被动土压力作用的变分极限平衡法研究
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摘要
在极限平衡法的框架内,引入Lagrange乘子,将被动土压力问题以变分学观点来描述,转化为含有两个函数自变量的泛函极值问题,进而根据力系几何关系将其转化为带有约束的函数极值问题。极值问题的计算结果表明,挡墙背后土体在被动临界状态时存在着沿某平面滑动和绕对数螺旋柱面转动两种破坏形式。在平面滑动破坏的情况下,土压力的大小完全等同于Coulomb(1776)和Rankine(1857)理论的结果;对于土体转动破坏情况,土压力的大小与Sokolovskii(1956)和Chen(1975)的结果比较吻合。与已有测试数据的对比表明,两类理论计算结果构成的有序实数对可以作为被动土压力大小的一个区间估计,包含了各种墙体变位模式下刚性挡墙上作用的被动土压力。
Using the limit equilibrium approach,the problem of passive earth pressure on rigid retaining wall is formulated in terms of variation calculus with Lagrange multipliers,and is transformed as the functional extreme-value problem of two variables.It is further transformed into the solution of extreme value of restrained functions incorporating the geometrical relations of the forces.Computation results show that there exist two kinds of failure modes-sliding along a plane surface(translation)and rotation along a log-spiral cylinder surface(rotation)when the soil behind retaining walls reaches critical passive state.In the translational failure mode,the passive earth pressure coincides with the theories of Coulomb(1776)and Rankine(1857);in rotational failure mode,the passive earth pressure agrees well with that of Chen(1975)and Sokolovskii(1956).Comparisons with measured data indicate that the ordered real number pair predicted by the two theoretical theories can be regarded as an interval estimation of passive pressure,which comprises the earth pressures on the rigid retaining wall with any wall movement.
Using the limit equilibrium approach,the problem of passive earth pressure on rigid retaining wall is formulated in terms of variation calculus with Lagrange multipliers,and is transformed as the functional extreme-value problem of two variables.It is further transformed into the solution of extreme value of restrained functions incorporating the geometrical relations of the forces.Computation results show that there exist two kinds of failure modes-sliding along a plane surface(translation)and rotation along a log-spiral cylinder surface(rotation)when the soil behind retaining walls reaches critical passive state.In the translational failure mode,the passive earth pressure coincides with the theories of Coulomb(1776)and Rankine(1857);in rotational failure mode,the passive earth pressure agrees well with that of Chen(1975)and Sokolovskii(1956).Comparisons with measured data indicate that the ordered real number pair predicted by the two theoretical theories can be regarded as an interval estimation of passive pressure,which comprises the earth pressures on the rigid retaining wall with any wall movement.
引文
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[28]栾茂田,金崇磐,林皋.非均质土体稳定性分析的广义极限平衡法及其应用[J].大连理工大学学报,1991,31(4):463～471.Luan Maotian,Jin Chongpan,Lin Gao.Generalized limit equilibrium method with applications for stability analysis of non-homogeneous soil structures and foundations[J].Journal of Dalian University of Technology,1991,31(4):463～471.(in Chinese)
[29]栾茂田,金崇磐,林皋.土体稳定分析极限平衡法改进及其应用[J].岩土工程学报,1992,14(增刊):20～29.Luan Maotian,Jin Chongpan,Lin Gao.Improved limit equilibrium method and its applications to stability analysis of soil masses[J].Chinese Journal of Geotechnical Engineering,1992,14(supplement):20～29.(in Chinese)
[30]栾茂田.非均质土建筑物与地基的动力稳定性研究[D].大连:大连理工大学,1989.Luan Maotian.Dynamic stability analyses of non-homogeneous soil structures and foundations[D].Dalian:Dalian University of Technology,1989.(in Chinese)
[31]Leshchinsky D,Boedeker R H.Geosynthetic reinforced soil structures[J].Journal of Geotechnical Engineering,ASCE,1989,115(10):1459～1478.
[32]Leshchinsky D.Slope stability analysis:generalized approach[J].Journal of Geotechnical Engineering,ASCE,1990,116(5):851～867.
