两种类型夹层锥壳的位移型统一理论及应用(Ⅰ)

本文使用锥壳的Donnell型位移基本方程组,通过引入位移函数和广义荷载,将两种类型的夹层锥壳之基本方程组化成为一个关于位移函数的八阶可解偏微分方程,由此可得到许多常见问题的新型控制方程.     

变厚度圆柱壳的轴对称弯曲问题

本文详细地研究了厚度h=h_0ξ~的圆柱壳的轴对称弯曲问题.文中通过引入一个位移函数H(ξ),将该问题的方程组化成一个关于H(ξ)的6阶常微分方程,用广义超几何函数给出问题的精确解.     

球面各向同性弹性力学的位移解法

本文引入三个位移函数（ｗ，Ｇ，ψ），将球面各向同性弹性力学运动方程，简化为关于ψ的二阶偏微分方程，和关于Ｗ和Ｇ的联立方程。在静力学问题中，联立方程可进一步简化，ｗ和Ｇ可用另一位移函数Ｆ表示，而Ｆ满足一个四阶偏微分方程。在球壳固有振动问题中，则简化为一个独立的二阶常微分方程，和另两个二阶的联立的常微分方程，证明了在多层球壳中，它们分别对应独立的两类振动。改进了常微分方程的解法，并计算了一个二层球壳的频率。     

常子午线曲率薄壳轴对称几何非线性问题的复变量方程

1.几何非线性问题的基本方程在本世纪初,Reissner H.和Meissner E.利用在线性薄壳理论中存在的静力-几何比拟关系,将线弹性薄壳轴对称问题,归结为以应力函数和转角为未知量的两个常微分方程。以后,人们利用这两个方程的相似性,引入复未知函数,把一些典型壳体的方程简化为一个二阶变系数常微分方程,为这些问题的求解带来极大的便利。本文将这一方法推广到薄壳大位移问题,导出用复未知函数表示的常子午线曲率壳体轴对称变形的非线性微分方程。从这个一般方程可以直接得到关于柱壳,锥壳,圆球壳,环壳和圆板几何非线性问     

浸没的球面各向同性球壳的自由振动

本文引入三个们移函数并用球面调和函数展开，可将球面各向同性弹性力学的基本方程转化成一个独立的二阶常微分方程和另一个耦合的二阶常微分方程组，采用液动压力表示流体与壳的相互作用，可以把无限大不可压缩流体中任意厚度球面各向同性球壳的自由振动频率计算归结为一个代数循征值问题，文中计算了若干种情况下球壳的频率，在各向同性情形与有关文献作了比较。     

含面内转动自由度的广义协调曲面矩形扁壳元

扁壳单元中引入结点转角自由度可以在不增加结点的情况下,增加位移场的阶次,提高计算精度,从而显著地提高单元性能。同时在单元中引入泡状位移场,能有效地扩大了单元位移场的解空间,所构造的单元具有计算精度高、对计算网格畸变不敏感的优良特性。本文利用广义协调薄板单元RGC-12的位移函数作为扁壳元的法向位移,利广义协调矩形膜元的位移函数作为扁壳面的切向位移,通过附加面内转动自由度构造了一个具有24个自由度的4结点广义协调曲面矩形扁壳元GRC-S24。在此基础上再增加一个广义泡状位移,又构造了一个具有更高计算精度的曲面矩形扁壳元GRC-S24M。并通过实例分析对这两个单元的收敛性和精度进行了验证。     

直角坐标下厚圆柱扁壳弯曲的一般解

基于直角坐标下考虑横向剪切变形情况下厚圆柱扁壳的几何方程、物理方程、平衡微分方程,建立了以3个中面位移和2个中面转角为独立变量的中厚圆柱扁壳弯曲的位移型基本微分方程.因该方程可退化为薄圆柱扁壳弯曲的基本微分方程,说明了其推导过程的正确性及一般性.此外,厚圆柱扁壳的位移型基本微分方程是一个10阶微分方程,对其使用双重三角...     

