Risk evaluation of colluvial cutting slope based on fuzzy analytic hierarchy process and multilevel fuzzy comprehensive evaluation

Abstract

The stability of a colluvial slope, which is different from a rock or soil slope, is determined by the properties of both the bedrock and the colluvium. Coupled with artificial excavation and environmental effects, the stability factors of such slopes are complicated. To rapidly and effectively evaluate the risk of a colluvial cutting slope, a risk evaluation system for this type of slope is established herein. First, an evaluation index system is established, and reasonable risk evaluation indices are selected. Second, the fuzzy analytic hierarchy process (FAHP) is applied, a fuzzy pairwise comparison matrix, that must satisfy a consistency test, is constructed, and the weight of each index is determined. Third, the risk evaluation grades are divided into 4 risk grades, and the risk evaluation criteria for each basic index are determined. Finally, the three-level fuzzy comprehensive evaluation (FCE) method is applied, the membership function for each index is constructed, the membership degree is calculated, and the risk grade of the colluvial cutting slope is determined. This risk evaluation system is used to evaluate the risks of 148 colluvial cutting slopes along the Xiaomengyang-Mohan highway in Yunnan, China. The results show that there are 24 slopes of low risk (grade I), 85 of medium risk (grade II), 22 of high risk (grade III), and 17 of very high risk (grade IV). The evaluation results obtained are in good agreement with the actual slope instability states: failure occurred in 15 out of 85 slopes of risk grade II, 13 out of 22 slopes of risk grade III, and 16 out of 17 slopes of risk grade IV. This application demonstrates that the proposed risk evaluation system for colluvial cutting slopes is universal, and stable and that the calculation results are objective.

Keywords

Colluvial cutting slope risk evaluation system fuzzy analytic hierarchy process fuzzy comprehensive evaluation

1 Introduction

Colluvial slopes are widely distributed around the world, for example, in Southwest China [1], the Western United States [2], southwestern Europe [3], Southeast Brazil [4], and northern Chile [5]. Colluvial slopes are usually formed through a strong downcutting period, followed by the long-term gravity-driven processes, rainfall or earthquake activity [6]. The stability of a colluvial slope, which is different from a rock slope (Fig. 1(a)) or soil slope (Fig. 1(b)), is determined by the properties of both the bedrock and the colluvium. The geological evolution of a colluvial slope is illustrated in Fig. 1(c).

Fig.1

Different types of slopes and the formation process of a colluvial cutting slope (a) rock slope; (b) soil slope; (c) the geological evolution of a colluvial slope; (d) colluvial cutting slope.

With the acceleration of urbanization and economic development, the number of civil engineering endeavors in mountainous areas, throughout which many colluvial slopes are present, is increasing; one example is the implementation of the Belt and Road Initiative in Southwest China, where highways are being constructed in mountainous areas with a gradually increasing frequency. Southwest China, which is situated around the Tibetan Plateau and the Yunnan-Guizhou Plateau, experienced a strong downcutting period in the Quaternary, resulting in severe deformation, the destruction of existing slopes and the formation of a large number of thick colluvial slopes [1]. Coupled with artificial excavation and environmental effects, the stability factors of colluvial slopes are complicated [7]. Therefore, to provide guidance for supporting measures and for the construction management of colluvial cutting slopes, thereby ensuring the safety of engineering constructions on colluvial cutting slopes, it is highly necessary to evaluate the risk of a colluvial cutting slope (Fig. 1(d)).

To date, many studies have been conducted to evaluate the risk and analyze the stability of colluvial slopes. Ohlmacher [8] used multiple logistic regression and geographic information system (GIS) technology to predict the landslide hazard of colluvial slopes in Northeast Kansas, USA. Meisina [9] adopted the stability index mapping (SINMAP) and shallow landsliding stability (SHALSTAB) models to analyze the stability of colluvial slopes in an area of the Oltrepo Pavese, Northern Apennines. Xiang [10] conducted geological surveys and employed an optimization method to search for the critical slip surface of a colluvial slope on the bank of the Three Gorges Reservoir. Zhou [11] analyzed the stability of a colluvial slope at the dam site of the Gushui hydropower station under an engineering disturbance. Choobbasti [12] developed artificial neural network (ANN) systems consisting of multilayer perceptron networks to predict the stability of a colluvial slope at a specified location. He [13] established the dynamic displacement destabilized criterion using damage mechanics and applied it to the typical rainfall-induced landslide that occurs on colluvial slopes in the Three Gorges Reservoir area. Matassoni [14] studied the seismic influence on colluvial slopes by measuring microtremors and installing a field mobile seismic station. Liang [15] combined electrical resistivity tomography (ERT) and terrestrial laser scanning (TLS) with traditional monitoring methods to monitor changes in water content and the deformation of the colluvial slope landslide in the Goulingping valley in the middle of the Bailong River catchment in Gansu Province, China.

In summary, most of the abovementioned studies were performed on individual slopes in particular areas relying on the acquisition of detailed parameters of rock and soil mechanics and measuring structural characteristics through monitoring or experimentation, as this information is also widely used for other kinds of slopes [16, 17]. In contrast, little research has been conducted on rapidly and effectively evaluating the risks for many colluvial slopes, such as colluvial cutting slopes along highways, due to the absence of detailed parameters. Numerous mechanical tests and engineering investigations have shown that colluvial slopes are not specifically defined and that their stability boundaries are not very clear and rather fuzzy; consequently, it is difficult to measure such slopes uniformly with classic mathematical models. Alternatively, considering the relevant factors of the evaluated objects and their attributes, comprehensive evaluation methods, such as the fuzzy evaluation method, gray correlation analysis, and reliability analysis, are suitable for analyzing the stability of slopes subjected to multivariate and multifactor influences and are thus widely used for the risk evaluations of rock and soil slopes [18 –23]. Nevertheless, no comprehensive evaluation method has been proposed for colluvial slopes.

The fuzzy evaluation method, while mature, is a still developing method based on fuzzy set theory [24] introduced to address uncertainties caused by imprecision or fuzziness. Many scholars have devoted themselves to studying fuzzy theory methods or algorithms [25 –31], and others have investigated the applications of fuzzy theory [32 –36]. In this paper, the fuzzy analytic hierarchy process (FAHP) and fuzzy comprehensive evaluation (FCE) methods are adopted. The analytic hierarchy process (AHP), which is a multilevel weight analysis and decision-making approach with high logic and practicability, was initially proposed by Saaty [37]. However, the AHP technique is greatly influenced by the subjectivity of human judgment; consequently, the FAHP, which combines the advantages of fuzzy methods with the AHP [38], was developed to consider the fuzziness of human judgment and to make the weight of the determined index more reasonable [39]. The FCE technique is a comprehensive decision-making method for solving complex multivariate problems [40], that can reasonably quantify the fuzzy index in the evaluation system [41]. These two methods have been applied for evaluation problems in several disciplines, such as petrochemistry [42], mining [43], education [44], transportation [45], aeroacoustics [46], electronic engineering [47], and medical science [48], accordingly, objective and effective evaluation results have been obtained in these practical applications. Therefore, the FAHP and FCE methods are selected herein to establish a risk evaluation system of colluvial cutting slopes.

