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。     

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.     

Postbuckling analysis and imperfection sensitivity of general shells by the finite element method

Buckling and imperfection sensitivity are the primary considerations in analysis and design of thin shell structures. The objective here is to develop accurate and efficient capabilities to predict the postbuckling behavior of shells, including imperfection sensitivity. The approach used is based on the Lyapunov–Schmidt–Koiter (LSK) decomposition and asymptotic expansion in conjunction with the finite element method. This LSK formulation for shells is derived and implemented in a finite element code. The method is applied to cylindrical and spherical shells. Cases of linear and nonlinear prebuckling behavior, coincident as well as non-coincident buckling modes, and modal interactions are studied. The results from the asymptotic analysis are compared to exact solutions obtained by numerically tracking the bifurcated equilibrium branches. The accuracy of the LSK asymptotic technique, its range of validity, and its limitations are illustrated.     

Torsional-mode transients in variable-thickness shells of revolution

For thin shells of revolution the existence of torsional-vibration modes, uncoupled from bending and extensional modes, has been established[1]. Here a linear second-order differential equation for the uncoupled torsional stress mode is obtained and its solution for impact loading of shells is sought. The mode-superposition method which utilizes the natural modes of vibration predicted by elementary theory, is, in general, not satisfactory for sharp impact loading as many modes are often required for convergence. Hence we employ two novel techniques for solving the impact problems. Firstly a formal asymptotic procedure, based on extensions to geometrical optics, is employed to generate asymptotic wavefront expansions. Rigorous justifications for this formal technique are provided in an appendix. Secondly a transform technique whereby solutions are sought in terms of Bessel functions is discussed and applied to particular impact loading problems. The Bessel function solutions found here can be used to determine the natural frequencies of the shells. Shells both finite and infinite in extent are discussed and reflections at a stress-free end are examined.     

Mechanical and thermal postbuckling of FGM thick circular cylindrical shells reinforced by FGM stiffener system using higher-order shear deformation theory

The postbuckling of the eccentrically stiffened circular cylindrical shells made of functionally graded materials(FGMs),subjected to the axial compressive load and external uniform pressure and filled inside by the elastic foundations in the thermal environments,is investigated with an analytical method.The shells are reinforced by FGM stringers and rings.The thermal elements of the shells and stiffeners in the fundamental equations are considered.The equilibrium and nonlinear stability equations in terms of the displacement components for the stiffened shells are derived with the third-order shear deformation theory and Leckhniskii smeared stiffener technique.The closed-form expressions for determining the buckling load and postbuckling load-deflection curves are obtained with the Galerkin method.The effects of the stiffeners,the foundations,the material and dimensional parameters,and the pre-existent axial compressive and thermal load are considered.     

Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction

Abstract

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely an overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect, as was done in a previous study by Byskov and Hutchinson, may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.     

Analysis of postbuckling strength of stringer stiffened cylindrical shells under axial compression

An analysis of the postbuckling strength of stringer stiffened cylindrical shells, subject to axial compression is described. The method used in this paper is based on plastic analysis extending Murray’s method which was used to analyse postbuckling behaviour of stiffened plates loaded axially and in bending. The mechanism tripping of stringer and crumpling of shell plate is described based on how the test specimens are deformed after buckling.Finally the theoretical analyses are compared with the experimental results of steel specimens. The theoretical results coincide quite well with the experimental data. It should therefore be possible to use the method described here to analyse postbuckling strength of stringer stiffened cylindrical shells and to estimate energy absorption capabilities in relation to collision studies.     

Describing the elastoplastic deformation of layered shells under axisymmetric loading with allowance for the third invariant of the stress deviator

A method for numerical analysis of the elastoplastic stress–strain state of thin layered shells of revolution under axisymmetric loading is proposed. Constitutive equations describing the elastoplastic deformation of isotropic materials with allowance for the stress mode are used. Numerical results are presented     

Buckling and postbuckling of stiffened cylindrical shells under axial compression

Buckling and postbuckling behaviors of perfect and imperfect,stringer andorthotropically stiffened cylindrical shells have been studied under axial compression.Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref.[1],a theoretical analysis is presented.The effects of material properties of stiffeners andskin,which are made of different materials,on the buckling load and postbuckling behaviorof stiffened cylindrical shells have also been discussed.     

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.     

