基于上限有限元法的节理岩质边坡稳定性分析
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摘要
上限原理有限元法不仅可以得到边坡的安全系数,还可以给出临界滑动面,且摆脱了极限平衡法假设条件过多、过于刚性的缺点,具有更严谨的理论基础,因此拥有更广阔的应用前景。但是,对于岩质边坡,由于其存在大量的节理面,造成应力不连续,这给直接运用传统的数值计算方法带来了困难。其实,每组的节理面均表示在某一特定的方位上边坡的材料特性比较差,只需在处理这些特定方位时,采用相应的软弱节理面的材料特征即可,这样便可有效地化解节理面难以模拟的缺陷。因此,基于传统的上限有限元法,采用四边形单元,通过对单元建立积分意义上的协调方程的弱形式,来得到可以调整单元内部速度场的线性化的协调方程,从而可以克服插值速度场为非线性的缺点。对于任意应力点,从空间方位出发,将方位进行离散,建立并推导出了基于方位离散线性化的上限有限元法,该方法在考虑含多组节理面的岩质边坡计算上有着更好的优势。两个算例结果表明:该方法与传统的上限有限元法一样,可以稳定地从极限解的上方收敛,满足上限性质,且对于含多组节理面的岩质边坡同样有很好的收敛性。
By the upper bound limit analysis method,both safety factor of slope and critical slip surface can be obtained simultaneously. Because of having a more rigorous theory basis and less hypothetical and rigid conditions than limit equilibrium method,the upper bound limit analysis method has a broad application prospect. But,because the rock slope has many joint planes,leading to much stress discontinuity,this brings difficulties to the direct application of traditional numerical methods. Actually,every joint surface expresses that the specific orientation has a relatively poor slope material property,and we could just take corresponding material property when considering these weak joint surfaces,thus the difficulty of simulating joint surface can be solved effectively. So,on the basis of traditional upper bound FE method and by adopting quadrilateral element,through establishing the weak form of compatibility equations by integral over the whole elements,the linear coordination equations which can adjust the internal velocity field of the elements can be obtained,and the shortcoming of nonlinear interpolation velocity field can be overcome. From the viewpoint of discrete spatial orientation for any stress point,the upper bound limit FE method based on the linearized spatial discretization is built and derived,which has the advantages for calculating the rock slope withmultiple joints. Two examples prove that as the same as the traditional upper bound limit analysis method,the proposed method can converge to an exact solution stably from the upward side to satisfy the upper bound property,and also has good convergence for the rock slopes with multiple joints.
By the upper bound limit analysis method,both safety factor of slope and critical slip surface can be obtained simultaneously. Because of having a more rigorous theory basis and less hypothetical and rigid conditions than limit equilibrium method,the upper bound limit analysis method has a broad application prospect. But,because the rock slope has many joint planes,leading to much stress discontinuity,this brings difficulties to the direct application of traditional numerical methods. Actually,every joint surface expresses that the specific orientation has a relatively poor slope material property,and we could just take corresponding material property when considering these weak joint surfaces,thus the difficulty of simulating joint surface can be solved effectively. So,on the basis of traditional upper bound FE method and by adopting quadrilateral element,through establishing the weak form of compatibility equations by integral over the whole elements,the linear coordination equations which can adjust the internal velocity field of the elements can be obtained,and the shortcoming of nonlinear interpolation velocity field can be overcome. From the viewpoint of discrete spatial orientation for any stress point,the upper bound limit FE method based on the linearized spatial discretization is built and derived,which has the advantages for calculating the rock slope withmultiple joints. Two examples prove that as the same as the traditional upper bound limit analysis method,the proposed method can converge to an exact solution stably from the upward side to satisfy the upper bound property,and also has good convergence for the rock slopes with multiple joints.
引文
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[13]冯树荣,赵海斌,蒋中明.节理岩体边坡稳定性分析新方法[J].岩土力学,2009,30(6):1639-1642.FENG Shu-rong,ZHAO Hai-bin,JIANG Zhong-ming.A New Method of Stability Analysis of Jointed Rock Slope[J].Rock and Soil Mechanics,2009,30(6):1639-1642.
