Fragility analysis of a long-span transmission tower

Abstract

Numerous transmission towers have collapsed due to experiencing strong winds; therefore, the purpose of this article is to investigate the collapse mechanism and the anti-collapse performance of a long-span transmission tower–line system. The detailed finite element model of a typical tower–line system is established in ABAQUS. A global damage index is proposed to quantitatively estimate the overall damage of the structure and define the collapse criteria. An incremental dynamic analysis is performed to obtain the collapse mechanism and the ultimate capacity of the structure. Subsequently, a fragility analysis for evaluating the anti-collapse performance is conducted due to the uncertainty of wind loads. Eventually, the influence of the wind attack angle and the length of the side spans on the fragility is discussed. The results demonstrate that the proposed global damage index is capable of quantitatively reflecting the overall damage and assessing the ultimate capacity of the structure. In addition, the uncertainty of the wind load has a significant influence on the ultimate capacity and the failure position. Furthermore, the results reveal that the wind attack angle and the length of the side spans have an apparent effect on the fragility of the structure.

Keywords

collapse mechanism fragility analysis global damage index transmission tower–line system wind load

Introduction

The transmission tower–line system mainly composed of a group of transmission towers and transmission lines is a complicated coupling system with the characteristics of a vast height, a large span, light weight, and a slender shape, which are widely used in the field of power transmission. If one tower collapses, then the entire power supply system is paralyzed, resulting in large-area power outages, which not only seriously affects our normal life but also causes immeasurable financial loss. Moreover, compared with other natural disasters, wind hazards have a wider range of impacts and are the number one cause of power line damage (Liang et al., 2015). The structural characteristics of the tower–line systems determine its vulnerability to wind loads. Over the past few decades, the collapse events of the systems under wind loads can be easily found and have led to severe financial loss. More specifically, Zhang (2006) reported the failures of 18 towers belonging to 500-kV lines and 57 towers belonging to 110-kV lines due to strong wind events in China. Kaňák et al. (2007) studied a downburst event that occurred in southwestern Slovakia in 2003 where at least 19 electricity transmission towers collapsed. The statistics show that, in the Americas, Australia, and South Africa, approximately 80% of the transmission tower failures are due to strong wind loadings, and such failures have led to large-area power outages (Dempsey and White, 1996; Savory et al., 2001).

Due to the high dependencies of our normal life on electricity, it is important for actual engineering and our society to study the collapse mechanism of transmission tower–line systems under wind loads, which could ensure the safe operation of the systems under high-intensity wind. During these years, this aspect has attracted increasing attention from experts and scholars, and numerous numerical (Asgarian et al., 2016; Tapia-Hernández et al., 2017; Zhang et al., 2013) and experimental (Fu et al., 2019; Li et al., 2018; Moon et al., 2009) studies have been conducted. However, almost all the studies mentioned above have been aimed at the collapse of a single tower, with a few studies related to the collapse of tower–line systems under wind loads. Nevertheless, the coupling effect of the lines and towers has a great influence on the response of the transmission towers which should not be neglected. As a consequence, the transmission tower–line system regarded as a whole for modeling and analysis is more accurate.

In addition, it should be noted that all the above researches have been primarily based on deterministic analyses. However, the wind load is random due to the existence of fluctuating wind components, and the anti-collapse performance of the transmission tower–line system under wind loads should be analyzed using a probabilistic method. As a consequence, it is necessary to study the fragility of the system under wind loads. To date, very limited research studies have been performed to investigate the fragility of the transmission tower–line system, and the analysis of fragility under wind loads is even rarer. Fu et al. (2016) presented a fragility analysis of transmission towers under wind and rain loads. Most studies on the fragility of tower–line systems have mainly been focused on seismic fragility (Long et al., 2018; Park et al., 2016; Tian et al., 2019). Under these conditions, it is crucial to analyze the fragility of the transmission tower–line system under wind loads and further explore its anti-collapse performance.

