# Engineering Application and Prediction of the Influence Area of the Rockfall Hazards

Authors: Xiaohui Liao; Xueliang Wang; Lihui Li; Haiyang Liu; Zhifa Yang; Zigan Chen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The identification of potential rockfall and the accurate prediction of its trajectory are critical in prevention and mitigation of rockfall hazard. It is an important precondition to assess the uncertainty of rockfall motion, study the effective identification technology of potential rockfall, predict the rockfall trajectory, and calculate the threatened area by rockfall hazards. In this study, field investigations and numerical simulations were carried out to identify potential rockfall on a weathered rock slope. As a case study, our calculations results show that the area of tensile stress concentration and plastic failure is the potential area where the rockmass will fall off the surface of the weathered rock slope. A mathematical model for calculating the rockfall influence area of the weathered rock slope was established based on the optimization theory, neural network technology, and genetic optimization algorithm. The rockfall influence area of the weathered rock slope was determined using maximum horizontal distance of rockfall in the specified slope cross sections and described on the topographic map using spline curves to form a closed possibly vulnerable area. As a case study, our calculations confirm that the distributions of the plastic failure and tensile stress areas obtained from the numerical simulations are consistent with the dangerous rock masses identified by field investigations at Guanyindong Slope that is a popular tourist scenic spot in Zhejiang Province, China. In this study, it has been indicated that the influence area can be used as the basis for the design of passive protection methods for rock slopes vulnerable to rockfall hazards.


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## Body

## 1. Introduction

Rockfall is a common geological disaster that occurs during any construction activities in the mountainous region. Because rockfalls are random and destructive, the trajectory of rockfalls has not been effectively predicted. These disasters can result in enormous losses to any construction projects and tourism sites in the hilly/mountainous areas.In recent years, many studies have been conducted on the controlling factors and motion characteristics of rockfalls using theoretical analysis [1], field investigations [2–4], statistical analysis [5], field tests [6–8], and numerical simulation analysis [9]. Previous models can be divided into three groups: empirical models [3, 10–12], analysis and calculation models based on the theory of kinematics [5, 10, 13–19], and mathematical models based on Geographic Information System (GIS) [20–22]. Empirical models usually require a great deal of field investigation and experiments for statistical analysis and thus consume a great amount of manpower and financial resources. Formulas based on empirical models are often only locally applicable. Both analysis and calculation models based on the theory of kinematics and mathematical models based on GIS require a large number of input parameters related to slope morphology and rolling blocks’ characteristics, which are usually not readily available.All of the above-mentioned studies satisfactorily explain the characteristics of the rockfall motion of the slope and provide a theoretical basis for the prevention and control of rockfall hazards. However, it is impossible to use empirical formulas or specific mathematical expression to accurately predict the potential trajectory of dangerous rockmasses after falling because the trajectory is closely related to the complex and variable slope parameters (e.g., the shape of the slope, the slope’s vegetation cover, and the shape and size of the blocks). Due to many factors controlling rockfall, the slope roughness exerts a limiting effect on the lateral motion of the rockfall, especially for steep rock slopes [15, 23]. In addition, several complex parameters are required to simulate the three-dimensional rockfall run-out, which increases the level of uncertainty. However, it is feasible to approximate the rockfall motion as a planar motion for a steep rock slope with a relatively smooth surface [24]. Taking the Guanyindong Rock Slope, which is located in a tourist scenic spot in Zhejiang Province, China, as an example, field investigations and numerical simulation analysis were used to identify potential dangerous rockmass. In addition, the neural network technique was used to calculate the influence area of rockfall. Based on field investigation and numerical simulation results, a corresponding mathematical model for determining the influence range was established, and the maximum run-out distance of the potential rockfall was computed using a genetic optimization algorithm. This study provides a basis for the design of passive preventative methods for preventing rockfall.


## 2. Calculation Theory of the Influence Area of Rockfall

Authors have established the model to calculate the run-out distance of rockfall. This model can calculate the maximum run-out distance of rockfall at a specific profile of the rock slope.


### 2.1. Basic Assumptions

For any profile of a steep slope that weathers easily, the slope’s shape, slope angle, the accumulation at the toe of the slope, and the amount of vegetation covering on the slope’s surface are all fixed constants. Historical rockfall data also shows that the mass of the rockfall block and its location usually change. Therefore, in this study, the following assumptions are made to establish a model for calculating the horizontal distance of the rockfall for any slope profile.(1)
The rockfall motion is a planar motion.(2)
The initial velocity of the dangerous rock mass at the start of the rockfall is zero.(3)
For any given profile of the slope, the mass of the dangerous rock mass and the position of the rockfall block are variables.


### 2.2. Establishment of the Local Coordinate System

Based on the above assumptions, a local coordinate system was established for the slope (see Figure1). The x axis was defined as the direction of the slope’s surface; the z axis was defined as the elevation direction of the slope; and the y axis was defined as the direction perpendicular to the paper.Figure 1
The coordinate system for the rockfall horizontal distance calculation model. (a) Weathered rock slope surface. (b) Cross profile I-I.
(a)
(b)


### 2.3. Basic Equations and Analysis

It is assumed that rockfall occurs for any sectionx=xi and that rockfall has a planar motion. Also, it is assumed that, at the beginning of the rockfall motion, the position of the dangerous rockmass v is xi,yi,zi, the mass of the potential rockfall is mv (kg), and the position after the rockfall of the dangerous rockmass is xj,yj,zj. For section x=xi, the dangerous rockmass v rockfall and the resulting horizontal distance of the motion are(1)Dv,x=xi=yj−yi=fz,mv,v=1,2,3,…,n1;i=1,2,3,…,n2.Iffxiz,mv=fz,mv+yi, then(2)yj=fxiz,mv.In the above formulas,Dv,x=xi is the rockfall horizontal distance (m) of the dangerous rockmass v at cross section x=xi of the slope, and f and fx are the rockfall motion functions.The calculation of the maximum horizontal distance of the rockfall at cross profilex=xi can be converted into an optimization problem expressed as follows:(3)maxyx=xi=maxfxiz,mv,s.txi≥0,z≥0,mv≥0,where v is the number of the dangerous rockmasses at profile x=xi; i.e., v=1,2,3,…,n1; and i is the number of the cross sections of the slope; i.e., i=1,2,3,…,n2.To solve the optimality theory problem expressed by (3), the position of the dangerous rockmass where the weathered slope may be rockfall, i.e., the range of the x and z values, is determined first. For convenience, a finite number of profiles were defined along the x direction of the slope, the maximum rockfall horizontal distance of each profile was calculated and marked on the topographic map, and a spline curve was used to connect all of the points to form an enclosed area; this area represents the influence area of the weathered slope rockfall.


### 2.4. Constraints

As shown in (2) and (3), the calculation of the influence area of the rockfall at profile x=xi requires that the position z and mass m (kg) of the rockfall dangerous rock mass, i.e., the constraints of the model in (3), be determined as follows.


#### 2.4.1. Position Constraints for the Rockfall Dangerous Rockmass

The rockfall and spalling of a rock slope are the results of the interaction of various external forces. Rockfall occurs due to a combination of the rockmass gravity and external forces, such as fissure water pressure, expansion pressure exerted by plant roots, and weathering, which is caused by high-steep slopes and steeply inclined fissures. Examples of this failure include brittle tensional cracks in the surface of the rockmass under low stress conditions, spalling of the surface rockmass due to the close relationship between the rockmass structure and the stress state of the rockmass, and cantilever failure, which is the formation of tensional cracks caused by the shear failure or plastic failure of the rockmass in the rock cavity area under high stress conditions. Therefore, the tensile stress distribution and the plastic failure inside the rockmass of steep weathered slopes are important conditions that can accelerate the occurrence of rockfall. Accordingly, the area where the slope may be rockfall can be identified and used as the position constraints needed to calculate the horizontal distance of the rockfall motion.FLAC3D is a 3D numerical simulation software developed by ITASCA that is commonly used in geotechnical engineering. FLAC3D uses the explicit difference scheme to solve the governing differential equation of the field. Therefore, it can satisfactorily simulate characteristics such as the internal stress state, the yield failure, and the plastic flow of the rockmass. In addition, it can be used to analyze the 3D spatial deformation behavior of a rockmass after it reaches the yield limit [25] and to determine the plastic failure zone and the tensile stress concentration of the rockmass. In order to determine the position range of the dangerous rockmass and where it may be rockfall, we used FLAC3D to calculate the tensile stress zone and plastic failure zone at any position on the slope’s surface. Based on this, the detailed slope failure conditions of the site can be used to determine the two zones, and the highly overlapping zone of the two can be used as the position constraints for the dangerous rockmass that may be rockfall.


#### 2.4.2. Mass Constraints of the Collapsing Dangerous Rockmass

The rock mass with potential rockfall was determined using the following methods:(1)
A detailed investigation of historical rockfall of the steep weathered slope was conducted. The shape, size, location, and distance of the rockfall block were counted and analyzed in the detailed investigation, so as to determine the volume for the historical rockfall mass.(2)
Unmanned aerial vehicle (UAV) tilt photography modeling and 3D laser scanning techniques were employed to establish a 3D numerical model of the slope mass and to identify and measure the size of the dangerous rockmass that could be rockfall.(3)
According to the Engineering Rockmass Test Method Standards (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013) [26], drill core sampling (ϕ 50 mm × 100 mm in size, 5 samples in each group) is conducted in different areas depending on types of rock present on the slope, and then laboratory tests were conducted to obtain the physical and mechanical parameters of the rock samples, including the natural density of the slope mass. The masses (m=ρv) of various dangerous rockmasses determined from steps 1 and 2 were calculated and ranked in ascending order; mmin,mmax is the mass constraint range of the rockfall dangerous rockmass.