[14]李泽,王均星.基于非线性规划的岩质变坡有限元塑性极限分析下限法研究[J].岩石力学与工程学报,2007,26(4):747-753.LI Ze,WANG Jun-xing.Lower Bound Limit Study on Plastic Limit Analysis of Rock Slope Using Finite Elements Based on Nonlinear Programming[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(4):747-753.
[15]陈炜,王均星.节理岩质边坡的块体元塑性极限分析下限法[J].岩土工程学报,2008,30(2):272-277.CHEN Wei,WANG Jun-xing.Plastic Lower Bound Limit Analysis Using Block Element Method for Jointed Rock Slope[J].Chinese Journal of Geotechnical Engineering,2008,30(2):272-277.
[16]孙聪,李春光,郑宏,等.同时考虑张拉及剪切破坏的边坡上限原理有限元法[J].岩石力学与工程学报,2015,34(增1):2783-2791.SUN Cong,LI Chun-guang,ZHENG Hong,et al.Linearized Upper Bound Limit Analysis Considering Tension and Shear Failures for Slope Stability Problems[J].Chinese Journal of Rock Mechanics and Engineering,2015,34(S1):2783-2791.
[17]孙训方,方孝淑,关来泰.材料力学(Ⅰ)[M].北京:高等教育出版社,2006.SUN Xun-fang,FANG Xiao-shu,GUAN Lai-tai.Mechanics of Materials(I)[M].Beijing:Higher Education Press,2006.
[18]孙聪,李春光,郑宏,等.基于四边形网格的边坡上限有限元法[J].岩石力学与工程学报,2015,34(1):114-120.SUN Cong,LI Chun-guang,ZHENG Hong,et al.Upper Bound Limit Analysis of Slopes Using Quadrilateral Finite Elements[J].Chinese Journal of Rock Mechanics and Engineering,2015,34(1):114-120.
[19]李国英,沈珠江.下限原理有限单元法及其在土工问题中的应用[J].岩土工程学报,1997,19(5):84-89.LI Guo-ying,SHEN Zhu-jiang.Lower Bound Finite Element Method and Its Application in Geotechnical Problems[J].Chinese Journal of Geotechnical Engineering,1997,19(5):84-89.
[2]MICHALOWSKI R L.Slope Stability Analysis:A Kinematical Approach[J].Geotechnique,1995,45(2):283-293.
[3]陈祖煜.土力学经典问题的极限分析上、下限解[J].岩土工程学报,2002,24(1):1-11.CHEN Zu-yu.Limit Analysis for the Classic Problems of Soil Mechanics[J].Chinese Journal of Geotechnical Engineering,2002,24(1):1-11.
[4]LYAMIN A V,SLOAN S W.Upper Bound Limit Analysis Using Finite Element and Linear Programming[J].International Journal for Numerical&Analytical Methods in Geomechanics,2002,26(2):61-77.
[5]SLOAN S W,KLEEMAN P W.Upper Bound Limit Analysis Using Discontinuous Velocity Fields[J].Computer Methods in Applied Mechanics and Engineering,1995,127(1):293-314.
[6]殷建华,陈健,李焯芬.岩土边坡稳定性的刚体有限元上限分析法[J].岩石力学与工程学报,2004,3(6):898-905.YIN Jian-hua,CHEN Jian,LI Zhuo-fen.Upper Limit Analysis of Stability of Rock and Soil Slopes Using Rigid Finite Elements[J].Chinese Journal of Rock Mechanics and Engineering,2004,3(6):898-905.
[7]殷建华,陈健,李焯芬.考虑孔隙水压力的土坡稳定性的刚体有限元上限分析[J].岩土工程学报,2003,25(3):273-277.YIN Jian-hua,CHEN Jian,LI Zhuo-fen.A Rigid Finite Element Method for Upper Bound Limit Analysis of Soil Slopes Subjected to Pore Water Pressure[J].Chinese Journal of Geotechnical Engineering,2003,25(3):273-277.
[8]高连生,赵炼恒,李亮,等.带台阶的多级边坡稳定性上限分析[J].公路交通科技,2014,31(6):1-10.GAO Lian-sheng,ZHAO Lian-heng,LI Liang,et al.Analysis on Upper Bound of Stability of Multi-stage Slope with Benches[J].Journal of Highway and Transportation Research and Development,2014,31(6):1-10.
