# Interval Nonprobabilistic Reliability Analysis for Ancient Landslide considering Strain-Softening Behavior: A Case Study

Authors: Zilong Zhou; Chenglong Lin; Xin Cai; Riyan Lan
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

Uncertainties in geotechnical parameters significantly affect the stability evaluation of an ancient landslide, especially when considering the strain-softening behavior. Due to the great difficulty in obtaining the probability density distribution of geoparameters, an interval nonprobability reliability analysis framework combined with numerical strain-softening constitutive relations was established in this paper. Interval variables were defined as the uncertain parameters in the strain-softening model. The interval nonprobabilistic reliability was defined as the minimum distance from the origin point to the failure surface in the standard normal space, which is the key index for describing the ability of a system to tolerate the variation of uncertain parameters. The proposed method was used to evaluate the reliability of Baishi ancient landslide. The parameter sensitivity analysis was also conducted. Through the proposed method, it is considered that Baishi ancient landslide is safe and stable, and the strain thresholdkr is the dominant parameter. The results calculated by the proposed method agree well with the actual situation. This indicates the proposed method is more applicable than the traditional probability method when the data are scare.


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## Body

## 1. Introduction

Considerable uncertainties exist in landslide engineering [1–5]. Traditional deterministic analysis cannot account for the uncertainty explicitly in most cases [6, 7]. Overestimation or conservative estimation of stability is very common [8]. Therefore, the reliability analysis considering the effect of uncertainties should be proposed for exactly evaluating the landslide stability.The reliability method has been developed for different fields since 1930s [9]. The reliability of engineering structure is defined as the ability to perform the predetermined function in the specified time under the specified conditions. In landslide engineering, the basic steps of reliability analysis are as follows [9, 10]: (1) determining the input variables; (2) determining the performance function of the limiting state; (3) calculating the reliability index. Great achievements have been made on landslide reliability calculation. The uncertainties in geomaterial properties and subsurface stratigraphic and other aspects of landslide engineering were well considered [11–13]. However, the strain-softening behavior, as a common characteristic in geomaterial deformation, was usually neglected [14, 15]. It is necessary to consider the strain-softening behavior to evaluate the slope stability accurately [16–19].Limited studies can be found to analyze the landslide reliability considering the strain-softening behavior. Terzaghi and Peck [20] firstly took notice of the strain-softening behavior of soil. Bishop [21] proposed the concept of progressive failure of slopes. Skempton [22] defined the average residual factor R over a slip surface as the proportion of slip surface length over which the shear strength has reduced to a residual value. In a long time, residual factor R has been the most commonly used parameter to describe the strain-softening behavior in reliability analysis. Grivas and Chowdhury [23] firstly developed the probabilistic reliability analysis for strain-softening slopes. Stability factor Fs was obtained by the limit equilibrium method with residual factor R, and a simple probabilistic reliability analysis was conducted under “φ=0” assumption (ignoring the internal friction angle). Chowdhury et al. [24] considered the correlations between shear parameters. In recent years, Metya et al. [25] conducted a probabilistic reliability analysis based on the first -order reliability method; the performance function (Fs−1) based on the Bishop simplified method was adapted to take strain-softening into account in terms of residual factor R. Bhattacharya et al. [26] regraded residual factor R as a random variable, and the influence of R was comprehensively analyzed and compared under different probability density distributions. These probabilistic reliability analyses can well describe the uncertainties in the strain-softening slope. However, these methods require the probability density distribution of uncertain parameters, which significantly affects the results of reliability evaluations. Usually, it is hard to obtain the adequate data describing the strain-softening behavior of landslide. The independent normal or log-normal distribution assumption can also cause some errors in the strain-softening relation. For examples, strength parameters may be negative based on normal distribution assumption. There are always some possibilities that the residual strength even exceeds the peak strength due to the long-tail curve in the independent normal or log-normal distribution assumption (cr>cp, φr>φp, in which cr and cp are the peak and residual cohesion, respectively, and φr and φp are the peak and residual frictional angle respectively). In addition, the randomness of some uncertain parameters needs further discussion. Fortunately, the interval theory may provide a new strategy. This new idea was usually used in structure engineering [27–29]. Reliability was used to describe the ability of a system to tolerate the variation of uncertain parameters instead of failure probability. Interval values can better describe the uncertainties in stain-softening relations when the data are scare. The bounds of the interval value can also guarantee the correct relations between peak and residual strength (cp>cr, φp>φr). Therefore, interval nonprobability reliability method has great application prospect in landslide engineering [30–32].In this paper, an interval nonprobabilistic reliability analysis framework combined with numerical strain-softening constitutive relations was established. The proposed method requires only the boundary values instead of specific probability density distribution functions of uncertain parameters describing the strain-softening behavior, greatly decreasing the demanding for data. Nonprobabilistic reliability indexη instead of a deterministic safety factor Fs or traditional probability of failure was used to describe the stability. The method was used to verify the stability of Baishi ancient landslide. Results of the proposed method were compared with the results of the traditional probabilistic method and the in situ investigation to prove the applicability. The sensitivity analysis was also discussed to make a reference for similar engineering.


## 2. Methodology

### 2.1. Interval Variables in Strain-Softening Model

To consider the uncertainties as well as the strain-softening behavior, numerical strain-softening constitutive model is commonly used [33]. In the simplified constitutive model, the strain-softening behavior is characterized by five parameters: peak cohesion cp, peak friction angle φp, residual cohesion cr, residual friction angle φr, and the threshold parameter when strength reduces from peak to residual kr.The uncertainty of different parameters needs to be confirmed. As shown in Figure1, kr was a parameter describing the malleability of strain-softening materials [33]. When kr increases from 0 to ∞, materials change from brittleness to malleability, and the strength of geomaterials changes from residual to peak state. Considering the noticeable malleability effect in soil deformation, kr is considered as a variable in this paper. kr is also greatly associated with the residual factor R [22]; they are the cause and effect in negative correlation. Peak and residual shear strength (cp, φp, cr, and φr) are the common parameters with great uncertainties describing the inherent quality of geomaterials [15, 26]. Therefore, all these parameters in numerical strain-softening constitutive model (cp, φp, cr, φr, and kr) are confirmed as interval variables to consider the uncertainties. It is noted that interval variables are assumed mutually independent.Figure 1
(a) Shear stress-displacement curves under a certain vertical stress. (b) Numerical strain-softening constitutive model (Conte et al.) [33].
(a)(b)


### 2.2. Determination of Safety FactorFs

The response interval ofFs controlled by interval variables is obtained based on the numerical model in FLAC3D. Each response of Fs against each group of interval variables is calculated by the shear strength reduction (SSR) method, which is commonly used in landslide engineering [34, 35]. The SSR method can truly represent stress-strain relations in the progressive failure process [33], thus very suitable for landslide with strain-softening behavior. In this method, the safety factor is defined as the ratio of the actual shear strength to the reduced shear strength at failure (equations. (1) and (2)). Zhang et al. [36] extended this method to solve Fs of a homogeneous slope with a strain-softening behavior:(1)cf′=c′Fs,(2)φf′=arctantanφ′Fs,where Fs is the safety factor; cf′ and φf′ are the real strength parameters at failure; and c′ and φ′ are the original strength parameters.Interval variablescp, φp, cr, φr, and kr are the input parameters. Specially, the strength reduction process is only conducted in the slip zones [37]. The accumulations and the bedrock layer remain unchanged with the Mohr–coulomb model. In other words, slip zones are considered as the overriding potential slip surface. The reason is that failures usually occur in local weaker materials firstly in the ancient landslide.


