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Reliability back analysis of landslide shear strength parameters based on a general nonlinear failure criterion
International Journal of Rock Mechanics & Mining Sciences 126 (2020) 104189
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International Journal of Rock Mechanics and Mining Sciences journal homepage: http://www.elsevier.com/locate/ijrmms
Reliability back analysis of landslide shear strength parameters based on a general nonlinear failure criterion Shi Zuo a, Lianheng Zhao a, b, *, Dongping Deng a, Zhibin Wang c, Zhigang Zhao d a
School of Civil Engineering, Central South University, Changsha, Hunan, 410075, China Key Laboratory of Heavy-haul Railway Engineering Structure, Ministry of Education, Central South University, Changsha, Hunan, 410075, China c School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China d China Railway Seven Engineering Group Co., Ltd, Zhengzhou, 450016, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: landslide Shear strength parameters Back analysis Nonlinear Mohr–Coulomb failure criterion Reliability
The shear strength parameters of a sliding surface are important parameters for the stability evaluation and engineering design of a landslide. To consider the randomness, uncertainty and the nonlinear characteristics of the geotechnical, the reliability back analysis method of determining shear strength parameters of a landslide based on nonlinear Mohr-Coulomb failure criteria is proposed in this study. By analyzing a slope with linear failure surface, the feasibility and validity of the proposed method are discussed in an engineering case. The proposed method provides a new choice and reference for the determination of the shear strength parameters of the sliding surface of a landslide.
1. Introduction Determining the shear strength parameters of a sliding surface is essential for the stability evaluation and engineering design of a land slide. Unreasonable values of shear strength parameters can cause some difficulties in design and regulation measure.1–3 In general, the method for determining shear strength parameters can be classified into the test method, engineering experience analogy method, and back analysis method.4,5 In comparison, the back analysis method can accurately consider factors affecting slope stability, and obtain shear strength pa rameters that concur with real situations,6–11 which has been received widely attention to evaluate the stability of a landslide. At present, the deterministic parameter back analysis method is still the main method used in engineering.4,5,8,12–16 However, adopting this deterministic method might be inaccurate due to the lack of the considerations of the variations in the geotechnical properties in time and space, and un certainties in the physical, model, and statistical results. To solve this problem, the reliability method is applied to the parameter back ana lysis.10,17–20 By using the mean parameter values of geotechnical ma terials as well as the covariance matrix of random variables, the variations and uncertainties are considered with the probability theory and mathematics statistics. Therefore, a more reasonable value that is fitted with the reality can be obtained.
Additionally, previous researches are based on linear failure crite rion of the geotechnical material. In fact, the nonlinear characteristics have been observed in many experiments of geotechnical medium, and the linear failure criterion is only a special case. Neglecting the nonlinear characteristics of the geotechnical material can affect slope stability analysis significantly.21–23 Hence, various nonlinear failure criteria were proposed to fit the curved failure envelopes of soils,21,24–28 and were widely use in the analysis the stability of slopes. However, there are limited investigations regarding parameter back analysis based on nonlinear failure criterion. In recent years, Cai,29 Sharifzadeh,30 Akın,1,11 and Kang31 back analyzed nonlinear strength parameters of a rock slope based on the nonlinear Hoek–Brown failure criterion. Huang32 back analyzed the strength parameters based on the linear Mohr–Coulomb failure criterion, the nonlinear Barton–Bandis failure criterion, and the nonlinear Hoek–Brown failure criterion, separately. These back analysis methods are based on the deterministic method, and no specific research has been performed on the parameter back analysis method based on the nonlinear Mohr–Coulomb failure criterion. Therefore, we herein propose a reliability back analysis method by combining the upper bound method of limit analysis and limit equilib rium method based on the nonlinear Mohr–Coulomb failure criterion. To illustrate the proposed method, an example of back analysis of a slope with linear failure surface is presented; subsequently, this method is
* Corresponding author. School of Civil Engineering, Central South University, Changsha, Hunan, 410075, China. E-mail address: [email protected] (L. Zhao). https://doi.org/10.1016/j.ijrmms.2019.104189 Received 11 April 2019; Received in revised form 17 August 2019; Accepted 20 December 2019 Available online 26 December 2019 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104189
applied to a slope with a broken line failure surface (a failure mechanism can be applied to any shape of the slope) to determine the nonlinear strength parameters. By combining with a specific engineering case, and comparing the back analysis results of the upper bound method of limit analysis and limit equilibrium method, we attempt to provide a more accurate and effective method for selection and reference to determine the values of the strength parameters.
ct ¼
� �1=ð1 m⋅σt ⋅tanϕt c0
∂τ c0 σn ¼ ⋅ 1þ ∂σ n mσ t σt
�ð1
mÞ
þ σ t ⋅tanϕt
mÞ=m
(4) (5)
The strength reduction technique is adopted in this section, and the reduced strength parameters are derived as follows: � cf ¼ ct =Fs (6) ϕf ¼ arctanðtanϕt =Fs Þ where the nonlinear strength parameters ct and ϕt can be acquired using equations (4) and (5), respectively. In addition, it can be observed in Fig. 1 that the strength values obtained by the tangent method under different stress states are different. This is where the nonlinear failure criterion is different from the linear failure criterion. In reality, the strength values are different under different stress states. That is why the nonlinear failure criterion has been proposed to better characterize the geotechnical characteris tics. In this paper, the stress state of different slices(σni) is calculated, and then the different strength values (cti, tanϕti) of each block are obtained by the tangent method.
(1)
where σ and τ are normal and shear stresses on the failure surface, respectively; c means the cohesion. ϕ means internal friction angle. Since many experimental results have indicated that the strength envelope of geotechnical materials is nonlinear, the power-law nonlinear failure criterion proposed by Zhang and Chen28 comprises a simple and practical expression, which can accurately present the nonlinear strength envelope of geotechnical materials. The power-law nonlinear failure criterion is presented as follows33–36:
2.1.1. Slope with linear failure surface As shown in Fig. 2, for a homogeneous slope with a linear failure surface, H is the slope height, α is the slope toe angle, β is the slope top angle, and the strength parameters of the slope geotechnical materials obey the nonlinear Mohr–Coulomb failure criterion. The slope becomes unstable with a linear failure surface under the action of gravity W, and an angle of θ exists between the failure surface and the horizontal plane. The slope body is considered a rigid wedge with velocity V, and the angle between the failure surface and the velocity is ϕt according to the associated flow rule. According to the virtual work equations and the strength reduction technique, the strength parameters are reduced by the same strength reduction factor Fs (cf ¼ ct =Fs ϕf ¼ arctanðtanϕt =Fs Þ), thus making the external work equal to the internal work. Subsequently, the implicit express of Fs can be expressed as follows: � �� � ct H cosϕf 1 2 γH ðcot θ cot αÞsin θ ϕf (7) Fs ¼ 2 sin θ
(2)
where c0, σt, and m are the nonlinear geotechnical parameters; c0 is the intercept of the curve with the vertical axis; σ t is the intercept of the curve with the horizontal axis, where the curve passes (0, c0) and ( σ t, 0); m is the nonlinear coefficient. When m 6¼ 1, equation (2) is the nonlinear Mohr–Coulomb failure criterion (Fig. 1); when m ¼ 1, equa tion (2) converts to the well-known linear Mohr–Coulomb failure cri terion.3 Slope stability analysis based on the nonlinear Mohr–Coulomb failure criterion. 2.1. Slope stability analysis by the upper bound method of limit analysis The basic principle is the virtual work equations when the upper bound method of the limit analysis is adopted to analyze the slope sta bility. Improving the yield strength of geotechnical materials does not reduce the limit load of the structure. Therefore, the tangent method is adopted when analyzing the slope stability by the upper bound method of limit analysis. By improving the geomaterial strength to acquire the upper bound solution, the tangential equation of equation (2) can be expressed as follows:
τ ¼ ct þ tanϕt ⋅σn
m
⋅ c0 ⋅
�
Currently, the linear Mohr–Coulomb failure criterion has been adopted widely in geotechnical engineering; it can be expressed as follows:
τ ¼ c0 ⋅ð1 þ σ n =σ t Þ1=m
1
tanϕt ¼
2. Nonlinear failure criterion
τ ¼ c þ σ ⋅tan ϕ
m
where γ is the unit weight. According to the upper bound method of limit analysis, the safety factor is the minimum of equation (7). To simplify the calculation, the problem to calculate the minimum of equation (7) is transformed into a nonlinear constrained optimization problem:
(3)
minF � s ¼ Fs ðθ; ϕt Þ 0 < ϕt < arctan½c0 =ðmσ0 Þ� s:t: β<θ<α
where tanϕt and ct are the gradient and intercept of the tangent line, respectively, as shown in Fig. 1. Further, ct and tanϕt can be expressed as follows36:
Fig. 1. Nonlinear Mohr–Coulomb failure criterion and its tangential line.
