Reliability Analysis of Transmission Tower Based on Unscented Transformation Under Ice and Wind Loads

Abstract

Due to the complexity of the transmission tower structure and the correlation between wind and ice loads in the actual project, it is difficult to analyze the reliability of transmission towers with traditional methods. To solve this problem, the unscented transformation (UT) principle is presented concisely and used in the reliability analysis of transmission towers in this paper. Moreover, the finite element model of the target transmission tower is created. The reliability indices of the transmission tower under various loading cases are evaluated using UT and analyzed relative to the outcomes of the Monte Carlo method (MCS). Lastly, by analyzing and validating a wine-cup shape tangent tower, the simulation results show that the UT yields reliability indices with less than 6% relative error compared with MCS results for the transmission towers with lower reliability, which are more important in engineering. Variations in error caused by the change in correlation coefficients among variables are small. Consequently, the efficiency of calculations is improved by the UT-based reliability calculations for transmission towers in the case of correlated variables, which better meet engineering application requirements. It is proved that the method of reliability analysis for transmission towers based on the UT is applicable.

1. Introduction

In recent years, the frequent occurrence of ice, wind, and other disasters in China has posed substantial challenges to the normal operation of transmission tower structures. Within the realm of intricate engineering structures, the methods profoundly influence the analysis of the safety and reliability for actual engineering structures. Consequently, the development of a simple and effective reliability analysis method for transmission towers is of considerable theoretical and practical importance.

When the performance function of the transmission tower and the distribution of the underlying random variables are known, the MCS can be utilized for structural reliability assessments. By assessing the reliability of transmission towers by the MCS, the impact of mechanical model uncertainties induced by topologies of structures was considered in [1]. In [2], the MCS was used to calculate each structural element of transmission lines, such as the tower element. However, the disadvantage of the MCS is that it requires a large computational cost [3], though it is often used as a relatively exact solution to verify or validate approximate analytics. Due to the arbitrary nature of random variables distribution and non-Gaussian distribution of the performance functions, the reliability index of the structure is impossible to compute directly. The JC method was proposed by Rackwitz and Fiessler [4], which is suitable and fashionable for calculating the structural reliability index with arbitrary distribution random variables. The JC method is used to normalize variables with arbitrary distribution in Gaussian distribution before the solution of the problem. For example, tower elements satisfying the minimum design requirement in Chinese codes are reliability calibrated by the JC method [5]. Nevertheless, the mapping relationship of the input–output quantities of random variables is highly nonlinear in the context of practical engineering, which makes it difficult to analyze the structure reliability using the JC method. It can be demonstrated that the wind load and icing are not independent under identical meteorological conditions [6]. Wang T. et al. assess the global reliability of coupled transmission tower-insulator-line systems considering soil–structure interaction (CTTILSs-SSI) subjected to the multi-hazard of wind and ice. Moreover, to investigate the influence of soil–structure interaction (SSI) and soil types on the global reliability of transmission tower-insulator-line systems (TTILSs), the global reliability of CTTILSs-SSI with various soil types is compared with that of a fixed-base system [7]. The high-order moments-based improved maximum entropy method (HM-IMEM) was proposed and extended in [8]; it was used to assess the wind resistance global reliability of the in-service transmission tower. To accurately predict the stability bearing capacity of cross-bracings with semi-rigid connections and to efficiently conduct its probabilistic assessment, a prediction method based on the hybrid model, which is a combination of particle swarm optimization (PSO) and a backpropagation neural network (BPNN), was proposed by [9].

