#### 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 threshold 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.

#### 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 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 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 was obtained by the limit equilibrium method with residual factor , and a simple probabilistic reliability analysis was conducted under “” 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 () based on the Bishop simplified method was adapted to take strain-softening into account in terms of residual factor . Bhattacharya et al. [26] regraded residual factor as a random variable, and the influence of 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 (, , in which and are the peak and residual cohesion, respectively, and and 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 (, ). 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 index instead of a deterministic safety factor 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 , peak friction angle , residual cohesion , residual friction angle , and the threshold parameter when strength reduces from peak to residual .

The uncertainty of different parameters needs to be confirmed. As shown in Figure 1, was a parameter describing the malleability of strain-softening materials [33]. When 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, is considered as a variable in this paper. is also greatly associated with the residual factor [22]; they are the cause and effect in negative correlation. Peak and residual shear strength (, , , and ) 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 (, , , , and ) are confirmed as interval variables to consider the uncertainties. It is noted that interval variables are assumed mutually independent.

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##### 2.2. Determination of Safety Factor

The response interval of controlled by interval variables is obtained based on the numerical model in . Each response of 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 of a homogeneous slope with a strain-softening behavior:where is the safety factor; and are the real strength parameters at failure; and and are the original strength parameters.

Interval variables , , , , and 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 Function

Performance function presenting the limit state of system is established based on as

However, numerical method cannot give an explicit relationship between and interval variables. In this paper, the explicit expression of 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], can be written as

For simplification, could be rewritten aswhere is the residual factor, , , is the residual safety factor, and is the peak safety factor.

In equation (5), each subitem () was assumed as a linear or quadratic polynomial for convenience. As introduced in Section 2.1, when , materials are brittle, strength of whole slip surface are in residual state, and then , . Therefore, the assumed expressions of each subitems are written aswhere are the undetermined coefficients.

Substituting equations (6)–(9) into equation (3), the performance function can be expressed as

After fitting a series of deterministic results ((, , , , ), ), the explicit expression of in equation (10) can be obtained.

##### 2.4. Reliability Analysis Based on Interval Theory

Interval nonprobability reliability index , 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).

For , where *M* is the performance function, () are the controlling interval variables, there are (*i* = 1, 2, …, *n*), where and are the lower and upper bounds of respectively. Then, the performance function *M* is also an interval to response, where and are the lower and upper bounds of , respectively. and 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):where is the failure plane, and this hypersurface divides the space into the failure domain and the security domain. When , the system is in failure state and the opposite it is in safe state. According to equations (11)–(13), if , then (*i* = 1, 2, …, *n*), , and the system is safe and reliable. If , then (*i* = 1, 2, …, *n*), < 0 and the system fails. If , then (*i* = 1, 2, …, *n*), or or 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 (Figure 3(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.

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##### 3.2. Parameter Collection

Some critical parameters used for simulation were collected from geological surveys or engineering experience as listed in Table 1.

##### 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 (; ) in strain-softening constitutive is well presented.

The numerical model in 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.

#### 4. Results and Discussion

##### 4.1. Stability Analysis

Table 3 presents some results of deterministic analysis of Baishi ancient landslide. Three examples are conducted. In example 1, safety factor is calculated in peak strength when the peak strength parameters () are set as the maximum, mean, and minimum values, respectively. In example 2, safety factor is calculated in residual strength when the residual strength parameters () are set as the maximum, mean, and minimum values, respectively. In example 3, safety factor is calculated considering the strain-softening behavior when 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.

Results in examples 1 and 2 indicate that, when is calculated only in peak or residual strength, uncertainties of shear strength parameters (; ) can cause great differences. The values of and 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 is calculated considering the strain-softening behavior, the results are significantly influenced by the value of . Also, is set as a variable. Moreover, calculation results of are 1.695, 1.095, and 1.414, respectively, when parameters (; ) 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

Table 4 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 groups of two-level design of five variables (, , and ); 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.