Fig. 2. Slope with linear failure surface. 2
(8)
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104189
2.1.2. Slope with broken line failure surface Fig. 3 shows a homogeneous slope with a broken failure surface. The slope body is divided into vertical slices. The number of slices are n. The strength parameters of the failure surface (c0, σt, m), and the strength parameters of the interface between different slices (c0’, σ t’, m’) obey the nonlinear Mohr–Coulomb failure. Based on the kinematically admissible velocity field, the velocity of each slice and each interface can be derived: � � � cos θm ϕ’tm ϕtðmþ1Þ 1 � V1 Vi ¼ Π im¼1 (9) cos θmþ1 ϕ’tm ϕtðmþ1Þ � Ui ¼ Π im¼1
� sinðθmþ1 θm Þ � U1 ’ cos θmþ1 ϕtm ϕtðmþ1Þ
Fig. 4. Shear failure slope.
(10)
2.2.1. Slope with linear failure surface As shown in Fig. 2, a homogeneous slope with a linear failure surface becomes unstable under the action of gravity W. L is the length of the failure surface. According to equation (13), the Fs can be expressed as follows:
Then the external work and the internal work can be calculated separately, and according to the virtual work equations and the strength reduction technique, equation (11) can be obtained: n X
Vi Wi sin θi i¼1
n 1 n X � X ϕfi ¼ c’fi cos ϕ’fi L’i Ui þ cfi cosϕfi Li Vi i¼1
(11)
RB
RB
i¼1
τf dl ¼ τdl A
Fs ¼ RAB
where cti and ϕti are the sliding surface strength parameters of the ith slice; Li is the length of the ith slice; c’ti and ϕ0 ti are the interface strength parameters of the ith interface; L’i is the length of the ith interfacecfi ¼
A
� � �1=m �1=m θ c0 1 þ σσnt dl c0 L 1 þ W Lcos σt ¼ RB W sin θ τdl A
(14)
2.2.2. Slope with broken failure surface When analyzing slope stability with a nonlinear failure surface based on the limit equilibrium method, the slice method is typically adopted, such as the Swedish, Bishop, or Janbu method. Through the hypothesis of the force between different slices, the safety factor of the slope can be calculated with the mechanical equilibrium condition. Equation (13) is applied to analyze the slope stability based on the limit equilibrium method. When calculating the driving shear force and the resistant shear force along the slip surface, the subsection integral method is to simplify the calculation with mathematical meaning. Therefore, it is not the strictly slice method, the slope body is considered as an integral. The accuracy of this method is verified by the subsequent analysis. As shown in Fig. 3, the driving shear force and the resistant shear force along the slip surface are calculated based on the subsection in tegral method:
cti =Fs c’fi ¼ c’ti =Fs ϕfi ¼ arctanðtanϕti =Fs Þ, ϕ’fi ¼ arctanðtan ϕiti =Fs Þ, Simi
larly, the problem to calculate Fs is transformed into a nonlinear con strained optimization problem as follows: � minFs ¼ Fs θi ; ϕti ; ϕ’ti 8 > > > 0 < ϕti < arctan½c0 =ðmσ0 Þ� < �� (12) � � s:t: 0 < ϕ’ti < arctan c’0 mσ’0 > > > : 0 < θi < α 2.2. Slope stability analysis by the limit equilibrium method When adopting the limit equilibrium method to analyze slope sta bility based on the nonlinear failure criterion, the mechanical equilib rium is considered. As shown in Fig. 4, the safety factor of a shear failure slope can be defined as the ratio of the resistant shear force to the driving shear force along the slip surface33: R τf dl Fs ¼ R (13) τdl
R An
n R P Ai
A
τf dl
A0
τdl
Fs ¼ R A0 n
Ai
τfi dl 1
Ai Ai
1
¼ i¼1 n R P i¼1
τi dl
n R P Ai
¼ i¼1
Ai
� �1=m � �1=m n P σni Wi cosθi 1 þ 1 þ c dl c L 0 0 i σt Li σ t 1 ¼ i¼1 P n R n P Ai τ dl Wi sinθi Ai 1 i i¼1
i¼1
(15)
3. The reliability back analysis method for nonlinear strength parameters
R where τ is the shear stress of the slip surface, and τdl is the driving shear force along the slip surface. τf can be calculated by equation (2) R when the nonlinear Mohr–Coulomb failure criterion is adopted, and τf dl is the resistant shear force along the slip surface.
The back analysis method is to calculate the shear strength param eters of slope by assuming that the slope is under limit state with the consideration of the stability states and sliding surface determined by investigation. It can be classified as the deterministic back analysis method and probability back analysis method19,20,37: 3.1. Deterministic back analysis method The traditional deterministic back analysis method aims to deter mine the shear strength parameters (c, ϕ) based on the linear Mohr–Coulomb failure criterion. Based on the landslide stable state, a corresponding safety factor (Fs) is given, typically equal to 1.0. The c-ϕ curves can be obtained with Fs as shown in Fig. 5. Generally, the parameter values can be determined by a single curve with one parameter given value first, or the intersection point of the curves is expected to be the back analysis values of the shear strength parameters when more than one curve is obtained. In a deterministic method, the
Fig. 3. Slope with broken line failure surface. 3
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objective. The values of parameters with minimum reliability index are expected to be the back analysis parameters values.42 In a probabilistic back analysis, multiple sets of parameters can be back calculated simultaneously; therefore, the problem of nonlinear strength parameter back analysis based on the nonlinear Mohr–Coulomb failure criterion can also be solved. The influence of uncertain factors can be considered by calculating the reliability index and failure probability when deter mining the parameters based on the probability back analysis method, which can reflect the actual status of the landslide comprehensively. The simplification process of probability back analysis method:
Fig. 5. Parameters back analysis by c – ϕ curves.
(1) To determine the basic information of the slope: stability state, failure criterion, parameter data, etc. (2) To calculate the factor of safety with appropriate analysis model and establish the reliability solution model. (3) To calculate the minimum reliability index and obtain the opti mized shear strength parameters.
form of the landslide and parameter values are thought to be certain. As for the nonlinear Mohr–Coulomb failure criterion, the failure envelope of the geotechnical materials is controlled by three parameters: c0, σt, and m. When the traditional deterministic back analysis method is to determine these three parameters, the c0 - σ t - m surface can be ob tained with a given Fs by changing the values of these three parameters. In theory, two parameters are assumed, and then, the other parameter value can be determined based on the c0 - σt - m surface. Other way is to obtain three c0 - σt - m surfaces by analyzing several landslide sections. The intersection area of these surfaces can be expected to be the back analysis parameter values.
