# Fuzzy Reliability Calculation Method of Highway Alignment

Authors: Fu Zhu; Ya Chen; Shizhong Zhao
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

Considering the randomness and fuzziness of highway route design parameters, based on the reliability theory and triangular fuzzy numbers, the fuzzy variables are transformed into random variables. Three new fuzzy reliability calculation methods of highway alignment are derived, namely, the equivalent density function method, the information entropy method, and the general density function method. The proposed calculation methods are applied to the reliability analysis of highway horizontal curve design. The results show that if the fuzzy variables are symmetric fuzzy numbers, the equivalent mean values obtained by the three methods are equal. If the fuzzy variables are asymmetric fuzzy numbers, the equivalent mean values and equivalent standard deviations of the three methods are different. Compared with the traditional reliability calculation method, the values obtained by the proposed method have a large reliability index, small failure probability, and high calculation accuracy. Compared with the other two methods, the general density function method has the fastest convergence and higher accuracy. The general density function method is then applied to the calculation of highway circular curve radius, under the condition of a superelevation failure rate of 5%, to generate the recommended values of the safety and reliability index.


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## Body

## 1. Introduction

Fuzzy mathematics basically deals with “either/or” problems, which makes up for the shortcomings of the “either/or” binary logic of deterministic mathematics. Therefore, fuzzy mathematics is developing very rapidly, and its application areas are expanding; it has been utilized in road engineering reliability design [1–4]. Highway alignment design refers to the comprehensive design of the combination of flat, vertical, and horizontal alignment. The alignment design should ensure the safety and comfort of vehicle driving, the visual and psychological responses of the driver, guide the driver’s line of sight, maintain the continuity and coordination of the alignment, and provide the driver with enough comfort and safety. The alignment design mainly includes requirements for horizontal curves, vertical curves, lane widths, and curve widening, as well as the requirements for conditions such as the superelevation and the sight distance. There are many factors that affect the function of highway alignment design (such as drivers, vehicles, and road environment), and the various influencing factors vary greatly in time and space with complicated relationships. Highway alignment design is a nondeterministic problem. The uncertainty stems from the randomness and fuzziness of the design parameters. Therefore, further research on the fuzzy reliability calculation method of highway alignment corresponding to the design specifications [5, 6] can provide the necessary technical means for the optimization of highway route design and the reliability assessment of highway projects.In order to improve the calculation accuracy of highway alignment reliability, scholars have investigated and studied the “uncertainty” factors based on reliability theory, and the research on the reliability of highway linearity is gradually developed from a system [7–10], which can be roughly divided into three categories: highway surroundings, moving vehicles, and drivers. Among them, the highway surroundings include buildings, bridges, tunnels, vegetation, weather, geology, sudden disasters, and so forth [11–13]. These factors are highly complex, and there are certain connections between them, which increases the degree of danger during the movement of vehicles and directly affects the reliability of highway alignment [14–18]. The structural reliability of vehicles traveling on the road and the characteristics of drivers indirectly affect the reliability of highway alignment [19, 20]. The current highway alignment reliability theory only considers the randomness of the influencing factors and does not consider the fuzzy nature of these influencing factors. Fuzzy reliability theory is a further development of reliability theory, which considers the fuzziness of influencing factors when applied to structural reliability calculation [21–25]. Therefore, considering the fuzzy nature of the influencing factors of highway alignment reliability, applying fuzzy reliability theory to highway alignment reliability calculation is of great significance to improve the accuracy of highway alignment reliability calculation.In this study, based on the structural reliability theory, the basic theory of the fuzzy reliability of highway alignment is proposed with fuzzy and random variables as the basic variables. The triangular membership function was introduced, and three methods of calculating the fuzzy reliability of highway alignment are derived, namely, the equivalent density function, information entropy, and general density function methods. The proposed calculation methods are applied to the analysis of the reliability of the superelevation design of the highway horizontal curve.


