挡土墙上被动土压力的变分求解方法
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摘要
基于滑楔体整体极限平衡方程,根据变分法原理推导了被动土压力泛函极值的变分模型,并引入拉格朗日乘子,将等周变分模型转化为含有两个函数自变量的泛函极值模型。依据欧拉方程、边界条件和横截条件,得到了滑裂面函数和滑裂面上的应力函数,函数泛函极值模型转化为两个未知量的函数优化模型。算例表明,对于一般土体,在作用点位置系数下界限处,滑裂面呈现对数螺旋曲面,此时被动土压力最小;当作用点位置上移时,被动土压力呈非线性增长,在作用点位置系数上界限处,滑裂面为平面,被动土压力达到最大,与库仑土压力理论解完全一致,但作用点在墙体的相对位置并非在墙高的1/3处。结果表明,被动土压力大小和作用点位置受坡面的起伏和坡面超载的不均匀性影响比较明显。
Considering an inclined rough retaining wall under the general conditions such as curvilinear fill,cohesive soil and uneven surface load,the functional extreme-value isoperimetric model about passive earth pressure is deduced based on the force equilibrium equations of the sliding mass.Then,the model can be transferred into a functional extreme-value problem with two undetermined functions by introducing Lagrange undetermined multiplier.According to Euler equations,Logarithmic spiral slip surface and normal stress distribution along the slip surface are obtained.Combined with the boundary conditions and transversality conditions,the conditional functional extremum problem of passive earth pressure involves searching the minimum of unconstrained optimizations of function with two unknown Lagrange multiplier.Results show that the passive earth pressure resultant force is minimal when the point of resultant force is on the lower bound and it increase nonlinearly as the point of resultant force moved up to the upper bound for general soil.Accordingly,the slip face evolves from logarithmic spiral face to plan.Although the magnitude of passive earth pressure reaches maximal that is the same with result calculated from Coulomb's theory,the application point of earth pressure is not at 1/3 height of the retaining wall.In addition,curvilinear fill and uneven surface load have significant effect on both the magnitude and location of application point of passive earth pressure.
Considering an inclined rough retaining wall under the general conditions such as curvilinear fill,cohesive soil and uneven surface load,the functional extreme-value isoperimetric model about passive earth pressure is deduced based on the force equilibrium equations of the sliding mass.Then,the model can be transferred into a functional extreme-value problem with two undetermined functions by introducing Lagrange undetermined multiplier.According to Euler equations,Logarithmic spiral slip surface and normal stress distribution along the slip surface are obtained.Combined with the boundary conditions and transversality conditions,the conditional functional extremum problem of passive earth pressure involves searching the minimum of unconstrained optimizations of function with two unknown Lagrange multiplier.Results show that the passive earth pressure resultant force is minimal when the point of resultant force is on the lower bound and it increase nonlinearly as the point of resultant force moved up to the upper bound for general soil.Accordingly,the slip face evolves from logarithmic spiral face to plan.Although the magnitude of passive earth pressure reaches maximal that is the same with result calculated from Coulomb's theory,the application point of earth pressure is not at 1/3 height of the retaining wall.In addition,curvilinear fill and uneven surface load have significant effect on both the magnitude and location of application point of passive earth pressure.
引文
[1]李永刚.挡土墙被动土压力研究[J].岩土力学,2003,24(2):273-276.LI Y G.Study on passive earth pressure of retaining wall[J].Rock and Soil Mechanics,2003,24(2):273-276.(in Chinese)
[2]王立国,王志凯.考虑土拱效应的挡土墙被动土压力分析[J].水利与建筑工程学报,2010,8(4):146-149.WANG L G,WANG Z K.Analysis on passive earth pressure for retaining walls considering soil arching[J].Journal of Water Resources and Architectural Engineering,2010,8(4):146-149.(in Chinese)
[3]侯键,夏唐代,孔祥冰,等.基于土拱效应原理求解挡土墙被动土压力[J].岩土力学,2012,33(10):2296-3000.HOU J,XIA T D,KONG X B,et al.Passive earth pressure on retaining walls calculated by principle of soil arching effect[J].Rock and Soil Mechanics,2012,33(10):2296-3000.(in Chinese)
[4]CHANG M F.Lateral earth pressures behind rotating walls[J].Canadian Geotechnical Journal,1997,34(2):498-509.