4. Case study To verify the feasibility and validity of the proposed nonlinear strength parameter back analysis method, a slope with a linear failure surface is based on a theoretical model. The analysis was performed with the upper bound method of limit analysis and the limit equilibrium method. The proposed method was applied to an actual case to perform the back analysis, to discuss the practicality of the proposed method.
3.2. Probability back analysis method The reliability theory is used to perform the probability parameter back analysis in this study. Slope reliability refers to the probability of a slope maintaining stability. When performing reliability analysis, the function of the limit equilibrium state is required. For slope engineering, the safety factor is typically used for the assessment of slope stability. It is suggested that the slope is in a critical ultimate state when Fs is at a critical value. When Fs � 1, the slope fails; when Fs ¼ 1, the slope is in the limit state, Therefore, the function equation is as follows38: Z ¼ gðX1 ; X2 ; X3 ⋅ ⋅ ⋅ Xn Þ ¼ Fs
1
4.1. Slope with linear failure surface 4.1.1. Deterministic back analysis method Without an existing slope containing more than three landslide sections and failure with a linear surface, a theoretical model is applied to verify the feasibility of the nonlinear strength parameter back analysis method. It is assumed that a homogeneous slope failure with a linear surface, which abides by the nonlinear Mohr–Coulomb failure criterion. With H ¼ 20 m, α ¼ 75� , β ¼ 0� , θ ¼ 45� , γ ¼ 20 kN/m3 and the c0 - σt - m surface under this condition is as shown in Fig. 6: As shown in Fig. 6, the back analysis results based on the limit equilibrium method coincide well with the results based on the upper bound method of limit analysis, thereby verifying the accuracy of these two methods of slope stability analysis with a linear failure surface. In addition, the back analysis values of the nonlinear strength parameters c0, σ t, and m on the surface can be determined. The assumption is that
(16)
where X1 ; X2 ; X3 ⋅⋅⋅Xn are random variables. When Z < 0, structural failure occurs; when Z ¼ 0, the structure is in a limit equilibrium state; when Z > 0, the structure is reliable. Apart from the failure probability, the reliability index is another index typically used to represent the reliability of the structure. The matrix expression of the reliability index is39–41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ min ðX μÞT C 1 ðX μÞ (17) x2F
This form is called the Hasofer–Lind reliability index, and it can also be expressed as follows: sffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi T X i μi X i μi β ¼ min ½R� 1 (18) x2F
σi
σi
where X is the random variables vector, μ is the mean vector of the variables, σ is the standard deviation vector of the variables, C is the covariance matrix, R is the correlation matrix consisting of correlation coefficients ρ: [1 ρc, ϕ; ρϕ, c 1], and F is the failure domain. Compared to the covariance matrix C, the correlation matrix R is easier to obtained, and its physical meaning is clearer. Therefore, equation (18) is adopted to calculate the reliability based on the first-order reliability method (FORM).17,37,39,40 The parameters are random variables with a specified distribution, rather than fixed values in the process of parameters inversion based on the reliability theory that can consider the uncertainty and randomness of the parameters. When the failure model is established based on the slope limit state, a series of geotechnical parameters values can be ob tained by assuming the critical factor of safety as the back analysis
Fig. 6. c0 - σ t - m surface of slope with linear failure surface. 4
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two parameter values are known and the other parameter value can be obtained based on the c0 - σt - m surface. 4.1.2. Probability back analysis method From the analysis above, we found that the three nonlinear strength parameters c0, σt, and m cannot be determined directly with only one landslide section. Hence, probability back analysis method can be applied to solve this problem. As only one landslide section is required to back analyze the multiple parameters simultaneously, the influence of uncertain factors can be considered by calculating the reliability index and failure probability when determining the parameters based on the probability back analysis method, which can reflect the actual status of the landslide comprehensively. In probability back analysis method, it is important to determine the variable coefficient of the strength parameters. Chen et al.43 reported the approximate range of the variable coefficient of friction coefficient and cohesion by summarizing the relevant literature. The variable co efficient range of friction coefficient and cohesion are 0.02–0.37 and 0.10–0.47, respectively. Considering the lack of test data, when per forming the analysis based on the nonlinear failure criterion, c0, σ t, and m are assumed to fit with the normal distribution, and mean values of μm ¼ 1.5, μc0 ¼ 70 kPa, and μσt ¼ 200 kPa. From the literature,43 the var iable coefficients are assigned as COVm ¼ 0.1, COVc0 ¼ 0.1, and COVσt ¼ 0.15. The correlation between different parameters is not taken into account in caculation. The reliability back analysis results calculated by the FORM are listed in Table 1: As shown in Table 1, the reliability back analysis results based on the limit equilibrium method coincide well with the results based on the upper bound method of limit analysis, thus verifying the accuracy of these two methods of slope stability analysis with linear failure surface, and the accuracy of the reliability back analysis method. When looking at prior information, the parameter determination problem with only one landslide section can be successfully solved by using the reliability back analysis method, and can also avoid the un certainty of manual parameter selection. The uncertainty and random ness of geotechnical materials can be considered by adopting this method, in which more precise parameter values can be obtained.
Fig. 7. The slope divided with linear failure surface into five slices.
Fig. 8. c0 - σ t - m surface with broken failure surface.
4.2. Slope with broken failure surface
Table 2 Reliability back analysis results with broken failure surface.
4.2.1. Case verification To confirm the accuracy of the slope stability analysis method based on the nonlinear Mohr–Coulomb failure criterion, the slope with a linear failure surface adopted in the previous section is divided into five slices as shown in Fig. 7. The parameter back analysis method based on a slope with a broken failure surface is applied to this case. After performing the procedure shown in Fig. 7, the deterministic back analysis and reliability back analysis methods are used based on the slope stability analysis method with a broken failure surface. The back analysis results are shown in Fig. 8 and Table 2. Dividing the slope body into five slices and calculating the driving shear force and the resistant shear force along the slip surface with the subsection integral method is a simplified approach. Therefore, the back analysis results acquired with a broken failure surface and linear failure surface are different. However, the difference is small, thus verifying the accuracy of the parameter back analysis method based on a slope with a broken failure surface. The back analysis results can be used as a
Reliability index
c0
σt
m
Upper bound method of limit analysis Limit equilibrium method
1.451
60.18
208.79
1.53
1.451
60.18
208.79
1.53
Reliability index
c0
σt
m
Upper bound method of limit analysis Limit equilibrium method
1.426
60.35
208.46
1.53
1.432
60.31
208.55
1.53
reference. By comparing the back analysis results with the upper bound method of limit analysis and the limit equilibrium method, it is found that the difference between these two methods is small, thus proving that these two slope stability analysis methods with broken failure surface based on the nonlinear Mohr–Coulomb failure criterion are correct. 4.2.2. Xianjiagou landslide An engineering case (Xianjiagou landslide) is used in this study to discuss the nonlinear strength parameters back analysis method. The landslide is located in Lingyan village, Dujiangyan city, Sichuang province (Fig. 9). The middle and rear of the slope is extremely steep, and the bedrock has a large angle. The slope body can easily slide along the bedrock surface under the action of self-weight. Because it was summer at that time, rainfall increased. The precipitation infiltrated into the soil easily owing to ground cracks, thus further reducing the shear strength of the soil. With the increase of the weight of the soil, the Xianjiagou landslide occurred (as shown in Fig. 10).
