# Analytical Solution of Ground Stress Induced by Shallow Tunneling with Arbitrary Distributed Loads on Ground Surface

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

## 3. General Formulas of the Exact Analytical Solutions of Ground Displacements and Stresses

_{1}(z) and ψ

_{1}(z), and they were obtained as follows:

_{1}(z) and ψ

_{1}(z), could be transformed to the functions in terms of ζ as follows:

_{0}, a

_{k}, b

_{k}, c

_{0}, c

_{k}, and d

_{k}could be calculated by the recursive relations obtained from the boundary conditions.

## 4. Recursive Relations and Laurent Series Coefficients for the Proposed Model

#### 4.1. The Outer Boundary of the Ground Surface

_{0}, c

_{k}, and d

_{k}satisfy the following recursive relations:

_{k}are in Fourier series terms in Equation (9), as seen in Appendix A.

#### 4.2. The Inner Boundary of the Tunnel Profile

_{0}, a

_{k}, and b

_{k}are calculated using the following equations integrating the related coefficients A

_{k}in Laurent series terms

_{k}variables are in Laurent series terms in Equations (13) and (14), seen in Appendix B.

## 5. An Application of the Derived Exact Analytic Solutions

#### 5.1. Prediction of the Distribution Characteristics of the Potential Plastic Zone

_{0}was 30 mm. The magnitudes of the distributed loads q were 100, 200, and 300 kN/m, and the ranges of the distributed loads (a

_{0}, b

_{0}) were (0 m, 7 m), (2 m, 9 m), and (4 m, 11 m).

#### 5.2. Influences of Different Magnitudes and Ranges of Arbitrary Distributed Loads on the Potential Plastic Zones

_{0}, b

_{0}) = (0 m, 7 m), (2 m, 9 m), (4 m, 11 m) on the distribution characteristics of the potential plastic zone induced by tunneling. It can be observed that when the range of distributed loads (a

_{0}, b

_{0}) was close enough to the tunnel, the tunneling-induced potential plastic zones around the tunnel and the potential plastic zones around the distributed loads coalesced.

#### 5.3. Influences of Different Tunnel Boundary Conditions on the Potential Plastic Zones

#### 5.4. Verification of the Potential Plastic Zone Obtained from the Derived Analytical Solutions with the Numerical Simulations

## 6. Conclusions

- (1)
- The potential plastic zones around the tunnel and around the ground surface structure coalesced when the magnitude of the distributed load q was big enough, while if the magnitude of the distributed load q was relatively small, the potential plastic zones separated into two parts.
- (2)
- If the range of the distributed loads (a
_{0}, b_{0}) was close enough to the tunnel, the tunneling-induced potential plastic zones around the tunnel and the potential plastic zones around the distributed loads coalesced. - (3)
- The results indicated that different symmetric tunnel boundary conditions greatly affected the distribution characteristics of the potential plastic zone.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

r | radius of a circular tunnel |

h | burial depth of the tunnel below the ground surface |

μ | shear modulus |

ν | Poisson’s ratio |

z | physical plane |

x, y | coordinate axis in physical plane |

ζ | mapped plane |

ξ, η | coordinate axis in mapped plane |

l_{1}, l_{2}, d_{1}, d_{2} | range of arbitrary distributed loads on ground surface |

q_{x}, q_{y} | magnitude of arbitrary distributed loads on ground surface |

σ_{xx}, σ_{yy}, σ_{xy} | components of stress in physical plane |

u_{x}, u_{y} | components of displacement in physical plane |

ω(ζ) | conformal transformation |

α | a parameter defines the conformal transformation |

φ_{1}(z), ψ_{1} (z) | analytic functions in physical plane |

φ(ζ), ψ(ζ) | analytic functions in mapped plane |

i | imaginary constant |

a_{0}, a_{k}, b_{k}, c_{0}, c_{k}, d_{k} | Laurent series coefficients related to boundary conditions |

κ | related parameter to Poisson’s ratio ν |

## Appendix A. Fourier Coefficients for Arbitrary Distributed Loads

## Appendix B. Fourier Coefficients for the Four Different Symmetric Boundary Conditions

## References

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**Figure 3.**Summarized boundary conditions of shallow tunnels [11].

**Figure 6.**Boundaries of potential plastic zones for different range of distributed loads (a

_{0}, b

_{0}).

**Figure 7.**Boundaries of potential plastic zones for different boundary conditions of shallow tunnel.

**Figure 8.**Verification of the potential plastic zone obtained from the derived analytical solutions with the numerical simulations.

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**MDPI and ACS Style**

Li, Z.; Wang, J.; Han, K.
Analytical Solution of Ground Stress Induced by Shallow Tunneling with Arbitrary Distributed Loads on Ground Surface. *Symmetry* **2019**, *11*, 823.
https://doi.org/10.3390/sym11060823

**AMA Style**

Li Z, Wang J, Han K.
Analytical Solution of Ground Stress Induced by Shallow Tunneling with Arbitrary Distributed Loads on Ground Surface. *Symmetry*. 2019; 11(6):823.
https://doi.org/10.3390/sym11060823

**Chicago/Turabian Style**

Li, Zejun, Jianchen Wang, and Kaihang Han.
2019. "Analytical Solution of Ground Stress Induced by Shallow Tunneling with Arbitrary Distributed Loads on Ground Surface" *Symmetry* 11, no. 6: 823.
https://doi.org/10.3390/sym11060823