# Reliability Calibration of Tower Members in Transmission Line System Crossing High-Speed Railway

Authors: Yunfen Feng; Shufei Gao; Zeyu Feng
Journal: Advances in Civil Engineering
 (2022)

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## Abstract

Due to the rapid development of high-speed railway in China, transmission line systems across the high-speed railway system have become more prevalent in recent years, which highlights the importance of reliable design for systems. The design is generally based on the structural reliability theory, in which the determination of target reliability level is the key. To determine the reliability level of a transmission line across high-speed railway, the statistical parameters of load effects and resistances of tower members are derived and reliability calibration of tower elements satisfying the minimum design requirement in Chinese codes is performed by JC method. Furthermore, the reliability level of a transmission line across high-speed railway is divided into three classes, in which Class 1 is the strongest. According to the calibration results, the minimum target reliability indices are recommended. The results show that the reliability level is similar to the reliability of tower elements in the U.S. but higher than that in Canada. The target reliability indices with values of 3.7, 3.2, and 2.7 are recommended for Class 1, Class 2, and Class 3, respectively.


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## Body

## 1. Introduction

With the rapid development of the national economy, the electrical power system has attained prominent achievements in China. The transmission line system, as the electric power carrier, is a lifeline project, which is becoming more and more complex. Therefore, it is necessary to present new requirements for the design philosophy. Meanwhile, the phenomenon of transmission line system crossing the high-speed railway system is increasing substantially with the rapid development of high-speed railway. To ensure the safe operation of high-speed railway, the higher performance requirements for transmission line crossing high-speed railway must be satisfied. Since transmission line systems and rail facilities in China were designed by using different design approaches with various material strengths, partial factors, or safety factors, the safety of transmission system and railway facility cannot be clearly identified only based on the design specifications or codes. For this reason, the identical criterion should be employed to determine the safety of high-speed railway and transmission line system.During the past 40 years, rapid and significant development has arisen in the field of structural safety. The primary theme in structural safety is reliability analysis, which can be defined as the consistent evaluation of structure safety using probability theory [1]. The uncertainties associated with load effects, material properties, physical dimension, and calculation model are fully taken into account in reliability analysis. Reliability analysis in the industries of buildings, bridges have been extensively carried out, and target reliability indices have been calibrated in the aforementioned industries. However, very little research effort has been conducted on reliability of transmission line system [2–5].As is known to all, reliability-based design method has been introduced to design standards or codes of electric system in many countries and applied to the design of overhead transmission line, such as “National Electric Safety Code (NESC C2-2002)” [6], “Guidelines for Electrical Transmission Line Structural Loading (ASCE 74–2009)” [7], “Canadian Electrical Code: Overhead Systems (CSA C22.3 No.1–2001)” [8], “Overhead Electrical Lines Exceeding AC 45 kV (EN 50341–1)” [9], and “Design Criteria of Overhead Transmission Lines (IEC 60826–2003)” [10]. The probability-based design method has also been applied to Chinese design codes of overhead transmission line, such as “Technical Regulation of Design for Tower and Pole Structures for Overhead Transmission Line (DL 5154–2012)” [11], “Code for Design of 110 kV∼750 kV Overhead Transmission Line (GB 50545–2010)” [12], and “Code for Designing of ±800 kV DC Overhead Transmission Line (GB 50790–2013)” [13]. In the field of rail engineering, reliability-based limit state design method is specified in Chinese standard “Unified Standard for Reliability Design of Railway Structures (GB 50216–2019)” [14].For ultra-high voltage (UHV) transmission line, in order to satisfy the safety requirement of transmission line across high-speed railway, target reliability indices of the components of transmission line satisfying the minimum requirements should be calibrated. The research is limited to calibration of reliability index and definition of reliability level of transmission tower for transmission line crossing high-speed railway.


## 2. Statistical Parameters of Load Effects and Resistance

The transmission line is composed of power transmission towers, insulators, fittings, conductors, and Earth wires. To serve its purpose, transmission line system must be safe against various loads, such as the weight of structure itself, wind load, and ice load. In the reliability analysis and design of overhead line structures, probabilistic distributions and statistical parameters of structural load effects and resistance capacity must be determined first.


### 2.1. Statistical Parameters of Load Effect

Load effects are the moments, shears, and axial forces resulting from the loads on the structure. Statistical distributions of load effects are consistent with that of the loads. Statistical parameters of the loads are defined in terms of the bias factor (the ratio of mean value to nominal value) and coefficient of variation (the ratio of standard deviation to the mean value, abbreviated as COV). It should be pointed out that loads are classified by their characteristics varying with time, that is, permanent load and variable loads such as wind load or ice load.


#### 2.1.1. Permanent Load

The major part of permanent load is the weight of towers, insulators, and fittings. Based on the existing researches, permanent load follows a normal distribution [15]. Based on the results of statistical analysis of building structure, the bias factor of permanent load kG = 1.06 and COV δG = 0.07 are suggested in the article [15].


#### 2.1.2. Wind Load

Wind load is one of the major variable loads acting on overhead transmission line. The action mechanism of wind load on overhead transmission line is very complicated. The same statistical parameters and probability distribution as that used in building structures are employed in this work. Wind load normally follows a Gumbel distribution. Two load cases will be considered as described below.(1) The Extreme Wind Load Case. 30 m/s was taken as reference wind speed in the extreme wind load case and the speed can be treated as standardized wind speed which is used to define the standard value of wind pressure Wk in Chinese code. The bias factor and COV of the extreme wind load are taken as kWT = 0.908 and δWT = 0.193, referencing the statistical parameters of building structure designed using Chinese codes [16].(2) The Combined Wind and Ice Load Case. 10 m/s is taken as the reference wind speed in the combined wind and ice load case. It is assumed that the wind speed of 10 m/s is the maximum wind speed for the condition of annual maximum ice thickness, so the average wind load effect can be estimated by the following equation:(1)μwt=μWT+lnt/TαWT=μWT+σWTlnt/T1.2826=0.908Wk+0.1752Wk×ln1/12/501.2826=0.034Wk,where μwt is the mean of wind load for the combined wind and ice load case; μWT and σWT are the mean and standard deviation of wind load in the extreme wind load case, respectively; αWT is the scale parameter of Gumbel distribution, taken as 1.2826; t is 1 month; and T is the return period of wind load, taken as 50 years.Hence, the bias factor of wind load equals 0.034 for the combined wind and ice load case. The COV of wind load is still assumed to be taken as 0.193.


#### 2.1.3. Ice Load

When drops of water in the air and wet snow are in contact with components of overhead transmission line, such as conductors, insulators, and fittings, the coagulated ice on power transmission conductors may occur. There are many natural factors affecting icing, such as weather, terrain, altitude, wind speed and direction, and cable conductor.Given that the icing thickness is not even, substantial uncertainties cannot be neglected in this problem. Ideally, the statistics of ice thickness used for reliability analysis should be obtained from the local weather stations. However, due to the lack of statistical data available related to the coagulated ice, the statistical results of observation station in Enshi of Hubei Province are used in the article [17]. Based on the previous statistical data [17] and engineering experience, ice load is assumed to follow a lognormal distribution with the bias factor kI = 1.1 and the COV δI = 0.3.


### 2.2. Statistical Parameters of Resistance

Resistance is the ability of components and structures to resist load effects. There is indeed a high degree of uncertainty associated with structural resistance, mainly from three aspects: the uncertainties in material properties, the uncertainties associated with geometric dimensioning, and calculation model.


#### 2.2.1. Strength Calculation of Axially Loaded Members

According to Chinese code “DL/T 5154–2012” [11], the resistance for strength calculation of axially loaded tower members can be expressed as(2)R=ΩΡmAf,where ΩP is the model error of Equation (2), m is the strength reduction factor; A is the cross-sectional area of tower members; and f is the material strength of tower members.The bias factor and COV can be, respectively, described by the following relationships:(3)kR=μRRk=kPmμAμfAkfk=kPmkAkf,(4)δR=δP2+δA2+δf2,in which kP, kA, and kf as well as δP, δA, and δf denote, respectively, the bias factor and COV of the calculation model, cross-sectional area, and material strength of tower members, as shown in Table 1 [18]; fk, Ak, and Rk are the characteristic value of steel strength, cross-sectional area of tower members, and resistance, respectively.Table 1
Statistical parameters of resistance for strength calculation of axially loaded members.
VariablesStatistical parameterskδΩP1.050.07A1.000.05f1.080.08Substitution of the parameters in Table1 into (3) and (4) can estimate the statistical parameters of resistance for strength calculation of axially loaded members of tower, namely, kR = 1.134 and δR = 0.117.


