Explicit Peck formula applied to ground displacement based on an elastic analytical solution for a shallow tunnel

Abstract

Using the complex variable method, an elastic analytical solution of the ground displacement caused by a shallow circular tunneling is derived. Non-symmetric deformation relative to the horizontal center line of the tunnel cross-section is used as a boundary condition. A comparison between the proposed analytical method and the Finite Element Method is carried out to validate the rationality of the obtained analytical solution. Two parameters in the Peck formula, namely the maximum settlement of the ground surface center and the width coefficient of settlement curve, are fitted and determined. We propose a modified Peck formula by considering three input parameters, namely the tunnel depth, tunnel radius, and the tunnel gap. The influence of these three parameters on the modified Peck formula is analyzed. The applicability of the modified Peck formula is further investigated by reference to six engineering projects. The ground surface displacement obtained by the explicit Peck formula is in good agreement with the field data, and the maximum error is only 1.3 cm. The proposed formula can quickly and reasonably predict the ground surface settlement caused by tunnelling.

Graphical abstract

Keywords

complex variable method / elastic analytical solution / maximum settlement formula / modified Peck formula

References

[1] Ng C, Boonyarak T, Mašín D. Effects of pillar depth and shielding on the interaction of crossing multitunnels. Journal of Geotechnical and Geoenvironmental Engineering, 2015, 141(6): 04015021

[2] Franza A, Marshall A M, Haji T, Abdelatif A O, Carbonari S, Morici M. A simplified elastic analysis of tunnel-piled structure interaction. Tunnelling and Underground Space Technology, 2017, 61(1): 104–121

[3] Zhao K, Janutolo M, Barla G. A completely 3D model for the simulation of mechanized tunnel excavation. Rock Mechanics and Rock Engineering, 2012, 45(4): 475–497

[4] Zhao K, Bonini M, Debernardi D, Janutolo M, Barla G, Chen G X. Computational modelling of the mechanised excavation of deep tunnels in weak rock. Computers and Geotechnics, 2015, 66: 158–171

[5] Zheng G, Zhang T Y, Diao Y. Mechanism and countermeasures of preceding tunnel distortion induced by succeeding EPBS tunneling in close proximity. Computers and Geotechnics, 2015, 66: 53–65

[6] Zhang M J, Li S H, Li P F. Numerical analysis of ground displacement and segmental stress and influence of yaw excavation loadings for a curved shield tunnel. Computers and Geotechnics, 2020, 118: 103325

[7] Zhu W S, Li Y, Li S C, Wang S G, Zhang Q B. Quasi-three-dimensional physical model tests on a cavern complex under high in-situ stresses. International Journal of Rock Mechanics and Mining Sciences, 2011, 48(2): 199–209

[8] Li P F, Zhao Y, Zhou X J. Displacement characteristics of high-speed railway tunnel construction in loess ground by using multi-step excavation method. Tunnelling and Underground Space Technology, 2016, 51: 41–55

[9] Yu H T, Cai C, Guan X F, Yuan Y. Analytical solution for long lined tunnels subjected to travelling loads. Tunnelling and Underground Space Technology, 2016, 58: 209–215

[10] Yu H T, Cai C, Bobet A, Zhao X, Yuan Y. Analytical solution for longitudinal bending stiffness of shield tunnels. Tunnelling and Underground Space Technology, 2019, 83: 27–34

[11] Zhang Z G, Huang M S, Zhang C P, Jiang K M, Wang Z W, Xi X G. Complex variable solution for twin tunneling-induced ground movements considering nonuniform convergence pattern. International Journal of Geomechanics, 2020, 20(6): 04020060

[12] Lu D C, Lin Q T, Tian Y, Du X L, Gong Q M. Formula for predicting ground settlement induced by tunnelling based on Gaussian function. Tunnelling and Underground Space Technology, 2020, 103: 103443

[13] Zheng G, Dai X, Diao Y, Zeng C F. Experimental and simplified model study of the development of ground settlement under hazards induced by loss of groundwater and sand. Natural Hazards, 2016, 82(3): 1869–1893

[14] Hasanpour R, Chakeri H, Ozcelik Y, Denek H. Evaluation of surface settlements in the Istanbul metro in terms of analytical, numerical and direct measurements. Bulletin of Engineering Geology and the Environment, 2012, 71(3): 499–510

[15] Zhang Z G, Pan Y T, Zhang M X, Lv X L, Jiang K M, Li S N. Complex variable analytical prediction for ground deformation and lining responses due to shield tunneling considering groundwater level variation in clays. Computers and Geotechnics, 2020, 120: 103443

[16] Sagaseta C. Analysis of undrained soil deformation due to ground loss. Geotechnique, 1987, 37(3): 301–320

[17] Verruijt A. A complex variable solution for a deforming circular tunnel in an elastic half-plane. International Journal for Numerical and Analytical Methods in Geomechanics, 1997, 21(2): 77–89

[18] PeckR B. Deep excavations and tunneling in soft ground. In: Proceedings of 7th International Conference SMFE. Mexico: A.A. Balkema, 1969, 225–290

