An analytical model based on complex variable theory is proposed to investigate ground responses due to shallow tunneling in multi-layered ground with an arbitrary ground surface load. The ground layers are assumed to be linear-elastic with full-stick contact between them. To solve the proposed multi-boundary problem, a series of analytic functions is introduced to accurately express the stresses and displacements contributed by different boundaries. Based on the principle of linear-elastic superposition, the multi-boundary problem is converted into a superposition of multiple single-boundary problems. The conformal mappings of different boundaries are independent of each other, which allows the stress and displacement fields to be obtained by the sum of components from each boundary. The analytical results are validated based on numerical and in situ monitoring results. The present model is superior to the classical model for analyzing ground responses of shallow tunneling in multi-layered ground; thus, it can be used with assurance to estimate the ground movement and surface building safety of shallow tunnel constructions beneath surface buildings. Moreover, the solution for the ground stress distribution can be used to estimate the safety of a single-layer composite ground.
0–12 m: extremely soft to soft clay (Cu = 15–25 kPa); > 12 m: extremely hard to hard clay
10/35
0.5/0.48
18.5/1.33
81
Tab.2
Fig.8
Fig.9
Fig.10
Fig.11
Fig.12
Fig.13
Fig.14
1
O Jenck, D Dias. 3D-finite difference analysis of the interaction between concrete building and shallow tunnelling. Geotechnique, 2004, 54(8): 519–528 https://doi.org/10.1680/geot.2004.54.8.519
2
Q Fang, D Zhang, L N Y Wong. Shallow tunnelling method (STM) for subway station construction in soft ground. Tunnelling and Underground Space Technology, 2012, 29: 10–30 https://doi.org/10.1016/j.tust.2011.12.007
3
D Zhang, Q Fang, Y Hou, P Li, L N Yuen Wong. Protection of buildings against damages as a result of adjacent large-span tunneling in shallowly buried soft ground. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 139(6): 903–913 https://doi.org/10.1061/(ASCE)GT.1943-5606.0000823
4
Q Fang, X Liu, K Zeng, X Zhang, M Zhou, J Du. Centrifuge modelling of tunnelling below existing twin tunnels with different types of support. Underground Space, 2022, 7(6): 1125–1138 https://doi.org/10.1016/j.undsp.2022.02.007
5
K J Shou, J A L Napier. A two-dimensional linear variation displacement discontinuity method for three-layered elastic media. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(6): 719–729 https://doi.org/10.1016/S0148-9062(99)00042-X
6
D Zhang, H Huang, Q Hu, F Jiang. Influence of multi-layered soil formation on shield tunnel lining behavior. Tunnelling and Underground Space Technology, 2015, 47: 123–135 https://doi.org/10.1016/j.tust.2014.12.011
7
Z Yuan, Z Cao, H Tang, Y Xu, T Wu. Analytical layer element with a circular cavity and its application in predicting ground vibrations from surface and underground moving sources. Computers and Geotechnics, 2021, 137: 104262 https://doi.org/10.1016/j.compgeo.2021.104262
8
J Zhang, Y Gao, X Liu, Z Zhang, Y Yuan, H Mang. A shield tunneling method for enlarging the diameter of existing tunnels: Experimental investigations. Tunnelling and Underground Space Technology, 2022, 128: 104605 https://doi.org/10.1016/j.tust.2022.104605
9
N A Do, D Dias. Tunnel lining design in multi-layered grounds. Tunnelling and Underground Space Technology, 2018, 81: 103–111 https://doi.org/10.1016/j.tust.2018.07.005
10
E Ieronymaki, A J Whittle, H H Einstein. Comparative study of the effects of three tunneling methods on ground movements in stiff clay. Tunnelling and Underground Space Technology, 2018, 74: 167–177 https://doi.org/10.1016/j.tust.2018.01.005
11
Z Chen, C He, G Xu, G Ma, D Wu. A case study on the asymmetric deformation characteristics and mechanical behavior of deep-buried tunnel in phyllite. Rock Mechanics and Rock Engineering, 2019, 52(11): 4527–4545 https://doi.org/10.1007/s00603-019-01836-2
12
