ZHU Jiang-hong. Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory[J]. Applied Mathematics and Mechanics, 2013, 34(4): 345-354. doi: 10.3879/j.issn.1000-0887.2013.04.003
Citation: ZHU Jiang-hong. Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory[J]. Applied Mathematics and Mechanics, 2013, 34(4): 345-354. doi: 10.3879/j.issn.1000-0887.2013.04.003

Analytic Functions in Stress Analysis of the Surrounding Rock for Caverns With the Complex Variable Theory

doi: 10.3879/j.issn.1000-0887.2013.04.003
  • Received Date: 2012-02-22
  • Rev Recd Date: 2013-04-03
  • Publish Date: 2013-04-15
  • Two analytic functions must be derived from the stress equation for the boundary condition, when the surrounding rock stress is analyzed in the tunnel with arbitrary excavation cross section with the complex variable theory. Based on the Harnack theorem, the stress equation for the boundary condition was transformed into an integral equation in the surrounding rock. The mapping function expressed with the finite Laurent series was introduced into the integral equation, and then the boundary with arbitrary excavation cross section was change into the boundary with the unit circle line for solving two analytic functions. The analyticity was studied for each integrand in the integral equation in discussed domain. Using above result, each integral term of the equation was solved with the Residue theorem and the general formulas of two analytic functions were obtained to calculate the surrounding rock stress in tunnel with arbitrary excavation cross section. Moreover, study examples of the analytic functions were provide for the circular tunnel and the elliptic tunnel and two general formulas for analytic function. The results were accord with the literatures. General formulas of two analytic function obtained with the Residue theorem are adapt to calculate the surrounding rock stress in the caverns with arbitrary excavation cross section. Compared to other methods, it is much simpler to obtain the analytical solutions and the calculating results are more precise for the actual caverns.
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