ORIGINAL PAPER An analytical method for quantifying the correlation among slope failure modes in spatially variable soils Dong Zheng 1 • Dian-Qing Li 1 • Zi-Jun Cao 1 • Xiao-Song Tang 1 • Kok-Kwang Phoon 2 Received: 13 April 2016 / Accepted: 16 July 2016 Ó Springer-Verlag Berlin Heidelberg 2016 Abstract An efficient analytical method for quantifying the correlation between performance functions of different slope failure modes in spatially variable soils is proposed, and its performance in slope system reliability analysis is investigated. First, a new correlation coefficient (NCC) is proposed to evaluate the correlation among slope failure modes considering spatial variability. For comparison and verification, the simulation-based correlation coefficient (SCC) is also presented. Second, appying these two types of correlation coefficients, the effects of soil spatial vari- ability on the representative slip surfaces (RSSs) and the system probability of slope failure are investigated using different system reliability methods, including a proba- bilistic network evaluation technique, a risk aggregation approach, and a bimodal bounds method. A single-layered cohesive slope is investigated to illustrate the validity of the proposed NCC. The results indicate that the proposed NCC can efficiently and accurately quantify the correlation among slope failure modes considering soil spatial vari- ability. The number of RSSs indicated by the NCC is in good agreement with the number obtained using the SCC. The system failure probabilities of slope stability obtained with the SCC and the NCC using a risk aggregation approach are generally comparable. Also, the system reli- ability bounds of slope stability obtained using the NCC are relatively close together and comparable to those obtained using the SCC. Thus, the NCC shows good per- formance when evaluating the correlation among slope failure modes, and was effectively applied to analyze a single-layered cohesive slope considering soil spatial variability. Keywords Slope  Spatial variability  Correlation coefficient  Failure mode  System reliability Introduction Geotechnical problems often involve multiple failure modes; that is, there may be several potential modes of failure, in which the occurrence of any one of the potential failure modes will cause nonperformance of the system (Ang and Tang 1984; Tang et al. 2015). Multiple failure modes can be found in several research fields, i.e., the stability of slopes (Fenton and Griffiths 2008; Huang et al. 2010, 2013), retaining walls (Li et al. 2015a), piles (Fan et al. 2014), and foundations (Li et al. 2015b). More importantly, different failure modes are often correlated (Chowdhury and Xu 1994, 1995; Low et al. 2011; Ji and Low 2012; Zhang et al. 2011, 2013; Cho 2013; Li et al. 2014; Zeng and Jimenez 2014; Zeng et al. 2015). For instance, if we consider two different failure modes of a soil slope, their factors of safety (FSs) are mainly influenced by the strength parameters (e.g., undrained shear strength S u , cohesion c, or friction angle /) at the bottom of each slip surface. Note that soil properties are spatially correlated and vary spatially (e.g., c and /), i.e., they differ in value from one point in space to another due to differences in geological deposition history and human activities (Elkateb et al. & Dian-Qing Li dianqing@whu.edu.cn 1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, 8 Donghu South Road, Wuhan 430072, China 2 Department of Civil and Environmental Engineering, National University of Singapore, Blk E1A, #07-03, 1 Engineering Drive 2, Singapore 117576, Singapore 123 Bull Eng Geol Environ DOI 10.1007/s10064-016-0923-1