[5]蒋波,应宏伟,谢康和,等.平动模式下挡土墙非极限状态主动土压力计算[J].中国公路学报,2005,18(2):24-27.JIANG B,YING H W,XIE K H,et al.Computation of active earth pressure under non-limit state for retaining wall with mode of translation[J].China Journal of Highway and Transport,2005,18(2):24-27.(in Chinese)
[6]杨泰华,贺怀建.黏性土基于强度折减法的位移土压力理论[J].岩土力学,2009,30(Sup2):185-188.YANG T H,HE H J.Displacement-earth pressure theory for cohesive soil based on shear strength reduction[J].Rock and Soil Mechanics,2009,30(Sup2):185-188.(in Chinese)
[7]杨泰华,贺怀建.考虑位移效应的土压力计算理论[J].岩土力学,2010,31(11):3635-3639,3650.YANG T H,HE H J.Earth pressure calculation theory considering displacement effects[J].Rock and Soil Mechanics,2010,31(11):3635-3639,3650.(in Chinese)
[8]杨泰华,贺怀建,俞晓.RTT变位模式下考虑位移影响的被动侧土压力的计算与分析[J].武汉科技大学学报(自然科学版),2010,33(6):656-659.YANG T H,HE H J,YU X.Calculation and analysis of passive earth pressure in the model of RTTmovement with displacement effects considered[J].Journal of Wuhan University of Science and Technology(Natural Science Editim)2010,33(6):656-659.(in Chinese)
[9]杨泰华,龚建伍,汤斌,等.不同变位模式下无黏性土非极限被动土压力计算分析[J].岩土力学,2013,34(10):2979-2983,2990.YANG T H,GONG J W,TANG B,et al.Calculation and analysis of unlimited passive earth pressure of cohesionless soil in different movement modes[J].Rock and Soil Mechanics,2013,34(10):2979-2983,2990.(in Chinese)
[10]程康,夏唐代,单华峰,等.考虑土拱效应的非饱和土被动土压力计算[J].水利与建筑工程学报,2017,15(5):63-68.CHENG K,XIA T D,SHAN H F,et al.Passive earth pressure calculation of unsaturated soil considering arching effect[J].Journal of Water Resources and Architectural Engineering,2017,15(5):63-68.(in Chinese)
[11]赵均海,周先成,李艳.基于双剪统一强度理论的非饱和土库仑被动土压力统一解[J].工业建筑,2015,45(10):101-105.ZHAO J H,ZHOU X C,LI Y.Unified solution of Coulomb’s passive earth pressure for unsaturated soils based on twin shear unified strength theory[J].Industrial Construction,2015,45(10):101-105.(in Chinese)
[12]KOPASCY J.Uber die bruchfl chen und bruchspannungen in den erdbauten[M].Budapest:Akadémiaikiadó,1955:81-99.
[13]KOPASCY J.Three-dimensional stress distribution and slip surfaces in earth works at rupture[C]//Proceedings of the 4th International Conference on Soil Mechanics and Foundations Engineering,London,Britain,1957.
[14]KOPASCY J.Distribution des contraintesàla rupture,forme de la surface de glissement et hauteur théorique des talus[C]//Proceedings of the 5th International Conference on Soil Mechanics and Foundations Engineering,Paris,France,1961:641-650.
[15]李兴高,刘维宁,张弥.关于库仑土压力理论的探讨[J].岩土工程学报,2005,27(6):677-681.LI X G,LIU W N,ZHANG M.Discussion on Coulomb earth pressure theory[J].Chinese Journal of Geotechnical Engineering,2005,27(6):677-681.(in Chinese)
[16]李兴高,刘维宁.Coulomb土压力理论的两种解法[J].岩土力学,2006,27(6):981-985.LI X G,LIU W N.Two methods to get the Coulomb earth pressure theory[J].Rock and Soil Mechanics,2006,27(6):981-985.(in Chinese)
[17]李兴高.刘维宁.被动土压力作用的变分极限平衡法研究[J].工程力学,2007,24(1):11-17.LI X G,LIU W N.Study on the passive earth pressure by variational limit equilibrium method[J].Engineering Mechanics,2007,24(1):11-17.(in Chinese)
[2]王立国,王志凯.考虑土拱效应的挡土墙被动土压力分析[J].水利与建筑工程学报,2010,8(4):146-149.WANG L G,WANG Z K.Analysis on passive earth pressure for retaining walls considering soil arching[J].Journal of Water Resources and Architectural Engineering,2010,8(4):146-149.(in Chinese)
[3]侯键,夏唐代,孔祥冰,等.基于土拱效应原理求解挡土墙被动土压力[J].岩土力学,2012,33(10):2296-3000.HOU J,XIA T D,KONG X B,et al.Passive earth pressure on retaining walls calculated by principle of soil arching effect[J].Rock and Soil Mechanics,2012,33(10):2296-3000.(in Chinese)
[4]CHANG M F.Lateral earth pressures behind rotating walls[J].Canadian Geotechnical Journal,1997,34(2):498-509.
