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M. Hicks, W. Spencer (2010)
Influence of heterogeneity on the reliability and failure of a long 3D slopeComputers and Geotechnics, 37
B. Low, W. Tang (2007)
Efficient Spreadsheet Algorithm for First-Order Reliability MethodJournal of Engineering Mechanics-asce, 133
D. Griffiths, G. Fenton (2004)
Probabilistic slope stability analysis by finite elementsJournal of Geotechnical and Geoenvironmental Engineering, 130
T. Nguyen, B. Indraratna (2021)
Rail track degradation under mud pumping evaluated through site and laboratory investigationsInternational Journal of Rail Transportation, 10
(2016)
VPO (2012) Flood risk in the netherlands: the method in brief
Wensheng Zhang, Q. Luo, Liang-wei Jiang, Teng-fei Wang, Chong Tang, Li Zhengtao (2021)
Improved Vanmarcke analytical model for 3D slope reliability analysisComputers and Geotechnics, 134
E. Vanmarcke (2014)
Multi-hazard Risk Assessment of Civil Infrastructure Systems, with a Focus on Long Linear Structures such as Levees
Z. Lomnicki, R. Barlow, F. Proschan, L. Hunter (1966)
Mathematical Theory of ReliabilityJournal of the Operational Research Society, 17
(2015)
ISO 2394 General principles on reliability of structures
M. Hicks, Yajun Li (2018)
Influence of length effect on embankment slope reliability in 3DInternational Journal for Numerical and Analytical Methods in Geomechanics, 42
(2015)
Technical code for risk management of railway construction engineering (Q/CR 9006-2014)
J. Duncan, S. Wright (2005)
Soil Strength and Slope Stability
(2014)
The national flood risk analysis for the Netherlands: Final report
R. Melchers (1987)
Structural Reliability: Analysis and Prediction
E Vanmarcke (1977)
Reliability of earth slopesJ Geotech Eng Div, 103
L. Zhang, J. Zhang, L. Zhang, W. Tang (2011)
Stability analysis of rainfall-induced slope failure: a review, 164
T. Orr (2019)
Honing safety and reliability aspects for the second generation of Eurocode 7Georisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 13
J. Christian, C. Ladd, G. Baecher (1994)
Reliability Applied to Slope Stability AnalysisJournal of Geotechnical Engineering, 120
J. Ching, Y. Hu, Zon-Yee Yang, Jang-Quang Shiau, Jeng-Cheung Chen, Yisa Li (2011)
Reliability-based design for allowable bearing capacity of footings on rock masses by considering angle of distortionInternational Journal of Rock Mechanics and Mining Sciences, 48
K. Phoon, J. Retief (2016)
Reliability of Geotechnical Structures in ISO2394
B. Cami, Sina Javankhoshdel, K. Phoon, J. Ching (2020)
Scale of Fluctuation for Spatially Varying Soils: Estimation Methods and ValuesASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, 6
J. Ching, K. Phoon, Z. Khan, Dongming Zhang, Hong-wei Huang (2020)
Role of municipal database in constructing site-specific multivariate probability distributionComputers and Geotechnics, 124
C. Reale, J. Xue, Zhan-cheng Pan, K. Gavin (2015)
Deterministic and Probabilistic Multi-Modal Analysis of Slope StabilityComputers and Geotechnics, 66
E. Vanmarcke (2010)
Random Fields: Analysis and Synthesis (Revised and Expanded New Edition)
T. Xiao, Dianqing Li, Z. Cao, S. Au, K. Phoon (2016)
Three-dimensional slope reliability and risk assessment using auxiliary random finite element methodComputers and Geotechnics, 79
(2002)
Engineering geology field manual
D. Tobutt, E. Richards (1979)
The reliability of earth slopesInternational Journal for Numerical and Analytical Methods in Geomechanics, 3
Xing-min Meng, Tian-jun Qi, Yan Zhao, T. Dijkstra, W. Shi, Yin-fei Luo, Yuan-zhao Wu, Xiao-jun Su, Fu-meng Zhao, Jin-hui Ma, Yi Zhang, Guan Chen, Donghong Chen, Mao-sheng Zhang (2021)
Deformation of the Zhangjiazhuang high-speed railway tunnel: an analysis of causal mechanisms using geomorphological surveys and D-InSAR monitoringJournal of Mountain Science, 18
K. Phoon, F. Kulhawy (1999)
Characterization of Geotechnical Variability
RE Barlow, F Proschan (1965)
10.1137/1.9781611971194Mathematical theory of reliability
R. Salgado, Sungmin Yoon (2003)
DYNAMIC CONE PENETRATION TEST (DCPT) FOR SUBGRADE ASSESSMENT
W. Fellenius (1936)
Calculation of stability of earth dam, 4
R. Jongejan, E. Calle (2013)
Calibrating semi-probabilistic safety assessments rules for flood defencesGeorisk: Assessment and Management of Risk for Engineered Systems and Geohazards, 7
Shunhua Zhou, Zhiyao Tian, Honggui Di, P. Guo, Longlong Fu (2020)
Investigation of a loess-mudstone landslide and the induced structural damage in a high-speed railway tunnelBulletin of Engineering Geology and the Environment, 79
H. El-Ramly, N. Morgenstern, D. Cruden (2003)
Probabilistic stability analysis of a tailings dyke on presheared clay-shaleCanadian Geotechnical Journal, 40
Zhan-ju Lin, F. Niu, Xiaolong Li, Li Anyuan, Ming-hao Liu, Jing Luo, Zhujie Shao (2018)
Characteristics and controlling factors of frost heave in high-speed railway subgrade, Northwest ChinaCold Regions Science and Technology
Sung-Eun Cho (2013)
First-order reliability analysis of slope considering multiple failure modesEngineering Geology, 154
R. Salgado, Dongwook Kim (2014)
Reliability Analysis of Load and Resistance Factor Design of SlopesJournal of Geotechnical and Geoenvironmental Engineering, 140
Science (2022)
Rail. Eng. Science (2022) 30(4):482–493 https://doi.org/10.1007/s40534-022-00274-1 System reliability evaluation of long railway subgrade slopes considering discrete instability 1,2,3 1,3 1,3 1,3 Wensheng Zhang · Qiang Luo · Tengfei Wang · Qi Wang · 1,3 1,3 Liangwei Jiang · Dehui Kong Received: 23 November 2021 / Revised: 28 March 2022 / Accepted: 28 March 2022 / Published online: 1 June 2022 © The Author(s) 2022 Abstract This paper develops a dual-indicator discrete Keywords Subgrade slope · Local instability · method (DDM) for evaluating the system reliability per- Segmentation · k-out-of-n system · System reliability · formance of long soil subgrade slopes. First, they are seg- Railway engineering mented into many slope sections using the random finite element method, to ensure each section statistically contains one potential local instability. Then, the k-out-of-n system 1 Introduction model is used to describe the relationship between the total number of sections n , the acceptable number of failure sec- Soil subgrade is widely used in railway engineering world- tions m , the reliability of sections R , and the system reli- wide, such as in the South Coast rail line in Australia [1] and sec ability R . Finally , m and R are jointly used to assess the the Lanzhou-Xinjiang rail line in China [2]. The instability sys sys system reliability performance. For cases lacking spatial data of subgrade slopes is one of the main risk sources of railway of soil properties, a simplified DDM is provided in which operations due to the potential for huge social and economic long subgrade slopes are segmented by the empirical value impacts and even passenger casualties. For example, the of section length and R is substituted by that of cross- Ingenheim derailment accident in France caused by a land- sec sections taken from them. The results show that (1) DDM slide due to heavy precipitation (see Fig. 1) caused 21 inju- can provide the probability that the actual number of local ries and serious damage to vehicles [3]. Rainfall is one of the instabilities does not exceed a desired threshold. (2) R most significant triggering factors for slope failures in many sys decreases with increasing n or decreasing R ; that is, it regions [4]. For long subgrade slopes composed of spatial sec is likely to encounter more local instabilities for longer or variable soils, the resisting moment of the potential local weaker subgrade slopes. n is negatively related to the hori- failure surface at different locations varies significantly, and zontal scale of fluctuation of soil properties and positively it is highly likely to encounter discretely distributed local related to the total length of subgrade slopes L . (3) When instability events along the length direction [5]. Since any L is sufficiently large, there is a considerable opportunity to local slope instability event may cause traffic disruption, the meet local instabilities even if R is large enough. reliability assessment of subgrade slopes should be treated sec as a system problem. For the system reliability evaluation of subgrade slopes, the key target is to estimate the probability that the actual number of local instability events does not exceed an acceptable threshold, such as 1. * Qiang Luo lqrock@swjtu.edu.cn Traditionally, deterministic plane strain analyses are 1 adopted to assess the stability level of a set of cross-sections School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China taken from long subgrade slope via the factor of safety ( F ) 2 [6, 7], in which characteristic values are used to represent Department of Civil and Environmental Engineering, Harbin Institute of Technology, Shenzhen 518055, China spatially variable soil parameters. When F is no smaller than 3 a predetermined value, such as 1.25, the entire subgrade MOE Key Laboratory of High-Speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China slope is deemed sufficiently safe. More recently, probabilistic Vol:.(1234567890) 1 3 System reliability evaluation of long railway subgrade slopes considering discrete… 483 indicator of the distance within which soil property values show a strong correlation, which is defined as the area under the autocorrelation function curve [16–18]. There are vari- ous methods to estimate the value of SOF using field testing data (e.g., cone penetration testing (CPT) data), such as the Sliding mass method of moment, maximum-likelihood estimation, and Bayesian analysis [19]. Based on three-dimensional RFEM, Hicks and Spencer [5] developed a numerical method to calculate the R that no local failure event will occur for sys long slopes consisting of statistically homogeneous soils. The RFEM is regarded as a versatile approach to predict the responses of large-scale subgrade slopes, mainly because Strasbourg-Vendenheim line, France March 2021 no prior assumption concerning the location and shape of failure surfaces is required. However, it is not trivial to carry Fig. 1 An example of local instability of subgrade slopes out a full three-dimensional probabilistic analysis for long earth slopes due to the immense computational expense. A VNK (Veiligheid Nederland in Kaart) method with less plane strain analyses have been developed and widely used, computational effort was proposed in the Netherlands when e.g., the first-order reliability method [8 –10] and the random assessing the system reliability of dike rings [20–22]. A finite element method (RFEM) [11]. The random variable or dike ring is divided into multiple adjacent segments within two-dimensional random field is used to describe the spatial which soil properties can be regarded as statistically homo- variability of soil property, and the reliability level of cross- geneous, and the length of each segment ranges from 150 to sections is evaluated in terms of reliability ( R ), probability 2000 m, with an average of 750 m [21–23]. The reliabilities of failure ( P ), or reliability index ( ) [12]. Nevertheless, an of all segments are combined to obtain the probability that infinite rupture surface in the length direction is assumed in no instability event occurs in any segment, i.e., system reli- the above two-dimensional analysis methods, which is not in ability, in which the series system model is adopted. A series line with reality. As soil parameters are naturally variable in system means failure of any segment triggers the loss of sys- three-dimensional space, the reliability of a cross-section is tem function [24, 25], which is applicable to flood defense not representative of that of long subgrade slopes, especially since breaches in any location will cause flooding and vast when there are large fluctuations in soil properties. Previous losses. The abovementioned three-dimensional approaches research has shown that the reliability of three-dimensional can be classified into two categories: continuous and discrete earth slopes can be largely lower than that of a cross-section methods. The former treats a long slope as an indivisible taken from them when the length is large enough, which is object, such as Vanmarcke and RFEM method. The latter known as the length effect [13– 15]. Thus, when the cross- treats a long slope as a discrete system composed of many sections taken from a long subgrade slope all have enough components, such as VNK method. They both use one single reliability, the reliability of this subgrade slope can still be indicator (i.e., R ) to describe the reliability level of long sys very low. In other words, there may be a considerable pos- slopes, which provides engineers with confidence that no sibility to encounter one or more local failure events. Param- local instability event will occur. The occurrence probability eters F or only provides engineers with rough confidence that the number of local failure events does not surpass a that no local sliding event will occur along subgrade slopes. tolerated quantity more than zero cannot be obtained. Over the past decades, three-dimensional probabilistic To evaluate the system reliability of long continuous analysis techniques have been developed to assess the reli- subgrade slopes considering multiple potential local insta- ability of long subgrade slopes, in which a three-dimensional bilities, this work proposes a dual-indicator discrete method random field is used to model spatially variable soil proper - (DDM) easy to be employed in practice. The properties of ties. For long slope composed of statistically homogeneous soils are assumed to be statistically homogeneous. First, a soils, using the local averaging theory and first-crossing long subgrade slope is equally divided into many adjacent theory, Vanmarcke [14, 15] proposed an analytical method sections based on the number of local instabilities predicted to compute the system reliability ( R ) that no local failure by RFEM. Each section statistically contains a potential sys event will occur. The sliding mass is assumed to be a cylin- instability. Then, the k-out-of-n system model is adopted to der bounded by two vertical planes in Vanmarcke method. describe the relationship between the acceptable number of Statistically, homogeneous soil properties mean that the failure sections, the reliability of sections, and the reliability scale of fluctuation (SOF) and the statistics such as mean of the entire subgrade slope. Finally, two indicators named and variance are constant in earth space [16]. The SOF is an acceptable failure ratio and system failure probability acc Rail. Eng. Science (2022) 30(4):482–493 1 3 484 W. Zhang et al. M Mo od de e 1 1 M Mo od de e 2 2 System reliability analysis of long subgrade slopes Entire Local instability