Abstract: Tunnelling and underground excavations involve the removal of soil or rock mass from their initial locations. This process may be accurately modelled by the unloading of a cavity from the in-situ stress state. In the past decades, for the purpose of controlling ground deformation and maintaining tunnel stability, cavity contraction theory had been used to predict ground reaction curves, to estimate ground settlements due to tunnelling and to design tunnel support systems. This paper presented a comprehensive review of analytical cavity contraction solutions for these typical applications. For illustration, a new finite strain analytical solution was derived for the unloading analysis of a cylindrical spherical cavity in non-associated Mohr Coulomb materials with a finite radial extent. This paper would be beneficial for the future development of cavity contraction theory and its application to practical tunnelling problems.
余海岁, 庄培芝. 岩土介质小孔收缩理论及其在隧道工程中的应用[J]. 隧道与地下工程灾害防治, 2019, 1(4): 13-32.
YU Haisui, ZHUANG Peizhi. Cavity contraction theory and its application to tunnelling. Hazard Control in Tunnelling and Underground Engineering, 2019, 1(4): 13-32.
洪开荣. 我国隧道及地下工程近两年的发展与展望[J]. 隧道建设, 2017, 37(2): 123-134. HONG Kairong. Development and prospects of tunnels and underground works in China in recent two years[J]. Tunnel Construction, 2017, 37(2): 123-134.
[2]
MAIR R J. Tunnelling and geotechnics: new horizons[J]. Géotechnique, 2008, 58(9): 695-736.
[3]
田四明, 赵勇, 石少帅,等. 中国铁路隧道建设期典型灾害防控方法现状、问题与对策[J]. 隧道与地下工程灾害防治, 2019(2): 24-48. TIAN Siming, ZHAO Yong, SHI Shaoshuai, et al. The status, problems and countermeasures of typical disaster prevention and control methods during the construction period of Chinese railway tunnels[J]. Hazard Control in Tunnelling and Underground Engineering, 2019(2): 24-48.
[4]
钱七虎. 地下工程建设安全面临的挑战与对策[J]. 岩石力学与工程学报, 2012,31(10): 1945-1956. QIAN Qihu. Challenges faced by underground projects construction safety and countermeasures[J]. Chinese Journal of Rock Mechanics and Engineering, 2012, 31(10): 1945-1956.
[5]
王梦恕, 张成平. 城市地下工程建设的事故分析及控制对策[J]. 建筑科学与工程学报,2008,25(2): 1-6. WANG Mengshu, ZHANG Chengping. Analysis of accident induced by urban underground project construction and its control measures[J]. Journal of Architecture and Civil Engineering, 2008, 25(2): 1-6.
GONZÁLEZ C, SAGASETA C. Patterns of soil deformations around tunnels. application to the extension of Madrid metro[J]. Computers and Geotechnics, 2001, 28(6/7): 445-468.
[8]
VERRUIJT A. A complex variable solution for a deforming circular tunnel in an elastic half-plane[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1997, 21(2): 77-89.
[9]
VERRUIJT A, BOOKER J R. Surface settlements due to deformation of a tunnel in an elastic half plane[J]. Géotechnique, 1996, 46(4): 753-756.
[10]
陈子萌. 围岩力学分析中的解析方法[M]. 北京: 煤炭工业出版社; 1994.
[11]
MAIR R J, TAYLOR R N. In prediction of clay behaviour around tunnels using plasticity solutions[C] //In Predictive Soil Mechanics: Proceedings of the Wroth Memorial Symposium. Oxford, UK:Thomas Telford, 1993: 449-463.
[12]
SAGASETA C. Analysis of undraind soil deformation due to ground loss[J].Géotechnique, 1987, 37(3): 301-320.
[13]
TERZAGHI K, RICHART F E Jr. Stresses in rock about cavities[J]. Géotechnique, 1952, 3(2): 57-90.
[14]
BROWN E T, BRAY J W, LADANYI B, et al. Ground response curves for rock tunnels[J].Journal of Geotechnical Engineering, 1983, 109(1): 15-39.
[15]
YU H S. Cavity expansion methods in geomechanics[M].Dordrecht, Netherlands: Springer Netherlands, 2000.
