不排水条件下的临坡条形基础承载力有限元极限分析
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摘要
为计算作用在不排水边坡上条形基础的极限承载力,首先,根据临坡条形基础承载特性,将基础假定为刚体且基底与土体之间没有摩擦力,同时将基础的埋置深度用其两侧的超载代替,并将其视为平面应变问题,建立计算模型。然后,通过有限单元对应力场和速度场进行离散,根据上、下限定理建立节点应力和节点速度的约束方程,将外力荷载或总的内能耗散作为目标函数,建立相应的数学规划模型,并采用优化算法求解该模型,得到严格的上、下限解,在优化计算过程中对Mohr-Coulomb准则进行了双曲线近似处理。最后,在此基础上分析了坡高H、基础与坡顶的距离L、超载q、坡角β以及黏聚力C_u对极限承载力系数p/γB的影响(γ为重度,B为坡顶基础宽度,p为基础承载力)。同时将计算结果与已有成果进行对比分析,验证了该方法的正确性,总结了3种极限破坏模式,并提供了常见不排水边坡的设计计算表格,以供实际工程设计参考。研究结果表明:p/γB随H的增大先急剧减小,然后基本稳定,趋近于一固定值,临界坡高为2倍基础宽度;p/γB随基础与坡顶的距离增大而增大,且趋向于一常数;p/γB随超载的增大而线性增大,且L/B越大,增长速率越快;p/γB与坡角基本成线性关系,其值随着坡角β的增大而减小;p/γB随着黏聚力系数(C_u/γB)的增大而线性增大。
In order to calculate the bearing capacity of strip footings on undrained slopes, first, according to the bearing characteristics of strip footing on slopes, the footing was assumed to be a smooth rigid body and there was no friction between the base and the soil, the footing buried depth was replaced by overloads on both sides, the problem was considered to be a plane strain problem, and a calculation model was established. Second, the stress field and velocity field was discreted by finite element, and the constraint equations of nodal stress and nodal velocity was established, according to the upper and lower bound theorems. The total internal energy dissipation or external force load was used as the objective function, a mathematical programing model was established and an appropriate algorithm was used to solve the function,and the strict upper and lower solutions were obtained. The hyperbolic approximation of the Mohr-Coulomb criterion was processed during the optimization calculation. At the same time, the results of this paper were compared with the existing results, the correctness of the method was verified. Three types of failure mechanics were summarized, and the design tables for undrained slopes were presented to provide reference for engineering practice. Finally, based on the numerical results, the effects of slope height H, the distance between the foundation and the top of the slope L, overload q, slope angle β, and cohesion C_u on the bearing capacity factor p/γB were evaluated. Where γ was unit of weight, B was the width of slope top foundation, p was bearing capacity of foundation. The results show that p/γB increases with an increase in H sharply and eventally apparoches a constant, the critical slope height is two times that of the footing width. p/γB increases with an increase in L/B and eventally apparoches a constant. p/γB increases linearly with overload q and the larger the L/B, the faster is the growth rate. The relationship between p/γB and slope angle β is linear, and it decreases with an increase in the slope angle β. p/γB is linear with the cohesion C_u/γB, and it increases with the increase in cohesion C_u/γB. 1 tab, 13 figs, 23 refs.
In order to calculate the bearing capacity of strip footings on undrained slopes, first, according to the bearing characteristics of strip footing on slopes, the footing was assumed to be a smooth rigid body and there was no friction between the base and the soil, the footing buried depth was replaced by overloads on both sides, the problem was considered to be a plane strain problem, and a calculation model was established. Second, the stress field and velocity field was discreted by finite element, and the constraint equations of nodal stress and nodal velocity was established, according to the upper and lower bound theorems. The total internal energy dissipation or external force load was used as the objective function, a mathematical programing model was established and an appropriate algorithm was used to solve the function,and the strict upper and lower solutions were obtained. The hyperbolic approximation of the Mohr-Coulomb criterion was processed during the optimization calculation. At the same time, the results of this paper were compared with the existing results, the correctness of the method was verified. Three types of failure mechanics were summarized, and the design tables for undrained slopes were presented to provide reference for engineering practice. Finally, based on the numerical results, the effects of slope height H, the distance between the foundation and the top of the slope L, overload q, slope angle β, and cohesion C_u on the bearing capacity factor p/γB were evaluated. Where γ was unit of weight, B was the width of slope top foundation, p was bearing capacity of foundation. The results show that p/γB increases with an increase in H sharply and eventally apparoches a constant, the critical slope height is two times that of the footing width. p/γB increases with an increase in L/B and eventally apparoches a constant. p/γB increases linearly with overload q and the larger the L/B, the faster is the growth rate. The relationship between p/γB and slope angle β is linear, and it decreases with an increase in the slope angle β. p/γB is linear with the cohesion C_u/γB, and it increases with the increase in cohesion C_u/γB. 1 tab, 13 figs, 23 refs.