Table 1 Reliability back analysis results of slope with linear failure surface. Stability analysis method
Stability analysis method
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nonlinear strength parameters back analysis method can be discussed. 4.2.2.1. Deterministic back analysis method. As for the Mohr–Coulomb failure criterion, two c-ϕ curves should be adopted with two different landslide sections, and the back analysis values can be determined by the intersection point of the two curves. For the nonlinear Mohr–Coulomb failure criterion, three parameters (c0, σt, and m) needs be known; in theory, three limit state equations are required. Therefore, three typical sections are selected for the nonlinear strength parameter back analysis (as shown in Fig. 11). The c0 - σt - m surfaces for different landslide sections are shown in Fig. 12: As shown in Figs. 12-a, 12-b, and 12-c, the back analysis results obtained by the limit equilibrium method are similar to the results ob tained by the upper bound method, and the results obtained by the limit equilibrium method are more conservative. These further prove that the slope stability analysis methods based on the upper bound method and the limit equilibrium method and the back analysis method are accurate and feasible. The computational efficiency is high with the nonlinear strength parameters based on the limit equilibrium method since more parameters are involved compared with the upper bound method. As shown in Figs. 12-d and 12-f, an intersection area of these three surfaces exists: m2[1.0–2.0], c02[5.0–35.0], and σ t2[0–200]. As these surfaces do not intersect at a point, the back analysis results cannot be determined directly from Fig. 12. However, the back analysis results can be expected to be part of the intersection area.
Fig. 9. Location of the landslide.
4.2.2.2. Probability back analysis method. From the analysis above, the back analysis results cannot be determined directly with the determin istic back analysis method. In addition, uncertainty and randomness exist in the geotechnical materials. Therefore, the reliability back analysis method is adopted. When considering the uncertainty and randomness of the geotechnical materials, only one landslide section is required to solve the back analysis problem, which is more accurate and effective than the deterministic back analysis method. Since the values of the nonlinear strength parameters that are not adopted in the existing survey, the necessary prior information for reliability back analysis is lacking, such as the distribution type, mean value, and standard deviation of the nonlinear strength parameters. However, the accurate prior information is important for the back analysis.19,37 Therefore, the intersection area obtained by the deter ministic back analysis method is treated as the optimal initial value of the reliability back analysis. Additionally, it indicates that when the upper bound method is used to analyze the slope stability, more un known quantities exist during the analysis. It is difficult to obtain the global optimal solution with the existing optimization method when the reliability back analysis method is adopted. Therefore, the reliability back analysis method based on the limit equilibrium method is adopted to determine the nonlinear strength parameters.
Fig. 10. Planar graph of the landslide.
Due to the lack of a method for acquiring the strength parameter values based on the nonlinear Mohr–Coulomb failure criterion in these exit exploration technologies, only the experimental values of tradi tional strength parameters (c, ϕ) are available in the reconnaissance report. The assumption in this paper is that the geotechnical materials of this slope obey the nonlinear Mohr–Coulomb failure criterion, the
Fig. 11. Different sections for Xianjiagou landslide. 6
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International Journal of Rock Mechanics and Mining Sciences 126 (2020) 104189
Fig. 12. c0 - σ t - m surfaces for different sections.
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According to the intersection area obtained by the deterministic back analysis method, c0, σt, and m are assumed to obey the normal distri bution with mean values of μm ¼ 1.5, μc0 ¼ 20 kPa, and μσt ¼ 100 kPa, respectively. Based on literature,43 the variable coefficients are given as COVm ¼ 0.1, COVc0 ¼ 0.1, and COVσt ¼ 0.15. The stability analysis for the slope with broken failure surface based on the limit equilibrium method is presented, and the nonlinear strength parameters reliability back analysis results for Sections A, B, and C of the Xianjiagou landslide are summarized in Table 3. According to the reliability back analysis results, we found that the back analysis results of different sections are similar. The mean values are m ¼ 1.228, c0 ¼ 22.414 kPa, and σt ¼ 74.416 kPa. Further, these mean values can be expected to be the reasonable parameter values, and can be used as a reference for the subsequent theoretical analysis and slope reinforcement design.
Table 3 Nonlinear strength parameters reliability back analysis results for Xianjiagou landslide. Sections
Reliability index
c0
σt
m
A B C Mean value
2.606 2.653 2.809 2.689
22.249 22.266 22.727 22.414
77.605 77.238 74.405 74.416
1.228 1.221 1.235 1.228
Table 4 Reliability back analysis results for Xianjiagou landslide in literature.44
4.2.2.3. Comparison with linear condition. Based on the linear Mohr–Coulomb failure criterion, the back analysis of the shear strength parameters for Sections A, B, and C of the Xianjiagou landslide has also been performed.44 The reliability back analysis results by the FORM are summarized in Table 4. The reliability back analysis results of different sections based on the linear Mohr–Coulomb failure criterion are similar. The mean values are c ¼ 30.4 kPa and ϕ ¼ 9.0� . Based on the mean values of the reliability back analysis results obtained from the linear and nonlinear Mohr–Coulomb failure criterion, the strength envelopes of the geotechnical materials are as shown in Fig. 13. The strength envelopes obtained from the back analysis results are different when different failure criterions are considered. Under different normal stresses, the curvature and intercept of the tangent line of the strength envelope based on the nonlinear Mohr–Coulomb failure criterion are different from those of the strength envelope based on the linear Mohr–Coulomb failure criterion. As shown in Fig. 13, when the geotechnical materials have been considered to be damaged according to the nonlinear Mohr–Coulomb failure criterion in the low normal stress section, the geotechnical materials are still considered as stable according to the linear Mohr–Coulomb failure criterion. However, the opposite applies in the high normal stress section. This difference will inevitably lead to different results in the theoretical analysis and slope reinforcement design. Therefore, the corresponding stress distribution along the slip surface must be considered in the geotechnical materials with obvious nonlinear characteristics. If the linear failure criterion is used in the calculation of any problems in this geotechnical material, it may cause unnecessary wastage and even potential safety hazard to the slope reinforcement.
Sections
Reliability index
c
ϕ
A B C Mean value
1.316 1.194 1.290 1.267
30.5 30.6 30.0 30.4
8.9 9.1 9.1 9.0
Fig. 13. Strength envelopes of different failure criteria.
deterministic and probability back analysis methods were presented and discussed; they could serve as reference for the theoretical analysis and slope reinforcement design 3) When certain prior information is ascertained, the parameter deter mination problem with only one landslide section could be solved well by adopting the reliability back analysis method, and could avoid the uncertainty of manual parameter selection. The uncer tainty and randomness of geotechnical materials could be considered by adopting this method to obtain more precise parameter values. 4) Choosing accurate failure criterion is crucial to theoretical analysis and slope reinforcement design. The methods to determine the most appropriate failure criteria, and to apply them to engineering prac tice accurately require further research.
5. Conclusions Based on the limit equilibrium method and the upper bound method of limit analysis, the stability analysis methods of slope with linear or broken failure surfaces were presented under the nonlinear Mohr–Coulomb failure criterion. Based on the reliability theory, the reliability back analysis method of nonlinear strength parameters was presented and discussed. The primary conclusions are as follows: 1) By adopting the deterministic and probability back analysis methods, the back analysis method for nonlinear strength parameters c0, σ t, and m were presented and discussed, which were important for determining the slope strength parameters. 2) By comparing the back analysis results with the upper bound method of limit analysis and the limit equilibrium method, and comparing the back analysis results acquired with broken and linear failure surfaces, the accuracy and feasibility of the nonlinear strength pa rameters back analysis method were verified. Using the Xianjiagou landslide as a case, the concrete implementation method of deter mining the nonlinear strength parameters by combining the
Acknowledgments This work was supported by the National Natural Science Foundation of China, China (No. 51978666, 51878668); the CSC scholarship, China (No. 201906370092); the Guizhou Provincial Department of Trans portation Foundation, China (No. 2018-123-040) and the Fundamental Research Funds for the Central Universities of Central South University, China (No. 2019zzts009). The financial supports are gratefully acknowledged.
