Fragility analysis and estimation of collapse status for transmission tower subjected to wind and rain loads

Structural Safety 58 (2016) 1–10

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Structural Safety journal homepage: www.elsevier.com/locate/strusafe

Fragility analysis and estimation of collapse status for transmission tower subjected to wind and rain loads Xing Fu ⇑, Hong-Nan Li, Gang Li Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116023, China State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116023, China

a r t i c l e

i n f o

Article history: Received 13 April 2015 Received in revised form 15 August 2015 Accepted 21 August 2015

Keywords: Transmission tower Wind and rain loads Equivalent basic wind speed Fragility analysis Nonlinear analysis Critical collapse curve

a b s t r a c t In this paper, the fragility analysis and concept of critical collapse curve for transmission tower subjected to wind and rain loads are presented to acquire the collapse equivalent basic wind speed and most unfavorable combinations of wind and rain loads corresponding to collapse status. The calculating method for wind and rain loads is simplified and the error analysis is performed to validate its effectiveness. The concept of equivalent basic wind speed is used to conduct the fragility analysis of transmission tower subjected to wind and rain loads which avoid the complex formula of rain load and the choice of different combinations of basic wind speed and rain intensity, and then the concept of critical collapse curve is proposed to evaluate the collapse status of transmission tower. At last the influence of wind attack angle and wind spectrum on the fragility and critical collapse curves is discussed, and results show that the wind attack angle and wind spectrum have a great influence on fragility and critical collapse curves. In this study, it can be seen that the use of equivalent basic wind speed make it possible to conduct the fragility analysis under wind and rain loads and the proposed concept of critical collapse curve is very convenient to evaluate the collapse status for structures subjected to wind and rain loads. In addition, the rain load has a great contribution to the tower collapse and should be paid more attention during severe gales and thunderstorms. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Transmission tower is a supporter of power consumer, and its collapse often causes great economic loss and many accidents. In recent years, the fact of many transmission tower-line systems collapsed during typhoons or hurricanes attracts most researchers’ attention. Typhoons or hurricanes are always accompanied by the strong rainfall, and the influence of rain load on the tower collapse has not been studied before. Therefore, considering the rain load, even the action of both wind and rain loads together is very necessary and significant. Wind-driven rain (WDR) is the rain that has a horizontal velocity component. Choi [1–3] made major breakthroughs in the numerical simulation of WDR by using computational fluid dynamic. Blocken and Carmeliet [4–6] have extended the Choi’s simulation technique by adding a temporal component and developing a new weighted data averaging technique, allowing for the determination of both the spatial and temporal distribution of WDR. Li et al. [7] proposed a new computational approach for ⇑ Corresponding author at: Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116023, China. http://dx.doi.org/10.1016/j.strusafe.2015.08.002 0167-4730/Ó 2015 Elsevier Ltd. All rights reserved.

the rain load on the transmission tower, and carried out the dynamic response analyses and experiments of the transmission tower under the wind and rain excitations. The results showed that the proposed approach agrees well with the wind tunnel test and the rain load influence on the transmission tower should not be ignored during the strong rainstorm. Fu et al. [8] modified the existed rain load model [7] by introducing velocity ratio of raindrop horizontal velocity to corresponding wind speed. Ibarra [9] proposed a methodology for evaluating the global incremental (sidesway) collapse based on a relative intensity measure instead of an Engineering Demand Parameter. The proposed method was applied to the development of collapse fragility curves. Nielson and DesRoches [10] gave an expanded methodology for the generation of analytical fragility curves for highway bridges, where the contribution of major components of the bridge, such as columns, bearings and abutments, to its overall bridge system fragility, was considered, which showed that the bridge as a system is more fragile than any one of individual components. Padgett and DesRoches [11] presented an analytical methodology for developing fragility curves to classify retrofitted bridge systems and results indicated the importance of evaluating the impact of retrofit not only on the targeted response quantity and component