The remainder of this paper is organized as follows. In Section 2, the concepts of fuzzy systems are presented, and the FAHP is introduced. In Section 3, the FCE technique is introduced. In Section 4, the risk evaluation system for colluvial cutting slopes is established. Section 5 presents an application of the proposed risk evaluation system, and the results are analyzed. Finally, conclusions are given in Section 6.

2 Fuzzy analytic hierarchy process

2.1 Fuzzy sets and fuzzy numbers

Assume a fuzzy set $\tilde{A} = {[x . μ_{\tilde{A}} (x)] | x \in X}$ is an ordered pair set, where X is a subset of real numbers R and $μ_{\tilde{A}} (x)$ is called the membership function, which assigns a membership from 0 to 1 to each object x. A fuzzy number is the normalized form of a fuzzy set. There are many kinds of fuzzy numbers; among them, triangular fuzzy numbers, trapezoidal fuzzy numbers and Gaussian fuzzy numbers are common. Because triangular fuzzy numbers are very intuitive and efficient, they are used to construct the comparison matrix in this paper. A triangular fuzzy number $\tilde{A} = (l . m . u)$ is shown in Fig. 2. The membership function can be expressed as Equation (1).

Fig.2

Triangular fuzzy number.

$μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - l}{m - l} & \begin{matrix} \end{matrix} & l \leq x \leq m \\ \frac{u - x}{u - m} & \begin{matrix} \end{matrix} & m \leq x \leq u \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix}$ (1)

Consider two triangular fuzzy numbers ${\tilde{A}}_{1} = (l_{1} . m_{1} . u_{1})$ and ${\tilde{A}}_{2} = (l_{2} . m_{2} . u_{2})$ , and one constant c. The operations of triangular fuzzy numbers [49] can be defined as Equation (2) ∼ Equation (6): ${\tilde{A}}_{1} \oplus {\tilde{A}}_{2} = (l_{1} + l_{2}, m_{1} + m_{2}, u_{1} + u_{2})$ (2) ${\tilde{A}}_{1} \otimes {\tilde{A}}_{2} = (l_{1} l_{2}, m_{1} m_{2}, u_{1} u_{2})$ (3) $c \otimes {\tilde{A}}_{2} = ({cl}_{2}, {cm}_{2}, {cu}_{2})$ (4) $\frac{{\tilde{A}}_{1}}{{\tilde{A}}_{2}} = (\frac{l_{1}}{u_{2}}, \frac{m_{1}}{m_{2}}, \frac{u_{1}}{l_{2}})$ (5) $\frac{1}{{\tilde{A}}_{1}} = (\frac{1}{u_{1}}, \frac{1}{m_{1}}, \frac{1}{l_{1}})$ (6) where ⊕ denotes the addition of two fuzzy numbers, and ⊗ denotes the multiplication of either two fuzzy numbers or one fuzzy number and one constant.

2.2 FAHP procedure

Several practical FAHP methods can be found in the literature. Among these different approaches, the FAHP proposed by Chang [38] is widely used in different applications because it boasts a lower computational complexity than other methods.

Assume that $\tilde{A} = ({\tilde{a}}_{ij})_{b \times b}$ is a fuzzy pairwise comparison matrix of b samples, where ${\tilde{a}}_{ij} = (l_{ij} . m_{ij} . u_{ij})$ . Chang’s method can be described as Equation (7) ∼ Equation (10). $D_{i} = \sum_{j = 1}^{b} {{\tilde{a}}_{ij} \otimes [\sum_{i = 1}^{b} \sum_{j = 1}^{b} {\tilde{a}}_{ij}]}^{- 1}$ (7) with

$\begin{matrix} \sum_{j = 1}^{b} {\tilde{a}}_{ij} = (\sum_{j = 1}^{b} l_{ij}, \sum_{j = 1}^{b} m_{ij}, \sum_{j = 1}^{b} u_{ij}) \\ i = 1, 2, . . ., b \end{matrix}$ (8) $\sum_{i = 1}^{b} \sum_{j = 1}^{b} {\tilde{a}}_{ij} = (\sum_{i = 1}^{b} \sum_{j = 1}^{b} l_{ij}, \sum_{i = 1}^{b} \sum_{j = 1}^{b} m_{ij}, \sum_{i = 1}^{b} \sum_{j = 1}^{b} u_{ij})$ (9)

$\begin{matrix} {[\sum_{i = 1}^{b} \sum_{j = 1}^{b} {\tilde{a}}_{ij}]}^{- 1} = (\frac{1}{\sum_{i = 1}^{b} \sum_{j = 1}^{b} u_{ij}}, \\ \frac{1}{\sum_{i = 1}^{b} \sum_{j = 1}^{b} m_{ij}}, \frac{1}{\sum_{i = 1}^{b} \sum_{j = 1}^{b} l_{ij}}) \end{matrix}$ (10) where D_i is the initial fuzzy weight of the i-th object. To obtain the weight of the i-th object, the degree of possibility is introduced. The degree of possibility P of D_i ≥ D_j is defined as Equation (11), and is shown in Fig. 3:

Fig.3

Comparison between D_i and D_j_.

$\begin{matrix} P (D_{i} \geq D_{j}) = \\ {\begin{matrix} 1 & \begin{matrix} \end{matrix} & m_{i} \geq m_{j} \\ \frac{l_{j} - u_{i}}{(m_{i} - u_{i}) - (m_{j} - l_{j})} & \begin{matrix} \end{matrix} & m_{i} \leq m_{j}, u_{i} \geq l_{j} \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix} \end{matrix}$ (11)

Assume that d_i (i = 1,2, ... ,b) is the minimum degree of possibility of D_i ≥ D_j (j = 1 , 2 , . . . , b , j ≠ i) , and is obtained from Equation (12): $d_{i} = min P (D_{i} \geq D_{j}, j = 1, 2, . . ., b, j \neq i)$ (12)

Take d_i as the initial weight of the i-th index. Then, the initial weight vector of b indices can be calculated and expressed as Equation (13): $W^{'} = {(d_{1}, d_{2}, . . ., d_{b})}^{T}$ (13)

Finally, the weight vector can be normalized as Equation (14): $W = {(W_{1}, W_{2}, . . ., W_{b})}^{T}$ (14)

2.3 Construction of a fuzzy pairwise comparison matrix

To objectively determine the weight of an index, the criterion of constructing fuzzy pairwise comparison matrix must be determined first. The method used in this paper was proposed by Kahraman [50], in this technique, linguistic descriptions can be transformed into triangular fuzzy numbers, as shown in Table 1.

