Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior

SHI Gaoping; ZHU Jianghong; YANG Jianhui

doi:10.21656/1000-0887.400220

Volume 41 Issue 5

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SHI Gaoping, ZHU Jianghong, YANG Jianhui. Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior[J]. Applied Mathematics and Mechanics, 2020, 41(5): 541-556. doi: 10.21656/1000-0887.400220

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Analytical Stress Solutions Around Tunnels With Arbitrary-Excavation Cross Sections Based on Conformal Mapping From a Unit Circle to the Tunnel Exterior

doi: 10.21656/1000-0887.400220

¹School of Machinery and Automobile Engineering, Zhejiang University of Water Resources and Electric Power,Hangzhou 310018, P.R.China;²School of Civil Engineering and Architecture, Zhejiang University of Science and Technology, Hangzhou 310023, P.R.China;³Hangzhou Dajiangdong Investment and Development Co., Ltd.,Hangzhou 311222, P.R.China

Received Date: 2019-07-18
Rev Recd Date: 2019-09-21
Publish Date: 2020-05-01

Abstract

Abstract

The stresses around tunnels with arbitrary-excavation cross sections were studied with the stress field simplified as a uniform stress field at an angle to the horizontal axis at infinity. The conformal mapping function was introduced to express the stress functions of surrounding rock with variable ζ instead of variable z. Then the Cauchy integral operator was applied to the stress boundary equation for tunnel wall. The general analytical solutions to stress functions and analytical solutions of surrounding rock stresses were obtained for tunnels with arbitrary-excavation cross sections according to the integrand analyticity and the residue theorem. Moreover, with the stress boundary conditions for elliptical tunnels substituted into the general analytical solutions, the obtained particular solutions to stress functions consisted with the results in literatures. The surrounding rock stresses were analyzed for 2-lane road tunnels based on the explicit general analytical solutions and the results accorded with engineering rules. The stress distribution of tunnel wall was analyzed with the ABAQUS finite element method, and the results coincided with the analytical solution. The analytical calculation can be expediently carried out for underground tunnels with arbitrary-excavation cross sections as long as their conformal mapping functions are presented from the unit circle exterior to the tunnel exterior.
- conformal mapping,
- arbitrary-excavation cross section,
- analytical solution,
- stress function of surrounding rock,
- residue theorem

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References

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