## 2. Basic Theory of Fuzzy Reliability

### 2.1. Reliability Theory

According to the Chinese specification [26] for highway reliability, the ability of a road to perform its intended function within a specified time and under specified conditions is referred to as the reliability of the highway. The probability of completing the intended function is called the reliability of the highway. Here, “specified time” refers to the design base period of the highway, “specified conditions” refers to the predetermined construction conditions and applicable conditions of the highway design, and the “intended function” refers to the various functional requirements of the highway. If the highway alignment exceeds a specific state and cannot meet the functional requirements of the design, this specific state is called the limit state. This state is the critical state that is used to judge the reliability of the highway; beyond this state, the probability that the road cannot complete the intended function is expressed by the probability of failure Pf.Suppose that the performance function for highway alignment design is(1)Z=S−D,where S is the capacity of the designed expressway, D is the load to be borne by the expressway, and both are random variables. The limit state equation for the highway alignment design when S and D are independent of each other is(2)Z=S−D=0.WhenZ>0, the route design parameters meet the design requirements; when Z=0, the route design parameters are at critical values; and when Z<0, the route design parameters do not meet the design requirements and are in a failed state.According to the random probability density formula, the probability of failure expression is obtained as(3)Pf=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞dfSsfDddsdd,where fSs, fDd, and fZz are the probability density functions of S, D, and Z, respectively.According to whether the variableZ obeys normal distribution, the calculation methods of reliability can be divided into two categories. If Z is nonnormal distribution, the Monte Carlo method and the first-order reliability method are generally adopted. If Z is normal distribution, the first-order second-moment and checking point methods are generally adopted [27–31]. Assume that the function Z obeys normal distribution, and its average and standard deviation are μZ and σZ, respectively. Then, the probability density function Z of fZz is(4)fZz=12πσZexp−z−μZ22σZ2.The distribution curve is shown in Figure1. Pf is the area of the shaded portion under the probability density curve in Figure 1.Figure 1
Relationship between reliability indicators and probability of failure.Lety=Z−μZ/σZ, Z can be transformed into a standard normal distribution variable, and its probability density function and cumulative distribution function, respectively, can be(5)ℓy=12πexp−y22,(6)Φy=∫−∞yℓydy.Combining equations (3), (5), (6), and (7), the following equation is obtained:(7)Pf=∫−∞−μZ/σZℓydy=Φ−μZσZ.To simplify the expression and meet the accuracy requirement, letβ=μZ/σZ, where β is called reliability index. Then, its corresponding equation to the probability of failure can be expressed as(8)Pf=Φ−β=1−Φβ.


### 2.2. Solution of Triangular Fuzzy Numbers

Fuzzy numbers are a special fuzzy set introduced by Zadeh [32] as a mathematical expression for a fuzzy description of real numbers. A fuzzy subset A is called fuzzy numbers when it is assumed that the domain of A˜ is the set of all real numbers R, and the following conditions are satisfied: (1) A˜ is a regular convex fuzzy set; and (2) A˜ is bounded.Triangular fuzzy numbers are a fuzzy set whose image is a triangle. Such fuzzy numbers are usually specified using the notationA˜m,m−δ,ϕ−m, where m is the core element corresponding to the value when A˜x is 1, m−δ is the length of the line segment from m to its left vertex, ϕ−m is the length of the line segment from m to the right vertex, δ is the lower bound value of the triangular fuzzy numbers A˜, and ϕ is the upper bound value of the triangular fuzzy numbers A˜, as shown in Figure 2.Figure 2
Triangular fuzzy numbers.Its membership function can be expressed as follows:(9)A˜x=0,x≤δ,1+x−mm−δ,δ<x<m,1−x−mϕ−m,m<x<ϕ,0,x≥ϕ.The parameters in the triangular fuzzy numbersA˜ can be calculated by the following equation:(10)δ=ψ−2ψ−ξ,ψ≥2ψ−ξ,0,ψ≤2ψ−ξ,(11)m=ψ,(12)ϕ=ψ+2ζ−ψ.ξ, ψ, and ζ in equations (10), (11), and (12) are the minimum, intermediate, and maximum values of the calculated parameters provided by the data, respectively.


## 2.1. Reliability Theory

According to the Chinese specification [26] for highway reliability, the ability of a road to perform its intended function within a specified time and under specified conditions is referred to as the reliability of the highway. The probability of completing the intended function is called the reliability of the highway. Here, “specified time” refers to the design base period of the highway, “specified conditions” refers to the predetermined construction conditions and applicable conditions of the highway design, and the “intended function” refers to the various functional requirements of the highway. If the highway alignment exceeds a specific state and cannot meet the functional requirements of the design, this specific state is called the limit state. This state is the critical state that is used to judge the reliability of the highway; beyond this state, the probability that the road cannot complete the intended function is expressed by the probability of failure Pf.Suppose that the performance function for highway alignment design is(1)Z=S−D,where S is the capacity of the designed expressway, D is the load to be borne by the expressway, and both are random variables. The limit state equation for the highway alignment design when S and D are independent of each other is(2)Z=S−D=0.WhenZ>0, the route design parameters meet the design requirements; when Z=0, the route design parameters are at critical values; and when Z<0, the route design parameters do not meet the design requirements and are in a failed state.According to the random probability density formula, the probability of failure expression is obtained as(3)Pf=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞dfSsfDddsdd,where fSs, fDd, and fZz are the probability density functions of S, D, and Z, respectively.According to whether the variableZ obeys normal distribution, the calculation methods of reliability can be divided into two categories. If Z is nonnormal distribution, the Monte Carlo method and the first-order reliability method are generally adopted. If Z is normal distribution, the first-order second-moment and checking point methods are generally adopted [27–31]. Assume that the function Z obeys normal distribution, and its average and standard deviation are μZ and σZ, respectively. Then, the probability density function Z of fZz is(4)fZz=12πσZexp−z−μZ22σZ2.The distribution curve is shown in Figure1. Pf is the area of the shaded portion under the probability density curve in Figure 1.Figure 1
Relationship between reliability indicators and probability of failure.Lety=Z−μZ/σZ, Z can be transformed into a standard normal distribution variable, and its probability density function and cumulative distribution function, respectively, can be(5)ℓy=12πexp−y22,(6)Φy=∫−∞yℓydy.Combining equations (3), (5), (6), and (7), the following equation is obtained:(7)Pf=∫−∞−μZ/σZℓydy=Φ−μZσZ.To simplify the expression and meet the accuracy requirement, letβ=μZ/σZ, where β is called reliability index. Then, its corresponding equation to the probability of failure can be expressed as(8)Pf=Φ−β=1−Φβ.