Although the collapse analysis of transmission towers under wind loads has been studied, there have been limited investigations on the tower–line system. It can be seen from the above reviews that research achievements on the fragility analysis of the transmission tower–line system under wind loads are even fewer. To fill the research gap, this research focuses on the collapse and fragility analyses of the transmission tower–line system under wind loads. The remainder of this article is organized as follows. Section “Numerical simulation of a long-span transmission tower–line system” provides the prototype and finite element (FE) model of the selected typical transmission tower–line system and introduces the process of simulating wind loads. Section “Collapse criteria” gives the collapse criterion based on the proposed global damage index (GDI). Considering the dynamic coupling effect between the towers and lines, section “Collapse damage analysis” analyzes the collapse mechanism of the structure. Section “Fragility analysis” performs the fragility analysis and investigates the influence of the wind attack angles and the length of side spans. Finally, section “Summary and conclusion” presents the major conclusions of this research.

Numerical simulation of a long-span transmission tower–line system

FE model

As this investigation is focused on the long-span transmission tower–line system, a 220-kV tower–line system crossing the Yellow River in Shandong Province, China, is selected for this case study. The system consists of two suspension-type towers with the same configuration (Towers 1 and 2) and three span lines (Spans 1–3), the lengths of which are 294, 1118, and 285 m, respectively. The suspension-type tower is 122 m with two cross arms at elevations of 102 and 112.5 m, respectively. The square base of the suspension-type tower is 25 m. The practical graph of the suspension-type towers is shown in Figure 1(a). In addition, Figure 1(b) shows the configuration of the suspension-type towers, in which Segments 1–10 are illustrated along the height of the tower.

Figure 1.

Practical graph (a) and configuration (b) of the suspension-type tower.

The transmission towers are composed of main and diagonal members, which were constructed by steel tubes of types Q345 and Q235 (which are very similar to types A441 and A242 in ASTM, respectively). The lines are divided into three layers, for which the types from top to bottom are two optical ground lines, two dual-core conductor lines, and four dual-core conductor lines adopted by OPGW-180, LHBGJ-400/95, and LHBGJ-400/95, respectively.

A detailed FE model of the system is established based on engineering data in ABAQUS (n.d.), which is illustrated in Figure 2. The suspension-type tower, including 1140 elements and 431 points, is modeled by B31 beam elements. Considering that the truss elements only transmit axial forces, the lines and insulators, including 1132 elements and 1140 points, are established by T3D2 (truss element). The lines are hinged to the ground; meanwhile, the foundations of the towers are assumed to be fixed, and the interaction between the foundation and the tower is neglected. As a matter of convenience, the X-, Y-, and Z-axes of the model are defined as the transverse, longitudinal, and vertical directions of the system, respectively. The first frequencies of the system in the X- and Y-directions are 1.02 and 1.05 Hz, respectively.

Figure 2.

Detail FE model of the transmission tower–line system.

Simulation of wind loads

The methods for wind load simulation are mainly the harmony superposition method and the linear filter method (Ballaben et al., 2017; Di Paola and Gullo, 2001). Compared to the harmony superposition method, the linear filter method has been widely used to simulate fluctuating wind speeds due to its advantages of less calculations and higher efficiency. Because of the vast volume of the structure, the number of calculations is extremely large when conducting wind load simulations. In view of this situation, the autoregressive (AR) model is employed to simulate the fluctuating wind speed. For a wind profile near the ground, the logarithmic profile is more accurate, and a correction is necessary when the altitude is more than 30 m (ESDU 85020, 1982). Therefore, the modified logarithmic wind profile established by the Engineering Sciences Data Unit (ESDU) (ESDU 85020, 1982) is employed in this article. The spectral density function is simulated according to the Kaimal spectrum, in which the turbulence scale varies with height (Miguel et al., 2012; Sørensen et al., 2002; Togbenou et al., 2016).

In addition, the fluctuating wind speed can be described as

{v (t)} = [C] {u (t)}

(1)

where $[C]$ is a lower triangular matrix determined by the cross-correlation function and ${u (t)}$ is a vector consisting of the uncorrelated fluctuating wind speed.

