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The occurrence of multiple critical slip surfaces with equal importance in rehabilitating and reinforcing slopes has been frequently observed in geotechnical engineering practices. The simultaneous determination of these potential slip surfaces is, however, not trivial. This paper presents a methodology based on the smoothed particle hydrodynamics (SPH) approach, which can simultaneously determine multiple failure slip surfaces and the debris flow process without previous knowledge or trial-and-error processes, and this methodology is validated against a slope with the presence of multiple critical slip surfaces. The proposed methodology serves as an efficient and effective alternative approach to traditional approaches, which involve cumbersome treatments performed by engineers based on their subjective experiences. The multiple sources of failure slip surfaces in slope stability are equivalent to multiple sources of initiation of slope failure, and it is found that SPH can provide a direct and systematic tool for identifying multiple failure slip surfaces. However, some minor potential problems are also found with the use of the SPH method in actual applications.

This paper explores the influence of local average size, Δz, on the failure probability, p f, of spatially variable slope failure along a given slip surface and along a combination of two slip surfaces. The spatial variability of statistical geotechnical property with depth is modeled by a lognormal random field with an exponentially decaying correlation structure. The probabilistic slope stability analysis incorporating random field theory with Monte Carlo simulation is adopted to conduct the sensitivity study. The variance reduction due to local averaging and the correlations between local averages are considered (denoted M1), and a simplified approach (denoted M2) neglecting the variance reduction and simulating correlations between local averages at the mid-points of local averages is compared with M1. The results of sensitivity study on an undrained slope have shown that the influence of Δz on p f of slope failure along a given slip surface hinges on the given slip surface. As the ratio of Δz to the scale of fluctuation, λ, varies in the range of 0.0 to 0.4, M1 and M2 can predict the results within 10% normalized discrepancy for the p f of slope failure along a given slip surface. As Δz/λ is greater than 0.4, the performance of M1 is superior to that of M2. As Δz increases, a larger value of Δz cannot identify the additional contribution to p f if slope failure along more than one slip surface is taken into account.

Owing to the complicated slope stratigraphy (e.g., multiple soil layers and multiple benches or gradients in side slopes), multiple failure surfaces for slope stability have been recognized in geotechnical discipline. This paper aims to develop a systematic and probabilistic approach to locate the multiple failure surfaces combining the traditional limit equilibrium method with Monte Carlo Simulation. Each of the multiple failure surfaces is selected from a large pool of failure surfaces and the correlation coefficient between two failure surfaces in factor of safety (FS) is adopted to characterize the extent to which two failure surfaces are correlated. After eliminating those highly correlated failure surfaces, the multiple failure surfaces can be gradually identified. The number of failure samples and the number of exclusive failure samples corresponding to each of multiple failure surfaces are determined within the proposed methodology. These data are reanalyzed to find the critical failure surface with the maximum failure probability, the critical failure surface with maximum simplified risk, and those failure surfaces dominating the risk of slope failure. The proposed approach is illustrated through two examples excerpted from the literature and validated against the results from the commercial software package and literature. The comparative study manifests that the critical failure surface with the minimum FS does not always coincide with that with the maximum failure probability and with the maximum simplified risk. In addition to FS, the failure surfaces should be received much attention. The proposed methodology provides an effective tool in decision making for slope stabilization and rehabilitation process.

This paper investigates, using the random field theory and Monte Carlo simulation, the effects of random field discretization on failure probability, p f, and failure mechanism of cohesive soil slope stability. The spatial sizes of the discretized elements in random field Δx, Δy in horizontal and vertical directions, respectively, are assigned a series of combinational values in order to model the discretization accuracy. The p f of deterministic critical slip surface (DCSS) and that of the slope system both are analyzed. The numerical simulation results have demonstrated that both the ratios of Δy/λ y (λ y = scale of fluctuation in vertical direction) and Δx/λ x (λ x = scale of fluctuation in horizontal direction) contribute in a similar manner to the accuracy of p f of DCSS. The effect of random field discretization on the p f can be negligible if both the ratios of Δx/λ x and Δy/λ y are no greater than 0.1. The normalized discrepancy tends to increase at a linear rate with Δy/λ y when Δx/λ x is larger than 0.1, and vice versa for p f of DCSS. The random field discretization tends to have more considerable influence on the p f of DCSS than on that of the slope system. The variation of p f versus λ x and λ y may exhibit opposite trends for the cases where the limit state functions of slope failure are defined on DCSS and on the slope system as well. Finally, the p f of slope system converges in a more rapid manner to that of DCSS than the failure mechanism does to DCSS as the spatial variability of soil property grows from significant to negligible.

