Dynamic reliability analysis of slopes based on the probability density evolution method

Dynamic reliability analysis of slopes based on the probability density evolution method

Soil Dynamics and Earthquake Engineering 94 (2017) 1–6 Contents lists available at ScienceDirect Soil Dynamics and Earthquake Engineering journal ho...

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Soil Dynamics and Earthquake Engineering 94 (2017) 1–6

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn

Dynamic reliability analysis of slopes based on the probability density evolution method

MARK



Yu Huanga,b, , Min Xionga a b

Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China

A R T I C L E I N F O

A BS T RAC T

Keywords: PDEM Slope stability Reliability Stochastic earthquake excitation Time history analysis

Earthquake ground motions display random behavior; therefore, it is necessary to investigate the seismic response of engineering structures using stochastic analysis methods. The probability density evolution method (PDEM) is applied in stochastic seismic response analysis and dynamic reliability evaluation of slope stability. Combining PDEM with finite-element dynamic time-series analysis, this study analyses the stochastic seismic response of a slope under random earthquake ground motion. Comparison between the results of the Monte Carlo stochastic simulation and PDEM analysis demonstrates the effectiveness and high precision of the PDEM method. We assess the stability of the slope under earthquake conditions using the safety factor criterion to obtain the stability probability of the slope. Compared with traditional equilibrium methods, the reliability analysis can directly reflect the failure probability and degree of safety of the slope, demonstrating a novel approach to slope stability assessment using a random dynamic method.

1. Introduction In the past few decades, the random reliability theory has been used to address geotechnical engineering problems. Using the Newmark model, Mankelow and Murphy [1] obtained the mean and standard deviation of the Newmark displacement with a Gaussian distribution. To investigate the spatial variation of soil properties, the random finite element method (FEM) was incorporated into slope stability analysis as a probabilistic analysis tool [2,3]. Based on the low-discrepancy sequence Monte Carlo (MC) method, Shinoda et al. [4] calculated the limit state exceedance probability of typical earth dams and geosynthetic-reinforced soil slopes under earthquake loading. For slopes under seismic loading, several stochastic analysis approaches were developed for stability analysis and slope reliability evaluation. Different models were developed for seismic reliability assessment of earth slopes with short term stability [5]. Peng et al. [6] proposed a neural network method to consider the reliability of earth slopes. Using stochastic parameters such as the internal friction angle, cohesion, and soil unit weight, the MC method was applied to analyze an infinite slope subjected to pseudo-static earthquake loading [7]. Most of the studies on slope stability based on reliability methods only investigated the probability of earth slope failure triggered by earthquakes based on the Newmark model and static or pseudo-static analysis. However, there are two aspects that limit the effectiveness of



these methods [8]. First, most of the above methods do not take into account the characteristics of the seismic ground motion and the seismic wave amplification by the soil mass. Second, most of the methods do not account for the dynamic nonlinear behavior of soil under the seismic excitation. Therefore, it is necessary to investigate the seismic stability of slopes from the viewpoint of stochastic vibration using dynamic time-history analysis. The traditional MC method [9,10] significantly improves our understanding of stochastic vibration analysis in the field of slope engineering; however, engineers are reluctant to adopt it in seismic assessments of slope stability because of certain limitations [11]. Therefore, in this study we analyze the slope stability under stochastic earthquake excitation based on the probability density evolution method (PDEM) [12,13]. By applying deterministic dynamic timehistory analyses of seismic excitation using FEM combined with the PDEM, we obtain the probability density functions of the seismic responses of the slope under stochastic earthquake ground motion, and calculate the seismic dynamic reliability precisely. This paper proposes an efficient method for dynamic analysis of slopes subjected to recently developed PDEM. It is completely different from the traditional stochastic reliability analysis method, the PDEM can combine the stochastic dynamic analysis and currently various advanced deterministic slope stability analysis methods. According to the PDEM, the stochastic seismic response analysis of slope is

Corresponding author at: Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China. E-mail address: [email protected] (Y. Huang).

http://dx.doi.org/10.1016/j.soildyn.2016.11.011 Received 15 June 2016; Received in revised form 3 November 2016; Accepted 27 November 2016 0267-7261/ © 2016 Elsevier Ltd. All rights reserved.