## 2.2. Solution of Triangular Fuzzy Numbers

Fuzzy numbers are a special fuzzy set introduced by Zadeh [32] as a mathematical expression for a fuzzy description of real numbers. A fuzzy subset A is called fuzzy numbers when it is assumed that the domain of A˜ is the set of all real numbers R, and the following conditions are satisfied: (1) A˜ is a regular convex fuzzy set; and (2) A˜ is bounded.Triangular fuzzy numbers are a fuzzy set whose image is a triangle. Such fuzzy numbers are usually specified using the notationA˜m,m−δ,ϕ−m, where m is the core element corresponding to the value when A˜x is 1, m−δ is the length of the line segment from m to its left vertex, ϕ−m is the length of the line segment from m to the right vertex, δ is the lower bound value of the triangular fuzzy numbers A˜, and ϕ is the upper bound value of the triangular fuzzy numbers A˜, as shown in Figure 2.Figure 2
Triangular fuzzy numbers.Its membership function can be expressed as follows:(9)A˜x=0,x≤δ,1+x−mm−δ,δ<x<m,1−x−mϕ−m,m<x<ϕ,0,x≥ϕ.The parameters in the triangular fuzzy numbersA˜ can be calculated by the following equation:(10)δ=ψ−2ψ−ξ,ψ≥2ψ−ξ,0,ψ≤2ψ−ξ,(11)m=ψ,(12)ϕ=ψ+2ζ−ψ.ξ, ψ, and ζ in equations (10), (11), and (12) are the minimum, intermediate, and maximum values of the calculated parameters provided by the data, respectively.


## 3. Calculation Method of Fuzzy Reliability

At present, the fuzzy reliability theory can be broadly classified into three main categories. The first category is profust reliability theory, which considers that all uncertain parameters can be included in the probability category and discusses the fuzziness of the existence of their failure probabilities. The second category is profust reliability theory, which considers that all uncertain parameters are fuzzy variables. The third category is profust reliability theory, which considers both randomness and fuzziness in uncertainty parameters. This theory emerges late and needs to be improved. When facing the coexistence of randomness and fuzziness, the result can not reflect the real safety situation of the road, and the fuzziness will be lost if the reality is directly calculated by the random probability formula.


### 3.1. Equivalent Density Function Method

Suppose thatA∈℘X, for ∀λ∈0,1, Aλ=defx|Ax≥λ. Aλ is called the λ-cut set of A, where λ is called the threshold or confidence level, and ℘ denotes the power set of the argument domain X [4]. According to the definition of the λ-cut set, it is known that the λ-cut set of a fuzzy set Aλ is a classical set consisting of members whose A˜x is not less than λ, and its eigenfunction is(13)χAλx=1,Ax≥λ,0,Ax<λ.Assume that the highway provides design indicators as the random variablesS, and the design indicators expected by drivers to be fuzzy variables D˜. From the definition of the λ-cut set in fuzzy mathematics, it is known that D˜λ∈aλ,bλ is a classical set under the threshold of λ (λ∈0,1), as shown in Figure 3.Figure 3
λ-cut set of fuzzy variables D˜.Suppose thatD˜λ follows a uniform distribution. Its probability density function fD˜λx1 at the threshold value λ is(14)fD˜λx1=1bλ−aλ.TakingfD˜λx1 as an integrand and integrating over λ, the equivalent probability density function is obtained as(15)f^D˜x1=∫0λ1bλ−aλdλ.Substituting the probability density functionf^D˜x1 into equation (3) yields the probability of failure Pf1 as(16)Pf1=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x1fSsf^D˜x1dsdx1.If the fuzzy variablesD˜ are triangular fuzzy numbers, by combining equations (3), (15), and (16), the mean μ^x1 and standard deviation σ^x1 of the equivalent random variables are derived as(17)μ^x1=m2+δ+ϕ4,(18)σ^x1=ϕ−mm−δ9+7δ+ϕ−2m2144.According to equation (16), the triangular fuzzy number equivalent normal random variable probability density function f^D˜x1 can be introduced as(19)f^x1=−ln1−λϕ−δ=12πσ^x1e−1/2x1−μx1/σ^x1.


