# Dynamic Reliability Analysis of Layered Slope Considering Soil Spatial Variability Subjected to Mainshock–Aftershock Sequence

^{1}

^{2}

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^{*}

## Abstract

**:**

_{P}) of a slope produced by the mainshock–aftershock sequence (MAS) is studied. A slope reliability analysis method is proposed based on the Newmark displacement method and the generalized probability density evolution method (GPDEM) to quantify the effect of the spatial variability of materials parameters on dynamic reliability. Firstly, the parameter random field is generated based on the spectral representation method, and the randomly generated parameters are assigned to the finite element model (FEM). In addition, the random simulation method of MAS considering the correlation between aftershock and mainshock is adopted based on the Copula function to generate the MAS. Then, the D

_{P}of slopes caused by the MAS considering the spatial variability is calculated based on the Newmark method. The impacts of the coefficient of variation (COV) and aftershock on the D

_{P}of slope is analyzed by means of mean values. Finally, the effect of COV and aftershock on the reliability of D

_{P}is explained from a probabilistic point of view based on the GPDEM. The results revealed that with the increase in the COV, the mean of the D

_{P}of the slope shows a trend of increasing gradually. The D

_{P}of slope is more sensitive to the coefficient of variation of friction angle (COV

_{F}). The mean D

_{P}of the slope induced by the MAS is larger compared to the single mainshock, and the PGA has a significant impact on the D

_{P}.

## 1. Introduction

^{2}in Central Taiwan [2]. Around 20,000 people perished in the 2008 Wenchuan earthquake as a result of a large number of landslides and slope instability issues, which made up nearly half of all earthquake fatalities [3,4]. In the Yushu earthquake in 2010, the earthquake created more than 2000 landslides, resulting in a direct economic loss of about CNY 600,000, 8 deaths, and 14 injuries [5]. These aforementioned disaster consequences show that reasonable consideration of dynamic response and sliding displacement of slopes induced by strong earthquakes is very essential for predicting the potential damage possibility of ground motion and conducting rapid seismic risk assessment.

_{P}caused by a mainshock–aftershock sequence needs further verification.

_{P}of the slope is obtained by a nonintrusive analysis. Firstly, the impact of COV on a slope’s dynamic stability is investigated from the mean value of D

_{P}, and then the influence of COV and PGA on a slope’s dynamic reliability is explained from a probabilistic point of view by combining the GPDEM. The flowchart of the evaluation framework is depicted in Figure 1.

## 2. Generalized Probability Density Evolution Method

_{1}, H

_{2}, …, H

_{n})

^{T}; n is the number of degrees of freedom. In addition, it is worth noting that it can be regarded as a variable Θ and t.

_{z}

_{,1}, H

_{z}

_{,2}, …, H

_{z},

_{m})

^{T}.

- (1)
- Point selection and probability assignment in probability space.

- (2)
- Deterministic solutions for dynamic systems.

- (3)
- Solving probability density evolution equation.

- (4)
- Cumulative summation.

## 3. Simulation of Random Field and Random Main Aftershock Sequence

#### 3.1. Spectral Representation Method

#### 3.2. Generation of Parametric Random Fields Based on Spectral Representation Method

#### 3.3. Random Simulation of Mainshock–Aftershock Sequence (MAS)

- (1)
- Establishment of a physical random function model of the MAS.
- (2)
- The real MASs are collected from the PEER to determine the physical parameters in the physical random function model of the mainshock–aftershock sequence.
- (3)
- Select a representative set of points of seismic parameters according to the GF difference. Then, establish the relevance between the aftershock and mainshock parameters based on the Copula theory.
- (4)
- Generate of a series of random MASs by using the narrowband harmonic superposition method.

## 4. Nonintrusive Analysis of Slope Dynamic Reliability

- (1)
- Establish the slope of the FE model, divide the model mesh, set the boundary conditions, define the load loading method, define the material properties, and assign the elements in the SIGMA/W module with the parameter averages. Then, establish the corresponding relationship between the elements, groups, and material properties. Additionally, establish the stability analysis model in SLOPE/W, and save the FEM as a file with the extension name of “.xml”.
- (2)
- The slope strength parameters are simulated by the spectral representation method. N groups of data of parameters will be generated, and the parameters in the “xml” file will be replaced in batches with the newly generated n groups of data through MATLAB programming to obtain n new “.xml” files.
- (3)
- Use the UE text editor software to directly use GeoStudio to batch calculate the stability of the n new “.xml” files obtained in step (2). Then, output the calculation result files corresponding to each group of parameters.
- (4)
- The calculation results corresponding to all parameter groups are extracted in batch, and the D
_{P}is statistically analyzed.