### 2.3. Performance FunctionM

Performance functionM presenting the limit state of system is established based on Fs as(3)M=Fs−1=gcp,φp,cr,φr,kr.However, numerical method cannot give an explicit relationship betweenM and interval variables. In this paper, the explicit expression of M was obtained by response surface fitting [38] via a series of deterministic tests. Usually, the explicit expression needs to be assumed in advance [10, 39, 40]. Some preliminary treatment is made to optimize the performance function in this paper. According to the expression obtained by the limit equilibrium method [26], Fs can be written as(4)Fs=Rcr+γz∗cos2i∗tanφrγz∗sini∗cosi+1−Rcp+γz∗cos2i∗tanφpγz∗sini∗cosi.For simplification,Fs could be rewritten as(5)Fs=w1Fp+w2Fr,where R is the residual factor, w1=1−R, w2=R, Fr=cr+γz∗cos2i∗tanφr/γz∗sini∗cosi is the residual safety factor, and Fp=cp+γz∗cos2i∗tanφp/γz∗sini∗cosi is the peak safety factor.In equation (5), each subitem (w1,Fp,w2,Fr) was assumed as a linear or quadratic polynomial for convenience. As introduced in Section 2.1, when kr=0, materials are brittle, strength of whole slip surface are in residual state, and then w1=0, w2=1. Therefore, the assumed expressions of each subitems are written as(6)Fp=a1+a2cp+a3φp+a4φp2,(7)Fr=b1+b2cr+b3φr+b4φr2,(8)w1=c1kr+c2kr2,(9)w2=1−c1kr+c2kr2,where a1,a2,a3,a4,b1,b2,b3,b4,c1,c2 are the undetermined coefficients.Substituting equations (6)–(9) into equation (3), the performance function can be expressed as(10)M=c1kr+c2kr2a1+a2cp+a3φp+a4φp2+1−c1kr−c2kr2b1+b2cr+b3φr+b4φr2−1.After fitting a series of deterministic results ((cp, φp, cr, φr, kr), M), the explicit expression of M in equation (10) can be obtained.


### 2.4. Reliability Analysis Based on Interval Theory

Interval nonprobability reliability indexη, representing the ability of a system to tolerate the variation of uncertain parameters, is defined as the minimum distance between original point and failure surface in the standard normal space [27–29] (as shown in Figure 2).Figure 2
Schematic diagram of interval nonprobability reliability.For∀M=gx1,x2,…,xn, where M is the performance function, (x1,x2,…,xn) are the controlling interval variables, there are ∀xi∈ximin,ximax (i = 1, 2, …, n), where ximin and ximax are the lower and upper bounds of xi respectively. Then, the performance function M is also an interval M=Mmin,Mmax to response, where Mmin and Mmax are the lower and upper bounds of M, respectively. Mc and Mr are defined as the mean and radius of M, respectively (equations (11) and (12)). Then, the interval nonprobability reliability index η is defined as in equation (13):(11)Mc=12Mmin+Mmax,(12)Mr=12Mmax−Mmin,(13)η=McMr,where M=gx1,x2,…,xn=0 is the failure plane, and this hypersurface divides the space into the failure domain and the security domain. When M=gx1,x2,…,xn<0, the system is in failure state and the opposite it is in safe state. According to equations (11)–(13), if η>1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn>0, and the system is safe and reliable. If η<1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn < 0 and the system fails. If −1≤η≤1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn<0 or M=gx1,x2,…,xn>0 or M=gx1,x2,…,xn=0 are all possible. As a result, the system could be safe or in failure. And, the larger the value of η, the safer the system [32].


## 2.1. Interval Variables in Strain-Softening Model

To consider the uncertainties as well as the strain-softening behavior, numerical strain-softening constitutive model is commonly used [33]. In the simplified constitutive model, the strain-softening behavior is characterized by five parameters: peak cohesion cp, peak friction angle φp, residual cohesion cr, residual friction angle φr, and the threshold parameter when strength reduces from peak to residual kr.The uncertainty of different parameters needs to be confirmed. As shown in Figure1, kr was a parameter describing the malleability of strain-softening materials [33]. When kr increases from 0 to ∞, materials change from brittleness to malleability, and the strength of geomaterials changes from residual to peak state. Considering the noticeable malleability effect in soil deformation, kr is considered as a variable in this paper. kr is also greatly associated with the residual factor R [22]; they are the cause and effect in negative correlation. Peak and residual shear strength (cp, φp, cr, and φr) are the common parameters with great uncertainties describing the inherent quality of geomaterials [15, 26]. Therefore, all these parameters in numerical strain-softening constitutive model (cp, φp, cr, φr, and kr) are confirmed as interval variables to consider the uncertainties. It is noted that interval variables are assumed mutually independent.Figure 1
(a) Shear stress-displacement curves under a certain vertical stress. (b) Numerical strain-softening constitutive model (Conte et al.) [33].
(a)(b)


## 2.2. Determination of Safety FactorFs

The response interval ofFs controlled by interval variables is obtained based on the numerical model in FLAC3D. Each response of Fs against each group of interval variables is calculated by the shear strength reduction (SSR) method, which is commonly used in landslide engineering [34, 35]. The SSR method can truly represent stress-strain relations in the progressive failure process [33], thus very suitable for landslide with strain-softening behavior. In this method, the safety factor is defined as the ratio of the actual shear strength to the reduced shear strength at failure (equations. (1) and (2)). Zhang et al. [36] extended this method to solve Fs of a homogeneous slope with a strain-softening behavior:(1)cf′=c′Fs,(2)φf′=arctantanφ′Fs,where Fs is the safety factor; cf′ and φf′ are the real strength parameters at failure; and c′ and φ′ are the original strength parameters.Interval variablescp, φp, cr, φr, and kr are the input parameters. Specially, the strength reduction process is only conducted in the slip zones [37]. The accumulations and the bedrock layer remain unchanged with the Mohr–coulomb model. In other words, slip zones are considered as the overriding potential slip surface. The reason is that failures usually occur in local weaker materials firstly in the ancient landslide.


## 2.3. Performance FunctionM

Performance functionM presenting the limit state of system is established based on Fs as(3)M=Fs−1=gcp,φp,cr,φr,kr.However, numerical method cannot give an explicit relationship betweenM and interval variables. In this paper, the explicit expression of M was obtained by response surface fitting [38] via a series of deterministic tests. Usually, the explicit expression needs to be assumed in advance [10, 39, 40]. Some preliminary treatment is made to optimize the performance function in this paper. According to the expression obtained by the limit equilibrium method [26], Fs can be written as(4)Fs=Rcr+γz∗cos2i∗tanφrγz∗sini∗cosi+1−Rcp+γz∗cos2i∗tanφpγz∗sini∗cosi.For simplification,Fs could be rewritten as(5)Fs=w1Fp+w2Fr,where R is the residual factor, w1=1−R, w2=R, Fr=cr+γz∗cos2i∗tanφr/γz∗sini∗cosi is the residual safety factor, and Fp=cp+γz∗cos2i∗tanφp/γz∗sini∗cosi is the peak safety factor.In equation (5), each subitem (w1,Fp,w2,Fr) was assumed as a linear or quadratic polynomial for convenience. As introduced in Section 2.1, when kr=0, materials are brittle, strength of whole slip surface are in residual state, and then w1=0, w2=1. Therefore, the assumed expressions of each subitems are written as(6)Fp=a1+a2cp+a3φp+a4φp2,(7)Fr=b1+b2cr+b3φr+b4φr2,(8)w1=c1kr+c2kr2,(9)w2=1−c1kr+c2kr2,where a1,a2,a3,a4,b1,b2,b3,b4,c1,c2 are the undetermined coefficients.Substituting equations (6)–(9) into equation (3), the performance function can be expressed as(10)M=c1kr+c2kr2a1+a2cp+a3φp+a4φp2+1−c1kr−c2kr2b1+b2cr+b3φr+b4φr2−1.After fitting a series of deterministic results ((cp, φp, cr, φr, kr), M), the explicit expression of M in equation (10) can be obtained.