The nonlinear-fitting function is used to solve the undetermined coefficients by least squares regression. Results of undetermined coefficients are listed in Table 5. The comparisons of in FLAC 3D obtained by the response function are shown in Figure 5. obtained from the response surface and that from FLAC 3D agree well. This means that the fitting of the performance function is valid.

Substituting the results of Table 5 into equations (3) and (6)–(9), we obtain the explicit expression of performance function as

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

The response result of induced by and , response result of induced by and , and response result of and induced by are shown in Figures 6(a)–6(c), respectively. These results indicate that has a greater influence on the peak safety factor than ; has a greater influence on the residual safety factor than . The nonlinear influence of shear strength parameters (; ) is not obvious on respective intervals. However, greatly affects the weight function and , and the nonlinear influence is obvious even on the small interval .

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Sensitivity of safety factor 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 ; however, the influence of and is not so obvious. The sensitivity to interval variables can be ranked as .

Under the assumptions in Section 2.3, and are equal to the weight functions to control the contributions of and to . That is to say, is equal to a weight coefficient to control the contributions of (; ) to . Therefore, the interval response of induced by each interval variables under different is shown in Figure 8 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 increasing, the interval response of induced by and changes slightly. In detail, the value of induced by decreases, induced by increases. However, with increasing, the interval response of induced by decreases and that induced by increases. The average level of interval response of increases with the growth of . Therefore, it is considered that all the interval variables are of great significance to safety factor , where is the most outstanding one.

##### 4.3. Reliability Analysis

###### 4.3.1. Results of Reliability

According to the result in equation (14), the monotony of performance function can be obtained. For , and . For (, , , ) on their interval, and . For each single interval variable, function is continuous monotonic. However, 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 ⟶ 0, which is unreal for an ancient landslide with prolonged dormancy. Therefore, in Baishi ancient landslide, we adopt , and the interval nonprobability reliability is solved as

Result of indicates that Baishi ancient landslide is reliable and safe () 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], , , , , and are considered as random variables. , , , and are assumed as log-normal distribution, but 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.

Results in Table 6 indicate that the reliability is greatly influenced by the probability density distribution. The probability reliability index 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, , where is the center value, is the radius of interval and . and are the controlling indexes of an interval.

The influence of interval radius to interval nonprobability reliability index is shown in Figure 9. Fix the center value of an interval variable at original value; change the radius 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 and decreases, significantly improves; the fall of the radius of and contributes little to the improvement of ; the drop in radius of even makes decrease. Therefore, the influence of variable radius is ranked as follows: . It is worth noting that decreasing variable radius of declines as well. The reason is that, when the radius of decreases to 0, eventually converges to a mean value. It indicates that the center value of interval variable is also of great importance.

The influence of interval center value to is shown in Figure 10. The radius 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 and increases, significantly increases. And, the line of grows faster. When the center value of interval variables increases, the growth of is not so obvious.

The interval response of induced by interval variables (, ) under different is shown in Figure 11. Set to 0.00, 0.02, 0.04, 0.06, 0.08, and 0.10, respectively. The radius 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 increasing, the interval response of induced by and changes slightly. In detail, the value of induced by decreases and that induced by increases. However, with increasing, the interval response of induced by quickly decreases and that induced by quickly increases. In summary, both the center value and range of all interval variables are important to . The is the most significant factor which should be preferentially determined.

#### 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 (, , , , and ) in the strain-softening numerical constitutive relation are set as interval variables. The interval nonprobability reliability index 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 index of Baishi is 1.2029 (), 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 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 that is the most significant variable controlling both safety factor and interval nonprobability reliability index . In addition, is equivalent to a weight coefficient that can affect the influence of (, ) and (, ). When increases, the influence of (, ) decreases but that of (, ) 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.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported by the National Basic Research Program of China (2015CB060200), the National Natural Science Foundation of China (41772313), Hunan Science and Technology Planning Project (2019RS3001), and the Graduated Students’ Research and Innovation Fund Project of Central South University (2019zzts987).