[33]Leshchinsky D,Huang C C.Generalized slope stability analysis:interpretation,modification and compassion[J].Journal of Geotechnical Engineering,ASCE,1992,118(10):1559～1576.
[34]Leshchinsky D,Huang C C.Generalized three-dimensional slope-stability analysis[J].Journal of Geotechnical Engineering,ASCE,1992,118(11):1748～1764.
[35]Soubra A H,Kastner R.Influence of seepage flow on the passive earth pressures[A].Proceeding of the International Conference on Retaining Structures[C],ICE,Cambridge,Britain,1992.67～76.
[36]Lemonnier P,Soubra A H,Kastner R.Variational displacement method for geosynthetically reinforced slope stability analysis:I.Local stability[J].Geotextiles and Geomembranes,1998,16(1):1～25.
[37]Lemonnier P,Soubra A H,Kastner R.Variational displacement method for geosynthetically reinforced slope stability analysis II.Global stability[J].Geotextiles and Geomembranes,1998,16(1):27～44.
[38]Soubra A H,Kastner R,Benmansour A.Passive earth pressures in the presence of hydraulic gradients[J].Gétechnique,1999,49(3):319～330.
[39]Soubra A H,Kastner R,Benmansour A.Influence of the seepage force on the passive earth pressures[A].Proseeding of the12th European Conference on Soil Mechanics and Geotechnical Engineering[C],Amsterdam,Netherland,1999,2:851～856.
[40]蔡蒨蒨,许履瑚,梁在中,沈永欢.实用数学手册[M].北京:科学出版社,2001.450～452.Cai Qianqian,Xu Luhu,Liang Zaizhong,Shen Yonghuan.Practical mathematics handbook[M].Beijing:Science Press,2001.450～452.(in Chinese)
[41]苏金明,阮沈勇.MATLAB6.1实用指南(下册)[M].北京:电子工业出版社,2002.40～52.Su Jinming,Ruan Shenyong.Practical guidebook of MATLAB6.1[M].Beijing:Electronic Industry Publishing House,2002.40～52.(in Chinese)
[42]Chen W H.Limit analysis and soil plasticity[M].Amsterdam,New York:Elsevier Scientific Publishing Corporation,1975.341～398.
[43]索科洛夫斯基B B.松散介质静力学[M].北京:地质出版社,1956.97～110.Sokolovskii B B.Statics of granular media[M].Beijing:Geology Press,1956.97～110.(in Chinese)
[2]Fang Y S,Chen T J,Wu B F.Passive earth pressures with various walls movements[J].Journal of Geotechnical Engineering,ASCE,1994,128(8):651～659.
[3]徐日庆,陈页开,杨仲轩,龚晓南.刚性挡墙被动土压力模型试验研究[J].岩土工程学报,2002,24(5):569～575.Xu Riqing,Chen Yekai,Yang Zhongxuan,Gong Xiaonan.Experimental research on the passive earth pressure acting on a rigid wall[J].Chinese Journal of Geotechnical Engineering,2002,24(5):569～575.(in Chinese)
[4]Fang Y S,Ho Y C,Chen T J.Passive earth pressure with critical state concept[J].Journal of Geotechnical and Geoenvironmental Engineering,ASCE,2002,123(3):250～259.
[5]Kopáscy J.über Die Bruchfl?chen und Bruchspannungen in Den Erdbauten.Gedenkbuch Für Prof.Dr.J.Jáky[M].(Editor K.Széchy),Budapest:Akadémiaikiadó,1955,81～99.
[6]Kopáscy J.Three-dimensional stress distribution and slip surfaces in earth works at rupture[A].Proceedings of the4th International Conference on Soil Mechanics and Foundations Engineering[C],London,Britain,1957,(1):339～342.
[7]Kopáscy J.Distribution des contraintesàla rupture,forme de la surface de glissement et hauteur théorique des talus[A].Proceedings of the5th International Conference on Soil Mechanics and Foundations Engineering[C],Paris,France,1961,(2):641～650.