First, the P-order AR filter is used to simulate the vector ${u (t)}$ , which can be expressed as

{u (t)} = \sum_{k = 1}^{P} φ_{k} {u (t - k Δ t)} + σ_{N} {N (t)}

(2)

where $Δ t$ is the time interval taken as 0.1 s, $N (t)$ represents the normally distributed random numbers with mean 0 and variance 1, P is the order of the AR model, and $φ_{k}$ are the AR parameters, defined using the following expression

{R_{u} (j Δ t)} = \sum_{k = 1}^{P} [R_{u} (j - k) Δ t] φ_{k} (j = 1, 2, \dots, P)

(3)

After determining the AR parameters $φ_{k}$ , $σ_{N}$ can be obtained

σ_{N}^{2} = R_{u} (0) - \sum_{k = 1}^{P} φ_{k} R_{u} (k Δ t)

(4)

The next step is determining the matrix $[C]$

[R] = σ_{u}^{2} [C] [I] {[C]}^{T}

(5)

where $σ_{u}$ is the root mean square error of the fluctuating wind speed and $[I]$ is the N-order identity matrix.

The elements in the matrix $[R]$ can be expressed as follows

R_{ij} = \int_{0}^{\infty} S (p_{i}, p_{j}, n) dn

(6)

where $S (p_{i}, p_{j}, n)$ is the cross-power spectrum of $p_{i}$ and $p_{j}$ .

Therefore, the element of the matrix $[C]$ can be obtained by

\begin{matrix} c_{jj} = \sqrt{(\frac{R_{jj}}{σ_{u}^{2}}) - \sum_{k = 1}^{j - 1} c_{jk}^{2}} \\ c_{ji} = (\frac{R_{ji}}{σ_{u}^{2}}) - \sum_{k = 1}^{i - 1} \frac{c_{jk} c_{ik}}{c_{ii}}, j \neq i \end{matrix}

(7)

The total wind speed $V (t)$ is the summation of the mean speed and fluctuating wind speed. The wind load acting on the structure is given by

F (t) = \frac{μ_{s} ρ_{a} V {(t)}^{2} A}{2}

(8)

where $μ_{S}$ is the shape coefficient taking a value of 2.3 (GB 50009-2001, 2006) and $A$ is the projected area of the structure in the windward direction.

It is impossible to calculate the wind load at all points in an actual simulation (Fu and Li, 2016). Hence, the tower is divided into 10 segments, as shown in Figure 1(b). The midpoints of each segment are regarded as the simulation points.

The simulation parameters for the fluctuating wind speed are as follows: (1) the terrain roughness is Class B, (2) the time interval is 0.1 s, (3) the design wind speed is 27 m/s, and (4) the order of the AR model is 4. Figure 3(a) shows the wind speed of Segment 1 when the structure is under the design wind speed. Figure 3(b) shows the comparison of the target wind power spectrum and the simulated spectrum of Segment 1, which shows a good agreement.

Figure 3.

Wind simulation results of Segment 1: (a) wind speed and (b) power spectral comparison.

Collapse criteria

Material constitutive model

The detailed material constitutive model is the precondition for accurately analyzing the collapse mechanism of the structure. Since the transmission tower is a type of high-rise structure for supporting the transmission lines, the slenderness ratio of the elements is generally large, leading to the facile buckling of the elements under strong wind. Therefore, when conducting the collapse analysis of the system, the buckling effect of elements should be taken into adequate account; otherwise, the capacity of the structure will be overestimated. For this reason, the Tian–Ma–Qu material model (Tian et al., 2017a) is used to simulate the nonlinear behavior of steel tubes, which is introduced into the user material routine VUMAT in ABAQUS. A comparison of the experimental (Black et al., 1980) and simulation results is shown in Figure 4, which fitted well. And the effectiveness of the material model has also been verified by experiments (Tian et al., 2017a, 2017b, 2018).

Figure 4.

Comparison of the experimental and simulation results: (a) Pipe 4 Std and (b) Pipe 3-1/2 Std.

Damage index

The failure of structures is usually caused by damage, which gradually accumulates to a certain threshold. Under cyclic loads, such as wind loads, the initial imperfections are developed inside the steel, that is, the steel becomes damaged, which causes the deterioration of its mechanical properties, and the damage continuously accumulates with an increase in the number of load cycles leading to the failure of the structure. Thus, it is necessary to correctly evaluate the effect of damage accumulation on the collapse analysis of transmission line systems under wind loads.