[19] Mair R J, Taylor R N, Bracegirdle A. Subsurface settlement profiles above tunnels in clays. Geotechnique, 1993, 43(2): 315–320

[20] Chou W I, Bobet A. Predictions of ground deformations in shallow tunnels in clay. Tunnelling and Underground Space Technology, 2002, 17(1): 3–19

[21] Park K H. Elastic solution for tunneling-induced ground movements in clays. International Journal of Geomechanics, 2004, 4(4): 310–318

[22] Yang X L, Wang J M. Ground movement prediction for tunnels using simplified procedure. Tunnelling and Underground Space Technology, 2011, 26(3): 462–471

[23] Fu J Y, Yang J S, Yan L, Abbas S M. An analytical solution for deforming twin-parallel tunnels in an elastic half plane. International Journal for Numerical and Analytical Methods in Geomechanics, 2015, 39(5): 524–538

[24] Fang Q, Song H R, Zhang D L. 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

[25] Zhang Z G, Huang M S, Xi X, Yuan X. Complex variable solutions for soil and liner deformation due to tunneling in clays. International Journal of Geomechanics, 2018, 18(7): 04018074

[26] Verruijt A. Deformations of an elastic half plane with a circular cavity. International Journal of Solids and Structures, 1998, 35(21): 2795–2804

[27] Lu D C, Kong F C, Du X L, Shen C P, Gong Q M, Li P F. A unified displacement function to analytically predict ground deformation of shallow tunnel. Tunnelling and Underground Space Technology, 2019, 88: 129–143

[28] Kong F C, Lu D C, Du X L, Shen C P. Displacement analytical prediction of shallow tunnel based on unified displacement function under slope boundary. International Journal for Numerical and Analytical Methods in Geomechanics, 2019, 43(1): 183–211

[29] Kong F C, Lu D C, Du X L, Shen C P. Elastic analytical solution of shallow tunnel owing to twin tunnelling based on a unified displacement function. Applied Mathematical Modelling, 2019, 68: 422–442

[30] Verruijt A, Strack O E. Buoyancy of tunnels in soft soils. Geotechnique, 2008, 58(6): 513–515

[31] Park K H. Analytical solution for tunnelling-induced ground movement in clays. Tunnelling and Underground Space Technology, 2005, 20(3): 249–261

[32] Wang L Z, Li L L, Lu X J. Complex variable solutions for tunneling-induced ground movement. International Journal of Geomechanics, 2009, 9(2): 63–72

[33] Fu J Y, Yang J S, Klapperich H, Wang S Y. Analytical prediction of ground movements due to a nonuniform deforming tunnel. International Journal of Geomechanics, 2016, 16(4): 04015089

[34] Lu D C, Kong F C, Du X L, Shen C P, Su C C, Wang J. Fractional viscoelastic analytical solution for the ground displacement of a shallow tunnel based on a time-dependent unified displacement function. Computers and Geotechnics, 2020, 117: 103284

[35] Marshall A M, Farrell R, Klar A, Mair R J. Tunnels in sands: The effect of size, depth and volume loss on greenfield displacements. Geotechnique, 2012, 62(5): 385–399

[36] Loganathan N, Poulos H G. Analytical prediction for tunneling induced ground movement in clays. Journal of Geotechnical and Geoenvironmental Engineering, 1998, 124(9): 846–856

[37] Lee Y J. Investigation of subsurface deformations associated with model tunnels in a granular mass. Tunnelling and Underground Space Technology, 2009, 24(6): 654–664

[38] Lin Q T, Meng X, Lu D C, Miao J B, Zhao Z H, Du X L. A unified empirical method for predicting both vertical and horizontal ground displacements induced by tunnel excavation. Canadian Geotechnical Journal, 2024, 61(7): 1468–1495

[39] Gonzalez C, Sagaseta C. Patterns of soil deformations around tunnels. Application to the extension of Madrid Metro. Computers and Geotechnics, 2001, 28(6): 445–468

[40] Pinto F, Whittle A J. Discussion of “Elastic solution for tunneling-induced ground movements in clays” by KH Park. International Journal of Geomechanics, 2006, 6(1): 72–73

[41] Lee K M, Rowe R K, Lo K Y. Subsidence owing to tunnelling. I. Estimating the gap parameter. Canadian Geotechnical Journal, 1992, 29(6): 929–940

[42] MuskhelishviliN I. Some Basic Problems of The Mathematical Theory of Elasticity. Groningen: Noordhoff, 1953

[43] Kong F C, Lu D C, Ma Y D, Tian T, Yu H T, Du X L. Novel hybrid method to predict the ground-displacement field caused by shallow tunnel excavation. Science China Technological Sciences, 2023, 66(1): 101–114

[44] Kong F C, Lu D C, Ma C, Shen C P, Yang X D, Du X L. Fractional viscoelastic solution of stratum displacement of a shallow tunnel under the surface slope condition. Underground Space, 2023, 10: 233–247

[45] Palmer J H L, Belshaw D J. Deformations and pore pressures in the vicinity of a precast, segmented, concrete-lined tunnel in clay. Canadian Geotechnical Journal, 1980, 17(2): 174–184

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