instability Step 1Step 2Step 3 Computation Segmentation Modelling System failure Division System model: parameters: -out-of-n system probability: P M Mo od de e 3 3 Fig. 2 Diagram of the process of DDM Before failure Entire instability After failure Classification criterion P are used to describe the reliability level of subgrade f,sys Mode 1: δ h slopes together. is the ratio of the acceptable number acc Mode 2: h≤δ L /2 of local failures to the total number of sections, and P f,sys Mode 3: L/2≤δ presents the possibility that the acceptable failure ratio is exceeded by actual failure ratios. Figure 2 presents the dia- gram of the process of DDM. It is noted that, for the system Fig. 3 Three failure modes of earth slope (after Hicks and Spencer [5]) reliability design of new subgrade slopes, engineers usually want the probability of no local instability to be greater than a prescribed value, e.g., 95%. But for the system reliabil- ity evaluation of existing subgrade slopes, engineers need zones, the failure occurs through weak and strong zones a more comprehensive understanding of the level of sys- along its entire length. This mode is analogous to a two- tem safety, not just the probability of no instability, but the dimensional deterministic analysis based on mean values occurrence probability of multiple instabilities. Actually, it of soil properties. is not rare that multiple local failure events are encountered Mode 2: For h ≤ 𝛿 < L∕2 , t he becomes large enough h h along a rail line in engineering practice. and local failure mass propagates much possibly through The rest of this paper is organized as follows. In Sect. 2, horizontal semi-continuous weaker zones. The reliability the segmentation method of long subgrade slope is estab- of a slope decreases with increasing L due to the increased lished. In Sect. 3, the k-out-of-n system is introduced to chance of encountering a zone weak enough to trigger a derive the expression of the system reliability of subgrade failure. slopes. In Sect. 4, a case study is provided to illustrate the Mode 3: For L∕2 ≤ , the soil presents a layered appear- use of DDM in current engineering practice, where spatial ance and the failure extends along the length of a slope, data of soil properties is unavailable. In Sect. 5, some con- resulting in a global failure. In this case, the failure surface cluding remarks are listed. develops along weak layers and appears analogous to that from a two-dimensional stochastic analysis. According to Phoon and Kulhawy [18], El-Ramly et al. 2 Segmentation of long subgrade slope [17], and others [26, 27], the horizontal SOF of soil proper- ties typically ranges between 10 and 100 m for various soil 2.1 Discrete instability modes of long subgrade slope types. In engineering practice, the subgrade slope height mostly varies from 3 to 20 m, whereas the length could Using the three-dimensional RFEM, Hicks and Spencer [5] extend from several to tens of kilometers. A prominent fea- identified three modes of instability of long soil slopes com- ture of subgrade slopes is the large value of L∕ and L∕h . posed of statistically homogeneous soils, as illustrated in Therefore, the instability mode for long subgrade slopes Fig. 3. The analyzed long slope has an angle of 45°, a height will theoretically be mode 2, which is in line with practical of 5 m, and a length of 100 m. The investigation shows that observations. the instability modes depend on the magnitude of the hori- zontal SOF of soil strength parameters relative to slope 2.2 Prediction of the number of local instabilities length L and height h , and has little relationship with the vertical SOF . In the study, ranges from 1 to 1000 m, For short soil slopes, it is less possible to encounter many v h and is fixed as 1 m. The following are general outlines of failure events owing to the lower probability of more than the three failure modes. one weak domain existing along the length direction [13]. Mode 1: For 𝛿 < h , as the is too small to allow a But, the occurrence probability rises with increasing slope h h rupture surface to develop through semi-continuous weaker length. For example, multiple local instabilities can be found Rail. Eng. Science (2022) 30(4):482–493 1 3 System reliability evaluation of long railway subgrade slopes considering discrete… 485 in long subgrade slopes along operating railway lines. The of discrete local instabilities for subgrade slopes with any prediction of the number of potential local instabilities is length, based on the study of Hicks and Li [13]. As the slope important for the system reliability evaluation of subgrade with a length of 500 m is long enough to eliminate the influ- slopes. A study by Hicks and Li [13] regarding mode 2 pro- ence of boundary conditions on the stability analysis results, vides a possible way to determine the number of local insta- the 95% quantiles in Fig. 4 could largely represent the upper bilities for long subgrade slopes. bound of the number of potential local instabilities. Hence, Using the three-dimensional RFEM, the discrete failure these statistical results are seen as a benchmark to predict the features of L = 500 m long slope composed of statistically number of potential failures for longer subgrade slopes. As homogeneous cohesive soils, with a height h = 5 m and an listed in Table 1, the N for 500 m long slope is extended as angle of 26.6°, were explored by Hicks and Li [13]. Four the number of potential local failures per kilometer ( N ) 1km values of were utilized, namely 12, 24, 50, and 100 m, by multiplying a factor of two. Then, the total number of respectively. The vertical SOF was taken as a constant of potential discrete local instabilities is expressed by 1 m since it has little effect on the discrete failure feature of n = N × L, (1) 1km long slopes [5]. The safety factor based on mean property values was fixed as 1.30. For each case of , 1000 Monte h where the total length L of subgrade slopes is in kilometers. Carlo simulations for the random field of soil properties For illustration, providing L = 100 km long subgrade were carried out, and the number of discretely distributed slope with similar geometries and soil properties to those local instabilities was counted. Hicks and Li [13] indicate in Hicks and Li [13], if = 24 m, the n will be 12 × 100 = that the number of local failures is significantly influenced 1200. It should be noted that the values shown in Table 1 by . There can be many discrete failures within the length h are only applicable to situations that slope geometries and when is small, and the number declines apparently with h soil properties are similar to the model used in the litera- the increase of . Based on the number of discrete failures h ture [13], not necessarily suitable for more general circum- of 1000 Monte Carlo realizations, as shown in Fig. 4, the stances. Further research is required to examine the effect 95% quantile of the failure number is around N = 9 as = 95 h of slope geometries as well as soil properties on the number 12 m, and N decreases to 6, 4, 3 as the grows up to 24, 95 h of potential local instabilities. 