### 3.2. Information Entropy Method

Shannon [33], the founder of information entropy, believes that information is inherently random and that information entropy is a measure of the degree of unconstrained random variables; the greater the uncertainty, the greater the entropy. With the growing understanding of information, it is recognized that on many occasions, information may not have probabilistic characteristics in a statistical sense, and its uncertainty cannot be described by probability theory. In this regard, Aldo Luca and Settimo Termini [34, 35] proposed the concept of “nonprobability entropy,” such as “fuzzy information entropy” and so forth. Probabilistic and fuzzy entropies are used to measure the degree of uncertainty of random and fuzzy variables, respectively.Assume that the highway provides design indicators as random variablesS and the design indicators expected by the driver to be fuzzy variables D˜. The equivalent random variables transformed by fuzzy variables are D^, and their probability density function is fD˜x2. According to the information entropy expression provided by Shannon, the information entropy expression H^x2 for the equivalent random variable of the highway D^ is derived as follows:(20)H^x2=−∫xfD˜x2lnfD˜x2dx2.Based on the fuzzy information entropy expression provided by Aldo Luca and Settimo Termini, the expression for the fuzzy information entropy of highway fuzzy variablesD˜ is derived as follows:(21)Gx2=−∫x2fD˜x2lnfD˜′x2dx2,where fD˜′x2=D˜x2/∫x2D˜x2dx2, and D˜x2 is the membership function of the fuzzy variables D˜.Suppose that the transformed random variables follow a normal distribution. According to equation (20), we can obtain the following equation:(22)H^x2=−∫x2fD˜x2lnfD˜x2dx2=ln2πeσx2.Combining equations (21) and (22), the equivalent variance of the transformation of the fuzzy variable into a normal random variable is deduced as(23)σ^x2=12πeGx2−0.5.Letμ^x2=m; the equivalent normal probability density function is(24)f^D˜x2=12πσ^x2e−1/2x2−m/σ^x22.Substituting the probability density functionf^D˜x2 into equation (3) yields the probability of failure Pf2 as(25)Pf2=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x2fSsf^D˜x2dsdx2.If the fuzzy variablesD˜ are the triangular fuzzy numbers, the equivalent variance of the triangular fuzzy numbers is deduced from Gx2=lnϕ−δ/2+0.5 as(26)σ^x2=ϕ−δ22π.


### 3.3. General Density Function Method

The generalized density function method is also known as the weighted average method. Here, the normalization principle is used, and the generalized density function of the transformed equivalent random variable is defined asfA˜x=A˜x/∫xA˜xdx, following the mathematical description of the probability density function of random variables. The generalized density function defined in this way retains the distribution information of the membership function of the original fuzzy variables but also satisfies the completeness and nonnegativity required by the probability density function. As the generalized density function has no change in the relative size of each function value within the range of values of the independent variable, it corresponds to the density size of the original membership function at a certain value, which still implies the fuzzy degree of the original fuzzy variables taking that value.Suppose that the highway provides the design indicators as random variablesS and the design indicators expected by drivers to be fuzzy variables D˜. The generalized density function of the fuzzy variables D˜ can be derived as(27)fD˜x3=D˜x3∫x3D˜x3dx3.Substituting the probability density functionf^D˜x3 into equation (3) yields the probability of failure Pf3 to be(28)Pf3=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x3fSsf^D˜x3dsdx3.If the fuzzy variablesD˜ are triangular fuzzy numbers, their equivalent mean value can be derived from the definition of the generalized density function as(29)μ^x3=m+δ+ϕ3.Taking equation (26) as the equivalent scalar variance of this method, the equivalent random variable probability density function can be introduced according to equation (27) as(30)f^x3=2D˜x3ϕ−δ=12πσ^x2e−1/2x3−μx3/σ^x2.By analyzing the equivalent means and variances of the three methods, the following conclusions were drawn:(1)
If the fuzzy variable is a symmetric fuzzy number, namely,δ+ϕ=2m, and by substituting it into equations (17) and (29), it can be derived as μ^x1=μ^x2=μ^x3=m. In this case, the equivalent means obtained by the three methods are equal; if the fuzzy variable is an asymmetric fuzzy number, the equivalent means and equivalent standard deviations of the three methods are different.(2)
For the equivalent density function method, the true number of logarithms in the equivalent probability density functionf^x1=−ln1−λ/ϕ−δ is equal to 0 when the degree of membership of the triangular fuzzy numbers λ=1, at which point the equivalent probability density function is meaningless.(3)
For the information entropy method, the equivalent mean value is taken as the value when the fuzzy number membership function is 1. If the fuzzy number is asymmetric, there is some error in this value.


## 3.1. Equivalent Density Function Method

Suppose thatA∈℘X, for ∀λ∈0,1, Aλ=defx|Ax≥λ. Aλ is called the λ-cut set of A, where λ is called the threshold or confidence level, and ℘ denotes the power set of the argument domain X [4]. According to the definition of the λ-cut set, it is known that the λ-cut set of a fuzzy set Aλ is a classical set consisting of members whose A˜x is not less than λ, and its eigenfunction is(13)χAλx=1,Ax≥λ,0,Ax<λ.Assume that the highway provides design indicators as the random variablesS, and the design indicators expected by drivers to be fuzzy variables D˜. From the definition of the λ-cut set in fuzzy mathematics, it is known that D˜λ∈aλ,bλ is a classical set under the threshold of λ (λ∈0,1), as shown in Figure 3.Figure 3
λ-cut set of fuzzy variables D˜.Suppose thatD˜λ follows a uniform distribution. Its probability density function fD˜λx1 at the threshold value λ is(14)fD˜λx1=1bλ−aλ.TakingfD˜λx1 as an integrand and integrating over λ, the equivalent probability density function is obtained as(15)f^D˜x1=∫0λ1bλ−aλdλ.Substituting the probability density functionf^D˜x1 into equation (3) yields the probability of failure Pf1 as(16)Pf1=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x1fSsf^D˜x1dsdx1.If the fuzzy variablesD˜ are triangular fuzzy numbers, by combining equations (3), (15), and (16), the mean μ^x1 and standard deviation σ^x1 of the equivalent random variables are derived as(17)μ^x1=m2+δ+ϕ4,(18)σ^x1=ϕ−mm−δ9+7δ+ϕ−2m2144.According to equation (16), the triangular fuzzy number equivalent normal random variable probability density function f^D˜x1 can be introduced as(19)f^x1=−ln1−λϕ−δ=12πσ^x1e−1/2x1−μx1/σ^x1.