#### 2.2.2. Stability Calculation of Axial Compression Members

Similarly, based on DL/T 5154–2012, the resistance for stability calculation of axial compression members of tower can be written as(5)R=ΩΡϕmΝAf,where ϕ is the stability coefficient of axial compression tower member and mN is the stability reduction factor of compression chord member.Therefore, the bias factor and COV of resistance can be, respectively, computed by(6)kR=μRRk=μΩpμAμϕmNμfAϕkmNfk=kPkϕkAkf,(7)δR=δP2+δϕ2+δA2+δf2,where kϕ and δϕ are the bias factor and COV of ϕ. Statistical parameters of ΩP in (6) and (7) are listed in Table 2 [18]. Statistical parameters of ϕ shown in Table 2 are derived by the following method.Table 2
Statistical parameters of resistance for stability calculation of axial compression members.
VariablesStatistical parameterskδΩP1.0700.096ϕ1.0250.068According to Chinese code DL/T5154-2012 [11], ϕ can be calculated as(8)ϕ=1−α1λ¯2λ¯≤0.21512λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯2λ¯>0.215,in which λ¯ is defined as(9)λ¯=λπfyE,where λ is slenderness ratio of tower member; fy is steel yield strength; E is elastic modulus of steel; and α1, α2, and α3 are factors specified in Chinese code DL/T 5154–2012 [11].Since the stability coefficientϕ shown in (8) is a piecewise function, the definition of probability distribution function and statistical parameters of ϕ are complicated. In order to determine the statistical parameters of the stability coefficient ϕ, steel yield strength fy is considered as random variable following a normal distribution with a bias factor of 1.08 and a COV of 0.08 [18]. Generally speaking, the variability of elastic modulus E is very small and a constant value of 2×105 MPa is adopted in the article.Then, 100,000 random numbers forfy is achieved through Monte Carlo simulation (MCS), which are substituted into (8), thereby obtaining the random values of ϕ. According to the aforementioned statistical analysis, the statistical parameters of ϕ for λ of 10, 50, and 100 are given in Table 3, in which Section a and Section b are classifications of a cross-section of members in Chinese code DL/T 5154–2012 [11]. The readers are referred to [11] for detailed information about the classification of section and steel classes (i.e. Q235 and Q345). For illustrative purposes, slenderness ratio λ = 50 of Section b is taken as an example, and the frequency histogram of ϕ/ϕk is depicted in Figure 1, where ϕk is the characteristic value of ϕ. The bias factor of ϕ is equal to 1.025, COV of ϕ is equal to 0.068, as listed in Table 3.Consequently, substituting statistical parameters of all random variables shown in (6) and (7) can obtain that the bias factor and COV of resistance for stability calculation of axial compression members are 1.185 and 0.150, respectively.Table 3
Bias factor and COV ofϕ/ϕk.
Section typeλQ235Q345Steel thicknessd > 16～40 mmSteel thicknessd > 40～60 mmSteel thicknessd > 16～35 mmSteel thicknessd > 35～50 mmBias factorCOVBias factorCOVBias factorCOVBias factorCOVSection a101.0900.0001.0880.0001.1090.0011.1040.001500.9940.0070.9940.0070.9900.0110.9910.0101000.9540.0510.9560.0490.9440.0680.9460.062Section b101.1520.0011.1480.0011.1860.0011.1770.001500.9900.0110.9900.0110.9840.0170.9860.0151000.9570.0490.9580.0480.9480.0600.9500.058Average value1.023—1.022—1.027—1.026—Grand average1.025Figure 1
Frequency histogram ofϕ/ϕk for slenderness ratio λ = 50 of Section b. (a) Q235: d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d >35∼50 mm.
(a)(b)(c)(d)


### 2.3. Summary of Statistical Parameters for Load Effects and Resistance

A summary of the statistical parameters is given in Table4 including bias factor, COV, and distribution type.Table 4
Statistical parameters of load effect and resistance of transmission tower.
VariablesStatistical parametersProbability distributionkδLoad effectPermanent load1.0600.070Normal distributionWind loadExtreme wind load case0.9080.193Gumbel distributionWind + ice load0.0340.193Ice load1.1000.300ResistanceStrength calculation1.1340.117Lognormal distributionStability calculation1.1850.150


## 2.1. Statistical Parameters of Load Effect

Load effects are the moments, shears, and axial forces resulting from the loads on the structure. Statistical distributions of load effects are consistent with that of the loads. Statistical parameters of the loads are defined in terms of the bias factor (the ratio of mean value to nominal value) and coefficient of variation (the ratio of standard deviation to the mean value, abbreviated as COV). It should be pointed out that loads are classified by their characteristics varying with time, that is, permanent load and variable loads such as wind load or ice load.


### 2.1.1. Permanent Load

The major part of permanent load is the weight of towers, insulators, and fittings. Based on the existing researches, permanent load follows a normal distribution [15]. Based on the results of statistical analysis of building structure, the bias factor of permanent load kG = 1.06 and COV δG = 0.07 are suggested in the article [15].


### 2.1.2. Wind Load

Wind load is one of the major variable loads acting on overhead transmission line. The action mechanism of wind load on overhead transmission line is very complicated. The same statistical parameters and probability distribution as that used in building structures are employed in this work. Wind load normally follows a Gumbel distribution. Two load cases will be considered as described below.(1) The Extreme Wind Load Case. 30 m/s was taken as reference wind speed in the extreme wind load case and the speed can be treated as standardized wind speed which is used to define the standard value of wind pressure Wk in Chinese code. The bias factor and COV of the extreme wind load are taken as kWT = 0.908 and δWT = 0.193, referencing the statistical parameters of building structure designed using Chinese codes [16].(2) The Combined Wind and Ice Load Case. 10 m/s is taken as the reference wind speed in the combined wind and ice load case. It is assumed that the wind speed of 10 m/s is the maximum wind speed for the condition of annual maximum ice thickness, so the average wind load effect can be estimated by the following equation:(1)μwt=μWT+lnt/TαWT=μWT+σWTlnt/T1.2826=0.908Wk+0.1752Wk×ln1/12/501.2826=0.034Wk,where μwt is the mean of wind load for the combined wind and ice load case; μWT and σWT are the mean and standard deviation of wind load in the extreme wind load case, respectively; αWT is the scale parameter of Gumbel distribution, taken as 1.2826; t is 1 month; and T is the return period of wind load, taken as 50 years.Hence, the bias factor of wind load equals 0.034 for the combined wind and ice load case. The COV of wind load is still assumed to be taken as 0.193.


### 2.1.3. Ice Load

When drops of water in the air and wet snow are in contact with components of overhead transmission line, such as conductors, insulators, and fittings, the coagulated ice on power transmission conductors may occur. There are many natural factors affecting icing, such as weather, terrain, altitude, wind speed and direction, and cable conductor.Given that the icing thickness is not even, substantial uncertainties cannot be neglected in this problem. Ideally, the statistics of ice thickness used for reliability analysis should be obtained from the local weather stations. However, due to the lack of statistical data available related to the coagulated ice, the statistical results of observation station in Enshi of Hubei Province are used in the article [17]. Based on the previous statistical data [17] and engineering experience, ice load is assumed to follow a lognormal distribution with the bias factor kI = 1.1 and the COV δI = 0.3.


## 2.1.1. Permanent Load

The major part of permanent load is the weight of towers, insulators, and fittings. Based on the existing researches, permanent load follows a normal distribution [15]. Based on the results of statistical analysis of building structure, the bias factor of permanent load kG = 1.06 and COV δG = 0.07 are suggested in the article [15].


## 2.1.2. Wind Load

Wind load is one of the major variable loads acting on overhead transmission line. The action mechanism of wind load on overhead transmission line is very complicated. The same statistical parameters and probability distribution as that used in building structures are employed in this work. Wind load normally follows a Gumbel distribution. Two load cases will be considered as described below.(1) The Extreme Wind Load Case. 30 m/s was taken as reference wind speed in the extreme wind load case and the speed can be treated as standardized wind speed which is used to define the standard value of wind pressure Wk in Chinese code. The bias factor and COV of the extreme wind load are taken as kWT = 0.908 and δWT = 0.193, referencing the statistical parameters of building structure designed using Chinese codes [16].(2) The Combined Wind and Ice Load Case. 10 m/s is taken as the reference wind speed in the combined wind and ice load case. It is assumed that the wind speed of 10 m/s is the maximum wind speed for the condition of annual maximum ice thickness, so the average wind load effect can be estimated by the following equation:(1)μwt=μWT+lnt/TαWT=μWT+σWTlnt/T1.2826=0.908Wk+0.1752Wk×ln1/12/501.2826=0.034Wk,where μwt is the mean of wind load for the combined wind and ice load case; μWT and σWT are the mean and standard deviation of wind load in the extreme wind load case, respectively; αWT is the scale parameter of Gumbel distribution, taken as 1.2826; t is 1 month; and T is the return period of wind load, taken as 50 years.Hence, the bias factor of wind load equals 0.034 for the combined wind and ice load case. The COV of wind load is still assumed to be taken as 0.193.