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vulnerability but also on the overall bridge fragility. Rota et al. [12] put forward a new analytical approach for the derivation of fragility curves for masonry buildings based on the nonlinear stochastic analysis of building prototypes. Billah and his colleagues [13] used fragility curves to assess the relative performance of various retrofit multicolumn bridge bents under both near-fault and far-field seismic ground motions. Until now, the seismic fragility analysis has been studied for a long time and its importance has been shown with significance. However, the fragility analysis under wind load is just in an initial stage. Lee and Rosowsky [14] investigated a fragility assessment for roof sheathing in light frame constructions built in strong wind regions and the presented fragility models that can be used to develop performance-based design guidelines for wood frame structures as well as tools for condition assessment and loss estimation for use with the existing building inventory. Lee and Rosowsky [15] described a procedure to develop fragility curves for wood frame structures subjected to lateral wind loads, and a quick analysis to develop approximate fragilities is conducted. For the simulation of the along-wind dynamic response of tall buildings under turbulent winds, Smith and Caracoglia [16] presented a numerical algorithm and then the fragility curves were derived to estimate the performance based on the proposed numerical algorithm. Herbin and Barbato [17] developed a methodology for the windborne debris impact fragility curves for building envelope components during hurricanes. Seo and Caracoglia [18] used fragility analysis to estimate life-cycle monetary loss of long-span bridges due to wind hazards. Shafieezadeh and his colleagues [19] established a probabilistic framework for the agedependent fragility analysis of wood utility poles against hurricanes and strong winds. Although the rain load formula has been proposed and validated, there are limited investigations on the rain load influence on transmission tower. The fragility analysis has been widely used in seismic engineering field [9–13,20–26]. Whereas, it can be seen from above reviews that research achievements of fragility analysis subjected to wind load are very limited, and the production of this approach for the transmission tower subjected to wind and rain loads is even fewer and has not been found to be published in literature so far. Given all this, a method of fragility analysis subjected to wind and rain loads using equivalent basic wind speed is proposed in this paper, and then the concept of critical collapse curve is also presented to obtain the most unfavorable combinations of basic wind speed and rain intensity corresponding to collapse status.

where qa is the air density taking 1:235 kg=m3 , ls is the body shape parameter, and A is the projected area of structure in the windward direction. The rain pressure for the specified rain diameter yields [7,8]:

Pr ðV a ; R; D; H; aÞ ¼ kqw SðcðH; D; aÞV a ; RÞnðD; RÞc3 ðH; D; aÞV 3a D3 ð4Þ where k is a factor taking 102.0 in 1=m, qw is the raindrop density taking 1000 kg=m3 , D is the raindrop diameter, nðD; RÞ is the raindrop spectrum which means the raindrop size distribution, c is the velocity ratio, and SðV r ; RÞ denotes the area of the normalized curve in Fig. 1 integrating from 0 to the time of Dt. Based on the Marshall–Palmer raindrop spectrum [27], the raindrop size distribution is expressed as:

nðD; RÞ ¼ n0 expðKDÞ

ð5Þ

where n0 ¼ 8  103 in 1=ðm3 mmÞ, K ¼ 4:1R0:21 in 1/mm, and R is the rain intensity (mm/h). Dt can be calculated by:

pffiffiffi 2 ffiffiffiffiffiffiffiffiffiffi Dt ¼ p 3 NðRÞV r

ð6Þ

where V r is the raindrop horizontal velocity and NðRÞ is the total R1 number of raindrops per unit volume taking 0 nðD; RÞdD. The velocity ratio is defined as the ratio of raindrop horizontal velocity to the corresponding wind speed, which can be expressed as [8]: 8  0:8  a  0:5008 > þ 1 ðH 6 150 mÞ  0:0167Þ D3 > 0:12 > ð0:2373H > > < Vr cðH; D; aÞ ¼ ¼ 1 ðH > 150 mÞ > Va > > > > : ð7Þ

The rain load for a specified rain intensity and wind speed can be derived from integrating Eq. (4) as:

Z Fr ¼ 0

1

Pr ðV a ; R; D; H; aÞAdD

ð8Þ

Wind load is very easy to calculate based on Eq. (3), and yet the form of Eqs. (4) and (8) is so complex that it can only be obtained by programming. Thus, it will be very meaningful to propose a simplified method to calculate wind and rain loads for its easy and wide applications.

2. Theoretical method for calculating wind and rain loads 3. Simplified method for calculating wind and rain loads The mean wind speed varying with altitude can be obtained by the exponential wind profile expression:


V a ¼ V 10

a

H 10

ð1Þ

where V 10 is the basic wind speed representing the mean wind speed during 10 min at the altitude of 10 m, H is the altitude, and a is the ground roughness coefficient. The wind pressure in free wind field and wind load acting on structures are written by:

Pw ¼

1 q V2 2 a a

and

F w ¼ ls


 1 qa V 2a A 2

3.1. Simplified method For simplifying the calculating process of wind and rain loads, the equivalent basic wind speed (EBWS) and equivalent ground roughness coefficient (EGRC) are employed, and the process of finding EBWS and EGRC is listed as below:

ð2Þ

ð3Þ Fig. 1. Schematic of normalized curve and SðV r ; RÞ.

