Seismic slope stability analysis using modified pseudo dynamic method with uniform random field of initial phases

Liang Xu; Hongliang Jing; Jiahui Wen; Liang Li; Ziyang Song

doi:10.1371/journal.pone.0330435

Abstract

This paper develops an enhanced pseudo dynamic method for conducting seismic stability analysis of slopes. The enhancement is justified by the development of uniform sampling approach and the incorporation of both different and correlated initial phases through a uniform random field. The advanced methodology, named by modified pseudo dynamic Bishop method (MPDBM) is verified against the existing research results and extended to homogeneous soil slope with different scales emphasizing the effect of scale of fluctuation on the minimum factor of safety (FS_min). The numerical results demonstrate that the identical assumption on initial phases in the traditional pseudo dynamic method underestimates the FS_min as compared to the current method considering the spatial variability of initial phases for the cases where the slope soil is subjected to its natural frequency. This underestimation grows significant as the slope scale enlarges.

Citation: Xu L, Jing H, Wen J, Li L, Song Z (2025) Seismic slope stability analysis using modified pseudo dynamic method with uniform random field of initial phases. PLoS One 20(8): e0330435. https://doi.org/10.1371/journal.pone.0330435

Editor: Jianguo Wang, China University of Mining and Technology, CHINA

Received: April 20, 2025; Accepted: July 31, 2025; Published: August 22, 2025

Copyright: © 2025 Xu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting information files.

Funding: The work described in the paper was supported by the Shandong Provincial Natural Science Foundation (Grant No. ZR2023ME007). The funding agency was responsible for data collection. The financial support is gratefully acknowledged.

Competing interests: The authors have declared that no competing interests exist.

1. Introduction

Slope failures caused by earthquakes have led to huge economic losses and casualties, and hence seismic slope stability analysis has triggered the worldwide attention [1–7]. Among the methods for seismic slope stability evaluation, the Factor of Safety (FS)-based one still remains fair popularity owing to its simplicity and easiness in decision making on slope stability. The FS-based methodology can be further classified into three categories: (1) pseudo static method(PSM), in which seismic forces are dealt with by constant forces acting on the centriod of sliding mass [8–10]; (2) pseuso dynamic method (PDM) where part of the dynamic characteristics of seismic forces are taken into account [11–14]; and (3) dynamic time history method, in which the full dynamic characteristics of seismic forces are properly considered. To put it in detail, the PSM, by adding an equivalent static force loaded at the center of sliding mass to the conventional limit equilibrium method, results in a pseudo static Factor of Safety(FS), based on which the seismic stability of slopes can be evaluated in a simplified manner. The PSM finds overwhelming popularity within the geotechnical practitioners, although it only focuses on the amplitude of the seismic force. The dynamic time history method, integrates the complicated soil dynamic constitutive model and the seismic acceleration records to obtain the distributions of stress, strain, and water pressure within the sliding mass, based on which the comprehensive assessment on seismic stability of slopes can be achieved in a systematic but inefficient manner as compared to PSM. As a proper balance between PSM and dynamic time history method, the PDM, mimics the seismic force using a sinusoidal wave stemming from the slope base with a specific initial phase [15,16] to properly address the dynamic characteristic of earthquake.

The PDM was originated by Steedman and Zeng and was validated against the seismic active thrust on a rigid retaining wall [15]. Unlike the constant or linearly varied seismic forces acting on the vertical components of sliding mass in pseudo static method, the concept of initial phase was introduced in PDM to address the variation of seismic forces acting on the vertical components of sliding mass. The seismic acceleration is regarded as a sinusoidal wave projecting from the slope base with a specific initial phase and the seismic acceleration acting on each of vertical components of the sliding mass can be directly determined with the propagation velocity of the sinusoidal wave in the original PDM [17]. The original concept of PDM was integrated with limit equilibrium method and limit analysis to evaluate the seismic slope stability [18–23]. For example, Pan et al. [18] calculated the reliability of slopes using limit analysis coupled with PDM highlighting the influence of slope geometry and seismic parameters on the probability of failure. Hazari et al. [24] combined the Swedish circle method with PDM to examine the slope stability and compared the results from the proposed method with those by the finite element method. Zhou and Qin [11] used the upper bound limit analysis method coupled with PDM to solve the FS of slopes emphasizing on the linear amplification on the amplitude of seismic acceleration.Their outputs have demonstrated that the PDM leads to larger FS than PSM does indicating that the PSM tends to underestimate the FS of the slope and thus to make conservative decision on slope stability. It was also noted in [11] that the increase in FS form the PDM fades out as the shear wave velocity increases, which was also confirmed by [23]. Despite the popularity of original PDM, it suffers from the incompatible stress boundary condition at the slope crest. To properly address this issue, Bellezza [16] developed a modified pseudo dynamic method(MPDM) by regarding the slope soil as a more realistic visco-elastic material(e.g., Kelvin-Voigt material represented by a purely elastic spring and a purely viscous dashpot connected in parallel). Since the origination of MPDM, it has been applied into seismic-related geotechnical problems. For example, Chanda et al [25] proposed a method for calculating slope stability based on MPDM using the horizontal slice method. Zhong and Yang [19] introduced a method for analyzing the stability of rock slopes using the MPDM within the framework of limit analysis.Zhou and Qin [26] analyzed the impact of soft bands on the seismic stability of slopes using finite element limit analysis combined with MPDM. Li et al. [27] used the upper bound theorem of limit analysis to analyze the stability of slopes reinforced with piles under the MPDM, finding that the FS of slope decreases significantly when the normalized frequency of the shear wave coincides with the natural frequency of the soil, which agrees well with that presented in [16].

