强度折减采样结合SVM代理模型的土质边坡可靠度高效分析

卢建; 曾鹏; 冯兵; 王鑫; 严祖龙

doi:10.19509/j.cnki.dzkq.tb20240756

强度折减采样结合SVM代理模型的土质边坡可靠度高效分析

doi: 10.19509/j.cnki.dzkq.tb20240756

卢建^a,,
曾鹏^b, ,,
冯兵^a,
王鑫^a,
严祖龙^a

a.
成都理工大学：环境与土木工程学院
b.
成都理工大学：地质灾害防治与地质环境保护国家重点实验室，成都 610059

基金项目: 国家自然科学基金项目（42377189）

详细信息

作者简介:
卢建：E-mail：1633580118@qq.com

通讯作者:
E-mail：zengpeng15@cdut.edu.cn

中图分类号: TU43
计量
- 文章访问数: 48
- PDF下载量: 1
- 被引次数: 0
出版历程
- 收稿日期: 2024-12-05
- 录用日期: 2025-06-30
- 修回日期: 2025-06-28
- 网络出版日期: 2025-06-30

Efficient reliability analysis of soil slopes by combining strength reduction sampling with SVM surrogate model

LU Jian^a
,,
ZENG Peng^{b
, ,},
FENG Bing^a,
WANG Xin^a,
YAN Zulong^a

a.
College of Environment and Civil Engineering, Chengdu University of Technology
b.
State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu 610059, China

More Information

Author Bio:
E-mail：1633580118@qq.com

Corresponding author: E-mail：zengpeng15@cdut.edu.cn

摘要

摘要:
边坡可靠度分析的高昂计算成本限制了其在工程实践中的广泛应用，亟需开发更加高效、准确的分析方法。本研究提出了一种有限差分强度折减法高效采样（strength reduction sampling，简称SRS）结合主动学习支持向量机（support vector machine，简称SVM）代理模型的新型土质边坡可靠度分析方法（SRS-SVM）。通过强度折减采样策略，高效生成邻近极限状态面高信息量的训练样本点，提升了SVM代理模型的训练效率。通过4个经典边坡案例对所提方法的高效性和准确性进行了验证。研究结果表明，本方法相比于传统的可靠度方法，如经典响应面（classical response surface method，简称CRSM）、径向基函数（radial basis function，简称RBF）和高斯过程回归结合强度折减采样（strength reduction sampling combined with Gaussian process regression，简称SRS-GPR）等，在计算效率（数值模型计算小于40次）和计算准确性（系统失稳概率相对误差绝对值小于1.5%）均表现出明显的优势，并且在处理功能函数非线性及复杂边坡时展现出较强的适应性。本研究提出的SRS-SVM方法将强度折减采样与主动学习SVM分类模型结合，在计算效率、计算精度以及处理复杂问题等方面表现优异，具有良好的工程应用前景，可为工程边坡风险评价与防治提供高效的技术支撑。
- 边坡稳定性 /
- 不确定性 /
- 强度折减采样 /
- SVM代理模型 /
- 可靠度分析
Abstract:
Objective
Landslides caused by slope instability have posed a considerable threat to human lives and property, especially under the background of rapid infrastructure development and drastic global climate change. Traditional slope stability evaluation methods usually ignore the randomness of soil parameters, and conventional reliability analysis approaches such as monte carlo simulation (MCS) suffer from excessively high computational costs, which seriously restrict their practical engineering applications.
Methods
To address these problems, this study develops a novel and efficient method for reliability analysis of soil slopes, termed strength reduction sampling-support vector machine (SRS-SVM), by integrating the finite difference strength reduction sampling strategy and an active learning support vector machine surrogate model. The proposed method employs strength reduction sampling to generate highly informative training points strictly near the limit state surface (LSS), where one single numerical model evaluation can produce three key samples, thus remarkably improving the training efficiency of the SVM surrogate model. Meanwhile, an improved active learning function based on k-fold cross-validation and jackknifing estimation is adopted to select points with high uncertainty close to the LSS, and a convergence criterion based on the stability of system failure probability is applied to balance computational accuracy and efficiency. Four typical slope cases, including a single-layer soil slope, a two-layer clay slope, a three-layer slope with four random variables, and a four-layer complex dam slope with six random variables, are adopted to verify the performance of the SRS-SVM method. The proposed method is compared with several classic reliability approaches, such as the classical response surface method (CRSM), radial basis function (RBF), and strength reduction sampling-gaussian process regression (SRS-GPR).
Results
The results demonstrate that the SRS-SVM method requires fewer than 40 numerical model evaluations for all cases, and the absolute relative errors of the system failure probability are controlled within 1.5%, showing overwhelming advantages over traditional methods in both computational efficiency and accuracy. Furthermore, the method presents strong adaptability in dealing with highly nonlinear performance functions and multi-variable complex slope conditions.
Conclusion
This study combines the high-efficiency sampling characteristic of strength reduction technique and the superior classification ability of SVM, providing a new high-performance solution for accurate and fast reliability analysis of soil slopes. The SRS-SVM method has broad application prospects in practical engineering risk assessment, disaster prevention and mitigation of slopes, and can effectively support the reliability-based design of geotechnical engineering under uncertain conditions.
- slope stability /
- uncertainty /
- strength reduction sampling /
- SVM surrogate model /
- reliability analysis