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Appendix A. Supplementary data
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Reliability back analysis of landslide shear strength parameters based on a general nonlinear failure criterion Shi Zuo a, Lianheng Zhao a, b, *, Dongping Deng a, Zhibin Wang c, Zhigang Zhao d a
School of Civil Engineering, Central South University, Changsha, Hunan, 410075, China Key Laboratory of Heavy-haul Railway Engineering Structure, Ministry of Education, Central South University, Changsha, Hunan, 410075, China c School of Civil Engineering, Hunan University of Science and Technology, Xiangtan, 411201, Hunan, China d China Railway Seven Engineering Group Co., Ltd, Zhengzhou, 450016, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: landslide Shear strength parameters Back analysis Nonlinear Mohr–Coulomb failure criterion Reliability
The shear strength parameters of a sliding surface are important parameters for the stability evaluation and engineering design of a landslide. To consider the randomness, uncertainty and the nonlinear characteristics of the geotechnical, the reliability back analysis method of determining shear strength parameters of a landslide based on nonlinear Mohr-Coulomb failure criteria is proposed in this study. By analyzing a slope with linear failure surface, the feasibility and validity of the proposed method are discussed in an engineering case. The proposed method provides a new choice and reference for the determination of the shear strength parameters of the sliding surface of a landslide.
1. Introduction Determining the shear strength parameters of a sliding surface is essential for the stability evaluation and engineering design of a land slide. Unreasonable values of shear strength parameters can cause some difficulties in design and regulation measure.1–3 In general, the method for determining shear strength parameters can be classified into the test method, engineering experience analogy method, and back analysis method.4,5 In comparison, the back analysis method can accurately consider factors affecting slope stability, and obtain shear strength pa rameters that concur with real situations,6–11 which has been received widely attention to evaluate the stability of a landslide. At present, the deterministic parameter back analysis method is still the main method used in engineering.4,5,8,12–16 However, adopting this deterministic method might be inaccurate due to the lack of the considerations of the variations in the geotechnical properties in time and space, and un certainties in the physical, model, and statistical results. To solve this problem, the reliability method is applied to the parameter back ana lysis.10,17–20 By using the mean parameter values of geotechnical ma terials as well as the covariance matrix of random variables, the variations and uncertainties are considered with the probability theory and mathematics statistics. Therefore, a more reasonable value that is fitted with the reality can be obtained.
Additionally, previous researches are based on linear failure crite rion of the geotechnical material. In fact, the nonlinear characteristics have been observed in many experiments of geotechnical medium, and the linear failure criterion is only a special case. Neglecting the nonlinear characteristics of the geotechnical material can affect slope stability analysis significantly.21–23 Hence, various nonlinear failure criteria were proposed to fit the curved failure envelopes of soils,21,24–28 and were widely use in the analysis the stability of slopes. However, there are limited investigations regarding parameter back analysis based on nonlinear failure criterion. In recent years, Cai,29 Sharifzadeh,30 Akın,1,11 and Kang31 back analyzed nonlinear strength parameters of a rock slope based on the nonlinear Hoek–Brown failure criterion. Huang32 back analyzed the strength parameters based on the linear Mohr–Coulomb failure criterion, the nonlinear Barton–Bandis failure criterion, and the nonlinear Hoek–Brown failure criterion, separately. These back analysis methods are based on the deterministic method, and no specific research has been performed on the parameter back analysis method based on the nonlinear Mohr–Coulomb failure criterion. Therefore, we herein propose a reliability back analysis method by combining the upper bound method of limit analysis and limit equilib rium method based on the nonlinear Mohr–Coulomb failure criterion. To illustrate the proposed method, an example of back analysis of a slope with linear failure surface is presented; subsequently, this method is
* Corresponding author. School of Civil Engineering, Central South University, Changsha, Hunan, 410075, China. E-mail address: [email protected] (L. Zhao). https://doi.org/10.1016/j.ijrmms.2019.104189 Received 11 April 2019; Received in revised form 17 August 2019; Accepted 20 December 2019 Available online 26 December 2019 1365-1609/© 2019 Elsevier Ltd. All rights reserved.
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applied to a slope with a broken line failure surface (a failure mechanism can be applied to any shape of the slope) to determine the nonlinear strength parameters. By combining with a specific engineering case, and comparing the back analysis results of the upper bound method of limit analysis and limit equilibrium method, we attempt to provide a more accurate and effective method for selection and reference to determine the values of the strength parameters.
ct ¼
� �1=ð1 m⋅σt ⋅tanϕt c0
∂τ c0 σn ¼ ⋅ 1þ ∂σ n mσ t σt
�ð1
mÞ
þ σ t ⋅tanϕt
mÞ=m
(4) (5)
The strength reduction technique is adopted in this section, and the reduced strength parameters are derived as follows: � cf ¼ ct =Fs (6) ϕf ¼ arctanðtanϕt =Fs Þ where the nonlinear strength parameters ct and ϕt can be acquired using equations (4) and (5), respectively. In addition, it can be observed in Fig. 1 that the strength values obtained by the tangent method under different stress states are different. This is where the nonlinear failure criterion is different from the linear failure criterion. In reality, the strength values are different under different stress states. That is why the nonlinear failure criterion has been proposed to better characterize the geotechnical characteris tics. In this paper, the stress state of different slices(σni) is calculated, and then the different strength values (cti, tanϕti) of each block are obtained by the tangent method.
(1)
where σ and τ are normal and shear stresses on the failure surface, respectively; c means the cohesion. ϕ means internal friction angle. Since many experimental results have indicated that the strength envelope of geotechnical materials is nonlinear, the power-law nonlinear failure criterion proposed by Zhang and Chen28 comprises a simple and practical expression, which can accurately present the nonlinear strength envelope of geotechnical materials. The power-law nonlinear failure criterion is presented as follows33–36:
2.1.1. Slope with linear failure surface As shown in Fig. 2, for a homogeneous slope with a linear failure surface, H is the slope height, α is the slope toe angle, β is the slope top angle, and the strength parameters of the slope geotechnical materials obey the nonlinear Mohr–Coulomb failure criterion. The slope becomes unstable with a linear failure surface under the action of gravity W, and an angle of θ exists between the failure surface and the horizontal plane. The slope body is considered a rigid wedge with velocity V, and the angle between the failure surface and the velocity is ϕt according to the associated flow rule. According to the virtual work equations and the strength reduction technique, the strength parameters are reduced by the same strength reduction factor Fs (cf ¼ ct =Fs ϕf ¼ arctanðtanϕt =Fs Þ), thus making the external work equal to the internal work. Subsequently, the implicit express of Fs can be expressed as follows: � �� � ct H cosϕf 1 2 γH ðcot θ cot αÞsin θ ϕf (7) Fs ¼ 2 sin θ
(2)
where c0, σt, and m are the nonlinear geotechnical parameters; c0 is the intercept of the curve with the vertical axis; σ t is the intercept of the curve with the horizontal axis, where the curve passes (0, c0) and ( σ t, 0); m is the nonlinear coefficient. When m 6¼ 1, equation (2) is the nonlinear Mohr–Coulomb failure criterion (Fig. 1); when m ¼ 1, equa tion (2) converts to the well-known linear Mohr–Coulomb failure cri terion.3 Slope stability analysis based on the nonlinear Mohr–Coulomb failure criterion. 2.1. Slope stability analysis by the upper bound method of limit analysis The basic principle is the virtual work equations when the upper bound method of the limit analysis is adopted to analyze the slope sta bility. Improving the yield strength of geotechnical materials does not reduce the limit load of the structure. Therefore, the tangent method is adopted when analyzing the slope stability by the upper bound method of limit analysis. By improving the geomaterial strength to acquire the upper bound solution, the tangential equation of equation (2) can be expressed as follows:
τ ¼ ct þ tanϕt ⋅σn
m
⋅ c0 ⋅
�
Currently, the linear Mohr–Coulomb failure criterion has been adopted widely in geotechnical engineering; it can be expressed as follows:
τ ¼ c0 ⋅ð1 þ σ n =σ t Þ1=m
1
tanϕt ¼
2. Nonlinear failure criterion
τ ¼ c þ σ ⋅tan ϕ
m
where γ is the unit weight. According to the upper bound method of limit analysis, the safety factor is the minimum of equation (7). To simplify the calculation, the problem to calculate the minimum of equation (7) is transformed into a nonlinear constrained optimization problem:
(3)
minF � s ¼ Fs ðθ; ϕt Þ 0 < ϕt < arctan½c0 =ðmσ0 Þ� s:t: β<θ<α
where tanϕt and ct are the gradient and intercept of the tangent line, respectively, as shown in Fig. 1. Further, ct and tanϕt can be expressed as follows36:
Fig. 1. Nonlinear Mohr–Coulomb failure criterion and its tangential line.
