Solving axisymmetric dynamic problems for reinforced shells of revolution on an elastic foundation

The dynamic behavior of reinforced shells of revolution in an elastic medium is modeled. Pasternak’s model is used. A problem of vibration of discretely reinforced shells of revolution is formulated and a numerical algorithm is developed to solve it. Results from an analysis of the dynamic behavior of a reinforced spherical shell on an elastic foundation are presented as an example Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 99–106, February 2009.     

Vibrations of a Semiinfinite,Orthotropic, Cylindrical Shell of Open Profile

Natural vibrations localized at the free edge of a semiinfinite, elastic, orthotropic, circular cylindrical shell of open profile are studied. The cylinder is hinged along the bounding generatrices. Dispersion equations are derived from the classical equations describing the dynamic equilibrium for orthotropic cylindrical shells. It is established that these dispersion equations and the dispersion equations for a semiinfinite orthotropic plate strip are in an asymptotic relationship. A procedure for analysis of the possible types of vibrations at the free edge of the cylinder is described. Approximate values of the dimensionless natural frequency and damping factor are determined for shells of different radii     

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.     

On the stability of nearly cylindrical long shells of revolution

In contrast to [1–4], where the stability problem was studied for shells of medium length, in the present paper we study the stability problem for nearly cylindrical long shells under the action of meridian forces uniformly distributed over their ends and under the action of the normal pressure distributed over the entire lateral surface of the shell. We consider the shells whose generating midsurface shape is determined by a parabolic function. The study is performed for nonaxisymmetric buckling modes by using an equation refined as compared with the equation given in [1]. We consider shells of both positive and negative Gaussian curvature. We assume that the shell ends are freely supported and obtain formulas for the critical load under both separate and joint action of the meridian forces and the pressure. In the specific case of a cylindrical shell under the action of longitudinal compression, the formulas thus obtained imply both the Euler formula and the Southwell-Timoshenko formula [5]. When solving the problem, we use the Bubnov-Galerkin method combined with the optimal approximation method [6].     

The axial symmetrical edge problems for thin-walled shells of revolution

In this paper, the uniformly valid asymptotic solutions for the axial symmetrical edge problems of thin-walled shells of revolution in bending are given.     

复合材料旋转壳自由振动分析的新方法

提出了一种半解析区域分解法来分析任意边界条件的复合材料层合旋转壳自由振动. 沿壳体旋转轴线将壳体分解为一些自由的层合壳段, 视位移边界界面为一种特殊的分区界面;采用分区广义变分和最小二乘加权残值法将壳体所有分区界面上的位移协调方程引入到壳体的能量泛函中, 使层合壳的振动分析问题归结为无约束泛函变分问题. 层合壳段位移变量采用Fourier 级数和Chebyshev 多项式展开. 以不同边界条件的层合圆柱壳、圆锥壳及球壳为例, 采用区域分解法分析了其自由振动, 并将计算结果与其他文献值进行了对比. 算例表明, 该方法具有高效率、高精度和收敛性好等优点.     

一类新型受任意载荷的杂交旋转壳单元

本文从弹性力学Reissner变分原理出发推导旋转壳曲线坐标系下内分，位移的二类变量广义变分原理，依据这个原理推导一类旋转壳坐标系中具有独立横向转角的受谐和外载荷下的杂交旋转壳单元，内力模式的选用使刚度矩阵的剪切部份在薄壳情况下能反映Kirchhoff假设，并使单元刚度矩阵满秩，从而保证单元无剪切自锁和零能模式，数例证明这类单元对中厚和薄旋转壳具有良好的通用性和较高的精度。     

Some applications of semi-analytical finite element method in static and dynamic analysis of structures

In this paper a kind of semi-analytical finite element method based on the transfer matrix analysis of shells of revolution is briefly formulated. Some recent investigations on its application are presented: (1) a reanalysis algorithm for improving the accuracy of free vibration analysis; (2) a kind of semi-analytical ring element for the stress analysis of a curved pipe of the slender torus shell type.     

A new kind of mixed elements for shells of revolution subjected to arbitrary loads

In this paper a generalized variational principle with two-field variables is derived from the Reissner principle of elasticity in the curvilinear coordinates of a revolution shell, based on which, a new kind of mixed elements with independent transverse rotations is formulated for revolution shells subjected to harmonic external loads. The resultant-stress interpolations are carefully selected so that the shear part of the element stiffness contains the Kirchhoff hypothesis for thin shells and element stiffness matrices have correct ranks. The elements are free from shear locking and spurious kinematic modes. Numerical examples show that the new elements have good generality and high accuracy for thin and moderately-thick revolution shells.     