Table 1
Linguistic descriptions and fuzzy numbers of importance

Linguistic descriptions of importance Triangular fuzzy numbers of importance Triangular fuzzy reciprocal numbers

Just equal (1, 1, 1) (1, 1, 1)

Equally important (1/2, 1, 3/2) (2/3, 1, 2)

In between (1, 3/2, 2) (1/2, 2/3, 1)

Weakly more important (3/2, 2, 5/2) (2/5, 1/2, 2/3)

In between (2, 5/2, 3) (1/3, 2/5, 1/2)

Strongly more important (5/2, 3, 7/2) (2/7, 1/3, 2/5)

In between (3, 7/2, 4) (1/4, 2/7, 1/3)

Very strongly more important (7/2, 4, 9/2) (2/9, 1/4, 2/7)

In between (4, 9/2, 5) (1/5, 2/9, 1/4)

Absolutely more important (9/2, 5, 11/2) (2/11, 1/5, 2/9)

Linguistic descriptions of importance	Triangular fuzzy numbers of importance	Triangular fuzzy reciprocal numbers
Just equal	(1, 1, 1)	(1, 1, 1)
Equally important	(1/2, 1, 3/2)	(2/3, 1, 2)
In between	(1, 3/2, 2)	(1/2, 2/3, 1)
Weakly more important	(3/2, 2, 5/2)	(2/5, 1/2, 2/3)
In between	(2, 5/2, 3)	(1/3, 2/5, 1/2)
Strongly more important	(5/2, 3, 7/2)	(2/7, 1/3, 2/5)
In between	(3, 7/2, 4)	(1/4, 2/7, 1/3)
Very strongly more important	(7/2, 4, 9/2)	(2/9, 1/4, 2/7)
In between	(4, 9/2, 5)	(1/5, 2/9, 1/4)
Absolutely more important	(9/2, 5, 11/2)	(2/11, 1/5, 2/9)

Consider a problem with n factors in one layer; a triangular fuzzy number is used to judge the relative importance between factors i and j. If one decision maker postulates that factor i is more important than factor j, the triangular fuzzy number can be obtained according to Table 1. If factor j is believed to be more important than factor i, then the reciprocal of the triangular fuzzy number is taken. Accordingly, the comparison matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ can be constructed as Equation (15): $\begin{matrix} \tilde{A} & = [\begin{matrix} 1 & {\tilde{a}}_{12} & \dots & {\tilde{a}}_{1 n} \\ {\tilde{a}}_{21} & 1 & \dots & {\tilde{a}}_{2 n} \\ \dots & \dots & \dots & \dots \\ {\tilde{a}}_{n 1} & {\tilde{a}}_{n 2} & \dots & 1 \end{matrix}] \\ = [\begin{matrix} 1 & {\tilde{a}}_{12} & . . . & {\tilde{a}}_{1 n} \\ 1 / {\tilde{a}}_{12} & 1 & . . . & {\tilde{a}}_{2 n} \\ . . . & . . . & . . . & . . . \\ 1 / {\tilde{a}}_{1 n} & 1 / {\tilde{a}}_{2 n} & . . . & 1 \end{matrix}] \end{matrix}$ (15)

2.4 Consistency test of the fuzzy pairwise comparison matrix

To ensure the quality of the decision, it is necessary to test the consistency of the fuzzy pairwise comparison matrix. For consistency testing, a fuzzy matrix is usually converted into a crisp matrix [51]. Several methods are available to convert a fuzzy matrix into a crisp matrix [52, 53]. The method presented by Chang [54] is used in this paper. A fuzzy triangular number ${\tilde{a}}_{ij} = (l_{ij} . m_{ij} . u_{ij})$ . can be converted into a crisp number ${(a_{ij}^{α})}^{λ}$ , following Equation (16) ∼ Equation (18):

$\begin{matrix} {(a_{ij}^{α})}^{λ} = [λ \cdot l_{ij}^{α} + (1 - λ) u_{ij}^{α}], 0 \leq λ \leq 1, \\ 0 \leq α \leq 1 \end{matrix}$ (16) with $l_{ij}^{α} = (m_{ij} - l_{ij}) \times α + l_{ij}$ (17) $u_{ij}^{α} = u_{ij} - (u_{ij} - m_{ij}) \times α$ (18) where $l_{ij}^{α}$ denotes the left-end value and $u_{ij}^{α}$ denotes the right-end value related to α. α is the degree of stability of the decision-making environment that takes any value from 0 to 1. The decision-making environment tends to become more stable as α increases; that is, the environment is the most unstable and the uncertainty is the highest when α = 0. λ is the degree of optimism of the decision maker and takes any value from 0 to 1. The decision maker has the most optimistic attitude when λ = 0 and the most pessimistic attitude when λ = 1.

The crisp matrix is obtained when all fuzzy numbers are converted into crisp numbers, as shown in Equation (19); the maximum eigenvalue λ_max of the crisp matrix can be obtained and the consistence index (CI) can be calculated from Equation (20). In addition, the random index (RI) of a matrix of n dimensions can be obtained from Table 2. Then, the consistency ratio (CR) can be calculated from Equation (21). If CR < 0.1, the crisp matrix and the original fuzzy matrix satisfy the consistency condition.

Table 2

Random index (RI) of matrix

n	3	4	5	6	7	8	9
RI	0.52	0.89	1.12	1.26	1.36	1.41	1.46

$[{(a_{ij}^{α})}^{λ}] = [\begin{matrix} 1 & {(a_{12}^{α})}^{λ} & . . . & {(a_{1 n}^{α})}^{λ} \\ {(a_{21}^{α})}^{λ} & 1 & . . . & {(a_{2 n}^{α})}^{λ} \\ . . . & . . . & . . . & . . . \\ {(a_{n 1}^{α})}^{λ} & {(a_{n 2}^{α})}^{λ} & . . . & 1 \end{matrix}]$ (19) $CI = \frac{λ_{max} - n}{n - 1}$ (20)

2.5 Construction of the representative matrix of all decision makers

Each individual judgment is different. Hence, the judgments of all decision makers must be brought together. In the conventional AHP, there are two basic methods for aggregating individual judgments into a group judgment: individual judgment aggregation and individual priority aggregation [55]. The most commonly used method is individual judgment aggregation, whose algorithm is a geometric average, which is chosen in this paper.

Considering a group of K decision makers judging the weights of n indices, a total of K fuzzy comparison matrices ${\tilde{A}}_{k} = ({\tilde{a}}_{ijk})_{n \times n}$ are obtained by pairwise comparison, where ${\tilde{a}}_{ijk} = (l_{ijk} . m_{ijk} . u_{ijk})$ . The representative matrix $\tilde{A} = ({\tilde{a}}_{ij})_{n \times n}$ , where ${\tilde{a}}_{ij} = (l_{ij} . m_{ij} . u_{ij})$ , can be obtained from Equation (22) ∼ Equation (24): $CR = \frac{CI}{RI (n)}$ (21) $l_{ij} = min_{k = 1, 2, . . ., K} (l_{ijk})$ (22) $m_{ij} = \sqrt[K]{\prod_{k = 1}^{K} m_{ijk}}$ (23) $u_{ij} = max_{k = 1, 2, . . ., K} (u_{ijk})$ (24)

3 Fuzzy comprehensive evaluation method

3.1 Membership function and membership degree

According to the mathematical classification, there are two types of basic evaluation indices: continuous indices and discrete indices. For a continuous index, a suitable distribution form of a membership function can be selected to obtain the membership degree according to the index characteristics. The most commonly used distribution forms are trapezoidal, triangular, and normal, among others. In contrast, it is difficult to obtain the membership function of a discrete index directly, and the membership degree can be evaluated only by experience. Thus, the corresponding membership degree is generally calculated by constructing a pairwise comparison matrix [56]. The relative importance obtained from the comparison matrix can be analyzed as the membership degree.