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(1) The first step is calculating the sum of wind and rain pressures with the ground roughness coefficient a, basic wind speed V 10 , rain intensity R, and altitude H, and then equivalent wind speed V  is derived with Eq. (2). If H is 10 m, the derived equivalent wind speed is EBWS V 10 . (2) If a, R and V 10 is fixed, the equivalent wind speed V  with different altitudes can be derived, and then EGRC a is obtained based on Eq. (1). (3) The results of first and second steps are fitted to obtain the fitting formulae of EBWS and EGRC. Based on the above process, the EBWS and EGRC are obtained and then the difference of equivalent and original values are calculated as illustrated in Figs. 2 and 3 to find the potential laws. In Fig. 2, it can be found that the difference of wind speed is directly proportional to basic wind speed and rain intensity and has nothing to do with ground roughness coefficient. Meanwhile all the curves are very similar, and therefore it’s assumed that all the curves in Fig. 2 have the same shape function but with different scale factors. The shape function can be expressed as the sum of two exponential functions with the first scale factor of 1:

f 1 ðV 10 Þ ¼ expð0:006462V 10 Þ  1:2486 expð0:2769V 10 Þ

ð9Þ

The scale factors of Eq. (9) for different rain intensities and ground roughness coefficients are illustrated in Fig. 4, which can be seen that the three curves overlap and the ground roughness coefficient has no effect on the scale factor of Eq. (9), and so all curves of Fig. 4 yield the same function as:

f 2 ðRÞ ¼ 0:009376R0:7087

ð10Þ

Fig. 4. The scale factor of Eq. (9) in Fig. 2.

Therefore, the fitting formula of EBWS can be expressed as:

V 10 ¼ V 10 þ f 1 ðV 10 Þf 2 ðRÞ ¼ V 10 þ 0:009376R0:7087 ðexpð0:006462V 10 Þ

ð11Þ

1:2486 expð0:2769V 10 ÞÞ In Eq. (11), it’s obvious that the EBWS contains the basic wind speed and rain intensity indicating that the EBWS is a fusion of wind and rainfall. In Fig. 3 it can be obviously found that the variation range of the difference is very small and the maximum difference is smaller than 0.01 indicating that EGRC and ground roughness coefficient are so close that it can be simplified as a ¼ a.

Fig. 2. The difference of EBWS and basic wind speed.

Fig. 3. The difference of EGRC and ground roughness coefficient.

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Fig. 5. Flow chart of simplified method for calculating rain load.

The process of calculating rain load based on the simplified method is summarized as below: (1) Calculate the EBWS V 10 with the rain intensity R and basic wind speed V 10 based on Eq. (11). (2) Calculate the wind load F w with the specified condition based on Eqs. (1) and (3). (3) Obtain the total wind and rain loads F total based on Eqs. (1) and (3) with V 10 instead of V 10 . (4) The rain load F r is derived by subtracting the wind load from total wind and rain loads: F r ¼ F total  F w . It can be seen from above process that the simplified method greatly simplifies the calculating process, and for most conditions only the first three steps are needed to obtain the total wind and rain loads. In order to show these steps clearly, the flow chart of the simplified method for calculating the rain load is illustrated in Fig. 5. Fig. 7. Simplified nodes.

3.2. Error analysis A 500 kV transmission tower with the height of 254 m and the material of Q235 and Q345 (which are very close to A242 and A441 in ASTM, respectively) is employed to validate the effectiveness of the simplified method. The cross-sections of tower members consist of steel tube and angle steel. The finite element modal is illustrated in Fig. 6, and simplified nodes applied by loads are shown in Fig. 7. The acquisition point of displacement is shown in Fig. 8.

Fig. 8. Sketch of acquisition point.

Fig. 6. Finite element model of transmission tower.