[5]蒋波,应宏伟,谢康和,等.平动模式下挡土墙非极限状态主动土压力计算[J].中国公路学报,2005,18(2):24-27.JIANG B,YING H W,XIE K H,et al.Computation of active earth pressure under non-limit state for retaining wall with mode of translation[J].China Journal of Highway and Transport,2005,18(2):24-27.(in Chinese)
[6]杨泰华,贺怀建.黏性土基于强度折减法的位移土压力理论[J].岩土力学,2009,30(Sup2):185-188.YANG T H,HE H J.Displacement-earth pressure theory for cohesive soil based on shear strength reduction[J].Rock and Soil Mechanics,2009,30(Sup2):185-188.(in Chinese)
[7]杨泰华,贺怀建.考虑位移效应的土压力计算理论[J].岩土力学,2010,31(11):3635-3639,3650.YANG T H,HE H J.Earth pressure calculation theory considering displacement effects[J].Rock and Soil Mechanics,2010,31(11):3635-3639,3650.(in Chinese)
[8]杨泰华,贺怀建,俞晓.RTT变位模式下考虑位移影响的被动侧土压力的计算与分析[J].武汉科技大学学报(自然科学版),2010,33(6):656-659.YANG T H,HE H J,YU X.Calculation and analysis of passive earth pressure in the model of RTTmovement with displacement effects considered[J].Journal of Wuhan University of Science and Technology(Natural Science Editim)2010,33(6):656-659.(in Chinese)
[9]杨泰华,龚建伍,汤斌,等.不同变位模式下无黏性土非极限被动土压力计算分析[J].岩土力学,2013,34(10):2979-2983,2990.YANG T H,GONG J W,TANG B,et al.Calculation and analysis of unlimited passive earth pressure of cohesionless soil in different movement modes[J].Rock and Soil Mechanics,2013,34(10):2979-2983,2990.(in Chinese)
[10]程康,夏唐代,单华峰,等.考虑土拱效应的非饱和土被动土压力计算[J].水利与建筑工程学报,2017,15(5):63-68.CHENG K,XIA T D,SHAN H F,et al.Passive earth pressure calculation of unsaturated soil considering arching effect[J].Journal of Water Resources and Architectural Engineering,2017,15(5):63-68.(in Chinese)
[11]赵均海,周先成,李艳.基于双剪统一强度理论的非饱和土库仑被动土压力统一解[J].工业建筑,2015,45(10):101-105.ZHAO J H,ZHOU X C,LI Y.Unified solution of Coulomb’s passive earth pressure for unsaturated soils based on twin shear unified strength theory[J].Industrial Construction,2015,45(10):101-105.(in Chinese)
[12]KOPASCY J.Uber die bruchfl chen und bruchspannungen in den erdbauten[M].Budapest:Akadémiaikiadó,1955:81-99.
[13]KOPASCY J.Three-dimensional stress distribution and slip surfaces in earth works at rupture[C]//Proceedings of the 4th International Conference on Soil Mechanics and Foundations Engineering,London,Britain,1957.
[14]KOPASCY J.Distribution des contraintesàla rupture,forme de la surface de glissement et hauteur théorique des talus[C]//Proceedings of the 5th International Conference on Soil Mechanics and Foundations Engineering,Paris,France,1961:641-650.
[15]李兴高,刘维宁,张弥.关于库仑土压力理论的探讨[J].岩土工程学报,2005,27(6):677-681.LI X G,LIU W N,ZHANG M.Discussion on Coulomb earth pressure theory[J].Chinese Journal of Geotechnical Engineering,2005,27(6):677-681.(in Chinese)
[16]李兴高,刘维宁.Coulomb土压力理论的两种解法[J].岩土力学,2006,27(6):981-985.LI X G,LIU W N.Two methods to get the Coulomb earth pressure theory[J].Rock and Soil Mechanics,2006,27(6):981-985.(in Chinese)
[17]李兴高.刘维宁.被动土压力作用的变分极限平衡法研究[J].工程力学,2007,24(1):11-17.LI X G,LIU W N.Study on the passive earth pressure by variational limit equilibrium method[J].Engineering Mechanics,2007,24(1):11-17.(in Chinese)