Nonlinear stability of advanced sandwich cylindrical shells comprising porous functionally graded material and carbon nanotube reinforced composite layers under elevated temperature

The nonlinear stability of sandwich cylindrical shells comprising porous functionally graded material(FGM) and carbon nanotube reinforced composite(CNTRC)layers subjected to uniform temperature rise is investigated. Two sandwich models corresponding to CNTRC and FGM face sheets are proposed. Carbon nanotubes(CNTs) in the CNTRC layer are embedded into a matrix according to functionally graded distributions. The effects of porosity in the FGM and the temperature dependence of properties of all constituent materials are considered. The effective properties of the porous FGM and CNTRC are determined by using the modified and extended versions of a linear mixture rule, respectively. The basic equations governing the stability problem of thin sandwich cylindrical shells are established within the framework of the Donnell shell theory including the von K'arm'an-Donnell nonlinearity. These equations are solved by using the multi-term analytical solutions and the Galerkin method for simply supported shells.The critical buckling temperatures and postbuckling paths are determined through an iteration procedure. The study reveals that the sandwich shell model with a CNTRC core layer and relatively thin porous FGM face sheets can have the best capacity of thermal load carrying. In addition, unlike the cases of mechanical loads, porosities have beneficial effects on the nonlinear stability of sandwich shells under the thermal load. It is suggested that an appropriate combination of advantages of FGM and CNTRC can result in optimal efficiency for advanced sandwich structures.     

加权残值法分析轴压圆柱薄壳后屈曲问题

本文首次应用了样条配点法分析了受到轴向压力的圆柱形薄壳的后屈问题,壳体的方程是L.H.Donnell的非线性正交异性圆柱形壳体方程,壳体的挠度试函数及应力函数试函数都是于轴向用了五次B样条函数基函数,周向用余弦函数。计算模型是周向为半个波长的壳块,可适应后屈曲实验变形跳跃现象。非线性代数方程组用了Newton—Rophson迭代法求解。由此所得的理论上的后屈曲曲线与国外近代实验相符。     

A boundary layer theory for the buckling of thin cylindrical shells under external pressure

Based on the boundary layer theory for the buckling of thin elastic shells suggested in ref. [14]. the buckling and postbuckling behavior of clamped circular cylindrical shells under lateral or hydrostatic pressure is studied applying singular perturbation method by taking deflection as perturbation parameter. The effects of initial geometric imperfection are also considered. Some numerical results for perfect and imperfect cylindrical shells are given. The analytical results obtained are compared with some experimental data in detail, which shows that both are rather coincident.     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

Studies on the torsional buckling of elastic cylindrical shells

The experimental phenomenon and theoretical analysis are given for the torsional buckling of elastic cylindrical shells. From the experiment, it is found that the postbuckling deformation doesn't occupy the whole length when the shell is longer. In the theoretical calculation, only the normal displacement boundary condition is taken into account. By comparing the present calculation results with the accurate result of Yamakis theory and the results of the present experiment, it is shown that the influence of the axial and circumference boundary condition is less important.     

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.     

Thermomechanical postbuckling of composite laminated cylindrical shells with local geometric imperfections

The effect of local geometric imperfections on the buckling and postbuckling of composite laminated cylindrical shells subjected to combined axial compression and uniform temperature loading was investigated. The two cases of compressive postbuckling of initially heated shells and of thermal postbuckling of initially compressed shells are considered. The formulations are based on a boundary layer theory of shell buckling, which includes the effects of the nonlinear prebuckling deformation, the nonlinear large deflection in the postbuckling range and the initial geometric imperfection of the shell. The analysis uses a singular perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of cross-ply laminated cylindrical shells with or without initial local imperfections, from which results for isotropic cylindrical shells follow as a limiting case. Typical results are presented in dimensionless graphical form for different parameters and loading conditions.     

Nonlinear vibration of corrugated shallow shells under uniform load

Based on the large deflection dynamic equations of axisymmetric shallow shells of revolution, the nonlinear forced vibration of a corrugated shallow shell under uniform load is investigated. The nonlinear partial differential equations of shallow shell are reduced to the nonlinear integral-differential equations by the method of Green's function. To solve the integral-differential equations, expansion method is used to obtain Green's function. Then the integral-differential equations are reduced to the form with degenerate core by expanding Green's function as series of characteristic function. Therefore, the integral-differential equations become nonlinear ordinary differential equations with regard to time. The amplitude-frequency response under harmonic force is obtained by considering single mode vibration. As a numerical example, forced vibration phenomena of shallow spherical shells with sinusoidal corrugation are studied. The obtained solutions are available for reference to design of corrugated shells     

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.     

Calculating the thermoelastic stress state of medium-thickness shells of revolution

Analytic and numerical analyses are carried out to ascertain whether the theories of thin and medium-thickness shells can be used to calculate the thermoelastic state of shells of revolution. It is shown that the theory of thin shells should be used in the case of thermal loading and the theory of medium-thickness shells in the case of mechanical loading __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 58–67, May 2008.