The UT is a method of estimating the statistical properties of variables by a finite set of parameters which was originally proposed in [10,11,12]. This method is more straightforward to implement than the traditional method of linear approximation methods for nonlinear mappings [11,13]. The use of UT is pervasive across a diverse range of disciplines to illustrate, automatic control, navigation and guidance, artificial intelligence, etc. [14]. For the purpose of improving accuracy, a family of modified unscented transformation methods is proposed. To find nonproduct sigma/cubature points with positive weights that can exactly integrate polynomial functions of desired order with respect to the Gaussian and uniform probability density functions, conjugate unscented transformation (CUT) were presented in [15]. In [16], a family of enhanced conjugate unscented transformation (ECUT) methods, including ECUT-4, ECUT-6, and ECUT-8 methods, was proposed for statistical moment estimation by combining the original CUT, variable transformation, and exact dimension reduction method. There are several advantages of this method, such as enhanced computational accuracy and efficiency, distribution independence, and ease of application. Furthermore, the covariance matrix of the variables can be used to consider the correlation coefficients between the random variables. The reliability analysis method for transmission towers based on the UT is presented in this paper.

2. Principle of the UT

The concept of the UT is estimating the statistical properties of variables by a finite set of parameters. The fundamental principle of the UT in two dimensions is shown in Figure 1 [17]. The samples were drawn deterministically from Γ_x in the UT, instead of randomly as in the MCS [18]. In the UT, the selected certainty sample points are transformed into a new set of points with unchanged weights via nonlinear mapping.

The fundamental steps can be summarized as follows: draw the set of σ-point about x; conduct nonlinear transformations; and numerically analyze for mean and variance of y. The aforementioned steps can be described in a feasible algorithm as follows:

For the sake of argument, consider an n-dimensional variable x. The random variable y is a function of x, with the following nonlinear mapping to x:

$$y=f(x)$$

(1)

The set of sample points Γ_x consists of sample points x⁽ⁱ⁾ and associated weights W:

$${\mathrm{\Gamma }}_x={[x^{(i)},W^{(i)}]}_{i=1}^{2n}$$

(2)

The set of σ-points contains 2n points, which are distributed symmetrically about the mean of x. The symmetric point set is capable of achieving only second-order accuracy. In order to make higher-order approximations, it is necessary to introduce new parameters into the σ point set. One method is to add new points equal to the mean value of x with weights W₀, and the improved 2n + 1 σ-points are defined as follows:

$$x_0={\mu }_x$$

(3)

$$x_i={\mu }_x+\gamma {(\sqrt{V_x})}_i,i=1,2,\dots ,n$$

(4)

$$x_i={\mu }_x-\gamma {(\sqrt{V_x})}_{i-n},i=n+1,n+2,\dots ,2n$$

(5)

$$\gamma =\sqrt{n+\lambda }$$

(6)

where µ_x is the mean matrix of x; V_x is the covariance matrix of x; and λ represents the scale parameter:

$$\lambda ={\alpha }^2\cdot (n-k)-n$$

(7)

where α is a positive scale factor; the constant k is determined as follows:

$$k=\left\{\begin{array}{l}3-n,n\le 3\\ 0,n>3\end{array}\right.$$

(8)

Nonlinear transformation of the set of σ-point by f is determined as follows:

$$Y_i=f(x_i)$$

(9)

The weights of the computed set of σ points remain unchanged.

Numerically analyze using the computed point set and weights for μ_y and V_y:

$${\mu }_y={\displaystyle \sum _{i=0}^{2n}{W}_i^{(\mu )}{Y}_i}$$

(10)

$$V_y={\displaystyle \sum _{i=0}^{2n}{W}_i^{(V)}(Y_i-{\mu }_y)\cdot {(Y_i-{\mu }_y)}^{\mathrm{T}}}$$

(11)

where W_i is the weight of the ith σ point; and W_i^(μ) and W_i^(V) are used to compute the mean and variance, respectively.

$$W_i^{(\mu )}={\displaystyle \frac{\lambda }{n+\lambda }}$$

(12)

$$W_0^{(V)}={\displaystyle \frac{\lambda }{n+\lambda }}+(1-{\alpha }^2+\eta )$$

(13)

$$W_i^{(\mu )}=W_i^{(V)}={\displaystyle \frac{1}{2\cdot (n+\lambda )}},i=1,2,\dots ,2n$$

(14)

where is the standard UT when α = 1 and η = 1.