## 2.4. Reliability Analysis Based on Interval Theory

Interval nonprobability reliability indexη, representing the ability of a system to tolerate the variation of uncertain parameters, is defined as the minimum distance between original point and failure surface in the standard normal space [27–29] (as shown in Figure 2).Figure 2
Schematic diagram of interval nonprobability reliability.For∀M=gx1,x2,…,xn, where M is the performance function, (x1,x2,…,xn) are the controlling interval variables, there are ∀xi∈ximin,ximax (i = 1, 2, …, n), where ximin and ximax are the lower and upper bounds of xi respectively. Then, the performance function M is also an interval M=Mmin,Mmax to response, where Mmin and Mmax are the lower and upper bounds of M, respectively. Mc and Mr are defined as the mean and radius of M, respectively (equations (11) and (12)). Then, the interval nonprobability reliability index η is defined as in equation (13):(11)Mc=12Mmin+Mmax,(12)Mr=12Mmax−Mmin,(13)η=McMr,where M=gx1,x2,…,xn=0 is the failure plane, and this hypersurface divides the space into the failure domain and the security domain. When M=gx1,x2,…,xn<0, the system is in failure state and the opposite it is in safe state. According to equations (11)–(13), if η>1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn>0, and the system is safe and reliable. If η<1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn < 0 and the system fails. If −1≤η≤1, then ∀xi∈ximin,ximax (i = 1, 2, …, n), M=gx1,x2,…,xn<0 or M=gx1,x2,…,xn>0 or M=gx1,x2,…,xn=0 are all possible. As a result, the system could be safe or in failure. And, the larger the value of η, the safer the system [32].


## 3. Case Study: Baishi Ancient Landslide

### 3.1. Geological Background

To verify the applicability of our reliability analysis method, we took a strain-softening landslide, i.e., Baishi ancient landslide, as an example. Baishi ancient landslide is located at the south mountains area in Guangxi province, China (Figure3(a)). Geographic, geomorphic, elevation, and other information is shown in Figure 3(b). Geologically, Baishi landslide is an accumulated ancient landslide with obvious ancient slip bands. The geomaterial distribution could be described as three uneven layers: (1) argillaceous sandstone bedrock downmost, (2) slip bands with strain-softening behavior in-between, and (3) soil-gravel accumulations uppermost.Figure 3
(a) Location and (b) digital elevation model of Baishi ancient landslide.
(a)(b)


### 3.2. Parameter Collection

Some critical parameters used for simulation were collected from geological surveys or engineering experience as listed in Table1.Table 1
Physical and mechanical parameters of Baishi ancient landslide.
ItemAccumulationsBedrockSlip bandsConstitutive relationMohr–CoulombMohr–CoulombStrain-softeningE (GPa)1100.1v0.30.30.3ρ (kg·m3)196021302450c (kPa)26130—φ (°)2942—cp (kPa)——12.5∼16.8φp (°)——17.9∼24.7cr (kPa)——3.1∼12.5φr (°)——10.6∼17.9Note: only the shear strength parameters of slip bands were collected in detail.


### 3.3. Interval Variables and Numerical Model in Case Study

According to in situ data and engineering experience [15, 33, 41], interval variables of Baishi ancient landslide are set in Table 2. Correlation relation of parameters (cp>cr; φp>φr) in strain-softening constitutive is well presented.Table 2
Interval variables in the case study.
Interval variablesUnitsRangecpkPa[12.5, 16.8]φp°[17.9, 24.7]crkPa[3.1, 12.5]φr°[10.6, 17.9]kr—[0, 0.1]Note: only the parameters of slip zones are considered with the strain-softening behavior and abstracted as interval variables.The numerical model inFLAC3D of Baishi ancient landslide is established on the real geological information. According to some preliminary computation and in situ surveys, profile A-A′ in Area I is chosen as the main research object (shown in Figure 4). In this model, the ancient slip zones are abstracted as a 2 m thick band. The mesh of slip zones is almost all quadrangles of 1 m × 1 m, and the grid division is enough for computation.Figure 4
Geological information and numerical model of the case study.


## 3.1. Geological Background

To verify the applicability of our reliability analysis method, we took a strain-softening landslide, i.e., Baishi ancient landslide, as an example. Baishi ancient landslide is located at the south mountains area in Guangxi province, China (Figure3(a)). Geographic, geomorphic, elevation, and other information is shown in Figure 3(b). Geologically, Baishi landslide is an accumulated ancient landslide with obvious ancient slip bands. The geomaterial distribution could be described as three uneven layers: (1) argillaceous sandstone bedrock downmost, (2) slip bands with strain-softening behavior in-between, and (3) soil-gravel accumulations uppermost.Figure 3
(a) Location and (b) digital elevation model of Baishi ancient landslide.
(a)(b)


## 3.2. Parameter Collection

Some critical parameters used for simulation were collected from geological surveys or engineering experience as listed in Table1.Table 1
Physical and mechanical parameters of Baishi ancient landslide.
ItemAccumulationsBedrockSlip bandsConstitutive relationMohr–CoulombMohr–CoulombStrain-softeningE (GPa)1100.1v0.30.30.3ρ (kg·m3)196021302450c (kPa)26130—φ (°)2942—cp (kPa)——12.5∼16.8φp (°)——17.9∼24.7cr (kPa)——3.1∼12.5φr (°)——10.6∼17.9Note: only the shear strength parameters of slip bands were collected in detail.


## 3.3. Interval Variables and Numerical Model in Case Study

According to in situ data and engineering experience [15, 33, 41], interval variables of Baishi ancient landslide are set in Table 2. Correlation relation of parameters (cp>cr; φp>φr) in strain-softening constitutive is well presented.Table 2
Interval variables in the case study.
Interval variablesUnitsRangecpkPa[12.5, 16.8]φp°[17.9, 24.7]crkPa[3.1, 12.5]φr°[10.6, 17.9]kr—[0, 0.1]Note: only the parameters of slip zones are considered with the strain-softening behavior and abstracted as interval variables.The numerical model inFLAC3D of Baishi ancient landslide is established on the real geological information. According to some preliminary computation and in situ surveys, profile A-A′ in Area I is chosen as the main research object (shown in Figure 4). In this model, the ancient slip zones are abstracted as a 2 m thick band. The mesh of slip zones is almost all quadrangles of 1 m × 1 m, and the grid division is enough for computation.Figure 4
Geological information and numerical model of the case study.


