Buckling and postbuckling behavior of half-loaded shallow spherical shells

Applying the Galerkin procedure to Marguerre's shallow shell equations, reasonably accurate solutions are obtained for the buckling and postbuckling behavior of clamped shallow spherical shells subjected to external pressure distributed over half the surface.For the geometrical parameter λ smaller than 4.5, the half-loaded shells do not buckle. The buckling loads for 6 ≦ λ ≦ 15 lie about 17–30% below those of uniformly loaded shells. However, the lowest loads after buckling are higher than those for uniformly loading. As λ keeps increasing, the buckling mode shows more and more waves along the circumferential direction. The amplitude of the asymmetrical deflection component Δ₁ becomes remarkably large immediately after buckling. After passing the postbuckling lowest equilibrium point, the sinusoidal deflection disappears and the deflection is changed such that only one location on the shell surface, including the non-loaded area, has been deflected almost symmetrically. The effects of initial imperfections are not so great as for the uniformly loaded case.     

Experimental investigation of the buckling of shallow spherical shells

In the present paper, the buckling behavior of clamped thin shallow spherical shells under external pressure is studied. Seventy-nine plastic shells formed by thermovacuum process were tested. The distributions of initial geometrical imperfections and vertical displacements were minutely measured with a differential transformer. It was possible to control the symmetrical initial geometrical imperfection of each specimen.Results indicate that the buckling phenomena of shallow spherical shells vary greatly with the symmetrical initial imperfection parameter η. In the case of the geometrical parameter λ larger than 5.5, the amplitude of the asymmetrical displacement component with the bifurcation buckling wave calculated by Huang becomes large immediately before buckling. The validity of Huang's theory for an initially perfect shell is experimentally demonstrated.     

旋转壳的数值传递函数方法

应用数值传递函数方法建立一种用于分析旋转壳静力、动力响应的截锥壳单元,在本方法中,单元的位移在环向展开为Fourier级数的形式,应用薄壳理论可以得到解耦的微分方程,通过Laplace变换可以将方程转化为频域内的常微分方程,将其表示为状态空间形式后,可以应用数值传递函数方法求解,对复杂的系统可以应用与有限元类似的方法,划分多个单元组合求解,文中给出了几种旋转壳的动力、静力问题的数值算例,并与其它方法进行了比较,表明本文方法具有精度高,计算方便等特点。     

THE SCHRÖDINGER EQUATION OF THIN SHELL THEORIES

This work is the continuation of the discussions of [50] and [51]. In this paper: (A) The Love-Kirchhoff equation of small deflection problem for elastic thin shell with constant curvature are classified as the same several solutions of Schrodinger equation, and we show clearly that its form in axisymmetric problem;(B) For example for the small deflection problem, we extract me general solution of the vibration problem of thin spherical shell with equal thickness by the force in central surface and axisymmetric external field, that this is distinct from ref. [50] in variable. Today the variable is a space-place, and is not time;(C) The von Kármán-Vlasov equation of large deflection problem for shallow shell are classified as the solutions of AKNS equations and in it the one-dimensional problem is classified as the solution of simple Schrodinger equation for eigenvalues problem, and we transform the large deflection of shallow shell from nonlinear problem into soluble linear problem.     

Twist buckling and the foldable cylinder: an exercise in origami

Basic mechanisms for the buckling of a thin cylindrical shell under torsional loading are reviewed from a post-buckling perspective. Deflections are considered so far into the large-deflection range that the shell is allowed to fold to a flat two-dimensional form, in a mechanism reminiscent of a deployable structure. Critical and initial post-buckling effects are explored through concepts of energy minimization and hidden symmetries. For comparisons with the final large-deflection folded shape, a truss element program is employed. It is shown that, as buckling develops, the mode shape must change to accommodate both the symmetry-breaking aspects of the predominately inwards deflection, and the rotation of peak and valley lines of the buckle pattern necessary to accommodate the geometry of the final folded shape.     

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.     