3. The Limit State Function

When the member is in compression and bending simultaneously, the failure type is instability failure and the limit state function is [19]:

$$G_1(x)=f_{\mathrm{y}}-{\displaystyle \frac{N}{\phi \cdot q_{\mathrm{N}}\cdot A}}-{\displaystyle \frac{R}{q_{\mathrm{R}}\cdot C}}$$

(15)

where f_y is the yield stress; N is the member axial tension; φ is the stability coefficient; q_N is the compression strength reduction factor; A is the gross cross-sectional area; R is the value of the bending moment; q_R is the stability strength reduction coefficient; and C is the sectional resistance moment.

When the member is in tension and bending simultaneously, the failure type is strength failure and the limit state function is [19]:

$$G_2(x)=f_{\mathrm{y}}-{\displaystyle \frac{N}{q\cdot A_{\mathrm{n}}}}-{\displaystyle \frac{R}{q_{\mathrm{R}}\cdot C}}$$

(16)

where q is the strength reduction factor; and A_n is the net cross-sectional area.

4. Wind Load and Ice Load

In the design of tall structures such as transmission towers, wind load is of particular importance. Wind load is also important for the ice load when it comes to the conditions for icing. It is necessary to take into account the wind load when considering ice cover formation.

4.1. Wind Load

The standard value of the wind on the tower is determined according to the following formula [20]:

$$P_{\mathrm{S}}={\delta }_{\mathrm{Z}}\cdot {\delta }_{\mathrm{S}}\cdot {\epsilon }_{\mathrm{Z}}\cdot C\cdot B_{\mathrm{f}}\cdot P_0$$

(17)

where δ_Z is the wind pressure height variation correction factor; δ_S is the shape factor; ε_Z is the dynamic response factor; C is the ice accretion wind load amplification factor; B_f is the value of the projected area of beam elements subject to wind pressure; and P₀ is the standard value of the basic wind pressure.

The standard value of the wind load on the conductor and the ground wire is calculated according to the following formula [20]:

$$P_{\mathrm{X}}=\chi \cdot P_0\cdot {\delta }_{\mathrm{Z}}\cdot {\delta }_{\mathrm{SC}}\cdot {\epsilon }_{\mathrm{C}}\cdot d\cdot L_{\mathrm{p}}\cdot C\cdot {\sin }^2\theta$$

(18)

where χ is the non-uniform wind pressure factor; δ_SC is the shape coefficient; ε_C is the wind load adjustment coefficient; d is the diameter of the conductor windbreak; L_p is the wind span of the transmission line; and θ is the angle between the wind direction and the conductor and ground wire.

4.2. Ice Load

The icing on transmission towers can be classified into five types: Glaze, Mixed Rime, Hoar Frost, Rime, Snow, and Fog [21,22,23]. Through the investigation and analysis of the site data of previous ice disasters in China, the type of icing was glazing with rain, and the density of the ice was 0.9 kg/cm³ [24].

In the finite element analysis of the reliability of transmission towers, there are three ways of simulating ice load:

(1)

The additional force simulation method. This method is used to simulate the ice load of the conductor by 10 equally spaced concentrations in ADINA [25]. The accuracy of this method is limited.

(2)

The additional ice unit method. The ice load of the conductor by the additional ice unit method is simulated in [26].

(3)

The equivalent density method. The icing of the conductor by the equivalent density method is simulated in [27,28].

In this paper, the icing load was simulated using the equivalent density method, assuming that the effect of icing on gravity was only considered and the effect of icing on stiffness was not considered. The calculations assume that the angle iron and the entire line of the same level conductor and the ground wire were uniformly covered with ice.

The equivalent density of the angle iron after icing is as follows:

$${\rho }^{\prime }={\displaystyle \frac{m+m_1}{S}}$$

(19)

where m is the unit mass of the non-iced angle iron; m₁ is the unit ice mass; and S is the cross-sectional area of the angle iron.

$$m_1={\rho }_1\cdot [2(b+2t)\cdot (a+2t)-{(a+2t)}^2-2b\cdot a+a^2]$$

(20)

where ρ₁ is the density of the ice, take 900 kg/m³; b is the width of the angle iron; a is the thickness of the angle iron; and t is the thickness of ice.