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.     

Geometrically nonlinear bending of thin-walled shells and plates under creep-damage conditions

Summary A phenomenological constitutive model for characterization of creep and damage processes in metals is applied to the simulation of mechanical behaviour of thin-walled shells and plates. Basic equations of the shell theory are formulated with geometrical nonlinearities at finite time-dependent deflections of shells and plates in moderate bending. Numerical solutions of initial/boundary-value problems have been obtained for rectangular thin plates (two-dimensional case) and axisymmetrically loaded shells of revolution (one-dimensional case). Based on the numerical examples for the two problems, the influence of geometrical nonlinearities on the creep deformation and damage evolution in shells and plates is discussed. Accepted for publication 30 October 1996     

球壳轴对称弯曲问题共轭二阶挠度微分方程

提出球壳轴对称弯曲问题共轭二阶挠度微分方程并给出了初等函数解. 球壳微分方程是薄壳理论三大壳之一旋转壳的典型方程. 共轭二阶挠度微分方程是球 壳中微分方程形式最简单的, 是人们最喜爱的挠度微分方程. 挠度微分方程满足边 界条件非常简单, 使球壳的计算得到很大的简化.     

Vibrations of a cantilever nonclosed orthotropic cylindrical membrane shell of variable curvature

We study natural vibrations of a cantilever nonclosed orthotropic cylindrical shell with an arbitrary plane directrix. We assume that the shell is hinged on two generators and its bending rigidity is zero (a membrane shell). We find the dispersion and characteristic equations for the characteristics of natural frequencies and damping ratios of the corresponding modal vibrations. Specific computations are performed for shells with directrix being a parabola and with various values of the curvature and length of the generator.     

Maximum stability formulas for reinforced cylindrical shells under external pressure

The problem of determining the reinforcement structure that maximizes the stability of a cylindrical shell subjected to external pressure was formulated in [1], where a numerical solution was obtained for a particular class of structures on the basis of a formula for the stability limit of a hinged anisotropic circular cylindrical shell of medium length in the membrane state. In the present article the stability limit is determined more accurately, without any constraint on the length of the shell, and the optimization is carried out over a broader class of structures.Translated from Zhurnal Prikladnoi Mekhaniki i Teknicheskoi Fiziki, No. 5, pp. 159–167, September–October, 1977.     

Stability of shells of revolution subject to long-term damage

The problem of bifurcational instability of shells of revolution made of homogeneous materials subject to long-term damage is formulated and solved     

Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

A mixed variational principle is derived by Hamilton’s method from the principle of minimum potential energy for thin anisotropic shells of revolution and is then used to derive a normal system of equations with complex coefficients. Discrete orthogonalization is used to solve this homogeneous system and the nonlinear system of equations that describes the precritical state of shells. A shell generated by revolving a circular arc around the axis parallel to its chord is analyzed for stability. The solution is compared with the approximate solution obtained assuming that the precritical state is membrane. It is established that the approximate problem formulation gives incorrect results for shells of negative Gaussian curvature     

Stability of shells of revolution made of a particulate composite with components subject to long-term damage

The problem of bifurcation instability of shells of revolution made of particulate composites with components subject to long-term damage is formulated and solved     

Solution of certain variational problems of thermal resilience for thin shells considering the selection of optimum conditions for localized heat treatment

One of the possible ways of stating and solving the selection problem for optimum temperature fields for localized axisymmetric heating of shells is investigated. The minimum of shell elastic energy is taken as the optimization criterion. An infinite cylindrical shell was considered in a similar formulation in [1], The corresponding variational problem is formulated for the functional of elastic energy with additional limitations imposed on the function of twist angle at specified shell sections. The variational problem is reduced to an isoperimetric by the use of singular functionals of the -function kind. The related Euler equation is obtained, and this together with the problem resolvent equation constitute a complete set of equations for determining the extremum temperature field with related stress-strain state of the shell. Cylindrical, conical, and spherical shells are considered separately. A numerical analysis is made for the simplest conditions of localized heating of cylindrical and conical shells.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 4, pp. 47–54, July–August, 1968.