General solution for u-shaped bellows overall-bending problems

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Axisymmetry problems of ring shells under arbitrary distributed loads

Extending Novozhilov's transformation, this paper simplifies successfully the axisymmetry problems of ring shells under arbitrary distributed loads, and the equation of the problems in complex form which is similar to Novozhilov's is obtained. The particular solution is obtained. Combining with Chien's general solution of homogeneous equation, this paper gives the general solution for the general symmetry problems of ring shells. Various examples of useful loads and closed ring shells are discussed respectively.     

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.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅰ)-GOVERNING EQUATION AND GENERAL SOLUTION

The overall bending of circular ring shells subjected to bending moments and lateral forces is discussed. The derivation of the equations was based upon the theory of flexible shells generalized by E.L. Axelrad and the assumption of the moderately slender ratio less than 1/3 (i.e., ratio between curvature radius of the meridian and distance from the meridional curvature center to the axis of revolution). The present general solution is an analytical one convergent in the whole domain of the shell and with the necessary integral constants for the boundary value problems. It can be used to calculate the stresses and displacements of the related bellows. The whole work is arranged into four parts: (Ⅰ) Governing equation and general solution; (Ⅱ) Calculation for Omega-shaped bellows; (Ⅲ) Calculation for C-shaped bellows; (Ⅳ) Calculation for U-shaped bellows. This paper is the first part.     

The ring shells under gravitative loads

This paper gives Novozlov’s equation (ref. [1]) in a simple style of ring shells with equal thickness under the actions of gravitative loads. By means of Fourier series, the special solution of the equation is established. Using the results of the homogeneous solution in ref. [2], we find out the general solution of the problem, and derive the expressions of stress and displacement.Two examples are given as the application of the above results.     

A Donnell type theory for finite deflection of stiffened thin conical shells composed of composite materials

A Donnell type theory is developed for finite deflection of closely stiffened truncatedlaminated composite conical shells under arbitrary loads by using the variational calculusand smeared-stiffener theory.The most general bending-stretching coupling and the effectof eccentricity of stiffeners are considered.The equilibrium equations,boundary conditionsand the equation of compatibility are derived.The new equations.of the mixed-type ofstiffened laminated composite conical shells are obtained in terms of the transversedeflection and stress function.The simplified equations are also given for some commonlyencountered cases.     

General solutions of axial symmetrical ring shells

This paper gives the general solutions of axial symmetrical ring shells for all values of slenderness ratio. This solution is newly brought out, and can be used to solve various practical problems, including corrugated tubes, thermal expansion joints, Borden tubes, etc.     

Subspace variational principle of rods and shells

This paper builds the general forms of subspace variational principles of rods andshells which are taken as the controlled equations of the constitutive theories developedfrom the three-dimensional (non-polar) continuum mechanics. And the constitutiveequations of rods and shells using the principles are satisfactory.     

一类偏微分方程组的一般解及其在壳体理论中的应用

本文给出了构造一类偏微分方程组一般解的方法,用这种方法构造的一般解是完备的。最后,利用这个方法构造了壳体中的柱壳平衡方程和锥壳运动方程的一般解。     

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     

The initial postbuckling behavior of ring stiffened shells under hydrostatic pressure

In this article the initial postbuckling behavior of ring stiffened cylindrical shells under hydrostatic pressure is analyzed by Koiter's theory. The nonlinear bending equations consistent with boundary conditions have been used in prebuckling state. The eigenvalue problem is solved by Galerkin's method. The obtained buckling loads are compared with the results which are based on classical stability theory. As calculating examples, three typical outside-stiffened cylinders with different ring stiffener parameters are chosen. The results show that the strength of stiffener not only influences buckling load obviously, but also changes the imperfection-sensitivity of cylindrical shells.     

Equation of axisymmetrical ring shells with variable wall thickness in complex quantity and its general solution

The purpose of this paper is to derive the equation of axisymmetrical ring shells with variable wall thickness in complex quantity and to give its general solution.     

用初参数法解C型波纹管在子午面内整体弯曲

波纹管是一种呈波纹状的管形薄壁结构,作为弹性敏感元件、密封元件和位移补偿器,在现代工业、国防和民用设备中的应用日益广泛.波纹管的计算,特别是在非轴对称受力和变形方面亟待完善.本文讨论了C型波纹管在子午面内整体弯曲的应力分布和端面角位移.从柔性圆环壳在子午面内整体弯曲的复变量方程出发,化边值问题为初值问题,用S.Gill法,得到了该波纹管的数值解,适用于0＜α＜1的范围.通过3个例子,将本文的数值解和已有的细环壳一般解(解析解)作了比较.结果表明,两者基本一致,并且,当环壳较细(α较小)时,两种计算结果更为接近.     