3.2 Procedure of the FCE method

Consider two discussion domains: a factor set U ={ u₁, u₂, . . . , u_n } (u_i is the evaluation factor) and an evaluation set V ={ v₁, v₂, . . . , v_m } (v_j is the evaluation grade). Each element in U is judged separately to obtain a membership matrix R by the membership function selected, the form of which can be expressed as Equation (25) [41]: $R = {(r_{ij})}_{n \times m} 0 \leq r_{ij} \leq 1$ (25) where r_ij denotes the membership degree of the i-th evaluation factor to the j-th evaluation grade.

Consider the weight vector of n indices W ={ W₁, W₂, . . . , W_n }, the normalized form of which satisfies the condition in Equation (26): $\sum_{i = 1}^{n} W_{i} = 1$ (26)

The fuzzy mapping of U∼V can be calculated as Equation (27): $B = W \cdot R$ (27) B ={ b₁, b₂, . . . , b_m } is the result of a fuzzy calculation, where b_i represents the membership degree of the evaluation object to the evaluation grade v_i. According to the maximum membership principle, the grade corresponding to the maximum membership degree in B is the final result of the FCE, A multilevel FCE is needed when the evaluation system is classified.

4 Establishment of a risk evaluation system for colluvial cutting slopes

4.1 Establishment of an evaluation index system

According to the characteristics of colluvial cutting slopes and the statistics in the relevant literature [57 –60], an evaluation index system is constructed herein. Corresponding to the applications of the FAHP, the index system can be divided into three layers: a target layer, a criterion layer and an index layer. The architecture of the system and its specific indices are shown in Fig. 4.

Fig.4

Evaluation index system of colluvial cutting slopes.

4.2 Determination of the weights of indices based on the FAHP

To obtain the weights of the indices of each layer, a group of 10 decision makers, including experts and geologists engaged in slope research, is formed. Taking the “Characteristics of rock and soil (U₁₂)” layer as an example, the fuzzy comparison matrices are constructed by these 10 decision makers, and the representative matrix is obtained by aggregating the 10 comparison matrices and then employing Equation (22) ∼ Equation (24), as shown in Table 3.

Table 3
Representative matrix of the subindices within the “Characteristics of rock and soil (U₁₂)” index

U₁₂₁ U₁₂₂ U₁₂₃ U₁₂₄ U₁₂₅ U₁₂₆

U₁₂₁ (1, 1, 1) (0.286, 0.434, 1) (0.286, 0.498, 1) (0.25, 0.398, 0.667) (0.222, 0.286, 0.5) (0.5, 1.521, 3)

U₁₂₂ (1, 2.302, 3.5) (1, 1, 1) (0.4, 0.668, 2.5) (0.333, 0.592, 1.5) (0.25, 0.4, 0.667) (1, 2.393, 3.5)

U₁₂₃ (1, 2.008, 3.5) (0.4, 1.496, 2.5) (1, 1, 1) (0.286, 0.492, 1) (0.286, 0.402, 1) (1, 2.311, 3.5)

U₁₂₄ (1.5, 2.512, 4) (0.667, 1.689, 3) (1, 2.031, 3.5) (1, 1, 1) (0.286, 0.495, 1) (1.5, 2.714, 4)

U₁₂₅ (2, 3.492, 4.5) (1.5, 2.502, 4) (1, 2.487, 3.5) (1, 2.019, 3.5) (1, 1, 1) (2.5, 3.901, 5)

U₁₂₆ (0.333, 0.658, 2) (0.286, 0.418, 1) (0.286, 0.433, 1) (0.25, 0.369, 0.667) (0.2, 0.256, 0.4) (1, 1, 1)

	U₁₂₁	U₁₂₂	U₁₂₃	U₁₂₄	U₁₂₅	U₁₂₆
U₁₂₁	(1, 1, 1)	(0.286, 0.434, 1)	(0.286, 0.498, 1)	(0.25, 0.398, 0.667)	(0.222, 0.286, 0.5)	(0.5, 1.521, 3)
U₁₂₂	(1, 2.302, 3.5)	(1, 1, 1)	(0.4, 0.668, 2.5)	(0.333, 0.592, 1.5)	(0.25, 0.4, 0.667)	(1, 2.393, 3.5)
U₁₂₃	(1, 2.008, 3.5)	(0.4, 1.496, 2.5)	(1, 1, 1)	(0.286, 0.492, 1)	(0.286, 0.402, 1)	(1, 2.311, 3.5)
U₁₂₄	(1.5, 2.512, 4)	(0.667, 1.689, 3)	(1, 2.031, 3.5)	(1, 1, 1)	(0.286, 0.495, 1)	(1.5, 2.714, 4)
U₁₂₅	(2, 3.492, 4.5)	(1.5, 2.502, 4)	(1, 2.487, 3.5)	(1, 2.019, 3.5)	(1, 1, 1)	(2.5, 3.901, 5)
U₁₂₆	(0.333, 0.658, 2)	(0.286, 0.418, 1)	(0.286, 0.433, 1)	(0.25, 0.369, 0.667)	(0.2, 0.256, 0.4)	(1, 1, 1)

Each of the 10 fuzzy comparison matrices should be tested for consistency, as should the representative matrix. The stability of the decision-making environment (α) and the personal attitude (λ) of the decision makers need to be determined first. In this paper, α= 0.5 is used to indicate that the environmental uncertainty is stable, and λ = 0.5 is used to indicate that the attitude of each decision maker is fair. The crisp matrix of the representative matrix of the “Characteristics of rock and soil (U₁₂)” layer can be obtained using Equation (16) ∼ Equation (18), as shown in Table 4.

Table 4

Crisp matrix of the subindices within the “Characteristics of rock and soil (U₁₂)” index

	U₁₂₁	U₁₂₂	U₁₂₃	U₁₂₄	U₁₂₅	U₁₂₆
U₁₂₁	1	0.539	0.570	0.428	0.324	1.636
U₁₂₂	2.276	1	1.059	0.754	0.429	2.322
U₁₂₃	2.129	1.473	1	0.568	0.523	2.281
U₁₂₄	2.631	1.761	2.141	1	0.569	2.732
U₁₂₅	3.371	2.626	2.369	2.135	1	3.826
U₁₂₆	0.912	0.530	0.538	0.413	0.278	1

CR = 0.0975 < 0.1 is obtained by employing Equation (20) ∼ Equation (21), which demonstrates that the representative matrix is reasonable.