The schematic diagram for defining the angle iron section is shown in Figure 2.

5. Model of Loads Framework of the Proposed Method

The technical route of the transmission tower reliability analysis method based on the UT is presented in Figure 3. The UT method is shown by the part of the red dotted frame in Figure 3. The main implementation steps are as follows.

(1)

The material strength, icing thickness, and wind pressure during ice cover are taken as random variables, and the mean matrix and covariance matrix are calculated using the mean, variance, and correlation coefficient of the random variables.

(2)

The 2n + 1 σ points are drawn deterministically based on the mean matrix, covariance matrix, and scale parameters.

(3)

The equivalent density of each angle iron for different ice thicknesses is calculated. The transmission tower is divided into ten segments and the wind load for each is applied separately. The finite element model of the transmission tower and load are created using the ANSYS.

(4)

The 2n + 1 Z-points are calculated based on the 2n + 1 σ points, functional function Z, and the finite element model.

(5)

The weights are computed in combination with the scale parameters.

(6)

The mean and variance of the Z-points are obtained based on the results of steps (4) and (5).

(7)

The reliability index β for the element are calculated according to the definition.

6. Example to Demonstrate

The wine-cup shape tangent tower was used as the study object in Example 1. The Q345 was used as the principal material expect support of ground wire, and the Q235 was used as inclined and transverse materials. The transmission tower was 32 m in height. The principal, inclined, and transverse materials of the transmission tower were simulated with the Beam189, and the auxiliary materials of transmission tower were simulated by LINK8.

Beam189 is accurate and requires more memory to calculate than Beam188 [29]. The schematic diagram of the shape of the Beam189 unit is shown in Figure 4. The orientations of load application are presented by the number ①~⑤ in Figure 4. The finite element model of the line wine-cup shape tangent tower is shown in Figure 5.

To validate the finite element model, static analysis under gravity of this model is presented in Figure 6.

Firstly, loading case 1 was considered as 10 mm ice thickness + x-direction wind load (20 m/s) + unbroken line. The random variables and their statistical properties [30] are summarized in

Table 1. Means and coefficient of variation in the random variables.

Name	Mean	COV	Distribution
Yield strength of the auxiliary materials (MPa)	258.50	0.110	Gaussian distribution
Yield strength of the principal material (MPa)	379.50	0.110	Gaussian distribution
Basic wind stress(N/mm2)	245.40	0.193	Gumbel distribution
Thickness of ice(mm)	10.00	0.181	Gumbel distribution

.

The ice load, weight, wind loads, boundary conditions, and conductor and ground wire loads were applied. The equivalent density of the angle steel after ice cover was calculated according to different angle steel sections in the mast model and assigned values to the equivalent material density corresponding to each section. The transmission tower was divided into 10 segments and the wind pressure was approximated as an equivalent concentrated force, then the size of the equivalent concentrated force for each segment was calculated separately and applied to the main material nodes. The conductor and ground wire loads were approximated as an equivalent concentrated force. The schematic diagram for the load application is presented in Figure 7.

First, it was assumed that the Pearson correlation ρ for the thickness of ice and wind stress was 0. Nine σ-points are shown in

Table 2. σ-points and Z-points.

Number	σ-Points
1	258.500	379.50	245.40	10.00
2	269.165	379.50	245.40	10.00
3	258.500	392.42	245.40	10.00
4	258.500	379.50	259.16	10.00
5	258.500	379.50	245.40	12.69
6	247.835	379.50	245.40	10.00
7	258.500	366.58	245.40	10.00
8	258.500	379.50	231.63	10.00
9	258.500	379.50	245.40	7.31

. The reliability index was calculated to be β = 1.978, according to the definition of the reliability indicator. The value of β was calculated by the MCS to be 1.838. The relative error between the two methods was 7.1%, which used the MCS as the standard value.

Then, the ρ between the thickness of ice and wind stress was considered. The reliability index β of the transmission towers and the relative error by the UT method and the MCS under different values of the correlation coefficients ρ are presented in

Table 3. The reliability index and the relative error under different values of ρ by UT and MCS.