### 2.5. Model Solution

#### 2.5.1. Calculation of the Horizontal Distance of the Rockfall

Due to the randomness of the rockfall, the function of the horizontal distance of the rockfall motiony=fxz,m is uncertain for any cross profile of the weathered slope; i.e., the functional mapping relationship between the horizontal distance of the rockfall y, the rockfall block position coordinate z, and the mass of the rockfall block m cannot be described by an exact expression. However, based on the uncertainty function’s mapping relationship, the BP (backpack) neural network can learn the rules of the corresponding function by analyzing the sample data; thus, the result closest to the expected output value can be obtained given an input value [27, 28]. As a result, the BP neural network can be used to express the function of the horizontal distance of the rockfall y=fxz,m; see Figure 2.Figure 2
The rockfall horizontal distance calculation model.(1) Establishment of the Network Model. We propose the fact that the structural model of the BP neural network of the horizontal distance of the rockfall function has the three hidden layer neural network structure shown in Figure 2.(2) The Input Layer. The input layer includes the position coordinate z of the dangerous rockmass and the mass m of the dangerous rockmass. The input layer can be expressed in vector form as X=z,mT=x1,x2T.(3) The Hidden Layer. The output vector of each hidden layer is expressed as U3=u13,u23,…,um33T, U2=u12,u22,…,um22T, and U1=u11,u21,…,um11T. The weight matrix from the input layer to the hidden layer is V1=v11,v21,…,vm11T, V2=v12,v22,…,vm22T, and V3=v13,v23,…,vm33T. The column vector vji is the weight vector corresponding to the jth neuron of the ith hidden layer. If the threshold of the neuron is Ti, then(4)uj3=fnetj,netj=∑i=0m2vij3ui2−Ti2,j=1,2,…,m3,(5)uj2=fnetj,netj=∑i=0m1vij2ui1−Ti1,j=1,2,…,m2.(6)uj1=fnetj,netj=∑i=02vij1xi−Ti0,j=1,2,…,m1.(4) The Output Layer. The output layer is the value of the horizontal distance of the rockfall motion, which can be expressed by the vector O=o1T. The rockfall horizontal distances from investigations of historical rockfalls or from numerical simulation are taken as the expected output value, which can be expressed by the vector d=d1T. This value can be used as the basis for the neural network, which can be used to calculate and adjust the weights of the output layer and the hidden layers. If the weight matrix between hidden layer 3 and the output layer is d=ω1T=ω11,ω21,…,ωm31, the output layer can be expressed as(7)o1=fnetj,netj=∑i=0m3ωi1ui3−Ti4.(5) The Transfer Function. The unipolar Sigmoid function was used for the transfer functions of both the output layer and the hidden layer [29]:(8)fx=11+e−x.In summary, when using the neural network model, as shown by (4) through (8), as the model for calculating the horizontal distance of the rockfall motion, as shown in (2), it is necessary to use a large number of rockfall motion statistical data points to train the neural network model. As a result, the model can be used to calculate the horizontal distance of the rockfall motion y of any dangerous rockmass. In the calculation, the rockfall motion sample data can be obtained from historical rockfall data from field investigations, the rockfall numerical simulation, and/or from physical model tests.


#### 2.5.2. Solution for the Influence Area of the Rockfall

The genetic optimization algorithm is a type of mathematical optimization algorithm formed by mimicking the biological evolution process of chromosomes, which undergo reproduction, crossover, and mutation to produce new, fitter chromosomes [30]. Because it is an effective method for solving the optimality problems, the optimality problem shown in (3) can be solved using the genetic optimization algorithm using the following steps:(1)
Randomly generate the initial populationP0 and then use the model expressed in (4)–(8) to calculate the rockfall horizontal distance of each individual mass. Perform the nondominated sorting of the population and give each individual a rank.(2)
Perform the binary tournament selection, crossover, and mutation ofP0 to obtain a new population Q0 and let t=0.(3)
CombinePt and Qt to form a new population Rt and then perform the nondominated sorting of Rt and select the best N individuals to generate a population Pt+1.(4)
Reproduce, cross over, and mutate the populationPt+1 to generate a population Qt+1 Qt + 1.(5)
If the number of iterations reaches the preset maximum number, the optimization ends. Otherwise, go to step 1.Based on the above-mentioned steps, a genetic algorithm (GA) program was written using MATLAB 2012b to calculate the maximum rockfall horizontal distance of any cross profile of the weathered slope mass.


## 2.1. Basic Assumptions

For any profile of a steep slope that weathers easily, the slope’s shape, slope angle, the accumulation at the toe of the slope, and the amount of vegetation covering on the slope’s surface are all fixed constants. Historical rockfall data also shows that the mass of the rockfall block and its location usually change. Therefore, in this study, the following assumptions are made to establish a model for calculating the horizontal distance of the rockfall for any slope profile.(1)
The rockfall motion is a planar motion.(2)
The initial velocity of the dangerous rock mass at the start of the rockfall is zero.(3)
For any given profile of the slope, the mass of the dangerous rock mass and the position of the rockfall block are variables.


## 2.2. Establishment of the Local Coordinate System

Based on the above assumptions, a local coordinate system was established for the slope (see Figure1). The x axis was defined as the direction of the slope’s surface; the z axis was defined as the elevation direction of the slope; and the y axis was defined as the direction perpendicular to the paper.Figure 1
The coordinate system for the rockfall horizontal distance calculation model. (a) Weathered rock slope surface. (b) Cross profile I-I.
(a)
(b)


## 2.3. Basic Equations and Analysis

It is assumed that rockfall occurs for any sectionx=xi and that rockfall has a planar motion. Also, it is assumed that, at the beginning of the rockfall motion, the position of the dangerous rockmass v is xi,yi,zi, the mass of the potential rockfall is mv (kg), and the position after the rockfall of the dangerous rockmass is xj,yj,zj. For section x=xi, the dangerous rockmass v rockfall and the resulting horizontal distance of the motion are(1)Dv,x=xi=yj−yi=fz,mv,v=1,2,3,…,n1;i=1,2,3,…,n2.Iffxiz,mv=fz,mv+yi, then(2)yj=fxiz,mv.In the above formulas,Dv,x=xi is the rockfall horizontal distance (m) of the dangerous rockmass v at cross section x=xi of the slope, and f and fx are the rockfall motion functions.The calculation of the maximum horizontal distance of the rockfall at cross profilex=xi can be converted into an optimization problem expressed as follows:(3)maxyx=xi=maxfxiz,mv,s.txi≥0,z≥0,mv≥0,where v is the number of the dangerous rockmasses at profile x=xi; i.e., v=1,2,3,…,n1; and i is the number of the cross sections of the slope; i.e., i=1,2,3,…,n2.To solve the optimality theory problem expressed by (3), the position of the dangerous rockmass where the weathered slope may be rockfall, i.e., the range of the x and z values, is determined first. For convenience, a finite number of profiles were defined along the x direction of the slope, the maximum rockfall horizontal distance of each profile was calculated and marked on the topographic map, and a spline curve was used to connect all of the points to form an enclosed area; this area represents the influence area of the weathered slope rockfall.


## 2.4. Constraints

As shown in (2) and (3), the calculation of the influence area of the rockfall at profile x=xi requires that the position z and mass m (kg) of the rockfall dangerous rock mass, i.e., the constraints of the model in (3), be determined as follows.


### 2.4.1. Position Constraints for the Rockfall Dangerous Rockmass

The rockfall and spalling of a rock slope are the results of the interaction of various external forces. Rockfall occurs due to a combination of the rockmass gravity and external forces, such as fissure water pressure, expansion pressure exerted by plant roots, and weathering, which is caused by high-steep slopes and steeply inclined fissures. Examples of this failure include brittle tensional cracks in the surface of the rockmass under low stress conditions, spalling of the surface rockmass due to the close relationship between the rockmass structure and the stress state of the rockmass, and cantilever failure, which is the formation of tensional cracks caused by the shear failure or plastic failure of the rockmass in the rock cavity area under high stress conditions. Therefore, the tensile stress distribution and the plastic failure inside the rockmass of steep weathered slopes are important conditions that can accelerate the occurrence of rockfall. Accordingly, the area where the slope may be rockfall can be identified and used as the position constraints needed to calculate the horizontal distance of the rockfall motion.FLAC3D is a 3D numerical simulation software developed by ITASCA that is commonly used in geotechnical engineering. FLAC3D uses the explicit difference scheme to solve the governing differential equation of the field. Therefore, it can satisfactorily simulate characteristics such as the internal stress state, the yield failure, and the plastic flow of the rockmass. In addition, it can be used to analyze the 3D spatial deformation behavior of a rockmass after it reaches the yield limit [25] and to determine the plastic failure zone and the tensile stress concentration of the rockmass. In order to determine the position range of the dangerous rockmass and where it may be rockfall, we used FLAC3D to calculate the tensile stress zone and plastic failure zone at any position on the slope’s surface. Based on this, the detailed slope failure conditions of the site can be used to determine the two zones, and the highly overlapping zone of the two can be used as the position constraints for the dangerous rockmass that may be rockfall.


### 2.4.2. Mass Constraints of the Collapsing Dangerous Rockmass

The rock mass with potential rockfall was determined using the following methods:(1)
A detailed investigation of historical rockfall of the steep weathered slope was conducted. The shape, size, location, and distance of the rockfall block were counted and analyzed in the detailed investigation, so as to determine the volume for the historical rockfall mass.(2)
Unmanned aerial vehicle (UAV) tilt photography modeling and 3D laser scanning techniques were employed to establish a 3D numerical model of the slope mass and to identify and measure the size of the dangerous rockmass that could be rockfall.(3)
According to the Engineering Rockmass Test Method Standards (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013) [26], drill core sampling (ϕ 50 mm × 100 mm in size, 5 samples in each group) is conducted in different areas depending on types of rock present on the slope, and then laboratory tests were conducted to obtain the physical and mechanical parameters of the rock samples, including the natural density of the slope mass. The masses (m=ρv) of various dangerous rockmasses determined from steps 1 and 2 were calculated and ranked in ascending order; mmin,mmax is the mass constraint range of the rockfall dangerous rockmass.


