阶梯折算法的收敛条件及其一般格式
The Convergent Condition and United Formula of Step Reduction Method
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摘要: 叶开沅教授创造了阶梯折算法[1].利用这个方法求解非均匀弹性力学问题,所得到的解可以用解析式表达,并具有计算量小、精度高的优点.本文通过数学上的推导,给出了阶梯折算法的收敛条件,并证明了当收敛条件满足时,所得到的解可一致收敛于精确解.文中还给出了阶梯折算法的一般格式及误差估计.由于采用矩阵形式表达,避免了以往冗长的数学表达式,使得解的形式非常简洁.文末给出算例,算例表明运用本文的理论,可以得到阶梯折算法的正确模式.Abstract: The step reduction method was first suggested by Prof. Yeh Kai-yuan[1]. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time, its calculating lime is very short and convergent speed very fast. In this paper, the convergent condition and united formula of step reduction method are given by mathematical method. It is proved that the solution of displacement and stress resultants obtained by this method can converge to exact solution uniformly, when the convergent condition is satisfied. By united formula, the analytic solution can be expressed as matrix form, and therefore the former complicated expression can be avoided. Two numerical examples are given at the end of this paper which indicate that, by the theory in this paper, a right model can be obtained for step reduction method.
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[1] 叶开沅,非均匀变厚度弹性力学的若干问题的一般解,Ⅳ.非均匀变厚度梁的弯曲,稳定和自由振动,兰州义学学报,力学专号,1 (1979), 133-157. [2] 叶开沅、许剑云,非均匀变厚度弹性体力学的若干问题的一般解,Ⅰ.在非均匀定常温度场下的非均匀变厚度高速旋转圆盘的弹塑性应力分析,兰州大学学报,力学专号,1 (1979),60-74. [3] 叶开沅、郭建虎,非均匀变草度弹性体力学的若干问题的一般解,在任意定常温度场和任意分布载荷下的任意轴对称非均匀、变厚度环形板的弯曲问题,兰州大学学报,力学专号,1 (1979),75-114 [4] 叶开沅、李金荣、李进潮,非均匀变厚度弹性叶力学的若干问题一般解,任意分布载荷下两对边简支单向非均匀变厚度矩形板的弯曲问题,兰州大学学报,力学专号,1 (1979),115-132. [5] 叶开沅、尚新春,任意分布载荷下两对边简支单向非均匀变厚度矩形板的弯曲问题的一般解—折线折算法,兰州大学学报,力学专号,19 (1983), 58-75. [6] 叶开沅、汤任基、甄继庆,非均匀变截面弹性圆环在任意载荷下的弯曲问题,应用数学和力学,2, 1 (1982), 1-12. [7] Reltorys, Karel, lrariational Methods in Mathematics, Science and Engineering, D.Reidel Publishing Company Press, Holland, sec, ed,(1980).
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