HTML全文

图 1 强度折减采样原理^[17]

c. 边坡黏聚力；φ. 边坡的内摩擦角；F_s. 边坡的稳定性系数；下同

Figure 1. Principle of strength reduction sampling ^[17]

下载: 全尺寸图片幻灯片

图 2 强度折减采样结合主动学习SVM代理模型的边坡可靠度分析流程

P_{f, s}. 边坡系统失稳概率；3−σ. 3倍标准差；ε_e. 交叉验证模型计算的系统失稳概率的最大相对误差

Figure 2. Slope reliability analysis process using strength reduction sampling combined with active learning SVM surrogate model

下载: 全尺寸图片幻灯片

图 3 单层土质边坡网格及几何尺寸(a)及SRS-SVM与SRS-GPR代理模型计算收敛过程(b)、采样与建模效果(c)

Figure 3. Mesh and geometric dimensions (a), convergence process (b) and sampling and modeling performance (c) of SRS-SVM and SRS-GPR surrogate models of single-layer soil slope

下载: 全尺寸图片幻灯片

图 4 2层黏土边坡网格及几何尺寸(a)及SRS-SVM与SRS-GPR代理模型计算收敛过程(b)和采样与建模效果(c)

Figure 4. Mesh and geometric dimensions (a), convergence process (b), sampling and modeling performance (c) of SRS-SVM and SRS-GPR surrogate models of two-layer clay slope

下载: 全尺寸图片幻灯片

图 5 3层土边坡网格及几何尺寸(a)及SRS-SVM与SRS-GPR代理模型计算收敛过程(b)

Figure 5. Mesh and geometric dimensions (a) and convergence process of SRS-SVM and SRS-GPR surrogate models (b) of three-layer soil slope

下载: 全尺寸图片幻灯片

图 6 4层边坡网格及几何尺寸(a)及SRS-SVM与SRS-GPR代理模型计算收敛过程(b)

Figure 6. Mesh and geometric dimensions of four-layer slope (a) and convergence process of SRS-SVM and SRS-GPR surrogate models (b)

下载: 全尺寸图片幻灯片

表 1 单层土质边坡参数统计信息

Table 1. Statistical information of parameters for single-layer soil slope

参数	均值	标准差	分布类型
密度/(kg·m⁻³)	1764	—	—
黏聚力$ c $/kPa	9.8	3	对数正态
内摩擦角$ \varphi $/(°)	10	2	对数正态

