An exact analytic method applied to nonhomogeneous ring- and stringer-stiffened cylindrical shells

In this paper, the nonlinear axial symmetric deformation problem of nonhomogeneous ring- and stringer-stiffened shells is first solved by the exact analytic method. An analytic expression of displacements and stress resultants is obtained and its convergence is proved. Displacements and stress resultants converge to exact solution uniformly. Finally, it is only necessary to solve a system of linear algebraic equations with two unknowns. Four numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by the exact analytic method.     

General solution for large deflection problems of nonhomogeneous circular plates on elastic foundation

In this paper, a new method is presented based on [1]. It can be used to solve the arbitrary nonlinear system of differential equations with variable coefficients. By this method, the general solution for large deformation of nonhomogeneous circular plates resting on an elastic foundation is derived. The convergence of the solution is proved. Finally, it is only necessary to solve a set of nonlinear algebraic equations with three unknowns. The solution obtained by the present method has large convergence range and the computation is simpler and more rapid than other numerical methods.Numerical examples given at the end of this paper indicate that satisfactory results of stress resullants and displacements can be obtained by the present method. The correctness of the theory in this paper is, confirmed.     

圆柱壳开孔的应力分析

本文研究圆柱壳开孔附近的应力分析问题。由于开孔问题的边界形状较为复杂,使用圆柱壳的 Morley 方程进行求解较为困难。本文在进行误差分析的基础上提出修正 Mo-rley 方程,并给出适合于开孔问题边界的一般解。以圆柱壳受轴向拉伸和内压为例分析了开孔附近的应力分布。给出了开孔率为 1/2以下的开孔解,并对 Donnell 方程解的适用范围给予了新的评价。     

The general solution for axial symmetrical bending of nonhomogeneous circular plates resting on an elastic foundation

In this paper,a new method,the exact analytic method,is presented on the basis of stepreduction method.By this method,the general solution for the bending of nonhomogenouscircular plates and circular plates with a circular hole at the center resting,on an elastfcfoundation is obtained under arbitrary axial symmetrical loads and boundary conditions.The uniform convergence of the solution is proved.This general solution can also be applieddirectly to the bending of circular plates without elastic foundation.Finally,it is onlynecessary to solve a set of binary linear algebraic equation.Numerical examples are givenat the end of this paper which indicate satisfactory results of stress resultants anddisplacements can be obtained by the present method.     

Free vibration of nonlinear deformation ring-and stringer-stiffened nonuniform cylindrical shell

Cylindrical shells stiffened with rings and stringers are used in many structural applications such as pipes conveying fluids or gases and acrospace. In this paper, the general solution is obtained for free vibration of nonlinear deformation ring-and stringer-stiffened cylindrical shell with arbitrary boundary condition by step reduction method^[1]. Finally, it is only necessary to solve a nonlinear algebraic equation. This equation is expressed as an analytic form. Its convergence is proved. Three numerical examples are given at the end of the paper which indicate that satisfactory results can be obtained by step reduction method. Project Supported by the Science and Technic Fund of the National Education Committee.     

On the stress concentration in thick cylindrical shells with an arbitrary cutout

1.IntroductionThefocusofmuch'researcllforalongtimehasbeenontheinvestigationofstressconcentrationsofthincylindricalshellsll--41.Thestressproblemisdiscussedinreferences[5,6].Whentheratioofthicknesshofshellstothecutoutsizeroisnotrathersmall,thegoverningequationofReissuer'sshallowshellincludingtheeffectoftransversesheardeformationsmustbeusedl7-sl.Thethickshelltheorymakesupweaknessesoftheclassicaltheoryofthinshells.Thetheoryofthickshellscanbeusedforanalyzingstressconcentrationsofcircularcylindrica…     

An exact element method for bending of nonhomogeneous thin plates

In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.     

A nonlinear composite shell theory

A general nonlinear theory for the dynamics of elastic anisotropic circular cylindrical shells undergoing small strains and moderate-rotation vibrations is presented. The theory fully accounts for extensionality and geometric nonlinearities by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvatures and strain-displacement expressions that contain the von Karman strains as a special case. Moreover, the linear part of the theory contains, as special cases, most of the classical linear theories when appropriate stress resultants and couples are defined. Parabolic distributions of the transverse shear strains are accounted for by using a third-order theory and hence shear correction factors are not required. Five third-order nonlinear partial differential equations describing the extension, bending, and shear vibrations of shells are obtained using the principle of virtual work and an asymptotic analysis. These equations show that laminated shells display linear elastic and nonlinear geometric couplings among all motions.     