Soil Dynamics and Earthquake Engineering 94 (2017) 1–6

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meanwhile, the stochastic earthquake ground motion can reflect uncertainties of intensity and frequency. With the bi-modulation function of intensity and frequency, the model also can respond the non-stationary characteristic of intensity and frequency. (2) For the determined θq , solve the dynamic equation (Eq. (1)) with the given earthquake excitations, and obtain the velocity of the seismic response. (3) Substitute the velocity into the PDEM equation and solve it by the finite difference method. (4) Calculate the sum of the results for q = 1, 2, …, npt , and obtain the required probability density function.

translated into serials of deterministic responses analyses. The time history dynamic method is used for the deterministic seismic calculation. It can reflect the characteristics of the seismic ground motion and the dynamic nonlinear behavior of soil under the seismic excitation are used in this paper for the seismic dynamic analysis of slope under the earthquake loading. The study addresses a new stochastic seismic dynamic method for the slope seismic stability analysis; the PDEM has been shown to be more than 14 times more efficient than traditional MC method. And this paper is the first succeeded in investigating the stability reliability of slope based on seismic dynamic time history. 2. The PDEM equation and dynamic reliability

The effectiveness and accuracy of this method for solving geotechnical engineering problems were verified by Huang et al. [10,13,14]. Finally, combining the data of an equivalent extreme event [14] with the PDEM solution, the seismic dynamic reliability can be obtained. For the PDEM, the extreme value of the each response of deterministic time history analysis makes up a virtual stochastic process, and which random process can be regarded as the input for the PDEM, therefore, solving the PDEM equation, the dynamic reliability is obtained. Here, should be emphasized that PDEM is a newly developed stochastic dynamic analysis method, it can be based on the existing deterministic methods (e.g. FEM, limit equilibrium method and so on) and which does not lead to the violation of objective physical law.

Without loss of generality, the dynamic balance equation of the soil slope subjected to earthquake motion can be expressed as follows [12]:

MÜ + CU̇ + f (U , U̇ ) = −MIxg̈ (Θ , t ); U̇ (t0 ) = x 0̇ , U (t0 ) = x 0 ,

(1)

where M and C are the mass and damping matrices, respectively and f is the nonlinear restoring force vector; Ü , U̇ , and U are the acceleration, velocity, and displacement vectors, respectively; I is the unit vector, ẍg is the earthquake ground motion process, and Θ is a random vector. The seismic response vector H is composed of the physical quantities which attract us being evaluated and can be described as

H = (H1, H2, …, Hm )T .

(2)

According to the probability conservation principle, the stochastic system is conservative and composed of (H (t ), Θ). Therefore, its joint probability pHΘ (h , θ, t ) satisfies the generalized probability density evolution equation [12,14] ∂pHΘ (h, θ, t ) ∂t

m

+

∑ Hj̇ (θ, t )

∂pHΘ (h, θ, t ) ∂hj

j =1

3. The stochastic ground motion According to the numerical steps for solving the PDEM equation, first we need to obtain the time series of the stochastic earthquake ground motion. The acceleration time series of the stochastic seismic ground motion are generated by the spectral representation of the ground motion and the stochastic function [15]. Based on the work of Cacciola and Deodatis [16], we introduce the generalized Clough and Penzien earthquake ground motion power spectrum density function model in the form of the following time evolution power spectrum [17]:

= 0. (3)

The initial condition of Eq. (3) is

pHΘ (h , θ, t0 ) = pΘ (θ, t ) δ (h − h 0 ),

(4)

where h 0 is the initial value of H (t ) and δ (⋅) is the Dirac function. The probability density function pH (H , t ) of H (t ) is given by

pH (h , t ) =

S Xg̈ (t , ω) = f (t ) 2 ⋅

∫ΩΘ pHΘ (h, θ, t )dθ.

(5)

(7)

In Eq. (3), the dimensions of the PDEM equation depend on the dimensions of the physical parameters being evaluated, and are unrelated to the dimensions of the primitive stochastic dynamic system. However, the dimensions of the classical density evolution equations cannot be smaller than those of the primitive stochastic dynamic system. Any physical parameters which attracted us (e.g. displacement and safety factor) can be chosen as the random variable in PDEM equation. For the slope stability analysis in this study, the selected physical parameter is the time series of the safety factor. The one-dimensional PDEM equation for the slope analysis can then be written as [10,14] ∂pFs Θ (Fs, θ, t ) ∂t

+ Fṡ (θ , t )