## 5. Model Establishment and Material Parameters

#### 5.1. Finite Element Model

#### 5.2. Calculation Parameters

_{C}and COV

_{F}) are set as 0.1, 0.2, and 0.3. The vertical and horizontal autocorrelation distances (l

_{h}and l

_{v}) are 20 m and 2 m, respectively. The horizontal dimension of the random field unit is 2 m, and the vertical dimension is 0.5 m. The ratios of the vertical and horizontal fluctuation ranges to the vertical and horizontal dimensions of the random field are ${\delta}_{\mathrm{h}}/{l}_{x}=20\sqrt{\pi}/2$ = 17.7 and ${\delta}_{\mathrm{v}}/{l}_{y}=2\sqrt{\pi}/0.5$ = 7.08, respectively, which are greater than the accuracy requirements (5.7~7.6) given by Ching and Phoon [48].

#### 5.3. Input of the Mainshock–Aftershock Sequence

## 6. Effect of Coefficient of Variation on Dynamic Reliability of Layered Soil Slope

_{P}is adopted to assess the dynamic stability of slope, so it is necessary to define the critical D

_{P}of the soil slope. According to previous research [6], three D

_{P}thresholds (0.05 m, 0.5 m, and 1 m) were used to assess the dynamic reliability of the layered slope. A total of 86 sets of random material parameters were generated based on spectral representation to explore the impact of spatial variability on slope dynamic reliability.

#### 6.1. Case 1: Clayey Soil–Gravel Soil–Sandy Soil–Foundation Soil

_{P}discrete points for Case 1 when the COV

_{C}values are 0.1, 0.2, and 0.3. It is significant that when the COV

_{C}is small, the D

_{P}is small and relatively concentrated. With the increase in COV

_{C}, the range of variation of soil cohesion increases, the distribution of discrete points of D

_{P}becomes more discrete, and the mean of D

_{P}gradually increases. When the COV

_{C}is 0.1, the mean of D

_{P}caused by the MAS is 0.63 m, while the mean D

_{P}for the slope subjected to the single mainshock is 0.339 m. It is obvious that the mean D

_{P}of the slope induced by MAS is wider than the D

_{P}caused by the single mainshock. Moreover, the mean D

_{P}values of the slope under the MAS are 0.668 m and 0.725 m, respectively, when the COV

_{C}is 0.2 and 0.3. At this time, the mean D

_{P}of the slope due to the single mainshock is 0.368 m and 0.42 m. The mean value of D

_{P}increases continuously along with the increment of COV

_{C}, and the discrepancy of D

_{P}also shows a gradual tendency to increase.

_{P}discrete points of slope under various PGA when the COV

_{C}is 0.3. When the PGA values are 0.4 g and 0.6 g, the mean D

_{P}values of the slope caused by the MAS are 0.386 m and 0.924 m, respectively. However, the mean D

_{P}values of the slope induced by the single mainshock are 0.224 m and 0.505 m. The D

_{P}of the slope constantly changed incrementally with the increase in PGA.

_{P}of slope under different COV

_{C}. When the COV

_{C}is 0.1, the maximum value of PDF is 3.98, the fluctuation region of D

_{P}is 0.4–1.0 m, and the D

_{P}is primarily focused around 0.45 m. When the COV

_{C}is 0.3, the maximum value of PDF is 1.2, the fluctuation region of D

_{P}of slope is 0–1.5 m, and the D

_{P}is relatively centralized around 0.8 m. With the growth of the COV

_{C}, the PDF value gradually decreases, the curve gradually shifts to the right, the D

_{P}distribution range is wider, and the failure probability of the slope is higher. Table 2 shows the reliability information of the slope when the cumulative slips are 0.05 m, 0.5 m, and 1 m under different COV

_{C}and PGA. The dynamic reliability of the slope caused by the MAS decreases by 13% with the COV

_{C}increasing from 0.1 to 0.3 when the PGA is 0.5 g, and the displacement threshold is 1 m. When the COV

_{C}is 0.3, the dynamic reliability of the slope under the action of the MAS is also reduced by 13% compared with the single mainshock. In addition, with the PGA increasing from 0.4 g to 0.6 g, the dynamic reliability of the slope induced by MAS decreased by 35%.

