# Predicting Ground Settlement Due to Symmetrical Tunneling through an Energy Conservation Method

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## Abstract

**:**

## 1. Introduction

## 2. Energy Conservation Method for Tunneling-Induced Ground Deformation

_{s}is the maximum engineering shear strain, which can be derived as follows:

_{x}is the strain in the horizontal direction (x-direction), ε

_{z}is the strain in the vertical direction (z-direction), γ

_{xz}is the shear strain in x-z plane, and u and v are the horizontal and vertical displacements of the ground.

_{x}and F

_{z}are the components of initial force in the horizontal and vertical directions along the tunnel boundary, respectively.

## 3. Numerical Validation of Proposed Method

^{3}. The contours of the vertical displacements under various conditions are shown in Figure 1, in which the profiles of ground displacements were similar. ΔE was calculated from all zones of the models, and ΔW was calculated from all grid points. ΔN was only calculated from particular grid points along the tunnel boundary. All results for the three overburden depths are shown in Table 1.

_{z}+ ε

_{x}, were always unchanged when the stress plane transformed to the direction of maximum shear stress, and ε

_{y}was always equal to zero.

## 4. Differences between Applications of the Method to Tunneling and to Excavating Foundations

## 5. Prediction of Ground Deformation Using Proposed Method

#### 5.1. Prediction Using the Differential of Empirical Solutions

_{0}is the tunnel depth as measured from the ground surface, ε

_{0}is the ground loss, R is the radius of the tunnel, and v

_{L}and u

_{L}are the vertical and horizontal displacements.

_{max}is the maximum subsurface settlement at depth z.

_{0}so as to simplify this equation.

_{0}is the only unknown in the equations for ΔW and ΔE. The equation for ΔW is a linear term about ε

_{0}, while the equation for ΔE is a quadratic term about ε

_{0}. The unknown ε

_{0}is equal to the expression for ΔE divided by that for ΔW. Consequently, ground displacements could be predicted once ground loss (ε

_{0}) was obtained. The maximum surface settlements were predicted in four different cases, including at depths of 15 m, 20 m, and 25 m. The results calculated using these two empirical methods were compared with the numerical results, and are shown in Table 3.

_{0}− z) in the expressions for vertical and horizontal displacement. When z was close to z

_{0}, the denominator was extremely small. Thus, the value of shear strain increased radically as the tunnel was excavated deeper. The energy in shearing the soil was extremely large. Accordingly, the obtained ground loss ε

_{0}was fairly small. On the contrary, the energy in shearing the soil would not increase at shallower tunnel depths. We supposed this method was suitable for shallow tunnels.

#### 5.2. Prediction Using Fitted Expressions of Numerical Results

_{s}is the distance to the point of inflexion, and they are both satisfied by exponential functions at depth z. Note that w

_{0}is the maximum surface settlement (x = 0, z = 0.5) which was later obtained. Accordingly, the fitted expressions were functions related to w

_{0}, x, and z.

_{s}(Mair et al. [4]). The soil outside the total width of the surface settlement trough was assumed to be stationary. However, the soil deformed at the same position in the numerical simulations even though the values were fairly small. Additionally, the deeper the buried soil was, the narrower the settlement trough became, and the larger the maximum vertical displacements were. This satisfied the conclusions presented by Fang et al. [18]. Therefore, the fitted expressions of vertical displacements were not exact at deeper depths. Most maximum values of the fitted expressions were smaller than those of the numerical results. Contrarily, the parameters w/w

_{0}and i

_{s}were both fitted exactly. The parameter w/w

_{0}increased exponentially with an increase in depth z

_{0}, while the parameter i

_{s}decreased exponentially.

_{0}. We perceived that the coefficients in the expressions changed regularly with overburden depth z

_{0}. Maximum engineering shear strain can probably be predicted by an accurate formula which has actual physical meaning. This formula is perhaps relative to the radius of the tunnel r and the overburden depth z

_{0}, and it can be studied further.

_{0}, and ΔE is a real number. Thus, all parameters were finally prepared for the calculation of w

_{0}. Using the Matlab software, the maximum surface settlements (w

_{0}) were calculated at depths of 15 m, 20 m and 25 m. All analytical results were compared with the numerical results, and are shown in Table 4.

_{0}, and the tunnel diameter r. The fitted expressions are worth studying further so as to calculate the surface and subsurface settlements.

