Dynamic probabilistic analysis of non-homogeneous slopes based on a simplified deterministic model

https://doi.org/10.1016/j.soildyn.2020.106563Get rights and content

Highlights

  • A new procedure for the probabilistic seismic stability analysis of slopes is proposed.

  • A variety of valuable results (e.g. failure probability and sensitivity index) are provided by the procedure.

  • Total computational time of a slope probabilistic seismic analysis can be significantly reduced by using the procedure.

  • A sensitivity analysis is performed to quantify the effect of 9 input parameters on onslope dynamic stability analysis.

Abstract

This article proposes an efficient procedure for the probabilistic seismic analysis of slopes and applies it to two slope examples including a real engineering project. The proposed procedure evaluates the stability condition of a non-homogeneous slope by using a discretization kinematic approach (DKA), and accounts for the time and space variations of a seismic loading by employing pseudo-dynamic method (PDM). Three uncertainty quantification techniques (Subset Simulation, Global Sensitivity Analysis and First Order Reliability Method), are implemented into the procedure in order to provide multiple probabilistic results (e.g. failure probability, reliability index, design point and sensitivity index) for slope seismic reliability problems. The introduced DKA is validated by comparing with two previous studies. Additionally, a variety of hypothetical cases were analyzed in order to provide some insights into the following two issues about slope seismic analyses: 1) the comparison of two seismic methods (PDM and pseudo-static method) in a probabilistic framework; 2) the accuracy of FORM for estimating failure probability. By benefiting from the high computational efficiency of the introduced DKA, two further studies related to sensitivity analysis and fragility curves are also carried out. The obtained results show that the proposed procedure is effective for slope probabilistic seismic analyses and is able to efficiently provide a variety of useful results.

Introduction

Earthquake landslide is one of the greatest geological hazards and has received worldwide attention because of its destructive influence. Studies about seismic stability analyses of slopes are often carried out in a deterministic context [[1], [2], [3]]. They usually used average geotechnical parameters to obtain the safety factor and considered it as an evaluation index to represent the stability condition. However, uncertainties and randomness of soil properties always exist due to the lack of enough measured geological and geotechnical data, the complex geological conditions, the inherent variability of soil parameters and measurement errors [4]. Probabilistic analyses, which can rationally account for these uncertainties, are thus more suitable for the safety assessment of slopes in engineering geology fields [5,6].

There are some seismic reliability analyses of slopes in recent years [[7], [8], [9], [10], [11], [12]]. Concerning the deterministic slope dynamic analysis model, the pseudo-static method (PSM) was commonly implemented in the existing works due to its simplicity [8,13]. The seismic loading on a soil mass is represented by a permanent body force in the PSM context. This technique may lead to biased results because it is unable to consider the nonlinear dynamic behavior and intensity-frequency characteristics of earthquakes presented in practical engineering, such as the duration of earthquake ground motions, amplification factor and frequency. A more rigorous approach is the Finite-Element (or Finite-Difference) dynamic time-history analysis method since it accounts for the characteristics of the seismic ground motion and the seismic wave amplification by soils [14,15]. However, this method, being very time-consuming, is not quite practical for an efficient reliability analysis given that a traditional reliability analysis such as Monte-Carlo Simulation (MCS) requires usually over thousands of simulations [9,16]. Alternatively, the pseudo-dynamic method (PDM) provides a good compromise between the above-mentioned methods (PSM and time-acceleration method) and was used in a significant number of studies to solve geotechnical problems considering seismic effects within a deterministic framework [1,16,17]. The PDM takes into account the time and space variations of a ground shaking and permits to obtain more realistic results compared with the PSM [16,17]. It is thus used in this work to evaluate the seismic effects in a slope stability analysis.

Besides, most of the existing reliability studies under seismic loadings only considered homogeneous soils. It is not in good accordance with a real geotechnical investigation, which showed that the soil exhibits spatial variability in its properties. There are mainly two kinds of non-homogeneity, namely depth-dependency (i.e. strength parameters vary along with depth) [18,19] and multi-layered cases [20,21]. Xiao et al. [9] employed a 2D finite difference numerical model to consider the shear strength spatial variation in a slope probabilistic seismic analysis. However, using numerical models suffers from a heavy computational burden compared to analytical methods due to the necessary time for the model preparation and calculation, especially for reliability analyses. In light of this, it is preferable to use analytical methods at least for the preliminary design stage since the induced estimate is relatively accurate for most cases and the computational time can be significantly reduced.

Limit Equilibrium Method (LEM) is the most traditional and popular method used for slope reliability analyses in practice due to its simple calculation procedure and computational efficiency [7,10,22,23]. However, LEM cannot account for the stress-strain relation of soils and may lead to biased results due to the assumptions made for the failure surface and inter-slice forces [24]. Another analytical method commonly adopted is Limit Analysis (LA). It was developed in a more rigorous way than LEM as it respects the plasticity theory [25,26]. The upper bound limit analysis, also known as the kinematic approach, which needs to search a kinematically admissible velocity field, is more popular and widespread compared to the lower bound limit analysis. The latter requires the definition of an admissible stress field which is often not easy to implement [19]. Stability analyses based on the kinematic approach were widely performed since they can give a rigorous upper bound solution [[27], [28], [29], [30], [31]]. However, it could inevitably introduce complex and tedious integral calculations for non-homogeneous cases if the traditional log-spiral mechanism of kinematic approach is considered. In order to avoid these shortcomings, a discretization kinematic approach (DKA) based on a ‘point-to-point’ technique was developed by Mollon et al. [27,28] and has been used in the context of slope stability analyses [1,32,33]. This technique permits to build a discretized mechanism based on the normality condition, i.e. the velocity vector is inclined with the tangential slope failure line by an angle equaling to the soil friction angle. It means that the failure surface can be easily generated even in cases with spatially varying friction angles, which can avoid the shortcomings of the failure surface assumptions. The efficiency of this method is also improved due to the failure mass discretization. This technique is implemented herein to consider complex cases with non-homogeneous soils.