按位移求解弹性力学平面问题的解析构造解研究

针对弹性力学平面问题偏微分方程组的位移法,引入多指数函数,提出了含未知参量的指数函数、三角函数和线性函数组合形式的位移函数解析构造解。建立了任意边界条件与未知参量之间所满足的非线性代数方程组,确定了边界节点条件和未知参量的数量关系。推导了具有对称位移边界的位移函数解析构造解。构建了位移函数构造解的精度判定方法。求解了具有对称位移边界条件的矩形板算例的位移解与误差分析。研究结果可为位移法理论和实际工程应用提供参考。     

扁球壳在均布压力作用下的非线性弯曲问题

本文应用逐步加裁法将圆底扁球壳在均布压力作用下的非线性微分方程组化为线性的微分方程组.然后以三次B样条函数为试函数,用配点法将此线性的微分方程组化成线性代数方程组.最后利用递推公式解此线性代数方程组,从而使问题得到解决.     

锥壳一般弯曲,稳定和振动问题的位移型统一方程及其应用

1.前言弹性锥壳的一般弯曲、稳定和振动问题,在实际工程中经常遇到,但对其研究基本上限于轴对称问题且都是以挠度函数和应力函数为基本未知量.我们认为,对于锥壳的特征值问题、弹性地基锥壳以及锥壳组合结构,则宜采用锥壳的位移解法.本文作者之一曾对锥壳一般弯曲问题的位移解法进行了系统的研究,以广义超几何函数给出了一般解.在应用文献[1]结果的基础上,本文通过引入一个广义载荷q_n(s,θ,t),得到了以位移函数U(s,θ,t)表示的弹性锥壳一般弯曲、稳定和振动(包括弹性地基影响)问题的统一型式的控制方程.文献[2]用级数给出了锥壳横向自由振动问题的解,但应指出,由于文献[2]中     

The exact solution for the general bending problems of conical shells on the elastic foundation

The general bending problem of conical shells on the elastic foundation (Winkler Medium) is not solved. In this paper, the displacement solution method for this problem is presented. From the governing differential equations in displacement form of conical shell and by introducing a displacement function U(s,θ), the differential equations are changed into an eight-order soluble partial differential equation about the displacement function U(s,θ) in which the coefficients are variable. At the same time, the expressions of the displacement and internal force components of the shell are also given by the displacement function U(s θ). As special cases of this paper, the displacement function introduced by V.S. Vlasov in circular cylindrical shell^[5], the basic equation of the cylindrical shell on the elastic foundation and that of the circular plates on the elastic foundation are directly derived.Under the arbitrary loads and boundary conditions, the general bending problem of the conical shell on the elastic foundation is reduced to find the displacement function U(s,θ).The general solution of the eight-order differential equation is obtained in series form. For the symmetric bending deformation of the conical shell on the elastic foundation, which has been widely usedinpractice,the detailed numerical results and boundary influence coefficients for edge loads have been obtained. These results have important meaning in analysis of conical shell combination construction on the elastic foundation,and provide a valuable judgement for the numerical solution accuracy of some of the same type of the existing problem.     

Displacement type unified equation for bending, buckling and vibration of conical shells and its applications

By using Donnell's simplication and starting from the displacement type equations of conical shells, and introducing a displacement functionU(s,,) (In the limit case, it will be reduced to cylindrical shell displacement function introduced by V. S. Vlasov) and a generalized loadq,(s,,),the equations of conical shells are changed into an eighth—order solvable partial differential equation about the displacement functionU(s,,). As a special case, the general bending problem of conical shells on Winkler foundation has been studied. Detailed numerical results and boundary coefficients for edge unit loads are obtained.The project supported by the National Natural Science Foundation of China.     