## 3.2. Information Entropy Method

Shannon [33], the founder of information entropy, believes that information is inherently random and that information entropy is a measure of the degree of unconstrained random variables; the greater the uncertainty, the greater the entropy. With the growing understanding of information, it is recognized that on many occasions, information may not have probabilistic characteristics in a statistical sense, and its uncertainty cannot be described by probability theory. In this regard, Aldo Luca and Settimo Termini [34, 35] proposed the concept of “nonprobability entropy,” such as “fuzzy information entropy” and so forth. Probabilistic and fuzzy entropies are used to measure the degree of uncertainty of random and fuzzy variables, respectively.Assume that the highway provides design indicators as random variablesS and the design indicators expected by the driver to be fuzzy variables D˜. The equivalent random variables transformed by fuzzy variables are D^, and their probability density function is fD˜x2. According to the information entropy expression provided by Shannon, the information entropy expression H^x2 for the equivalent random variable of the highway D^ is derived as follows:(20)H^x2=−∫xfD˜x2lnfD˜x2dx2.Based on the fuzzy information entropy expression provided by Aldo Luca and Settimo Termini, the expression for the fuzzy information entropy of highway fuzzy variablesD˜ is derived as follows:(21)Gx2=−∫x2fD˜x2lnfD˜′x2dx2,where fD˜′x2=D˜x2/∫x2D˜x2dx2, and D˜x2 is the membership function of the fuzzy variables D˜.Suppose that the transformed random variables follow a normal distribution. According to equation (20), we can obtain the following equation:(22)H^x2=−∫x2fD˜x2lnfD˜x2dx2=ln2πeσx2.Combining equations (21) and (22), the equivalent variance of the transformation of the fuzzy variable into a normal random variable is deduced as(23)σ^x2=12πeGx2−0.5.Letμ^x2=m; the equivalent normal probability density function is(24)f^D˜x2=12πσ^x2e−1/2x2−m/σ^x22.Substituting the probability density functionf^D˜x2 into equation (3) yields the probability of failure Pf2 as(25)Pf2=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x2fSsf^D˜x2dsdx2.If the fuzzy variablesD˜ are the triangular fuzzy numbers, the equivalent variance of the triangular fuzzy numbers is deduced from Gx2=lnϕ−δ/2+0.5 as(26)σ^x2=ϕ−δ22π.


## 3.3. General Density Function Method

The generalized density function method is also known as the weighted average method. Here, the normalization principle is used, and the generalized density function of the transformed equivalent random variable is defined asfA˜x=A˜x/∫xA˜xdx, following the mathematical description of the probability density function of random variables. The generalized density function defined in this way retains the distribution information of the membership function of the original fuzzy variables but also satisfies the completeness and nonnegativity required by the probability density function. As the generalized density function has no change in the relative size of each function value within the range of values of the independent variable, it corresponds to the density size of the original membership function at a certain value, which still implies the fuzzy degree of the original fuzzy variables taking that value.Suppose that the highway provides the design indicators as random variablesS and the design indicators expected by drivers to be fuzzy variables D˜. The generalized density function of the fuzzy variables D˜ can be derived as(27)fD˜x3=D˜x3∫x3D˜x3dx3.Substituting the probability density functionf^D˜x3 into equation (3) yields the probability of failure Pf3 to be(28)Pf3=PZ<0=∫−∞0fZzdz=∫−∞+∞∫−∞x3fSsf^D˜x3dsdx3.If the fuzzy variablesD˜ are triangular fuzzy numbers, their equivalent mean value can be derived from the definition of the generalized density function as(29)μ^x3=m+δ+ϕ3.Taking equation (26) as the equivalent scalar variance of this method, the equivalent random variable probability density function can be introduced according to equation (27) as(30)f^x3=2D˜x3ϕ−δ=12πσ^x2e−1/2x3−μx3/σ^x2.By analyzing the equivalent means and variances of the three methods, the following conclusions were drawn:(1)
If the fuzzy variable is a symmetric fuzzy number, namely,δ+ϕ=2m, and by substituting it into equations (17) and (29), it can be derived as μ^x1=μ^x2=μ^x3=m. In this case, the equivalent means obtained by the three methods are equal; if the fuzzy variable is an asymmetric fuzzy number, the equivalent means and equivalent standard deviations of the three methods are different.(2)
For the equivalent density function method, the true number of logarithms in the equivalent probability density functionf^x1=−ln1−λ/ϕ−δ is equal to 0 when the degree of membership of the triangular fuzzy numbers λ=1, at which point the equivalent probability density function is meaningless.(3)
For the information entropy method, the equivalent mean value is taken as the value when the fuzzy number membership function is 1. If the fuzzy number is asymmetric, there is some error in this value.