∂pFs Θ (Fs, θ, t ) ∂Fs

= 0,

ωg4 (t ) + 4ξg2 (t ) ωg2 (t ) ω2 ω4 ⋅ ⋅S0 (t ), 2 2 2 [ω − ωg2 (t )] + 4ξg2 (t ) ωg2 (t ) ω2 ⎡ω2 − ω2 (t ) ⎤ + 4ξ 2 (t ) ω2 (t ) ω2 ⎢ ⎥ f f f ⎣ ⎦

where f (t ) is the intensity modulation function, represented by d ⎡t ⎛ t ⎞⎤ f (t ) = ⎢ exp ⎜1 − ⎟ ⎥ . ⎝ c ⎠⎦ ⎣c

(8)

c = 7s is the arrival time of the peak ground acceleration and d = 2 is the parameter that controls the shape of f (t ) and effectively regulates the enhancement and attenuation of the stochastic earthquake process. In the generalized Clough and Penzien power spectrum model, the following parameters reflect the non-stationary characteristics of the earthquake ground motions [15,17]

(6)

ωg (t ) = ω0 − a

where Fs is the dynamic seismic time series of the safety factor of the slope, which can be calculated by the formulas presented in the following section. Eq. (3) can be solved by the following numerical method [10].

t T

ωf (t ) = 0.1ωg (t )

t ξg (t ) = ξ0 + b , T

ξ f (t ) = ξg (t ).

(9) (10)

In Eq. (9), the site parameters ωg (t ) and ξg (t ) are the dominant angular frequency and damping ratio, respectively, which change with time; the initial values of ωg (t ) and ξg (t ) are ω0 = 11s−1 and ξ0 = 0.85; a = 8s−1 and b = 0.15 are the variance ratios of the site parameters; ω0 , ξ0 , a and b can be determined according to site classification and the earthquake characteristics; T = 20s is the duration of the time series. In Eq. (10), ωf (t ), and ξ f (t ) are the corresponding filtering parameters. The site parameters and filtering parameters are linear functions of time; thus, ωf (t ) and ξ f (t ) are parameters that vary in time within a certain range.

(1) Select the representative discretized points θq(q = 1, 2, …, npt ) in the basic random variable space Θ and determine the corresponding probability. In this paper, the random source is the earthquake ground motion; therefore, the basic random variables relate to the earthquake ground motion parameters. In this paper, the randomness of the earthquake ground motion mainly embodies in peak acceleration value (PGA) and the frequency spectrum distribution. The randomness is characterized by the basic random variables; 2

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Y. Huang, M. Xiong

20

10 30

10

30

Fig. 2. Sketch of the soil slope (unit: m).

natural unit weight γ = 17.8kN/ m3, cohesion force c = 20kPa , internal friction angle φ = 25o , elastic modulus E = 20MPa , Poisson ratio μ = 0.3, dilatancy angle ψ = 28o [19]. Based on the above mentioned description, the PDEM can be in light of the existing deterministic methods [20,21], and the PDEM can translate the stochastic seismic investigation into a series of deterministic dynamic seismic analyses. Based on the seismic dynamic time history analysis method, the seismic responses of the slope for the deterministic dynamic analysis are obtained by the 2D FEM of the Geostudio 2007 software. For the left and right boundary conditions, the number of degrees of freedom in the x direction is fixed, and in the base of the slope the number of degrees of freedom in the x and y directions is fixed. The soil material is characterized using the Mohr-Coulomb model. According to previous studies of slope stability using finite element analyses [22,23], the safety factor of the slope can be defined as follows,

Fs =

∑ Sr , ∑ Sm

(12)

where Sr is the incremental resisting shear strength, Sm is the mobilized shear force, and Sr and Sm are the shear forces along the slip slice which can be calculated as

The time-evolution power spectrum also includes the parameter S0 (t ) which is the spectrum parameter that reflects the strength of the earthquake, expressed as [17] 2 amax ⎛ r 2πωg (t ) ⎜2ξg (t ) + ⎝

1 ⎞ ⎟ 2ξg (t ) ⎠

(13)

Sm = τn β .

(14)

In Eq. (13), τ is the shear strength at the center of the slip slice, β is the base length of the slice, c and φ are the cohesion forces of the slice and the internal friction angle, respectively. σn and τm are the normal stress and shear stress at the center of the slice, respectively, calculated as

Fig. 1. (a) Simulated time series of seismic ground motion. (b) Second-order numerical statistical results between simulated seismic time series and stochastic earthquake ground motion.