NONLINEAR THEORY OF MULTILAYER SANDWICH SHELLS AND ITS APPLICATION (Ⅱ)──FUNDAMENTAL EQUATIONS FOR ORTHOTROPIC SHALLOW SHELLS

This paper applied the simplified theory for multilayer sandwich shells undergoing moderate/small rotations in Ref. [1] to shallow shells. The equilibrium equations and boundary conditions of large deflection of orthotropic and the special case, isotropic shells, are presented.     

A new displacement-type stability equation and general stability analysis of laminated composite circular conical shells with triangular grid stiffeners

In this paper,based on the theory of Donnell-type shallow shell,a new displacement-type stability equations is first developed for laminated composite circular conical shellswith triangular grid stiffeners by using the variational calculus and generalized smeared-stiffener theory.The most general bending stretching couplings,the effect of eccentricity ofstiffeners are considered.Then,for general stability of composite triangular grid stiffenedconical shells without twist coupling terms,the approximate formulas are obtained forcritical external pressure by using Galerkin‘s procedure.Numerical examples for a certainC/E composite conical shells with inside triangular grid stiffeners are calculated and theresults are in good agreement with the experimental data.Finally,the influence of someparameters on critical external pressure is studied.The stability equations developed andthe formulas for critical external pressure obtained in this paper should be very useful in theastronautical engineering design.     

NONLINEAR THEORY OF MULTILAYER SANDWICH SHELLS AND ITS APPLICATION(Ⅲ)──LARGE DEFLECTION AND POSTBUCKLING OF SHALLOW SHELLS

In this paper, exact solutions of large deflection of multilayer sandwich shallow shells under transverse forces and different boundary conditions are presented. Exact results of postbuckling of multilayer sandwich plates, shallow cylindrical shells and nonlinear deflection of general shallow shells such as spherical shells under inplane edge forces are also obtained by the same procedure.     

The effect of transverse shear deformation on stress concentration factors for shallow shells with a small circular hole

Simplified equations are derived for the analysis of stress concentration for shear-deformable shallow shells with a small hole.General solutions of the equations are obtained,in terms of series,for shallow spherical shells and shallow circular cylindrical shells with asmall circular hole.Approximate explicit solutions and numerical results are obtianed forthe stress concentration factors of shallow circular cylindrical shells with a small hole onwhich uniform pressure is acting.     

Dynamics of the geometrically non-linear analysis of anisotropic laminated cylindrical shells

A general approach, based on shearable shell theory, to predict the influence of geometric non-linearities on the natural frequencies of an elastic anisotropic laminated cylindrical shell incorporating large displacements and rotations is presented in this paper. The effects of shear deformations and rotary inertia are taken into account in the equations of motion. The hybrid finite element approach and shearable shell theory are used to determine the shape function matrix. The analytical solution is divided into two parts. In part one, the displacement functions are obtained by the exact solution of the equilibrium equations of a cylindrical shell based on shearable shell theory instead of the usually used and more arbitrary interpolating polynomials. The mass and linear stiffness matrices are derived by exact analytical integration. In part two, the modal coefficients are obtained, using Green's exact strain-displacement relations, for these displacement functions. The second- and third-order non-linear stiffness matrices are then calculated by precise analytical integration and superimposed on the linear part of equations to establish the non-linear modal equations. Comparison with available results is satisfactorily good.     

GENERAL SOLUTION OF THE OVERALL BENDING OF FLEXIBLE CIRCULAR RING SHELLS WITH MODERATELY SLENDER RATIO AND APPLICATIONS TO THE BELLOWS (Ⅱ)- CALCULATION FOR OMEGA-SHAPED BELLOWS

is one of the applications of (Ⅰ), in which the angular stiffness, the lateral stiffness and the corresponding stress distributions of Omega-shaped bellows were calculated, and the present results were compared with those of the other theories and experiments. It is shown that the non-homogeneous solution of (Ⅰ) can solve the pure bending problem of the bellows by itself, and be more effective than by the theory of slender ring shells; but if a lateral slide of the bellows support exists the non-homogeneous solution will no longer entirely satisfy the boundary conditions of the problem, in this case the homogeneous solution of (Ⅰ) should be included, that is to say, the full solution of (Ⅰ) can meet all the requirements.     

On the different formulations in linear bifurcation analysis of laminated cylindrical shells

The stability pattern of shells is governed by a set of nonlinear partial differential equations. The solution procedure can be simplified, and fast and accurate predictions of the critical buckling load obtained, with the aid of a multilevel approach. Under this approach the lower levels are implemented by means of the perturbation technique, with the nonlinear prebuckling deformation disregarded, and a linear set of equations solved for each state. It turns out, however, that in these circumstances the prediction may differ depending on the chosen formulation. In an attempt to find the reasons for these differences, the linear bifurcation buckling behavior of laminated cylindrical shells was examined via two well-known formulations, with u–v–w and w–F as the unknowns. A third, mixed formulation, was found the most reliable in predicting the buckling behavior.