任意多孔条件下围岩力学分析的非迭代解析方法

A non-iterative analytical method for mechanical analysis of surrounding rock with arbitrary shape holes

多孔条件下的围岩力学分析对隧道施工安全具有重要意义,目前常用的Schwarz交替法在求解多孔问题时,其精度受迭代次数影响且计算过程复杂,不适合工程应用。为此,基于解析函数的多级数展开原理建立任意多孔条件下围岩力学分析的非迭代求解方法。该方法将解析函数在每一个孔洞上单独进行Laurent级数展开,从而精确地表达了整个多连通区域内的应力与位移场。通过对每个孔洞单独进行保角变换,建立各个孔口的应力与位移边界条件。最后,利用Fourier变换技术将边界条件转化到频域上进行分析,从而求解解析函数中所有的待定系数。该方法中每个孔口的保角变换都相互独立,因而可以处理各种复杂形状的多孔问题。整个求解过程不需要进行任何迭代即可获得精确的应力与位移解答,求解过程简单高效、适合隧道工程应用。

The mechanical analysis of surrounding rock with arbitrary shape holes is of great significance for the safety of tunnel construction. In traditional analytical methods,the Schwarz alternating method is usually used to solve the multiple-hole problem,but its accuracy and efficiency are seriously affected by the number of iterations. Therefore,the Schwarz alternating method is not suitable for engineering applications. To solve this problem,in this paper,a non-iterative analytical method for the mechanical analysis of surrounding rock with arbitrary shape holes was proposed based on the theory of multiple Laurent series expansions. This method introduces several groups of analytic functions to accurately reformulate the stress and displacement fields of the multiply-connected domain,and establishes the stress and displacement boundary conditions by performing the conformal mapping on each hole individually. The conformal transformations of different holes are independent of each other,which makes it possible to handle the complex boundaries. Finally,the Fourier transform technique is used to transform the boundary conditions into the frequency equations,and then all unknowns in the analytic function can be uniquely determined. The proposed method can directly obtain the accurate solution of stress and displacement without any iteration,and hence is simple,efficient and suitable for engineering applications.