[8]Chen W F,Snitbhan N.On slip surface and slope stability analysis[J].Soils and Foundations,1975,15(3):41～49.
[9]Biernatowski K.Stability of slopes in variational and probabilistic solutions[A].Proceeding of the6th European Conference on Soil Mechanics and Foundation Engineering[C],Vienna,Austria,1976,(I/1-1):3～7.
[10]Baker R,Garber M.Variational Analysis of the Stability of Slopes[A].Proceedings of the9th International Conference on Soil Mechanics and Foundations Engineering[C],Tokyo,Japan,1977,(2):9～12.
[11]Baker R,Garber M.Discussion of‘On slip surface and slope stability analysis.’by Chen W F,Snitbhan N[J].Soils and Foundations,1977,17(1):65～68.
[12]Baker R,Garber M.Theoretical analysis of the stability of slopes[J].Géotechnique,1978,28(4):395～411.
[13]Garber M,Baker R.Bearing capacity by variational method[J].Journal of Geotechnical Engineering Division,ASCE,1977,103(11):1209～1225.
[14]Garber M,Baker R.Extreme-value problems of limiting equilibrium[J].Journal of Geotechnical Engineering Division,ASCE,1979,105(10):1155～1171.
[15]Castillo E,Luce?o,A.One application of the calculus of variations to bearing capacity of foundations.Proceedings of the2nd International Conference on Applied Numerical Modelling[C].Madrid,Spain,1978.353～363.
[16]De Jong De Josselin G.Application of the calculus of variations to the vertical cut-off in cohesive frictionless soil[J].Géotechnique,1980,30(1):1～16.
[17]Baker R.Determination of the critical slip surface in slope stability computations[J].International Journal for Numerical and Analytical Methods in Geomechanics,1980,4(4):333～359.
[18]Baker R.Tensile strength,tension cracks and stability of slopes[J].Soils and Foundations,1981,21(2):1～17.
[19]Ugai K.Three-dimensional stability analysis of vertical cohesive slopes[J].Soils and Foundations,1985,25(3):41～48.
[20]Leshchinsky D,Reinschmidt A J.Stability of membrane reinforced slopes[J].Journal of Geotechnical Engineering,ASCE,1985,111(11):1285～1300.
[21]Leshchinsky D,Baker R,Silver M L.Three-dimensional analysis of slope stability[J].International Journal for Numerical and Analytical Methods in Geomechanics,1985,9(2):199～233.
[22]Spencer A J M,O’Mahony T C.An application of the calculus of variations to rectilinear flow of granular materials[J].International Journal for Numerical and Analytical Methods in Geomechanics,1985,9(2):225～235.
[23]Leshchinsky D,San K C.Pseudostatic seismic stability of slopes:Design charts[J].Journal of Geotechnical Engineering,ASCE,1994,120(9):1514～1532.
[24]Leshchinsky D,Baker R.Three-dimensional slope stability:End effects[J].Soils and Foundations,1986,26(4):98～100.