To accurately analyze the bearing capacity of the system, the damage index is introduced to evaluate the damage degree of the elements, and the expression is as follows

D = (1 - β) \frac{ε_{m}^{p} - ε_{0}^{p}}{ε_{u}^{p} - ε_{0}^{p}} + β \sum_{i = 1}^{N} \frac{ε_{i}^{p} - ε_{0}^{p}}{ε_{u}^{p} - ε_{0}^{p}}

(9)

where $ε_{m}^{p}$ is the maximum normal strain value during the analysis, $ε_{i}^{p}$ is the highest normal strain during the ith half loading cycle, $ε_{u}^{p}$ is the ultimate normal strain of the material, $β$ is the weight coefficient for the cumulative damage taken as 0.0268, and N is the total number of half loading cycles.

A collapse simulation is a complicated process to conduct, and the selection of the collapse criteria is a key step in the collapse analysis. The performance index used to judge the collapse of the structures is usually the horizontal tip displacement (Fu et al., 2016). However, for the transmission tower–line system, which is a special structure with both high and flexible characteristics, large rotational deformations may exist. The collapse of the system is an overall behavior that requires the contribution of all elements. As a consequence, a single index, such as tip displacement, is not sufficient to demonstrate the contribution of every element in the structure. Along this line of consideration, the damage index of elements mentioned above also cannot reflect the overall damage of the transmission tower completely; in other words, the failure of one element does not mean the collapse of the whole tower.

Therefore, it is necessary to select an index that can reflect the overall structure to judge the collapse of the system. Based on the abovementioned damage index of elements, the GDI connected with each member of the tower, which was utilized to estimate the collapse using the weighted mean method, can be calculated via the next two steps. First, the segmental damage index (SDI) of Segment i can be calculated as follows (Barbosa et al., 2017; Elenas, 2000; Kostinakis et al., 2015)

SD I_{i} = \frac{\sum_{j = 1}^{n} w_{j} D_{j}}{\sum_{j = 1}^{n} w_{j}}

(10)

where n is the total number of elements in Segment i, D_j is the damage index of the jth element in Segment i, and $w_{j}$ is the weighting factor of the jth element in Segment i, which is assumed to be proportional to the damage index D_j (Park et al., 1985; Powell and Allahabadi, 1988). Therefore, the SDI can be simplified as follows

SD I_{i} = \frac{\sum_{j = 1}^{n} D_{j}^{2}}{\sum_{j = 1}^{n} D_{j}}

(11)

The next step is to calculate the GDI based on the SDI. The GDI can be expressed as a sum of the SDI values by a weighted method

GDI = \sum_{i = 1}^{m} λ_{i} SD I_{i}

(12)

where m is the total number of segments and $λ_{i}$ is the weighting factor corresponding to Segment i, which reflects the relative importance of the segment in the transmission tower. There are many methods for selecting the values of the weighting factor $λ_{i}$ . Park and Ang (1985) defined the weighting factor according to the energy-absorbing contribution factor, which is closely correlated with the distribution of the absorbed energy. Chung et al. (1989) introduced the position of the segment as the weighting factor to reflect the influence of the position of the segment on global damage. Furthermore, Ou et al. (2000) considered the position of the segment and the damage to define the weighting factor, which is more comprehensive. Therefore, the third factor is utilized in this article, which can be expressed as follows

w_{i} = \frac{(m + 1 - i) SD I_{i}}{\sum_{k = 1}^{m} (m + 1 - k) SD I_{k}}

(13)

After obtaining the GDI, the last step is to classify the collapse threshold value of the structure. Based on previous studies (Ghobarah et al., 1999; Park et al., 1985), the collapse threshold value is 0.8.

Collapse damage analysis

In the previous section, the detailed FE model considering the coupling effect between the towers and lines is established, and the GDI is proposed. In this section, the collapse of the transmission tower–line system under wind loads is simulated using the incremental dynamic analysis (IDA) method, and the aforementioned GDI is utilized to estimate whether the towers collapse and determine the critical wind speed, which is defined as the wind speed corresponding to the appearance of the collapse of the structure. The critical wind speed reflects the anti-collapse performance of the system. Note that the wind loads are applied along the transverse direction of the system. The wind loads utilized in this section are generated by MATLAB based on the abovementioned method. Using constant intensity measure (IM) steps with an increment of 0.2 m/s (Vamvatsikos and Cornell, 2002), a suite of wind load samples is simulated to conduct collapse analyses of the structure based on the IDA method. The critical wind speed is 30.2 m/s, and the collapse state of the system is illustrated in Figure 5. In addition, the different damage levels are plotted in different colors; for example, red represents completely failed. The damage level of Tower 1 is much more severe than that of Tower 2, and, moreover, the initial failure member appears in Segment 6 of Tower 1. Therefore, the following discussion is mainly focused on Tower 1 in this section.