The literature review on the PDM and the MPDM has found that the FS from the PDM and MPDM is dependent on the initial phase and the most dangerous initial phase must be determined to obtain the minimum FS (denoted herein by FS_min), based on which the slope stability is evaluated by PDM and MPDM in the previous studies. However, it must be noted that the initial phase at different locations of slope base is assumed to be identical in the existing studies. This assumption may be applicable to small scale slopes. However, for large scale slopes, the initial phase at the slope base may be intuitively different but correlated within one seismic motion regarding the influencing factors as epicentral distance, epicenter depth, and the propagation velocity of the seismic wave. In addition, the seismic slope stability was investigated considering the different arrival times of seismic waves at various locations [28–30], which justified the necessity of incorporating the different but correlated initial phase in seismic slope stability using PDM or MPDM. Therefore, how to evaluate the seismic slope stability considering both the difference and correlation in initial phase using MPDM remains an open question.

This paper begins with the combination of MPDM with simplified Bishop Method, followed by the simulation of initial phases using random field theory. Then, the proposed methodology, named modified pseudo dynamic Bishop method(MPDBM) is described in detail and is validated against the previous research outputs. Then, The proposed methodology is elucidated through homogeneous soil slopes with different scales to investigate the effect of damping ratio, scale of fluctuation, and slope scale on the seismic slope stability. Finally, discussions and conclusions are drawn based on the findings.

2. The pseudo dynamic Bishop method for seismic slope stability

2.1 Calculation of horizontal seismic force by MPDM

The MPDM developed in [16] is adopted herein to calculate the horizontal seismic force acting on the sliding mass. Fig 1 demonstrates the distribution of horizontal seismic force, which is proportional to the distribution of propagated acceleration with initial phase t = i₁ at the slope base. The selected circular sliding surface has a center point of O and radius of R. The sliding mass is composed of the soils between the slope surface in black line and circular sliding surface in red line as shown in Fig 1. The sliding mass is divided into a finite number(e.g., m in Fig 1) of vertical slices for the easy implementation of limit equilibrium method. Define an axis Y with origin at the slope crest and its positive direction is downward. The Y coordinate of the slope base is H. Consider slice i for example, b_i is the width of slice i; γ_i is the average unit weight of the slice i; α_i is the inclination angle of slice i with respect to horizontal direction. An arbitrary horizontal sub-element of slice i with Y coordinate of Y_i has a height of dY_i and a width of b_i. Y_i0 and Y_i1 respectively denote the Y coordinates of top and bottom centers of slice i; Following the assumptions adopted by MPDM, the arrived seismic motion is subsequently propagated through the visco-elastic slope soils in a wave like that shown in pink. Note that the angular frequency of seismic motion is ω and the propagation velocity of the shear wave in the slope soil is V_s. The damping ratio of soil is D. The horizontal seismic force acting on the aforementioned sub-element of slice i, denoted by F_hi(t), is:

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Fig 1. The diagram of horizontal seismic force in MPDM.

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(1)

where k_h is the horizontal seismic coefficient at the slope base; Four variables, denoted by C_sz,S_sz,Cs,Ss are defined to facilitate the expression of Eq. (1). They are dependent on the ω, D, Vs, and Yi.

(2)

(3)

(4)

(5)

where k_s1 and k_s2 are defined as:

(6)

(7)

The resultant sliding moment acting on the slice i from the horizontal seismic force can be determined by integration:

(8)

where is the sliding moment acting on the slice i from the horizontal seismic force. The resultant sliding moment acting on the whole sliding mass can be summed as:

(9)

2.2 Pseudo Dynamic Bishop Method(PDBM) for slope stability

The traditional simplified Bishop method [31] can be extended to a pseudo dynamic Bishop method for slope stability, where the contribution of horizontal seismic force is quantified using the MPDM as described in Eq.(9). Refer to Fig 2, where the sliding soil is divided into m slices. G_i is the weight of slice i, E_Ri is the horizontal inter-slice force between slice i and slice i + 1, E_Li is the horizontal interslice force between slice i and slice i-1,N_i and T_i are respectively the normal and shear force at the base of slice i, F_hi(t) is the distributed horizontal seismic force. Thanks to the horizontal direction for the F_hi(t), the force equilibrium equation in vertical direction in traditional simplified Bishop method can still be available.