The total wind and rain loads are generated based on both the integration formula of Eq. (8) and the simplified method of Eq. (11). The harmony superposition method [28,29] is used to generate the fluctuating wind speed, and simulation parameters for the fluctuating wind speed here are as follows: (1) the power-law exponent is 0.12; (2) the total time of simulating the fluctuating wind speed is 300 s; (3) the horizontal fluctuating wind spectrum takes the Davenport spectrum which is recommended by Chinese standard [30]; and (4) basic wind speed takes: 5, 10, 15 and 20 m/s, and rain intensity takes 0, 40, 80, 120, 160 and 200 mm/h. Then the response of transmission tower subjected to wind and rain loads is calculated and the average displacement of the acquisition point is illustrated in Fig. 9, which can be found that the blue and black lines are very close. Meanwhile Table 1 gives the relative errors of simplified method relative to integration formula, and the maximum relative error of average displacement can only reach to 1.05% indicating that the simplified method can satisfy the accuracy in practical engineering.

X. Fu et al. / Structural Safety 58 (2016) 1–10

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Fig. 9. Average tower tip displacements based on integration formula and simplified method.

Table 1 Relative errors of simplified method relative to integration formula. R (mm/h)

0

40

80

120

160

200

V10 (m/s) 5 10 15 20

0.00% 0.00% 0.00% 0.00%

0.24% 0.33% 0.27% 0.23%

0.33% 0.53% 0.46% 0.40%

0.40% 0.72% 0.62% 0.54%

0.47% 0.89% 0.77% 0.64%

0.53% 1.05% 0.88% 0.75%

4. Fragility analysis and critical collapse curve 4.1. Fragility analysis subjected to wind and rain loads There are many methods to calculate fragility curves of structures [9,10,12,13,24,25], and most of them are sourced from the seismic fragility analysis. Collapse fragility curves can be defined by the cumulative distribution function (CDF), and consequently a relation between the load and the collapse probability is established. The collapse data can be obtained based on the incremental dynamic analysis (IDA), and the lognormal collapse fragility is defined by two parameters, median collapse intensity and the standard deviation of the natural logarithm. The median collapse intensity corresponds to a 50% probability of collapse. The slope of the lognormal distribution is measured by the standard deviation, and reflects the dispersion in results due to record-to-record variability. In the FEMA for the seismic fragility analysis, the record-to-record variability, the standard deviation of the natural logarithm, is set to a fixed value [25]. Nevertheless, there’s little discussion on the fragility analysis subjected to wind and rain loads. Hence, the standard deviation of the natural logarithm for the fragility analysis subjected to these loads must be calculated by the simulation results. The collapse intensities are assumed to conform to the lognormal distribution. So the collapse EBWSs need to be taken log when conducting the fragility analysis subjected to wind and rain loads, and then the median value u and standard deviation r are calculated. Finally, the collapse probability for the specified EBWS V 10 can be obtained by:

1 P ¼ Fðxju; rÞ ¼ pffiffiffiffiffiffiffi r 2p

Z

x

e 1

ðtuÞ2 2r2

dt

ð12Þ

where x is the log of the EBWS taking lnðV 10 Þ. Since considering the basic wind speed and rain intensity together is complicated, the EBWS is used here to simplify the analytical process. Based on the method of calculating rain load in Section 2, the total wind and rain loads can be derived by Eqs. (3) and

(11), and the simplified calculating process is so similar to the method of calculating wind load in which the EBWS merely contains the component of rain intensity. Therefore, the EBWS can be considered as the basic wind speed when conducting the nonlinear analysis to obtain the limited displacement of structural collapse. The calculating process of fragility curve induced by wind and rain loads is described as below: (1) Conduct the nonlinear analysis for structures under the wind load and obtain the relation of the EBWS and displacement. (2) Perform the IDA for structures with varying the value of EBWS, and obtain the curve of maximum displacement of structural tip and EBWS. Repeating above analysis, 20 EBWSs that the first time reaches or exceeds the limited displacement of structural collapse are obtained. (3) Plot fragility curves based on the 20 EBWSs in step 2, and then the collapse EBWS corresponding to 10% collapse probability is derived exceeding which the structure is considered to collapse. 4.2. Definition of critical collapse curve The collapse EBWS can be obtained by the fragility analysis. However, there are countless combinations of basic wind speed and rain intensity that can satisfy the collapse EBWS to be seen obviously from Eq. (11). Hence, the critical collapse curve (CCC) is defined to compare different combinations and judge the collapse status of structure. The calculating method of CCC is given as below: (1) Calculate the collapse EBWS corresponding to the 10% collapse probability of collapse fragility curve. (2) Substitute the collapse EBWS obtained from the first step into Eq. (11), and the Newton iterative method is used to solve the equation to acquire the combinations of basic wind speed and its corresponding rain intensity. (3) The CCC can be derived by plotting different combinations of basic wind speed and its corresponding rain intensity given in step 2. The process of calculating fragility curve and CCC for structures subjected to wind and rain loads is summarized as shown in Fig. 10. Any tall structure can draw the CCC like Fig. 11. For any combination of basic wind speed and rain intensity, we can draw its coordinate point in Fig. 11. If the point is below the CCC, the structure is