As for the slope reliability evaluation, the MCS is popular due to its robustness and conceptual simplicity, whereas it lacks of computational efficiency especially for cases with small failure probabilities [34]. An improved sampling method - Subset Simulation (SS) was proposed by Au and Beck [35], which reduces the variance of MCS estimator with limited evaluations. It expresses a small failure probability as a product of larger conditional failure probabilities for some intermediate failure events. This method has been widely employed in the structure reliability analyses [34,36]. However, the above-mentioned probabilistic studies mainly focused on estimating the failure probability or reliability index but often ignored other useful and interesting results. It is well recognized that providing as many results as possible in a probabilistic analysis could be beneficial for practical engineering designs. For example, the design point obtained by First Order Reliability Method (FORM) permits to know how much margin there is with respect to the current mean values, and the index of Global Sensitivity Analysis (GSA) gives a measure about the effect of each input variable on the model response variation and permits to reduce the problem dimension for complex cases [36]. These results could provide complementary information to the safety factor which is the sole evaluation index in a deterministic analysis.

In this research, an efficient procedure (named as DP-SGF, which combines DKA, PDM, SS, GSA and FORM) is proposed for reliability and sensitivity analysis of slope stability under seismic loadings. Compared to the previous studies, the main advantages of this procedure and the major contributions of this work are as follows: (1) A variety of useful results (i.e. failure probability, reliability index, sensitivity index, design point and importance factor) can be provided by the DP-SGF but with only a limited computation time. The procedure is thus recommended to be used in an initial design stage in order to have a quick estimation of the target results and help the decision making; (2) Various sensitivity analysis methods are applied to slope seismic problems, which were rarely performed in the previous works. The obtained sensitivity indices allow quantifying the importance of each considered uncertainty parameter and making a rank of them; (3) The soil non-homogeneity and intensity-frequency characteristics of earthquakes can be accounted for in the introduced computational model DKA-PDM. DKA is able to consider the non-homogeneity of soils due to the employed discretized mechanism while the PDM is more realistic than the commonly used PSM since the time and space variations of a ground shaking are ignored in the PSM.

This paper starts with a brief presentation of the employed probabilistic methods. The main principles of the PDM and DKA, as well as the optimization process of the safety factor calculation, are then clarified. The probabilistic analysis procedure is proposed based on the presented deterministic and probabilistic methods and is detailed via a flowchart. The accuracy and efficiency of the DP-SGF are highlighted by comparing it with two existing studies and the reference method MCS. The analyses show that the DP-SGF can efficiently provide a variety of valuable results. Besides, two comparison studies about PDM-PSM and SS-FORM are conducted in order to discuss their performance in a probabilistic analysis. Moreover, two discussions which include sensitivity analysis of several parameters (about soil and seismic intensity), and slope fragility curves are further performed herein in order to demonstrate some potential improvements/extensions of the proposed probabilistic procedure.

Section snippets

Presentation of the employed probabilistic approaches

This section aims at providing a brief introduction of the probabilistic methods employed in this study, which include SS, FORM and GSA.

The proposed procedure DP-SGF

The seismic stability analysis of a non-homogeneous slope is carried out by using PDM and DKA in this study. This section aims at presenting the main principles of the two techniques and the optimization process of the safety factor Fs. Additionally, a probabilistic analysis procedure which combines the presented deterministic model and the above-mentioned probabilistic analysis methods is proposed and detailed via a flowchart.

Application of the DP-SGF to two slope examples

This section aims to present the application of the proposed procedure DP-SGF to two slope examples, and show its capacity of providing a variety of useful results. One slope's shear strength parameters vary linearly with depth and the other one is a real case named Congress Street cut which is a two-stage slope with four soil layers [44]. Additionally, the introduced discretization kinematic approach DKA for slope stability analysis is validated for both the two slopes by comparing with the

Discussion

This section aims to provide some discussions on two issues that may influence the results of a slope probabilistic seismic analysis, and demonstrates some further capacities of the DP-SGF. Firstly, two commonly used seismic analysis methods (PSM and PDM) are compared in a probabilistic context. It is followed by a survey on the FORM accuracy of Pf estimation when seismic loadings are considered. Then, by benefiting from the high computational efficiency of the DKA-PDM, two further studies are

Summary and conclusion

A new efficient procedure named DP-SGF, which combines three probabilistic analysis methods (SS, GSA and FORM) with the deterministic model DKA-PDM, is proposed to perform the slope reliability and sensitivity analyses under the conditions of non-homogeneous soils and seismic loadings. The DP-SGF requires fewer calls to the deterministic model than the traditional approach MCS and has the capacity of providing a variety of interesting results (failure probability, reliability index, sensitivity

Author declaration

Tingting Zhang: Methodology, Software, Visualization, Writing - Original Draft. Xiangfeng Guo: Conceptualization, Formal analysis, Writing - Review & Editing. Daniel Dias: Resources, Supervision, Writing - Review & Editing. Zhibin Sun: Software, Supervision, Writing - Review & Editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The first author thanks gratefully the China Scholarship Council for providing a PhD Scholarship (CSC No. 201906690049). The financial support is greatly appreciated.

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