50, and 100 m, respectively. Besides, basedon the failure feature of mode 3, the number of failure events will be 1 2.3 Segmentation of long subgrade slope when = L∕2 and L. Theoretically, to get the number of potential local insta- Since soil properties are statistically homogeneous in bilities for subgrade slopes, the full-size RFEM model can three-dimensional space, a subgrade slope with n predicted be built and analyzed. However, it is extremely difficult to potential local instabilities is equally divided into n adjacent apply the RFEM to all subgrade slopes encountered in prac- sections. Each section statistically contains no more than tice due to the formidable computational cost. A simple yet one potential local instability event, with a confidence of effective way is established herein to predict the number 95%. The stability status of arbitrary two sections would be largely independent, and the failure probabilities of any two sections are assumed to be independent in this work. In ( (a a) ) ( (b b) ) ( (c c) ) ( (d d) ) reality, the stability status of two adjacent sections may be =12 m =24 m =50 m =100 m related, and strictly speaking, this dependency should be N =9 N =6 N =4 N =3 95 95 95 95 measured and considered when analyzing the system reli- ability of subgrade slopes. However, this relevance is hard to be determined. In addition, based on probability theory, Table 1 Results of numerical simulation (including data from Hicks and Li [13]) (m) N N L (m) l (m) h 95 1km sec sli 12 9 18 55.6 13.3 ± 8.0 0246 8 02 4 68 024 68 02468 10 24 6 12 83.3 17.7 ± 9.9 Number of discrete failures 50 4 8 125.0 27.4 ± 15.2 100 3 6 166.7 47.5 ± 27.3 Fig. 4 The number of discrete failures at different (after Hicks and Li [13]) “A ± B” stands for “mean ± standard deviation” Rail. Eng. Science (2022) 30(4):482–493 1 3 Frequency 486 W. Zhang et al. when the reliability of components is assumed to be inde- Then, the system normal state of subgrade slope can also pendent, the results of system reliability analysis are on the be defined by ≤ , and the system failure state means act acc conservative side. Based on the value of N , the average 𝜂 >𝜂 . The system reliability R stands for the probabil- 95 act acc sys lengths L of subgrade slope sections are computed and ity that ≤ , and the P presents the probability that sec act acc f,sys listed in Table 1. For instance, suppose the = 24 m, then 𝜂 >𝜂 . If all sections have the same failure probability h act acc L = 500/6 83.3 m. Since n is obtained according to the P , the system failure probability is given by sec f,sec 95% quantile of the RFEM simulation data, L represents sec m n i n−i a minimum length within which one potential failure event P = 1 − P 1 − P , (2) f,sys f,sec f,sec could occur in a statistical sense. i=0 Besides, the means and standard deviations of the length of sliding masses ( l ) are added in Table 1. L is approxi- where = n!∕[i!(n − i)!] is the binomial coefficient. sli sec mately four folds of the mean of l . The ratio of L ∕ sli sec h In engineering practice, failure probabilities of different is negatively related to , decreasing from 4.5 to 3.5, 2.5, subgrade slope sections are generally different. To compute and 1.5 when increases from 12 to 24, 50, and 100 m, the system failure probability P , all possible combinations f,sys respectively. of section statuses fulfilling the system failure state need to be considered. These combinations are referred to as system failure scenarios. The total number of system failure scenarios 3 System reliability evaluation of long subgrade is given by slope n n n n Q = = + +⋯ + . (3) 3.1 k‑out‑of‑ n model and system reliability formula j 0 1 m j=0 Based on the standard of ISO2394 [28], system reliability Let status vector S represent a specific combination of sec- is defined as the ability of a structure with more than one tion statuses. Each element of the vector corresponds to a sec- structural member to fulfill specified requirements. That is, tion with specified status, where 1 stands for the normal state the system reliability analysis depends on the definition of and 0 stands for the failure state. The length of S equals n . For a specified requirement. For a subgrade slope consisting of instance, if the allowable failure number is m = 0 , the number n sections, it is required that at least k ( 0 < k ≤ n ) sections of possible combinations is Q = = 1 , and the status vec- are normal, and accordingly, the maximum tolerated failure section is m = n − k ( 0 ≤ m < n ). This can be modeled by tor will be S = (1, 1, … ,1,1) where “T” stands for trans- the k-out-of-n system [25], as shown in Fig. 5. If m = 0 , n n pose. Similarly, as m = 1 , there are Q = + = n + 1 0 1 the system is called the series system [24, 25], which is a special case of k-out-of-n system. Letting the actual num- possibilities of combinations, including S = (1, 1, … ,1,1) , S = (1, 1, … ,1,0) , etc. All above status vectors form the ber of failure sections be m , the actual failure ratio and act 2 acceptable failure ratio can be computed by = m ∕n and status matrix : sce act act = m∕n , respectively, where and are normalized acc act acc = S , S ,⋯ , S . (4) sce 1 2 Q variables ranging from 0 to 1. n×Q contains all possible combinations of section statuses sce satisfying system failure state, with a size of n × Q . For any status vector S ( q = 1, 2,… , Q ), the sets of subscripts of nor- mal and failure sections are denoted by Nor and Fai , respec- q q Slope section 1 tively. Take S as an example, the Nor will be {1, 2, … , n} and 1 1 Fai will be empty. Then, the occurrence probability of the q -th system failure scenario is expressed as Slope section 2 Judgement P = P 1 − P . f,q f,sec f,sec (5) Fai Nor System normal: Then, the system failure probability is the sum of P : normal sections f ,q System failure: normal sections Slope section n P = P . (6) f,sys f,q q=1 Fig. 5 Schematic of k-out-of-n system model Rail. Eng. Science (2022) 30(4):482–493 1 3 System reliability evaluation of long railway subgrade slopes considering discrete… 487 Equation (2) is a special case of Eq. (6) where all compo- 1.0 =0.010 nents have the same failure probability P . f,sec n=1000 (i.e., m=1) 0.8 ( ) Reliability index 3.2 Two indicators describing system reliability Insensitive Sensitive Insensitive 0.6 The acceptable failure ratio and the system failure prob- acc nterval nterval nterval ability P should be used synergistically to describe the f,sys reliability level of long subgrade slopes. Among them, acc 0.4 reflects the engineers’ expectations and requirements for the P β reliability of subgrade slopes, and P quantifies the pos- Point P f,sys sys 0.2 sibility that is exceeded by the actual failure ratio . A 0.006 2.51 0.043 acc act R ≥95% sys For instance, if a P = 0.6% (i.e., system reliability index B 0.0162.14 0.922 f,sys 0.05 = 2.5) is obtained when setting = 1%, one has a 0.0 sys acc 0.0100.015 0.0200.159 99.4% confidence that is no greater than 1%. act (4.00) (2.58) (2.33) (2.17) (2.05) (1.00) For subgrade slopes with different degrees of failure con- Section failure probability, P sequence, different values of should be adopted. The , acc second generation of Eurocode 7, to be published in the early 2020s [29], classifies geotechnical structures into three Fig. 6 P versus P f ,sys f ,sec geotechnical categories (GC) that combine the consequence class (CC) and the geotechnical complexity class (GCC). For GC1, GC2, and GC3, the values of 1.0%, 0.5%, and 0.1% = 0.010 are utilized for illustration ( m = n × = 1 ) . acc acc are suggested for , as shown in Table 2. The = 1% In Fig. 6, P has an S-shaped curve and keeps rising acc acc f,sys means that only one component is allowed to fail in every with the increase of P , indicating that weak compo- f,sec 1000 components in a k-out-of-n system. For some special nents always result in poor performance of the system. structures such as monumental structures or structures that P grows up sharply in the middle interval of P , e.g., f,sys f,sec will have unbearable consequences of failing, the series sys- from 0.006 ( = 2.51 ) to 0.016 ( = 2.14 ). This inter- sec sec tem model should be utilized, i.e., = 0. val of P is called sensitive interval (SI). However, when acc f,sec −5 −3 the value of P is very small ( 3.17 × 10 –6.00 × 10 ) f,sec 3.3 Factors influencing system failure probability or very large (0.016–0.159), the P increases slowly. f,sys These two intervals are called insensitive intervals (II). If This section investigates the influence of the number of sec- the section failure probability P lies in the SI, a minor f,sec tions n , section failure probability P , and the acceptable change of P can trigger a considerable mutation in sys- f,sec f,sec failure ratio on the system failure probability P , based tem safety, i.e., P , suggesting that the section reliability acc f,sys f,sys on Eq. (2). evaluation requires higher precision than that when P f,sec is located out of SI. 3.3.1 Influence of section reliability on system failure probability 3.3.2 Influence of the number of sections on system failure Figure 6 shows the system failure probability P versus probability f,sys −5 section failure probability P . A range of 3.17 × 10 f,sec −1 –1.59 × 10 for P is considered, corresponding to a Figure 7 shows the influence of section number n on the f,sec range of 4.0–1.0 of section reliability index . The total system failure probability P , where the is fixed as sec f,sys acc section number n = 1000 and the acceptable failure ratio 0.1% and n = 1000 , 2000 , 3000 are used for illustration. A larger n produces a steeper P curve in the middle range of f,sys P , which leads to a narrower sensitive interval and wider f,sec Table 2 Selection of acceptable failure ratio insensitive intervals. That is, for a system with more com- CC GCC/ acc ponents, the P is more sensitive to P located in SI and f,sys f,sec Lower (GCC1) Normal (GCC2) Higher (GCC3) less sensitive to P located in II. Besides, for a given value f,sec of P , the P with a larger n is not necessarily larger than f,sec f,sys CC4 (highest) GC3/0.1% GC3/0.1% GC3/0.1% that with a smaller n (see vertical dashed lines L1 and L2). CC3 (higher) GC2/0.5% GC3/0.1% GC3/0.1% Moreover, the same P may be observed even if different f,sys CC2 (normal) GC2/0.5% GC2/0.5% GC3/0.1% section numbers n are used (see the point of intersection), CC1 (lower) GC1/1.0% GC2/0.5% GC2/0.5% because of the different values of m that they have. Rail. Eng. Science (2022) 30(4):482–493 1 3 System failure probability, P sys 488 W. Zhang et al. 1.0 1.0 0.8 0.8 0.6 0.6 n = 1000 = 0.010 oint f ( ) Reliability index 0.4 0.4 n=1000 (m=1) intersection η =0.001(m=1) n=2000 (m=2) η =0.005(m=5) n=3000 (m=3) 0.2 0.2 ) Re η =0.010(m=10) liability index R ≥95% R ≥95% L1 L2 0.0 0.0 0.005 0.010 0.015 0.020 0.159 0.005 0.010 0.0150.020 0.159 (4.00) (2.58) (2.33) (2.17) (2.05) (1.00) (4.00) (2.58) (2.33) (2.17) (2.05) (1.00) Section failure probability, P Section failure probability, P Fig. 8 Influence of on P acc f ,sys Fig. 7 Influence of n on P f ,sys 3.3.3 Influence of acceptable failure ratio on system failure site investigation reports. Besides, subgrade slopes are not probability always continuous and soil properties are not always statisti- cally homogeneous. To apply DDM at the current stage, the Figure 8 shows the influence of the acceptable failure ratio segmentation of long subgrade slopes and the calculation of on a system failure probability P , where n is fixed sectional slope reliabilities need to be reasonably simplie fi d. acc f,sys at 1000. has a positive effect on the width of SI. That This section shows a possible way to apply DDM to a built acc is, the larger the is, the less sensitive of P to P high-speed railway project, in which only the point statistics acc f,sys f ,sec becomes. In addition, the P sharply decreases with the of soil properties (i.e., mean and standard deviation) and f,sys increase of . Taking P = 0.002 for example, the P slope geometries are accessible. acc f ,sec f ,sys decreases from 64.3% (point C) to 2.4% (point D) when increases from 0.001 ( m = 1 ) to 0.005 ( m = 5 ). In other 4.1 Problem description acc words, one has the confidence of 35.8% that m does not act exceed 1, whereas one has the confidence of 97.6% that m The high-speed railway project to be assessed began in act does not exceed 5. Figure 8 also provides useful informa- 2015 and was currently under construction at the time of tion for guiding the design of long subgrade slopes. For writing this article. The total length of the railway line instance, if one hopes a subgrade slope has system reliability is 211 km with mileage from K5 + 000 to K216 + 000, of 95% ( = 1.65 ), the failure probability of each section of which the total length of subgrades is around 206 km, sys −3 is required to be less than 6.21 × 10 ( = 2.50 ) w h e n including 46.91 km of cuttings, 13.18 km of embankments, sec the acceptable failure ratio = 0.010 (point E). However, and 5.91 km to be constructed. The rail line is located in acc −3 the P is required to be less than 2.64 × 10 ( = 2.80 ) elevated portions of the temperate zone, with elevation rang- f ,sec sec when the = 0.005 (point F). In other words, one can save ing from 1200 to 2000 m. The region along the route has a acc lots of investment to improve the safety level of each section semi-arid climate featuring dry and warm to hot summers by slightly accepting more local instability events. When and cold winters. Due to such topography and climate con- considering the economic criterion, the k-out-of-n system ditions, water infiltration and seepage rarely happen, and model provides an alternative approach to design the system the likelihood of rainfall-induced slope instability events is reliability of subgrade slopes. extremely low. In the site investigation report, three types of slopes were not considered since they have little effect on railway 4 Case study safety, including (1) a slope height less than 2 m, (2) a slope gradient less than 1:3, and (3) a distance larger than 3 m The spatial parameters of soil properties such as SOF are between the toe of cutting slopes and the ditch of the engi- generally not required in the current design specifications neered structure. Table 3 shows the number and geometries of railway engineering, leading to a lack of such data in of 2345 surveyed cross-sectional soil slopes. Rock slopes Rail. Eng. Science (2022) 30(4):482–493 1 3 System failure probability, P System failure probability, P System reliability evaluation of long railway subgrade slopes considering discrete… 489 Table 3 Mileage and the Slope type Mileage Number of Gradient range Height range (m) number of cross-sections cross-sections Embankment K5 + 000–K15 + 000 135 1:2–1:3 2–16.62 K15 + 000–K19 + 721 51 1:2–1:3 2–16.65 K19 + 721–K25 + 000 91 1:1.83–1:3 2–23.80 K25 + 000–K40 + 000 69 1:1.67–1:3 2–12.46 K40 + 000–K51 + 000 85 1:1.67–1:3 2–8.27 K51 + 000–K62 + 000 182 1:1.5–1:3 2–11.17 K62 + 000–K102 + 300 N.A. N.A. N.A. K102 + 300–K118 + 250 118 1:2–1:3 2–7.99 K118 + 250–K130 + 000 82 1:1.85–1:3 2–14.54 K130 + 000–K140 + 000 111 1:1.6–1:3 2–9.14 K140 + 000–K155 + 000 268 1:1.8–1:3 2–7.28 