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X. Fu et al. / Structural Safety 58 (2016) 1–10

Calculating limited

Calculating EBWSs

displacement of structural

that the first time

Plotting fragility

collapse through

reaching limited

curves

nonlinear analysis

displacement

Calculating the Plotting CCC

different combinations of

Calculating collapse EBWS

wind and rain

Fig. 10. Flow chart of calculating fragility curve and CCC subjected to wind and rain loads.

safe, that is to say the structure is in safety region; otherwise, the structure will collapse, that is to say the structure is in collapse region. 5. Case study of a 500 kV transmission tower 5.1. Nonlinear analysis of transmission tower For the transmission tower, the immediate cause of tower collapse is not the material strength, but the member buckling due to the present of initial eccentricity. The nonlinear time history analysis is time-consuming especially on complex load conditions. At present the nonlinear analysis of transmission tower [31–34] is only used to obtain the limited displacement of tower collapse, and the criterion of equal displacement [35], which assumes that the structure will collapse if the linear dynamic displacement is equal to the corresponding static nonlinear buckling displacement, is used to conduct the fragility analysis. Therefore, the computation time-consuming is reduced greatly due to the linear simulation. The employed transmission tower is illustrated in Fig. 6. The elastic modulus of Q235 and Q345 takes 2.06 GPa, and the yield strength equals to 235 MPa and 345 MPa, respectively. The Bilinear Isotropic Hardening Plasticity model is used to simulate the constitutive model of steel material. The acquisition point of displacement is the same as Fig. 8, and the results of nonlinear analysis for different EBWSs are illustrated in Fig. 12. Although results in Fig. 12 are obtained under the pure wind load, the vertical axis represents the EBWS which already contains the rain intensity essentially, and this can be comprehended by Eq. (11). In Fig. 12 there’s an obvious buckling point, and the maximum EBWS that the tower keeps elasticity is 75 m/ s with the corresponding tower tip displacement of 1.367 m. 5.2. Fragility curve and CCC for transmission tower subjected to wind and rain loads Based on the criterion of equal displacement, in which assuming the dynamic buckling displacement be equal to the static

Fig. 12. Nonlinear analysis of transmission tower.

buckling displacement and the structure would collapse if the dynamic displacement achieves the static buckling displacement, the limited displacement of tower collapse for fragility analysis is 1.367 m. Once the tower tip displacement reaches or exceeds the limited displacement, it can be thought as tower collapse. Simulation parameters for the fluctuating wind speed here are as follows: (1) the power-law exponent is 0.30; (2) the total time of simulating the fluctuating wind speed is 300 s; and (3) the horizontal fluctuating wind spectrum takes the Davenport spectrum. The IDA is employed to obtain the tower response under different EBWSs, and the increment of EBWS is set to 0.1 m/s. The maximum tower tip displacements for different EBWSs are calculated and illustrated in Fig. 13. In order to avoid the data overlapping, only a small part of the data is given here. In Fig. 13, the vertical thick line represents the limited displacement of tower collapse obtained by the nonlinear analysis, and the EBWSs corresponding to the point which of these curves reach or exceed the limited displacement first time are obtained. Based on the 20 EBWSs and the fragility theory, the collapse fragility curve of transmission tower subjected to wind and rain loads in the longitudinal direction is obtained as shown in Fig. 14. In Fig. 14, the collapse EBWS with the percentage of 10% is 43.18 m/s indicating that the transmission tower is considered to collapse if the EBWS reaches or exceeds 43.18 m/s. The CCC with the EBWS of 43.18 m/s is obtained and shown in Fig. 15. In Fig. 15, there’s a small gap between the curve and horizontal axis. The reason is the error of simplified method of calculating rain load. The relative errors in Eq. (8) and the simplified method are so small that the small gap in Fig. 15 can be ignored and it can satisfy the accuracy in practical engineering. The design basic wind speed of the tower is 42 m/s, so it’s obvious that the tower is safe without considering the rain intensity. However, if there’s a precipitation and the rain intensity is larger than 619 mm/h, the transmission tower will be very dangerous. In Guadeloupe, Caribbean, 38.1 mm of rain falls in 1 min in 1970 corresponding to

Fig. 11. Schematic plot of CCC.