## 2.1.3. Ice Load

When drops of water in the air and wet snow are in contact with components of overhead transmission line, such as conductors, insulators, and fittings, the coagulated ice on power transmission conductors may occur. There are many natural factors affecting icing, such as weather, terrain, altitude, wind speed and direction, and cable conductor.Given that the icing thickness is not even, substantial uncertainties cannot be neglected in this problem. Ideally, the statistics of ice thickness used for reliability analysis should be obtained from the local weather stations. However, due to the lack of statistical data available related to the coagulated ice, the statistical results of observation station in Enshi of Hubei Province are used in the article [17]. Based on the previous statistical data [17] and engineering experience, ice load is assumed to follow a lognormal distribution with the bias factor kI = 1.1 and the COV δI = 0.3.


## 2.2. Statistical Parameters of Resistance

Resistance is the ability of components and structures to resist load effects. There is indeed a high degree of uncertainty associated with structural resistance, mainly from three aspects: the uncertainties in material properties, the uncertainties associated with geometric dimensioning, and calculation model.


### 2.2.1. Strength Calculation of Axially Loaded Members

According to Chinese code “DL/T 5154–2012” [11], the resistance for strength calculation of axially loaded tower members can be expressed as(2)R=ΩΡmAf,where ΩP is the model error of Equation (2), m is the strength reduction factor; A is the cross-sectional area of tower members; and f is the material strength of tower members.The bias factor and COV can be, respectively, described by the following relationships:(3)kR=μRRk=kPmμAμfAkfk=kPmkAkf,(4)δR=δP2+δA2+δf2,in which kP, kA, and kf as well as δP, δA, and δf denote, respectively, the bias factor and COV of the calculation model, cross-sectional area, and material strength of tower members, as shown in Table 1 [18]; fk, Ak, and Rk are the characteristic value of steel strength, cross-sectional area of tower members, and resistance, respectively.Table 1
Statistical parameters of resistance for strength calculation of axially loaded members.
VariablesStatistical parameterskδΩP1.050.07A1.000.05f1.080.08Substitution of the parameters in Table1 into (3) and (4) can estimate the statistical parameters of resistance for strength calculation of axially loaded members of tower, namely, kR = 1.134 and δR = 0.117.


### 2.2.2. Stability Calculation of Axial Compression Members

Similarly, based on DL/T 5154–2012, the resistance for stability calculation of axial compression members of tower can be written as(5)R=ΩΡϕmΝAf,where ϕ is the stability coefficient of axial compression tower member and mN is the stability reduction factor of compression chord member.Therefore, the bias factor and COV of resistance can be, respectively, computed by(6)kR=μRRk=μΩpμAμϕmNμfAϕkmNfk=kPkϕkAkf,(7)δR=δP2+δϕ2+δA2+δf2,where kϕ and δϕ are the bias factor and COV of ϕ. Statistical parameters of ΩP in (6) and (7) are listed in Table 2 [18]. Statistical parameters of ϕ shown in Table 2 are derived by the following method.Table 2
Statistical parameters of resistance for stability calculation of axial compression members.
VariablesStatistical parameterskδΩP1.0700.096ϕ1.0250.068According to Chinese code DL/T5154-2012 [11], ϕ can be calculated as(8)ϕ=1−α1λ¯2λ¯≤0.21512λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯2λ¯>0.215,in which λ¯ is defined as(9)λ¯=λπfyE,where λ is slenderness ratio of tower member; fy is steel yield strength; E is elastic modulus of steel; and α1, α2, and α3 are factors specified in Chinese code DL/T 5154–2012 [11].Since the stability coefficientϕ shown in (8) is a piecewise function, the definition of probability distribution function and statistical parameters of ϕ are complicated. In order to determine the statistical parameters of the stability coefficient ϕ, steel yield strength fy is considered as random variable following a normal distribution with a bias factor of 1.08 and a COV of 0.08 [18]. Generally speaking, the variability of elastic modulus E is very small and a constant value of 2×105 MPa is adopted in the article.Then, 100,000 random numbers forfy is achieved through Monte Carlo simulation (MCS), which are substituted into (8), thereby obtaining the random values of ϕ. According to the aforementioned statistical analysis, the statistical parameters of ϕ for λ of 10, 50, and 100 are given in Table 3, in which Section a and Section b are classifications of a cross-section of members in Chinese code DL/T 5154–2012 [11]. The readers are referred to [11] for detailed information about the classification of section and steel classes (i.e. Q235 and Q345). For illustrative purposes, slenderness ratio λ = 50 of Section b is taken as an example, and the frequency histogram of ϕ/ϕk is depicted in Figure 1, where ϕk is the characteristic value of ϕ. The bias factor of ϕ is equal to 1.025, COV of ϕ is equal to 0.068, as listed in Table 3.Consequently, substituting statistical parameters of all random variables shown in (6) and (7) can obtain that the bias factor and COV of resistance for stability calculation of axial compression members are 1.185 and 0.150, respectively.Table 3
Bias factor and COV ofϕ/ϕk.
Section typeλQ235Q345Steel thicknessd > 16～40 mmSteel thicknessd > 40～60 mmSteel thicknessd > 16～35 mmSteel thicknessd > 35～50 mmBias factorCOVBias factorCOVBias factorCOVBias factorCOVSection a101.0900.0001.0880.0001.1090.0011.1040.001500.9940.0070.9940.0070.9900.0110.9910.0101000.9540.0510.9560.0490.9440.0680.9460.062Section b101.1520.0011.1480.0011.1860.0011.1770.001500.9900.0110.9900.0110.9840.0170.9860.0151000.9570.0490.9580.0480.9480.0600.9500.058Average value1.023—1.022—1.027—1.026—Grand average1.025Figure 1
Frequency histogram ofϕ/ϕk for slenderness ratio λ = 50 of Section b. (a) Q235: d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d >35∼50 mm.
(a)(b)(c)(d)


## 2.2.1. Strength Calculation of Axially Loaded Members

According to Chinese code “DL/T 5154–2012” [11], the resistance for strength calculation of axially loaded tower members can be expressed as(2)R=ΩΡmAf,where ΩP is the model error of Equation (2), m is the strength reduction factor; A is the cross-sectional area of tower members; and f is the material strength of tower members.The bias factor and COV can be, respectively, described by the following relationships:(3)kR=μRRk=kPmμAμfAkfk=kPmkAkf,(4)δR=δP2+δA2+δf2,in which kP, kA, and kf as well as δP, δA, and δf denote, respectively, the bias factor and COV of the calculation model, cross-sectional area, and material strength of tower members, as shown in Table 1 [18]; fk, Ak, and Rk are the characteristic value of steel strength, cross-sectional area of tower members, and resistance, respectively.Table 1
Statistical parameters of resistance for strength calculation of axially loaded members.
VariablesStatistical parameterskδΩP1.050.07A1.000.05f1.080.08Substitution of the parameters in Table1 into (3) and (4) can estimate the statistical parameters of resistance for strength calculation of axially loaded members of tower, namely, kR = 1.134 and δR = 0.117.