_{P}discrete points for Case 1 when the COV

_{F}values are 0.1, 0.2, and 0.3. As the COV

_{F}is small, the D

_{P}for the slope is low and concentrated. With the increase in COV

_{F}, the fluctuating region increases and the mean value of D

_{P}of slope gradually increases and becomes more discrete. When the COV

_{F}is 0.1, the mean D

_{P}of the slope under the MAS is 0.674 m and the mean D

_{P}of the slope under the single mainshock is 0.372 m. The mean D

_{P}of the slope induced by the MAS is larger than that under the single mainshock. When the COV

_{F}values are 0.2 and 0.3, the mean D

_{P}values of the slope caused by the MAS are 0.743 m and 0.795 m, respectively. At this time, the mean D

_{P}values of the slope due to the single mainshock are 0.409 m and 0.432 m. With increasing COV

_{F}, the mean value of D

_{P}of the slope continuously increases.

_{P}discrete points of slope under different PGA when the COV

_{F}is 0.3. When the PGA values are 0.4 g and 0.6 g, the mean D

_{P}values of the slope subjected to the MAS are 0.41 m and 1.165 m, respectively. However, the mean D

_{P}value of for Case 2 caused by the single mainshock are 0.246 m and 0.648 m. Obviously, the D

_{P}of for the slope increases continuously with the growth of PGA.

_{P}of the slope under different COV

_{F}values. The maximum PDF value is 1.89 when the COV

_{F}is 0.1, and the D

_{P}of the slope is principally around 0.7 m. With the growth of the COV

_{F}, the PDF value gradually decreases, and the D

_{P}of the slope is more widely distributed. Table 3 shows the information of dynamic reliability for the slope when the cumulative slip is 0.05 m, 0.5 m, and 1 m under different COV

_{F}and PGA. When the PGA is 0.5 g and the threshold is 1 m, the dynamic reliability of the slope induced by the MAS decreases by 25% with the COV

_{F}increasing from 0.1 to 0.3. When the COV

_{F}is 0.3, the dynamic reliability of the slope under the action of the MAS is also reduced by 31% compared with the single mainshock. In addition, with the PGA increasing from 0.4 g to 0.6 g, the dynamic reliability of the slope under the MAS decreased by 17%.

_{C}and COV

_{F}, the dynamic reliability gradually decreases, and the failure probability gradually increases under different displacement thresholds. In contrast, the dynamic reliability of slopes is more sensitive to COV

_{F}. Additionally, the dynamic reliability of slopes is more sensitive to the COV

_{F}. Similar conclusions were also obtained by Huang et al. [1], but the impact of aftershocks was not considered in their research.

#### 6.2. Case 2: Clayey Soil–Sandy Soil–Gravel Soil–Foundation Soil

_{P}dispersion points for Case 2 when the COV

_{C}values are 0.1, 0.2, and 0.3, respectively. The D

_{P}of the slope is smaller and more concentrated when the COV

_{C}is small. With the increase in the COV

_{C}, the D

_{P}of the slope is gradually increased and became more discrete. When the COV

_{C}is 0.1, the mean of D

_{P}caused by the MAS is 0.293 m, and the mean of D

_{P}of the slope induced by the single mainshock is 0.247 m. When the COV

_{C}values are 0.2 and 0.3, the mean of D

_{P}subjected to the MAS is 0.36 m and 0.458 m, respectively. At this time, the mean of D

_{P}values under a single mainshock are 0.28 m and 0.335 m. The mean of D

_{P}increases continuously, and the difference shows a trend of increasing gradually with the increase in the COV

_{C}.

_{P}discrete points of the slope under different PGA when the COV

_{C}is 0.3. When the PGA is 0.4 g and 0.6 g, the mean of D

_{P}values under the MAS are 0.261 m and 0.881 m, respectively. However, the mean of D

_{P}values under a single mainshock are 0.173 m and 0.577 m. The D

_{P}of the slope is raised step by step with the increase in PGA.

_{P}of slope under different COV

_{C}. The maximum PDF value is 3.2 when the COV

_{C}is 0.1, and the D

_{P}is mainly concentrated around 0.2–0.4 m. As the COV

_{C}improves, the PDF value gradually decreases, the curve gradually shifts to the right, and the D

_{P}of the slope is more widely distributed. Table 4 shows the information of dynamic reliability of D

_{P}for the slope when the cumulative slip is 0.05 m, 0.5 m, and 1 m under different COV

_{C}and PGA values. When the PGA is 0.5 g and the displacement threshold is 1 m, the reliability of the slope subjected to the MAS decreases by 7% with the COV

_{C}increasing from 0.1 to 0.3. When the COV

_{C}is 0.3, the dynamic reliability of the slope under the action of the MAS is also reduced by 7% compared with the single mainshock. In addition, with the PGA increasing from 0.4 g to 0.6 g, the reliability of D

_{P}of the slope produced by the MAS decreased by 8%.

