Elastic buckling of an axially compressed cylindrical panel with three edges simply supported and one edge free

The paper is devoted to a circular cylindrical panel with three edges simply supported and one edge free. This panel is axially compressed by a load uniformly distributed at both ends. The Donnell equations for linear buckling of shells are assumed and the adequate boundary conditions are defined. The deflection function and the force function–the Airy function are formulated with respect to the boundary conditions. The Donnell equations are reduced to a generalized eigenvalue problem with the use of Galerkin method. The results of numerical investigation are presented in figures.     

Large displacement stability analysis of elastic–plastic unsymmetrical sandwich cylindrical shells

The paper presents stability analysis of elastic–plastic, free supported sandwich cylindrical shell with unsymmetrical faces, loaded by longitudinal forces, transversal pressure, and shear. The J₂ incremental Prandtl–Reuss plastic flow theory constitutive relations were used in the analysis and it was assumed that geometrical strain-displacement relations are nonlinear ones. It was also assumed that the shell faces have different thicknesses and these are made of different isotropic compressible materials with linear stress-hardening. Active loading processes are accepted in the analysis, the stability equations are derived using strain energy formulation. Ritz method is used to solve the equations and an iterative computational algorithm was elaborated to get numerical results.     

网格圆柱扁壳的稳定性

本文用将网格壳转化为连续壳的方法建立了网格圆柱扁亮屋盖的稳定平衡微分方程,所得的方程是属于异性圆柱扁壳的方程,并用方法求得简支边界条件的网格圆柱扁壳的临界荷载计算公式。     

Shallow cylindrical tuned liquid damper for vibration control of high‐rise structures

A new simplified model for the application of a shallow cylindrical tuned liquid damper (TLD) in structural vibration control is presented in this paper. The dynamic properties of a shallow liquid in cylindrical containers subjected to forced horizontal oscillation are analysed directly from the continuity and momentum equations of fluids. Following some practical assumptions, the nonlinear partial differential equations describing the wave movement of a shallow liquid in cylindrical containers is established and a numerical procedure for the solution of these equations is proposed using the finite element method. The formula for determining the control force provided by the shallow cylindrical TLD is presented and the effect of several parameters on the control efficiency of a shallow cylindrical TLD controlled structure under wind action is investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

利用Hamilton函数分析圆柱壳动态扭转屈曲

在Hamilton函数中考虑应力波的影响,研究冲击扭转载荷作用下弹性圆柱壳的动态屈曲。采用辛方法将基本方程转化为对偶变量的Hamilton典型方程。在辛空间将临界扭转和屈曲模式分别简化为求解辛本征值和本征解问题。主要影响因素有冲击时间、边界条件和厚度等,通过数值算例对这些因素进行了详细讨论。研究结果表明,边界条件的影响有限(自由边界条件除外)。在壳体自由端能够观察到局部动态屈曲模式。新的分析和数值模拟结果可作为壳结构的安全设计准则。     

Natural vibrations of loaded noncircular cylindrical shells containing a quiescent fluid

The paper deals with a solution of three-dimensional problems of natural vibrations and stability of loaded cylindrical shells with circular and arbitrary cross sections containing a quiescent ideal compressible fluid. A mathematical formulation of the problem has been developed based on the variational principle of virtual displacements taking into account the pre-stressed undeformed state caused by the action of static forces on the shell. The motion of potential compressible non-viscous fluid is described by a wave equation, which is transformed using the Bubnov–Galerkin method. The solution of the problem reduces to the computation of complex eigenvalues of a coupled system of two equations. Based on the developed finite element algorithm several numerical examples have been considered to analyze the influence of fluid levels, ratio of ellipse semi-axes, shell thickness and boundary conditions on the natural frequencies and vibration modes of circular and elliptical cylindrical shells loaded by mechanical forces. It has been found that the value of the external uniformly distributed pressure giving rise to instability does not depend on the level of fluid in the shell. The results allow us to conclude that the dynamic characteristics of the system are specified not only by the equivalent added mass of the fluid but also by hydroelastic interaction at the wetted surface.     

Dynamic torsional buckling of cylindrical shells in Hamiltonian system

By considering the effect of stress waves in a Hamiltonian system, this paper treats dynamic buckling of an elastic cylindrical shell which is subjected to an impact torsional load. A symplectic analytical approach is employed to convert the fundamental equations to the Hamiltonian canonical equations in dual variables. In a symplectic space, the critical torsion and buckling mode are reduced to solving the symplectic eigenvalue and eigensolution, respectively. The primary influence factors, such as the impact time, boundary conditions and thickness, are discussed in detail through some numerical examples. It is found that boundary conditions have limited influence except free boundary condition in the context of the scope in this paper. The localization of dynamic buckling patterns can be observed at the free end of the shell. The new analytical and numerical results serve as guidelines for safer designs of shell structures.     