## 2.4.1. Position Constraints for the Rockfall Dangerous Rockmass

The rockfall and spalling of a rock slope are the results of the interaction of various external forces. Rockfall occurs due to a combination of the rockmass gravity and external forces, such as fissure water pressure, expansion pressure exerted by plant roots, and weathering, which is caused by high-steep slopes and steeply inclined fissures. Examples of this failure include brittle tensional cracks in the surface of the rockmass under low stress conditions, spalling of the surface rockmass due to the close relationship between the rockmass structure and the stress state of the rockmass, and cantilever failure, which is the formation of tensional cracks caused by the shear failure or plastic failure of the rockmass in the rock cavity area under high stress conditions. Therefore, the tensile stress distribution and the plastic failure inside the rockmass of steep weathered slopes are important conditions that can accelerate the occurrence of rockfall. Accordingly, the area where the slope may be rockfall can be identified and used as the position constraints needed to calculate the horizontal distance of the rockfall motion.FLAC3D is a 3D numerical simulation software developed by ITASCA that is commonly used in geotechnical engineering. FLAC3D uses the explicit difference scheme to solve the governing differential equation of the field. Therefore, it can satisfactorily simulate characteristics such as the internal stress state, the yield failure, and the plastic flow of the rockmass. In addition, it can be used to analyze the 3D spatial deformation behavior of a rockmass after it reaches the yield limit [25] and to determine the plastic failure zone and the tensile stress concentration of the rockmass. In order to determine the position range of the dangerous rockmass and where it may be rockfall, we used FLAC3D to calculate the tensile stress zone and plastic failure zone at any position on the slope’s surface. Based on this, the detailed slope failure conditions of the site can be used to determine the two zones, and the highly overlapping zone of the two can be used as the position constraints for the dangerous rockmass that may be rockfall.


## 2.4.2. Mass Constraints of the Collapsing Dangerous Rockmass

The rock mass with potential rockfall was determined using the following methods:(1)
A detailed investigation of historical rockfall of the steep weathered slope was conducted. The shape, size, location, and distance of the rockfall block were counted and analyzed in the detailed investigation, so as to determine the volume for the historical rockfall mass.(2)
Unmanned aerial vehicle (UAV) tilt photography modeling and 3D laser scanning techniques were employed to establish a 3D numerical model of the slope mass and to identify and measure the size of the dangerous rockmass that could be rockfall.(3)
According to the Engineering Rockmass Test Method Standards (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013) [26], drill core sampling (ϕ 50 mm × 100 mm in size, 5 samples in each group) is conducted in different areas depending on types of rock present on the slope, and then laboratory tests were conducted to obtain the physical and mechanical parameters of the rock samples, including the natural density of the slope mass. The masses (m=ρv) of various dangerous rockmasses determined from steps 1 and 2 were calculated and ranked in ascending order; mmin,mmax is the mass constraint range of the rockfall dangerous rockmass.


## 2.5. Model Solution

### 2.5.1. Calculation of the Horizontal Distance of the Rockfall

Due to the randomness of the rockfall, the function of the horizontal distance of the rockfall motiony=fxz,m is uncertain for any cross profile of the weathered slope; i.e., the functional mapping relationship between the horizontal distance of the rockfall y, the rockfall block position coordinate z, and the mass of the rockfall block m cannot be described by an exact expression. However, based on the uncertainty function’s mapping relationship, the BP (backpack) neural network can learn the rules of the corresponding function by analyzing the sample data; thus, the result closest to the expected output value can be obtained given an input value [27, 28]. As a result, the BP neural network can be used to express the function of the horizontal distance of the rockfall y=fxz,m; see Figure 2.Figure 2
The rockfall horizontal distance calculation model.(1) Establishment of the Network Model. We propose the fact that the structural model of the BP neural network of the horizontal distance of the rockfall function has the three hidden layer neural network structure shown in Figure 2.(2) The Input Layer. The input layer includes the position coordinate z of the dangerous rockmass and the mass m of the dangerous rockmass. The input layer can be expressed in vector form as X=z,mT=x1,x2T.(3) The Hidden Layer. The output vector of each hidden layer is expressed as U3=u13,u23,…,um33T, U2=u12,u22,…,um22T, and U1=u11,u21,…,um11T. The weight matrix from the input layer to the hidden layer is V1=v11,v21,…,vm11T, V2=v12,v22,…,vm22T, and V3=v13,v23,…,vm33T. The column vector vji is the weight vector corresponding to the jth neuron of the ith hidden layer. If the threshold of the neuron is Ti, then(4)uj3=fnetj,netj=∑i=0m2vij3ui2−Ti2,j=1,2,…,m3,(5)uj2=fnetj,netj=∑i=0m1vij2ui1−Ti1,j=1,2,…,m2.(6)uj1=fnetj,netj=∑i=02vij1xi−Ti0,j=1,2,…,m1.(4) The Output Layer. The output layer is the value of the horizontal distance of the rockfall motion, which can be expressed by the vector O=o1T. The rockfall horizontal distances from investigations of historical rockfalls or from numerical simulation are taken as the expected output value, which can be expressed by the vector d=d1T. This value can be used as the basis for the neural network, which can be used to calculate and adjust the weights of the output layer and the hidden layers. If the weight matrix between hidden layer 3 and the output layer is d=ω1T=ω11,ω21,…,ωm31, the output layer can be expressed as(7)o1=fnetj,netj=∑i=0m3ωi1ui3−Ti4.(5) The Transfer Function. The unipolar Sigmoid function was used for the transfer functions of both the output layer and the hidden layer [29]:(8)fx=11+e−x.In summary, when using the neural network model, as shown by (4) through (8), as the model for calculating the horizontal distance of the rockfall motion, as shown in (2), it is necessary to use a large number of rockfall motion statistical data points to train the neural network model. As a result, the model can be used to calculate the horizontal distance of the rockfall motion y of any dangerous rockmass. In the calculation, the rockfall motion sample data can be obtained from historical rockfall data from field investigations, the rockfall numerical simulation, and/or from physical model tests.


### 2.5.2. Solution for the Influence Area of the Rockfall

The genetic optimization algorithm is a type of mathematical optimization algorithm formed by mimicking the biological evolution process of chromosomes, which undergo reproduction, crossover, and mutation to produce new, fitter chromosomes [30]. Because it is an effective method for solving the optimality problems, the optimality problem shown in (3) can be solved using the genetic optimization algorithm using the following steps:(1)
Randomly generate the initial populationP0 and then use the model expressed in (4)–(8) to calculate the rockfall horizontal distance of each individual mass. Perform the nondominated sorting of the population and give each individual a rank.(2)
Perform the binary tournament selection, crossover, and mutation ofP0 to obtain a new population Q0 and let t=0.(3)
CombinePt and Qt to form a new population Rt and then perform the nondominated sorting of Rt and select the best N individuals to generate a population Pt+1.(4)
Reproduce, cross over, and mutate the populationPt+1 to generate a population Qt+1 Qt + 1.(5)
If the number of iterations reaches the preset maximum number, the optimization ends. Otherwise, go to step 1.Based on the above-mentioned steps, a genetic algorithm (GA) program was written using MATLAB 2012b to calculate the maximum rockfall horizontal distance of any cross profile of the weathered slope mass.


## 2.5.1. Calculation of the Horizontal Distance of the Rockfall

Due to the randomness of the rockfall, the function of the horizontal distance of the rockfall motiony=fxz,m is uncertain for any cross profile of the weathered slope; i.e., the functional mapping relationship between the horizontal distance of the rockfall y, the rockfall block position coordinate z, and the mass of the rockfall block m cannot be described by an exact expression. However, based on the uncertainty function’s mapping relationship, the BP (backpack) neural network can learn the rules of the corresponding function by analyzing the sample data; thus, the result closest to the expected output value can be obtained given an input value [27, 28]. As a result, the BP neural network can be used to express the function of the horizontal distance of the rockfall y=fxz,m; see Figure 2.Figure 2
The rockfall horizontal distance calculation model.(1) Establishment of the Network Model. We propose the fact that the structural model of the BP neural network of the horizontal distance of the rockfall function has the three hidden layer neural network structure shown in Figure 2.(2) The Input Layer. The input layer includes the position coordinate z of the dangerous rockmass and the mass m of the dangerous rockmass. The input layer can be expressed in vector form as X=z,mT=x1,x2T.(3) The Hidden Layer. The output vector of each hidden layer is expressed as U3=u13,u23,…,um33T, U2=u12,u22,…,um22T, and U1=u11,u21,…,um11T. The weight matrix from the input layer to the hidden layer is V1=v11,v21,…,vm11T, V2=v12,v22,…,vm22T, and V3=v13,v23,…,vm33T. The column vector vji is the weight vector corresponding to the jth neuron of the ith hidden layer. If the threshold of the neuron is Ti, then(4)uj3=fnetj,netj=∑i=0m2vij3ui2−Ti2,j=1,2,…,m3,(5)uj2=fnetj,netj=∑i=0m1vij2ui1−Ti1,j=1,2,…,m2.(6)uj1=fnetj,netj=∑i=02vij1xi−Ti0,j=1,2,…,m1.(4) The Output Layer. The output layer is the value of the horizontal distance of the rockfall motion, which can be expressed by the vector O=o1T. The rockfall horizontal distances from investigations of historical rockfalls or from numerical simulation are taken as the expected output value, which can be expressed by the vector d=d1T. This value can be used as the basis for the neural network, which can be used to calculate and adjust the weights of the output layer and the hidden layers. If the weight matrix between hidden layer 3 and the output layer is d=ω1T=ω11,ω21,…,ωm31, the output layer can be expressed as(7)o1=fnetj,netj=∑i=0m3ωi1ui3−Ti4.(5) The Transfer Function. The unipolar Sigmoid function was used for the transfer functions of both the output layer and the hidden layer [29]:(8)fx=11+e−x.In summary, when using the neural network model, as shown by (4) through (8), as the model for calculating the horizontal distance of the rockfall motion, as shown in (2), it is necessary to use a large number of rockfall motion statistical data points to train the neural network model. As a result, the model can be used to calculate the horizontal distance of the rockfall motion y of any dangerous rockmass. In the calculation, the rockfall motion sample data can be obtained from historical rockfall data from field investigations, the rockfall numerical simulation, and/or from physical model tests.