[9]王均星,王汉辉,张优秀,等.非均质土坡的有限元塑性极限分析[J].岩土力学,2004,25(3):415-421.WANG Jun-xing,WANG Han-hui,ZHANG You-xiu,et al.Plastic Limit Analysis of Heterogeneous Soil Slope Using Finite Elements[J].Rock and Soil Mechanics,2004,25(3):415-421.
[10]黄阜,杨小礼,赵炼恒,等.基于Hoek-Brown破坏准则的浅埋条形锚板抗拔力上限分析[J].岩土力学,2012,33(1):179-184,190.HUANG Fu,YANG Xiao-li,ZHAO Lian-heng,et al.Upper Bound Solution of Ultimate Pullout Capacity of Strip Plate Anchor Based on Hoek-Brown Failure Criterion[J].Rock and Soil Mechanics,2012,33(1):179-184,190.
[11]王根龙,伍法权,祁生文,等.加锚岩质边坡稳定性评价的极限分析上限法研究[J].岩石力学与工程学报,2007,26(12):2556-2563.WANG Gen-long,WU Fa-quan,QI Sheng-wen,et al.Research on Limit Analysis Upper Bound Method for Stability Evaluation of Anchored Rock Slope[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(12):2556-2563.
[12]赵尚毅,郑颖人,邓卫东.用有限元强度折减法进行节理岩质边坡稳定性分析[J].岩石力学与工程学报,2003,22(2):254-260.ZHAO Shang-yi,ZHENG Ying-ren,DENG Wei-dong.Stability Analysis on Jointed Rock Slope by Strength Reduction FEM[J].Chinese Journal of Rock Mechanics and Engineering,2003,22(2):254-260.
[13]冯树荣,赵海斌,蒋中明.节理岩体边坡稳定性分析新方法[J].岩土力学,2009,30(6):1639-1642.FENG Shu-rong,ZHAO Hai-bin,JIANG Zhong-ming.A New Method of Stability Analysis of Jointed Rock Slope[J].Rock and Soil Mechanics,2009,30(6):1639-1642.
[14]李泽,王均星.基于非线性规划的岩质变坡有限元塑性极限分析下限法研究[J].岩石力学与工程学报,2007,26(4):747-753.LI Ze,WANG Jun-xing.Lower Bound Limit Study on Plastic Limit Analysis of Rock Slope Using Finite Elements Based on Nonlinear Programming[J].Chinese Journal of Rock Mechanics and Engineering,2007,26(4):747-753.
[15]陈炜,王均星.节理岩质边坡的块体元塑性极限分析下限法[J].岩土工程学报,2008,30(2):272-277.CHEN Wei,WANG Jun-xing.Plastic Lower Bound Limit Analysis Using Block Element Method for Jointed Rock Slope[J].Chinese Journal of Geotechnical Engineering,2008,30(2):272-277.
[16]孙聪,李春光,郑宏,等.同时考虑张拉及剪切破坏的边坡上限原理有限元法[J].岩石力学与工程学报,2015,34(增1):2783-2791.SUN Cong,LI Chun-guang,ZHENG Hong,et al.Linearized Upper Bound Limit Analysis Considering Tension and Shear Failures for Slope Stability Problems[J].Chinese Journal of Rock Mechanics and Engineering,2015,34(S1):2783-2791.
[17]孙训方,方孝淑,关来泰.材料力学(Ⅰ)[M].北京:高等教育出版社,2006.SUN Xun-fang,FANG Xiao-shu,GUAN Lai-tai.Mechanics of Materials(I)[M].Beijing:Higher Education Press,2006.
[18]孙聪,李春光,郑宏,等.基于四边形网格的边坡上限有限元法[J].岩石力学与工程学报,2015,34(1):114-120.SUN Cong,LI Chun-guang,ZHENG Hong,et al.Upper Bound Limit Analysis of Slopes Using Quadrilateral Finite Elements[J].Chinese Journal of Rock Mechanics and Engineering,2015,34(1):114-120.
[19]李国英,沈珠江.下限原理有限单元法及其在土工问题中的应用[J].岩土工程学报,1997,19(5):84-89.LI Guo-ying,SHEN Zhu-jiang.Lower Bound Finite Element Method and Its Application in Geotechnical Problems[J].Chinese Journal of Geotechnical Engineering,1997,19(5):84-89.