外压作用下偏心加劲功能梯度圆柱壳的非线性屈曲和后屈曲分析

通过解析法研究外压作用下功能梯度加劲薄圆柱壳的非线性屈曲和后屈曲性能。通过其在内部的偏心环和纵梁对壳体进行加固,假定壳体和加固件的材料性能在厚度方向为连续梯度。根据VonKarman理论中的刚度法和传统的壳理论推导出基本关系和平衡方程,可更准确地选择三种关于挠曲的近似公式,且使用盖勒金法得出的显式表达式可以推测出临界荷载和后屈曲压力-挠曲曲线。数值结果显示了加固件能有效地增强壳体稳定性。     

Elastic buckling of horizontal barrelled shells filled with liquid—Numerical analysis

In the paper an alternative for horizontal cylindrical tank is presented. The proposed solution has the form of a barrelled tank in which the classical cylindrical shell is replaced by the barrelled one. The geometry of this shape is described. The buckling behaviour of the horizontal barrelled shell filled with liquid is analysed. The liquid level and the negative internal pressure which appear during emptying of the tank are also taken into consideration. On the basis of numerical analyses number of plots as well as analytical formulae are elaborated as a tool for design of barrelled tank with respect of stability criterion. Analyses are made on three families of barrelled shells of different capacities. In each family the capacity and length are constant. Advantages of barrelled tanks in comparison with cylindrical one are discussed.     

Free vibration of a clamped-free circular cylindrical shell partially filled with liquid—Part I: Theoretical analysis

Theoretical analyses are presented for the linear free vibration of a clamped-free cylindrical shell partially filled with an incompressible, inviscid liquid. For the vibration of the shell itself, the dynamic version of the Donnell equations was used and the problem was solved with the modified Galerkin procedure, taking the effect of the axisymmetric deformation due to the static liquid pressure into consideration. Concerning the vibration relevant to the liquid motion, the solution for the velocity potential was assumed as a sum of two sets of linear combinations of the suitable harmonic function, the unknown parameters of which were imposed to satisfy both boundary conditions along the wetted shell wall and the free liquid surface in a sense of appropriate series expansions. The procedure stated in the foregoing leads to a determinantal equation for the determination of the natural frequencies of the present shell-liquid system. To compare with the experimental results which will be stated in a companion paper,¹⁴ detailed numerical results will be presented in another companion paper¹³ on the free vibration characteristics of the two test cylinders partially filled with water.     

偏心加筋板的弹性屈曲分析

本文在同时计及板和偏心筋条的面内和面外位移的情况下,建立了能适应各种筋条尺寸和位置的离散型偏心加筋板稳定性控制方程式,采用半解析数值方法把因筋条偏心而产生的三个耦合偏微分方程转化为常系数微分方程,并采用加权残值方法导出齐次方程组,最终化为广义特征值问题予以求解.数值计算结果表明,本文方法简便可靠,精度较好.     

Non-uniform torsional behavior and stability of thin-walled elastic beams with arbitrary cross sections

In this paper, the analysis of the behavior of thin-wa lled beams, derived from Proki ’s work, is carried out by using beam theory with a single warping function valid for arbitrary form of cross sections, without any distinction between open and closed profiles and without using sectorial coordinates. The finite element method is used and numerical examples show the accuracy of the solution by comparison with other numerical or analytical results. For the stability analysis, analytical and numerical calculations of critical loads are given for beams submitted to bending moment and centrally applied forces. Equilibrium equations are established from the principle of virtual work. Critical loads are calculated by considering that a structure already in equilibrium reaches instability if there is one or more than one equilibrium position for the same loading. Results with this formulation are compared to those obtained with classical warping functions.     

含圆柱形衬砌直角域压电体的动力反平面特性

探讨了直角域附近一圆形衬砌对压电材料的反平面动力学行为的影响.通过利用镜像法对直角域情况进行等效处理到全空间中,在复数域中利用衬砌周围的边界条件构造方程求解未知系数,并通过编程进行最终的数值计算得出压电材料在衬砌周围的动应力和电场集中系数.在数值计算中,重点讨论了衬砌和基体为不同材料、衬砌和基体电极化方向相同和不同时,它们随压电材料物理系数、结构几何参数,以及入射波的频率变化的计算结果.结果表明,在SH波的作用下,压电材料在衬砌周围的动应力集中系数和电场集中系数都有一定的相似性,并且沿衬砌周围有规律的分布,然而在入射波的频率较低时二者的值相对较大,同时计算结果还与全空间衬砌下的计算结果有一定的相似性.     