## 4. Engineering Example and Results Analysis

### 4.1. Engineering Example

A specific highway was constructed as an early main artery linking north–south traffic in China and was fully opened in the 1990s. In this study, the selected engineering section (K115–K287) was built as a two-way four-lane highway with a design speed of 120 km/h and asphalt concrete pavement. Based on Xiaolei Zhang’s data description of the accident section, the parameters of vehicle running speed were obtained and presented in Table1 [36].Table 1
Parameters for calculating the operating speed of triangular fuzzy numbers.
Calculation parametersMinimum valueMedian valueMaximum valueOperating speedkm/h109.495117.095122.75The data in Table1 were substituted into equations (10)–(12) to calculate the trigonometric membership function of the operating speed as follows:(31)V˜x=0,x≤101.895,1+x−117.09515.2,101.895<x<117.095,1−x−117.09511.31,117.095<x<128.405,0,x≥128.405.According to the U.S. Green Book [5] on sideway force coefficients, the corresponding relationship between operating speed and the sideway force coefficients, at a superelevation value of 6%, is shown in Table 2.Table 2
Lateral force coefficients under different operating speed conditions.
Operating speedkm/h406080100120Sideway force coefficient0.230.170.140.120.09According to [37] and others, the value of the sideway force coefficient is mainly determined by the range of friction coefficient between the highway surface and tires and the feeling of comfort of the driver and passengers. The degree of road damage and dry/wet conditions, the quality of the vehicle tires, and the physical condition of the person are all random factors. Therefore, to simplify the solution process, it was assumed that the sideway force coefficient also follows a normal distribution; the sideway force coefficient, at an operating speed of 120 km/h in Table 2, was taken as the mean value, and the standard deviation was taken as 0.005 [36].The fuzzy reliability function for superelevation is derived from the superelevation formula,ih=V2/127R−μ, as(32)Z=iS−V˜2127R−μ,where iS is the superelevation design of the highway and takes the value of 6%. V˜ is the fuzzy operating speed, and its membership function is V˜x. R is the radius of the circular curve, which, according to the U.S. Green Book [6], takes the value of 756m. μ is the sideway force coefficient, and its probability density function is(33)f^xμ=10.01πe−1/2xμ−0.9/0.0052.The equivalent density function method was used to calculate the equivalent mean and standard deviation of the triangular membership function of the fuzzy operating speed according to (1) and (2). The equivalent mean and standard deviation were μ^v1=116.122 and σ^v1=4.454, respectively. Its equivalent normal random variable probability density function is(34)f^x1=18.908πe−1/2x1−116.122/4.4542.Using the information entropy method, the fuzzy information entropy of the triangular membership function of the fuzzy operating speed was calculated according to equation (21) as Gv=3.084. According to μ^v2=m and equation (23), the equivalent mean and standard deviation were found to be μ^v2=117.095 and σ^v2=2.982, respectively.Its equivalent normal random variable probability density function is(35)f^x2=15.964πe−1/2x2−117.095/2.9822.The general density function method was used to calculate the equivalent mean and standard deviation of the fuzzy operating speed according to equations (27) and (22), and they were obtained as μ^v3=115.798 and σ^v2=2.982, respectively.Its equivalent normal random variable probability density function is(36)f^x3=15.964πe−1/2x2−115.789/2.9822.According to the reliability theory, the improved first-order second-moment method [38] is used to calculate the reliability of highway alignment. After several iterations of programming in Matlab, the results of the previous and subsequent iterations are subtracted, and when the difference is less than the allowable error εε≤1×10−6, it is considered to be convergent. Three methods to calculate the reliability index and probability of failure of superelevation are obtained, as shown in Table 3.Table 3
Reliability index and failure probability.
Calculation methodReliability indexProbability of failureEquivalent density function method0.79540.2132Information entropy method0.81020.2089General density function method1.17250.1204


