轴对称载荷下正交异性圆环壳的一般解

本文导出了在Kirchhoff假设下受轴对称载荷时、材料正交异性的圆环壳的复变量方程,给出了该方程的一般解,该解可用于α=a/R<1,其中a为环壳截面的半径,R为环壳的总体半径,文中并举例说明其应用,结果表明本文所提出的解是很有效的。     

The R-function method used to solve nonlinear bending problems for orthotropic shallow shells on an elastic foundation

The paper proposes a method to solve geometrically nonlinear bending problems for thin orthotropic shallow shells and plates interacting with a Winkler–Pasternak foundation under transverse loading. This method is based on Ritz’s variational method and the R-function method. The developed algorithm and software are used to solve a number of test problems and to study complex-shaped shells. The effect of the shape of shells, the boundary conditions, the stiffness of the foundation, and the load distribution on the behavior of isotropic and orthotropic shells undergoing geometrically nonlinear bending is studied     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.     

AN ANALYTICAL SOLUTION TO LARGE DEFLECTION EQUATIONS OF SIMPLY－SUPPORTED RECTANGULAR HYPERBOLOIDAL SHALLOW SHELLS OF ORTHOTROPIC COMPOSITES

ＡＮＡＮＡＬＹＴＩＣＡＬＳＯＬＵＴＩＯＮＴＯＬＡＲＧＥＤＥＦＬＥＣＴＩＯＮＥＱＵＡＴＩＯＮＳＯＦＳＩＭＰＬＹ－ＳＵＰＰＯＲＴＥＤＲＥＣＴＡＮＧＵＬＡＲＨＹＰＥＲＢＯＬＯＩＤＡＬＳＨＡＬＬＯＷＳＨＥＬＬＳＯＦＯＲＴＨＯＴＲＯＰＩＣＣＯＭＰＯＳＩＴＥＳＤ...     

Solving Stress Problems for Elastic Systems of Anisotropic Shells of Revolution with Regard for Transverse Shear and Reduction

An approach is proposed for refined solution of stress problems for elastic systems consisting of coaxial shells of revolution. Transverse shear and reduction are taken into account. Multivariant calculations made for orthotropic cylindrical shells with elliptical end-plates allow us to analyze the influence of the semiaxis ratio and intermediate supports on the stress–strain state of the shell systems under consideration     

Analysis of Stresses and Displacements in Orthotropic Noncircular Cylindrical Shells Under Centrifugal Loads

This paper addresses the class of stress–strain problems for thin orthotropic cylindrical shells of arbitrary cross section under centrifugal loads. Separating out the variables for a simply supported shell yields a system of ordinary differential equations for which the boundary-value problem is solved by a stable numerical method. Study is made of the distribution of displacements in shells of elliptical cross section versus the ratio of ellipse exes and the eccentricity of the axis of revolution relative to the geometrical axis of symmetry     

正交异性扁壳的等价关系式和不变量

提出各向同性扁壳比拟法,分析满足条件D_3=D_(12)=(D_1D_2)~(1/2)的正交异性扁壳大挠度弯曲和超屈曲问题,导出了正交异性扁壳与各向同性扁壳之间,两种不同正交异性扁壳之间坐标变量、扁壳厚度和曲率半径、荷载、挠度、转角、弯矩、扭矩、中面应力的等价关系式,还证明了等价正交异性扁壳的几个等价不变量。     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

Thermal gradient effect on the free vibration of a cardioid cross-section cylindrical shell

For the first time, a new cross-section profile and efficient method are developed for the vibration analysis of isotropic and orthotropic cylindrical shells having circumferentially varying profile of a cardioid cross-section expressed as an arbitrary function, under thermal gradient effect. The governing equations of orthotropic cylindrical shells with varying thermal gradient around its circumference are derived as a boundary-value problem and solved numerically as an initial-value problem, based on the framework of Flügge's shell theory, transfer matrix approach and Romberg integration method. As a semi-analytical procedure, the trigonometric functions are used with Fourier's approach to approximate the solution in the longitudinal direction and also to reduce the two-dimensional problem to one-dimensional one. The thermal gradient is assumed to arise due to the variation of Young's moduli and shear modulus, along the circumferential direction of the shell. The results are obtained to indicate the effects of cardioid cross-section on the natural frequencies and corresponding mode shapes in the thermal environment as well as the sensitivity of the vibration behavior to the thermal gradient ratio and the orthotropy of the shell is also investigated for different types of vibration modes. In general, close agreement between the obtained results and those of other researchers has been found.     