Figure 5.

Collapse of the long-span transmission tower–line system.

Figure 6 illustrates the time-history curves of the horizontal and vertical tip displacement of Tower 1 and the damage states of the structure at key moments (t₁, t₂, t₃, t₄) during the collapse process. Note that different colors represent different damage levels, and the buckling elements are highlighted in red.

Figure 6.

Time-history curves of the tip displacement and damage states of the structure at specific time points.

When t = 91.1 s, Diagonal Element 673 is slightly damaged. There is no element failed until t₃ (i.e. 101 s), and, more specifically, the damage index D of the members is not up to 1, which is defined as the threshold of member buckling. The horizontal and vertical tip displacements of Tower 1 stay at a relatively lower level in this period, and the transmission tower is almost free of deformation, which means that the tower is nearly elastic. When t = 101 s, the damage index D of Diagonal Element 673 reaches 1.0; note that Element 673 is highlighted in Figure 6. The member buckling initiates on Diagonal Element 673 in Segment 6, which is defined as the initial failure element of the structure. The buckling of Diagonal Element 673 leads to a rapid decrease in its bearing capacity, which causes internal force redistribution within the transmission tower. Thus, after that point, it can be observed from Figure 6 that the slope of the horizontal displacement curve obviously increases; as a result, the horizontal displacement continues to grow. When t reaches t₄ (i.e. 106.9 s), several elements fail, and this failure propagates rapidly in the structure. Notably, there are visible deformations in several segments. Subsequently, the displacement curves begin to increase sharply. As the time reaches t₅ (i.e. 109.2 s), with the failure of a substantial number of elements, the path of force transfer is broken, leading to the collapse of the tower. As a consequence, the horizontal and vertical tip displacements show a prominent divergence trend.

Figure 7 shows the GDI of Tower 1 under wind loads. It is found that the GDI is a kind of monotonically increasing function, owing to the fact that the damage is irreversible. Note that the GDI varies from 0 to 1 and generally does not reach 1. According to previous studies, several damage levels are defined to investigate the detailed damage process of the transmission tower. The thresholds of slight damage, moderate damage, extensive damage, and collapse are 0.2, 0.4, 0.6, and 0.8, respectively (Ghobarah et al., 1999; Park et al., 1985). When t = 91.1 s, the GDI is slightly more than 0.2, which means that the tower is slightly damaged. When t = 101 s, with the failure of Diagonal Element 673, the GDI almost comes to 0.6, and the structure is moderately damaged. When t = 106.9 s, a number of elements fail, and the GDI is 0.7, which implies that the tower stays in an extensive damage stage. When t = 109.2 s, the GDI is 0.8, which is the threshold of collapse.

Figure 7.

GDI of Tower 1.

Fragility analysis

Because the wind load is a kind of random load, the uncertainty of the load may seriously affect the simulation results. In the above section, the case of a collapse simulation is given. Furthermore, considering the randomness of a wind load, the collapse of transmission tower–line systems should be studied via a probabilistic method, and fragility analysis is a powerful tool in this aspect. In this section, based on the IDA method, a fragility simulation of the system is conducted, and the influence of the wind attack angle and the length of the side span are investigated.

Fragility analysis under a wind attack angle of 0°

The fragility of the system can be defined as the conditional probability that the structure collapses when the wind load demand is equal to or greater than the ultimate capacity with a given wind IM (i.e. the wind speed), which can be calculated as follows

P (C | IM = x) = P (GDI \geq GD I_{u} | IM = x)

(14)

where $P (C | IM = x)$ is the collapse probability of the system with a certain IM, GDI is the global damage index proposed in this study, and GDI_u is the threshold of the collapse taken as 0.8 (Ghobarah et al., 1999; Park et al., 1985).