X. Fu et al. / Structural Safety 58 (2016) 1–10

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Fig. 13. Maximum displacement of tower tip.

Fig. 16. Schematic diagram of wind attack angle. Fig. 14. Collapse fragility curve of transmission tower induced by wind and rain loads.

Fig. 15. CCC for basic wind speed and rain intensity.

2286 mm/h, the heaviest rainfall ever on record. In Xi’an, China, 59.1 mm of rain falls in 5 min in 1973 corresponding to 709.2 mm/h [8]. Based on the two cases, it can be seen that the transient rain intensity can be very large, and the rain intensity of 619 mm/h can occur in natural for extreme conditions. Thus, the contribution of rain load on the tower collapse cannot be neglected and should be paid more attention. Although the fragility curve and CCC are obtained, the influence of wind attack angle and wind spectrum has not been considered yet. Different wind attack angles can lead to different load distributions in two horizontal directions which will result in different responses and wind spectrum decides the energy distribution of fluctuating wind. Thereby it’s very necessary to study the influence of wind attack angle and wind spectrum on the tower response and fragility curve as well as the CCC.

5.3. Influence of wind attack angle and wind spectrum on fragility curve and CCC 5.3.1. Influence of wind attack angle For a fixed structure, the wind may come from its any directions, so the wind attack angle will change with the variation of wind direction, which is shown in Fig. 16. Due to the variation of wind attack angle in natural environment, it’s essential to study the influence of wind attack angle on the fragility curve and CCC for transmission tower subjected to wind and rain loads. Since the transmission tower is symmetrical in both X and Y directions, the wind attack angle takes from 0° to 90° with the increment of 15°. The fragility curves of transmission tower with different wind attack angles are illustrated in Fig. 17. It’s obvious that the distances between any two curves are very big, which indicates that the wind attack angle has a great influence on the fragility curve. Firstly the fragility curve moves to right with the increase of wind attack angle, while the curve move to left since the wind attack angle reaches 60°. The left-most fragility curve corresponds to the wind attack angle of 0°, and the right-most curve corresponds to the wind attack angle of 60°. If the wind and rain loads acting on the tower are fixed, when the wind attack angle is 0°, the tower response is the largest; on the contrary when the wind attack angle is 60°, the tower response is the smallest. The reason is that the projected areas in the two horizontal directions are different and the difference is quite large. The EBWSs corresponding to the percentage of 10% in Fig. 17 are captured, and these speeds are the collapse EBWS for different wind attack angles as shown in Fig. 18. In Fig. 18, it’s obvious that the wind attack angle has a significant impact on the collapse EBWS and the most favorable and unfavorable wind attack angles are 60° and 0° respectively, which have the same conclusions to Fig. 17.

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Fig. 17. Fragility curves subjected to wind and rain loads for different wind attack angles.

fSðf Þ 4x2D ¼ 4=3 u2 ð1 þ x2 Þ

ð13Þ

D

xD ¼ 1200

f V 10

where u is the surface shear wind speed, and f is the fluctuating frequency. In 1970, Harris [37] proposed a modified wind spectrum based on the Davenport spectrum written by:

fSðf Þ 4xH ¼ 5=6 u2 ð2 þ x2 Þ

ð14Þ

H

Fig. 18. Collapse EBWS for different wind attack angles.

xH ¼

1800f V 10

The CCCs for different wind attack angles subjected to wind and rain loads are obtained as illustrated in Fig. 19, in which the curve corresponding to the wind attack angle of 0° is at the bottom, and the curve corresponding to the angle of 60° is at the top. The lower the curve position is, the lower the load capacity of tower has. Hence, the CCC with the wind attack angle of 0° plays the role in control which meets the conclusion of Figs. 17 and 18.