## 4. Results and Discussion

### 4.1. Stability Analysis

Table3 presents some results of deterministic analysis of Baishi ancient landslide. Three examples are conducted. In example 1, safety factor Fs is calculated in peak strength when the peak strength parameters (cp,φp) are set as the maximum, mean, and minimum values, respectively. In example 2, safety factor Fs is calculated in residual strength when the residual strength parameters (cr,φr) are set as the maximum, mean, and minimum values, respectively. In example 3, safety factor Fs is calculated considering the strain-softening behavior when kr is set as the maximum, mean, and minimum values, respectively. In these three examples, all the parameters are set as mean values except for the assigned parameters. In examples 1 and 2, results are also calculated by the Spencer method to make a comparison with that by the SSR method.Table 3
Comparison ofFs considering three different strength states.
ItemVariableFs (SSR)Fs (spencer)Relative errors (%)Example 1cp,φpMax.1.9971.9970.000Mean1.6951.6980.177Min.1.4021.4060.284Example 2cr,φrMax.1.4021.4060.284Mean1.0951.0920.274Min.0.7930.7940.126Example 3krMax.1.535——Mean1.414——Min.1.097——Results in examples 1 and 2 indicate that, whenFs is calculated only in peak or residual strength, uncertainties of shear strength parameters (cp,φp; cr,φr) can cause great differences. The values of cp,φp and cr,φr should be thus considered as variables. Results calculated by SSR and Spencer methods are in good agreement, which indicates that the numerical model and method in this paper are valid. Results in example 3 indicate that when Fs is calculated considering the strain-softening behavior, the results are significantly influenced by the value of kr. Also, kr is set as a variable. Moreover, calculation results of Fs are 1.695, 1.095, and 1.414, respectively, when parameters (cp,φp; cr,φr) are set as the mean value of peak strength and residual strength and in the case of considering the strain-softening behavior. Result indicates that the strain-softening behavior should be considered in the stability evaluation of Baishi ancient landslide.


### 4.2. Performance Function Analysis

#### 4.2.1. Results of Performance Function

Table4 presents some results of deterministic tests for fitting the performance function. The central composite design was used to conduct the deterministic stability computation. As shown in Table 3, numbers 1–32 are the 25 groups of two-level design of five variables (cp,φp, cr,φr, and kr); numbers 33–37 are 5 groups of extension tests for nonlinear influence checkout; numbers 38–42 are 5 groups of random tests on the limited interval.Table 4
Results of central composite design in FLAC 3D.
Numbercp (kPa)φp (°)cr (kPa)φr (°)krFs-FLAC 3D116.8024.7012.5017.900.001.408212.5024.7012.5017.900.001.405316.8017.9012.5017.900.001.408412.5017.9012.5017.900.001.405516.8024.703.1017.900.001.364612.5024.703.1017.900.001.364716.8017.903.1017.900.001.364812.5017.903.1017.900.001.359916.8024.7012.5010.600.000.8521012.5024.7012.5010.600.000.8521116.8017.9012.5010.600.000.8521212.5017.9012.5010.600.000.8531316.8024.703.1010.600.000.7931412.5024.703.1010.600.000.7931516.8017.903.1010.600.000.7931612.5017.903.1010.600.000.7961716.8024.7012.5017.900.101.8341812.5024.7012.5017.900.101.8171916.8017.9012.5017.900.101.4212012.5017.9012.5017.900.101.4022116.8024.703.1017.900.101.8212212.5024.703.1017.900.101.8042316.8017.903.1017.900.101.4062412.5017.903.1017.900.101.3922516.8024.7012.5010.600.101.7142612.5024.7012.5010.600.101.6972716.8017.9012.5010.600.101.2782812.5017.9012.5010.600.101.2602916.8024.703.1010.600.101.7013012.5024.703.1010.600.101.6813116.8017.903.1010.600.101.2673212.5017.903.1010.600.101.2493314.6521.307.8014.250.001.0973414.6521.307.8014.250.041.3643514.6521.307.8014.250.051.4143614.6521.307.8014.250.061.4543714.6521.307.8014.250.101.5353812.6418.223.9111.390.010.9283913.6619.906.1912.910.041.2614015.1421.969.2615.140.071.5614116.1323.6310.3616.610.091.7314216.3324.2311.4717.590.101.798The nonlinear-fitting function is used to solve the undetermined coefficients by least squares regression. Results of undetermined coefficients are listed in Table5. The comparisons of Fs in FLAC 3D obtained by the response function are shown in Figure 5. Fs obtained from the response surface and that from FLAC 3D agree well. This means that the fitting of the performance function is valid.Table 5
Results of undetermined coefficients by regression.
CoefficientValuea10.350237a20.005312a30.035093a40.001088b10.065479b20.005458b30.061833b40.00053c114.46171c2−67.99980Figure 5
Results ofFs comparison between the response surface and FLAC 3D.Substituting the results of Table5 into equations (3) and (6)–(9), we obtain the explicit expression of performance function as(14)M=14.46171kr−67.9998kr20.350237+0.005312cp+0.035093φp+0.001088φp2+1−14.46171kr+67.9998kr20.065479+0.005458cr+0.061833φr+0.00053φr2−1,(15)Fp=0.350237+0.005312cp+0.035093φp+0.001088φp2,(16)Fr=0.065479+0.005458cr+0.061833φr+0.00053φr2,(17)w1=14.46171kr−67.9998kr2,(18)w2=1−14.46171kr+67.9998kr2.


#### 4.2.2. Sensitivity of Safety Factor to Variables

The response result ofFp induced by cp and φp, response result of Fr induced by cr and φr, and response result of w1 and w2 induced by kr are shown in Figures 6(a)–6(c), respectively. These results indicate that φp has a greater influence on the peak safety factor Fp than cp; φr has a greater influence on the residual safety factor Fr than cr. The nonlinear influence of shear strength parameters (cp,φp; cr,φr) is not obvious on respective intervals. However, kr greatly affects the weight function w1 and w2, and the nonlinear influence is obvious even on the small interval kr=0,0.1.Figure 6
(a) Response result ofFp induced by cp and φp. (b) Response result of Fr induced by cr and φr. (c) Response results of w1 and w2 induced by kr.
(a)(b)(c)Sensitivity of safety factorFs to variables is shown in Figure 7. Each interval variable is analyzed with the others fixed at the mean value. Results indicate that all the interval variables have some influence to safety factor Fs; however, the influence of cp and cr is not so obvious. The sensitivity to interval variables can be ranked as kr>φp>φr>cr>cp.Figure 7
Interval response ofFs induced by each interval variables.Under the assumptions in Section2.3, w1 and w2 are equal to the weight functions to control the contributions of Fp and Fr to Fs. That is to say, kr is equal to a weight coefficient to control the contributions of (cp,φp; cr,φr) to Fs. Therefore, the interval response of Fs induced by each interval variables under different kr is shown in Figure 8kr values are set to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. When an interval variable is analyzed, others are fixed at the mean value. Results indicate that, with kr increasing, the interval response of Fs induced by cr and cp changes slightly. In detail, the value of Fs induced by cr decreases, induced by φp increases. However, with kr increasing, the interval response of Fs induced by φr decreases and that induced by φp increases. The average level of interval response of Fs increases with the growth of kr. Therefore, it is considered that all the interval variables are of great significance to safety factor Fs, where kr is the most outstanding one.Figure 8
Interval response ofFs induced by the interval variables (cp,φp; cr,φr) under different kr values.