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表 2 单层土质边坡各代理模型预测结果

Table 2. Prediction results of each surrogate model for single-layer soil slope

方法	数值模型调用次数/次	有效样本点个数/个	系统失稳概率/%	与LHS的相对误差/%	数据来源
SRS-SVM	7	21	7.47	0.26	本研究
SRS-GPR	19	38	7.43	−0.26	本研究
CRSM	15	15	6.94	−6.80	ZHANG等^[33]
ARBF	36	36	7.41	−0.53	张天龙等^[3]
BCM	79	79	7.84	5.23	ZENG等^[11]
LHS参考值	—	—	7.45	—	本研究
注：LHS为拉丁超立方抽样；下同

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表 3 2层黏土边坡参数统计信息

Table 3. Statistical information of parameters for two-layer clay slope

土层	密度/(kg·m⁻³)	黏聚力$ c $/kPa
土层	密度/(kg·m⁻³)	均值	标准差	分布类型
上层	1900	120	36	对数正态
下层	1900	160	48	对数正态

下载: 导出CSV

表 4 2层黏土边坡各代理模型预测结果

Table 4. Prediction results of each surrogate model for two-layer clay slope

方法	数值模型调用次数/次	训练点个数/个	系统失稳概率/%	与LHS的相对误差/%	数据来源
SRS-SVM	15	45	0.445	1.13	本文
SRS-GPR	27	54	0.432	−1.81	本文
LSSVM	30	30	0.410	6.81	KANG等^[39]
Kriging	100	100	0.458	4.09	ZHANG等^[33]
CRSM	—	—	0.411	4.31	JI和LOW^[40]
LHS参考值	—	—	0.440	—	本文

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表 5 3层土边坡参数统计信息

Table 5. Statistical information of parameters for three-layer soil slope

土层	密度/（kg·m⁻³）	黏聚力$ c $/kPa		内摩擦角$ \varphi $/(°)
土层	密度/（kg·m⁻³）	均值	标准差	均值	标准差	分布类型
第1层	1950	0	—	38	—	—
第2层		5.3	1.59	23	4.6	正态
第3层		7.2	2.16	20	4	正态

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表 6 3层土边坡各代理模型预测结果

Table 6. Prediction results of each surrogate model for three-layer soil slope

方法	数值模型调用次数/次	训练点个数/个	系统失稳概率/%	与LHS的相对误差/%	数据来源
SRS-SVM	27	81	2.04	−0.48	本研究
SRS-GPR	34	68	2.10	2.43	本研究
CRSM	60	60	2.37	15.61	Hu等^[41]
ARBF	73	73	2.40	17.07	张天龙^[12]
ASVM	100	100	2.39	16.58	张天龙^[12]
LHS参考值	—	—	2.05	—	本研究

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表 7 4层边坡参数统计信息

Table 7. Statistical information of parameters for a four-layer slope

随机变量	均值	标准差	分布类型
坝体填土重度/（kg·m⁻³）	2000	110	正态分布
坝体填土内摩擦角/（°）	30	1.79
硬质黏土高度/m	4	0.48
海成黏土黏聚力/kPa	34.5	3.95
湖积黏土黏聚力/kPa	31.2	6.31
底部3层黏土总厚度/m	18.5	1

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表 8 4层边坡各代理模型预测结果

Table 8. Prediction results of each surrogate model for a four-layer slope

方法	数值模型调用次数/次	训练点个数/个	系统失稳概率/%	与LHS的相对误差/%	数据来源
SRS-SVM	37	111	7.17	−1.10	本研究
SRS-GPR	54	108	7.07	−2.48	本研究
AK	117	117	8.40	15.86	ZENG等^[11]
BCM	270	270	8.49	17.10	ZENG等^[11]
RSM	—	—	6.81	−6.07	XU等^[43]
LHS参考值	—	—	7.25	—	本研究

下载: 导出CSV

参考文献(43)