Accurate equations for laminated composite deep thick shells

This paper derives accurate equations of elastic deformation for laminated composite deep, thick shells. The equations include shells with a pre-twist and accurate force and moment resultants which are considerably different than those used for plates. This is due to the fact that the stresses over the thickness of the shell have to be integrated on a trapezoidal-like cross-section of a shell element to obtain the stress resultants. Numerical results are obtained and showed that accurate stress resultants are needed for laminated composite deep thick shells, especially if the curvature is not spherical. A consistent set of equations of motion, energy functionals and boundary conditions are also derived. These may be used in obtaining exact solutions or approximate ones like the Ritz or finite element methods.     

含孔曲板弹性波散射与动应力分析

基于敞口浅柱壳弹性波动方程及摄动方法，对无限大含孔曲板弹性波散射及动应力问题进行了分析研究，将经典薄板弯曲波动问题的分析解作为本问题的主项，给出了在稳态波下孔洞附近散射波的零阶渐近解。建立了求解含孔曲板弹性波散射与动应力问题的边界积分方程法，利用积分方程法可获得问题的近似分析解。并给出了无限大曲板圆孔附近动应力集中系数的数值结果，且对计算结果进行了分析与讨论。     

Axisymmetric problem of a nonhomogeneous elastic layer

Summary The paper deals with a theoretical treatment of elastic behavior for a medium with nonhomogeneous materials property, which is defined by the relation , i.e., shear modulus of elasticity G varies with the dimensionless axial coordinate by the power product form, arbitrarily. Fundamental differential equation for such nonhomogeneous medium has been already proposed in [5]. It is given by a second-order partial differential equation. However, it was found that the fundamental equation is not sufficient in general to solve several kinds of boundary-value problems. On the other hand, it is shown in the present paper making use of the fundamental equations system for a nonhomogeneous medium, which has been proposed in our previous paper [7], it is possible to solve axisymmetric problems for a thick plate (layer) subjected to an arbitrarily distributed load or a concentrated load on its surfaces. Numerical calculations are carried out for several cases, taking into account the variation of the nonhomogeneous parameter m. The numerical results for displacements stress and components are shown in graphical form. Accepted for publication 25 March 1997     

STRESS CONCENTRATIONS IN CYLINDRICAL SHELLS WITH LARGE OPENINGS

Based on Donnell's shallow shell equation, a new method is given in this paper to ana-lyze theoretical solutions of stress concentrations about cylindrical shells with large openings. With themethod of complex variable function, a series of conformal mapping functions are obtained from dif-ferent cutouts' boundary curves in the developed plane of a cylindrical shell to the unit circle. And,the general expressions for the equations of a cylindrical shell, including the solutions of stress concen-trations meeting the boundary conditions of the large openings' edges, are given in the mapping plane.Furthermore, by applying the orthogonal function expansion technique, the problem can be summa-rized into the solution of infinite algebraic equation series. Finally, numerical results are obtained forstress concentration factors at the cutout's edge with various opening's ratios and different loadingconditions. This new method, at the same time, gives a possibility to the research of cylindrical shellswith large non-circular openings or with nozzles.     

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.     

Large-strain constitutive laws for the cylindrical deformation of shells

Expressions are developed for the strain energy and stress resultants in the case of large strain-large bending cylindrical deformations of shells of non-linear materials. The deformation variables are the extension and curvature of the material reference surface. The Mooney material is used as an example where appropriate.The influence of the large deformations on the constitutive laws is evaluated. Of special interest is the effect of cross terms and of the ability to express the stress resultants as partial derivatives of the strain energy with respect to appropriate deformation variables.     