Determination of the Axisymmetric Geometrically Nonlinear Thermoviscoelastoplastic State of Laminated Medium-Thickness Shells

A method is developed to determine the axisymmetric geometrically nonlinear thermoelastoviscoplastic stress–strain state of branched laminated medium-thickness shells of revolution. The method is based on the hypotheses of a rectilinear element for the whole set of layers. The shells are subject to loads that cause a meridional stress state and torsion. They can consist of isotropic layers, which deform beyond the elastic limit, and elastic orthotropic layers. The relations of thermoviscoplastic theory, which describe simple processes of loading, are employed as the equations of state for the isotropic layers. The solution of the problem is reduced to numerical integration of systems of differential equations. The geometrically nonlinear elastoplastic state of a two-layer corrugated shell of medium thickness is calculated as an example     

Spline-approximation solution of two-dimensional problems of statics for orthotropic conical shells in a refined formulation

An approach to solving two-dimensional stress-strain problems for orthotropic conical shells of variable thickness in a refined formulation is developed. The approach is based on the spline-approximation and the stable discrete-orthogonalization method, which is used to solve the one-dimensional problem. The dependence of the deflection and stresses on the thickness and apex angle of a shell of constant volume is given as an example __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 11, pp. 43–54, November 2007.     

On the analyses of vibration and stability for orthotropic circular cylindrical shells conveying fluid

Based on flügge's thin orthotropic shell equations and a linear potential flow theory, buckling instability and flutter-type instability of orthotropic circular cylindrical shells conveying fluid are studied. By means of the expansion in beam-mode functions and Galerkin's method they can be reduced to solving a generalized complex eigenvalue problem. In calculating the generalized forces defined as Eq. (23), a direct numerical integration technique, which has proven to be more efficient than others, is used. For clamped-clamped orthotropic shells several numerical examples show that the effects of axial elastic modulus on the dynamic behaviors are quite different from those of circumferential elastic modulus.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

On the non-classical mathematical models of coupled problems of thermo-elasticity for multi-layer shallow shells with initial imperfections

Mathematical modeling of evolutionary states of non-homogeneous multi-layer shallow shells with orthotropic initial imperfections belongs to one of the most important and necessary steps while constructing numerous technical devices, as well as aviation and ship structural members. In first part of the paper fundamental hypotheses are formulated which allow us to derive Hamilton–Ostrogradsky equations. The latter yield equations governing shell behavior within the applied hypotheses and modified Pelekh–Sheremetev conditions. The aim of second part of the paper is to formulate fundamental hypotheses in order to construct coupled boundary problems of thermo-elasticity which are used in non-classical mathematical models for multi-layer shallow shells with initial imperfections. In addition, a coupled problem for multi-layer shell taking into account a 3D heat transfer equation is formulated. Third part of the paper introduces necessary phase spaces for the second boundary value problem for evolutionary equations, defining the coupled problem of thermo-elasticity for a simply supported shallow shell. The theorem regarding uniqueness of the mentioned boundary value problem is proved. It is also proved that the approximate solution regarding the second boundary value problem defining condition for the thermo-mechanical evolution for rectangular shallow homogeneous and isotropic shells can be found using the Bubnov–Galerkin method.     

Effect of transverse shear and material orthotropy in a cracked spherical cap

The elastostatic problem for a relatively thin-walled spherical cap containing a through crack is considered. The problem is formulated for a specially orthotropic material within the confines of a linearized, shallow shell theory. The theory used is equivalent to Reissner's theory of flat plates and hence permits the consideration of all five physical conditions on the shell boundaries separately. The solution of the problem is reduced to that of a pair of singular integral equations and the asymptotic stress state around the crack tips is investigated. The numerical solution of the problem is given for an isotropic shell and for two specially orthotropic shells. The results indicate that the material orthotropy as well as the shell curvature and thickness may have a considerable effect on the stress intensity factors at the crack tips.     

Asymptotic homogenization modeling of smart composite generally orthotropic grid-reinforced shells: Part I – theory

We develop in this paper a comprehensive micromechanical model for the analysis of thin smart composite grid-reinforced shells with an embedded periodic grid of generally orthotropic cylindrical reinforcements that may also exhibit piezoelectric properties. The original boundary value problem which characterizes the thermopiezoelastic behavior of the smart shell is decoupled via the asymptotic homogenization technique into three simpler problems the solution of which permits the determination of the effective elastic, piezoelectric and thermal expansion coefficients. The general orthotropy of the constituent materials is very important from the practical viewpoint and it renders the resulting analysis a lot more complicated. In Part II of this work the model is applied to the analysis of several practically important examples including cylindrical reinforced smart composite shells and multi-layer smart shells.     

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。     

Stress–strain state of nonthin orthotropic spherical shells of variable thickness

The stress–strain state of an orthotropic spherical shell with thickness varying in two coordinate directions is analyzed. Different boundary conditions are considered, and a refined problem statement is used. A numerical analytic method based on spline-approximation and discrete orthogonalization is developed. The stress–strain state of spherical orthotropic shells with variable thickness is studied     

Stress Analysis of Biconvex Laminated Orthotropic Shells that are Shallow to a Variable Degree

The stress–strain state of biconvex laminated orthotropic shells is analyzed against the degree of shallowness and the parameters of orthotropy. Numerical values of deflections and stresses are obtained by solving two-dimensional boundary-value problems using spline-functions and the discrete-orthogonalization method. The effect of the rise of shells on the displacement and stress fields is analyzed for different parameters of orthotropy