[16]
LOGANATHAN N, POULOS H G. Analytical prediction for tunneling-induced ground movements in clays[J]. Journal of Geotechnical and Geoenvironmental Engineering, 1998, 124(9): 846-856.
[17]
PINTO F, WHITTLE A J. Ground movements due to shallow tunnels in soft ground. I: analytical solutions[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2014, 140(4): 04013040.
[18]
MARSHALL A M. Tunnel-pile interaction analysis using cavity expansion methods[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2012, 138(10): 1237-1246.
[19]
OGAWA T, LO K Y. Effects of dilatancy and yield criteria on displacements around tunnels[J].Canadian Geotechnical Journal, 1987, 24(1): 100-113.
[20]
CARRANZA-TORRES C, FAIRHURST C. Application of the convergence-confinement method of tunnel design to rock masses that satisfy the hoek-brown failure criterion[J].Tunnelling and Underground Space Technology, 2000, 15(2): 187-213.
[21]
BRADY B H, BROWN E T. Rock mechanics for underground mining [M].3rd Ed. London, UK: Kluwer Academic Publishers, 2004.
[22]
PANET M. Calcul des tunnels par la Me'thode de convergence—confinement[M].Paris, France: Presses de l’Ecole Nationale des Ponts et Chausse'es, 1995.
[23]
FENNER R. Untersuchungen zur erkenntnis des gebirgsdrucks[J]. Glückauf, 1938(74): 681-695, 705-715.
[24]
张治国, 师敏之, 张成平, 等. 类矩形盾构隧道开挖引起邻近地下管线变形研究[J]. 岩石力学与工程学报, 2019, 38(4): 852-864. ZHANG Zhiguo, SHI Minzhi, ZHANG Chengping, et al. Research on deformation of adjacent underground pipelines caused by excavation of quasi-rectangular shields[J]. Chinese Journal of Rock Mechanics and Engineering, 2019, 38(4): 852-864.
WANG H N, UTILI S, JIANG M J. An analytical approach for the sequential excavation of axisymmetric lined tunnels in viscoelastic rock[J].International Journal of Rock Mechanics and Mining Sciences,2014, 68: 85-106.
[27]
SAVIN G N. Stress distribution around holes[M].Washington, D.C., USA: National Aeronautics and Space Administration, 1970.
[28]
MUSKHELISHVILI N I. Some basic problems of the mathematical theory of elasticity[M].Dordrecht, Netherlands:Springer Netherlands, 1977.
[29]
LEE K M, ROWE R K, LO K Y. Subsidence owing to tunnelling I: estimating the gap parameter[J]. Canadian Geotechnical Journal, 1992, 29(6): 929-940.
[30]
VLACHOPOULOS N, DIEDERICHS M S. Improved longitudinal displacement profiles for convergence confinement analysis of deep tunnels[J]. Rock Mechanics and Rock Engineering, 2009, 42(2): 131-146.
[31]
VLACHOPOULOS N, DIEDERICHS M S. Appropriate uses and practical limitations of 2D numerical analysis of tunnels and tunnel support response[J]. Geotechnical and Geological Engineering, 2014, 32(2): 469-488.
[32]
HUANG M S, LI S, YU J, et al. Continuous field based upper bound analysis for three-dimensional tunnel face stability in undrained clay[J]. Computers and Geotechnics, 2018, 94: 207-213.
[33]
SLOAN S W, ASSADI A. Undrained stability of a square tunnel in a soil whose strength increases linearly with depth[J]. Computers and Geotechnics, 1991, 12(4): 321-346.
[34]
GONZÁLEZ-NICIEZA C, ÁLVAREZ-VIGIL A E, MENÉNDEZ-DÍAZ A, et al. Influence of the depth and shape of a tunnel in the application of the convergence-confinement method[J].Tunnelling and Underground Space Technology, 2008, 23(1): 25-37.
[35]
ROWE R K, LO K Y, KACK G J. A method of estimating surface settlement above tunnels constructed in soft ground[J]. Canadian Geotechnical Journal, 1983, 20(1): 11-22.
[36]
KARAKUS M. Appraising the methods accounting for 3D tunnelling effects in 2D plane strain FE analysis[J].Tunnelling and Underground Space Technology, 2007, 22(1): 47-56.