## 2.5.2. Solution for the Influence Area of the Rockfall

The genetic optimization algorithm is a type of mathematical optimization algorithm formed by mimicking the biological evolution process of chromosomes, which undergo reproduction, crossover, and mutation to produce new, fitter chromosomes [30]. Because it is an effective method for solving the optimality problems, the optimality problem shown in (3) can be solved using the genetic optimization algorithm using the following steps:(1)
Randomly generate the initial populationP0 and then use the model expressed in (4)–(8) to calculate the rockfall horizontal distance of each individual mass. Perform the nondominated sorting of the population and give each individual a rank.(2)
Perform the binary tournament selection, crossover, and mutation ofP0 to obtain a new population Q0 and let t=0.(3)
CombinePt and Qt to form a new population Rt and then perform the nondominated sorting of Rt and select the best N individuals to generate a population Pt+1.(4)
Reproduce, cross over, and mutate the populationPt+1 to generate a population Qt+1 Qt + 1.(5)
If the number of iterations reaches the preset maximum number, the optimization ends. Otherwise, go to step 1.Based on the above-mentioned steps, a genetic algorithm (GA) program was written using MATLAB 2012b to calculate the maximum rockfall horizontal distance of any cross profile of the weathered slope mass.


## 3. Case Study

Multiple episodes of rockfall have occurred historically on the Guanyindong Slope, which is a tourist scenic spot in Zhejiang Province, China. Future rockfall poses a threat to the lives and property of tourists. Taking the Guanyindong Slope as an example, we used the above calculation method to determine the influence area of the Guanyindong Slope, providing a basis for determining and designing disaster prevention and reduction measures for the Guanyindong Slope.


### 3.1. Overall Characteristics of the Guanyindong Slope

The geographical coordinates of the Guanyindong Slope are 120°36′31.2″ E and 28°42′11.6″ N. The high-steep weathered slope mass has an elevation of approximately 126 m, a slope of 70–88°, and slope aspect towards 310°. There is a 19 × 17 × 10 m3 rock cave (the Guanyin cavity) at the bottom of the slope (Figure 3(a)). Inside this rock cave there is a Guanyin statue, which integrates the natural and cultural landscapes and merges the religious and cultural characteristics of the region, making it one of the most important tourist attractions. In the middle of the Guanyindong Slope, there is a weak layer (with an elevation of 230–245 m) that contains a number of cavities formed by weathering (W1–W4 in Figure 3(b)).Figure 3
The overall characteristics of the Guanyindong Slope. (a) The overall characteristics of the Guanyindong Slope; (b) the main planar structural distribution of the Guanyindong Slope; and (c) profile I-I.
(a)
(b)
(c)
(d)The lithology of the slope is composed of Upper Jurassic Jiuliping Formation altered tuff (J3j1) (Figure4). To further illustrate the microscopic characteristics of the rock, one of the microscopic images of rock samples is shown in Figure 4(b). In addition, due to weathering and flowing water, part of the surface the rock block is grayish black; however, the fresh surface of the rock is grayish white. The bedrock on the surface of the slope is exposed with less vegetation cover (Figure 3(a)). In general, the rockmass is severely weathered with well-developed structural planes (Figure 3(b)). The main controlling structural planes include regional crack structural plane F3 in the steep slope at the Guanyindong entrance (orientation of N40E/80 NW) and the oblique normal fault bedding F8 at the cavity’s entrance (orientation of N58W/60 SW). In addition, the slope has gently inclined bedding with the orientation of N85W/10 NE, and a weak intercalated layer has formed between the planes, with steeply inclined cracks with orientation s of N32E/88 NW and N33E/72 NW. We also observed that these structural planes cut each other within the rockmass to form large, outwardly inclined blocks, such as block GYD1 (62 m long × 8 m wide × 2 m thick) (Figures 3(c) and 3(d)).Figure 4
Rock type and its microscopic characteristics of the Guanyindong Slope. (a) The altered tuff samples; (b) microscopic images of the altered tuff.
(a)
(b)When exposed to external factors such as heavy rainfall, the cracks continue to expand and gradually expand the plastic failure zone of the rockmass and ultimately decrease its strength. Eventually, this will trigger failure phenomena. For example, spalling and rockfall of the rockmass surface as well as local instability and overturning of the block may occur. Based on our field investigations (Figure5), there have been multiple episodes of rockfall and rolling in this location, with fragments as large as 2 m × 1.5 m × 1 m and as small as 0.05 m × 0.1 m × 0.1 m.Figure 5
Historical rockfall survey results for the Guanyindong Slope.Under the continuous weathering, the rockfall and falling blocks from the surface of the weathered rock slope are uncertain. Thus, it is difficult to predict the position and size of the rockfall block and to track its movement accurately.


### 3.2. Rockfall Mathematical Model of the Weathered Slope

The rockfall mathematical model of the Guanyindong Slope can be expressed in (3). Based on the field investigations, a local coordinate system was established for the Guanyindong Slope mass (Figures 6(a) and 6(b)).Figure 6
The local coordinate system for the Guanyindong Slope. (a) The local coordinate system; (b) the local coordinate system of the cross sectionx = 12 m.
(a)
(b)


### 3.3. Constraints

#### 3.3.1. Position and Numerical Analysis of Dangerous Rockmass that May Be Rockfall

Based on the description in Section2.2, to determine the position of the dangerous rockmass where rockfall may occur on the Guanyindong Slope, i.e., to determine the ranges of the x and z values, we used both FLAC3D numerical simulation analysis and geological field investigations.(1) Establishment of the FLAC3D Numerical Model. The Mohr–Coulomb constitutive model was used in the calculation model. The 3D numerical model was 445 m × 467 m × 258 m, and the model mesh contained 126,438 elements and 31,799 nodes. The upper boundary of the model is the surface, which is set as the free boundary, and the lower boundary is located approximately 45 m below the bottom of the Guanyindong Slope. The model is fixed in the z direction and around the boundary in the one-way horizontal axial direction (Figure 7(a)).Figure 7
Identification of potential rockfall areas. (a) The minimum principal stress; (b) the maximum principal stress; (c) the plastic failure area; and (d) field investigation results for the failure area.
(a)
(b)
(c)
(d)(2) Selection of Calculation Parameters. To obtain the calculation parameters, four groups of rock samples (ϕ 50 mm × 100 mm in size, 5 samples in each group) were collected by drilling into the Guanyindong Slope. According to the Standard for Test Methods of Engineering Rockmass (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013), a triaxial rock testing machine (TAW2000) was used to conduct the laboratory tests. Based on the field investigation (Figure 7(d)), we used the orthogonal test method [31] to determine the distribution of the plastic failure of the rockmass using the theory of displacement back analysis [32] (Figure 7(c)). Then, the calculated parameters were used as the physical and mechanical parameters of the Guanyindong rockmass, as shown in Table 1.Table 1
The physical and mechanical parameters of rock samples from Guanyindong Slope.
Rock
Densityρkg/m
Tensile strengthσtPa
CohesioncPa
Modulus of elasticityEPa
Poisson’s ratio
Angle of frictionφ°
Tuff (rockmass)
2875
8.25
×
10
6
2.05
×
10
7
7.70
×
10
9
0.18
46.51(3) Results of the Numerical Simulation Analysis. By numerical simulation, we obtained the results as follows (Figure 7). First, the tensile stress concentration region is mainly distributed in the filling body of the weak intercalated layer between the rockmass above the Guanyindong Slope and the gently inclined bedding (5°∠10°). Second, the plastic failure zone is mainly concentrated in the vicinity of the filling body of the weak intercalated layer and in the cracks (orientation of 302°∠88°) in the gently inclined bedding.


#### 3.3.2. Field Investigation Results

The field investigations provided the following information. Because of continuous weathering, the filling body of the weak intercalated layer was between the gently inclined bedding (orientation of N85W/10 NE) of the Guanyindong Slope experienced rockfall and spalling in multiple locations (see spalling trace locations A and D in Figure7(d)), forming multiple cavities. The steeply inclined cracks (orientation of N32E/88 NW) and bedding (orientation of N85W/10 NE) cut the rockmass, forming a number of dangerous rockmasses, leading to multiple historical episodes of rockfall and spalling (see spalling trace locations B in Figure 7(d)). Multiple historical events of rockfall and spalling also occurred in the upper part of the Guanyindong Slope (see spalling trace locations C in Figure 7(d)). Authors observed that because areas where the spalling of steep rocks occurred often exhibit fresh grayish-white surfaces, their traces tend to be obvious and are easy to be identified (Figure 7(d)).


#### 3.3.3. Delineation of Potential Rockfall Areas

The rockmass failure distribution obtained from the aforementioned field investigations (Figure7(d)) is in a good agreement with the plastic failure zone and tensile stress distribution zone (Figures 7(a)–7(c)) determined from the numerical simulation. According to Section 2.2.1, we can delineate the potential rockfall area of the Guanyindong Slope mass, as shown in Figure 7(d). The value range is(9)x,zx∈0,127,z∈0,60.


### 3.4. Estimation of the Mass of the Potentially Rockfall Dangerous Rockmass

According to Section 2.2.2, the field investigations and a combination of UAV and 3D laser scanning were carried out. The results revealed that the volume of historical rockfall blocks from the Guanyindong Slope was from 0.0005 m3 to 3 m3. Based on the Guanyindong Slope’s rock density of 2875 kg/m3 (Table 1), we estimate the value range of the mass m of the potentially rockfall rockmass to be(10)m∈1.4,9487.