受轴向冲击薄壁圆管的几何畸变相似律研究

对于受轴向冲击载荷作用的薄壁圆管动态响应的相似律问题,由于圆管的薄壁特性导致厚度无法与高度和半径按相同的比例进行结构缩放,从而产生模型的几何畸变,此时传统的相似律已无法描述原型与畸变模型之间的动态响应规律。基于薄壁圆管轴向冲击问题的控制方程,通过能量守恒和量纲分析,推导了考虑几何畸变条件下轴向冲击载荷作用的理想弹塑性薄壁圆管动态响应的相似律。通过在给定应变与应变率区间上建立比例模型预测的流动屈服应力与原型流动屈服应力的最佳逼近关系,将几何畸变相似律进一步推广至包含应变率和应变硬化的材料。通过数值方法验证了提出的几何畸变模型相似律的适用性。分析结果表明,提出的考虑厚度畸变的受轴向冲击薄壁圆管的相似律可用于预测原型结构的冲击动态响应,并显著降低比例模型与原型结构平均载荷和能量的偏差。     

A general nonlinear theory of elastic shells

This paper presents a general nonlinear theory of elastic shells for large deflections and finite strains in reference to a certain natural state. By expanding the displacement components into power series in the coordinate θ³ normal to the undeformed middle surface of shells, the expansions of the Cauchy-Green strain tensors are expressed in terms of these expanded displacement components. Through the modified Hellinger-Reissner variational principle for a three-dimensional elastic continuum, a set of the fundamental shell equations is derived in terms of the expanded Cauchy-Green strain tensors and Kirchhoff stress resultants. The Love-Kirchhoff hypothesis is not assumed and higher order stretching and bending are taken into consideration. For elastic shells of isotropic materials, assuming the strain-energy to be an analytic function of the strain measures, general nonlinear constitutive equations are then derived. Thus, a complete and consistent two-dimensional shell theory incorporating the geometrical and physical nonlinearities is established. The classical theories of shells are directly derivable from the present results by proper truncations of the series.     

Analysis of the stress state of a flattened conical shell of moderate thickness with a structurally weakening hole

Conclusions We determined the relationship between the nature of the stress distribution on the hole surface in a flexible plate as a function of thickness. We observed a great difference between the stress densities in flattened, thin and moderate-thickness conical shells and the stress concentrations near holes in thin cylindrical shells and thin, almost cylindrical, conical shells. The stress distribution near the hole in flattened conical shells of moderate thickness is similar to the stress distribution near the holes in flexible, thick plates. During loading of conical shells by an axial force, the lowest stress concentration factor near the holes is obtained when the axis of the hole is parallel to the shell axis. As the thickness of the shell is increased, the stress concentration factor near the holes increases.Kiev University. Ukrainian Institute of Water Management Engineers, Rovno. Translated from Prikladnaya Mekhanika, Vol. 24, No. 9, pp. 65–70, September, 1988.     

On approximate large strain relations for a shell of revolution

A series of geometric and constitutive relations is studied for large axisymmetric strain of elastic shells of revolution. The kinematic assumption employs a modified Kirchhoff hypothesis which accounts for thickness changes but neglects transverse shear deformation. Calculations are presented for cylindrical and spherical shells composed of incompressible materials with two types of strain energy density function: Mooney-Rivlin (rubber) and exponential (biological tissue). Comparison of results for large bending at a clamped edge demonstrates the accuracy and limitations of various approximations for stress and strain. The computations indicate that the stress resultants are quite sensitive to the details of the asymmetric motion of points relative to the reference surface.     

Asymptotic Analysis of the Eversion of Nonlinearly Elastic Shells

In this paper we study the eversion of axisymmetric, strictly convex, nonlinearly elastic shells within a general, geometrically exact theory in which the shell can suffer flexure, extension, and shear. Each such theory is endowed with a thickness parameter ε². For such shells with free or with fixed hinged edges, we give conditions on ε and the data ensuring that there is an everted state under zero applied load, we show how to approximate it effectively with an asymptotic series in ε whose error we can estimate, we determine the qualitative properties of the everted state, paying particular attention to the formation of a lip near the edge, and we give specific formulas for the shape, the strains, and the stress resultants. This revised version was published online in August 2006 with corrections to the Cover Date.     

Stress and displacement of ring shells under centrifugal force

Using the results of refs. [1] and [2] about the general axial symmetrical problem, this paper calculates the stress and displacement of ring shells under centrifugal force. The solution is given in Fourier series form.In the paper the examples of open ring shells and close ring shells are given respectively.