When the representative matrix satisfies the consistency condition, the weights of the indices can be calculated. Applying Chang’s method, the initial fuzzy weight of each subindex in the “Characteristics of rock and soil (U₁₂)” index can be calculated using Equation (7) ∼ Equation (10) from Table 3 as follows: $\begin{matrix} D_{1} & = (2.544, 4.138, 7.167) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.033, 0.086, 0.258) \end{matrix}$ $\begin{matrix} D_{2} & = (3.983, 7.355, 12.667) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.052, 0.153, 0.456) \end{matrix}$ $\begin{matrix} D_{3} & = (3.971, 7.710, 12.5) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.052, 0.16, 0.45) \end{matrix}$ $\begin{matrix} D_{4} & = (5.952, 10.441, 16.5) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.078, 0.217, 0.593) \end{matrix}$ $\begin{matrix} D_{5} & = (9, 15.401, 21.5) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.118, 0.32, 0.773) \end{matrix}$ $\begin{matrix} D_{6} & = (2.355, 3.133, 6.067) \otimes \\ (\frac{1}{76.4}, \frac{1}{48.178}, \frac{1}{27.806}) \\ = (0.031, 0.065, 0.218) \end{matrix}$

Then, the values of Di (i = 1, 2, ... , 6) are individually compared, and the degree of possibility P(Di > Dj) is obtained by using Equation (11), as shown in Table 5.

Table 5

Values of P(D_i > D_j)

P(D_i > D_j)	i	1	2	3	4	5	6
j
1	1	1	1	1	1	0.899
2	0.755	1	1	1	1	0.655
3	0.735	0.982	1	1	1	0.636
4	0.579	0.855	0.868	1	1	0.480
5	0.375	0.669	0.675	0.822	1	0.283
6	1	1	1	1	1	1

The minimum degree of probability of each index relative to the other indices is obtained from Table 5 as follows: $d_{1} = min P (D_{1} \geq D_{j}, j = 2, 3, 4, 5, 6) = 0.375$ $d_{2} = min P (D_{2} \geq D_{j}, j = 1, 3, 4, 5, 6) = 0.669$ $d_{3} = min P (D_{3} \geq D_{j}, j = 1, 2, 4, 5, 6) = 0.675$ $d_{4} = min P (D_{4} \geq D_{j}, j = 1, 2, 3, 5, 6) = 0.822$ $d_{5} = min P (D_{5} \geq D_{j}, j = 1, 2, 3, 4, 6) = 1$ $d_{6} = min P (D_{6} \geq D_{j}, j = 1, 2, 3, 4, 5) = 0.283$

The initial weights of the subindices within the “Characteristics of rock and soil (U12)” layer can be expressed as follows: $W^{'} = (0.375, 0.669, 0.675, 0.822, 1, 0.283)^{T}$

The weights of the indices are then normalized and expressed as follows: $W = (0.098, 0175, 0.177, 0.215, 0.262 0.074)^{T}$

Similarly, the representative matrices for the indices of all other layers are obtained, and the results are shown in Table 6 ∼ Table 10. The images of fuzzy numbers of importance pertaining to the representative matrices are shown in Fig. 5.

Table 6

Representative matrix of the subindices within the “Terrain and geomorphology (U₁₁)” index

	U ₁₁₁	U ₁₁₂	U ₁₁₃	U ₁₁₄
U ₁₁₁	(1, 1, 1)	(0.286, 0.452, 1)	(0.5, 1.503, 3)	(0.667, 1.721, 3)
U ₁₁₂	(1, 2.212, 3.5)	(1, 1, 1)	(1.5, 2.916, 4)	(3, 4.126, 5)
U ₁₁₃	(0.333, 0.665, 2)	(0.25, 0.343, 0.667)	(1, 1, 1)	(0.667, 1.702, 3)
U ₁₁₄	(0.333, 0.581, 1.5)	(0.2, 0.242, 0.333)	(0.333, 0.588, 1.5)	(1, 1, 1)

Fig.5

Fuzzy numbers of importance pertaining to representative matrices. (a) comparison with U₁; (b) comparison with U₁₁; (c) comparison with U₂₁; (d) comparison with U₁₁₁; (e) comparison with U₁₂₁; (f) comparison with U_211 .

Table 7

Representative matrix of the subindices within the “Risk evaluation of colluvial cutting slope (U)” index

	U ₁	U ₂
U ₁	(1, 1, 1)	(1, 2.015, 3.5)
U ₂	(0.286, 0.496, 1)	(1, 1, 1)

Table 8

Representative matrix of the subindices within the “Root hazard source (U₁)” index

	U ₁₁	U ₁₂
U ₁₁	(1, 1, 1)	(0.286, 0.474, 1)
U ₁₂	(1, 2.112, 3.5)	(1, 1, 1)

Table 9

Representative matrix of the subindices within the “State hazard source (U₂)” index

	U ₂₁	U ₂₂	U ₂₃
U ₂₁	(1, 1, 1)	(2, 3.201, 4.5)	(2, 3.322, 4.5)
U ₂₂	(0.222, 0.312, 0.5)	(1, 1, 1)	(0.4, 1.498, 2.5)
U ₂₃	(0.222, 0.301, 0.5)	(0.4, 0.668, 2.5)	(1, 1, 1)

Table 10

Representative matrix of the subindices within the “Artificial excavation (U₂₁)” index

	U ₂₁₁	U ₂₁₂	U ₂₁₃
U ₂₁₁	(1, 1, 1)	(0.286, 0.467, 1)	(1, 2.232, 3.5)
U ₂₁₂	(1, 2.143, 3.5)	(1, 1, 1)	(1.5, 2.919, 4)
U ₂₁₃	(0.286, 0.448, 1)	(0.25, 0.343, 0.667)	(1, 1, 1)

Next, the weights of the subindices within the indices ranging from U to U₂₁ are calculated, and the results are shown in Table 11. The normalized weights of all the basic indices are illustrated graphically in Fig. 6. The weight determination results show that the “slope structure (U₁₂₅)” is the most influential index for the stability of colluvial cutting slopes, followed by the “crushed stone content of colluvium (U₁₂₄)”, which is one of the typical characteristics of colluvial cutting slopes differentiating them from rock slopes and soil slopes. The least influential factor is the “vegetation development characteristics (U₁₁₄)”. These results suggest that the “slope structure (U₁₂₅)” and the “crushed stone content of colluvium (U₁₂₄)” should be investigated and determined with a considerable accuracy in the risk evaluation of a colluvial cutting slope.

Table 11

Weights of the indices within each layer

Target layer	Criterion layer	Weight	Index layer	Weight	Index layer	Weight	Normalized weight
Risk evaluation of colluvial cutting slope (U)	Root hazard source (U₁)	0.679	Terrain and geomorphology (U₁₁)	0.314	Slope height (U₁₁₁)	0.257	0.055
					Slope angle (U₁₁₂)	0.422	0.090
				Slope shape (U₁₁₃)	0.211	0.045
					Vegetation development characteristics (U₁₁₄)	0.110	0.023
			Characteristics of rock and soil (U₁₂)	0.686	Bedrock lithology (U₁₂₁)	0.098	0.046
					Weathering degree of bedrock (U₁₂₂)	0.175	0.082
					Thickness of colluvium (U₁₂₃)	0.177	0.082
					Crushed stone content of colluvium (U₁₂₄)	0.215	0.100
					Slope structure (U₁₂₅)	0.262	0.122
					Water table (U₁₂₆)	0.074	0.034
	State hazard source (U₂)	0.321	Artificial excavation (U₂₁)	0.600	Excavation height (U₂₁₁)	0.359	0.069
					Excavation angle (U₂₁₂)	0.495	0.095
					Excavation mode (U₂₁₃)	0.146	0.028
			Rainfall (U₂₂)	0.211	Rainfall intensity (U₂₂₁)	1.000	0.068
			Earthquake (U₂₃)	0.189	Seismic intensity (U₂₃₁)	1.000	0.061

Fig.6

Normalized weights of the basic indices.