Fig. 2. Slope with linear failure surface. 2
(8)
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2.1.2. Slope with broken line failure surface Fig. 3 shows a homogeneous slope with a broken failure surface. The slope body is divided into vertical slices. The number of slices are n. The strength parameters of the failure surface (c0, σt, m), and the strength parameters of the interface between different slices (c0’, σ t’, m’) obey the nonlinear Mohr–Coulomb failure. Based on the kinematically admissible velocity field, the velocity of each slice and each interface can be derived: � � � cos θm ϕ’tm ϕtðmþ1Þ 1 � V1 Vi ¼ Π im¼1 (9) cos θmþ1 ϕ’tm ϕtðmþ1Þ � Ui ¼ Π im¼1
� sinðθmþ1 θm Þ � U1 ’ cos θmþ1 ϕtm ϕtðmþ1Þ
Fig. 4. Shear failure slope.
(10)
2.2.1. Slope with linear failure surface As shown in Fig. 2, a homogeneous slope with a linear failure surface becomes unstable under the action of gravity W. L is the length of the failure surface. According to equation (13), the Fs can be expressed as follows:
Then the external work and the internal work can be calculated separately, and according to the virtual work equations and the strength reduction technique, equation (11) can be obtained: n X
Vi Wi sin θi i¼1
n 1 n X � X ϕfi ¼ c’fi cos ϕ’fi L’i Ui þ cfi cosϕfi Li Vi i¼1
(11)
RB
RB
i¼1
τf dl ¼ τdl A
Fs ¼ RAB
where cti and ϕti are the sliding surface strength parameters of the ith slice; Li is the length of the ith slice; c’ti and ϕ0 ti are the interface strength parameters of the ith interface; L’i is the length of the ith interfacecfi ¼
A
� � �1=m �1=m θ c0 1 þ σσnt dl c0 L 1 þ W Lcos σt ¼ RB W sin θ τdl A
(14)
2.2.2. Slope with broken failure surface When analyzing slope stability with a nonlinear failure surface based on the limit equilibrium method, the slice method is typically adopted, such as the Swedish, Bishop, or Janbu method. Through the hypothesis of the force between different slices, the safety factor of the slope can be calculated with the mechanical equilibrium condition. Equation (13) is applied to analyze the slope stability based on the limit equilibrium method. When calculating the driving shear force and the resistant shear force along the slip surface, the subsection integral method is to simplify the calculation with mathematical meaning. Therefore, it is not the strictly slice method, the slope body is considered as an integral. The accuracy of this method is verified by the subsequent analysis. As shown in Fig. 3, the driving shear force and the resistant shear force along the slip surface are calculated based on the subsection in tegral method:
cti =Fs c’fi ¼ c’ti =Fs ϕfi ¼ arctanðtanϕti =Fs Þ, ϕ’fi ¼ arctanðtan ϕiti =Fs Þ, Simi
larly, the problem to calculate Fs is transformed into a nonlinear con strained optimization problem as follows: � minFs ¼ Fs θi ; ϕti ; ϕ’ti 8 > > > 0 < ϕti < arctan½c0 =ðmσ0 Þ� < �� (12) � � s:t: 0 < ϕ’ti < arctan c’0 mσ’0 > > > : 0 < θi < α 2.2. Slope stability analysis by the limit equilibrium method When adopting the limit equilibrium method to analyze slope sta bility based on the nonlinear failure criterion, the mechanical equilib rium is considered. As shown in Fig. 4, the safety factor of a shear failure slope can be defined as the ratio of the resistant shear force to the driving shear force along the slip surface33: R τf dl Fs ¼ R (13) τdl
R An
n R P Ai
A
τf dl
A0
τdl
Fs ¼ R A0 n
Ai
τfi dl 1
Ai Ai
1
¼ i¼1 n R P i¼1
τi dl
n R P Ai
¼ i¼1
Ai
� �1=m � �1=m n P σni Wi cosθi 1 þ 1 þ c dl c L 0 0 i σt Li σ t 1 ¼ i¼1 P n R n P Ai τ dl Wi sinθi Ai 1 i i¼1
i¼1
(15)
3. The reliability back analysis method for nonlinear strength parameters
R where τ is the shear stress of the slip surface, and τdl is the driving shear force along the slip surface. τf can be calculated by equation (2) R when the nonlinear Mohr–Coulomb failure criterion is adopted, and τf dl is the resistant shear force along the slip surface.
The back analysis method is to calculate the shear strength param eters of slope by assuming that the slope is under limit state with the consideration of the stability states and sliding surface determined by investigation. It can be classified as the deterministic back analysis method and probability back analysis method19,20,37: 3.1. Deterministic back analysis method The traditional deterministic back analysis method aims to deter mine the shear strength parameters (c, ϕ) based on the linear Mohr–Coulomb failure criterion. Based on the landslide stable state, a corresponding safety factor (Fs) is given, typically equal to 1.0. The c-ϕ curves can be obtained with Fs as shown in Fig. 5. Generally, the parameter values can be determined by a single curve with one parameter given value first, or the intersection point of the curves is expected to be the back analysis values of the shear strength parameters when more than one curve is obtained. In a deterministic method, the
Fig. 3. Slope with broken line failure surface. 3
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objective. The values of parameters with minimum reliability index are expected to be the back analysis parameters values.42 In a probabilistic back analysis, multiple sets of parameters can be back calculated simultaneously; therefore, the problem of nonlinear strength parameter back analysis based on the nonlinear Mohr–Coulomb failure criterion can also be solved. The influence of uncertain factors can be considered by calculating the reliability index and failure probability when deter mining the parameters based on the probability back analysis method, which can reflect the actual status of the landslide comprehensively. The simplification process of probability back analysis method:
Fig. 5. Parameters back analysis by c – ϕ curves.
(1) To determine the basic information of the slope: stability state, failure criterion, parameter data, etc. (2) To calculate the factor of safety with appropriate analysis model and establish the reliability solution model. (3) To calculate the minimum reliability index and obtain the opti mized shear strength parameters.
form of the landslide and parameter values are thought to be certain. As for the nonlinear Mohr–Coulomb failure criterion, the failure envelope of the geotechnical materials is controlled by three parameters: c0, σt, and m. When the traditional deterministic back analysis method is to determine these three parameters, the c0 - σ t - m surface can be ob tained with a given Fs by changing the values of these three parameters. In theory, two parameters are assumed, and then, the other parameter value can be determined based on the c0 - σt - m surface. Other way is to obtain three c0 - σt - m surfaces by analyzing several landslide sections. The intersection area of these surfaces can be expected to be the back analysis parameter values.