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Fig 2. Forces acting on slice i.

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Hence, the traditional simplified Bishop method is extended by directly incorporating the contributions of sliding moment resulting from F_hi(t). The PDBM for slope stability is:

(10)

where FS(t) is the Factor of Safety at initial phase t; φ_i and c_i is the internal friction angle and cohesion at the base of slice i.

Obviously, the calculation of FS using the PDBM depends on initial phase t, as indicated by Eq.(10). Previous studies [13,23] have to alter the initial phase t and to find the FS_min of PDM or MPDM for seismic slope stability. However, the initial phase ts at different locations(e.g., t₁, t₂, and t_m as shown in Fig 3) on the slope base are assumed to be identical(e.g., t₁ = t₂ = …, t_m = t). This identical assumption is applicable to the small scale slope where the slope width is negligible as compared to the epicentral distance. As will be discussed in case studies, the initial phase at different locations on the slope base may be varied for medium and large scale slopes. As compared to Fig 3, Fig 4 shows the seismic force distribution of a slope in the case of different initial phases. The arrows below the slope base in different colors indicate the different initial phases of the seismic wave that reaches the base. The comparison between the seismic force distribution of the sliding mass illustrates the difference between identical assumption (used in PDM and MPDM) and different initial phase assumption(will be considered in this study) justifies the necessity of incorporating variations in initial phase in seismic slope stability using PDM and MPDM.

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Fig 3. Schematic diagram of identical assumption on initial phases.

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Fig 4. Schematic diagram of different initial phases.

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3. The simulation of both different and correlated initial phases

3.1 Influencing factors of both different and correlated initial phases

In this section, the influences of epicentral horizontal distance(denoted by E_h), the epicentral depth(denoted by E_d), and the propagation velocity of seismic wave within the slope base(denoted by V_p) are demonstrated and discussed in detail for the further understanding of the importance of both different and correlated initial phases.

Fig 5 illustrates the influence of E_h on the initial phase for the given combinations of E_d and V_p. As depicted in Fig 5, variations in the E_h for a specific location of the slope base accordingly result in the disparity in its initial phase. Consider slice i for example, its projection point in the slope base is S. The respective E_h of point S is E_h1, E_h2, and E_h3 in Fig 5. Given the same V_p, the initial phase of S varies accordingly. In addition, the initial phases at adjacent points S₁ and S₂ correlate to that at point S. This correlation weakens as the distance between point S and the focus point, see point Q, for example increases. It is intuitively derived from Fig 5 that the correlation in initial phase at point S and Q is sorted in descending order as the distance between S and epicenter increases.

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Fig 5. Diagram of influence of E_h on initial phase.

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Apart from E_h, E_d which ranges from several hundred meters to tens of kilometers exhibits significant influence on the initial phase [32]. Fig 6 demonstrates the variation trend of initial phase at point S and the correlation trend of initial phases between point S and its adjacent points S₁ and S₂, and Q as well under the influence of E_d. Refer to Fig 6, for the same E_h of point S, the E_d is E_d1, E_d2, E_d3, respectively. Given the same V_p, the initial phase of S varies accordingly. It is noticed from Fig 7 that for the same E_h and E_d of point S, the respective V_p is V_p1, V_p2, V_p3 in Fig 7. Given the same E_h and E_d, the initial phase of S varies accordingly. It can be concluded from Figs 5–7 that the initial phase of point S is different but highly correlated with those of adjacent points S₁ and S₂, and the correlations diminish as the relative distance between two points increases. The variation and correlation trends of initial phases coincide with those concepts of random field theory, where the soil properties at two different locations are different but correlated and the correlation decreases as the relative distance between two locations increases [33]. To properly simulate both different and correlated initial phases, the random field theory will be briefly reviewed in the next section.

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Fig 6. Diagram of influence of E_d on initial pahse.

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Fig 7. Diagram of influence of V_p on initial phase.

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3.2 Simulation of correlated initial phases using random field theory

The random field theory was originated by Vanmarcke [33] to simulate the spatial variability of soil properties in geotechnical engineering. Since the initiation of random field theory, it has been widely used to investigate the influence of spatial variability of soil properties on the performance of geotechnical structures, e.g., slope, foundation, and embankments [33–45]. The fundamentals of the random field theory are the autocorrelation function and the scale of fluctuation. Since the precise determination of the autocorrelation function is nontrivial, the previous studies [46–52] can be referred to for preliminary studies. As for the determination of the scale of fluctuation for the initial phase, in situ or laboratory model tests can be referred to, where the accelerations of at different locations on the slope base are to be measured.The determination of the autocorrelation function and scale of fluctuation is not given in this study and the influence of scale of fluctuation of the initial phase on seismic slope stability is focused on. Consider the stationary random field of initial phase for example, the exponential autocorrelation function is assumed for simplicity and the scale of fluctuation, denoted by λ, is selected to reflect the integrated influence of E_d, E_h,and V_p on the correlation of initial phases. The initial phase, denoted by P, is modeled by one dimensional random field spatially varying along the horizontal direction. The value of P at the same width is assumed to be fully correlated. The spatial variability with width is simulated by a homogeneous uniform random field with an exponentially decaying correlation structure.