4.3 Constructing of the membership functions for basic indices

Combined with the characteristics of colluvial cutting slopes and the relevant literature [61], a risk evaluation set is established with four grades in this study: V = {Low, Medium, High, Very high}. For continuous indices, including the slope height, slope angle, thickness of colluvium, crushed stone content of colluvium, excavation height, and excavation angle, membership functions with a triangular form are constructed in this paper according to the risk evaluation criteria. Due to differences in their dimension and length, the raw data must be normalized. In contrast, for discrete indices, including the slope shape, vegetation development characteristics, bedrock lithology, weathering degree of bedrock, slope structure, water table, excavation mode, rainfall intensity, and seismic intensity, the importance comparison matrices are constructed by experts according to the risk evaluation criteria. The risk evaluation criteria for the basic indices are shown in Table 12. The values related to the continuous indices in brackets represent the normalized values, and the values related to the discrete indices in brackets are classification grades.

Table 12
Risk evaluation criteria for the basic indices

Basic indices Evaluation criteria for risk grades

Low (I) Medium (II) High (III) Very high (IV)

Slope height U₁₁₁ (m) <30 (0.50) 35 (0.58) 50 (0.83) >60 (1.00)

Slope angle U₁₁₂ (°) <20 (0.33) 30 (0.50) 20 (0.83) >60 (1.00)

Slope shape U₁₁₃ The vertical and cross sections of the slope are both either straight or concave (I) The vertical section is straight or concave, and the cross section is convex (II) The vertical section is convex, and the cross section is straight or concave (III) The vertical and cross sections are both either convex or complex (IV)

Vegetation development characteristics U₁₁₄ Trees, shrubs and grass are all well developed (I) Trees are well developed, shrubs and grass are not (II) Shrubs and grass are well developed, trees are not (III) Trees, shrubs and grass are all not developed (IV)

Bedrock lithology U₁₂₁ Extremely hard rock (I) Hard rock (II) Soft rock (III) Extremely soft rock (IV)

Weathering degree of bedrock U₁₂₂ Unweathered or weakly weathered (I) Moderately weathered (II) Strongly weathered (III) Fully weathered (IV)

Thickness of colluvium U₁₂₃ (m) <5 (0.20) 10 (0.40) 20 (0.80) >25 (1.0)

Crushed stone content of colluvium U₁₂₄ (%) >75 (1.0) 32.5 (0.83) 37.5 (0.50) <25 (0.33)

Slope structure U₁₂₅ Advantageous (I) General (II) Disadvantageous (III) Extremely disadvantageous (IV)

Water table U₁₂₆ (H is slope height) <0.25 H (I) (0.25∼0.5)H (II) (0.5∼0.75)H (III) >0.75 H (IV)

Excavation height U₂₁₁ (m) <20 (0.40) 25 (0.50) 40 (0.80) >50 (1.0)

Excavation angle U₂₁₂ (°) <20 (0.33) 30 (0.50) 50 (0.83) >60 (1.0)

Excavation mode U₂₁₃ Direct excavation (I) Partial drilling excavation (II) Drilling excavation (III) Blasting excavation (IV)

Rainfall intensity U₂₂₁ (mm/d) <10 (I) 10∼25 (II) 25∼50 (III) >50 (IV)

Seismic intensity U₂₃₁ <6 (I) 6∼7 (II) 7∼8 (III) >8 (IV)

Basic indices	Evaluation criteria for risk grades
Slope height U₁₁₁ (m)	<30 (0.50)	35 (0.58)	50 (0.83)	>60 (1.00)
Slope angle U₁₁₂ (°)	<20 (0.33)	30 (0.50)	20 (0.83)	>60 (1.00)
Slope shape U₁₁₃	The vertical and cross sections of the slope are both either straight or concave (I)	The vertical section is straight or concave, and the cross section is convex (II)	The vertical section is convex, and the cross section is straight or concave (III)	The vertical and cross sections are both either convex or complex (IV)
Vegetation development characteristics U₁₁₄	Trees, shrubs and grass are all well developed (I)	Trees are well developed, shrubs and grass are not (II)	Shrubs and grass are well developed, trees are not (III)	Trees, shrubs and grass are all not developed (IV)
Bedrock lithology U₁₂₁	Extremely hard rock (I)	Hard rock (II)	Soft rock (III)	Extremely soft rock (IV)
Weathering degree of bedrock U₁₂₂	Unweathered or weakly weathered (I)	Moderately weathered (II)	Strongly weathered (III)	Fully weathered (IV)
Thickness of colluvium U₁₂₃ (m)	<5 (0.20)	10 (0.40)	20 (0.80)	>25 (1.0)
Crushed stone content of colluvium U₁₂₄ (%)	>75 (1.0)	32.5 (0.83)	37.5 (0.50)	<25 (0.33)
Slope structure U₁₂₅	Advantageous (I)	General (II)	Disadvantageous (III)	Extremely disadvantageous (IV)
Water table U₁₂₆ (H is slope height)	<0.25 H (I)	(0.25∼0.5)H (II)	(0.5∼0.75)H (III)	>0.75 H (IV)
Excavation height U₂₁₁ (m)	<20 (0.40)	25 (0.50)	40 (0.80)	>50 (1.0)
Excavation angle U₂₁₂ (°)	<20 (0.33)	30 (0.50)	50 (0.83)	>60 (1.0)
Excavation mode U₂₁₃	Direct excavation (I)	Partial drilling excavation (II)	Drilling excavation (III)	Blasting excavation (IV)
Rainfall intensity U₂₂₁ (mm/d)	<10 (I)	10∼25 (II)	25∼50 (III)	>50 (IV)
Seismic intensity U₂₃₁	<6 (I)	6∼7 (II)	7∼8 (III)	>8 (IV)

Taking the continuous “slope height (U₁₁₁)” index as an example, the membership functions corresponding to each risk grade Low (I), Medium (II), High (III), Very high (IV) are constructed following Equation (28) ∼ Equation (31):

$V_{I U_{111}} (x) = {\begin{matrix} 1 & \begin{matrix} \end{matrix} & x \leq 0 . 5 \\ \frac{x - 0.58}{0.5 - 0.58} & \begin{matrix} \end{matrix} & 0.5 < x \leq 0.58 \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix}$ (28)

$V_{II U_{111}} (x) = {\begin{matrix} \frac{x - 0.5}{0.58 - 0.5} & \begin{matrix} \end{matrix} & 0.5 < x \leq 0 . 58 \\ \frac{0.83 - x}{0.83 - 0.58} & \begin{matrix} \end{matrix} & 0.58 < x \leq 0.83 \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix}$ (29)

$V_{III U_{111}} (x) = {\begin{matrix} \frac{x - 0.58}{0.83 - 0.58} & \begin{matrix} \end{matrix} & 0.58 < x \leq 0 . 83 \\ \frac{1 - x}{1 - 0.83} & \begin{matrix} \end{matrix} & 0.83 < x \leq 1 \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix}$ (30)

$V_{IV U_{111}} (x) = {\begin{matrix} 1 & \begin{matrix} \end{matrix} & x > 1 \\ \frac{x - 0.83}{1 - 0.83} & \begin{matrix} \end{matrix} & 0.83 < x \leq 1 \\ 0 & \begin{matrix} \end{matrix} & otherwise \end{matrix}$ (31)

The membership functions for the other continuous indices are constructed similarly, and images of the membership functions are shown in Fig. 7:

Fig.7

Membership functions for continuous indices (a) U₁₁₁; (b) U₁₁₂; (c) U₁₂₃; (d) U₁₂₄; (e) U₂₁₁; (f) U₂₁₂. 1-membership of low grade; 2-membership of medium grade; 3-membership of high grade; 4-membership of very high grade.

