Limit analysis method for slope stability based on discretization of rigid blocks
WANG Xiao-gang, LIN Xing-chao
State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, China Institute of Water Resources and Hydropower Research, Beijing 100048, China
Abstract：The classic slope stability analysis problem is solved by abandoning the traditional and direct solutions for safety factors of slopes, such as the limit equilibrium equation and the introduction of assumptions. Instead, the general construction method for constructing a slope limit state mechanism is adopted, and the plastic (upper and lower) limit theorems are taken into account. Moreover, taking the interfacial forces and velocities as the main variables and the basic requirements for statically admissible stress field or kinematically admissible displacement field as the constraints, and on the precondition that all assumptions are omitted, the solution for the safety factor problem of slopes is converted into an upper- and lower-bound optimization problem. Furthermore, a complete and united limit analysis method for slope stability is established by gradually shifting the upper- and lower-limit values toward the real solution for the safety factor. To conclude, the proposed method can provide a robust theoretical basis for slope stability analysis due to the omission of assumptions and address the bottleneck resulting from the method of expanding the 2D slope stability analysis to its 3D form. The accuracy and realiability of the calculated results as well as the rationality and feasibility of its engineering applications are validated through 6 representative examples.
汪小刚, 林兴超. 基于刚性块体离散的边坡稳定极限分析法[J]. 岩土工程学报, 2022, 44(9): 1587-1597.
WANG Xiao-gang, LIN Xing-chao. Limit analysis method for slope stability based on discretization of rigid blocks. Chinese J. Geot. Eng., 2022, 44(9): 1587-1597.
[1] 周健, 崔积弘, 贾敏才, 等. 静力触探试验的离散元数值模拟研究[J]. 岩土工程学报, 2007, 29(11): 1604-1610. (ZHOU Jian, CUI Ji-hong, JIA Min-cai, et al.Numerical simulation of cone penetration test by discrete element method[J]. Chinese Journal of Geotechnical Engineering, 2007, 29(11): 1604-1610. (in Chinese)) [2] CHEN Z Y, WANG X G, HABERFIELD C, et al.A three-dimensional slope stability analysis method using the upper bound theorem part I: theory and methods[J]. International Journal of Rock Mechanics and Mining Sciences, 2001, 38(3): 369-378. [3] CHEN Z Y, WANG J, WANG Y J, et al.A three-dimensional slope stability analysis method using the upper bound theorem Part II: numerical approaches, applications and extensions[J]. International Journal of Rock Mechanics and Mining Sciences, 2001, 38(3): 379-397. [4] 陈祖煜, 弥宏亮, 汪小刚. 边坡稳定三维分析的极限平衡方法[J]. 岩土工程学报, 2001, 23(5): 525-529. (CHEN Zu-yu, MI Hong-liang, WANG Xiao-gang.A three-dimensional limit equilibrium method for slope stability analysis[J]. Chinese Journal of Geotechnical Engineering, 2001, 23(5): 525-529. (in Chinese)) [5] JANBU N.Application of composite slip surface for stability analysis[C]// Proceedings of Europan Conference on Stability of Earth Slopes, Stockholm, Sweden, 1954. [6] BISHOP A W.The use of the slip circle in the stability analysis of slopes[J]. Géotechnique, 1955, 5(1): 7-17. [7] MORGENSTERN N R, PRICE V E.The analysis of the stability of general slip surfaces[J]. Géotechnique, 1965, 15(1): 79-93. [8] SPENCER E.A method of analysis of the stability of embankments assuming parallel inter-slice forces[J]. Géotechnique, 1967, 17(1): 11-26. [9] HOEK E, BRAY J W.Rock slope engineering[M]. London: The Institution of Mining and Metallurgy, 1977. [10] SARMA S K.Stability analysis of embankments and slopes[J]. Journal of the Geotechnical Engineering Division, 1979, 105(12): 1511-1524. [11] WANG X G, LIN X C, LI X, et al.3D slope stability analysis method based on Pan's maximum principle[J]. Landslides, 2020, 17(5): 1163-1176. [12] WANG X G, LIN X C, LI X, et al.3D slope stability analysis method based on Pan's maximum principle[J]. Landslides, 2020, 17(5): 1163-1176. [13] 冯树仁, 丰定祥, 葛修润, 等. 边坡稳定性的三维极限平衡分析方法及应用[J]. 岩土工程学报, 1999, 21(6): 657-661. (FENG Shu-ren, FENG Ding-xiang, GE Xiu-run, et al.3D limit equilibrium method for slope stability and its application[J]. Chinese Journal of Geotechnical Engineering, 1999, 21(6): 657-661. (in Chinese)) [14] 郑榕明, 朱禄娟, 谷兆祺. 