The Architectural Association of Japan has modified the Karman spectrum as [38]:

5.3.2. Influence of wind spectrum In the most of wind spectra, Davenport, Karman, Harris, and Simiu spectra are widely used. However, the wind spectrum describes the energy distribution of fluctuating wind in the frequency domain, and there are big differences among the fluctuating wind speeds generated by different wind spectra. Hence, it’s necessary to study the influence of wind spectrum on the fragility curve and CCC. In 1960s, Davenport [36] summarized a fluctuating wind spectrum based on the meteorological data as follows:

where ru is the root mean square error of fluctuating wind speed, and Lu ðHÞ is the turbulence integral scale. In 1974, Simiu [39] gave a horizontal fluctuating wind spectrum associated with altitude:

fSðf Þ

r2u

xK ¼

¼

4xK

ð15Þ

fLu ðHÞ VðHÞ

fSðH; f Þ 200xS ¼ u2 ð1 þ 50xS Þ5=3 xS ¼

5=6

6:677ð1 þ 70:8x2K Þ

fH VðHÞ

Fig. 19. CCCs for different wind attack angles.

ð16Þ

X. Fu et al. / Structural Safety 58 (2016) 1–10

9

Fig. 20. The fragility curves of transmission tower based on the four wind spectra.

Fig. 21. CCCs of transmission tower based on the four wind spectra.

In this section, the study on the influence of wind spectrum on the fragility curve and CCC with the wind attack angle of 0°, which is the most unfavorable condition, is conducted. Using the four wind spectra, the fragility curve and CCC of transmission tower subjected to wind and rain loads are calculated and shown in Figs. 20 and 21, respectively. It can be seen from Fig. 20 that the wind spectrum has a great influence on the fragility curve. The fragility curves of the Davenport and Harris spectra are so close since the Harris spectrum is derived from Davenport spectrum and the two spectra are very similar. In addition, the wind spectrum corresponding to the left fragility curve has more energy than the right one, because more energy of wind spectrum is needed for smaller EBWS conditions to blow down the transmission tower. In Fig. 21, the similar conclusions can be summarized as Fig. 20. The biggest basic wind speed of the Simiu spectrum corresponding to the rain intensity of 0 mm/h is 30.3 m/s. However, the design basic wind speed of tower is 42 m/s illustrating that the tower is not safe if employing the Simiu spectrum. 6. Summary and conclusions A simplified method is presented to simplify the calculating process of wind and rain loads and then the error analysis is performed to validate its effectiveness. A methodology of fragility analysis for transmission tower subjected to wind and rain loads is proposed and conducted based on the existing achievement of fragility analysis. The concept of EBWS is used to conduct the fragility analysis subjected to wind and rain loads which avoid the complex formula of rain load and the choice of different combinations of basic wind speed and rain intensity. Although only the wind load is considered when conducting the fragility analysis, the concept of EBWS already contains rain load essentially. On the basis of fragility analysis subjected to wind and rain loads, the concept of CCC is given, and the calculating process is introduced in details. It’s very easy to evaluate the collapse status

for structures subjected to wind and rain loads using the CCC. For any combination of basic wind speed and rain intensity, if the coordinate point is below the CCC, the structure is safe; while, if the coordinate point is on or above the CCC, the structure will collapse. The CCC presents a method to compare different combinations of basic wind speed and rain intensity, and judge the collapse status. The effects of wind attack angle and wind spectrum on the tower response and fragility curve as well as CCC are studied and some conclusions are obtained as follows: (1) The wind attack angle and wind spectrum have great influence on the fragility and critical collapse curves. (2) The most favorable and unfavorable wind attack angles for the employed transmission tower are 60° and 0°, respectively. (3) The Simiu spectrum has the biggest energy distribution of fluctuating wind than the rest three spectra and the best way to decrease the influence of wind spectrum is to employ the local measured wind spectrum. (4) The rain load has a great contribution to the tower collapse and should be paid more attention during severe gales and thunderstorms.

Acknowledgement This research was supported by the National Natural Science Foundation of China (Grant No. 51421064). References [1] Choi EC. Simulation of wind-driven-rain around a building. J Wind Eng Ind Aerodyn 1993;46:721–9. [2] Choi EC. Determination of wind-driven-rain intensity on building faces. J Wind Eng Ind Aerodyn 1994;51:55–69. [3] Choi EC. Numerical modelling of gust effect on wind-driven rain. J Wind Eng Ind Aerodyn 1997;72:107–16.

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