K155 + 000–K165 + 560 230 1:2–1:3 2–8.55 K165 + 560–K175 + 000 150 1:1.92–1:3 2–5.14 K175 + 000–K185 + 000 55 1:1.8–1:3 2–5.25 K185 + 000–K195 + 000 200 1:1.92–1:3 2–4.23 K195 + 000–K205 + 000 132 1:1.8–1:3 2–8.65 K205 + 000–K216 + 000 158 1:1.8–1:3 2–9.60 Cutting Soil slope 228 1:0.6–1:2.8 2–15.45 Rock slope 346 1:0.35–1:6.88 2–30.95 Soil slope 2345 N.A. N.A. In total Rock slope 346 N.A. N.A. N.A. stands for not available are not considered in the current system reliability evalua- standard deviation ( ) of friction angle, and the coefficient tion of subgrade slopes. The point statistics of soil strength of variation of cohesion (V ) and friction angle (V ). The parameters are listed in Table 4, including the mean ( ) unit weight of soil embankments is approximately 19 kN/ 3 3 and standard deviation ( ) of cohesion, the mean ( ) and m , and = 20–22 kN/m for soil cuttings. The variability Table 4 Mileage and the Mileage (kPa) (°) (kPa) (°) V V c c c statistics of soil strength parameters K5 + 000–K15 + 000 16.52 40.42 10.17 3.02 0.62 0.07 K15 + 000–K19 + 721 14.75 40.04 6.74 4.51 0.46 0.11 K19 + 721–K25 + 000 17.71 41.81 7.72 3.21 0.44 0.08 K25 + 000–K40 + 000 22.53 44.33 7.93 2.07 0.35 0.05 K40 + 000–K51 + 000 23.87 37.93 6.31 3.68 0.26 0.10 K51 + 000–K62 + 000 21.29 43.88 6.37 4.68 0.30 0.11 K62 + 000–K102 + 300 N.A. N.A. N.A. N.A. N.A. N.A. K102 + 300–K118 + 250 20.94 43.53 9.58 3.43 0.46 0.08 K118 + 250–K130 + 000 25.22 44.90 7.14 1.78 0.28 0.04 K130 + 000–K140 + 000 17.44 34.03 7.95 2.57 0.46 0.08 K140 + 000–K155 + 000 14.08 39.74 7.22 3.50 0.51 0.09 K155 + 000–K165 + 560 12.49 37.3 5.74 2.60 0.46 0.07 K165 + 560–K175 + 000 10.50 38.31 5.19 2.12 0.49 0.06 K175 + 000–K185 + 000 15.37 44.57 7.23 4.00 0.47 0.09 K185 + 000–K195 + 000 9.55 42.20 7.09 4.51 0.74 0.11 K195 + 000–K205 + 000 4.13 37.66 3.33 1.99 0.81 0.05 K205 + 000–K216 + 000 4.63 37.61 3.76 1.49 0.81 0.04 N.A. stands for not available Rail. Eng. Science (2022) 30(4):482–493 1 3 490 W. Zhang et al. of is rationally neglected. The site investigation was car- investigation reports. Based on slope geometries and the ried out between 2015 and 2018. For soil samples collected point statistics of soil properties, only the reliability analy- in embankments, direct shear tests were performed on these sis of cross-sectional slope can be carried out. samples to obtain their strength properties. Dynamic cone Whether or not considering the spatial variability of soil penetration testing was used on-site to evaluate the mechani- parameters, the reliability obtained by three-dimensional cal properties of soil samples in cut sections [30], and the probabilistic analysis ( R ) is generally greater than that sec original data were converted to cohesion and friction angles of cross-sectional analysis ( R ) due to end effect [ 15, 32]. cs [31] to facilitate the probabilistic analysis of soil slope sta- The end effect means that the resisting moment provided bility. During the investigation, local instabilities of two by two end parts of a three-dimensional sliding mass is distant cut slopes were reported. larger than the driving moment provided by it. That is, using two-dimensional slope reliability analysis results 4.2 Segmentation of long subgrade slope instead of three-dimensional slope reliability analysis results is on the conservative side. In this case study, the There are two stages to divide the subgrade slope into mul- reliability of slope sections will be substituted by that of tiple sections. In the first stage, the subgrade is divided into cross-sections taken from them, i.e., letting R = R . The sec cs many segments in the following principles: (1) the boundary histograms of the reliability index ( ) of slope sections sec between subgrade slope and structure, such as bridge, (2) a are shown in Fig. 9. In the two situations, the mean of sec change in the type of slope, i.e., cutting and embankment, is 4.49, and the mean and standard deviation (s.d.) of sec (3) an apparent change of slope height, i.e., greater than are almost the same; the slope with 𝛽 > 2 accounts for sec 1 m, and (4) an apparent change in the statistics of strength more than 99.5%, and the slope with 𝛽 > 3 accounts for sec parameters such that they can no longer be regarded as sta- more than 97.5%. The number of slopes with 𝛽 < 2 for sec tistically homogeneous. In the second stage, continuous sub- the two situations is only 5 and 9, respectively. For most grade slope segments composed of statistically homogene- engineered slopes, = 2–3 is usually thought to be large ous soils, they are further divided into multiple sections as enough to ensure stability, corresponding to a failure prob- per Sect. 2. Nevertheless, the SOF of soil properties is not ability of 2.3%–0.13%. provided in the survey report, and the number of sections cannot be directly computed. Based on the work of Hicks and Li [13], the of soil properties typically ranges from 10 to 100 m, and L is sec around 3.5 to 1.5 folds of as increases from 12 to ( (a a) ) h h 0.25 100 m. That is, L varies from 35 to 150 m when is sec h L =100 m unknown. For slopes with different parameters, these ratios 0.20 n=987 may change. To divide continuous long subgrade slopes into mean=4.49 0.15 s.d.=0.88 multiple sections, a simplified yet practical DDM is pro - posed herein. The value of 100 m is adopted as the empiri- 0.10 β >2: 99.5% cal value of L . For a segment, the residual length less sec >3: 97.6% 0.05 sec than 100 m will be ignored. Discretely distributed subgrade segments shorter than 100 m are regarded as individual sec- 0.00 1 2 8 tions. In total, there are 997 slope sections. For comparison, L = 50 m is also analyzed. The corresponding total num- Sectional reliability index, β sec ber of slope sections is 1980. ( (b b) ) 0.25 L =50 m 0.20 4.3 Computation of sectional slope reliability n=1980 mean=4.49 0.15 s.d.=0.89 For the reliability analysis of long soil slopes, Vanmarcke 0.10 [15] proposed an analytical method using local average β >2: 99.5% theory in which sliding mass is assumed to be a cylinder β >3: 98.4% sec 0.05 cut by two vertical planes. Zhang et al. [32] improved the Vanmarcke method by changing vertical planes to ellip- 0.00 1 2 8 soidal surfaces which is more in line with reality. They can also be analyzed using RFEM [33]. However, all the Sectional reliability index, β above methods require the horizontal and vertical scale of fluctuations, which are not available in most current Fig. 9 Histogram of β : a L = 100 m; b L = 50 m sec sec sec Rail. Eng. Science (2022) 30(4):482–493 1 3 Frequency Frequency System reliability evaluation of long railway subgrade slopes considering discrete… 491 4.4 Evaluation of system reliability Frequently 0.3 -1 Figure 10 shows the occurrence probability P versus the Occasionally actual number of failure sections m , which changes from 0.0 act 3 0 to 5. For the situation of L = 100 m, the probability -2 Seldom sec 10 P of the event that no local instability will occur is 63%, 0.0 i.e., m = 0, and P = 29% for the event that m = 1. P -3 act act 10 Rarely decreases nonlinearly with increasing m . When the actual act 0.0003 number of failure sections reaches 4, the occurrence prob- Almost never -4 −3 ability is only 8.5 × 10 . That is, the most possible result is that the actual number of failure sections equals zero. The -5 L =50 m probability of the event that the total number of failure sec- L =100 m tions is no greater than 2 reaches