THE DISPLACEMENT SOLUTION OF CONICAL SHELL AND ITS APPLICATION

In this paper,the displacement solution method of the conical shell is presented.Fromthe differential equations in displacement form of conical shell and by introducing adisplacement function,U(s,θ),the differential equations are changed into an eight-ordersoluble partial differential equation about the displacement function U(s,θ)in which thecoefficients are variable.At the same time,the expressions of the displacement and internalforce components of the shell are also given by the displacement function.As special casesof this paper,the displacement function introduced by V.Z.Vlasov in circular cylindricalshell,the basic equation of the cylindrical shell and that of the circular plate are directlyderived.Under the arbitrary loads and boundary conditions,the general bending problem of theconical shell is reduced to finding the displacement function U(s,θ),and the generalsolution of the governing equation is obtained in generalized hypergeometric function,Forthe axisymmetric bending deformation of the     

载流圆锥薄壳的磁弹性应力与变形分析

在建立旋转壳体的非线性磁弹性运动方程的基础上,研究了电磁场和机械载荷联合作用下载流圆锥薄壳的磁弹性效应．通过算例,得到了载流圆锥薄壳的位移及应力与通电电流强度之间的关系．解决了圆锥薄壳顶点处的奇异性问题,给出了轴对称条件下的数值解．计算结果表明：改变通电电流强度,可以改变载流圆锥薄壳的应力与变形状态,达到控制圆锥薄壳的受力与变形的目的．     

To the solution of the thermoelasticity problem for conical shells

We consider the stress-strain state of thin conical shells in the case of arbitary distribution of the temperature field over the shell. We obtain equations of the general theory based on the classical Kirchhoff-Love hypotheses alone. But since these equations are very complicated, attempts to construct exact solutions by analytic methods encounter considerable or insurmountable difficulties. Therefore, the present paper deals with boundary value problems posed for simplified differential equations. The total stress-strain state is constructed by “gluing” together the solutions of these equations. Such an approach (the asymptotic synthesis method) turns out to be efficient in studying not only shells of positive and zero curvature [1, 2] and cylindrical shells [3] but also conical shells [4, 5]. Here we illustrate it by an example of an arbitrary temperature field, and the problem is reduced to solving differential equations with polynomial coefficients and with right-hand side containing the Heaviside function, the delta function, and their derivatives.     

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。     

圆锥壳自由振动传递函数解

本文在线性弹性理论基础上，给出了一种求解圆锥薄壳自由振动的渐进传递函数解法，壳体的三个位移分量，外力和边界条件首先沿环向展开的Ｆｏｕｒｉｅｒ级数，然后关于时间变量进行Ｌａｐｌａｃｅ变换，这样就将壳体的控制方程化为一系列含复参数ｓ的变系数常微分方程组，通过定义状态变量。得到了壳体动力学问题的状态空间控制微分方程，引入一小参数，并利用摄动技术就可以得到微分方程的渐进传递函数解，将各于锥段的解进行综合，     

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。     

Free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines

Free vibrations of layered conical shell frusta of differently varying thickness are studied using the spline function approximation technique. The equations of motion for layered conical shells, in the longitudinal, circumferential and transverse displacement components, are derived using extension of Love’s first approximation theory. Assuming the displacement components in a separable form, a system of coupled equations on three displacement functions are obtained. Since no closed form solutions are generally possible, a numerical solution procedure is adopted in which the displacement functions are approximated by cubic and quintic splines. A generalized eigenvalue problem is obtained which is solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The vibrations of two-layered conical shells, made up of several types of layer materials and supported differently at the ends are considered. Linear, sinusoidal and exponential variations in thickness of layers are assumed. Parametric studies are made on the variation of frequency parameter with respect to the relative layer thickness, cone angle, length ratio, type of thickness variation and thickness variation parameter. The effect of neglecting the coupling between bending and stretching is also analysed.     

Asymptotic method for analysis of a conical shell under local loading

Differential equations of the general theory describing the stress-strain state of conical shells are very complicated, and when computing the exact solution of the problem by analytic methods one encounters severe or even so far insurmountable difficulties. Therefore, in the present paper we develop an approach based on the method of asymptotic synthesis of the stressed state, which has already proved efficient when solving similar problems for cylindrical shells [1, 2]. We essentially use the fourth-order differential equations obtained by Kan [3], which describe the ground state and the boundary effect. Earlier, such equations have already been used to solve problems concerning force and thermal actions on weakly conical shells [4–6]. By applying the asymptotic synthesis method to these equations, we manage to obtain sufficiently accurate closed-form solutions.