以及船舶上常见的薄壁结构，其中性面可看作是一条动曲线沿着另一条曲线扫掠所形成的曲面．双曲率壳的非线性理论不断更新推动着双曲率壳力学行为的研究．随着工程实际应用的不断改进，如功能梯度材料(FGM)，加筋壳，弹性地基模型等的引入，双曲率壳在强度、变形和稳定性等方面的研究得到了进一步促进．本文首先回顾了双曲率壳结构非线性力学基本理论发展过程，主要阐述了经典的二维板壳理论，如Donnell 薄壳理论，一阶剪切变形壳理论，高阶剪切变形壳理论，和三维板壳理论的理论体系及基本公式，并对几种理论之间的联系和应用进行了总结和讨论，简述了近几十年来国内外学者在双曲率壳非线性弯曲、稳定性和振动等方面的最新研究成果，最后对双曲率壳体研究目前的局限性和未来的研究方向进行了探讨．     

Nonlinear Bending Theory of Diagonal Square Pyramid Reticulated Shallow Shells

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Nonlinear axisymmetric thermomechanical response of piezo-FGM shallow spherical shells

Thermomechanical instability of shallow spherical shells made of functionally graded material (FGM) and surface-bonded piezoelectric actuators is studied in this paper. The governing equations are based on the classical shell theory of shells and the Sanders nonlinear kinematics equations. It is assumed that the property of the FGMs varies continuously through the thickness of the shell according to a power law distribution of the volume fraction of the constituent materials. The constituent materials of the functionally graded shell are assumed to be mixture of ceramic and metal. The analytical solutions are obtained for uniform external pressure, thermal loading, and constant applied actuator voltage.     

On the theory of elastic shells made from a material with voids

In this paper we present a theory for porous elastic shells using the model of Cosserat surfaces. We employ the Nunziato–Cowin theory of elastic materials with voids and introduce two scalar fields to describe the porosity of the shell: one field characterizes the volume fraction variations along the middle surface, while the other accounts for the changes in volume fraction along the shell thickness. Starting from the basic principles, we first deduce the equations of the nonlinear theory of Cosserat shells with voids. Then, in the context of the linear theory, we prove the uniqueness of solution for the boundary initial value problem. In the case of an isotropic and homogeneous material, we determine the constitutive coefficients for Cosserat shells, by comparison with the results derived from the three-dimensional theory of elastic media with voids. To this aim, we solve two elastostatic problems concerning rectangular plates with voids: the pure bending problem and the extensional deformation under hydrostatic pressure.     

大跨度双层网壳的非线性动态响应

网壳结构在大跨度结构中得到广泛应用．在建立了双层网格扁球壳的非线性强迫振动微分方程的基础上,研究了在边缘滑动固定的边界条件下,双层网格扁球壳的非线性动态响应问题．用突变理论建立了该网壳的尖点突变模型,得出了突变的临界方程,并阐述了网壳参数对该结构动态屈曲的影响．     

球形厚壳的应力分析

文中将球形厚壳视为三维弹性体，按三维弹性理论问题求解其应力及位移。     

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.     

Green函数法在轴对称板壳理论中的应用

本文从轴对称板壳理论的基本方程出发,通过建立Green函数,导出了轴对称线载荷下解的一般表述式,由此可以求出任意轴对称载荷下的解,然后本文分别讨论了圆板和扁球壳受线载问题的解,文中的结果适用于各种边界条件。