up to 98.30%. Thus, if -6 the acceptable number of failure sections m is 2, the system 23 4 failure probability will be P = 1.70%. For the situation of f,sys Acceptable number of failure sections m L = 50 m, the decreasing law of P is also applicable. The sec probability of the event that the total number of failure sec- Fig. 11 Results of system failure probability tions is no greater than 2 reaches up to 97.00%. Figure 11 shows the curve of system failure probability than 0, or having a 0.90% confidence that the actual number P versus the acceptable number of failure sections m , f,sys where P decreases sharply with the increase of m from of failure sections is greater than 2, etc. The full data is f,sys listed in Table 5. Based on Figs. 9 and 11, although most 0 to 5. In the Chinese code for risk management of railway construction engineering [34], the occurrence probability cross-sectional slope reliability indices are larger than 3, there is still a considerable probability to encounter several of an event is divided into five categories, corresponding to events with different frequencies, including frequently ( P ≥ local instability events along the railway line. This corrobo- rates the length effect in the system reliability analysis of 0.3), occasionally (0.03 ≤ P < 0.3), seldom (0.003 ≤ P < 0.03), rarely (0.0003 ≤ P < 0.003), and almost never ( P < long side slopes. For the situation of L = 50 m, the P sec f,sys is always larger than that of the situation of L = 100 m, 0.0003), as displayed in Fig. 11. For the situation of L = sec sec 100 m, the acceptable number of failure sections varying because the number of components in the former system is twice the latter one. from 0 to 5 can be frequently, occasionally, seldom, rarely, and almost never exceeded by actual values m . The system There are two aspects of simplifications when applying act DDM to the real subgrade slope system lacking spatial data. reliability level can be described as having a 36.6% confi- dence that the actual number of failure sections is greater One is the segmentation of subgrade slopes based on empiri- cal L , and the other is using R to substitute R For the sec cs sec former simplification, the prediction of system reliability may be conservative ononconservative depending on the 0.7 0.63 magnitude of L (e.g., 50 m) relative to the actual section L =100 m sec 0.6 length determined by rigorous approaches. Based on the L =50 m sec case study, when making L = 50 and 100 m respectively, sec 0.5 0.48 the difference between P rapidly increases with increas- f,sys ing m (see Fig. 10). That is, the determination of L has a sec 0.4 0.36 0.29 Table 5 Calculation results of system failure probability 0.3 -2 6.33 10 m P f ,sys 0.2 -3 8.53x10 -4 L = 100 m L = 50 m 0.13 x 8.0610 sec sec -5 4.30 10 0.1 0 0.366020 0.524180 -2 2.9710 -3 -4 4.901 x 0 6.51 0 x 0 1 0.072730 0.164840 0.0 2 0.009380 0.035330 2 345 6 3 0.000850 0.005590 Actual number of instability sections, m 4 0.000044 0.000689 5 0.000001 0.000039 Fig. 10 Histogram of occurrence probability Rail. Eng. Science (2022) 30(4):482–493 1 3 Occurrence probability, P System failure probability, P , 492 W. Zhang et al. considerable impact on prediction results, and an unreason- analyzing or designing the stability of subgrade slopes, sys- able value of L may lead to wrong outcomes. The latter tem reliability should be given sufficient attention in addition sec simplification is conservative since R is always greater to the traditional cross-sectional reliability. sec than R . Based on Zhang et al. [32], the ratio R ∕R A case study is provided to show the application of DDM cs sec cs decreases with the increase of the length of sliding mass or when lacking spatial data of soil proprieties. The DDM is the horizontal SOFs of shear strength parameters. For slopes simplified in two aspects. First, subgrade slopes are seg- with typical geometry and soil parameters, the ratio mostly mented into many sections based on the empirical length of ranges from 1 to 2. slope sections L . Second, the reliability of sections R sec sec For subgrade slopes that lack essential spatial data, the is substituted by that of cross-sections R taken from them. cs computation of system reliability on the basis of empiri- The first simplifying measures may lead to conservative or cal L and R provides decision-makers helpful additional unconservative results, and the second will make the sys- sec cs information to get as comprehensive a picture of long sub- tem reliability analysis results conservative. Nevertheless, grade slope systems as possible, although the two simplify- by cautiously estimating the lower and upper values of L , sec ing measures bring uncertainties to the prediction of system one can roughly evaluate the bounds of system reliability safety level. Actually, various methods have been developed using the simplified DDM. It is highly recommended that to characterize this parameter from soil data, particularly the spatial data of soil properties be determined in future CPT measurements [19]. Thus, to assess the system safety site investigations. of long subgrade slopes using DDM, one of the greatest In many cases, subgrades, bridges, and tunnels together challenges lies in incorporating the survey of SOF into site form the foundation of a railroad. In addition to subgrade investigation specifications. Another major challenge stems slopes, there are usually a large number of natural slopes from the specificity of soil properties at different sites, which along the line, and their instability will also have a great is a distinctive and fundamental feature of geotechnical engi- impact on rigid structures as well as transition sections [36, neering practice. To overcome this limitation, the establish- 37]. In a broad sense, all slopes along a rail line can be ment of a global database of SOFs will be a meaningful regarded as the k-out-of-n system. By analyzing the reli- work direction [35]. ability of this system, the overall landslide risk of a line can be assessed. Acknowledgements The work was supported by the National Natural 5 Concluding remarks Science Foundation of China (Nos. 52078435 and 51878560). Profes- sor Kok-Kwang Phoon and Dr. Chong Tang from National University This paper develops a DDM method to evaluate the system of Singapore and Dr. Jifeng Lian from Xihua University are greatly reliability performance of long subgrade slopes. DDM is appreciated for their useful suggestions in the process of preparing this manuscript. The first author would greatly like to thank the financial an improvement of the VNK method in three aspects. First, support from the open research fund of MOE Key Laboratory of High- long subgrade slopes are segmented into multiple adjacent Speed Railway Engineering. sections based on the number of potential local instabili- ties predicted by the random finite element method. Second, Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adap- two indicators are adopted to describe the system reliability tation, distribution and reproduction in any medium or format, as long performance, namely the acceptable number of failure sec- as you give appropriate credit to the original author(s) and the source, tions m and the system reliability R . The former charac- sys provide a link to the Creative Commons licence, and indicate if changes terizes the designer’s expectation of system safety, and the were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated latter expresses the confidence that m will not be exceeded. otherwise in a credit line to the material. If material is not included in Third, the k-out-of-n system model with redundancy features the article’s Creative Commons licence and your intended use is not is introduced to establish the relationship between the toler- permitted by statutory regulation or exceeds the permitted use, you will ance threshold m and the system reliability R . need to obtain permission directly from the copyright holder. To view a sys copy of this licence, visithttp:// creat iveco mmons. org/ licen ses/ by/4. 