Generally, the fragility of the structure obeys a lognormal distribution (Fu and Li, 2018)

\begin{matrix} P (GDI \geq GD I_{u} | IM = x) = \\ \int_{0}^{IM} \frac{1}{im \sqrt{2 π} ξ_{IM}} \exp {- \frac{{[\ln (im) - λ_{IM}]}^{2}}{2 ξ_{IM}^{2}}} d (im) \end{matrix}

(15)

where $λ_{IM}$ and $ξ_{IM}$ are the logarithmic normal distribution median intensity and the standard deviation of the IM, respectively.

Notably, the fragility of the structure can be evaluated using fragility curves that reflect the relationship between the wind speed and the collapse probability (Fu and Li, 2018). To gain insight into the fragility of the structure, fragility analysis is conducted based on the IDA method. The detailed calculation processes of the fragility analysis are exhibited in the flowchart (Figure 8).

Figure 8.

Calculation processes of the fragility analysis.

First, 20 groups of wind loads are generated using MATLAB with an increment of 0.2 m/s. Note that the wind loads are applied along the transverse direction of the structure. The collapse criterion is the GDI of the tower more than 0.8 (Ghobarah et al., 1999; Park et al., 1985), as mentioned above. Then, the collapse analyses of the system under 20 groups of wind samples are conducted by the IDA method. As a result, the critical wind speeds are obtained, which are used to illustrate the fragility curve fitted by the lognormal distribution.

The critical wind speeds and damage positions under 20 groups of wind samples that input along the transverse direction of system are tabulated in Table 1. The results demonstrate that the damage positions mainly appear on Segments 5 and 6. More specifically, 7 samples collapsed in Segment 5, and 13 samples collapsed in Segment 6. In addition, the probability of collapse in Segment 6 is greater than that in Segment 5; in other words, Segment 6 is the most vulnerable region of the transmission tower.

Table 1.

Critical wind speeds and damage positions.

ID	Critical wind speed (m/s)	Damage position	ID	Critical wind speed (m/s)	Damage position
A 1	30.2	Segment 6	A 11	31.2	Segment 6
A 2	30.2	Segment 6	A 12	31.4	Segment 6
A 3	29.8	Segment 5	A 13	32.2	Segment 6
A 4	31.0	Segment 5	A 14	29.4	Segment 6
A 5	29.0	Segment 5	A 15	28.4	Segment 5
A 6	30.0	Segment 5	A 16	30.2	Segment 6
A 7	29.8	Segment 6	A 17	28.6	Segment 6
A 8	31.2	Segment 6	A 18	31.2	Segment 5
A 9	31.0	Segment 6	A 19	30.4	Segment 6
A 10	30.2	Segment 5	A 20	31.0	Segment 6

According to the critical wind speeds and the fragility theory mentioned above, the fragility curve of the system when the wind loads input along the transverse direction is depicted in Figure 9. Notably, the horizontal and vertical axes represent the probability of collapse and the wind speed, respectively. In other words, the fragility curve reflects the relationship between the collapse probability and wind speed. In a fragility analysis, the IM corresponding to the probability of 10% is typically regarded as the critical value that determines whether the structure is predicted to collapse (FEMA-P695, 2009). As shown in Figure 9, the wind speed corresponding to the probability of 10% is 29.2 m/s, that is, the structure is regarded as collapsed if the wind speed is more than 29.2 m/s.

Figure 9.

Fragility curve of the transmission tower–line system.

Influence of the wind attack angle

In the last section, the fragility of the structure under the wind loads which input along the transverse direction is investigated. However, the wind may come from not only the transverse direction but also any other directions that lead to different wind attack angles applied to the system. Furthermore, different wind attack angles can lead to different load distributions in two horizontal directions that result in different responses (Deng et al., 2016). If the influence of the wind attack angle is ignored, then it is possible to misestimate the capacity of the structure. Thus, it is essential to investigate the influence of wind attack angle on the fragility of the transmission tower–line system. Considering the symmetry of the structure and the recommendation of the Chinese standard (DL/T5154-2002, 2002), four potentially unfavorable wind attack angles are selected herein: 0°, 45°, 60°, and 90°. The definition of the wind attack angle $θ$ is illustrated in Figure 10.

Figure 10.

Schematic of the wind attack angle.