ρ	UT	MCS	Relative Error
−0.9	2.067	1.898	8.2%
−0.7	2.055	1.884	8.3%
−0.5	2.023	1.865	7.8%
−0.3	2.001	1.851	7.5%
0	1.978	1.838	7.1%
0.3	1.955	1.822	6.8%
0.5	1.932	1.806	6.5%
0.7	1.897	1.789	5.7%
0.9	1.866	1.776	4.8%

.

To facilitate the interpretation of the results, a comparison of the UT and the MCS calculations is shown in Figure 8. The analysis revealed that when the positive correlation between wind pressure and ice thickness was more pronounced, the β of the target tower was lower, and the relative error of the two methods was smaller.

To further explore the influence of the reliability index and correlation coefficient on the relative errors of the two methods, consider the different working cases of transmission towers. Loading case 2: 11.5 mm ice thickness + wind of 20 m/s in x-direction; loading case 3: 14 mm ice thickness + wind of 20 m/s in x-direction; and loading case 4: 11.5 mm ice thickness + wind of 21 m/s in x-direction. The results of the calculations for Cases 2, 3, and 4 are presented in

Table 4. Variation in reliability index with value of correlation coefficient under different loading cases.

ρ	Loading Case 2		Loading Case 3		Loading Case 4
	UT	MCS	UT	MCS	UT	MCS
−0.9	1.620	1.511	1.267	1.192	0.897	0.852
−0.7	1.597	1.495	1.233	1.165	0.884	0.837
−0.5	1.582	1.479	1.210	1.146	0.857	0.811
−0.3	1.549	1.458	1.169	1.115	0.812	0.784
0	1.529	1.437	1.155	1.096	0.781	0.753
0.3	1.512	1.415	1.121	1.073	0.754	0.728
0.5	1.490	1.398	1.102	1.059	0.736	0.715
0.7	1.456	1.373	1.080	1.040	0.725	0.703
0.9	1.429	1.356	1.068	1.033	0.699	0.684

. The relative error under different values of ρ between two methods are presented in

Table 5. The relative error under different values of ρ by the UT and MCS.

ρ	Loading Case 2	Loading Case 3	Loading Case 4
−0.9	6.7%	5.9%	5.0%
−0.7	6.4%	5.5%	5.3%
−0.5	6.5%	5.3%	5.4%
−0.3	5.9%	4.6%	3.5%
0	6.0%	5.1%	3.6%
0.3	6.4%	4.3%	3.5%
0.5	6.2%	3.9%	2.8%
0.7	5.7%	3.7%	3.0%
0.9	5.1%	3.3%	2.1%

.

A dot plot of the calculation results for different values of the correlation coefficient for loading cases 1, 2, 3, and 4 are presented in Figure 9. There is a positive correlation between the reliability of the transmission tower structure and the relative error of the UT and MCS calculation results. The relative error histogram for the two methods with different values of the correlation coefficient for four loading cases is given in Figure 10. The general trend indicates that the greater the positive correlation, the smaller the relative error between the two methods when the mean of the ice thickness and the wind pressure is held constant. The relative errors were smaller under loading cases with lower reliability.

The computational time and number of computations for the two computational methods on the same computer are presented in

Table 6. Number of calculations and calculation time of UT and MCS.

Method	UT	MCS
Number of calculations	11	20,000
Calculation time	19 s	47 s

.

7. Conclusions

The UT-based reliability analysis of transmission towers is proposed and the principle of the UT is introduced in this paper. The conclusions are drawn as follows:

(1)

Equivalent normalization is not required for the UT.

(2)

The results of the example indicate that the error of the calculations of the UT and MCS is within 6% when the reliability index is below 1.3. Engineering is more focused on transmission towers and loading cases with lower reliability. Consequently, the analysis method presented in this paper is a reasonable approach.

(3)

The calculation process can be simplified by the application of the UT when there is a correlation between the variables.

(4)

The 2n + 1 calculations are only needed for n-dimensional variables by the UT. The computation of the UT is more efficient than MCS.