[1]	陈应显, 王璞, 周萌, 等. 断层面分割露天矿边坡精细化建模及稳定性分析[J]. 地质科技通报, 2026, 45(1): 314-323. doi: 10.19509/j.cnki.dzkq.tb20240273 CHEN Y X, WANG P, ZHOU M, et al. Refined modeling and stability analysis of open-pit mine slopes segmented by fault planes[J]. Bulletin of Geological Science and Technology, 2026, 45(1): 314-323. (in Chinese with English abstract doi: 10.19509/j.cnki.dzkq.tb20240273
[2]	蔡征龙, 孟永东, 秦毅, 等. 基于改进Green-Ampt模型的滑坡稳定时变可靠度分析[J]. 岩土力学, 2022, 43(1): 268-276. CAI Z L, MENG Y D, QIN Y, et al. Time-varying reliability of landslide stability based on improved Green-Ampt model[J]. Rock and Soil Mechanics, 2022, 43(1): 268-276. (in Chinese with English abstract
[3]	张天龙, 曾鹏, 李天斌, 等. 基于主动学习径向基函数的边坡系统可靠度分析[J]. 岩土力学, 2020, 41(9): 3098-3108. doi: 10.16285/j.rsm.2019.1695 ZHANG T L, ZENG P, LI T B, et al. System reliability analyses of slopes based on active-learning radial basis function[J]. Rock and Soil Mechanics, 2020, 41(9): 3098-3108. (in Chinese with English abstract doi: 10.16285/j.rsm.2019.1695
[4]	KURNAZ T F, ERDEN C, DAĞDEVIREN U, et al. Comparison of machine learning algorithms for slope stability prediction using an automated machine learning approach[J]. Natural Hazards, 2024, 120(8): 6991-7014. doi: 10.1007/s11069-024-06490-8
[5]	ZHANG W G, GU X, HONG L, et al. Comprehensive review of machine learning in geotechnical reliability analysis: Algorithms, applications and further challenges[J]. Applied Soft Computing, 2023, 136: 110066. doi: 10.1016/j.asoc.2023.110066
[6]	姬建, 崔红志, 佟斌, 等. 基于物理过程不确定性的降雨诱发浅层滑坡易发性快速区划: GIS-FORM技术开发与应用[J]. 岩石力学与工程学报, 2024, 43(4): 838-850. JI J, CUI H Z, TONG B, et al. Fast zoning of rainfall-induced shallow landslide susceptibility based on physical process uncertainty: Development and application of GIS-FORM[J]. Chinese Journal of Rock Mechanics and Engineering, 2024, 43(4): 838-850. (in Chinese with English abstract
[7]	高玉峰, 王玉杰, 张飞, 等. 边坡工程与堤坝工程研究进展[J]. 土木工程学报, 2024, 57(8): 97-118. doi: 10.15951/j.tmgcxb.23110915 GAO Y F, WANG Y J, ZHANG F, et al. State-of-the-art review on slope and embankment dams[J]. China Civil Engineering Journal, 2024, 57(8): 97-118. (in Chinese with English abstract doi: 10.15951/j.tmgcxb.23110915
[8]	何婷婷, 尚岳全, 吕庆, 等. 边坡可靠度分析的支持向量机法[J]. 岩土力学, 2013, 34(11): 3269-3276. HE T T, SHANG Y Q, LYU Q, et al. Slope reliability analysis using support vector machine[J]. Rock and Soil Mechanics, 2013, 34(11): 3269-3276. (in Chinese with English abstract
[9]	苏永华, 罗正东, 张盼凤, 等. 基于Kriging的边坡稳定可靠度主动搜索法[J]. 岩土工程学报, 2013, 35(10): 1863-1869. SU Y H, LUO Z D, ZHANG P F, et al. Active searching algorithm for slope stability reliability based on Kriging model[J]. Chinese Journal of Geotechnical Engineering, 2013, 35(10): 1863-1869. (in Chinese with English abstract
[10]	万增勇, 李苹, 杨关, 等. 基于代理模型的土质边坡可靠度分析[J]. 武汉大学学报(工学版), 2023, 56(10): 1217-1223. doi: 10.3969/j.issn.1006-4869.2019.06.015 WAN Z Y, LI P, YANG G, et al. The reliability analysis of soil slope based on surrogate model[J]. Engineering Journal of Wuhan University, 2023, 56(10): 1217-1223. (in Chinese with English abstract doi: 10.3969/j.issn.1006-4869.2019.06.015