_{P}discrete points for Case 2 when the COV

_{F}values are 0.1, 0.2, and 0.3. At a small COV

_{F}value, the D

_{P}of the slope is small and concentrated. The variation range of soil increases with the increase in COV

_{F}, so that the mean of D

_{P}gradually increases and becomes more discrete. When the COV

_{F}is 0.1, the mean of D

_{P}subjected to the MAS is 0.299 m, and the mean of D

_{P}caused by the single mainshock is 0.247 m. The mean of D

_{P}under the MAS is larger than that caused by a single mainshock. When the COV

_{F}values are 0.2 and 0.3, the mean of D

_{P}produced by MAS is 0.404 m and 0.553 m, respectively. At this time, the mean of D

_{P}values due to a single mainshock are 0.293 m and 0.364 m. The mean D

_{P}of the slope keeps increasing with the growth of COV

_{F}, and the discrepancy of D

_{P}also displays a gradual increasing tendency.

_{P}discrete points of slope under the action of different PGA when the COV

_{F}is 0.3. When the PGA values are 0.4 g and 0.6 g, the mean of D

_{P}values of the slope produced by the MAS are 0.261 m and 0.881 m, respectively. However, the mean D

_{P}values of the slope due to the single mainshock are only 0.173 m and 0.577 m. The D

_{P}of the slope increases continuously with the increase in PGA.

_{P}of the slope under different COV

_{F}. The maximum PDF value is 3.2 when the COV

_{F}is 0.1, and the D

_{P}of the slope is mainly focused around 0–0.5 m. The PDF value gradually decreases, the curve gradually shifts to the right with the increase in the COV

_{F}, and the D

_{P}of the slope is more widely distributed. Table 5 lists the dynamic reliability of the slope when the cumulative slips are 0.05 m, 0.5 m, and 1 m under COV

_{F}and PGA. When the PGA is 0.5 g and the threshold is 1 m, the dynamic reliability of the slope induced by the MAS decreases by 17% with the COV

_{F}increasing from 0.1 to 0.3. When the COV

_{F}is 0.3, the dynamic reliability of the slope subjected to the MAS is also reduced by 14% compared with the single mainshock. In addition, the dynamic reliability of the slope under the MAS decreased by 14% with the PGA increasing from 0.4 g to 0.6 g.

_{F}has a significantly greater influence on the dynamic reliability of the slope. However, due to the different distribution of soil layers in the layered slopes, the influence of the COV on the failure probability of a slope is different. Compared with Case 2, the lower layer located on the slope is a sandy soil with poorer properties and its thickness is relatively deep. Therefore, the dynamic reliability of Case 1 is more significantly affected by the COV. The influence of different soil layers on the dynamic reliability and sliding surface position of slopes has been discussed in our previous research and detailed content can be found in [6].

## 7. Conclusions

_{P}of layered soil slope is calculated by nonintrusive reliability analysis, and the influence of the COV

_{C}and COV

_{F}on the dynamic reliability of slope is compared. The conclusions of this study are as follows:

- (1)
- A reliability analysis method for D
_{P}of the slope is established based on the GPDEM and Newmark methods. Combined with the noninvasive stochastic analysis method, the failure probability of a slope can be quickly obtained. - (2)
- According to the stochastic dynamic calculation results of the layered soil slope, COV
_{C}and COV_{F}have a significant impact on the D_{P}of the slope induced by the MAS. The mean of D_{P}of the slope also presents a trend of increasing gradually with an increase in the COV_{C}and COV_{F}values. In contrast, the D_{P}of slope is more sensitive to the COV_{F}. - (3)
- Affected by the randomness and nonlinearity of the materials, the PDF curve has nonuniform single or double peaks. As the COV increases, the PDF curve becomes lower and wider, and the failure probability of the layered soil slope increases. When the D
_{P}threshold is 1 m and PGA is 0.5 g, the dynamic reliability of the soil slope is continuously reduced, and the failure probability is even increased by about 20% with the COV increasing from 0.1 to 0.3. - (4)
- The impact of aftershocks on the D
_{P}of the soil slope cannot be ignored. The mean of D_{P}of the slope induced by the MSA is larger than that under a single mainshock. The dynamic reliability of the slope caused by the MAS can even be reduced by 7–30% compared with a single mainshock when the displacement threshold is 1 m and the COV_{C}is 0.3. Additionally, the impact of aftershocks on the D_{P}of slope increases with an increase in PGA.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Huang, Y.; Zhao, L.; Li, X. Slope-Dynamic Reliability Analysis Considering Spatial Variability of Soil Parameters. Int. J. Geomech.
**2020**, 20, 04020068. [Google Scholar] [CrossRef] - Huang, J. Chi-Chi earthquake induced landslides in Taiwan. Earthq. Eng. Eng. Seismol.
**2000**, 2, 25–33. [Google Scholar] - Xu, C.; Xu, X.; Dai, F. Three (nearly) complete inventories of landslides triggered by the May 12, 2008 Wenchuan Mw 7.9 earthquake of China and their spatial distribution statistical analysis. Landsides
**2014**, 11, 441–461. [Google Scholar] [CrossRef][Green Version] - Yin, Y.; Zheng, X.; Li, P.; Li, S. Catastrophic landslides associated with the M8.0 Wenchuan earthquake. Bull. Eng. Geol. Environ.
**2011**, 70, 35–50. [Google Scholar] [CrossRef] - Xu, C.; Xu, X. Statistical analysis of landslides caused by the Mw 6.9 Yushu, China, earthquake of April 14, 2010. Nat. Hazards
**2014**, 72, 871–893. [Google Scholar] [CrossRef] - Wang, G.; Pang, R.; Xu, B.; Yu, X. Permanent displacement reliability analysis of soil slopes subjected to mainshock-aftershock sequences. Comput. Geotech.
**2023**, 153, 105069. [Google Scholar] [CrossRef] - Zhang, S.; Wang, G.; Sa, W. Damage evaluation of concrete gravity dams under mainshock-aftershock seismic sequences. Soil Dyn. Earthq. Eng.
**2013**, 50, 16–27. [Google Scholar] [CrossRef] - Liu, J.; Yi, G.; Zhang, Z.; Guan, Z.; Ruan, X.; Long, F.; Du, F. Introduction to the Lushan, Sichuan M7.0 earthquake on 20 April 2013. Chin. J. Geophys. Chin. Ed.
**2013**, 56, 1404–1407. [Google Scholar] [CrossRef] - Kim, B.; Shin, M. A model for estimating horizontal aftershock ground motions for active crustal regions. Soil Dyn. Earthq. Eng.
**2017**, 92, 165–175. [Google Scholar] [CrossRef] - Pang, R.; Xu, B.; Zhou, Y.; Zhang, X.; Wang, X. Fragility analysis of high CFRDs subjected to mainshock-aftershock sequences based on plastic failure. Eng. Struct.
**2020**, 206, 110152. [Google Scholar] [CrossRef] - Zhou, Z.; Yu, X.; Lu, D. Identifying Optimal Intensity Measures for Predicting Damage Potential of Mainshock-Aftershock Sequences. Appl. Sci.
**2020**, 10, 6795. [Google Scholar] [CrossRef] - Shen, J.; Chen, J.; Ding, G. Random field model of sequential ground motions. Bull. Earthq. Eng.
**2020**, 18, 5119–5141. [Google Scholar] [CrossRef] - Jibson, R.W. Methods for assessing the stability of slopes during earthquakes—A retrospective. Eng. Geol.
**2011**, 122, 43–50. [Google Scholar] [CrossRef] - Li, J.; Chen, J. The principle of preservation of probability and the generalized density evolution equation. Struct. Saf.
**2008**, 30, 65–77. [Google Scholar] [CrossRef] - Su, H.; Fu, Z.; Gao, A.; Wen, Z. Numerical simulation of soil levee slope instability using particle-flow code method. Nat. Hazards Rev.
**2019**, 20, 04019001. [Google Scholar] [CrossRef] - Su, H.; Hu, J.; Yang, M.J. Evaluation method for slope stability under multianchor support. Nat. Hazards Rev.
**2014**, 16, 04014033. [Google Scholar] [CrossRef] - Chousianitis, K.; Del Gaudio, V.; Sabatakakis, N.; Kavoura, K.; Drakatos, G.; Bathrellos, G.D.; Skilodimou, H.D. Assessment of earthquake-induced landslide hazard in Greece; from Arias intensity to spatial distribution of slope resistance demand. Bull. Seismol. Soc. Am.
**2016**, 106, 174–188. [Google Scholar] [CrossRef] - Du, W.; Wang, G. A one-step Newmark displacement model for probabilistic seismic slope displacement hazard analysis. Eng. Geol.
**2016**, 205, 12–23. [Google Scholar] [CrossRef] - Wang, M.; Li, D.; Du, W. Probabilistic seismic displacement hazard assessment of earth slopes incorporating spatially random soil parameters. J. Geotech. Geoenvironmental Eng.
**2021**, 147, 04021119. [Google Scholar] [CrossRef] - Li, Y.; Pang, R.; Xu, B.; Wang, X.; Fan, Q.; Jiang, F. GPDEM-based stochastic seismic response analysis of high concrete-faced rockfill dam with spatial variability of rockfill properties based on plastic deformation. Comput. Geotech.
