AN EXACT ELEMENT METHOD FOR PLANE PROBLEM

In this paper,based on the step reduction method and exact analytic method a newmethod—exact element method for constructing finite element,is presented.Since the newmethod doesn’t need the variational principle,it can be applied to solve non-positive andpositive definite partial differential equations with arbitrary variable coefficient.By thismethod,a quadrilateral noncompatible element with 8 degrees of freedom is derived for thesolution of plane problem.Since Jacobi’s transformation is not applied,the present elementmay degenerate into a triangle element.It is convenient to use the element in engineering.Inthis paper,the convergence is proved.Numerical examples are given at the end of this paper,which indicate satisfactory results of stress and displacements can be obtained and havehigher numerical precision in nodes.     

A high convergent precision exact analytic method for differential equation with variable coefficients

The exact analytic method was given by[1].It can be used for arbitrary variable coefficient differential equations and the solution obtained can have the second order convergent precision.In this paper,a new high precision algorithm is given based on[1],through a bending problem of variable cross-section beams.It can have the fourth convergent precision without increasing computation work.The present computation method is not only simple but also fast.The numerical examples are given at the end of this paper which indicate that the high convergent precision can be obtained using only a few elements.The correctness of the theory in this paper is confirmed.     

An exact element method for bending of nonhomogeneous thin plates

In this paper, based on the step reduction method, a new method, the exact element method for constructing finite element, is presented. Since the new method doesn't need the variational principle, it can be applied to solve non-positive and positive definite partial differential equations with arbitrary variable coefficient. By this method, a triangle noncompatible element with 6 degrees of freedom is derived to solve the bending of nonhomogeneous plate. The convergence of displacements and stress resultants which have satisfactory numerical precision is proved. Numerical examples are given at the end of this paper, which indicate satisfactory results of stress resultants and displacements can be obtained by the present method.     

Substructure computational algorithm for exact analytic method

In[1],the exact analytic method for the solution of differential equation with variablecoefficients was suggested and an analytic expression of solution was given by initialparameter algorithm.But to some problems such as the bending,free vibration andbuckling of nonhomogeneous long cylinders,it is difficult to obtain their solutions by theinitial parameter algorithm on computer.In this paper,the substructure computationalalgorithm for the exact analytic method is presented through the bending of non-homogeneous long cylindrical shell.This substructure algorithm can be applied to solve theproblems which can not be calculated by the initial parameter algorithm on computer.Finally,the problems can be reduced to solving a low order system of algebraic equationslike the initial parameter algorithm.Numerical examples are given and compared with theinitial para-algorithm at the end of the paper,which confirms the correcthess of thesubstructure computational algorithm.     

精确解析法的子结构算法

文[1]提出精确解析法,用以求解任意变系数常微分方程,并利用初参数算法给出一个解的解析表达式.但利用初参数算法,对某一类问题,如长柱壳弯曲和振动等,它们的解将难以在计算机上得到.本文通过非均匀轴对称长圆柱壳弯曲问题,给出精确解析法的子结构算法,它能够计算初参数算法在计算机上不能解决的问题.问题最后和初参数算法一样能归结为求解一个低阶代数方程组.文末给出算例,表明本文算法的正确性,并和初参数算法作了比较.     

The general solution on nonlinear axial symmetrical deformation of nonhomogeneous cylindrical shells

In this paper, the general solution on nonlinear axial symmetrical deformation of nonhomogeneous cylindrical shells is obtained by step reduction method⁽¹⁾. The general formula of displacements and stress resultants, which is used to solve the bending problems of nonhomogeneous cylindrical shells under arbitrary axial symmetric loads, is derived. Its uniform convergence is proved. Finally, it is only necessary to solve one set of binary linear algebraic equations. A numerical example is given at the end of the paper which indicates satisfactory results of displacement and stress resultants can be obtained and converge to the exact solution.     

非均匀圆柱壳非线性轴对称变形一般解

本文利用阶梯折算法[1],得到了非均匀圆柱壳非线性轴对称变形的一般解.文中导出了在任意轴对称载荷下求解非均匀圆柱壳非线性弯曲的位移和内力的一般公式,并给出一致收敛于精确解的证明.问题最后归结为求解二元一次代数方程组,文末给出算例.算例表明,无论内力和位移都可得到满意的结果,并收敛于精确解.     

静不定梁的等强度设计

本文提出了用最小余能原理对静不定梁进行等强度设计的方法.并就任意分布载荷,集中力或集中力偶作用下的单跨或多跨梁给出了沿梁轴刚度变化的解析表达式.对具有固定宽度可变高度的矩形截面梁和可变外层厚度的夹层梁给出了算例.与已有结果的比较,显示了本方法的有效性.     

非均匀变厚度圆盘的定常热传导

本文采用阶梯折算法,求得了任意非均匀变厚度圆盘定常热传导问题的一般解.并通过算例,对阶梯折算法的误差进行了分析.结果表明,该方法对于求解各类变系数常微分方程是十分有效的.     

在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题

本文得出了在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题的解析解.我们假设温度沿壳体厚度直线分布.在第一部分.我们研究了常用边界条件下的中心开孔圆底扁球壳的自由振动.作为例子,我们计算了一边缘夹紧的扁球壳的自然基频(m=0),所得结果与E.Reissner[1]的结果作了比较.频率方程的解法是钱伟长[2]提出来的.这将在附录3中介绍.在第二部分,我们研究了在任意谐温度场和任意谐法向动载荷联合作用下的中心开孔圆底扁球壳的强迫振动.在第三部分,我们研究了在任意不定常温度场和任意法向动载荷联合作用下的具有初始条件的上述壳体的强迫振动.在附录1和2中,我们讨论了如何用应力函数来表示位移边界条件和m=1情形的边界条件.