This paper develops a quantitative approach for risk evaluation of slope failure based on a finite number of Representative Slip Surfaces (RSS) with Limit Equilibrium Method (LEM). A second order polynomial response surface function (RSF) is calibrated for each RSS in order to improve the computational efficiency. The volume of sliding mass associated with each RSS is a measure of consequence of slope failure along this RSS. The risk of slope failure is evaluated by Monte Carlo Simulation (MCS) and the contribution of each RSS to risk of slope failure is quantified. Thus, the risk of slope failure is de-aggregated into each RSS. The proposed methodology is illustrated through a three-layer cohesive soil slope. The de-aggregations of probability of slope failure, p f , and risk of slope failure, R, are conducted at different scale of fluctuations, and the results for de-aggregation of p f are compared with those for de-aggregation of R. The comparison indicates that the group of key RSS that dominate p f differ considerably from those that dominate R. These different groups of key RSS may lead to different strategies in the subsequent risk mitigation, if any. The proposed approach provides an efficient and quantitative tool for identifying the key group of RSS in the planning of slope risk mitigation.

This paper develops an empirical approach to slope reliability considering 2D spatial variability of soil properties with length-based representative slip surfaces. The correlation coefficient between factors of safety for two slip surfaces is analogous to that between two slip surfaces which is quantified based on the lengths within different elements of the slip surface. A large number of potential slip surfaces are categorized into separate slip surface groups within each of which the correlation coefficient between any two slip surfaces is higher than the threshold value. A slip surface with the minimum factor of safety is considered to be the representative slip surface for each of slip surface groups. The factor of safety is minimized among all the representative slip surfaces, and the minimum one is regarded as the output for each set of random samples generated by Monte Carlo simulation (MCS). The methodology is validated through comparison with other methods. It is found that slope failure probability, p f, increases as the scale of fluctuation, λ y (in vertical direction), and λ x (in horizontal direction) increase. The effect of non-circular failure mechanism was studied at different λ x and λ y values. Significant larger failure probabilities are provided with non-circular failure mechanism than with circular failure mechanism. For 2D spatial variability case, two orders of magnitude larger failure probability is noticed.

Summary This paper develops a multiple response surfaces approach to approximate the limit state function for slope failure by second-order polynomial functions, to incorporate the variation of the most probable slip surfaces, and to evaluate the slope failure probability pf. The proposed methodology was illustrated through a cohesive soil slope example. It is shown that the pf values estimated from multiple response surfaces agree well with those pf values that have been obtained by searching a large number of potential slip surfaces in each Monte Carlo simulation (MCS) sample. The variation of number of the most probable slip surfaces is studied at different scale of fluctuation (λ) values. It is found that when full correlation assumed for each of random fields (i.e., spatial variability is ignored), the number of the most probable slip surfaces is equal to the number of random fields (in this study, it is 3). When the spatial variability grows significantly, the number of the most probable slip surfaces or number of multiple response surfaces firstly increases evidently to a higher value and then varies slightly. In addition, the contribution of a specific most probable slip surface varies dramatically at different spatial variability level, and therefore, the variation of the most probable slip surfaces should be accounted for in the reliability analysis. The multiple response surfaces approach developed in this paper provides a limit equilibrium method and MCS-based means to incorporate such a variation of the most probable slip surfaces in slope reliability analysis. Copyright © 2014 John Wiley & Sons, Ltd.

This paper develops a risk de-aggregation and system reliability approach to evaluate the slope failure probability, pf, using representative slip surfaces together with MCS. An efficient procedure is developed to strategically select the candidate representative slip surfaces, and a risk de-aggregation approach is proposed to quantify contribution of each candidate representative slip surface to the pf, identify the representative slip surfaces, and determine how many representative slip surfaces are needed for estimating the pf with reasonable accuracy. Risk de-aggregation is performed by collecting the failure samples generated in MCS and analyzing them statistically. The proposed methodology is illustrated through a cohesive soil slope example and validated against results from previous studies. When compared with the previous studies, the proposed approach substantially improves the computational efficiency in probabilistic slope stability analysis. The proposed approach is used to explore the effect of spatial variability on the pf. It is found that, when spatial variability is ignored or perfect correlation assumed, the pf of the whole slope system can be solely attributed to a single representative slip surface. In this case, it is theoretically appropriate to use only one slip surface in the reliability analysis. As the spatial variability becomes growingly significant, the number of representative slip surfaces increases, and all representative slip surfaces (i.e., failure modes) contribute more equally to the overall system risk. The variation of failure modes has substantial effect on the pf, and all representative surfaces have to be incorporated properly in the reliability analysis. The risk de-aggregation and system reliability approach developed in this paper provides a practical and efficient means to incorporate such a variation of failure modes in probabilistic slope stability analysis.

The Mg 2− x Al 4+1/2 x Li 1/2 x Si 5 O 18 (0.1≤ x ≤1) ceramics with the substitution of (Li 1/2 Al 1/2 ) 2+ for Mg 2+ were synthesized by the sol–gel method. The characterization of the modified cordierite included X-ray diffraction, SEM, EDS and infrared radiation. The crystal structure of Mg 2 Al 4 Si 5 O 18 with the substitution of (Li 1/2 Al 1/2 ) 2+ for Mg 2+ changed and the amount of secondary phase increased with increasing the x value from 0.1 to 1. High infrared emissivity over 0.9 in the band of 8–14 μm at room temperature was obtained in Mg 2− x Al 4+1/2 x Li 1/2 x Si 5 O 18 ( x =0.1). The material based on cordierite with x =0.1 sintered at 1200 °C maintained a single phase, compact microstructure and good infrared emissivity with potential use in infrared heating.