引文
[1] BOWLES L E.Foundation analysis and design[M].Bothell:McGraw-Hill Publishing Company,2001.
[2] MEYERHOF G.The ultimate bearing capacity of foundations on slope[M].Geotechnique,1951,2(4):301-332.
[3] GRAHAM J,ANDREWS M,SHIELDS D H.Stress characteristics for shallow footings in cohesionless slopes[J].Canadian Geotechnical Journal,1988,25(2):238-249.
[4] 杨峰,阳军生,张学民,等.斜坡地基单侧滑移破坏模式及承载力上限解[J].工程力学,2010,27(6):162-168.YANG Feng,YANG Jun-sheng,ZHANG Xue-min,et al.One-side slip failure mechanism and upper bound solution for bearing capacity of foundation on slope[J].Engineering Mechanics,2010,27(6):162-168.
[5] 王志斌,李亮,杨晓礼.极限分析上限法在山区斜坡地基上高填方路堤极限承载力研究中的应用[J].塑性工程学报,2007,14(2):118-122.WANG Zhi-bin,LI Liang,YANG Xiao-li.Application of upper-bound theorem of limit analysis to ultimate capacity of embankments on mountain slope foundation[J].Journal of Plasticity Engineering,2007,14(2):118-122.
[6] 尉学勇,王晓谋,怀超.斜坡地基极限承载力上限解计算与分析[J].岩土工程学报,2010,32(3):381-387.YU Xue-yong,WANG Xiao-mou,HUAI Chao.Calculation and analysis of upper limit solution of ultimate bearing capacity of slope ground[J].Chinese Journal of Geotechnical Engineering,2010,32(3):381-387.
[7] 胡卫东,曹文贵.基于非对称破坏模式的临坡地基承载力上限极限分析方法[J].中国公路学报,2014,27(6):1-9.HU Wei-dong,CAO Wen-gui.Upper limit analysis method for ultimate bearing capacity of ground foundation adjacent to slope based on asymmetry failure mode[J].China Journal of Highway and Transport,2014,27(6):1-9.
[8] 胡卫东,曹文贵.基于双侧非对称破坏模式的临坡地基承载力极限平衡分析方法[J].土木工程学报,2015,48(1):120-128.HU Wei-dong,CAO Wen-gui.The limit equilibrium method for ultimate bearing capacity of ground foundation adjacent to slope based on bilateral asymmetry failure mode[J].China Civil Engineer Journal,2015,48(1):120-128.
[9] 胡卫东,曹文贵,袁青松.基于非对称双侧破坏模式的临坡地基承载力上限分析[J].岩土力学,2016,37(10):2787-2794.HU Wei-dong,CAO Wen-gui,YUAN Qing-song.An upper-bound limit analysis of ultimate bearing capacity of ground foundation adjacent to slope based on asymmetric and bilateral failure mode[J].Rock and Soil Mechanics,2016,37(10):2787-2794.
[10] 马庆宏,朱大勇,雷先顺,等.无黏性土斜坡地基承载力模型试验研究[J].岩土工程学报,2014,36(7):1271-1280.MA Qing-hong,ZHU Da-yong,LEI Xian-shun,et al.Model tests on bearing capacity of footing on sand slopes[J].Chinese Journal of Geotechnical Engineering,2014,36(7):1271-1280.
[11] IP K W.Bearing capacity for foundation near slope[D].Montreal:Concordia University,2005.
[12] 姜鹏明,盛欢,陆长锋,等.边坡极限承载力的严密解[J].工程力学,2009,26(增1):77-80.JIANG Peng-ming,SHENG Huan,LU Chang-feng,et al.Rigorous solution of the slope ultimate bearing capacity[J].Engineering Mechanics,2009,26(S1):77-80.
[13] 王红雨,杨敏.临近基坑条形基础地基承载力简化上限解[J].同济大学学报:自然科学版,2006,34(3):319-324.WANG Hong-yu,YANG Min.Approximate upper-bound solution for bearing capacity of strip footings near excavations[J].Journal of Tongji University:Natural Science,2006,34(3):319-324.
[14] 许锡昌,陈善雄,姜领发.斜坡地基附加应力分布规律模型试验及数值模拟[J].岩土力学,2015,36(增2):267-273.XU Xi-chang,CHEN Shan-xiong,JIANG Ling-fa.Experimental study and numerical simulation on additional stress distribution of foundation adjacent to slopes[J].Rock and Soil Mechanics,2015,36(S2):267-273.