4. Case study To verify the feasibility and validity of the proposed nonlinear strength parameter back analysis method, a slope with a linear failure surface is based on a theoretical model. The analysis was performed with the upper bound method of limit analysis and the limit equilibrium method. The proposed method was applied to an actual case to perform the back analysis, to discuss the practicality of the proposed method.
3.2. Probability back analysis method The reliability theory is used to perform the probability parameter back analysis in this study. Slope reliability refers to the probability of a slope maintaining stability. When performing reliability analysis, the function of the limit equilibrium state is required. For slope engineering, the safety factor is typically used for the assessment of slope stability. It is suggested that the slope is in a critical ultimate state when Fs is at a critical value. When Fs � 1, the slope fails; when Fs ¼ 1, the slope is in the limit state, Therefore, the function equation is as follows38: Z ¼ gðX1 ; X2 ; X3 ⋅ ⋅ ⋅ Xn Þ ¼ Fs
1
4.1. Slope with linear failure surface 4.1.1. Deterministic back analysis method Without an existing slope containing more than three landslide sections and failure with a linear surface, a theoretical model is applied to verify the feasibility of the nonlinear strength parameter back analysis method. It is assumed that a homogeneous slope failure with a linear surface, which abides by the nonlinear Mohr–Coulomb failure criterion. With H ¼ 20 m, α ¼ 75� , β ¼ 0� , θ ¼ 45� , γ ¼ 20 kN/m3 and the c0 - σt - m surface under this condition is as shown in Fig. 6: As shown in Fig. 6, the back analysis results based on the limit equilibrium method coincide well with the results based on the upper bound method of limit analysis, thereby verifying the accuracy of these two methods of slope stability analysis with a linear failure surface. In addition, the back analysis values of the nonlinear strength parameters c0, σ t, and m on the surface can be determined. The assumption is that
(16)
where X1 ; X2 ; X3 ⋅⋅⋅Xn are random variables. When Z < 0, structural failure occurs; when Z ¼ 0, the structure is in a limit equilibrium state; when Z > 0, the structure is reliable. Apart from the failure probability, the reliability index is another index typically used to represent the reliability of the structure. The matrix expression of the reliability index is39–41 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi β ¼ min ðX μÞT C 1 ðX μÞ (17) x2F
This form is called the Hasofer–Lind reliability index, and it can also be expressed as follows: sffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ffiffi T X i μi X i μi β ¼ min ½R� 1 (18) x2F
σi
σi
where X is the random variables vector, μ is the mean vector of the variables, σ is the standard deviation vector of the variables, C is the covariance matrix, R is the correlation matrix consisting of correlation coefficients ρ: [1 ρc, ϕ; ρϕ, c 1], and F is the failure domain. Compared to the covariance matrix C, the correlation matrix R is easier to obtained, and its physical meaning is clearer. Therefore, equation (18) is adopted to calculate the reliability based on the first-order reliability method (FORM).17,37,39,40 The parameters are random variables with a specified distribution, rather than fixed values in the process of parameters inversion based on the reliability theory that can consider the uncertainty and randomness of the parameters. When the failure model is established based on the slope limit state, a series of geotechnical parameters values can be ob tained by assuming the critical factor of safety as the back analysis
Fig. 6. c0 - σ t - m surface of slope with linear failure surface. 4
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two parameter values are known and the other parameter value can be obtained based on the c0 - σt - m surface. 4.1.2. Probability back analysis method From the analysis above, we found that the three nonlinear strength parameters c0, σt, and m cannot be determined directly with only one landslide section. Hence, probability back analysis method can be applied to solve this problem. As only one landslide section is required to back analyze the multiple parameters simultaneously, the influence of uncertain factors can be considered by calculating the reliability index and failure probability when determining the parameters based on the probability back analysis method, which can reflect the actual status of the landslide comprehensively. In probability back analysis method, it is important to determine the variable coefficient of the strength parameters. Chen et al.43 reported the approximate range of the variable coefficient of friction coefficient and cohesion by summarizing the relevant literature. The variable co efficient range of friction coefficient and cohesion are 0.02–0.37 and 0.10–0.47, respectively. Considering the lack of test data, when per forming the analysis based on the nonlinear failure criterion, c0, σ t, and m are assumed to fit with the normal distribution, and mean values of μm ¼ 1.5, μc0 ¼ 70 kPa, and μσt ¼ 200 kPa. From the literature,43 the var iable coefficients are assigned as COVm ¼ 0.1, COVc0 ¼ 0.1, and COVσt ¼ 0.15. The correlation between different parameters is not taken into account in caculation. The reliability back analysis results calculated by the FORM are listed in Table 1: As shown in Table 1, the reliability back analysis results based on the limit equilibrium method coincide well with the results based on the upper bound method of limit analysis, thus verifying the accuracy of these two methods of slope stability analysis with linear failure surface, and the accuracy of the reliability back analysis method. When looking at prior information, the parameter determination problem with only one landslide section can be successfully solved by using the reliability back analysis method, and can also avoid the un certainty of manual parameter selection. The uncertainty and random ness of geotechnical materials can be considered by adopting this method, in which more precise parameter values can be obtained.
Fig. 7. The slope divided with linear failure surface into five slices.
Fig. 8. c0 - σ t - m surface with broken failure surface.
4.2. Slope with broken failure surface
Table 2 Reliability back analysis results with broken failure surface.
4.2.1. Case verification To confirm the accuracy of the slope stability analysis method based on the nonlinear Mohr–Coulomb failure criterion, the slope with a linear failure surface adopted in the previous section is divided into five slices as shown in Fig. 7. The parameter back analysis method based on a slope with a broken failure surface is applied to this case. After performing the procedure shown in Fig. 7, the deterministic back analysis and reliability back analysis methods are used based on the slope stability analysis method with a broken failure surface. The back analysis results are shown in Fig. 8 and Table 2. Dividing the slope body into five slices and calculating the driving shear force and the resistant shear force along the slip surface with the subsection integral method is a simplified approach. Therefore, the back analysis results acquired with a broken failure surface and linear failure surface are different. However, the difference is small, thus verifying the accuracy of the parameter back analysis method based on a slope with a broken failure surface. The back analysis results can be used as a
Reliability index
c0
σt
m
Upper bound method of limit analysis Limit equilibrium method
1.451
60.18
208.79
1.53
1.451
60.18
208.79
1.53
Reliability index
c0
σt
m
Upper bound method of limit analysis Limit equilibrium method
1.426
60.35
208.46
1.53
1.432
60.31
208.55
1.53
reference. By comparing the back analysis results with the upper bound method of limit analysis and the limit equilibrium method, it is found that the difference between these two methods is small, thus proving that these two slope stability analysis methods with broken failure surface based on the nonlinear Mohr–Coulomb failure criterion are correct. 4.2.2. Xianjiagou landslide An engineering case (Xianjiagou landslide) is used in this study to discuss the nonlinear strength parameters back analysis method. The landslide is located in Lingyan village, Dujiangyan city, Sichuang province (Fig. 9). The middle and rear of the slope is extremely steep, and the bedrock has a large angle. The slope body can easily slide along the bedrock surface under the action of self-weight. Because it was summer at that time, rainfall increased. The precipitation infiltrated into the soil easily owing to ground cracks, thus further reducing the shear strength of the soil. With the increase of the weight of the soil, the Xianjiagou landslide occurred (as shown in Fig. 10).