H Zheng, P Li, G Ma, Q Zhang. Experimental investigation of mechanical characteristics for linings of twins tunnels with asymmetric cross-section. Tunnelling and Underground Space Technology, 2022, 119: 104209 https://doi.org/10.1016/j.tust.2021.104209
13
P Li, F Wang, C Zhang, Z Li. Face stability analysis of a shallow tunnel in the saturated and multilayered soils in short-term condition. Computers and Geotechnics, 2019, 107: 25–35 https://doi.org/10.1016/j.compgeo.2018.11.011
14
D Zhang, S Chen, R Wang, D Zhang, B Li. Behaviour of a large-diameter shield tunnel through multi-layered strata. Tunnelling and Underground Space Technology, 2021, 116: 104062 https://doi.org/10.1016/j.tust.2021.104062
15
R B Peck. Deep excavations and tunneling in soft ground. In: Proceedings of 7th International Conference on Soil Mechanics and Foundation Engineering. Mexico City: Sociedad Mexicana de Mecanica de Suelo, 1969, 225–290
16
A R Selby. Surface movements caused by tunnelling in two-layer soil. Geological Society Engineering Geology Special Publication, 1988, 5(1): 71–77 https://doi.org/10.1144/GSL.ENG.1988.005.01.05
17
Z Sun, D Zhang, A Li, S Lu, Q Tai, Z Chu. Model test and numerical analysis for the face failure mechanism of large cross-section tunnels under different ground conditions. Tunnelling and Underground Space Technology, 2022, 130: 104735 https://doi.org/10.1016/j.tust.2022.104735
18
J Zhang, X Liu, T Ren, Y Shi, Y Yuan. Numerical analysis of tunnel segments strengthened by steel-concrete composites. Underground Space, 2022, 7(6): 1115–1124 https://doi.org/10.1016/j.undsp.2022.02.004
19
Q Di, P Li, M Zhang, J Wu. Influence of permeability anisotropy of seepage flow on the tunnel face stability. Underground Space, 2022, 8: 1–14 https://doi.org/10.1016/j.undsp.2022.04.009
20
Q Di, P Li, M Zhang, X Cui. Investigation of progressive settlement of sandy cobble strata for shield tunnels with different burial depths. Engineering Failure Analysis, 2022, 141: 106708 https://doi.org/10.1016/j.engfailanal.2022.106708
21
Q Di, P Li, M Zhang, X Cui. Influence of relative density on deformation and failure characteristics induced by tunnel face instability in sandy cobble strata. Engineering Failure Analysis, 2022, 141: 106641 https://doi.org/10.1016/j.engfailanal.2022.106641
22
Z Sun, D Zhang, Q Fang, G Dui, Q Tai, F Sun. Analysis of the interaction between tunnel support and surrounding rock considering pre-reinforcement. Tunnelling and Underground Space Technology, 2021, 115: 104074 https://doi.org/10.1016/j.tust.2021.104074
23
Z Sun, D Zhang, Q Fang, D Liu, G Dui. Displacement process analysis of deep tunnels with grouted rockbolts considering bolt installation time and bolt length. Computers and Geotechnics, 2021, 140: 104437 https://doi.org/10.1016/j.compgeo.2021.104437
24
Z Sun, D Zhang, Q Fang, G Dui, Z Chu. Analytical solutions for deep tunnels in strain-softening rocks modeled by different elastic strain definitions with the unified strength theory. Science China. Technological Sciences, 2022, 65(10): 2503–2519 https://doi.org/10.1007/s11431-022-2158-9
25
Z Zhang, M Huang, M Zhang. Theoretical prediction of ground movements induced by tunnelling in multi-layered soils. Tunnelling and Underground Space Technology, 2011, 26(2): 345–355 https://doi.org/10.1016/j.tust.2010.11.005
26
D M Zymnis, I Chatzigiannelis, A J Whittle. Effect of anisotropy in ground movements caused by tunnelling. Geotechnique, 2013, 63(13): 1083–1102 https://doi.org/10.1680/geot.12.P.056
27
L Cao, D Zhang, Q Fang. Semi-analytical prediction for tunnelling-induced ground movements in multi-layered clayey soils. Tunnelling and Underground Space Technology, 2020, 102: 103446 https://doi.org/10.1016/j.tust.2020.103446
28
A Verruijt, J R Booker. Surface settlements due to deformation of a tunnel in an elastic half plane. Geotechnique, 1996, 46(4): 753–756 https://doi.org/10.1680/geot.1996.46.4.753
Q Fang, H Song, D Zhang. Complex variable analysis for stress distribution of an underwater tunnel in an elastic half plane. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(16): 1821–1835 https://doi.org/10.1002/nag.2375
31