[15] 赵明华,胡啸,张锐.临坡地基承载力极限分析上限有限元数值模拟[J].岩土力学,2016,37(4):1137-1143,1152.ZHAO Ming-hua,HU Xiao,ZHANG Rui.Numerical simulation of the bearing capacity of a foundation near slope using the upper bound finite element method[J].Rock and Soil Mechanics,2016,37(4):1137-1143,1152.
[16] 尹鑫,周海祚,郑刚.地震作用下临近边坡的条形基础极限承载力研究[J].岩土工程学报,2017,39(增2):95-98.YIN Xin,ZHOU Hai-zuo,ZHENG Gang.Seismic bearing capacity of strip footings adjacent to slopes[J].Chinese Journal of Geotechnical Engineering,2017,39(S2):95-98.
[17] ZHOU Hai-zuo,ZHENG Gang,YIN Xin,et al.The bearing capacity and failure mechanism of a vertically load strip footing placed on the top of slopes[J].Computers and Strucures,2018,94:12-21.
[18] ABBO A J,SLOAN S W.A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion[J].Computers and Structures,1995,54(3):427-441.
[19] LYAMIN A V,SLOAN S W.Upper bound limit analysis using linear finite elements and non-linear programming[J].International Journal for Numerical and Analytical Methods in Geomechanics,2002,26(2):181-216.
[20] LYAMIN A V,SLOAN S W,KRABBENHOFT K,et al.Lower bound limit analysis with adaptive remeshing[J].International Journal for Numerical Methods in Engineering,2005,63(14):1961-1974.
[21] 赵明华,张锐,刘猛,等.下限分析有限单元法的非线性规划求解[J].岩土力学,2015,36(12):3589-3597.ZHAO Ming-hua,ZHANG Rui,LIU Meng,et al.Nonlinear programming of lower bound finite element method[J].Rock and Soil Mechanics,2015,36(12):3589-3597.
[22] 赵明华,张锐.有限元上限分析网格自适应方法及其工程应用[J].岩土工程学报,2016,38(3):537-545.ZHAO Ming-hua,ZHANG Rui.Adaptive mesh refinement of upper bound finite element method and its applications in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):537-545.
[23] SHIAU J S,MERIFIELD R S,LYAMIN A V,et al.Undrained stability of footings on slopes[J].International Journal of Geomechanics,2011,11(5):381-390.
[2] MEYERHOF G.The ultimate bearing capacity of foundations on slope[M].Geotechnique,1951,2(4):301-332.
[3] GRAHAM J,ANDREWS M,SHIELDS D H.Stress characteristics for shallow footings in cohesionless slopes[J].Canadian Geotechnical Journal,1988,25(2):238-249.
[4] 杨峰,阳军生,张学民,等.斜坡地基单侧滑移破坏模式及承载力上限解[J].工程力学,2010,27(6):162-168.YANG Feng,YANG Jun-sheng,ZHANG Xue-min,et al.One-side slip failure mechanism and upper bound solution for bearing capacity of foundation on slope[J].Engineering Mechanics,2010,27(6):162-168.
[5] 王志斌,李亮,杨晓礼.极限分析上限法在山区斜坡地基上高填方路堤极限承载力研究中的应用[J].塑性工程学报,2007,14(2):118-122.WANG Zhi-bin,LI Liang,YANG Xiao-li.Application of upper-bound theorem of limit analysis to ultimate capacity of embankments on mountain slope foundation[J].Journal of Plasticity Engineering,2007,14(2):118-122.
[6] 尉学勇,王晓谋,怀超.斜坡地基极限承载力上限解计算与分析[J].岩土工程学报,2010,32(3):381-387.YU Xue-yong,WANG Xiao-mou,HUAI Chao.Calculation and analysis of upper limit solution of ultimate bearing capacity of slope ground[J].Chinese Journal of Geotechnical Engineering,2010,32(3):381-387.
[7] 胡卫东,曹文贵.基于非对称破坏模式的临坡地基承载力上限极限分析方法[J].中国公路学报,2014,27(6):1-9.HU Wei-dong,CAO Wen-gui.Upper limit analysis method for ultimate bearing capacity of ground foundation adjacent to slope based on asymmetry failure mode[J].China Journal of Highway and Transport,2014,27(6):1-9.
[8] 胡卫东,曹文贵.基于双侧非对称破坏模式的临坡地基承载力极限平衡分析方法[J].土木工程学报,2015,48(1):120-128.HU Wei-dong,CAO Wen-gui.The limit equilibrium method for ultimate bearing capacity of ground foundation adjacent to slope based on bilateral asymmetry failure mode[J].China Civil Engineer Journal,2015,48(1):120-128.