[25]栾茂田,金崇磐,林皋.非均质地基上建筑物的抗震稳定性[A].第一届全国计算岩土力学研讨会论文集[C],峨眉山,四川,1987.222～232.Luan Maotian,Jin Chongpan,Lin Gao.Aseismic stability of buildings on non-homogeneous foundation[A].Proceeding of the First National Conference on Computational Geomechanics[C].Emeishan,Sichuan,1987.222～232.(in Chinese)
[26]栾茂田,金崇磐,林皋.非均质地基上浅基础的极限承载力[J].岩土工程学报,1988,10(4):14～27.Luan Maotian,Jin Chongpan,Lin Gao.Ultimate bearing capacity of shallow footings on non-homogeneous soil foundations[J].Chinese Journal of Geotechnical Engineering,1988,10(4):14～27.(in Chinese)
[27]栾茂田,金崇磐,林皋.非均布地震载荷作用下土坡的稳定性[J].水利学报,1990,35(1):65～72.Luan Maotian,Jin Chongpan,Lin Gao.Soil slope stability under non-uniform earthquake load[J].Journal of Hydraulic Engineering,1990,35(1):65～72.(in Chinese)
[28]栾茂田,金崇磐,林皋.非均质土体稳定性分析的广义极限平衡法及其应用[J].大连理工大学学报,1991,31(4):463～471.Luan Maotian,Jin Chongpan,Lin Gao.Generalized limit equilibrium method with applications for stability analysis of non-homogeneous soil structures and foundations[J].Journal of Dalian University of Technology,1991,31(4):463～471.(in Chinese)
[29]栾茂田,金崇磐,林皋.土体稳定分析极限平衡法改进及其应用[J].岩土工程学报,1992,14(增刊):20～29.Luan Maotian,Jin Chongpan,Lin Gao.Improved limit equilibrium method and its applications to stability analysis of soil masses[J].Chinese Journal of Geotechnical Engineering,1992,14(supplement):20～29.(in Chinese)
[30]栾茂田.非均质土建筑物与地基的动力稳定性研究[D].大连:大连理工大学,1989.Luan Maotian.Dynamic stability analyses of non-homogeneous soil structures and foundations[D].Dalian:Dalian University of Technology,1989.(in Chinese)
[31]Leshchinsky D,Boedeker R H.Geosynthetic reinforced soil structures[J].Journal of Geotechnical Engineering,ASCE,1989,115(10):1459～1478.
[32]Leshchinsky D.Slope stability analysis:generalized approach[J].Journal of Geotechnical Engineering,ASCE,1990,116(5):851～867.
[33]Leshchinsky D,Huang C C.Generalized slope stability analysis:interpretation,modification and compassion[J].Journal of Geotechnical Engineering,ASCE,1992,118(10):1559～1576.
[34]Leshchinsky D,Huang C C.Generalized three-dimensional slope-stability analysis[J].Journal of Geotechnical Engineering,ASCE,1992,118(11):1748～1764.
[35]Soubra A H,Kastner R.Influence of seepage flow on the passive earth pressures[A].Proceeding of the International Conference on Retaining Structures[C],ICE,Cambridge,Britain,1992.67～76.
[36]Lemonnier P,Soubra A H,Kastner R.Variational displacement method for geosynthetically reinforced slope stability analysis:I.Local stability[J].Geotextiles and Geomembranes,1998,16(1):1～25.
[37]Lemonnier P,Soubra A H,Kastner R.Variational displacement method for geosynthetically reinforced slope stability analysis II.Global stability[J].Geotextiles and Geomembranes,1998,16(1):27～44.
[38]Soubra A H,Kastner R,Benmansour A.Passive earth pressures in the presence of hydraulic gradients[J].Gétechnique,1999,49(3):319～330.
[39]Soubra A H,Kastner R,Benmansour A.Influence of the seepage force on the passive earth pressures[A].Proseeding of the12th European Conference on Soil Mechanics and Geotechnical Engineering[C],Amsterdam,Netherland,1999,2:851～856.
[40]蔡蒨蒨,许履瑚,梁在中,沈永欢.实用数学手册[M].北京:科学出版社,2001.450～452.Cai Qianqian,Xu Luhu,Liang Zaizhong,Shen Yonghuan.Practical mathematics handbook[M].Beijing:Science Press,2001.450～452.(in Chinese)
[41]苏金明,阮沈勇.MATLAB6.1实用指南(下册)[M].北京:电子工业出版社,2002.40～52.Su Jinming,Ruan Shenyong.Practical guidebook of MATLAB6.1[M].Beijing:Electronic Industry Publishing House,2002.40～52.(in Chinese)
[42]Chen W H.Limit analysis and soil plasticity[M].Amsterdam,New York:Elsevier Scientific Publishing Corporation,1975.341～398.
[43]索科洛夫斯基B B.松散介质静力学[M].北京:地质出版社,1956.97～110.Sokolovskii B B.Statics of granular media[M].Beijing:Geology Press,1956.97～110.(in Chinese)
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