S0 (t ) =

Sr = τβ = (c + σn tan φ) β ,

σn =

σx + σy 2

+

(σx − σy )cos 2θ

τn = τxy cos 2θ −

, (11)

2

(σx − σy )sin 2θ 2

+ τxy cos 2θ ,

,

(15) (16)

where σx and σy are the stresses in the x and y directions at the center of the base, respectively; τxy is the shear stress at the center of the base; θ is the angle between the direction of the normal stress and the x axis positive direction. For the dynamic seismic time-series analysis, all the above stresses also change with time. Once these parameters are known, the safety factor can be calculated according to Eq. (12). Here need explanation is that the time history of factor of safety is calculated by the Bishop slice method [24] according to the Eq. (12) in at the each time step in the duration of the earthquake ground acceleration. For a certainty seismic acceleration time history, although the slice surface is determined, the stresses of the slide elements are not immutable and change with every time step. Therefore, the factor of the safety is also the time history and changes with time, the specific calculation of the factor of safety is carried out by the GeoStudio software. In order to assessment the stability under a series of the determined acceleration time history samples, the minimum value of each factor of safety time history is selected as the stability index. Although the high accuracy of the PDEM was demonstrated using a single degree of freedom oscillation system with stochastic excitation [13], here we calculate the dynamic seismic reliability of the slope under earthquake loading using the MC method and compare the slope simulation results of the PDEM and MC to evaluate the accuracy of the

220cm/ s 2

in which, a max = is the average value of the peak ground acceleration, determined from the seismic signal, r = 2.9 is the effective peak factor which is determined from the optimum fitting of the seismic response spectrum. All the above seismic ground motion parameters in the generalized Clough and Penzien evolution power spectrum model are selected according to the Chinese seismic code for Shanghai, China [18]. Based on the spectral representation and stochastic function method [17], the acceleration time series are generated with assigned probabilities which satisfy the same set system, and the probability summation of the set is equal to 1. In our model, the acceleration time series consisted of 610 samples (Fig. 1). The second-order statistics between the time series and the stochastic earthquake ground motion confirm the effectiveness and accuracy of the spectral representation and the simulated function of the stochastic seismic ground motion. 4. Numerical case The 10 m high soil slope model used in our simulation is shown in the Fig. 2. The main physical mechanical parameters of the soil are: 3

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PDEM calculation. In seismic response analysis, there is broad agreement that the MC simulation seems to be the only universal method with adequate precision for arbitrary-dimension nonlinear stochastic dynamic system [9,10]. In practice, the MC simulation method is selected as an accurate approach to check and verify the other numerical solution (e.g. PDEM) and approximate analytical results for the static and dynamic responses [25,26]. To enable a fair comparison of the computational efficiency between the MC method and PDEM, the calculation times of the two methods are compared. Performing the dynamic time-series analysis on the same computer, the MC stochastic simulation took 166.67 h while the PDEM required 12.17 h to calculate the probability data. The comparison of computation time demonstrates that the PDEM is much more efficient (about 14 times to MC) with the same precision level. Therefore, the PDEM has a significant computational efficiency with an acceptable degree of accuracy. This method can be directly applied in seismic design and assessment of practical slope engineering. The PDEM also makes the stochastic seismic dynamic response analysis based on dynamic time history procedure be realized in the field of slope engineering. After a series of deterministic dynamic seismic time-series analyses, we substitute the dynamic seismic safety factor of the time series into the PDEM equation as the response velocities, and solve the PDEM equation by the finite difference method. The stochastic responses and the dynamic reliability of the desired physical parameter can then be obtained. The stochastic seismic responses are illustrated in the Figs. 3 and 4. Fig. 3a shows the mean and standard deviation of the safety factor.