Taking the discrete “Slope shape U₁₁₃” index as an example, the importance comparison matrices corresponding to each classification grade are constructed following Equation (32) ∼ Equation (35):

$P_{I U_{113}} = [\begin{matrix} 1 & 2 & 3 & 5 \\ 1 / 2 & 1 & 2 & 3 \\ 1 / 3 & 1 / 2 & 1 & 2 \\ 1 / 5 & 1 / 3 & 1 / 2 & 1 \end{matrix}]$ (32)

$P_{II U_{113}} = [\begin{matrix} 1 & 1 / 3 & 2 & 3 \\ 3 & 1 & 2 & 5 \\ 1 / 2 & 1 / 2 & 1 & 3 \\ 1 / 3 & 1 / 5 & 1 / 3 & 1 \end{matrix}]$ (33)

$P_{III U_{113}} = [\begin{matrix} 1 & 1 / 3 & 1 / 4 & 1 / 3 \\ 3 & 1 & 1 / 3 & 1 \\ 4 & 3 & 1 & 3 \\ 3 & 1 & 1 / 3 & 1 \end{matrix}]$ (34)

$P_{IV U_{113}} = [\begin{matrix} 1 & 1 / 2 & 1 / 3 & 1 / 7 \\ 2 & 1 & 1 / 2 & 1 / 5 \\ 3 & 2 & 1 & 1 / 3 \\ 7 & 5 & 3 & 1 \end{matrix}]$ (35)

The membership degree to each risk grade can be calculated from the importance comparison matrix determined by the selected classification grade. The membership degree values of the discrete indices to each classification grade are thus obtained, as shown in Table 13.

Table 13

Membership degree values of discrete indices

Classification grade	Risk grade	U ₁₁₃	U ₁₁₄	U ₁₂₁	U ₁₂₂	U ₁₂₅	U ₁₂₆	U ₂₁₃	U ₂₂₁	U ₂₃₁
I	I	0.483	0.483	0.637	0.565	0.669	0.507	0.558	0.546	0.546
	II	0.272	0.272	0.258	0.262	0.243	0.263	0.290	0.304	0.304
	III	0.157	0.157	0.105	0.118	0.088	0.152	0.095	0.100	0.100
	IV	0.088	0.088	0	0.055	0	0.078	0.056	0.050	0.050
II	I	0.248	0.235	0.303	0.170	0.278	0.248	0.284	0.288	0.288
	II	0.483	0.449	0.487	0.473	0.467	0.483	0.487	0.477	0.477
	III	0.191	0.235	0.140	0.284	0.160	0.191	0.165	0.154	0.154
	IV	0.078	0.082	0.071	0.073	0.095	0.078	0.064	0.081	0.081
III	I	0.085	0.122	0.073	0.073	0.122	0.085	0.078	0.084	0.084
	II	0.204	0.227	0.170	0.170	0.227	0.204	0.240	0.243	0.243
	III	0.507	0.424	0.473	0.473	0.424	0.507	0.496	0.502	0.502
	IV	0.204	0.227	0.284	0.284	0.227	0.204	0.186	0.172	0.172
IV	I	0.072	0.088	0.072	0.072	0.067	0.072	0.085	0.092	0.092
	II	0.123	0.157	0.123	0.123	0.107	0.123	0.133	0.136	0.136
	III	0.218	0.272	0.218	0.218	0.177	0.218	0.257	0.261	0.261
	IV	0.587	0.483	0.587	0.587	0.649	0.587	0.526	0.511	0.511

4.4 Risk evaluation process of colluvial cutting slopes based on a multilevel FCE

Based on the membership function constructed for each basic index, the corresponding membership degree to each evaluation grade can be obtained when given a value to the index, and the membership degree matrix of basic indices R reflecting the fuzzy relation between U and V can be obtained.

The risk evaluation system of colluvial cutting slopes based on the FCE method is segmented into three levels according to the index system structure. The membership degree vector of the first-level B ₁ can be calculated by combining the weight vector of basic indices W ₁ in the index layer with the membership degree matrix of basic indices R following Equation (36): $B_{1} = W_{1} \cdot R = {b_{11}, b_{12}, b_{13}, b_{14}}$ (36) The membership degree vectors of the second-level B ₂ and third-level B ₃ can be obtained respectively by Equation (37) and Equation (38): $B_{2} = W_{2} \cdot B_{1} = {b_{21}, b_{22}, b_{23}, b_{24}}$ (37) $B_{3} = W_{3} \cdot B_{2} = {b_{31}, b_{32}, b_{33}, b_{34}}$ (38) where W ₂ denotes the weight vector of the upper indices in the index layer and W ₃ represents the weight vector of the indices in the criterion layer.

The membership degree vector of the third-level B ₃ is the membership degree of the evaluation object to each risk grade. According to the maximum membership degree principle, b_3i = max {b₃₁, b₃₂, b₃₃, b₃₄}, and the risk grade related to b_3i is the result of the slope evaluated.

The risk evaluation system process is illustrated in Fig. 8.

Fig.8

Risk evaluation system process.

5 Case application and results

The Xiaomengyang-Mohan section of the Kunmo Expressway located in Jinghong, Xishuangbanna, Mengla County, Yunnan Province, is a part of the planned AH3 line of the Asian Highway Network that starts in Ulan Ude, Russia, and enters China via Ulaanbaatar, Mongolia, and then stretches from Beijing to Chiang Mai, Thailand, via Shanghai, Hangzhou, Kunming, Mohan, South Pylon, and Laos. The Xiaomengyang-Mohan highway is the last part of the Kunming-Bangkok passage in China. The total length of the highway reconstruction and extension project is 156.043 kilometers. The geographical location of the project is shown in Fig. 9.

Fig.9

Geographical location of the project.