[37]
DO N A, DIAS D. A comparison of 2D and 3D numerical simulations of tunnelling in soft soils[J]. Environmental Earth Sciences, 2017, 76(3): 102.
[38]
AFTES. Analysis of tunnel stability by the convergence-confinement method[J].Underground Space. 1978, 4(4): 221-223.
[39]
ROWE R K, KACK G J. A theoretical examination of the settlements induced by tunnelling: four case histories[J]. Canadian Geotechnical Journal, 1983, 20(2): 299-314.
[40]
阿比尔的, 郑颖人, 冯夏庭, 等. 隧道特征线法的修正与发展[J].岩石力学与工程学报, 2015, 34(增刊1): 3067-3073. ABI Erdi, ZHENG Yingren, FENG Xiating, et al. Modification and development of characteristic line method of tunnel[J]. Chinese Journal of Rock Mechanics and Engineering, 2015, 34(Suppl.1): 3067-3073.
[41]
ALEJANO L R, RODRÍGUEZ-DONO A, VEIGA M. Plastic radii and longitudinal deformation profiles of tunnels excavated in strain-softening rock masses[J]. Tunnelling and Underground Space Technology, 2012, 30: 169-182.
[42]
崔岚, 郑俊杰, 苗晨曦, 等. 隧道纵向变形曲线与围岩特征曲线耦合分析[J].岩土工程学报, 2014, 36(4): 707-715. CUI Lan, ZHENG Junjie, MIAO Chenxi, et al. Coupling analysis of longitudinal deformation profile and ground reaction curve[J]. Chinese Journal of Geotechnical Engineering, 2014, 36(4): 707-715.
[43]
FRANZA A, MARSHALL A M. Empirical and semi-analytical methods for evaluating tunnelling-induced ground movements in sands[J].Tunnelling and Underground Space Technology, 2019, 88: 47-62.
[44]
PANET M, GUENOT A. In Analysis of convergence behind the face of a tunnel[C] //Proceedings of Tunnelling82. London, UK:Pergamon Press, 1983:197-204.
[45]
CORBETTA F, BERNAUD D, NGUYEN Minh D. Contribution à la méthode convergence-confinement par le principe de la similitude[J].Revue Française De Géotechnique, 1991(54): 5-11.
[46]
UNLU T, GERCEK H. Effect of poisson's ratio on the normalized radial displacements occurring around the face of a circular tunnel[J]. Tunnelling and Underground Space Technology, 2003, 18(5): 547-553.
[47]
张常光, 赵均海, 张庆贺. 基于统一强度理论的深埋圆形岩石隧道收敛限制分析[J].岩土工程学报, 2012, 34(1): 110-114. ZHANG Changguang, ZHAO Junhai, ZHANG Qinghe. Convergence-confinement analysis of deep circular rock tunnels based on unified strength theory[J]. Chinese Journal of Geotechnical Engineering, 2012, 34(1): 110-114.
[48]
FANG Y, CHEN Z, TAO L, et al. Model tests on longitudinal surface settlement caused by shield tunnelling in sandy soil[J]. Sustainable Cities and Society, 2019, 47: 101504.
[49]
FARGNOLI V, BOLDINI D, AMOROSI A. TBM tunnelling-induced settlements in coarse-grained soils: the case of the new Milan underground line 5[J]. Tunnelling and Underground Space Technology, 2013, 38: 336-347.
[50]
MOUSIVAND M, MALEKI M, NEKOOEI M, et al. Application of convergence-confinement method in analysis of shallow non-circular tunnels[J].Geotechnical and Geological Engineering, 2017, 35(3): 1185-1198.
[51]
SONG S G, LI S C, LI L P, et al. Study on longitudinal deformation profile of rock mass in a subsea tunnel[J]. Marine Georesources & Geotechnology, 2016, 34(4): 376-383.
[52]
LUO Y B, CHEN J X, CHEN Y, et al. Longitudinal deformation profile of a tunnel in weak rock mass by using the back analysis method[J].Tunnelling and Underground Space Technology, 2018, 71: 478-493.
[53]
VRAKAS A, ANAGNOSTOU G. A finite strain closed-form solution for the elastoplastic ground response curve in tunnelling[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(11): 1131-1148.