[11]	ZENG P, ZHANG T L, LI T B, et al. Binary classification method for efficient and accurate system reliability analyses of layered soil slopes[J]. Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 2022, 16(3): 435-451.
[12]	张天龙. 基于主动学习代理模型和强度折减法的土质滑坡失稳概率评价方法研究[D]. 成都: 成都理工大学, 2020. ZHANG T L. System probability of failure assessment of soil landslides using active-learning surrogate models and strength reduction method[D]. Chengdu: Chengdu University of Technology, 2020. (in Chinese with English abstract
[13]	邓志平, 钟敏, 潘敏, 等. 考虑参数空间变异性和基于高效代理模型的边坡可靠度分析[J]. 岩土工程学报, 2024, 46(2): 273-281. DENG Z P, ZHONG M, PAN M, et al. Slope reliability analysis considering spatial variability of parameters based on efficient surrogate model[J]. Chinese Journal of Geotechnical Engineering, 2024, 46(2): 273-281. (in Chinese with English abstract
[14]	CHO S E. Probabilistic stability analyses of slopes using the ANN-based response surface[J]. Computers and Geotechnics, 2009, 36(5): 787-797. doi: 10.1016/j.compgeo.2009.01.003
[15]	胡添翼, 戴波, 何启, 等. 基于随机森林分类算法的边坡稳定预测模型[J]. 人民黄河, 2017, 39(5): 115-118. doi: 10.3969/j.issn.1000-1379.2017.05.027 HU T Y, DAI B, HE Q, et al. Slope stability forecasting model based on Radom forest classification algorithm[J]. Yellow River, 2017, 39(5): 115-118. (in Chinese with English abstract doi: 10.3969/j.issn.1000-1379.2017.05.027
[16]	LIU Y D, LI X Y, LIU X, et al. A combined shear strength reduction and surrogate model method for efficient reliability analysis of slopes[J]. Computers and Geotechnics, 2022, 152: 105021. doi: 10.1016/j.compgeo.2022.105021
[17]	刘亚栋, 刘贤, 黎学优, 等. 基于强度折减采样与高斯过程回归的空间变异土坡自适应可靠度分析[J]. 岩土工程学报, 2024, 46(5): 978-987. doi: 10.11779/CJGE20230065 LIU Y D, LIU X, LI X Y, et al. Adaptive reliability analysis of spatially variable soil slopes using strength reduction sampling and Gaussian process regression[J]. Chinese Journal of Geotechnical Engineering, 2024, 46(5): 978-987. (in Chinese with English abstract doi: 10.11779/CJGE20230065
[18]	陈家豪, 张燕, 杜明芳, 等. 基于优化极限学习机模型的边坡稳定性预测研究[J]. 金属矿山, 2024(6): 191-198. doi: 10.19614/j.cnki.jsks.202406026 CHEN J H, ZHANG Y, DU M F, et al. Study on slope stability prediction based on optimized extreme learning machine model[J]. Metal Mine, 2024(6): 191-198. (in Chinese with English abstract doi: 10.19614/j.cnki.jsks.202406026
[19]	LIN S, ZHENG H, HAN C, et al. Evaluation and prediction of slope stability using machine learning approaches[J]. Frontiers of Structural and Civil Engineering, 2021, 15(4): 821-833. doi: 10.1007/s11709-021-0742-8
[20]	武博强, 王勇, 张葆永, 等. 土质边坡数值模型稳定性的求解精度研究[J]. 公路交通科技, 2024, 41(1): 71-78. doi: 10.3969/j.issn.1002-0268.2024.01.009 WU B Q, WANG Y, ZHANG B Y, et al. Study on solution accuracy of numerical model stability with soil slope[J]. Journal of Highway and Transportation Research and Development, 2024, 41(1): 71-78. (in Chinese with English abstract doi: 10.3969/j.issn.1002-0268.2024.01.009