To generate the correlated initial phases using the random field theory, the mid-point method [39,53,54] is adopted to discretize the slope base domain. As shown in Fig 8, the W-m-wide slope base is divided into n 1-m-wide sub-bases. The initial phase within each sub-base is represented by an entry in a P vector with a length of W. P is assumed to be a uniformly distributed random variable within lower limit I_l and upper limit I_u. Let P(w_i) be the initial phase at width w_i and be the inverse cumulative distribution function. The correlation η_ij between and at respective width w_i and w_j can be given by:

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Fig 8. Discretization of random field variable P.

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(11)

where λ is the scale of fluctuation. when , and are effectively uncorrected [34]. In contrast, when , and are highly correlated. In this example, the value of λ varies from 4 m to infinity for consideration of different spatial correlations.

Consider, for example, covariance matrix decomposition method is adopted to generate the horizontal uniform random field of P, denoted by .

(12)

where = a standard Gaussian vector with n independent components; = a n-by-n lower-triangular matrix determined from Cholesky decomposition of the correlation matrix = .

4. The proposed methodology

The proposed methodology, which combines PDBM with correlated initial phase R(P) to assess the seismic stability of soil slopes, is called MPDBM. Fig 9 depicts the flowchart of the proposed methodology. It is seen from Fig 9 that MPDMB consists of four parts. That is, Part 1:Search the critical sliding surface; Part 2: Determine the seismic parameters; Part 3: Generation of R(P); Part 4: Evaluate the seismic slope stability using MPDBM.

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Fig 9. Flowchart of MPDBM.

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In Part 1, the slope model is established in Geostudio and the critical sliding surface with minimum FS is searched, based on which the seismic slope stability is evaluated. Part 2 involves the selection of seismic parameters such as the angular frequency of seismic wave ω, the shear wave velocity in the slope soils Vs, the damping ratio D and scale of fluctuation in the random field of R(P). Part 3 is the generation of R(P), based on which a series of seismic FS can be calculated using the MPDBM. In Part 4, the minimum seismic FS is selected from the series of FS determined in Part 3 to evaluate the seismic stability of slopes.

5. Validation of MPDBM

Before the proposed MPDBM can be extended to case studies, it should be validated against the results from the previous research [16]. The maximum active thrust acting on the rigid retaining wall and the corresponding distribution of horizontal seismic acceleration were investigated using the MPDM in [16]. Since the distribution of horizontal seismic acceleration is dependent only on ω, Vs, H, and D, the slope stability problem is adopted to investigate the distribution of horizontal seismic acceleration leading to minimum FS, which are comparable with the results from [16].

As shown in Fig 10, a homogeneous slope with slope height of 12m and slope angle of 30.96° is considered. The unit weight of the slope soil is 18kN/m³. The respective cohesion and frictional angle are 8 kPa and 20°. As a preliminary approach to evaluate the seismic slope stability, the slope stability without considering seismic forces is encouraged. In this paper, the Bishop method is selected in SLOPE/W to calculate the FS for the specific circular sliding surface. The circular sliding surface with minimum FS is defined as the critical sliding surface. The minimum FS without seismic forces is determined to be 1.104 and the critical sliding surface is plotted in Fig 10 in red line. The proposed MPDBM is used to calculate the minimum seismic FS of the critical sliding surface and the corresponding distribution of seismic horizontal acceleration. To simplify the comparison, a specific vertical slice with a width of b_h, denoted by slice_h in Fig 10 is focused on to study the distribution of horizontal seismic acceleration with the normalized depth. To ensure identical comparison with the result from [16], the same parameters are adopted.That is, Nf = 2, and D = 0.1, k_h = 0.2. The normalized frequency(Nf) is defined as:

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Fig 10. The slope model and the specific vertical slice.

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(13)

The 100 realizations of R(P) are generated following the descriptions in Section 3 under a set of scale of fluctuations(λ = 50, 500 and +∞m). Refer to Section 3, the slope base is 50m wide and it is discretized into 50 1-m-wide sub-bases. Each of 100 realizations of R(P) is regarded as the input for the initial phases of 50 sub-bases. The corresponding FS is calculated using Eq.(10). As a result, 100 FSs are obtained and the minimum one is found to be 0.701,0.689 and 0.683 under λ = 50, 500 and +∞m, respectively. The realization of R(P) leading to the minimum FS is plotted versus the normalized depth. Fig 11 summarizes the distributions of horizontal seismic accelerations (a_h) under λ = 50, 500, and +∞m, respectively and the results form the Bellezza (2014) as well. As shown in Fig 11, the distribution of a_h obtained at λ=+∞m (i.e., the initial phases are fully correlated as in MPDM) agrees well with that from [16]. The consistent comparison demonstrates the effectiveness of the proposed MPDBM. That is, the proposed MPDBM degenerates into MPDM when the spatial variability is not considered (λ=+∞m). As λ decreases,i.e., the spatial variability of initial phases turns to be significant, the distribution of horizontal seismic acceleration differs from that without considering spatial variability of initial phases.