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

- Peck, R.B. Deep excavation and tunnelling in soft ground. In Proceedings of the 7th International Conference on Soil Mechanics and Foundation Engineering, Mexico City, Mexico, 25–29 August 1969; Sociedad Mexicana de Mecanica: Mexico City, Mexico, 1969; Volume 3, pp. 225–290. [Google Scholar]
- O’Reilly, M.P.; New, B.M. Settlements above tunnels in the United Kingdom—Their magnitude and prediction. In Tunnelling ’82, 3rd ed.; Institute of Mining and Metallurgy: London, UK, 1982; pp. 173–181. ISBN 090048862X. [Google Scholar]
- Clough, G.W.; Schmidt, B. Design and performance of excavations and tunnels in soft clay. In Soft Clay Engineering; Brand, E.W., Brenner, R.P., Eds.; Elsevier: Amsterdam, The Netherlands, 1981; pp. 569–634. ISBN 9780444417848. [Google Scholar]
- Mair, R.J.; Taylor, R.N.; Bracegirdle, A. Subsurface settlement profiles above tunnels in clays. Géotechnique
**1993**, 43, 315–320. [Google Scholar] [CrossRef] - Verruijt, A.; Booker, J.R. Surface settlements due to deformation of a tunnel in an elastic half plane. Géotechnique
**1996**, 46, 753–756. [Google Scholar] [CrossRef] - Sagaseta, C. Analysis of undrained soil deformation due to ground loss. Géotechnique
**1987**, 37, 301–320. [Google Scholar] [CrossRef] - Loganathan, N.; Poulos, H.G. Analytical prediction of tunneling-induced ground movements in clays. ASCE J. Geotech. Geoenviron. Eng.
**1998**, 124, 846–856. [Google Scholar] [CrossRef] - Bobet, A. Analytical solutions of shallow tunnels in saturated ground. ASCE J. Eng. Mech.
**2001**, 127, 1258–1266. [Google Scholar] [CrossRef] - Potts, D.M.; Zdravkovic, L.; Zdravković, L. Finite Element Analysis in Geotechnical Engineering: Application; Thomas Telford: London, UK, 2001; ISBN 9780727727831. [Google Scholar]
- Chou, W.; Bobet, A. Response by the authors to Verruijt, A.; Sagaseta, C.; Strack, O.E. Discussion to the paper: Chou, W.; Bobet, A. Prediction s of ground deformations in shallow tunnels in clay. Tunn. Undergr. Space Technol.
**2003**, 18, 95–97. [Google Scholar] [CrossRef] - Chou, W.; Bobet, A. Prediction s of ground deformations in shallow tunnels in clay. Tunn. Undergr. Space Technol.
**2002**, 17, 3–19. [Google Scholar] [CrossRef] - Osman, A.S.; Bolton, M.D. Simple plasticity- based prediction of the undrained settlement of shallow circular foundations on clay. Géotechnique
**2005**, 55, 435–447. [Google Scholar] [CrossRef] [Green Version] - Osman, A.S.; Bolton, M.D. A new design method for retaining walls in clay. Can. Geotech. J.
**2004**, 41, 451–466. [Google Scholar] [CrossRef] - Osman, A.S.; Bolton, M.D.; Mair, R.J. Predicting 2D ground movements around tunnels in undrained clay. Géotechnique
**2006**, 56, 597–604. [Google Scholar] [CrossRef] [Green Version] - Itasca Consulting Group, Inc. FLAC3D—Fast Lagrangian Analysis of Continua in 3 Dimensions, Version 5.0; Itasca: Minneapolis, MN, USA, 2012. [Google Scholar]
- Grant, R.J.; Taylor, R.N. Tunnelling-induced ground movements in clay. Proc. Inst. Civ. Eng. Geotech. Eng.
**2000**, 143, 43–55. [Google Scholar] [CrossRef] - Mair, R.J.; Taylor, R.N. Bored tunneling in urban environment. In Proceedings of the 14th International Conference on Soil Mechanics and Foundation Engineering, Hamburg, Germany, 6–12 September 1997; Volume 4, pp. 2353–2385. [Google Scholar]
- Fang, Y.S.; Wu, C.T.; Chen, S.F.; Liu, C. An estimation of subsurface settlement due to shield tunneling. Tunn. Undergr. Space Technol.
**2014**, 44, 121–129. [Google Scholar] [CrossRef]