[54]
PARK K, KIM Y. Analytical solution for a circular opening in an elastic-brittle-plastic rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2006, 43(4): 616-622.
[55]
YU H S, ROWE R K. Plasticity solutions for soil behaviour around contracting cavities and tunnels[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 1999, 23(12): 1245-1279.
[56]
YU H S, ZHUANG P Z, MO P Q. A unified critical state model for geomaterials with an application to tunnelling[J].Journal of Rock Mechanics and Geotechnical Engineering, 2019, 11(3): 464-480.
[57]
CARRANZA-TORRES C, FAIRHURST C. The elasto-plastic response of underground excavations in rock masses that satisfy the Hoek-Brown failure criterion[J]. International Journal of Rock Mechanics and Mining Sciences, 1999, 36(6): 777-809.
[58]
DETOURNAY E. Elastoplastic model of a deep tunnel for a rock with variable dilatancy[J].Rock Mechanics and Rock Engineering, 1986, 19(2): 99-108.
[59]
SHARAN S K. Exact and approximate solutions for displacements around circular openings in elastic-brittle-plastic Hoek-Brown rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2005, 42(4): 542-549.
[60]
MO P Q, YU H S. Undrained cavity-contraction analysis for prediction of soil behavior around tunnels[J].International Journal of Geomechanics, 2017, 17(5): 04016121.
[61]
ALONSO E, ALEJANO L R, VARAS F, et al. Ground response curves for rock masses exhibiting strain-softening behaviour[J].International Journal for Numerical and Analytical Methods in Geomechanics, 2003, 27(13): 1153-1185.
[62]
CHEN S L, ABOUSLEIMAN Y N, MURALEETHARAN K K.Closed-form elastoplastic solution for the wellbore problem in strain hardening/softening rock formations[J]. International Journal of Geomechanics, 2012, 12(4): 494-507.
[63]
范鹏贤, 王明洋, 李文培. 岩土介质中圆形隧洞围岩压力理论分析进展[J].现代隧道技术,2010,47(2): 1-7,23. FAN Pengxian WANG Mingyang, LI Wenpei. Progress in theoretical analysis of ground response to circular excavations in rock & soil medium[J]. Modern Tunnelling Technology, 2010, 47(2): 1-7, 23.
[64]
VRAKAS A, ANAGNOSTOU G. A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences[J]. Géotechnique, 2015, 65(11): 936-944.
[65]
SHARAN S K. Analytical solutions for stresses and displacements around a circular opening in a generalized Hoek-Brown rock[J]. International Journal of Rock Mechanics and Mining Sciences, 2008, 45(1): 78-85.
[66]
MASSINAS S A, SAKELLARIOU M G. Closed-form solution for plastic zone formation around a circular tunnel in half-space obeying Mohr-Coulomb criterion[J].Géotechnique, 2009, 59(8): 691-701.
[67]
OSMAN A S, BOLTON M D, MAIR R J. Predicting 2D ground movements around tunnels in undrained clay[J]. Géotechnique, 2006, 56(9): 597-604.
[68]
MAIR R J, GUNN M J, O'REILLY M P. In ground movement around shallow tunnels in soft clay[C] //Proceedings of the 10th International Conference on Soil Mechanics and Foundation Engineering. Stockholm, Sweden:[s.n.] , 1981:323-328.
[69]
GRANT R J. Movements around a tunnel in two-layer ground[D].London, UK: City University London, 1998.
[70]
FRANZA A, MARSHALL A M, ZHOU B. Greenfield tunnelling in sands: the effects of soil density and relative depth[J]. Géotechnique, 2019, 69(4): 297-307.
[71]
ZHUANG P Z, YANG H, YU H S, et al. Plasticity solutions for predictions of undrained stability and ground deformation of shallow tunnels in clay[J].Tunnelling and Underground Space Technology, 2019.
[72]
HOEK E, BROWN ET. Underground excavations in rock[M]. London, UK: the Institute of Mining and Metallurgy, 1980.