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Fig 11. Distribution of horizontal acceleration versus normalized depth.

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6. Case studies

The proposed methodology is illustrated through the homogeneous soil slope shown in Fig 10. The critical sliding surface depicted in red line in Fig 10 is based on to evaluate the seismic stability of the slope. How the variation of critical sliding surface influences the seismic stability will be discussed in Section 6.5. For seismic parameters, the k_h = 0.1 and D = 0.2 are specified.

6.1 Determine the minimum FS in MPDM using uniform sampling approach

Following the previous research outputs of MPDM [16,27], The normalized frequency is reckoned as a crucial factor. Bellezza [16] has pointed out that Nf = 0.5π, where the slope soil is subjected to its natural frequency, leads to maximum seismic forces and minimum FS. Hence, at Nf = 0.5π, the variations of FS(t) with initial phase t is demonstrated and an equivalent uniform sampling approach, which is easily implemented into MPDBM, is described and validated. Consider ω = 2π (i.e.,T = 1s), the initial phase t in Eq.(10) is assumed to be a set of values ranging between 0 and T with equal increment ∆T. Equation (10) is used to calculate the FS(t) under Nf = 0.5π, 1.5π, and 2.0π, respectively.

The respective variations of FS(t) with t under Nf = 0.5π, 1.5π, and 2.0π are plotted in Fig 12. For the case of Nf = 0.5π, the FS(t) decreases from 0.91 to a minimum of 0.71 as t varies from 0 to 0.19, and it increases to to a maximum of 2.18 as t ranges from 0.19 to.0.69, and finally it decreases to the original value of 0.91(t = 0) as t increases from 0.69 to 1.0. This variation trend indicates that the FS(t) is a periodical function with a period of T = 1s. The minimum FS value 0.71 is adopted for assessing the slope stability. Similar variation trends have been noticed for the cases of Nf = 1.5π, and 1.5π. It is clearly noticed that the respective minimum FS values of FS(t) are 0.71, 1.02 and 1.03 with t = 0.19s, 0.29s and 0.46s under Nf = 0.5π, 1.5π, and 2.0π. It is demonstrated that the minimum FS value and the corresponding initial phase t are dependent on Nf. The determination of minimum FS value is of importance. An mathematical method, where the derivative of active thrust to initial phase t is adopted, is presented in [16]. However, in this study, refer to Eq.(10), the derivative of FS to initial phase t is not easily dealt with as compared to that in [16]. Therefore, a set of tentative initial phase t must be tried and compared to find the minimum FS value as does in Fig 12. To facilitate the implementation of MPDBM, an alternative approach, named uniform sampling approach, is proposed in this study to find the minimum FS value and the corresponding initial phase t. The initial phase t is assumed to be a uniformly distributed random variable within the range between 0 and T. A finite number of random samples are generated within the range between 0 and T, consider N_u random samples for example, each of N_u samples is substituted into Eq.(10) and a FS can be obtained. Finally, a total of N_u FS values are available. The minimum one among the N_u FS values is the minimum FS value. Fig 13 demonstrates the variation of FS_min with different N_u values at different λ under Nf = 0.5π.

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Fig 12. Variation of FS(t) with t.

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Fig 13. Variation of FS_min with Nu for different λs at Nf = 0.5π.

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Consider λ = 4m as an example, the respective FS_min is 1.05, 1.07, 0.84, 0.87, 0.86, 0.80, 0.86, 0.81, 0.85, 0.84, 0.82, 0.85, 0.79, 0.79, 0.79 and 0.79 for N_u = 3, 5, 10, 20, 30, 40, 50, 80, 100, 150, 200, 300, 500, 1000, 2000 and 5000. It is noticed that once N_u is greater than 500, the FS_min remains unchanged at 0.79. As λ increases, e.g., λ = 20m, the respective FS_min is 1.01, 0.89, 0.81, 0.77, 0.87, 0.75, 0.79, 0.75, 0.76, 0.74, 0.74, 0.74, 0.74, 0.74, 0.74 and 0.74 for N_u = 3, 5, 10, 20, 30, 40, 50, 80, 100, 150, 200, 300, 500, 1000, 2000 and 5000. When N_u is greater than 200, the FS_min remains unchanged at 0.74. For the rest of λ cases, similar variation trends have been noticed. However, it is worthy to note that a larger λ relates to a smaller minimum N_u beyond which the FS_min obtained by uniform sampling approach is not varied. It implies that as the spatial variability of initial phases grows significant, more samples are required in uniform sampling approach, and vice versa. When it comes to an extreme case, i.e., λ=∞m, the minimum N_u is 35 and the final unfluctuated FS_min is 0.71, which coincides with that found in the traditional method shown in Fig 12. This identical comparison justifies the effectiveness of the uniform sampling approach. As a result, the uniform sampling approach is adopted in MPDBM.