### 4.3. Reliability Analysis

#### 4.3.1. Results of Reliability

According to the result in equation (14), the monotony of performance function M can be obtained. For ∀kr∈0,0.1, w1∈0,0.7662 and w2∈0.2338,1. For ∀(cp, φp, cr, φr) on their interval, Fp∈1.3934,1.9701 and Fr∈0.7974,1.4103. For each single interval variable, function M is continuous monotonic. However, kr is at least more than 0.04 in Baishi ancient landslide according to the literature [41]. In fact, the strength of whole slip surface is in residual state when kr ⟶ 0, which is unreal for an ancient landslide with prolonged dormancy. Therefore, in Baishi ancient landslide, we adopt kr∈0.04,0.1, and the interval nonprobability reliability is solved as(19)Mmin=0.0773,Mmax=0.8392,η=McMr=1/2Mmin+Mmax1/2Mmax−Mmin=0.5×0.0773+0.83920.5×0.8392−0,0773=1.2029.Result ofη indicates that Baishi ancient landslide is reliable and safe (η>1) under the natural condition, agreeing well with the actual situation. In fact, the planned highway began the construction several months ago, and this landslide was also safe during the disturbance of construction.The reliability analysis is also conducted by the traditional probabilistic method to make a comparison. According to the literature [26], cp, φp, cr, φr, and kr are considered as random variables. cp, φp, cr, and φr are assumed as log-normal distribution, but kr is assumed as beta distribution. Mean values are fixed as the same as the mean of intervals in this paper, and coefficients of variation are chosen from the literature [26]. The probability reliability index is also defined as the minimum distance from the origin point to the failure surface in the standard normal space. Some results are listed in Table 6.Table 6
Results in probabilistic reliability calculations under different variable distributions.
COV of log-normal variablesBeta distribution parameterspf (%)PRI-ηINPRI-ηcpcrφpφrqr0.20.2551610.850.94431.20290.50.510101634.760.83640.20.2556610.711.40160.50.510106634.911.35670.20.2556211.362.10750.50.510106235.751.7697Note: COV: coefficient of variation,q and r are the parameters describing the probability density distribution, pf is the probability of failure, PRI-η is the probability reliability index, and INPRI-η is interval nonprobability of failure. More details are shown in the literature [26].Results in Table6 indicate that the reliability is greatly influenced by the probability density distribution. The probability reliability index η can be little to 0.8364 with nearly 10% probability of failure. It can be also large to 2.1075 but with nearly 35% probability of failure. These results cannot be used in landslide engineering. However, bounds of interval variables can well solve this problem. The robustness of interval nonprobability reliability makes it more suitable to evaluate a landslide with the strain-softening behavior when the probability density is unknown.


#### 4.3.2. Sensitivity of Interval Nonprobability Reliability Index on Variables

For each interval variable,xi=xic+δixir, where xic is the center value, xir is the radius of interval and δi=−1,1. xic and xir are the controlling indexes of an interval.The influence of interval radius to interval nonprobability reliability indexη is shown in Figure 9. Fix the center value xic of an interval variable at original value; change the radius xir from 0 to original maximum radius, presented by axis from 0 to 1 (Figure 9), while other interval variables are set as usual. Results indicate that when the interval radius of kr and φr decreases, η significantly improves; the fall of the radius of cp and cr contributes little to the improvement of η; the drop in radius of φp even makes η decrease. Therefore, the influence of variable radius is ranked as follows: kr>φr>cr>cp>φp. It is worth noting that decreasing variable radius of φp declines η as well. The reason is that, when the radius of φp decreases to 0, φp eventually converges to a mean value. It indicates that the center value xic of interval variable is also of great importance.Figure 9
Results of reliability indexη influenced by the interval radius of five variables.The influence of interval center value toη is shown in Figure 10. The radius xir of an interval variable is fixed at 0; the interval center value changes from minimum to maximum on the interval, presented by axis from −1 to 1 (Figure 10), while other interval variables are set as usual. Results indicate that when the center value of interval variables kr and φr increases, η significantly increases. And, the line of φr grows faster. When the center value of interval variables cr,cp,φp increases, the growth of η is not so obvious.Figure 10
Results of reliability indexη influenced by the interval center of five variables.The interval response ofη induced by interval variables (cp,φp, cr,φr) under different kr is shown in Figure 11. Set kr to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. The radius xir of an interval variable is fixed at 0; the interval center value is changed from the minimum value to the maximum on the interval, while other interval variables are set as usual. Results in Figure 11 indicate that, with kr increasing, the interval response of η induced by cr and cp changes slightly. In detail, the value of η induced by cr decreases and that induced by φp increases. However, with kr increasing, the interval response of η induced by φr quickly decreases and that induced by φp quickly increases. In summary, both the center value and range of all interval variables are important to η. The kr is the most significant factor which should be preferentially determined.Figure 11
Interval response of reliability indexη induced by interval variables (cp,φp, cr,φr) under different kr values.


## 4.1. Stability Analysis

Table3 presents some results of deterministic analysis of Baishi ancient landslide. Three examples are conducted. In example 1, safety factor Fs is calculated in peak strength when the peak strength parameters (cp,φp) are set as the maximum, mean, and minimum values, respectively. In example 2, safety factor Fs is calculated in residual strength when the residual strength parameters (cr,φr) are set as the maximum, mean, and minimum values, respectively. In example 3, safety factor Fs is calculated considering the strain-softening behavior when kr is set as the maximum, mean, and minimum values, respectively. In these three examples, all the parameters are set as mean values except for the assigned parameters. In examples 1 and 2, results are also calculated by the Spencer method to make a comparison with that by the SSR method.Table 3
Comparison ofFs considering three different strength states.
ItemVariableFs (SSR)Fs (spencer)Relative errors (%)Example 1cp,φpMax.1.9971.9970.000Mean1.6951.6980.177Min.1.4021.4060.284Example 2cr,φrMax.1.4021.4060.284Mean1.0951.0920.274Min.0.7930.7940.126Example 3krMax.1.535——Mean1.414——Min.1.097——Results in examples 1 and 2 indicate that, whenFs is calculated only in peak or residual strength, uncertainties of shear strength parameters (cp,φp; cr,φr) can cause great differences. The values of cp,φp and cr,φr should be thus considered as variables. Results calculated by SSR and Spencer methods are in good agreement, which indicates that the numerical model and method in this paper are valid. Results in example 3 indicate that when Fs is calculated considering the strain-softening behavior, the results are significantly influenced by the value of kr. Also, kr is set as a variable. Moreover, calculation results of Fs are 1.695, 1.095, and 1.414, respectively, when parameters (cp,φp; cr,φr) are set as the mean value of peak strength and residual strength and in the case of considering the strain-softening behavior. Result indicates that the strain-softening behavior should be considered in the stability evaluation of Baishi ancient landslide.