**2021**, 139, 104416. [Google Scholar] [CrossRef] - Pang, R.; Xu, B.; Kong, X.; Zhou, Y.; Zou, D. Seismic performance evaluation of high CFRD slopes subjected to near-fault ground motions based on generalized probability density evolution method. Eng. Geol.
**2018**, 246, 391–401. [Google Scholar] [CrossRef] - Pang, R.; Xu, B.; Zou, D.; Kong, X. Stochastic seismic performance assessment of high CFRDs based on generalized probability density evolution method. Comput. Geotech.
**2018**, 97, 233–245. [Google Scholar] [CrossRef] - Chen, X.; Gao, R.; Gong, W.; Li, Y.; Qiu, J. Random seismic response and dynamic fuzzy reliability analysis of bedding rock slopes based on Pseudoexcitation method. Int. J. Geomech.
**2018**, 18, 04017165. [Google Scholar] [CrossRef] - Liu, L.; Cheng, Y. System reliability analysis of soil slopes using an advanced Kriging metamodel and quasi-Monte Carlo simulation. Int. J. Geomech.
**2018**, 18, 06018019. [Google Scholar] [CrossRef] - Hasofer, A.M.; Lind, N.C. Exact and invariant second-moment code format. J. Eng. Mech. Div.
**1974**, 100, 111–121. [Google Scholar] [CrossRef] - Metropolis, N.; Ulam, S. The monte carlo method. J. Am. Stat. Assoc.
**1949**, 44, 335–341. [Google Scholar] [CrossRef] - Wong, F.S. Slope reliability and response surface method. J. Geotech. Eng.
**1985**, 111, 32–53. [Google Scholar] [CrossRef] - Pasculli, A.; Calista, M.; Sciarra, N. Variability of local stress states resulting from the application of Monte Carlo and finite difference methods to the stability study of a selected slope. Eng. Geol.
**2018**, 245, 370–389. [Google Scholar] [CrossRef] - Li, J.; Chen, J.B. Probability density evolution method for dynamic response analysis of structures with uncertain parameters. Comput. Mech.
**2004**, 34, 400–409. [Google Scholar] [CrossRef] - Li, J.; Chen, J.B. The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems. Probabilistic Eng. Mech.
**2006**, 21, 442–453. [Google Scholar] [CrossRef] - Huang, Y.; Xiong, M. Dynamic reliability analysis of slopes based on the probability density evolution method. Soil Dyn. Earthq. Eng.
**2017**, 94, 1–6. [Google Scholar] [CrossRef] - Huang, Y.; Hu, H.; Xiong, M. Probability density evolution method for seismic displacement-based assessment of earth retaining structures. Eng. Geol.
**2018**, 234, 167–173. [Google Scholar] [CrossRef] - Pang, R.; Xu, B.; Zhou, Y.; Song, L. Seismic time-history response and system reliability analysis of slopes considering uncertainty of multi-parameters and earthquake excitations. Comput. Geotech.
**2021**, 136, 104245. [Google Scholar] [CrossRef] - Xu, B.; Pang, R.; Zhou, Y. Verification of stochastic seismic analysis method and seismic performance evaluation based on multi-indices for high CFRDs. Eng. Geol.
**2020**, 264, 105412. [Google Scholar] [CrossRef] - Bai, T.; Hu, X.; Gu, F. Practice of searching a noncircular critical slip surface in a slope with soil variability. Int. J. Geomech.
**2019**, 19, 04018199. [Google Scholar] [CrossRef] - Hu, H.; Huang, Y.; Zhao, L. Probabilistic Seismic-Stability Analysis of Slopes Considering the Coupling Effect of Random Ground Motions and Spatially-Variable Soil Properties. Nat. Hazards Rev.
**2020**, 21, 04020028. [Google Scholar] [CrossRef] - Li, D.; Qi, X.; Phoon, K.; Zhang, L.; Zhou, C. Effect of spatially variable shear strength parameters with linearly increasing mean trend on reliability of infinite slopes. Struct. Saf.
**2014**, 49, 45–55. [Google Scholar] [CrossRef] - Hariri-Ardebili, M.A.; Seyed-Kolbadi, S.M.; Saouma, V.E.; Salamon, J.; Rajagopalan, B. Random finite element method for the seismic analysis of gravity dams. Eng. Struct.
**2018**, 171, 405–420. [Google Scholar] [CrossRef] - Zhang, J.; Zhang, M.; Tang, W. New methods for system reliability analysis of soil slopes. Can. Geotech. J.
**2011**, 48, 1138–1148. [Google Scholar] [CrossRef] - Cornell, A. First-Order Uncertainty Analysis of Soil Deformation and Stability; Publication of University of Hong Kong: Hong Kong, China, 1972. [Google Scholar]
- Vanmarcke, E. Reliability of earth slopes. J. Geotechnical Eng.
**1977**, 103, 1247–1265. [Google Scholar] [CrossRef] - Der Kiureghian, A.; Ke, J. The stochastic finite element method in structural reliability. Probabilistic Eng. Mech.
**1988**, 3, 83–91. [Google Scholar] [CrossRef] - Shinozuka, M.; Deodatis, G. Simulation of multi-dimensional Gaussian stochastic fields by spectral representation. Appl. Mech. Rev.
**1996**, 49, 29–53. [Google Scholar] [CrossRef] - Matthies, H.G.; Brenner, C.E.; Bucher, C.G.; Guedes Soares, C. Uncertainties in probabilistic numerical analysis of structures and solids-Stochastic finite elements. Struct. Saf.
**1997**, 19, 283–336. [Google Scholar] [CrossRef] - Ghiocel, D.; Ghanem, R. Stochastic finite-element analysis of seismic soil-structure interaction. J. Eng. Mech.
**2002**, 128, 66–77. [Google Scholar] [CrossRef] - Pang, R.; Zhou, Y.; Chen, G.; Jing, M. Stochastic mainshock-aftershock simulation and its applications in dynamic reliability of structural systems via DPIM. J. Eng. Mech.
**2023**, 149, 04022096. [Google Scholar] [CrossRef] - Huang, L. Dynamic Stability Analysis of Layered Soil Slope under Earthquake; Southwest Jiaotong University: Chengdu, China, 2017. (In Chinese) [Google Scholar]
- Ching, J.; Phoon, K. Effect of element sizes in random field finite element simulations of soil shear strength. Comput. Struct.
**2013**, 126, 120–134. [Google Scholar] [CrossRef]