6.2 Effect of λ on seismic slope stability

The proposed methodology is used to calculate the minimum FS value for a variety of λ values in order to investigate the effect of λ on seismic slope stability. A set of λ values, e.g., 4, 20, 50, 500, 1000,and +∞ are assumed. Starting with an initial value of 0.1π, Nf is increased by equal step of π/200 until it reaches 2π, leading to a total of 400 individual Nf values. For each combination of λ and Nf, following the description in Section 3.2, the-50-m-wide slope base is divided into n = 50 1-m-wide sub-bases, Equation(11) is used to establish the correlation matrix L, and finally Nu samples (i.e., Nu realizations of R(P)) are generated in accordance with Eq.(12). It is noted that I₁ = 0 and I_u = T in this study. Thereafter, the proposed uniform sampling approach is used to calculate the minimum FS value. The full combinations of six λ values and 400 Nf values yield 2,400 minimum FS values.

The results are summarized in Fig 14. For λ = ∞ m, the respective FS_min is 0.91, 0.71, 1.04, 1.02, and 1.03 under Nf = 0.1π, 0.5π, 1.0π, 1.5π, and 2.0π. Two local minima are observed. One is found at Nf = 0.5π and the other is at 1.5π, which is consistent with the observations from MPDM in [16,27]. The minimum FS_min is 0.71 at Nf = 0.5π. Quite similar variation trends of FS_min with Nf are noticed for the cases of λ = 4, 20, 50, 500, and 1000 m. However, the respective minimum FS_min are 0.79, 0.75, 0.74, 0.72 and 0.72 at Nf = 103π/200, 99π/200, 19π/40 and 99π/200 for λ = 4, 20, 50, 500 and 1000m. It is worth noting that the respective minimum FS_min is in the vicinity of 0.5π. To explore the relationship between λ and the minimum FS_min in detail, Fig 15 depicts the FS_min for different λ values at Nf = π/10, π/2, π, 3π/2, 2π. For example, at Nf = π/2, the respective FS_min is 0.79, 0.74, 0.72, 0.71, 0.71for λ = 4, 20, 50, 500, 1000 and ∞ m. It is observed that as λ increases from 4m to 500m, the FS_min decreases dramatically from 0.79 to 0.71, and it varies slightly around 0.71 as λ is greater than 500m. The difference between 0.79(λ = 4m) and 0.71(λ=+∞m) can be up to 11%, which significantly underestimates the seismic slope stability and tends to result in a conservative slope design if spatial variability of initial phase is not considered.

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Fig 14. Variations of FS_min with Nf at different λs (small scale slope).

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Fig 15. Variations of FS_min with λ at different Nfs (small scale slope).

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It is worth noting that the variation trend of FS_min with respect to λ is likely to differs from that at Nf = 0.5π. For example, as illustrated in Fig 15, at Nf = 3π/2, the respective FS_min is 1.02, 0.99, 0.99, 0.99, 1.00 and 1.02 for λ = 4, 20, 50, 500, 1000 and ∞ m. It is seen that the FS_min drops firstly and then it increases to a value close to the initial value. The minimum FS_min is observed at λ = 20m, which is clearly different from that observed at Nf = 0.5π(the minimum FS_min is at λ = +∞m). The results from the MPDBM without considering spatial variability of initial phases can be either conservative or unconservative depending on the Nf value.

6.3 Effect of D on seismic slope stability

D is assumed to be constant at 0.2 in the previous sections 6.2 and 6.3. To investigate the effect of D on the seismic slope stability, D = 0.1 is specified and the FS_min at each combination of three λ(4, 50, and +∞m) and 400 Nf values is calculated. The results from D = 0.2 are compared with those from D = 0.1. Fig 16 compares the variation trend of the FS_min with respect to Nf for D = 0.1 and 0.2 at λ = +∞ m. It is observed from Fig 16 that the variation trend of FS_min at D = 0.1 agrees fairly well with that at D = 0.2. It is interesting to note that the FS_min calculated at D = 0.1 is lower than that at D = 0.2 as Nf ranges between 0.1 and 0.685π. The most significant difference is found at Nf = 0.5π (the Nf leading to the first local minimum), where the difference in FS_min reaches 0.19 = 0.72-0.53.As Nf varies from 0.685π to 1.31π, the FS_min calculated at D = 0.1 is slightly greater than that at D = 0.2 with a maximum difference of 0.02 in FS_min at Nf = 1.145π. When Nf increases from 1.145π to 2.0π, the FS_min calculated at D = 0.1 is lower than that at D = 0.2 with a maximum difference of 0.05 = 1.02-0.97 in FS_min at Nf = 1.5π. The comparisons at λ = 50 m and λ = 4m are shown in Fig 17 and Fig 18, respectively. Very similar variation trends have been noticed at λ = 4 and 50 m. It is concluded from the comparisons that a smaller D tends to yield a lower FS_min especially at Nf values corresponding to the natural frequency of slope soil. It must be noted that a smaller D also leads to a slightly greater FS_min when Nf ranges between 0.5π and 1.5π in this study.