## 4.2. Performance Function Analysis

### 4.2.1. Results of Performance Function

Table4 presents some results of deterministic tests for fitting the performance function. The central composite design was used to conduct the deterministic stability computation. As shown in Table 3, numbers 1–32 are the 25 groups of two-level design of five variables (cp,φp, cr,φr, and kr); numbers 33–37 are 5 groups of extension tests for nonlinear influence checkout; numbers 38–42 are 5 groups of random tests on the limited interval.Table 4
Results of central composite design in FLAC 3D.
Numbercp (kPa)φp (°)cr (kPa)φr (°)krFs-FLAC 3D116.8024.7012.5017.900.001.408212.5024.7012.5017.900.001.405316.8017.9012.5017.900.001.408412.5017.9012.5017.900.001.405516.8024.703.1017.900.001.364612.5024.703.1017.900.001.364716.8017.903.1017.900.001.364812.5017.903.1017.900.001.359916.8024.7012.5010.600.000.8521012.5024.7012.5010.600.000.8521116.8017.9012.5010.600.000.8521212.5017.9012.5010.600.000.8531316.8024.703.1010.600.000.7931412.5024.703.1010.600.000.7931516.8017.903.1010.600.000.7931612.5017.903.1010.600.000.7961716.8024.7012.5017.900.101.8341812.5024.7012.5017.900.101.8171916.8017.9012.5017.900.101.4212012.5017.9012.5017.900.101.4022116.8024.703.1017.900.101.8212212.5024.703.1017.900.101.8042316.8017.903.1017.900.101.4062412.5017.903.1017.900.101.3922516.8024.7012.5010.600.101.7142612.5024.7012.5010.600.101.6972716.8017.9012.5010.600.101.2782812.5017.9012.5010.600.101.2602916.8024.703.1010.600.101.7013012.5024.703.1010.600.101.6813116.8017.903.1010.600.101.2673212.5017.903.1010.600.101.2493314.6521.307.8014.250.001.0973414.6521.307.8014.250.041.3643514.6521.307.8014.250.051.4143614.6521.307.8014.250.061.4543714.6521.307.8014.250.101.5353812.6418.223.9111.390.010.9283913.6619.906.1912.910.041.2614015.1421.969.2615.140.071.5614116.1323.6310.3616.610.091.7314216.3324.2311.4717.590.101.798The nonlinear-fitting function is used to solve the undetermined coefficients by least squares regression. Results of undetermined coefficients are listed in Table5. The comparisons of Fs in FLAC 3D obtained by the response function are shown in Figure 5. Fs obtained from the response surface and that from FLAC 3D agree well. This means that the fitting of the performance function is valid.Table 5
Results of undetermined coefficients by regression.
CoefficientValuea10.350237a20.005312a30.035093a40.001088b10.065479b20.005458b30.061833b40.00053c114.46171c2−67.99980Figure 5
Results ofFs comparison between the response surface and FLAC 3D.Substituting the results of Table5 into equations (3) and (6)–(9), we obtain the explicit expression of performance function as(14)M=14.46171kr−67.9998kr20.350237+0.005312cp+0.035093φp+0.001088φp2+1−14.46171kr+67.9998kr20.065479+0.005458cr+0.061833φr+0.00053φr2−1,(15)Fp=0.350237+0.005312cp+0.035093φp+0.001088φp2,(16)Fr=0.065479+0.005458cr+0.061833φr+0.00053φr2,(17)w1=14.46171kr−67.9998kr2,(18)w2=1−14.46171kr+67.9998kr2.


### 4.2.2. Sensitivity of Safety Factor to Variables

The response result ofFp induced by cp and φp, response result of Fr induced by cr and φr, and response result of w1 and w2 induced by kr are shown in Figures 6(a)–6(c), respectively. These results indicate that φp has a greater influence on the peak safety factor Fp than cp; φr has a greater influence on the residual safety factor Fr than cr. The nonlinear influence of shear strength parameters (cp,φp; cr,φr) is not obvious on respective intervals. However, kr greatly affects the weight function w1 and w2, and the nonlinear influence is obvious even on the small interval kr=0,0.1.Figure 6
(a) Response result ofFp induced by cp and φp. (b) Response result of Fr induced by cr and φr. (c) Response results of w1 and w2 induced by kr.
(a)(b)(c)Sensitivity of safety factorFs to variables is shown in Figure 7. Each interval variable is analyzed with the others fixed at the mean value. Results indicate that all the interval variables have some influence to safety factor Fs; however, the influence of cp and cr is not so obvious. The sensitivity to interval variables can be ranked as kr>φp>φr>cr>cp.Figure 7
Interval response ofFs induced by each interval variables.Under the assumptions in Section2.3, w1 and w2 are equal to the weight functions to control the contributions of Fp and Fr to Fs. That is to say, kr is equal to a weight coefficient to control the contributions of (cp,φp; cr,φr) to Fs. Therefore, the interval response of Fs induced by each interval variables under different kr is shown in Figure 8kr values are set to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. When an interval variable is analyzed, others are fixed at the mean value. Results indicate that, with kr increasing, the interval response of Fs induced by cr and cp changes slightly. In detail, the value of Fs induced by cr decreases, induced by φp increases. However, with kr increasing, the interval response of Fs induced by φr decreases and that induced by φp increases. The average level of interval response of Fs increases with the growth of kr. Therefore, it is considered that all the interval variables are of great significance to safety factor Fs, where kr is the most outstanding one.Figure 8
Interval response ofFs induced by the interval variables (cp,φp; cr,φr) under different kr values.


## 4.2.1. Results of Performance Function

Table4 presents some results of deterministic tests for fitting the performance function. The central composite design was used to conduct the deterministic stability computation. As shown in Table 3, numbers 1–32 are the 25 groups of two-level design of five variables (cp,φp, cr,φr, and kr); numbers 33–37 are 5 groups of extension tests for nonlinear influence checkout; numbers 38–42 are 5 groups of random tests on the limited interval.Table 4
Results of central composite design in FLAC 3D.
Numbercp (kPa)φp (°)cr (kPa)φr (°)krFs-FLAC 3D116.8024.7012.5017.900.001.408212.5024.7012.5017.900.001.405316.8017.9012.5017.900.001.408412.5017.9012.5017.900.001.405516.8024.703.1017.900.001.364612.5024.703.1017.900.001.364716.8017.903.1017.900.001.364812.5017.903.1017.900.001.359916.8024.7012.5010.600.000.8521012.5024.7012.5010.600.000.8521116.8017.9012.5010.600.000.8521212.5017.9012.5010.600.000.8531316.8024.703.1010.600.000.7931412.5024.703.1010.600.000.7931516.8017.903.1010.600.000.7931612.5017.903.1010.600.000.7961716.8024.7012.5017.900.101.8341812.5024.7012.5017.900.101.8171916.8017.9012.5017.900.101.4212012.5017.9012.5017.900.101.4022116.8024.703.1017.900.101.8212212.5024.703.1017.900.101.8042316.8017.903.1017.900.101.4062412.5017.903.1017.900.101.3922516.8024.7012.5010.600.101.7142612.5024.7012.5010.600.101.6972716.8017.9012.5010.600.101.2782812.5017.9012.5010.600.101.2602916.8024.703.1010.600.101.7013012.5024.703.1010.600.101.6813116.8017.903.1010.600.101.2673212.5017.903.1010.600.101.2493314.6521.307.8014.250.001.0973414.6521.307.8014.250.041.3643514.6521.307.8014.250.051.4143614.6521.307.8014.250.061.4543714.6521.307.8014.250.101.5353812.6418.223.9111.390.010.9283913.6619.906.1912.910.041.2614015.1421.969.2615.140.071.5614116.1323.6310.3616.610.091.7314216.3324.2311.4717.590.101.798The nonlinear-fitting function is used to solve the undetermined coefficients by least squares regression. Results of undetermined coefficients are listed in Table5. The comparisons of Fs in FLAC 3D obtained by the response function are shown in Figure 5. Fs obtained from the response surface and that from FLAC 3D agree well. This means that the fitting of the performance function is valid.Table 5
Results of undetermined coefficients by regression.
CoefficientValuea10.350237a20.005312a30.035093a40.001088b10.065479b20.005458b30.061833b40.00053c114.46171c2−67.99980Figure 5
Results ofFs comparison between the response surface and FLAC 3D.Substituting the results of Table5 into equations (3) and (6)–(9), we obtain the explicit expression of performance function as(14)M=14.46171kr−67.9998kr20.350237+0.005312cp+0.035093φp+0.001088φp2+1−14.46171kr+67.9998kr20.065479+0.005458cr+0.061833φr+0.00053φr2−1,(15)Fp=0.350237+0.005312cp+0.035093φp+0.001088φp2,(16)Fr=0.065479+0.005458cr+0.061833φr+0.00053φr2,(17)w1=14.46171kr−67.9998kr2,(18)w2=1−14.46171kr+67.9998kr2.