[21]	褚雪松, 庞峰, 李亮, 等. 边坡稳定有限元强度折减法与极限平衡法对比[J]. 人民黄河, 2011, 33(10): 93-95. CHU X S, PANG F, LI L, et al. Comparative study on FEM strength reduction method and limit equilibrium method for the slope stability analysis[J]. Yellow River, 2011, 33(10): 93-95. (in Chinese with English abstract
[22]	MA J Z, ZHANG J, HUANG H W, et al. Identification of representative slip surfaces for reliability analysis of soil slopes based on shear strength reduction[J]. Computers and Geotechnics, 2017, 85: 199-206. doi: 10.1016/j.compgeo.2016.12.033
[23]	徐文刚, 余旭荣, 年廷凯, 等. 基于FLAC^3D的三维边坡稳定性强度折减法计算效率改进算法及其应用[J]. 吉林大学学报(地球科学版), 2021, 51(5): 1347-1355. doi: 10.13278/j.cnki.jjuese.20200292 XU W G, YU X R, NIAN T K, et al. Optimization and application of FLAC^3D strength-reduction computation in three-dimension slope stability analysis[J]. Journal of Jilin University (Earth Science Edition), 2021, 51(5): 1347-1355. (in Chinese with English abstract doi: 10.13278/j.cnki.jjuese.20200292
[24]	程刚, 吴勇飞, 曹德胜, 等. 人工智能算法在滑坡监测与预测技术中的研究与应用[J]. 地质科技通报, 2025, 44(5): 302-316. doi: 10.19509/j.cnki.dzkq.tb20250115 CHENG G, WU Y F, CAO D S, et al. Research and application of artificial intelligence algorithms in landslide monitoring and prediction technology[J]. Bulletin of Geological Science and Technology, 2025, 44(5): 302-316. (in Chinese with English abstract doi: 10.19509/j.cnki.dzkq.tb20250115
[25]	赵国彦, 邹景煜, 王猛. 基于混沌粒子群改进支持向量机对露天矿边坡稳定性的分类预测[J]. 矿冶工程, 2024, 44(2): 8-12. doi: 10.3969/j.issn.0253-6099.2024.02.003 ZHAO G Y, ZOU J Y, WANG M. Classification and prediction of slope stability of open-pit mine with support vector machine based on chaotic particle swarm optimization[J]. Mining and Metallurgical Engineering, 2024, 44(2): 8-12. (in Chinese with English abstract doi: 10.3969/j.issn.0253-6099.2024.02.003
[26]	曹文庚, 潘登, 徐郅杰, 等. 河南省滑坡灾害易发性制图研究: 多种机器学习模型的对比[J]. 地质科技通报, 2025, 44(1): 101-111. doi: 10.19509/j.cnki.dzkq.tb20230338 CAO W G, PAN D, XU Z J, et al. Landslide disaster vulnerability mapping study in Henan Province: Comparison of different machine learning models[J]. Bulletin of Geological Science and Technology, 2025, 44(1): 101-111. (in Chinese with English abstract doi: 10.19509/j.cnki.dzkq.tb20230338
[27]	LI X Y, LIU Y D, YANG Z Y, et al. Efficient slope reliability analysis using adaptive classification-based sampling method[J]. Bulletin of Engineering Geology and the Environment, 2021, 80(12): 8977-8993. doi: 10.1007/s10064-021-02476-z
[28]	PAN Q J, DIAS D. An efficient reliability method combining adaptive Support Vector Machine and Monte Carlo Simulation[J]. Structural Safety, 2017, 67: 85-95. doi: 10.1016/j.strusafe.2017.04.006
[29]	LI X, GONG C L, GU L X, et al. A sequential surrogate method for reliability analysis based on radial basis function[J]. Structural Safety, 2018, 73: 42-53. doi: 10.1016/j.strusafe.2018.02.005
[30]	KLEIJNEN J P C, VAN BEERS W C M. Application-driven sequential designs for simulation experiments: Kriging metamodelling[J]. Journal of the Operational Research Society, 2004, 55(8): 876-883. doi: 10.1057/palgrave.jors.2601747