D Zhang, T Xu, H Fang, Q Fang, L Cao, M Wen. Analytical modeling of complex contact behavior between rock mass and lining structure. Journal of Rock Mechanics and Geotechnical Engineering, 2022, 14(3): 813–824 https://doi.org/10.1016/j.jrmge.2021.10.007
32
H Fang, D Zhang, Q Fang. A semi-analytical method for frictional contact analysis between rock mass and concrete linings. Applied Mathematical Modelling, 2022, 105: 17–28 https://doi.org/10.1016/j.apm.2021.12.030
33
Z ZhangM HuangY PanZ LiS MaY Zhang. Time-dependent analyses for ground movement and stress field induced by tunnelling considering rainfall infiltration mechanics. Tunnelling and Underground Space Technology, 2022, 122, 104378
34
H Fang, D Zhang, Q Fang, M Wen. A generalized complex variable method for multiple tunnels at great depth considering the interaction between linings and surrounding rock. Computers and Geotechnics, 2021, 129: 103891 https://doi.org/10.1016/j.compgeo.2020.103891
35
H Katebi, A H Rezaei, M Hajialilue-Bonab, A Tarifard. Assessment the influence of ground stratification, tunnel and surface buildings specifications on shield tunnel lining loads (by FEM). Tunnelling and Underground Space Technology, 2015, 49: 67–78 https://doi.org/10.1016/j.tust.2015.04.004
36
H Huang, D Zhang. Resilience analysis of shield tunnel lining under extreme surcharge: Characterization and field application. Tunnelling and Underground Space Technology, 2016, 51: 301–312 https://doi.org/10.1016/j.tust.2015.10.044
37
H P Simon, G Marte. Effect of surface loading on the hydro-mechanical response of a tunnel in saturated ground. Underground Space, 2016, 1(1): 1–19 https://doi.org/10.1016/j.undsp.2016.06.001
38
H Wang, X Chen, M Jiang, F Song, L Wu. The analytical predictions on displacement and stress around shallow tunnels subjected to surcharge loadings. Tunnelling and Underground Space Technology, 2018, 71: 403–427 https://doi.org/10.1016/j.tust.2017.09.015
39
X Gao, H Wang, M Jiang. Analytical solutions for the displacement and stress of lined circular tunnel subjected to surcharge loadings in semi-infinite ground. Applied Mathematical Modelling, 2021, 89: 771–791 https://doi.org/10.1016/j.apm.2020.07.061
40
Z Zhang, M Huang, Y Pan, K Jiang, Z Li, S Ma, Y Zhang. Analytical prediction of time-dependent behavior for tunneling-induced ground movements and stresses subjected to surcharge loading based on rheological mechanics. Computers and Geotechnics, 2021, 129: 103858 https://doi.org/10.1016/j.compgeo.2020.103858
41
K H Park. Analytical solution for tunnelling-induced ground movement in clays. Tunnelling and Underground Space Technology, 2005, 20(3): 249–261 https://doi.org/10.1016/j.tust.2004.08.009
42
K M Lee, R K Rowe, K Y Lo. Subsidence owing to tunnelling. I. Estimating the gap parameter. Canadian Geotechnical Journal, 1992, 29(6): 929–940 https://doi.org/10.1139/t92-104
43
N Loganathan, H G Poulos. Analytical prediction for tunneling-induced ground movements in clays. Journal of Geotechnical and Geoenvironmental Engineering, 1998, 124(9): 846–856 https://doi.org/10.1061/(ASCE)1090-0241(1998)124:9(846
44
P B Attewell, I W Farmer. Ground deformations resulting from shield tunnelling in London clay. Canadian Geotechnical Journal, 1974, 11(3): 380–395 https://doi.org/10.1139/t74-039
45
R K Rowe, G J Kack. Theoretical examination of the settlements induced by tunneling: Four case histories. Canadian Geotechnical Journal, 1983, 20(2): 299–314 https://doi.org/10.1139/t83-033
46
N Phienwej. Ground movements in shield tunnelling in Bangkok soils. In: Proceedings of 14th International Conference on Soil Mechanics and Foundation Engineering. Hamburg: International Society for Soil Mechanics and Foundation Engineering, 1997, 1469–1472
47
B ZengD HuangN PengF Chen. Analogous stochastic medium theory method (ASMTM) for predicting soil displacement induced by general and special-section shield tunnel construction. Chinses Journal of Rock Mechanics and Engineering, 2018, 37: 4356–4366 (in Chinese)
48
Z SunD ZhangQ FangN HuangfuZ Chu. Convergenceconfinement analysis for tunnels with combined bolt–cable system considering the effects of intermediate principal stress. Acta Geotechnica, 2022 (in press)