0/. Parametric analyses are conducted to investigate the influ- ence of impact factors on system reliability. The total num- ber of slope sections n is negatively related to the horizontal SOF of soil properties and positively related to the length of References subgrade slopes. R decreases with increasing n , decreas- sys ing m , or decreasing reliability of slope sections R . Sub- sec 1. Nguyen TT, Indraratna B (2022) Rail track degradation under mud grade slopes with larger length or weaker soil strength face pumping evaluated through site and laboratory investigations. Int a higher risk to encounter more local instabilities, and there J Rail Transp 10(1):44–71 may still be a considerable possibility to encounter local instabilities even if all sections have high reliability. When Rail. Eng. Science (2022) 30(4):482–493 1 3 System reliability evaluation of long railway subgrade slopes considering discrete… 493 2. Lin Z, Niu F, Li X et al (2018) Characteristics and controlling 21. VPO (2012) Flood risk in the netherlands: the method in brief. factors of frost heave in high-speed railway subgrade, Northwest Utrecht, Netherlands: VNK Project Office China. Cold Reg Sci Technol 153:33–44 22. VPO (2014) The national flood risk analysis for the Netherlands: 3. Wikipedia (2021) Ingenheim derailment https://en. wikip edia. or g/ Final report. Utrecht, Netherlands: VNK Project Office wiki/ Ingen heim_ derai lment. Accessed 24 Feb 2022 23. Jongejan RB, Calle EOF (2013) Calibrating semi-probabilistic 4. Zhang LL, Zhang J, Zhang LM et al (2011) Stability analysis of safety assessments rules for flood defences. Georisk Assess rainfallinduced slope failure: a review. Proc Inst Civ Eng Geotech Manag Risk Eng Syst Geohazards 7(2):88–98 Eng 164(5):299–316 24. Melchers RE, Beck AT (2017) Structural reliability analysis and 5. Hicks MA, Spencer WA (2010) Influence of heterogeneity on prediction, 3rd edn. John Wiley Sons, Hoboken, NJ, USA. https:// the reliability and failure of a long 3D slope. Comput Geotech doi. org/ 10. 1002/ 97811 19266 105 37(7–8):948–955 25. Barlow RE, Proschan F (1965) Mathematical theory of reliability. 6. Duncan JM, Wright SG, Brandon TL (2014) Soil Strength and John Wiley Sons, New York, NY, USA. https://doi. or g/10. 1137/1. Slope Stability, 2nd edn. John Wiley & Sons, Hoboken, NJ, USA97816 11971 194 7. Fellenius W (1936) Calculation of stability of earth dam. In: Pro- 26. Salgado R, Kim D (2014) Reliability analysis of load and ceedings of the Second Congress of Large Dams, Washington, resistance factor design of slopes. J Geotech Geoenviron Eng DC, vol 4, pp 445–462 140(1):57–73 8. Cho SE (2013) First-order reliability analysis of slope considering 27. Ching JY, Hu YG, Yang ZY et al (2011) Reliability-based multiple failure modes. Eng Geol 154:98–105 design for allowable bearing capacity of footings on rock masses 9. Low BK, Tang WH (2007) Efficient spreadsheet algorithm for by considering angle of distortion. Int J Rock Mech Min Sci first-order reliability method. J Eng Mech 133(12):1378–1387 48(5):728–740 10. Christian JT, Ladd CC, Baecher GB (1994) Reliability applied to 28. ISO (2015) ISO 2394 General principles on reliability of struc- slope stability analysis. J Geotech Eng 120(12):2180–2207 tures. Geneva, Switzerland: International Organization for 11. Griffiths DV, Fenton GA (2004) Probabilistic slope stability analy - Standardization sis by finite elements. J Geotech Geoenviron Eng 130(5):507–518 29. Orr TLL (2019) Honing safety and reliability aspects for the sec- 12. Reale C, Xue J, Pan Z et al (2015) Deterministic and probabil- ond generation of Eurocode 7. Georisk Assess Manag Risk Eng istic multi-modal analysis of slope stability. Comput Geotech Syst Geohazards 13(3):205–213 66:172–179 30. Salgado R, Yoon S (2003) Dynamic cone penetration test (DCPT) 13. Hicks MA, Li YJ (2018) Influence of length effect on embank - for subgrade assessment. Indiana: Purdue University, West ment slope reliability in 3D. Int J Numer Anal Methods Geomech Lafayette 42(7):891–915 31. USBR (2002) Engineering geology field manual, 2nd edn. US 14. Vanmarcke E (2014) Multi-hazard risk assessment of civil infra- Department of the Interior Bureau of Reclamation, Washington, structure systems with a focus on long linear structures such as DC levees. In: Liu XL, Ang AH-S (eds) Sustainable development 32. Zhang W, Luo Q, Jiang L et al (2021) Improved Vanmarcke’s ana- of critical infrastructure. International Conference on Sustain- lytical model for 3D slope reliability analysis. Comput Geotech able Development of Critical Infrastructure, May 16–18, 2014, 134:104106 Shanghai, China. Reston, VA, USA: American Society of Civil 33. Xiao T, Li DQ, Cao ZJ et al (2016) Three-dimensional slope reli- Engineers, 2014: 37–56 ability and risk assessment using auxiliary random finite element 15. Vanmarcke E (1977) Reliability of earth slopes. J Geotech Eng method. Comput Geotech 79:146–158 Div 103(11):1247–1265 34. Office of China Railway (2015) Technical code for risk manage- 16. Vanmarcke E (2010) Random fields: analysis and synthesis ment of railway construction engineering (Q/CR 9006-2014). (revised and expanded new edition). World Scientific, Singapore. China Railway Publishing House, Beijing (in Chinese) https:// doi. org/ 10. 1142/ 5807 35. Ching JY, Phoon KK, Khan Z et al (2020) Role of municipal 17. El-Ramly H, Morgenstern NR, Cruden DM (2003) Probabilistic database in constructing site-specific multivariate probability dis- stability analysis of a tailings dyke on presheared clay shale. Can tribution. Comput Geotech 124:103623 Geotech J 40(1):192–208 36. Meng XM, Qi TJ, Zhao Y et al (2021) Deformation of the 18. Phoon KK, Kulhawy FH (1999) Characterization of geotechnical Zhangjiazhuang high-speed railway tunnel: an analysis of causal variability. Can Geotech J 36(4):612–624 mechanisms using geomorphological surveys and D-InSAR moni- 19. Cami B, Javankhoshdel S, Asce AM et al (2020) Scale of fluc- toring. J Mt Sci 18(7):1920–1936 tuation for spatially varying soils: estimation methods and val- 37. Zhou S, Tian Z, Di H et al (2020) Investigation of a loess-mud- ues. ASCE ASME J Risk Uncertain Eng Syst Part A Civ Eng stone landslide and the induced structural damage in a high-speed 6(4):03120002 railway tunnel. Bull Eng Geol Environ 79(5):2201–2212 20. Phoon KK, Retief JV (2016) Reliability of geotechnical structures in ISO2394. CRC Press/Balkema, Leiden, Netherlands Rail. Eng. Science (2022) 30(4):482–493 1 3
Railway Engineering Science – Springer Journals
Published: Dec 1, 2022
Keywords: Subgrade slope; Local instability; Segmentation; k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-out-of-n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document} system; System reliability; Railway engineering
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