The surface layer of the highway route area is mainly composed of Quaternary colluvial and residual product. The colluvial slope layer is distributed mainly atop the slope and valley slope and is composed of silty clay, gravel soil, and massive stone soil. Most of the basement rocks are composed of loosely cemented fuchsia conglomerate intercalated with lithic sandstone, argillaceous siltstone and gray-yellow quartz sandstone. Field photos of typical features are shown in Fig. 10, and the typical geological profile is provided in Fig. 11. According to meteorological data, the annual rainfall within the route area is 1291.2 ∼ 1943.9 mm. According to the relevant statistics, the route area is a multi-seismic area, the seismic intensity is VII∼VIII magnitude. Hence, the complex factors of instability for these colluvial slopes represents considerable hidden risks to slope cutting engineering.

Fig.10

Typical field photos of colluvial slopes.

Fig.11

Typical geological profile of the colluvial slope in the project area.

The risk evaluation system established in this paper is used to evaluate the risks of 148 colluvial cutting slopes along the Xiaomengyang-Mohan highway section. The results are as follows. There are 24 slopes of low risk (grade I), accounting for 16.2% of the total; 85 of medium risk (grade II), accounting for 57.4%; 22 of high risk (grade III), accounting for 14.9%; and 17 of very high risk (grade IV), accounting for 11.5%, as shown in Fig. 12. According to the site construction data, the failure or local failure slopes all have a risk grade of II, III, or IV: failure occurred in 15 out of 85 slopes of risk grade II, accounting for 18%; 13 out of 22 slopes of risk grade III, accounting for 59%; and 16 out of 17 slopes of risk grade IV, accounting for 94%, as shown in Fig. 13. The risk grades of some typical slopes are shown in Table 14. Evidently, the evaluation results obtained by applying the risk evaluation system proposed in this paper are in good agreement with the actual slope instability states. These evaluation results show that the risk grades of colluvial cutting slopes in different locations along this section of highway are different due to the discrete spatial distribution of rock and soil masses, but most of the slopes are of risk grade II, indicating that this section displays medium engineering geological conditions. According to expert suggestions and the technical safety risk evaluation guide for high cutting slope engineering construction, different engineering protection measures should be adopted for slopes of different risk grades: frame beams or retaining wall engineering constructions are recommended for slopes of risk grade I; grouting engineering is recommended for slopes of risk grade II; anchorage engineering employing anchor cables is recommended for slopes of risk grade III; and anti-slide piles are recommended for slopes of risk grade IV. Furthermore, slopes of risk grades III and IV should be monitored more closely, and protective measures should be adjusted according to the monitoring results to ensure the safety and stability of colluvial slopes along the highway.

Fig.12

Slope grade distribution along the Xiaomengyang-Mohan highway.

Fig.13

Distribution of failure slopes along the Xiaomengyang-Mohan highway.

Table 14

Evaluation results of typical colluvial slopes along the Xiaomengyang-Mohan highway

Slope serial number	Evaluation membership degree				Evaluation result	Actual state
	I	II	III	IV
15	0.2133	0.2172	0.261	0.3085	IV	Failure
30	0.3034	0.2608	0.2149	0.2209	I	Stable
83	0.2332	0.2909	0.3035	0.1725	III	Local failure
94	0.084	0.2791	0.3895	0.2474	III	Failure
131	0.1578	0.3936	0.2221	0.2265	II	Stable
143	0.2044	0.3759	0.3042	0.1155	II	Local failure

6 Conclusion

This paper presents an evaluation index system for colluvial cutting slopes and establishes a risk evaluation system for such slopes based on the FAHP and FCE methods. This evaluation system is then applied to evaluate the risks of 148 colluvial cutting slopes along the Xiaomengyang-Mohan highway. According to the result, some conclusions can be summarized as follows:

In the process of evaluating the risk of a colluvial cutting slope, the weights of indices are decided by 10 experts; considering the subjectivity and variability of human judgment, the FAHP resolves this problem well.

The system of colluvial cutting slopes is complicated, as it contains many influencing factors on the slope stability, and fuzziness and uncertainty are evident in the process of quantization during the evaluation. The three-level FCE applied in this paper works well for this evaluation.

The evaluation results obtained in this paper are in good agreement with the actual slope instability states of real-world sites. These findings show that the risk evaluation system of colluvial cutting slopes established in this paper has a high efficiency and a high accuracy, and it can provide decision-making advice to engineers and guidance for supporting measures and construction management of colluvial cutting slopes.

The future work is to further discuss the advantages and disadvantages of the risk evaluation system established in this paper in detail by comparing it with other methods and investigate more effective methods to ascertain the main influencing factors and weight determinations to broadly adapt the risk evaluation system proposed herein to more types of engineering risk evaluations.

Footnotes

Acknowledgments

The work reported in this paper is financially supported by the National Natural Science Foundation of China (No. 51779250, No. U1402231, No. 51679232), the National Key Basic Research Program of China (973 Program) under Grant No. 2015CB057905, and the Traffic Science, Technology and Education Project of Yunnan Province [2017] No. 33.

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Basic indices	Evaluation criteria for risk grades
	Low (I)	Medium (II)	High (III)	Very high (IV)
Slope height U₁₁₁ (m)	<30 (0.50)	35 (0.58)	50 (0.83)	>60 (1.00)
Slope angle U₁₁₂ (°)	<20 (0.33)	30 (0.50)	20 (0.83)	>60 (1.00)
Slope shape U₁₁₃	The vertical and cross sections of the slope are both either straight or concave (I)	The vertical section is straight or concave, and the cross section is convex (II)	The vertical section is convex, and the cross section is straight or concave (III)	The vertical and cross sections are both either convex or complex (IV)
Vegetation development characteristics U₁₁₄	Trees, shrubs and grass are all well developed (I)	Trees are well developed, shrubs and grass are not (II)	Shrubs and grass are well developed, trees are not (III)	Trees, shrubs and grass are all not developed (IV)
Bedrock lithology U₁₂₁	Extremely hard rock (I)	Hard rock (II)	Soft rock (III)	Extremely soft rock (IV)
Weathering degree of bedrock U₁₂₂	Unweathered or weakly weathered (I)	Moderately weathered (II)	Strongly weathered (III)	Fully weathered (IV)
Thickness of colluvium U₁₂₃ (m)	<5 (0.20)	10 (0.40)	20 (0.80)	>25 (1.0)
Crushed stone content of colluvium U₁₂₄ (%)	>75 (1.0)	32.5 (0.83)	37.5 (0.50)	<25 (0.33)
Slope structure U₁₂₅	Advantageous (I)	General (II)	Disadvantageous (III)	Extremely disadvantageous (IV)
Water table U₁₂₆ (H is slope height)	<0.25 H (I)	(0.25∼0.5)H (II)	(0.5∼0.75)H (III)	>0.75 H (IV)
Excavation height U₂₁₁ (m)	<20 (0.40)	25 (0.50)	40 (0.80)	>50 (1.0)
Excavation angle U₂₁₂ (°)	<20 (0.33)	30 (0.50)	50 (0.83)	>60 (1.0)
Excavation mode U₂₁₃	Direct excavation (I)	Partial drilling excavation (II)	Drilling excavation (III)	Blasting excavation (IV)
Rainfall intensity U₂₂₁ (mm/d)	<10 (I)	10∼25 (II)	25∼50 (III)	>50 (IV)
Seismic intensity U₂₃₁	<6 (I)	6∼7 (II)	7∼8 (III)	>8 (IV)