**Figure 1.**Vertical displacements under various conditions: (

**a**) overburden depth of 15 m, (

**b**) overburden depth of 20 mn (

**c**) and overburden depth of 25 m.

**Figure 2.**Behavior of soil due to tunneling under various conditions: (

**a**) overburden depth of 15 m, (

**b**) overburden depth of 20 m, (

**c**) and overburden depth of 25 m. (

**d**) Direction of maximum shear stress.

**Figure 4.**(

**a**) Vertical displacements and fitted curves at various depths up to 15 m. (

**b**) Parameter w/w

_{0}and its fitted curve at various depths. (

**c**) Parameter i

_{s}and its fitted curve at various depths.

**Figure 5.**(

**a**) Vertical displacements and fitted curves at various depths up to 20 m. (

**b**) Parameter w/w

_{0}and its fitted curve at various depths. (

**c**) Parameter i

_{s}and its fitted curve at various depths.

**Figure 6.**(

**a**) Vertical displacements and fitted curves at various depths up to 25 m. (

**b**) Parameter w/w

_{0}and its fitted curve at various depths. (

**c**) Parameter i

_{s}and its fitted curve at various depths.

**Figure 7.**(

**a**) Maximum engineering shear strains and fitted curves at various depths up to 15 m. (

**b**) Parameter A and its fitted curve at various depths. (

**c**) Parameter B and its fitted curve at various depths.

**Figure 8.**(

**a**) Maximum engineering shear strains and fitted curves at various depths up to 20 m. (

**b**) Parameter A and its fitted curve at various depths. (

**c**) Parameter B and its fitted curve at various depths.

**Figure 9.**(

**a**) Maximum engineering shear strains and fitted curves at various depths up to 25 m. (

**b**) Parameter A and its fitted curve at various depths. (

**c**) Parameter B and its fitted curve at various depths.

Maximum Settlement of Surface (mm) | Maximum Settlement of Crown (mm) | ΔE (kN·m) | ΔW (kN·m) | ΔN (kN·m) | Δ (kN·m) | Error (%) | |
---|---|---|---|---|---|---|---|

Depth 15 m | 8.9 | 21.0 | 187.3 | 199.9 | 32.4 | 19.8 | 10.6 |

Depth 20 m | 9.9 | 28.9 | 415.6 | 456.8 | 56.6 | 15.4 | 3.7 |

Depth 25 m | 10.6 | 36.8 | 677.6 | 753.2 | 87.9 | 12.3 | 1.8 |

Shear Strains | Volume Strains | The Ratio (%) | |
---|---|---|---|

Depth 15 m | 4.554 | 0.111 | 2.443 |

Depth 20 m | 6.531 | −0.079 | 0.121 |

Depth 25 m | 8.620 | −0.036 | 0.419 |

Numerical Method (mm) | Mair et al.’s [4] Method (mm) | Loganathan and Poulos’s [7] Method (mm) | |
---|---|---|---|

Depth 15 m | 8.6 | 5.1 | 8.2 |

Depth 20 m | 9.9 | 8.0 | 5.0 |

Depth 25 m | 10.7 | 11.8 | 3.3 |

Numerical Result (mm) | Analytical Result (mm) | Error (%) | |
---|---|---|---|

Depth 15 m | 8.6 | 8.9 | 3.5 |

Depth 20 m | 9.9 | 10.3 | 4.0 |

Depth 25 m | 10.7 | 11.1 | 3.7 |

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## Share and Cite

**MDPI and ACS Style**

Liu, X.; Fang, Q.; Zhou, Q.; Liu, Y.
Predicting Ground Settlement Due to Symmetrical Tunneling through an Energy Conservation Method. *Symmetry* **2018**, *10*, 186.
https://doi.org/10.3390/sym10060186

**AMA Style**

Liu X, Fang Q, Zhou Q, Liu Y.
Predicting Ground Settlement Due to Symmetrical Tunneling through an Energy Conservation Method. *Symmetry*. 2018; 10(6):186.
https://doi.org/10.3390/sym10060186

**Chicago/Turabian Style**

Liu, Xiang, Qian Fang, Qiushuang Zhou, and Yan Liu.
2018. "Predicting Ground Settlement Due to Symmetrical Tunneling through an Energy Conservation Method" *Symmetry* 10, no. 6: 186.
https://doi.org/10.3390/sym10060186