Note that only Tower 1 and a portion of lines are illustrated in Figure 10, and other parts of the structure are omitted due to the symmetry of the structure. The positive direction of the Y-axis is defined as a wind attack angle of 0°, and the wind attack angle increases gradually with a counterclockwise direction. The wind loads under different wind attack angles are calculated based on the Chinese standard (DL/T5154-2002, 2002).

As shown in Figure 11(a), the fragility curves of the system under four potentially unfavorable wind attack angles are obtained. As mentioned above, in the fragility analysis, the IM corresponding to the probability of 10% is typically regarded as the critical value that determines whether the structure is predicted to collapse (FEMA-P695, 2009). The critical wind speeds of the system under wind attack angles of 0°, 45°, 60°, and 90° are 38.8, 33.1, 31.5, and 29.2 m/s, respectively. These facts demonstrate that the wind attack angle has an obvious effect on the critical wind speed and fragility of the structure. The major cause of this effect is that the wind load distribution varies with the change of the wind attack angle, which leads to different responses of the system. In addition, the critical wind speed decreases with the increasing wind attack angle, which indicates that the structure is more likely to collapse with the increasing wind attack angle. The maximum and minimum values of critical wind speeds are obtained under the wind attack angles of 0° and 90°, respectively. Namely, the most unfavorable and favorable wind attack angles are 90° and 0°, respectively.

Figure 11.

Influence of (a) the wind attack angle and (b) the length of the side span.

Nevertheless, the lengths of the two side spans are 285 and 294 m, respectively, which are not perfectly equal. Although the difference between the two side spans is not very large, further research is still needed to better investigate the influence of the length of the side spans on the fragility. The influence of the length of side spans on the fragility of the system is studied by switching the direction of wind loads. More specifically, the loads of the fragility under the wind attack angle of 0° input along the wind attack angle of 180° and the fragility of the system under the wind attack angle of 180° are conducted. The comparison of the fragility curves of the system under the wind attack angles of 0° and 180° is shown in Figure 11(b).

As shown in Figure 11(b), the distance between the two fragility curves is significant, which indicates that the fragility of the system is sensitive to the length of the two side spans. The wind speeds corresponding to the probability of 10% under the wind attack angles of 0° and 180° are 38.8 and 39.8 m/s, respectively, which determine the critical wind speed as mentioned above. The critical wind speed under a wind attack angle of 180° is more than 0°, which demonstrates that the structure is more likely to collapse under a wind attack angle of 0°.

Summary and conclusion

This study focuses on the collapse and fragility analyses of a typical long-span transmission tower–line system under wind loads. A new index GDI is proposed to estimate the collapse of the system; the dynamic coupling effect between the tower and lines is taken into account during the collapse and fragility analyses of the system under wind loads using the IDA method. The critical wind speed is utilized to reflect the anti-collapse capacity of the structure. Furthermore, the influence of the wind attack angle and the length of side spans is also investigated. Based on the numerical results obtained from this investigation, the following significant conclusions are drawn:

The proposed GDI is capable of reflecting the damage process of the whole structure and quantitatively assessing the ultimate capacity of the transmission tower–line system.

The damage position of the transmission tower most likely appears in Segment 6, which can be defined as the weakest position of the structure.

The wind attack angle and length of the side spans have an obvious effect on the fragility and critical wind speed of the long-span transmission tower–line system.

The most favorable and unfavorable wind attack angles are 180° and 90°, and the corresponding critical wind speeds are 39.8 and 29.2 m/s, respectively.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by the Young Scholars Program of Shandong University (Grant No. 2017WLJH33), the National Natural Science Foundation of China (Grant No. 51708089), the China Postdoctoral Science Foundation (Grant Nos 2017M620101 and 2019T120207), and the Fundamental Research Funds for the Central Universities (Grant No. DUT19RC(4)021).

ORCID iD

Xing Fu

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Fragility analysis of a long-span transmission tower–line system under wind loads

Abstract

Keywords

Introduction

Numerical simulation of a long-span transmission tower–line system

FE model

Simulation of wind loads

Collapse criteria

Material constitutive model

Damage index

Collapse damage analysis

Fragility analysis

Fragility analysis under a wind attack angle of 0°

Influence of the wind attack angle

Summary and conclusion

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

References