## 4.2.2. Sensitivity of Safety Factor to Variables

The response result ofFp induced by cp and φp, response result of Fr induced by cr and φr, and response result of w1 and w2 induced by kr are shown in Figures 6(a)–6(c), respectively. These results indicate that φp has a greater influence on the peak safety factor Fp than cp; φr has a greater influence on the residual safety factor Fr than cr. The nonlinear influence of shear strength parameters (cp,φp; cr,φr) is not obvious on respective intervals. However, kr greatly affects the weight function w1 and w2, and the nonlinear influence is obvious even on the small interval kr=0,0.1.Figure 6
(a) Response result ofFp induced by cp and φp. (b) Response result of Fr induced by cr and φr. (c) Response results of w1 and w2 induced by kr.
(a)(b)(c)Sensitivity of safety factorFs to variables is shown in Figure 7. Each interval variable is analyzed with the others fixed at the mean value. Results indicate that all the interval variables have some influence to safety factor Fs; however, the influence of cp and cr is not so obvious. The sensitivity to interval variables can be ranked as kr>φp>φr>cr>cp.Figure 7
Interval response ofFs induced by each interval variables.Under the assumptions in Section2.3, w1 and w2 are equal to the weight functions to control the contributions of Fp and Fr to Fs. That is to say, kr is equal to a weight coefficient to control the contributions of (cp,φp; cr,φr) to Fs. Therefore, the interval response of Fs induced by each interval variables under different kr is shown in Figure 8kr values are set to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. When an interval variable is analyzed, others are fixed at the mean value. Results indicate that, with kr increasing, the interval response of Fs induced by cr and cp changes slightly. In detail, the value of Fs induced by cr decreases, induced by φp increases. However, with kr increasing, the interval response of Fs induced by φr decreases and that induced by φp increases. The average level of interval response of Fs increases with the growth of kr. Therefore, it is considered that all the interval variables are of great significance to safety factor Fs, where kr is the most outstanding one.Figure 8
Interval response ofFs induced by the interval variables (cp,φp; cr,φr) under different kr values.


## 4.3. Reliability Analysis

### 4.3.1. Results of Reliability

According to the result in equation (14), the monotony of performance function M can be obtained. For ∀kr∈0,0.1, w1∈0,0.7662 and w2∈0.2338,1. For ∀(cp, φp, cr, φr) on their interval, Fp∈1.3934,1.9701 and Fr∈0.7974,1.4103. For each single interval variable, function M is continuous monotonic. However, kr is at least more than 0.04 in Baishi ancient landslide according to the literature [41]. In fact, the strength of whole slip surface is in residual state when kr ⟶ 0, which is unreal for an ancient landslide with prolonged dormancy. Therefore, in Baishi ancient landslide, we adopt kr∈0.04,0.1, and the interval nonprobability reliability is solved as(19)Mmin=0.0773,Mmax=0.8392,η=McMr=1/2Mmin+Mmax1/2Mmax−Mmin=0.5×0.0773+0.83920.5×0.8392−0,0773=1.2029.Result ofη indicates that Baishi ancient landslide is reliable and safe (η>1) under the natural condition, agreeing well with the actual situation. In fact, the planned highway began the construction several months ago, and this landslide was also safe during the disturbance of construction.The reliability analysis is also conducted by the traditional probabilistic method to make a comparison. According to the literature [26], cp, φp, cr, φr, and kr are considered as random variables. cp, φp, cr, and φr are assumed as log-normal distribution, but kr is assumed as beta distribution. Mean values are fixed as the same as the mean of intervals in this paper, and coefficients of variation are chosen from the literature [26]. The probability reliability index is also defined as the minimum distance from the origin point to the failure surface in the standard normal space. Some results are listed in Table 6.Table 6
Results in probabilistic reliability calculations under different variable distributions.
COV of log-normal variablesBeta distribution parameterspf (%)PRI-ηINPRI-ηcpcrφpφrqr0.20.2551610.850.94431.20290.50.510101634.760.83640.20.2556610.711.40160.50.510106634.911.35670.20.2556211.362.10750.50.510106235.751.7697Note: COV: coefficient of variation,q and r are the parameters describing the probability density distribution, pf is the probability of failure, PRI-η is the probability reliability index, and INPRI-η is interval nonprobability of failure. More details are shown in the literature [26].Results in Table6 indicate that the reliability is greatly influenced by the probability density distribution. The probability reliability index η can be little to 0.8364 with nearly 10% probability of failure. It can be also large to 2.1075 but with nearly 35% probability of failure. These results cannot be used in landslide engineering. However, bounds of interval variables can well solve this problem. The robustness of interval nonprobability reliability makes it more suitable to evaluate a landslide with the strain-softening behavior when the probability density is unknown.


### 4.3.2. Sensitivity of Interval Nonprobability Reliability Index on Variables

For each interval variable,xi=xic+δixir, where xic is the center value, xir is the radius of interval and δi=−1,1. xic and xir are the controlling indexes of an interval.The influence of interval radius to interval nonprobability reliability indexη is shown in Figure 9. Fix the center value xic of an interval variable at original value; change the radius xir from 0 to original maximum radius, presented by axis from 0 to 1 (Figure 9), while other interval variables are set as usual. Results indicate that when the interval radius of kr and φr decreases, η significantly improves; the fall of the radius of cp and cr contributes little to the improvement of η; the drop in radius of φp even makes η decrease. Therefore, the influence of variable radius is ranked as follows: kr>φr>cr>cp>φp. It is worth noting that decreasing variable radius of φp declines η as well. The reason is that, when the radius of φp decreases to 0, φp eventually converges to a mean value. It indicates that the center value xic of interval variable is also of great importance.Figure 9
Results of reliability indexη influenced by the interval radius of five variables.The influence of interval center value toη is shown in Figure 10. The radius xir of an interval variable is fixed at 0; the interval center value changes from minimum to maximum on the interval, presented by axis from −1 to 1 (Figure 10), while other interval variables are set as usual. Results indicate that when the center value of interval variables kr and φr increases, η significantly increases. And, the line of φr grows faster. When the center value of interval variables cr,cp,φp increases, the growth of η is not so obvious.Figure 10
Results of reliability indexη influenced by the interval center of five variables.The interval response ofη induced by interval variables (cp,φp, cr,φr) under different kr is shown in Figure 11. Set kr to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. The radius xir of an interval variable is fixed at 0; the interval center value is changed from the minimum value to the maximum on the interval, while other interval variables are set as usual. Results in Figure 11 indicate that, with kr increasing, the interval response of η induced by cr and cp changes slightly. In detail, the value of η induced by cr decreases and that induced by φp increases. However, with kr increasing, the interval response of η induced by φr quickly decreases and that induced by φp quickly increases. In summary, both the center value and range of all interval variables are important to η. The kr is the most significant factor which should be preferentially determined.Figure 11
Interval response of reliability indexη induced by interval variables (cp,φp, cr,φr) under different kr values.