## 2.2.2. Stability Calculation of Axial Compression Members

Similarly, based on DL/T 5154–2012, the resistance for stability calculation of axial compression members of tower can be written as(5)R=ΩΡϕmΝAf,where ϕ is the stability coefficient of axial compression tower member and mN is the stability reduction factor of compression chord member.Therefore, the bias factor and COV of resistance can be, respectively, computed by(6)kR=μRRk=μΩpμAμϕmNμfAϕkmNfk=kPkϕkAkf,(7)δR=δP2+δϕ2+δA2+δf2,where kϕ and δϕ are the bias factor and COV of ϕ. Statistical parameters of ΩP in (6) and (7) are listed in Table 2 [18]. Statistical parameters of ϕ shown in Table 2 are derived by the following method.Table 2
Statistical parameters of resistance for stability calculation of axial compression members.
VariablesStatistical parameterskδΩP1.0700.096ϕ1.0250.068According to Chinese code DL/T5154-2012 [11], ϕ can be calculated as(8)ϕ=1−α1λ¯2λ¯≤0.21512λ¯2α2+α3λ¯+λ¯2−α2+α3λ¯+λ¯22−4λ¯2λ¯>0.215,in which λ¯ is defined as(9)λ¯=λπfyE,where λ is slenderness ratio of tower member; fy is steel yield strength; E is elastic modulus of steel; and α1, α2, and α3 are factors specified in Chinese code DL/T 5154–2012 [11].Since the stability coefficientϕ shown in (8) is a piecewise function, the definition of probability distribution function and statistical parameters of ϕ are complicated. In order to determine the statistical parameters of the stability coefficient ϕ, steel yield strength fy is considered as random variable following a normal distribution with a bias factor of 1.08 and a COV of 0.08 [18]. Generally speaking, the variability of elastic modulus E is very small and a constant value of 2×105 MPa is adopted in the article.Then, 100,000 random numbers forfy is achieved through Monte Carlo simulation (MCS), which are substituted into (8), thereby obtaining the random values of ϕ. According to the aforementioned statistical analysis, the statistical parameters of ϕ for λ of 10, 50, and 100 are given in Table 3, in which Section a and Section b are classifications of a cross-section of members in Chinese code DL/T 5154–2012 [11]. The readers are referred to [11] for detailed information about the classification of section and steel classes (i.e. Q235 and Q345). For illustrative purposes, slenderness ratio λ = 50 of Section b is taken as an example, and the frequency histogram of ϕ/ϕk is depicted in Figure 1, where ϕk is the characteristic value of ϕ. The bias factor of ϕ is equal to 1.025, COV of ϕ is equal to 0.068, as listed in Table 3.Consequently, substituting statistical parameters of all random variables shown in (6) and (7) can obtain that the bias factor and COV of resistance for stability calculation of axial compression members are 1.185 and 0.150, respectively.Table 3
Bias factor and COV ofϕ/ϕk.
Section typeλQ235Q345Steel thicknessd > 16～40 mmSteel thicknessd > 40～60 mmSteel thicknessd > 16～35 mmSteel thicknessd > 35～50 mmBias factorCOVBias factorCOVBias factorCOVBias factorCOVSection a101.0900.0001.0880.0001.1090.0011.1040.001500.9940.0070.9940.0070.9900.0110.9910.0101000.9540.0510.9560.0490.9440.0680.9460.062Section b101.1520.0011.1480.0011.1860.0011.1770.001500.9900.0110.9900.0110.9840.0170.9860.0151000.9570.0490.9580.0480.9480.0600.9500.058Average value1.023—1.022—1.027—1.026—Grand average1.025Figure 1
Frequency histogram ofϕ/ϕk for slenderness ratio λ = 50 of Section b. (a) Q235: d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d >35∼50 mm.
(a)(b)(c)(d)


## 2.3. Summary of Statistical Parameters for Load Effects and Resistance

A summary of the statistical parameters is given in Table4 including bias factor, COV, and distribution type.Table 4
Statistical parameters of load effect and resistance of transmission tower.
VariablesStatistical parametersProbability distributionkδLoad effectPermanent load1.0600.070Normal distributionWind loadExtreme wind load case0.9080.193Gumbel distributionWind + ice load0.0340.193Ice load1.1000.300ResistanceStrength calculation1.1340.117Lognormal distributionStability calculation1.1850.150


## 3. Reliability Calibration

### 3.1. Primary Expressions in Chinese Design Codes

The reliability indices for structures or components designed according to Chinese codes representing the minimum design requirements can be determined by calibration of reliability. Based on Chinese codes “DLT5154-2012” [11] and “GB 50545–2010” [12], design expression with partial safety factor for ultimate limit state is written as(10)γ0γGSGk+ψ∑γQiSQik≤Rd,where γ0 is the coefficient for importance of structure, not less than 1.1 for the important transmission line, 0.9 for the temporary transmission line, 1.0 for the other transmission line;γG is the partial safety factor of permanent load, not more than 1.1 in favorable conditions, 1.2 in unfavorable conditions; γQi is the partial safety factor of variable load Qi taken as 1.4; SGk, SQik are, respectively, characteristic value of permanent load effect and variable load effect; Rd is design value of components or structures, and different formulas are adopted for different tower members in Chinese code GB 50545–2010 [12]; Ψ is combination coefficient of variable load, 1.0 for normal operation condition, 0.9 for the design condition of conductor breaking, installation, and uneven icing, and 0.75 for checking calculation.According to (10), the design value of resistance satisfying the minimum design requirements specified by Chinese codes can be expressed by(11)Rd=γ0γGSGk+ψ∑γQiSQik.


### 3.2. Determination of Reliability Index

#### 3.2.1. Expression of Resistance

(1) Strength Calculation of Axially Loaded Members. The strength of tower member in axial stress given by Chinese codes is written as [11](12)NAn≤mfd,where An is the net cross-sectional area of tower member and fd is the design value of steel strength.The axial force should meet the minimum design requirements described by (12) and can be expressed as follows:(13)N=Rd=mAnfd.Substitution of (13) into (11) leads to(14)γ0γGNGk+ψ∑γQiNQik=mAnfd,in which NGk and NQik are, respectively, the characteristic value of permanent load and variable loads.(2) Stability Calculation of Axial Compression Members. Correspondingly, the stability calculation of axial compression member should satisfy the following expression given by Chinese code “DL/T 5154–2012” [11]:(15)NϕkAn≤mNfd,where ϕk is the characteristic value of stability coefficient ϕ for stability calculation of tower member in axial compression.Then, Equation (11) can be rewritten by the following formula:(16)γ0γGNGk+ψ∑γQiNQik=ϕkmNAnfd.