**Figure 4.**Acceleration change curve of the mainshock–aftershock sequence: (

**a**) horizontal; (

**b**) vertical.

**Figure 5.**Distribution of discrete points of D

_{P}under different COV

_{C}(Case 1): (

**a**) COV

_{C}= 0.1; (

**b**) COV

_{C}= 0.2; (

**c**) COV

_{C}= 0.3.

**Figure 6.**Distribution of discrete points of D

_{P}under different PGA when COV

_{C}=0.3 (Case 1): (

**a**) PGA = 0.4 g; (

**b**) PGA = 0.6 g.

**Figure 8.**Probability distribution of D

_{P}with different PGA when COV

_{C}= 0.3 (Case 1): (

**a**) PDF; (

**b**) CDF.

**Figure 9.**Distribution of discrete points of D

_{P}under different COV

_{F}(Case 1): (

**a**) COV

_{F}= 0.1; (

**b**) COV

_{F}= 0.2; (

**c**) COV

_{F}= 0.3.

**Figure 10.**Distribution of discrete points of D

_{P}under different PGA when COV

_{F}= 0.3 (Case 1): (

**a**) PGA = 0.4 g; (

**b**) PGA = 0.6 g.

**Figure 12.**Probability distribution of D

_{P}with different PGA when COV

_{F}= 0.3 (Case 1): (

**a**) PDF; (

**b**) CDF.

**Figure 13.**Distribution of discrete points of D

_{P}under different COV

_{C}(Case 2): (

**a**) COV

_{C}= 0.1; (

**b**) COV

_{C}= 0.2; (

**c**) COV

_{C}= 0.3.

**Figure 14.**Distribution of discrete points of D

_{P}under different PGA when COV

_{C}= 0.3 (Case 2): (

**a**) PGA = 0.4 g; (

**b**) PGA = 0.6 g.

**Figure 16.**Probability distribution of D

_{P}with different PGA when COV

_{C}= 0.3 (Case 2): (

**a**) horizontal; (

**b**) vertical.

**Figure 17.**Distribution of discrete points of D

_{P}under different COV

_{F}(Case 2): (

**a**) COV

_{F}= 0.1; (

**b**) COV

_{F}= 0.2; (

**c**) COV

_{F}= 0.3.