### 4.2. Results Analysis

The reliability index and failure probability are calculated by the three methods of equivalent density function method, information entropy method, and general density function method. It is compared with the calculation results of the improved first-order second moment method in the traditional reliability calculation method. The results are shown in Figure4.Figure 4
Comparison of new and traditional reliability calculation methods.
(a)(b)It can be seen from Figure4 that compared with the traditional method, the reliability index calculated by the three methods is high, and the failure probability is small. It can be seen that the fuzzy reliability calculation method has higher accuracy than the traditional reliability calculation method. The calculation results of the three methods are compared with each other. The failure probability of the general density function method is about 10% lower than that of the information entropy method and the equivalent density function method, so the calculation accuracy of the general density function method is higher.The variation law of reliability indexes and probability of failure calculated by the three methods under the different radius of circular curve conditions are shown in Figure5.Figure 5
Calculation results of reliability index and probability of failure under the different radius of circular curve conditions.
(a)(b)As shown in Figure5, the general density function method is better than the equivalent density function method and the information entropy method, with faster computational convergence and higher accuracy. When the radius of the circular curve is less than 754 m, the equivalent density function method is more accurate; when the radius of the circular curve is greater than 754 m, the information entropy method has higher accuracy and converges faster than the equivalent density function method.Comparing the regulations in the Design Specification for Highway Alignment of China [5] and the U.S. Green Book [6] on the radius of a circular curve, it was found that the corresponding radius of a circular curve is not the same when the same superelevation value is taken. The generalized density function method was used to calculate the probability of failure in both the Chinese specification and the U.S. Green book at the same superelevation value under different circular curve radius conditions, as shown in Table 4.Table 4
Calculation of the probability of failure of superelevation.
Superelevation value (%)ChinaUnited StatesRadius of circular curve (m)Probability of failure (%)Radius of circular curve (m)Probability of failure (%)48100.2546——67100.21737560.033786500.0866670.0348105700.16325970.035612——5400.0373Note: “—” indicates that the relevant data are not given in the specification.As can be seen from Table4, the radius of the circular curve specified in the U.S. Green Book is larger than that specified in the Chinese specification under the same superelevation value. The calculated probability of failure is smaller than that of the latter, indicating that the radius of the circular curve specified in the Chinese specification is smaller under the same probability of failure of superelevation. The variability of the probability of failure of the Chinese specification is found to be greater when comparing the results of the calculation of the probability of failure of the Chinese specification under different superelevation values. When the radius of the circular curve is 650 and 570 m in particular, the probability of failure of superelevation differs by 13.13%. In the U.S. Green Book, the probability of failure of superelevation is stable at approximately 3%, with little variability. Thus, the radius of the circular curve specified in the Chinese specification is open to question.Under the condition of 5% probability of failure of superelevation, the generalized density function method was used to calculate the radius of the circular curve under different superelevation values; it was compared with the traditional reliability calculation methods. The results are shown in Figure6.Figure 6
Results of the radius of the circular curve.As can be seen from Figure6, the radius of the circular curve calculated by the generalized density function method under the condition of 5% probability of failure of superelevation [38] takes a smaller value than that of the traditional reliability methods, indicating that the generalized density function method converges more quickly. Table 5 provides the recommended values of the radius of the circular curve under the condition of 5% probability of failure of superelevation.Table 5
Recommended values of the radius of circular curves under different superelevation conditions.
Superelevation value (%)4681012Radius of circular curve (m)861747660591535


## 4.1. Engineering Example

A specific highway was constructed as an early main artery linking north–south traffic in China and was fully opened in the 1990s. In this study, the selected engineering section (K115–K287) was built as a two-way four-lane highway with a design speed of 120 km/h and asphalt concrete pavement. Based on Xiaolei Zhang’s data description of the accident section, the parameters of vehicle running speed were obtained and presented in Table1 [36].Table 1
Parameters for calculating the operating speed of triangular fuzzy numbers.
Calculation parametersMinimum valueMedian valueMaximum valueOperating speedkm/h109.495117.095122.75The data in Table1 were substituted into equations (10)–(12) to calculate the trigonometric membership function of the operating speed as follows:(31)V˜x=0,x≤101.895,1+x−117.09515.2,101.895<x<117.095,1−x−117.09511.31,117.095<x<128.405,0,x≥128.405.According to the U.S. Green Book [5] on sideway force coefficients, the corresponding relationship between operating speed and the sideway force coefficients, at a superelevation value of 6%, is shown in Table 2.Table 2
Lateral force coefficients under different operating speed conditions.
Operating speedkm/h406080100120Sideway force coefficient0.230.170.140.120.09According to [37] and others, the value of the sideway force coefficient is mainly determined by the range of friction coefficient between the highway surface and tires and the feeling of comfort of the driver and passengers. The degree of road damage and dry/wet conditions, the quality of the vehicle tires, and the physical condition of the person are all random factors. Therefore, to simplify the solution process, it was assumed that the sideway force coefficient also follows a normal distribution; the sideway force coefficient, at an operating speed of 120 km/h in Table 2, was taken as the mean value, and the standard deviation was taken as 0.005 [36].The fuzzy reliability function for superelevation is derived from the superelevation formula,ih=V2/127R−μ, as(32)Z=iS−V˜2127R−μ,where iS is the superelevation design of the highway and takes the value of 6%. V˜ is the fuzzy operating speed, and its membership function is V˜x. R is the radius of the circular curve, which, according to the U.S. Green Book [6], takes the value of 756m. μ is the sideway force coefficient, and its probability density function is(33)f^xμ=10.01πe−1/2xμ−0.9/0.0052.The equivalent density function method was used to calculate the equivalent mean and standard deviation of the triangular membership function of the fuzzy operating speed according to (1) and (2). The equivalent mean and standard deviation were μ^v1=116.122 and σ^v1=4.454, respectively. Its equivalent normal random variable probability density function is(34)f^x1=18.908πe−1/2x1−116.122/4.4542.Using the information entropy method, the fuzzy information entropy of the triangular membership function of the fuzzy operating speed was calculated according to equation (21) as Gv=3.084. According to μ^v2=m and equation (23), the equivalent mean and standard deviation were found to be μ^v2=117.095 and σ^v2=2.982, respectively.Its equivalent normal random variable probability density function is(35)f^x2=15.964πe−1/2x2−117.095/2.9822.The general density function method was used to calculate the equivalent mean and standard deviation of the fuzzy operating speed according to equations (27) and (22), and they were obtained as μ^v3=115.798 and σ^v2=2.982, respectively.Its equivalent normal random variable probability density function is(36)f^x3=15.964πe−1/2x2−115.789/2.9822.According to the reliability theory, the improved first-order second-moment method [38] is used to calculate the reliability of highway alignment. After several iterations of programming in Matlab, the results of the previous and subsequent iterations are subtracted, and when the difference is less than the allowable error εε≤1×10−6, it is considered to be convergent. Three methods to calculate the reliability index and probability of failure of superelevation are obtained, as shown in Table 3.Table 3
Reliability index and failure probability.
Calculation methodReliability indexProbability of failureEquivalent density function method0.79540.2132Information entropy method0.81020.2089General density function method1.17250.1204