[9] 胡卫东,曹文贵,袁青松.基于非对称双侧破坏模式的临坡地基承载力上限分析[J].岩土力学,2016,37(10):2787-2794.HU Wei-dong,CAO Wen-gui,YUAN Qing-song.An upper-bound limit analysis of ultimate bearing capacity of ground foundation adjacent to slope based on asymmetric and bilateral failure mode[J].Rock and Soil Mechanics,2016,37(10):2787-2794.
[10] 马庆宏,朱大勇,雷先顺,等.无黏性土斜坡地基承载力模型试验研究[J].岩土工程学报,2014,36(7):1271-1280.MA Qing-hong,ZHU Da-yong,LEI Xian-shun,et al.Model tests on bearing capacity of footing on sand slopes[J].Chinese Journal of Geotechnical Engineering,2014,36(7):1271-1280.
[11] IP K W.Bearing capacity for foundation near slope[D].Montreal:Concordia University,2005.
[12] 姜鹏明,盛欢,陆长锋,等.边坡极限承载力的严密解[J].工程力学,2009,26(增1):77-80.JIANG Peng-ming,SHENG Huan,LU Chang-feng,et al.Rigorous solution of the slope ultimate bearing capacity[J].Engineering Mechanics,2009,26(S1):77-80.
[13] 王红雨,杨敏.临近基坑条形基础地基承载力简化上限解[J].同济大学学报:自然科学版,2006,34(3):319-324.WANG Hong-yu,YANG Min.Approximate upper-bound solution for bearing capacity of strip footings near excavations[J].Journal of Tongji University:Natural Science,2006,34(3):319-324.
[14] 许锡昌,陈善雄,姜领发.斜坡地基附加应力分布规律模型试验及数值模拟[J].岩土力学,2015,36(增2):267-273.XU Xi-chang,CHEN Shan-xiong,JIANG Ling-fa.Experimental study and numerical simulation on additional stress distribution of foundation adjacent to slopes[J].Rock and Soil Mechanics,2015,36(S2):267-273.
[15] 赵明华,胡啸,张锐.临坡地基承载力极限分析上限有限元数值模拟[J].岩土力学,2016,37(4):1137-1143,1152.ZHAO Ming-hua,HU Xiao,ZHANG Rui.Numerical simulation of the bearing capacity of a foundation near slope using the upper bound finite element method[J].Rock and Soil Mechanics,2016,37(4):1137-1143,1152.
[16] 尹鑫,周海祚,郑刚.地震作用下临近边坡的条形基础极限承载力研究[J].岩土工程学报,2017,39(增2):95-98.YIN Xin,ZHOU Hai-zuo,ZHENG Gang.Seismic bearing capacity of strip footings adjacent to slopes[J].Chinese Journal of Geotechnical Engineering,2017,39(S2):95-98.
[17] ZHOU Hai-zuo,ZHENG Gang,YIN Xin,et al.The bearing capacity and failure mechanism of a vertically load strip footing placed on the top of slopes[J].Computers and Strucures,2018,94:12-21.
[18] ABBO A J,SLOAN S W.A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion[J].Computers and Structures,1995,54(3):427-441.
[19] LYAMIN A V,SLOAN S W.Upper bound limit analysis using linear finite elements and non-linear programming[J].International Journal for Numerical and Analytical Methods in Geomechanics,2002,26(2):181-216.
[20] LYAMIN A V,SLOAN S W,KRABBENHOFT K,et al.Lower bound limit analysis with adaptive remeshing[J].International Journal for Numerical Methods in Engineering,2005,63(14):1961-1974.
[21] 赵明华,张锐,刘猛,等.下限分析有限单元法的非线性规划求解[J].岩土力学,2015,36(12):3589-3597.ZHAO Ming-hua,ZHANG Rui,LIU Meng,et al.Nonlinear programming of lower bound finite element method[J].Rock and Soil Mechanics,2015,36(12):3589-3597.
[22] 赵明华,张锐.有限元上限分析网格自适应方法及其工程应用[J].岩土工程学报,2016,38(3):537-545.ZHAO Ming-hua,ZHANG Rui.Adaptive mesh refinement of upper bound finite element method and its applications in geotechnical engineering[J].Chinese Journal of Geotechnical Engineering,2016,38(3):537-545.
[23] SHIAU J S,MERIFIELD R S,LYAMIN A V,et al.Undrained stability of footings on slopes[J].International Journal of Geomechanics,2011,11(5):381-390.
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