6 5

PDF

4

2 1 0 1.4 1.2

bilit

Mean

0.8

act

2.2

or

2.4

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PDEM MC 0

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10

12

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Time(

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PDEM MC 0.05

0

2.8

s)

2

0.9

1.11

Std.D

yF

2.6

(a)

1.12

Fig. 4. (a) Time evolution PDF surface of stability factor. (b) Contour plot of the PDF surface shown in (a). 0

2

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18

They are of the same order of magnitude, indicating that the safety factor has a significant variation characteristic. The comparison of the results of the MC method and PDEM of second-order statistics of the safety factor shows a very small relative error, which demonstrates the high precision of the PDEM for stochastic dynamic analysis based on dynamic time-series calculation. It should be noted that the PDEM computation time is much shorter than the MC stochastic simulation time. The variability of the safety factor can be clearly observed in the PDFs of the stability factor at different points in time (Fig. 3b). The shape of the instantaneous PDFs is complex and changes with time, illustrating the PDF's evolution with time. It is difficult to describe the PDFs using conventional regular distributions, such as normal distribution or logarithmic normal distribution. The PDFs show a double peak or multi-peak distribution which has an important impact on the dynamic reliability of the slope. Fig. 4 depicts the changes of the PDFs of the slope safety factor with time. The dynamic seismic response of the slope is extremely variable. The surface depicting the time evolution of the stability factor PDF appears as peaks and valleys, with the peaks reflecting the larger PDF values corresponding to higher probabilities. Fig. 4b shows the contour of the PDF surface from Fig. 4a; the safety factor is distributed mainly in the interval 0.9–1.2 within the time interval of 2–3 s. Thus, the contour of the PDF surface also indicates

20

Time(s) (a) 6 PDF at 2 sec PDF at 3 sec PDF at 10 sec

5

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PDF

3 1

Sta

1.13

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0 0.5

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Stability Factor (b) Fig. 3. (a) Mean and standard deviation of the stability factor. (b) Typical probability density functions (PDFs) of stability factor at three points in time.

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the earth slope is 0.9318. According to the seismic dynamic reliability obtained here, the soil slope will be stable under the condition of 7 degree seismic fortifications with relative intensity.

9 8

5. Conclusion

7

PDF

6

The stability of a slope under earthquake loading is a complex problem, especially considering the stochastic characteristic of earthquake ground motion; the stability factor is also random. Using a stochastic earthquake ground motion as the loading, the recently developed stochastic vibration method PDEM based on dynamic time-series analysis is introduced in this paper for dynamic seismic slope stability analysis. We present a numerical case study to evaluate the stability of a slope considering the dynamic reliability, and provide an accurate quantitative description of the stability of the slope. Compared with the MC method, the PDEM is a cost-effective, efficient, and feasible analysis method for seismic dynamic analyses of geotechnical engineering problems; our results demonstrate the benefits of analyzing geotechnical problems using stochastic methods. The methodology presented in this paper is a novel approach for accurate and reliable seismic modelling and evaluation of stochastic vibration and reliability that may be useful for geotechnical engineers.

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Acknowledgements

CDF

0.6 0.5

This work was supported by the National Science Fund for Distinguished Young Scholars of China (Grant No. 41625011).