均布载荷作用下带边缘大波纹膜片的非线性弯曲

采用轴对称旋转壳体的简化Reissner方程，研究了在均布载荷作用下具有硬中心的带边缘大波纹膜片的非线性弯曲问题．应用积分方程方法，获得了具有夹紧固定和滑动固定两种外边界的膜片的特征关系，即荷载-中心挠度曲线．作为算例，给出了夹紧固定膜片中的应力分布．     

Nonlinear axisymmetric bending and stability of thin spherical shallow shell with variable thickness under uniformly distributed loads

In this paper, the nonlinear bending and stability of thin spherical shallow shell with variable thickness under uniformly distributed loads are investigated by a new modified iteration method proposed by Prof. Yeh Kai-yuan and the author^[1]. Deflections and critical loads have been calculated and the numerical results obtained have been given in figures and tabular forms. It is shown that the final equation determining the central deflection and the load obtained coincides with the cusp catastrophe manifold. Projects Supported by the Science Fund of the Chinese Academy of Sciences     

Large deflections of deep orthotropic spherical shells under radial concentrated load: asymptotic solution

Nonlinear behavior of deep orthotropic spherical shells under inward radial concentrated load is studied. The singular perturbation method is developed and applied to Reissner’s equations describing axially symmetric large deflections of thin shells of revolution. A small parameter proportional to the ratio of shell thickness to the sphere radius is used. The simple asymptotic formulas describing load–deflection diagrams, maximum bending and membrane stresses of the structure are derived. The influence of boundary conditions on the behavior of the shell by large deflections is considered. Obtained asymptotic solution is in close agreement with the experimental and numerical results and has the same accuracy (in the asymptotic meaning) as the given equations of nonlinear theory of thin shells.     

Effect of Initial Imperfections on the Flexural Eigenvibrations of Cylindrical Shells

The effect of small initial deviations from the ideal circular shape of a shell on the frequencies and modes of flexural eigenvibrations is studied with the use of the linear theory of thin shallow shells. It is assumed that the initial deviations are responsible for interaction between flexural and radial vibrations of the shell. The modal equations are derived by the Bubnov—Galerkin method. It is shown that the initial deviations from the ideal circular shape split the flexural vibration spectrum, and the fundamental frequency decreases compared to that of the ideal shell.     

Finite deformation of an ideal rigid-plastic membrane

We consider finite deformation of a membrane which is a cylinder in the initial state under a pressure uniformly distributed over the inner surface. The problem is solved using the model of deformation of an incompressible rigid-plastic material with full plasticity. Exact analytical relations are obtained for the pressure and kinematic characteristics as functions of the rotation angle of the normal on the contour of the shell. Elongations and displacements are found as functions of the radial coordinate. The time of reaching the maximum value of the external pressure is determined. It is shown that the change in the membrane thickness along the radial coordinate is constant.     

集中载荷作用下具有光滑中心波纹膜片的非线性分析

波纹膜片是一种薄壳弹性体，由于它的参数很多，又相互制约，所以使得它的设计很复杂。在大多数位移式仪器仪表中，要求波纹膜片产生至少和膜片厚度是同样数量级的弹性位移。这就要求使用薄壳的几何非线性理论进行分析。大多数学者研究波纹膜片的弯曲问题，是采用扁壳理论讨论具有浅波纹的膜片。而工程实际中，经常遇到深波纹膜片，这就要求从一般壳体大挠度方程进行求解。本文采用轴对称旋转壳体的简化Re-issner方程。研究了在中心集中载荷作用下具有光滑中心波纹膜片的非线性弯曲问题。应用积分方程方法，可以获得膜片的特征关系（载荷-中心挠度关系）和应力分布。文末给出实例计算的数值结果。     

Stability of cylindrical shells with initial imperfections under the action of external pressure

We use the equations of nonlinear theory of shallow shells to solve the problem of stability of thin elastic isotropic cylindrical shells, with small initial shape imperfections, that are under the action of external uniform pressure. The problem solution is constructed by the Rayleigh-Ritz method with the approximation of the shell midsurface displacement by double functional sums in trigonometric and beam functions. The system of nonlinear algebraic equations is solved by using the methods of continuation with respect to a close-to-best parameter. For the initial imperfections of the shells, we use their normalized deflections from the limit points of overcritical branches of the loading trajectories. We consider various cases of the shell fixation and support under loading by lateral and hydrostatic uniform pressure. We also construct the range of values of the critical pressure, which, with the maximal deviation of the shell shape from the cylindrical shape up to 30%, covers practically all known experimental data.     