#### 3.2.2. Calculation of Reliability Index

The variable loads of transmission tower include wind, ice, conductor, or ground wire broken loads, but for the sake of simplicity, only the extreme wind load case and the combined wind and ice load case are considered in this work. As shown in Table4, in addition to the statistical parameters of resistance, the function and statistical parameters of load effects are taken as the same values for both strength and stability calculation of axially loaded tower members. Therefore, for illustrative purpose, calculation of reliability index for strength calculation of axially loaded tower members is taken as example.(1) Extreme Wind Load Case. The performance function of transmission tower for strength calculation of axially loaded member can be expressed as follows:(17)Z=R−NG−NW,where R is the resistance; NG is the axial force resulting from permanent load acting on tower member; and NW is the axial force resulting from the extreme wind load acting on tower member.The mean and standard deviation of permanent load and wind load can be expressed as follows:(18)μNG=kNGNGk,σNG=δNGkNGNGk,(19)μNW=kNWρWNGk,σNW=δNWkNWρWNGk,where kNG, kNW, δNG, and δNW are the bias factors and COV of permanent load and wind load, respectively; ρWis the wind load effect ratio, which is the ratio of the characteristic value of wind load to the characteristic value of permanent load and can be expressed as ρW=NWk/NGk.The mean of resistance can be expressed as follows:(20)μR=kRRk=kRmAnfdfkfd.The characteristic valuefk and the design value fd of steel strength for various steel classes and thicknesses are summarized in Table 5.Table 5
Characteristic values and design values of steel strength.
Steel thickness (mm)Q235Q345fk (MPa)fd (MPa)fk (MPa)fd (MPa)d≤16235215345310d>16∼40235205345295d>40∼60235200345265d>60∼100235190345250Wind loads on power transmission tower include the wind loads acting on the tower members as well as conductors, overhead ground wires, insulators, and fittings. When determining wind loads acting on conductors and ground wires designed by Chinese code “GB50545-2010,” wind load adjustment factorβc shown in Table 6 should be taken into account. Experience shows that wind loads acting on conductors, overhead ground wires, insulators, and fittings can be taken half of the total wind loads on tower. Therefore, by substituting (14) into (20), the following equations can be obtained:(21)μR=kRfkfdγGNGk+0.51+βcγQ1NWk=kRfkfdγG+0.51+βcγQ1ρWNGk.Table 6
Wind load adjustment coefficientβc.
Wind speed (m/s)<2020～2727～31.5≥31.5βc1.01.11.21.3As seen in (18), (19), and (21), the mean and standard deviation of structure resistance, permanent load effect, and wind load effect are proportional to the characteristic value of permanent load NGk. It is well known that the value of reliability index depends on the wind load effect ratio ρW rather than the specific value of load effects and resistance. Analyses of numerous actual power transmission towers have shown that the ratio ρW tends to lie within a wider range of values ranging from 0.1 to 100.The most widely used approach for reliability analysis is JC method. Then reliability indicesβ of tower members for various steel classes with different wind load adjustment factors βc are calculated according to JC method in this work, as shown in Figures 2 and 3. The average values of reliability indices of various steel classes for different wind load adjustment factors βc are summarized in Table 7.Figure 2
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 3
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 7
The average value of reliability indices of different steel class.
Component typeβcQ235Q345d > 16∼40 mmd > 40∼60 mmd > 16∼35 mmd > 35∼50 mmStrength calculation of axially loaded member1.02.89182.79072.90922.95631.13.03992.93963.05713.10391.23.18153.08193.19853.24501.33.31713.21813.33413.3803Stability calculation of axially loaded member1.02.79722.70492.81302.85611.12.93582.84412.95152.99441.23.06832.97723.08403.12661.33.19543.10473.21093.2533It can be seen from Figures2 and 3 that the reliability indices of power transmission tower members relate directly to the ratio ρW. For ρW ≤ 1.0, the reliability indices for strength calculation of axially loaded tower members increase with the increased ρW; meanwhile, for ρW > 1.0, the reliability indices decrease with the increased ρW. As shown in Table 7, the average values of reliability indices increase with the wind load adjustment factor βc, while the average values of reliability indices for stability calculation of axial compression members are always lower than those for strength calculation.(2) The Combined Wind and Ice Load Case. Axially loaded tower members in strength calculation are still taken as examples herein. In the combined wind and ice load case, the performance function of transmission tower member for strength calculation of axially loaded member can be expressed as follows:(22)Z=R−NG−NW−N1.So, the mean values and standard deviations of wind load and ice load in the combined wind and ice load case are written as follows:(23)μNWI=kNWIρWINGk,σNWI=δNWIkNWIρWINGk,μNI=kNIρINGk,σNI=δNIkNIρINGk,where kNWI,kNI, δNWI, and δNI are, respectively, the bias factors and COV of wind load and ice load in the combined wind and ice load case; ρWI is the wind load effect ratio, ρWI=NWk/NGk; ρI is the ice load effect ratio,ρI=NIk/NGk.According to Table6, βc for the combined wind and ice load case is equal to 1, since the wind speed of 10 m/s is assumed to be the maximum wind speed for the condition of annual maximum ice thickness. Therefore, the mean of resistance for stability calculation of axial compression member is defined as:(24)μR=kRfkfdγG+γQ1ρWI+γQ2ρINGk,where γQ2 is the partial factor of ice load.For the combined wind and ice load case, ice loads would dominate the reliability of tower. In accordance with the analysis results of actual power transmission towers, the ratioρWI tends to lie within the range 0.02∼10, while ρI ranges from 0.02 to 5.To compute the reliability indices of tower members in the combined wind and ice load case, JC method is employed in the present article. The reliability indices for differentρWI and ρI are depicted in Figures 4 and 5; meanwhile, the average values of reliability indices for various steel classes are shown in Table 8.Figure 4
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 5
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 8
Reliability indices of member of different steel classes.
ClassSteel thicknessStrength calculationStability calculationQ235d > 16∼40 mm3.35263.2025d > 40 ∼60 mm3.25263.1125Q345d > 16∼35 mm3.36973.218d > 35∼50 mm3.41653.2602As can be seen in Figures4 and 5, when the ice load effect ratio ρI is taken a fixed value, the reliability indices increase with the ratio ρWI, while the reliability indices decrease with ρI in the case of a fixed wind load effect ratio ρWI. Since the reliability in the combined wind and ice load case is governed mainly by ρI, a higher reliability level is produced for ρWI > 0.1 and ρI < 0.1, as compared with the extreme wind load case. However, the average reliability indices of axial compression member under stability are lower than those of axially loaded member for strength calculation, as shown in Table 8.


## 3.1. Primary Expressions in Chinese Design Codes

The reliability indices for structures or components designed according to Chinese codes representing the minimum design requirements can be determined by calibration of reliability. Based on Chinese codes “DLT5154-2012” [11] and “GB 50545–2010” [12], design expression with partial safety factor for ultimate limit state is written as(10)γ0γGSGk+ψ∑γQiSQik≤Rd,where γ0 is the coefficient for importance of structure, not less than 1.1 for the important transmission line, 0.9 for the temporary transmission line, 1.0 for the other transmission line;γG is the partial safety factor of permanent load, not more than 1.1 in favorable conditions, 1.2 in unfavorable conditions; γQi is the partial safety factor of variable load Qi taken as 1.4; SGk, SQik are, respectively, characteristic value of permanent load effect and variable load effect; Rd is design value of components or structures, and different formulas are adopted for different tower members in Chinese code GB 50545–2010 [12]; Ψ is combination coefficient of variable load, 1.0 for normal operation condition, 0.9 for the design condition of conductor breaking, installation, and uneven icing, and 0.75 for checking calculation.According to (10), the design value of resistance satisfying the minimum design requirements specified by Chinese codes can be expressed by(11)Rd=γ0γGSGk+ψ∑γQiSQik.


## 3.2. Determination of Reliability Index

### 3.2.1. Expression of Resistance

(1) Strength Calculation of Axially Loaded Members. The strength of tower member in axial stress given by Chinese codes is written as [11](12)NAn≤mfd,where An is the net cross-sectional area of tower member and fd is the design value of steel strength.The axial force should meet the minimum design requirements described by (12) and can be expressed as follows:(13)N=Rd=mAnfd.Substitution of (13) into (11) leads to(14)γ0γGNGk+ψ∑γQiNQik=mAnfd,in which NGk and NQik are, respectively, the characteristic value of permanent load and variable loads.(2) Stability Calculation of Axial Compression Members. Correspondingly, the stability calculation of axial compression member should satisfy the following expression given by Chinese code “DL/T 5154–2012” [11]:(15)NϕkAn≤mNfd,where ϕk is the characteristic value of stability coefficient ϕ for stability calculation of tower member in axial compression.Then, Equation (11) can be rewritten by the following formula:(16)γ0γGNGk+ψ∑γQiNQik=ϕkmNAnfd.