Table 1 Reliability back analysis results of slope with linear failure surface. Stability analysis method
Stability analysis method
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nonlinear strength parameters back analysis method can be discussed. 4.2.2.1. Deterministic back analysis method. As for the Mohr–Coulomb failure criterion, two c-ϕ curves should be adopted with two different landslide sections, and the back analysis values can be determined by the intersection point of the two curves. For the nonlinear Mohr–Coulomb failure criterion, three parameters (c0, σt, and m) needs be known; in theory, three limit state equations are required. Therefore, three typical sections are selected for the nonlinear strength parameter back analysis (as shown in Fig. 11). The c0 - σt - m surfaces for different landslide sections are shown in Fig. 12: As shown in Figs. 12-a, 12-b, and 12-c, the back analysis results obtained by the limit equilibrium method are similar to the results ob tained by the upper bound method, and the results obtained by the limit equilibrium method are more conservative. These further prove that the slope stability analysis methods based on the upper bound method and the limit equilibrium method and the back analysis method are accurate and feasible. The computational efficiency is high with the nonlinear strength parameters based on the limit equilibrium method since more parameters are involved compared with the upper bound method. As shown in Figs. 12-d and 12-f, an intersection area of these three surfaces exists: m2[1.0–2.0], c02[5.0–35.0], and σ t2[0–200]. As these surfaces do not intersect at a point, the back analysis results cannot be determined directly from Fig. 12. However, the back analysis results can be expected to be part of the intersection area.
Fig. 9. Location of the landslide.
4.2.2.2. Probability back analysis method. From the analysis above, the back analysis results cannot be determined directly with the determin istic back analysis method. In addition, uncertainty and randomness exist in the geotechnical materials. Therefore, the reliability back analysis method is adopted. When considering the uncertainty and randomness of the geotechnical materials, only one landslide section is required to solve the back analysis problem, which is more accurate and effective than the deterministic back analysis method. Since the values of the nonlinear strength parameters that are not adopted in the existing survey, the necessary prior information for reliability back analysis is lacking, such as the distribution type, mean value, and standard deviation of the nonlinear strength parameters. However, the accurate prior information is important for the back analysis.19,37 Therefore, the intersection area obtained by the deter ministic back analysis method is treated as the optimal initial value of the reliability back analysis. Additionally, it indicates that when the upper bound method is used to analyze the slope stability, more un known quantities exist during the analysis. It is difficult to obtain the global optimal solution with the existing optimization method when the reliability back analysis method is adopted. Therefore, the reliability back analysis method based on the limit equilibrium method is adopted to determine the nonlinear strength parameters.
Fig. 10. Planar graph of the landslide.
Due to the lack of a method for acquiring the strength parameter values based on the nonlinear Mohr–Coulomb failure criterion in these exit exploration technologies, only the experimental values of tradi tional strength parameters (c, ϕ) are available in the reconnaissance report. The assumption in this paper is that the geotechnical materials of this slope obey the nonlinear Mohr–Coulomb failure criterion, the
Fig. 11. Different sections for Xianjiagou landslide. 6
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Fig. 12. c0 - σ t - m surfaces for different sections.
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According to the intersection area obtained by the deterministic back analysis method, c0, σt, and m are assumed to obey the normal distri bution with mean values of μm ¼ 1.5, μc0 ¼ 20 kPa, and μσt ¼ 100 kPa, respectively. Based on literature,43 the variable coefficients are given as COVm ¼ 0.1, COVc0 ¼ 0.1, and COVσt ¼ 0.15. The stability analysis for the slope with broken failure surface based on the limit equilibrium method is presented, and the nonlinear strength parameters reliability back analysis results for Sections A, B, and C of the Xianjiagou landslide are summarized in Table 3. According to the reliability back analysis results, we found that the back analysis results of different sections are similar. The mean values are m ¼ 1.228, c0 ¼ 22.414 kPa, and σt ¼ 74.416 kPa. Further, these mean values can be expected to be the reasonable parameter values, and can be used as a reference for the subsequent theoretical analysis and slope reinforcement design.
Table 3 Nonlinear strength parameters reliability back analysis results for Xianjiagou landslide. Sections
Reliability index
c0
σt
m
A B C Mean value
2.606 2.653 2.809 2.689
22.249 22.266 22.727 22.414
77.605 77.238 74.405 74.416
1.228 1.221 1.235 1.228
Table 4 Reliability back analysis results for Xianjiagou landslide in literature.44
4.2.2.3. Comparison with linear condition. Based on the linear Mohr–Coulomb failure criterion, the back analysis of the shear strength parameters for Sections A, B, and C of the Xianjiagou landslide has also been performed.44 The reliability back analysis results by the FORM are summarized in Table 4. The reliability back analysis results of different sections based on the linear Mohr–Coulomb failure criterion are similar. The mean values are c ¼ 30.4 kPa and ϕ ¼ 9.0� . Based on the mean values of the reliability back analysis results obtained from the linear and nonlinear Mohr–Coulomb failure criterion, the strength envelopes of the geotechnical materials are as shown in Fig. 13. The strength envelopes obtained from the back analysis results are different when different failure criterions are considered. Under different normal stresses, the curvature and intercept of the tangent line of the strength envelope based on the nonlinear Mohr–Coulomb failure criterion are different from those of the strength envelope based on the linear Mohr–Coulomb failure criterion. As shown in Fig. 13, when the geotechnical materials have been considered to be damaged according to the nonlinear Mohr–Coulomb failure criterion in the low normal stress section, the geotechnical materials are still considered as stable according to the linear Mohr–Coulomb failure criterion. However, the opposite applies in the high normal stress section. This difference will inevitably lead to different results in the theoretical analysis and slope reinforcement design. Therefore, the corresponding stress distribution along the slip surface must be considered in the geotechnical materials with obvious nonlinear characteristics. If the linear failure criterion is used in the calculation of any problems in this geotechnical material, it may cause unnecessary wastage and even potential safety hazard to the slope reinforcement.
Sections
Reliability index
c
ϕ
A B C Mean value
1.316 1.194 1.290 1.267
30.5 30.6 30.0 30.4
8.9 9.1 9.1 9.0
Fig. 13. Strength envelopes of different failure criteria.
deterministic and probability back analysis methods were presented and discussed; they could serve as reference for the theoretical analysis and slope reinforcement design 3) When certain prior information is ascertained, the parameter deter mination problem with only one landslide section could be solved well by adopting the reliability back analysis method, and could avoid the uncertainty of manual parameter selection. The uncer tainty and randomness of geotechnical materials could be considered by adopting this method to obtain more precise parameter values. 4) Choosing accurate failure criterion is crucial to theoretical analysis and slope reinforcement design. The methods to determine the most appropriate failure criteria, and to apply them to engineering prac tice accurately require further research.
5. Conclusions Based on the limit equilibrium method and the upper bound method of limit analysis, the stability analysis methods of slope with linear or broken failure surfaces were presented under the nonlinear Mohr–Coulomb failure criterion. Based on the reliability theory, the reliability back analysis method of nonlinear strength parameters was presented and discussed. The primary conclusions are as follows: 1) By adopting the deterministic and probability back analysis methods, the back analysis method for nonlinear strength parameters c0, σ t, and m were presented and discussed, which were important for determining the slope strength parameters. 2) By comparing the back analysis results with the upper bound method of limit analysis and the limit equilibrium method, and comparing the back analysis results acquired with broken and linear failure surfaces, the accuracy and feasibility of the nonlinear strength pa rameters back analysis method were verified. Using the Xianjiagou landslide as a case, the concrete implementation method of deter mining the nonlinear strength parameters by combining the
Acknowledgments This work was supported by the National Natural Science Foundation of China, China (No. 51978666, 51878668); the CSC scholarship, China (No. 201906370092); the Guizhou Provincial Department of Trans portation Foundation, China (No. 2018-123-040) and the Fundamental Research Funds for the Central Universities of Central South University, China (No. 2019zzts009). The financial supports are gratefully acknowledged.
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Appendix A. Supplementary data
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