### 3.5. Calculation of the Influence Area of the Rockfall

#### 3.5.1. Calculation Model

According to the above analysis, the mathematical model for determining the rockfall area of the Guanyindong Slope is(11)maxy=maxfxz,mv,s.tx∈0,127,z∈0,60,mv∈1.4,9487.The procedure for solving the calculation model is as follows:(1)
Ten cross sections within the value range ofx, i.e., x=12,24,36,48,60,72,84,96,108,120, were defined.(2)
The objective functiony=fxz,mv in (11)fd11 is an uncertainty function, which was represented using the neural network model described in Section 2.3.1.(3)
Using the historical rockfall data, the normal and tangential coefficients of restitution [33–35] and the friction angle of the Guanyindong Slope were determined using the theory of displacement back analysis (Figure 8) [32]. Based on the parameters, RocFall2D software was used to simulate possible rockfall events for each cross profile and to obtain sample data for these rockfall events. In order to reduce the error of slope geometry to numerical simulation, according to the principle of aerial photogrammetry [36], the slope image data are acquired by using the tilt photography of five routes including East, South, West, North, and orthoimage with the help of PHANTOM 4 RTK Unmanned Aerial Vehicle (UAV). Among them, the UAV camera of the East, South, West, and North routes forms a 45°angle with the horizontal plane, and the UAV camera of the orthoimage route forms a 90°angle with the horizontal plane. Using context Capture Center Master and other software applications, combined with the principle of reverse modeling [37], the collected image data is generated into a three-dimensional model and 3D point cloud, and then the position, shape, size, and other data of the slope section are identified (Figures 1 and 8). The computer program for the neural network model, which was constructed in training step (2), was written using MATLAB. Then, the trained neural network was used to predict possible rockfall events.Figure 8
The back analysis mechanical parameters of the rockfall motion. (a) Historical rockfall event 1; (b) back analysis results for historical rockfall event 1; (c) historical rockfall event 2; and (d) back analysis results for historical rockfall event 2.
(a)
(b)
(c)
(d)Using the GA described in Section 2.3.2, a relevant program was prepared using MATLAB 2012b. Then, the calculation model represented by (11) was solved to obtain the maximum possible rockfall horizontal distance of each cross profile.


#### 3.5.2. Calculation Model Solution

According to the model solving procedure described in Sections3.5.1 and 2.3.1, the RocFall™ was used to obtain neural network training data for each cross section of the Guanyindong Slope. Because the historical rockfall data for the Guanyindong Slope obtained from the field investigations was not sufficient to satisfy the requirements of the neural network training, we obtained the normal and tangential restitution coefficients and the friction angle of the rockmass based on the theory of displacement back analysis [32] using historical rockfall data (see Figure 8). Then, numerical simulations of the possible rockfall events of each cross profile of the slope were carried out to obtain the necessary corresponding sample training data.(1) Determination of the Simulation Parameters of the Rockfall Motion. The historical rockfall events were back-calculated as shown in Figures 8(a) and 7(b). The size of the rockfall rock was 1.8 m × 1.5 m × 0.75 m, and the obtained parameters for the rockfall of the rockmass included a normal restitution coefficient of 0.53, a tangential restitution coefficient of 0.75, and a friction angle of 46.5°.The rockfall motion was verified using back-calculated the historical rockfall events shown in Figures8(c) and 7(d). The size of the rockfall rock was 2 m × 1.5 m × 1 m. The simulation results by rockfall are consistent with the field investigations and confirm the validity of the above parameters.(2) Acquisition of Sample Data. According to the rockfall parameters obtained from the above, we used the uniform design table to design a rockfall simulation test for each cross profile and to obtain training sample data for the neural network of each cross profile. Table 2 shows the orthogonal factor level table of the cross profile x=12m designed using the uniform design table U30∗3012. Table 3 presents the experimental arrangement and results of the numerical simulation test, i.e., the training sample data of cross profile x=12m (the coordinate system is shown in Figure 6(b)). Due to space limitations, only data for cross profile x=12m is reported.Table 2
The orthogonal factor levels of cross section.
Level
Factor
Z
M
1
6
1.40
2
7.86
328.49
3
9.72
655.58
4
11.59
982.67
5
13.45
1309.76
6
15.31
1636.85
7
17.17
1963.94
8
19.03
2291.03
9
20.90
2618.12
10
22.76
2945.21
11
24.62
3272.30
12
26.48
3599.39
13
28.34
3926.48
14
30.21
4253.57
15
32.07
4580.66
16
33.93
4907.74
17
35.79
5234.83
18
37.66
5561.92
19
39.52
5889.01
20
41.38
6216.10
21
43.24
6543.19
22
45.10
6870.28
23
46.97
7197.37
24
48.83
7524.46
25
50.69
7851.55
26
52.55
8178.64
27
54.41
8505.73
28
56.28
8832.82
29
58.14
9159.91
30
60
9487.00
Z is the elevation of rockfall rock. M is mass of rockfall rock.Table 3
The experimental arrangements and results.
Experiment
Factor
Experimental resulty
Z
M
1
6
6870.28
−8.111
2
7.86
3926.48
−8.161
3
9.72
982.67
−8.030
4
11.59
8178.64
−10.884
5
13.45
5234.83
−15.378
6
15.31
2291.03
−6.621
7
17.17
9487.00
−6.211
8
19.03
6543.19
−5.252
9
20.90
3599.39
75.662
10
22.76
655.58
78.820
11
24.62
7851.55
87.112
12
26.48
4907.74
89.953
13
28.34
1963.94
86.387
14
30.21
9159.91
89.403
15
32.07
6216.10
91.772
16
33.93
8832.82
99.755
17
35.79
6870.28
102.602
18
37.66
7524.46
104.472
19
39.52
4580.66
−20.093
20
41.38
1636.85
96.014
21
43.24
8832.82
96.387
22
45.10
5889.01
−18.076
23
46.97
2945.21
−24.929
24
48.83
1.40
−25.123
25
50.69
7197.37
−26.288
26
52.55
4253.57
−27.473
27
54.41
1309.76
−28.655
28
56.28
8505.73
−29.959
29
58.14
5561.92
−30.994
30
60
2618.12
−33.952
Z is the elevation of rockfall rock. M is mass of rockfall rock.The experimental arrangement and results are shown in Table3.(3) Calculation of the Influence Area of the Rockfall of Each Cross Section. The GA calculation program was created using MATLAB 2012b according to the method described in Section 2.3.2. Then, it was used to solve the maximum influence area of each profile and the height Z and mass M of the corresponding rockfall rockmass. The calculation results are shown in Table 4.Table 4
The calculation results for the influence range of each cross section of the Guanyindong Slope.
Slope section positionx (m)
Elevation of rockfall rockZ (m)
Mass of rockfall rockM (kg)
Position of rockfall rocky (m)
12
38.705
4986.204
111.616
24
60
4196.453
102.397
36
41.841
7240.130
51.735
48
37.205
2151.947
70.18
60
60
7601.522
88.684
72
60
4351.76
90.131
84
51.640
2836.387
76.176
96
60
3658.595
90.930
108
60
3527.153
87.308
120
59.456
4974.903
117.478


## 3.1. Overall Characteristics of the Guanyindong Slope

The geographical coordinates of the Guanyindong Slope are 120°36′31.2″ E and 28°42′11.6″ N. The high-steep weathered slope mass has an elevation of approximately 126 m, a slope of 70–88°, and slope aspect towards 310°. There is a 19 × 17 × 10 m3 rock cave (the Guanyin cavity) at the bottom of the slope (Figure 3(a)). Inside this rock cave there is a Guanyin statue, which integrates the natural and cultural landscapes and merges the religious and cultural characteristics of the region, making it one of the most important tourist attractions. In the middle of the Guanyindong Slope, there is a weak layer (with an elevation of 230–245 m) that contains a number of cavities formed by weathering (W1–W4 in Figure 3(b)).Figure 3
The overall characteristics of the Guanyindong Slope. (a) The overall characteristics of the Guanyindong Slope; (b) the main planar structural distribution of the Guanyindong Slope; and (c) profile I-I.
(a)
(b)
(c)
(d)The lithology of the slope is composed of Upper Jurassic Jiuliping Formation altered tuff (J3j1) (Figure4). To further illustrate the microscopic characteristics of the rock, one of the microscopic images of rock samples is shown in Figure 4(b). In addition, due to weathering and flowing water, part of the surface the rock block is grayish black; however, the fresh surface of the rock is grayish white. The bedrock on the surface of the slope is exposed with less vegetation cover (Figure 3(a)). In general, the rockmass is severely weathered with well-developed structural planes (Figure 3(b)). The main controlling structural planes include regional crack structural plane F3 in the steep slope at the Guanyindong entrance (orientation of N40E/80 NW) and the oblique normal fault bedding F8 at the cavity’s entrance (orientation of N58W/60 SW). In addition, the slope has gently inclined bedding with the orientation of N85W/10 NE, and a weak intercalated layer has formed between the planes, with steeply inclined cracks with orientation s of N32E/88 NW and N33E/72 NW. We also observed that these structural planes cut each other within the rockmass to form large, outwardly inclined blocks, such as block GYD1 (62 m long × 8 m wide × 2 m thick) (Figures 3(c) and 3(d)).Figure 4
Rock type and its microscopic characteristics of the Guanyindong Slope. (a) The altered tuff samples; (b) microscopic images of the altered tuff.
(a)
(b)When exposed to external factors such as heavy rainfall, the cracks continue to expand and gradually expand the plastic failure zone of the rockmass and ultimately decrease its strength. Eventually, this will trigger failure phenomena. For example, spalling and rockfall of the rockmass surface as well as local instability and overturning of the block may occur. Based on our field investigations (Figure5), there have been multiple episodes of rockfall and rolling in this location, with fragments as large as 2 m × 1.5 m × 1 m and as small as 0.05 m × 0.1 m × 0.1 m.Figure 5
Historical rockfall survey results for the Guanyindong Slope.Under the continuous weathering, the rockfall and falling blocks from the surface of the weathered rock slope are uncertain. Thus, it is difficult to predict the position and size of the rockfall block and to track its movement accurately.


