主应力及其方向余弦计算的进一步研究

Further Study on the Calculation of Principal Stresses and Their Direction Cosines

国内外计算主应力及其方向余弦有两种方法:计算公式法和雅可比方法。首先介绍线性代数中特征值和特征向呈的计算以及弹性力学中主应力及其方向余弦的计算,通过二祥的对比论述了两种方法的数学渊源并且发现:目前采用计算公式法或雅可比方法计算主应力方向余弦,实际上均只计算了两组主应力方向余弦解答中的一组解答,只有将另一组解答补齐,才能得到完整的计算结果。其次利用笔者编写的N层弹性体系力学计算程序NESCP中的计算公式法和国际上著名的BISAR计算机程序中的雅可比方法对各种力学状况下的几十个算例进行了数值计算并列出了4个算例。计算结果表明:在各种力学状况下两种方法的计算结果十分接近,无论是采用计算公式法或是采用雅可比方法均能得到足够精确的主应力及其方向余弦的计算结果。

Up to now,there are two methods for calculating principal stresses and their direction cosines at home and abroad:calculating formulation method and Jacobi method.Firstly,the calculation of characteristic value and characteristic vector in linear algebra and the calculation of principal stresses and their direction cosines in elasticity were introduced.By comparing the two calculation processes,the mathematical origin of the two methods was discussed.It is found that in the process that calculating formulation method or Jacobi method is used to calculate the direction cosines of principal stresses at present,in fact,only one set of solutions of two sets of direction cosine solutions of principal stresses are calculated.Only when the other set of solutions is complemented the complete calculation results can be obtained.Secondly,some dozens of numerical examples under various mechanical conditions were calculated by using the calculating formulation method in NESCP(N-layer Elastic System Computer Program)computer program compiled by author and Jacobi method in BISAR(Bitumen Structures Analysis in Roads)computer program well known in the world,and four examples were listed.The calculation results show that the calculation results of the above two methods are very close under various mechanical conditions.Whether the calculating formulation method or Jacobi method is used,the calculation results of the principal stresses and their direction cosines can be obtained accurately enough.