**Figure 18.**Distribution of discrete points of D

_{P}under different PGA when COV

_{F}= 0.3 (Case 2): (

**a**) PGA = 0.4 g; (

**b**) PGA = 0.6 g.

**Figure 20.**Probability distribution of D

_{P}with different PGA when COV

_{F}= 0.3 (Case 2): (

**a**) PDF; (

**b**) CDF.

Materials | c (kPa) | φ (°) | E (MPa) | γ (kN/m^{3}) | v |
---|---|---|---|---|---|

Clayey soil | 70.24 | 24.00 | 86 | 22.16 | 0.35 |

Sandy soil | 13.65 | 32.50 | 60 | 17.23 | 0.32 |

Gravelly soil | 18.23 | 38.50 | 73 | 19.55 | 0.3 |

Foundation soil | 200 | 35.02 | 800 | 25.14 | 0.25 |

Type of Ground Motion | D_{P} | COV_{C} | PGA | ||||
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 g | 0.5 g | 0.6 g | ||

Mainshock–aftershock sequence | 0.05 m | 0 | 0 | 0.016 | 0.035 | 0.016 | 0.003 |

0.5 m | 0.09 | 0.21 | 0.26 | 0.77 | 0.26 | 0.15 | |

1 m | 1 | 0.96 | 0.87 | 0.98 | 0.87 | 0.63 | |

Single mainshock | 0.05 m | 0 | 0 | 0.016 | 0.035 | 0.016 | 0.003 |

0.5 m | 0.99 | 0.92 | 0.84 | 0.95 | 0.84 | 0.58 | |

1 m | 1 | 1 | 1 | 1 | 1 | 1 |

Type of Ground Motion | D_{P} | COV_{F} | PGA | ||||
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 g | 0.5 g | 0.6 g | ||

Mainshock–aftershock sequence | 0.05 m | 0.01 | 0.03 | 0.16 | 0.23 | 0.16 | 0.04 |

0.5 m | 0.23 | 0.38 | 0.32 | 0.61 | 0.32 | 0.24 | |

1 m | 0.92 | 0.71 | 0.67 | 0.99 | 0.67 | 0.38 | |

Single mainshock | 0.05 m | 0.01 | 0.03 | 0.16 | 0.26 | 0.1 | 0.04 |

0.5 m | 0.89 | 0.88 | 0.61 | 0.89 | 0.61 | 0.38 | |

1 m | 1 | 1 | 0.98 | 1 | 0.98 | 0.83 |

Type of Ground Motion | D_{P} | COV_{C} | PGA | ||||
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 g | 0.5 g | 0.6 g | ||

Mainshock–aftershock sequence | 0.05 m | 0 | 0.02 | 0.11 | 0.3 | 0.11 | 0.1 |

0.5 m | 0.97 | 0.72 | 0.63 | 0.94 | 0.63 | 0.37 | |

1 m | 1 | 1 | 0.93 | 1 | 0.93 | 0.62 | |

Single mainshock | 0.05 m | 0 | 0.03 | 0.11 | 0.33 | 0.11 | 0.1 |

0.5 m | 1 | 0.96 | 0.78 | 0.98 | 0.78 | 0.52 | |

1 m | 1 | 1 | 1 | 1 | 1 | 0.92 |

Type of Ground Motion | D_{P} | COV_{F} | PGA | ||||
---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 g | 0.5 g | 0.6 g | ||

Mainshock–aftershock sequence | 0.05 m | 0 | 0.03 | 0.18 | 0.4 | 0.18 | 0.05 |

0.5 m | 0.95 | 0.74 | 0.62 | 0.83 | 0.62 | 0.4 | |

1 m | 1 | 0.94 | 0.83 | 0.94 | 0.83 | 0.65 | |

Single mainshock | 0.05 m | 0 | 0.03 | 0.19 | 0.4 | 0.19 | 0.06 |

0.5 m | 1 | 0.87 | 0.74 | 0.95 | 0.74 | 0.51 | |

1 m | 1 | 1 | 0.97 | 1 | 0.97 | 0.86 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, H.; Wang, G.; Yu, X.; Pang, R.
Dynamic Reliability Analysis of Layered Slope Considering Soil Spatial Variability Subjected to Mainshock–Aftershock Sequence. *Water* **2023**, *15*, 1540.
https://doi.org/10.3390/w15081540

**AMA Style**

Zhou H, Wang G, Yu X, Pang R.
Dynamic Reliability Analysis of Layered Slope Considering Soil Spatial Variability Subjected to Mainshock–Aftershock Sequence. *Water*. 2023; 15(8):1540.
https://doi.org/10.3390/w15081540

**Chicago/Turabian Style**

Zhou, Huaiming, Gan Wang, Xiang Yu, and Rui Pang.
2023. "Dynamic Reliability Analysis of Layered Slope Considering Soil Spatial Variability Subjected to Mainshock–Aftershock Sequence" *Water* 15, no. 8: 1540.
https://doi.org/10.3390/w15081540