### 3.2.2. Calculation of Reliability Index

The variable loads of transmission tower include wind, ice, conductor, or ground wire broken loads, but for the sake of simplicity, only the extreme wind load case and the combined wind and ice load case are considered in this work. As shown in Table4, in addition to the statistical parameters of resistance, the function and statistical parameters of load effects are taken as the same values for both strength and stability calculation of axially loaded tower members. Therefore, for illustrative purpose, calculation of reliability index for strength calculation of axially loaded tower members is taken as example.(1) Extreme Wind Load Case. The performance function of transmission tower for strength calculation of axially loaded member can be expressed as follows:(17)Z=R−NG−NW,where R is the resistance; NG is the axial force resulting from permanent load acting on tower member; and NW is the axial force resulting from the extreme wind load acting on tower member.The mean and standard deviation of permanent load and wind load can be expressed as follows:(18)μNG=kNGNGk,σNG=δNGkNGNGk,(19)μNW=kNWρWNGk,σNW=δNWkNWρWNGk,where kNG, kNW, δNG, and δNW are the bias factors and COV of permanent load and wind load, respectively; ρWis the wind load effect ratio, which is the ratio of the characteristic value of wind load to the characteristic value of permanent load and can be expressed as ρW=NWk/NGk.The mean of resistance can be expressed as follows:(20)μR=kRRk=kRmAnfdfkfd.The characteristic valuefk and the design value fd of steel strength for various steel classes and thicknesses are summarized in Table 5.Table 5
Characteristic values and design values of steel strength.
Steel thickness (mm)Q235Q345fk (MPa)fd (MPa)fk (MPa)fd (MPa)d≤16235215345310d>16∼40235205345295d>40∼60235200345265d>60∼100235190345250Wind loads on power transmission tower include the wind loads acting on the tower members as well as conductors, overhead ground wires, insulators, and fittings. When determining wind loads acting on conductors and ground wires designed by Chinese code “GB50545-2010,” wind load adjustment factorβc shown in Table 6 should be taken into account. Experience shows that wind loads acting on conductors, overhead ground wires, insulators, and fittings can be taken half of the total wind loads on tower. Therefore, by substituting (14) into (20), the following equations can be obtained:(21)μR=kRfkfdγGNGk+0.51+βcγQ1NWk=kRfkfdγG+0.51+βcγQ1ρWNGk.Table 6
Wind load adjustment coefficientβc.
Wind speed (m/s)<2020～2727～31.5≥31.5βc1.01.11.21.3As seen in (18), (19), and (21), the mean and standard deviation of structure resistance, permanent load effect, and wind load effect are proportional to the characteristic value of permanent load NGk. It is well known that the value of reliability index depends on the wind load effect ratio ρW rather than the specific value of load effects and resistance. Analyses of numerous actual power transmission towers have shown that the ratio ρW tends to lie within a wider range of values ranging from 0.1 to 100.The most widely used approach for reliability analysis is JC method. Then reliability indicesβ of tower members for various steel classes with different wind load adjustment factors βc are calculated according to JC method in this work, as shown in Figures 2 and 3. The average values of reliability indices of various steel classes for different wind load adjustment factors βc are summarized in Table 7.Figure 2
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 3
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 7
The average value of reliability indices of different steel class.
Component typeβcQ235Q345d > 16∼40 mmd > 40∼60 mmd > 16∼35 mmd > 35∼50 mmStrength calculation of axially loaded member1.02.89182.79072.90922.95631.13.03992.93963.05713.10391.23.18153.08193.19853.24501.33.31713.21813.33413.3803Stability calculation of axially loaded member1.02.79722.70492.81302.85611.12.93582.84412.95152.99441.23.06832.97723.08403.12661.33.19543.10473.21093.2533It can be seen from Figures2 and 3 that the reliability indices of power transmission tower members relate directly to the ratio ρW. For ρW ≤ 1.0, the reliability indices for strength calculation of axially loaded tower members increase with the increased ρW; meanwhile, for ρW > 1.0, the reliability indices decrease with the increased ρW. As shown in Table 7, the average values of reliability indices increase with the wind load adjustment factor βc, while the average values of reliability indices for stability calculation of axial compression members are always lower than those for strength calculation.(2) The Combined Wind and Ice Load Case. Axially loaded tower members in strength calculation are still taken as examples herein. In the combined wind and ice load case, the performance function of transmission tower member for strength calculation of axially loaded member can be expressed as follows:(22)Z=R−NG−NW−N1.So, the mean values and standard deviations of wind load and ice load in the combined wind and ice load case are written as follows:(23)μNWI=kNWIρWINGk,σNWI=δNWIkNWIρWINGk,μNI=kNIρINGk,σNI=δNIkNIρINGk,where kNWI,kNI, δNWI, and δNI are, respectively, the bias factors and COV of wind load and ice load in the combined wind and ice load case; ρWI is the wind load effect ratio, ρWI=NWk/NGk; ρI is the ice load effect ratio,ρI=NIk/NGk.According to Table6, βc for the combined wind and ice load case is equal to 1, since the wind speed of 10 m/s is assumed to be the maximum wind speed for the condition of annual maximum ice thickness. Therefore, the mean of resistance for stability calculation of axial compression member is defined as:(24)μR=kRfkfdγG+γQ1ρWI+γQ2ρINGk,where γQ2 is the partial factor of ice load.For the combined wind and ice load case, ice loads would dominate the reliability of tower. In accordance with the analysis results of actual power transmission towers, the ratioρWI tends to lie within the range 0.02∼10, while ρI ranges from 0.02 to 5.To compute the reliability indices of tower members in the combined wind and ice load case, JC method is employed in the present article. The reliability indices for differentρWI and ρI are depicted in Figures 4 and 5; meanwhile, the average values of reliability indices for various steel classes are shown in Table 8.Figure 4
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 5
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 8
Reliability indices of member of different steel classes.
ClassSteel thicknessStrength calculationStability calculationQ235d > 16∼40 mm3.35263.2025d > 40 ∼60 mm3.25263.1125Q345d > 16∼35 mm3.36973.218d > 35∼50 mm3.41653.2602As can be seen in Figures4 and 5, when the ice load effect ratio ρI is taken a fixed value, the reliability indices increase with the ratio ρWI, while the reliability indices decrease with ρI in the case of a fixed wind load effect ratio ρWI. Since the reliability in the combined wind and ice load case is governed mainly by ρI, a higher reliability level is produced for ρWI > 0.1 and ρI < 0.1, as compared with the extreme wind load case. However, the average reliability indices of axial compression member under stability are lower than those of axially loaded member for strength calculation, as shown in Table 8.


## 3.2.1. Expression of Resistance

(1) Strength Calculation of Axially Loaded Members. The strength of tower member in axial stress given by Chinese codes is written as [11](12)NAn≤mfd,where An is the net cross-sectional area of tower member and fd is the design value of steel strength.The axial force should meet the minimum design requirements described by (12) and can be expressed as follows:(13)N=Rd=mAnfd.Substitution of (13) into (11) leads to(14)γ0γGNGk+ψ∑γQiNQik=mAnfd,in which NGk and NQik are, respectively, the characteristic value of permanent load and variable loads.(2) Stability Calculation of Axial Compression Members. Correspondingly, the stability calculation of axial compression member should satisfy the following expression given by Chinese code “DL/T 5154–2012” [11]:(15)NϕkAn≤mNfd,where ϕk is the characteristic value of stability coefficient ϕ for stability calculation of tower member in axial compression.Then, Equation (11) can be rewritten by the following formula:(16)γ0γGNGk+ψ∑γQiNQik=ϕkmNAnfd.


## 3.2.2. Calculation of Reliability Index

The variable loads of transmission tower include wind, ice, conductor, or ground wire broken loads, but for the sake of simplicity, only the extreme wind load case and the combined wind and ice load case are considered in this work. As shown in Table4, in addition to the statistical parameters of resistance, the function and statistical parameters of load effects are taken as the same values for both strength and stability calculation of axially loaded tower members. Therefore, for illustrative purpose, calculation of reliability index for strength calculation of axially loaded tower members is taken as example.(1) Extreme Wind Load Case. The performance function of transmission tower for strength calculation of axially loaded member can be expressed as follows:(17)Z=R−NG−NW,where R is the resistance; NG is the axial force resulting from permanent load acting on tower member; and NW is the axial force resulting from the extreme wind load acting on tower member.The mean and standard deviation of permanent load and wind load can be expressed as follows:(18)μNG=kNGNGk,σNG=δNGkNGNGk,(19)μNW=kNWρWNGk,σNW=δNWkNWρWNGk,where kNG, kNW, δNG, and δNW are the bias factors and COV of permanent load and wind load, respectively; ρWis the wind load effect ratio, which is the ratio of the characteristic value of wind load to the characteristic value of permanent load and can be expressed as ρW=NWk/NGk.The mean of resistance can be expressed as follows:(20)μR=kRRk=kRmAnfdfkfd.The characteristic valuefk and the design value fd of steel strength for various steel classes and thicknesses are summarized in Table 5.Table 5
Characteristic values and design values of steel strength.
Steel thickness (mm)Q235Q345fk (MPa)fd (MPa)fk (MPa)fd (MPa)d≤16235215345310d>16∼40235205345295d>40∼60235200345265d>60∼100235190345250Wind loads on power transmission tower include the wind loads acting on the tower members as well as conductors, overhead ground wires, insulators, and fittings. When determining wind loads acting on conductors and ground wires designed by Chinese code “GB50545-2010,” wind load adjustment factorβc shown in Table 6 should be taken into account. Experience shows that wind loads acting on conductors, overhead ground wires, insulators, and fittings can be taken half of the total wind loads on tower. Therefore, by substituting (14) into (20), the following equations can be obtained:(21)μR=kRfkfdγGNGk+0.51+βcγQ1NWk=kRfkfdγG+0.51+βcγQ1ρWNGk.Table 6
Wind load adjustment coefficientβc.
Wind speed (m/s)<2020～2727～31.5≥31.5βc1.01.11.21.3As seen in (18), (19), and (21), the mean and standard deviation of structure resistance, permanent load effect, and wind load effect are proportional to the characteristic value of permanent load NGk. It is well known that the value of reliability index depends on the wind load effect ratio ρW rather than the specific value of load effects and resistance. Analyses of numerous actual power transmission towers have shown that the ratio ρW tends to lie within a wider range of values ranging from 0.1 to 100.The most widely used approach for reliability analysis is JC method. Then reliability indicesβ of tower members for various steel classes with different wind load adjustment factors βc are calculated according to JC method in this work, as shown in Figures 2 and 3. The average values of reliability indices of various steel classes for different wind load adjustment factors βc are summarized in Table 7.Figure 2
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 3
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 7
The average value of reliability indices of different steel class.
Component typeβcQ235Q345d > 16∼40 mmd > 40∼60 mmd > 16∼35 mmd > 35∼50 mmStrength calculation of axially loaded member1.02.89182.79072.90922.95631.13.03992.93963.05713.10391.23.18153.08193.19853.24501.33.31713.21813.33413.3803Stability calculation of axially loaded member1.02.79722.70492.81302.85611.12.93582.84412.95152.99441.23.06832.97723.08403.12661.33.19543.10473.21093.2533It can be seen from Figures2 and 3 that the reliability indices of power transmission tower members relate directly to the ratio ρW. For ρW ≤ 1.0, the reliability indices for strength calculation of axially loaded tower members increase with the increased ρW; meanwhile, for ρW > 1.0, the reliability indices decrease with the increased ρW. As shown in Table 7, the average values of reliability indices increase with the wind load adjustment factor βc, while the average values of reliability indices for stability calculation of axial compression members are always lower than those for strength calculation.(2) The Combined Wind and Ice Load Case. Axially loaded tower members in strength calculation are still taken as examples herein. In the combined wind and ice load case, the performance function of transmission tower member for strength calculation of axially loaded member can be expressed as follows:(22)Z=R−NG−NW−N1.So, the mean values and standard deviations of wind load and ice load in the combined wind and ice load case are written as follows:(23)μNWI=kNWIρWINGk,σNWI=δNWIkNWIρWINGk,μNI=kNIρINGk,σNI=δNIkNIρINGk,where kNWI,kNI, δNWI, and δNI are, respectively, the bias factors and COV of wind load and ice load in the combined wind and ice load case; ρWI is the wind load effect ratio, ρWI=NWk/NGk; ρI is the ice load effect ratio,ρI=NIk/NGk.According to Table6, βc for the combined wind and ice load case is equal to 1, since the wind speed of 10 m/s is assumed to be the maximum wind speed for the condition of annual maximum ice thickness. Therefore, the mean of resistance for stability calculation of axial compression member is defined as:(24)μR=kRfkfdγG+γQ1ρWI+γQ2ρINGk,where γQ2 is the partial factor of ice load.For the combined wind and ice load case, ice loads would dominate the reliability of tower. In accordance with the analysis results of actual power transmission towers, the ratioρWI tends to lie within the range 0.02∼10, while ρI ranges from 0.02 to 5.To compute the reliability indices of tower members in the combined wind and ice load case, JC method is employed in the present article. The reliability indices for differentρWI and ρI are depicted in Figures 4 and 5; meanwhile, the average values of reliability indices for various steel classes are shown in Table 8.Figure 4
Reliability indices for strength calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Figure 5
Reliability indices for stability calculation of axially loaded member. (a) Q235:d > 16∼40 mm, (b) Q235: d > 40∼60 mm, (c) Q345: d > 16∼35 mm, and (d) Q345: d > 35∼50 mm.
(a)(b)(c)(d)Table 8
Reliability indices of member of different steel classes.
ClassSteel thicknessStrength calculationStability calculationQ235d > 16∼40 mm3.35263.2025d > 40 ∼60 mm3.25263.1125Q345d > 16∼35 mm3.36973.218d > 35∼50 mm3.41653.2602As can be seen in Figures4 and 5, when the ice load effect ratio ρI is taken a fixed value, the reliability indices increase with the ratio ρWI, while the reliability indices decrease with ρI in the case of a fixed wind load effect ratio ρWI. Since the reliability in the combined wind and ice load case is governed mainly by ρI, a higher reliability level is produced for ρWI > 0.1 and ρI < 0.1, as compared with the extreme wind load case. However, the average reliability indices of axial compression member under stability are lower than those of axially loaded member for strength calculation, as shown in Table 8.