含裂纹损伤充液圆柱壳的振动响应求解方法

薄壁圆柱壳流体冲击振动响应是一个复杂的流固耦合(FSI)动力学问题,对于薄壳状态监测与缺陷识别具有重要意义。基于Flügge壳体应力理论,得到壳体运动的高阶偏微分方程组(PDE),利用波传播方法获得圆柱壳系统振动响应。将壳体周围流体定义为理想声学介质,通过亥姆霍兹方程描述声压场,得到流固耦合条件下的薄壁圆柱壳受迫振动响应演变规律。针对薄壳裂纹损伤识别问题,基于断裂力学理论建立局部柔度矩阵,结合呼吸型线弹簧模型(LSM),构造裂纹附近应力及位移连续条件,获得含裂纹损伤充液圆柱壳的振动响应,进而给出一种基于振动能量流的裂纹损伤识别方法。研究结果表明：充液圆柱壳耦合系统在非线性激励下,位移响应在沿轴向、周向和径向的传播特性差异明显;裂纹的存在会导致结构局部柔度的降低和耦合系统固有频率下降;归一化输入功率流能够有效地对充液圆柱壳耦合系统进行结构裂纹损伤识别。研究结果可为充液薄壳振动响应方面的研究提供有益参考,也可为流固耦合条件下的结构裂纹损伤识别方面提供技术支持。     

爆炸加载下装配垫片对金属柱壳膨胀断裂影响研究

采用多普勒光纤探针测速技术（Doppler pins system，DPS，又称全光纤位移干涉测速技术）和高速摄影技术，研究装配垫片对金属柱壳膨胀断裂的影响，获得了有无垫片对应柱壳外表面位置的速度曲线和垫片对柱壳膨胀断裂影响明显的高速摄影图像。实验结果表明:与无垫片区域相比，垫片区域的柱壳外表面经历了先凸起后内凹的过程，导致垫片对应柱壳的径向运动位移发生反复错位，最终低于无垫片区域约0.34 mm，该位移差可能导致柱壳发生径向剪切断裂；实验结果还表明，在垫片与间隙交界处两侧（沿垫片方向约7.5°、沿间隙方向约9°）处各增加了一条裂纹，该断裂模式既不同于环向拉伸断裂，也不同于45°的剪切断裂，而是由垫片/间隙边界产生的两束稀疏应力波传到柱壳外表面引起的扰动影响所致，这个新的断裂模式与柱壳材料的动态力学性能密切相关。数值模拟结果表明，装配垫片对柱壳断裂机制影响不仅包含该处附加的质量效应，还应考虑炸药通过垫片后作用在柱壳上的冲击加载幅值变化、冲击加载时序与其他部位不同步的差异，以及垫片/间隙交界处引起的表面波传播对柱壳断裂模式的后续发展行为的影响。     

绕径旋转横观各向同性圆球壳的变形和应力分析

本文用三维弹性力学理论研究绕径旋转横观各向同性圆球壳的变形和应力分布，给出任意厚度圆球壳的位移和应务的封闭形式解以及数值结果。研究表明球壳上的位移和应力大小与旋转角速度的平方成正比，球面内的正应力远大于径向正应力和剪应力。     

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。     

Use of the Riesz method to calculate axisymmetric vibrations of composite shells of revolution supported by rings and filled with a liquid

We consider the axisymmetric vibrations of a composite structure shaped as a system of thin shells of revolution connected by rings and filled with an ideal incompressible liquid. The structure is divided into independent shell blocks and frame rings. According to the Riesz method, the displacements of each free block treated as a momentless shell are represented as a series in prescribed functions supplemented with local functions of the shell boundary bending.     

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.