## 3.2. Rockfall Mathematical Model of the Weathered Slope

The rockfall mathematical model of the Guanyindong Slope can be expressed in (3). Based on the field investigations, a local coordinate system was established for the Guanyindong Slope mass (Figures 6(a) and 6(b)).Figure 6
The local coordinate system for the Guanyindong Slope. (a) The local coordinate system; (b) the local coordinate system of the cross sectionx = 12 m.
(a)
(b)


## 3.3. Constraints

### 3.3.1. Position and Numerical Analysis of Dangerous Rockmass that May Be Rockfall

Based on the description in Section2.2, to determine the position of the dangerous rockmass where rockfall may occur on the Guanyindong Slope, i.e., to determine the ranges of the x and z values, we used both FLAC3D numerical simulation analysis and geological field investigations.(1) Establishment of the FLAC3D Numerical Model. The Mohr–Coulomb constitutive model was used in the calculation model. The 3D numerical model was 445 m × 467 m × 258 m, and the model mesh contained 126,438 elements and 31,799 nodes. The upper boundary of the model is the surface, which is set as the free boundary, and the lower boundary is located approximately 45 m below the bottom of the Guanyindong Slope. The model is fixed in the z direction and around the boundary in the one-way horizontal axial direction (Figure 7(a)).Figure 7
Identification of potential rockfall areas. (a) The minimum principal stress; (b) the maximum principal stress; (c) the plastic failure area; and (d) field investigation results for the failure area.
(a)
(b)
(c)
(d)(2) Selection of Calculation Parameters. To obtain the calculation parameters, four groups of rock samples (ϕ 50 mm × 100 mm in size, 5 samples in each group) were collected by drilling into the Guanyindong Slope. According to the Standard for Test Methods of Engineering Rockmass (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013), a triaxial rock testing machine (TAW2000) was used to conduct the laboratory tests. Based on the field investigation (Figure 7(d)), we used the orthogonal test method [31] to determine the distribution of the plastic failure of the rockmass using the theory of displacement back analysis [32] (Figure 7(c)). Then, the calculated parameters were used as the physical and mechanical parameters of the Guanyindong rockmass, as shown in Table 1.Table 1
The physical and mechanical parameters of rock samples from Guanyindong Slope.
Rock
Densityρkg/m
Tensile strengthσtPa
CohesioncPa
Modulus of elasticityEPa
Poisson’s ratio
Angle of frictionφ°
Tuff (rockmass)
2875
8.25
×
10
6
2.05
×
10
7
7.70
×
10
9
0.18
46.51(3) Results of the Numerical Simulation Analysis. By numerical simulation, we obtained the results as follows (Figure 7). First, the tensile stress concentration region is mainly distributed in the filling body of the weak intercalated layer between the rockmass above the Guanyindong Slope and the gently inclined bedding (5°∠10°). Second, the plastic failure zone is mainly concentrated in the vicinity of the filling body of the weak intercalated layer and in the cracks (orientation of 302°∠88°) in the gently inclined bedding.


### 3.3.2. Field Investigation Results

The field investigations provided the following information. Because of continuous weathering, the filling body of the weak intercalated layer was between the gently inclined bedding (orientation of N85W/10 NE) of the Guanyindong Slope experienced rockfall and spalling in multiple locations (see spalling trace locations A and D in Figure7(d)), forming multiple cavities. The steeply inclined cracks (orientation of N32E/88 NW) and bedding (orientation of N85W/10 NE) cut the rockmass, forming a number of dangerous rockmasses, leading to multiple historical episodes of rockfall and spalling (see spalling trace locations B in Figure 7(d)). Multiple historical events of rockfall and spalling also occurred in the upper part of the Guanyindong Slope (see spalling trace locations C in Figure 7(d)). Authors observed that because areas where the spalling of steep rocks occurred often exhibit fresh grayish-white surfaces, their traces tend to be obvious and are easy to be identified (Figure 7(d)).


### 3.3.3. Delineation of Potential Rockfall Areas

The rockmass failure distribution obtained from the aforementioned field investigations (Figure7(d)) is in a good agreement with the plastic failure zone and tensile stress distribution zone (Figures 7(a)–7(c)) determined from the numerical simulation. According to Section 2.2.1, we can delineate the potential rockfall area of the Guanyindong Slope mass, as shown in Figure 7(d). The value range is(9)x,zx∈0,127,z∈0,60.


## 3.3.1. Position and Numerical Analysis of Dangerous Rockmass that May Be Rockfall

Based on the description in Section2.2, to determine the position of the dangerous rockmass where rockfall may occur on the Guanyindong Slope, i.e., to determine the ranges of the x and z values, we used both FLAC3D numerical simulation analysis and geological field investigations.(1) Establishment of the FLAC3D Numerical Model. The Mohr–Coulomb constitutive model was used in the calculation model. The 3D numerical model was 445 m × 467 m × 258 m, and the model mesh contained 126,438 elements and 31,799 nodes. The upper boundary of the model is the surface, which is set as the free boundary, and the lower boundary is located approximately 45 m below the bottom of the Guanyindong Slope. The model is fixed in the z direction and around the boundary in the one-way horizontal axial direction (Figure 7(a)).Figure 7
Identification of potential rockfall areas. (a) The minimum principal stress; (b) the maximum principal stress; (c) the plastic failure area; and (d) field investigation results for the failure area.
(a)
(b)
(c)
(d)(2) Selection of Calculation Parameters. To obtain the calculation parameters, four groups of rock samples (ϕ 50 mm × 100 mm in size, 5 samples in each group) were collected by drilling into the Guanyindong Slope. According to the Standard for Test Methods of Engineering Rockmass (GB/T 50266-2013) (Ministry of Housing and Urban Rural Development of People’s Republic of China 2013), a triaxial rock testing machine (TAW2000) was used to conduct the laboratory tests. Based on the field investigation (Figure 7(d)), we used the orthogonal test method [31] to determine the distribution of the plastic failure of the rockmass using the theory of displacement back analysis [32] (Figure 7(c)). Then, the calculated parameters were used as the physical and mechanical parameters of the Guanyindong rockmass, as shown in Table 1.Table 1
The physical and mechanical parameters of rock samples from Guanyindong Slope.
Rock
Densityρkg/m
Tensile strengthσtPa
CohesioncPa
Modulus of elasticityEPa
Poisson’s ratio
Angle of frictionφ°
Tuff (rockmass)
2875
8.25
×
10
6
2.05
×
10
7
7.70
×
10
9
0.18
46.51(3) Results of the Numerical Simulation Analysis. By numerical simulation, we obtained the results as follows (Figure 7). First, the tensile stress concentration region is mainly distributed in the filling body of the weak intercalated layer between the rockmass above the Guanyindong Slope and the gently inclined bedding (5°∠10°). Second, the plastic failure zone is mainly concentrated in the vicinity of the filling body of the weak intercalated layer and in the cracks (orientation of 302°∠88°) in the gently inclined bedding.


## 3.3.2. Field Investigation Results

The field investigations provided the following information. Because of continuous weathering, the filling body of the weak intercalated layer was between the gently inclined bedding (orientation of N85W/10 NE) of the Guanyindong Slope experienced rockfall and spalling in multiple locations (see spalling trace locations A and D in Figure7(d)), forming multiple cavities. The steeply inclined cracks (orientation of N32E/88 NW) and bedding (orientation of N85W/10 NE) cut the rockmass, forming a number of dangerous rockmasses, leading to multiple historical episodes of rockfall and spalling (see spalling trace locations B in Figure 7(d)). Multiple historical events of rockfall and spalling also occurred in the upper part of the Guanyindong Slope (see spalling trace locations C in Figure 7(d)). Authors observed that because areas where the spalling of steep rocks occurred often exhibit fresh grayish-white surfaces, their traces tend to be obvious and are easy to be identified (Figure 7(d)).


## 3.3.3. Delineation of Potential Rockfall Areas

The rockmass failure distribution obtained from the aforementioned field investigations (Figure7(d)) is in a good agreement with the plastic failure zone and tensile stress distribution zone (Figures 7(a)–7(c)) determined from the numerical simulation. According to Section 2.2.1, we can delineate the potential rockfall area of the Guanyindong Slope mass, as shown in Figure 7(d). The value range is(9)x,zx∈0,127,z∈0,60.


## 3.4. Estimation of the Mass of the Potentially Rockfall Dangerous Rockmass

According to Section 2.2.2, the field investigations and a combination of UAV and 3D laser scanning were carried out. The results revealed that the volume of historical rockfall blocks from the Guanyindong Slope was from 0.0005 m3 to 3 m3. Based on the Guanyindong Slope’s rock density of 2875 kg/m3 (Table 1), we estimate the value range of the mass m of the potentially rockfall rockmass to be(10)m∈1.4,9487.


## 3.5. Calculation of the Influence Area of the Rockfall

### 3.5.1. Calculation Model

According to the above analysis, the mathematical model for determining the rockfall area of the Guanyindong Slope is(11)maxy=maxfxz,mv,s.tx∈0,127,z∈0,60,mv∈1.4,9487.The procedure for solving the calculation model is as follows:(1)
Ten cross sections within the value range ofx, i.e., x=12,24,36,48,60,72,84,96,108,120, were defined.(2)
The objective functiony=fxz,mv in (11)fd11 is an uncertainty function, which was represented using the neural network model described in Section 2.3.1.(3)
Using the historical rockfall data, the normal and tangential coefficients of restitution [33–35] and the friction angle of the Guanyindong Slope were determined using the theory of displacement back analysis (Figure 8) [32]. Based on the parameters, RocFall2D software was used to simulate possible rockfall events for each cross profile and to obtain sample data for these rockfall events. In order to reduce the error of slope geometry to numerical simulation, according to the principle of aerial photogrammetry [36], the slope image data are acquired by using the tilt photography of five routes including East, South, West, North, and orthoimage with the help of PHANTOM 4 RTK Unmanned Aerial Vehicle (UAV). Among them, the UAV camera of the East, South, West, and North routes forms a 45°angle with the horizontal plane, and the UAV camera of the orthoimage route forms a 90°angle with the horizontal plane. Using context Capture Center Master and other software applications, combined with the principle of reverse modeling [37], the collected image data is generated into a three-dimensional model and 3D point cloud, and then the position, shape, size, and other data of the slope section are identified (Figures 1 and 8). The computer program for the neural network model, which was constructed in training step (2), was written using MATLAB. Then, the trained neural network was used to predict possible rockfall events.Figure 8
The back analysis mechanical parameters of the rockfall motion. (a) Historical rockfall event 1; (b) back analysis results for historical rockfall event 1; (c) historical rockfall event 2; and (d) back analysis results for historical rockfall event 2.
(a)
(b)
(c)
(d)Using the GA described in Section 2.3.2, a relevant program was prepared using MATLAB 2012b. Then, the calculation model represented by (11) was solved to obtain the maximum possible rockfall horizontal distance of each cross profile.


