竖向非轴对称位移边界条件下柱孔收缩问题弹性解析
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摘要
针对竖向非轴对称位移边界条件下浅埋隧道开挖围岩位移和地表位移计算,基于虚拟镜像技术的源-源法,将竖向地表的剪应力修正为0,并在应力修正之前获得围岩的弹性解。对于柱孔和镜像柱孔联合作用在水平地表边界产生的竖向正应力和剪应力,采用调和函数及相应的应力函数解答进行应力修正,获得第2和第3部分围岩位移解。通过Boussinesq解答和积分思路,将竖向地表边界的水平正应力修正为0。根据线弹性叠加原理,将以上位移进行叠加,得到竖向非轴对称位移边界条件下浅埋隧道开挖围岩位移的最终解。由围岩位移最终解推导出水平和竖向地表边界的地表沉降和侧向位移。通过与数值模拟方法的比较,验证本文理论方法的可靠性。
A new approach of displacements of the surrounding rock for shallow tunnels excavated under non-axisymmetric displacement boundary conditions on a vertical surface was investigated in this study. In the proposed approach, by using the source-source method of virtual image technique, the shear stress of the vertical ground surface was revised to be zero, and elastic solutions of surrounding rock were obtained before stress revision. For the vertical normal stress and shear stress produced by the combined action of cylindrical cavity and the mirror cylindrical cavity on the horizontal surface boundary, the solution of the stress was corrected by the harmonic function and the corresponding stress function solution, and the displacement of the second and third parts of the surrounding rock were obtained. Based on the Boussinesq's solutions and integral method, the horizontal normal stress of vertical ground surface was revised to be zero. Finally, based on the principle of linear superposition, the above displacements were superimposed to get the final solution of the displacement of the surrounding rock under the vertical non-axisymmetric displacement boundary condition. Furthermore, the ground settlements and lateral displacements of the horizontal and vertical ground surfaces were derived by the proposed approach. The proposed approach was well verified by comparing with the numerical method.
A new approach of displacements of the surrounding rock for shallow tunnels excavated under non-axisymmetric displacement boundary conditions on a vertical surface was investigated in this study. In the proposed approach, by using the source-source method of virtual image technique, the shear stress of the vertical ground surface was revised to be zero, and elastic solutions of surrounding rock were obtained before stress revision. For the vertical normal stress and shear stress produced by the combined action of cylindrical cavity and the mirror cylindrical cavity on the horizontal surface boundary, the solution of the stress was corrected by the harmonic function and the corresponding stress function solution, and the displacement of the second and third parts of the surrounding rock were obtained. Based on the Boussinesq's solutions and integral method, the horizontal normal stress of vertical ground surface was revised to be zero. Finally, based on the principle of linear superposition, the above displacements were superimposed to get the final solution of the displacement of the surrounding rock under the vertical non-axisymmetric displacement boundary condition. Furthermore, the ground settlements and lateral displacements of the horizontal and vertical ground surfaces were derived by the proposed approach. The proposed approach was well verified by comparing with the numerical method.
引文
[1]Bobet A.Analytical solutions for shallow tunnels in saturated ground[J].Journal of Engineering Mechanics,2001,127(12):1258-1266.
[2]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.
[3]SHI C,CAO C,LEI M.An analysis of the ground deformation caused by shield tunnel construction combining an elastic half-space model and stochastic medium theory[J].KSCE Journal of Civil Engineering,2017,21(5):1933-1944.
[4]Carter J P,Yeung S K.Analysis of cylindrical cavity expansion in a strain weakening material[J].Computers&Geotechnics,1985,1(3):161-180.
[5]Carter J P,Booker J R,Yeung S K.Cavity expansion in cohesive frictional soils[J].Géotechnique,1986,36(3):349-358.
[6]Collins I F,Yu H S.Undrained cavity expansions in critical state soils[J].International Journal for Numerical and Analytical Methods in Geomechanics,1996,20(7):489-516.
[7]Farmer I W,Glossop N H.Settlement associated with removal of compressed air pressure during tunnelling in alluvial clay[J].Geotechnique,1979,29(1):67-72.
[8]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.
[9]Carranza-Torres C,Reich T.Analytical and numerical study of the stability of shallow underground circular openings in cohesive ground[J].Engineering Geology,2017(226):70-92.
[10]Park K H.Elastic solution for tunneling-induced ground movements in clays[J].International Journal of Geomechanics,2004,4(4):310-318.
[11]Sagaseta C.Analysis of undrained soil deformation due to ground loss[J].Geotechnique,1987,37(3):301-320.
[12]Verrjijt A.Deformations of an elastic half plane with circular cavity[J].International Journal of Solids Structures,1998,35(21):2795-2804.
[13]Yasuda N,Tsukada K,Asakura T.Elastic solutions for circular tunnel with void behind lining[J].Tunnelling&Underground Space Technology,2017(70):274-285.
[14]YU H S,Carter J P.Rigorous similarity solutions for cavity expansion in cohesive-frictional soils[J].International Journal of Geomechanics,2002,2(2):233-258.
[15]LI J P,ZHANG Y G,CHEN H B,et al.Analytical solutions of spherical cavity expansion near a slope due to pile installation[J].Journal of Applied Mathematics,2013(4):1-11.
[16]Boussinesq J.Application des potentiels a l’equilibre et des mouvements des solides elastiques[M].Paris:Gauthier-Villars,1885.
[2]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.
[3]SHI C,CAO C,LEI M.An analysis of the ground deformation caused by shield tunnel construction combining an elastic half-space model and stochastic medium theory[J].KSCE Journal of Civil Engineering,2017,21(5):1933-1944.
[4]Carter J P,Yeung S K.Analysis of cylindrical cavity expansion in a strain weakening material[J].Computers&Geotechnics,1985,1(3):161-180.
[5]Carter J P,Booker J R,Yeung S K.Cavity expansion in cohesive frictional soils[J].Géotechnique,1986,36(3):349-358.
[6]Collins I F,Yu H S.Undrained cavity expansions in critical state soils[J].International Journal for Numerical and Analytical Methods in Geomechanics,1996,20(7):489-516.
[7]Farmer I W,Glossop N H.Settlement associated with removal of compressed air pressure during tunnelling in alluvial clay[J].Geotechnique,1979,29(1):67-72.
[8]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.
[9]Carranza-Torres C,Reich T.Analytical and numerical study of the stability of shallow underground circular openings in cohesive ground[J].Engineering Geology,2017(226):70-92.
[10]Park K H.Elastic solution for tunneling-induced ground movements in clays[J].International Journal of Geomechanics,2004,4(4):310-318.
[11]Sagaseta C.Analysis of undrained soil deformation due to ground loss[J].Geotechnique,1987,37(3):301-320.
[12]Verrjijt A.Deformations of an elastic half plane with circular cavity[J].International Journal of Solids Structures,1998,35(21):2795-2804.
[13]Yasuda N,Tsukada K,Asakura T.Elastic solutions for circular tunnel with void behind lining[J].Tunnelling&Underground Space Technology,2017(70):274-285.
[14]YU H S,Carter J P.Rigorous similarity solutions for cavity expansion in cohesive-frictional soils[J].International Journal of Geomechanics,2002,2(2):233-258.
[15]LI J P,ZHANG Y G,CHEN H B,et al.Analytical solutions of spherical cavity expansion near a slope due to pile installation[J].Journal of Applied Mathematics,2013(4):1-11.
[16]Boussinesq J.Application des potentiels a l’equilibre et des mouvements des solides elastiques[M].Paris:Gauthier-Villars,1885.