[31]	ECHARD B, GAYTON N, LEMAIRE M. AK-MCS: An active learning reliability method combining Kriging and Monte Carlo Simulation[J]. Structural Safety, 2011, 33(2): 145-154. doi: 10.1016/j.strusafe.2011.01.002
[32]	XIAO N C, ZUO M J, ZHOU C N. A new adaptive sequential sampling method to construct surrogate models for efficient reliability analysis[J]. Reliability Engineering & System Safety, 2018, 169: 330-338. doi: 10.1016/j.ress.2017.09.008
[33]	ZHANG J, HUANG H W, PHOON K K. Application of the Kriging-based response surface method to the system reliability of soil slopes[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2013, 139(4): 651-655. doi: 10.1061/(ASCE)GT.1943-5606.0000801
[34]	CHOWDHURY R, RAO B N. Probabilistic stability assessment of slopes using high dimensional model representation[J]. Computers and Geotechnics, 2010, 37(7/8): 876-884. doi: 10.1016/j.compgeo.2010.07.007
[35]	ZHANG J, CHEN H Z, HUANG H W, et al. Efficient response surface method for practical geotechnical reliability analysis[J]. Computers and Geotechnics, 2015, 69: 496-505. doi: 10.1016/j.compgeo.2015.06.010
[36]	CHING J, PHOON K K, HU Y G. Efficient evaluation of reliability for slopes with circular slip surfaces using importance sampling[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2009, 135(6): 768-777. doi: 10.1061/(ASCE)GT.1943-5606.0000035
[37]	WANG T, JI J, YIN X, et al. Efficient probabilistic slope stability analysis using conditional probability-based weighted low-discrepancy simulation[J]. Computers and Geotechnics, 2024, 174: 106615. doi: 10.1016/j.compgeo.2024.106615
[38]	LIU Y B, REN W Z, LIU C C, et al. A search method for probabilistic critical slip surfaces with arbitrary shapes and its application in slope reliability analysis[J]. Natural Hazards, 2021, 107(2): 1657-1679. doi: 10.1007/s11069-021-04651-7
[39]	KANG F, LI J S, LI J J. System reliability analysis of slopes using least squares support vector machines with particle swarm optimization[J]. Neurocomputing, 2016, 209: 46-56. doi: 10.1016/j.neucom.2015.11.122
[40]	JI J, LOW B K. Stratified response surfaces for system probabilistic evaluation of slopes[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2012, 138(11): 1398-1406. doi: 10.1061/(ASCE)GT.1943-5606.0000711
[41]	HU C, LEI R D, GU Q H. Application of supplementary sampling method in slope reliability analysis[J]. International Journal of Geomechanics, 2023, 23(7): 04023097. doi: 10.1061/IJGNAI.GMENG-8209
[42]	CHRISTIAN J T, LADD C C, BAECHER G B. Reliability applied to slope stability analysis[J]. Journal of Geotechnical Engineering, 1994, 120(12): 2180-2207. doi: 10.1061/(ASCE)0733-9410(1994)120:12(2180)
[43]	XU B, LOW B K. Probabilistic stability analyses of embankments based on finite-element method[J]. Journal of Geotechnical and Geoenvironmental Engineering, 2006, 132(11): 1444-1454. doi: 10.1061/(ASCE)1090-0241(2006)132:11(1444)