## 4.3.1. Results of Reliability

According to the result in equation (14), the monotony of performance function M can be obtained. For ∀kr∈0,0.1, w1∈0,0.7662 and w2∈0.2338,1. For ∀(cp, φp, cr, φr) on their interval, Fp∈1.3934,1.9701 and Fr∈0.7974,1.4103. For each single interval variable, function M is continuous monotonic. However, kr is at least more than 0.04 in Baishi ancient landslide according to the literature [41]. In fact, the strength of whole slip surface is in residual state when kr ⟶ 0, which is unreal for an ancient landslide with prolonged dormancy. Therefore, in Baishi ancient landslide, we adopt kr∈0.04,0.1, and the interval nonprobability reliability is solved as(19)Mmin=0.0773,Mmax=0.8392,η=McMr=1/2Mmin+Mmax1/2Mmax−Mmin=0.5×0.0773+0.83920.5×0.8392−0,0773=1.2029.Result ofη indicates that Baishi ancient landslide is reliable and safe (η>1) under the natural condition, agreeing well with the actual situation. In fact, the planned highway began the construction several months ago, and this landslide was also safe during the disturbance of construction.The reliability analysis is also conducted by the traditional probabilistic method to make a comparison. According to the literature [26], cp, φp, cr, φr, and kr are considered as random variables. cp, φp, cr, and φr are assumed as log-normal distribution, but kr is assumed as beta distribution. Mean values are fixed as the same as the mean of intervals in this paper, and coefficients of variation are chosen from the literature [26]. The probability reliability index is also defined as the minimum distance from the origin point to the failure surface in the standard normal space. Some results are listed in Table 6.Table 6
Results in probabilistic reliability calculations under different variable distributions.
COV of log-normal variablesBeta distribution parameterspf (%)PRI-ηINPRI-ηcpcrφpφrqr0.20.2551610.850.94431.20290.50.510101634.760.83640.20.2556610.711.40160.50.510106634.911.35670.20.2556211.362.10750.50.510106235.751.7697Note: COV: coefficient of variation,q and r are the parameters describing the probability density distribution, pf is the probability of failure, PRI-η is the probability reliability index, and INPRI-η is interval nonprobability of failure. More details are shown in the literature [26].Results in Table6 indicate that the reliability is greatly influenced by the probability density distribution. The probability reliability index η can be little to 0.8364 with nearly 10% probability of failure. It can be also large to 2.1075 but with nearly 35% probability of failure. These results cannot be used in landslide engineering. However, bounds of interval variables can well solve this problem. The robustness of interval nonprobability reliability makes it more suitable to evaluate a landslide with the strain-softening behavior when the probability density is unknown.


## 4.3.2. Sensitivity of Interval Nonprobability Reliability Index on Variables

For each interval variable,xi=xic+δixir, where xic is the center value, xir is the radius of interval and δi=−1,1. xic and xir are the controlling indexes of an interval.The influence of interval radius to interval nonprobability reliability indexη is shown in Figure 9. Fix the center value xic of an interval variable at original value; change the radius xir from 0 to original maximum radius, presented by axis from 0 to 1 (Figure 9), while other interval variables are set as usual. Results indicate that when the interval radius of kr and φr decreases, η significantly improves; the fall of the radius of cp and cr contributes little to the improvement of η; the drop in radius of φp even makes η decrease. Therefore, the influence of variable radius is ranked as follows: kr>φr>cr>cp>φp. It is worth noting that decreasing variable radius of φp declines η as well. The reason is that, when the radius of φp decreases to 0, φp eventually converges to a mean value. It indicates that the center value xic of interval variable is also of great importance.Figure 9
Results of reliability indexη influenced by the interval radius of five variables.The influence of interval center value toη is shown in Figure 10. The radius xir of an interval variable is fixed at 0; the interval center value changes from minimum to maximum on the interval, presented by axis from −1 to 1 (Figure 10), while other interval variables are set as usual. Results indicate that when the center value of interval variables kr and φr increases, η significantly increases. And, the line of φr grows faster. When the center value of interval variables cr,cp,φp increases, the growth of η is not so obvious.Figure 10
Results of reliability indexη influenced by the interval center of five variables.The interval response ofη induced by interval variables (cp,φp, cr,φr) under different kr is shown in Figure 11. Set kr to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. The radius xir of an interval variable is fixed at 0; the interval center value is changed from the minimum value to the maximum on the interval, while other interval variables are set as usual. Results in Figure 11 indicate that, with kr increasing, the interval response of η induced by cr and cp changes slightly. In detail, the value of η induced by cr decreases and that induced by φp increases. However, with kr increasing, the interval response of η induced by φr quickly decreases and that induced by φp quickly increases. In summary, both the center value and range of all interval variables are important to η. The kr is the most significant factor which should be preferentially determined.Figure 11
Interval response of reliability indexη induced by interval variables (cp,φp, cr,φr) under different kr values.


## 5. Conclusions

In the present study, an interval nonprobability reliability analysis framework combined with numerical strain-softening constitutive relations was established to evaluate the reliability of Baishi ancient landslide. The main conclusions can be drawn as follows:(1)
The uncertainties of geomaterial properties significantly affect the stability evaluation of the ancient landslide. To correctly and effectively consider these uncertainties when data are scarce, an interval nonprobability reliability analysis framework combined with numerical strain-softening constitutive relations is established.(2)
The proposed reliability analysis method is verified by Baishi ancient landslide as a case study. Uncertain parameters (cp, φp, cr, φr, and kr) in the strain-softening numerical constitutive relation are set as interval variables. The interval nonprobability reliability index η is defined as the minimum distance from the origin point to the failure surface in the standard normal space. This index is used to describe the ability of a system to tolerate the variation of uncertain parameters.(3)
The interval nonprobability reliability indexη of Baishi is 1.2029 (>1), indicating that Baishi ancient landslide is safe under the nature condition. The results calculated by the proposed method agree with the reality. However, the reliability index η in the probability method ranges greatly; Baishi ancient landslide holds a 35% probability of failure. Therefore, the interval nonprobability method is more suitable.(4)
The sensitivity analysis indicates thatkr is the most significant variable controlling both safety factor Fs and interval nonprobability reliability index η. In addition, kr is equivalent to a weight coefficient that can affect the influence of (cr, φr) and (cp, φp). When kr increases, the influence of (cr, φr) decreases but that of (cp, φp) increases.(5)
Interval nonprobability reliability method combined with numerical strain-softening constitutive relation can accurately present the relation of parameters between peak and residual strength with a few data and obtain robust results. It provides a great complement for traditional probabilistic methods.



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