## 4.2. Results Analysis

The reliability index and failure probability are calculated by the three methods of equivalent density function method, information entropy method, and general density function method. It is compared with the calculation results of the improved first-order second moment method in the traditional reliability calculation method. The results are shown in Figure4.Figure 4
Comparison of new and traditional reliability calculation methods.
(a)(b)It can be seen from Figure4 that compared with the traditional method, the reliability index calculated by the three methods is high, and the failure probability is small. It can be seen that the fuzzy reliability calculation method has higher accuracy than the traditional reliability calculation method. The calculation results of the three methods are compared with each other. The failure probability of the general density function method is about 10% lower than that of the information entropy method and the equivalent density function method, so the calculation accuracy of the general density function method is higher.The variation law of reliability indexes and probability of failure calculated by the three methods under the different radius of circular curve conditions are shown in Figure5.Figure 5
Calculation results of reliability index and probability of failure under the different radius of circular curve conditions.
(a)(b)As shown in Figure5, the general density function method is better than the equivalent density function method and the information entropy method, with faster computational convergence and higher accuracy. When the radius of the circular curve is less than 754 m, the equivalent density function method is more accurate; when the radius of the circular curve is greater than 754 m, the information entropy method has higher accuracy and converges faster than the equivalent density function method.Comparing the regulations in the Design Specification for Highway Alignment of China [5] and the U.S. Green Book [6] on the radius of a circular curve, it was found that the corresponding radius of a circular curve is not the same when the same superelevation value is taken. The generalized density function method was used to calculate the probability of failure in both the Chinese specification and the U.S. Green book at the same superelevation value under different circular curve radius conditions, as shown in Table 4.Table 4
Calculation of the probability of failure of superelevation.
Superelevation value (%)ChinaUnited StatesRadius of circular curve (m)Probability of failure (%)Radius of circular curve (m)Probability of failure (%)48100.2546——67100.21737560.033786500.0866670.0348105700.16325970.035612——5400.0373Note: “—” indicates that the relevant data are not given in the specification.As can be seen from Table4, the radius of the circular curve specified in the U.S. Green Book is larger than that specified in the Chinese specification under the same superelevation value. The calculated probability of failure is smaller than that of the latter, indicating that the radius of the circular curve specified in the Chinese specification is smaller under the same probability of failure of superelevation. The variability of the probability of failure of the Chinese specification is found to be greater when comparing the results of the calculation of the probability of failure of the Chinese specification under different superelevation values. When the radius of the circular curve is 650 and 570 m in particular, the probability of failure of superelevation differs by 13.13%. In the U.S. Green Book, the probability of failure of superelevation is stable at approximately 3%, with little variability. Thus, the radius of the circular curve specified in the Chinese specification is open to question.Under the condition of 5% probability of failure of superelevation, the generalized density function method was used to calculate the radius of the circular curve under different superelevation values; it was compared with the traditional reliability calculation methods. The results are shown in Figure6.Figure 6
Results of the radius of the circular curve.As can be seen from Figure6, the radius of the circular curve calculated by the generalized density function method under the condition of 5% probability of failure of superelevation [38] takes a smaller value than that of the traditional reliability methods, indicating that the generalized density function method converges more quickly. Table 5 provides the recommended values of the radius of the circular curve under the condition of 5% probability of failure of superelevation.Table 5
Recommended values of the radius of circular curves under different superelevation conditions.
Superelevation value (%)4681012Radius of circular curve (m)861747660591535


## 5. Conclusion

In this study, the current research status of fuzzy reliability theory in the field of structural engineering was examined in detail. The shortcomings of the existing highway alignment reliability theory were analyzed, and a fuzzy reliability calculation method for highway alignment was derived and applied to the superelevation reliability analysis of highway horizontal curves on the basis of considering the randomness and fuzzy nature of highway alignment. The following conclusions can be drawn:(1)
Compared with the traditional highway alignment reliability calculation method, the highway alignment reliability calculation method based on fuzzy reliability theory has a higher reliability index and less failure probability, and the fuzziness is considered in the calculation method to make the calculation accuracy higher.(2)
When the general density function, the equivalent density function, and the information entropy methods are used for fuzzy reliability analysis of highway alignment, the general density function method converges faster and has higher accuracy.(3)
The probability of failure of superelevation calculated by the general density function method with the specified value of the radius of the circular curve in the Chinese specification is high and has poor stability. Under the condition of 5% probability of failure of superelevation, the recommended value of the radius of the highway circular curve calculated by the general density function method can make the highway alignment design safer and more reliable.



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*Source: 1904382-2022-05-16.xml*