非对称旋转破坏的三维Bishop边坡稳定算法[J]. 岩土工程学报, 2002, 24(6): 706-709. (CHENG Yung-ming, ZHU Lu-juan, GU Zhao-qi.Rotational failure of 3D non-symmetric slope predicted by Bishop's method[J]. Chinese Journal of Geotechnical Engineering, 2002, 24(6): 706-709. (in Chinese)) [15] 张均锋, 王思莹, 祈涛. 边坡稳定分析的三维Spencer法[J]. 岩石力学与工程学报, 2005, 24(19): 3434-3439. (ZHANG Jun-feng, WANG Si-ying, QI Tao.Three-dimensional spencer method for slope stability analysis[J]. Chinese Journal of Rock Mechanics and Engineering, 2005, 24(19): 3434-3439. (in Chinese)) [16] 李同录, 王艳霞, 邓宏科. 一种改进的三维边坡稳定性分析方法[J]. 岩土工程学报, 2003, 25(5): 611-614. (LI Tong-lu, WANG Yan-xia, DENG Hong-ke.An improved method for three-dimensional slope stability analysis[J]. Chinese Journal of Geotechnical Engineering, 2003, 25(5): 611-614. (in Chinese)) [17] 朱大勇, 钱七虎. 三维边坡严格与准严格极限平衡解答及工程应用[J]. 岩石力学与工程学报, 2007, 26(8): 1513-1528. (ZHU Da-yong, QIAN Qi-hu.Rigorous and quasi-rigorous limit equilibrium solutions of 3D slope stability and application to engineering[J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(8): 1513-1528. (in Chinese)) [18] 郑宏. 严格三维极限平衡法[J]. 岩石力学与工程学报, 2007, 26(8): 1529-1537. (ZHENG Hong.A rigorous three-dimensional limit equilibrium method[J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(8): 1529-1537. (in Chinese)) [19] 陈昌富, 朱剑锋. 基于Morgenstern-Price法边坡三维稳定性分析[J]. 岩石力学与工程学报, 2010, 29(7): 1473-1480. (CHEN Chang-fu, ZHU Jian-feng.A three-dimensional slope stability analysis procedure based on Morgenstern-Price method[J]. Chinese Journal of Rock Mechanics and Engineering, 2010, 29(7): 1473-1480. (in Chinese)) [20] 郭明伟, 葛修润, 李春光, 等. 边坡和坝基抗滑稳定分析的三维矢量和法及其工程应用[J]. 岩石力学与工程学报, 2010, 29(1): 8-20. (GUO Ming-wei, GE Xiu-run, LI Chun-guang, et al.Three-dimensional vector sum method employed in slope and dam foundation stability analyses and its applications to practical engineering[J]. Chinese Journal of Rock Mechanics and Engineering, 2010, 29(1): 8-20. (in Chinese)) [21] 凌道盛, 戚顺超, 陈锋, 等. 一种基于Morgenstern-Price法假定的三维边坡稳定性分析法[J]. 岩石力学与工程学报, 2013, 32(1): 107-116. (LING Dao-sheng, QI Shun-chao, CHEN Feng, et al.A limit equilibrium method based on Morgenstern-Price method for 3D slope stability analysis[J]. Chinese Journal of Rock Mechanics and Engineering, 2013, 32(1): 107-116. (in Chinese)) [22] MCCOOK D K.Limit analysis and soil plasticity[J]. Soil Science Society of America Journal, 1976, 40: 4. [23] LE C V, NGUYEN-XUAN H, NGUYEN-DANG H.Upper and lower bound limit analysis of plates using FEM and second-order cone programming[J]. Computers & Structures, 2010, 88(1/2): 65-73. [24] SLOAN S W, KLEEMAN P W.Upper bound limit analysis using discontinuous velocity fields[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 127(1/2/3/4): 293-314. [25] LYAMIN A V, SLOAN S W.Upper bound limit analysis using linear finite elements and nonlinear programming[C]// Civil-Comp Proceedings, Finite Elements: Techniques and Developments. Stirlingshire, UK: Civil-Comp Press, 2002: 61-77. [26] YIN J, CHEN J, LEE C.Upper limit analysis of stability of rock and soil slopes using rigid finite elements[J]. Chinese Journal of Rock Mechanics & Engineering, 2004, 23(6): 898-905. [27] LIU F T, ZHAO J D.Upper bound limit analysis using radial point interpolation meshless method and nonlinear programming[J]. International Journal of Mechanical Sciences, 2013, 70: 26-38. [28] YU S B, ZHANG X, SLOAN S W.A 3D upper bound limit analysis using radial point interpolation meshless method and second-order cone programming[J]. International Journal for Numerical Methods in Engineering, 2016, 108(13): 1686-1704. [29] YUAN S, DU J N.Effective stress-based upper bound limit analysis of unsaturated soils using the weak form quadrature element method[J]. Computers and Geotechnics, 2018, 98: 172-180. [30] HUANG M S, FAN X P, WANG H R.Three-dimensional upper bound stability analysis of slopes with weak interlayer based on rotational-translational mechanisms[J]. Engineering Geology, 2017, 223: 82-91. [31] ZHANG X.Three-dimensional stability analysis of concave slopes in plan view[J]. Journal of Geotechnical Engineering, 1988, 114(6): 658-671.