柱坐标系下地下水非稳定流模拟的有限层法

在柱坐标系下,利用贝塞耳函数和基函数提出了满足地下水流动边界条件的降深试探函数;采用伽辽金法原理,推导了层状非均质各向异性越流承压含水层中地下水非稳定流的有限层方程,建立了地下水流动的有限层分析方法。在此基础上,编制了相应的计算程序实现了有限层方程的求解。经典算例的解析解与有限层解的对比分析验证了本文方法的正确性和有效性。     

Soil-pile interaction in inhomogeneous transversely isotropic media

Through a rigorous mathematical formulation, this paper considers the axisymmetric interaction analysis of a thin-walled cylindrical pile with finite length embedded in an exponentially graded transversely isotropic half-space under axial loads at the top of pile. The action of pile to the embedding medium is represented with axial and radial ring loads and the relevant set of Green's functions for the pile and semi-infinite soil medium has been developed. By satisfying the compatibility conditions between the two interacting media, the interaction analysis is shown to be reducible to a pair of Fredholm integral equations. Due to the complex nature of mathematical formulations and the intrinsic singularity of stress transfer, the analytical solution of obtained boundary integral equations is not possible and a suitable numerical scheme has been developed. By means of adaptive gradient shape functions which are capable of smoothly capturing regular and singular solutions, the given Fredholm integral equations are discretized in the common boundary of interacting media and solved numerically for the interfacial stress components. The numerical procedure is verified through comparisons with previous studies and the effect of inhomogeneity of soil layer on different responses of the interaction process is discussed by presenting several numerical results.     

Nonlinear mechanical,thermal and thermo-mechanical postbuckling of imperfect eccentrically stiffened thin FGM cylindrical panels on elastic foundations

This paper presents an analytical approach to investigate the nonlinear stability analysis of eccentrically stiffened thin FGM cylindrical panels on elastic foundations subjected to mechanical loads, thermal loads and the combination of these loads. The material properties are assumed to be temperature-dependent and graded in the thickness direction according to a simple power law distribution. Governing equations are derived basing on the classical shell theory incorporating von Karman–Donnell type nonlinearity, initial geometrical imperfection, the Lekhnitsky smeared stiffeners technique and Pasternak type elastic foundations. Explicit relations of load–deflection curves for FGM cylindrical panels are determined by applying stress function and Galerkin method. The effects of material and geometrical properties, imperfection, elastic foundations and stiffeners on the buckling and postbuckling of the FGM panels are discussed in detail. The obtained results are validated by comparing with those in the literature..     

Elasticity solution for static analysis of laminated cylindrical panel using differential quadrature method

The three-dimensional (3D) solution for static analysis of cross-ply cylindrical panel is presented using differential quadrature method (DQM) and Fourier series approach. The panel is assumed to be simply-supported at one pair of opposite edges and arbitrary conditions at the other pair. Applying the DQM to the governing differential equations and to the boundary conditions along the axial or circumferential directions, new state equations about state variables at discrete points are derived. Displacement and stress fields according to the various kinds of edges conditions are obtained by solving these state equations. The method is validated by comparing numerical results for the simply-supported edges with that available in the literature.     

Multicriteria optimization of cold-formed thin-walled beams with generalized open shape under different loads

The paper presents an analysis of the optimal design of cold-formed beams with generalized open shapes under pure bending, uniformly distributed loads, concentrated loads and axial loads with constant bending moment. The optimization problem includes the cross section area as the first objective function and the deflection of a beam as the second one. The geometric parameters of cross sections are selected as design variables. The set of constraints includes global stability condition, selected forms of local stability conditions, strength condition and technological and constructional requirements in a form of geometric relations. The strength and stability conditions are formulated and analytically solved using mathematical equations. The optimization problem is formulated and solved with help of the Pareto concept of optimality. The numerical procedure, based on the Messac normalized constraint method, include discrete, continuous and discrete-continuous sets of design variables. Results of the numerical analysis for different loads of beams with monosymmetrical cross section shapes are presented in tables.     

岩体中地下多层圆形结构动力响应分析

在城市地铁隧道结构以及其它地下工程选型方面,常常采用圆形结构。根据地下圆形结构和围岩的特点,建立平面冲击波载荷作用下有岩石中多层圆形结构体系的动力广义泛函,并采用变分原理推导出体系的线性和非线性动力微分方程组。采用该方法对承受平面应力波的有软回填层的岩石中多层圆形结构进行弹塑性动力响应计算,研究地下多层圆形结构的动力特性和回填层厚度对结构体系动力响应的影响。给出软回填层削波减振的最佳厚度和围岩与回填材料的最佳声阻抗比。该研究结果适用于平面应力波荷载或地震荷载下,地下工程结构选型及结构动力学分析。     

A study of the dynamic stability of anisotropic thin-walled beams

The dynamic stability of an anisotropic thin-walled beam subjected to harmonically varying axial compression is studied in this paper. The analysis is based on the assumption that the shape of the cross-section of the beam does not change during deformation. The shear of the middle surface of the beam is taken into account. The dynamic equations are reduced to a set of coupled Mathieu equations by using the Galerkin method. Then Hill's infinite-determinant method and a shooting-type numerical method are used to investigate the Mathieu equations to determine the stability regions of the beam in the parameter space. Two examples, one of which is a complete circular tube and the other is a circular tube with a longitudinal seam, are given.