[73]
Us Department of Transportation.Improved design of tunnel supports: volume 1: simplified analysis for ground-structure interaction in tunneling[R].SCHWARTZ C W, EINSTEIN H H.UMTA-MA-06-0100-80-4.[S.l.] :Urban Mass Transportation Administration, 1980.
[74]
KIM H J, EISENSTEIN Z. Prediction of tunnel lining loads using correction factors[J].Engineering Geology, 2006, 85(3/4): 302-312.
[75]
CAQUOT A, KERISEL J. Traité de mécanique des sols[M].Paris, France:Gauthier-Villars, 1966.
[76]
MAIR R J. Centrifugal modelling of tunnel construction in soft clay[D].Cambridge, UK: Cambridge University, 1979.
[77]
WU B R, LEE C J. Ground movements and collapse mechanisms induced by tunneling in clayey soil[J]. International Journal of Physical Modelling in Geotechnics, 2003, 3(4): 15-29.
[78]
WILSON D W, ABBO A J, SLOAN S W, et al. Undrained stability of a circular tunnel where the shear strength increases linearly with depth[J]. Canadian Geotechnical Journal, 2011, 48(9): 1328-1342.
[79]
SLOAN S W, ASSADI A. In stability of shallow tunnels in soft ground[C] //Proceedings of the Wroth Memorial Symposium Thoms Telford: in Predictive Soil Mechanics.Oxford, UK:[s.n.] , 1993.
[80]
郑颖人, 王永甫. 隧道稳定性分析与设计方法讲座之一: 隧道围岩压力理论进展与破坏机制研究[J].隧道建设, 2013, 33(6): 423-430. ZHENG Yingren, WANG Yongfu. Evolution of rock mass pressure theory and researches on tunnel failure mechanism[J]. Tunnel Construction, 2013, 33(6): 423-430.
[81]
TIMOSHENKO S P, GOODIER J N. Theory of elasticity[M]. 3rd Ed. London, UK: McGraw-Hill, Inc, 1970.
[82]
GALIN L A. Plane elastic-plastic problem: plastic regions around circular holes in plates and beams[J]. Prikladnaia Matematikai Mekhanika, 1946, 10: 365-386.
[83]
YARUSHINA V M, DABROWSKI M, PODLADCHIKOV Y Y. An analytical benchmark with combined pressure and shear loading for elastoplastic numerical models[J]. Geochemistry, Geophysics, Geosystems, 2010, 11(8):1-16.
[84]
DETOURNAY E, ST JOHN C M. Design charts for a deep circular tunnel under non-uniform loading[J]. Rock Mechanics and Rock Engineering, 1988, 21(2): 119-137.
[85]
TOKAR G. Generalization of Galin's problem to frictional materials and discontinuous stress fields[J]. International Journal of Solids and Structures, 1990, 26(2): 129-147.
[86]
DETOURNAY E. An approximate statical solution of the elastoplastic interface for the problem of Galin with a cohesive-frictional material[J].International Journal of Solids and Structures, 1986, 22(12): 1435-1454.
[87]
ZHUANG P Z, YU H S. Two-dimensional elastoplastic analysis of cylindrical cavity problems in Tresca materials[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2019, 43(8): 1612-1633.
[88]
ZHOU H, KONG G, LIU H, et al. A semi-analytical solution for cylindrical cavity expansion in elastic-perfectly plastic soil under biaxial in situ stress field[J].Géotechnique, 2016, 66(9): 786-788.
[89]
KERCHMAN V I, ERLIKHMAN F M. A variational method of solving an elastic-plastic problem for a body with a circular hole[J]. Journal of Applied Mathematics and Mechanics, 1988, 52(1): 105-110.
[90]
BRADFORD I D R, DURBAN D. Stress and deformation fields around a cylindrical cavity embedded in a pressure-sensitive elastoplastic medium[J]. Journal of Applied Mechanics, 1998, 65(2): 374-379.
[91]
DETOURNAY E, FAIRHURST C. Two-dimensional elastoplastic analysis of a long, cylindrical cavity under non-hydrostatic loading[J]. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 1987, 24(4): 197-211.
[92]
ZHUANG P Z, YU H S. A unified analytical solution for elastic-plastic stress analysis of a cylindrical cavity in Mohr-Coulomb materials under biaxial in situ stresses[J].Géotechnique, 2019, 69(4): 369-376.