### 3.5.2. Calculation Model Solution

According to the model solving procedure described in Sections3.5.1 and 2.3.1, the RocFall™ was used to obtain neural network training data for each cross section of the Guanyindong Slope. Because the historical rockfall data for the Guanyindong Slope obtained from the field investigations was not sufficient to satisfy the requirements of the neural network training, we obtained the normal and tangential restitution coefficients and the friction angle of the rockmass based on the theory of displacement back analysis [32] using historical rockfall data (see Figure 8). Then, numerical simulations of the possible rockfall events of each cross profile of the slope were carried out to obtain the necessary corresponding sample training data.(1) Determination of the Simulation Parameters of the Rockfall Motion. The historical rockfall events were back-calculated as shown in Figures 8(a) and 7(b). The size of the rockfall rock was 1.8 m × 1.5 m × 0.75 m, and the obtained parameters for the rockfall of the rockmass included a normal restitution coefficient of 0.53, a tangential restitution coefficient of 0.75, and a friction angle of 46.5°.The rockfall motion was verified using back-calculated the historical rockfall events shown in Figures8(c) and 7(d). The size of the rockfall rock was 2 m × 1.5 m × 1 m. The simulation results by rockfall are consistent with the field investigations and confirm the validity of the above parameters.(2) Acquisition of Sample Data. According to the rockfall parameters obtained from the above, we used the uniform design table to design a rockfall simulation test for each cross profile and to obtain training sample data for the neural network of each cross profile. Table 2 shows the orthogonal factor level table of the cross profile x=12m designed using the uniform design table U30∗3012. Table 3 presents the experimental arrangement and results of the numerical simulation test, i.e., the training sample data of cross profile x=12m (the coordinate system is shown in Figure 6(b)). Due to space limitations, only data for cross profile x=12m is reported.Table 2
The orthogonal factor levels of cross section.
Level
Factor
Z
M
1
6
1.40
2
7.86
328.49
3
9.72
655.58
4
11.59
982.67
5
13.45
1309.76
6
15.31
1636.85
7
17.17
1963.94
8
19.03
2291.03
9
20.90
2618.12
10
22.76
2945.21
11
24.62
3272.30
12
26.48
3599.39
13
28.34
3926.48
14
30.21
4253.57
15
32.07
4580.66
16
33.93
4907.74
17
35.79
5234.83
18
37.66
5561.92
19
39.52
5889.01
20
41.38
6216.10
21
43.24
6543.19
22
45.10
6870.28
23
46.97
7197.37
24
48.83
7524.46
25
50.69
7851.55
26
52.55
8178.64
27
54.41
8505.73
28
56.28
8832.82
29
58.14
9159.91
30
60
9487.00
Z is the elevation of rockfall rock. M is mass of rockfall rock.Table 3
The experimental arrangements and results.
Experiment
Factor
Experimental resulty
Z
M
1
6
6870.28
−8.111
2
7.86
3926.48
−8.161
3
9.72
982.67
−8.030
4
11.59
8178.64
−10.884
5
13.45
5234.83
−15.378
6
15.31
2291.03
−6.621
7
17.17
9487.00
−6.211
8
19.03
6543.19
−5.252
9
20.90
3599.39
75.662
10
22.76
655.58
78.820
11
24.62
7851.55
87.112
12
26.48
4907.74
89.953
13
28.34
1963.94
86.387
14
30.21
9159.91
89.403
15
32.07
6216.10
91.772
16
33.93
8832.82
99.755
17
35.79
6870.28
102.602
18
37.66
7524.46
104.472
19
39.52
4580.66
−20.093
20
41.38
1636.85
96.014
21
43.24
8832.82
96.387
22
45.10
5889.01
−18.076
23
46.97
2945.21
−24.929
24
48.83
1.40
−25.123
25
50.69
7197.37
−26.288
26
52.55
4253.57
−27.473
27
54.41
1309.76
−28.655
28
56.28
8505.73
−29.959
29
58.14
5561.92
−30.994
30
60
2618.12
−33.952
Z is the elevation of rockfall rock. M is mass of rockfall rock.The experimental arrangement and results are shown in Table3.(3) Calculation of the Influence Area of the Rockfall of Each Cross Section. The GA calculation program was created using MATLAB 2012b according to the method described in Section 2.3.2. Then, it was used to solve the maximum influence area of each profile and the height Z and mass M of the corresponding rockfall rockmass. The calculation results are shown in Table 4.Table 4
The calculation results for the influence range of each cross section of the Guanyindong Slope.
Slope section positionx (m)
Elevation of rockfall rockZ (m)
Mass of rockfall rockM (kg)
Position of rockfall rocky (m)
12
38.705
4986.204
111.616
24
60
4196.453
102.397
36
41.841
7240.130
51.735
48
37.205
2151.947
70.18
60
60
7601.522
88.684
72
60
4351.76
90.131
84
51.640
2836.387
76.176
96
60
3658.595
90.930
108
60
3527.153
87.308
120
59.456
4974.903
117.478


## 3.5.1. Calculation Model

According to the above analysis, the mathematical model for determining the rockfall area of the Guanyindong Slope is(11)maxy=maxfxz,mv,s.tx∈0,127,z∈0,60,mv∈1.4,9487.The procedure for solving the calculation model is as follows:(1)
Ten cross sections within the value range ofx, i.e., x=12,24,36,48,60,72,84,96,108,120, were defined.(2)
The objective functiony=fxz,mv in (11)fd11 is an uncertainty function, which was represented using the neural network model described in Section 2.3.1.(3)
Using the historical rockfall data, the normal and tangential coefficients of restitution [33–35] and the friction angle of the Guanyindong Slope were determined using the theory of displacement back analysis (Figure 8) [32]. Based on the parameters, RocFall2D software was used to simulate possible rockfall events for each cross profile and to obtain sample data for these rockfall events. In order to reduce the error of slope geometry to numerical simulation, according to the principle of aerial photogrammetry [36], the slope image data are acquired by using the tilt photography of five routes including East, South, West, North, and orthoimage with the help of PHANTOM 4 RTK Unmanned Aerial Vehicle (UAV). Among them, the UAV camera of the East, South, West, and North routes forms a 45°angle with the horizontal plane, and the UAV camera of the orthoimage route forms a 90°angle with the horizontal plane. Using context Capture Center Master and other software applications, combined with the principle of reverse modeling [37], the collected image data is generated into a three-dimensional model and 3D point cloud, and then the position, shape, size, and other data of the slope section are identified (Figures 1 and 8). The computer program for the neural network model, which was constructed in training step (2), was written using MATLAB. Then, the trained neural network was used to predict possible rockfall events.Figure 8
The back analysis mechanical parameters of the rockfall motion. (a) Historical rockfall event 1; (b) back analysis results for historical rockfall event 1; (c) historical rockfall event 2; and (d) back analysis results for historical rockfall event 2.
(a)
(b)
(c)
(d)Using the GA described in Section 2.3.2, a relevant program was prepared using MATLAB 2012b. Then, the calculation model represented by (11) was solved to obtain the maximum possible rockfall horizontal distance of each cross profile.


## 3.5.2. Calculation Model Solution

According to the model solving procedure described in Sections3.5.1 and 2.3.1, the RocFall™ was used to obtain neural network training data for each cross section of the Guanyindong Slope. Because the historical rockfall data for the Guanyindong Slope obtained from the field investigations was not sufficient to satisfy the requirements of the neural network training, we obtained the normal and tangential restitution coefficients and the friction angle of the rockmass based on the theory of displacement back analysis [32] using historical rockfall data (see Figure 8). Then, numerical simulations of the possible rockfall events of each cross profile of the slope were carried out to obtain the necessary corresponding sample training data.(1) Determination of the Simulation Parameters of the Rockfall Motion. The historical rockfall events were back-calculated as shown in Figures 8(a) and 7(b). The size of the rockfall rock was 1.8 m × 1.5 m × 0.75 m, and the obtained parameters for the rockfall of the rockmass included a normal restitution coefficient of 0.53, a tangential restitution coefficient of 0.75, and a friction angle of 46.5°.The rockfall motion was verified using back-calculated the historical rockfall events shown in Figures8(c) and 7(d). The size of the rockfall rock was 2 m × 1.5 m × 1 m. The simulation results by rockfall are consistent with the field investigations and confirm the validity of the above parameters.(2) Acquisition of Sample Data. According to the rockfall parameters obtained from the above, we used the uniform design table to design a rockfall simulation test for each cross profile and to obtain training sample data for the neural network of each cross profile. Table 2 shows the orthogonal factor level table of the cross profile x=12m designed using the uniform design table U30∗3012. Table 3 presents the experimental arrangement and results of the numerical simulation test, i.e., the training sample data of cross profile x=12m (the coordinate system is shown in Figure 6(b)). Due to space limitations, only data for cross profile x=12m is reported.Table 2
The orthogonal factor levels of cross section.
Level
Factor
Z
M
1
6
1.40
2
7.86
328.49
3
9.72
655.58
4
11.59
982.67
5
13.45
1309.76
6
15.31
1636.85
7
17.17
1963.94
8
19.03
2291.03
9
20.90
2618.12
10
22.76
2945.21
11
24.62
3272.30
12
26.48
3599.39
13
28.34
3926.48
14
30.21
4253.57
15
32.07
4580.66
16
33.93
4907.74
17
35.79
5234.83
18
37.66
5561.92
19
39.52
5889.01
20
41.38
6216.10
21
43.24
6543.19
22
45.10
6870.28
23
46.97
7197.37
24
48.83
7524.46
25
50.69
7851.55
26
52.55
8178.64
27
54.41
8505.73
28
56.28
8832.82
29
58.14
9159.91
30
60
9487.00
Z is the elevation of rockfall rock. M is mass of rockfall rock.Table 3
The experimental arrangements and results.
Experiment
Factor
Experimental resulty
Z
M
1
6
6870.28
−8.111
2
7.86
3926.48
−8.161
3
9.72
982.67
−8.030
4
11.59
8178.64
−10.884
5
13.45
5234.83
−15.378
6
15.31
2291.03
−6.621
7
17.17
9487.00
−6.211
8
19.03
6543.19
−5.252
9
20.90
3599.39
75.662
10
22.76
655.58
78.820
11
24.62
7851.55
87.112
12
26.48
4907.74
89.953
13
28.34
1963.94
86.387
14
30.21
9159.91
89.403
15
32.07
6216.10
91.772
16
33.93
8832.82
99.755
17
35.79
6870.28
102.602
18
37.66
7524.46
104.472
19
39.52
4580.66
−20.093
20
41.38
1636.85
96.014
21
43.24
8832.82
96.387
22
45.10
5889.01
−18.076
23
46.97
2945.21
−24.929
24
48.83
1.40
−25.123
25
50.69
7197.37
−26.288
26
52.55
4253.57
−27.473
27
54.41
1309.76
−28.655
28
56.28
8505.73
−29.959
29
58.14
5561.92
−30.994
30
60
2618.12
−33.952
Z is the elevation of rockfall rock. M is mass of rockfall rock.The experimental arrangement and results are shown in Table3.(3) Calculation of the Influence Area of the Rockfall of Each Cross Section. The GA calculation program was created using MATLAB 2012b according to the method described in Section 2.3.2. Then, it was used to solve the maximum influence area of each profile and the height Z and mass M of the corresponding rockfall rockmass. The calculation results are shown in Table 4.Table 4
The calculation results for the influence range of each cross section of the Guanyindong Slope.
Slope section positionx (m)
Elevation of rockfall rockZ (m)
Mass of rockfall rockM (kg)
Position of rockfall rocky (m)
12
38.705
4986.204
111.616
24
60
4196.453
102.397
36
41.841
7240.130
51.735
48
37.205
2151.947
70.18
60
60
7601.522
88.684
72
60
4351.76
90.131
84
51.640
2836.387
76.176
96
60
3658.595
90.930
108
60
3527.153
87.308
120
59.456
4974.903
117.478