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References

0.2

[1] Mankelow JM, MURPHY W. Using GIS in the probabilistic assessment of earthquake triggered landslide hazards. J Earthq Eng 1998;2(4):593–623. [2] Griffiths DV, Fenton GA. Probabilistic slope stability analysis by finite elements. J Geotech Geoenviron Eng 2004;130(5):507–18. [3] Griffiths DV, Fenton GA. The random finite element method (rfem) in slope stability analysis. probabilisticmethods in geotechnical engineering. New York, Vienna: Springer; 2007. [4] Shinoda M, Horii K, Yonezawa T, Tateyama M, Koseki J. Reliability-based seismic deformation analysis of reinforced soil slopes. Soils Found 2006;46(4):477–90. [5] Al-Homoud AS, Tahtamoni WW. Seismic reliability analysis of earth slopes under short term stability conditions. Geotech Geol Eng 2002;20(3):201–33. [6] Peng HS, Deng J, Gu DS. Earth slope reliability analysis under seismic loadings using neural network. J Cent South Univ 2005;12(5):606–10. [7] Johari A, Khodaparast AR. Analytical stochastic analysis of seismic stability of infinite slope. Soil Dyn Earthq Eng 2015;79:17–21. [8] Lin ML, Wang KL. Seismic slope behavior in a large-scale shaking table model test. Eng Geol 2006;86(2–3):118–33. [9] Shinozuka M. Monte Carlo solution of structural dynamics. Comput Struct 1972;2(5–6):855–74. [10] J. Li and J.B. Chen, The principle of preservation of probability and the generalized density evolution equation, Struct Saf 30 (1), 65-77 [11] El-Ramly H, Morgenstern NR, Cruden DM. Probabilistic slope stability analysis for practice. Can Geotech J 2002;39(3):665–83. [12] Li J, Liu ZJ, Chen JB. Orthogonal expansion of ground motion and PDEM-based seismic response analysis of nonlinear structures. Earthq Eng Eng Vib 2009;8(3):313–28. [13] Huang Y, Xiong M, Zhou HB. Ground seismic response analysis based on the probability density evolution method. Eng Geol 2015;198:30–9. [14] Li J, Chen JB. Stochastic dynamics of structures. Singapore: John Wiley & Sons; 2009. [15] Liu ZJ, Zeng B, Zhou YH, Tian B. Probabilistic model of ground motion processes and seismic dynamic reliability analysis of the gravity dam. Hydraul Eng J 2014;45(9):1066–74, [in Chinese]. [16] Cacciola P, Deodatis G. A method for generating fully non-stationary and spectrumcompatible ground motion vector processes. Soil Dyn Earthq Eng 2011;31(3):351–60. [17] Liu ZJ, Liu W, Peng YB. Random function based spectral representation of stationary and non-stationary stochastic processes. Probabilist Eng Mech 2016;45:115–26. [18] Code for Seismic Design of Buildings (GB50011-2010), 2010. [in Chinese]. [19] GEO-SLOPE International Ltd. Dynamic Modeling with QUAKE/W 2007. [20] Huang Y, Sawada K, Moriguchi S, Yashima A, Zhang F. Numerical assessment of the effect of reinforcement on the performance of reinforced soil dikes. Geotext Geomembranes 2006;24(3):169–74. [21] Huang Y, Zhang WJ, Xu Q, Xie P, Hao L. Run-out analysis of flow-like landslides triggered by the Ms 8.0 2008 Wenchuan earthquake using smoothed particle

0.1 0

1

1.05

1.1

1.15

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1.35

1.4

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Stability Factor (b)

Fig. 5. Stability factor PDF. (a) and cumulative distribution function (CDF) (b) of the extreme-value event.

the variability of the safety factor. Generally, the probability surface describes a stationary flow containing many peaks (Fig. 4b). Further research is needed to understand the processes and mechanism governing the changes in the probability densities of nonlinear stochastic dynamic systems; however, our results demonstrate the need to further investigate the stability of the slopes under earthquake loading. In order to obtain the seismic dynamic reliability of the stability factor of the soil slope, the cumulative distribution function (CDF) of the factor of safety can be calculated by integration of PDF. Therefore, the PDF of the minimum value should be obtained by the same above mentioned process of PDEM. Due to the minimum value of the safety factor is not the time history, the virtual stochastic process should be constructed in light the equivalent extreme event. Finally, by simulating an equivalent extreme event [10,14] of the stability factor and applying the PDEM, the dynamic reliability can be obtained. Fig. 5a and b show the PDF and the cumulative distribution function of the stability factor for an extreme event. It should be emphasized that the Fig. 5(b) is the PDF of the minimum value of the factor of safety, and the Fig. 5(b) is quite different from the Fig. 3(b) which shows the PDFs of the factor time history change and evolve with time. The Fig. 3(b) indicates that the PDFs with different time instants corresponding to probability distributions, and they are not always the Gaussian distribution. For the general soil slope engineering stability problem, the dynamic failure probability corresponding to a stability factor of Fs = 1.2 [27] is 0.0682; therefore, the seismic dynamic reliability of

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[25] Schuëller GI. Developments in stochastic structural mechanics. Arch Appl Mech 2006;75(10):755–73. [26] Schuëller GI. A state-of-the-art report on computational stochastic mechanics. Probabilist. Eng Mech 1997;12(4):197–321. [27] Design Specification for Slopes of Hydropower and Water Conservancy Project (DL/T 5353–2006), 2007[n Chinese].

hydrodynamics. Landslides 2012;9(2):275–83. [22] Liu HL, Fei K, Gao YF. Time history analysis method of slope seismic stability. Rock Soil Mech 2003;24:553–60, [in Chinese]. [23] Zienkiewicz OC, Humpheson C, Lewis RW. Associated and non-associated viscoplasticity and plasticity in soil mechanics. Geotechnique 1975;25(4):671–89. [24] Bishop AW. The use of the slip circle in the stability analysis of slopes. Geotechnique 1955;5(1):7–17.

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