[93]
FU J Y, YANG J S, KLAPPERICH H, et al. Analytical prediction of ground movements due to a nonuniform deforming tunnel[J]. International Journal of Geomechanics, 2016, 16(4): 04015089.
[94]
ZHANG Z G, ZHANG M X, JIANG Y J, et al. Analytical prediction for ground movements and liner internal forces induced by shallow tunnels considering non-uniform convergence pattern and ground-liner interaction mechanism[J]. Soils and Foundations, 2017, 57(2): 211-226.
[95]
CHOU W I, BOBET A. Predictions of ground deformations in shallow tunnels in clay[J].Tunnelling and Underground Space Technology, 2002, 17(1): 3-19.
[96]
PARK K H. Elastic solution for tunneling-induced ground movements in clays[J].International Journal of Geomechanics, 2004, 4(4): 310-318.
[97]
STRACK O E, VERRUIJT A. A complex variable solution for a deforming buoyant tunnel in a heavy elastic half-plane[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2002, 26(12): 1235-1252.
[98]
ZHANG Z G, HUANG M S, XI X G, et al. Complex variable solutions for soil and liner deformation due to tunneling in clays[J]. International Journal of Geomechanics, 2018, 18(7): 04018074.
[99]
BOBET A. Analytical solutions for shallow tunnels in saturated ground[J]. Journal of Engineering Mechanics, 2001, 127(12): 1258-1266.
[100]
LITWINISZYN J. In the theories and model research of movements of ground masses[C] // Proceedings of the European Congress on Ground Movement. Leeds, UK:[s.n.] , 1957:209.
YANG J S, LIU B C, WANG M C. Modeling of tunneling-induced ground surface movements using stochastic medium theory[J]. Tunnelling and Underground Space Technology, 2004, 19(2): 113-123.
[104]
YANG X L, WANG J M. Ground movement prediction for tunnels using simplified procedure[J].Tunnelling and Underground Space Technology, 2011, 26(3): 462-471.
[105]
VERRUIJT A, SAGASETA C, STRACK O E. Discussion of “predictions of ground deformations in shallow tunnels in clay”[J]. Tunnelling and Underground Space Technology, 2003, 18(1): 93-94.
[106]
WANG H N, UTILI S, JIANG M J, et al. Analytical solutions for tunnels of elliptical cross-section in rheological rock accounting for sequential excavation[J].Rock Mechanics and Rock Engineering, 2015, 48(5): 1997-2029.
[107]
PECK R B. In deep excavations and tunneling in soft ground[C] //Proceedings of the 7th International Conference on Soil Mechanics and Foundation Engineering. Mexico City, Mexico:[s.n.] , 1969:311-375.
[108]
SCHMIDT B. Settlements and ground movements associated with tunnelling in soil[D].Illinois, USA: University of Illinois at Urbana-Champaign, 1969.
[109]
VORSTER T E, KLAR A, SOGA K, et al. Estimating the effects of tunneling on existing pipelines[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2005, 131(11): 1399-1410.
[110]
ZHANG Q J, WU K, CUI S S, et al. Surface settlement induced by subway tunnel construction based on modified peck formula[J]. Geotechnical and Geological Engineering, 2019, 37(4): 2823-2835.
[111]
ATTEWELL P, WOODMAN J P. Predicting the dynamics of ground settlement and its derivatives caused by tunnelling in soil[J].Ground Engineering, 1982, 15(8): 13-22.
[112]
NEW B M, BOWERS K H. Ground movement model validation at the heathrow express trial tunnel[C] //Tunnelling’94. Boston, USA:Springer US, 1994:301-329.
[113]
REED M B. Stresses and displacements around a cylindrical cavity in soft rock[J].IMA Journal of Applied Mathematics, 1986, 36(3): 223-245.
[114]
CHADWICK P. The quasi-static expansion of a spherical cavity in metals and ideal soils[J].The Quarterly Journal of Mechanics and Applied Mathematics, 1959, 12(1): 52-71.
[115]
YU H S, HOULSBY G T. Finite cavity expansion in dilatant soils: loading analysis[J].Géotechnique, 1991, 41(2): 173-183.
2019, Vol. 1 