## 4. Target Reliability Index of Transmission Tower for Transmission Line System Crossing High-Speed Railway

### 4.1. Reference Value of Reliability Indexβ0

The reliability analysis is carried out for transmission tower designed by Chinese codes “GB 50545–2010” and “DL/T 5154–2012,” and average values of the above-mentioned reliability indices shown in Tables7 and 8 are listed in Table 9.Table 9
Summary of reliability calibration results of transmission tower.
TypesLoad caseβc = 1.0βc = 1.1βc = 1.2βc = 1.3Average valueStrength calculationExtreme wind load2.88703.03513.17673.31243.1028Wind load + ice load3.3479———3.3479Stability calculationExtreme wind load2.79282.93153.06403.19112.9949Wind load + ice load3.1983———3.1983It can be seen from Table9 that the average values of reliability indices for strength calculation of axially loaded member are 3.1028 in the extreme wind load case and 3.3479 in the combined wind and ice load case, respectively. The average reliability indices for stability of axial compression member are, respectively, 2.9949 in the extreme wind load case and 3.1983 in the combined wind and ice load case. Based on the analysis and adjustment of the average reliability indices, the reference value of reliability index of transmission tower is presented for transmission line employing 50-year design reference period and crossing high-speed railway in this work, as shown in Table 10. The reference value shown in Table 10 represents the reliability level of transmission tower meeting the minimum design requirements specified by “GB 50545–2010” and “DL/T 5154–2012.”Table 10
Reference valueβ0 of transmission tower.
Types of tower memberLoad caseβ0Strength calculationExtreme wind load3.10Wind load + ice load3.40Stability calculationExtreme wind load3.00Wind load + ice load3.20Average value3.18


### 4.2. Target Reliability IndexβT

#### 4.2.1. Safety Classes

In general, it is difficult to quantificationally calculate the loss due to structural failure. Therefore, safety classes of engineering structure are usually defined by a qualitative analysis method combined with engineering experience. Safety of engineering structure can be classified into three classes in design codes of many countries, such as Eurocode “Basis of structural design (EN 1900:2002)” [19] and Chinese standard “Unified standard for reliability design of engineering structures (GB 50153–2008)” [15]. Therefore, Classes 1, 2, and 3 are specified for transmission line crossing high-speed railway, with Class 1 being the strongest. Tension section across high-speed railway is defined as Class 1, tension section not crossing high-speed railway is defined as Class 2, and temporary line is defined as Class 3.


#### 4.2.2. Recommendation of Target Reliability Index

The numerical values of the reliability are often described on the basis of the reliability indexβ defined by β = −Φ−1(Pf), in which Pf is the failure probability. The relationship between β and Pf is given in Table 11. It is well recognized that the safety degree of engineering structure is medium and high for 10–3 < Pf < 10–4 and 10–4 < Pf < 10−5 [20].Table 11
Relationship betweenβ and Pf.
β2.73.13.23.53.74.04.2Pf3.5 × 10−31 × 10−36.40 × 10−42.33 × 10−41.1 × 10−43.17 × 10−51.3 × 10−5Calibration method is a very simple and practicable method applied to define the structural safety level. Therefore, target levels for reliability are often based on calibration [20]. Target reliability indices βT (i.e., recommended minimum values for reliability index) stipulated in Eurocode “EN 1900:2002” [19] and Chinese standard “GB 50153–2008” [15] are listed in Table 12.Table 12
Target reliability indices stipulated in EN 1900:2002 and GB 50153–2008 (ultimate limit state, 50 year reference period).
CodesSafety classesβTEN 1900:2002 [19]RC34.3RC23.8RC13.3GB 50153–2008 [15]Ductile failureClass 13.7Class 23.2Class 32.7Brittle failureClass 14.2Class 23.7Class 33.2As can be seen in Table12, there is a difference of about 0.5 for reliability indices between the adjacent safety classes, while the probability of failure can approximately differ by an order of magnitude.Based on the above section, the acceptable safety level of transmission tower Class 2 should be in agreement with the average reference values described in Table10. Thus, a target reliability index of 3.2 is suggested for Class 2. Then a reliability index of 3.7 is recommended for Class 1 in this study for the reason that adjacent safety classes have a difference of about 0.5 for reliability indices. Since transmission line of Class 3 is temporary, functional failure does not lead to serious consequences, reliability can be reduced accordingly. Consequently, a reliability index of transmission tower for Class 3 can be 2.7.


## 4.1. Reference Value of Reliability Indexβ0

The reliability analysis is carried out for transmission tower designed by Chinese codes “GB 50545–2010” and “DL/T 5154–2012,” and average values of the above-mentioned reliability indices shown in Tables7 and 8 are listed in Table 9.Table 9
Summary of reliability calibration results of transmission tower.
TypesLoad caseβc = 1.0βc = 1.1βc = 1.2βc = 1.3Average valueStrength calculationExtreme wind load2.88703.03513.17673.31243.1028Wind load + ice load3.3479———3.3479Stability calculationExtreme wind load2.79282.93153.06403.19112.9949Wind load + ice load3.1983———3.1983It can be seen from Table9 that the average values of reliability indices for strength calculation of axially loaded member are 3.1028 in the extreme wind load case and 3.3479 in the combined wind and ice load case, respectively. The average reliability indices for stability of axial compression member are, respectively, 2.9949 in the extreme wind load case and 3.1983 in the combined wind and ice load case. Based on the analysis and adjustment of the average reliability indices, the reference value of reliability index of transmission tower is presented for transmission line employing 50-year design reference period and crossing high-speed railway in this work, as shown in Table 10. The reference value shown in Table 10 represents the reliability level of transmission tower meeting the minimum design requirements specified by “GB 50545–2010” and “DL/T 5154–2012.”Table 10
Reference valueβ0 of transmission tower.
Types of tower memberLoad caseβ0Strength calculationExtreme wind load3.10Wind load + ice load3.40Stability calculationExtreme wind load3.00Wind load + ice load3.20Average value3.18


## 4.2. Target Reliability IndexβT

### 4.2.1. Safety Classes

In general, it is difficult to quantificationally calculate the loss due to structural failure. Therefore, safety classes of engineering structure are usually defined by a qualitative analysis method combined with engineering experience. Safety of engineering structure can be classified into three classes in design codes of many countries, such as Eurocode “Basis of structural design (EN 1900:2002)” [19] and Chinese standard “Unified standard for reliability design of engineering structures (GB 50153–2008)” [15]. Therefore, Classes 1, 2, and 3 are specified for transmission line crossing high-speed railway, with Class 1 being the strongest. Tension section across high-speed railway is defined as Class 1, tension section not crossing high-speed railway is defined as Class 2, and temporary line is defined as Class 3.