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Fig 16. Effect of D on FS_min at λ=∞m.

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Fig 17. Effect of D on FS_min at λ = 50m.

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Fig 18. Effect of D on FS_min at λ = 4m.

https://doi.org/10.1371/journal.pone.0330435.g018

6.4 Effect of λ on seismic stability of slope with different scale

Two slopes with different scales are considered in this section. The first one is shown in Fig 19. It is a medium scale homogeneous slope with slope height = 212m and slope angle = 22.08°.The cohesion c, internal friction angle φ, soil unit weight and damping ratio of soil are 18 kPa, 20°, 18kN/m³ and 0.2,respectively. The other one, a large scale slope, has a slope height = 800m and slope angle = 14.53°,which is shown in Fig 20. The cohesion c, internal friction angle φ, soil unit weight and damping ratio of soil are 20 kPa, 16°, 19kN/m³ and 0.2,respectively. Similarly, the Bishop method is used to calculate the FS for a given slip circular surface.The respective minimum FS without seismic forces is determined to be 1.10 and 1.21 and the corresponding critical sliding surfaces are plotted in red line in Figs 19 and 20 for the medium and large scale slope.

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Fig 19. Homogeneous soil slope(medium scale).

https://doi.org/10.1371/journal.pone.0330435.g019

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Fig 20. Homogeneous soil slope(large scale).

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Similar to the treatment in Section 6.2, 2400 combinations of λ and Nf are considered. The variation trends of FS_min with Nf and λ are shown in Figs 21–24 for two slopes with different scales. It can be noticed that the variation trends found in Figs 21 and 23 agree with those demonstrated in Fig 14. Fig 22 plots the variation of FS_min with λ under Nf = 0.5πfor the medium scale slope. It is seen that the FS_min drops from 0.91 to 0.60 as λ values increase from 4 to +∞m for medium scale slope. The difference between 0.91(λ = 4m) and 0.60(λ=+∞m) reaches 50%, which is significantly higher than 14% as compared to small scale slope. This comparison indicates a more pronounced effect of λ on medium scale slope.

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Fig 21. Variations of FS_min with Nf at different λs for medium scale slope.

https://doi.org/10.1371/journal.pone.0330435.g021

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Fig 22. Variations of FS_min with λ at Nf = 0.5π for medium scale slope.

https://doi.org/10.1371/journal.pone.0330435.g022

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Fig 23. Variations of FS_min with Nf at different λs for large scale slope.

https://doi.org/10.1371/journal.pone.0330435.g023

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Fig 24. Variations of FS_min with λ at Nf = 0.5π for large scale slope.

https://doi.org/10.1371/journal.pone.0330435.g024

Fig 24 plots the variation of FS_min with λ under Nf = 0.5π for the large scale slope. In the case of large scale slope, the FS_min drops from 1.08 to 0.65 as λ values increase from 4 to +∞m. The discrepancy between 1.08(λ = 4m) and 0.65(λ=+∞m) arrives at 67%, which is significantly higher than 11% (small scale slope) and slightly higher than 50%(medium scale slope). The effect of of λ on seismic slope stability grows significant as the slope scale increases. As a result, the proposed MPDBM is recommended for seismic stability analysis of slopes with medium and large scale slopes.

6.5 The influence of critical sliding surface

In the previous sections, the critical sliding surface without seismic forces is focused. To see how the varied critical sliding surfaces influence the seismic stability, the critical sliding surface with pseudo static forces (a_h = 0.1g), defined as pseudo static critical sliding surface is incorporated into the analysis. The locations of the two mentioned critical sliding surfaces are compared in Fig 25. The FS_mins for the pseudo static critical sliding surface at each combination of two λ(4 and +∞m) and 400 Nf values are calculated.