## 4. Discussion of the Calculation Results

For comparison, we used Rocscience RocFall 6.0 to conduct a numerical simulation of the calculation results again. The results indicate that the deviation between the results of the rockfall numerical simulation and those of the GA calculations is only 0.096–2.491 m (Figure9). In particular, the smallest and largest deviations occur for cross profiles x=72m and x=120m, respectively (Figure 9). Therefore, the calculation results are believed to be reliable.Figure 9
The comparison between the calculation results and the numerical simulation results.


### 4.1. Application of the Calculation Results

According to the above calculation and analysis, we obtained a safe horizontal distance for the engineering application of the Guanyindong Slope. This is shown as follows:(12)y=maxfxz,mv.Next, the maximum horizontal distance values of the potential rockfall of the different cross profiles were connected to form a spline curve in order to obtain the rockfall influence area on the Guanyindong Slope (Figure10). The recorded historical rockfall events (labeled as historical rockfalls A to M in Figures 5 and 10) were identified in the field investigations and the positions of the rockfall of the dangerous rockmasses all fall within the calculated influence area of the rockfall. It should be noted that the department in charge of this scenic spot now closes the Guanyindong scenic spot due to the potential rockfalls that would threaten the safety of tourists. Then, the optimized design was carried out based on the above research results.Figure 10
The rockfall influence area of the Guanyindong Slope.


### 4.2. Countermeasures

According to Figure10, to ensure the safety of tourists in the Guanyindong scenic spot, the following disaster prevention and mitigation measures should be considered; see Figure 11(a).(1)
Abandon the original 151 m long trail into the Guanyindong scenic spot and replace it with shrubs and other vegetations to keep people out of the influence area of potential rockfall. Then, construct a new tourist route (the improved tourist route (South) or the improved tourist route (North)) (as shown in Figure10).(2)
The improved tourist route (North) leads to the cobblestone path at Wanxian Bridge. First, it extends to the northeast, and then it turns and extends to the southeast, which avoids the potential rockfall influence area. In addition, a tunnel can be constructed in this location at an elevation of approximately 200 m to provide access to the northern end of the Guanyindong scenic area. The improved tourist route (North) is 202 m long, including a 41.8 m long and 2 m wide tunnel. This scheme is 51 m longer than the original tourist route. In addition, we recommend that a concrete canopy be added to protect tourists from rockfalls caused by the steep weathered slope at the entrance.(3)
The improved tourist route (South) follows the original cobblestone path (the southward extension of Wanxian Bridge) but then follows a new route on the southwestern side of the influence area of the rockfall. A tunnel can be constructed in this location at an elevation of approximately 200 m to access the southern side of the Guanyindong scenic area. This route is 203 m long, including the 107 m long and 2 m wide tunnel. This scheme is 52 m longer than the original tourist route. A concrete canopy should also be installed at the tunnel’s entrance to ensure the safety of tourists.(4)
The original terrace in front of the Guanyindong scenic spot is within the influence area of the rockfall. Historical records include multiple rock rockfall and rockfall events occurring in this area. For the safety of the tourists, this terrace should be abandoned and converted into landscaped vegetation. Visitors should be prohibited from entering this area.(5)
For areas with severe weathering such as weak regolith at an elevation of about 250 m, a new method for surface vegetalization of slope reinforced with anchoring bar and shotcrete structure [38] can be adopted for reinforcement. The prestressed bolt was implanted into the risk rockmass (GYD1). The initial elevation of the bolt layout is 210 m, and a total of 7 layers were arranged with each layer having a height of 5 m. The horizontal spacing of the bolt arrangement was 5 m. The dip angle was 30°, and the prestress was 400 KN. The length of bolt was 15 m, the length of free section was 5 m, and the length of anchorage section was 8 m. The bolt was made of 4×7ϕ5mm steel strand (Figure 11(a)). Numerical simulation results show that the displacements of monitoring point were improved obviously after reinforcement (Figure 11(b)).Figure 11
Engineering reinforcement measures. (a) Engineering reinforcement plan; (b) deformation monitoring results.


## 4.1. Application of the Calculation Results

According to the above calculation and analysis, we obtained a safe horizontal distance for the engineering application of the Guanyindong Slope. This is shown as follows:(12)y=maxfxz,mv.Next, the maximum horizontal distance values of the potential rockfall of the different cross profiles were connected to form a spline curve in order to obtain the rockfall influence area on the Guanyindong Slope (Figure10). The recorded historical rockfall events (labeled as historical rockfalls A to M in Figures 5 and 10) were identified in the field investigations and the positions of the rockfall of the dangerous rockmasses all fall within the calculated influence area of the rockfall. It should be noted that the department in charge of this scenic spot now closes the Guanyindong scenic spot due to the potential rockfalls that would threaten the safety of tourists. Then, the optimized design was carried out based on the above research results.Figure 10
The rockfall influence area of the Guanyindong Slope.


## 4.2. Countermeasures

According to Figure10, to ensure the safety of tourists in the Guanyindong scenic spot, the following disaster prevention and mitigation measures should be considered; see Figure 11(a).(1)
Abandon the original 151 m long trail into the Guanyindong scenic spot and replace it with shrubs and other vegetations to keep people out of the influence area of potential rockfall. Then, construct a new tourist route (the improved tourist route (South) or the improved tourist route (North)) (as shown in Figure10).(2)
The improved tourist route (North) leads to the cobblestone path at Wanxian Bridge. First, it extends to the northeast, and then it turns and extends to the southeast, which avoids the potential rockfall influence area. In addition, a tunnel can be constructed in this location at an elevation of approximately 200 m to provide access to the northern end of the Guanyindong scenic area. The improved tourist route (North) is 202 m long, including a 41.8 m long and 2 m wide tunnel. This scheme is 51 m longer than the original tourist route. In addition, we recommend that a concrete canopy be added to protect tourists from rockfalls caused by the steep weathered slope at the entrance.(3)
The improved tourist route (South) follows the original cobblestone path (the southward extension of Wanxian Bridge) but then follows a new route on the southwestern side of the influence area of the rockfall. A tunnel can be constructed in this location at an elevation of approximately 200 m to access the southern side of the Guanyindong scenic area. This route is 203 m long, including the 107 m long and 2 m wide tunnel. This scheme is 52 m longer than the original tourist route. A concrete canopy should also be installed at the tunnel’s entrance to ensure the safety of tourists.(4)
The original terrace in front of the Guanyindong scenic spot is within the influence area of the rockfall. Historical records include multiple rock rockfall and rockfall events occurring in this area. For the safety of the tourists, this terrace should be abandoned and converted into landscaped vegetation. Visitors should be prohibited from entering this area.(5)
For areas with severe weathering such as weak regolith at an elevation of about 250 m, a new method for surface vegetalization of slope reinforced with anchoring bar and shotcrete structure [38] can be adopted for reinforcement. The prestressed bolt was implanted into the risk rockmass (GYD1). The initial elevation of the bolt layout is 210 m, and a total of 7 layers were arranged with each layer having a height of 5 m. The horizontal spacing of the bolt arrangement was 5 m. The dip angle was 30°, and the prestress was 400 KN. The length of bolt was 15 m, the length of free section was 5 m, and the length of anchorage section was 8 m. The bolt was made of 4×7ϕ5mm steel strand (Figure 11(a)). Numerical simulation results show that the displacements of monitoring point were improved obviously after reinforcement (Figure 11(b)).Figure 11
Engineering reinforcement measures. (a) Engineering reinforcement plan; (b) deformation monitoring results.


## 5. Conclusions

Since predicting the rockfall trajectory and calculating the influence area of rockfall are of profound importance in studying the uncertainty of rockfall and the active protection against rockfall hazards, a reliable estimate of the influence area of rockfall is crucial to the design of passive preventative methods for preventing rockfall. We studied the calculation method of the influence area of rockfall, by taking an engineering case in a tourist scenic spot in Zhejiang Province, China.(1)
The results of field investigations and numerical simulation indicate that the area of tensile stress concentration and plastic failure is the potential rockfall area where the rockmass will fall from the surface of weathered rock slope.(2)
The normal and tangential restitution coefficients and the friction angle of the weathered rock slope for the rockfall numerical simulation were obtained based on back analysis of historical rockfall events. The numerical simulation results based on these parameters can be used as learning samples for the BP neural network. Then, the motion and influence area of the potential rockfall can be effectively predicted. However, the RocFall2D numerical simulation results still deviated slightly from the historical data.(3)
The prediction of the influence area based on optimization theory, neural network technology, and the genetic optimization algorithm provides a new way to eliminate rockfall uncertainties and can be used as a guidance in the prevention of rockfall hazards.



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*Source: 1067215-2020-05-07.xml*