### 4.2.2. Recommendation of Target Reliability Index

The numerical values of the reliability are often described on the basis of the reliability indexβ defined by β = −Φ−1(Pf), in which Pf is the failure probability. The relationship between β and Pf is given in Table 11. It is well recognized that the safety degree of engineering structure is medium and high for 10–3 < Pf < 10–4 and 10–4 < Pf < 10−5 [20].Table 11
Relationship betweenβ and Pf.
β2.73.13.23.53.74.04.2Pf3.5 × 10−31 × 10−36.40 × 10−42.33 × 10−41.1 × 10−43.17 × 10−51.3 × 10−5Calibration method is a very simple and practicable method applied to define the structural safety level. Therefore, target levels for reliability are often based on calibration [20]. Target reliability indices βT (i.e., recommended minimum values for reliability index) stipulated in Eurocode “EN 1900:2002” [19] and Chinese standard “GB 50153–2008” [15] are listed in Table 12.Table 12
Target reliability indices stipulated in EN 1900:2002 and GB 50153–2008 (ultimate limit state, 50 year reference period).
CodesSafety classesβTEN 1900:2002 [19]RC34.3RC23.8RC13.3GB 50153–2008 [15]Ductile failureClass 13.7Class 23.2Class 32.7Brittle failureClass 14.2Class 23.7Class 33.2As can be seen in Table12, there is a difference of about 0.5 for reliability indices between the adjacent safety classes, while the probability of failure can approximately differ by an order of magnitude.Based on the above section, the acceptable safety level of transmission tower Class 2 should be in agreement with the average reference values described in Table10. Thus, a target reliability index of 3.2 is suggested for Class 2. Then a reliability index of 3.7 is recommended for Class 1 in this study for the reason that adjacent safety classes have a difference of about 0.5 for reliability indices. Since transmission line of Class 3 is temporary, functional failure does not lead to serious consequences, reliability can be reduced accordingly. Consequently, a reliability index of transmission tower for Class 3 can be 2.7.


## 4.2.1. Safety Classes

In general, it is difficult to quantificationally calculate the loss due to structural failure. Therefore, safety classes of engineering structure are usually defined by a qualitative analysis method combined with engineering experience. Safety of engineering structure can be classified into three classes in design codes of many countries, such as Eurocode “Basis of structural design (EN 1900:2002)” [19] and Chinese standard “Unified standard for reliability design of engineering structures (GB 50153–2008)” [15]. Therefore, Classes 1, 2, and 3 are specified for transmission line crossing high-speed railway, with Class 1 being the strongest. Tension section across high-speed railway is defined as Class 1, tension section not crossing high-speed railway is defined as Class 2, and temporary line is defined as Class 3.


## 4.2.2. Recommendation of Target Reliability Index

The numerical values of the reliability are often described on the basis of the reliability indexβ defined by β = −Φ−1(Pf), in which Pf is the failure probability. The relationship between β and Pf is given in Table 11. It is well recognized that the safety degree of engineering structure is medium and high for 10–3 < Pf < 10–4 and 10–4 < Pf < 10−5 [20].Table 11
Relationship betweenβ and Pf.
β2.73.13.23.53.74.04.2Pf3.5 × 10−31 × 10−36.40 × 10−42.33 × 10−41.1 × 10−43.17 × 10−51.3 × 10−5Calibration method is a very simple and practicable method applied to define the structural safety level. Therefore, target levels for reliability are often based on calibration [20]. Target reliability indices βT (i.e., recommended minimum values for reliability index) stipulated in Eurocode “EN 1900:2002” [19] and Chinese standard “GB 50153–2008” [15] are listed in Table 12.Table 12
Target reliability indices stipulated in EN 1900:2002 and GB 50153–2008 (ultimate limit state, 50 year reference period).
CodesSafety classesβTEN 1900:2002 [19]RC34.3RC23.8RC13.3GB 50153–2008 [15]Ductile failureClass 13.7Class 23.2Class 32.7Brittle failureClass 14.2Class 23.7Class 33.2As can be seen in Table12, there is a difference of about 0.5 for reliability indices between the adjacent safety classes, while the probability of failure can approximately differ by an order of magnitude.Based on the above section, the acceptable safety level of transmission tower Class 2 should be in agreement with the average reference values described in Table10. Thus, a target reliability index of 3.2 is suggested for Class 2. Then a reliability index of 3.7 is recommended for Class 1 in this study for the reason that adjacent safety classes have a difference of about 0.5 for reliability indices. Since transmission line of Class 3 is temporary, functional failure does not lead to serious consequences, reliability can be reduced accordingly. Consequently, a reliability index of transmission tower for Class 3 can be 2.7.


## 5. Discussion

tIn order to determine the reliability of transmission line across high-speed railway, reliability calibration for steel angle tower of transmission line satisfying the minimum design requirements specified by Chinese codes was performed. The reliability indices of the tower for the extreme wind load case and the combined wind and ice load case are shown in Figures2–5, respectively. Based on the aforementioned results, some discussions are provided as follows:(1)
In the extreme wind load case, reliability indices for strength calculation of axially loaded member approximately range from 2.60 to 3.61, whereas the indices for stability calculation of axial compression member range from 2.38 to 3.40, as can be seen in Figures2 and 3. Correspondingly, reliability indices in the combined wind and ice load case are 1.32 to 6.31 and 1.39 to 8.09. The results indicate a wider range of reliability, probably due to the wide ranges of ρw and ρI. Furthermore, the ice load effect ratio ρI plays an important role in the reliability of tower, as shown in Figures 4 and 5.(2)
The reliability calibration results for the American National Electrical Safety Code (NESC C2-2002) and the Canadian Standard Association (CSA C22.3 No.1–2001) were presented in Ref. [4]. The reliability indices of transmission tower designed by NESC in the extreme wind load ranges from 2.36 to 3.01, whereas the indices for the combined wind with ice load case ranges from 2.29 to 3.91. Correspondingly, the reliability indices of transmission tower designed by CSA are 0.85 to 2.00 and 1.68 to 2.78, respectively.It can be seen that the reliability level of transmission tower designed by Chinese code is about similar with that specified in NESC and higher than that specified by CAS for extreme wind load case and the combined wind and ice load case.(3)
There are some deficiencies in statistical parameters used for the reliability analysis in this work. Ideally, the statistical data of various load effects, such as wind load and ice load used for the reliability analysis should be obtained from the field. However, due to the lack of such information required, an alternative approach was used in this work. For instance, parameters of building structure stipulated in Chinese code were used as statistical parameters of permanent load and wind load in the extreme wind load case. Statistical parameters of wind load in the combined wind and ice load were derived by 50 year return annual maximum wind speed. Parameters of ice load were determined based on a statistical analysis of limited field data. Statistical parameters of resistance of tower for stability calculation were derived by considering the uncertainty of stability coefficient.(4)
The calibration results show that the reliability level of tower satisfying the minimum requirements in Chinese code needs to be improved. In this study, the recommended target reliability indices were given based on the average reliability indices obtained from reliability calibration and the recommended values of engineering structures suggested in the codes of many countries.


## 6. Conclusions

Based on the analysis results obtained, the following conclusions can be drawn:(1)
The reliability indices achieved lie within a large range, which may be caused by a wide range ofρw and ρI. The reliability level of transmission tower is similar with that of American standards but higher than that of Canadian specifications.(2)
The average reference value of reliability index was taken as approximately 3.2. The acceptable safety level of transmission tower Class 2 should be in agreement with this value.(3)
Classes 1, 2, and 3 were specified for transmission line crossing high-speed railway, with Class 1 being the strongest. For tower of transmission line across high-speed railway, the target reliability indices of Class 1, Class 2, and Class 3 were recommended as 3.7, 3.2, and 2.7.



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