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Fig 25. Locations of different critical sliding surfaces of large scale slope.

https://doi.org/10.1371/journal.pone.0330435.g025

The results from critical sliding surface without seismic forces are compared with those from pseudo static critical sliding surface. Fig 26 compares the variation trends of the FS_min with respect to Nf at λ = +∞ m between critical sliding surface without seismic forces and pseudo static critical sliding surface. It is observed from Fig 26 that the variation trend of FS_min for pseudo static critical surface closely aligns with that for critical surface without seismic force. However, it is noted from Fig 27 that significant discrepancy in FS_min has been observed for each of Nfs at λ = 4 m. Specifically,when Nf = 0.5π, the FS_min from critical sliding surface without seismic forces is 1.088, whereas it decreases to 0.968 if the pseudo static critical sliding surface is used. Therefore, it must be noted that the critical sliding surface should be carefully selected. Based on the current comparisons, it is found that the choice of critical sliding surface tends to significantly influence the seismic stability at λ = 4 m.

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Fig 26. Variations of FS_min with Nf for large scale slope under different slip surfaces at λ = +∞.

https://doi.org/10.1371/journal.pone.0330435.g026

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Fig 27. Variations of FS_min with Nf for large scale slope under different slip surfaces at λ = 4m.

https://doi.org/10.1371/journal.pone.0330435.g027

7. Discussion

The proposed MPDBM emphasizes two issues in the use of pseudo dynamic method for assessing seismic slope stability. The first one is how to find the initial phase leading to the minimum FS for a given sliding surface. The other one is how to consider both the difference and the correlation in the initial phases. To address the first issue, a uniform sampling approach is originated and validated as an alternative to the traditional method, where a finite number of individual initial phases are tried. The advantage of the uniform sampling approach beyond the traditional method lies in its time efficiency, especially for low frequency seismic waves. It is interesting to note that the first issue has scarcely been dealt with. One exception is the research from [16], where a mathematical approach is adopted to directly determine the initial phase. Compared to the mathematical approach, the uniform sampling approach exhibits broader applicability because it does not require mandatory derivatives of geotechnical performance indicators (e.g., FS, displacement) with respect to the initial phase.

The second issue stems from the uncertainties and randomness in seismic motions. The seismic uncertainty is fairly complicated and only epicentral horizontal distance, epicentral depth, and the propagation velocity of the seismic wave in the slope base is briefly discussed to illustrate the existence of second issue. The initial phase behaves somewhat like the spatial variability of soil properties owing to the fact that both the difference and the correlation in the initial phase must be properly dealt with. Therefore, uniform random field of initial phase is adopted herein to simulate the spatial variability in the horizontal direction along the slope base. It is worth noting that the initial phase is assumed to be uniformly distributed for preliminary analysis. Besides, the uniform sampling approach and the uniform random field of initial phase coincide in the uniform essence of the initial phase.

For simplicity, the circular sliding surface and the simplified Bishop method are adopted to evaluate the seismic slope stability for simplicity and for the homogeneous soil slopes. However, the uniform sampling approach and the uniform random field can be easily combined with existing methods for assessing slope stability including but not limited to limit equilibrium methods, limit analysis, and strength reduction method. As a result, the proposed methodology is versatile although only homogeneous soil slopes are discussed in this study.

The proposed MPDBM is implemented within a commercial software package, SLOPE/W, and effects of the scale of fluctuation, damping ratio, and the slope scale on the seismic FS are investigated, providing a basis for further analysis of key findings.

8. Conclusions

The proposed MPDBM is developed to address the limitations of traditional pseudo dynamic methods in accounting for initial phase variability, and its application and key findings are summarized as follows:

(1). The proposed MPDBM is able to delineate the difference in seismic FS attributed to the spatial variability of initial phase and serve as an effective and alternative tool for conducting seismic slope stability using pseudo dynamic method.
(2). The traditional method ignoring the spatial variability of initial phase underestimates the FS for the cases where the slope soil is subjected to its natural frequency as compared to the proposed MPDBM with scale of fluctuation = 4m(i.e., the spatial variability of initial phase is significant). The underestimation turns to be profound for large scale slopes.
(3). For the cases, where the frequency of the seismic wave deviates from the natural frequency of slope soil, the consideration of the spatial variability of initial phase may yield smaller FS than the traditional method ignoring the spatial variability of initial phase.

Supporting information

S1 File. Supporting information mainly includes the data of figure.

https://doi.org/10.1371/journal.pone.0330435.s001

(XLSX)

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Figures

Abstract

1. Introduction

2. The pseudo dynamic Bishop method for seismic slope stability

2.1 Calculation of horizontal seismic force by MPDM

2.2 Pseudo Dynamic Bishop Method(PDBM) for slope stability

3. The simulation of both different and correlated initial phases

3.1 Influencing factors of both different and correlated initial phases

3.2 Simulation of correlated initial phases using random field theory

4. The proposed methodology

5. Validation of MPDBM

6. Case studies

6.1 Determine the minimum FS in MPDM using uniform sampling approach

6.2 Effect of λ on seismic slope stability

6.3 Effect of D on seismic slope stability

6.4 Effect of λ on seismic stability of slope with different scale

6.5 The influence of critical sliding surface

7. Discussion

8. Conclusions

Supporting information

S1 File. Supporting information mainly includes the data of figure.

References